LMFDB - Embedded newform 1176.4.a.u.1.2

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{505}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 126 $ x^2 - x - 126
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-10.7361$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +6.73610 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +6.73610 q^{5} +9.00000 q^{9} -53.6805 q^{11} -1.73610 q^{13} +20.2083 q^{15} +46.9444 q^{17} +20.1527 q^{19} +118.944 q^{23} -79.6249 q^{25} +27.0000 q^{27} -103.681 q^{29} +157.528 q^{31} -161.042 q^{33} -37.7361 q^{37} -5.20831 q^{39} +287.250 q^{41} +504.875 q^{43} +60.6249 q^{45} +220.500 q^{47} +140.833 q^{51} +292.319 q^{53} -361.597 q^{55} +60.4582 q^{57} +595.875 q^{59} +265.389 q^{61} -11.6946 q^{65} +936.735 q^{67} +356.833 q^{69} -545.944 q^{71} -299.374 q^{73} -238.875 q^{75} -940.333 q^{79} +81.0000 q^{81} -611.319 q^{83} +316.222 q^{85} -311.042 q^{87} +1176.19 q^{89} +472.583 q^{93} +135.751 q^{95} -1482.49 q^{97} -483.125 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 9 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 9 * q^5 + 18 * q^9	$ 2 q + 6 q^{3} - 9 q^{5} + 18 q^{9} + 5 q^{11} + 19 q^{13} - 27 q^{15} + 4 q^{17} - 117 q^{19} + 148 q^{23} + 43 q^{25} + 54 q^{27} - 95 q^{29} + 360 q^{31} + 15 q^{33} - 53 q^{37} + 57 q^{39} + 170 q^{41} + 403 q^{43} - 81 q^{45} - 368 q^{47} + 12 q^{51} + 697 q^{53} - 1285 q^{55} - 351 q^{57} + 585 q^{59} + 1160 q^{61} - 338 q^{65} + 233 q^{67} + 444 q^{69} + 616 q^{71} + 817 q^{73} + 129 q^{75} - 802 q^{79} + 162 q^{81} + 283 q^{83} + 992 q^{85} - 285 q^{87} + 1858 q^{89} + 1080 q^{93} + 2294 q^{95} - 1729 q^{97} + 45 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 9 * q^5 + 18 * q^9 + 5 * q^11 + 19 * q^13 - 27 * q^15 + 4 * q^17 - 117 * q^19 + 148 * q^23 + 43 * q^25 + 54 * q^27 - 95 * q^29 + 360 * q^31 + 15 * q^33 - 53 * q^37 + 57 * q^39 + 170 * q^41 + 403 * q^43 - 81 * q^45 - 368 * q^47 + 12 * q^51 + 697 * q^53 - 1285 * q^55 - 351 * q^57 + 585 * q^59 + 1160 * q^61 - 338 * q^65 + 233 * q^67 + 444 * q^69 + 616 * q^71 + 817 * q^73 + 129 * q^75 - 802 * q^79 + 162 * q^81 + 283 * q^83 + 992 * q^85 - 285 * q^87 + 1858 * q^89 + 1080 * q^93 + 2294 * q^95 - 1729 * q^97 + 45 * 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
$5$	6.73610	0.602495	0.301248	−	0.953546i	$-0.402597\pi$
$5$	6.73610	0.602495	0.301248	+	0.953546i	$0.402597\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−53.6805	−1.47139	−0.735695	−	0.677313i	$-0.763144\pi$
$11$	−53.6805	−1.47139	−0.735695	+	0.677313i	$0.763144\pi$
$12$	0	0
$13$	−1.73610	−0.0370391	−0.0185195	−	0.999828i	$-0.505895\pi$
$13$	−1.73610	−0.0370391	−0.0185195	+	0.999828i	$0.505895\pi$
$14$	0	0
$15$	20.2083	0.347851
$16$	0	0
$17$	46.9444	0.669747	0.334873	−	0.942263i	$-0.391306\pi$
$17$	46.9444	0.669747	0.334873	+	0.942263i	$0.391306\pi$
$18$	0	0
$19$	20.1527	0.243334	0.121667	−	0.992571i	$-0.461176\pi$
$19$	20.1527	0.243334	0.121667	+	0.992571i	$0.461176\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	118.944	1.07833	0.539166	−	0.842200i	$-0.318740\pi$
$23$	118.944	1.07833	0.539166	+	0.842200i	$0.318740\pi$
$24$	0	0
$25$	−79.6249	−0.636999
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−103.681	−0.663896	−0.331948	−	0.943298i	$-0.607706\pi$
$29$	−103.681	−0.663896	−0.331948	+	0.943298i	$0.607706\pi$
$30$	0	0
$31$	157.528	0.912672	0.456336	−	0.889808i	$-0.349162\pi$
$31$	157.528	0.912672	0.456336	+	0.889808i	$0.349162\pi$
$32$	0	0
$33$	−161.042	−0.849507
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−37.7361	−0.167670	−0.0838348	−	0.996480i	$-0.526717\pi$
$37$	−37.7361	−0.167670	−0.0838348	+	0.996480i	$0.526717\pi$
$38$	0	0
$39$	−5.20831	−0.0213845
$40$	0	0
$41$	287.250	1.09417	0.547084	−	0.837078i	$-0.315738\pi$
$41$	287.250	1.09417	0.547084	+	0.837078i	$0.315738\pi$
$42$	0	0
$43$	504.875	1.79053	0.895264	−	0.445537i	$-0.146987\pi$
$43$	504.875	1.79053	0.895264	+	0.445537i	$0.146987\pi$
$44$	0	0
$45$	60.6249	0.200832
$46$	0	0
$47$	220.500	0.684323	0.342162	−	0.939641i	$-0.388841\pi$
$47$	220.500	0.684323	0.342162	+	0.939641i	$0.388841\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	140.833	0.386678
$52$	0	0
$53$	292.319	0.757607	0.378803	−	0.925477i	$-0.376336\pi$
$53$	292.319	0.757607	0.378803	+	0.925477i	$0.376336\pi$
$54$	0	0
$55$	−361.597	−0.886505
$56$	0	0
$57$	60.4582	0.140489
$58$	0	0
$59$	595.875	1.31485	0.657426	−	0.753519i	$-0.271645\pi$
$59$	595.875	1.31485	0.657426	+	0.753519i	$0.271645\pi$
$60$	0	0
$61$	265.389	0.557043	0.278521	−	0.960430i	$-0.410156\pi$
$61$	265.389	0.557043	0.278521	+	0.960430i	$0.410156\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11.6946	−0.0223159
$66$	0	0
$67$	936.735	1.70807	0.854033	−	0.520218i	$-0.174149\pi$
$67$	936.735	1.70807	0.854033	+	0.520218i	$0.174149\pi$
$68$	0	0
$69$	356.833	0.622575
$70$	0	0
$71$	−545.944	−0.912558	−0.456279	−	0.889837i	$-0.650818\pi$
$71$	−545.944	−0.912558	−0.456279	+	0.889837i	$0.650818\pi$
$72$	0	0
$73$	−299.374	−0.479988	−0.239994	−	0.970774i	$-0.577145\pi$
$73$	−299.374	−0.479988	−0.239994	+	0.970774i	$0.577145\pi$
$74$	0	0
$75$	−238.875	−0.367772
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−940.333	−1.33919	−0.669593	−	0.742728i	$-0.733532\pi$
$79$	−940.333	−1.33919	−0.669593	+	0.742728i	$0.733532\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−611.319	−0.808445	−0.404223	−	0.914661i	$-0.632458\pi$
$83$	−611.319	−0.808445	−0.404223	+	0.914661i	$0.632458\pi$
$84$	0	0
$85$	316.222	0.403519
$86$	0	0
$87$	−311.042	−0.383301
$88$	0	0
$89$	1176.19	1.40086	0.700429	−	0.713722i	$-0.252992\pi$
$89$	1176.19	1.40086	0.700429	+	0.713722i	$0.252992\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	472.583	0.526931
$94$	0	0
$95$	135.751	0.146608
$96$	0	0
$97$	−1482.49	−1.55179	−0.775895	−	0.630862i	$-0.782701\pi$
$97$	−1482.49	−1.55179	−0.775895	+	0.630862i	$0.782701\pi$
$98$	0	0
$99$	−483.125	−0.490463
$100$	0	0
$101$	−617.944	−0.608789	−0.304395	−	0.952546i	$-0.598454\pi$
$101$	−617.944	−0.608789	−0.304395	+	0.952546i	$0.598454\pi$
$102$	0	0
$103$	816.764	0.781341	0.390670	−	0.920531i	$-0.372243\pi$
$103$	816.764	0.781341	0.390670	+	0.920531i	$0.372243\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1685.38	1.52272	0.761361	−	0.648328i	$-0.224531\pi$
$107$	1685.38	1.52272	0.761361	+	0.648328i	$0.224531\pi$
$108$	0	0
$109$	91.5968	0.0804898	0.0402449	−	0.999190i	$-0.487186\pi$
$109$	91.5968	0.0804898	0.0402449	+	0.999190i	$0.487186\pi$
$110$	0	0
$111$	−113.208	−0.0968041
$112$	0	0
$113$	614.194	0.511314	0.255657	−	0.966767i	$-0.417708\pi$
$113$	614.194	0.511314	0.255657	+	0.966767i	$0.417708\pi$
$114$	0	0
$115$	801.222	0.649690
$116$	0	0
$117$	−15.6249	−0.0123464
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1550.60	1.16499
$122$	0	0
$123$	861.750	0.631718
$124$	0	0
$125$	−1378.37	−0.986284
$126$	0	0
$127$	234.278	0.163691	0.0818457	−	0.996645i	$-0.473919\pi$
$127$	234.278	0.163691	0.0818457	+	0.996645i	$0.473919\pi$
$128$	0	0
$129$	1514.62	1.03376
$130$	0	0
$131$	−2664.76	−1.77726	−0.888631	−	0.458622i	$-0.848343\pi$
$131$	−2664.76	−1.77726	−0.888631	+	0.458622i	$0.848343\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	181.875	0.115950
$136$	0	0
$137$	−1046.31	−0.652496	−0.326248	−	0.945284i	$-0.605784\pi$
$137$	−1046.31	−0.652496	−0.326248	+	0.945284i	$0.605784\pi$
$138$	0	0
$139$	−647.265	−0.394966	−0.197483	−	0.980306i	$-0.563277\pi$
$139$	−647.265	−0.394966	−0.197483	+	0.980306i	$0.563277\pi$
$140$	0	0
$141$	661.499	0.395094
$142$	0	0
$143$	93.1949	0.0544989
$144$	0	0
$145$	−698.403	−0.399994
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1326.81	0.729505	0.364752	−	0.931105i	$-0.381154\pi$
$149$	1326.81	0.729505	0.364752	+	0.931105i	$0.381154\pi$
$150$	0	0
$151$	3683.24	1.98502	0.992508	−	0.122178i	$-0.0389879\pi$
$151$	3683.24	1.98502	0.992508	+	0.122178i	$0.0389879\pi$
$152$	0	0
$153$	422.500	0.223249
$154$	0	0
$155$	1061.12	0.549881
$156$	0	0
$157$	2818.25	1.43262	0.716308	−	0.697784i	$-0.245831\pi$
$157$	2818.25	1.43262	0.716308	+	0.697784i	$0.245831\pi$
$158$	0	0
$159$	876.958	0.437405
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4106.61	1.97334	0.986670	−	0.162733i	$-0.0520308\pi$
$163$	4106.61	1.97334	0.986670	+	0.162733i	$0.0520308\pi$
$164$	0	0
$165$	−1084.79	−0.511824
$166$	0	0
$167$	−1045.56	−0.484476	−0.242238	−	0.970217i	$-0.577881\pi$
$167$	−1045.56	−0.484476	−0.242238	+	0.970217i	$0.577881\pi$
$168$	0	0
$169$	−2193.99	−0.998628
$170$	0	0
$171$	181.374	0.0811114
$172$	0	0
$173$	1910.31	0.839525	0.419763	−	0.907634i	$-0.362113\pi$
$173$	1910.31	0.839525	0.419763	+	0.907634i	$0.362113\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1787.62	0.759130
$178$	0	0
$179$	231.140	0.0965151	0.0482576	−	0.998835i	$-0.484633\pi$
$179$	231.140	0.0965151	0.0482576	+	0.998835i	$0.484633\pi$
$180$	0	0
$181$	−3308.40	−1.35863	−0.679314	−	0.733848i	$-0.737722\pi$
$181$	−3308.40	−1.35863	−0.679314	+	0.733848i	$0.737722\pi$
$182$	0	0
$183$	796.167	0.321609
$184$	0	0
$185$	−254.194	−0.101020
$186$	0	0
$187$	−2520.00	−0.985458
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1800.75	0.682187	0.341093	−	0.940029i	$-0.389203\pi$
$191$	1800.75	0.682187	0.341093	+	0.940029i	$0.389203\pi$
$192$	0	0
$193$	−1283.36	−0.478644	−0.239322	−	0.970940i	$-0.576925\pi$
$193$	−1283.36	−0.478644	−0.239322	+	0.970940i	$0.576925\pi$
$194$	0	0
$195$	−35.0837	−0.0128841
$196$	0	0
$197$	60.7514	0.0219714	0.0109857	−	0.999940i	$-0.496503\pi$
$197$	60.7514	0.0219714	0.0109857	+	0.999940i	$0.496503\pi$
$198$	0	0
$199$	−3860.39	−1.37515	−0.687577	−	0.726112i	$-0.741325\pi$
$199$	−3860.39	−1.37515	−0.687577	+	0.726112i	$0.741325\pi$
$200$	0	0
$201$	2810.21	0.986153
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1934.94	0.659231
$206$	0	0
$207$	1070.50	0.359444
$208$	0	0
$209$	−1081.81	−0.358039
$210$	0	0
$211$	491.637	0.160406	0.0802031	−	0.996779i	$-0.474443\pi$
$211$	491.637	0.160406	0.0802031	+	0.996779i	$0.474443\pi$
$212$	0	0
$213$	−1637.83	−0.526866
$214$	0	0
$215$	3400.89	1.07878
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−898.123	−0.277121
$220$	0	0
$221$	−81.5003	−0.0248068
$222$	0	0
$223$	−823.376	−0.247253	−0.123626	−	0.992329i	$-0.539452\pi$
$223$	−823.376	−0.247253	−0.123626	+	0.992329i	$0.539452\pi$
$224$	0	0
$225$	−716.624	−0.212333
$226$	0	0
$227$	3758.87	1.09905	0.549527	−	0.835476i	$-0.314808\pi$
$227$	3758.87	1.09905	0.549527	+	0.835476i	$0.314808\pi$
$228$	0	0
$229$	2440.10	0.704132	0.352066	−	0.935975i	$-0.385479\pi$
$229$	2440.10	0.704132	0.352066	+	0.935975i	$0.385479\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1949.86	−0.548239	−0.274120	−	0.961696i	$-0.588386\pi$
$233$	−1949.86	−0.548239	−0.274120	+	0.961696i	$0.588386\pi$
$234$	0	0
$235$	1485.31	0.412301
$236$	0	0
$237$	−2821.00	−0.773180
$238$	0	0
$239$	2705.64	0.732273	0.366137	−	0.930561i	$-0.380680\pi$
$239$	2705.64	0.732273	0.366137	+	0.930561i	$0.380680\pi$
$240$	0	0
$241$	−604.211	−0.161496	−0.0807482	−	0.996735i	$-0.525731\pi$
$241$	−604.211	−0.161496	−0.0807482	+	0.996735i	$0.525731\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−34.9872	−0.00901288
$248$	0	0
$249$	−1833.96	−0.466756
$250$	0	0
$251$	1631.82	0.410356	0.205178	−	0.978725i	$-0.434223\pi$
$251$	1631.82	0.410356	0.205178	+	0.978725i	$0.434223\pi$
$252$	0	0
$253$	−6385.00	−1.58665
$254$	0	0
$255$	948.667	0.232972
$256$	0	0
$257$	7106.55	1.72488	0.862441	−	0.506158i	$-0.168935\pi$
$257$	7106.55	1.72488	0.862441	+	0.506158i	$0.168935\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−933.125	−0.221299
$262$	0	0
$263$	5570.25	1.30599	0.652996	−	0.757361i	$-0.273512\pi$
$263$	5570.25	1.30599	0.652996	+	0.757361i	$0.273512\pi$
$264$	0	0
$265$	1969.09	0.456455
$266$	0	0
$267$	3528.58	0.808786
$268$	0	0
$269$	−3913.18	−0.886955	−0.443477	−	0.896286i	$-0.646255\pi$
$269$	−3913.18	−0.886955	−0.443477	+	0.896286i	$0.646255\pi$
$270$	0	0
$271$	−6072.51	−1.36118	−0.680588	−	0.732666i	$-0.738276\pi$
$271$	−6072.51	−1.36118	−0.680588	+	0.732666i	$0.738276\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4274.31	0.937274
$276$	0	0
$277$	6140.62	1.33196	0.665982	−	0.745968i	$-0.268013\pi$
$277$	6140.62	1.33196	0.665982	+	0.745968i	$0.268013\pi$
$278$	0	0
$279$	1417.75	0.304224
$280$	0	0
$281$	−365.751	−0.0776472	−0.0388236	−	0.999246i	$-0.512361\pi$
$281$	−365.751	−0.0776472	−0.0388236	+	0.999246i	$0.512361\pi$
$282$	0	0
$283$	710.348	0.149208	0.0746039	−	0.997213i	$-0.476231\pi$
$283$	710.348	0.149208	0.0746039	+	0.997213i	$0.476231\pi$
$284$	0	0
$285$	407.252	0.0846440
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2709.22	−0.551440
$290$	0	0
$291$	−4447.46	−0.895926
$292$	0	0
$293$	−296.373	−0.0590932	−0.0295466	−	0.999563i	$-0.509406\pi$
$293$	−296.373	−0.0590932	−0.0295466	+	0.999563i	$0.509406\pi$
$294$	0	0
$295$	4013.87	0.792192
$296$	0	0
$297$	−1449.37	−0.283169
$298$	0	0
$299$	−206.500	−0.0399404
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1853.83	−0.351485
$304$	0	0
$305$	1787.69	0.335616
$306$	0	0
$307$	−8596.87	−1.59821	−0.799103	−	0.601194i	$-0.794692\pi$
$307$	−8596.87	−1.59821	−0.799103	+	0.601194i	$0.794692\pi$
$308$	0	0
$309$	2450.29	0.451107
$310$	0	0
$311$	10475.3	1.90997	0.954984	−	0.296658i	$-0.0958723\pi$
$311$	10475.3	1.90997	0.954984	+	0.296658i	$0.0958723\pi$
$312$	0	0
$313$	−5496.52	−0.992594	−0.496297	−	0.868153i	$-0.665307\pi$
$313$	−5496.52	−0.992594	−0.496297	+	0.868153i	$0.665307\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6556.37	1.16165	0.580824	−	0.814029i	$-0.302730\pi$
$317$	6556.37	1.16165	0.580824	+	0.814029i	$0.302730\pi$
$318$	0	0
$319$	5565.62	0.976850
$320$	0	0
$321$	5056.13	0.879145
$322$	0	0
$323$	946.057	0.162972
$324$	0	0
$325$	138.237	0.0235939
$326$	0	0
$327$	274.790	0.0464708
$328$	0	0
$329$	0	0
$330$	0	0
$331$	102.376	0.0170003	0.00850017	−	0.999964i	$-0.497294\pi$
$331$	102.376	0.0170003	0.00850017	+	0.999964i	$0.497294\pi$
$332$	0	0
$333$	−339.625	−0.0558899
$334$	0	0
$335$	6309.95	1.02910
$336$	0	0
$337$	1333.50	0.215549	0.107775	−	0.994175i	$-0.465627\pi$
$337$	1333.50	0.215549	0.107775	+	0.994175i	$0.465627\pi$
$338$	0	0
$339$	1842.58	0.295208
$340$	0	0
$341$	−8456.17	−1.34290
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2403.67	0.375099
$346$	0	0
$347$	−1101.00	−0.170331	−0.0851655	−	0.996367i	$-0.527142\pi$
$347$	−1101.00	−0.170331	−0.0851655	+	0.996367i	$0.527142\pi$
$348$	0	0
$349$	−9467.89	−1.45216	−0.726081	−	0.687609i	$-0.758660\pi$
$349$	−9467.89	−1.45216	−0.726081	+	0.687609i	$0.758660\pi$
$350$	0	0
$351$	−46.8748	−0.00712818
$352$	0	0
$353$	1684.69	0.254015	0.127007	−	0.991902i	$-0.459463\pi$
$353$	1684.69	0.254015	0.127007	+	0.991902i	$0.459463\pi$
$354$	0	0
$355$	−3677.53	−0.549812
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6635.25	−0.975474	−0.487737	−	0.872991i	$-0.662177\pi$
$359$	−6635.25	−0.975474	−0.487737	+	0.872991i	$0.662177\pi$
$360$	0	0
$361$	−6452.87	−0.940788
$362$	0	0
$363$	4651.79	0.672605
$364$	0	0
$365$	−2016.62	−0.289191
$366$	0	0
$367$	7121.97	1.01298	0.506490	−	0.862246i	$-0.330943\pi$
$367$	7121.97	1.01298	0.506490	+	0.862246i	$0.330943\pi$
$368$	0	0
$369$	2585.25	0.364723
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−13621.3	−1.89085	−0.945424	−	0.325843i	$-0.894352\pi$
$373$	−13621.3	−1.89085	−0.945424	+	0.325843i	$0.894352\pi$
$374$	0	0
$375$	−4135.12	−0.569432
$376$	0	0
$377$	180.000	0.0245901
$378$	0	0
$379$	−3331.51	−0.451525	−0.225763	−	0.974182i	$-0.572487\pi$
$379$	−3331.51	−0.451525	−0.225763	+	0.974182i	$0.572487\pi$
$380$	0	0
$381$	702.834	0.0945073
$382$	0	0
$383$	−3275.75	−0.437031	−0.218515	−	0.975833i	$-0.570121\pi$
$383$	−3275.75	−0.437031	−0.218515	+	0.975833i	$0.570121\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4543.87	0.596842
$388$	0	0
$389$	−14975.4	−1.95188	−0.975941	−	0.218035i	$-0.930035\pi$
$389$	−14975.4	−1.95188	−0.975941	+	0.218035i	$0.930035\pi$
$390$	0	0
$391$	5583.78	0.722209
$392$	0	0
$393$	−7994.29	−1.02610
$394$	0	0
$395$	−6334.18	−0.806853
$396$	0	0
$397$	10758.6	1.36010	0.680050	−	0.733166i	$-0.261958\pi$
$397$	10758.6	1.36010	0.680050	+	0.733166i	$0.261958\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2607.00	−0.324657	−0.162328	−	0.986737i	$-0.551900\pi$
$401$	−2607.00	−0.324657	−0.162328	+	0.986737i	$0.551900\pi$
$402$	0	0
$403$	−273.484	−0.0338045
$404$	0	0
$405$	545.624	0.0669439
$406$	0	0
$407$	2025.69	0.246707
$408$	0	0
$409$	−6214.64	−0.751330	−0.375665	−	0.926756i	$-0.622586\pi$
$409$	−6214.64	−0.751330	−0.375665	+	0.926756i	$0.622586\pi$
$410$	0	0
$411$	−3138.92	−0.376719
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4117.91	−0.487085
$416$	0	0
$417$	−1941.79	−0.228034
$418$	0	0
$419$	13111.4	1.52872	0.764358	−	0.644792i	$-0.223056\pi$
$419$	13111.4	1.52872	0.764358	+	0.644792i	$0.223056\pi$
$420$	0	0
$421$	−8410.12	−0.973596	−0.486798	−	0.873514i	$-0.661835\pi$
$421$	−8410.12	−0.973596	−0.486798	+	0.873514i	$0.661835\pi$
$422$	0	0
$423$	1984.50	0.228108
$424$	0	0
$425$	−3737.95	−0.426628
$426$	0	0
$427$	0	0
$428$	0	0
$429$	279.585	0.0314650
$430$	0	0
$431$	7783.31	0.869858	0.434929	−	0.900465i	$-0.356774\pi$
$431$	7783.31	0.869858	0.434929	+	0.900465i	$0.356774\pi$
$432$	0	0
$433$	−13870.1	−1.53939	−0.769695	−	0.638412i	$-0.779591\pi$
$433$	−13870.1	−1.53939	−0.769695	+	0.638412i	$0.779591\pi$
$434$	0	0
$435$	−2095.21	−0.230937
$436$	0	0
$437$	2397.05	0.262395
$438$	0	0
$439$	−5679.63	−0.617480	−0.308740	−	0.951146i	$-0.599907\pi$
$439$	−5679.63	−0.617480	−0.308740	+	0.951146i	$0.599907\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2316.93	−0.248489	−0.124245	−	0.992252i	$-0.539651\pi$
$443$	−2316.93	−0.248489	−0.124245	+	0.992252i	$0.539651\pi$
$444$	0	0
$445$	7922.97	0.844010
$446$	0	0
$447$	3980.42	0.421180
$448$	0	0
$449$	9526.86	1.00134	0.500669	−	0.865639i	$-0.333088\pi$
$449$	9526.86	1.00134	0.500669	+	0.865639i	$0.333088\pi$
$450$	0	0
$451$	−15419.7	−1.60995
$452$	0	0
$453$	11049.7	1.14605
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16627.3	−1.70195	−0.850974	−	0.525208i	$-0.823987\pi$
$457$	−16627.3	−1.70195	−0.850974	+	0.525208i	$0.823987\pi$
$458$	0	0
$459$	1267.50	0.128893
$460$	0	0
$461$	11338.3	1.14550	0.572749	−	0.819730i	$-0.305877\pi$
$461$	11338.3	1.14550	0.572749	+	0.819730i	$0.305877\pi$
$462$	0	0
$463$	9207.87	0.924246	0.462123	−	0.886816i	$-0.347088\pi$
$463$	9207.87	0.924246	0.462123	+	0.886816i	$0.347088\pi$
$464$	0	0
$465$	3183.37	0.317474
$466$	0	0
$467$	17215.7	1.70588	0.852941	−	0.522007i	$-0.174816\pi$
$467$	17215.7	1.70588	0.852941	+	0.522007i	$0.174816\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8454.74	0.827121
$472$	0	0
$473$	−27101.9	−2.63456
$474$	0	0
$475$	−1604.66	−0.155004
$476$	0	0
$477$	2630.88	0.252536
$478$	0	0
$479$	−10293.8	−0.981911	−0.490956	−	0.871185i	$-0.663352\pi$
$479$	−10293.8	−0.981911	−0.490956	+	0.871185i	$0.663352\pi$
$480$	0	0
$481$	65.5137	0.00621033
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9986.18	−0.934946
$486$	0	0
$487$	11109.0	1.03367	0.516833	−	0.856086i	$-0.327111\pi$
$487$	11109.0	1.03367	0.516833	+	0.856086i	$0.327111\pi$
$488$	0	0
$489$	12319.8	1.13931
$490$	0	0
$491$	−6573.88	−0.604226	−0.302113	−	0.953272i	$-0.597692\pi$
$491$	−6573.88	−0.604226	−0.302113	+	0.953272i	$0.597692\pi$
$492$	0	0
$493$	−4867.22	−0.444642
$494$	0	0
$495$	−3254.38	−0.295502
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−16841.4	−1.51087	−0.755435	−	0.655223i	$-0.772575\pi$
$499$	−16841.4	−1.51087	−0.755435	+	0.655223i	$0.772575\pi$
$500$	0	0
$501$	−3136.67	−0.279712
$502$	0	0
$503$	18436.6	1.63429	0.817146	−	0.576431i	$-0.195555\pi$
$503$	18436.6	1.63429	0.817146	+	0.576431i	$0.195555\pi$
$504$	0	0
$505$	−4162.53	−0.366793
$506$	0	0
$507$	−6581.96	−0.576558
$508$	0	0
$509$	−20825.2	−1.81348	−0.906738	−	0.421694i	$-0.861436\pi$
$509$	−20825.2	−1.81348	−0.906738	+	0.421694i	$0.861436\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	544.123	0.0468297
$514$	0	0
$515$	5501.80	0.470754
$516$	0	0
$517$	−11836.5	−1.00691
$518$	0	0
$519$	5730.92	0.484700
$520$	0	0
$521$	−14384.7	−1.20961	−0.604805	−	0.796373i	$-0.706749\pi$
$521$	−14384.7	−1.20961	−0.604805	+	0.796373i	$0.706749\pi$
$522$	0	0
$523$	−6487.49	−0.542405	−0.271203	−	0.962522i	$-0.587421\pi$
$523$	−6487.49	−0.542405	−0.271203	+	0.962522i	$0.587421\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7395.05	0.611259
$528$	0	0
$529$	1980.77	0.162799
$530$	0	0
$531$	5362.87	0.438284
$532$	0	0
$533$	−498.695	−0.0405270
$534$	0	0
$535$	11352.9	0.917433
$536$	0	0
$537$	693.420	0.0557230
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14416.6	1.14569	0.572844	−	0.819664i	$-0.305840\pi$
$541$	14416.6	1.14569	0.572844	+	0.819664i	$0.305840\pi$
$542$	0	0
$543$	−9925.21	−0.784404
$544$	0	0
$545$	617.006	0.0484947
$546$	0	0
$547$	−4881.38	−0.381559	−0.190780	−	0.981633i	$-0.561102\pi$
$547$	−4881.38	−0.381559	−0.190780	+	0.981633i	$0.561102\pi$
$548$	0	0
$549$	2388.50	0.185681
$550$	0	0
$551$	−2089.44	−0.161549
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−762.583	−0.0583240
$556$	0	0
$557$	−3153.13	−0.239860	−0.119930	−	0.992782i	$-0.538267\pi$
$557$	−3153.13	−0.239860	−0.119930	+	0.992782i	$0.538267\pi$
$558$	0	0
$559$	−876.514	−0.0663195
$560$	0	0
$561$	−7560.00	−0.568954
$562$	0	0
$563$	−6795.87	−0.508724	−0.254362	−	0.967109i	$-0.581866\pi$
$563$	−6795.87	−0.508724	−0.254362	+	0.967109i	$0.581866\pi$
$564$	0	0
$565$	4137.28	0.308065
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10708.3	−0.788955	−0.394477	−	0.918906i	$-0.629074\pi$
$569$	−10708.3	−0.788955	−0.394477	+	0.918906i	$0.629074\pi$
$570$	0	0
$571$	6531.66	0.478706	0.239353	−	0.970933i	$-0.423065\pi$
$571$	6531.66	0.478706	0.239353	+	0.970933i	$0.423065\pi$
$572$	0	0
$573$	5402.25	0.393861
$574$	0	0
$575$	−9470.94	−0.686896
$576$	0	0
$577$	−17461.9	−1.25987	−0.629937	−	0.776646i	$-0.716919\pi$
$577$	−17461.9	−1.25987	−0.629937	+	0.776646i	$0.716919\pi$
$578$	0	0
$579$	−3850.08	−0.276345
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−15691.9	−1.11473
$584$	0	0
$585$	−105.251	−0.00743863
$586$	0	0
$587$	−9254.42	−0.650717	−0.325358	−	0.945591i	$-0.605485\pi$
$587$	−9254.42	−0.650717	−0.325358	+	0.945591i	$0.605485\pi$
$588$	0	0
$589$	3174.61	0.222084
$590$	0	0
$591$	182.254	0.0126852
$592$	0	0
$593$	21019.2	1.45557	0.727786	−	0.685804i	$-0.240549\pi$
$593$	21019.2	1.45557	0.727786	+	0.685804i	$0.240549\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11581.2	−0.793945
$598$	0	0
$599$	−26971.7	−1.83979	−0.919894	−	0.392167i	$-0.871725\pi$
$599$	−26971.7	−1.83979	−0.919894	+	0.392167i	$0.871725\pi$
$600$	0	0
$601$	26510.3	1.79930	0.899648	−	0.436616i	$-0.143823\pi$
$601$	26510.3	1.79930	0.899648	+	0.436616i	$0.143823\pi$
$602$	0	0
$603$	8430.62	0.569355
$604$	0	0
$605$	10445.0	0.701899
$606$	0	0
$607$	−29412.0	−1.96671	−0.983356	−	0.181687i	$-0.941844\pi$
$607$	−29412.0	−1.96671	−0.983356	+	0.181687i	$0.941844\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−382.810	−0.0253467
$612$	0	0
$613$	−6068.64	−0.399853	−0.199926	−	0.979811i	$-0.564070\pi$
$613$	−6068.64	−0.399853	−0.199926	+	0.979811i	$0.564070\pi$
$614$	0	0
$615$	5804.83	0.380607
$616$	0	0
$617$	1904.56	0.124270	0.0621352	−	0.998068i	$-0.480209\pi$
$617$	1904.56	0.124270	0.0621352	+	0.998068i	$0.480209\pi$
$618$	0	0
$619$	406.478	0.0263937	0.0131969	−	0.999913i	$-0.495799\pi$
$619$	406.478	0.0263937	0.0131969	+	0.999913i	$0.495799\pi$
$620$	0	0
$621$	3211.50	0.207525
$622$	0	0
$623$	0	0
$624$	0	0
$625$	668.244	0.0427676
$626$	0	0
$627$	−3245.42	−0.206714
$628$	0	0
$629$	−1771.50	−0.112296
$630$	0	0
$631$	−15294.2	−0.964903	−0.482452	−	0.875923i	$-0.660254\pi$
$631$	−15294.2	−0.964903	−0.482452	+	0.875923i	$0.660254\pi$
$632$	0	0
$633$	1474.91	0.0926105
$634$	0	0
$635$	1578.12	0.0986233
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4913.49	−0.304186
$640$	0	0
$641$	−15722.2	−0.968782	−0.484391	−	0.874852i	$-0.660959\pi$
$641$	−15722.2	−0.968782	−0.484391	+	0.874852i	$0.660959\pi$
$642$	0	0
$643$	−18529.9	−1.13646	−0.568232	−	0.822868i	$-0.692372\pi$
$643$	−18529.9	−1.13646	−0.568232	+	0.822868i	$0.692372\pi$
$644$	0	0
$645$	10202.7	0.622836
$646$	0	0
$647$	8638.95	0.524934	0.262467	−	0.964941i	$-0.415464\pi$
$647$	8638.95	0.524934	0.262467	+	0.964941i	$0.415464\pi$
$648$	0	0
$649$	−31986.9	−1.93466
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27884.4	1.67106	0.835528	−	0.549448i	$-0.185162\pi$
$653$	27884.4	1.67106	0.835528	+	0.549448i	$0.185162\pi$
$654$	0	0
$655$	−17950.1	−1.07079
$656$	0	0
$657$	−2694.37	−0.159996
$658$	0	0
$659$	−5767.55	−0.340928	−0.170464	−	0.985364i	$-0.554527\pi$
$659$	−5767.55	−0.340928	−0.170464	+	0.985364i	$0.554527\pi$
$660$	0	0
$661$	−4090.12	−0.240677	−0.120338	−	0.992733i	$-0.538398\pi$
$661$	−4090.12	−0.240677	−0.120338	+	0.992733i	$0.538398\pi$
$662$	0	0
$663$	−244.501	−0.0143222
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12332.2	−0.715900
$668$	0	0
$669$	−2470.13	−0.142751
$670$	0	0
$671$	−14246.2	−0.819627
$672$	0	0
$673$	−21667.7	−1.24106	−0.620528	−	0.784185i	$-0.713082\pi$
$673$	−21667.7	−1.24106	−0.620528	+	0.784185i	$0.713082\pi$
$674$	0	0
$675$	−2149.87	−0.122591
$676$	0	0
$677$	15318.9	0.869648	0.434824	−	0.900515i	$-0.356811\pi$
$677$	15318.9	0.869648	0.434824	+	0.900515i	$0.356811\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	11276.6	0.634539
$682$	0	0
$683$	9981.36	0.559189	0.279595	−	0.960118i	$-0.409800\pi$
$683$	9981.36	0.559189	0.279595	+	0.960118i	$0.409800\pi$
$684$	0	0
$685$	−7048.02	−0.393126
$686$	0	0
$687$	7320.29	0.406531
$688$	0	0
$689$	−507.497	−0.0280611
$690$	0	0
$691$	−23243.1	−1.27961	−0.639805	−	0.768537i	$-0.720985\pi$
$691$	−23243.1	−1.27961	−0.639805	+	0.768537i	$0.720985\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4360.04	−0.237965
$696$	0	0
$697$	13484.8	0.732815
$698$	0	0
$699$	−5849.59	−0.316526
$700$	0	0
$701$	−11295.6	−0.608598	−0.304299	−	0.952577i	$-0.598422\pi$
$701$	−11295.6	−0.608598	−0.304299	+	0.952577i	$0.598422\pi$
$702$	0	0
$703$	−760.485	−0.0407998
$704$	0	0
$705$	4455.93	0.238042
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6868.11	0.363804	0.181902	−	0.983317i	$-0.441775\pi$
$709$	6868.11	0.363804	0.181902	+	0.983317i	$0.441775\pi$
$710$	0	0
$711$	−8463.00	−0.446395
$712$	0	0
$713$	18737.1	0.984163
$714$	0	0
$715$	627.770	0.0328353
$716$	0	0
$717$	8116.92	0.422778
$718$	0	0
$719$	31479.5	1.63281	0.816404	−	0.577481i	$-0.195964\pi$
$719$	31479.5	1.63281	0.816404	+	0.577481i	$0.195964\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1812.63	−0.0932400
$724$	0	0
$725$	8255.55	0.422901
$726$	0	0
$727$	−26045.8	−1.32873	−0.664363	−	0.747410i	$-0.731297\pi$
$727$	−26045.8	−1.32873	−0.664363	+	0.747410i	$0.731297\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	23701.0	1.19920
$732$	0	0
$733$	3538.40	0.178300	0.0891499	−	0.996018i	$-0.471585\pi$
$733$	3538.40	0.178300	0.0891499	+	0.996018i	$0.471585\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−50284.4	−2.51323
$738$	0	0
$739$	10426.9	0.519024	0.259512	−	0.965740i	$-0.416438\pi$
$739$	10426.9	0.519024	0.259512	+	0.965740i	$0.416438\pi$
$740$	0	0
$741$	−104.962	−0.00520359
$742$	0	0
$743$	2518.88	0.124372	0.0621862	−	0.998065i	$-0.480193\pi$
$743$	2518.88	0.124372	0.0621862	+	0.998065i	$0.480193\pi$
$744$	0	0
$745$	8937.50	0.439523
$746$	0	0
$747$	−5501.87	−0.269482
$748$	0	0
$749$	0	0
$750$	0	0
$751$	14820.3	0.720105	0.360053	−	0.932932i	$-0.382759\pi$
$751$	14820.3	0.720105	0.360053	+	0.932932i	$0.382759\pi$
$752$	0	0
$753$	4895.46	0.236919
$754$	0	0
$755$	24810.7	1.19596
$756$	0	0
$757$	8892.84	0.426969	0.213485	−	0.976946i	$-0.431519\pi$
$757$	8892.84	0.426969	0.213485	+	0.976946i	$0.431519\pi$
$758$	0	0
$759$	−19155.0	−0.916050
$760$	0	0
$761$	−22795.0	−1.08583	−0.542916	−	0.839787i	$-0.682680\pi$
$761$	−22795.0	−1.08583	−0.542916	+	0.839787i	$0.682680\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2846.00	0.134506
$766$	0	0
$767$	−1034.50	−0.0487009
$768$	0	0
$769$	13025.5	0.610809	0.305405	−	0.952223i	$-0.401208\pi$
$769$	13025.5	0.610809	0.305405	+	0.952223i	$0.401208\pi$
$770$	0	0
$771$	21319.7	0.995861
$772$	0	0
$773$	−26467.0	−1.23150	−0.615751	−	0.787941i	$-0.711147\pi$
$773$	−26467.0	−1.23150	−0.615751	+	0.787941i	$0.711147\pi$
$774$	0	0
$775$	−12543.1	−0.581371
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5788.87	0.266249
$780$	0	0
$781$	29306.5	1.34273
$782$	0	0
$783$	−2799.37	−0.127767
$784$	0	0
$785$	18984.0	0.863144
$786$	0	0
$787$	−33216.6	−1.50450	−0.752252	−	0.658876i	$-0.771032\pi$
$787$	−33216.6	−1.50450	−0.752252	+	0.658876i	$0.771032\pi$
$788$	0	0
$789$	16710.7	0.754015
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−460.743	−0.0206324
$794$	0	0
$795$	5907.28	0.263534
$796$	0	0
$797$	15363.9	0.682830	0.341415	−	0.939913i	$-0.389094\pi$
$797$	15363.9	0.682830	0.341415	+	0.939913i	$0.389094\pi$
$798$	0	0
$799$	10351.2	0.458323
$800$	0	0
$801$	10585.7	0.466953
$802$	0	0
$803$	16070.6	0.706249
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11739.5	−0.512083
$808$	0	0
$809$	26742.7	1.16220	0.581102	−	0.813831i	$-0.302622\pi$
$809$	26742.7	1.16220	0.581102	+	0.813831i	$0.302622\pi$
$810$	0	0
$811$	23651.0	1.02404	0.512022	−	0.858973i	$-0.328897\pi$
$811$	23651.0	1.02404	0.512022	+	0.858973i	$0.328897\pi$
$812$	0	0
$813$	−18217.5	−0.785876
$814$	0	0
$815$	27662.5	1.18893
$816$	0	0
$817$	10174.6	0.435697
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16974.1	0.721558	0.360779	−	0.932651i	$-0.382511\pi$
$821$	16974.1	0.721558	0.360779	+	0.932651i	$0.382511\pi$
$822$	0	0
$823$	22328.9	0.945730	0.472865	−	0.881135i	$-0.343220\pi$
$823$	22328.9	0.945730	0.472865	+	0.881135i	$0.343220\pi$
$824$	0	0
$825$	12822.9	0.541135
$826$	0	0
$827$	15731.8	0.661485	0.330743	−	0.943721i	$-0.392701\pi$
$827$	15731.8	0.661485	0.330743	+	0.943721i	$0.392701\pi$
$828$	0	0
$829$	38025.4	1.59309	0.796547	−	0.604576i	$-0.206658\pi$
$829$	38025.4	1.59309	0.796547	+	0.604576i	$0.206658\pi$
$830$	0	0
$831$	18421.9	0.769010
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−7042.97	−0.291895
$836$	0	0
$837$	4253.25	0.175644
$838$	0	0
$839$	−16546.5	−0.680868	−0.340434	−	0.940268i	$-0.610574\pi$
$839$	−16546.5	−0.680868	−0.340434	+	0.940268i	$0.610574\pi$
$840$	0	0
$841$	−13639.4	−0.559242
$842$	0	0
$843$	−1097.25	−0.0448296
$844$	0	0
$845$	−14778.9	−0.601669
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2131.04	0.0861452
$850$	0	0
$851$	−4488.50	−0.180803
$852$	0	0
$853$	−13511.9	−0.542365	−0.271182	−	0.962528i	$-0.587415\pi$
$853$	−13511.9	−0.542365	−0.271182	+	0.962528i	$0.587415\pi$
$854$	0	0
$855$	1221.76	0.0488692
$856$	0	0
$857$	−17178.2	−0.684711	−0.342355	−	0.939570i	$-0.611225\pi$
$857$	−17178.2	−0.684711	−0.342355	+	0.939570i	$0.611225\pi$
$858$	0	0
$859$	35914.3	1.42652	0.713260	−	0.700899i	$-0.247218\pi$
$859$	35914.3	1.42652	0.713260	+	0.700899i	$0.247218\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−882.508	−0.0348099	−0.0174049	−	0.999849i	$-0.505540\pi$
$863$	−882.508	−0.0348099	−0.0174049	+	0.999849i	$0.505540\pi$
$864$	0	0
$865$	12868.0	0.505810
$866$	0	0
$867$	−8127.67	−0.318374
$868$	0	0
$869$	50477.6	1.97046
$870$	0	0
$871$	−1626.27	−0.0632652
$872$	0	0
$873$	−13342.4	−0.517263
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17420.1	0.670737	0.335368	−	0.942087i	$-0.391139\pi$
$877$	17420.1	0.670737	0.335368	+	0.942087i	$0.391139\pi$
$878$	0	0
$879$	−889.120	−0.0341175
$880$	0	0
$881$	−38097.8	−1.45692	−0.728460	−	0.685088i	$-0.759764\pi$
$881$	−38097.8	−1.45692	−0.728460	+	0.685088i	$0.759764\pi$
$882$	0	0
$883$	−13870.6	−0.528633	−0.264317	−	0.964436i	$-0.585146\pi$
$883$	−13870.6	−0.528633	−0.264317	+	0.964436i	$0.585146\pi$
$884$	0	0
$885$	12041.6	0.457372
$886$	0	0
$887$	17467.8	0.661230	0.330615	−	0.943766i	$-0.392744\pi$
$887$	17467.8	0.661230	0.330615	+	0.943766i	$0.392744\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4348.12	−0.163488
$892$	0	0
$893$	4443.67	0.166519
$894$	0	0
$895$	1556.98	0.0581499
$896$	0	0
$897$	−619.499	−0.0230596
$898$	0	0
$899$	−16332.6	−0.605919
$900$	0	0
$901$	13722.8	0.507405
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−22285.7	−0.818567
$906$	0	0
$907$	−28536.3	−1.04469	−0.522344	−	0.852735i	$-0.674942\pi$
$907$	−28536.3	−1.04469	−0.522344	+	0.852735i	$0.674942\pi$
$908$	0	0
$909$	−5561.49	−0.202930
$910$	0	0
$911$	−50235.7	−1.82698	−0.913492	−	0.406857i	$-0.866625\pi$
$911$	−50235.7	−1.82698	−0.913492	+	0.406857i	$0.866625\pi$
$912$	0	0
$913$	32815.9	1.18954
$914$	0	0
$915$	5363.07	0.193768
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5317.08	0.190854	0.0954268	−	0.995436i	$-0.469578\pi$
$919$	5317.08	0.190854	0.0954268	+	0.995436i	$0.469578\pi$
$920$	0	0
$921$	−25790.6	−0.922725
$922$	0	0
$923$	947.814	0.0338003
$924$	0	0
$925$	3004.73	0.106805
$926$	0	0
$927$	7350.87	0.260447
$928$	0	0
$929$	−25493.1	−0.900325	−0.450162	−	0.892947i	$-0.648634\pi$
$929$	−25493.1	−0.900325	−0.450162	+	0.892947i	$0.648634\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	31425.9	1.10272
$934$	0	0
$935$	−16975.0	−0.593734
$936$	0	0
$937$	−12691.4	−0.442486	−0.221243	−	0.975219i	$-0.571011\pi$
$937$	−12691.4	−0.442486	−0.221243	+	0.975219i	$0.571011\pi$
$938$	0	0
$939$	−16489.6	−0.573074
$940$	0	0
$941$	15378.4	0.532753	0.266377	−	0.963869i	$-0.414174\pi$
$941$	15378.4	0.532753	0.266377	+	0.963869i	$0.414174\pi$
$942$	0	0
$943$	34166.8	1.17988
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50442.9	1.73091	0.865455	−	0.500986i	$-0.167029\pi$
$947$	50442.9	1.73091	0.865455	+	0.500986i	$0.167029\pi$
$948$	0	0
$949$	519.745	0.0177783
$950$	0	0
$951$	19669.1	0.670678
$952$	0	0
$953$	31787.3	1.08047	0.540237	−	0.841513i	$-0.318335\pi$
$953$	31787.3	1.08047	0.540237	+	0.841513i	$0.318335\pi$
$954$	0	0
$955$	12130.0	0.411014
$956$	0	0
$957$	16696.9	0.563984
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−4975.99	−0.167030
$962$	0	0
$963$	15168.4	0.507574
$964$	0	0
$965$	−8644.84	−0.288381
$966$	0	0
$967$	1005.61	0.0334417	0.0167209	−	0.999860i	$-0.494677\pi$
$967$	1005.61	0.0334417	0.0167209	+	0.999860i	$0.494677\pi$
$968$	0	0
$969$	2838.17	0.0940921
$970$	0	0
$971$	−7070.91	−0.233693	−0.116847	−	0.993150i	$-0.537279\pi$
$971$	−7070.91	−0.233693	−0.116847	+	0.993150i	$0.537279\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	414.711	0.0136219
$976$	0	0
$977$	−11097.1	−0.363384	−0.181692	−	0.983355i	$-0.558157\pi$
$977$	−11097.1	−0.363384	−0.181692	+	0.983355i	$0.558157\pi$
$978$	0	0
$979$	−63138.7	−2.06121
$980$	0	0
$981$	824.371	0.0268299
$982$	0	0
$983$	17775.5	0.576755	0.288377	−	0.957517i	$-0.406884\pi$
$983$	17775.5	0.576755	0.288377	+	0.957517i	$0.406884\pi$
$984$	0	0
$985$	409.228	0.0132376
$986$	0	0
$987$	0	0
$988$	0	0
$989$	60052.0	1.93078
$990$	0	0
$991$	4691.86	0.150395	0.0751977	−	0.997169i	$-0.476041\pi$
$991$	4691.86	0.150395	0.0751977	+	0.997169i	$0.476041\pi$
$992$	0	0
$993$	307.129	0.00981515
$994$	0	0
$995$	−26004.0	−0.828523
$996$	0	0
$997$	8471.39	0.269099	0.134549	−	0.990907i	$-0.457041\pi$
$997$	8471.39	0.269099	0.134549	+	0.990907i	$0.457041\pi$
$998$	0	0
$999$	−1018.87	−0.0322680

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.4.a.u.1.2		2
4.3	odd	2		2352.4.a.bo.1.2		2
7.3	odd	6		168.4.q.d.121.2	yes	4
7.5	odd	6		168.4.q.d.25.2	✓	4
7.6	odd	2		1176.4.a.r.1.1		2
21.5	even	6		504.4.s.f.361.1		4
21.17	even	6		504.4.s.f.289.1		4
28.3	even	6		336.4.q.g.289.2		4
28.19	even	6		336.4.q.g.193.2		4
28.27	even	2		2352.4.a.cc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.d.25.2	✓	4	7.5	odd	6
168.4.q.d.121.2	yes	4	7.3	odd	6
336.4.q.g.193.2		4	28.19	even	6
336.4.q.g.289.2		4	28.3	even	6
504.4.s.f.289.1		4	21.17	even	6
504.4.s.f.361.1		4	21.5	even	6
1176.4.a.r.1.1		2	7.6	odd	2
1176.4.a.u.1.2		2	1.1	even	1	trivial
2352.4.a.bo.1.2		2	4.3	odd	2
2352.4.a.cc.1.1		2	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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +6.73610 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +6.73610 q^{5} +9.00000 q^{9} -53.6805 q^{11} -1.73610 q^{13} +20.2083 q^{15} +46.9444 q^{17} +20.1527 q^{19} +118.944 q^{23} -79.6249 q^{25} +27.0000 q^{27} -103.681 q^{29} +157.528 q^{31} -161.042 q^{33} -37.7361 q^{37} -5.20831 q^{39} +287.250 q^{41} +504.875 q^{43} +60.6249 q^{45} +220.500 q^{47} +140.833 q^{51} +292.319 q^{53} -361.597 q^{55} +60.4582 q^{57} +595.875 q^{59} +265.389 q^{61} -11.6946 q^{65} +936.735 q^{67} +356.833 q^{69} -545.944 q^{71} -299.374 q^{73} -238.875 q^{75} -940.333 q^{79} +81.0000 q^{81} -611.319 q^{83} +316.222 q^{85} -311.042 q^{87} +1176.19 q^{89} +472.583 q^{93} +135.751 q^{95} -1482.49 q^{97} -483.125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 9 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 9 * q^5 + 18 * q^9	\( 2 q + 6 q^{3} - 9 q^{5} + 18 q^{9} + 5 q^{11} + 19 q^{13} - 27 q^{15} + 4 q^{17} - 117 q^{19} + 148 q^{23} + 43 q^{25} + 54 q^{27} - 95 q^{29} + 360 q^{31} + 15 q^{33} - 53 q^{37} + 57 q^{39} + 170 q^{41} + 403 q^{43} - 81 q^{45} - 368 q^{47} + 12 q^{51} + 697 q^{53} - 1285 q^{55} - 351 q^{57} + 585 q^{59} + 1160 q^{61} - 338 q^{65} + 233 q^{67} + 444 q^{69} + 616 q^{71} + 817 q^{73} + 129 q^{75} - 802 q^{79} + 162 q^{81} + 283 q^{83} + 992 q^{85} - 285 q^{87} + 1858 q^{89} + 1080 q^{93} + 2294 q^{95} - 1729 q^{97} + 45 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 9 * q^5 + 18 * q^9 + 5 * q^11 + 19 * q^13 - 27 * q^15 + 4 * q^17 - 117 * q^19 + 148 * q^23 + 43 * q^25 + 54 * q^27 - 95 * q^29 + 360 * q^31 + 15 * q^33 - 53 * q^37 + 57 * q^39 + 170 * q^41 + 403 * q^43 - 81 * q^45 - 368 * q^47 + 12 * q^51 + 697 * q^53 - 1285 * q^55 - 351 * q^57 + 585 * q^59 + 1160 * q^61 - 338 * q^65 + 233 * q^67 + 444 * q^69 + 616 * q^71 + 817 * q^73 + 129 * q^75 - 802 * q^79 + 162 * q^81 + 283 * q^83 + 992 * q^85 - 285 * q^87 + 1858 * q^89 + 1080 * q^93 + 2294 * q^95 - 1729 * q^97 + 45 * q^99

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