LMFDB - Embedded newform 4008.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + \cdots + 242 $ x^11 - x^10 - 33x^9 + 22x^8 + 417x^7 - 151x^6 - 2470x^5 + 272x^4 + 6584x^3 + 292x^2 - 5687*x + 242
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.11942$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.11942 q^{5} -0.802640 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.11942 q^{5} -0.802640 q^{7} +1.00000 q^{9} -1.53063 q^{11} +6.15404 q^{13} +2.11942 q^{15} +2.02477 q^{17} -8.09372 q^{19} +0.802640 q^{21} -1.69737 q^{23} -0.508068 q^{25} -1.00000 q^{27} +8.39635 q^{29} +1.83232 q^{31} +1.53063 q^{33} +1.70113 q^{35} -9.93315 q^{37} -6.15404 q^{39} -2.89174 q^{41} +2.94539 q^{43} -2.11942 q^{45} -7.81191 q^{47} -6.35577 q^{49} -2.02477 q^{51} +9.25602 q^{53} +3.24405 q^{55} +8.09372 q^{57} +5.76308 q^{59} -0.0936477 q^{61} -0.802640 q^{63} -13.0430 q^{65} -8.66806 q^{67} +1.69737 q^{69} -8.00704 q^{71} +1.52554 q^{73} +0.508068 q^{75} +1.22855 q^{77} +15.5706 q^{79} +1.00000 q^{81} +13.8255 q^{83} -4.29133 q^{85} -8.39635 q^{87} +11.9663 q^{89} -4.93948 q^{91} -1.83232 q^{93} +17.1540 q^{95} -13.7852 q^{97} -1.53063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9	$ 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9} - q^{11} + 10 q^{13} - 10 q^{15} + 17 q^{17} + 2 q^{19} + q^{21} - 3 q^{23} + 21 q^{25} - 11 q^{27} + 17 q^{29} - 15 q^{31} + q^{33} + 11 q^{35} + 4 q^{37} - 10 q^{39} + 16 q^{41} + 10 q^{43} + 10 q^{45} - 16 q^{47} + 22 q^{49} - 17 q^{51} + 42 q^{53} - 5 q^{55} - 2 q^{57} - 2 q^{59} + 12 q^{61} - q^{63} + 10 q^{65} - q^{67} + 3 q^{69} - 9 q^{71} + 24 q^{73} - 21 q^{75} + 22 q^{77} - 30 q^{79} + 11 q^{81} + 16 q^{83} + 25 q^{85} - 17 q^{87} + 37 q^{89} + q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97} - q^{99}+O(q^{100}) $ 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9 - q^11 + 10 * q^13 - 10 * q^15 + 17 * q^17 + 2 * q^19 + q^21 - 3 * q^23 + 21 * q^25 - 11 * q^27 + 17 * q^29 - 15 * q^31 + q^33 + 11 * q^35 + 4 * q^37 - 10 * q^39 + 16 * q^41 + 10 * q^43 + 10 * q^45 - 16 * q^47 + 22 * q^49 - 17 * q^51 + 42 * q^53 - 5 * q^55 - 2 * q^57 - 2 * q^59 + 12 * q^61 - q^63 + 10 * q^65 - q^67 + 3 * q^69 - 9 * q^71 + 24 * q^73 - 21 * q^75 + 22 * q^77 - 30 * q^79 + 11 * q^81 + 16 * q^83 + 25 * q^85 - 17 * q^87 + 37 * q^89 + q^91 + 15 * q^93 + 5 * q^95 + 4 * q^97 - q^99

Display 100 coefficients 10 coefficients

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
$5$	−2.11942	−0.947832	−0.473916	−	0.880570i	$-0.657160\pi$
$5$	−2.11942	−0.947832	−0.473916	+	0.880570i	$0.657160\pi$
$6$	0	0
$7$	−0.802640	−0.303369	−0.151685	−	0.988429i	$-0.548470\pi$
$7$	−0.802640	−0.303369	−0.151685	+	0.988429i	$0.548470\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.53063	−0.461503	−0.230752	−	0.973013i	$-0.574118\pi$
$11$	−1.53063	−0.461503	−0.230752	+	0.973013i	$0.574118\pi$
$12$	0	0
$13$	6.15404	1.70682	0.853412	−	0.521236i	$-0.174529\pi$
$13$	6.15404	1.70682	0.853412	+	0.521236i	$0.174529\pi$
$14$	0	0
$15$	2.11942	0.547231
$16$	0	0
$17$	2.02477	0.491079	0.245539	−	0.969387i	$-0.421035\pi$
$17$	2.02477	0.491079	0.245539	+	0.969387i	$0.421035\pi$
$18$	0	0
$19$	−8.09372	−1.85683	−0.928414	−	0.371548i	$-0.878827\pi$
$19$	−8.09372	−1.85683	−0.928414	+	0.371548i	$0.878827\pi$
$20$	0	0
$21$	0.802640	0.175150
$22$	0	0
$23$	−1.69737	−0.353927	−0.176963	−	0.984217i	$-0.556627\pi$
$23$	−1.69737	−0.353927	−0.176963	+	0.984217i	$0.556627\pi$
$24$	0	0
$25$	−0.508068	−0.101614
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.39635	1.55916	0.779582	−	0.626301i	$-0.215432\pi$
$29$	8.39635	1.55916	0.779582	+	0.626301i	$0.215432\pi$
$30$	0	0
$31$	1.83232	0.329095	0.164548	−	0.986369i	$-0.447384\pi$
$31$	1.83232	0.329095	0.164548	+	0.986369i	$0.447384\pi$
$32$	0	0
$33$	1.53063	0.266449
$34$	0	0
$35$	1.70113	0.287543
$36$	0	0
$37$	−9.93315	−1.63300	−0.816500	−	0.577345i	$-0.804089\pi$
$37$	−9.93315	−1.63300	−0.816500	+	0.577345i	$0.804089\pi$
$38$	0	0
$39$	−6.15404	−0.985436
$40$	0	0
$41$	−2.89174	−0.451614	−0.225807	−	0.974172i	$-0.572502\pi$
$41$	−2.89174	−0.451614	−0.225807	+	0.974172i	$0.572502\pi$
$42$	0	0
$43$	2.94539	0.449168	0.224584	−	0.974455i	$-0.427898\pi$
$43$	2.94539	0.449168	0.224584	+	0.974455i	$0.427898\pi$
$44$	0	0
$45$	−2.11942	−0.315944
$46$	0	0
$47$	−7.81191	−1.13948	−0.569742	−	0.821823i	$-0.692957\pi$
$47$	−7.81191	−1.13948	−0.569742	+	0.821823i	$0.692957\pi$
$48$	0	0
$49$	−6.35577	−0.907967
$50$	0	0
$51$	−2.02477	−0.283524
$52$	0	0
$53$	9.25602	1.27141	0.635706	−	0.771931i	$-0.280709\pi$
$53$	9.25602	1.27141	0.635706	+	0.771931i	$0.280709\pi$
$54$	0	0
$55$	3.24405	0.437428
$56$	0	0
$57$	8.09372	1.07204
$58$	0	0
$59$	5.76308	0.750289	0.375144	−	0.926966i	$-0.377593\pi$
$59$	5.76308	0.750289	0.375144	+	0.926966i	$0.377593\pi$
$60$	0	0
$61$	−0.0936477	−0.0119904	−0.00599518	−	0.999982i	$-0.501908\pi$
$61$	−0.0936477	−0.0119904	−0.00599518	+	0.999982i	$0.501908\pi$
$62$	0	0
$63$	−0.802640	−0.101123
$64$	0	0
$65$	−13.0430	−1.61778
$66$	0	0
$67$	−8.66806	−1.05897	−0.529486	−	0.848319i	$-0.677615\pi$
$67$	−8.66806	−1.05897	−0.529486	+	0.848319i	$0.677615\pi$
$68$	0	0
$69$	1.69737	0.204340
$70$	0	0
$71$	−8.00704	−0.950261	−0.475130	−	0.879915i	$-0.657599\pi$
$71$	−8.00704	−0.950261	−0.475130	+	0.879915i	$0.657599\pi$
$72$	0	0
$73$	1.52554	0.178552	0.0892758	−	0.996007i	$-0.471545\pi$
$73$	1.52554	0.178552	0.0892758	+	0.996007i	$0.471545\pi$
$74$	0	0
$75$	0.508068	0.0586667
$76$	0	0
$77$	1.22855	0.140006
$78$	0	0
$79$	15.5706	1.75183	0.875917	−	0.482461i	$-0.160257\pi$
$79$	15.5706	1.75183	0.875917	+	0.482461i	$0.160257\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.8255	1.51755	0.758775	−	0.651352i	$-0.225798\pi$
$83$	13.8255	1.51755	0.758775	+	0.651352i	$0.225798\pi$
$84$	0	0
$85$	−4.29133	−0.465460
$86$	0	0
$87$	−8.39635	−0.900183
$88$	0	0
$89$	11.9663	1.26842	0.634211	−	0.773160i	$-0.281325\pi$
$89$	11.9663	1.26842	0.634211	+	0.773160i	$0.281325\pi$
$90$	0	0
$91$	−4.93948	−0.517798
$92$	0	0
$93$	−1.83232	−0.190003
$94$	0	0
$95$	17.1540	1.75996
$96$	0	0
$97$	−13.7852	−1.39968	−0.699840	−	0.714300i	$-0.746745\pi$
$97$	−13.7852	−1.39968	−0.699840	+	0.714300i	$0.746745\pi$
$98$	0	0
$99$	−1.53063	−0.153834
$100$	0	0
$101$	13.2301	1.31644	0.658220	−	0.752825i	$-0.271309\pi$
$101$	13.2301	1.31644	0.658220	+	0.752825i	$0.271309\pi$
$102$	0	0
$103$	−9.58267	−0.944209	−0.472104	−	0.881543i	$-0.656505\pi$
$103$	−9.58267	−0.944209	−0.472104	+	0.881543i	$0.656505\pi$
$104$	0	0
$105$	−1.70113	−0.166013
$106$	0	0
$107$	3.19020	0.308408	0.154204	−	0.988039i	$-0.450719\pi$
$107$	3.19020	0.308408	0.154204	+	0.988039i	$0.450719\pi$
$108$	0	0
$109$	−13.5473	−1.29760	−0.648800	−	0.760959i	$-0.724729\pi$
$109$	−13.5473	−1.29760	−0.648800	+	0.760959i	$0.724729\pi$
$110$	0	0
$111$	9.93315	0.942813
$112$	0	0
$113$	6.42224	0.604154	0.302077	−	0.953284i	$-0.402320\pi$
$113$	6.42224	0.604154	0.302077	+	0.953284i	$0.402320\pi$
$114$	0	0
$115$	3.59744	0.335463
$116$	0	0
$117$	6.15404	0.568942
$118$	0	0
$119$	−1.62516	−0.148978
$120$	0	0
$121$	−8.65716	−0.787015
$122$	0	0
$123$	2.89174	0.260740
$124$	0	0
$125$	11.6739	1.04415
$126$	0	0
$127$	6.93176	0.615094	0.307547	−	0.951533i	$-0.400492\pi$
$127$	6.93176	0.615094	0.307547	+	0.951533i	$0.400492\pi$
$128$	0	0
$129$	−2.94539	−0.259327
$130$	0	0
$131$	−5.62848	−0.491762	−0.245881	−	0.969300i	$-0.579077\pi$
$131$	−5.62848	−0.491762	−0.245881	+	0.969300i	$0.579077\pi$
$132$	0	0
$133$	6.49634	0.563304
$134$	0	0
$135$	2.11942	0.182410
$136$	0	0
$137$	1.30884	0.111822	0.0559109	−	0.998436i	$-0.482194\pi$
$137$	1.30884	0.111822	0.0559109	+	0.998436i	$0.482194\pi$
$138$	0	0
$139$	16.9293	1.43593	0.717963	−	0.696082i	$-0.245075\pi$
$139$	16.9293	1.43593	0.717963	+	0.696082i	$0.245075\pi$
$140$	0	0
$141$	7.81191	0.657882
$142$	0	0
$143$	−9.41959	−0.787705
$144$	0	0
$145$	−17.7954	−1.47783
$146$	0	0
$147$	6.35577	0.524215
$148$	0	0
$149$	23.9654	1.96332	0.981660	−	0.190642i	$-0.0610570\pi$
$149$	23.9654	1.96332	0.981660	+	0.190642i	$0.0610570\pi$
$150$	0	0
$151$	4.74051	0.385777	0.192888	−	0.981221i	$-0.438214\pi$
$151$	4.74051	0.385777	0.192888	+	0.981221i	$0.438214\pi$
$152$	0	0
$153$	2.02477	0.163693
$154$	0	0
$155$	−3.88346	−0.311927
$156$	0	0
$157$	12.2918	0.980991	0.490495	−	0.871444i	$-0.336816\pi$
$157$	12.2918	0.980991	0.490495	+	0.871444i	$0.336816\pi$
$158$	0	0
$159$	−9.25602	−0.734050
$160$	0	0
$161$	1.36238	0.107371
$162$	0	0
$163$	14.4990	1.13565	0.567823	−	0.823151i	$-0.307786\pi$
$163$	14.4990	1.13565	0.567823	+	0.823151i	$0.307786\pi$
$164$	0	0
$165$	−3.24405	−0.252549
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	24.8723	1.91325
$170$	0	0
$171$	−8.09372	−0.618943
$172$	0	0
$173$	−14.2287	−1.08179	−0.540893	−	0.841091i	$-0.681914\pi$
$173$	−14.2287	−1.08179	−0.540893	+	0.841091i	$0.681914\pi$
$174$	0	0
$175$	0.407796	0.0308265
$176$	0	0
$177$	−5.76308	−0.433180
$178$	0	0
$179$	5.46854	0.408738	0.204369	−	0.978894i	$-0.434486\pi$
$179$	5.46854	0.408738	0.204369	+	0.978894i	$0.434486\pi$
$180$	0	0
$181$	8.71174	0.647538	0.323769	−	0.946136i	$-0.395050\pi$
$181$	8.71174	0.647538	0.323769	+	0.946136i	$0.395050\pi$
$182$	0	0
$183$	0.0936477	0.00692263
$184$	0	0
$185$	21.0525	1.54781
$186$	0	0
$187$	−3.09918	−0.226634
$188$	0	0
$189$	0.802640	0.0583834
$190$	0	0
$191$	−6.84326	−0.495161	−0.247580	−	0.968867i	$-0.579635\pi$
$191$	−6.84326	−0.495161	−0.247580	+	0.968867i	$0.579635\pi$
$192$	0	0
$193$	11.6905	0.841499	0.420750	−	0.907177i	$-0.361767\pi$
$193$	11.6905	0.841499	0.420750	+	0.907177i	$0.361767\pi$
$194$	0	0
$195$	13.0430	0.934028
$196$	0	0
$197$	20.4537	1.45727	0.728633	−	0.684904i	$-0.240156\pi$
$197$	20.4537	1.45727	0.728633	+	0.684904i	$0.240156\pi$
$198$	0	0
$199$	19.5826	1.38817	0.694086	−	0.719892i	$-0.255809\pi$
$199$	19.5826	1.38817	0.694086	+	0.719892i	$0.255809\pi$
$200$	0	0
$201$	8.66806	0.611398
$202$	0	0
$203$	−6.73924	−0.473002
$204$	0	0
$205$	6.12881	0.428055
$206$	0	0
$207$	−1.69737	−0.117976
$208$	0	0
$209$	12.3885	0.856932
$210$	0	0
$211$	−12.0070	−0.826598	−0.413299	−	0.910595i	$-0.635624\pi$
$211$	−12.0070	−0.826598	−0.413299	+	0.910595i	$0.635624\pi$
$212$	0	0
$213$	8.00704	0.548633
$214$	0	0
$215$	−6.24252	−0.425736
$216$	0	0
$217$	−1.47070	−0.0998373
$218$	0	0
$219$	−1.52554	−0.103087
$220$	0	0
$221$	12.4605	0.838185
$222$	0	0
$223$	20.0988	1.34591	0.672957	−	0.739681i	$-0.265024\pi$
$223$	20.0988	1.34591	0.672957	+	0.739681i	$0.265024\pi$
$224$	0	0
$225$	−0.508068	−0.0338712
$226$	0	0
$227$	14.1963	0.942242	0.471121	−	0.882068i	$-0.343849\pi$
$227$	14.1963	0.942242	0.471121	+	0.882068i	$0.343849\pi$
$228$	0	0
$229$	19.7750	1.30677	0.653386	−	0.757025i	$-0.273348\pi$
$229$	19.7750	1.30677	0.653386	+	0.757025i	$0.273348\pi$
$230$	0	0
$231$	−1.22855	−0.0808324
$232$	0	0
$233$	9.37536	0.614200	0.307100	−	0.951677i	$-0.400641\pi$
$233$	9.37536	0.614200	0.307100	+	0.951677i	$0.400641\pi$
$234$	0	0
$235$	16.5567	1.08004
$236$	0	0
$237$	−15.5706	−1.01142
$238$	0	0
$239$	−11.0467	−0.714551	−0.357276	−	0.933999i	$-0.616294\pi$
$239$	−11.0467	−0.714551	−0.357276	+	0.933999i	$0.616294\pi$
$240$	0	0
$241$	3.14465	0.202565	0.101282	−	0.994858i	$-0.467705\pi$
$241$	3.14465	0.202565	0.101282	+	0.994858i	$0.467705\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	13.4705	0.860601
$246$	0	0
$247$	−49.8091	−3.16928
$248$	0	0
$249$	−13.8255	−0.876158
$250$	0	0
$251$	19.4249	1.22609	0.613044	−	0.790049i	$-0.289945\pi$
$251$	19.4249	1.22609	0.613044	+	0.790049i	$0.289945\pi$
$252$	0	0
$253$	2.59806	0.163338
$254$	0	0
$255$	4.29133	0.268734
$256$	0	0
$257$	24.3783	1.52068	0.760338	−	0.649528i	$-0.225033\pi$
$257$	24.3783	1.52068	0.760338	+	0.649528i	$0.225033\pi$
$258$	0	0
$259$	7.97274	0.495402
$260$	0	0
$261$	8.39635	0.519721
$262$	0	0
$263$	−19.1381	−1.18010	−0.590051	−	0.807366i	$-0.700893\pi$
$263$	−19.1381	−1.18010	−0.590051	+	0.807366i	$0.700893\pi$
$264$	0	0
$265$	−19.6174	−1.20509
$266$	0	0
$267$	−11.9663	−0.732324
$268$	0	0
$269$	−31.8583	−1.94244	−0.971218	−	0.238192i	$-0.923445\pi$
$269$	−31.8583	−1.94244	−0.971218	+	0.238192i	$0.923445\pi$
$270$	0	0
$271$	−27.6866	−1.68184	−0.840922	−	0.541157i	$-0.817986\pi$
$271$	−27.6866	−1.68184	−0.840922	+	0.541157i	$0.817986\pi$
$272$	0	0
$273$	4.93948	0.298951
$274$	0	0
$275$	0.777666	0.0468951
$276$	0	0
$277$	16.5531	0.994582	0.497291	−	0.867584i	$-0.334328\pi$
$277$	16.5531	0.994582	0.497291	+	0.867584i	$0.334328\pi$
$278$	0	0
$279$	1.83232	0.109698
$280$	0	0
$281$	−16.1221	−0.961766	−0.480883	−	0.876785i	$-0.659684\pi$
$281$	−16.1221	−0.961766	−0.480883	+	0.876785i	$0.659684\pi$
$282$	0	0
$283$	−15.4620	−0.919118	−0.459559	−	0.888147i	$-0.651992\pi$
$283$	−15.4620	−0.919118	−0.459559	+	0.888147i	$0.651992\pi$
$284$	0	0
$285$	−17.1540	−1.01611
$286$	0	0
$287$	2.32103	0.137006
$288$	0	0
$289$	−12.9003	−0.758842
$290$	0	0
$291$	13.7852	0.808105
$292$	0	0
$293$	−13.3824	−0.781806	−0.390903	−	0.920432i	$-0.627837\pi$
$293$	−13.3824	−0.781806	−0.390903	+	0.920432i	$0.627837\pi$
$294$	0	0
$295$	−12.2144	−0.711148
$296$	0	0
$297$	1.53063	0.0888164
$298$	0	0
$299$	−10.4457	−0.604091
$300$	0	0
$301$	−2.36409	−0.136264
$302$	0	0
$303$	−13.2301	−0.760048
$304$	0	0
$305$	0.198479	0.0113648
$306$	0	0
$307$	−10.3905	−0.593018	−0.296509	−	0.955030i	$-0.595822\pi$
$307$	−10.3905	−0.593018	−0.296509	+	0.955030i	$0.595822\pi$
$308$	0	0
$309$	9.58267	0.545139
$310$	0	0
$311$	5.04038	0.285814	0.142907	−	0.989736i	$-0.454355\pi$
$311$	5.04038	0.285814	0.142907	+	0.989736i	$0.454355\pi$
$312$	0	0
$313$	12.2609	0.693025	0.346513	−	0.938045i	$-0.387366\pi$
$313$	12.2609	0.693025	0.346513	+	0.938045i	$0.387366\pi$
$314$	0	0
$315$	1.70113	0.0958477
$316$	0	0
$317$	−15.7166	−0.882731	−0.441365	−	0.897327i	$-0.645506\pi$
$317$	−15.7166	−0.882731	−0.441365	+	0.897327i	$0.645506\pi$
$318$	0	0
$319$	−12.8517	−0.719559
$320$	0	0
$321$	−3.19020	−0.178059
$322$	0	0
$323$	−16.3879	−0.911848
$324$	0	0
$325$	−3.12668	−0.173437
$326$	0	0
$327$	13.5473	0.749169
$328$	0	0
$329$	6.27015	0.345685
$330$	0	0
$331$	−18.2687	−1.00414	−0.502069	−	0.864827i	$-0.667428\pi$
$331$	−18.2687	−1.00414	−0.502069	+	0.864827i	$0.667428\pi$
$332$	0	0
$333$	−9.93315	−0.544334
$334$	0	0
$335$	18.3712	1.00373
$336$	0	0
$337$	20.6321	1.12390	0.561951	−	0.827171i	$-0.310051\pi$
$337$	20.6321	1.12390	0.561951	+	0.827171i	$0.310051\pi$
$338$	0	0
$339$	−6.42224	−0.348808
$340$	0	0
$341$	−2.80462	−0.151878
$342$	0	0
$343$	10.7199	0.578819
$344$	0	0
$345$	−3.59744	−0.193680
$346$	0	0
$347$	7.45323	0.400110	0.200055	−	0.979785i	$-0.435888\pi$
$347$	7.45323	0.400110	0.200055	+	0.979785i	$0.435888\pi$
$348$	0	0
$349$	20.1869	1.08058	0.540289	−	0.841480i	$-0.318315\pi$
$349$	20.1869	1.08058	0.540289	+	0.841480i	$0.318315\pi$
$350$	0	0
$351$	−6.15404	−0.328479
$352$	0	0
$353$	5.40011	0.287419	0.143709	−	0.989620i	$-0.454097\pi$
$353$	5.40011	0.287419	0.143709	+	0.989620i	$0.454097\pi$
$354$	0	0
$355$	16.9703	0.900688
$356$	0	0
$357$	1.62516	0.0860126
$358$	0	0
$359$	14.1097	0.744680	0.372340	−	0.928096i	$-0.378555\pi$
$359$	14.1097	0.744680	0.372340	+	0.928096i	$0.378555\pi$
$360$	0	0
$361$	46.5084	2.44781
$362$	0	0
$363$	8.65716	0.454383
$364$	0	0
$365$	−3.23327	−0.169237
$366$	0	0
$367$	8.40423	0.438697	0.219349	−	0.975647i	$-0.429607\pi$
$367$	8.40423	0.438697	0.219349	+	0.975647i	$0.429607\pi$
$368$	0	0
$369$	−2.89174	−0.150538
$370$	0	0
$371$	−7.42925	−0.385707
$372$	0	0
$373$	−23.3605	−1.20956	−0.604780	−	0.796393i	$-0.706739\pi$
$373$	−23.3605	−1.20956	−0.604780	+	0.796393i	$0.706739\pi$
$374$	0	0
$375$	−11.6739	−0.602837
$376$	0	0
$377$	51.6715	2.66122
$378$	0	0
$379$	35.8583	1.84192	0.920959	−	0.389659i	$-0.127407\pi$
$379$	35.8583	1.84192	0.920959	+	0.389659i	$0.127407\pi$
$380$	0	0
$381$	−6.93176	−0.355125
$382$	0	0
$383$	−1.69504	−0.0866123	−0.0433061	−	0.999062i	$-0.513789\pi$
$383$	−1.69504	−0.0866123	−0.0433061	+	0.999062i	$0.513789\pi$
$384$	0	0
$385$	−2.60380	−0.132702
$386$	0	0
$387$	2.94539	0.149723
$388$	0	0
$389$	14.2244	0.721203	0.360602	−	0.932720i	$-0.382571\pi$
$389$	14.2244	0.721203	0.360602	+	0.932720i	$0.382571\pi$
$390$	0	0
$391$	−3.43679	−0.173806
$392$	0	0
$393$	5.62848	0.283919
$394$	0	0
$395$	−33.0007	−1.66045
$396$	0	0
$397$	−13.0556	−0.655240	−0.327620	−	0.944810i	$-0.606247\pi$
$397$	−13.0556	−0.655240	−0.327620	+	0.944810i	$0.606247\pi$
$398$	0	0
$399$	−6.49634	−0.325224
$400$	0	0
$401$	32.8074	1.63832	0.819162	−	0.573563i	$-0.194439\pi$
$401$	32.8074	1.63832	0.819162	+	0.573563i	$0.194439\pi$
$402$	0	0
$403$	11.2762	0.561708
$404$	0	0
$405$	−2.11942	−0.105315
$406$	0	0
$407$	15.2040	0.753635
$408$	0	0
$409$	−15.4482	−0.763865	−0.381932	−	0.924190i	$-0.624741\pi$
$409$	−15.4482	−0.763865	−0.381932	+	0.924190i	$0.624741\pi$
$410$	0	0
$411$	−1.30884	−0.0645603
$412$	0	0
$413$	−4.62567	−0.227615
$414$	0	0
$415$	−29.3021	−1.43838
$416$	0	0
$417$	−16.9293	−0.829032
$418$	0	0
$419$	−1.69199	−0.0826592	−0.0413296	−	0.999146i	$-0.513159\pi$
$419$	−1.69199	−0.0826592	−0.0413296	+	0.999146i	$0.513159\pi$
$420$	0	0
$421$	28.1615	1.37251	0.686254	−	0.727362i	$-0.259254\pi$
$421$	28.1615	1.37251	0.686254	+	0.727362i	$0.259254\pi$
$422$	0	0
$423$	−7.81191	−0.379828
$424$	0	0
$425$	−1.02872	−0.0499003
$426$	0	0
$427$	0.0751653	0.00363750
$428$	0	0
$429$	9.41959	0.454782
$430$	0	0
$431$	−15.1892	−0.731636	−0.365818	−	0.930686i	$-0.619211\pi$
$431$	−15.1892	−0.731636	−0.365818	+	0.930686i	$0.619211\pi$
$432$	0	0
$433$	−8.76398	−0.421170	−0.210585	−	0.977576i	$-0.567537\pi$
$433$	−8.76398	−0.421170	−0.210585	+	0.977576i	$0.567537\pi$
$434$	0	0
$435$	17.7954	0.853223
$436$	0	0
$437$	13.7381	0.657181
$438$	0	0
$439$	−32.6631	−1.55892	−0.779462	−	0.626450i	$-0.784507\pi$
$439$	−32.6631	−1.55892	−0.779462	+	0.626450i	$0.784507\pi$
$440$	0	0
$441$	−6.35577	−0.302656
$442$	0	0
$443$	10.5636	0.501894	0.250947	−	0.968001i	$-0.419258\pi$
$443$	10.5636	0.501894	0.250947	+	0.968001i	$0.419258\pi$
$444$	0	0
$445$	−25.3615	−1.20225
$446$	0	0
$447$	−23.9654	−1.13352
$448$	0	0
$449$	4.46969	0.210938	0.105469	−	0.994423i	$-0.466366\pi$
$449$	4.46969	0.210938	0.105469	+	0.994423i	$0.466366\pi$
$450$	0	0
$451$	4.42620	0.208421
$452$	0	0
$453$	−4.74051	−0.222728
$454$	0	0
$455$	10.4688	0.490786
$456$	0	0
$457$	22.9914	1.07549	0.537745	−	0.843107i	$-0.319276\pi$
$457$	22.9914	1.07549	0.537745	+	0.843107i	$0.319276\pi$
$458$	0	0
$459$	−2.02477	−0.0945081
$460$	0	0
$461$	7.77260	0.362006	0.181003	−	0.983483i	$-0.442066\pi$
$461$	7.77260	0.362006	0.181003	+	0.983483i	$0.442066\pi$
$462$	0	0
$463$	−18.5847	−0.863705	−0.431853	−	0.901944i	$-0.642140\pi$
$463$	−18.5847	−0.863705	−0.431853	+	0.901944i	$0.642140\pi$
$464$	0	0
$465$	3.88346	0.180091
$466$	0	0
$467$	16.4040	0.759088	0.379544	−	0.925174i	$-0.376081\pi$
$467$	16.4040	0.759088	0.379544	+	0.925174i	$0.376081\pi$
$468$	0	0
$469$	6.95733	0.321260
$470$	0	0
$471$	−12.2918	−0.566375
$472$	0	0
$473$	−4.50832	−0.207293
$474$	0	0
$475$	4.11217	0.188679
$476$	0	0
$477$	9.25602	0.423804
$478$	0	0
$479$	4.17459	0.190742	0.0953709	−	0.995442i	$-0.469596\pi$
$479$	4.17459	0.190742	0.0953709	+	0.995442i	$0.469596\pi$
$480$	0	0
$481$	−61.1291	−2.78725
$482$	0	0
$483$	−1.36238	−0.0619904
$484$	0	0
$485$	29.2167	1.32666
$486$	0	0
$487$	−8.01182	−0.363050	−0.181525	−	0.983386i	$-0.558103\pi$
$487$	−8.01182	−0.363050	−0.181525	+	0.983386i	$0.558103\pi$
$488$	0	0
$489$	−14.4990	−0.655666
$490$	0	0
$491$	19.5821	0.883730	0.441865	−	0.897082i	$-0.354317\pi$
$491$	19.5821	0.883730	0.441865	+	0.897082i	$0.354317\pi$
$492$	0	0
$493$	17.0007	0.765672
$494$	0	0
$495$	3.24405	0.145809
$496$	0	0
$497$	6.42677	0.288280
$498$	0	0
$499$	1.25191	0.0560432	0.0280216	−	0.999607i	$-0.491079\pi$
$499$	1.25191	0.0560432	0.0280216	+	0.999607i	$0.491079\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−20.5341	−0.915572	−0.457786	−	0.889062i	$-0.651357\pi$
$503$	−20.5341	−0.915572	−0.457786	+	0.889062i	$0.651357\pi$
$504$	0	0
$505$	−28.0400	−1.24777
$506$	0	0
$507$	−24.8723	−1.10462
$508$	0	0
$509$	−21.7060	−0.962102	−0.481051	−	0.876693i	$-0.659745\pi$
$509$	−21.7060	−0.962102	−0.481051	+	0.876693i	$0.659745\pi$
$510$	0	0
$511$	−1.22446	−0.0541670
$512$	0	0
$513$	8.09372	0.357347
$514$	0	0
$515$	20.3097	0.894952
$516$	0	0
$517$	11.9572	0.525876
$518$	0	0
$519$	14.2287	0.624570
$520$	0	0
$521$	31.3324	1.37270	0.686350	−	0.727272i	$-0.259212\pi$
$521$	31.3324	1.37270	0.686350	+	0.727272i	$0.259212\pi$
$522$	0	0
$523$	13.8582	0.605979	0.302989	−	0.952994i	$-0.402015\pi$
$523$	13.8582	0.605979	0.302989	+	0.952994i	$0.402015\pi$
$524$	0	0
$525$	−0.407796	−0.0177977
$526$	0	0
$527$	3.71003	0.161612
$528$	0	0
$529$	−20.1189	−0.874736
$530$	0	0
$531$	5.76308	0.250096
$532$	0	0
$533$	−17.7959	−0.770826
$534$	0	0
$535$	−6.76136	−0.292319
$536$	0	0
$537$	−5.46854	−0.235985
$538$	0	0
$539$	9.72835	0.419030
$540$	0	0
$541$	8.41986	0.361998	0.180999	−	0.983483i	$-0.442067\pi$
$541$	8.41986	0.361998	0.180999	+	0.983483i	$0.442067\pi$
$542$	0	0
$543$	−8.71174	−0.373856
$544$	0	0
$545$	28.7125	1.22991
$546$	0	0
$547$	−32.8513	−1.40462	−0.702310	−	0.711871i	$-0.747848\pi$
$547$	−32.8513	−1.40462	−0.702310	+	0.711871i	$0.747848\pi$
$548$	0	0
$549$	−0.0936477	−0.00399679
$550$	0	0
$551$	−67.9577	−2.89510
$552$	0	0
$553$	−12.4976	−0.531453
$554$	0	0
$555$	−21.0525	−0.893629
$556$	0	0
$557$	−41.3326	−1.75132	−0.875659	−	0.482929i	$-0.839573\pi$
$557$	−41.3326	−1.75132	−0.875659	+	0.482929i	$0.839573\pi$
$558$	0	0
$559$	18.1261	0.766651
$560$	0	0
$561$	3.09918	0.130847
$562$	0	0
$563$	29.5494	1.24536	0.622680	−	0.782477i	$-0.286044\pi$
$563$	29.5494	1.24536	0.622680	+	0.782477i	$0.286044\pi$
$564$	0	0
$565$	−13.6114	−0.572637
$566$	0	0
$567$	−0.802640	−0.0337077
$568$	0	0
$569$	28.3368	1.18794	0.593971	−	0.804487i	$-0.297559\pi$
$569$	28.3368	1.18794	0.593971	+	0.804487i	$0.297559\pi$
$570$	0	0
$571$	26.1544	1.09453	0.547265	−	0.836960i	$-0.315669\pi$
$571$	26.1544	1.09453	0.547265	+	0.836960i	$0.315669\pi$
$572$	0	0
$573$	6.84326	0.285881
$574$	0	0
$575$	0.862382	0.0359638
$576$	0	0
$577$	7.89822	0.328807	0.164404	−	0.986393i	$-0.447430\pi$
$577$	7.89822	0.328807	0.164404	+	0.986393i	$0.447430\pi$
$578$	0	0
$579$	−11.6905	−0.485840
$580$	0	0
$581$	−11.0969	−0.460378
$582$	0	0
$583$	−14.1676	−0.586761
$584$	0	0
$585$	−13.0430	−0.539261
$586$	0	0
$587$	35.8279	1.47878	0.739388	−	0.673280i	$-0.235115\pi$
$587$	35.8279	1.47878	0.739388	+	0.673280i	$0.235115\pi$
$588$	0	0
$589$	−14.8303	−0.611073
$590$	0	0
$591$	−20.4537	−0.841353
$592$	0	0
$593$	−21.5505	−0.884974	−0.442487	−	0.896775i	$-0.645904\pi$
$593$	−21.5505	−0.884974	−0.442487	+	0.896775i	$0.645904\pi$
$594$	0	0
$595$	3.44439	0.141206
$596$	0	0
$597$	−19.5826	−0.801462
$598$	0	0
$599$	−16.3883	−0.669608	−0.334804	−	0.942288i	$-0.608670\pi$
$599$	−16.3883	−0.669608	−0.334804	+	0.942288i	$0.608670\pi$
$600$	0	0
$601$	−17.5147	−0.714439	−0.357220	−	0.934020i	$-0.616275\pi$
$601$	−17.5147	−0.714439	−0.357220	+	0.934020i	$0.616275\pi$
$602$	0	0
$603$	−8.66806	−0.352991
$604$	0	0
$605$	18.3481	0.745958
$606$	0	0
$607$	6.07723	0.246667	0.123334	−	0.992365i	$-0.460641\pi$
$607$	6.07723	0.246667	0.123334	+	0.992365i	$0.460641\pi$
$608$	0	0
$609$	6.73924	0.273088
$610$	0	0
$611$	−48.0749	−1.94490
$612$	0	0
$613$	−18.9440	−0.765140	−0.382570	−	0.923927i	$-0.624961\pi$
$613$	−18.9440	−0.765140	−0.382570	+	0.923927i	$0.624961\pi$
$614$	0	0
$615$	−6.12881	−0.247137
$616$	0	0
$617$	−24.1057	−0.970458	−0.485229	−	0.874387i	$-0.661264\pi$
$617$	−24.1057	−0.970458	−0.485229	+	0.874387i	$0.661264\pi$
$618$	0	0
$619$	10.4384	0.419555	0.209777	−	0.977749i	$-0.432726\pi$
$619$	10.4384	0.419555	0.209777	+	0.977749i	$0.432726\pi$
$620$	0	0
$621$	1.69737	0.0681133
$622$	0	0
$623$	−9.60460	−0.384800
$624$	0	0
$625$	−22.2015	−0.888061
$626$	0	0
$627$	−12.3885	−0.494750
$628$	0	0
$629$	−20.1123	−0.801932
$630$	0	0
$631$	−7.14705	−0.284519	−0.142260	−	0.989829i	$-0.545437\pi$
$631$	−7.14705	−0.284519	−0.142260	+	0.989829i	$0.545437\pi$
$632$	0	0
$633$	12.0070	0.477237
$634$	0	0
$635$	−14.6913	−0.583006
$636$	0	0
$637$	−39.1137	−1.54974
$638$	0	0
$639$	−8.00704	−0.316754
$640$	0	0
$641$	−41.9265	−1.65600	−0.827998	−	0.560731i	$-0.810520\pi$
$641$	−41.9265	−1.65600	−0.827998	+	0.560731i	$0.810520\pi$
$642$	0	0
$643$	41.0956	1.62065	0.810327	−	0.585978i	$-0.199289\pi$
$643$	41.0956	1.62065	0.810327	+	0.585978i	$0.199289\pi$
$644$	0	0
$645$	6.24252	0.245799
$646$	0	0
$647$	44.0967	1.73362	0.866810	−	0.498638i	$-0.166166\pi$
$647$	44.0967	1.73362	0.866810	+	0.498638i	$0.166166\pi$
$648$	0	0
$649$	−8.82116	−0.346261
$650$	0	0
$651$	1.47070	0.0576411
$652$	0	0
$653$	−18.0236	−0.705318	−0.352659	−	0.935752i	$-0.614722\pi$
$653$	−18.0236	−0.705318	−0.352659	+	0.935752i	$0.614722\pi$
$654$	0	0
$655$	11.9291	0.466108
$656$	0	0
$657$	1.52554	0.0595172
$658$	0	0
$659$	−3.10967	−0.121135	−0.0605677	−	0.998164i	$-0.519291\pi$
$659$	−3.10967	−0.121135	−0.0605677	+	0.998164i	$0.519291\pi$
$660$	0	0
$661$	49.6200	1.93000	0.964998	−	0.262259i	$-0.0844674\pi$
$661$	49.6200	1.93000	0.964998	+	0.262259i	$0.0844674\pi$
$662$	0	0
$663$	−12.4605	−0.483926
$664$	0	0
$665$	−13.7685	−0.533918
$666$	0	0
$667$	−14.2517	−0.551830
$668$	0	0
$669$	−20.0988	−0.777064
$670$	0	0
$671$	0.143340	0.00553359
$672$	0	0
$673$	15.6182	0.602039	0.301019	−	0.953618i	$-0.402673\pi$
$673$	15.6182	0.602039	0.301019	+	0.953618i	$0.402673\pi$
$674$	0	0
$675$	0.508068	0.0195556
$676$	0	0
$677$	49.8253	1.91494	0.957471	−	0.288529i	$-0.0931663\pi$
$677$	49.8253	1.91494	0.957471	+	0.288529i	$0.0931663\pi$
$678$	0	0
$679$	11.0646	0.424620
$680$	0	0
$681$	−14.1963	−0.544004
$682$	0	0
$683$	11.6091	0.444210	0.222105	−	0.975023i	$-0.428707\pi$
$683$	11.6091	0.444210	0.222105	+	0.975023i	$0.428707\pi$
$684$	0	0
$685$	−2.77398	−0.105988
$686$	0	0
$687$	−19.7750	−0.754465
$688$	0	0
$689$	56.9620	2.17008
$690$	0	0
$691$	−18.2300	−0.693501	−0.346751	−	0.937957i	$-0.612715\pi$
$691$	−18.2300	−0.693501	−0.346751	+	0.937957i	$0.612715\pi$
$692$	0	0
$693$	1.22855	0.0466686
$694$	0	0
$695$	−35.8803	−1.36102
$696$	0	0
$697$	−5.85511	−0.221778
$698$	0	0
$699$	−9.37536	−0.354609
$700$	0	0
$701$	22.9534	0.866937	0.433468	−	0.901169i	$-0.357290\pi$
$701$	22.9534	0.866937	0.433468	+	0.901169i	$0.357290\pi$
$702$	0	0
$703$	80.3962	3.03220
$704$	0	0
$705$	−16.5567	−0.623562
$706$	0	0
$707$	−10.6190	−0.399368
$708$	0	0
$709$	−43.5154	−1.63425	−0.817127	−	0.576458i	$-0.804434\pi$
$709$	−43.5154	−1.63425	−0.817127	+	0.576458i	$0.804434\pi$
$710$	0	0
$711$	15.5706	0.583945
$712$	0	0
$713$	−3.11014	−0.116476
$714$	0	0
$715$	19.9640	0.746613
$716$	0	0
$717$	11.0467	0.412546
$718$	0	0
$719$	50.0902	1.86805	0.934026	−	0.357206i	$-0.116271\pi$
$719$	50.0902	1.86805	0.934026	+	0.357206i	$0.116271\pi$
$720$	0	0
$721$	7.69143	0.286444
$722$	0	0
$723$	−3.14465	−0.116951
$724$	0	0
$725$	−4.26592	−0.158432
$726$	0	0
$727$	−25.8400	−0.958354	−0.479177	−	0.877718i	$-0.659065\pi$
$727$	−25.8400	−0.958354	−0.479177	+	0.877718i	$0.659065\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.96374	0.220577
$732$	0	0
$733$	12.2488	0.452420	0.226210	−	0.974079i	$-0.427366\pi$
$733$	12.2488	0.452420	0.226210	+	0.974079i	$0.427366\pi$
$734$	0	0
$735$	−13.4705	−0.496868
$736$	0	0
$737$	13.2676	0.488719
$738$	0	0
$739$	−47.4080	−1.74393	−0.871966	−	0.489567i	$-0.837155\pi$
$739$	−47.4080	−1.74393	−0.871966	+	0.489567i	$0.837155\pi$
$740$	0	0
$741$	49.8091	1.82978
$742$	0	0
$743$	0.720168	0.0264204	0.0132102	−	0.999913i	$-0.495795\pi$
$743$	0.720168	0.0264204	0.0132102	+	0.999913i	$0.495795\pi$
$744$	0	0
$745$	−50.7926	−1.86090
$746$	0	0
$747$	13.8255	0.505850
$748$	0	0
$749$	−2.56058	−0.0935614
$750$	0	0
$751$	16.7424	0.610940	0.305470	−	0.952202i	$-0.401186\pi$
$751$	16.7424	0.610940	0.305470	+	0.952202i	$0.401186\pi$
$752$	0	0
$753$	−19.4249	−0.707882
$754$	0	0
$755$	−10.0471	−0.365652
$756$	0	0
$757$	−36.0703	−1.31100	−0.655498	−	0.755197i	$-0.727541\pi$
$757$	−36.0703	−1.31100	−0.655498	+	0.755197i	$0.727541\pi$
$758$	0	0
$759$	−2.59806	−0.0943035
$760$	0	0
$761$	−10.3511	−0.375226	−0.187613	−	0.982243i	$-0.560075\pi$
$761$	−10.3511	−0.375226	−0.187613	+	0.982243i	$0.560075\pi$
$762$	0	0
$763$	10.8736	0.393652
$764$	0	0
$765$	−4.29133	−0.155153
$766$	0	0
$767$	35.4662	1.28061
$768$	0	0
$769$	−27.2909	−0.984134	−0.492067	−	0.870557i	$-0.663758\pi$
$769$	−27.2909	−0.984134	−0.492067	+	0.870557i	$0.663758\pi$
$770$	0	0
$771$	−24.3783	−0.877962
$772$	0	0
$773$	44.8338	1.61256	0.806279	−	0.591535i	$-0.201478\pi$
$773$	44.8338	1.61256	0.806279	+	0.591535i	$0.201478\pi$
$774$	0	0
$775$	−0.930946	−0.0334406
$776$	0	0
$777$	−7.97274	−0.286021
$778$	0	0
$779$	23.4050	0.838570
$780$	0	0
$781$	12.2558	0.438548
$782$	0	0
$783$	−8.39635	−0.300061
$784$	0	0
$785$	−26.0514	−0.929815
$786$	0	0
$787$	−21.8489	−0.778830	−0.389415	−	0.921062i	$-0.627323\pi$
$787$	−21.8489	−0.778830	−0.389415	+	0.921062i	$0.627323\pi$
$788$	0	0
$789$	19.1381	0.681333
$790$	0	0
$791$	−5.15475	−0.183282
$792$	0	0
$793$	−0.576312	−0.0204654
$794$	0	0
$795$	19.6174	0.695756
$796$	0	0
$797$	19.1416	0.678030	0.339015	−	0.940781i	$-0.389906\pi$
$797$	19.1416	0.678030	0.339015	+	0.940781i	$0.389906\pi$
$798$	0	0
$799$	−15.8173	−0.559577
$800$	0	0
$801$	11.9663	0.422807
$802$	0	0
$803$	−2.33505	−0.0824021
$804$	0	0
$805$	−2.88745	−0.101769
$806$	0	0
$807$	31.8583	1.12147
$808$	0	0
$809$	−47.0992	−1.65592	−0.827960	−	0.560787i	$-0.810499\pi$
$809$	−47.0992	−1.65592	−0.827960	+	0.560787i	$0.810499\pi$
$810$	0	0
$811$	31.1864	1.09510	0.547552	−	0.836772i	$-0.315560\pi$
$811$	31.1864	1.09510	0.547552	+	0.836772i	$0.315560\pi$
$812$	0	0
$813$	27.6866	0.971013
$814$	0	0
$815$	−30.7294	−1.07640
$816$	0	0
$817$	−23.8392	−0.834028
$818$	0	0
$819$	−4.93948	−0.172599
$820$	0	0
$821$	−26.0414	−0.908851	−0.454425	−	0.890785i	$-0.650155\pi$
$821$	−26.0414	−0.908851	−0.454425	+	0.890785i	$0.650155\pi$
$822$	0	0
$823$	−32.8099	−1.14368	−0.571841	−	0.820364i	$-0.693771\pi$
$823$	−32.8099	−1.14368	−0.571841	+	0.820364i	$0.693771\pi$
$824$	0	0
$825$	−0.777666	−0.0270749
$826$	0	0
$827$	−7.60622	−0.264494	−0.132247	−	0.991217i	$-0.542219\pi$
$827$	−7.60622	−0.264494	−0.132247	+	0.991217i	$0.542219\pi$
$828$	0	0
$829$	20.5841	0.714917	0.357458	−	0.933929i	$-0.383643\pi$
$829$	20.5841	0.714917	0.357458	+	0.933929i	$0.383643\pi$
$830$	0	0
$831$	−16.5531	−0.574222
$832$	0	0
$833$	−12.8690	−0.445883
$834$	0	0
$835$	−2.11942	−0.0733455
$836$	0	0
$837$	−1.83232	−0.0633344
$838$	0	0
$839$	−51.8511	−1.79010	−0.895050	−	0.445966i	$-0.852860\pi$
$839$	−51.8511	−1.79010	−0.895050	+	0.445966i	$0.852860\pi$
$840$	0	0
$841$	41.4987	1.43099
$842$	0	0
$843$	16.1221	0.555276
$844$	0	0
$845$	−52.7147	−1.81344
$846$	0	0
$847$	6.94858	0.238756
$848$	0	0
$849$	15.4620	0.530653
$850$	0	0
$851$	16.8603	0.577963
$852$	0	0
$853$	−39.0857	−1.33827	−0.669135	−	0.743141i	$-0.733335\pi$
$853$	−39.0857	−1.33827	−0.669135	+	0.743141i	$0.733335\pi$
$854$	0	0
$855$	17.1540	0.586654
$856$	0	0
$857$	−28.2522	−0.965077	−0.482538	−	0.875875i	$-0.660285\pi$
$857$	−28.2522	−0.965077	−0.482538	+	0.875875i	$0.660285\pi$
$858$	0	0
$859$	−51.1161	−1.74406	−0.872030	−	0.489453i	$-0.837196\pi$
$859$	−51.1161	−1.74406	−0.872030	+	0.489453i	$0.837196\pi$
$860$	0	0
$861$	−2.32103	−0.0791004
$862$	0	0
$863$	−5.24574	−0.178567	−0.0892835	−	0.996006i	$-0.528458\pi$
$863$	−5.24574	−0.178567	−0.0892835	+	0.996006i	$0.528458\pi$
$864$	0	0
$865$	30.1565	1.02535
$866$	0	0
$867$	12.9003	0.438118
$868$	0	0
$869$	−23.8330	−0.808478
$870$	0	0
$871$	−53.3437	−1.80748
$872$	0	0
$873$	−13.7852	−0.466560
$874$	0	0
$875$	−9.36993	−0.316762
$876$	0	0
$877$	−3.27093	−0.110451	−0.0552257	−	0.998474i	$-0.517588\pi$
$877$	−3.27093	−0.110451	−0.0552257	+	0.998474i	$0.517588\pi$
$878$	0	0
$879$	13.3824	0.451376
$880$	0	0
$881$	7.23440	0.243733	0.121867	−	0.992546i	$-0.461112\pi$
$881$	7.23440	0.243733	0.121867	+	0.992546i	$0.461112\pi$
$882$	0	0
$883$	5.28825	0.177964	0.0889820	−	0.996033i	$-0.471639\pi$
$883$	5.28825	0.177964	0.0889820	+	0.996033i	$0.471639\pi$
$884$	0	0
$885$	12.2144	0.410582
$886$	0	0
$887$	−4.41900	−0.148376	−0.0741878	−	0.997244i	$-0.523636\pi$
$887$	−4.41900	−0.148376	−0.0741878	+	0.997244i	$0.523636\pi$
$888$	0	0
$889$	−5.56370	−0.186601
$890$	0	0
$891$	−1.53063	−0.0512781
$892$	0	0
$893$	63.2275	2.11583
$894$	0	0
$895$	−11.5901	−0.387415
$896$	0	0
$897$	10.4457	0.348772
$898$	0	0
$899$	15.3848	0.513113
$900$	0	0
$901$	18.7413	0.624363
$902$	0	0
$903$	2.36409	0.0786719
$904$	0	0
$905$	−18.4638	−0.613758
$906$	0	0
$907$	27.3464	0.908022	0.454011	−	0.890996i	$-0.349993\pi$
$907$	27.3464	0.908022	0.454011	+	0.890996i	$0.349993\pi$
$908$	0	0
$909$	13.2301	0.438814
$910$	0	0
$911$	21.3819	0.708415	0.354207	−	0.935167i	$-0.384751\pi$
$911$	21.3819	0.708415	0.354207	+	0.935167i	$0.384751\pi$
$912$	0	0
$913$	−21.1618	−0.700355
$914$	0	0
$915$	−0.198479	−0.00656150
$916$	0	0
$917$	4.51764	0.149186
$918$	0	0
$919$	50.1624	1.65471	0.827353	−	0.561682i	$-0.189846\pi$
$919$	50.1624	1.65471	0.827353	+	0.561682i	$0.189846\pi$
$920$	0	0
$921$	10.3905	0.342379
$922$	0	0
$923$	−49.2757	−1.62193
$924$	0	0
$925$	5.04672	0.165935
$926$	0	0
$927$	−9.58267	−0.314736
$928$	0	0
$929$	−46.5655	−1.52777	−0.763883	−	0.645355i	$-0.776709\pi$
$929$	−46.5655	−1.52777	−0.763883	+	0.645355i	$0.776709\pi$
$930$	0	0
$931$	51.4418	1.68594
$932$	0	0
$933$	−5.04038	−0.165015
$934$	0	0
$935$	6.56845	0.214811
$936$	0	0
$937$	23.1861	0.757457	0.378728	−	0.925508i	$-0.376361\pi$
$937$	23.1861	0.757457	0.378728	+	0.925508i	$0.376361\pi$
$938$	0	0
$939$	−12.2609	−0.400118
$940$	0	0
$941$	−28.8102	−0.939186	−0.469593	−	0.882883i	$-0.655599\pi$
$941$	−28.8102	−0.939186	−0.469593	+	0.882883i	$0.655599\pi$
$942$	0	0
$943$	4.90837	0.159838
$944$	0	0
$945$	−1.70113	−0.0553377
$946$	0	0
$947$	27.1110	0.880989	0.440495	−	0.897755i	$-0.354803\pi$
$947$	27.1110	0.880989	0.440495	+	0.897755i	$0.354803\pi$
$948$	0	0
$949$	9.38827	0.304756
$950$	0	0
$951$	15.7166	0.509645
$952$	0	0
$953$	54.3342	1.76006	0.880028	−	0.474922i	$-0.157524\pi$
$953$	54.3342	1.76006	0.880028	+	0.474922i	$0.157524\pi$
$954$	0	0
$955$	14.5037	0.469329
$956$	0	0
$957$	12.8517	0.415438
$958$	0	0
$959$	−1.05053	−0.0339233
$960$	0	0
$961$	−27.6426	−0.891696
$962$	0	0
$963$	3.19020	0.102803
$964$	0	0
$965$	−24.7770	−0.797600
$966$	0	0
$967$	33.3646	1.07293	0.536466	−	0.843922i	$-0.319759\pi$
$967$	33.3646	1.07293	0.536466	+	0.843922i	$0.319759\pi$
$968$	0	0
$969$	16.3879	0.526456
$970$	0	0
$971$	56.5247	1.81396	0.906982	−	0.421170i	$-0.138380\pi$
$971$	56.5247	1.81396	0.906982	+	0.421170i	$0.138380\pi$
$972$	0	0
$973$	−13.5881	−0.435616
$974$	0	0
$975$	3.12668	0.100134
$976$	0	0
$977$	15.7001	0.502292	0.251146	−	0.967949i	$-0.419193\pi$
$977$	15.7001	0.502292	0.251146	+	0.967949i	$0.419193\pi$
$978$	0	0
$979$	−18.3160	−0.585381
$980$	0	0
$981$	−13.5473	−0.432533
$982$	0	0
$983$	16.5877	0.529066	0.264533	−	0.964377i	$-0.414782\pi$
$983$	16.5877	0.529066	0.264533	+	0.964377i	$0.414782\pi$
$984$	0	0
$985$	−43.3500	−1.38124
$986$	0	0
$987$	−6.27015	−0.199581
$988$	0	0
$989$	−4.99943	−0.158973
$990$	0	0
$991$	−34.8301	−1.10641	−0.553207	−	0.833044i	$-0.686596\pi$
$991$	−34.8301	−1.10641	−0.553207	+	0.833044i	$0.686596\pi$
$992$	0	0
$993$	18.2687	0.579740
$994$	0	0
$995$	−41.5037	−1.31576
$996$	0	0
$997$	−45.7106	−1.44767	−0.723834	−	0.689974i	$-0.757622\pi$
$997$	−45.7106	−1.44767	−0.723834	+	0.689974i	$0.757622\pi$
$998$	0	0
$999$	9.93315	0.314271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.k.1.2	✓	11
4.3	odd	2		8016.2.a.be.1.2		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.k.1.2	✓	11	1.1	even	1	trivial
8016.2.a.be.1.2		11	4.3	odd	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.11942 q^{5} -0.802640 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.11942 q^{5} -0.802640 q^{7} +1.00000 q^{9} -1.53063 q^{11} +6.15404 q^{13} +2.11942 q^{15} +2.02477 q^{17} -8.09372 q^{19} +0.802640 q^{21} -1.69737 q^{23} -0.508068 q^{25} -1.00000 q^{27} +8.39635 q^{29} +1.83232 q^{31} +1.53063 q^{33} +1.70113 q^{35} -9.93315 q^{37} -6.15404 q^{39} -2.89174 q^{41} +2.94539 q^{43} -2.11942 q^{45} -7.81191 q^{47} -6.35577 q^{49} -2.02477 q^{51} +9.25602 q^{53} +3.24405 q^{55} +8.09372 q^{57} +5.76308 q^{59} -0.0936477 q^{61} -0.802640 q^{63} -13.0430 q^{65} -8.66806 q^{67} +1.69737 q^{69} -8.00704 q^{71} +1.52554 q^{73} +0.508068 q^{75} +1.22855 q^{77} +15.5706 q^{79} +1.00000 q^{81} +13.8255 q^{83} -4.29133 q^{85} -8.39635 q^{87} +11.9663 q^{89} -4.93948 q^{91} -1.83232 q^{93} +17.1540 q^{95} -13.7852 q^{97} -1.53063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9	\( 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9} - q^{11} + 10 q^{13} - 10 q^{15} + 17 q^{17} + 2 q^{19} + q^{21} - 3 q^{23} + 21 q^{25} - 11 q^{27} + 17 q^{29} - 15 q^{31} + q^{33} + 11 q^{35} + 4 q^{37} - 10 q^{39} + 16 q^{41} + 10 q^{43} + 10 q^{45} - 16 q^{47} + 22 q^{49} - 17 q^{51} + 42 q^{53} - 5 q^{55} - 2 q^{57} - 2 q^{59} + 12 q^{61} - q^{63} + 10 q^{65} - q^{67} + 3 q^{69} - 9 q^{71} + 24 q^{73} - 21 q^{75} + 22 q^{77} - 30 q^{79} + 11 q^{81} + 16 q^{83} + 25 q^{85} - 17 q^{87} + 37 q^{89} + q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97} - q^{99}+O(q^{100}) \) 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9 - q^11 + 10 * q^13 - 10 * q^15 + 17 * q^17 + 2 * q^19 + q^21 - 3 * q^23 + 21 * q^25 - 11 * q^27 + 17 * q^29 - 15 * q^31 + q^33 + 11 * q^35 + 4 * q^37 - 10 * q^39 + 16 * q^41 + 10 * q^43 + 10 * q^45 - 16 * q^47 + 22 * q^49 - 17 * q^51 + 42 * q^53 - 5 * q^55 - 2 * q^57 - 2 * q^59 + 12 * q^61 - q^63 + 10 * q^65 - q^67 + 3 * q^69 - 9 * q^71 + 24 * q^73 - 21 * q^75 + 22 * q^77 - 30 * q^79 + 11 * q^81 + 16 * q^83 + 25 * q^85 - 17 * q^87 + 37 * q^89 + q^91 + 15 * q^93 + 5 * q^95 + 4 * q^97 - q^99

Introduction

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