LMFDB - Embedded newform 1176.4.a.be.1.4

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:	$4$
Coefficient field:	4.4.391168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 40x^{2} + 382 $ x^4 - 40*x^2 + 382
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.96955$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +14.6418 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +14.6418 q^{5} +9.00000 q^{9} -11.4490 q^{11} +20.8071 q^{13} +43.9254 q^{15} -18.8331 q^{17} -91.0178 q^{19} +28.9336 q^{23} +89.3825 q^{25} +27.0000 q^{27} +281.235 q^{29} +276.010 q^{31} -34.3469 q^{33} +250.567 q^{37} +62.4214 q^{39} -12.1422 q^{41} +65.2121 q^{43} +131.776 q^{45} -199.886 q^{47} -56.4992 q^{51} -64.1260 q^{53} -167.633 q^{55} -273.054 q^{57} +69.0791 q^{59} +656.210 q^{61} +304.654 q^{65} -419.714 q^{67} +86.8007 q^{69} +1022.46 q^{71} +103.280 q^{73} +268.148 q^{75} +260.616 q^{79} +81.0000 q^{81} -879.313 q^{83} -275.750 q^{85} +843.705 q^{87} +1480.70 q^{89} +828.029 q^{93} -1332.67 q^{95} +177.673 q^{97} -103.041 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 72 q^{17} + 32 q^{19} + 8 q^{23} + 164 q^{25} + 108 q^{27} + 144 q^{29} + 48 q^{31} + 120 q^{33} + 48 q^{37} + 144 q^{39} + 72 q^{41} + 512 q^{43} + 72 q^{45} + 160 q^{47} + 216 q^{51} + 536 q^{53} - 336 q^{55} + 96 q^{57} + 240 q^{59} + 896 q^{61} - 136 q^{65} + 1088 q^{67} + 24 q^{69} + 1288 q^{71} + 1488 q^{73} + 492 q^{75} + 416 q^{79} + 324 q^{81} - 112 q^{83} - 1512 q^{85} + 432 q^{87} + 3160 q^{89} + 144 q^{93} - 240 q^{95} + 2384 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 + 72 * q^17 + 32 * q^19 + 8 * q^23 + 164 * q^25 + 108 * q^27 + 144 * q^29 + 48 * q^31 + 120 * q^33 + 48 * q^37 + 144 * q^39 + 72 * q^41 + 512 * q^43 + 72 * q^45 + 160 * q^47 + 216 * q^51 + 536 * q^53 - 336 * q^55 + 96 * q^57 + 240 * q^59 + 896 * q^61 - 136 * q^65 + 1088 * q^67 + 24 * q^69 + 1288 * q^71 + 1488 * q^73 + 492 * q^75 + 416 * q^79 + 324 * q^81 - 112 * q^83 - 1512 * q^85 + 432 * q^87 + 3160 * q^89 + 144 * q^93 - 240 * q^95 + 2384 * 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
$5$	14.6418	1.30960	0.654802	−	0.755801i	$-0.272752\pi$
$5$	14.6418	1.30960	0.654802	+	0.755801i	$0.272752\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.4490	−0.313817	−0.156909	−	0.987613i	$-0.550153\pi$
$11$	−11.4490	−0.313817	−0.156909	+	0.987613i	$0.550153\pi$
$12$	0	0
$13$	20.8071	0.443913	0.221956	−	0.975057i	$-0.428756\pi$
$13$	20.8071	0.443913	0.221956	+	0.975057i	$0.428756\pi$
$14$	0	0
$15$	43.9254	0.756100
$16$	0	0
$17$	−18.8331	−0.268688	−0.134344	−	0.990935i	$-0.542893\pi$
$17$	−18.8331	−0.268688	−0.134344	+	0.990935i	$0.542893\pi$
$18$	0	0
$19$	−91.0178	−1.09900	−0.549498	−	0.835495i	$-0.685181\pi$
$19$	−91.0178	−1.09900	−0.549498	+	0.835495i	$0.685181\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	28.9336	0.262307	0.131154	−	0.991362i	$-0.458132\pi$
$23$	28.9336	0.262307	0.131154	+	0.991362i	$0.458132\pi$
$24$	0	0
$25$	89.3825	0.715060
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	281.235	1.80083	0.900414	−	0.435034i	$-0.143264\pi$
$29$	281.235	1.80083	0.900414	+	0.435034i	$0.143264\pi$
$30$	0	0
$31$	276.010	1.59912	0.799561	−	0.600585i	$-0.205065\pi$
$31$	276.010	1.59912	0.799561	+	0.600585i	$0.205065\pi$
$32$	0	0
$33$	−34.3469	−0.181182
$34$	0	0
$35$	0	0
$36$	0	0
$37$	250.567	1.11332	0.556662	−	0.830739i	$-0.312082\pi$
$37$	250.567	1.11332	0.556662	+	0.830739i	$0.312082\pi$
$38$	0	0
$39$	62.4214	0.256293
$40$	0	0
$41$	−12.1422	−0.0462512	−0.0231256	−	0.999733i	$-0.507362\pi$
$41$	−12.1422	−0.0462512	−0.0231256	+	0.999733i	$0.507362\pi$
$42$	0	0
$43$	65.2121	0.231273	0.115637	−	0.993292i	$-0.463109\pi$
$43$	65.2121	0.231273	0.115637	+	0.993292i	$0.463109\pi$
$44$	0	0
$45$	131.776	0.436534
$46$	0	0
$47$	−199.886	−0.620350	−0.310175	−	0.950680i	$-0.600388\pi$
$47$	−199.886	−0.620350	−0.310175	+	0.950680i	$0.600388\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−56.4992	−0.155127
$52$	0	0
$53$	−64.1260	−0.166196	−0.0830980	−	0.996541i	$-0.526481\pi$
$53$	−64.1260	−0.166196	−0.0830980	+	0.996541i	$0.526481\pi$
$54$	0	0
$55$	−167.633	−0.410976
$56$	0	0
$57$	−273.054	−0.634506
$58$	0	0
$59$	69.0791	0.152429	0.0762147	−	0.997091i	$-0.475717\pi$
$59$	69.0791	0.152429	0.0762147	+	0.997091i	$0.475717\pi$
$60$	0	0
$61$	656.210	1.37736	0.688681	−	0.725064i	$-0.258190\pi$
$61$	656.210	1.37736	0.688681	+	0.725064i	$0.258190\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	304.654	0.581349
$66$	0	0
$67$	−419.714	−0.765317	−0.382658	−	0.923890i	$-0.624991\pi$
$67$	−419.714	−0.765317	−0.382658	+	0.923890i	$0.624991\pi$
$68$	0	0
$69$	86.8007	0.151443
$70$	0	0
$71$	1022.46	1.70907	0.854536	−	0.519393i	$-0.173842\pi$
$71$	1022.46	1.70907	0.854536	+	0.519393i	$0.173842\pi$
$72$	0	0
$73$	103.280	0.165590	0.0827948	−	0.996567i	$-0.473615\pi$
$73$	103.280	0.165590	0.0827948	+	0.996567i	$0.473615\pi$
$74$	0	0
$75$	268.148	0.412840
$76$	0	0
$77$	0	0
$78$	0	0
$79$	260.616	0.371159	0.185579	−	0.982629i	$-0.440584\pi$
$79$	260.616	0.371159	0.185579	+	0.982629i	$0.440584\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−879.313	−1.16286	−0.581428	−	0.813598i	$-0.697506\pi$
$83$	−879.313	−1.16286	−0.581428	+	0.813598i	$0.697506\pi$
$84$	0	0
$85$	−275.750	−0.351874
$86$	0	0
$87$	843.705	1.03971
$88$	0	0
$89$	1480.70	1.76353	0.881763	−	0.471692i	$-0.156357\pi$
$89$	1480.70	1.76353	0.881763	+	0.471692i	$0.156357\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	828.029	0.923254
$94$	0	0
$95$	−1332.67	−1.43925
$96$	0	0
$97$	177.673	0.185979	0.0929895	−	0.995667i	$-0.470358\pi$
$97$	177.673	0.185979	0.0929895	+	0.995667i	$0.470358\pi$
$98$	0	0
$99$	−103.041	−0.104606
$100$	0	0
$101$	−872.534	−0.859608	−0.429804	−	0.902922i	$-0.641417\pi$
$101$	−872.534	−0.859608	−0.429804	+	0.902922i	$0.641417\pi$
$102$	0	0
$103$	942.292	0.901425	0.450713	−	0.892669i	$-0.351170\pi$
$103$	942.292	0.901425	0.450713	+	0.892669i	$0.351170\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1162.43	−1.05024	−0.525121	−	0.851028i	$-0.675980\pi$
$107$	−1162.43	−1.05024	−0.525121	+	0.851028i	$0.675980\pi$
$108$	0	0
$109$	−35.2015	−0.0309330	−0.0154665	−	0.999880i	$-0.504923\pi$
$109$	−35.2015	−0.0309330	−0.0154665	+	0.999880i	$0.504923\pi$
$110$	0	0
$111$	751.701	0.642777
$112$	0	0
$113$	−420.632	−0.350175	−0.175087	−	0.984553i	$-0.556021\pi$
$113$	−420.632	−0.350175	−0.175087	+	0.984553i	$0.556021\pi$
$114$	0	0
$115$	423.640	0.343519
$116$	0	0
$117$	187.264	0.147971
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1199.92	−0.901519
$122$	0	0
$123$	−36.4267	−0.0267032
$124$	0	0
$125$	−521.504	−0.373158
$126$	0	0
$127$	−722.609	−0.504891	−0.252445	−	0.967611i	$-0.581235\pi$
$127$	−722.609	−0.504891	−0.252445	+	0.967611i	$0.581235\pi$
$128$	0	0
$129$	195.636	0.133526
$130$	0	0
$131$	799.224	0.533042	0.266521	−	0.963829i	$-0.414126\pi$
$131$	799.224	0.533042	0.266521	+	0.963829i	$0.414126\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	395.329	0.252033
$136$	0	0
$137$	−2951.13	−1.84038	−0.920191	−	0.391470i	$-0.871967\pi$
$137$	−2951.13	−1.84038	−0.920191	+	0.391470i	$0.871967\pi$
$138$	0	0
$139$	−102.191	−0.0623579	−0.0311789	−	0.999514i	$-0.509926\pi$
$139$	−102.191	−0.0623579	−0.0311789	+	0.999514i	$0.509926\pi$
$140$	0	0
$141$	−599.659	−0.358159
$142$	0	0
$143$	−238.220	−0.139307
$144$	0	0
$145$	4117.79	2.35837
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1798.70	0.988963	0.494482	−	0.869188i	$-0.335358\pi$
$149$	1798.70	0.988963	0.494482	+	0.869188i	$0.335358\pi$
$150$	0	0
$151$	−3196.81	−1.72286	−0.861432	−	0.507873i	$-0.830432\pi$
$151$	−3196.81	−1.72286	−0.861432	+	0.507873i	$0.830432\pi$
$152$	0	0
$153$	−169.498	−0.0895626
$154$	0	0
$155$	4041.28	2.09422
$156$	0	0
$157$	3476.27	1.76711	0.883556	−	0.468326i	$-0.155143\pi$
$157$	3476.27	1.76711	0.883556	+	0.468326i	$0.155143\pi$
$158$	0	0
$159$	−192.378	−0.0959533
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3789.94	1.82117	0.910587	−	0.413318i	$-0.135630\pi$
$163$	3789.94	1.82117	0.910587	+	0.413318i	$0.135630\pi$
$164$	0	0
$165$	−502.900	−0.237277
$166$	0	0
$167$	2711.44	1.25639	0.628195	−	0.778056i	$-0.283794\pi$
$167$	2711.44	1.25639	0.628195	+	0.778056i	$0.283794\pi$
$168$	0	0
$169$	−1764.06	−0.802942
$170$	0	0
$171$	−819.161	−0.366332
$172$	0	0
$173$	−2504.48	−1.10065	−0.550324	−	0.834951i	$-0.685496\pi$
$173$	−2504.48	−1.10065	−0.550324	+	0.834951i	$0.685496\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	207.237	0.0880052
$178$	0	0
$179$	−2089.31	−0.872413	−0.436207	−	0.899847i	$-0.643678\pi$
$179$	−2089.31	−0.872413	−0.436207	+	0.899847i	$0.643678\pi$
$180$	0	0
$181$	4081.29	1.67602	0.838010	−	0.545655i	$-0.183719\pi$
$181$	4081.29	1.67602	0.838010	+	0.545655i	$0.183719\pi$
$182$	0	0
$183$	1968.63	0.795220
$184$	0	0
$185$	3668.75	1.45801
$186$	0	0
$187$	215.619	0.0843189
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2056.23	0.778970	0.389485	−	0.921033i	$-0.372653\pi$
$191$	2056.23	0.778970	0.389485	+	0.921033i	$0.372653\pi$
$192$	0	0
$193$	−1489.48	−0.555520	−0.277760	−	0.960650i	$-0.589592\pi$
$193$	−1489.48	−0.555520	−0.277760	+	0.960650i	$0.589592\pi$
$194$	0	0
$195$	913.963	0.335642
$196$	0	0
$197$	1591.66	0.575640	0.287820	−	0.957684i	$-0.407069\pi$
$197$	1591.66	0.575640	0.287820	+	0.957684i	$0.407069\pi$
$198$	0	0
$199$	−4303.32	−1.53294	−0.766468	−	0.642282i	$-0.777988\pi$
$199$	−4303.32	−1.53294	−0.766468	+	0.642282i	$0.777988\pi$
$200$	0	0
$201$	−1259.14	−0.441856
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−177.784	−0.0605707
$206$	0	0
$207$	260.402	0.0874358
$208$	0	0
$209$	1042.06	0.344884
$210$	0	0
$211$	2209.01	0.720733	0.360367	−	0.932811i	$-0.382652\pi$
$211$	2209.01	0.720733	0.360367	+	0.932811i	$0.382652\pi$
$212$	0	0
$213$	3067.39	0.986733
$214$	0	0
$215$	954.824	0.302876
$216$	0	0
$217$	0	0
$218$	0	0
$219$	309.841	0.0956033
$220$	0	0
$221$	−391.863	−0.119274
$222$	0	0
$223$	−4731.91	−1.42095	−0.710476	−	0.703721i	$-0.751520\pi$
$223$	−4731.91	−1.42095	−0.710476	+	0.703721i	$0.751520\pi$
$224$	0	0
$225$	804.443	0.238353
$226$	0	0
$227$	622.308	0.181956	0.0909780	−	0.995853i	$-0.471001\pi$
$227$	622.308	0.181956	0.0909780	+	0.995853i	$0.471001\pi$
$228$	0	0
$229$	530.619	0.153119	0.0765595	−	0.997065i	$-0.475606\pi$
$229$	530.619	0.153119	0.0765595	+	0.997065i	$0.475606\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3189.28	0.896723	0.448362	−	0.893852i	$-0.352008\pi$
$233$	3189.28	0.896723	0.448362	+	0.893852i	$0.352008\pi$
$234$	0	0
$235$	−2926.70	−0.812412
$236$	0	0
$237$	781.847	0.214289
$238$	0	0
$239$	1515.36	0.410127	0.205064	−	0.978749i	$-0.434260\pi$
$239$	1515.36	0.410127	0.205064	+	0.978749i	$0.434260\pi$
$240$	0	0
$241$	1235.38	0.330200	0.165100	−	0.986277i	$-0.447205\pi$
$241$	1235.38	0.330200	0.165100	+	0.986277i	$0.447205\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1893.82	−0.487858
$248$	0	0
$249$	−2637.94	−0.671376
$250$	0	0
$251$	2165.31	0.544514	0.272257	−	0.962225i	$-0.412230\pi$
$251$	2165.31	0.544514	0.272257	+	0.962225i	$0.412230\pi$
$252$	0	0
$253$	−331.259	−0.0823166
$254$	0	0
$255$	−827.251	−0.203155
$256$	0	0
$257$	3947.43	0.958108	0.479054	−	0.877786i	$-0.340980\pi$
$257$	3947.43	0.958108	0.479054	+	0.877786i	$0.340980\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2531.11	0.600276
$262$	0	0
$263$	−997.979	−0.233985	−0.116992	−	0.993133i	$-0.537325\pi$
$263$	−997.979	−0.233985	−0.116992	+	0.993133i	$0.537325\pi$
$264$	0	0
$265$	−938.921	−0.217651
$266$	0	0
$267$	4442.10	1.01817
$268$	0	0
$269$	6029.26	1.36658	0.683291	−	0.730146i	$-0.260548\pi$
$269$	6029.26	1.36658	0.683291	+	0.730146i	$0.260548\pi$
$270$	0	0
$271$	3119.65	0.699280	0.349640	−	0.936884i	$-0.386304\pi$
$271$	3119.65	0.699280	0.349640	+	0.936884i	$0.386304\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1023.34	−0.224398
$276$	0	0
$277$	−2003.80	−0.434645	−0.217323	−	0.976100i	$-0.569732\pi$
$277$	−2003.80	−0.434645	−0.217323	+	0.976100i	$0.569732\pi$
$278$	0	0
$279$	2484.09	0.533041
$280$	0	0
$281$	−8915.35	−1.89269	−0.946344	−	0.323162i	$-0.895254\pi$
$281$	−8915.35	−1.89269	−0.946344	+	0.323162i	$0.895254\pi$
$282$	0	0
$283$	−3239.09	−0.680366	−0.340183	−	0.940359i	$-0.610489\pi$
$283$	−3239.09	−0.680366	−0.340183	+	0.940359i	$0.610489\pi$
$284$	0	0
$285$	−3998.00	−0.830951
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4558.32	−0.927807
$290$	0	0
$291$	533.019	0.107375
$292$	0	0
$293$	5997.02	1.19573	0.597867	−	0.801595i	$-0.296015\pi$
$293$	5997.02	1.19573	0.597867	+	0.801595i	$0.296015\pi$
$294$	0	0
$295$	1011.44	0.199622
$296$	0	0
$297$	−309.122	−0.0603942
$298$	0	0
$299$	602.025	0.116442
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2617.60	−0.496295
$304$	0	0
$305$	9608.10	1.80380
$306$	0	0
$307$	−9676.32	−1.79888	−0.899441	−	0.437042i	$-0.856026\pi$
$307$	−9676.32	−1.79888	−0.899441	+	0.437042i	$0.856026\pi$
$308$	0	0
$309$	2826.88	0.520438
$310$	0	0
$311$	1458.55	0.265938	0.132969	−	0.991120i	$-0.457549\pi$
$311$	1458.55	0.265938	0.132969	+	0.991120i	$0.457549\pi$
$312$	0	0
$313$	10115.7	1.82674	0.913371	−	0.407128i	$-0.133470\pi$
$313$	10115.7	1.82674	0.913371	+	0.407128i	$0.133470\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8516.05	1.50886	0.754431	−	0.656379i	$-0.227913\pi$
$317$	8516.05	1.50886	0.754431	+	0.656379i	$0.227913\pi$
$318$	0	0
$319$	−3219.85	−0.565131
$320$	0	0
$321$	−3487.28	−0.606357
$322$	0	0
$323$	1714.15	0.295287
$324$	0	0
$325$	1859.80	0.317424
$326$	0	0
$327$	−105.605	−0.0178592
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2800.56	−0.465053	−0.232527	−	0.972590i	$-0.574699\pi$
$331$	−2800.56	−0.465053	−0.232527	+	0.972590i	$0.574699\pi$
$332$	0	0
$333$	2255.10	0.371108
$334$	0	0
$335$	−6145.37	−1.00226
$336$	0	0
$337$	−6972.73	−1.12709	−0.563544	−	0.826086i	$-0.690563\pi$
$337$	−6972.73	−1.12709	−0.563544	+	0.826086i	$0.690563\pi$
$338$	0	0
$339$	−1261.90	−0.202173
$340$	0	0
$341$	−3160.02	−0.501832
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1270.92	0.198331
$346$	0	0
$347$	−704.074	−0.108924	−0.0544621	−	0.998516i	$-0.517344\pi$
$347$	−704.074	−0.108924	−0.0544621	+	0.998516i	$0.517344\pi$
$348$	0	0
$349$	−5350.53	−0.820652	−0.410326	−	0.911939i	$-0.634585\pi$
$349$	−5350.53	−0.820652	−0.410326	+	0.911939i	$0.634585\pi$
$350$	0	0
$351$	561.793	0.0854310
$352$	0	0
$353$	1230.18	0.185485	0.0927423	−	0.995690i	$-0.470437\pi$
$353$	1230.18	0.185485	0.0927423	+	0.995690i	$0.470437\pi$
$354$	0	0
$355$	14970.7	2.23820
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5402.57	−0.794253	−0.397126	−	0.917764i	$-0.629993\pi$
$359$	−5402.57	−0.794253	−0.397126	+	0.917764i	$0.629993\pi$
$360$	0	0
$361$	1425.25	0.207792
$362$	0	0
$363$	−3599.76	−0.520492
$364$	0	0
$365$	1512.21	0.216857
$366$	0	0
$367$	−13258.7	−1.88583	−0.942913	−	0.333040i	$-0.891926\pi$
$367$	−13258.7	−1.88583	−0.942913	+	0.333040i	$0.891926\pi$
$368$	0	0
$369$	−109.280	−0.0154171
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6424.77	−0.891856	−0.445928	−	0.895069i	$-0.647126\pi$
$373$	−6424.77	−0.891856	−0.445928	+	0.895069i	$0.647126\pi$
$374$	0	0
$375$	−1564.51	−0.215443
$376$	0	0
$377$	5851.70	0.799410
$378$	0	0
$379$	−11167.5	−1.51355	−0.756773	−	0.653678i	$-0.773225\pi$
$379$	−11167.5	−1.51355	−0.756773	+	0.653678i	$0.773225\pi$
$380$	0	0
$381$	−2167.83	−0.291499
$382$	0	0
$383$	−8758.23	−1.16847	−0.584236	−	0.811584i	$-0.698606\pi$
$383$	−8758.23	−1.16847	−0.584236	+	0.811584i	$0.698606\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	586.909	0.0770911
$388$	0	0
$389$	12054.8	1.57122	0.785610	−	0.618723i	$-0.212350\pi$
$389$	12054.8	1.57122	0.785610	+	0.618723i	$0.212350\pi$
$390$	0	0
$391$	−544.908	−0.0704788
$392$	0	0
$393$	2397.67	0.307752
$394$	0	0
$395$	3815.88	0.486071
$396$	0	0
$397$	−4650.10	−0.587863	−0.293932	−	0.955826i	$-0.594964\pi$
$397$	−4650.10	−0.587863	−0.293932	+	0.955826i	$0.594964\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7378.14	0.918819	0.459410	−	0.888225i	$-0.348061\pi$
$401$	7378.14	0.918819	0.459410	+	0.888225i	$0.348061\pi$
$402$	0	0
$403$	5742.97	0.709871
$404$	0	0
$405$	1185.99	0.145511
$406$	0	0
$407$	−2868.73	−0.349380
$408$	0	0
$409$	−3727.49	−0.450642	−0.225321	−	0.974285i	$-0.572343\pi$
$409$	−3727.49	−0.450642	−0.225321	+	0.974285i	$0.572343\pi$
$410$	0	0
$411$	−8853.40	−1.06254
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12874.7	−1.52288
$416$	0	0
$417$	−306.574	−0.0360023
$418$	0	0
$419$	−11146.7	−1.29964	−0.649822	−	0.760087i	$-0.725156\pi$
$419$	−11146.7	−1.29964	−0.649822	+	0.760087i	$0.725156\pi$
$420$	0	0
$421$	−11777.7	−1.36344	−0.681719	−	0.731614i	$-0.738768\pi$
$421$	−11777.7	−1.36344	−0.681719	+	0.731614i	$0.738768\pi$
$422$	0	0
$423$	−1798.98	−0.206783
$424$	0	0
$425$	−1683.35	−0.192128
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−714.660	−0.0804292
$430$	0	0
$431$	−5574.28	−0.622978	−0.311489	−	0.950250i	$-0.600828\pi$
$431$	−5574.28	−0.622978	−0.311489	+	0.950250i	$0.600828\pi$
$432$	0	0
$433$	240.867	0.0267328	0.0133664	−	0.999911i	$-0.495745\pi$
$433$	240.867	0.0267328	0.0133664	+	0.999911i	$0.495745\pi$
$434$	0	0
$435$	12353.4	1.36161
$436$	0	0
$437$	−2633.47	−0.288275
$438$	0	0
$439$	−9433.79	−1.02563	−0.512813	−	0.858500i	$-0.671397\pi$
$439$	−9433.79	−1.02563	−0.512813	+	0.858500i	$0.671397\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9053.72	−0.971005	−0.485502	−	0.874235i	$-0.661363\pi$
$443$	−9053.72	−0.971005	−0.485502	+	0.874235i	$0.661363\pi$
$444$	0	0
$445$	21680.1	2.30952
$446$	0	0
$447$	5396.11	0.570978
$448$	0	0
$449$	2726.05	0.286526	0.143263	−	0.989685i	$-0.454241\pi$
$449$	2726.05	0.286526	0.143263	+	0.989685i	$0.454241\pi$
$450$	0	0
$451$	139.016	0.0145144
$452$	0	0
$453$	−9590.42	−0.994696
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10628.5	1.08792	0.543960	−	0.839111i	$-0.316924\pi$
$457$	10628.5	1.08792	0.543960	+	0.839111i	$0.316924\pi$
$458$	0	0
$459$	−508.493	−0.0517090
$460$	0	0
$461$	11324.0	1.14406	0.572032	−	0.820232i	$-0.306155\pi$
$461$	11324.0	1.14406	0.572032	+	0.820232i	$0.306155\pi$
$462$	0	0
$463$	11711.0	1.17550	0.587751	−	0.809042i	$-0.300014\pi$
$463$	11711.0	1.17550	0.587751	+	0.809042i	$0.300014\pi$
$464$	0	0
$465$	12123.8	1.20910
$466$	0	0
$467$	−15480.6	−1.53396	−0.766980	−	0.641671i	$-0.778241\pi$
$467$	−15480.6	−1.53396	−0.766980	+	0.641671i	$0.778241\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	10428.8	1.02024
$472$	0	0
$473$	−746.611	−0.0725776
$474$	0	0
$475$	−8135.41	−0.785848
$476$	0	0
$477$	−577.134	−0.0553987
$478$	0	0
$479$	4747.39	0.452847	0.226423	−	0.974029i	$-0.427297\pi$
$479$	4747.39	0.452847	0.226423	+	0.974029i	$0.427297\pi$
$480$	0	0
$481$	5213.58	0.494218
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2601.45	0.243559
$486$	0	0
$487$	−7970.85	−0.741671	−0.370835	−	0.928699i	$-0.620929\pi$
$487$	−7970.85	−0.741671	−0.370835	+	0.928699i	$0.620929\pi$
$488$	0	0
$489$	11369.8	1.05145
$490$	0	0
$491$	13019.1	1.19662	0.598311	−	0.801264i	$-0.295839\pi$
$491$	13019.1	1.19662	0.598311	+	0.801264i	$0.295839\pi$
$492$	0	0
$493$	−5296.52	−0.483860
$494$	0	0
$495$	−1508.70	−0.136992
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−15490.8	−1.38971	−0.694854	−	0.719151i	$-0.744531\pi$
$499$	−15490.8	−1.38971	−0.694854	+	0.719151i	$0.744531\pi$
$500$	0	0
$501$	8134.31	0.725377
$502$	0	0
$503$	8646.86	0.766490	0.383245	−	0.923647i	$-0.374806\pi$
$503$	8646.86	0.766490	0.383245	+	0.923647i	$0.374806\pi$
$504$	0	0
$505$	−12775.5	−1.12574
$506$	0	0
$507$	−5292.19	−0.463579
$508$	0	0
$509$	−10252.0	−0.892753	−0.446376	−	0.894845i	$-0.647286\pi$
$509$	−10252.0	−0.892753	−0.446376	+	0.894845i	$0.647286\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2457.48	−0.211502
$514$	0	0
$515$	13796.9	1.18051
$516$	0	0
$517$	2288.49	0.194676
$518$	0	0
$519$	−7513.45	−0.635460
$520$	0	0
$521$	−10025.3	−0.843023	−0.421512	−	0.906823i	$-0.638500\pi$
$521$	−10025.3	−0.843023	−0.421512	+	0.906823i	$0.638500\pi$
$522$	0	0
$523$	−186.782	−0.0156164	−0.00780822	−	0.999970i	$-0.502485\pi$
$523$	−186.782	−0.0156164	−0.00780822	+	0.999970i	$0.502485\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5198.11	−0.429665
$528$	0	0
$529$	−11329.8	−0.931195
$530$	0	0
$531$	621.712	0.0508098
$532$	0	0
$533$	−252.646	−0.0205315
$534$	0	0
$535$	−17020.0	−1.37540
$536$	0	0
$537$	−6267.92	−0.503688
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9729.13	−0.773176	−0.386588	−	0.922253i	$-0.626346\pi$
$541$	−9729.13	−0.773176	−0.386588	+	0.922253i	$0.626346\pi$
$542$	0	0
$543$	12243.9	0.967651
$544$	0	0
$545$	−515.414	−0.0405099
$546$	0	0
$547$	−19387.1	−1.51541	−0.757707	−	0.652595i	$-0.773680\pi$
$547$	−19387.1	−1.51541	−0.757707	+	0.652595i	$0.773680\pi$
$548$	0	0
$549$	5905.89	0.459121
$550$	0	0
$551$	−25597.4	−1.97910
$552$	0	0
$553$	0	0
$554$	0	0
$555$	11006.3	0.841783
$556$	0	0
$557$	13182.0	1.00276	0.501381	−	0.865227i	$-0.332826\pi$
$557$	13182.0	1.00276	0.501381	+	0.865227i	$0.332826\pi$
$558$	0	0
$559$	1356.88	0.102665
$560$	0	0
$561$	646.857	0.0486815
$562$	0	0
$563$	−14211.0	−1.06381	−0.531903	−	0.846805i	$-0.678523\pi$
$563$	−14211.0	−1.06381	−0.531903	+	0.846805i	$0.678523\pi$
$564$	0	0
$565$	−6158.81	−0.458590
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−855.367	−0.0630208	−0.0315104	−	0.999503i	$-0.510032\pi$
$569$	−855.367	−0.0630208	−0.0315104	+	0.999503i	$0.510032\pi$
$570$	0	0
$571$	16827.3	1.23327	0.616637	−	0.787247i	$-0.288495\pi$
$571$	16827.3	1.23327	0.616637	+	0.787247i	$0.288495\pi$
$572$	0	0
$573$	6168.68	0.449738
$574$	0	0
$575$	2586.16	0.187566
$576$	0	0
$577$	3961.45	0.285818	0.142909	−	0.989736i	$-0.454354\pi$
$577$	3961.45	0.285818	0.142909	+	0.989736i	$0.454354\pi$
$578$	0	0
$579$	−4468.45	−0.320730
$580$	0	0
$581$	0	0
$582$	0	0
$583$	734.176	0.0521552
$584$	0	0
$585$	2741.89	0.193783
$586$	0	0
$587$	−130.006	−0.00914124	−0.00457062	−	0.999990i	$-0.501455\pi$
$587$	−130.006	−0.00914124	−0.00457062	+	0.999990i	$0.501455\pi$
$588$	0	0
$589$	−25121.8	−1.75743
$590$	0	0
$591$	4774.98	0.332346
$592$	0	0
$593$	−11730.9	−0.812361	−0.406180	−	0.913793i	$-0.633140\pi$
$593$	−11730.9	−0.812361	−0.406180	+	0.913793i	$0.633140\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12910.0	−0.885041
$598$	0	0
$599$	−2349.76	−0.160281	−0.0801407	−	0.996784i	$-0.525537\pi$
$599$	−2349.76	−0.160281	−0.0801407	+	0.996784i	$0.525537\pi$
$600$	0	0
$601$	−21900.5	−1.48642	−0.743210	−	0.669059i	$-0.766698\pi$
$601$	−21900.5	−1.48642	−0.743210	+	0.669059i	$0.766698\pi$
$602$	0	0
$603$	−3777.42	−0.255106
$604$	0	0
$605$	−17569.0	−1.18063
$606$	0	0
$607$	23463.6	1.56896	0.784480	−	0.620154i	$-0.212930\pi$
$607$	23463.6	1.56896	0.784480	+	0.620154i	$0.212930\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4159.07	−0.275381
$612$	0	0
$613$	−9012.69	−0.593832	−0.296916	−	0.954904i	$-0.595958\pi$
$613$	−9012.69	−0.593832	−0.296916	+	0.954904i	$0.595958\pi$
$614$	0	0
$615$	−533.353	−0.0349705
$616$	0	0
$617$	18605.8	1.21401	0.607004	−	0.794699i	$-0.292371\pi$
$617$	18605.8	1.21401	0.607004	+	0.794699i	$0.292371\pi$
$618$	0	0
$619$	−7917.28	−0.514091	−0.257045	−	0.966399i	$-0.582749\pi$
$619$	−7917.28	−0.514091	−0.257045	+	0.966399i	$0.582749\pi$
$620$	0	0
$621$	781.207	0.0504811
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−18808.6	−1.20375
$626$	0	0
$627$	3126.18	0.199119
$628$	0	0
$629$	−4718.95	−0.299136
$630$	0	0
$631$	−10620.3	−0.670025	−0.335013	−	0.942214i	$-0.608741\pi$
$631$	−10620.3	−0.670025	−0.335013	+	0.942214i	$0.608741\pi$
$632$	0	0
$633$	6627.04	0.416116
$634$	0	0
$635$	−10580.3	−0.661207
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9202.17	0.569690
$640$	0	0
$641$	−6334.25	−0.390309	−0.195154	−	0.980773i	$-0.562521\pi$
$641$	−6334.25	−0.390309	−0.195154	+	0.980773i	$0.562521\pi$
$642$	0	0
$643$	25268.3	1.54974	0.774872	−	0.632118i	$-0.217814\pi$
$643$	25268.3	1.54974	0.774872	+	0.632118i	$0.217814\pi$
$644$	0	0
$645$	2864.47	0.174866
$646$	0	0
$647$	−2288.39	−0.139051	−0.0695256	−	0.997580i	$-0.522149\pi$
$647$	−2288.39	−0.139051	−0.0695256	+	0.997580i	$0.522149\pi$
$648$	0	0
$649$	−790.884	−0.0478350
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12721.2	−0.762357	−0.381179	−	0.924501i	$-0.624482\pi$
$653$	−12721.2	−0.762357	−0.381179	+	0.924501i	$0.624482\pi$
$654$	0	0
$655$	11702.1	0.698073
$656$	0	0
$657$	929.523	0.0551966
$658$	0	0
$659$	15931.0	0.941707	0.470853	−	0.882211i	$-0.343946\pi$
$659$	15931.0	0.941707	0.470853	+	0.882211i	$0.343946\pi$
$660$	0	0
$661$	−3790.79	−0.223063	−0.111531	−	0.993761i	$-0.535576\pi$
$661$	−3790.79	−0.223063	−0.111531	+	0.993761i	$0.535576\pi$
$662$	0	0
$663$	−1175.59	−0.0688628
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8137.13	0.472370
$668$	0	0
$669$	−14195.7	−0.820387
$670$	0	0
$671$	−7512.92	−0.432240
$672$	0	0
$673$	−27903.0	−1.59819	−0.799096	−	0.601203i	$-0.794688\pi$
$673$	−27903.0	−1.59819	−0.799096	+	0.601203i	$0.794688\pi$
$674$	0	0
$675$	2413.33	0.137613
$676$	0	0
$677$	12845.9	0.729257	0.364629	−	0.931153i	$-0.381196\pi$
$677$	12845.9	0.729257	0.364629	+	0.931153i	$0.381196\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1866.92	0.105052
$682$	0	0
$683$	−27871.5	−1.56146	−0.780728	−	0.624870i	$-0.785152\pi$
$683$	−27871.5	−1.56146	−0.780728	+	0.624870i	$0.785152\pi$
$684$	0	0
$685$	−43209.9	−2.41017
$686$	0	0
$687$	1591.86	0.0884033
$688$	0	0
$689$	−1334.28	−0.0737765
$690$	0	0
$691$	12700.3	0.699190	0.349595	−	0.936901i	$-0.386319\pi$
$691$	12700.3	0.699190	0.349595	+	0.936901i	$0.386319\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1496.26	−0.0816641
$696$	0	0
$697$	228.676	0.0124271
$698$	0	0
$699$	9567.83	0.517723
$700$	0	0
$701$	−18491.7	−0.996320	−0.498160	−	0.867085i	$-0.665991\pi$
$701$	−18491.7	−0.996320	−0.498160	+	0.867085i	$0.665991\pi$
$702$	0	0
$703$	−22806.1	−1.22354
$704$	0	0
$705$	−8780.10	−0.469046
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10589.8	−0.560944	−0.280472	−	0.959862i	$-0.590491\pi$
$709$	−10589.8	−0.560944	−0.280472	+	0.959862i	$0.590491\pi$
$710$	0	0
$711$	2345.54	0.123720
$712$	0	0
$713$	7985.95	0.419462
$714$	0	0
$715$	−3487.97	−0.182437
$716$	0	0
$717$	4546.07	0.236787
$718$	0	0
$719$	4494.41	0.233120	0.116560	−	0.993184i	$-0.462813\pi$
$719$	4494.41	0.233120	0.116560	+	0.993184i	$0.462813\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3706.15	0.190641
$724$	0	0
$725$	25137.5	1.28770
$726$	0	0
$727$	−3251.31	−0.165866	−0.0829328	−	0.996555i	$-0.526429\pi$
$727$	−3251.31	−0.165866	−0.0829328	+	0.996555i	$0.526429\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1228.15	−0.0621403
$732$	0	0
$733$	17052.9	0.859293	0.429647	−	0.902997i	$-0.358638\pi$
$733$	17052.9	0.859293	0.429647	+	0.902997i	$0.358638\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4805.29	0.240170
$738$	0	0
$739$	33680.4	1.67653	0.838265	−	0.545264i	$-0.183571\pi$
$739$	33680.4	1.67653	0.838265	+	0.545264i	$0.183571\pi$
$740$	0	0
$741$	−5681.47	−0.281665
$742$	0	0
$743$	−18689.5	−0.922813	−0.461407	−	0.887189i	$-0.652655\pi$
$743$	−18689.5	−0.922813	−0.461407	+	0.887189i	$0.652655\pi$
$744$	0	0
$745$	26336.3	1.29515
$746$	0	0
$747$	−7913.81	−0.387619
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25028.2	−1.21610	−0.608052	−	0.793897i	$-0.708049\pi$
$751$	−25028.2	−1.21610	−0.608052	+	0.793897i	$0.708049\pi$
$752$	0	0
$753$	6495.93	0.314375
$754$	0	0
$755$	−46807.0	−2.25627
$756$	0	0
$757$	−18630.2	−0.894488	−0.447244	−	0.894412i	$-0.647594\pi$
$757$	−18630.2	−0.894488	−0.447244	+	0.894412i	$0.647594\pi$
$758$	0	0
$759$	−993.778	−0.0475255
$760$	0	0
$761$	28611.1	1.36288	0.681440	−	0.731874i	$-0.261354\pi$
$761$	28611.1	1.36288	0.681440	+	0.731874i	$0.261354\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2481.75	−0.117291
$766$	0	0
$767$	1437.34	0.0676654
$768$	0	0
$769$	15072.5	0.706798	0.353399	−	0.935473i	$-0.385026\pi$
$769$	15072.5	0.706798	0.353399	+	0.935473i	$0.385026\pi$
$770$	0	0
$771$	11842.3	0.553164
$772$	0	0
$773$	−32904.3	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$773$	−32904.3	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$774$	0	0
$775$	24670.4	1.14347
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1105.16	0.0508299
$780$	0	0
$781$	−11706.1	−0.536336
$782$	0	0
$783$	7593.34	0.346570
$784$	0	0
$785$	50898.9	2.31422
$786$	0	0
$787$	31033.3	1.40562	0.702808	−	0.711380i	$-0.251929\pi$
$787$	31033.3	1.40562	0.702808	+	0.711380i	$0.251929\pi$
$788$	0	0
$789$	−2993.94	−0.135091
$790$	0	0
$791$	0	0
$792$	0	0
$793$	13653.9	0.611428
$794$	0	0
$795$	−2816.76	−0.125661
$796$	0	0
$797$	5906.64	0.262514	0.131257	−	0.991348i	$-0.458099\pi$
$797$	5906.64	0.262514	0.131257	+	0.991348i	$0.458099\pi$
$798$	0	0
$799$	3764.48	0.166680
$800$	0	0
$801$	13326.3	0.587842
$802$	0	0
$803$	−1182.45	−0.0519649
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18087.8	0.788997
$808$	0	0
$809$	−9652.69	−0.419494	−0.209747	−	0.977756i	$-0.567264\pi$
$809$	−9652.69	−0.419494	−0.209747	+	0.977756i	$0.567264\pi$
$810$	0	0
$811$	29469.2	1.27596	0.637980	−	0.770053i	$-0.279770\pi$
$811$	29469.2	1.27596	0.637980	+	0.770053i	$0.279770\pi$
$812$	0	0
$813$	9358.94	0.403730
$814$	0	0
$815$	55491.6	2.38501
$816$	0	0
$817$	−5935.47	−0.254169
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−667.350	−0.0283687	−0.0141843	−	0.999899i	$-0.504515\pi$
$821$	−667.350	−0.0283687	−0.0141843	+	0.999899i	$0.504515\pi$
$822$	0	0
$823$	−18030.5	−0.763673	−0.381836	−	0.924230i	$-0.624708\pi$
$823$	−18030.5	−0.763673	−0.381836	+	0.924230i	$0.624708\pi$
$824$	0	0
$825$	−3070.01	−0.129556
$826$	0	0
$827$	−2362.88	−0.0993535	−0.0496767	−	0.998765i	$-0.515819\pi$
$827$	−2362.88	−0.0993535	−0.0496767	+	0.998765i	$0.515819\pi$
$828$	0	0
$829$	−26128.9	−1.09468	−0.547342	−	0.836909i	$-0.684360\pi$
$829$	−26128.9	−1.09468	−0.547342	+	0.836909i	$0.684360\pi$
$830$	0	0
$831$	−6011.40	−0.250942
$832$	0	0
$833$	0	0
$834$	0	0
$835$	39700.3	1.64537
$836$	0	0
$837$	7452.26	0.307751
$838$	0	0
$839$	39017.9	1.60554	0.802769	−	0.596290i	$-0.203359\pi$
$839$	39017.9	1.60554	0.802769	+	0.596290i	$0.203359\pi$
$840$	0	0
$841$	54704.1	2.24298
$842$	0	0
$843$	−26746.0	−1.09274
$844$	0	0
$845$	−25829.1	−1.05153
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9717.26	−0.392810
$850$	0	0
$851$	7249.80	0.292033
$852$	0	0
$853$	−24487.4	−0.982922	−0.491461	−	0.870900i	$-0.663537\pi$
$853$	−24487.4	−0.982922	−0.491461	+	0.870900i	$0.663537\pi$
$854$	0	0
$855$	−11994.0	−0.479750
$856$	0	0
$857$	2351.90	0.0937450	0.0468725	−	0.998901i	$-0.485075\pi$
$857$	2351.90	0.0937450	0.0468725	+	0.998901i	$0.485075\pi$
$858$	0	0
$859$	5413.01	0.215005	0.107503	−	0.994205i	$-0.465715\pi$
$859$	5413.01	0.215005	0.107503	+	0.994205i	$0.465715\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32772.7	1.29270	0.646348	−	0.763043i	$-0.276295\pi$
$863$	32772.7	1.29270	0.646348	+	0.763043i	$0.276295\pi$
$864$	0	0
$865$	−36670.1	−1.44141
$866$	0	0
$867$	−13674.9	−0.535670
$868$	0	0
$869$	−2983.78	−0.116476
$870$	0	0
$871$	−8733.05	−0.339734
$872$	0	0
$873$	1599.06	0.0619930
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−40592.0	−1.56294	−0.781468	−	0.623946i	$-0.785529\pi$
$877$	−40592.0	−1.56294	−0.781468	+	0.623946i	$0.785529\pi$
$878$	0	0
$879$	17991.1	0.690357
$880$	0	0
$881$	21588.5	0.825578	0.412789	−	0.910827i	$-0.364555\pi$
$881$	21588.5	0.825578	0.412789	+	0.910827i	$0.364555\pi$
$882$	0	0
$883$	11223.7	0.427757	0.213878	−	0.976860i	$-0.431390\pi$
$883$	11223.7	0.427757	0.213878	+	0.976860i	$0.431390\pi$
$884$	0	0
$885$	3034.33	0.115252
$886$	0	0
$887$	−36557.9	−1.38387	−0.691936	−	0.721958i	$-0.743242\pi$
$887$	−36557.9	−1.38387	−0.691936	+	0.721958i	$0.743242\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−927.365	−0.0348686
$892$	0	0
$893$	18193.2	0.681762
$894$	0	0
$895$	−30591.2	−1.14252
$896$	0	0
$897$	1806.08	0.0672276
$898$	0	0
$899$	77623.5	2.87974
$900$	0	0
$901$	1207.69	0.0446548
$902$	0	0
$903$	0	0
$904$	0	0
$905$	59757.4	2.19492
$906$	0	0
$907$	−19252.0	−0.704798	−0.352399	−	0.935850i	$-0.614634\pi$
$907$	−19252.0	−0.704798	−0.352399	+	0.935850i	$0.614634\pi$
$908$	0	0
$909$	−7852.81	−0.286536
$910$	0	0
$911$	4910.11	0.178572	0.0892861	−	0.996006i	$-0.471541\pi$
$911$	4910.11	0.178572	0.0892861	+	0.996006i	$0.471541\pi$
$912$	0	0
$913$	10067.2	0.364925
$914$	0	0
$915$	28824.3	1.04142
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1908.48	−0.0685039	−0.0342520	−	0.999413i	$-0.510905\pi$
$919$	−1908.48	−0.0685039	−0.0342520	+	0.999413i	$0.510905\pi$
$920$	0	0
$921$	−29029.0	−1.03859
$922$	0	0
$923$	21274.5	0.758678
$924$	0	0
$925$	22396.3	0.796093
$926$	0	0
$927$	8480.63	0.300475
$928$	0	0
$929$	30463.0	1.07584	0.537921	−	0.842995i	$-0.319210\pi$
$929$	30463.0	1.07584	0.537921	+	0.842995i	$0.319210\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4375.65	0.153540
$934$	0	0
$935$	3157.05	0.110424
$936$	0	0
$937$	468.146	0.0163219	0.00816097	−	0.999967i	$-0.497402\pi$
$937$	468.146	0.0163219	0.00816097	+	0.999967i	$0.497402\pi$
$938$	0	0
$939$	30347.0	1.05467
$940$	0	0
$941$	−8443.67	−0.292514	−0.146257	−	0.989247i	$-0.546723\pi$
$941$	−8443.67	−0.292514	−0.146257	+	0.989247i	$0.546723\pi$
$942$	0	0
$943$	−351.319	−0.0121320
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7269.71	−0.249455	−0.124727	−	0.992191i	$-0.539806\pi$
$947$	−7269.71	−0.249455	−0.124727	+	0.992191i	$0.539806\pi$
$948$	0	0
$949$	2148.97	0.0735074
$950$	0	0
$951$	25548.2	0.871142
$952$	0	0
$953$	25806.9	0.877195	0.438598	−	0.898684i	$-0.355475\pi$
$953$	25806.9	0.877195	0.438598	+	0.898684i	$0.355475\pi$
$954$	0	0
$955$	30106.9	1.02014
$956$	0	0
$957$	−9659.54	−0.326279
$958$	0	0
$959$	0	0
$960$	0	0
$961$	46390.3	1.55719
$962$	0	0
$963$	−10461.8	−0.350081
$964$	0	0
$965$	−21808.7	−0.727511
$966$	0	0
$967$	7155.61	0.237962	0.118981	−	0.992897i	$-0.462037\pi$
$967$	7155.61	0.237962	0.118981	+	0.992897i	$0.462037\pi$
$968$	0	0
$969$	5142.44	0.170484
$970$	0	0
$971$	29970.2	0.990515	0.495257	−	0.868746i	$-0.335074\pi$
$971$	29970.2	0.990515	0.495257	+	0.868746i	$0.335074\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	5579.39	0.183265
$976$	0	0
$977$	−46223.3	−1.51363	−0.756813	−	0.653631i	$-0.773245\pi$
$977$	−46223.3	−1.51363	−0.756813	+	0.653631i	$0.773245\pi$
$978$	0	0
$979$	−16952.5	−0.553425
$980$	0	0
$981$	−316.814	−0.0103110
$982$	0	0
$983$	41135.8	1.33472	0.667360	−	0.744736i	$-0.267424\pi$
$983$	41135.8	1.33472	0.667360	+	0.744736i	$0.267424\pi$
$984$	0	0
$985$	23304.8	0.753860
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1886.82	0.0606647
$990$	0	0
$991$	−4228.11	−0.135530	−0.0677651	−	0.997701i	$-0.521587\pi$
$991$	−4228.11	−0.135530	−0.0677651	+	0.997701i	$0.521587\pi$
$992$	0	0
$993$	−8401.67	−0.268499
$994$	0	0
$995$	−63008.4	−2.00754
$996$	0	0
$997$	−38767.3	−1.23147	−0.615734	−	0.787954i	$-0.711140\pi$
$997$	−38767.3	−1.23147	−0.615734	+	0.787954i	$0.711140\pi$
$998$	0	0
$999$	6765.31	0.214259

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.4.a.be.1.4	yes	4
4.3	odd	2		2352.4.a.cn.1.4		4
7.6	odd	2		1176.4.a.z.1.1	✓	4
28.27	even	2		2352.4.a.co.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.z.1.1	✓	4	7.6	odd	2
1176.4.a.be.1.4	yes	4	1.1	even	1	trivial
2352.4.a.cn.1.4		4	4.3	odd	2
2352.4.a.co.1.1		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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +14.6418 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +14.6418 q^{5} +9.00000 q^{9} -11.4490 q^{11} +20.8071 q^{13} +43.9254 q^{15} -18.8331 q^{17} -91.0178 q^{19} +28.9336 q^{23} +89.3825 q^{25} +27.0000 q^{27} +281.235 q^{29} +276.010 q^{31} -34.3469 q^{33} +250.567 q^{37} +62.4214 q^{39} -12.1422 q^{41} +65.2121 q^{43} +131.776 q^{45} -199.886 q^{47} -56.4992 q^{51} -64.1260 q^{53} -167.633 q^{55} -273.054 q^{57} +69.0791 q^{59} +656.210 q^{61} +304.654 q^{65} -419.714 q^{67} +86.8007 q^{69} +1022.46 q^{71} +103.280 q^{73} +268.148 q^{75} +260.616 q^{79} +81.0000 q^{81} -879.313 q^{83} -275.750 q^{85} +843.705 q^{87} +1480.70 q^{89} +828.029 q^{93} -1332.67 q^{95} +177.673 q^{97} -103.041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9	\( 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9} + 40 q^{11} + 48 q^{13} + 24 q^{15} + 72 q^{17} + 32 q^{19} + 8 q^{23} + 164 q^{25} + 108 q^{27} + 144 q^{29} + 48 q^{31} + 120 q^{33} + 48 q^{37} + 144 q^{39} + 72 q^{41} + 512 q^{43} + 72 q^{45} + 160 q^{47} + 216 q^{51} + 536 q^{53} - 336 q^{55} + 96 q^{57} + 240 q^{59} + 896 q^{61} - 136 q^{65} + 1088 q^{67} + 24 q^{69} + 1288 q^{71} + 1488 q^{73} + 492 q^{75} + 416 q^{79} + 324 q^{81} - 112 q^{83} - 1512 q^{85} + 432 q^{87} + 3160 q^{89} + 144 q^{93} - 240 q^{95} + 2384 q^{97} + 360 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9 + 40 * q^11 + 48 * q^13 + 24 * q^15 + 72 * q^17 + 32 * q^19 + 8 * q^23 + 164 * q^25 + 108 * q^27 + 144 * q^29 + 48 * q^31 + 120 * q^33 + 48 * q^37 + 144 * q^39 + 72 * q^41 + 512 * q^43 + 72 * q^45 + 160 * q^47 + 216 * q^51 + 536 * q^53 - 336 * q^55 + 96 * q^57 + 240 * q^59 + 896 * q^61 - 136 * q^65 + 1088 * q^67 + 24 * q^69 + 1288 * q^71 + 1488 * q^73 + 492 * q^75 + 416 * q^79 + 324 * q^81 - 112 * q^83 - 1512 * q^85 + 432 * q^87 + 3160 * q^89 + 144 * q^93 - 240 * q^95 + 2384 * q^97 + 360 * q^99

Introduction

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