LMFDB - Embedded newform 8016.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.825785$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.77037 q^{5} -1.86579 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.77037 q^{5} -1.86579 q^{7} +1.00000 q^{9} -2.28773 q^{13} +2.77037 q^{15} +3.65157 q^{17} -4.84388 q^{19} -1.86579 q^{21} +4.84388 q^{23} +2.67495 q^{25} +1.00000 q^{27} +0.459260 q^{29} +6.70967 q^{31} -5.16893 q^{35} +9.26582 q^{37} -2.28773 q^{39} -6.03619 q^{41} +2.63616 q^{43} +2.77037 q^{45} -0.325049 q^{47} -3.51883 q^{49} +3.65157 q^{51} +7.61425 q^{53} -4.84388 q^{57} -7.69426 q^{59} +3.54074 q^{61} -1.86579 q^{63} -6.33786 q^{65} +11.1082 q^{67} +4.84388 q^{69} -8.14702 q^{71} +12.1162 q^{73} +2.67495 q^{75} -4.73305 q^{79} +1.00000 q^{81} +5.50342 q^{83} +10.1162 q^{85} +0.459260 q^{87} +7.16893 q^{89} +4.26842 q^{91} +6.70967 q^{93} -13.4193 q^{95} -3.21569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9} + 8 q^{13} + 5 q^{15} + 10 q^{17} + 2 q^{19} - q^{21} - 2 q^{23} + 5 q^{25} + 4 q^{27} + 14 q^{29} - q^{31} - 5 q^{35} + 5 q^{37} + 8 q^{39} + 14 q^{41} - 2 q^{43} + 5 q^{45} - 7 q^{47} + 13 q^{49} + 10 q^{51} + 3 q^{53} + 2 q^{57} + 5 q^{59} + 2 q^{61} - q^{63} + 6 q^{65} + 7 q^{67} - 2 q^{69} - 2 q^{71} + 2 q^{73} + 5 q^{75} + 10 q^{79} + 4 q^{81} - 13 q^{83} - 6 q^{85} + 14 q^{87} + 13 q^{89} + 30 q^{91} - q^{93} + 2 q^{95} + 5 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9 + 8 * q^13 + 5 * q^15 + 10 * q^17 + 2 * q^19 - q^21 - 2 * q^23 + 5 * q^25 + 4 * q^27 + 14 * q^29 - q^31 - 5 * q^35 + 5 * q^37 + 8 * q^39 + 14 * q^41 - 2 * q^43 + 5 * q^45 - 7 * q^47 + 13 * q^49 + 10 * q^51 + 3 * q^53 + 2 * q^57 + 5 * q^59 + 2 * q^61 - q^63 + 6 * q^65 + 7 * q^67 - 2 * q^69 - 2 * q^71 + 2 * q^73 + 5 * q^75 + 10 * q^79 + 4 * q^81 - 13 * q^83 - 6 * q^85 + 14 * q^87 + 13 * q^89 + 30 * q^91 - q^93 + 2 * q^95 + 5 * q^97

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.77037	1.23895	0.619474	−	0.785017i	$-0.287346\pi$
$5$	2.77037	1.23895	0.619474	+	0.785017i	$0.287346\pi$
$6$	0	0
$7$	−1.86579	−0.705202	−0.352601	−	0.935774i	$-0.614703\pi$
$7$	−1.86579	−0.705202	−0.352601	+	0.935774i	$0.614703\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.28773	−0.634502	−0.317251	−	0.948342i	$-0.602760\pi$
$13$	−2.28773	−0.634502	−0.317251	+	0.948342i	$0.602760\pi$
$14$	0	0
$15$	2.77037	0.715306
$16$	0	0
$17$	3.65157	0.885636	0.442818	−	0.896612i	$-0.353979\pi$
$17$	3.65157	0.885636	0.442818	+	0.896612i	$0.353979\pi$
$18$	0	0
$19$	−4.84388	−1.11126	−0.555631	−	0.831429i	$-0.687523\pi$
$19$	−4.84388	−1.11126	−0.555631	+	0.831429i	$0.687523\pi$
$20$	0	0
$21$	−1.86579	−0.407149
$22$	0	0
$23$	4.84388	1.01002	0.505009	−	0.863114i	$-0.331489\pi$
$23$	4.84388	1.01002	0.505009	+	0.863114i	$0.331489\pi$
$24$	0	0
$25$	2.67495	0.534990
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.459260	0.0852824	0.0426412	−	0.999090i	$-0.486423\pi$
$29$	0.459260	0.0852824	0.0426412	+	0.999090i	$0.486423\pi$
$30$	0	0
$31$	6.70967	1.20509	0.602546	−	0.798084i	$-0.294153\pi$
$31$	6.70967	1.20509	0.602546	+	0.798084i	$0.294153\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.16893	−0.873708
$36$	0	0
$37$	9.26582	1.52329	0.761646	−	0.647994i	$-0.224392\pi$
$37$	9.26582	1.52329	0.761646	+	0.647994i	$0.224392\pi$
$38$	0	0
$39$	−2.28773	−0.366330
$40$	0	0
$41$	−6.03619	−0.942694	−0.471347	−	0.881948i	$-0.656232\pi$
$41$	−6.03619	−0.942694	−0.471347	+	0.881948i	$0.656232\pi$
$42$	0	0
$43$	2.63616	0.402011	0.201005	−	0.979590i	$-0.435579\pi$
$43$	2.63616	0.402011	0.201005	+	0.979590i	$0.435579\pi$
$44$	0	0
$45$	2.77037	0.412982
$46$	0	0
$47$	−0.325049	−0.0474133	−0.0237067	−	0.999719i	$-0.507547\pi$
$47$	−0.325049	−0.0474133	−0.0237067	+	0.999719i	$0.507547\pi$
$48$	0	0
$49$	−3.51883	−0.502690
$50$	0	0
$51$	3.65157	0.511322
$52$	0	0
$53$	7.61425	1.04590	0.522949	−	0.852364i	$-0.324832\pi$
$53$	7.61425	1.04590	0.522949	+	0.852364i	$0.324832\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.84388	−0.641587
$58$	0	0
$59$	−7.69426	−1.00171	−0.500854	−	0.865532i	$-0.666980\pi$
$59$	−7.69426	−1.00171	−0.500854	+	0.865532i	$0.666980\pi$
$60$	0	0
$61$	3.54074	0.453345	0.226673	−	0.973971i	$-0.427215\pi$
$61$	3.54074	0.453345	0.226673	+	0.973971i	$0.427215\pi$
$62$	0	0
$63$	−1.86579	−0.235067
$64$	0	0
$65$	−6.33786	−0.786114
$66$	0	0
$67$	11.1082	1.35709	0.678543	−	0.734561i	$-0.262612\pi$
$67$	11.1082	1.35709	0.678543	+	0.734561i	$0.262612\pi$
$68$	0	0
$69$	4.84388	0.583134
$70$	0	0
$71$	−8.14702	−0.966873	−0.483436	−	0.875379i	$-0.660612\pi$
$71$	−8.14702	−0.966873	−0.483436	+	0.875379i	$0.660612\pi$
$72$	0	0
$73$	12.1162	1.41809	0.709047	−	0.705161i	$-0.249125\pi$
$73$	12.1162	1.41809	0.709047	+	0.705161i	$0.249125\pi$
$74$	0	0
$75$	2.67495	0.308877
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.73305	−0.532510	−0.266255	−	0.963903i	$-0.585786\pi$
$79$	−4.73305	−0.532510	−0.266255	+	0.963903i	$0.585786\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.50342	0.604079	0.302039	−	0.953295i	$-0.402333\pi$
$83$	5.50342	0.604079	0.302039	+	0.953295i	$0.402333\pi$
$84$	0	0
$85$	10.1162	1.09726
$86$	0	0
$87$	0.459260	0.0492378
$88$	0	0
$89$	7.16893	0.759905	0.379952	−	0.925006i	$-0.375940\pi$
$89$	7.16893	0.759905	0.379952	+	0.925006i	$0.375940\pi$
$90$	0	0
$91$	4.26842	0.447452
$92$	0	0
$93$	6.70967	0.695760
$94$	0	0
$95$	−13.4193	−1.37679
$96$	0	0
$97$	−3.21569	−0.326504	−0.163252	−	0.986584i	$-0.552198\pi$
$97$	−3.21569	−0.326504	−0.163252	+	0.986584i	$0.552198\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.53667	−0.948934	−0.474467	−	0.880273i	$-0.657359\pi$
$101$	−9.53667	−0.948934	−0.474467	+	0.880273i	$0.657359\pi$
$102$	0	0
$103$	13.1615	1.29684	0.648420	−	0.761283i	$-0.275430\pi$
$103$	13.1615	1.29684	0.648420	+	0.761283i	$0.275430\pi$
$104$	0	0
$105$	−5.16893	−0.504436
$106$	0	0
$107$	9.22556	0.891868	0.445934	−	0.895066i	$-0.352872\pi$
$107$	9.22556	0.891868	0.445934	+	0.895066i	$0.352872\pi$
$108$	0	0
$109$	−5.90605	−0.565697	−0.282849	−	0.959165i	$-0.591279\pi$
$109$	−5.90605	−0.565697	−0.282849	+	0.959165i	$0.591279\pi$
$110$	0	0
$111$	9.26582	0.879472
$112$	0	0
$113$	−1.41397	−0.133015	−0.0665074	−	0.997786i	$-0.521186\pi$
$113$	−1.41397	−0.133015	−0.0665074	+	0.997786i	$0.521186\pi$
$114$	0	0
$115$	13.4193	1.25136
$116$	0	0
$117$	−2.28773	−0.211501
$118$	0	0
$119$	−6.81306	−0.624552
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−6.03619	−0.544265
$124$	0	0
$125$	−6.44125	−0.576123
$126$	0	0
$127$	1.71877	0.152516	0.0762582	−	0.997088i	$-0.475703\pi$
$127$	1.71877	0.152516	0.0762582	+	0.997088i	$0.475703\pi$
$128$	0	0
$129$	2.63616	0.232101
$130$	0	0
$131$	6.47857	0.566035	0.283018	−	0.959115i	$-0.408665\pi$
$131$	6.47857	0.566035	0.283018	+	0.959115i	$0.408665\pi$
$132$	0	0
$133$	9.03766	0.783664
$134$	0	0
$135$	2.77037	0.238435
$136$	0	0
$137$	20.1598	1.72237	0.861185	−	0.508292i	$-0.169723\pi$
$137$	20.1598	1.72237	0.861185	+	0.508292i	$0.169723\pi$
$138$	0	0
$139$	11.4234	0.968921	0.484460	−	0.874813i	$-0.339016\pi$
$139$	11.4234	0.968921	0.484460	+	0.874813i	$0.339016\pi$
$140$	0	0
$141$	−0.325049	−0.0273741
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.27232	0.105660
$146$	0	0
$147$	−3.51883	−0.290228
$148$	0	0
$149$	1.22963	0.100735	0.0503676	−	0.998731i	$-0.483961\pi$
$149$	1.22963	0.100735	0.0503676	+	0.998731i	$0.483961\pi$
$150$	0	0
$151$	21.3085	1.73406	0.867031	−	0.498254i	$-0.166025\pi$
$151$	21.3085	1.73406	0.867031	+	0.498254i	$0.166025\pi$
$152$	0	0
$153$	3.65157	0.295212
$154$	0	0
$155$	18.5883	1.49305
$156$	0	0
$157$	7.68776	0.613550	0.306775	−	0.951782i	$-0.400750\pi$
$157$	7.68776	0.613550	0.306775	+	0.951782i	$0.400750\pi$
$158$	0	0
$159$	7.61425	0.603849
$160$	0	0
$161$	−9.03766	−0.712267
$162$	0	0
$163$	23.3020	1.82515	0.912577	−	0.408905i	$-0.134089\pi$
$163$	23.3020	1.82515	0.912577	+	0.408905i	$0.134089\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−7.76630	−0.597407
$170$	0	0
$171$	−4.84388	−0.370421
$172$	0	0
$173$	−3.54074	−0.269197	−0.134599	−	0.990900i	$-0.542975\pi$
$173$	−3.54074	−0.269197	−0.134599	+	0.990900i	$0.542975\pi$
$174$	0	0
$175$	−4.99090	−0.377276
$176$	0	0
$177$	−7.69426	−0.578336
$178$	0	0
$179$	13.6878	1.02307	0.511535	−	0.859262i	$-0.329077\pi$
$179$	13.6878	1.02307	0.511535	+	0.859262i	$0.329077\pi$
$180$	0	0
$181$	−21.4969	−1.59785	−0.798927	−	0.601428i	$-0.794598\pi$
$181$	−21.4969	−1.59785	−0.798927	+	0.601428i	$0.794598\pi$
$182$	0	0
$183$	3.54074	0.261739
$184$	0	0
$185$	25.6697	1.88728
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.86579	−0.135716
$190$	0	0
$191$	−20.9729	−1.51754	−0.758772	−	0.651356i	$-0.774200\pi$
$191$	−20.9729	−1.51754	−0.758772	+	0.651356i	$0.774200\pi$
$192$	0	0
$193$	2.57546	0.185386	0.0926928	−	0.995695i	$-0.470453\pi$
$193$	2.57546	0.185386	0.0926928	+	0.995695i	$0.470453\pi$
$194$	0	0
$195$	−6.33786	−0.453863
$196$	0	0
$197$	−3.36384	−0.239664	−0.119832	−	0.992794i	$-0.538236\pi$
$197$	−3.36384	−0.239664	−0.119832	+	0.992794i	$0.538236\pi$
$198$	0	0
$199$	8.57546	0.607898	0.303949	−	0.952688i	$-0.401695\pi$
$199$	8.57546	0.607898	0.303949	+	0.952688i	$0.401695\pi$
$200$	0	0
$201$	11.1082	0.783514
$202$	0	0
$203$	−0.856882	−0.0601413
$204$	0	0
$205$	−16.7225	−1.16795
$206$	0	0
$207$	4.84388	0.336673
$208$	0	0
$209$	0	0
$210$	0	0
$211$	11.1561	0.768019	0.384010	−	0.923329i	$-0.374543\pi$
$211$	11.1561	0.768019	0.384010	+	0.923329i	$0.374543\pi$
$212$	0	0
$213$	−8.14702	−0.558224
$214$	0	0
$215$	7.30314	0.498070
$216$	0	0
$217$	−12.5188	−0.849833
$218$	0	0
$219$	12.1162	0.818737
$220$	0	0
$221$	−8.35380	−0.561937
$222$	0	0
$223$	1.86579	0.124943	0.0624713	−	0.998047i	$-0.480102\pi$
$223$	1.86579	0.124943	0.0624713	+	0.998047i	$0.480102\pi$
$224$	0	0
$225$	2.67495	0.178330
$226$	0	0
$227$	−14.2259	−0.944206	−0.472103	−	0.881543i	$-0.656505\pi$
$227$	−14.2259	−0.944206	−0.472103	+	0.881543i	$0.656505\pi$
$228$	0	0
$229$	−8.65304	−0.571809	−0.285904	−	0.958258i	$-0.592294\pi$
$229$	−8.65304	−0.571809	−0.285904	+	0.958258i	$0.592294\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.81903	0.512241	0.256121	−	0.966645i	$-0.417556\pi$
$233$	7.81903	0.512241	0.256121	+	0.966645i	$0.417556\pi$
$234$	0	0
$235$	−0.900507	−0.0587426
$236$	0	0
$237$	−4.73305	−0.307445
$238$	0	0
$239$	−8.57546	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$239$	−8.57546	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$240$	0	0
$241$	4.19084	0.269956	0.134978	−	0.990849i	$-0.456904\pi$
$241$	4.19084	0.269956	0.134978	+	0.990849i	$0.456904\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−9.74846	−0.622806
$246$	0	0
$247$	11.0815	0.705098
$248$	0	0
$249$	5.50342	0.348765
$250$	0	0
$251$	−4.46220	−0.281652	−0.140826	−	0.990034i	$-0.544976\pi$
$251$	−4.46220	−0.281652	−0.140826	+	0.990034i	$0.544976\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	10.1162	0.633501
$256$	0	0
$257$	19.5769	1.22118	0.610588	−	0.791949i	$-0.290933\pi$
$257$	19.5769	1.22118	0.610588	+	0.791949i	$0.290933\pi$
$258$	0	0
$259$	−17.2881	−1.07423
$260$	0	0
$261$	0.459260	0.0284275
$262$	0	0
$263$	17.4013	1.07301	0.536506	−	0.843897i	$-0.319744\pi$
$263$	17.4013	1.07301	0.536506	+	0.843897i	$0.319744\pi$
$264$	0	0
$265$	21.0943	1.29581
$266$	0	0
$267$	7.16893	0.438731
$268$	0	0
$269$	7.37371	0.449583	0.224791	−	0.974407i	$-0.427830\pi$
$269$	7.37371	0.449583	0.224791	+	0.974407i	$0.427830\pi$
$270$	0	0
$271$	−14.9963	−0.910958	−0.455479	−	0.890246i	$-0.650532\pi$
$271$	−14.9963	−0.910958	−0.455479	+	0.890246i	$0.650532\pi$
$272$	0	0
$273$	4.26842	0.258337
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.4379	0.627152	0.313576	−	0.949563i	$-0.398473\pi$
$277$	10.4379	0.627152	0.313576	+	0.949563i	$0.398473\pi$
$278$	0	0
$279$	6.70967	0.401697
$280$	0	0
$281$	7.91255	0.472023	0.236012	−	0.971750i	$-0.424160\pi$
$281$	7.91255	0.472023	0.236012	+	0.971750i	$0.424160\pi$
$282$	0	0
$283$	−21.0507	−1.25133	−0.625666	−	0.780091i	$-0.715173\pi$
$283$	−21.0507	−1.25133	−0.625666	+	0.780091i	$0.715173\pi$
$284$	0	0
$285$	−13.4193	−0.794893
$286$	0	0
$287$	11.2623	0.664790
$288$	0	0
$289$	−3.66604	−0.215650
$290$	0	0
$291$	−3.21569	−0.188507
$292$	0	0
$293$	19.5407	1.14158	0.570791	−	0.821095i	$-0.306637\pi$
$293$	19.5407	1.14158	0.570791	+	0.821095i	$0.306637\pi$
$294$	0	0
$295$	−21.3159	−1.24106
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−11.0815	−0.640859
$300$	0	0
$301$	−4.91852	−0.283499
$302$	0	0
$303$	−9.53667	−0.547867
$304$	0	0
$305$	9.80916	0.561671
$306$	0	0
$307$	−2.99497	−0.170932	−0.0854659	−	0.996341i	$-0.527238\pi$
$307$	−2.99497	−0.170932	−0.0854659	+	0.996341i	$0.527238\pi$
$308$	0	0
$309$	13.1615	0.748731
$310$	0	0
$311$	13.2543	0.751583	0.375791	−	0.926704i	$-0.377371\pi$
$311$	13.2543	0.751583	0.375791	+	0.926704i	$0.377371\pi$
$312$	0	0
$313$	25.8318	1.46010	0.730051	−	0.683393i	$-0.239496\pi$
$313$	25.8318	1.46010	0.730051	+	0.683393i	$0.239496\pi$
$314$	0	0
$315$	−5.16893	−0.291236
$316$	0	0
$317$	−28.6917	−1.61148	−0.805742	−	0.592267i	$-0.798233\pi$
$317$	−28.6917	−1.61148	−0.805742	+	0.592267i	$0.798233\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	9.22556	0.514920
$322$	0	0
$323$	−17.6878	−0.984173
$324$	0	0
$325$	−6.11956	−0.339452
$326$	0	0
$327$	−5.90605	−0.326605
$328$	0	0
$329$	0.606474	0.0334360
$330$	0	0
$331$	−0.445321	−0.0244770	−0.0122385	−	0.999925i	$-0.503896\pi$
$331$	−0.445321	−0.0244770	−0.0122385	+	0.999925i	$0.503896\pi$
$332$	0	0
$333$	9.26582	0.507764
$334$	0	0
$335$	30.7739	1.68136
$336$	0	0
$337$	−33.6259	−1.83172	−0.915860	−	0.401497i	$-0.868490\pi$
$337$	−33.6259	−1.83172	−0.915860	+	0.401497i	$0.868490\pi$
$338$	0	0
$339$	−1.41397	−0.0767961
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.6259	1.05970
$344$	0	0
$345$	13.4193	0.722473
$346$	0	0
$347$	−9.08798	−0.487868	−0.243934	−	0.969792i	$-0.578438\pi$
$347$	−9.08798	−0.487868	−0.243934	+	0.969792i	$0.578438\pi$
$348$	0	0
$349$	−2.18434	−0.116925	−0.0584624	−	0.998290i	$-0.518620\pi$
$349$	−2.18434	−0.116925	−0.0584624	+	0.998290i	$0.518620\pi$
$350$	0	0
$351$	−2.28773	−0.122110
$352$	0	0
$353$	15.9126	0.846940	0.423470	−	0.905910i	$-0.360812\pi$
$353$	15.9126	0.846940	0.423470	+	0.905910i	$0.360812\pi$
$354$	0	0
$355$	−22.5703	−1.19790
$356$	0	0
$357$	−6.81306	−0.360585
$358$	0	0
$359$	−10.9601	−0.578451	−0.289225	−	0.957261i	$-0.593398\pi$
$359$	−10.9601	−0.578451	−0.289225	+	0.957261i	$0.593398\pi$
$360$	0	0
$361$	4.46316	0.234903
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	33.5664	1.75694
$366$	0	0
$367$	−13.2723	−0.692809	−0.346405	−	0.938085i	$-0.612598\pi$
$367$	−13.2723	−0.692809	−0.346405	+	0.938085i	$0.612598\pi$
$368$	0	0
$369$	−6.03619	−0.314231
$370$	0	0
$371$	−14.2066	−0.737569
$372$	0	0
$373$	−31.2736	−1.61929	−0.809643	−	0.586922i	$-0.800339\pi$
$373$	−31.2736	−1.61929	−0.809643	+	0.586922i	$0.800339\pi$
$374$	0	0
$375$	−6.44125	−0.332625
$376$	0	0
$377$	−1.05066	−0.0541118
$378$	0	0
$379$	11.9989	0.616340	0.308170	−	0.951331i	$-0.400283\pi$
$379$	11.9989	0.616340	0.308170	+	0.951331i	$0.400283\pi$
$380$	0	0
$381$	1.71877	0.0880554
$382$	0	0
$383$	−10.9215	−0.558061	−0.279030	−	0.960282i	$-0.590013\pi$
$383$	−10.9215	−0.558061	−0.279030	+	0.960282i	$0.590013\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.63616	0.134004
$388$	0	0
$389$	27.3148	1.38492	0.692458	−	0.721458i	$-0.256528\pi$
$389$	27.3148	1.38492	0.692458	+	0.721458i	$0.256528\pi$
$390$	0	0
$391$	17.6878	0.894508
$392$	0	0
$393$	6.47857	0.326801
$394$	0	0
$395$	−13.1123	−0.659751
$396$	0	0
$397$	9.57156	0.480383	0.240191	−	0.970726i	$-0.422790\pi$
$397$	9.57156	0.480383	0.240191	+	0.970726i	$0.422790\pi$
$398$	0	0
$399$	9.03766	0.452449
$400$	0	0
$401$	3.22313	0.160955	0.0804777	−	0.996756i	$-0.474355\pi$
$401$	3.22313	0.160955	0.0804777	+	0.996756i	$0.474355\pi$
$402$	0	0
$403$	−15.3499	−0.764633
$404$	0	0
$405$	2.77037	0.137661
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−4.41250	−0.218184	−0.109092	−	0.994032i	$-0.534794\pi$
$409$	−4.41250	−0.218184	−0.109092	+	0.994032i	$0.534794\pi$
$410$	0	0
$411$	20.1598	0.994411
$412$	0	0
$413$	14.3559	0.706406
$414$	0	0
$415$	15.2465	0.748422
$416$	0	0
$417$	11.4234	0.559407
$418$	0	0
$419$	1.11230	0.0543394	0.0271697	−	0.999631i	$-0.491351\pi$
$419$	1.11230	0.0543394	0.0271697	+	0.999631i	$0.491351\pi$
$420$	0	0
$421$	−9.88674	−0.481850	−0.240925	−	0.970544i	$-0.577451\pi$
$421$	−9.88674	−0.481850	−0.240925	+	0.970544i	$0.577451\pi$
$422$	0	0
$423$	−0.325049	−0.0158044
$424$	0	0
$425$	9.76777	0.473806
$426$	0	0
$427$	−6.60628	−0.319700
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.165029	0.00794917	0.00397459	−	0.999992i	$-0.498735\pi$
$431$	0.165029	0.00794917	0.00397459	+	0.999992i	$0.498735\pi$
$432$	0	0
$433$	1.78335	0.0857024	0.0428512	−	0.999081i	$-0.486356\pi$
$433$	1.78335	0.0857024	0.0428512	+	0.999081i	$0.486356\pi$
$434$	0	0
$435$	1.27232	0.0610031
$436$	0	0
$437$	−23.4632	−1.12240
$438$	0	0
$439$	−12.1914	−0.581861	−0.290931	−	0.956744i	$-0.593965\pi$
$439$	−12.1914	−0.581861	−0.290931	+	0.956744i	$0.593965\pi$
$440$	0	0
$441$	−3.51883	−0.167563
$442$	0	0
$443$	−0.919815	−0.0437017	−0.0218509	−	0.999761i	$-0.506956\pi$
$443$	−0.919815	−0.0437017	−0.0218509	+	0.999761i	$0.506956\pi$
$444$	0	0
$445$	19.8606	0.941482
$446$	0	0
$447$	1.22963	0.0581595
$448$	0	0
$449$	5.25951	0.248212	0.124106	−	0.992269i	$-0.460394\pi$
$449$	5.25951	0.248212	0.124106	+	0.992269i	$0.460394\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	21.3085	1.00116
$454$	0	0
$455$	11.8251	0.554369
$456$	0	0
$457$	−23.1847	−1.08453	−0.542267	−	0.840206i	$-0.682434\pi$
$457$	−23.1847	−1.08453	−0.542267	+	0.840206i	$0.682434\pi$
$458$	0	0
$459$	3.65157	0.170441
$460$	0	0
$461$	−26.5008	−1.23427	−0.617133	−	0.786859i	$-0.711706\pi$
$461$	−26.5008	−1.23427	−0.617133	+	0.786859i	$0.711706\pi$
$462$	0	0
$463$	−33.3553	−1.55015	−0.775075	−	0.631869i	$-0.782288\pi$
$463$	−33.3553	−1.55015	−0.775075	+	0.631869i	$0.782288\pi$
$464$	0	0
$465$	18.5883	0.862010
$466$	0	0
$467$	30.1886	1.39696	0.698480	−	0.715629i	$-0.253860\pi$
$467$	30.1886	1.39696	0.698480	+	0.715629i	$0.253860\pi$
$468$	0	0
$469$	−20.7256	−0.957020
$470$	0	0
$471$	7.68776	0.354233
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−12.9571	−0.594514
$476$	0	0
$477$	7.61425	0.348632
$478$	0	0
$479$	10.5398	0.481575	0.240787	−	0.970578i	$-0.422594\pi$
$479$	10.5398	0.481575	0.240787	+	0.970578i	$0.422594\pi$
$480$	0	0
$481$	−21.1977	−0.966531
$482$	0	0
$483$	−9.03766	−0.411228
$484$	0	0
$485$	−8.90865	−0.404521
$486$	0	0
$487$	9.00147	0.407896	0.203948	−	0.978982i	$-0.434623\pi$
$487$	9.00147	0.407896	0.203948	+	0.978982i	$0.434623\pi$
$488$	0	0
$489$	23.3020	1.05375
$490$	0	0
$491$	−36.6200	−1.65264	−0.826318	−	0.563204i	$-0.809568\pi$
$491$	−36.6200	−1.65264	−0.826318	+	0.563204i	$0.809568\pi$
$492$	0	0
$493$	1.67702	0.0755291
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.2006	0.681841
$498$	0	0
$499$	3.13924	0.140532	0.0702659	−	0.997528i	$-0.477615\pi$
$499$	3.13924	0.140532	0.0702659	+	0.997528i	$0.477615\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−22.0029	−0.981063	−0.490531	−	0.871424i	$-0.663197\pi$
$503$	−22.0029	−0.981063	−0.490531	+	0.871424i	$0.663197\pi$
$504$	0	0
$505$	−26.4201	−1.17568
$506$	0	0
$507$	−7.76630	−0.344913
$508$	0	0
$509$	−3.18694	−0.141259	−0.0706293	−	0.997503i	$-0.522501\pi$
$509$	−3.18694	−0.141259	−0.0706293	+	0.997503i	$0.522501\pi$
$510$	0	0
$511$	−22.6063	−1.00004
$512$	0	0
$513$	−4.84388	−0.213862
$514$	0	0
$515$	36.4622	1.60672
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−3.54074	−0.155421
$520$	0	0
$521$	15.6983	0.687756	0.343878	−	0.939014i	$-0.388259\pi$
$521$	15.6983	0.687756	0.343878	+	0.939014i	$0.388259\pi$
$522$	0	0
$523$	23.0325	1.00714	0.503569	−	0.863955i	$-0.332020\pi$
$523$	23.0325	1.00714	0.503569	+	0.863955i	$0.332020\pi$
$524$	0	0
$525$	−4.99090	−0.217821
$526$	0	0
$527$	24.5008	1.06727
$528$	0	0
$529$	0.463158	0.0201373
$530$	0	0
$531$	−7.69426	−0.333902
$532$	0	0
$533$	13.8092	0.598141
$534$	0	0
$535$	25.5582	1.10498
$536$	0	0
$537$	13.6878	0.590670
$538$	0	0
$539$	0	0
$540$	0	0
$541$	34.0269	1.46293	0.731466	−	0.681878i	$-0.238837\pi$
$541$	34.0269	1.46293	0.731466	+	0.681878i	$0.238837\pi$
$542$	0	0
$543$	−21.4969	−0.922521
$544$	0	0
$545$	−16.3619	−0.700869
$546$	0	0
$547$	−16.6620	−0.712414	−0.356207	−	0.934407i	$-0.615930\pi$
$547$	−16.6620	−0.712414	−0.356207	+	0.934407i	$0.615930\pi$
$548$	0	0
$549$	3.54074	0.151115
$550$	0	0
$551$	−2.22460	−0.0947711
$552$	0	0
$553$	8.83087	0.375527
$554$	0	0
$555$	25.6697	1.08962
$556$	0	0
$557$	−8.87470	−0.376033	−0.188016	−	0.982166i	$-0.560206\pi$
$557$	−8.87470	−0.376033	−0.188016	+	0.982166i	$0.560206\pi$
$558$	0	0
$559$	−6.03082	−0.255076
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.3885	0.479969	0.239984	−	0.970777i	$-0.422858\pi$
$563$	11.3885	0.479969	0.239984	+	0.970777i	$0.422858\pi$
$564$	0	0
$565$	−3.91721	−0.164798
$566$	0	0
$567$	−1.86579	−0.0783558
$568$	0	0
$569$	−23.6304	−0.990639	−0.495319	−	0.868711i	$-0.664949\pi$
$569$	−23.6304	−0.990639	−0.495319	+	0.868711i	$0.664949\pi$
$570$	0	0
$571$	4.46333	0.186785	0.0933923	−	0.995629i	$-0.470229\pi$
$571$	4.46333	0.186785	0.0933923	+	0.995629i	$0.470229\pi$
$572$	0	0
$573$	−20.9729	−0.876155
$574$	0	0
$575$	12.9571	0.540350
$576$	0	0
$577$	28.1319	1.17115	0.585574	−	0.810619i	$-0.300869\pi$
$577$	28.1319	1.17115	0.585574	+	0.810619i	$0.300869\pi$
$578$	0	0
$579$	2.57546	0.107032
$580$	0	0
$581$	−10.2682	−0.425998
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6.33786	−0.262038
$586$	0	0
$587$	7.58100	0.312901	0.156451	−	0.987686i	$-0.449995\pi$
$587$	7.58100	0.312901	0.156451	+	0.987686i	$0.449995\pi$
$588$	0	0
$589$	−32.5008	−1.33917
$590$	0	0
$591$	−3.36384	−0.138370
$592$	0	0
$593$	−14.1832	−0.582434	−0.291217	−	0.956657i	$-0.594060\pi$
$593$	−14.1832	−0.582434	−0.291217	+	0.956657i	$0.594060\pi$
$594$	0	0
$595$	−18.8747	−0.773787
$596$	0	0
$597$	8.57546	0.350970
$598$	0	0
$599$	−5.91255	−0.241580	−0.120790	−	0.992678i	$-0.538543\pi$
$599$	−5.91255	−0.241580	−0.120790	+	0.992678i	$0.538543\pi$
$600$	0	0
$601$	20.3150	0.828666	0.414333	−	0.910125i	$-0.364015\pi$
$601$	20.3150	0.828666	0.414333	+	0.910125i	$0.364015\pi$
$602$	0	0
$603$	11.1082	0.452362
$604$	0	0
$605$	−30.4741	−1.23895
$606$	0	0
$607$	−18.3403	−0.744409	−0.372205	−	0.928151i	$-0.621398\pi$
$607$	−18.3403	−0.744409	−0.372205	+	0.928151i	$0.621398\pi$
$608$	0	0
$609$	−0.856882	−0.0347226
$610$	0	0
$611$	0.743625	0.0300838
$612$	0	0
$613$	9.45830	0.382017	0.191009	−	0.981588i	$-0.438824\pi$
$613$	9.45830	0.382017	0.191009	+	0.981588i	$0.438824\pi$
$614$	0	0
$615$	−16.7225	−0.674315
$616$	0	0
$617$	15.5537	0.626170	0.313085	−	0.949725i	$-0.398637\pi$
$617$	15.5537	0.626170	0.313085	+	0.949725i	$0.398637\pi$
$618$	0	0
$619$	−17.7841	−0.714805	−0.357402	−	0.933951i	$-0.616338\pi$
$619$	−17.7841	−0.714805	−0.357402	+	0.933951i	$0.616338\pi$
$620$	0	0
$621$	4.84388	0.194378
$622$	0	0
$623$	−13.3757	−0.535887
$624$	0	0
$625$	−31.2194	−1.24878
$626$	0	0
$627$	0	0
$628$	0	0
$629$	33.8348	1.34908
$630$	0	0
$631$	−8.32505	−0.331415	−0.165707	−	0.986175i	$-0.552991\pi$
$631$	−8.32505	−0.331415	−0.165707	+	0.986175i	$0.552991\pi$
$632$	0	0
$633$	11.1561	0.443416
$634$	0	0
$635$	4.76164	0.188960
$636$	0	0
$637$	8.05013	0.318958
$638$	0	0
$639$	−8.14702	−0.322291
$640$	0	0
$641$	−46.4286	−1.83382	−0.916910	−	0.399094i	$-0.869325\pi$
$641$	−46.4286	−1.83382	−0.916910	+	0.399094i	$0.869325\pi$
$642$	0	0
$643$	−24.4839	−0.965552	−0.482776	−	0.875744i	$-0.660372\pi$
$643$	−24.4839	−0.965552	−0.482776	+	0.875744i	$0.660372\pi$
$644$	0	0
$645$	7.30314	0.287561
$646$	0	0
$647$	−32.1110	−1.26241	−0.631207	−	0.775615i	$-0.717440\pi$
$647$	−32.1110	−1.26241	−0.631207	+	0.775615i	$0.717440\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−12.5188	−0.490652
$652$	0	0
$653$	−32.3100	−1.26439	−0.632194	−	0.774810i	$-0.717845\pi$
$653$	−32.3100	−1.26439	−0.632194	+	0.774810i	$0.717845\pi$
$654$	0	0
$655$	17.9480	0.701288
$656$	0	0
$657$	12.1162	0.472698
$658$	0	0
$659$	17.6634	0.688070	0.344035	−	0.938957i	$-0.388206\pi$
$659$	17.6634	0.688070	0.344035	+	0.938957i	$0.388206\pi$
$660$	0	0
$661$	48.1800	1.87398	0.936992	−	0.349350i	$-0.113597\pi$
$661$	48.1800	1.87398	0.936992	+	0.349350i	$0.113597\pi$
$662$	0	0
$663$	−8.35380	−0.324435
$664$	0	0
$665$	25.0377	0.970919
$666$	0	0
$667$	2.22460	0.0861368
$668$	0	0
$669$	1.86579	0.0721356
$670$	0	0
$671$	0	0
$672$	0	0
$673$	37.5634	1.44796	0.723982	−	0.689819i	$-0.242310\pi$
$673$	37.5634	1.44796	0.723982	+	0.689819i	$0.242310\pi$
$674$	0	0
$675$	2.67495	0.102959
$676$	0	0
$677$	−16.0776	−0.617912	−0.308956	−	0.951076i	$-0.599980\pi$
$677$	−16.0776	−0.617912	−0.308956	+	0.951076i	$0.599980\pi$
$678$	0	0
$679$	5.99980	0.230251
$680$	0	0
$681$	−14.2259	−0.545137
$682$	0	0
$683$	−30.5847	−1.17029	−0.585146	−	0.810928i	$-0.698963\pi$
$683$	−30.5847	−1.17029	−0.585146	+	0.810928i	$0.698963\pi$
$684$	0	0
$685$	55.8502	2.13393
$686$	0	0
$687$	−8.65304	−0.330134
$688$	0	0
$689$	−17.4193	−0.663624
$690$	0	0
$691$	8.16390	0.310569	0.155285	−	0.987870i	$-0.450371\pi$
$691$	8.16390	0.310569	0.155285	+	0.987870i	$0.450371\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	31.6471	1.20044
$696$	0	0
$697$	−22.0416	−0.834884
$698$	0	0
$699$	7.81903	0.295743
$700$	0	0
$701$	−10.7948	−0.407716	−0.203858	−	0.979000i	$-0.565348\pi$
$701$	−10.7948	−0.407716	−0.203858	+	0.979000i	$0.565348\pi$
$702$	0	0
$703$	−44.8825	−1.69278
$704$	0	0
$705$	−0.900507	−0.0339151
$706$	0	0
$707$	17.7934	0.669190
$708$	0	0
$709$	36.6462	1.37628	0.688138	−	0.725580i	$-0.258428\pi$
$709$	36.6462	1.37628	0.688138	+	0.725580i	$0.258428\pi$
$710$	0	0
$711$	−4.73305	−0.177503
$712$	0	0
$713$	32.5008	1.21717
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.57546	−0.320256
$718$	0	0
$719$	0.797115	0.0297274	0.0148637	−	0.999890i	$-0.495269\pi$
$719$	0.797115	0.0297274	0.0148637	+	0.999890i	$0.495269\pi$
$720$	0	0
$721$	−24.5566	−0.914535
$722$	0	0
$723$	4.19084	0.155859
$724$	0	0
$725$	1.22850	0.0456252
$726$	0	0
$727$	−43.8401	−1.62594	−0.812970	−	0.582305i	$-0.802151\pi$
$727$	−43.8401	−1.62594	−0.812970	+	0.582305i	$0.802151\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.62612	0.356035
$732$	0	0
$733$	47.4917	1.75415	0.877073	−	0.480357i	$-0.159493\pi$
$733$	47.4917	1.75415	0.877073	+	0.480357i	$0.159493\pi$
$734$	0	0
$735$	−9.74846	−0.359577
$736$	0	0
$737$	0	0
$738$	0	0
$739$	36.7600	1.35224	0.676118	−	0.736793i	$-0.263661\pi$
$739$	36.7600	1.35224	0.676118	+	0.736793i	$0.263661\pi$
$740$	0	0
$741$	11.0815	0.407088
$742$	0	0
$743$	−37.7830	−1.38612	−0.693062	−	0.720878i	$-0.743739\pi$
$743$	−37.7830	−1.38612	−0.693062	+	0.720878i	$0.743739\pi$
$744$	0	0
$745$	3.40653	0.124806
$746$	0	0
$747$	5.50342	0.201360
$748$	0	0
$749$	−17.2129	−0.628947
$750$	0	0
$751$	0.746053	0.0272239	0.0136119	−	0.999907i	$-0.495667\pi$
$751$	0.746053	0.0272239	0.0136119	+	0.999907i	$0.495667\pi$
$752$	0	0
$753$	−4.46220	−0.162612
$754$	0	0
$755$	59.0325	2.14841
$756$	0	0
$757$	−43.9458	−1.59724	−0.798618	−	0.601838i	$-0.794435\pi$
$757$	−43.9458	−1.59724	−0.798618	+	0.601838i	$0.794435\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.4360	1.53831	0.769153	−	0.639065i	$-0.220679\pi$
$761$	42.4360	1.53831	0.769153	+	0.639065i	$0.220679\pi$
$762$	0	0
$763$	11.0194	0.398931
$764$	0	0
$765$	10.1162	0.365752
$766$	0	0
$767$	17.6024	0.635585
$768$	0	0
$769$	−10.4006	−0.375054	−0.187527	−	0.982259i	$-0.560047\pi$
$769$	−10.4006	−0.375054	−0.187527	+	0.982259i	$0.560047\pi$
$770$	0	0
$771$	19.5769	0.705046
$772$	0	0
$773$	−29.5960	−1.06450	−0.532248	−	0.846589i	$-0.678653\pi$
$773$	−29.5960	−1.06450	−0.532248	+	0.846589i	$0.678653\pi$
$774$	0	0
$775$	17.9480	0.644712
$776$	0	0
$777$	−17.2881	−0.620206
$778$	0	0
$779$	29.2386	1.04758
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0.459260	0.0164126
$784$	0	0
$785$	21.2979	0.760156
$786$	0	0
$787$	−34.4531	−1.22812	−0.614061	−	0.789259i	$-0.710465\pi$
$787$	−34.4531	−1.22812	−0.614061	+	0.789259i	$0.710465\pi$
$788$	0	0
$789$	17.4013	0.619504
$790$	0	0
$791$	2.63817	0.0938024
$792$	0	0
$793$	−8.10025	−0.287648
$794$	0	0
$795$	21.0943	0.748137
$796$	0	0
$797$	19.5239	0.691571	0.345785	−	0.938314i	$-0.387613\pi$
$797$	19.5239	0.691571	0.345785	+	0.938314i	$0.387613\pi$
$798$	0	0
$799$	−1.18694	−0.0419909
$800$	0	0
$801$	7.16893	0.253302
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−25.0377	−0.882461
$806$	0	0
$807$	7.37371	0.259567
$808$	0	0
$809$	−13.8220	−0.485954	−0.242977	−	0.970032i	$-0.578124\pi$
$809$	−13.8220	−0.485954	−0.242977	+	0.970032i	$0.578124\pi$
$810$	0	0
$811$	14.3579	0.504173	0.252087	−	0.967705i	$-0.418883\pi$
$811$	14.3579	0.504173	0.252087	+	0.967705i	$0.418883\pi$
$812$	0	0
$813$	−14.9963	−0.525942
$814$	0	0
$815$	64.5552	2.26127
$816$	0	0
$817$	−12.7692	−0.446739
$818$	0	0
$819$	4.26842	0.149151
$820$	0	0
$821$	−0.819201	−0.0285903	−0.0142952	−	0.999898i	$-0.504550\pi$
$821$	−0.819201	−0.0285903	−0.0142952	+	0.999898i	$0.504550\pi$
$822$	0	0
$823$	−2.98553	−0.104069	−0.0520344	−	0.998645i	$-0.516571\pi$
$823$	−2.98553	−0.104069	−0.0520344	+	0.998645i	$0.516571\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.14461	0.178896	0.0894478	−	0.995992i	$-0.471490\pi$
$827$	5.14461	0.178896	0.0894478	+	0.995992i	$0.471490\pi$
$828$	0	0
$829$	36.5878	1.27075	0.635374	−	0.772204i	$-0.280846\pi$
$829$	36.5878	1.27075	0.635374	+	0.772204i	$0.280846\pi$
$830$	0	0
$831$	10.4379	0.362086
$832$	0	0
$833$	−12.8492	−0.445200
$834$	0	0
$835$	2.77037	0.0958726
$836$	0	0
$837$	6.70967	0.231920
$838$	0	0
$839$	−40.3484	−1.39298	−0.696491	−	0.717566i	$-0.745256\pi$
$839$	−40.3484	−1.39298	−0.696491	+	0.717566i	$0.745256\pi$
$840$	0	0
$841$	−28.7891	−0.992727
$842$	0	0
$843$	7.91255	0.272523
$844$	0	0
$845$	−21.5155	−0.740156
$846$	0	0
$847$	20.5237	0.705202
$848$	0	0
$849$	−21.0507	−0.722457
$850$	0	0
$851$	44.8825	1.53855
$852$	0	0
$853$	23.1977	0.794273	0.397137	−	0.917759i	$-0.370004\pi$
$853$	23.1977	0.794273	0.397137	+	0.917759i	$0.370004\pi$
$854$	0	0
$855$	−13.4193	−0.458932
$856$	0	0
$857$	18.2095	0.622026	0.311013	−	0.950406i	$-0.399332\pi$
$857$	18.2095	0.622026	0.311013	+	0.950406i	$0.399332\pi$
$858$	0	0
$859$	−26.6170	−0.908161	−0.454080	−	0.890961i	$-0.650032\pi$
$859$	−26.6170	−0.908161	−0.454080	+	0.890961i	$0.650032\pi$
$860$	0	0
$861$	11.2623	0.383817
$862$	0	0
$863$	−43.9617	−1.49647	−0.748237	−	0.663432i	$-0.769099\pi$
$863$	−43.9617	−1.49647	−0.748237	+	0.663432i	$0.769099\pi$
$864$	0	0
$865$	−9.80916	−0.333521
$866$	0	0
$867$	−3.66604	−0.124505
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−25.4126	−0.861073
$872$	0	0
$873$	−3.21569	−0.108835
$874$	0	0
$875$	12.0180	0.406283
$876$	0	0
$877$	−9.53780	−0.322069	−0.161034	−	0.986949i	$-0.551483\pi$
$877$	−9.53780	−0.322069	−0.161034	+	0.986949i	$0.551483\pi$
$878$	0	0
$879$	19.5407	0.659093
$880$	0	0
$881$	37.0430	1.24801	0.624006	−	0.781420i	$-0.285504\pi$
$881$	37.0430	1.24801	0.624006	+	0.781420i	$0.285504\pi$
$882$	0	0
$883$	19.3629	0.651614	0.325807	−	0.945436i	$-0.394364\pi$
$883$	19.3629	0.651614	0.325807	+	0.945436i	$0.394364\pi$
$884$	0	0
$885$	−21.3159	−0.716528
$886$	0	0
$887$	44.5673	1.49642	0.748212	−	0.663460i	$-0.230913\pi$
$887$	44.5673	1.49642	0.748212	+	0.663460i	$0.230913\pi$
$888$	0	0
$889$	−3.20687	−0.107555
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.57450	0.0526886
$894$	0	0
$895$	37.9202	1.26753
$896$	0	0
$897$	−11.0815	−0.370000
$898$	0	0
$899$	3.08148	0.102773
$900$	0	0
$901$	27.8040	0.926284
$902$	0	0
$903$	−4.91852	−0.163678
$904$	0	0
$905$	−59.5544	−1.97966
$906$	0	0
$907$	−14.3846	−0.477633	−0.238817	−	0.971065i	$-0.576760\pi$
$907$	−14.3846	−0.477633	−0.238817	+	0.971065i	$0.576760\pi$
$908$	0	0
$909$	−9.53667	−0.316311
$910$	0	0
$911$	8.95988	0.296854	0.148427	−	0.988923i	$-0.452579\pi$
$911$	8.95988	0.296854	0.148427	+	0.988923i	$0.452579\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	9.80916	0.324281
$916$	0	0
$917$	−12.0876	−0.399169
$918$	0	0
$919$	1.95932	0.0646319	0.0323159	−	0.999478i	$-0.489712\pi$
$919$	1.95932	0.0646319	0.0323159	+	0.999478i	$0.489712\pi$
$920$	0	0
$921$	−2.99497	−0.0986876
$922$	0	0
$923$	18.6382	0.613483
$924$	0	0
$925$	24.7856	0.814946
$926$	0	0
$927$	13.1615	0.432280
$928$	0	0
$929$	−5.17187	−0.169684	−0.0848418	−	0.996394i	$-0.527038\pi$
$929$	−5.17187	−0.169684	−0.0848418	+	0.996394i	$0.527038\pi$
$930$	0	0
$931$	17.0448	0.558620
$932$	0	0
$933$	13.2543	0.433927
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.5625	−0.606409	−0.303204	−	0.952926i	$-0.598057\pi$
$937$	−18.5625	−0.606409	−0.303204	+	0.952926i	$0.598057\pi$
$938$	0	0
$939$	25.8318	0.842990
$940$	0	0
$941$	−31.7244	−1.03418	−0.517092	−	0.855930i	$-0.672986\pi$
$941$	−31.7244	−1.03418	−0.517092	+	0.855930i	$0.672986\pi$
$942$	0	0
$943$	−29.2386	−0.952139
$944$	0	0
$945$	−5.16893	−0.168145
$946$	0	0
$947$	29.5686	0.960851	0.480425	−	0.877036i	$-0.340482\pi$
$947$	29.5686	0.960851	0.480425	+	0.877036i	$0.340482\pi$
$948$	0	0
$949$	−27.7186	−0.899783
$950$	0	0
$951$	−28.6917	−0.930391
$952$	0	0
$953$	45.9337	1.48794	0.743969	−	0.668214i	$-0.232941\pi$
$953$	45.9337	1.48794	0.743969	+	0.668214i	$0.232941\pi$
$954$	0	0
$955$	−58.1026	−1.88016
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−37.6140	−1.21462
$960$	0	0
$961$	14.0196	0.452247
$962$	0	0
$963$	9.22556	0.297289
$964$	0	0
$965$	7.13497	0.229683
$966$	0	0
$967$	−58.4340	−1.87911	−0.939555	−	0.342398i	$-0.888761\pi$
$967$	−58.4340	−1.87911	−0.939555	+	0.342398i	$0.888761\pi$
$968$	0	0
$969$	−17.6878	−0.568213
$970$	0	0
$971$	41.4158	1.32910	0.664548	−	0.747246i	$-0.268624\pi$
$971$	41.4158	1.32910	0.664548	+	0.747246i	$0.268624\pi$
$972$	0	0
$973$	−21.3137	−0.683285
$974$	0	0
$975$	−6.11956	−0.195983
$976$	0	0
$977$	1.11179	0.0355692	0.0177846	−	0.999842i	$-0.494339\pi$
$977$	1.11179	0.0355692	0.0177846	+	0.999842i	$0.494339\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−5.90605	−0.188566
$982$	0	0
$983$	54.5294	1.73922	0.869608	−	0.493742i	$-0.164371\pi$
$983$	54.5294	1.73922	0.869608	+	0.493742i	$0.164371\pi$
$984$	0	0
$985$	−9.31908	−0.296931
$986$	0	0
$987$	0.606474	0.0193043
$988$	0	0
$989$	12.7692	0.406038
$990$	0	0
$991$	33.2060	1.05482	0.527411	−	0.849610i	$-0.323163\pi$
$991$	33.2060	1.05482	0.527411	+	0.849610i	$0.323163\pi$
$992$	0	0
$993$	−0.445321	−0.0141318
$994$	0	0
$995$	23.7572	0.753154
$996$	0	0
$997$	−6.03376	−0.191091	−0.0955455	−	0.995425i	$-0.530460\pi$
$997$	−6.03376	−0.191091	−0.0955455	+	0.995425i	$0.530460\pi$
$998$	0	0
$999$	9.26582	0.293157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.o.1.3		4
4.3	odd	2		1002.2.a.i.1.3	✓	4
12.11	even	2		3006.2.a.s.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.i.1.3	✓	4	4.3	odd	2
3006.2.a.s.1.2		4	12.11	even	2
8016.2.a.o.1.3		4	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.77037 q^{5} -1.86579 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.77037 q^{5} -1.86579 q^{7} +1.00000 q^{9} -2.28773 q^{13} +2.77037 q^{15} +3.65157 q^{17} -4.84388 q^{19} -1.86579 q^{21} +4.84388 q^{23} +2.67495 q^{25} +1.00000 q^{27} +0.459260 q^{29} +6.70967 q^{31} -5.16893 q^{35} +9.26582 q^{37} -2.28773 q^{39} -6.03619 q^{41} +2.63616 q^{43} +2.77037 q^{45} -0.325049 q^{47} -3.51883 q^{49} +3.65157 q^{51} +7.61425 q^{53} -4.84388 q^{57} -7.69426 q^{59} +3.54074 q^{61} -1.86579 q^{63} -6.33786 q^{65} +11.1082 q^{67} +4.84388 q^{69} -8.14702 q^{71} +12.1162 q^{73} +2.67495 q^{75} -4.73305 q^{79} +1.00000 q^{81} +5.50342 q^{83} +10.1162 q^{85} +0.459260 q^{87} +7.16893 q^{89} +4.26842 q^{91} +6.70967 q^{93} -13.4193 q^{95} -3.21569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9} + 8 q^{13} + 5 q^{15} + 10 q^{17} + 2 q^{19} - q^{21} - 2 q^{23} + 5 q^{25} + 4 q^{27} + 14 q^{29} - q^{31} - 5 q^{35} + 5 q^{37} + 8 q^{39} + 14 q^{41} - 2 q^{43} + 5 q^{45} - 7 q^{47} + 13 q^{49} + 10 q^{51} + 3 q^{53} + 2 q^{57} + 5 q^{59} + 2 q^{61} - q^{63} + 6 q^{65} + 7 q^{67} - 2 q^{69} - 2 q^{71} + 2 q^{73} + 5 q^{75} + 10 q^{79} + 4 q^{81} - 13 q^{83} - 6 q^{85} + 14 q^{87} + 13 q^{89} + 30 q^{91} - q^{93} + 2 q^{95} + 5 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9 + 8 * q^13 + 5 * q^15 + 10 * q^17 + 2 * q^19 - q^21 - 2 * q^23 + 5 * q^25 + 4 * q^27 + 14 * q^29 - q^31 - 5 * q^35 + 5 * q^37 + 8 * q^39 + 14 * q^41 - 2 * q^43 + 5 * q^45 - 7 * q^47 + 13 * q^49 + 10 * q^51 + 3 * q^53 + 2 * q^57 + 5 * q^59 + 2 * q^61 - q^63 + 6 * q^65 + 7 * q^67 - 2 * q^69 - 2 * q^71 + 2 * q^73 + 5 * q^75 + 10 * q^79 + 4 * q^81 - 13 * q^83 - 6 * q^85 + 14 * q^87 + 13 * q^89 + 30 * q^91 - q^93 + 2 * q^95 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more