# Properties

 Label 7920.2.a.cj.1.3 Level $7920$ Weight $2$ Character 7920.1 Self dual yes Analytic conductor $63.242$ 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] = [7920,2,Mod(1,7920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7920.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: $$63.2415184009$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ 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 $$\chi$$ $$=$$ 7920.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} +4.42864 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} +4.42864 q^{7} +1.00000 q^{11} -0.622216 q^{13} +5.18421 q^{17} -7.05086 q^{19} +8.85728 q^{23} +1.00000 q^{25} +7.80642 q^{29} -2.75557 q^{31} -4.42864 q^{35} -2.00000 q^{37} +0.193576 q^{41} -5.67307 q^{43} -2.75557 q^{47} +12.6128 q^{49} +10.8573 q^{53} -1.00000 q^{55} -4.85728 q^{59} +6.85728 q^{61} +0.622216 q^{65} +1.24443 q^{67} +2.75557 q^{71} +4.23506 q^{73} +4.42864 q^{77} -8.56199 q^{79} +0.133353 q^{83} -5.18421 q^{85} -5.61285 q^{89} -2.75557 q^{91} +7.05086 q^{95} +7.24443 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5}+O(q^{10})$$ 3 * q - 3 * q^5 $$3 q - 3 q^{5} + 3 q^{11} - 2 q^{13} + 2 q^{17} - 8 q^{19} + 3 q^{25} + 10 q^{29} - 8 q^{31} - 6 q^{37} + 14 q^{41} - 4 q^{43} - 8 q^{47} + 11 q^{49} + 6 q^{53} - 3 q^{55} + 12 q^{59} - 6 q^{61} + 2 q^{65} + 4 q^{67} + 8 q^{71} - 14 q^{73} - 12 q^{79} - 2 q^{85} + 10 q^{89} - 8 q^{91} + 8 q^{95} + 22 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 + 3 * q^11 - 2 * q^13 + 2 * q^17 - 8 * q^19 + 3 * q^25 + 10 * q^29 - 8 * q^31 - 6 * q^37 + 14 * q^41 - 4 * q^43 - 8 * q^47 + 11 * q^49 + 6 * q^53 - 3 * q^55 + 12 * q^59 - 6 * q^61 + 2 * q^65 + 4 * q^67 + 8 * q^71 - 14 * q^73 - 12 * q^79 - 2 * q^85 + 10 * q^89 - 8 * q^91 + 8 * q^95 + 22 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −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 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ −0.622216 −0.172572 −0.0862858 0.996270i $$-0.527500\pi$$
−0.0862858 + 0.996270i $$0.527500\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.18421 1.25736 0.628678 0.777666i $$-0.283597\pi$$
0.628678 + 0.777666i $$0.283597\pi$$
$$18$$ 0 0
$$19$$ −7.05086 −1.61758 −0.808789 0.588100i $$-0.799876\pi$$
−0.808789 + 0.588100i $$0.799876\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.85728 1.84687 0.923435 0.383754i $$-0.125369\pi$$
0.923435 + 0.383754i $$0.125369\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 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 $$-0.579595\pi$$
−0.247457 + 0.968899i $$0.579595\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.42864 −0.748577
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 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 $$0.642392\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −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
$$52$$ 0 0
$$53$$ 10.8573 1.49136 0.745681 0.666303i $$-0.232124\pi$$
0.745681 + 0.666303i $$0.232124\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.85728 −0.632364 −0.316182 0.948699i $$-0.602401\pi$$
−0.316182 + 0.948699i $$0.602401\pi$$
$$60$$ 0 0
$$61$$ 6.85728 0.877985 0.438992 0.898491i $$-0.355336\pi$$
0.438992 + 0.898491i $$0.355336\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0.622216 0.0771764
$$66$$ 0 0
$$67$$ 1.24443 0.152031 0.0760157 0.997107i $$-0.475780\pi$$
0.0760157 + 0.997107i $$0.475780\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 2.75557 0.327026 0.163513 0.986541i $$-0.447717\pi$$
0.163513 + 0.986541i $$0.447717\pi$$
$$72$$ 0 0
$$73$$ 4.23506 0.495677 0.247838 0.968801i $$-0.420280\pi$$
0.247838 + 0.968801i $$0.420280\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.42864 0.504690
$$78$$ 0 0
$$79$$ −8.56199 −0.963299 −0.481650 0.876364i $$-0.659962\pi$$
−0.481650 + 0.876364i $$0.659962\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 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
$$86$$ 0 0
$$87$$ 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
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 7.05086 0.723402
$$96$$ 0 0
$$97$$ 7.24443 0.735561 0.367780 0.929913i $$-0.380118\pi$$
0.367780 + 0.929913i $$0.380118\pi$$
$$98$$ 0 0
$$99$$ 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 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 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
$$112$$ 0 0
$$113$$ 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
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 22.9590 2.10465
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 15.1842 1.34738 0.673690 0.739014i $$-0.264708\pi$$
0.673690 + 0.739014i $$0.264708\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 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
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 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 $$0.772441\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −0.622216 −0.0520323
$$144$$ 0 0
$$145$$ −7.80642 −0.648288
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 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 $$-0.335026\pi$$
0.495387 + 0.868672i $$0.335026\pi$$
$$152$$ 0 0
$$153$$ 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 $$0.236224\pi$$
$$158$$ 0 0
$$159$$ 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 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −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
$$172$$ 0 0
$$173$$ 9.18421 0.698262 0.349131 0.937074i $$-0.386477\pi$$
0.349131 + 0.937074i $$0.386477\pi$$
$$174$$ 0 0
$$175$$ 4.42864 0.334774
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −25.3274 −1.89306 −0.946530 0.322617i $$-0.895437\pi$$
−0.946530 + 0.322617i $$0.895437\pi$$
$$180$$ 0 0
$$181$$ −13.6128 −1.01184 −0.505918 0.862582i $$-0.668846\pi$$
−0.505918 + 0.862582i $$0.668846\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 5.18421 0.379107
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.10171 0.441504 0.220752 0.975330i $$-0.429149\pi$$
0.220752 + 0.975330i $$0.429149\pi$$
$$192$$ 0 0
$$193$$ 18.3368 1.31991 0.659955 0.751305i $$-0.270575\pi$$
0.659955 + 0.751305i $$0.270575\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.69535 0.477024 0.238512 0.971140i $$-0.423340\pi$$
0.238512 + 0.971140i $$0.423340\pi$$
$$198$$ 0 0
$$199$$ −14.1017 −0.999644 −0.499822 0.866128i $$-0.666601\pi$$
−0.499822 + 0.866128i $$0.666601\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 34.5718 2.42647
$$204$$ 0 0
$$205$$ −0.193576 −0.0135199
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −7.05086 −0.487718
$$210$$ 0 0
$$211$$ 10.6637 0.734120 0.367060 0.930197i $$-0.380364\pi$$
0.367060 + 0.930197i $$0.380364\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 5.67307 0.386900
$$216$$ 0 0
$$217$$ −12.2034 −0.828422
$$218$$ 0 0
$$219$$ 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 $$0.595841\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 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
$$232$$ 0 0
$$233$$ 4.32693 0.283467 0.141733 0.989905i $$-0.454732\pi$$
0.141733 + 0.989905i $$0.454732\pi$$
$$234$$ 0 0
$$235$$ 2.75557 0.179753
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −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 $$-0.513805\pi$$
−0.0433563 + 0.999060i $$0.513805\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −12.6128 −0.805805
$$246$$ 0 0
$$247$$ 4.38715 0.279148
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 22.7556 1.43632 0.718159 0.695879i $$-0.244985\pi$$
0.718159 + 0.695879i $$0.244985\pi$$
$$252$$ 0 0
$$253$$ 8.85728 0.556852
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 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
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −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
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −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 $$0.646931\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ 14.6035 0.877438 0.438719 0.898624i $$-0.355432\pi$$
0.438719 + 0.898624i $$0.355432\pi$$
$$278$$ 0 0
$$279$$ 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 $$0.799432\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0.857279 0.0506036
$$288$$ 0 0
$$289$$ 9.87601 0.580942
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 2.81579 0.164500 0.0822502 0.996612i $$-0.473789\pi$$
0.0822502 + 0.996612i $$0.473789\pi$$
$$294$$ 0 0
$$295$$ 4.85728 0.282802
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −5.51114 −0.318717
$$300$$ 0 0
$$301$$ −25.1240 −1.44812
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −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
$$310$$ 0 0
$$311$$ 19.8796 1.12727 0.563633 0.826025i $$-0.309403\pi$$
0.563633 + 0.826025i $$0.309403\pi$$
$$312$$ 0 0
$$313$$ −15.7146 −0.888239 −0.444120 0.895967i $$-0.646483\pi$$
−0.444120 + 0.895967i $$0.646483\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −16.4889 −0.926107 −0.463053 0.886330i $$-0.653246\pi$$
−0.463053 + 0.886330i $$0.653246\pi$$
$$318$$ 0 0
$$319$$ 7.80642 0.437076
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −36.5531 −2.03387
$$324$$ 0 0
$$325$$ −0.622216 −0.0345143
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −12.2034 −0.672796
$$330$$ 0 0
$$331$$ −15.3461 −0.843500 −0.421750 0.906712i $$-0.638584\pi$$
−0.421750 + 0.906712i $$0.638584\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −1.24443 −0.0679905
$$336$$ 0 0
$$337$$ 28.2351 1.53806 0.769031 0.639212i $$-0.220739\pi$$
0.769031 + 0.639212i $$0.220739\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2.75557 −0.149222
$$342$$ 0 0
$$343$$ 24.8573 1.34217
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 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
$$352$$ 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
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 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
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −4.23506 −0.221673
$$366$$ 0 0
$$367$$ −33.7975 −1.76422 −0.882108 0.471046i $$-0.843876\pi$$
−0.882108 + 0.471046i $$0.843876\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 48.0830 2.49634
$$372$$ 0 0
$$373$$ 33.9496 1.75784 0.878922 0.476965i $$-0.158263\pi$$
0.878922 + 0.476965i $$0.158263\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.85728 −0.250163
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −14.6351 −0.747820 −0.373910 0.927465i $$-0.621983\pi$$
−0.373910 + 0.927465i $$0.621983\pi$$
$$384$$ 0 0
$$385$$ −4.42864 −0.225704
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 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
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 8.56199 0.430801
$$396$$ 0 0
$$397$$ −12.7556 −0.640184 −0.320092 0.947387i $$-0.603714\pi$$
−0.320092 + 0.947387i $$0.603714\pi$$
$$398$$ 0 0
$$399$$ 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
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −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
$$412$$ 0 0
$$413$$ −21.5111 −1.05849
$$414$$ 0 0
$$415$$ −0.133353 −0.00654605
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 15.6128 0.762738 0.381369 0.924423i $$-0.375453\pi$$
0.381369 + 0.924423i $$0.375453\pi$$
$$420$$ 0 0
$$421$$ 7.89829 0.384939 0.192470 0.981303i $$-0.438350\pi$$
0.192470 + 0.981303i $$0.438350\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 5.18421 0.251471
$$426$$ 0 0
$$427$$ 30.3684 1.46963
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 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 $$0.386964\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −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
$$442$$ 0 0
$$443$$ −13.1240 −0.623539 −0.311770 0.950158i $$-0.600922\pi$$
−0.311770 + 0.950158i $$0.600922\pi$$
$$444$$ 0 0
$$445$$ 5.61285 0.266074
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 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
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 2.75557 0.129183
$$456$$ 0 0
$$457$$ −23.4608 −1.09745 −0.548724 0.836004i $$-0.684886\pi$$
−0.548724 + 0.836004i $$0.684886\pi$$
$$458$$ 0 0
$$459$$ 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 $$0.351586\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 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
$$472$$ 0 0
$$473$$ −5.67307 −0.260848
$$474$$ 0 0
$$475$$ −7.05086 −0.323515
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −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
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −7.24443 −0.328953
$$486$$ 0 0
$$487$$ 11.5299 0.522468 0.261234 0.965275i $$-0.415871\pi$$
0.261234 + 0.965275i $$0.415871\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 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
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 12.2034 0.547398
$$498$$ 0 0
$$499$$ 25.3274 1.13381 0.566905 0.823783i $$-0.308141\pi$$
0.566905 + 0.823783i $$0.308141\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 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
$$508$$ 0 0
$$509$$ 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
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −11.6128 −0.511723
$$516$$ 0 0
$$517$$ −2.75557 −0.121190
$$518$$ 0 0
$$519$$ 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 $$0.546895\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −14.2854 −0.622284
$$528$$ 0 0
$$529$$ 55.4514 2.41093
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −0.120446 −0.00521710
$$534$$ 0 0
$$535$$ −2.62222 −0.113368
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 12.6128 0.543274
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 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 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −55.0420 −2.34487
$$552$$ 0 0
$$553$$ −37.9180 −1.61244
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 20.7971 0.881200 0.440600 0.897704i $$-0.354766\pi$$
0.440600 + 0.897704i $$0.354766\pi$$
$$558$$ 0 0
$$559$$ 3.52987 0.149298
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 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
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 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 $$-0.778138\pi$$
−0.766772 + 0.641919i $$0.778138\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 8.85728 0.369374
$$576$$ 0 0
$$577$$ 4.22216 0.175771 0.0878853 0.996131i $$-0.471989\pi$$
0.0878853 + 0.996131i $$0.471989\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0.590573 0.0245011
$$582$$ 0 0
$$583$$ 10.8573 0.449663
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 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
$$592$$ 0 0
$$593$$ −27.9398 −1.14735 −0.573675 0.819083i $$-0.694483\pi$$
−0.573675 + 0.819083i $$0.694483\pi$$
$$594$$ 0 0
$$595$$ −22.9590 −0.941227
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −31.2257 −1.27585 −0.637924 0.770100i $$-0.720206\pi$$
−0.637924 + 0.770100i $$0.720206\pi$$
$$600$$ 0 0
$$601$$ −8.75557 −0.357147 −0.178574 0.983927i $$-0.557148\pi$$
−0.178574 + 0.983927i $$0.557148\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −1.00000 −0.0406558
$$606$$ 0 0
$$607$$ 15.1842 0.616308 0.308154 0.951336i $$-0.400289\pi$$
0.308154 + 0.951336i $$0.400289\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.71456 0.0693636
$$612$$ 0 0
$$613$$ 42.7239 1.72560 0.862802 0.505543i $$-0.168708\pi$$
0.862802 + 0.505543i $$0.168708\pi$$
$$614$$ 0 0
$$615$$ 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 $$-0.615037\pi$$
−0.353584 + 0.935403i $$0.615037\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −24.8573 −0.995886
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −10.3684 −0.413416
$$630$$ 0 0
$$631$$ −15.8163 −0.629636 −0.314818 0.949152i $$-0.601943\pi$$
−0.314818 + 0.949152i $$0.601943\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −15.1842 −0.602567
$$636$$ 0 0
$$637$$ −7.84791 −0.310946
$$638$$ 0 0
$$639$$ 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 $$0.616177\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −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
$$652$$ 0 0
$$653$$ −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 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −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 $$0.681192\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 31.2257 1.21088
$$666$$ 0 0
$$667$$ 69.1437 2.67725
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 6.85728 0.264722
$$672$$ 0 0
$$673$$ −9.86665 −0.380331 −0.190166 0.981752i $$-0.560903\pi$$
−0.190166 + 0.981752i $$0.560903\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 5.65433 0.217314 0.108657 0.994079i $$-0.465345\pi$$
0.108657 + 0.994079i $$0.465345\pi$$
$$678$$ 0 0
$$679$$ 32.0830 1.23123
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 34.1847 1.30804 0.654020 0.756477i $$-0.273081\pi$$
0.654020 + 0.756477i $$0.273081\pi$$
$$684$$ 0 0
$$685$$ −0.488863 −0.0186785
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −6.75557 −0.257367
$$690$$ 0 0
$$691$$ 19.2257 0.731380 0.365690 0.930737i $$-0.380833\pi$$
0.365690 + 0.930737i $$0.380833\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 17.8064 0.675436
$$696$$ 0 0
$$697$$ 1.00354 0.0380118
$$698$$ 0 0
$$699$$ 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
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −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
$$712$$ 0 0
$$713$$ −24.4068 −0.914043
$$714$$ 0 0
$$715$$ 0.622216 0.0232695
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 23.8163 0.888197 0.444098 0.895978i $$-0.353524\pi$$
0.444098 + 0.895978i $$0.353524\pi$$
$$720$$ 0 0
$$721$$ 51.4291 1.91532
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 7.80642 0.289923
$$726$$ 0 0
$$727$$ 32.9403 1.22169 0.610843 0.791752i $$-0.290831\pi$$
0.610843 + 0.791752i $$0.290831\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −29.4104 −1.08778
$$732$$ 0 0
$$733$$ −29.8666 −1.10315 −0.551575 0.834125i $$-0.685973\pi$$
−0.551575 + 0.834125i $$0.685973\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.24443 0.0458392
$$738$$ 0 0
$$739$$ 5.06959 0.186488 0.0932440 0.995643i $$-0.470276\pi$$
0.0932440 + 0.995643i $$0.470276\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 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
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 11.6128 0.424324
$$750$$ 0 0
$$751$$ 6.63512 0.242119 0.121060 0.992645i $$-0.461371\pi$$
0.121060 + 0.992645i $$0.461371\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −12.1748 −0.443088
$$756$$ 0 0
$$757$$ 8.75557 0.318227 0.159113 0.987260i $$-0.449136\pi$$
0.159113 + 0.987260i $$0.449136\pi$$
$$758$$ 0 0
$$759$$ 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
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 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
$$772$$ 0 0
$$773$$ −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
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1.36488 −0.0489018
$$780$$ 0 0
$$781$$ 2.75557 0.0986020
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −18.4701 −0.659227
$$786$$ 0 0
$$787$$ 11.2672 0.401632 0.200816 0.979629i $$-0.435641\pi$$
0.200816 + 0.979629i $$0.435641\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 26.5718 0.944786
$$792$$ 0 0
$$793$$ −4.26671 −0.151515
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −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
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 4.23506 0.149452
$$804$$ 0 0
$$805$$ −39.2257 −1.38252
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 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 $$-0.538005\pi$$
−0.119112 + 0.992881i $$0.538005\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −10.1017 −0.353847
$$816$$ 0 0
$$817$$ 40.0000 1.39942
$$818$$ 0 0
$$819$$ 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 $$0.761943\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 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
$$832$$ 0 0
$$833$$ 65.3876 2.26555
$$834$$ 0 0
$$835$$ 16.3368 0.565357
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 3.34614 0.115522 0.0577608 0.998330i $$-0.481604\pi$$
0.0577608 + 0.998330i $$0.481604\pi$$
$$840$$ 0 0
$$841$$ 31.9403 1.10139
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 12.6128 0.433895
$$846$$ 0 0
$$847$$ 4.42864 0.152170
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −17.7146 −0.607247
$$852$$ 0 0
$$853$$ −26.4197 −0.904595 −0.452297 0.891867i $$-0.649395\pi$$
−0.452297 + 0.891867i $$0.649395\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −38.7783 −1.32464 −0.662321 0.749220i $$-0.730429\pi$$
−0.662321 + 0.749220i $$0.730429\pi$$
$$858$$ 0 0
$$859$$ 27.3087 0.931760 0.465880 0.884848i $$-0.345738\pi$$
0.465880 + 0.884848i $$0.345738\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −49.5308 −1.68605 −0.843024 0.537875i $$-0.819227\pi$$
−0.843024 + 0.537875i $$0.819227\pi$$
$$864$$ 0 0
$$865$$ −9.18421 −0.312272
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −8.56199 −0.290446
$$870$$ 0 0
$$871$$ −0.774305 −0.0262363
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −4.42864 −0.149715
$$876$$ 0 0
$$877$$ −4.50177 −0.152014 −0.0760070 0.997107i $$-0.524217\pi$$
−0.0760070 + 0.997107i $$0.524217\pi$$
$$878$$ 0 0
$$879$$ 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 $$0.330029\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 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
$$892$$ 0 0
$$893$$ 19.4291 0.650171
$$894$$ 0 0
$$895$$ 25.3274 0.846602
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −21.5111 −0.717437
$$900$$ 0 0
$$901$$ 56.2864 1.87517
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 13.6128 0.452506
$$906$$ 0 0
$$907$$ 53.2641 1.76861 0.884303 0.466913i $$-0.154634\pi$$
0.884303 + 0.466913i $$0.154634\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −0.590573 −0.0195665 −0.00978327 0.999952i $$-0.503114\pi$$
−0.00978327 + 0.999952i $$0.503114\pi$$
$$912$$ 0 0
$$913$$ 0.133353 0.00441334
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 5.51114 0.181994
$$918$$ 0 0
$$919$$ −55.8707 −1.84300 −0.921502 0.388375i $$-0.873037\pi$$
−0.921502 + 0.388375i $$0.873037\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −1.71456 −0.0564354
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −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
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −5.18421 −0.169542
$$936$$ 0 0
$$937$$ −27.8479 −0.909752 −0.454876 0.890555i $$-0.650316\pi$$
−0.454876 + 0.890555i $$0.650316\pi$$
$$938$$ 0 0
$$939$$ 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
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 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
$$952$$ 0 0
$$953$$ 8.71408 0.282277 0.141138 0.989990i $$-0.454924\pi$$
0.141138 + 0.989990i $$0.454924\pi$$
$$954$$ 0 0
$$955$$ −6.10171 −0.197447
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 2.16500 0.0699114
$$960$$ 0 0
$$961$$ −23.4068 −0.755059
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −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
$$970$$ 0 0
$$971$$ −57.1437 −1.83383 −0.916914 0.399085i $$-0.869328\pi$$
−0.916914 + 0.399085i $$0.869328\pi$$
$$972$$ 0 0
$$973$$ −78.8582 −2.52808
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 16.2480 0.519819 0.259909 0.965633i $$-0.416307\pi$$
0.259909 + 0.965633i $$0.416307\pi$$
$$978$$ 0 0
$$979$$ −5.61285 −0.179387
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 1.12399 0.0358496 0.0179248 0.999839i $$-0.494294\pi$$
0.0179248 + 0.999839i $$0.494294\pi$$
$$984$$ 0 0
$$985$$ −6.69535 −0.213331
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −50.2480 −1.59779
$$990$$ 0 0
$$991$$ 53.6513 1.70429 0.852144 0.523307i $$-0.175302\pi$$
0.852144 + 0.523307i $$0.175302\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 14.1017 0.447054
$$996$$ 0 0
$$997$$ −35.7275 −1.13150 −0.565750 0.824577i $$-0.691413\pi$$
−0.565750 + 0.824577i $$0.691413\pi$$
$$998$$ 0 0
$$999$$ 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 7920.2.a.cj.1.3 3
3.2 odd 2 2640.2.a.be.1.3 3
4.3 odd 2 495.2.a.e.1.1 3
12.11 even 2 165.2.a.c.1.3 3
20.3 even 4 2475.2.c.r.199.5 6
20.7 even 4 2475.2.c.r.199.2 6
20.19 odd 2 2475.2.a.bb.1.3 3
44.43 even 2 5445.2.a.z.1.3 3
60.23 odd 4 825.2.c.g.199.2 6
60.47 odd 4 825.2.c.g.199.5 6
60.59 even 2 825.2.a.k.1.1 3
84.83 odd 2 8085.2.a.bk.1.3 3
132.131 odd 2 1815.2.a.m.1.1 3
660.659 odd 2 9075.2.a.cf.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 3 12.11 even 2
495.2.a.e.1.1 3 4.3 odd 2
825.2.a.k.1.1 3 60.59 even 2
825.2.c.g.199.2 6 60.23 odd 4
825.2.c.g.199.5 6 60.47 odd 4
1815.2.a.m.1.1 3 132.131 odd 2
2475.2.a.bb.1.3 3 20.19 odd 2
2475.2.c.r.199.2 6 20.7 even 4
2475.2.c.r.199.5 6 20.3 even 4
2640.2.a.be.1.3 3 3.2 odd 2
5445.2.a.z.1.3 3 44.43 even 2
7920.2.a.cj.1.3 3 1.1 even 1 trivial
8085.2.a.bk.1.3 3 84.83 odd 2
9075.2.a.cf.1.3 3 660.659 odd 2