LMFDB - Embedded newform 8004.2.a.f.1.4

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 17x^{7} + 4x^{6} + 75x^{5} + x^{4} - 118x^{3} - 26x^{2} + 60x + 24 $ x^9 - x^8 - 17x^7 + 4x^6 + 75x^5 + x^4 - 118x^3 - 26x^2 + 60x + 24
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			1.4
Root			$1.05365$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.05365 q^{5} +3.84396 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.05365 q^{5} +3.84396 q^{7} +1.00000 q^{9} -2.99831 q^{11} -4.11036 q^{13} -1.05365 q^{15} +3.37201 q^{17} -3.86805 q^{19} +3.84396 q^{21} -1.00000 q^{23} -3.88983 q^{25} +1.00000 q^{27} -1.00000 q^{29} +1.19131 q^{31} -2.99831 q^{33} -4.05018 q^{35} +2.51215 q^{37} -4.11036 q^{39} +3.71037 q^{41} -5.77433 q^{43} -1.05365 q^{45} +6.95307 q^{47} +7.77604 q^{49} +3.37201 q^{51} -7.64083 q^{53} +3.15916 q^{55} -3.86805 q^{57} +8.37197 q^{59} -9.45949 q^{61} +3.84396 q^{63} +4.33087 q^{65} -5.84419 q^{67} -1.00000 q^{69} -13.2308 q^{71} -8.75519 q^{73} -3.88983 q^{75} -11.5254 q^{77} +11.5676 q^{79} +1.00000 q^{81} -11.3438 q^{83} -3.55290 q^{85} -1.00000 q^{87} -4.62287 q^{89} -15.8001 q^{91} +1.19131 q^{93} +4.07556 q^{95} +5.25662 q^{97} -2.99831 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41} - 19 q^{43} - q^{45} - 3 q^{47} - 6 q^{49} - 2 q^{51} - 9 q^{53} - 7 q^{55} - 11 q^{57} - 2 q^{59} - 25 q^{61} - 5 q^{63} - 12 q^{65} - 20 q^{67} - 9 q^{69} + 9 q^{71} - 11 q^{73} - 10 q^{75} - 19 q^{77} + 4 q^{79} + 9 q^{81} - 9 q^{83} - 50 q^{85} - 9 q^{87} - 29 q^{89} - 38 q^{91} + 23 q^{95} - 43 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * 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
$5$	−1.05365	−0.471205	−0.235603	−	0.971849i	$-0.575706\pi$
$5$	−1.05365	−0.471205	−0.235603	+	0.971849i	$0.575706\pi$
$6$	0	0
$7$	3.84396	1.45288	0.726440	−	0.687229i	$-0.241173\pi$
$7$	3.84396	1.45288	0.726440	+	0.687229i	$0.241173\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.99831	−0.904025	−0.452012	−	0.892012i	$-0.649294\pi$
$11$	−2.99831	−0.904025	−0.452012	+	0.892012i	$0.649294\pi$
$12$	0	0
$13$	−4.11036	−1.14001	−0.570005	−	0.821641i	$-0.693059\pi$
$13$	−4.11036	−1.14001	−0.570005	+	0.821641i	$0.693059\pi$
$14$	0	0
$15$	−1.05365	−0.272050
$16$	0	0
$17$	3.37201	0.817832	0.408916	−	0.912572i	$-0.365907\pi$
$17$	3.37201	0.817832	0.408916	+	0.912572i	$0.365907\pi$
$18$	0	0
$19$	−3.86805	−0.887392	−0.443696	−	0.896177i	$-0.646333\pi$
$19$	−3.86805	−0.887392	−0.443696	+	0.896177i	$0.646333\pi$
$20$	0	0
$21$	3.84396	0.838821
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.88983	−0.777966
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.19131	0.213966	0.106983	−	0.994261i	$-0.465881\pi$
$31$	1.19131	0.213966	0.106983	+	0.994261i	$0.465881\pi$
$32$	0	0
$33$	−2.99831	−0.521939
$34$	0	0
$35$	−4.05018	−0.684605
$36$	0	0
$37$	2.51215	0.412995	0.206497	−	0.978447i	$-0.433794\pi$
$37$	2.51215	0.412995	0.206497	+	0.978447i	$0.433794\pi$
$38$	0	0
$39$	−4.11036	−0.658185
$40$	0	0
$41$	3.71037	0.579463	0.289732	−	0.957108i	$-0.406434\pi$
$41$	3.71037	0.579463	0.289732	+	0.957108i	$0.406434\pi$
$42$	0	0
$43$	−5.77433	−0.880578	−0.440289	−	0.897856i	$-0.645124\pi$
$43$	−5.77433	−0.880578	−0.440289	+	0.897856i	$0.645124\pi$
$44$	0	0
$45$	−1.05365	−0.157068
$46$	0	0
$47$	6.95307	1.01421	0.507105	−	0.861884i	$-0.330716\pi$
$47$	6.95307	1.01421	0.507105	+	0.861884i	$0.330716\pi$
$48$	0	0
$49$	7.77604	1.11086
$50$	0	0
$51$	3.37201	0.472175
$52$	0	0
$53$	−7.64083	−1.04955	−0.524774	−	0.851241i	$-0.675850\pi$
$53$	−7.64083	−1.04955	−0.524774	+	0.851241i	$0.675850\pi$
$54$	0	0
$55$	3.15916	0.425981
$56$	0	0
$57$	−3.86805	−0.512336
$58$	0	0
$59$	8.37197	1.08994	0.544969	−	0.838456i	$-0.316541\pi$
$59$	8.37197	1.08994	0.544969	+	0.838456i	$0.316541\pi$
$60$	0	0
$61$	−9.45949	−1.21116	−0.605582	−	0.795783i	$-0.707059\pi$
$61$	−9.45949	−1.21116	−0.605582	+	0.795783i	$0.707059\pi$
$62$	0	0
$63$	3.84396	0.484294
$64$	0	0
$65$	4.33087	0.537179
$66$	0	0
$67$	−5.84419	−0.713981	−0.356991	−	0.934108i	$-0.616197\pi$
$67$	−5.84419	−0.713981	−0.356991	+	0.934108i	$0.616197\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−13.2308	−1.57021	−0.785106	−	0.619361i	$-0.787392\pi$
$71$	−13.2308	−1.57021	−0.785106	+	0.619361i	$0.787392\pi$
$72$	0	0
$73$	−8.75519	−1.02472	−0.512359	−	0.858771i	$-0.671228\pi$
$73$	−8.75519	−1.02472	−0.512359	+	0.858771i	$0.671228\pi$
$74$	0	0
$75$	−3.88983	−0.449159
$76$	0	0
$77$	−11.5254	−1.31344
$78$	0	0
$79$	11.5676	1.30146	0.650728	−	0.759311i	$-0.274464\pi$
$79$	11.5676	1.30146	0.650728	+	0.759311i	$0.274464\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.3438	−1.24514	−0.622571	−	0.782563i	$-0.713912\pi$
$83$	−11.3438	−1.24514	−0.622571	+	0.782563i	$0.713912\pi$
$84$	0	0
$85$	−3.55290	−0.385367
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−4.62287	−0.490023	−0.245012	−	0.969520i	$-0.578792\pi$
$89$	−4.62287	−0.490023	−0.245012	+	0.969520i	$0.578792\pi$
$90$	0	0
$91$	−15.8001	−1.65630
$92$	0	0
$93$	1.19131	0.123534
$94$	0	0
$95$	4.07556	0.418143
$96$	0	0
$97$	5.25662	0.533729	0.266864	−	0.963734i	$-0.414012\pi$
$97$	5.25662	0.533729	0.266864	+	0.963734i	$0.414012\pi$
$98$	0	0
$99$	−2.99831	−0.301342
$100$	0	0
$101$	−6.12186	−0.609148	−0.304574	−	0.952489i	$-0.598514\pi$
$101$	−6.12186	−0.609148	−0.304574	+	0.952489i	$0.598514\pi$
$102$	0	0
$103$	−6.81197	−0.671203	−0.335601	−	0.942004i	$-0.608940\pi$
$103$	−6.81197	−0.671203	−0.335601	+	0.942004i	$0.608940\pi$
$104$	0	0
$105$	−4.05018	−0.395257
$106$	0	0
$107$	−10.0631	−0.972833	−0.486416	−	0.873727i	$-0.661696\pi$
$107$	−10.0631	−0.972833	−0.486416	+	0.873727i	$0.661696\pi$
$108$	0	0
$109$	−15.3271	−1.46807	−0.734037	−	0.679109i	$-0.762366\pi$
$109$	−15.3271	−1.46807	−0.734037	+	0.679109i	$0.762366\pi$
$110$	0	0
$111$	2.51215	0.238443
$112$	0	0
$113$	14.1970	1.33554	0.667769	−	0.744369i	$-0.267250\pi$
$113$	14.1970	1.33554	0.667769	+	0.744369i	$0.267250\pi$
$114$	0	0
$115$	1.05365	0.0982531
$116$	0	0
$117$	−4.11036	−0.380003
$118$	0	0
$119$	12.9619	1.18821
$120$	0	0
$121$	−2.01013	−0.182739
$122$	0	0
$123$	3.71037	0.334553
$124$	0	0
$125$	9.36674	0.837787
$126$	0	0
$127$	2.53496	0.224941	0.112471	−	0.993655i	$-0.464124\pi$
$127$	2.53496	0.224941	0.112471	+	0.993655i	$0.464124\pi$
$128$	0	0
$129$	−5.77433	−0.508402
$130$	0	0
$131$	−1.70481	−0.148950	−0.0744748	−	0.997223i	$-0.523728\pi$
$131$	−1.70481	−0.148950	−0.0744748	+	0.997223i	$0.523728\pi$
$132$	0	0
$133$	−14.8686	−1.28927
$134$	0	0
$135$	−1.05365	−0.0906835
$136$	0	0
$137$	6.05269	0.517116	0.258558	−	0.965996i	$-0.416753\pi$
$137$	6.05269	0.517116	0.258558	+	0.965996i	$0.416753\pi$
$138$	0	0
$139$	1.50865	0.127962	0.0639811	−	0.997951i	$-0.479620\pi$
$139$	1.50865	0.127962	0.0639811	+	0.997951i	$0.479620\pi$
$140$	0	0
$141$	6.95307	0.585554
$142$	0	0
$143$	12.3241	1.03060
$144$	0	0
$145$	1.05365	0.0875006
$146$	0	0
$147$	7.77604	0.641357
$148$	0	0
$149$	−8.73792	−0.715838	−0.357919	−	0.933753i	$-0.616514\pi$
$149$	−8.73792	−0.715838	−0.357919	+	0.933753i	$0.616514\pi$
$150$	0	0
$151$	−10.9690	−0.892644	−0.446322	−	0.894872i	$-0.647266\pi$
$151$	−10.9690	−0.892644	−0.446322	+	0.894872i	$0.647266\pi$
$152$	0	0
$153$	3.37201	0.272611
$154$	0	0
$155$	−1.25522	−0.100822
$156$	0	0
$157$	2.48510	0.198332	0.0991662	−	0.995071i	$-0.468382\pi$
$157$	2.48510	0.198332	0.0991662	+	0.995071i	$0.468382\pi$
$158$	0	0
$159$	−7.64083	−0.605957
$160$	0	0
$161$	−3.84396	−0.302947
$162$	0	0
$163$	6.86595	0.537783	0.268891	−	0.963171i	$-0.413343\pi$
$163$	6.86595	0.537783	0.268891	+	0.963171i	$0.413343\pi$
$164$	0	0
$165$	3.15916	0.245940
$166$	0	0
$167$	−21.9485	−1.69842	−0.849211	−	0.528053i	$-0.822922\pi$
$167$	−21.9485	−1.69842	−0.849211	+	0.528053i	$0.822922\pi$
$168$	0	0
$169$	3.89510	0.299623
$170$	0	0
$171$	−3.86805	−0.295797
$172$	0	0
$173$	15.2665	1.16069	0.580345	−	0.814370i	$-0.302918\pi$
$173$	15.2665	1.16069	0.580345	+	0.814370i	$0.302918\pi$
$174$	0	0
$175$	−14.9524	−1.13029
$176$	0	0
$177$	8.37197	0.629276
$178$	0	0
$179$	−4.39056	−0.328166	−0.164083	−	0.986447i	$-0.552467\pi$
$179$	−4.39056	−0.328166	−0.164083	+	0.986447i	$0.552467\pi$
$180$	0	0
$181$	−5.77641	−0.429357	−0.214679	−	0.976685i	$-0.568870\pi$
$181$	−5.77641	−0.429357	−0.214679	+	0.976685i	$0.568870\pi$
$182$	0	0
$183$	−9.45949	−0.699265
$184$	0	0
$185$	−2.64692	−0.194605
$186$	0	0
$187$	−10.1103	−0.739340
$188$	0	0
$189$	3.84396	0.279607
$190$	0	0
$191$	14.9901	1.08465	0.542323	−	0.840170i	$-0.317545\pi$
$191$	14.9901	1.08465	0.542323	+	0.840170i	$0.317545\pi$
$192$	0	0
$193$	−19.3122	−1.39012	−0.695062	−	0.718950i	$-0.744623\pi$
$193$	−19.3122	−1.39012	−0.695062	+	0.718950i	$0.744623\pi$
$194$	0	0
$195$	4.33087	0.310140
$196$	0	0
$197$	20.3886	1.45263	0.726314	−	0.687363i	$-0.241232\pi$
$197$	20.3886	1.45263	0.726314	+	0.687363i	$0.241232\pi$
$198$	0	0
$199$	−8.14767	−0.577573	−0.288786	−	0.957394i	$-0.593252\pi$
$199$	−8.14767	−0.577573	−0.288786	+	0.957394i	$0.593252\pi$
$200$	0	0
$201$	−5.84419	−0.412217
$202$	0	0
$203$	−3.84396	−0.269793
$204$	0	0
$205$	−3.90942	−0.273046
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	11.5976	0.802224
$210$	0	0
$211$	−8.44276	−0.581224	−0.290612	−	0.956841i	$-0.593859\pi$
$211$	−8.44276	−0.581224	−0.290612	+	0.956841i	$0.593859\pi$
$212$	0	0
$213$	−13.2308	−0.906563
$214$	0	0
$215$	6.08411	0.414933
$216$	0	0
$217$	4.57937	0.310868
$218$	0	0
$219$	−8.75519	−0.591621
$220$	0	0
$221$	−13.8602	−0.932336
$222$	0	0
$223$	−7.74336	−0.518534	−0.259267	−	0.965806i	$-0.583481\pi$
$223$	−7.74336	−0.518534	−0.259267	+	0.965806i	$0.583481\pi$
$224$	0	0
$225$	−3.88983	−0.259322
$226$	0	0
$227$	5.22412	0.346737	0.173369	−	0.984857i	$-0.444535\pi$
$227$	5.22412	0.346737	0.173369	+	0.984857i	$0.444535\pi$
$228$	0	0
$229$	−24.2214	−1.60059	−0.800296	−	0.599605i	$-0.795324\pi$
$229$	−24.2214	−1.60059	−0.800296	+	0.599605i	$0.795324\pi$
$230$	0	0
$231$	−11.5254	−0.758315
$232$	0	0
$233$	11.9952	0.785832	0.392916	−	0.919574i	$-0.371466\pi$
$233$	11.9952	0.785832	0.392916	+	0.919574i	$0.371466\pi$
$234$	0	0
$235$	−7.32608	−0.477901
$236$	0	0
$237$	11.5676	0.751396
$238$	0	0
$239$	10.3057	0.666618	0.333309	−	0.942818i	$-0.391835\pi$
$239$	10.3057	0.666618	0.333309	+	0.942818i	$0.391835\pi$
$240$	0	0
$241$	24.0342	1.54818	0.774088	−	0.633078i	$-0.218209\pi$
$241$	24.0342	1.54818	0.774088	+	0.633078i	$0.218209\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−8.19320	−0.523444
$246$	0	0
$247$	15.8991	1.01164
$248$	0	0
$249$	−11.3438	−0.718884
$250$	0	0
$251$	−30.6106	−1.93212	−0.966062	−	0.258309i	$-0.916835\pi$
$251$	−30.6106	−1.93212	−0.966062	+	0.258309i	$0.916835\pi$
$252$	0	0
$253$	2.99831	0.188502
$254$	0	0
$255$	−3.55290	−0.222491
$256$	0	0
$257$	−11.2912	−0.704328	−0.352164	−	0.935938i	$-0.614554\pi$
$257$	−11.2912	−0.704328	−0.352164	+	0.935938i	$0.614554\pi$
$258$	0	0
$259$	9.65660	0.600032
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	2.27348	0.140189	0.0700944	−	0.997540i	$-0.477670\pi$
$263$	2.27348	0.140189	0.0700944	+	0.997540i	$0.477670\pi$
$264$	0	0
$265$	8.05074	0.494553
$266$	0	0
$267$	−4.62287	−0.282915
$268$	0	0
$269$	19.7673	1.20524	0.602618	−	0.798030i	$-0.294124\pi$
$269$	19.7673	1.20524	0.602618	+	0.798030i	$0.294124\pi$
$270$	0	0
$271$	−7.93328	−0.481912	−0.240956	−	0.970536i	$-0.577461\pi$
$271$	−7.93328	−0.481912	−0.240956	+	0.970536i	$0.577461\pi$
$272$	0	0
$273$	−15.8001	−0.956264
$274$	0	0
$275$	11.6629	0.703300
$276$	0	0
$277$	−0.212742	−0.0127824	−0.00639122	−	0.999980i	$-0.502034\pi$
$277$	−0.212742	−0.0127824	−0.00639122	+	0.999980i	$0.502034\pi$
$278$	0	0
$279$	1.19131	0.0713221
$280$	0	0
$281$	22.1414	1.32085	0.660424	−	0.750893i	$-0.270377\pi$
$281$	22.1414	1.32085	0.660424	+	0.750893i	$0.270377\pi$
$282$	0	0
$283$	13.9321	0.828175	0.414087	−	0.910237i	$-0.364101\pi$
$283$	13.9321	0.828175	0.414087	+	0.910237i	$0.364101\pi$
$284$	0	0
$285$	4.07556	0.241415
$286$	0	0
$287$	14.2625	0.841891
$288$	0	0
$289$	−5.62957	−0.331151
$290$	0	0
$291$	5.25662	0.308148
$292$	0	0
$293$	11.2436	0.656856	0.328428	−	0.944529i	$-0.393481\pi$
$293$	11.2436	0.656856	0.328428	+	0.944529i	$0.393481\pi$
$294$	0	0
$295$	−8.82110	−0.513584
$296$	0	0
$297$	−2.99831	−0.173980
$298$	0	0
$299$	4.11036	0.237709
$300$	0	0
$301$	−22.1963	−1.27937
$302$	0	0
$303$	−6.12186	−0.351692
$304$	0	0
$305$	9.96696	0.570706
$306$	0	0
$307$	−7.87090	−0.449216	−0.224608	−	0.974449i	$-0.572110\pi$
$307$	−7.87090	−0.449216	−0.224608	+	0.974449i	$0.572110\pi$
$308$	0	0
$309$	−6.81197	−0.387519
$310$	0	0
$311$	−18.6227	−1.05600	−0.527999	−	0.849245i	$-0.677057\pi$
$311$	−18.6227	−1.05600	−0.527999	+	0.849245i	$0.677057\pi$
$312$	0	0
$313$	−28.2165	−1.59489	−0.797445	−	0.603391i	$-0.793816\pi$
$313$	−28.2165	−1.59489	−0.797445	+	0.603391i	$0.793816\pi$
$314$	0	0
$315$	−4.05018	−0.228202
$316$	0	0
$317$	11.5300	0.647589	0.323794	−	0.946127i	$-0.395041\pi$
$317$	11.5300	0.647589	0.323794	+	0.946127i	$0.395041\pi$
$318$	0	0
$319$	2.99831	0.167873
$320$	0	0
$321$	−10.0631	−0.561665
$322$	0	0
$323$	−13.0431	−0.725737
$324$	0	0
$325$	15.9886	0.886889
$326$	0	0
$327$	−15.3271	−0.847593
$328$	0	0
$329$	26.7273	1.47353
$330$	0	0
$331$	−15.8270	−0.869933	−0.434966	−	0.900447i	$-0.643240\pi$
$331$	−15.8270	−0.869933	−0.434966	+	0.900447i	$0.643240\pi$
$332$	0	0
$333$	2.51215	0.137665
$334$	0	0
$335$	6.15771	0.336432
$336$	0	0
$337$	35.0902	1.91149	0.955744	−	0.294201i	$-0.0950534\pi$
$337$	35.0902	1.91149	0.955744	+	0.294201i	$0.0950534\pi$
$338$	0	0
$339$	14.1970	0.771073
$340$	0	0
$341$	−3.57193	−0.193431
$342$	0	0
$343$	2.98307	0.161070
$344$	0	0
$345$	1.05365	0.0567264
$346$	0	0
$347$	7.04063	0.377961	0.188980	−	0.981981i	$-0.439482\pi$
$347$	7.04063	0.377961	0.188980	+	0.981981i	$0.439482\pi$
$348$	0	0
$349$	25.9595	1.38958	0.694790	−	0.719213i	$-0.255497\pi$
$349$	25.9595	1.38958	0.694790	+	0.719213i	$0.255497\pi$
$350$	0	0
$351$	−4.11036	−0.219395
$352$	0	0
$353$	0.603506	0.0321214	0.0160607	−	0.999871i	$-0.494888\pi$
$353$	0.603506	0.0321214	0.0160607	+	0.999871i	$0.494888\pi$
$354$	0	0
$355$	13.9406	0.739892
$356$	0	0
$357$	12.9619	0.686015
$358$	0	0
$359$	25.3454	1.33768	0.668840	−	0.743407i	$-0.266791\pi$
$359$	25.3454	1.33768	0.668840	+	0.743407i	$0.266791\pi$
$360$	0	0
$361$	−4.03819	−0.212536
$362$	0	0
$363$	−2.01013	−0.105505
$364$	0	0
$365$	9.22488	0.482852
$366$	0	0
$367$	−12.9327	−0.675082	−0.337541	−	0.941311i	$-0.609595\pi$
$367$	−12.9327	−0.675082	−0.337541	+	0.941311i	$0.609595\pi$
$368$	0	0
$369$	3.71037	0.193154
$370$	0	0
$371$	−29.3711	−1.52487
$372$	0	0
$373$	−19.3620	−1.00253	−0.501263	−	0.865295i	$-0.667131\pi$
$373$	−19.3620	−1.00253	−0.501263	+	0.865295i	$0.667131\pi$
$374$	0	0
$375$	9.36674	0.483696
$376$	0	0
$377$	4.11036	0.211695
$378$	0	0
$379$	19.4039	0.996709	0.498355	−	0.866973i	$-0.333938\pi$
$379$	19.4039	0.996709	0.498355	+	0.866973i	$0.333938\pi$
$380$	0	0
$381$	2.53496	0.129870
$382$	0	0
$383$	−14.4367	−0.737679	−0.368839	−	0.929493i	$-0.620245\pi$
$383$	−14.4367	−0.737679	−0.368839	+	0.929493i	$0.620245\pi$
$384$	0	0
$385$	12.1437	0.618900
$386$	0	0
$387$	−5.77433	−0.293526
$388$	0	0
$389$	−3.47564	−0.176222	−0.0881109	−	0.996111i	$-0.528083\pi$
$389$	−3.47564	−0.176222	−0.0881109	+	0.996111i	$0.528083\pi$
$390$	0	0
$391$	−3.37201	−0.170530
$392$	0	0
$393$	−1.70481	−0.0859961
$394$	0	0
$395$	−12.1882	−0.613253
$396$	0	0
$397$	−10.5852	−0.531258	−0.265629	−	0.964075i	$-0.585580\pi$
$397$	−10.5852	−0.531258	−0.265629	+	0.964075i	$0.585580\pi$
$398$	0	0
$399$	−14.8686	−0.744363
$400$	0	0
$401$	−4.26171	−0.212820	−0.106410	−	0.994322i	$-0.533936\pi$
$401$	−4.26171	−0.212820	−0.106410	+	0.994322i	$0.533936\pi$
$402$	0	0
$403$	−4.89673	−0.243924
$404$	0	0
$405$	−1.05365	−0.0523561
$406$	0	0
$407$	−7.53220	−0.373357
$408$	0	0
$409$	−23.7368	−1.17371	−0.586854	−	0.809693i	$-0.699634\pi$
$409$	−23.7368	−1.17371	−0.586854	+	0.809693i	$0.699634\pi$
$410$	0	0
$411$	6.05269	0.298557
$412$	0	0
$413$	32.1815	1.58355
$414$	0	0
$415$	11.9523	0.586718
$416$	0	0
$417$	1.50865	0.0738790
$418$	0	0
$419$	−8.02309	−0.391953	−0.195977	−	0.980609i	$-0.562788\pi$
$419$	−8.02309	−0.391953	−0.195977	+	0.980609i	$0.562788\pi$
$420$	0	0
$421$	−21.5402	−1.04981	−0.524904	−	0.851162i	$-0.675899\pi$
$421$	−21.5402	−1.04981	−0.524904	+	0.851162i	$0.675899\pi$
$422$	0	0
$423$	6.95307	0.338070
$424$	0	0
$425$	−13.1165	−0.636245
$426$	0	0
$427$	−36.3619	−1.75968
$428$	0	0
$429$	12.3241	0.595016
$430$	0	0
$431$	12.3878	0.596699	0.298349	−	0.954457i	$-0.403564\pi$
$431$	12.3878	0.596699	0.298349	+	0.954457i	$0.403564\pi$
$432$	0	0
$433$	−31.9196	−1.53396	−0.766979	−	0.641672i	$-0.778241\pi$
$433$	−31.9196	−1.53396	−0.766979	+	0.641672i	$0.778241\pi$
$434$	0	0
$435$	1.05365	0.0505185
$436$	0	0
$437$	3.86805	0.185034
$438$	0	0
$439$	2.95020	0.140805	0.0704027	−	0.997519i	$-0.477572\pi$
$439$	2.95020	0.140805	0.0704027	+	0.997519i	$0.477572\pi$
$440$	0	0
$441$	7.77604	0.370288
$442$	0	0
$443$	−1.78186	−0.0846587	−0.0423294	−	0.999104i	$-0.513478\pi$
$443$	−1.78186	−0.0846587	−0.0423294	+	0.999104i	$0.513478\pi$
$444$	0	0
$445$	4.87087	0.230901
$446$	0	0
$447$	−8.73792	−0.413289
$448$	0	0
$449$	13.0594	0.616309	0.308154	−	0.951336i	$-0.400289\pi$
$449$	13.0594	0.616309	0.308154	+	0.951336i	$0.400289\pi$
$450$	0	0
$451$	−11.1249	−0.523849
$452$	0	0
$453$	−10.9690	−0.515368
$454$	0	0
$455$	16.6477	0.780456
$456$	0	0
$457$	−34.6605	−1.62135	−0.810676	−	0.585495i	$-0.800900\pi$
$457$	−34.6605	−1.62135	−0.810676	+	0.585495i	$0.800900\pi$
$458$	0	0
$459$	3.37201	0.157392
$460$	0	0
$461$	−3.90563	−0.181903	−0.0909517	−	0.995855i	$-0.528991\pi$
$461$	−3.90563	−0.181903	−0.0909517	+	0.995855i	$0.528991\pi$
$462$	0	0
$463$	−1.90167	−0.0883782	−0.0441891	−	0.999023i	$-0.514070\pi$
$463$	−1.90167	−0.0883782	−0.0441891	+	0.999023i	$0.514070\pi$
$464$	0	0
$465$	−1.25522	−0.0582096
$466$	0	0
$467$	19.9394	0.922686	0.461343	−	0.887222i	$-0.347368\pi$
$467$	19.9394	0.922686	0.461343	+	0.887222i	$0.347368\pi$
$468$	0	0
$469$	−22.4648	−1.03733
$470$	0	0
$471$	2.48510	0.114507
$472$	0	0
$473$	17.3132	0.796064
$474$	0	0
$475$	15.0461	0.690360
$476$	0	0
$477$	−7.64083	−0.349850
$478$	0	0
$479$	2.63168	0.120245	0.0601223	−	0.998191i	$-0.480851\pi$
$479$	2.63168	0.120245	0.0601223	+	0.998191i	$0.480851\pi$
$480$	0	0
$481$	−10.3258	−0.470818
$482$	0	0
$483$	−3.84396	−0.174906
$484$	0	0
$485$	−5.53862	−0.251496
$486$	0	0
$487$	−25.3381	−1.14818	−0.574090	−	0.818792i	$-0.694644\pi$
$487$	−25.3381	−1.14818	−0.574090	+	0.818792i	$0.694644\pi$
$488$	0	0
$489$	6.86595	0.310489
$490$	0	0
$491$	−7.67502	−0.346369	−0.173184	−	0.984889i	$-0.555406\pi$
$491$	−7.67502	−0.346369	−0.173184	+	0.984889i	$0.555406\pi$
$492$	0	0
$493$	−3.37201	−0.151868
$494$	0	0
$495$	3.15916	0.141994
$496$	0	0
$497$	−50.8589	−2.28133
$498$	0	0
$499$	31.0752	1.39112	0.695558	−	0.718470i	$-0.255157\pi$
$499$	31.0752	1.39112	0.695558	+	0.718470i	$0.255157\pi$
$500$	0	0
$501$	−21.9485	−0.980585
$502$	0	0
$503$	−32.9729	−1.47019	−0.735094	−	0.677965i	$-0.762862\pi$
$503$	−32.9729	−1.47019	−0.735094	+	0.677965i	$0.762862\pi$
$504$	0	0
$505$	6.45028	0.287034
$506$	0	0
$507$	3.89510	0.172987
$508$	0	0
$509$	29.0485	1.28755	0.643777	−	0.765213i	$-0.277366\pi$
$509$	29.0485	1.28755	0.643777	+	0.765213i	$0.277366\pi$
$510$	0	0
$511$	−33.6546	−1.48879
$512$	0	0
$513$	−3.86805	−0.170779
$514$	0	0
$515$	7.17740	0.316274
$516$	0	0
$517$	−20.8475	−0.916870
$518$	0	0
$519$	15.2665	0.670125
$520$	0	0
$521$	−2.94797	−0.129153	−0.0645766	−	0.997913i	$-0.520570\pi$
$521$	−2.94797	−0.129153	−0.0645766	+	0.997913i	$0.520570\pi$
$522$	0	0
$523$	−0.732472	−0.0320288	−0.0160144	−	0.999872i	$-0.505098\pi$
$523$	−0.732472	−0.0320288	−0.0160144	+	0.999872i	$0.505098\pi$
$524$	0	0
$525$	−14.9524	−0.652574
$526$	0	0
$527$	4.01712	0.174988
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	8.37197	0.363313
$532$	0	0
$533$	−15.2510	−0.660594
$534$	0	0
$535$	10.6029	0.458404
$536$	0	0
$537$	−4.39056	−0.189467
$538$	0	0
$539$	−23.3150	−1.00425
$540$	0	0
$541$	−6.53411	−0.280923	−0.140462	−	0.990086i	$-0.544859\pi$
$541$	−6.53411	−0.280923	−0.140462	+	0.990086i	$0.544859\pi$
$542$	0	0
$543$	−5.77641	−0.247889
$544$	0	0
$545$	16.1494	0.691764
$546$	0	0
$547$	−13.4802	−0.576371	−0.288185	−	0.957575i	$-0.593052\pi$
$547$	−13.4802	−0.576371	−0.288185	+	0.957575i	$0.593052\pi$
$548$	0	0
$549$	−9.45949	−0.403721
$550$	0	0
$551$	3.86805	0.164784
$552$	0	0
$553$	44.4654	1.89086
$554$	0	0
$555$	−2.64692	−0.112355
$556$	0	0
$557$	−41.5475	−1.76043	−0.880213	−	0.474579i	$-0.842600\pi$
$557$	−41.5475	−1.76043	−0.880213	+	0.474579i	$0.842600\pi$
$558$	0	0
$559$	23.7346	1.00387
$560$	0	0
$561$	−10.1103	−0.426858
$562$	0	0
$563$	7.16298	0.301884	0.150942	−	0.988543i	$-0.451769\pi$
$563$	7.16298	0.301884	0.150942	+	0.988543i	$0.451769\pi$
$564$	0	0
$565$	−14.9586	−0.629312
$566$	0	0
$567$	3.84396	0.161431
$568$	0	0
$569$	−4.04783	−0.169694	−0.0848469	−	0.996394i	$-0.527040\pi$
$569$	−4.04783	−0.169694	−0.0848469	+	0.996394i	$0.527040\pi$
$570$	0	0
$571$	39.6968	1.66126	0.830631	−	0.556824i	$-0.187980\pi$
$571$	39.6968	1.66126	0.830631	+	0.556824i	$0.187980\pi$
$572$	0	0
$573$	14.9901	0.626221
$574$	0	0
$575$	3.88983	0.162217
$576$	0	0
$577$	47.7703	1.98870	0.994352	−	0.106132i	$-0.0338466\pi$
$577$	47.7703	1.98870	0.994352	+	0.106132i	$0.0338466\pi$
$578$	0	0
$579$	−19.3122	−0.802589
$580$	0	0
$581$	−43.6051	−1.80904
$582$	0	0
$583$	22.9096	0.948818
$584$	0	0
$585$	4.33087	0.179060
$586$	0	0
$587$	−32.5855	−1.34495	−0.672474	−	0.740120i	$-0.734768\pi$
$587$	−32.5855	−1.34495	−0.672474	+	0.740120i	$0.734768\pi$
$588$	0	0
$589$	−4.60806	−0.189872
$590$	0	0
$591$	20.3886	0.838675
$592$	0	0
$593$	−20.0888	−0.824948	−0.412474	−	0.910969i	$-0.635335\pi$
$593$	−20.0888	−0.824948	−0.412474	+	0.910969i	$0.635335\pi$
$594$	0	0
$595$	−13.6572	−0.559892
$596$	0	0
$597$	−8.14767	−0.333462
$598$	0	0
$599$	−5.48348	−0.224049	−0.112025	−	0.993705i	$-0.535734\pi$
$599$	−5.48348	−0.224049	−0.112025	+	0.993705i	$0.535734\pi$
$600$	0	0
$601$	0.842597	0.0343703	0.0171851	−	0.999852i	$-0.494530\pi$
$601$	0.842597	0.0343703	0.0171851	+	0.999852i	$0.494530\pi$
$602$	0	0
$603$	−5.84419	−0.237994
$604$	0	0
$605$	2.11797	0.0861077
$606$	0	0
$607$	−11.2200	−0.455408	−0.227704	−	0.973730i	$-0.573122\pi$
$607$	−11.2200	−0.455408	−0.227704	+	0.973730i	$0.573122\pi$
$608$	0	0
$609$	−3.84396	−0.155765
$610$	0	0
$611$	−28.5797	−1.15621
$612$	0	0
$613$	26.5229	1.07125	0.535625	−	0.844456i	$-0.320076\pi$
$613$	26.5229	1.07125	0.535625	+	0.844456i	$0.320076\pi$
$614$	0	0
$615$	−3.90942	−0.157643
$616$	0	0
$617$	−20.8210	−0.838222	−0.419111	−	0.907935i	$-0.637658\pi$
$617$	−20.8210	−0.838222	−0.419111	+	0.907935i	$0.637658\pi$
$618$	0	0
$619$	2.65829	0.106846	0.0534229	−	0.998572i	$-0.482987\pi$
$619$	2.65829	0.106846	0.0534229	+	0.998572i	$0.482987\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−17.7701	−0.711945
$624$	0	0
$625$	9.57991	0.383196
$626$	0	0
$627$	11.5976	0.463164
$628$	0	0
$629$	8.47098	0.337760
$630$	0	0
$631$	32.4686	1.29255	0.646276	−	0.763103i	$-0.276325\pi$
$631$	32.4686	1.29255	0.646276	+	0.763103i	$0.276325\pi$
$632$	0	0
$633$	−8.44276	−0.335570
$634$	0	0
$635$	−2.67095	−0.105993
$636$	0	0
$637$	−31.9624	−1.26639
$638$	0	0
$639$	−13.2308	−0.523404
$640$	0	0
$641$	−18.2086	−0.719198	−0.359599	−	0.933107i	$-0.617087\pi$
$641$	−18.2086	−0.719198	−0.359599	+	0.933107i	$0.617087\pi$
$642$	0	0
$643$	23.0680	0.909714	0.454857	−	0.890564i	$-0.349690\pi$
$643$	23.0680	0.909714	0.454857	+	0.890564i	$0.349690\pi$
$644$	0	0
$645$	6.08411	0.239562
$646$	0	0
$647$	21.7437	0.854833	0.427416	−	0.904055i	$-0.359424\pi$
$647$	21.7437	0.854833	0.427416	+	0.904055i	$0.359424\pi$
$648$	0	0
$649$	−25.1018	−0.985330
$650$	0	0
$651$	4.57937	0.179479
$652$	0	0
$653$	−24.2489	−0.948932	−0.474466	−	0.880274i	$-0.657359\pi$
$653$	−24.2489	−0.948932	−0.474466	+	0.880274i	$0.657359\pi$
$654$	0	0
$655$	1.79626	0.0701859
$656$	0	0
$657$	−8.75519	−0.341573
$658$	0	0
$659$	−22.6732	−0.883224	−0.441612	−	0.897206i	$-0.645593\pi$
$659$	−22.6732	−0.883224	−0.441612	+	0.897206i	$0.645593\pi$
$660$	0	0
$661$	−28.6994	−1.11628	−0.558138	−	0.829748i	$-0.688484\pi$
$661$	−28.6994	−1.11628	−0.558138	+	0.829748i	$0.688484\pi$
$662$	0	0
$663$	−13.8602	−0.538285
$664$	0	0
$665$	15.6663	0.607513
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−7.74336	−0.299376
$670$	0	0
$671$	28.3625	1.09492
$672$	0	0
$673$	24.5075	0.944694	0.472347	−	0.881413i	$-0.343407\pi$
$673$	24.5075	0.944694	0.472347	+	0.881413i	$0.343407\pi$
$674$	0	0
$675$	−3.88983	−0.149720
$676$	0	0
$677$	13.1650	0.505973	0.252986	−	0.967470i	$-0.418587\pi$
$677$	13.1650	0.505973	0.252986	+	0.967470i	$0.418587\pi$
$678$	0	0
$679$	20.2062	0.775444
$680$	0	0
$681$	5.22412	0.200189
$682$	0	0
$683$	−36.2915	−1.38866	−0.694328	−	0.719659i	$-0.744298\pi$
$683$	−36.2915	−1.38866	−0.694328	+	0.719659i	$0.744298\pi$
$684$	0	0
$685$	−6.37739	−0.243668
$686$	0	0
$687$	−24.2214	−0.924103
$688$	0	0
$689$	31.4066	1.19650
$690$	0	0
$691$	−49.6214	−1.88769	−0.943843	−	0.330396i	$-0.892818\pi$
$691$	−49.6214	−1.88769	−0.943843	+	0.330396i	$0.892818\pi$
$692$	0	0
$693$	−11.5254	−0.437813
$694$	0	0
$695$	−1.58959	−0.0602965
$696$	0	0
$697$	12.5114	0.473903
$698$	0	0
$699$	11.9952	0.453700
$700$	0	0
$701$	41.6098	1.57158	0.785791	−	0.618493i	$-0.212256\pi$
$701$	41.6098	1.57158	0.785791	+	0.618493i	$0.212256\pi$
$702$	0	0
$703$	−9.71712	−0.366488
$704$	0	0
$705$	−7.32608	−0.275916
$706$	0	0
$707$	−23.5322	−0.885019
$708$	0	0
$709$	46.9149	1.76193	0.880964	−	0.473184i	$-0.156895\pi$
$709$	46.9149	1.76193	0.880964	+	0.473184i	$0.156895\pi$
$710$	0	0
$711$	11.5676	0.433819
$712$	0	0
$713$	−1.19131	−0.0446151
$714$	0	0
$715$	−12.9853	−0.485623
$716$	0	0
$717$	10.3057	0.384872
$718$	0	0
$719$	36.3871	1.35701	0.678504	−	0.734597i	$-0.262629\pi$
$719$	36.3871	1.35701	0.678504	+	0.734597i	$0.262629\pi$
$720$	0	0
$721$	−26.1849	−0.975178
$722$	0	0
$723$	24.0342	0.893840
$724$	0	0
$725$	3.88983	0.144465
$726$	0	0
$727$	−26.3432	−0.977016	−0.488508	−	0.872559i	$-0.662459\pi$
$727$	−26.3432	−0.977016	−0.488508	+	0.872559i	$0.662459\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.4711	−0.720164
$732$	0	0
$733$	−8.78194	−0.324368	−0.162184	−	0.986761i	$-0.551854\pi$
$733$	−8.78194	−0.324368	−0.162184	+	0.986761i	$0.551854\pi$
$734$	0	0
$735$	−8.19320	−0.302211
$736$	0	0
$737$	17.5227	0.645457
$738$	0	0
$739$	−15.7102	−0.577908	−0.288954	−	0.957343i	$-0.593307\pi$
$739$	−15.7102	−0.577908	−0.288954	+	0.957343i	$0.593307\pi$
$740$	0	0
$741$	15.8991	0.584068
$742$	0	0
$743$	32.1896	1.18092	0.590460	−	0.807067i	$-0.298946\pi$
$743$	32.1896	1.18092	0.590460	+	0.807067i	$0.298946\pi$
$744$	0	0
$745$	9.20668	0.337307
$746$	0	0
$747$	−11.3438	−0.415048
$748$	0	0
$749$	−38.6820	−1.41341
$750$	0	0
$751$	39.0576	1.42523	0.712617	−	0.701554i	$-0.247510\pi$
$751$	39.0576	1.42523	0.712617	+	0.701554i	$0.247510\pi$
$752$	0	0
$753$	−30.6106	−1.11551
$754$	0	0
$755$	11.5574	0.420618
$756$	0	0
$757$	−47.0573	−1.71033	−0.855163	−	0.518360i	$-0.826543\pi$
$757$	−47.0573	−1.71033	−0.855163	+	0.518360i	$0.826543\pi$
$758$	0	0
$759$	2.99831	0.108832
$760$	0	0
$761$	−28.7873	−1.04354	−0.521770	−	0.853086i	$-0.674728\pi$
$761$	−28.7873	−1.04354	−0.521770	+	0.853086i	$0.674728\pi$
$762$	0	0
$763$	−58.9170	−2.13294
$764$	0	0
$765$	−3.55290	−0.128456
$766$	0	0
$767$	−34.4118	−1.24254
$768$	0	0
$769$	4.09103	0.147526	0.0737632	−	0.997276i	$-0.476499\pi$
$769$	4.09103	0.147526	0.0737632	+	0.997276i	$0.476499\pi$
$770$	0	0
$771$	−11.2912	−0.406644
$772$	0	0
$773$	−13.9030	−0.500055	−0.250028	−	0.968239i	$-0.580440\pi$
$773$	−13.9030	−0.500055	−0.250028	+	0.968239i	$0.580440\pi$
$774$	0	0
$775$	−4.63401	−0.166458
$776$	0	0
$777$	9.65660	0.346429
$778$	0	0
$779$	−14.3519	−0.514211
$780$	0	0
$781$	39.6702	1.41951
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−2.61842	−0.0934553
$786$	0	0
$787$	19.6286	0.699684	0.349842	−	0.936809i	$-0.386235\pi$
$787$	19.6286	0.699684	0.349842	+	0.936809i	$0.386235\pi$
$788$	0	0
$789$	2.27348	0.0809380
$790$	0	0
$791$	54.5725	1.94038
$792$	0	0
$793$	38.8819	1.38074
$794$	0	0
$795$	8.05074	0.285530
$796$	0	0
$797$	18.9729	0.672056	0.336028	−	0.941852i	$-0.390916\pi$
$797$	18.9729	0.672056	0.336028	+	0.941852i	$0.390916\pi$
$798$	0	0
$799$	23.4458	0.829453
$800$	0	0
$801$	−4.62287	−0.163341
$802$	0	0
$803$	26.2508	0.926370
$804$	0	0
$805$	4.05018	0.142750
$806$	0	0
$807$	19.7673	0.695843
$808$	0	0
$809$	38.0811	1.33886	0.669430	−	0.742875i	$-0.266538\pi$
$809$	38.0811	1.33886	0.669430	+	0.742875i	$0.266538\pi$
$810$	0	0
$811$	9.82539	0.345016	0.172508	−	0.985008i	$-0.444813\pi$
$811$	9.82539	0.345016	0.172508	+	0.985008i	$0.444813\pi$
$812$	0	0
$813$	−7.93328	−0.278232
$814$	0	0
$815$	−7.23429	−0.253406
$816$	0	0
$817$	22.3354	0.781417
$818$	0	0
$819$	−15.8001	−0.552100
$820$	0	0
$821$	−24.1342	−0.842291	−0.421145	−	0.906993i	$-0.638372\pi$
$821$	−24.1342	−0.842291	−0.421145	+	0.906993i	$0.638372\pi$
$822$	0	0
$823$	−4.29620	−0.149756	−0.0748781	−	0.997193i	$-0.523857\pi$
$823$	−4.29620	−0.149756	−0.0748781	+	0.997193i	$0.523857\pi$
$824$	0	0
$825$	11.6629	0.406051
$826$	0	0
$827$	17.1668	0.596949	0.298475	−	0.954418i	$-0.403522\pi$
$827$	17.1668	0.596949	0.298475	+	0.954418i	$0.403522\pi$
$828$	0	0
$829$	13.9519	0.484570	0.242285	−	0.970205i	$-0.422103\pi$
$829$	13.9519	0.484570	0.242285	+	0.970205i	$0.422103\pi$
$830$	0	0
$831$	−0.212742	−0.00737995
$832$	0	0
$833$	26.2209	0.908499
$834$	0	0
$835$	23.1259	0.800305
$836$	0	0
$837$	1.19131	0.0411778
$838$	0	0
$839$	−27.7385	−0.957638	−0.478819	−	0.877914i	$-0.658935\pi$
$839$	−27.7385	−0.957638	−0.478819	+	0.877914i	$0.658935\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	22.1414	0.762592
$844$	0	0
$845$	−4.10405	−0.141184
$846$	0	0
$847$	−7.72688	−0.265499
$848$	0	0
$849$	13.9321	0.478147
$850$	0	0
$851$	−2.51215	−0.0861153
$852$	0	0
$853$	−34.2858	−1.17392	−0.586961	−	0.809615i	$-0.699676\pi$
$853$	−34.2858	−1.17392	−0.586961	+	0.809615i	$0.699676\pi$
$854$	0	0
$855$	4.07556	0.139381
$856$	0	0
$857$	−20.5939	−0.703474	−0.351737	−	0.936099i	$-0.614409\pi$
$857$	−20.5939	−0.703474	−0.351737	+	0.936099i	$0.614409\pi$
$858$	0	0
$859$	30.8542	1.05273	0.526365	−	0.850259i	$-0.323555\pi$
$859$	30.8542	1.05273	0.526365	+	0.850259i	$0.323555\pi$
$860$	0	0
$861$	14.2625	0.486066
$862$	0	0
$863$	25.0641	0.853192	0.426596	−	0.904442i	$-0.359713\pi$
$863$	25.0641	0.853192	0.426596	+	0.904442i	$0.359713\pi$
$864$	0	0
$865$	−16.0855	−0.546924
$866$	0	0
$867$	−5.62957	−0.191190
$868$	0	0
$869$	−34.6832	−1.17655
$870$	0	0
$871$	24.0217	0.813946
$872$	0	0
$873$	5.25662	0.177910
$874$	0	0
$875$	36.0054	1.21720
$876$	0	0
$877$	20.3360	0.686697	0.343349	−	0.939208i	$-0.388439\pi$
$877$	20.3360	0.686697	0.343349	+	0.939208i	$0.388439\pi$
$878$	0	0
$879$	11.2436	0.379236
$880$	0	0
$881$	30.7783	1.03695	0.518474	−	0.855093i	$-0.326500\pi$
$881$	30.7783	1.03695	0.518474	+	0.855093i	$0.326500\pi$
$882$	0	0
$883$	−17.5357	−0.590123	−0.295062	−	0.955478i	$-0.595340\pi$
$883$	−17.5357	−0.590123	−0.295062	+	0.955478i	$0.595340\pi$
$884$	0	0
$885$	−8.82110	−0.296518
$886$	0	0
$887$	7.80627	0.262109	0.131054	−	0.991375i	$-0.458164\pi$
$887$	7.80627	0.262109	0.131054	+	0.991375i	$0.458164\pi$
$888$	0	0
$889$	9.74428	0.326813
$890$	0	0
$891$	−2.99831	−0.100447
$892$	0	0
$893$	−26.8948	−0.900001
$894$	0	0
$895$	4.62610	0.154634
$896$	0	0
$897$	4.11036	0.137241
$898$	0	0
$899$	−1.19131	−0.0397325
$900$	0	0
$901$	−25.7649	−0.858354
$902$	0	0
$903$	−22.1963	−0.738647
$904$	0	0
$905$	6.08629	0.202315
$906$	0	0
$907$	−6.11510	−0.203048	−0.101524	−	0.994833i	$-0.532372\pi$
$907$	−6.11510	−0.203048	−0.101524	+	0.994833i	$0.532372\pi$
$908$	0	0
$909$	−6.12186	−0.203049
$910$	0	0
$911$	−17.9663	−0.595250	−0.297625	−	0.954683i	$-0.596194\pi$
$911$	−17.9663	−0.595250	−0.297625	+	0.954683i	$0.596194\pi$
$912$	0	0
$913$	34.0122	1.12564
$914$	0	0
$915$	9.96696	0.329497
$916$	0	0
$917$	−6.55321	−0.216406
$918$	0	0
$919$	29.8869	0.985877	0.492938	−	0.870064i	$-0.335923\pi$
$919$	29.8869	0.985877	0.492938	+	0.870064i	$0.335923\pi$
$920$	0	0
$921$	−7.87090	−0.259355
$922$	0	0
$923$	54.3836	1.79006
$924$	0	0
$925$	−9.77183	−0.321296
$926$	0	0
$927$	−6.81197	−0.223734
$928$	0	0
$929$	−38.0968	−1.24992	−0.624958	−	0.780659i	$-0.714884\pi$
$929$	−38.0968	−1.24992	−0.624958	+	0.780659i	$0.714884\pi$
$930$	0	0
$931$	−30.0781	−0.985770
$932$	0	0
$933$	−18.6227	−0.609680
$934$	0	0
$935$	10.6527	0.348381
$936$	0	0
$937$	−5.67470	−0.185384	−0.0926922	−	0.995695i	$-0.529547\pi$
$937$	−5.67470	−0.185384	−0.0926922	+	0.995695i	$0.529547\pi$
$938$	0	0
$939$	−28.2165	−0.920810
$940$	0	0
$941$	−17.3940	−0.567028	−0.283514	−	0.958968i	$-0.591500\pi$
$941$	−17.3940	−0.567028	−0.283514	+	0.958968i	$0.591500\pi$
$942$	0	0
$943$	−3.71037	−0.120826
$944$	0	0
$945$	−4.05018	−0.131752
$946$	0	0
$947$	32.8386	1.06711	0.533555	−	0.845765i	$-0.320856\pi$
$947$	32.8386	1.06711	0.533555	+	0.845765i	$0.320856\pi$
$948$	0	0
$949$	35.9870	1.16819
$950$	0	0
$951$	11.5300	0.373885
$952$	0	0
$953$	−1.30468	−0.0422628	−0.0211314	−	0.999777i	$-0.506727\pi$
$953$	−1.30468	−0.0422628	−0.0211314	+	0.999777i	$0.506727\pi$
$954$	0	0
$955$	−15.7943	−0.511091
$956$	0	0
$957$	2.99831	0.0969216
$958$	0	0
$959$	23.2663	0.751308
$960$	0	0
$961$	−29.5808	−0.954218
$962$	0	0
$963$	−10.0631	−0.324278
$964$	0	0
$965$	20.3483	0.655034
$966$	0	0
$967$	47.8353	1.53828	0.769140	−	0.639080i	$-0.220685\pi$
$967$	47.8353	1.53828	0.769140	+	0.639080i	$0.220685\pi$
$968$	0	0
$969$	−13.0431	−0.419005
$970$	0	0
$971$	15.0117	0.481748	0.240874	−	0.970556i	$-0.422566\pi$
$971$	15.0117	0.481748	0.240874	+	0.970556i	$0.422566\pi$
$972$	0	0
$973$	5.79920	0.185914
$974$	0	0
$975$	15.9886	0.512045
$976$	0	0
$977$	−5.20746	−0.166601	−0.0833007	−	0.996524i	$-0.526546\pi$
$977$	−5.20746	−0.166601	−0.0833007	+	0.996524i	$0.526546\pi$
$978$	0	0
$979$	13.8608	0.442993
$980$	0	0
$981$	−15.3271	−0.489358
$982$	0	0
$983$	38.8861	1.24027	0.620137	−	0.784493i	$-0.287077\pi$
$983$	38.8861	1.24027	0.620137	+	0.784493i	$0.287077\pi$
$984$	0	0
$985$	−21.4824	−0.684486
$986$	0	0
$987$	26.7273	0.850740
$988$	0	0
$989$	5.77433	0.183613
$990$	0	0
$991$	34.4319	1.09377	0.546883	−	0.837209i	$-0.315814\pi$
$991$	34.4319	1.09377	0.546883	+	0.837209i	$0.315814\pi$
$992$	0	0
$993$	−15.8270	−0.502256
$994$	0	0
$995$	8.58476	0.272155
$996$	0	0
$997$	21.4908	0.680622	0.340311	−	0.940313i	$-0.389468\pi$
$997$	21.4908	0.680622	0.340311	+	0.940313i	$0.389468\pi$
$998$	0	0
$999$	2.51215	0.0794809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.f.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.f.1.4	✓	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.05365 q^{5} +3.84396 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.05365 q^{5} +3.84396 q^{7} +1.00000 q^{9} -2.99831 q^{11} -4.11036 q^{13} -1.05365 q^{15} +3.37201 q^{17} -3.86805 q^{19} +3.84396 q^{21} -1.00000 q^{23} -3.88983 q^{25} +1.00000 q^{27} -1.00000 q^{29} +1.19131 q^{31} -2.99831 q^{33} -4.05018 q^{35} +2.51215 q^{37} -4.11036 q^{39} +3.71037 q^{41} -5.77433 q^{43} -1.05365 q^{45} +6.95307 q^{47} +7.77604 q^{49} +3.37201 q^{51} -7.64083 q^{53} +3.15916 q^{55} -3.86805 q^{57} +8.37197 q^{59} -9.45949 q^{61} +3.84396 q^{63} +4.33087 q^{65} -5.84419 q^{67} -1.00000 q^{69} -13.2308 q^{71} -8.75519 q^{73} -3.88983 q^{75} -11.5254 q^{77} +11.5676 q^{79} +1.00000 q^{81} -11.3438 q^{83} -3.55290 q^{85} -1.00000 q^{87} -4.62287 q^{89} -15.8001 q^{91} +1.19131 q^{93} +4.07556 q^{95} +5.25662 q^{97} -2.99831 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41} - 19 q^{43} - q^{45} - 3 q^{47} - 6 q^{49} - 2 q^{51} - 9 q^{53} - 7 q^{55} - 11 q^{57} - 2 q^{59} - 25 q^{61} - 5 q^{63} - 12 q^{65} - 20 q^{67} - 9 q^{69} + 9 q^{71} - 11 q^{73} - 10 q^{75} - 19 q^{77} + 4 q^{79} + 9 q^{81} - 9 q^{83} - 50 q^{85} - 9 q^{87} - 29 q^{89} - 38 q^{91} + 23 q^{95} - 43 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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