LMFDB - Embedded newform 3856.2.a.h.1.6

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:	$6$
Coefficient field:	6.6.131357120.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 26x^{2} - 30x - 13 $ x^6 - 2x^5 - 10x^4 + 16x^3 + 26x^2 - 30*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 482)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.34780$ 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)

$f(q)$	$=$	$q+2.34780 q^{3} -3.81130 q^{5} -0.407599 q^{7} +2.51218 q^{9} +O(q^{10})$	$q+2.34780 q^{3} -3.81130 q^{5} -0.407599 q^{7} +2.51218 q^{9} +4.85999 q^{11} -2.78733 q^{13} -8.94818 q^{15} -1.04869 q^{17} +2.71468 q^{19} -0.956962 q^{21} -8.67128 q^{23} +9.52598 q^{25} -1.14530 q^{27} -4.08819 q^{29} +0.325219 q^{31} +11.4103 q^{33} +1.55348 q^{35} +11.4919 q^{37} -6.54411 q^{39} +0.439895 q^{41} -9.74430 q^{43} -9.57467 q^{45} +4.85474 q^{47} -6.83386 q^{49} -2.46212 q^{51} -8.80287 q^{53} -18.5228 q^{55} +6.37354 q^{57} -8.47943 q^{59} -6.37079 q^{61} -1.02396 q^{63} +10.6234 q^{65} +3.46878 q^{67} -20.3585 q^{69} +6.11292 q^{71} +2.77180 q^{73} +22.3651 q^{75} -1.98092 q^{77} -8.48606 q^{79} -10.2255 q^{81} -15.0594 q^{83} +3.99686 q^{85} -9.59827 q^{87} +5.55559 q^{89} +1.13611 q^{91} +0.763550 q^{93} -10.3465 q^{95} -12.8685 q^{97} +12.2092 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9	$ 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9} + 4 q^{11} + 9 q^{13} - 5 q^{15} + q^{17} - 10 q^{19} + 8 q^{21} - 9 q^{23} + 13 q^{25} - 8 q^{27} - q^{29} - 14 q^{31} + 10 q^{33} + 3 q^{35} + 20 q^{37} - 13 q^{39} - 4 q^{41} - 19 q^{43} - 6 q^{45} - q^{47} + 14 q^{49} - 5 q^{51} - 3 q^{53} - 11 q^{55} + 12 q^{57} + q^{59} + 4 q^{61} - 14 q^{63} + 5 q^{65} - 5 q^{67} - 15 q^{69} - 2 q^{71} + 15 q^{73} + 11 q^{75} - 6 q^{77} - 12 q^{79} - 18 q^{81} + 8 q^{83} - 32 q^{85} - 11 q^{87} - 24 q^{89} - q^{91} - 18 q^{93} - 9 q^{95} - 12 q^{97} + 40 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9 + 4 * q^11 + 9 * q^13 - 5 * q^15 + q^17 - 10 * q^19 + 8 * q^21 - 9 * q^23 + 13 * q^25 - 8 * q^27 - q^29 - 14 * q^31 + 10 * q^33 + 3 * q^35 + 20 * q^37 - 13 * q^39 - 4 * q^41 - 19 * q^43 - 6 * q^45 - q^47 + 14 * q^49 - 5 * q^51 - 3 * q^53 - 11 * q^55 + 12 * q^57 + q^59 + 4 * q^61 - 14 * q^63 + 5 * q^65 - 5 * q^67 - 15 * q^69 - 2 * q^71 + 15 * q^73 + 11 * q^75 - 6 * q^77 - 12 * q^79 - 18 * q^81 + 8 * q^83 - 32 * q^85 - 11 * q^87 - 24 * q^89 - q^91 - 18 * q^93 - 9 * q^95 - 12 * q^97 + 40 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.34780	1.35551	0.677753	−	0.735290i	$-0.262954\pi$
$3$	2.34780	1.35551	0.677753	+	0.735290i	$0.262954\pi$
$4$	0	0
$5$	−3.81130	−1.70446	−0.852232	−	0.523164i	$-0.824751\pi$
$5$	−3.81130	−1.70446	−0.852232	+	0.523164i	$0.824751\pi$
$6$	0	0
$7$	−0.407599	−0.154058	−0.0770289	−	0.997029i	$-0.524543\pi$
$7$	−0.407599	−0.154058	−0.0770289	+	0.997029i	$0.524543\pi$
$8$	0	0
$9$	2.51218	0.837394
$10$	0	0
$11$	4.85999	1.46534	0.732670	−	0.680584i	$-0.238274\pi$
$11$	4.85999	1.46534	0.732670	+	0.680584i	$0.238274\pi$
$12$	0	0
$13$	−2.78733	−0.773067	−0.386534	−	0.922275i	$-0.626328\pi$
$13$	−2.78733	−0.773067	−0.386534	+	0.922275i	$0.626328\pi$
$14$	0	0
$15$	−8.94818	−2.31041
$16$	0	0
$17$	−1.04869	−0.254344	−0.127172	−	0.991881i	$-0.540590\pi$
$17$	−1.04869	−0.254344	−0.127172	+	0.991881i	$0.540590\pi$
$18$	0	0
$19$	2.71468	0.622791	0.311396	−	0.950280i	$-0.399204\pi$
$19$	2.71468	0.622791	0.311396	+	0.950280i	$0.399204\pi$
$20$	0	0
$21$	−0.956962	−0.208826
$22$	0	0
$23$	−8.67128	−1.80809	−0.904044	−	0.427440i	$-0.859415\pi$
$23$	−8.67128	−1.80809	−0.904044	+	0.427440i	$0.859415\pi$
$24$	0	0
$25$	9.52598	1.90520
$26$	0	0
$27$	−1.14530	−0.220413
$28$	0	0
$29$	−4.08819	−0.759158	−0.379579	−	0.925159i	$-0.623931\pi$
$29$	−4.08819	−0.759158	−0.379579	+	0.925159i	$0.623931\pi$
$30$	0	0
$31$	0.325219	0.0584110	0.0292055	−	0.999573i	$-0.490702\pi$
$31$	0.325219	0.0584110	0.0292055	+	0.999573i	$0.490702\pi$
$32$	0	0
$33$	11.4103	1.98628
$34$	0	0
$35$	1.55348	0.262586
$36$	0	0
$37$	11.4919	1.88925	0.944627	−	0.328145i	$-0.106423\pi$
$37$	11.4919	1.88925	0.944627	+	0.328145i	$0.106423\pi$
$38$	0	0
$39$	−6.54411	−1.04790
$40$	0	0
$41$	0.439895	0.0687000	0.0343500	−	0.999410i	$-0.489064\pi$
$41$	0.439895	0.0687000	0.0343500	+	0.999410i	$0.489064\pi$
$42$	0	0
$43$	−9.74430	−1.48599	−0.742996	−	0.669296i	$-0.766596\pi$
$43$	−9.74430	−1.48599	−0.742996	+	0.669296i	$0.766596\pi$
$44$	0	0
$45$	−9.57467	−1.42731
$46$	0	0
$47$	4.85474	0.708136	0.354068	−	0.935220i	$-0.384798\pi$
$47$	4.85474	0.708136	0.354068	+	0.935220i	$0.384798\pi$
$48$	0	0
$49$	−6.83386	−0.976266
$50$	0	0
$51$	−2.46212	−0.344765
$52$	0	0
$53$	−8.80287	−1.20917	−0.604584	−	0.796542i	$-0.706661\pi$
$53$	−8.80287	−1.20917	−0.604584	+	0.796542i	$0.706661\pi$
$54$	0	0
$55$	−18.5228	−2.49762
$56$	0	0
$57$	6.37354	0.844197
$58$	0	0
$59$	−8.47943	−1.10393	−0.551964	−	0.833868i	$-0.686121\pi$
$59$	−8.47943	−1.10393	−0.551964	+	0.833868i	$0.686121\pi$
$60$	0	0
$61$	−6.37079	−0.815696	−0.407848	−	0.913050i	$-0.633721\pi$
$61$	−6.37079	−0.815696	−0.407848	+	0.913050i	$0.633721\pi$
$62$	0	0
$63$	−1.02396	−0.129007
$64$	0	0
$65$	10.6234	1.31767
$66$	0	0
$67$	3.46878	0.423779	0.211889	−	0.977294i	$-0.432038\pi$
$67$	3.46878	0.423779	0.211889	+	0.977294i	$0.432038\pi$
$68$	0	0
$69$	−20.3585	−2.45087
$70$	0	0
$71$	6.11292	0.725470	0.362735	−	0.931892i	$-0.381843\pi$
$71$	6.11292	0.725470	0.362735	+	0.931892i	$0.381843\pi$
$72$	0	0
$73$	2.77180	0.324414	0.162207	−	0.986757i	$-0.448139\pi$
$73$	2.77180	0.324414	0.162207	+	0.986757i	$0.448139\pi$
$74$	0	0
$75$	22.3651	2.58250
$76$	0	0
$77$	−1.98092	−0.225747
$78$	0	0
$79$	−8.48606	−0.954756	−0.477378	−	0.878698i	$-0.658413\pi$
$79$	−8.48606	−0.954756	−0.477378	+	0.878698i	$0.658413\pi$
$80$	0	0
$81$	−10.2255	−1.13617
$82$	0	0
$83$	−15.0594	−1.65298	−0.826489	−	0.562952i	$-0.809665\pi$
$83$	−15.0594	−1.65298	−0.826489	+	0.562952i	$0.809665\pi$
$84$	0	0
$85$	3.99686	0.433521
$86$	0	0
$87$	−9.59827	−1.02904
$88$	0	0
$89$	5.55559	0.588892	0.294446	−	0.955668i	$-0.404865\pi$
$89$	5.55559	0.588892	0.294446	+	0.955668i	$0.404865\pi$
$90$	0	0
$91$	1.13611	0.119097
$92$	0	0
$93$	0.763550	0.0791765
$94$	0	0
$95$	−10.3465	−1.06152
$96$	0	0
$97$	−12.8685	−1.30660	−0.653301	−	0.757098i	$-0.726616\pi$
$97$	−12.8685	−1.30660	−0.653301	+	0.757098i	$0.726616\pi$
$98$	0	0
$99$	12.2092	1.22707
$100$	0	0
$101$	−6.52011	−0.648776	−0.324388	−	0.945924i	$-0.605158\pi$
$101$	−6.52011	−0.648776	−0.324388	+	0.945924i	$0.605158\pi$
$102$	0	0
$103$	3.90343	0.384616	0.192308	−	0.981335i	$-0.438403\pi$
$103$	3.90343	0.384616	0.192308	+	0.981335i	$0.438403\pi$
$104$	0	0
$105$	3.64726	0.355937
$106$	0	0
$107$	−2.59961	−0.251313	−0.125657	−	0.992074i	$-0.540104\pi$
$107$	−2.59961	−0.251313	−0.125657	+	0.992074i	$0.540104\pi$
$108$	0	0
$109$	1.89496	0.181505	0.0907524	−	0.995873i	$-0.471073\pi$
$109$	1.89496	0.181505	0.0907524	+	0.995873i	$0.471073\pi$
$110$	0	0
$111$	26.9807	2.56089
$112$	0	0
$113$	5.32601	0.501029	0.250514	−	0.968113i	$-0.419400\pi$
$113$	5.32601	0.501029	0.250514	+	0.968113i	$0.419400\pi$
$114$	0	0
$115$	33.0488	3.08182
$116$	0	0
$117$	−7.00229	−0.647362
$118$	0	0
$119$	0.427444	0.0391837
$120$	0	0
$121$	12.6195	1.14722
$122$	0	0
$123$	1.03279	0.0931232
$124$	0	0
$125$	−17.2499	−1.54287
$126$	0	0
$127$	0.340598	0.0302232	0.0151116	−	0.999886i	$-0.495190\pi$
$127$	0.340598	0.0302232	0.0151116	+	0.999886i	$0.495190\pi$
$128$	0	0
$129$	−22.8777	−2.01427
$130$	0	0
$131$	−7.55165	−0.659791	−0.329896	−	0.944017i	$-0.607014\pi$
$131$	−7.55165	−0.659791	−0.329896	+	0.944017i	$0.607014\pi$
$132$	0	0
$133$	−1.10650	−0.0959458
$134$	0	0
$135$	4.36508	0.375687
$136$	0	0
$137$	−21.5761	−1.84337	−0.921683	−	0.387943i	$-0.873186\pi$
$137$	−21.5761	−1.84337	−0.921683	+	0.387943i	$0.873186\pi$
$138$	0	0
$139$	21.1211	1.79147	0.895733	−	0.444592i	$-0.146651\pi$
$139$	21.1211	1.79147	0.895733	+	0.444592i	$0.146651\pi$
$140$	0	0
$141$	11.3980	0.959882
$142$	0	0
$143$	−13.5464	−1.13281
$144$	0	0
$145$	15.5813	1.29396
$146$	0	0
$147$	−16.0446	−1.32333
$148$	0	0
$149$	−3.93654	−0.322494	−0.161247	−	0.986914i	$-0.551552\pi$
$149$	−3.93654	−0.322494	−0.161247	+	0.986914i	$0.551552\pi$
$150$	0	0
$151$	−8.17568	−0.665327	−0.332664	−	0.943046i	$-0.607947\pi$
$151$	−8.17568	−0.665327	−0.332664	+	0.943046i	$0.607947\pi$
$152$	0	0
$153$	−2.63450	−0.212986
$154$	0	0
$155$	−1.23951	−0.0995595
$156$	0	0
$157$	0.0864784	0.00690172	0.00345086	−	0.999994i	$-0.498902\pi$
$157$	0.0864784	0.00690172	0.00345086	+	0.999994i	$0.498902\pi$
$158$	0	0
$159$	−20.6674	−1.63903
$160$	0	0
$161$	3.53440	0.278550
$162$	0	0
$163$	−18.9831	−1.48687	−0.743435	−	0.668809i	$-0.766805\pi$
$163$	−18.9831	−1.48687	−0.743435	+	0.668809i	$0.766805\pi$
$164$	0	0
$165$	−43.4880	−3.38554
$166$	0	0
$167$	−9.43290	−0.729940	−0.364970	−	0.931019i	$-0.618921\pi$
$167$	−9.43290	−0.729940	−0.364970	+	0.931019i	$0.618921\pi$
$168$	0	0
$169$	−5.23077	−0.402367
$170$	0	0
$171$	6.81978	0.521522
$172$	0	0
$173$	18.2796	1.38977	0.694887	−	0.719119i	$-0.255454\pi$
$173$	18.2796	1.38977	0.694887	+	0.719119i	$0.255454\pi$
$174$	0	0
$175$	−3.88278	−0.293510
$176$	0	0
$177$	−19.9080	−1.49638
$178$	0	0
$179$	−12.8522	−0.960617	−0.480309	−	0.877100i	$-0.659475\pi$
$179$	−12.8522	−0.960617	−0.480309	+	0.877100i	$0.659475\pi$
$180$	0	0
$181$	21.3281	1.58531	0.792654	−	0.609672i	$-0.208699\pi$
$181$	21.3281	1.58531	0.792654	+	0.609672i	$0.208699\pi$
$182$	0	0
$183$	−14.9574	−1.10568
$184$	0	0
$185$	−43.7990	−3.22017
$186$	0	0
$187$	−5.09661	−0.372701
$188$	0	0
$189$	0.466823	0.0339564
$190$	0	0
$191$	−17.7870	−1.28702	−0.643510	−	0.765438i	$-0.722523\pi$
$191$	−17.7870	−1.28702	−0.643510	+	0.765438i	$0.722523\pi$
$192$	0	0
$193$	−14.3394	−1.03217	−0.516087	−	0.856536i	$-0.672612\pi$
$193$	−14.3394	−1.03217	−0.516087	+	0.856536i	$0.672612\pi$
$194$	0	0
$195$	24.9416	1.78610
$196$	0	0
$197$	25.1458	1.79156	0.895782	−	0.444494i	$-0.146616\pi$
$197$	25.1458	1.79156	0.895782	+	0.444494i	$0.146616\pi$
$198$	0	0
$199$	−10.2795	−0.728698	−0.364349	−	0.931263i	$-0.618708\pi$
$199$	−10.2795	−0.728698	−0.364349	+	0.931263i	$0.618708\pi$
$200$	0	0
$201$	8.14401	0.574434
$202$	0	0
$203$	1.66634	0.116954
$204$	0	0
$205$	−1.67657	−0.117097
$206$	0	0
$207$	−21.7838	−1.51408
$208$	0	0
$209$	13.1933	0.912601
$210$	0	0
$211$	−28.0422	−1.93050	−0.965252	−	0.261320i	$-0.915842\pi$
$211$	−28.0422	−1.93050	−0.965252	+	0.261320i	$0.915842\pi$
$212$	0	0
$213$	14.3519	0.983378
$214$	0	0
$215$	37.1384	2.53282
$216$	0	0
$217$	−0.132559	−0.00899868
$218$	0	0
$219$	6.50763	0.439745
$220$	0	0
$221$	2.92305	0.196625
$222$	0	0
$223$	12.9060	0.864247	0.432123	−	0.901814i	$-0.357764\pi$
$223$	12.9060	0.864247	0.432123	+	0.901814i	$0.357764\pi$
$224$	0	0
$225$	23.9310	1.59540
$226$	0	0
$227$	24.6376	1.63525	0.817627	−	0.575748i	$-0.195289\pi$
$227$	24.6376	1.63525	0.817627	+	0.575748i	$0.195289\pi$
$228$	0	0
$229$	−8.09344	−0.534830	−0.267415	−	0.963582i	$-0.586169\pi$
$229$	−8.09344	−0.534830	−0.267415	+	0.963582i	$0.586169\pi$
$230$	0	0
$231$	−4.65082	−0.306001
$232$	0	0
$233$	10.8370	0.709958	0.354979	−	0.934874i	$-0.384488\pi$
$233$	10.8370	0.709958	0.354979	+	0.934874i	$0.384488\pi$
$234$	0	0
$235$	−18.5028	−1.20699
$236$	0	0
$237$	−19.9236	−1.29418
$238$	0	0
$239$	7.75943	0.501916	0.250958	−	0.967998i	$-0.419254\pi$
$239$	7.75943	0.501916	0.250958	+	0.967998i	$0.419254\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	−20.5715	−1.31966
$244$	0	0
$245$	26.0459	1.66401
$246$	0	0
$247$	−7.56673	−0.481460
$248$	0	0
$249$	−35.3564	−2.24062
$250$	0	0
$251$	−22.6364	−1.42880	−0.714399	−	0.699738i	$-0.753300\pi$
$251$	−22.6364	−1.42880	−0.714399	+	0.699738i	$0.753300\pi$
$252$	0	0
$253$	−42.1423	−2.64946
$254$	0	0
$255$	9.38385	0.587640
$256$	0	0
$257$	0.167485	0.0104474	0.00522370	−	0.999986i	$-0.498337\pi$
$257$	0.167485	0.0104474	0.00522370	+	0.999986i	$0.498337\pi$
$258$	0	0
$259$	−4.68408	−0.291054
$260$	0	0
$261$	−10.2703	−0.635714
$262$	0	0
$263$	11.1147	0.685363	0.342682	−	0.939452i	$-0.388665\pi$
$263$	11.1147	0.685363	0.342682	+	0.939452i	$0.388665\pi$
$264$	0	0
$265$	33.5504	2.06098
$266$	0	0
$267$	13.0434	0.798246
$268$	0	0
$269$	−9.61340	−0.586140	−0.293070	−	0.956091i	$-0.594677\pi$
$269$	−9.61340	−0.586140	−0.293070	+	0.956091i	$0.594677\pi$
$270$	0	0
$271$	−25.3811	−1.54179	−0.770895	−	0.636962i	$-0.780191\pi$
$271$	−25.3811	−1.54179	−0.770895	+	0.636962i	$0.780191\pi$
$272$	0	0
$273$	2.66737	0.161437
$274$	0	0
$275$	46.2961	2.79176
$276$	0	0
$277$	23.2504	1.39698	0.698492	−	0.715618i	$-0.253855\pi$
$277$	23.2504	1.39698	0.698492	+	0.715618i	$0.253855\pi$
$278$	0	0
$279$	0.817009	0.0489131
$280$	0	0
$281$	−28.3271	−1.68985	−0.844926	−	0.534883i	$-0.820356\pi$
$281$	−28.3271	−1.68985	−0.844926	+	0.534883i	$0.820356\pi$
$282$	0	0
$283$	11.6889	0.694833	0.347416	−	0.937711i	$-0.387059\pi$
$283$	11.6889	0.694833	0.347416	+	0.937711i	$0.387059\pi$
$284$	0	0
$285$	−24.2915	−1.43890
$286$	0	0
$287$	−0.179300	−0.0105838
$288$	0	0
$289$	−15.9003	−0.935309
$290$	0	0
$291$	−30.2128	−1.77111
$292$	0	0
$293$	25.5328	1.49164	0.745821	−	0.666146i	$-0.232057\pi$
$293$	25.5328	1.49164	0.745821	+	0.666146i	$0.232057\pi$
$294$	0	0
$295$	32.3176	1.88161
$296$	0	0
$297$	−5.56615	−0.322981
$298$	0	0
$299$	24.1698	1.39777
$300$	0	0
$301$	3.97176	0.228929
$302$	0	0
$303$	−15.3079	−0.879419
$304$	0	0
$305$	24.2810	1.39032
$306$	0	0
$307$	20.6109	1.17633	0.588163	−	0.808743i	$-0.299851\pi$
$307$	20.6109	1.17633	0.588163	+	0.808743i	$0.299851\pi$
$308$	0	0
$309$	9.16448	0.521349
$310$	0	0
$311$	−26.3769	−1.49569	−0.747847	−	0.663871i	$-0.768912\pi$
$311$	−26.3769	−1.49569	−0.747847	+	0.663871i	$0.768912\pi$
$312$	0	0
$313$	20.2220	1.14301	0.571507	−	0.820597i	$-0.306359\pi$
$313$	20.2220	1.14301	0.571507	+	0.820597i	$0.306359\pi$
$314$	0	0
$315$	3.90262	0.219888
$316$	0	0
$317$	18.1125	1.01730	0.508651	−	0.860973i	$-0.330144\pi$
$317$	18.1125	1.01730	0.508651	+	0.860973i	$0.330144\pi$
$318$	0	0
$319$	−19.8685	−1.11242
$320$	0	0
$321$	−6.10337	−0.340657
$322$	0	0
$323$	−2.84686	−0.158403
$324$	0	0
$325$	−26.5521	−1.47285
$326$	0	0
$327$	4.44900	0.246031
$328$	0	0
$329$	−1.97878	−0.109094
$330$	0	0
$331$	25.9799	1.42798	0.713992	−	0.700154i	$-0.246885\pi$
$331$	25.9799	1.42798	0.713992	+	0.700154i	$0.246885\pi$
$332$	0	0
$333$	28.8697	1.58205
$334$	0	0
$335$	−13.2205	−0.722316
$336$	0	0
$337$	29.6685	1.61615	0.808073	−	0.589082i	$-0.200510\pi$
$337$	29.6685	1.61615	0.808073	+	0.589082i	$0.200510\pi$
$338$	0	0
$339$	12.5044	0.679147
$340$	0	0
$341$	1.58056	0.0855921
$342$	0	0
$343$	5.63866	0.304459
$344$	0	0
$345$	77.5921	4.17742
$346$	0	0
$347$	5.43400	0.291713	0.145856	−	0.989306i	$-0.453406\pi$
$347$	5.43400	0.291713	0.145856	+	0.989306i	$0.453406\pi$
$348$	0	0
$349$	−10.7514	−0.575510	−0.287755	−	0.957704i	$-0.592909\pi$
$349$	−10.7514	−0.575510	−0.287755	+	0.957704i	$0.592909\pi$
$350$	0	0
$351$	3.19234	0.170394
$352$	0	0
$353$	−16.6773	−0.887645	−0.443822	−	0.896115i	$-0.646378\pi$
$353$	−16.6773	−0.887645	−0.443822	+	0.896115i	$0.646378\pi$
$354$	0	0
$355$	−23.2981	−1.23654
$356$	0	0
$357$	1.00355	0.0531138
$358$	0	0
$359$	−15.4699	−0.816469	−0.408234	−	0.912877i	$-0.633855\pi$
$359$	−15.4699	−0.816469	−0.408234	+	0.912877i	$0.633855\pi$
$360$	0	0
$361$	−11.6305	−0.612131
$362$	0	0
$363$	29.6280	1.55507
$364$	0	0
$365$	−10.5641	−0.552952
$366$	0	0
$367$	20.6403	1.07741	0.538706	−	0.842494i	$-0.318913\pi$
$367$	20.6403	1.07741	0.538706	+	0.842494i	$0.318913\pi$
$368$	0	0
$369$	1.10510	0.0575290
$370$	0	0
$371$	3.58804	0.186282
$372$	0	0
$373$	13.3245	0.689914	0.344957	−	0.938618i	$-0.387893\pi$
$373$	13.3245	0.689914	0.344957	+	0.938618i	$0.387893\pi$
$374$	0	0
$375$	−40.4993	−2.09137
$376$	0	0
$377$	11.3952	0.586880
$378$	0	0
$379$	19.3631	0.994614	0.497307	−	0.867575i	$-0.334322\pi$
$379$	19.3631	0.994614	0.497307	+	0.867575i	$0.334322\pi$
$380$	0	0
$381$	0.799658	0.0409677
$382$	0	0
$383$	−19.0127	−0.971503	−0.485751	−	0.874097i	$-0.661454\pi$
$383$	−19.0127	−0.971503	−0.485751	+	0.874097i	$0.661454\pi$
$384$	0	0
$385$	7.54989	0.384778
$386$	0	0
$387$	−24.4794	−1.24436
$388$	0	0
$389$	6.76286	0.342891	0.171445	−	0.985194i	$-0.445156\pi$
$389$	6.76286	0.342891	0.171445	+	0.985194i	$0.445156\pi$
$390$	0	0
$391$	9.09348	0.459877
$392$	0	0
$393$	−17.7298	−0.894350
$394$	0	0
$395$	32.3429	1.62735
$396$	0	0
$397$	24.0133	1.20519	0.602596	−	0.798047i	$-0.294133\pi$
$397$	24.0133	1.20519	0.602596	+	0.798047i	$0.294133\pi$
$398$	0	0
$399$	−2.59785	−0.130055
$400$	0	0
$401$	2.68925	0.134295	0.0671474	−	0.997743i	$-0.478610\pi$
$401$	2.68925	0.134295	0.0671474	+	0.997743i	$0.478610\pi$
$402$	0	0
$403$	−0.906494	−0.0451557
$404$	0	0
$405$	38.9724	1.93655
$406$	0	0
$407$	55.8504	2.76840
$408$	0	0
$409$	8.26430	0.408643	0.204322	−	0.978904i	$-0.434501\pi$
$409$	8.26430	0.408643	0.204322	+	0.978904i	$0.434501\pi$
$410$	0	0
$411$	−50.6563	−2.49869
$412$	0	0
$413$	3.45621	0.170069
$414$	0	0
$415$	57.3957	2.81744
$416$	0	0
$417$	49.5881	2.42834
$418$	0	0
$419$	16.5813	0.810048	0.405024	−	0.914306i	$-0.367263\pi$
$419$	16.5813	0.810048	0.405024	+	0.914306i	$0.367263\pi$
$420$	0	0
$421$	−18.1121	−0.882731	−0.441366	−	0.897327i	$-0.645506\pi$
$421$	−18.1121	−0.882731	−0.441366	+	0.897327i	$0.645506\pi$
$422$	0	0
$423$	12.1960	0.592989
$424$	0	0
$425$	−9.98979	−0.484576
$426$	0	0
$427$	2.59673	0.125664
$428$	0	0
$429$	−31.8043	−1.53553
$430$	0	0
$431$	9.68282	0.466405	0.233202	−	0.972428i	$-0.425080\pi$
$431$	9.68282	0.466405	0.233202	+	0.972428i	$0.425080\pi$
$432$	0	0
$433$	26.4467	1.27095	0.635473	−	0.772123i	$-0.280805\pi$
$433$	26.4467	1.27095	0.635473	+	0.772123i	$0.280805\pi$
$434$	0	0
$435$	36.5818	1.75397
$436$	0	0
$437$	−23.5398	−1.12606
$438$	0	0
$439$	17.6829	0.843960	0.421980	−	0.906605i	$-0.361335\pi$
$439$	17.6829	0.843960	0.421980	+	0.906605i	$0.361335\pi$
$440$	0	0
$441$	−17.1679	−0.817519
$442$	0	0
$443$	8.39543	0.398879	0.199440	−	0.979910i	$-0.436088\pi$
$443$	8.39543	0.398879	0.199440	+	0.979910i	$0.436088\pi$
$444$	0	0
$445$	−21.1740	−1.00374
$446$	0	0
$447$	−9.24222	−0.437142
$448$	0	0
$449$	29.1627	1.37627	0.688137	−	0.725581i	$-0.258429\pi$
$449$	29.1627	1.37627	0.688137	+	0.725581i	$0.258429\pi$
$450$	0	0
$451$	2.13788	0.100669
$452$	0	0
$453$	−19.1949	−0.901855
$454$	0	0
$455$	−4.33007	−0.202997
$456$	0	0
$457$	1.35070	0.0631832	0.0315916	−	0.999501i	$-0.489942\pi$
$457$	1.35070	0.0631832	0.0315916	+	0.999501i	$0.489942\pi$
$458$	0	0
$459$	1.20107	0.0560609
$460$	0	0
$461$	9.13738	0.425570	0.212785	−	0.977099i	$-0.431747\pi$
$461$	9.13738	0.425570	0.212785	+	0.977099i	$0.431747\pi$
$462$	0	0
$463$	21.4323	0.996042	0.498021	−	0.867165i	$-0.334060\pi$
$463$	21.4323	0.996042	0.498021	+	0.867165i	$0.334060\pi$
$464$	0	0
$465$	−2.91012	−0.134953
$466$	0	0
$467$	0.431773	0.0199801	0.00999004	−	0.999950i	$-0.496820\pi$
$467$	0.431773	0.0199801	0.00999004	+	0.999950i	$0.496820\pi$
$468$	0	0
$469$	−1.41387	−0.0652864
$470$	0	0
$471$	0.203034	0.00935532
$472$	0	0
$473$	−47.3571	−2.17748
$474$	0	0
$475$	25.8600	1.18654
$476$	0	0
$477$	−22.1144	−1.01255
$478$	0	0
$479$	−14.3331	−0.654896	−0.327448	−	0.944869i	$-0.606189\pi$
$479$	−14.3331	−0.654896	−0.327448	+	0.944869i	$0.606189\pi$
$480$	0	0
$481$	−32.0317	−1.46052
$482$	0	0
$483$	8.29808	0.377576
$484$	0	0
$485$	49.0458	2.22706
$486$	0	0
$487$	−4.52148	−0.204888	−0.102444	−	0.994739i	$-0.532666\pi$
$487$	−4.52148	−0.204888	−0.102444	+	0.994739i	$0.532666\pi$
$488$	0	0
$489$	−44.5685	−2.01546
$490$	0	0
$491$	−26.8044	−1.20967	−0.604833	−	0.796352i	$-0.706760\pi$
$491$	−26.8044	−1.20967	−0.604833	+	0.796352i	$0.706760\pi$
$492$	0	0
$493$	4.28724	0.193088
$494$	0	0
$495$	−46.5328	−2.09149
$496$	0	0
$497$	−2.49162	−0.111764
$498$	0	0
$499$	20.7924	0.930796	0.465398	−	0.885102i	$-0.345911\pi$
$499$	20.7924	0.930796	0.465398	+	0.885102i	$0.345911\pi$
$500$	0	0
$501$	−22.1466	−0.989437
$502$	0	0
$503$	6.70010	0.298743	0.149371	−	0.988781i	$-0.452275\pi$
$503$	6.70010	0.298743	0.149371	+	0.988781i	$0.452275\pi$
$504$	0	0
$505$	24.8501	1.10581
$506$	0	0
$507$	−12.2808	−0.545410
$508$	0	0
$509$	−24.9632	−1.10648	−0.553238	−	0.833023i	$-0.686608\pi$
$509$	−24.9632	−1.10648	−0.553238	+	0.833023i	$0.686608\pi$
$510$	0	0
$511$	−1.12978	−0.0499785
$512$	0	0
$513$	−3.10913	−0.137272
$514$	0	0
$515$	−14.8771	−0.655564
$516$	0	0
$517$	23.5939	1.03766
$518$	0	0
$519$	42.9170	1.88385
$520$	0	0
$521$	−5.71399	−0.250335	−0.125167	−	0.992136i	$-0.539947\pi$
$521$	−5.71399	−0.250335	−0.125167	+	0.992136i	$0.539947\pi$
$522$	0	0
$523$	28.2721	1.23625	0.618125	−	0.786080i	$-0.287892\pi$
$523$	28.2721	1.23625	0.618125	+	0.786080i	$0.287892\pi$
$524$	0	0
$525$	−9.11600	−0.397855
$526$	0	0
$527$	−0.341053	−0.0148565
$528$	0	0
$529$	52.1911	2.26918
$530$	0	0
$531$	−21.3019	−0.924423
$532$	0	0
$533$	−1.22613	−0.0531097
$534$	0	0
$535$	9.90787	0.428355
$536$	0	0
$537$	−30.1744	−1.30212
$538$	0	0
$539$	−33.2125	−1.43056
$540$	0	0
$541$	5.25457	0.225912	0.112956	−	0.993600i	$-0.463968\pi$
$541$	5.25457	0.225912	0.112956	+	0.993600i	$0.463968\pi$
$542$	0	0
$543$	50.0743	2.14889
$544$	0	0
$545$	−7.22227	−0.309368
$546$	0	0
$547$	−31.8626	−1.36235	−0.681173	−	0.732122i	$-0.738530\pi$
$547$	−31.8626	−1.36235	−0.681173	+	0.732122i	$0.738530\pi$
$548$	0	0
$549$	−16.0046	−0.683059
$550$	0	0
$551$	−11.0981	−0.472797
$552$	0	0
$553$	3.45891	0.147088
$554$	0	0
$555$	−102.831	−4.36495
$556$	0	0
$557$	6.07474	0.257395	0.128698	−	0.991684i	$-0.458920\pi$
$557$	6.07474	0.257395	0.128698	+	0.991684i	$0.458920\pi$
$558$	0	0
$559$	27.1606	1.14877
$560$	0	0
$561$	−11.9658	−0.505198
$562$	0	0
$563$	−14.6444	−0.617189	−0.308595	−	0.951194i	$-0.599859\pi$
$563$	−14.6444	−0.617189	−0.308595	+	0.951194i	$0.599859\pi$
$564$	0	0
$565$	−20.2990	−0.853986
$566$	0	0
$567$	4.16790	0.175035
$568$	0	0
$569$	14.0462	0.588847	0.294424	−	0.955675i	$-0.404872\pi$
$569$	14.0462	0.588847	0.294424	+	0.955675i	$0.404872\pi$
$570$	0	0
$571$	−40.3221	−1.68743	−0.843714	−	0.536792i	$-0.819636\pi$
$571$	−40.3221	−1.68743	−0.843714	+	0.536792i	$0.819636\pi$
$572$	0	0
$573$	−41.7603	−1.74456
$574$	0	0
$575$	−82.6025	−3.44476
$576$	0	0
$577$	44.4769	1.85160	0.925799	−	0.378017i	$-0.123394\pi$
$577$	44.4769	1.85160	0.925799	+	0.378017i	$0.123394\pi$
$578$	0	0
$579$	−33.6661	−1.39912
$580$	0	0
$581$	6.13817	0.254654
$582$	0	0
$583$	−42.7818	−1.77184
$584$	0	0
$585$	26.6878	1.10340
$586$	0	0
$587$	0.0232592	0.000960012 0	0.000480006	−	1.00000i	$-0.499847\pi$
$587$	0.0232592	0.000960012 0	0.000480006		1.00000i	$0.499847\pi$
$588$	0	0
$589$	0.882867	0.0363779
$590$	0	0
$591$	59.0374	2.42847
$592$	0	0
$593$	−12.6779	−0.520619	−0.260309	−	0.965525i	$-0.583825\pi$
$593$	−12.6779	−0.520619	−0.260309	+	0.965525i	$0.583825\pi$
$594$	0	0
$595$	−1.62912	−0.0667873
$596$	0	0
$597$	−24.1344	−0.987754
$598$	0	0
$599$	−7.15033	−0.292155	−0.146077	−	0.989273i	$-0.546665\pi$
$599$	−7.15033	−0.292155	−0.146077	+	0.989273i	$0.546665\pi$
$600$	0	0
$601$	−9.90416	−0.403999	−0.202000	−	0.979386i	$-0.564744\pi$
$601$	−9.90416	−0.403999	−0.202000	+	0.979386i	$0.564744\pi$
$602$	0	0
$603$	8.71420	0.354870
$604$	0	0
$605$	−48.0965	−1.95540
$606$	0	0
$607$	−45.2965	−1.83853	−0.919264	−	0.393641i	$-0.871215\pi$
$607$	−45.2965	−1.83853	−0.919264	+	0.393641i	$0.871215\pi$
$608$	0	0
$609$	3.91224	0.158532
$610$	0	0
$611$	−13.5318	−0.547437
$612$	0	0
$613$	23.9811	0.968588	0.484294	−	0.874905i	$-0.339077\pi$
$613$	23.9811	0.968588	0.484294	+	0.874905i	$0.339077\pi$
$614$	0	0
$615$	−3.93625	−0.158725
$616$	0	0
$617$	−23.1091	−0.930338	−0.465169	−	0.885222i	$-0.654006\pi$
$617$	−23.1091	−0.930338	−0.465169	+	0.885222i	$0.654006\pi$
$618$	0	0
$619$	33.1906	1.33404	0.667021	−	0.745039i	$-0.267569\pi$
$619$	33.1906	1.33404	0.667021	+	0.745039i	$0.267569\pi$
$620$	0	0
$621$	9.93123	0.398527
$622$	0	0
$623$	−2.26445	−0.0907234
$624$	0	0
$625$	18.1144	0.724576
$626$	0	0
$627$	30.9753	1.23704
$628$	0	0
$629$	−12.0514	−0.480521
$630$	0	0
$631$	−47.6969	−1.89878	−0.949391	−	0.314097i	$-0.898298\pi$
$631$	−47.6969	−1.89878	−0.949391	+	0.314097i	$0.898298\pi$
$632$	0	0
$633$	−65.8376	−2.61681
$634$	0	0
$635$	−1.29812	−0.0515144
$636$	0	0
$637$	19.0483	0.754720
$638$	0	0
$639$	15.3568	0.607504
$640$	0	0
$641$	14.1477	0.558802	0.279401	−	0.960174i	$-0.409864\pi$
$641$	14.1477	0.558802	0.279401	+	0.960174i	$0.409864\pi$
$642$	0	0
$643$	−14.6138	−0.576310	−0.288155	−	0.957584i	$-0.593042\pi$
$643$	−14.6138	−0.576310	−0.288155	+	0.957584i	$0.593042\pi$
$644$	0	0
$645$	87.1937	3.43325
$646$	0	0
$647$	−10.0049	−0.393333	−0.196667	−	0.980470i	$-0.563012\pi$
$647$	−10.0049	−0.393333	−0.196667	+	0.980470i	$0.563012\pi$
$648$	0	0
$649$	−41.2099	−1.61763
$650$	0	0
$651$	−0.311222	−0.0121978
$652$	0	0
$653$	−6.43798	−0.251938	−0.125969	−	0.992034i	$-0.540204\pi$
$653$	−6.43798	−0.251938	−0.125969	+	0.992034i	$0.540204\pi$
$654$	0	0
$655$	28.7816	1.12459
$656$	0	0
$657$	6.96325	0.271662
$658$	0	0
$659$	−14.4481	−0.562818	−0.281409	−	0.959588i	$-0.590802\pi$
$659$	−14.4481	−0.562818	−0.281409	+	0.959588i	$0.590802\pi$
$660$	0	0
$661$	−32.2750	−1.25535	−0.627675	−	0.778475i	$-0.715993\pi$
$661$	−32.2750	−1.25535	−0.627675	+	0.778475i	$0.715993\pi$
$662$	0	0
$663$	6.86274	0.266527
$664$	0	0
$665$	4.21720	0.163536
$666$	0	0
$667$	35.4498	1.37262
$668$	0	0
$669$	30.3007	1.17149
$670$	0	0
$671$	−30.9619	−1.19527
$672$	0	0
$673$	33.1396	1.27744	0.638718	−	0.769441i	$-0.279465\pi$
$673$	33.1396	1.27744	0.638718	+	0.769441i	$0.279465\pi$
$674$	0	0
$675$	−10.9101	−0.419931
$676$	0	0
$677$	−28.9944	−1.11435	−0.557173	−	0.830397i	$-0.688114\pi$
$677$	−28.9944	−1.11435	−0.557173	+	0.830397i	$0.688114\pi$
$678$	0	0
$679$	5.24520	0.201292
$680$	0	0
$681$	57.8442	2.21659
$682$	0	0
$683$	−1.80765	−0.0691678	−0.0345839	−	0.999402i	$-0.511011\pi$
$683$	−1.80765	−0.0691678	−0.0345839	+	0.999402i	$0.511011\pi$
$684$	0	0
$685$	82.2327	3.14195
$686$	0	0
$687$	−19.0018	−0.724964
$688$	0	0
$689$	24.5366	0.934768
$690$	0	0
$691$	37.0383	1.40900	0.704502	−	0.709702i	$-0.251170\pi$
$691$	37.0383	1.40900	0.704502	+	0.709702i	$0.251170\pi$
$692$	0	0
$693$	−4.97644	−0.189039
$694$	0	0
$695$	−80.4987	−3.05349
$696$	0	0
$697$	−0.461313	−0.0174735
$698$	0	0
$699$	25.4432	0.962352
$700$	0	0
$701$	27.7059	1.04644	0.523218	−	0.852199i	$-0.324731\pi$
$701$	27.7059	1.04644	0.523218	+	0.852199i	$0.324731\pi$
$702$	0	0
$703$	31.1968	1.17661
$704$	0	0
$705$	−43.4410	−1.63608
$706$	0	0
$707$	2.65759	0.0999489
$708$	0	0
$709$	−31.3766	−1.17837	−0.589187	−	0.807997i	$-0.700552\pi$
$709$	−31.3766	−1.17837	−0.589187	+	0.807997i	$0.700552\pi$
$710$	0	0
$711$	−21.3185	−0.799507
$712$	0	0
$713$	−2.82006	−0.105612
$714$	0	0
$715$	51.6294	1.93083
$716$	0	0
$717$	18.2176	0.680350
$718$	0	0
$719$	10.5400	0.393074	0.196537	−	0.980496i	$-0.437030\pi$
$719$	10.5400	0.393074	0.196537	+	0.980496i	$0.437030\pi$
$720$	0	0
$721$	−1.59103	−0.0592531
$722$	0	0
$723$	2.34780	0.0873158
$724$	0	0
$725$	−38.9440	−1.44634
$726$	0	0
$727$	10.4004	0.385730	0.192865	−	0.981225i	$-0.438222\pi$
$727$	10.4004	0.385730	0.192865	+	0.981225i	$0.438222\pi$
$728$	0	0
$729$	−17.6215	−0.652646
$730$	0	0
$731$	10.2187	0.377954
$732$	0	0
$733$	−3.49971	−0.129265	−0.0646323	−	0.997909i	$-0.520587\pi$
$733$	−3.49971	−0.129265	−0.0646323	+	0.997909i	$0.520587\pi$
$734$	0	0
$735$	61.1506	2.25557
$736$	0	0
$737$	16.8582	0.620980
$738$	0	0
$739$	−9.90482	−0.364355	−0.182177	−	0.983266i	$-0.558314\pi$
$739$	−9.90482	−0.364355	−0.182177	+	0.983266i	$0.558314\pi$
$740$	0	0
$741$	−17.7652	−0.652621
$742$	0	0
$743$	−47.2127	−1.73206	−0.866032	−	0.499988i	$-0.833338\pi$
$743$	−47.2127	−1.73206	−0.866032	+	0.499988i	$0.833338\pi$
$744$	0	0
$745$	15.0033	0.549679
$746$	0	0
$747$	−37.8318	−1.38419
$748$	0	0
$749$	1.05960	0.0387168
$750$	0	0
$751$	−13.7635	−0.502238	−0.251119	−	0.967956i	$-0.580799\pi$
$751$	−13.7635	−0.502238	−0.251119	+	0.967956i	$0.580799\pi$
$752$	0	0
$753$	−53.1459	−1.93674
$754$	0	0
$755$	31.1599	1.13403
$756$	0	0
$757$	−32.1537	−1.16865	−0.584323	−	0.811521i	$-0.698640\pi$
$757$	−32.1537	−1.16865	−0.584323	+	0.811521i	$0.698640\pi$
$758$	0	0
$759$	−98.9418	−3.59136
$760$	0	0
$761$	−0.885275	−0.0320912	−0.0160456	−	0.999871i	$-0.505108\pi$
$761$	−0.885275	−0.0320912	−0.0160456	+	0.999871i	$0.505108\pi$
$762$	0	0
$763$	−0.772385	−0.0279622
$764$	0	0
$765$	10.0408	0.363028
$766$	0	0
$767$	23.6350	0.853411
$768$	0	0
$769$	−10.2356	−0.369104	−0.184552	−	0.982823i	$-0.559083\pi$
$769$	−10.2356	−0.369104	−0.184552	+	0.982823i	$0.559083\pi$
$770$	0	0
$771$	0.393221	0.0141615
$772$	0	0
$773$	−3.93686	−0.141599	−0.0707996	−	0.997491i	$-0.522555\pi$
$773$	−3.93686	−0.141599	−0.0707996	+	0.997491i	$0.522555\pi$
$774$	0	0
$775$	3.09803	0.111284
$776$	0	0
$777$	−10.9973	−0.394526
$778$	0	0
$779$	1.19417	0.0427857
$780$	0	0
$781$	29.7087	1.06306
$782$	0	0
$783$	4.68221	0.167329
$784$	0	0
$785$	−0.329595	−0.0117637
$786$	0	0
$787$	31.8254	1.13445	0.567227	−	0.823562i	$-0.308016\pi$
$787$	31.8254	1.13445	0.567227	+	0.823562i	$0.308016\pi$
$788$	0	0
$789$	26.0952	0.929013
$790$	0	0
$791$	−2.17087	−0.0771874
$792$	0	0
$793$	17.7575	0.630588
$794$	0	0
$795$	78.7697	2.79367
$796$	0	0
$797$	18.8466	0.667581	0.333790	−	0.942647i	$-0.391672\pi$
$797$	18.8466	0.667581	0.333790	+	0.942647i	$0.391672\pi$
$798$	0	0
$799$	−5.09111	−0.180110
$800$	0	0
$801$	13.9567	0.493134
$802$	0	0
$803$	13.4709	0.475377
$804$	0	0
$805$	−13.4707	−0.474778
$806$	0	0
$807$	−22.5704	−0.794515
$808$	0	0
$809$	−47.7190	−1.67771	−0.838855	−	0.544355i	$-0.816774\pi$
$809$	−47.7190	−1.67771	−0.838855	+	0.544355i	$0.816774\pi$
$810$	0	0
$811$	10.0133	0.351616	0.175808	−	0.984425i	$-0.443746\pi$
$811$	10.0133	0.351616	0.175808	+	0.984425i	$0.443746\pi$
$812$	0	0
$813$	−59.5897	−2.08990
$814$	0	0
$815$	72.3501	2.53431
$816$	0	0
$817$	−26.4527	−0.925462
$818$	0	0
$819$	2.85412	0.0997312
$820$	0	0
$821$	32.1295	1.12133	0.560663	−	0.828044i	$-0.310546\pi$
$821$	32.1295	1.12133	0.560663	+	0.828044i	$0.310546\pi$
$822$	0	0
$823$	46.2827	1.61331	0.806656	−	0.591021i	$-0.201275\pi$
$823$	46.2827	1.61331	0.806656	+	0.591021i	$0.201275\pi$
$824$	0	0
$825$	108.694	3.78425
$826$	0	0
$827$	5.50701	0.191498	0.0957488	−	0.995406i	$-0.469475\pi$
$827$	5.50701	0.191498	0.0957488	+	0.995406i	$0.469475\pi$
$828$	0	0
$829$	33.5088	1.16381	0.581904	−	0.813257i	$-0.302308\pi$
$829$	33.5088	1.16381	0.581904	+	0.813257i	$0.302308\pi$
$830$	0	0
$831$	54.5875	1.89362
$832$	0	0
$833$	7.16660	0.248308
$834$	0	0
$835$	35.9516	1.24416
$836$	0	0
$837$	−0.372474	−0.0128746
$838$	0	0
$839$	−40.0484	−1.38262	−0.691312	−	0.722556i	$-0.742967\pi$
$839$	−40.0484	−1.38262	−0.691312	+	0.722556i	$0.742967\pi$
$840$	0	0
$841$	−12.2867	−0.423679
$842$	0	0
$843$	−66.5065	−2.29060
$844$	0	0
$845$	19.9360	0.685819
$846$	0	0
$847$	−5.14367	−0.176739
$848$	0	0
$849$	27.4432	0.941849
$850$	0	0
$851$	−99.6494	−3.41594
$852$	0	0
$853$	48.4427	1.65865	0.829324	−	0.558768i	$-0.188726\pi$
$853$	48.4427	1.65865	0.829324	+	0.558768i	$0.188726\pi$
$854$	0	0
$855$	−25.9922	−0.888914
$856$	0	0
$857$	35.5466	1.21425	0.607124	−	0.794607i	$-0.292323\pi$
$857$	35.5466	1.21425	0.607124	+	0.794607i	$0.292323\pi$
$858$	0	0
$859$	36.3859	1.24147	0.620735	−	0.784020i	$-0.286834\pi$
$859$	36.3859	1.24147	0.620735	+	0.784020i	$0.286834\pi$
$860$	0	0
$861$	−0.420962	−0.0143464
$862$	0	0
$863$	−36.3696	−1.23803	−0.619017	−	0.785377i	$-0.712469\pi$
$863$	−36.3696	−1.23803	−0.619017	+	0.785377i	$0.712469\pi$
$864$	0	0
$865$	−69.6691	−2.36882
$866$	0	0
$867$	−37.3307	−1.26782
$868$	0	0
$869$	−41.2421	−1.39904
$870$	0	0
$871$	−9.66865	−0.327610
$872$	0	0
$873$	−32.3281	−1.09414
$874$	0	0
$875$	7.03102	0.237692
$876$	0	0
$877$	−45.0135	−1.52000	−0.759999	−	0.649924i	$-0.774801\pi$
$877$	−45.0135	−1.52000	−0.759999	+	0.649924i	$0.774801\pi$
$878$	0	0
$879$	59.9460	2.02193
$880$	0	0
$881$	56.3116	1.89719	0.948593	−	0.316499i	$-0.102507\pi$
$881$	56.3116	1.89719	0.948593	+	0.316499i	$0.102507\pi$
$882$	0	0
$883$	−12.2425	−0.411993	−0.205997	−	0.978553i	$-0.566044\pi$
$883$	−12.2425	−0.411993	−0.205997	+	0.978553i	$0.566044\pi$
$884$	0	0
$885$	75.8755	2.55053
$886$	0	0
$887$	−19.9539	−0.669987	−0.334994	−	0.942220i	$-0.608734\pi$
$887$	−19.9539	−0.669987	−0.334994	+	0.942220i	$0.608734\pi$
$888$	0	0
$889$	−0.138827	−0.00465612
$890$	0	0
$891$	−49.6957	−1.66487
$892$	0	0
$893$	13.1791	0.441021
$894$	0	0
$895$	48.9835	1.63734
$896$	0	0
$897$	56.7459	1.89469
$898$	0	0
$899$	−1.32956	−0.0443432
$900$	0	0
$901$	9.23148	0.307545
$902$	0	0
$903$	9.32492	0.310314
$904$	0	0
$905$	−81.2878	−2.70210
$906$	0	0
$907$	45.4569	1.50937	0.754685	−	0.656087i	$-0.227790\pi$
$907$	45.4569	1.50937	0.754685	+	0.656087i	$0.227790\pi$
$908$	0	0
$909$	−16.3797	−0.543281
$910$	0	0
$911$	6.85309	0.227053	0.113526	−	0.993535i	$-0.463785\pi$
$911$	6.85309	0.227053	0.113526	+	0.993535i	$0.463785\pi$
$912$	0	0
$913$	−73.1882	−2.42218
$914$	0	0
$915$	57.0069	1.88459
$916$	0	0
$917$	3.07804	0.101646
$918$	0	0
$919$	−18.9140	−0.623915	−0.311958	−	0.950096i	$-0.600985\pi$
$919$	−18.9140	−0.623915	−0.311958	+	0.950096i	$0.600985\pi$
$920$	0	0
$921$	48.3903	1.59452
$922$	0	0
$923$	−17.0387	−0.560837
$924$	0	0
$925$	109.471	3.59940
$926$	0	0
$927$	9.80611	0.322075
$928$	0	0
$929$	−3.28463	−0.107765	−0.0538826	−	0.998547i	$-0.517160\pi$
$929$	−3.28463	−0.107765	−0.0538826	+	0.998547i	$0.517160\pi$
$930$	0	0
$931$	−18.5518	−0.608010
$932$	0	0
$933$	−61.9277	−2.02742
$934$	0	0
$935$	19.4247	0.635256
$936$	0	0
$937$	2.79976	0.0914643	0.0457322	−	0.998954i	$-0.485438\pi$
$937$	2.79976	0.0914643	0.0457322	+	0.998954i	$0.485438\pi$
$938$	0	0
$939$	47.4772	1.54936
$940$	0	0
$941$	−13.0622	−0.425816	−0.212908	−	0.977072i	$-0.568293\pi$
$941$	−13.0622	−0.425816	−0.212908	+	0.977072i	$0.568293\pi$
$942$	0	0
$943$	−3.81445	−0.124216
$944$	0	0
$945$	−1.77920	−0.0578775
$946$	0	0
$947$	35.5843	1.15633	0.578167	−	0.815918i	$-0.303768\pi$
$947$	35.5843	1.15633	0.578167	+	0.815918i	$0.303768\pi$
$948$	0	0
$949$	−7.72592	−0.250794
$950$	0	0
$951$	42.5247	1.37896
$952$	0	0
$953$	−43.4229	−1.40661	−0.703303	−	0.710890i	$-0.748292\pi$
$953$	−43.4229	−1.40661	−0.703303	+	0.710890i	$0.748292\pi$
$954$	0	0
$955$	67.7914	2.19368
$956$	0	0
$957$	−46.6474	−1.50790
$958$	0	0
$959$	8.79437	0.283985
$960$	0	0
$961$	−30.8942	−0.996588
$962$	0	0
$963$	−6.53068	−0.210448
$964$	0	0
$965$	54.6518	1.75930
$966$	0	0
$967$	56.2588	1.80916	0.904581	−	0.426302i	$-0.140184\pi$
$967$	56.2588	1.80916	0.904581	+	0.426302i	$0.140184\pi$
$968$	0	0
$969$	−6.68387	−0.214717
$970$	0	0
$971$	38.1291	1.22362	0.611811	−	0.791004i	$-0.290441\pi$
$971$	38.1291	1.22362	0.611811	+	0.791004i	$0.290441\pi$
$972$	0	0
$973$	−8.60892	−0.275989
$974$	0	0
$975$	−62.3391	−1.99645
$976$	0	0
$977$	−20.2896	−0.649121	−0.324561	−	0.945865i	$-0.605216\pi$
$977$	−20.2896	−0.649121	−0.324561	+	0.945865i	$0.605216\pi$
$978$	0	0
$979$	27.0001	0.862927
$980$	0	0
$981$	4.76050	0.151991
$982$	0	0
$983$	−45.1636	−1.44050	−0.720248	−	0.693717i	$-0.755972\pi$
$983$	−45.1636	−1.44050	−0.720248	+	0.693717i	$0.755972\pi$
$984$	0	0
$985$	−95.8381	−3.05366
$986$	0	0
$987$	−4.64580	−0.147877
$988$	0	0
$989$	84.4955	2.68680
$990$	0	0
$991$	−23.7345	−0.753953	−0.376976	−	0.926223i	$-0.623036\pi$
$991$	−23.7345	−0.753953	−0.376976	+	0.926223i	$0.623036\pi$
$992$	0	0
$993$	60.9957	1.93564
$994$	0	0
$995$	39.1784	1.24204
$996$	0	0
$997$	−22.1793	−0.702426	−0.351213	−	0.936296i	$-0.614231\pi$
$997$	−22.1793	−0.702426	−0.351213	+	0.936296i	$0.614231\pi$
$998$	0	0
$999$	−13.1617	−0.416417

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.h.1.6		6
4.3	odd	2		482.2.a.d.1.1	✓	6
12.11	even	2		4338.2.a.u.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
482.2.a.d.1.1	✓	6	4.3	odd	2
3856.2.a.h.1.6		6	1.1	even	1	trivial
4338.2.a.u.1.6		6	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3856 = 2^{4} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3856.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34780 q^{3} -3.81130 q^{5} -0.407599 q^{7} +2.51218 q^{9} +O(q^{10})\)	\(q+2.34780 q^{3} -3.81130 q^{5} -0.407599 q^{7} +2.51218 q^{9} +4.85999 q^{11} -2.78733 q^{13} -8.94818 q^{15} -1.04869 q^{17} +2.71468 q^{19} -0.956962 q^{21} -8.67128 q^{23} +9.52598 q^{25} -1.14530 q^{27} -4.08819 q^{29} +0.325219 q^{31} +11.4103 q^{33} +1.55348 q^{35} +11.4919 q^{37} -6.54411 q^{39} +0.439895 q^{41} -9.74430 q^{43} -9.57467 q^{45} +4.85474 q^{47} -6.83386 q^{49} -2.46212 q^{51} -8.80287 q^{53} -18.5228 q^{55} +6.37354 q^{57} -8.47943 q^{59} -6.37079 q^{61} -1.02396 q^{63} +10.6234 q^{65} +3.46878 q^{67} -20.3585 q^{69} +6.11292 q^{71} +2.77180 q^{73} +22.3651 q^{75} -1.98092 q^{77} -8.48606 q^{79} -10.2255 q^{81} -15.0594 q^{83} +3.99686 q^{85} -9.59827 q^{87} +5.55559 q^{89} +1.13611 q^{91} +0.763550 q^{93} -10.3465 q^{95} -12.8685 q^{97} +12.2092 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9	\( 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9} + 4 q^{11} + 9 q^{13} - 5 q^{15} + q^{17} - 10 q^{19} + 8 q^{21} - 9 q^{23} + 13 q^{25} - 8 q^{27} - q^{29} - 14 q^{31} + 10 q^{33} + 3 q^{35} + 20 q^{37} - 13 q^{39} - 4 q^{41} - 19 q^{43} - 6 q^{45} - q^{47} + 14 q^{49} - 5 q^{51} - 3 q^{53} - 11 q^{55} + 12 q^{57} + q^{59} + 4 q^{61} - 14 q^{63} + 5 q^{65} - 5 q^{67} - 15 q^{69} - 2 q^{71} + 15 q^{73} + 11 q^{75} - 6 q^{77} - 12 q^{79} - 18 q^{81} + 8 q^{83} - 32 q^{85} - 11 q^{87} - 24 q^{89} - q^{91} - 18 q^{93} - 9 q^{95} - 12 q^{97} + 40 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9 + 4 * q^11 + 9 * q^13 - 5 * q^15 + q^17 - 10 * q^19 + 8 * q^21 - 9 * q^23 + 13 * q^25 - 8 * q^27 - q^29 - 14 * q^31 + 10 * q^33 + 3 * q^35 + 20 * q^37 - 13 * q^39 - 4 * q^41 - 19 * q^43 - 6 * q^45 - q^47 + 14 * q^49 - 5 * q^51 - 3 * q^53 - 11 * q^55 + 12 * q^57 + q^59 + 4 * q^61 - 14 * q^63 + 5 * q^65 - 5 * q^67 - 15 * q^69 - 2 * q^71 + 15 * q^73 + 11 * q^75 - 6 * q^77 - 12 * q^79 - 18 * q^81 + 8 * q^83 - 32 * q^85 - 11 * q^87 - 24 * q^89 - q^91 - 18 * q^93 - 9 * q^95 - 12 * q^97 + 40 * q^99

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