LMFDB - Embedded newform 3856.2.a.n.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3856,2,Mod(1,3856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3856, 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("3856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3856 = 2^{4} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3856.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:	$30.7903150194$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.63125$ of defining polynomial
Character	$\chi$	$=$	3856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.16790 q^{3} +1.75438 q^{5} -5.06139 q^{7} -1.63601 q^{9} +O(q^{10})$	$q+1.16790 q^{3} +1.75438 q^{5} -5.06139 q^{7} -1.63601 q^{9} -1.08118 q^{11} +3.01110 q^{13} +2.04893 q^{15} +2.47710 q^{17} +7.12459 q^{19} -5.91119 q^{21} -5.33139 q^{23} -1.92217 q^{25} -5.41439 q^{27} -6.80248 q^{29} +8.37871 q^{31} -1.26271 q^{33} -8.87957 q^{35} -2.09398 q^{37} +3.51666 q^{39} -10.6564 q^{41} +5.49791 q^{43} -2.87018 q^{45} -4.90386 q^{47} +18.6176 q^{49} +2.89301 q^{51} -4.04410 q^{53} -1.89680 q^{55} +8.32081 q^{57} -5.64831 q^{59} -2.03784 q^{61} +8.28048 q^{63} +5.28260 q^{65} -7.65097 q^{67} -6.22653 q^{69} -12.2631 q^{71} +0.733715 q^{73} -2.24490 q^{75} +5.47229 q^{77} -6.86216 q^{79} -1.41544 q^{81} -5.58812 q^{83} +4.34576 q^{85} -7.94461 q^{87} +9.62798 q^{89} -15.2403 q^{91} +9.78549 q^{93} +12.4992 q^{95} -10.9185 q^{97} +1.76883 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	$ 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) $ 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * 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
$3$	1.16790	0.674287	0.337144	−	0.941453i	$-0.390539\pi$
$3$	1.16790	0.674287	0.337144	+	0.941453i	$0.390539\pi$
$4$	0	0
$5$	1.75438	0.784580	0.392290	−	0.919842i	$-0.371683\pi$
$5$	1.75438	0.784580	0.392290	+	0.919842i	$0.371683\pi$
$6$	0	0
$7$	−5.06139	−1.91302	−0.956512	−	0.291693i	$-0.905782\pi$
$7$	−5.06139	−1.91302	−0.956512	+	0.291693i	$0.905782\pi$
$8$	0	0
$9$	−1.63601	−0.545337
$10$	0	0
$11$	−1.08118	−0.325989	−0.162995	−	0.986627i	$-0.552115\pi$
$11$	−1.08118	−0.325989	−0.162995	+	0.986627i	$0.552115\pi$
$12$	0	0
$13$	3.01110	0.835129	0.417565	−	0.908647i	$-0.362884\pi$
$13$	3.01110	0.835129	0.417565	+	0.908647i	$0.362884\pi$
$14$	0	0
$15$	2.04893	0.529032
$16$	0	0
$17$	2.47710	0.600785	0.300393	−	0.953816i	$-0.402882\pi$
$17$	2.47710	0.600785	0.300393	+	0.953816i	$0.402882\pi$
$18$	0	0
$19$	7.12459	1.63449	0.817247	−	0.576287i	$-0.195499\pi$
$19$	7.12459	1.63449	0.817247	+	0.576287i	$0.195499\pi$
$20$	0	0
$21$	−5.91119	−1.28993
$22$	0	0
$23$	−5.33139	−1.11167	−0.555836	−	0.831292i	$-0.687602\pi$
$23$	−5.33139	−1.11167	−0.555836	+	0.831292i	$0.687602\pi$
$24$	0	0
$25$	−1.92217	−0.384434
$26$	0	0
$27$	−5.41439	−1.04200
$28$	0	0
$29$	−6.80248	−1.26319	−0.631595	−	0.775299i	$-0.717599\pi$
$29$	−6.80248	−1.26319	−0.631595	+	0.775299i	$0.717599\pi$
$30$	0	0
$31$	8.37871	1.50486	0.752430	−	0.658672i	$-0.228881\pi$
$31$	8.37871	1.50486	0.752430	+	0.658672i	$0.228881\pi$
$32$	0	0
$33$	−1.26271	−0.219810
$34$	0	0
$35$	−8.87957	−1.50092
$36$	0	0
$37$	−2.09398	−0.344249	−0.172124	−	0.985075i	$-0.555063\pi$
$37$	−2.09398	−0.344249	−0.172124	+	0.985075i	$0.555063\pi$
$38$	0	0
$39$	3.51666	0.563117
$40$	0	0
$41$	−10.6564	−1.66425	−0.832124	−	0.554589i	$-0.812875\pi$
$41$	−10.6564	−1.66425	−0.832124	+	0.554589i	$0.812875\pi$
$42$	0	0
$43$	5.49791	0.838423	0.419211	−	0.907889i	$-0.362307\pi$
$43$	5.49791	0.838423	0.419211	+	0.907889i	$0.362307\pi$
$44$	0	0
$45$	−2.87018	−0.427861
$46$	0	0
$47$	−4.90386	−0.715302	−0.357651	−	0.933855i	$-0.616422\pi$
$47$	−4.90386	−0.715302	−0.357651	+	0.933855i	$0.616422\pi$
$48$	0	0
$49$	18.6176	2.65966
$50$	0	0
$51$	2.89301	0.405102
$52$	0	0
$53$	−4.04410	−0.555500	−0.277750	−	0.960653i	$-0.589589\pi$
$53$	−4.04410	−0.555500	−0.277750	+	0.960653i	$0.589589\pi$
$54$	0	0
$55$	−1.89680	−0.255765
$56$	0	0
$57$	8.32081	1.10212
$58$	0	0
$59$	−5.64831	−0.735347	−0.367673	−	0.929955i	$-0.619846\pi$
$59$	−5.64831	−0.735347	−0.367673	+	0.929955i	$0.619846\pi$
$60$	0	0
$61$	−2.03784	−0.260919	−0.130459	−	0.991454i	$-0.541645\pi$
$61$	−2.03784	−0.260919	−0.130459	+	0.991454i	$0.541645\pi$
$62$	0	0
$63$	8.28048	1.04324
$64$	0	0
$65$	5.28260	0.655226
$66$	0	0
$67$	−7.65097	−0.934715	−0.467358	−	0.884068i	$-0.654794\pi$
$67$	−7.65097	−0.934715	−0.467358	+	0.884068i	$0.654794\pi$
$68$	0	0
$69$	−6.22653	−0.749586
$70$	0	0
$71$	−12.2631	−1.45536	−0.727682	−	0.685915i	$-0.759402\pi$
$71$	−12.2631	−1.45536	−0.727682	+	0.685915i	$0.759402\pi$
$72$	0	0
$73$	0.733715	0.0858748	0.0429374	−	0.999078i	$-0.486328\pi$
$73$	0.733715	0.0858748	0.0429374	+	0.999078i	$0.486328\pi$
$74$	0	0
$75$	−2.24490	−0.259219
$76$	0	0
$77$	5.47229	0.623625
$78$	0	0
$79$	−6.86216	−0.772053	−0.386027	−	0.922488i	$-0.626153\pi$
$79$	−6.86216	−0.772053	−0.386027	+	0.922488i	$0.626153\pi$
$80$	0	0
$81$	−1.41544	−0.157271
$82$	0	0
$83$	−5.58812	−0.613376	−0.306688	−	0.951810i	$-0.599221\pi$
$83$	−5.58812	−0.613376	−0.306688	+	0.951810i	$0.599221\pi$
$84$	0	0
$85$	4.34576	0.471364
$86$	0	0
$87$	−7.94461	−0.851752
$88$	0	0
$89$	9.62798	1.02056	0.510282	−	0.860007i	$-0.329541\pi$
$89$	9.62798	1.02056	0.510282	+	0.860007i	$0.329541\pi$
$90$	0	0
$91$	−15.2403	−1.59762
$92$	0	0
$93$	9.78549	1.01471
$94$	0	0
$95$	12.4992	1.28239
$96$	0	0
$97$	−10.9185	−1.10861	−0.554303	−	0.832315i	$-0.687015\pi$
$97$	−10.9185	−1.10861	−0.554303	+	0.832315i	$0.687015\pi$
$98$	0	0
$99$	1.76883	0.177774
$100$	0	0
$101$	−13.9401	−1.38709	−0.693545	−	0.720413i	$-0.743952\pi$
$101$	−13.9401	−1.38709	−0.693545	+	0.720413i	$0.743952\pi$
$102$	0	0
$103$	7.14621	0.704137	0.352068	−	0.935974i	$-0.385478\pi$
$103$	7.14621	0.704137	0.352068	+	0.935974i	$0.385478\pi$
$104$	0	0
$105$	−10.3704	−1.01205
$106$	0	0
$107$	−17.8929	−1.72977	−0.864886	−	0.501968i	$-0.832610\pi$
$107$	−17.8929	−1.72977	−0.864886	+	0.501968i	$0.832610\pi$
$108$	0	0
$109$	−14.9297	−1.43001	−0.715004	−	0.699120i	$-0.753575\pi$
$109$	−14.9297	−1.43001	−0.715004	+	0.699120i	$0.753575\pi$
$110$	0	0
$111$	−2.44556	−0.232123
$112$	0	0
$113$	0.859306	0.0808367	0.0404184	−	0.999183i	$-0.487131\pi$
$113$	0.859306	0.0808367	0.0404184	+	0.999183i	$0.487131\pi$
$114$	0	0
$115$	−9.35326	−0.872196
$116$	0	0
$117$	−4.92619	−0.455427
$118$	0	0
$119$	−12.5376	−1.14932
$120$	0	0
$121$	−9.83104	−0.893731
$122$	0	0
$123$	−12.4456	−1.12218
$124$	0	0
$125$	−12.1441	−1.08620
$126$	0	0
$127$	3.01886	0.267880	0.133940	−	0.990989i	$-0.457237\pi$
$127$	3.01886	0.267880	0.133940	+	0.990989i	$0.457237\pi$
$128$	0	0
$129$	6.42100	0.565338
$130$	0	0
$131$	2.17519	0.190047	0.0950234	−	0.995475i	$-0.469707\pi$
$131$	2.17519	0.190047	0.0950234	+	0.995475i	$0.469707\pi$
$132$	0	0
$133$	−36.0603	−3.12683
$134$	0	0
$135$	−9.49888	−0.817533
$136$	0	0
$137$	15.9150	1.35971	0.679854	−	0.733347i	$-0.262043\pi$
$137$	15.9150	1.35971	0.679854	+	0.733347i	$0.262043\pi$
$138$	0	0
$139$	15.0721	1.27840	0.639201	−	0.769040i	$-0.279265\pi$
$139$	15.0721	1.27840	0.639201	+	0.769040i	$0.279265\pi$
$140$	0	0
$141$	−5.72722	−0.482319
$142$	0	0
$143$	−3.25555	−0.272243
$144$	0	0
$145$	−11.9341	−0.991073
$146$	0	0
$147$	21.7435	1.79338
$148$	0	0
$149$	−1.36774	−0.112050	−0.0560250	−	0.998429i	$-0.517843\pi$
$149$	−1.36774	−0.112050	−0.0560250	+	0.998429i	$0.517843\pi$
$150$	0	0
$151$	−17.1854	−1.39853	−0.699264	−	0.714864i	$-0.746489\pi$
$151$	−17.1854	−1.39853	−0.699264	+	0.714864i	$0.746489\pi$
$152$	0	0
$153$	−4.05256	−0.327630
$154$	0	0
$155$	14.6994	1.18068
$156$	0	0
$157$	12.1279	0.967909	0.483955	−	0.875093i	$-0.339200\pi$
$157$	12.1279	0.967909	0.483955	+	0.875093i	$0.339200\pi$
$158$	0	0
$159$	−4.72310	−0.374566
$160$	0	0
$161$	26.9842	2.12665
$162$	0	0
$163$	−10.7083	−0.838737	−0.419369	−	0.907816i	$-0.637749\pi$
$163$	−10.7083	−0.838737	−0.419369	+	0.907816i	$0.637749\pi$
$164$	0	0
$165$	−2.21527	−0.172459
$166$	0	0
$167$	10.5539	0.816684	0.408342	−	0.912829i	$-0.366107\pi$
$167$	10.5539	0.816684	0.408342	+	0.912829i	$0.366107\pi$
$168$	0	0
$169$	−3.93327	−0.302559
$170$	0	0
$171$	−11.6559	−0.891350
$172$	0	0
$173$	−2.07621	−0.157851	−0.0789255	−	0.996881i	$-0.525149\pi$
$173$	−2.07621	−0.157851	−0.0789255	+	0.996881i	$0.525149\pi$
$174$	0	0
$175$	9.72883	0.735431
$176$	0	0
$177$	−6.59665	−0.495835
$178$	0	0
$179$	3.06922	0.229405	0.114702	−	0.993400i	$-0.463409\pi$
$179$	3.06922	0.229405	0.114702	+	0.993400i	$0.463409\pi$
$180$	0	0
$181$	−0.391403	−0.0290927	−0.0145464	−	0.999894i	$-0.504630\pi$
$181$	−0.391403	−0.0290927	−0.0145464	+	0.999894i	$0.504630\pi$
$182$	0	0
$183$	−2.37999	−0.175934
$184$	0	0
$185$	−3.67363	−0.270091
$186$	0	0
$187$	−2.67820	−0.195849
$188$	0	0
$189$	27.4043	1.99337
$190$	0	0
$191$	−4.99018	−0.361077	−0.180538	−	0.983568i	$-0.557784\pi$
$191$	−4.99018	−0.361077	−0.180538	+	0.983568i	$0.557784\pi$
$192$	0	0
$193$	13.5209	0.973254	0.486627	−	0.873610i	$-0.338227\pi$
$193$	13.5209	0.973254	0.486627	+	0.873610i	$0.338227\pi$
$194$	0	0
$195$	6.16955	0.441810
$196$	0	0
$197$	5.86751	0.418043	0.209021	−	0.977911i	$-0.432972\pi$
$197$	5.86751	0.418043	0.209021	+	0.977911i	$0.432972\pi$
$198$	0	0
$199$	−12.5774	−0.891585	−0.445792	−	0.895136i	$-0.647078\pi$
$199$	−12.5774	−0.891585	−0.445792	+	0.895136i	$0.647078\pi$
$200$	0	0
$201$	−8.93557	−0.630266
$202$	0	0
$203$	34.4300	2.41651
$204$	0	0
$205$	−18.6953	−1.30574
$206$	0	0
$207$	8.72221	0.606236
$208$	0	0
$209$	−7.70299	−0.532827
$210$	0	0
$211$	−6.00657	−0.413509	−0.206755	−	0.978393i	$-0.566290\pi$
$211$	−6.00657	−0.413509	−0.206755	+	0.978393i	$0.566290\pi$
$212$	0	0
$213$	−14.3221	−0.981333
$214$	0	0
$215$	9.64539	0.657810
$216$	0	0
$217$	−42.4079	−2.87884
$218$	0	0
$219$	0.856905	0.0579043
$220$	0	0
$221$	7.45880	0.501733
$222$	0	0
$223$	25.7624	1.72518	0.862590	−	0.505904i	$-0.168841\pi$
$223$	25.7624	1.72518	0.862590	+	0.505904i	$0.168841\pi$
$224$	0	0
$225$	3.14469	0.209646
$226$	0	0
$227$	−5.03720	−0.334331	−0.167165	−	0.985929i	$-0.553461\pi$
$227$	−5.03720	−0.334331	−0.167165	+	0.985929i	$0.553461\pi$
$228$	0	0
$229$	21.1245	1.39594	0.697972	−	0.716125i	$-0.254086\pi$
$229$	21.1245	1.39594	0.697972	+	0.716125i	$0.254086\pi$
$230$	0	0
$231$	6.39108	0.420502
$232$	0	0
$233$	20.9468	1.37227	0.686135	−	0.727474i	$-0.259306\pi$
$233$	20.9468	1.37227	0.686135	+	0.727474i	$0.259306\pi$
$234$	0	0
$235$	−8.60322	−0.561212
$236$	0	0
$237$	−8.01431	−0.520586
$238$	0	0
$239$	7.83541	0.506831	0.253415	−	0.967358i	$-0.418446\pi$
$239$	7.83541	0.506831	0.253415	+	0.967358i	$0.418446\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	14.5901	0.935955
$244$	0	0
$245$	32.6623	2.08672
$246$	0	0
$247$	21.4529	1.36501
$248$	0	0
$249$	−6.52636	−0.413591
$250$	0	0
$251$	7.01648	0.442876	0.221438	−	0.975174i	$-0.428925\pi$
$251$	7.01648	0.442876	0.221438	+	0.975174i	$0.428925\pi$
$252$	0	0
$253$	5.76421	0.362393
$254$	0	0
$255$	5.07542	0.317835
$256$	0	0
$257$	11.3057	0.705229	0.352615	−	0.935769i	$-0.385293\pi$
$257$	11.3057	0.705229	0.352615	+	0.935769i	$0.385293\pi$
$258$	0	0
$259$	10.5985	0.658556
$260$	0	0
$261$	11.1289	0.688864
$262$	0	0
$263$	−19.8667	−1.22503	−0.612516	−	0.790458i	$-0.709842\pi$
$263$	−19.8667	−1.22503	−0.612516	+	0.790458i	$0.709842\pi$
$264$	0	0
$265$	−7.09487	−0.435834
$266$	0	0
$267$	11.2445	0.688153
$268$	0	0
$269$	−20.2533	−1.23486	−0.617431	−	0.786625i	$-0.711827\pi$
$269$	−20.2533	−1.23486	−0.617431	+	0.786625i	$0.711827\pi$
$270$	0	0
$271$	4.23892	0.257496	0.128748	−	0.991677i	$-0.458904\pi$
$271$	4.23892	0.257496	0.128748	+	0.991677i	$0.458904\pi$
$272$	0	0
$273$	−17.7992	−1.07726
$274$	0	0
$275$	2.07822	0.125321
$276$	0	0
$277$	−0.369469	−0.0221992	−0.0110996	−	0.999938i	$-0.503533\pi$
$277$	−0.369469	−0.0221992	−0.0110996	+	0.999938i	$0.503533\pi$
$278$	0	0
$279$	−13.7077	−0.820656
$280$	0	0
$281$	24.6918	1.47299	0.736494	−	0.676444i	$-0.236480\pi$
$281$	24.6918	1.47299	0.736494	+	0.676444i	$0.236480\pi$
$282$	0	0
$283$	−0.0696775	−0.00414190	−0.00207095	−	0.999998i	$-0.500659\pi$
$283$	−0.0696775	−0.00414190	−0.00207095	+	0.999998i	$0.500659\pi$
$284$	0	0
$285$	14.5978	0.864700
$286$	0	0
$287$	53.9361	3.18375
$288$	0	0
$289$	−10.8640	−0.639057
$290$	0	0
$291$	−12.7517	−0.747518
$292$	0	0
$293$	−7.59140	−0.443494	−0.221747	−	0.975104i	$-0.571176\pi$
$293$	−7.59140	−0.443494	−0.221747	+	0.975104i	$0.571176\pi$
$294$	0	0
$295$	−9.90925	−0.576939
$296$	0	0
$297$	5.85395	0.339681
$298$	0	0
$299$	−16.0534	−0.928389
$300$	0	0
$301$	−27.8270	−1.60392
$302$	0	0
$303$	−16.2806	−0.935297
$304$	0	0
$305$	−3.57514	−0.204712
$306$	0	0
$307$	20.6437	1.17820	0.589100	−	0.808060i	$-0.299482\pi$
$307$	20.6437	1.17820	0.589100	+	0.808060i	$0.299482\pi$
$308$	0	0
$309$	8.34605	0.474790
$310$	0	0
$311$	−24.0882	−1.36592	−0.682959	−	0.730456i	$-0.739307\pi$
$311$	−24.0882	−1.36592	−0.682959	+	0.730456i	$0.739307\pi$
$312$	0	0
$313$	16.3804	0.925876	0.462938	−	0.886391i	$-0.346795\pi$
$313$	16.3804	0.925876	0.462938	+	0.886391i	$0.346795\pi$
$314$	0	0
$315$	14.5271	0.818508
$316$	0	0
$317$	−1.59833	−0.0897714	−0.0448857	−	0.998992i	$-0.514292\pi$
$317$	−1.59833	−0.0897714	−0.0448857	+	0.998992i	$0.514292\pi$
$318$	0	0
$319$	7.35473	0.411786
$320$	0	0
$321$	−20.8971	−1.16636
$322$	0	0
$323$	17.6483	0.981980
$324$	0	0
$325$	−5.78784	−0.321052
$326$	0	0
$327$	−17.4364	−0.964236
$328$	0	0
$329$	24.8203	1.36839
$330$	0	0
$331$	26.9070	1.47894	0.739472	−	0.673188i	$-0.235075\pi$
$331$	26.9070	1.47894	0.739472	+	0.673188i	$0.235075\pi$
$332$	0	0
$333$	3.42578	0.187732
$334$	0	0
$335$	−13.4227	−0.733359
$336$	0	0
$337$	−10.8794	−0.592640	−0.296320	−	0.955089i	$-0.595760\pi$
$337$	−10.8794	−0.592640	−0.296320	+	0.955089i	$0.595760\pi$
$338$	0	0
$339$	1.00358	0.0545072
$340$	0	0
$341$	−9.05892	−0.490568
$342$	0	0
$343$	−58.8013	−3.17497
$344$	0	0
$345$	−10.9237	−0.588110
$346$	0	0
$347$	−12.9007	−0.692546	−0.346273	−	0.938134i	$-0.612553\pi$
$347$	−12.9007	−0.692546	−0.346273	+	0.938134i	$0.612553\pi$
$348$	0	0
$349$	−10.7803	−0.577058	−0.288529	−	0.957471i	$-0.593166\pi$
$349$	−10.7803	−0.577058	−0.288529	+	0.957471i	$0.593166\pi$
$350$	0	0
$351$	−16.3033	−0.870205
$352$	0	0
$353$	6.56859	0.349611	0.174805	−	0.984603i	$-0.444070\pi$
$353$	6.56859	0.349611	0.174805	+	0.984603i	$0.444070\pi$
$354$	0	0
$355$	−21.5141	−1.14185
$356$	0	0
$357$	−14.6426	−0.774970
$358$	0	0
$359$	−15.8305	−0.835504	−0.417752	−	0.908561i	$-0.637182\pi$
$359$	−15.8305	−0.835504	−0.417752	+	0.908561i	$0.637182\pi$
$360$	0	0
$361$	31.7598	1.67157
$362$	0	0
$363$	−11.4817	−0.602631
$364$	0	0
$365$	1.28721	0.0673757
$366$	0	0
$367$	5.63657	0.294226	0.147113	−	0.989120i	$-0.453002\pi$
$367$	5.63657	0.294226	0.147113	+	0.989120i	$0.453002\pi$
$368$	0	0
$369$	17.4340	0.907576
$370$	0	0
$371$	20.4687	1.06268
$372$	0	0
$373$	1.29504	0.0670548	0.0335274	−	0.999438i	$-0.489326\pi$
$373$	1.29504	0.0670548	0.0335274	+	0.999438i	$0.489326\pi$
$374$	0	0
$375$	−14.1831	−0.732410
$376$	0	0
$377$	−20.4830	−1.05493
$378$	0	0
$379$	13.8155	0.709656	0.354828	−	0.934932i	$-0.384539\pi$
$379$	13.8155	0.709656	0.354828	+	0.934932i	$0.384539\pi$
$380$	0	0
$381$	3.52572	0.180628
$382$	0	0
$383$	−12.0143	−0.613900	−0.306950	−	0.951726i	$-0.599308\pi$
$383$	−12.0143	−0.613900	−0.306950	+	0.951726i	$0.599308\pi$
$384$	0	0
$385$	9.60044	0.489284
$386$	0	0
$387$	−8.99463	−0.457223
$388$	0	0
$389$	4.86222	0.246524	0.123262	−	0.992374i	$-0.460664\pi$
$389$	4.86222	0.246524	0.123262	+	0.992374i	$0.460664\pi$
$390$	0	0
$391$	−13.2064	−0.667876
$392$	0	0
$393$	2.54040	0.128146
$394$	0	0
$395$	−12.0388	−0.605738
$396$	0	0
$397$	9.17854	0.460658	0.230329	−	0.973113i	$-0.426020\pi$
$397$	9.17854	0.460658	0.230329	+	0.973113i	$0.426020\pi$
$398$	0	0
$399$	−42.1148	−2.10838
$400$	0	0
$401$	−28.0569	−1.40109	−0.700547	−	0.713607i	$-0.747060\pi$
$401$	−28.0569	−1.40109	−0.700547	+	0.713607i	$0.747060\pi$
$402$	0	0
$403$	25.2291	1.25675
$404$	0	0
$405$	−2.48321	−0.123392
$406$	0	0
$407$	2.26398	0.112221
$408$	0	0
$409$	−11.6170	−0.574422	−0.287211	−	0.957867i	$-0.592728\pi$
$409$	−11.6170	−0.574422	−0.287211	+	0.957867i	$0.592728\pi$
$410$	0	0
$411$	18.5871	0.916834
$412$	0	0
$413$	28.5883	1.40674
$414$	0	0
$415$	−9.80365	−0.481242
$416$	0	0
$417$	17.6027	0.862010
$418$	0	0
$419$	−15.5789	−0.761080	−0.380540	−	0.924765i	$-0.624262\pi$
$419$	−15.5789	−0.761080	−0.380540	+	0.924765i	$0.624262\pi$
$420$	0	0
$421$	7.53596	0.367280	0.183640	−	0.982994i	$-0.441212\pi$
$421$	7.53596	0.367280	0.183640	+	0.982994i	$0.441212\pi$
$422$	0	0
$423$	8.02277	0.390081
$424$	0	0
$425$	−4.76140	−0.230962
$426$	0	0
$427$	10.3143	0.499144
$428$	0	0
$429$	−3.80216	−0.183570
$430$	0	0
$431$	−24.2144	−1.16636	−0.583182	−	0.812341i	$-0.698193\pi$
$431$	−24.2144	−1.16636	−0.583182	+	0.812341i	$0.698193\pi$
$432$	0	0
$433$	1.90821	0.0917025	0.0458512	−	0.998948i	$-0.485400\pi$
$433$	1.90821	0.0917025	0.0458512	+	0.998948i	$0.485400\pi$
$434$	0	0
$435$	−13.9378	−0.668268
$436$	0	0
$437$	−37.9840	−1.81702
$438$	0	0
$439$	14.9135	0.711784	0.355892	−	0.934527i	$-0.384177\pi$
$439$	14.9135	0.711784	0.355892	+	0.934527i	$0.384177\pi$
$440$	0	0
$441$	−30.4586	−1.45041
$442$	0	0
$443$	23.0698	1.09608	0.548039	−	0.836453i	$-0.315375\pi$
$443$	23.0698	1.09608	0.548039	+	0.836453i	$0.315375\pi$
$444$	0	0
$445$	16.8911	0.800714
$446$	0	0
$447$	−1.59739	−0.0755539
$448$	0	0
$449$	14.5420	0.686277	0.343139	−	0.939285i	$-0.388510\pi$
$449$	14.5420	0.686277	0.343139	+	0.939285i	$0.388510\pi$
$450$	0	0
$451$	11.5215	0.542527
$452$	0	0
$453$	−20.0708	−0.943009
$454$	0	0
$455$	−26.7373	−1.25346
$456$	0	0
$457$	11.8002	0.551991	0.275995	−	0.961159i	$-0.410993\pi$
$457$	11.8002	0.551991	0.275995	+	0.961159i	$0.410993\pi$
$458$	0	0
$459$	−13.4120	−0.626019
$460$	0	0
$461$	15.0722	0.701985	0.350992	−	0.936378i	$-0.385844\pi$
$461$	15.0722	0.701985	0.350992	+	0.936378i	$0.385844\pi$
$462$	0	0
$463$	−24.3295	−1.13069	−0.565344	−	0.824855i	$-0.691257\pi$
$463$	−24.3295	−1.13069	−0.565344	+	0.824855i	$0.691257\pi$
$464$	0	0
$465$	17.1674	0.796120
$466$	0	0
$467$	−29.7743	−1.37779	−0.688895	−	0.724862i	$-0.741904\pi$
$467$	−29.7743	−1.37779	−0.688895	+	0.724862i	$0.741904\pi$
$468$	0	0
$469$	38.7245	1.78813
$470$	0	0
$471$	14.1641	0.652649
$472$	0	0
$473$	−5.94424	−0.273317
$474$	0	0
$475$	−13.6947	−0.628354
$476$	0	0
$477$	6.61619	0.302934
$478$	0	0
$479$	10.1987	0.465991	0.232996	−	0.972478i	$-0.425147\pi$
$479$	10.1987	0.465991	0.232996	+	0.972478i	$0.425147\pi$
$480$	0	0
$481$	−6.30520	−0.287492
$482$	0	0
$483$	31.5149	1.43398
$484$	0	0
$485$	−19.1551	−0.869790
$486$	0	0
$487$	24.9246	1.12944	0.564720	−	0.825282i	$-0.308984\pi$
$487$	24.9246	1.12944	0.564720	+	0.825282i	$0.308984\pi$
$488$	0	0
$489$	−12.5062	−0.565550
$490$	0	0
$491$	26.8710	1.21267	0.606335	−	0.795209i	$-0.292639\pi$
$491$	26.8710	1.21267	0.606335	+	0.795209i	$0.292639\pi$
$492$	0	0
$493$	−16.8504	−0.758905
$494$	0	0
$495$	3.10319	0.139478
$496$	0	0
$497$	62.0684	2.78415
$498$	0	0
$499$	37.6147	1.68387	0.841933	−	0.539582i	$-0.181418\pi$
$499$	37.6147	1.68387	0.841933	+	0.539582i	$0.181418\pi$
$500$	0	0
$501$	12.3259	0.550680
$502$	0	0
$503$	38.0897	1.69834	0.849168	−	0.528123i	$-0.177104\pi$
$503$	38.0897	1.69834	0.849168	+	0.528123i	$0.177104\pi$
$504$	0	0
$505$	−24.4561	−1.08828
$506$	0	0
$507$	−4.59367	−0.204012
$508$	0	0
$509$	−37.4493	−1.65991	−0.829954	−	0.557831i	$-0.811634\pi$
$509$	−37.4493	−1.65991	−0.829954	+	0.557831i	$0.811634\pi$
$510$	0	0
$511$	−3.71362	−0.164281
$512$	0	0
$513$	−38.5754	−1.70314
$514$	0	0
$515$	12.5371	0.552452
$516$	0	0
$517$	5.30198	0.233181
$518$	0	0
$519$	−2.42480	−0.106437
$520$	0	0
$521$	14.8226	0.649391	0.324695	−	0.945819i	$-0.394738\pi$
$521$	14.8226	0.649391	0.324695	+	0.945819i	$0.394738\pi$
$522$	0	0
$523$	−23.8624	−1.04343	−0.521715	−	0.853120i	$-0.674708\pi$
$523$	−23.8624	−1.04343	−0.521715	+	0.853120i	$0.674708\pi$
$524$	0	0
$525$	11.3623	0.495891
$526$	0	0
$527$	20.7549	0.904098
$528$	0	0
$529$	5.42372	0.235814
$530$	0	0
$531$	9.24069	0.401012
$532$	0	0
$533$	−32.0875	−1.38986
$534$	0	0
$535$	−31.3909	−1.35715
$536$	0	0
$537$	3.58454	0.154685
$538$	0	0
$539$	−20.1291	−0.867021
$540$	0	0
$541$	−12.9272	−0.555782	−0.277891	−	0.960613i	$-0.589635\pi$
$541$	−12.9272	−0.555782	−0.277891	+	0.960613i	$0.589635\pi$
$542$	0	0
$543$	−0.457119	−0.0196168
$544$	0	0
$545$	−26.1923	−1.12196
$546$	0	0
$547$	9.03128	0.386150	0.193075	−	0.981184i	$-0.438154\pi$
$547$	9.03128	0.386150	0.193075	+	0.981184i	$0.438154\pi$
$548$	0	0
$549$	3.33393	0.142289
$550$	0	0
$551$	−48.4649	−2.06467
$552$	0	0
$553$	34.7320	1.47696
$554$	0	0
$555$	−4.29043	−0.182119
$556$	0	0
$557$	−4.33616	−0.183729	−0.0918644	−	0.995772i	$-0.529283\pi$
$557$	−4.33616	−0.183729	−0.0918644	+	0.995772i	$0.529283\pi$
$558$	0	0
$559$	16.5547	0.700191
$560$	0	0
$561$	−3.12787	−0.132059
$562$	0	0
$563$	−6.56701	−0.276767	−0.138383	−	0.990379i	$-0.544191\pi$
$563$	−6.56701	−0.276767	−0.138383	+	0.990379i	$0.544191\pi$
$564$	0	0
$565$	1.50755	0.0634229
$566$	0	0
$567$	7.16407	0.300863
$568$	0	0
$569$	18.1137	0.759366	0.379683	−	0.925117i	$-0.376033\pi$
$569$	18.1137	0.759366	0.379683	+	0.925117i	$0.376033\pi$
$570$	0	0
$571$	−6.26298	−0.262098	−0.131049	−	0.991376i	$-0.541834\pi$
$571$	−6.26298	−0.262098	−0.131049	+	0.991376i	$0.541834\pi$
$572$	0	0
$573$	−5.82803	−0.243470
$574$	0	0
$575$	10.2478	0.427364
$576$	0	0
$577$	−1.76988	−0.0736810	−0.0368405	−	0.999321i	$-0.511729\pi$
$577$	−1.76988	−0.0736810	−0.0368405	+	0.999321i	$0.511729\pi$
$578$	0	0
$579$	15.7910	0.656253
$580$	0	0
$581$	28.2836	1.17340
$582$	0	0
$583$	4.37241	0.181087
$584$	0	0
$585$	−8.64239	−0.357319
$586$	0	0
$587$	−11.9082	−0.491502	−0.245751	−	0.969333i	$-0.579035\pi$
$587$	−11.9082	−0.491502	−0.245751	+	0.969333i	$0.579035\pi$
$588$	0	0
$589$	59.6949	2.45969
$590$	0	0
$591$	6.85266	0.281881
$592$	0	0
$593$	−34.2932	−1.40825	−0.704126	−	0.710075i	$-0.748661\pi$
$593$	−34.2932	−1.40825	−0.704126	+	0.710075i	$0.748661\pi$
$594$	0	0
$595$	−21.9956	−0.901731
$596$	0	0
$597$	−14.6891	−0.601184
$598$	0	0
$599$	18.2245	0.744632	0.372316	−	0.928106i	$-0.378564\pi$
$599$	18.2245	0.744632	0.372316	+	0.928106i	$0.378564\pi$
$600$	0	0
$601$	−19.2591	−0.785595	−0.392797	−	0.919625i	$-0.628493\pi$
$601$	−19.2591	−0.785595	−0.392797	+	0.919625i	$0.628493\pi$
$602$	0	0
$603$	12.5171	0.509735
$604$	0	0
$605$	−17.2473	−0.701204
$606$	0	0
$607$	−39.9799	−1.62273	−0.811367	−	0.584536i	$-0.801276\pi$
$607$	−39.9799	−1.62273	−0.811367	+	0.584536i	$0.801276\pi$
$608$	0	0
$609$	40.2108	1.62942
$610$	0	0
$611$	−14.7660	−0.597369
$612$	0	0
$613$	−18.3233	−0.740070	−0.370035	−	0.929018i	$-0.620654\pi$
$613$	−18.3233	−0.740070	−0.370035	+	0.929018i	$0.620654\pi$
$614$	0	0
$615$	−21.8342	−0.880442
$616$	0	0
$617$	12.3769	0.498273	0.249137	−	0.968468i	$-0.419853\pi$
$617$	12.3769	0.498273	0.249137	+	0.968468i	$0.419853\pi$
$618$	0	0
$619$	8.87878	0.356868	0.178434	−	0.983952i	$-0.442897\pi$
$619$	8.87878	0.356868	0.178434	+	0.983952i	$0.442897\pi$
$620$	0	0
$621$	28.8662	1.15836
$622$	0	0
$623$	−48.7309	−1.95236
$624$	0	0
$625$	−11.6944	−0.467777
$626$	0	0
$627$	−8.99632	−0.359278
$628$	0	0
$629$	−5.18701	−0.206820
$630$	0	0
$631$	−34.5564	−1.37567	−0.687835	−	0.725867i	$-0.741439\pi$
$631$	−34.5564	−1.37567	−0.687835	+	0.725867i	$0.741439\pi$
$632$	0	0
$633$	−7.01507	−0.278824
$634$	0	0
$635$	5.29621	0.210174
$636$	0	0
$637$	56.0596	2.22116
$638$	0	0
$639$	20.0626	0.793664
$640$	0	0
$641$	−1.70265	−0.0672505	−0.0336253	−	0.999435i	$-0.510705\pi$
$641$	−1.70265	−0.0672505	−0.0336253	+	0.999435i	$0.510705\pi$
$642$	0	0
$643$	−40.7018	−1.60512	−0.802561	−	0.596571i	$-0.796530\pi$
$643$	−40.7018	−1.60512	−0.802561	+	0.596571i	$0.796530\pi$
$644$	0	0
$645$	11.2648	0.443553
$646$	0	0
$647$	30.3915	1.19481	0.597406	−	0.801939i	$-0.296198\pi$
$647$	30.3915	1.19481	0.597406	+	0.801939i	$0.296198\pi$
$648$	0	0
$649$	6.10685	0.239715
$650$	0	0
$651$	−49.5282	−1.94116
$652$	0	0
$653$	6.95225	0.272062	0.136031	−	0.990705i	$-0.456565\pi$
$653$	6.95225	0.272062	0.136031	+	0.990705i	$0.456565\pi$
$654$	0	0
$655$	3.81609	0.149107
$656$	0	0
$657$	−1.20037	−0.0468307
$658$	0	0
$659$	38.2304	1.48924	0.744622	−	0.667486i	$-0.232630\pi$
$659$	38.2304	1.48924	0.744622	+	0.667486i	$0.232630\pi$
$660$	0	0
$661$	35.0225	1.36222	0.681109	−	0.732182i	$-0.261498\pi$
$661$	35.0225	1.36222	0.681109	+	0.732182i	$0.261498\pi$
$662$	0	0
$663$	8.71113	0.338312
$664$	0	0
$665$	−63.2633	−2.45325
$666$	0	0
$667$	36.2667	1.40425
$668$	0	0
$669$	30.0879	1.16327
$670$	0	0
$671$	2.20328	0.0850566
$672$	0	0
$673$	−19.3765	−0.746909	−0.373454	−	0.927649i	$-0.621827\pi$
$673$	−19.3765	−0.746909	−0.373454	+	0.927649i	$0.621827\pi$
$674$	0	0
$675$	10.4074	0.400580
$676$	0	0
$677$	−3.01738	−0.115967	−0.0579837	−	0.998318i	$-0.518467\pi$
$677$	−3.01738	−0.115967	−0.0579837	+	0.998318i	$0.518467\pi$
$678$	0	0
$679$	55.2627	2.12079
$680$	0	0
$681$	−5.88294	−0.225435
$682$	0	0
$683$	5.60316	0.214399	0.107199	−	0.994238i	$-0.465812\pi$
$683$	5.60316	0.214399	0.107199	+	0.994238i	$0.465812\pi$
$684$	0	0
$685$	27.9209	1.06680
$686$	0	0
$687$	24.6713	0.941267
$688$	0	0
$689$	−12.1772	−0.463914
$690$	0	0
$691$	−49.0976	−1.86776	−0.933881	−	0.357583i	$-0.883601\pi$
$691$	−49.0976	−1.86776	−0.933881	+	0.357583i	$0.883601\pi$
$692$	0	0
$693$	−8.95272	−0.340086
$694$	0	0
$695$	26.4422	1.00301
$696$	0	0
$697$	−26.3970	−0.999856
$698$	0	0
$699$	24.4637	0.925304
$700$	0	0
$701$	14.0248	0.529709	0.264855	−	0.964288i	$-0.414676\pi$
$701$	14.0248	0.529709	0.264855	+	0.964288i	$0.414676\pi$
$702$	0	0
$703$	−14.9188	−0.562673
$704$	0	0
$705$	−10.0477	−0.378418
$706$	0	0
$707$	70.5561	2.65354
$708$	0	0
$709$	−10.0074	−0.375835	−0.187918	−	0.982185i	$-0.560174\pi$
$709$	−10.0074	−0.375835	−0.187918	+	0.982185i	$0.560174\pi$
$710$	0	0
$711$	11.2266	0.421029
$712$	0	0
$713$	−44.6702	−1.67291
$714$	0	0
$715$	−5.71146	−0.213596
$716$	0	0
$717$	9.15098	0.341749
$718$	0	0
$719$	−32.3086	−1.20491	−0.602454	−	0.798154i	$-0.705810\pi$
$719$	−32.3086	−1.20491	−0.602454	+	0.798154i	$0.705810\pi$
$720$	0	0
$721$	−36.1697	−1.34703
$722$	0	0
$723$	1.16790	0.0434347
$724$	0	0
$725$	13.0755	0.485612
$726$	0	0
$727$	−14.7153	−0.545762	−0.272881	−	0.962048i	$-0.587977\pi$
$727$	−14.7153	−0.545762	−0.272881	+	0.962048i	$0.587977\pi$
$728$	0	0
$729$	21.2861	0.788373
$730$	0	0
$731$	13.6189	0.503712
$732$	0	0
$733$	28.2634	1.04393	0.521966	−	0.852966i	$-0.325199\pi$
$733$	28.2634	1.04393	0.521966	+	0.852966i	$0.325199\pi$
$734$	0	0
$735$	38.1463	1.40705
$736$	0	0
$737$	8.27210	0.304707
$738$	0	0
$739$	34.0536	1.25268	0.626340	−	0.779550i	$-0.284552\pi$
$739$	34.0536	1.25268	0.626340	+	0.779550i	$0.284552\pi$
$740$	0	0
$741$	25.0548	0.920411
$742$	0	0
$743$	47.2361	1.73293	0.866463	−	0.499241i	$-0.166388\pi$
$743$	47.2361	1.73293	0.866463	+	0.499241i	$0.166388\pi$
$744$	0	0
$745$	−2.39954	−0.0879122
$746$	0	0
$747$	9.14222	0.334496
$748$	0	0
$749$	90.5629	3.30910
$750$	0	0
$751$	−36.7588	−1.34135	−0.670674	−	0.741752i	$-0.733995\pi$
$751$	−36.7588	−1.34135	−0.670674	+	0.741752i	$0.733995\pi$
$752$	0	0
$753$	8.19454	0.298626
$754$	0	0
$755$	−30.1496	−1.09726
$756$	0	0
$757$	7.43315	0.270162	0.135081	−	0.990835i	$-0.456870\pi$
$757$	7.43315	0.270162	0.135081	+	0.990835i	$0.456870\pi$
$758$	0	0
$759$	6.73202	0.244357
$760$	0	0
$761$	1.06982	0.0387809	0.0193905	−	0.999812i	$-0.493827\pi$
$761$	1.06982	0.0387809	0.0193905	+	0.999812i	$0.493827\pi$
$762$	0	0
$763$	75.5651	2.73564
$764$	0	0
$765$	−7.10972	−0.257052
$766$	0	0
$767$	−17.0076	−0.614109
$768$	0	0
$769$	−26.2212	−0.945562	−0.472781	−	0.881180i	$-0.656750\pi$
$769$	−26.2212	−0.945562	−0.472781	+	0.881180i	$0.656750\pi$
$770$	0	0
$771$	13.2039	0.475527
$772$	0	0
$773$	−15.9051	−0.572065	−0.286033	−	0.958220i	$-0.592337\pi$
$773$	−15.9051	−0.572065	−0.286033	+	0.958220i	$0.592337\pi$
$774$	0	0
$775$	−16.1053	−0.578519
$776$	0	0
$777$	12.3779	0.444056
$778$	0	0
$779$	−75.9225	−2.72020
$780$	0	0
$781$	13.2587	0.474433
$782$	0	0
$783$	36.8313	1.31624
$784$	0	0
$785$	21.2768	0.759403
$786$	0	0
$787$	44.7474	1.59507	0.797536	−	0.603272i	$-0.206137\pi$
$787$	44.7474	1.59507	0.797536	+	0.603272i	$0.206137\pi$
$788$	0	0
$789$	−23.2023	−0.826023
$790$	0	0
$791$	−4.34928	−0.154643
$792$	0	0
$793$	−6.13614	−0.217901
$794$	0	0
$795$	−8.28609	−0.293877
$796$	0	0
$797$	−20.2003	−0.715530	−0.357765	−	0.933812i	$-0.616461\pi$
$797$	−20.2003	−0.715530	−0.357765	+	0.933812i	$0.616461\pi$
$798$	0	0
$799$	−12.1474	−0.429743
$800$	0	0
$801$	−15.7515	−0.556551
$802$	0	0
$803$	−0.793280	−0.0279943
$804$	0	0
$805$	47.3405	1.66853
$806$	0	0
$807$	−23.6538	−0.832652
$808$	0	0
$809$	31.3338	1.10164	0.550818	−	0.834625i	$-0.314316\pi$
$809$	31.3338	1.10164	0.550818	+	0.834625i	$0.314316\pi$
$810$	0	0
$811$	28.2192	0.990909	0.495454	−	0.868634i	$-0.335002\pi$
$811$	28.2192	0.990909	0.495454	+	0.868634i	$0.335002\pi$
$812$	0	0
$813$	4.95064	0.173626
$814$	0	0
$815$	−18.7863	−0.658057
$816$	0	0
$817$	39.1704	1.37040
$818$	0	0
$819$	24.9334	0.871242
$820$	0	0
$821$	−36.8115	−1.28473	−0.642365	−	0.766399i	$-0.722047\pi$
$821$	−36.8115	−1.28473	−0.642365	+	0.766399i	$0.722047\pi$
$822$	0	0
$823$	34.8680	1.21542	0.607711	−	0.794158i	$-0.292088\pi$
$823$	34.8680	1.21542	0.607711	+	0.794158i	$0.292088\pi$
$824$	0	0
$825$	2.42715	0.0845024
$826$	0	0
$827$	11.6305	0.404431	0.202216	−	0.979341i	$-0.435186\pi$
$827$	11.6305	0.404431	0.202216	+	0.979341i	$0.435186\pi$
$828$	0	0
$829$	−45.8232	−1.59150	−0.795752	−	0.605622i	$-0.792924\pi$
$829$	−45.8232	−1.59150	−0.795752	+	0.605622i	$0.792924\pi$
$830$	0	0
$831$	−0.431502	−0.0149686
$832$	0	0
$833$	46.1178	1.59789
$834$	0	0
$835$	18.5155	0.640755
$836$	0	0
$837$	−45.3656	−1.56807
$838$	0	0
$839$	3.48632	0.120361	0.0601806	−	0.998188i	$-0.480832\pi$
$839$	3.48632	0.120361	0.0601806	+	0.998188i	$0.480832\pi$
$840$	0	0
$841$	17.2737	0.595646
$842$	0	0
$843$	28.8375	0.993216
$844$	0	0
$845$	−6.90043	−0.237382
$846$	0	0
$847$	49.7587	1.70973
$848$	0	0
$849$	−0.0813764	−0.00279283
$850$	0	0
$851$	11.1638	0.382692
$852$	0	0
$853$	−44.6130	−1.52752	−0.763760	−	0.645500i	$-0.776649\pi$
$853$	−44.6130	−1.52752	−0.763760	+	0.645500i	$0.776649\pi$
$854$	0	0
$855$	−20.4488	−0.699336
$856$	0	0
$857$	21.1188	0.721404	0.360702	−	0.932681i	$-0.382537\pi$
$857$	21.1188	0.721404	0.360702	+	0.932681i	$0.382537\pi$
$858$	0	0
$859$	36.4020	1.24202	0.621011	−	0.783802i	$-0.286722\pi$
$859$	36.4020	1.24202	0.621011	+	0.783802i	$0.286722\pi$
$860$	0	0
$861$	62.9920	2.14676
$862$	0	0
$863$	4.01817	0.136780	0.0683900	−	0.997659i	$-0.478214\pi$
$863$	4.01817	0.136780	0.0683900	+	0.997659i	$0.478214\pi$
$864$	0	0
$865$	−3.64244	−0.123847
$866$	0	0
$867$	−12.6880	−0.430908
$868$	0	0
$869$	7.41925	0.251681
$870$	0	0
$871$	−23.0379	−0.780608
$872$	0	0
$873$	17.8628	0.604563
$874$	0	0
$875$	61.4659	2.07793
$876$	0	0
$877$	48.0234	1.62163	0.810817	−	0.585300i	$-0.199023\pi$
$877$	48.0234	1.62163	0.810817	+	0.585300i	$0.199023\pi$
$878$	0	0
$879$	−8.86599	−0.299042
$880$	0	0
$881$	−56.8752	−1.91617	−0.958087	−	0.286479i	$-0.907515\pi$
$881$	−56.8752	−1.91617	−0.958087	+	0.286479i	$0.907515\pi$
$882$	0	0
$883$	−44.7065	−1.50449	−0.752246	−	0.658882i	$-0.771030\pi$
$883$	−44.7065	−1.50449	−0.752246	+	0.658882i	$0.771030\pi$
$884$	0	0
$885$	−11.5730	−0.389022
$886$	0	0
$887$	−5.64866	−0.189663	−0.0948317	−	0.995493i	$-0.530231\pi$
$887$	−5.64866	−0.189663	−0.0948317	+	0.995493i	$0.530231\pi$
$888$	0	0
$889$	−15.2796	−0.512462
$890$	0	0
$891$	1.53035	0.0512686
$892$	0	0
$893$	−34.9380	−1.16916
$894$	0	0
$895$	5.38457	0.179986
$896$	0	0
$897$	−18.7487	−0.626001
$898$	0	0
$899$	−56.9960	−1.90092
$900$	0	0
$901$	−10.0176	−0.333736
$902$	0	0
$903$	−32.4992	−1.08150
$904$	0	0
$905$	−0.686667	−0.0228256
$906$	0	0
$907$	−25.2839	−0.839538	−0.419769	−	0.907631i	$-0.637889\pi$
$907$	−25.2839	−0.839538	−0.419769	+	0.907631i	$0.637889\pi$
$908$	0	0
$909$	22.8061	0.756431
$910$	0	0
$911$	−27.3571	−0.906380	−0.453190	−	0.891414i	$-0.649714\pi$
$911$	−27.3571	−0.906380	−0.453190	+	0.891414i	$0.649714\pi$
$912$	0	0
$913$	6.04178	0.199954
$914$	0	0
$915$	−4.17540	−0.138034
$916$	0	0
$917$	−11.0095	−0.363564
$918$	0	0
$919$	44.3463	1.46285	0.731424	−	0.681923i	$-0.238856\pi$
$919$	44.3463	1.46285	0.731424	+	0.681923i	$0.238856\pi$
$920$	0	0
$921$	24.1098	0.794445
$922$	0	0
$923$	−36.9255	−1.21542
$924$	0	0
$925$	4.02499	0.132341
$926$	0	0
$927$	−11.6913	−0.383992
$928$	0	0
$929$	−33.2552	−1.09107	−0.545535	−	0.838088i	$-0.683673\pi$
$929$	−33.2552	−1.09107	−0.545535	+	0.838088i	$0.683673\pi$
$930$	0	0
$931$	132.643	4.34720
$932$	0	0
$933$	−28.1326	−0.921021
$934$	0	0
$935$	−4.69857	−0.153660
$936$	0	0
$937$	−51.9272	−1.69639	−0.848194	−	0.529686i	$-0.822310\pi$
$937$	−51.9272	−1.69639	−0.848194	+	0.529686i	$0.822310\pi$
$938$	0	0
$939$	19.1307	0.624306
$940$	0	0
$941$	21.0604	0.686549	0.343274	−	0.939235i	$-0.388464\pi$
$941$	21.0604	0.686549	0.343274	+	0.939235i	$0.388464\pi$
$942$	0	0
$943$	56.8134	1.85010
$944$	0	0
$945$	48.0775	1.56396
$946$	0	0
$947$	22.4529	0.729621	0.364810	−	0.931082i	$-0.381134\pi$
$947$	22.4529	0.729621	0.364810	+	0.931082i	$0.381134\pi$
$948$	0	0
$949$	2.20929	0.0717166
$950$	0	0
$951$	−1.86669	−0.0605317
$952$	0	0
$953$	29.3558	0.950927	0.475463	−	0.879736i	$-0.342280\pi$
$953$	29.3558	0.950927	0.475463	+	0.879736i	$0.342280\pi$
$954$	0	0
$955$	−8.75465	−0.283294
$956$	0	0
$957$	8.58958	0.277662
$958$	0	0
$959$	−80.5519	−2.60116
$960$	0	0
$961$	39.2028	1.26461
$962$	0	0
$963$	29.2730	0.943309
$964$	0	0
$965$	23.7207	0.763596
$966$	0	0
$967$	−26.9397	−0.866324	−0.433162	−	0.901316i	$-0.642602\pi$
$967$	−26.9397	−0.866324	−0.433162	+	0.901316i	$0.642602\pi$
$968$	0	0
$969$	20.6115	0.662136
$970$	0	0
$971$	38.3427	1.23047	0.615237	−	0.788342i	$-0.289060\pi$
$971$	38.3427	1.23047	0.615237	+	0.788342i	$0.289060\pi$
$972$	0	0
$973$	−76.2859	−2.44561
$974$	0	0
$975$	−6.75962	−0.216481
$976$	0	0
$977$	−13.7841	−0.440991	−0.220496	−	0.975388i	$-0.570767\pi$
$977$	−13.7841	−0.440991	−0.220496	+	0.975388i	$0.570767\pi$
$978$	0	0
$979$	−10.4096	−0.332693
$980$	0	0
$981$	24.4252	0.779836
$982$	0	0
$983$	−14.8201	−0.472688	−0.236344	−	0.971669i	$-0.575949\pi$
$983$	−14.8201	−0.472688	−0.236344	+	0.971669i	$0.575949\pi$
$984$	0	0
$985$	10.2938	0.327988
$986$	0	0
$987$	28.9877	0.922688
$988$	0	0
$989$	−29.3115	−0.932051
$990$	0	0
$991$	52.3768	1.66380	0.831902	−	0.554923i	$-0.187252\pi$
$991$	52.3768	1.66380	0.831902	+	0.554923i	$0.187252\pi$
$992$	0	0
$993$	31.4247	0.997233
$994$	0	0
$995$	−22.0654	−0.699520
$996$	0	0
$997$	23.1646	0.733630	0.366815	−	0.930294i	$-0.380448\pi$
$997$	23.1646	0.733630	0.366815	+	0.930294i	$0.380448\pi$
$998$	0	0
$999$	11.3377	0.358708

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.n.1.8		12
4.3	odd	2		241.2.a.b.1.9	✓	12
12.11	even	2		2169.2.a.h.1.4		12
20.19	odd	2		6025.2.a.h.1.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.9	✓	12	4.3	odd	2
2169.2.a.h.1.4		12	12.11	even	2
3856.2.a.n.1.8		12	1.1	even	1	trivial
6025.2.a.h.1.4		12	20.19	odd	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:	\( N \)	\(=\)	\( 3856 = 2^{4} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3856.a (trivial)

\(f(q)\)	\(=\)	\(q+1.16790 q^{3} +1.75438 q^{5} -5.06139 q^{7} -1.63601 q^{9} +O(q^{10})\)	\(q+1.16790 q^{3} +1.75438 q^{5} -5.06139 q^{7} -1.63601 q^{9} -1.08118 q^{11} +3.01110 q^{13} +2.04893 q^{15} +2.47710 q^{17} +7.12459 q^{19} -5.91119 q^{21} -5.33139 q^{23} -1.92217 q^{25} -5.41439 q^{27} -6.80248 q^{29} +8.37871 q^{31} -1.26271 q^{33} -8.87957 q^{35} -2.09398 q^{37} +3.51666 q^{39} -10.6564 q^{41} +5.49791 q^{43} -2.87018 q^{45} -4.90386 q^{47} +18.6176 q^{49} +2.89301 q^{51} -4.04410 q^{53} -1.89680 q^{55} +8.32081 q^{57} -5.64831 q^{59} -2.03784 q^{61} +8.28048 q^{63} +5.28260 q^{65} -7.65097 q^{67} -6.22653 q^{69} -12.2631 q^{71} +0.733715 q^{73} -2.24490 q^{75} +5.47229 q^{77} -6.86216 q^{79} -1.41544 q^{81} -5.58812 q^{83} +4.34576 q^{85} -7.94461 q^{87} +9.62798 q^{89} -15.2403 q^{91} +9.78549 q^{93} +12.4992 q^{95} -10.9185 q^{97} +1.76883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * q^99

Introduction

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