LMFDB - Embedded newform 4008.2.a.h.1.3

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 23x^{6} - 3x^{5} + 163x^{4} + 13x^{3} - 418x^{2} + 4x + 269 $ x^8 - 23x^6 - 3x^5 + 163x^4 + 13x^3 - 418x^2 + 4x + 269
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			1.3
Root			$-2.12535$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.12535 q^{5} +2.08900 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.12535 q^{5} +2.08900 q^{7} +1.00000 q^{9} -3.99111 q^{11} +3.23865 q^{13} +2.12535 q^{15} -1.33878 q^{17} +2.48827 q^{19} -2.08900 q^{21} -1.08900 q^{23} -0.482879 q^{25} -1.00000 q^{27} +5.80941 q^{29} -9.89064 q^{31} +3.99111 q^{33} -4.43987 q^{35} +7.35179 q^{37} -3.23865 q^{39} -10.3049 q^{41} +5.90969 q^{43} -2.12535 q^{45} +11.5733 q^{47} -2.63606 q^{49} +1.33878 q^{51} +5.50865 q^{53} +8.48251 q^{55} -2.48827 q^{57} -13.9870 q^{59} -2.75964 q^{61} +2.08900 q^{63} -6.88327 q^{65} +8.87335 q^{67} +1.08900 q^{69} -0.828726 q^{71} -0.0949232 q^{73} +0.482879 q^{75} -8.33744 q^{77} +15.8260 q^{79} +1.00000 q^{81} -8.15253 q^{83} +2.84538 q^{85} -5.80941 q^{87} -5.10300 q^{89} +6.76556 q^{91} +9.89064 q^{93} -5.28845 q^{95} +15.8568 q^{97} -3.99111 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 - q^7 + 8 * q^9	$ 8 q - 8 q^{3} - q^{7} + 8 q^{9} - 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} + 9 q^{23} + 6 q^{25} - 8 q^{27} + 17 q^{29} - 23 q^{31} + 3 q^{33} - 15 q^{35} + 8 q^{37} + 8 q^{39} - 8 q^{41} - 2 q^{43} - 34 q^{47} + 5 q^{49} + 7 q^{51} + 12 q^{53} - 7 q^{55} - 16 q^{59} - 2 q^{61} - q^{63} - 14 q^{65} + 21 q^{67} - 9 q^{69} - 29 q^{71} - 38 q^{73} - 6 q^{75} + 20 q^{77} - 12 q^{79} + 8 q^{81} - 32 q^{83} + 23 q^{85} - 17 q^{87} + 11 q^{89} - 5 q^{91} + 23 q^{93} - 67 q^{95} + 8 q^{97} - 3 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 - q^7 + 8 * q^9 - 3 * q^11 - 8 * q^13 - 7 * q^17 + q^21 + 9 * q^23 + 6 * q^25 - 8 * q^27 + 17 * q^29 - 23 * q^31 + 3 * q^33 - 15 * q^35 + 8 * q^37 + 8 * q^39 - 8 * q^41 - 2 * q^43 - 34 * q^47 + 5 * q^49 + 7 * q^51 + 12 * q^53 - 7 * q^55 - 16 * q^59 - 2 * q^61 - q^63 - 14 * q^65 + 21 * q^67 - 9 * q^69 - 29 * q^71 - 38 * q^73 - 6 * q^75 + 20 * q^77 - 12 * q^79 + 8 * q^81 - 32 * q^83 + 23 * q^85 - 17 * q^87 + 11 * q^89 - 5 * q^91 + 23 * q^93 - 67 * q^95 + 8 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.12535	−0.950486	−0.475243	−	0.879855i	$-0.657640\pi$
$5$	−2.12535	−0.950486	−0.475243	+	0.879855i	$0.657640\pi$
$6$	0	0
$7$	2.08900	0.789569	0.394785	−	0.918774i	$-0.370819\pi$
$7$	2.08900	0.789569	0.394785	+	0.918774i	$0.370819\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.99111	−1.20336	−0.601682	−	0.798736i	$-0.705503\pi$
$11$	−3.99111	−1.20336	−0.601682	+	0.798736i	$0.705503\pi$
$12$	0	0
$13$	3.23865	0.898240	0.449120	−	0.893471i	$-0.351738\pi$
$13$	3.23865	0.898240	0.449120	+	0.893471i	$0.351738\pi$
$14$	0	0
$15$	2.12535	0.548764
$16$	0	0
$17$	−1.33878	−0.324702	−0.162351	−	0.986733i	$-0.551908\pi$
$17$	−1.33878	−0.324702	−0.162351	+	0.986733i	$0.551908\pi$
$18$	0	0
$19$	2.48827	0.570848	0.285424	−	0.958401i	$-0.407866\pi$
$19$	2.48827	0.570848	0.285424	+	0.958401i	$0.407866\pi$
$20$	0	0
$21$	−2.08900	−0.455858
$22$	0	0
$23$	−1.08900	−0.227073	−0.113537	−	0.993534i	$-0.536218\pi$
$23$	−1.08900	−0.227073	−0.113537	+	0.993534i	$0.536218\pi$
$24$	0	0
$25$	−0.482879	−0.0965758
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.80941	1.07878	0.539390	−	0.842056i	$-0.318655\pi$
$29$	5.80941	1.07878	0.539390	+	0.842056i	$0.318655\pi$
$30$	0	0
$31$	−9.89064	−1.77641	−0.888205	−	0.459447i	$-0.848048\pi$
$31$	−9.89064	−1.77641	−0.888205	+	0.459447i	$0.848048\pi$
$32$	0	0
$33$	3.99111	0.694763
$34$	0	0
$35$	−4.43987	−0.750475
$36$	0	0
$37$	7.35179	1.20863	0.604314	−	0.796746i	$-0.293447\pi$
$37$	7.35179	1.20863	0.604314	+	0.796746i	$0.293447\pi$
$38$	0	0
$39$	−3.23865	−0.518599
$40$	0	0
$41$	−10.3049	−1.60935	−0.804675	−	0.593716i	$-0.797660\pi$
$41$	−10.3049	−1.60935	−0.804675	+	0.593716i	$0.797660\pi$
$42$	0	0
$43$	5.90969	0.901220	0.450610	−	0.892721i	$-0.351207\pi$
$43$	5.90969	0.901220	0.450610	+	0.892721i	$0.351207\pi$
$44$	0	0
$45$	−2.12535	−0.316829
$46$	0	0
$47$	11.5733	1.68814	0.844070	−	0.536234i	$-0.180153\pi$
$47$	11.5733	1.68814	0.844070	+	0.536234i	$0.180153\pi$
$48$	0	0
$49$	−2.63606	−0.376580
$50$	0	0
$51$	1.33878	0.187467
$52$	0	0
$53$	5.50865	0.756671	0.378336	−	0.925669i	$-0.376497\pi$
$53$	5.50865	0.756671	0.378336	+	0.925669i	$0.376497\pi$
$54$	0	0
$55$	8.48251	1.14378
$56$	0	0
$57$	−2.48827	−0.329579
$58$	0	0
$59$	−13.9870	−1.82096	−0.910478	−	0.413558i	$-0.864286\pi$
$59$	−13.9870	−1.82096	−0.910478	+	0.413558i	$0.864286\pi$
$60$	0	0
$61$	−2.75964	−0.353336	−0.176668	−	0.984270i	$-0.556532\pi$
$61$	−2.75964	−0.353336	−0.176668	+	0.984270i	$0.556532\pi$
$62$	0	0
$63$	2.08900	0.263190
$64$	0	0
$65$	−6.88327	−0.853765
$66$	0	0
$67$	8.87335	1.08405	0.542026	−	0.840362i	$-0.317657\pi$
$67$	8.87335	1.08405	0.542026	+	0.840362i	$0.317657\pi$
$68$	0	0
$69$	1.08900	0.131101
$70$	0	0
$71$	−0.828726	−0.0983516	−0.0491758	−	0.998790i	$-0.515659\pi$
$71$	−0.828726	−0.0983516	−0.0491758	+	0.998790i	$0.515659\pi$
$72$	0	0
$73$	−0.0949232	−0.0111099	−0.00555496	−	0.999985i	$-0.501768\pi$
$73$	−0.0949232	−0.0111099	−0.00555496	+	0.999985i	$0.501768\pi$
$74$	0	0
$75$	0.482879	0.0557581
$76$	0	0
$77$	−8.33744	−0.950139
$78$	0	0
$79$	15.8260	1.78057	0.890285	−	0.455404i	$-0.150505\pi$
$79$	15.8260	1.78057	0.890285	+	0.455404i	$0.150505\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.15253	−0.894856	−0.447428	−	0.894320i	$-0.647660\pi$
$83$	−8.15253	−0.894856	−0.447428	+	0.894320i	$0.647660\pi$
$84$	0	0
$85$	2.84538	0.308624
$86$	0	0
$87$	−5.80941	−0.622834
$88$	0	0
$89$	−5.10300	−0.540917	−0.270459	−	0.962732i	$-0.587175\pi$
$89$	−5.10300	−0.540917	−0.270459	+	0.962732i	$0.587175\pi$
$90$	0	0
$91$	6.76556	0.709223
$92$	0	0
$93$	9.89064	1.02561
$94$	0	0
$95$	−5.28845	−0.542583
$96$	0	0
$97$	15.8568	1.61001	0.805005	−	0.593268i	$-0.202163\pi$
$97$	15.8568	1.61001	0.805005	+	0.593268i	$0.202163\pi$
$98$	0	0
$99$	−3.99111	−0.401121
$100$	0	0
$101$	−8.72977	−0.868645	−0.434322	−	0.900758i	$-0.643012\pi$
$101$	−8.72977	−0.868645	−0.434322	+	0.900758i	$0.643012\pi$
$102$	0	0
$103$	−19.1263	−1.88457	−0.942287	−	0.334807i	$-0.891329\pi$
$103$	−19.1263	−1.88457	−0.942287	+	0.334807i	$0.891329\pi$
$104$	0	0
$105$	4.43987	0.433287
$106$	0	0
$107$	−3.36186	−0.325004	−0.162502	−	0.986708i	$-0.551956\pi$
$107$	−3.36186	−0.325004	−0.162502	+	0.986708i	$0.551956\pi$
$108$	0	0
$109$	−10.7988	−1.03434	−0.517171	−	0.855882i	$-0.673015\pi$
$109$	−10.7988	−1.03434	−0.517171	+	0.855882i	$0.673015\pi$
$110$	0	0
$111$	−7.35179	−0.697801
$112$	0	0
$113$	−4.16836	−0.392127	−0.196063	−	0.980591i	$-0.562816\pi$
$113$	−4.16836	−0.392127	−0.196063	+	0.980591i	$0.562816\pi$
$114$	0	0
$115$	2.31452	0.215830
$116$	0	0
$117$	3.23865	0.299413
$118$	0	0
$119$	−2.79671	−0.256374
$120$	0	0
$121$	4.92893	0.448085
$122$	0	0
$123$	10.3049	0.929159
$124$	0	0
$125$	11.6530	1.04228
$126$	0	0
$127$	−21.8072	−1.93507	−0.967537	−	0.252730i	$-0.918672\pi$
$127$	−21.8072	−1.93507	−0.967537	+	0.252730i	$0.918672\pi$
$128$	0	0
$129$	−5.90969	−0.520320
$130$	0	0
$131$	−19.2842	−1.68487	−0.842433	−	0.538801i	$-0.818877\pi$
$131$	−19.2842	−1.68487	−0.842433	+	0.538801i	$0.818877\pi$
$132$	0	0
$133$	5.19801	0.450724
$134$	0	0
$135$	2.12535	0.182921
$136$	0	0
$137$	10.5983	0.905471	0.452735	−	0.891645i	$-0.350448\pi$
$137$	10.5983	0.905471	0.452735	+	0.891645i	$0.350448\pi$
$138$	0	0
$139$	2.68531	0.227765	0.113883	−	0.993494i	$-0.463671\pi$
$139$	2.68531	0.227765	0.113883	+	0.993494i	$0.463671\pi$
$140$	0	0
$141$	−11.5733	−0.974648
$142$	0	0
$143$	−12.9258	−1.08091
$144$	0	0
$145$	−12.3470	−1.02537
$146$	0	0
$147$	2.63606	0.217419
$148$	0	0
$149$	1.98625	0.162720	0.0813598	−	0.996685i	$-0.474074\pi$
$149$	1.98625	0.162720	0.0813598	+	0.996685i	$0.474074\pi$
$150$	0	0
$151$	−21.0079	−1.70960	−0.854799	−	0.518959i	$-0.826320\pi$
$151$	−21.0079	−1.70960	−0.854799	+	0.518959i	$0.826320\pi$
$152$	0	0
$153$	−1.33878	−0.108234
$154$	0	0
$155$	21.0211	1.68845
$156$	0	0
$157$	−3.27178	−0.261116	−0.130558	−	0.991441i	$-0.541677\pi$
$157$	−3.27178	−0.261116	−0.130558	+	0.991441i	$0.541677\pi$
$158$	0	0
$159$	−5.50865	−0.436864
$160$	0	0
$161$	−2.27493	−0.179290
$162$	0	0
$163$	−5.72813	−0.448662	−0.224331	−	0.974513i	$-0.572020\pi$
$163$	−5.72813	−0.448662	−0.224331	+	0.974513i	$0.572020\pi$
$164$	0	0
$165$	−8.48251	−0.660362
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−2.51114	−0.193165
$170$	0	0
$171$	2.48827	0.190283
$172$	0	0
$173$	−5.82828	−0.443116	−0.221558	−	0.975147i	$-0.571114\pi$
$173$	−5.82828	−0.443116	−0.221558	+	0.975147i	$0.571114\pi$
$174$	0	0
$175$	−1.00874	−0.0762533
$176$	0	0
$177$	13.9870	1.05133
$178$	0	0
$179$	−5.47026	−0.408867	−0.204433	−	0.978880i	$-0.565535\pi$
$179$	−5.47026	−0.408867	−0.204433	+	0.978880i	$0.565535\pi$
$180$	0	0
$181$	25.5026	1.89559	0.947797	−	0.318875i	$-0.103305\pi$
$181$	25.5026	1.89559	0.947797	+	0.318875i	$0.103305\pi$
$182$	0	0
$183$	2.75964	0.203999
$184$	0	0
$185$	−15.6251	−1.14878
$186$	0	0
$187$	5.34321	0.390734
$188$	0	0
$189$	−2.08900	−0.151953
$190$	0	0
$191$	−27.2999	−1.97535	−0.987675	−	0.156522i	$-0.949972\pi$
$191$	−27.2999	−1.97535	−0.987675	+	0.156522i	$0.949972\pi$
$192$	0	0
$193$	−18.3998	−1.32445	−0.662223	−	0.749306i	$-0.730387\pi$
$193$	−18.3998	−1.32445	−0.662223	+	0.749306i	$0.730387\pi$
$194$	0	0
$195$	6.88327	0.492921
$196$	0	0
$197$	17.3315	1.23482	0.617408	−	0.786643i	$-0.288183\pi$
$197$	17.3315	1.23482	0.617408	+	0.786643i	$0.288183\pi$
$198$	0	0
$199$	14.1446	1.00268	0.501341	−	0.865250i	$-0.332840\pi$
$199$	14.1446	1.00268	0.501341	+	0.865250i	$0.332840\pi$
$200$	0	0
$201$	−8.87335	−0.625878
$202$	0	0
$203$	12.1359	0.851772
$204$	0	0
$205$	21.9015	1.52966
$206$	0	0
$207$	−1.08900	−0.0756910
$208$	0	0
$209$	−9.93095	−0.686938
$210$	0	0
$211$	−9.22231	−0.634890	−0.317445	−	0.948277i	$-0.602825\pi$
$211$	−9.22231	−0.634890	−0.317445	+	0.948277i	$0.602825\pi$
$212$	0	0
$213$	0.828726	0.0567834
$214$	0	0
$215$	−12.5602	−0.856597
$216$	0	0
$217$	−20.6616	−1.40260
$218$	0	0
$219$	0.0949232	0.00641432
$220$	0	0
$221$	−4.33584	−0.291660
$222$	0	0
$223$	−19.7400	−1.32189	−0.660944	−	0.750435i	$-0.729844\pi$
$223$	−19.7400	−1.32189	−0.660944	+	0.750435i	$0.729844\pi$
$224$	0	0
$225$	−0.482879	−0.0321919
$226$	0	0
$227$	−5.79089	−0.384355	−0.192177	−	0.981360i	$-0.561555\pi$
$227$	−5.79089	−0.384355	−0.192177	+	0.981360i	$0.561555\pi$
$228$	0	0
$229$	1.43319	0.0947079	0.0473540	−	0.998878i	$-0.484921\pi$
$229$	1.43319	0.0947079	0.0473540	+	0.998878i	$0.484921\pi$
$230$	0	0
$231$	8.33744	0.548563
$232$	0	0
$233$	−13.0534	−0.855157	−0.427579	−	0.903978i	$-0.640633\pi$
$233$	−13.0534	−0.855157	−0.427579	+	0.903978i	$0.640633\pi$
$234$	0	0
$235$	−24.5973	−1.60455
$236$	0	0
$237$	−15.8260	−1.02801
$238$	0	0
$239$	16.3163	1.05541	0.527706	−	0.849427i	$-0.323052\pi$
$239$	16.3163	1.05541	0.527706	+	0.849427i	$0.323052\pi$
$240$	0	0
$241$	−24.4205	−1.57306	−0.786531	−	0.617550i	$-0.788125\pi$
$241$	−24.4205	−1.57306	−0.786531	+	0.617550i	$0.788125\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.60256	0.357934
$246$	0	0
$247$	8.05864	0.512759
$248$	0	0
$249$	8.15253	0.516645
$250$	0	0
$251$	−9.91841	−0.626045	−0.313022	−	0.949746i	$-0.601341\pi$
$251$	−9.91841	−0.626045	−0.313022	+	0.949746i	$0.601341\pi$
$252$	0	0
$253$	4.34633	0.273252
$254$	0	0
$255$	−2.84538	−0.178184
$256$	0	0
$257$	5.97086	0.372452	0.186226	−	0.982507i	$-0.440374\pi$
$257$	5.97086	0.372452	0.186226	+	0.982507i	$0.440374\pi$
$258$	0	0
$259$	15.3579	0.954295
$260$	0	0
$261$	5.80941	0.359593
$262$	0	0
$263$	−3.79794	−0.234191	−0.117096	−	0.993121i	$-0.537358\pi$
$263$	−3.79794	−0.234191	−0.117096	+	0.993121i	$0.537358\pi$
$264$	0	0
$265$	−11.7078	−0.719205
$266$	0	0
$267$	5.10300	0.312299
$268$	0	0
$269$	18.1289	1.10534	0.552670	−	0.833400i	$-0.313609\pi$
$269$	18.1289	1.10534	0.552670	+	0.833400i	$0.313609\pi$
$270$	0	0
$271$	−27.8658	−1.69273	−0.846364	−	0.532605i	$-0.821213\pi$
$271$	−27.8658	−1.69273	−0.846364	+	0.532605i	$0.821213\pi$
$272$	0	0
$273$	−6.76556	−0.409470
$274$	0	0
$275$	1.92722	0.116216
$276$	0	0
$277$	−7.45511	−0.447934	−0.223967	−	0.974597i	$-0.571901\pi$
$277$	−7.45511	−0.447934	−0.223967	+	0.974597i	$0.571901\pi$
$278$	0	0
$279$	−9.89064	−0.592137
$280$	0	0
$281$	2.09663	0.125075	0.0625373	−	0.998043i	$-0.480081\pi$
$281$	2.09663	0.125075	0.0625373	+	0.998043i	$0.480081\pi$
$282$	0	0
$283$	2.62190	0.155856	0.0779279	−	0.996959i	$-0.475170\pi$
$283$	2.62190	0.155856	0.0779279	+	0.996959i	$0.475170\pi$
$284$	0	0
$285$	5.28845	0.313261
$286$	0	0
$287$	−21.5269	−1.27069
$288$	0	0
$289$	−15.2077	−0.894569
$290$	0	0
$291$	−15.8568	−0.929539
$292$	0	0
$293$	−9.86788	−0.576488	−0.288244	−	0.957557i	$-0.593071\pi$
$293$	−9.86788	−0.576488	−0.288244	+	0.957557i	$0.593071\pi$
$294$	0	0
$295$	29.7274	1.73079
$296$	0	0
$297$	3.99111	0.231588
$298$	0	0
$299$	−3.52690	−0.203966
$300$	0	0
$301$	12.3454	0.711576
$302$	0	0
$303$	8.72977	0.501512
$304$	0	0
$305$	5.86521	0.335841
$306$	0	0
$307$	22.4538	1.28151	0.640753	−	0.767747i	$-0.278622\pi$
$307$	22.4538	1.28151	0.640753	+	0.767747i	$0.278622\pi$
$308$	0	0
$309$	19.1263	1.08806
$310$	0	0
$311$	−19.7217	−1.11831	−0.559157	−	0.829062i	$-0.688875\pi$
$311$	−19.7217	−1.11831	−0.559157	+	0.829062i	$0.688875\pi$
$312$	0	0
$313$	22.4601	1.26952	0.634761	−	0.772709i	$-0.281099\pi$
$313$	22.4601	1.26952	0.634761	+	0.772709i	$0.281099\pi$
$314$	0	0
$315$	−4.43987	−0.250158
$316$	0	0
$317$	−10.6210	−0.596534	−0.298267	−	0.954482i	$-0.596409\pi$
$317$	−10.6210	−0.596534	−0.298267	+	0.954482i	$0.596409\pi$
$318$	0	0
$319$	−23.1860	−1.29817
$320$	0	0
$321$	3.36186	0.187641
$322$	0	0
$323$	−3.33124	−0.185355
$324$	0	0
$325$	−1.56388	−0.0867483
$326$	0	0
$327$	10.7988	0.597177
$328$	0	0
$329$	24.1767	1.33290
$330$	0	0
$331$	−23.2975	−1.28055	−0.640273	−	0.768147i	$-0.721179\pi$
$331$	−23.2975	−1.28055	−0.640273	+	0.768147i	$0.721179\pi$
$332$	0	0
$333$	7.35179	0.402876
$334$	0	0
$335$	−18.8590	−1.03038
$336$	0	0
$337$	−7.74585	−0.421943	−0.210972	−	0.977492i	$-0.567663\pi$
$337$	−7.74585	−0.421943	−0.210972	+	0.977492i	$0.567663\pi$
$338$	0	0
$339$	4.16836	0.226394
$340$	0	0
$341$	39.4746	2.13767
$342$	0	0
$343$	−20.1298	−1.08691
$344$	0	0
$345$	−2.31452	−0.124609
$346$	0	0
$347$	−30.3702	−1.63036	−0.815178	−	0.579210i	$-0.803361\pi$
$347$	−30.3702	−1.63036	−0.815178	+	0.579210i	$0.803361\pi$
$348$	0	0
$349$	−22.3020	−1.19380	−0.596900	−	0.802316i	$-0.703601\pi$
$349$	−22.3020	−1.19380	−0.596900	+	0.802316i	$0.703601\pi$
$350$	0	0
$351$	−3.23865	−0.172866
$352$	0	0
$353$	−11.2540	−0.598990	−0.299495	−	0.954098i	$-0.596818\pi$
$353$	−11.2540	−0.598990	−0.299495	+	0.954098i	$0.596818\pi$
$354$	0	0
$355$	1.76133	0.0934819
$356$	0	0
$357$	2.79671	0.148018
$358$	0	0
$359$	−26.4040	−1.39355	−0.696775	−	0.717289i	$-0.745383\pi$
$359$	−26.4040	−1.39355	−0.696775	+	0.717289i	$0.745383\pi$
$360$	0	0
$361$	−12.8085	−0.674132
$362$	0	0
$363$	−4.92893	−0.258702
$364$	0	0
$365$	0.201745	0.0105598
$366$	0	0
$367$	6.04475	0.315533	0.157767	−	0.987476i	$-0.449571\pi$
$367$	6.04475	0.315533	0.157767	+	0.987476i	$0.449571\pi$
$368$	0	0
$369$	−10.3049	−0.536450
$370$	0	0
$371$	11.5076	0.597444
$372$	0	0
$373$	28.2217	1.46127	0.730633	−	0.682771i	$-0.239225\pi$
$373$	28.2217	1.46127	0.730633	+	0.682771i	$0.239225\pi$
$374$	0	0
$375$	−11.6530	−0.601761
$376$	0	0
$377$	18.8147	0.969004
$378$	0	0
$379$	17.6625	0.907262	0.453631	−	0.891190i	$-0.350128\pi$
$379$	17.6625	0.907262	0.453631	+	0.891190i	$0.350128\pi$
$380$	0	0
$381$	21.8072	1.11722
$382$	0	0
$383$	−9.85617	−0.503627	−0.251813	−	0.967776i	$-0.581027\pi$
$383$	−9.85617	−0.503627	−0.251813	+	0.967776i	$0.581027\pi$
$384$	0	0
$385$	17.7200	0.903094
$386$	0	0
$387$	5.90969	0.300407
$388$	0	0
$389$	13.8070	0.700044	0.350022	−	0.936741i	$-0.386174\pi$
$389$	13.8070	0.700044	0.350022	+	0.936741i	$0.386174\pi$
$390$	0	0
$391$	1.45794	0.0737310
$392$	0	0
$393$	19.2842	0.972758
$394$	0	0
$395$	−33.6359	−1.69241
$396$	0	0
$397$	−19.6133	−0.984366	−0.492183	−	0.870492i	$-0.663801\pi$
$397$	−19.6133	−0.984366	−0.492183	+	0.870492i	$0.663801\pi$
$398$	0	0
$399$	−5.19801	−0.260226
$400$	0	0
$401$	36.4301	1.81923	0.909617	−	0.415448i	$-0.136375\pi$
$401$	36.4301	1.81923	0.909617	+	0.415448i	$0.136375\pi$
$402$	0	0
$403$	−32.0323	−1.59564
$404$	0	0
$405$	−2.12535	−0.105610
$406$	0	0
$407$	−29.3418	−1.45442
$408$	0	0
$409$	17.4129	0.861010	0.430505	−	0.902588i	$-0.358335\pi$
$409$	17.4129	0.861010	0.430505	+	0.902588i	$0.358335\pi$
$410$	0	0
$411$	−10.5983	−0.522774
$412$	0	0
$413$	−29.2190	−1.43777
$414$	0	0
$415$	17.3270	0.850548
$416$	0	0
$417$	−2.68531	−0.131500
$418$	0	0
$419$	−11.8440	−0.578620	−0.289310	−	0.957236i	$-0.593426\pi$
$419$	−11.8440	−0.578620	−0.289310	+	0.957236i	$0.593426\pi$
$420$	0	0
$421$	0.454844	0.0221677	0.0110839	−	0.999939i	$-0.496472\pi$
$421$	0.454844	0.0221677	0.0110839	+	0.999939i	$0.496472\pi$
$422$	0	0
$423$	11.5733	0.562713
$424$	0	0
$425$	0.646468	0.0313583
$426$	0	0
$427$	−5.76491	−0.278983
$428$	0	0
$429$	12.9258	0.624064
$430$	0	0
$431$	−26.2521	−1.26452	−0.632260	−	0.774756i	$-0.717873\pi$
$431$	−26.2521	−1.26452	−0.632260	+	0.774756i	$0.717873\pi$
$432$	0	0
$433$	−1.98898	−0.0955843	−0.0477922	−	0.998857i	$-0.515219\pi$
$433$	−1.98898	−0.0955843	−0.0477922	+	0.998857i	$0.515219\pi$
$434$	0	0
$435$	12.3470	0.591995
$436$	0	0
$437$	−2.70974	−0.129624
$438$	0	0
$439$	−29.9321	−1.42858	−0.714290	−	0.699850i	$-0.753250\pi$
$439$	−29.9321	−1.42858	−0.714290	+	0.699850i	$0.753250\pi$
$440$	0	0
$441$	−2.63606	−0.125527
$442$	0	0
$443$	−2.28558	−0.108591	−0.0542956	−	0.998525i	$-0.517291\pi$
$443$	−2.28558	−0.108591	−0.0542956	+	0.998525i	$0.517291\pi$
$444$	0	0
$445$	10.8457	0.514134
$446$	0	0
$447$	−1.98625	−0.0939462
$448$	0	0
$449$	12.0752	0.569865	0.284933	−	0.958548i	$-0.408029\pi$
$449$	12.0752	0.569865	0.284933	+	0.958548i	$0.408029\pi$
$450$	0	0
$451$	41.1278	1.93663
$452$	0	0
$453$	21.0079	0.987037
$454$	0	0
$455$	−14.3792	−0.674107
$456$	0	0
$457$	32.1051	1.50181	0.750906	−	0.660409i	$-0.229617\pi$
$457$	32.1051	1.50181	0.750906	+	0.660409i	$0.229617\pi$
$458$	0	0
$459$	1.33878	0.0624888
$460$	0	0
$461$	−7.07193	−0.329372	−0.164686	−	0.986346i	$-0.552661\pi$
$461$	−7.07193	−0.329372	−0.164686	+	0.986346i	$0.552661\pi$
$462$	0	0
$463$	11.6733	0.542502	0.271251	−	0.962509i	$-0.412563\pi$
$463$	11.6733	0.542502	0.271251	+	0.962509i	$0.412563\pi$
$464$	0	0
$465$	−21.0211	−0.974829
$466$	0	0
$467$	−11.4693	−0.530734	−0.265367	−	0.964147i	$-0.585493\pi$
$467$	−11.4693	−0.530734	−0.265367	+	0.964147i	$0.585493\pi$
$468$	0	0
$469$	18.5365	0.855934
$470$	0	0
$471$	3.27178	0.150756
$472$	0	0
$473$	−23.5862	−1.08450
$474$	0	0
$475$	−1.20153	−0.0551301
$476$	0	0
$477$	5.50865	0.252224
$478$	0	0
$479$	22.0875	1.00921	0.504603	−	0.863352i	$-0.331639\pi$
$479$	22.0875	1.00921	0.504603	+	0.863352i	$0.331639\pi$
$480$	0	0
$481$	23.8099	1.08564
$482$	0	0
$483$	2.27493	0.103513
$484$	0	0
$485$	−33.7012	−1.53029
$486$	0	0
$487$	38.3362	1.73718	0.868589	−	0.495533i	$-0.165027\pi$
$487$	38.3362	1.73718	0.868589	+	0.495533i	$0.165027\pi$
$488$	0	0
$489$	5.72813	0.259035
$490$	0	0
$491$	28.8672	1.30276	0.651379	−	0.758753i	$-0.274191\pi$
$491$	28.8672	1.30276	0.651379	+	0.758753i	$0.274191\pi$
$492$	0	0
$493$	−7.77752	−0.350282
$494$	0	0
$495$	8.48251	0.381260
$496$	0	0
$497$	−1.73121	−0.0776554
$498$	0	0
$499$	14.1802	0.634794	0.317397	−	0.948293i	$-0.397191\pi$
$499$	14.1802	0.634794	0.317397	+	0.948293i	$0.397191\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	34.5400	1.54006	0.770032	−	0.638006i	$-0.220240\pi$
$503$	34.5400	1.54006	0.770032	+	0.638006i	$0.220240\pi$
$504$	0	0
$505$	18.5538	0.825635
$506$	0	0
$507$	2.51114	0.111524
$508$	0	0
$509$	29.1481	1.29197	0.645984	−	0.763351i	$-0.276447\pi$
$509$	29.1481	1.29197	0.645984	+	0.763351i	$0.276447\pi$
$510$	0	0
$511$	−0.198295	−0.00877205
$512$	0	0
$513$	−2.48827	−0.109860
$514$	0	0
$515$	40.6502	1.79126
$516$	0	0
$517$	−46.1903	−2.03145
$518$	0	0
$519$	5.82828	0.255833
$520$	0	0
$521$	11.9483	0.523463	0.261731	−	0.965141i	$-0.415707\pi$
$521$	11.9483	0.523463	0.261731	+	0.965141i	$0.415707\pi$
$522$	0	0
$523$	41.6693	1.82207	0.911036	−	0.412327i	$-0.135284\pi$
$523$	41.6693	1.82207	0.911036	+	0.412327i	$0.135284\pi$
$524$	0	0
$525$	1.00874	0.0440249
$526$	0	0
$527$	13.2414	0.576803
$528$	0	0
$529$	−21.8141	−0.948438
$530$	0	0
$531$	−13.9870	−0.606985
$532$	0	0
$533$	−33.3739	−1.44558
$534$	0	0
$535$	7.14514	0.308912
$536$	0	0
$537$	5.47026	0.236059
$538$	0	0
$539$	10.5208	0.453163
$540$	0	0
$541$	−16.9098	−0.727007	−0.363504	−	0.931593i	$-0.618420\pi$
$541$	−16.9098	−0.727007	−0.363504	+	0.931593i	$0.618420\pi$
$542$	0	0
$543$	−25.5026	−1.09442
$544$	0	0
$545$	22.9513	0.983127
$546$	0	0
$547$	24.2529	1.03698	0.518489	−	0.855085i	$-0.326495\pi$
$547$	24.2529	1.03698	0.518489	+	0.855085i	$0.326495\pi$
$548$	0	0
$549$	−2.75964	−0.117779
$550$	0	0
$551$	14.4554	0.615820
$552$	0	0
$553$	33.0607	1.40588
$554$	0	0
$555$	15.6251	0.663251
$556$	0	0
$557$	−28.4990	−1.20754	−0.603771	−	0.797158i	$-0.706336\pi$
$557$	−28.4990	−1.20754	−0.603771	+	0.797158i	$0.706336\pi$
$558$	0	0
$559$	19.1394	0.809512
$560$	0	0
$561$	−5.34321	−0.225590
$562$	0	0
$563$	−16.8070	−0.708332	−0.354166	−	0.935183i	$-0.615235\pi$
$563$	−16.8070	−0.708332	−0.354166	+	0.935183i	$0.615235\pi$
$564$	0	0
$565$	8.85924	0.372711
$566$	0	0
$567$	2.08900	0.0877299
$568$	0	0
$569$	−1.50924	−0.0632708	−0.0316354	−	0.999499i	$-0.510072\pi$
$569$	−1.50924	−0.0632708	−0.0316354	+	0.999499i	$0.510072\pi$
$570$	0	0
$571$	19.7918	0.828262	0.414131	−	0.910217i	$-0.364086\pi$
$571$	19.7918	0.828262	0.414131	+	0.910217i	$0.364086\pi$
$572$	0	0
$573$	27.2999	1.14047
$574$	0	0
$575$	0.525857	0.0219298
$576$	0	0
$577$	−28.2655	−1.17671	−0.588354	−	0.808604i	$-0.700224\pi$
$577$	−28.2655	−1.17671	−0.588354	+	0.808604i	$0.700224\pi$
$578$	0	0
$579$	18.3998	0.764670
$580$	0	0
$581$	−17.0307	−0.706551
$582$	0	0
$583$	−21.9856	−0.910551
$584$	0	0
$585$	−6.88327	−0.284588
$586$	0	0
$587$	38.5338	1.59046	0.795230	−	0.606307i	$-0.207350\pi$
$587$	38.5338	1.59046	0.795230	+	0.606307i	$0.207350\pi$
$588$	0	0
$589$	−24.6106	−1.01406
$590$	0	0
$591$	−17.3315	−0.712922
$592$	0	0
$593$	1.07271	0.0440510	0.0220255	−	0.999757i	$-0.492989\pi$
$593$	1.07271	0.0440510	0.0220255	+	0.999757i	$0.492989\pi$
$594$	0	0
$595$	5.94400	0.243680
$596$	0	0
$597$	−14.1446	−0.578899
$598$	0	0
$599$	9.42396	0.385053	0.192526	−	0.981292i	$-0.438332\pi$
$599$	9.42396	0.385053	0.192526	+	0.981292i	$0.438332\pi$
$600$	0	0
$601$	11.8272	0.482443	0.241221	−	0.970470i	$-0.422452\pi$
$601$	11.8272	0.482443	0.241221	+	0.970470i	$0.422452\pi$
$602$	0	0
$603$	8.87335	0.361351
$604$	0	0
$605$	−10.4757	−0.425899
$606$	0	0
$607$	−5.80749	−0.235719	−0.117859	−	0.993030i	$-0.537603\pi$
$607$	−5.80749	−0.235719	−0.117859	+	0.993030i	$0.537603\pi$
$608$	0	0
$609$	−12.1359	−0.491771
$610$	0	0
$611$	37.4819	1.51635
$612$	0	0
$613$	14.0851	0.568890	0.284445	−	0.958692i	$-0.408191\pi$
$613$	14.0851	0.568890	0.284445	+	0.958692i	$0.408191\pi$
$614$	0	0
$615$	−21.9015	−0.883152
$616$	0	0
$617$	−12.8152	−0.515922	−0.257961	−	0.966155i	$-0.583051\pi$
$617$	−12.8152	−0.515922	−0.257961	+	0.966155i	$0.583051\pi$
$618$	0	0
$619$	−45.2871	−1.82024	−0.910122	−	0.414341i	$-0.864012\pi$
$619$	−45.2871	−1.82024	−0.910122	+	0.414341i	$0.864012\pi$
$620$	0	0
$621$	1.08900	0.0437002
$622$	0	0
$623$	−10.6602	−0.427091
$624$	0	0
$625$	−22.3524	−0.894097
$626$	0	0
$627$	9.93095	0.396604
$628$	0	0
$629$	−9.84243	−0.392443
$630$	0	0
$631$	−2.43728	−0.0970264	−0.0485132	−	0.998823i	$-0.515448\pi$
$631$	−2.43728	−0.0970264	−0.0485132	+	0.998823i	$0.515448\pi$
$632$	0	0
$633$	9.22231	0.366554
$634$	0	0
$635$	46.3479	1.83926
$636$	0	0
$637$	−8.53728	−0.338260
$638$	0	0
$639$	−0.828726	−0.0327839
$640$	0	0
$641$	37.4275	1.47830	0.739149	−	0.673542i	$-0.235228\pi$
$641$	37.4275	1.47830	0.739149	+	0.673542i	$0.235228\pi$
$642$	0	0
$643$	−32.1270	−1.26697	−0.633483	−	0.773757i	$-0.718375\pi$
$643$	−32.1270	−1.26697	−0.633483	+	0.773757i	$0.718375\pi$
$644$	0	0
$645$	12.5602	0.494557
$646$	0	0
$647$	−43.5139	−1.71071	−0.855353	−	0.518045i	$-0.826660\pi$
$647$	−43.5139	−1.71071	−0.855353	+	0.518045i	$0.826660\pi$
$648$	0	0
$649$	55.8237	2.19127
$650$	0	0
$651$	20.6616	0.809791
$652$	0	0
$653$	30.4074	1.18993	0.594966	−	0.803751i	$-0.297166\pi$
$653$	30.4074	1.18993	0.594966	+	0.803751i	$0.297166\pi$
$654$	0	0
$655$	40.9856	1.60144
$656$	0	0
$657$	−0.0949232	−0.00370331
$658$	0	0
$659$	−11.7890	−0.459235	−0.229617	−	0.973281i	$-0.573747\pi$
$659$	−11.7890	−0.459235	−0.229617	+	0.973281i	$0.573747\pi$
$660$	0	0
$661$	38.6952	1.50507	0.752534	−	0.658553i	$-0.228831\pi$
$661$	38.6952	1.50507	0.752534	+	0.658553i	$0.228831\pi$
$662$	0	0
$663$	4.33584	0.168390
$664$	0	0
$665$	−11.0476	−0.428407
$666$	0	0
$667$	−6.32647	−0.244962
$668$	0	0
$669$	19.7400	0.763192
$670$	0	0
$671$	11.0140	0.425192
$672$	0	0
$673$	−48.5648	−1.87203	−0.936017	−	0.351955i	$-0.885517\pi$
$673$	−48.5648	−1.87203	−0.936017	+	0.351955i	$0.885517\pi$
$674$	0	0
$675$	0.482879	0.0185860
$676$	0	0
$677$	42.0847	1.61745	0.808724	−	0.588189i	$-0.200159\pi$
$677$	42.0847	1.61745	0.808724	+	0.588189i	$0.200159\pi$
$678$	0	0
$679$	33.1248	1.27121
$680$	0	0
$681$	5.79089	0.221907
$682$	0	0
$683$	−32.7145	−1.25179	−0.625893	−	0.779909i	$-0.715265\pi$
$683$	−32.7145	−1.25179	−0.625893	+	0.779909i	$0.715265\pi$
$684$	0	0
$685$	−22.5250	−0.860637
$686$	0	0
$687$	−1.43319	−0.0546797
$688$	0	0
$689$	17.8406	0.679672
$690$	0	0
$691$	23.5864	0.897268	0.448634	−	0.893715i	$-0.351911\pi$
$691$	23.5864	0.897268	0.448634	+	0.893715i	$0.351911\pi$
$692$	0	0
$693$	−8.33744	−0.316713
$694$	0	0
$695$	−5.70723	−0.216488
$696$	0	0
$697$	13.7959	0.522558
$698$	0	0
$699$	13.0534	0.493725
$700$	0	0
$701$	−43.8121	−1.65476	−0.827381	−	0.561641i	$-0.810170\pi$
$701$	−43.8121	−1.65476	−0.827381	+	0.561641i	$0.810170\pi$
$702$	0	0
$703$	18.2932	0.689943
$704$	0	0
$705$	24.5973	0.926389
$706$	0	0
$707$	−18.2365	−0.685855
$708$	0	0
$709$	34.2772	1.28731	0.643653	−	0.765317i	$-0.277418\pi$
$709$	34.2772	1.28731	0.643653	+	0.765317i	$0.277418\pi$
$710$	0	0
$711$	15.8260	0.593523
$712$	0	0
$713$	10.7709	0.403375
$714$	0	0
$715$	27.4719	1.02739
$716$	0	0
$717$	−16.3163	−0.609342
$718$	0	0
$719$	−41.3946	−1.54376	−0.771879	−	0.635770i	$-0.780683\pi$
$719$	−41.3946	−1.54376	−0.771879	+	0.635770i	$0.780683\pi$
$720$	0	0
$721$	−39.9550	−1.48800
$722$	0	0
$723$	24.4205	0.908208
$724$	0	0
$725$	−2.80524	−0.104184
$726$	0	0
$727$	0.263467	0.00977146	0.00488573	−	0.999988i	$-0.498445\pi$
$727$	0.263467	0.00977146	0.00488573	+	0.999988i	$0.498445\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.91177	−0.292628
$732$	0	0
$733$	−43.2404	−1.59712	−0.798561	−	0.601914i	$-0.794405\pi$
$733$	−43.2404	−1.59712	−0.798561	+	0.601914i	$0.794405\pi$
$734$	0	0
$735$	−5.60256	−0.206654
$736$	0	0
$737$	−35.4145	−1.30451
$738$	0	0
$739$	−1.46955	−0.0540582	−0.0270291	−	0.999635i	$-0.508605\pi$
$739$	−1.46955	−0.0540582	−0.0270291	+	0.999635i	$0.508605\pi$
$740$	0	0
$741$	−8.05864	−0.296041
$742$	0	0
$743$	42.5588	1.56133	0.780666	−	0.624949i	$-0.214880\pi$
$743$	42.5588	1.56133	0.780666	+	0.624949i	$0.214880\pi$
$744$	0	0
$745$	−4.22147	−0.154663
$746$	0	0
$747$	−8.15253	−0.298285
$748$	0	0
$749$	−7.02295	−0.256613
$750$	0	0
$751$	11.2612	0.410925	0.205463	−	0.978665i	$-0.434130\pi$
$751$	11.2612	0.410925	0.205463	+	0.978665i	$0.434130\pi$
$752$	0	0
$753$	9.91841	0.361447
$754$	0	0
$755$	44.6492	1.62495
$756$	0	0
$757$	−36.1221	−1.31288	−0.656440	−	0.754378i	$-0.727939\pi$
$757$	−36.1221	−1.31288	−0.656440	+	0.754378i	$0.727939\pi$
$758$	0	0
$759$	−4.34633	−0.157762
$760$	0	0
$761$	26.3050	0.953556	0.476778	−	0.879024i	$-0.341805\pi$
$761$	26.3050	0.953556	0.476778	+	0.879024i	$0.341805\pi$
$762$	0	0
$763$	−22.5588	−0.816684
$764$	0	0
$765$	2.84538	0.102875
$766$	0	0
$767$	−45.2991	−1.63566
$768$	0	0
$769$	−34.5596	−1.24625	−0.623126	−	0.782121i	$-0.714138\pi$
$769$	−34.5596	−1.24625	−0.623126	+	0.782121i	$0.714138\pi$
$770$	0	0
$771$	−5.97086	−0.215035
$772$	0	0
$773$	48.4403	1.74228	0.871139	−	0.491037i	$-0.163382\pi$
$773$	48.4403	1.74228	0.871139	+	0.491037i	$0.163382\pi$
$774$	0	0
$775$	4.77598	0.171558
$776$	0	0
$777$	−15.3579	−0.550963
$778$	0	0
$779$	−25.6413	−0.918694
$780$	0	0
$781$	3.30753	0.118353
$782$	0	0
$783$	−5.80941	−0.207611
$784$	0	0
$785$	6.95368	0.248187
$786$	0	0
$787$	−51.6378	−1.84069	−0.920345	−	0.391108i	$-0.872092\pi$
$787$	−51.6378	−1.84069	−0.920345	+	0.391108i	$0.872092\pi$
$788$	0	0
$789$	3.79794	0.135210
$790$	0	0
$791$	−8.70773	−0.309611
$792$	0	0
$793$	−8.93752	−0.317381
$794$	0	0
$795$	11.7078	0.415233
$796$	0	0
$797$	−32.9526	−1.16724	−0.583620	−	0.812027i	$-0.698364\pi$
$797$	−32.9526	−1.16724	−0.583620	+	0.812027i	$0.698364\pi$
$798$	0	0
$799$	−15.4941	−0.548141
$800$	0	0
$801$	−5.10300	−0.180306
$802$	0	0
$803$	0.378849	0.0133693
$804$	0	0
$805$	4.83504	0.170413
$806$	0	0
$807$	−18.1289	−0.638168
$808$	0	0
$809$	18.0323	0.633981	0.316991	−	0.948429i	$-0.397328\pi$
$809$	18.0323	0.633981	0.316991	+	0.948429i	$0.397328\pi$
$810$	0	0
$811$	25.2580	0.886928	0.443464	−	0.896292i	$-0.353749\pi$
$811$	25.2580	0.886928	0.443464	+	0.896292i	$0.353749\pi$
$812$	0	0
$813$	27.8658	0.977297
$814$	0	0
$815$	12.1743	0.426447
$816$	0	0
$817$	14.7049	0.514460
$818$	0	0
$819$	6.76556	0.236408
$820$	0	0
$821$	−10.5226	−0.367241	−0.183621	−	0.982997i	$-0.558782\pi$
$821$	−10.5226	−0.367241	−0.183621	+	0.982997i	$0.558782\pi$
$822$	0	0
$823$	27.6407	0.963493	0.481746	−	0.876311i	$-0.340003\pi$
$823$	27.6407	0.963493	0.481746	+	0.876311i	$0.340003\pi$
$824$	0	0
$825$	−1.92722	−0.0670973
$826$	0	0
$827$	16.0893	0.559478	0.279739	−	0.960076i	$-0.409752\pi$
$827$	16.0893	0.559478	0.279739	+	0.960076i	$0.409752\pi$
$828$	0	0
$829$	−52.4489	−1.82163	−0.910813	−	0.412818i	$-0.864544\pi$
$829$	−52.4489	−1.82163	−0.910813	+	0.412818i	$0.864544\pi$
$830$	0	0
$831$	7.45511	0.258615
$832$	0	0
$833$	3.52910	0.122276
$834$	0	0
$835$	−2.12535	−0.0735508
$836$	0	0
$837$	9.89064	0.341870
$838$	0	0
$839$	−50.3239	−1.73737	−0.868687	−	0.495361i	$-0.835036\pi$
$839$	−50.3239	−1.73737	−0.868687	+	0.495361i	$0.835036\pi$
$840$	0	0
$841$	4.74925	0.163767
$842$	0	0
$843$	−2.09663	−0.0722119
$844$	0	0
$845$	5.33705	0.183600
$846$	0	0
$847$	10.2966	0.353794
$848$	0	0
$849$	−2.62190	−0.0899833
$850$	0	0
$851$	−8.00613	−0.274447
$852$	0	0
$853$	46.8272	1.60333	0.801666	−	0.597773i	$-0.203947\pi$
$853$	46.8272	1.60333	0.801666	+	0.597773i	$0.203947\pi$
$854$	0	0
$855$	−5.28845	−0.180861
$856$	0	0
$857$	−22.7908	−0.778518	−0.389259	−	0.921128i	$-0.627269\pi$
$857$	−22.7908	−0.778518	−0.389259	+	0.921128i	$0.627269\pi$
$858$	0	0
$859$	50.9399	1.73805	0.869024	−	0.494771i	$-0.164748\pi$
$859$	50.9399	1.73805	0.869024	+	0.494771i	$0.164748\pi$
$860$	0	0
$861$	21.5269	0.733635
$862$	0	0
$863$	−21.1311	−0.719312	−0.359656	−	0.933085i	$-0.617106\pi$
$863$	−21.1311	−0.719312	−0.359656	+	0.933085i	$0.617106\pi$
$864$	0	0
$865$	12.3871	0.421175
$866$	0	0
$867$	15.2077	0.516480
$868$	0	0
$869$	−63.1634	−2.14267
$870$	0	0
$871$	28.7377	0.973739
$872$	0	0
$873$	15.8568	0.536670
$874$	0	0
$875$	24.3433	0.822953
$876$	0	0
$877$	27.6881	0.934961	0.467480	−	0.884003i	$-0.345162\pi$
$877$	27.6881	0.934961	0.467480	+	0.884003i	$0.345162\pi$
$878$	0	0
$879$	9.86788	0.332835
$880$	0	0
$881$	−0.856756	−0.0288649	−0.0144324	−	0.999896i	$-0.504594\pi$
$881$	−0.856756	−0.0288649	−0.0144324	+	0.999896i	$0.504594\pi$
$882$	0	0
$883$	−21.6775	−0.729507	−0.364753	−	0.931104i	$-0.618847\pi$
$883$	−21.6775	−0.729507	−0.364753	+	0.931104i	$0.618847\pi$
$884$	0	0
$885$	−29.7274	−0.999274
$886$	0	0
$887$	−6.19244	−0.207922	−0.103961	−	0.994581i	$-0.533152\pi$
$887$	−6.19244	−0.207922	−0.103961	+	0.994581i	$0.533152\pi$
$888$	0	0
$889$	−45.5553	−1.52788
$890$	0	0
$891$	−3.99111	−0.133707
$892$	0	0
$893$	28.7975	0.963671
$894$	0	0
$895$	11.6262	0.388622
$896$	0	0
$897$	3.52690	0.117760
$898$	0	0
$899$	−57.4588	−1.91636
$900$	0	0
$901$	−7.37486	−0.245692
$902$	0	0
$903$	−12.3454	−0.410828
$904$	0	0
$905$	−54.2020	−1.80174
$906$	0	0
$907$	47.2992	1.57054	0.785272	−	0.619152i	$-0.212523\pi$
$907$	47.2992	1.57054	0.785272	+	0.619152i	$0.212523\pi$
$908$	0	0
$909$	−8.72977	−0.289548
$910$	0	0
$911$	29.3033	0.970862	0.485431	−	0.874275i	$-0.338663\pi$
$911$	29.3033	0.970862	0.485431	+	0.874275i	$0.338663\pi$
$912$	0	0
$913$	32.5376	1.07684
$914$	0	0
$915$	−5.86521	−0.193898
$916$	0	0
$917$	−40.2847	−1.33032
$918$	0	0
$919$	−1.40652	−0.0463967	−0.0231984	−	0.999731i	$-0.507385\pi$
$919$	−1.40652	−0.0463967	−0.0231984	+	0.999731i	$0.507385\pi$
$920$	0	0
$921$	−22.4538	−0.739878
$922$	0	0
$923$	−2.68395	−0.0883434
$924$	0	0
$925$	−3.55003	−0.116724
$926$	0	0
$927$	−19.1263	−0.628191
$928$	0	0
$929$	15.9126	0.522075	0.261038	−	0.965329i	$-0.415935\pi$
$929$	15.9126	0.522075	0.261038	+	0.965329i	$0.415935\pi$
$930$	0	0
$931$	−6.55923	−0.214970
$932$	0	0
$933$	19.7217	0.645659
$934$	0	0
$935$	−11.3562	−0.371387
$936$	0	0
$937$	29.0686	0.949630	0.474815	−	0.880086i	$-0.342515\pi$
$937$	29.0686	0.949630	0.474815	+	0.880086i	$0.342515\pi$
$938$	0	0
$939$	−22.4601	−0.732959
$940$	0	0
$941$	11.0504	0.360231	0.180116	−	0.983645i	$-0.442353\pi$
$941$	11.0504	0.360231	0.180116	+	0.983645i	$0.442353\pi$
$942$	0	0
$943$	11.2220	0.365440
$944$	0	0
$945$	4.43987	0.144429
$946$	0	0
$947$	41.8317	1.35935	0.679674	−	0.733514i	$-0.262121\pi$
$947$	41.8317	1.35935	0.679674	+	0.733514i	$0.262121\pi$
$948$	0	0
$949$	−0.307423	−0.00997938
$950$	0	0
$951$	10.6210	0.344409
$952$	0	0
$953$	−1.30392	−0.0422380	−0.0211190	−	0.999777i	$-0.506723\pi$
$953$	−1.30392	−0.0422380	−0.0211190	+	0.999777i	$0.506723\pi$
$954$	0	0
$955$	58.0218	1.87754
$956$	0	0
$957$	23.1860	0.749496
$958$	0	0
$959$	22.1398	0.714932
$960$	0	0
$961$	66.8247	2.15564
$962$	0	0
$963$	−3.36186	−0.108335
$964$	0	0
$965$	39.1061	1.25887
$966$	0	0
$967$	40.5523	1.30408	0.652038	−	0.758186i	$-0.273914\pi$
$967$	40.5523	1.30408	0.652038	+	0.758186i	$0.273914\pi$
$968$	0	0
$969$	3.33124	0.107015
$970$	0	0
$971$	−36.8286	−1.18189	−0.590944	−	0.806713i	$-0.701244\pi$
$971$	−36.8286	−1.18189	−0.590944	+	0.806713i	$0.701244\pi$
$972$	0	0
$973$	5.60963	0.179836
$974$	0	0
$975$	1.56388	0.0500842
$976$	0	0
$977$	−26.3225	−0.842133	−0.421066	−	0.907030i	$-0.638344\pi$
$977$	−26.3225	−0.842133	−0.421066	+	0.907030i	$0.638344\pi$
$978$	0	0
$979$	20.3666	0.650920
$980$	0	0
$981$	−10.7988	−0.344780
$982$	0	0
$983$	39.3825	1.25611	0.628054	−	0.778170i	$-0.283852\pi$
$983$	39.3825	1.25611	0.628054	+	0.778170i	$0.283852\pi$
$984$	0	0
$985$	−36.8355	−1.17368
$986$	0	0
$987$	−24.1767	−0.769552
$988$	0	0
$989$	−6.43568	−0.204643
$990$	0	0
$991$	−59.0907	−1.87708	−0.938539	−	0.345173i	$-0.887820\pi$
$991$	−59.0907	−1.87708	−0.938539	+	0.345173i	$0.887820\pi$
$992$	0	0
$993$	23.2975	0.739324
$994$	0	0
$995$	−30.0622	−0.953036
$996$	0	0
$997$	12.7840	0.404873	0.202436	−	0.979295i	$-0.435114\pi$
$997$	12.7840	0.404873	0.202436	+	0.979295i	$0.435114\pi$
$998$	0	0
$999$	−7.35179	−0.232600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.h.1.3	✓	8
4.3	odd	2		8016.2.a.y.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.h.1.3	✓	8	1.1	even	1	trivial
8016.2.a.y.1.3		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.12535 q^{5} +2.08900 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.12535 q^{5} +2.08900 q^{7} +1.00000 q^{9} -3.99111 q^{11} +3.23865 q^{13} +2.12535 q^{15} -1.33878 q^{17} +2.48827 q^{19} -2.08900 q^{21} -1.08900 q^{23} -0.482879 q^{25} -1.00000 q^{27} +5.80941 q^{29} -9.89064 q^{31} +3.99111 q^{33} -4.43987 q^{35} +7.35179 q^{37} -3.23865 q^{39} -10.3049 q^{41} +5.90969 q^{43} -2.12535 q^{45} +11.5733 q^{47} -2.63606 q^{49} +1.33878 q^{51} +5.50865 q^{53} +8.48251 q^{55} -2.48827 q^{57} -13.9870 q^{59} -2.75964 q^{61} +2.08900 q^{63} -6.88327 q^{65} +8.87335 q^{67} +1.08900 q^{69} -0.828726 q^{71} -0.0949232 q^{73} +0.482879 q^{75} -8.33744 q^{77} +15.8260 q^{79} +1.00000 q^{81} -8.15253 q^{83} +2.84538 q^{85} -5.80941 q^{87} -5.10300 q^{89} +6.76556 q^{91} +9.89064 q^{93} -5.28845 q^{95} +15.8568 q^{97} -3.99111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 - q^7 + 8 * q^9	\( 8 q - 8 q^{3} - q^{7} + 8 q^{9} - 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} + 9 q^{23} + 6 q^{25} - 8 q^{27} + 17 q^{29} - 23 q^{31} + 3 q^{33} - 15 q^{35} + 8 q^{37} + 8 q^{39} - 8 q^{41} - 2 q^{43} - 34 q^{47} + 5 q^{49} + 7 q^{51} + 12 q^{53} - 7 q^{55} - 16 q^{59} - 2 q^{61} - q^{63} - 14 q^{65} + 21 q^{67} - 9 q^{69} - 29 q^{71} - 38 q^{73} - 6 q^{75} + 20 q^{77} - 12 q^{79} + 8 q^{81} - 32 q^{83} + 23 q^{85} - 17 q^{87} + 11 q^{89} - 5 q^{91} + 23 q^{93} - 67 q^{95} + 8 q^{97} - 3 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 - q^7 + 8 * q^9 - 3 * q^11 - 8 * q^13 - 7 * q^17 + q^21 + 9 * q^23 + 6 * q^25 - 8 * q^27 + 17 * q^29 - 23 * q^31 + 3 * q^33 - 15 * q^35 + 8 * q^37 + 8 * q^39 - 8 * q^41 - 2 * q^43 - 34 * q^47 + 5 * q^49 + 7 * q^51 + 12 * q^53 - 7 * q^55 - 16 * q^59 - 2 * q^61 - q^63 - 14 * q^65 + 21 * q^67 - 9 * q^69 - 29 * q^71 - 38 * q^73 - 6 * q^75 + 20 * q^77 - 12 * q^79 + 8 * q^81 - 32 * q^83 + 23 * q^85 - 17 * q^87 + 11 * q^89 - 5 * q^91 + 23 * q^93 - 67 * q^95 + 8 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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