Properties

 Label 2475.2.a.bb.1.3 Level $2475$ Weight $2$ Character 2475.1 Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$19.7629745003$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.90321 q^{2} +1.62222 q^{4} +4.42864 q^{7} -0.719004 q^{8} +O(q^{10})$$ $$q+1.90321 q^{2} +1.62222 q^{4} +4.42864 q^{7} -0.719004 q^{8} -1.00000 q^{11} +0.622216 q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421 q^{17} +7.05086 q^{19} -1.90321 q^{22} +8.85728 q^{23} +1.18421 q^{26} +7.18421 q^{28} +7.80642 q^{29} +2.75557 q^{31} -7.34122 q^{32} -9.86665 q^{34} +2.00000 q^{37} +13.4193 q^{38} +0.193576 q^{41} -5.67307 q^{43} -1.62222 q^{44} +16.8573 q^{46} -2.75557 q^{47} +12.6128 q^{49} +1.00937 q^{52} -10.8573 q^{53} -3.18421 q^{56} +14.8573 q^{58} +4.85728 q^{59} +6.85728 q^{61} +5.24443 q^{62} -4.74620 q^{64} +1.24443 q^{67} -8.40990 q^{68} -2.75557 q^{71} -4.23506 q^{73} +3.80642 q^{74} +11.4380 q^{76} -4.42864 q^{77} +8.56199 q^{79} +0.368416 q^{82} +0.133353 q^{83} -10.7971 q^{86} +0.719004 q^{88} -5.61285 q^{89} +2.75557 q^{91} +14.3684 q^{92} -5.24443 q^{94} -7.24443 q^{97} +24.0049 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 5 q^{4} - 9 q^{8}+O(q^{10})$$ 3 * q - q^2 + 5 * q^4 - 9 * q^8 $$3 q - q^{2} + 5 q^{4} - 9 q^{8} - 3 q^{11} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} + 8 q^{19} + q^{22} - 10 q^{26} + 8 q^{28} + 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34} + 6 q^{37} + 14 q^{41} - 4 q^{43} - 5 q^{44} + 24 q^{46} - 8 q^{47} + 11 q^{49} + 30 q^{52} - 6 q^{53} + 4 q^{56} + 18 q^{58} - 12 q^{59} - 6 q^{61} + 16 q^{62} + 13 q^{64} + 4 q^{67} + 42 q^{68} - 8 q^{71} + 14 q^{73} - 2 q^{74} + 48 q^{76} + 12 q^{79} - 26 q^{82} + 8 q^{86} + 9 q^{88} + 10 q^{89} + 8 q^{91} + 16 q^{92} - 16 q^{94} - 22 q^{97} + 39 q^{98}+O(q^{100})$$ 3 * q - q^2 + 5 * q^4 - 9 * q^8 - 3 * q^11 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 + 8 * q^19 + q^22 - 10 * q^26 + 8 * q^28 + 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 + 6 * q^37 + 14 * q^41 - 4 * q^43 - 5 * q^44 + 24 * q^46 - 8 * q^47 + 11 * q^49 + 30 * q^52 - 6 * q^53 + 4 * q^56 + 18 * q^58 - 12 * q^59 - 6 * q^61 + 16 * q^62 + 13 * q^64 + 4 * q^67 + 42 * q^68 - 8 * q^71 + 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 26 * q^82 + 8 * q^86 + 9 * q^88 + 10 * q^89 + 8 * q^91 + 16 * q^92 - 16 * q^94 - 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
$$5$$ 0 0
$$6$$ 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
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 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 $$-0.200124\pi$$
0.808789 + 0.588100i $$0.200124\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −1.90321 −0.405766
$$23$$ 8.85728 1.84687 0.923435 0.383754i $$-0.125369\pi$$
0.923435 + 0.383754i $$0.125369\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.18421 0.232242
$$27$$ 0 0
$$28$$ 7.18421 1.35769
$$29$$ 7.80642 1.44962 0.724808 0.688951i $$-0.241928\pi$$
0.724808 + 0.688951i $$0.241928\pi$$
$$30$$ 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
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 13.4193 2.17689
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0.193576 0.0302315 0.0151158 0.999886i $$-0.495188\pi$$
0.0151158 + 0.999886i $$0.495188\pi$$
$$42$$ 0 0
$$43$$ −5.67307 −0.865135 −0.432568 0.901602i $$-0.642392\pi$$
−0.432568 + 0.901602i $$0.642392\pi$$
$$44$$ −1.62222 −0.244558
$$45$$ 0 0
$$46$$ 16.8573 2.48547
$$47$$ −2.75557 −0.401941 −0.200971 0.979597i $$-0.564410\pi$$
−0.200971 + 0.979597i $$0.564410\pi$$
$$48$$ 0 0
$$49$$ 12.6128 1.80184
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00937 0.139974
$$53$$ −10.8573 −1.49136 −0.745681 0.666303i $$-0.767876\pi$$
−0.745681 + 0.666303i $$0.767876\pi$$
$$54$$ 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 $$-0.355336\pi$$
0.438992 + 0.898491i $$0.355336\pi$$
$$62$$ 5.24443 0.666043
$$63$$ 0 0
$$64$$ −4.74620 −0.593275
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 1.24443 0.152031 0.0760157 0.997107i $$-0.475780\pi$$
0.0760157 + 0.997107i $$0.475780\pi$$
$$68$$ −8.40990 −1.01985
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −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
$$77$$ −4.42864 −0.504690
$$78$$ 0 0
$$79$$ 8.56199 0.963299 0.481650 0.876364i $$-0.340038\pi$$
0.481650 + 0.876364i $$0.340038\pi$$
$$80$$ 0 0
$$81$$ 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
$$85$$ 0 0
$$86$$ −10.7971 −1.16428
$$87$$ 0 0
$$88$$ 0.719004 0.0766461
$$89$$ −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
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −7.24443 −0.735561 −0.367780 0.929913i $$-0.619882\pi$$
−0.367780 + 0.929913i $$0.619882\pi$$
$$98$$ 24.0049 2.42486
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.66370 −0.464056 −0.232028 0.972709i $$-0.574536\pi$$
−0.232028 + 0.972709i $$0.574536\pi$$
$$102$$ 0 0
$$103$$ 11.6128 1.14425 0.572124 0.820167i $$-0.306120\pi$$
0.572124 + 0.820167i $$0.306120\pi$$
$$104$$ −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 $$-0.893120\pi$$
−0.944156 + 0.329499i $$0.893120\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −20.4286 −1.93032
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 12.6637 1.17580
$$117$$ 0 0
$$118$$ 9.24443 0.851019
$$119$$ −22.9590 −2.10465
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 13.0509 1.18157
$$123$$ 0 0
$$124$$ 4.47013 0.401429
$$125$$ 0 0
$$126$$ 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
$$130$$ 0 0
$$131$$ −1.24443 −0.108726 −0.0543632 0.998521i $$-0.517313\pi$$
−0.0543632 + 0.998521i $$0.517313\pi$$
$$132$$ 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 $$-0.506648\pi$$
−0.0208832 + 0.999782i $$0.506648\pi$$
$$138$$ 0 0
$$139$$ 17.8064 1.51032 0.755161 0.655540i $$-0.227559\pi$$
0.755161 + 0.655540i $$0.227559\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −5.24443 −0.440103
$$143$$ −0.622216 −0.0520323
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −8.06022 −0.667069
$$147$$ 0 0
$$148$$ 3.24443 0.266691
$$149$$ 1.43801 0.117806 0.0589031 0.998264i $$-0.481240\pi$$
0.0589031 + 0.998264i $$0.481240\pi$$
$$150$$ 0 0
$$151$$ −12.1748 −0.990774 −0.495387 0.868672i $$-0.664974\pi$$
−0.495387 + 0.868672i $$0.664974\pi$$
$$152$$ −5.06959 −0.411198
$$153$$ 0 0
$$154$$ −8.42864 −0.679199
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −18.4701 −1.47408 −0.737038 0.675851i $$-0.763776\pi$$
−0.737038 + 0.675851i $$0.763776\pi$$
$$158$$ 16.2953 1.29638
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 39.2257 3.09142
$$162$$ 0 0
$$163$$ 10.1017 0.791227 0.395614 0.918417i $$-0.370532\pi$$
0.395614 + 0.918417i $$0.370532\pi$$
$$164$$ 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
$$170$$ 0 0
$$171$$ 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
$$175$$ 0 0
$$176$$ 4.61285 0.347706
$$177$$ 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
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 5.18421 0.379107
$$188$$ −4.47013 −0.326017
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −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 0
$$201$$ 0 0
$$202$$ −8.87601 −0.624514
$$203$$ 34.5718 2.42647
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 22.1017 1.53990
$$207$$ 0 0
$$208$$ −2.87019 −0.199012
$$209$$ −7.05086 −0.487718
$$210$$ 0 0
$$211$$ −10.6637 −0.734120 −0.367060 0.930197i $$-0.619636\pi$$
−0.367060 + 0.930197i $$0.619636\pi$$
$$212$$ −17.6128 −1.20966
$$213$$ 0 0
$$214$$ 4.99063 0.341153
$$215$$ 0 0
$$216$$ 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 $$-0.595841\pi$$
−0.296564 + 0.955013i $$0.595841\pi$$
$$224$$ −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 $$0.375807\pi$$
$$230$$ 0 0
$$231$$ 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
$$235$$ 0 0
$$236$$ 7.87955 0.512915
$$237$$ 0 0
$$238$$ −43.6958 −2.83238
$$239$$ 3.34614 0.216444 0.108222 0.994127i $$-0.465484\pi$$
0.108222 + 0.994127i $$0.465484\pi$$
$$240$$ 0 0
$$241$$ −1.34614 −0.0867126 −0.0433563 0.999060i $$-0.513805\pi$$
−0.0433563 + 0.999060i $$0.513805\pi$$
$$242$$ 1.90321 0.122343
$$243$$ 0 0
$$244$$ 11.1240 0.712140
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.38715 0.279148
$$248$$ −1.98126 −0.125810
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −22.7556 −1.43632 −0.718159 0.695879i $$-0.755015\pi$$
−0.718159 + 0.695879i $$0.755015\pi$$
$$252$$ 0 0
$$253$$ −8.85728 −0.556852
$$254$$ 28.8988 1.81327
$$255$$ 0 0
$$256$$ 20.2444 1.26528
$$257$$ −6.85728 −0.427745 −0.213873 0.976862i $$-0.568608\pi$$
−0.213873 + 0.976862i $$0.568608\pi$$
$$258$$ 0 0
$$259$$ 8.85728 0.550365
$$260$$ 0 0
$$261$$ 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
$$265$$ 0 0
$$266$$ 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 $$-0.353069\pi$$
0.445378 + 0.895343i $$0.353069\pi$$
$$272$$ 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
$$280$$ 0 0
$$281$$ 0.193576 0.0115478 0.00577389 0.999983i $$-0.498162\pi$$
0.00577389 + 0.999983i $$0.498162\pi$$
$$282$$ 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
$$286$$ −1.18421 −0.0700237
$$287$$ 0.857279 0.0506036
$$288$$ 0 0
$$289$$ 9.87601 0.580942
$$290$$ 0 0
$$291$$ 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
$$295$$ 0 0
$$296$$ −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
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 24.4286 1.39422 0.697108 0.716966i $$-0.254470\pi$$
0.697108 + 0.716966i $$0.254470\pi$$
$$308$$ −7.18421 −0.409358
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −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 $$-0.353517\pi$$
0.444120 + 0.895967i $$0.353517\pi$$
$$314$$ −35.1526 −1.98377
$$315$$ 0 0
$$316$$ 13.8894 0.781340
$$317$$ 16.4889 0.926107 0.463053 0.886330i $$-0.346754\pi$$
0.463053 + 0.886330i $$0.346754\pi$$
$$318$$ 0 0
$$319$$ −7.80642 −0.437076
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 74.6548 4.16035
$$323$$ −36.5531 −2.03387
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 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
$$335$$ 0 0
$$336$$ 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
$$340$$ 0 0
$$341$$ −2.75557 −0.149222
$$342$$ 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 $$-0.456048\pi$$
0.137642 + 0.990482i $$0.456048\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 7.34122 0.391288
$$353$$ −9.34614 −0.497445 −0.248722 0.968575i $$-0.580011\pi$$
−0.248722 + 0.968575i $$0.580011\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −9.10525 −0.482577
$$357$$ 0 0
$$358$$ 48.2034 2.54763
$$359$$ −10.7556 −0.567657 −0.283829 0.958875i $$-0.591605\pi$$
−0.283829 + 0.958875i $$0.591605\pi$$
$$360$$ 0 0
$$361$$ 30.7146 1.61656
$$362$$ −25.9081 −1.36170
$$363$$ 0 0
$$364$$ 4.47013 0.234298
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −33.7975 −1.76422 −0.882108 0.471046i $$-0.843876\pi$$
−0.882108 + 0.471046i $$0.843876\pi$$
$$368$$ −40.8573 −2.12983
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −48.0830 −2.49634
$$372$$ 0 0
$$373$$ −33.9496 −1.75784 −0.878922 0.476965i $$-0.841737\pi$$
−0.878922 + 0.476965i $$0.841737\pi$$
$$374$$ 9.86665 0.510192
$$375$$ 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 $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −11.6128 −0.594165
$$383$$ −14.6351 −0.747820 −0.373910 0.927465i $$-0.621983\pi$$
−0.373910 + 0.927465i $$0.621983\pi$$
$$384$$ 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
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 12.7556 0.640184 0.320092 0.947387i $$-0.396286\pi$$
0.320092 + 0.947387i $$0.396286\pi$$
$$398$$ 26.8385 1.34529
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −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
$$407$$ −2.00000 −0.0991363
$$408$$ 0 0
$$409$$ −7.12399 −0.352258 −0.176129 0.984367i $$-0.556358\pi$$
−0.176129 + 0.984367i $$0.556358\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 18.8385 0.928108
$$413$$ 21.5111 1.05849
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −4.56782 −0.223956
$$417$$ 0 0
$$418$$ −13.4193 −0.656358
$$419$$ −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
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 30.3684 1.46963
$$428$$ 4.25380 0.205615
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −34.3051 −1.65242 −0.826210 0.563362i $$-0.809508\pi$$
−0.826210 + 0.563362i $$0.809508\pi$$
$$432$$ 0 0
$$433$$ −14.4701 −0.695390 −0.347695 0.937608i $$-0.613036\pi$$
−0.347695 + 0.937608i $$0.613036\pi$$
$$434$$ 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 $$-0.652505\pi$$
−0.460988 + 0.887406i $$0.652505\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −6.13918 −0.292011
$$443$$ −13.1240 −0.623539 −0.311770 0.950158i $$-0.600922\pi$$
−0.311770 + 0.950158i $$0.600922\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −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
$$451$$ −0.193576 −0.00911514
$$452$$ −9.73329 −0.457816
$$453$$ 0 0
$$454$$ 25.4608 1.19493
$$455$$ 0 0
$$456$$ 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
$$460$$ 0 0
$$461$$ 28.8671 1.34448 0.672238 0.740335i $$-0.265333\pi$$
0.672238 + 0.740335i $$0.265333\pi$$
$$462$$ 0 0
$$463$$ 19.3461 0.899091 0.449546 0.893257i $$-0.351586\pi$$
0.449546 + 0.893257i $$0.351586\pi$$
$$464$$ −36.0098 −1.67172
$$465$$ 0 0
$$466$$ −8.23506 −0.381482
$$467$$ 3.14272 0.145428 0.0727139 0.997353i $$-0.476834\pi$$
0.0727139 + 0.997353i $$0.476834\pi$$
$$468$$ 0 0
$$469$$ 5.51114 0.254481
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −3.49240 −0.160751
$$473$$ 5.67307 0.260848
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −37.2444 −1.70710
$$477$$ 0 0
$$478$$ 6.36842 0.291285
$$479$$ 24.8573 1.13576 0.567879 0.823112i $$-0.307764\pi$$
0.567879 + 0.823112i $$0.307764\pi$$
$$480$$ 0 0
$$481$$ 1.24443 0.0567412
$$482$$ −2.56199 −0.116696
$$483$$ 0 0
$$484$$ 1.62222 0.0737371
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 11.5299 0.522468 0.261234 0.965275i $$-0.415871\pi$$
0.261234 + 0.965275i $$0.415871\pi$$
$$488$$ −4.93041 −0.223189
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −16.3872 −0.739542 −0.369771 0.929123i $$-0.620564\pi$$
−0.369771 + 0.929123i $$0.620564\pi$$
$$492$$ 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 $$0.691859\pi$$
$$500$$ 0 0
$$501$$ 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
$$505$$ 0 0
$$506$$ −16.8573 −0.749397
$$507$$ 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
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 2.75557 0.121190
$$518$$ 16.8573 0.740666
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 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
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 50.6548 2.19616
$$533$$ 0.120446 0.00521710
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −0.894751 −0.0386473
$$537$$ 0 0
$$538$$ −16.1561 −0.696539
$$539$$ −12.6128 −0.543274
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 27.9081 1.19876
$$543$$ 0 0
$$544$$ 38.0584 1.63174
$$545$$ 0 0
$$546$$ 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
$$560$$ 0 0
$$561$$ 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
$$565$$ 0 0
$$566$$ −51.7373 −2.17468
$$567$$ 0 0
$$568$$ 1.98126 0.0831320
$$569$$ 7.33630 0.307554 0.153777 0.988106i $$-0.450856\pi$$
0.153777 + 0.988106i $$0.450856\pi$$
$$570$$ 0 0
$$571$$ 36.6450 1.53354 0.766772 0.641919i $$-0.221862\pi$$
0.766772 + 0.641919i $$0.221862\pi$$
$$572$$ −1.00937 −0.0422038
$$573$$ 0 0
$$574$$ 1.63158 0.0681010
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −4.22216 −0.175771 −0.0878853 0.996131i $$-0.528011\pi$$
−0.0878853 + 0.996131i $$0.528011\pi$$
$$578$$ 18.7961 0.781817
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0.590573 0.0245011
$$582$$ 0 0
$$583$$ 10.8573 0.449663
$$584$$ 3.04503 0.126004
$$585$$ 0 0
$$586$$ −5.35905 −0.221380
$$587$$ 34.3684 1.41854 0.709268 0.704939i $$-0.249026\pi$$
0.709268 + 0.704939i $$0.249026\pi$$
$$588$$ 0 0
$$589$$ 19.4291 0.800563
$$590$$ 0 0
$$591$$ 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
$$595$$ 0 0
$$596$$ 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 $$-0.557148\pi$$
−0.178574 + 0.983927i $$0.557148\pi$$
$$602$$ −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
$$610$$ 0 0
$$611$$ −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
$$616$$ 3.18421 0.128295
$$617$$ −3.51114 −0.141353 −0.0706765 0.997499i $$-0.522516\pi$$
−0.0706765 + 0.997499i $$0.522516\pi$$
$$618$$ 0 0
$$619$$ 17.5941 0.707167 0.353584 0.935403i $$-0.384963\pi$$
0.353584 + 0.935403i $$0.384963\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −37.8350 −1.51705
$$623$$ −24.8573 −0.995886
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 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
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 7.84791 0.310946
$$638$$ −14.8573 −0.588205
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −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 $$-0.616177\pi$$
−0.356931 + 0.934131i $$0.616177\pi$$
$$644$$ 63.6325 2.50747
$$645$$ 0 0
$$646$$ −69.5683 −2.73713
$$647$$ −47.0420 −1.84941 −0.924705 0.380684i $$-0.875689\pi$$
−0.924705 + 0.380684i $$0.875689\pi$$
$$648$$ 0 0
$$649$$ −4.85728 −0.190665
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 16.3872 0.641770
$$653$$ 30.0830 1.17724 0.588619 0.808411i $$-0.299672\pi$$
0.588619 + 0.808411i $$0.299672\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −0.892937 −0.0348633
$$657$$ 0 0
$$658$$ −23.2257 −0.905432
$$659$$ 10.2854 0.400664 0.200332 0.979728i $$-0.435798\pi$$
0.200332 + 0.979728i $$0.435798\pi$$
$$660$$ 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
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 69.1437 2.67725
$$668$$ −26.5018 −1.02538
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −6.85728 −0.264722
$$672$$ 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
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −5.24443 −0.200820
$$683$$ 34.1847 1.30804 0.654020 0.756477i $$-0.273081\pi$$
0.654020 + 0.756477i $$0.273081\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 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
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −1.00354 −0.0380118
$$698$$ 9.78769 0.370469
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 29.9081 1.12961 0.564807 0.825223i $$-0.308950\pi$$
0.564807 + 0.825223i $$0.308950\pi$$
$$702$$ 0 0
$$703$$ 14.1017 0.531856
$$704$$ 4.74620 0.178879
$$705$$ 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 $$0.592929\pi$$
$$710$$ 0 0
$$711$$ 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
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 32.9403 1.22169 0.610843 0.791752i $$-0.290831\pi$$
0.610843 + 0.791752i $$0.290831\pi$$
$$728$$ −1.98126 −0.0734305
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 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
$$737$$ −1.24443 −0.0458392
$$738$$ 0 0
$$739$$ −5.06959 −0.186488 −0.0932440 0.995643i $$-0.529724\pi$$
−0.0932440 + 0.995643i $$0.529724\pi$$
$$740$$ 0 0
$$741$$ 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
$$745$$ 0 0
$$746$$ −64.6133 −2.36566
$$747$$ 0 0
$$748$$ 8.40990 0.307497
$$749$$ 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
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −8.75557 −0.318227 −0.159113 0.987260i $$-0.550864\pi$$
−0.159113 + 0.987260i $$0.550864\pi$$
$$758$$ −38.0642 −1.38256
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −3.15257 −0.114280 −0.0571402 0.998366i $$-0.518198\pi$$
−0.0571402 + 0.998366i $$0.518198\pi$$
$$762$$ 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 $$-0.674868\pi$$
−0.522144 + 0.852857i $$0.674868\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −29.7462 −1.07059
$$773$$ 29.1427 1.04819 0.524095 0.851660i $$-0.324404\pi$$
0.524095 + 0.851660i $$0.324404\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 5.20877 0.186984
$$777$$ 0 0
$$778$$ 10.6824 0.382984
$$779$$ 1.36488 0.0489018
$$780$$ 0 0
$$781$$ 2.75557 0.0986020
$$782$$ −87.3916 −3.12512
$$783$$ 0 0
$$784$$ −58.1811 −2.07790
$$785$$ 0 0
$$786$$ 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
$$790$$ 0 0
$$791$$ −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 $$-0.233310\pi$$
0.743195 + 0.669075i $$0.233310\pi$$
$$798$$ 0 0
$$799$$ 14.2854 0.505383
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −3.80642 −0.134409
$$803$$ 4.23506 0.149452
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 3.26317 0.114940
$$807$$ 0 0
$$808$$ 3.35322 0.117966
$$809$$ 27.8064 0.977622 0.488811 0.872390i $$-0.337431\pi$$
0.488811 + 0.872390i $$0.337431\pi$$
$$810$$ 0 0
$$811$$ 6.78415 0.238224 0.119112 0.992881i $$-0.461995\pi$$
0.119112 + 0.992881i $$0.461995\pi$$
$$812$$ 56.0830 1.96813
$$813$$ 0 0
$$814$$ −3.80642 −0.133415
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −40.0000 −1.39942
$$818$$ −13.5585 −0.474060
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −3.62269 −0.126433 −0.0632164 0.998000i $$-0.520136\pi$$
−0.0632164 + 0.998000i $$0.520136\pi$$
$$822$$ 0 0
$$823$$ −42.0642 −1.46627 −0.733134 0.680085i $$-0.761943\pi$$
−0.733134 + 0.680085i $$0.761943\pi$$
$$824$$ −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 0
$$831$$ 0 0
$$832$$ −2.95316 −0.102382
$$833$$ −65.3876 −2.26555
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −11.4380 −0.395592
$$837$$ 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
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4.42864 0.152170
$$848$$ 50.0830 1.71986
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 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 $$0.654262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −65.2899 −2.22378
$$863$$ −49.5308 −1.68605 −0.843024 0.537875i $$-0.819227\pi$$
−0.843024 + 0.537875i $$0.819227\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −27.5397 −0.935838
$$867$$ 0 0
$$868$$ 19.7966 0.671940
$$869$$ −8.56199 −0.290446
$$870$$ 0 0
$$871$$ 0.774305 0.0262363
$$872$$ 14.1748 0.480021
$$873$$ 0 0
$$874$$ 118.858 4.02044
$$875$$ 0 0
$$876$$ 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 $$-0.330029\pi$$
0.508963 + 0.860789i $$0.330029\pi$$
$$884$$ −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
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −14.3684 −0.481090
$$893$$ −19.4291 −0.650171
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 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
$$902$$ −0.368416 −0.0122669
$$903$$ 0 0
$$904$$ 4.31402 0.143482
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 53.2641 1.76861 0.884303 0.466913i $$-0.154634\pi$$
0.884303 + 0.466913i $$0.154634\pi$$
$$908$$ 21.7017 0.720195
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0.590573 0.0195665 0.00978327 0.999952i $$-0.496886\pi$$
0.00978327 + 0.999952i $$0.496886\pi$$
$$912$$ 0 0
$$913$$ −0.133353 −0.00441334
$$914$$ 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 $$-0.126963\pi$$
0.921502 + 0.388375i $$0.126963\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 54.9403 1.80936
$$923$$ −1.71456 −0.0564354
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 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
$$940$$ 0 0
$$941$$ −10.4157 −0.339543 −0.169772 0.985483i $$-0.554303\pi$$
−0.169772 + 0.985483i $$0.554303\pi$$
$$942$$ 0 0
$$943$$ 1.71456 0.0558337
$$944$$ −22.4059 −0.729250
$$945$$ 0 0
$$946$$ 10.7971 0.351043
$$947$$ 8.47013 0.275242 0.137621 0.990485i $$-0.456054\pi$$
0.137621 + 0.990485i $$0.456054\pi$$
$$948$$ 0 0
$$949$$ −2.63512 −0.0855397
$$950$$ 0 0
$$951$$ 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
$$955$$ 0 0
$$956$$ 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
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −44.2449 −1.42282 −0.711410 0.702777i $$-0.751943\pi$$
−0.711410 + 0.702777i $$0.751943\pi$$
$$968$$ −0.719004 −0.0231097
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 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 $$-0.583693\pi$$
−0.259909 + 0.965633i $$0.583693\pi$$
$$978$$ 0 0
$$979$$ 5.61285 0.179387
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −31.1882 −0.995256
$$983$$ 1.12399 0.0358496 0.0179248 0.999839i $$-0.494294\pi$$
0.0179248 + 0.999839i $$0.494294\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −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
$$995$$ 0 0
$$996$$ 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 2475.2.a.bb.1.3 3
3.2 odd 2 825.2.a.k.1.1 3
5.2 odd 4 2475.2.c.r.199.5 6
5.3 odd 4 2475.2.c.r.199.2 6
5.4 even 2 495.2.a.e.1.1 3
15.2 even 4 825.2.c.g.199.2 6
15.8 even 4 825.2.c.g.199.5 6
15.14 odd 2 165.2.a.c.1.3 3
20.19 odd 2 7920.2.a.cj.1.3 3
33.32 even 2 9075.2.a.cf.1.3 3
55.54 odd 2 5445.2.a.z.1.3 3
60.59 even 2 2640.2.a.be.1.3 3
105.104 even 2 8085.2.a.bk.1.3 3
165.164 even 2 1815.2.a.m.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 3 15.14 odd 2
495.2.a.e.1.1 3 5.4 even 2
825.2.a.k.1.1 3 3.2 odd 2
825.2.c.g.199.2 6 15.2 even 4
825.2.c.g.199.5 6 15.8 even 4
1815.2.a.m.1.1 3 165.164 even 2
2475.2.a.bb.1.3 3 1.1 even 1 trivial
2475.2.c.r.199.2 6 5.3 odd 4
2475.2.c.r.199.5 6 5.2 odd 4
2640.2.a.be.1.3 3 60.59 even 2
5445.2.a.z.1.3 3 55.54 odd 2
7920.2.a.cj.1.3 3 20.19 odd 2
8085.2.a.bk.1.3 3 105.104 even 2
9075.2.a.cf.1.3 3 33.32 even 2