LMFDB - Embedded newform 1632.2.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1632,2,Mod(1,1632)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1632, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1632.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1632 = 2^{5} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1632.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:	$13.0315856099$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	1632.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.77846 q^{5} -0.941367 q^{7} +1.00000 q^{9} -2.71982 q^{11} -0.719824 q^{13} -3.77846 q^{15} +1.00000 q^{17} +2.71982 q^{19} +0.941367 q^{21} +1.77846 q^{23} +9.27674 q^{25} -1.00000 q^{27} +8.61555 q^{29} +8.05520 q^{31} +2.71982 q^{33} -3.55691 q^{35} +2.94137 q^{37} +0.719824 q^{39} -8.27674 q^{41} -0.837090 q^{43} +3.77846 q^{45} +1.88273 q^{47} -6.11383 q^{49} -1.00000 q^{51} -1.55691 q^{53} -10.2767 q^{55} -2.71982 q^{57} +0.443086 q^{59} -2.49828 q^{61} -0.941367 q^{63} -2.71982 q^{65} +12.9966 q^{67} -1.77846 q^{69} -6.61555 q^{71} +7.43965 q^{73} -9.27674 q^{75} +2.56035 q^{77} -2.82410 q^{79} +1.00000 q^{81} +17.2311 q^{83} +3.77846 q^{85} -8.61555 q^{87} +9.11383 q^{89} +0.677618 q^{91} -8.05520 q^{93} +10.2767 q^{95} +8.11727 q^{97} -2.71982 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} + q^{11} + 7 q^{13} - 3 q^{15} + 3 q^{17} - q^{19} + 2 q^{21} - 3 q^{23} + 2 q^{25} - 3 q^{27} + 10 q^{29} - 10 q^{31} - q^{33} + 6 q^{35} + 8 q^{37} - 7 q^{39}+ \cdots + q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 + q^11 + 7 * q^13 - 3 * q^15 + 3 * q^17 - q^19 + 2 * q^21 - 3 * q^23 + 2 * q^25 - 3 * q^27 + 10 * q^29 - 10 * q^31 - q^33 + 6 * q^35 + 8 * q^37 - 7 * q^39 + q^41 + 5 * q^43 + 3 * q^45 + 4 * q^47 + 15 * q^49 - 3 * q^51 + 12 * q^53 - 5 * q^55 + q^57 + 18 * q^59 + 10 * q^61 - 2 * q^63 + q^65 + 4 * q^67 + 3 * q^69 - 4 * q^71 + 4 * q^73 - 2 * q^75 + 26 * q^77 - 6 * q^79 + 3 * q^81 + 20 * q^83 + 3 * q^85 - 10 * q^87 - 6 * q^89 + 22 * q^91 + 10 * q^93 + 5 * q^95 + 26 * q^97 + 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 (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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.77846	1.68978	0.844889	−	0.534942i	$-0.179667\pi$
$5$	3.77846	1.68978	0.844889	+	0.534942i	$0.179667\pi$
$6$	0	0
$7$	−0.941367	−0.355803	−0.177902	−	0.984048i	$-0.556931\pi$
$7$	−0.941367	−0.355803	−0.177902	+	0.984048i	$0.556931\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.71982	−0.820058	−0.410029	−	0.912073i	$-0.634481\pi$
$11$	−2.71982	−0.820058	−0.410029	+	0.912073i	$0.634481\pi$
$12$	0	0
$13$	−0.719824	−0.199643	−0.0998216	−	0.995005i	$-0.531827\pi$
$13$	−0.719824	−0.199643	−0.0998216	+	0.995005i	$0.531827\pi$
$14$	0	0
$15$	−3.77846	−0.975593
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.71982	0.623970	0.311985	−	0.950087i	$-0.399006\pi$
$19$	2.71982	0.623970	0.311985	+	0.950087i	$0.399006\pi$
$20$	0	0
$21$	0.941367	0.205423
$22$	0	0
$23$	1.77846	0.370834	0.185417	−	0.982660i	$-0.440636\pi$
$23$	1.77846	0.370834	0.185417	+	0.982660i	$0.440636\pi$
$24$	0	0
$25$	9.27674	1.85535
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.61555	1.59987	0.799933	−	0.600089i	$-0.204868\pi$
$29$	8.61555	1.59987	0.799933	+	0.600089i	$0.204868\pi$
$30$	0	0
$31$	8.05520	1.44676	0.723378	−	0.690452i	$-0.242588\pi$
$31$	8.05520	1.44676	0.723378	+	0.690452i	$0.242588\pi$
$32$	0	0
$33$	2.71982	0.473461
$34$	0	0
$35$	−3.55691	−0.601228
$36$	0	0
$37$	2.94137	0.483558	0.241779	−	0.970331i	$-0.422269\pi$
$37$	2.94137	0.483558	0.241779	+	0.970331i	$0.422269\pi$
$38$	0	0
$39$	0.719824	0.115264
$40$	0	0
$41$	−8.27674	−1.29261	−0.646305	−	0.763079i	$-0.723686\pi$
$41$	−8.27674	−1.29261	−0.646305	+	0.763079i	$0.723686\pi$
$42$	0	0
$43$	−0.837090	−0.127655	−0.0638275	−	0.997961i	$-0.520331\pi$
$43$	−0.837090	−0.127655	−0.0638275	+	0.997961i	$0.520331\pi$
$44$	0	0
$45$	3.77846	0.563259
$46$	0	0
$47$	1.88273	0.274625	0.137312	−	0.990528i	$-0.456154\pi$
$47$	1.88273	0.274625	0.137312	+	0.990528i	$0.456154\pi$
$48$	0	0
$49$	−6.11383	−0.873404
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−1.55691	−0.213859	−0.106929	−	0.994267i	$-0.534102\pi$
$53$	−1.55691	−0.213859	−0.106929	+	0.994267i	$0.534102\pi$
$54$	0	0
$55$	−10.2767	−1.38572
$56$	0	0
$57$	−2.71982	−0.360249
$58$	0	0
$59$	0.443086	0.0576849	0.0288424	−	0.999584i	$-0.490818\pi$
$59$	0.443086	0.0576849	0.0288424	+	0.999584i	$0.490818\pi$
$60$	0	0
$61$	−2.49828	−0.319872	−0.159936	−	0.987127i	$-0.551129\pi$
$61$	−2.49828	−0.319872	−0.159936	+	0.987127i	$0.551129\pi$
$62$	0	0
$63$	−0.941367	−0.118601
$64$	0	0
$65$	−2.71982	−0.337353
$66$	0	0
$67$	12.9966	1.58778	0.793891	−	0.608060i	$-0.208052\pi$
$67$	12.9966	1.58778	0.793891	+	0.608060i	$0.208052\pi$
$68$	0	0
$69$	−1.77846	−0.214101
$70$	0	0
$71$	−6.61555	−0.785121	−0.392561	−	0.919726i	$-0.628411\pi$
$71$	−6.61555	−0.785121	−0.392561	+	0.919726i	$0.628411\pi$
$72$	0	0
$73$	7.43965	0.870745	0.435372	−	0.900250i	$-0.356617\pi$
$73$	7.43965	0.870745	0.435372	+	0.900250i	$0.356617\pi$
$74$	0	0
$75$	−9.27674	−1.07119
$76$	0	0
$77$	2.56035	0.291779
$78$	0	0
$79$	−2.82410	−0.317736	−0.158868	−	0.987300i	$-0.550784\pi$
$79$	−2.82410	−0.317736	−0.158868	+	0.987300i	$0.550784\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.2311	1.89136	0.945679	−	0.325101i	$-0.105398\pi$
$83$	17.2311	1.89136	0.945679	+	0.325101i	$0.105398\pi$
$84$	0	0
$85$	3.77846	0.409831
$86$	0	0
$87$	−8.61555	−0.923684
$88$	0	0
$89$	9.11383	0.966064	0.483032	−	0.875603i	$-0.339535\pi$
$89$	9.11383	0.966064	0.483032	+	0.875603i	$0.339535\pi$
$90$	0	0
$91$	0.677618	0.0710337
$92$	0	0
$93$	−8.05520	−0.835285
$94$	0	0
$95$	10.2767	1.05437
$96$	0	0
$97$	8.11727	0.824184	0.412092	−	0.911142i	$-0.364798\pi$
$97$	8.11727	0.824184	0.412092	+	0.911142i	$0.364798\pi$
$98$	0	0
$99$	−2.71982	−0.273353
$100$	0	0
$101$	9.32238	0.927612	0.463806	−	0.885937i	$-0.346484\pi$
$101$	9.32238	0.927612	0.463806	+	0.885937i	$0.346484\pi$
$102$	0	0
$103$	−10.7198	−1.05626	−0.528128	−	0.849165i	$-0.677106\pi$
$103$	−10.7198	−1.05626	−0.528128	+	0.849165i	$0.677106\pi$
$104$	0	0
$105$	3.55691	0.347119
$106$	0	0
$107$	16.1595	1.56220	0.781098	−	0.624409i	$-0.214660\pi$
$107$	16.1595	1.56220	0.781098	+	0.624409i	$0.214660\pi$
$108$	0	0
$109$	−3.17590	−0.304196	−0.152098	−	0.988365i	$-0.548603\pi$
$109$	−3.17590	−0.304196	−0.152098	+	0.988365i	$0.548603\pi$
$110$	0	0
$111$	−2.94137	−0.279182
$112$	0	0
$113$	−0.954357	−0.0897783	−0.0448892	−	0.998992i	$-0.514293\pi$
$113$	−0.954357	−0.0897783	−0.0448892	+	0.998992i	$0.514293\pi$
$114$	0	0
$115$	6.71982	0.626627
$116$	0	0
$117$	−0.719824	−0.0665477
$118$	0	0
$119$	−0.941367	−0.0862950
$120$	0	0
$121$	−3.60256	−0.327505
$122$	0	0
$123$	8.27674	0.746288
$124$	0	0
$125$	16.1595	1.44535
$126$	0	0
$127$	4.60256	0.408411	0.204205	−	0.978928i	$-0.434539\pi$
$127$	4.60256	0.408411	0.204205	+	0.978928i	$0.434539\pi$
$128$	0	0
$129$	0.837090	0.0737017
$130$	0	0
$131$	−0.159472	−0.0139331	−0.00696656	−	0.999976i	$-0.502218\pi$
$131$	−0.159472	−0.0139331	−0.00696656	+	0.999976i	$0.502218\pi$
$132$	0	0
$133$	−2.56035	−0.222011
$134$	0	0
$135$	−3.77846	−0.325198
$136$	0	0
$137$	−18.5535	−1.58513	−0.792565	−	0.609787i	$-0.791255\pi$
$137$	−18.5535	−1.58513	−0.792565	+	0.609787i	$0.791255\pi$
$138$	0	0
$139$	−2.11727	−0.179584	−0.0897921	−	0.995961i	$-0.528620\pi$
$139$	−2.11727	−0.179584	−0.0897921	+	0.995961i	$0.528620\pi$
$140$	0	0
$141$	−1.88273	−0.158555
$142$	0	0
$143$	1.95779	0.163719
$144$	0	0
$145$	32.5535	2.70342
$146$	0	0
$147$	6.11383	0.504260
$148$	0	0
$149$	−3.43965	−0.281787	−0.140893	−	0.990025i	$-0.544998\pi$
$149$	−3.43965	−0.281787	−0.140893	+	0.990025i	$0.544998\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	30.4362	2.44470
$156$	0	0
$157$	15.3906	1.22830	0.614150	−	0.789189i	$-0.289499\pi$
$157$	15.3906	1.22830	0.614150	+	0.789189i	$0.289499\pi$
$158$	0	0
$159$	1.55691	0.123471
$160$	0	0
$161$	−1.67418	−0.131944
$162$	0	0
$163$	−16.5535	−1.29657	−0.648284	−	0.761398i	$-0.724513\pi$
$163$	−16.5535	−1.29657	−0.648284	+	0.761398i	$0.724513\pi$
$164$	0	0
$165$	10.2767	0.800043
$166$	0	0
$167$	−20.4492	−1.58241	−0.791203	−	0.611553i	$-0.790545\pi$
$167$	−20.4492	−1.58241	−0.791203	+	0.611553i	$0.790545\pi$
$168$	0	0
$169$	−12.4819	−0.960143
$170$	0	0
$171$	2.71982	0.207990
$172$	0	0
$173$	18.8923	1.43635	0.718177	−	0.695861i	$-0.244977\pi$
$173$	18.8923	1.43635	0.718177	+	0.695861i	$0.244977\pi$
$174$	0	0
$175$	−8.73281	−0.660139
$176$	0	0
$177$	−0.443086	−0.0333044
$178$	0	0
$179$	24.5535	1.83521	0.917606	−	0.397490i	$-0.130119\pi$
$179$	24.5535	1.83521	0.917606	+	0.397490i	$0.130119\pi$
$180$	0	0
$181$	−15.9379	−1.18466	−0.592328	−	0.805697i	$-0.701791\pi$
$181$	−15.9379	−1.18466	−0.592328	+	0.805697i	$0.701791\pi$
$182$	0	0
$183$	2.49828	0.184678
$184$	0	0
$185$	11.1138	0.817105
$186$	0	0
$187$	−2.71982	−0.198893
$188$	0	0
$189$	0.941367	0.0684744
$190$	0	0
$191$	6.11727	0.442630	0.221315	−	0.975202i	$-0.428965\pi$
$191$	6.11727	0.442630	0.221315	+	0.975202i	$0.428965\pi$
$192$	0	0
$193$	15.2311	1.09636	0.548179	−	0.836361i	$-0.315321\pi$
$193$	15.2311	1.09636	0.548179	+	0.836361i	$0.315321\pi$
$194$	0	0
$195$	2.71982	0.194771
$196$	0	0
$197$	12.7750	0.910182	0.455091	−	0.890445i	$-0.349607\pi$
$197$	12.7750	0.910182	0.455091	+	0.890445i	$0.349607\pi$
$198$	0	0
$199$	−8.94137	−0.633837	−0.316918	−	0.948453i	$-0.602648\pi$
$199$	−8.94137	−0.633837	−0.316918	+	0.948453i	$0.602648\pi$
$200$	0	0
$201$	−12.9966	−0.916707
$202$	0	0
$203$	−8.11039	−0.569238
$204$	0	0
$205$	−31.2733	−2.18422
$206$	0	0
$207$	1.77846	0.123611
$208$	0	0
$209$	−7.39744	−0.511692
$210$	0	0
$211$	−7.55691	−0.520239	−0.260120	−	0.965576i	$-0.583762\pi$
$211$	−7.55691	−0.520239	−0.260120	+	0.965576i	$0.583762\pi$
$212$	0	0
$213$	6.61555	0.453290
$214$	0	0
$215$	−3.16291	−0.215709
$216$	0	0
$217$	−7.58289	−0.514760
$218$	0	0
$219$	−7.43965	−0.502725
$220$	0	0
$221$	−0.719824	−0.0484206
$222$	0	0
$223$	−19.5078	−1.30634	−0.653171	−	0.757211i	$-0.726562\pi$
$223$	−19.5078	−1.30634	−0.653171	+	0.757211i	$0.726562\pi$
$224$	0	0
$225$	9.27674	0.618449
$226$	0	0
$227$	1.72326	0.114377	0.0571885	−	0.998363i	$-0.481786\pi$
$227$	1.72326	0.114377	0.0571885	+	0.998363i	$0.481786\pi$
$228$	0	0
$229$	−20.8793	−1.37974	−0.689871	−	0.723932i	$-0.742333\pi$
$229$	−20.8793	−1.37974	−0.689871	+	0.723932i	$0.742333\pi$
$230$	0	0
$231$	−2.56035	−0.168459
$232$	0	0
$233$	11.5991	0.759884	0.379942	−	0.925010i	$-0.375944\pi$
$233$	11.5991	0.759884	0.379942	+	0.925010i	$0.375944\pi$
$234$	0	0
$235$	7.11383	0.464055
$236$	0	0
$237$	2.82410	0.183445
$238$	0	0
$239$	14.4362	0.933801	0.466900	−	0.884310i	$-0.345371\pi$
$239$	14.4362	0.933801	0.466900	+	0.884310i	$0.345371\pi$
$240$	0	0
$241$	11.8827	0.765434	0.382717	−	0.923866i	$-0.374988\pi$
$241$	11.8827	0.765434	0.382717	+	0.923866i	$0.374988\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−23.1008	−1.47586
$246$	0	0
$247$	−1.95779	−0.124571
$248$	0	0
$249$	−17.2311	−1.09198
$250$	0	0
$251$	27.1138	1.71141	0.855705	−	0.517464i	$-0.173124\pi$
$251$	27.1138	1.71141	0.855705	+	0.517464i	$0.173124\pi$
$252$	0	0
$253$	−4.83709	−0.304105
$254$	0	0
$255$	−3.77846	−0.236616
$256$	0	0
$257$	−24.8793	−1.55193	−0.775964	−	0.630777i	$-0.782736\pi$
$257$	−24.8793	−1.55193	−0.775964	+	0.630777i	$0.782736\pi$
$258$	0	0
$259$	−2.76891	−0.172051
$260$	0	0
$261$	8.61555	0.533289
$262$	0	0
$263$	24.9966	1.54135	0.770677	−	0.637226i	$-0.219918\pi$
$263$	24.9966	1.54135	0.770677	+	0.637226i	$0.219918\pi$
$264$	0	0
$265$	−5.88273	−0.361373
$266$	0	0
$267$	−9.11383	−0.557757
$268$	0	0
$269$	−21.8957	−1.33501	−0.667503	−	0.744607i	$-0.732637\pi$
$269$	−21.8957	−1.33501	−0.667503	+	0.744607i	$0.732637\pi$
$270$	0	0
$271$	4.60256	0.279585	0.139793	−	0.990181i	$-0.455356\pi$
$271$	4.60256	0.279585	0.139793	+	0.990181i	$0.455356\pi$
$272$	0	0
$273$	−0.677618	−0.0410113
$274$	0	0
$275$	−25.2311	−1.52149
$276$	0	0
$277$	−7.82754	−0.470311	−0.235156	−	0.971958i	$-0.575560\pi$
$277$	−7.82754	−0.470311	−0.235156	+	0.971958i	$0.575560\pi$
$278$	0	0
$279$	8.05520	0.482252
$280$	0	0
$281$	25.1138	1.49817	0.749083	−	0.662476i	$-0.230495\pi$
$281$	25.1138	1.49817	0.749083	+	0.662476i	$0.230495\pi$
$282$	0	0
$283$	−3.32238	−0.197495	−0.0987475	−	0.995113i	$-0.531484\pi$
$283$	−3.32238	−0.197495	−0.0987475	+	0.995113i	$0.531484\pi$
$284$	0	0
$285$	−10.2767	−0.608741
$286$	0	0
$287$	7.79145	0.459915
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−8.11727	−0.475843
$292$	0	0
$293$	−29.4328	−1.71948	−0.859740	−	0.510731i	$-0.829375\pi$
$293$	−29.4328	−1.71948	−0.859740	+	0.510731i	$0.829375\pi$
$294$	0	0
$295$	1.67418	0.0974746
$296$	0	0
$297$	2.71982	0.157820
$298$	0	0
$299$	−1.28018	−0.0740345
$300$	0	0
$301$	0.788009	0.0454201
$302$	0	0
$303$	−9.32238	−0.535557
$304$	0	0
$305$	−9.43965	−0.540513
$306$	0	0
$307$	−15.3484	−0.875977	−0.437989	−	0.898981i	$-0.644309\pi$
$307$	−15.3484	−0.875977	−0.437989	+	0.898981i	$0.644309\pi$
$308$	0	0
$309$	10.7198	0.609829
$310$	0	0
$311$	−31.6121	−1.79256	−0.896279	−	0.443490i	$-0.853740\pi$
$311$	−31.6121	−1.79256	−0.896279	+	0.443490i	$0.853740\pi$
$312$	0	0
$313$	−15.4656	−0.874169	−0.437084	−	0.899420i	$-0.643989\pi$
$313$	−15.4656	−0.874169	−0.437084	+	0.899420i	$0.643989\pi$
$314$	0	0
$315$	−3.55691	−0.200409
$316$	0	0
$317$	−22.6087	−1.26983	−0.634915	−	0.772582i	$-0.718965\pi$
$317$	−22.6087	−1.26983	−0.634915	+	0.772582i	$0.718965\pi$
$318$	0	0
$319$	−23.4328	−1.31198
$320$	0	0
$321$	−16.1595	−0.901934
$322$	0	0
$323$	2.71982	0.151335
$324$	0	0
$325$	−6.67762	−0.370408
$326$	0	0
$327$	3.17590	0.175628
$328$	0	0
$329$	−1.77234	−0.0977124
$330$	0	0
$331$	−29.3906	−1.61545	−0.807726	−	0.589558i	$-0.799302\pi$
$331$	−29.3906	−1.61545	−0.807726	+	0.589558i	$0.799302\pi$
$332$	0	0
$333$	2.94137	0.161186
$334$	0	0
$335$	49.1070	2.68300
$336$	0	0
$337$	−7.67418	−0.418039	−0.209020	−	0.977911i	$-0.567027\pi$
$337$	−7.67418	−0.418039	−0.209020	+	0.977911i	$0.567027\pi$
$338$	0	0
$339$	0.954357	0.0518335
$340$	0	0
$341$	−21.9087	−1.18642
$342$	0	0
$343$	12.3449	0.666563
$344$	0	0
$345$	−6.71982	−0.361783
$346$	0	0
$347$	−0.234533	−0.0125904	−0.00629519	−	0.999980i	$-0.502004\pi$
$347$	−0.234533	−0.0125904	−0.00629519	+	0.999980i	$0.502004\pi$
$348$	0	0
$349$	8.48529	0.454207	0.227104	−	0.973871i	$-0.427074\pi$
$349$	8.48529	0.454207	0.227104	+	0.973871i	$0.427074\pi$
$350$	0	0
$351$	0.719824	0.0384214
$352$	0	0
$353$	−16.8793	−0.898394	−0.449197	−	0.893433i	$-0.648290\pi$
$353$	−16.8793	−0.898394	−0.449197	+	0.893433i	$0.648290\pi$
$354$	0	0
$355$	−24.9966	−1.32668
$356$	0	0
$357$	0.941367	0.0498224
$358$	0	0
$359$	30.2277	1.59535	0.797677	−	0.603084i	$-0.206062\pi$
$359$	30.2277	1.59535	0.797677	+	0.603084i	$0.206062\pi$
$360$	0	0
$361$	−11.6026	−0.610661
$362$	0	0
$363$	3.60256	0.189085
$364$	0	0
$365$	28.1104	1.47137
$366$	0	0
$367$	3.50172	0.182788	0.0913941	−	0.995815i	$-0.470868\pi$
$367$	3.50172	0.182788	0.0913941	+	0.995815i	$0.470868\pi$
$368$	0	0
$369$	−8.27674	−0.430870
$370$	0	0
$371$	1.46563	0.0760916
$372$	0	0
$373$	−22.7620	−1.17857	−0.589287	−	0.807924i	$-0.700591\pi$
$373$	−22.7620	−1.17857	−0.589287	+	0.807924i	$0.700591\pi$
$374$	0	0
$375$	−16.1595	−0.834472
$376$	0	0
$377$	−6.20168	−0.319403
$378$	0	0
$379$	28.7880	1.47874	0.739370	−	0.673299i	$-0.235123\pi$
$379$	28.7880	1.47874	0.739370	+	0.673299i	$0.235123\pi$
$380$	0	0
$381$	−4.60256	−0.235796
$382$	0	0
$383$	−28.6639	−1.46466	−0.732328	−	0.680952i	$-0.761566\pi$
$383$	−28.6639	−1.46466	−0.732328	+	0.680952i	$0.761566\pi$
$384$	0	0
$385$	9.67418	0.493042
$386$	0	0
$387$	−0.837090	−0.0425517
$388$	0	0
$389$	−22.7880	−1.15540	−0.577699	−	0.816250i	$-0.696049\pi$
$389$	−22.7880	−1.15540	−0.577699	+	0.816250i	$0.696049\pi$
$390$	0	0
$391$	1.77846	0.0899404
$392$	0	0
$393$	0.159472	0.00804429
$394$	0	0
$395$	−10.6707	−0.536903
$396$	0	0
$397$	−18.0812	−0.907468	−0.453734	−	0.891137i	$-0.649908\pi$
$397$	−18.0812	−0.907468	−0.453734	+	0.891137i	$0.649908\pi$
$398$	0	0
$399$	2.56035	0.128178
$400$	0	0
$401$	1.71639	0.0857122	0.0428561	−	0.999081i	$-0.486354\pi$
$401$	1.71639	0.0857122	0.0428561	+	0.999081i	$0.486354\pi$
$402$	0	0
$403$	−5.79832	−0.288835
$404$	0	0
$405$	3.77846	0.187753
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−25.1629	−1.24423	−0.622113	−	0.782928i	$-0.713726\pi$
$409$	−25.1629	−1.24423	−0.622113	+	0.782928i	$0.713726\pi$
$410$	0	0
$411$	18.5535	0.915175
$412$	0	0
$413$	−0.417106	−0.0205245
$414$	0	0
$415$	65.1070	3.19597
$416$	0	0
$417$	2.11727	0.103683
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	21.5078	1.04823	0.524114	−	0.851648i	$-0.324397\pi$
$421$	21.5078	1.04823	0.524114	+	0.851648i	$0.324397\pi$
$422$	0	0
$423$	1.88273	0.0915416
$424$	0	0
$425$	9.27674	0.449988
$426$	0	0
$427$	2.35180	0.113812
$428$	0	0
$429$	−1.95779	−0.0945232
$430$	0	0
$431$	−4.73281	−0.227972	−0.113986	−	0.993482i	$-0.536362\pi$
$431$	−4.73281	−0.227972	−0.113986	+	0.993482i	$0.536362\pi$
$432$	0	0
$433$	10.2836	0.494199	0.247099	−	0.968990i	$-0.420523\pi$
$433$	10.2836	0.494199	0.247099	+	0.968990i	$0.420523\pi$
$434$	0	0
$435$	−32.5535	−1.56082
$436$	0	0
$437$	4.83709	0.231389
$438$	0	0
$439$	−25.7294	−1.22800	−0.613998	−	0.789308i	$-0.710440\pi$
$439$	−25.7294	−1.22800	−0.613998	+	0.789308i	$0.710440\pi$
$440$	0	0
$441$	−6.11383	−0.291135
$442$	0	0
$443$	23.3484	1.10931	0.554657	−	0.832079i	$-0.312849\pi$
$443$	23.3484	1.10931	0.554657	+	0.832079i	$0.312849\pi$
$444$	0	0
$445$	34.4362	1.63243
$446$	0	0
$447$	3.43965	0.162690
$448$	0	0
$449$	−28.6448	−1.35183	−0.675915	−	0.736980i	$-0.736251\pi$
$449$	−28.6448	−1.35183	−0.675915	+	0.736980i	$0.736251\pi$
$450$	0	0
$451$	22.5113	1.06001
$452$	0	0
$453$	20.0000	0.939682
$454$	0	0
$455$	2.56035	0.120031
$456$	0	0
$457$	17.7164	0.828738	0.414369	−	0.910109i	$-0.364002\pi$
$457$	17.7164	0.828738	0.414369	+	0.910109i	$0.364002\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	13.8759	0.646263	0.323132	−	0.946354i	$-0.395264\pi$
$461$	13.8759	0.646263	0.323132	+	0.946354i	$0.395264\pi$
$462$	0	0
$463$	36.1104	1.67819	0.839096	−	0.543983i	$-0.183084\pi$
$463$	36.1104	1.67819	0.839096	+	0.543983i	$0.183084\pi$
$464$	0	0
$465$	−30.4362	−1.41145
$466$	0	0
$467$	−30.4622	−1.40962	−0.704811	−	0.709395i	$-0.748968\pi$
$467$	−30.4622	−1.40962	−0.704811	+	0.709395i	$0.748968\pi$
$468$	0	0
$469$	−12.2345	−0.564938
$470$	0	0
$471$	−15.3906	−0.709160
$472$	0	0
$473$	2.27674	0.104685
$474$	0	0
$475$	25.2311	1.15768
$476$	0	0
$477$	−1.55691	−0.0712862
$478$	0	0
$479$	−10.5665	−0.482794	−0.241397	−	0.970426i	$-0.577606\pi$
$479$	−10.5665	−0.482794	−0.241397	+	0.970426i	$0.577606\pi$
$480$	0	0
$481$	−2.11727	−0.0965390
$482$	0	0
$483$	1.67418	0.0761779
$484$	0	0
$485$	30.6707	1.39269
$486$	0	0
$487$	−5.28629	−0.239545	−0.119772	−	0.992801i	$-0.538216\pi$
$487$	−5.28629	−0.239545	−0.119772	+	0.992801i	$0.538216\pi$
$488$	0	0
$489$	16.5535	0.748574
$490$	0	0
$491$	−14.4622	−0.652669	−0.326335	−	0.945254i	$-0.605814\pi$
$491$	−14.4622	−0.652669	−0.326335	+	0.945254i	$0.605814\pi$
$492$	0	0
$493$	8.61555	0.388025
$494$	0	0
$495$	−10.2767	−0.461905
$496$	0	0
$497$	6.22766	0.279349
$498$	0	0
$499$	−1.23109	−0.0551114	−0.0275557	−	0.999620i	$-0.508772\pi$
$499$	−1.23109	−0.0551114	−0.0275557	+	0.999620i	$0.508772\pi$
$500$	0	0
$501$	20.4492	0.913603
$502$	0	0
$503$	4.33881	0.193458	0.0967290	−	0.995311i	$-0.469162\pi$
$503$	4.33881	0.193458	0.0967290	+	0.995311i	$0.469162\pi$
$504$	0	0
$505$	35.2242	1.56746
$506$	0	0
$507$	12.4819	0.554339
$508$	0	0
$509$	14.9706	0.663559	0.331780	−	0.943357i	$-0.392351\pi$
$509$	14.9706	0.663559	0.331780	+	0.943357i	$0.392351\pi$
$510$	0	0
$511$	−7.00344	−0.309814
$512$	0	0
$513$	−2.71982	−0.120083
$514$	0	0
$515$	−40.5044	−1.78484
$516$	0	0
$517$	−5.12070	−0.225208
$518$	0	0
$519$	−18.8923	−0.829279
$520$	0	0
$521$	25.7164	1.12666	0.563328	−	0.826234i	$-0.309521\pi$
$521$	25.7164	1.12666	0.563328	+	0.826234i	$0.309521\pi$
$522$	0	0
$523$	28.9966	1.26793	0.633966	−	0.773361i	$-0.281426\pi$
$523$	28.9966	1.26793	0.633966	+	0.773361i	$0.281426\pi$
$524$	0	0
$525$	8.73281	0.381131
$526$	0	0
$527$	8.05520	0.350890
$528$	0	0
$529$	−19.8371	−0.862482
$530$	0	0
$531$	0.443086	0.0192283
$532$	0	0
$533$	5.95779	0.258061
$534$	0	0
$535$	61.0579	2.63976
$536$	0	0
$537$	−24.5535	−1.05956
$538$	0	0
$539$	16.6285	0.716242
$540$	0	0
$541$	−34.8172	−1.49691	−0.748455	−	0.663186i	$-0.769204\pi$
$541$	−34.8172	−1.49691	−0.748455	+	0.663186i	$0.769204\pi$
$542$	0	0
$543$	15.9379	0.683962
$544$	0	0
$545$	−12.0000	−0.514024
$546$	0	0
$547$	−24.4431	−1.04511	−0.522555	−	0.852605i	$-0.675021\pi$
$547$	−24.4431	−1.04511	−0.522555	+	0.852605i	$0.675021\pi$
$548$	0	0
$549$	−2.49828	−0.106624
$550$	0	0
$551$	23.4328	0.998270
$552$	0	0
$553$	2.65851	0.113052
$554$	0	0
$555$	−11.1138	−0.471756
$556$	0	0
$557$	1.00344	0.0425170	0.0212585	−	0.999774i	$-0.493233\pi$
$557$	1.00344	0.0425170	0.0212585	+	0.999774i	$0.493233\pi$
$558$	0	0
$559$	0.602558	0.0254855
$560$	0	0
$561$	2.71982	0.114831
$562$	0	0
$563$	10.9053	0.459603	0.229801	−	0.973238i	$-0.426192\pi$
$563$	10.9053	0.459603	0.229801	+	0.973238i	$0.426192\pi$
$564$	0	0
$565$	−3.60600	−0.151705
$566$	0	0
$567$	−0.941367	−0.0395337
$568$	0	0
$569$	−1.34836	−0.0565262	−0.0282631	−	0.999601i	$-0.508998\pi$
$569$	−1.34836	−0.0565262	−0.0282631	+	0.999601i	$0.508998\pi$
$570$	0	0
$571$	12.3189	0.515532	0.257766	−	0.966207i	$-0.417014\pi$
$571$	12.3189	0.515532	0.257766	+	0.966207i	$0.417014\pi$
$572$	0	0
$573$	−6.11727	−0.255552
$574$	0	0
$575$	16.4983	0.688026
$576$	0	0
$577$	33.2993	1.38627	0.693134	−	0.720809i	$-0.256230\pi$
$577$	33.2993	1.38627	0.693134	+	0.720809i	$0.256230\pi$
$578$	0	0
$579$	−15.2311	−0.632983
$580$	0	0
$581$	−16.2208	−0.672951
$582$	0	0
$583$	4.23453	0.175376
$584$	0	0
$585$	−2.71982	−0.112451
$586$	0	0
$587$	22.4622	0.927114	0.463557	−	0.886067i	$-0.346573\pi$
$587$	22.4622	0.927114	0.463557	+	0.886067i	$0.346573\pi$
$588$	0	0
$589$	21.9087	0.902733
$590$	0	0
$591$	−12.7750	−0.525494
$592$	0	0
$593$	1.21199	0.0497705	0.0248853	−	0.999690i	$-0.492078\pi$
$593$	1.21199	0.0497705	0.0248853	+	0.999690i	$0.492078\pi$
$594$	0	0
$595$	−3.55691	−0.145819
$596$	0	0
$597$	8.94137	0.365946
$598$	0	0
$599$	−37.3415	−1.52573	−0.762866	−	0.646557i	$-0.776208\pi$
$599$	−37.3415	−1.52573	−0.762866	+	0.646557i	$0.776208\pi$
$600$	0	0
$601$	−20.2277	−0.825103	−0.412552	−	0.910934i	$-0.635362\pi$
$601$	−20.2277	−0.825103	−0.412552	+	0.910934i	$0.635362\pi$
$602$	0	0
$603$	12.9966	0.529261
$604$	0	0
$605$	−13.6121	−0.553411
$606$	0	0
$607$	−22.4914	−0.912898	−0.456449	−	0.889750i	$-0.650879\pi$
$607$	−22.4914	−0.912898	−0.456449	+	0.889750i	$0.650879\pi$
$608$	0	0
$609$	8.11039	0.328650
$610$	0	0
$611$	−1.35524	−0.0548270
$612$	0	0
$613$	3.62510	0.146416	0.0732082	−	0.997317i	$-0.476676\pi$
$613$	3.62510	0.146416	0.0732082	+	0.997317i	$0.476676\pi$
$614$	0	0
$615$	31.2733	1.26106
$616$	0	0
$617$	−9.23797	−0.371907	−0.185953	−	0.982559i	$-0.559537\pi$
$617$	−9.23797	−0.371907	−0.185953	+	0.982559i	$0.559537\pi$
$618$	0	0
$619$	−5.67418	−0.228065	−0.114032	−	0.993477i	$-0.536377\pi$
$619$	−5.67418	−0.228065	−0.114032	+	0.993477i	$0.536377\pi$
$620$	0	0
$621$	−1.77846	−0.0713670
$622$	0	0
$623$	−8.57946	−0.343729
$624$	0	0
$625$	14.6742	0.586967
$626$	0	0
$627$	7.39744	0.295425
$628$	0	0
$629$	2.94137	0.117280
$630$	0	0
$631$	−30.3871	−1.20969	−0.604846	−	0.796342i	$-0.706765\pi$
$631$	−30.3871	−1.20969	−0.604846	+	0.796342i	$0.706765\pi$
$632$	0	0
$633$	7.55691	0.300360
$634$	0	0
$635$	17.3906	0.690124
$636$	0	0
$637$	4.40088	0.174369
$638$	0	0
$639$	−6.61555	−0.261707
$640$	0	0
$641$	8.04221	0.317648	0.158824	−	0.987307i	$-0.449230\pi$
$641$	8.04221	0.317648	0.158824	+	0.987307i	$0.449230\pi$
$642$	0	0
$643$	−5.67418	−0.223768	−0.111884	−	0.993721i	$-0.535688\pi$
$643$	−5.67418	−0.223768	−0.111884	+	0.993721i	$0.535688\pi$
$644$	0	0
$645$	3.16291	0.124539
$646$	0	0
$647$	−2.76891	−0.108857	−0.0544284	−	0.998518i	$-0.517334\pi$
$647$	−2.76891	−0.108857	−0.0544284	+	0.998518i	$0.517334\pi$
$648$	0	0
$649$	−1.20512	−0.0473049
$650$	0	0
$651$	7.58289	0.297197
$652$	0	0
$653$	−18.4492	−0.721973	−0.360986	−	0.932571i	$-0.617560\pi$
$653$	−18.4492	−0.721973	−0.360986	+	0.932571i	$0.617560\pi$
$654$	0	0
$655$	−0.602558	−0.0235439
$656$	0	0
$657$	7.43965	0.290248
$658$	0	0
$659$	5.46563	0.212911	0.106455	−	0.994317i	$-0.466050\pi$
$659$	5.46563	0.212911	0.106455	+	0.994317i	$0.466050\pi$
$660$	0	0
$661$	−13.7424	−0.534516	−0.267258	−	0.963625i	$-0.586118\pi$
$661$	−13.7424	−0.534516	−0.267258	+	0.963625i	$0.586118\pi$
$662$	0	0
$663$	0.719824	0.0279556
$664$	0	0
$665$	−9.67418	−0.375149
$666$	0	0
$667$	15.3224	0.593285
$668$	0	0
$669$	19.5078	0.754216
$670$	0	0
$671$	6.79488	0.262314
$672$	0	0
$673$	32.4362	1.25032	0.625162	−	0.780495i	$-0.285033\pi$
$673$	32.4362	1.25032	0.625162	+	0.780495i	$0.285033\pi$
$674$	0	0
$675$	−9.27674	−0.357062
$676$	0	0
$677$	3.56990	0.137203	0.0686013	−	0.997644i	$-0.478146\pi$
$677$	3.56990	0.137203	0.0686013	+	0.997644i	$0.478146\pi$
$678$	0	0
$679$	−7.64133	−0.293247
$680$	0	0
$681$	−1.72326	−0.0660355
$682$	0	0
$683$	−36.9215	−1.41276	−0.706381	−	0.707832i	$-0.749673\pi$
$683$	−36.9215	−1.41276	−0.706381	+	0.707832i	$0.749673\pi$
$684$	0	0
$685$	−70.1035	−2.67852
$686$	0	0
$687$	20.8793	0.796595
$688$	0	0
$689$	1.12070	0.0426954
$690$	0	0
$691$	30.1035	1.14519	0.572596	−	0.819838i	$-0.305936\pi$
$691$	30.1035	1.14519	0.572596	+	0.819838i	$0.305936\pi$
$692$	0	0
$693$	2.56035	0.0972597
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	−8.27674	−0.313504
$698$	0	0
$699$	−11.5991	−0.438719
$700$	0	0
$701$	−5.79145	−0.218740	−0.109370	−	0.994001i	$-0.534883\pi$
$701$	−5.79145	−0.218740	−0.109370	+	0.994001i	$0.534883\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	−7.11383	−0.267922
$706$	0	0
$707$	−8.77578	−0.330047
$708$	0	0
$709$	−30.9414	−1.16203	−0.581014	−	0.813894i	$-0.697344\pi$
$709$	−30.9414	−1.16203	−0.581014	+	0.813894i	$0.697344\pi$
$710$	0	0
$711$	−2.82410	−0.105912
$712$	0	0
$713$	14.3258	0.536506
$714$	0	0
$715$	7.39744	0.276649
$716$	0	0
$717$	−14.4362	−0.539130
$718$	0	0
$719$	−47.5370	−1.77283	−0.886416	−	0.462889i	$-0.846813\pi$
$719$	−47.5370	−1.77283	−0.886416	+	0.462889i	$0.846813\pi$
$720$	0	0
$721$	10.0913	0.375819
$722$	0	0
$723$	−11.8827	−0.441924
$724$	0	0
$725$	79.9242	2.96831
$726$	0	0
$727$	−12.8862	−0.477922	−0.238961	−	0.971029i	$-0.576807\pi$
$727$	−12.8862	−0.477922	−0.238961	+	0.971029i	$0.576807\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.837090	−0.0309609
$732$	0	0
$733$	9.23797	0.341212	0.170606	−	0.985339i	$-0.445427\pi$
$733$	9.23797	0.341212	0.170606	+	0.985339i	$0.445427\pi$
$734$	0	0
$735$	23.1008	0.852087
$736$	0	0
$737$	−35.3484	−1.30207
$738$	0	0
$739$	−35.9249	−1.32152	−0.660760	−	0.750597i	$-0.729766\pi$
$739$	−35.9249	−1.32152	−0.660760	+	0.750597i	$0.729766\pi$
$740$	0	0
$741$	1.95779	0.0719214
$742$	0	0
$743$	52.3741	1.92142	0.960710	−	0.277553i	$-0.0895233\pi$
$743$	52.3741	1.92142	0.960710	+	0.277553i	$0.0895233\pi$
$744$	0	0
$745$	−12.9966	−0.476157
$746$	0	0
$747$	17.2311	0.630453
$748$	0	0
$749$	−15.2120	−0.555834
$750$	0	0
$751$	−37.8138	−1.37984	−0.689922	−	0.723883i	$-0.742355\pi$
$751$	−37.8138	−1.37984	−0.689922	+	0.723883i	$0.742355\pi$
$752$	0	0
$753$	−27.1138	−0.988083
$754$	0	0
$755$	−75.5691	−2.75024
$756$	0	0
$757$	30.6026	1.11227	0.556134	−	0.831092i	$-0.312284\pi$
$757$	30.6026	1.11227	0.556134	+	0.831092i	$0.312284\pi$
$758$	0	0
$759$	4.83709	0.175575
$760$	0	0
$761$	14.6516	0.531121	0.265561	−	0.964094i	$-0.414443\pi$
$761$	14.6516	0.531121	0.265561	+	0.964094i	$0.414443\pi$
$762$	0	0
$763$	2.98969	0.108234
$764$	0	0
$765$	3.77846	0.136610
$766$	0	0
$767$	−0.318944	−0.0115164
$768$	0	0
$769$	−53.6182	−1.93352	−0.966761	−	0.255681i	$-0.917700\pi$
$769$	−53.6182	−1.93352	−0.966761	+	0.255681i	$0.917700\pi$
$770$	0	0
$771$	24.8793	0.896006
$772$	0	0
$773$	37.7655	1.35833	0.679165	−	0.733986i	$-0.262342\pi$
$773$	37.7655	1.35833	0.679165	+	0.733986i	$0.262342\pi$
$774$	0	0
$775$	74.7259	2.68423
$776$	0	0
$777$	2.76891	0.0993339
$778$	0	0
$779$	−22.5113	−0.806550
$780$	0	0
$781$	17.9931	0.643845
$782$	0	0
$783$	−8.61555	−0.307895
$784$	0	0
$785$	58.1526	2.07556
$786$	0	0
$787$	19.7914	0.705489	0.352744	−	0.935720i	$-0.385249\pi$
$787$	19.7914	0.705489	0.352744	+	0.935720i	$0.385249\pi$
$788$	0	0
$789$	−24.9966	−0.889901
$790$	0	0
$791$	0.898400	0.0319434
$792$	0	0
$793$	1.79832	0.0638603
$794$	0	0
$795$	5.88273	0.208639
$796$	0	0
$797$	50.0122	1.77152	0.885762	−	0.464140i	$-0.153636\pi$
$797$	50.0122	1.77152	0.885762	+	0.464140i	$0.153636\pi$
$798$	0	0
$799$	1.88273	0.0666063
$800$	0	0
$801$	9.11383	0.322021
$802$	0	0
$803$	−20.2345	−0.714061
$804$	0	0
$805$	−6.32582	−0.222956
$806$	0	0
$807$	21.8957	0.770766
$808$	0	0
$809$	−15.8077	−0.555768	−0.277884	−	0.960615i	$-0.589633\pi$
$809$	−15.8077	−0.555768	−0.277884	+	0.960615i	$0.589633\pi$
$810$	0	0
$811$	−44.6898	−1.56927	−0.784636	−	0.619956i	$-0.787150\pi$
$811$	−44.6898	−1.56927	−0.784636	+	0.619956i	$0.787150\pi$
$812$	0	0
$813$	−4.60256	−0.161419
$814$	0	0
$815$	−62.5466	−2.19091
$816$	0	0
$817$	−2.27674	−0.0796530
$818$	0	0
$819$	0.677618	0.0236779
$820$	0	0
$821$	−29.6872	−1.03609	−0.518045	−	0.855354i	$-0.673340\pi$
$821$	−29.6872	−1.03609	−0.518045	+	0.855354i	$0.673340\pi$
$822$	0	0
$823$	−7.84664	−0.273517	−0.136758	−	0.990604i	$-0.543668\pi$
$823$	−7.84664	−0.273517	−0.136758	+	0.990604i	$0.543668\pi$
$824$	0	0
$825$	25.2311	0.878434
$826$	0	0
$827$	−43.5078	−1.51292	−0.756458	−	0.654043i	$-0.773072\pi$
$827$	−43.5078	−1.51292	−0.756458	+	0.654043i	$0.773072\pi$
$828$	0	0
$829$	17.7655	0.617020	0.308510	−	0.951221i	$-0.400170\pi$
$829$	17.7655	0.617020	0.308510	+	0.951221i	$0.400170\pi$
$830$	0	0
$831$	7.82754	0.271534
$832$	0	0
$833$	−6.11383	−0.211832
$834$	0	0
$835$	−77.2664	−2.67391
$836$	0	0
$837$	−8.05520	−0.278428
$838$	0	0
$839$	1.98701	0.0685992	0.0342996	−	0.999412i	$-0.489080\pi$
$839$	1.98701	0.0685992	0.0342996	+	0.999412i	$0.489080\pi$
$840$	0	0
$841$	45.2277	1.55957
$842$	0	0
$843$	−25.1138	−0.864966
$844$	0	0
$845$	−47.1621	−1.62243
$846$	0	0
$847$	3.39133	0.116527
$848$	0	0
$849$	3.32238	0.114024
$850$	0	0
$851$	5.23109	0.179320
$852$	0	0
$853$	53.1430	1.81958	0.909792	−	0.415065i	$-0.136241\pi$
$853$	53.1430	1.81958	0.909792	+	0.415065i	$0.136241\pi$
$854$	0	0
$855$	10.2767	0.351457
$856$	0	0
$857$	49.3346	1.68524	0.842619	−	0.538510i	$-0.181013\pi$
$857$	49.3346	1.68524	0.842619	+	0.538510i	$0.181013\pi$
$858$	0	0
$859$	5.77234	0.196950	0.0984749	−	0.995140i	$-0.468604\pi$
$859$	5.77234	0.196950	0.0984749	+	0.995140i	$0.468604\pi$
$860$	0	0
$861$	−7.79145	−0.265532
$862$	0	0
$863$	38.9637	1.32634	0.663170	−	0.748469i	$-0.269211\pi$
$863$	38.9637	1.32634	0.663170	+	0.748469i	$0.269211\pi$
$864$	0	0
$865$	71.3837	2.42712
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	7.68106	0.260562
$870$	0	0
$871$	−9.35524	−0.316990
$872$	0	0
$873$	8.11727	0.274728
$874$	0	0
$875$	−15.2120	−0.514259
$876$	0	0
$877$	11.9379	0.403115	0.201558	−	0.979477i	$-0.435400\pi$
$877$	11.9379	0.403115	0.201558	+	0.979477i	$0.435400\pi$
$878$	0	0
$879$	29.4328	0.992743
$880$	0	0
$881$	−3.23109	−0.108858	−0.0544292	−	0.998518i	$-0.517334\pi$
$881$	−3.23109	−0.108858	−0.0544292	+	0.998518i	$0.517334\pi$
$882$	0	0
$883$	4.49217	0.151173	0.0755867	−	0.997139i	$-0.475917\pi$
$883$	4.49217	0.151173	0.0755867	+	0.997139i	$0.475917\pi$
$884$	0	0
$885$	−1.67418	−0.0562770
$886$	0	0
$887$	−0.671504	−0.0225469	−0.0112735	−	0.999936i	$-0.503589\pi$
$887$	−0.671504	−0.0225469	−0.0112735	+	0.999936i	$0.503589\pi$
$888$	0	0
$889$	−4.33270	−0.145314
$890$	0	0
$891$	−2.71982	−0.0911175
$892$	0	0
$893$	5.12070	0.171358
$894$	0	0
$895$	92.7743	3.10110
$896$	0	0
$897$	1.28018	0.0427438
$898$	0	0
$899$	69.3999	2.31462
$900$	0	0
$901$	−1.55691	−0.0518683
$902$	0	0
$903$	−0.788009	−0.0262233
$904$	0	0
$905$	−60.2208	−2.00181
$906$	0	0
$907$	−13.4656	−0.447119	−0.223559	−	0.974690i	$-0.571768\pi$
$907$	−13.4656	−0.447119	−0.223559	+	0.974690i	$0.571768\pi$
$908$	0	0
$909$	9.32238	0.309204
$910$	0	0
$911$	45.2372	1.49878	0.749388	−	0.662131i	$-0.230348\pi$
$911$	45.2372	1.49878	0.749388	+	0.662131i	$0.230348\pi$
$912$	0	0
$913$	−46.8655	−1.55102
$914$	0	0
$915$	9.43965	0.312065
$916$	0	0
$917$	0.150122	0.00495745
$918$	0	0
$919$	−8.04908	−0.265515	−0.132757	−	0.991149i	$-0.542383\pi$
$919$	−8.04908	−0.265515	−0.132757	+	0.991149i	$0.542383\pi$
$920$	0	0
$921$	15.3484	0.505746
$922$	0	0
$923$	4.76203	0.156744
$924$	0	0
$925$	27.2863	0.897168
$926$	0	0
$927$	−10.7198	−0.352085
$928$	0	0
$929$	−8.59568	−0.282015	−0.141008	−	0.990009i	$-0.545034\pi$
$929$	−8.59568	−0.282015	−0.141008	+	0.990009i	$0.545034\pi$
$930$	0	0
$931$	−16.6285	−0.544978
$932$	0	0
$933$	31.6121	1.03493
$934$	0	0
$935$	−10.2767	−0.336085
$936$	0	0
$937$	19.9931	0.653147	0.326573	−	0.945172i	$-0.394106\pi$
$937$	19.9931	0.653147	0.326573	+	0.945172i	$0.394106\pi$
$938$	0	0
$939$	15.4656	0.504702
$940$	0	0
$941$	40.7259	1.32763	0.663814	−	0.747898i	$-0.268937\pi$
$941$	40.7259	1.32763	0.663814	+	0.747898i	$0.268937\pi$
$942$	0	0
$943$	−14.7198	−0.479343
$944$	0	0
$945$	3.55691	0.115706
$946$	0	0
$947$	0.344923	0.0112085	0.00560425	−	0.999984i	$-0.498216\pi$
$947$	0.344923	0.0112085	0.00560425	+	0.999984i	$0.498216\pi$
$948$	0	0
$949$	−5.35524	−0.173838
$950$	0	0
$951$	22.6087	0.733136
$952$	0	0
$953$	39.0225	1.26406	0.632032	−	0.774942i	$-0.282221\pi$
$953$	39.0225	1.26406	0.632032	+	0.774942i	$0.282221\pi$
$954$	0	0
$955$	23.1138	0.747946
$956$	0	0
$957$	23.4328	0.757474
$958$	0	0
$959$	17.4656	0.563995
$960$	0	0
$961$	33.8862	1.09310
$962$	0	0
$963$	16.1595	0.520732
$964$	0	0
$965$	57.5500	1.85260
$966$	0	0
$967$	−51.4097	−1.65322	−0.826612	−	0.562773i	$-0.809735\pi$
$967$	−51.4097	−1.65322	−0.826612	+	0.562773i	$0.809735\pi$
$968$	0	0
$969$	−2.71982	−0.0873733
$970$	0	0
$971$	54.8915	1.76155	0.880776	−	0.473532i	$-0.157021\pi$
$971$	54.8915	1.76155	0.880776	+	0.473532i	$0.157021\pi$
$972$	0	0
$973$	1.99312	0.0638966
$974$	0	0
$975$	6.67762	0.213855
$976$	0	0
$977$	16.6967	0.534175	0.267088	−	0.963672i	$-0.413939\pi$
$977$	16.6967	0.534175	0.267088	+	0.963672i	$0.413939\pi$
$978$	0	0
$979$	−24.7880	−0.792228
$980$	0	0
$981$	−3.17590	−0.101399
$982$	0	0
$983$	−2.24752	−0.0716848	−0.0358424	−	0.999357i	$-0.511411\pi$
$983$	−2.24752	−0.0716848	−0.0358424	+	0.999357i	$0.511411\pi$
$984$	0	0
$985$	48.2699	1.53801
$986$	0	0
$987$	1.77234	0.0564143
$988$	0	0
$989$	−1.48873	−0.0473388
$990$	0	0
$991$	7.05863	0.224225	0.112112	−	0.993696i	$-0.464238\pi$
$991$	7.05863	0.224225	0.112112	+	0.993696i	$0.464238\pi$
$992$	0	0
$993$	29.3906	0.932681
$994$	0	0
$995$	−33.7846	−1.07104
$996$	0	0
$997$	−1.61211	−0.0510560	−0.0255280	−	0.999674i	$-0.508127\pi$
$997$	−1.61211	−0.0510560	−0.0255280	+	0.999674i	$0.508127\pi$
$998$	0	0
$999$	−2.94137	−0.0930607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1632.2.a.r.1.3	✓	3
3.2	odd	2		4896.2.a.ba.1.1		3
4.3	odd	2		1632.2.a.t.1.3	yes	3
8.3	odd	2		3264.2.a.bq.1.1		3
8.5	even	2		3264.2.a.bs.1.1		3
12.11	even	2		4896.2.a.bb.1.1		3
24.5	odd	2		9792.2.a.dk.1.3		3
24.11	even	2		9792.2.a.dl.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1632.2.a.r.1.3	✓	3	1.1	even	1	trivial
1632.2.a.t.1.3	yes	3	4.3	odd	2
3264.2.a.bq.1.1		3	8.3	odd	2
3264.2.a.bs.1.1		3	8.5	even	2
4896.2.a.ba.1.1		3	3.2	odd	2
4896.2.a.bb.1.1		3	12.11	even	2
9792.2.a.dk.1.3		3	24.5	odd	2
9792.2.a.dl.1.3		3	24.11	even	2

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:	\( N \)	\(=\)	\( 1632 = 2^{5} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1632.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.77846 q^{5} -0.941367 q^{7} +1.00000 q^{9} -2.71982 q^{11} -0.719824 q^{13} -3.77846 q^{15} +1.00000 q^{17} +2.71982 q^{19} +0.941367 q^{21} +1.77846 q^{23} +9.27674 q^{25} -1.00000 q^{27} +8.61555 q^{29} +8.05520 q^{31} +2.71982 q^{33} -3.55691 q^{35} +2.94137 q^{37} +0.719824 q^{39} -8.27674 q^{41} -0.837090 q^{43} +3.77846 q^{45} +1.88273 q^{47} -6.11383 q^{49} -1.00000 q^{51} -1.55691 q^{53} -10.2767 q^{55} -2.71982 q^{57} +0.443086 q^{59} -2.49828 q^{61} -0.941367 q^{63} -2.71982 q^{65} +12.9966 q^{67} -1.77846 q^{69} -6.61555 q^{71} +7.43965 q^{73} -9.27674 q^{75} +2.56035 q^{77} -2.82410 q^{79} +1.00000 q^{81} +17.2311 q^{83} +3.77846 q^{85} -8.61555 q^{87} +9.11383 q^{89} +0.677618 q^{91} -8.05520 q^{93} +10.2767 q^{95} +8.11727 q^{97} -2.71982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} + q^{11} + 7 q^{13} - 3 q^{15} + 3 q^{17} - q^{19} + 2 q^{21} - 3 q^{23} + 2 q^{25} - 3 q^{27} + 10 q^{29} - 10 q^{31} - q^{33} + 6 q^{35} + 8 q^{37} - 7 q^{39}+ \cdots + q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 + q^11 + 7 * q^13 - 3 * q^15 + 3 * q^17 - q^19 + 2 * q^21 - 3 * q^23 + 2 * q^25 - 3 * q^27 + 10 * q^29 - 10 * q^31 - q^33 + 6 * q^35 + 8 * q^37 - 7 * q^39 + q^41 + 5 * q^43 + 3 * q^45 + 4 * q^47 + 15 * q^49 - 3 * q^51 + 12 * q^53 - 5 * q^55 + q^57 + 18 * q^59 + 10 * q^61 - 2 * q^63 + q^65 + 4 * q^67 + 3 * q^69 - 4 * q^71 + 4 * q^73 - 2 * q^75 + 26 * q^77 - 6 * q^79 + 3 * q^81 + 20 * q^83 + 3 * q^85 - 10 * q^87 - 6 * q^89 + 22 * q^91 + 10 * q^93 + 5 * q^95 + 26 * q^97 + q^99

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