LMFDB - Embedded newform 2004.2.a.b.1.2

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:	$1$
Dimension:	$5$
Coefficient field:	5.5.161121.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 3x^{2} + 5x + 1 $ x^5 - x^4 - 6x^3 + 3x^2 + 5*x + 1
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			1.2
Root			$2.56399$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.85181 q^{5} +2.41580 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.85181 q^{5} +2.41580 q^{7} +1.00000 q^{9} -2.49614 q^{11} +0.116176 q^{13} -1.85181 q^{15} -6.12798 q^{17} -3.34227 q^{19} +2.41580 q^{21} -8.19583 q^{23} -1.57081 q^{25} +1.00000 q^{27} +0.0936338 q^{29} +1.44840 q^{31} -2.49614 q^{33} -4.47360 q^{35} -0.230021 q^{37} +0.116176 q^{39} -4.49763 q^{41} +5.35233 q^{43} -1.85181 q^{45} +7.01021 q^{47} -1.16392 q^{49} -6.12798 q^{51} +3.67773 q^{53} +4.62237 q^{55} -3.34227 q^{57} -1.76972 q^{59} +8.66144 q^{61} +2.41580 q^{63} -0.215135 q^{65} -15.1723 q^{67} -8.19583 q^{69} -8.51567 q^{71} +4.74977 q^{73} -1.57081 q^{75} -6.03017 q^{77} -3.24808 q^{79} +1.00000 q^{81} -0.437500 q^{83} +11.3478 q^{85} +0.0936338 q^{87} +11.9663 q^{89} +0.280657 q^{91} +1.44840 q^{93} +6.18925 q^{95} +5.72224 q^{97} -2.49614 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - 7 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 - 7 * q^5 - 2 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} - 7 q^{5} - 2 q^{7} + 5 q^{9} - 5 q^{11} - 8 q^{13} - 7 q^{15} - 7 q^{17} + 2 q^{19} - 2 q^{21} - 13 q^{23} + 2 q^{25} + 5 q^{27} - 11 q^{29} - 12 q^{31} - 5 q^{33} - 12 q^{35} - 7 q^{37} - 8 q^{39} - 12 q^{41} - 7 q^{45} - 19 q^{47} - 9 q^{49} - 7 q^{51} - 21 q^{53} - q^{55} + 2 q^{57} - 7 q^{59} - 6 q^{61} - 2 q^{63} + 14 q^{65} + 10 q^{67} - 13 q^{69} - 35 q^{71} - 8 q^{73} + 2 q^{75} - 6 q^{77} + 5 q^{81} - 11 q^{83} + 5 q^{85} - 11 q^{87} - 32 q^{89} + 5 q^{91} - 12 q^{93} - 19 q^{95} + 11 q^{97} - 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - 7 * q^5 - 2 * q^7 + 5 * q^9 - 5 * q^11 - 8 * q^13 - 7 * q^15 - 7 * q^17 + 2 * q^19 - 2 * q^21 - 13 * q^23 + 2 * q^25 + 5 * q^27 - 11 * q^29 - 12 * q^31 - 5 * q^33 - 12 * q^35 - 7 * q^37 - 8 * q^39 - 12 * q^41 - 7 * q^45 - 19 * q^47 - 9 * q^49 - 7 * q^51 - 21 * q^53 - q^55 + 2 * q^57 - 7 * q^59 - 6 * q^61 - 2 * q^63 + 14 * q^65 + 10 * q^67 - 13 * q^69 - 35 * q^71 - 8 * q^73 + 2 * q^75 - 6 * q^77 + 5 * q^81 - 11 * q^83 + 5 * q^85 - 11 * q^87 - 32 * q^89 + 5 * q^91 - 12 * q^93 - 19 * q^95 + 11 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.85181	−0.828154	−0.414077	−	0.910242i	$-0.635896\pi$
$5$	−1.85181	−0.828154	−0.414077	+	0.910242i	$0.635896\pi$
$6$	0	0
$7$	2.41580	0.913086	0.456543	−	0.889701i	$-0.349087\pi$
$7$	2.41580	0.913086	0.456543	+	0.889701i	$0.349087\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.49614	−0.752614	−0.376307	−	0.926495i	$-0.622806\pi$
$11$	−2.49614	−0.752614	−0.376307	+	0.926495i	$0.622806\pi$
$12$	0	0
$13$	0.116176	0.0322213	0.0161107	−	0.999870i	$-0.494872\pi$
$13$	0.116176	0.0322213	0.0161107	+	0.999870i	$0.494872\pi$
$14$	0	0
$15$	−1.85181	−0.478135
$16$	0	0
$17$	−6.12798	−1.48625	−0.743127	−	0.669151i	$-0.766658\pi$
$17$	−6.12798	−1.48625	−0.743127	+	0.669151i	$0.766658\pi$
$18$	0	0
$19$	−3.34227	−0.766770	−0.383385	−	0.923589i	$-0.625242\pi$
$19$	−3.34227	−0.766770	−0.383385	+	0.923589i	$0.625242\pi$
$20$	0	0
$21$	2.41580	0.527170
$22$	0	0
$23$	−8.19583	−1.70895	−0.854475	−	0.519493i	$-0.826121\pi$
$23$	−8.19583	−1.70895	−0.854475	+	0.519493i	$0.826121\pi$
$24$	0	0
$25$	−1.57081	−0.314161
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.0936338	0.0173874	0.00869368	−	0.999962i	$-0.497233\pi$
$29$	0.0936338	0.0173874	0.00869368	+	0.999962i	$0.497233\pi$
$30$	0	0
$31$	1.44840	0.260140	0.130070	−	0.991505i	$-0.458480\pi$
$31$	1.44840	0.260140	0.130070	+	0.991505i	$0.458480\pi$
$32$	0	0
$33$	−2.49614	−0.434522
$34$	0	0
$35$	−4.47360	−0.756176
$36$	0	0
$37$	−0.230021	−0.0378152	−0.0189076	−	0.999821i	$-0.506019\pi$
$37$	−0.230021	−0.0378152	−0.0189076	+	0.999821i	$0.506019\pi$
$38$	0	0
$39$	0.116176	0.0186030
$40$	0	0
$41$	−4.49763	−0.702411	−0.351206	−	0.936298i	$-0.614228\pi$
$41$	−4.49763	−0.702411	−0.351206	+	0.936298i	$0.614228\pi$
$42$	0	0
$43$	5.35233	0.816223	0.408111	−	0.912932i	$-0.366188\pi$
$43$	5.35233	0.816223	0.408111	+	0.912932i	$0.366188\pi$
$44$	0	0
$45$	−1.85181	−0.276051
$46$	0	0
$47$	7.01021	1.02254	0.511272	−	0.859419i	$-0.329174\pi$
$47$	7.01021	1.02254	0.511272	+	0.859419i	$0.329174\pi$
$48$	0	0
$49$	−1.16392	−0.166274
$50$	0	0
$51$	−6.12798	−0.858089
$52$	0	0
$53$	3.67773	0.505176	0.252588	−	0.967574i	$-0.418718\pi$
$53$	3.67773	0.505176	0.252588	+	0.967574i	$0.418718\pi$
$54$	0	0
$55$	4.62237	0.623280
$56$	0	0
$57$	−3.34227	−0.442695
$58$	0	0
$59$	−1.76972	−0.230398	−0.115199	−	0.993342i	$-0.536751\pi$
$59$	−1.76972	−0.230398	−0.115199	+	0.993342i	$0.536751\pi$
$60$	0	0
$61$	8.66144	1.10898	0.554492	−	0.832189i	$-0.312913\pi$
$61$	8.66144	1.10898	0.554492	+	0.832189i	$0.312913\pi$
$62$	0	0
$63$	2.41580	0.304362
$64$	0	0
$65$	−0.215135	−0.0266842
$66$	0	0
$67$	−15.1723	−1.85359	−0.926795	−	0.375569i	$-0.877447\pi$
$67$	−15.1723	−1.85359	−0.926795	+	0.375569i	$0.877447\pi$
$68$	0	0
$69$	−8.19583	−0.986662
$70$	0	0
$71$	−8.51567	−1.01062	−0.505312	−	0.862937i	$-0.668623\pi$
$71$	−8.51567	−1.01062	−0.505312	+	0.862937i	$0.668623\pi$
$72$	0	0
$73$	4.74977	0.555918	0.277959	−	0.960593i	$-0.410342\pi$
$73$	4.74977	0.555918	0.277959	+	0.960593i	$0.410342\pi$
$74$	0	0
$75$	−1.57081	−0.181381
$76$	0	0
$77$	−6.03017	−0.687201
$78$	0	0
$79$	−3.24808	−0.365437	−0.182719	−	0.983165i	$-0.558490\pi$
$79$	−3.24808	−0.365437	−0.182719	+	0.983165i	$0.558490\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.437500	−0.0480219	−0.0240109	−	0.999712i	$-0.507644\pi$
$83$	−0.437500	−0.0480219	−0.0240109	+	0.999712i	$0.507644\pi$
$84$	0	0
$85$	11.3478	1.23085
$86$	0	0
$87$	0.0936338	0.0100386
$88$	0	0
$89$	11.9663	1.26842	0.634212	−	0.773159i	$-0.281325\pi$
$89$	11.9663	1.26842	0.634212	+	0.773159i	$0.281325\pi$
$90$	0	0
$91$	0.280657	0.0294209
$92$	0	0
$93$	1.44840	0.150192
$94$	0	0
$95$	6.18925	0.635004
$96$	0	0
$97$	5.72224	0.581005	0.290503	−	0.956874i	$-0.406178\pi$
$97$	5.72224	0.581005	0.290503	+	0.956874i	$0.406178\pi$
$98$	0	0
$99$	−2.49614	−0.250871
$100$	0	0
$101$	−18.5958	−1.85035	−0.925176	−	0.379539i	$-0.876083\pi$
$101$	−18.5958	−1.85035	−0.925176	+	0.379539i	$0.876083\pi$
$102$	0	0
$103$	−19.3000	−1.90169	−0.950843	−	0.309674i	$-0.899780\pi$
$103$	−19.3000	−1.90169	−0.950843	+	0.309674i	$0.899780\pi$
$104$	0	0
$105$	−4.47360	−0.436578
$106$	0	0
$107$	−18.2861	−1.76779	−0.883893	−	0.467689i	$-0.845087\pi$
$107$	−18.2861	−1.76779	−0.883893	+	0.467689i	$0.845087\pi$
$108$	0	0
$109$	−16.0245	−1.53487	−0.767435	−	0.641127i	$-0.778467\pi$
$109$	−16.0245	−1.53487	−0.767435	+	0.641127i	$0.778467\pi$
$110$	0	0
$111$	−0.230021	−0.0218326
$112$	0	0
$113$	−9.02419	−0.848924	−0.424462	−	0.905446i	$-0.639537\pi$
$113$	−9.02419	−0.848924	−0.424462	+	0.905446i	$0.639537\pi$
$114$	0	0
$115$	15.1771	1.41527
$116$	0	0
$117$	0.116176	0.0107404
$118$	0	0
$119$	−14.8040	−1.35708
$120$	0	0
$121$	−4.76930	−0.433572
$122$	0	0
$123$	−4.49763	−0.405537
$124$	0	0
$125$	12.1679	1.08833
$126$	0	0
$127$	−0.925332	−0.0821099	−0.0410550	−	0.999157i	$-0.513072\pi$
$127$	−0.925332	−0.0821099	−0.0410550	+	0.999157i	$0.513072\pi$
$128$	0	0
$129$	5.35233	0.471246
$130$	0	0
$131$	2.88590	0.252142	0.126071	−	0.992021i	$-0.459763\pi$
$131$	2.88590	0.252142	0.126071	+	0.992021i	$0.459763\pi$
$132$	0	0
$133$	−8.07426	−0.700127
$134$	0	0
$135$	−1.85181	−0.159378
$136$	0	0
$137$	18.1131	1.54750	0.773752	−	0.633488i	$-0.218377\pi$
$137$	18.1131	1.54750	0.773752	+	0.633488i	$0.218377\pi$
$138$	0	0
$139$	14.8150	1.25659	0.628294	−	0.777976i	$-0.283753\pi$
$139$	14.8150	1.25659	0.628294	+	0.777976i	$0.283753\pi$
$140$	0	0
$141$	7.01021	0.590366
$142$	0	0
$143$	−0.289991	−0.0242502
$144$	0	0
$145$	−0.173392	−0.0143994
$146$	0	0
$147$	−1.16392	−0.0959983
$148$	0	0
$149$	−0.577645	−0.0473225	−0.0236613	−	0.999720i	$-0.507532\pi$
$149$	−0.577645	−0.0473225	−0.0236613	+	0.999720i	$0.507532\pi$
$150$	0	0
$151$	−1.65587	−0.134753	−0.0673766	−	0.997728i	$-0.521463\pi$
$151$	−1.65587	−0.134753	−0.0673766	+	0.997728i	$0.521463\pi$
$152$	0	0
$153$	−6.12798	−0.495418
$154$	0	0
$155$	−2.68215	−0.215436
$156$	0	0
$157$	−4.20297	−0.335434	−0.167717	−	0.985835i	$-0.553639\pi$
$157$	−4.20297	−0.335434	−0.167717	+	0.985835i	$0.553639\pi$
$158$	0	0
$159$	3.67773	0.291663
$160$	0	0
$161$	−19.7995	−1.56042
$162$	0	0
$163$	20.7509	1.62534	0.812668	−	0.582727i	$-0.198014\pi$
$163$	20.7509	1.62534	0.812668	+	0.582727i	$0.198014\pi$
$164$	0	0
$165$	4.62237	0.359851
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.9865	−0.998962
$170$	0	0
$171$	−3.34227	−0.255590
$172$	0	0
$173$	−14.4252	−1.09673	−0.548363	−	0.836240i	$-0.684749\pi$
$173$	−14.4252	−1.09673	−0.548363	+	0.836240i	$0.684749\pi$
$174$	0	0
$175$	−3.79475	−0.286856
$176$	0	0
$177$	−1.76972	−0.133020
$178$	0	0
$179$	21.8860	1.63583	0.817916	−	0.575337i	$-0.195129\pi$
$179$	21.8860	1.63583	0.817916	+	0.575337i	$0.195129\pi$
$180$	0	0
$181$	−10.9551	−0.814284	−0.407142	−	0.913365i	$-0.633475\pi$
$181$	−10.9551	−0.814284	−0.407142	+	0.913365i	$0.633475\pi$
$182$	0	0
$183$	8.66144	0.640272
$184$	0	0
$185$	0.425955	0.0313168
$186$	0	0
$187$	15.2963	1.11857
$188$	0	0
$189$	2.41580	0.175723
$190$	0	0
$191$	4.40538	0.318762	0.159381	−	0.987217i	$-0.449050\pi$
$191$	4.40538	0.318762	0.159381	+	0.987217i	$0.449050\pi$
$192$	0	0
$193$	2.81738	0.202799	0.101400	−	0.994846i	$-0.467668\pi$
$193$	2.81738	0.202799	0.101400	+	0.994846i	$0.467668\pi$
$194$	0	0
$195$	−0.215135	−0.0154061
$196$	0	0
$197$	20.7266	1.47671	0.738353	−	0.674414i	$-0.235604\pi$
$197$	20.7266	1.47671	0.738353	+	0.674414i	$0.235604\pi$
$198$	0	0
$199$	−0.813799	−0.0576887	−0.0288444	−	0.999584i	$-0.509183\pi$
$199$	−0.813799	−0.0576887	−0.0288444	+	0.999584i	$0.509183\pi$
$200$	0	0
$201$	−15.1723	−1.07017
$202$	0	0
$203$	0.226200	0.0158762
$204$	0	0
$205$	8.32874	0.581705
$206$	0	0
$207$	−8.19583	−0.569650
$208$	0	0
$209$	8.34278	0.577082
$210$	0	0
$211$	27.3088	1.88002	0.940009	−	0.341151i	$-0.110817\pi$
$211$	27.3088	1.88002	0.940009	+	0.341151i	$0.110817\pi$
$212$	0	0
$213$	−8.51567	−0.583484
$214$	0	0
$215$	−9.91149	−0.675958
$216$	0	0
$217$	3.49903	0.237530
$218$	0	0
$219$	4.74977	0.320960
$220$	0	0
$221$	−0.711923	−0.0478891
$222$	0	0
$223$	−0.297772	−0.0199403	−0.00997015	−	0.999950i	$-0.503174\pi$
$223$	−0.297772	−0.0199403	−0.00997015	+	0.999950i	$0.503174\pi$
$224$	0	0
$225$	−1.57081	−0.104720
$226$	0	0
$227$	−2.46412	−0.163550	−0.0817748	−	0.996651i	$-0.526059\pi$
$227$	−2.46412	−0.163550	−0.0817748	+	0.996651i	$0.526059\pi$
$228$	0	0
$229$	7.52769	0.497444	0.248722	−	0.968575i	$-0.419989\pi$
$229$	7.52769	0.497444	0.248722	+	0.968575i	$0.419989\pi$
$230$	0	0
$231$	−6.03017	−0.396756
$232$	0	0
$233$	7.47360	0.489612	0.244806	−	0.969572i	$-0.421276\pi$
$233$	7.47360	0.489612	0.244806	+	0.969572i	$0.421276\pi$
$234$	0	0
$235$	−12.9816	−0.846824
$236$	0	0
$237$	−3.24808	−0.210985
$238$	0	0
$239$	−12.2797	−0.794306	−0.397153	−	0.917752i	$-0.630002\pi$
$239$	−12.2797	−0.794306	−0.397153	+	0.917752i	$0.630002\pi$
$240$	0	0
$241$	−12.7966	−0.824299	−0.412150	−	0.911116i	$-0.635222\pi$
$241$	−12.7966	−0.824299	−0.412150	+	0.911116i	$0.635222\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.15535	0.137700
$246$	0	0
$247$	−0.388291	−0.0247064
$248$	0	0
$249$	−0.437500	−0.0277254
$250$	0	0
$251$	13.7924	0.870571	0.435285	−	0.900292i	$-0.356647\pi$
$251$	13.7924	0.870571	0.435285	+	0.900292i	$0.356647\pi$
$252$	0	0
$253$	20.4579	1.28618
$254$	0	0
$255$	11.3478	0.710630
$256$	0	0
$257$	−2.03436	−0.126900	−0.0634501	−	0.997985i	$-0.520210\pi$
$257$	−2.03436	−0.126900	−0.0634501	+	0.997985i	$0.520210\pi$
$258$	0	0
$259$	−0.555684	−0.0345285
$260$	0	0
$261$	0.0936338	0.00579579
$262$	0	0
$263$	22.6680	1.39777	0.698886	−	0.715233i	$-0.253680\pi$
$263$	22.6680	1.39777	0.698886	+	0.715233i	$0.253680\pi$
$264$	0	0
$265$	−6.81046	−0.418363
$266$	0	0
$267$	11.9663	0.732325
$268$	0	0
$269$	−23.5210	−1.43410	−0.717052	−	0.697020i	$-0.754509\pi$
$269$	−23.5210	−1.43410	−0.717052	+	0.697020i	$0.754509\pi$
$270$	0	0
$271$	15.5238	0.943005	0.471503	−	0.881865i	$-0.343712\pi$
$271$	15.5238	0.943005	0.471503	+	0.881865i	$0.343712\pi$
$272$	0	0
$273$	0.280657	0.0169861
$274$	0	0
$275$	3.92095	0.236442
$276$	0	0
$277$	9.69430	0.582474	0.291237	−	0.956651i	$-0.405933\pi$
$277$	9.69430	0.582474	0.291237	+	0.956651i	$0.405933\pi$
$278$	0	0
$279$	1.44840	0.0867132
$280$	0	0
$281$	−6.65022	−0.396719	−0.198359	−	0.980129i	$-0.563561\pi$
$281$	−6.65022	−0.396719	−0.198359	+	0.980129i	$0.563561\pi$
$282$	0	0
$283$	20.1057	1.19516	0.597580	−	0.801809i	$-0.296129\pi$
$283$	20.1057	1.19516	0.597580	+	0.801809i	$0.296129\pi$
$284$	0	0
$285$	6.18925	0.366620
$286$	0	0
$287$	−10.8654	−0.641362
$288$	0	0
$289$	20.5521	1.20895
$290$	0	0
$291$	5.72224	0.335443
$292$	0	0
$293$	2.72431	0.159156	0.0795781	−	0.996829i	$-0.474643\pi$
$293$	2.72431	0.159156	0.0795781	+	0.996829i	$0.474643\pi$
$294$	0	0
$295$	3.27718	0.190805
$296$	0	0
$297$	−2.49614	−0.144841
$298$	0	0
$299$	−0.952157	−0.0550646
$300$	0	0
$301$	12.9302	0.745282
$302$	0	0
$303$	−18.5958	−1.06830
$304$	0	0
$305$	−16.0393	−0.918410
$306$	0	0
$307$	−7.63757	−0.435899	−0.217950	−	0.975960i	$-0.569937\pi$
$307$	−7.63757	−0.435899	−0.217950	+	0.975960i	$0.569937\pi$
$308$	0	0
$309$	−19.3000	−1.09794
$310$	0	0
$311$	19.6655	1.11513	0.557565	−	0.830133i	$-0.311736\pi$
$311$	19.6655	1.11513	0.557565	+	0.830133i	$0.311736\pi$
$312$	0	0
$313$	−27.6680	−1.56389	−0.781944	−	0.623349i	$-0.785772\pi$
$313$	−27.6680	−1.56389	−0.781944	+	0.623349i	$0.785772\pi$
$314$	0	0
$315$	−4.47360	−0.252059
$316$	0	0
$317$	8.67973	0.487502	0.243751	−	0.969838i	$-0.421622\pi$
$317$	8.67973	0.487502	0.243751	+	0.969838i	$0.421622\pi$
$318$	0	0
$319$	−0.233723	−0.0130860
$320$	0	0
$321$	−18.2861	−1.02063
$322$	0	0
$323$	20.4814	1.13962
$324$	0	0
$325$	−0.182490	−0.0101227
$326$	0	0
$327$	−16.0245	−0.886158
$328$	0	0
$329$	16.9353	0.933671
$330$	0	0
$331$	12.1719	0.669027	0.334513	−	0.942391i	$-0.391428\pi$
$331$	12.1719	0.669027	0.334513	+	0.942391i	$0.391428\pi$
$332$	0	0
$333$	−0.230021	−0.0126051
$334$	0	0
$335$	28.0962	1.53506
$336$	0	0
$337$	−14.9558	−0.814692	−0.407346	−	0.913274i	$-0.633546\pi$
$337$	−14.9558	−0.814692	−0.407346	+	0.913274i	$0.633546\pi$
$338$	0	0
$339$	−9.02419	−0.490127
$340$	0	0
$341$	−3.61539	−0.195785
$342$	0	0
$343$	−19.7224	−1.06491
$344$	0	0
$345$	15.1771	0.817108
$346$	0	0
$347$	−18.7269	−1.00531	−0.502655	−	0.864487i	$-0.667644\pi$
$347$	−18.7269	−1.00531	−0.502655	+	0.864487i	$0.667644\pi$
$348$	0	0
$349$	15.2459	0.816094	0.408047	−	0.912961i	$-0.366210\pi$
$349$	15.2459	0.816094	0.408047	+	0.912961i	$0.366210\pi$
$350$	0	0
$351$	0.116176	0.00620100
$352$	0	0
$353$	−6.76872	−0.360262	−0.180131	−	0.983643i	$-0.557652\pi$
$353$	−6.76872	−0.360262	−0.180131	+	0.983643i	$0.557652\pi$
$354$	0	0
$355$	15.7694	0.836952
$356$	0	0
$357$	−14.8040	−0.783509
$358$	0	0
$359$	−10.6775	−0.563537	−0.281769	−	0.959482i	$-0.590921\pi$
$359$	−10.6775	−0.563537	−0.281769	+	0.959482i	$0.590921\pi$
$360$	0	0
$361$	−7.82920	−0.412063
$362$	0	0
$363$	−4.76930	−0.250323
$364$	0	0
$365$	−8.79566	−0.460386
$366$	0	0
$367$	−4.63745	−0.242073	−0.121036	−	0.992648i	$-0.538622\pi$
$367$	−4.63745	−0.242073	−0.121036	+	0.992648i	$0.538622\pi$
$368$	0	0
$369$	−4.49763	−0.234137
$370$	0	0
$371$	8.88466	0.461269
$372$	0	0
$373$	11.0186	0.570521	0.285260	−	0.958450i	$-0.407920\pi$
$373$	11.0186	0.570521	0.285260	+	0.958450i	$0.407920\pi$
$374$	0	0
$375$	12.1679	0.628346
$376$	0	0
$377$	0.0108780	0.000560244 0
$378$	0	0
$379$	23.2728	1.19544	0.597721	−	0.801705i	$-0.296073\pi$
$379$	23.2728	1.19544	0.597721	+	0.801705i	$0.296073\pi$
$380$	0	0
$381$	−0.925332	−0.0474062
$382$	0	0
$383$	−33.6147	−1.71763	−0.858816	−	0.512284i	$-0.828799\pi$
$383$	−33.6147	−1.71763	−0.858816	+	0.512284i	$0.828799\pi$
$384$	0	0
$385$	11.1667	0.569108
$386$	0	0
$387$	5.35233	0.272074
$388$	0	0
$389$	−22.6735	−1.14959	−0.574795	−	0.818298i	$-0.694918\pi$
$389$	−22.6735	−1.14959	−0.574795	+	0.818298i	$0.694918\pi$
$390$	0	0
$391$	50.2239	2.53993
$392$	0	0
$393$	2.88590	0.145574
$394$	0	0
$395$	6.01482	0.302638
$396$	0	0
$397$	−32.1120	−1.61166	−0.805828	−	0.592149i	$-0.798280\pi$
$397$	−32.1120	−1.61166	−0.805828	+	0.592149i	$0.798280\pi$
$398$	0	0
$399$	−8.07426	−0.404219
$400$	0	0
$401$	−5.31642	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$401$	−5.31642	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$402$	0	0
$403$	0.168268	0.00838205
$404$	0	0
$405$	−1.85181	−0.0920171
$406$	0	0
$407$	0.574164	0.0284603
$408$	0	0
$409$	−28.0684	−1.38789	−0.693947	−	0.720026i	$-0.744130\pi$
$409$	−28.0684	−1.38789	−0.693947	+	0.720026i	$0.744130\pi$
$410$	0	0
$411$	18.1131	0.893452
$412$	0	0
$413$	−4.27529	−0.210373
$414$	0	0
$415$	0.810166	0.0397695
$416$	0	0
$417$	14.8150	0.725492
$418$	0	0
$419$	5.57807	0.272506	0.136253	−	0.990674i	$-0.456494\pi$
$419$	5.57807	0.272506	0.136253	+	0.990674i	$0.456494\pi$
$420$	0	0
$421$	6.45061	0.314383	0.157192	−	0.987568i	$-0.449756\pi$
$421$	6.45061	0.314383	0.157192	+	0.987568i	$0.449756\pi$
$422$	0	0
$423$	7.01021	0.340848
$424$	0	0
$425$	9.62587	0.466923
$426$	0	0
$427$	20.9243	1.01260
$428$	0	0
$429$	−0.289991	−0.0140009
$430$	0	0
$431$	−31.4574	−1.51525	−0.757624	−	0.652692i	$-0.773640\pi$
$431$	−31.4574	−1.51525	−0.757624	+	0.652692i	$0.773640\pi$
$432$	0	0
$433$	9.89285	0.475420	0.237710	−	0.971336i	$-0.423603\pi$
$433$	9.89285	0.475420	0.237710	+	0.971336i	$0.423603\pi$
$434$	0	0
$435$	−0.173392	−0.00831350
$436$	0	0
$437$	27.3927	1.31037
$438$	0	0
$439$	15.9324	0.760412	0.380206	−	0.924902i	$-0.375853\pi$
$439$	15.9324	0.760412	0.380206	+	0.924902i	$0.375853\pi$
$440$	0	0
$441$	−1.16392	−0.0554247
$442$	0	0
$443$	34.7291	1.65003	0.825014	−	0.565112i	$-0.191167\pi$
$443$	34.7291	1.65003	0.825014	+	0.565112i	$0.191167\pi$
$444$	0	0
$445$	−22.1593	−1.05045
$446$	0	0
$447$	−0.577645	−0.0273217
$448$	0	0
$449$	−22.3182	−1.05326	−0.526631	−	0.850094i	$-0.676545\pi$
$449$	−22.3182	−1.05326	−0.526631	+	0.850094i	$0.676545\pi$
$450$	0	0
$451$	11.2267	0.528645
$452$	0	0
$453$	−1.65587	−0.0777998
$454$	0	0
$455$	−0.519723	−0.0243650
$456$	0	0
$457$	3.44665	0.161227	0.0806137	−	0.996745i	$-0.474312\pi$
$457$	3.44665	0.161227	0.0806137	+	0.996745i	$0.474312\pi$
$458$	0	0
$459$	−6.12798	−0.286030
$460$	0	0
$461$	−9.03541	−0.420821	−0.210411	−	0.977613i	$-0.567480\pi$
$461$	−9.03541	−0.420821	−0.210411	+	0.977613i	$0.567480\pi$
$462$	0	0
$463$	−12.9438	−0.601551	−0.300776	−	0.953695i	$-0.597245\pi$
$463$	−12.9438	−0.601551	−0.300776	+	0.953695i	$0.597245\pi$
$464$	0	0
$465$	−2.68215	−0.124382
$466$	0	0
$467$	−15.5010	−0.717303	−0.358651	−	0.933472i	$-0.616763\pi$
$467$	−15.5010	−0.717303	−0.358651	+	0.933472i	$0.616763\pi$
$468$	0	0
$469$	−36.6532	−1.69249
$470$	0	0
$471$	−4.20297	−0.193663
$472$	0	0
$473$	−13.3602	−0.614300
$474$	0	0
$475$	5.25007	0.240890
$476$	0	0
$477$	3.67773	0.168392
$478$	0	0
$479$	−12.8336	−0.586381	−0.293190	−	0.956054i	$-0.594717\pi$
$479$	−12.8336	−0.586381	−0.293190	+	0.956054i	$0.594717\pi$
$480$	0	0
$481$	−0.0267229	−0.00121846
$482$	0	0
$483$	−19.7995	−0.900908
$484$	0	0
$485$	−10.5965	−0.481162
$486$	0	0
$487$	4.39313	0.199072	0.0995359	−	0.995034i	$-0.468264\pi$
$487$	4.39313	0.199072	0.0995359	+	0.995034i	$0.468264\pi$
$488$	0	0
$489$	20.7509	0.938388
$490$	0	0
$491$	16.0337	0.723593	0.361796	−	0.932257i	$-0.382164\pi$
$491$	16.0337	0.723593	0.361796	+	0.932257i	$0.382164\pi$
$492$	0	0
$493$	−0.573786	−0.0258420
$494$	0	0
$495$	4.62237	0.207760
$496$	0	0
$497$	−20.5721	−0.922786
$498$	0	0
$499$	32.1588	1.43963	0.719813	−	0.694168i	$-0.244228\pi$
$499$	32.1588	1.43963	0.719813	+	0.694168i	$0.244228\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	7.88618	0.351627	0.175814	−	0.984423i	$-0.443744\pi$
$503$	7.88618	0.351627	0.175814	+	0.984423i	$0.443744\pi$
$504$	0	0
$505$	34.4359	1.53238
$506$	0	0
$507$	−12.9865	−0.576751
$508$	0	0
$509$	24.9425	1.10556	0.552779	−	0.833328i	$-0.313568\pi$
$509$	24.9425	1.10556	0.552779	+	0.833328i	$0.313568\pi$
$510$	0	0
$511$	11.4745	0.507601
$512$	0	0
$513$	−3.34227	−0.147565
$514$	0	0
$515$	35.7399	1.57489
$516$	0	0
$517$	−17.4985	−0.769581
$518$	0	0
$519$	−14.4252	−0.633196
$520$	0	0
$521$	3.76759	0.165061	0.0825305	−	0.996589i	$-0.473700\pi$
$521$	3.76759	0.165061	0.0825305	+	0.996589i	$0.473700\pi$
$522$	0	0
$523$	22.7409	0.994389	0.497195	−	0.867639i	$-0.334364\pi$
$523$	22.7409	0.994389	0.497195	+	0.867639i	$0.334364\pi$
$524$	0	0
$525$	−3.79475	−0.165616
$526$	0	0
$527$	−8.87574	−0.386633
$528$	0	0
$529$	44.1717	1.92051
$530$	0	0
$531$	−1.76972	−0.0767993
$532$	0	0
$533$	−0.522515	−0.0226326
$534$	0	0
$535$	33.8624	1.46400
$536$	0	0
$537$	21.8860	0.944449
$538$	0	0
$539$	2.90530	0.125140
$540$	0	0
$541$	10.6993	0.460000	0.230000	−	0.973191i	$-0.426127\pi$
$541$	10.6993	0.460000	0.230000	+	0.973191i	$0.426127\pi$
$542$	0	0
$543$	−10.9551	−0.470127
$544$	0	0
$545$	29.6743	1.27111
$546$	0	0
$547$	−20.8098	−0.889763	−0.444881	−	0.895590i	$-0.646754\pi$
$547$	−20.8098	−0.889763	−0.444881	+	0.895590i	$0.646754\pi$
$548$	0	0
$549$	8.66144	0.369661
$550$	0	0
$551$	−0.312950	−0.0133321
$552$	0	0
$553$	−7.84670	−0.333676
$554$	0	0
$555$	0.425955	0.0180808
$556$	0	0
$557$	−21.7669	−0.922292	−0.461146	−	0.887324i	$-0.652562\pi$
$557$	−21.7669	−0.922292	−0.461146	+	0.887324i	$0.652562\pi$
$558$	0	0
$559$	0.621811	0.0262998
$560$	0	0
$561$	15.2963	0.645810
$562$	0	0
$563$	−22.6695	−0.955404	−0.477702	−	0.878522i	$-0.658530\pi$
$563$	−22.6695	−0.955404	−0.477702	+	0.878522i	$0.658530\pi$
$564$	0	0
$565$	16.7111	0.703040
$566$	0	0
$567$	2.41580	0.101454
$568$	0	0
$569$	20.9889	0.879898	0.439949	−	0.898023i	$-0.354996\pi$
$569$	20.9889	0.879898	0.439949	+	0.898023i	$0.354996\pi$
$570$	0	0
$571$	31.4599	1.31656	0.658278	−	0.752775i	$-0.271285\pi$
$571$	31.4599	1.31656	0.658278	+	0.752775i	$0.271285\pi$
$572$	0	0
$573$	4.40538	0.184038
$574$	0	0
$575$	12.8741	0.536885
$576$	0	0
$577$	−25.6636	−1.06839	−0.534196	−	0.845361i	$-0.679385\pi$
$577$	−25.6636	−1.06839	−0.534196	+	0.845361i	$0.679385\pi$
$578$	0	0
$579$	2.81738	0.117086
$580$	0	0
$581$	−1.05691	−0.0438481
$582$	0	0
$583$	−9.18013	−0.380202
$584$	0	0
$585$	−0.215135	−0.00889474
$586$	0	0
$587$	27.5089	1.13542	0.567708	−	0.823230i	$-0.307830\pi$
$587$	27.5089	1.13542	0.567708	+	0.823230i	$0.307830\pi$
$588$	0	0
$589$	−4.84094	−0.199467
$590$	0	0
$591$	20.7266	0.852577
$592$	0	0
$593$	−27.4088	−1.12554	−0.562772	−	0.826612i	$-0.690265\pi$
$593$	−27.4088	−1.12554	−0.562772	+	0.826612i	$0.690265\pi$
$594$	0	0
$595$	27.4141	1.12387
$596$	0	0
$597$	−0.813799	−0.0333066
$598$	0	0
$599$	35.6007	1.45460	0.727302	−	0.686317i	$-0.240774\pi$
$599$	35.6007	1.45460	0.727302	+	0.686317i	$0.240774\pi$
$600$	0	0
$601$	−4.45143	−0.181578	−0.0907888	−	0.995870i	$-0.528939\pi$
$601$	−4.45143	−0.181578	−0.0907888	+	0.995870i	$0.528939\pi$
$602$	0	0
$603$	−15.1723	−0.617863
$604$	0	0
$605$	8.83182	0.359065
$606$	0	0
$607$	−39.6466	−1.60921	−0.804604	−	0.593812i	$-0.797622\pi$
$607$	−39.6466	−1.60921	−0.804604	+	0.593812i	$0.797622\pi$
$608$	0	0
$609$	0.226200	0.00916610
$610$	0	0
$611$	0.814417	0.0329478
$612$	0	0
$613$	37.0509	1.49647	0.748237	−	0.663432i	$-0.230901\pi$
$613$	37.0509	1.49647	0.748237	+	0.663432i	$0.230901\pi$
$614$	0	0
$615$	8.32874	0.335847
$616$	0	0
$617$	35.9525	1.44739	0.723697	−	0.690118i	$-0.242441\pi$
$617$	35.9525	1.44739	0.723697	+	0.690118i	$0.242441\pi$
$618$	0	0
$619$	41.1996	1.65595	0.827976	−	0.560763i	$-0.189492\pi$
$619$	41.1996	1.65595	0.827976	+	0.560763i	$0.189492\pi$
$620$	0	0
$621$	−8.19583	−0.328887
$622$	0	0
$623$	28.9081	1.15818
$624$	0	0
$625$	−14.6785	−0.587142
$626$	0	0
$627$	8.34278	0.333178
$628$	0	0
$629$	1.40956	0.0562030
$630$	0	0
$631$	−25.7109	−1.02354	−0.511768	−	0.859124i	$-0.671009\pi$
$631$	−25.7109	−1.02354	−0.511768	+	0.859124i	$0.671009\pi$
$632$	0	0
$633$	27.3088	1.08543
$634$	0	0
$635$	1.71354	0.0679996
$636$	0	0
$637$	−0.135219	−0.00535757
$638$	0	0
$639$	−8.51567	−0.336875
$640$	0	0
$641$	2.43782	0.0962879	0.0481440	−	0.998840i	$-0.484669\pi$
$641$	2.43782	0.0962879	0.0481440	+	0.998840i	$0.484669\pi$
$642$	0	0
$643$	30.5238	1.20374	0.601870	−	0.798594i	$-0.294423\pi$
$643$	30.5238	1.20374	0.601870	+	0.798594i	$0.294423\pi$
$644$	0	0
$645$	−9.91149	−0.390265
$646$	0	0
$647$	−3.26910	−0.128521	−0.0642607	−	0.997933i	$-0.520469\pi$
$647$	−3.26910	−0.128521	−0.0642607	+	0.997933i	$0.520469\pi$
$648$	0	0
$649$	4.41746	0.173401
$650$	0	0
$651$	3.49903	0.137138
$652$	0	0
$653$	5.24631	0.205304	0.102652	−	0.994717i	$-0.467267\pi$
$653$	5.24631	0.205304	0.102652	+	0.994717i	$0.467267\pi$
$654$	0	0
$655$	−5.34413	−0.208812
$656$	0	0
$657$	4.74977	0.185306
$658$	0	0
$659$	−24.5139	−0.954925	−0.477462	−	0.878652i	$-0.658443\pi$
$659$	−24.5139	−0.954925	−0.477462	+	0.878652i	$0.658443\pi$
$660$	0	0
$661$	−45.1848	−1.75749	−0.878743	−	0.477295i	$-0.841617\pi$
$661$	−45.1848	−1.75749	−0.878743	+	0.477295i	$0.841617\pi$
$662$	0	0
$663$	−0.711923	−0.0276488
$664$	0	0
$665$	14.9520	0.579813
$666$	0	0
$667$	−0.767407	−0.0297141
$668$	0	0
$669$	−0.297772	−0.0115125
$670$	0	0
$671$	−21.6202	−0.834637
$672$	0	0
$673$	26.7834	1.03243	0.516213	−	0.856460i	$-0.327341\pi$
$673$	26.7834	1.03243	0.516213	+	0.856460i	$0.327341\pi$
$674$	0	0
$675$	−1.57081	−0.0604603
$676$	0	0
$677$	43.3493	1.66605	0.833025	−	0.553235i	$-0.186607\pi$
$677$	43.3493	1.66605	0.833025	+	0.553235i	$0.186607\pi$
$678$	0	0
$679$	13.8238	0.530508
$680$	0	0
$681$	−2.46412	−0.0944253
$682$	0	0
$683$	−10.0366	−0.384040	−0.192020	−	0.981391i	$-0.561504\pi$
$683$	−10.0366	−0.384040	−0.192020	+	0.981391i	$0.561504\pi$
$684$	0	0
$685$	−33.5420	−1.28157
$686$	0	0
$687$	7.52769	0.287199
$688$	0	0
$689$	0.427263	0.0162774
$690$	0	0
$691$	−17.9872	−0.684266	−0.342133	−	0.939652i	$-0.611149\pi$
$691$	−17.9872	−0.684266	−0.342133	+	0.939652i	$0.611149\pi$
$692$	0	0
$693$	−6.03017	−0.229067
$694$	0	0
$695$	−27.4345	−1.04065
$696$	0	0
$697$	27.5614	1.04396
$698$	0	0
$699$	7.47360	0.282677
$700$	0	0
$701$	30.7815	1.16260	0.581301	−	0.813689i	$-0.302544\pi$
$701$	30.7815	1.16260	0.581301	+	0.813689i	$0.302544\pi$
$702$	0	0
$703$	0.768793	0.0289956
$704$	0	0
$705$	−12.9816	−0.488914
$706$	0	0
$707$	−44.9237	−1.68953
$708$	0	0
$709$	−6.09943	−0.229069	−0.114534	−	0.993419i	$-0.536538\pi$
$709$	−6.09943	−0.229069	−0.114534	+	0.993419i	$0.536538\pi$
$710$	0	0
$711$	−3.24808	−0.121812
$712$	0	0
$713$	−11.8708	−0.444565
$714$	0	0
$715$	0.537007	0.0200829
$716$	0	0
$717$	−12.2797	−0.458593
$718$	0	0
$719$	−24.6860	−0.920632	−0.460316	−	0.887755i	$-0.652264\pi$
$719$	−24.6860	−0.920632	−0.460316	+	0.887755i	$0.652264\pi$
$720$	0	0
$721$	−46.6249	−1.73640
$722$	0	0
$723$	−12.7966	−0.475909
$724$	0	0
$725$	−0.147081	−0.00546243
$726$	0	0
$727$	36.0712	1.33781	0.668903	−	0.743349i	$-0.266764\pi$
$727$	36.0712	1.33781	0.668903	+	0.743349i	$0.266764\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−32.7990	−1.21311
$732$	0	0
$733$	−30.1781	−1.11465	−0.557327	−	0.830293i	$-0.688173\pi$
$733$	−30.1781	−1.11465	−0.557327	+	0.830293i	$0.688173\pi$
$734$	0	0
$735$	2.15535	0.0795014
$736$	0	0
$737$	37.8721	1.39504
$738$	0	0
$739$	−11.5288	−0.424093	−0.212047	−	0.977260i	$-0.568013\pi$
$739$	−11.5288	−0.424093	−0.212047	+	0.977260i	$0.568013\pi$
$740$	0	0
$741$	−0.388291	−0.0142642
$742$	0	0
$743$	−18.5601	−0.680904	−0.340452	−	0.940262i	$-0.610580\pi$
$743$	−18.5601	−0.680904	−0.340452	+	0.940262i	$0.610580\pi$
$744$	0	0
$745$	1.06969	0.0391903
$746$	0	0
$747$	−0.437500	−0.0160073
$748$	0	0
$749$	−44.1756	−1.61414
$750$	0	0
$751$	−2.26069	−0.0824937	−0.0412469	−	0.999149i	$-0.513133\pi$
$751$	−2.26069	−0.0824937	−0.0412469	+	0.999149i	$0.513133\pi$
$752$	0	0
$753$	13.7924	0.502624
$754$	0	0
$755$	3.06636	0.111596
$756$	0	0
$757$	−45.7056	−1.66120	−0.830598	−	0.556872i	$-0.812001\pi$
$757$	−45.7056	−1.66120	−0.830598	+	0.556872i	$0.812001\pi$
$758$	0	0
$759$	20.4579	0.742576
$760$	0	0
$761$	−21.2344	−0.769745	−0.384873	−	0.922970i	$-0.625755\pi$
$761$	−21.2344	−0.769745	−0.384873	+	0.922970i	$0.625755\pi$
$762$	0	0
$763$	−38.7120	−1.40147
$764$	0	0
$765$	11.3478	0.410282
$766$	0	0
$767$	−0.205598	−0.00742373
$768$	0	0
$769$	24.5451	0.885118	0.442559	−	0.896739i	$-0.354071\pi$
$769$	24.5451	0.885118	0.442559	+	0.896739i	$0.354071\pi$
$770$	0	0
$771$	−2.03436	−0.0732658
$772$	0	0
$773$	−29.6966	−1.06811	−0.534056	−	0.845449i	$-0.679333\pi$
$773$	−29.6966	−1.06811	−0.534056	+	0.845449i	$0.679333\pi$
$774$	0	0
$775$	−2.27515	−0.0817257
$776$	0	0
$777$	−0.555684	−0.0199351
$778$	0	0
$779$	15.0323	0.538588
$780$	0	0
$781$	21.2563	0.760609
$782$	0	0
$783$	0.0936338	0.00334620
$784$	0	0
$785$	7.78310	0.277791
$786$	0	0
$787$	−1.16617	−0.0415694	−0.0207847	−	0.999784i	$-0.506616\pi$
$787$	−1.16617	−0.0415694	−0.0207847	+	0.999784i	$0.506616\pi$
$788$	0	0
$789$	22.6680	0.807004
$790$	0	0
$791$	−21.8006	−0.775141
$792$	0	0
$793$	1.00625	0.0357330
$794$	0	0
$795$	−6.81046	−0.241542
$796$	0	0
$797$	−3.69459	−0.130869	−0.0654345	−	0.997857i	$-0.520843\pi$
$797$	−3.69459	−0.130869	−0.0654345	+	0.997857i	$0.520843\pi$
$798$	0	0
$799$	−42.9584	−1.51976
$800$	0	0
$801$	11.9663	0.422808
$802$	0	0
$803$	−11.8561	−0.418392
$804$	0	0
$805$	36.6648	1.29227
$806$	0	0
$807$	−23.5210	−0.827980
$808$	0	0
$809$	7.83155	0.275343	0.137671	−	0.990478i	$-0.456038\pi$
$809$	7.83155	0.275343	0.137671	+	0.990478i	$0.456038\pi$
$810$	0	0
$811$	−19.6708	−0.690734	−0.345367	−	0.938468i	$-0.612246\pi$
$811$	−19.6708	−0.690734	−0.345367	+	0.938468i	$0.612246\pi$
$812$	0	0
$813$	15.5238	0.544444
$814$	0	0
$815$	−38.4267	−1.34603
$816$	0	0
$817$	−17.8890	−0.625855
$818$	0	0
$819$	0.280657	0.00980695
$820$	0	0
$821$	29.8119	1.04044	0.520222	−	0.854031i	$-0.325849\pi$
$821$	29.8119	1.04044	0.520222	+	0.854031i	$0.325849\pi$
$822$	0	0
$823$	26.6385	0.928559	0.464280	−	0.885689i	$-0.346313\pi$
$823$	26.6385	0.928559	0.464280	+	0.885689i	$0.346313\pi$
$824$	0	0
$825$	3.92095	0.136510
$826$	0	0
$827$	−30.1035	−1.04680	−0.523401	−	0.852086i	$-0.675337\pi$
$827$	−30.1035	−1.04680	−0.523401	+	0.852086i	$0.675337\pi$
$828$	0	0
$829$	−25.4482	−0.883852	−0.441926	−	0.897051i	$-0.645705\pi$
$829$	−25.4482	−0.883852	−0.441926	+	0.897051i	$0.645705\pi$
$830$	0	0
$831$	9.69430	0.336292
$832$	0	0
$833$	7.13247	0.247125
$834$	0	0
$835$	1.85181	0.0640845
$836$	0	0
$837$	1.44840	0.0500639
$838$	0	0
$839$	1.40852	0.0486275	0.0243138	−	0.999704i	$-0.492260\pi$
$839$	1.40852	0.0486275	0.0243138	+	0.999704i	$0.492260\pi$
$840$	0	0
$841$	−28.9912	−0.999698
$842$	0	0
$843$	−6.65022	−0.229046
$844$	0	0
$845$	24.0485	0.827294
$846$	0	0
$847$	−11.5217	−0.395889
$848$	0	0
$849$	20.1057	0.690026
$850$	0	0
$851$	1.88521	0.0646243
$852$	0	0
$853$	−27.2693	−0.933682	−0.466841	−	0.884341i	$-0.654608\pi$
$853$	−27.2693	−0.933682	−0.466841	+	0.884341i	$0.654608\pi$
$854$	0	0
$855$	6.18925	0.211668
$856$	0	0
$857$	8.80951	0.300927	0.150464	−	0.988616i	$-0.451923\pi$
$857$	8.80951	0.300927	0.150464	+	0.988616i	$0.451923\pi$
$858$	0	0
$859$	−28.9321	−0.987152	−0.493576	−	0.869703i	$-0.664310\pi$
$859$	−28.9321	−0.987152	−0.493576	+	0.869703i	$0.664310\pi$
$860$	0	0
$861$	−10.8654	−0.370291
$862$	0	0
$863$	−1.47181	−0.0501008	−0.0250504	−	0.999686i	$-0.507975\pi$
$863$	−1.47181	−0.0501008	−0.0250504	+	0.999686i	$0.507975\pi$
$864$	0	0
$865$	26.7127	0.908259
$866$	0	0
$867$	20.5521	0.697987
$868$	0	0
$869$	8.10765	0.275033
$870$	0	0
$871$	−1.76265	−0.0597251
$872$	0	0
$873$	5.72224	0.193668
$874$	0	0
$875$	29.3951	0.993737
$876$	0	0
$877$	−37.7379	−1.27432	−0.637159	−	0.770732i	$-0.719891\pi$
$877$	−37.7379	−1.27432	−0.637159	+	0.770732i	$0.719891\pi$
$878$	0	0
$879$	2.72431	0.0918888
$880$	0	0
$881$	−24.7811	−0.834897	−0.417449	−	0.908700i	$-0.637076\pi$
$881$	−24.7811	−0.834897	−0.417449	+	0.908700i	$0.637076\pi$
$882$	0	0
$883$	36.9451	1.24330	0.621651	−	0.783295i	$-0.286462\pi$
$883$	36.9451	1.24330	0.621651	+	0.783295i	$0.286462\pi$
$884$	0	0
$885$	3.27718	0.110161
$886$	0	0
$887$	46.6482	1.56629	0.783146	−	0.621837i	$-0.213614\pi$
$887$	46.6482	1.56629	0.783146	+	0.621837i	$0.213614\pi$
$888$	0	0
$889$	−2.23542	−0.0749734
$890$	0	0
$891$	−2.49614	−0.0836238
$892$	0	0
$893$	−23.4301	−0.784057
$894$	0	0
$895$	−40.5286	−1.35472
$896$	0	0
$897$	−0.952157	−0.0317916
$898$	0	0
$899$	0.135619	0.00452314
$900$	0	0
$901$	−22.5371	−0.750819
$902$	0	0
$903$	12.9302	0.430288
$904$	0	0
$905$	20.2867	0.674353
$906$	0	0
$907$	−43.9961	−1.46087	−0.730433	−	0.682985i	$-0.760681\pi$
$907$	−43.9961	−1.46087	−0.730433	+	0.682985i	$0.760681\pi$
$908$	0	0
$909$	−18.5958	−0.616784
$910$	0	0
$911$	−16.0348	−0.531258	−0.265629	−	0.964075i	$-0.585580\pi$
$911$	−16.0348	−0.531258	−0.265629	+	0.964075i	$0.585580\pi$
$912$	0	0
$913$	1.09206	0.0361419
$914$	0	0
$915$	−16.0393	−0.530244
$916$	0	0
$917$	6.97174	0.230227
$918$	0	0
$919$	−5.37241	−0.177219	−0.0886096	−	0.996066i	$-0.528242\pi$
$919$	−5.37241	−0.177219	−0.0886096	+	0.996066i	$0.528242\pi$
$920$	0	0
$921$	−7.63757	−0.251667
$922$	0	0
$923$	−0.989314	−0.0325637
$924$	0	0
$925$	0.361318	0.0118801
$926$	0	0
$927$	−19.3000	−0.633895
$928$	0	0
$929$	−23.0096	−0.754920	−0.377460	−	0.926026i	$-0.623202\pi$
$929$	−23.0096	−0.754920	−0.377460	+	0.926026i	$0.623202\pi$
$930$	0	0
$931$	3.89013	0.127494
$932$	0	0
$933$	19.6655	0.643820
$934$	0	0
$935$	−28.3258	−0.926352
$936$	0	0
$937$	−50.5567	−1.65162	−0.825808	−	0.563951i	$-0.809281\pi$
$937$	−50.5567	−1.65162	−0.825808	+	0.563951i	$0.809281\pi$
$938$	0	0
$939$	−27.6680	−0.902911
$940$	0	0
$941$	16.0633	0.523648	0.261824	−	0.965116i	$-0.415676\pi$
$941$	16.0633	0.523648	0.261824	+	0.965116i	$0.415676\pi$
$942$	0	0
$943$	36.8618	1.20039
$944$	0	0
$945$	−4.47360	−0.145526
$946$	0	0
$947$	10.2063	0.331659	0.165830	−	0.986154i	$-0.446970\pi$
$947$	10.2063	0.331659	0.165830	+	0.986154i	$0.446970\pi$
$948$	0	0
$949$	0.551808	0.0179124
$950$	0	0
$951$	8.67973	0.281459
$952$	0	0
$953$	−10.7589	−0.348515	−0.174257	−	0.984700i	$-0.555752\pi$
$953$	−10.7589	−0.348515	−0.174257	+	0.984700i	$0.555752\pi$
$954$	0	0
$955$	−8.15792	−0.263984
$956$	0	0
$957$	−0.233723	−0.00755519
$958$	0	0
$959$	43.7575	1.41300
$960$	0	0
$961$	−28.9022	−0.932327
$962$	0	0
$963$	−18.2861	−0.589262
$964$	0	0
$965$	−5.21724	−0.167949
$966$	0	0
$967$	−32.7046	−1.05171	−0.525855	−	0.850574i	$-0.676255\pi$
$967$	−32.7046	−1.05171	−0.525855	+	0.850574i	$0.676255\pi$
$968$	0	0
$969$	20.4814	0.657957
$970$	0	0
$971$	−5.93206	−0.190369	−0.0951845	−	0.995460i	$-0.530344\pi$
$971$	−5.93206	−0.190369	−0.0951845	+	0.995460i	$0.530344\pi$
$972$	0	0
$973$	35.7900	1.14737
$974$	0	0
$975$	−0.182490	−0.00584434
$976$	0	0
$977$	36.2266	1.15899	0.579496	−	0.814975i	$-0.303249\pi$
$977$	36.2266	1.15899	0.579496	+	0.814975i	$0.303249\pi$
$978$	0	0
$979$	−29.8695	−0.954634
$980$	0	0
$981$	−16.0245	−0.511623
$982$	0	0
$983$	6.41022	0.204454	0.102227	−	0.994761i	$-0.467403\pi$
$983$	6.41022	0.204454	0.102227	+	0.994761i	$0.467403\pi$
$984$	0	0
$985$	−38.3816	−1.22294
$986$	0	0
$987$	16.9353	0.539055
$988$	0	0
$989$	−43.8668	−1.39488
$990$	0	0
$991$	−11.8990	−0.377986	−0.188993	−	0.981978i	$-0.560522\pi$
$991$	−11.8990	−0.377986	−0.188993	+	0.981978i	$0.560522\pi$
$992$	0	0
$993$	12.1719	0.386263
$994$	0	0
$995$	1.50700	0.0477751
$996$	0	0
$997$	−12.7858	−0.404930	−0.202465	−	0.979290i	$-0.564895\pi$
$997$	−12.7858	−0.404930	−0.202465	+	0.979290i	$0.564895\pi$
$998$	0	0
$999$	−0.230021	−0.00727754

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2004.2.a.b.1.2	✓	5
3.2	odd	2		6012.2.a.f.1.4		5
4.3	odd	2		8016.2.a.q.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.b.1.2	✓	5	1.1	even	1	trivial
6012.2.a.f.1.4		5	3.2	odd	2
8016.2.a.q.1.2		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.85181 q^{5} +2.41580 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.85181 q^{5} +2.41580 q^{7} +1.00000 q^{9} -2.49614 q^{11} +0.116176 q^{13} -1.85181 q^{15} -6.12798 q^{17} -3.34227 q^{19} +2.41580 q^{21} -8.19583 q^{23} -1.57081 q^{25} +1.00000 q^{27} +0.0936338 q^{29} +1.44840 q^{31} -2.49614 q^{33} -4.47360 q^{35} -0.230021 q^{37} +0.116176 q^{39} -4.49763 q^{41} +5.35233 q^{43} -1.85181 q^{45} +7.01021 q^{47} -1.16392 q^{49} -6.12798 q^{51} +3.67773 q^{53} +4.62237 q^{55} -3.34227 q^{57} -1.76972 q^{59} +8.66144 q^{61} +2.41580 q^{63} -0.215135 q^{65} -15.1723 q^{67} -8.19583 q^{69} -8.51567 q^{71} +4.74977 q^{73} -1.57081 q^{75} -6.03017 q^{77} -3.24808 q^{79} +1.00000 q^{81} -0.437500 q^{83} +11.3478 q^{85} +0.0936338 q^{87} +11.9663 q^{89} +0.280657 q^{91} +1.44840 q^{93} +6.18925 q^{95} +5.72224 q^{97} -2.49614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 7 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 - 7 * q^5 - 2 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} - 7 q^{5} - 2 q^{7} + 5 q^{9} - 5 q^{11} - 8 q^{13} - 7 q^{15} - 7 q^{17} + 2 q^{19} - 2 q^{21} - 13 q^{23} + 2 q^{25} + 5 q^{27} - 11 q^{29} - 12 q^{31} - 5 q^{33} - 12 q^{35} - 7 q^{37} - 8 q^{39} - 12 q^{41} - 7 q^{45} - 19 q^{47} - 9 q^{49} - 7 q^{51} - 21 q^{53} - q^{55} + 2 q^{57} - 7 q^{59} - 6 q^{61} - 2 q^{63} + 14 q^{65} + 10 q^{67} - 13 q^{69} - 35 q^{71} - 8 q^{73} + 2 q^{75} - 6 q^{77} + 5 q^{81} - 11 q^{83} + 5 q^{85} - 11 q^{87} - 32 q^{89} + 5 q^{91} - 12 q^{93} - 19 q^{95} + 11 q^{97} - 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - 7 * q^5 - 2 * q^7 + 5 * q^9 - 5 * q^11 - 8 * q^13 - 7 * q^15 - 7 * q^17 + 2 * q^19 - 2 * q^21 - 13 * q^23 + 2 * q^25 + 5 * q^27 - 11 * q^29 - 12 * q^31 - 5 * q^33 - 12 * q^35 - 7 * q^37 - 8 * q^39 - 12 * q^41 - 7 * q^45 - 19 * q^47 - 9 * q^49 - 7 * q^51 - 21 * q^53 - q^55 + 2 * q^57 - 7 * q^59 - 6 * q^61 - 2 * q^63 + 14 * q^65 + 10 * q^67 - 13 * q^69 - 35 * q^71 - 8 * q^73 + 2 * q^75 - 6 * q^77 + 5 * q^81 - 11 * q^83 + 5 * q^85 - 11 * q^87 - 32 * q^89 + 5 * q^91 - 12 * q^93 - 19 * q^95 + 11 * q^97 - 5 * q^99

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