LMFDB - Embedded newform 8004.2.a.i.1.11

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			1.11
Root			$1.38601$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.38601 q^{5} -3.43605 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.38601 q^{5} -3.43605 q^{7} +1.00000 q^{9} +2.64786 q^{11} +3.17941 q^{13} -1.38601 q^{15} -7.44723 q^{17} +7.20390 q^{19} +3.43605 q^{21} +1.00000 q^{23} -3.07898 q^{25} -1.00000 q^{27} +1.00000 q^{29} -6.70556 q^{31} -2.64786 q^{33} -4.76239 q^{35} +2.03449 q^{37} -3.17941 q^{39} +7.14729 q^{41} -3.08585 q^{43} +1.38601 q^{45} +5.04741 q^{47} +4.80643 q^{49} +7.44723 q^{51} +9.24810 q^{53} +3.66995 q^{55} -7.20390 q^{57} +0.888525 q^{59} -0.116322 q^{61} -3.43605 q^{63} +4.40669 q^{65} -7.88452 q^{67} -1.00000 q^{69} -15.4402 q^{71} +8.68948 q^{73} +3.07898 q^{75} -9.09817 q^{77} +10.3464 q^{79} +1.00000 q^{81} +2.69423 q^{83} -10.3219 q^{85} -1.00000 q^{87} +15.8946 q^{89} -10.9246 q^{91} +6.70556 q^{93} +9.98468 q^{95} +8.50807 q^{97} +2.64786 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.38601	0.619842	0.309921	−	0.950762i	$-0.399697\pi$
$5$	1.38601	0.619842	0.309921	+	0.950762i	$0.399697\pi$
$6$	0	0
$7$	−3.43605	−1.29870	−0.649352	−	0.760488i	$-0.724960\pi$
$7$	−3.43605	−1.29870	−0.649352	+	0.760488i	$0.724960\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.64786	0.798359	0.399180	−	0.916873i	$-0.369295\pi$
$11$	2.64786	0.798359	0.399180	+	0.916873i	$0.369295\pi$
$12$	0	0
$13$	3.17941	0.881810	0.440905	−	0.897554i	$-0.354658\pi$
$13$	3.17941	0.881810	0.440905	+	0.897554i	$0.354658\pi$
$14$	0	0
$15$	−1.38601	−0.357866
$16$	0	0
$17$	−7.44723	−1.80622	−0.903110	−	0.429410i	$-0.858721\pi$
$17$	−7.44723	−1.80622	−0.903110	+	0.429410i	$0.858721\pi$
$18$	0	0
$19$	7.20390	1.65269	0.826345	−	0.563165i	$-0.190416\pi$
$19$	7.20390	1.65269	0.826345	+	0.563165i	$0.190416\pi$
$20$	0	0
$21$	3.43605	0.749807
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.07898	−0.615796
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−6.70556	−1.20435	−0.602177	−	0.798363i	$-0.705700\pi$
$31$	−6.70556	−1.20435	−0.602177	+	0.798363i	$0.705700\pi$
$32$	0	0
$33$	−2.64786	−0.460933
$34$	0	0
$35$	−4.76239	−0.804991
$36$	0	0
$37$	2.03449	0.334468	0.167234	−	0.985917i	$-0.446517\pi$
$37$	2.03449	0.334468	0.167234	+	0.985917i	$0.446517\pi$
$38$	0	0
$39$	−3.17941	−0.509113
$40$	0	0
$41$	7.14729	1.11622	0.558110	−	0.829767i	$-0.311527\pi$
$41$	7.14729	1.11622	0.558110	+	0.829767i	$0.311527\pi$
$42$	0	0
$43$	−3.08585	−0.470587	−0.235294	−	0.971924i	$-0.575605\pi$
$43$	−3.08585	−0.470587	−0.235294	+	0.971924i	$0.575605\pi$
$44$	0	0
$45$	1.38601	0.206614
$46$	0	0
$47$	5.04741	0.736240	0.368120	−	0.929778i	$-0.380002\pi$
$47$	5.04741	0.736240	0.368120	+	0.929778i	$0.380002\pi$
$48$	0	0
$49$	4.80643	0.686633
$50$	0	0
$51$	7.44723	1.04282
$52$	0	0
$53$	9.24810	1.27032	0.635162	−	0.772379i	$-0.280933\pi$
$53$	9.24810	1.27032	0.635162	+	0.772379i	$0.280933\pi$
$54$	0	0
$55$	3.66995	0.494856
$56$	0	0
$57$	−7.20390	−0.954181
$58$	0	0
$59$	0.888525	0.115676	0.0578381	−	0.998326i	$-0.481579\pi$
$59$	0.888525	0.115676	0.0578381	+	0.998326i	$0.481579\pi$
$60$	0	0
$61$	−0.116322	−0.0148936	−0.00744678	−	0.999972i	$-0.502370\pi$
$61$	−0.116322	−0.0148936	−0.00744678	+	0.999972i	$0.502370\pi$
$62$	0	0
$63$	−3.43605	−0.432901
$64$	0	0
$65$	4.40669	0.546583
$66$	0	0
$67$	−7.88452	−0.963247	−0.481623	−	0.876378i	$-0.659953\pi$
$67$	−7.88452	−0.963247	−0.481623	+	0.876378i	$0.659953\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−15.4402	−1.83241	−0.916205	−	0.400709i	$-0.868764\pi$
$71$	−15.4402	−1.83241	−0.916205	+	0.400709i	$0.868764\pi$
$72$	0	0
$73$	8.68948	1.01703	0.508513	−	0.861054i	$-0.330195\pi$
$73$	8.68948	1.01703	0.508513	+	0.861054i	$0.330195\pi$
$74$	0	0
$75$	3.07898	0.355530
$76$	0	0
$77$	−9.09817	−1.03683
$78$	0	0
$79$	10.3464	1.16406	0.582030	−	0.813168i	$-0.302259\pi$
$79$	10.3464	1.16406	0.582030	+	0.813168i	$0.302259\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.69423	0.295731	0.147865	−	0.989008i	$-0.452760\pi$
$83$	2.69423	0.295731	0.147865	+	0.989008i	$0.452760\pi$
$84$	0	0
$85$	−10.3219	−1.11957
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	15.8946	1.68483	0.842414	−	0.538831i	$-0.181134\pi$
$89$	15.8946	1.68483	0.842414	+	0.538831i	$0.181134\pi$
$90$	0	0
$91$	−10.9246	−1.14521
$92$	0	0
$93$	6.70556	0.695334
$94$	0	0
$95$	9.98468	1.02441
$96$	0	0
$97$	8.50807	0.863863	0.431932	−	0.901906i	$-0.357832\pi$
$97$	8.50807	0.863863	0.431932	+	0.901906i	$0.357832\pi$
$98$	0	0
$99$	2.64786	0.266120
$100$	0	0
$101$	−13.6985	−1.36305	−0.681526	−	0.731794i	$-0.738683\pi$
$101$	−13.6985	−1.36305	−0.681526	+	0.731794i	$0.738683\pi$
$102$	0	0
$103$	0.0167442	0.00164985	0.000824925	−	1.00000i	$-0.499737\pi$
$103$	0.0167442	0.00164985	0.000824925		1.00000i	$0.499737\pi$
$104$	0	0
$105$	4.76239	0.464762
$106$	0	0
$107$	−11.9335	−1.15366	−0.576828	−	0.816866i	$-0.695710\pi$
$107$	−11.9335	−1.15366	−0.576828	+	0.816866i	$0.695710\pi$
$108$	0	0
$109$	2.32636	0.222825	0.111412	−	0.993774i	$-0.464463\pi$
$109$	2.32636	0.222825	0.111412	+	0.993774i	$0.464463\pi$
$110$	0	0
$111$	−2.03449	−0.193105
$112$	0	0
$113$	−4.35571	−0.409750	−0.204875	−	0.978788i	$-0.565679\pi$
$113$	−4.35571	−0.409750	−0.204875	+	0.978788i	$0.565679\pi$
$114$	0	0
$115$	1.38601	0.129246
$116$	0	0
$117$	3.17941	0.293937
$118$	0	0
$119$	25.5891	2.34574
$120$	0	0
$121$	−3.98885	−0.362623
$122$	0	0
$123$	−7.14729	−0.644450
$124$	0	0
$125$	−11.1975	−1.00154
$126$	0	0
$127$	−0.680671	−0.0603998	−0.0301999	−	0.999544i	$-0.509614\pi$
$127$	−0.680671	−0.0603998	−0.0301999	+	0.999544i	$0.509614\pi$
$128$	0	0
$129$	3.08585	0.271694
$130$	0	0
$131$	3.31610	0.289729	0.144864	−	0.989452i	$-0.453725\pi$
$131$	3.31610	0.289729	0.144864	+	0.989452i	$0.453725\pi$
$132$	0	0
$133$	−24.7530	−2.14635
$134$	0	0
$135$	−1.38601	−0.119289
$136$	0	0
$137$	−5.69652	−0.486686	−0.243343	−	0.969940i	$-0.578244\pi$
$137$	−5.69652	−0.486686	−0.243343	+	0.969940i	$0.578244\pi$
$138$	0	0
$139$	14.7590	1.25184	0.625920	−	0.779887i	$-0.284723\pi$
$139$	14.7590	1.25184	0.625920	+	0.779887i	$0.284723\pi$
$140$	0	0
$141$	−5.04741	−0.425068
$142$	0	0
$143$	8.41863	0.704001
$144$	0	0
$145$	1.38601	0.115102
$146$	0	0
$147$	−4.80643	−0.396428
$148$	0	0
$149$	−0.160060	−0.0131126	−0.00655630	−	0.999979i	$-0.502087\pi$
$149$	−0.160060	−0.0131126	−0.00655630	+	0.999979i	$0.502087\pi$
$150$	0	0
$151$	2.17562	0.177050	0.0885248	−	0.996074i	$-0.471785\pi$
$151$	2.17562	0.177050	0.0885248	+	0.996074i	$0.471785\pi$
$152$	0	0
$153$	−7.44723	−0.602073
$154$	0	0
$155$	−9.29396	−0.746509
$156$	0	0
$157$	12.7136	1.01466	0.507330	−	0.861752i	$-0.330633\pi$
$157$	12.7136	1.01466	0.507330	+	0.861752i	$0.330633\pi$
$158$	0	0
$159$	−9.24810	−0.733422
$160$	0	0
$161$	−3.43605	−0.270799
$162$	0	0
$163$	−20.9400	−1.64014	−0.820072	−	0.572260i	$-0.806067\pi$
$163$	−20.9400	−1.64014	−0.820072	+	0.572260i	$0.806067\pi$
$164$	0	0
$165$	−3.66995	−0.285705
$166$	0	0
$167$	−2.32471	−0.179892	−0.0899459	−	0.995947i	$-0.528669\pi$
$167$	−2.32471	−0.179892	−0.0899459	+	0.995947i	$0.528669\pi$
$168$	0	0
$169$	−2.89134	−0.222411
$170$	0	0
$171$	7.20390	0.550896
$172$	0	0
$173$	16.3745	1.24493	0.622464	−	0.782648i	$-0.286132\pi$
$173$	16.3745	1.24493	0.622464	+	0.782648i	$0.286132\pi$
$174$	0	0
$175$	10.5795	0.799737
$176$	0	0
$177$	−0.888525	−0.0667856
$178$	0	0
$179$	−1.66977	−0.124804	−0.0624022	−	0.998051i	$-0.519876\pi$
$179$	−1.66977	−0.124804	−0.0624022	+	0.998051i	$0.519876\pi$
$180$	0	0
$181$	8.94963	0.665221	0.332610	−	0.943064i	$-0.392071\pi$
$181$	8.94963	0.665221	0.332610	+	0.943064i	$0.392071\pi$
$182$	0	0
$183$	0.116322	0.00859880
$184$	0	0
$185$	2.81982	0.207317
$186$	0	0
$187$	−19.7192	−1.44201
$188$	0	0
$189$	3.43605	0.249936
$190$	0	0
$191$	−9.07286	−0.656489	−0.328244	−	0.944593i	$-0.606457\pi$
$191$	−9.07286	−0.656489	−0.328244	+	0.944593i	$0.606457\pi$
$192$	0	0
$193$	9.95677	0.716704	0.358352	−	0.933587i	$-0.383339\pi$
$193$	9.95677	0.716704	0.358352	+	0.933587i	$0.383339\pi$
$194$	0	0
$195$	−4.40669	−0.315570
$196$	0	0
$197$	−4.20922	−0.299894	−0.149947	−	0.988694i	$-0.547910\pi$
$197$	−4.20922	−0.299894	−0.149947	+	0.988694i	$0.547910\pi$
$198$	0	0
$199$	13.5522	0.960689	0.480344	−	0.877080i	$-0.340512\pi$
$199$	13.5522	0.960689	0.480344	+	0.877080i	$0.340512\pi$
$200$	0	0
$201$	7.88452	0.556131
$202$	0	0
$203$	−3.43605	−0.241163
$204$	0	0
$205$	9.90621	0.691880
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	19.0749	1.31944
$210$	0	0
$211$	7.74403	0.533121	0.266561	−	0.963818i	$-0.414113\pi$
$211$	7.74403	0.533121	0.266561	+	0.963818i	$0.414113\pi$
$212$	0	0
$213$	15.4402	1.05794
$214$	0	0
$215$	−4.27701	−0.291690
$216$	0	0
$217$	23.0406	1.56410
$218$	0	0
$219$	−8.68948	−0.587180
$220$	0	0
$221$	−23.6778	−1.59274
$222$	0	0
$223$	−14.9462	−1.00087	−0.500436	−	0.865774i	$-0.666827\pi$
$223$	−14.9462	−1.00087	−0.500436	+	0.865774i	$0.666827\pi$
$224$	0	0
$225$	−3.07898	−0.205265
$226$	0	0
$227$	19.4919	1.29372	0.646861	−	0.762608i	$-0.276081\pi$
$227$	19.4919	1.29372	0.646861	+	0.762608i	$0.276081\pi$
$228$	0	0
$229$	12.4043	0.819697	0.409848	−	0.912154i	$-0.365582\pi$
$229$	12.4043	0.819697	0.409848	+	0.912154i	$0.365582\pi$
$230$	0	0
$231$	9.09817	0.598615
$232$	0	0
$233$	21.6702	1.41966	0.709830	−	0.704373i	$-0.248772\pi$
$233$	21.6702	1.41966	0.709830	+	0.704373i	$0.248772\pi$
$234$	0	0
$235$	6.99575	0.456352
$236$	0	0
$237$	−10.3464	−0.672070
$238$	0	0
$239$	−7.20633	−0.466139	−0.233069	−	0.972460i	$-0.574877\pi$
$239$	−7.20633	−0.466139	−0.233069	+	0.972460i	$0.574877\pi$
$240$	0	0
$241$	21.5021	1.38507	0.692536	−	0.721383i	$-0.256493\pi$
$241$	21.5021	1.38507	0.692536	+	0.721383i	$0.256493\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.66176	0.425604
$246$	0	0
$247$	22.9042	1.45736
$248$	0	0
$249$	−2.69423	−0.170740
$250$	0	0
$251$	2.47047	0.155934	0.0779672	−	0.996956i	$-0.475157\pi$
$251$	2.47047	0.155934	0.0779672	+	0.996956i	$0.475157\pi$
$252$	0	0
$253$	2.64786	0.166469
$254$	0	0
$255$	10.3219	0.646384
$256$	0	0
$257$	14.7725	0.921482	0.460741	−	0.887535i	$-0.347584\pi$
$257$	14.7725	0.921482	0.460741	+	0.887535i	$0.347584\pi$
$258$	0	0
$259$	−6.99060	−0.434375
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	15.8393	0.976694	0.488347	−	0.872649i	$-0.337600\pi$
$263$	15.8393	0.976694	0.488347	+	0.872649i	$0.337600\pi$
$264$	0	0
$265$	12.8179	0.787400
$266$	0	0
$267$	−15.8946	−0.972736
$268$	0	0
$269$	9.31464	0.567924	0.283962	−	0.958836i	$-0.408351\pi$
$269$	9.31464	0.567924	0.283962	+	0.958836i	$0.408351\pi$
$270$	0	0
$271$	8.34005	0.506622	0.253311	−	0.967385i	$-0.418480\pi$
$271$	8.34005	0.506622	0.253311	+	0.967385i	$0.418480\pi$
$272$	0	0
$273$	10.9246	0.661188
$274$	0	0
$275$	−8.15270	−0.491626
$276$	0	0
$277$	14.0290	0.842921	0.421461	−	0.906847i	$-0.361518\pi$
$277$	14.0290	0.842921	0.421461	+	0.906847i	$0.361518\pi$
$278$	0	0
$279$	−6.70556	−0.401451
$280$	0	0
$281$	12.9642	0.773382	0.386691	−	0.922209i	$-0.373618\pi$
$281$	12.9642	0.773382	0.386691	+	0.922209i	$0.373618\pi$
$282$	0	0
$283$	−32.1659	−1.91207	−0.956033	−	0.293260i	$-0.905260\pi$
$283$	−32.1659	−1.91207	−0.956033	+	0.293260i	$0.905260\pi$
$284$	0	0
$285$	−9.98468	−0.591441
$286$	0	0
$287$	−24.5584	−1.44964
$288$	0	0
$289$	38.4613	2.26243
$290$	0	0
$291$	−8.50807	−0.498752
$292$	0	0
$293$	−21.7425	−1.27021	−0.635104	−	0.772427i	$-0.719043\pi$
$293$	−21.7425	−1.27021	−0.635104	+	0.772427i	$0.719043\pi$
$294$	0	0
$295$	1.23150	0.0717009
$296$	0	0
$297$	−2.64786	−0.153644
$298$	0	0
$299$	3.17941	0.183870
$300$	0	0
$301$	10.6031	0.611154
$302$	0	0
$303$	13.6985	0.786958
$304$	0	0
$305$	−0.161224	−0.00923165
$306$	0	0
$307$	33.5501	1.91481	0.957403	−	0.288754i	$-0.0932409\pi$
$307$	33.5501	1.91481	0.957403	+	0.288754i	$0.0932409\pi$
$308$	0	0
$309$	−0.0167442	−0.000952542 0
$310$	0	0
$311$	15.8244	0.897317	0.448658	−	0.893703i	$-0.351902\pi$
$311$	15.8244	0.897317	0.448658	+	0.893703i	$0.351902\pi$
$312$	0	0
$313$	0.361470	0.0204315	0.0102157	−	0.999948i	$-0.496748\pi$
$313$	0.361470	0.0204315	0.0102157	+	0.999948i	$0.496748\pi$
$314$	0	0
$315$	−4.76239	−0.268330
$316$	0	0
$317$	8.33691	0.468248	0.234124	−	0.972207i	$-0.424778\pi$
$317$	8.33691	0.468248	0.234124	+	0.972207i	$0.424778\pi$
$318$	0	0
$319$	2.64786	0.148252
$320$	0	0
$321$	11.9335	0.666064
$322$	0	0
$323$	−53.6492	−2.98512
$324$	0	0
$325$	−9.78934	−0.543015
$326$	0	0
$327$	−2.32636	−0.128648
$328$	0	0
$329$	−17.3431	−0.956158
$330$	0	0
$331$	−15.6444	−0.859894	−0.429947	−	0.902854i	$-0.641468\pi$
$331$	−15.6444	−0.859894	−0.429947	+	0.902854i	$0.641468\pi$
$332$	0	0
$333$	2.03449	0.111489
$334$	0	0
$335$	−10.9280	−0.597061
$336$	0	0
$337$	−1.64739	−0.0897390	−0.0448695	−	0.998993i	$-0.514287\pi$
$337$	−1.64739	−0.0897390	−0.0448695	+	0.998993i	$0.514287\pi$
$338$	0	0
$339$	4.35571	0.236569
$340$	0	0
$341$	−17.7554	−0.961507
$342$	0	0
$343$	7.53721	0.406971
$344$	0	0
$345$	−1.38601	−0.0746202
$346$	0	0
$347$	33.2002	1.78228	0.891141	−	0.453727i	$-0.149906\pi$
$347$	33.2002	1.78228	0.891141	+	0.453727i	$0.149906\pi$
$348$	0	0
$349$	8.30391	0.444498	0.222249	−	0.974990i	$-0.428660\pi$
$349$	8.30391	0.444498	0.222249	+	0.974990i	$0.428660\pi$
$350$	0	0
$351$	−3.17941	−0.169704
$352$	0	0
$353$	−18.7231	−0.996532	−0.498266	−	0.867024i	$-0.666030\pi$
$353$	−18.7231	−0.996532	−0.498266	+	0.867024i	$0.666030\pi$
$354$	0	0
$355$	−21.4002	−1.13580
$356$	0	0
$357$	−25.5891	−1.35432
$358$	0	0
$359$	33.7762	1.78264	0.891319	−	0.453376i	$-0.149780\pi$
$359$	33.7762	1.78264	0.891319	+	0.453376i	$0.149780\pi$
$360$	0	0
$361$	32.8962	1.73138
$362$	0	0
$363$	3.98885	0.209360
$364$	0	0
$365$	12.0437	0.630395
$366$	0	0
$367$	−18.5205	−0.966761	−0.483381	−	0.875410i	$-0.660591\pi$
$367$	−18.5205	−0.966761	−0.483381	+	0.875410i	$0.660591\pi$
$368$	0	0
$369$	7.14729	0.372073
$370$	0	0
$371$	−31.7769	−1.64978
$372$	0	0
$373$	2.87668	0.148949	0.0744744	−	0.997223i	$-0.476272\pi$
$373$	2.87668	0.148949	0.0744744	+	0.997223i	$0.476272\pi$
$374$	0	0
$375$	11.1975	0.578238
$376$	0	0
$377$	3.17941	0.163748
$378$	0	0
$379$	−35.1025	−1.80309	−0.901546	−	0.432684i	$-0.857567\pi$
$379$	−35.1025	−1.80309	−0.901546	+	0.432684i	$0.857567\pi$
$380$	0	0
$381$	0.680671	0.0348718
$382$	0	0
$383$	−18.5022	−0.945419	−0.472710	−	0.881218i	$-0.656724\pi$
$383$	−18.5022	−0.945419	−0.472710	+	0.881218i	$0.656724\pi$
$384$	0	0
$385$	−12.6101	−0.642672
$386$	0	0
$387$	−3.08585	−0.156862
$388$	0	0
$389$	28.8868	1.46462	0.732311	−	0.680971i	$-0.238442\pi$
$389$	28.8868	1.46462	0.732311	+	0.680971i	$0.238442\pi$
$390$	0	0
$391$	−7.44723	−0.376623
$392$	0	0
$393$	−3.31610	−0.167275
$394$	0	0
$395$	14.3402	0.721533
$396$	0	0
$397$	1.98400	0.0995742	0.0497871	−	0.998760i	$-0.484146\pi$
$397$	1.98400	0.0995742	0.0497871	+	0.998760i	$0.484146\pi$
$398$	0	0
$399$	24.7530	1.23920
$400$	0	0
$401$	15.3643	0.767255	0.383628	−	0.923488i	$-0.374675\pi$
$401$	15.3643	0.767255	0.383628	+	0.923488i	$0.374675\pi$
$402$	0	0
$403$	−21.3197	−1.06201
$404$	0	0
$405$	1.38601	0.0688713
$406$	0	0
$407$	5.38703	0.267025
$408$	0	0
$409$	−12.0754	−0.597090	−0.298545	−	0.954396i	$-0.596501\pi$
$409$	−12.0754	−0.597090	−0.298545	+	0.954396i	$0.596501\pi$
$410$	0	0
$411$	5.69652	0.280989
$412$	0	0
$413$	−3.05302	−0.150229
$414$	0	0
$415$	3.73423	0.183306
$416$	0	0
$417$	−14.7590	−0.722751
$418$	0	0
$419$	10.9620	0.535529	0.267764	−	0.963484i	$-0.413715\pi$
$419$	10.9620	0.535529	0.267764	+	0.963484i	$0.413715\pi$
$420$	0	0
$421$	1.61154	0.0785419	0.0392709	−	0.999229i	$-0.487496\pi$
$421$	1.61154	0.0785419	0.0392709	+	0.999229i	$0.487496\pi$
$422$	0	0
$423$	5.04741	0.245413
$424$	0	0
$425$	22.9299	1.11226
$426$	0	0
$427$	0.399689	0.0193423
$428$	0	0
$429$	−8.41863	−0.406455
$430$	0	0
$431$	14.7932	0.712566	0.356283	−	0.934378i	$-0.384044\pi$
$431$	14.7932	0.712566	0.356283	+	0.934378i	$0.384044\pi$
$432$	0	0
$433$	26.8890	1.29220	0.646101	−	0.763252i	$-0.276399\pi$
$433$	26.8890	1.29220	0.646101	+	0.763252i	$0.276399\pi$
$434$	0	0
$435$	−1.38601	−0.0664540
$436$	0	0
$437$	7.20390	0.344610
$438$	0	0
$439$	−6.15227	−0.293632	−0.146816	−	0.989164i	$-0.546902\pi$
$439$	−6.15227	−0.293632	−0.146816	+	0.989164i	$0.546902\pi$
$440$	0	0
$441$	4.80643	0.228878
$442$	0	0
$443$	9.14045	0.434276	0.217138	−	0.976141i	$-0.430328\pi$
$443$	9.14045	0.434276	0.217138	+	0.976141i	$0.430328\pi$
$444$	0	0
$445$	22.0301	1.04433
$446$	0	0
$447$	0.160060	0.00757057
$448$	0	0
$449$	−12.0735	−0.569782	−0.284891	−	0.958560i	$-0.591957\pi$
$449$	−12.0735	−0.569782	−0.284891	+	0.958560i	$0.591957\pi$
$450$	0	0
$451$	18.9250	0.891144
$452$	0	0
$453$	−2.17562	−0.102220
$454$	0	0
$455$	−15.1416	−0.709850
$456$	0	0
$457$	39.8810	1.86555	0.932776	−	0.360456i	$-0.117379\pi$
$457$	39.8810	1.86555	0.932776	+	0.360456i	$0.117379\pi$
$458$	0	0
$459$	7.44723	0.347607
$460$	0	0
$461$	−7.10698	−0.331005	−0.165502	−	0.986209i	$-0.552925\pi$
$461$	−7.10698	−0.331005	−0.165502	+	0.986209i	$0.552925\pi$
$462$	0	0
$463$	15.7900	0.733821	0.366911	−	0.930256i	$-0.380415\pi$
$463$	15.7900	0.733821	0.366911	+	0.930256i	$0.380415\pi$
$464$	0	0
$465$	9.29396	0.430997
$466$	0	0
$467$	−30.7866	−1.42463	−0.712316	−	0.701859i	$-0.752354\pi$
$467$	−30.7866	−1.42463	−0.712316	+	0.701859i	$0.752354\pi$
$468$	0	0
$469$	27.0916	1.25097
$470$	0	0
$471$	−12.7136	−0.585814
$472$	0	0
$473$	−8.17088	−0.375698
$474$	0	0
$475$	−22.1807	−1.01772
$476$	0	0
$477$	9.24810	0.423441
$478$	0	0
$479$	25.9678	1.18650	0.593249	−	0.805019i	$-0.297845\pi$
$479$	25.9678	1.18650	0.593249	+	0.805019i	$0.297845\pi$
$480$	0	0
$481$	6.46847	0.294937
$482$	0	0
$483$	3.43605	0.156346
$484$	0	0
$485$	11.7923	0.535459
$486$	0	0
$487$	23.8307	1.07987	0.539936	−	0.841706i	$-0.318448\pi$
$487$	23.8307	1.07987	0.539936	+	0.841706i	$0.318448\pi$
$488$	0	0
$489$	20.9400	0.946938
$490$	0	0
$491$	23.1326	1.04396	0.521981	−	0.852957i	$-0.325193\pi$
$491$	23.1326	1.04396	0.521981	+	0.852957i	$0.325193\pi$
$492$	0	0
$493$	−7.44723	−0.335406
$494$	0	0
$495$	3.66995	0.164952
$496$	0	0
$497$	53.0532	2.37976
$498$	0	0
$499$	−27.2255	−1.21878	−0.609389	−	0.792871i	$-0.708585\pi$
$499$	−27.2255	−1.21878	−0.609389	+	0.792871i	$0.708585\pi$
$500$	0	0
$501$	2.32471	0.103861
$502$	0	0
$503$	−29.0909	−1.29710	−0.648550	−	0.761172i	$-0.724624\pi$
$503$	−29.0909	−1.29710	−0.648550	+	0.761172i	$0.724624\pi$
$504$	0	0
$505$	−18.9862	−0.844877
$506$	0	0
$507$	2.89134	0.128409
$508$	0	0
$509$	5.92211	0.262493	0.131246	−	0.991350i	$-0.458102\pi$
$509$	5.92211	0.262493	0.131246	+	0.991350i	$0.458102\pi$
$510$	0	0
$511$	−29.8575	−1.32082
$512$	0	0
$513$	−7.20390	−0.318060
$514$	0	0
$515$	0.0232075	0.00102265
$516$	0	0
$517$	13.3648	0.587784
$518$	0	0
$519$	−16.3745	−0.718760
$520$	0	0
$521$	35.4649	1.55375	0.776873	−	0.629658i	$-0.216805\pi$
$521$	35.4649	1.55375	0.776873	+	0.629658i	$0.216805\pi$
$522$	0	0
$523$	−38.7489	−1.69437	−0.847186	−	0.531296i	$-0.821705\pi$
$523$	−38.7489	−1.69437	−0.847186	+	0.531296i	$0.821705\pi$
$524$	0	0
$525$	−10.5795	−0.461728
$526$	0	0
$527$	49.9379	2.17533
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.888525	0.0385587
$532$	0	0
$533$	22.7242	0.984294
$534$	0	0
$535$	−16.5399	−0.715084
$536$	0	0
$537$	1.66977	0.0720559
$538$	0	0
$539$	12.7267	0.548180
$540$	0	0
$541$	29.2132	1.25598	0.627988	−	0.778223i	$-0.283879\pi$
$541$	29.2132	1.25598	0.627988	+	0.778223i	$0.283879\pi$
$542$	0	0
$543$	−8.94963	−0.384065
$544$	0	0
$545$	3.22435	0.138116
$546$	0	0
$547$	−32.1886	−1.37629	−0.688143	−	0.725575i	$-0.741574\pi$
$547$	−32.1886	−1.37629	−0.688143	+	0.725575i	$0.741574\pi$
$548$	0	0
$549$	−0.116322	−0.00496452
$550$	0	0
$551$	7.20390	0.306897
$552$	0	0
$553$	−35.5507	−1.51177
$554$	0	0
$555$	−2.81982	−0.119695
$556$	0	0
$557$	30.0309	1.27245	0.636226	−	0.771503i	$-0.280495\pi$
$557$	30.0309	1.27245	0.636226	+	0.771503i	$0.280495\pi$
$558$	0	0
$559$	−9.81118	−0.414969
$560$	0	0
$561$	19.7192	0.832546
$562$	0	0
$563$	−3.77593	−0.159137	−0.0795683	−	0.996829i	$-0.525354\pi$
$563$	−3.77593	−0.159137	−0.0795683	+	0.996829i	$0.525354\pi$
$564$	0	0
$565$	−6.03705	−0.253980
$566$	0	0
$567$	−3.43605	−0.144300
$568$	0	0
$569$	−10.4927	−0.439878	−0.219939	−	0.975514i	$-0.570586\pi$
$569$	−10.4927	−0.439878	−0.219939	+	0.975514i	$0.570586\pi$
$570$	0	0
$571$	−0.129941	−0.00543785	−0.00271893	−	0.999996i	$-0.500865\pi$
$571$	−0.129941	−0.00543785	−0.00271893	+	0.999996i	$0.500865\pi$
$572$	0	0
$573$	9.07286	0.379024
$574$	0	0
$575$	−3.07898	−0.128402
$576$	0	0
$577$	−20.0833	−0.836080	−0.418040	−	0.908429i	$-0.637283\pi$
$577$	−20.0833	−0.836080	−0.418040	+	0.908429i	$0.637283\pi$
$578$	0	0
$579$	−9.95677	−0.413789
$580$	0	0
$581$	−9.25752	−0.384067
$582$	0	0
$583$	24.4877	1.01417
$584$	0	0
$585$	4.40669	0.182194
$586$	0	0
$587$	30.5687	1.26170	0.630852	−	0.775903i	$-0.282705\pi$
$587$	30.5687	1.26170	0.630852	+	0.775903i	$0.282705\pi$
$588$	0	0
$589$	−48.3062	−1.99042
$590$	0	0
$591$	4.20922	0.173144
$592$	0	0
$593$	4.39665	0.180549	0.0902744	−	0.995917i	$-0.471226\pi$
$593$	4.39665	0.180549	0.0902744	+	0.995917i	$0.471226\pi$
$594$	0	0
$595$	35.4667	1.45399
$596$	0	0
$597$	−13.5522	−0.554654
$598$	0	0
$599$	32.6995	1.33607	0.668033	−	0.744132i	$-0.267137\pi$
$599$	32.6995	1.33607	0.668033	+	0.744132i	$0.267137\pi$
$600$	0	0
$601$	−9.56886	−0.390322	−0.195161	−	0.980771i	$-0.562523\pi$
$601$	−9.56886	−0.390322	−0.195161	+	0.980771i	$0.562523\pi$
$602$	0	0
$603$	−7.88452	−0.321082
$604$	0	0
$605$	−5.52858	−0.224769
$606$	0	0
$607$	−44.1392	−1.79155	−0.895777	−	0.444504i	$-0.853380\pi$
$607$	−44.1392	−1.79155	−0.895777	+	0.444504i	$0.853380\pi$
$608$	0	0
$609$	3.43605	0.139236
$610$	0	0
$611$	16.0478	0.649224
$612$	0	0
$613$	40.3468	1.62959	0.814796	−	0.579748i	$-0.196849\pi$
$613$	40.3468	1.62959	0.814796	+	0.579748i	$0.196849\pi$
$614$	0	0
$615$	−9.90621	−0.399457
$616$	0	0
$617$	−1.67027	−0.0672427	−0.0336213	−	0.999435i	$-0.510704\pi$
$617$	−1.67027	−0.0672427	−0.0336213	+	0.999435i	$0.510704\pi$
$618$	0	0
$619$	18.2055	0.731740	0.365870	−	0.930666i	$-0.380771\pi$
$619$	18.2055	0.731740	0.365870	+	0.930666i	$0.380771\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−54.6147	−2.18809
$624$	0	0
$625$	−0.124984	−0.00499936
$626$	0	0
$627$	−19.0749	−0.761779
$628$	0	0
$629$	−15.1513	−0.604122
$630$	0	0
$631$	−26.8565	−1.06914	−0.534569	−	0.845125i	$-0.679526\pi$
$631$	−26.8565	−1.06914	−0.534569	+	0.845125i	$0.679526\pi$
$632$	0	0
$633$	−7.74403	−0.307798
$634$	0	0
$635$	−0.943416	−0.0374383
$636$	0	0
$637$	15.2816	0.605480
$638$	0	0
$639$	−15.4402	−0.610803
$640$	0	0
$641$	49.0780	1.93846	0.969232	−	0.246151i	$-0.0791658\pi$
$641$	49.0780	1.93846	0.969232	+	0.246151i	$0.0791658\pi$
$642$	0	0
$643$	−20.7188	−0.817070	−0.408535	−	0.912743i	$-0.633960\pi$
$643$	−20.7188	−0.817070	−0.408535	+	0.912743i	$0.633960\pi$
$644$	0	0
$645$	4.27701	0.168407
$646$	0	0
$647$	−17.3265	−0.681176	−0.340588	−	0.940213i	$-0.610626\pi$
$647$	−17.3265	−0.681176	−0.340588	+	0.940213i	$0.610626\pi$
$648$	0	0
$649$	2.35269	0.0923511
$650$	0	0
$651$	−23.0406	−0.903033
$652$	0	0
$653$	−6.74821	−0.264078	−0.132039	−	0.991245i	$-0.542152\pi$
$653$	−6.74821	−0.264078	−0.132039	+	0.991245i	$0.542152\pi$
$654$	0	0
$655$	4.59614	0.179586
$656$	0	0
$657$	8.68948	0.339009
$658$	0	0
$659$	48.8832	1.90422	0.952110	−	0.305757i	$-0.0989094\pi$
$659$	48.8832	1.90422	0.952110	+	0.305757i	$0.0989094\pi$
$660$	0	0
$661$	−30.5438	−1.18802	−0.594009	−	0.804458i	$-0.702456\pi$
$661$	−30.5438	−1.18802	−0.594009	+	0.804458i	$0.702456\pi$
$662$	0	0
$663$	23.6778	0.919570
$664$	0	0
$665$	−34.3078	−1.33040
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	14.9462	0.577854
$670$	0	0
$671$	−0.308005	−0.0118904
$672$	0	0
$673$	−22.5888	−0.870736	−0.435368	−	0.900253i	$-0.643382\pi$
$673$	−22.5888	−0.870736	−0.435368	+	0.900253i	$0.643382\pi$
$674$	0	0
$675$	3.07898	0.118510
$676$	0	0
$677$	−48.9364	−1.88078	−0.940389	−	0.340100i	$-0.889539\pi$
$677$	−48.9364	−1.88078	−0.940389	+	0.340100i	$0.889539\pi$
$678$	0	0
$679$	−29.2341	−1.12190
$680$	0	0
$681$	−19.4919	−0.746931
$682$	0	0
$683$	35.1962	1.34675	0.673373	−	0.739303i	$-0.264845\pi$
$683$	35.1962	1.34675	0.673373	+	0.739303i	$0.264845\pi$
$684$	0	0
$685$	−7.89542	−0.301669
$686$	0	0
$687$	−12.4043	−0.473252
$688$	0	0
$689$	29.4035	1.12018
$690$	0	0
$691$	35.9630	1.36810	0.684049	−	0.729436i	$-0.260217\pi$
$691$	35.9630	1.36810	0.684049	+	0.729436i	$0.260217\pi$
$692$	0	0
$693$	−9.09817	−0.345611
$694$	0	0
$695$	20.4561	0.775943
$696$	0	0
$697$	−53.2275	−2.01614
$698$	0	0
$699$	−21.6702	−0.819641
$700$	0	0
$701$	2.71185	0.102425	0.0512126	−	0.998688i	$-0.483691\pi$
$701$	2.71185	0.102425	0.0512126	+	0.998688i	$0.483691\pi$
$702$	0	0
$703$	14.6563	0.552771
$704$	0	0
$705$	−6.99575	−0.263475
$706$	0	0
$707$	47.0687	1.77020
$708$	0	0
$709$	45.8632	1.72243	0.861214	−	0.508243i	$-0.169705\pi$
$709$	45.8632	1.72243	0.861214	+	0.508243i	$0.169705\pi$
$710$	0	0
$711$	10.3464	0.388020
$712$	0	0
$713$	−6.70556	−0.251125
$714$	0	0
$715$	11.6683	0.436369
$716$	0	0
$717$	7.20633	0.269125
$718$	0	0
$719$	18.9186	0.705546	0.352773	−	0.935709i	$-0.385239\pi$
$719$	18.9186	0.705546	0.352773	+	0.935709i	$0.385239\pi$
$720$	0	0
$721$	−0.0575337	−0.00214267
$722$	0	0
$723$	−21.5021	−0.799672
$724$	0	0
$725$	−3.07898	−0.114350
$726$	0	0
$727$	8.18289	0.303487	0.151743	−	0.988420i	$-0.451511\pi$
$727$	8.18289	0.303487	0.151743	+	0.988420i	$0.451511\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	22.9810	0.849984
$732$	0	0
$733$	−37.2184	−1.37469	−0.687347	−	0.726329i	$-0.741225\pi$
$733$	−37.2184	−1.37469	−0.687347	+	0.726329i	$0.741225\pi$
$734$	0	0
$735$	−6.66176	−0.245723
$736$	0	0
$737$	−20.8771	−0.769017
$738$	0	0
$739$	10.4788	0.385468	0.192734	−	0.981251i	$-0.438265\pi$
$739$	10.4788	0.385468	0.192734	+	0.981251i	$0.438265\pi$
$740$	0	0
$741$	−22.9042	−0.841406
$742$	0	0
$743$	−32.5439	−1.19392	−0.596959	−	0.802272i	$-0.703625\pi$
$743$	−32.5439	−1.19392	−0.596959	+	0.802272i	$0.703625\pi$
$744$	0	0
$745$	−0.221844	−0.00812774
$746$	0	0
$747$	2.69423	0.0985769
$748$	0	0
$749$	41.0041	1.49826
$750$	0	0
$751$	−15.1513	−0.552878	−0.276439	−	0.961031i	$-0.589154\pi$
$751$	−15.1513	−0.552878	−0.276439	+	0.961031i	$0.589154\pi$
$752$	0	0
$753$	−2.47047	−0.0900288
$754$	0	0
$755$	3.01543	0.109743
$756$	0	0
$757$	−51.6875	−1.87861	−0.939307	−	0.343079i	$-0.888530\pi$
$757$	−51.6875	−1.87861	−0.939307	+	0.343079i	$0.888530\pi$
$758$	0	0
$759$	−2.64786	−0.0961111
$760$	0	0
$761$	5.36043	0.194316	0.0971578	−	0.995269i	$-0.469025\pi$
$761$	5.36043	0.194316	0.0971578	+	0.995269i	$0.469025\pi$
$762$	0	0
$763$	−7.99348	−0.289384
$764$	0	0
$765$	−10.3219	−0.373190
$766$	0	0
$767$	2.82499	0.102004
$768$	0	0
$769$	−13.3418	−0.481117	−0.240559	−	0.970635i	$-0.577331\pi$
$769$	−13.3418	−0.481117	−0.240559	+	0.970635i	$0.577331\pi$
$770$	0	0
$771$	−14.7725	−0.532018
$772$	0	0
$773$	49.0812	1.76533	0.882665	−	0.470003i	$-0.155747\pi$
$773$	49.0812	1.76533	0.882665	+	0.470003i	$0.155747\pi$
$774$	0	0
$775$	20.6463	0.741636
$776$	0	0
$777$	6.99060	0.250786
$778$	0	0
$779$	51.4884	1.84476
$780$	0	0
$781$	−40.8834	−1.46292
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	17.6212	0.628928
$786$	0	0
$787$	−17.5113	−0.624211	−0.312106	−	0.950047i	$-0.601034\pi$
$787$	−17.5113	−0.624211	−0.312106	+	0.950047i	$0.601034\pi$
$788$	0	0
$789$	−15.8393	−0.563895
$790$	0	0
$791$	14.9664	0.532145
$792$	0	0
$793$	−0.369837	−0.0131333
$794$	0	0
$795$	−12.8179	−0.454606
$796$	0	0
$797$	−34.3432	−1.21650	−0.608248	−	0.793747i	$-0.708128\pi$
$797$	−34.3432	−1.21650	−0.608248	+	0.793747i	$0.708128\pi$
$798$	0	0
$799$	−37.5892	−1.32981
$800$	0	0
$801$	15.8946	0.561609
$802$	0	0
$803$	23.0085	0.811952
$804$	0	0
$805$	−4.76239	−0.167852
$806$	0	0
$807$	−9.31464	−0.327891
$808$	0	0
$809$	−23.6894	−0.832874	−0.416437	−	0.909165i	$-0.636721\pi$
$809$	−23.6894	−0.832874	−0.416437	+	0.909165i	$0.636721\pi$
$810$	0	0
$811$	47.4858	1.66745	0.833727	−	0.552177i	$-0.186203\pi$
$811$	47.4858	1.66745	0.833727	+	0.552177i	$0.186203\pi$
$812$	0	0
$813$	−8.34005	−0.292498
$814$	0	0
$815$	−29.0230	−1.01663
$816$	0	0
$817$	−22.2302	−0.777735
$818$	0	0
$819$	−10.9246	−0.381737
$820$	0	0
$821$	−13.3373	−0.465474	−0.232737	−	0.972540i	$-0.574768\pi$
$821$	−13.3373	−0.465474	−0.232737	+	0.972540i	$0.574768\pi$
$822$	0	0
$823$	−9.97366	−0.347660	−0.173830	−	0.984776i	$-0.555614\pi$
$823$	−9.97366	−0.347660	−0.173830	+	0.984776i	$0.555614\pi$
$824$	0	0
$825$	8.15270	0.283841
$826$	0	0
$827$	32.4815	1.12949	0.564746	−	0.825265i	$-0.308974\pi$
$827$	32.4815	1.12949	0.564746	+	0.825265i	$0.308974\pi$
$828$	0	0
$829$	39.4553	1.37034	0.685170	−	0.728383i	$-0.259728\pi$
$829$	39.4553	1.37034	0.685170	+	0.728383i	$0.259728\pi$
$830$	0	0
$831$	−14.0290	−0.486661
$832$	0	0
$833$	−35.7946	−1.24021
$834$	0	0
$835$	−3.22207	−0.111504
$836$	0	0
$837$	6.70556	0.231778
$838$	0	0
$839$	−10.7235	−0.370218	−0.185109	−	0.982718i	$-0.559264\pi$
$839$	−10.7235	−0.370218	−0.185109	+	0.982718i	$0.559264\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−12.9642	−0.446512
$844$	0	0
$845$	−4.00743	−0.137860
$846$	0	0
$847$	13.7059	0.470940
$848$	0	0
$849$	32.1659	1.10393
$850$	0	0
$851$	2.03449	0.0697413
$852$	0	0
$853$	18.1546	0.621602	0.310801	−	0.950475i	$-0.399403\pi$
$853$	18.1546	0.621602	0.310801	+	0.950475i	$0.399403\pi$
$854$	0	0
$855$	9.98468	0.341469
$856$	0	0
$857$	0.491319	0.0167831	0.00839156	−	0.999965i	$-0.497329\pi$
$857$	0.491319	0.0167831	0.00839156	+	0.999965i	$0.497329\pi$
$858$	0	0
$859$	13.4023	0.457281	0.228641	−	0.973511i	$-0.426572\pi$
$859$	13.4023	0.457281	0.228641	+	0.973511i	$0.426572\pi$
$860$	0	0
$861$	24.5584	0.836949
$862$	0	0
$863$	−7.60501	−0.258877	−0.129439	−	0.991587i	$-0.541318\pi$
$863$	−7.60501	−0.258877	−0.129439	+	0.991587i	$0.541318\pi$
$864$	0	0
$865$	22.6952	0.771659
$866$	0	0
$867$	−38.4613	−1.30621
$868$	0	0
$869$	27.3958	0.929337
$870$	0	0
$871$	−25.0681	−0.849401
$872$	0	0
$873$	8.50807	0.287954
$874$	0	0
$875$	38.4753	1.30070
$876$	0	0
$877$	−24.4204	−0.824617	−0.412309	−	0.911044i	$-0.635277\pi$
$877$	−24.4204	−0.824617	−0.412309	+	0.911044i	$0.635277\pi$
$878$	0	0
$879$	21.7425	0.733355
$880$	0	0
$881$	23.8176	0.802436	0.401218	−	0.915983i	$-0.368587\pi$
$881$	23.8176	0.802436	0.401218	+	0.915983i	$0.368587\pi$
$882$	0	0
$883$	36.3543	1.22342	0.611709	−	0.791083i	$-0.290482\pi$
$883$	36.3543	1.22342	0.611709	+	0.791083i	$0.290482\pi$
$884$	0	0
$885$	−1.23150	−0.0413965
$886$	0	0
$887$	24.1787	0.811840	0.405920	−	0.913909i	$-0.366951\pi$
$887$	24.1787	0.811840	0.405920	+	0.913909i	$0.366951\pi$
$888$	0	0
$889$	2.33882	0.0784415
$890$	0	0
$891$	2.64786	0.0887066
$892$	0	0
$893$	36.3610	1.21678
$894$	0	0
$895$	−2.31431	−0.0773590
$896$	0	0
$897$	−3.17941	−0.106157
$898$	0	0
$899$	−6.70556	−0.223643
$900$	0	0
$901$	−68.8728	−2.29448
$902$	0	0
$903$	−10.6031	−0.352850
$904$	0	0
$905$	12.4043	0.412332
$906$	0	0
$907$	−52.8003	−1.75320	−0.876602	−	0.481216i	$-0.840195\pi$
$907$	−52.8003	−1.75320	−0.876602	+	0.481216i	$0.840195\pi$
$908$	0	0
$909$	−13.6985	−0.454351
$910$	0	0
$911$	20.9718	0.694827	0.347414	−	0.937712i	$-0.387060\pi$
$911$	20.9718	0.694827	0.347414	+	0.937712i	$0.387060\pi$
$912$	0	0
$913$	7.13395	0.236099
$914$	0	0
$915$	0.161224	0.00532990
$916$	0	0
$917$	−11.3943	−0.376272
$918$	0	0
$919$	−50.6431	−1.67056	−0.835282	−	0.549822i	$-0.814695\pi$
$919$	−50.6431	−1.67056	−0.835282	+	0.549822i	$0.814695\pi$
$920$	0	0
$921$	−33.5501	−1.10551
$922$	0	0
$923$	−49.0906	−1.61584
$924$	0	0
$925$	−6.26415	−0.205964
$926$	0	0
$927$	0.0167442	0.000549950 0
$928$	0	0
$929$	51.3435	1.68452	0.842262	−	0.539068i	$-0.181224\pi$
$929$	51.3435	1.68452	0.842262	+	0.539068i	$0.181224\pi$
$930$	0	0
$931$	34.6251	1.13479
$932$	0	0
$933$	−15.8244	−0.518066
$934$	0	0
$935$	−27.3310	−0.893819
$936$	0	0
$937$	−38.6986	−1.26423	−0.632114	−	0.774876i	$-0.717812\pi$
$937$	−38.6986	−1.26423	−0.632114	+	0.774876i	$0.717812\pi$
$938$	0	0
$939$	−0.361470	−0.0117961
$940$	0	0
$941$	14.1476	0.461198	0.230599	−	0.973049i	$-0.425931\pi$
$941$	14.1476	0.461198	0.230599	+	0.973049i	$0.425931\pi$
$942$	0	0
$943$	7.14729	0.232748
$944$	0	0
$945$	4.76239	0.154921
$946$	0	0
$947$	18.4717	0.600248	0.300124	−	0.953900i	$-0.402972\pi$
$947$	18.4717	0.600248	0.300124	+	0.953900i	$0.402972\pi$
$948$	0	0
$949$	27.6274	0.896824
$950$	0	0
$951$	−8.33691	−0.270343
$952$	0	0
$953$	35.7365	1.15762	0.578810	−	0.815462i	$-0.303517\pi$
$953$	35.7365	1.15762	0.578810	+	0.815462i	$0.303517\pi$
$954$	0	0
$955$	−12.5751	−0.406919
$956$	0	0
$957$	−2.64786	−0.0855931
$958$	0	0
$959$	19.5735	0.632062
$960$	0	0
$961$	13.9645	0.450468
$962$	0	0
$963$	−11.9335	−0.384552
$964$	0	0
$965$	13.8002	0.444243
$966$	0	0
$967$	12.4558	0.400551	0.200275	−	0.979740i	$-0.435816\pi$
$967$	12.4558	0.400551	0.200275	+	0.979740i	$0.435816\pi$
$968$	0	0
$969$	53.6492	1.72346
$970$	0	0
$971$	33.1287	1.06315	0.531576	−	0.847011i	$-0.321600\pi$
$971$	33.1287	1.06315	0.531576	+	0.847011i	$0.321600\pi$
$972$	0	0
$973$	−50.7126	−1.62577
$974$	0	0
$975$	9.78934	0.313510
$976$	0	0
$977$	−50.0824	−1.60228	−0.801140	−	0.598478i	$-0.795773\pi$
$977$	−50.0824	−1.60228	−0.801140	+	0.598478i	$0.795773\pi$
$978$	0	0
$979$	42.0867	1.34510
$980$	0	0
$981$	2.32636	0.0742749
$982$	0	0
$983$	−1.89456	−0.0604271	−0.0302136	−	0.999543i	$-0.509619\pi$
$983$	−1.89456	−0.0604271	−0.0302136	+	0.999543i	$0.509619\pi$
$984$	0	0
$985$	−5.83401	−0.185887
$986$	0	0
$987$	17.3431	0.552038
$988$	0	0
$989$	−3.08585	−0.0981243
$990$	0	0
$991$	29.7752	0.945841	0.472921	−	0.881105i	$-0.343200\pi$
$991$	29.7752	0.945841	0.472921	+	0.881105i	$0.343200\pi$
$992$	0	0
$993$	15.6444	0.496460
$994$	0	0
$995$	18.7834	0.595475
$996$	0	0
$997$	−0.654663	−0.0207334	−0.0103667	−	0.999946i	$-0.503300\pi$
$997$	−0.654663	−0.0207334	−0.0103667	+	0.999946i	$0.503300\pi$
$998$	0	0
$999$	−2.03449	−0.0643683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.i.1.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.i.1.11	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.38601 q^{5} -3.43605 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.38601 q^{5} -3.43605 q^{7} +1.00000 q^{9} +2.64786 q^{11} +3.17941 q^{13} -1.38601 q^{15} -7.44723 q^{17} +7.20390 q^{19} +3.43605 q^{21} +1.00000 q^{23} -3.07898 q^{25} -1.00000 q^{27} +1.00000 q^{29} -6.70556 q^{31} -2.64786 q^{33} -4.76239 q^{35} +2.03449 q^{37} -3.17941 q^{39} +7.14729 q^{41} -3.08585 q^{43} +1.38601 q^{45} +5.04741 q^{47} +4.80643 q^{49} +7.44723 q^{51} +9.24810 q^{53} +3.66995 q^{55} -7.20390 q^{57} +0.888525 q^{59} -0.116322 q^{61} -3.43605 q^{63} +4.40669 q^{65} -7.88452 q^{67} -1.00000 q^{69} -15.4402 q^{71} +8.68948 q^{73} +3.07898 q^{75} -9.09817 q^{77} +10.3464 q^{79} +1.00000 q^{81} +2.69423 q^{83} -10.3219 q^{85} -1.00000 q^{87} +15.8946 q^{89} -10.9246 q^{91} +6.70556 q^{93} +9.98468 q^{95} +8.50807 q^{97} +2.64786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 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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