LMFDB - Embedded newform 644.2.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [644,2,Mod(1,644)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(644, 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("644.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 644 = 2^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	644.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:	$5.14236589017$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.8580816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 12x^{3} + 10x^{2} + 20x + 6 $ x^5 - x^4 - 12x^3 + 10x^2 + 20*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.09295$ of defining polynomial
Character	$\chi$	$=$	644.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.09295 q^{3} +1.68712 q^{5} +1.00000 q^{7} +6.56635 q^{9} +O(q^{10})$	$q+3.09295 q^{3} +1.68712 q^{5} +1.00000 q^{7} +6.56635 q^{9} +1.03228 q^{11} -4.59863 q^{13} +5.21819 q^{15} -1.50568 q^{17} -7.62471 q^{19} +3.09295 q^{21} +1.00000 q^{23} -2.15362 q^{25} +11.0305 q^{27} -1.99380 q^{29} -6.28575 q^{31} +3.19280 q^{33} +1.68712 q^{35} +7.43881 q^{37} -14.2233 q^{39} +2.41273 q^{41} +9.75845 q^{43} +11.0782 q^{45} -5.47270 q^{47} +1.00000 q^{49} -4.65700 q^{51} -14.0089 q^{53} +1.74159 q^{55} -23.5829 q^{57} +11.1304 q^{59} -0.248311 q^{61} +6.56635 q^{63} -7.75845 q^{65} +6.60483 q^{67} +3.09295 q^{69} -3.58727 q^{71} +2.21303 q^{73} -6.66104 q^{75} +1.03228 q^{77} +5.84534 q^{79} +14.4179 q^{81} -7.50338 q^{83} -2.54027 q^{85} -6.16672 q^{87} +7.81292 q^{89} -4.59863 q^{91} -19.4415 q^{93} -12.8638 q^{95} +14.0713 q^{97} +6.77833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} + 4 q^{5} + 5 q^{7} + 10 q^{9}+O(q^{10}) $ 5 * q + q^3 + 4 * q^5 + 5 * q^7 + 10 * q^9	$ 5 q + q^{3} + 4 q^{5} + 5 q^{7} + 10 q^{9} + 2 q^{11} + 3 q^{13} - 6 q^{15} + 4 q^{17} + 6 q^{19} + q^{21} + 5 q^{23} + 15 q^{25} + q^{27} + 5 q^{29} - q^{31} + 4 q^{35} + 22 q^{37} - q^{39} + 15 q^{41} + 12 q^{43} + 36 q^{45} - 21 q^{47} + 5 q^{49} + 24 q^{51} + 2 q^{53} - 22 q^{55} - 34 q^{57} - 12 q^{61} + 10 q^{63} - 2 q^{65} + 22 q^{67} + q^{69} - 15 q^{71} + 17 q^{73} - 47 q^{75} + 2 q^{77} + 2 q^{79} + 37 q^{81} - 16 q^{83} - 8 q^{85} - 33 q^{87} - 24 q^{89} + 3 q^{91} + 5 q^{93} - 8 q^{95} + 38 q^{97} - 36 q^{99}+O(q^{100}) $ 5 * q + q^3 + 4 * q^5 + 5 * q^7 + 10 * q^9 + 2 * q^11 + 3 * q^13 - 6 * q^15 + 4 * q^17 + 6 * q^19 + q^21 + 5 * q^23 + 15 * q^25 + q^27 + 5 * q^29 - q^31 + 4 * q^35 + 22 * q^37 - q^39 + 15 * q^41 + 12 * q^43 + 36 * q^45 - 21 * q^47 + 5 * q^49 + 24 * q^51 + 2 * q^53 - 22 * q^55 - 34 * q^57 - 12 * q^61 + 10 * q^63 - 2 * q^65 + 22 * q^67 + q^69 - 15 * q^71 + 17 * q^73 - 47 * q^75 + 2 * q^77 + 2 * q^79 + 37 * q^81 - 16 * q^83 - 8 * q^85 - 33 * q^87 - 24 * q^89 + 3 * q^91 + 5 * q^93 - 8 * q^95 + 38 * q^97 - 36 * 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
$3$	3.09295	1.78572	0.892858	−	0.450338i	$-0.148696\pi$
$3$	3.09295	1.78572	0.892858	+	0.450338i	$0.148696\pi$
$4$	0	0
$5$	1.68712	0.754504	0.377252	−	0.926111i	$-0.376869\pi$
$5$	1.68712	0.754504	0.377252	+	0.926111i	$0.376869\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	6.56635	2.18878
$10$	0	0
$11$	1.03228	0.311245	0.155623	−	0.987817i	$-0.450262\pi$
$11$	1.03228	0.311245	0.155623	+	0.987817i	$0.450262\pi$
$12$	0	0
$13$	−4.59863	−1.27543	−0.637716	−	0.770272i	$-0.720120\pi$
$13$	−4.59863	−1.27543	−0.637716	+	0.770272i	$0.720120\pi$
$14$	0	0
$15$	5.21819	1.34733
$16$	0	0
$17$	−1.50568	−0.365181	−0.182591	−	0.983189i	$-0.558448\pi$
$17$	−1.50568	−0.365181	−0.182591	+	0.983189i	$0.558448\pi$
$18$	0	0
$19$	−7.62471	−1.74923	−0.874615	−	0.484819i	$-0.838886\pi$
$19$	−7.62471	−1.74923	−0.874615	+	0.484819i	$0.838886\pi$
$20$	0	0
$21$	3.09295	0.674937
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.15362	−0.430724
$26$	0	0
$27$	11.0305	2.12283
$28$	0	0
$29$	−1.99380	−0.370239	−0.185120	−	0.982716i	$-0.559267\pi$
$29$	−1.99380	−0.370239	−0.185120	+	0.982716i	$0.559267\pi$
$30$	0	0
$31$	−6.28575	−1.12895	−0.564477	−	0.825449i	$-0.690922\pi$
$31$	−6.28575	−1.12895	−0.564477	+	0.825449i	$0.690922\pi$
$32$	0	0
$33$	3.19280	0.555796
$34$	0	0
$35$	1.68712	0.285176
$36$	0	0
$37$	7.43881	1.22293	0.611466	−	0.791270i	$-0.290580\pi$
$37$	7.43881	1.22293	0.611466	+	0.791270i	$0.290580\pi$
$38$	0	0
$39$	−14.2233	−2.27756
$40$	0	0
$41$	2.41273	0.376805	0.188403	−	0.982092i	$-0.439669\pi$
$41$	2.41273	0.376805	0.188403	+	0.982092i	$0.439669\pi$
$42$	0	0
$43$	9.75845	1.48815	0.744075	−	0.668096i	$-0.232890\pi$
$43$	9.75845	1.48815	0.744075	+	0.668096i	$0.232890\pi$
$44$	0	0
$45$	11.0782	1.65145
$46$	0	0
$47$	−5.47270	−0.798275	−0.399138	−	0.916891i	$-0.630690\pi$
$47$	−5.47270	−0.798275	−0.399138	+	0.916891i	$0.630690\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.65700	−0.652110
$52$	0	0
$53$	−14.0089	−1.92427	−0.962137	−	0.272567i	$-0.912127\pi$
$53$	−14.0089	−1.92427	−0.962137	+	0.272567i	$0.912127\pi$
$54$	0	0
$55$	1.74159	0.234836
$56$	0	0
$57$	−23.5829	−3.12363
$58$	0	0
$59$	11.1304	1.44905	0.724527	−	0.689246i	$-0.242058\pi$
$59$	11.1304	1.44905	0.724527	+	0.689246i	$0.242058\pi$
$60$	0	0
$61$	−0.248311	−0.0317930	−0.0158965	−	0.999874i	$-0.505060\pi$
$61$	−0.248311	−0.0317930	−0.0158965	+	0.999874i	$0.505060\pi$
$62$	0	0
$63$	6.56635	0.827282
$64$	0	0
$65$	−7.75845	−0.962318
$66$	0	0
$67$	6.60483	0.806909	0.403454	−	0.915000i	$-0.367809\pi$
$67$	6.60483	0.806909	0.403454	+	0.915000i	$0.367809\pi$
$68$	0	0
$69$	3.09295	0.372348
$70$	0	0
$71$	−3.58727	−0.425731	−0.212865	−	0.977082i	$-0.568280\pi$
$71$	−3.58727	−0.425731	−0.212865	+	0.977082i	$0.568280\pi$
$72$	0	0
$73$	2.21303	0.259015	0.129508	−	0.991578i	$-0.458660\pi$
$73$	2.21303	0.259015	0.129508	+	0.991578i	$0.458660\pi$
$74$	0	0
$75$	−6.66104	−0.769151
$76$	0	0
$77$	1.03228	0.117640
$78$	0	0
$79$	5.84534	0.657652	0.328826	−	0.944391i	$-0.393347\pi$
$79$	5.84534	0.657652	0.328826	+	0.944391i	$0.393347\pi$
$80$	0	0
$81$	14.4179	1.60199
$82$	0	0
$83$	−7.50338	−0.823603	−0.411801	−	0.911274i	$-0.635100\pi$
$83$	−7.50338	−0.823603	−0.411801	+	0.911274i	$0.635100\pi$
$84$	0	0
$85$	−2.54027	−0.275531
$86$	0	0
$87$	−6.16672	−0.661142
$88$	0	0
$89$	7.81292	0.828168	0.414084	−	0.910239i	$-0.364102\pi$
$89$	7.81292	0.828168	0.414084	+	0.910239i	$0.364102\pi$
$90$	0	0
$91$	−4.59863	−0.482068
$92$	0	0
$93$	−19.4415	−2.01599
$94$	0	0
$95$	−12.8638	−1.31980
$96$	0	0
$97$	14.0713	1.42873	0.714364	−	0.699775i	$-0.246716\pi$
$97$	14.0713	1.42873	0.714364	+	0.699775i	$0.246716\pi$
$98$	0	0
$99$	6.77833	0.681248
$100$	0	0
$101$	15.3186	1.52426	0.762129	−	0.647425i	$-0.224154\pi$
$101$	15.3186	1.52426	0.762129	+	0.647425i	$0.224154\pi$
$102$	0	0
$103$	−4.36288	−0.429888	−0.214944	−	0.976626i	$-0.568957\pi$
$103$	−4.36288	−0.429888	−0.214944	+	0.976626i	$0.568957\pi$
$104$	0	0
$105$	5.21819	0.509243
$106$	0	0
$107$	−6.18730	−0.598149	−0.299074	−	0.954230i	$-0.596678\pi$
$107$	−6.18730	−0.598149	−0.299074	+	0.954230i	$0.596678\pi$
$108$	0	0
$109$	12.1859	1.16720	0.583599	−	0.812042i	$-0.301644\pi$
$109$	12.1859	1.16720	0.583599	+	0.812042i	$0.301644\pi$
$110$	0	0
$111$	23.0079	2.18381
$112$	0	0
$113$	15.5691	1.46461	0.732307	−	0.680974i	$-0.238443\pi$
$113$	15.5691	1.46461	0.732307	+	0.680974i	$0.238443\pi$
$114$	0	0
$115$	1.68712	0.157325
$116$	0	0
$117$	−30.1962	−2.79164
$118$	0	0
$119$	−1.50568	−0.138026
$120$	0	0
$121$	−9.93439	−0.903126
$122$	0	0
$123$	7.46245	0.672867
$124$	0	0
$125$	−12.0690	−1.07949
$126$	0	0
$127$	−7.47486	−0.663286	−0.331643	−	0.943405i	$-0.607603\pi$
$127$	−7.47486	−0.663286	−0.331643	+	0.943405i	$0.607603\pi$
$128$	0	0
$129$	30.1824	2.65741
$130$	0	0
$131$	−20.5930	−1.79922	−0.899609	−	0.436696i	$-0.856148\pi$
$131$	−20.5930	−1.79922	−0.899609	+	0.436696i	$0.856148\pi$
$132$	0	0
$133$	−7.62471	−0.661147
$134$	0	0
$135$	18.6099	1.60168
$136$	0	0
$137$	6.61335	0.565017	0.282508	−	0.959265i	$-0.408833\pi$
$137$	6.61335	0.565017	0.282508	+	0.959265i	$0.408833\pi$
$138$	0	0
$139$	1.98298	0.168194	0.0840970	−	0.996458i	$-0.473199\pi$
$139$	1.98298	0.168194	0.0840970	+	0.996458i	$0.473199\pi$
$140$	0	0
$141$	−16.9268	−1.42549
$142$	0	0
$143$	−4.74709	−0.396972
$144$	0	0
$145$	−3.36378	−0.279347
$146$	0	0
$147$	3.09295	0.255102
$148$	0	0
$149$	5.63712	0.461811	0.230905	−	0.972976i	$-0.425831\pi$
$149$	5.63712	0.461811	0.230905	+	0.972976i	$0.425831\pi$
$150$	0	0
$151$	−21.0350	−1.71180	−0.855902	−	0.517138i	$-0.826998\pi$
$151$	−21.0350	−1.71180	−0.855902	+	0.517138i	$0.826998\pi$
$152$	0	0
$153$	−9.88682	−0.799302
$154$	0	0
$155$	−10.6048	−0.851801
$156$	0	0
$157$	2.75719	0.220048	0.110024	−	0.993929i	$-0.464907\pi$
$157$	2.75719	0.220048	0.110024	+	0.993929i	$0.464907\pi$
$158$	0	0
$159$	−43.3289	−3.43621
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	8.40033	0.657964	0.328982	−	0.944336i	$-0.393294\pi$
$163$	8.40033	0.657964	0.328982	+	0.944336i	$0.393294\pi$
$164$	0	0
$165$	5.38665	0.419350
$166$	0	0
$167$	−15.1905	−1.17548	−0.587738	−	0.809051i	$-0.699981\pi$
$167$	−15.1905	−1.17548	−0.587738	+	0.809051i	$0.699981\pi$
$168$	0	0
$169$	8.14742	0.626724
$170$	0	0
$171$	−50.0665	−3.82868
$172$	0	0
$173$	−23.9444	−1.82046	−0.910228	−	0.414108i	$-0.864094\pi$
$173$	−23.9444	−1.82046	−0.910228	+	0.414108i	$0.864094\pi$
$174$	0	0
$175$	−2.15362	−0.162798
$176$	0	0
$177$	34.4258	2.58760
$178$	0	0
$179$	8.66320	0.647518	0.323759	−	0.946140i	$-0.395053\pi$
$179$	8.66320	0.647518	0.323759	+	0.946140i	$0.395053\pi$
$180$	0	0
$181$	−1.62702	−0.120935	−0.0604676	−	0.998170i	$-0.519259\pi$
$181$	−1.62702	−0.120935	−0.0604676	+	0.998170i	$0.519259\pi$
$182$	0	0
$183$	−0.768014	−0.0567732
$184$	0	0
$185$	12.5502	0.922708
$186$	0	0
$187$	−1.55429	−0.113661
$188$	0	0
$189$	11.0305	0.802354
$190$	0	0
$191$	−7.30968	−0.528910	−0.264455	−	0.964398i	$-0.585192\pi$
$191$	−7.30968	−0.528910	−0.264455	+	0.964398i	$0.585192\pi$
$192$	0	0
$193$	24.5306	1.76575	0.882875	−	0.469608i	$-0.155605\pi$
$193$	24.5306	1.76575	0.882875	+	0.469608i	$0.155605\pi$
$194$	0	0
$195$	−23.9965	−1.71843
$196$	0	0
$197$	3.97288	0.283056	0.141528	−	0.989934i	$-0.454799\pi$
$197$	3.97288	0.283056	0.141528	+	0.989934i	$0.454799\pi$
$198$	0	0
$199$	5.06813	0.359270	0.179635	−	0.983733i	$-0.442508\pi$
$199$	5.06813	0.359270	0.179635	+	0.983733i	$0.442508\pi$
$200$	0	0
$201$	20.4284	1.44091
$202$	0	0
$203$	−1.99380	−0.139937
$204$	0	0
$205$	4.07057	0.284301
$206$	0	0
$207$	6.56635	0.456393
$208$	0	0
$209$	−7.87087	−0.544439
$210$	0	0
$211$	−5.44773	−0.375037	−0.187519	−	0.982261i	$-0.560045\pi$
$211$	−5.44773	−0.375037	−0.187519	+	0.982261i	$0.560045\pi$
$212$	0	0
$213$	−11.0953	−0.760234
$214$	0	0
$215$	16.4637	1.12282
$216$	0	0
$217$	−6.28575	−0.426705
$218$	0	0
$219$	6.84478	0.462528
$220$	0	0
$221$	6.92407	0.465763
$222$	0	0
$223$	−3.30598	−0.221385	−0.110692	−	0.993855i	$-0.535307\pi$
$223$	−3.30598	−0.221385	−0.110692	+	0.993855i	$0.535307\pi$
$224$	0	0
$225$	−14.1414	−0.942761
$226$	0	0
$227$	−6.38561	−0.423828	−0.211914	−	0.977288i	$-0.567970\pi$
$227$	−6.38561	−0.423828	−0.211914	+	0.977288i	$0.567970\pi$
$228$	0	0
$229$	14.8661	0.982381	0.491191	−	0.871052i	$-0.336562\pi$
$229$	14.8661	0.982381	0.491191	+	0.871052i	$0.336562\pi$
$230$	0	0
$231$	3.19280	0.210071
$232$	0	0
$233$	−30.3114	−1.98576	−0.992881	−	0.119106i	$-0.961997\pi$
$233$	−30.3114	−1.98576	−0.992881	+	0.119106i	$0.961997\pi$
$234$	0	0
$235$	−9.23311	−0.602302
$236$	0	0
$237$	18.0793	1.17438
$238$	0	0
$239$	−16.5341	−1.06950	−0.534750	−	0.845010i	$-0.679594\pi$
$239$	−16.5341	−1.06950	−0.534750	+	0.845010i	$0.679594\pi$
$240$	0	0
$241$	−6.70294	−0.431775	−0.215887	−	0.976418i	$-0.569264\pi$
$241$	−6.70294	−0.431775	−0.215887	+	0.976418i	$0.569264\pi$
$242$	0	0
$243$	11.5022	0.737866
$244$	0	0
$245$	1.68712	0.107786
$246$	0	0
$247$	35.0633	2.23102
$248$	0	0
$249$	−23.2076	−1.47072
$250$	0	0
$251$	−20.2008	−1.27506	−0.637532	−	0.770423i	$-0.720045\pi$
$251$	−20.2008	−1.27506	−0.637532	+	0.770423i	$0.720045\pi$
$252$	0	0
$253$	1.03228	0.0648991
$254$	0	0
$255$	−7.85692	−0.492020
$256$	0	0
$257$	26.6643	1.66327	0.831637	−	0.555319i	$-0.187404\pi$
$257$	26.6643	1.66327	0.831637	+	0.555319i	$0.187404\pi$
$258$	0	0
$259$	7.43881	0.462225
$260$	0	0
$261$	−13.0920	−0.810373
$262$	0	0
$263$	17.8543	1.10094	0.550470	−	0.834855i	$-0.314448\pi$
$263$	17.8543	1.10094	0.550470	+	0.834855i	$0.314448\pi$
$264$	0	0
$265$	−23.6348	−1.45187
$266$	0	0
$267$	24.1650	1.47887
$268$	0	0
$269$	−3.71780	−0.226678	−0.113339	−	0.993556i	$-0.536155\pi$
$269$	−3.71780	−0.226678	−0.113339	+	0.993556i	$0.536155\pi$
$270$	0	0
$271$	−4.69402	−0.285142	−0.142571	−	0.989785i	$-0.545537\pi$
$271$	−4.69402	−0.285142	−0.142571	+	0.989785i	$0.545537\pi$
$272$	0	0
$273$	−14.2233	−0.860836
$274$	0	0
$275$	−2.22315	−0.134061
$276$	0	0
$277$	25.6185	1.53927	0.769634	−	0.638486i	$-0.220439\pi$
$277$	25.6185	1.53927	0.769634	+	0.638486i	$0.220439\pi$
$278$	0	0
$279$	−41.2745	−2.47104
$280$	0	0
$281$	7.08833	0.422854	0.211427	−	0.977394i	$-0.432189\pi$
$281$	7.08833	0.422854	0.211427	+	0.977394i	$0.432189\pi$
$282$	0	0
$283$	1.68928	0.100417	0.0502087	−	0.998739i	$-0.484011\pi$
$283$	1.68928	0.100417	0.0502087	+	0.998739i	$0.484011\pi$
$284$	0	0
$285$	−39.7872	−2.35679
$286$	0	0
$287$	2.41273	0.142419
$288$	0	0
$289$	−14.7329	−0.866643
$290$	0	0
$291$	43.5219	2.55130
$292$	0	0
$293$	6.82622	0.398792	0.199396	−	0.979919i	$-0.436102\pi$
$293$	6.82622	0.398792	0.199396	+	0.979919i	$0.436102\pi$
$294$	0	0
$295$	18.7783	1.09332
$296$	0	0
$297$	11.3866	0.660720
$298$	0	0
$299$	−4.59863	−0.265946
$300$	0	0
$301$	9.75845	0.562468
$302$	0	0
$303$	47.3797	2.72189
$304$	0	0
$305$	−0.418931	−0.0239879
$306$	0	0
$307$	12.0589	0.688239	0.344120	−	0.938926i	$-0.388177\pi$
$307$	12.0589	0.688239	0.344120	+	0.938926i	$0.388177\pi$
$308$	0	0
$309$	−13.4942	−0.767657
$310$	0	0
$311$	29.6166	1.67940	0.839702	−	0.543047i	$-0.182730\pi$
$311$	29.6166	1.67940	0.839702	+	0.543047i	$0.182730\pi$
$312$	0	0
$313$	19.9421	1.12719	0.563596	−	0.826051i	$-0.309418\pi$
$313$	19.9421	1.12719	0.563596	+	0.826051i	$0.309418\pi$
$314$	0	0
$315$	11.0782	0.624188
$316$	0	0
$317$	10.7659	0.604675	0.302337	−	0.953201i	$-0.402233\pi$
$317$	10.7659	0.604675	0.302337	+	0.953201i	$0.402233\pi$
$318$	0	0
$319$	−2.05816	−0.115235
$320$	0	0
$321$	−19.1370	−1.06812
$322$	0	0
$323$	11.4804	0.638786
$324$	0	0
$325$	9.90370	0.549359
$326$	0	0
$327$	37.6904	2.08428
$328$	0	0
$329$	−5.47270	−0.301720
$330$	0	0
$331$	9.71000	0.533710	0.266855	−	0.963737i	$-0.414016\pi$
$331$	9.71000	0.533710	0.266855	+	0.963737i	$0.414016\pi$
$332$	0	0
$333$	48.8458	2.67673
$334$	0	0
$335$	11.1432	0.608816
$336$	0	0
$337$	28.9295	1.57589	0.787946	−	0.615745i	$-0.211145\pi$
$337$	28.9295	1.57589	0.787946	+	0.615745i	$0.211145\pi$
$338$	0	0
$339$	48.1544	2.61539
$340$	0	0
$341$	−6.48868	−0.351382
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	5.21819	0.280938
$346$	0	0
$347$	−24.3254	−1.30585	−0.652927	−	0.757421i	$-0.726459\pi$
$347$	−24.3254	−1.30585	−0.652927	+	0.757421i	$0.726459\pi$
$348$	0	0
$349$	17.5873	0.941425	0.470713	−	0.882287i	$-0.343997\pi$
$349$	17.5873	0.941425	0.470713	+	0.882287i	$0.343997\pi$
$350$	0	0
$351$	−50.7254	−2.70752
$352$	0	0
$353$	−24.5952	−1.30907	−0.654534	−	0.756033i	$-0.727135\pi$
$353$	−24.5952	−1.30907	−0.654534	+	0.756033i	$0.727135\pi$
$354$	0	0
$355$	−6.05216	−0.321215
$356$	0	0
$357$	−4.65700	−0.246474
$358$	0	0
$359$	8.71125	0.459762	0.229881	−	0.973219i	$-0.426166\pi$
$359$	8.71125	0.459762	0.229881	+	0.973219i	$0.426166\pi$
$360$	0	0
$361$	39.1363	2.05980
$362$	0	0
$363$	−30.7266	−1.61273
$364$	0	0
$365$	3.73365	0.195428
$366$	0	0
$367$	−32.9646	−1.72074	−0.860370	−	0.509670i	$-0.829768\pi$
$367$	−32.9646	−1.72074	−0.860370	+	0.509670i	$0.829768\pi$
$368$	0	0
$369$	15.8428	0.824744
$370$	0	0
$371$	−14.0089	−0.727307
$372$	0	0
$373$	−28.2743	−1.46399	−0.731994	−	0.681311i	$-0.761410\pi$
$373$	−28.2743	−1.46399	−0.731994	+	0.681311i	$0.761410\pi$
$374$	0	0
$375$	−37.3289	−1.92766
$376$	0	0
$377$	9.16874	0.472214
$378$	0	0
$379$	30.6797	1.57591	0.787956	−	0.615732i	$-0.211140\pi$
$379$	30.6797	1.57591	0.787956	+	0.615732i	$0.211140\pi$
$380$	0	0
$381$	−23.1194	−1.18444
$382$	0	0
$383$	−18.6992	−0.955487	−0.477743	−	0.878499i	$-0.658545\pi$
$383$	−18.6992	−0.955487	−0.477743	+	0.878499i	$0.658545\pi$
$384$	0	0
$385$	1.74159	0.0887596
$386$	0	0
$387$	64.0774	3.25724
$388$	0	0
$389$	27.6780	1.40333	0.701665	−	0.712507i	$-0.252440\pi$
$389$	27.6780	1.40333	0.701665	+	0.712507i	$0.252440\pi$
$390$	0	0
$391$	−1.50568	−0.0761455
$392$	0	0
$393$	−63.6931	−3.21289
$394$	0	0
$395$	9.86180	0.496201
$396$	0	0
$397$	17.3039	0.868457	0.434228	−	0.900803i	$-0.357021\pi$
$397$	17.3039	0.868457	0.434228	+	0.900803i	$0.357021\pi$
$398$	0	0
$399$	−23.5829	−1.18062
$400$	0	0
$401$	−5.24511	−0.261928	−0.130964	−	0.991387i	$-0.541807\pi$
$401$	−5.24511	−0.261928	−0.130964	+	0.991387i	$0.541807\pi$
$402$	0	0
$403$	28.9059	1.43990
$404$	0	0
$405$	24.3247	1.20871
$406$	0	0
$407$	7.67896	0.380632
$408$	0	0
$409$	18.3092	0.905334	0.452667	−	0.891680i	$-0.350473\pi$
$409$	18.3092	0.905334	0.452667	+	0.891680i	$0.350473\pi$
$410$	0	0
$411$	20.4548	1.00896
$412$	0	0
$413$	11.1304	0.547691
$414$	0	0
$415$	−12.6591	−0.621411
$416$	0	0
$417$	6.13325	0.300347
$418$	0	0
$419$	1.66795	0.0814849	0.0407424	−	0.999170i	$-0.487028\pi$
$419$	1.66795	0.0814849	0.0407424	+	0.999170i	$0.487028\pi$
$420$	0	0
$421$	11.8276	0.576443	0.288222	−	0.957564i	$-0.406936\pi$
$421$	11.8276	0.576443	0.288222	+	0.957564i	$0.406936\pi$
$422$	0	0
$423$	−35.9357	−1.74725
$424$	0	0
$425$	3.24266	0.157292
$426$	0	0
$427$	−0.248311	−0.0120166
$428$	0	0
$429$	−14.6825	−0.708879
$430$	0	0
$431$	13.9979	0.674256	0.337128	−	0.941459i	$-0.390545\pi$
$431$	13.9979	0.674256	0.337128	+	0.941459i	$0.390545\pi$
$432$	0	0
$433$	−0.561952	−0.0270057	−0.0135028	−	0.999909i	$-0.504298\pi$
$433$	−0.561952	−0.0270057	−0.0135028	+	0.999909i	$0.504298\pi$
$434$	0	0
$435$	−10.4040	−0.498834
$436$	0	0
$437$	−7.62471	−0.364740
$438$	0	0
$439$	6.94681	0.331553	0.165777	−	0.986163i	$-0.446987\pi$
$439$	6.94681	0.331553	0.165777	+	0.986163i	$0.446987\pi$
$440$	0	0
$441$	6.56635	0.312683
$442$	0	0
$443$	−17.6518	−0.838664	−0.419332	−	0.907833i	$-0.637736\pi$
$443$	−17.6518	−0.838664	−0.419332	+	0.907833i	$0.637736\pi$
$444$	0	0
$445$	13.1813	0.624856
$446$	0	0
$447$	17.4353	0.824663
$448$	0	0
$449$	−24.2966	−1.14663	−0.573315	−	0.819335i	$-0.694343\pi$
$449$	−24.2966	−1.14663	−0.573315	+	0.819335i	$0.694343\pi$
$450$	0	0
$451$	2.49062	0.117279
$452$	0	0
$453$	−65.0602	−3.05680
$454$	0	0
$455$	−7.75845	−0.363722
$456$	0	0
$457$	−29.5045	−1.38016	−0.690081	−	0.723732i	$-0.742425\pi$
$457$	−29.5045	−1.38016	−0.690081	+	0.723732i	$0.742425\pi$
$458$	0	0
$459$	−16.6085	−0.775217
$460$	0	0
$461$	−35.5817	−1.65721	−0.828603	−	0.559837i	$-0.810864\pi$
$461$	−35.5817	−1.65721	−0.828603	+	0.559837i	$0.810864\pi$
$462$	0	0
$463$	−26.6237	−1.23731	−0.618653	−	0.785664i	$-0.712322\pi$
$463$	−26.6237	−1.23731	−0.618653	+	0.785664i	$0.712322\pi$
$464$	0	0
$465$	−32.8002	−1.52107
$466$	0	0
$467$	9.87226	0.456834	0.228417	−	0.973563i	$-0.426645\pi$
$467$	9.87226	0.456834	0.228417	+	0.973563i	$0.426645\pi$
$468$	0	0
$469$	6.60483	0.304983
$470$	0	0
$471$	8.52786	0.392943
$472$	0	0
$473$	10.0735	0.463180
$474$	0	0
$475$	16.4207	0.753435
$476$	0	0
$477$	−91.9875	−4.21182
$478$	0	0
$479$	10.6949	0.488664	0.244332	−	0.969692i	$-0.421431\pi$
$479$	10.6949	0.488664	0.244332	+	0.969692i	$0.421431\pi$
$480$	0	0
$481$	−34.2084	−1.55977
$482$	0	0
$483$	3.09295	0.140734
$484$	0	0
$485$	23.7401	1.07798
$486$	0	0
$487$	18.8367	0.853572	0.426786	−	0.904353i	$-0.359646\pi$
$487$	18.8367	0.853572	0.426786	+	0.904353i	$0.359646\pi$
$488$	0	0
$489$	25.9818	1.17494
$490$	0	0
$491$	42.5963	1.92234	0.961172	−	0.275952i	$-0.0889929\pi$
$491$	42.5963	1.92234	0.961172	+	0.275952i	$0.0889929\pi$
$492$	0	0
$493$	3.00202	0.135204
$494$	0	0
$495$	11.4359	0.514004
$496$	0	0
$497$	−3.58727	−0.160911
$498$	0	0
$499$	−11.8591	−0.530885	−0.265442	−	0.964127i	$-0.585518\pi$
$499$	−11.8591	−0.530885	−0.265442	+	0.964127i	$0.585518\pi$
$500$	0	0
$501$	−46.9835	−2.09907
$502$	0	0
$503$	18.3821	0.819618	0.409809	−	0.912171i	$-0.365595\pi$
$503$	18.3821	0.819618	0.409809	+	0.912171i	$0.365595\pi$
$504$	0	0
$505$	25.8443	1.15006
$506$	0	0
$507$	25.1996	1.11915
$508$	0	0
$509$	−32.9364	−1.45988	−0.729940	−	0.683511i	$-0.760452\pi$
$509$	−32.9364	−1.45988	−0.729940	+	0.683511i	$0.760452\pi$
$510$	0	0
$511$	2.21303	0.0978985
$512$	0	0
$513$	−84.1047	−3.71331
$514$	0	0
$515$	−7.36072	−0.324352
$516$	0	0
$517$	−5.64938	−0.248459
$518$	0	0
$519$	−74.0587	−3.25082
$520$	0	0
$521$	4.05140	0.177495	0.0887475	−	0.996054i	$-0.471714\pi$
$521$	4.05140	0.177495	0.0887475	+	0.996054i	$0.471714\pi$
$522$	0	0
$523$	−42.1697	−1.84395	−0.921976	−	0.387247i	$-0.873426\pi$
$523$	−42.1697	−1.84395	−0.921976	+	0.387247i	$0.873426\pi$
$524$	0	0
$525$	−6.66104	−0.290712
$526$	0	0
$527$	9.46434	0.412273
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	73.0861	3.17166
$532$	0	0
$533$	−11.0953	−0.480589
$534$	0	0
$535$	−10.4387	−0.451306
$536$	0	0
$537$	26.7949	1.15628
$538$	0	0
$539$	1.03228	0.0444636
$540$	0	0
$541$	36.6551	1.57593	0.787963	−	0.615722i	$-0.211136\pi$
$541$	36.6551	1.57593	0.787963	+	0.615722i	$0.211136\pi$
$542$	0	0
$543$	−5.03228	−0.215956
$544$	0	0
$545$	20.5591	0.880655
$546$	0	0
$547$	29.5270	1.26248	0.631242	−	0.775586i	$-0.282545\pi$
$547$	29.5270	1.26248	0.631242	+	0.775586i	$0.282545\pi$
$548$	0	0
$549$	−1.63050	−0.0695879
$550$	0	0
$551$	15.2021	0.647633
$552$	0	0
$553$	5.84534	0.248569
$554$	0	0
$555$	38.8171	1.64769
$556$	0	0
$557$	−1.58287	−0.0670683	−0.0335342	−	0.999438i	$-0.510676\pi$
$557$	−1.58287	−0.0670683	−0.0335342	+	0.999438i	$0.510676\pi$
$558$	0	0
$559$	−44.8755	−1.89803
$560$	0	0
$561$	−4.80734	−0.202966
$562$	0	0
$563$	−4.12913	−0.174022	−0.0870111	−	0.996207i	$-0.527732\pi$
$563$	−4.12913	−0.174022	−0.0870111	+	0.996207i	$0.527732\pi$
$564$	0	0
$565$	26.2669	1.10506
$566$	0	0
$567$	14.4179	0.605494
$568$	0	0
$569$	−25.7996	−1.08158	−0.540788	−	0.841159i	$-0.681874\pi$
$569$	−25.7996	−1.08158	−0.540788	+	0.841159i	$0.681874\pi$
$570$	0	0
$571$	−35.2733	−1.47614	−0.738070	−	0.674724i	$-0.764263\pi$
$571$	−35.2733	−1.47614	−0.738070	+	0.674724i	$0.764263\pi$
$572$	0	0
$573$	−22.6085	−0.944483
$574$	0	0
$575$	−2.15362	−0.0898121
$576$	0	0
$577$	−36.4660	−1.51810	−0.759050	−	0.651032i	$-0.774336\pi$
$577$	−36.4660	−1.51810	−0.759050	+	0.651032i	$0.774336\pi$
$578$	0	0
$579$	75.8719	3.15313
$580$	0	0
$581$	−7.50338	−0.311293
$582$	0	0
$583$	−14.4612	−0.598921
$584$	0	0
$585$	−50.9447	−2.10630
$586$	0	0
$587$	−5.59097	−0.230764	−0.115382	−	0.993321i	$-0.536809\pi$
$587$	−5.59097	−0.230764	−0.115382	+	0.993321i	$0.536809\pi$
$588$	0	0
$589$	47.9271	1.97480
$590$	0	0
$591$	12.2879	0.505457
$592$	0	0
$593$	6.32996	0.259940	0.129970	−	0.991518i	$-0.458512\pi$
$593$	6.32996	0.259940	0.129970	+	0.991518i	$0.458512\pi$
$594$	0	0
$595$	−2.54027	−0.104141
$596$	0	0
$597$	15.6755	0.641555
$598$	0	0
$599$	−24.5839	−1.00447	−0.502236	−	0.864731i	$-0.667489\pi$
$599$	−24.5839	−1.00447	−0.502236	+	0.864731i	$0.667489\pi$
$600$	0	0
$601$	3.05089	0.124448	0.0622242	−	0.998062i	$-0.480181\pi$
$601$	3.05089	0.124448	0.0622242	+	0.998062i	$0.480181\pi$
$602$	0	0
$603$	43.3696	1.76615
$604$	0	0
$605$	−16.7605	−0.681412
$606$	0	0
$607$	−41.7404	−1.69419	−0.847096	−	0.531441i	$-0.821651\pi$
$607$	−41.7404	−1.69419	−0.847096	+	0.531441i	$0.821651\pi$
$608$	0	0
$609$	−6.16672	−0.249888
$610$	0	0
$611$	25.1669	1.01815
$612$	0	0
$613$	−29.8181	−1.20434	−0.602172	−	0.798367i	$-0.705698\pi$
$613$	−29.8181	−1.20434	−0.602172	+	0.798367i	$0.705698\pi$
$614$	0	0
$615$	12.5901	0.507681
$616$	0	0
$617$	10.0671	0.405286	0.202643	−	0.979253i	$-0.435047\pi$
$617$	10.0671	0.405286	0.202643	+	0.979253i	$0.435047\pi$
$618$	0	0
$619$	45.8912	1.84452	0.922262	−	0.386565i	$-0.126338\pi$
$619$	45.8912	1.84452	0.922262	+	0.386565i	$0.126338\pi$
$620$	0	0
$621$	11.0305	0.442640
$622$	0	0
$623$	7.81292	0.313018
$624$	0	0
$625$	−9.59383	−0.383753
$626$	0	0
$627$	−24.3442	−0.972214
$628$	0	0
$629$	−11.2005	−0.446592
$630$	0	0
$631$	−2.90455	−0.115628	−0.0578141	−	0.998327i	$-0.518413\pi$
$631$	−2.90455	−0.115628	−0.0578141	+	0.998327i	$0.518413\pi$
$632$	0	0
$633$	−16.8496	−0.669711
$634$	0	0
$635$	−12.6110	−0.500452
$636$	0	0
$637$	−4.59863	−0.182204
$638$	0	0
$639$	−23.5553	−0.931832
$640$	0	0
$641$	−12.6244	−0.498632	−0.249316	−	0.968422i	$-0.580206\pi$
$641$	−12.6244	−0.498632	−0.249316	+	0.968422i	$0.580206\pi$
$642$	0	0
$643$	41.8207	1.64925	0.824623	−	0.565683i	$-0.191387\pi$
$643$	41.8207	1.64925	0.824623	+	0.565683i	$0.191387\pi$
$644$	0	0
$645$	50.9214	2.00503
$646$	0	0
$647$	9.90154	0.389270	0.194635	−	0.980876i	$-0.437648\pi$
$647$	9.90154	0.389270	0.194635	+	0.980876i	$0.437648\pi$
$648$	0	0
$649$	11.4897	0.451011
$650$	0	0
$651$	−19.4415	−0.761974
$652$	0	0
$653$	1.04101	0.0407377	0.0203689	−	0.999793i	$-0.493516\pi$
$653$	1.04101	0.0407377	0.0203689	+	0.999793i	$0.493516\pi$
$654$	0	0
$655$	−34.7429	−1.35752
$656$	0	0
$657$	14.5315	0.566928
$658$	0	0
$659$	−27.8915	−1.08650	−0.543249	−	0.839571i	$-0.682806\pi$
$659$	−27.8915	−1.08650	−0.543249	+	0.839571i	$0.682806\pi$
$660$	0	0
$661$	15.5672	0.605493	0.302747	−	0.953071i	$-0.402096\pi$
$661$	15.5672	0.605493	0.302747	+	0.953071i	$0.402096\pi$
$662$	0	0
$663$	21.4158	0.831721
$664$	0	0
$665$	−12.8638	−0.498838
$666$	0	0
$667$	−1.99380	−0.0772002
$668$	0	0
$669$	−10.2252	−0.395330
$670$	0	0
$671$	−0.256327	−0.00989541
$672$	0	0
$673$	12.3546	0.476235	0.238117	−	0.971236i	$-0.423470\pi$
$673$	12.3546	0.476235	0.238117	+	0.971236i	$0.423470\pi$
$674$	0	0
$675$	−23.7556	−0.914353
$676$	0	0
$677$	23.5113	0.903611	0.451806	−	0.892116i	$-0.350780\pi$
$677$	23.5113	0.903611	0.451806	+	0.892116i	$0.350780\pi$
$678$	0	0
$679$	14.0713	0.540008
$680$	0	0
$681$	−19.7504	−0.756836
$682$	0	0
$683$	30.0602	1.15022	0.575110	−	0.818076i	$-0.304959\pi$
$683$	30.0602	1.15022	0.575110	+	0.818076i	$0.304959\pi$
$684$	0	0
$685$	11.1575	0.426307
$686$	0	0
$687$	45.9802	1.75425
$688$	0	0
$689$	64.4219	2.45428
$690$	0	0
$691$	−26.9501	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$691$	−26.9501	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$692$	0	0
$693$	6.77833	0.257488
$694$	0	0
$695$	3.34552	0.126903
$696$	0	0
$697$	−3.63280	−0.137602
$698$	0	0
$699$	−93.7516	−3.54601
$700$	0	0
$701$	15.3480	0.579688	0.289844	−	0.957074i	$-0.406397\pi$
$701$	15.3480	0.579688	0.289844	+	0.957074i	$0.406397\pi$
$702$	0	0
$703$	−56.7188	−2.13919
$704$	0	0
$705$	−28.5576	−1.07554
$706$	0	0
$707$	15.3186	0.576115
$708$	0	0
$709$	−14.6979	−0.551989	−0.275995	−	0.961159i	$-0.589007\pi$
$709$	−14.6979	−0.551989	−0.275995	+	0.961159i	$0.589007\pi$
$710$	0	0
$711$	38.3825	1.43946
$712$	0	0
$713$	−6.28575	−0.235403
$714$	0	0
$715$	−8.00892	−0.299517
$716$	0	0
$717$	−51.1391	−1.90982
$718$	0	0
$719$	50.6159	1.88766	0.943828	−	0.330437i	$-0.107196\pi$
$719$	50.6159	1.88766	0.943828	+	0.330437i	$0.107196\pi$
$720$	0	0
$721$	−4.36288	−0.162482
$722$	0	0
$723$	−20.7319	−0.771027
$724$	0	0
$725$	4.29388	0.159471
$726$	0	0
$727$	−30.1746	−1.11911	−0.559557	−	0.828792i	$-0.689029\pi$
$727$	−30.1746	−1.11911	−0.559557	+	0.828792i	$0.689029\pi$
$728$	0	0
$729$	−7.67792	−0.284367
$730$	0	0
$731$	−14.6931	−0.543444
$732$	0	0
$733$	−9.09081	−0.335777	−0.167888	−	0.985806i	$-0.553695\pi$
$733$	−9.09081	−0.335777	−0.167888	+	0.985806i	$0.553695\pi$
$734$	0	0
$735$	5.21819	0.192476
$736$	0	0
$737$	6.81806	0.251147
$738$	0	0
$739$	35.8506	1.31878	0.659392	−	0.751799i	$-0.270814\pi$
$739$	35.8506	1.31878	0.659392	+	0.751799i	$0.270814\pi$
$740$	0	0
$741$	108.449	3.98397
$742$	0	0
$743$	48.6330	1.78417	0.892086	−	0.451866i	$-0.149241\pi$
$743$	48.6330	1.78417	0.892086	+	0.451866i	$0.149241\pi$
$744$	0	0
$745$	9.51050	0.348438
$746$	0	0
$747$	−49.2698	−1.80269
$748$	0	0
$749$	−6.18730	−0.226079
$750$	0	0
$751$	−45.7127	−1.66808	−0.834040	−	0.551703i	$-0.813978\pi$
$751$	−45.7127	−1.66808	−0.834040	+	0.551703i	$0.813978\pi$
$752$	0	0
$753$	−62.4802	−2.27690
$754$	0	0
$755$	−35.4886	−1.29156
$756$	0	0
$757$	39.1647	1.42346	0.711732	−	0.702451i	$-0.247911\pi$
$757$	39.1647	1.42346	0.711732	+	0.702451i	$0.247911\pi$
$758$	0	0
$759$	3.19280	0.115891
$760$	0	0
$761$	−9.27015	−0.336043	−0.168021	−	0.985783i	$-0.553738\pi$
$761$	−9.27015	−0.336043	−0.168021	+	0.985783i	$0.553738\pi$
$762$	0	0
$763$	12.1859	0.441159
$764$	0	0
$765$	−16.6803	−0.603077
$766$	0	0
$767$	−51.1846	−1.84817
$768$	0	0
$769$	25.0074	0.901790	0.450895	−	0.892577i	$-0.351105\pi$
$769$	25.0074	0.901790	0.450895	+	0.892577i	$0.351105\pi$
$770$	0	0
$771$	82.4715	2.97014
$772$	0	0
$773$	48.4476	1.74254	0.871270	−	0.490805i	$-0.163297\pi$
$773$	48.4476	1.74254	0.871270	+	0.490805i	$0.163297\pi$
$774$	0	0
$775$	13.5371	0.486268
$776$	0	0
$777$	23.0079	0.825403
$778$	0	0
$779$	−18.3964	−0.659118
$780$	0	0
$781$	−3.70308	−0.132507
$782$	0	0
$783$	−21.9927	−0.785954
$784$	0	0
$785$	4.65172	0.166027
$786$	0	0
$787$	−2.55227	−0.0909785	−0.0454892	−	0.998965i	$-0.514485\pi$
$787$	−2.55227	−0.0909785	−0.0454892	+	0.998965i	$0.514485\pi$
$788$	0	0
$789$	55.2224	1.96597
$790$	0	0
$791$	15.5691	0.553572
$792$	0	0
$793$	1.14189	0.0405497
$794$	0	0
$795$	−73.1012	−2.59263
$796$	0	0
$797$	−21.0148	−0.744382	−0.372191	−	0.928156i	$-0.621393\pi$
$797$	−21.0148	−0.744382	−0.372191	+	0.928156i	$0.621393\pi$
$798$	0	0
$799$	8.24014	0.291515
$800$	0	0
$801$	51.3024	1.81268
$802$	0	0
$803$	2.28447	0.0806172
$804$	0	0
$805$	1.68712	0.0594632
$806$	0	0
$807$	−11.4990	−0.404783
$808$	0	0
$809$	−22.8004	−0.801619	−0.400809	−	0.916161i	$-0.631271\pi$
$809$	−22.8004	−0.801619	−0.400809	+	0.916161i	$0.631271\pi$
$810$	0	0
$811$	−0.488666	−0.0171594	−0.00857970	−	0.999963i	$-0.502731\pi$
$811$	−0.488666	−0.0171594	−0.00857970	+	0.999963i	$0.502731\pi$
$812$	0	0
$813$	−14.5184	−0.509182
$814$	0	0
$815$	14.1724	0.496437
$816$	0	0
$817$	−74.4054	−2.60312
$818$	0	0
$819$	−30.1962	−1.05514
$820$	0	0
$821$	4.77694	0.166716	0.0833581	−	0.996520i	$-0.473435\pi$
$821$	4.77694	0.166716	0.0833581	+	0.996520i	$0.473435\pi$
$822$	0	0
$823$	−1.28003	−0.0446191	−0.0223096	−	0.999751i	$-0.507102\pi$
$823$	−1.28003	−0.0446191	−0.0223096	+	0.999751i	$0.507102\pi$
$824$	0	0
$825$	−6.87608	−0.239394
$826$	0	0
$827$	−9.53963	−0.331725	−0.165863	−	0.986149i	$-0.553041\pi$
$827$	−9.53963	−0.331725	−0.165863	+	0.986149i	$0.553041\pi$
$828$	0	0
$829$	18.7436	0.650993	0.325496	−	0.945543i	$-0.394469\pi$
$829$	18.7436	0.650993	0.325496	+	0.945543i	$0.394469\pi$
$830$	0	0
$831$	79.2368	2.74869
$832$	0	0
$833$	−1.50568	−0.0521687
$834$	0	0
$835$	−25.6282	−0.886901
$836$	0	0
$837$	−69.3353	−2.39658
$838$	0	0
$839$	−41.3275	−1.42678	−0.713392	−	0.700765i	$-0.752842\pi$
$839$	−41.3275	−1.42678	−0.713392	+	0.700765i	$0.752842\pi$
$840$	0	0
$841$	−25.0248	−0.862923
$842$	0	0
$843$	21.9239	0.755098
$844$	0	0
$845$	13.7457	0.472866
$846$	0	0
$847$	−9.93439	−0.341350
$848$	0	0
$849$	5.22486	0.179317
$850$	0	0
$851$	7.43881	0.254999
$852$	0	0
$853$	−18.2746	−0.625710	−0.312855	−	0.949801i	$-0.601285\pi$
$853$	−18.2746	−0.625710	−0.312855	+	0.949801i	$0.601285\pi$
$854$	0	0
$855$	−84.4683	−2.88876
$856$	0	0
$857$	−8.70477	−0.297349	−0.148675	−	0.988886i	$-0.547501\pi$
$857$	−8.70477	−0.297349	−0.148675	+	0.988886i	$0.547501\pi$
$858$	0	0
$859$	−31.5148	−1.07527	−0.537636	−	0.843177i	$-0.680683\pi$
$859$	−31.5148	−1.07527	−0.537636	+	0.843177i	$0.680683\pi$
$860$	0	0
$861$	7.46245	0.254320
$862$	0	0
$863$	−31.9386	−1.08720	−0.543601	−	0.839344i	$-0.682939\pi$
$863$	−31.9386	−1.08720	−0.543601	+	0.839344i	$0.682939\pi$
$864$	0	0
$865$	−40.3970	−1.37354
$866$	0	0
$867$	−45.5682	−1.54758
$868$	0	0
$869$	6.03405	0.204691
$870$	0	0
$871$	−30.3732	−1.02916
$872$	0	0
$873$	92.3973	3.12717
$874$	0	0
$875$	−12.0690	−0.408008
$876$	0	0
$877$	−18.4725	−0.623772	−0.311886	−	0.950120i	$-0.600961\pi$
$877$	−18.4725	−0.623772	−0.311886	+	0.950120i	$0.600961\pi$
$878$	0	0
$879$	21.1132	0.712130
$880$	0	0
$881$	−33.8108	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$881$	−33.8108	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$882$	0	0
$883$	−6.05996	−0.203934	−0.101967	−	0.994788i	$-0.532514\pi$
$883$	−6.05996	−0.203934	−0.101967	+	0.994788i	$0.532514\pi$
$884$	0	0
$885$	58.0805	1.95235
$886$	0	0
$887$	−31.8696	−1.07008	−0.535038	−	0.844828i	$-0.679703\pi$
$887$	−31.8696	−1.07008	−0.535038	+	0.844828i	$0.679703\pi$
$888$	0	0
$889$	−7.47486	−0.250699
$890$	0	0
$891$	14.8833	0.498611
$892$	0	0
$893$	41.7278	1.39637
$894$	0	0
$895$	14.6159	0.488555
$896$	0	0
$897$	−14.2233	−0.474904
$898$	0	0
$899$	12.5325	0.417983
$900$	0	0
$901$	21.0930	0.702709
$902$	0	0
$903$	30.1824	1.00441
$904$	0	0
$905$	−2.74498	−0.0912461
$906$	0	0
$907$	−31.3215	−1.04001	−0.520007	−	0.854162i	$-0.674071\pi$
$907$	−31.3215	−1.04001	−0.520007	+	0.854162i	$0.674071\pi$
$908$	0	0
$909$	100.587	3.33627
$910$	0	0
$911$	18.9429	0.627607	0.313803	−	0.949488i	$-0.398397\pi$
$911$	18.9429	0.627607	0.313803	+	0.949488i	$0.398397\pi$
$912$	0	0
$913$	−7.74561	−0.256342
$914$	0	0
$915$	−1.29573	−0.0428356
$916$	0	0
$917$	−20.5930	−0.680041
$918$	0	0
$919$	−4.81238	−0.158746	−0.0793729	−	0.996845i	$-0.525292\pi$
$919$	−4.81238	−0.158746	−0.0793729	+	0.996845i	$0.525292\pi$
$920$	0	0
$921$	37.2977	1.22900
$922$	0	0
$923$	16.4965	0.542990
$924$	0	0
$925$	−16.0204	−0.526746
$926$	0	0
$927$	−28.6482	−0.940931
$928$	0	0
$929$	−51.4767	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$929$	−51.4767	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$930$	0	0
$931$	−7.62471	−0.249890
$932$	0	0
$933$	91.6027	2.99894
$934$	0	0
$935$	−2.62228	−0.0857576
$936$	0	0
$937$	17.4514	0.570113	0.285057	−	0.958511i	$-0.407988\pi$
$937$	17.4514	0.570113	0.285057	+	0.958511i	$0.407988\pi$
$938$	0	0
$939$	61.6798	2.01284
$940$	0	0
$941$	4.97254	0.162100	0.0810500	−	0.996710i	$-0.474173\pi$
$941$	4.97254	0.162100	0.0810500	+	0.996710i	$0.474173\pi$
$942$	0	0
$943$	2.41273	0.0785693
$944$	0	0
$945$	18.6099	0.605379
$946$	0	0
$947$	−3.63732	−0.118197	−0.0590985	−	0.998252i	$-0.518823\pi$
$947$	−3.63732	−0.118197	−0.0590985	+	0.998252i	$0.518823\pi$
$948$	0	0
$949$	−10.1769	−0.330356
$950$	0	0
$951$	33.2985	1.07978
$952$	0	0
$953$	−25.8287	−0.836675	−0.418337	−	0.908292i	$-0.637387\pi$
$953$	−25.8287	−0.836675	−0.418337	+	0.908292i	$0.637387\pi$
$954$	0	0
$955$	−12.3323	−0.399064
$956$	0	0
$957$	−6.36580	−0.205777
$958$	0	0
$959$	6.61335	0.213556
$960$	0	0
$961$	8.51070	0.274539
$962$	0	0
$963$	−40.6280	−1.30922
$964$	0	0
$965$	41.3861	1.33227
$966$	0	0
$967$	45.9124	1.47644	0.738221	−	0.674559i	$-0.235666\pi$
$967$	45.9124	1.47644	0.738221	+	0.674559i	$0.235666\pi$
$968$	0	0
$969$	35.5083	1.14069
$970$	0	0
$971$	29.2526	0.938762	0.469381	−	0.882996i	$-0.344477\pi$
$971$	29.2526	0.938762	0.469381	+	0.882996i	$0.344477\pi$
$972$	0	0
$973$	1.98298	0.0635713
$974$	0	0
$975$	30.6317	0.980999
$976$	0	0
$977$	−35.8731	−1.14768	−0.573841	−	0.818967i	$-0.694547\pi$
$977$	−35.8731	−1.14768	−0.573841	+	0.818967i	$0.694547\pi$
$978$	0	0
$979$	8.06515	0.257763
$980$	0	0
$981$	80.0169	2.55474
$982$	0	0
$983$	−57.1687	−1.82340	−0.911699	−	0.410860i	$-0.865229\pi$
$983$	−57.1687	−1.82340	−0.911699	+	0.410860i	$0.865229\pi$
$984$	0	0
$985$	6.70273	0.213567
$986$	0	0
$987$	−16.9268	−0.538786
$988$	0	0
$989$	9.75845	0.310301
$990$	0	0
$991$	41.9142	1.33145	0.665725	−	0.746197i	$-0.268123\pi$
$991$	41.9142	1.33145	0.665725	+	0.746197i	$0.268123\pi$
$992$	0	0
$993$	30.0326	0.953055
$994$	0	0
$995$	8.55055	0.271071
$996$	0	0
$997$	−41.0447	−1.29990	−0.649950	−	0.759977i	$-0.725210\pi$
$997$	−41.0447	−1.29990	−0.649950	+	0.759977i	$0.725210\pi$
$998$	0	0
$999$	82.0541	2.59608

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	644.2.a.c.1.5	✓	5
3.2	odd	2		5796.2.a.s.1.3		5
4.3	odd	2		2576.2.a.bc.1.1		5
7.6	odd	2		4508.2.a.g.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.2.a.c.1.5	✓	5	1.1	even	1	trivial
2576.2.a.bc.1.1		5	4.3	odd	2
4508.2.a.g.1.1		5	7.6	odd	2
5796.2.a.s.1.3		5	3.2	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:	\( N \)	\(=\)	\( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	644.a (trivial)

\(f(q)\)	\(=\)	\(q+3.09295 q^{3} +1.68712 q^{5} +1.00000 q^{7} +6.56635 q^{9} +O(q^{10})\)	\(q+3.09295 q^{3} +1.68712 q^{5} +1.00000 q^{7} +6.56635 q^{9} +1.03228 q^{11} -4.59863 q^{13} +5.21819 q^{15} -1.50568 q^{17} -7.62471 q^{19} +3.09295 q^{21} +1.00000 q^{23} -2.15362 q^{25} +11.0305 q^{27} -1.99380 q^{29} -6.28575 q^{31} +3.19280 q^{33} +1.68712 q^{35} +7.43881 q^{37} -14.2233 q^{39} +2.41273 q^{41} +9.75845 q^{43} +11.0782 q^{45} -5.47270 q^{47} +1.00000 q^{49} -4.65700 q^{51} -14.0089 q^{53} +1.74159 q^{55} -23.5829 q^{57} +11.1304 q^{59} -0.248311 q^{61} +6.56635 q^{63} -7.75845 q^{65} +6.60483 q^{67} +3.09295 q^{69} -3.58727 q^{71} +2.21303 q^{73} -6.66104 q^{75} +1.03228 q^{77} +5.84534 q^{79} +14.4179 q^{81} -7.50338 q^{83} -2.54027 q^{85} -6.16672 q^{87} +7.81292 q^{89} -4.59863 q^{91} -19.4415 q^{93} -12.8638 q^{95} +14.0713 q^{97} +6.77833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} + 4 q^{5} + 5 q^{7} + 10 q^{9}+O(q^{10}) \) 5 * q + q^3 + 4 * q^5 + 5 * q^7 + 10 * q^9	\( 5 q + q^{3} + 4 q^{5} + 5 q^{7} + 10 q^{9} + 2 q^{11} + 3 q^{13} - 6 q^{15} + 4 q^{17} + 6 q^{19} + q^{21} + 5 q^{23} + 15 q^{25} + q^{27} + 5 q^{29} - q^{31} + 4 q^{35} + 22 q^{37} - q^{39} + 15 q^{41} + 12 q^{43} + 36 q^{45} - 21 q^{47} + 5 q^{49} + 24 q^{51} + 2 q^{53} - 22 q^{55} - 34 q^{57} - 12 q^{61} + 10 q^{63} - 2 q^{65} + 22 q^{67} + q^{69} - 15 q^{71} + 17 q^{73} - 47 q^{75} + 2 q^{77} + 2 q^{79} + 37 q^{81} - 16 q^{83} - 8 q^{85} - 33 q^{87} - 24 q^{89} + 3 q^{91} + 5 q^{93} - 8 q^{95} + 38 q^{97} - 36 q^{99}+O(q^{100}) \) 5 * q + q^3 + 4 * q^5 + 5 * q^7 + 10 * q^9 + 2 * q^11 + 3 * q^13 - 6 * q^15 + 4 * q^17 + 6 * q^19 + q^21 + 5 * q^23 + 15 * q^25 + q^27 + 5 * q^29 - q^31 + 4 * q^35 + 22 * q^37 - q^39 + 15 * q^41 + 12 * q^43 + 36 * q^45 - 21 * q^47 + 5 * q^49 + 24 * q^51 + 2 * q^53 - 22 * q^55 - 34 * q^57 - 12 * q^61 + 10 * q^63 - 2 * q^65 + 22 * q^67 + q^69 - 15 * q^71 + 17 * q^73 - 47 * q^75 + 2 * q^77 + 2 * q^79 + 37 * q^81 - 16 * q^83 - 8 * q^85 - 33 * q^87 - 24 * q^89 + 3 * q^91 + 5 * q^93 - 8 * q^95 + 38 * q^97 - 36 * q^99

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