LMFDB - Embedded newform 1176.4.a.bc.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:	$4$
Coefficient field:	4.4.145408.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} + 142 $ x^4 - 24*x^2 + 142
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.66254$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} -16.6019 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -16.6019 q^{5} +9.00000 q^{9} -49.2481 q^{11} +64.4774 q^{13} -49.8056 q^{15} +132.429 q^{17} -82.0605 q^{19} +82.0466 q^{23} +150.622 q^{25} +27.0000 q^{27} +157.717 q^{29} -185.291 q^{31} -147.744 q^{33} -51.9002 q^{37} +193.432 q^{39} -49.4929 q^{41} +313.788 q^{43} -149.417 q^{45} -553.259 q^{47} +397.287 q^{51} -619.868 q^{53} +817.611 q^{55} -246.181 q^{57} -712.989 q^{59} -287.421 q^{61} -1070.45 q^{65} +226.636 q^{67} +246.140 q^{69} +55.3834 q^{71} +799.071 q^{73} +451.867 q^{75} -120.313 q^{79} +81.0000 q^{81} -857.862 q^{83} -2198.57 q^{85} +473.151 q^{87} -377.167 q^{89} -555.873 q^{93} +1362.36 q^{95} -1265.20 q^{97} -443.233 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} - 8 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 - 8 * q^5 + 36 * q^9	$ 4 q + 12 q^{3} - 8 q^{5} + 36 q^{9} - 40 q^{11} - 48 q^{13} - 24 q^{15} + 152 q^{17} - 224 q^{19} - 8 q^{23} - 28 q^{25} + 108 q^{27} - 144 q^{29} - 400 q^{31} - 120 q^{33} - 304 q^{37} - 144 q^{39} + 152 q^{41} + 160 q^{43} - 72 q^{45} - 544 q^{47} + 456 q^{51} - 1320 q^{53} - 16 q^{55} - 672 q^{57} - 1040 q^{59} - 896 q^{61} - 648 q^{65} - 416 q^{67} - 24 q^{69} + 248 q^{71} + 752 q^{73} - 84 q^{75} + 864 q^{79} + 324 q^{81} - 1456 q^{83} - 1608 q^{85} - 432 q^{87} - 2936 q^{89} - 1200 q^{93} - 80 q^{95} - 144 q^{97} - 360 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 - 8 * q^5 + 36 * q^9 - 40 * q^11 - 48 * q^13 - 24 * q^15 + 152 * q^17 - 224 * q^19 - 8 * q^23 - 28 * q^25 + 108 * q^27 - 144 * q^29 - 400 * q^31 - 120 * q^33 - 304 * q^37 - 144 * q^39 + 152 * q^41 + 160 * q^43 - 72 * q^45 - 544 * q^47 + 456 * q^51 - 1320 * q^53 - 16 * q^55 - 672 * q^57 - 1040 * q^59 - 896 * q^61 - 648 * q^65 - 416 * q^67 - 24 * q^69 + 248 * q^71 + 752 * q^73 - 84 * q^75 + 864 * q^79 + 324 * q^81 - 1456 * q^83 - 1608 * q^85 - 432 * q^87 - 2936 * q^89 - 1200 * q^93 - 80 * q^95 - 144 * q^97 - 360 * 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
$5$	−16.6019	−1.48492	−0.742458	−	0.669892i	$-0.766340\pi$
$5$	−16.6019	−1.48492	−0.742458	+	0.669892i	$0.766340\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−49.2481	−1.34990	−0.674949	−	0.737865i	$-0.735834\pi$
$11$	−49.2481	−1.34990	−0.674949	+	0.737865i	$0.735834\pi$
$12$	0	0
$13$	64.4774	1.37560	0.687801	−	0.725900i	$-0.258576\pi$
$13$	64.4774	1.37560	0.687801	+	0.725900i	$0.258576\pi$
$14$	0	0
$15$	−49.8056	−0.857317
$16$	0	0
$17$	132.429	1.88934	0.944669	−	0.328024i	$-0.106383\pi$
$17$	132.429	1.88934	0.944669	+	0.328024i	$0.106383\pi$
$18$	0	0
$19$	−82.0605	−0.990840	−0.495420	−	0.868654i	$-0.664986\pi$
$19$	−82.0605	−0.990840	−0.495420	+	0.868654i	$0.664986\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	82.0466	0.743822	0.371911	−	0.928268i	$-0.378703\pi$
$23$	82.0466	0.743822	0.371911	+	0.928268i	$0.378703\pi$
$24$	0	0
$25$	150.622	1.20498
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	157.717	1.00991	0.504954	−	0.863146i	$-0.331509\pi$
$29$	157.717	1.00991	0.504954	+	0.863146i	$0.331509\pi$
$30$	0	0
$31$	−185.291	−1.07352	−0.536762	−	0.843734i	$-0.680353\pi$
$31$	−185.291	−1.07352	−0.536762	+	0.843734i	$0.680353\pi$
$32$	0	0
$33$	−147.744	−0.779363
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−51.9002	−0.230604	−0.115302	−	0.993331i	$-0.536784\pi$
$37$	−51.9002	−0.230604	−0.115302	+	0.993331i	$0.536784\pi$
$38$	0	0
$39$	193.432	0.794204
$40$	0	0
$41$	−49.4929	−0.188524	−0.0942622	−	0.995547i	$-0.530049\pi$
$41$	−49.4929	−0.188524	−0.0942622	+	0.995547i	$0.530049\pi$
$42$	0	0
$43$	313.788	1.11284	0.556421	−	0.830901i	$-0.312174\pi$
$43$	313.788	1.11284	0.556421	+	0.830901i	$0.312174\pi$
$44$	0	0
$45$	−149.417	−0.494972
$46$	0	0
$47$	−553.259	−1.71705	−0.858523	−	0.512775i	$-0.828618\pi$
$47$	−553.259	−1.71705	−0.858523	+	0.512775i	$0.828618\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	397.287	1.09081
$52$	0	0
$53$	−619.868	−1.60652	−0.803259	−	0.595630i	$-0.796903\pi$
$53$	−619.868	−1.60652	−0.803259	+	0.595630i	$0.796903\pi$
$54$	0	0
$55$	817.611	2.00448
$56$	0	0
$57$	−246.181	−0.572062
$58$	0	0
$59$	−712.989	−1.57328	−0.786638	−	0.617414i	$-0.788180\pi$
$59$	−712.989	−1.57328	−0.786638	+	0.617414i	$0.788180\pi$
$60$	0	0
$61$	−287.421	−0.603287	−0.301644	−	0.953421i	$-0.597535\pi$
$61$	−287.421	−0.603287	−0.301644	+	0.953421i	$0.597535\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1070.45	−2.04265
$66$	0	0
$67$	226.636	0.413254	0.206627	−	0.978420i	$-0.433751\pi$
$67$	226.636	0.413254	0.206627	+	0.978420i	$0.433751\pi$
$68$	0	0
$69$	246.140	0.429446
$70$	0	0
$71$	55.3834	0.0925746	0.0462873	−	0.998928i	$-0.485261\pi$
$71$	55.3834	0.0925746	0.0462873	+	0.998928i	$0.485261\pi$
$72$	0	0
$73$	799.071	1.28115	0.640577	−	0.767894i	$-0.278695\pi$
$73$	799.071	1.28115	0.640577	+	0.767894i	$0.278695\pi$
$74$	0	0
$75$	451.867	0.695694
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−120.313	−0.171345	−0.0856725	−	0.996323i	$-0.527304\pi$
$79$	−120.313	−0.171345	−0.0856725	+	0.996323i	$0.527304\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−857.862	−1.13449	−0.567244	−	0.823550i	$-0.691991\pi$
$83$	−857.862	−1.13449	−0.567244	+	0.823550i	$0.691991\pi$
$84$	0	0
$85$	−2198.57	−2.80551
$86$	0	0
$87$	473.151	0.583071
$88$	0	0
$89$	−377.167	−0.449209	−0.224604	−	0.974450i	$-0.572109\pi$
$89$	−377.167	−0.449209	−0.224604	+	0.974450i	$0.572109\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−555.873	−0.619799
$94$	0	0
$95$	1362.36	1.47132
$96$	0	0
$97$	−1265.20	−1.32435	−0.662174	−	0.749351i	$-0.730366\pi$
$97$	−1265.20	−1.32435	−0.662174	+	0.749351i	$0.730366\pi$
$98$	0	0
$99$	−443.233	−0.449966
$100$	0	0
$101$	−2011.24	−1.98144	−0.990722	−	0.135906i	$-0.956605\pi$
$101$	−2011.24	−1.98144	−0.990722	+	0.135906i	$0.956605\pi$
$102$	0	0
$103$	−38.9204	−0.0372325	−0.0186162	−	0.999827i	$-0.505926\pi$
$103$	−38.9204	−0.0372325	−0.0186162	+	0.999827i	$0.505926\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1293.26	−1.16845	−0.584223	−	0.811593i	$-0.698601\pi$
$107$	−1293.26	−1.16845	−0.584223	+	0.811593i	$0.698601\pi$
$108$	0	0
$109$	1403.08	1.23294	0.616470	−	0.787378i	$-0.288562\pi$
$109$	1403.08	1.23294	0.616470	+	0.787378i	$0.288562\pi$
$110$	0	0
$111$	−155.701	−0.133139
$112$	0	0
$113$	−448.950	−0.373749	−0.186875	−	0.982384i	$-0.559836\pi$
$113$	−448.950	−0.373749	−0.186875	+	0.982384i	$0.559836\pi$
$114$	0	0
$115$	−1362.13	−1.10451
$116$	0	0
$117$	580.297	0.458534
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1094.38	0.822222
$122$	0	0
$123$	−148.479	−0.108845
$124$	0	0
$125$	−425.377	−0.304375
$126$	0	0
$127$	−634.373	−0.443240	−0.221620	−	0.975133i	$-0.571134\pi$
$127$	−634.373	−0.443240	−0.221620	+	0.975133i	$0.571134\pi$
$128$	0	0
$129$	941.363	0.642499
$130$	0	0
$131$	−627.531	−0.418532	−0.209266	−	0.977859i	$-0.567107\pi$
$131$	−627.531	−0.418532	−0.209266	+	0.977859i	$0.567107\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−448.251	−0.285772
$136$	0	0
$137$	1795.53	1.11973	0.559863	−	0.828585i	$-0.310854\pi$
$137$	1795.53	1.11973	0.559863	+	0.828585i	$0.310854\pi$
$138$	0	0
$139$	565.754	0.345228	0.172614	−	0.984990i	$-0.444779\pi$
$139$	565.754	0.345228	0.172614	+	0.984990i	$0.444779\pi$
$140$	0	0
$141$	−1659.78	−0.991337
$142$	0	0
$143$	−3175.39	−1.85692
$144$	0	0
$145$	−2618.40	−1.49963
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−583.578	−0.320863	−0.160432	−	0.987047i	$-0.551289\pi$
$149$	−583.578	−0.320863	−0.160432	+	0.987047i	$0.551289\pi$
$150$	0	0
$151$	874.848	0.471484	0.235742	−	0.971816i	$-0.424248\pi$
$151$	874.848	0.471484	0.235742	+	0.971816i	$0.424248\pi$
$152$	0	0
$153$	1191.86	0.629780
$154$	0	0
$155$	3076.18	1.59409
$156$	0	0
$157$	−203.881	−0.103640	−0.0518201	−	0.998656i	$-0.516502\pi$
$157$	−203.881	−0.103640	−0.0518201	+	0.998656i	$0.516502\pi$
$158$	0	0
$159$	−1859.61	−0.927524
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2329.09	−1.11919	−0.559596	−	0.828765i	$-0.689044\pi$
$163$	−2329.09	−1.11919	−0.559596	+	0.828765i	$0.689044\pi$
$164$	0	0
$165$	2452.83	1.15729
$166$	0	0
$167$	−2742.47	−1.27077	−0.635385	−	0.772196i	$-0.719159\pi$
$167$	−2742.47	−1.27077	−0.635385	+	0.772196i	$0.719159\pi$
$168$	0	0
$169$	1960.34	0.892279
$170$	0	0
$171$	−738.544	−0.330280
$172$	0	0
$173$	2797.24	1.22931	0.614654	−	0.788796i	$-0.289295\pi$
$173$	2797.24	1.22931	0.614654	+	0.788796i	$0.289295\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2138.97	−0.908332
$178$	0	0
$179$	−1363.49	−0.569340	−0.284670	−	0.958626i	$-0.591884\pi$
$179$	−1363.49	−0.569340	−0.284670	+	0.958626i	$0.591884\pi$
$180$	0	0
$181$	−1845.23	−0.757763	−0.378882	−	0.925445i	$-0.623691\pi$
$181$	−1845.23	−0.757763	−0.378882	+	0.925445i	$0.623691\pi$
$182$	0	0
$183$	−862.264	−0.348308
$184$	0	0
$185$	861.640	0.342427
$186$	0	0
$187$	−6521.88	−2.55041
$188$	0	0
$189$	0	0
$190$	0	0
$191$	496.967	0.188268	0.0941342	−	0.995560i	$-0.469992\pi$
$191$	496.967	0.188268	0.0941342	+	0.995560i	$0.469992\pi$
$192$	0	0
$193$	3290.12	1.22709	0.613544	−	0.789660i	$-0.289743\pi$
$193$	3290.12	1.22709	0.613544	+	0.789660i	$0.289743\pi$
$194$	0	0
$195$	−3211.34	−1.17933
$196$	0	0
$197$	−2989.83	−1.08130	−0.540651	−	0.841247i	$-0.681822\pi$
$197$	−2989.83	−1.08130	−0.540651	+	0.841247i	$0.681822\pi$
$198$	0	0
$199$	4746.57	1.69083	0.845417	−	0.534108i	$-0.179352\pi$
$199$	4746.57	1.69083	0.845417	+	0.534108i	$0.179352\pi$
$200$	0	0
$201$	679.909	0.238592
$202$	0	0
$203$	0	0
$204$	0	0
$205$	821.676	0.279943
$206$	0	0
$207$	738.419	0.247941
$208$	0	0
$209$	4041.32	1.33753
$210$	0	0
$211$	5034.43	1.64258	0.821289	−	0.570512i	$-0.193255\pi$
$211$	5034.43	1.64258	0.821289	+	0.570512i	$0.193255\pi$
$212$	0	0
$213$	166.150	0.0534480
$214$	0	0
$215$	−5209.46	−1.65248
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2397.21	0.739674
$220$	0	0
$221$	8538.68	2.59898
$222$	0	0
$223$	2448.32	0.735208	0.367604	−	0.929982i	$-0.380178\pi$
$223$	2448.32	0.735208	0.367604	+	0.929982i	$0.380178\pi$
$224$	0	0
$225$	1355.60	0.401659
$226$	0	0
$227$	3573.33	1.04480	0.522401	−	0.852700i	$-0.325036\pi$
$227$	3573.33	1.04480	0.522401	+	0.852700i	$0.325036\pi$
$228$	0	0
$229$	−3378.69	−0.974978	−0.487489	−	0.873129i	$-0.662087\pi$
$229$	−3378.69	−0.974978	−0.487489	+	0.873129i	$0.662087\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3130.51	0.880199	0.440100	−	0.897949i	$-0.354943\pi$
$233$	3130.51	0.880199	0.440100	+	0.897949i	$0.354943\pi$
$234$	0	0
$235$	9185.14	2.54967
$236$	0	0
$237$	−360.939	−0.0989261
$238$	0	0
$239$	−3964.01	−1.07285	−0.536424	−	0.843949i	$-0.680225\pi$
$239$	−3964.01	−1.07285	−0.536424	+	0.843949i	$0.680225\pi$
$240$	0	0
$241$	−3173.03	−0.848102	−0.424051	−	0.905638i	$-0.639392\pi$
$241$	−3173.03	−0.848102	−0.424051	+	0.905638i	$0.639392\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5291.05	−1.36300
$248$	0	0
$249$	−2573.59	−0.654997
$250$	0	0
$251$	−2509.92	−0.631174	−0.315587	−	0.948897i	$-0.602201\pi$
$251$	−2509.92	−0.631174	−0.315587	+	0.948897i	$0.602201\pi$
$252$	0	0
$253$	−4040.64	−1.00408
$254$	0	0
$255$	−6595.71	−1.61976
$256$	0	0
$257$	−5695.39	−1.38237	−0.691184	−	0.722679i	$-0.742911\pi$
$257$	−5695.39	−1.38237	−0.691184	+	0.722679i	$0.742911\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1419.45	0.336636
$262$	0	0
$263$	−6420.02	−1.50523	−0.752614	−	0.658462i	$-0.771208\pi$
$263$	−6420.02	−1.50523	−0.752614	+	0.658462i	$0.771208\pi$
$264$	0	0
$265$	10291.0	2.38555
$266$	0	0
$267$	−1131.50	−0.259351
$268$	0	0
$269$	−290.460	−0.0658351	−0.0329175	−	0.999458i	$-0.510480\pi$
$269$	−290.460	−0.0658351	−0.0329175	+	0.999458i	$0.510480\pi$
$270$	0	0
$271$	3681.21	0.825157	0.412578	−	0.910922i	$-0.364628\pi$
$271$	3681.21	0.825157	0.412578	+	0.910922i	$0.364628\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7417.86	−1.62660
$276$	0	0
$277$	−5427.03	−1.17718	−0.588590	−	0.808432i	$-0.700317\pi$
$277$	−5427.03	−1.17718	−0.588590	+	0.808432i	$0.700317\pi$
$278$	0	0
$279$	−1667.62	−0.357841
$280$	0	0
$281$	−5237.25	−1.11184	−0.555922	−	0.831235i	$-0.687634\pi$
$281$	−5237.25	−1.11184	−0.555922	+	0.831235i	$0.687634\pi$
$282$	0	0
$283$	−8302.83	−1.74400	−0.872000	−	0.489506i	$-0.837177\pi$
$283$	−8302.83	−1.74400	−0.872000	+	0.489506i	$0.837177\pi$
$284$	0	0
$285$	4087.07	0.849464
$286$	0	0
$287$	0	0
$288$	0	0
$289$	12624.5	2.56960
$290$	0	0
$291$	−3795.60	−0.764612
$292$	0	0
$293$	−2773.58	−0.553018	−0.276509	−	0.961011i	$-0.589178\pi$
$293$	−2773.58	−0.553018	−0.276509	+	0.961011i	$0.589178\pi$
$294$	0	0
$295$	11837.0	2.33618
$296$	0	0
$297$	−1329.70	−0.259788
$298$	0	0
$299$	5290.15	1.02320
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−6033.72	−1.14399
$304$	0	0
$305$	4771.73	0.895832
$306$	0	0
$307$	−6528.52	−1.21369	−0.606844	−	0.794821i	$-0.707565\pi$
$307$	−6528.52	−1.21369	−0.606844	+	0.794821i	$0.707565\pi$
$308$	0	0
$309$	−116.761	−0.0214962
$310$	0	0
$311$	−4350.89	−0.793300	−0.396650	−	0.917970i	$-0.629827\pi$
$311$	−4350.89	−0.793300	−0.396650	+	0.917970i	$0.629827\pi$
$312$	0	0
$313$	−5411.10	−0.977168	−0.488584	−	0.872517i	$-0.662486\pi$
$313$	−5411.10	−0.977168	−0.488584	+	0.872517i	$0.662486\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7013.79	−1.24269	−0.621346	−	0.783536i	$-0.713414\pi$
$317$	−7013.79	−1.24269	−0.621346	+	0.783536i	$0.713414\pi$
$318$	0	0
$319$	−7767.27	−1.36327
$320$	0	0
$321$	−3879.77	−0.674603
$322$	0	0
$323$	−10867.2	−1.87203
$324$	0	0
$325$	9711.73	1.65757
$326$	0	0
$327$	4209.23	0.711839
$328$	0	0
$329$	0	0
$330$	0	0
$331$	142.799	0.0237128	0.0118564	−	0.999930i	$-0.496226\pi$
$331$	142.799	0.0237128	0.0118564	+	0.999930i	$0.496226\pi$
$332$	0	0
$333$	−467.102	−0.0768679
$334$	0	0
$335$	−3762.59	−0.613648
$336$	0	0
$337$	−1489.10	−0.240702	−0.120351	−	0.992731i	$-0.538402\pi$
$337$	−1489.10	−0.240702	−0.120351	+	0.992731i	$0.538402\pi$
$338$	0	0
$339$	−1346.85	−0.215784
$340$	0	0
$341$	9125.23	1.44915
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4086.38	−0.637691
$346$	0	0
$347$	4017.06	0.621461	0.310730	−	0.950498i	$-0.399426\pi$
$347$	4017.06	0.621461	0.310730	+	0.950498i	$0.399426\pi$
$348$	0	0
$349$	−4235.87	−0.649687	−0.324844	−	0.945768i	$-0.605312\pi$
$349$	−4235.87	−0.649687	−0.324844	+	0.945768i	$0.605312\pi$
$350$	0	0
$351$	1740.89	0.264735
$352$	0	0
$353$	11399.1	1.71873	0.859366	−	0.511361i	$-0.170858\pi$
$353$	11399.1	1.71873	0.859366	+	0.511361i	$0.170858\pi$
$354$	0	0
$355$	−919.468	−0.137466
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−713.980	−0.104965	−0.0524825	−	0.998622i	$-0.516713\pi$
$359$	−713.980	−0.104965	−0.0524825	+	0.998622i	$0.516713\pi$
$360$	0	0
$361$	−125.080	−0.0182359
$362$	0	0
$363$	3283.13	0.474710
$364$	0	0
$365$	−13266.1	−1.90241
$366$	0	0
$367$	−12235.8	−1.74033	−0.870167	−	0.492757i	$-0.835989\pi$
$367$	−12235.8	−1.74033	−0.870167	+	0.492757i	$0.835989\pi$
$368$	0	0
$369$	−445.437	−0.0628415
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−9060.32	−1.25771	−0.628854	−	0.777523i	$-0.716476\pi$
$373$	−9060.32	−1.25771	−0.628854	+	0.777523i	$0.716476\pi$
$374$	0	0
$375$	−1276.13	−0.175731
$376$	0	0
$377$	10169.2	1.38923
$378$	0	0
$379$	10607.9	1.43770	0.718852	−	0.695163i	$-0.244668\pi$
$379$	10607.9	1.43770	0.718852	+	0.695163i	$0.244668\pi$
$380$	0	0
$381$	−1903.12	−0.255905
$382$	0	0
$383$	4281.14	0.571164	0.285582	−	0.958354i	$-0.407813\pi$
$383$	4281.14	0.571164	0.285582	+	0.958354i	$0.407813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2824.09	0.370947
$388$	0	0
$389$	−4334.27	−0.564926	−0.282463	−	0.959278i	$-0.591151\pi$
$389$	−4334.27	−0.564926	−0.282463	+	0.959278i	$0.591151\pi$
$390$	0	0
$391$	10865.4	1.40533
$392$	0	0
$393$	−1882.59	−0.241639
$394$	0	0
$395$	1997.42	0.254433
$396$	0	0
$397$	−12347.5	−1.56097	−0.780485	−	0.625175i	$-0.785028\pi$
$397$	−12347.5	−1.56097	−0.780485	+	0.625175i	$0.785028\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8321.66	1.03632	0.518159	−	0.855284i	$-0.326617\pi$
$401$	8321.66	1.03632	0.518159	+	0.855284i	$0.326617\pi$
$402$	0	0
$403$	−11947.1	−1.47674
$404$	0	0
$405$	−1344.75	−0.164991
$406$	0	0
$407$	2555.99	0.311291
$408$	0	0
$409$	15242.4	1.84276	0.921382	−	0.388659i	$-0.127062\pi$
$409$	15242.4	1.84276	0.921382	+	0.388659i	$0.127062\pi$
$410$	0	0
$411$	5386.59	0.646474
$412$	0	0
$413$	0	0
$414$	0	0
$415$	14242.1	1.68462
$416$	0	0
$417$	1697.26	0.199317
$418$	0	0
$419$	8251.84	0.962121	0.481061	−	0.876687i	$-0.340252\pi$
$419$	8251.84	0.962121	0.481061	+	0.876687i	$0.340252\pi$
$420$	0	0
$421$	−1825.97	−0.211383	−0.105691	−	0.994399i	$-0.533706\pi$
$421$	−1825.97	−0.211383	−0.105691	+	0.994399i	$0.533706\pi$
$422$	0	0
$423$	−4979.34	−0.572349
$424$	0	0
$425$	19946.8	2.27661
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9526.18	−1.07209
$430$	0	0
$431$	−11399.2	−1.27397	−0.636986	−	0.770876i	$-0.719819\pi$
$431$	−11399.2	−1.27397	−0.636986	+	0.770876i	$0.719819\pi$
$432$	0	0
$433$	2484.03	0.275692	0.137846	−	0.990454i	$-0.455982\pi$
$433$	2484.03	0.275692	0.137846	+	0.990454i	$0.455982\pi$
$434$	0	0
$435$	−7855.20	−0.865812
$436$	0	0
$437$	−6732.78	−0.737008
$438$	0	0
$439$	−11572.5	−1.25814	−0.629071	−	0.777348i	$-0.716565\pi$
$439$	−11572.5	−1.25814	−0.629071	+	0.777348i	$0.716565\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11072.6	1.18753	0.593766	−	0.804638i	$-0.297640\pi$
$443$	11072.6	1.18753	0.593766	+	0.804638i	$0.297640\pi$
$444$	0	0
$445$	6261.67	0.667038
$446$	0	0
$447$	−1750.73	−0.185250
$448$	0	0
$449$	12248.0	1.28735	0.643673	−	0.765300i	$-0.277410\pi$
$449$	12248.0	1.28735	0.643673	+	0.765300i	$0.277410\pi$
$450$	0	0
$451$	2437.43	0.254489
$452$	0	0
$453$	2624.54	0.272211
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12667.5	1.29663	0.648315	−	0.761372i	$-0.275474\pi$
$457$	12667.5	1.29663	0.648315	+	0.761372i	$0.275474\pi$
$458$	0	0
$459$	3575.58	0.363603
$460$	0	0
$461$	11444.0	1.15618	0.578089	−	0.815973i	$-0.303798\pi$
$461$	11444.0	1.15618	0.578089	+	0.815973i	$0.303798\pi$
$462$	0	0
$463$	5577.93	0.559888	0.279944	−	0.960016i	$-0.409684\pi$
$463$	5577.93	0.559888	0.279944	+	0.960016i	$0.409684\pi$
$464$	0	0
$465$	9228.53	0.920350
$466$	0	0
$467$	13494.5	1.33716	0.668578	−	0.743642i	$-0.266903\pi$
$467$	13494.5	1.33716	0.668578	+	0.743642i	$0.266903\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−611.644	−0.0598366
$472$	0	0
$473$	−15453.5	−1.50222
$474$	0	0
$475$	−12360.1	−1.19394
$476$	0	0
$477$	−5578.82	−0.535506
$478$	0	0
$479$	−5321.13	−0.507576	−0.253788	−	0.967260i	$-0.581677\pi$
$479$	−5321.13	−0.507576	−0.253788	+	0.967260i	$0.581677\pi$
$480$	0	0
$481$	−3346.39	−0.317219
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21004.7	1.96655
$486$	0	0
$487$	5915.38	0.550414	0.275207	−	0.961385i	$-0.411254\pi$
$487$	5915.38	0.550414	0.275207	+	0.961385i	$0.411254\pi$
$488$	0	0
$489$	−6987.27	−0.646166
$490$	0	0
$491$	20748.9	1.90710	0.953548	−	0.301240i	$-0.0974004\pi$
$491$	20748.9	1.90710	0.953548	+	0.301240i	$0.0974004\pi$
$492$	0	0
$493$	20886.3	1.90806
$494$	0	0
$495$	7358.50	0.668162
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10673.6	0.957546	0.478773	−	0.877939i	$-0.341082\pi$
$499$	10673.6	0.957546	0.478773	+	0.877939i	$0.341082\pi$
$500$	0	0
$501$	−8227.41	−0.733679
$502$	0	0
$503$	−18494.4	−1.63941	−0.819706	−	0.572784i	$-0.805863\pi$
$503$	−18494.4	−1.63941	−0.819706	+	0.572784i	$0.805863\pi$
$504$	0	0
$505$	33390.3	2.94228
$506$	0	0
$507$	5881.01	0.515158
$508$	0	0
$509$	−5766.36	−0.502140	−0.251070	−	0.967969i	$-0.580782\pi$
$509$	−5766.36	−0.502140	−0.251070	+	0.967969i	$0.580782\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2215.63	−0.190687
$514$	0	0
$515$	646.152	0.0552871
$516$	0	0
$517$	27247.0	2.31784
$518$	0	0
$519$	8391.73	0.709742
$520$	0	0
$521$	4394.77	0.369555	0.184778	−	0.982780i	$-0.440844\pi$
$521$	4394.77	0.369555	0.184778	+	0.982780i	$0.440844\pi$
$522$	0	0
$523$	1063.00	0.0888752	0.0444376	−	0.999012i	$-0.485850\pi$
$523$	1063.00	0.0888752	0.0444376	+	0.999012i	$0.485850\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24537.9	−2.02825
$528$	0	0
$529$	−5435.35	−0.446729
$530$	0	0
$531$	−6416.90	−0.524425
$532$	0	0
$533$	−3191.18	−0.259334
$534$	0	0
$535$	21470.5	1.73505
$536$	0	0
$537$	−4090.47	−0.328709
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4002.20	−0.318056	−0.159028	−	0.987274i	$-0.550836\pi$
$541$	−4002.20	−0.318056	−0.159028	+	0.987274i	$0.550836\pi$
$542$	0	0
$543$	−5535.70	−0.437495
$544$	0	0
$545$	−23293.7	−1.83081
$546$	0	0
$547$	13365.7	1.04475	0.522373	−	0.852717i	$-0.325047\pi$
$547$	13365.7	1.04475	0.522373	+	0.852717i	$0.325047\pi$
$548$	0	0
$549$	−2586.79	−0.201096
$550$	0	0
$551$	−12942.3	−1.00066
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2584.92	0.197700
$556$	0	0
$557$	5675.70	0.431754	0.215877	−	0.976421i	$-0.430739\pi$
$557$	5675.70	0.431754	0.215877	+	0.976421i	$0.430739\pi$
$558$	0	0
$559$	20232.2	1.53083
$560$	0	0
$561$	−19565.6	−1.47248
$562$	0	0
$563$	−281.060	−0.0210396	−0.0105198	−	0.999945i	$-0.503349\pi$
$563$	−281.060	−0.0210396	−0.0105198	+	0.999945i	$0.503349\pi$
$564$	0	0
$565$	7453.41	0.554987
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6736.73	0.496342	0.248171	−	0.968716i	$-0.420171\pi$
$569$	6736.73	0.496342	0.248171	+	0.968716i	$0.420171\pi$
$570$	0	0
$571$	−16445.8	−1.20532	−0.602659	−	0.797999i	$-0.705892\pi$
$571$	−16445.8	−1.20532	−0.602659	+	0.797999i	$0.705892\pi$
$572$	0	0
$573$	1490.90	0.108697
$574$	0	0
$575$	12358.0	0.896289
$576$	0	0
$577$	6097.01	0.439900	0.219950	−	0.975511i	$-0.429411\pi$
$577$	6097.01	0.439900	0.219950	+	0.975511i	$0.429411\pi$
$578$	0	0
$579$	9870.36	0.708460
$580$	0	0
$581$	0	0
$582$	0	0
$583$	30527.4	2.16863
$584$	0	0
$585$	−9634.01	−0.680884
$586$	0	0
$587$	−19531.2	−1.37332	−0.686659	−	0.726980i	$-0.740923\pi$
$587$	−19531.2	−1.37332	−0.686659	+	0.726980i	$0.740923\pi$
$588$	0	0
$589$	15205.1	1.06369
$590$	0	0
$591$	−8969.48	−0.624290
$592$	0	0
$593$	17701.7	1.22583	0.612917	−	0.790147i	$-0.289996\pi$
$593$	17701.7	1.22583	0.612917	+	0.790147i	$0.289996\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14239.7	0.976203
$598$	0	0
$599$	−16786.0	−1.14500	−0.572501	−	0.819904i	$-0.694027\pi$
$599$	−16786.0	−1.14500	−0.572501	+	0.819904i	$0.694027\pi$
$600$	0	0
$601$	6281.32	0.426324	0.213162	−	0.977017i	$-0.431624\pi$
$601$	6281.32	0.426324	0.213162	+	0.977017i	$0.431624\pi$
$602$	0	0
$603$	2039.73	0.137751
$604$	0	0
$605$	−18168.7	−1.22093
$606$	0	0
$607$	−2452.50	−0.163993	−0.0819967	−	0.996633i	$-0.526130\pi$
$607$	−2452.50	−0.163993	−0.0819967	+	0.996633i	$0.526130\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−35672.7	−2.36197
$612$	0	0
$613$	14792.7	0.974670	0.487335	−	0.873215i	$-0.337969\pi$
$613$	14792.7	0.974670	0.487335	+	0.873215i	$0.337969\pi$
$614$	0	0
$615$	2465.03	0.161625
$616$	0	0
$617$	−15044.2	−0.981614	−0.490807	−	0.871268i	$-0.663298\pi$
$617$	−15044.2	−0.981614	−0.490807	+	0.871268i	$0.663298\pi$
$618$	0	0
$619$	8948.32	0.581039	0.290520	−	0.956869i	$-0.406172\pi$
$619$	8948.32	0.581039	0.290520	+	0.956869i	$0.406172\pi$
$620$	0	0
$621$	2215.26	0.143149
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11765.7	−0.753006
$626$	0	0
$627$	12124.0	0.772225
$628$	0	0
$629$	−6873.09	−0.435688
$630$	0	0
$631$	−27769.9	−1.75198	−0.875991	−	0.482327i	$-0.839792\pi$
$631$	−27769.9	−1.75198	−0.875991	+	0.482327i	$0.839792\pi$
$632$	0	0
$633$	15103.3	0.948343
$634$	0	0
$635$	10531.8	0.658174
$636$	0	0
$637$	0	0
$638$	0	0
$639$	498.451	0.0308582
$640$	0	0
$641$	−15807.9	−0.974063	−0.487031	−	0.873384i	$-0.661920\pi$
$641$	−15807.9	−0.974063	−0.487031	+	0.873384i	$0.661920\pi$
$642$	0	0
$643$	21911.4	1.34386	0.671929	−	0.740616i	$-0.265466\pi$
$643$	21911.4	1.34386	0.671929	+	0.740616i	$0.265466\pi$
$644$	0	0
$645$	−15628.4	−0.954058
$646$	0	0
$647$	11245.6	0.683323	0.341662	−	0.939823i	$-0.389010\pi$
$647$	11245.6	0.683323	0.341662	+	0.939823i	$0.389010\pi$
$648$	0	0
$649$	35113.4	2.12376
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8668.43	0.519482	0.259741	−	0.965678i	$-0.416363\pi$
$653$	8668.43	0.519482	0.259741	+	0.965678i	$0.416363\pi$
$654$	0	0
$655$	10418.2	0.621484
$656$	0	0
$657$	7191.64	0.427051
$658$	0	0
$659$	−12625.9	−0.746334	−0.373167	−	0.927764i	$-0.621728\pi$
$659$	−12625.9	−0.746334	−0.373167	+	0.927764i	$0.621728\pi$
$660$	0	0
$661$	−1884.10	−0.110867	−0.0554334	−	0.998462i	$-0.517654\pi$
$661$	−1884.10	−0.110867	−0.0554334	+	0.998462i	$0.517654\pi$
$662$	0	0
$663$	25616.0	1.50052
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12940.2	0.751192
$668$	0	0
$669$	7344.95	0.424473
$670$	0	0
$671$	14155.0	0.814376
$672$	0	0
$673$	−25932.5	−1.48533	−0.742663	−	0.669665i	$-0.766438\pi$
$673$	−25932.5	−1.48533	−0.742663	+	0.669665i	$0.766438\pi$
$674$	0	0
$675$	4066.80	0.231898
$676$	0	0
$677$	18226.4	1.03471	0.517354	−	0.855772i	$-0.326917\pi$
$677$	18226.4	1.03471	0.517354	+	0.855772i	$0.326917\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	10720.0	0.603217
$682$	0	0
$683$	−10680.6	−0.598362	−0.299181	−	0.954196i	$-0.596713\pi$
$683$	−10680.6	−0.598362	−0.299181	+	0.954196i	$0.596713\pi$
$684$	0	0
$685$	−29809.2	−1.66270
$686$	0	0
$687$	−10136.1	−0.562904
$688$	0	0
$689$	−39967.5	−2.20993
$690$	0	0
$691$	−20500.4	−1.12862	−0.564308	−	0.825565i	$-0.690857\pi$
$691$	−20500.4	−1.12862	−0.564308	+	0.825565i	$0.690857\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9392.58	−0.512634
$696$	0	0
$697$	−6554.30	−0.356186
$698$	0	0
$699$	9391.53	0.508183
$700$	0	0
$701$	13903.9	0.749137	0.374568	−	0.927199i	$-0.377791\pi$
$701$	13903.9	0.749137	0.374568	+	0.927199i	$0.377791\pi$
$702$	0	0
$703$	4258.95	0.228491
$704$	0	0
$705$	27555.4	1.47205
$706$	0	0
$707$	0	0
$708$	0	0
$709$	18055.3	0.956391	0.478195	−	0.878253i	$-0.341291\pi$
$709$	18055.3	0.956391	0.478195	+	0.878253i	$0.341291\pi$
$710$	0	0
$711$	−1082.82	−0.0571150
$712$	0	0
$713$	−15202.5	−0.798510
$714$	0	0
$715$	52717.5	2.75737
$716$	0	0
$717$	−11892.0	−0.619409
$718$	0	0
$719$	9649.29	0.500498	0.250249	−	0.968182i	$-0.419488\pi$
$719$	9649.29	0.500498	0.250249	+	0.968182i	$0.419488\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9519.08	−0.489652
$724$	0	0
$725$	23755.7	1.21692
$726$	0	0
$727$	−12451.5	−0.635214	−0.317607	−	0.948222i	$-0.602879\pi$
$727$	−12451.5	−0.635214	−0.317607	+	0.948222i	$0.602879\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	41554.6	2.10253
$732$	0	0
$733$	−21799.1	−1.09845	−0.549227	−	0.835673i	$-0.685078\pi$
$733$	−21799.1	−1.09845	−0.549227	+	0.835673i	$0.685078\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11161.4	−0.557850
$738$	0	0
$739$	−15158.8	−0.754568	−0.377284	−	0.926098i	$-0.623142\pi$
$739$	−15158.8	−0.754568	−0.377284	+	0.926098i	$0.623142\pi$
$740$	0	0
$741$	−15873.1	−0.786929
$742$	0	0
$743$	5094.25	0.251534	0.125767	−	0.992060i	$-0.459861\pi$
$743$	5094.25	0.251534	0.125767	+	0.992060i	$0.459861\pi$
$744$	0	0
$745$	9688.49	0.476455
$746$	0	0
$747$	−7720.76	−0.378163
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−13213.6	−0.642038	−0.321019	−	0.947073i	$-0.604025\pi$
$751$	−13213.6	−0.642038	−0.321019	+	0.947073i	$0.604025\pi$
$752$	0	0
$753$	−7529.76	−0.364408
$754$	0	0
$755$	−14524.1	−0.700114
$756$	0	0
$757$	32248.9	1.54836	0.774180	−	0.632966i	$-0.218163\pi$
$757$	32248.9	1.54836	0.774180	+	0.632966i	$0.218163\pi$
$758$	0	0
$759$	−12121.9	−0.579707
$760$	0	0
$761$	15715.4	0.748596	0.374298	−	0.927308i	$-0.377884\pi$
$761$	15715.4	0.748596	0.374298	+	0.927308i	$0.377884\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−19787.1	−0.935170
$766$	0	0
$767$	−45971.7	−2.16420
$768$	0	0
$769$	24907.4	1.16799	0.583994	−	0.811758i	$-0.301489\pi$
$769$	24907.4	1.16799	0.583994	+	0.811758i	$0.301489\pi$
$770$	0	0
$771$	−17086.2	−0.798111
$772$	0	0
$773$	−22027.2	−1.02492	−0.512461	−	0.858710i	$-0.671266\pi$
$773$	−22027.2	−1.02492	−0.512461	+	0.858710i	$0.671266\pi$
$774$	0	0
$775$	−27908.9	−1.29357
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4061.41	0.186798
$780$	0	0
$781$	−2727.53	−0.124966
$782$	0	0
$783$	4258.36	0.194357
$784$	0	0
$785$	3384.81	0.153897
$786$	0	0
$787$	−38388.7	−1.73877	−0.869384	−	0.494137i	$-0.835484\pi$
$787$	−38388.7	−1.73877	−0.869384	+	0.494137i	$0.835484\pi$
$788$	0	0
$789$	−19260.0	−0.869044
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−18532.2	−0.829883
$794$	0	0
$795$	30872.9	1.37730
$796$	0	0
$797$	28628.1	1.27235	0.636174	−	0.771546i	$-0.280516\pi$
$797$	28628.1	1.27235	0.636174	+	0.771546i	$0.280516\pi$
$798$	0	0
$799$	−73267.6	−3.24408
$800$	0	0
$801$	−3394.50	−0.149736
$802$	0	0
$803$	−39352.8	−1.72943
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−871.379	−0.0380099
$808$	0	0
$809$	−11542.2	−0.501608	−0.250804	−	0.968038i	$-0.580695\pi$
$809$	−11542.2	−0.501608	−0.250804	+	0.968038i	$0.580695\pi$
$810$	0	0
$811$	−29374.0	−1.27184	−0.635919	−	0.771756i	$-0.719379\pi$
$811$	−29374.0	−1.27184	−0.635919	+	0.771756i	$0.719379\pi$
$812$	0	0
$813$	11043.6	0.476404
$814$	0	0
$815$	38667.2	1.66191
$816$	0	0
$817$	−25749.6	−1.10265
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17945.2	−0.762839	−0.381419	−	0.924402i	$-0.624565\pi$
$821$	−17945.2	−0.762839	−0.381419	+	0.924402i	$0.624565\pi$
$822$	0	0
$823$	42524.3	1.80110	0.900549	−	0.434755i	$-0.143165\pi$
$823$	42524.3	1.80110	0.900549	+	0.434755i	$0.143165\pi$
$824$	0	0
$825$	−22253.6	−0.939116
$826$	0	0
$827$	−43291.0	−1.82029	−0.910143	−	0.414293i	$-0.864029\pi$
$827$	−43291.0	−1.82029	−0.910143	+	0.414293i	$0.864029\pi$
$828$	0	0
$829$	−28580.9	−1.19741	−0.598707	−	0.800968i	$-0.704319\pi$
$829$	−28580.9	−1.19741	−0.598707	+	0.800968i	$0.704319\pi$
$830$	0	0
$831$	−16281.1	−0.679645
$832$	0	0
$833$	0	0
$834$	0	0
$835$	45530.1	1.88699
$836$	0	0
$837$	−5002.85	−0.206600
$838$	0	0
$839$	11526.0	0.474281	0.237141	−	0.971475i	$-0.423790\pi$
$839$	11526.0	0.474281	0.237141	+	0.971475i	$0.423790\pi$
$840$	0	0
$841$	485.697	0.0199146
$842$	0	0
$843$	−15711.7	−0.641923
$844$	0	0
$845$	−32545.3	−1.32496
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−24908.5	−1.00690
$850$	0	0
$851$	−4258.23	−0.171528
$852$	0	0
$853$	25692.6	1.03130	0.515650	−	0.856799i	$-0.327551\pi$
$853$	25692.6	1.03130	0.515650	+	0.856799i	$0.327551\pi$
$854$	0	0
$855$	12261.2	0.490438
$856$	0	0
$857$	−33824.0	−1.34820	−0.674100	−	0.738640i	$-0.735468\pi$
$857$	−33824.0	−1.34820	−0.674100	+	0.738640i	$0.735468\pi$
$858$	0	0
$859$	13254.7	0.526479	0.263240	−	0.964730i	$-0.415209\pi$
$859$	13254.7	0.526479	0.263240	+	0.964730i	$0.415209\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5166.82	0.203801	0.101901	−	0.994795i	$-0.467508\pi$
$863$	5166.82	0.203801	0.101901	+	0.994795i	$0.467508\pi$
$864$	0	0
$865$	−46439.5	−1.82542
$866$	0	0
$867$	37873.4	1.48356
$868$	0	0
$869$	5925.18	0.231298
$870$	0	0
$871$	14612.9	0.568473
$872$	0	0
$873$	−11386.8	−0.441449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−25268.0	−0.972907	−0.486453	−	0.873707i	$-0.661710\pi$
$877$	−25268.0	−0.972907	−0.486453	+	0.873707i	$0.661710\pi$
$878$	0	0
$879$	−8320.74	−0.319285
$880$	0	0
$881$	42652.8	1.63111	0.815555	−	0.578680i	$-0.196432\pi$
$881$	42652.8	1.63111	0.815555	+	0.578680i	$0.196432\pi$
$882$	0	0
$883$	31070.9	1.18416	0.592082	−	0.805877i	$-0.298306\pi$
$883$	31070.9	1.18416	0.592082	+	0.805877i	$0.298306\pi$
$884$	0	0
$885$	35510.9	1.34880
$886$	0	0
$887$	36035.6	1.36410	0.682050	−	0.731306i	$-0.261089\pi$
$887$	36035.6	1.36410	0.682050	+	0.731306i	$0.261089\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3989.10	−0.149989
$892$	0	0
$893$	45400.7	1.70132
$894$	0	0
$895$	22636.5	0.845423
$896$	0	0
$897$	15870.5	0.590746
$898$	0	0
$899$	−29223.5	−1.08416
$900$	0	0
$901$	−82088.6	−3.03526
$902$	0	0
$903$	0	0
$904$	0	0
$905$	30634.3	1.12522
$906$	0	0
$907$	17945.5	0.656967	0.328484	−	0.944510i	$-0.393462\pi$
$907$	17945.5	0.656967	0.328484	+	0.944510i	$0.393462\pi$
$908$	0	0
$909$	−18101.2	−0.660481
$910$	0	0
$911$	17672.5	0.642717	0.321359	−	0.946958i	$-0.395861\pi$
$911$	17672.5	0.642717	0.321359	+	0.946958i	$0.395861\pi$
$912$	0	0
$913$	42248.1	1.53144
$914$	0	0
$915$	14315.2	0.517209
$916$	0	0
$917$	0	0
$918$	0	0
$919$	30422.3	1.09199	0.545995	−	0.837788i	$-0.316152\pi$
$919$	30422.3	1.09199	0.545995	+	0.837788i	$0.316152\pi$
$920$	0	0
$921$	−19585.6	−0.700723
$922$	0	0
$923$	3570.98	0.127346
$924$	0	0
$925$	−7817.32	−0.277872
$926$	0	0
$927$	−350.284	−0.0124108
$928$	0	0
$929$	16017.6	0.565683	0.282841	−	0.959167i	$-0.408723\pi$
$929$	16017.6	0.565683	0.282841	+	0.959167i	$0.408723\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−13052.7	−0.458012
$934$	0	0
$935$	108275.	3.78715
$936$	0	0
$937$	−7370.43	−0.256970	−0.128485	−	0.991711i	$-0.541011\pi$
$937$	−7370.43	−0.256970	−0.128485	+	0.991711i	$0.541011\pi$
$938$	0	0
$939$	−16233.3	−0.564168
$940$	0	0
$941$	14999.7	0.519634	0.259817	−	0.965658i	$-0.416338\pi$
$941$	14999.7	0.519634	0.259817	+	0.965658i	$0.416338\pi$
$942$	0	0
$943$	−4060.73	−0.140229
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12699.6	0.435776	0.217888	−	0.975974i	$-0.430083\pi$
$947$	12699.6	0.435776	0.217888	+	0.975974i	$0.430083\pi$
$948$	0	0
$949$	51522.0	1.76236
$950$	0	0
$951$	−21041.4	−0.717469
$952$	0	0
$953$	5347.41	0.181762	0.0908811	−	0.995862i	$-0.471032\pi$
$953$	5347.41	0.181762	0.0908811	+	0.995862i	$0.471032\pi$
$954$	0	0
$955$	−8250.59	−0.279563
$956$	0	0
$957$	−23301.8	−0.787086
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4541.70	0.152452
$962$	0	0
$963$	−11639.3	−0.389482
$964$	0	0
$965$	−54622.2	−1.82212
$966$	0	0
$967$	38787.2	1.28988	0.644939	−	0.764234i	$-0.276883\pi$
$967$	38787.2	1.28988	0.644939	+	0.764234i	$0.276883\pi$
$968$	0	0
$969$	−32601.6	−1.08082
$970$	0	0
$971$	−28064.9	−0.927545	−0.463772	−	0.885954i	$-0.653504\pi$
$971$	−28064.9	−0.927545	−0.463772	+	0.885954i	$0.653504\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	29135.2	0.956998
$976$	0	0
$977$	20896.8	0.684287	0.342144	−	0.939648i	$-0.388847\pi$
$977$	20896.8	0.684287	0.342144	+	0.939648i	$0.388847\pi$
$978$	0	0
$979$	18574.8	0.606386
$980$	0	0
$981$	12627.7	0.410980
$982$	0	0
$983$	−1791.08	−0.0581147	−0.0290573	−	0.999578i	$-0.509251\pi$
$983$	−1791.08	−0.0581147	−0.0290573	+	0.999578i	$0.509251\pi$
$984$	0	0
$985$	49636.7	1.60564
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25745.2	0.827755
$990$	0	0
$991$	22843.9	0.732251	0.366126	−	0.930565i	$-0.380684\pi$
$991$	22843.9	0.732251	0.366126	+	0.930565i	$0.380684\pi$
$992$	0	0
$993$	428.396	0.0136906
$994$	0	0
$995$	−78802.0	−2.51075
$996$	0	0
$997$	−12875.8	−0.409007	−0.204504	−	0.978866i	$-0.565558\pi$
$997$	−12875.8	−0.409007	−0.204504	+	0.978866i	$0.565558\pi$
$998$	0	0
$999$	−1401.30	−0.0443797

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.4.a.bc.1.1	yes	4
4.3	odd	2		2352.4.a.ck.1.1		4
7.6	odd	2		1176.4.a.bb.1.4	✓	4
28.27	even	2		2352.4.a.cr.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.bb.1.4	✓	4	7.6	odd	2
1176.4.a.bc.1.1	yes	4	1.1	even	1	trivial
2352.4.a.ck.1.1		4	4.3	odd	2
2352.4.a.cr.1.4		4	28.27	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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1176.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -16.6019 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -16.6019 q^{5} +9.00000 q^{9} -49.2481 q^{11} +64.4774 q^{13} -49.8056 q^{15} +132.429 q^{17} -82.0605 q^{19} +82.0466 q^{23} +150.622 q^{25} +27.0000 q^{27} +157.717 q^{29} -185.291 q^{31} -147.744 q^{33} -51.9002 q^{37} +193.432 q^{39} -49.4929 q^{41} +313.788 q^{43} -149.417 q^{45} -553.259 q^{47} +397.287 q^{51} -619.868 q^{53} +817.611 q^{55} -246.181 q^{57} -712.989 q^{59} -287.421 q^{61} -1070.45 q^{65} +226.636 q^{67} +246.140 q^{69} +55.3834 q^{71} +799.071 q^{73} +451.867 q^{75} -120.313 q^{79} +81.0000 q^{81} -857.862 q^{83} -2198.57 q^{85} +473.151 q^{87} -377.167 q^{89} -555.873 q^{93} +1362.36 q^{95} -1265.20 q^{97} -443.233 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} - 8 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 - 8 * q^5 + 36 * q^9	\( 4 q + 12 q^{3} - 8 q^{5} + 36 q^{9} - 40 q^{11} - 48 q^{13} - 24 q^{15} + 152 q^{17} - 224 q^{19} - 8 q^{23} - 28 q^{25} + 108 q^{27} - 144 q^{29} - 400 q^{31} - 120 q^{33} - 304 q^{37} - 144 q^{39} + 152 q^{41} + 160 q^{43} - 72 q^{45} - 544 q^{47} + 456 q^{51} - 1320 q^{53} - 16 q^{55} - 672 q^{57} - 1040 q^{59} - 896 q^{61} - 648 q^{65} - 416 q^{67} - 24 q^{69} + 248 q^{71} + 752 q^{73} - 84 q^{75} + 864 q^{79} + 324 q^{81} - 1456 q^{83} - 1608 q^{85} - 432 q^{87} - 2936 q^{89} - 1200 q^{93} - 80 q^{95} - 144 q^{97} - 360 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 - 8 * q^5 + 36 * q^9 - 40 * q^11 - 48 * q^13 - 24 * q^15 + 152 * q^17 - 224 * q^19 - 8 * q^23 - 28 * q^25 + 108 * q^27 - 144 * q^29 - 400 * q^31 - 120 * q^33 - 304 * q^37 - 144 * q^39 + 152 * q^41 + 160 * q^43 - 72 * q^45 - 544 * q^47 + 456 * q^51 - 1320 * q^53 - 16 * q^55 - 672 * q^57 - 1040 * q^59 - 896 * q^61 - 648 * q^65 - 416 * q^67 - 24 * q^69 + 248 * q^71 + 752 * q^73 - 84 * q^75 + 864 * q^79 + 324 * q^81 - 1456 * q^83 - 1608 * q^85 - 432 * q^87 - 2936 * q^89 - 1200 * q^93 - 80 * q^95 - 144 * q^97 - 360 * q^99

Introduction

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