# Properties

 Label 2368.2.a.s.1.2 Level $2368$ Weight $2$ Character 2368.1 Self dual yes Analytic conductor $18.909$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, 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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2368 = 2^{6} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2368.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: $$18.9085751986$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 2368.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.302776 q^{3} -1.30278 q^{5} +4.60555 q^{7} -2.90833 q^{9} +O(q^{10})$$ $$q+0.302776 q^{3} -1.30278 q^{5} +4.60555 q^{7} -2.90833 q^{9} -1.30278 q^{11} +2.30278 q^{13} -0.394449 q^{15} -6.00000 q^{17} -2.00000 q^{19} +1.39445 q^{21} -6.90833 q^{23} -3.30278 q^{25} -1.78890 q^{27} -6.90833 q^{29} +3.30278 q^{31} -0.394449 q^{33} -6.00000 q^{35} -1.00000 q^{37} +0.697224 q^{39} -0.908327 q^{41} +6.60555 q^{43} +3.78890 q^{45} -2.60555 q^{47} +14.2111 q^{49} -1.81665 q^{51} +6.00000 q^{53} +1.69722 q^{55} -0.605551 q^{57} -3.39445 q^{59} +10.5139 q^{61} -13.3944 q^{63} -3.00000 q^{65} -14.5139 q^{67} -2.09167 q^{69} +6.00000 q^{71} -8.69722 q^{73} -1.00000 q^{75} -6.00000 q^{77} -16.1194 q^{79} +8.18335 q^{81} -17.2111 q^{83} +7.81665 q^{85} -2.09167 q^{87} +5.21110 q^{89} +10.6056 q^{91} +1.00000 q^{93} +2.60555 q^{95} +12.4222 q^{97} +3.78890 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} + q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 + q^5 + 2 * q^7 + 5 * q^9 $$2 q - 3 q^{3} + q^{5} + 2 q^{7} + 5 q^{9} + q^{11} + q^{13} - 8 q^{15} - 12 q^{17} - 4 q^{19} + 10 q^{21} - 3 q^{23} - 3 q^{25} - 18 q^{27} - 3 q^{29} + 3 q^{31} - 8 q^{33} - 12 q^{35} - 2 q^{37} + 5 q^{39} + 9 q^{41} + 6 q^{43} + 22 q^{45} + 2 q^{47} + 14 q^{49} + 18 q^{51} + 12 q^{53} + 7 q^{55} + 6 q^{57} - 14 q^{59} + 3 q^{61} - 34 q^{63} - 6 q^{65} - 11 q^{67} - 15 q^{69} + 12 q^{71} - 21 q^{73} - 2 q^{75} - 12 q^{77} - 7 q^{79} + 38 q^{81} - 20 q^{83} - 6 q^{85} - 15 q^{87} - 4 q^{89} + 14 q^{91} + 2 q^{93} - 2 q^{95} - 4 q^{97} + 22 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 + q^5 + 2 * q^7 + 5 * q^9 + q^11 + q^13 - 8 * q^15 - 12 * q^17 - 4 * q^19 + 10 * q^21 - 3 * q^23 - 3 * q^25 - 18 * q^27 - 3 * q^29 + 3 * q^31 - 8 * q^33 - 12 * q^35 - 2 * q^37 + 5 * q^39 + 9 * q^41 + 6 * q^43 + 22 * q^45 + 2 * q^47 + 14 * q^49 + 18 * q^51 + 12 * q^53 + 7 * q^55 + 6 * q^57 - 14 * q^59 + 3 * q^61 - 34 * q^63 - 6 * q^65 - 11 * q^67 - 15 * q^69 + 12 * q^71 - 21 * q^73 - 2 * q^75 - 12 * q^77 - 7 * q^79 + 38 * q^81 - 20 * q^83 - 6 * q^85 - 15 * q^87 - 4 * q^89 + 14 * q^91 + 2 * q^93 - 2 * q^95 - 4 * q^97 + 22 * q^99

## 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
$$3$$ 0.302776 0.174808 0.0874038 0.996173i $$-0.472143\pi$$
0.0874038 + 0.996173i $$0.472143\pi$$
$$4$$ 0 0
$$5$$ −1.30278 −0.582619 −0.291309 0.956629i $$-0.594091\pi$$
−0.291309 + 0.956629i $$0.594091\pi$$
$$6$$ 0 0
$$7$$ 4.60555 1.74073 0.870367 0.492403i $$-0.163881\pi$$
0.870367 + 0.492403i $$0.163881\pi$$
$$8$$ 0 0
$$9$$ −2.90833 −0.969442
$$10$$ 0 0
$$11$$ −1.30278 −0.392802 −0.196401 0.980524i $$-0.562925\pi$$
−0.196401 + 0.980524i $$0.562925\pi$$
$$12$$ 0 0
$$13$$ 2.30278 0.638675 0.319338 0.947641i $$-0.396540\pi$$
0.319338 + 0.947641i $$0.396540\pi$$
$$14$$ 0 0
$$15$$ −0.394449 −0.101846
$$16$$ 0 0
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 1.39445 0.304294
$$22$$ 0 0
$$23$$ −6.90833 −1.44049 −0.720243 0.693722i $$-0.755970\pi$$
−0.720243 + 0.693722i $$0.755970\pi$$
$$24$$ 0 0
$$25$$ −3.30278 −0.660555
$$26$$ 0 0
$$27$$ −1.78890 −0.344273
$$28$$ 0 0
$$29$$ −6.90833 −1.28284 −0.641422 0.767188i $$-0.721655\pi$$
−0.641422 + 0.767188i $$0.721655\pi$$
$$30$$ 0 0
$$31$$ 3.30278 0.593196 0.296598 0.955002i $$-0.404148\pi$$
0.296598 + 0.955002i $$0.404148\pi$$
$$32$$ 0 0
$$33$$ −0.394449 −0.0686647
$$34$$ 0 0
$$35$$ −6.00000 −1.01419
$$36$$ 0 0
$$37$$ −1.00000 −0.164399
$$38$$ 0 0
$$39$$ 0.697224 0.111645
$$40$$ 0 0
$$41$$ −0.908327 −0.141857 −0.0709284 0.997481i $$-0.522596\pi$$
−0.0709284 + 0.997481i $$0.522596\pi$$
$$42$$ 0 0
$$43$$ 6.60555 1.00734 0.503669 0.863897i $$-0.331983\pi$$
0.503669 + 0.863897i $$0.331983\pi$$
$$44$$ 0 0
$$45$$ 3.78890 0.564815
$$46$$ 0 0
$$47$$ −2.60555 −0.380059 −0.190029 0.981778i $$-0.560858\pi$$
−0.190029 + 0.981778i $$0.560858\pi$$
$$48$$ 0 0
$$49$$ 14.2111 2.03016
$$50$$ 0 0
$$51$$ −1.81665 −0.254382
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 1.69722 0.228854
$$56$$ 0 0
$$57$$ −0.605551 −0.0802072
$$58$$ 0 0
$$59$$ −3.39445 −0.441920 −0.220960 0.975283i $$-0.570919\pi$$
−0.220960 + 0.975283i $$0.570919\pi$$
$$60$$ 0 0
$$61$$ 10.5139 1.34616 0.673082 0.739568i $$-0.264970\pi$$
0.673082 + 0.739568i $$0.264970\pi$$
$$62$$ 0 0
$$63$$ −13.3944 −1.68754
$$64$$ 0 0
$$65$$ −3.00000 −0.372104
$$66$$ 0 0
$$67$$ −14.5139 −1.77315 −0.886576 0.462583i $$-0.846923\pi$$
−0.886576 + 0.462583i $$0.846923\pi$$
$$68$$ 0 0
$$69$$ −2.09167 −0.251808
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ −8.69722 −1.01793 −0.508967 0.860786i $$-0.669972\pi$$
−0.508967 + 0.860786i $$0.669972\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −6.00000 −0.683763
$$78$$ 0 0
$$79$$ −16.1194 −1.81358 −0.906789 0.421585i $$-0.861474\pi$$
−0.906789 + 0.421585i $$0.861474\pi$$
$$80$$ 0 0
$$81$$ 8.18335 0.909261
$$82$$ 0 0
$$83$$ −17.2111 −1.88916 −0.944582 0.328276i $$-0.893533\pi$$
−0.944582 + 0.328276i $$0.893533\pi$$
$$84$$ 0 0
$$85$$ 7.81665 0.847835
$$86$$ 0 0
$$87$$ −2.09167 −0.224251
$$88$$ 0 0
$$89$$ 5.21110 0.552376 0.276188 0.961104i $$-0.410929\pi$$
0.276188 + 0.961104i $$0.410929\pi$$
$$90$$ 0 0
$$91$$ 10.6056 1.11176
$$92$$ 0 0
$$93$$ 1.00000 0.103695
$$94$$ 0 0
$$95$$ 2.60555 0.267324
$$96$$ 0 0
$$97$$ 12.4222 1.26128 0.630642 0.776074i $$-0.282792\pi$$
0.630642 + 0.776074i $$0.282792\pi$$
$$98$$ 0 0
$$99$$ 3.78890 0.380799
$$100$$ 0 0
$$101$$ −16.4222 −1.63407 −0.817035 0.576588i $$-0.804384\pi$$
−0.817035 + 0.576588i $$0.804384\pi$$
$$102$$ 0 0
$$103$$ 3.30278 0.325432 0.162716 0.986673i $$-0.447975\pi$$
0.162716 + 0.986673i $$0.447975\pi$$
$$104$$ 0 0
$$105$$ −1.81665 −0.177287
$$106$$ 0 0
$$107$$ −4.30278 −0.415965 −0.207983 0.978133i $$-0.566690\pi$$
−0.207983 + 0.978133i $$0.566690\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −0.302776 −0.0287382
$$112$$ 0 0
$$113$$ 11.2111 1.05465 0.527326 0.849663i $$-0.323195\pi$$
0.527326 + 0.849663i $$0.323195\pi$$
$$114$$ 0 0
$$115$$ 9.00000 0.839254
$$116$$ 0 0
$$117$$ −6.69722 −0.619159
$$118$$ 0 0
$$119$$ −27.6333 −2.53314
$$120$$ 0 0
$$121$$ −9.30278 −0.845707
$$122$$ 0 0
$$123$$ −0.275019 −0.0247977
$$124$$ 0 0
$$125$$ 10.8167 0.967471
$$126$$ 0 0
$$127$$ −4.78890 −0.424946 −0.212473 0.977167i $$-0.568152\pi$$
−0.212473 + 0.977167i $$0.568152\pi$$
$$128$$ 0 0
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ −3.39445 −0.296574 −0.148287 0.988944i $$-0.547376\pi$$
−0.148287 + 0.988944i $$0.547376\pi$$
$$132$$ 0 0
$$133$$ −9.21110 −0.798704
$$134$$ 0 0
$$135$$ 2.33053 0.200580
$$136$$ 0 0
$$137$$ −9.90833 −0.846525 −0.423263 0.906007i $$-0.639115\pi$$
−0.423263 + 0.906007i $$0.639115\pi$$
$$138$$ 0 0
$$139$$ −8.90833 −0.755594 −0.377797 0.925888i $$-0.623318\pi$$
−0.377797 + 0.925888i $$0.623318\pi$$
$$140$$ 0 0
$$141$$ −0.788897 −0.0664372
$$142$$ 0 0
$$143$$ −3.00000 −0.250873
$$144$$ 0 0
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ 4.30278 0.354887
$$148$$ 0 0
$$149$$ 1.81665 0.148826 0.0744130 0.997228i $$-0.476292\pi$$
0.0744130 + 0.997228i $$0.476292\pi$$
$$150$$ 0 0
$$151$$ −13.3944 −1.09002 −0.545012 0.838428i $$-0.683475\pi$$
−0.545012 + 0.838428i $$0.683475\pi$$
$$152$$ 0 0
$$153$$ 17.4500 1.41075
$$154$$ 0 0
$$155$$ −4.30278 −0.345607
$$156$$ 0 0
$$157$$ −7.21110 −0.575509 −0.287754 0.957704i $$-0.592909\pi$$
−0.287754 + 0.957704i $$0.592909\pi$$
$$158$$ 0 0
$$159$$ 1.81665 0.144070
$$160$$ 0 0
$$161$$ −31.8167 −2.50750
$$162$$ 0 0
$$163$$ 20.4222 1.59959 0.799795 0.600273i $$-0.204941\pi$$
0.799795 + 0.600273i $$0.204941\pi$$
$$164$$ 0 0
$$165$$ 0.513878 0.0400054
$$166$$ 0 0
$$167$$ 12.5139 0.968353 0.484176 0.874970i $$-0.339119\pi$$
0.484176 + 0.874970i $$0.339119\pi$$
$$168$$ 0 0
$$169$$ −7.69722 −0.592094
$$170$$ 0 0
$$171$$ 5.81665 0.444811
$$172$$ 0 0
$$173$$ 23.2111 1.76471 0.882354 0.470587i $$-0.155958\pi$$
0.882354 + 0.470587i $$0.155958\pi$$
$$174$$ 0 0
$$175$$ −15.2111 −1.14985
$$176$$ 0 0
$$177$$ −1.02776 −0.0772509
$$178$$ 0 0
$$179$$ −7.81665 −0.584244 −0.292122 0.956381i $$-0.594361\pi$$
−0.292122 + 0.956381i $$0.594361\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ 3.18335 0.235320
$$184$$ 0 0
$$185$$ 1.30278 0.0957820
$$186$$ 0 0
$$187$$ 7.81665 0.571610
$$188$$ 0 0
$$189$$ −8.23886 −0.599289
$$190$$ 0 0
$$191$$ 12.5139 0.905472 0.452736 0.891644i $$-0.350448\pi$$
0.452736 + 0.891644i $$0.350448\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ −0.908327 −0.0650466
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −2.42221 −0.171706 −0.0858528 0.996308i $$-0.527361\pi$$
−0.0858528 + 0.996308i $$0.527361\pi$$
$$200$$ 0 0
$$201$$ −4.39445 −0.309961
$$202$$ 0 0
$$203$$ −31.8167 −2.23309
$$204$$ 0 0
$$205$$ 1.18335 0.0826485
$$206$$ 0 0
$$207$$ 20.0917 1.39647
$$208$$ 0 0
$$209$$ 2.60555 0.180230
$$210$$ 0 0
$$211$$ −6.69722 −0.461056 −0.230528 0.973066i $$-0.574045\pi$$
−0.230528 + 0.973066i $$0.574045\pi$$
$$212$$ 0 0
$$213$$ 1.81665 0.124475
$$214$$ 0 0
$$215$$ −8.60555 −0.586894
$$216$$ 0 0
$$217$$ 15.2111 1.03260
$$218$$ 0 0
$$219$$ −2.63331 −0.177942
$$220$$ 0 0
$$221$$ −13.8167 −0.929409
$$222$$ 0 0
$$223$$ 15.8167 1.05916 0.529581 0.848260i $$-0.322349\pi$$
0.529581 + 0.848260i $$0.322349\pi$$
$$224$$ 0 0
$$225$$ 9.60555 0.640370
$$226$$ 0 0
$$227$$ 7.81665 0.518810 0.259405 0.965769i $$-0.416474\pi$$
0.259405 + 0.965769i $$0.416474\pi$$
$$228$$ 0 0
$$229$$ −17.3944 −1.14946 −0.574729 0.818344i $$-0.694892\pi$$
−0.574729 + 0.818344i $$0.694892\pi$$
$$230$$ 0 0
$$231$$ −1.81665 −0.119527
$$232$$ 0 0
$$233$$ −9.51388 −0.623275 −0.311637 0.950201i $$-0.600877\pi$$
−0.311637 + 0.950201i $$0.600877\pi$$
$$234$$ 0 0
$$235$$ 3.39445 0.221429
$$236$$ 0 0
$$237$$ −4.88057 −0.317027
$$238$$ 0 0
$$239$$ 0.513878 0.0332400 0.0166200 0.999862i $$-0.494709\pi$$
0.0166200 + 0.999862i $$0.494709\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 0 0
$$243$$ 7.84441 0.503219
$$244$$ 0 0
$$245$$ −18.5139 −1.18281
$$246$$ 0 0
$$247$$ −4.60555 −0.293044
$$248$$ 0 0
$$249$$ −5.21110 −0.330240
$$250$$ 0 0
$$251$$ 6.78890 0.428511 0.214256 0.976778i $$-0.431267\pi$$
0.214256 + 0.976778i $$0.431267\pi$$
$$252$$ 0 0
$$253$$ 9.00000 0.565825
$$254$$ 0 0
$$255$$ 2.36669 0.148208
$$256$$ 0 0
$$257$$ −11.2111 −0.699329 −0.349665 0.936875i $$-0.613704\pi$$
−0.349665 + 0.936875i $$0.613704\pi$$
$$258$$ 0 0
$$259$$ −4.60555 −0.286175
$$260$$ 0 0
$$261$$ 20.0917 1.24364
$$262$$ 0 0
$$263$$ −7.81665 −0.481996 −0.240998 0.970526i $$-0.577475\pi$$
−0.240998 + 0.970526i $$0.577475\pi$$
$$264$$ 0 0
$$265$$ −7.81665 −0.480173
$$266$$ 0 0
$$267$$ 1.57779 0.0965595
$$268$$ 0 0
$$269$$ 6.78890 0.413926 0.206963 0.978349i $$-0.433642\pi$$
0.206963 + 0.978349i $$0.433642\pi$$
$$270$$ 0 0
$$271$$ 6.42221 0.390121 0.195061 0.980791i $$-0.437510\pi$$
0.195061 + 0.980791i $$0.437510\pi$$
$$272$$ 0 0
$$273$$ 3.21110 0.194345
$$274$$ 0 0
$$275$$ 4.30278 0.259467
$$276$$ 0 0
$$277$$ 25.1194 1.50928 0.754640 0.656139i $$-0.227811\pi$$
0.754640 + 0.656139i $$0.227811\pi$$
$$278$$ 0 0
$$279$$ −9.60555 −0.575069
$$280$$ 0 0
$$281$$ −12.0000 −0.715860 −0.357930 0.933748i $$-0.616517\pi$$
−0.357930 + 0.933748i $$0.616517\pi$$
$$282$$ 0 0
$$283$$ −17.3944 −1.03399 −0.516996 0.855988i $$-0.672950\pi$$
−0.516996 + 0.855988i $$0.672950\pi$$
$$284$$ 0 0
$$285$$ 0.788897 0.0467303
$$286$$ 0 0
$$287$$ −4.18335 −0.246935
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 3.76114 0.220482
$$292$$ 0 0
$$293$$ 25.0278 1.46214 0.731069 0.682304i $$-0.239022\pi$$
0.731069 + 0.682304i $$0.239022\pi$$
$$294$$ 0 0
$$295$$ 4.42221 0.257471
$$296$$ 0 0
$$297$$ 2.33053 0.135231
$$298$$ 0 0
$$299$$ −15.9083 −0.920002
$$300$$ 0 0
$$301$$ 30.4222 1.75351
$$302$$ 0 0
$$303$$ −4.97224 −0.285648
$$304$$ 0 0
$$305$$ −13.6972 −0.784301
$$306$$ 0 0
$$307$$ −7.09167 −0.404743 −0.202372 0.979309i $$-0.564865\pi$$
−0.202372 + 0.979309i $$0.564865\pi$$
$$308$$ 0 0
$$309$$ 1.00000 0.0568880
$$310$$ 0 0
$$311$$ 5.09167 0.288722 0.144361 0.989525i $$-0.453887\pi$$
0.144361 + 0.989525i $$0.453887\pi$$
$$312$$ 0 0
$$313$$ 27.0278 1.52770 0.763850 0.645394i $$-0.223307\pi$$
0.763850 + 0.645394i $$0.223307\pi$$
$$314$$ 0 0
$$315$$ 17.4500 0.983194
$$316$$ 0 0
$$317$$ 5.21110 0.292685 0.146342 0.989234i $$-0.453250\pi$$
0.146342 + 0.989234i $$0.453250\pi$$
$$318$$ 0 0
$$319$$ 9.00000 0.503903
$$320$$ 0 0
$$321$$ −1.30278 −0.0727138
$$322$$ 0 0
$$323$$ 12.0000 0.667698
$$324$$ 0 0
$$325$$ −7.60555 −0.421880
$$326$$ 0 0
$$327$$ −0.605551 −0.0334871
$$328$$ 0 0
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −1.21110 −0.0665682 −0.0332841 0.999446i $$-0.510597\pi$$
−0.0332841 + 0.999446i $$0.510597\pi$$
$$332$$ 0 0
$$333$$ 2.90833 0.159375
$$334$$ 0 0
$$335$$ 18.9083 1.03307
$$336$$ 0 0
$$337$$ −19.1194 −1.04150 −0.520751 0.853709i $$-0.674348\pi$$
−0.520751 + 0.853709i $$0.674348\pi$$
$$338$$ 0 0
$$339$$ 3.39445 0.184361
$$340$$ 0 0
$$341$$ −4.30278 −0.233008
$$342$$ 0 0
$$343$$ 33.2111 1.79323
$$344$$ 0 0
$$345$$ 2.72498 0.146708
$$346$$ 0 0
$$347$$ −31.8167 −1.70801 −0.854004 0.520267i $$-0.825832\pi$$
−0.854004 + 0.520267i $$0.825832\pi$$
$$348$$ 0 0
$$349$$ 22.2389 1.19042 0.595209 0.803571i $$-0.297069\pi$$
0.595209 + 0.803571i $$0.297069\pi$$
$$350$$ 0 0
$$351$$ −4.11943 −0.219879
$$352$$ 0 0
$$353$$ 31.8167 1.69343 0.846715 0.532047i $$-0.178577\pi$$
0.846715 + 0.532047i $$0.178577\pi$$
$$354$$ 0 0
$$355$$ −7.81665 −0.414865
$$356$$ 0 0
$$357$$ −8.36669 −0.442812
$$358$$ 0 0
$$359$$ −11.2111 −0.591699 −0.295850 0.955235i $$-0.595603\pi$$
−0.295850 + 0.955235i $$0.595603\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ −2.81665 −0.147836
$$364$$ 0 0
$$365$$ 11.3305 0.593067
$$366$$ 0 0
$$367$$ −17.8167 −0.930022 −0.465011 0.885305i $$-0.653950\pi$$
−0.465011 + 0.885305i $$0.653950\pi$$
$$368$$ 0 0
$$369$$ 2.64171 0.137522
$$370$$ 0 0
$$371$$ 27.6333 1.43465
$$372$$ 0 0
$$373$$ −3.81665 −0.197619 −0.0988094 0.995106i $$-0.531503\pi$$
−0.0988094 + 0.995106i $$0.531503\pi$$
$$374$$ 0 0
$$375$$ 3.27502 0.169121
$$376$$ 0 0
$$377$$ −15.9083 −0.819321
$$378$$ 0 0
$$379$$ 15.3305 0.787477 0.393738 0.919223i $$-0.371182\pi$$
0.393738 + 0.919223i $$0.371182\pi$$
$$380$$ 0 0
$$381$$ −1.44996 −0.0742838
$$382$$ 0 0
$$383$$ 20.8444 1.06510 0.532550 0.846399i $$-0.321234\pi$$
0.532550 + 0.846399i $$0.321234\pi$$
$$384$$ 0 0
$$385$$ 7.81665 0.398374
$$386$$ 0 0
$$387$$ −19.2111 −0.976555
$$388$$ 0 0
$$389$$ 11.8806 0.602369 0.301184 0.953566i $$-0.402618\pi$$
0.301184 + 0.953566i $$0.402618\pi$$
$$390$$ 0 0
$$391$$ 41.4500 2.09621
$$392$$ 0 0
$$393$$ −1.02776 −0.0518435
$$394$$ 0 0
$$395$$ 21.0000 1.05662
$$396$$ 0 0
$$397$$ −27.8167 −1.39608 −0.698039 0.716060i $$-0.745944\pi$$
−0.698039 + 0.716060i $$0.745944\pi$$
$$398$$ 0 0
$$399$$ −2.78890 −0.139620
$$400$$ 0 0
$$401$$ 13.8167 0.689971 0.344985 0.938608i $$-0.387884\pi$$
0.344985 + 0.938608i $$0.387884\pi$$
$$402$$ 0 0
$$403$$ 7.60555 0.378859
$$404$$ 0 0
$$405$$ −10.6611 −0.529753
$$406$$ 0 0
$$407$$ 1.30278 0.0645762
$$408$$ 0 0
$$409$$ −5.02776 −0.248607 −0.124303 0.992244i $$-0.539670\pi$$
−0.124303 + 0.992244i $$0.539670\pi$$
$$410$$ 0 0
$$411$$ −3.00000 −0.147979
$$412$$ 0 0
$$413$$ −15.6333 −0.769265
$$414$$ 0 0
$$415$$ 22.4222 1.10066
$$416$$ 0 0
$$417$$ −2.69722 −0.132084
$$418$$ 0 0
$$419$$ 25.1472 1.22852 0.614260 0.789104i $$-0.289455\pi$$
0.614260 + 0.789104i $$0.289455\pi$$
$$420$$ 0 0
$$421$$ −28.7250 −1.39997 −0.699985 0.714158i $$-0.746810\pi$$
−0.699985 + 0.714158i $$0.746810\pi$$
$$422$$ 0 0
$$423$$ 7.57779 0.368445
$$424$$ 0 0
$$425$$ 19.8167 0.961249
$$426$$ 0 0
$$427$$ 48.4222 2.34331
$$428$$ 0 0
$$429$$ −0.908327 −0.0438544
$$430$$ 0 0
$$431$$ −5.21110 −0.251010 −0.125505 0.992093i $$-0.540055\pi$$
−0.125505 + 0.992093i $$0.540055\pi$$
$$432$$ 0 0
$$433$$ −11.9361 −0.573612 −0.286806 0.957989i $$-0.592593\pi$$
−0.286806 + 0.957989i $$0.592593\pi$$
$$434$$ 0 0
$$435$$ 2.72498 0.130653
$$436$$ 0 0
$$437$$ 13.8167 0.660940
$$438$$ 0 0
$$439$$ −9.33053 −0.445322 −0.222661 0.974896i $$-0.571474\pi$$
−0.222661 + 0.974896i $$0.571474\pi$$
$$440$$ 0 0
$$441$$ −41.3305 −1.96812
$$442$$ 0 0
$$443$$ −0.275019 −0.0130666 −0.00653328 0.999979i $$-0.502080\pi$$
−0.00653328 + 0.999979i $$0.502080\pi$$
$$444$$ 0 0
$$445$$ −6.78890 −0.321825
$$446$$ 0 0
$$447$$ 0.550039 0.0260159
$$448$$ 0 0
$$449$$ −0.788897 −0.0372304 −0.0186152 0.999827i $$-0.505926\pi$$
−0.0186152 + 0.999827i $$0.505926\pi$$
$$450$$ 0 0
$$451$$ 1.18335 0.0557216
$$452$$ 0 0
$$453$$ −4.05551 −0.190545
$$454$$ 0 0
$$455$$ −13.8167 −0.647735
$$456$$ 0 0
$$457$$ 4.60555 0.215439 0.107719 0.994181i $$-0.465645\pi$$
0.107719 + 0.994181i $$0.465645\pi$$
$$458$$ 0 0
$$459$$ 10.7334 0.500991
$$460$$ 0 0
$$461$$ 16.4222 0.764858 0.382429 0.923985i $$-0.375088\pi$$
0.382429 + 0.923985i $$0.375088\pi$$
$$462$$ 0 0
$$463$$ 30.3028 1.40829 0.704145 0.710056i $$-0.251331\pi$$
0.704145 + 0.710056i $$0.251331\pi$$
$$464$$ 0 0
$$465$$ −1.30278 −0.0604148
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ −66.8444 −3.08659
$$470$$ 0 0
$$471$$ −2.18335 −0.100603
$$472$$ 0 0
$$473$$ −8.60555 −0.395684
$$474$$ 0 0
$$475$$ 6.60555 0.303083
$$476$$ 0 0
$$477$$ −17.4500 −0.798979
$$478$$ 0 0
$$479$$ 12.1194 0.553751 0.276875 0.960906i $$-0.410701\pi$$
0.276875 + 0.960906i $$0.410701\pi$$
$$480$$ 0 0
$$481$$ −2.30278 −0.104998
$$482$$ 0 0
$$483$$ −9.63331 −0.438331
$$484$$ 0 0
$$485$$ −16.1833 −0.734848
$$486$$ 0 0
$$487$$ −22.7889 −1.03266 −0.516332 0.856389i $$-0.672703\pi$$
−0.516332 + 0.856389i $$0.672703\pi$$
$$488$$ 0 0
$$489$$ 6.18335 0.279621
$$490$$ 0 0
$$491$$ 14.7250 0.664529 0.332265 0.943186i $$-0.392187\pi$$
0.332265 + 0.943186i $$0.392187\pi$$
$$492$$ 0 0
$$493$$ 41.4500 1.86681
$$494$$ 0 0
$$495$$ −4.93608 −0.221860
$$496$$ 0 0
$$497$$ 27.6333 1.23952
$$498$$ 0 0
$$499$$ −8.23886 −0.368822 −0.184411 0.982849i $$-0.559038\pi$$
−0.184411 + 0.982849i $$0.559038\pi$$
$$500$$ 0 0
$$501$$ 3.78890 0.169275
$$502$$ 0 0
$$503$$ −24.5139 −1.09302 −0.546510 0.837453i $$-0.684044\pi$$
−0.546510 + 0.837453i $$0.684044\pi$$
$$504$$ 0 0
$$505$$ 21.3944 0.952040
$$506$$ 0 0
$$507$$ −2.33053 −0.103503
$$508$$ 0 0
$$509$$ 25.8167 1.14430 0.572152 0.820148i $$-0.306109\pi$$
0.572152 + 0.820148i $$0.306109\pi$$
$$510$$ 0 0
$$511$$ −40.0555 −1.77195
$$512$$ 0 0
$$513$$ 3.57779 0.157964
$$514$$ 0 0
$$515$$ −4.30278 −0.189603
$$516$$ 0 0
$$517$$ 3.39445 0.149288
$$518$$ 0 0
$$519$$ 7.02776 0.308484
$$520$$ 0 0
$$521$$ −9.63331 −0.422043 −0.211021 0.977481i $$-0.567679\pi$$
−0.211021 + 0.977481i $$0.567679\pi$$
$$522$$ 0 0
$$523$$ −32.2389 −1.40971 −0.704853 0.709353i $$-0.748987\pi$$
−0.704853 + 0.709353i $$0.748987\pi$$
$$524$$ 0 0
$$525$$ −4.60555 −0.201003
$$526$$ 0 0
$$527$$ −19.8167 −0.863227
$$528$$ 0 0
$$529$$ 24.7250 1.07500
$$530$$ 0 0
$$531$$ 9.87217 0.428416
$$532$$ 0 0
$$533$$ −2.09167 −0.0906004
$$534$$ 0 0
$$535$$ 5.60555 0.242349
$$536$$ 0 0
$$537$$ −2.36669 −0.102130
$$538$$ 0 0
$$539$$ −18.5139 −0.797449
$$540$$ 0 0
$$541$$ 20.9361 0.900113 0.450056 0.893000i $$-0.351404\pi$$
0.450056 + 0.893000i $$0.351404\pi$$
$$542$$ 0 0
$$543$$ −6.05551 −0.259867
$$544$$ 0 0
$$545$$ 2.60555 0.111610
$$546$$ 0 0
$$547$$ 13.3944 0.572705 0.286353 0.958124i $$-0.407557\pi$$
0.286353 + 0.958124i $$0.407557\pi$$
$$548$$ 0 0
$$549$$ −30.5778 −1.30503
$$550$$ 0 0
$$551$$ 13.8167 0.588609
$$552$$ 0 0
$$553$$ −74.2389 −3.15696
$$554$$ 0 0
$$555$$ 0.394449 0.0167434
$$556$$ 0 0
$$557$$ 6.51388 0.276002 0.138001 0.990432i $$-0.455932\pi$$
0.138001 + 0.990432i $$0.455932\pi$$
$$558$$ 0 0
$$559$$ 15.2111 0.643361
$$560$$ 0 0
$$561$$ 2.36669 0.0999218
$$562$$ 0 0
$$563$$ −44.0555 −1.85672 −0.928359 0.371684i $$-0.878780\pi$$
−0.928359 + 0.371684i $$0.878780\pi$$
$$564$$ 0 0
$$565$$ −14.6056 −0.614460
$$566$$ 0 0
$$567$$ 37.6888 1.58278
$$568$$ 0 0
$$569$$ −10.4222 −0.436922 −0.218461 0.975846i $$-0.570104\pi$$
−0.218461 + 0.975846i $$0.570104\pi$$
$$570$$ 0 0
$$571$$ 20.3028 0.849645 0.424822 0.905277i $$-0.360337\pi$$
0.424822 + 0.905277i $$0.360337\pi$$
$$572$$ 0 0
$$573$$ 3.78890 0.158283
$$574$$ 0 0
$$575$$ 22.8167 0.951520
$$576$$ 0 0
$$577$$ −28.2389 −1.17560 −0.587800 0.809007i $$-0.700006\pi$$
−0.587800 + 0.809007i $$0.700006\pi$$
$$578$$ 0 0
$$579$$ −1.21110 −0.0503317
$$580$$ 0 0
$$581$$ −79.2666 −3.28853
$$582$$ 0 0
$$583$$ −7.81665 −0.323733
$$584$$ 0 0
$$585$$ 8.72498 0.360734
$$586$$ 0 0
$$587$$ 2.36669 0.0976838 0.0488419 0.998807i $$-0.484447\pi$$
0.0488419 + 0.998807i $$0.484447\pi$$
$$588$$ 0 0
$$589$$ −6.60555 −0.272177
$$590$$ 0 0
$$591$$ 1.81665 0.0747272
$$592$$ 0 0
$$593$$ 36.5139 1.49945 0.749723 0.661752i $$-0.230187\pi$$
0.749723 + 0.661752i $$0.230187\pi$$
$$594$$ 0 0
$$595$$ 36.0000 1.47586
$$596$$ 0 0
$$597$$ −0.733385 −0.0300154
$$598$$ 0 0
$$599$$ −35.2111 −1.43869 −0.719343 0.694655i $$-0.755557\pi$$
−0.719343 + 0.694655i $$0.755557\pi$$
$$600$$ 0 0
$$601$$ −20.6972 −0.844257 −0.422129 0.906536i $$-0.638717\pi$$
−0.422129 + 0.906536i $$0.638717\pi$$
$$602$$ 0 0
$$603$$ 42.2111 1.71897
$$604$$ 0 0
$$605$$ 12.1194 0.492725
$$606$$ 0 0
$$607$$ −31.5139 −1.27911 −0.639554 0.768746i $$-0.720881\pi$$
−0.639554 + 0.768746i $$0.720881\pi$$
$$608$$ 0 0
$$609$$ −9.63331 −0.390361
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ 0 0
$$613$$ 8.18335 0.330522 0.165261 0.986250i $$-0.447153\pi$$
0.165261 + 0.986250i $$0.447153\pi$$
$$614$$ 0 0
$$615$$ 0.358288 0.0144476
$$616$$ 0 0
$$617$$ 47.5694 1.91507 0.957536 0.288314i $$-0.0930948\pi$$
0.957536 + 0.288314i $$0.0930948\pi$$
$$618$$ 0 0
$$619$$ 2.69722 0.108411 0.0542053 0.998530i $$-0.482737\pi$$
0.0542053 + 0.998530i $$0.482737\pi$$
$$620$$ 0 0
$$621$$ 12.3583 0.495921
$$622$$ 0 0
$$623$$ 24.0000 0.961540
$$624$$ 0 0
$$625$$ 2.42221 0.0968882
$$626$$ 0 0
$$627$$ 0.788897 0.0315055
$$628$$ 0 0
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 18.3028 0.728622 0.364311 0.931277i $$-0.381305\pi$$
0.364311 + 0.931277i $$0.381305\pi$$
$$632$$ 0 0
$$633$$ −2.02776 −0.0805961
$$634$$ 0 0
$$635$$ 6.23886 0.247582
$$636$$ 0 0
$$637$$ 32.7250 1.29661
$$638$$ 0 0
$$639$$ −17.4500 −0.690310
$$640$$ 0 0
$$641$$ −2.48612 −0.0981959 −0.0490980 0.998794i $$-0.515635\pi$$
−0.0490980 + 0.998794i $$0.515635\pi$$
$$642$$ 0 0
$$643$$ 29.8167 1.17585 0.587927 0.808914i $$-0.299944\pi$$
0.587927 + 0.808914i $$0.299944\pi$$
$$644$$ 0 0
$$645$$ −2.60555 −0.102593
$$646$$ 0 0
$$647$$ −25.9361 −1.01965 −0.509826 0.860277i $$-0.670290\pi$$
−0.509826 + 0.860277i $$0.670290\pi$$
$$648$$ 0 0
$$649$$ 4.42221 0.173587
$$650$$ 0 0
$$651$$ 4.60555 0.180506
$$652$$ 0 0
$$653$$ −6.90833 −0.270344 −0.135172 0.990822i $$-0.543159\pi$$
−0.135172 + 0.990822i $$0.543159\pi$$
$$654$$ 0 0
$$655$$ 4.42221 0.172790
$$656$$ 0 0
$$657$$ 25.2944 0.986827
$$658$$ 0 0
$$659$$ 42.1194 1.64074 0.820370 0.571833i $$-0.193767\pi$$
0.820370 + 0.571833i $$0.193767\pi$$
$$660$$ 0 0
$$661$$ 12.4861 0.485654 0.242827 0.970070i $$-0.421925\pi$$
0.242827 + 0.970070i $$0.421925\pi$$
$$662$$ 0 0
$$663$$ −4.18335 −0.162468
$$664$$ 0 0
$$665$$ 12.0000 0.465340
$$666$$ 0 0
$$667$$ 47.7250 1.84792
$$668$$ 0 0
$$669$$ 4.78890 0.185149
$$670$$ 0 0
$$671$$ −13.6972 −0.528775
$$672$$ 0 0
$$673$$ 24.3028 0.936803 0.468402 0.883516i $$-0.344830\pi$$
0.468402 + 0.883516i $$0.344830\pi$$
$$674$$ 0 0
$$675$$ 5.90833 0.227412
$$676$$ 0 0
$$677$$ 36.2389 1.39277 0.696386 0.717667i $$-0.254790\pi$$
0.696386 + 0.717667i $$0.254790\pi$$
$$678$$ 0 0
$$679$$ 57.2111 2.19556
$$680$$ 0 0
$$681$$ 2.36669 0.0906918
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 12.9083 0.493202
$$686$$ 0 0
$$687$$ −5.26662 −0.200934
$$688$$ 0 0
$$689$$ 13.8167 0.526373
$$690$$ 0 0
$$691$$ −8.00000 −0.304334 −0.152167 0.988355i $$-0.548625\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 0 0
$$693$$ 17.4500 0.662869
$$694$$ 0 0
$$695$$ 11.6056 0.440224
$$696$$ 0 0
$$697$$ 5.44996 0.206432
$$698$$ 0 0
$$699$$ −2.88057 −0.108953
$$700$$ 0 0
$$701$$ −14.8806 −0.562031 −0.281016 0.959703i $$-0.590671\pi$$
−0.281016 + 0.959703i $$0.590671\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ 0 0
$$705$$ 1.02776 0.0387075
$$706$$ 0 0
$$707$$ −75.6333 −2.84448
$$708$$ 0 0
$$709$$ 1.66947 0.0626982 0.0313491 0.999508i $$-0.490020\pi$$
0.0313491 + 0.999508i $$0.490020\pi$$
$$710$$ 0 0
$$711$$ 46.8806 1.75816
$$712$$ 0 0
$$713$$ −22.8167 −0.854490
$$714$$ 0 0
$$715$$ 3.90833 0.146163
$$716$$ 0 0
$$717$$ 0.155590 0.00581061
$$718$$ 0 0
$$719$$ −8.36669 −0.312025 −0.156012 0.987755i $$-0.549864\pi$$
−0.156012 + 0.987755i $$0.549864\pi$$
$$720$$ 0 0
$$721$$ 15.2111 0.566491
$$722$$ 0 0
$$723$$ 2.42221 0.0900828
$$724$$ 0 0
$$725$$ 22.8167 0.847389
$$726$$ 0 0
$$727$$ 29.9083 1.10924 0.554619 0.832104i $$-0.312864\pi$$
0.554619 + 0.832104i $$0.312864\pi$$
$$728$$ 0 0
$$729$$ −22.1749 −0.821294
$$730$$ 0 0
$$731$$ −39.6333 −1.46589
$$732$$ 0 0
$$733$$ −29.6333 −1.09453 −0.547266 0.836959i $$-0.684331\pi$$
−0.547266 + 0.836959i $$0.684331\pi$$
$$734$$ 0 0
$$735$$ −5.60555 −0.206764
$$736$$ 0 0
$$737$$ 18.9083 0.696497
$$738$$ 0 0
$$739$$ 42.3305 1.55715 0.778577 0.627549i $$-0.215942\pi$$
0.778577 + 0.627549i $$0.215942\pi$$
$$740$$ 0 0
$$741$$ −1.39445 −0.0512264
$$742$$ 0 0
$$743$$ 35.4500 1.30053 0.650266 0.759706i $$-0.274657\pi$$
0.650266 + 0.759706i $$0.274657\pi$$
$$744$$ 0 0
$$745$$ −2.36669 −0.0867089
$$746$$ 0 0
$$747$$ 50.0555 1.83144
$$748$$ 0 0
$$749$$ −19.8167 −0.724085
$$750$$ 0 0
$$751$$ 14.0000 0.510867 0.255434 0.966827i $$-0.417782\pi$$
0.255434 + 0.966827i $$0.417782\pi$$
$$752$$ 0 0
$$753$$ 2.05551 0.0749070
$$754$$ 0 0
$$755$$ 17.4500 0.635069
$$756$$ 0 0
$$757$$ −9.30278 −0.338115 −0.169058 0.985606i $$-0.554072\pi$$
−0.169058 + 0.985606i $$0.554072\pi$$
$$758$$ 0 0
$$759$$ 2.72498 0.0989105
$$760$$ 0 0
$$761$$ 42.1194 1.52683 0.763414 0.645909i $$-0.223522\pi$$
0.763414 + 0.645909i $$0.223522\pi$$
$$762$$ 0 0
$$763$$ −9.21110 −0.333464
$$764$$ 0 0
$$765$$ −22.7334 −0.821927
$$766$$ 0 0
$$767$$ −7.81665 −0.282243
$$768$$ 0 0
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ −3.39445 −0.122248
$$772$$ 0 0
$$773$$ 50.0555 1.80037 0.900186 0.435506i $$-0.143431\pi$$
0.900186 + 0.435506i $$0.143431\pi$$
$$774$$ 0 0
$$775$$ −10.9083 −0.391839
$$776$$ 0 0
$$777$$ −1.39445 −0.0500256
$$778$$ 0 0
$$779$$ 1.81665 0.0650884
$$780$$ 0 0
$$781$$ −7.81665 −0.279702
$$782$$ 0 0
$$783$$ 12.3583 0.441649
$$784$$ 0 0
$$785$$ 9.39445 0.335302
$$786$$ 0 0
$$787$$ −25.2111 −0.898679 −0.449339 0.893361i $$-0.648341\pi$$
−0.449339 + 0.893361i $$0.648341\pi$$
$$788$$ 0 0
$$789$$ −2.36669 −0.0842565
$$790$$ 0 0
$$791$$ 51.6333 1.83587
$$792$$ 0 0
$$793$$ 24.2111 0.859761
$$794$$ 0 0
$$795$$ −2.36669 −0.0839379
$$796$$ 0 0
$$797$$ 17.3305 0.613879 0.306939 0.951729i $$-0.400695\pi$$
0.306939 + 0.951729i $$0.400695\pi$$
$$798$$ 0 0
$$799$$ 15.6333 0.553067
$$800$$ 0 0
$$801$$ −15.1556 −0.535496
$$802$$ 0 0
$$803$$ 11.3305 0.399846
$$804$$ 0 0
$$805$$ 41.4500 1.46092
$$806$$ 0 0
$$807$$ 2.05551 0.0723575
$$808$$ 0 0
$$809$$ 29.4500 1.03541 0.517703 0.855561i $$-0.326787\pi$$
0.517703 + 0.855561i $$0.326787\pi$$
$$810$$ 0 0
$$811$$ −54.1472 −1.90136 −0.950682 0.310166i $$-0.899615\pi$$
−0.950682 + 0.310166i $$0.899615\pi$$
$$812$$ 0 0
$$813$$ 1.94449 0.0681961
$$814$$ 0 0
$$815$$ −26.6056 −0.931952
$$816$$ 0 0
$$817$$ −13.2111 −0.462198
$$818$$ 0 0
$$819$$ −30.8444 −1.07779
$$820$$ 0 0
$$821$$ −11.2111 −0.391270 −0.195635 0.980677i $$-0.562677\pi$$
−0.195635 + 0.980677i $$0.562677\pi$$
$$822$$ 0 0
$$823$$ −12.8444 −0.447728 −0.223864 0.974620i $$-0.571867\pi$$
−0.223864 + 0.974620i $$0.571867\pi$$
$$824$$ 0 0
$$825$$ 1.30278 0.0453568
$$826$$ 0 0
$$827$$ −27.3944 −0.952598 −0.476299 0.879283i $$-0.658022\pi$$
−0.476299 + 0.879283i $$0.658022\pi$$
$$828$$ 0 0
$$829$$ −4.72498 −0.164105 −0.0820527 0.996628i $$-0.526148\pi$$
−0.0820527 + 0.996628i $$0.526148\pi$$
$$830$$ 0 0
$$831$$ 7.60555 0.263834
$$832$$ 0 0
$$833$$ −85.2666 −2.95431
$$834$$ 0 0
$$835$$ −16.3028 −0.564181
$$836$$ 0 0
$$837$$ −5.90833 −0.204222
$$838$$ 0 0
$$839$$ −49.0278 −1.69263 −0.846313 0.532686i $$-0.821183\pi$$
−0.846313 + 0.532686i $$0.821183\pi$$
$$840$$ 0 0
$$841$$ 18.7250 0.645689
$$842$$ 0 0
$$843$$ −3.63331 −0.125138
$$844$$ 0 0
$$845$$ 10.0278 0.344965
$$846$$ 0 0
$$847$$ −42.8444 −1.47215
$$848$$ 0 0
$$849$$ −5.26662 −0.180750
$$850$$ 0 0
$$851$$ 6.90833 0.236814
$$852$$ 0 0
$$853$$ 11.5416 0.395178 0.197589 0.980285i $$-0.436689\pi$$
0.197589 + 0.980285i $$0.436689\pi$$
$$854$$ 0 0
$$855$$ −7.57779 −0.259155
$$856$$ 0 0
$$857$$ −14.8444 −0.507075 −0.253538 0.967326i $$-0.581594\pi$$
−0.253538 + 0.967326i $$0.581594\pi$$
$$858$$ 0 0
$$859$$ 24.0555 0.820764 0.410382 0.911914i $$-0.365395\pi$$
0.410382 + 0.911914i $$0.365395\pi$$
$$860$$ 0 0
$$861$$ −1.26662 −0.0431661
$$862$$ 0 0
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 0 0
$$865$$ −30.2389 −1.02815
$$866$$ 0 0
$$867$$ 5.75274 0.195373
$$868$$ 0 0
$$869$$ 21.0000 0.712376
$$870$$ 0 0
$$871$$ −33.4222 −1.13247
$$872$$ 0 0
$$873$$ −36.1278 −1.22274
$$874$$ 0 0
$$875$$ 49.8167 1.68411
$$876$$ 0 0
$$877$$ −7.21110 −0.243502 −0.121751 0.992561i $$-0.538851\pi$$
−0.121751 + 0.992561i $$0.538851\pi$$
$$878$$ 0 0
$$879$$ 7.57779 0.255593
$$880$$ 0 0
$$881$$ 25.5416 0.860520 0.430260 0.902705i $$-0.358422\pi$$
0.430260 + 0.902705i $$0.358422\pi$$
$$882$$ 0 0
$$883$$ 2.42221 0.0815137 0.0407568 0.999169i $$-0.487023\pi$$
0.0407568 + 0.999169i $$0.487023\pi$$
$$884$$ 0 0
$$885$$ 1.33894 0.0450078
$$886$$ 0 0
$$887$$ 28.4222 0.954324 0.477162 0.878815i $$-0.341665\pi$$
0.477162 + 0.878815i $$0.341665\pi$$
$$888$$ 0 0
$$889$$ −22.0555 −0.739718
$$890$$ 0 0
$$891$$ −10.6611 −0.357159
$$892$$ 0 0
$$893$$ 5.21110 0.174383
$$894$$ 0 0
$$895$$ 10.1833 0.340392
$$896$$ 0 0
$$897$$ −4.81665 −0.160823
$$898$$ 0 0
$$899$$ −22.8167 −0.760978
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 9.21110 0.306526
$$904$$ 0 0
$$905$$ 26.0555 0.866115
$$906$$ 0 0
$$907$$ −26.0000 −0.863316 −0.431658 0.902037i $$-0.642071\pi$$
−0.431658 + 0.902037i $$0.642071\pi$$
$$908$$ 0 0
$$909$$ 47.7611 1.58414
$$910$$ 0 0
$$911$$ −46.4222 −1.53804 −0.769018 0.639227i $$-0.779254\pi$$
−0.769018 + 0.639227i $$0.779254\pi$$
$$912$$ 0 0
$$913$$ 22.4222 0.742067
$$914$$ 0 0
$$915$$ −4.14719 −0.137102
$$916$$ 0 0
$$917$$ −15.6333 −0.516257
$$918$$ 0 0
$$919$$ −38.4222 −1.26743 −0.633716 0.773566i $$-0.718471\pi$$
−0.633716 + 0.773566i $$0.718471\pi$$
$$920$$ 0 0
$$921$$ −2.14719 −0.0707522
$$922$$ 0 0
$$923$$ 13.8167 0.454781
$$924$$ 0 0
$$925$$ 3.30278 0.108595
$$926$$ 0 0
$$927$$ −9.60555 −0.315488
$$928$$ 0 0
$$929$$ −36.5139 −1.19798 −0.598991 0.800756i $$-0.704431\pi$$
−0.598991 + 0.800756i $$0.704431\pi$$
$$930$$ 0 0
$$931$$ −28.4222 −0.931500
$$932$$ 0 0
$$933$$ 1.54163 0.0504708
$$934$$ 0 0
$$935$$ −10.1833 −0.333031
$$936$$ 0 0
$$937$$ −28.9083 −0.944394 −0.472197 0.881493i $$-0.656539\pi$$
−0.472197 + 0.881493i $$0.656539\pi$$
$$938$$ 0 0
$$939$$ 8.18335 0.267053
$$940$$ 0 0
$$941$$ 7.81665 0.254816 0.127408 0.991850i $$-0.459334\pi$$
0.127408 + 0.991850i $$0.459334\pi$$
$$942$$ 0 0
$$943$$ 6.27502 0.204343
$$944$$ 0 0
$$945$$ 10.7334 0.349157
$$946$$ 0 0
$$947$$ −39.6333 −1.28791 −0.643955 0.765064i $$-0.722707\pi$$
−0.643955 + 0.765064i $$0.722707\pi$$
$$948$$ 0 0
$$949$$ −20.0278 −0.650128
$$950$$ 0 0
$$951$$ 1.57779 0.0511635
$$952$$ 0 0
$$953$$ −18.7527 −0.607461 −0.303730 0.952758i $$-0.598232\pi$$
−0.303730 + 0.952758i $$0.598232\pi$$
$$954$$ 0 0
$$955$$ −16.3028 −0.527545
$$956$$ 0 0
$$957$$ 2.72498 0.0880861
$$958$$ 0 0
$$959$$ −45.6333 −1.47358
$$960$$ 0 0
$$961$$ −20.0917 −0.648118
$$962$$ 0 0
$$963$$ 12.5139 0.403254
$$964$$ 0 0
$$965$$ 5.21110 0.167751
$$966$$ 0 0
$$967$$ 25.7250 0.827260 0.413630 0.910445i $$-0.364261\pi$$
0.413630 + 0.910445i $$0.364261\pi$$
$$968$$ 0 0
$$969$$ 3.63331 0.116719
$$970$$ 0 0
$$971$$ −31.5416 −1.01222 −0.506110 0.862469i $$-0.668917\pi$$
−0.506110 + 0.862469i $$0.668917\pi$$
$$972$$ 0 0
$$973$$ −41.0278 −1.31529
$$974$$ 0 0
$$975$$ −2.30278 −0.0737478
$$976$$ 0 0
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 0 0
$$979$$ −6.78890 −0.216974
$$980$$ 0 0
$$981$$ 5.81665 0.185711
$$982$$ 0 0
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ 0 0
$$985$$ −7.81665 −0.249059
$$986$$ 0 0
$$987$$ −3.63331 −0.115649
$$988$$ 0 0
$$989$$ −45.6333 −1.45105
$$990$$ 0 0
$$991$$ 54.3028 1.72498 0.862492 0.506070i $$-0.168902\pi$$
0.862492 + 0.506070i $$0.168902\pi$$
$$992$$ 0 0
$$993$$ −0.366692 −0.0116366
$$994$$ 0 0
$$995$$ 3.15559 0.100039
$$996$$ 0 0
$$997$$ 23.5778 0.746716 0.373358 0.927687i $$-0.378206\pi$$
0.373358 + 0.927687i $$0.378206\pi$$
$$998$$ 0 0
$$999$$ 1.78890 0.0565982
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 2368.2.a.s.1.2 2
4.3 odd 2 2368.2.a.ba.1.1 2
8.3 odd 2 592.2.a.f.1.2 2
8.5 even 2 74.2.a.a.1.1 2
24.5 odd 2 666.2.a.j.1.1 2
24.11 even 2 5328.2.a.bf.1.1 2
40.13 odd 4 1850.2.b.i.149.3 4
40.29 even 2 1850.2.a.u.1.2 2
40.37 odd 4 1850.2.b.i.149.2 4
56.13 odd 2 3626.2.a.a.1.2 2
88.21 odd 2 8954.2.a.p.1.1 2
296.221 even 2 2738.2.a.l.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.1 2 8.5 even 2
592.2.a.f.1.2 2 8.3 odd 2
666.2.a.j.1.1 2 24.5 odd 2
1850.2.a.u.1.2 2 40.29 even 2
1850.2.b.i.149.2 4 40.37 odd 4
1850.2.b.i.149.3 4 40.13 odd 4
2368.2.a.s.1.2 2 1.1 even 1 trivial
2368.2.a.ba.1.1 2 4.3 odd 2
2738.2.a.l.1.1 2 296.221 even 2
3626.2.a.a.1.2 2 56.13 odd 2
5328.2.a.bf.1.1 2 24.11 even 2
8954.2.a.p.1.1 2 88.21 odd 2