LMFDB - Embedded newform 8004.2.a.i.1.1

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			1.1
Root			$-4.28632$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -4.28632 q^{5} +1.70119 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.28632 q^{5} +1.70119 q^{7} +1.00000 q^{9} +4.51446 q^{11} -1.48859 q^{13} +4.28632 q^{15} -1.55072 q^{17} -1.31263 q^{19} -1.70119 q^{21} +1.00000 q^{23} +13.3726 q^{25} -1.00000 q^{27} +1.00000 q^{29} -2.61274 q^{31} -4.51446 q^{33} -7.29183 q^{35} -0.322680 q^{37} +1.48859 q^{39} -7.70149 q^{41} +4.70934 q^{43} -4.28632 q^{45} +13.0439 q^{47} -4.10597 q^{49} +1.55072 q^{51} +7.24304 q^{53} -19.3504 q^{55} +1.31263 q^{57} +9.70167 q^{59} -4.23430 q^{61} +1.70119 q^{63} +6.38057 q^{65} -3.71674 q^{67} -1.00000 q^{69} +11.9916 q^{71} -10.7328 q^{73} -13.3726 q^{75} +7.67994 q^{77} -17.6430 q^{79} +1.00000 q^{81} +11.1598 q^{83} +6.64689 q^{85} -1.00000 q^{87} -17.7972 q^{89} -2.53236 q^{91} +2.61274 q^{93} +5.62635 q^{95} -0.650043 q^{97} +4.51446 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.28632	−1.91690	−0.958451	−	0.285257i	$-0.907921\pi$
$5$	−4.28632	−1.91690	−0.958451	+	0.285257i	$0.907921\pi$
$6$	0	0
$7$	1.70119	0.642988	0.321494	−	0.946912i	$-0.395815\pi$
$7$	1.70119	0.642988	0.321494	+	0.946912i	$0.395815\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.51446	1.36116	0.680581	−	0.732673i	$-0.261728\pi$
$11$	4.51446	1.36116	0.680581	+	0.732673i	$0.261728\pi$
$12$	0	0
$13$	−1.48859	−0.412860	−0.206430	−	0.978461i	$-0.566185\pi$
$13$	−1.48859	−0.412860	−0.206430	+	0.978461i	$0.566185\pi$
$14$	0	0
$15$	4.28632	1.10672
$16$	0	0
$17$	−1.55072	−0.376105	−0.188052	−	0.982159i	$-0.560217\pi$
$17$	−1.55072	−0.376105	−0.188052	+	0.982159i	$0.560217\pi$
$18$	0	0
$19$	−1.31263	−0.301138	−0.150569	−	0.988600i	$-0.548111\pi$
$19$	−1.31263	−0.301138	−0.150569	+	0.988600i	$0.548111\pi$
$20$	0	0
$21$	−1.70119	−0.371229
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	13.3726	2.67451
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.61274	−0.469262	−0.234631	−	0.972084i	$-0.575388\pi$
$31$	−2.61274	−0.469262	−0.234631	+	0.972084i	$0.575388\pi$
$32$	0	0
$33$	−4.51446	−0.785867
$34$	0	0
$35$	−7.29183	−1.23254
$36$	0	0
$37$	−0.322680	−0.0530483	−0.0265241	−	0.999648i	$-0.508444\pi$
$37$	−0.322680	−0.0530483	−0.0265241	+	0.999648i	$0.508444\pi$
$38$	0	0
$39$	1.48859	0.238365
$40$	0	0
$41$	−7.70149	−1.20277	−0.601385	−	0.798959i	$-0.705384\pi$
$41$	−7.70149	−1.20277	−0.601385	+	0.798959i	$0.705384\pi$
$42$	0	0
$43$	4.70934	0.718168	0.359084	−	0.933305i	$-0.383089\pi$
$43$	4.70934	0.718168	0.359084	+	0.933305i	$0.383089\pi$
$44$	0	0
$45$	−4.28632	−0.638967
$46$	0	0
$47$	13.0439	1.90264	0.951322	−	0.308199i	$-0.0997263\pi$
$47$	13.0439	1.90264	0.951322	+	0.308199i	$0.0997263\pi$
$48$	0	0
$49$	−4.10597	−0.586567
$50$	0	0
$51$	1.55072	0.217144
$52$	0	0
$53$	7.24304	0.994908	0.497454	−	0.867490i	$-0.334268\pi$
$53$	7.24304	0.994908	0.497454	+	0.867490i	$0.334268\pi$
$54$	0	0
$55$	−19.3504	−2.60921
$56$	0	0
$57$	1.31263	0.173862
$58$	0	0
$59$	9.70167	1.26305	0.631525	−	0.775356i	$-0.282429\pi$
$59$	9.70167	1.26305	0.631525	+	0.775356i	$0.282429\pi$
$60$	0	0
$61$	−4.23430	−0.542146	−0.271073	−	0.962559i	$-0.587378\pi$
$61$	−4.23430	−0.542146	−0.271073	+	0.962559i	$0.587378\pi$
$62$	0	0
$63$	1.70119	0.214329
$64$	0	0
$65$	6.38057	0.791412
$66$	0	0
$67$	−3.71674	−0.454072	−0.227036	−	0.973886i	$-0.572903\pi$
$67$	−3.71674	−0.454072	−0.227036	+	0.973886i	$0.572903\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	11.9916	1.42314	0.711570	−	0.702616i	$-0.247985\pi$
$71$	11.9916	1.42314	0.711570	+	0.702616i	$0.247985\pi$
$72$	0	0
$73$	−10.7328	−1.25618	−0.628092	−	0.778139i	$-0.716164\pi$
$73$	−10.7328	−1.25618	−0.628092	+	0.778139i	$0.716164\pi$
$74$	0	0
$75$	−13.3726	−1.54413
$76$	0	0
$77$	7.67994	0.875210
$78$	0	0
$79$	−17.6430	−1.98500	−0.992498	−	0.122262i	$-0.960985\pi$
$79$	−17.6430	−1.98500	−0.992498	+	0.122262i	$0.960985\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.1598	1.22495	0.612473	−	0.790491i	$-0.290175\pi$
$83$	11.1598	1.22495	0.612473	+	0.790491i	$0.290175\pi$
$84$	0	0
$85$	6.64689	0.720956
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−17.7972	−1.88650	−0.943251	−	0.332080i	$-0.892250\pi$
$89$	−17.7972	−1.88650	−0.943251	+	0.332080i	$0.892250\pi$
$90$	0	0
$91$	−2.53236	−0.265464
$92$	0	0
$93$	2.61274	0.270929
$94$	0	0
$95$	5.62635	0.577252
$96$	0	0
$97$	−0.650043	−0.0660019	−0.0330009	−	0.999455i	$-0.510506\pi$
$97$	−0.650043	−0.0660019	−0.0330009	+	0.999455i	$0.510506\pi$
$98$	0	0
$99$	4.51446	0.453721
$100$	0	0
$101$	−16.9798	−1.68956	−0.844778	−	0.535117i	$-0.820267\pi$
$101$	−16.9798	−1.68956	−0.844778	+	0.535117i	$0.820267\pi$
$102$	0	0
$103$	−19.0401	−1.87607	−0.938037	−	0.346535i	$-0.887358\pi$
$103$	−19.0401	−1.87607	−0.938037	+	0.346535i	$0.887358\pi$
$104$	0	0
$105$	7.29183	0.711610
$106$	0	0
$107$	10.3417	0.999766	0.499883	−	0.866093i	$-0.333376\pi$
$107$	10.3417	0.999766	0.499883	+	0.866093i	$0.333376\pi$
$108$	0	0
$109$	−5.48832	−0.525685	−0.262843	−	0.964839i	$-0.584660\pi$
$109$	−5.48832	−0.525685	−0.262843	+	0.964839i	$0.584660\pi$
$110$	0	0
$111$	0.322680	0.0306274
$112$	0	0
$113$	5.27014	0.495773	0.247886	−	0.968789i	$-0.420264\pi$
$113$	5.27014	0.495773	0.247886	+	0.968789i	$0.420264\pi$
$114$	0	0
$115$	−4.28632	−0.399702
$116$	0	0
$117$	−1.48859	−0.137620
$118$	0	0
$119$	−2.63806	−0.241831
$120$	0	0
$121$	9.38037	0.852761
$122$	0	0
$123$	7.70149	0.694420
$124$	0	0
$125$	−35.8876	−3.20988
$126$	0	0
$127$	−7.46610	−0.662509	−0.331255	−	0.943541i	$-0.607472\pi$
$127$	−7.46610	−0.662509	−0.331255	+	0.943541i	$0.607472\pi$
$128$	0	0
$129$	−4.70934	−0.414634
$130$	0	0
$131$	11.3089	0.988059	0.494030	−	0.869445i	$-0.335523\pi$
$131$	11.3089	0.988059	0.494030	+	0.869445i	$0.335523\pi$
$132$	0	0
$133$	−2.23303	−0.193628
$134$	0	0
$135$	4.28632	0.368908
$136$	0	0
$137$	−9.30951	−0.795365	−0.397683	−	0.917523i	$-0.630186\pi$
$137$	−9.30951	−0.795365	−0.397683	+	0.917523i	$0.630186\pi$
$138$	0	0
$139$	18.9397	1.60645	0.803224	−	0.595677i	$-0.203116\pi$
$139$	18.9397	1.60645	0.803224	+	0.595677i	$0.203116\pi$
$140$	0	0
$141$	−13.0439	−1.09849
$142$	0	0
$143$	−6.72017	−0.561969
$144$	0	0
$145$	−4.28632	−0.355960
$146$	0	0
$147$	4.10597	0.338654
$148$	0	0
$149$	0.200164	0.0163981	0.00819906	−	0.999966i	$-0.497390\pi$
$149$	0.200164	0.0163981	0.00819906	+	0.999966i	$0.497390\pi$
$150$	0	0
$151$	16.1714	1.31601	0.658004	−	0.753015i	$-0.271401\pi$
$151$	16.1714	1.31601	0.658004	+	0.753015i	$0.271401\pi$
$152$	0	0
$153$	−1.55072	−0.125368
$154$	0	0
$155$	11.1991	0.899530
$156$	0	0
$157$	−6.72841	−0.536986	−0.268493	−	0.963282i	$-0.586526\pi$
$157$	−6.72841	−0.536986	−0.268493	+	0.963282i	$0.586526\pi$
$158$	0	0
$159$	−7.24304	−0.574410
$160$	0	0
$161$	1.70119	0.134072
$162$	0	0
$163$	−2.55383	−0.200032	−0.100016	−	0.994986i	$-0.531889\pi$
$163$	−2.55383	−0.200032	−0.100016	+	0.994986i	$0.531889\pi$
$164$	0	0
$165$	19.3504	1.50643
$166$	0	0
$167$	−0.272414	−0.0210800	−0.0105400	−	0.999944i	$-0.503355\pi$
$167$	−0.272414	−0.0210800	−0.0105400	+	0.999944i	$0.503355\pi$
$168$	0	0
$169$	−10.7841	−0.829547
$170$	0	0
$171$	−1.31263	−0.100379
$172$	0	0
$173$	17.0168	1.29376	0.646881	−	0.762591i	$-0.276073\pi$
$173$	17.0168	1.29376	0.646881	+	0.762591i	$0.276073\pi$
$174$	0	0
$175$	22.7492	1.71968
$176$	0	0
$177$	−9.70167	−0.729222
$178$	0	0
$179$	−12.3164	−0.920569	−0.460285	−	0.887771i	$-0.652253\pi$
$179$	−12.3164	−0.920569	−0.460285	+	0.887771i	$0.652253\pi$
$180$	0	0
$181$	−4.16472	−0.309561	−0.154781	−	0.987949i	$-0.549467\pi$
$181$	−4.16472	−0.309561	−0.154781	+	0.987949i	$0.549467\pi$
$182$	0	0
$183$	4.23430	0.313008
$184$	0	0
$185$	1.38311	0.101688
$186$	0	0
$187$	−7.00067	−0.511939
$188$	0	0
$189$	−1.70119	−0.123743
$190$	0	0
$191$	−18.8725	−1.36557	−0.682784	−	0.730621i	$-0.739231\pi$
$191$	−18.8725	−1.36557	−0.682784	+	0.730621i	$0.739231\pi$
$192$	0	0
$193$	−7.72554	−0.556097	−0.278048	−	0.960567i	$-0.589688\pi$
$193$	−7.72554	−0.556097	−0.278048	+	0.960567i	$0.589688\pi$
$194$	0	0
$195$	−6.38057	−0.456922
$196$	0	0
$197$	20.0387	1.42770	0.713851	−	0.700298i	$-0.246949\pi$
$197$	20.0387	1.42770	0.713851	+	0.700298i	$0.246949\pi$
$198$	0	0
$199$	26.1404	1.85304	0.926520	−	0.376244i	$-0.122785\pi$
$199$	26.1404	1.85304	0.926520	+	0.376244i	$0.122785\pi$
$200$	0	0
$201$	3.71674	0.262158
$202$	0	0
$203$	1.70119	0.119400
$204$	0	0
$205$	33.0111	2.30559
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−5.92581	−0.409897
$210$	0	0
$211$	5.36990	0.369679	0.184840	−	0.982769i	$-0.440823\pi$
$211$	5.36990	0.369679	0.184840	+	0.982769i	$0.440823\pi$
$212$	0	0
$213$	−11.9916	−0.821650
$214$	0	0
$215$	−20.1858	−1.37666
$216$	0	0
$217$	−4.44476	−0.301730
$218$	0	0
$219$	10.7328	0.725258
$220$	0	0
$221$	2.30838	0.155279
$222$	0	0
$223$	21.2451	1.42268	0.711339	−	0.702849i	$-0.248089\pi$
$223$	21.2451	1.42268	0.711339	+	0.702849i	$0.248089\pi$
$224$	0	0
$225$	13.3726	0.891505
$226$	0	0
$227$	0.120251	0.00798134	0.00399067	−	0.999992i	$-0.498730\pi$
$227$	0.120251	0.00798134	0.00399067	+	0.999992i	$0.498730\pi$
$228$	0	0
$229$	25.2154	1.66628	0.833141	−	0.553060i	$-0.186540\pi$
$229$	25.2154	1.66628	0.833141	+	0.553060i	$0.186540\pi$
$230$	0	0
$231$	−7.67994	−0.505303
$232$	0	0
$233$	15.5036	1.01567	0.507837	−	0.861453i	$-0.330445\pi$
$233$	15.5036	1.01567	0.507837	+	0.861453i	$0.330445\pi$
$234$	0	0
$235$	−55.9102	−3.64718
$236$	0	0
$237$	17.6430	1.14604
$238$	0	0
$239$	−4.11436	−0.266136	−0.133068	−	0.991107i	$-0.542483\pi$
$239$	−4.11436	−0.266136	−0.133068	+	0.991107i	$0.542483\pi$
$240$	0	0
$241$	8.29590	0.534386	0.267193	−	0.963643i	$-0.413904\pi$
$241$	8.29590	0.534386	0.267193	+	0.963643i	$0.413904\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	17.5995	1.12439
$246$	0	0
$247$	1.95396	0.124328
$248$	0	0
$249$	−11.1598	−0.707223
$250$	0	0
$251$	24.4531	1.54347	0.771733	−	0.635946i	$-0.219390\pi$
$251$	24.4531	1.54347	0.771733	+	0.635946i	$0.219390\pi$
$252$	0	0
$253$	4.51446	0.283822
$254$	0	0
$255$	−6.64689	−0.416244
$256$	0	0
$257$	−6.87202	−0.428665	−0.214332	−	0.976761i	$-0.568758\pi$
$257$	−6.87202	−0.428665	−0.214332	+	0.976761i	$0.568758\pi$
$258$	0	0
$259$	−0.548939	−0.0341094
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	14.2428	0.878246	0.439123	−	0.898427i	$-0.355289\pi$
$263$	14.2428	0.878246	0.439123	+	0.898427i	$0.355289\pi$
$264$	0	0
$265$	−31.0460	−1.90714
$266$	0	0
$267$	17.7972	1.08917
$268$	0	0
$269$	−23.4638	−1.43061	−0.715306	−	0.698811i	$-0.753713\pi$
$269$	−23.4638	−1.43061	−0.715306	+	0.698811i	$0.753713\pi$
$270$	0	0
$271$	21.3651	1.29784	0.648918	−	0.760859i	$-0.275222\pi$
$271$	21.3651	1.29784	0.648918	+	0.760859i	$0.275222\pi$
$272$	0	0
$273$	2.53236	0.153266
$274$	0	0
$275$	60.3700	3.64045
$276$	0	0
$277$	−20.2990	−1.21965	−0.609824	−	0.792537i	$-0.708760\pi$
$277$	−20.2990	−1.21965	−0.609824	+	0.792537i	$0.708760\pi$
$278$	0	0
$279$	−2.61274	−0.156421
$280$	0	0
$281$	13.9991	0.835117	0.417558	−	0.908650i	$-0.362886\pi$
$281$	13.9991	0.835117	0.417558	+	0.908650i	$0.362886\pi$
$282$	0	0
$283$	23.4813	1.39582	0.697909	−	0.716186i	$-0.254114\pi$
$283$	23.4813	1.39582	0.697909	+	0.716186i	$0.254114\pi$
$284$	0	0
$285$	−5.62635	−0.333276
$286$	0	0
$287$	−13.1017	−0.773367
$288$	0	0
$289$	−14.5953	−0.858545
$290$	0	0
$291$	0.650043	0.0381062
$292$	0	0
$293$	6.87158	0.401442	0.200721	−	0.979648i	$-0.435672\pi$
$293$	6.87158	0.401442	0.200721	+	0.979648i	$0.435672\pi$
$294$	0	0
$295$	−41.5845	−2.42114
$296$	0	0
$297$	−4.51446	−0.261956
$298$	0	0
$299$	−1.48859	−0.0860872
$300$	0	0
$301$	8.01147	0.461773
$302$	0	0
$303$	16.9798	0.975465
$304$	0	0
$305$	18.1496	1.03924
$306$	0	0
$307$	5.69689	0.325139	0.162569	−	0.986697i	$-0.448022\pi$
$307$	5.69689	0.325139	0.162569	+	0.986697i	$0.448022\pi$
$308$	0	0
$309$	19.0401	1.08315
$310$	0	0
$311$	−8.45195	−0.479266	−0.239633	−	0.970864i	$-0.577027\pi$
$311$	−8.45195	−0.479266	−0.239633	+	0.970864i	$0.577027\pi$
$312$	0	0
$313$	−29.2225	−1.65175	−0.825877	−	0.563850i	$-0.809319\pi$
$313$	−29.2225	−1.65175	−0.825877	+	0.563850i	$0.809319\pi$
$314$	0	0
$315$	−7.29183	−0.410848
$316$	0	0
$317$	−3.03917	−0.170697	−0.0853485	−	0.996351i	$-0.527200\pi$
$317$	−3.03917	−0.170697	−0.0853485	+	0.996351i	$0.527200\pi$
$318$	0	0
$319$	4.51446	0.252761
$320$	0	0
$321$	−10.3417	−0.577215
$322$	0	0
$323$	2.03552	0.113259
$324$	0	0
$325$	−19.9062	−1.10420
$326$	0	0
$327$	5.48832	0.303505
$328$	0	0
$329$	22.1900	1.22338
$330$	0	0
$331$	33.0230	1.81511	0.907555	−	0.419933i	$-0.137946\pi$
$331$	33.0230	1.81511	0.907555	+	0.419933i	$0.137946\pi$
$332$	0	0
$333$	−0.322680	−0.0176828
$334$	0	0
$335$	15.9311	0.870411
$336$	0	0
$337$	33.0649	1.80116	0.900580	−	0.434690i	$-0.143142\pi$
$337$	33.0649	1.80116	0.900580	+	0.434690i	$0.143142\pi$
$338$	0	0
$339$	−5.27014	−0.286235
$340$	0	0
$341$	−11.7951	−0.638742
$342$	0	0
$343$	−18.8933	−1.02014
$344$	0	0
$345$	4.28632	0.230768
$346$	0	0
$347$	26.7918	1.43826	0.719130	−	0.694875i	$-0.244540\pi$
$347$	26.7918	1.43826	0.719130	+	0.694875i	$0.244540\pi$
$348$	0	0
$349$	−12.6622	−0.677790	−0.338895	−	0.940824i	$-0.610053\pi$
$349$	−12.6622	−0.677790	−0.338895	+	0.940824i	$0.610053\pi$
$350$	0	0
$351$	1.48859	0.0794549
$352$	0	0
$353$	2.30737	0.122809	0.0614046	−	0.998113i	$-0.480442\pi$
$353$	2.30737	0.122809	0.0614046	+	0.998113i	$0.480442\pi$
$354$	0	0
$355$	−51.3998	−2.72802
$356$	0	0
$357$	2.63806	0.139621
$358$	0	0
$359$	30.5132	1.61043	0.805213	−	0.592986i	$-0.202051\pi$
$359$	30.5132	1.61043	0.805213	+	0.592986i	$0.202051\pi$
$360$	0	0
$361$	−17.2770	−0.909316
$362$	0	0
$363$	−9.38037	−0.492342
$364$	0	0
$365$	46.0044	2.40798
$366$	0	0
$367$	−8.40674	−0.438828	−0.219414	−	0.975632i	$-0.570415\pi$
$367$	−8.40674	−0.438828	−0.219414	+	0.975632i	$0.570415\pi$
$368$	0	0
$369$	−7.70149	−0.400923
$370$	0	0
$371$	12.3218	0.639714
$372$	0	0
$373$	13.9437	0.721977	0.360988	−	0.932570i	$-0.382439\pi$
$373$	13.9437	0.721977	0.360988	+	0.932570i	$0.382439\pi$
$374$	0	0
$375$	35.8876	1.85323
$376$	0	0
$377$	−1.48859	−0.0766661
$378$	0	0
$379$	29.7324	1.52725	0.763624	−	0.645661i	$-0.223418\pi$
$379$	29.7324	1.52725	0.763624	+	0.645661i	$0.223418\pi$
$380$	0	0
$381$	7.46610	0.382500
$382$	0	0
$383$	5.87971	0.300439	0.150219	−	0.988653i	$-0.452002\pi$
$383$	5.87971	0.300439	0.150219	+	0.988653i	$0.452002\pi$
$384$	0	0
$385$	−32.9187	−1.67769
$386$	0	0
$387$	4.70934	0.239389
$388$	0	0
$389$	−3.17532	−0.160995	−0.0804976	−	0.996755i	$-0.525651\pi$
$389$	−3.17532	−0.160995	−0.0804976	+	0.996755i	$0.525651\pi$
$390$	0	0
$391$	−1.55072	−0.0784233
$392$	0	0
$393$	−11.3089	−0.570456
$394$	0	0
$395$	75.6237	3.80504
$396$	0	0
$397$	4.94465	0.248165	0.124082	−	0.992272i	$-0.460401\pi$
$397$	4.94465	0.248165	0.124082	+	0.992272i	$0.460401\pi$
$398$	0	0
$399$	2.23303	0.111791
$400$	0	0
$401$	32.4870	1.62232	0.811161	−	0.584823i	$-0.198836\pi$
$401$	32.4870	1.62232	0.811161	+	0.584823i	$0.198836\pi$
$402$	0	0
$403$	3.88929	0.193739
$404$	0	0
$405$	−4.28632	−0.212989
$406$	0	0
$407$	−1.45673	−0.0722073
$408$	0	0
$409$	−6.19666	−0.306405	−0.153202	−	0.988195i	$-0.548959\pi$
$409$	−6.19666	−0.306405	−0.153202	+	0.988195i	$0.548959\pi$
$410$	0	0
$411$	9.30951	0.459204
$412$	0	0
$413$	16.5043	0.812125
$414$	0	0
$415$	−47.8345	−2.34810
$416$	0	0
$417$	−18.9397	−0.927483
$418$	0	0
$419$	28.8722	1.41050	0.705250	−	0.708959i	$-0.250835\pi$
$419$	28.8722	1.41050	0.705250	+	0.708959i	$0.250835\pi$
$420$	0	0
$421$	36.9336	1.80003	0.900016	−	0.435857i	$-0.143554\pi$
$421$	36.9336	1.80003	0.900016	+	0.435857i	$0.143554\pi$
$422$	0	0
$423$	13.0439	0.634215
$424$	0	0
$425$	−20.7371	−1.00590
$426$	0	0
$427$	−7.20332	−0.348593
$428$	0	0
$429$	6.72017	0.324453
$430$	0	0
$431$	−18.7749	−0.904354	−0.452177	−	0.891928i	$-0.649353\pi$
$431$	−18.7749	−0.904354	−0.452177	+	0.891928i	$0.649353\pi$
$432$	0	0
$433$	25.2249	1.21223	0.606116	−	0.795377i	$-0.292727\pi$
$433$	25.2249	1.21223	0.606116	+	0.795377i	$0.292727\pi$
$434$	0	0
$435$	4.28632	0.205514
$436$	0	0
$437$	−1.31263	−0.0627916
$438$	0	0
$439$	24.5690	1.17261	0.586306	−	0.810090i	$-0.300582\pi$
$439$	24.5690	1.17261	0.586306	+	0.810090i	$0.300582\pi$
$440$	0	0
$441$	−4.10597	−0.195522
$442$	0	0
$443$	−23.7431	−1.12807	−0.564034	−	0.825752i	$-0.690751\pi$
$443$	−23.7431	−1.12807	−0.564034	+	0.825752i	$0.690751\pi$
$444$	0	0
$445$	76.2847	3.61624
$446$	0	0
$447$	−0.200164	−0.00946745
$448$	0	0
$449$	27.8149	1.31266	0.656332	−	0.754472i	$-0.272107\pi$
$449$	27.8149	1.31266	0.656332	+	0.754472i	$0.272107\pi$
$450$	0	0
$451$	−34.7681	−1.63716
$452$	0	0
$453$	−16.1714	−0.759797
$454$	0	0
$455$	10.8545	0.508868
$456$	0	0
$457$	41.9187	1.96088	0.980438	−	0.196829i	$-0.0630644\pi$
$457$	41.9187	1.96088	0.980438	+	0.196829i	$0.0630644\pi$
$458$	0	0
$459$	1.55072	0.0723814
$460$	0	0
$461$	4.48238	0.208765	0.104383	−	0.994537i	$-0.466713\pi$
$461$	4.48238	0.208765	0.104383	+	0.994537i	$0.466713\pi$
$462$	0	0
$463$	21.4327	0.996064	0.498032	−	0.867159i	$-0.334056\pi$
$463$	21.4327	0.996064	0.498032	+	0.867159i	$0.334056\pi$
$464$	0	0
$465$	−11.1991	−0.519344
$466$	0	0
$467$	−23.7109	−1.09721	−0.548605	−	0.836081i	$-0.684841\pi$
$467$	−23.7109	−1.09721	−0.548605	+	0.836081i	$0.684841\pi$
$468$	0	0
$469$	−6.32286	−0.291962
$470$	0	0
$471$	6.72841	0.310029
$472$	0	0
$473$	21.2601	0.977543
$474$	0	0
$475$	−17.5532	−0.805397
$476$	0	0
$477$	7.24304	0.331636
$478$	0	0
$479$	−7.90312	−0.361103	−0.180551	−	0.983566i	$-0.557788\pi$
$479$	−7.90312	−0.361103	−0.180551	+	0.983566i	$0.557788\pi$
$480$	0	0
$481$	0.480337	0.0219015
$482$	0	0
$483$	−1.70119	−0.0774066
$484$	0	0
$485$	2.78629	0.126519
$486$	0	0
$487$	−40.5725	−1.83852	−0.919259	−	0.393654i	$-0.871211\pi$
$487$	−40.5725	−1.83852	−0.919259	+	0.393654i	$0.871211\pi$
$488$	0	0
$489$	2.55383	0.115488
$490$	0	0
$491$	32.5970	1.47108	0.735541	−	0.677480i	$-0.236928\pi$
$491$	32.5970	1.47108	0.735541	+	0.677480i	$0.236928\pi$
$492$	0	0
$493$	−1.55072	−0.0698409
$494$	0	0
$495$	−19.3504	−0.869738
$496$	0	0
$497$	20.3999	0.915061
$498$	0	0
$499$	16.3779	0.733177	0.366589	−	0.930383i	$-0.380526\pi$
$499$	16.3779	0.733177	0.366589	+	0.930383i	$0.380526\pi$
$500$	0	0
$501$	0.272414	0.0121706
$502$	0	0
$503$	−38.6217	−1.72206	−0.861028	−	0.508557i	$-0.830179\pi$
$503$	−38.6217	−1.72206	−0.861028	+	0.508557i	$0.830179\pi$
$504$	0	0
$505$	72.7810	3.23871
$506$	0	0
$507$	10.7841	0.478939
$508$	0	0
$509$	14.7904	0.655571	0.327786	−	0.944752i	$-0.393698\pi$
$509$	14.7904	0.655571	0.327786	+	0.944752i	$0.393698\pi$
$510$	0	0
$511$	−18.2585	−0.807711
$512$	0	0
$513$	1.31263	0.0579540
$514$	0	0
$515$	81.6119	3.59625
$516$	0	0
$517$	58.8860	2.58981
$518$	0	0
$519$	−17.0168	−0.746954
$520$	0	0
$521$	17.4435	0.764212	0.382106	−	0.924119i	$-0.375199\pi$
$521$	17.4435	0.764212	0.382106	+	0.924119i	$0.375199\pi$
$522$	0	0
$523$	2.98273	0.130426	0.0652129	−	0.997871i	$-0.479227\pi$
$523$	2.98273	0.130426	0.0652129	+	0.997871i	$0.479227\pi$
$524$	0	0
$525$	−22.7492	−0.992858
$526$	0	0
$527$	4.05163	0.176492
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	9.70167	0.421016
$532$	0	0
$533$	11.4643	0.496575
$534$	0	0
$535$	−44.3277	−1.91645
$536$	0	0
$537$	12.3164	0.531491
$538$	0	0
$539$	−18.5362	−0.798412
$540$	0	0
$541$	−3.89722	−0.167554	−0.0837772	−	0.996485i	$-0.526698\pi$
$541$	−3.89722	−0.167554	−0.0837772	+	0.996485i	$0.526698\pi$
$542$	0	0
$543$	4.16472	0.178725
$544$	0	0
$545$	23.5247	1.00769
$546$	0	0
$547$	−25.3710	−1.08479	−0.542393	−	0.840125i	$-0.682482\pi$
$547$	−25.3710	−1.08479	−0.542393	+	0.840125i	$0.682482\pi$
$548$	0	0
$549$	−4.23430	−0.180715
$550$	0	0
$551$	−1.31263	−0.0559199
$552$	0	0
$553$	−30.0141	−1.27633
$554$	0	0
$555$	−1.38311	−0.0587098
$556$	0	0
$557$	−19.5331	−0.827646	−0.413823	−	0.910357i	$-0.635807\pi$
$557$	−19.5331	−0.827646	−0.413823	+	0.910357i	$0.635807\pi$
$558$	0	0
$559$	−7.01027	−0.296503
$560$	0	0
$561$	7.00067	0.295568
$562$	0	0
$563$	27.8534	1.17388	0.586939	−	0.809631i	$-0.300333\pi$
$563$	27.8534	1.17388	0.586939	+	0.809631i	$0.300333\pi$
$564$	0	0
$565$	−22.5895	−0.950348
$566$	0	0
$567$	1.70119	0.0714431
$568$	0	0
$569$	18.4536	0.773617	0.386808	−	0.922160i	$-0.373578\pi$
$569$	18.4536	0.773617	0.386808	+	0.922160i	$0.373578\pi$
$570$	0	0
$571$	22.9586	0.960789	0.480394	−	0.877053i	$-0.340494\pi$
$571$	22.9586	0.960789	0.480394	+	0.877053i	$0.340494\pi$
$572$	0	0
$573$	18.8725	0.788411
$574$	0	0
$575$	13.3726	0.557675
$576$	0	0
$577$	−16.2346	−0.675854	−0.337927	−	0.941172i	$-0.609726\pi$
$577$	−16.2346	−0.675854	−0.337927	+	0.941172i	$0.609726\pi$
$578$	0	0
$579$	7.72554	0.321063
$580$	0	0
$581$	18.9849	0.787626
$582$	0	0
$583$	32.6984	1.35423
$584$	0	0
$585$	6.38057	0.263804
$586$	0	0
$587$	18.1911	0.750828	0.375414	−	0.926857i	$-0.377500\pi$
$587$	18.1911	0.750828	0.375414	+	0.926857i	$0.377500\pi$
$588$	0	0
$589$	3.42956	0.141313
$590$	0	0
$591$	−20.0387	−0.824284
$592$	0	0
$593$	−43.8123	−1.79915	−0.899577	−	0.436762i	$-0.856125\pi$
$593$	−43.8123	−1.79915	−0.899577	+	0.436762i	$0.856125\pi$
$594$	0	0
$595$	11.3076	0.463566
$596$	0	0
$597$	−26.1404	−1.06985
$598$	0	0
$599$	−30.0723	−1.22872	−0.614359	−	0.789026i	$-0.710586\pi$
$599$	−30.0723	−1.22872	−0.614359	+	0.789026i	$0.710586\pi$
$600$	0	0
$601$	−4.69148	−0.191370	−0.0956848	−	0.995412i	$-0.530504\pi$
$601$	−4.69148	−0.191370	−0.0956848	+	0.995412i	$0.530504\pi$
$602$	0	0
$603$	−3.71674	−0.151357
$604$	0	0
$605$	−40.2073	−1.63466
$606$	0	0
$607$	−2.04644	−0.0830626	−0.0415313	−	0.999137i	$-0.513224\pi$
$607$	−2.04644	−0.0830626	−0.0415313	+	0.999137i	$0.513224\pi$
$608$	0	0
$609$	−1.70119	−0.0689355
$610$	0	0
$611$	−19.4169	−0.785525
$612$	0	0
$613$	−24.0994	−0.973368	−0.486684	−	0.873578i	$-0.661794\pi$
$613$	−24.0994	−0.973368	−0.486684	+	0.873578i	$0.661794\pi$
$614$	0	0
$615$	−33.0111	−1.33113
$616$	0	0
$617$	−35.8902	−1.44488	−0.722442	−	0.691431i	$-0.756980\pi$
$617$	−35.8902	−1.44488	−0.722442	+	0.691431i	$0.756980\pi$
$618$	0	0
$619$	−20.8832	−0.839368	−0.419684	−	0.907670i	$-0.637859\pi$
$619$	−20.8832	−0.839368	−0.419684	+	0.907670i	$0.637859\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−30.2764	−1.21300
$624$	0	0
$625$	86.9628	3.47851
$626$	0	0
$627$	5.92581	0.236654
$628$	0	0
$629$	0.500387	0.0199517
$630$	0	0
$631$	−28.5474	−1.13645	−0.568226	−	0.822872i	$-0.692370\pi$
$631$	−28.5474	−1.13645	−0.568226	+	0.822872i	$0.692370\pi$
$632$	0	0
$633$	−5.36990	−0.213434
$634$	0	0
$635$	32.0021	1.26997
$636$	0	0
$637$	6.11209	0.242170
$638$	0	0
$639$	11.9916	0.474380
$640$	0	0
$641$	26.2128	1.03534	0.517672	−	0.855579i	$-0.326799\pi$
$641$	26.2128	1.03534	0.517672	+	0.855579i	$0.326799\pi$
$642$	0	0
$643$	−35.7568	−1.41011	−0.705055	−	0.709153i	$-0.749078\pi$
$643$	−35.7568	−1.41011	−0.705055	+	0.709153i	$0.749078\pi$
$644$	0	0
$645$	20.1858	0.794814
$646$	0	0
$647$	−26.4374	−1.03936	−0.519681	−	0.854360i	$-0.673949\pi$
$647$	−26.4374	−1.03936	−0.519681	+	0.854360i	$0.673949\pi$
$648$	0	0
$649$	43.7978	1.71921
$650$	0	0
$651$	4.44476	0.174204
$652$	0	0
$653$	−13.1035	−0.512780	−0.256390	−	0.966573i	$-0.582533\pi$
$653$	−13.1035	−0.512780	−0.256390	+	0.966573i	$0.582533\pi$
$654$	0	0
$655$	−48.4734	−1.89401
$656$	0	0
$657$	−10.7328	−0.418728
$658$	0	0
$659$	−8.19408	−0.319196	−0.159598	−	0.987182i	$-0.551020\pi$
$659$	−8.19408	−0.319196	−0.159598	+	0.987182i	$0.551020\pi$
$660$	0	0
$661$	−10.1152	−0.393434	−0.196717	−	0.980460i	$-0.563028\pi$
$661$	−10.1152	−0.393434	−0.196717	+	0.980460i	$0.563028\pi$
$662$	0	0
$663$	−2.30838	−0.0896501
$664$	0	0
$665$	9.57147	0.371166
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−21.2451	−0.821383
$670$	0	0
$671$	−19.1156	−0.737948
$672$	0	0
$673$	31.4242	1.21132	0.605658	−	0.795725i	$-0.292910\pi$
$673$	31.4242	1.21132	0.605658	+	0.795725i	$0.292910\pi$
$674$	0	0
$675$	−13.3726	−0.514711
$676$	0	0
$677$	−36.4259	−1.39996	−0.699980	−	0.714162i	$-0.746808\pi$
$677$	−36.4259	−1.39996	−0.699980	+	0.714162i	$0.746808\pi$
$678$	0	0
$679$	−1.10584	−0.0424384
$680$	0	0
$681$	−0.120251	−0.00460803
$682$	0	0
$683$	−29.6658	−1.13513	−0.567566	−	0.823328i	$-0.692115\pi$
$683$	−29.6658	−1.13513	−0.567566	+	0.823328i	$0.692115\pi$
$684$	0	0
$685$	39.9036	1.52464
$686$	0	0
$687$	−25.2154	−0.962029
$688$	0	0
$689$	−10.7819	−0.410758
$690$	0	0
$691$	31.1234	1.18399	0.591996	−	0.805941i	$-0.298340\pi$
$691$	31.1234	1.18399	0.591996	+	0.805941i	$0.298340\pi$
$692$	0	0
$693$	7.67994	0.291737
$694$	0	0
$695$	−81.1818	−3.07940
$696$	0	0
$697$	11.9429	0.452368
$698$	0	0
$699$	−15.5036	−0.586400
$700$	0	0
$701$	21.0718	0.795870	0.397935	−	0.917414i	$-0.369727\pi$
$701$	21.0718	0.795870	0.397935	+	0.917414i	$0.369727\pi$
$702$	0	0
$703$	0.423559	0.0159748
$704$	0	0
$705$	55.9102	2.10570
$706$	0	0
$707$	−28.8858	−1.08636
$708$	0	0
$709$	−4.39042	−0.164886	−0.0824429	−	0.996596i	$-0.526272\pi$
$709$	−4.39042	−0.164886	−0.0824429	+	0.996596i	$0.526272\pi$
$710$	0	0
$711$	−17.6430	−0.661665
$712$	0	0
$713$	−2.61274	−0.0978479
$714$	0	0
$715$	28.8048	1.07724
$716$	0	0
$717$	4.11436	0.153653
$718$	0	0
$719$	30.6811	1.14421	0.572106	−	0.820180i	$-0.306127\pi$
$719$	30.6811	1.14421	0.572106	+	0.820180i	$0.306127\pi$
$720$	0	0
$721$	−32.3907	−1.20629
$722$	0	0
$723$	−8.29590	−0.308528
$724$	0	0
$725$	13.3726	0.496645
$726$	0	0
$727$	−19.9250	−0.738976	−0.369488	−	0.929236i	$-0.620467\pi$
$727$	−19.9250	−0.738976	−0.369488	+	0.929236i	$0.620467\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.30287	−0.270106
$732$	0	0
$733$	−20.5753	−0.759965	−0.379982	−	0.924994i	$-0.624070\pi$
$733$	−20.5753	−0.759965	−0.379982	+	0.924994i	$0.624070\pi$
$734$	0	0
$735$	−17.5995	−0.649167
$736$	0	0
$737$	−16.7791	−0.618065
$738$	0	0
$739$	−19.8600	−0.730560	−0.365280	−	0.930898i	$-0.619027\pi$
$739$	−19.8600	−0.730560	−0.365280	+	0.930898i	$0.619027\pi$
$740$	0	0
$741$	−1.95396	−0.0717806
$742$	0	0
$743$	13.3852	0.491056	0.245528	−	0.969389i	$-0.421039\pi$
$743$	13.3852	0.491056	0.245528	+	0.969389i	$0.421039\pi$
$744$	0	0
$745$	−0.857970	−0.0314336
$746$	0	0
$747$	11.1598	0.408315
$748$	0	0
$749$	17.5931	0.642837
$750$	0	0
$751$	−26.0496	−0.950565	−0.475282	−	0.879833i	$-0.657654\pi$
$751$	−26.0496	−0.950565	−0.475282	+	0.879833i	$0.657654\pi$
$752$	0	0
$753$	−24.4531	−0.891121
$754$	0	0
$755$	−69.3157	−2.52266
$756$	0	0
$757$	−28.7077	−1.04340	−0.521700	−	0.853129i	$-0.674702\pi$
$757$	−28.7077	−1.04340	−0.521700	+	0.853129i	$0.674702\pi$
$758$	0	0
$759$	−4.51446	−0.163865
$760$	0	0
$761$	10.9034	0.395247	0.197624	−	0.980278i	$-0.436678\pi$
$761$	10.9034	0.395247	0.197624	+	0.980278i	$0.436678\pi$
$762$	0	0
$763$	−9.33665	−0.338009
$764$	0	0
$765$	6.64689	0.240319
$766$	0	0
$767$	−14.4418	−0.521462
$768$	0	0
$769$	−35.0411	−1.26361	−0.631807	−	0.775126i	$-0.717686\pi$
$769$	−35.0411	−1.26361	−0.631807	+	0.775126i	$0.717686\pi$
$770$	0	0
$771$	6.87202	0.247490
$772$	0	0
$773$	−3.19141	−0.114787	−0.0573936	−	0.998352i	$-0.518279\pi$
$773$	−3.19141	−0.114787	−0.0573936	+	0.998352i	$0.518279\pi$
$774$	0	0
$775$	−34.9391	−1.25505
$776$	0	0
$777$	0.548939	0.0196931
$778$	0	0
$779$	10.1092	0.362200
$780$	0	0
$781$	54.1356	1.93712
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	28.8402	1.02935
$786$	0	0
$787$	5.29740	0.188832	0.0944160	−	0.995533i	$-0.469902\pi$
$787$	5.29740	0.188832	0.0944160	+	0.995533i	$0.469902\pi$
$788$	0	0
$789$	−14.2428	−0.507056
$790$	0	0
$791$	8.96549	0.318776
$792$	0	0
$793$	6.30312	0.223830
$794$	0	0
$795$	31.0460	1.10109
$796$	0	0
$797$	10.2239	0.362150	0.181075	−	0.983469i	$-0.442042\pi$
$797$	10.2239	0.362150	0.181075	+	0.983469i	$0.442042\pi$
$798$	0	0
$799$	−20.2274	−0.715594
$800$	0	0
$801$	−17.7972	−0.628834
$802$	0	0
$803$	−48.4530	−1.70987
$804$	0	0
$805$	−7.29183	−0.257003
$806$	0	0
$807$	23.4638	0.825964
$808$	0	0
$809$	44.0830	1.54988	0.774939	−	0.632036i	$-0.217781\pi$
$809$	44.0830	1.54988	0.774939	+	0.632036i	$0.217781\pi$
$810$	0	0
$811$	29.5966	1.03928	0.519638	−	0.854387i	$-0.326067\pi$
$811$	29.5966	1.03928	0.519638	+	0.854387i	$0.326067\pi$
$812$	0	0
$813$	−21.3651	−0.749306
$814$	0	0
$815$	10.9466	0.383441
$816$	0	0
$817$	−6.18162	−0.216267
$818$	0	0
$819$	−2.53236	−0.0884879
$820$	0	0
$821$	0.314106	0.0109624	0.00548119	−	0.999985i	$-0.498255\pi$
$821$	0.314106	0.0109624	0.00548119	+	0.999985i	$0.498255\pi$
$822$	0	0
$823$	1.68347	0.0586819	0.0293410	−	0.999569i	$-0.490659\pi$
$823$	1.68347	0.0586819	0.0293410	+	0.999569i	$0.490659\pi$
$824$	0	0
$825$	−60.3700	−2.10181
$826$	0	0
$827$	−31.8277	−1.10676	−0.553379	−	0.832930i	$-0.686662\pi$
$827$	−31.8277	−1.10676	−0.553379	+	0.832930i	$0.686662\pi$
$828$	0	0
$829$	42.0120	1.45914	0.729568	−	0.683908i	$-0.239721\pi$
$829$	42.0120	1.45914	0.729568	+	0.683908i	$0.239721\pi$
$830$	0	0
$831$	20.2990	0.704164
$832$	0	0
$833$	6.36721	0.220611
$834$	0	0
$835$	1.16766	0.0404084
$836$	0	0
$837$	2.61274	0.0903095
$838$	0	0
$839$	30.5412	1.05440	0.527200	−	0.849742i	$-0.323242\pi$
$839$	30.5412	1.05440	0.527200	+	0.849742i	$0.323242\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−13.9991	−0.482155
$844$	0	0
$845$	46.2242	1.59016
$846$	0	0
$847$	15.9577	0.548315
$848$	0	0
$849$	−23.4813	−0.805876
$850$	0	0
$851$	−0.322680	−0.0110613
$852$	0	0
$853$	−10.3151	−0.353181	−0.176591	−	0.984284i	$-0.556507\pi$
$853$	−10.3151	−0.353181	−0.176591	+	0.984284i	$0.556507\pi$
$854$	0	0
$855$	5.62635	0.192417
$856$	0	0
$857$	28.7351	0.981571	0.490786	−	0.871280i	$-0.336710\pi$
$857$	28.7351	0.981571	0.490786	+	0.871280i	$0.336710\pi$
$858$	0	0
$859$	22.6931	0.774278	0.387139	−	0.922021i	$-0.373463\pi$
$859$	22.6931	0.774278	0.387139	+	0.922021i	$0.373463\pi$
$860$	0	0
$861$	13.1017	0.446503
$862$	0	0
$863$	26.0771	0.887676	0.443838	−	0.896107i	$-0.353617\pi$
$863$	26.0771	0.887676	0.443838	+	0.896107i	$0.353617\pi$
$864$	0	0
$865$	−72.9395	−2.48002
$866$	0	0
$867$	14.5953	0.495681
$868$	0	0
$869$	−79.6488	−2.70190
$870$	0	0
$871$	5.53268	0.187468
$872$	0	0
$873$	−0.650043	−0.0220006
$874$	0	0
$875$	−61.0514	−2.06391
$876$	0	0
$877$	−9.89634	−0.334176	−0.167088	−	0.985942i	$-0.553436\pi$
$877$	−9.89634	−0.334176	−0.167088	+	0.985942i	$0.553436\pi$
$878$	0	0
$879$	−6.87158	−0.231773
$880$	0	0
$881$	43.1270	1.45298	0.726492	−	0.687175i	$-0.241149\pi$
$881$	43.1270	1.45298	0.726492	+	0.687175i	$0.241149\pi$
$882$	0	0
$883$	−43.1136	−1.45089	−0.725445	−	0.688280i	$-0.758366\pi$
$883$	−43.1136	−1.45089	−0.725445	+	0.688280i	$0.758366\pi$
$884$	0	0
$885$	41.5845	1.39785
$886$	0	0
$887$	−20.2874	−0.681185	−0.340593	−	0.940211i	$-0.610628\pi$
$887$	−20.2874	−0.681185	−0.340593	+	0.940211i	$0.610628\pi$
$888$	0	0
$889$	−12.7012	−0.425985
$890$	0	0
$891$	4.51446	0.151240
$892$	0	0
$893$	−17.1218	−0.572958
$894$	0	0
$895$	52.7920	1.76464
$896$	0	0
$897$	1.48859	0.0497025
$898$	0	0
$899$	−2.61274	−0.0871398
$900$	0	0
$901$	−11.2319	−0.374190
$902$	0	0
$903$	−8.01147	−0.266605
$904$	0	0
$905$	17.8513	0.593398
$906$	0	0
$907$	3.56974	0.118531	0.0592656	−	0.998242i	$-0.481124\pi$
$907$	3.56974	0.118531	0.0592656	+	0.998242i	$0.481124\pi$
$908$	0	0
$909$	−16.9798	−0.563185
$910$	0	0
$911$	16.8315	0.557651	0.278826	−	0.960342i	$-0.410055\pi$
$911$	16.8315	0.557651	0.278826	+	0.960342i	$0.410055\pi$
$912$	0	0
$913$	50.3805	1.66735
$914$	0	0
$915$	−18.1496	−0.600006
$916$	0	0
$917$	19.2385	0.635310
$918$	0	0
$919$	7.69555	0.253853	0.126926	−	0.991912i	$-0.459489\pi$
$919$	7.69555	0.253853	0.126926	+	0.991912i	$0.459489\pi$
$920$	0	0
$921$	−5.69689	−0.187719
$922$	0	0
$923$	−17.8505	−0.587557
$924$	0	0
$925$	−4.31506	−0.141878
$926$	0	0
$927$	−19.0401	−0.625358
$928$	0	0
$929$	35.4882	1.16433	0.582165	−	0.813071i	$-0.302206\pi$
$929$	35.4882	1.16433	0.582165	+	0.813071i	$0.302206\pi$
$930$	0	0
$931$	5.38961	0.176637
$932$	0	0
$933$	8.45195	0.276704
$934$	0	0
$935$	30.0071	0.981338
$936$	0	0
$937$	31.7653	1.03773	0.518864	−	0.854857i	$-0.326355\pi$
$937$	31.7653	1.03773	0.518864	+	0.854857i	$0.326355\pi$
$938$	0	0
$939$	29.2225	0.953640
$940$	0	0
$941$	48.4539	1.57955	0.789777	−	0.613394i	$-0.210196\pi$
$941$	48.4539	1.57955	0.789777	+	0.613394i	$0.210196\pi$
$942$	0	0
$943$	−7.70149	−0.250795
$944$	0	0
$945$	7.29183	0.237203
$946$	0	0
$947$	40.5567	1.31792	0.658958	−	0.752180i	$-0.270997\pi$
$947$	40.5567	1.31792	0.658958	+	0.752180i	$0.270997\pi$
$948$	0	0
$949$	15.9768	0.518628
$950$	0	0
$951$	3.03917	0.0985520
$952$	0	0
$953$	−18.7158	−0.606263	−0.303131	−	0.952949i	$-0.598032\pi$
$953$	−18.7158	−0.606263	−0.303131	+	0.952949i	$0.598032\pi$
$954$	0	0
$955$	80.8937	2.61766
$956$	0	0
$957$	−4.51446	−0.145932
$958$	0	0
$959$	−15.8372	−0.511410
$960$	0	0
$961$	−24.1736	−0.779793
$962$	0	0
$963$	10.3417	0.333255
$964$	0	0
$965$	33.1142	1.06598
$966$	0	0
$967$	−22.1539	−0.712421	−0.356210	−	0.934406i	$-0.615931\pi$
$967$	−22.1539	−0.712421	−0.356210	+	0.934406i	$0.615931\pi$
$968$	0	0
$969$	−2.03552	−0.0653903
$970$	0	0
$971$	−12.2222	−0.392228	−0.196114	−	0.980581i	$-0.562832\pi$
$971$	−12.2222	−0.392228	−0.196114	+	0.980581i	$0.562832\pi$
$972$	0	0
$973$	32.2200	1.03293
$974$	0	0
$975$	19.9062	0.637510
$976$	0	0
$977$	−29.0812	−0.930391	−0.465196	−	0.885208i	$-0.654016\pi$
$977$	−29.0812	−0.930391	−0.465196	+	0.885208i	$0.654016\pi$
$978$	0	0
$979$	−80.3449	−2.56783
$980$	0	0
$981$	−5.48832	−0.175228
$982$	0	0
$983$	39.2793	1.25281	0.626407	−	0.779496i	$-0.284525\pi$
$983$	39.2793	1.25281	0.626407	+	0.779496i	$0.284525\pi$
$984$	0	0
$985$	−85.8926	−2.73676
$986$	0	0
$987$	−22.1900	−0.706317
$988$	0	0
$989$	4.70934	0.149748
$990$	0	0
$991$	23.4921	0.746250	0.373125	−	0.927781i	$-0.378286\pi$
$991$	23.4921	0.746250	0.373125	+	0.927781i	$0.378286\pi$
$992$	0	0
$993$	−33.0230	−1.04795
$994$	0	0
$995$	−112.046	−3.55210
$996$	0	0
$997$	46.0294	1.45777	0.728883	−	0.684638i	$-0.240040\pi$
$997$	46.0294	1.45777	0.728883	+	0.684638i	$0.240040\pi$
$998$	0	0
$999$	0.322680	0.0102091

Twists

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

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

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.28632 q^{5} +1.70119 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.28632 q^{5} +1.70119 q^{7} +1.00000 q^{9} +4.51446 q^{11} -1.48859 q^{13} +4.28632 q^{15} -1.55072 q^{17} -1.31263 q^{19} -1.70119 q^{21} +1.00000 q^{23} +13.3726 q^{25} -1.00000 q^{27} +1.00000 q^{29} -2.61274 q^{31} -4.51446 q^{33} -7.29183 q^{35} -0.322680 q^{37} +1.48859 q^{39} -7.70149 q^{41} +4.70934 q^{43} -4.28632 q^{45} +13.0439 q^{47} -4.10597 q^{49} +1.55072 q^{51} +7.24304 q^{53} -19.3504 q^{55} +1.31263 q^{57} +9.70167 q^{59} -4.23430 q^{61} +1.70119 q^{63} +6.38057 q^{65} -3.71674 q^{67} -1.00000 q^{69} +11.9916 q^{71} -10.7328 q^{73} -13.3726 q^{75} +7.67994 q^{77} -17.6430 q^{79} +1.00000 q^{81} +11.1598 q^{83} +6.64689 q^{85} -1.00000 q^{87} -17.7972 q^{89} -2.53236 q^{91} +2.61274 q^{93} +5.62635 q^{95} -0.650043 q^{97} +4.51446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 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

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