LMFDB - Embedded newform 225.4.b.i.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$13.2754297513$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.58114 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.4.b.i.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16228i q^{2} -2.00000 q^{4} +15.0000i q^{7} -18.9737i q^{8} +O(q^{10})$	$q-3.16228i q^{2} -2.00000 q^{4} +15.0000i q^{7} -18.9737i q^{8} -63.2456 q^{11} -35.0000i q^{13} +47.4342 q^{14} -76.0000 q^{16} -88.5438i q^{17} -91.0000 q^{19} +200.000i q^{22} -113.842i q^{23} -110.680 q^{26} -30.0000i q^{28} -63.2456 q^{29} -147.000 q^{31} +88.5438i q^{32} -280.000 q^{34} +370.000i q^{37} +287.767i q^{38} +442.719 q^{41} -335.000i q^{43} +126.491 q^{44} -360.000 q^{46} +177.088i q^{47} +118.000 q^{49} +70.0000i q^{52} +88.5438i q^{53} +284.605 q^{56} +200.000i q^{58} -885.438 q^{59} +427.000 q^{61} +464.855i q^{62} -328.000 q^{64} +15.0000i q^{67} +177.088i q^{68} +63.2456 q^{71} +70.0000i q^{73} +1170.04 q^{74} +182.000 q^{76} -948.683i q^{77} +876.000 q^{79} -1400.00i q^{82} -531.263i q^{83} -1059.36 q^{86} +1200.00i q^{88} +525.000 q^{91} +227.684i q^{92} +560.000 q^{94} -1085.00i q^{97} -373.149i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} - 304 q^{16} - 364 q^{19} - 588 q^{31} - 1120 q^{34} - 1440 q^{46} + 472 q^{49} + 1708 q^{61} - 1312 q^{64} + 728 q^{76} + 3504 q^{79} + 2100 q^{91} + 2240 q^{94}+O(q^{100}) $ 4 * q - 8 * q^4 - 304 * q^16 - 364 * q^19 - 588 * q^31 - 1120 * q^34 - 1440 * q^46 + 472 * q^49 + 1708 * q^61 - 1312 * q^64 + 728 * q^76 + 3504 * q^79 + 2100 * q^91 + 2240 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 3.16228i	− 1.11803i	−0.829156	−	0.559017i	$-0.811179\pi$
$2$	− 3.16228i	− 1.11803i	0.829156	−	0.559017i	$-0.188821\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.250000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	15.0000i	0.809924i	0.914334	+	0.404962i	$0.132715\pi$
$7$	15.0000i	0.809924i	−0.914334	+	0.404962i	$0.867285\pi$
$8$	− 18.9737i	− 0.838525i
$9$	0	0
$10$	0	0
$11$	−63.2456	−1.73357	−0.866784	−	0.498683i	$-0.833817\pi$
$11$	−63.2456	−1.73357	−0.866784	+	0.498683i	$0.833817\pi$
$12$	0	0
$13$	− 35.0000i	− 0.746712i	−0.927688	−	0.373356i	$-0.878207\pi$
$13$	− 35.0000i	− 0.746712i	0.927688	−	0.373356i	$-0.121793\pi$
$14$	47.4342	0.905522
$15$	0	0
$16$	−76.0000	−1.18750
$17$	− 88.5438i	− 1.26324i	−0.775280	−	0.631618i	$-0.782391\pi$
$17$	− 88.5438i	− 1.26324i	0.775280	−	0.631618i	$-0.217609\pi$
$18$	0	0
$19$	−91.0000	−1.09878	−0.549390	−	0.835566i	$-0.685140\pi$
$19$	−91.0000	−1.09878	−0.549390	+	0.835566i	$0.685140\pi$
$20$	0	0
$21$	0	0
$22$	200.000i	1.93819i
$23$	− 113.842i	− 1.03207i	−0.856566	−	0.516037i	$-0.827407\pi$
$23$	− 113.842i	− 1.03207i	0.856566	−	0.516037i	$-0.172593\pi$
$24$	0	0
$25$	0	0
$26$	−110.680	−0.834849
$27$	0	0
$28$	− 30.0000i	− 0.202481i
$29$	−63.2456	−0.404979	−0.202490	−	0.979284i	$-0.564903\pi$
$29$	−63.2456	−0.404979	−0.202490	+	0.979284i	$0.564903\pi$
$30$	0	0
$31$	−147.000	−0.851677	−0.425838	−	0.904799i	$-0.640021\pi$
$31$	−147.000	−0.851677	−0.425838	+	0.904799i	$0.640021\pi$
$32$	88.5438i	0.489140i
$33$	0	0
$34$	−280.000	−1.41234
$35$	0	0
$36$	0	0
$37$	370.000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	370.000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	287.767i	1.22847i
$39$	0	0
$40$	0	0
$41$	442.719	1.68637	0.843184	−	0.537625i	$-0.180679\pi$
$41$	442.719	1.68637	0.843184	+	0.537625i	$0.180679\pi$
$42$	0	0
$43$	− 335.000i	− 1.18807i	−0.804439	−	0.594035i	$-0.797534\pi$
$43$	− 335.000i	− 1.18807i	0.804439	−	0.594035i	$-0.202466\pi$
$44$	126.491	0.433392
$45$	0	0
$46$	−360.000	−1.15389
$47$	177.088i	0.549593i	0.961502	+	0.274797i	$0.0886105\pi$
$47$	177.088i	0.549593i	−0.961502	+	0.274797i	$0.911390\pi$
$48$	0	0
$49$	118.000	0.344023
$50$	0	0
$51$	0	0
$52$	70.0000i	0.186678i
$53$	88.5438i	0.229480i	0.993396	+	0.114740i	$0.0366034\pi$
$53$	88.5438i	0.229480i	−0.993396	+	0.114740i	$0.963397\pi$
$54$	0	0
$55$	0	0
$56$	284.605	0.679142
$57$	0	0
$58$	200.000i	0.452781i
$59$	−885.438	−1.95380	−0.976900	−	0.213698i	$-0.931449\pi$
$59$	−885.438	−1.95380	−0.976900	+	0.213698i	$0.931449\pi$
$60$	0	0
$61$	427.000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	427.000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	464.855i	0.952204i
$63$	0	0
$64$	−328.000	−0.640625
$65$	0	0
$66$	0	0
$67$	15.0000i	0.0273514i	0.999906	+	0.0136757i	$0.00435324\pi$
$67$	15.0000i	0.0273514i	−0.999906	+	0.0136757i	$0.995647\pi$
$68$	177.088i	0.315809i
$69$	0	0
$70$	0	0
$71$	63.2456	0.105716	0.0528582	−	0.998602i	$-0.483167\pi$
$71$	63.2456	0.105716	0.0528582	+	0.998602i	$0.483167\pi$
$72$	0	0
$73$	70.0000i	0.112231i	0.998424	+	0.0561156i	$0.0178715\pi$
$73$	70.0000i	0.112231i	−0.998424	+	0.0561156i	$0.982128\pi$
$74$	1170.04	1.83804
$75$	0	0
$76$	182.000	0.274695
$77$	− 948.683i	− 1.40406i
$78$	0	0
$79$	876.000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$79$	876.000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$80$	0	0
$81$	0	0
$82$	− 1400.00i	− 1.88542i
$83$	− 531.263i	− 0.702574i	−0.936268	−	0.351287i	$-0.885744\pi$
$83$	− 531.263i	− 0.702574i	0.936268	−	0.351287i	$-0.114256\pi$
$84$	0	0
$85$	0	0
$86$	−1059.36	−1.32830
$87$	0	0
$88$	1200.00i	1.45364i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	525.000	0.604780
$92$	227.684i	0.258018i
$93$	0	0
$94$	560.000	0.614464
$95$	0	0
$96$	0	0
$97$	− 1085.00i	− 1.13572i	−0.823124	−	0.567861i	$-0.807771\pi$
$97$	− 1085.00i	− 1.13572i	0.823124	−	0.567861i	$-0.192229\pi$
$98$	− 373.149i	− 0.384630i
$99$	0	0
$100$	0	0
$101$	−1328.16	−1.30848	−0.654240	−	0.756287i	$-0.727011\pi$
$101$	−1328.16	−1.30848	−0.654240	+	0.756287i	$0.727011\pi$
$102$	0	0
$103$	− 1540.00i	− 1.47321i	−0.676323	−	0.736605i	$-0.736428\pi$
$103$	− 1540.00i	− 1.47321i	0.676323	−	0.736605i	$-0.263572\pi$
$104$	−664.078	−0.626137
$105$	0	0
$106$	280.000	0.256566
$107$	− 796.894i	− 0.719987i	−0.932955	−	0.359994i	$-0.882779\pi$
$107$	− 796.894i	− 0.719987i	0.932955	−	0.359994i	$-0.117221\pi$
$108$	0	0
$109$	811.000	0.712658	0.356329	−	0.934361i	$-0.384028\pi$
$109$	811.000	0.712658	0.356329	+	0.934361i	$0.384028\pi$
$110$	0	0
$111$	0	0
$112$	− 1140.00i	− 0.961785i
$113$	− 1619.09i	− 1.34788i	−0.738785	−	0.673942i	$-0.764600\pi$
$113$	− 1619.09i	− 1.34788i	0.738785	−	0.673942i	$-0.235400\pi$
$114$	0	0
$115$	0	0
$116$	126.491	0.101245
$117$	0	0
$118$	2800.00i	2.18441i
$119$	1328.16	1.02313
$120$	0	0
$121$	2669.00	2.00526
$122$	− 1350.29i	− 1.00205i
$123$	0	0
$124$	294.000	0.212919
$125$	0	0
$126$	0	0
$127$	260.000i	0.181664i	0.995866	+	0.0908318i	$0.0289526\pi$
$127$	260.000i	0.181664i	−0.995866	+	0.0908318i	$0.971047\pi$
$128$	1745.58i	1.20538i
$129$	0	0
$130$	0	0
$131$	1328.16	0.885814	0.442907	−	0.896568i	$-0.353947\pi$
$131$	1328.16	0.885814	0.442907	+	0.896568i	$0.353947\pi$
$132$	0	0
$133$	− 1365.00i	− 0.889929i
$134$	47.4342	0.0305798
$135$	0	0
$136$	−1680.00	−1.05926
$137$	2656.31i	1.65653i	0.560339	+	0.828263i	$0.310671\pi$
$137$	2656.31i	1.65653i	−0.560339	+	0.828263i	$0.689329\pi$
$138$	0	0
$139$	784.000	0.478403	0.239201	−	0.970970i	$-0.423114\pi$
$139$	784.000	0.478403	0.239201	+	0.970970i	$0.423114\pi$
$140$	0	0
$141$	0	0
$142$	− 200.000i	− 0.118195i
$143$	2213.59i	1.29448i
$144$	0	0
$145$	0	0
$146$	221.359	0.125478
$147$	0	0
$148$	− 740.000i	− 0.410997i
$149$	−2150.35	−1.18230	−0.591152	−	0.806560i	$-0.701327\pi$
$149$	−2150.35	−1.18230	−0.591152	+	0.806560i	$0.701327\pi$
$150$	0	0
$151$	−797.000	−0.429529	−0.214765	−	0.976666i	$-0.568898\pi$
$151$	−797.000	−0.429529	−0.214765	+	0.976666i	$0.568898\pi$
$152$	1726.60i	0.921356i
$153$	0	0
$154$	−3000.00	−1.56978
$155$	0	0
$156$	0	0
$157$	1225.00i	0.622711i	0.950293	+	0.311356i	$0.100783\pi$
$157$	1225.00i	0.622711i	−0.950293	+	0.311356i	$0.899217\pi$
$158$	− 2770.16i	− 1.39482i
$159$	0	0
$160$	0	0
$161$	1707.63	0.835901
$162$	0	0
$163$	275.000i	0.132145i	0.997815	+	0.0660726i	$0.0210469\pi$
$163$	275.000i	0.132145i	−0.997815	+	0.0660726i	$0.978953\pi$
$164$	−885.438	−0.421592
$165$	0	0
$166$	−1680.00	−0.785502
$167$	− 265.631i	− 0.123085i	−0.998104	−	0.0615424i	$-0.980398\pi$
$167$	− 265.631i	− 0.123085i	0.998104	−	0.0615424i	$-0.0196019\pi$
$168$	0	0
$169$	972.000	0.442421
$170$	0	0
$171$	0	0
$172$	670.000i	0.297018i
$173$	3984.47i	1.75106i	0.483162	+	0.875531i	$0.339488\pi$
$173$	3984.47i	1.75106i	−0.483162	+	0.875531i	$0.660512\pi$
$174$	0	0
$175$	0	0
$176$	4806.66	2.05861
$177$	0	0
$178$	0	0
$179$	1770.88	0.739449	0.369725	−	0.929141i	$-0.379452\pi$
$179$	1770.88	0.739449	0.369725	+	0.929141i	$0.379452\pi$
$180$	0	0
$181$	−1687.00	−0.692783	−0.346391	−	0.938090i	$-0.612593\pi$
$181$	−1687.00	−0.692783	−0.346391	+	0.938090i	$0.612593\pi$
$182$	− 1660.20i	− 0.676164i
$183$	0	0
$184$	−2160.00	−0.865420
$185$	0	0
$186$	0	0
$187$	5600.00i	2.18991i
$188$	− 354.175i	− 0.137398i
$189$	0	0
$190$	0	0
$191$	−2213.59	−0.838587	−0.419293	−	0.907851i	$-0.637722\pi$
$191$	−2213.59	−0.838587	−0.419293	+	0.907851i	$0.637722\pi$
$192$	0	0
$193$	− 3625.00i	− 1.35199i	−0.736908	−	0.675993i	$-0.763715\pi$
$193$	− 3625.00i	− 1.35199i	0.736908	−	0.675993i	$-0.236285\pi$
$194$	−3431.07	−1.26978
$195$	0	0
$196$	−236.000	−0.0860058
$197$	1745.58i	0.631306i	0.948875	+	0.315653i	$0.102224\pi$
$197$	1745.58i	0.631306i	−0.948875	+	0.315653i	$0.897776\pi$
$198$	0	0
$199$	−3199.00	−1.13955	−0.569777	−	0.821800i	$-0.692970\pi$
$199$	−3199.00	−1.13955	−0.569777	+	0.821800i	$0.692970\pi$
$200$	0	0
$201$	0	0
$202$	4200.00i	1.46293i
$203$	− 948.683i	− 0.328003i
$204$	0	0
$205$	0	0
$206$	−4869.91	−1.64710
$207$	0	0
$208$	2660.00i	0.886720i
$209$	5755.35	1.90481
$210$	0	0
$211$	887.000	0.289401	0.144700	−	0.989476i	$-0.453778\pi$
$211$	887.000	0.289401	0.144700	+	0.989476i	$0.453778\pi$
$212$	− 177.088i	− 0.0573699i
$213$	0	0
$214$	−2520.00	−0.804970
$215$	0	0
$216$	0	0
$217$	− 2205.00i	− 0.689793i
$218$	− 2564.61i	− 0.796776i
$219$	0	0
$220$	0	0
$221$	−3099.03	−0.943274
$222$	0	0
$223$	− 3535.00i	− 1.06153i	−0.847519	−	0.530765i	$-0.821905\pi$
$223$	− 3535.00i	− 1.06153i	0.847519	−	0.530765i	$-0.178095\pi$
$224$	−1328.16	−0.396166
$225$	0	0
$226$	−5120.00	−1.50698
$227$	88.5438i	0.0258892i	0.999916	+	0.0129446i	$0.00412052\pi$
$227$	88.5438i	0.0258892i	−0.999916	+	0.0129446i	$0.995879\pi$
$228$	0	0
$229$	5019.00	1.44832	0.724159	−	0.689633i	$-0.242228\pi$
$229$	5019.00	1.44832	0.724159	+	0.689633i	$0.242228\pi$
$230$	0	0
$231$	0	0
$232$	1200.00i	0.339586i
$233$	− 3605.00i	− 1.01361i	−0.862061	−	0.506805i	$-0.830826\pi$
$233$	− 3605.00i	− 1.01361i	0.862061	−	0.506805i	$-0.169174\pi$
$234$	0	0
$235$	0	0
$236$	1770.88	0.488450
$237$	0	0
$238$	− 4200.00i	− 1.14389i
$239$	442.719	0.119821	0.0599103	−	0.998204i	$-0.480919\pi$
$239$	442.719	0.119821	0.0599103	+	0.998204i	$0.480919\pi$
$240$	0	0
$241$	−623.000	−0.166518	−0.0832592	−	0.996528i	$-0.526533\pi$
$241$	−623.000	−0.166518	−0.0832592	+	0.996528i	$0.526533\pi$
$242$	− 8440.12i	− 2.24195i
$243$	0	0
$244$	−854.000	−0.224065
$245$	0	0
$246$	0	0
$247$	3185.00i	0.820472i
$248$	2789.13i	0.714153i
$249$	0	0
$250$	0	0
$251$	−2656.31	−0.667988	−0.333994	−	0.942575i	$-0.608397\pi$
$251$	−2656.31	−0.667988	−0.333994	+	0.942575i	$0.608397\pi$
$252$	0	0
$253$	7200.00i	1.78917i
$254$	822.192	0.203106
$255$	0	0
$256$	2896.00	0.707031
$257$	− 3718.84i	− 0.902626i	−0.892366	−	0.451313i	$-0.850956\pi$
$257$	− 3718.84i	− 0.902626i	0.892366	−	0.451313i	$-0.149044\pi$
$258$	0	0
$259$	−5550.00	−1.33151
$260$	0	0
$261$	0	0
$262$	− 4200.00i	− 0.990370i
$263$	− 2896.65i	− 0.679144i	−0.940580	−	0.339572i	$-0.889718\pi$
$263$	− 2896.65i	− 0.679144i	0.940580	−	0.339572i	$-0.110282\pi$
$264$	0	0
$265$	0	0
$266$	−4316.51	−0.994970
$267$	0	0
$268$	− 30.0000i	− 0.00683784i
$269$	442.719	0.100346	0.0501729	−	0.998741i	$-0.484023\pi$
$269$	442.719	0.100346	0.0501729	+	0.998741i	$0.484023\pi$
$270$	0	0
$271$	−308.000	−0.0690394	−0.0345197	−	0.999404i	$-0.510990\pi$
$271$	−308.000	−0.0690394	−0.0345197	+	0.999404i	$0.510990\pi$
$272$	6729.33i	1.50009i
$273$	0	0
$274$	8400.00	1.85205
$275$	0	0
$276$	0	0
$277$	− 7485.00i	− 1.62357i	−0.583953	−	0.811787i	$-0.698495\pi$
$277$	− 7485.00i	− 1.62357i	0.583953	−	0.811787i	$-0.301505\pi$
$278$	− 2479.23i	− 0.534871i
$279$	0	0
$280$	0	0
$281$	948.683	0.201401	0.100701	−	0.994917i	$-0.467892\pi$
$281$	948.683	0.201401	0.100701	+	0.994917i	$0.467892\pi$
$282$	0	0
$283$	− 525.000i	− 0.110276i	−0.998479	−	0.0551378i	$-0.982440\pi$
$283$	− 525.000i	− 0.110276i	0.998479	−	0.0551378i	$-0.0175598\pi$
$284$	−126.491	−0.0264291
$285$	0	0
$286$	7000.00	1.44727
$287$	6640.78i	1.36583i
$288$	0	0
$289$	−2927.00	−0.595766
$290$	0	0
$291$	0	0
$292$	− 140.000i	− 0.0280578i
$293$	6375.15i	1.27113i	0.772048	+	0.635564i	$0.219232\pi$
$293$	6375.15i	1.27113i	−0.772048	+	0.635564i	$0.780768\pi$
$294$	0	0
$295$	0	0
$296$	7020.26	1.37853
$297$	0	0
$298$	6800.00i	1.32186i
$299$	−3984.47	−0.770662
$300$	0	0
$301$	5025.00	0.962246
$302$	2520.34i	0.480228i
$303$	0	0
$304$	6916.00	1.30480
$305$	0	0
$306$	0	0
$307$	− 2695.00i	− 0.501016i	−0.968115	−	0.250508i	$-0.919402\pi$
$307$	− 2695.00i	− 0.501016i	0.968115	−	0.250508i	$-0.0805976\pi$
$308$	1897.37i	0.351015i
$309$	0	0
$310$	0	0
$311$	5312.63	0.968654	0.484327	−	0.874887i	$-0.339064\pi$
$311$	5312.63	0.968654	0.484327	+	0.874887i	$0.339064\pi$
$312$	0	0
$313$	2555.00i	0.461397i	0.973025	+	0.230698i	$0.0741010\pi$
$313$	2555.00i	0.461397i	−0.973025	+	0.230698i	$0.925899\pi$
$314$	3873.79	0.696212
$315$	0	0
$316$	−1752.00	−0.311891
$317$	− 7462.98i	− 1.32228i	−0.750263	−	0.661140i	$-0.770073\pi$
$317$	− 7462.98i	− 1.32228i	0.750263	−	0.661140i	$-0.229927\pi$
$318$	0	0
$319$	4000.00	0.702060
$320$	0	0
$321$	0	0
$322$	− 5400.00i	− 0.934566i
$323$	8057.48i	1.38802i
$324$	0	0
$325$	0	0
$326$	869.626	0.147743
$327$	0	0
$328$	− 8400.00i	− 1.41406i
$329$	−2656.31	−0.445129
$330$	0	0
$331$	−5088.00	−0.844900	−0.422450	−	0.906386i	$-0.638830\pi$
$331$	−5088.00	−0.844900	−0.422450	+	0.906386i	$0.638830\pi$
$332$	1062.53i	0.175644i
$333$	0	0
$334$	−840.000	−0.137613
$335$	0	0
$336$	0	0
$337$	− 11485.0i	− 1.85646i	−0.372004	−	0.928231i	$-0.621329\pi$
$337$	− 11485.0i	− 1.85646i	0.372004	−	0.928231i	$-0.378671\pi$
$338$	− 3073.73i	− 0.494642i
$339$	0	0
$340$	0	0
$341$	9297.10	1.47644
$342$	0	0
$343$	6915.00i	1.08856i
$344$	−6356.18	−0.996227
$345$	0	0
$346$	12600.0	1.95775
$347$	6400.45i	0.990185i	0.868840	+	0.495092i	$0.164866\pi$
$347$	6400.45i	0.990185i	−0.868840	+	0.495092i	$0.835134\pi$
$348$	0	0
$349$	9674.00	1.48377	0.741887	−	0.670525i	$-0.233931\pi$
$349$	9674.00	1.48377	0.741887	+	0.670525i	$0.233931\pi$
$350$	0	0
$351$	0	0
$352$	− 5600.00i	− 0.847957i
$353$	− 5578.26i	− 0.841078i	−0.907274	−	0.420539i	$-0.861841\pi$
$353$	− 5578.26i	− 0.841078i	0.907274	−	0.420539i	$-0.138159\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	− 5600.00i	− 0.826730i
$359$	−8917.62	−1.31101	−0.655507	−	0.755189i	$-0.727545\pi$
$359$	−8917.62	−1.31101	−0.655507	+	0.755189i	$0.727545\pi$
$360$	0	0
$361$	1422.00	0.207319
$362$	5334.76i	0.774555i
$363$	0	0
$364$	−1050.00	−0.151195
$365$	0	0
$366$	0	0
$367$	8505.00i	1.20969i	0.796342	+	0.604847i	$0.206766\pi$
$367$	8505.00i	1.20969i	−0.796342	+	0.604847i	$0.793234\pi$
$368$	8651.99i	1.22559i
$369$	0	0
$370$	0	0
$371$	−1328.16	−0.185861
$372$	0	0
$373$	6775.00i	0.940472i	0.882541	+	0.470236i	$0.155831\pi$
$373$	6775.00i	0.940472i	−0.882541	+	0.470236i	$0.844169\pi$
$374$	17708.8	2.44839
$375$	0	0
$376$	3360.00	0.460848
$377$	2213.59i	0.302403i
$378$	0	0
$379$	−9241.00	−1.25245	−0.626225	−	0.779643i	$-0.715401\pi$
$379$	−9241.00	−1.25245	−0.626225	+	0.779643i	$0.715401\pi$
$380$	0	0
$381$	0	0
$382$	7000.00i	0.937568i
$383$	− 9031.46i	− 1.20493i	−0.798147	−	0.602463i	$-0.794186\pi$
$383$	− 9031.46i	− 1.20493i	0.798147	−	0.602463i	$-0.205814\pi$
$384$	0	0
$385$	0	0
$386$	−11463.3	−1.51157
$387$	0	0
$388$	2170.00i	0.283931i
$389$	−4363.94	−0.568794	−0.284397	−	0.958707i	$-0.591793\pi$
$389$	−4363.94	−0.568794	−0.284397	+	0.958707i	$0.591793\pi$
$390$	0	0
$391$	−10080.0	−1.30375
$392$	− 2238.89i	− 0.288472i
$393$	0	0
$394$	5520.00	0.705821
$395$	0	0
$396$	0	0
$397$	− 4795.00i	− 0.606182i	−0.952962	−	0.303091i	$-0.901981\pi$
$397$	− 4795.00i	− 0.606182i	0.952962	−	0.303091i	$-0.0980186\pi$
$398$	10116.1i	1.27406i
$399$	0	0
$400$	0	0
$401$	−7020.26	−0.874252	−0.437126	−	0.899400i	$-0.644004\pi$
$401$	−7020.26	−0.874252	−0.437126	+	0.899400i	$0.644004\pi$
$402$	0	0
$403$	5145.00i	0.635957i
$404$	2656.31	0.327120
$405$	0	0
$406$	−3000.00	−0.366718
$407$	− 23400.9i	− 2.84997i
$408$	0	0
$409$	−6881.00	−0.831891	−0.415946	−	0.909389i	$-0.636549\pi$
$409$	−6881.00	−0.831891	−0.415946	+	0.909389i	$0.636549\pi$
$410$	0	0
$411$	0	0
$412$	3080.00i	0.368303i
$413$	− 13281.6i	− 1.58243i
$414$	0	0
$415$	0	0
$416$	3099.03	0.365247
$417$	0	0
$418$	− 18200.0i	− 2.12964i
$419$	12838.8	1.49694	0.748471	−	0.663167i	$-0.230788\pi$
$419$	12838.8	1.49694	0.748471	+	0.663167i	$0.230788\pi$
$420$	0	0
$421$	6418.00	0.742979	0.371490	−	0.928437i	$-0.378847\pi$
$421$	6418.00	0.742979	0.371490	+	0.928437i	$0.378847\pi$
$422$	− 2804.94i	− 0.323560i
$423$	0	0
$424$	1680.00	0.192425
$425$	0	0
$426$	0	0
$427$	6405.00i	0.725901i
$428$	1593.79i	0.179997i
$429$	0	0
$430$	0	0
$431$	−1328.16	−0.148434	−0.0742170	−	0.997242i	$-0.523646\pi$
$431$	−1328.16	−0.148434	−0.0742170	+	0.997242i	$0.523646\pi$
$432$	0	0
$433$	8155.00i	0.905091i	0.891741	+	0.452545i	$0.149484\pi$
$433$	8155.00i	0.905091i	−0.891741	+	0.452545i	$0.850516\pi$
$434$	−6972.82	−0.771212
$435$	0	0
$436$	−1622.00	−0.178164
$437$	10359.6i	1.13402i
$438$	0	0
$439$	−14749.0	−1.60349	−0.801744	−	0.597667i	$-0.796094\pi$
$439$	−14749.0	−1.60349	−0.801744	+	0.597667i	$0.796094\pi$
$440$	0	0
$441$	0	0
$442$	9800.00i	1.05461i
$443$	− 11156.5i	− 1.19653i	−0.801299	−	0.598264i	$-0.795857\pi$
$443$	− 11156.5i	− 1.19653i	0.801299	−	0.598264i	$-0.204143\pi$
$444$	0	0
$445$	0	0
$446$	−11178.7	−1.18683
$447$	0	0
$448$	− 4920.00i	− 0.518857i
$449$	−9233.85	−0.970540	−0.485270	−	0.874364i	$-0.661279\pi$
$449$	−9233.85	−0.970540	−0.485270	+	0.874364i	$0.661279\pi$
$450$	0	0
$451$	−28000.0	−2.92343
$452$	3238.17i	0.336971i
$453$	0	0
$454$	280.000	0.0289450
$455$	0	0
$456$	0	0
$457$	− 1030.00i	− 0.105430i	−0.998610	−	0.0527148i	$-0.983213\pi$
$457$	− 1030.00i	− 0.105430i	0.998610	−	0.0527148i	$-0.0167874\pi$
$458$	− 15871.5i	− 1.61927i
$459$	0	0
$460$	0	0
$461$	−9297.10	−0.939282	−0.469641	−	0.882858i	$-0.655617\pi$
$461$	−9297.10	−0.939282	−0.469641	+	0.882858i	$0.655617\pi$
$462$	0	0
$463$	− 11940.0i	− 1.19849i	−0.800567	−	0.599243i	$-0.795468\pi$
$463$	− 11940.0i	− 1.19849i	0.800567	−	0.599243i	$-0.204532\pi$
$464$	4806.66	0.480913
$465$	0	0
$466$	−11400.0	−1.13325
$467$	− 12130.5i	− 1.20200i	−0.799250	−	0.600998i	$-0.794770\pi$
$467$	− 12130.5i	− 1.20200i	0.799250	−	0.600998i	$-0.205230\pi$
$468$	0	0
$469$	−225.000	−0.0221525
$470$	0	0
$471$	0	0
$472$	16800.0i	1.63831i
$473$	21187.3i	2.05960i
$474$	0	0
$475$	0	0
$476$	−2656.31	−0.255781
$477$	0	0
$478$	− 1400.00i	− 0.133963i
$479$	−2656.31	−0.253382	−0.126691	−	0.991942i	$-0.540436\pi$
$479$	−2656.31	−0.253382	−0.126691	+	0.991942i	$0.540436\pi$
$480$	0	0
$481$	12950.0	1.22759
$482$	1970.10i	0.186173i
$483$	0	0
$484$	−5338.00	−0.501315
$485$	0	0
$486$	0	0
$487$	14125.0i	1.31430i	0.753759	+	0.657151i	$0.228239\pi$
$487$	14125.0i	1.31430i	−0.753759	+	0.657151i	$0.771761\pi$
$488$	− 8101.76i	− 0.751535i
$489$	0	0
$490$	0	0
$491$	4047.72	0.372038	0.186019	−	0.982546i	$-0.440441\pi$
$491$	4047.72	0.372038	0.186019	+	0.982546i	$0.440441\pi$
$492$	0	0
$493$	5600.00i	0.511585i
$494$	10071.9	0.917316
$495$	0	0
$496$	11172.0	1.01137
$497$	948.683i	0.0856223i
$498$	0	0
$499$	−1279.00	−0.114741	−0.0573706	−	0.998353i	$-0.518272\pi$
$499$	−1279.00	−0.114741	−0.0573706	+	0.998353i	$0.518272\pi$
$500$	0	0
$501$	0	0
$502$	8400.00i	0.746833i
$503$	16380.6i	1.45204i	0.687675	+	0.726019i	$0.258631\pi$
$503$	16380.6i	1.45204i	−0.687675	+	0.726019i	$0.741369\pi$
$504$	0	0
$505$	0	0
$506$	22768.4	2.00035
$507$	0	0
$508$	− 520.000i	− 0.0454159i
$509$	4427.19	0.385524	0.192762	−	0.981246i	$-0.438255\pi$
$509$	4427.19	0.385524	0.192762	+	0.981246i	$0.438255\pi$
$510$	0	0
$511$	−1050.00	−0.0908988
$512$	4806.66i	0.414895i
$513$	0	0
$514$	−11760.0	−1.00917
$515$	0	0
$516$	0	0
$517$	− 11200.0i	− 0.952757i
$518$	17550.6i	1.48867i
$519$	0	0
$520$	0	0
$521$	11068.0	0.930704	0.465352	−	0.885126i	$-0.345928\pi$
$521$	11068.0	0.930704	0.465352	+	0.885126i	$0.345928\pi$
$522$	0	0
$523$	− 10745.0i	− 0.898367i	−0.893439	−	0.449184i	$-0.851715\pi$
$523$	− 10745.0i	− 0.898367i	0.893439	−	0.449184i	$-0.148285\pi$
$524$	−2656.31	−0.221453
$525$	0	0
$526$	−9160.00	−0.759306
$527$	13015.9i	1.07587i
$528$	0	0
$529$	−793.000	−0.0651763
$530$	0	0
$531$	0	0
$532$	2730.00i	0.222482i
$533$	− 15495.2i	− 1.25923i
$534$	0	0
$535$	0	0
$536$	284.605	0.0229348
$537$	0	0
$538$	− 1400.00i	− 0.112190i
$539$	−7462.98	−0.596388
$540$	0	0
$541$	−9423.00	−0.748847	−0.374424	−	0.927258i	$-0.622159\pi$
$541$	−9423.00	−0.748847	−0.374424	+	0.927258i	$0.622159\pi$
$542$	973.982i	0.0771884i
$543$	0	0
$544$	7840.00	0.617899
$545$	0	0
$546$	0	0
$547$	9080.00i	0.709749i	0.934914	+	0.354875i	$0.115476\pi$
$547$	9080.00i	0.709749i	−0.934914	+	0.354875i	$0.884524\pi$
$548$	− 5312.63i	− 0.414132i
$549$	0	0
$550$	0	0
$551$	5755.35	0.444984
$552$	0	0
$553$	13140.0i	1.01043i
$554$	−23669.6	−1.81521
$555$	0	0
$556$	−1568.00	−0.119601
$557$	8348.41i	0.635069i	0.948247	+	0.317535i	$0.102855\pi$
$557$	8348.41i	0.635069i	−0.948247	+	0.317535i	$0.897145\pi$
$558$	0	0
$559$	−11725.0	−0.887146
$560$	0	0
$561$	0	0
$562$	− 3000.00i	− 0.225173i
$563$	7703.31i	0.576653i	0.957532	+	0.288327i	$0.0930989\pi$
$563$	7703.31i	0.576653i	−0.957532	+	0.288327i	$0.906901\pi$
$564$	0	0
$565$	0	0
$566$	−1660.20	−0.123292
$567$	0	0
$568$	− 1200.00i	− 0.0886459i
$569$	−6261.31	−0.461314	−0.230657	−	0.973035i	$-0.574088\pi$
$569$	−6261.31	−0.461314	−0.230657	+	0.973035i	$0.574088\pi$
$570$	0	0
$571$	587.000	0.0430213	0.0215107	−	0.999769i	$-0.493152\pi$
$571$	587.000	0.0430213	0.0215107	+	0.999769i	$0.493152\pi$
$572$	− 4427.19i	− 0.323619i
$573$	0	0
$574$	21000.0	1.52704
$575$	0	0
$576$	0	0
$577$	− 15995.0i	− 1.15404i	−0.816730	−	0.577020i	$-0.804216\pi$
$577$	− 15995.0i	− 1.15404i	0.816730	−	0.577020i	$-0.195784\pi$
$578$	9255.99i	0.666087i
$579$	0	0
$580$	0	0
$581$	7968.94	0.569032
$582$	0	0
$583$	− 5600.00i	− 0.397819i
$584$	1328.16	0.0941088
$585$	0	0
$586$	20160.0	1.42116
$587$	20453.6i	1.43818i	0.694918	+	0.719089i	$0.255441\pi$
$587$	20453.6i	1.43818i	−0.694918	+	0.719089i	$0.744559\pi$
$588$	0	0
$589$	13377.0	0.935806
$590$	0	0
$591$	0	0
$592$	− 28120.0i	− 1.95224i
$593$	11599.2i	0.803244i	0.915806	+	0.401622i	$0.131553\pi$
$593$	11599.2i	0.803244i	−0.915806	+	0.401622i	$0.868447\pi$
$594$	0	0
$595$	0	0
$596$	4300.70	0.295576
$597$	0	0
$598$	12600.0i	0.861626i
$599$	19100.2	1.30286	0.651428	−	0.758710i	$-0.274170\pi$
$599$	19100.2	1.30286	0.651428	+	0.758710i	$0.274170\pi$
$600$	0	0
$601$	3143.00	0.213321	0.106660	−	0.994296i	$-0.465984\pi$
$601$	3143.00	0.213321	0.106660	+	0.994296i	$0.465984\pi$
$602$	− 15890.4i	− 1.07582i
$603$	0	0
$604$	1594.00	0.107382
$605$	0	0
$606$	0	0
$607$	2660.00i	0.177868i	0.996038	+	0.0889342i	$0.0283461\pi$
$607$	2660.00i	0.177868i	−0.996038	+	0.0889342i	$0.971654\pi$
$608$	− 8057.48i	− 0.537457i
$609$	0	0
$610$	0	0
$611$	6198.06	0.410388
$612$	0	0
$613$	6670.00i	0.439476i	0.975559	+	0.219738i	$0.0705202\pi$
$613$	6670.00i	0.439476i	−0.975559	+	0.219738i	$0.929480\pi$
$614$	−8522.34	−0.560152
$615$	0	0
$616$	−18000.0	−1.17734
$617$	− 11042.7i	− 0.720521i	−0.932852	−	0.360260i	$-0.882688\pi$
$617$	− 11042.7i	− 0.720521i	0.932852	−	0.360260i	$-0.117312\pi$
$618$	0	0
$619$	−5579.00	−0.362260	−0.181130	−	0.983459i	$-0.557975\pi$
$619$	−5579.00	−0.362260	−0.181130	+	0.983459i	$0.557975\pi$
$620$	0	0
$621$	0	0
$622$	− 16800.0i	− 1.08299i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	8079.62	0.515857
$627$	0	0
$628$	− 2450.00i	− 0.155678i
$629$	32761.2	2.07675
$630$	0	0
$631$	23617.0	1.48998	0.744990	−	0.667075i	$-0.232454\pi$
$631$	23617.0	1.48998	0.744990	+	0.667075i	$0.232454\pi$
$632$	− 16620.9i	− 1.04612i
$633$	0	0
$634$	−23600.0	−1.47835
$635$	0	0
$636$	0	0
$637$	− 4130.00i	− 0.256886i
$638$	− 12649.1i	− 0.784926i
$639$	0	0
$640$	0	0
$641$	−31496.3	−1.94076	−0.970381	−	0.241579i	$-0.922335\pi$
$641$	−31496.3	−1.94076	−0.970381	+	0.241579i	$0.922335\pi$
$642$	0	0
$643$	14280.0i	0.875814i	0.899020	+	0.437907i	$0.144280\pi$
$643$	14280.0i	0.875814i	−0.899020	+	0.437907i	$0.855720\pi$
$644$	−3415.26	−0.208975
$645$	0	0
$646$	25480.0	1.55185
$647$	12396.1i	0.753234i	0.926369	+	0.376617i	$0.122913\pi$
$647$	12396.1i	0.753234i	−0.926369	+	0.376617i	$0.877087\pi$
$648$	0	0
$649$	56000.0	3.38705
$650$	0	0
$651$	0	0
$652$	− 550.000i	− 0.0330363i
$653$	− 12042.0i	− 0.721651i	−0.932633	−	0.360825i	$-0.882495\pi$
$653$	− 12042.0i	− 0.721651i	0.932633	−	0.360825i	$-0.117505\pi$
$654$	0	0
$655$	0	0
$656$	−33646.6	−2.00256
$657$	0	0
$658$	8400.00i	0.497669i
$659$	5755.35	0.340207	0.170104	−	0.985426i	$-0.445590\pi$
$659$	5755.35	0.340207	0.170104	+	0.985426i	$0.445590\pi$
$660$	0	0
$661$	25438.0	1.49686	0.748429	−	0.663215i	$-0.230808\pi$
$661$	25438.0	1.49686	0.748429	+	0.663215i	$0.230808\pi$
$662$	16089.7i	0.944626i
$663$	0	0
$664$	−10080.0	−0.589126
$665$	0	0
$666$	0	0
$667$	7200.00i	0.417969i
$668$	531.263i	0.0307712i
$669$	0	0
$670$	0	0
$671$	−27005.9	−1.55372
$672$	0	0
$673$	− 15150.0i	− 0.867741i	−0.900975	−	0.433870i	$-0.857148\pi$
$673$	− 15150.0i	− 0.867741i	0.900975	−	0.433870i	$-0.142852\pi$
$674$	−36318.8	−2.07559
$675$	0	0
$676$	−1944.00	−0.110605
$677$	− 20984.9i	− 1.19131i	−0.803242	−	0.595653i	$-0.796893\pi$
$677$	− 20984.9i	− 1.19131i	0.803242	−	0.595653i	$-0.203107\pi$
$678$	0	0
$679$	16275.0	0.919849
$680$	0	0
$681$	0	0
$682$	− 29400.0i	− 1.65071i
$683$	− 1973.26i	− 0.110549i	−0.998471	−	0.0552743i	$-0.982397\pi$
$683$	− 1973.26i	− 0.110549i	0.998471	−	0.0552743i	$-0.0176033\pi$
$684$	0	0
$685$	0	0
$686$	21867.2	1.21704
$687$	0	0
$688$	25460.0i	1.41083i
$689$	3099.03	0.171355
$690$	0	0
$691$	−20552.0	−1.13145	−0.565727	−	0.824592i	$-0.691404\pi$
$691$	−20552.0	−1.13145	−0.565727	+	0.824592i	$0.691404\pi$
$692$	− 7968.94i	− 0.437765i
$693$	0	0
$694$	20240.0	1.10706
$695$	0	0
$696$	0	0
$697$	− 39200.0i	− 2.13028i
$698$	− 30591.9i	− 1.65891i
$699$	0	0
$700$	0	0
$701$	8411.66	0.453215	0.226608	−	0.973986i	$-0.427236\pi$
$701$	8411.66	0.453215	0.226608	+	0.973986i	$0.427236\pi$
$702$	0	0
$703$	− 33670.0i	− 1.80638i
$704$	20744.5	1.11057
$705$	0	0
$706$	−17640.0	−0.940354
$707$	− 19922.3i	− 1.05977i
$708$	0	0
$709$	−17281.0	−0.915376	−0.457688	−	0.889113i	$-0.651322\pi$
$709$	−17281.0	−0.915376	−0.457688	+	0.889113i	$0.651322\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16734.8i	0.878993i
$714$	0	0
$715$	0	0
$716$	−3541.75	−0.184862
$717$	0	0
$718$	28200.0i	1.46576i
$719$	−9297.10	−0.482230	−0.241115	−	0.970497i	$-0.577513\pi$
$719$	−9297.10	−0.482230	−0.241115	+	0.970497i	$0.577513\pi$
$720$	0	0
$721$	23100.0	1.19319
$722$	− 4496.76i	− 0.231790i
$723$	0	0
$724$	3374.00	0.173196
$725$	0	0
$726$	0	0
$727$	30415.0i	1.55162i	0.630965	+	0.775811i	$0.282659\pi$
$727$	30415.0i	1.55162i	−0.630965	+	0.775811i	$0.717341\pi$
$728$	− 9961.17i	− 0.507123i
$729$	0	0
$730$	0	0
$731$	−29662.2	−1.50081
$732$	0	0
$733$	− 18550.0i	− 0.934734i	−0.884063	−	0.467367i	$-0.845203\pi$
$733$	− 18550.0i	− 0.934734i	0.884063	−	0.467367i	$-0.154797\pi$
$734$	26895.2	1.35248
$735$	0	0
$736$	10080.0	0.504828
$737$	− 948.683i	− 0.0474155i
$738$	0	0
$739$	−24016.0	−1.19546	−0.597729	−	0.801699i	$-0.703930\pi$
$739$	−24016.0	−1.19546	−0.597729	+	0.801699i	$0.703930\pi$
$740$	0	0
$741$	0	0
$742$	4200.00i	0.207799i
$743$	− 12396.1i	− 0.612072i	−0.952020	−	0.306036i	$-0.900997\pi$
$743$	− 12396.1i	− 0.612072i	0.952020	−	0.306036i	$-0.0990029\pi$
$744$	0	0
$745$	0	0
$746$	21424.4	1.05148
$747$	0	0
$748$	− 11200.0i	− 0.547477i
$749$	11953.4	0.583135
$750$	0	0
$751$	−7772.00	−0.377636	−0.188818	−	0.982012i	$-0.560466\pi$
$751$	−7772.00	−0.377636	−0.188818	+	0.982012i	$0.560466\pi$
$752$	− 13458.7i	− 0.652642i
$753$	0	0
$754$	7000.00	0.338097
$755$	0	0
$756$	0	0
$757$	− 3865.00i	− 0.185569i	−0.995686	−	0.0927846i	$-0.970423\pi$
$757$	− 3865.00i	− 0.185569i	0.995686	−	0.0927846i	$-0.0295768\pi$
$758$	29222.6i	1.40028i
$759$	0	0
$760$	0	0
$761$	15937.9	0.759195	0.379598	−	0.925152i	$-0.376062\pi$
$761$	15937.9	0.759195	0.379598	+	0.925152i	$0.376062\pi$
$762$	0	0
$763$	12165.0i	0.577199i
$764$	4427.19	0.209647
$765$	0	0
$766$	−28560.0	−1.34715
$767$	30990.3i	1.45893i
$768$	0	0
$769$	7441.00	0.348933	0.174466	−	0.984663i	$-0.444180\pi$
$769$	7441.00	0.348933	0.174466	+	0.984663i	$0.444180\pi$
$770$	0	0
$771$	0	0
$772$	7250.00i	0.337996i
$773$	25766.2i	1.19890i	0.800413	+	0.599448i	$0.204613\pi$
$773$	25766.2i	1.19890i	−0.800413	+	0.599448i	$0.795387\pi$
$774$	0	0
$775$	0	0
$776$	−20586.4	−0.952332
$777$	0	0
$778$	13800.0i	0.635931i
$779$	−40287.4	−1.85295
$780$	0	0
$781$	−4000.00	−0.183267
$782$	31875.8i	1.45764i
$783$	0	0
$784$	−8968.00	−0.408528
$785$	0	0
$786$	0	0
$787$	6125.00i	0.277424i	0.990333	+	0.138712i	$0.0442962\pi$
$787$	6125.00i	0.277424i	−0.990333	+	0.138712i	$0.955704\pi$
$788$	− 3491.15i	− 0.157826i
$789$	0	0
$790$	0	0
$791$	24286.3	1.09168
$792$	0	0
$793$	− 14945.0i	− 0.669247i
$794$	−15163.1	−0.677732
$795$	0	0
$796$	6398.00	0.284888
$797$	20010.9i	0.889363i	0.895689	+	0.444681i	$0.146683\pi$
$797$	20010.9i	0.889363i	−0.895689	+	0.444681i	$0.853317\pi$
$798$	0	0
$799$	15680.0	0.694266
$800$	0	0
$801$	0	0
$802$	22200.0i	0.977443i
$803$	− 4427.19i	− 0.194561i
$804$	0	0
$805$	0	0
$806$	16269.9	0.711022
$807$	0	0
$808$	25200.0i	1.09719i
$809$	−2719.56	−0.118189	−0.0590943	−	0.998252i	$-0.518821\pi$
$809$	−2719.56	−0.118189	−0.0590943	+	0.998252i	$0.518821\pi$
$810$	0	0
$811$	28623.0	1.23932	0.619661	−	0.784870i	$-0.287270\pi$
$811$	28623.0	1.23932	0.619661	+	0.784870i	$0.287270\pi$
$812$	1897.37i	0.0820006i
$813$	0	0
$814$	−74000.0	−3.18636
$815$	0	0
$816$	0	0
$817$	30485.0i	1.30543i
$818$	21759.6i	0.930083i
$819$	0	0
$820$	0	0
$821$	26563.1	1.12918	0.564592	−	0.825370i	$-0.309034\pi$
$821$	26563.1	1.12918	0.564592	+	0.825370i	$0.309034\pi$
$822$	0	0
$823$	− 1135.00i	− 0.0480724i	−0.999711	−	0.0240362i	$-0.992348\pi$
$823$	− 1135.00i	− 0.0480724i	0.999711	−	0.0240362i	$-0.00765170\pi$
$824$	−29219.4	−1.23532
$825$	0	0
$826$	−42000.0	−1.76921
$827$	− 999.280i	− 0.0420174i	−0.999779	−	0.0210087i	$-0.993312\pi$
$827$	− 999.280i	− 0.0420174i	0.999779	−	0.0210087i	$-0.00668776\pi$
$828$	0	0
$829$	1974.00	0.0827019	0.0413509	−	0.999145i	$-0.486834\pi$
$829$	1974.00	0.0827019	0.0413509	+	0.999145i	$0.486834\pi$
$830$	0	0
$831$	0	0
$832$	11480.0i	0.478362i
$833$	− 10448.2i	− 0.434583i
$834$	0	0
$835$	0	0
$836$	−11510.7	−0.476203
$837$	0	0
$838$	− 40600.0i	− 1.67363i
$839$	42058.3	1.73065	0.865324	−	0.501213i	$-0.167113\pi$
$839$	42058.3	1.73065	0.865324	+	0.501213i	$0.167113\pi$
$840$	0	0
$841$	−20389.0	−0.835992
$842$	− 20295.5i	− 0.830676i
$843$	0	0
$844$	−1774.00	−0.0723502
$845$	0	0
$846$	0	0
$847$	40035.0i	1.62411i
$848$	− 6729.33i	− 0.272507i
$849$	0	0
$850$	0	0
$851$	42121.5	1.69672
$852$	0	0
$853$	− 36645.0i	− 1.47093i	−0.677564	−	0.735464i	$-0.736964\pi$
$853$	− 36645.0i	− 1.47093i	0.677564	−	0.735464i	$-0.263036\pi$
$854$	20254.4	0.811582
$855$	0	0
$856$	−15120.0	−0.603728
$857$	10802.3i	0.430573i	0.976551	+	0.215286i	$0.0690685\pi$
$857$	10802.3i	0.430573i	−0.976551	+	0.215286i	$0.930931\pi$
$858$	0	0
$859$	−20104.0	−0.798533	−0.399266	−	0.916835i	$-0.630735\pi$
$859$	−20104.0	−0.798533	−0.399266	+	0.916835i	$0.630735\pi$
$860$	0	0
$861$	0	0
$862$	4200.00i	0.165954i
$863$	2150.35i	0.0848189i	0.999100	+	0.0424095i	$0.0135034\pi$
$863$	2150.35i	0.0848189i	−0.999100	+	0.0424095i	$0.986497\pi$
$864$	0	0
$865$	0	0
$866$	25788.4	1.01192
$867$	0	0
$868$	4410.00i	0.172448i
$869$	−55403.1	−2.16274
$870$	0	0
$871$	525.000	0.0204236
$872$	− 15387.6i	− 0.597582i
$873$	0	0
$874$	32760.0	1.26788
$875$	0	0
$876$	0	0
$877$	− 21975.0i	− 0.846115i	−0.906103	−	0.423058i	$-0.860957\pi$
$877$	− 21975.0i	− 0.846115i	0.906103	−	0.423058i	$-0.139043\pi$
$878$	46640.4i	1.79275i
$879$	0	0
$880$	0	0
$881$	885.438	0.0338606	0.0169303	−	0.999857i	$-0.494611\pi$
$881$	885.438	0.0338606	0.0169303	+	0.999857i	$0.494611\pi$
$882$	0	0
$883$	− 34915.0i	− 1.33067i	−0.746544	−	0.665336i	$-0.768288\pi$
$883$	− 34915.0i	− 1.33067i	0.746544	−	0.665336i	$-0.231712\pi$
$884$	6198.06	0.235818
$885$	0	0
$886$	−35280.0	−1.33776
$887$	− 49407.4i	− 1.87028i	−0.354277	−	0.935140i	$-0.615273\pi$
$887$	− 49407.4i	− 1.87028i	0.354277	−	0.935140i	$-0.384727\pi$
$888$	0	0
$889$	−3900.00	−0.147134
$890$	0	0
$891$	0	0
$892$	7070.00i	0.265382i
$893$	− 16115.0i	− 0.603882i
$894$	0	0
$895$	0	0
$896$	−26183.7	−0.976266
$897$	0	0
$898$	29200.0i	1.08510i
$899$	9297.10	0.344912
$900$	0	0
$901$	7840.00	0.289887
$902$	88543.8i	3.26850i
$903$	0	0
$904$	−30720.0	−1.13023
$905$	0	0
$906$	0	0
$907$	17880.0i	0.654571i	0.944926	+	0.327285i	$0.106134\pi$
$907$	17880.0i	0.654571i	−0.944926	+	0.327285i	$0.893866\pi$
$908$	− 177.088i	− 0.00647231i
$909$	0	0
$910$	0	0
$911$	−17329.3	−0.630236	−0.315118	−	0.949053i	$-0.602044\pi$
$911$	−17329.3	−0.630236	−0.315118	+	0.949053i	$0.602044\pi$
$912$	0	0
$913$	33600.0i	1.21796i
$914$	−3257.15	−0.117874
$915$	0	0
$916$	−10038.0	−0.362080
$917$	19922.3i	0.717442i
$918$	0	0
$919$	25829.0	0.927117	0.463558	−	0.886066i	$-0.346572\pi$
$919$	25829.0	0.927117	0.463558	+	0.886066i	$0.346572\pi$
$920$	0	0
$921$	0	0
$922$	29400.0i	1.05015i
$923$	− 2213.59i	− 0.0789397i
$924$	0	0
$925$	0	0
$926$	−37757.6	−1.33995
$927$	0	0
$928$	− 5600.00i	− 0.198092i
$929$	−3099.03	−0.109447	−0.0547233	−	0.998502i	$-0.517428\pi$
$929$	−3099.03	−0.109447	−0.0547233	+	0.998502i	$0.517428\pi$
$930$	0	0
$931$	−10738.0	−0.378006
$932$	7209.99i	0.253403i
$933$	0	0
$934$	−38360.0	−1.34387
$935$	0	0
$936$	0	0
$937$	40005.0i	1.39478i	0.716693	+	0.697389i	$0.245655\pi$
$937$	40005.0i	1.39478i	−0.716693	+	0.697389i	$0.754345\pi$
$938$	711.512i	0.0247673i
$939$	0	0
$940$	0	0
$941$	−30104.9	−1.04292	−0.521462	−	0.853275i	$-0.674613\pi$
$941$	−30104.9	−1.04292	−0.521462	+	0.853275i	$0.674613\pi$
$942$	0	0
$943$	− 50400.0i	− 1.74046i
$944$	67293.3	2.32014
$945$	0	0
$946$	67000.0	2.30270
$947$	− 43297.9i	− 1.48574i	−0.669437	−	0.742868i	$-0.733465\pi$
$947$	− 43297.9i	− 1.48574i	0.669437	−	0.742868i	$-0.266535\pi$
$948$	0	0
$949$	2450.00	0.0838044
$950$	0	0
$951$	0	0
$952$	− 25200.0i	− 0.857917i
$953$	− 53682.8i	− 1.82472i	−0.409390	−	0.912360i	$-0.634258\pi$
$953$	− 53682.8i	− 1.82472i	0.409390	−	0.912360i	$-0.365742\pi$
$954$	0	0
$955$	0	0
$956$	−885.438	−0.0299551
$957$	0	0
$958$	8400.00i	0.283290i
$959$	−39844.7	−1.34166
$960$	0	0
$961$	−8182.00	−0.274647
$962$	− 40951.5i	− 1.37248i
$963$	0	0
$964$	1246.00	0.0416296
$965$	0	0
$966$	0	0
$967$	− 20540.0i	− 0.683063i	−0.939870	−	0.341531i	$-0.889054\pi$
$967$	− 20540.0i	− 0.683063i	0.939870	−	0.341531i	$-0.110946\pi$
$968$	− 50640.7i	− 1.68146i
$969$	0	0
$970$	0	0
$971$	−10625.3	−0.351164	−0.175582	−	0.984465i	$-0.556181\pi$
$971$	−10625.3	−0.351164	−0.175582	+	0.984465i	$0.556181\pi$
$972$	0	0
$973$	11760.0i	0.387470i
$974$	44667.2	1.46943
$975$	0	0
$976$	−32452.0	−1.06431
$977$	− 28359.3i	− 0.928654i	−0.885664	−	0.464327i	$-0.846296\pi$
$977$	− 28359.3i	− 0.928654i	0.885664	−	0.464327i	$-0.153704\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	− 12800.0i	− 0.415952i
$983$	35063.3i	1.13769i	0.822446	+	0.568844i	$0.192609\pi$
$983$	35063.3i	1.13769i	−0.822446	+	0.568844i	$0.807391\pi$
$984$	0	0
$985$	0	0
$986$	17708.8	0.571969
$987$	0	0
$988$	− 6370.00i	− 0.205118i
$989$	−38137.1	−1.22618
$990$	0	0
$991$	−18283.0	−0.586053	−0.293027	−	0.956104i	$-0.594662\pi$
$991$	−18283.0	−0.586053	−0.293027	+	0.956104i	$0.594662\pi$
$992$	− 13015.9i	− 0.416589i
$993$	0	0
$994$	3000.00	0.0957286
$995$	0	0
$996$	0	0
$997$	− 13230.0i	− 0.420259i	−0.977674	−	0.210130i	$-0.932611\pi$
$997$	− 13230.0i	− 0.420259i	0.977674	−	0.210130i	$-0.0673886\pi$
$998$	4044.55i	0.128285i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.b.i.199.2		4
3.2	odd	2	inner	225.4.b.i.199.4		4
5.2	odd	4		225.4.a.l.1.2	yes	2
5.3	odd	4		225.4.a.m.1.1	yes	2
5.4	even	2	inner	225.4.b.i.199.3		4
15.2	even	4		225.4.a.l.1.1	✓	2
15.8	even	4		225.4.a.m.1.2	yes	2
15.14	odd	2	inner	225.4.b.i.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.a.l.1.1	✓	2	15.2	even	4
225.4.a.l.1.2	yes	2	5.2	odd	4
225.4.a.m.1.1	yes	2	5.3	odd	4
225.4.a.m.1.2	yes	2	15.8	even	4
225.4.b.i.199.1		4	15.14	odd	2	inner
225.4.b.i.199.2		4	1.1	even	1	trivial
225.4.b.i.199.3		4	5.4	even	2	inner
225.4.b.i.199.4		4	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.16228i q^{2} -2.00000 q^{4} +15.0000i q^{7} -18.9737i q^{8} +O(q^{10})\)	\(q-3.16228i q^{2} -2.00000 q^{4} +15.0000i q^{7} -18.9737i q^{8} -63.2456 q^{11} -35.0000i q^{13} +47.4342 q^{14} -76.0000 q^{16} -88.5438i q^{17} -91.0000 q^{19} +200.000i q^{22} -113.842i q^{23} -110.680 q^{26} -30.0000i q^{28} -63.2456 q^{29} -147.000 q^{31} +88.5438i q^{32} -280.000 q^{34} +370.000i q^{37} +287.767i q^{38} +442.719 q^{41} -335.000i q^{43} +126.491 q^{44} -360.000 q^{46} +177.088i q^{47} +118.000 q^{49} +70.0000i q^{52} +88.5438i q^{53} +284.605 q^{56} +200.000i q^{58} -885.438 q^{59} +427.000 q^{61} +464.855i q^{62} -328.000 q^{64} +15.0000i q^{67} +177.088i q^{68} +63.2456 q^{71} +70.0000i q^{73} +1170.04 q^{74} +182.000 q^{76} -948.683i q^{77} +876.000 q^{79} -1400.00i q^{82} -531.263i q^{83} -1059.36 q^{86} +1200.00i q^{88} +525.000 q^{91} +227.684i q^{92} +560.000 q^{94} -1085.00i q^{97} -373.149i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} - 304 q^{16} - 364 q^{19} - 588 q^{31} - 1120 q^{34} - 1440 q^{46} + 472 q^{49} + 1708 q^{61} - 1312 q^{64} + 728 q^{76} + 3504 q^{79} + 2100 q^{91} + 2240 q^{94}+O(q^{100}) \) 4 * q - 8 * q^4 - 304 * q^16 - 364 * q^19 - 588 * q^31 - 1120 * q^34 - 1440 * q^46 + 472 * q^49 + 1708 * q^61 - 1312 * q^64 + 728 * q^76 + 3504 * q^79 + 2100 * q^91 + 2240 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more