LMFDB - Embedded newform 2004.2.a.b.1.4

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			1.4
Root			$-2.07823$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.614948 q^{5} -3.46328 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.614948 q^{5} -3.46328 q^{7} +1.00000 q^{9} +3.80906 q^{11} -2.93060 q^{13} -0.614948 q^{15} +3.15646 q^{17} -5.58256 q^{19} -3.46328 q^{21} -0.574360 q^{23} -4.62184 q^{25} +1.00000 q^{27} -3.25128 q^{29} -1.74532 q^{31} +3.80906 q^{33} +2.12974 q^{35} -4.35964 q^{37} -2.93060 q^{39} -1.51130 q^{41} +9.97984 q^{43} -0.614948 q^{45} -8.29318 q^{47} +4.99433 q^{49} +3.15646 q^{51} +0.465048 q^{53} -2.34237 q^{55} -5.58256 q^{57} -9.58387 q^{59} -6.22999 q^{61} -3.46328 q^{63} +1.80217 q^{65} +3.41132 q^{67} -0.574360 q^{69} -8.58319 q^{71} -9.90115 q^{73} -4.62184 q^{75} -13.1918 q^{77} +10.2406 q^{79} +1.00000 q^{81} -8.39859 q^{83} -1.94106 q^{85} -3.25128 q^{87} -7.12992 q^{89} +10.1495 q^{91} -1.74532 q^{93} +3.43299 q^{95} +4.70410 q^{97} +3.80906 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.614948	−0.275013	−0.137507	−	0.990501i	$-0.543909\pi$
$5$	−0.614948	−0.275013	−0.137507	+	0.990501i	$0.543909\pi$
$6$	0	0
$7$	−3.46328	−1.30900	−0.654499	−	0.756063i	$-0.727120\pi$
$7$	−3.46328	−1.30900	−0.654499	+	0.756063i	$0.727120\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.80906	1.14847	0.574237	−	0.818689i	$-0.305299\pi$
$11$	3.80906	1.14847	0.574237	+	0.818689i	$0.305299\pi$
$12$	0	0
$13$	−2.93060	−0.812801	−0.406401	−	0.913695i	$-0.633216\pi$
$13$	−2.93060	−0.812801	−0.406401	+	0.913695i	$0.633216\pi$
$14$	0	0
$15$	−0.614948	−0.158779
$16$	0	0
$17$	3.15646	0.765555	0.382778	−	0.923841i	$-0.374968\pi$
$17$	3.15646	0.765555	0.382778	+	0.923841i	$0.374968\pi$
$18$	0	0
$19$	−5.58256	−1.28073	−0.640363	−	0.768072i	$-0.721216\pi$
$19$	−5.58256	−1.28073	−0.640363	+	0.768072i	$0.721216\pi$
$20$	0	0
$21$	−3.46328	−0.755750
$22$	0	0
$23$	−0.574360	−0.119762	−0.0598811	−	0.998206i	$-0.519072\pi$
$23$	−0.574360	−0.119762	−0.0598811	+	0.998206i	$0.519072\pi$
$24$	0	0
$25$	−4.62184	−0.924368
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.25128	−0.603748	−0.301874	−	0.953348i	$-0.597612\pi$
$29$	−3.25128	−0.603748	−0.301874	+	0.953348i	$0.597612\pi$
$30$	0	0
$31$	−1.74532	−0.313468	−0.156734	−	0.987641i	$-0.550097\pi$
$31$	−1.74532	−0.313468	−0.156734	+	0.987641i	$0.550097\pi$
$32$	0	0
$33$	3.80906	0.663072
$34$	0	0
$35$	2.12974	0.359992
$36$	0	0
$37$	−4.35964	−0.716720	−0.358360	−	0.933583i	$-0.616664\pi$
$37$	−4.35964	−0.716720	−0.358360	+	0.933583i	$0.616664\pi$
$38$	0	0
$39$	−2.93060	−0.469271
$40$	0	0
$41$	−1.51130	−0.236026	−0.118013	−	0.993012i	$-0.537652\pi$
$41$	−1.51130	−0.236026	−0.118013	+	0.993012i	$0.537652\pi$
$42$	0	0
$43$	9.97984	1.52191	0.760956	−	0.648804i	$-0.224730\pi$
$43$	9.97984	1.52191	0.760956	+	0.648804i	$0.224730\pi$
$44$	0	0
$45$	−0.614948	−0.0916711
$46$	0	0
$47$	−8.29318	−1.20968	−0.604842	−	0.796345i	$-0.706764\pi$
$47$	−8.29318	−1.20968	−0.604842	+	0.796345i	$0.706764\pi$
$48$	0	0
$49$	4.99433	0.713476
$50$	0	0
$51$	3.15646	0.441993
$52$	0	0
$53$	0.465048	0.0638792	0.0319396	−	0.999490i	$-0.489832\pi$
$53$	0.465048	0.0638792	0.0319396	+	0.999490i	$0.489832\pi$
$54$	0	0
$55$	−2.34237	−0.315845
$56$	0	0
$57$	−5.58256	−0.739428
$58$	0	0
$59$	−9.58387	−1.24771	−0.623857	−	0.781539i	$-0.714435\pi$
$59$	−9.58387	−1.24771	−0.623857	+	0.781539i	$0.714435\pi$
$60$	0	0
$61$	−6.22999	−0.797668	−0.398834	−	0.917023i	$-0.630585\pi$
$61$	−6.22999	−0.797668	−0.398834	+	0.917023i	$0.630585\pi$
$62$	0	0
$63$	−3.46328	−0.436333
$64$	0	0
$65$	1.80217	0.223531
$66$	0	0
$67$	3.41132	0.416759	0.208380	−	0.978048i	$-0.433181\pi$
$67$	3.41132	0.416759	0.208380	+	0.978048i	$0.433181\pi$
$68$	0	0
$69$	−0.574360	−0.0691448
$70$	0	0
$71$	−8.58319	−1.01864	−0.509319	−	0.860578i	$-0.670103\pi$
$71$	−8.58319	−1.01864	−0.509319	+	0.860578i	$0.670103\pi$
$72$	0	0
$73$	−9.90115	−1.15884	−0.579421	−	0.815028i	$-0.696721\pi$
$73$	−9.90115	−1.15884	−0.579421	+	0.815028i	$0.696721\pi$
$74$	0	0
$75$	−4.62184	−0.533684
$76$	0	0
$77$	−13.1918	−1.50335
$78$	0	0
$79$	10.2406	1.15216	0.576078	−	0.817395i	$-0.304583\pi$
$79$	10.2406	1.15216	0.576078	+	0.817395i	$0.304583\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.39859	−0.921865	−0.460932	−	0.887435i	$-0.652485\pi$
$83$	−8.39859	−0.921865	−0.460932	+	0.887435i	$0.652485\pi$
$84$	0	0
$85$	−1.94106	−0.210538
$86$	0	0
$87$	−3.25128	−0.348574
$88$	0	0
$89$	−7.12992	−0.755770	−0.377885	−	0.925853i	$-0.623348\pi$
$89$	−7.12992	−0.755770	−0.377885	+	0.925853i	$0.623348\pi$
$90$	0	0
$91$	10.1495	1.06396
$92$	0	0
$93$	−1.74532	−0.180981
$94$	0	0
$95$	3.43299	0.352217
$96$	0	0
$97$	4.70410	0.477629	0.238815	−	0.971065i	$-0.423241\pi$
$97$	4.70410	0.477629	0.238815	+	0.971065i	$0.423241\pi$
$98$	0	0
$99$	3.80906	0.382825
$100$	0	0
$101$	−6.74351	−0.671004	−0.335502	−	0.942039i	$-0.608906\pi$
$101$	−6.74351	−0.671004	−0.335502	+	0.942039i	$0.608906\pi$
$102$	0	0
$103$	−3.37572	−0.332620	−0.166310	−	0.986074i	$-0.553185\pi$
$103$	−3.37572	−0.332620	−0.166310	+	0.986074i	$0.553185\pi$
$104$	0	0
$105$	2.12974	0.207841
$106$	0	0
$107$	−6.87891	−0.665010	−0.332505	−	0.943102i	$-0.607894\pi$
$107$	−6.87891	−0.665010	−0.332505	+	0.943102i	$0.607894\pi$
$108$	0	0
$109$	−12.2964	−1.17778	−0.588891	−	0.808213i	$-0.700435\pi$
$109$	−12.2964	−1.17778	−0.588891	+	0.808213i	$0.700435\pi$
$110$	0	0
$111$	−4.35964	−0.413798
$112$	0	0
$113$	5.04942	0.475009	0.237505	−	0.971386i	$-0.423671\pi$
$113$	5.04942	0.475009	0.237505	+	0.971386i	$0.423671\pi$
$114$	0	0
$115$	0.353202	0.0329362
$116$	0	0
$117$	−2.93060	−0.270934
$118$	0	0
$119$	−10.9317	−1.00211
$120$	0	0
$121$	3.50891	0.318992
$122$	0	0
$123$	−1.51130	−0.136269
$124$	0	0
$125$	5.91693	0.529227
$126$	0	0
$127$	8.43089	0.748121	0.374060	−	0.927404i	$-0.377965\pi$
$127$	8.43089	0.748121	0.374060	+	0.927404i	$0.377965\pi$
$128$	0	0
$129$	9.97984	0.878676
$130$	0	0
$131$	7.65327	0.668669	0.334335	−	0.942454i	$-0.391488\pi$
$131$	7.65327	0.668669	0.334335	+	0.942454i	$0.391488\pi$
$132$	0	0
$133$	19.3340	1.67647
$134$	0	0
$135$	−0.614948	−0.0529263
$136$	0	0
$137$	−5.27810	−0.450938	−0.225469	−	0.974250i	$-0.572391\pi$
$137$	−5.27810	−0.450938	−0.225469	+	0.974250i	$0.572391\pi$
$138$	0	0
$139$	5.53478	0.469454	0.234727	−	0.972061i	$-0.424580\pi$
$139$	5.53478	0.469454	0.234727	+	0.972061i	$0.424580\pi$
$140$	0	0
$141$	−8.29318	−0.698412
$142$	0	0
$143$	−11.1628	−0.933481
$144$	0	0
$145$	1.99937	0.166039
$146$	0	0
$147$	4.99433	0.411926
$148$	0	0
$149$	13.1803	1.07977	0.539886	−	0.841738i	$-0.318467\pi$
$149$	13.1803	1.07977	0.539886	+	0.841738i	$0.318467\pi$
$150$	0	0
$151$	−2.29363	−0.186653	−0.0933266	−	0.995636i	$-0.529750\pi$
$151$	−2.29363	−0.186653	−0.0933266	+	0.995636i	$0.529750\pi$
$152$	0	0
$153$	3.15646	0.255185
$154$	0	0
$155$	1.07328	0.0862080
$156$	0	0
$157$	−14.6203	−1.16682	−0.583412	−	0.812177i	$-0.698283\pi$
$157$	−14.6203	−1.16682	−0.583412	+	0.812177i	$0.698283\pi$
$158$	0	0
$159$	0.465048	0.0368807
$160$	0	0
$161$	1.98917	0.156769
$162$	0	0
$163$	15.6158	1.22312	0.611560	−	0.791198i	$-0.290542\pi$
$163$	15.6158	1.22312	0.611560	+	0.791198i	$0.290542\pi$
$164$	0	0
$165$	−2.34237	−0.182353
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−4.41160	−0.339354
$170$	0	0
$171$	−5.58256	−0.426909
$172$	0	0
$173$	13.8874	1.05584	0.527919	−	0.849295i	$-0.322972\pi$
$173$	13.8874	1.05584	0.527919	+	0.849295i	$0.322972\pi$
$174$	0	0
$175$	16.0067	1.21000
$176$	0	0
$177$	−9.58387	−0.720368
$178$	0	0
$179$	3.21586	0.240364	0.120182	−	0.992752i	$-0.461652\pi$
$179$	3.21586	0.240364	0.120182	+	0.992752i	$0.461652\pi$
$180$	0	0
$181$	7.33183	0.544971	0.272485	−	0.962160i	$-0.412154\pi$
$181$	7.33183	0.544971	0.272485	+	0.962160i	$0.412154\pi$
$182$	0	0
$183$	−6.22999	−0.460534
$184$	0	0
$185$	2.68095	0.197107
$186$	0	0
$187$	12.0231	0.879220
$188$	0	0
$189$	−3.46328	−0.251917
$190$	0	0
$191$	2.33599	0.169026	0.0845130	−	0.996422i	$-0.473067\pi$
$191$	2.33599	0.169026	0.0845130	+	0.996422i	$0.473067\pi$
$192$	0	0
$193$	9.73301	0.700597	0.350299	−	0.936638i	$-0.386080\pi$
$193$	9.73301	0.700597	0.350299	+	0.936638i	$0.386080\pi$
$194$	0	0
$195$	1.80217	0.129056
$196$	0	0
$197$	−19.0432	−1.35677	−0.678386	−	0.734705i	$-0.737320\pi$
$197$	−19.0432	−1.35677	−0.678386	+	0.734705i	$0.737320\pi$
$198$	0	0
$199$	5.26986	0.373570	0.186785	−	0.982401i	$-0.440193\pi$
$199$	5.26986	0.373570	0.186785	+	0.982401i	$0.440193\pi$
$200$	0	0
$201$	3.41132	0.240616
$202$	0	0
$203$	11.2601	0.790305
$204$	0	0
$205$	0.929372	0.0649102
$206$	0	0
$207$	−0.574360	−0.0399208
$208$	0	0
$209$	−21.2643	−1.47088
$210$	0	0
$211$	−1.49086	−0.102635	−0.0513176	−	0.998682i	$-0.516342\pi$
$211$	−1.49086	−0.102635	−0.0513176	+	0.998682i	$0.516342\pi$
$212$	0	0
$213$	−8.58319	−0.588110
$214$	0	0
$215$	−6.13709	−0.418546
$216$	0	0
$217$	6.04453	0.410330
$218$	0	0
$219$	−9.90115	−0.669058
$220$	0	0
$221$	−9.25032	−0.622244
$222$	0	0
$223$	−0.343506	−0.0230029	−0.0115014	−	0.999934i	$-0.503661\pi$
$223$	−0.343506	−0.0230029	−0.0115014	+	0.999934i	$0.503661\pi$
$224$	0	0
$225$	−4.62184	−0.308123
$226$	0	0
$227$	8.12470	0.539256	0.269628	−	0.962965i	$-0.413099\pi$
$227$	8.12470	0.539256	0.269628	+	0.962965i	$0.413099\pi$
$228$	0	0
$229$	5.95619	0.393596	0.196798	−	0.980444i	$-0.436946\pi$
$229$	5.95619	0.393596	0.196798	+	0.980444i	$0.436946\pi$
$230$	0	0
$231$	−13.1918	−0.867959
$232$	0	0
$233$	0.870260	0.0570126	0.0285063	−	0.999594i	$-0.490925\pi$
$233$	0.870260	0.0570126	0.0285063	+	0.999594i	$0.490925\pi$
$234$	0	0
$235$	5.09988	0.332679
$236$	0	0
$237$	10.2406	0.665197
$238$	0	0
$239$	13.0177	0.842046	0.421023	−	0.907050i	$-0.361671\pi$
$239$	13.0177	0.842046	0.421023	+	0.907050i	$0.361671\pi$
$240$	0	0
$241$	−2.82934	−0.182254	−0.0911270	−	0.995839i	$-0.529047\pi$
$241$	−2.82934	−0.182254	−0.0911270	+	0.995839i	$0.529047\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.07126	−0.196215
$246$	0	0
$247$	16.3602	1.04098
$248$	0	0
$249$	−8.39859	−0.532239
$250$	0	0
$251$	−16.4580	−1.03882	−0.519410	−	0.854525i	$-0.673848\pi$
$251$	−16.4580	−1.03882	−0.519410	+	0.854525i	$0.673848\pi$
$252$	0	0
$253$	−2.18777	−0.137544
$254$	0	0
$255$	−1.94106	−0.121554
$256$	0	0
$257$	24.5773	1.53309	0.766545	−	0.642191i	$-0.221974\pi$
$257$	24.5773	1.53309	0.766545	+	0.642191i	$0.221974\pi$
$258$	0	0
$259$	15.0987	0.938185
$260$	0	0
$261$	−3.25128	−0.201249
$262$	0	0
$263$	24.6483	1.51988	0.759939	−	0.649994i	$-0.225229\pi$
$263$	24.6483	1.51988	0.759939	+	0.649994i	$0.225229\pi$
$264$	0	0
$265$	−0.285980	−0.0175676
$266$	0	0
$267$	−7.12992	−0.436344
$268$	0	0
$269$	−4.52549	−0.275924	−0.137962	−	0.990438i	$-0.544055\pi$
$269$	−4.52549	−0.275924	−0.137962	+	0.990438i	$0.544055\pi$
$270$	0	0
$271$	−22.1048	−1.34277	−0.671385	−	0.741109i	$-0.734300\pi$
$271$	−22.1048	−1.34277	−0.671385	+	0.741109i	$0.734300\pi$
$272$	0	0
$273$	10.1495	0.614275
$274$	0	0
$275$	−17.6048	−1.06161
$276$	0	0
$277$	−14.4555	−0.868548	−0.434274	−	0.900781i	$-0.642995\pi$
$277$	−14.4555	−0.868548	−0.434274	+	0.900781i	$0.642995\pi$
$278$	0	0
$279$	−1.74532	−0.104489
$280$	0	0
$281$	3.60180	0.214866	0.107433	−	0.994212i	$-0.465737\pi$
$281$	3.60180	0.214866	0.107433	+	0.994212i	$0.465737\pi$
$282$	0	0
$283$	−22.0928	−1.31328	−0.656639	−	0.754205i	$-0.728023\pi$
$283$	−22.0928	−1.31328	−0.656639	+	0.754205i	$0.728023\pi$
$284$	0	0
$285$	3.43299	0.203352
$286$	0	0
$287$	5.23407	0.308957
$288$	0	0
$289$	−7.03673	−0.413926
$290$	0	0
$291$	4.70410	0.275759
$292$	0	0
$293$	−0.403322	−0.0235623	−0.0117812	−	0.999931i	$-0.503750\pi$
$293$	−0.403322	−0.0235623	−0.0117812	+	0.999931i	$0.503750\pi$
$294$	0	0
$295$	5.89358	0.343138
$296$	0	0
$297$	3.80906	0.221024
$298$	0	0
$299$	1.68322	0.0973430
$300$	0	0
$301$	−34.5630	−1.99218
$302$	0	0
$303$	−6.74351	−0.387404
$304$	0	0
$305$	3.83112	0.219369
$306$	0	0
$307$	28.5615	1.63009	0.815045	−	0.579397i	$-0.196712\pi$
$307$	28.5615	1.63009	0.815045	+	0.579397i	$0.196712\pi$
$308$	0	0
$309$	−3.37572	−0.192038
$310$	0	0
$311$	−14.0740	−0.798061	−0.399031	−	0.916938i	$-0.630653\pi$
$311$	−14.0740	−0.798061	−0.399031	+	0.916938i	$0.630653\pi$
$312$	0	0
$313$	13.3132	0.752504	0.376252	−	0.926517i	$-0.377213\pi$
$313$	13.3132	0.752504	0.376252	+	0.926517i	$0.377213\pi$
$314$	0	0
$315$	2.12974	0.119997
$316$	0	0
$317$	−23.0614	−1.29526	−0.647630	−	0.761955i	$-0.724240\pi$
$317$	−23.0614	−1.29526	−0.647630	+	0.761955i	$0.724240\pi$
$318$	0	0
$319$	−12.3843	−0.693388
$320$	0	0
$321$	−6.87891	−0.383944
$322$	0	0
$323$	−17.6211	−0.980467
$324$	0	0
$325$	13.5447	0.751327
$326$	0	0
$327$	−12.2964	−0.679993
$328$	0	0
$329$	28.7216	1.58347
$330$	0	0
$331$	31.1825	1.71394	0.856972	−	0.515363i	$-0.172343\pi$
$331$	31.1825	1.71394	0.856972	+	0.515363i	$0.172343\pi$
$332$	0	0
$333$	−4.35964	−0.238907
$334$	0	0
$335$	−2.09779	−0.114614
$336$	0	0
$337$	−0.817442	−0.0445289	−0.0222645	−	0.999752i	$-0.507088\pi$
$337$	−0.817442	−0.0445289	−0.0222645	+	0.999752i	$0.507088\pi$
$338$	0	0
$339$	5.04942	0.274247
$340$	0	0
$341$	−6.64802	−0.360010
$342$	0	0
$343$	6.94619	0.375059
$344$	0	0
$345$	0.353202	0.0190157
$346$	0	0
$347$	−9.97471	−0.535471	−0.267735	−	0.963492i	$-0.586275\pi$
$347$	−9.97471	−0.535471	−0.267735	+	0.963492i	$0.586275\pi$
$348$	0	0
$349$	−13.6700	−0.731740	−0.365870	−	0.930666i	$-0.619229\pi$
$349$	−13.6700	−0.731740	−0.365870	+	0.930666i	$0.619229\pi$
$350$	0	0
$351$	−2.93060	−0.156424
$352$	0	0
$353$	−0.0887759	−0.00472506	−0.00236253	−	0.999997i	$-0.500752\pi$
$353$	−0.0887759	−0.00472506	−0.00236253	+	0.999997i	$0.500752\pi$
$354$	0	0
$355$	5.27822	0.280139
$356$	0	0
$357$	−10.9317	−0.578569
$358$	0	0
$359$	−6.69628	−0.353416	−0.176708	−	0.984263i	$-0.556545\pi$
$359$	−6.69628	−0.353416	−0.176708	+	0.984263i	$0.556545\pi$
$360$	0	0
$361$	12.1650	0.640261
$362$	0	0
$363$	3.50891	0.184170
$364$	0	0
$365$	6.08870	0.318697
$366$	0	0
$367$	26.5833	1.38764	0.693819	−	0.720150i	$-0.255927\pi$
$367$	26.5833	1.38764	0.693819	+	0.720150i	$0.255927\pi$
$368$	0	0
$369$	−1.51130	−0.0786752
$370$	0	0
$371$	−1.61059	−0.0836178
$372$	0	0
$373$	25.1865	1.30411	0.652053	−	0.758173i	$-0.273908\pi$
$373$	25.1865	1.30411	0.652053	+	0.758173i	$0.273908\pi$
$374$	0	0
$375$	5.91693	0.305549
$376$	0	0
$377$	9.52820	0.490727
$378$	0	0
$379$	−15.4647	−0.794370	−0.397185	−	0.917739i	$-0.630013\pi$
$379$	−15.4647	−0.794370	−0.397185	+	0.917739i	$0.630013\pi$
$380$	0	0
$381$	8.43089	0.431928
$382$	0	0
$383$	30.0700	1.53651	0.768253	−	0.640147i	$-0.221126\pi$
$383$	30.0700	1.53651	0.768253	+	0.640147i	$0.221126\pi$
$384$	0	0
$385$	8.11230	0.413441
$386$	0	0
$387$	9.97984	0.507304
$388$	0	0
$389$	−26.8151	−1.35958	−0.679790	−	0.733407i	$-0.737929\pi$
$389$	−26.8151	−1.35958	−0.679790	+	0.733407i	$0.737929\pi$
$390$	0	0
$391$	−1.81295	−0.0916846
$392$	0	0
$393$	7.65327	0.386056
$394$	0	0
$395$	−6.29743	−0.316858
$396$	0	0
$397$	−10.7692	−0.540490	−0.270245	−	0.962792i	$-0.587105\pi$
$397$	−10.7692	−0.540490	−0.270245	+	0.962792i	$0.587105\pi$
$398$	0	0
$399$	19.3340	0.967910
$400$	0	0
$401$	17.6845	0.883121	0.441561	−	0.897231i	$-0.354425\pi$
$401$	17.6845	0.883121	0.441561	+	0.897231i	$0.354425\pi$
$402$	0	0
$403$	5.11483	0.254788
$404$	0	0
$405$	−0.614948	−0.0305570
$406$	0	0
$407$	−16.6061	−0.823134
$408$	0	0
$409$	−20.1733	−0.997506	−0.498753	−	0.866744i	$-0.666209\pi$
$409$	−20.1733	−0.997506	−0.498753	+	0.866744i	$0.666209\pi$
$410$	0	0
$411$	−5.27810	−0.260349
$412$	0	0
$413$	33.1917	1.63325
$414$	0	0
$415$	5.16470	0.253525
$416$	0	0
$417$	5.53478	0.271039
$418$	0	0
$419$	7.91248	0.386550	0.193275	−	0.981145i	$-0.438089\pi$
$419$	7.91248	0.386550	0.193275	+	0.981145i	$0.438089\pi$
$420$	0	0
$421$	7.95900	0.387898	0.193949	−	0.981012i	$-0.437870\pi$
$421$	7.95900	0.387898	0.193949	+	0.981012i	$0.437870\pi$
$422$	0	0
$423$	−8.29318	−0.403228
$424$	0	0
$425$	−14.5887	−0.707654
$426$	0	0
$427$	21.5762	1.04415
$428$	0	0
$429$	−11.1628	−0.538945
$430$	0	0
$431$	6.43209	0.309823	0.154912	−	0.987928i	$-0.450491\pi$
$431$	6.43209	0.309823	0.154912	+	0.987928i	$0.450491\pi$
$432$	0	0
$433$	25.3350	1.21752	0.608761	−	0.793354i	$-0.291667\pi$
$433$	25.3350	1.21752	0.608761	+	0.793354i	$0.291667\pi$
$434$	0	0
$435$	1.99937	0.0958625
$436$	0	0
$437$	3.20640	0.153383
$438$	0	0
$439$	−5.63416	−0.268904	−0.134452	−	0.990920i	$-0.542927\pi$
$439$	−5.63416	−0.268904	−0.134452	+	0.990920i	$0.542927\pi$
$440$	0	0
$441$	4.99433	0.237825
$442$	0	0
$443$	34.5091	1.63958	0.819789	−	0.572666i	$-0.194091\pi$
$443$	34.5091	1.63958	0.819789	+	0.572666i	$0.194091\pi$
$444$	0	0
$445$	4.38453	0.207847
$446$	0	0
$447$	13.1803	0.623406
$448$	0	0
$449$	−37.4839	−1.76897	−0.884487	−	0.466565i	$-0.845491\pi$
$449$	−37.4839	−1.76897	−0.884487	+	0.466565i	$0.845491\pi$
$450$	0	0
$451$	−5.75663	−0.271069
$452$	0	0
$453$	−2.29363	−0.107764
$454$	0	0
$455$	−6.24141	−0.292602
$456$	0	0
$457$	8.87783	0.415287	0.207644	−	0.978205i	$-0.433421\pi$
$457$	8.87783	0.415287	0.207644	+	0.978205i	$0.433421\pi$
$458$	0	0
$459$	3.15646	0.147331
$460$	0	0
$461$	9.67760	0.450731	0.225365	−	0.974274i	$-0.427642\pi$
$461$	9.67760	0.450731	0.225365	+	0.974274i	$0.427642\pi$
$462$	0	0
$463$	8.66347	0.402626	0.201313	−	0.979527i	$-0.435479\pi$
$463$	8.66347	0.402626	0.201313	+	0.979527i	$0.435479\pi$
$464$	0	0
$465$	1.07328	0.0497722
$466$	0	0
$467$	31.1541	1.44164	0.720819	−	0.693124i	$-0.243766\pi$
$467$	31.1541	1.44164	0.720819	+	0.693124i	$0.243766\pi$
$468$	0	0
$469$	−11.8144	−0.545537
$470$	0	0
$471$	−14.6203	−0.673666
$472$	0	0
$473$	38.0138	1.74787
$474$	0	0
$475$	25.8017	1.18386
$476$	0	0
$477$	0.465048	0.0212931
$478$	0	0
$479$	6.78094	0.309829	0.154915	−	0.987928i	$-0.450490\pi$
$479$	6.78094	0.309829	0.154915	+	0.987928i	$0.450490\pi$
$480$	0	0
$481$	12.7763	0.582551
$482$	0	0
$483$	1.98917	0.0905104
$484$	0	0
$485$	−2.89278	−0.131354
$486$	0	0
$487$	3.19422	0.144744	0.0723719	−	0.997378i	$-0.476943\pi$
$487$	3.19422	0.144744	0.0723719	+	0.997378i	$0.476943\pi$
$488$	0	0
$489$	15.6158	0.706169
$490$	0	0
$491$	36.1947	1.63344	0.816722	−	0.577031i	$-0.195789\pi$
$491$	36.1947	1.63344	0.816722	+	0.577031i	$0.195789\pi$
$492$	0	0
$493$	−10.2626	−0.462202
$494$	0	0
$495$	−2.34237	−0.105282
$496$	0	0
$497$	29.7260	1.33339
$498$	0	0
$499$	−19.5019	−0.873026	−0.436513	−	0.899698i	$-0.643787\pi$
$499$	−19.5019	−0.873026	−0.436513	+	0.899698i	$0.643787\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−23.7924	−1.06085	−0.530426	−	0.847731i	$-0.677968\pi$
$503$	−23.7924	−1.06085	−0.530426	+	0.847731i	$0.677968\pi$
$504$	0	0
$505$	4.14691	0.184535
$506$	0	0
$507$	−4.41160	−0.195926
$508$	0	0
$509$	−17.6743	−0.783399	−0.391700	−	0.920093i	$-0.628113\pi$
$509$	−17.6743	−0.783399	−0.391700	+	0.920093i	$0.628113\pi$
$510$	0	0
$511$	34.2905	1.51692
$512$	0	0
$513$	−5.58256	−0.246476
$514$	0	0
$515$	2.07589	0.0914748
$516$	0	0
$517$	−31.5892	−1.38929
$518$	0	0
$519$	13.8874	0.609588
$520$	0	0
$521$	−41.0601	−1.79887	−0.899437	−	0.437050i	$-0.856023\pi$
$521$	−41.0601	−1.79887	−0.899437	+	0.437050i	$0.856023\pi$
$522$	0	0
$523$	2.50620	0.109588	0.0547942	−	0.998498i	$-0.482550\pi$
$523$	2.50620	0.109588	0.0547942	+	0.998498i	$0.482550\pi$
$524$	0	0
$525$	16.0067	0.698591
$526$	0	0
$527$	−5.50904	−0.239977
$528$	0	0
$529$	−22.6701	−0.985657
$530$	0	0
$531$	−9.58387	−0.415904
$532$	0	0
$533$	4.42902	0.191842
$534$	0	0
$535$	4.23018	0.182886
$536$	0	0
$537$	3.21586	0.138774
$538$	0	0
$539$	19.0237	0.819409
$540$	0	0
$541$	36.2521	1.55860	0.779299	−	0.626652i	$-0.215575\pi$
$541$	36.2521	1.55860	0.779299	+	0.626652i	$0.215575\pi$
$542$	0	0
$543$	7.33183	0.314639
$544$	0	0
$545$	7.56165	0.323906
$546$	0	0
$547$	10.5354	0.450461	0.225230	−	0.974306i	$-0.427686\pi$
$547$	10.5354	0.450461	0.225230	+	0.974306i	$0.427686\pi$
$548$	0	0
$549$	−6.22999	−0.265889
$550$	0	0
$551$	18.1505	0.773236
$552$	0	0
$553$	−35.4660	−1.50817
$554$	0	0
$555$	2.68095	0.113800
$556$	0	0
$557$	−17.9650	−0.761200	−0.380600	−	0.924740i	$-0.624283\pi$
$557$	−17.9650	−0.761200	−0.380600	+	0.924740i	$0.624283\pi$
$558$	0	0
$559$	−29.2469	−1.23701
$560$	0	0
$561$	12.0231	0.507618
$562$	0	0
$563$	−9.50941	−0.400774	−0.200387	−	0.979717i	$-0.564220\pi$
$563$	−9.50941	−0.400774	−0.200387	+	0.979717i	$0.564220\pi$
$564$	0	0
$565$	−3.10513	−0.130634
$566$	0	0
$567$	−3.46328	−0.145444
$568$	0	0
$569$	−18.4813	−0.774778	−0.387389	−	0.921916i	$-0.626623\pi$
$569$	−18.4813	−0.774778	−0.387389	+	0.921916i	$0.626623\pi$
$570$	0	0
$571$	12.7478	0.533478	0.266739	−	0.963769i	$-0.414054\pi$
$571$	12.7478	0.533478	0.266739	+	0.963769i	$0.414054\pi$
$572$	0	0
$573$	2.33599	0.0975872
$574$	0	0
$575$	2.65460	0.110704
$576$	0	0
$577$	−36.1464	−1.50480	−0.752398	−	0.658709i	$-0.771103\pi$
$577$	−36.1464	−1.50480	−0.752398	+	0.658709i	$0.771103\pi$
$578$	0	0
$579$	9.73301	0.404490
$580$	0	0
$581$	29.0867	1.20672
$582$	0	0
$583$	1.77139	0.0733636
$584$	0	0
$585$	1.80217	0.0745104
$586$	0	0
$587$	−44.8985	−1.85316	−0.926580	−	0.376098i	$-0.877266\pi$
$587$	−44.8985	−1.85316	−0.926580	+	0.376098i	$0.877266\pi$
$588$	0	0
$589$	9.74334	0.401468
$590$	0	0
$591$	−19.0432	−0.783333
$592$	0	0
$593$	−24.0796	−0.988832	−0.494416	−	0.869225i	$-0.664618\pi$
$593$	−24.0796	−0.988832	−0.494416	+	0.869225i	$0.664618\pi$
$594$	0	0
$595$	6.72245	0.275594
$596$	0	0
$597$	5.26986	0.215681
$598$	0	0
$599$	4.07062	0.166321	0.0831604	−	0.996536i	$-0.473499\pi$
$599$	4.07062	0.166321	0.0831604	+	0.996536i	$0.473499\pi$
$600$	0	0
$601$	−13.0899	−0.533949	−0.266974	−	0.963704i	$-0.586024\pi$
$601$	−13.0899	−0.533949	−0.266974	+	0.963704i	$0.586024\pi$
$602$	0	0
$603$	3.41132	0.138920
$604$	0	0
$605$	−2.15780	−0.0877269
$606$	0	0
$607$	29.3314	1.19053	0.595263	−	0.803531i	$-0.297048\pi$
$607$	29.3314	1.19053	0.595263	+	0.803531i	$0.297048\pi$
$608$	0	0
$609$	11.2601	0.456283
$610$	0	0
$611$	24.3040	0.983233
$612$	0	0
$613$	−0.550896	−0.0222505	−0.0111252	−	0.999938i	$-0.503541\pi$
$613$	−0.550896	−0.0222505	−0.0111252	+	0.999938i	$0.503541\pi$
$614$	0	0
$615$	0.929372	0.0374759
$616$	0	0
$617$	−47.2384	−1.90175	−0.950874	−	0.309578i	$-0.899812\pi$
$617$	−47.2384	−1.90175	−0.950874	+	0.309578i	$0.899812\pi$
$618$	0	0
$619$	−2.51048	−0.100905	−0.0504524	−	0.998726i	$-0.516066\pi$
$619$	−2.51048	−0.100905	−0.0504524	+	0.998726i	$0.516066\pi$
$620$	0	0
$621$	−0.574360	−0.0230483
$622$	0	0
$623$	24.6929	0.989301
$624$	0	0
$625$	19.4706	0.778823
$626$	0	0
$627$	−21.2643	−0.849214
$628$	0	0
$629$	−13.7610	−0.548689
$630$	0	0
$631$	42.8135	1.70438	0.852190	−	0.523233i	$-0.175274\pi$
$631$	42.8135	1.70438	0.852190	+	0.523233i	$0.175274\pi$
$632$	0	0
$633$	−1.49086	−0.0592564
$634$	0	0
$635$	−5.18456	−0.205743
$636$	0	0
$637$	−14.6364	−0.579915
$638$	0	0
$639$	−8.58319	−0.339546
$640$	0	0
$641$	−38.7594	−1.53090	−0.765452	−	0.643493i	$-0.777485\pi$
$641$	−38.7594	−1.53090	−0.765452	+	0.643493i	$0.777485\pi$
$642$	0	0
$643$	30.1097	1.18741	0.593705	−	0.804682i	$-0.297664\pi$
$643$	30.1097	1.18741	0.593705	+	0.804682i	$0.297664\pi$
$644$	0	0
$645$	−6.13709	−0.241648
$646$	0	0
$647$	14.3050	0.562389	0.281195	−	0.959651i	$-0.409269\pi$
$647$	14.3050	0.562389	0.281195	+	0.959651i	$0.409269\pi$
$648$	0	0
$649$	−36.5055	−1.43297
$650$	0	0
$651$	6.04453	0.236904
$652$	0	0
$653$	21.0547	0.823933	0.411966	−	0.911199i	$-0.364842\pi$
$653$	21.0547	0.823933	0.411966	+	0.911199i	$0.364842\pi$
$654$	0	0
$655$	−4.70637	−0.183893
$656$	0	0
$657$	−9.90115	−0.386281
$658$	0	0
$659$	−4.02197	−0.156674	−0.0783368	−	0.996927i	$-0.524961\pi$
$659$	−4.02197	−0.156674	−0.0783368	+	0.996927i	$0.524961\pi$
$660$	0	0
$661$	19.5974	0.762252	0.381126	−	0.924523i	$-0.375536\pi$
$661$	19.5974	0.762252	0.381126	+	0.924523i	$0.375536\pi$
$662$	0	0
$663$	−9.25032	−0.359253
$664$	0	0
$665$	−11.8894	−0.461051
$666$	0	0
$667$	1.86741	0.0723062
$668$	0	0
$669$	−0.343506	−0.0132807
$670$	0	0
$671$	−23.7304	−0.916101
$672$	0	0
$673$	−18.6277	−0.718046	−0.359023	−	0.933329i	$-0.616890\pi$
$673$	−18.6277	−0.718046	−0.359023	+	0.933329i	$0.616890\pi$
$674$	0	0
$675$	−4.62184	−0.177895
$676$	0	0
$677$	−19.2425	−0.739549	−0.369774	−	0.929122i	$-0.620565\pi$
$677$	−19.2425	−0.739549	−0.369774	+	0.929122i	$0.620565\pi$
$678$	0	0
$679$	−16.2916	−0.625216
$680$	0	0
$681$	8.12470	0.311339
$682$	0	0
$683$	36.3752	1.39186	0.695929	−	0.718111i	$-0.254993\pi$
$683$	36.3752	1.39186	0.695929	+	0.718111i	$0.254993\pi$
$684$	0	0
$685$	3.24576	0.124014
$686$	0	0
$687$	5.95619	0.227243
$688$	0	0
$689$	−1.36287	−0.0519211
$690$	0	0
$691$	−31.4677	−1.19709	−0.598543	−	0.801091i	$-0.704254\pi$
$691$	−31.4677	−1.19709	−0.598543	+	0.801091i	$0.704254\pi$
$692$	0	0
$693$	−13.1918	−0.501117
$694$	0	0
$695$	−3.40360	−0.129106
$696$	0	0
$697$	−4.77037	−0.180691
$698$	0	0
$699$	0.870260	0.0329162
$700$	0	0
$701$	−38.1706	−1.44168	−0.720841	−	0.693100i	$-0.756244\pi$
$701$	−38.1706	−1.44168	−0.720841	+	0.693100i	$0.756244\pi$
$702$	0	0
$703$	24.3379	0.917922
$704$	0	0
$705$	5.09988	0.192072
$706$	0	0
$707$	23.3547	0.878343
$708$	0	0
$709$	12.0061	0.450900	0.225450	−	0.974255i	$-0.427615\pi$
$709$	12.0061	0.450900	0.225450	+	0.974255i	$0.427615\pi$
$710$	0	0
$711$	10.2406	0.384052
$712$	0	0
$713$	1.00244	0.0375417
$714$	0	0
$715$	6.86455	0.256720
$716$	0	0
$717$	13.0177	0.486155
$718$	0	0
$719$	18.4358	0.687539	0.343769	−	0.939054i	$-0.388296\pi$
$719$	18.4358	0.687539	0.343769	+	0.939054i	$0.388296\pi$
$720$	0	0
$721$	11.6911	0.435399
$722$	0	0
$723$	−2.82934	−0.105224
$724$	0	0
$725$	15.0269	0.558085
$726$	0	0
$727$	−14.4468	−0.535802	−0.267901	−	0.963446i	$-0.586330\pi$
$727$	−14.4468	−0.535802	−0.267901	+	0.963446i	$0.586330\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.5010	1.16511
$732$	0	0
$733$	44.5104	1.64403	0.822014	−	0.569467i	$-0.192850\pi$
$733$	44.5104	1.64403	0.822014	+	0.569467i	$0.192850\pi$
$734$	0	0
$735$	−3.07126	−0.113285
$736$	0	0
$737$	12.9939	0.478637
$738$	0	0
$739$	29.3056	1.07802	0.539011	−	0.842298i	$-0.318798\pi$
$739$	29.3056	1.07802	0.539011	+	0.842298i	$0.318798\pi$
$740$	0	0
$741$	16.3602	0.601008
$742$	0	0
$743$	9.42691	0.345840	0.172920	−	0.984936i	$-0.444680\pi$
$743$	9.42691	0.345840	0.172920	+	0.984936i	$0.444680\pi$
$744$	0	0
$745$	−8.10520	−0.296951
$746$	0	0
$747$	−8.39859	−0.307288
$748$	0	0
$749$	23.8236	0.870496
$750$	0	0
$751$	32.0724	1.17034	0.585170	−	0.810911i	$-0.301028\pi$
$751$	32.0724	1.17034	0.585170	+	0.810911i	$0.301028\pi$
$752$	0	0
$753$	−16.4580	−0.599763
$754$	0	0
$755$	1.41047	0.0513321
$756$	0	0
$757$	−17.9811	−0.653533	−0.326767	−	0.945105i	$-0.605959\pi$
$757$	−17.9811	−0.653533	−0.326767	+	0.945105i	$0.605959\pi$
$758$	0	0
$759$	−2.18777	−0.0794110
$760$	0	0
$761$	34.0598	1.23467	0.617333	−	0.786702i	$-0.288213\pi$
$761$	34.0598	1.23467	0.617333	+	0.786702i	$0.288213\pi$
$762$	0	0
$763$	42.5859	1.54171
$764$	0	0
$765$	−1.94106	−0.0701793
$766$	0	0
$767$	28.0865	1.01414
$768$	0	0
$769$	1.20670	0.0435147	0.0217574	−	0.999763i	$-0.493074\pi$
$769$	1.20670	0.0435147	0.0217574	+	0.999763i	$0.493074\pi$
$770$	0	0
$771$	24.5773	0.885129
$772$	0	0
$773$	−15.9957	−0.575327	−0.287663	−	0.957732i	$-0.592878\pi$
$773$	−15.9957	−0.575327	−0.287663	+	0.957732i	$0.592878\pi$
$774$	0	0
$775$	8.06658	0.289760
$776$	0	0
$777$	15.0987	0.541661
$778$	0	0
$779$	8.43693	0.302284
$780$	0	0
$781$	−32.6938	−1.16988
$782$	0	0
$783$	−3.25128	−0.116191
$784$	0	0
$785$	8.99070	0.320892
$786$	0	0
$787$	−32.7244	−1.16650	−0.583250	−	0.812292i	$-0.698219\pi$
$787$	−32.7244	−1.16650	−0.583250	+	0.812292i	$0.698219\pi$
$788$	0	0
$789$	24.6483	0.877502
$790$	0	0
$791$	−17.4876	−0.621786
$792$	0	0
$793$	18.2576	0.648346
$794$	0	0
$795$	−0.285980	−0.0101427
$796$	0	0
$797$	−31.2345	−1.10638	−0.553191	−	0.833054i	$-0.686590\pi$
$797$	−31.2345	−1.10638	−0.553191	+	0.833054i	$0.686590\pi$
$798$	0	0
$799$	−26.1771	−0.926080
$800$	0	0
$801$	−7.12992	−0.251923
$802$	0	0
$803$	−37.7140	−1.33090
$804$	0	0
$805$	−1.22324	−0.0431135
$806$	0	0
$807$	−4.52549	−0.159305
$808$	0	0
$809$	−46.8880	−1.64849	−0.824247	−	0.566230i	$-0.808401\pi$
$809$	−46.8880	−1.64849	−0.824247	+	0.566230i	$0.808401\pi$
$810$	0	0
$811$	−40.2325	−1.41275	−0.706377	−	0.707836i	$-0.749672\pi$
$811$	−40.2325	−1.41275	−0.706377	+	0.707836i	$0.749672\pi$
$812$	0	0
$813$	−22.1048	−0.775248
$814$	0	0
$815$	−9.60288	−0.336374
$816$	0	0
$817$	−55.7130	−1.94915
$818$	0	0
$819$	10.1495	0.354652
$820$	0	0
$821$	39.3186	1.37223	0.686115	−	0.727494i	$-0.259315\pi$
$821$	39.3186	1.37223	0.686115	+	0.727494i	$0.259315\pi$
$822$	0	0
$823$	−4.64345	−0.161860	−0.0809302	−	0.996720i	$-0.525789\pi$
$823$	−4.64345	−0.161860	−0.0809302	+	0.996720i	$0.525789\pi$
$824$	0	0
$825$	−17.6048	−0.612922
$826$	0	0
$827$	19.5624	0.680251	0.340125	−	0.940380i	$-0.389530\pi$
$827$	19.5624	0.680251	0.340125	+	0.940380i	$0.389530\pi$
$828$	0	0
$829$	3.01629	0.104760	0.0523801	−	0.998627i	$-0.483319\pi$
$829$	3.01629	0.104760	0.0523801	+	0.998627i	$0.483319\pi$
$830$	0	0
$831$	−14.4555	−0.501457
$832$	0	0
$833$	15.7644	0.546205
$834$	0	0
$835$	0.614948	0.0212812
$836$	0	0
$837$	−1.74532	−0.0603270
$838$	0	0
$839$	−25.4908	−0.880041	−0.440020	−	0.897988i	$-0.645029\pi$
$839$	−25.4908	−0.880041	−0.440020	+	0.897988i	$0.645029\pi$
$840$	0	0
$841$	−18.4292	−0.635489
$842$	0	0
$843$	3.60180	0.124053
$844$	0	0
$845$	2.71291	0.0933268
$846$	0	0
$847$	−12.1523	−0.417559
$848$	0	0
$849$	−22.0928	−0.758222
$850$	0	0
$851$	2.50400	0.0858360
$852$	0	0
$853$	22.8504	0.782382	0.391191	−	0.920310i	$-0.372063\pi$
$853$	22.8504	0.782382	0.391191	+	0.920310i	$0.372063\pi$
$854$	0	0
$855$	3.43299	0.117406
$856$	0	0
$857$	13.9103	0.475167	0.237584	−	0.971367i	$-0.423645\pi$
$857$	13.9103	0.475167	0.237584	+	0.971367i	$0.423645\pi$
$858$	0	0
$859$	−33.0864	−1.12889	−0.564447	−	0.825469i	$-0.690910\pi$
$859$	−33.0864	−1.12889	−0.564447	+	0.825469i	$0.690910\pi$
$860$	0	0
$861$	5.23407	0.178377
$862$	0	0
$863$	−48.6677	−1.65667	−0.828334	−	0.560234i	$-0.810711\pi$
$863$	−48.6677	−1.65667	−0.828334	+	0.560234i	$0.810711\pi$
$864$	0	0
$865$	−8.54002	−0.290369
$866$	0	0
$867$	−7.03673	−0.238980
$868$	0	0
$869$	39.0069	1.32322
$870$	0	0
$871$	−9.99721	−0.338742
$872$	0	0
$873$	4.70410	0.159210
$874$	0	0
$875$	−20.4920	−0.692757
$876$	0	0
$877$	21.5965	0.729264	0.364632	−	0.931152i	$-0.381195\pi$
$877$	21.5965	0.729264	0.364632	+	0.931152i	$0.381195\pi$
$878$	0	0
$879$	−0.403322	−0.0136037
$880$	0	0
$881$	−1.38305	−0.0465960	−0.0232980	−	0.999729i	$-0.507417\pi$
$881$	−1.38305	−0.0465960	−0.0232980	+	0.999729i	$0.507417\pi$
$882$	0	0
$883$	−24.0085	−0.807951	−0.403976	−	0.914770i	$-0.632372\pi$
$883$	−24.0085	−0.807951	−0.403976	+	0.914770i	$0.632372\pi$
$884$	0	0
$885$	5.89358	0.198111
$886$	0	0
$887$	−38.6184	−1.29668	−0.648339	−	0.761352i	$-0.724536\pi$
$887$	−38.6184	−1.29668	−0.648339	+	0.761352i	$0.724536\pi$
$888$	0	0
$889$	−29.1986	−0.979289
$890$	0	0
$891$	3.80906	0.127608
$892$	0	0
$893$	46.2972	1.54928
$894$	0	0
$895$	−1.97759	−0.0661034
$896$	0	0
$897$	1.68322	0.0562010
$898$	0	0
$899$	5.67452	0.189256
$900$	0	0
$901$	1.46791	0.0489030
$902$	0	0
$903$	−34.5630	−1.15019
$904$	0	0
$905$	−4.50870	−0.149874
$906$	0	0
$907$	33.3349	1.10687	0.553433	−	0.832893i	$-0.313317\pi$
$907$	33.3349	1.10687	0.553433	+	0.832893i	$0.313317\pi$
$908$	0	0
$909$	−6.74351	−0.223668
$910$	0	0
$911$	−10.7100	−0.354838	−0.177419	−	0.984135i	$-0.556775\pi$
$911$	−10.7100	−0.354838	−0.177419	+	0.984135i	$0.556775\pi$
$912$	0	0
$913$	−31.9907	−1.05874
$914$	0	0
$915$	3.83112	0.126653
$916$	0	0
$917$	−26.5054	−0.875287
$918$	0	0
$919$	−56.1186	−1.85118	−0.925591	−	0.378525i	$-0.876431\pi$
$919$	−56.1186	−1.85118	−0.925591	+	0.378525i	$0.876431\pi$
$920$	0	0
$921$	28.5615	0.941133
$922$	0	0
$923$	25.1539	0.827950
$924$	0	0
$925$	20.1495	0.662513
$926$	0	0
$927$	−3.37572	−0.110873
$928$	0	0
$929$	29.8268	0.978587	0.489293	−	0.872119i	$-0.337255\pi$
$929$	29.8268	0.978587	0.489293	+	0.872119i	$0.337255\pi$
$930$	0	0
$931$	−27.8812	−0.913768
$932$	0	0
$933$	−14.0740	−0.460761
$934$	0	0
$935$	−7.39361	−0.241797
$936$	0	0
$937$	23.8975	0.780696	0.390348	−	0.920667i	$-0.372355\pi$
$937$	23.8975	0.780696	0.390348	+	0.920667i	$0.372355\pi$
$938$	0	0
$939$	13.3132	0.434458
$940$	0	0
$941$	15.7999	0.515061	0.257530	−	0.966270i	$-0.417091\pi$
$941$	15.7999	0.515061	0.257530	+	0.966270i	$0.417091\pi$
$942$	0	0
$943$	0.868031	0.0282670
$944$	0	0
$945$	2.12974	0.0692805
$946$	0	0
$947$	−50.5289	−1.64197	−0.820985	−	0.570950i	$-0.806575\pi$
$947$	−50.5289	−1.64197	−0.820985	+	0.570950i	$0.806575\pi$
$948$	0	0
$949$	29.0163	0.941909
$950$	0	0
$951$	−23.0614	−0.747818
$952$	0	0
$953$	−14.2502	−0.461609	−0.230804	−	0.973000i	$-0.574136\pi$
$953$	−14.2502	−0.461609	−0.230804	+	0.973000i	$0.574136\pi$
$954$	0	0
$955$	−1.43651	−0.0464844
$956$	0	0
$957$	−12.3843	−0.400328
$958$	0	0
$959$	18.2795	0.590277
$960$	0	0
$961$	−27.9539	−0.901738
$962$	0	0
$963$	−6.87891	−0.221670
$964$	0	0
$965$	−5.98530	−0.192673
$966$	0	0
$967$	25.5040	0.820152	0.410076	−	0.912051i	$-0.365502\pi$
$967$	25.5040	0.820152	0.410076	+	0.912051i	$0.365502\pi$
$968$	0	0
$969$	−17.6211	−0.566073
$970$	0	0
$971$	−6.48205	−0.208019	−0.104009	−	0.994576i	$-0.533167\pi$
$971$	−6.48205	−0.208019	−0.104009	+	0.994576i	$0.533167\pi$
$972$	0	0
$973$	−19.1685	−0.614514
$974$	0	0
$975$	13.5447	0.433779
$976$	0	0
$977$	−17.9008	−0.572698	−0.286349	−	0.958125i	$-0.592442\pi$
$977$	−17.9008	−0.572698	−0.286349	+	0.958125i	$0.592442\pi$
$978$	0	0
$979$	−27.1582	−0.867981
$980$	0	0
$981$	−12.2964	−0.392594
$982$	0	0
$983$	−54.4682	−1.73727	−0.868634	−	0.495455i	$-0.835001\pi$
$983$	−54.4682	−1.73727	−0.868634	+	0.495455i	$0.835001\pi$
$984$	0	0
$985$	11.7106	0.373130
$986$	0	0
$987$	28.7216	0.914220
$988$	0	0
$989$	−5.73202	−0.182268
$990$	0	0
$991$	−39.3981	−1.25152	−0.625760	−	0.780016i	$-0.715211\pi$
$991$	−39.3981	−1.25152	−0.625760	+	0.780016i	$0.715211\pi$
$992$	0	0
$993$	31.1825	0.989546
$994$	0	0
$995$	−3.24069	−0.102737
$996$	0	0
$997$	14.2166	0.450245	0.225122	−	0.974331i	$-0.427722\pi$
$997$	14.2166	0.450245	0.225122	+	0.974331i	$0.427722\pi$
$998$	0	0
$999$	−4.35964	−0.137933

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.614948 q^{5} -3.46328 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.614948 q^{5} -3.46328 q^{7} +1.00000 q^{9} +3.80906 q^{11} -2.93060 q^{13} -0.614948 q^{15} +3.15646 q^{17} -5.58256 q^{19} -3.46328 q^{21} -0.574360 q^{23} -4.62184 q^{25} +1.00000 q^{27} -3.25128 q^{29} -1.74532 q^{31} +3.80906 q^{33} +2.12974 q^{35} -4.35964 q^{37} -2.93060 q^{39} -1.51130 q^{41} +9.97984 q^{43} -0.614948 q^{45} -8.29318 q^{47} +4.99433 q^{49} +3.15646 q^{51} +0.465048 q^{53} -2.34237 q^{55} -5.58256 q^{57} -9.58387 q^{59} -6.22999 q^{61} -3.46328 q^{63} +1.80217 q^{65} +3.41132 q^{67} -0.574360 q^{69} -8.58319 q^{71} -9.90115 q^{73} -4.62184 q^{75} -13.1918 q^{77} +10.2406 q^{79} +1.00000 q^{81} -8.39859 q^{83} -1.94106 q^{85} -3.25128 q^{87} -7.12992 q^{89} +10.1495 q^{91} -1.74532 q^{93} +3.43299 q^{95} +4.70410 q^{97} +3.80906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 7 q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) 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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