LMFDB - Embedded newform 9200.2.a.de.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 13x^{6} + 38x^{5} + 41x^{4} - 123x^{3} + 15x^{2} + 32x - 8 $ x^8 - 3x^7 - 13x^6 + 38x^5 + 41x^4 - 123x^3 + 15x^2 + 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.54092$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54092 q^{3} -0.780573 q^{7} +3.45626 q^{9} +O(q^{10})$	$q-2.54092 q^{3} -0.780573 q^{7} +3.45626 q^{9} -5.16287 q^{11} +3.34898 q^{13} -6.62869 q^{17} -6.40858 q^{19} +1.98337 q^{21} -1.00000 q^{23} -1.15931 q^{27} -0.0877744 q^{29} -3.29521 q^{31} +13.1184 q^{33} -4.68291 q^{37} -8.50947 q^{39} -5.75800 q^{41} -2.62914 q^{43} -5.13068 q^{47} -6.39071 q^{49} +16.8429 q^{51} +8.80399 q^{53} +16.2837 q^{57} -2.98687 q^{59} -2.05559 q^{61} -2.69786 q^{63} +3.60245 q^{67} +2.54092 q^{69} -11.5286 q^{71} -10.4673 q^{73} +4.03000 q^{77} -3.43878 q^{79} -7.42306 q^{81} -3.62938 q^{83} +0.223028 q^{87} +14.3109 q^{89} -2.61412 q^{91} +8.37284 q^{93} -0.427584 q^{97} -17.8442 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9	$ 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} - 7 q^{11} + 11 q^{13} - 7 q^{17} - 11 q^{19} - 8 q^{23} + 12 q^{27} + 22 q^{29} - 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} + 22 q^{43} + 4 q^{47} + 39 q^{49} + 19 q^{51} + 4 q^{53} - 32 q^{59} + 17 q^{61} + 44 q^{63} - 4 q^{67} - 3 q^{69} - 15 q^{71} + 6 q^{73} + 18 q^{77} + 2 q^{79} + 24 q^{81} + 36 q^{83} - 4 q^{87} + 46 q^{89} + 35 q^{91} + 20 q^{93} + 3 q^{97} - 61 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9 - 7 * q^11 + 11 * q^13 - 7 * q^17 - 11 * q^19 - 8 * q^23 + 12 * q^27 + 22 * q^29 - 9 * q^31 - 9 * q^33 + 4 * q^37 + 7 * q^41 + 22 * q^43 + 4 * q^47 + 39 * q^49 + 19 * q^51 + 4 * q^53 - 32 * q^59 + 17 * q^61 + 44 * q^63 - 4 * q^67 - 3 * q^69 - 15 * q^71 + 6 * q^73 + 18 * q^77 + 2 * q^79 + 24 * q^81 + 36 * q^83 - 4 * q^87 + 46 * q^89 + 35 * q^91 + 20 * q^93 + 3 * q^97 - 61 * 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
$3$	−2.54092	−1.46700	−0.733499	−	0.679690i	$-0.762114\pi$
$3$	−2.54092	−1.46700	−0.733499	+	0.679690i	$0.762114\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.780573	−0.295029	−0.147514	−	0.989060i	$-0.547127\pi$
$7$	−0.780573	−0.295029	−0.147514	+	0.989060i	$0.547127\pi$
$8$	0	0
$9$	3.45626	1.15209
$10$	0	0
$11$	−5.16287	−1.55666	−0.778332	−	0.627853i	$-0.783934\pi$
$11$	−5.16287	−1.55666	−0.778332	+	0.627853i	$0.783934\pi$
$12$	0	0
$13$	3.34898	0.928839	0.464419	−	0.885615i	$-0.346263\pi$
$13$	3.34898	0.928839	0.464419	+	0.885615i	$0.346263\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.62869	−1.60769	−0.803847	−	0.594836i	$-0.797217\pi$
$17$	−6.62869	−1.60769	−0.803847	+	0.594836i	$0.797217\pi$
$18$	0	0
$19$	−6.40858	−1.47023	−0.735114	−	0.677943i	$-0.762872\pi$
$19$	−6.40858	−1.47023	−0.735114	+	0.677943i	$0.762872\pi$
$20$	0	0
$21$	1.98337	0.432807
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.15931	−0.223109
$28$	0	0
$29$	−0.0877744	−0.0162993	−0.00814965	−	0.999967i	$-0.502594\pi$
$29$	−0.0877744	−0.0162993	−0.00814965	+	0.999967i	$0.502594\pi$
$30$	0	0
$31$	−3.29521	−0.591837	−0.295918	−	0.955213i	$-0.595626\pi$
$31$	−3.29521	−0.591837	−0.295918	+	0.955213i	$0.595626\pi$
$32$	0	0
$33$	13.1184	2.28362
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.68291	−0.769865	−0.384933	−	0.922945i	$-0.625775\pi$
$37$	−4.68291	−0.769865	−0.384933	+	0.922945i	$0.625775\pi$
$38$	0	0
$39$	−8.50947	−1.36261
$40$	0	0
$41$	−5.75800	−0.899249	−0.449624	−	0.893218i	$-0.648442\pi$
$41$	−5.75800	−0.899249	−0.449624	+	0.893218i	$0.648442\pi$
$42$	0	0
$43$	−2.62914	−0.400940	−0.200470	−	0.979700i	$-0.564247\pi$
$43$	−2.62914	−0.400940	−0.200470	+	0.979700i	$0.564247\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.13068	−0.748387	−0.374193	−	0.927351i	$-0.622080\pi$
$47$	−5.13068	−0.748387	−0.374193	+	0.927351i	$0.622080\pi$
$48$	0	0
$49$	−6.39071	−0.912958
$50$	0	0
$51$	16.8429	2.35848
$52$	0	0
$53$	8.80399	1.20932	0.604660	−	0.796483i	$-0.293309\pi$
$53$	8.80399	1.20932	0.604660	+	0.796483i	$0.293309\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	16.2837	2.15682
$58$	0	0
$59$	−2.98687	−0.388858	−0.194429	−	0.980917i	$-0.562285\pi$
$59$	−2.98687	−0.388858	−0.194429	+	0.980917i	$0.562285\pi$
$60$	0	0
$61$	−2.05559	−0.263191	−0.131596	−	0.991303i	$-0.542010\pi$
$61$	−2.05559	−0.263191	−0.131596	+	0.991303i	$0.542010\pi$
$62$	0	0
$63$	−2.69786	−0.339898
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.60245	0.440109	0.220054	−	0.975488i	$-0.429376\pi$
$67$	3.60245	0.440109	0.220054	+	0.975488i	$0.429376\pi$
$68$	0	0
$69$	2.54092	0.305890
$70$	0	0
$71$	−11.5286	−1.36819	−0.684096	−	0.729392i	$-0.739803\pi$
$71$	−11.5286	−1.36819	−0.684096	+	0.729392i	$0.739803\pi$
$72$	0	0
$73$	−10.4673	−1.22510	−0.612551	−	0.790431i	$-0.709857\pi$
$73$	−10.4673	−1.22510	−0.612551	+	0.790431i	$0.709857\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.03000	0.459261
$78$	0	0
$79$	−3.43878	−0.386893	−0.193447	−	0.981111i	$-0.561967\pi$
$79$	−3.43878	−0.386893	−0.193447	+	0.981111i	$0.561967\pi$
$80$	0	0
$81$	−7.42306	−0.824785
$82$	0	0
$83$	−3.62938	−0.398376	−0.199188	−	0.979961i	$-0.563830\pi$
$83$	−3.62938	−0.398376	−0.199188	+	0.979961i	$0.563830\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.223028	0.0239111
$88$	0	0
$89$	14.3109	1.51695	0.758477	−	0.651700i	$-0.225944\pi$
$89$	14.3109	1.51695	0.758477	+	0.651700i	$0.225944\pi$
$90$	0	0
$91$	−2.61412	−0.274034
$92$	0	0
$93$	8.37284	0.868223
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.427584	−0.0434146	−0.0217073	−	0.999764i	$-0.506910\pi$
$97$	−0.427584	−0.0434146	−0.0217073	+	0.999764i	$0.506910\pi$
$98$	0	0
$99$	−17.8442	−1.79341
$100$	0	0
$101$	−7.99434	−0.795467	−0.397733	−	0.917501i	$-0.630203\pi$
$101$	−7.99434	−0.795467	−0.397733	+	0.917501i	$0.630203\pi$
$102$	0	0
$103$	10.7780	1.06199	0.530993	−	0.847376i	$-0.321819\pi$
$103$	10.7780	1.06199	0.530993	+	0.847376i	$0.321819\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.1623	1.46580	0.732898	−	0.680339i	$-0.238167\pi$
$107$	15.1623	1.46580	0.732898	+	0.680339i	$0.238167\pi$
$108$	0	0
$109$	−16.5127	−1.58163	−0.790816	−	0.612054i	$-0.790344\pi$
$109$	−16.5127	−1.58163	−0.790816	+	0.612054i	$0.790344\pi$
$110$	0	0
$111$	11.8989	1.12939
$112$	0	0
$113$	14.5233	1.36623	0.683117	−	0.730309i	$-0.260624\pi$
$113$	14.5233	1.36623	0.683117	+	0.730309i	$0.260624\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11.5749	1.07010
$118$	0	0
$119$	5.17418	0.474316
$120$	0	0
$121$	15.6552	1.42320
$122$	0	0
$123$	14.6306	1.31920
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−16.4865	−1.46294	−0.731471	−	0.681873i	$-0.761166\pi$
$127$	−16.4865	−1.46294	−0.731471	+	0.681873i	$0.761166\pi$
$128$	0	0
$129$	6.68042	0.588178
$130$	0	0
$131$	−4.94157	−0.431747	−0.215874	−	0.976421i	$-0.569260\pi$
$131$	−4.94157	−0.431747	−0.215874	+	0.976421i	$0.569260\pi$
$132$	0	0
$133$	5.00236	0.433760
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.9516	−1.53371	−0.766854	−	0.641821i	$-0.778179\pi$
$137$	−17.9516	−1.53371	−0.766854	+	0.641821i	$0.778179\pi$
$138$	0	0
$139$	17.4812	1.48274	0.741368	−	0.671099i	$-0.234177\pi$
$139$	17.4812	1.48274	0.741368	+	0.671099i	$0.234177\pi$
$140$	0	0
$141$	13.0366	1.09788
$142$	0	0
$143$	−17.2903	−1.44589
$144$	0	0
$145$	0	0
$146$	0	0
$147$	16.2382	1.33931
$148$	0	0
$149$	−22.0079	−1.80295	−0.901477	−	0.432827i	$-0.857516\pi$
$149$	−22.0079	−1.80295	−0.901477	+	0.432827i	$0.857516\pi$
$150$	0	0
$151$	−7.40875	−0.602916	−0.301458	−	0.953480i	$-0.597473\pi$
$151$	−7.40875	−0.602916	−0.301458	+	0.953480i	$0.597473\pi$
$152$	0	0
$153$	−22.9104	−1.85220
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.19240	−0.414399	−0.207199	−	0.978299i	$-0.566435\pi$
$157$	−5.19240	−0.414399	−0.207199	+	0.978299i	$0.566435\pi$
$158$	0	0
$159$	−22.3702	−1.77407
$160$	0	0
$161$	0.780573	0.0615178
$162$	0	0
$163$	11.5618	0.905588	0.452794	−	0.891615i	$-0.350427\pi$
$163$	11.5618	0.905588	0.452794	+	0.891615i	$0.350427\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.4882	−1.50804	−0.754019	−	0.656852i	$-0.771888\pi$
$167$	−19.4882	−1.50804	−0.754019	+	0.656852i	$0.771888\pi$
$168$	0	0
$169$	−1.78436	−0.137258
$170$	0	0
$171$	−22.1497	−1.69383
$172$	0	0
$173$	18.9062	1.43741	0.718706	−	0.695314i	$-0.244735\pi$
$173$	18.9062	1.43741	0.718706	+	0.695314i	$0.244735\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.58939	0.570454
$178$	0	0
$179$	−16.5404	−1.23628	−0.618142	−	0.786066i	$-0.712114\pi$
$179$	−16.5404	−1.23628	−0.618142	+	0.786066i	$0.712114\pi$
$180$	0	0
$181$	5.51586	0.409990	0.204995	−	0.978763i	$-0.434282\pi$
$181$	5.51586	0.409990	0.204995	+	0.978763i	$0.434282\pi$
$182$	0	0
$183$	5.22308	0.386101
$184$	0	0
$185$	0	0
$186$	0	0
$187$	34.2231	2.50264
$188$	0	0
$189$	0.904924	0.0658235
$190$	0	0
$191$	−24.0346	−1.73909	−0.869543	−	0.493858i	$-0.835586\pi$
$191$	−24.0346	−1.73909	−0.869543	+	0.493858i	$0.835586\pi$
$192$	0	0
$193$	−15.2163	−1.09530	−0.547648	−	0.836709i	$-0.684477\pi$
$193$	−15.2163	−1.09530	−0.547648	+	0.836709i	$0.684477\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.82490	0.628748	0.314374	−	0.949299i	$-0.398205\pi$
$197$	8.82490	0.628748	0.314374	+	0.949299i	$0.398205\pi$
$198$	0	0
$199$	−15.3228	−1.08620	−0.543101	−	0.839668i	$-0.682750\pi$
$199$	−15.3228	−1.08620	−0.543101	+	0.839668i	$0.682750\pi$
$200$	0	0
$201$	−9.15352	−0.645639
$202$	0	0
$203$	0.0685144	0.00480877
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.45626	−0.240226
$208$	0	0
$209$	33.0866	2.28865
$210$	0	0
$211$	−4.67170	−0.321613	−0.160806	−	0.986986i	$-0.551409\pi$
$211$	−4.67170	−0.321613	−0.160806	+	0.986986i	$0.551409\pi$
$212$	0	0
$213$	29.2932	2.00714
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.57215	0.174609
$218$	0	0
$219$	26.5965	1.79722
$220$	0	0
$221$	−22.1993	−1.49329
$222$	0	0
$223$	20.7813	1.39162	0.695810	−	0.718226i	$-0.255046\pi$
$223$	20.7813	1.39162	0.695810	+	0.718226i	$0.255046\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.7696	−0.913920	−0.456960	−	0.889487i	$-0.651062\pi$
$227$	−13.7696	−0.913920	−0.456960	+	0.889487i	$0.651062\pi$
$228$	0	0
$229$	18.6065	1.22955	0.614777	−	0.788701i	$-0.289246\pi$
$229$	18.6065	1.22955	0.614777	+	0.788701i	$0.289246\pi$
$230$	0	0
$231$	−10.2399	−0.673735
$232$	0	0
$233$	−0.451458	−0.0295760	−0.0147880	−	0.999891i	$-0.504707\pi$
$233$	−0.451458	−0.0295760	−0.0147880	+	0.999891i	$0.504707\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.73766	0.567572
$238$	0	0
$239$	−12.5522	−0.811937	−0.405969	−	0.913887i	$-0.633066\pi$
$239$	−12.5522	−0.811937	−0.405969	+	0.913887i	$0.633066\pi$
$240$	0	0
$241$	16.7582	1.07949	0.539745	−	0.841828i	$-0.318521\pi$
$241$	16.7582	1.07949	0.539745	+	0.841828i	$0.318521\pi$
$242$	0	0
$243$	22.3393	1.43307
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−21.4622	−1.36561
$248$	0	0
$249$	9.22194	0.584417
$250$	0	0
$251$	6.77866	0.427865	0.213933	−	0.976848i	$-0.431373\pi$
$251$	6.77866	0.427865	0.213933	+	0.976848i	$0.431373\pi$
$252$	0	0
$253$	5.16287	0.324587
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.18182	−0.447990	−0.223995	−	0.974590i	$-0.571910\pi$
$257$	−7.18182	−0.447990	−0.223995	+	0.974590i	$0.571910\pi$
$258$	0	0
$259$	3.65535	0.227133
$260$	0	0
$261$	−0.303371	−0.0187782
$262$	0	0
$263$	18.2061	1.12264	0.561318	−	0.827600i	$-0.310294\pi$
$263$	18.2061	1.12264	0.561318	+	0.827600i	$0.310294\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−36.3628	−2.22537
$268$	0	0
$269$	−5.56736	−0.339448	−0.169724	−	0.985492i	$-0.554288\pi$
$269$	−5.56736	−0.339448	−0.169724	+	0.985492i	$0.554288\pi$
$270$	0	0
$271$	−3.85908	−0.234422	−0.117211	−	0.993107i	$-0.537395\pi$
$271$	−3.85908	−0.234422	−0.117211	+	0.993107i	$0.537395\pi$
$272$	0	0
$273$	6.64226	0.402008
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.01542	0.361431	0.180716	−	0.983535i	$-0.442159\pi$
$277$	6.01542	0.361431	0.180716	+	0.983535i	$0.442159\pi$
$278$	0	0
$279$	−11.3891	−0.681846
$280$	0	0
$281$	−16.8891	−1.00752	−0.503761	−	0.863843i	$-0.668051\pi$
$281$	−16.8891	−1.00752	−0.503761	+	0.863843i	$0.668051\pi$
$282$	0	0
$283$	1.62178	0.0964047	0.0482023	−	0.998838i	$-0.484651\pi$
$283$	1.62178	0.0964047	0.0482023	+	0.998838i	$0.484651\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.49454	0.265304
$288$	0	0
$289$	26.9395	1.58468
$290$	0	0
$291$	1.08645	0.0636891
$292$	0	0
$293$	−28.3659	−1.65716	−0.828578	−	0.559874i	$-0.810849\pi$
$293$	−28.3659	−1.65716	−0.828578	+	0.559874i	$0.810849\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.98535	0.347305
$298$	0	0
$299$	−3.34898	−0.193676
$300$	0	0
$301$	2.05223	0.118289
$302$	0	0
$303$	20.3130	1.16695
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5.54137	−0.316262	−0.158131	−	0.987418i	$-0.550547\pi$
$307$	−5.54137	−0.316262	−0.158131	+	0.987418i	$0.550547\pi$
$308$	0	0
$309$	−27.3860	−1.55793
$310$	0	0
$311$	−9.09153	−0.515533	−0.257767	−	0.966207i	$-0.582987\pi$
$311$	−9.09153	−0.515533	−0.257767	+	0.966207i	$0.582987\pi$
$312$	0	0
$313$	−24.6354	−1.39248	−0.696238	−	0.717811i	$-0.745144\pi$
$313$	−24.6354	−1.39248	−0.696238	+	0.717811i	$0.745144\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.3406	−1.31094	−0.655469	−	0.755222i	$-0.727529\pi$
$317$	−23.3406	−1.31094	−0.655469	+	0.755222i	$0.727529\pi$
$318$	0	0
$319$	0.453168	0.0253725
$320$	0	0
$321$	−38.5262	−2.15032
$322$	0	0
$323$	42.4805	2.36368
$324$	0	0
$325$	0	0
$326$	0	0
$327$	41.9575	2.32025
$328$	0	0
$329$	4.00487	0.220796
$330$	0	0
$331$	−24.6239	−1.35345	−0.676725	−	0.736236i	$-0.736601\pi$
$331$	−24.6239	−1.35345	−0.676725	+	0.736236i	$0.736601\pi$
$332$	0	0
$333$	−16.1853	−0.886950
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.9958	0.816875	0.408438	−	0.912786i	$-0.366074\pi$
$337$	14.9958	0.816875	0.408438	+	0.912786i	$0.366074\pi$
$338$	0	0
$339$	−36.9024	−2.00426
$340$	0	0
$341$	17.0127	0.921290
$342$	0	0
$343$	10.4524	0.564378
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.76366	0.470458	0.235229	−	0.971940i	$-0.424416\pi$
$347$	8.76366	0.470458	0.235229	+	0.971940i	$0.424416\pi$
$348$	0	0
$349$	33.1731	1.77572	0.887859	−	0.460116i	$-0.152192\pi$
$349$	33.1731	1.77572	0.887859	+	0.460116i	$0.152192\pi$
$350$	0	0
$351$	−3.88249	−0.207232
$352$	0	0
$353$	−23.8513	−1.26948	−0.634738	−	0.772727i	$-0.718892\pi$
$353$	−23.8513	−1.26948	−0.634738	+	0.772727i	$0.718892\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−13.1472	−0.695821
$358$	0	0
$359$	8.53298	0.450353	0.225177	−	0.974318i	$-0.427704\pi$
$359$	8.53298	0.450353	0.225177	+	0.974318i	$0.427704\pi$
$360$	0	0
$361$	22.0699	1.16157
$362$	0	0
$363$	−39.7786	−2.08783
$364$	0	0
$365$	0	0
$366$	0	0
$367$	23.3388	1.21828	0.609138	−	0.793064i	$-0.291515\pi$
$367$	23.3388	1.21828	0.609138	+	0.793064i	$0.291515\pi$
$368$	0	0
$369$	−19.9011	−1.03601
$370$	0	0
$371$	−6.87216	−0.356785
$372$	0	0
$373$	26.3073	1.36214	0.681070	−	0.732218i	$-0.261515\pi$
$373$	26.3073	1.36214	0.681070	+	0.732218i	$0.261515\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.293955	−0.0151394
$378$	0	0
$379$	33.0601	1.69818	0.849091	−	0.528246i	$-0.177150\pi$
$379$	33.0601	1.69818	0.849091	+	0.528246i	$0.177150\pi$
$380$	0	0
$381$	41.8908	2.14613
$382$	0	0
$383$	−23.7177	−1.21192	−0.605958	−	0.795496i	$-0.707210\pi$
$383$	−23.7177	−1.21192	−0.605958	+	0.795496i	$0.707210\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.08697	−0.461917
$388$	0	0
$389$	4.39287	0.222728	0.111364	−	0.993780i	$-0.464478\pi$
$389$	4.39287	0.222728	0.111364	+	0.993780i	$0.464478\pi$
$390$	0	0
$391$	6.62869	0.335227
$392$	0	0
$393$	12.5561	0.633372
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.668437	0.0335479	0.0167740	−	0.999859i	$-0.494660\pi$
$397$	0.668437	0.0335479	0.0167740	+	0.999859i	$0.494660\pi$
$398$	0	0
$399$	−12.7106	−0.636325
$400$	0	0
$401$	11.1310	0.555857	0.277928	−	0.960602i	$-0.410352\pi$
$401$	11.1310	0.555857	0.277928	+	0.960602i	$0.410352\pi$
$402$	0	0
$403$	−11.0356	−0.549721
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.1772	1.19842
$408$	0	0
$409$	−17.7667	−0.878506	−0.439253	−	0.898364i	$-0.644757\pi$
$409$	−17.7667	−0.878506	−0.439253	+	0.898364i	$0.644757\pi$
$410$	0	0
$411$	45.6135	2.24995
$412$	0	0
$413$	2.33147	0.114724
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−44.4182	−2.17517
$418$	0	0
$419$	37.4277	1.82846	0.914231	−	0.405194i	$-0.132796\pi$
$419$	37.4277	1.82846	0.914231	+	0.405194i	$0.132796\pi$
$420$	0	0
$421$	−27.0798	−1.31979	−0.659895	−	0.751358i	$-0.729399\pi$
$421$	−27.0798	−1.31979	−0.659895	+	0.751358i	$0.729399\pi$
$422$	0	0
$423$	−17.7329	−0.862205
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.60454	0.0776490
$428$	0	0
$429$	43.9333	2.12112
$430$	0	0
$431$	33.3003	1.60402	0.802009	−	0.597312i	$-0.203765\pi$
$431$	33.3003	1.60402	0.802009	+	0.597312i	$0.203765\pi$
$432$	0	0
$433$	2.71773	0.130606	0.0653028	−	0.997865i	$-0.479199\pi$
$433$	2.71773	0.130606	0.0653028	+	0.997865i	$0.479199\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.40858	0.306564
$438$	0	0
$439$	16.7478	0.799331	0.399665	−	0.916661i	$-0.369126\pi$
$439$	16.7478	0.799331	0.399665	+	0.916661i	$0.369126\pi$
$440$	0	0
$441$	−22.0879	−1.05181
$442$	0	0
$443$	−11.6392	−0.552993	−0.276497	−	0.961015i	$-0.589173\pi$
$443$	−11.6392	−0.552993	−0.276497	+	0.961015i	$0.589173\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	55.9201	2.64493
$448$	0	0
$449$	10.4132	0.491430	0.245715	−	0.969342i	$-0.420977\pi$
$449$	10.4132	0.491430	0.245715	+	0.969342i	$0.420977\pi$
$450$	0	0
$451$	29.7278	1.39983
$452$	0	0
$453$	18.8250	0.884476
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.66798	−0.311915	−0.155957	−	0.987764i	$-0.549846\pi$
$457$	−6.66798	−0.311915	−0.155957	+	0.987764i	$0.549846\pi$
$458$	0	0
$459$	7.68469	0.358690
$460$	0	0
$461$	−10.1142	−0.471064	−0.235532	−	0.971867i	$-0.575683\pi$
$461$	−10.1142	−0.471064	−0.235532	+	0.971867i	$0.575683\pi$
$462$	0	0
$463$	25.4957	1.18489	0.592443	−	0.805612i	$-0.298164\pi$
$463$	25.4957	1.18489	0.592443	+	0.805612i	$0.298164\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.17713	−0.424667	−0.212334	−	0.977197i	$-0.568106\pi$
$467$	−9.17713	−0.424667	−0.212334	+	0.977197i	$0.568106\pi$
$468$	0	0
$469$	−2.81197	−0.129845
$470$	0	0
$471$	13.1935	0.607922
$472$	0	0
$473$	13.5739	0.624128
$474$	0	0
$475$	0	0
$476$	0	0
$477$	30.4288	1.39324
$478$	0	0
$479$	22.6115	1.03315	0.516574	−	0.856243i	$-0.327207\pi$
$479$	22.6115	1.03315	0.516574	+	0.856243i	$0.327207\pi$
$480$	0	0
$481$	−15.6829	−0.715081
$482$	0	0
$483$	−1.98337	−0.0902465
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−6.32242	−0.286496	−0.143248	−	0.989687i	$-0.545755\pi$
$487$	−6.32242	−0.286496	−0.143248	+	0.989687i	$0.545755\pi$
$488$	0	0
$489$	−29.3775	−1.32850
$490$	0	0
$491$	−20.5410	−0.927002	−0.463501	−	0.886096i	$-0.653407\pi$
$491$	−20.5410	−0.927002	−0.463501	+	0.886096i	$0.653407\pi$
$492$	0	0
$493$	0.581830	0.0262043
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.99891	0.403656
$498$	0	0
$499$	33.1588	1.48439	0.742196	−	0.670183i	$-0.233784\pi$
$499$	33.1588	1.48439	0.742196	+	0.670183i	$0.233784\pi$
$500$	0	0
$501$	49.5178	2.21229
$502$	0	0
$503$	16.9045	0.753736	0.376868	−	0.926267i	$-0.377001\pi$
$503$	16.9045	0.753736	0.376868	+	0.926267i	$0.377001\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.53391	0.201358
$508$	0	0
$509$	28.6358	1.26926	0.634631	−	0.772816i	$-0.281152\pi$
$509$	28.6358	1.26926	0.634631	+	0.772816i	$0.281152\pi$
$510$	0	0
$511$	8.17048	0.361441
$512$	0	0
$513$	7.42951	0.328021
$514$	0	0
$515$	0	0
$516$	0	0
$517$	26.4890	1.16499
$518$	0	0
$519$	−48.0391	−2.10868
$520$	0	0
$521$	38.4207	1.68324	0.841622	−	0.540068i	$-0.181601\pi$
$521$	38.4207	1.68324	0.841622	+	0.540068i	$0.181601\pi$
$522$	0	0
$523$	38.6386	1.68955	0.844773	−	0.535124i	$-0.179735\pi$
$523$	38.6386	1.68955	0.844773	+	0.535124i	$0.179735\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.8429	0.951492
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−10.3234	−0.447997
$532$	0	0
$533$	−19.2834	−0.835257
$534$	0	0
$535$	0	0
$536$	0	0
$537$	42.0277	1.81363
$538$	0	0
$539$	32.9944	1.42117
$540$	0	0
$541$	4.42943	0.190436	0.0952180	−	0.995456i	$-0.469645\pi$
$541$	4.42943	0.190436	0.0952180	+	0.995456i	$0.469645\pi$
$542$	0	0
$543$	−14.0153	−0.601455
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.1695	1.20444	0.602220	−	0.798330i	$-0.294283\pi$
$547$	28.1695	1.20444	0.602220	+	0.798330i	$0.294283\pi$
$548$	0	0
$549$	−7.10464	−0.303219
$550$	0	0
$551$	0.562509	0.0239637
$552$	0	0
$553$	2.68422	0.114145
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.1942	0.516687	0.258343	−	0.966053i	$-0.416823\pi$
$557$	12.1942	0.516687	0.258343	+	0.966053i	$0.416823\pi$
$558$	0	0
$559$	−8.80492	−0.372409
$560$	0	0
$561$	−86.9579	−3.67137
$562$	0	0
$563$	−15.8362	−0.667415	−0.333708	−	0.942677i	$-0.608300\pi$
$563$	−15.8362	−0.667415	−0.333708	+	0.942677i	$0.608300\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.79424	0.243335
$568$	0	0
$569$	0.674355	0.0282704	0.0141352	−	0.999900i	$-0.495500\pi$
$569$	0.674355	0.0282704	0.0141352	+	0.999900i	$0.495500\pi$
$570$	0	0
$571$	39.9107	1.67021	0.835105	−	0.550090i	$-0.185407\pi$
$571$	39.9107	1.67021	0.835105	+	0.550090i	$0.185407\pi$
$572$	0	0
$573$	61.0700	2.55124
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−0.727423	−0.0302830	−0.0151415	−	0.999885i	$-0.504820\pi$
$577$	−0.727423	−0.0302830	−0.0151415	+	0.999885i	$0.504820\pi$
$578$	0	0
$579$	38.6635	1.60680
$580$	0	0
$581$	2.83299	0.117532
$582$	0	0
$583$	−45.4538	−1.88251
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.4075	1.58525	0.792623	−	0.609712i	$-0.208715\pi$
$587$	38.4075	1.58525	0.792623	+	0.609712i	$0.208715\pi$
$588$	0	0
$589$	21.1176	0.870135
$590$	0	0
$591$	−22.4233	−0.922373
$592$	0	0
$593$	−9.65052	−0.396299	−0.198150	−	0.980172i	$-0.563493\pi$
$593$	−9.65052	−0.396299	−0.198150	+	0.980172i	$0.563493\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	38.9338	1.59346
$598$	0	0
$599$	−35.5951	−1.45438	−0.727189	−	0.686437i	$-0.759174\pi$
$599$	−35.5951	−1.45438	−0.727189	+	0.686437i	$0.759174\pi$
$600$	0	0
$601$	−2.48461	−0.101349	−0.0506747	−	0.998715i	$-0.516137\pi$
$601$	−2.48461	−0.101349	−0.0506747	+	0.998715i	$0.516137\pi$
$602$	0	0
$603$	12.4510	0.507043
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15.0165	−0.609500	−0.304750	−	0.952432i	$-0.598573\pi$
$607$	−15.0165	−0.609500	−0.304750	+	0.952432i	$0.598573\pi$
$608$	0	0
$609$	−0.174089	−0.00705445
$610$	0	0
$611$	−17.1825	−0.695131
$612$	0	0
$613$	−4.41556	−0.178343	−0.0891714	−	0.996016i	$-0.528422\pi$
$613$	−4.41556	−0.178343	−0.0891714	+	0.996016i	$0.528422\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.0132	−0.443373	−0.221686	−	0.975118i	$-0.571156\pi$
$617$	−11.0132	−0.443373	−0.221686	+	0.975118i	$0.571156\pi$
$618$	0	0
$619$	−28.9235	−1.16253	−0.581267	−	0.813713i	$-0.697443\pi$
$619$	−28.9235	−1.16253	−0.581267	+	0.813713i	$0.697443\pi$
$620$	0	0
$621$	1.15931	0.0465214
$622$	0	0
$623$	−11.1707	−0.447545
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−84.0704	−3.35745
$628$	0	0
$629$	31.0415	1.23771
$630$	0	0
$631$	13.6238	0.542357	0.271178	−	0.962529i	$-0.412587\pi$
$631$	13.6238	0.542357	0.271178	+	0.962529i	$0.412587\pi$
$632$	0	0
$633$	11.8704	0.471806
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−21.4023	−0.847991
$638$	0	0
$639$	−39.8458	−1.57627
$640$	0	0
$641$	−28.1924	−1.11353	−0.556766	−	0.830670i	$-0.687958\pi$
$641$	−28.1924	−1.11353	−0.556766	+	0.830670i	$0.687958\pi$
$642$	0	0
$643$	−18.5514	−0.731596	−0.365798	−	0.930694i	$-0.619204\pi$
$643$	−18.5514	−0.731596	−0.365798	+	0.930694i	$0.619204\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.6338	0.535999	0.268000	−	0.963419i	$-0.413637\pi$
$647$	13.6338	0.535999	0.268000	+	0.963419i	$0.413637\pi$
$648$	0	0
$649$	15.4208	0.605320
$650$	0	0
$651$	−6.53562	−0.256151
$652$	0	0
$653$	−37.6196	−1.47217	−0.736084	−	0.676890i	$-0.763327\pi$
$653$	−37.6196	−1.47217	−0.736084	+	0.676890i	$0.763327\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−36.1776	−1.41142
$658$	0	0
$659$	−11.7947	−0.459455	−0.229727	−	0.973255i	$-0.573783\pi$
$659$	−11.7947	−0.459455	−0.229727	+	0.973255i	$0.573783\pi$
$660$	0	0
$661$	−6.49839	−0.252758	−0.126379	−	0.991982i	$-0.540336\pi$
$661$	−6.49839	−0.252758	−0.126379	+	0.991982i	$0.540336\pi$
$662$	0	0
$663$	56.4066	2.19065
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.0877744	0.00339864
$668$	0	0
$669$	−52.8036	−2.04150
$670$	0	0
$671$	10.6127	0.409700
$672$	0	0
$673$	45.7524	1.76362	0.881812	−	0.471601i	$-0.156324\pi$
$673$	45.7524	1.76362	0.881812	+	0.471601i	$0.156324\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.3462	−0.974133	−0.487066	−	0.873365i	$-0.661933\pi$
$677$	−25.3462	−0.974133	−0.487066	+	0.873365i	$0.661933\pi$
$678$	0	0
$679$	0.333760	0.0128085
$680$	0	0
$681$	34.9874	1.34072
$682$	0	0
$683$	−24.3192	−0.930548	−0.465274	−	0.885167i	$-0.654044\pi$
$683$	−24.3192	−0.930548	−0.465274	+	0.885167i	$0.654044\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−47.2776	−1.80375
$688$	0	0
$689$	29.4843	1.12326
$690$	0	0
$691$	−13.4950	−0.513374	−0.256687	−	0.966495i	$-0.582631\pi$
$691$	−13.4950	−0.513374	−0.256687	+	0.966495i	$0.582631\pi$
$692$	0	0
$693$	13.9287	0.529107
$694$	0	0
$695$	0	0
$696$	0	0
$697$	38.1680	1.44572
$698$	0	0
$699$	1.14712	0.0433879
$700$	0	0
$701$	25.8467	0.976217	0.488109	−	0.872783i	$-0.337687\pi$
$701$	25.8467	0.976217	0.488109	+	0.872783i	$0.337687\pi$
$702$	0	0
$703$	30.0108	1.13188
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.24017	0.234686
$708$	0	0
$709$	−44.9373	−1.68766	−0.843828	−	0.536613i	$-0.819704\pi$
$709$	−44.9373	−1.68766	−0.843828	+	0.536613i	$0.819704\pi$
$710$	0	0
$711$	−11.8853	−0.445734
$712$	0	0
$713$	3.29521	0.123406
$714$	0	0
$715$	0	0
$716$	0	0
$717$	31.8942	1.19111
$718$	0	0
$719$	−25.2188	−0.940502	−0.470251	−	0.882533i	$-0.655837\pi$
$719$	−25.2188	−0.940502	−0.470251	+	0.882533i	$0.655837\pi$
$720$	0	0
$721$	−8.41300	−0.313317
$722$	0	0
$723$	−42.5812	−1.58361
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.6009	0.689868	0.344934	−	0.938627i	$-0.387901\pi$
$727$	18.6009	0.689868	0.344934	+	0.938627i	$0.387901\pi$
$728$	0	0
$729$	−34.4931	−1.27752
$730$	0	0
$731$	17.4277	0.644588
$732$	0	0
$733$	30.0268	1.10907	0.554533	−	0.832162i	$-0.312897\pi$
$733$	30.0268	1.10907	0.554533	+	0.832162i	$0.312897\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.5990	−0.685101
$738$	0	0
$739$	21.5513	0.792779	0.396389	−	0.918082i	$-0.370263\pi$
$739$	21.5513	0.792779	0.396389	+	0.918082i	$0.370263\pi$
$740$	0	0
$741$	54.5336	2.00334
$742$	0	0
$743$	−0.818910	−0.0300429	−0.0150215	−	0.999887i	$-0.504782\pi$
$743$	−0.818910	−0.0300429	−0.0150215	+	0.999887i	$0.504782\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12.5441	−0.458963
$748$	0	0
$749$	−11.8353	−0.432452
$750$	0	0
$751$	40.9390	1.49388	0.746942	−	0.664889i	$-0.231521\pi$
$751$	40.9390	1.49388	0.746942	+	0.664889i	$0.231521\pi$
$752$	0	0
$753$	−17.2240	−0.627678
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.99781	0.217994	0.108997	−	0.994042i	$-0.465236\pi$
$757$	5.99781	0.217994	0.108997	+	0.994042i	$0.465236\pi$
$758$	0	0
$759$	−13.1184	−0.476168
$760$	0	0
$761$	35.9456	1.30303	0.651513	−	0.758637i	$-0.274135\pi$
$761$	35.9456	1.30303	0.651513	+	0.758637i	$0.274135\pi$
$762$	0	0
$763$	12.8894	0.466627
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.0030	−0.361186
$768$	0	0
$769$	−52.5292	−1.89425	−0.947126	−	0.320861i	$-0.896028\pi$
$769$	−52.5292	−1.89425	−0.947126	+	0.320861i	$0.896028\pi$
$770$	0	0
$771$	18.2484	0.657201
$772$	0	0
$773$	−10.2492	−0.368638	−0.184319	−	0.982866i	$-0.559008\pi$
$773$	−10.2492	−0.368638	−0.184319	+	0.982866i	$0.559008\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−9.28794	−0.333203
$778$	0	0
$779$	36.9006	1.32210
$780$	0	0
$781$	59.5206	2.12982
$782$	0	0
$783$	0.101758	0.00363652
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24.2959	0.866057	0.433028	−	0.901380i	$-0.357445\pi$
$787$	24.2959	0.866057	0.433028	+	0.901380i	$0.357445\pi$
$788$	0	0
$789$	−46.2602	−1.64691
$790$	0	0
$791$	−11.3365	−0.403079
$792$	0	0
$793$	−6.88412	−0.244462
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.7075	−0.450122	−0.225061	−	0.974345i	$-0.572258\pi$
$797$	−12.7075	−0.450122	−0.225061	+	0.974345i	$0.572258\pi$
$798$	0	0
$799$	34.0097	1.20318
$800$	0	0
$801$	49.4622	1.74766
$802$	0	0
$803$	54.0412	1.90707
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.1462	0.497970
$808$	0	0
$809$	39.2057	1.37840	0.689200	−	0.724571i	$-0.257962\pi$
$809$	39.2057	1.37840	0.689200	+	0.724571i	$0.257962\pi$
$810$	0	0
$811$	8.34215	0.292932	0.146466	−	0.989216i	$-0.453210\pi$
$811$	8.34215	0.292932	0.146466	+	0.989216i	$0.453210\pi$
$812$	0	0
$813$	9.80560	0.343897
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.8490	0.589473
$818$	0	0
$819$	−9.03507	−0.315711
$820$	0	0
$821$	−44.1654	−1.54138	−0.770691	−	0.637209i	$-0.780089\pi$
$821$	−44.1654	−1.54138	−0.770691	+	0.637209i	$0.780089\pi$
$822$	0	0
$823$	28.3933	0.989727	0.494863	−	0.868971i	$-0.335218\pi$
$823$	28.3933	0.989727	0.494863	+	0.868971i	$0.335218\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.1333	−0.873970	−0.436985	−	0.899469i	$-0.643954\pi$
$827$	−25.1333	−0.873970	−0.436985	+	0.899469i	$0.643954\pi$
$828$	0	0
$829$	0.477200	0.0165739	0.00828693	−	0.999966i	$-0.497362\pi$
$829$	0.477200	0.0165739	0.00828693	+	0.999966i	$0.497362\pi$
$830$	0	0
$831$	−15.2847	−0.530219
$832$	0	0
$833$	42.3620	1.46776
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.82016	0.132044
$838$	0	0
$839$	18.7742	0.648159	0.324079	−	0.946030i	$-0.394945\pi$
$839$	18.7742	0.648159	0.324079	+	0.946030i	$0.394945\pi$
$840$	0	0
$841$	−28.9923	−0.999734
$842$	0	0
$843$	42.9139	1.47803
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−12.2200	−0.419885
$848$	0	0
$849$	−4.12080	−0.141426
$850$	0	0
$851$	4.68291	0.160528
$852$	0	0
$853$	−1.41061	−0.0482983	−0.0241492	−	0.999708i	$-0.507688\pi$
$853$	−1.41061	−0.0482983	−0.0241492	+	0.999708i	$0.507688\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.0802	−1.16416	−0.582079	−	0.813132i	$-0.697761\pi$
$857$	−34.0802	−1.16416	−0.582079	+	0.813132i	$0.697761\pi$
$858$	0	0
$859$	−2.81779	−0.0961418	−0.0480709	−	0.998844i	$-0.515307\pi$
$859$	−2.81779	−0.0961418	−0.0480709	+	0.998844i	$0.515307\pi$
$860$	0	0
$861$	−11.4203	−0.389201
$862$	0	0
$863$	−43.8527	−1.49276	−0.746382	−	0.665517i	$-0.768211\pi$
$863$	−43.8527	−1.49276	−0.746382	+	0.665517i	$0.768211\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−68.4511	−2.32472
$868$	0	0
$869$	17.7540	0.602263
$870$	0	0
$871$	12.0645	0.408790
$872$	0	0
$873$	−1.47784	−0.0500173
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.803651	0.0271374	0.0135687	−	0.999908i	$-0.495681\pi$
$877$	0.803651	0.0271374	0.0135687	+	0.999908i	$0.495681\pi$
$878$	0	0
$879$	72.0754	2.43104
$880$	0	0
$881$	51.3971	1.73161	0.865806	−	0.500379i	$-0.166806\pi$
$881$	51.3971	1.73161	0.865806	+	0.500379i	$0.166806\pi$
$882$	0	0
$883$	26.7742	0.901024	0.450512	−	0.892770i	$-0.351241\pi$
$883$	26.7742	0.901024	0.450512	+	0.892770i	$0.351241\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.02598	0.101602	0.0508012	−	0.998709i	$-0.483822\pi$
$887$	3.02598	0.101602	0.0508012	+	0.998709i	$0.483822\pi$
$888$	0	0
$889$	12.8689	0.431610
$890$	0	0
$891$	38.3243	1.28391
$892$	0	0
$893$	32.8804	1.10030
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.50947	0.284123
$898$	0	0
$899$	0.289235	0.00964652
$900$	0	0
$901$	−58.3589	−1.94422
$902$	0	0
$903$	−5.21456	−0.173530
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−36.1018	−1.19874	−0.599370	−	0.800472i	$-0.704582\pi$
$907$	−36.1018	−1.19874	−0.599370	+	0.800472i	$0.704582\pi$
$908$	0	0
$909$	−27.6305	−0.916446
$910$	0	0
$911$	−55.7984	−1.84868	−0.924342	−	0.381566i	$-0.875385\pi$
$911$	−55.7984	−1.84868	−0.924342	+	0.381566i	$0.875385\pi$
$912$	0	0
$913$	18.7380	0.620137
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.85726	0.127378
$918$	0	0
$919$	38.9936	1.28628	0.643141	−	0.765748i	$-0.277631\pi$
$919$	38.9936	1.28628	0.643141	+	0.765748i	$0.277631\pi$
$920$	0	0
$921$	14.0801	0.463957
$922$	0	0
$923$	−38.6090	−1.27083
$924$	0	0
$925$	0	0
$926$	0	0
$927$	37.2515	1.22350
$928$	0	0
$929$	−51.1409	−1.67788	−0.838939	−	0.544226i	$-0.816823\pi$
$929$	−51.1409	−1.67788	−0.838939	+	0.544226i	$0.816823\pi$
$930$	0	0
$931$	40.9553	1.34226
$932$	0	0
$933$	23.1008	0.756287
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−45.7344	−1.49408	−0.747039	−	0.664780i	$-0.768525\pi$
$937$	−45.7344	−1.49408	−0.747039	+	0.664780i	$0.768525\pi$
$938$	0	0
$939$	62.5965	2.04276
$940$	0	0
$941$	46.9452	1.53037	0.765184	−	0.643811i	$-0.222648\pi$
$941$	46.9452	1.53037	0.765184	+	0.643811i	$0.222648\pi$
$942$	0	0
$943$	5.75800	0.187506
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.3471	0.888660	0.444330	−	0.895863i	$-0.353442\pi$
$947$	27.3471	0.888660	0.444330	+	0.895863i	$0.353442\pi$
$948$	0	0
$949$	−35.0547	−1.13792
$950$	0	0
$951$	59.3065	1.92314
$952$	0	0
$953$	−39.4077	−1.27654	−0.638270	−	0.769813i	$-0.720350\pi$
$953$	−39.4077	−1.27654	−0.638270	+	0.769813i	$0.720350\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.15146	−0.0372215
$958$	0	0
$959$	14.0125	0.452488
$960$	0	0
$961$	−20.1416	−0.649730
$962$	0	0
$963$	52.4048	1.68872
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.59634	−0.179966	−0.0899831	−	0.995943i	$-0.528681\pi$
$967$	−5.59634	−0.179966	−0.0899831	+	0.995943i	$0.528681\pi$
$968$	0	0
$969$	−107.939	−3.46751
$970$	0	0
$971$	19.2387	0.617398	0.308699	−	0.951160i	$-0.400106\pi$
$971$	19.2387	0.617398	0.308699	+	0.951160i	$0.400106\pi$
$972$	0	0
$973$	−13.6453	−0.437450
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.1594	−1.31681	−0.658403	−	0.752666i	$-0.728768\pi$
$977$	−41.1594	−1.31681	−0.658403	+	0.752666i	$0.728768\pi$
$978$	0	0
$979$	−73.8854	−2.36139
$980$	0	0
$981$	−57.0722	−1.82218
$982$	0	0
$983$	−53.9616	−1.72111	−0.860554	−	0.509359i	$-0.829883\pi$
$983$	−53.9616	−1.72111	−0.860554	+	0.509359i	$0.829883\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−10.1760	−0.323907
$988$	0	0
$989$	2.62914	0.0836017
$990$	0	0
$991$	−50.4344	−1.60210	−0.801051	−	0.598597i	$-0.795725\pi$
$991$	−50.4344	−1.60210	−0.801051	+	0.598597i	$0.795725\pi$
$992$	0	0
$993$	62.5672	1.98551
$994$	0	0
$995$	0	0
$996$	0	0
$997$	44.7163	1.41618	0.708090	−	0.706122i	$-0.249557\pi$
$997$	44.7163	1.41618	0.708090	+	0.706122i	$0.249557\pi$
$998$	0	0
$999$	5.42893	0.171764

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.de.1.1		8
4.3	odd	2		4600.2.a.bj.1.8		8
5.2	odd	4		1840.2.e.h.369.14		16
5.3	odd	4		1840.2.e.h.369.3		16
5.4	even	2		9200.2.a.dd.1.8		8
20.3	even	4		920.2.e.c.369.14	yes	16
20.7	even	4		920.2.e.c.369.3	✓	16
20.19	odd	2		4600.2.a.bk.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.3	✓	16	20.7	even	4
920.2.e.c.369.14	yes	16	20.3	even	4
1840.2.e.h.369.3		16	5.3	odd	4
1840.2.e.h.369.14		16	5.2	odd	4
4600.2.a.bj.1.8		8	4.3	odd	2
4600.2.a.bk.1.1		8	20.19	odd	2
9200.2.a.dd.1.8		8	5.4	even	2
9200.2.a.de.1.1		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q-2.54092 q^{3} -0.780573 q^{7} +3.45626 q^{9} +O(q^{10})\)	\(q-2.54092 q^{3} -0.780573 q^{7} +3.45626 q^{9} -5.16287 q^{11} +3.34898 q^{13} -6.62869 q^{17} -6.40858 q^{19} +1.98337 q^{21} -1.00000 q^{23} -1.15931 q^{27} -0.0877744 q^{29} -3.29521 q^{31} +13.1184 q^{33} -4.68291 q^{37} -8.50947 q^{39} -5.75800 q^{41} -2.62914 q^{43} -5.13068 q^{47} -6.39071 q^{49} +16.8429 q^{51} +8.80399 q^{53} +16.2837 q^{57} -2.98687 q^{59} -2.05559 q^{61} -2.69786 q^{63} +3.60245 q^{67} +2.54092 q^{69} -11.5286 q^{71} -10.4673 q^{73} +4.03000 q^{77} -3.43878 q^{79} -7.42306 q^{81} -3.62938 q^{83} +0.223028 q^{87} +14.3109 q^{89} -2.61412 q^{91} +8.37284 q^{93} -0.427584 q^{97} -17.8442 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9	\( 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} - 7 q^{11} + 11 q^{13} - 7 q^{17} - 11 q^{19} - 8 q^{23} + 12 q^{27} + 22 q^{29} - 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} + 22 q^{43} + 4 q^{47} + 39 q^{49} + 19 q^{51} + 4 q^{53} - 32 q^{59} + 17 q^{61} + 44 q^{63} - 4 q^{67} - 3 q^{69} - 15 q^{71} + 6 q^{73} + 18 q^{77} + 2 q^{79} + 24 q^{81} + 36 q^{83} - 4 q^{87} + 46 q^{89} + 35 q^{91} + 20 q^{93} + 3 q^{97} - 61 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9 - 7 * q^11 + 11 * q^13 - 7 * q^17 - 11 * q^19 - 8 * q^23 + 12 * q^27 + 22 * q^29 - 9 * q^31 - 9 * q^33 + 4 * q^37 + 7 * q^41 + 22 * q^43 + 4 * q^47 + 39 * q^49 + 19 * q^51 + 4 * q^53 - 32 * q^59 + 17 * q^61 + 44 * q^63 - 4 * q^67 - 3 * q^69 - 15 * q^71 + 6 * q^73 + 18 * q^77 + 2 * q^79 + 24 * q^81 + 36 * q^83 - 4 * q^87 + 46 * q^89 + 35 * q^91 + 20 * q^93 + 3 * q^97 - 61 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more