LMFDB - Embedded newform 3344.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 $ x^7 - x^6 - 14x^5 + 10x^4 + 59x^3 - 27x^2 - 66*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.45416$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.71116 q^{3} +2.59296 q^{5} +2.00933 q^{7} -0.0719365 q^{9} +1.00000 q^{11} -4.45988 q^{13} -4.43696 q^{15} +4.54435 q^{17} -1.00000 q^{19} -3.43828 q^{21} -7.48175 q^{23} +1.72344 q^{25} +5.25657 q^{27} -3.17425 q^{29} -9.34529 q^{31} -1.71116 q^{33} +5.21010 q^{35} -6.84486 q^{37} +7.63157 q^{39} +0.644299 q^{41} +8.07470 q^{43} -0.186528 q^{45} -11.6485 q^{47} -2.96260 q^{49} -7.77611 q^{51} -5.53442 q^{53} +2.59296 q^{55} +1.71116 q^{57} -9.16829 q^{59} +9.45268 q^{61} -0.144544 q^{63} -11.5643 q^{65} -0.113535 q^{67} +12.8025 q^{69} +9.84191 q^{71} -2.38027 q^{73} -2.94907 q^{75} +2.00933 q^{77} +2.01251 q^{79} -8.77902 q^{81} +2.90426 q^{83} +11.7833 q^{85} +5.43164 q^{87} -8.82031 q^{89} -8.96136 q^{91} +15.9913 q^{93} -2.59296 q^{95} +11.0100 q^{97} -0.0719365 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) $ 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * q^99

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
$3$	−1.71116	−0.987938	−0.493969	−	0.869480i	$-0.664454\pi$
$3$	−1.71116	−0.987938	−0.493969	+	0.869480i	$0.664454\pi$
$4$	0	0
$5$	2.59296	1.15961	0.579803	−	0.814756i	$-0.303129\pi$
$5$	2.59296	1.15961	0.579803	+	0.814756i	$0.303129\pi$
$6$	0	0
$7$	2.00933	0.759454	0.379727	−	0.925099i	$-0.376018\pi$
$7$	2.00933	0.759454	0.379727	+	0.925099i	$0.376018\pi$
$8$	0	0
$9$	−0.0719365	−0.0239788
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.45988	−1.23695	−0.618475	−	0.785805i	$-0.712249\pi$
$13$	−4.45988	−1.23695	−0.618475	+	0.785805i	$0.712249\pi$
$14$	0	0
$15$	−4.43696	−1.14562
$16$	0	0
$17$	4.54435	1.10217	0.551084	−	0.834450i	$-0.314214\pi$
$17$	4.54435	1.10217	0.551084	+	0.834450i	$0.314214\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.43828	−0.750294
$22$	0	0
$23$	−7.48175	−1.56005	−0.780026	−	0.625747i	$-0.784794\pi$
$23$	−7.48175	−1.56005	−0.780026	+	0.625747i	$0.784794\pi$
$24$	0	0
$25$	1.72344	0.344687
$26$	0	0
$27$	5.25657	1.01163
$28$	0	0
$29$	−3.17425	−0.589443	−0.294722	−	0.955583i	$-0.595227\pi$
$29$	−3.17425	−0.589443	−0.294722	+	0.955583i	$0.595227\pi$
$30$	0	0
$31$	−9.34529	−1.67846	−0.839232	−	0.543774i	$-0.816995\pi$
$31$	−9.34529	−1.67846	−0.839232	+	0.543774i	$0.816995\pi$
$32$	0	0
$33$	−1.71116	−0.297874
$34$	0	0
$35$	5.21010	0.880668
$36$	0	0
$37$	−6.84486	−1.12529	−0.562644	−	0.826699i	$-0.690216\pi$
$37$	−6.84486	−1.12529	−0.562644	+	0.826699i	$0.690216\pi$
$38$	0	0
$39$	7.63157	1.22203
$40$	0	0
$41$	0.644299	0.100623	0.0503113	−	0.998734i	$-0.483979\pi$
$41$	0.644299	0.100623	0.0503113	+	0.998734i	$0.483979\pi$
$42$	0	0
$43$	8.07470	1.23138	0.615690	−	0.787988i	$-0.288877\pi$
$43$	8.07470	1.23138	0.615690	+	0.787988i	$0.288877\pi$
$44$	0	0
$45$	−0.186528	−0.0278060
$46$	0	0
$47$	−11.6485	−1.69911	−0.849557	−	0.527497i	$-0.823131\pi$
$47$	−11.6485	−1.69911	−0.849557	+	0.527497i	$0.823131\pi$
$48$	0	0
$49$	−2.96260	−0.423229
$50$	0	0
$51$	−7.77611	−1.08887
$52$	0	0
$53$	−5.53442	−0.760211	−0.380105	−	0.924943i	$-0.624112\pi$
$53$	−5.53442	−0.760211	−0.380105	+	0.924943i	$0.624112\pi$
$54$	0	0
$55$	2.59296	0.349635
$56$	0	0
$57$	1.71116	0.226648
$58$	0	0
$59$	−9.16829	−1.19361	−0.596805	−	0.802386i	$-0.703563\pi$
$59$	−9.16829	−1.19361	−0.596805	+	0.802386i	$0.703563\pi$
$60$	0	0
$61$	9.45268	1.21029	0.605146	−	0.796115i	$-0.293115\pi$
$61$	9.45268	1.21029	0.605146	+	0.796115i	$0.293115\pi$
$62$	0	0
$63$	−0.144544	−0.0182108
$64$	0	0
$65$	−11.5643	−1.43437
$66$	0	0
$67$	−0.113535	−0.0138705	−0.00693524	−	0.999976i	$-0.502208\pi$
$67$	−0.113535	−0.0138705	−0.00693524	+	0.999976i	$0.502208\pi$
$68$	0	0
$69$	12.8025	1.54123
$70$	0	0
$71$	9.84191	1.16802	0.584010	−	0.811746i	$-0.301483\pi$
$71$	9.84191	1.16802	0.584010	+	0.811746i	$0.301483\pi$
$72$	0	0
$73$	−2.38027	−0.278590	−0.139295	−	0.990251i	$-0.544484\pi$
$73$	−2.38027	−0.278590	−0.139295	+	0.990251i	$0.544484\pi$
$74$	0	0
$75$	−2.94907	−0.340530
$76$	0	0
$77$	2.00933	0.228984
$78$	0	0
$79$	2.01251	0.226425	0.113213	−	0.993571i	$-0.463886\pi$
$79$	2.01251	0.226425	0.113213	+	0.993571i	$0.463886\pi$
$80$	0	0
$81$	−8.77902	−0.975446
$82$	0	0
$83$	2.90426	0.318784	0.159392	−	0.987215i	$-0.449047\pi$
$83$	2.90426	0.318784	0.159392	+	0.987215i	$0.449047\pi$
$84$	0	0
$85$	11.7833	1.27808
$86$	0	0
$87$	5.43164	0.582333
$88$	0	0
$89$	−8.82031	−0.934951	−0.467475	−	0.884006i	$-0.654836\pi$
$89$	−8.82031	−0.934951	−0.467475	+	0.884006i	$0.654836\pi$
$90$	0	0
$91$	−8.96136	−0.939406
$92$	0	0
$93$	15.9913	1.65822
$94$	0	0
$95$	−2.59296	−0.266032
$96$	0	0
$97$	11.0100	1.11790	0.558949	−	0.829202i	$-0.311205\pi$
$97$	11.0100	1.11790	0.558949	+	0.829202i	$0.311205\pi$
$98$	0	0
$99$	−0.0719365	−0.00722989
$100$	0	0
$101$	0.864844	0.0860552	0.0430276	−	0.999074i	$-0.486300\pi$
$101$	0.864844	0.0860552	0.0430276	+	0.999074i	$0.486300\pi$
$102$	0	0
$103$	−2.64855	−0.260969	−0.130485	−	0.991450i	$-0.541653\pi$
$103$	−2.64855	−0.260969	−0.130485	+	0.991450i	$0.541653\pi$
$104$	0	0
$105$	−8.91531	−0.870045
$106$	0	0
$107$	0.180382	0.0174382	0.00871910	−	0.999962i	$-0.497225\pi$
$107$	0.180382	0.0174382	0.00871910	+	0.999962i	$0.497225\pi$
$108$	0	0
$109$	6.60778	0.632910	0.316455	−	0.948607i	$-0.397507\pi$
$109$	6.60778	0.632910	0.316455	+	0.948607i	$0.397507\pi$
$110$	0	0
$111$	11.7126	1.11172
$112$	0	0
$113$	4.22599	0.397548	0.198774	−	0.980045i	$-0.436304\pi$
$113$	4.22599	0.397548	0.198774	+	0.980045i	$0.436304\pi$
$114$	0	0
$115$	−19.3999	−1.80905
$116$	0	0
$117$	0.320828	0.0296606
$118$	0	0
$119$	9.13110	0.837046
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.10250	−0.0994089
$124$	0	0
$125$	−8.49599	−0.759905
$126$	0	0
$127$	−14.0808	−1.24947	−0.624734	−	0.780838i	$-0.714793\pi$
$127$	−14.0808	−1.24947	−0.624734	+	0.780838i	$0.714793\pi$
$128$	0	0
$129$	−13.8171	−1.21653
$130$	0	0
$131$	−10.1097	−0.883290	−0.441645	−	0.897190i	$-0.645605\pi$
$131$	−10.1097	−0.883290	−0.441645	+	0.897190i	$0.645605\pi$
$132$	0	0
$133$	−2.00933	−0.174231
$134$	0	0
$135$	13.6301	1.17309
$136$	0	0
$137$	−2.86731	−0.244971	−0.122485	−	0.992470i	$-0.539086\pi$
$137$	−2.86731	−0.244971	−0.122485	+	0.992470i	$0.539086\pi$
$138$	0	0
$139$	21.3049	1.80706	0.903529	−	0.428526i	$-0.140967\pi$
$139$	21.3049	1.80706	0.903529	+	0.428526i	$0.140967\pi$
$140$	0	0
$141$	19.9325	1.67862
$142$	0	0
$143$	−4.45988	−0.372954
$144$	0	0
$145$	−8.23070	−0.683522
$146$	0	0
$147$	5.06949	0.418124
$148$	0	0
$149$	−7.28860	−0.597105	−0.298553	−	0.954393i	$-0.596504\pi$
$149$	−7.28860	−0.597105	−0.298553	+	0.954393i	$0.596504\pi$
$150$	0	0
$151$	−22.4316	−1.82546	−0.912731	−	0.408562i	$-0.866031\pi$
$151$	−22.4316	−1.82546	−0.912731	+	0.408562i	$0.866031\pi$
$152$	0	0
$153$	−0.326905	−0.0264287
$154$	0	0
$155$	−24.2320	−1.94636
$156$	0	0
$157$	15.9630	1.27399	0.636994	−	0.770869i	$-0.280178\pi$
$157$	15.9630	1.27399	0.636994	+	0.770869i	$0.280178\pi$
$158$	0	0
$159$	9.47027	0.751041
$160$	0	0
$161$	−15.0333	−1.18479
$162$	0	0
$163$	2.04841	0.160444	0.0802218	−	0.996777i	$-0.474437\pi$
$163$	2.04841	0.160444	0.0802218	+	0.996777i	$0.474437\pi$
$164$	0	0
$165$	−4.43696	−0.345417
$166$	0	0
$167$	6.99086	0.540969	0.270485	−	0.962724i	$-0.412816\pi$
$167$	6.99086	0.540969	0.270485	+	0.962724i	$0.412816\pi$
$168$	0	0
$169$	6.89056	0.530043
$170$	0	0
$171$	0.0719365	0.00550112
$172$	0	0
$173$	−19.3508	−1.47122	−0.735609	−	0.677407i	$-0.763104\pi$
$173$	−19.3508	−1.47122	−0.735609	+	0.677407i	$0.763104\pi$
$174$	0	0
$175$	3.46295	0.261774
$176$	0	0
$177$	15.6884	1.17921
$178$	0	0
$179$	22.8792	1.71007	0.855037	−	0.518567i	$-0.173534\pi$
$179$	22.8792	1.71007	0.855037	+	0.518567i	$0.173534\pi$
$180$	0	0
$181$	−1.33552	−0.0992682	−0.0496341	−	0.998767i	$-0.515806\pi$
$181$	−1.33552	−0.0992682	−0.0496341	+	0.998767i	$0.515806\pi$
$182$	0	0
$183$	−16.1750	−1.19569
$184$	0	0
$185$	−17.7485	−1.30489
$186$	0	0
$187$	4.54435	0.332316
$188$	0	0
$189$	10.5622	0.768285
$190$	0	0
$191$	−0.690960	−0.0499961	−0.0249981	−	0.999687i	$-0.507958\pi$
$191$	−0.690960	−0.0499961	−0.0249981	+	0.999687i	$0.507958\pi$
$192$	0	0
$193$	−19.3391	−1.39206	−0.696031	−	0.718012i	$-0.745052\pi$
$193$	−19.3391	−1.39206	−0.696031	+	0.718012i	$0.745052\pi$
$194$	0	0
$195$	19.7883	1.41707
$196$	0	0
$197$	−12.7676	−0.909653	−0.454826	−	0.890580i	$-0.650299\pi$
$197$	−12.7676	−0.909653	−0.454826	+	0.890580i	$0.650299\pi$
$198$	0	0
$199$	1.78947	0.126852	0.0634261	−	0.997987i	$-0.479797\pi$
$199$	1.78947	0.126852	0.0634261	+	0.997987i	$0.479797\pi$
$200$	0	0
$201$	0.194276	0.0137032
$202$	0	0
$203$	−6.37811	−0.447655
$204$	0	0
$205$	1.67064	0.116683
$206$	0	0
$207$	0.538210	0.0374082
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−10.5987	−0.729644	−0.364822	−	0.931077i	$-0.618870\pi$
$211$	−10.5987	−0.729644	−0.364822	+	0.931077i	$0.618870\pi$
$212$	0	0
$213$	−16.8411	−1.15393
$214$	0	0
$215$	20.9374	1.42792
$216$	0	0
$217$	−18.7777	−1.27472
$218$	0	0
$219$	4.07302	0.275229
$220$	0	0
$221$	−20.2673	−1.36333
$222$	0	0
$223$	3.22342	0.215856	0.107928	−	0.994159i	$-0.465578\pi$
$223$	3.22342	0.215856	0.107928	+	0.994159i	$0.465578\pi$
$224$	0	0
$225$	−0.123978	−0.00826520
$226$	0	0
$227$	19.1441	1.27064	0.635320	−	0.772249i	$-0.280868\pi$
$227$	19.1441	1.27064	0.635320	+	0.772249i	$0.280868\pi$
$228$	0	0
$229$	−29.4688	−1.94735	−0.973676	−	0.227935i	$-0.926803\pi$
$229$	−29.4688	−1.94735	−0.973676	+	0.227935i	$0.926803\pi$
$230$	0	0
$231$	−3.43828	−0.226222
$232$	0	0
$233$	19.5192	1.27875	0.639374	−	0.768896i	$-0.279194\pi$
$233$	19.5192	1.27875	0.639374	+	0.768896i	$0.279194\pi$
$234$	0	0
$235$	−30.2042	−1.97030
$236$	0	0
$237$	−3.44372	−0.223694
$238$	0	0
$239$	−11.1392	−0.720535	−0.360267	−	0.932849i	$-0.617315\pi$
$239$	−11.1392	−0.720535	−0.360267	+	0.932849i	$0.617315\pi$
$240$	0	0
$241$	−16.7483	−1.07885	−0.539427	−	0.842033i	$-0.681359\pi$
$241$	−16.7483	−1.07885	−0.539427	+	0.842033i	$0.681359\pi$
$242$	0	0
$243$	−0.747423	−0.0479472
$244$	0	0
$245$	−7.68191	−0.490779
$246$	0	0
$247$	4.45988	0.283776
$248$	0	0
$249$	−4.96965	−0.314939
$250$	0	0
$251$	−30.1448	−1.90272	−0.951360	−	0.308081i	$-0.900313\pi$
$251$	−30.1448	−1.90272	−0.951360	+	0.308081i	$0.900313\pi$
$252$	0	0
$253$	−7.48175	−0.470373
$254$	0	0
$255$	−20.1631	−1.26266
$256$	0	0
$257$	−11.6015	−0.723680	−0.361840	−	0.932240i	$-0.617851\pi$
$257$	−11.6015	−0.723680	−0.361840	+	0.932240i	$0.617851\pi$
$258$	0	0
$259$	−13.7536	−0.854605
$260$	0	0
$261$	0.228344	0.0141342
$262$	0	0
$263$	3.88242	0.239401	0.119700	−	0.992810i	$-0.461807\pi$
$263$	3.88242	0.239401	0.119700	+	0.992810i	$0.461807\pi$
$264$	0	0
$265$	−14.3505	−0.881545
$266$	0	0
$267$	15.0929	0.923673
$268$	0	0
$269$	23.7542	1.44832	0.724160	−	0.689632i	$-0.242228\pi$
$269$	23.7542	1.44832	0.724160	+	0.689632i	$0.242228\pi$
$270$	0	0
$271$	−17.3686	−1.05507	−0.527533	−	0.849535i	$-0.676883\pi$
$271$	−17.3686	−1.05507	−0.527533	+	0.849535i	$0.676883\pi$
$272$	0	0
$273$	15.3343	0.928075
$274$	0	0
$275$	1.72344	0.103927
$276$	0	0
$277$	6.47702	0.389167	0.194583	−	0.980886i	$-0.437665\pi$
$277$	6.47702	0.389167	0.194583	+	0.980886i	$0.437665\pi$
$278$	0	0
$279$	0.672267	0.0402476
$280$	0	0
$281$	−11.8438	−0.706542	−0.353271	−	0.935521i	$-0.614931\pi$
$281$	−11.8438	−0.706542	−0.353271	+	0.935521i	$0.614931\pi$
$282$	0	0
$283$	−11.1566	−0.663192	−0.331596	−	0.943421i	$-0.607587\pi$
$283$	−11.1566	−0.663192	−0.331596	+	0.943421i	$0.607587\pi$
$284$	0	0
$285$	4.43696	0.262823
$286$	0	0
$287$	1.29461	0.0764183
$288$	0	0
$289$	3.65116	0.214774
$290$	0	0
$291$	−18.8399	−1.10441
$292$	0	0
$293$	−25.7289	−1.50310	−0.751551	−	0.659675i	$-0.770694\pi$
$293$	−25.7289	−1.50310	−0.751551	+	0.659675i	$0.770694\pi$
$294$	0	0
$295$	−23.7730	−1.38412
$296$	0	0
$297$	5.25657	0.305017
$298$	0	0
$299$	33.3677	1.92970
$300$	0	0
$301$	16.2247	0.935178
$302$	0	0
$303$	−1.47989	−0.0850172
$304$	0	0
$305$	24.5104	1.40346
$306$	0	0
$307$	−0.843227	−0.0481255	−0.0240627	−	0.999710i	$-0.507660\pi$
$307$	−0.843227	−0.0481255	−0.0240627	+	0.999710i	$0.507660\pi$
$308$	0	0
$309$	4.53209	0.257821
$310$	0	0
$311$	3.89732	0.220997	0.110498	−	0.993876i	$-0.464755\pi$
$311$	3.89732	0.220997	0.110498	+	0.993876i	$0.464755\pi$
$312$	0	0
$313$	8.00036	0.452207	0.226104	−	0.974103i	$-0.427401\pi$
$313$	8.00036	0.452207	0.226104	+	0.974103i	$0.427401\pi$
$314$	0	0
$315$	−0.374797	−0.0211174
$316$	0	0
$317$	24.3173	1.36579	0.682897	−	0.730515i	$-0.260720\pi$
$317$	24.3173	1.36579	0.682897	+	0.730515i	$0.260720\pi$
$318$	0	0
$319$	−3.17425	−0.177724
$320$	0	0
$321$	−0.308663	−0.0172279
$322$	0	0
$323$	−4.54435	−0.252855
$324$	0	0
$325$	−7.68633	−0.426361
$326$	0	0
$327$	−11.3070	−0.625276
$328$	0	0
$329$	−23.4057	−1.29040
$330$	0	0
$331$	−1.00272	−0.0551145	−0.0275573	−	0.999620i	$-0.508773\pi$
$331$	−1.00272	−0.0551145	−0.0275573	+	0.999620i	$0.508773\pi$
$332$	0	0
$333$	0.492395	0.0269831
$334$	0	0
$335$	−0.294391	−0.0160843
$336$	0	0
$337$	−24.3229	−1.32495	−0.662477	−	0.749082i	$-0.730495\pi$
$337$	−24.3229	−1.32495	−0.662477	+	0.749082i	$0.730495\pi$
$338$	0	0
$339$	−7.23134	−0.392752
$340$	0	0
$341$	−9.34529	−0.506076
$342$	0	0
$343$	−20.0181	−1.08088
$344$	0	0
$345$	33.1962	1.78723
$346$	0	0
$347$	14.4536	0.775908	0.387954	−	0.921679i	$-0.373182\pi$
$347$	14.4536	0.775908	0.387954	+	0.921679i	$0.373182\pi$
$348$	0	0
$349$	34.9365	1.87011	0.935055	−	0.354503i	$-0.115350\pi$
$349$	34.9365	1.87011	0.935055	+	0.354503i	$0.115350\pi$
$350$	0	0
$351$	−23.4437	−1.25133
$352$	0	0
$353$	1.66429	0.0885813	0.0442906	−	0.999019i	$-0.485897\pi$
$353$	1.66429	0.0885813	0.0442906	+	0.999019i	$0.485897\pi$
$354$	0	0
$355$	25.5197	1.35444
$356$	0	0
$357$	−15.6248	−0.826950
$358$	0	0
$359$	−11.6721	−0.616032	−0.308016	−	0.951381i	$-0.599665\pi$
$359$	−11.6721	−0.616032	−0.308016	+	0.951381i	$0.599665\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.71116	−0.0898125
$364$	0	0
$365$	−6.17195	−0.323054
$366$	0	0
$367$	1.44184	0.0752633	0.0376317	−	0.999292i	$-0.488019\pi$
$367$	1.44184	0.0752633	0.0376317	+	0.999292i	$0.488019\pi$
$368$	0	0
$369$	−0.0463486	−0.00241281
$370$	0	0
$371$	−11.1205	−0.577345
$372$	0	0
$373$	7.90630	0.409372	0.204686	−	0.978828i	$-0.434383\pi$
$373$	7.90630	0.409372	0.204686	+	0.978828i	$0.434383\pi$
$374$	0	0
$375$	14.5380	0.750739
$376$	0	0
$377$	14.1568	0.729111
$378$	0	0
$379$	25.0206	1.28522	0.642611	−	0.766192i	$-0.277851\pi$
$379$	25.0206	1.28522	0.642611	+	0.766192i	$0.277851\pi$
$380$	0	0
$381$	24.0945	1.23440
$382$	0	0
$383$	−18.3767	−0.939005	−0.469503	−	0.882931i	$-0.655567\pi$
$383$	−18.3767	−0.939005	−0.469503	+	0.882931i	$0.655567\pi$
$384$	0	0
$385$	5.21010	0.265531
$386$	0	0
$387$	−0.580866	−0.0295271
$388$	0	0
$389$	16.0386	0.813188	0.406594	−	0.913609i	$-0.366716\pi$
$389$	16.0386	0.813188	0.406594	+	0.913609i	$0.366716\pi$
$390$	0	0
$391$	−33.9997	−1.71944
$392$	0	0
$393$	17.2993	0.872636
$394$	0	0
$395$	5.21836	0.262564
$396$	0	0
$397$	−28.7655	−1.44370	−0.721849	−	0.692051i	$-0.756707\pi$
$397$	−28.7655	−1.44370	−0.721849	+	0.692051i	$0.756707\pi$
$398$	0	0
$399$	3.43828	0.172129
$400$	0	0
$401$	−22.0831	−1.10278	−0.551389	−	0.834249i	$-0.685902\pi$
$401$	−22.0831	−1.10278	−0.551389	+	0.834249i	$0.685902\pi$
$402$	0	0
$403$	41.6789	2.07617
$404$	0	0
$405$	−22.7636	−1.13113
$406$	0	0
$407$	−6.84486	−0.339287
$408$	0	0
$409$	6.78338	0.335416	0.167708	−	0.985837i	$-0.446363\pi$
$409$	6.78338	0.335416	0.167708	+	0.985837i	$0.446363\pi$
$410$	0	0
$411$	4.90642	0.242016
$412$	0	0
$413$	−18.4221	−0.906492
$414$	0	0
$415$	7.53063	0.369664
$416$	0	0
$417$	−36.4561	−1.78526
$418$	0	0
$419$	−18.6261	−0.909943	−0.454971	−	0.890506i	$-0.650351\pi$
$419$	−18.6261	−0.909943	−0.454971	+	0.890506i	$0.650351\pi$
$420$	0	0
$421$	−1.65361	−0.0805921	−0.0402960	−	0.999188i	$-0.512830\pi$
$421$	−1.65361	−0.0805921	−0.0402960	+	0.999188i	$0.512830\pi$
$422$	0	0
$423$	0.837955	0.0407428
$424$	0	0
$425$	7.83191	0.379903
$426$	0	0
$427$	18.9935	0.919161
$428$	0	0
$429$	7.63157	0.368456
$430$	0	0
$431$	22.0877	1.06393	0.531964	−	0.846767i	$-0.321454\pi$
$431$	22.0877	1.06393	0.531964	+	0.846767i	$0.321454\pi$
$432$	0	0
$433$	−17.5885	−0.845249	−0.422625	−	0.906305i	$-0.638891\pi$
$433$	−17.5885	−0.845249	−0.422625	+	0.906305i	$0.638891\pi$
$434$	0	0
$435$	14.0840	0.675278
$436$	0	0
$437$	7.48175	0.357900
$438$	0	0
$439$	−9.46754	−0.451861	−0.225931	−	0.974143i	$-0.572542\pi$
$439$	−9.46754	−0.451861	−0.225931	+	0.974143i	$0.572542\pi$
$440$	0	0
$441$	0.213119	0.0101485
$442$	0	0
$443$	14.0458	0.667337	0.333669	−	0.942690i	$-0.391713\pi$
$443$	14.0458	0.667337	0.333669	+	0.942690i	$0.391713\pi$
$444$	0	0
$445$	−22.8707	−1.08418
$446$	0	0
$447$	12.4719	0.589903
$448$	0	0
$449$	18.8516	0.889660	0.444830	−	0.895615i	$-0.353264\pi$
$449$	18.8516	0.889660	0.444830	+	0.895615i	$0.353264\pi$
$450$	0	0
$451$	0.644299	0.0303389
$452$	0	0
$453$	38.3841	1.80344
$454$	0	0
$455$	−23.2365	−1.08934
$456$	0	0
$457$	22.1917	1.03808	0.519041	−	0.854749i	$-0.326289\pi$
$457$	22.1917	1.03808	0.519041	+	0.854749i	$0.326289\pi$
$458$	0	0
$459$	23.8877	1.11498
$460$	0	0
$461$	29.0705	1.35395	0.676973	−	0.736008i	$-0.263291\pi$
$461$	29.0705	1.35395	0.676973	+	0.736008i	$0.263291\pi$
$462$	0	0
$463$	−21.1214	−0.981594	−0.490797	−	0.871274i	$-0.663294\pi$
$463$	−21.1214	−0.981594	−0.490797	+	0.871274i	$0.663294\pi$
$464$	0	0
$465$	41.4647	1.92288
$466$	0	0
$467$	6.37732	0.295107	0.147554	−	0.989054i	$-0.452860\pi$
$467$	6.37732	0.295107	0.147554	+	0.989054i	$0.452860\pi$
$468$	0	0
$469$	−0.228129	−0.0105340
$470$	0	0
$471$	−27.3153	−1.25862
$472$	0	0
$473$	8.07470	0.371275
$474$	0	0
$475$	−1.72344	−0.0790767
$476$	0	0
$477$	0.398127	0.0182290
$478$	0	0
$479$	−29.3832	−1.34255	−0.671276	−	0.741208i	$-0.734253\pi$
$479$	−29.3832	−1.34255	−0.671276	+	0.741208i	$0.734253\pi$
$480$	0	0
$481$	30.5273	1.39193
$482$	0	0
$483$	25.7243	1.17050
$484$	0	0
$485$	28.5485	1.29632
$486$	0	0
$487$	34.3525	1.55666	0.778331	−	0.627854i	$-0.216066\pi$
$487$	34.3525	1.55666	0.778331	+	0.627854i	$0.216066\pi$
$488$	0	0
$489$	−3.50515	−0.158508
$490$	0	0
$491$	−23.0123	−1.03853	−0.519264	−	0.854614i	$-0.673794\pi$
$491$	−23.0123	−1.03853	−0.519264	+	0.854614i	$0.673794\pi$
$492$	0	0
$493$	−14.4249	−0.649666
$494$	0	0
$495$	−0.186528	−0.00838383
$496$	0	0
$497$	19.7756	0.887058
$498$	0	0
$499$	18.9873	0.849988	0.424994	−	0.905196i	$-0.360276\pi$
$499$	18.9873	0.849988	0.424994	+	0.905196i	$0.360276\pi$
$500$	0	0
$501$	−11.9625	−0.534444
$502$	0	0
$503$	33.3366	1.48641	0.743203	−	0.669066i	$-0.233306\pi$
$503$	33.3366	1.48641	0.743203	+	0.669066i	$0.233306\pi$
$504$	0	0
$505$	2.24251	0.0997902
$506$	0	0
$507$	−11.7908	−0.523650
$508$	0	0
$509$	10.1122	0.448217	0.224109	−	0.974564i	$-0.428053\pi$
$509$	10.1122	0.448217	0.224109	+	0.974564i	$0.428053\pi$
$510$	0	0
$511$	−4.78274	−0.211576
$512$	0	0
$513$	−5.25657	−0.232083
$514$	0	0
$515$	−6.86758	−0.302622
$516$	0	0
$517$	−11.6485	−0.512302
$518$	0	0
$519$	33.1124	1.45347
$520$	0	0
$521$	24.9735	1.09411	0.547055	−	0.837097i	$-0.315749\pi$
$521$	24.9735	1.09411	0.547055	+	0.837097i	$0.315749\pi$
$522$	0	0
$523$	27.5980	1.20678	0.603388	−	0.797448i	$-0.293817\pi$
$523$	27.5980	1.20678	0.603388	+	0.797448i	$0.293817\pi$
$524$	0	0
$525$	−5.92566	−0.258617
$526$	0	0
$527$	−42.4683	−1.84995
$528$	0	0
$529$	32.9765	1.43376
$530$	0	0
$531$	0.659535	0.0286214
$532$	0	0
$533$	−2.87350	−0.124465
$534$	0	0
$535$	0.467724	0.0202215
$536$	0	0
$537$	−39.1500	−1.68945
$538$	0	0
$539$	−2.96260	−0.127608
$540$	0	0
$541$	8.78281	0.377603	0.188801	−	0.982015i	$-0.439540\pi$
$541$	8.78281	0.377603	0.188801	+	0.982015i	$0.439540\pi$
$542$	0	0
$543$	2.28528	0.0980708
$544$	0	0
$545$	17.1337	0.733927
$546$	0	0
$547$	−20.2388	−0.865347	−0.432674	−	0.901551i	$-0.642430\pi$
$547$	−20.2388	−0.865347	−0.432674	+	0.901551i	$0.642430\pi$
$548$	0	0
$549$	−0.679993	−0.0290214
$550$	0	0
$551$	3.17425	0.135228
$552$	0	0
$553$	4.04379	0.171959
$554$	0	0
$555$	30.3704	1.28915
$556$	0	0
$557$	21.6180	0.915985	0.457993	−	0.888956i	$-0.348569\pi$
$557$	21.6180	0.915985	0.457993	+	0.888956i	$0.348569\pi$
$558$	0	0
$559$	−36.0122	−1.52316
$560$	0	0
$561$	−7.77611	−0.328308
$562$	0	0
$563$	2.72687	0.114924	0.0574620	−	0.998348i	$-0.481699\pi$
$563$	2.72687	0.114924	0.0574620	+	0.998348i	$0.481699\pi$
$564$	0	0
$565$	10.9578	0.460999
$566$	0	0
$567$	−17.6399	−0.740807
$568$	0	0
$569$	18.8953	0.792131	0.396065	−	0.918222i	$-0.370375\pi$
$569$	18.8953	0.792131	0.396065	+	0.918222i	$0.370375\pi$
$570$	0	0
$571$	−32.7597	−1.37095	−0.685475	−	0.728096i	$-0.740405\pi$
$571$	−32.7597	−1.37095	−0.685475	+	0.728096i	$0.740405\pi$
$572$	0	0
$573$	1.18234	0.0493931
$574$	0	0
$575$	−12.8943	−0.537730
$576$	0	0
$577$	−39.3665	−1.63885	−0.819425	−	0.573186i	$-0.805707\pi$
$577$	−39.3665	−1.63885	−0.819425	+	0.573186i	$0.805707\pi$
$578$	0	0
$579$	33.0923	1.37527
$580$	0	0
$581$	5.83561	0.242102
$582$	0	0
$583$	−5.53442	−0.229212
$584$	0	0
$585$	0.831895	0.0343946
$586$	0	0
$587$	46.0274	1.89976	0.949878	−	0.312621i	$-0.101207\pi$
$587$	46.0274	1.89976	0.949878	+	0.312621i	$0.101207\pi$
$588$	0	0
$589$	9.34529	0.385066
$590$	0	0
$591$	21.8474	0.898680
$592$	0	0
$593$	−11.6575	−0.478715	−0.239358	−	0.970932i	$-0.576937\pi$
$593$	−11.6575	−0.478715	−0.239358	+	0.970932i	$0.576937\pi$
$594$	0	0
$595$	23.6766	0.970644
$596$	0	0
$597$	−3.06207	−0.125322
$598$	0	0
$599$	36.7024	1.49962	0.749811	−	0.661653i	$-0.230145\pi$
$599$	36.7024	1.49962	0.749811	+	0.661653i	$0.230145\pi$
$600$	0	0
$601$	−22.7859	−0.929456	−0.464728	−	0.885454i	$-0.653848\pi$
$601$	−22.7859	−0.929456	−0.464728	+	0.885454i	$0.653848\pi$
$602$	0	0
$603$	0.00816730	0.000332598 0
$604$	0	0
$605$	2.59296	0.105419
$606$	0	0
$607$	−18.4924	−0.750584	−0.375292	−	0.926907i	$-0.622458\pi$
$607$	−18.4924	−0.750584	−0.375292	+	0.926907i	$0.622458\pi$
$608$	0	0
$609$	10.9140	0.442256
$610$	0	0
$611$	51.9511	2.10172
$612$	0	0
$613$	21.1249	0.853229	0.426614	−	0.904434i	$-0.359706\pi$
$613$	21.1249	0.853229	0.426614	+	0.904434i	$0.359706\pi$
$614$	0	0
$615$	−2.85873	−0.115275
$616$	0	0
$617$	24.6204	0.991178	0.495589	−	0.868557i	$-0.334952\pi$
$617$	24.6204	0.991178	0.495589	+	0.868557i	$0.334952\pi$
$618$	0	0
$619$	8.89942	0.357698	0.178849	−	0.983877i	$-0.442763\pi$
$619$	8.89942	0.357698	0.178849	+	0.983877i	$0.442763\pi$
$620$	0	0
$621$	−39.3283	−1.57819
$622$	0	0
$623$	−17.7229	−0.710052
$624$	0	0
$625$	−30.6470	−1.22588
$626$	0	0
$627$	1.71116	0.0683371
$628$	0	0
$629$	−31.1055	−1.24026
$630$	0	0
$631$	−15.2980	−0.609005	−0.304503	−	0.952512i	$-0.598490\pi$
$631$	−15.2980	−0.609005	−0.304503	+	0.952512i	$0.598490\pi$
$632$	0	0
$633$	18.1360	0.720843
$634$	0	0
$635$	−36.5109	−1.44889
$636$	0	0
$637$	13.2129	0.523513
$638$	0	0
$639$	−0.707993	−0.0280078
$640$	0	0
$641$	−28.4765	−1.12475	−0.562377	−	0.826881i	$-0.690113\pi$
$641$	−28.4765	−1.12475	−0.562377	+	0.826881i	$0.690113\pi$
$642$	0	0
$643$	12.5264	0.493992	0.246996	−	0.969017i	$-0.420557\pi$
$643$	12.5264	0.493992	0.246996	+	0.969017i	$0.420557\pi$
$644$	0	0
$645$	−35.8272	−1.41069
$646$	0	0
$647$	17.3701	0.682889	0.341445	−	0.939902i	$-0.389084\pi$
$647$	17.3701	0.682889	0.341445	+	0.939902i	$0.389084\pi$
$648$	0	0
$649$	−9.16829	−0.359887
$650$	0	0
$651$	32.1317	1.25934
$652$	0	0
$653$	−32.5362	−1.27324	−0.636621	−	0.771177i	$-0.719668\pi$
$653$	−32.5362	−1.27324	−0.636621	+	0.771177i	$0.719668\pi$
$654$	0	0
$655$	−26.2141	−1.02427
$656$	0	0
$657$	0.171228	0.00668025
$658$	0	0
$659$	−48.1592	−1.87602	−0.938008	−	0.346614i	$-0.887332\pi$
$659$	−48.1592	−1.87602	−0.938008	+	0.346614i	$0.887332\pi$
$660$	0	0
$661$	−6.19327	−0.240890	−0.120445	−	0.992720i	$-0.538432\pi$
$661$	−6.19327	−0.240890	−0.120445	+	0.992720i	$0.538432\pi$
$662$	0	0
$663$	34.6805	1.34688
$664$	0	0
$665$	−5.21010	−0.202039
$666$	0	0
$667$	23.7489	0.919562
$668$	0	0
$669$	−5.51578	−0.213252
$670$	0	0
$671$	9.45268	0.364917
$672$	0	0
$673$	22.2416	0.857351	0.428676	−	0.903459i	$-0.358980\pi$
$673$	22.2416	0.857351	0.428676	+	0.903459i	$0.358980\pi$
$674$	0	0
$675$	9.05937	0.348695
$676$	0	0
$677$	45.2780	1.74018	0.870088	−	0.492897i	$-0.164062\pi$
$677$	45.2780	1.74018	0.870088	+	0.492897i	$0.164062\pi$
$678$	0	0
$679$	22.1227	0.848992
$680$	0	0
$681$	−32.7586	−1.25531
$682$	0	0
$683$	36.8106	1.40852	0.704259	−	0.709943i	$-0.251279\pi$
$683$	36.8106	1.40852	0.704259	+	0.709943i	$0.251279\pi$
$684$	0	0
$685$	−7.43482	−0.284070
$686$	0	0
$687$	50.4258	1.92386
$688$	0	0
$689$	24.6829	0.940342
$690$	0	0
$691$	35.0477	1.33328	0.666638	−	0.745382i	$-0.267733\pi$
$691$	35.0477	1.33328	0.666638	+	0.745382i	$0.267733\pi$
$692$	0	0
$693$	−0.144544	−0.00549077
$694$	0	0
$695$	55.2428	2.09548
$696$	0	0
$697$	2.92793	0.110903
$698$	0	0
$699$	−33.4005	−1.26332
$700$	0	0
$701$	−29.2997	−1.10664	−0.553318	−	0.832970i	$-0.686639\pi$
$701$	−29.2997	−1.10664	−0.553318	+	0.832970i	$0.686639\pi$
$702$	0	0
$703$	6.84486	0.258159
$704$	0	0
$705$	51.6841	1.94654
$706$	0	0
$707$	1.73776	0.0653550
$708$	0	0
$709$	34.3627	1.29052	0.645259	−	0.763964i	$-0.276750\pi$
$709$	34.3627	1.29052	0.645259	+	0.763964i	$0.276750\pi$
$710$	0	0
$711$	−0.144773	−0.00542941
$712$	0	0
$713$	69.9191	2.61849
$714$	0	0
$715$	−11.5643	−0.432480
$716$	0	0
$717$	19.0609	0.711843
$718$	0	0
$719$	−23.2839	−0.868345	−0.434172	−	0.900830i	$-0.642959\pi$
$719$	−23.2839	−0.868345	−0.434172	+	0.900830i	$0.642959\pi$
$720$	0	0
$721$	−5.32180	−0.198194
$722$	0	0
$723$	28.6590	1.06584
$724$	0	0
$725$	−5.47062	−0.203174
$726$	0	0
$727$	10.5331	0.390652	0.195326	−	0.980738i	$-0.437424\pi$
$727$	10.5331	0.390652	0.195326	+	0.980738i	$0.437424\pi$
$728$	0	0
$729$	27.6160	1.02282
$730$	0	0
$731$	36.6943	1.35719
$732$	0	0
$733$	46.7251	1.72583	0.862915	−	0.505349i	$-0.168636\pi$
$733$	46.7251	1.72583	0.862915	+	0.505349i	$0.168636\pi$
$734$	0	0
$735$	13.1450	0.484859
$736$	0	0
$737$	−0.113535	−0.00418211
$738$	0	0
$739$	−6.14537	−0.226061	−0.113031	−	0.993592i	$-0.536056\pi$
$739$	−6.14537	−0.226061	−0.113031	+	0.993592i	$0.536056\pi$
$740$	0	0
$741$	−7.63157	−0.280353
$742$	0	0
$743$	−5.00527	−0.183625	−0.0918127	−	0.995776i	$-0.529266\pi$
$743$	−5.00527	−0.183625	−0.0918127	+	0.995776i	$0.529266\pi$
$744$	0	0
$745$	−18.8990	−0.692407
$746$	0	0
$747$	−0.208922	−0.00764406
$748$	0	0
$749$	0.362447	0.0132435
$750$	0	0
$751$	−45.1888	−1.64896	−0.824482	−	0.565889i	$-0.808533\pi$
$751$	−45.1888	−1.64896	−0.824482	+	0.565889i	$0.808533\pi$
$752$	0	0
$753$	51.5825	1.87977
$754$	0	0
$755$	−58.1643	−2.11682
$756$	0	0
$757$	−2.72001	−0.0988606	−0.0494303	−	0.998778i	$-0.515741\pi$
$757$	−2.72001	−0.0988606	−0.0494303	+	0.998778i	$0.515741\pi$
$758$	0	0
$759$	12.8025	0.464700
$760$	0	0
$761$	−9.10142	−0.329926	−0.164963	−	0.986300i	$-0.552751\pi$
$761$	−9.10142	−0.329926	−0.164963	+	0.986300i	$0.552751\pi$
$762$	0	0
$763$	13.2772	0.480666
$764$	0	0
$765$	−0.847651	−0.0306469
$766$	0	0
$767$	40.8895	1.47644
$768$	0	0
$769$	37.2427	1.34301	0.671503	−	0.741002i	$-0.265649\pi$
$769$	37.2427	1.34301	0.671503	+	0.741002i	$0.265649\pi$
$770$	0	0
$771$	19.8520	0.714951
$772$	0	0
$773$	−14.7996	−0.532306	−0.266153	−	0.963931i	$-0.585753\pi$
$773$	−14.7996	−0.532306	−0.266153	+	0.963931i	$0.585753\pi$
$774$	0	0
$775$	−16.1060	−0.578545
$776$	0	0
$777$	23.5345	0.844297
$778$	0	0
$779$	−0.644299	−0.0230844
$780$	0	0
$781$	9.84191	0.352171
$782$	0	0
$783$	−16.6857	−0.596297
$784$	0	0
$785$	41.3915	1.47732
$786$	0	0
$787$	43.0866	1.53587	0.767936	−	0.640527i	$-0.221284\pi$
$787$	43.0866	1.53587	0.767936	+	0.640527i	$0.221284\pi$
$788$	0	0
$789$	−6.64344	−0.236513
$790$	0	0
$791$	8.49140	0.301919
$792$	0	0
$793$	−42.1579	−1.49707
$794$	0	0
$795$	24.5560	0.870912
$796$	0	0
$797$	29.2827	1.03724	0.518622	−	0.855003i	$-0.326445\pi$
$797$	29.2827	1.03724	0.518622	+	0.855003i	$0.326445\pi$
$798$	0	0
$799$	−52.9351	−1.87271
$800$	0	0
$801$	0.634502	0.0224190
$802$	0	0
$803$	−2.38027	−0.0839980
$804$	0	0
$805$	−38.9807	−1.37389
$806$	0	0
$807$	−40.6472	−1.43085
$808$	0	0
$809$	−34.4478	−1.21112	−0.605560	−	0.795800i	$-0.707051\pi$
$809$	−34.4478	−1.21112	−0.605560	+	0.795800i	$0.707051\pi$
$810$	0	0
$811$	−41.0171	−1.44031	−0.720153	−	0.693815i	$-0.755928\pi$
$811$	−41.0171	−1.44031	−0.720153	+	0.693815i	$0.755928\pi$
$812$	0	0
$813$	29.7204	1.04234
$814$	0	0
$815$	5.31144	0.186052
$816$	0	0
$817$	−8.07470	−0.282498
$818$	0	0
$819$	0.644649	0.0225259
$820$	0	0
$821$	14.6696	0.511973	0.255987	−	0.966680i	$-0.417600\pi$
$821$	14.6696	0.511973	0.255987	+	0.966680i	$0.417600\pi$
$822$	0	0
$823$	−8.16556	−0.284634	−0.142317	−	0.989821i	$-0.545455\pi$
$823$	−8.16556	−0.284634	−0.142317	+	0.989821i	$0.545455\pi$
$824$	0	0
$825$	−2.94907	−0.102674
$826$	0	0
$827$	21.7995	0.758043	0.379021	−	0.925388i	$-0.376261\pi$
$827$	21.7995	0.758043	0.379021	+	0.925388i	$0.376261\pi$
$828$	0	0
$829$	−13.3293	−0.462944	−0.231472	−	0.972842i	$-0.574354\pi$
$829$	−13.3293	−0.462944	−0.231472	+	0.972842i	$0.574354\pi$
$830$	0	0
$831$	−11.0832	−0.384472
$832$	0	0
$833$	−13.4631	−0.466470
$834$	0	0
$835$	18.1270	0.627311
$836$	0	0
$837$	−49.1242	−1.69798
$838$	0	0
$839$	−24.2904	−0.838598	−0.419299	−	0.907848i	$-0.637724\pi$
$839$	−24.2904	−0.838598	−0.419299	+	0.907848i	$0.637724\pi$
$840$	0	0
$841$	−18.9241	−0.652557
$842$	0	0
$843$	20.2666	0.698019
$844$	0	0
$845$	17.8669	0.614641
$846$	0	0
$847$	2.00933	0.0690413
$848$	0	0
$849$	19.0907	0.655193
$850$	0	0
$851$	51.2115	1.75551
$852$	0	0
$853$	−4.18601	−0.143326	−0.0716631	−	0.997429i	$-0.522831\pi$
$853$	−4.18601	−0.143326	−0.0716631	+	0.997429i	$0.522831\pi$
$854$	0	0
$855$	0.186528	0.00637914
$856$	0	0
$857$	−14.1228	−0.482425	−0.241212	−	0.970472i	$-0.577545\pi$
$857$	−14.1228	−0.482425	−0.241212	+	0.970472i	$0.577545\pi$
$858$	0	0
$859$	−24.1260	−0.823169	−0.411584	−	0.911372i	$-0.635024\pi$
$859$	−24.1260	−0.823169	−0.411584	+	0.911372i	$0.635024\pi$
$860$	0	0
$861$	−2.21528	−0.0754966
$862$	0	0
$863$	28.2661	0.962188	0.481094	−	0.876669i	$-0.340240\pi$
$863$	28.2661	0.962188	0.481094	+	0.876669i	$0.340240\pi$
$864$	0	0
$865$	−50.1759	−1.70603
$866$	0	0
$867$	−6.24771	−0.212183
$868$	0	0
$869$	2.01251	0.0682697
$870$	0	0
$871$	0.506352	0.0171571
$872$	0	0
$873$	−0.792022	−0.0268059
$874$	0	0
$875$	−17.0712	−0.577113
$876$	0	0
$877$	−28.4762	−0.961573	−0.480787	−	0.876838i	$-0.659649\pi$
$877$	−28.4762	−0.961573	−0.480787	+	0.876838i	$0.659649\pi$
$878$	0	0
$879$	44.0263	1.48497
$880$	0	0
$881$	13.0669	0.440234	0.220117	−	0.975473i	$-0.429356\pi$
$881$	13.0669	0.440234	0.220117	+	0.975473i	$0.429356\pi$
$882$	0	0
$883$	−15.9805	−0.537786	−0.268893	−	0.963170i	$-0.586658\pi$
$883$	−15.9805	−0.537786	−0.268893	+	0.963170i	$0.586658\pi$
$884$	0	0
$885$	40.6794	1.36742
$886$	0	0
$887$	7.47191	0.250882	0.125441	−	0.992101i	$-0.459965\pi$
$887$	7.47191	0.250882	0.125441	+	0.992101i	$0.459965\pi$
$888$	0	0
$889$	−28.2929	−0.948914
$890$	0	0
$891$	−8.77902	−0.294108
$892$	0	0
$893$	11.6485	0.389803
$894$	0	0
$895$	59.3249	1.98301
$896$	0	0
$897$	−57.0974	−1.90643
$898$	0	0
$899$	29.6643	0.989359
$900$	0	0
$901$	−25.1504	−0.837880
$902$	0	0
$903$	−27.7631	−0.923897
$904$	0	0
$905$	−3.46294	−0.115112
$906$	0	0
$907$	−48.3649	−1.60593	−0.802966	−	0.596025i	$-0.796746\pi$
$907$	−48.3649	−1.60593	−0.802966	+	0.596025i	$0.796746\pi$
$908$	0	0
$909$	−0.0622139	−0.00206350
$910$	0	0
$911$	−33.2104	−1.10031	−0.550154	−	0.835063i	$-0.685431\pi$
$911$	−33.2104	−1.10031	−0.550154	+	0.835063i	$0.685431\pi$
$912$	0	0
$913$	2.90426	0.0961170
$914$	0	0
$915$	−41.9412	−1.38653
$916$	0	0
$917$	−20.3137	−0.670818
$918$	0	0
$919$	−55.5155	−1.83129	−0.915644	−	0.401990i	$-0.868319\pi$
$919$	−55.5155	−1.83129	−0.915644	+	0.401990i	$0.868319\pi$
$920$	0	0
$921$	1.44289	0.0475450
$922$	0	0
$923$	−43.8938	−1.44478
$924$	0	0
$925$	−11.7967	−0.387873
$926$	0	0
$927$	0.190527	0.00625774
$928$	0	0
$929$	−19.2731	−0.632330	−0.316165	−	0.948704i	$-0.602395\pi$
$929$	−19.2731	−0.632330	−0.316165	+	0.948704i	$0.602395\pi$
$930$	0	0
$931$	2.96260	0.0970954
$932$	0	0
$933$	−6.66893	−0.218331
$934$	0	0
$935$	11.7833	0.385356
$936$	0	0
$937$	33.0983	1.08127	0.540637	−	0.841256i	$-0.318183\pi$
$937$	33.0983	1.08127	0.540637	+	0.841256i	$0.318183\pi$
$938$	0	0
$939$	−13.6899	−0.446753
$940$	0	0
$941$	−6.91821	−0.225527	−0.112764	−	0.993622i	$-0.535970\pi$
$941$	−6.91821	−0.225527	−0.112764	+	0.993622i	$0.535970\pi$
$942$	0	0
$943$	−4.82048	−0.156977
$944$	0	0
$945$	27.3873	0.890908
$946$	0	0
$947$	−47.4763	−1.54277	−0.771386	−	0.636367i	$-0.780436\pi$
$947$	−47.4763	−1.54277	−0.771386	+	0.636367i	$0.780436\pi$
$948$	0	0
$949$	10.6157	0.344601
$950$	0	0
$951$	−41.6107	−1.34932
$952$	0	0
$953$	−12.5758	−0.407370	−0.203685	−	0.979036i	$-0.565292\pi$
$953$	−12.5758	−0.407370	−0.203685	+	0.979036i	$0.565292\pi$
$954$	0	0
$955$	−1.79163	−0.0579758
$956$	0	0
$957$	5.43164	0.175580
$958$	0	0
$959$	−5.76136	−0.186044
$960$	0	0
$961$	56.3345	1.81724
$962$	0	0
$963$	−0.0129761	−0.000418148 0
$964$	0	0
$965$	−50.1456	−1.61424
$966$	0	0
$967$	12.1925	0.392084	0.196042	−	0.980595i	$-0.437191\pi$
$967$	12.1925	0.392084	0.196042	+	0.980595i	$0.437191\pi$
$968$	0	0
$969$	7.77611	0.249805
$970$	0	0
$971$	15.2425	0.489155	0.244577	−	0.969630i	$-0.421351\pi$
$971$	15.2425	0.489155	0.244577	+	0.969630i	$0.421351\pi$
$972$	0	0
$973$	42.8085	1.37238
$974$	0	0
$975$	13.1525	0.421218
$976$	0	0
$977$	−17.7548	−0.568027	−0.284013	−	0.958820i	$-0.591666\pi$
$977$	−17.7548	−0.568027	−0.284013	+	0.958820i	$0.591666\pi$
$978$	0	0
$979$	−8.82031	−0.281898
$980$	0	0
$981$	−0.475340	−0.0151764
$982$	0	0
$983$	41.1104	1.31122	0.655609	−	0.755100i	$-0.272412\pi$
$983$	41.1104	1.31122	0.655609	+	0.755100i	$0.272412\pi$
$984$	0	0
$985$	−33.1058	−1.05484
$986$	0	0
$987$	40.0509	1.27483
$988$	0	0
$989$	−60.4129	−1.92102
$990$	0	0
$991$	18.6736	0.593185	0.296593	−	0.955004i	$-0.404150\pi$
$991$	18.6736	0.593185	0.296593	+	0.955004i	$0.404150\pi$
$992$	0	0
$993$	1.71581	0.0544497
$994$	0	0
$995$	4.64002	0.147099
$996$	0	0
$997$	−51.5507	−1.63263	−0.816313	−	0.577609i	$-0.803986\pi$
$997$	−51.5507	−1.63263	−0.816313	+	0.577609i	$0.803986\pi$
$998$	0	0
$999$	−35.9805	−1.13837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.ba.1.3		7
4.3	odd	2		209.2.a.d.1.5	✓	7
12.11	even	2		1881.2.a.p.1.3		7
20.19	odd	2		5225.2.a.n.1.3		7
44.43	even	2		2299.2.a.q.1.3		7
76.75	even	2		3971.2.a.i.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.5	✓	7	4.3	odd	2
1881.2.a.p.1.3		7	12.11	even	2
2299.2.a.q.1.3		7	44.43	even	2
3344.2.a.ba.1.3		7	1.1	even	1	trivial
3971.2.a.i.1.3		7	76.75	even	2
5225.2.a.n.1.3		7	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-1.71116 q^{3} +2.59296 q^{5} +2.00933 q^{7} -0.0719365 q^{9} +1.00000 q^{11} -4.45988 q^{13} -4.43696 q^{15} +4.54435 q^{17} -1.00000 q^{19} -3.43828 q^{21} -7.48175 q^{23} +1.72344 q^{25} +5.25657 q^{27} -3.17425 q^{29} -9.34529 q^{31} -1.71116 q^{33} +5.21010 q^{35} -6.84486 q^{37} +7.63157 q^{39} +0.644299 q^{41} +8.07470 q^{43} -0.186528 q^{45} -11.6485 q^{47} -2.96260 q^{49} -7.77611 q^{51} -5.53442 q^{53} +2.59296 q^{55} +1.71116 q^{57} -9.16829 q^{59} +9.45268 q^{61} -0.144544 q^{63} -11.5643 q^{65} -0.113535 q^{67} +12.8025 q^{69} +9.84191 q^{71} -2.38027 q^{73} -2.94907 q^{75} +2.00933 q^{77} +2.01251 q^{79} -8.77902 q^{81} +2.90426 q^{83} +11.7833 q^{85} +5.43164 q^{87} -8.82031 q^{89} -8.96136 q^{91} +15.9913 q^{93} -2.59296 q^{95} +11.0100 q^{97} -0.0719365 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) \) 7 * q - 2 * q^3 + 2 * q^5 - 10 * q^7 + 11 * q^9 + 7 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 7 * q^19 - 14 * q^21 - 10 * q^23 + 9 * q^25 + 4 * q^27 - 18 * q^29 - 24 * q^31 - 2 * q^33 - 8 * q^35 - 24 * q^39 - 12 * q^41 - 2 * q^43 - 4 * q^45 - 8 * q^47 + 17 * q^49 + 24 * q^51 + 2 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 + 14 * q^61 - 14 * q^65 - 8 * q^67 - 6 * q^69 - 10 * q^71 - 6 * q^73 - 26 * q^75 - 10 * q^77 - 52 * q^79 - q^81 + 10 * q^83 - 12 * q^85 - 6 * q^87 - 12 * q^91 + 2 * q^93 - 2 * q^95 - 24 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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