LMFDB - Embedded newform 2004.2.a.d.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2004,2,Mod(1,2004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2004 = 2^{2} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2004.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:	$16.0020205651$
Analytic rank:	$0$
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} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 $ x^9 - 29x^7 - 7x^6 + 266x^5 + 69x^4 - 901x^3 - 199x^2 + 875*x + 391
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.42964$ of defining polynomial
Character	$\chi$	$=$	2004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.42964 q^{5} +3.44225 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.42964 q^{5} +3.44225 q^{7} +1.00000 q^{9} -3.41215 q^{11} -2.00499 q^{13} +3.42964 q^{15} -3.07729 q^{17} -0.565935 q^{19} +3.44225 q^{21} +7.40517 q^{23} +6.76244 q^{25} +1.00000 q^{27} +9.73140 q^{29} -2.62390 q^{31} -3.41215 q^{33} +11.8057 q^{35} -0.527120 q^{37} -2.00499 q^{39} +3.19921 q^{41} +7.98923 q^{43} +3.42964 q^{45} -6.96499 q^{47} +4.84911 q^{49} -3.07729 q^{51} +2.40664 q^{53} -11.7025 q^{55} -0.565935 q^{57} +6.23937 q^{59} -14.3204 q^{61} +3.44225 q^{63} -6.87641 q^{65} +0.623373 q^{67} +7.40517 q^{69} -13.4942 q^{71} +8.13677 q^{73} +6.76244 q^{75} -11.7455 q^{77} -4.25787 q^{79} +1.00000 q^{81} -15.2864 q^{83} -10.5540 q^{85} +9.73140 q^{87} +3.69141 q^{89} -6.90169 q^{91} -2.62390 q^{93} -1.94095 q^{95} +10.6187 q^{97} -3.41215 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9} + 7 q^{11} + 6 q^{13} + 9 q^{15} + 7 q^{17} + 2 q^{19} + 2 q^{21} + 19 q^{23} + 22 q^{25} + 9 q^{27} + 13 q^{29} + 12 q^{31} + 7 q^{33} + 4 q^{35} + 15 q^{37} + 6 q^{39} + 18 q^{41} - 6 q^{43} + 9 q^{45} + 25 q^{47} + 19 q^{49} + 7 q^{51} + 17 q^{53} - 3 q^{55} + 2 q^{57} + 3 q^{59} + 14 q^{61} + 2 q^{63} + 14 q^{65} - 4 q^{67} + 19 q^{69} + 17 q^{71} - 20 q^{73} + 22 q^{75} + 14 q^{77} - 8 q^{79} + 9 q^{81} - q^{83} + 5 q^{85} + 13 q^{87} + 36 q^{89} - 41 q^{91} + 12 q^{93} + 5 q^{95} + 31 q^{97} + 7 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9 + 7 * q^11 + 6 * q^13 + 9 * q^15 + 7 * q^17 + 2 * q^19 + 2 * q^21 + 19 * q^23 + 22 * q^25 + 9 * q^27 + 13 * q^29 + 12 * q^31 + 7 * q^33 + 4 * q^35 + 15 * q^37 + 6 * q^39 + 18 * q^41 - 6 * q^43 + 9 * q^45 + 25 * q^47 + 19 * q^49 + 7 * q^51 + 17 * q^53 - 3 * q^55 + 2 * q^57 + 3 * q^59 + 14 * q^61 + 2 * q^63 + 14 * q^65 - 4 * q^67 + 19 * q^69 + 17 * q^71 - 20 * q^73 + 22 * q^75 + 14 * q^77 - 8 * q^79 + 9 * q^81 - q^83 + 5 * q^85 + 13 * q^87 + 36 * q^89 - 41 * q^91 + 12 * q^93 + 5 * q^95 + 31 * q^97 + 7 * 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
$5$	3.42964	1.53378	0.766891	−	0.641777i	$-0.221803\pi$
$5$	3.42964	1.53378	0.766891	+	0.641777i	$0.221803\pi$
$6$	0	0
$7$	3.44225	1.30105	0.650525	−	0.759485i	$-0.274549\pi$
$7$	3.44225	1.30105	0.650525	+	0.759485i	$0.274549\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.41215	−1.02880	−0.514401	−	0.857549i	$-0.671986\pi$
$11$	−3.41215	−1.02880	−0.514401	+	0.857549i	$0.671986\pi$
$12$	0	0
$13$	−2.00499	−0.556085	−0.278042	−	0.960569i	$-0.589686\pi$
$13$	−2.00499	−0.556085	−0.278042	+	0.960569i	$0.589686\pi$
$14$	0	0
$15$	3.42964	0.885530
$16$	0	0
$17$	−3.07729	−0.746353	−0.373176	−	0.927760i	$-0.621731\pi$
$17$	−3.07729	−0.746353	−0.373176	+	0.927760i	$0.621731\pi$
$18$	0	0
$19$	−0.565935	−0.129834	−0.0649172	−	0.997891i	$-0.520678\pi$
$19$	−0.565935	−0.129834	−0.0649172	+	0.997891i	$0.520678\pi$
$20$	0	0
$21$	3.44225	0.751161
$22$	0	0
$23$	7.40517	1.54409	0.772043	−	0.635571i	$-0.219235\pi$
$23$	7.40517	1.54409	0.772043	+	0.635571i	$0.219235\pi$
$24$	0	0
$25$	6.76244	1.35249
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.73140	1.80708	0.903538	−	0.428508i	$-0.140961\pi$
$29$	9.73140	1.80708	0.903538	+	0.428508i	$0.140961\pi$
$30$	0	0
$31$	−2.62390	−0.471266	−0.235633	−	0.971842i	$-0.575716\pi$
$31$	−2.62390	−0.471266	−0.235633	+	0.971842i	$0.575716\pi$
$32$	0	0
$33$	−3.41215	−0.593980
$34$	0	0
$35$	11.8057	1.99553
$36$	0	0
$37$	−0.527120	−0.0866580	−0.0433290	−	0.999061i	$-0.513796\pi$
$37$	−0.527120	−0.0866580	−0.0433290	+	0.999061i	$0.513796\pi$
$38$	0	0
$39$	−2.00499	−0.321056
$40$	0	0
$41$	3.19921	0.499632	0.249816	−	0.968293i	$-0.419630\pi$
$41$	3.19921	0.499632	0.249816	+	0.968293i	$0.419630\pi$
$42$	0	0
$43$	7.98923	1.21835	0.609173	−	0.793037i	$-0.291501\pi$
$43$	7.98923	1.21835	0.609173	+	0.793037i	$0.291501\pi$
$44$	0	0
$45$	3.42964	0.511261
$46$	0	0
$47$	−6.96499	−1.01595	−0.507974	−	0.861372i	$-0.669605\pi$
$47$	−6.96499	−1.01595	−0.507974	+	0.861372i	$0.669605\pi$
$48$	0	0
$49$	4.84911	0.692729
$50$	0	0
$51$	−3.07729	−0.430907
$52$	0	0
$53$	2.40664	0.330577	0.165288	−	0.986245i	$-0.447144\pi$
$53$	2.40664	0.330577	0.165288	+	0.986245i	$0.447144\pi$
$54$	0	0
$55$	−11.7025	−1.57796
$56$	0	0
$57$	−0.565935	−0.0749599
$58$	0	0
$59$	6.23937	0.812297	0.406149	−	0.913807i	$-0.366872\pi$
$59$	6.23937	0.812297	0.406149	+	0.913807i	$0.366872\pi$
$60$	0	0
$61$	−14.3204	−1.83354	−0.916769	−	0.399418i	$-0.869212\pi$
$61$	−14.3204	−1.83354	−0.916769	+	0.399418i	$0.869212\pi$
$62$	0	0
$63$	3.44225	0.433683
$64$	0	0
$65$	−6.87641	−0.852913
$66$	0	0
$67$	0.623373	0.0761572	0.0380786	−	0.999275i	$-0.487876\pi$
$67$	0.623373	0.0761572	0.0380786	+	0.999275i	$0.487876\pi$
$68$	0	0
$69$	7.40517	0.891478
$70$	0	0
$71$	−13.4942	−1.60147	−0.800735	−	0.599018i	$-0.795558\pi$
$71$	−13.4942	−1.60147	−0.800735	+	0.599018i	$0.795558\pi$
$72$	0	0
$73$	8.13677	0.952336	0.476168	−	0.879354i	$-0.342025\pi$
$73$	8.13677	0.952336	0.476168	+	0.879354i	$0.342025\pi$
$74$	0	0
$75$	6.76244	0.780860
$76$	0	0
$77$	−11.7455	−1.33852
$78$	0	0
$79$	−4.25787	−0.479048	−0.239524	−	0.970890i	$-0.576991\pi$
$79$	−4.25787	−0.479048	−0.239524	+	0.970890i	$0.576991\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.2864	−1.67790	−0.838951	−	0.544207i	$-0.816831\pi$
$83$	−15.2864	−1.67790	−0.838951	+	0.544207i	$0.816831\pi$
$84$	0	0
$85$	−10.5540	−1.14474
$86$	0	0
$87$	9.73140	1.04332
$88$	0	0
$89$	3.69141	0.391289	0.195645	−	0.980675i	$-0.437320\pi$
$89$	3.69141	0.391289	0.195645	+	0.980675i	$0.437320\pi$
$90$	0	0
$91$	−6.90169	−0.723494
$92$	0	0
$93$	−2.62390	−0.272086
$94$	0	0
$95$	−1.94095	−0.199138
$96$	0	0
$97$	10.6187	1.07816	0.539082	−	0.842253i	$-0.318771\pi$
$97$	10.6187	1.07816	0.539082	+	0.842253i	$0.318771\pi$
$98$	0	0
$99$	−3.41215	−0.342934
$100$	0	0
$101$	15.4086	1.53321	0.766604	−	0.642120i	$-0.221945\pi$
$101$	15.4086	1.53321	0.766604	+	0.642120i	$0.221945\pi$
$102$	0	0
$103$	0.208803	0.0205740	0.0102870	−	0.999947i	$-0.496725\pi$
$103$	0.208803	0.0205740	0.0102870	+	0.999947i	$0.496725\pi$
$104$	0	0
$105$	11.8057	1.15212
$106$	0	0
$107$	−15.4023	−1.48900	−0.744498	−	0.667624i	$-0.767311\pi$
$107$	−15.4023	−1.48900	−0.744498	+	0.667624i	$0.767311\pi$
$108$	0	0
$109$	−18.6798	−1.78920	−0.894600	−	0.446867i	$-0.852540\pi$
$109$	−18.6798	−1.78920	−0.894600	+	0.446867i	$0.852540\pi$
$110$	0	0
$111$	−0.527120	−0.0500320
$112$	0	0
$113$	7.05501	0.663679	0.331840	−	0.943336i	$-0.392331\pi$
$113$	7.05501	0.663679	0.331840	+	0.943336i	$0.392331\pi$
$114$	0	0
$115$	25.3971	2.36829
$116$	0	0
$117$	−2.00499	−0.185362
$118$	0	0
$119$	−10.5928	−0.971042
$120$	0	0
$121$	0.642792	0.0584356
$122$	0	0
$123$	3.19921	0.288463
$124$	0	0
$125$	6.04455	0.540641
$126$	0	0
$127$	2.32500	0.206310	0.103155	−	0.994665i	$-0.467106\pi$
$127$	2.32500	0.206310	0.103155	+	0.994665i	$0.467106\pi$
$128$	0	0
$129$	7.98923	0.703413
$130$	0	0
$131$	10.3974	0.908423	0.454211	−	0.890894i	$-0.349921\pi$
$131$	10.3974	0.908423	0.454211	+	0.890894i	$0.349921\pi$
$132$	0	0
$133$	−1.94809	−0.168921
$134$	0	0
$135$	3.42964	0.295177
$136$	0	0
$137$	3.14496	0.268692	0.134346	−	0.990934i	$-0.457107\pi$
$137$	3.14496	0.268692	0.134346	+	0.990934i	$0.457107\pi$
$138$	0	0
$139$	−4.67040	−0.396138	−0.198069	−	0.980188i	$-0.563467\pi$
$139$	−4.67040	−0.396138	−0.198069	+	0.980188i	$0.563467\pi$
$140$	0	0
$141$	−6.96499	−0.586558
$142$	0	0
$143$	6.84134	0.572102
$144$	0	0
$145$	33.3752	2.77166
$146$	0	0
$147$	4.84911	0.399947
$148$	0	0
$149$	17.4241	1.42744	0.713721	−	0.700430i	$-0.247009\pi$
$149$	17.4241	1.42744	0.713721	+	0.700430i	$0.247009\pi$
$150$	0	0
$151$	2.51771	0.204888	0.102444	−	0.994739i	$-0.467334\pi$
$151$	2.51771	0.204888	0.102444	+	0.994739i	$0.467334\pi$
$152$	0	0
$153$	−3.07729	−0.248784
$154$	0	0
$155$	−8.99903	−0.722820
$156$	0	0
$157$	−12.3224	−0.983436	−0.491718	−	0.870754i	$-0.663631\pi$
$157$	−12.3224	−0.983436	−0.491718	+	0.870754i	$0.663631\pi$
$158$	0	0
$159$	2.40664	0.190859
$160$	0	0
$161$	25.4905	2.00893
$162$	0	0
$163$	−17.4945	−1.37027	−0.685137	−	0.728414i	$-0.740258\pi$
$163$	−17.4945	−1.37027	−0.685137	+	0.728414i	$0.740258\pi$
$164$	0	0
$165$	−11.7025	−0.911036
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−8.98001	−0.690770
$170$	0	0
$171$	−0.565935	−0.0432781
$172$	0	0
$173$	−8.48906	−0.645411	−0.322706	−	0.946499i	$-0.604592\pi$
$173$	−8.48906	−0.645411	−0.322706	+	0.946499i	$0.604592\pi$
$174$	0	0
$175$	23.2780	1.75965
$176$	0	0
$177$	6.23937	0.468980
$178$	0	0
$179$	−9.45927	−0.707019	−0.353510	−	0.935431i	$-0.615012\pi$
$179$	−9.45927	−0.707019	−0.353510	+	0.935431i	$0.615012\pi$
$180$	0	0
$181$	−2.41594	−0.179575	−0.0897877	−	0.995961i	$-0.528619\pi$
$181$	−2.41594	−0.179575	−0.0897877	+	0.995961i	$0.528619\pi$
$182$	0	0
$183$	−14.3204	−1.05859
$184$	0	0
$185$	−1.80783	−0.132915
$186$	0	0
$187$	10.5002	0.767850
$188$	0	0
$189$	3.44225	0.250387
$190$	0	0
$191$	5.50321	0.398198	0.199099	−	0.979979i	$-0.436198\pi$
$191$	5.50321	0.398198	0.199099	+	0.979979i	$0.436198\pi$
$192$	0	0
$193$	−15.1701	−1.09197	−0.545983	−	0.837796i	$-0.683844\pi$
$193$	−15.1701	−1.09197	−0.545983	+	0.837796i	$0.683844\pi$
$194$	0	0
$195$	−6.87641	−0.492430
$196$	0	0
$197$	7.92869	0.564896	0.282448	−	0.959283i	$-0.408854\pi$
$197$	7.92869	0.564896	0.282448	+	0.959283i	$0.408854\pi$
$198$	0	0
$199$	−22.8595	−1.62047	−0.810233	−	0.586107i	$-0.800660\pi$
$199$	−22.8595	−1.62047	−0.810233	+	0.586107i	$0.800660\pi$
$200$	0	0
$201$	0.623373	0.0439694
$202$	0	0
$203$	33.4979	2.35109
$204$	0	0
$205$	10.9721	0.766327
$206$	0	0
$207$	7.40517	0.514695
$208$	0	0
$209$	1.93106	0.133574
$210$	0	0
$211$	−9.58607	−0.659932	−0.329966	−	0.943993i	$-0.607037\pi$
$211$	−9.58607	−0.659932	−0.329966	+	0.943993i	$0.607037\pi$
$212$	0	0
$213$	−13.4942	−0.924609
$214$	0	0
$215$	27.4002	1.86868
$216$	0	0
$217$	−9.03212	−0.613141
$218$	0	0
$219$	8.13677	0.549832
$220$	0	0
$221$	6.16995	0.415035
$222$	0	0
$223$	11.0930	0.742842	0.371421	−	0.928464i	$-0.378871\pi$
$223$	11.0930	0.742842	0.371421	+	0.928464i	$0.378871\pi$
$224$	0	0
$225$	6.76244	0.450830
$226$	0	0
$227$	−13.3442	−0.885686	−0.442843	−	0.896599i	$-0.646030\pi$
$227$	−13.3442	−0.885686	−0.442843	+	0.896599i	$0.646030\pi$
$228$	0	0
$229$	8.71943	0.576196	0.288098	−	0.957601i	$-0.406977\pi$
$229$	8.71943	0.576196	0.288098	+	0.957601i	$0.406977\pi$
$230$	0	0
$231$	−11.7455	−0.772797
$232$	0	0
$233$	23.4773	1.53805	0.769026	−	0.639218i	$-0.220742\pi$
$233$	23.4773	1.53805	0.769026	+	0.639218i	$0.220742\pi$
$234$	0	0
$235$	−23.8874	−1.55824
$236$	0	0
$237$	−4.25787	−0.276579
$238$	0	0
$239$	−23.7397	−1.53559	−0.767797	−	0.640693i	$-0.778647\pi$
$239$	−23.7397	−1.53559	−0.767797	+	0.640693i	$0.778647\pi$
$240$	0	0
$241$	2.51503	0.162007	0.0810037	−	0.996714i	$-0.474187\pi$
$241$	2.51503	0.162007	0.0810037	+	0.996714i	$0.474187\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	16.6307	1.06250
$246$	0	0
$247$	1.13470	0.0721989
$248$	0	0
$249$	−15.2864	−0.968737
$250$	0	0
$251$	9.11605	0.575400	0.287700	−	0.957721i	$-0.407109\pi$
$251$	9.11605	0.575400	0.287700	+	0.957721i	$0.407109\pi$
$252$	0	0
$253$	−25.2676	−1.58856
$254$	0	0
$255$	−10.5540	−0.660918
$256$	0	0
$257$	29.6735	1.85098	0.925490	−	0.378773i	$-0.123654\pi$
$257$	29.6735	1.85098	0.925490	+	0.378773i	$0.123654\pi$
$258$	0	0
$259$	−1.81448	−0.112746
$260$	0	0
$261$	9.73140	0.602359
$262$	0	0
$263$	−6.20229	−0.382450	−0.191225	−	0.981546i	$-0.561246\pi$
$263$	−6.20229	−0.382450	−0.191225	+	0.981546i	$0.561246\pi$
$264$	0	0
$265$	8.25390	0.507033
$266$	0	0
$267$	3.69141	0.225911
$268$	0	0
$269$	−5.92390	−0.361186	−0.180593	−	0.983558i	$-0.557802\pi$
$269$	−5.92390	−0.361186	−0.180593	+	0.983558i	$0.557802\pi$
$270$	0	0
$271$	−5.59031	−0.339587	−0.169793	−	0.985480i	$-0.554310\pi$
$271$	−5.59031	−0.339587	−0.169793	+	0.985480i	$0.554310\pi$
$272$	0	0
$273$	−6.90169	−0.417709
$274$	0	0
$275$	−23.0745	−1.39144
$276$	0	0
$277$	−30.4989	−1.83250	−0.916251	−	0.400606i	$-0.868800\pi$
$277$	−30.4989	−1.83250	−0.916251	+	0.400606i	$0.868800\pi$
$278$	0	0
$279$	−2.62390	−0.157089
$280$	0	0
$281$	−26.0428	−1.55359	−0.776793	−	0.629756i	$-0.783155\pi$
$281$	−26.0428	−1.55359	−0.776793	+	0.629756i	$0.783155\pi$
$282$	0	0
$283$	−7.81579	−0.464601	−0.232300	−	0.972644i	$-0.574625\pi$
$283$	−7.81579	−0.464601	−0.232300	+	0.972644i	$0.574625\pi$
$284$	0	0
$285$	−1.94095	−0.114972
$286$	0	0
$287$	11.0125	0.650046
$288$	0	0
$289$	−7.53028	−0.442957
$290$	0	0
$291$	10.6187	0.622478
$292$	0	0
$293$	15.2331	0.889929	0.444965	−	0.895548i	$-0.353216\pi$
$293$	15.2331	0.889929	0.444965	+	0.895548i	$0.353216\pi$
$294$	0	0
$295$	21.3988	1.24589
$296$	0	0
$297$	−3.41215	−0.197993
$298$	0	0
$299$	−14.8473	−0.858642
$300$	0	0
$301$	27.5010	1.58513
$302$	0	0
$303$	15.4086	0.885198
$304$	0	0
$305$	−49.1138	−2.81225
$306$	0	0
$307$	22.5939	1.28950	0.644750	−	0.764394i	$-0.276962\pi$
$307$	22.5939	1.28950	0.644750	+	0.764394i	$0.276962\pi$
$308$	0	0
$309$	0.208803	0.0118784
$310$	0	0
$311$	−18.3609	−1.04115	−0.520576	−	0.853815i	$-0.674283\pi$
$311$	−18.3609	−1.04115	−0.520576	+	0.853815i	$0.674283\pi$
$312$	0	0
$313$	−11.7189	−0.662389	−0.331194	−	0.943563i	$-0.607452\pi$
$313$	−11.7189	−0.662389	−0.331194	+	0.943563i	$0.607452\pi$
$314$	0	0
$315$	11.8057	0.665176
$316$	0	0
$317$	5.56779	0.312718	0.156359	−	0.987700i	$-0.450024\pi$
$317$	5.56779	0.312718	0.156359	+	0.987700i	$0.450024\pi$
$318$	0	0
$319$	−33.2050	−1.85912
$320$	0	0
$321$	−15.4023	−0.859673
$322$	0	0
$323$	1.74155	0.0969023
$324$	0	0
$325$	−13.5586	−0.752098
$326$	0	0
$327$	−18.6798	−1.03300
$328$	0	0
$329$	−23.9753	−1.32180
$330$	0	0
$331$	−13.9422	−0.766334	−0.383167	−	0.923679i	$-0.625167\pi$
$331$	−13.9422	−0.766334	−0.383167	+	0.923679i	$0.625167\pi$
$332$	0	0
$333$	−0.527120	−0.0288860
$334$	0	0
$335$	2.13795	0.116809
$336$	0	0
$337$	−20.7120	−1.12825	−0.564127	−	0.825688i	$-0.690787\pi$
$337$	−20.7120	−1.12825	−0.564127	+	0.825688i	$0.690787\pi$
$338$	0	0
$339$	7.05501	0.383176
$340$	0	0
$341$	8.95315	0.484840
$342$	0	0
$343$	−7.40392	−0.399774
$344$	0	0
$345$	25.3971	1.36733
$346$	0	0
$347$	0.782314	0.0419968	0.0209984	−	0.999780i	$-0.493316\pi$
$347$	0.782314	0.0419968	0.0209984	+	0.999780i	$0.493316\pi$
$348$	0	0
$349$	8.12106	0.434710	0.217355	−	0.976093i	$-0.430257\pi$
$349$	8.12106	0.434710	0.217355	+	0.976093i	$0.430257\pi$
$350$	0	0
$351$	−2.00499	−0.107019
$352$	0	0
$353$	24.1568	1.28574	0.642869	−	0.765976i	$-0.277744\pi$
$353$	24.1568	1.28574	0.642869	+	0.765976i	$0.277744\pi$
$354$	0	0
$355$	−46.2804	−2.45631
$356$	0	0
$357$	−10.5928	−0.560631
$358$	0	0
$359$	18.9721	1.00131	0.500654	−	0.865647i	$-0.333093\pi$
$359$	18.9721	1.00131	0.500654	+	0.865647i	$0.333093\pi$
$360$	0	0
$361$	−18.6797	−0.983143
$362$	0	0
$363$	0.642792	0.0337378
$364$	0	0
$365$	27.9062	1.46068
$366$	0	0
$367$	24.9269	1.30117	0.650587	−	0.759432i	$-0.274523\pi$
$367$	24.9269	1.30117	0.650587	+	0.759432i	$0.274523\pi$
$368$	0	0
$369$	3.19921	0.166544
$370$	0	0
$371$	8.28425	0.430097
$372$	0	0
$373$	−2.81458	−0.145733	−0.0728666	−	0.997342i	$-0.523215\pi$
$373$	−2.81458	−0.145733	−0.0728666	+	0.997342i	$0.523215\pi$
$374$	0	0
$375$	6.04455	0.312139
$376$	0	0
$377$	−19.5114	−1.00489
$378$	0	0
$379$	0.592271	0.0304229	0.0152114	−	0.999884i	$-0.495158\pi$
$379$	0.592271	0.0304229	0.0152114	+	0.999884i	$0.495158\pi$
$380$	0	0
$381$	2.32500	0.119113
$382$	0	0
$383$	30.2931	1.54790	0.773952	−	0.633244i	$-0.218277\pi$
$383$	30.2931	1.54790	0.773952	+	0.633244i	$0.218277\pi$
$384$	0	0
$385$	−40.2828	−2.05300
$386$	0	0
$387$	7.98923	0.406115
$388$	0	0
$389$	27.8198	1.41052	0.705261	−	0.708948i	$-0.250830\pi$
$389$	27.8198	1.41052	0.705261	+	0.708948i	$0.250830\pi$
$390$	0	0
$391$	−22.7879	−1.15243
$392$	0	0
$393$	10.3974	0.524478
$394$	0	0
$395$	−14.6030	−0.734756
$396$	0	0
$397$	−0.292346	−0.0146724	−0.00733622	−	0.999973i	$-0.502335\pi$
$397$	−0.292346	−0.0146724	−0.00733622	+	0.999973i	$0.502335\pi$
$398$	0	0
$399$	−1.94809	−0.0975266
$400$	0	0
$401$	2.65795	0.132732	0.0663659	−	0.997795i	$-0.478860\pi$
$401$	2.65795	0.132732	0.0663659	+	0.997795i	$0.478860\pi$
$402$	0	0
$403$	5.26090	0.262064
$404$	0	0
$405$	3.42964	0.170420
$406$	0	0
$407$	1.79861	0.0891540
$408$	0	0
$409$	33.7254	1.66761	0.833807	−	0.552056i	$-0.186156\pi$
$409$	33.7254	1.66761	0.833807	+	0.552056i	$0.186156\pi$
$410$	0	0
$411$	3.14496	0.155130
$412$	0	0
$413$	21.4775	1.05684
$414$	0	0
$415$	−52.4269	−2.57354
$416$	0	0
$417$	−4.67040	−0.228710
$418$	0	0
$419$	−23.3953	−1.14293	−0.571466	−	0.820625i	$-0.693625\pi$
$419$	−23.3953	−1.14293	−0.571466	+	0.820625i	$0.693625\pi$
$420$	0	0
$421$	29.0353	1.41510	0.707548	−	0.706665i	$-0.249801\pi$
$421$	29.0353	1.41510	0.707548	+	0.706665i	$0.249801\pi$
$422$	0	0
$423$	−6.96499	−0.338649
$424$	0	0
$425$	−20.8100	−1.00943
$426$	0	0
$427$	−49.2944	−2.38552
$428$	0	0
$429$	6.84134	0.330303
$430$	0	0
$431$	17.9561	0.864917	0.432458	−	0.901654i	$-0.357646\pi$
$431$	17.9561	0.864917	0.432458	+	0.901654i	$0.357646\pi$
$432$	0	0
$433$	−9.77953	−0.469974	−0.234987	−	0.971998i	$-0.575505\pi$
$433$	−9.77953	−0.469974	−0.234987	+	0.971998i	$0.575505\pi$
$434$	0	0
$435$	33.3752	1.60022
$436$	0	0
$437$	−4.19085	−0.200475
$438$	0	0
$439$	−26.6654	−1.27267	−0.636335	−	0.771413i	$-0.719550\pi$
$439$	−26.6654	−1.27267	−0.636335	+	0.771413i	$0.719550\pi$
$440$	0	0
$441$	4.84911	0.230910
$442$	0	0
$443$	33.8389	1.60774	0.803868	−	0.594807i	$-0.202772\pi$
$443$	33.8389	1.60774	0.803868	+	0.594807i	$0.202772\pi$
$444$	0	0
$445$	12.6602	0.600152
$446$	0	0
$447$	17.4241	0.824134
$448$	0	0
$449$	12.1207	0.572011	0.286006	−	0.958228i	$-0.407672\pi$
$449$	12.1207	0.572011	0.286006	+	0.958228i	$0.407672\pi$
$450$	0	0
$451$	−10.9162	−0.514023
$452$	0	0
$453$	2.51771	0.118292
$454$	0	0
$455$	−23.6703	−1.10968
$456$	0	0
$457$	33.2110	1.55355	0.776773	−	0.629780i	$-0.216855\pi$
$457$	33.2110	1.55355	0.776773	+	0.629780i	$0.216855\pi$
$458$	0	0
$459$	−3.07729	−0.143636
$460$	0	0
$461$	−8.65829	−0.403257	−0.201628	−	0.979462i	$-0.564623\pi$
$461$	−8.65829	−0.403257	−0.201628	+	0.979462i	$0.564623\pi$
$462$	0	0
$463$	−9.11138	−0.423442	−0.211721	−	0.977330i	$-0.567907\pi$
$463$	−9.11138	−0.423442	−0.211721	+	0.977330i	$0.567907\pi$
$464$	0	0
$465$	−8.99903	−0.417320
$466$	0	0
$467$	−5.77095	−0.267048	−0.133524	−	0.991046i	$-0.542629\pi$
$467$	−5.77095	−0.267048	−0.133524	+	0.991046i	$0.542629\pi$
$468$	0	0
$469$	2.14581	0.0990843
$470$	0	0
$471$	−12.3224	−0.567787
$472$	0	0
$473$	−27.2605	−1.25344
$474$	0	0
$475$	−3.82710	−0.175600
$476$	0	0
$477$	2.40664	0.110192
$478$	0	0
$479$	−13.0611	−0.596777	−0.298388	−	0.954445i	$-0.596449\pi$
$479$	−13.0611	−0.596777	−0.298388	+	0.954445i	$0.596449\pi$
$480$	0	0
$481$	1.05687	0.0481892
$482$	0	0
$483$	25.4905	1.15986
$484$	0	0
$485$	36.4183	1.65367
$486$	0	0
$487$	4.46374	0.202272	0.101136	−	0.994873i	$-0.467752\pi$
$487$	4.46374	0.202272	0.101136	+	0.994873i	$0.467752\pi$
$488$	0	0
$489$	−17.4945	−0.791128
$490$	0	0
$491$	−32.0860	−1.44802	−0.724010	−	0.689790i	$-0.757703\pi$
$491$	−32.0860	−1.44802	−0.724010	+	0.689790i	$0.757703\pi$
$492$	0	0
$493$	−29.9464	−1.34872
$494$	0	0
$495$	−11.7025	−0.525987
$496$	0	0
$497$	−46.4506	−2.08359
$498$	0	0
$499$	−1.83022	−0.0819321	−0.0409660	−	0.999161i	$-0.513044\pi$
$499$	−1.83022	−0.0819321	−0.0409660	+	0.999161i	$0.513044\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−38.9557	−1.73695	−0.868475	−	0.495734i	$-0.834899\pi$
$503$	−38.9557	−1.73695	−0.868475	+	0.495734i	$0.834899\pi$
$504$	0	0
$505$	52.8458	2.35161
$506$	0	0
$507$	−8.98001	−0.398816
$508$	0	0
$509$	−29.2174	−1.29504	−0.647520	−	0.762049i	$-0.724194\pi$
$509$	−29.2174	−1.29504	−0.647520	+	0.762049i	$0.724194\pi$
$510$	0	0
$511$	28.0088	1.23904
$512$	0	0
$513$	−0.565935	−0.0249866
$514$	0	0
$515$	0.716119	0.0315560
$516$	0	0
$517$	23.7656	1.04521
$518$	0	0
$519$	−8.48906	−0.372629
$520$	0	0
$521$	−4.48938	−0.196683	−0.0983416	−	0.995153i	$-0.531354\pi$
$521$	−4.48938	−0.196683	−0.0983416	+	0.995153i	$0.531354\pi$
$522$	0	0
$523$	−5.38078	−0.235285	−0.117643	−	0.993056i	$-0.537534\pi$
$523$	−5.38078	−0.235285	−0.117643	+	0.993056i	$0.537534\pi$
$524$	0	0
$525$	23.2780	1.01594
$526$	0	0
$527$	8.07450	0.351731
$528$	0	0
$529$	31.8366	1.38420
$530$	0	0
$531$	6.23937	0.270766
$532$	0	0
$533$	−6.41438	−0.277838
$534$	0	0
$535$	−52.8244	−2.28380
$536$	0	0
$537$	−9.45927	−0.408198
$538$	0	0
$539$	−16.5459	−0.712682
$540$	0	0
$541$	6.00067	0.257989	0.128995	−	0.991645i	$-0.458825\pi$
$541$	6.00067	0.257989	0.128995	+	0.991645i	$0.458825\pi$
$542$	0	0
$543$	−2.41594	−0.103678
$544$	0	0
$545$	−64.0650	−2.74424
$546$	0	0
$547$	−3.34004	−0.142810	−0.0714050	−	0.997447i	$-0.522748\pi$
$547$	−3.34004	−0.142810	−0.0714050	+	0.997447i	$0.522748\pi$
$548$	0	0
$549$	−14.3204	−0.611179
$550$	0	0
$551$	−5.50734	−0.234621
$552$	0	0
$553$	−14.6567	−0.623265
$554$	0	0
$555$	−1.80783	−0.0767382
$556$	0	0
$557$	−14.2270	−0.602816	−0.301408	−	0.953495i	$-0.597457\pi$
$557$	−14.2270	−0.602816	−0.301408	+	0.953495i	$0.597457\pi$
$558$	0	0
$559$	−16.0183	−0.677504
$560$	0	0
$561$	10.5002	0.443318
$562$	0	0
$563$	19.8750	0.837633	0.418817	−	0.908071i	$-0.362445\pi$
$563$	19.8750	0.837633	0.418817	+	0.908071i	$0.362445\pi$
$564$	0	0
$565$	24.1962	1.01794
$566$	0	0
$567$	3.44225	0.144561
$568$	0	0
$569$	38.8695	1.62950	0.814748	−	0.579816i	$-0.196876\pi$
$569$	38.8695	1.62950	0.814748	+	0.579816i	$0.196876\pi$
$570$	0	0
$571$	−45.0691	−1.88608	−0.943042	−	0.332674i	$-0.892049\pi$
$571$	−45.0691	−1.88608	−0.943042	+	0.332674i	$0.892049\pi$
$572$	0	0
$573$	5.50321	0.229900
$574$	0	0
$575$	50.0771	2.08836
$576$	0	0
$577$	33.1251	1.37902	0.689509	−	0.724277i	$-0.257827\pi$
$577$	33.1251	1.37902	0.689509	+	0.724277i	$0.257827\pi$
$578$	0	0
$579$	−15.1701	−0.630447
$580$	0	0
$581$	−52.6197	−2.18303
$582$	0	0
$583$	−8.21181	−0.340099
$584$	0	0
$585$	−6.87641	−0.284304
$586$	0	0
$587$	14.4500	0.596415	0.298207	−	0.954501i	$-0.403611\pi$
$587$	14.4500	0.596415	0.298207	+	0.954501i	$0.403611\pi$
$588$	0	0
$589$	1.48496	0.0611866
$590$	0	0
$591$	7.92869	0.326143
$592$	0	0
$593$	−29.0136	−1.19145	−0.595724	−	0.803189i	$-0.703135\pi$
$593$	−29.0136	−1.19145	−0.595724	+	0.803189i	$0.703135\pi$
$594$	0	0
$595$	−36.3296	−1.48937
$596$	0	0
$597$	−22.8595	−0.935577
$598$	0	0
$599$	22.5523	0.921461	0.460731	−	0.887540i	$-0.347587\pi$
$599$	22.5523	0.921461	0.460731	+	0.887540i	$0.347587\pi$
$600$	0	0
$601$	21.5721	0.879943	0.439972	−	0.898012i	$-0.354989\pi$
$601$	21.5721	0.879943	0.439972	+	0.898012i	$0.354989\pi$
$602$	0	0
$603$	0.623373	0.0253857
$604$	0	0
$605$	2.20454	0.0896275
$606$	0	0
$607$	16.5729	0.672674	0.336337	−	0.941742i	$-0.390812\pi$
$607$	16.5729	0.672674	0.336337	+	0.941742i	$0.390812\pi$
$608$	0	0
$609$	33.4979	1.35741
$610$	0	0
$611$	13.9647	0.564953
$612$	0	0
$613$	−28.6472	−1.15705	−0.578524	−	0.815665i	$-0.696371\pi$
$613$	−28.6472	−1.15705	−0.578524	+	0.815665i	$0.696371\pi$
$614$	0	0
$615$	10.9721	0.442439
$616$	0	0
$617$	22.2824	0.897054	0.448527	−	0.893769i	$-0.351949\pi$
$617$	22.2824	0.897054	0.448527	+	0.893769i	$0.351949\pi$
$618$	0	0
$619$	1.46731	0.0589760	0.0294880	−	0.999565i	$-0.490612\pi$
$619$	1.46731	0.0589760	0.0294880	+	0.999565i	$0.490612\pi$
$620$	0	0
$621$	7.40517	0.297159
$622$	0	0
$623$	12.7068	0.509086
$624$	0	0
$625$	−13.0816	−0.523263
$626$	0	0
$627$	1.93106	0.0771190
$628$	0	0
$629$	1.62210	0.0646775
$630$	0	0
$631$	21.5061	0.856144	0.428072	−	0.903745i	$-0.359193\pi$
$631$	21.5061	0.856144	0.428072	+	0.903745i	$0.359193\pi$
$632$	0	0
$633$	−9.58607	−0.381012
$634$	0	0
$635$	7.97391	0.316435
$636$	0	0
$637$	−9.72242	−0.385216
$638$	0	0
$639$	−13.4942	−0.533824
$640$	0	0
$641$	2.14974	0.0849096	0.0424548	−	0.999098i	$-0.486482\pi$
$641$	2.14974	0.0849096	0.0424548	+	0.999098i	$0.486482\pi$
$642$	0	0
$643$	24.0361	0.947890	0.473945	−	0.880554i	$-0.342830\pi$
$643$	24.0361	0.947890	0.473945	+	0.880554i	$0.342830\pi$
$644$	0	0
$645$	27.4002	1.07888
$646$	0	0
$647$	−36.1627	−1.42170	−0.710851	−	0.703343i	$-0.751690\pi$
$647$	−36.1627	−1.42170	−0.710851	+	0.703343i	$0.751690\pi$
$648$	0	0
$649$	−21.2897	−0.835694
$650$	0	0
$651$	−9.03212	−0.353997
$652$	0	0
$653$	−15.4966	−0.606428	−0.303214	−	0.952923i	$-0.598060\pi$
$653$	−15.4966	−0.606428	−0.303214	+	0.952923i	$0.598060\pi$
$654$	0	0
$655$	35.6593	1.39332
$656$	0	0
$657$	8.13677	0.317445
$658$	0	0
$659$	−24.9559	−0.972142	−0.486071	−	0.873919i	$-0.661570\pi$
$659$	−24.9559	−0.972142	−0.486071	+	0.873919i	$0.661570\pi$
$660$	0	0
$661$	−8.12692	−0.316100	−0.158050	−	0.987431i	$-0.550521\pi$
$661$	−8.12692	−0.316100	−0.158050	+	0.987431i	$0.550521\pi$
$662$	0	0
$663$	6.16995	0.239621
$664$	0	0
$665$	−6.68126	−0.259088
$666$	0	0
$667$	72.0627	2.79028
$668$	0	0
$669$	11.0930	0.428880
$670$	0	0
$671$	48.8634	1.88635
$672$	0	0
$673$	−40.7000	−1.56887	−0.784434	−	0.620212i	$-0.787046\pi$
$673$	−40.7000	−1.56887	−0.784434	+	0.620212i	$0.787046\pi$
$674$	0	0
$675$	6.76244	0.260287
$676$	0	0
$677$	−0.855237	−0.0328694	−0.0164347	−	0.999865i	$-0.505232\pi$
$677$	−0.855237	−0.0328694	−0.0164347	+	0.999865i	$0.505232\pi$
$678$	0	0
$679$	36.5522	1.40274
$680$	0	0
$681$	−13.3442	−0.511351
$682$	0	0
$683$	−17.1386	−0.655791	−0.327895	−	0.944714i	$-0.606339\pi$
$683$	−17.1386	−0.655791	−0.327895	+	0.944714i	$0.606339\pi$
$684$	0	0
$685$	10.7861	0.412116
$686$	0	0
$687$	8.71943	0.332667
$688$	0	0
$689$	−4.82529	−0.183829
$690$	0	0
$691$	−49.0132	−1.86455	−0.932275	−	0.361749i	$-0.882180\pi$
$691$	−49.0132	−1.86455	−0.932275	+	0.361749i	$0.882180\pi$
$692$	0	0
$693$	−11.7455	−0.446174
$694$	0	0
$695$	−16.0178	−0.607589
$696$	0	0
$697$	−9.84489	−0.372902
$698$	0	0
$699$	23.4773	0.887995
$700$	0	0
$701$	−19.8877	−0.751148	−0.375574	−	0.926792i	$-0.622554\pi$
$701$	−19.8877	−0.751148	−0.375574	+	0.926792i	$0.622554\pi$
$702$	0	0
$703$	0.298316	0.0112512
$704$	0	0
$705$	−23.8874	−0.899652
$706$	0	0
$707$	53.0401	1.99478
$708$	0	0
$709$	19.6916	0.739532	0.369766	−	0.929125i	$-0.379438\pi$
$709$	19.6916	0.739532	0.369766	+	0.929125i	$0.379438\pi$
$710$	0	0
$711$	−4.25787	−0.159683
$712$	0	0
$713$	−19.4304	−0.727675
$714$	0	0
$715$	23.4634	0.877480
$716$	0	0
$717$	−23.7397	−0.886576
$718$	0	0
$719$	−14.0200	−0.522859	−0.261429	−	0.965223i	$-0.584194\pi$
$719$	−14.0200	−0.522859	−0.261429	+	0.965223i	$0.584194\pi$
$720$	0	0
$721$	0.718752	0.0267677
$722$	0	0
$723$	2.51503	0.0935350
$724$	0	0
$725$	65.8080	2.44405
$726$	0	0
$727$	6.39045	0.237009	0.118504	−	0.992954i	$-0.462190\pi$
$727$	6.39045	0.237009	0.118504	+	0.992954i	$0.462190\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.5852	−0.909316
$732$	0	0
$733$	27.2071	1.00492	0.502459	−	0.864601i	$-0.332429\pi$
$733$	27.2071	1.00492	0.502459	+	0.864601i	$0.332429\pi$
$734$	0	0
$735$	16.6307	0.613432
$736$	0	0
$737$	−2.12705	−0.0783507
$738$	0	0
$739$	−31.7809	−1.16908	−0.584540	−	0.811365i	$-0.698725\pi$
$739$	−31.7809	−1.16908	−0.584540	+	0.811365i	$0.698725\pi$
$740$	0	0
$741$	1.13470	0.0416841
$742$	0	0
$743$	−25.6426	−0.940735	−0.470368	−	0.882471i	$-0.655879\pi$
$743$	−25.6426	−0.940735	−0.470368	+	0.882471i	$0.655879\pi$
$744$	0	0
$745$	59.7586	2.18938
$746$	0	0
$747$	−15.2864	−0.559301
$748$	0	0
$749$	−53.0186	−1.93726
$750$	0	0
$751$	−13.3662	−0.487739	−0.243870	−	0.969808i	$-0.578417\pi$
$751$	−13.3662	−0.487739	−0.243870	+	0.969808i	$0.578417\pi$
$752$	0	0
$753$	9.11605	0.332207
$754$	0	0
$755$	8.63483	0.314254
$756$	0	0
$757$	46.5562	1.69211	0.846057	−	0.533092i	$-0.178970\pi$
$757$	46.5562	1.69211	0.846057	+	0.533092i	$0.178970\pi$
$758$	0	0
$759$	−25.2676	−0.917155
$760$	0	0
$761$	30.9214	1.12090	0.560450	−	0.828188i	$-0.310628\pi$
$761$	30.9214	1.12090	0.560450	+	0.828188i	$0.310628\pi$
$762$	0	0
$763$	−64.3006	−2.32784
$764$	0	0
$765$	−10.5540	−0.381581
$766$	0	0
$767$	−12.5099	−0.451706
$768$	0	0
$769$	−28.7870	−1.03809	−0.519044	−	0.854748i	$-0.673712\pi$
$769$	−28.7870	−1.03809	−0.519044	+	0.854748i	$0.673712\pi$
$770$	0	0
$771$	29.6735	1.06866
$772$	0	0
$773$	49.9587	1.79689	0.898445	−	0.439086i	$-0.144698\pi$
$773$	49.9587	1.79689	0.898445	+	0.439086i	$0.144698\pi$
$774$	0	0
$775$	−17.7440	−0.637382
$776$	0	0
$777$	−1.81448	−0.0650941
$778$	0	0
$779$	−1.81054	−0.0648694
$780$	0	0
$781$	46.0444	1.64760
$782$	0	0
$783$	9.73140	0.347772
$784$	0	0
$785$	−42.2615	−1.50838
$786$	0	0
$787$	20.4420	0.728680	0.364340	−	0.931266i	$-0.381295\pi$
$787$	20.4420	0.728680	0.364340	+	0.931266i	$0.381295\pi$
$788$	0	0
$789$	−6.20229	−0.220807
$790$	0	0
$791$	24.2851	0.863480
$792$	0	0
$793$	28.7123	1.01960
$794$	0	0
$795$	8.25390	0.292736
$796$	0	0
$797$	−43.6420	−1.54588	−0.772939	−	0.634481i	$-0.781214\pi$
$797$	−43.6420	−1.54588	−0.772939	+	0.634481i	$0.781214\pi$
$798$	0	0
$799$	21.4333	0.758256
$800$	0	0
$801$	3.69141	0.130430
$802$	0	0
$803$	−27.7639	−0.979767
$804$	0	0
$805$	87.4232	3.08126
$806$	0	0
$807$	−5.92390	−0.208531
$808$	0	0
$809$	−34.1990	−1.20237	−0.601187	−	0.799108i	$-0.705305\pi$
$809$	−34.1990	−1.20237	−0.601187	+	0.799108i	$0.705305\pi$
$810$	0	0
$811$	18.4143	0.646613	0.323306	−	0.946294i	$-0.395206\pi$
$811$	18.4143	0.646613	0.323306	+	0.946294i	$0.395206\pi$
$812$	0	0
$813$	−5.59031	−0.196061
$814$	0	0
$815$	−59.9998	−2.10170
$816$	0	0
$817$	−4.52139	−0.158183
$818$	0	0
$819$	−6.90169	−0.241165
$820$	0	0
$821$	13.9748	0.487724	0.243862	−	0.969810i	$-0.421586\pi$
$821$	13.9748	0.487724	0.243862	+	0.969810i	$0.421586\pi$
$822$	0	0
$823$	−38.7321	−1.35012	−0.675058	−	0.737765i	$-0.735881\pi$
$823$	−38.7321	−1.35012	−0.675058	+	0.737765i	$0.735881\pi$
$824$	0	0
$825$	−23.0745	−0.803351
$826$	0	0
$827$	15.2856	0.531531	0.265766	−	0.964038i	$-0.414375\pi$
$827$	15.2856	0.531531	0.265766	+	0.964038i	$0.414375\pi$
$828$	0	0
$829$	15.8583	0.550783	0.275391	−	0.961332i	$-0.411193\pi$
$829$	15.8583	0.550783	0.275391	+	0.961332i	$0.411193\pi$
$830$	0	0
$831$	−30.4989	−1.05800
$832$	0	0
$833$	−14.9221	−0.517021
$834$	0	0
$835$	3.42964	0.118688
$836$	0	0
$837$	−2.62390	−0.0906952
$838$	0	0
$839$	11.9835	0.413716	0.206858	−	0.978371i	$-0.433676\pi$
$839$	11.9835	0.413716	0.206858	+	0.978371i	$0.433676\pi$
$840$	0	0
$841$	65.7002	2.26552
$842$	0	0
$843$	−26.0428	−0.896963
$844$	0	0
$845$	−30.7982	−1.05949
$846$	0	0
$847$	2.21265	0.0760276
$848$	0	0
$849$	−7.81579	−0.268237
$850$	0	0
$851$	−3.90341	−0.133807
$852$	0	0
$853$	−41.5858	−1.42387	−0.711936	−	0.702245i	$-0.752181\pi$
$853$	−41.5858	−1.42387	−0.711936	+	0.702245i	$0.752181\pi$
$854$	0	0
$855$	−1.94095	−0.0663793
$856$	0	0
$857$	20.9894	0.716985	0.358492	−	0.933533i	$-0.383291\pi$
$857$	20.9894	0.716985	0.358492	+	0.933533i	$0.383291\pi$
$858$	0	0
$859$	4.03076	0.137528	0.0687638	−	0.997633i	$-0.478095\pi$
$859$	4.03076	0.137528	0.0687638	+	0.997633i	$0.478095\pi$
$860$	0	0
$861$	11.0125	0.375304
$862$	0	0
$863$	−13.1107	−0.446292	−0.223146	−	0.974785i	$-0.571633\pi$
$863$	−13.1107	−0.446292	−0.223146	+	0.974785i	$0.571633\pi$
$864$	0	0
$865$	−29.1144	−0.989921
$866$	0	0
$867$	−7.53028	−0.255742
$868$	0	0
$869$	14.5285	0.492846
$870$	0	0
$871$	−1.24986	−0.0423499
$872$	0	0
$873$	10.6187	0.359388
$874$	0	0
$875$	20.8069	0.703400
$876$	0	0
$877$	−10.7825	−0.364098	−0.182049	−	0.983289i	$-0.558273\pi$
$877$	−10.7825	−0.364098	−0.182049	+	0.983289i	$0.558273\pi$
$878$	0	0
$879$	15.2331	0.513801
$880$	0	0
$881$	31.6377	1.06590	0.532951	−	0.846146i	$-0.321083\pi$
$881$	31.6377	1.06590	0.532951	+	0.846146i	$0.321083\pi$
$882$	0	0
$883$	−36.5870	−1.23125	−0.615625	−	0.788039i	$-0.711097\pi$
$883$	−36.5870	−1.23125	−0.615625	+	0.788039i	$0.711097\pi$
$884$	0	0
$885$	21.3988	0.719313
$886$	0	0
$887$	53.4262	1.79387	0.896937	−	0.442157i	$-0.145787\pi$
$887$	53.4262	1.79387	0.896937	+	0.442157i	$0.145787\pi$
$888$	0	0
$889$	8.00323	0.268420
$890$	0	0
$891$	−3.41215	−0.114311
$892$	0	0
$893$	3.94173	0.131905
$894$	0	0
$895$	−32.4419	−1.08441
$896$	0	0
$897$	−14.8473	−0.495737
$898$	0	0
$899$	−25.5342	−0.851614
$900$	0	0
$901$	−7.40592	−0.246727
$902$	0	0
$903$	27.5010	0.915175
$904$	0	0
$905$	−8.28580	−0.275429
$906$	0	0
$907$	−54.6924	−1.81603	−0.908015	−	0.418938i	$-0.862403\pi$
$907$	−54.6924	−1.81603	−0.908015	+	0.418938i	$0.862403\pi$
$908$	0	0
$909$	15.4086	0.511070
$910$	0	0
$911$	−0.683962	−0.0226607	−0.0113303	−	0.999936i	$-0.503607\pi$
$911$	−0.683962	−0.0226607	−0.0113303	+	0.999936i	$0.503607\pi$
$912$	0	0
$913$	52.1596	1.72623
$914$	0	0
$915$	−49.1138	−1.62365
$916$	0	0
$917$	35.7904	1.18190
$918$	0	0
$919$	48.8179	1.61035	0.805177	−	0.593035i	$-0.202070\pi$
$919$	48.8179	1.61035	0.805177	+	0.593035i	$0.202070\pi$
$920$	0	0
$921$	22.5939	0.744493
$922$	0	0
$923$	27.0558	0.890553
$924$	0	0
$925$	−3.56462	−0.117204
$926$	0	0
$927$	0.208803	0.00685799
$928$	0	0
$929$	18.6013	0.610288	0.305144	−	0.952306i	$-0.401295\pi$
$929$	18.6013	0.610288	0.305144	+	0.952306i	$0.401295\pi$
$930$	0	0
$931$	−2.74428	−0.0899401
$932$	0	0
$933$	−18.3609	−0.601110
$934$	0	0
$935$	36.0119	1.17771
$936$	0	0
$937$	27.6778	0.904193	0.452096	−	0.891969i	$-0.350676\pi$
$937$	27.6778	0.904193	0.452096	+	0.891969i	$0.350676\pi$
$938$	0	0
$939$	−11.7189	−0.382430
$940$	0	0
$941$	4.40771	0.143687	0.0718437	−	0.997416i	$-0.477112\pi$
$941$	4.40771	0.143687	0.0718437	+	0.997416i	$0.477112\pi$
$942$	0	0
$943$	23.6907	0.771474
$944$	0	0
$945$	11.8057	0.384039
$946$	0	0
$947$	16.0931	0.522954	0.261477	−	0.965210i	$-0.415790\pi$
$947$	16.0931	0.522954	0.261477	+	0.965210i	$0.415790\pi$
$948$	0	0
$949$	−16.3142	−0.529580
$950$	0	0
$951$	5.56779	0.180548
$952$	0	0
$953$	−8.08542	−0.261912	−0.130956	−	0.991388i	$-0.541805\pi$
$953$	−8.08542	−0.261912	−0.130956	+	0.991388i	$0.541805\pi$
$954$	0	0
$955$	18.8740	0.610750
$956$	0	0
$957$	−33.2050	−1.07337
$958$	0	0
$959$	10.8258	0.349582
$960$	0	0
$961$	−24.1152	−0.777908
$962$	0	0
$963$	−15.4023	−0.496332
$964$	0	0
$965$	−52.0280	−1.67484
$966$	0	0
$967$	45.0879	1.44993	0.724965	−	0.688786i	$-0.241856\pi$
$967$	45.0879	1.44993	0.724965	+	0.688786i	$0.241856\pi$
$968$	0	0
$969$	1.74155	0.0559466
$970$	0	0
$971$	39.5386	1.26885	0.634427	−	0.772983i	$-0.281236\pi$
$971$	39.5386	1.26885	0.634427	+	0.772983i	$0.281236\pi$
$972$	0	0
$973$	−16.0767	−0.515395
$974$	0	0
$975$	−13.5586	−0.434224
$976$	0	0
$977$	−53.9173	−1.72497	−0.862483	−	0.506086i	$-0.831092\pi$
$977$	−53.9173	−1.72497	−0.862483	+	0.506086i	$0.831092\pi$
$978$	0	0
$979$	−12.5957	−0.402559
$980$	0	0
$981$	−18.6798	−0.596400
$982$	0	0
$983$	32.3887	1.03304	0.516520	−	0.856275i	$-0.327227\pi$
$983$	32.3887	1.03304	0.516520	+	0.856275i	$0.327227\pi$
$984$	0	0
$985$	27.1926	0.866428
$986$	0	0
$987$	−23.9753	−0.763141
$988$	0	0
$989$	59.1616	1.88123
$990$	0	0
$991$	41.4568	1.31692	0.658459	−	0.752617i	$-0.271209\pi$
$991$	41.4568	1.31692	0.658459	+	0.752617i	$0.271209\pi$
$992$	0	0
$993$	−13.9422	−0.442443
$994$	0	0
$995$	−78.3999	−2.48544
$996$	0	0
$997$	−34.5191	−1.09323	−0.546615	−	0.837384i	$-0.684084\pi$
$997$	−34.5191	−1.09323	−0.546615	+	0.837384i	$0.684084\pi$
$998$	0	0
$999$	−0.527120	−0.0166773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2004.2.a.d.1.7	✓	9
3.2	odd	2		6012.2.a.h.1.3		9
4.3	odd	2		8016.2.a.bb.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.d.1.7	✓	9	1.1	even	1	trivial
6012.2.a.h.1.3		9	3.2	odd	2
8016.2.a.bb.1.7		9	4.3	odd	2

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:	\( N \)	\(=\)	\( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.42964 q^{5} +3.44225 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.42964 q^{5} +3.44225 q^{7} +1.00000 q^{9} -3.41215 q^{11} -2.00499 q^{13} +3.42964 q^{15} -3.07729 q^{17} -0.565935 q^{19} +3.44225 q^{21} +7.40517 q^{23} +6.76244 q^{25} +1.00000 q^{27} +9.73140 q^{29} -2.62390 q^{31} -3.41215 q^{33} +11.8057 q^{35} -0.527120 q^{37} -2.00499 q^{39} +3.19921 q^{41} +7.98923 q^{43} +3.42964 q^{45} -6.96499 q^{47} +4.84911 q^{49} -3.07729 q^{51} +2.40664 q^{53} -11.7025 q^{55} -0.565935 q^{57} +6.23937 q^{59} -14.3204 q^{61} +3.44225 q^{63} -6.87641 q^{65} +0.623373 q^{67} +7.40517 q^{69} -13.4942 q^{71} +8.13677 q^{73} +6.76244 q^{75} -11.7455 q^{77} -4.25787 q^{79} +1.00000 q^{81} -15.2864 q^{83} -10.5540 q^{85} +9.73140 q^{87} +3.69141 q^{89} -6.90169 q^{91} -2.62390 q^{93} -1.94095 q^{95} +10.6187 q^{97} -3.41215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} + 9 q^{5} + 2 q^{7} + 9 q^{9} + 7 q^{11} + 6 q^{13} + 9 q^{15} + 7 q^{17} + 2 q^{19} + 2 q^{21} + 19 q^{23} + 22 q^{25} + 9 q^{27} + 13 q^{29} + 12 q^{31} + 7 q^{33} + 4 q^{35} + 15 q^{37} + 6 q^{39} + 18 q^{41} - 6 q^{43} + 9 q^{45} + 25 q^{47} + 19 q^{49} + 7 q^{51} + 17 q^{53} - 3 q^{55} + 2 q^{57} + 3 q^{59} + 14 q^{61} + 2 q^{63} + 14 q^{65} - 4 q^{67} + 19 q^{69} + 17 q^{71} - 20 q^{73} + 22 q^{75} + 14 q^{77} - 8 q^{79} + 9 q^{81} - q^{83} + 5 q^{85} + 13 q^{87} + 36 q^{89} - 41 q^{91} + 12 q^{93} + 5 q^{95} + 31 q^{97} + 7 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 + 9 * q^5 + 2 * q^7 + 9 * q^9 + 7 * q^11 + 6 * q^13 + 9 * q^15 + 7 * q^17 + 2 * q^19 + 2 * q^21 + 19 * q^23 + 22 * q^25 + 9 * q^27 + 13 * q^29 + 12 * q^31 + 7 * q^33 + 4 * q^35 + 15 * q^37 + 6 * q^39 + 18 * q^41 - 6 * q^43 + 9 * q^45 + 25 * q^47 + 19 * q^49 + 7 * q^51 + 17 * q^53 - 3 * q^55 + 2 * q^57 + 3 * q^59 + 14 * q^61 + 2 * q^63 + 14 * q^65 - 4 * q^67 + 19 * q^69 + 17 * q^71 - 20 * q^73 + 22 * q^75 + 14 * q^77 - 8 * q^79 + 9 * q^81 - q^83 + 5 * q^85 + 13 * q^87 + 36 * q^89 - 41 * q^91 + 12 * q^93 + 5 * q^95 + 31 * q^97 + 7 * q^99

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