LMFDB - Embedded newform 1840.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13955077.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 14x^{3} - x^{2} + 32x + 16 $ x^5 - 14x^3 - x^2 + 32x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.568386$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.568386 q^{3} -1.00000 q^{5} -4.73770 q^{7} -2.67694 q^{9} +O(q^{10})$	$q+0.568386 q^{3} -1.00000 q^{5} -4.73770 q^{7} -2.67694 q^{9} +0.360532 q^{11} +5.26123 q^{13} -0.568386 q^{15} +0.370852 q^{17} +4.60586 q^{19} -2.69284 q^{21} +1.00000 q^{23} +1.00000 q^{25} -3.22669 q^{27} +0.939238 q^{29} -9.66662 q^{31} +0.204921 q^{33} +4.73770 q^{35} +3.26862 q^{37} +2.99041 q^{39} +5.29977 q^{41} +2.67694 q^{45} +1.25491 q^{47} +15.4458 q^{49} +0.210787 q^{51} +10.9278 q^{53} -0.360532 q^{55} +2.61790 q^{57} +9.66955 q^{59} +9.71441 q^{61} +12.6825 q^{63} -5.26123 q^{65} +7.07001 q^{67} +0.568386 q^{69} -11.3747 q^{71} +0.745086 q^{73} +0.568386 q^{75} -1.70809 q^{77} -0.415709 q^{79} +6.19680 q^{81} +9.26862 q^{83} -0.370852 q^{85} +0.533850 q^{87} +12.6122 q^{89} -24.9261 q^{91} -5.49437 q^{93} -4.60586 q^{95} +14.0404 q^{97} -0.965121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 5 * q - 5 * q^5 + 2 * q^7 + 13 * q^9	$ 5 q - 5 q^{5} + 2 q^{7} + 13 q^{9} + q^{11} + 4 q^{13} + 4 q^{17} - 7 q^{19} + 6 q^{21} + 5 q^{23} + 5 q^{25} - 3 q^{27} + 4 q^{29} - 19 q^{31} + 17 q^{33} - 2 q^{35} + 15 q^{37} - 19 q^{39} + 25 q^{41} - 13 q^{45} + 11 q^{47} + 25 q^{49} - 19 q^{51} + 3 q^{53} - q^{55} + 48 q^{57} + q^{59} - 5 q^{61} + 41 q^{63} - 4 q^{65} - 9 q^{67} - q^{71} - q^{73} + 18 q^{77} + 2 q^{79} + 57 q^{81} + 45 q^{83} - 4 q^{85} + 9 q^{87} + 6 q^{89} - 11 q^{91} - 39 q^{93} + 7 q^{95} + 25 q^{97} + 65 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 + 2 * q^7 + 13 * q^9 + q^11 + 4 * q^13 + 4 * q^17 - 7 * q^19 + 6 * q^21 + 5 * q^23 + 5 * q^25 - 3 * q^27 + 4 * q^29 - 19 * q^31 + 17 * q^33 - 2 * q^35 + 15 * q^37 - 19 * q^39 + 25 * q^41 - 13 * q^45 + 11 * q^47 + 25 * q^49 - 19 * q^51 + 3 * q^53 - q^55 + 48 * q^57 + q^59 - 5 * q^61 + 41 * q^63 - 4 * q^65 - 9 * q^67 - q^71 - q^73 + 18 * q^77 + 2 * q^79 + 57 * q^81 + 45 * q^83 - 4 * q^85 + 9 * q^87 + 6 * q^89 - 11 * q^91 - 39 * q^93 + 7 * q^95 + 25 * q^97 + 65 * 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
$3$	0.568386	0.328158	0.164079	−	0.986447i	$-0.447535\pi$
$3$	0.568386	0.328158	0.164079	+	0.986447i	$0.447535\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.73770	−1.79068	−0.895341	−	0.445381i	$-0.853068\pi$
$7$	−4.73770	−1.79068	−0.895341	+	0.445381i	$0.853068\pi$
$8$	0	0
$9$	−2.67694	−0.892312
$10$	0	0
$11$	0.360532	0.108704	0.0543522	−	0.998522i	$-0.482691\pi$
$11$	0.360532	0.108704	0.0543522	+	0.998522i	$0.482691\pi$
$12$	0	0
$13$	5.26123	1.45920	0.729601	−	0.683873i	$-0.239706\pi$
$13$	5.26123	1.45920	0.729601	+	0.683873i	$0.239706\pi$
$14$	0	0
$15$	−0.568386	−0.146757
$16$	0	0
$17$	0.370852	0.0899449	0.0449724	−	0.998988i	$-0.485680\pi$
$17$	0.370852	0.0899449	0.0449724	+	0.998988i	$0.485680\pi$
$18$	0	0
$19$	4.60586	1.05666	0.528328	−	0.849040i	$-0.322819\pi$
$19$	4.60586	1.05666	0.528328	+	0.849040i	$0.322819\pi$
$20$	0	0
$21$	−2.69284	−0.587626
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.22669	−0.620977
$28$	0	0
$29$	0.939238	0.174412	0.0872061	−	0.996190i	$-0.472206\pi$
$29$	0.939238	0.174412	0.0872061	+	0.996190i	$0.472206\pi$
$30$	0	0
$31$	−9.66662	−1.73618	−0.868088	−	0.496411i	$-0.834651\pi$
$31$	−9.66662	−1.73618	−0.868088	+	0.496411i	$0.834651\pi$
$32$	0	0
$33$	0.204921	0.0356722
$34$	0	0
$35$	4.73770	0.800817
$36$	0	0
$37$	3.26862	0.537357	0.268679	−	0.963230i	$-0.413413\pi$
$37$	3.26862	0.537357	0.268679	+	0.963230i	$0.413413\pi$
$38$	0	0
$39$	2.99041	0.478849
$40$	0	0
$41$	5.29977	0.827685	0.413843	−	0.910348i	$-0.364186\pi$
$41$	5.29977	0.827685	0.413843	+	0.910348i	$0.364186\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	2.67694	0.399054
$46$	0	0
$47$	1.25491	0.183048	0.0915240	−	0.995803i	$-0.470826\pi$
$47$	1.25491	0.183048	0.0915240	+	0.995803i	$0.470826\pi$
$48$	0	0
$49$	15.4458	2.20654
$50$	0	0
$51$	0.210787	0.0295161
$52$	0	0
$53$	10.9278	1.50106	0.750528	−	0.660839i	$-0.229799\pi$
$53$	10.9278	1.50106	0.750528	+	0.660839i	$0.229799\pi$
$54$	0	0
$55$	−0.360532	−0.0486141
$56$	0	0
$57$	2.61790	0.346750
$58$	0	0
$59$	9.66955	1.25887	0.629434	−	0.777054i	$-0.283287\pi$
$59$	9.66955	1.25887	0.629434	+	0.777054i	$0.283287\pi$
$60$	0	0
$61$	9.71441	1.24380	0.621901	−	0.783096i	$-0.286361\pi$
$61$	9.71441	1.24380	0.621901	+	0.783096i	$0.286361\pi$
$62$	0	0
$63$	12.6825	1.59785
$64$	0	0
$65$	−5.26123	−0.652575
$66$	0	0
$67$	7.07001	0.863739	0.431870	−	0.901936i	$-0.357854\pi$
$67$	7.07001	0.863739	0.431870	+	0.901936i	$0.357854\pi$
$68$	0	0
$69$	0.568386	0.0684257
$70$	0	0
$71$	−11.3747	−1.34993	−0.674965	−	0.737850i	$-0.735841\pi$
$71$	−11.3747	−1.34993	−0.674965	+	0.737850i	$0.735841\pi$
$72$	0	0
$73$	0.745086	0.0872057	0.0436029	−	0.999049i	$-0.486116\pi$
$73$	0.745086	0.0872057	0.0436029	+	0.999049i	$0.486116\pi$
$74$	0	0
$75$	0.568386	0.0656316
$76$	0	0
$77$	−1.70809	−0.194655
$78$	0	0
$79$	−0.415709	−0.0467709	−0.0233854	−	0.999727i	$-0.507444\pi$
$79$	−0.415709	−0.0467709	−0.0233854	+	0.999727i	$0.507444\pi$
$80$	0	0
$81$	6.19680	0.688534
$82$	0	0
$83$	9.26862	1.01736	0.508681	−	0.860955i	$-0.330133\pi$
$83$	9.26862	1.01736	0.508681	+	0.860955i	$0.330133\pi$
$84$	0	0
$85$	−0.370852	−0.0402246
$86$	0	0
$87$	0.533850	0.0572347
$88$	0	0
$89$	12.6122	1.33689	0.668444	−	0.743763i	$-0.266961\pi$
$89$	12.6122	1.33689	0.668444	+	0.743763i	$0.266961\pi$
$90$	0	0
$91$	−24.9261	−2.61297
$92$	0	0
$93$	−5.49437	−0.569740
$94$	0	0
$95$	−4.60586	−0.472551
$96$	0	0
$97$	14.0404	1.42559	0.712793	−	0.701374i	$-0.247430\pi$
$97$	14.0404	1.42559	0.712793	+	0.701374i	$0.247430\pi$
$98$	0	0
$99$	−0.965121	−0.0969983
$100$	0	0
$101$	−12.7440	−1.26808	−0.634038	−	0.773302i	$-0.718604\pi$
$101$	−12.7440	−1.26808	−0.634038	+	0.773302i	$0.718604\pi$
$102$	0	0
$103$	3.31347	0.326486	0.163243	−	0.986586i	$-0.447805\pi$
$103$	3.31347	0.326486	0.163243	+	0.986586i	$0.447805\pi$
$104$	0	0
$105$	2.69284	0.262795
$106$	0	0
$107$	8.13184	0.786135	0.393067	−	0.919510i	$-0.371414\pi$
$107$	8.13184	0.786135	0.393067	+	0.919510i	$0.371414\pi$
$108$	0	0
$109$	−5.79508	−0.555068	−0.277534	−	0.960716i	$-0.589517\pi$
$109$	−5.79508	−0.555068	−0.277534	+	0.960716i	$0.589517\pi$
$110$	0	0
$111$	1.85784	0.176338
$112$	0	0
$113$	−7.60724	−0.715629	−0.357815	−	0.933793i	$-0.616478\pi$
$113$	−7.60724	−0.715629	−0.357815	+	0.933793i	$0.616478\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−14.0840	−1.30206
$118$	0	0
$119$	−1.75699	−0.161063
$120$	0	0
$121$	−10.8700	−0.988183
$122$	0	0
$123$	3.01232	0.271611
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.13077	0.366547	0.183273	−	0.983062i	$-0.441331\pi$
$127$	4.13077	0.366547	0.183273	+	0.983062i	$0.441331\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.7582	1.55154	0.775770	−	0.631016i	$-0.217362\pi$
$131$	17.7582	1.55154	0.775770	+	0.631016i	$0.217362\pi$
$132$	0	0
$133$	−21.8212	−1.89213
$134$	0	0
$135$	3.22669	0.277709
$136$	0	0
$137$	6.86954	0.586905	0.293452	−	0.955974i	$-0.405196\pi$
$137$	6.86954	0.586905	0.293452	+	0.955974i	$0.405196\pi$
$138$	0	0
$139$	−20.6635	−1.75266	−0.876330	−	0.481712i	$-0.840015\pi$
$139$	−20.6635	−1.75266	−0.876330	+	0.481712i	$0.840015\pi$
$140$	0	0
$141$	0.713276	0.0600686
$142$	0	0
$143$	1.89684	0.158622
$144$	0	0
$145$	−0.939238	−0.0779995
$146$	0	0
$147$	8.77917	0.724094
$148$	0	0
$149$	−16.4763	−1.34979	−0.674896	−	0.737912i	$-0.735812\pi$
$149$	−16.4763	−1.34979	−0.674896	+	0.737912i	$0.735812\pi$
$150$	0	0
$151$	8.89224	0.723640	0.361820	−	0.932248i	$-0.382156\pi$
$151$	8.89224	0.723640	0.361820	+	0.932248i	$0.382156\pi$
$152$	0	0
$153$	−0.992748	−0.0802589
$154$	0	0
$155$	9.66662	0.776441
$156$	0	0
$157$	−6.62249	−0.528532	−0.264266	−	0.964450i	$-0.585130\pi$
$157$	−6.62249	−0.528532	−0.264266	+	0.964450i	$0.585130\pi$
$158$	0	0
$159$	6.21124	0.492583
$160$	0	0
$161$	−4.73770	−0.373383
$162$	0	0
$163$	16.0779	1.25932	0.629658	−	0.776872i	$-0.283195\pi$
$163$	16.0779	1.25932	0.629658	+	0.776872i	$0.283195\pi$
$164$	0	0
$165$	−0.204921	−0.0159531
$166$	0	0
$167$	−13.8763	−1.07378	−0.536891	−	0.843651i	$-0.680402\pi$
$167$	−13.8763	−1.07378	−0.536891	+	0.843651i	$0.680402\pi$
$168$	0	0
$169$	14.6805	1.12927
$170$	0	0
$171$	−12.3296	−0.942867
$172$	0	0
$173$	−2.05518	−0.156252	−0.0781261	−	0.996943i	$-0.524894\pi$
$173$	−2.05518	−0.156252	−0.0781261	+	0.996943i	$0.524894\pi$
$174$	0	0
$175$	−4.73770	−0.358136
$176$	0	0
$177$	5.49604	0.413108
$178$	0	0
$179$	−17.6478	−1.31906	−0.659531	−	0.751678i	$-0.729245\pi$
$179$	−17.6478	−1.31906	−0.659531	+	0.751678i	$0.729245\pi$
$180$	0	0
$181$	2.85477	0.212193	0.106097	−	0.994356i	$-0.466165\pi$
$181$	2.85477	0.212193	0.106097	+	0.994356i	$0.466165\pi$
$182$	0	0
$183$	5.52153	0.408164
$184$	0	0
$185$	−3.26862	−0.240313
$186$	0	0
$187$	0.133704	0.00977741
$188$	0	0
$189$	15.2871	1.11197
$190$	0	0
$191$	−1.13677	−0.0822540	−0.0411270	−	0.999154i	$-0.513095\pi$
$191$	−1.13677	−0.0822540	−0.0411270	+	0.999154i	$0.513095\pi$
$192$	0	0
$193$	−26.4694	−1.90531	−0.952654	−	0.304055i	$-0.901659\pi$
$193$	−26.4694	−1.90531	−0.952654	+	0.304055i	$0.901659\pi$
$194$	0	0
$195$	−2.99041	−0.214148
$196$	0	0
$197$	5.44424	0.387886	0.193943	−	0.981013i	$-0.437872\pi$
$197$	5.44424	0.387886	0.193943	+	0.981013i	$0.437872\pi$
$198$	0	0
$199$	18.6122	1.31938	0.659691	−	0.751537i	$-0.270687\pi$
$199$	18.6122	1.31938	0.659691	+	0.751537i	$0.270687\pi$
$200$	0	0
$201$	4.01850	0.283443
$202$	0	0
$203$	−4.44983	−0.312317
$204$	0	0
$205$	−5.29977	−0.370152
$206$	0	0
$207$	−2.67694	−0.186060
$208$	0	0
$209$	1.66056	0.114863
$210$	0	0
$211$	−6.10003	−0.419944	−0.209972	−	0.977707i	$-0.567337\pi$
$211$	−6.10003	−0.419944	−0.209972	+	0.977707i	$0.567337\pi$
$212$	0	0
$213$	−6.46523	−0.442990
$214$	0	0
$215$	0	0
$216$	0	0
$217$	45.7975	3.10894
$218$	0	0
$219$	0.423497	0.0286173
$220$	0	0
$221$	1.95114	0.131248
$222$	0	0
$223$	16.4136	1.09913	0.549567	−	0.835450i	$-0.314793\pi$
$223$	16.4136	1.09913	0.549567	+	0.835450i	$0.314793\pi$
$224$	0	0
$225$	−2.67694	−0.178462
$226$	0	0
$227$	−1.21171	−0.0804240	−0.0402120	−	0.999191i	$-0.512803\pi$
$227$	−1.21171	−0.0804240	−0.0402120	+	0.999191i	$0.512803\pi$
$228$	0	0
$229$	−22.8293	−1.50860	−0.754300	−	0.656529i	$-0.772024\pi$
$229$	−22.8293	−1.50860	−0.754300	+	0.656529i	$0.772024\pi$
$230$	0	0
$231$	−0.970856	−0.0638776
$232$	0	0
$233$	21.9420	1.43747	0.718735	−	0.695284i	$-0.244722\pi$
$233$	21.9420	1.43747	0.718735	+	0.695284i	$0.244722\pi$
$234$	0	0
$235$	−1.25491	−0.0818616
$236$	0	0
$237$	−0.236283	−0.0153482
$238$	0	0
$239$	8.58274	0.555172	0.277586	−	0.960701i	$-0.410466\pi$
$239$	8.58274	0.555172	0.277586	+	0.960701i	$0.410466\pi$
$240$	0	0
$241$	15.3857	0.991079	0.495540	−	0.868585i	$-0.334970\pi$
$241$	15.3857	0.991079	0.495540	+	0.868585i	$0.334970\pi$
$242$	0	0
$243$	13.2023	0.846925
$244$	0	0
$245$	−15.4458	−0.986795
$246$	0	0
$247$	24.2325	1.54187
$248$	0	0
$249$	5.26815	0.333856
$250$	0	0
$251$	8.85477	0.558908	0.279454	−	0.960159i	$-0.409847\pi$
$251$	8.85477	0.558908	0.279454	+	0.960159i	$0.409847\pi$
$252$	0	0
$253$	0.360532	0.0226664
$254$	0	0
$255$	−0.210787	−0.0132000
$256$	0	0
$257$	−22.6008	−1.40980	−0.704899	−	0.709308i	$-0.749008\pi$
$257$	−22.6008	−1.40980	−0.704899	+	0.709308i	$0.749008\pi$
$258$	0	0
$259$	−15.4857	−0.962236
$260$	0	0
$261$	−2.51428	−0.155630
$262$	0	0
$263$	7.73590	0.477016	0.238508	−	0.971141i	$-0.423342\pi$
$263$	7.73590	0.477016	0.238508	+	0.971141i	$0.423342\pi$
$264$	0	0
$265$	−10.9278	−0.671292
$266$	0	0
$267$	7.16858	0.438710
$268$	0	0
$269$	−3.87154	−0.236052	−0.118026	−	0.993011i	$-0.537657\pi$
$269$	−3.87154	−0.236052	−0.118026	+	0.993011i	$0.537657\pi$
$270$	0	0
$271$	−4.24705	−0.257990	−0.128995	−	0.991645i	$-0.541175\pi$
$271$	−4.24705	−0.257990	−0.128995	+	0.991645i	$0.541175\pi$
$272$	0	0
$273$	−14.1677	−0.857466
$274$	0	0
$275$	0.360532	0.0217409
$276$	0	0
$277$	−7.26754	−0.436664	−0.218332	−	0.975875i	$-0.570062\pi$
$277$	−7.26754	−0.436664	−0.218332	+	0.975875i	$0.570062\pi$
$278$	0	0
$279$	25.8769	1.54921
$280$	0	0
$281$	−13.0131	−0.776297	−0.388148	−	0.921597i	$-0.626885\pi$
$281$	−13.0131	−0.776297	−0.388148	+	0.921597i	$0.626885\pi$
$282$	0	0
$283$	−17.8342	−1.06013	−0.530067	−	0.847956i	$-0.677833\pi$
$283$	−17.8342	−1.06013	−0.530067	+	0.847956i	$0.677833\pi$
$284$	0	0
$285$	−2.61790	−0.155071
$286$	0	0
$287$	−25.1087	−1.48212
$288$	0	0
$289$	−16.8625	−0.991910
$290$	0	0
$291$	7.98037	0.467818
$292$	0	0
$293$	7.32291	0.427809	0.213905	−	0.976855i	$-0.431382\pi$
$293$	7.32291	0.427809	0.213905	+	0.976855i	$0.431382\pi$
$294$	0	0
$295$	−9.66955	−0.562983
$296$	0	0
$297$	−1.16333	−0.0675030
$298$	0	0
$299$	5.26123	0.304265
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.24352	−0.416129
$304$	0	0
$305$	−9.71441	−0.556245
$306$	0	0
$307$	29.1127	1.66155	0.830775	−	0.556608i	$-0.187897\pi$
$307$	29.1127	1.66155	0.830775	+	0.556608i	$0.187897\pi$
$308$	0	0
$309$	1.88333	0.107139
$310$	0	0
$311$	−0.458912	−0.0260225	−0.0130113	−	0.999915i	$-0.504142\pi$
$311$	−0.458912	−0.0260225	−0.0130113	+	0.999915i	$0.504142\pi$
$312$	0	0
$313$	−17.8279	−1.00769	−0.503846	−	0.863794i	$-0.668082\pi$
$313$	−17.8279	−1.00769	−0.503846	+	0.863794i	$0.668082\pi$
$314$	0	0
$315$	−12.6825	−0.714579
$316$	0	0
$317$	23.8799	1.34123	0.670614	−	0.741807i	$-0.266031\pi$
$317$	23.8799	1.34123	0.670614	+	0.741807i	$0.266031\pi$
$318$	0	0
$319$	0.338625	0.0189594
$320$	0	0
$321$	4.62203	0.257976
$322$	0	0
$323$	1.70809	0.0950407
$324$	0	0
$325$	5.26123	0.291840
$326$	0	0
$327$	−3.29384	−0.182150
$328$	0	0
$329$	−5.94540	−0.327781
$330$	0	0
$331$	−30.0214	−1.65013	−0.825063	−	0.565041i	$-0.808860\pi$
$331$	−30.0214	−1.65013	−0.825063	+	0.565041i	$0.808860\pi$
$332$	0	0
$333$	−8.74988	−0.479490
$334$	0	0
$335$	−7.07001	−0.386276
$336$	0	0
$337$	11.9440	0.650631	0.325316	−	0.945605i	$-0.394529\pi$
$337$	11.9440	0.650631	0.325316	+	0.945605i	$0.394529\pi$
$338$	0	0
$339$	−4.32385	−0.234839
$340$	0	0
$341$	−3.48512	−0.188730
$342$	0	0
$343$	−40.0136	−2.16053
$344$	0	0
$345$	−0.568386	−0.0306009
$346$	0	0
$347$	0.873132	0.0468722	0.0234361	−	0.999725i	$-0.492539\pi$
$347$	0.873132	0.0468722	0.0234361	+	0.999725i	$0.492539\pi$
$348$	0	0
$349$	−18.8868	−1.01099	−0.505493	−	0.862831i	$-0.668689\pi$
$349$	−18.8868	−1.01099	−0.505493	+	0.862831i	$0.668689\pi$
$350$	0	0
$351$	−16.9764	−0.906131
$352$	0	0
$353$	−5.43875	−0.289475	−0.144738	−	0.989470i	$-0.546234\pi$
$353$	−5.43875	−0.289475	−0.144738	+	0.989470i	$0.546234\pi$
$354$	0	0
$355$	11.3747	0.603707
$356$	0	0
$357$	−0.998646	−0.0528540
$358$	0	0
$359$	23.4229	1.23622	0.618108	−	0.786093i	$-0.287899\pi$
$359$	23.4229	1.23622	0.618108	+	0.786093i	$0.287899\pi$
$360$	0	0
$361$	2.21390	0.116521
$362$	0	0
$363$	−6.17837	−0.324280
$364$	0	0
$365$	−0.745086	−0.0389996
$366$	0	0
$367$	−27.2014	−1.41990	−0.709951	−	0.704252i	$-0.751283\pi$
$367$	−27.2014	−1.41990	−0.709951	+	0.704252i	$0.751283\pi$
$368$	0	0
$369$	−14.1872	−0.738554
$370$	0	0
$371$	−51.7728	−2.68791
$372$	0	0
$373$	20.0431	1.03779	0.518897	−	0.854837i	$-0.326343\pi$
$373$	20.0431	1.03779	0.518897	+	0.854837i	$0.326343\pi$
$374$	0	0
$375$	−0.568386	−0.0293513
$376$	0	0
$377$	4.94155	0.254503
$378$	0	0
$379$	12.8364	0.659362	0.329681	−	0.944092i	$-0.393059\pi$
$379$	12.8364	0.659362	0.329681	+	0.944092i	$0.393059\pi$
$380$	0	0
$381$	2.34787	0.120285
$382$	0	0
$383$	30.5405	1.56055	0.780273	−	0.625439i	$-0.215080\pi$
$383$	30.5405	1.56055	0.780273	+	0.625439i	$0.215080\pi$
$384$	0	0
$385$	1.70809	0.0870524
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.3392	1.63966	0.819831	−	0.572605i	$-0.194067\pi$
$389$	32.3392	1.63966	0.819831	+	0.572605i	$0.194067\pi$
$390$	0	0
$391$	0.370852	0.0187548
$392$	0	0
$393$	10.0935	0.509150
$394$	0	0
$395$	0.415709	0.0209166
$396$	0	0
$397$	28.9147	1.45119	0.725594	−	0.688123i	$-0.241565\pi$
$397$	28.9147	1.45119	0.725594	+	0.688123i	$0.241565\pi$
$398$	0	0
$399$	−12.4028	−0.620919
$400$	0	0
$401$	24.2862	1.21279	0.606397	−	0.795162i	$-0.292614\pi$
$401$	24.2862	1.21279	0.606397	+	0.795162i	$0.292614\pi$
$402$	0	0
$403$	−50.8583	−2.53343
$404$	0	0
$405$	−6.19680	−0.307922
$406$	0	0
$407$	1.17844	0.0584131
$408$	0	0
$409$	31.8371	1.57424	0.787122	−	0.616797i	$-0.211570\pi$
$409$	31.8371	1.57424	0.787122	+	0.616797i	$0.211570\pi$
$410$	0	0
$411$	3.90455	0.192597
$412$	0	0
$413$	−45.8114	−2.25423
$414$	0	0
$415$	−9.26862	−0.454978
$416$	0	0
$417$	−11.7449	−0.575149
$418$	0	0
$419$	21.4880	1.04976	0.524879	−	0.851177i	$-0.324110\pi$
$419$	21.4880	1.04976	0.524879	+	0.851177i	$0.324110\pi$
$420$	0	0
$421$	−13.2930	−0.647859	−0.323930	−	0.946081i	$-0.605004\pi$
$421$	−13.2930	−0.647859	−0.323930	+	0.946081i	$0.605004\pi$
$422$	0	0
$423$	−3.35933	−0.163336
$424$	0	0
$425$	0.370852	0.0179890
$426$	0	0
$427$	−46.0239	−2.22725
$428$	0	0
$429$	1.07814	0.0520530
$430$	0	0
$431$	0.534460	0.0257440	0.0128720	−	0.999917i	$-0.495903\pi$
$431$	0.534460	0.0257440	0.0128720	+	0.999917i	$0.495903\pi$
$432$	0	0
$433$	16.3377	0.785140	0.392570	−	0.919722i	$-0.371586\pi$
$433$	16.3377	0.785140	0.392570	+	0.919722i	$0.371586\pi$
$434$	0	0
$435$	−0.533850	−0.0255962
$436$	0	0
$437$	4.60586	0.220328
$438$	0	0
$439$	−28.7573	−1.37251	−0.686255	−	0.727361i	$-0.740746\pi$
$439$	−28.7573	−1.37251	−0.686255	+	0.727361i	$0.740746\pi$
$440$	0	0
$441$	−41.3474	−1.96892
$442$	0	0
$443$	15.6542	0.743751	0.371876	−	0.928283i	$-0.378715\pi$
$443$	15.6542	0.743751	0.371876	+	0.928283i	$0.378715\pi$
$444$	0	0
$445$	−12.6122	−0.597874
$446$	0	0
$447$	−9.36491	−0.442945
$448$	0	0
$449$	38.4302	1.81363	0.906817	−	0.421525i	$-0.138505\pi$
$449$	38.4302	1.81363	0.906817	+	0.421525i	$0.138505\pi$
$450$	0	0
$451$	1.91074	0.0899730
$452$	0	0
$453$	5.05422	0.237468
$454$	0	0
$455$	24.9261	1.16855
$456$	0	0
$457$	−37.2199	−1.74107	−0.870536	−	0.492104i	$-0.836228\pi$
$457$	−37.2199	−1.74107	−0.870536	+	0.492104i	$0.836228\pi$
$458$	0	0
$459$	−1.19663	−0.0558537
$460$	0	0
$461$	−3.25491	−0.151596	−0.0757982	−	0.997123i	$-0.524150\pi$
$461$	−3.25491	−0.151596	−0.0757982	+	0.997123i	$0.524150\pi$
$462$	0	0
$463$	−29.9534	−1.39205	−0.696027	−	0.718016i	$-0.745050\pi$
$463$	−29.9534	−1.39205	−0.696027	+	0.718016i	$0.745050\pi$
$464$	0	0
$465$	5.49437	0.254795
$466$	0	0
$467$	39.3302	1.81999	0.909993	−	0.414623i	$-0.136087\pi$
$467$	39.3302	1.81999	0.909993	+	0.414623i	$0.136087\pi$
$468$	0	0
$469$	−33.4956	−1.54668
$470$	0	0
$471$	−3.76413	−0.173442
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.60586	0.211331
$476$	0	0
$477$	−29.2532	−1.33941
$478$	0	0
$479$	−8.18660	−0.374055	−0.187028	−	0.982355i	$-0.559885\pi$
$479$	−8.18660	−0.374055	−0.187028	+	0.982355i	$0.559885\pi$
$480$	0	0
$481$	17.1969	0.784113
$482$	0	0
$483$	−2.69284	−0.122529
$484$	0	0
$485$	−14.0404	−0.637542
$486$	0	0
$487$	31.9963	1.44989	0.724946	−	0.688806i	$-0.241865\pi$
$487$	31.9963	1.44989	0.724946	+	0.688806i	$0.241865\pi$
$488$	0	0
$489$	9.13844	0.413255
$490$	0	0
$491$	7.12584	0.321585	0.160792	−	0.986988i	$-0.448595\pi$
$491$	7.12584	0.321585	0.160792	+	0.986988i	$0.448595\pi$
$492$	0	0
$493$	0.348319	0.0156875
$494$	0	0
$495$	0.965121	0.0433790
$496$	0	0
$497$	53.8899	2.41729
$498$	0	0
$499$	−23.4219	−1.04851	−0.524254	−	0.851562i	$-0.675656\pi$
$499$	−23.4219	−1.04851	−0.524254	+	0.851562i	$0.675656\pi$
$500$	0	0
$501$	−7.88711	−0.352370
$502$	0	0
$503$	3.94170	0.175752	0.0878758	−	0.996131i	$-0.471992\pi$
$503$	3.94170	0.175752	0.0878758	+	0.996131i	$0.471992\pi$
$504$	0	0
$505$	12.7440	0.567101
$506$	0	0
$507$	8.34421	0.370579
$508$	0	0
$509$	38.5237	1.70753	0.853766	−	0.520656i	$-0.174313\pi$
$509$	38.5237	1.70753	0.853766	+	0.520656i	$0.174313\pi$
$510$	0	0
$511$	−3.52999	−0.156158
$512$	0	0
$513$	−14.8617	−0.656159
$514$	0	0
$515$	−3.31347	−0.146009
$516$	0	0
$517$	0.452436	0.0198981
$518$	0	0
$519$	−1.16813	−0.0512754
$520$	0	0
$521$	−39.2428	−1.71926	−0.859630	−	0.510917i	$-0.829306\pi$
$521$	−39.2428	−1.71926	−0.859630	+	0.510917i	$0.829306\pi$
$522$	0	0
$523$	39.0190	1.70618	0.853090	−	0.521763i	$-0.174726\pi$
$523$	39.0190	1.70618	0.853090	+	0.521763i	$0.174726\pi$
$524$	0	0
$525$	−2.69284	−0.117525
$526$	0	0
$527$	−3.58489	−0.156160
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−25.8848	−1.12330
$532$	0	0
$533$	27.8833	1.20776
$534$	0	0
$535$	−8.13184	−0.351570
$536$	0	0
$537$	−10.0308	−0.432860
$538$	0	0
$539$	5.56870	0.239861
$540$	0	0
$541$	2.54061	0.109230	0.0546148	−	0.998508i	$-0.482607\pi$
$541$	2.54061	0.109230	0.0546148	+	0.998508i	$0.482607\pi$
$542$	0	0
$543$	1.62261	0.0696329
$544$	0	0
$545$	5.79508	0.248234
$546$	0	0
$547$	−6.78776	−0.290224	−0.145112	−	0.989415i	$-0.546354\pi$
$547$	−6.78776	−0.290224	−0.145112	+	0.989415i	$0.546354\pi$
$548$	0	0
$549$	−26.0049	−1.10986
$550$	0	0
$551$	4.32600	0.184294
$552$	0	0
$553$	1.96950	0.0837517
$554$	0	0
$555$	−1.85784	−0.0788607
$556$	0	0
$557$	−24.7086	−1.04694	−0.523468	−	0.852045i	$-0.675362\pi$
$557$	−24.7086	−1.04694	−0.523468	+	0.852045i	$0.675362\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.0759955	0.00320853
$562$	0	0
$563$	28.1423	1.18606	0.593029	−	0.805181i	$-0.297932\pi$
$563$	28.1423	1.18606	0.593029	+	0.805181i	$0.297932\pi$
$564$	0	0
$565$	7.60724	0.320039
$566$	0	0
$567$	−29.3586	−1.23294
$568$	0	0
$569$	41.1938	1.72694	0.863468	−	0.504404i	$-0.168288\pi$
$569$	41.1938	1.72694	0.863468	+	0.504404i	$0.168288\pi$
$570$	0	0
$571$	−32.4687	−1.35877	−0.679387	−	0.733780i	$-0.737754\pi$
$571$	−32.4687	−1.35877	−0.679387	+	0.733780i	$0.737754\pi$
$572$	0	0
$573$	−0.646126	−0.0269923
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	6.61770	0.275498	0.137749	−	0.990467i	$-0.456013\pi$
$577$	6.61770	0.275498	0.137749	+	0.990467i	$0.456013\pi$
$578$	0	0
$579$	−15.0448	−0.625242
$580$	0	0
$581$	−43.9119	−1.82177
$582$	0	0
$583$	3.93984	0.163171
$584$	0	0
$585$	14.0840	0.582301
$586$	0	0
$587$	8.18008	0.337628	0.168814	−	0.985648i	$-0.446006\pi$
$587$	8.18008	0.337628	0.168814	+	0.985648i	$0.446006\pi$
$588$	0	0
$589$	−44.5230	−1.83454
$590$	0	0
$591$	3.09443	0.127288
$592$	0	0
$593$	29.7489	1.22164	0.610821	−	0.791768i	$-0.290839\pi$
$593$	29.7489	1.22164	0.610821	+	0.791768i	$0.290839\pi$
$594$	0	0
$595$	1.75699	0.0720294
$596$	0	0
$597$	10.5789	0.432966
$598$	0	0
$599$	−39.4069	−1.61012	−0.805061	−	0.593192i	$-0.797868\pi$
$599$	−39.4069	−1.61012	−0.805061	+	0.593192i	$0.797868\pi$
$600$	0	0
$601$	−34.3183	−1.39987	−0.699937	−	0.714205i	$-0.746788\pi$
$601$	−34.3183	−1.39987	−0.699937	+	0.714205i	$0.746788\pi$
$602$	0	0
$603$	−18.9260	−0.770725
$604$	0	0
$605$	10.8700	0.441929
$606$	0	0
$607$	11.7570	0.477200	0.238600	−	0.971118i	$-0.423312\pi$
$607$	11.7570	0.477200	0.238600	+	0.971118i	$0.423312\pi$
$608$	0	0
$609$	−2.52922	−0.102489
$610$	0	0
$611$	6.60239	0.267104
$612$	0	0
$613$	5.86862	0.237031	0.118516	−	0.992952i	$-0.462186\pi$
$613$	5.86862	0.237031	0.118516	+	0.992952i	$0.462186\pi$
$614$	0	0
$615$	−3.01232	−0.121468
$616$	0	0
$617$	−8.45842	−0.340523	−0.170262	−	0.985399i	$-0.554461\pi$
$617$	−8.45842	−0.340523	−0.170262	+	0.985399i	$0.554461\pi$
$618$	0	0
$619$	−0.851047	−0.0342065	−0.0171032	−	0.999854i	$-0.505444\pi$
$619$	−0.851047	−0.0342065	−0.0171032	+	0.999854i	$0.505444\pi$
$620$	0	0
$621$	−3.22669	−0.129483
$622$	0	0
$623$	−59.7527	−2.39394
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.943838	0.0376933
$628$	0	0
$629$	1.21217	0.0483325
$630$	0	0
$631$	−19.7131	−0.784768	−0.392384	−	0.919802i	$-0.628350\pi$
$631$	−19.7131	−0.784768	−0.392384	+	0.919802i	$0.628350\pi$
$632$	0	0
$633$	−3.46717	−0.137808
$634$	0	0
$635$	−4.13077	−0.163925
$636$	0	0
$637$	81.2638	3.21979
$638$	0	0
$639$	30.4494	1.20456
$640$	0	0
$641$	−34.3593	−1.35711	−0.678555	−	0.734550i	$-0.737393\pi$
$641$	−34.3593	−1.35711	−0.678555	+	0.734550i	$0.737393\pi$
$642$	0	0
$643$	24.2817	0.957578	0.478789	−	0.877930i	$-0.341076\pi$
$643$	24.2817	0.957578	0.478789	+	0.877930i	$0.341076\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.1465	1.18518	0.592590	−	0.805504i	$-0.298105\pi$
$647$	30.1465	1.18518	0.592590	+	0.805504i	$0.298105\pi$
$648$	0	0
$649$	3.48618	0.136845
$650$	0	0
$651$	26.0307	1.02022
$652$	0	0
$653$	−7.17298	−0.280700	−0.140350	−	0.990102i	$-0.544823\pi$
$653$	−7.17298	−0.280700	−0.140350	+	0.990102i	$0.544823\pi$
$654$	0	0
$655$	−17.7582	−0.693870
$656$	0	0
$657$	−1.99455	−0.0778148
$658$	0	0
$659$	12.8521	0.500645	0.250323	−	0.968162i	$-0.419463\pi$
$659$	12.8521	0.500645	0.250323	+	0.968162i	$0.419463\pi$
$660$	0	0
$661$	3.72426	0.144857	0.0724285	−	0.997374i	$-0.476925\pi$
$661$	3.72426	0.144857	0.0724285	+	0.997374i	$0.476925\pi$
$662$	0	0
$663$	1.10900	0.0430700
$664$	0	0
$665$	21.8212	0.846188
$666$	0	0
$667$	0.939238	0.0363675
$668$	0	0
$669$	9.32924	0.360689
$670$	0	0
$671$	3.50235	0.135207
$672$	0	0
$673$	13.5204	0.521175	0.260587	−	0.965450i	$-0.416084\pi$
$673$	13.5204	0.521175	0.260587	+	0.965450i	$0.416084\pi$
$674$	0	0
$675$	−3.22669	−0.124195
$676$	0	0
$677$	−17.8534	−0.686161	−0.343081	−	0.939306i	$-0.611470\pi$
$677$	−17.8534	−0.686161	−0.343081	+	0.939306i	$0.611470\pi$
$678$	0	0
$679$	−66.5192	−2.55277
$680$	0	0
$681$	−0.688719	−0.0263918
$682$	0	0
$683$	−15.8200	−0.605337	−0.302669	−	0.953096i	$-0.597878\pi$
$683$	−15.8200	−0.605337	−0.302669	+	0.953096i	$0.597878\pi$
$684$	0	0
$685$	−6.86954	−0.262472
$686$	0	0
$687$	−12.9758	−0.495059
$688$	0	0
$689$	57.4939	2.19034
$690$	0	0
$691$	−9.45476	−0.359676	−0.179838	−	0.983696i	$-0.557557\pi$
$691$	−9.45476	−0.359676	−0.179838	+	0.983696i	$0.557557\pi$
$692$	0	0
$693$	4.57245	0.173693
$694$	0	0
$695$	20.6635	0.783813
$696$	0	0
$697$	1.96543	0.0744460
$698$	0	0
$699$	12.4715	0.471717
$700$	0	0
$701$	29.0096	1.09568	0.547838	−	0.836584i	$-0.315451\pi$
$701$	29.0096	1.09568	0.547838	+	0.836584i	$0.315451\pi$
$702$	0	0
$703$	15.0548	0.567801
$704$	0	0
$705$	−0.713276	−0.0268635
$706$	0	0
$707$	60.3773	2.27072
$708$	0	0
$709$	−8.22423	−0.308868	−0.154434	−	0.988003i	$-0.549355\pi$
$709$	−8.22423	−0.308868	−0.154434	+	0.988003i	$0.549355\pi$
$710$	0	0
$711$	1.11283	0.0417342
$712$	0	0
$713$	−9.66662	−0.362018
$714$	0	0
$715$	−1.89684	−0.0709378
$716$	0	0
$717$	4.87831	0.182184
$718$	0	0
$719$	12.5468	0.467917	0.233958	−	0.972247i	$-0.424832\pi$
$719$	12.5468	0.467917	0.233958	+	0.972247i	$0.424832\pi$
$720$	0	0
$721$	−15.6982	−0.584633
$722$	0	0
$723$	8.74501	0.325230
$724$	0	0
$725$	0.939238	0.0348824
$726$	0	0
$727$	29.1952	1.08279	0.541395	−	0.840768i	$-0.317896\pi$
$727$	29.1952	1.08279	0.541395	+	0.840768i	$0.317896\pi$
$728$	0	0
$729$	−11.0864	−0.410609
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−36.0414	−1.33122	−0.665611	−	0.746299i	$-0.731829\pi$
$733$	−36.0414	−1.33122	−0.665611	+	0.746299i	$0.731829\pi$
$734$	0	0
$735$	−8.77917	−0.323825
$736$	0	0
$737$	2.54896	0.0938923
$738$	0	0
$739$	−0.728002	−0.0267800	−0.0133900	−	0.999910i	$-0.504262\pi$
$739$	−0.728002	−0.0267800	−0.0133900	+	0.999910i	$0.504262\pi$
$740$	0	0
$741$	13.7734	0.505978
$742$	0	0
$743$	32.8880	1.20655	0.603273	−	0.797535i	$-0.293863\pi$
$743$	32.8880	1.20655	0.603273	+	0.797535i	$0.293863\pi$
$744$	0	0
$745$	16.4763	0.603646
$746$	0	0
$747$	−24.8115	−0.907805
$748$	0	0
$749$	−38.5262	−1.40772
$750$	0	0
$751$	−14.3337	−0.523044	−0.261522	−	0.965197i	$-0.584224\pi$
$751$	−14.3337	−0.523044	−0.261522	+	0.965197i	$0.584224\pi$
$752$	0	0
$753$	5.03293	0.183410
$754$	0	0
$755$	−8.89224	−0.323622
$756$	0	0
$757$	−0.0587519	−0.00213537	−0.00106769	−	0.999999i	$-0.500340\pi$
$757$	−0.0587519	−0.00213537	−0.00106769	+	0.999999i	$0.500340\pi$
$758$	0	0
$759$	0.204921	0.00743817
$760$	0	0
$761$	45.7135	1.65711	0.828556	−	0.559906i	$-0.189163\pi$
$761$	45.7135	1.65711	0.828556	+	0.559906i	$0.189163\pi$
$762$	0	0
$763$	27.4553	0.993950
$764$	0	0
$765$	0.992748	0.0358929
$766$	0	0
$767$	50.8737	1.83694
$768$	0	0
$769$	−39.3949	−1.42061	−0.710307	−	0.703892i	$-0.751444\pi$
$769$	−39.3949	−1.42061	−0.710307	+	0.703892i	$0.751444\pi$
$770$	0	0
$771$	−12.8460	−0.462636
$772$	0	0
$773$	21.1799	0.761788	0.380894	−	0.924619i	$-0.375616\pi$
$773$	21.1799	0.761788	0.380894	+	0.924619i	$0.375616\pi$
$774$	0	0
$775$	−9.66662	−0.347235
$776$	0	0
$777$	−8.80187	−0.315765
$778$	0	0
$779$	24.4100	0.874578
$780$	0	0
$781$	−4.10094	−0.146743
$782$	0	0
$783$	−3.03063	−0.108306
$784$	0	0
$785$	6.62249	0.236367
$786$	0	0
$787$	−21.5154	−0.766941	−0.383470	−	0.923553i	$-0.625271\pi$
$787$	−21.5154	−0.766941	−0.383470	+	0.923553i	$0.625271\pi$
$788$	0	0
$789$	4.39698	0.156537
$790$	0	0
$791$	36.0408	1.28146
$792$	0	0
$793$	51.1097	1.81496
$794$	0	0
$795$	−6.21124	−0.220290
$796$	0	0
$797$	−54.7227	−1.93838	−0.969189	−	0.246320i	$-0.920779\pi$
$797$	−54.7227	−1.93838	−0.969189	+	0.246320i	$0.920779\pi$
$798$	0	0
$799$	0.465388	0.0164642
$800$	0	0
$801$	−33.7620	−1.19292
$802$	0	0
$803$	0.268627	0.00947965
$804$	0	0
$805$	4.73770	0.166982
$806$	0	0
$807$	−2.20053	−0.0774623
$808$	0	0
$809$	−43.5554	−1.53133	−0.765663	−	0.643242i	$-0.777589\pi$
$809$	−43.5554	−1.53133	−0.765663	+	0.643242i	$0.777589\pi$
$810$	0	0
$811$	13.1837	0.462944	0.231472	−	0.972842i	$-0.425646\pi$
$811$	13.1837	0.462944	0.231472	+	0.972842i	$0.425646\pi$
$812$	0	0
$813$	−2.41397	−0.0846615
$814$	0	0
$815$	−16.0779	−0.563183
$816$	0	0
$817$	0	0
$818$	0	0
$819$	66.7256	2.33158
$820$	0	0
$821$	−3.63328	−0.126803	−0.0634013	−	0.997988i	$-0.520195\pi$
$821$	−3.63328	−0.126803	−0.0634013	+	0.997988i	$0.520195\pi$
$822$	0	0
$823$	39.5044	1.37704	0.688518	−	0.725220i	$-0.258262\pi$
$823$	39.5044	1.37704	0.688518	+	0.725220i	$0.258262\pi$
$824$	0	0
$825$	0.204921	0.00713444
$826$	0	0
$827$	−48.3819	−1.68240	−0.841202	−	0.540721i	$-0.818152\pi$
$827$	−48.3819	−1.68240	−0.841202	+	0.540721i	$0.818152\pi$
$828$	0	0
$829$	−44.5864	−1.54855	−0.774275	−	0.632850i	$-0.781885\pi$
$829$	−44.5864	−1.54855	−0.774275	+	0.632850i	$0.781885\pi$
$830$	0	0
$831$	−4.13077	−0.143295
$832$	0	0
$833$	5.72810	0.198467
$834$	0	0
$835$	13.8763	0.480210
$836$	0	0
$837$	31.1912	1.07813
$838$	0	0
$839$	29.2324	1.00921	0.504606	−	0.863350i	$-0.331638\pi$
$839$	29.2324	1.00921	0.504606	+	0.863350i	$0.331638\pi$
$840$	0	0
$841$	−28.1178	−0.969580
$842$	0	0
$843$	−7.39647	−0.254748
$844$	0	0
$845$	−14.6805	−0.505025
$846$	0	0
$847$	51.4989	1.76952
$848$	0	0
$849$	−10.1367	−0.347891
$850$	0	0
$851$	3.26862	0.112047
$852$	0	0
$853$	50.5599	1.73114	0.865569	−	0.500790i	$-0.166957\pi$
$853$	50.5599	1.73114	0.865569	+	0.500790i	$0.166957\pi$
$854$	0	0
$855$	12.3296	0.421663
$856$	0	0
$857$	−21.9568	−0.750030	−0.375015	−	0.927019i	$-0.622362\pi$
$857$	−21.9568	−0.750030	−0.375015	+	0.927019i	$0.622362\pi$
$858$	0	0
$859$	−13.7671	−0.469726	−0.234863	−	0.972029i	$-0.575464\pi$
$859$	−13.7671	−0.469726	−0.234863	+	0.972029i	$0.575464\pi$
$860$	0	0
$861$	−14.2714	−0.486370
$862$	0	0
$863$	26.5748	0.904618	0.452309	−	0.891861i	$-0.350600\pi$
$863$	26.5748	0.904618	0.452309	+	0.891861i	$0.350600\pi$
$864$	0	0
$865$	2.05518	0.0698781
$866$	0	0
$867$	−9.58439	−0.325503
$868$	0	0
$869$	−0.149876	−0.00508420
$870$	0	0
$871$	37.1969	1.26037
$872$	0	0
$873$	−37.5853	−1.27207
$874$	0	0
$875$	4.73770	0.160163
$876$	0	0
$877$	45.9414	1.55133	0.775665	−	0.631145i	$-0.217415\pi$
$877$	45.9414	1.55133	0.775665	+	0.631145i	$0.217415\pi$
$878$	0	0
$879$	4.16224	0.140389
$880$	0	0
$881$	−23.9054	−0.805392	−0.402696	−	0.915334i	$-0.631927\pi$
$881$	−23.9054	−0.805392	−0.402696	+	0.915334i	$0.631927\pi$
$882$	0	0
$883$	−52.4002	−1.76341	−0.881703	−	0.471804i	$-0.843603\pi$
$883$	−52.4002	−1.76341	−0.881703	+	0.471804i	$0.843603\pi$
$884$	0	0
$885$	−5.49604	−0.184747
$886$	0	0
$887$	32.7608	1.10000	0.550000	−	0.835165i	$-0.314628\pi$
$887$	32.7608	1.10000	0.550000	+	0.835165i	$0.314628\pi$
$888$	0	0
$889$	−19.5703	−0.656368
$890$	0	0
$891$	2.23415	0.0748467
$892$	0	0
$893$	5.77995	0.193419
$894$	0	0
$895$	17.6478	0.589902
$896$	0	0
$897$	2.99041	0.0998469
$898$	0	0
$899$	−9.07926	−0.302810
$900$	0	0
$901$	4.05262	0.135012
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.85477	−0.0948957
$906$	0	0
$907$	−20.9426	−0.695388	−0.347694	−	0.937608i	$-0.613035\pi$
$907$	−20.9426	−0.695388	−0.347694	+	0.937608i	$0.613035\pi$
$908$	0	0
$909$	34.1149	1.13152
$910$	0	0
$911$	−40.6418	−1.34652	−0.673262	−	0.739404i	$-0.735107\pi$
$911$	−40.6418	−1.34652	−0.673262	+	0.739404i	$0.735107\pi$
$912$	0	0
$913$	3.34163	0.110592
$914$	0	0
$915$	−5.52153	−0.182536
$916$	0	0
$917$	−84.1330	−2.77831
$918$	0	0
$919$	50.3241	1.66004	0.830019	−	0.557735i	$-0.188330\pi$
$919$	50.3241	1.66004	0.830019	+	0.557735i	$0.188330\pi$
$920$	0	0
$921$	16.5473	0.545251
$922$	0	0
$923$	−59.8449	−1.96982
$924$	0	0
$925$	3.26862	0.107471
$926$	0	0
$927$	−8.86996	−0.291328
$928$	0	0
$929$	24.1954	0.793825	0.396912	−	0.917856i	$-0.370082\pi$
$929$	24.1954	0.793825	0.396912	+	0.917856i	$0.370082\pi$
$930$	0	0
$931$	71.1411	2.33155
$932$	0	0
$933$	−0.260839	−0.00853949
$934$	0	0
$935$	−0.133704	−0.00437259
$936$	0	0
$937$	−37.2988	−1.21850	−0.609250	−	0.792978i	$-0.708529\pi$
$937$	−37.2988	−1.21850	−0.609250	+	0.792978i	$0.708529\pi$
$938$	0	0
$939$	−10.1331	−0.330682
$940$	0	0
$941$	16.6467	0.542667	0.271334	−	0.962485i	$-0.412535\pi$
$941$	16.6467	0.542667	0.271334	+	0.962485i	$0.412535\pi$
$942$	0	0
$943$	5.29977	0.172584
$944$	0	0
$945$	−15.2871	−0.497289
$946$	0	0
$947$	30.1868	0.980938	0.490469	−	0.871459i	$-0.336825\pi$
$947$	30.1868	0.980938	0.490469	+	0.871459i	$0.336825\pi$
$948$	0	0
$949$	3.92007	0.127251
$950$	0	0
$951$	13.5730	0.440134
$952$	0	0
$953$	34.6992	1.12402	0.562008	−	0.827132i	$-0.310029\pi$
$953$	34.6992	1.12402	0.562008	+	0.827132i	$0.310029\pi$
$954$	0	0
$955$	1.13677	0.0367851
$956$	0	0
$957$	0.192470	0.00622167
$958$	0	0
$959$	−32.5458	−1.05096
$960$	0	0
$961$	62.4435	2.01431
$962$	0	0
$963$	−21.7684	−0.701478
$964$	0	0
$965$	26.4694	0.852080
$966$	0	0
$967$	11.4788	0.369133	0.184566	−	0.982820i	$-0.440912\pi$
$967$	11.4788	0.369133	0.184566	+	0.982820i	$0.440912\pi$
$968$	0	0
$969$	0.970856	0.0311884
$970$	0	0
$971$	46.8441	1.50330	0.751650	−	0.659563i	$-0.229258\pi$
$971$	46.8441	1.50330	0.751650	+	0.659563i	$0.229258\pi$
$972$	0	0
$973$	97.8977	3.13846
$974$	0	0
$975$	2.99041	0.0957698
$976$	0	0
$977$	39.3737	1.25968	0.629838	−	0.776727i	$-0.283121\pi$
$977$	39.3737	1.25968	0.629838	+	0.776727i	$0.283121\pi$
$978$	0	0
$979$	4.54709	0.145326
$980$	0	0
$981$	15.5131	0.495294
$982$	0	0
$983$	−42.3505	−1.35077	−0.675385	−	0.737465i	$-0.736023\pi$
$983$	−42.3505	−1.35077	−0.675385	+	0.737465i	$0.736023\pi$
$984$	0	0
$985$	−5.44424	−0.173468
$986$	0	0
$987$	−3.37929	−0.107564
$988$	0	0
$989$	0	0
$990$	0	0
$991$	2.00104	0.0635652	0.0317826	−	0.999495i	$-0.489882\pi$
$991$	2.00104	0.0635652	0.0317826	+	0.999495i	$0.489882\pi$
$992$	0	0
$993$	−17.0638	−0.541502
$994$	0	0
$995$	−18.6122	−0.590045
$996$	0	0
$997$	1.97936	0.0626869	0.0313435	−	0.999509i	$-0.490021\pi$
$997$	1.97936	0.0626869	0.0313435	+	0.999509i	$0.490021\pi$
$998$	0	0
$999$	−10.5468	−0.333687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.a.v.1.3		5
4.3	odd	2		920.2.a.j.1.3	✓	5
5.4	even	2		9200.2.a.cu.1.3		5
8.3	odd	2		7360.2.a.co.1.3		5
8.5	even	2		7360.2.a.cp.1.3		5
12.11	even	2		8280.2.a.bs.1.5		5
20.3	even	4		4600.2.e.u.4049.5		10
20.7	even	4		4600.2.e.u.4049.6		10
20.19	odd	2		4600.2.a.be.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.3	✓	5	4.3	odd	2
1840.2.a.v.1.3		5	1.1	even	1	trivial
4600.2.a.be.1.3		5	20.19	odd	2
4600.2.e.u.4049.5		10	20.3	even	4
4600.2.e.u.4049.6		10	20.7	even	4
7360.2.a.co.1.3		5	8.3	odd	2
7360.2.a.cp.1.3		5	8.5	even	2
8280.2.a.bs.1.5		5	12.11	even	2
9200.2.a.cu.1.3		5	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q+0.568386 q^{3} -1.00000 q^{5} -4.73770 q^{7} -2.67694 q^{9} +O(q^{10})\)	\(q+0.568386 q^{3} -1.00000 q^{5} -4.73770 q^{7} -2.67694 q^{9} +0.360532 q^{11} +5.26123 q^{13} -0.568386 q^{15} +0.370852 q^{17} +4.60586 q^{19} -2.69284 q^{21} +1.00000 q^{23} +1.00000 q^{25} -3.22669 q^{27} +0.939238 q^{29} -9.66662 q^{31} +0.204921 q^{33} +4.73770 q^{35} +3.26862 q^{37} +2.99041 q^{39} +5.29977 q^{41} +2.67694 q^{45} +1.25491 q^{47} +15.4458 q^{49} +0.210787 q^{51} +10.9278 q^{53} -0.360532 q^{55} +2.61790 q^{57} +9.66955 q^{59} +9.71441 q^{61} +12.6825 q^{63} -5.26123 q^{65} +7.07001 q^{67} +0.568386 q^{69} -11.3747 q^{71} +0.745086 q^{73} +0.568386 q^{75} -1.70809 q^{77} -0.415709 q^{79} +6.19680 q^{81} +9.26862 q^{83} -0.370852 q^{85} +0.533850 q^{87} +12.6122 q^{89} -24.9261 q^{91} -5.49437 q^{93} -4.60586 q^{95} +14.0404 q^{97} -0.965121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 5 * q - 5 * q^5 + 2 * q^7 + 13 * q^9	\( 5 q - 5 q^{5} + 2 q^{7} + 13 q^{9} + q^{11} + 4 q^{13} + 4 q^{17} - 7 q^{19} + 6 q^{21} + 5 q^{23} + 5 q^{25} - 3 q^{27} + 4 q^{29} - 19 q^{31} + 17 q^{33} - 2 q^{35} + 15 q^{37} - 19 q^{39} + 25 q^{41} - 13 q^{45} + 11 q^{47} + 25 q^{49} - 19 q^{51} + 3 q^{53} - q^{55} + 48 q^{57} + q^{59} - 5 q^{61} + 41 q^{63} - 4 q^{65} - 9 q^{67} - q^{71} - q^{73} + 18 q^{77} + 2 q^{79} + 57 q^{81} + 45 q^{83} - 4 q^{85} + 9 q^{87} + 6 q^{89} - 11 q^{91} - 39 q^{93} + 7 q^{95} + 25 q^{97} + 65 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 + 2 * q^7 + 13 * q^9 + q^11 + 4 * q^13 + 4 * q^17 - 7 * q^19 + 6 * q^21 + 5 * q^23 + 5 * q^25 - 3 * q^27 + 4 * q^29 - 19 * q^31 + 17 * q^33 - 2 * q^35 + 15 * q^37 - 19 * q^39 + 25 * q^41 - 13 * q^45 + 11 * q^47 + 25 * q^49 - 19 * q^51 + 3 * q^53 - q^55 + 48 * q^57 + q^59 - 5 * q^61 + 41 * q^63 - 4 * q^65 - 9 * q^67 - q^71 - q^73 + 18 * q^77 + 2 * q^79 + 57 * q^81 + 45 * q^83 - 4 * q^85 + 9 * q^87 + 6 * q^89 - 11 * q^91 - 39 * q^93 + 7 * q^95 + 25 * q^97 + 65 * q^99

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