LMFDB - Embedded newform 4008.2.a.k.1.9

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			1.9
Root			$-2.34947$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.34947 q^{5} +3.54581 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.34947 q^{5} +3.54581 q^{7} +1.00000 q^{9} -5.22121 q^{11} +5.27416 q^{13} -3.34947 q^{15} +6.86903 q^{17} +5.25090 q^{19} -3.54581 q^{21} +2.32838 q^{23} +6.21896 q^{25} -1.00000 q^{27} -0.922527 q^{29} -10.9921 q^{31} +5.22121 q^{33} +11.8766 q^{35} -10.2033 q^{37} -5.27416 q^{39} +2.58506 q^{41} +4.38272 q^{43} +3.34947 q^{45} +6.30219 q^{47} +5.57274 q^{49} -6.86903 q^{51} +10.7064 q^{53} -17.4883 q^{55} -5.25090 q^{57} +10.3875 q^{59} +10.7958 q^{61} +3.54581 q^{63} +17.6657 q^{65} -8.19936 q^{67} -2.32838 q^{69} -3.27323 q^{71} -9.70252 q^{73} -6.21896 q^{75} -18.5134 q^{77} -14.2308 q^{79} +1.00000 q^{81} -12.2173 q^{83} +23.0076 q^{85} +0.922527 q^{87} +1.83867 q^{89} +18.7011 q^{91} +10.9921 q^{93} +17.5878 q^{95} -10.0691 q^{97} -5.22121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.34947	1.49793	0.748965	−	0.662610i	$-0.230551\pi$
$5$	3.34947	1.49793	0.748965	+	0.662610i	$0.230551\pi$
$6$	0	0
$7$	3.54581	1.34019	0.670094	−	0.742276i	$-0.266254\pi$
$7$	3.54581	1.34019	0.670094	+	0.742276i	$0.266254\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.22121	−1.57425	−0.787127	−	0.616791i	$-0.788433\pi$
$11$	−5.22121	−1.57425	−0.787127	+	0.616791i	$0.788433\pi$
$12$	0	0
$13$	5.27416	1.46279	0.731394	−	0.681955i	$-0.238870\pi$
$13$	5.27416	1.46279	0.731394	+	0.681955i	$0.238870\pi$
$14$	0	0
$15$	−3.34947	−0.864830
$16$	0	0
$17$	6.86903	1.66598	0.832992	−	0.553285i	$-0.186626\pi$
$17$	6.86903	1.66598	0.832992	+	0.553285i	$0.186626\pi$
$18$	0	0
$19$	5.25090	1.20464	0.602320	−	0.798255i	$-0.294243\pi$
$19$	5.25090	1.20464	0.602320	+	0.798255i	$0.294243\pi$
$20$	0	0
$21$	−3.54581	−0.773758
$22$	0	0
$23$	2.32838	0.485500	0.242750	−	0.970089i	$-0.421950\pi$
$23$	2.32838	0.485500	0.242750	+	0.970089i	$0.421950\pi$
$24$	0	0
$25$	6.21896	1.24379
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.922527	−0.171309	−0.0856545	−	0.996325i	$-0.527298\pi$
$29$	−0.922527	−0.171309	−0.0856545	+	0.996325i	$0.527298\pi$
$30$	0	0
$31$	−10.9921	−1.97425	−0.987123	−	0.159963i	$-0.948863\pi$
$31$	−10.9921	−1.97425	−0.987123	+	0.159963i	$0.948863\pi$
$32$	0	0
$33$	5.22121	0.908896
$34$	0	0
$35$	11.8766	2.00751
$36$	0	0
$37$	−10.2033	−1.67742	−0.838708	−	0.544581i	$-0.816689\pi$
$37$	−10.2033	−1.67742	−0.838708	+	0.544581i	$0.816689\pi$
$38$	0	0
$39$	−5.27416	−0.844541
$40$	0	0
$41$	2.58506	0.403719	0.201859	−	0.979415i	$-0.435302\pi$
$41$	2.58506	0.403719	0.201859	+	0.979415i	$0.435302\pi$
$42$	0	0
$43$	4.38272	0.668359	0.334179	−	0.942509i	$-0.391541\pi$
$43$	4.38272	0.668359	0.334179	+	0.942509i	$0.391541\pi$
$44$	0	0
$45$	3.34947	0.499310
$46$	0	0
$47$	6.30219	0.919270	0.459635	−	0.888108i	$-0.347980\pi$
$47$	6.30219	0.919270	0.459635	+	0.888108i	$0.347980\pi$
$48$	0	0
$49$	5.57274	0.796106
$50$	0	0
$51$	−6.86903	−0.961857
$52$	0	0
$53$	10.7064	1.47064	0.735319	−	0.677721i	$-0.237032\pi$
$53$	10.7064	1.47064	0.735319	+	0.677721i	$0.237032\pi$
$54$	0	0
$55$	−17.4883	−2.35812
$56$	0	0
$57$	−5.25090	−0.695499
$58$	0	0
$59$	10.3875	1.35234	0.676169	−	0.736747i	$-0.263639\pi$
$59$	10.3875	1.35234	0.676169	+	0.736747i	$0.263639\pi$
$60$	0	0
$61$	10.7958	1.38226	0.691129	−	0.722732i	$-0.257114\pi$
$61$	10.7958	1.38226	0.691129	+	0.722732i	$0.257114\pi$
$62$	0	0
$63$	3.54581	0.446730
$64$	0	0
$65$	17.6657	2.19115
$66$	0	0
$67$	−8.19936	−1.00171	−0.500856	−	0.865531i	$-0.666981\pi$
$67$	−8.19936	−1.00171	−0.500856	+	0.865531i	$0.666981\pi$
$68$	0	0
$69$	−2.32838	−0.280304
$70$	0	0
$71$	−3.27323	−0.388460	−0.194230	−	0.980956i	$-0.562221\pi$
$71$	−3.27323	−0.388460	−0.194230	+	0.980956i	$0.562221\pi$
$72$	0	0
$73$	−9.70252	−1.13559	−0.567797	−	0.823169i	$-0.692204\pi$
$73$	−9.70252	−1.13559	−0.567797	+	0.823169i	$0.692204\pi$
$74$	0	0
$75$	−6.21896	−0.718104
$76$	0	0
$77$	−18.5134	−2.10980
$78$	0	0
$79$	−14.2308	−1.60109	−0.800543	−	0.599275i	$-0.795456\pi$
$79$	−14.2308	−1.60109	−0.800543	+	0.599275i	$0.795456\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.2173	−1.34103	−0.670513	−	0.741898i	$-0.733926\pi$
$83$	−12.2173	−1.34103	−0.670513	+	0.741898i	$0.733926\pi$
$84$	0	0
$85$	23.0076	2.49553
$86$	0	0
$87$	0.922527	0.0989053
$88$	0	0
$89$	1.83867	0.194899	0.0974494	−	0.995240i	$-0.468932\pi$
$89$	1.83867	0.194899	0.0974494	+	0.995240i	$0.468932\pi$
$90$	0	0
$91$	18.7011	1.96041
$92$	0	0
$93$	10.9921	1.13983
$94$	0	0
$95$	17.5878	1.80447
$96$	0	0
$97$	−10.0691	−1.02237	−0.511183	−	0.859472i	$-0.670792\pi$
$97$	−10.0691	−1.02237	−0.511183	+	0.859472i	$0.670792\pi$
$98$	0	0
$99$	−5.22121	−0.524752
$100$	0	0
$101$	3.62392	0.360593	0.180297	−	0.983612i	$-0.442294\pi$
$101$	3.62392	0.360593	0.180297	+	0.983612i	$0.442294\pi$
$102$	0	0
$103$	1.34478	0.132505	0.0662527	−	0.997803i	$-0.478896\pi$
$103$	1.34478	0.132505	0.0662527	+	0.997803i	$0.478896\pi$
$104$	0	0
$105$	−11.8766	−1.15904
$106$	0	0
$107$	5.14056	0.496957	0.248479	−	0.968637i	$-0.420069\pi$
$107$	5.14056	0.496957	0.248479	+	0.968637i	$0.420069\pi$
$108$	0	0
$109$	−8.05520	−0.771549	−0.385774	−	0.922593i	$-0.626066\pi$
$109$	−8.05520	−0.771549	−0.385774	+	0.922593i	$0.626066\pi$
$110$	0	0
$111$	10.2033	0.968457
$112$	0	0
$113$	−18.0162	−1.69482	−0.847412	−	0.530936i	$-0.821841\pi$
$113$	−18.0162	−1.69482	−0.847412	+	0.530936i	$0.821841\pi$
$114$	0	0
$115$	7.79884	0.727245
$116$	0	0
$117$	5.27416	0.487596
$118$	0	0
$119$	24.3563	2.23273
$120$	0	0
$121$	16.2611	1.47828
$122$	0	0
$123$	−2.58506	−0.233087
$124$	0	0
$125$	4.08289	0.365185
$126$	0	0
$127$	−6.40394	−0.568258	−0.284129	−	0.958786i	$-0.591704\pi$
$127$	−6.40394	−0.568258	−0.284129	+	0.958786i	$0.591704\pi$
$128$	0	0
$129$	−4.38272	−0.385877
$130$	0	0
$131$	17.2197	1.50449	0.752246	−	0.658882i	$-0.228970\pi$
$131$	17.2197	1.50449	0.752246	+	0.658882i	$0.228970\pi$
$132$	0	0
$133$	18.6187	1.61445
$134$	0	0
$135$	−3.34947	−0.288277
$136$	0	0
$137$	−16.8321	−1.43806	−0.719032	−	0.694977i	$-0.755415\pi$
$137$	−16.8321	−1.43806	−0.719032	+	0.694977i	$0.755415\pi$
$138$	0	0
$139$	−9.08472	−0.770556	−0.385278	−	0.922801i	$-0.625894\pi$
$139$	−9.08472	−0.770556	−0.385278	+	0.922801i	$0.625894\pi$
$140$	0	0
$141$	−6.30219	−0.530741
$142$	0	0
$143$	−27.5375	−2.30280
$144$	0	0
$145$	−3.08998	−0.256609
$146$	0	0
$147$	−5.57274	−0.459632
$148$	0	0
$149$	−10.5718	−0.866077	−0.433038	−	0.901376i	$-0.642559\pi$
$149$	−10.5718	−0.866077	−0.433038	+	0.901376i	$0.642559\pi$
$150$	0	0
$151$	9.42466	0.766968	0.383484	−	0.923548i	$-0.374724\pi$
$151$	9.42466	0.766968	0.383484	+	0.923548i	$0.374724\pi$
$152$	0	0
$153$	6.86903	0.555328
$154$	0	0
$155$	−36.8179	−2.95728
$156$	0	0
$157$	−11.0413	−0.881195	−0.440597	−	0.897705i	$-0.645233\pi$
$157$	−11.0413	−0.881195	−0.440597	+	0.897705i	$0.645233\pi$
$158$	0	0
$159$	−10.7064	−0.849073
$160$	0	0
$161$	8.25598	0.650662
$162$	0	0
$163$	0.204447	0.0160135	0.00800676	−	0.999968i	$-0.497451\pi$
$163$	0.204447	0.0160135	0.00800676	+	0.999968i	$0.497451\pi$
$164$	0	0
$165$	17.4883	1.36146
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	14.8168	1.13975
$170$	0	0
$171$	5.25090	0.401547
$172$	0	0
$173$	−5.27958	−0.401399	−0.200700	−	0.979653i	$-0.564322\pi$
$173$	−5.27958	−0.401399	−0.200700	+	0.979653i	$0.564322\pi$
$174$	0	0
$175$	22.0512	1.66692
$176$	0	0
$177$	−10.3875	−0.780773
$178$	0	0
$179$	19.1223	1.42926	0.714632	−	0.699501i	$-0.246594\pi$
$179$	19.1223	1.42926	0.714632	+	0.699501i	$0.246594\pi$
$180$	0	0
$181$	24.5626	1.82572	0.912861	−	0.408270i	$-0.133868\pi$
$181$	24.5626	1.82572	0.912861	+	0.408270i	$0.133868\pi$
$182$	0	0
$183$	−10.7958	−0.798046
$184$	0	0
$185$	−34.1758	−2.51265
$186$	0	0
$187$	−35.8647	−2.62268
$188$	0	0
$189$	−3.54581	−0.257919
$190$	0	0
$191$	11.5678	0.837016	0.418508	−	0.908213i	$-0.362553\pi$
$191$	11.5678	0.837016	0.418508	+	0.908213i	$0.362553\pi$
$192$	0	0
$193$	−4.06662	−0.292722	−0.146361	−	0.989231i	$-0.546756\pi$
$193$	−4.06662	−0.292722	−0.146361	+	0.989231i	$0.546756\pi$
$194$	0	0
$195$	−17.6657	−1.26506
$196$	0	0
$197$	24.7058	1.76021	0.880106	−	0.474777i	$-0.157471\pi$
$197$	24.7058	1.76021	0.880106	+	0.474777i	$0.157471\pi$
$198$	0	0
$199$	−6.95811	−0.493247	−0.246624	−	0.969111i	$-0.579321\pi$
$199$	−6.95811	−0.493247	−0.246624	+	0.969111i	$0.579321\pi$
$200$	0	0
$201$	8.19936	0.578338
$202$	0	0
$203$	−3.27110	−0.229586
$204$	0	0
$205$	8.65859	0.604742
$206$	0	0
$207$	2.32838	0.161833
$208$	0	0
$209$	−27.4161	−1.89641
$210$	0	0
$211$	−12.0883	−0.832196	−0.416098	−	0.909320i	$-0.636603\pi$
$211$	−12.0883	−0.832196	−0.416098	+	0.909320i	$0.636603\pi$
$212$	0	0
$213$	3.27323	0.224278
$214$	0	0
$215$	14.6798	1.00115
$216$	0	0
$217$	−38.9760	−2.64586
$218$	0	0
$219$	9.70252	0.655635
$220$	0	0
$221$	36.2284	2.43698
$222$	0	0
$223$	6.74178	0.451463	0.225732	−	0.974190i	$-0.427523\pi$
$223$	6.74178	0.451463	0.225732	+	0.974190i	$0.427523\pi$
$224$	0	0
$225$	6.21896	0.414598
$226$	0	0
$227$	−16.6461	−1.10484	−0.552419	−	0.833566i	$-0.686295\pi$
$227$	−16.6461	−1.10484	−0.552419	+	0.833566i	$0.686295\pi$
$228$	0	0
$229$	4.34626	0.287209	0.143604	−	0.989635i	$-0.454131\pi$
$229$	4.34626	0.287209	0.143604	+	0.989635i	$0.454131\pi$
$230$	0	0
$231$	18.5134	1.21809
$232$	0	0
$233$	15.4375	1.01135	0.505674	−	0.862725i	$-0.331244\pi$
$233$	15.4375	1.01135	0.505674	+	0.862725i	$0.331244\pi$
$234$	0	0
$235$	21.1090	1.37700
$236$	0	0
$237$	14.2308	0.924388
$238$	0	0
$239$	3.91114	0.252990	0.126495	−	0.991967i	$-0.459627\pi$
$239$	3.91114	0.252990	0.126495	+	0.991967i	$0.459627\pi$
$240$	0	0
$241$	7.89534	0.508584	0.254292	−	0.967128i	$-0.418158\pi$
$241$	7.89534	0.508584	0.254292	+	0.967128i	$0.418158\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	18.6657	1.19251
$246$	0	0
$247$	27.6941	1.76213
$248$	0	0
$249$	12.2173	0.774241
$250$	0	0
$251$	−9.57349	−0.604273	−0.302137	−	0.953265i	$-0.597700\pi$
$251$	−9.57349	−0.604273	−0.302137	+	0.953265i	$0.597700\pi$
$252$	0	0
$253$	−12.1570	−0.764301
$254$	0	0
$255$	−23.0076	−1.44079
$256$	0	0
$257$	−22.3318	−1.39302	−0.696511	−	0.717546i	$-0.745265\pi$
$257$	−22.3318	−1.39302	−0.696511	+	0.717546i	$0.745265\pi$
$258$	0	0
$259$	−36.1790	−2.24805
$260$	0	0
$261$	−0.922527	−0.0571030
$262$	0	0
$263$	19.8026	1.22108	0.610539	−	0.791986i	$-0.290953\pi$
$263$	19.8026	1.22108	0.610539	+	0.791986i	$0.290953\pi$
$264$	0	0
$265$	35.8608	2.20291
$266$	0	0
$267$	−1.83867	−0.112525
$268$	0	0
$269$	13.4173	0.818068	0.409034	−	0.912519i	$-0.365866\pi$
$269$	13.4173	0.818068	0.409034	+	0.912519i	$0.365866\pi$
$270$	0	0
$271$	−25.1007	−1.52476	−0.762379	−	0.647130i	$-0.775969\pi$
$271$	−25.1007	−1.52476	−0.762379	+	0.647130i	$0.775969\pi$
$272$	0	0
$273$	−18.7011	−1.13184
$274$	0	0
$275$	−32.4705	−1.95805
$276$	0	0
$277$	−3.87307	−0.232710	−0.116355	−	0.993208i	$-0.537121\pi$
$277$	−3.87307	−0.232710	−0.116355	+	0.993208i	$0.537121\pi$
$278$	0	0
$279$	−10.9921	−0.658082
$280$	0	0
$281$	−3.69139	−0.220210	−0.110105	−	0.993920i	$-0.535119\pi$
$281$	−3.69139	−0.220210	−0.110105	+	0.993920i	$0.535119\pi$
$282$	0	0
$283$	7.75352	0.460899	0.230449	−	0.973084i	$-0.425980\pi$
$283$	7.75352	0.460899	0.230449	+	0.973084i	$0.425980\pi$
$284$	0	0
$285$	−17.5878	−1.04181
$286$	0	0
$287$	9.16613	0.541059
$288$	0	0
$289$	30.1836	1.77550
$290$	0	0
$291$	10.0691	0.590263
$292$	0	0
$293$	−1.01242	−0.0591460	−0.0295730	−	0.999563i	$-0.509415\pi$
$293$	−1.01242	−0.0591460	−0.0295730	+	0.999563i	$0.509415\pi$
$294$	0	0
$295$	34.7927	2.02571
$296$	0	0
$297$	5.22121	0.302965
$298$	0	0
$299$	12.2802	0.710184
$300$	0	0
$301$	15.5403	0.895727
$302$	0	0
$303$	−3.62392	−0.208189
$304$	0	0
$305$	36.1601	2.07052
$306$	0	0
$307$	25.9773	1.48260	0.741301	−	0.671173i	$-0.234209\pi$
$307$	25.9773	1.48260	0.741301	+	0.671173i	$0.234209\pi$
$308$	0	0
$309$	−1.34478	−0.0765020
$310$	0	0
$311$	−10.2568	−0.581608	−0.290804	−	0.956783i	$-0.593923\pi$
$311$	−10.2568	−0.581608	−0.290804	+	0.956783i	$0.593923\pi$
$312$	0	0
$313$	13.2574	0.749351	0.374675	−	0.927156i	$-0.377754\pi$
$313$	13.2574	0.749351	0.374675	+	0.927156i	$0.377754\pi$
$314$	0	0
$315$	11.8766	0.669169
$316$	0	0
$317$	21.0416	1.18181	0.590907	−	0.806739i	$-0.298770\pi$
$317$	21.0416	1.18181	0.590907	+	0.806739i	$0.298770\pi$
$318$	0	0
$319$	4.81671	0.269684
$320$	0	0
$321$	−5.14056	−0.286918
$322$	0	0
$323$	36.0686	2.00691
$324$	0	0
$325$	32.7998	1.81941
$326$	0	0
$327$	8.05520	0.445454
$328$	0	0
$329$	22.3464	1.23199
$330$	0	0
$331$	11.4570	0.629735	0.314868	−	0.949136i	$-0.398040\pi$
$331$	11.4570	0.629735	0.314868	+	0.949136i	$0.398040\pi$
$332$	0	0
$333$	−10.2033	−0.559139
$334$	0	0
$335$	−27.4635	−1.50049
$336$	0	0
$337$	−4.81048	−0.262044	−0.131022	−	0.991379i	$-0.541826\pi$
$337$	−4.81048	−0.262044	−0.131022	+	0.991379i	$0.541826\pi$
$338$	0	0
$339$	18.0162	0.978507
$340$	0	0
$341$	57.3923	3.10797
$342$	0	0
$343$	−5.06078	−0.273256
$344$	0	0
$345$	−7.79884	−0.419875
$346$	0	0
$347$	10.5296	0.565258	0.282629	−	0.959229i	$-0.408793\pi$
$347$	10.5296	0.565258	0.282629	+	0.959229i	$0.408793\pi$
$348$	0	0
$349$	−8.11284	−0.434271	−0.217135	−	0.976142i	$-0.569671\pi$
$349$	−8.11284	−0.434271	−0.217135	+	0.976142i	$0.569671\pi$
$350$	0	0
$351$	−5.27416	−0.281514
$352$	0	0
$353$	16.7942	0.893867	0.446933	−	0.894567i	$-0.352516\pi$
$353$	16.7942	0.893867	0.446933	+	0.894567i	$0.352516\pi$
$354$	0	0
$355$	−10.9636	−0.581886
$356$	0	0
$357$	−24.3563	−1.28907
$358$	0	0
$359$	5.05875	0.266991	0.133495	−	0.991049i	$-0.457380\pi$
$359$	5.05875	0.266991	0.133495	+	0.991049i	$0.457380\pi$
$360$	0	0
$361$	8.57200	0.451158
$362$	0	0
$363$	−16.2611	−0.853484
$364$	0	0
$365$	−32.4983	−1.70104
$366$	0	0
$367$	18.3665	0.958723	0.479362	−	0.877617i	$-0.340868\pi$
$367$	18.3665	0.958723	0.479362	+	0.877617i	$0.340868\pi$
$368$	0	0
$369$	2.58506	0.134573
$370$	0	0
$371$	37.9628	1.97093
$372$	0	0
$373$	−12.5025	−0.647355	−0.323678	−	0.946167i	$-0.604919\pi$
$373$	−12.5025	−0.647355	−0.323678	+	0.946167i	$0.604919\pi$
$374$	0	0
$375$	−4.08289	−0.210839
$376$	0	0
$377$	−4.86555	−0.250589
$378$	0	0
$379$	−20.8449	−1.07073	−0.535365	−	0.844621i	$-0.679826\pi$
$379$	−20.8449	−1.07073	−0.535365	+	0.844621i	$0.679826\pi$
$380$	0	0
$381$	6.40394	0.328084
$382$	0	0
$383$	−31.4740	−1.60825	−0.804123	−	0.594463i	$-0.797365\pi$
$383$	−31.4740	−1.60825	−0.804123	+	0.594463i	$0.797365\pi$
$384$	0	0
$385$	−62.0101	−3.16033
$386$	0	0
$387$	4.38272	0.222786
$388$	0	0
$389$	8.57981	0.435014	0.217507	−	0.976059i	$-0.430208\pi$
$389$	8.57981	0.435014	0.217507	+	0.976059i	$0.430208\pi$
$390$	0	0
$391$	15.9937	0.808836
$392$	0	0
$393$	−17.2197	−0.868619
$394$	0	0
$395$	−47.6656	−2.39831
$396$	0	0
$397$	20.3543	1.02155	0.510776	−	0.859714i	$-0.329358\pi$
$397$	20.3543	1.02155	0.510776	+	0.859714i	$0.329358\pi$
$398$	0	0
$399$	−18.6187	−0.932100
$400$	0	0
$401$	16.1277	0.805378	0.402689	−	0.915337i	$-0.368076\pi$
$401$	16.1277	0.805378	0.402689	+	0.915337i	$0.368076\pi$
$402$	0	0
$403$	−57.9743	−2.88790
$404$	0	0
$405$	3.34947	0.166437
$406$	0	0
$407$	53.2737	2.64068
$408$	0	0
$409$	−18.1245	−0.896199	−0.448099	−	0.893984i	$-0.647899\pi$
$409$	−18.1245	−0.896199	−0.448099	+	0.893984i	$0.647899\pi$
$410$	0	0
$411$	16.8321	0.830267
$412$	0	0
$413$	36.8321	1.81239
$414$	0	0
$415$	−40.9216	−2.00876
$416$	0	0
$417$	9.08472	0.444881
$418$	0	0
$419$	−23.7981	−1.16261	−0.581307	−	0.813684i	$-0.697459\pi$
$419$	−23.7981	−1.16261	−0.581307	+	0.813684i	$0.697459\pi$
$420$	0	0
$421$	−14.4408	−0.703800	−0.351900	−	0.936038i	$-0.614464\pi$
$421$	−14.4408	−0.703800	−0.351900	+	0.936038i	$0.614464\pi$
$422$	0	0
$423$	6.30219	0.306423
$424$	0	0
$425$	42.7183	2.07214
$426$	0	0
$427$	38.2797	1.85249
$428$	0	0
$429$	27.5375	1.32952
$430$	0	0
$431$	−12.8595	−0.619420	−0.309710	−	0.950831i	$-0.600232\pi$
$431$	−12.8595	−0.619420	−0.309710	+	0.950831i	$0.600232\pi$
$432$	0	0
$433$	16.3060	0.783618	0.391809	−	0.920047i	$-0.371849\pi$
$433$	16.3060	0.783618	0.391809	+	0.920047i	$0.371849\pi$
$434$	0	0
$435$	3.08998	0.148153
$436$	0	0
$437$	12.2261	0.584853
$438$	0	0
$439$	10.2780	0.490544	0.245272	−	0.969454i	$-0.421123\pi$
$439$	10.2780	0.490544	0.245272	+	0.969454i	$0.421123\pi$
$440$	0	0
$441$	5.57274	0.265369
$442$	0	0
$443$	−32.8390	−1.56023	−0.780113	−	0.625638i	$-0.784839\pi$
$443$	−32.8390	−1.56023	−0.780113	+	0.625638i	$0.784839\pi$
$444$	0	0
$445$	6.15858	0.291945
$446$	0	0
$447$	10.5718	0.500030
$448$	0	0
$449$	37.3707	1.76363	0.881817	−	0.471592i	$-0.156321\pi$
$449$	37.3707	1.76363	0.881817	+	0.471592i	$0.156321\pi$
$450$	0	0
$451$	−13.4972	−0.635556
$452$	0	0
$453$	−9.42466	−0.442809
$454$	0	0
$455$	62.6390	2.93656
$456$	0	0
$457$	32.2479	1.50849	0.754246	−	0.656592i	$-0.228003\pi$
$457$	32.2479	1.50849	0.754246	+	0.656592i	$0.228003\pi$
$458$	0	0
$459$	−6.86903	−0.320619
$460$	0	0
$461$	−27.8346	−1.29639	−0.648193	−	0.761476i	$-0.724475\pi$
$461$	−27.8346	−1.29639	−0.648193	+	0.761476i	$0.724475\pi$
$462$	0	0
$463$	6.04994	0.281164	0.140582	−	0.990069i	$-0.455103\pi$
$463$	6.04994	0.281164	0.140582	+	0.990069i	$0.455103\pi$
$464$	0	0
$465$	36.8179	1.70739
$466$	0	0
$467$	17.8938	0.828028	0.414014	−	0.910271i	$-0.364126\pi$
$467$	17.8938	0.828028	0.414014	+	0.910271i	$0.364126\pi$
$468$	0	0
$469$	−29.0733	−1.34248
$470$	0	0
$471$	11.0413	0.508758
$472$	0	0
$473$	−22.8831	−1.05217
$474$	0	0
$475$	32.6552	1.49832
$476$	0	0
$477$	10.7064	0.490213
$478$	0	0
$479$	−14.3503	−0.655683	−0.327841	−	0.944733i	$-0.606321\pi$
$479$	−14.3503	−0.655683	−0.327841	+	0.944733i	$0.606321\pi$
$480$	0	0
$481$	−53.8140	−2.45371
$482$	0	0
$483$	−8.25598	−0.375660
$484$	0	0
$485$	−33.7263	−1.53143
$486$	0	0
$487$	−16.1796	−0.733170	−0.366585	−	0.930385i	$-0.619473\pi$
$487$	−16.1796	−0.733170	−0.366585	+	0.930385i	$0.619473\pi$
$488$	0	0
$489$	−0.204447	−0.00924541
$490$	0	0
$491$	33.6911	1.52046	0.760228	−	0.649656i	$-0.225087\pi$
$491$	33.6911	1.52046	0.760228	+	0.649656i	$0.225087\pi$
$492$	0	0
$493$	−6.33687	−0.285398
$494$	0	0
$495$	−17.4883	−0.786041
$496$	0	0
$497$	−11.6062	−0.520610
$498$	0	0
$499$	−24.5927	−1.10092	−0.550461	−	0.834861i	$-0.685548\pi$
$499$	−24.5927	−1.10092	−0.550461	+	0.834861i	$0.685548\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	8.11857	0.361989	0.180995	−	0.983484i	$-0.442068\pi$
$503$	8.11857	0.361989	0.180995	+	0.983484i	$0.442068\pi$
$504$	0	0
$505$	12.1382	0.540143
$506$	0	0
$507$	−14.8168	−0.658035
$508$	0	0
$509$	5.74908	0.254823	0.127412	−	0.991850i	$-0.459333\pi$
$509$	5.74908	0.254823	0.127412	+	0.991850i	$0.459333\pi$
$510$	0	0
$511$	−34.4032	−1.52191
$512$	0	0
$513$	−5.25090	−0.231833
$514$	0	0
$515$	4.50431	0.198484
$516$	0	0
$517$	−32.9051	−1.44716
$518$	0	0
$519$	5.27958	0.231748
$520$	0	0
$521$	−32.3615	−1.41778	−0.708891	−	0.705318i	$-0.750804\pi$
$521$	−32.3615	−1.41778	−0.708891	+	0.705318i	$0.750804\pi$
$522$	0	0
$523$	36.4106	1.59213	0.796063	−	0.605214i	$-0.206912\pi$
$523$	36.4106	1.59213	0.796063	+	0.605214i	$0.206912\pi$
$524$	0	0
$525$	−22.0512	−0.962395
$526$	0	0
$527$	−75.5053	−3.28906
$528$	0	0
$529$	−17.5787	−0.764289
$530$	0	0
$531$	10.3875	0.450779
$532$	0	0
$533$	13.6340	0.590555
$534$	0	0
$535$	17.2182	0.744407
$536$	0	0
$537$	−19.1223	−0.825186
$538$	0	0
$539$	−29.0965	−1.25327
$540$	0	0
$541$	−17.8083	−0.765637	−0.382818	−	0.923824i	$-0.625047\pi$
$541$	−17.8083	−0.765637	−0.382818	+	0.923824i	$0.625047\pi$
$542$	0	0
$543$	−24.5626	−1.05408
$544$	0	0
$545$	−26.9807	−1.15573
$546$	0	0
$547$	−11.2379	−0.480500	−0.240250	−	0.970711i	$-0.577229\pi$
$547$	−11.2379	−0.480500	−0.240250	+	0.970711i	$0.577229\pi$
$548$	0	0
$549$	10.7958	0.460752
$550$	0	0
$551$	−4.84410	−0.206366
$552$	0	0
$553$	−50.4595	−2.14576
$554$	0	0
$555$	34.1758	1.45068
$556$	0	0
$557$	−6.44916	−0.273260	−0.136630	−	0.990622i	$-0.543627\pi$
$557$	−6.44916	−0.273260	−0.136630	+	0.990622i	$0.543627\pi$
$558$	0	0
$559$	23.1152	0.977668
$560$	0	0
$561$	35.8647	1.51421
$562$	0	0
$563$	13.4201	0.565590	0.282795	−	0.959180i	$-0.408738\pi$
$563$	13.4201	0.565590	0.282795	+	0.959180i	$0.408738\pi$
$564$	0	0
$565$	−60.3449	−2.53873
$566$	0	0
$567$	3.54581	0.148910
$568$	0	0
$569$	−13.4681	−0.564613	−0.282306	−	0.959324i	$-0.591099\pi$
$569$	−13.4681	−0.564613	−0.282306	+	0.959324i	$0.591099\pi$
$570$	0	0
$571$	19.5925	0.819919	0.409959	−	0.912104i	$-0.365543\pi$
$571$	19.5925	0.819919	0.409959	+	0.912104i	$0.365543\pi$
$572$	0	0
$573$	−11.5678	−0.483251
$574$	0	0
$575$	14.4801	0.603862
$576$	0	0
$577$	−21.3611	−0.889274	−0.444637	−	0.895711i	$-0.646667\pi$
$577$	−21.3611	−0.889274	−0.444637	+	0.895711i	$0.646667\pi$
$578$	0	0
$579$	4.06662	0.169003
$580$	0	0
$581$	−43.3203	−1.79723
$582$	0	0
$583$	−55.9004	−2.31516
$584$	0	0
$585$	17.6657	0.730385
$586$	0	0
$587$	−28.4585	−1.17461	−0.587305	−	0.809366i	$-0.699811\pi$
$587$	−28.4585	−1.17461	−0.587305	+	0.809366i	$0.699811\pi$
$588$	0	0
$589$	−57.7187	−2.37826
$590$	0	0
$591$	−24.7058	−1.01626
$592$	0	0
$593$	−14.7257	−0.604711	−0.302355	−	0.953195i	$-0.597773\pi$
$593$	−14.7257	−0.604711	−0.302355	+	0.953195i	$0.597773\pi$
$594$	0	0
$595$	81.5806	3.34448
$596$	0	0
$597$	6.95811	0.284777
$598$	0	0
$599$	24.5093	1.00142	0.500712	−	0.865614i	$-0.333072\pi$
$599$	24.5093	1.00142	0.500712	+	0.865614i	$0.333072\pi$
$600$	0	0
$601$	−34.6537	−1.41355	−0.706777	−	0.707436i	$-0.749852\pi$
$601$	−34.6537	−1.41355	−0.706777	+	0.707436i	$0.749852\pi$
$602$	0	0
$603$	−8.19936	−0.333904
$604$	0	0
$605$	54.4659	2.21436
$606$	0	0
$607$	−9.43186	−0.382827	−0.191414	−	0.981509i	$-0.561307\pi$
$607$	−9.43186	−0.382827	−0.191414	+	0.981509i	$0.561307\pi$
$608$	0	0
$609$	3.27110	0.132552
$610$	0	0
$611$	33.2388	1.34470
$612$	0	0
$613$	36.6729	1.48121	0.740603	−	0.671943i	$-0.234540\pi$
$613$	36.6729	1.48121	0.740603	+	0.671943i	$0.234540\pi$
$614$	0	0
$615$	−8.65859	−0.349148
$616$	0	0
$617$	−8.27551	−0.333159	−0.166580	−	0.986028i	$-0.553272\pi$
$617$	−8.27551	−0.333159	−0.166580	+	0.986028i	$0.553272\pi$
$618$	0	0
$619$	26.8480	1.07911	0.539556	−	0.841950i	$-0.318592\pi$
$619$	26.8480	1.07911	0.539556	+	0.841950i	$0.318592\pi$
$620$	0	0
$621$	−2.32838	−0.0934346
$622$	0	0
$623$	6.51957	0.261201
$624$	0	0
$625$	−17.4193	−0.696772
$626$	0	0
$627$	27.4161	1.09489
$628$	0	0
$629$	−70.0870	−2.79455
$630$	0	0
$631$	−8.42971	−0.335581	−0.167791	−	0.985823i	$-0.553663\pi$
$631$	−8.42971	−0.335581	−0.167791	+	0.985823i	$0.553663\pi$
$632$	0	0
$633$	12.0883	0.480468
$634$	0	0
$635$	−21.4498	−0.851210
$636$	0	0
$637$	29.3915	1.16453
$638$	0	0
$639$	−3.27323	−0.129487
$640$	0	0
$641$	43.5628	1.72063	0.860313	−	0.509766i	$-0.170268\pi$
$641$	43.5628	1.72063	0.860313	+	0.509766i	$0.170268\pi$
$642$	0	0
$643$	−10.3839	−0.409500	−0.204750	−	0.978814i	$-0.565638\pi$
$643$	−10.3839	−0.409500	−0.204750	+	0.978814i	$0.565638\pi$
$644$	0	0
$645$	−14.6798	−0.578017
$646$	0	0
$647$	13.4049	0.527003	0.263501	−	0.964659i	$-0.415123\pi$
$647$	13.4049	0.527003	0.263501	+	0.964659i	$0.415123\pi$
$648$	0	0
$649$	−54.2354	−2.12892
$650$	0	0
$651$	38.9760	1.52759
$652$	0	0
$653$	24.6733	0.965539	0.482770	−	0.875747i	$-0.339631\pi$
$653$	24.6733	0.965539	0.482770	+	0.875747i	$0.339631\pi$
$654$	0	0
$655$	57.6769	2.25362
$656$	0	0
$657$	−9.70252	−0.378531
$658$	0	0
$659$	33.9385	1.32206	0.661028	−	0.750362i	$-0.270121\pi$
$659$	33.9385	1.32206	0.661028	+	0.750362i	$0.270121\pi$
$660$	0	0
$661$	−20.3567	−0.791783	−0.395891	−	0.918297i	$-0.629564\pi$
$661$	−20.3567	−0.791783	−0.395891	+	0.918297i	$0.629564\pi$
$662$	0	0
$663$	−36.2284	−1.40699
$664$	0	0
$665$	62.3628	2.41833
$666$	0	0
$667$	−2.14799	−0.0831706
$668$	0	0
$669$	−6.74178	−0.260652
$670$	0	0
$671$	−56.3670	−2.17602
$672$	0	0
$673$	−23.5647	−0.908352	−0.454176	−	0.890912i	$-0.650066\pi$
$673$	−23.5647	−0.908352	−0.454176	+	0.890912i	$0.650066\pi$
$674$	0	0
$675$	−6.21896	−0.239368
$676$	0	0
$677$	20.6458	0.793482	0.396741	−	0.917931i	$-0.370141\pi$
$677$	20.6458	0.793482	0.396741	+	0.917931i	$0.370141\pi$
$678$	0	0
$679$	−35.7032	−1.37016
$680$	0	0
$681$	16.6461	0.637879
$682$	0	0
$683$	−15.2872	−0.584950	−0.292475	−	0.956273i	$-0.594479\pi$
$683$	−15.2872	−0.584950	−0.292475	+	0.956273i	$0.594479\pi$
$684$	0	0
$685$	−56.3787	−2.15412
$686$	0	0
$687$	−4.34626	−0.165820
$688$	0	0
$689$	56.4673	2.15123
$690$	0	0
$691$	1.05755	0.0402310	0.0201155	−	0.999798i	$-0.493597\pi$
$691$	1.05755	0.0402310	0.0201155	+	0.999798i	$0.493597\pi$
$692$	0	0
$693$	−18.5134	−0.703266
$694$	0	0
$695$	−30.4290	−1.15424
$696$	0	0
$697$	17.7569	0.672589
$698$	0	0
$699$	−15.4375	−0.583902
$700$	0	0
$701$	−27.2205	−1.02810	−0.514051	−	0.857759i	$-0.671856\pi$
$701$	−27.2205	−1.02810	−0.514051	+	0.857759i	$0.671856\pi$
$702$	0	0
$703$	−53.5767	−2.02068
$704$	0	0
$705$	−21.1090	−0.795012
$706$	0	0
$707$	12.8497	0.483263
$708$	0	0
$709$	3.84618	0.144446	0.0722231	−	0.997388i	$-0.476991\pi$
$709$	3.84618	0.144446	0.0722231	+	0.997388i	$0.476991\pi$
$710$	0	0
$711$	−14.2308	−0.533695
$712$	0	0
$713$	−25.5938	−0.958497
$714$	0	0
$715$	−92.2361	−3.44943
$716$	0	0
$717$	−3.91114	−0.146064
$718$	0	0
$719$	31.7863	1.18543	0.592715	−	0.805412i	$-0.298056\pi$
$719$	31.7863	1.18543	0.592715	+	0.805412i	$0.298056\pi$
$720$	0	0
$721$	4.76834	0.177582
$722$	0	0
$723$	−7.89534	−0.293631
$724$	0	0
$725$	−5.73716	−0.213073
$726$	0	0
$727$	11.8047	0.437813	0.218907	−	0.975746i	$-0.429751\pi$
$727$	11.8047	0.437813	0.218907	+	0.975746i	$0.429751\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.1050	1.11348
$732$	0	0
$733$	−15.2574	−0.563546	−0.281773	−	0.959481i	$-0.590923\pi$
$733$	−15.2574	−0.563546	−0.281773	+	0.959481i	$0.590923\pi$
$734$	0	0
$735$	−18.6657	−0.688496
$736$	0	0
$737$	42.8106	1.57695
$738$	0	0
$739$	−39.0262	−1.43560	−0.717802	−	0.696248i	$-0.754852\pi$
$739$	−39.0262	−1.43560	−0.717802	+	0.696248i	$0.754852\pi$
$740$	0	0
$741$	−27.6941	−1.01737
$742$	0	0
$743$	−9.20570	−0.337724	−0.168862	−	0.985640i	$-0.554009\pi$
$743$	−9.20570	−0.337724	−0.168862	+	0.985640i	$0.554009\pi$
$744$	0	0
$745$	−35.4100	−1.29732
$746$	0	0
$747$	−12.2173	−0.447008
$748$	0	0
$749$	18.2274	0.666016
$750$	0	0
$751$	−38.3585	−1.39972	−0.699861	−	0.714279i	$-0.746755\pi$
$751$	−38.3585	−1.39972	−0.699861	+	0.714279i	$0.746755\pi$
$752$	0	0
$753$	9.57349	0.348877
$754$	0	0
$755$	31.5676	1.14886
$756$	0	0
$757$	37.1329	1.34962	0.674810	−	0.737992i	$-0.264226\pi$
$757$	37.1329	1.34962	0.674810	+	0.737992i	$0.264226\pi$
$758$	0	0
$759$	12.1570	0.441269
$760$	0	0
$761$	−25.9072	−0.939137	−0.469568	−	0.882896i	$-0.655590\pi$
$761$	−25.9072	−0.939137	−0.469568	+	0.882896i	$0.655590\pi$
$762$	0	0
$763$	−28.5622	−1.03402
$764$	0	0
$765$	23.0076	0.831843
$766$	0	0
$767$	54.7854	1.97818
$768$	0	0
$769$	−4.98095	−0.179618	−0.0898089	−	0.995959i	$-0.528626\pi$
$769$	−4.98095	−0.179618	−0.0898089	+	0.995959i	$0.528626\pi$
$770$	0	0
$771$	22.3318	0.804262
$772$	0	0
$773$	3.71962	0.133785	0.0668927	−	0.997760i	$-0.478691\pi$
$773$	3.71962	0.133785	0.0668927	+	0.997760i	$0.478691\pi$
$774$	0	0
$775$	−68.3597	−2.45555
$776$	0	0
$777$	36.1790	1.29791
$778$	0	0
$779$	13.5739	0.486336
$780$	0	0
$781$	17.0902	0.611536
$782$	0	0
$783$	0.922527	0.0329684
$784$	0	0
$785$	−36.9827	−1.31997
$786$	0	0
$787$	−1.36356	−0.0486056	−0.0243028	−	0.999705i	$-0.507737\pi$
$787$	−1.36356	−0.0486056	−0.0243028	+	0.999705i	$0.507737\pi$
$788$	0	0
$789$	−19.8026	−0.704990
$790$	0	0
$791$	−63.8821	−2.27138
$792$	0	0
$793$	56.9386	2.02195
$794$	0	0
$795$	−35.8608	−1.27185
$796$	0	0
$797$	−35.5615	−1.25965	−0.629826	−	0.776736i	$-0.716874\pi$
$797$	−35.5615	−1.25965	−0.629826	+	0.776736i	$0.716874\pi$
$798$	0	0
$799$	43.2900	1.53149
$800$	0	0
$801$	1.83867	0.0649663
$802$	0	0
$803$	50.6589	1.78771
$804$	0	0
$805$	27.6532	0.974646
$806$	0	0
$807$	−13.4173	−0.472312
$808$	0	0
$809$	−27.8127	−0.977842	−0.488921	−	0.872328i	$-0.662609\pi$
$809$	−27.8127	−0.977842	−0.488921	+	0.872328i	$0.662609\pi$
$810$	0	0
$811$	−54.8403	−1.92570	−0.962851	−	0.270033i	$-0.912965\pi$
$811$	−54.8403	−1.92570	−0.962851	+	0.270033i	$0.912965\pi$
$812$	0	0
$813$	25.1007	0.880320
$814$	0	0
$815$	0.684789	0.0239871
$816$	0	0
$817$	23.0133	0.805132
$818$	0	0
$819$	18.7011	0.653471
$820$	0	0
$821$	−20.8424	−0.727404	−0.363702	−	0.931515i	$-0.618487\pi$
$821$	−20.8424	−0.727404	−0.363702	+	0.931515i	$0.618487\pi$
$822$	0	0
$823$	14.4050	0.502125	0.251063	−	0.967971i	$-0.419220\pi$
$823$	14.4050	0.502125	0.251063	+	0.967971i	$0.419220\pi$
$824$	0	0
$825$	32.4705	1.13048
$826$	0	0
$827$	−18.2464	−0.634490	−0.317245	−	0.948344i	$-0.602758\pi$
$827$	−18.2464	−0.634490	−0.317245	+	0.948344i	$0.602758\pi$
$828$	0	0
$829$	−18.2942	−0.635385	−0.317693	−	0.948194i	$-0.602908\pi$
$829$	−18.2942	−0.635385	−0.317693	+	0.948194i	$0.602908\pi$
$830$	0	0
$831$	3.87307	0.134355
$832$	0	0
$833$	38.2793	1.32630
$834$	0	0
$835$	3.34947	0.115913
$836$	0	0
$837$	10.9921	0.379944
$838$	0	0
$839$	−1.08068	−0.0373092	−0.0186546	−	0.999826i	$-0.505938\pi$
$839$	−1.08068	−0.0373092	−0.0186546	+	0.999826i	$0.505938\pi$
$840$	0	0
$841$	−28.1489	−0.970653
$842$	0	0
$843$	3.69139	0.127138
$844$	0	0
$845$	49.6283	1.70727
$846$	0	0
$847$	57.6585	1.98117
$848$	0	0
$849$	−7.75352	−0.266100
$850$	0	0
$851$	−23.7572	−0.814386
$852$	0	0
$853$	−25.1917	−0.862546	−0.431273	−	0.902221i	$-0.641935\pi$
$853$	−25.1917	−0.862546	−0.431273	+	0.902221i	$0.641935\pi$
$854$	0	0
$855$	17.5878	0.601489
$856$	0	0
$857$	−40.0093	−1.36669	−0.683345	−	0.730095i	$-0.739476\pi$
$857$	−40.0093	−1.36669	−0.683345	+	0.730095i	$0.739476\pi$
$858$	0	0
$859$	−55.1265	−1.88089	−0.940445	−	0.339945i	$-0.889591\pi$
$859$	−55.1265	−1.88089	−0.940445	+	0.339945i	$0.889591\pi$
$860$	0	0
$861$	−9.16613	−0.312381
$862$	0	0
$863$	−40.1875	−1.36800	−0.684000	−	0.729482i	$-0.739761\pi$
$863$	−40.1875	−1.36800	−0.684000	+	0.729482i	$0.739761\pi$
$864$	0	0
$865$	−17.6838	−0.601268
$866$	0	0
$867$	−30.1836	−1.02509
$868$	0	0
$869$	74.3019	2.52052
$870$	0	0
$871$	−43.2447	−1.46529
$872$	0	0
$873$	−10.0691	−0.340789
$874$	0	0
$875$	14.4771	0.489416
$876$	0	0
$877$	26.9315	0.909412	0.454706	−	0.890642i	$-0.349744\pi$
$877$	26.9315	0.909412	0.454706	+	0.890642i	$0.349744\pi$
$878$	0	0
$879$	1.01242	0.0341480
$880$	0	0
$881$	43.4956	1.46540	0.732701	−	0.680550i	$-0.238259\pi$
$881$	43.4956	1.46540	0.732701	+	0.680550i	$0.238259\pi$
$882$	0	0
$883$	54.9488	1.84917	0.924587	−	0.380970i	$-0.124410\pi$
$883$	54.9488	1.84917	0.924587	+	0.380970i	$0.124410\pi$
$884$	0	0
$885$	−34.7927	−1.16954
$886$	0	0
$887$	−45.4217	−1.52511	−0.762556	−	0.646923i	$-0.776056\pi$
$887$	−45.4217	−1.52511	−0.762556	+	0.646923i	$0.776056\pi$
$888$	0	0
$889$	−22.7071	−0.761572
$890$	0	0
$891$	−5.22121	−0.174917
$892$	0	0
$893$	33.0922	1.10739
$894$	0	0
$895$	64.0495	2.14094
$896$	0	0
$897$	−12.2802	−0.410025
$898$	0	0
$899$	10.1405	0.338206
$900$	0	0
$901$	73.5426	2.45006
$902$	0	0
$903$	−15.5403	−0.517148
$904$	0	0
$905$	82.2717	2.73480
$906$	0	0
$907$	3.43231	0.113968	0.0569840	−	0.998375i	$-0.481852\pi$
$907$	3.43231	0.113968	0.0569840	+	0.998375i	$0.481852\pi$
$908$	0	0
$909$	3.62392	0.120198
$910$	0	0
$911$	−16.2715	−0.539098	−0.269549	−	0.962987i	$-0.586875\pi$
$911$	−16.2715	−0.539098	−0.269549	+	0.962987i	$0.586875\pi$
$912$	0	0
$913$	63.7892	2.11112
$914$	0	0
$915$	−36.1601	−1.19542
$916$	0	0
$917$	61.0577	2.01630
$918$	0	0
$919$	47.6271	1.57107	0.785537	−	0.618815i	$-0.212387\pi$
$919$	47.6271	1.57107	0.785537	+	0.618815i	$0.212387\pi$
$920$	0	0
$921$	−25.9773	−0.855980
$922$	0	0
$923$	−17.2635	−0.568236
$924$	0	0
$925$	−63.4541	−2.08636
$926$	0	0
$927$	1.34478	0.0441685
$928$	0	0
$929$	10.6196	0.348417	0.174208	−	0.984709i	$-0.444263\pi$
$929$	10.6196	0.348417	0.174208	+	0.984709i	$0.444263\pi$
$930$	0	0
$931$	29.2619	0.959021
$932$	0	0
$933$	10.2568	0.335792
$934$	0	0
$935$	−120.128	−3.92860
$936$	0	0
$937$	−32.3052	−1.05537	−0.527683	−	0.849441i	$-0.676939\pi$
$937$	−32.3052	−1.05537	−0.527683	+	0.849441i	$0.676939\pi$
$938$	0	0
$939$	−13.2574	−0.432638
$940$	0	0
$941$	−34.9102	−1.13804	−0.569020	−	0.822324i	$-0.692677\pi$
$941$	−34.9102	−1.13804	−0.569020	+	0.822324i	$0.692677\pi$
$942$	0	0
$943$	6.01900	0.196006
$944$	0	0
$945$	−11.8766	−0.386345
$946$	0	0
$947$	1.60729	0.0522299	0.0261149	−	0.999659i	$-0.491686\pi$
$947$	1.60729	0.0522299	0.0261149	+	0.999659i	$0.491686\pi$
$948$	0	0
$949$	−51.1726	−1.66113
$950$	0	0
$951$	−21.0416	−0.682321
$952$	0	0
$953$	−17.1682	−0.556133	−0.278067	−	0.960562i	$-0.589694\pi$
$953$	−17.1682	−0.556133	−0.278067	+	0.960562i	$0.589694\pi$
$954$	0	0
$955$	38.7460	1.25379
$956$	0	0
$957$	−4.81671	−0.155702
$958$	0	0
$959$	−59.6834	−1.92728
$960$	0	0
$961$	89.8271	2.89765
$962$	0	0
$963$	5.14056	0.165652
$964$	0	0
$965$	−13.6210	−0.438476
$966$	0	0
$967$	−19.4971	−0.626984	−0.313492	−	0.949591i	$-0.601499\pi$
$967$	−19.4971	−0.626984	−0.313492	+	0.949591i	$0.601499\pi$
$968$	0	0
$969$	−36.0686	−1.15869
$970$	0	0
$971$	−20.8748	−0.669903	−0.334952	−	0.942235i	$-0.608720\pi$
$971$	−20.8748	−0.669903	−0.334952	+	0.942235i	$0.608720\pi$
$972$	0	0
$973$	−32.2126	−1.03269
$974$	0	0
$975$	−32.7998	−1.05043
$976$	0	0
$977$	−52.7385	−1.68725	−0.843627	−	0.536930i	$-0.819584\pi$
$977$	−52.7385	−1.68725	−0.843627	+	0.536930i	$0.819584\pi$
$978$	0	0
$979$	−9.60009	−0.306820
$980$	0	0
$981$	−8.05520	−0.257183
$982$	0	0
$983$	−17.3260	−0.552613	−0.276307	−	0.961070i	$-0.589110\pi$
$983$	−17.3260	−0.552613	−0.276307	+	0.961070i	$0.589110\pi$
$984$	0	0
$985$	82.7512	2.63667
$986$	0	0
$987$	−22.3464	−0.711293
$988$	0	0
$989$	10.2046	0.324488
$990$	0	0
$991$	46.9567	1.49163	0.745814	−	0.666154i	$-0.232061\pi$
$991$	46.9567	1.49163	0.745814	+	0.666154i	$0.232061\pi$
$992$	0	0
$993$	−11.4570	−0.363578
$994$	0	0
$995$	−23.3060	−0.738850
$996$	0	0
$997$	−31.3277	−0.992159	−0.496080	−	0.868277i	$-0.665228\pi$
$997$	−31.3277	−0.992159	−0.496080	+	0.868277i	$0.665228\pi$
$998$	0	0
$999$	10.2033	0.322819

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.k.1.9	✓	11
4.3	odd	2		8016.2.a.be.1.9		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.k.1.9	✓	11	1.1	even	1	trivial
8016.2.a.be.1.9		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.34947 q^{5} +3.54581 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.34947 q^{5} +3.54581 q^{7} +1.00000 q^{9} -5.22121 q^{11} +5.27416 q^{13} -3.34947 q^{15} +6.86903 q^{17} +5.25090 q^{19} -3.54581 q^{21} +2.32838 q^{23} +6.21896 q^{25} -1.00000 q^{27} -0.922527 q^{29} -10.9921 q^{31} +5.22121 q^{33} +11.8766 q^{35} -10.2033 q^{37} -5.27416 q^{39} +2.58506 q^{41} +4.38272 q^{43} +3.34947 q^{45} +6.30219 q^{47} +5.57274 q^{49} -6.86903 q^{51} +10.7064 q^{53} -17.4883 q^{55} -5.25090 q^{57} +10.3875 q^{59} +10.7958 q^{61} +3.54581 q^{63} +17.6657 q^{65} -8.19936 q^{67} -2.32838 q^{69} -3.27323 q^{71} -9.70252 q^{73} -6.21896 q^{75} -18.5134 q^{77} -14.2308 q^{79} +1.00000 q^{81} -12.2173 q^{83} +23.0076 q^{85} +0.922527 q^{87} +1.83867 q^{89} +18.7011 q^{91} +10.9921 q^{93} +17.5878 q^{95} -10.0691 q^{97} -5.22121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9}+O(q^{10}) \) 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

L-functions

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