LMFDB - Embedded newform 4008.2.a.j.1.5

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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 $ x^10 - 26x^8 - 3x^7 + 220x^6 + 42x^5 - 675x^4 - 67x^3 + 628x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.108417$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.10842 q^{5} -3.58606 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.10842 q^{5} -3.58606 q^{7} +1.00000 q^{9} +5.82693 q^{11} -1.93527 q^{13} +1.10842 q^{15} +2.35109 q^{17} -8.34063 q^{19} +3.58606 q^{21} +8.17815 q^{23} -3.77141 q^{25} -1.00000 q^{27} -1.11196 q^{29} +6.08391 q^{31} -5.82693 q^{33} +3.97485 q^{35} +1.22871 q^{37} +1.93527 q^{39} -0.437215 q^{41} -5.51348 q^{43} -1.10842 q^{45} +1.26678 q^{47} +5.85981 q^{49} -2.35109 q^{51} +2.37338 q^{53} -6.45867 q^{55} +8.34063 q^{57} +2.91215 q^{59} -2.63174 q^{61} -3.58606 q^{63} +2.14509 q^{65} +10.0579 q^{67} -8.17815 q^{69} +9.30176 q^{71} +1.33390 q^{73} +3.77141 q^{75} -20.8957 q^{77} -6.20776 q^{79} +1.00000 q^{81} +6.69681 q^{83} -2.60599 q^{85} +1.11196 q^{87} -6.61497 q^{89} +6.93999 q^{91} -6.08391 q^{93} +9.24489 q^{95} -15.7311 q^{97} +5.82693 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9	$ 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9} - q^{11} - 6 q^{13} + 10 q^{15} - 9 q^{17} + 2 q^{19} - q^{21} + 7 q^{23} + 12 q^{25} - 10 q^{27} - 13 q^{29} + 23 q^{31} + q^{33} + q^{35} - 6 q^{37} + 6 q^{39} - 12 q^{41} - 10 q^{45} + 10 q^{47} + 7 q^{49} + 9 q^{51} - 26 q^{53} + 11 q^{55} - 2 q^{57} - 10 q^{59} - 10 q^{61} + q^{63} - 22 q^{65} - 5 q^{67} - 7 q^{69} + 25 q^{71} - 8 q^{73} - 12 q^{75} - 46 q^{77} + 26 q^{79} + 10 q^{81} - 14 q^{83} + 9 q^{85} + 13 q^{87} - 31 q^{89} - 3 q^{91} - 23 q^{93} - 5 q^{95} - 32 q^{97} - q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9 - q^11 - 6 * q^13 + 10 * q^15 - 9 * q^17 + 2 * q^19 - q^21 + 7 * q^23 + 12 * q^25 - 10 * q^27 - 13 * q^29 + 23 * q^31 + q^33 + q^35 - 6 * q^37 + 6 * q^39 - 12 * q^41 - 10 * q^45 + 10 * q^47 + 7 * q^49 + 9 * q^51 - 26 * q^53 + 11 * q^55 - 2 * q^57 - 10 * q^59 - 10 * q^61 + q^63 - 22 * q^65 - 5 * q^67 - 7 * q^69 + 25 * q^71 - 8 * q^73 - 12 * q^75 - 46 * q^77 + 26 * q^79 + 10 * q^81 - 14 * q^83 + 9 * q^85 + 13 * q^87 - 31 * q^89 - 3 * q^91 - 23 * q^93 - 5 * q^95 - 32 * 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
$5$	−1.10842	−0.495699	−0.247850	−	0.968799i	$-0.579724\pi$
$5$	−1.10842	−0.495699	−0.247850	+	0.968799i	$0.579724\pi$
$6$	0	0
$7$	−3.58606	−1.35540	−0.677701	−	0.735337i	$-0.737024\pi$
$7$	−3.58606	−1.35540	−0.677701	+	0.735337i	$0.737024\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.82693	1.75689	0.878443	−	0.477847i	$-0.158582\pi$
$11$	5.82693	1.75689	0.878443	+	0.477847i	$0.158582\pi$
$12$	0	0
$13$	−1.93527	−0.536748	−0.268374	−	0.963315i	$-0.586486\pi$
$13$	−1.93527	−0.536748	−0.268374	+	0.963315i	$0.586486\pi$
$14$	0	0
$15$	1.10842	0.286192
$16$	0	0
$17$	2.35109	0.570224	0.285112	−	0.958494i	$-0.407969\pi$
$17$	2.35109	0.570224	0.285112	+	0.958494i	$0.407969\pi$
$18$	0	0
$19$	−8.34063	−1.91347	−0.956736	−	0.290959i	$-0.906026\pi$
$19$	−8.34063	−1.91347	−0.956736	+	0.290959i	$0.906026\pi$
$20$	0	0
$21$	3.58606	0.782542
$22$	0	0
$23$	8.17815	1.70526	0.852631	−	0.522513i	$-0.175005\pi$
$23$	8.17815	1.70526	0.852631	+	0.522513i	$0.175005\pi$
$24$	0	0
$25$	−3.77141	−0.754282
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.11196	−0.206486	−0.103243	−	0.994656i	$-0.532922\pi$
$29$	−1.11196	−0.206486	−0.103243	+	0.994656i	$0.532922\pi$
$30$	0	0
$31$	6.08391	1.09270	0.546351	−	0.837556i	$-0.316016\pi$
$31$	6.08391	1.09270	0.546351	+	0.837556i	$0.316016\pi$
$32$	0	0
$33$	−5.82693	−1.01434
$34$	0	0
$35$	3.97485	0.671872
$36$	0	0
$37$	1.22871	0.201998	0.100999	−	0.994887i	$-0.467796\pi$
$37$	1.22871	0.201998	0.100999	+	0.994887i	$0.467796\pi$
$38$	0	0
$39$	1.93527	0.309891
$40$	0	0
$41$	−0.437215	−0.0682815	−0.0341408	−	0.999417i	$-0.510869\pi$
$41$	−0.437215	−0.0682815	−0.0341408	+	0.999417i	$0.510869\pi$
$42$	0	0
$43$	−5.51348	−0.840797	−0.420399	−	0.907340i	$-0.638110\pi$
$43$	−5.51348	−0.840797	−0.420399	+	0.907340i	$0.638110\pi$
$44$	0	0
$45$	−1.10842	−0.165233
$46$	0	0
$47$	1.26678	0.184779	0.0923893	−	0.995723i	$-0.470550\pi$
$47$	1.26678	0.184779	0.0923893	+	0.995723i	$0.470550\pi$
$48$	0	0
$49$	5.85981	0.837115
$50$	0	0
$51$	−2.35109	−0.329219
$52$	0	0
$53$	2.37338	0.326008	0.163004	−	0.986625i	$-0.447882\pi$
$53$	2.37338	0.326008	0.163004	+	0.986625i	$0.447882\pi$
$54$	0	0
$55$	−6.45867	−0.870887
$56$	0	0
$57$	8.34063	1.10474
$58$	0	0
$59$	2.91215	0.379129	0.189565	−	0.981868i	$-0.439292\pi$
$59$	2.91215	0.379129	0.189565	+	0.981868i	$0.439292\pi$
$60$	0	0
$61$	−2.63174	−0.336960	−0.168480	−	0.985705i	$-0.553886\pi$
$61$	−2.63174	−0.336960	−0.168480	+	0.985705i	$0.553886\pi$
$62$	0	0
$63$	−3.58606	−0.451801
$64$	0	0
$65$	2.14509	0.266065
$66$	0	0
$67$	10.0579	1.22877	0.614386	−	0.789006i	$-0.289404\pi$
$67$	10.0579	1.22877	0.614386	+	0.789006i	$0.289404\pi$
$68$	0	0
$69$	−8.17815	−0.984534
$70$	0	0
$71$	9.30176	1.10392	0.551958	−	0.833872i	$-0.313881\pi$
$71$	9.30176	1.10392	0.551958	+	0.833872i	$0.313881\pi$
$72$	0	0
$73$	1.33390	0.156121	0.0780606	−	0.996949i	$-0.475127\pi$
$73$	1.33390	0.156121	0.0780606	+	0.996949i	$0.475127\pi$
$74$	0	0
$75$	3.77141	0.435485
$76$	0	0
$77$	−20.8957	−2.38129
$78$	0	0
$79$	−6.20776	−0.698427	−0.349214	−	0.937043i	$-0.613551\pi$
$79$	−6.20776	−0.698427	−0.349214	+	0.937043i	$0.613551\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.69681	0.735071	0.367535	−	0.930010i	$-0.380202\pi$
$83$	6.69681	0.735071	0.367535	+	0.930010i	$0.380202\pi$
$84$	0	0
$85$	−2.60599	−0.282660
$86$	0	0
$87$	1.11196	0.119215
$88$	0	0
$89$	−6.61497	−0.701186	−0.350593	−	0.936528i	$-0.614020\pi$
$89$	−6.61497	−0.701186	−0.350593	+	0.936528i	$0.614020\pi$
$90$	0	0
$91$	6.93999	0.727509
$92$	0	0
$93$	−6.08391	−0.630872
$94$	0	0
$95$	9.24489	0.948506
$96$	0	0
$97$	−15.7311	−1.59726	−0.798628	−	0.601825i	$-0.794440\pi$
$97$	−15.7311	−1.59726	−0.798628	+	0.601825i	$0.794440\pi$
$98$	0	0
$99$	5.82693	0.585629
$100$	0	0
$101$	1.62087	0.161282	0.0806411	−	0.996743i	$-0.474303\pi$
$101$	1.62087	0.161282	0.0806411	+	0.996743i	$0.474303\pi$
$102$	0	0
$103$	2.78462	0.274376	0.137188	−	0.990545i	$-0.456194\pi$
$103$	2.78462	0.274376	0.137188	+	0.990545i	$0.456194\pi$
$104$	0	0
$105$	−3.97485	−0.387905
$106$	0	0
$107$	−13.0772	−1.26422	−0.632110	−	0.774879i	$-0.717811\pi$
$107$	−13.0772	−1.26422	−0.632110	+	0.774879i	$0.717811\pi$
$108$	0	0
$109$	−1.53097	−0.146640	−0.0733202	−	0.997308i	$-0.523359\pi$
$109$	−1.53097	−0.146640	−0.0733202	+	0.997308i	$0.523359\pi$
$110$	0	0
$111$	−1.22871	−0.116624
$112$	0	0
$113$	−18.1720	−1.70948	−0.854741	−	0.519055i	$-0.826284\pi$
$113$	−18.1720	−1.70948	−0.854741	+	0.519055i	$0.826284\pi$
$114$	0	0
$115$	−9.06480	−0.845297
$116$	0	0
$117$	−1.93527	−0.178916
$118$	0	0
$119$	−8.43116	−0.772883
$120$	0	0
$121$	22.9532	2.08665
$122$	0	0
$123$	0.437215	0.0394223
$124$	0	0
$125$	9.72238	0.869596
$126$	0	0
$127$	−13.5455	−1.20197	−0.600986	−	0.799260i	$-0.705225\pi$
$127$	−13.5455	−1.20197	−0.600986	+	0.799260i	$0.705225\pi$
$128$	0	0
$129$	5.51348	0.485434
$130$	0	0
$131$	−11.5340	−1.00773	−0.503866	−	0.863782i	$-0.668089\pi$
$131$	−11.5340	−1.00773	−0.503866	+	0.863782i	$0.668089\pi$
$132$	0	0
$133$	29.9100	2.59352
$134$	0	0
$135$	1.10842	0.0953974
$136$	0	0
$137$	−14.8636	−1.26988	−0.634941	−	0.772560i	$-0.718976\pi$
$137$	−14.8636	−1.26988	−0.634941	+	0.772560i	$0.718976\pi$
$138$	0	0
$139$	1.83355	0.155520	0.0777600	−	0.996972i	$-0.475223\pi$
$139$	1.83355	0.155520	0.0777600	+	0.996972i	$0.475223\pi$
$140$	0	0
$141$	−1.26678	−0.106682
$142$	0	0
$143$	−11.2767	−0.943005
$144$	0	0
$145$	1.23252	0.102355
$146$	0	0
$147$	−5.85981	−0.483309
$148$	0	0
$149$	−16.6162	−1.36125	−0.680625	−	0.732632i	$-0.738292\pi$
$149$	−16.6162	−1.36125	−0.680625	+	0.732632i	$0.738292\pi$
$150$	0	0
$151$	10.5836	0.861278	0.430639	−	0.902524i	$-0.358288\pi$
$151$	10.5836	0.861278	0.430639	+	0.902524i	$0.358288\pi$
$152$	0	0
$153$	2.35109	0.190075
$154$	0	0
$155$	−6.74351	−0.541652
$156$	0	0
$157$	−21.3072	−1.70050	−0.850249	−	0.526380i	$-0.823549\pi$
$157$	−21.3072	−1.70050	−0.850249	+	0.526380i	$0.823549\pi$
$158$	0	0
$159$	−2.37338	−0.188221
$160$	0	0
$161$	−29.3273	−2.31132
$162$	0	0
$163$	−18.5920	−1.45624	−0.728119	−	0.685451i	$-0.759605\pi$
$163$	−18.5920	−1.45624	−0.728119	+	0.685451i	$0.759605\pi$
$164$	0	0
$165$	6.45867	0.502807
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−9.25472	−0.711902
$170$	0	0
$171$	−8.34063	−0.637824
$172$	0	0
$173$	7.60016	0.577829	0.288915	−	0.957355i	$-0.406706\pi$
$173$	7.60016	0.577829	0.288915	+	0.957355i	$0.406706\pi$
$174$	0	0
$175$	13.5245	1.02236
$176$	0	0
$177$	−2.91215	−0.218890
$178$	0	0
$179$	4.08215	0.305114	0.152557	−	0.988295i	$-0.451249\pi$
$179$	4.08215	0.305114	0.152557	+	0.988295i	$0.451249\pi$
$180$	0	0
$181$	−5.91060	−0.439331	−0.219666	−	0.975575i	$-0.570497\pi$
$181$	−5.91060	−0.439331	−0.219666	+	0.975575i	$0.570497\pi$
$182$	0	0
$183$	2.63174	0.194544
$184$	0	0
$185$	−1.36192	−0.100130
$186$	0	0
$187$	13.6997	1.00182
$188$	0	0
$189$	3.58606	0.260847
$190$	0	0
$191$	3.06512	0.221784	0.110892	−	0.993832i	$-0.464629\pi$
$191$	3.06512	0.221784	0.110892	+	0.993832i	$0.464629\pi$
$192$	0	0
$193$	5.34361	0.384642	0.192321	−	0.981332i	$-0.438399\pi$
$193$	5.34361	0.384642	0.192321	+	0.981332i	$0.438399\pi$
$194$	0	0
$195$	−2.14509	−0.153613
$196$	0	0
$197$	−15.9344	−1.13528	−0.567640	−	0.823277i	$-0.692143\pi$
$197$	−15.9344	−1.13528	−0.567640	+	0.823277i	$0.692143\pi$
$198$	0	0
$199$	−2.74400	−0.194517	−0.0972584	−	0.995259i	$-0.531007\pi$
$199$	−2.74400	−0.194517	−0.0972584	+	0.995259i	$0.531007\pi$
$200$	0	0
$201$	−10.0579	−0.709431
$202$	0	0
$203$	3.98756	0.279872
$204$	0	0
$205$	0.484617	0.0338471
$206$	0	0
$207$	8.17815	0.568421
$208$	0	0
$209$	−48.6003	−3.36175
$210$	0	0
$211$	2.39097	0.164601	0.0823004	−	0.996608i	$-0.473773\pi$
$211$	2.39097	0.164601	0.0823004	+	0.996608i	$0.473773\pi$
$212$	0	0
$213$	−9.30176	−0.637346
$214$	0	0
$215$	6.11123	0.416782
$216$	0	0
$217$	−21.8173	−1.48105
$218$	0	0
$219$	−1.33390	−0.0901367
$220$	0	0
$221$	−4.55001	−0.306066
$222$	0	0
$223$	−9.20998	−0.616746	−0.308373	−	0.951266i	$-0.599784\pi$
$223$	−9.20998	−0.616746	−0.308373	+	0.951266i	$0.599784\pi$
$224$	0	0
$225$	−3.77141	−0.251427
$226$	0	0
$227$	−25.9471	−1.72217	−0.861085	−	0.508461i	$-0.830215\pi$
$227$	−25.9471	−1.72217	−0.861085	+	0.508461i	$0.830215\pi$
$228$	0	0
$229$	−10.7591	−0.710981	−0.355491	−	0.934680i	$-0.615686\pi$
$229$	−10.7591	−0.710981	−0.355491	+	0.934680i	$0.615686\pi$
$230$	0	0
$231$	20.8957	1.37484
$232$	0	0
$233$	−14.5686	−0.954419	−0.477210	−	0.878789i	$-0.658352\pi$
$233$	−14.5686	−0.954419	−0.477210	+	0.878789i	$0.658352\pi$
$234$	0	0
$235$	−1.40412	−0.0915946
$236$	0	0
$237$	6.20776	0.403237
$238$	0	0
$239$	16.4810	1.06607	0.533034	−	0.846094i	$-0.321052\pi$
$239$	16.4810	1.06607	0.533034	+	0.846094i	$0.321052\pi$
$240$	0	0
$241$	−10.3175	−0.664610	−0.332305	−	0.943172i	$-0.607826\pi$
$241$	−10.3175	−0.664610	−0.332305	+	0.943172i	$0.607826\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.49511	−0.414957
$246$	0	0
$247$	16.1414	1.02705
$248$	0	0
$249$	−6.69681	−0.424393
$250$	0	0
$251$	−24.4320	−1.54214	−0.771068	−	0.636753i	$-0.780277\pi$
$251$	−24.4320	−1.54214	−0.771068	+	0.636753i	$0.780277\pi$
$252$	0	0
$253$	47.6536	2.99595
$254$	0	0
$255$	2.60599	0.163194
$256$	0	0
$257$	20.7309	1.29316	0.646578	−	0.762848i	$-0.276199\pi$
$257$	20.7309	1.29316	0.646578	+	0.762848i	$0.276199\pi$
$258$	0	0
$259$	−4.40622	−0.273789
$260$	0	0
$261$	−1.11196	−0.0688288
$262$	0	0
$263$	22.3616	1.37888	0.689439	−	0.724344i	$-0.257857\pi$
$263$	22.3616	1.37888	0.689439	+	0.724344i	$0.257857\pi$
$264$	0	0
$265$	−2.63069	−0.161602
$266$	0	0
$267$	6.61497	0.404830
$268$	0	0
$269$	22.2023	1.35370	0.676849	−	0.736122i	$-0.263345\pi$
$269$	22.2023	1.35370	0.676849	+	0.736122i	$0.263345\pi$
$270$	0	0
$271$	3.34350	0.203103	0.101552	−	0.994830i	$-0.467619\pi$
$271$	3.34350	0.203103	0.101552	+	0.994830i	$0.467619\pi$
$272$	0	0
$273$	−6.93999	−0.420028
$274$	0	0
$275$	−21.9758	−1.32519
$276$	0	0
$277$	19.0759	1.14616	0.573079	−	0.819500i	$-0.305749\pi$
$277$	19.0759	1.14616	0.573079	+	0.819500i	$0.305749\pi$
$278$	0	0
$279$	6.08391	0.364234
$280$	0	0
$281$	−8.66664	−0.517008	−0.258504	−	0.966010i	$-0.583230\pi$
$281$	−8.66664	−0.517008	−0.258504	+	0.966010i	$0.583230\pi$
$282$	0	0
$283$	−14.1300	−0.839943	−0.419972	−	0.907537i	$-0.637960\pi$
$283$	−14.1300	−0.839943	−0.419972	+	0.907537i	$0.637960\pi$
$284$	0	0
$285$	−9.24489	−0.547620
$286$	0	0
$287$	1.56788	0.0925489
$288$	0	0
$289$	−11.4724	−0.674845
$290$	0	0
$291$	15.7311	0.922176
$292$	0	0
$293$	−28.4168	−1.66013	−0.830063	−	0.557669i	$-0.811696\pi$
$293$	−28.4168	−1.66013	−0.830063	+	0.557669i	$0.811696\pi$
$294$	0	0
$295$	−3.22787	−0.187934
$296$	0	0
$297$	−5.82693	−0.338113
$298$	0	0
$299$	−15.8269	−0.915296
$300$	0	0
$301$	19.7716	1.13962
$302$	0	0
$303$	−1.62087	−0.0931163
$304$	0	0
$305$	2.91706	0.167031
$306$	0	0
$307$	12.5787	0.717903	0.358951	−	0.933356i	$-0.383134\pi$
$307$	12.5787	0.717903	0.358951	+	0.933356i	$0.383134\pi$
$308$	0	0
$309$	−2.78462	−0.158411
$310$	0	0
$311$	19.3481	1.09713	0.548566	−	0.836107i	$-0.315174\pi$
$311$	19.3481	1.09713	0.548566	+	0.836107i	$0.315174\pi$
$312$	0	0
$313$	−6.51682	−0.368353	−0.184176	−	0.982893i	$-0.558962\pi$
$313$	−6.51682	−0.368353	−0.184176	+	0.982893i	$0.558962\pi$
$314$	0	0
$315$	3.97485	0.223957
$316$	0	0
$317$	−1.39781	−0.0785091	−0.0392545	−	0.999229i	$-0.512498\pi$
$317$	−1.39781	−0.0785091	−0.0392545	+	0.999229i	$0.512498\pi$
$318$	0	0
$319$	−6.47933	−0.362773
$320$	0	0
$321$	13.0772	0.729898
$322$	0	0
$323$	−19.6096	−1.09111
$324$	0	0
$325$	7.29871	0.404859
$326$	0	0
$327$	1.53097	0.0846628
$328$	0	0
$329$	−4.54274	−0.250449
$330$	0	0
$331$	−12.1562	−0.668167	−0.334084	−	0.942543i	$-0.608427\pi$
$331$	−12.1562	−0.668167	−0.334084	+	0.942543i	$0.608427\pi$
$332$	0	0
$333$	1.22871	0.0673328
$334$	0	0
$335$	−11.1484	−0.609101
$336$	0	0
$337$	−32.2776	−1.75827	−0.879137	−	0.476569i	$-0.841880\pi$
$337$	−32.2776	−1.75827	−0.879137	+	0.476569i	$0.841880\pi$
$338$	0	0
$339$	18.1720	0.986970
$340$	0	0
$341$	35.4505	1.91975
$342$	0	0
$343$	4.08880	0.220774
$344$	0	0
$345$	9.06480	0.488033
$346$	0	0
$347$	34.8053	1.86845	0.934223	−	0.356690i	$-0.116095\pi$
$347$	34.8053	1.86845	0.934223	+	0.356690i	$0.116095\pi$
$348$	0	0
$349$	−21.6820	−1.16061	−0.580305	−	0.814399i	$-0.697067\pi$
$349$	−21.6820	−1.16061	−0.580305	+	0.814399i	$0.697067\pi$
$350$	0	0
$351$	1.93527	0.103297
$352$	0	0
$353$	29.6786	1.57963	0.789816	−	0.613344i	$-0.210176\pi$
$353$	29.6786	1.57963	0.789816	+	0.613344i	$0.210176\pi$
$354$	0	0
$355$	−10.3102	−0.547210
$356$	0	0
$357$	8.43116	0.446224
$358$	0	0
$359$	19.1227	1.00926	0.504630	−	0.863336i	$-0.331629\pi$
$359$	19.1227	1.00926	0.504630	+	0.863336i	$0.331629\pi$
$360$	0	0
$361$	50.5661	2.66137
$362$	0	0
$363$	−22.9532	−1.20473
$364$	0	0
$365$	−1.47852	−0.0773892
$366$	0	0
$367$	−27.7060	−1.44624	−0.723120	−	0.690723i	$-0.757293\pi$
$367$	−27.7060	−1.44624	−0.723120	+	0.690723i	$0.757293\pi$
$368$	0	0
$369$	−0.437215	−0.0227605
$370$	0	0
$371$	−8.51106	−0.441872
$372$	0	0
$373$	−0.0776589	−0.00402103	−0.00201051	−	0.999998i	$-0.500640\pi$
$373$	−0.0776589	−0.00402103	−0.00201051	+	0.999998i	$0.500640\pi$
$374$	0	0
$375$	−9.72238	−0.502062
$376$	0	0
$377$	2.15195	0.110831
$378$	0	0
$379$	32.2133	1.65469	0.827343	−	0.561698i	$-0.189852\pi$
$379$	32.2133	1.65469	0.827343	+	0.561698i	$0.189852\pi$
$380$	0	0
$381$	13.5455	0.693958
$382$	0	0
$383$	−26.6228	−1.36036	−0.680180	−	0.733045i	$-0.738098\pi$
$383$	−26.6228	−1.36036	−0.680180	+	0.733045i	$0.738098\pi$
$384$	0	0
$385$	23.1612	1.18040
$386$	0	0
$387$	−5.51348	−0.280266
$388$	0	0
$389$	15.6359	0.792771	0.396386	−	0.918084i	$-0.370264\pi$
$389$	15.6359	0.792771	0.396386	+	0.918084i	$0.370264\pi$
$390$	0	0
$391$	19.2276	0.972382
$392$	0	0
$393$	11.5340	0.581814
$394$	0	0
$395$	6.88078	0.346210
$396$	0	0
$397$	−15.2250	−0.764118	−0.382059	−	0.924138i	$-0.624785\pi$
$397$	−15.2250	−0.764118	−0.382059	+	0.924138i	$0.624785\pi$
$398$	0	0
$399$	−29.9100	−1.49737
$400$	0	0
$401$	16.5045	0.824198	0.412099	−	0.911139i	$-0.364796\pi$
$401$	16.5045	0.824198	0.412099	+	0.911139i	$0.364796\pi$
$402$	0	0
$403$	−11.7740	−0.586506
$404$	0	0
$405$	−1.10842	−0.0550777
$406$	0	0
$407$	7.15960	0.354888
$408$	0	0
$409$	0.771326	0.0381396	0.0190698	−	0.999818i	$-0.493930\pi$
$409$	0.771326	0.0381396	0.0190698	+	0.999818i	$0.493930\pi$
$410$	0	0
$411$	14.8636	0.733167
$412$	0	0
$413$	−10.4431	−0.513872
$414$	0	0
$415$	−7.42286	−0.364374
$416$	0	0
$417$	−1.83355	−0.0897895
$418$	0	0
$419$	−23.9285	−1.16899	−0.584493	−	0.811399i	$-0.698706\pi$
$419$	−23.9285	−1.16899	−0.584493	+	0.811399i	$0.698706\pi$
$420$	0	0
$421$	30.3244	1.47792	0.738960	−	0.673749i	$-0.235317\pi$
$421$	30.3244	1.47792	0.738960	+	0.673749i	$0.235317\pi$
$422$	0	0
$423$	1.26678	0.0615929
$424$	0	0
$425$	−8.86694	−0.430110
$426$	0	0
$427$	9.43757	0.456716
$428$	0	0
$429$	11.2767	0.544444
$430$	0	0
$431$	−3.40550	−0.164037	−0.0820187	−	0.996631i	$-0.526137\pi$
$431$	−3.40550	−0.164037	−0.0820187	+	0.996631i	$0.526137\pi$
$432$	0	0
$433$	−31.7810	−1.52730	−0.763649	−	0.645632i	$-0.776594\pi$
$433$	−31.7810	−1.52730	−0.763649	+	0.645632i	$0.776594\pi$
$434$	0	0
$435$	−1.23252	−0.0590947
$436$	0	0
$437$	−68.2109	−3.26297
$438$	0	0
$439$	30.5689	1.45897	0.729487	−	0.683995i	$-0.239759\pi$
$439$	30.5689	1.45897	0.729487	+	0.683995i	$0.239759\pi$
$440$	0	0
$441$	5.85981	0.279038
$442$	0	0
$443$	2.56432	0.121834	0.0609172	−	0.998143i	$-0.480597\pi$
$443$	2.56432	0.121834	0.0609172	+	0.998143i	$0.480597\pi$
$444$	0	0
$445$	7.33215	0.347577
$446$	0	0
$447$	16.6162	0.785918
$448$	0	0
$449$	3.18976	0.150534	0.0752671	−	0.997163i	$-0.476019\pi$
$449$	3.18976	0.150534	0.0752671	+	0.997163i	$0.476019\pi$
$450$	0	0
$451$	−2.54762	−0.119963
$452$	0	0
$453$	−10.5836	−0.497259
$454$	0	0
$455$	−7.69241	−0.360626
$456$	0	0
$457$	−1.77487	−0.0830251	−0.0415126	−	0.999138i	$-0.513218\pi$
$457$	−1.77487	−0.0830251	−0.0415126	+	0.999138i	$0.513218\pi$
$458$	0	0
$459$	−2.35109	−0.109740
$460$	0	0
$461$	11.2106	0.522128	0.261064	−	0.965321i	$-0.415927\pi$
$461$	11.2106	0.522128	0.261064	+	0.965321i	$0.415927\pi$
$462$	0	0
$463$	3.01144	0.139953	0.0699767	−	0.997549i	$-0.477708\pi$
$463$	3.01144	0.139953	0.0699767	+	0.997549i	$0.477708\pi$
$464$	0	0
$465$	6.74351	0.312723
$466$	0	0
$467$	6.78730	0.314079	0.157039	−	0.987592i	$-0.449805\pi$
$467$	6.78730	0.314079	0.157039	+	0.987592i	$0.449805\pi$
$468$	0	0
$469$	−36.0683	−1.66548
$470$	0	0
$471$	21.3072	0.981783
$472$	0	0
$473$	−32.1267	−1.47719
$474$	0	0
$475$	31.4559	1.44330
$476$	0	0
$477$	2.37338	0.108669
$478$	0	0
$479$	−31.4903	−1.43883	−0.719414	−	0.694581i	$-0.755590\pi$
$479$	−31.4903	−1.43883	−0.719414	+	0.694581i	$0.755590\pi$
$480$	0	0
$481$	−2.37788	−0.108422
$482$	0	0
$483$	29.3273	1.33444
$484$	0	0
$485$	17.4367	0.791759
$486$	0	0
$487$	−8.22731	−0.372815	−0.186408	−	0.982472i	$-0.559684\pi$
$487$	−8.22731	−0.372815	−0.186408	+	0.982472i	$0.559684\pi$
$488$	0	0
$489$	18.5920	0.840759
$490$	0	0
$491$	25.0905	1.13232	0.566159	−	0.824296i	$-0.308429\pi$
$491$	25.0905	1.13232	0.566159	+	0.824296i	$0.308429\pi$
$492$	0	0
$493$	−2.61433	−0.117743
$494$	0	0
$495$	−6.45867	−0.290296
$496$	0	0
$497$	−33.3566	−1.49625
$498$	0	0
$499$	−23.0183	−1.03044	−0.515220	−	0.857058i	$-0.672290\pi$
$499$	−23.0183	−1.03044	−0.515220	+	0.857058i	$0.672290\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−19.4340	−0.866520	−0.433260	−	0.901269i	$-0.642637\pi$
$503$	−19.4340	−0.866520	−0.433260	+	0.901269i	$0.642637\pi$
$504$	0	0
$505$	−1.79660	−0.0799475
$506$	0	0
$507$	9.25472	0.411017
$508$	0	0
$509$	−33.3745	−1.47930	−0.739650	−	0.672992i	$-0.765009\pi$
$509$	−33.3745	−1.47930	−0.739650	+	0.672992i	$0.765009\pi$
$510$	0	0
$511$	−4.78344	−0.211607
$512$	0	0
$513$	8.34063	0.368248
$514$	0	0
$515$	−3.08651	−0.136008
$516$	0	0
$517$	7.38143	0.324635
$518$	0	0
$519$	−7.60016	−0.333610
$520$	0	0
$521$	−6.47375	−0.283620	−0.141810	−	0.989894i	$-0.545292\pi$
$521$	−6.47375	−0.283620	−0.141810	+	0.989894i	$0.545292\pi$
$522$	0	0
$523$	−21.0452	−0.920243	−0.460121	−	0.887856i	$-0.652194\pi$
$523$	−21.0452	−0.920243	−0.460121	+	0.887856i	$0.652194\pi$
$524$	0	0
$525$	−13.5245	−0.590257
$526$	0	0
$527$	14.3038	0.623085
$528$	0	0
$529$	43.8822	1.90792
$530$	0	0
$531$	2.91215	0.126376
$532$	0	0
$533$	0.846130	0.0366499
$534$	0	0
$535$	14.4950	0.626673
$536$	0	0
$537$	−4.08215	−0.176158
$538$	0	0
$539$	34.1447	1.47072
$540$	0	0
$541$	−44.0678	−1.89462	−0.947312	−	0.320312i	$-0.896212\pi$
$541$	−44.0678	−1.89462	−0.947312	+	0.320312i	$0.896212\pi$
$542$	0	0
$543$	5.91060	0.253648
$544$	0	0
$545$	1.69695	0.0726895
$546$	0	0
$547$	−4.66539	−0.199478	−0.0997389	−	0.995014i	$-0.531801\pi$
$547$	−4.66539	−0.199478	−0.0997389	+	0.995014i	$0.531801\pi$
$548$	0	0
$549$	−2.63174	−0.112320
$550$	0	0
$551$	9.27447	0.395106
$552$	0	0
$553$	22.2614	0.946650
$554$	0	0
$555$	1.36192	0.0578103
$556$	0	0
$557$	−12.0759	−0.511674	−0.255837	−	0.966720i	$-0.582351\pi$
$557$	−12.0759	−0.511674	−0.255837	+	0.966720i	$0.582351\pi$
$558$	0	0
$559$	10.6701	0.451296
$560$	0	0
$561$	−13.6997	−0.578401
$562$	0	0
$563$	−30.9102	−1.30271	−0.651354	−	0.758774i	$-0.725799\pi$
$563$	−30.9102	−1.30271	−0.651354	+	0.758774i	$0.725799\pi$
$564$	0	0
$565$	20.1422	0.847389
$566$	0	0
$567$	−3.58606	−0.150600
$568$	0	0
$569$	−2.21913	−0.0930306	−0.0465153	−	0.998918i	$-0.514812\pi$
$569$	−2.21913	−0.0930306	−0.0465153	+	0.998918i	$0.514812\pi$
$570$	0	0
$571$	−16.6261	−0.695779	−0.347890	−	0.937535i	$-0.613102\pi$
$571$	−16.6261	−0.695779	−0.347890	+	0.937535i	$0.613102\pi$
$572$	0	0
$573$	−3.06512	−0.128047
$574$	0	0
$575$	−30.8432	−1.28625
$576$	0	0
$577$	−20.3134	−0.845656	−0.422828	−	0.906210i	$-0.638963\pi$
$577$	−20.3134	−0.845656	−0.422828	+	0.906210i	$0.638963\pi$
$578$	0	0
$579$	−5.34361	−0.222073
$580$	0	0
$581$	−24.0152	−0.996316
$582$	0	0
$583$	13.8295	0.572760
$584$	0	0
$585$	2.14509	0.0886885
$586$	0	0
$587$	−4.97202	−0.205217	−0.102609	−	0.994722i	$-0.532719\pi$
$587$	−4.97202	−0.205217	−0.102609	+	0.994722i	$0.532719\pi$
$588$	0	0
$589$	−50.7436	−2.09086
$590$	0	0
$591$	15.9344	0.655454
$592$	0	0
$593$	25.1873	1.03432	0.517159	−	0.855889i	$-0.326989\pi$
$593$	25.1873	1.03432	0.517159	+	0.855889i	$0.326989\pi$
$594$	0	0
$595$	9.34524	0.383117
$596$	0	0
$597$	2.74400	0.112304
$598$	0	0
$599$	−6.20186	−0.253401	−0.126701	−	0.991941i	$-0.540439\pi$
$599$	−6.20186	−0.253401	−0.126701	+	0.991941i	$0.540439\pi$
$600$	0	0
$601$	35.9069	1.46467	0.732337	−	0.680942i	$-0.238429\pi$
$601$	35.9069	1.46467	0.732337	+	0.680942i	$0.238429\pi$
$602$	0	0
$603$	10.0579	0.409590
$604$	0	0
$605$	−25.4417	−1.03435
$606$	0	0
$607$	8.05818	0.327071	0.163536	−	0.986537i	$-0.447710\pi$
$607$	8.05818	0.327071	0.163536	+	0.986537i	$0.447710\pi$
$608$	0	0
$609$	−3.98756	−0.161584
$610$	0	0
$611$	−2.45156	−0.0991795
$612$	0	0
$613$	−28.4034	−1.14720	−0.573601	−	0.819135i	$-0.694454\pi$
$613$	−28.4034	−1.14720	−0.573601	+	0.819135i	$0.694454\pi$
$614$	0	0
$615$	−0.484617	−0.0195416
$616$	0	0
$617$	−2.66256	−0.107191	−0.0535953	−	0.998563i	$-0.517068\pi$
$617$	−2.66256	−0.107191	−0.0535953	+	0.998563i	$0.517068\pi$
$618$	0	0
$619$	−39.0923	−1.57125	−0.785626	−	0.618702i	$-0.787659\pi$
$619$	−39.0923	−1.57125	−0.785626	+	0.618702i	$0.787659\pi$
$620$	0	0
$621$	−8.17815	−0.328178
$622$	0	0
$623$	23.7217	0.950389
$624$	0	0
$625$	8.08060	0.323224
$626$	0	0
$627$	48.6003	1.94091
$628$	0	0
$629$	2.88881	0.115184
$630$	0	0
$631$	26.2314	1.04426	0.522129	−	0.852867i	$-0.325138\pi$
$631$	26.2314	1.04426	0.522129	+	0.852867i	$0.325138\pi$
$632$	0	0
$633$	−2.39097	−0.0950324
$634$	0	0
$635$	15.0141	0.595816
$636$	0	0
$637$	−11.3403	−0.449320
$638$	0	0
$639$	9.30176	0.367972
$640$	0	0
$641$	14.4177	0.569464	0.284732	−	0.958607i	$-0.408095\pi$
$641$	14.4177	0.569464	0.284732	+	0.958607i	$0.408095\pi$
$642$	0	0
$643$	−20.1947	−0.796402	−0.398201	−	0.917298i	$-0.630365\pi$
$643$	−20.1947	−0.796402	−0.398201	+	0.917298i	$0.630365\pi$
$644$	0	0
$645$	−6.11123	−0.240629
$646$	0	0
$647$	23.8303	0.936865	0.468432	−	0.883499i	$-0.344819\pi$
$647$	23.8303	0.936865	0.468432	+	0.883499i	$0.344819\pi$
$648$	0	0
$649$	16.9689	0.666087
$650$	0	0
$651$	21.8173	0.855086
$652$	0	0
$653$	14.2191	0.556436	0.278218	−	0.960518i	$-0.410256\pi$
$653$	14.2191	0.556436	0.278218	+	0.960518i	$0.410256\pi$
$654$	0	0
$655$	12.7845	0.499532
$656$	0	0
$657$	1.33390	0.0520404
$658$	0	0
$659$	43.8749	1.70912	0.854562	−	0.519350i	$-0.173826\pi$
$659$	43.8749	1.70912	0.854562	+	0.519350i	$0.173826\pi$
$660$	0	0
$661$	38.1442	1.48364	0.741819	−	0.670600i	$-0.233963\pi$
$661$	38.1442	1.48364	0.741819	+	0.670600i	$0.233963\pi$
$662$	0	0
$663$	4.55001	0.176708
$664$	0	0
$665$	−33.1527	−1.28561
$666$	0	0
$667$	−9.09380	−0.352113
$668$	0	0
$669$	9.20998	0.356078
$670$	0	0
$671$	−15.3350	−0.592000
$672$	0	0
$673$	−36.3580	−1.40150	−0.700749	−	0.713408i	$-0.747151\pi$
$673$	−36.3580	−1.40150	−0.700749	+	0.713408i	$0.747151\pi$
$674$	0	0
$675$	3.77141	0.145162
$676$	0	0
$677$	−31.4917	−1.21032	−0.605162	−	0.796103i	$-0.706892\pi$
$677$	−31.4917	−1.21032	−0.605162	+	0.796103i	$0.706892\pi$
$678$	0	0
$679$	56.4128	2.16492
$680$	0	0
$681$	25.9471	0.994295
$682$	0	0
$683$	−7.77327	−0.297436	−0.148718	−	0.988880i	$-0.547515\pi$
$683$	−7.77327	−0.297436	−0.148718	+	0.988880i	$0.547515\pi$
$684$	0	0
$685$	16.4751	0.629480
$686$	0	0
$687$	10.7591	0.410485
$688$	0	0
$689$	−4.59313	−0.174984
$690$	0	0
$691$	24.7869	0.942937	0.471469	−	0.881883i	$-0.343724\pi$
$691$	24.7869	0.942937	0.471469	+	0.881883i	$0.343724\pi$
$692$	0	0
$693$	−20.8957	−0.793763
$694$	0	0
$695$	−2.03234	−0.0770911
$696$	0	0
$697$	−1.02793	−0.0389358
$698$	0	0
$699$	14.5686	0.551034
$700$	0	0
$701$	17.1454	0.647571	0.323786	−	0.946130i	$-0.395044\pi$
$701$	17.1454	0.647571	0.323786	+	0.946130i	$0.395044\pi$
$702$	0	0
$703$	−10.2482	−0.386518
$704$	0	0
$705$	1.40412	0.0528822
$706$	0	0
$707$	−5.81252	−0.218602
$708$	0	0
$709$	−22.0504	−0.828120	−0.414060	−	0.910250i	$-0.635890\pi$
$709$	−22.0504	−0.828120	−0.414060	+	0.910250i	$0.635890\pi$
$710$	0	0
$711$	−6.20776	−0.232809
$712$	0	0
$713$	49.7552	1.86335
$714$	0	0
$715$	12.4993	0.467447
$716$	0	0
$717$	−16.4810	−0.615495
$718$	0	0
$719$	16.6766	0.621933	0.310967	−	0.950421i	$-0.399347\pi$
$719$	16.6766	0.621933	0.310967	+	0.950421i	$0.399347\pi$
$720$	0	0
$721$	−9.98579	−0.371890
$722$	0	0
$723$	10.3175	0.383713
$724$	0	0
$725$	4.19367	0.155749
$726$	0	0
$727$	32.6201	1.20981	0.604906	−	0.796297i	$-0.293211\pi$
$727$	32.6201	1.20981	0.604906	+	0.796297i	$0.293211\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.9627	−0.479443
$732$	0	0
$733$	−9.93774	−0.367059	−0.183529	−	0.983014i	$-0.558752\pi$
$733$	−9.93774	−0.367059	−0.183529	+	0.983014i	$0.558752\pi$
$734$	0	0
$735$	6.49511	0.239576
$736$	0	0
$737$	58.6069	2.15881
$738$	0	0
$739$	15.1776	0.558316	0.279158	−	0.960245i	$-0.409945\pi$
$739$	15.1776	0.558316	0.279158	+	0.960245i	$0.409945\pi$
$740$	0	0
$741$	−16.1414	−0.592968
$742$	0	0
$743$	18.2366	0.669034	0.334517	−	0.942390i	$-0.391427\pi$
$743$	18.2366	0.669034	0.334517	+	0.942390i	$0.391427\pi$
$744$	0	0
$745$	18.4176	0.674770
$746$	0	0
$747$	6.69681	0.245024
$748$	0	0
$749$	46.8956	1.71353
$750$	0	0
$751$	20.7074	0.755625	0.377813	−	0.925882i	$-0.376676\pi$
$751$	20.7074	0.755625	0.377813	+	0.925882i	$0.376676\pi$
$752$	0	0
$753$	24.4320	0.890352
$754$	0	0
$755$	−11.7310	−0.426935
$756$	0	0
$757$	−23.2734	−0.845887	−0.422943	−	0.906156i	$-0.639003\pi$
$757$	−23.2734	−0.845887	−0.422943	+	0.906156i	$0.639003\pi$
$758$	0	0
$759$	−47.6536	−1.72971
$760$	0	0
$761$	−19.0198	−0.689469	−0.344734	−	0.938700i	$-0.612031\pi$
$761$	−19.0198	−0.689469	−0.344734	+	0.938700i	$0.612031\pi$
$762$	0	0
$763$	5.49015	0.198757
$764$	0	0
$765$	−2.60599	−0.0942199
$766$	0	0
$767$	−5.63579	−0.203497
$768$	0	0
$769$	−9.30332	−0.335486	−0.167743	−	0.985831i	$-0.553648\pi$
$769$	−9.30332	−0.335486	−0.167743	+	0.985831i	$0.553648\pi$
$770$	0	0
$771$	−20.7309	−0.746604
$772$	0	0
$773$	−0.524571	−0.0188675	−0.00943376	−	0.999956i	$-0.503003\pi$
$773$	−0.524571	−0.0188675	−0.00943376	+	0.999956i	$0.503003\pi$
$774$	0	0
$775$	−22.9449	−0.824206
$776$	0	0
$777$	4.40622	0.158072
$778$	0	0
$779$	3.64665	0.130655
$780$	0	0
$781$	54.2007	1.93945
$782$	0	0
$783$	1.11196	0.0397383
$784$	0	0
$785$	23.6172	0.842936
$786$	0	0
$787$	−40.4621	−1.44232	−0.721159	−	0.692769i	$-0.756390\pi$
$787$	−40.4621	−1.44232	−0.721159	+	0.692769i	$0.756390\pi$
$788$	0	0
$789$	−22.3616	−0.796095
$790$	0	0
$791$	65.1660	2.31704
$792$	0	0
$793$	5.09313	0.180862
$794$	0	0
$795$	2.63069	0.0933010
$796$	0	0
$797$	−22.5180	−0.797628	−0.398814	−	0.917032i	$-0.630578\pi$
$797$	−22.5180	−0.797628	−0.398814	+	0.917032i	$0.630578\pi$
$798$	0	0
$799$	2.97831	0.105365
$800$	0	0
$801$	−6.61497	−0.233729
$802$	0	0
$803$	7.77255	0.274287
$804$	0	0
$805$	32.5069	1.14572
$806$	0	0
$807$	−22.2023	−0.781558
$808$	0	0
$809$	4.88970	0.171913	0.0859563	−	0.996299i	$-0.472605\pi$
$809$	4.88970	0.171913	0.0859563	+	0.996299i	$0.472605\pi$
$810$	0	0
$811$	−24.3224	−0.854074	−0.427037	−	0.904234i	$-0.640443\pi$
$811$	−24.3224	−0.854074	−0.427037	+	0.904234i	$0.640443\pi$
$812$	0	0
$813$	−3.34350	−0.117262
$814$	0	0
$815$	20.6077	0.721856
$816$	0	0
$817$	45.9858	1.60884
$818$	0	0
$819$	6.93999	0.242503
$820$	0	0
$821$	−38.1388	−1.33105	−0.665526	−	0.746375i	$-0.731793\pi$
$821$	−38.1388	−1.33105	−0.665526	+	0.746375i	$0.731793\pi$
$822$	0	0
$823$	49.6247	1.72981	0.864904	−	0.501937i	$-0.167379\pi$
$823$	49.6247	1.72981	0.864904	+	0.501937i	$0.167379\pi$
$824$	0	0
$825$	21.9758	0.765098
$826$	0	0
$827$	−0.755830	−0.0262828	−0.0131414	−	0.999914i	$-0.504183\pi$
$827$	−0.755830	−0.0262828	−0.0131414	+	0.999914i	$0.504183\pi$
$828$	0	0
$829$	46.2544	1.60648	0.803240	−	0.595655i	$-0.203107\pi$
$829$	46.2544	1.60648	0.803240	+	0.595655i	$0.203107\pi$
$830$	0	0
$831$	−19.0759	−0.661734
$832$	0	0
$833$	13.7770	0.477343
$834$	0	0
$835$	1.10842	0.0383584
$836$	0	0
$837$	−6.08391	−0.210291
$838$	0	0
$839$	20.6518	0.712979	0.356490	−	0.934299i	$-0.383973\pi$
$839$	20.6518	0.712979	0.356490	+	0.934299i	$0.383973\pi$
$840$	0	0
$841$	−27.7635	−0.957363
$842$	0	0
$843$	8.66664	0.298495
$844$	0	0
$845$	10.2581	0.352889
$846$	0	0
$847$	−82.3113	−2.82825
$848$	0	0
$849$	14.1300	0.484942
$850$	0	0
$851$	10.0486	0.344460
$852$	0	0
$853$	6.14218	0.210304	0.105152	−	0.994456i	$-0.466467\pi$
$853$	6.14218	0.210304	0.105152	+	0.994456i	$0.466467\pi$
$854$	0	0
$855$	9.24489	0.316169
$856$	0	0
$857$	−2.59797	−0.0887449	−0.0443724	−	0.999015i	$-0.514129\pi$
$857$	−2.59797	−0.0887449	−0.0443724	+	0.999015i	$0.514129\pi$
$858$	0	0
$859$	6.04659	0.206307	0.103153	−	0.994665i	$-0.467107\pi$
$859$	6.04659	0.206307	0.103153	+	0.994665i	$0.467107\pi$
$860$	0	0
$861$	−1.56788	−0.0534331
$862$	0	0
$863$	−31.2858	−1.06498	−0.532490	−	0.846436i	$-0.678744\pi$
$863$	−31.2858	−1.06498	−0.532490	+	0.846436i	$0.678744\pi$
$864$	0	0
$865$	−8.42415	−0.286430
$866$	0	0
$867$	11.4724	0.389622
$868$	0	0
$869$	−36.1722	−1.22706
$870$	0	0
$871$	−19.4648	−0.659540
$872$	0	0
$873$	−15.7311	−0.532419
$874$	0	0
$875$	−34.8650	−1.17865
$876$	0	0
$877$	21.9914	0.742596	0.371298	−	0.928514i	$-0.378913\pi$
$877$	21.9914	0.742596	0.371298	+	0.928514i	$0.378913\pi$
$878$	0	0
$879$	28.4168	0.958475
$880$	0	0
$881$	46.0980	1.55308	0.776540	−	0.630067i	$-0.216973\pi$
$881$	46.0980	1.55308	0.776540	+	0.630067i	$0.216973\pi$
$882$	0	0
$883$	−43.2394	−1.45512	−0.727561	−	0.686043i	$-0.759346\pi$
$883$	−43.2394	−1.45512	−0.727561	+	0.686043i	$0.759346\pi$
$884$	0	0
$885$	3.22787	0.108504
$886$	0	0
$887$	−27.7504	−0.931766	−0.465883	−	0.884846i	$-0.654263\pi$
$887$	−27.7504	−0.931766	−0.465883	+	0.884846i	$0.654263\pi$
$888$	0	0
$889$	48.5750	1.62915
$890$	0	0
$891$	5.82693	0.195210
$892$	0	0
$893$	−10.5657	−0.353569
$894$	0	0
$895$	−4.52472	−0.151245
$896$	0	0
$897$	15.8269	0.528446
$898$	0	0
$899$	−6.76508	−0.225628
$900$	0	0
$901$	5.58003	0.185898
$902$	0	0
$903$	−19.7716	−0.657959
$904$	0	0
$905$	6.55141	0.217776
$906$	0	0
$907$	16.8017	0.557892	0.278946	−	0.960307i	$-0.410015\pi$
$907$	16.8017	0.557892	0.278946	+	0.960307i	$0.410015\pi$
$908$	0	0
$909$	1.62087	0.0537607
$910$	0	0
$911$	5.00813	0.165927	0.0829633	−	0.996553i	$-0.473562\pi$
$911$	5.00813	0.165927	0.0829633	+	0.996553i	$0.473562\pi$
$912$	0	0
$913$	39.0219	1.29144
$914$	0	0
$915$	−2.91706	−0.0964352
$916$	0	0
$917$	41.3617	1.36588
$918$	0	0
$919$	54.4008	1.79452	0.897259	−	0.441506i	$-0.145555\pi$
$919$	54.4008	1.79452	0.897259	+	0.441506i	$0.145555\pi$
$920$	0	0
$921$	−12.5787	−0.414481
$922$	0	0
$923$	−18.0014	−0.592524
$924$	0	0
$925$	−4.63396	−0.152364
$926$	0	0
$927$	2.78462	0.0914588
$928$	0	0
$929$	48.9043	1.60450	0.802249	−	0.596989i	$-0.203636\pi$
$929$	48.9043	1.60450	0.802249	+	0.596989i	$0.203636\pi$
$930$	0	0
$931$	−48.8745	−1.60180
$932$	0	0
$933$	−19.3481	−0.633429
$934$	0	0
$935$	−15.1849	−0.496601
$936$	0	0
$937$	28.0717	0.917063	0.458531	−	0.888678i	$-0.348376\pi$
$937$	28.0717	0.917063	0.458531	+	0.888678i	$0.348376\pi$
$938$	0	0
$939$	6.51682	0.212668
$940$	0	0
$941$	32.6738	1.06513	0.532567	−	0.846388i	$-0.321227\pi$
$941$	32.6738	1.06513	0.532567	+	0.846388i	$0.321227\pi$
$942$	0	0
$943$	−3.57561	−0.116438
$944$	0	0
$945$	−3.97485	−0.129302
$946$	0	0
$947$	57.3226	1.86273	0.931367	−	0.364082i	$-0.118617\pi$
$947$	57.3226	1.86273	0.931367	+	0.364082i	$0.118617\pi$
$948$	0	0
$949$	−2.58146	−0.0837977
$950$	0	0
$951$	1.39781	0.0453272
$952$	0	0
$953$	26.1665	0.847615	0.423808	−	0.905752i	$-0.360693\pi$
$953$	26.1665	0.847615	0.423808	+	0.905752i	$0.360693\pi$
$954$	0	0
$955$	−3.39743	−0.109938
$956$	0	0
$957$	6.47933	0.209447
$958$	0	0
$959$	53.3017	1.72120
$960$	0	0
$961$	6.01398	0.193999
$962$	0	0
$963$	−13.0772	−0.421407
$964$	0	0
$965$	−5.92295	−0.190667
$966$	0	0
$967$	−21.9688	−0.706468	−0.353234	−	0.935535i	$-0.614918\pi$
$967$	−21.9688	−0.706468	−0.353234	+	0.935535i	$0.614918\pi$
$968$	0	0
$969$	19.6096	0.629951
$970$	0	0
$971$	−10.7987	−0.346548	−0.173274	−	0.984874i	$-0.555435\pi$
$971$	−10.7987	−0.346548	−0.173274	+	0.984874i	$0.555435\pi$
$972$	0	0
$973$	−6.57523	−0.210792
$974$	0	0
$975$	−7.29871	−0.233746
$976$	0	0
$977$	19.8767	0.635911	0.317955	−	0.948106i	$-0.397004\pi$
$977$	19.8767	0.635911	0.317955	+	0.948106i	$0.397004\pi$
$978$	0	0
$979$	−38.5450	−1.23190
$980$	0	0
$981$	−1.53097	−0.0488801
$982$	0	0
$983$	57.3457	1.82904	0.914522	−	0.404536i	$-0.132567\pi$
$983$	57.3457	1.82904	0.914522	+	0.404536i	$0.132567\pi$
$984$	0	0
$985$	17.6620	0.562757
$986$	0	0
$987$	4.54274	0.144597
$988$	0	0
$989$	−45.0900	−1.43378
$990$	0	0
$991$	−24.9058	−0.791160	−0.395580	−	0.918431i	$-0.629456\pi$
$991$	−24.9058	−0.791160	−0.395580	+	0.918431i	$0.629456\pi$
$992$	0	0
$993$	12.1562	0.385767
$994$	0	0
$995$	3.04149	0.0964218
$996$	0	0
$997$	14.7340	0.466631	0.233316	−	0.972401i	$-0.425042\pi$
$997$	14.7340	0.466631	0.233316	+	0.972401i	$0.425042\pi$
$998$	0	0
$999$	−1.22871	−0.0388746

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.j.1.5	✓	10
4.3	odd	2		8016.2.a.bd.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.j.1.5	✓	10	1.1	even	1	trivial
8016.2.a.bd.1.5		10	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.10842 q^{5} -3.58606 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.10842 q^{5} -3.58606 q^{7} +1.00000 q^{9} +5.82693 q^{11} -1.93527 q^{13} +1.10842 q^{15} +2.35109 q^{17} -8.34063 q^{19} +3.58606 q^{21} +8.17815 q^{23} -3.77141 q^{25} -1.00000 q^{27} -1.11196 q^{29} +6.08391 q^{31} -5.82693 q^{33} +3.97485 q^{35} +1.22871 q^{37} +1.93527 q^{39} -0.437215 q^{41} -5.51348 q^{43} -1.10842 q^{45} +1.26678 q^{47} +5.85981 q^{49} -2.35109 q^{51} +2.37338 q^{53} -6.45867 q^{55} +8.34063 q^{57} +2.91215 q^{59} -2.63174 q^{61} -3.58606 q^{63} +2.14509 q^{65} +10.0579 q^{67} -8.17815 q^{69} +9.30176 q^{71} +1.33390 q^{73} +3.77141 q^{75} -20.8957 q^{77} -6.20776 q^{79} +1.00000 q^{81} +6.69681 q^{83} -2.60599 q^{85} +1.11196 q^{87} -6.61497 q^{89} +6.93999 q^{91} -6.08391 q^{93} +9.24489 q^{95} -15.7311 q^{97} +5.82693 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9} - q^{11} - 6 q^{13} + 10 q^{15} - 9 q^{17} + 2 q^{19} - q^{21} + 7 q^{23} + 12 q^{25} - 10 q^{27} - 13 q^{29} + 23 q^{31} + q^{33} + q^{35} - 6 q^{37} + 6 q^{39} - 12 q^{41} - 10 q^{45} + 10 q^{47} + 7 q^{49} + 9 q^{51} - 26 q^{53} + 11 q^{55} - 2 q^{57} - 10 q^{59} - 10 q^{61} + q^{63} - 22 q^{65} - 5 q^{67} - 7 q^{69} + 25 q^{71} - 8 q^{73} - 12 q^{75} - 46 q^{77} + 26 q^{79} + 10 q^{81} - 14 q^{83} + 9 q^{85} + 13 q^{87} - 31 q^{89} - 3 q^{91} - 23 q^{93} - 5 q^{95} - 32 q^{97} - q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9 - q^11 - 6 * q^13 + 10 * q^15 - 9 * q^17 + 2 * q^19 - q^21 + 7 * q^23 + 12 * q^25 - 10 * q^27 - 13 * q^29 + 23 * q^31 + q^33 + q^35 - 6 * q^37 + 6 * q^39 - 12 * q^41 - 10 * q^45 + 10 * q^47 + 7 * q^49 + 9 * q^51 - 26 * q^53 + 11 * q^55 - 2 * q^57 - 10 * q^59 - 10 * q^61 + q^63 - 22 * q^65 - 5 * q^67 - 7 * q^69 + 25 * q^71 - 8 * q^73 - 12 * q^75 - 46 * q^77 + 26 * q^79 + 10 * q^81 - 14 * q^83 + 9 * q^85 + 13 * q^87 - 31 * q^89 - 3 * q^91 - 23 * q^93 - 5 * q^95 - 32 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more