LMFDB - Embedded newform 1176.4.a.bd.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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 152x^{2} - 177x + 2922 $ x^4 - 2x^3 - 152x^2 - 177*x + 2922
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-8.69515$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +0.128591 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +0.128591 q^{5} +9.00000 q^{9} +54.0066 q^{11} -50.2350 q^{13} +0.385773 q^{15} -131.487 q^{17} +91.5094 q^{19} +179.487 q^{23} -124.983 q^{25} +27.0000 q^{27} -69.8961 q^{29} +326.847 q^{31} +162.020 q^{33} +301.698 q^{37} -150.705 q^{39} +296.048 q^{41} -144.302 q^{43} +1.15732 q^{45} -360.085 q^{47} -394.462 q^{51} +1.83513 q^{53} +6.94477 q^{55} +274.528 q^{57} -53.2386 q^{59} -108.121 q^{61} -6.45978 q^{65} +842.007 q^{67} +538.462 q^{69} -241.111 q^{71} -206.983 q^{73} -374.950 q^{75} +559.962 q^{79} +81.0000 q^{81} +986.652 q^{83} -16.9081 q^{85} -209.688 q^{87} -443.366 q^{89} +980.541 q^{93} +11.7673 q^{95} -740.815 q^{97} +486.059 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 4 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 4 * q^5 + 36 * q^9	$ 4 q + 12 q^{3} + 4 q^{5} + 36 q^{9} + 14 q^{11} + 22 q^{13} + 12 q^{15} + 96 q^{17} - 26 q^{19} + 96 q^{23} + 110 q^{25} + 108 q^{27} - 76 q^{29} + 238 q^{31} + 42 q^{33} + 562 q^{37} + 66 q^{39} + 428 q^{41} - 258 q^{43} + 36 q^{45} - 80 q^{47} + 288 q^{51} + 1476 q^{55} - 78 q^{57} + 262 q^{59} - 276 q^{61} + 2196 q^{65} + 150 q^{67} + 288 q^{69} - 848 q^{71} - 218 q^{73} + 330 q^{75} + 1762 q^{79} + 324 q^{81} + 3450 q^{83} + 1452 q^{85} - 228 q^{87} - 344 q^{89} + 714 q^{93} + 2004 q^{95} - 622 q^{97} + 126 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 + 4 * q^5 + 36 * q^9 + 14 * q^11 + 22 * q^13 + 12 * q^15 + 96 * q^17 - 26 * q^19 + 96 * q^23 + 110 * q^25 + 108 * q^27 - 76 * q^29 + 238 * q^31 + 42 * q^33 + 562 * q^37 + 66 * q^39 + 428 * q^41 - 258 * q^43 + 36 * q^45 - 80 * q^47 + 288 * q^51 + 1476 * q^55 - 78 * q^57 + 262 * q^59 - 276 * q^61 + 2196 * q^65 + 150 * q^67 + 288 * q^69 - 848 * q^71 - 218 * q^73 + 330 * q^75 + 1762 * q^79 + 324 * q^81 + 3450 * q^83 + 1452 * q^85 - 228 * q^87 - 344 * q^89 + 714 * q^93 + 2004 * q^95 - 622 * q^97 + 126 * 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
$5$	0.128591	0.0115015	0.00575077	−	0.999983i	$-0.498169\pi$
$5$	0.128591	0.0115015	0.00575077	+	0.999983i	$0.498169\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	54.0066	1.48033	0.740163	−	0.672427i	$-0.234748\pi$
$11$	54.0066	1.48033	0.740163	+	0.672427i	$0.234748\pi$
$12$	0	0
$13$	−50.2350	−1.07175	−0.535873	−	0.844299i	$-0.680017\pi$
$13$	−50.2350	−1.07175	−0.535873	+	0.844299i	$0.680017\pi$
$14$	0	0
$15$	0.385773	0.00664042
$16$	0	0
$17$	−131.487	−1.87590	−0.937951	−	0.346767i	$-0.887279\pi$
$17$	−131.487	−1.87590	−0.937951	+	0.346767i	$0.887279\pi$
$18$	0	0
$19$	91.5094	1.10493	0.552466	−	0.833536i	$-0.313687\pi$
$19$	91.5094	1.10493	0.552466	+	0.833536i	$0.313687\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	179.487	1.62720	0.813602	−	0.581423i	$-0.197504\pi$
$23$	179.487	1.62720	0.813602	+	0.581423i	$0.197504\pi$
$24$	0	0
$25$	−124.983	−0.999868
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−69.8961	−0.447565	−0.223782	−	0.974639i	$-0.571841\pi$
$29$	−69.8961	−0.447565	−0.223782	+	0.974639i	$0.571841\pi$
$30$	0	0
$31$	326.847	1.89366	0.946829	−	0.321736i	$-0.104266\pi$
$31$	326.847	1.89366	0.946829	+	0.321736i	$0.104266\pi$
$32$	0	0
$33$	162.020	0.854667
$34$	0	0
$35$	0	0
$36$	0	0
$37$	301.698	1.34051	0.670254	−	0.742132i	$-0.266185\pi$
$37$	301.698	1.34051	0.670254	+	0.742132i	$0.266185\pi$
$38$	0	0
$39$	−150.705	−0.618773
$40$	0	0
$41$	296.048	1.12768	0.563840	−	0.825884i	$-0.309324\pi$
$41$	296.048	1.12768	0.563840	+	0.825884i	$0.309324\pi$
$42$	0	0
$43$	−144.302	−0.511764	−0.255882	−	0.966708i	$-0.582366\pi$
$43$	−144.302	−0.511764	−0.255882	+	0.966708i	$0.582366\pi$
$44$	0	0
$45$	1.15732	0.00383385
$46$	0	0
$47$	−360.085	−1.11753	−0.558764	−	0.829326i	$-0.688724\pi$
$47$	−360.085	−1.11753	−0.558764	+	0.829326i	$0.688724\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−394.462	−1.08305
$52$	0	0
$53$	1.83513	0.00475613	0.00237807	−	0.999997i	$-0.499243\pi$
$53$	1.83513	0.00475613	0.00237807	+	0.999997i	$0.499243\pi$
$54$	0	0
$55$	6.94477	0.0170260
$56$	0	0
$57$	274.528	0.637932
$58$	0	0
$59$	−53.2386	−0.117476	−0.0587379	−	0.998273i	$-0.518708\pi$
$59$	−53.2386	−0.117476	−0.0587379	+	0.998273i	$0.518708\pi$
$60$	0	0
$61$	−108.121	−0.226942	−0.113471	−	0.993541i	$-0.536197\pi$
$61$	−108.121	−0.226942	−0.113471	+	0.993541i	$0.536197\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.45978	−0.0123267
$66$	0	0
$67$	842.007	1.53534	0.767668	−	0.640848i	$-0.221417\pi$
$67$	842.007	1.53534	0.767668	+	0.640848i	$0.221417\pi$
$68$	0	0
$69$	538.462	0.939466
$70$	0	0
$71$	−241.111	−0.403023	−0.201511	−	0.979486i	$-0.564585\pi$
$71$	−241.111	−0.403023	−0.201511	+	0.979486i	$0.564585\pi$
$72$	0	0
$73$	−206.983	−0.331857	−0.165929	−	0.986138i	$-0.553062\pi$
$73$	−206.983	−0.331857	−0.165929	+	0.986138i	$0.553062\pi$
$74$	0	0
$75$	−374.950	−0.577274
$76$	0	0
$77$	0	0
$78$	0	0
$79$	559.962	0.797477	0.398738	−	0.917065i	$-0.369448\pi$
$79$	559.962	0.797477	0.398738	+	0.917065i	$0.369448\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	986.652	1.30481	0.652404	−	0.757871i	$-0.273760\pi$
$83$	986.652	1.30481	0.652404	+	0.757871i	$0.273760\pi$
$84$	0	0
$85$	−16.9081	−0.0215758
$86$	0	0
$87$	−209.688	−0.258402
$88$	0	0
$89$	−443.366	−0.528053	−0.264026	−	0.964515i	$-0.585051\pi$
$89$	−443.366	−0.528053	−0.264026	+	0.964515i	$0.585051\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	980.541	1.09330
$94$	0	0
$95$	11.7673	0.0127084
$96$	0	0
$97$	−740.815	−0.775447	−0.387723	−	0.921776i	$-0.626738\pi$
$97$	−740.815	−0.775447	−0.387723	+	0.921776i	$0.626738\pi$
$98$	0	0
$99$	486.059	0.493442
$100$	0	0
$101$	743.777	0.732758	0.366379	−	0.930466i	$-0.380597\pi$
$101$	743.777	0.732758	0.366379	+	0.930466i	$0.380597\pi$
$102$	0	0
$103$	104.651	0.100112	0.0500559	−	0.998746i	$-0.484060\pi$
$103$	104.651	0.100112	0.0500559	+	0.998746i	$0.484060\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	513.700	0.464124	0.232062	−	0.972701i	$-0.425453\pi$
$107$	513.700	0.464124	0.232062	+	0.972701i	$0.425453\pi$
$108$	0	0
$109$	975.038	0.856805	0.428402	−	0.903588i	$-0.359077\pi$
$109$	975.038	0.856805	0.428402	+	0.903588i	$0.359077\pi$
$110$	0	0
$111$	905.093	0.773942
$112$	0	0
$113$	1926.07	1.60345	0.801723	−	0.597696i	$-0.203917\pi$
$113$	1926.07	1.60345	0.801723	+	0.597696i	$0.203917\pi$
$114$	0	0
$115$	23.0805	0.0187153
$116$	0	0
$117$	−452.115	−0.357248
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1585.71	1.19137
$122$	0	0
$123$	888.143	0.651066
$124$	0	0
$125$	−32.1457	−0.0230016
$126$	0	0
$127$	−1125.95	−0.786709	−0.393355	−	0.919387i	$-0.628686\pi$
$127$	−1125.95	−0.786709	−0.393355	+	0.919387i	$0.628686\pi$
$128$	0	0
$129$	−432.906	−0.295467
$130$	0	0
$131$	1493.24	0.995918	0.497959	−	0.867201i	$-0.334083\pi$
$131$	1493.24	0.995918	0.497959	+	0.867201i	$0.334083\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.47196	0.00221347
$136$	0	0
$137$	1460.77	0.910965	0.455483	−	0.890245i	$-0.349467\pi$
$137$	1460.77	0.910965	0.455483	+	0.890245i	$0.349467\pi$
$138$	0	0
$139$	2225.85	1.35823	0.679116	−	0.734031i	$-0.262363\pi$
$139$	2225.85	1.35823	0.679116	+	0.734031i	$0.262363\pi$
$140$	0	0
$141$	−1080.26	−0.645206
$142$	0	0
$143$	−2713.02	−1.58653
$144$	0	0
$145$	−8.98802	−0.00514769
$146$	0	0
$147$	0	0
$148$	0	0
$149$	394.681	0.217003	0.108502	−	0.994096i	$-0.465395\pi$
$149$	394.681	0.217003	0.108502	+	0.994096i	$0.465395\pi$
$150$	0	0
$151$	−3124.20	−1.68373	−0.841867	−	0.539685i	$-0.818543\pi$
$151$	−3124.20	−1.68373	−0.841867	+	0.539685i	$0.818543\pi$
$152$	0	0
$153$	−1183.39	−0.625301
$154$	0	0
$155$	42.0296	0.0217800
$156$	0	0
$157$	−3600.71	−1.83037	−0.915183	−	0.403038i	$-0.867954\pi$
$157$	−3600.71	−1.83037	−0.915183	+	0.403038i	$0.867954\pi$
$158$	0	0
$159$	5.50540	0.00274596
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1974.02	0.948573	0.474287	−	0.880370i	$-0.342706\pi$
$163$	1974.02	0.948573	0.474287	+	0.880370i	$0.342706\pi$
$164$	0	0
$165$	20.8343	0.00982999
$166$	0	0
$167$	1067.09	0.494453	0.247227	−	0.968958i	$-0.420481\pi$
$167$	1067.09	0.494453	0.247227	+	0.968958i	$0.420481\pi$
$168$	0	0
$169$	326.559	0.148638
$170$	0	0
$171$	823.584	0.368310
$172$	0	0
$173$	−274.655	−0.120703	−0.0603515	−	0.998177i	$-0.519222\pi$
$173$	−274.655	−0.120703	−0.0603515	+	0.998177i	$0.519222\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−159.716	−0.0678247
$178$	0	0
$179$	554.332	0.231468	0.115734	−	0.993280i	$-0.463078\pi$
$179$	554.332	0.231468	0.115734	+	0.993280i	$0.463078\pi$
$180$	0	0
$181$	−685.436	−0.281481	−0.140741	−	0.990047i	$-0.544948\pi$
$181$	−685.436	−0.281481	−0.140741	+	0.990047i	$0.544948\pi$
$182$	0	0
$183$	−324.363	−0.131025
$184$	0	0
$185$	38.7956	0.0154179
$186$	0	0
$187$	−7101.18	−2.77695
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4898.87	1.85586	0.927932	−	0.372750i	$-0.121585\pi$
$191$	4898.87	1.85586	0.927932	+	0.372750i	$0.121585\pi$
$192$	0	0
$193$	3140.26	1.17120	0.585598	−	0.810601i	$-0.300860\pi$
$193$	3140.26	1.17120	0.585598	+	0.810601i	$0.300860\pi$
$194$	0	0
$195$	−19.3793	−0.00711684
$196$	0	0
$197$	−227.412	−0.0822460	−0.0411230	−	0.999154i	$-0.513094\pi$
$197$	−227.412	−0.0822460	−0.0411230	+	0.999154i	$0.513094\pi$
$198$	0	0
$199$	−1214.50	−0.432631	−0.216316	−	0.976323i	$-0.569404\pi$
$199$	−1214.50	−0.432631	−0.216316	+	0.976323i	$0.569404\pi$
$200$	0	0
$201$	2526.02	0.886427
$202$	0	0
$203$	0	0
$204$	0	0
$205$	38.0691	0.0129700
$206$	0	0
$207$	1615.39	0.542401
$208$	0	0
$209$	4942.11	1.63566
$210$	0	0
$211$	5116.07	1.66922	0.834608	−	0.550844i	$-0.185694\pi$
$211$	5116.07	1.66922	0.834608	+	0.550844i	$0.185694\pi$
$212$	0	0
$213$	−723.333	−0.232685
$214$	0	0
$215$	−18.5560	−0.00588608
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−620.950	−0.191598
$220$	0	0
$221$	6605.27	2.01049
$222$	0	0
$223$	119.384	0.0358499	0.0179250	−	0.999839i	$-0.494294\pi$
$223$	119.384	0.0358499	0.0179250	+	0.999839i	$0.494294\pi$
$224$	0	0
$225$	−1124.85	−0.333289
$226$	0	0
$227$	1992.00	0.582439	0.291220	−	0.956656i	$-0.405939\pi$
$227$	1992.00	0.582439	0.291220	+	0.956656i	$0.405939\pi$
$228$	0	0
$229$	703.476	0.203000	0.101500	−	0.994836i	$-0.467636\pi$
$229$	703.476	0.203000	0.101500	+	0.994836i	$0.467636\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	961.783	0.270423	0.135211	−	0.990817i	$-0.456829\pi$
$233$	961.783	0.270423	0.135211	+	0.990817i	$0.456829\pi$
$234$	0	0
$235$	−46.3038	−0.0128533
$236$	0	0
$237$	1679.89	0.460423
$238$	0	0
$239$	−4464.71	−1.20836	−0.604179	−	0.796848i	$-0.706499\pi$
$239$	−4464.71	−1.20836	−0.604179	+	0.796848i	$0.706499\pi$
$240$	0	0
$241$	−435.311	−0.116352	−0.0581761	−	0.998306i	$-0.518528\pi$
$241$	−435.311	−0.116352	−0.0581761	+	0.998306i	$0.518528\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4596.98	−1.18421
$248$	0	0
$249$	2959.95	0.753331
$250$	0	0
$251$	863.003	0.217021	0.108510	−	0.994095i	$-0.465392\pi$
$251$	863.003	0.217021	0.108510	+	0.994095i	$0.465392\pi$
$252$	0	0
$253$	9693.49	2.40879
$254$	0	0
$255$	−50.7243	−0.0124568
$256$	0	0
$257$	−536.693	−0.130264	−0.0651322	−	0.997877i	$-0.520747\pi$
$257$	−536.693	−0.130264	−0.0651322	+	0.997877i	$0.520747\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−629.065	−0.149188
$262$	0	0
$263$	371.908	0.0871971	0.0435986	−	0.999049i	$-0.486118\pi$
$263$	371.908	0.0871971	0.0435986	+	0.999049i	$0.486118\pi$
$264$	0	0
$265$	0.235982	5.47029e−5 0
$266$	0	0
$267$	−1330.10	−0.304871
$268$	0	0
$269$	−4505.61	−1.02123	−0.510616	−	0.859809i	$-0.670583\pi$
$269$	−4505.61	−1.02123	−0.510616	+	0.859809i	$0.670583\pi$
$270$	0	0
$271$	−396.115	−0.0887908	−0.0443954	−	0.999014i	$-0.514136\pi$
$271$	−396.115	−0.0887908	−0.0443954	+	0.999014i	$0.514136\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6749.93	−1.48013
$276$	0	0
$277$	−6508.74	−1.41181	−0.705907	−	0.708305i	$-0.749460\pi$
$277$	−6508.74	−1.41181	−0.705907	+	0.708305i	$0.749460\pi$
$278$	0	0
$279$	2941.62	0.631220
$280$	0	0
$281$	2785.80	0.591413	0.295707	−	0.955279i	$-0.404445\pi$
$281$	2785.80	0.591413	0.295707	+	0.955279i	$0.404445\pi$
$282$	0	0
$283$	0.611846	0.000128518 0	6.42588e−5	−	1.00000i	$-0.499980\pi$
$283$	0.611846	0.000128518 0	6.42588e−5		1.00000i	$0.499980\pi$
$284$	0	0
$285$	35.3019	0.00733720
$286$	0	0
$287$	0	0
$288$	0	0
$289$	12375.9	2.51901
$290$	0	0
$291$	−2222.44	−0.447704
$292$	0	0
$293$	−4145.98	−0.826657	−0.413329	−	0.910582i	$-0.635634\pi$
$293$	−4145.98	−0.826657	−0.413329	+	0.910582i	$0.635634\pi$
$294$	0	0
$295$	−6.84601	−0.00135115
$296$	0	0
$297$	1458.18	0.284889
$298$	0	0
$299$	−9016.55	−1.74395
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2231.33	0.423058
$304$	0	0
$305$	−13.9034	−0.00261019
$306$	0	0
$307$	1960.53	0.364473	0.182236	−	0.983255i	$-0.441666\pi$
$307$	1960.53	0.364473	0.182236	+	0.983255i	$0.441666\pi$
$308$	0	0
$309$	313.952	0.0577996
$310$	0	0
$311$	5207.57	0.949499	0.474749	−	0.880121i	$-0.342539\pi$
$311$	5207.57	0.949499	0.474749	+	0.880121i	$0.342539\pi$
$312$	0	0
$313$	−3991.71	−0.720846	−0.360423	−	0.932789i	$-0.617368\pi$
$313$	−3991.71	−0.720846	−0.360423	+	0.932789i	$0.617368\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1805.04	0.319814	0.159907	−	0.987132i	$-0.448881\pi$
$317$	1805.04	0.319814	0.159907	+	0.987132i	$0.448881\pi$
$318$	0	0
$319$	−3774.85	−0.662543
$320$	0	0
$321$	1541.10	0.267962
$322$	0	0
$323$	−12032.3	−2.07274
$324$	0	0
$325$	6278.55	1.07160
$326$	0	0
$327$	2925.11	0.494676
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6906.18	1.14682	0.573411	−	0.819268i	$-0.305620\pi$
$331$	6906.18	1.14682	0.573411	+	0.819268i	$0.305620\pi$
$332$	0	0
$333$	2715.28	0.446836
$334$	0	0
$335$	108.275	0.0176587
$336$	0	0
$337$	−6081.36	−0.983006	−0.491503	−	0.870876i	$-0.663552\pi$
$337$	−6081.36	−0.983006	−0.491503	+	0.870876i	$0.663552\pi$
$338$	0	0
$339$	5778.21	0.925750
$340$	0	0
$341$	17651.9	2.80323
$342$	0	0
$343$	0	0
$344$	0	0
$345$	69.2414	0.0108053
$346$	0	0
$347$	−7787.02	−1.20469	−0.602347	−	0.798234i	$-0.705768\pi$
$347$	−7787.02	−1.20469	−0.602347	+	0.798234i	$0.705768\pi$
$348$	0	0
$349$	−1928.25	−0.295751	−0.147875	−	0.989006i	$-0.547243\pi$
$349$	−1928.25	−0.295751	−0.147875	+	0.989006i	$0.547243\pi$
$350$	0	0
$351$	−1356.35	−0.206258
$352$	0	0
$353$	−3404.27	−0.513289	−0.256645	−	0.966506i	$-0.582617\pi$
$353$	−3404.27	−0.513289	−0.256645	+	0.966506i	$0.582617\pi$
$354$	0	0
$355$	−31.0047	−0.00463538
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4417.05	0.649368	0.324684	−	0.945823i	$-0.394742\pi$
$359$	4417.05	0.649368	0.324684	+	0.945823i	$0.394742\pi$
$360$	0	0
$361$	1514.97	0.220873
$362$	0	0
$363$	4757.13	0.687837
$364$	0	0
$365$	−26.6162	−0.00381687
$366$	0	0
$367$	−5076.65	−0.722068	−0.361034	−	0.932553i	$-0.617576\pi$
$367$	−5076.65	−0.722068	−0.361034	+	0.932553i	$0.617576\pi$
$368$	0	0
$369$	2664.43	0.375893
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6876.64	−0.954582	−0.477291	−	0.878745i	$-0.658381\pi$
$373$	−6876.64	−0.954582	−0.477291	+	0.878745i	$0.658381\pi$
$374$	0	0
$375$	−96.4370	−0.0132800
$376$	0	0
$377$	3511.23	0.479676
$378$	0	0
$379$	−9285.61	−1.25850	−0.629248	−	0.777205i	$-0.716637\pi$
$379$	−9285.61	−1.25850	−0.629248	+	0.777205i	$0.716637\pi$
$380$	0	0
$381$	−3377.85	−0.454207
$382$	0	0
$383$	−7681.65	−1.02484	−0.512420	−	0.858735i	$-0.671251\pi$
$383$	−7681.65	−1.02484	−0.512420	+	0.858735i	$0.671251\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1298.72	−0.170588
$388$	0	0
$389$	8627.72	1.12453	0.562266	−	0.826956i	$-0.309930\pi$
$389$	8627.72	1.12453	0.562266	+	0.826956i	$0.309930\pi$
$390$	0	0
$391$	−23600.3	−3.05247
$392$	0	0
$393$	4479.73	0.574993
$394$	0	0
$395$	72.0062	0.00917221
$396$	0	0
$397$	5735.61	0.725094	0.362547	−	0.931966i	$-0.381907\pi$
$397$	5735.61	0.725094	0.362547	+	0.931966i	$0.381907\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7217.79	−0.898850	−0.449425	−	0.893318i	$-0.648371\pi$
$401$	−7217.79	−0.898850	−0.449425	+	0.893318i	$0.648371\pi$
$402$	0	0
$403$	−16419.2	−2.02952
$404$	0	0
$405$	10.4159	0.00127795
$406$	0	0
$407$	16293.7	1.98439
$408$	0	0
$409$	−10802.9	−1.30604	−0.653019	−	0.757341i	$-0.726498\pi$
$409$	−10802.9	−1.30604	−0.653019	+	0.757341i	$0.726498\pi$
$410$	0	0
$411$	4382.32	0.525946
$412$	0	0
$413$	0	0
$414$	0	0
$415$	126.875	0.0150073
$416$	0	0
$417$	6677.55	0.784175
$418$	0	0
$419$	13257.1	1.54571	0.772856	−	0.634582i	$-0.218828\pi$
$419$	13257.1	1.54571	0.772856	+	0.634582i	$0.218828\pi$
$420$	0	0
$421$	−6252.11	−0.723774	−0.361887	−	0.932222i	$-0.617867\pi$
$421$	−6252.11	−0.723774	−0.361887	+	0.932222i	$0.617867\pi$
$422$	0	0
$423$	−3240.77	−0.372510
$424$	0	0
$425$	16433.7	1.87565
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−8139.07	−0.915986
$430$	0	0
$431$	4949.08	0.553106	0.276553	−	0.960999i	$-0.410808\pi$
$431$	4949.08	0.553106	0.276553	+	0.960999i	$0.410808\pi$
$432$	0	0
$433$	−16602.8	−1.84267	−0.921337	−	0.388764i	$-0.872902\pi$
$433$	−16602.8	−1.84267	−0.921337	+	0.388764i	$0.872902\pi$
$434$	0	0
$435$	−26.9641	−0.00297202
$436$	0	0
$437$	16424.8	1.79795
$438$	0	0
$439$	−4708.38	−0.511887	−0.255944	−	0.966692i	$-0.582386\pi$
$439$	−4708.38	−0.511887	−0.255944	+	0.966692i	$0.582386\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1700.44	−0.182370	−0.0911852	−	0.995834i	$-0.529066\pi$
$443$	−1700.44	−0.182370	−0.0911852	+	0.995834i	$0.529066\pi$
$444$	0	0
$445$	−57.0129	−0.00607342
$446$	0	0
$447$	1184.04	0.125287
$448$	0	0
$449$	10050.2	1.05635	0.528173	−	0.849137i	$-0.322877\pi$
$449$	10050.2	1.05635	0.528173	+	0.849137i	$0.322877\pi$
$450$	0	0
$451$	15988.5	1.66933
$452$	0	0
$453$	−9372.60	−0.972104
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10991.9	−1.12512	−0.562560	−	0.826757i	$-0.690183\pi$
$457$	−10991.9	−1.12512	−0.562560	+	0.826757i	$0.690183\pi$
$458$	0	0
$459$	−3550.16	−0.361018
$460$	0	0
$461$	−548.440	−0.0554086	−0.0277043	−	0.999616i	$-0.508820\pi$
$461$	−548.440	−0.0554086	−0.0277043	+	0.999616i	$0.508820\pi$
$462$	0	0
$463$	4028.04	0.404317	0.202159	−	0.979353i	$-0.435204\pi$
$463$	4028.04	0.404317	0.202159	+	0.979353i	$0.435204\pi$
$464$	0	0
$465$	126.089	0.0125747
$466$	0	0
$467$	2070.69	0.205182	0.102591	−	0.994724i	$-0.467287\pi$
$467$	2070.69	0.205182	0.102591	+	0.994724i	$0.467287\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10802.1	−1.05676
$472$	0	0
$473$	−7793.26	−0.757579
$474$	0	0
$475$	−11437.2	−1.10479
$476$	0	0
$477$	16.5162	0.00158538
$478$	0	0
$479$	−178.785	−0.0170541	−0.00852704	−	0.999964i	$-0.502714\pi$
$479$	−178.785	−0.0170541	−0.00852704	+	0.999964i	$0.502714\pi$
$480$	0	0
$481$	−15155.8	−1.43668
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−95.2622	−0.00891883
$486$	0	0
$487$	−16597.0	−1.54431	−0.772156	−	0.635433i	$-0.780822\pi$
$487$	−16597.0	−1.54431	−0.772156	+	0.635433i	$0.780822\pi$
$488$	0	0
$489$	5922.07	0.547659
$490$	0	0
$491$	−4547.22	−0.417949	−0.208975	−	0.977921i	$-0.567013\pi$
$491$	−4547.22	−0.417949	−0.208975	+	0.977921i	$0.567013\pi$
$492$	0	0
$493$	9190.45	0.839588
$494$	0	0
$495$	62.5029	0.00567535
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1025.97	0.0920414	0.0460207	−	0.998940i	$-0.485346\pi$
$499$	1025.97	0.0920414	0.0460207	+	0.998940i	$0.485346\pi$
$500$	0	0
$501$	3201.26	0.285473
$502$	0	0
$503$	6745.26	0.597925	0.298962	−	0.954265i	$-0.403359\pi$
$503$	6745.26	0.597925	0.298962	+	0.954265i	$0.403359\pi$
$504$	0	0
$505$	95.6431	0.00842785
$506$	0	0
$507$	979.676	0.0858164
$508$	0	0
$509$	−6001.02	−0.522575	−0.261287	−	0.965261i	$-0.584147\pi$
$509$	−6001.02	−0.522575	−0.261287	+	0.965261i	$0.584147\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2470.75	0.212644
$514$	0	0
$515$	13.4571	0.00115144
$516$	0	0
$517$	−19447.0	−1.65431
$518$	0	0
$519$	−823.964	−0.0696879
$520$	0	0
$521$	4695.21	0.394820	0.197410	−	0.980321i	$-0.436747\pi$
$521$	4695.21	0.394820	0.197410	+	0.980321i	$0.436747\pi$
$522$	0	0
$523$	771.767	0.0645258	0.0322629	−	0.999479i	$-0.489729\pi$
$523$	771.767	0.0645258	0.0322629	+	0.999479i	$0.489729\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42976.2	−3.55232
$528$	0	0
$529$	20048.7	1.64779
$530$	0	0
$531$	−479.147	−0.0391586
$532$	0	0
$533$	−14872.0	−1.20859
$534$	0	0
$535$	66.0573	0.00533814
$536$	0	0
$537$	1663.00	0.133638
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3385.66	0.269059	0.134529	−	0.990910i	$-0.457048\pi$
$541$	3385.66	0.269059	0.134529	+	0.990910i	$0.457048\pi$
$542$	0	0
$543$	−2056.31	−0.162513
$544$	0	0
$545$	125.381	0.00985457
$546$	0	0
$547$	4988.75	0.389952	0.194976	−	0.980808i	$-0.437537\pi$
$547$	4988.75	0.389952	0.194976	+	0.980808i	$0.437537\pi$
$548$	0	0
$549$	−973.090	−0.0756475
$550$	0	0
$551$	−6396.15	−0.494529
$552$	0	0
$553$	0	0
$554$	0	0
$555$	116.387	0.00890153
$556$	0	0
$557$	−20207.3	−1.53718	−0.768591	−	0.639740i	$-0.779042\pi$
$557$	−20207.3	−1.53718	−0.768591	+	0.639740i	$0.779042\pi$
$558$	0	0
$559$	7249.02	0.548481
$560$	0	0
$561$	−21303.5	−1.60327
$562$	0	0
$563$	−11690.4	−0.875118	−0.437559	−	0.899190i	$-0.644157\pi$
$563$	−11690.4	−0.875118	−0.437559	+	0.899190i	$0.644157\pi$
$564$	0	0
$565$	247.675	0.0184421
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17928.9	−1.32095	−0.660473	−	0.750849i	$-0.729644\pi$
$569$	−17928.9	−1.32095	−0.660473	+	0.750849i	$0.729644\pi$
$570$	0	0
$571$	−15673.7	−1.14873	−0.574364	−	0.818600i	$-0.694751\pi$
$571$	−15673.7	−1.14873	−0.574364	+	0.818600i	$0.694751\pi$
$572$	0	0
$573$	14696.6	1.07148
$574$	0	0
$575$	−22432.9	−1.62699
$576$	0	0
$577$	−13653.0	−0.985064	−0.492532	−	0.870294i	$-0.663929\pi$
$577$	−13653.0	−0.985064	−0.492532	+	0.870294i	$0.663929\pi$
$578$	0	0
$579$	9420.78	0.676191
$580$	0	0
$581$	0	0
$582$	0	0
$583$	99.1093	0.00704063
$584$	0	0
$585$	−58.1380	−0.00410891
$586$	0	0
$587$	−18021.5	−1.26717	−0.633583	−	0.773675i	$-0.718417\pi$
$587$	−18021.5	−1.26717	−0.633583	+	0.773675i	$0.718417\pi$
$588$	0	0
$589$	29909.6	2.09236
$590$	0	0
$591$	−682.237	−0.0474847
$592$	0	0
$593$	21137.3	1.46375	0.731877	−	0.681437i	$-0.238645\pi$
$593$	21137.3	1.46375	0.731877	+	0.681437i	$0.238645\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3643.50	−0.249780
$598$	0	0
$599$	9369.59	0.639117	0.319559	−	0.947567i	$-0.396465\pi$
$599$	9369.59	0.639117	0.319559	+	0.947567i	$0.396465\pi$
$600$	0	0
$601$	−22750.0	−1.54408	−0.772040	−	0.635573i	$-0.780764\pi$
$601$	−22750.0	−1.54408	−0.772040	+	0.635573i	$0.780764\pi$
$602$	0	0
$603$	7578.06	0.511779
$604$	0	0
$605$	203.908	0.0137026
$606$	0	0
$607$	−5973.65	−0.399445	−0.199722	−	0.979853i	$-0.564004\pi$
$607$	−5973.65	−0.399445	−0.199722	+	0.979853i	$0.564004\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18088.9	1.19771
$612$	0	0
$613$	−23347.4	−1.53832	−0.769162	−	0.639054i	$-0.779326\pi$
$613$	−23347.4	−1.53832	−0.769162	+	0.639054i	$0.779326\pi$
$614$	0	0
$615$	114.207	0.00748826
$616$	0	0
$617$	28199.1	1.83996	0.919979	−	0.391968i	$-0.128206\pi$
$617$	28199.1	1.83996	0.919979	+	0.391968i	$0.128206\pi$
$618$	0	0
$619$	−2975.56	−0.193211	−0.0966057	−	0.995323i	$-0.530799\pi$
$619$	−2975.56	−0.193211	−0.0966057	+	0.995323i	$0.530799\pi$
$620$	0	0
$621$	4846.16	0.313155
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15618.8	0.999603
$626$	0	0
$627$	14826.3	0.944348
$628$	0	0
$629$	−39669.4	−2.51466
$630$	0	0
$631$	−2631.33	−0.166009	−0.0830044	−	0.996549i	$-0.526452\pi$
$631$	−2631.33	−0.166009	−0.0830044	+	0.996549i	$0.526452\pi$
$632$	0	0
$633$	15348.2	0.963723
$634$	0	0
$635$	−144.787	−0.00904837
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2170.00	−0.134341
$640$	0	0
$641$	14494.2	0.893117	0.446559	−	0.894754i	$-0.352649\pi$
$641$	14494.2	0.893117	0.446559	+	0.894754i	$0.352649\pi$
$642$	0	0
$643$	−15176.0	−0.930767	−0.465383	−	0.885109i	$-0.654084\pi$
$643$	−15176.0	−0.930767	−0.465383	+	0.885109i	$0.654084\pi$
$644$	0	0
$645$	−55.6679	−0.00339833
$646$	0	0
$647$	−7569.18	−0.459931	−0.229965	−	0.973199i	$-0.573861\pi$
$647$	−7569.18	−0.459931	−0.229965	+	0.973199i	$0.573861\pi$
$648$	0	0
$649$	−2875.23	−0.173903
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13933.5	−0.835007	−0.417504	−	0.908675i	$-0.637095\pi$
$653$	−13933.5	−0.835007	−0.417504	+	0.908675i	$0.637095\pi$
$654$	0	0
$655$	192.018	0.0114546
$656$	0	0
$657$	−1862.85	−0.110619
$658$	0	0
$659$	−17015.5	−1.00581	−0.502904	−	0.864342i	$-0.667735\pi$
$659$	−17015.5	−1.00581	−0.502904	+	0.864342i	$0.667735\pi$
$660$	0	0
$661$	−16535.4	−0.972999	−0.486500	−	0.873681i	$-0.661726\pi$
$661$	−16535.4	−0.972999	−0.486500	+	0.873681i	$0.661726\pi$
$662$	0	0
$663$	19815.8	1.16076
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12545.5	−0.728279
$668$	0	0
$669$	358.152	0.0206980
$670$	0	0
$671$	−5839.25	−0.335949
$672$	0	0
$673$	4571.05	0.261814	0.130907	−	0.991395i	$-0.458211\pi$
$673$	4571.05	0.261814	0.130907	+	0.991395i	$0.458211\pi$
$674$	0	0
$675$	−3374.55	−0.192425
$676$	0	0
$677$	26284.5	1.49217	0.746083	−	0.665853i	$-0.231932\pi$
$677$	26284.5	1.49217	0.746083	+	0.665853i	$0.231932\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5976.00	0.336271
$682$	0	0
$683$	6797.22	0.380803	0.190402	−	0.981706i	$-0.439021\pi$
$683$	6797.22	0.380803	0.190402	+	0.981706i	$0.439021\pi$
$684$	0	0
$685$	187.842	0.0104775
$686$	0	0
$687$	2110.43	0.117202
$688$	0	0
$689$	−92.1880	−0.00509737
$690$	0	0
$691$	27758.7	1.52821	0.764103	−	0.645094i	$-0.223182\pi$
$691$	27758.7	1.52821	0.764103	+	0.645094i	$0.223182\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	286.225	0.0156218
$696$	0	0
$697$	−38926.5	−2.11542
$698$	0	0
$699$	2885.35	0.156129
$700$	0	0
$701$	−21638.3	−1.16586	−0.582929	−	0.812523i	$-0.698093\pi$
$701$	−21638.3	−1.16586	−0.582929	+	0.812523i	$0.698093\pi$
$702$	0	0
$703$	27608.2	1.48117
$704$	0	0
$705$	−138.911	−0.00742086
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−16288.7	−0.862815	−0.431408	−	0.902157i	$-0.641983\pi$
$709$	−16288.7	−0.862815	−0.431408	+	0.902157i	$0.641983\pi$
$710$	0	0
$711$	5039.66	0.265826
$712$	0	0
$713$	58664.8	3.08137
$714$	0	0
$715$	−348.871	−0.0182476
$716$	0	0
$717$	−13394.1	−0.697646
$718$	0	0
$719$	−3391.74	−0.175926	−0.0879628	−	0.996124i	$-0.528036\pi$
$719$	−3391.74	−0.175926	−0.0879628	+	0.996124i	$0.528036\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1305.93	−0.0671760
$724$	0	0
$725$	8735.86	0.447506
$726$	0	0
$727$	23158.7	1.18144	0.590722	−	0.806875i	$-0.298843\pi$
$727$	23158.7	1.18144	0.590722	+	0.806875i	$0.298843\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	18973.9	0.960020
$732$	0	0
$733$	−27629.2	−1.39223	−0.696116	−	0.717929i	$-0.745090\pi$
$733$	−27629.2	−1.39223	−0.696116	+	0.717929i	$0.745090\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	45473.9	2.27280
$738$	0	0
$739$	−9461.65	−0.470978	−0.235489	−	0.971877i	$-0.575669\pi$
$739$	−9461.65	−0.470978	−0.235489	+	0.971877i	$0.575669\pi$
$740$	0	0
$741$	−13790.9	−0.683701
$742$	0	0
$743$	−14316.9	−0.706913	−0.353457	−	0.935451i	$-0.614994\pi$
$743$	−14316.9	−0.706913	−0.353457	+	0.935451i	$0.614994\pi$
$744$	0	0
$745$	50.7524	0.00249587
$746$	0	0
$747$	8879.86	0.434936
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−7486.07	−0.363742	−0.181871	−	0.983322i	$-0.558215\pi$
$751$	−7486.07	−0.363742	−0.181871	+	0.983322i	$0.558215\pi$
$752$	0	0
$753$	2589.01	0.125297
$754$	0	0
$755$	−401.744	−0.0193655
$756$	0	0
$757$	−17416.8	−0.836227	−0.418114	−	0.908395i	$-0.637309\pi$
$757$	−17416.8	−0.836227	−0.418114	+	0.908395i	$0.637309\pi$
$758$	0	0
$759$	29080.5	1.39072
$760$	0	0
$761$	−33231.0	−1.58295	−0.791474	−	0.611203i	$-0.790686\pi$
$761$	−33231.0	−1.58295	−0.791474	+	0.611203i	$0.790686\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−152.173	−0.00719192
$766$	0	0
$767$	2674.44	0.125904
$768$	0	0
$769$	13714.2	0.643103	0.321552	−	0.946892i	$-0.395796\pi$
$769$	13714.2	0.643103	0.321552	+	0.946892i	$0.395796\pi$
$770$	0	0
$771$	−1610.08	−0.0752082
$772$	0	0
$773$	15209.3	0.707685	0.353842	−	0.935305i	$-0.384875\pi$
$773$	15209.3	0.707685	0.353842	+	0.935305i	$0.384875\pi$
$774$	0	0
$775$	−40850.5	−1.89341
$776$	0	0
$777$	0	0
$778$	0	0
$779$	27091.1	1.24601
$780$	0	0
$781$	−13021.6	−0.596605
$782$	0	0
$783$	−1887.20	−0.0861339
$784$	0	0
$785$	−463.019	−0.0210520
$786$	0	0
$787$	−13221.5	−0.598850	−0.299425	−	0.954120i	$-0.596795\pi$
$787$	−13221.5	−0.598850	−0.299425	+	0.954120i	$0.596795\pi$
$788$	0	0
$789$	1115.72	0.0503433
$790$	0	0
$791$	0	0
$792$	0	0
$793$	5431.47	0.243224
$794$	0	0
$795$	0.707946	3.15827e−5 0
$796$	0	0
$797$	13373.9	0.594387	0.297194	−	0.954817i	$-0.403949\pi$
$797$	13373.9	0.594387	0.297194	+	0.954817i	$0.403949\pi$
$798$	0	0
$799$	47346.6	2.09637
$800$	0	0
$801$	−3990.29	−0.176018
$802$	0	0
$803$	−11178.5	−0.491257
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−13516.8	−0.589609
$808$	0	0
$809$	−11229.3	−0.488011	−0.244005	−	0.969774i	$-0.578461\pi$
$809$	−11229.3	−0.488011	−0.244005	+	0.969774i	$0.578461\pi$
$810$	0	0
$811$	−4301.42	−0.186243	−0.0931215	−	0.995655i	$-0.529685\pi$
$811$	−4301.42	−0.186243	−0.0931215	+	0.995655i	$0.529685\pi$
$812$	0	0
$813$	−1188.35	−0.0512634
$814$	0	0
$815$	253.842	0.0109101
$816$	0	0
$817$	−13205.0	−0.565464
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−40910.6	−1.73909	−0.869543	−	0.493857i	$-0.835587\pi$
$821$	−40910.6	−1.73909	−0.869543	+	0.493857i	$0.835587\pi$
$822$	0	0
$823$	28043.2	1.18776	0.593879	−	0.804554i	$-0.297596\pi$
$823$	28043.2	1.18776	0.593879	+	0.804554i	$0.297596\pi$
$824$	0	0
$825$	−20249.8	−0.854554
$826$	0	0
$827$	17598.9	0.739993	0.369996	−	0.929033i	$-0.379359\pi$
$827$	17598.9	0.739993	0.369996	+	0.929033i	$0.379359\pi$
$828$	0	0
$829$	−14565.5	−0.610231	−0.305116	−	0.952315i	$-0.598695\pi$
$829$	−14565.5	−0.610231	−0.305116	+	0.952315i	$0.598695\pi$
$830$	0	0
$831$	−19526.2	−0.815111
$832$	0	0
$833$	0	0
$834$	0	0
$835$	137.218	0.00568697
$836$	0	0
$837$	8824.86	0.364435
$838$	0	0
$839$	711.056	0.0292591	0.0146295	−	0.999893i	$-0.495343\pi$
$839$	711.056	0.0292591	0.0146295	+	0.999893i	$0.495343\pi$
$840$	0	0
$841$	−19503.5	−0.799686
$842$	0	0
$843$	8357.41	0.341453
$844$	0	0
$845$	41.9925	0.00170957
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.83554	7.41996e−5 0
$850$	0	0
$851$	54150.9	2.18128
$852$	0	0
$853$	−7459.48	−0.299423	−0.149712	−	0.988730i	$-0.547834\pi$
$853$	−7459.48	−0.299423	−0.149712	+	0.988730i	$0.547834\pi$
$854$	0	0
$855$	105.906	0.00423614
$856$	0	0
$857$	3560.92	0.141936	0.0709678	−	0.997479i	$-0.477391\pi$
$857$	3560.92	0.141936	0.0709678	+	0.997479i	$0.477391\pi$
$858$	0	0
$859$	13519.1	0.536981	0.268491	−	0.963282i	$-0.413475\pi$
$859$	13519.1	0.536981	0.268491	+	0.963282i	$0.413475\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20395.9	0.804502	0.402251	−	0.915530i	$-0.368228\pi$
$863$	20395.9	0.804502	0.402251	+	0.915530i	$0.368228\pi$
$864$	0	0
$865$	−35.3182	−0.00138827
$866$	0	0
$867$	37127.7	1.45435
$868$	0	0
$869$	30241.6	1.18053
$870$	0	0
$871$	−42298.3	−1.64549
$872$	0	0
$873$	−6667.33	−0.258482
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7197.26	0.277120	0.138560	−	0.990354i	$-0.455753\pi$
$877$	7197.26	0.277120	0.138560	+	0.990354i	$0.455753\pi$
$878$	0	0
$879$	−12437.9	−0.477271
$880$	0	0
$881$	11588.3	0.443156	0.221578	−	0.975143i	$-0.428879\pi$
$881$	11588.3	0.443156	0.221578	+	0.975143i	$0.428879\pi$
$882$	0	0
$883$	21959.0	0.836898	0.418449	−	0.908240i	$-0.362574\pi$
$883$	21959.0	0.836898	0.418449	+	0.908240i	$0.362574\pi$
$884$	0	0
$885$	−20.5380	−0.000780088 0
$886$	0	0
$887$	12518.9	0.473894	0.236947	−	0.971523i	$-0.423853\pi$
$887$	12518.9	0.473894	0.236947	+	0.971523i	$0.423853\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4374.53	0.164481
$892$	0	0
$893$	−32951.2	−1.23479
$894$	0	0
$895$	71.2822	0.00266224
$896$	0	0
$897$	−27049.6	−1.00687
$898$	0	0
$899$	−22845.3	−0.847535
$900$	0	0
$901$	−241.297	−0.00892204
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−88.1410	−0.00323747
$906$	0	0
$907$	28270.0	1.03494	0.517470	−	0.855701i	$-0.326874\pi$
$907$	28270.0	1.03494	0.517470	+	0.855701i	$0.326874\pi$
$908$	0	0
$909$	6693.99	0.244253
$910$	0	0
$911$	−7908.58	−0.287621	−0.143811	−	0.989605i	$-0.545936\pi$
$911$	−7908.58	−0.287621	−0.143811	+	0.989605i	$0.545936\pi$
$912$	0	0
$913$	53285.7	1.93154
$914$	0	0
$915$	−41.7102	−0.00150699
$916$	0	0
$917$	0	0
$918$	0	0
$919$	31450.6	1.12890	0.564451	−	0.825467i	$-0.309088\pi$
$919$	31450.6	1.12890	0.564451	+	0.825467i	$0.309088\pi$
$920$	0	0
$921$	5881.58	0.210428
$922$	0	0
$923$	12112.2	0.431938
$924$	0	0
$925$	−37707.2	−1.34033
$926$	0	0
$927$	941.855	0.0333706
$928$	0	0
$929$	−3469.19	−0.122519	−0.0612596	−	0.998122i	$-0.519512\pi$
$929$	−3469.19	−0.122519	−0.0612596	+	0.998122i	$0.519512\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15622.7	0.548193
$934$	0	0
$935$	−913.148	−0.0319392
$936$	0	0
$937$	50935.7	1.77588	0.887939	−	0.459962i	$-0.152137\pi$
$937$	50935.7	1.77588	0.887939	+	0.459962i	$0.152137\pi$
$938$	0	0
$939$	−11975.1	−0.416181
$940$	0	0
$941$	47894.9	1.65922	0.829612	−	0.558341i	$-0.188562\pi$
$941$	47894.9	1.65922	0.829612	+	0.558341i	$0.188562\pi$
$942$	0	0
$943$	53136.7	1.83496
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10538.9	0.361634	0.180817	−	0.983517i	$-0.442126\pi$
$947$	10538.9	0.361634	0.180817	+	0.983517i	$0.442126\pi$
$948$	0	0
$949$	10397.8	0.355667
$950$	0	0
$951$	5415.11	0.184645
$952$	0	0
$953$	31123.5	1.05791	0.528955	−	0.848650i	$-0.322584\pi$
$953$	31123.5	1.05791	0.528955	+	0.848650i	$0.322584\pi$
$954$	0	0
$955$	629.951	0.0213453
$956$	0	0
$957$	−11324.6	−0.382519
$958$	0	0
$959$	0	0
$960$	0	0
$961$	77037.9	2.58594
$962$	0	0
$963$	4623.30	0.154708
$964$	0	0
$965$	403.810	0.0134706
$966$	0	0
$967$	−1626.14	−0.0540776	−0.0270388	−	0.999634i	$-0.508608\pi$
$967$	−1626.14	−0.0540776	−0.0270388	+	0.999634i	$0.508608\pi$
$968$	0	0
$969$	−36096.9	−1.19670
$970$	0	0
$971$	35500.1	1.17328	0.586638	−	0.809849i	$-0.300451\pi$
$971$	35500.1	1.17328	0.586638	+	0.809849i	$0.300451\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	18835.6	0.618691
$976$	0	0
$977$	26570.2	0.870068	0.435034	−	0.900414i	$-0.356736\pi$
$977$	26570.2	0.870068	0.435034	+	0.900414i	$0.356736\pi$
$978$	0	0
$979$	−23944.7	−0.781691
$980$	0	0
$981$	8775.34	0.285602
$982$	0	0
$983$	−53851.6	−1.74730	−0.873652	−	0.486552i	$-0.838254\pi$
$983$	−53851.6	−1.74730	−0.873652	+	0.486552i	$0.838254\pi$
$984$	0	0
$985$	−29.2432	−0.000945955 0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−25900.4	−0.832745
$990$	0	0
$991$	2829.75	0.0907063	0.0453531	−	0.998971i	$-0.485559\pi$
$991$	2829.75	0.0907063	0.0453531	+	0.998971i	$0.485559\pi$
$992$	0	0
$993$	20718.6	0.662118
$994$	0	0
$995$	−156.174	−0.00497593
$996$	0	0
$997$	12904.3	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$997$	12904.3	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$998$	0	0
$999$	8145.83	0.257981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.4.a.bd.1.2		4
4.3	odd	2		2352.4.a.cm.1.2		4
7.2	even	3		168.4.q.f.25.3	✓	8
7.4	even	3		168.4.q.f.121.3	yes	8
7.6	odd	2		1176.4.a.ba.1.3		4
21.2	odd	6		504.4.s.j.361.2		8
21.11	odd	6		504.4.s.j.289.2		8
28.11	odd	6		336.4.q.m.289.3		8
28.23	odd	6		336.4.q.m.193.3		8
28.27	even	2		2352.4.a.cp.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.f.25.3	✓	8	7.2	even	3
168.4.q.f.121.3	yes	8	7.4	even	3
336.4.q.m.193.3		8	28.23	odd	6
336.4.q.m.289.3		8	28.11	odd	6
504.4.s.j.289.2		8	21.11	odd	6
504.4.s.j.361.2		8	21.2	odd	6
1176.4.a.ba.1.3		4	7.6	odd	2
1176.4.a.bd.1.2		4	1.1	even	1	trivial
2352.4.a.cm.1.2		4	4.3	odd	2
2352.4.a.cp.1.3		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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +0.128591 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +0.128591 q^{5} +9.00000 q^{9} +54.0066 q^{11} -50.2350 q^{13} +0.385773 q^{15} -131.487 q^{17} +91.5094 q^{19} +179.487 q^{23} -124.983 q^{25} +27.0000 q^{27} -69.8961 q^{29} +326.847 q^{31} +162.020 q^{33} +301.698 q^{37} -150.705 q^{39} +296.048 q^{41} -144.302 q^{43} +1.15732 q^{45} -360.085 q^{47} -394.462 q^{51} +1.83513 q^{53} +6.94477 q^{55} +274.528 q^{57} -53.2386 q^{59} -108.121 q^{61} -6.45978 q^{65} +842.007 q^{67} +538.462 q^{69} -241.111 q^{71} -206.983 q^{73} -374.950 q^{75} +559.962 q^{79} +81.0000 q^{81} +986.652 q^{83} -16.9081 q^{85} -209.688 q^{87} -443.366 q^{89} +980.541 q^{93} +11.7673 q^{95} -740.815 q^{97} +486.059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 4 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 4 * q^5 + 36 * q^9	\( 4 q + 12 q^{3} + 4 q^{5} + 36 q^{9} + 14 q^{11} + 22 q^{13} + 12 q^{15} + 96 q^{17} - 26 q^{19} + 96 q^{23} + 110 q^{25} + 108 q^{27} - 76 q^{29} + 238 q^{31} + 42 q^{33} + 562 q^{37} + 66 q^{39} + 428 q^{41} - 258 q^{43} + 36 q^{45} - 80 q^{47} + 288 q^{51} + 1476 q^{55} - 78 q^{57} + 262 q^{59} - 276 q^{61} + 2196 q^{65} + 150 q^{67} + 288 q^{69} - 848 q^{71} - 218 q^{73} + 330 q^{75} + 1762 q^{79} + 324 q^{81} + 3450 q^{83} + 1452 q^{85} - 228 q^{87} - 344 q^{89} + 714 q^{93} + 2004 q^{95} - 622 q^{97} + 126 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 4 * q^5 + 36 * q^9 + 14 * q^11 + 22 * q^13 + 12 * q^15 + 96 * q^17 - 26 * q^19 + 96 * q^23 + 110 * q^25 + 108 * q^27 - 76 * q^29 + 238 * q^31 + 42 * q^33 + 562 * q^37 + 66 * q^39 + 428 * q^41 - 258 * q^43 + 36 * q^45 - 80 * q^47 + 288 * q^51 + 1476 * q^55 - 78 * q^57 + 262 * q^59 - 276 * q^61 + 2196 * q^65 + 150 * q^67 + 288 * q^69 - 848 * q^71 - 218 * q^73 + 330 * q^75 + 1762 * q^79 + 324 * q^81 + 3450 * q^83 + 1452 * q^85 - 228 * q^87 - 344 * q^89 + 714 * q^93 + 2004 * q^95 - 622 * q^97 + 126 * q^99

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