# Properties

 Label 1875.2.a.l.1.5 Level $1875$ Weight $2$ Character 1875.1 Self dual yes Analytic conductor $14.972$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.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: $$14.9719503790$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.46840000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9$$ x^6 - x^5 - 11*x^4 + 8*x^3 + 31*x^2 - 15*x - 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$2.13324$$ of defining polynomial Character $$\chi$$ $$=$$ 1875.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.13324 q^{2} -1.00000 q^{3} +2.55073 q^{4} -2.13324 q^{6} -2.16876 q^{7} +1.17484 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.13324 q^{2} -1.00000 q^{3} +2.55073 q^{4} -2.13324 q^{6} -2.16876 q^{7} +1.17484 q^{8} +1.00000 q^{9} -2.50913 q^{11} -2.55073 q^{12} +4.33379 q^{13} -4.62650 q^{14} -2.59524 q^{16} +6.77562 q^{17} +2.13324 q^{18} +6.83602 q^{19} +2.16876 q^{21} -5.35259 q^{22} +1.67843 q^{23} -1.17484 q^{24} +9.24504 q^{26} -1.00000 q^{27} -5.53193 q^{28} +4.44927 q^{29} +6.56295 q^{31} -7.88596 q^{32} +2.50913 q^{33} +14.4541 q^{34} +2.55073 q^{36} -7.97720 q^{37} +14.5829 q^{38} -4.33379 q^{39} +11.2249 q^{41} +4.62650 q^{42} +4.25487 q^{43} -6.40012 q^{44} +3.58050 q^{46} -4.98652 q^{47} +2.59524 q^{48} -2.29646 q^{49} -6.77562 q^{51} +11.0543 q^{52} +8.21338 q^{53} -2.13324 q^{54} -2.54795 q^{56} -6.83602 q^{57} +9.49138 q^{58} +3.67416 q^{59} +4.93960 q^{61} +14.0004 q^{62} -2.16876 q^{63} -11.6322 q^{64} +5.35259 q^{66} -11.5812 q^{67} +17.2828 q^{68} -1.67843 q^{69} -2.30251 q^{71} +1.17484 q^{72} +1.11599 q^{73} -17.0173 q^{74} +17.4368 q^{76} +5.44172 q^{77} -9.24504 q^{78} +7.78306 q^{79} +1.00000 q^{81} +23.9454 q^{82} -9.87708 q^{83} +5.53193 q^{84} +9.07667 q^{86} -4.44927 q^{87} -2.94783 q^{88} -1.24025 q^{89} -9.39897 q^{91} +4.28122 q^{92} -6.56295 q^{93} -10.6375 q^{94} +7.88596 q^{96} -15.0488 q^{97} -4.89892 q^{98} -2.50913 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{2} - 6 q^{3} + 11 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10})$$ 6 * q + q^2 - 6 * q^3 + 11 * q^4 - q^6 - 2 * q^7 + 6 * q^8 + 6 * q^9 $$6 q + q^{2} - 6 q^{3} + 11 q^{4} - q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9} - 11 q^{12} - 4 q^{14} + 17 q^{16} + 2 q^{17} + q^{18} - 2 q^{19} + 2 q^{21} + 9 q^{22} - q^{23} - 6 q^{24} + 37 q^{26} - 6 q^{27} - 44 q^{28} + 31 q^{29} - 2 q^{31} + 33 q^{32} + 37 q^{34} + 11 q^{36} - 22 q^{37} + 27 q^{38} + 33 q^{41} + 4 q^{42} - 3 q^{43} - 11 q^{44} - 12 q^{46} - 6 q^{47} - 17 q^{48} + 4 q^{49} - 2 q^{51} - 33 q^{52} + 14 q^{53} - q^{54} - 30 q^{56} + 2 q^{57} - q^{58} - 8 q^{59} + 34 q^{61} + 31 q^{62} - 2 q^{63} + 12 q^{64} - 9 q^{66} + 2 q^{67} + 27 q^{68} + q^{69} - 3 q^{71} + 6 q^{72} - 36 q^{73} + 36 q^{74} + 27 q^{76} + 16 q^{77} - 37 q^{78} + 25 q^{79} + 6 q^{81} + 36 q^{82} - 12 q^{83} + 44 q^{84} - 30 q^{86} - 31 q^{87} + 56 q^{88} + 18 q^{89} + 28 q^{91} + 3 q^{92} + 2 q^{93} - 50 q^{94} - 33 q^{96} + 7 q^{97} + 15 q^{98}+O(q^{100})$$ 6 * q + q^2 - 6 * q^3 + 11 * q^4 - q^6 - 2 * q^7 + 6 * q^8 + 6 * q^9 - 11 * q^12 - 4 * q^14 + 17 * q^16 + 2 * q^17 + q^18 - 2 * q^19 + 2 * q^21 + 9 * q^22 - q^23 - 6 * q^24 + 37 * q^26 - 6 * q^27 - 44 * q^28 + 31 * q^29 - 2 * q^31 + 33 * q^32 + 37 * q^34 + 11 * q^36 - 22 * q^37 + 27 * q^38 + 33 * q^41 + 4 * q^42 - 3 * q^43 - 11 * q^44 - 12 * q^46 - 6 * q^47 - 17 * q^48 + 4 * q^49 - 2 * q^51 - 33 * q^52 + 14 * q^53 - q^54 - 30 * q^56 + 2 * q^57 - q^58 - 8 * q^59 + 34 * q^61 + 31 * q^62 - 2 * q^63 + 12 * q^64 - 9 * q^66 + 2 * q^67 + 27 * q^68 + q^69 - 3 * q^71 + 6 * q^72 - 36 * q^73 + 36 * q^74 + 27 * q^76 + 16 * q^77 - 37 * q^78 + 25 * q^79 + 6 * q^81 + 36 * q^82 - 12 * q^83 + 44 * q^84 - 30 * q^86 - 31 * q^87 + 56 * q^88 + 18 * q^89 + 28 * q^91 + 3 * q^92 + 2 * q^93 - 50 * q^94 - 33 * q^96 + 7 * q^97 + 15 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.13324 1.50843 0.754216 0.656627i $$-0.228017\pi$$
0.754216 + 0.656627i $$0.228017\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 2.55073 1.27536
$$5$$ 0 0
$$6$$ −2.13324 −0.870893
$$7$$ −2.16876 −0.819716 −0.409858 0.912149i $$-0.634422\pi$$
−0.409858 + 0.912149i $$0.634422\pi$$
$$8$$ 1.17484 0.415369
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.50913 −0.756532 −0.378266 0.925697i $$-0.623480\pi$$
−0.378266 + 0.925697i $$0.623480\pi$$
$$12$$ −2.55073 −0.736332
$$13$$ 4.33379 1.20198 0.600989 0.799257i $$-0.294774\pi$$
0.600989 + 0.799257i $$0.294774\pi$$
$$14$$ −4.62650 −1.23648
$$15$$ 0 0
$$16$$ −2.59524 −0.648810
$$17$$ 6.77562 1.64333 0.821665 0.569971i $$-0.193046\pi$$
0.821665 + 0.569971i $$0.193046\pi$$
$$18$$ 2.13324 0.502810
$$19$$ 6.83602 1.56829 0.784145 0.620578i $$-0.213102\pi$$
0.784145 + 0.620578i $$0.213102\pi$$
$$20$$ 0 0
$$21$$ 2.16876 0.473263
$$22$$ −5.35259 −1.14118
$$23$$ 1.67843 0.349977 0.174989 0.984570i $$-0.444011\pi$$
0.174989 + 0.984570i $$0.444011\pi$$
$$24$$ −1.17484 −0.239813
$$25$$ 0 0
$$26$$ 9.24504 1.81310
$$27$$ −1.00000 −0.192450
$$28$$ −5.53193 −1.04544
$$29$$ 4.44927 0.826209 0.413104 0.910684i $$-0.364444\pi$$
0.413104 + 0.910684i $$0.364444\pi$$
$$30$$ 0 0
$$31$$ 6.56295 1.17874 0.589371 0.807863i $$-0.299376\pi$$
0.589371 + 0.807863i $$0.299376\pi$$
$$32$$ −7.88596 −1.39405
$$33$$ 2.50913 0.436784
$$34$$ 14.4541 2.47885
$$35$$ 0 0
$$36$$ 2.55073 0.425122
$$37$$ −7.97720 −1.31144 −0.655722 0.755002i $$-0.727636\pi$$
−0.655722 + 0.755002i $$0.727636\pi$$
$$38$$ 14.5829 2.36566
$$39$$ −4.33379 −0.693962
$$40$$ 0 0
$$41$$ 11.2249 1.75303 0.876517 0.481371i $$-0.159861\pi$$
0.876517 + 0.481371i $$0.159861\pi$$
$$42$$ 4.62650 0.713885
$$43$$ 4.25487 0.648861 0.324431 0.945909i $$-0.394827\pi$$
0.324431 + 0.945909i $$0.394827\pi$$
$$44$$ −6.40012 −0.964854
$$45$$ 0 0
$$46$$ 3.58050 0.527916
$$47$$ −4.98652 −0.727358 −0.363679 0.931524i $$-0.618479\pi$$
−0.363679 + 0.931524i $$0.618479\pi$$
$$48$$ 2.59524 0.374590
$$49$$ −2.29646 −0.328066
$$50$$ 0 0
$$51$$ −6.77562 −0.948777
$$52$$ 11.0543 1.53296
$$53$$ 8.21338 1.12820 0.564098 0.825708i $$-0.309224\pi$$
0.564098 + 0.825708i $$0.309224\pi$$
$$54$$ −2.13324 −0.290298
$$55$$ 0 0
$$56$$ −2.54795 −0.340484
$$57$$ −6.83602 −0.905453
$$58$$ 9.49138 1.24628
$$59$$ 3.67416 0.478335 0.239168 0.970978i $$-0.423125\pi$$
0.239168 + 0.970978i $$0.423125\pi$$
$$60$$ 0 0
$$61$$ 4.93960 0.632451 0.316226 0.948684i $$-0.397584\pi$$
0.316226 + 0.948684i $$0.397584\pi$$
$$62$$ 14.0004 1.77805
$$63$$ −2.16876 −0.273239
$$64$$ −11.6322 −1.45402
$$65$$ 0 0
$$66$$ 5.35259 0.658859
$$67$$ −11.5812 −1.41487 −0.707436 0.706778i $$-0.750148\pi$$
−0.707436 + 0.706778i $$0.750148\pi$$
$$68$$ 17.2828 2.09584
$$69$$ −1.67843 −0.202059
$$70$$ 0 0
$$71$$ −2.30251 −0.273257 −0.136629 0.990622i $$-0.543627\pi$$
−0.136629 + 0.990622i $$0.543627\pi$$
$$72$$ 1.17484 0.138456
$$73$$ 1.11599 0.130617 0.0653084 0.997865i $$-0.479197\pi$$
0.0653084 + 0.997865i $$0.479197\pi$$
$$74$$ −17.0173 −1.97822
$$75$$ 0 0
$$76$$ 17.4368 2.00014
$$77$$ 5.44172 0.620141
$$78$$ −9.24504 −1.04679
$$79$$ 7.78306 0.875663 0.437831 0.899057i $$-0.355747\pi$$
0.437831 + 0.899057i $$0.355747\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 23.9454 2.64433
$$83$$ −9.87708 −1.08415 −0.542075 0.840330i $$-0.682361\pi$$
−0.542075 + 0.840330i $$0.682361\pi$$
$$84$$ 5.53193 0.603583
$$85$$ 0 0
$$86$$ 9.07667 0.978763
$$87$$ −4.44927 −0.477012
$$88$$ −2.94783 −0.314240
$$89$$ −1.24025 −0.131466 −0.0657329 0.997837i $$-0.520939\pi$$
−0.0657329 + 0.997837i $$0.520939\pi$$
$$90$$ 0 0
$$91$$ −9.39897 −0.985280
$$92$$ 4.28122 0.446348
$$93$$ −6.56295 −0.680546
$$94$$ −10.6375 −1.09717
$$95$$ 0 0
$$96$$ 7.88596 0.804857
$$97$$ −15.0488 −1.52797 −0.763985 0.645234i $$-0.776760\pi$$
−0.763985 + 0.645234i $$0.776760\pi$$
$$98$$ −4.89892 −0.494866
$$99$$ −2.50913 −0.252177
$$100$$ 0 0
$$101$$ −17.0211 −1.69366 −0.846830 0.531864i $$-0.821492\pi$$
−0.846830 + 0.531864i $$0.821492\pi$$
$$102$$ −14.4541 −1.43116
$$103$$ −10.0859 −0.993793 −0.496897 0.867810i $$-0.665527\pi$$
−0.496897 + 0.867810i $$0.665527\pi$$
$$104$$ 5.09151 0.499264
$$105$$ 0 0
$$106$$ 17.5212 1.70180
$$107$$ −5.67218 −0.548351 −0.274175 0.961680i $$-0.588405\pi$$
−0.274175 + 0.961680i $$0.588405\pi$$
$$108$$ −2.55073 −0.245444
$$109$$ 8.34557 0.799361 0.399680 0.916655i $$-0.369121\pi$$
0.399680 + 0.916655i $$0.369121\pi$$
$$110$$ 0 0
$$111$$ 7.97720 0.757163
$$112$$ 5.62846 0.531839
$$113$$ 14.8224 1.39438 0.697188 0.716888i $$-0.254434\pi$$
0.697188 + 0.716888i $$0.254434\pi$$
$$114$$ −14.5829 −1.36581
$$115$$ 0 0
$$116$$ 11.3489 1.05372
$$117$$ 4.33379 0.400659
$$118$$ 7.83788 0.721536
$$119$$ −14.6947 −1.34706
$$120$$ 0 0
$$121$$ −4.70425 −0.427659
$$122$$ 10.5374 0.954009
$$123$$ −11.2249 −1.01211
$$124$$ 16.7403 1.50332
$$125$$ 0 0
$$126$$ −4.62650 −0.412162
$$127$$ 0.502113 0.0445553 0.0222777 0.999752i $$-0.492908\pi$$
0.0222777 + 0.999752i $$0.492908\pi$$
$$128$$ −9.04239 −0.799242
$$129$$ −4.25487 −0.374620
$$130$$ 0 0
$$131$$ 6.05788 0.529280 0.264640 0.964347i $$-0.414747\pi$$
0.264640 + 0.964347i $$0.414747\pi$$
$$132$$ 6.40012 0.557059
$$133$$ −14.8257 −1.28555
$$134$$ −24.7056 −2.13424
$$135$$ 0 0
$$136$$ 7.96027 0.682588
$$137$$ 1.94014 0.165757 0.0828786 0.996560i $$-0.473589\pi$$
0.0828786 + 0.996560i $$0.473589\pi$$
$$138$$ −3.58050 −0.304793
$$139$$ 19.2023 1.62872 0.814358 0.580363i $$-0.197089\pi$$
0.814358 + 0.580363i $$0.197089\pi$$
$$140$$ 0 0
$$141$$ 4.98652 0.419940
$$142$$ −4.91181 −0.412190
$$143$$ −10.8741 −0.909335
$$144$$ −2.59524 −0.216270
$$145$$ 0 0
$$146$$ 2.38068 0.197026
$$147$$ 2.29646 0.189409
$$148$$ −20.3477 −1.67257
$$149$$ 9.49138 0.777564 0.388782 0.921330i $$-0.372896\pi$$
0.388782 + 0.921330i $$0.372896\pi$$
$$150$$ 0 0
$$151$$ −12.5747 −1.02332 −0.511659 0.859189i $$-0.670969\pi$$
−0.511659 + 0.859189i $$0.670969\pi$$
$$152$$ 8.03123 0.651419
$$153$$ 6.77562 0.547776
$$154$$ 11.6085 0.935440
$$155$$ 0 0
$$156$$ −11.0543 −0.885055
$$157$$ 18.7991 1.50033 0.750165 0.661250i $$-0.229974\pi$$
0.750165 + 0.661250i $$0.229974\pi$$
$$158$$ 16.6032 1.32088
$$159$$ −8.21338 −0.651364
$$160$$ 0 0
$$161$$ −3.64012 −0.286882
$$162$$ 2.13324 0.167603
$$163$$ 7.29949 0.571740 0.285870 0.958268i $$-0.407717\pi$$
0.285870 + 0.958268i $$0.407717\pi$$
$$164$$ 28.6317 2.23576
$$165$$ 0 0
$$166$$ −21.0702 −1.63537
$$167$$ −14.7991 −1.14519 −0.572594 0.819839i $$-0.694063\pi$$
−0.572594 + 0.819839i $$0.694063\pi$$
$$168$$ 2.54795 0.196579
$$169$$ 5.78176 0.444750
$$170$$ 0 0
$$171$$ 6.83602 0.522763
$$172$$ 10.8530 0.827535
$$173$$ −2.29460 −0.174455 −0.0872276 0.996188i $$-0.527801\pi$$
−0.0872276 + 0.996188i $$0.527801\pi$$
$$174$$ −9.49138 −0.719540
$$175$$ 0 0
$$176$$ 6.51180 0.490845
$$177$$ −3.67416 −0.276167
$$178$$ −2.64575 −0.198307
$$179$$ 13.6637 1.02127 0.510636 0.859797i $$-0.329410\pi$$
0.510636 + 0.859797i $$0.329410\pi$$
$$180$$ 0 0
$$181$$ 8.71878 0.648062 0.324031 0.946046i $$-0.394962\pi$$
0.324031 + 0.946046i $$0.394962\pi$$
$$182$$ −20.0503 −1.48623
$$183$$ −4.93960 −0.365146
$$184$$ 1.97189 0.145370
$$185$$ 0 0
$$186$$ −14.0004 −1.02656
$$187$$ −17.0009 −1.24323
$$188$$ −12.7193 −0.927647
$$189$$ 2.16876 0.157754
$$190$$ 0 0
$$191$$ 11.6080 0.839925 0.419963 0.907541i $$-0.362043\pi$$
0.419963 + 0.907541i $$0.362043\pi$$
$$192$$ 11.6322 0.839481
$$193$$ −14.8653 −1.07002 −0.535012 0.844844i $$-0.679693\pi$$
−0.535012 + 0.844844i $$0.679693\pi$$
$$194$$ −32.1027 −2.30484
$$195$$ 0 0
$$196$$ −5.85766 −0.418404
$$197$$ 22.8292 1.62651 0.813256 0.581906i $$-0.197693\pi$$
0.813256 + 0.581906i $$0.197693\pi$$
$$198$$ −5.35259 −0.380392
$$199$$ 5.87697 0.416607 0.208304 0.978064i $$-0.433206\pi$$
0.208304 + 0.978064i $$0.433206\pi$$
$$200$$ 0 0
$$201$$ 11.5812 0.816876
$$202$$ −36.3101 −2.55477
$$203$$ −9.64942 −0.677256
$$204$$ −17.2828 −1.21004
$$205$$ 0 0
$$206$$ −21.5157 −1.49907
$$207$$ 1.67843 0.116659
$$208$$ −11.2472 −0.779855
$$209$$ −17.1525 −1.18646
$$210$$ 0 0
$$211$$ −24.6884 −1.69962 −0.849810 0.527090i $$-0.823283\pi$$
−0.849810 + 0.527090i $$0.823283\pi$$
$$212$$ 20.9501 1.43886
$$213$$ 2.30251 0.157765
$$214$$ −12.1002 −0.827149
$$215$$ 0 0
$$216$$ −1.17484 −0.0799378
$$217$$ −14.2335 −0.966232
$$218$$ 17.8031 1.20578
$$219$$ −1.11599 −0.0754116
$$220$$ 0 0
$$221$$ 29.3641 1.97525
$$222$$ 17.0173 1.14213
$$223$$ −3.31060 −0.221694 −0.110847 0.993837i $$-0.535356\pi$$
−0.110847 + 0.993837i $$0.535356\pi$$
$$224$$ 17.1028 1.14273
$$225$$ 0 0
$$226$$ 31.6198 2.10332
$$227$$ −2.70185 −0.179328 −0.0896641 0.995972i $$-0.528579\pi$$
−0.0896641 + 0.995972i $$0.528579\pi$$
$$228$$ −17.4368 −1.15478
$$229$$ −9.33473 −0.616856 −0.308428 0.951248i $$-0.599803\pi$$
−0.308428 + 0.951248i $$0.599803\pi$$
$$230$$ 0 0
$$231$$ −5.44172 −0.358039
$$232$$ 5.22718 0.343181
$$233$$ 9.52491 0.623998 0.311999 0.950083i $$-0.399002\pi$$
0.311999 + 0.950083i $$0.399002\pi$$
$$234$$ 9.24504 0.604367
$$235$$ 0 0
$$236$$ 9.37179 0.610052
$$237$$ −7.78306 −0.505564
$$238$$ −31.3474 −2.03195
$$239$$ 7.84648 0.507546 0.253773 0.967264i $$-0.418328\pi$$
0.253773 + 0.967264i $$0.418328\pi$$
$$240$$ 0 0
$$241$$ 1.42783 0.0919747 0.0459874 0.998942i $$-0.485357\pi$$
0.0459874 + 0.998942i $$0.485357\pi$$
$$242$$ −10.0353 −0.645095
$$243$$ −1.00000 −0.0641500
$$244$$ 12.5996 0.806606
$$245$$ 0 0
$$246$$ −23.9454 −1.52670
$$247$$ 29.6259 1.88505
$$248$$ 7.71042 0.489612
$$249$$ 9.87708 0.625935
$$250$$ 0 0
$$251$$ −17.5762 −1.10940 −0.554701 0.832050i $$-0.687167\pi$$
−0.554701 + 0.832050i $$0.687167\pi$$
$$252$$ −5.53193 −0.348479
$$253$$ −4.21141 −0.264769
$$254$$ 1.07113 0.0672087
$$255$$ 0 0
$$256$$ 3.97476 0.248423
$$257$$ −19.2890 −1.20322 −0.601608 0.798791i $$-0.705473\pi$$
−0.601608 + 0.798791i $$0.705473\pi$$
$$258$$ −9.07667 −0.565089
$$259$$ 17.3007 1.07501
$$260$$ 0 0
$$261$$ 4.44927 0.275403
$$262$$ 12.9229 0.798382
$$263$$ −22.2432 −1.37157 −0.685786 0.727803i $$-0.740541\pi$$
−0.685786 + 0.727803i $$0.740541\pi$$
$$264$$ 2.94783 0.181426
$$265$$ 0 0
$$266$$ −31.6268 −1.93917
$$267$$ 1.24025 0.0759018
$$268$$ −29.5406 −1.80448
$$269$$ 25.1785 1.53516 0.767581 0.640951i $$-0.221460\pi$$
0.767581 + 0.640951i $$0.221460\pi$$
$$270$$ 0 0
$$271$$ −5.70091 −0.346306 −0.173153 0.984895i $$-0.555395\pi$$
−0.173153 + 0.984895i $$0.555395\pi$$
$$272$$ −17.5843 −1.06621
$$273$$ 9.39897 0.568852
$$274$$ 4.13879 0.250033
$$275$$ 0 0
$$276$$ −4.28122 −0.257699
$$277$$ 19.1611 1.15128 0.575638 0.817704i $$-0.304754\pi$$
0.575638 + 0.817704i $$0.304754\pi$$
$$278$$ 40.9631 2.45681
$$279$$ 6.56295 0.392914
$$280$$ 0 0
$$281$$ 4.36015 0.260105 0.130052 0.991507i $$-0.458485\pi$$
0.130052 + 0.991507i $$0.458485\pi$$
$$282$$ 10.6375 0.633451
$$283$$ −24.9081 −1.48063 −0.740317 0.672258i $$-0.765324\pi$$
−0.740317 + 0.672258i $$0.765324\pi$$
$$284$$ −5.87307 −0.348503
$$285$$ 0 0
$$286$$ −23.1970 −1.37167
$$287$$ −24.3441 −1.43699
$$288$$ −7.88596 −0.464684
$$289$$ 28.9090 1.70053
$$290$$ 0 0
$$291$$ 15.0488 0.882174
$$292$$ 2.84659 0.166584
$$293$$ −13.5651 −0.792484 −0.396242 0.918146i $$-0.629686\pi$$
−0.396242 + 0.918146i $$0.629686\pi$$
$$294$$ 4.89892 0.285711
$$295$$ 0 0
$$296$$ −9.37194 −0.544733
$$297$$ 2.50913 0.145595
$$298$$ 20.2474 1.17290
$$299$$ 7.27397 0.420665
$$300$$ 0 0
$$301$$ −9.22780 −0.531882
$$302$$ −26.8250 −1.54360
$$303$$ 17.0211 0.977835
$$304$$ −17.7411 −1.01752
$$305$$ 0 0
$$306$$ 14.4541 0.826283
$$307$$ −11.9672 −0.683007 −0.341504 0.939880i $$-0.610936\pi$$
−0.341504 + 0.939880i $$0.610936\pi$$
$$308$$ 13.8803 0.790906
$$309$$ 10.0859 0.573767
$$310$$ 0 0
$$311$$ 7.66452 0.434615 0.217308 0.976103i $$-0.430273\pi$$
0.217308 + 0.976103i $$0.430273\pi$$
$$312$$ −5.09151 −0.288250
$$313$$ 26.3840 1.49131 0.745655 0.666332i $$-0.232137\pi$$
0.745655 + 0.666332i $$0.232137\pi$$
$$314$$ 40.1030 2.26315
$$315$$ 0 0
$$316$$ 19.8525 1.11679
$$317$$ −10.8880 −0.611529 −0.305765 0.952107i $$-0.598912\pi$$
−0.305765 + 0.952107i $$0.598912\pi$$
$$318$$ −17.5212 −0.982537
$$319$$ −11.1638 −0.625053
$$320$$ 0 0
$$321$$ 5.67218 0.316590
$$322$$ −7.76526 −0.432741
$$323$$ 46.3183 2.57722
$$324$$ 2.55073 0.141707
$$325$$ 0 0
$$326$$ 15.5716 0.862431
$$327$$ −8.34557 −0.461511
$$328$$ 13.1875 0.728155
$$329$$ 10.8146 0.596227
$$330$$ 0 0
$$331$$ −32.2137 −1.77062 −0.885312 0.464998i $$-0.846055\pi$$
−0.885312 + 0.464998i $$0.846055\pi$$
$$332$$ −25.1938 −1.38269
$$333$$ −7.97720 −0.437148
$$334$$ −31.5701 −1.72744
$$335$$ 0 0
$$336$$ −5.62846 −0.307058
$$337$$ −3.47169 −0.189115 −0.0945576 0.995519i $$-0.530144\pi$$
−0.0945576 + 0.995519i $$0.530144\pi$$
$$338$$ 12.3339 0.670875
$$339$$ −14.8224 −0.805043
$$340$$ 0 0
$$341$$ −16.4673 −0.891755
$$342$$ 14.5829 0.788553
$$343$$ 20.1618 1.08864
$$344$$ 4.99879 0.269517
$$345$$ 0 0
$$346$$ −4.89494 −0.263154
$$347$$ −10.9805 −0.589465 −0.294733 0.955580i $$-0.595231\pi$$
−0.294733 + 0.955580i $$0.595231\pi$$
$$348$$ −11.3489 −0.608364
$$349$$ −26.3158 −1.40865 −0.704325 0.709878i $$-0.748750\pi$$
−0.704325 + 0.709878i $$0.748750\pi$$
$$350$$ 0 0
$$351$$ −4.33379 −0.231321
$$352$$ 19.7869 1.05465
$$353$$ −16.3668 −0.871119 −0.435559 0.900160i $$-0.643449\pi$$
−0.435559 + 0.900160i $$0.643449\pi$$
$$354$$ −7.83788 −0.416579
$$355$$ 0 0
$$356$$ −3.16353 −0.167667
$$357$$ 14.6947 0.777727
$$358$$ 29.1480 1.54052
$$359$$ −17.4871 −0.922934 −0.461467 0.887157i $$-0.652677\pi$$
−0.461467 + 0.887157i $$0.652677\pi$$
$$360$$ 0 0
$$361$$ 27.7311 1.45953
$$362$$ 18.5993 0.977557
$$363$$ 4.70425 0.246909
$$364$$ −23.9742 −1.25659
$$365$$ 0 0
$$366$$ −10.5374 −0.550798
$$367$$ 8.29594 0.433044 0.216522 0.976278i $$-0.430529\pi$$
0.216522 + 0.976278i $$0.430529\pi$$
$$368$$ −4.35593 −0.227068
$$369$$ 11.2249 0.584345
$$370$$ 0 0
$$371$$ −17.8129 −0.924799
$$372$$ −16.7403 −0.867945
$$373$$ −19.1295 −0.990486 −0.495243 0.868755i $$-0.664921\pi$$
−0.495243 + 0.868755i $$0.664921\pi$$
$$374$$ −36.2671 −1.87533
$$375$$ 0 0
$$376$$ −5.85836 −0.302122
$$377$$ 19.2822 0.993085
$$378$$ 4.62650 0.237962
$$379$$ −27.0952 −1.39179 −0.695894 0.718145i $$-0.744992\pi$$
−0.695894 + 0.718145i $$0.744992\pi$$
$$380$$ 0 0
$$381$$ −0.502113 −0.0257240
$$382$$ 24.7627 1.26697
$$383$$ 19.8026 1.01187 0.505933 0.862573i $$-0.331148\pi$$
0.505933 + 0.862573i $$0.331148\pi$$
$$384$$ 9.04239 0.461443
$$385$$ 0 0
$$386$$ −31.7112 −1.61406
$$387$$ 4.25487 0.216287
$$388$$ −38.3853 −1.94872
$$389$$ −23.0130 −1.16680 −0.583402 0.812183i $$-0.698279\pi$$
−0.583402 + 0.812183i $$0.698279\pi$$
$$390$$ 0 0
$$391$$ 11.3724 0.575128
$$392$$ −2.69798 −0.136269
$$393$$ −6.05788 −0.305580
$$394$$ 48.7002 2.45348
$$395$$ 0 0
$$396$$ −6.40012 −0.321618
$$397$$ −20.5745 −1.03260 −0.516301 0.856407i $$-0.672691\pi$$
−0.516301 + 0.856407i $$0.672691\pi$$
$$398$$ 12.5370 0.628423
$$399$$ 14.8257 0.742214
$$400$$ 0 0
$$401$$ −13.1542 −0.656889 −0.328444 0.944523i $$-0.606524\pi$$
−0.328444 + 0.944523i $$0.606524\pi$$
$$402$$ 24.7056 1.23220
$$403$$ 28.4425 1.41682
$$404$$ −43.4161 −2.16003
$$405$$ 0 0
$$406$$ −20.5846 −1.02159
$$407$$ 20.0159 0.992150
$$408$$ −7.96027 −0.394092
$$409$$ 2.76531 0.136736 0.0683679 0.997660i $$-0.478221\pi$$
0.0683679 + 0.997660i $$0.478221\pi$$
$$410$$ 0 0
$$411$$ −1.94014 −0.0956999
$$412$$ −25.7264 −1.26745
$$413$$ −7.96839 −0.392099
$$414$$ 3.58050 0.175972
$$415$$ 0 0
$$416$$ −34.1761 −1.67562
$$417$$ −19.2023 −0.940340
$$418$$ −36.5904 −1.78970
$$419$$ −9.70852 −0.474292 −0.237146 0.971474i $$-0.576212\pi$$
−0.237146 + 0.971474i $$0.576212\pi$$
$$420$$ 0 0
$$421$$ −4.21745 −0.205546 −0.102773 0.994705i $$-0.532772\pi$$
−0.102773 + 0.994705i $$0.532772\pi$$
$$422$$ −52.6664 −2.56376
$$423$$ −4.98652 −0.242453
$$424$$ 9.64942 0.468617
$$425$$ 0 0
$$426$$ 4.91181 0.237978
$$427$$ −10.7128 −0.518430
$$428$$ −14.4682 −0.699347
$$429$$ 10.8741 0.525005
$$430$$ 0 0
$$431$$ 18.9785 0.914164 0.457082 0.889425i $$-0.348895\pi$$
0.457082 + 0.889425i $$0.348895\pi$$
$$432$$ 2.59524 0.124863
$$433$$ 13.7054 0.658640 0.329320 0.944218i $$-0.393181\pi$$
0.329320 + 0.944218i $$0.393181\pi$$
$$434$$ −30.3635 −1.45750
$$435$$ 0 0
$$436$$ 21.2873 1.01948
$$437$$ 11.4738 0.548866
$$438$$ −2.38068 −0.113753
$$439$$ −33.3843 −1.59335 −0.796673 0.604410i $$-0.793409\pi$$
−0.796673 + 0.604410i $$0.793409\pi$$
$$440$$ 0 0
$$441$$ −2.29646 −0.109355
$$442$$ 62.6409 2.97952
$$443$$ 11.6290 0.552512 0.276256 0.961084i $$-0.410906\pi$$
0.276256 + 0.961084i $$0.410906\pi$$
$$444$$ 20.3477 0.965659
$$445$$ 0 0
$$446$$ −7.06231 −0.334410
$$447$$ −9.49138 −0.448927
$$448$$ 25.2275 1.19189
$$449$$ −13.3509 −0.630069 −0.315034 0.949080i $$-0.602016\pi$$
−0.315034 + 0.949080i $$0.602016\pi$$
$$450$$ 0 0
$$451$$ −28.1647 −1.32623
$$452$$ 37.8080 1.77834
$$453$$ 12.5747 0.590813
$$454$$ −5.76371 −0.270504
$$455$$ 0 0
$$456$$ −8.03123 −0.376097
$$457$$ −23.3042 −1.09013 −0.545063 0.838395i $$-0.683494\pi$$
−0.545063 + 0.838395i $$0.683494\pi$$
$$458$$ −19.9132 −0.930485
$$459$$ −6.77562 −0.316259
$$460$$ 0 0
$$461$$ 5.03269 0.234396 0.117198 0.993109i $$-0.462609\pi$$
0.117198 + 0.993109i $$0.462609\pi$$
$$462$$ −11.6085 −0.540077
$$463$$ −12.7759 −0.593746 −0.296873 0.954917i $$-0.595944\pi$$
−0.296873 + 0.954917i $$0.595944\pi$$
$$464$$ −11.5469 −0.536052
$$465$$ 0 0
$$466$$ 20.3190 0.941257
$$467$$ 4.86284 0.225025 0.112513 0.993650i $$-0.464110\pi$$
0.112513 + 0.993650i $$0.464110\pi$$
$$468$$ 11.0543 0.510987
$$469$$ 25.1169 1.15979
$$470$$ 0 0
$$471$$ −18.7991 −0.866216
$$472$$ 4.31655 0.198685
$$473$$ −10.6760 −0.490884
$$474$$ −16.6032 −0.762609
$$475$$ 0 0
$$476$$ −37.4823 −1.71800
$$477$$ 8.21338 0.376065
$$478$$ 16.7385 0.765599
$$479$$ 4.79202 0.218953 0.109476 0.993989i $$-0.465083\pi$$
0.109476 + 0.993989i $$0.465083\pi$$
$$480$$ 0 0
$$481$$ −34.5715 −1.57633
$$482$$ 3.04591 0.138738
$$483$$ 3.64012 0.165631
$$484$$ −11.9993 −0.545421
$$485$$ 0 0
$$486$$ −2.13324 −0.0967659
$$487$$ 9.16949 0.415509 0.207755 0.978181i $$-0.433384\pi$$
0.207755 + 0.978181i $$0.433384\pi$$
$$488$$ 5.80325 0.262701
$$489$$ −7.29949 −0.330094
$$490$$ 0 0
$$491$$ 31.9054 1.43987 0.719935 0.694042i $$-0.244172\pi$$
0.719935 + 0.694042i $$0.244172\pi$$
$$492$$ −28.6317 −1.29081
$$493$$ 30.1466 1.35773
$$494$$ 63.1992 2.84347
$$495$$ 0 0
$$496$$ −17.0324 −0.764778
$$497$$ 4.99359 0.223993
$$498$$ 21.0702 0.944179
$$499$$ 9.07297 0.406162 0.203081 0.979162i $$-0.434905\pi$$
0.203081 + 0.979162i $$0.434905\pi$$
$$500$$ 0 0
$$501$$ 14.7991 0.661175
$$502$$ −37.4944 −1.67346
$$503$$ 38.7485 1.72771 0.863856 0.503739i $$-0.168043\pi$$
0.863856 + 0.503739i $$0.168043\pi$$
$$504$$ −2.54795 −0.113495
$$505$$ 0 0
$$506$$ −8.98396 −0.399386
$$507$$ −5.78176 −0.256777
$$508$$ 1.28075 0.0568243
$$509$$ 32.9367 1.45989 0.729947 0.683503i $$-0.239545\pi$$
0.729947 + 0.683503i $$0.239545\pi$$
$$510$$ 0 0
$$511$$ −2.42032 −0.107069
$$512$$ 26.5639 1.17397
$$513$$ −6.83602 −0.301818
$$514$$ −41.1482 −1.81497
$$515$$ 0 0
$$516$$ −10.8530 −0.477777
$$517$$ 12.5118 0.550270
$$518$$ 36.9065 1.62158
$$519$$ 2.29460 0.100722
$$520$$ 0 0
$$521$$ −17.7521 −0.777735 −0.388867 0.921294i $$-0.627134\pi$$
−0.388867 + 0.921294i $$0.627134\pi$$
$$522$$ 9.49138 0.415426
$$523$$ −4.98678 −0.218056 −0.109028 0.994039i $$-0.534774\pi$$
−0.109028 + 0.994039i $$0.534774\pi$$
$$524$$ 15.4520 0.675025
$$525$$ 0 0
$$526$$ −47.4501 −2.06892
$$527$$ 44.4681 1.93706
$$528$$ −6.51180 −0.283390
$$529$$ −20.1829 −0.877516
$$530$$ 0 0
$$531$$ 3.67416 0.159445
$$532$$ −37.8164 −1.63955
$$533$$ 48.6463 2.10711
$$534$$ 2.64575 0.114493
$$535$$ 0 0
$$536$$ −13.6061 −0.587693
$$537$$ −13.6637 −0.589632
$$538$$ 53.7120 2.31569
$$539$$ 5.76214 0.248193
$$540$$ 0 0
$$541$$ −3.49960 −0.150460 −0.0752298 0.997166i $$-0.523969\pi$$
−0.0752298 + 0.997166i $$0.523969\pi$$
$$542$$ −12.1614 −0.522378
$$543$$ −8.71878 −0.374159
$$544$$ −53.4322 −2.29089
$$545$$ 0 0
$$546$$ 20.0503 0.858073
$$547$$ −17.4640 −0.746709 −0.373354 0.927689i $$-0.621792\pi$$
−0.373354 + 0.927689i $$0.621792\pi$$
$$548$$ 4.94877 0.211401
$$549$$ 4.93960 0.210817
$$550$$ 0 0
$$551$$ 30.4153 1.29574
$$552$$ −1.97189 −0.0839291
$$553$$ −16.8796 −0.717795
$$554$$ 40.8752 1.73662
$$555$$ 0 0
$$556$$ 48.9798 2.07721
$$557$$ 6.02774 0.255403 0.127702 0.991813i $$-0.459240\pi$$
0.127702 + 0.991813i $$0.459240\pi$$
$$558$$ 14.0004 0.592683
$$559$$ 18.4397 0.779917
$$560$$ 0 0
$$561$$ 17.0009 0.717780
$$562$$ 9.30126 0.392350
$$563$$ 9.19823 0.387659 0.193830 0.981035i $$-0.437909\pi$$
0.193830 + 0.981035i $$0.437909\pi$$
$$564$$ 12.7193 0.535577
$$565$$ 0 0
$$566$$ −53.1351 −2.23343
$$567$$ −2.16876 −0.0910795
$$568$$ −2.70508 −0.113503
$$569$$ 9.06517 0.380032 0.190016 0.981781i $$-0.439146\pi$$
0.190016 + 0.981781i $$0.439146\pi$$
$$570$$ 0 0
$$571$$ −6.34688 −0.265609 −0.132804 0.991142i $$-0.542398\pi$$
−0.132804 + 0.991142i $$0.542398\pi$$
$$572$$ −27.7368 −1.15973
$$573$$ −11.6080 −0.484931
$$574$$ −51.9320 −2.16760
$$575$$ 0 0
$$576$$ −11.6322 −0.484675
$$577$$ 2.41418 0.100504 0.0502519 0.998737i $$-0.483998\pi$$
0.0502519 + 0.998737i $$0.483998\pi$$
$$578$$ 61.6700 2.56513
$$579$$ 14.8653 0.617779
$$580$$ 0 0
$$581$$ 21.4211 0.888695
$$582$$ 32.1027 1.33070
$$583$$ −20.6085 −0.853516
$$584$$ 1.31111 0.0542541
$$585$$ 0 0
$$586$$ −28.9378 −1.19541
$$587$$ 4.50976 0.186138 0.0930689 0.995660i $$-0.470332\pi$$
0.0930689 + 0.995660i $$0.470332\pi$$
$$588$$ 5.85766 0.241566
$$589$$ 44.8645 1.84861
$$590$$ 0 0
$$591$$ −22.8292 −0.939067
$$592$$ 20.7027 0.850877
$$593$$ −33.2824 −1.36674 −0.683371 0.730071i $$-0.739487\pi$$
−0.683371 + 0.730071i $$0.739487\pi$$
$$594$$ 5.35259 0.219620
$$595$$ 0 0
$$596$$ 24.2099 0.991678
$$597$$ −5.87697 −0.240528
$$598$$ 15.5172 0.634544
$$599$$ 39.1061 1.59783 0.798915 0.601444i $$-0.205408\pi$$
0.798915 + 0.601444i $$0.205408\pi$$
$$600$$ 0 0
$$601$$ −28.8076 −1.17509 −0.587543 0.809193i $$-0.699905\pi$$
−0.587543 + 0.809193i $$0.699905\pi$$
$$602$$ −19.6852 −0.802307
$$603$$ −11.5812 −0.471624
$$604$$ −32.0747 −1.30510
$$605$$ 0 0
$$606$$ 36.3101 1.47500
$$607$$ −23.0933 −0.937327 −0.468663 0.883377i $$-0.655264\pi$$
−0.468663 + 0.883377i $$0.655264\pi$$
$$608$$ −53.9085 −2.18628
$$609$$ 9.64942 0.391014
$$610$$ 0 0
$$611$$ −21.6105 −0.874268
$$612$$ 17.2828 0.698615
$$613$$ −18.2228 −0.736013 −0.368007 0.929823i $$-0.619960\pi$$
−0.368007 + 0.929823i $$0.619960\pi$$
$$614$$ −25.5291 −1.03027
$$615$$ 0 0
$$616$$ 6.39315 0.257587
$$617$$ 15.4484 0.621930 0.310965 0.950421i $$-0.399348\pi$$
0.310965 + 0.950421i $$0.399348\pi$$
$$618$$ 21.5157 0.865488
$$619$$ 16.5042 0.663359 0.331680 0.943392i $$-0.392385\pi$$
0.331680 + 0.943392i $$0.392385\pi$$
$$620$$ 0 0
$$621$$ −1.67843 −0.0673531
$$622$$ 16.3503 0.655587
$$623$$ 2.68980 0.107765
$$624$$ 11.2472 0.450249
$$625$$ 0 0
$$626$$ 56.2835 2.24954
$$627$$ 17.1525 0.685004
$$628$$ 47.9514 1.91347
$$629$$ −54.0505 −2.15513
$$630$$ 0 0
$$631$$ −38.1237 −1.51768 −0.758841 0.651276i $$-0.774234\pi$$
−0.758841 + 0.651276i $$0.774234\pi$$
$$632$$ 9.14386 0.363723
$$633$$ 24.6884 0.981276
$$634$$ −23.2267 −0.922450
$$635$$ 0 0
$$636$$ −20.9501 −0.830726
$$637$$ −9.95240 −0.394329
$$638$$ −23.8151 −0.942850
$$639$$ −2.30251 −0.0910858
$$640$$ 0 0
$$641$$ 38.9299 1.53764 0.768819 0.639466i $$-0.220845\pi$$
0.768819 + 0.639466i $$0.220845\pi$$
$$642$$ 12.1002 0.477555
$$643$$ 40.5346 1.59853 0.799265 0.600979i $$-0.205222\pi$$
0.799265 + 0.600979i $$0.205222\pi$$
$$644$$ −9.28496 −0.365879
$$645$$ 0 0
$$646$$ 98.8082 3.88755
$$647$$ 31.8752 1.25315 0.626573 0.779363i $$-0.284457\pi$$
0.626573 + 0.779363i $$0.284457\pi$$
$$648$$ 1.17484 0.0461521
$$649$$ −9.21896 −0.361876
$$650$$ 0 0
$$651$$ 14.2335 0.557855
$$652$$ 18.6190 0.729177
$$653$$ −19.5285 −0.764210 −0.382105 0.924119i $$-0.624801\pi$$
−0.382105 + 0.924119i $$0.624801\pi$$
$$654$$ −17.8031 −0.696158
$$655$$ 0 0
$$656$$ −29.1313 −1.13738
$$657$$ 1.11599 0.0435389
$$658$$ 23.0701 0.899367
$$659$$ 47.6562 1.85642 0.928211 0.372055i $$-0.121347\pi$$
0.928211 + 0.372055i $$0.121347\pi$$
$$660$$ 0 0
$$661$$ 3.91738 0.152368 0.0761842 0.997094i $$-0.475726\pi$$
0.0761842 + 0.997094i $$0.475726\pi$$
$$662$$ −68.7196 −2.67086
$$663$$ −29.3641 −1.14041
$$664$$ −11.6040 −0.450322
$$665$$ 0 0
$$666$$ −17.0173 −0.659408
$$667$$ 7.46779 0.289154
$$668$$ −37.7485 −1.46053
$$669$$ 3.31060 0.127995
$$670$$ 0 0
$$671$$ −12.3941 −0.478470
$$672$$ −17.1028 −0.659754
$$673$$ 28.1866 1.08651 0.543257 0.839566i $$-0.317191\pi$$
0.543257 + 0.839566i $$0.317191\pi$$
$$674$$ −7.40597 −0.285267
$$675$$ 0 0
$$676$$ 14.7477 0.567219
$$677$$ −9.50611 −0.365350 −0.182675 0.983173i $$-0.558476\pi$$
−0.182675 + 0.983173i $$0.558476\pi$$
$$678$$ −31.6198 −1.21435
$$679$$ 32.6372 1.25250
$$680$$ 0 0
$$681$$ 2.70185 0.103535
$$682$$ −35.1288 −1.34515
$$683$$ 7.49565 0.286813 0.143407 0.989664i $$-0.454194\pi$$
0.143407 + 0.989664i $$0.454194\pi$$
$$684$$ 17.4368 0.666714
$$685$$ 0 0
$$686$$ 43.0101 1.64213
$$687$$ 9.33473 0.356142
$$688$$ −11.0424 −0.420987
$$689$$ 35.5951 1.35607
$$690$$ 0 0
$$691$$ −0.745188 −0.0283483 −0.0141741 0.999900i $$-0.504512\pi$$
−0.0141741 + 0.999900i $$0.504512\pi$$
$$692$$ −5.85290 −0.222494
$$693$$ 5.44172 0.206714
$$694$$ −23.4241 −0.889167
$$695$$ 0 0
$$696$$ −5.22718 −0.198136
$$697$$ 76.0556 2.88081
$$698$$ −56.1379 −2.12485
$$699$$ −9.52491 −0.360265
$$700$$ 0 0
$$701$$ 31.6488 1.19536 0.597679 0.801735i $$-0.296090\pi$$
0.597679 + 0.801735i $$0.296090\pi$$
$$702$$ −9.24504 −0.348931
$$703$$ −54.5323 −2.05672
$$704$$ 29.1867 1.10002
$$705$$ 0 0
$$706$$ −34.9145 −1.31402
$$707$$ 36.9147 1.38832
$$708$$ −9.37179 −0.352214
$$709$$ −16.0181 −0.601571 −0.300785 0.953692i $$-0.597249\pi$$
−0.300785 + 0.953692i $$0.597249\pi$$
$$710$$ 0 0
$$711$$ 7.78306 0.291888
$$712$$ −1.45709 −0.0546068
$$713$$ 11.0155 0.412532
$$714$$ 31.3474 1.17315
$$715$$ 0 0
$$716$$ 34.8524 1.30250
$$717$$ −7.84648 −0.293032
$$718$$ −37.3043 −1.39218
$$719$$ 14.3316 0.534478 0.267239 0.963630i $$-0.413889\pi$$
0.267239 + 0.963630i $$0.413889\pi$$
$$720$$ 0 0
$$721$$ 21.8739 0.814628
$$722$$ 59.1573 2.20161
$$723$$ −1.42783 −0.0531016
$$724$$ 22.2393 0.826515
$$725$$ 0 0
$$726$$ 10.0353 0.372445
$$727$$ −18.3551 −0.680755 −0.340377 0.940289i $$-0.610555\pi$$
−0.340377 + 0.940289i $$0.610555\pi$$
$$728$$ −11.0423 −0.409254
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 28.8294 1.06629
$$732$$ −12.5996 −0.465694
$$733$$ −0.665145 −0.0245677 −0.0122838 0.999925i $$-0.503910\pi$$
−0.0122838 + 0.999925i $$0.503910\pi$$
$$734$$ 17.6973 0.653218
$$735$$ 0 0
$$736$$ −13.2360 −0.487887
$$737$$ 29.0588 1.07040
$$738$$ 23.9454 0.881443
$$739$$ −32.8332 −1.20779 −0.603895 0.797064i $$-0.706385\pi$$
−0.603895 + 0.797064i $$0.706385\pi$$
$$740$$ 0 0
$$741$$ −29.6259 −1.08833
$$742$$ −37.9992 −1.39500
$$743$$ 29.6004 1.08593 0.542967 0.839754i $$-0.317301\pi$$
0.542967 + 0.839754i $$0.317301\pi$$
$$744$$ −7.71042 −0.282678
$$745$$ 0 0
$$746$$ −40.8078 −1.49408
$$747$$ −9.87708 −0.361383
$$748$$ −43.3648 −1.58557
$$749$$ 12.3016 0.449492
$$750$$ 0 0
$$751$$ 24.9709 0.911199 0.455600 0.890185i $$-0.349425\pi$$
0.455600 + 0.890185i $$0.349425\pi$$
$$752$$ 12.9412 0.471917
$$753$$ 17.5762 0.640514
$$754$$ 41.1337 1.49800
$$755$$ 0 0
$$756$$ 5.53193 0.201194
$$757$$ 11.2173 0.407700 0.203850 0.979002i $$-0.434655\pi$$
0.203850 + 0.979002i $$0.434655\pi$$
$$758$$ −57.8007 −2.09942
$$759$$ 4.21141 0.152864
$$760$$ 0 0
$$761$$ 19.3105 0.700004 0.350002 0.936749i $$-0.386181\pi$$
0.350002 + 0.936749i $$0.386181\pi$$
$$762$$ −1.07113 −0.0388029
$$763$$ −18.0996 −0.655249
$$764$$ 29.6089 1.07121
$$765$$ 0 0
$$766$$ 42.2438 1.52633
$$767$$ 15.9231 0.574948
$$768$$ −3.97476 −0.143427
$$769$$ −19.3372 −0.697318 −0.348659 0.937250i $$-0.613363\pi$$
−0.348659 + 0.937250i $$0.613363\pi$$
$$770$$ 0 0
$$771$$ 19.2890 0.694677
$$772$$ −37.9172 −1.36467
$$773$$ −7.79217 −0.280265 −0.140132 0.990133i $$-0.544753\pi$$
−0.140132 + 0.990133i $$0.544753\pi$$
$$774$$ 9.07667 0.326254
$$775$$ 0 0
$$776$$ −17.6799 −0.634671
$$777$$ −17.3007 −0.620658
$$778$$ −49.0923 −1.76004
$$779$$ 76.7336 2.74926
$$780$$ 0 0
$$781$$ 5.77730 0.206728
$$782$$ 24.2601 0.867540
$$783$$ −4.44927 −0.159004
$$784$$ 5.95987 0.212853
$$785$$ 0 0
$$786$$ −12.9229 −0.460946
$$787$$ 9.07666 0.323548 0.161774 0.986828i $$-0.448278\pi$$
0.161774 + 0.986828i $$0.448278\pi$$
$$788$$ 58.2311 2.07440
$$789$$ 22.2432 0.791877
$$790$$ 0 0
$$791$$ −32.1463 −1.14299
$$792$$ −2.94783 −0.104747
$$793$$ 21.4072 0.760192
$$794$$ −43.8903 −1.55761
$$795$$ 0 0
$$796$$ 14.9906 0.531326
$$797$$ 22.9954 0.814541 0.407270 0.913308i $$-0.366481\pi$$
0.407270 + 0.913308i $$0.366481\pi$$
$$798$$ 31.6268 1.11958
$$799$$ −33.7867 −1.19529
$$800$$ 0 0
$$801$$ −1.24025 −0.0438219
$$802$$ −28.0611 −0.990872
$$803$$ −2.80017 −0.0988158
$$804$$ 29.5406 1.04182
$$805$$ 0 0
$$806$$ 60.6747 2.13718
$$807$$ −25.1785 −0.886327
$$808$$ −19.9970 −0.703493
$$809$$ −50.6773 −1.78172 −0.890860 0.454278i $$-0.849897\pi$$
−0.890860 + 0.454278i $$0.849897\pi$$
$$810$$ 0 0
$$811$$ 26.7614 0.939720 0.469860 0.882741i $$-0.344304\pi$$
0.469860 + 0.882741i $$0.344304\pi$$
$$812$$ −24.6130 −0.863749
$$813$$ 5.70091 0.199940
$$814$$ 42.6987 1.49659
$$815$$ 0 0
$$816$$ 17.5843 0.615575
$$817$$ 29.0864 1.01760
$$818$$ 5.89908 0.206256
$$819$$ −9.39897 −0.328427
$$820$$ 0 0
$$821$$ −0.832947 −0.0290700 −0.0145350 0.999894i $$-0.504627\pi$$
−0.0145350 + 0.999894i $$0.504627\pi$$
$$822$$ −4.13879 −0.144357
$$823$$ 6.67034 0.232514 0.116257 0.993219i $$-0.462910\pi$$
0.116257 + 0.993219i $$0.462910\pi$$
$$824$$ −11.8493 −0.412791
$$825$$ 0 0
$$826$$ −16.9985 −0.591454
$$827$$ −1.34470 −0.0467599 −0.0233799 0.999727i $$-0.507443\pi$$
−0.0233799 + 0.999727i $$0.507443\pi$$
$$828$$ 4.28122 0.148783
$$829$$ −24.7770 −0.860541 −0.430270 0.902700i $$-0.641582\pi$$
−0.430270 + 0.902700i $$0.641582\pi$$
$$830$$ 0 0
$$831$$ −19.1611 −0.664690
$$832$$ −50.4115 −1.74770
$$833$$ −15.5600 −0.539121
$$834$$ −40.9631 −1.41844
$$835$$ 0 0
$$836$$ −43.7513 −1.51317
$$837$$ −6.56295 −0.226849
$$838$$ −20.7106 −0.715437
$$839$$ −6.72534 −0.232184 −0.116092 0.993238i $$-0.537037\pi$$
−0.116092 + 0.993238i $$0.537037\pi$$
$$840$$ 0 0
$$841$$ −9.20399 −0.317379
$$842$$ −8.99685 −0.310052
$$843$$ −4.36015 −0.150171
$$844$$ −62.9734 −2.16763
$$845$$ 0 0
$$846$$ −10.6375 −0.365723
$$847$$ 10.2024 0.350559
$$848$$ −21.3157 −0.731984
$$849$$ 24.9081 0.854844
$$850$$ 0 0
$$851$$ −13.3892 −0.458975
$$852$$ 5.87307 0.201208
$$853$$ −16.7244 −0.572632 −0.286316 0.958135i $$-0.592431\pi$$
−0.286316 + 0.958135i $$0.592431\pi$$
$$854$$ −22.8531 −0.782016
$$855$$ 0 0
$$856$$ −6.66391 −0.227768
$$857$$ 12.9930 0.443831 0.221915 0.975066i $$-0.428769\pi$$
0.221915 + 0.975066i $$0.428769\pi$$
$$858$$ 23.1970 0.791933
$$859$$ −36.5682 −1.24769 −0.623845 0.781548i $$-0.714430\pi$$
−0.623845 + 0.781548i $$0.714430\pi$$
$$860$$ 0 0
$$861$$ 24.3441 0.829646
$$862$$ 40.4859 1.37895
$$863$$ −46.3384 −1.57738 −0.788688 0.614794i $$-0.789239\pi$$
−0.788688 + 0.614794i $$0.789239\pi$$
$$864$$ 7.88596 0.268286
$$865$$ 0 0
$$866$$ 29.2370 0.993513
$$867$$ −28.9090 −0.981802
$$868$$ −36.3058 −1.23230
$$869$$ −19.5287 −0.662467
$$870$$ 0 0
$$871$$ −50.1906 −1.70064
$$872$$ 9.80472 0.332030
$$873$$ −15.0488 −0.509323
$$874$$ 24.4764 0.827926
$$875$$ 0 0
$$876$$ −2.84659 −0.0961773
$$877$$ −44.2548 −1.49438 −0.747190 0.664611i $$-0.768597\pi$$
−0.747190 + 0.664611i $$0.768597\pi$$
$$878$$ −71.2169 −2.40345
$$879$$ 13.5651 0.457541
$$880$$ 0 0
$$881$$ −42.2495 −1.42342 −0.711712 0.702472i $$-0.752080\pi$$
−0.711712 + 0.702472i $$0.752080\pi$$
$$882$$ −4.89892 −0.164955
$$883$$ 3.49289 0.117545 0.0587726 0.998271i $$-0.481281\pi$$
0.0587726 + 0.998271i $$0.481281\pi$$
$$884$$ 74.9000 2.51916
$$885$$ 0 0
$$886$$ 24.8076 0.833427
$$887$$ 14.7275 0.494501 0.247250 0.968952i $$-0.420473\pi$$
0.247250 + 0.968952i $$0.420473\pi$$
$$888$$ 9.37194 0.314502
$$889$$ −1.08896 −0.0365227
$$890$$ 0 0
$$891$$ −2.50913 −0.0840591
$$892$$ −8.44444 −0.282741
$$893$$ −34.0879 −1.14071
$$894$$ −20.2474 −0.677175
$$895$$ 0 0
$$896$$ 19.6108 0.655151
$$897$$ −7.27397 −0.242871
$$898$$ −28.4808 −0.950416
$$899$$ 29.2004 0.973886
$$900$$ 0 0
$$901$$ 55.6508 1.85400
$$902$$ −60.0823 −2.00052
$$903$$ 9.22780 0.307082
$$904$$ 17.4140 0.579180
$$905$$ 0 0
$$906$$ 26.8250 0.891200
$$907$$ 7.33785 0.243649 0.121825 0.992552i $$-0.461125\pi$$
0.121825 + 0.992552i $$0.461125\pi$$
$$908$$ −6.89169 −0.228709
$$909$$ −17.0211 −0.564553
$$910$$ 0 0
$$911$$ −53.5731 −1.77496 −0.887479 0.460849i $$-0.847545\pi$$
−0.887479 + 0.460849i $$0.847545\pi$$
$$912$$ 17.7411 0.587466
$$913$$ 24.7829 0.820195
$$914$$ −49.7136 −1.64438
$$915$$ 0 0
$$916$$ −23.8104 −0.786717
$$917$$ −13.1381 −0.433859
$$918$$ −14.4541 −0.477055
$$919$$ −43.2016 −1.42509 −0.712545 0.701627i $$-0.752458\pi$$
−0.712545 + 0.701627i $$0.752458\pi$$
$$920$$ 0 0
$$921$$ 11.9672 0.394334
$$922$$ 10.7359 0.353570
$$923$$ −9.97859 −0.328449
$$924$$ −13.8803 −0.456630
$$925$$ 0 0
$$926$$ −27.2541 −0.895625
$$927$$ −10.0859 −0.331264
$$928$$ −35.0868 −1.15178
$$929$$ −14.8420 −0.486951 −0.243476 0.969907i $$-0.578288\pi$$
−0.243476 + 0.969907i $$0.578288\pi$$
$$930$$ 0 0
$$931$$ −15.6987 −0.514503
$$932$$ 24.2955 0.795824
$$933$$ −7.66452 −0.250925
$$934$$ 10.3736 0.339435
$$935$$ 0 0
$$936$$ 5.09151 0.166421
$$937$$ −3.75611 −0.122707 −0.0613534 0.998116i $$-0.519542\pi$$
−0.0613534 + 0.998116i $$0.519542\pi$$
$$938$$ 53.5805 1.74947
$$939$$ −26.3840 −0.861008
$$940$$ 0 0
$$941$$ 59.9208 1.95336 0.976682 0.214692i $$-0.0688747\pi$$
0.976682 + 0.214692i $$0.0688747\pi$$
$$942$$ −40.1030 −1.30663
$$943$$ 18.8402 0.613521
$$944$$ −9.53532 −0.310348
$$945$$ 0 0
$$946$$ −22.7746 −0.740465
$$947$$ −24.4212 −0.793581 −0.396791 0.917909i $$-0.629876\pi$$
−0.396791 + 0.917909i $$0.629876\pi$$
$$948$$ −19.8525 −0.644779
$$949$$ 4.83647 0.156998
$$950$$ 0 0
$$951$$ 10.8880 0.353067
$$952$$ −17.2639 −0.559528
$$953$$ −36.6466 −1.18710 −0.593550 0.804797i $$-0.702274\pi$$
−0.593550 + 0.804797i $$0.702274\pi$$
$$954$$ 17.5212 0.567268
$$955$$ 0 0
$$956$$ 20.0142 0.647307
$$957$$ 11.1638 0.360875
$$958$$ 10.2225 0.330275
$$959$$ −4.20770 −0.135874
$$960$$ 0 0
$$961$$ 12.0723 0.389431
$$962$$ −73.7495 −2.37778
$$963$$ −5.67218 −0.182784
$$964$$ 3.64201 0.117301
$$965$$ 0 0
$$966$$ 7.76526 0.249843
$$967$$ −47.4395 −1.52555 −0.762776 0.646663i $$-0.776164\pi$$
−0.762776 + 0.646663i $$0.776164\pi$$
$$968$$ −5.52674 −0.177636
$$969$$ −46.3183 −1.48796
$$970$$ 0 0
$$971$$ −5.68782 −0.182531 −0.0912655 0.995827i $$-0.529091\pi$$
−0.0912655 + 0.995827i $$0.529091\pi$$
$$972$$ −2.55073 −0.0818147
$$973$$ −41.6452 −1.33508
$$974$$ 19.5608 0.626767
$$975$$ 0 0
$$976$$ −12.8194 −0.410340
$$977$$ 2.29901 0.0735518 0.0367759 0.999324i $$-0.488291\pi$$
0.0367759 + 0.999324i $$0.488291\pi$$
$$978$$ −15.5716 −0.497925
$$979$$ 3.11194 0.0994581
$$980$$ 0 0
$$981$$ 8.34557 0.266454
$$982$$ 68.0620 2.17194
$$983$$ 2.35879 0.0752336 0.0376168 0.999292i $$-0.488023\pi$$
0.0376168 + 0.999292i $$0.488023\pi$$
$$984$$ −13.1875 −0.420401
$$985$$ 0 0
$$986$$ 64.3100 2.04805
$$987$$ −10.8146 −0.344232
$$988$$ 75.5676 2.40413
$$989$$ 7.14150 0.227087
$$990$$ 0 0
$$991$$ −7.76677 −0.246720 −0.123360 0.992362i $$-0.539367\pi$$
−0.123360 + 0.992362i $$0.539367\pi$$
$$992$$ −51.7552 −1.64323
$$993$$ 32.2137 1.02227
$$994$$ 10.6526 0.337879
$$995$$ 0 0
$$996$$ 25.1938 0.798295
$$997$$ −8.35947 −0.264747 −0.132374 0.991200i $$-0.542260\pi$$
−0.132374 + 0.991200i $$0.542260\pi$$
$$998$$ 19.3549 0.612668
$$999$$ 7.97720 0.252388
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 1875.2.a.l.1.5 yes 6
3.2 odd 2 5625.2.a.o.1.2 6
5.2 odd 4 1875.2.b.e.1249.10 12
5.3 odd 4 1875.2.b.e.1249.3 12
5.4 even 2 1875.2.a.i.1.2 6
15.14 odd 2 5625.2.a.r.1.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.i.1.2 6 5.4 even 2
1875.2.a.l.1.5 yes 6 1.1 even 1 trivial
1875.2.b.e.1249.3 12 5.3 odd 4
1875.2.b.e.1249.10 12 5.2 odd 4
5625.2.a.o.1.2 6 3.2 odd 2
5625.2.a.r.1.5 6 15.14 odd 2