LMFDB - Embedded newform 5200.2.a.cb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5200,2,Mod(1,5200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5200 = 2^{4} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5200.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:	$41.5222090511$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	5200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.17009 q^{3} +1.70928 q^{7} +7.04945 q^{9} +O(q^{10})$	$q-3.17009 q^{3} +1.70928 q^{7} +7.04945 q^{9} +2.53919 q^{11} -1.00000 q^{13} +0.921622 q^{17} +0.539189 q^{19} -5.41855 q^{21} -2.82991 q^{23} -12.8371 q^{27} -5.12783 q^{29} -0.879362 q^{31} -8.04945 q^{33} -6.04945 q^{37} +3.17009 q^{39} +1.26180 q^{41} -6.43188 q^{43} -5.70928 q^{47} -4.07838 q^{49} -2.92162 q^{51} +8.49693 q^{53} -1.70928 q^{57} +4.72261 q^{59} +8.04945 q^{61} +12.0494 q^{63} -7.86603 q^{67} +8.97107 q^{69} +14.4813 q^{71} -1.95055 q^{73} +4.34017 q^{77} -0.496928 q^{79} +19.5464 q^{81} -8.63090 q^{83} +16.2557 q^{87} +12.8371 q^{89} -1.70928 q^{91} +2.78765 q^{93} +5.91548 q^{97} +17.8999 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 4 * q^3 - 2 * q^7 + 3 * q^9	$ 3 q - 4 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{11} - 3 q^{13} + 6 q^{17} - 2 q^{21} - 14 q^{23} - 10 q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{33} + 4 q^{39} - 4 q^{41} - 6 q^{43} - 10 q^{47} - 9 q^{49} - 12 q^{51} + 8 q^{53} + 2 q^{57} + 8 q^{59} + 6 q^{61} + 18 q^{63} - 10 q^{67} + 12 q^{69} + 12 q^{71} - 24 q^{73} + 2 q^{77} + 16 q^{79} + 23 q^{81} - 22 q^{83} + 6 q^{87} + 10 q^{89} + 2 q^{91} - 2 q^{93} - 14 q^{97} + 8 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 - 2 * q^7 + 3 * q^9 + 6 * q^11 - 3 * q^13 + 6 * q^17 - 2 * q^21 - 14 * q^23 - 10 * q^27 + 6 * q^29 + 10 * q^31 - 6 * q^33 + 4 * q^39 - 4 * q^41 - 6 * q^43 - 10 * q^47 - 9 * q^49 - 12 * q^51 + 8 * q^53 + 2 * q^57 + 8 * q^59 + 6 * q^61 + 18 * q^63 - 10 * q^67 + 12 * q^69 + 12 * q^71 - 24 * q^73 + 2 * q^77 + 16 * q^79 + 23 * q^81 - 22 * q^83 + 6 * q^87 + 10 * q^89 + 2 * q^91 - 2 * q^93 - 14 * q^97 + 8 * 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
$3$	−3.17009	−1.83025	−0.915125	−	0.403170i	$-0.867908\pi$
$3$	−3.17009	−1.83025	−0.915125	+	0.403170i	$0.867908\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.70928	0.646045	0.323023	−	0.946391i	$-0.395301\pi$
$7$	1.70928	0.646045	0.323023	+	0.946391i	$0.395301\pi$
$8$	0	0
$9$	7.04945	2.34982
$10$	0	0
$11$	2.53919	0.765594	0.382797	−	0.923832i	$-0.374961\pi$
$11$	2.53919	0.765594	0.382797	+	0.923832i	$0.374961\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.921622	0.223526	0.111763	−	0.993735i	$-0.464350\pi$
$17$	0.921622	0.223526	0.111763	+	0.993735i	$0.464350\pi$
$18$	0	0
$19$	0.539189	0.123698	0.0618492	−	0.998086i	$-0.480300\pi$
$19$	0.539189	0.123698	0.0618492	+	0.998086i	$0.480300\pi$
$20$	0	0
$21$	−5.41855	−1.18242
$22$	0	0
$23$	−2.82991	−0.590078	−0.295039	−	0.955485i	$-0.595333\pi$
$23$	−2.82991	−0.590078	−0.295039	+	0.955485i	$0.595333\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−12.8371	−2.47050
$28$	0	0
$29$	−5.12783	−0.952213	−0.476107	−	0.879388i	$-0.657952\pi$
$29$	−5.12783	−0.952213	−0.476107	+	0.879388i	$0.657952\pi$
$30$	0	0
$31$	−0.879362	−0.157938	−0.0789690	−	0.996877i	$-0.525163\pi$
$31$	−0.879362	−0.157938	−0.0789690	+	0.996877i	$0.525163\pi$
$32$	0	0
$33$	−8.04945	−1.40123
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.04945	−0.994523	−0.497262	−	0.867601i	$-0.665661\pi$
$37$	−6.04945	−0.994523	−0.497262	+	0.867601i	$0.665661\pi$
$38$	0	0
$39$	3.17009	0.507620
$40$	0	0
$41$	1.26180	0.197059	0.0985297	−	0.995134i	$-0.468586\pi$
$41$	1.26180	0.197059	0.0985297	+	0.995134i	$0.468586\pi$
$42$	0	0
$43$	−6.43188	−0.980853	−0.490426	−	0.871483i	$-0.663159\pi$
$43$	−6.43188	−0.980853	−0.490426	+	0.871483i	$0.663159\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.70928	−0.832783	−0.416392	−	0.909185i	$-0.636705\pi$
$47$	−5.70928	−0.832783	−0.416392	+	0.909185i	$0.636705\pi$
$48$	0	0
$49$	−4.07838	−0.582625
$50$	0	0
$51$	−2.92162	−0.409109
$52$	0	0
$53$	8.49693	1.16714	0.583571	−	0.812062i	$-0.301655\pi$
$53$	8.49693	1.16714	0.583571	+	0.812062i	$0.301655\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.70928	−0.226399
$58$	0	0
$59$	4.72261	0.614831	0.307415	−	0.951575i	$-0.400536\pi$
$59$	4.72261	0.614831	0.307415	+	0.951575i	$0.400536\pi$
$60$	0	0
$61$	8.04945	1.03063	0.515313	−	0.857002i	$-0.327676\pi$
$61$	8.04945	1.03063	0.515313	+	0.857002i	$0.327676\pi$
$62$	0	0
$63$	12.0494	1.51809
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.86603	−0.960989	−0.480494	−	0.876998i	$-0.659543\pi$
$67$	−7.86603	−0.960989	−0.480494	+	0.876998i	$0.659543\pi$
$68$	0	0
$69$	8.97107	1.07999
$70$	0	0
$71$	14.4813	1.71862	0.859309	−	0.511457i	$-0.170894\pi$
$71$	14.4813	1.71862	0.859309	+	0.511457i	$0.170894\pi$
$72$	0	0
$73$	−1.95055	−0.228295	−0.114147	−	0.993464i	$-0.536414\pi$
$73$	−1.95055	−0.228295	−0.114147	+	0.993464i	$0.536414\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.34017	0.494609
$78$	0	0
$79$	−0.496928	−0.0559088	−0.0279544	−	0.999609i	$-0.508899\pi$
$79$	−0.496928	−0.0559088	−0.0279544	+	0.999609i	$0.508899\pi$
$80$	0	0
$81$	19.5464	2.17182
$82$	0	0
$83$	−8.63090	−0.947364	−0.473682	−	0.880696i	$-0.657075\pi$
$83$	−8.63090	−0.947364	−0.473682	+	0.880696i	$0.657075\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	16.2557	1.74279
$88$	0	0
$89$	12.8371	1.36073	0.680365	−	0.732873i	$-0.261821\pi$
$89$	12.8371	1.36073	0.680365	+	0.732873i	$0.261821\pi$
$90$	0	0
$91$	−1.70928	−0.179181
$92$	0	0
$93$	2.78765	0.289066
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.91548	0.600626	0.300313	−	0.953841i	$-0.402909\pi$
$97$	5.91548	0.600626	0.300313	+	0.953841i	$0.402909\pi$
$98$	0	0
$99$	17.8999	1.79901
$100$	0	0
$101$	−16.4391	−1.63575	−0.817874	−	0.575397i	$-0.804848\pi$
$101$	−16.4391	−1.63575	−0.817874	+	0.575397i	$0.804848\pi$
$102$	0	0
$103$	10.1906	1.00411	0.502055	−	0.864836i	$-0.332577\pi$
$103$	10.1906	1.00411	0.502055	+	0.864836i	$0.332577\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.75154	−0.942717	−0.471358	−	0.881942i	$-0.656236\pi$
$107$	−9.75154	−0.942717	−0.471358	+	0.881942i	$0.656236\pi$
$108$	0	0
$109$	−16.8638	−1.61526	−0.807628	−	0.589693i	$-0.799249\pi$
$109$	−16.8638	−1.61526	−0.807628	+	0.589693i	$0.799249\pi$
$110$	0	0
$111$	19.1773	1.82023
$112$	0	0
$113$	−11.7587	−1.10617	−0.553084	−	0.833126i	$-0.686549\pi$
$113$	−11.7587	−1.10617	−0.553084	+	0.833126i	$0.686549\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.04945	−0.651722
$118$	0	0
$119$	1.57531	0.144408
$120$	0	0
$121$	−4.55252	−0.413865
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.0072	1.59788	0.798940	−	0.601411i	$-0.205395\pi$
$127$	18.0072	1.59788	0.798940	+	0.601411i	$0.205395\pi$
$128$	0	0
$129$	20.3896	1.79521
$130$	0	0
$131$	−14.2557	−1.24552	−0.622761	−	0.782412i	$-0.713989\pi$
$131$	−14.2557	−1.24552	−0.622761	+	0.782412i	$0.713989\pi$
$132$	0	0
$133$	0.921622	0.0799148
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.7854	1.17776	0.588882	−	0.808219i	$-0.299568\pi$
$137$	13.7854	1.17776	0.588882	+	0.808219i	$0.299568\pi$
$138$	0	0
$139$	6.65368	0.564358	0.282179	−	0.959362i	$-0.408943\pi$
$139$	6.65368	0.564358	0.282179	+	0.959362i	$0.408943\pi$
$140$	0	0
$141$	18.0989	1.52420
$142$	0	0
$143$	−2.53919	−0.212338
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.9288	1.06635
$148$	0	0
$149$	−9.07838	−0.743730	−0.371865	−	0.928287i	$-0.621282\pi$
$149$	−9.07838	−0.743730	−0.371865	+	0.928287i	$0.621282\pi$
$150$	0	0
$151$	−3.27739	−0.266711	−0.133355	−	0.991068i	$-0.542575\pi$
$151$	−3.27739	−0.266711	−0.133355	+	0.991068i	$0.542575\pi$
$152$	0	0
$153$	6.49693	0.525246
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.8371	1.02451	0.512256	−	0.858833i	$-0.328810\pi$
$157$	12.8371	1.02451	0.512256	+	0.858833i	$0.328810\pi$
$158$	0	0
$159$	−26.9360	−2.13616
$160$	0	0
$161$	−4.83710	−0.381217
$162$	0	0
$163$	−12.0494	−0.943786	−0.471893	−	0.881656i	$-0.656429\pi$
$163$	−12.0494	−0.943786	−0.471893	+	0.881656i	$0.656429\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.72979	−0.675532	−0.337766	−	0.941230i	$-0.609671\pi$
$167$	−8.72979	−0.675532	−0.337766	+	0.941230i	$0.609671\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.80098	0.290669
$172$	0	0
$173$	0.863763	0.0656707	0.0328354	−	0.999461i	$-0.489546\pi$
$173$	0.863763	0.0656707	0.0328354	+	0.999461i	$0.489546\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.9711	−1.12529
$178$	0	0
$179$	−19.9155	−1.48855	−0.744276	−	0.667872i	$-0.767205\pi$
$179$	−19.9155	−1.48855	−0.744276	+	0.667872i	$0.767205\pi$
$180$	0	0
$181$	14.3896	1.06957	0.534786	−	0.844987i	$-0.320392\pi$
$181$	14.3896	1.06957	0.534786	+	0.844987i	$0.320392\pi$
$182$	0	0
$183$	−25.5174	−1.88630
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.34017	0.171130
$188$	0	0
$189$	−21.9421	−1.59606
$190$	0	0
$191$	−1.47641	−0.106829	−0.0534146	−	0.998572i	$-0.517010\pi$
$191$	−1.47641	−0.106829	−0.0534146	+	0.998572i	$0.517010\pi$
$192$	0	0
$193$	−17.7321	−1.27638	−0.638191	−	0.769878i	$-0.720317\pi$
$193$	−17.7321	−1.27638	−0.638191	+	0.769878i	$0.720317\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	5.39189	0.382221	0.191110	−	0.981569i	$-0.438791\pi$
$199$	5.39189	0.382221	0.191110	+	0.981569i	$0.438791\pi$
$200$	0	0
$201$	24.9360	1.75885
$202$	0	0
$203$	−8.76487	−0.615173
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−19.9493	−1.38657
$208$	0	0
$209$	1.36910	0.0947028
$210$	0	0
$211$	22.7526	1.56635	0.783176	−	0.621800i	$-0.213598\pi$
$211$	22.7526	1.56635	0.783176	+	0.621800i	$0.213598\pi$
$212$	0	0
$213$	−45.9071	−3.14550
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.50307	−0.102035
$218$	0	0
$219$	6.18342	0.417837
$220$	0	0
$221$	−0.921622	−0.0619950
$222$	0	0
$223$	−8.76099	−0.586679	−0.293340	−	0.956008i	$-0.594767\pi$
$223$	−8.76099	−0.586679	−0.293340	+	0.956008i	$0.594767\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.2267	1.14338	0.571689	−	0.820470i	$-0.306288\pi$
$227$	17.2267	1.14338	0.571689	+	0.820470i	$0.306288\pi$
$228$	0	0
$229$	3.07838	0.203425	0.101712	−	0.994814i	$-0.467568\pi$
$229$	3.07838	0.203425	0.101712	+	0.994814i	$0.467568\pi$
$230$	0	0
$231$	−13.7587	−0.905258
$232$	0	0
$233$	18.9360	1.24054	0.620269	−	0.784389i	$-0.287023\pi$
$233$	18.9360	1.24054	0.620269	+	0.784389i	$0.287023\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.57531	0.102327
$238$	0	0
$239$	−6.63809	−0.429382	−0.214691	−	0.976682i	$-0.568874\pi$
$239$	−6.63809	−0.429382	−0.214691	+	0.976682i	$0.568874\pi$
$240$	0	0
$241$	−9.47641	−0.610429	−0.305215	−	0.952284i	$-0.598728\pi$
$241$	−9.47641	−0.610429	−0.305215	+	0.952284i	$0.598728\pi$
$242$	0	0
$243$	−23.4524	−1.50447
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.539189	−0.0343078
$248$	0	0
$249$	27.3607	1.73391
$250$	0	0
$251$	−29.4596	−1.85947	−0.929736	−	0.368226i	$-0.879965\pi$
$251$	−29.4596	−1.85947	−0.929736	+	0.368226i	$0.879965\pi$
$252$	0	0
$253$	−7.18568	−0.451760
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.4657	−1.27662	−0.638309	−	0.769781i	$-0.720366\pi$
$257$	−20.4657	−1.27662	−0.638309	+	0.769781i	$0.720366\pi$
$258$	0	0
$259$	−10.3402	−0.642507
$260$	0	0
$261$	−36.1483	−2.23753
$262$	0	0
$263$	−9.14342	−0.563808	−0.281904	−	0.959443i	$-0.590966\pi$
$263$	−9.14342	−0.563808	−0.281904	+	0.959443i	$0.590966\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−40.6947	−2.49048
$268$	0	0
$269$	11.3919	0.694576	0.347288	−	0.937759i	$-0.387103\pi$
$269$	11.3919	0.694576	0.347288	+	0.937759i	$0.387103\pi$
$270$	0	0
$271$	21.1350	1.28386	0.641930	−	0.766763i	$-0.278134\pi$
$271$	21.1350	1.28386	0.641930	+	0.766763i	$0.278134\pi$
$272$	0	0
$273$	5.41855	0.327946
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.0784	−0.785804	−0.392902	−	0.919580i	$-0.628529\pi$
$277$	−13.0784	−0.785804	−0.392902	+	0.919580i	$0.628529\pi$
$278$	0	0
$279$	−6.19902	−0.371125
$280$	0	0
$281$	−0.680346	−0.0405860	−0.0202930	−	0.999794i	$-0.506460\pi$
$281$	−0.680346	−0.0405860	−0.0202930	+	0.999794i	$0.506460\pi$
$282$	0	0
$283$	19.2956	1.14701	0.573504	−	0.819203i	$-0.305584\pi$
$283$	19.2956	1.14701	0.573504	+	0.819203i	$0.305584\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.15676	0.127309
$288$	0	0
$289$	−16.1506	−0.950036
$290$	0	0
$291$	−18.7526	−1.09930
$292$	0	0
$293$	−9.46800	−0.553126	−0.276563	−	0.960996i	$-0.589196\pi$
$293$	−9.46800	−0.553126	−0.276563	+	0.960996i	$0.589196\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−32.5958	−1.89140
$298$	0	0
$299$	2.82991	0.163658
$300$	0	0
$301$	−10.9939	−0.633675
$302$	0	0
$303$	52.1133	2.99383
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.264063	0.0150709	0.00753543	−	0.999972i	$-0.497601\pi$
$307$	0.264063	0.0150709	0.00753543	+	0.999972i	$0.497601\pi$
$308$	0	0
$309$	−32.3051	−1.83777
$310$	0	0
$311$	−13.0472	−0.739838	−0.369919	−	0.929064i	$-0.620615\pi$
$311$	−13.0472	−0.739838	−0.369919	+	0.929064i	$0.620615\pi$
$312$	0	0
$313$	−33.7009	−1.90489	−0.952443	−	0.304718i	$-0.901438\pi$
$313$	−33.7009	−1.90489	−0.952443	+	0.304718i	$0.901438\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.9506	−0.783541	−0.391771	−	0.920063i	$-0.628137\pi$
$317$	−13.9506	−0.783541	−0.391771	+	0.920063i	$0.628137\pi$
$318$	0	0
$319$	−13.0205	−0.729009
$320$	0	0
$321$	30.9132	1.72541
$322$	0	0
$323$	0.496928	0.0276498
$324$	0	0
$325$	0	0
$326$	0	0
$327$	53.4596	2.95632
$328$	0	0
$329$	−9.75872	−0.538016
$330$	0	0
$331$	18.4547	1.01436	0.507180	−	0.861840i	$-0.330688\pi$
$331$	18.4547	1.01436	0.507180	+	0.861840i	$0.330688\pi$
$332$	0	0
$333$	−42.6453	−2.33695
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−15.8576	−0.863820	−0.431910	−	0.901917i	$-0.642160\pi$
$337$	−15.8576	−0.863820	−0.431910	+	0.901917i	$0.642160\pi$
$338$	0	0
$339$	37.2762	2.02456
$340$	0	0
$341$	−2.23287	−0.120916
$342$	0	0
$343$	−18.9360	−1.02245
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.72487	−0.522059	−0.261029	−	0.965331i	$-0.584062\pi$
$347$	−9.72487	−0.522059	−0.261029	+	0.965331i	$0.584062\pi$
$348$	0	0
$349$	−30.9093	−1.65454	−0.827269	−	0.561805i	$-0.810107\pi$
$349$	−30.9093	−1.65454	−0.827269	+	0.561805i	$0.810107\pi$
$350$	0	0
$351$	12.8371	0.685194
$352$	0	0
$353$	5.95055	0.316716	0.158358	−	0.987382i	$-0.449380\pi$
$353$	5.95055	0.316716	0.158358	+	0.987382i	$0.449380\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.99386	−0.264303
$358$	0	0
$359$	−10.9783	−0.579410	−0.289705	−	0.957116i	$-0.593557\pi$
$359$	−10.9783	−0.579410	−0.289705	+	0.957116i	$0.593557\pi$
$360$	0	0
$361$	−18.7093	−0.984699
$362$	0	0
$363$	14.4319	0.757477
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.3740	−0.541520	−0.270760	−	0.962647i	$-0.587275\pi$
$367$	−10.3740	−0.541520	−0.270760	+	0.962647i	$0.587275\pi$
$368$	0	0
$369$	8.89496	0.463053
$370$	0	0
$371$	14.5236	0.754027
$372$	0	0
$373$	23.9877	1.24204	0.621018	−	0.783796i	$-0.286719\pi$
$373$	23.9877	1.24204	0.621018	+	0.783796i	$0.286719\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.12783	0.264096
$378$	0	0
$379$	−29.7575	−1.52854	−0.764270	−	0.644896i	$-0.776901\pi$
$379$	−29.7575	−1.52854	−0.764270	+	0.644896i	$0.776901\pi$
$380$	0	0
$381$	−57.0843	−2.92452
$382$	0	0
$383$	−12.4163	−0.634442	−0.317221	−	0.948352i	$-0.602750\pi$
$383$	−12.4163	−0.634442	−0.317221	+	0.948352i	$0.602750\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−45.3412	−2.30482
$388$	0	0
$389$	−16.8371	−0.853675	−0.426837	−	0.904328i	$-0.640372\pi$
$389$	−16.8371	−0.853675	−0.426837	+	0.904328i	$0.640372\pi$
$390$	0	0
$391$	−2.60811	−0.131898
$392$	0	0
$393$	45.1917	2.27962
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.89269	−0.195369	−0.0976843	−	0.995217i	$-0.531144\pi$
$397$	−3.89269	−0.195369	−0.0976843	+	0.995217i	$0.531144\pi$
$398$	0	0
$399$	−2.92162	−0.146264
$400$	0	0
$401$	−9.10504	−0.454684	−0.227342	−	0.973815i	$-0.573004\pi$
$401$	−9.10504	−0.454684	−0.227342	+	0.973815i	$0.573004\pi$
$402$	0	0
$403$	0.879362	0.0438041
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.3607	−0.761401
$408$	0	0
$409$	−19.4186	−0.960186	−0.480093	−	0.877218i	$-0.659397\pi$
$409$	−19.4186	−0.960186	−0.480093	+	0.877218i	$0.659397\pi$
$410$	0	0
$411$	−43.7009	−2.15560
$412$	0	0
$413$	8.07223	0.397209
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−21.0928	−1.03292
$418$	0	0
$419$	16.7792	0.819720	0.409860	−	0.912149i	$-0.365578\pi$
$419$	16.7792	0.819720	0.409860	+	0.912149i	$0.365578\pi$
$420$	0	0
$421$	−19.0205	−0.927003	−0.463502	−	0.886096i	$-0.653407\pi$
$421$	−19.0205	−0.927003	−0.463502	+	0.886096i	$0.653407\pi$
$422$	0	0
$423$	−40.2472	−1.95689
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.7587	0.665831
$428$	0	0
$429$	8.04945	0.388631
$430$	0	0
$431$	−8.02997	−0.386790	−0.193395	−	0.981121i	$-0.561950\pi$
$431$	−8.02997	−0.386790	−0.193395	+	0.981121i	$0.561950\pi$
$432$	0	0
$433$	13.0472	0.627008	0.313504	−	0.949587i	$-0.398497\pi$
$433$	13.0472	0.627008	0.313504	+	0.949587i	$0.398497\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.52586	−0.0729917
$438$	0	0
$439$	7.70086	0.367542	0.183771	−	0.982969i	$-0.441169\pi$
$439$	7.70086	0.367542	0.183771	+	0.982969i	$0.441169\pi$
$440$	0	0
$441$	−28.7503	−1.36906
$442$	0	0
$443$	−6.39084	−0.303638	−0.151819	−	0.988408i	$-0.548513\pi$
$443$	−6.39084	−0.303638	−0.151819	+	0.988408i	$0.548513\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	28.7792	1.36121
$448$	0	0
$449$	31.6163	1.49207	0.746034	−	0.665908i	$-0.231956\pi$
$449$	31.6163	1.49207	0.746034	+	0.665908i	$0.231956\pi$
$450$	0	0
$451$	3.20394	0.150867
$452$	0	0
$453$	10.3896	0.488147
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.6430	1.66731	0.833655	−	0.552286i	$-0.186244\pi$
$457$	35.6430	1.66731	0.833655	+	0.552286i	$0.186244\pi$
$458$	0	0
$459$	−11.8310	−0.552222
$460$	0	0
$461$	−14.9795	−0.697664	−0.348832	−	0.937185i	$-0.613422\pi$
$461$	−14.9795	−0.697664	−0.348832	+	0.937185i	$0.613422\pi$
$462$	0	0
$463$	−9.09663	−0.422756	−0.211378	−	0.977404i	$-0.567795\pi$
$463$	−9.09663	−0.422756	−0.211378	+	0.977404i	$0.567795\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.87709	0.0868616	0.0434308	−	0.999056i	$-0.486171\pi$
$467$	1.87709	0.0868616	0.0434308	+	0.999056i	$0.486171\pi$
$468$	0	0
$469$	−13.4452	−0.620842
$470$	0	0
$471$	−40.6947	−1.87511
$472$	0	0
$473$	−16.3318	−0.750935
$474$	0	0
$475$	0	0
$476$	0	0
$477$	59.8987	2.74257
$478$	0	0
$479$	15.7431	0.719322	0.359661	−	0.933083i	$-0.382892\pi$
$479$	15.7431	0.719322	0.359661	+	0.933083i	$0.382892\pi$
$480$	0	0
$481$	6.04945	0.275831
$482$	0	0
$483$	15.3340	0.697723
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.94441	0.224053	0.112026	−	0.993705i	$-0.464266\pi$
$487$	4.94441	0.224053	0.112026	+	0.993705i	$0.464266\pi$
$488$	0	0
$489$	38.1978	1.72736
$490$	0	0
$491$	−39.4863	−1.78199	−0.890995	−	0.454014i	$-0.849992\pi$
$491$	−39.4863	−1.78199	−0.890995	+	0.454014i	$0.849992\pi$
$492$	0	0
$493$	−4.72592	−0.212845
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.7526	1.11030
$498$	0	0
$499$	−1.67089	−0.0747993	−0.0373997	−	0.999300i	$-0.511907\pi$
$499$	−1.67089	−0.0747993	−0.0373997	+	0.999300i	$0.511907\pi$
$500$	0	0
$501$	27.6742	1.23639
$502$	0	0
$503$	9.08557	0.405105	0.202553	−	0.979271i	$-0.435076\pi$
$503$	9.08557	0.405105	0.202553	+	0.979271i	$0.435076\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.17009	−0.140788
$508$	0	0
$509$	−19.5441	−0.866277	−0.433139	−	0.901327i	$-0.642594\pi$
$509$	−19.5441	−0.866277	−0.433139	+	0.901327i	$0.642594\pi$
$510$	0	0
$511$	−3.33403	−0.147489
$512$	0	0
$513$	−6.92162	−0.305597
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−14.4969	−0.637574
$518$	0	0
$519$	−2.73820	−0.120194
$520$	0	0
$521$	6.50534	0.285004	0.142502	−	0.989795i	$-0.454485\pi$
$521$	6.50534	0.285004	0.142502	+	0.989795i	$0.454485\pi$
$522$	0	0
$523$	36.5452	1.59801	0.799004	−	0.601326i	$-0.205361\pi$
$523$	36.5452	1.59801	0.799004	+	0.601326i	$0.205361\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.810439	−0.0353033
$528$	0	0
$529$	−14.9916	−0.651808
$530$	0	0
$531$	33.2918	1.44474
$532$	0	0
$533$	−1.26180	−0.0546544
$534$	0	0
$535$	0	0
$536$	0	0
$537$	63.1338	2.72442
$538$	0	0
$539$	−10.3558	−0.446055
$540$	0	0
$541$	20.3402	0.874492	0.437246	−	0.899342i	$-0.355954\pi$
$541$	20.3402	0.874492	0.437246	+	0.899342i	$0.355954\pi$
$542$	0	0
$543$	−45.6163	−1.95758
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−11.5948	−0.495757	−0.247879	−	0.968791i	$-0.579733\pi$
$547$	−11.5948	−0.495757	−0.247879	+	0.968791i	$0.579733\pi$
$548$	0	0
$549$	56.7442	2.42178
$550$	0	0
$551$	−2.76487	−0.117787
$552$	0	0
$553$	−0.849388	−0.0361196
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.7298	−0.454636	−0.227318	−	0.973821i	$-0.572996\pi$
$557$	−10.7298	−0.454636	−0.227318	+	0.973821i	$0.572996\pi$
$558$	0	0
$559$	6.43188	0.272040
$560$	0	0
$561$	−7.41855	−0.313211
$562$	0	0
$563$	10.2485	0.431921	0.215961	−	0.976402i	$-0.430712\pi$
$563$	10.2485	0.431921	0.215961	+	0.976402i	$0.430712\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	33.4101	1.40309
$568$	0	0
$569$	−8.84551	−0.370823	−0.185412	−	0.982661i	$-0.559362\pi$
$569$	−8.84551	−0.370823	−0.185412	+	0.982661i	$0.559362\pi$
$570$	0	0
$571$	−9.29299	−0.388900	−0.194450	−	0.980912i	$-0.562292\pi$
$571$	−9.29299	−0.388900	−0.194450	+	0.980912i	$0.562292\pi$
$572$	0	0
$573$	4.68035	0.195524
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19.5259	0.812872	0.406436	−	0.913679i	$-0.366771\pi$
$577$	19.5259	0.812872	0.406436	+	0.913679i	$0.366771\pi$
$578$	0	0
$579$	56.2122	2.33610
$580$	0	0
$581$	−14.7526	−0.612040
$582$	0	0
$583$	21.5753	0.893558
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.5029	0.928794	0.464397	−	0.885627i	$-0.346271\pi$
$587$	22.5029	0.928794	0.464397	+	0.885627i	$0.346271\pi$
$588$	0	0
$589$	−0.474142	−0.0195367
$590$	0	0
$591$	−6.34017	−0.260800
$592$	0	0
$593$	−4.43907	−0.182291	−0.0911454	−	0.995838i	$-0.529053\pi$
$593$	−4.43907	−0.182291	−0.0911454	+	0.995838i	$0.529053\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.0928	−0.699560
$598$	0	0
$599$	−33.3607	−1.36308	−0.681540	−	0.731780i	$-0.738690\pi$
$599$	−33.3607	−1.36308	−0.681540	+	0.731780i	$0.738690\pi$
$600$	0	0
$601$	13.3197	0.543320	0.271660	−	0.962393i	$-0.412427\pi$
$601$	13.3197	0.543320	0.271660	+	0.962393i	$0.412427\pi$
$602$	0	0
$603$	−55.4512	−2.25815
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14.1184	0.573047	0.286523	−	0.958073i	$-0.407500\pi$
$607$	14.1184	0.573047	0.286523	+	0.958073i	$0.407500\pi$
$608$	0	0
$609$	27.7854	1.12592
$610$	0	0
$611$	5.70928	0.230973
$612$	0	0
$613$	26.8104	1.08286	0.541432	−	0.840745i	$-0.317882\pi$
$613$	26.8104	1.08286	0.541432	+	0.840745i	$0.317882\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.8950	−0.599649	−0.299824	−	0.953994i	$-0.596928\pi$
$617$	−14.8950	−0.599649	−0.299824	+	0.953994i	$0.596928\pi$
$618$	0	0
$619$	−45.3184	−1.82150	−0.910751	−	0.412956i	$-0.864496\pi$
$619$	−45.3184	−1.82150	−0.910751	+	0.412956i	$0.864496\pi$
$620$	0	0
$621$	36.3279	1.45779
$622$	0	0
$623$	21.9421	0.879093
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.34017	−0.173330
$628$	0	0
$629$	−5.57531	−0.222302
$630$	0	0
$631$	−37.8876	−1.50828	−0.754141	−	0.656713i	$-0.771946\pi$
$631$	−37.8876	−1.50828	−0.754141	+	0.656713i	$0.771946\pi$
$632$	0	0
$633$	−72.1276	−2.86682
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.07838	0.161591
$638$	0	0
$639$	102.085	4.03844
$640$	0	0
$641$	−8.47027	−0.334555	−0.167278	−	0.985910i	$-0.553498\pi$
$641$	−8.47027	−0.334555	−0.167278	+	0.985910i	$0.553498\pi$
$642$	0	0
$643$	34.1750	1.34773	0.673865	−	0.738854i	$-0.264633\pi$
$643$	34.1750	1.34773	0.673865	+	0.738854i	$0.264633\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.8238	−0.543468	−0.271734	−	0.962372i	$-0.587597\pi$
$647$	−13.8238	−0.543468	−0.271734	+	0.962372i	$0.587597\pi$
$648$	0	0
$649$	11.9916	0.470711
$650$	0	0
$651$	4.76487	0.186750
$652$	0	0
$653$	42.8781	1.67795	0.838976	−	0.544169i	$-0.183155\pi$
$653$	42.8781	1.67795	0.838976	+	0.544169i	$0.183155\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.7503	−0.536451
$658$	0	0
$659$	23.2495	0.905672	0.452836	−	0.891594i	$-0.350412\pi$
$659$	23.2495	0.905672	0.452836	+	0.891594i	$0.350412\pi$
$660$	0	0
$661$	27.0661	1.05275	0.526374	−	0.850253i	$-0.323551\pi$
$661$	27.0661	1.05275	0.526374	+	0.850253i	$0.323551\pi$
$662$	0	0
$663$	2.92162	0.113466
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.5113	0.561880
$668$	0	0
$669$	27.7731	1.07377
$670$	0	0
$671$	20.4391	0.789042
$672$	0	0
$673$	−16.1711	−0.623351	−0.311676	−	0.950189i	$-0.600890\pi$
$673$	−16.1711	−0.623351	−0.311676	+	0.950189i	$0.600890\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−43.1194	−1.65721	−0.828607	−	0.559831i	$-0.810866\pi$
$677$	−43.1194	−1.65721	−0.828607	+	0.559831i	$0.810866\pi$
$678$	0	0
$679$	10.1112	0.388032
$680$	0	0
$681$	−54.6102	−2.09267
$682$	0	0
$683$	−17.7093	−0.677627	−0.338813	−	0.940854i	$-0.610026\pi$
$683$	−17.7093	−0.677627	−0.338813	+	0.940854i	$0.610026\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−9.75872	−0.372319
$688$	0	0
$689$	−8.49693	−0.323707
$690$	0	0
$691$	24.8794	0.946456	0.473228	−	0.880940i	$-0.343089\pi$
$691$	24.8794	0.946456	0.473228	+	0.880940i	$0.343089\pi$
$692$	0	0
$693$	30.5958	1.16224
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.16290	0.0440479
$698$	0	0
$699$	−60.0288	−2.27050
$700$	0	0
$701$	33.0661	1.24889	0.624445	−	0.781069i	$-0.285325\pi$
$701$	33.0661	1.24889	0.624445	+	0.781069i	$0.285325\pi$
$702$	0	0
$703$	−3.26180	−0.123021
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−28.0989	−1.05677
$708$	0	0
$709$	2.18342	0.0820000	0.0410000	−	0.999159i	$-0.486946\pi$
$709$	2.18342	0.0820000	0.0410000	+	0.999159i	$0.486946\pi$
$710$	0	0
$711$	−3.50307	−0.131375
$712$	0	0
$713$	2.48852	0.0931957
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.0433	0.785877
$718$	0	0
$719$	−5.20847	−0.194243	−0.0971216	−	0.995273i	$-0.530964\pi$
$719$	−5.20847	−0.194243	−0.0971216	+	0.995273i	$0.530964\pi$
$720$	0	0
$721$	17.4186	0.648701
$722$	0	0
$723$	30.0410	1.11724
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.52464	−0.130721	−0.0653607	−	0.997862i	$-0.520820\pi$
$727$	−3.52464	−0.130721	−0.0653607	+	0.997862i	$0.520820\pi$
$728$	0	0
$729$	15.7070	0.581741
$730$	0	0
$731$	−5.92777	−0.219246
$732$	0	0
$733$	21.8310	0.806345	0.403172	−	0.915124i	$-0.367907\pi$
$733$	21.8310	0.806345	0.403172	+	0.915124i	$0.367907\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.9733	−0.735727
$738$	0	0
$739$	−50.3533	−1.85228	−0.926139	−	0.377184i	$-0.876893\pi$
$739$	−50.3533	−1.85228	−0.926139	+	0.377184i	$0.876893\pi$
$740$	0	0
$741$	1.70928	0.0627918
$742$	0	0
$743$	30.7877	1.12949	0.564745	−	0.825266i	$-0.308975\pi$
$743$	30.7877	1.12949	0.564745	+	0.825266i	$0.308975\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−60.8431	−2.22613
$748$	0	0
$749$	−16.6681	−0.609038
$750$	0	0
$751$	10.6225	0.387620	0.193810	−	0.981039i	$-0.437915\pi$
$751$	10.6225	0.387620	0.193810	+	0.981039i	$0.437915\pi$
$752$	0	0
$753$	93.3894	3.40330
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.98562	0.290242	0.145121	−	0.989414i	$-0.453643\pi$
$757$	7.98562	0.290242	0.145121	+	0.989414i	$0.453643\pi$
$758$	0	0
$759$	22.7792	0.826834
$760$	0	0
$761$	−48.9360	−1.77393	−0.886964	−	0.461838i	$-0.847190\pi$
$761$	−48.9360	−1.77393	−0.886964	+	0.461838i	$0.847190\pi$
$762$	0	0
$763$	−28.8248	−1.04353
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.72261	−0.170523
$768$	0	0
$769$	−7.99547	−0.288324	−0.144162	−	0.989554i	$-0.546049\pi$
$769$	−7.99547	−0.288324	−0.144162	+	0.989554i	$0.546049\pi$
$770$	0	0
$771$	64.8781	2.33653
$772$	0	0
$773$	−26.6141	−0.957242	−0.478621	−	0.878022i	$-0.658863\pi$
$773$	−26.6141	−0.957242	−0.478621	+	0.878022i	$0.658863\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	32.7792	1.17595
$778$	0	0
$779$	0.680346	0.0243759
$780$	0	0
$781$	36.7708	1.31576
$782$	0	0
$783$	65.8264	2.35244
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9.25792	0.330009	0.165005	−	0.986293i	$-0.447236\pi$
$787$	9.25792	0.330009	0.165005	+	0.986293i	$0.447236\pi$
$788$	0	0
$789$	28.9854	1.03191
$790$	0	0
$791$	−20.0989	−0.714634
$792$	0	0
$793$	−8.04945	−0.285844
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.9421	−0.564700	−0.282350	−	0.959312i	$-0.591114\pi$
$797$	−15.9421	−0.564700	−0.282350	+	0.959312i	$0.591114\pi$
$798$	0	0
$799$	−5.26180	−0.186149
$800$	0	0
$801$	90.4945	3.19747
$802$	0	0
$803$	−4.95282	−0.174781
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−36.1133	−1.27125
$808$	0	0
$809$	17.9239	0.630170	0.315085	−	0.949063i	$-0.397967\pi$
$809$	17.9239	0.630170	0.315085	+	0.949063i	$0.397967\pi$
$810$	0	0
$811$	−7.43415	−0.261048	−0.130524	−	0.991445i	$-0.541666\pi$
$811$	−7.43415	−0.261048	−0.130524	+	0.991445i	$0.541666\pi$
$812$	0	0
$813$	−66.9998	−2.34979
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.46800	−0.121330
$818$	0	0
$819$	−12.0494	−0.421042
$820$	0	0
$821$	20.4801	0.714761	0.357380	−	0.933959i	$-0.383670\pi$
$821$	20.4801	0.714761	0.357380	+	0.933959i	$0.383670\pi$
$822$	0	0
$823$	−3.75154	−0.130770	−0.0653852	−	0.997860i	$-0.520828\pi$
$823$	−3.75154	−0.130770	−0.0653852	+	0.997860i	$0.520828\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.1483	−1.67428	−0.837141	−	0.546987i	$-0.815775\pi$
$827$	−48.1483	−1.67428	−0.837141	+	0.546987i	$0.815775\pi$
$828$	0	0
$829$	36.5608	1.26981	0.634904	−	0.772591i	$-0.281040\pi$
$829$	36.5608	1.26981	0.634904	+	0.772591i	$0.281040\pi$
$830$	0	0
$831$	41.4596	1.43822
$832$	0	0
$833$	−3.75872	−0.130232
$834$	0	0
$835$	0	0
$836$	0	0
$837$	11.2885	0.390186
$838$	0	0
$839$	−45.2294	−1.56149	−0.780746	−	0.624849i	$-0.785161\pi$
$839$	−45.2294	−1.56149	−0.780746	+	0.624849i	$0.785161\pi$
$840$	0	0
$841$	−2.70540	−0.0932896
$842$	0	0
$843$	2.15676	0.0742826
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.78151	−0.267376
$848$	0	0
$849$	−61.1689	−2.09931
$850$	0	0
$851$	17.1194	0.586846
$852$	0	0
$853$	37.2534	1.27553	0.637766	−	0.770230i	$-0.279859\pi$
$853$	37.2534	1.27553	0.637766	+	0.770230i	$0.279859\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.8371	−0.370188	−0.185094	−	0.982721i	$-0.559259\pi$
$857$	−10.8371	−0.370188	−0.185094	+	0.982721i	$0.559259\pi$
$858$	0	0
$859$	−14.6081	−0.498422	−0.249211	−	0.968449i	$-0.580171\pi$
$859$	−14.6081	−0.498422	−0.249211	+	0.968449i	$0.580171\pi$
$860$	0	0
$861$	−6.83710	−0.233008
$862$	0	0
$863$	−10.3440	−0.352116	−0.176058	−	0.984380i	$-0.556335\pi$
$863$	−10.3440	−0.352116	−0.176058	+	0.984380i	$0.556335\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	51.1988	1.73880
$868$	0	0
$869$	−1.26180	−0.0428035
$870$	0	0
$871$	7.86603	0.266530
$872$	0	0
$873$	41.7009	1.41136
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.0677	−1.28545	−0.642727	−	0.766095i	$-0.722197\pi$
$877$	−38.0677	−1.28545	−0.642727	+	0.766095i	$0.722197\pi$
$878$	0	0
$879$	30.0144	1.01236
$880$	0	0
$881$	12.0494	0.405956	0.202978	−	0.979183i	$-0.434938\pi$
$881$	12.0494	0.405956	0.202978	+	0.979183i	$0.434938\pi$
$882$	0	0
$883$	−0.320699	−0.0107924	−0.00539619	−	0.999985i	$-0.501718\pi$
$883$	−0.320699	−0.0107924	−0.00539619	+	0.999985i	$0.501718\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.62144	0.121596	0.0607981	−	0.998150i	$-0.480635\pi$
$887$	3.62144	0.121596	0.0607981	+	0.998150i	$0.480635\pi$
$888$	0	0
$889$	30.7792	1.03230
$890$	0	0
$891$	49.6319	1.66273
$892$	0	0
$893$	−3.07838	−0.103014
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.97107	−0.299535
$898$	0	0
$899$	4.50921	0.150391
$900$	0	0
$901$	7.83096	0.260887
$902$	0	0
$903$	34.8515	1.15978
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10.9333	0.363036	0.181518	−	0.983388i	$-0.441899\pi$
$907$	10.9333	0.363036	0.181518	+	0.983388i	$0.441899\pi$
$908$	0	0
$909$	−115.886	−3.84371
$910$	0	0
$911$	37.5897	1.24540	0.622701	−	0.782460i	$-0.286035\pi$
$911$	37.5897	1.24540	0.622701	+	0.782460i	$0.286035\pi$
$912$	0	0
$913$	−21.9155	−0.725297
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−24.3668	−0.804664
$918$	0	0
$919$	−33.6742	−1.11081	−0.555405	−	0.831580i	$-0.687437\pi$
$919$	−33.6742	−1.11081	−0.555405	+	0.831580i	$0.687437\pi$
$920$	0	0
$921$	−0.837101	−0.0275834
$922$	0	0
$923$	−14.4813	−0.476659
$924$	0	0
$925$	0	0
$926$	0	0
$927$	71.8381	2.35947
$928$	0	0
$929$	−43.2039	−1.41748	−0.708738	−	0.705472i	$-0.750735\pi$
$929$	−43.2039	−1.41748	−0.708738	+	0.705472i	$0.750735\pi$
$930$	0	0
$931$	−2.19902	−0.0720698
$932$	0	0
$933$	41.3607	1.35409
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27.5630	0.900445	0.450222	−	0.892916i	$-0.351345\pi$
$937$	27.5630	0.900445	0.450222	+	0.892916i	$0.351345\pi$
$938$	0	0
$939$	106.835	3.48642
$940$	0	0
$941$	58.1666	1.89618	0.948088	−	0.318007i	$-0.103013\pi$
$941$	58.1666	1.89618	0.948088	+	0.318007i	$0.103013\pi$
$942$	0	0
$943$	−3.57077	−0.116280
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.5152	−1.57653	−0.788266	−	0.615335i	$-0.789021\pi$
$947$	−48.5152	−1.57653	−0.788266	+	0.615335i	$0.789021\pi$
$948$	0	0
$949$	1.95055	0.0633176
$950$	0	0
$951$	44.2245	1.43408
$952$	0	0
$953$	−23.0349	−0.746173	−0.373087	−	0.927796i	$-0.621701\pi$
$953$	−23.0349	−0.746173	−0.373087	+	0.927796i	$0.621701\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	41.2762	1.33427
$958$	0	0
$959$	23.5630	0.760890
$960$	0	0
$961$	−30.2267	−0.975056
$962$	0	0
$963$	−68.7429	−2.21521
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−54.9998	−1.76868	−0.884338	−	0.466848i	$-0.845389\pi$
$967$	−54.9998	−1.76868	−0.884338	+	0.466848i	$0.845389\pi$
$968$	0	0
$969$	−1.57531	−0.0506061
$970$	0	0
$971$	−9.70540	−0.311461	−0.155731	−	0.987800i	$-0.549773\pi$
$971$	−9.70540	−0.311461	−0.155731	+	0.987800i	$0.549773\pi$
$972$	0	0
$973$	11.3730	0.364601
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.2062	1.03037	0.515184	−	0.857080i	$-0.327724\pi$
$977$	32.2062	1.03037	0.515184	+	0.857080i	$0.327724\pi$
$978$	0	0
$979$	32.5958	1.04177
$980$	0	0
$981$	−118.880	−3.79555
$982$	0	0
$983$	−44.0782	−1.40588	−0.702938	−	0.711251i	$-0.748129\pi$
$983$	−44.0782	−1.40588	−0.702938	+	0.711251i	$0.748129\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	30.9360	0.984704
$988$	0	0
$989$	18.2017	0.578779
$990$	0	0
$991$	−12.0677	−0.383343	−0.191672	−	0.981459i	$-0.561391\pi$
$991$	−12.0677	−0.383343	−0.191672	+	0.981459i	$0.561391\pi$
$992$	0	0
$993$	−58.5029	−1.85653
$994$	0	0
$995$	0	0
$996$	0	0
$997$	45.7587	1.44919	0.724597	−	0.689173i	$-0.242026\pi$
$997$	45.7587	1.44919	0.724597	+	0.689173i	$0.242026\pi$
$998$	0	0
$999$	77.6574	2.45697

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5200.2.a.cb.1.1		3
4.3	odd	2		325.2.a.k.1.2		3
5.2	odd	4		1040.2.d.c.209.6		6
5.3	odd	4		1040.2.d.c.209.1		6
5.4	even	2		5200.2.a.cj.1.3		3
12.11	even	2		2925.2.a.bf.1.2		3
20.3	even	4		65.2.b.a.14.2	✓	6
20.7	even	4		65.2.b.a.14.5	yes	6
20.19	odd	2		325.2.a.j.1.2		3
52.51	odd	2		4225.2.a.ba.1.2		3
60.23	odd	4		585.2.c.b.469.5		6
60.47	odd	4		585.2.c.b.469.2		6
60.59	even	2		2925.2.a.bj.1.2		3
260.3	even	12		845.2.n.f.529.5		12
260.7	odd	12		845.2.l.e.699.3		12
260.23	even	12		845.2.n.g.529.2		12
260.43	even	12		845.2.n.g.484.5		12
260.47	odd	4		845.2.d.a.844.3		6
260.63	odd	12		845.2.l.d.654.3		12
260.67	odd	12		845.2.l.d.654.4		12
260.83	odd	4		845.2.d.a.844.4		6
260.87	even	12		845.2.n.f.484.5		12
260.103	even	4		845.2.b.c.339.5		6
260.107	even	12		845.2.n.f.529.2		12
260.123	odd	12		845.2.l.e.699.4		12
260.127	even	12		845.2.n.g.529.5		12
260.147	even	12		845.2.n.g.484.2		12
260.163	odd	12		845.2.l.d.699.4		12
260.167	odd	12		845.2.l.e.654.4		12
260.187	odd	4		845.2.d.b.844.3		6
260.203	odd	4		845.2.d.b.844.4		6
260.207	even	4		845.2.b.c.339.2		6
260.223	odd	12		845.2.l.e.654.3		12
260.227	odd	12		845.2.l.d.699.3		12
260.243	even	12		845.2.n.f.484.2		12
260.259	odd	2		4225.2.a.bh.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.b.a.14.2	✓	6	20.3	even	4
65.2.b.a.14.5	yes	6	20.7	even	4
325.2.a.j.1.2		3	20.19	odd	2
325.2.a.k.1.2		3	4.3	odd	2
585.2.c.b.469.2		6	60.47	odd	4
585.2.c.b.469.5		6	60.23	odd	4
845.2.b.c.339.2		6	260.207	even	4
845.2.b.c.339.5		6	260.103	even	4
845.2.d.a.844.3		6	260.47	odd	4
845.2.d.a.844.4		6	260.83	odd	4
845.2.d.b.844.3		6	260.187	odd	4
845.2.d.b.844.4		6	260.203	odd	4
845.2.l.d.654.3		12	260.63	odd	12
845.2.l.d.654.4		12	260.67	odd	12
845.2.l.d.699.3		12	260.227	odd	12
845.2.l.d.699.4		12	260.163	odd	12
845.2.l.e.654.3		12	260.223	odd	12
845.2.l.e.654.4		12	260.167	odd	12
845.2.l.e.699.3		12	260.7	odd	12
845.2.l.e.699.4		12	260.123	odd	12
845.2.n.f.484.2		12	260.243	even	12
845.2.n.f.484.5		12	260.87	even	12
845.2.n.f.529.2		12	260.107	even	12
845.2.n.f.529.5		12	260.3	even	12
845.2.n.g.484.2		12	260.147	even	12
845.2.n.g.484.5		12	260.43	even	12
845.2.n.g.529.2		12	260.23	even	12
845.2.n.g.529.5		12	260.127	even	12
1040.2.d.c.209.1		6	5.3	odd	4
1040.2.d.c.209.6		6	5.2	odd	4
2925.2.a.bf.1.2		3	12.11	even	2
2925.2.a.bj.1.2		3	60.59	even	2
4225.2.a.ba.1.2		3	52.51	odd	2
4225.2.a.bh.1.2		3	260.259	odd	2
5200.2.a.cb.1.1		3	1.1	even	1	trivial
5200.2.a.cj.1.3		3	5.4	even	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:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

\(f(q)\)	\(=\)	\(q-3.17009 q^{3} +1.70928 q^{7} +7.04945 q^{9} +O(q^{10})\)	\(q-3.17009 q^{3} +1.70928 q^{7} +7.04945 q^{9} +2.53919 q^{11} -1.00000 q^{13} +0.921622 q^{17} +0.539189 q^{19} -5.41855 q^{21} -2.82991 q^{23} -12.8371 q^{27} -5.12783 q^{29} -0.879362 q^{31} -8.04945 q^{33} -6.04945 q^{37} +3.17009 q^{39} +1.26180 q^{41} -6.43188 q^{43} -5.70928 q^{47} -4.07838 q^{49} -2.92162 q^{51} +8.49693 q^{53} -1.70928 q^{57} +4.72261 q^{59} +8.04945 q^{61} +12.0494 q^{63} -7.86603 q^{67} +8.97107 q^{69} +14.4813 q^{71} -1.95055 q^{73} +4.34017 q^{77} -0.496928 q^{79} +19.5464 q^{81} -8.63090 q^{83} +16.2557 q^{87} +12.8371 q^{89} -1.70928 q^{91} +2.78765 q^{93} +5.91548 q^{97} +17.8999 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 4 * q^3 - 2 * q^7 + 3 * q^9	\( 3 q - 4 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{11} - 3 q^{13} + 6 q^{17} - 2 q^{21} - 14 q^{23} - 10 q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{33} + 4 q^{39} - 4 q^{41} - 6 q^{43} - 10 q^{47} - 9 q^{49} - 12 q^{51} + 8 q^{53} + 2 q^{57} + 8 q^{59} + 6 q^{61} + 18 q^{63} - 10 q^{67} + 12 q^{69} + 12 q^{71} - 24 q^{73} + 2 q^{77} + 16 q^{79} + 23 q^{81} - 22 q^{83} + 6 q^{87} + 10 q^{89} + 2 q^{91} - 2 q^{93} - 14 q^{97} + 8 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 - 2 * q^7 + 3 * q^9 + 6 * q^11 - 3 * q^13 + 6 * q^17 - 2 * q^21 - 14 * q^23 - 10 * q^27 + 6 * q^29 + 10 * q^31 - 6 * q^33 + 4 * q^39 - 4 * q^41 - 6 * q^43 - 10 * q^47 - 9 * q^49 - 12 * q^51 + 8 * q^53 + 2 * q^57 + 8 * q^59 + 6 * q^61 + 18 * q^63 - 10 * q^67 + 12 * q^69 + 12 * q^71 - 24 * q^73 + 2 * q^77 + 16 * q^79 + 23 * q^81 - 22 * q^83 + 6 * q^87 + 10 * q^89 + 2 * q^91 - 2 * q^93 - 14 * q^97 + 8 * q^99

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