LMFDB - Embedded newform 8036.2.a.p.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 110x^{6} - 154x^{5} - 282x^{4} + 256x^{3} + 253x^{2} - 126x - 49 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 110x^6 - 154x^5 - 282x^4 + 256x^3 + 253x^2 - 126*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.58799$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58799 q^{3} +3.78481 q^{5} +3.69770 q^{9} +O(q^{10})$	$q+2.58799 q^{3} +3.78481 q^{5} +3.69770 q^{9} -3.59611 q^{11} +3.80530 q^{13} +9.79504 q^{15} +4.90120 q^{17} +0.726528 q^{19} +2.67934 q^{23} +9.32475 q^{25} +1.80564 q^{27} -1.31419 q^{29} +4.60193 q^{31} -9.30670 q^{33} +6.36856 q^{37} +9.84809 q^{39} +1.00000 q^{41} -6.79538 q^{43} +13.9951 q^{45} -9.58539 q^{47} +12.6843 q^{51} +3.00523 q^{53} -13.6106 q^{55} +1.88025 q^{57} +4.98541 q^{59} -2.28749 q^{61} +14.4023 q^{65} -5.58747 q^{67} +6.93410 q^{69} -1.69196 q^{71} +0.259201 q^{73} +24.1324 q^{75} +2.89788 q^{79} -6.42011 q^{81} -13.2415 q^{83} +18.5501 q^{85} -3.40110 q^{87} -17.3487 q^{89} +11.9098 q^{93} +2.74977 q^{95} +1.39398 q^{97} -13.2973 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * 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
$3$	2.58799	1.49418	0.747089	−	0.664724i	$-0.231451\pi$
$3$	2.58799	1.49418	0.747089	+	0.664724i	$0.231451\pi$
$4$	0	0
$5$	3.78481	1.69262	0.846308	−	0.532694i	$-0.178820\pi$
$5$	3.78481	1.69262	0.846308	+	0.532694i	$0.178820\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.69770	1.23257
$10$	0	0
$11$	−3.59611	−1.08427	−0.542134	−	0.840292i	$-0.682383\pi$
$11$	−3.59611	−1.08427	−0.542134	+	0.840292i	$0.682383\pi$
$12$	0	0
$13$	3.80530	1.05540	0.527700	−	0.849431i	$-0.323054\pi$
$13$	3.80530	1.05540	0.527700	+	0.849431i	$0.323054\pi$
$14$	0	0
$15$	9.79504	2.52907
$16$	0	0
$17$	4.90120	1.18872	0.594358	−	0.804201i	$-0.297406\pi$
$17$	4.90120	1.18872	0.594358	+	0.804201i	$0.297406\pi$
$18$	0	0
$19$	0.726528	0.166677	0.0833384	−	0.996521i	$-0.473442\pi$
$19$	0.726528	0.166677	0.0833384	+	0.996521i	$0.473442\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.67934	0.558681	0.279340	−	0.960192i	$-0.409884\pi$
$23$	2.67934	0.558681	0.279340	+	0.960192i	$0.409884\pi$
$24$	0	0
$25$	9.32475	1.86495
$26$	0	0
$27$	1.80564	0.347496
$28$	0	0
$29$	−1.31419	−0.244038	−0.122019	−	0.992528i	$-0.538937\pi$
$29$	−1.31419	−0.244038	−0.122019	+	0.992528i	$0.538937\pi$
$30$	0	0
$31$	4.60193	0.826531	0.413266	−	0.910611i	$-0.364388\pi$
$31$	4.60193	0.826531	0.413266	+	0.910611i	$0.364388\pi$
$32$	0	0
$33$	−9.30670	−1.62009
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.36856	1.04698	0.523492	−	0.852031i	$-0.324629\pi$
$37$	6.36856	1.04698	0.523492	+	0.852031i	$0.324629\pi$
$38$	0	0
$39$	9.84809	1.57696
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−6.79538	−1.03629	−0.518143	−	0.855294i	$-0.673376\pi$
$43$	−6.79538	−1.03629	−0.518143	+	0.855294i	$0.673376\pi$
$44$	0	0
$45$	13.9951	2.08626
$46$	0	0
$47$	−9.58539	−1.39817	−0.699086	−	0.715037i	$-0.746410\pi$
$47$	−9.58539	−1.39817	−0.699086	+	0.715037i	$0.746410\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	12.6843	1.77615
$52$	0	0
$53$	3.00523	0.412800	0.206400	−	0.978468i	$-0.433825\pi$
$53$	3.00523	0.412800	0.206400	+	0.978468i	$0.433825\pi$
$54$	0	0
$55$	−13.6106	−1.83525
$56$	0	0
$57$	1.88025	0.249045
$58$	0	0
$59$	4.98541	0.649045	0.324522	−	0.945878i	$-0.394796\pi$
$59$	4.98541	0.649045	0.324522	+	0.945878i	$0.394796\pi$
$60$	0	0
$61$	−2.28749	−0.292883	−0.146442	−	0.989219i	$-0.546782\pi$
$61$	−2.28749	−0.292883	−0.146442	+	0.989219i	$0.546782\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.4023	1.78639
$66$	0	0
$67$	−5.58747	−0.682618	−0.341309	−	0.939951i	$-0.610870\pi$
$67$	−5.58747	−0.682618	−0.341309	+	0.939951i	$0.610870\pi$
$68$	0	0
$69$	6.93410	0.834768
$70$	0	0
$71$	−1.69196	−0.200799	−0.100400	−	0.994947i	$-0.532012\pi$
$71$	−1.69196	−0.200799	−0.100400	+	0.994947i	$0.532012\pi$
$72$	0	0
$73$	0.259201	0.0303371	0.0151686	−	0.999885i	$-0.495172\pi$
$73$	0.259201	0.0303371	0.0151686	+	0.999885i	$0.495172\pi$
$74$	0	0
$75$	24.1324	2.78657
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.89788	0.326037	0.163019	−	0.986623i	$-0.447877\pi$
$79$	2.89788	0.326037	0.163019	+	0.986623i	$0.447877\pi$
$80$	0	0
$81$	−6.42011	−0.713346
$82$	0	0
$83$	−13.2415	−1.45344	−0.726722	−	0.686931i	$-0.758957\pi$
$83$	−13.2415	−1.45344	−0.726722	+	0.686931i	$0.758957\pi$
$84$	0	0
$85$	18.5501	2.01204
$86$	0	0
$87$	−3.40110	−0.364636
$88$	0	0
$89$	−17.3487	−1.83895	−0.919477	−	0.393143i	$-0.871388\pi$
$89$	−17.3487	−1.83895	−0.919477	+	0.393143i	$0.871388\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	11.9098	1.23498
$94$	0	0
$95$	2.74977	0.282120
$96$	0	0
$97$	1.39398	0.141537	0.0707687	−	0.997493i	$-0.477455\pi$
$97$	1.39398	0.141537	0.0707687	+	0.997493i	$0.477455\pi$
$98$	0	0
$99$	−13.2973	−1.33643
$100$	0	0
$101$	−20.0334	−1.99340	−0.996699	−	0.0811850i	$-0.974130\pi$
$101$	−20.0334	−1.99340	−0.996699	+	0.0811850i	$0.974130\pi$
$102$	0	0
$103$	11.2645	1.10993	0.554963	−	0.831875i	$-0.312732\pi$
$103$	11.2645	1.10993	0.554963	+	0.831875i	$0.312732\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.9772	−1.06121	−0.530605	−	0.847619i	$-0.678035\pi$
$107$	−10.9772	−1.06121	−0.530605	+	0.847619i	$0.678035\pi$
$108$	0	0
$109$	−17.3625	−1.66302	−0.831511	−	0.555508i	$-0.812524\pi$
$109$	−17.3625	−1.66302	−0.831511	+	0.555508i	$0.812524\pi$
$110$	0	0
$111$	16.4818	1.56438
$112$	0	0
$113$	14.3203	1.34714	0.673569	−	0.739124i	$-0.264760\pi$
$113$	14.3203	1.34714	0.673569	+	0.739124i	$0.264760\pi$
$114$	0	0
$115$	10.1408	0.945632
$116$	0	0
$117$	14.0709	1.30085
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.93200	0.175636
$122$	0	0
$123$	2.58799	0.233351
$124$	0	0
$125$	16.3683	1.46403
$126$	0	0
$127$	12.8284	1.13834	0.569169	−	0.822221i	$-0.307265\pi$
$127$	12.8284	1.13834	0.569169	+	0.822221i	$0.307265\pi$
$128$	0	0
$129$	−17.5864	−1.54839
$130$	0	0
$131$	−16.7714	−1.46532	−0.732660	−	0.680594i	$-0.761722\pi$
$131$	−16.7714	−1.46532	−0.732660	+	0.680594i	$0.761722\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.83401	0.588178
$136$	0	0
$137$	12.3762	1.05737	0.528684	−	0.848818i	$-0.322686\pi$
$137$	12.3762	1.05737	0.528684	+	0.848818i	$0.322686\pi$
$138$	0	0
$139$	−15.5902	−1.32235	−0.661173	−	0.750233i	$-0.729941\pi$
$139$	−15.5902	−1.32235	−0.661173	+	0.750233i	$0.729941\pi$
$140$	0	0
$141$	−24.8069	−2.08912
$142$	0	0
$143$	−13.6843	−1.14434
$144$	0	0
$145$	−4.97394	−0.413063
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.5452	1.35543	0.677717	−	0.735323i	$-0.262970\pi$
$149$	16.5452	1.35543	0.677717	+	0.735323i	$0.262970\pi$
$150$	0	0
$151$	4.20029	0.341815	0.170907	−	0.985287i	$-0.445330\pi$
$151$	4.20029	0.341815	0.170907	+	0.985287i	$0.445330\pi$
$152$	0	0
$153$	18.1232	1.46517
$154$	0	0
$155$	17.4174	1.39900
$156$	0	0
$157$	18.4113	1.46939	0.734693	−	0.678400i	$-0.237326\pi$
$157$	18.4113	1.46939	0.734693	+	0.678400i	$0.237326\pi$
$158$	0	0
$159$	7.77750	0.616796
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0.747023	0.0585114	0.0292557	−	0.999572i	$-0.490686\pi$
$163$	0.747023	0.0585114	0.0292557	+	0.999572i	$0.490686\pi$
$164$	0	0
$165$	−35.2240	−2.74219
$166$	0	0
$167$	3.55850	0.275365	0.137683	−	0.990476i	$-0.456035\pi$
$167$	3.55850	0.275365	0.137683	+	0.990476i	$0.456035\pi$
$168$	0	0
$169$	1.48032	0.113871
$170$	0	0
$171$	2.68648	0.205440
$172$	0	0
$173$	12.3522	0.939122	0.469561	−	0.882900i	$-0.344412\pi$
$173$	12.3522	0.939122	0.469561	+	0.882900i	$0.344412\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.9022	0.969788
$178$	0	0
$179$	19.1605	1.43212	0.716062	−	0.698037i	$-0.245943\pi$
$179$	19.1605	1.43212	0.716062	+	0.698037i	$0.245943\pi$
$180$	0	0
$181$	18.1901	1.35206	0.676031	−	0.736873i	$-0.263698\pi$
$181$	18.1901	1.35206	0.676031	+	0.736873i	$0.263698\pi$
$182$	0	0
$183$	−5.92001	−0.437620
$184$	0	0
$185$	24.1037	1.77214
$186$	0	0
$187$	−17.6252	−1.28889
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.5726	−0.765003	−0.382502	−	0.923955i	$-0.624937\pi$
$191$	−10.5726	−0.765003	−0.382502	+	0.923955i	$0.624937\pi$
$192$	0	0
$193$	−6.02663	−0.433806	−0.216903	−	0.976193i	$-0.569596\pi$
$193$	−6.02663	−0.433806	−0.216903	+	0.976193i	$0.569596\pi$
$194$	0	0
$195$	37.2731	2.66918
$196$	0	0
$197$	−8.15800	−0.581233	−0.290617	−	0.956840i	$-0.593860\pi$
$197$	−8.15800	−0.581233	−0.290617	+	0.956840i	$0.593860\pi$
$198$	0	0
$199$	−11.2537	−0.797751	−0.398875	−	0.917005i	$-0.630599\pi$
$199$	−11.2537	−0.797751	−0.398875	+	0.917005i	$0.630599\pi$
$200$	0	0
$201$	−14.4603	−1.01995
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.78481	0.264342
$206$	0	0
$207$	9.90739	0.688611
$208$	0	0
$209$	−2.61267	−0.180722
$210$	0	0
$211$	11.5779	0.797055	0.398527	−	0.917156i	$-0.369521\pi$
$211$	11.5779	0.797055	0.398527	+	0.917156i	$0.369521\pi$
$212$	0	0
$213$	−4.37879	−0.300030
$214$	0	0
$215$	−25.7192	−1.75403
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.670809	0.0453291
$220$	0	0
$221$	18.6505	1.25457
$222$	0	0
$223$	−16.3491	−1.09482	−0.547410	−	0.836865i	$-0.684386\pi$
$223$	−16.3491	−1.09482	−0.547410	+	0.836865i	$0.684386\pi$
$224$	0	0
$225$	34.4801	2.29868
$226$	0	0
$227$	3.86697	0.256660	0.128330	−	0.991732i	$-0.459038\pi$
$227$	3.86697	0.256660	0.128330	+	0.991732i	$0.459038\pi$
$228$	0	0
$229$	26.6084	1.75833	0.879167	−	0.476514i	$-0.158100\pi$
$229$	26.6084	1.75833	0.879167	+	0.476514i	$0.158100\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.62705	0.499665	0.249832	−	0.968289i	$-0.419624\pi$
$233$	7.62705	0.499665	0.249832	+	0.968289i	$0.419624\pi$
$234$	0	0
$235$	−36.2788	−2.36657
$236$	0	0
$237$	7.49969	0.487157
$238$	0	0
$239$	28.2707	1.82868	0.914340	−	0.404948i	$-0.132710\pi$
$239$	28.2707	1.82868	0.914340	+	0.404948i	$0.132710\pi$
$240$	0	0
$241$	27.1444	1.74852	0.874261	−	0.485457i	$-0.161347\pi$
$241$	27.1444	1.74852	0.874261	+	0.485457i	$0.161347\pi$
$242$	0	0
$243$	−22.0321	−1.41336
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.76466	0.175911
$248$	0	0
$249$	−34.2689	−2.17170
$250$	0	0
$251$	15.8143	0.998188	0.499094	−	0.866548i	$-0.333666\pi$
$251$	15.8143	0.998188	0.499094	+	0.866548i	$0.333666\pi$
$252$	0	0
$253$	−9.63519	−0.605759
$254$	0	0
$255$	48.0075	3.00634
$256$	0	0
$257$	−26.8521	−1.67499	−0.837493	−	0.546448i	$-0.815980\pi$
$257$	−26.8521	−1.67499	−0.837493	+	0.546448i	$0.815980\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−4.85947	−0.300793
$262$	0	0
$263$	−6.32193	−0.389827	−0.194913	−	0.980820i	$-0.562443\pi$
$263$	−6.32193	−0.389827	−0.194913	+	0.980820i	$0.562443\pi$
$264$	0	0
$265$	11.3742	0.698711
$266$	0	0
$267$	−44.8982	−2.74772
$268$	0	0
$269$	4.15268	0.253193	0.126597	−	0.991954i	$-0.459595\pi$
$269$	4.15268	0.253193	0.126597	+	0.991954i	$0.459595\pi$
$270$	0	0
$271$	−3.95643	−0.240336	−0.120168	−	0.992754i	$-0.538343\pi$
$271$	−3.95643	−0.240336	−0.120168	+	0.992754i	$0.538343\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−33.5328	−2.02211
$276$	0	0
$277$	2.26011	0.135797	0.0678985	−	0.997692i	$-0.478371\pi$
$277$	2.26011	0.135797	0.0678985	+	0.997692i	$0.478371\pi$
$278$	0	0
$279$	17.0166	1.01875
$280$	0	0
$281$	14.7417	0.879414	0.439707	−	0.898141i	$-0.355082\pi$
$281$	14.7417	0.879414	0.439707	+	0.898141i	$0.355082\pi$
$282$	0	0
$283$	−0.419276	−0.0249234	−0.0124617	−	0.999922i	$-0.503967\pi$
$283$	−0.419276	−0.0249234	−0.0124617	+	0.999922i	$0.503967\pi$
$284$	0	0
$285$	7.11637	0.421537
$286$	0	0
$287$	0	0
$288$	0	0
$289$	7.02174	0.413044
$290$	0	0
$291$	3.60761	0.211482
$292$	0	0
$293$	10.3460	0.604419	0.302209	−	0.953242i	$-0.402276\pi$
$293$	10.3460	0.604419	0.302209	+	0.953242i	$0.402276\pi$
$294$	0	0
$295$	18.8688	1.09858
$296$	0	0
$297$	−6.49329	−0.376779
$298$	0	0
$299$	10.1957	0.589632
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−51.8463	−2.97849
$304$	0	0
$305$	−8.65771	−0.495739
$306$	0	0
$307$	25.0647	1.43052	0.715259	−	0.698860i	$-0.246309\pi$
$307$	25.0647	1.43052	0.715259	+	0.698860i	$0.246309\pi$
$308$	0	0
$309$	29.1525	1.65843
$310$	0	0
$311$	−1.52589	−0.0865253	−0.0432627	−	0.999064i	$-0.513775\pi$
$311$	−1.52589	−0.0865253	−0.0432627	+	0.999064i	$0.513775\pi$
$312$	0	0
$313$	−5.46504	−0.308902	−0.154451	−	0.988000i	$-0.549361\pi$
$313$	−5.46504	−0.308902	−0.154451	+	0.988000i	$0.549361\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.45336	−0.474788	−0.237394	−	0.971413i	$-0.576293\pi$
$317$	−8.45336	−0.474788	−0.237394	+	0.971413i	$0.576293\pi$
$318$	0	0
$319$	4.72595	0.264603
$320$	0	0
$321$	−28.4090	−1.58564
$322$	0	0
$323$	3.56086	0.198131
$324$	0	0
$325$	35.4835	1.96827
$326$	0	0
$327$	−44.9339	−2.48485
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.7945	0.978075	0.489037	−	0.872263i	$-0.337348\pi$
$331$	17.7945	0.978075	0.489037	+	0.872263i	$0.337348\pi$
$332$	0	0
$333$	23.5490	1.29048
$334$	0	0
$335$	−21.1475	−1.15541
$336$	0	0
$337$	6.12448	0.333622	0.166811	−	0.985989i	$-0.446653\pi$
$337$	6.12448	0.333622	0.166811	+	0.985989i	$0.446653\pi$
$338$	0	0
$339$	37.0608	2.01286
$340$	0	0
$341$	−16.5490	−0.896181
$342$	0	0
$343$	0	0
$344$	0	0
$345$	26.2442	1.41294
$346$	0	0
$347$	24.2673	1.30273	0.651367	−	0.758762i	$-0.274196\pi$
$347$	24.2673	1.30273	0.651367	+	0.758762i	$0.274196\pi$
$348$	0	0
$349$	6.76708	0.362233	0.181117	−	0.983462i	$-0.442029\pi$
$349$	6.76708	0.362233	0.181117	+	0.983462i	$0.442029\pi$
$350$	0	0
$351$	6.87102	0.366748
$352$	0	0
$353$	10.4903	0.558343	0.279171	−	0.960241i	$-0.409940\pi$
$353$	10.4903	0.558343	0.279171	+	0.960241i	$0.409940\pi$
$354$	0	0
$355$	−6.40375	−0.339876
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−36.6271	−1.93310	−0.966552	−	0.256469i	$-0.917441\pi$
$359$	−36.6271	−1.93310	−0.966552	+	0.256469i	$0.917441\pi$
$360$	0	0
$361$	−18.4722	−0.972219
$362$	0	0
$363$	4.99999	0.262431
$364$	0	0
$365$	0.981024	0.0513491
$366$	0	0
$367$	−9.14736	−0.477488	−0.238744	−	0.971083i	$-0.576736\pi$
$367$	−9.14736	−0.477488	−0.238744	+	0.971083i	$0.576736\pi$
$368$	0	0
$369$	3.69770	0.192495
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−11.8244	−0.612242	−0.306121	−	0.951993i	$-0.599031\pi$
$373$	−11.8244	−0.612242	−0.306121	+	0.951993i	$0.599031\pi$
$374$	0	0
$375$	42.3611	2.18752
$376$	0	0
$377$	−5.00087	−0.257558
$378$	0	0
$379$	−17.2821	−0.887720	−0.443860	−	0.896096i	$-0.646391\pi$
$379$	−17.2821	−0.887720	−0.443860	+	0.896096i	$0.646391\pi$
$380$	0	0
$381$	33.1998	1.70088
$382$	0	0
$383$	−28.3230	−1.44724	−0.723619	−	0.690200i	$-0.757523\pi$
$383$	−28.3230	−1.44724	−0.723619	+	0.690200i	$0.757523\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−25.1273	−1.27729
$388$	0	0
$389$	−32.3079	−1.63807	−0.819037	−	0.573741i	$-0.805492\pi$
$389$	−32.3079	−1.63807	−0.819037	+	0.573741i	$0.805492\pi$
$390$	0	0
$391$	13.1320	0.664112
$392$	0	0
$393$	−43.4041	−2.18945
$394$	0	0
$395$	10.9679	0.551856
$396$	0	0
$397$	22.4571	1.12709	0.563545	−	0.826085i	$-0.309437\pi$
$397$	22.4571	1.12709	0.563545	+	0.826085i	$0.309437\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.0371	−0.950669	−0.475335	−	0.879805i	$-0.657673\pi$
$401$	−19.0371	−0.950669	−0.475335	+	0.879805i	$0.657673\pi$
$402$	0	0
$403$	17.5117	0.872322
$404$	0	0
$405$	−24.2989	−1.20742
$406$	0	0
$407$	−22.9020	−1.13521
$408$	0	0
$409$	−17.7091	−0.875660	−0.437830	−	0.899058i	$-0.644253\pi$
$409$	−17.7091	−0.875660	−0.437830	+	0.899058i	$0.644253\pi$
$410$	0	0
$411$	32.0295	1.57990
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−50.1165	−2.46012
$416$	0	0
$417$	−40.3474	−1.97582
$418$	0	0
$419$	−5.55936	−0.271593	−0.135796	−	0.990737i	$-0.543359\pi$
$419$	−5.55936	−0.271593	−0.135796	+	0.990737i	$0.543359\pi$
$420$	0	0
$421$	24.8300	1.21014	0.605071	−	0.796172i	$-0.293145\pi$
$421$	24.8300	1.21014	0.605071	+	0.796172i	$0.293145\pi$
$422$	0	0
$423$	−35.4439	−1.72334
$424$	0	0
$425$	45.7025	2.21689
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−35.4148	−1.70984
$430$	0	0
$431$	−16.6624	−0.802598	−0.401299	−	0.915947i	$-0.631441\pi$
$431$	−16.6624	−0.802598	−0.401299	+	0.915947i	$0.631441\pi$
$432$	0	0
$433$	−27.0719	−1.30099	−0.650496	−	0.759510i	$-0.725439\pi$
$433$	−27.0719	−1.30099	−0.650496	+	0.759510i	$0.725439\pi$
$434$	0	0
$435$	−12.8725	−0.617190
$436$	0	0
$437$	1.94661	0.0931191
$438$	0	0
$439$	−9.77885	−0.466719	−0.233360	−	0.972391i	$-0.574972\pi$
$439$	−9.77885	−0.466719	−0.233360	+	0.972391i	$0.574972\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−33.3426	−1.58416	−0.792078	−	0.610420i	$-0.791001\pi$
$443$	−33.3426	−1.58416	−0.792078	+	0.610420i	$0.791001\pi$
$444$	0	0
$445$	−65.6613	−3.11264
$446$	0	0
$447$	42.8188	2.02526
$448$	0	0
$449$	−19.5152	−0.920981	−0.460490	−	0.887665i	$-0.652326\pi$
$449$	−19.5152	−0.920981	−0.460490	+	0.887665i	$0.652326\pi$
$450$	0	0
$451$	−3.59611	−0.169334
$452$	0	0
$453$	10.8703	0.510732
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.0689	−0.471001	−0.235500	−	0.971874i	$-0.575673\pi$
$457$	−10.0689	−0.471001	−0.235500	+	0.971874i	$0.575673\pi$
$458$	0	0
$459$	8.84981	0.413074
$460$	0	0
$461$	−10.8841	−0.506923	−0.253462	−	0.967345i	$-0.581569\pi$
$461$	−10.8841	−0.506923	−0.253462	+	0.967345i	$0.581569\pi$
$462$	0	0
$463$	−7.76854	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$463$	−7.76854	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$464$	0	0
$465$	45.0761	2.09035
$466$	0	0
$467$	−3.61280	−0.167180	−0.0835902	−	0.996500i	$-0.526639\pi$
$467$	−3.61280	−0.167180	−0.0835902	+	0.996500i	$0.526639\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	47.6484	2.19552
$472$	0	0
$473$	24.4369	1.12361
$474$	0	0
$475$	6.77469	0.310844
$476$	0	0
$477$	11.1124	0.508803
$478$	0	0
$479$	26.6680	1.21849	0.609247	−	0.792981i	$-0.291472\pi$
$479$	26.6680	1.21849	0.609247	+	0.792981i	$0.291472\pi$
$480$	0	0
$481$	24.2343	1.10499
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.27595	0.239568
$486$	0	0
$487$	13.5887	0.615763	0.307882	−	0.951425i	$-0.400380\pi$
$487$	13.5887	0.615763	0.307882	+	0.951425i	$0.400380\pi$
$488$	0	0
$489$	1.93329	0.0874264
$490$	0	0
$491$	−25.9300	−1.17021	−0.585103	−	0.810959i	$-0.698946\pi$
$491$	−25.9300	−1.17021	−0.585103	+	0.810959i	$0.698946\pi$
$492$	0	0
$493$	−6.44108	−0.290092
$494$	0	0
$495$	−50.3278	−2.26207
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10.5967	0.474371	0.237186	−	0.971464i	$-0.423775\pi$
$499$	10.5967	0.474371	0.237186	+	0.971464i	$0.423775\pi$
$500$	0	0
$501$	9.20938	0.411445
$502$	0	0
$503$	−1.10153	−0.0491148	−0.0245574	−	0.999698i	$-0.507818\pi$
$503$	−1.10153	−0.0491148	−0.0245574	+	0.999698i	$0.507818\pi$
$504$	0	0
$505$	−75.8225	−3.37406
$506$	0	0
$507$	3.83107	0.170144
$508$	0	0
$509$	11.6687	0.517208	0.258604	−	0.965983i	$-0.416738\pi$
$509$	11.6687	0.517208	0.258604	+	0.965983i	$0.416738\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.31185	0.0579196
$514$	0	0
$515$	42.6340	1.87868
$516$	0	0
$517$	34.4701	1.51599
$518$	0	0
$519$	31.9674	1.40322
$520$	0	0
$521$	3.47208	0.152115	0.0760573	−	0.997103i	$-0.475767\pi$
$521$	3.47208	0.152115	0.0760573	+	0.997103i	$0.475767\pi$
$522$	0	0
$523$	−18.1404	−0.793223	−0.396611	−	0.917987i	$-0.629814\pi$
$523$	−18.1404	−0.793223	−0.396611	+	0.917987i	$0.629814\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.5550	0.982510
$528$	0	0
$529$	−15.8211	−0.687876
$530$	0	0
$531$	18.4345	0.799991
$532$	0	0
$533$	3.80530	0.164826
$534$	0	0
$535$	−41.5467	−1.79622
$536$	0	0
$537$	49.5872	2.13985
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−31.7015	−1.36295	−0.681477	−	0.731840i	$-0.738662\pi$
$541$	−31.7015	−1.36295	−0.681477	+	0.731840i	$0.738662\pi$
$542$	0	0
$543$	47.0759	2.02022
$544$	0	0
$545$	−65.7135	−2.81486
$546$	0	0
$547$	14.3885	0.615207	0.307603	−	0.951515i	$-0.400473\pi$
$547$	14.3885	0.615207	0.307603	+	0.951515i	$0.400473\pi$
$548$	0	0
$549$	−8.45846	−0.360998
$550$	0	0
$551$	−0.954792	−0.0406755
$552$	0	0
$553$	0	0
$554$	0	0
$555$	62.3803	2.64790
$556$	0	0
$557$	2.52585	0.107024	0.0535118	−	0.998567i	$-0.482959\pi$
$557$	2.52585	0.107024	0.0535118	+	0.998567i	$0.482959\pi$
$558$	0	0
$559$	−25.8585	−1.09370
$560$	0	0
$561$	−45.6140	−1.92582
$562$	0	0
$563$	−10.6852	−0.450325	−0.225163	−	0.974321i	$-0.572291\pi$
$563$	−10.6852	−0.450325	−0.225163	+	0.974321i	$0.572291\pi$
$564$	0	0
$565$	54.1995	2.28019
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19.0959	−0.800541	−0.400271	−	0.916397i	$-0.631084\pi$
$569$	−19.0959	−0.800541	−0.400271	+	0.916397i	$0.631084\pi$
$570$	0	0
$571$	−38.3286	−1.60400	−0.802001	−	0.597323i	$-0.796231\pi$
$571$	−38.3286	−1.60400	−0.802001	+	0.597323i	$0.796231\pi$
$572$	0	0
$573$	−27.3617	−1.14305
$574$	0	0
$575$	24.9842	1.04191
$576$	0	0
$577$	−26.9868	−1.12347	−0.561737	−	0.827316i	$-0.689867\pi$
$577$	−26.9868	−1.12347	−0.561737	+	0.827316i	$0.689867\pi$
$578$	0	0
$579$	−15.5969	−0.648184
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−10.8071	−0.447585
$584$	0	0
$585$	53.2555	2.20184
$586$	0	0
$587$	29.5370	1.21912	0.609560	−	0.792740i	$-0.291346\pi$
$587$	29.5370	1.21912	0.609560	+	0.792740i	$0.291346\pi$
$588$	0	0
$589$	3.34343	0.137764
$590$	0	0
$591$	−21.1128	−0.868466
$592$	0	0
$593$	−28.4935	−1.17009	−0.585044	−	0.811002i	$-0.698923\pi$
$593$	−28.4935	−1.17009	−0.585044	+	0.811002i	$0.698923\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−29.1244	−1.19198
$598$	0	0
$599$	34.9502	1.42803	0.714014	−	0.700132i	$-0.246875\pi$
$599$	34.9502	1.42803	0.714014	+	0.700132i	$0.246875\pi$
$600$	0	0
$601$	15.2328	0.621360	0.310680	−	0.950515i	$-0.399443\pi$
$601$	15.2328	0.621360	0.310680	+	0.950515i	$0.399443\pi$
$602$	0	0
$603$	−20.6608	−0.841372
$604$	0	0
$605$	7.31223	0.297284
$606$	0	0
$607$	15.9275	0.646476	0.323238	−	0.946318i	$-0.395228\pi$
$607$	15.9275	0.646476	0.323238	+	0.946318i	$0.395228\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−36.4753	−1.47563
$612$	0	0
$613$	12.2603	0.495189	0.247594	−	0.968864i	$-0.420360\pi$
$613$	12.2603	0.495189	0.247594	+	0.968864i	$0.420360\pi$
$614$	0	0
$615$	9.79504	0.394974
$616$	0	0
$617$	−35.8405	−1.44288	−0.721441	−	0.692475i	$-0.756520\pi$
$617$	−35.8405	−1.44288	−0.721441	+	0.692475i	$0.756520\pi$
$618$	0	0
$619$	2.41570	0.0970953	0.0485476	−	0.998821i	$-0.484541\pi$
$619$	2.41570	0.0970953	0.0485476	+	0.998821i	$0.484541\pi$
$620$	0	0
$621$	4.83793	0.194139
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15.3272	0.613090
$626$	0	0
$627$	−6.76157	−0.270031
$628$	0	0
$629$	31.2136	1.24457
$630$	0	0
$631$	−30.1822	−1.20154	−0.600768	−	0.799423i	$-0.705138\pi$
$631$	−30.1822	−1.20154	−0.600768	+	0.799423i	$0.705138\pi$
$632$	0	0
$633$	29.9635	1.19094
$634$	0	0
$635$	48.5531	1.92677
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.25638	−0.247498
$640$	0	0
$641$	−16.9868	−0.670938	−0.335469	−	0.942051i	$-0.608895\pi$
$641$	−16.9868	−0.670938	−0.335469	+	0.942051i	$0.608895\pi$
$642$	0	0
$643$	−0.216985	−0.00855705	−0.00427853	−	0.999991i	$-0.501362\pi$
$643$	−0.216985	−0.00855705	−0.00427853	+	0.999991i	$0.501362\pi$
$644$	0	0
$645$	−66.5610	−2.62084
$646$	0	0
$647$	19.3789	0.761863	0.380931	−	0.924603i	$-0.375603\pi$
$647$	19.3789	0.761863	0.380931	+	0.924603i	$0.375603\pi$
$648$	0	0
$649$	−17.9281	−0.703738
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.7962	1.67475	0.837373	−	0.546632i	$-0.184090\pi$
$653$	42.7962	1.67475	0.837373	+	0.546632i	$0.184090\pi$
$654$	0	0
$655$	−63.4763	−2.48023
$656$	0	0
$657$	0.958446	0.0373925
$658$	0	0
$659$	9.67207	0.376770	0.188385	−	0.982095i	$-0.439675\pi$
$659$	9.67207	0.376770	0.188385	+	0.982095i	$0.439675\pi$
$660$	0	0
$661$	5.74266	0.223363	0.111682	−	0.993744i	$-0.464376\pi$
$661$	5.74266	0.223363	0.111682	+	0.993744i	$0.464376\pi$
$662$	0	0
$663$	48.2674	1.87455
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.52115	−0.136339
$668$	0	0
$669$	−42.3115	−1.63586
$670$	0	0
$671$	8.22607	0.317564
$672$	0	0
$673$	−9.67554	−0.372965	−0.186482	−	0.982458i	$-0.559709\pi$
$673$	−9.67554	−0.372965	−0.186482	+	0.982458i	$0.559709\pi$
$674$	0	0
$675$	16.8372	0.648063
$676$	0	0
$677$	−20.8582	−0.801646	−0.400823	−	0.916155i	$-0.631276\pi$
$677$	−20.8582	−0.801646	−0.400823	+	0.916155i	$0.631276\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	10.0077	0.383495
$682$	0	0
$683$	30.5389	1.16854	0.584270	−	0.811560i	$-0.301381\pi$
$683$	30.5389	1.16854	0.584270	+	0.811560i	$0.301381\pi$
$684$	0	0
$685$	46.8414	1.78972
$686$	0	0
$687$	68.8624	2.62726
$688$	0	0
$689$	11.4358	0.435669
$690$	0	0
$691$	−46.9732	−1.78694	−0.893472	−	0.449119i	$-0.851738\pi$
$691$	−46.9732	−1.78694	−0.893472	+	0.449119i	$0.851738\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−59.0060	−2.23823
$696$	0	0
$697$	4.90120	0.185646
$698$	0	0
$699$	19.7388	0.746588
$700$	0	0
$701$	32.8159	1.23944	0.619719	−	0.784824i	$-0.287247\pi$
$701$	32.8159	1.23944	0.619719	+	0.784824i	$0.287247\pi$
$702$	0	0
$703$	4.62693	0.174508
$704$	0	0
$705$	−93.8893	−3.53608
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−16.6674	−0.625957	−0.312979	−	0.949760i	$-0.601327\pi$
$709$	−16.6674	−0.625957	−0.312979	+	0.949760i	$0.601327\pi$
$710$	0	0
$711$	10.7155	0.401862
$712$	0	0
$713$	12.3301	0.461767
$714$	0	0
$715$	−51.7923	−1.93692
$716$	0	0
$717$	73.1643	2.73237
$718$	0	0
$719$	−5.33891	−0.199108	−0.0995538	−	0.995032i	$-0.531742\pi$
$719$	−5.33891	−0.199108	−0.0995538	+	0.995032i	$0.531742\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	70.2493	2.61260
$724$	0	0
$725$	−12.2545	−0.455119
$726$	0	0
$727$	−16.2839	−0.603938	−0.301969	−	0.953318i	$-0.597644\pi$
$727$	−16.2839	−0.603938	−0.301969	+	0.953318i	$0.597644\pi$
$728$	0	0
$729$	−37.7586	−1.39847
$730$	0	0
$731$	−33.3055	−1.23185
$732$	0	0
$733$	−35.7554	−1.32066	−0.660328	−	0.750978i	$-0.729583\pi$
$733$	−35.7554	−1.32066	−0.660328	+	0.750978i	$0.729583\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20.0931	0.740140
$738$	0	0
$739$	13.2364	0.486910	0.243455	−	0.969912i	$-0.421719\pi$
$739$	13.2364	0.486910	0.243455	+	0.969912i	$0.421719\pi$
$740$	0	0
$741$	7.15491	0.262842
$742$	0	0
$743$	39.0119	1.43121	0.715604	−	0.698506i	$-0.246151\pi$
$743$	39.0119	1.43121	0.715604	+	0.698506i	$0.246151\pi$
$744$	0	0
$745$	62.6203	2.29423
$746$	0	0
$747$	−48.9631	−1.79147
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18.0871	0.660008	0.330004	−	0.943980i	$-0.392950\pi$
$751$	18.0871	0.660008	0.330004	+	0.943980i	$0.392950\pi$
$752$	0	0
$753$	40.9272	1.49147
$754$	0	0
$755$	15.8973	0.578561
$756$	0	0
$757$	−3.57240	−0.129841	−0.0649205	−	0.997890i	$-0.520679\pi$
$757$	−3.57240	−0.129841	−0.0649205	+	0.997890i	$0.520679\pi$
$758$	0	0
$759$	−24.9358	−0.905112
$760$	0	0
$761$	8.71186	0.315804	0.157902	−	0.987455i	$-0.449527\pi$
$761$	8.71186	0.315804	0.157902	+	0.987455i	$0.449527\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	68.5926	2.47997
$766$	0	0
$767$	18.9710	0.685002
$768$	0	0
$769$	4.38267	0.158043	0.0790216	−	0.996873i	$-0.474820\pi$
$769$	4.38267	0.158043	0.0790216	+	0.996873i	$0.474820\pi$
$770$	0	0
$771$	−69.4930	−2.50273
$772$	0	0
$773$	32.7723	1.17874	0.589370	−	0.807863i	$-0.299376\pi$
$773$	32.7723	1.17874	0.589370	+	0.807863i	$0.299376\pi$
$774$	0	0
$775$	42.9119	1.54144
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.726528	0.0260306
$780$	0	0
$781$	6.08449	0.217720
$782$	0	0
$783$	−2.37295	−0.0848023
$784$	0	0
$785$	69.6834	2.48711
$786$	0	0
$787$	−35.7887	−1.27573	−0.637865	−	0.770148i	$-0.720182\pi$
$787$	−35.7887	−1.27573	−0.637865	+	0.770148i	$0.720182\pi$
$788$	0	0
$789$	−16.3611	−0.582470
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8.70460	−0.309109
$794$	0	0
$795$	29.4363	1.04400
$796$	0	0
$797$	46.6956	1.65404	0.827022	−	0.562170i	$-0.190033\pi$
$797$	46.6956	1.65404	0.827022	+	0.562170i	$0.190033\pi$
$798$	0	0
$799$	−46.9799	−1.66203
$800$	0	0
$801$	−64.1501	−2.26663
$802$	0	0
$803$	−0.932113	−0.0328936
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.7471	0.378315
$808$	0	0
$809$	−52.6645	−1.85159	−0.925793	−	0.378030i	$-0.876602\pi$
$809$	−52.6645	−1.85159	−0.925793	+	0.378030i	$0.876602\pi$
$810$	0	0
$811$	51.2790	1.80065	0.900324	−	0.435220i	$-0.143329\pi$
$811$	51.2790	1.80065	0.900324	+	0.435220i	$0.143329\pi$
$812$	0	0
$813$	−10.2392	−0.359105
$814$	0	0
$815$	2.82734	0.0990373
$816$	0	0
$817$	−4.93703	−0.172725
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.6346	−0.545651	−0.272825	−	0.962064i	$-0.587958\pi$
$821$	−15.6346	−0.545651	−0.272825	+	0.962064i	$0.587958\pi$
$822$	0	0
$823$	41.5894	1.44972	0.724858	−	0.688898i	$-0.241905\pi$
$823$	41.5894	1.44972	0.724858	+	0.688898i	$0.241905\pi$
$824$	0	0
$825$	−86.7827	−3.02138
$826$	0	0
$827$	20.1201	0.699645	0.349823	−	0.936816i	$-0.386242\pi$
$827$	20.1201	0.699645	0.349823	+	0.936816i	$0.386242\pi$
$828$	0	0
$829$	−54.9561	−1.90870	−0.954352	−	0.298684i	$-0.903452\pi$
$829$	−54.9561	−1.90870	−0.954352	+	0.298684i	$0.903452\pi$
$830$	0	0
$831$	5.84916	0.202905
$832$	0	0
$833$	0	0
$834$	0	0
$835$	13.4682	0.466088
$836$	0	0
$837$	8.30944	0.287216
$838$	0	0
$839$	−32.4924	−1.12176	−0.560881	−	0.827896i	$-0.689537\pi$
$839$	−32.4924	−1.12176	−0.560881	+	0.827896i	$0.689537\pi$
$840$	0	0
$841$	−27.2729	−0.940445
$842$	0	0
$843$	38.1513	1.31400
$844$	0	0
$845$	5.60274	0.192740
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.08508	−0.0372400
$850$	0	0
$851$	17.0635	0.584930
$852$	0	0
$853$	−20.7697	−0.711140	−0.355570	−	0.934650i	$-0.615713\pi$
$853$	−20.7697	−0.711140	−0.355570	+	0.934650i	$0.615713\pi$
$854$	0	0
$855$	10.1678	0.347732
$856$	0	0
$857$	−40.2776	−1.37586	−0.687928	−	0.725779i	$-0.741479\pi$
$857$	−40.2776	−1.37586	−0.687928	+	0.725779i	$0.741479\pi$
$858$	0	0
$859$	−48.9883	−1.67146	−0.835730	−	0.549141i	$-0.814955\pi$
$859$	−48.9883	−1.67146	−0.835730	+	0.549141i	$0.814955\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.1659	−0.924739	−0.462370	−	0.886687i	$-0.653001\pi$
$863$	−27.1659	−0.924739	−0.462370	+	0.886687i	$0.653001\pi$
$864$	0	0
$865$	46.7508	1.58957
$866$	0	0
$867$	18.1722	0.617160
$868$	0	0
$869$	−10.4211	−0.353511
$870$	0	0
$871$	−21.2620	−0.720436
$872$	0	0
$873$	5.15453	0.174454
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−40.8059	−1.37792	−0.688959	−	0.724800i	$-0.741932\pi$
$877$	−40.8059	−1.37792	−0.688959	+	0.724800i	$0.741932\pi$
$878$	0	0
$879$	26.7753	0.903109
$880$	0	0
$881$	47.3129	1.59401	0.797005	−	0.603972i	$-0.206416\pi$
$881$	47.3129	1.59401	0.797005	+	0.603972i	$0.206416\pi$
$882$	0	0
$883$	38.7892	1.30536	0.652680	−	0.757633i	$-0.273644\pi$
$883$	38.7892	1.30536	0.652680	+	0.757633i	$0.273644\pi$
$884$	0	0
$885$	48.8323	1.64148
$886$	0	0
$887$	−2.43098	−0.0816243	−0.0408122	−	0.999167i	$-0.512995\pi$
$887$	−2.43098	−0.0816243	−0.0408122	+	0.999167i	$0.512995\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	23.0874	0.773458
$892$	0	0
$893$	−6.96405	−0.233043
$894$	0	0
$895$	72.5188	2.42404
$896$	0	0
$897$	26.3864	0.881015
$898$	0	0
$899$	−6.04779	−0.201705
$900$	0	0
$901$	14.7292	0.490701
$902$	0	0
$903$	0	0
$904$	0	0
$905$	68.8462	2.28852
$906$	0	0
$907$	45.0206	1.49488	0.747442	−	0.664327i	$-0.231282\pi$
$907$	45.0206	1.49488	0.747442	+	0.664327i	$0.231282\pi$
$908$	0	0
$909$	−74.0775	−2.45700
$910$	0	0
$911$	45.0034	1.49103	0.745515	−	0.666489i	$-0.232204\pi$
$911$	45.0034	1.49103	0.745515	+	0.666489i	$0.232204\pi$
$912$	0	0
$913$	47.6179	1.57592
$914$	0	0
$915$	−22.4061	−0.740722
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−44.8526	−1.47955	−0.739775	−	0.672854i	$-0.765068\pi$
$919$	−44.8526	−1.47955	−0.739775	+	0.672854i	$0.765068\pi$
$920$	0	0
$921$	64.8672	2.13745
$922$	0	0
$923$	−6.43843	−0.211924
$924$	0	0
$925$	59.3852	1.95257
$926$	0	0
$927$	41.6528	1.36806
$928$	0	0
$929$	31.8648	1.04545	0.522724	−	0.852502i	$-0.324916\pi$
$929$	31.8648	1.04545	0.522724	+	0.852502i	$0.324916\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3.94899	−0.129284
$934$	0	0
$935$	−66.7081	−2.18159
$936$	0	0
$937$	26.1454	0.854134	0.427067	−	0.904220i	$-0.359547\pi$
$937$	26.1454	0.854134	0.427067	+	0.904220i	$0.359547\pi$
$938$	0	0
$939$	−14.1435	−0.461555
$940$	0	0
$941$	−49.8800	−1.62604	−0.813021	−	0.582234i	$-0.802179\pi$
$941$	−49.8800	−1.62604	−0.813021	+	0.582234i	$0.802179\pi$
$942$	0	0
$943$	2.67934	0.0872512
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.0360	−0.651081	−0.325541	−	0.945528i	$-0.605546\pi$
$947$	−20.0360	−0.651081	−0.325541	+	0.945528i	$0.605546\pi$
$948$	0	0
$949$	0.986336	0.0320178
$950$	0	0
$951$	−21.8772	−0.709418
$952$	0	0
$953$	48.9363	1.58520	0.792601	−	0.609741i	$-0.208726\pi$
$953$	48.9363	1.58520	0.792601	+	0.609741i	$0.208726\pi$
$954$	0	0
$955$	−40.0151	−1.29486
$956$	0	0
$957$	12.2307	0.395363
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−9.82223	−0.316846
$962$	0	0
$963$	−40.5906	−1.30801
$964$	0	0
$965$	−22.8096	−0.734268
$966$	0	0
$967$	−1.69266	−0.0544324	−0.0272162	−	0.999630i	$-0.508664\pi$
$967$	−1.69266	−0.0544324	−0.0272162	+	0.999630i	$0.508664\pi$
$968$	0	0
$969$	9.21547	0.296043
$970$	0	0
$971$	26.8308	0.861042	0.430521	−	0.902580i	$-0.358330\pi$
$971$	26.8308	0.861042	0.430521	+	0.902580i	$0.358330\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	91.8310	2.94095
$976$	0	0
$977$	54.1829	1.73347	0.866733	−	0.498772i	$-0.166216\pi$
$977$	54.1829	1.73347	0.866733	+	0.498772i	$0.166216\pi$
$978$	0	0
$979$	62.3877	1.99392
$980$	0	0
$981$	−64.2012	−2.04979
$982$	0	0
$983$	−48.8385	−1.55771	−0.778853	−	0.627207i	$-0.784198\pi$
$983$	−48.8385	−1.55771	−0.778853	+	0.627207i	$0.784198\pi$
$984$	0	0
$985$	−30.8764	−0.983805
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.2071	−0.578953
$990$	0	0
$991$	31.6007	1.00383	0.501914	−	0.864917i	$-0.332629\pi$
$991$	31.6007	1.00383	0.501914	+	0.864917i	$0.332629\pi$
$992$	0	0
$993$	46.0520	1.46142
$994$	0	0
$995$	−42.5929	−1.35029
$996$	0	0
$997$	−21.2713	−0.673668	−0.336834	−	0.941564i	$-0.609356\pi$
$997$	−21.2713	−0.673668	−0.336834	+	0.941564i	$0.609356\pi$
$998$	0	0
$999$	11.4993	0.363823

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.p.1.8	yes	10
7.6	odd	2		8036.2.a.o.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.o.1.3	✓	10	7.6	odd	2
8036.2.a.p.1.8	yes	10	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58799 q^{3} +3.78481 q^{5} +3.69770 q^{9} +O(q^{10})\)	\(q+2.58799 q^{3} +3.78481 q^{5} +3.69770 q^{9} -3.59611 q^{11} +3.80530 q^{13} +9.79504 q^{15} +4.90120 q^{17} +0.726528 q^{19} +2.67934 q^{23} +9.32475 q^{25} +1.80564 q^{27} -1.31419 q^{29} +4.60193 q^{31} -9.30670 q^{33} +6.36856 q^{37} +9.84809 q^{39} +1.00000 q^{41} -6.79538 q^{43} +13.9951 q^{45} -9.58539 q^{47} +12.6843 q^{51} +3.00523 q^{53} -13.6106 q^{55} +1.88025 q^{57} +4.98541 q^{59} -2.28749 q^{61} +14.4023 q^{65} -5.58747 q^{67} +6.93410 q^{69} -1.69196 q^{71} +0.259201 q^{73} +24.1324 q^{75} +2.89788 q^{79} -6.42011 q^{81} -13.2415 q^{83} +18.5501 q^{85} -3.40110 q^{87} -17.3487 q^{89} +11.9098 q^{93} +2.74977 q^{95} +1.39398 q^{97} -13.2973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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