LMFDB - Embedded newform 1176.4.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,4,Mod(1,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1176.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1176.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:	$69.3862461668$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.58461.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 65x - 126 $ x^3 - x^2 - 65*x - 126
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.16736$ of defining polynomial
Character	$\chi$	$=$	1176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -10.1566 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -10.1566 q^{5} +9.00000 q^{9} -37.8005 q^{11} +74.0881 q^{13} +30.4699 q^{15} +44.6694 q^{17} -139.367 q^{19} +57.8200 q^{23} -21.8428 q^{25} -27.0000 q^{27} +39.5039 q^{29} +254.837 q^{31} +113.401 q^{33} +18.1048 q^{37} -222.264 q^{39} +420.819 q^{41} -382.346 q^{43} -91.4097 q^{45} +484.901 q^{47} -134.008 q^{51} +36.0875 q^{53} +383.925 q^{55} +418.102 q^{57} +145.831 q^{59} -78.1359 q^{61} -752.486 q^{65} -273.053 q^{67} -173.460 q^{69} +717.642 q^{71} -1108.28 q^{73} +65.5283 q^{75} -1085.66 q^{79} +81.0000 q^{81} -404.777 q^{83} -453.691 q^{85} -118.512 q^{87} -1354.42 q^{89} -764.510 q^{93} +1415.50 q^{95} +681.847 q^{97} -340.204 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 11 q^{5} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 + 11 * q^5 + 27 * q^9	$ 3 q - 9 q^{3} + 11 q^{5} + 27 q^{9} - 19 q^{11} + 22 q^{13} - 33 q^{15} + 104 q^{17} - 202 q^{19} - 280 q^{23} + 186 q^{25} - 81 q^{27} - 73 q^{29} + 131 q^{31} + 57 q^{33} - 326 q^{37} - 66 q^{39} + 516 q^{41} + 36 q^{43} + 99 q^{45} + 126 q^{47} - 312 q^{51} - 385 q^{53} + 611 q^{55} + 606 q^{57} - 285 q^{59} + 34 q^{61} - 920 q^{65} - 100 q^{67} + 840 q^{69} + 34 q^{71} + 108 q^{73} - 558 q^{75} - 2463 q^{79} + 243 q^{81} + 115 q^{83} - 500 q^{85} + 219 q^{87} + 110 q^{89} - 393 q^{93} - 2064 q^{95} + 2941 q^{97} - 171 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 + 11 * q^5 + 27 * q^9 - 19 * q^11 + 22 * q^13 - 33 * q^15 + 104 * q^17 - 202 * q^19 - 280 * q^23 + 186 * q^25 - 81 * q^27 - 73 * q^29 + 131 * q^31 + 57 * q^33 - 326 * q^37 - 66 * q^39 + 516 * q^41 + 36 * q^43 + 99 * q^45 + 126 * q^47 - 312 * q^51 - 385 * q^53 + 611 * q^55 + 606 * q^57 - 285 * q^59 + 34 * q^61 - 920 * q^65 - 100 * q^67 + 840 * q^69 + 34 * q^71 + 108 * q^73 - 558 * q^75 - 2463 * q^79 + 243 * q^81 + 115 * q^83 - 500 * q^85 + 219 * q^87 + 110 * q^89 - 393 * q^93 - 2064 * q^95 + 2941 * q^97 - 171 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−10.1566	−0.908437	−0.454218	−	0.890890i	$-0.650081\pi$
$5$	−10.1566	−0.908437	−0.454218	+	0.890890i	$0.650081\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−37.8005	−1.03612	−0.518058	−	0.855346i	$-0.673345\pi$
$11$	−37.8005	−1.03612	−0.518058	+	0.855346i	$0.673345\pi$
$12$	0	0
$13$	74.0881	1.58064	0.790321	−	0.612693i	$-0.209914\pi$
$13$	74.0881	1.58064	0.790321	+	0.612693i	$0.209914\pi$
$14$	0	0
$15$	30.4699	0.524486
$16$	0	0
$17$	44.6694	0.637290	0.318645	−	0.947874i	$-0.396772\pi$
$17$	44.6694	0.637290	0.318645	+	0.947874i	$0.396772\pi$
$18$	0	0
$19$	−139.367	−1.68279	−0.841397	−	0.540418i	$-0.818266\pi$
$19$	−139.367	−1.68279	−0.841397	+	0.540418i	$0.818266\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	57.8200	0.524187	0.262094	−	0.965042i	$-0.415587\pi$
$23$	57.8200	0.524187	0.262094	+	0.965042i	$0.415587\pi$
$24$	0	0
$25$	−21.8428	−0.174742
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	39.5039	0.252955	0.126477	−	0.991969i	$-0.459633\pi$
$29$	39.5039	0.252955	0.126477	+	0.991969i	$0.459633\pi$
$30$	0	0
$31$	254.837	1.47645	0.738226	−	0.674553i	$-0.235664\pi$
$31$	254.837	1.47645	0.738226	+	0.674553i	$0.235664\pi$
$32$	0	0
$33$	113.401	0.598201
$34$	0	0
$35$	0	0
$36$	0	0
$37$	18.1048	0.0804435	0.0402217	−	0.999191i	$-0.487194\pi$
$37$	18.1048	0.0804435	0.0402217	+	0.999191i	$0.487194\pi$
$38$	0	0
$39$	−222.264	−0.912584
$40$	0	0
$41$	420.819	1.60295	0.801475	−	0.598028i	$-0.204049\pi$
$41$	420.819	1.60295	0.801475	+	0.598028i	$0.204049\pi$
$42$	0	0
$43$	−382.346	−1.35598	−0.677991	−	0.735070i	$-0.737149\pi$
$43$	−382.346	−1.35598	−0.677991	+	0.735070i	$0.737149\pi$
$44$	0	0
$45$	−91.4097	−0.302812
$46$	0	0
$47$	484.901	1.50490	0.752448	−	0.658651i	$-0.228873\pi$
$47$	484.901	1.50490	0.752448	+	0.658651i	$0.228873\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−134.008	−0.367940
$52$	0	0
$53$	36.0875	0.0935283	0.0467642	−	0.998906i	$-0.485109\pi$
$53$	36.0875	0.0935283	0.0467642	+	0.998906i	$0.485109\pi$
$54$	0	0
$55$	383.925	0.941245
$56$	0	0
$57$	418.102	0.971561
$58$	0	0
$59$	145.831	0.321790	0.160895	−	0.986972i	$-0.448562\pi$
$59$	145.831	0.321790	0.160895	+	0.986972i	$0.448562\pi$
$60$	0	0
$61$	−78.1359	−0.164005	−0.0820023	−	0.996632i	$-0.526131\pi$
$61$	−78.1359	−0.164005	−0.0820023	+	0.996632i	$0.526131\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−752.486	−1.43591
$66$	0	0
$67$	−273.053	−0.497891	−0.248946	−	0.968517i	$-0.580084\pi$
$67$	−273.053	−0.497891	−0.248946	+	0.968517i	$0.580084\pi$
$68$	0	0
$69$	−173.460	−0.302640
$70$	0	0
$71$	717.642	1.19956	0.599778	−	0.800167i	$-0.295256\pi$
$71$	717.642	1.19956	0.599778	+	0.800167i	$0.295256\pi$
$72$	0	0
$73$	−1108.28	−1.77691	−0.888453	−	0.458968i	$-0.848219\pi$
$73$	−1108.28	−1.77691	−0.888453	+	0.458968i	$0.848219\pi$
$74$	0	0
$75$	65.5283	0.100887
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1085.66	−1.54615	−0.773076	−	0.634314i	$-0.781283\pi$
$79$	−1085.66	−1.54615	−0.773076	+	0.634314i	$0.781283\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−404.777	−0.535301	−0.267651	−	0.963516i	$-0.586247\pi$
$83$	−404.777	−0.535301	−0.267651	+	0.963516i	$0.586247\pi$
$84$	0	0
$85$	−453.691	−0.578938
$86$	0	0
$87$	−118.512	−0.146043
$88$	0	0
$89$	−1354.42	−1.61312	−0.806561	−	0.591151i	$-0.798674\pi$
$89$	−1354.42	−1.61312	−0.806561	+	0.591151i	$0.798674\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−764.510	−0.852430
$94$	0	0
$95$	1415.50	1.52871
$96$	0	0
$97$	681.847	0.713723	0.356861	−	0.934157i	$-0.383847\pi$
$97$	681.847	0.713723	0.356861	+	0.934157i	$0.383847\pi$
$98$	0	0
$99$	−340.204	−0.345372
$100$	0	0
$101$	798.533	0.786703	0.393352	−	0.919388i	$-0.371316\pi$
$101$	798.533	0.786703	0.393352	+	0.919388i	$0.371316\pi$
$102$	0	0
$103$	−453.726	−0.434048	−0.217024	−	0.976166i	$-0.569635\pi$
$103$	−453.726	−0.434048	−0.217024	+	0.976166i	$0.569635\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−911.741	−0.823751	−0.411875	−	0.911240i	$-0.635126\pi$
$107$	−911.741	−0.823751	−0.411875	+	0.911240i	$0.635126\pi$
$108$	0	0
$109$	22.8441	0.0200740	0.0100370	−	0.999950i	$-0.496805\pi$
$109$	22.8441	0.0200740	0.0100370	+	0.999950i	$0.496805\pi$
$110$	0	0
$111$	−54.3144	−0.0464441
$112$	0	0
$113$	462.542	0.385065	0.192532	−	0.981291i	$-0.438330\pi$
$113$	462.542	0.385065	0.192532	+	0.981291i	$0.438330\pi$
$114$	0	0
$115$	−587.257	−0.476191
$116$	0	0
$117$	666.793	0.526880
$118$	0	0
$119$	0	0
$120$	0	0
$121$	97.8742	0.0735344
$122$	0	0
$123$	−1262.46	−0.925464
$124$	0	0
$125$	1491.43	1.06718
$126$	0	0
$127$	−1792.58	−1.25249	−0.626244	−	0.779627i	$-0.715408\pi$
$127$	−1792.58	−1.25249	−0.626244	+	0.779627i	$0.715408\pi$
$128$	0	0
$129$	1147.04	0.782877
$130$	0	0
$131$	−580.446	−0.387128	−0.193564	−	0.981088i	$-0.562005\pi$
$131$	−580.446	−0.387128	−0.193564	+	0.981088i	$0.562005\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	274.229	0.174829
$136$	0	0
$137$	−2850.82	−1.77783	−0.888913	−	0.458077i	$-0.848538\pi$
$137$	−2850.82	−1.77783	−0.888913	+	0.458077i	$0.848538\pi$
$138$	0	0
$139$	187.777	0.114583	0.0572915	−	0.998357i	$-0.481754\pi$
$139$	187.777	0.114583	0.0572915	+	0.998357i	$0.481754\pi$
$140$	0	0
$141$	−1454.70	−0.868852
$142$	0	0
$143$	−2800.56	−1.63773
$144$	0	0
$145$	−401.226	−0.229793
$146$	0	0
$147$	0	0
$148$	0	0
$149$	497.684	0.273637	0.136818	−	0.990596i	$-0.456312\pi$
$149$	497.684	0.273637	0.136818	+	0.990596i	$0.456312\pi$
$150$	0	0
$151$	−968.646	−0.522035	−0.261018	−	0.965334i	$-0.584058\pi$
$151$	−968.646	−0.522035	−0.261018	+	0.965334i	$0.584058\pi$
$152$	0	0
$153$	402.025	0.212430
$154$	0	0
$155$	−2588.28	−1.34126
$156$	0	0
$157$	−176.120	−0.0895281	−0.0447640	−	0.998998i	$-0.514254\pi$
$157$	−176.120	−0.0895281	−0.0447640	+	0.998998i	$0.514254\pi$
$158$	0	0
$159$	−108.263	−0.0539986
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−18.7766	−0.00902267	−0.00451133	−	0.999990i	$-0.501436\pi$
$163$	−18.7766	−0.00902267	−0.00451133	+	0.999990i	$0.501436\pi$
$164$	0	0
$165$	−1151.78	−0.543428
$166$	0	0
$167$	1325.80	0.614334	0.307167	−	0.951656i	$-0.400619\pi$
$167$	1325.80	0.614334	0.307167	+	0.951656i	$0.400619\pi$
$168$	0	0
$169$	3292.05	1.49843
$170$	0	0
$171$	−1254.31	−0.560931
$172$	0	0
$173$	−3701.94	−1.62690	−0.813449	−	0.581636i	$-0.802413\pi$
$173$	−3701.94	−1.62690	−0.813449	+	0.581636i	$0.802413\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−437.494	−0.185785
$178$	0	0
$179$	−4363.41	−1.82199	−0.910996	−	0.412416i	$-0.864685\pi$
$179$	−4363.41	−1.82199	−0.910996	+	0.412416i	$0.864685\pi$
$180$	0	0
$181$	−130.885	−0.0537494	−0.0268747	−	0.999639i	$-0.508556\pi$
$181$	−130.885	−0.0537494	−0.0268747	+	0.999639i	$0.508556\pi$
$182$	0	0
$183$	234.408	0.0946881
$184$	0	0
$185$	−183.884	−0.0730778
$186$	0	0
$187$	−1688.53	−0.660306
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4592.05	−1.73963	−0.869814	−	0.493379i	$-0.835761\pi$
$191$	−4592.05	−1.73963	−0.869814	+	0.493379i	$0.835761\pi$
$192$	0	0
$193$	843.489	0.314589	0.157294	−	0.987552i	$-0.449723\pi$
$193$	843.489	0.314589	0.157294	+	0.987552i	$0.449723\pi$
$194$	0	0
$195$	2257.46	0.829025
$196$	0	0
$197$	1881.79	0.680570	0.340285	−	0.940322i	$-0.389476\pi$
$197$	1881.79	0.680570	0.340285	+	0.940322i	$0.389476\pi$
$198$	0	0
$199$	−1546.11	−0.550759	−0.275380	−	0.961336i	$-0.588804\pi$
$199$	−1546.11	−0.550759	−0.275380	+	0.961336i	$0.588804\pi$
$200$	0	0
$201$	819.159	0.287458
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4274.11	−1.45618
$206$	0	0
$207$	520.380	0.174729
$208$	0	0
$209$	5268.15	1.74357
$210$	0	0
$211$	−1520.42	−0.496068	−0.248034	−	0.968751i	$-0.579784\pi$
$211$	−1520.42	−0.496068	−0.248034	+	0.968751i	$0.579784\pi$
$212$	0	0
$213$	−2152.93	−0.692564
$214$	0	0
$215$	3883.35	1.23182
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3324.83	1.02590
$220$	0	0
$221$	3309.47	1.00733
$222$	0	0
$223$	−4735.42	−1.42201	−0.711003	−	0.703189i	$-0.751759\pi$
$223$	−4735.42	−1.42201	−0.711003	+	0.703189i	$0.751759\pi$
$224$	0	0
$225$	−196.585	−0.0582474
$226$	0	0
$227$	−2285.64	−0.668296	−0.334148	−	0.942521i	$-0.608449\pi$
$227$	−2285.64	−0.668296	−0.334148	+	0.942521i	$0.608449\pi$
$228$	0	0
$229$	479.209	0.138284	0.0691420	−	0.997607i	$-0.477974\pi$
$229$	479.209	0.138284	0.0691420	+	0.997607i	$0.477974\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5152.71	1.44878	0.724389	−	0.689392i	$-0.242122\pi$
$233$	5152.71	1.44878	0.724389	+	0.689392i	$0.242122\pi$
$234$	0	0
$235$	−4924.97	−1.36710
$236$	0	0
$237$	3256.97	0.892671
$238$	0	0
$239$	2378.94	0.643852	0.321926	−	0.946765i	$-0.395670\pi$
$239$	2378.94	0.643852	0.321926	+	0.946765i	$0.395670\pi$
$240$	0	0
$241$	−22.2202	−0.00593914	−0.00296957	−	0.999996i	$-0.500945\pi$
$241$	−22.2202	−0.00593914	−0.00296957	+	0.999996i	$0.500945\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10325.5	−2.65989
$248$	0	0
$249$	1214.33	0.309056
$250$	0	0
$251$	4845.08	1.21840	0.609201	−	0.793016i	$-0.291490\pi$
$251$	4845.08	1.21840	0.609201	+	0.793016i	$0.291490\pi$
$252$	0	0
$253$	−2185.62	−0.543118
$254$	0	0
$255$	1361.07	0.334250
$256$	0	0
$257$	4714.05	1.14418	0.572090	−	0.820191i	$-0.306133\pi$
$257$	4714.05	1.14418	0.572090	+	0.820191i	$0.306133\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	355.535	0.0843182
$262$	0	0
$263$	−3481.06	−0.816165	−0.408082	−	0.912945i	$-0.633802\pi$
$263$	−3481.06	−0.816165	−0.408082	+	0.912945i	$0.633802\pi$
$264$	0	0
$265$	−366.528	−0.0849646
$266$	0	0
$267$	4063.25	0.931336
$268$	0	0
$269$	297.092	0.0673384	0.0336692	−	0.999433i	$-0.489281\pi$
$269$	297.092	0.0673384	0.0336692	+	0.999433i	$0.489281\pi$
$270$	0	0
$271$	812.504	0.182126	0.0910629	−	0.995845i	$-0.470974\pi$
$271$	812.504	0.182126	0.0910629	+	0.995845i	$0.470974\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	825.667	0.181053
$276$	0	0
$277$	7209.01	1.56371	0.781855	−	0.623460i	$-0.214274\pi$
$277$	7209.01	1.56371	0.781855	+	0.623460i	$0.214274\pi$
$278$	0	0
$279$	2293.53	0.492151
$280$	0	0
$281$	−7861.53	−1.66897	−0.834483	−	0.551034i	$-0.814233\pi$
$281$	−7861.53	−1.66897	−0.834483	+	0.551034i	$0.814233\pi$
$282$	0	0
$283$	2051.70	0.430957	0.215479	−	0.976509i	$-0.430869\pi$
$283$	2051.70	0.430957	0.215479	+	0.976509i	$0.430869\pi$
$284$	0	0
$285$	−4246.51	−0.882602
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2917.64	−0.593861
$290$	0	0
$291$	−2045.54	−0.412068
$292$	0	0
$293$	−6247.59	−1.24569	−0.622847	−	0.782344i	$-0.714024\pi$
$293$	−6247.59	−1.24569	−0.622847	+	0.782344i	$0.714024\pi$
$294$	0	0
$295$	−1481.15	−0.292326
$296$	0	0
$297$	1020.61	0.199400
$298$	0	0
$299$	4283.77	0.828552
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2395.60	−0.454203
$304$	0	0
$305$	793.598	0.148988
$306$	0	0
$307$	−9872.98	−1.83544	−0.917721	−	0.397225i	$-0.869973\pi$
$307$	−9872.98	−1.83544	−0.917721	+	0.397225i	$0.869973\pi$
$308$	0	0
$309$	1361.18	0.250598
$310$	0	0
$311$	1584.47	0.288898	0.144449	−	0.989512i	$-0.453859\pi$
$311$	1584.47	0.288898	0.144449	+	0.989512i	$0.453859\pi$
$312$	0	0
$313$	−7403.93	−1.33704	−0.668522	−	0.743693i	$-0.733073\pi$
$313$	−7403.93	−1.33704	−0.668522	+	0.743693i	$0.733073\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10223.8	−1.81143	−0.905716	−	0.423884i	$-0.860666\pi$
$317$	−10223.8	−1.81143	−0.905716	+	0.423884i	$0.860666\pi$
$318$	0	0
$319$	−1493.26	−0.262090
$320$	0	0
$321$	2735.22	0.475593
$322$	0	0
$323$	−6225.46	−1.07243
$324$	0	0
$325$	−1618.29	−0.276205
$326$	0	0
$327$	−68.5322	−0.0115897
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6847.01	1.13700	0.568498	−	0.822684i	$-0.307525\pi$
$331$	6847.01	1.13700	0.568498	+	0.822684i	$0.307525\pi$
$332$	0	0
$333$	162.943	0.0268145
$334$	0	0
$335$	2773.30	0.452303
$336$	0	0
$337$	6735.78	1.08879	0.544394	−	0.838830i	$-0.316760\pi$
$337$	6735.78	1.08879	0.544394	+	0.838830i	$0.316760\pi$
$338$	0	0
$339$	−1387.63	−0.222317
$340$	0	0
$341$	−9632.94	−1.52977
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1761.77	0.274929
$346$	0	0
$347$	−7776.58	−1.20308	−0.601540	−	0.798843i	$-0.705446\pi$
$347$	−7776.58	−1.20308	−0.601540	+	0.798843i	$0.705446\pi$
$348$	0	0
$349$	3043.15	0.466750	0.233375	−	0.972387i	$-0.425023\pi$
$349$	3043.15	0.466750	0.233375	+	0.972387i	$0.425023\pi$
$350$	0	0
$351$	−2000.38	−0.304195
$352$	0	0
$353$	2410.49	0.363449	0.181724	−	0.983350i	$-0.441832\pi$
$353$	2410.49	0.363449	0.181724	+	0.983350i	$0.441832\pi$
$354$	0	0
$355$	−7288.83	−1.08972
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11356.6	1.66957	0.834786	−	0.550574i	$-0.185591\pi$
$359$	11356.6	1.66957	0.834786	+	0.550574i	$0.185591\pi$
$360$	0	0
$361$	12564.3	1.83179
$362$	0	0
$363$	−293.623	−0.0424551
$364$	0	0
$365$	11256.4	1.61421
$366$	0	0
$367$	−1860.34	−0.264602	−0.132301	−	0.991210i	$-0.542237\pi$
$367$	−1860.34	−0.264602	−0.132301	+	0.991210i	$0.542237\pi$
$368$	0	0
$369$	3787.37	0.534317
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3507.96	0.486957	0.243479	−	0.969906i	$-0.421711\pi$
$373$	3507.96	0.486957	0.243479	+	0.969906i	$0.421711\pi$
$374$	0	0
$375$	−4474.29	−0.616136
$376$	0	0
$377$	2926.77	0.399831
$378$	0	0
$379$	−766.351	−0.103865	−0.0519324	−	0.998651i	$-0.516538\pi$
$379$	−766.351	−0.103865	−0.0519324	+	0.998651i	$0.516538\pi$
$380$	0	0
$381$	5377.74	0.723124
$382$	0	0
$383$	13914.7	1.85642	0.928212	−	0.372053i	$-0.121346\pi$
$383$	13914.7	1.85642	0.928212	+	0.372053i	$0.121346\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3441.12	−0.451994
$388$	0	0
$389$	−4598.31	−0.599340	−0.299670	−	0.954043i	$-0.596877\pi$
$389$	−4598.31	−0.599340	−0.299670	+	0.954043i	$0.596877\pi$
$390$	0	0
$391$	2582.79	0.334059
$392$	0	0
$393$	1741.34	0.223509
$394$	0	0
$395$	11026.6	1.40458
$396$	0	0
$397$	−12735.2	−1.60998	−0.804991	−	0.593287i	$-0.797830\pi$
$397$	−12735.2	−1.60998	−0.804991	+	0.593287i	$0.797830\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1802.56	0.224477	0.112239	−	0.993681i	$-0.464198\pi$
$401$	1802.56	0.224477	0.112239	+	0.993681i	$0.464198\pi$
$402$	0	0
$403$	18880.4	2.33374
$404$	0	0
$405$	−822.687	−0.100937
$406$	0	0
$407$	−684.369	−0.0833487
$408$	0	0
$409$	8336.30	1.00783	0.503916	−	0.863753i	$-0.331892\pi$
$409$	8336.30	1.00783	0.503916	+	0.863753i	$0.331892\pi$
$410$	0	0
$411$	8552.46	1.02643
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4111.17	0.486288
$416$	0	0
$417$	−563.331	−0.0661546
$418$	0	0
$419$	−11940.4	−1.39219	−0.696093	−	0.717952i	$-0.745080\pi$
$419$	−11940.4	−1.39219	−0.696093	+	0.717952i	$0.745080\pi$
$420$	0	0
$421$	−3876.28	−0.448737	−0.224368	−	0.974504i	$-0.572032\pi$
$421$	−3876.28	−0.448737	−0.224368	+	0.974504i	$0.572032\pi$
$422$	0	0
$423$	4364.11	0.501632
$424$	0	0
$425$	−975.705	−0.111362
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8401.69	0.945542
$430$	0	0
$431$	−6252.60	−0.698786	−0.349393	−	0.936976i	$-0.613612\pi$
$431$	−6252.60	−0.698786	−0.349393	+	0.936976i	$0.613612\pi$
$432$	0	0
$433$	−12189.7	−1.35289	−0.676446	−	0.736493i	$-0.736481\pi$
$433$	−12189.7	−1.35289	−0.676446	+	0.736493i	$0.736481\pi$
$434$	0	0
$435$	1203.68	0.132671
$436$	0	0
$437$	−8058.22	−0.882098
$438$	0	0
$439$	8281.78	0.900382	0.450191	−	0.892932i	$-0.351356\pi$
$439$	8281.78	0.900382	0.450191	+	0.892932i	$0.351356\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	626.486	0.0671902	0.0335951	−	0.999436i	$-0.489304\pi$
$443$	626.486	0.0671902	0.0335951	+	0.999436i	$0.489304\pi$
$444$	0	0
$445$	13756.3	1.46542
$446$	0	0
$447$	−1493.05	−0.157984
$448$	0	0
$449$	12789.8	1.34430	0.672148	−	0.740416i	$-0.265372\pi$
$449$	12789.8	1.34430	0.672148	+	0.740416i	$0.265372\pi$
$450$	0	0
$451$	−15907.2	−1.66084
$452$	0	0
$453$	2905.94	0.301397
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10708.8	1.09614	0.548069	−	0.836433i	$-0.315363\pi$
$457$	10708.8	1.09614	0.548069	+	0.836433i	$0.315363\pi$
$458$	0	0
$459$	−1206.08	−0.122647
$460$	0	0
$461$	2549.45	0.257570	0.128785	−	0.991673i	$-0.458892\pi$
$461$	2549.45	0.257570	0.128785	+	0.991673i	$0.458892\pi$
$462$	0	0
$463$	−13906.1	−1.39584	−0.697919	−	0.716177i	$-0.745890\pi$
$463$	−13906.1	−1.39584	−0.697919	+	0.716177i	$0.745890\pi$
$464$	0	0
$465$	7764.85	0.774379
$466$	0	0
$467$	5827.33	0.577423	0.288712	−	0.957416i	$-0.406773\pi$
$467$	5827.33	0.577423	0.288712	+	0.957416i	$0.406773\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	528.360	0.0516891
$472$	0	0
$473$	14452.9	1.40495
$474$	0	0
$475$	3044.17	0.294055
$476$	0	0
$477$	324.788	0.0311761
$478$	0	0
$479$	3813.03	0.363720	0.181860	−	0.983324i	$-0.441788\pi$
$479$	3813.03	0.363720	0.181860	+	0.983324i	$0.441788\pi$
$480$	0	0
$481$	1341.35	0.127152
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6925.27	−0.648372
$486$	0	0
$487$	−5783.76	−0.538167	−0.269083	−	0.963117i	$-0.586721\pi$
$487$	−5783.76	−0.538167	−0.269083	+	0.963117i	$0.586721\pi$
$488$	0	0
$489$	56.3297	0.00520924
$490$	0	0
$491$	−10840.9	−0.996419	−0.498209	−	0.867057i	$-0.666009\pi$
$491$	−10840.9	−0.996419	−0.498209	+	0.867057i	$0.666009\pi$
$492$	0	0
$493$	1764.62	0.161206
$494$	0	0
$495$	3455.33	0.313748
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12738.1	−1.14276	−0.571379	−	0.820686i	$-0.693591\pi$
$499$	−12738.1	−1.14276	−0.571379	+	0.820686i	$0.693591\pi$
$500$	0	0
$501$	−3977.41	−0.354686
$502$	0	0
$503$	5014.31	0.444487	0.222243	−	0.974991i	$-0.428662\pi$
$503$	5014.31	0.444487	0.222243	+	0.974991i	$0.428662\pi$
$504$	0	0
$505$	−8110.41	−0.714670
$506$	0	0
$507$	−9876.14	−0.865117
$508$	0	0
$509$	14977.5	1.30425	0.652126	−	0.758111i	$-0.273877\pi$
$509$	14977.5	1.30425	0.652126	+	0.758111i	$0.273877\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3762.92	0.323854
$514$	0	0
$515$	4608.33	0.394306
$516$	0	0
$517$	−18329.5	−1.55925
$518$	0	0
$519$	11105.8	0.939290
$520$	0	0
$521$	−3122.30	−0.262553	−0.131277	−	0.991346i	$-0.541908\pi$
$521$	−3122.30	−0.262553	−0.131277	+	0.991346i	$0.541908\pi$
$522$	0	0
$523$	9649.39	0.806765	0.403383	−	0.915031i	$-0.367834\pi$
$523$	9649.39	0.806765	0.403383	+	0.915031i	$0.367834\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11383.4	0.940929
$528$	0	0
$529$	−8823.85	−0.725228
$530$	0	0
$531$	1312.48	0.107263
$532$	0	0
$533$	31177.7	2.53369
$534$	0	0
$535$	9260.22	0.748326
$536$	0	0
$537$	13090.2	1.05193
$538$	0	0
$539$	0	0
$540$	0	0
$541$	15861.6	1.26052	0.630262	−	0.776382i	$-0.282947\pi$
$541$	15861.6	1.26052	0.630262	+	0.776382i	$0.282947\pi$
$542$	0	0
$543$	392.656	0.0310322
$544$	0	0
$545$	−232.019	−0.0182360
$546$	0	0
$547$	7477.29	0.584471	0.292236	−	0.956346i	$-0.405601\pi$
$547$	7477.29	0.584471	0.292236	+	0.956346i	$0.405601\pi$
$548$	0	0
$549$	−703.223	−0.0546682
$550$	0	0
$551$	−5505.55	−0.425670
$552$	0	0
$553$	0	0
$554$	0	0
$555$	551.651	0.0421915
$556$	0	0
$557$	−18170.4	−1.38224	−0.691119	−	0.722741i	$-0.742882\pi$
$557$	−18170.4	−1.38224	−0.691119	+	0.722741i	$0.742882\pi$
$558$	0	0
$559$	−28327.3	−2.14332
$560$	0	0
$561$	5065.58	0.381228
$562$	0	0
$563$	−467.613	−0.0350045	−0.0175023	−	0.999847i	$-0.505571\pi$
$563$	−467.613	−0.0350045	−0.0175023	+	0.999847i	$0.505571\pi$
$564$	0	0
$565$	−4697.87	−0.349807
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26449.8	1.94874	0.974368	−	0.224958i	$-0.0722245\pi$
$569$	26449.8	1.94874	0.974368	+	0.224958i	$0.0722245\pi$
$570$	0	0
$571$	8770.96	0.642826	0.321413	−	0.946939i	$-0.395842\pi$
$571$	8770.96	0.642826	0.321413	+	0.946939i	$0.395842\pi$
$572$	0	0
$573$	13776.1	1.00438
$574$	0	0
$575$	−1262.95	−0.0915976
$576$	0	0
$577$	3036.56	0.219088	0.109544	−	0.993982i	$-0.465061\pi$
$577$	3036.56	0.219088	0.109544	+	0.993982i	$0.465061\pi$
$578$	0	0
$579$	−2530.47	−0.181628
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1364.12	−0.0969061
$584$	0	0
$585$	−6772.37	−0.478638
$586$	0	0
$587$	−1620.33	−0.113932	−0.0569661	−	0.998376i	$-0.518143\pi$
$587$	−1620.33	−0.113932	−0.0569661	+	0.998376i	$0.518143\pi$
$588$	0	0
$589$	−35515.9	−2.48456
$590$	0	0
$591$	−5645.38	−0.392927
$592$	0	0
$593$	−27146.4	−1.87988	−0.939940	−	0.341340i	$-0.889119\pi$
$593$	−27146.4	−1.87988	−0.939940	+	0.341340i	$0.889119\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4638.34	0.317981
$598$	0	0
$599$	19211.0	1.31042	0.655209	−	0.755448i	$-0.272580\pi$
$599$	19211.0	1.31042	0.655209	+	0.755448i	$0.272580\pi$
$600$	0	0
$601$	−24463.9	−1.66041	−0.830204	−	0.557460i	$-0.811776\pi$
$601$	−24463.9	−1.66041	−0.830204	+	0.557460i	$0.811776\pi$
$602$	0	0
$603$	−2457.48	−0.165964
$604$	0	0
$605$	−994.073	−0.0668013
$606$	0	0
$607$	−20270.4	−1.35544	−0.677718	−	0.735322i	$-0.737031\pi$
$607$	−20270.4	−1.35544	−0.677718	+	0.735322i	$0.737031\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35925.4	2.37870
$612$	0	0
$613$	15584.8	1.02686	0.513429	−	0.858132i	$-0.328375\pi$
$613$	15584.8	1.02686	0.513429	+	0.858132i	$0.328375\pi$
$614$	0	0
$615$	12822.3	0.840725
$616$	0	0
$617$	2276.13	0.148515	0.0742573	−	0.997239i	$-0.476341\pi$
$617$	2276.13	0.148515	0.0742573	+	0.997239i	$0.476341\pi$
$618$	0	0
$619$	−14493.2	−0.941081	−0.470540	−	0.882378i	$-0.655941\pi$
$619$	−14493.2	−0.941081	−0.470540	+	0.882378i	$0.655941\pi$
$620$	0	0
$621$	−1561.14	−0.100880
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−12417.5	−0.794723
$626$	0	0
$627$	−15804.4	−1.00665
$628$	0	0
$629$	808.731	0.0512658
$630$	0	0
$631$	8517.07	0.537336	0.268668	−	0.963233i	$-0.413417\pi$
$631$	8517.07	0.537336	0.268668	+	0.963233i	$0.413417\pi$
$632$	0	0
$633$	4561.27	0.286405
$634$	0	0
$635$	18206.6	1.13781
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6458.78	0.399852
$640$	0	0
$641$	21914.3	1.35033	0.675167	−	0.737665i	$-0.264072\pi$
$641$	21914.3	1.35033	0.675167	+	0.737665i	$0.264072\pi$
$642$	0	0
$643$	−26799.5	−1.64365	−0.821826	−	0.569739i	$-0.807044\pi$
$643$	−26799.5	−1.64365	−0.821826	+	0.569739i	$0.807044\pi$
$644$	0	0
$645$	−11650.1	−0.711194
$646$	0	0
$647$	32176.4	1.95515	0.977577	−	0.210578i	$-0.0675347\pi$
$647$	32176.4	1.95515	0.977577	+	0.210578i	$0.0675347\pi$
$648$	0	0
$649$	−5512.49	−0.333411
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−270.878	−0.0162332	−0.00811660	−	0.999967i	$-0.502584\pi$
$653$	−270.878	−0.0162332	−0.00811660	+	0.999967i	$0.502584\pi$
$654$	0	0
$655$	5895.38	0.351682
$656$	0	0
$657$	−9974.50	−0.592302
$658$	0	0
$659$	−913.255	−0.0539838	−0.0269919	−	0.999636i	$-0.508593\pi$
$659$	−913.255	−0.0539838	−0.0269919	+	0.999636i	$0.508593\pi$
$660$	0	0
$661$	17642.8	1.03816	0.519080	−	0.854726i	$-0.326275\pi$
$661$	17642.8	1.03816	0.519080	+	0.854726i	$0.326275\pi$
$662$	0	0
$663$	−9928.42	−0.581581
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2284.11	0.132596
$668$	0	0
$669$	14206.3	0.820995
$670$	0	0
$671$	2953.57	0.169928
$672$	0	0
$673$	9839.88	0.563595	0.281798	−	0.959474i	$-0.409069\pi$
$673$	9839.88	0.563595	0.281798	+	0.959474i	$0.409069\pi$
$674$	0	0
$675$	589.755	0.0336292
$676$	0	0
$677$	27471.3	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$677$	27471.3	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6856.92	0.385841
$682$	0	0
$683$	−6363.61	−0.356511	−0.178255	−	0.983984i	$-0.557045\pi$
$683$	−6363.61	−0.356511	−0.178255	+	0.983984i	$0.557045\pi$
$684$	0	0
$685$	28954.7	1.61504
$686$	0	0
$687$	−1437.63	−0.0798383
$688$	0	0
$689$	2673.65	0.147835
$690$	0	0
$691$	−16512.5	−0.909068	−0.454534	−	0.890729i	$-0.650194\pi$
$691$	−16512.5	−0.909068	−0.454534	+	0.890729i	$0.650194\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1907.18	−0.104091
$696$	0	0
$697$	18797.8	1.02154
$698$	0	0
$699$	−15458.1	−0.836452
$700$	0	0
$701$	−32633.2	−1.75826	−0.879128	−	0.476585i	$-0.841874\pi$
$701$	−32633.2	−1.75826	−0.879128	+	0.476585i	$0.841874\pi$
$702$	0	0
$703$	−2523.22	−0.135370
$704$	0	0
$705$	14774.9	0.789298
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−22641.5	−1.19932	−0.599660	−	0.800255i	$-0.704698\pi$
$709$	−22641.5	−1.19932	−0.599660	+	0.800255i	$0.704698\pi$
$710$	0	0
$711$	−9770.92	−0.515384
$712$	0	0
$713$	14734.7	0.773937
$714$	0	0
$715$	28444.3	1.48777
$716$	0	0
$717$	−7136.81	−0.371728
$718$	0	0
$719$	−28786.0	−1.49310	−0.746549	−	0.665330i	$-0.768291\pi$
$719$	−28786.0	−1.49310	−0.746549	+	0.665330i	$0.768291\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	66.6607	0.00342896
$724$	0	0
$725$	−862.875	−0.0442019
$726$	0	0
$727$	−15302.1	−0.780639	−0.390319	−	0.920679i	$-0.627635\pi$
$727$	−15302.1	−0.780639	−0.390319	+	0.920679i	$0.627635\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−17079.2	−0.864155
$732$	0	0
$733$	16451.3	0.828983	0.414491	−	0.910053i	$-0.363960\pi$
$733$	16451.3	0.828983	0.414491	+	0.910053i	$0.363960\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10321.5	0.515873
$738$	0	0
$739$	−3878.19	−0.193047	−0.0965233	−	0.995331i	$-0.530772\pi$
$739$	−3878.19	−0.193047	−0.0965233	+	0.995331i	$0.530772\pi$
$740$	0	0
$741$	30976.4	1.53569
$742$	0	0
$743$	−7702.39	−0.380314	−0.190157	−	0.981754i	$-0.560900\pi$
$743$	−7702.39	−0.380314	−0.190157	+	0.981754i	$0.560900\pi$
$744$	0	0
$745$	−5054.79	−0.248582
$746$	0	0
$747$	−3642.99	−0.178434
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7068.66	0.343461	0.171730	−	0.985144i	$-0.445064\pi$
$751$	7068.66	0.343461	0.171730	+	0.985144i	$0.445064\pi$
$752$	0	0
$753$	−14535.2	−0.703445
$754$	0	0
$755$	9838.19	0.474236
$756$	0	0
$757$	21371.0	1.02608	0.513039	−	0.858365i	$-0.328520\pi$
$757$	21371.0	1.02608	0.513039	+	0.858365i	$0.328520\pi$
$758$	0	0
$759$	6556.87	0.313569
$760$	0	0
$761$	−16307.2	−0.776786	−0.388393	−	0.921494i	$-0.626970\pi$
$761$	−16307.2	−0.776786	−0.388393	+	0.921494i	$0.626970\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4083.22	−0.192979
$766$	0	0
$767$	10804.4	0.508634
$768$	0	0
$769$	−40553.7	−1.90169	−0.950847	−	0.309660i	$-0.899785\pi$
$769$	−40553.7	−1.90169	−0.950847	+	0.309660i	$0.899785\pi$
$770$	0	0
$771$	−14142.2	−0.660593
$772$	0	0
$773$	11220.5	0.522089	0.261045	−	0.965327i	$-0.415933\pi$
$773$	11220.5	0.522089	0.261045	+	0.965327i	$0.415933\pi$
$774$	0	0
$775$	−5566.34	−0.257999
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−58648.5	−2.69743
$780$	0	0
$781$	−27127.2	−1.24288
$782$	0	0
$783$	−1066.60	−0.0486812
$784$	0	0
$785$	1788.79	0.0813306
$786$	0	0
$787$	7836.87	0.354961	0.177480	−	0.984124i	$-0.443205\pi$
$787$	7836.87	0.354961	0.177480	+	0.984124i	$0.443205\pi$
$788$	0	0
$789$	10443.2	0.471213
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5788.94	−0.259232
$794$	0	0
$795$	1099.58	0.0490543
$796$	0	0
$797$	6521.90	0.289859	0.144929	−	0.989442i	$-0.453704\pi$
$797$	6521.90	0.289859	0.144929	+	0.989442i	$0.453704\pi$
$798$	0	0
$799$	21660.3	0.959056
$800$	0	0
$801$	−12189.7	−0.537707
$802$	0	0
$803$	41893.4	1.84108
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−891.277	−0.0388779
$808$	0	0
$809$	−22997.8	−0.999457	−0.499728	−	0.866182i	$-0.666567\pi$
$809$	−22997.8	−0.999457	−0.499728	+	0.866182i	$0.666567\pi$
$810$	0	0
$811$	−5253.94	−0.227485	−0.113743	−	0.993510i	$-0.536284\pi$
$811$	−5253.94	−0.227485	−0.113743	+	0.993510i	$0.536284\pi$
$812$	0	0
$813$	−2437.51	−0.105150
$814$	0	0
$815$	190.707	0.00819653
$816$	0	0
$817$	53286.6	2.28184
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14044.6	−0.597029	−0.298515	−	0.954405i	$-0.596491\pi$
$821$	−14044.6	−0.597029	−0.298515	+	0.954405i	$0.596491\pi$
$822$	0	0
$823$	13677.4	0.579300	0.289650	−	0.957133i	$-0.406461\pi$
$823$	13677.4	0.579300	0.289650	+	0.957133i	$0.406461\pi$
$824$	0	0
$825$	−2477.00	−0.104531
$826$	0	0
$827$	3206.14	0.134811	0.0674054	−	0.997726i	$-0.478528\pi$
$827$	3206.14	0.134811	0.0674054	+	0.997726i	$0.478528\pi$
$828$	0	0
$829$	−30876.6	−1.29359	−0.646796	−	0.762663i	$-0.723892\pi$
$829$	−30876.6	−1.29359	−0.646796	+	0.762663i	$0.723892\pi$
$830$	0	0
$831$	−21627.0	−0.902808
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13465.7	−0.558083
$836$	0	0
$837$	−6880.59	−0.284143
$838$	0	0
$839$	11040.6	0.454306	0.227153	−	0.973859i	$-0.427058\pi$
$839$	11040.6	0.454306	0.227153	+	0.973859i	$0.427058\pi$
$840$	0	0
$841$	−22828.4	−0.936014
$842$	0	0
$843$	23584.6	0.963578
$844$	0	0
$845$	−33436.1	−1.36123
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−6155.10	−0.248813
$850$	0	0
$851$	1046.82	0.0421674
$852$	0	0
$853$	12245.3	0.491524	0.245762	−	0.969330i	$-0.420962\pi$
$853$	12245.3	0.491524	0.245762	+	0.969330i	$0.420962\pi$
$854$	0	0
$855$	12739.5	0.509570
$856$	0	0
$857$	−18764.2	−0.747925	−0.373962	−	0.927444i	$-0.622001\pi$
$857$	−18764.2	−0.747925	−0.373962	+	0.927444i	$0.622001\pi$
$858$	0	0
$859$	−30566.2	−1.21409	−0.607046	−	0.794667i	$-0.707646\pi$
$859$	−30566.2	−1.21409	−0.607046	+	0.794667i	$0.707646\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14027.6	−0.553307	−0.276654	−	0.960970i	$-0.589225\pi$
$863$	−14027.6	−0.553307	−0.276654	+	0.960970i	$0.589225\pi$
$864$	0	0
$865$	37599.3	1.47793
$866$	0	0
$867$	8752.92	0.342866
$868$	0	0
$869$	41038.3	1.60199
$870$	0	0
$871$	−20230.0	−0.786988
$872$	0	0
$873$	6136.62	0.237908
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−33737.9	−1.29903	−0.649514	−	0.760349i	$-0.725028\pi$
$877$	−33737.9	−1.29903	−0.649514	+	0.760349i	$0.725028\pi$
$878$	0	0
$879$	18742.8	0.719201
$880$	0	0
$881$	−8512.70	−0.325539	−0.162770	−	0.986664i	$-0.552043\pi$
$881$	−8512.70	−0.325539	−0.162770	+	0.986664i	$0.552043\pi$
$882$	0	0
$883$	−27302.9	−1.04056	−0.520281	−	0.853995i	$-0.674173\pi$
$883$	−27302.9	−1.04056	−0.520281	+	0.853995i	$0.674173\pi$
$884$	0	0
$885$	4443.46	0.168774
$886$	0	0
$887$	25804.0	0.976789	0.488395	−	0.872623i	$-0.337583\pi$
$887$	25804.0	0.976789	0.488395	+	0.872623i	$0.337583\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3061.84	−0.115124
$892$	0	0
$893$	−67579.4	−2.53243
$894$	0	0
$895$	44317.5	1.65516
$896$	0	0
$897$	−12851.3	−0.478365
$898$	0	0
$899$	10067.0	0.373476
$900$	0	0
$901$	1612.01	0.0596047
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1329.36	0.0488279
$906$	0	0
$907$	29536.1	1.08129	0.540645	−	0.841251i	$-0.318180\pi$
$907$	29536.1	1.08129	0.540645	+	0.841251i	$0.318180\pi$
$908$	0	0
$909$	7186.80	0.262234
$910$	0	0
$911$	−19543.9	−0.710777	−0.355389	−	0.934719i	$-0.615651\pi$
$911$	−19543.9	−0.710777	−0.355389	+	0.934719i	$0.615651\pi$
$912$	0	0
$913$	15300.7	0.554634
$914$	0	0
$915$	−2380.79	−0.0860182
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8300.26	−0.297933	−0.148967	−	0.988842i	$-0.547595\pi$
$919$	−8300.26	−0.297933	−0.148967	+	0.988842i	$0.547595\pi$
$920$	0	0
$921$	29618.9	1.05969
$922$	0	0
$923$	53168.7	1.89607
$924$	0	0
$925$	−395.459	−0.0140569
$926$	0	0
$927$	−4083.54	−0.144683
$928$	0	0
$929$	−11835.8	−0.417999	−0.208999	−	0.977916i	$-0.567021\pi$
$929$	−11835.8	−0.417999	−0.208999	+	0.977916i	$0.567021\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−4753.42	−0.166795
$934$	0	0
$935$	17149.7	0.599846
$936$	0	0
$937$	19702.4	0.686926	0.343463	−	0.939166i	$-0.388400\pi$
$937$	19702.4	0.686926	0.343463	+	0.939166i	$0.388400\pi$
$938$	0	0
$939$	22211.8	0.771942
$940$	0	0
$941$	−32940.6	−1.14116	−0.570581	−	0.821241i	$-0.693282\pi$
$941$	−32940.6	−1.14116	−0.570581	+	0.821241i	$0.693282\pi$
$942$	0	0
$943$	24331.8	0.840246
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37692.0	1.29337	0.646687	−	0.762755i	$-0.276154\pi$
$947$	37692.0	1.29337	0.646687	+	0.762755i	$0.276154\pi$
$948$	0	0
$949$	−82110.2	−2.80865
$950$	0	0
$951$	30671.3	1.04583
$952$	0	0
$953$	2904.61	0.0987298	0.0493649	−	0.998781i	$-0.484280\pi$
$953$	2904.61	0.0987298	0.0493649	+	0.998781i	$0.484280\pi$
$954$	0	0
$955$	46639.8	1.58034
$956$	0	0
$957$	4479.79	0.151318
$958$	0	0
$959$	0	0
$960$	0	0
$961$	35150.7	1.17991
$962$	0	0
$963$	−8205.67	−0.274584
$964$	0	0
$965$	−8567.01	−0.285784
$966$	0	0
$967$	−7954.56	−0.264531	−0.132266	−	0.991214i	$-0.542225\pi$
$967$	−7954.56	−0.264531	−0.132266	+	0.991214i	$0.542225\pi$
$968$	0	0
$969$	18676.4	0.619166
$970$	0	0
$971$	15318.3	0.506268	0.253134	−	0.967431i	$-0.418539\pi$
$971$	15318.3	0.506268	0.253134	+	0.967431i	$0.418539\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4854.87	0.159467
$976$	0	0
$977$	−34372.3	−1.12555	−0.562777	−	0.826609i	$-0.690267\pi$
$977$	−34372.3	−1.12555	−0.562777	+	0.826609i	$0.690267\pi$
$978$	0	0
$979$	51197.5	1.67138
$980$	0	0
$981$	205.597	0.00669133
$982$	0	0
$983$	15356.6	0.498271	0.249136	−	0.968469i	$-0.419854\pi$
$983$	15356.6	0.498271	0.249136	+	0.968469i	$0.419854\pi$
$984$	0	0
$985$	−19112.7	−0.618255
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−22107.3	−0.710789
$990$	0	0
$991$	−50048.5	−1.60428	−0.802141	−	0.597134i	$-0.796306\pi$
$991$	−50048.5	−1.60428	−0.802141	+	0.597134i	$0.796306\pi$
$992$	0	0
$993$	−20541.0	−0.656445
$994$	0	0
$995$	15703.3	0.500330
$996$	0	0
$997$	7695.26	0.244445	0.122222	−	0.992503i	$-0.460998\pi$
$997$	7695.26	0.244445	0.122222	+	0.992503i	$0.460998\pi$
$998$	0	0
$999$	−488.829	−0.0154814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.4.a.x.1.1		3
4.3	odd	2		2352.4.a.cj.1.1		3
7.3	odd	6		168.4.q.e.121.1	yes	6
7.5	odd	6		168.4.q.e.25.1	✓	6
7.6	odd	2		1176.4.a.y.1.3		3
21.5	even	6		504.4.s.g.361.3		6
21.17	even	6		504.4.s.g.289.3		6
28.3	even	6		336.4.q.l.289.1		6
28.19	even	6		336.4.q.l.193.1		6
28.27	even	2		2352.4.a.ch.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.e.25.1	✓	6	7.5	odd	6
168.4.q.e.121.1	yes	6	7.3	odd	6
336.4.q.l.193.1		6	28.19	even	6
336.4.q.l.289.1		6	28.3	even	6
504.4.s.g.289.3		6	21.17	even	6
504.4.s.g.361.3		6	21.5	even	6
1176.4.a.x.1.1		3	1.1	even	1	trivial
1176.4.a.y.1.3		3	7.6	odd	2
2352.4.a.ch.1.3		3	28.27	even	2
2352.4.a.cj.1.1		3	4.3	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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1176.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -10.1566 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -10.1566 q^{5} +9.00000 q^{9} -37.8005 q^{11} +74.0881 q^{13} +30.4699 q^{15} +44.6694 q^{17} -139.367 q^{19} +57.8200 q^{23} -21.8428 q^{25} -27.0000 q^{27} +39.5039 q^{29} +254.837 q^{31} +113.401 q^{33} +18.1048 q^{37} -222.264 q^{39} +420.819 q^{41} -382.346 q^{43} -91.4097 q^{45} +484.901 q^{47} -134.008 q^{51} +36.0875 q^{53} +383.925 q^{55} +418.102 q^{57} +145.831 q^{59} -78.1359 q^{61} -752.486 q^{65} -273.053 q^{67} -173.460 q^{69} +717.642 q^{71} -1108.28 q^{73} +65.5283 q^{75} -1085.66 q^{79} +81.0000 q^{81} -404.777 q^{83} -453.691 q^{85} -118.512 q^{87} -1354.42 q^{89} -764.510 q^{93} +1415.50 q^{95} +681.847 q^{97} -340.204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 11 q^{5} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 + 11 * q^5 + 27 * q^9	\( 3 q - 9 q^{3} + 11 q^{5} + 27 q^{9} - 19 q^{11} + 22 q^{13} - 33 q^{15} + 104 q^{17} - 202 q^{19} - 280 q^{23} + 186 q^{25} - 81 q^{27} - 73 q^{29} + 131 q^{31} + 57 q^{33} - 326 q^{37} - 66 q^{39} + 516 q^{41} + 36 q^{43} + 99 q^{45} + 126 q^{47} - 312 q^{51} - 385 q^{53} + 611 q^{55} + 606 q^{57} - 285 q^{59} + 34 q^{61} - 920 q^{65} - 100 q^{67} + 840 q^{69} + 34 q^{71} + 108 q^{73} - 558 q^{75} - 2463 q^{79} + 243 q^{81} + 115 q^{83} - 500 q^{85} + 219 q^{87} + 110 q^{89} - 393 q^{93} - 2064 q^{95} + 2941 q^{97} - 171 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 + 11 * q^5 + 27 * q^9 - 19 * q^11 + 22 * q^13 - 33 * q^15 + 104 * q^17 - 202 * q^19 - 280 * q^23 + 186 * q^25 - 81 * q^27 - 73 * q^29 + 131 * q^31 + 57 * q^33 - 326 * q^37 - 66 * q^39 + 516 * q^41 + 36 * q^43 + 99 * q^45 + 126 * q^47 - 312 * q^51 - 385 * q^53 + 611 * q^55 + 606 * q^57 - 285 * q^59 + 34 * q^61 - 920 * q^65 - 100 * q^67 + 840 * q^69 + 34 * q^71 + 108 * q^73 - 558 * q^75 - 2463 * q^79 + 243 * q^81 + 115 * q^83 - 500 * q^85 + 219 * q^87 + 110 * q^89 - 393 * q^93 - 2064 * q^95 + 2941 * q^97 - 171 * q^99

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