LMFDB - Embedded newform 8004.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} + \cdots - 208 $ x^16 - 5x^15 - 43x^14 + 234x^13 + 634x^12 - 4048x^11 - 3483x^10 + 32512x^9 - 137x^8 - 121665x^7 + 60168x^6 + 172218x^5 - 138024x^4 - 17844x^3 + 14352x^2 + 72*x - 208
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			1.5
Root			$-1.83485$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.83485 q^{5} +1.71276 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.83485 q^{5} +1.71276 q^{7} +1.00000 q^{9} -0.324628 q^{11} +2.24187 q^{13} +1.83485 q^{15} +3.03434 q^{17} -6.93143 q^{19} -1.71276 q^{21} +1.00000 q^{23} -1.63333 q^{25} -1.00000 q^{27} +1.00000 q^{29} +0.153463 q^{31} +0.324628 q^{33} -3.14266 q^{35} +7.96250 q^{37} -2.24187 q^{39} +5.42941 q^{41} +4.81157 q^{43} -1.83485 q^{45} -4.18978 q^{47} -4.06644 q^{49} -3.03434 q^{51} +11.5727 q^{53} +0.595642 q^{55} +6.93143 q^{57} -2.10296 q^{59} -1.10334 q^{61} +1.71276 q^{63} -4.11350 q^{65} -3.92964 q^{67} -1.00000 q^{69} -11.2432 q^{71} -6.02817 q^{73} +1.63333 q^{75} -0.556010 q^{77} -9.02404 q^{79} +1.00000 q^{81} -0.162205 q^{83} -5.56755 q^{85} -1.00000 q^{87} +9.59572 q^{89} +3.83980 q^{91} -0.153463 q^{93} +12.7181 q^{95} -1.19093 q^{97} -0.324628 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * q^97 + 5 * 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
$5$	−1.83485	−0.820569	−0.410284	−	0.911958i	$-0.634571\pi$
$5$	−1.83485	−0.820569	−0.410284	+	0.911958i	$0.634571\pi$
$6$	0	0
$7$	1.71276	0.647363	0.323682	−	0.946166i	$-0.395079\pi$
$7$	1.71276	0.647363	0.323682	+	0.946166i	$0.395079\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.324628	−0.0978789	−0.0489395	−	0.998802i	$-0.515584\pi$
$11$	−0.324628	−0.0978789	−0.0489395	+	0.998802i	$0.515584\pi$
$12$	0	0
$13$	2.24187	0.621784	0.310892	−	0.950445i	$-0.399372\pi$
$13$	2.24187	0.621784	0.310892	+	0.950445i	$0.399372\pi$
$14$	0	0
$15$	1.83485	0.473756
$16$	0	0
$17$	3.03434	0.735935	0.367967	−	0.929839i	$-0.380054\pi$
$17$	3.03434	0.735935	0.367967	+	0.929839i	$0.380054\pi$
$18$	0	0
$19$	−6.93143	−1.59018	−0.795089	−	0.606492i	$-0.792576\pi$
$19$	−6.93143	−1.59018	−0.795089	+	0.606492i	$0.792576\pi$
$20$	0	0
$21$	−1.71276	−0.373755
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−1.63333	−0.326667
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.153463	0.0275627	0.0137813	−	0.999905i	$-0.495613\pi$
$31$	0.153463	0.0275627	0.0137813	+	0.999905i	$0.495613\pi$
$32$	0	0
$33$	0.324628	0.0565104
$34$	0	0
$35$	−3.14266	−0.531206
$36$	0	0
$37$	7.96250	1.30903	0.654514	−	0.756050i	$-0.272873\pi$
$37$	7.96250	1.30903	0.654514	+	0.756050i	$0.272873\pi$
$38$	0	0
$39$	−2.24187	−0.358987
$40$	0	0
$41$	5.42941	0.847931	0.423966	−	0.905678i	$-0.360638\pi$
$41$	5.42941	0.847931	0.423966	+	0.905678i	$0.360638\pi$
$42$	0	0
$43$	4.81157	0.733758	0.366879	−	0.930269i	$-0.380426\pi$
$43$	4.81157	0.733758	0.366879	+	0.930269i	$0.380426\pi$
$44$	0	0
$45$	−1.83485	−0.273523
$46$	0	0
$47$	−4.18978	−0.611142	−0.305571	−	0.952169i	$-0.598847\pi$
$47$	−4.18978	−0.611142	−0.305571	+	0.952169i	$0.598847\pi$
$48$	0	0
$49$	−4.06644	−0.580921
$50$	0	0
$51$	−3.03434	−0.424892
$52$	0	0
$53$	11.5727	1.58963	0.794817	−	0.606849i	$-0.207567\pi$
$53$	11.5727	1.58963	0.794817	+	0.606849i	$0.207567\pi$
$54$	0	0
$55$	0.595642	0.0803164
$56$	0	0
$57$	6.93143	0.918090
$58$	0	0
$59$	−2.10296	−0.273782	−0.136891	−	0.990586i	$-0.543711\pi$
$59$	−2.10296	−0.273782	−0.136891	+	0.990586i	$0.543711\pi$
$60$	0	0
$61$	−1.10334	−0.141268	−0.0706342	−	0.997502i	$-0.522502\pi$
$61$	−1.10334	−0.141268	−0.0706342	+	0.997502i	$0.522502\pi$
$62$	0	0
$63$	1.71276	0.215788
$64$	0	0
$65$	−4.11350	−0.510217
$66$	0	0
$67$	−3.92964	−0.480082	−0.240041	−	0.970763i	$-0.577161\pi$
$67$	−3.92964	−0.480082	−0.240041	+	0.970763i	$0.577161\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−11.2432	−1.33432	−0.667161	−	0.744913i	$-0.732491\pi$
$71$	−11.2432	−1.33432	−0.667161	+	0.744913i	$0.732491\pi$
$72$	0	0
$73$	−6.02817	−0.705544	−0.352772	−	0.935709i	$-0.614761\pi$
$73$	−6.02817	−0.705544	−0.352772	+	0.935709i	$0.614761\pi$
$74$	0	0
$75$	1.63333	0.188601
$76$	0	0
$77$	−0.556010	−0.0633632
$78$	0	0
$79$	−9.02404	−1.01528	−0.507642	−	0.861568i	$-0.669483\pi$
$79$	−9.02404	−1.01528	−0.507642	+	0.861568i	$0.669483\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.162205	−0.0178043	−0.00890215	−	0.999960i	$-0.502834\pi$
$83$	−0.162205	−0.0178043	−0.00890215	+	0.999960i	$0.502834\pi$
$84$	0	0
$85$	−5.56755	−0.603885
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	9.59572	1.01714	0.508572	−	0.861019i	$-0.330174\pi$
$89$	9.59572	1.01714	0.508572	+	0.861019i	$0.330174\pi$
$90$	0	0
$91$	3.83980	0.402520
$92$	0	0
$93$	−0.153463	−0.0159133
$94$	0	0
$95$	12.7181	1.30485
$96$	0	0
$97$	−1.19093	−0.120920	−0.0604601	−	0.998171i	$-0.519257\pi$
$97$	−1.19093	−0.120920	−0.0604601	+	0.998171i	$0.519257\pi$
$98$	0	0
$99$	−0.324628	−0.0326263
$100$	0	0
$101$	1.51959	0.151205	0.0756025	−	0.997138i	$-0.475912\pi$
$101$	1.51959	0.151205	0.0756025	+	0.997138i	$0.475912\pi$
$102$	0	0
$103$	19.7934	1.95030	0.975149	−	0.221548i	$-0.0711109\pi$
$103$	19.7934	1.95030	0.975149	+	0.221548i	$0.0711109\pi$
$104$	0	0
$105$	3.14266	0.306692
$106$	0	0
$107$	17.5955	1.70102	0.850511	−	0.525957i	$-0.176293\pi$
$107$	17.5955	1.70102	0.850511	+	0.525957i	$0.176293\pi$
$108$	0	0
$109$	15.6414	1.49817	0.749086	−	0.662473i	$-0.230493\pi$
$109$	15.6414	1.49817	0.749086	+	0.662473i	$0.230493\pi$
$110$	0	0
$111$	−7.96250	−0.755767
$112$	0	0
$113$	−8.49236	−0.798894	−0.399447	−	0.916756i	$-0.630798\pi$
$113$	−8.49236	−0.798894	−0.399447	+	0.916756i	$0.630798\pi$
$114$	0	0
$115$	−1.83485	−0.171100
$116$	0	0
$117$	2.24187	0.207261
$118$	0	0
$119$	5.19710	0.476417
$120$	0	0
$121$	−10.8946	−0.990420
$122$	0	0
$123$	−5.42941	−0.489553
$124$	0	0
$125$	12.1712	1.08862
$126$	0	0
$127$	2.78311	0.246961	0.123480	−	0.992347i	$-0.460594\pi$
$127$	2.78311	0.246961	0.123480	+	0.992347i	$0.460594\pi$
$128$	0	0
$129$	−4.81157	−0.423635
$130$	0	0
$131$	−3.49691	−0.305526	−0.152763	−	0.988263i	$-0.548817\pi$
$131$	−3.49691	−0.305526	−0.152763	+	0.988263i	$0.548817\pi$
$132$	0	0
$133$	−11.8719	−1.02942
$134$	0	0
$135$	1.83485	0.157919
$136$	0	0
$137$	−17.9808	−1.53620	−0.768102	−	0.640328i	$-0.778799\pi$
$137$	−17.9808	−1.53620	−0.768102	+	0.640328i	$0.778799\pi$
$138$	0	0
$139$	−8.22207	−0.697387	−0.348694	−	0.937237i	$-0.613375\pi$
$139$	−8.22207	−0.697387	−0.348694	+	0.937237i	$0.613375\pi$
$140$	0	0
$141$	4.18978	0.352843
$142$	0	0
$143$	−0.727774	−0.0608596
$144$	0	0
$145$	−1.83485	−0.152376
$146$	0	0
$147$	4.06644	0.335395
$148$	0	0
$149$	−1.98612	−0.162709	−0.0813546	−	0.996685i	$-0.525925\pi$
$149$	−1.98612	−0.162709	−0.0813546	+	0.996685i	$0.525925\pi$
$150$	0	0
$151$	−8.62021	−0.701503	−0.350752	−	0.936469i	$-0.614074\pi$
$151$	−8.62021	−0.701503	−0.350752	+	0.936469i	$0.614074\pi$
$152$	0	0
$153$	3.03434	0.245312
$154$	0	0
$155$	−0.281580	−0.0226171
$156$	0	0
$157$	−3.47315	−0.277188	−0.138594	−	0.990349i	$-0.544258\pi$
$157$	−3.47315	−0.277188	−0.138594	+	0.990349i	$0.544258\pi$
$158$	0	0
$159$	−11.5727	−0.917776
$160$	0	0
$161$	1.71276	0.134985
$162$	0	0
$163$	14.9038	1.16735	0.583676	−	0.811987i	$-0.301614\pi$
$163$	14.9038	1.16735	0.583676	+	0.811987i	$0.301614\pi$
$164$	0	0
$165$	−0.595642	−0.0463707
$166$	0	0
$167$	5.58054	0.431835	0.215917	−	0.976412i	$-0.430726\pi$
$167$	5.58054	0.431835	0.215917	+	0.976412i	$0.430726\pi$
$168$	0	0
$169$	−7.97400	−0.613384
$170$	0	0
$171$	−6.93143	−0.530060
$172$	0	0
$173$	−17.3399	−1.31833	−0.659164	−	0.751999i	$-0.729090\pi$
$173$	−17.3399	−1.31833	−0.659164	+	0.751999i	$0.729090\pi$
$174$	0	0
$175$	−2.79751	−0.211472
$176$	0	0
$177$	2.10296	0.158068
$178$	0	0
$179$	−8.86053	−0.662267	−0.331134	−	0.943584i	$-0.607431\pi$
$179$	−8.86053	−0.662267	−0.331134	+	0.943584i	$0.607431\pi$
$180$	0	0
$181$	−24.0277	−1.78596	−0.892981	−	0.450094i	$-0.851391\pi$
$181$	−24.0277	−1.78596	−0.892981	+	0.450094i	$0.851391\pi$
$182$	0	0
$183$	1.10334	0.0815613
$184$	0	0
$185$	−14.6100	−1.07415
$186$	0	0
$187$	−0.985030	−0.0720325
$188$	0	0
$189$	−1.71276	−0.124585
$190$	0	0
$191$	20.0670	1.45200	0.725998	−	0.687696i	$-0.241378\pi$
$191$	20.0670	1.45200	0.725998	+	0.687696i	$0.241378\pi$
$192$	0	0
$193$	14.5004	1.04376	0.521881	−	0.853019i	$-0.325231\pi$
$193$	14.5004	1.04376	0.521881	+	0.853019i	$0.325231\pi$
$194$	0	0
$195$	4.11350	0.294574
$196$	0	0
$197$	25.3961	1.80940	0.904698	−	0.426054i	$-0.140097\pi$
$197$	25.3961	1.80940	0.904698	+	0.426054i	$0.140097\pi$
$198$	0	0
$199$	−22.9534	−1.62713	−0.813563	−	0.581477i	$-0.802475\pi$
$199$	−22.9534	−1.62713	−0.813563	+	0.581477i	$0.802475\pi$
$200$	0	0
$201$	3.92964	0.277176
$202$	0	0
$203$	1.71276	0.120212
$204$	0	0
$205$	−9.96214	−0.695786
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	2.25013	0.155645
$210$	0	0
$211$	−14.9031	−1.02597	−0.512985	−	0.858398i	$-0.671460\pi$
$211$	−14.9031	−1.02597	−0.512985	+	0.858398i	$0.671460\pi$
$212$	0	0
$213$	11.2432	0.770372
$214$	0	0
$215$	−8.82850	−0.602099
$216$	0	0
$217$	0.262845	0.0178431
$218$	0	0
$219$	6.02817	0.407346
$220$	0	0
$221$	6.80260	0.457593
$222$	0	0
$223$	3.21974	0.215610	0.107805	−	0.994172i	$-0.465618\pi$
$223$	3.21974	0.215610	0.107805	+	0.994172i	$0.465618\pi$
$224$	0	0
$225$	−1.63333	−0.108889
$226$	0	0
$227$	22.1194	1.46811	0.734057	−	0.679088i	$-0.237625\pi$
$227$	22.1194	1.46811	0.734057	+	0.679088i	$0.237625\pi$
$228$	0	0
$229$	16.6341	1.09921	0.549605	−	0.835425i	$-0.314778\pi$
$229$	16.6341	1.09921	0.549605	+	0.835425i	$0.314778\pi$
$230$	0	0
$231$	0.556010	0.0365828
$232$	0	0
$233$	9.19416	0.602329	0.301165	−	0.953572i	$-0.402625\pi$
$233$	9.19416	0.602329	0.301165	+	0.953572i	$0.402625\pi$
$234$	0	0
$235$	7.68761	0.501484
$236$	0	0
$237$	9.02404	0.586175
$238$	0	0
$239$	14.3208	0.926334	0.463167	−	0.886271i	$-0.346713\pi$
$239$	14.3208	0.926334	0.463167	+	0.886271i	$0.346713\pi$
$240$	0	0
$241$	19.6949	1.26866	0.634329	−	0.773063i	$-0.281276\pi$
$241$	19.6949	1.26866	0.634329	+	0.773063i	$0.281276\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	7.46131	0.476685
$246$	0	0
$247$	−15.5394	−0.988748
$248$	0	0
$249$	0.162205	0.0102793
$250$	0	0
$251$	19.1162	1.20660	0.603301	−	0.797514i	$-0.293852\pi$
$251$	19.1162	1.20660	0.603301	+	0.797514i	$0.293852\pi$
$252$	0	0
$253$	−0.324628	−0.0204092
$254$	0	0
$255$	5.56755	0.348653
$256$	0	0
$257$	−10.5799	−0.659954	−0.329977	−	0.943989i	$-0.607041\pi$
$257$	−10.5799	−0.659954	−0.329977	+	0.943989i	$0.607041\pi$
$258$	0	0
$259$	13.6379	0.847416
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	3.00001	0.184988	0.0924942	−	0.995713i	$-0.470516\pi$
$263$	3.00001	0.184988	0.0924942	+	0.995713i	$0.470516\pi$
$264$	0	0
$265$	−21.2342	−1.30440
$266$	0	0
$267$	−9.59572	−0.587249
$268$	0	0
$269$	10.4320	0.636048	0.318024	−	0.948083i	$-0.396981\pi$
$269$	10.4320	0.636048	0.318024	+	0.948083i	$0.396981\pi$
$270$	0	0
$271$	−16.1350	−0.980129	−0.490064	−	0.871686i	$-0.663027\pi$
$271$	−16.1350	−0.980129	−0.490064	+	0.871686i	$0.663027\pi$
$272$	0	0
$273$	−3.83980	−0.232395
$274$	0	0
$275$	0.530225	0.0319738
$276$	0	0
$277$	21.6156	1.29876	0.649378	−	0.760466i	$-0.275029\pi$
$277$	21.6156	1.29876	0.649378	+	0.760466i	$0.275029\pi$
$278$	0	0
$279$	0.153463	0.00918757
$280$	0	0
$281$	26.5768	1.58544	0.792721	−	0.609585i	$-0.208664\pi$
$281$	26.5768	1.58544	0.792721	+	0.609585i	$0.208664\pi$
$282$	0	0
$283$	2.73513	0.162587	0.0812933	−	0.996690i	$-0.474095\pi$
$283$	2.73513	0.162587	0.0812933	+	0.996690i	$0.474095\pi$
$284$	0	0
$285$	−12.7181	−0.753356
$286$	0	0
$287$	9.29929	0.548920
$288$	0	0
$289$	−7.79280	−0.458400
$290$	0	0
$291$	1.19093	0.0698133
$292$	0	0
$293$	8.83703	0.516265	0.258132	−	0.966110i	$-0.416893\pi$
$293$	8.83703	0.516265	0.258132	+	0.966110i	$0.416893\pi$
$294$	0	0
$295$	3.85861	0.224657
$296$	0	0
$297$	0.324628	0.0188368
$298$	0	0
$299$	2.24187	0.129651
$300$	0	0
$301$	8.24108	0.475008
$302$	0	0
$303$	−1.51959	−0.0872982
$304$	0	0
$305$	2.02446	0.115920
$306$	0	0
$307$	6.01045	0.343034	0.171517	−	0.985181i	$-0.445133\pi$
$307$	6.01045	0.343034	0.171517	+	0.985181i	$0.445133\pi$
$308$	0	0
$309$	−19.7934	−1.12601
$310$	0	0
$311$	25.6664	1.45541	0.727703	−	0.685893i	$-0.240588\pi$
$311$	25.6664	1.45541	0.727703	+	0.685893i	$0.240588\pi$
$312$	0	0
$313$	−17.3009	−0.977907	−0.488954	−	0.872310i	$-0.662621\pi$
$313$	−17.3009	−0.977907	−0.488954	+	0.872310i	$0.662621\pi$
$314$	0	0
$315$	−3.14266	−0.177069
$316$	0	0
$317$	22.3014	1.25257	0.626285	−	0.779594i	$-0.284575\pi$
$317$	22.3014	1.25257	0.626285	+	0.779594i	$0.284575\pi$
$318$	0	0
$319$	−0.324628	−0.0181757
$320$	0	0
$321$	−17.5955	−0.982086
$322$	0	0
$323$	−21.0323	−1.17027
$324$	0	0
$325$	−3.66173	−0.203116
$326$	0	0
$327$	−15.6414	−0.864969
$328$	0	0
$329$	−7.17610	−0.395631
$330$	0	0
$331$	−19.5061	−1.07215	−0.536076	−	0.844169i	$-0.680094\pi$
$331$	−19.5061	−1.07215	−0.536076	+	0.844169i	$0.680094\pi$
$332$	0	0
$333$	7.96250	0.436342
$334$	0	0
$335$	7.21030	0.393941
$336$	0	0
$337$	15.5982	0.849687	0.424844	−	0.905267i	$-0.360329\pi$
$337$	15.5982	0.849687	0.424844	+	0.905267i	$0.360329\pi$
$338$	0	0
$339$	8.49236	0.461242
$340$	0	0
$341$	−0.0498182	−0.00269781
$342$	0	0
$343$	−18.9542	−1.02343
$344$	0	0
$345$	1.83485	0.0987849
$346$	0	0
$347$	31.9578	1.71559	0.857793	−	0.513996i	$-0.171835\pi$
$347$	31.9578	1.71559	0.857793	+	0.513996i	$0.171835\pi$
$348$	0	0
$349$	30.7086	1.64380	0.821898	−	0.569635i	$-0.192915\pi$
$349$	30.7086	1.64380	0.821898	+	0.569635i	$0.192915\pi$
$350$	0	0
$351$	−2.24187	−0.119662
$352$	0	0
$353$	4.64406	0.247178	0.123589	−	0.992333i	$-0.460560\pi$
$353$	4.64406	0.247178	0.123589	+	0.992333i	$0.460560\pi$
$354$	0	0
$355$	20.6296	1.09490
$356$	0	0
$357$	−5.19710	−0.275060
$358$	0	0
$359$	−12.3358	−0.651059	−0.325530	−	0.945532i	$-0.605543\pi$
$359$	−12.3358	−0.651059	−0.325530	+	0.945532i	$0.605543\pi$
$360$	0	0
$361$	29.0447	1.52867
$362$	0	0
$363$	10.8946	0.571819
$364$	0	0
$365$	11.0608	0.578948
$366$	0	0
$367$	11.7419	0.612923	0.306462	−	0.951883i	$-0.400855\pi$
$367$	11.7419	0.612923	0.306462	+	0.951883i	$0.400855\pi$
$368$	0	0
$369$	5.42941	0.282644
$370$	0	0
$371$	19.8213	1.02907
$372$	0	0
$373$	8.28462	0.428961	0.214481	−	0.976728i	$-0.431194\pi$
$373$	8.28462	0.428961	0.214481	+	0.976728i	$0.431194\pi$
$374$	0	0
$375$	−12.1712	−0.628516
$376$	0	0
$377$	2.24187	0.115462
$378$	0	0
$379$	−12.8329	−0.659182	−0.329591	−	0.944124i	$-0.606911\pi$
$379$	−12.8329	−0.659182	−0.329591	+	0.944124i	$0.606911\pi$
$380$	0	0
$381$	−2.78311	−0.142583
$382$	0	0
$383$	36.8683	1.88388	0.941940	−	0.335781i	$-0.109000\pi$
$383$	36.8683	1.88388	0.941940	+	0.335781i	$0.109000\pi$
$384$	0	0
$385$	1.02019	0.0519939
$386$	0	0
$387$	4.81157	0.244586
$388$	0	0
$389$	19.2985	0.978470	0.489235	−	0.872152i	$-0.337276\pi$
$389$	19.2985	0.978470	0.489235	+	0.872152i	$0.337276\pi$
$390$	0	0
$391$	3.03434	0.153453
$392$	0	0
$393$	3.49691	0.176396
$394$	0	0
$395$	16.5577	0.833111
$396$	0	0
$397$	22.0507	1.10669	0.553346	−	0.832952i	$-0.313351\pi$
$397$	22.0507	1.10669	0.553346	+	0.832952i	$0.313351\pi$
$398$	0	0
$399$	11.8719	0.594338
$400$	0	0
$401$	−8.46339	−0.422642	−0.211321	−	0.977417i	$-0.567777\pi$
$401$	−8.46339	−0.422642	−0.211321	+	0.977417i	$0.567777\pi$
$402$	0	0
$403$	0.344044	0.0171380
$404$	0	0
$405$	−1.83485	−0.0911743
$406$	0	0
$407$	−2.58485	−0.128126
$408$	0	0
$409$	1.84815	0.0913853	0.0456926	−	0.998956i	$-0.485451\pi$
$409$	1.84815	0.0913853	0.0456926	+	0.998956i	$0.485451\pi$
$410$	0	0
$411$	17.9808	0.886928
$412$	0	0
$413$	−3.60187	−0.177237
$414$	0	0
$415$	0.297621	0.0146097
$416$	0	0
$417$	8.22207	0.402637
$418$	0	0
$419$	−2.32702	−0.113683	−0.0568413	−	0.998383i	$-0.518103\pi$
$419$	−2.32702	−0.113683	−0.0568413	+	0.998383i	$0.518103\pi$
$420$	0	0
$421$	14.8437	0.723435	0.361718	−	0.932288i	$-0.382190\pi$
$421$	14.8437	0.723435	0.361718	+	0.932288i	$0.382190\pi$
$422$	0	0
$423$	−4.18978	−0.203714
$424$	0	0
$425$	−4.95609	−0.240406
$426$	0	0
$427$	−1.88976	−0.0914519
$428$	0	0
$429$	0.727774	0.0351373
$430$	0	0
$431$	6.17218	0.297303	0.148652	−	0.988890i	$-0.452507\pi$
$431$	6.17218	0.297303	0.148652	+	0.988890i	$0.452507\pi$
$432$	0	0
$433$	−18.8245	−0.904649	−0.452325	−	0.891853i	$-0.649405\pi$
$433$	−18.8245	−0.904649	−0.452325	+	0.891853i	$0.649405\pi$
$434$	0	0
$435$	1.83485	0.0879742
$436$	0	0
$437$	−6.93143	−0.331575
$438$	0	0
$439$	−24.0226	−1.14654	−0.573268	−	0.819368i	$-0.694325\pi$
$439$	−24.0226	−1.14654	−0.573268	+	0.819368i	$0.694325\pi$
$440$	0	0
$441$	−4.06644	−0.193640
$442$	0	0
$443$	10.3626	0.492340	0.246170	−	0.969227i	$-0.420828\pi$
$443$	10.3626	0.492340	0.246170	+	0.969227i	$0.420828\pi$
$444$	0	0
$445$	−17.6067	−0.834637
$446$	0	0
$447$	1.98612	0.0939402
$448$	0	0
$449$	3.43394	0.162058	0.0810288	−	0.996712i	$-0.474179\pi$
$449$	3.43394	0.162058	0.0810288	+	0.996712i	$0.474179\pi$
$450$	0	0
$451$	−1.76254	−0.0829946
$452$	0	0
$453$	8.62021	0.405013
$454$	0	0
$455$	−7.04545	−0.330296
$456$	0	0
$457$	17.5875	0.822711	0.411355	−	0.911475i	$-0.365056\pi$
$457$	17.5875	0.822711	0.411355	+	0.911475i	$0.365056\pi$
$458$	0	0
$459$	−3.03434	−0.141631
$460$	0	0
$461$	22.4078	1.04364	0.521818	−	0.853057i	$-0.325254\pi$
$461$	22.4078	1.04364	0.521818	+	0.853057i	$0.325254\pi$
$462$	0	0
$463$	27.2511	1.26647	0.633234	−	0.773960i	$-0.281727\pi$
$463$	27.2511	1.26647	0.633234	+	0.773960i	$0.281727\pi$
$464$	0	0
$465$	0.281580	0.0130580
$466$	0	0
$467$	−11.7419	−0.543351	−0.271675	−	0.962389i	$-0.587578\pi$
$467$	−11.7419	−0.543351	−0.271675	+	0.962389i	$0.587578\pi$
$468$	0	0
$469$	−6.73055	−0.310788
$470$	0	0
$471$	3.47315	0.160034
$472$	0	0
$473$	−1.56197	−0.0718194
$474$	0	0
$475$	11.3213	0.519459
$476$	0	0
$477$	11.5727	0.529878
$478$	0	0
$479$	−35.2411	−1.61020	−0.805102	−	0.593136i	$-0.797890\pi$
$479$	−35.2411	−1.61020	−0.805102	+	0.593136i	$0.797890\pi$
$480$	0	0
$481$	17.8509	0.813932
$482$	0	0
$483$	−1.71276	−0.0779334
$484$	0	0
$485$	2.18517	0.0992234
$486$	0	0
$487$	−30.4349	−1.37914	−0.689569	−	0.724220i	$-0.742200\pi$
$487$	−30.4349	−1.37914	−0.689569	+	0.724220i	$0.742200\pi$
$488$	0	0
$489$	−14.9038	−0.673971
$490$	0	0
$491$	11.8872	0.536461	0.268231	−	0.963355i	$-0.413561\pi$
$491$	11.8872	0.536461	0.268231	+	0.963355i	$0.413561\pi$
$492$	0	0
$493$	3.03434	0.136660
$494$	0	0
$495$	0.595642	0.0267721
$496$	0	0
$497$	−19.2569	−0.863792
$498$	0	0
$499$	10.5750	0.473401	0.236701	−	0.971583i	$-0.423934\pi$
$499$	10.5750	0.473401	0.236701	+	0.971583i	$0.423934\pi$
$500$	0	0
$501$	−5.58054	−0.249320
$502$	0	0
$503$	19.6881	0.877851	0.438926	−	0.898523i	$-0.355359\pi$
$503$	19.6881	0.877851	0.438926	+	0.898523i	$0.355359\pi$
$504$	0	0
$505$	−2.78822	−0.124074
$506$	0	0
$507$	7.97400	0.354138
$508$	0	0
$509$	31.6510	1.40291	0.701453	−	0.712716i	$-0.252535\pi$
$509$	31.6510	1.40291	0.701453	+	0.712716i	$0.252535\pi$
$510$	0	0
$511$	−10.3248	−0.456743
$512$	0	0
$513$	6.93143	0.306030
$514$	0	0
$515$	−36.3178	−1.60035
$516$	0	0
$517$	1.36012	0.0598180
$518$	0	0
$519$	17.3399	0.761137
$520$	0	0
$521$	−18.5910	−0.814488	−0.407244	−	0.913319i	$-0.633510\pi$
$521$	−18.5910	−0.814488	−0.407244	+	0.913319i	$0.633510\pi$
$522$	0	0
$523$	−28.7656	−1.25783	−0.628916	−	0.777473i	$-0.716501\pi$
$523$	−28.7656	−1.25783	−0.628916	+	0.777473i	$0.716501\pi$
$524$	0	0
$525$	2.79751	0.122093
$526$	0	0
$527$	0.465657	0.0202844
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.10296	−0.0912608
$532$	0	0
$533$	12.1721	0.527230
$534$	0	0
$535$	−32.2851	−1.39581
$536$	0	0
$537$	8.86053	0.382360
$538$	0	0
$539$	1.32008	0.0568599
$540$	0	0
$541$	−18.9433	−0.814436	−0.407218	−	0.913331i	$-0.633501\pi$
$541$	−18.9433	−0.814436	−0.407218	+	0.913331i	$0.633501\pi$
$542$	0	0
$543$	24.0277	1.03113
$544$	0	0
$545$	−28.6995	−1.22935
$546$	0	0
$547$	43.4893	1.85947	0.929733	−	0.368233i	$-0.120037\pi$
$547$	43.4893	1.85947	0.929733	+	0.368233i	$0.120037\pi$
$548$	0	0
$549$	−1.10334	−0.0470894
$550$	0	0
$551$	−6.93143	−0.295289
$552$	0	0
$553$	−15.4560	−0.657258
$554$	0	0
$555$	14.6100	0.620159
$556$	0	0
$557$	−2.35638	−0.0998432	−0.0499216	−	0.998753i	$-0.515897\pi$
$557$	−2.35638	−0.0998432	−0.0499216	+	0.998753i	$0.515897\pi$
$558$	0	0
$559$	10.7869	0.456239
$560$	0	0
$561$	0.985030	0.0415880
$562$	0	0
$563$	7.13063	0.300520	0.150260	−	0.988647i	$-0.451989\pi$
$563$	7.13063	0.300520	0.150260	+	0.988647i	$0.451989\pi$
$564$	0	0
$565$	15.5822	0.655547
$566$	0	0
$567$	1.71276	0.0719293
$568$	0	0
$569$	−40.3302	−1.69073	−0.845365	−	0.534189i	$-0.820617\pi$
$569$	−40.3302	−1.69073	−0.845365	+	0.534189i	$0.820617\pi$
$570$	0	0
$571$	24.3196	1.01774	0.508872	−	0.860842i	$-0.330063\pi$
$571$	24.3196	1.01774	0.508872	+	0.860842i	$0.330063\pi$
$572$	0	0
$573$	−20.0670	−0.838311
$574$	0	0
$575$	−1.63333	−0.0681147
$576$	0	0
$577$	37.7001	1.56948	0.784738	−	0.619827i	$-0.212797\pi$
$577$	37.7001	1.56948	0.784738	+	0.619827i	$0.212797\pi$
$578$	0	0
$579$	−14.5004	−0.602616
$580$	0	0
$581$	−0.277818	−0.0115258
$582$	0	0
$583$	−3.75682	−0.155592
$584$	0	0
$585$	−4.11350	−0.170072
$586$	0	0
$587$	20.4809	0.845336	0.422668	−	0.906285i	$-0.361094\pi$
$587$	20.4809	0.845336	0.422668	+	0.906285i	$0.361094\pi$
$588$	0	0
$589$	−1.06372	−0.0438296
$590$	0	0
$591$	−25.3961	−1.04466
$592$	0	0
$593$	−1.99892	−0.0820859	−0.0410430	−	0.999157i	$-0.513068\pi$
$593$	−1.99892	−0.0820859	−0.0410430	+	0.999157i	$0.513068\pi$
$594$	0	0
$595$	−9.53589	−0.390933
$596$	0	0
$597$	22.9534	0.939422
$598$	0	0
$599$	12.8750	0.526057	0.263029	−	0.964788i	$-0.415279\pi$
$599$	12.8750	0.526057	0.263029	+	0.964788i	$0.415279\pi$
$600$	0	0
$601$	15.6499	0.638373	0.319187	−	0.947692i	$-0.396590\pi$
$601$	15.6499	0.638373	0.319187	+	0.947692i	$0.396590\pi$
$602$	0	0
$603$	−3.92964	−0.160027
$604$	0	0
$605$	19.9900	0.812708
$606$	0	0
$607$	45.3361	1.84013	0.920067	−	0.391762i	$-0.128134\pi$
$607$	45.3361	1.84013	0.920067	+	0.391762i	$0.128134\pi$
$608$	0	0
$609$	−1.71276	−0.0694046
$610$	0	0
$611$	−9.39297	−0.379999
$612$	0	0
$613$	8.87478	0.358449	0.179224	−	0.983808i	$-0.442641\pi$
$613$	8.87478	0.358449	0.179224	+	0.983808i	$0.442641\pi$
$614$	0	0
$615$	9.96214	0.401712
$616$	0	0
$617$	−28.6912	−1.15506	−0.577532	−	0.816368i	$-0.695984\pi$
$617$	−28.6912	−1.15506	−0.577532	+	0.816368i	$0.695984\pi$
$618$	0	0
$619$	5.93342	0.238484	0.119242	−	0.992865i	$-0.461954\pi$
$619$	5.93342	0.238484	0.119242	+	0.992865i	$0.461954\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	16.4352	0.658462
$624$	0	0
$625$	−14.1655	−0.566622
$626$	0	0
$627$	−2.25013	−0.0898617
$628$	0	0
$629$	24.1609	0.963359
$630$	0	0
$631$	−0.435051	−0.0173191	−0.00865955	−	0.999963i	$-0.502756\pi$
$631$	−0.435051	−0.0173191	−0.00865955	+	0.999963i	$0.502756\pi$
$632$	0	0
$633$	14.9031	0.592344
$634$	0	0
$635$	−5.10658	−0.202648
$636$	0	0
$637$	−9.11646	−0.361207
$638$	0	0
$639$	−11.2432	−0.444774
$640$	0	0
$641$	3.55955	0.140594	0.0702970	−	0.997526i	$-0.477605\pi$
$641$	3.55955	0.140594	0.0702970	+	0.997526i	$0.477605\pi$
$642$	0	0
$643$	−22.6722	−0.894105	−0.447052	−	0.894508i	$-0.647526\pi$
$643$	−22.6722	−0.894105	−0.447052	+	0.894508i	$0.647526\pi$
$644$	0	0
$645$	8.82850	0.347622
$646$	0	0
$647$	−32.2678	−1.26858	−0.634289	−	0.773096i	$-0.718707\pi$
$647$	−32.2678	−1.26858	−0.634289	+	0.773096i	$0.718707\pi$
$648$	0	0
$649$	0.682679	0.0267975
$650$	0	0
$651$	−0.262845	−0.0103017
$652$	0	0
$653$	17.4321	0.682172	0.341086	−	0.940032i	$-0.389205\pi$
$653$	17.4321	0.682172	0.341086	+	0.940032i	$0.389205\pi$
$654$	0	0
$655$	6.41630	0.250705
$656$	0	0
$657$	−6.02817	−0.235181
$658$	0	0
$659$	−44.7795	−1.74436	−0.872181	−	0.489183i	$-0.837295\pi$
$659$	−44.7795	−1.74436	−0.872181	+	0.489183i	$0.837295\pi$
$660$	0	0
$661$	11.7323	0.456333	0.228166	−	0.973622i	$-0.426727\pi$
$661$	11.7323	0.456333	0.228166	+	0.973622i	$0.426727\pi$
$662$	0	0
$663$	−6.80260	−0.264191
$664$	0	0
$665$	21.7831	0.844713
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−3.21974	−0.124482
$670$	0	0
$671$	0.358175	0.0138272
$672$	0	0
$673$	28.5123	1.09907	0.549534	−	0.835471i	$-0.314805\pi$
$673$	28.5123	1.09907	0.549534	+	0.835471i	$0.314805\pi$
$674$	0	0
$675$	1.63333	0.0628671
$676$	0	0
$677$	−45.2187	−1.73790	−0.868948	−	0.494904i	$-0.835203\pi$
$677$	−45.2187	−1.73790	−0.868948	+	0.494904i	$0.835203\pi$
$678$	0	0
$679$	−2.03977	−0.0782793
$680$	0	0
$681$	−22.1194	−0.847616
$682$	0	0
$683$	26.4359	1.01154	0.505771	−	0.862668i	$-0.331208\pi$
$683$	26.4359	1.01154	0.505771	+	0.862668i	$0.331208\pi$
$684$	0	0
$685$	32.9920	1.26056
$686$	0	0
$687$	−16.6341	−0.634629
$688$	0	0
$689$	25.9446	0.988409
$690$	0	0
$691$	−15.8905	−0.604503	−0.302252	−	0.953228i	$-0.597738\pi$
$691$	−15.8905	−0.604503	−0.302252	+	0.953228i	$0.597738\pi$
$692$	0	0
$693$	−0.556010	−0.0211211
$694$	0	0
$695$	15.0862	0.572254
$696$	0	0
$697$	16.4747	0.624022
$698$	0	0
$699$	−9.19416	−0.347755
$700$	0	0
$701$	47.6090	1.79817	0.899084	−	0.437776i	$-0.144234\pi$
$701$	47.6090	1.79817	0.899084	+	0.437776i	$0.144234\pi$
$702$	0	0
$703$	−55.1915	−2.08159
$704$	0	0
$705$	−7.68761	−0.289532
$706$	0	0
$707$	2.60270	0.0978845
$708$	0	0
$709$	−4.09427	−0.153764	−0.0768818	−	0.997040i	$-0.524496\pi$
$709$	−4.09427	−0.153764	−0.0768818	+	0.997040i	$0.524496\pi$
$710$	0	0
$711$	−9.02404	−0.338428
$712$	0	0
$713$	0.153463	0.00574722
$714$	0	0
$715$	1.33536	0.0499395
$716$	0	0
$717$	−14.3208	−0.534819
$718$	0	0
$719$	−10.7902	−0.402408	−0.201204	−	0.979549i	$-0.564485\pi$
$719$	−10.7902	−0.402408	−0.201204	+	0.979549i	$0.564485\pi$
$720$	0	0
$721$	33.9013	1.26255
$722$	0	0
$723$	−19.6949	−0.732460
$724$	0	0
$725$	−1.63333	−0.0606605
$726$	0	0
$727$	46.1386	1.71119	0.855593	−	0.517648i	$-0.173192\pi$
$727$	46.1386	1.71119	0.855593	+	0.517648i	$0.173192\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.5999	0.539998
$732$	0	0
$733$	−29.2622	−1.08082	−0.540412	−	0.841401i	$-0.681732\pi$
$733$	−29.2622	−1.08082	−0.540412	+	0.841401i	$0.681732\pi$
$734$	0	0
$735$	−7.46131	−0.275214
$736$	0	0
$737$	1.27567	0.0469899
$738$	0	0
$739$	36.8527	1.35565	0.677824	−	0.735224i	$-0.262923\pi$
$739$	36.8527	1.35565	0.677824	+	0.735224i	$0.262923\pi$
$740$	0	0
$741$	15.5394	0.570854
$742$	0	0
$743$	35.5920	1.30574	0.652872	−	0.757469i	$-0.273564\pi$
$743$	35.5920	1.30574	0.652872	+	0.757469i	$0.273564\pi$
$744$	0	0
$745$	3.64423	0.133514
$746$	0	0
$747$	−0.162205	−0.00593477
$748$	0	0
$749$	30.1369	1.10118
$750$	0	0
$751$	−41.8311	−1.52644	−0.763220	−	0.646139i	$-0.776383\pi$
$751$	−41.8311	−1.52644	−0.763220	+	0.646139i	$0.776383\pi$
$752$	0	0
$753$	−19.1162	−0.696632
$754$	0	0
$755$	15.8168	0.575632
$756$	0	0
$757$	28.0872	1.02085	0.510424	−	0.859923i	$-0.329488\pi$
$757$	28.0872	1.02085	0.510424	+	0.859923i	$0.329488\pi$
$758$	0	0
$759$	0.324628	0.0117832
$760$	0	0
$761$	24.7936	0.898768	0.449384	−	0.893339i	$-0.351643\pi$
$761$	24.7936	0.898768	0.449384	+	0.893339i	$0.351643\pi$
$762$	0	0
$763$	26.7899	0.969861
$764$	0	0
$765$	−5.56755	−0.201295
$766$	0	0
$767$	−4.71458	−0.170233
$768$	0	0
$769$	11.8990	0.429087	0.214544	−	0.976714i	$-0.431174\pi$
$769$	11.8990	0.429087	0.214544	+	0.976714i	$0.431174\pi$
$770$	0	0
$771$	10.5799	0.381025
$772$	0	0
$773$	−32.6895	−1.17576	−0.587881	−	0.808948i	$-0.700038\pi$
$773$	−32.6895	−1.17576	−0.587881	+	0.808948i	$0.700038\pi$
$774$	0	0
$775$	−0.250656	−0.00900382
$776$	0	0
$777$	−13.6379	−0.489256
$778$	0	0
$779$	−37.6336	−1.34836
$780$	0	0
$781$	3.64985	0.130602
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	6.37270	0.227452
$786$	0	0
$787$	−52.1372	−1.85849	−0.929246	−	0.369463i	$-0.879542\pi$
$787$	−52.1372	−1.85849	−0.929246	+	0.369463i	$0.879542\pi$
$788$	0	0
$789$	−3.00001	−0.106803
$790$	0	0
$791$	−14.5454	−0.517175
$792$	0	0
$793$	−2.47355	−0.0878384
$794$	0	0
$795$	21.2342	0.753098
$796$	0	0
$797$	23.5361	0.833691	0.416846	−	0.908977i	$-0.363136\pi$
$797$	23.5361	0.833691	0.416846	+	0.908977i	$0.363136\pi$
$798$	0	0
$799$	−12.7132	−0.449761
$800$	0	0
$801$	9.59572	0.339048
$802$	0	0
$803$	1.95691	0.0690579
$804$	0	0
$805$	−3.14266	−0.110764
$806$	0	0
$807$	−10.4320	−0.367222
$808$	0	0
$809$	−26.4692	−0.930608	−0.465304	−	0.885151i	$-0.654055\pi$
$809$	−26.4692	−0.930608	−0.465304	+	0.885151i	$0.654055\pi$
$810$	0	0
$811$	18.2602	0.641202	0.320601	−	0.947214i	$-0.396115\pi$
$811$	18.2602	0.641202	0.320601	+	0.947214i	$0.396115\pi$
$812$	0	0
$813$	16.1350	0.565877
$814$	0	0
$815$	−27.3461	−0.957893
$816$	0	0
$817$	−33.3511	−1.16681
$818$	0	0
$819$	3.83980	0.134173
$820$	0	0
$821$	−33.2407	−1.16011	−0.580055	−	0.814577i	$-0.696969\pi$
$821$	−33.2407	−1.16011	−0.580055	+	0.814577i	$0.696969\pi$
$822$	0	0
$823$	29.2349	1.01906	0.509531	−	0.860452i	$-0.329819\pi$
$823$	29.2349	1.01906	0.509531	+	0.860452i	$0.329819\pi$
$824$	0	0
$825$	−0.530225	−0.0184601
$826$	0	0
$827$	1.88879	0.0656798	0.0328399	−	0.999461i	$-0.489545\pi$
$827$	1.88879	0.0656798	0.0328399	+	0.999461i	$0.489545\pi$
$828$	0	0
$829$	−27.2251	−0.945568	−0.472784	−	0.881178i	$-0.656751\pi$
$829$	−27.2251	−0.945568	−0.472784	+	0.881178i	$0.656751\pi$
$830$	0	0
$831$	−21.6156	−0.749837
$832$	0	0
$833$	−12.3390	−0.427520
$834$	0	0
$835$	−10.2394	−0.354350
$836$	0	0
$837$	−0.153463	−0.00530444
$838$	0	0
$839$	−52.2304	−1.80320	−0.901598	−	0.432576i	$-0.857605\pi$
$839$	−52.2304	−1.80320	−0.901598	+	0.432576i	$0.857605\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−26.5768	−0.915355
$844$	0	0
$845$	14.6311	0.503324
$846$	0	0
$847$	−18.6599	−0.641161
$848$	0	0
$849$	−2.73513	−0.0938694
$850$	0	0
$851$	7.96250	0.272951
$852$	0	0
$853$	−3.03850	−0.104036	−0.0520182	−	0.998646i	$-0.516565\pi$
$853$	−3.03850	−0.104036	−0.0520182	+	0.998646i	$0.516565\pi$
$854$	0	0
$855$	12.7181	0.434950
$856$	0	0
$857$	−17.0934	−0.583899	−0.291949	−	0.956434i	$-0.594304\pi$
$857$	−17.0934	−0.583899	−0.291949	+	0.956434i	$0.594304\pi$
$858$	0	0
$859$	8.49832	0.289959	0.144980	−	0.989435i	$-0.453688\pi$
$859$	8.49832	0.289959	0.144980	+	0.989435i	$0.453688\pi$
$860$	0	0
$861$	−9.29929	−0.316919
$862$	0	0
$863$	−4.51995	−0.153861	−0.0769304	−	0.997036i	$-0.524512\pi$
$863$	−4.51995	−0.153861	−0.0769304	+	0.997036i	$0.524512\pi$
$864$	0	0
$865$	31.8161	1.08178
$866$	0	0
$867$	7.79280	0.264657
$868$	0	0
$869$	2.92945	0.0993749
$870$	0	0
$871$	−8.80977	−0.298508
$872$	0	0
$873$	−1.19093	−0.0403067
$874$	0	0
$875$	20.8463	0.704734
$876$	0	0
$877$	−3.65379	−0.123380	−0.0616899	−	0.998095i	$-0.519649\pi$
$877$	−3.65379	−0.123380	−0.0616899	+	0.998095i	$0.519649\pi$
$878$	0	0
$879$	−8.83703	−0.298066
$880$	0	0
$881$	10.5490	0.355405	0.177703	−	0.984084i	$-0.443133\pi$
$881$	10.5490	0.355405	0.177703	+	0.984084i	$0.443133\pi$
$882$	0	0
$883$	28.5933	0.962240	0.481120	−	0.876655i	$-0.340230\pi$
$883$	28.5933	0.962240	0.481120	+	0.876655i	$0.340230\pi$
$884$	0	0
$885$	−3.85861	−0.129706
$886$	0	0
$887$	−11.9607	−0.401603	−0.200801	−	0.979632i	$-0.564355\pi$
$887$	−11.9607	−0.401603	−0.200801	+	0.979632i	$0.564355\pi$
$888$	0	0
$889$	4.76680	0.159873
$890$	0	0
$891$	−0.324628	−0.0108754
$892$	0	0
$893$	29.0412	0.971826
$894$	0	0
$895$	16.2577	0.543436
$896$	0	0
$897$	−2.24187	−0.0748540
$898$	0	0
$899$	0.153463	0.00511826
$900$	0	0
$901$	35.1155	1.16987
$902$	0	0
$903$	−8.24108	−0.274246
$904$	0	0
$905$	44.0871	1.46551
$906$	0	0
$907$	−52.7883	−1.75281	−0.876403	−	0.481578i	$-0.840064\pi$
$907$	−52.7883	−1.75281	−0.876403	+	0.481578i	$0.840064\pi$
$908$	0	0
$909$	1.51959	0.0504016
$910$	0	0
$911$	38.6152	1.27938	0.639690	−	0.768633i	$-0.279063\pi$
$911$	38.6152	1.27938	0.639690	+	0.768633i	$0.279063\pi$
$912$	0	0
$913$	0.0526562	0.00174267
$914$	0	0
$915$	−2.02446	−0.0669267
$916$	0	0
$917$	−5.98938	−0.197787
$918$	0	0
$919$	−5.06764	−0.167166	−0.0835829	−	0.996501i	$-0.526636\pi$
$919$	−5.06764	−0.167166	−0.0835829	+	0.996501i	$0.526636\pi$
$920$	0	0
$921$	−6.01045	−0.198051
$922$	0	0
$923$	−25.2058	−0.829661
$924$	0	0
$925$	−13.0054	−0.427616
$926$	0	0
$927$	19.7934	0.650100
$928$	0	0
$929$	32.9328	1.08049	0.540246	−	0.841507i	$-0.318331\pi$
$929$	32.9328	1.08049	0.540246	+	0.841507i	$0.318331\pi$
$930$	0	0
$931$	28.1863	0.923768
$932$	0	0
$933$	−25.6664	−0.840279
$934$	0	0
$935$	1.80738	0.0591076
$936$	0	0
$937$	8.94072	0.292081	0.146040	−	0.989279i	$-0.453347\pi$
$937$	8.94072	0.292081	0.146040	+	0.989279i	$0.453347\pi$
$938$	0	0
$939$	17.3009	0.564595
$940$	0	0
$941$	−20.7057	−0.674987	−0.337493	−	0.941328i	$-0.609579\pi$
$941$	−20.7057	−0.674987	−0.337493	+	0.941328i	$0.609579\pi$
$942$	0	0
$943$	5.42941	0.176806
$944$	0	0
$945$	3.14266	0.102231
$946$	0	0
$947$	−18.3482	−0.596237	−0.298119	−	0.954529i	$-0.596359\pi$
$947$	−18.3482	−0.596237	−0.298119	+	0.954529i	$0.596359\pi$
$948$	0	0
$949$	−13.5144	−0.438696
$950$	0	0
$951$	−22.3014	−0.723171
$952$	0	0
$953$	34.3438	1.11250	0.556252	−	0.831013i	$-0.312239\pi$
$953$	34.3438	1.11250	0.556252	+	0.831013i	$0.312239\pi$
$954$	0	0
$955$	−36.8199	−1.19146
$956$	0	0
$957$	0.324628	0.0104937
$958$	0	0
$959$	−30.7968	−0.994482
$960$	0	0
$961$	−30.9764	−0.999240
$962$	0	0
$963$	17.5955	0.567007
$964$	0	0
$965$	−26.6060	−0.856478
$966$	0	0
$967$	−55.1712	−1.77419	−0.887094	−	0.461589i	$-0.847279\pi$
$967$	−55.1712	−1.77419	−0.887094	+	0.461589i	$0.847279\pi$
$968$	0	0
$969$	21.0323	0.675655
$970$	0	0
$971$	46.7876	1.50149	0.750743	−	0.660594i	$-0.229696\pi$
$971$	46.7876	1.50149	0.750743	+	0.660594i	$0.229696\pi$
$972$	0	0
$973$	−14.0825	−0.451463
$974$	0	0
$975$	3.66173	0.117269
$976$	0	0
$977$	31.2119	0.998558	0.499279	−	0.866441i	$-0.333598\pi$
$977$	31.2119	0.998558	0.499279	+	0.866441i	$0.333598\pi$
$978$	0	0
$979$	−3.11504	−0.0995570
$980$	0	0
$981$	15.6414	0.499390
$982$	0	0
$983$	−45.7017	−1.45766	−0.728830	−	0.684695i	$-0.759935\pi$
$983$	−45.7017	−1.45766	−0.728830	+	0.684695i	$0.759935\pi$
$984$	0	0
$985$	−46.5979	−1.48473
$986$	0	0
$987$	7.17610	0.228418
$988$	0	0
$989$	4.81157	0.152999
$990$	0	0
$991$	20.4553	0.649783	0.324892	−	0.945751i	$-0.394672\pi$
$991$	20.4553	0.649783	0.324892	+	0.945751i	$0.394672\pi$
$992$	0	0
$993$	19.5061	0.619008
$994$	0	0
$995$	42.1161	1.33517
$996$	0	0
$997$	−14.7908	−0.468428	−0.234214	−	0.972185i	$-0.575252\pi$
$997$	−14.7908	−0.468428	−0.234214	+	0.972185i	$0.575252\pi$
$998$	0	0
$999$	−7.96250	−0.251922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.i.1.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.i.1.5	✓	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.83485 q^{5} +1.71276 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.83485 q^{5} +1.71276 q^{7} +1.00000 q^{9} -0.324628 q^{11} +2.24187 q^{13} +1.83485 q^{15} +3.03434 q^{17} -6.93143 q^{19} -1.71276 q^{21} +1.00000 q^{23} -1.63333 q^{25} -1.00000 q^{27} +1.00000 q^{29} +0.153463 q^{31} +0.324628 q^{33} -3.14266 q^{35} +7.96250 q^{37} -2.24187 q^{39} +5.42941 q^{41} +4.81157 q^{43} -1.83485 q^{45} -4.18978 q^{47} -4.06644 q^{49} -3.03434 q^{51} +11.5727 q^{53} +0.595642 q^{55} +6.93143 q^{57} -2.10296 q^{59} -1.10334 q^{61} +1.71276 q^{63} -4.11350 q^{65} -3.92964 q^{67} -1.00000 q^{69} -11.2432 q^{71} -6.02817 q^{73} +1.63333 q^{75} -0.556010 q^{77} -9.02404 q^{79} +1.00000 q^{81} -0.162205 q^{83} -5.56755 q^{85} -1.00000 q^{87} +9.59572 q^{89} +3.83980 q^{91} -0.153463 q^{93} +12.7181 q^{95} -1.19093 q^{97} -0.324628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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