LMFDB - Embedded newform 4008.2.a.l.1.8

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 $ x^12 - 4x^11 - 31x^10 + 131x^9 + 309x^8 - 1453x^7 - 1072x^6 + 6350x^5 + 1411x^4 - 11022x^3 - 2450x^2 + 6960*x + 3008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.56937$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.56937 q^{5} +3.21458 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.56937 q^{5} +3.21458 q^{7} +1.00000 q^{9} -5.96544 q^{11} +2.81457 q^{13} +1.56937 q^{15} +4.33096 q^{17} +4.82798 q^{19} +3.21458 q^{21} +3.36422 q^{23} -2.53708 q^{25} +1.00000 q^{27} -8.76511 q^{29} +2.34104 q^{31} -5.96544 q^{33} +5.04486 q^{35} +7.28651 q^{37} +2.81457 q^{39} -1.39154 q^{41} +2.86513 q^{43} +1.56937 q^{45} +9.87896 q^{47} +3.33349 q^{49} +4.33096 q^{51} -11.2692 q^{53} -9.36199 q^{55} +4.82798 q^{57} -1.96751 q^{59} -3.76751 q^{61} +3.21458 q^{63} +4.41711 q^{65} +11.6949 q^{67} +3.36422 q^{69} -4.01255 q^{71} +8.49415 q^{73} -2.53708 q^{75} -19.1764 q^{77} +3.69376 q^{79} +1.00000 q^{81} +8.33948 q^{83} +6.79689 q^{85} -8.76511 q^{87} +12.3598 q^{89} +9.04765 q^{91} +2.34104 q^{93} +7.57689 q^{95} +0.582982 q^{97} -5.96544 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * 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
$5$	1.56937	0.701844	0.350922	−	0.936405i	$-0.385868\pi$
$5$	1.56937	0.701844	0.350922	+	0.936405i	$0.385868\pi$
$6$	0	0
$7$	3.21458	1.21500	0.607498	−	0.794321i	$-0.292173\pi$
$7$	3.21458	1.21500	0.607498	+	0.794321i	$0.292173\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.96544	−1.79865	−0.899325	−	0.437282i	$-0.855941\pi$
$11$	−5.96544	−1.79865	−0.899325	+	0.437282i	$0.855941\pi$
$12$	0	0
$13$	2.81457	0.780622	0.390311	−	0.920683i	$-0.372368\pi$
$13$	2.81457	0.780622	0.390311	+	0.920683i	$0.372368\pi$
$14$	0	0
$15$	1.56937	0.405210
$16$	0	0
$17$	4.33096	1.05041	0.525207	−	0.850975i	$-0.323988\pi$
$17$	4.33096	1.05041	0.525207	+	0.850975i	$0.323988\pi$
$18$	0	0
$19$	4.82798	1.10761	0.553807	−	0.832645i	$-0.313175\pi$
$19$	4.82798	1.10761	0.553807	+	0.832645i	$0.313175\pi$
$20$	0	0
$21$	3.21458	0.701478
$22$	0	0
$23$	3.36422	0.701489	0.350744	−	0.936471i	$-0.385929\pi$
$23$	3.36422	0.701489	0.350744	+	0.936471i	$0.385929\pi$
$24$	0	0
$25$	−2.53708	−0.507415
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.76511	−1.62764	−0.813820	−	0.581117i	$-0.802616\pi$
$29$	−8.76511	−1.62764	−0.813820	+	0.581117i	$0.802616\pi$
$30$	0	0
$31$	2.34104	0.420463	0.210231	−	0.977652i	$-0.432578\pi$
$31$	2.34104	0.420463	0.210231	+	0.977652i	$0.432578\pi$
$32$	0	0
$33$	−5.96544	−1.03845
$34$	0	0
$35$	5.04486	0.852737
$36$	0	0
$37$	7.28651	1.19789	0.598947	−	0.800788i	$-0.295586\pi$
$37$	7.28651	1.19789	0.598947	+	0.800788i	$0.295586\pi$
$38$	0	0
$39$	2.81457	0.450692
$40$	0	0
$41$	−1.39154	−0.217322	−0.108661	−	0.994079i	$-0.534656\pi$
$41$	−1.39154	−0.217322	−0.108661	+	0.994079i	$0.534656\pi$
$42$	0	0
$43$	2.86513	0.436928	0.218464	−	0.975845i	$-0.429895\pi$
$43$	2.86513	0.436928	0.218464	+	0.975845i	$0.429895\pi$
$44$	0	0
$45$	1.56937	0.233948
$46$	0	0
$47$	9.87896	1.44099	0.720497	−	0.693458i	$-0.243914\pi$
$47$	9.87896	1.44099	0.720497	+	0.693458i	$0.243914\pi$
$48$	0	0
$49$	3.33349	0.476213
$50$	0	0
$51$	4.33096	0.606456
$52$	0	0
$53$	−11.2692	−1.54794	−0.773970	−	0.633223i	$-0.781732\pi$
$53$	−11.2692	−1.54794	−0.773970	+	0.633223i	$0.781732\pi$
$54$	0	0
$55$	−9.36199	−1.26237
$56$	0	0
$57$	4.82798	0.639481
$58$	0	0
$59$	−1.96751	−0.256148	−0.128074	−	0.991765i	$-0.540879\pi$
$59$	−1.96751	−0.256148	−0.128074	+	0.991765i	$0.540879\pi$
$60$	0	0
$61$	−3.76751	−0.482381	−0.241190	−	0.970478i	$-0.577538\pi$
$61$	−3.76751	−0.482381	−0.241190	+	0.970478i	$0.577538\pi$
$62$	0	0
$63$	3.21458	0.404998
$64$	0	0
$65$	4.41711	0.547875
$66$	0	0
$67$	11.6949	1.42875	0.714377	−	0.699761i	$-0.246710\pi$
$67$	11.6949	1.42875	0.714377	+	0.699761i	$0.246710\pi$
$68$	0	0
$69$	3.36422	0.405005
$70$	0	0
$71$	−4.01255	−0.476202	−0.238101	−	0.971240i	$-0.576525\pi$
$71$	−4.01255	−0.476202	−0.238101	+	0.971240i	$0.576525\pi$
$72$	0	0
$73$	8.49415	0.994165	0.497082	−	0.867703i	$-0.334405\pi$
$73$	8.49415	0.994165	0.497082	+	0.867703i	$0.334405\pi$
$74$	0	0
$75$	−2.53708	−0.292956
$76$	0	0
$77$	−19.1764	−2.18535
$78$	0	0
$79$	3.69376	0.415580	0.207790	−	0.978173i	$-0.433373\pi$
$79$	3.69376	0.415580	0.207790	+	0.978173i	$0.433373\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.33948	0.915377	0.457688	−	0.889113i	$-0.348677\pi$
$83$	8.33948	0.915377	0.457688	+	0.889113i	$0.348677\pi$
$84$	0	0
$85$	6.79689	0.737226
$86$	0	0
$87$	−8.76511	−0.939718
$88$	0	0
$89$	12.3598	1.31013	0.655067	−	0.755571i	$-0.272640\pi$
$89$	12.3598	1.31013	0.655067	+	0.755571i	$0.272640\pi$
$90$	0	0
$91$	9.04765	0.948452
$92$	0	0
$93$	2.34104	0.242754
$94$	0	0
$95$	7.57689	0.777372
$96$	0	0
$97$	0.582982	0.0591929	0.0295964	−	0.999562i	$-0.490578\pi$
$97$	0.582982	0.0591929	0.0295964	+	0.999562i	$0.490578\pi$
$98$	0	0
$99$	−5.96544	−0.599550
$100$	0	0
$101$	1.44938	0.144218	0.0721091	−	0.997397i	$-0.477027\pi$
$101$	1.44938	0.144218	0.0721091	+	0.997397i	$0.477027\pi$
$102$	0	0
$103$	−14.7825	−1.45657	−0.728284	−	0.685276i	$-0.759682\pi$
$103$	−14.7825	−1.45657	−0.728284	+	0.685276i	$0.759682\pi$
$104$	0	0
$105$	5.04486	0.492328
$106$	0	0
$107$	−10.3485	−1.00043	−0.500214	−	0.865902i	$-0.666745\pi$
$107$	−10.3485	−1.00043	−0.500214	+	0.865902i	$0.666745\pi$
$108$	0	0
$109$	−2.71160	−0.259724	−0.129862	−	0.991532i	$-0.541453\pi$
$109$	−2.71160	−0.259724	−0.129862	+	0.991532i	$0.541453\pi$
$110$	0	0
$111$	7.28651	0.691605
$112$	0	0
$113$	−0.419192	−0.0394343	−0.0197171	−	0.999806i	$-0.506277\pi$
$113$	−0.419192	−0.0394343	−0.0197171	+	0.999806i	$0.506277\pi$
$114$	0	0
$115$	5.27971	0.492336
$116$	0	0
$117$	2.81457	0.260207
$118$	0	0
$119$	13.9222	1.27625
$120$	0	0
$121$	24.5865	2.23514
$122$	0	0
$123$	−1.39154	−0.125471
$124$	0	0
$125$	−11.8285	−1.05797
$126$	0	0
$127$	−14.5074	−1.28733	−0.643663	−	0.765309i	$-0.722586\pi$
$127$	−14.5074	−1.28733	−0.643663	+	0.765309i	$0.722586\pi$
$128$	0	0
$129$	2.86513	0.252260
$130$	0	0
$131$	19.6189	1.71411	0.857054	−	0.515226i	$-0.172292\pi$
$131$	19.6189	1.71411	0.857054	+	0.515226i	$0.172292\pi$
$132$	0	0
$133$	15.5199	1.34575
$134$	0	0
$135$	1.56937	0.135070
$136$	0	0
$137$	−7.89669	−0.674660	−0.337330	−	0.941387i	$-0.609524\pi$
$137$	−7.89669	−0.674660	−0.337330	+	0.941387i	$0.609524\pi$
$138$	0	0
$139$	−6.36468	−0.539845	−0.269923	−	0.962882i	$-0.586998\pi$
$139$	−6.36468	−0.539845	−0.269923	+	0.962882i	$0.586998\pi$
$140$	0	0
$141$	9.87896	0.831958
$142$	0	0
$143$	−16.7902	−1.40406
$144$	0	0
$145$	−13.7557	−1.14235
$146$	0	0
$147$	3.33349	0.274942
$148$	0	0
$149$	9.16385	0.750732	0.375366	−	0.926877i	$-0.377517\pi$
$149$	9.16385	0.750732	0.375366	+	0.926877i	$0.377517\pi$
$150$	0	0
$151$	16.2998	1.32646	0.663228	−	0.748418i	$-0.269186\pi$
$151$	16.2998	1.32646	0.663228	+	0.748418i	$0.269186\pi$
$152$	0	0
$153$	4.33096	0.350138
$154$	0	0
$155$	3.67396	0.295099
$156$	0	0
$157$	18.2988	1.46040	0.730200	−	0.683233i	$-0.239427\pi$
$157$	18.2988	1.46040	0.730200	+	0.683233i	$0.239427\pi$
$158$	0	0
$159$	−11.2692	−0.893703
$160$	0	0
$161$	10.8145	0.852306
$162$	0	0
$163$	−9.82475	−0.769534	−0.384767	−	0.923014i	$-0.625718\pi$
$163$	−9.82475	−0.769534	−0.384767	+	0.923014i	$0.625718\pi$
$164$	0	0
$165$	−9.36199	−0.728830
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−5.07819	−0.390630
$170$	0	0
$171$	4.82798	0.369205
$172$	0	0
$173$	−19.1154	−1.45332	−0.726658	−	0.687000i	$-0.758927\pi$
$173$	−19.1154	−1.45332	−0.726658	+	0.687000i	$0.758927\pi$
$174$	0	0
$175$	−8.15562	−0.616507
$176$	0	0
$177$	−1.96751	−0.147887
$178$	0	0
$179$	−8.40836	−0.628471	−0.314235	−	0.949345i	$-0.601748\pi$
$179$	−8.40836	−0.628471	−0.314235	+	0.949345i	$0.601748\pi$
$180$	0	0
$181$	−20.3287	−1.51102	−0.755510	−	0.655137i	$-0.772611\pi$
$181$	−20.3287	−1.51102	−0.755510	+	0.655137i	$0.772611\pi$
$182$	0	0
$183$	−3.76751	−0.278503
$184$	0	0
$185$	11.4352	0.840735
$186$	0	0
$187$	−25.8361	−1.88932
$188$	0	0
$189$	3.21458	0.233826
$190$	0	0
$191$	24.2261	1.75294	0.876471	−	0.481454i	$-0.159891\pi$
$191$	24.2261	1.75294	0.876471	+	0.481454i	$0.159891\pi$
$192$	0	0
$193$	−12.2851	−0.884301	−0.442151	−	0.896941i	$-0.645784\pi$
$193$	−12.2851	−0.884301	−0.442151	+	0.896941i	$0.645784\pi$
$194$	0	0
$195$	4.41711	0.316316
$196$	0	0
$197$	−15.3618	−1.09448	−0.547240	−	0.836976i	$-0.684322\pi$
$197$	−15.3618	−1.09448	−0.547240	+	0.836976i	$0.684322\pi$
$198$	0	0
$199$	8.14641	0.577483	0.288742	−	0.957407i	$-0.406763\pi$
$199$	8.14641	0.577483	0.288742	+	0.957407i	$0.406763\pi$
$200$	0	0
$201$	11.6949	0.824892
$202$	0	0
$203$	−28.1761	−1.97757
$204$	0	0
$205$	−2.18384	−0.152526
$206$	0	0
$207$	3.36422	0.233830
$208$	0	0
$209$	−28.8010	−1.99221
$210$	0	0
$211$	−12.7610	−0.878503	−0.439251	−	0.898364i	$-0.644756\pi$
$211$	−12.7610	−0.878503	−0.439251	+	0.898364i	$0.644756\pi$
$212$	0	0
$213$	−4.01255	−0.274936
$214$	0	0
$215$	4.49645	0.306655
$216$	0	0
$217$	7.52544	0.510860
$218$	0	0
$219$	8.49415	0.573981
$220$	0	0
$221$	12.1898	0.819975
$222$	0	0
$223$	9.88563	0.661991	0.330995	−	0.943632i	$-0.392616\pi$
$223$	9.88563	0.661991	0.330995	+	0.943632i	$0.392616\pi$
$224$	0	0
$225$	−2.53708	−0.169138
$226$	0	0
$227$	−0.485281	−0.0322093	−0.0161046	−	0.999870i	$-0.505126\pi$
$227$	−0.485281	−0.0322093	−0.0161046	+	0.999870i	$0.505126\pi$
$228$	0	0
$229$	−28.3959	−1.87646	−0.938228	−	0.346019i	$-0.887533\pi$
$229$	−28.3959	−1.87646	−0.938228	+	0.346019i	$0.887533\pi$
$230$	0	0
$231$	−19.1764	−1.26171
$232$	0	0
$233$	−26.5196	−1.73736	−0.868679	−	0.495376i	$-0.835030\pi$
$233$	−26.5196	−1.73736	−0.868679	+	0.495376i	$0.835030\pi$
$234$	0	0
$235$	15.5037	1.01135
$236$	0	0
$237$	3.69376	0.239935
$238$	0	0
$239$	13.4261	0.868464	0.434232	−	0.900801i	$-0.357020\pi$
$239$	13.4261	0.868464	0.434232	+	0.900801i	$0.357020\pi$
$240$	0	0
$241$	11.4392	0.736865	0.368432	−	0.929655i	$-0.379895\pi$
$241$	11.4392	0.736865	0.368432	+	0.929655i	$0.379895\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.23149	0.334228
$246$	0	0
$247$	13.5887	0.864627
$248$	0	0
$249$	8.33948	0.528493
$250$	0	0
$251$	−16.4258	−1.03679	−0.518395	−	0.855141i	$-0.673470\pi$
$251$	−16.4258	−1.03679	−0.518395	+	0.855141i	$0.673470\pi$
$252$	0	0
$253$	−20.0691	−1.26173
$254$	0	0
$255$	6.79689	0.425638
$256$	0	0
$257$	17.6649	1.10190	0.550952	−	0.834537i	$-0.314265\pi$
$257$	17.6649	1.10190	0.550952	+	0.834537i	$0.314265\pi$
$258$	0	0
$259$	23.4230	1.45544
$260$	0	0
$261$	−8.76511	−0.542547
$262$	0	0
$263$	7.60215	0.468768	0.234384	−	0.972144i	$-0.424693\pi$
$263$	7.60215	0.468768	0.234384	+	0.972144i	$0.424693\pi$
$264$	0	0
$265$	−17.6855	−1.08641
$266$	0	0
$267$	12.3598	0.756406
$268$	0	0
$269$	−7.33835	−0.447427	−0.223714	−	0.974655i	$-0.571818\pi$
$269$	−7.33835	−0.447427	−0.223714	+	0.974655i	$0.571818\pi$
$270$	0	0
$271$	30.7629	1.86871	0.934356	−	0.356340i	$-0.115975\pi$
$271$	30.7629	1.86871	0.934356	+	0.356340i	$0.115975\pi$
$272$	0	0
$273$	9.04765	0.547589
$274$	0	0
$275$	15.1348	0.912662
$276$	0	0
$277$	15.0041	0.901508	0.450754	−	0.892648i	$-0.351155\pi$
$277$	15.0041	0.901508	0.450754	+	0.892648i	$0.351155\pi$
$278$	0	0
$279$	2.34104	0.140154
$280$	0	0
$281$	10.2001	0.608488	0.304244	−	0.952594i	$-0.401596\pi$
$281$	10.2001	0.608488	0.304244	+	0.952594i	$0.401596\pi$
$282$	0	0
$283$	2.65541	0.157847	0.0789237	−	0.996881i	$-0.474852\pi$
$283$	2.65541	0.157847	0.0789237	+	0.996881i	$0.474852\pi$
$284$	0	0
$285$	7.57689	0.448816
$286$	0	0
$287$	−4.47321	−0.264045
$288$	0	0
$289$	1.75725	0.103368
$290$	0	0
$291$	0.582982	0.0341750
$292$	0	0
$293$	19.0956	1.11558	0.557789	−	0.829983i	$-0.311650\pi$
$293$	19.0956	1.11558	0.557789	+	0.829983i	$0.311650\pi$
$294$	0	0
$295$	−3.08775	−0.179776
$296$	0	0
$297$	−5.96544	−0.346150
$298$	0	0
$299$	9.46885	0.547597
$300$	0	0
$301$	9.21017	0.530865
$302$	0	0
$303$	1.44938	0.0832644
$304$	0	0
$305$	−5.91263	−0.338556
$306$	0	0
$307$	2.29511	0.130989	0.0654944	−	0.997853i	$-0.479138\pi$
$307$	2.29511	0.130989	0.0654944	+	0.997853i	$0.479138\pi$
$308$	0	0
$309$	−14.7825	−0.840950
$310$	0	0
$311$	−0.877859	−0.0497788	−0.0248894	−	0.999690i	$-0.507923\pi$
$311$	−0.877859	−0.0497788	−0.0248894	+	0.999690i	$0.507923\pi$
$312$	0	0
$313$	−18.9459	−1.07089	−0.535443	−	0.844571i	$-0.679855\pi$
$313$	−18.9459	−1.07089	−0.535443	+	0.844571i	$0.679855\pi$
$314$	0	0
$315$	5.04486	0.284246
$316$	0	0
$317$	32.8653	1.84590	0.922950	−	0.384920i	$-0.125771\pi$
$317$	32.8653	1.84590	0.922950	+	0.384920i	$0.125771\pi$
$318$	0	0
$319$	52.2878	2.92755
$320$	0	0
$321$	−10.3485	−0.577598
$322$	0	0
$323$	20.9098	1.16345
$324$	0	0
$325$	−7.14078	−0.396099
$326$	0	0
$327$	−2.71160	−0.149952
$328$	0	0
$329$	31.7566	1.75080
$330$	0	0
$331$	6.81707	0.374700	0.187350	−	0.982293i	$-0.440010\pi$
$331$	6.81707	0.374700	0.187350	+	0.982293i	$0.440010\pi$
$332$	0	0
$333$	7.28651	0.399298
$334$	0	0
$335$	18.3536	1.00276
$336$	0	0
$337$	−14.6194	−0.796372	−0.398186	−	0.917305i	$-0.630360\pi$
$337$	−14.6194	−0.796372	−0.398186	+	0.917305i	$0.630360\pi$
$338$	0	0
$339$	−0.419192	−0.0227674
$340$	0	0
$341$	−13.9653	−0.756265
$342$	0	0
$343$	−11.7863	−0.636398
$344$	0	0
$345$	5.27971	0.284250
$346$	0	0
$347$	18.2051	0.977301	0.488651	−	0.872480i	$-0.337489\pi$
$347$	18.2051	0.977301	0.488651	+	0.872480i	$0.337489\pi$
$348$	0	0
$349$	12.8709	0.688964	0.344482	−	0.938793i	$-0.388055\pi$
$349$	12.8709	0.688964	0.344482	+	0.938793i	$0.388055\pi$
$350$	0	0
$351$	2.81457	0.150231
$352$	0	0
$353$	12.9234	0.687841	0.343921	−	0.938999i	$-0.388245\pi$
$353$	12.9234	0.687841	0.343921	+	0.938999i	$0.388245\pi$
$354$	0	0
$355$	−6.29718	−0.334220
$356$	0	0
$357$	13.9222	0.736841
$358$	0	0
$359$	−21.7806	−1.14954	−0.574769	−	0.818316i	$-0.694908\pi$
$359$	−21.7806	−1.14954	−0.574769	+	0.818316i	$0.694908\pi$
$360$	0	0
$361$	4.30936	0.226808
$362$	0	0
$363$	24.5865	1.29046
$364$	0	0
$365$	13.3305	0.697749
$366$	0	0
$367$	−18.0839	−0.943970	−0.471985	−	0.881607i	$-0.656462\pi$
$367$	−18.0839	−0.943970	−0.471985	+	0.881607i	$0.656462\pi$
$368$	0	0
$369$	−1.39154	−0.0724407
$370$	0	0
$371$	−36.2256	−1.88074
$372$	0	0
$373$	−28.5524	−1.47839	−0.739193	−	0.673494i	$-0.764793\pi$
$373$	−28.5524	−1.47839	−0.739193	+	0.673494i	$0.764793\pi$
$374$	0	0
$375$	−11.8285	−0.610819
$376$	0	0
$377$	−24.6700	−1.27057
$378$	0	0
$379$	29.6698	1.52404	0.762019	−	0.647555i	$-0.224208\pi$
$379$	29.6698	1.52404	0.762019	+	0.647555i	$0.224208\pi$
$380$	0	0
$381$	−14.5074	−0.743237
$382$	0	0
$383$	26.8107	1.36996	0.684982	−	0.728560i	$-0.259810\pi$
$383$	26.8107	1.36996	0.684982	+	0.728560i	$0.259810\pi$
$384$	0	0
$385$	−30.0948	−1.53377
$386$	0	0
$387$	2.86513	0.145643
$388$	0	0
$389$	−27.5430	−1.39649	−0.698243	−	0.715861i	$-0.746035\pi$
$389$	−27.5430	−1.39649	−0.698243	+	0.715861i	$0.746035\pi$
$390$	0	0
$391$	14.5703	0.736853
$392$	0	0
$393$	19.6189	0.989641
$394$	0	0
$395$	5.79687	0.291672
$396$	0	0
$397$	−15.7322	−0.789575	−0.394787	−	0.918772i	$-0.629182\pi$
$397$	−15.7322	−0.789575	−0.394787	+	0.918772i	$0.629182\pi$
$398$	0	0
$399$	15.5199	0.776966
$400$	0	0
$401$	−6.71138	−0.335150	−0.167575	−	0.985859i	$-0.553594\pi$
$401$	−6.71138	−0.335150	−0.167575	+	0.985859i	$0.553594\pi$
$402$	0	0
$403$	6.58902	0.328222
$404$	0	0
$405$	1.56937	0.0779827
$406$	0	0
$407$	−43.4673	−2.15459
$408$	0	0
$409$	−25.3626	−1.25410	−0.627049	−	0.778980i	$-0.715737\pi$
$409$	−25.3626	−1.25410	−0.627049	+	0.778980i	$0.715737\pi$
$410$	0	0
$411$	−7.89669	−0.389515
$412$	0	0
$413$	−6.32470	−0.311218
$414$	0	0
$415$	13.0877	0.642452
$416$	0	0
$417$	−6.36468	−0.311680
$418$	0	0
$419$	−4.52335	−0.220980	−0.110490	−	0.993877i	$-0.535242\pi$
$419$	−4.52335	−0.220980	−0.110490	+	0.993877i	$0.535242\pi$
$420$	0	0
$421$	25.4294	1.23935	0.619677	−	0.784857i	$-0.287264\pi$
$421$	25.4294	1.23935	0.619677	+	0.784857i	$0.287264\pi$
$422$	0	0
$423$	9.87896	0.480331
$424$	0	0
$425$	−10.9880	−0.532995
$426$	0	0
$427$	−12.1110	−0.586090
$428$	0	0
$429$	−16.7902	−0.810637
$430$	0	0
$431$	−29.8778	−1.43917	−0.719583	−	0.694407i	$-0.755667\pi$
$431$	−29.8778	−1.43917	−0.719583	+	0.694407i	$0.755667\pi$
$432$	0	0
$433$	17.1484	0.824100	0.412050	−	0.911161i	$-0.364813\pi$
$433$	17.1484	0.824100	0.412050	+	0.911161i	$0.364813\pi$
$434$	0	0
$435$	−13.7557	−0.659536
$436$	0	0
$437$	16.2424	0.776979
$438$	0	0
$439$	−15.7954	−0.753872	−0.376936	−	0.926239i	$-0.623022\pi$
$439$	−15.7954	−0.753872	−0.376936	+	0.926239i	$0.623022\pi$
$440$	0	0
$441$	3.33349	0.158738
$442$	0	0
$443$	22.8494	1.08561	0.542804	−	0.839860i	$-0.317363\pi$
$443$	22.8494	1.08561	0.542804	+	0.839860i	$0.317363\pi$
$444$	0	0
$445$	19.3971	0.919509
$446$	0	0
$447$	9.16385	0.433435
$448$	0	0
$449$	−15.5104	−0.731982	−0.365991	−	0.930618i	$-0.619270\pi$
$449$	−15.5104	−0.731982	−0.365991	+	0.930618i	$0.619270\pi$
$450$	0	0
$451$	8.30116	0.390886
$452$	0	0
$453$	16.2998	0.765830
$454$	0	0
$455$	14.1991	0.665665
$456$	0	0
$457$	36.1847	1.69265	0.846325	−	0.532666i	$-0.178810\pi$
$457$	36.1847	1.69265	0.846325	+	0.532666i	$0.178810\pi$
$458$	0	0
$459$	4.33096	0.202152
$460$	0	0
$461$	−12.1329	−0.565085	−0.282542	−	0.959255i	$-0.591178\pi$
$461$	−12.1329	−0.565085	−0.282542	+	0.959255i	$0.591178\pi$
$462$	0	0
$463$	24.5598	1.14139	0.570696	−	0.821161i	$-0.306673\pi$
$463$	24.5598	1.14139	0.570696	+	0.821161i	$0.306673\pi$
$464$	0	0
$465$	3.67396	0.170376
$466$	0	0
$467$	−22.6629	−1.04872	−0.524358	−	0.851498i	$-0.675695\pi$
$467$	−22.6629	−1.04872	−0.524358	+	0.851498i	$0.675695\pi$
$468$	0	0
$469$	37.5940	1.73593
$470$	0	0
$471$	18.2988	0.843163
$472$	0	0
$473$	−17.0918	−0.785880
$474$	0	0
$475$	−12.2489	−0.562020
$476$	0	0
$477$	−11.2692	−0.515980
$478$	0	0
$479$	11.5978	0.529915	0.264958	−	0.964260i	$-0.414642\pi$
$479$	11.5978	0.529915	0.264958	+	0.964260i	$0.414642\pi$
$480$	0	0
$481$	20.5084	0.935102
$482$	0	0
$483$	10.8145	0.492079
$484$	0	0
$485$	0.914916	0.0415442
$486$	0	0
$487$	−28.9911	−1.31371	−0.656856	−	0.754016i	$-0.728114\pi$
$487$	−28.9911	−1.31371	−0.656856	+	0.754016i	$0.728114\pi$
$488$	0	0
$489$	−9.82475	−0.444291
$490$	0	0
$491$	−37.7088	−1.70177	−0.850887	−	0.525349i	$-0.823935\pi$
$491$	−37.7088	−1.70177	−0.850887	+	0.525349i	$0.823935\pi$
$492$	0	0
$493$	−37.9614	−1.70969
$494$	0	0
$495$	−9.36199	−0.420790
$496$	0	0
$497$	−12.8987	−0.578584
$498$	0	0
$499$	22.0629	0.987669	0.493835	−	0.869556i	$-0.335595\pi$
$499$	22.0629	0.987669	0.493835	+	0.869556i	$0.335595\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−32.1216	−1.43223	−0.716117	−	0.697981i	$-0.754082\pi$
$503$	−32.1216	−1.43223	−0.716117	+	0.697981i	$0.754082\pi$
$504$	0	0
$505$	2.27461	0.101219
$506$	0	0
$507$	−5.07819	−0.225530
$508$	0	0
$509$	−21.3655	−0.947009	−0.473504	−	0.880791i	$-0.657011\pi$
$509$	−21.3655	−0.947009	−0.473504	+	0.880791i	$0.657011\pi$
$510$	0	0
$511$	27.3051	1.20791
$512$	0	0
$513$	4.82798	0.213160
$514$	0	0
$515$	−23.1993	−1.02228
$516$	0	0
$517$	−58.9324	−2.59184
$518$	0	0
$519$	−19.1154	−0.839072
$520$	0	0
$521$	4.55492	0.199555	0.0997774	−	0.995010i	$-0.468187\pi$
$521$	4.55492	0.199555	0.0997774	+	0.995010i	$0.468187\pi$
$522$	0	0
$523$	2.01218	0.0879864	0.0439932	−	0.999032i	$-0.485992\pi$
$523$	2.01218	0.0879864	0.0439932	+	0.999032i	$0.485992\pi$
$524$	0	0
$525$	−8.15562	−0.355940
$526$	0	0
$527$	10.1389	0.441660
$528$	0	0
$529$	−11.6820	−0.507913
$530$	0	0
$531$	−1.96751	−0.0853826
$532$	0	0
$533$	−3.91659	−0.169646
$534$	0	0
$535$	−16.2407	−0.702145
$536$	0	0
$537$	−8.40836	−0.362848
$538$	0	0
$539$	−19.8858	−0.856541
$540$	0	0
$541$	17.1502	0.737343	0.368672	−	0.929560i	$-0.379813\pi$
$541$	17.1502	0.737343	0.368672	+	0.929560i	$0.379813\pi$
$542$	0	0
$543$	−20.3287	−0.872388
$544$	0	0
$545$	−4.25550	−0.182286
$546$	0	0
$547$	−5.92215	−0.253213	−0.126606	−	0.991953i	$-0.540408\pi$
$547$	−5.92215	−0.253213	−0.126606	+	0.991953i	$0.540408\pi$
$548$	0	0
$549$	−3.76751	−0.160794
$550$	0	0
$551$	−42.3177	−1.80280
$552$	0	0
$553$	11.8739	0.504928
$554$	0	0
$555$	11.4352	0.485399
$556$	0	0
$557$	20.1694	0.854603	0.427301	−	0.904109i	$-0.359464\pi$
$557$	20.1694	0.854603	0.427301	+	0.904109i	$0.359464\pi$
$558$	0	0
$559$	8.06410	0.341075
$560$	0	0
$561$	−25.8361	−1.09080
$562$	0	0
$563$	−35.6864	−1.50400	−0.752002	−	0.659161i	$-0.770911\pi$
$563$	−35.6864	−1.50400	−0.752002	+	0.659161i	$0.770911\pi$
$564$	0	0
$565$	−0.657868	−0.0276767
$566$	0	0
$567$	3.21458	0.134999
$568$	0	0
$569$	−34.0630	−1.42800	−0.713998	−	0.700148i	$-0.753117\pi$
$569$	−34.0630	−1.42800	−0.713998	+	0.700148i	$0.753117\pi$
$570$	0	0
$571$	34.9133	1.46108	0.730538	−	0.682872i	$-0.239270\pi$
$571$	34.9133	1.46108	0.730538	+	0.682872i	$0.239270\pi$
$572$	0	0
$573$	24.2261	1.01206
$574$	0	0
$575$	−8.53529	−0.355946
$576$	0	0
$577$	15.2801	0.636117	0.318059	−	0.948071i	$-0.396969\pi$
$577$	15.2801	0.636117	0.318059	+	0.948071i	$0.396969\pi$
$578$	0	0
$579$	−12.2851	−0.510552
$580$	0	0
$581$	26.8079	1.11218
$582$	0	0
$583$	67.2256	2.78420
$584$	0	0
$585$	4.41711	0.182625
$586$	0	0
$587$	4.37405	0.180536	0.0902682	−	0.995917i	$-0.471228\pi$
$587$	4.37405	0.180536	0.0902682	+	0.995917i	$0.471228\pi$
$588$	0	0
$589$	11.3025	0.465710
$590$	0	0
$591$	−15.3618	−0.631899
$592$	0	0
$593$	−33.2932	−1.36719	−0.683595	−	0.729862i	$-0.739584\pi$
$593$	−33.2932	−1.36719	−0.683595	+	0.729862i	$0.739584\pi$
$594$	0	0
$595$	21.8491	0.895726
$596$	0	0
$597$	8.14641	0.333410
$598$	0	0
$599$	8.35724	0.341468	0.170734	−	0.985317i	$-0.445386\pi$
$599$	8.35724	0.341468	0.170734	+	0.985317i	$0.445386\pi$
$600$	0	0
$601$	−33.2266	−1.35534	−0.677671	−	0.735365i	$-0.737011\pi$
$601$	−33.2266	−1.35534	−0.677671	+	0.735365i	$0.737011\pi$
$602$	0	0
$603$	11.6949	0.476251
$604$	0	0
$605$	38.5854	1.56872
$606$	0	0
$607$	−17.0042	−0.690180	−0.345090	−	0.938570i	$-0.612152\pi$
$607$	−17.0042	−0.690180	−0.345090	+	0.938570i	$0.612152\pi$
$608$	0	0
$609$	−28.1761	−1.14175
$610$	0	0
$611$	27.8050	1.12487
$612$	0	0
$613$	−4.21522	−0.170251	−0.0851255	−	0.996370i	$-0.527129\pi$
$613$	−4.21522	−0.170251	−0.0851255	+	0.996370i	$0.527129\pi$
$614$	0	0
$615$	−2.18384	−0.0880610
$616$	0	0
$617$	−45.4867	−1.83123	−0.915613	−	0.402061i	$-0.868294\pi$
$617$	−45.4867	−1.83123	−0.915613	+	0.402061i	$0.868294\pi$
$618$	0	0
$619$	17.5216	0.704252	0.352126	−	0.935953i	$-0.385459\pi$
$619$	17.5216	0.704252	0.352126	+	0.935953i	$0.385459\pi$
$620$	0	0
$621$	3.36422	0.135002
$622$	0	0
$623$	39.7314	1.59181
$624$	0	0
$625$	−5.87787	−0.235115
$626$	0	0
$627$	−28.8010	−1.15020
$628$	0	0
$629$	31.5576	1.25828
$630$	0	0
$631$	−26.8300	−1.06809	−0.534043	−	0.845458i	$-0.679328\pi$
$631$	−26.8300	−1.06809	−0.534043	+	0.845458i	$0.679328\pi$
$632$	0	0
$633$	−12.7610	−0.507204
$634$	0	0
$635$	−22.7675	−0.903501
$636$	0	0
$637$	9.38236	0.371743
$638$	0	0
$639$	−4.01255	−0.158734
$640$	0	0
$641$	2.25848	0.0892044	0.0446022	−	0.999005i	$-0.485798\pi$
$641$	2.25848	0.0892044	0.0446022	+	0.999005i	$0.485798\pi$
$642$	0	0
$643$	−14.0840	−0.555418	−0.277709	−	0.960665i	$-0.589575\pi$
$643$	−14.0840	−0.555418	−0.277709	+	0.960665i	$0.589575\pi$
$644$	0	0
$645$	4.49645	0.177047
$646$	0	0
$647$	18.9010	0.743076	0.371538	−	0.928418i	$-0.378831\pi$
$647$	18.9010	0.743076	0.371538	+	0.928418i	$0.378831\pi$
$648$	0	0
$649$	11.7371	0.460720
$650$	0	0
$651$	7.52544	0.294945
$652$	0	0
$653$	−19.4383	−0.760680	−0.380340	−	0.924847i	$-0.624193\pi$
$653$	−19.4383	−0.760680	−0.380340	+	0.924847i	$0.624193\pi$
$654$	0	0
$655$	30.7893	1.20304
$656$	0	0
$657$	8.49415	0.331388
$658$	0	0
$659$	−21.8476	−0.851061	−0.425531	−	0.904944i	$-0.639912\pi$
$659$	−21.8476	−0.851061	−0.425531	+	0.904944i	$0.639912\pi$
$660$	0	0
$661$	−8.02092	−0.311978	−0.155989	−	0.987759i	$-0.549856\pi$
$661$	−8.02092	−0.311978	−0.155989	+	0.987759i	$0.549856\pi$
$662$	0	0
$663$	12.1898	0.473413
$664$	0	0
$665$	24.3565	0.944503
$666$	0	0
$667$	−29.4878	−1.14177
$668$	0	0
$669$	9.88563	0.382201
$670$	0	0
$671$	22.4749	0.867634
$672$	0	0
$673$	8.07660	0.311330	0.155665	−	0.987810i	$-0.450248\pi$
$673$	8.07660	0.311330	0.155665	+	0.987810i	$0.450248\pi$
$674$	0	0
$675$	−2.53708	−0.0976521
$676$	0	0
$677$	−26.4185	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$677$	−26.4185	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$678$	0	0
$679$	1.87404	0.0719191
$680$	0	0
$681$	−0.485281	−0.0185960
$682$	0	0
$683$	−50.8346	−1.94513	−0.972565	−	0.232630i	$-0.925267\pi$
$683$	−50.8346	−1.94513	−0.972565	+	0.232630i	$0.925267\pi$
$684$	0	0
$685$	−12.3928	−0.473506
$686$	0	0
$687$	−28.3959	−1.08337
$688$	0	0
$689$	−31.7179	−1.20836
$690$	0	0
$691$	1.96602	0.0747910	0.0373955	−	0.999301i	$-0.488094\pi$
$691$	1.96602	0.0747910	0.0373955	+	0.999301i	$0.488094\pi$
$692$	0	0
$693$	−19.1764	−0.728450
$694$	0	0
$695$	−9.98855	−0.378887
$696$	0	0
$697$	−6.02671	−0.228278
$698$	0	0
$699$	−26.5196	−1.00306
$700$	0	0
$701$	42.9796	1.62332	0.811659	−	0.584132i	$-0.198565\pi$
$701$	42.9796	1.62332	0.811659	+	0.584132i	$0.198565\pi$
$702$	0	0
$703$	35.1791	1.32680
$704$	0	0
$705$	15.5037	0.583905
$706$	0	0
$707$	4.65913	0.175224
$708$	0	0
$709$	17.9765	0.675121	0.337561	−	0.941304i	$-0.390398\pi$
$709$	17.9765	0.675121	0.337561	+	0.941304i	$0.390398\pi$
$710$	0	0
$711$	3.69376	0.138527
$712$	0	0
$713$	7.87577	0.294950
$714$	0	0
$715$	−26.3500	−0.985434
$716$	0	0
$717$	13.4261	0.501408
$718$	0	0
$719$	−51.1045	−1.90588	−0.952938	−	0.303165i	$-0.901957\pi$
$719$	−51.1045	−1.90588	−0.952938	+	0.303165i	$0.901957\pi$
$720$	0	0
$721$	−47.5196	−1.76972
$722$	0	0
$723$	11.4392	0.425429
$724$	0	0
$725$	22.2377	0.825889
$726$	0	0
$727$	−21.8541	−0.810523	−0.405262	−	0.914201i	$-0.632820\pi$
$727$	−21.8541	−0.810523	−0.405262	+	0.914201i	$0.632820\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.4088	0.458955
$732$	0	0
$733$	−26.5674	−0.981290	−0.490645	−	0.871360i	$-0.663239\pi$
$733$	−26.5674	−0.981290	−0.490645	+	0.871360i	$0.663239\pi$
$734$	0	0
$735$	5.23149	0.192966
$736$	0	0
$737$	−69.7650	−2.56983
$738$	0	0
$739$	48.6605	1.79001	0.895003	−	0.446060i	$-0.147173\pi$
$739$	48.6605	1.79001	0.895003	+	0.446060i	$0.147173\pi$
$740$	0	0
$741$	13.5887	0.499193
$742$	0	0
$743$	7.70496	0.282668	0.141334	−	0.989962i	$-0.454861\pi$
$743$	7.70496	0.282668	0.141334	+	0.989962i	$0.454861\pi$
$744$	0	0
$745$	14.3815	0.526896
$746$	0	0
$747$	8.33948	0.305126
$748$	0	0
$749$	−33.2661	−1.21552
$750$	0	0
$751$	−8.01994	−0.292652	−0.146326	−	0.989236i	$-0.546745\pi$
$751$	−8.01994	−0.292652	−0.146326	+	0.989236i	$0.546745\pi$
$752$	0	0
$753$	−16.4258	−0.598591
$754$	0	0
$755$	25.5804	0.930965
$756$	0	0
$757$	23.5836	0.857161	0.428580	−	0.903504i	$-0.359014\pi$
$757$	23.5836	0.857161	0.428580	+	0.903504i	$0.359014\pi$
$758$	0	0
$759$	−20.0691	−0.728462
$760$	0	0
$761$	−34.8994	−1.26510	−0.632551	−	0.774518i	$-0.717992\pi$
$761$	−34.8994	−1.26510	−0.632551	+	0.774518i	$0.717992\pi$
$762$	0	0
$763$	−8.71663	−0.315563
$764$	0	0
$765$	6.79689	0.245742
$766$	0	0
$767$	−5.53769	−0.199954
$768$	0	0
$769$	13.0487	0.470549	0.235275	−	0.971929i	$-0.424401\pi$
$769$	13.0487	0.470549	0.235275	+	0.971929i	$0.424401\pi$
$770$	0	0
$771$	17.6649	0.636185
$772$	0	0
$773$	7.14375	0.256943	0.128471	−	0.991713i	$-0.458993\pi$
$773$	7.14375	0.256943	0.128471	+	0.991713i	$0.458993\pi$
$774$	0	0
$775$	−5.93939	−0.213349
$776$	0	0
$777$	23.4230	0.840296
$778$	0	0
$779$	−6.71832	−0.240709
$780$	0	0
$781$	23.9367	0.856521
$782$	0	0
$783$	−8.76511	−0.313239
$784$	0	0
$785$	28.7176	1.02497
$786$	0	0
$787$	−7.36950	−0.262694	−0.131347	−	0.991336i	$-0.541930\pi$
$787$	−7.36950	−0.262694	−0.131347	+	0.991336i	$0.541930\pi$
$788$	0	0
$789$	7.60215	0.270644
$790$	0	0
$791$	−1.34752	−0.0479124
$792$	0	0
$793$	−10.6039	−0.376557
$794$	0	0
$795$	−17.6855	−0.627240
$796$	0	0
$797$	−11.3558	−0.402243	−0.201122	−	0.979566i	$-0.564459\pi$
$797$	−11.3558	−0.402243	−0.201122	+	0.979566i	$0.564459\pi$
$798$	0	0
$799$	42.7854	1.51364
$800$	0	0
$801$	12.3598	0.436711
$802$	0	0
$803$	−50.6714	−1.78815
$804$	0	0
$805$	16.9720	0.598186
$806$	0	0
$807$	−7.33835	−0.258322
$808$	0	0
$809$	40.4208	1.42112	0.710560	−	0.703637i	$-0.248442\pi$
$809$	40.4208	1.42112	0.710560	+	0.703637i	$0.248442\pi$
$810$	0	0
$811$	−38.6275	−1.35639	−0.678197	−	0.734880i	$-0.737238\pi$
$811$	−38.6275	−1.35639	−0.678197	+	0.734880i	$0.737238\pi$
$812$	0	0
$813$	30.7629	1.07890
$814$	0	0
$815$	−15.4187	−0.540093
$816$	0	0
$817$	13.8328	0.483947
$818$	0	0
$819$	9.04765	0.316151
$820$	0	0
$821$	12.1383	0.423631	0.211815	−	0.977310i	$-0.432062\pi$
$821$	12.1383	0.423631	0.211815	+	0.977310i	$0.432062\pi$
$822$	0	0
$823$	36.6833	1.27870	0.639350	−	0.768915i	$-0.279203\pi$
$823$	36.6833	1.27870	0.639350	+	0.768915i	$0.279203\pi$
$824$	0	0
$825$	15.1348	0.526925
$826$	0	0
$827$	−29.4355	−1.02357	−0.511787	−	0.859113i	$-0.671016\pi$
$827$	−29.4355	−1.02357	−0.511787	+	0.859113i	$0.671016\pi$
$828$	0	0
$829$	−8.55858	−0.297252	−0.148626	−	0.988894i	$-0.547485\pi$
$829$	−8.55858	−0.297252	−0.148626	+	0.988894i	$0.547485\pi$
$830$	0	0
$831$	15.0041	0.520486
$832$	0	0
$833$	14.4372	0.500221
$834$	0	0
$835$	1.56937	0.0543103
$836$	0	0
$837$	2.34104	0.0809181
$838$	0	0
$839$	49.4086	1.70578	0.852888	−	0.522094i	$-0.174849\pi$
$839$	49.4086	1.70578	0.852888	+	0.522094i	$0.174849\pi$
$840$	0	0
$841$	47.8271	1.64921
$842$	0	0
$843$	10.2001	0.351311
$844$	0	0
$845$	−7.96956	−0.274161
$846$	0	0
$847$	79.0352	2.71568
$848$	0	0
$849$	2.65541	0.0911333
$850$	0	0
$851$	24.5134	0.840310
$852$	0	0
$853$	−28.6271	−0.980173	−0.490087	−	0.871674i	$-0.663035\pi$
$853$	−28.6271	−0.980173	−0.490087	+	0.871674i	$0.663035\pi$
$854$	0	0
$855$	7.57689	0.259124
$856$	0	0
$857$	−25.9474	−0.886346	−0.443173	−	0.896436i	$-0.646147\pi$
$857$	−25.9474	−0.886346	−0.443173	+	0.896436i	$0.646147\pi$
$858$	0	0
$859$	−31.5220	−1.07552	−0.537758	−	0.843099i	$-0.680729\pi$
$859$	−31.5220	−1.07552	−0.537758	+	0.843099i	$0.680729\pi$
$860$	0	0
$861$	−4.47321	−0.152447
$862$	0	0
$863$	24.3259	0.828065	0.414032	−	0.910262i	$-0.364120\pi$
$863$	24.3259	0.828065	0.414032	+	0.910262i	$0.364120\pi$
$864$	0	0
$865$	−29.9991	−1.02000
$866$	0	0
$867$	1.75725	0.0596793
$868$	0	0
$869$	−22.0349	−0.747483
$870$	0	0
$871$	32.9160	1.11532
$872$	0	0
$873$	0.582982	0.0197310
$874$	0	0
$875$	−38.0235	−1.28543
$876$	0	0
$877$	−33.5288	−1.13219	−0.566094	−	0.824341i	$-0.691546\pi$
$877$	−33.5288	−1.13219	−0.566094	+	0.824341i	$0.691546\pi$
$878$	0	0
$879$	19.0956	0.644079
$880$	0	0
$881$	30.6148	1.03144	0.515719	−	0.856758i	$-0.327525\pi$
$881$	30.6148	1.03144	0.515719	+	0.856758i	$0.327525\pi$
$882$	0	0
$883$	−25.0058	−0.841511	−0.420756	−	0.907174i	$-0.638235\pi$
$883$	−25.0058	−0.841511	−0.420756	+	0.907174i	$0.638235\pi$
$884$	0	0
$885$	−3.08775	−0.103794
$886$	0	0
$887$	13.2595	0.445212	0.222606	−	0.974909i	$-0.428544\pi$
$887$	13.2595	0.445212	0.222606	+	0.974909i	$0.428544\pi$
$888$	0	0
$889$	−46.6352	−1.56409
$890$	0	0
$891$	−5.96544	−0.199850
$892$	0	0
$893$	47.6954	1.59606
$894$	0	0
$895$	−13.1958	−0.441088
$896$	0	0
$897$	9.46885	0.316156
$898$	0	0
$899$	−20.5194	−0.684362
$900$	0	0
$901$	−48.8064	−1.62598
$902$	0	0
$903$	9.21017	0.306495
$904$	0	0
$905$	−31.9033	−1.06050
$906$	0	0
$907$	−22.6968	−0.753636	−0.376818	−	0.926287i	$-0.622982\pi$
$907$	−22.6968	−0.753636	−0.376818	+	0.926287i	$0.622982\pi$
$908$	0	0
$909$	1.44938	0.0480727
$910$	0	0
$911$	2.15372	0.0713559	0.0356780	−	0.999363i	$-0.488641\pi$
$911$	2.15372	0.0713559	0.0356780	+	0.999363i	$0.488641\pi$
$912$	0	0
$913$	−49.7487	−1.64644
$914$	0	0
$915$	−5.91263	−0.195465
$916$	0	0
$917$	63.0663	2.08263
$918$	0	0
$919$	9.83614	0.324464	0.162232	−	0.986753i	$-0.448131\pi$
$919$	9.83614	0.324464	0.162232	+	0.986753i	$0.448131\pi$
$920$	0	0
$921$	2.29511	0.0756264
$922$	0	0
$923$	−11.2936	−0.371734
$924$	0	0
$925$	−18.4864	−0.607830
$926$	0	0
$927$	−14.7825	−0.485523
$928$	0	0
$929$	14.1814	0.465277	0.232639	−	0.972563i	$-0.425264\pi$
$929$	14.1814	0.465277	0.232639	+	0.972563i	$0.425264\pi$
$930$	0	0
$931$	16.0940	0.527461
$932$	0	0
$933$	−0.877859	−0.0287398
$934$	0	0
$935$	−40.5465	−1.32601
$936$	0	0
$937$	−7.82726	−0.255705	−0.127853	−	0.991793i	$-0.540808\pi$
$937$	−7.82726	−0.255705	−0.127853	+	0.991793i	$0.540808\pi$
$938$	0	0
$939$	−18.9459	−0.618276
$940$	0	0
$941$	−41.7609	−1.36137	−0.680683	−	0.732578i	$-0.738317\pi$
$941$	−41.7609	−1.36137	−0.680683	+	0.732578i	$0.738317\pi$
$942$	0	0
$943$	−4.68145	−0.152449
$944$	0	0
$945$	5.04486	0.164109
$946$	0	0
$947$	27.7498	0.901747	0.450874	−	0.892588i	$-0.351113\pi$
$947$	27.7498	0.901747	0.450874	+	0.892588i	$0.351113\pi$
$948$	0	0
$949$	23.9074	0.776067
$950$	0	0
$951$	32.8653	1.06573
$952$	0	0
$953$	−24.8241	−0.804130	−0.402065	−	0.915611i	$-0.631707\pi$
$953$	−24.8241	−0.804130	−0.402065	+	0.915611i	$0.631707\pi$
$954$	0	0
$955$	38.0198	1.23029
$956$	0	0
$957$	52.2878	1.69022
$958$	0	0
$959$	−25.3845	−0.819708
$960$	0	0
$961$	−25.5195	−0.823211
$962$	0	0
$963$	−10.3485	−0.333476
$964$	0	0
$965$	−19.2799	−0.620641
$966$	0	0
$967$	21.3164	0.685489	0.342744	−	0.939429i	$-0.388644\pi$
$967$	21.3164	0.685489	0.342744	+	0.939429i	$0.388644\pi$
$968$	0	0
$969$	20.9098	0.671719
$970$	0	0
$971$	36.8650	1.18305	0.591527	−	0.806285i	$-0.298525\pi$
$971$	36.8650	1.18305	0.591527	+	0.806285i	$0.298525\pi$
$972$	0	0
$973$	−20.4597	−0.655909
$974$	0	0
$975$	−7.14078	−0.228688
$976$	0	0
$977$	17.1237	0.547837	0.273918	−	0.961753i	$-0.411680\pi$
$977$	17.1237	0.547837	0.273918	+	0.961753i	$0.411680\pi$
$978$	0	0
$979$	−73.7316	−2.35647
$980$	0	0
$981$	−2.71160	−0.0865746
$982$	0	0
$983$	−19.0292	−0.606936	−0.303468	−	0.952842i	$-0.598145\pi$
$983$	−19.0292	−0.606936	−0.303468	+	0.952842i	$0.598145\pi$
$984$	0	0
$985$	−24.1083	−0.768155
$986$	0	0
$987$	31.7566	1.01083
$988$	0	0
$989$	9.63893	0.306500
$990$	0	0
$991$	28.2266	0.896648	0.448324	−	0.893871i	$-0.352021\pi$
$991$	28.2266	0.896648	0.448324	+	0.893871i	$0.352021\pi$
$992$	0	0
$993$	6.81707	0.216333
$994$	0	0
$995$	12.7847	0.405303
$996$	0	0
$997$	−30.3558	−0.961379	−0.480689	−	0.876891i	$-0.659613\pi$
$997$	−30.3558	−0.961379	−0.480689	+	0.876891i	$0.659613\pi$
$998$	0	0
$999$	7.28651	0.230535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.l.1.8	✓	12
4.3	odd	2		8016.2.a.bf.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.8	✓	12	1.1	even	1	trivial
8016.2.a.bf.1.8		12	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.56937 q^{5} +3.21458 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.56937 q^{5} +3.21458 q^{7} +1.00000 q^{9} -5.96544 q^{11} +2.81457 q^{13} +1.56937 q^{15} +4.33096 q^{17} +4.82798 q^{19} +3.21458 q^{21} +3.36422 q^{23} -2.53708 q^{25} +1.00000 q^{27} -8.76511 q^{29} +2.34104 q^{31} -5.96544 q^{33} +5.04486 q^{35} +7.28651 q^{37} +2.81457 q^{39} -1.39154 q^{41} +2.86513 q^{43} +1.56937 q^{45} +9.87896 q^{47} +3.33349 q^{49} +4.33096 q^{51} -11.2692 q^{53} -9.36199 q^{55} +4.82798 q^{57} -1.96751 q^{59} -3.76751 q^{61} +3.21458 q^{63} +4.41711 q^{65} +11.6949 q^{67} +3.36422 q^{69} -4.01255 q^{71} +8.49415 q^{73} -2.53708 q^{75} -19.1764 q^{77} +3.69376 q^{79} +1.00000 q^{81} +8.33948 q^{83} +6.79689 q^{85} -8.76511 q^{87} +12.3598 q^{89} +9.04765 q^{91} +2.34104 q^{93} +7.57689 q^{95} +0.582982 q^{97} -5.96544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * q^97 + q^99

Introduction

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