LMFDB - Embedded newform 2352.4.a.cm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
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	$\chi$	$=$	2352.1

$q$-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	$-0.765252\pi$
$11$	−54.0066	−1.48033	−0.740163	+	0.672427i	$0.765252\pi$
$12$	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	$-0.686313\pi$
$19$	−91.5094	−1.10493	−0.552466	+	0.833536i	$0.686313\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−179.487	−1.62720	−0.813602	−	0.581423i	$-0.802496\pi$
$23$	−179.487	−1.62720	−0.813602	+	0.581423i	$0.802496\pi$
$24$	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	$-0.895734\pi$
$31$	−326.847	−1.89366	−0.946829	+	0.321736i	$0.895734\pi$
$32$	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	$-0.417634\pi$
$43$	144.302	0.511764	0.255882	+	0.966708i	$0.417634\pi$
$44$	0	0
$45$	1.15732	0.00383385
$46$	0	0
$47$	360.085	1.11753	0.558764	−	0.829326i	$-0.311276\pi$
$47$	360.085	1.11753	0.558764	+	0.829326i	$0.311276\pi$
$48$	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	$-0.481292\pi$
$59$	53.2386	0.117476	0.0587379	+	0.998273i	$0.481292\pi$
$60$	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	$-0.778583\pi$
$67$	−842.007	−1.53534	−0.767668	+	0.640848i	$0.778583\pi$
$68$	0	0
$69$	538.462	0.939466
$70$	0	0
$71$	241.111	0.403023	0.201511	−	0.979486i	$-0.435415\pi$
$71$	241.111	0.403023	0.201511	+	0.979486i	$0.435415\pi$
$72$	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	$-0.630552\pi$
$79$	−559.962	−0.797477	−0.398738	+	0.917065i	$0.630552\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−986.652	−1.30481	−0.652404	−	0.757871i	$-0.726240\pi$
$83$	−986.652	−1.30481	−0.652404	+	0.757871i	$0.726240\pi$
$84$	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	$-0.515940\pi$
$103$	−104.651	−0.100112	−0.0500559	+	0.998746i	$0.515940\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−513.700	−0.464124	−0.232062	−	0.972701i	$-0.574547\pi$
$107$	−513.700	−0.464124	−0.232062	+	0.972701i	$0.574547\pi$
$108$	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	$-0.371314\pi$
$127$	1125.95	0.786709	0.393355	+	0.919387i	$0.371314\pi$
$128$	0	0
$129$	−432.906	−0.295467
$130$	0	0
$131$	−1493.24	−0.995918	−0.497959	−	0.867201i	$-0.665917\pi$
$131$	−1493.24	−0.995918	−0.497959	+	0.867201i	$0.665917\pi$
$132$	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	$-0.737637\pi$
$139$	−2225.85	−1.35823	−0.679116	+	0.734031i	$0.737637\pi$
$140$	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	$-0.181457\pi$
$151$	3124.20	1.68373	0.841867	+	0.539685i	$0.181457\pi$
$152$	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	$-0.657294\pi$
$163$	−1974.02	−0.948573	−0.474287	+	0.880370i	$0.657294\pi$
$164$	0	0
$165$	20.8343	0.00982999
$166$	0	0
$167$	−1067.09	−0.494453	−0.247227	−	0.968958i	$-0.579519\pi$
$167$	−1067.09	−0.494453	−0.247227	+	0.968958i	$0.579519\pi$
$168$	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	$-0.536922\pi$
$179$	−554.332	−0.231468	−0.115734	+	0.993280i	$0.536922\pi$
$180$	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	$-0.878415\pi$
$191$	−4898.87	−1.85586	−0.927932	+	0.372750i	$0.878415\pi$
$192$	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	$-0.430596\pi$
$199$	1214.50	0.432631	0.216316	+	0.976323i	$0.430596\pi$
$200$	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	$-0.814306\pi$
$211$	−5116.07	−1.66922	−0.834608	+	0.550844i	$0.814306\pi$
$212$	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	$-0.505706\pi$
$223$	−119.384	−0.0358499	−0.0179250	+	0.999839i	$0.505706\pi$
$224$	0	0
$225$	−1124.85	−0.333289
$226$	0	0
$227$	−1992.00	−0.582439	−0.291220	−	0.956656i	$-0.594061\pi$
$227$	−1992.00	−0.582439	−0.291220	+	0.956656i	$0.594061\pi$
$228$	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	$-0.293501\pi$
$239$	4464.71	1.20836	0.604179	+	0.796848i	$0.293501\pi$
$240$	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	$-0.534608\pi$
$251$	−863.003	−0.217021	−0.108510	+	0.994095i	$0.534608\pi$
$252$	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	$-0.513882\pi$
$263$	−371.908	−0.0871971	−0.0435986	+	0.999049i	$0.513882\pi$
$264$	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	$-0.485864\pi$
$271$	396.115	0.0887908	0.0443954	+	0.999014i	$0.485864\pi$
$272$	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	$-0.500020\pi$
$283$	−0.611846	−0.000128518 0	−6.42588e−5		1.00000i	$0.500020\pi$
$284$	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	$-0.558334\pi$
$307$	−1960.53	−0.364473	−0.182236	+	0.983255i	$0.558334\pi$
$308$	0	0
$309$	313.952	0.0577996
$310$	0	0
$311$	−5207.57	−0.949499	−0.474749	−	0.880121i	$-0.657461\pi$
$311$	−5207.57	−0.949499	−0.474749	+	0.880121i	$0.657461\pi$
$312$	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	$-0.694380\pi$
$331$	−6906.18	−1.14682	−0.573411	+	0.819268i	$0.694380\pi$
$332$	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	$-0.294232\pi$
$347$	7787.02	1.20469	0.602347	+	0.798234i	$0.294232\pi$
$348$	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	$-0.605258\pi$
$359$	−4417.05	−0.649368	−0.324684	+	0.945823i	$0.605258\pi$
$360$	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	$-0.382424\pi$
$367$	5076.65	0.722068	0.361034	+	0.932553i	$0.382424\pi$
$368$	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	$-0.283363\pi$
$379$	9285.61	1.25850	0.629248	+	0.777205i	$0.283363\pi$
$380$	0	0
$381$	−3377.85	−0.454207
$382$	0	0
$383$	7681.65	1.02484	0.512420	−	0.858735i	$-0.328749\pi$
$383$	7681.65	1.02484	0.512420	+	0.858735i	$0.328749\pi$
$384$	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	$-0.781172\pi$
$419$	−13257.1	−1.54571	−0.772856	+	0.634582i	$0.781172\pi$
$420$	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	$-0.589192\pi$
$431$	−4949.08	−0.553106	−0.276553	+	0.960999i	$0.589192\pi$
$432$	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	$-0.417614\pi$
$439$	4708.38	0.511887	0.255944	+	0.966692i	$0.417614\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1700.44	0.182370	0.0911852	−	0.995834i	$-0.470934\pi$
$443$	1700.44	0.182370	0.0911852	+	0.995834i	$0.470934\pi$
$444$	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	$-0.564796\pi$
$463$	−4028.04	−0.404317	−0.202159	+	0.979353i	$0.564796\pi$
$464$	0	0
$465$	126.089	0.0125747
$466$	0	0
$467$	−2070.69	−0.205182	−0.102591	−	0.994724i	$-0.532713\pi$
$467$	−2070.69	−0.205182	−0.102591	+	0.994724i	$0.532713\pi$
$468$	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	$-0.497286\pi$
$479$	178.785	0.0170541	0.00852704	+	0.999964i	$0.497286\pi$
$480$	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	$-0.219178\pi$
$487$	16597.0	1.54431	0.772156	+	0.635433i	$0.219178\pi$
$488$	0	0
$489$	5922.07	0.547659
$490$	0	0
$491$	4547.22	0.417949	0.208975	−	0.977921i	$-0.432987\pi$
$491$	4547.22	0.417949	0.208975	+	0.977921i	$0.432987\pi$
$492$	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	$-0.514654\pi$
$499$	−1025.97	−0.0920414	−0.0460207	+	0.998940i	$0.514654\pi$
$500$	0	0
$501$	3201.26	0.285473
$502$	0	0
$503$	−6745.26	−0.597925	−0.298962	−	0.954265i	$-0.596641\pi$
$503$	−6745.26	−0.597925	−0.298962	+	0.954265i	$0.596641\pi$
$504$	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	$-0.510271\pi$
$523$	−771.767	−0.0645258	−0.0322629	+	0.999479i	$0.510271\pi$
$524$	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	$-0.562463\pi$
$547$	−4988.75	−0.389952	−0.194976	+	0.980808i	$0.562463\pi$
$548$	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	$-0.355843\pi$
$563$	11690.4	0.875118	0.437559	+	0.899190i	$0.355843\pi$
$564$	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	$-0.305249\pi$
$571$	15673.7	1.14873	0.574364	+	0.818600i	$0.305249\pi$
$572$	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	$-0.281583\pi$
$587$	18021.5	1.26717	0.633583	+	0.773675i	$0.281583\pi$
$588$	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	$-0.603535\pi$
$599$	−9369.59	−0.639117	−0.319559	+	0.947567i	$0.603535\pi$
$600$	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	$-0.435996\pi$
$607$	5973.65	0.399445	0.199722	+	0.979853i	$0.435996\pi$
$608$	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	$-0.469201\pi$
$619$	2975.56	0.193211	0.0966057	+	0.995323i	$0.469201\pi$
$620$	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	$-0.473548\pi$
$631$	2631.33	0.166009	0.0830044	+	0.996549i	$0.473548\pi$
$632$	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	$-0.345916\pi$
$643$	15176.0	0.930767	0.465383	+	0.885109i	$0.345916\pi$
$644$	0	0
$645$	−55.6679	−0.00339833
$646$	0	0
$647$	7569.18	0.459931	0.229965	−	0.973199i	$-0.426139\pi$
$647$	7569.18	0.459931	0.229965	+	0.973199i	$0.426139\pi$
$648$	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	$-0.332265\pi$
$659$	17015.5	1.00581	0.502904	+	0.864342i	$0.332265\pi$
$660$	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	$-0.560979\pi$
$683$	−6797.22	−0.380803	−0.190402	+	0.981706i	$0.560979\pi$
$684$	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	$-0.776818\pi$
$691$	−27758.7	−1.52821	−0.764103	+	0.645094i	$0.776818\pi$
$692$	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	$-0.471964\pi$
$719$	3391.74	0.175926	0.0879628	+	0.996124i	$0.471964\pi$
$720$	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	$-0.701157\pi$
$727$	−23158.7	−1.18144	−0.590722	+	0.806875i	$0.701157\pi$
$728$	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	$-0.424331\pi$
$739$	9461.65	0.470978	0.235489	+	0.971877i	$0.424331\pi$
$740$	0	0
$741$	−13790.9	−0.683701
$742$	0	0
$743$	14316.9	0.706913	0.353457	−	0.935451i	$-0.385006\pi$
$743$	14316.9	0.706913	0.353457	+	0.935451i	$0.385006\pi$
$744$	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	$-0.441785\pi$
$751$	7486.07	0.363742	0.181871	+	0.983322i	$0.441785\pi$
$752$	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	$-0.403205\pi$
$787$	13221.5	0.598850	0.299425	+	0.954120i	$0.403205\pi$
$788$	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	$-0.470315\pi$
$811$	4301.42	0.186243	0.0931215	+	0.995655i	$0.470315\pi$
$812$	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	$-0.702404\pi$
$823$	−28043.2	−1.18776	−0.593879	+	0.804554i	$0.702404\pi$
$824$	0	0
$825$	−20249.8	−0.854554
$826$	0	0
$827$	−17598.9	−0.739993	−0.369996	−	0.929033i	$-0.620641\pi$
$827$	−17598.9	−0.739993	−0.369996	+	0.929033i	$0.620641\pi$
$828$	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	$-0.504657\pi$
$839$	−711.056	−0.0292591	−0.0146295	+	0.999893i	$0.504657\pi$
$840$	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	$-0.586525\pi$
$859$	−13519.1	−0.536981	−0.268491	+	0.963282i	$0.586525\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−20395.9	−0.804502	−0.402251	−	0.915530i	$-0.631772\pi$
$863$	−20395.9	−0.804502	−0.402251	+	0.915530i	$0.631772\pi$
$864$	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	$-0.637426\pi$
$883$	−21959.0	−0.836898	−0.418449	+	0.908240i	$0.637426\pi$
$884$	0	0
$885$	−20.5380	−0.000780088 0
$886$	0	0
$887$	−12518.9	−0.473894	−0.236947	−	0.971523i	$-0.576147\pi$
$887$	−12518.9	−0.473894	−0.236947	+	0.971523i	$0.576147\pi$
$888$	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	$-0.673126\pi$
$907$	−28270.0	−1.03494	−0.517470	+	0.855701i	$0.673126\pi$
$908$	0	0
$909$	6693.99	0.244253
$910$	0	0
$911$	7908.58	0.287621	0.143811	−	0.989605i	$-0.454064\pi$
$911$	7908.58	0.287621	0.143811	+	0.989605i	$0.454064\pi$
$912$	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	$-0.690912\pi$
$919$	−31450.6	−1.12890	−0.564451	+	0.825467i	$0.690912\pi$
$920$	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	$-0.557874\pi$
$947$	−10538.9	−0.361634	−0.180817	+	0.983517i	$0.557874\pi$
$948$	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	$-0.491392\pi$
$967$	1626.14	0.0540776	0.0270388	+	0.999634i	$0.491392\pi$
$968$	0	0
$969$	−36096.9	−1.19670
$970$	0	0
$971$	−35500.1	−1.17328	−0.586638	−	0.809849i	$-0.699549\pi$
$971$	−35500.1	−1.17328	−0.586638	+	0.809849i	$0.699549\pi$
$972$	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	$-0.161746\pi$
$983$	53851.6	1.74730	0.873652	+	0.486552i	$0.161746\pi$
$984$	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	$-0.514441\pi$
$991$	−2829.75	−0.0907063	−0.0453531	+	0.998971i	$0.514441\pi$
$992$	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	2352.4.a.cm.1.2		4
4.3	odd	2		1176.4.a.bd.1.2		4
7.2	even	3		336.4.q.m.193.3		8
7.4	even	3		336.4.q.m.289.3		8
7.6	odd	2		2352.4.a.cp.1.3		4
28.11	odd	6		168.4.q.f.121.3	yes	8
28.23	odd	6		168.4.q.f.25.3	✓	8
28.27	even	2		1176.4.a.ba.1.3		4
84.11	even	6		504.4.s.j.289.2		8
84.23	even	6		504.4.s.j.361.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.f.25.3	✓	8	28.23	odd	6
168.4.q.f.121.3	yes	8	28.11	odd	6
336.4.q.m.193.3		8	7.2	even	3
336.4.q.m.289.3		8	7.4	even	3
504.4.s.j.289.2		8	84.11	even	6
504.4.s.j.361.2		8	84.23	even	6
1176.4.a.ba.1.3		4	28.27	even	2
1176.4.a.bd.1.2		4	4.3	odd	2
2352.4.a.cm.1.2		4	1.1	even	1	trivial
2352.4.a.cp.1.3		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.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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