LMFDB - Embedded newform 8004.2.a.i.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:	$0$
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			1.4
Root			$-2.58465$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.58465 q^{5} -3.26725 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.58465 q^{5} -3.26725 q^{7} +1.00000 q^{9} -2.90115 q^{11} +5.43566 q^{13} +2.58465 q^{15} -2.76865 q^{17} -6.15044 q^{19} +3.26725 q^{21} +1.00000 q^{23} +1.68042 q^{25} -1.00000 q^{27} +1.00000 q^{29} -1.72826 q^{31} +2.90115 q^{33} +8.44471 q^{35} -4.97381 q^{37} -5.43566 q^{39} -6.02032 q^{41} -10.9734 q^{43} -2.58465 q^{45} -3.19760 q^{47} +3.67493 q^{49} +2.76865 q^{51} +5.41370 q^{53} +7.49845 q^{55} +6.15044 q^{57} -3.48319 q^{59} -3.77242 q^{61} -3.26725 q^{63} -14.0493 q^{65} -6.87084 q^{67} -1.00000 q^{69} +3.47332 q^{71} -5.99833 q^{73} -1.68042 q^{75} +9.47878 q^{77} +0.609370 q^{79} +1.00000 q^{81} -6.42111 q^{83} +7.15600 q^{85} -1.00000 q^{87} -15.0768 q^{89} -17.7597 q^{91} +1.72826 q^{93} +15.8967 q^{95} -6.60560 q^{97} -2.90115 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.58465	−1.15589	−0.577946	−	0.816075i	$-0.696145\pi$
$5$	−2.58465	−1.15589	−0.577946	+	0.816075i	$0.696145\pi$
$6$	0	0
$7$	−3.26725	−1.23491	−0.617453	−	0.786608i	$-0.711835\pi$
$7$	−3.26725	−1.23491	−0.617453	+	0.786608i	$0.711835\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.90115	−0.874729	−0.437364	−	0.899284i	$-0.644088\pi$
$11$	−2.90115	−0.874729	−0.437364	+	0.899284i	$0.644088\pi$
$12$	0	0
$13$	5.43566	1.50758	0.753791	−	0.657114i	$-0.228223\pi$
$13$	5.43566	1.50758	0.753791	+	0.657114i	$0.228223\pi$
$14$	0	0
$15$	2.58465	0.667354
$16$	0	0
$17$	−2.76865	−0.671497	−0.335748	−	0.941952i	$-0.608989\pi$
$17$	−2.76865	−0.671497	−0.335748	+	0.941952i	$0.608989\pi$
$18$	0	0
$19$	−6.15044	−1.41101	−0.705504	−	0.708706i	$-0.749279\pi$
$19$	−6.15044	−1.41101	−0.705504	+	0.708706i	$0.749279\pi$
$20$	0	0
$21$	3.26725	0.712973
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.68042	0.336084
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.72826	−0.310404	−0.155202	−	0.987883i	$-0.549603\pi$
$31$	−1.72826	−0.310404	−0.155202	+	0.987883i	$0.549603\pi$
$32$	0	0
$33$	2.90115	0.505025
$34$	0	0
$35$	8.44471	1.42742
$36$	0	0
$37$	−4.97381	−0.817689	−0.408845	−	0.912604i	$-0.634068\pi$
$37$	−4.97381	−0.817689	−0.408845	+	0.912604i	$0.634068\pi$
$38$	0	0
$39$	−5.43566	−0.870403
$40$	0	0
$41$	−6.02032	−0.940217	−0.470108	−	0.882609i	$-0.655785\pi$
$41$	−6.02032	−0.940217	−0.470108	+	0.882609i	$0.655785\pi$
$42$	0	0
$43$	−10.9734	−1.67342	−0.836712	−	0.547643i	$-0.815525\pi$
$43$	−10.9734	−1.67342	−0.836712	+	0.547643i	$0.815525\pi$
$44$	0	0
$45$	−2.58465	−0.385297
$46$	0	0
$47$	−3.19760	−0.466419	−0.233209	−	0.972427i	$-0.574923\pi$
$47$	−3.19760	−0.466419	−0.233209	+	0.972427i	$0.574923\pi$
$48$	0	0
$49$	3.67493	0.524990
$50$	0	0
$51$	2.76865	0.387689
$52$	0	0
$53$	5.41370	0.743628	0.371814	−	0.928307i	$-0.378736\pi$
$53$	5.41370	0.743628	0.371814	+	0.928307i	$0.378736\pi$
$54$	0	0
$55$	7.49845	1.01109
$56$	0	0
$57$	6.15044	0.814646
$58$	0	0
$59$	−3.48319	−0.453473	−0.226737	−	0.973956i	$-0.572806\pi$
$59$	−3.48319	−0.453473	−0.226737	+	0.973956i	$0.572806\pi$
$60$	0	0
$61$	−3.77242	−0.483009	−0.241504	−	0.970400i	$-0.577641\pi$
$61$	−3.77242	−0.483009	−0.241504	+	0.970400i	$0.577641\pi$
$62$	0	0
$63$	−3.26725	−0.411635
$64$	0	0
$65$	−14.0493	−1.74260
$66$	0	0
$67$	−6.87084	−0.839407	−0.419703	−	0.907661i	$-0.637866\pi$
$67$	−6.87084	−0.839407	−0.419703	+	0.907661i	$0.637866\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	3.47332	0.412207	0.206104	−	0.978530i	$-0.433922\pi$
$71$	3.47332	0.412207	0.206104	+	0.978530i	$0.433922\pi$
$72$	0	0
$73$	−5.99833	−0.702052	−0.351026	−	0.936366i	$-0.614167\pi$
$73$	−5.99833	−0.702052	−0.351026	+	0.936366i	$0.614167\pi$
$74$	0	0
$75$	−1.68042	−0.194038
$76$	0	0
$77$	9.47878	1.08021
$78$	0	0
$79$	0.609370	0.0685595	0.0342797	−	0.999412i	$-0.489086\pi$
$79$	0.609370	0.0685595	0.0342797	+	0.999412i	$0.489086\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.42111	−0.704809	−0.352404	−	0.935848i	$-0.614636\pi$
$83$	−6.42111	−0.704809	−0.352404	+	0.935848i	$0.614636\pi$
$84$	0	0
$85$	7.15600	0.776177
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−15.0768	−1.59814	−0.799069	−	0.601239i	$-0.794674\pi$
$89$	−15.0768	−1.59814	−0.799069	+	0.601239i	$0.794674\pi$
$90$	0	0
$91$	−17.7597	−1.86172
$92$	0	0
$93$	1.72826	0.179212
$94$	0	0
$95$	15.8967	1.63097
$96$	0	0
$97$	−6.60560	−0.670697	−0.335349	−	0.942094i	$-0.608854\pi$
$97$	−6.60560	−0.670697	−0.335349	+	0.942094i	$0.608854\pi$
$98$	0	0
$99$	−2.90115	−0.291576
$100$	0	0
$101$	15.5916	1.55142	0.775710	−	0.631090i	$-0.217392\pi$
$101$	15.5916	1.55142	0.775710	+	0.631090i	$0.217392\pi$
$102$	0	0
$103$	−15.2178	−1.49945	−0.749727	−	0.661747i	$-0.769815\pi$
$103$	−15.2178	−1.49945	−0.749727	+	0.661747i	$0.769815\pi$
$104$	0	0
$105$	−8.44471	−0.824119
$106$	0	0
$107$	−3.87036	−0.374162	−0.187081	−	0.982345i	$-0.559903\pi$
$107$	−3.87036	−0.374162	−0.187081	+	0.982345i	$0.559903\pi$
$108$	0	0
$109$	9.19716	0.880928	0.440464	−	0.897770i	$-0.354814\pi$
$109$	9.19716	0.880928	0.440464	+	0.897770i	$0.354814\pi$
$110$	0	0
$111$	4.97381	0.472093
$112$	0	0
$113$	8.86530	0.833977	0.416988	−	0.908912i	$-0.363086\pi$
$113$	8.86530	0.833977	0.416988	+	0.908912i	$0.363086\pi$
$114$	0	0
$115$	−2.58465	−0.241020
$116$	0	0
$117$	5.43566	0.502527
$118$	0	0
$119$	9.04588	0.829235
$120$	0	0
$121$	−2.58335	−0.234850
$122$	0	0
$123$	6.02032	0.542834
$124$	0	0
$125$	8.57995	0.767414
$126$	0	0
$127$	−0.0386548	−0.00343006	−0.00171503	−	0.999999i	$-0.500546\pi$
$127$	−0.0386548	−0.00343006	−0.00171503	+	0.999999i	$0.500546\pi$
$128$	0	0
$129$	10.9734	0.966152
$130$	0	0
$131$	−18.2687	−1.59615	−0.798073	−	0.602561i	$-0.794147\pi$
$131$	−18.2687	−1.59615	−0.798073	+	0.602561i	$0.794147\pi$
$132$	0	0
$133$	20.0950	1.74246
$134$	0	0
$135$	2.58465	0.222451
$136$	0	0
$137$	3.12032	0.266587	0.133294	−	0.991077i	$-0.457445\pi$
$137$	3.12032	0.266587	0.133294	+	0.991077i	$0.457445\pi$
$138$	0	0
$139$	−7.77344	−0.659335	−0.329667	−	0.944097i	$-0.606937\pi$
$139$	−7.77344	−0.659335	−0.329667	+	0.944097i	$0.606937\pi$
$140$	0	0
$141$	3.19760	0.269287
$142$	0	0
$143$	−15.7697	−1.31872
$144$	0	0
$145$	−2.58465	−0.214644
$146$	0	0
$147$	−3.67493	−0.303103
$148$	0	0
$149$	−16.7712	−1.37395	−0.686975	−	0.726681i	$-0.741062\pi$
$149$	−16.7712	−1.37395	−0.686975	+	0.726681i	$0.741062\pi$
$150$	0	0
$151$	−12.6678	−1.03089	−0.515445	−	0.856923i	$-0.672373\pi$
$151$	−12.6678	−1.03089	−0.515445	+	0.856923i	$0.672373\pi$
$152$	0	0
$153$	−2.76865	−0.223832
$154$	0	0
$155$	4.46695	0.358794
$156$	0	0
$157$	−12.1569	−0.970229	−0.485115	−	0.874451i	$-0.661222\pi$
$157$	−12.1569	−0.970229	−0.485115	+	0.874451i	$0.661222\pi$
$158$	0	0
$159$	−5.41370	−0.429334
$160$	0	0
$161$	−3.26725	−0.257496
$162$	0	0
$163$	−9.91818	−0.776852	−0.388426	−	0.921480i	$-0.626981\pi$
$163$	−9.91818	−0.776852	−0.388426	+	0.921480i	$0.626981\pi$
$164$	0	0
$165$	−7.49845	−0.583754
$166$	0	0
$167$	16.7134	1.29332	0.646662	−	0.762776i	$-0.276164\pi$
$167$	16.7134	1.29332	0.646662	+	0.762776i	$0.276164\pi$
$168$	0	0
$169$	16.5464	1.27280
$170$	0	0
$171$	−6.15044	−0.470336
$172$	0	0
$173$	4.59989	0.349723	0.174862	−	0.984593i	$-0.444052\pi$
$173$	4.59989	0.349723	0.174862	+	0.984593i	$0.444052\pi$
$174$	0	0
$175$	−5.49036	−0.415032
$176$	0	0
$177$	3.48319	0.261813
$178$	0	0
$179$	20.7314	1.54953	0.774767	−	0.632247i	$-0.217867\pi$
$179$	20.7314	1.54953	0.774767	+	0.632247i	$0.217867\pi$
$180$	0	0
$181$	22.6705	1.68509	0.842543	−	0.538629i	$-0.181058\pi$
$181$	22.6705	1.68509	0.842543	+	0.538629i	$0.181058\pi$
$182$	0	0
$183$	3.77242	0.278865
$184$	0	0
$185$	12.8556	0.945160
$186$	0	0
$187$	8.03227	0.587377
$188$	0	0
$189$	3.26725	0.237658
$190$	0	0
$191$	2.86879	0.207578	0.103789	−	0.994599i	$-0.466903\pi$
$191$	2.86879	0.207578	0.103789	+	0.994599i	$0.466903\pi$
$192$	0	0
$193$	−9.67593	−0.696488	−0.348244	−	0.937404i	$-0.613222\pi$
$193$	−9.67593	−0.696488	−0.348244	+	0.937404i	$0.613222\pi$
$194$	0	0
$195$	14.0493	1.00609
$196$	0	0
$197$	4.60951	0.328414	0.164207	−	0.986426i	$-0.447493\pi$
$197$	4.60951	0.328414	0.164207	+	0.986426i	$0.447493\pi$
$198$	0	0
$199$	17.3597	1.23059	0.615297	−	0.788296i	$-0.289036\pi$
$199$	17.3597	1.23059	0.615297	+	0.788296i	$0.289036\pi$
$200$	0	0
$201$	6.87084	0.484632
$202$	0	0
$203$	−3.26725	−0.229316
$204$	0	0
$205$	15.5604	1.08679
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	17.8433	1.23425
$210$	0	0
$211$	−6.77916	−0.466696	−0.233348	−	0.972393i	$-0.574968\pi$
$211$	−6.77916	−0.466696	−0.233348	+	0.972393i	$0.574968\pi$
$212$	0	0
$213$	−3.47332	−0.237988
$214$	0	0
$215$	28.3624	1.93430
$216$	0	0
$217$	5.64666	0.383320
$218$	0	0
$219$	5.99833	0.405330
$220$	0	0
$221$	−15.0495	−1.01234
$222$	0	0
$223$	−12.4694	−0.835014	−0.417507	−	0.908674i	$-0.637096\pi$
$223$	−12.4694	−0.835014	−0.417507	+	0.908674i	$0.637096\pi$
$224$	0	0
$225$	1.68042	0.112028
$226$	0	0
$227$	29.2499	1.94139	0.970693	−	0.240321i	$-0.0772527\pi$
$227$	29.2499	1.94139	0.970693	+	0.240321i	$0.0772527\pi$
$228$	0	0
$229$	−27.7176	−1.83163	−0.915817	−	0.401596i	$-0.868456\pi$
$229$	−27.7176	−1.83163	−0.915817	+	0.401596i	$0.868456\pi$
$230$	0	0
$231$	−9.47878	−0.623658
$232$	0	0
$233$	20.4677	1.34089	0.670443	−	0.741961i	$-0.266104\pi$
$233$	20.4677	1.34089	0.670443	+	0.741961i	$0.266104\pi$
$234$	0	0
$235$	8.26469	0.539129
$236$	0	0
$237$	−0.609370	−0.0395828
$238$	0	0
$239$	2.62599	0.169861	0.0849306	−	0.996387i	$-0.472933\pi$
$239$	2.62599	0.169861	0.0849306	+	0.996387i	$0.472933\pi$
$240$	0	0
$241$	−8.14128	−0.524426	−0.262213	−	0.965010i	$-0.584452\pi$
$241$	−8.14128	−0.524426	−0.262213	+	0.965010i	$0.584452\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−9.49842	−0.606832
$246$	0	0
$247$	−33.4317	−2.12721
$248$	0	0
$249$	6.42111	0.406922
$250$	0	0
$251$	13.8908	0.876778	0.438389	−	0.898785i	$-0.355549\pi$
$251$	13.8908	0.876778	0.438389	+	0.898785i	$0.355549\pi$
$252$	0	0
$253$	−2.90115	−0.182394
$254$	0	0
$255$	−7.15600	−0.448126
$256$	0	0
$257$	22.3718	1.39551	0.697757	−	0.716334i	$-0.254181\pi$
$257$	22.3718	1.39551	0.697757	+	0.716334i	$0.254181\pi$
$258$	0	0
$259$	16.2507	1.00977
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−24.8644	−1.53321	−0.766603	−	0.642121i	$-0.778055\pi$
$263$	−24.8644	−1.53321	−0.766603	+	0.642121i	$0.778055\pi$
$264$	0	0
$265$	−13.9925	−0.859553
$266$	0	0
$267$	15.0768	0.922686
$268$	0	0
$269$	−13.3151	−0.811838	−0.405919	−	0.913909i	$-0.633048\pi$
$269$	−13.3151	−0.811838	−0.405919	+	0.913909i	$0.633048\pi$
$270$	0	0
$271$	−13.0537	−0.792959	−0.396479	−	0.918044i	$-0.629768\pi$
$271$	−13.0537	−0.792959	−0.396479	+	0.918044i	$0.629768\pi$
$272$	0	0
$273$	17.7597	1.07486
$274$	0	0
$275$	−4.87515	−0.293983
$276$	0	0
$277$	31.3272	1.88227	0.941134	−	0.338033i	$-0.109762\pi$
$277$	31.3272	1.88227	0.941134	+	0.338033i	$0.109762\pi$
$278$	0	0
$279$	−1.72826	−0.103468
$280$	0	0
$281$	−8.97961	−0.535679	−0.267839	−	0.963464i	$-0.586310\pi$
$281$	−8.97961	−0.535679	−0.267839	+	0.963464i	$0.586310\pi$
$282$	0	0
$283$	2.83888	0.168754	0.0843769	−	0.996434i	$-0.473110\pi$
$283$	2.83888	0.168754	0.0843769	+	0.996434i	$0.473110\pi$
$284$	0	0
$285$	−15.8967	−0.941642
$286$	0	0
$287$	19.6699	1.16108
$288$	0	0
$289$	−9.33456	−0.549092
$290$	0	0
$291$	6.60560	0.387227
$292$	0	0
$293$	−17.3913	−1.01601	−0.508005	−	0.861354i	$-0.669617\pi$
$293$	−17.3913	−1.01601	−0.508005	+	0.861354i	$0.669617\pi$
$294$	0	0
$295$	9.00284	0.524166
$296$	0	0
$297$	2.90115	0.168342
$298$	0	0
$299$	5.43566	0.314353
$300$	0	0
$301$	35.8528	2.06652
$302$	0	0
$303$	−15.5916	−0.895712
$304$	0	0
$305$	9.75039	0.558306
$306$	0	0
$307$	18.8046	1.07323	0.536617	−	0.843826i	$-0.319702\pi$
$307$	18.8046	1.07323	0.536617	+	0.843826i	$0.319702\pi$
$308$	0	0
$309$	15.2178	0.865710
$310$	0	0
$311$	−10.6986	−0.606661	−0.303330	−	0.952885i	$-0.598099\pi$
$311$	−10.6986	−0.606661	−0.303330	+	0.952885i	$0.598099\pi$
$312$	0	0
$313$	26.6605	1.50694	0.753471	−	0.657481i	$-0.228378\pi$
$313$	26.6605	1.50694	0.753471	+	0.657481i	$0.228378\pi$
$314$	0	0
$315$	8.44471	0.475805
$316$	0	0
$317$	−24.9984	−1.40405	−0.702024	−	0.712153i	$-0.747720\pi$
$317$	−24.9984	−1.40405	−0.702024	+	0.712153i	$0.747720\pi$
$318$	0	0
$319$	−2.90115	−0.162433
$320$	0	0
$321$	3.87036	0.216022
$322$	0	0
$323$	17.0284	0.947487
$324$	0	0
$325$	9.13421	0.506675
$326$	0	0
$327$	−9.19716	−0.508604
$328$	0	0
$329$	10.4474	0.575983
$330$	0	0
$331$	35.7206	1.96338	0.981692	−	0.190476i	$-0.0610030\pi$
$331$	35.7206	1.96338	0.981692	+	0.190476i	$0.0610030\pi$
$332$	0	0
$333$	−4.97381	−0.272563
$334$	0	0
$335$	17.7587	0.970263
$336$	0	0
$337$	13.0895	0.713028	0.356514	−	0.934290i	$-0.383965\pi$
$337$	13.0895	0.713028	0.356514	+	0.934290i	$0.383965\pi$
$338$	0	0
$339$	−8.86530	−0.481497
$340$	0	0
$341$	5.01393	0.271520
$342$	0	0
$343$	10.8638	0.586592
$344$	0	0
$345$	2.58465	0.139153
$346$	0	0
$347$	1.35007	0.0724758	0.0362379	−	0.999343i	$-0.488463\pi$
$347$	1.35007	0.0724758	0.0362379	+	0.999343i	$0.488463\pi$
$348$	0	0
$349$	3.78246	0.202470	0.101235	−	0.994863i	$-0.467721\pi$
$349$	3.78246	0.202470	0.101235	+	0.994863i	$0.467721\pi$
$350$	0	0
$351$	−5.43566	−0.290134
$352$	0	0
$353$	−27.7802	−1.47859	−0.739295	−	0.673381i	$-0.764841\pi$
$353$	−27.7802	−1.47859	−0.739295	+	0.673381i	$0.764841\pi$
$354$	0	0
$355$	−8.97732	−0.476467
$356$	0	0
$357$	−9.04588	−0.478759
$358$	0	0
$359$	33.7405	1.78075	0.890377	−	0.455224i	$-0.150441\pi$
$359$	33.7405	1.78075	0.890377	+	0.455224i	$0.150441\pi$
$360$	0	0
$361$	18.8279	0.990944
$362$	0	0
$363$	2.58335	0.135591
$364$	0	0
$365$	15.5036	0.811496
$366$	0	0
$367$	−1.27364	−0.0664836	−0.0332418	−	0.999447i	$-0.510583\pi$
$367$	−1.27364	−0.0664836	−0.0332418	+	0.999447i	$0.510583\pi$
$368$	0	0
$369$	−6.02032	−0.313406
$370$	0	0
$371$	−17.6879	−0.918310
$372$	0	0
$373$	19.2000	0.994137	0.497069	−	0.867711i	$-0.334410\pi$
$373$	19.2000	0.994137	0.497069	+	0.867711i	$0.334410\pi$
$374$	0	0
$375$	−8.57995	−0.443067
$376$	0	0
$377$	5.43566	0.279951
$378$	0	0
$379$	0.333289	0.0171199	0.00855995	−	0.999963i	$-0.497275\pi$
$379$	0.333289	0.0171199	0.00855995	+	0.999963i	$0.497275\pi$
$380$	0	0
$381$	0.0386548	0.00198035
$382$	0	0
$383$	1.46228	0.0747188	0.0373594	−	0.999302i	$-0.488105\pi$
$383$	1.46228	0.0747188	0.0373594	+	0.999302i	$0.488105\pi$
$384$	0	0
$385$	−24.4993	−1.24860
$386$	0	0
$387$	−10.9734	−0.557808
$388$	0	0
$389$	18.3472	0.930241	0.465120	−	0.885247i	$-0.346011\pi$
$389$	18.3472	0.930241	0.465120	+	0.885247i	$0.346011\pi$
$390$	0	0
$391$	−2.76865	−0.140017
$392$	0	0
$393$	18.2687	0.921535
$394$	0	0
$395$	−1.57501	−0.0792473
$396$	0	0
$397$	7.65239	0.384062	0.192031	−	0.981389i	$-0.438493\pi$
$397$	7.65239	0.384062	0.192031	+	0.981389i	$0.438493\pi$
$398$	0	0
$399$	−20.0950	−1.00601
$400$	0	0
$401$	16.3401	0.815986	0.407993	−	0.912985i	$-0.366229\pi$
$401$	16.3401	0.815986	0.407993	+	0.912985i	$0.366229\pi$
$402$	0	0
$403$	−9.39423	−0.467960
$404$	0	0
$405$	−2.58465	−0.128432
$406$	0	0
$407$	14.4298	0.715256
$408$	0	0
$409$	0.101086	0.00499837	0.00249919	−	0.999997i	$-0.499204\pi$
$409$	0.101086	0.00499837	0.00249919	+	0.999997i	$0.499204\pi$
$410$	0	0
$411$	−3.12032	−0.153914
$412$	0	0
$413$	11.3805	0.559996
$414$	0	0
$415$	16.5963	0.814682
$416$	0	0
$417$	7.77344	0.380667
$418$	0	0
$419$	10.3659	0.506409	0.253204	−	0.967413i	$-0.418516\pi$
$419$	10.3659	0.506409	0.253204	+	0.967413i	$0.418516\pi$
$420$	0	0
$421$	5.97694	0.291298	0.145649	−	0.989336i	$-0.453473\pi$
$421$	5.97694	0.291298	0.145649	+	0.989336i	$0.453473\pi$
$422$	0	0
$423$	−3.19760	−0.155473
$424$	0	0
$425$	−4.65250	−0.225680
$426$	0	0
$427$	12.3254	0.596470
$428$	0	0
$429$	15.7697	0.761366
$430$	0	0
$431$	−7.85116	−0.378177	−0.189088	−	0.981960i	$-0.560553\pi$
$431$	−7.85116	−0.378177	−0.189088	+	0.981960i	$0.560553\pi$
$432$	0	0
$433$	12.7222	0.611389	0.305694	−	0.952130i	$-0.401111\pi$
$433$	12.7222	0.611389	0.305694	+	0.952130i	$0.401111\pi$
$434$	0	0
$435$	2.58465	0.123925
$436$	0	0
$437$	−6.15044	−0.294216
$438$	0	0
$439$	23.4129	1.11744	0.558718	−	0.829357i	$-0.311293\pi$
$439$	23.4129	1.11744	0.558718	+	0.829357i	$0.311293\pi$
$440$	0	0
$441$	3.67493	0.174997
$442$	0	0
$443$	−5.14557	−0.244474	−0.122237	−	0.992501i	$-0.539007\pi$
$443$	−5.14557	−0.244474	−0.122237	+	0.992501i	$0.539007\pi$
$444$	0	0
$445$	38.9683	1.84727
$446$	0	0
$447$	16.7712	0.793250
$448$	0	0
$449$	−16.5786	−0.782391	−0.391195	−	0.920308i	$-0.627938\pi$
$449$	−16.5786	−0.782391	−0.391195	+	0.920308i	$0.627938\pi$
$450$	0	0
$451$	17.4658	0.822434
$452$	0	0
$453$	12.6678	0.595184
$454$	0	0
$455$	45.9026	2.15195
$456$	0	0
$457$	−11.3367	−0.530307	−0.265154	−	0.964206i	$-0.585423\pi$
$457$	−11.3367	−0.530307	−0.265154	+	0.964206i	$0.585423\pi$
$458$	0	0
$459$	2.76865	0.129230
$460$	0	0
$461$	−35.1134	−1.63539	−0.817697	−	0.575649i	$-0.804750\pi$
$461$	−35.1134	−1.63539	−0.817697	+	0.575649i	$0.804750\pi$
$462$	0	0
$463$	−28.4019	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$463$	−28.4019	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$464$	0	0
$465$	−4.46695	−0.207150
$466$	0	0
$467$	12.3209	0.570143	0.285072	−	0.958506i	$-0.407983\pi$
$467$	12.3209	0.570143	0.285072	+	0.958506i	$0.407983\pi$
$468$	0	0
$469$	22.4488	1.03659
$470$	0	0
$471$	12.1569	0.560162
$472$	0	0
$473$	31.8354	1.46379
$474$	0	0
$475$	−10.3353	−0.474218
$476$	0	0
$477$	5.41370	0.247876
$478$	0	0
$479$	−1.37261	−0.0627162	−0.0313581	−	0.999508i	$-0.509983\pi$
$479$	−1.37261	−0.0627162	−0.0313581	+	0.999508i	$0.509983\pi$
$480$	0	0
$481$	−27.0360	−1.23273
$482$	0	0
$483$	3.26725	0.148665
$484$	0	0
$485$	17.0732	0.775253
$486$	0	0
$487$	19.9010	0.901803	0.450901	−	0.892574i	$-0.351103\pi$
$487$	19.9010	0.901803	0.450901	+	0.892574i	$0.351103\pi$
$488$	0	0
$489$	9.91818	0.448516
$490$	0	0
$491$	−0.0888982	−0.00401192	−0.00200596	−	0.999998i	$-0.500639\pi$
$491$	−0.0888982	−0.00401192	−0.00200596	+	0.999998i	$0.500639\pi$
$492$	0	0
$493$	−2.76865	−0.124694
$494$	0	0
$495$	7.49845	0.337030
$496$	0	0
$497$	−11.3482	−0.509037
$498$	0	0
$499$	16.6820	0.746790	0.373395	−	0.927672i	$-0.378194\pi$
$499$	16.6820	0.746790	0.373395	+	0.927672i	$0.378194\pi$
$500$	0	0
$501$	−16.7134	−0.746701
$502$	0	0
$503$	−3.99675	−0.178206	−0.0891032	−	0.996022i	$-0.528400\pi$
$503$	−3.99675	−0.178206	−0.0891032	+	0.996022i	$0.528400\pi$
$504$	0	0
$505$	−40.2988	−1.79327
$506$	0	0
$507$	−16.5464	−0.734853
$508$	0	0
$509$	9.87300	0.437613	0.218807	−	0.975768i	$-0.429784\pi$
$509$	9.87300	0.437613	0.218807	+	0.975768i	$0.429784\pi$
$510$	0	0
$511$	19.5981	0.866967
$512$	0	0
$513$	6.15044	0.271549
$514$	0	0
$515$	39.3327	1.73321
$516$	0	0
$517$	9.27672	0.407990
$518$	0	0
$519$	−4.59989	−0.201913
$520$	0	0
$521$	−25.7062	−1.12621	−0.563104	−	0.826386i	$-0.690393\pi$
$521$	−25.7062	−1.12621	−0.563104	+	0.826386i	$0.690393\pi$
$522$	0	0
$523$	−20.9459	−0.915902	−0.457951	−	0.888978i	$-0.651416\pi$
$523$	−20.9459	−0.915902	−0.457951	+	0.888978i	$0.651416\pi$
$524$	0	0
$525$	5.49036	0.239619
$526$	0	0
$527$	4.78495	0.208436
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.48319	−0.151158
$532$	0	0
$533$	−32.7245	−1.41745
$534$	0	0
$535$	10.0035	0.432490
$536$	0	0
$537$	−20.7314	−0.894624
$538$	0	0
$539$	−10.6615	−0.459224
$540$	0	0
$541$	−19.9499	−0.857715	−0.428857	−	0.903372i	$-0.641084\pi$
$541$	−19.9499	−0.857715	−0.428857	+	0.903372i	$0.641084\pi$
$542$	0	0
$543$	−22.6705	−0.972885
$544$	0	0
$545$	−23.7714	−1.01826
$546$	0	0
$547$	−17.5884	−0.752026	−0.376013	−	0.926614i	$-0.622705\pi$
$547$	−17.5884	−0.752026	−0.376013	+	0.926614i	$0.622705\pi$
$548$	0	0
$549$	−3.77242	−0.161003
$550$	0	0
$551$	−6.15044	−0.262018
$552$	0	0
$553$	−1.99096	−0.0846644
$554$	0	0
$555$	−12.8556	−0.545688
$556$	0	0
$557$	39.4366	1.67098	0.835492	−	0.549503i	$-0.185183\pi$
$557$	39.4366	1.67098	0.835492	+	0.549503i	$0.185183\pi$
$558$	0	0
$559$	−59.6476	−2.52282
$560$	0	0
$561$	−8.03227	−0.339123
$562$	0	0
$563$	29.4911	1.24290	0.621452	−	0.783453i	$-0.286543\pi$
$563$	29.4911	1.24290	0.621452	+	0.783453i	$0.286543\pi$
$564$	0	0
$565$	−22.9137	−0.963987
$566$	0	0
$567$	−3.26725	−0.137212
$568$	0	0
$569$	3.32457	0.139373	0.0696866	−	0.997569i	$-0.477800\pi$
$569$	3.32457	0.139373	0.0696866	+	0.997569i	$0.477800\pi$
$570$	0	0
$571$	−36.9084	−1.54457	−0.772284	−	0.635278i	$-0.780886\pi$
$571$	−36.9084	−1.54457	−0.772284	+	0.635278i	$0.780886\pi$
$572$	0	0
$573$	−2.86879	−0.119845
$574$	0	0
$575$	1.68042	0.0700784
$576$	0	0
$577$	25.8580	1.07648	0.538242	−	0.842790i	$-0.319089\pi$
$577$	25.8580	1.07648	0.538242	+	0.842790i	$0.319089\pi$
$578$	0	0
$579$	9.67593	0.402118
$580$	0	0
$581$	20.9794	0.870372
$582$	0	0
$583$	−15.7059	−0.650473
$584$	0	0
$585$	−14.0493	−0.580867
$586$	0	0
$587$	−38.2916	−1.58046	−0.790232	−	0.612808i	$-0.790040\pi$
$587$	−38.2916	−1.58046	−0.790232	+	0.612808i	$0.790040\pi$
$588$	0	0
$589$	10.6296	0.437983
$590$	0	0
$591$	−4.60951	−0.189610
$592$	0	0
$593$	−9.36472	−0.384563	−0.192281	−	0.981340i	$-0.561589\pi$
$593$	−9.36472	−0.384563	−0.192281	+	0.981340i	$0.561589\pi$
$594$	0	0
$595$	−23.3805	−0.958505
$596$	0	0
$597$	−17.3597	−0.710483
$598$	0	0
$599$	−5.91152	−0.241538	−0.120769	−	0.992681i	$-0.538536\pi$
$599$	−5.91152	−0.241538	−0.120769	+	0.992681i	$0.538536\pi$
$600$	0	0
$601$	18.1326	0.739643	0.369821	−	0.929103i	$-0.379419\pi$
$601$	18.1326	0.739643	0.369821	+	0.929103i	$0.379419\pi$
$602$	0	0
$603$	−6.87084	−0.279802
$604$	0	0
$605$	6.67706	0.271461
$606$	0	0
$607$	−47.2698	−1.91862	−0.959311	−	0.282353i	$-0.908885\pi$
$607$	−47.2698	−1.91862	−0.959311	+	0.282353i	$0.908885\pi$
$608$	0	0
$609$	3.26725	0.132396
$610$	0	0
$611$	−17.3811	−0.703164
$612$	0	0
$613$	−41.3828	−1.67144	−0.835718	−	0.549159i	$-0.814948\pi$
$613$	−41.3828	−1.67144	−0.835718	+	0.549159i	$0.814948\pi$
$614$	0	0
$615$	−15.5604	−0.627457
$616$	0	0
$617$	5.22450	0.210331	0.105165	−	0.994455i	$-0.466463\pi$
$617$	5.22450	0.210331	0.105165	+	0.994455i	$0.466463\pi$
$618$	0	0
$619$	−2.08761	−0.0839080	−0.0419540	−	0.999120i	$-0.513358\pi$
$619$	−2.08761	−0.0839080	−0.0419540	+	0.999120i	$0.513358\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	49.2597	1.97355
$624$	0	0
$625$	−30.5783	−1.22313
$626$	0	0
$627$	−17.8433	−0.712594
$628$	0	0
$629$	13.7708	0.549076
$630$	0	0
$631$	0.758471	0.0301943	0.0150971	−	0.999886i	$-0.495194\pi$
$631$	0.758471	0.0301943	0.0150971	+	0.999886i	$0.495194\pi$
$632$	0	0
$633$	6.77916	0.269447
$634$	0	0
$635$	0.0999093	0.00396478
$636$	0	0
$637$	19.9757	0.791466
$638$	0	0
$639$	3.47332	0.137402
$640$	0	0
$641$	9.04373	0.357206	0.178603	−	0.983921i	$-0.442842\pi$
$641$	9.04373	0.357206	0.178603	+	0.983921i	$0.442842\pi$
$642$	0	0
$643$	37.8286	1.49181	0.745907	−	0.666050i	$-0.232016\pi$
$643$	37.8286	1.49181	0.745907	+	0.666050i	$0.232016\pi$
$644$	0	0
$645$	−28.3624	−1.11677
$646$	0	0
$647$	9.03911	0.355364	0.177682	−	0.984088i	$-0.443140\pi$
$647$	9.03911	0.355364	0.177682	+	0.984088i	$0.443140\pi$
$648$	0	0
$649$	10.1053	0.396666
$650$	0	0
$651$	−5.64666	−0.221310
$652$	0	0
$653$	−3.81709	−0.149374	−0.0746871	−	0.997207i	$-0.523796\pi$
$653$	−3.81709	−0.149374	−0.0746871	+	0.997207i	$0.523796\pi$
$654$	0	0
$655$	47.2183	1.84497
$656$	0	0
$657$	−5.99833	−0.234017
$658$	0	0
$659$	−9.64692	−0.375791	−0.187895	−	0.982189i	$-0.560167\pi$
$659$	−9.64692	−0.375791	−0.187895	+	0.982189i	$0.560167\pi$
$660$	0	0
$661$	−19.7624	−0.768667	−0.384334	−	0.923194i	$-0.625569\pi$
$661$	−19.7624	−0.768667	−0.384334	+	0.923194i	$0.625569\pi$
$662$	0	0
$663$	15.0495	0.584473
$664$	0	0
$665$	−51.9387	−2.01410
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	12.4694	0.482095
$670$	0	0
$671$	10.9443	0.422502
$672$	0	0
$673$	−29.6832	−1.14420	−0.572101	−	0.820183i	$-0.693871\pi$
$673$	−29.6832	−1.14420	−0.572101	+	0.820183i	$0.693871\pi$
$674$	0	0
$675$	−1.68042	−0.0646795
$676$	0	0
$677$	19.7488	0.759008	0.379504	−	0.925190i	$-0.376095\pi$
$677$	19.7488	0.759008	0.379504	+	0.925190i	$0.376095\pi$
$678$	0	0
$679$	21.5822	0.828247
$680$	0	0
$681$	−29.2499	−1.12086
$682$	0	0
$683$	13.3720	0.511667	0.255834	−	0.966721i	$-0.417650\pi$
$683$	13.3720	0.511667	0.255834	+	0.966721i	$0.417650\pi$
$684$	0	0
$685$	−8.06495	−0.308146
$686$	0	0
$687$	27.7176	1.05749
$688$	0	0
$689$	29.4270	1.12108
$690$	0	0
$691$	−49.0254	−1.86501	−0.932507	−	0.361153i	$-0.882383\pi$
$691$	−49.0254	−1.86501	−0.932507	+	0.361153i	$0.882383\pi$
$692$	0	0
$693$	9.47878	0.360069
$694$	0	0
$695$	20.0916	0.762119
$696$	0	0
$697$	16.6682	0.631353
$698$	0	0
$699$	−20.4677	−0.774161
$700$	0	0
$701$	0.916175	0.0346035	0.0173017	−	0.999850i	$-0.494492\pi$
$701$	0.916175	0.0346035	0.0173017	+	0.999850i	$0.494492\pi$
$702$	0	0
$703$	30.5911	1.15377
$704$	0	0
$705$	−8.26469	−0.311266
$706$	0	0
$707$	−50.9416	−1.91586
$708$	0	0
$709$	−4.11514	−0.154547	−0.0772736	−	0.997010i	$-0.524621\pi$
$709$	−4.11514	−0.154547	−0.0772736	+	0.997010i	$0.524621\pi$
$710$	0	0
$711$	0.609370	0.0228532
$712$	0	0
$713$	−1.72826	−0.0647238
$714$	0	0
$715$	40.7591	1.52430
$716$	0	0
$717$	−2.62599	−0.0980694
$718$	0	0
$719$	−43.0447	−1.60530	−0.802648	−	0.596453i	$-0.796576\pi$
$719$	−43.0447	−1.60530	−0.802648	+	0.596453i	$0.796576\pi$
$720$	0	0
$721$	49.7204	1.85168
$722$	0	0
$723$	8.14128	0.302778
$724$	0	0
$725$	1.68042	0.0624093
$726$	0	0
$727$	−8.96336	−0.332433	−0.166216	−	0.986089i	$-0.553155\pi$
$727$	−8.96336	−0.332433	−0.166216	+	0.986089i	$0.553155\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.3815	1.12370
$732$	0	0
$733$	−10.5056	−0.388034	−0.194017	−	0.980998i	$-0.562152\pi$
$733$	−10.5056	−0.388034	−0.194017	+	0.980998i	$0.562152\pi$
$734$	0	0
$735$	9.49842	0.350355
$736$	0	0
$737$	19.9333	0.734253
$738$	0	0
$739$	52.9028	1.94606	0.973030	−	0.230679i	$-0.0740946\pi$
$739$	52.9028	1.94606	0.973030	+	0.230679i	$0.0740946\pi$
$740$	0	0
$741$	33.4317	1.22815
$742$	0	0
$743$	−11.0250	−0.404469	−0.202234	−	0.979337i	$-0.564820\pi$
$743$	−11.0250	−0.404469	−0.202234	+	0.979337i	$0.564820\pi$
$744$	0	0
$745$	43.3477	1.58814
$746$	0	0
$747$	−6.42111	−0.234936
$748$	0	0
$749$	12.6454	0.462054
$750$	0	0
$751$	−45.1829	−1.64875	−0.824374	−	0.566046i	$-0.808473\pi$
$751$	−45.1829	−1.64875	−0.824374	+	0.566046i	$0.808473\pi$
$752$	0	0
$753$	−13.8908	−0.506208
$754$	0	0
$755$	32.7418	1.19160
$756$	0	0
$757$	5.11999	0.186089	0.0930446	−	0.995662i	$-0.470340\pi$
$757$	5.11999	0.186089	0.0930446	+	0.995662i	$0.470340\pi$
$758$	0	0
$759$	2.90115	0.105305
$760$	0	0
$761$	47.5479	1.72361	0.861805	−	0.507240i	$-0.169334\pi$
$761$	47.5479	1.72361	0.861805	+	0.507240i	$0.169334\pi$
$762$	0	0
$763$	−30.0494	−1.08786
$764$	0	0
$765$	7.15600	0.258726
$766$	0	0
$767$	−18.9335	−0.683648
$768$	0	0
$769$	−32.5122	−1.17242	−0.586210	−	0.810159i	$-0.699381\pi$
$769$	−32.5122	−1.17242	−0.586210	+	0.810159i	$0.699381\pi$
$770$	0	0
$771$	−22.3718	−0.805701
$772$	0	0
$773$	−23.6598	−0.850984	−0.425492	−	0.904962i	$-0.639899\pi$
$773$	−23.6598	−0.850984	−0.425492	+	0.904962i	$0.639899\pi$
$774$	0	0
$775$	−2.90420	−0.104322
$776$	0	0
$777$	−16.2507	−0.582990
$778$	0	0
$779$	37.0277	1.32665
$780$	0	0
$781$	−10.0766	−0.360569
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	31.4214	1.12148
$786$	0	0
$787$	−35.3648	−1.26062	−0.630310	−	0.776344i	$-0.717072\pi$
$787$	−35.3648	−1.26062	−0.630310	+	0.776344i	$0.717072\pi$
$788$	0	0
$789$	24.8644	0.885197
$790$	0	0
$791$	−28.9652	−1.02988
$792$	0	0
$793$	−20.5056	−0.728175
$794$	0	0
$795$	13.9925	0.496263
$796$	0	0
$797$	4.02243	0.142482	0.0712408	−	0.997459i	$-0.477304\pi$
$797$	4.02243	0.142482	0.0712408	+	0.997459i	$0.477304\pi$
$798$	0	0
$799$	8.85306	0.313199
$800$	0	0
$801$	−15.0768	−0.532713
$802$	0	0
$803$	17.4020	0.614105
$804$	0	0
$805$	8.44471	0.297637
$806$	0	0
$807$	13.3151	0.468715
$808$	0	0
$809$	31.0035	1.09003	0.545013	−	0.838428i	$-0.316525\pi$
$809$	31.0035	1.09003	0.545013	+	0.838428i	$0.316525\pi$
$810$	0	0
$811$	9.52266	0.334386	0.167193	−	0.985924i	$-0.446530\pi$
$811$	9.52266	0.334386	0.167193	+	0.985924i	$0.446530\pi$
$812$	0	0
$813$	13.0537	0.457815
$814$	0	0
$815$	25.6350	0.897956
$816$	0	0
$817$	67.4911	2.36122
$818$	0	0
$819$	−17.7597	−0.620574
$820$	0	0
$821$	1.87751	0.0655257	0.0327628	−	0.999463i	$-0.489569\pi$
$821$	1.87751	0.0655257	0.0327628	+	0.999463i	$0.489569\pi$
$822$	0	0
$823$	−26.7916	−0.933897	−0.466948	−	0.884285i	$-0.654647\pi$
$823$	−26.7916	−0.933897	−0.466948	+	0.884285i	$0.654647\pi$
$824$	0	0
$825$	4.87515	0.169731
$826$	0	0
$827$	25.2439	0.877818	0.438909	−	0.898532i	$-0.355365\pi$
$827$	25.2439	0.877818	0.438909	+	0.898532i	$0.355365\pi$
$828$	0	0
$829$	43.8248	1.52210	0.761050	−	0.648693i	$-0.224684\pi$
$829$	43.8248	1.52210	0.761050	+	0.648693i	$0.224684\pi$
$830$	0	0
$831$	−31.3272	−1.08673
$832$	0	0
$833$	−10.1746	−0.352529
$834$	0	0
$835$	−43.1984	−1.49494
$836$	0	0
$837$	1.72826	0.0597374
$838$	0	0
$839$	50.4257	1.74089	0.870445	−	0.492266i	$-0.163831\pi$
$839$	50.4257	1.74089	0.870445	+	0.492266i	$0.163831\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	8.97961	0.309274
$844$	0	0
$845$	−42.7668	−1.47122
$846$	0	0
$847$	8.44045	0.290017
$848$	0	0
$849$	−2.83888	−0.0974300
$850$	0	0
$851$	−4.97381	−0.170500
$852$	0	0
$853$	−28.0675	−0.961014	−0.480507	−	0.876991i	$-0.659547\pi$
$853$	−28.0675	−0.961014	−0.480507	+	0.876991i	$0.659547\pi$
$854$	0	0
$855$	15.8967	0.543657
$856$	0	0
$857$	−10.2561	−0.350341	−0.175170	−	0.984538i	$-0.556048\pi$
$857$	−10.2561	−0.350341	−0.175170	+	0.984538i	$0.556048\pi$
$858$	0	0
$859$	41.7755	1.42536	0.712681	−	0.701488i	$-0.247481\pi$
$859$	41.7755	1.42536	0.712681	+	0.701488i	$0.247481\pi$
$860$	0	0
$861$	−19.6699	−0.670349
$862$	0	0
$863$	−13.2911	−0.452433	−0.226217	−	0.974077i	$-0.572636\pi$
$863$	−13.2911	−0.452433	−0.226217	+	0.974077i	$0.572636\pi$
$864$	0	0
$865$	−11.8891	−0.404242
$866$	0	0
$867$	9.33456	0.317018
$868$	0	0
$869$	−1.76787	−0.0599709
$870$	0	0
$871$	−37.3476	−1.26547
$872$	0	0
$873$	−6.60560	−0.223566
$874$	0	0
$875$	−28.0329	−0.947684
$876$	0	0
$877$	−28.6871	−0.968694	−0.484347	−	0.874876i	$-0.660943\pi$
$877$	−28.6871	−0.968694	−0.484347	+	0.874876i	$0.660943\pi$
$878$	0	0
$879$	17.3913	0.586594
$880$	0	0
$881$	−13.6373	−0.459451	−0.229726	−	0.973255i	$-0.573783\pi$
$881$	−13.6373	−0.459451	−0.229726	+	0.973255i	$0.573783\pi$
$882$	0	0
$883$	−30.8554	−1.03837	−0.519183	−	0.854663i	$-0.673764\pi$
$883$	−30.8554	−1.03837	−0.519183	+	0.854663i	$0.673764\pi$
$884$	0	0
$885$	−9.00284	−0.302627
$886$	0	0
$887$	20.2891	0.681242	0.340621	−	0.940201i	$-0.389363\pi$
$887$	20.2891	0.681242	0.340621	+	0.940201i	$0.389363\pi$
$888$	0	0
$889$	0.126295	0.00423580
$890$	0	0
$891$	−2.90115	−0.0971921
$892$	0	0
$893$	19.6667	0.658120
$894$	0	0
$895$	−53.5833	−1.79109
$896$	0	0
$897$	−5.43566	−0.181492
$898$	0	0
$899$	−1.72826	−0.0576407
$900$	0	0
$901$	−14.9886	−0.499344
$902$	0	0
$903$	−35.8528	−1.19311
$904$	0	0
$905$	−58.5954	−1.94778
$906$	0	0
$907$	−7.37111	−0.244754	−0.122377	−	0.992484i	$-0.539052\pi$
$907$	−7.37111	−0.244754	−0.122377	+	0.992484i	$0.539052\pi$
$908$	0	0
$909$	15.5916	0.517140
$910$	0	0
$911$	29.3900	0.973735	0.486867	−	0.873476i	$-0.338140\pi$
$911$	29.3900	0.973735	0.486867	+	0.873476i	$0.338140\pi$
$912$	0	0
$913$	18.6286	0.616516
$914$	0	0
$915$	−9.75039	−0.322338
$916$	0	0
$917$	59.6885	1.97109
$918$	0	0
$919$	−47.9846	−1.58287	−0.791433	−	0.611256i	$-0.790665\pi$
$919$	−47.9846	−1.58287	−0.791433	+	0.611256i	$0.790665\pi$
$920$	0	0
$921$	−18.8046	−0.619632
$922$	0	0
$923$	18.8798	0.621436
$924$	0	0
$925$	−8.35810	−0.274813
$926$	0	0
$927$	−15.2178	−0.499818
$928$	0	0
$929$	−11.9401	−0.391741	−0.195871	−	0.980630i	$-0.562753\pi$
$929$	−11.9401	−0.391741	−0.195871	+	0.980630i	$0.562753\pi$
$930$	0	0
$931$	−22.6025	−0.740766
$932$	0	0
$933$	10.6986	0.350256
$934$	0	0
$935$	−20.7606	−0.678944
$936$	0	0
$937$	6.48683	0.211915	0.105958	−	0.994371i	$-0.466209\pi$
$937$	6.48683	0.211915	0.105958	+	0.994371i	$0.466209\pi$
$938$	0	0
$939$	−26.6605	−0.870034
$940$	0	0
$941$	53.7386	1.75183	0.875915	−	0.482466i	$-0.160259\pi$
$941$	53.7386	1.75183	0.875915	+	0.482466i	$0.160259\pi$
$942$	0	0
$943$	−6.02032	−0.196049
$944$	0	0
$945$	−8.44471	−0.274706
$946$	0	0
$947$	1.43493	0.0466291	0.0233145	−	0.999728i	$-0.492578\pi$
$947$	1.43493	0.0466291	0.0233145	+	0.999728i	$0.492578\pi$
$948$	0	0
$949$	−32.6049	−1.05840
$950$	0	0
$951$	24.9984	0.810628
$952$	0	0
$953$	−47.4944	−1.53850	−0.769248	−	0.638950i	$-0.779369\pi$
$953$	−47.4944	−1.53850	−0.769248	+	0.638950i	$0.779369\pi$
$954$	0	0
$955$	−7.41482	−0.239938
$956$	0	0
$957$	2.90115	0.0937807
$958$	0	0
$959$	−10.1949	−0.329210
$960$	0	0
$961$	−28.0131	−0.903649
$962$	0	0
$963$	−3.87036	−0.124721
$964$	0	0
$965$	25.0089	0.805065
$966$	0	0
$967$	9.45128	0.303933	0.151966	−	0.988386i	$-0.451439\pi$
$967$	9.45128	0.303933	0.151966	+	0.988386i	$0.451439\pi$
$968$	0	0
$969$	−17.0284	−0.547032
$970$	0	0
$971$	2.14235	0.0687514	0.0343757	−	0.999409i	$-0.489056\pi$
$971$	2.14235	0.0687514	0.0343757	+	0.999409i	$0.489056\pi$
$972$	0	0
$973$	25.3978	0.814216
$974$	0	0
$975$	−9.13421	−0.292529
$976$	0	0
$977$	58.8448	1.88261	0.941306	−	0.337555i	$-0.109600\pi$
$977$	58.8448	1.88261	0.941306	+	0.337555i	$0.109600\pi$
$978$	0	0
$979$	43.7400	1.39794
$980$	0	0
$981$	9.19716	0.293643
$982$	0	0
$983$	−55.4775	−1.76946	−0.884730	−	0.466105i	$-0.845657\pi$
$983$	−55.4775	−1.76946	−0.884730	+	0.466105i	$0.845657\pi$
$984$	0	0
$985$	−11.9140	−0.379611
$986$	0	0
$987$	−10.4474	−0.332544
$988$	0	0
$989$	−10.9734	−0.348933
$990$	0	0
$991$	2.28446	0.0725683	0.0362841	−	0.999342i	$-0.488448\pi$
$991$	2.28446	0.0725683	0.0362841	+	0.999342i	$0.488448\pi$
$992$	0	0
$993$	−35.7206	−1.13356
$994$	0	0
$995$	−44.8687	−1.42243
$996$	0	0
$997$	36.7888	1.16511	0.582557	−	0.812790i	$-0.302052\pi$
$997$	36.7888	1.16511	0.582557	+	0.812790i	$0.302052\pi$
$998$	0	0
$999$	4.97381	0.157364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.i.1.4	✓	16

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.58465 q^{5} -3.26725 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.58465 q^{5} -3.26725 q^{7} +1.00000 q^{9} -2.90115 q^{11} +5.43566 q^{13} +2.58465 q^{15} -2.76865 q^{17} -6.15044 q^{19} +3.26725 q^{21} +1.00000 q^{23} +1.68042 q^{25} -1.00000 q^{27} +1.00000 q^{29} -1.72826 q^{31} +2.90115 q^{33} +8.44471 q^{35} -4.97381 q^{37} -5.43566 q^{39} -6.02032 q^{41} -10.9734 q^{43} -2.58465 q^{45} -3.19760 q^{47} +3.67493 q^{49} +2.76865 q^{51} +5.41370 q^{53} +7.49845 q^{55} +6.15044 q^{57} -3.48319 q^{59} -3.77242 q^{61} -3.26725 q^{63} -14.0493 q^{65} -6.87084 q^{67} -1.00000 q^{69} +3.47332 q^{71} -5.99833 q^{73} -1.68042 q^{75} +9.47878 q^{77} +0.609370 q^{79} +1.00000 q^{81} -6.42111 q^{83} +7.15600 q^{85} -1.00000 q^{87} -15.0768 q^{89} -17.7597 q^{91} +1.72826 q^{93} +15.8967 q^{95} -6.60560 q^{97} -2.90115 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 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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