LMFDB - Embedded newform 2448.4.a.bi.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2448,4,Mod(1,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, 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("2448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2448.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:	$144.436675694$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2636.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 14x - 4 $ x^3 - 14*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.287410$ of defining polynomial
Character	$\chi$	$=$	2448.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.9174 q^{5} -26.1222 q^{7} +O(q^{10})$	$q+11.9174 q^{5} -26.1222 q^{7} -3.24412 q^{11} -20.0515 q^{13} +17.0000 q^{17} -57.3466 q^{19} +77.0438 q^{23} +17.0243 q^{25} +286.162 q^{29} +8.54816 q^{31} -311.309 q^{35} +357.982 q^{37} -194.467 q^{41} +74.2619 q^{43} +23.6130 q^{47} +339.369 q^{49} -104.330 q^{53} -38.6614 q^{55} +249.363 q^{59} -370.384 q^{61} -238.961 q^{65} -939.650 q^{67} -520.197 q^{71} +348.741 q^{73} +84.7434 q^{77} +953.827 q^{79} -1414.28 q^{83} +202.596 q^{85} +486.132 q^{89} +523.788 q^{91} -683.422 q^{95} -685.281 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 8 q^{5} - 22 q^{7}+O(q^{10}) $ 3 * q + 8 * q^5 - 22 * q^7	$ 3 q + 8 q^{5} - 22 q^{7} - 28 q^{11} + 30 q^{13} + 51 q^{17} - 80 q^{19} + 142 q^{23} - 223 q^{25} + 456 q^{29} - 230 q^{31} - 332 q^{35} + 356 q^{37} + 294 q^{41} - 556 q^{43} + 640 q^{47} - 269 q^{49} - 302 q^{53} - 76 q^{55} + 636 q^{59} - 84 q^{61} - 408 q^{65} - 1008 q^{67} - 402 q^{71} + 838 q^{73} + 504 q^{77} + 594 q^{79} - 2396 q^{83} + 136 q^{85} + 170 q^{89} + 1016 q^{91} - 472 q^{95} - 270 q^{97}+O(q^{100}) $ 3 * q + 8 * q^5 - 22 * q^7 - 28 * q^11 + 30 * q^13 + 51 * q^17 - 80 * q^19 + 142 * q^23 - 223 * q^25 + 456 * q^29 - 230 * q^31 - 332 * q^35 + 356 * q^37 + 294 * q^41 - 556 * q^43 + 640 * q^47 - 269 * q^49 - 302 * q^53 - 76 * q^55 + 636 * q^59 - 84 * q^61 - 408 * q^65 - 1008 * q^67 - 402 * q^71 + 838 * q^73 + 504 * q^77 + 594 * q^79 - 2396 * q^83 + 136 * q^85 + 170 * q^89 + 1016 * q^91 - 472 * q^95 - 270 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	11.9174	1.06592	0.532962	−	0.846139i	$-0.321079\pi$
$5$	11.9174	1.06592	0.532962	+	0.846139i	$0.321079\pi$
$6$	0	0
$7$	−26.1222	−1.41047	−0.705233	−	0.708975i	$-0.749158\pi$
$7$	−26.1222	−1.41047	−0.705233	+	0.708975i	$0.749158\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.24412	−0.0889216	−0.0444608	−	0.999011i	$-0.514157\pi$
$11$	−3.24412	−0.0889216	−0.0444608	+	0.999011i	$0.514157\pi$
$12$	0	0
$13$	−20.0515	−0.427790	−0.213895	−	0.976857i	$-0.568615\pi$
$13$	−20.0515	−0.427790	−0.213895	+	0.976857i	$0.568615\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−57.3466	−0.692432	−0.346216	−	0.938155i	$-0.612534\pi$
$19$	−57.3466	−0.692432	−0.346216	+	0.938155i	$0.612534\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	77.0438	0.698467	0.349233	−	0.937036i	$-0.386442\pi$
$23$	77.0438	0.698467	0.349233	+	0.937036i	$0.386442\pi$
$24$	0	0
$25$	17.0243	0.136195
$26$	0	0
$27$	0	0
$28$	0	0
$29$	286.162	1.83238	0.916190	−	0.400744i	$-0.131248\pi$
$29$	286.162	1.83238	0.916190	+	0.400744i	$0.131248\pi$
$30$	0	0
$31$	8.54816	0.0495256	0.0247628	−	0.999693i	$-0.492117\pi$
$31$	8.54816	0.0495256	0.0247628	+	0.999693i	$0.492117\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−311.309	−1.50345
$36$	0	0
$37$	357.982	1.59059	0.795296	−	0.606221i	$-0.207315\pi$
$37$	357.982	1.59059	0.795296	+	0.606221i	$0.207315\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−194.467	−0.740748	−0.370374	−	0.928883i	$-0.620770\pi$
$41$	−194.467	−0.740748	−0.370374	+	0.928883i	$0.620770\pi$
$42$	0	0
$43$	74.2619	0.263368	0.131684	−	0.991292i	$-0.457962\pi$
$43$	74.2619	0.263368	0.131684	+	0.991292i	$0.457962\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	23.6130	0.0732831	0.0366416	−	0.999328i	$-0.488334\pi$
$47$	23.6130	0.0732831	0.0366416	+	0.999328i	$0.488334\pi$
$48$	0	0
$49$	339.369	0.989415
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−104.330	−0.270393	−0.135197	−	0.990819i	$-0.543167\pi$
$53$	−104.330	−0.270393	−0.135197	+	0.990819i	$0.543167\pi$
$54$	0	0
$55$	−38.6614	−0.0947837
$56$	0	0
$57$	0	0
$58$	0	0
$59$	249.363	0.550243	0.275122	−	0.961409i	$-0.411282\pi$
$59$	249.363	0.550243	0.275122	+	0.961409i	$0.411282\pi$
$60$	0	0
$61$	−370.384	−0.777424	−0.388712	−	0.921359i	$-0.627080\pi$
$61$	−370.384	−0.777424	−0.388712	+	0.921359i	$0.627080\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−238.961	−0.455992
$66$	0	0
$67$	−939.650	−1.71338	−0.856691	−	0.515830i	$-0.827483\pi$
$67$	−939.650	−1.71338	−0.856691	+	0.515830i	$0.827483\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−520.197	−0.869522	−0.434761	−	0.900546i	$-0.643167\pi$
$71$	−520.197	−0.869522	−0.434761	+	0.900546i	$0.643167\pi$
$72$	0	0
$73$	348.741	0.559137	0.279568	−	0.960126i	$-0.409809\pi$
$73$	348.741	0.559137	0.279568	+	0.960126i	$0.409809\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	84.7434	0.125421
$78$	0	0
$79$	953.827	1.35840	0.679202	−	0.733951i	$-0.262326\pi$
$79$	953.827	1.35840	0.679202	+	0.733951i	$0.262326\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1414.28	−1.87033	−0.935166	−	0.354211i	$-0.884750\pi$
$83$	−1414.28	−1.87033	−0.935166	+	0.354211i	$0.884750\pi$
$84$	0	0
$85$	202.596	0.258525
$86$	0	0
$87$	0	0
$88$	0	0
$89$	486.132	0.578987	0.289493	−	0.957180i	$-0.406513\pi$
$89$	486.132	0.578987	0.289493	+	0.957180i	$0.406513\pi$
$90$	0	0
$91$	523.788	0.603384
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−683.422	−0.738080
$96$	0	0
$97$	−685.281	−0.717317	−0.358659	−	0.933469i	$-0.616766\pi$
$97$	−685.281	−0.717317	−0.358659	+	0.933469i	$0.616766\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−864.755	−0.851944	−0.425972	−	0.904736i	$-0.640068\pi$
$101$	−864.755	−0.851944	−0.425972	+	0.904736i	$0.640068\pi$
$102$	0	0
$103$	−1880.91	−1.79933	−0.899665	−	0.436580i	$-0.856190\pi$
$103$	−1880.91	−1.79933	−0.899665	+	0.436580i	$0.856190\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−32.8149	−0.0296480	−0.0148240	−	0.999890i	$-0.504719\pi$
$107$	−32.8149	−0.0296480	−0.0148240	+	0.999890i	$0.504719\pi$
$108$	0	0
$109$	528.727	0.464613	0.232307	−	0.972643i	$-0.425373\pi$
$109$	528.727	0.464613	0.232307	+	0.972643i	$0.425373\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−414.691	−0.345229	−0.172614	−	0.984989i	$-0.555221\pi$
$113$	−414.691	−0.345229	−0.172614	+	0.984989i	$0.555221\pi$
$114$	0	0
$115$	918.161	0.744513
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−444.077	−0.342088
$120$	0	0
$121$	−1320.48	−0.992093
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1286.79	−0.920751
$126$	0	0
$127$	−596.093	−0.416494	−0.208247	−	0.978076i	$-0.566776\pi$
$127$	−596.093	−0.416494	−0.208247	+	0.978076i	$0.566776\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	121.819	0.0812472	0.0406236	−	0.999175i	$-0.487066\pi$
$131$	121.819	0.0812472	0.0406236	+	0.999175i	$0.487066\pi$
$132$	0	0
$133$	1498.02	0.976652
$134$	0	0
$135$	0	0
$136$	0	0
$137$	897.365	0.559614	0.279807	−	0.960056i	$-0.409730\pi$
$137$	897.365	0.559614	0.279807	+	0.960056i	$0.409730\pi$
$138$	0	0
$139$	2113.61	1.28974	0.644871	−	0.764292i	$-0.276911\pi$
$139$	2113.61	1.28974	0.644871	+	0.764292i	$0.276911\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	65.0493	0.0380398
$144$	0	0
$145$	3410.31	1.95318
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2580.76	−1.41895	−0.709476	−	0.704729i	$-0.751068\pi$
$149$	−2580.76	−1.41895	−0.709476	+	0.704729i	$0.751068\pi$
$150$	0	0
$151$	−1342.77	−0.723662	−0.361831	−	0.932244i	$-0.617848\pi$
$151$	−1342.77	−0.723662	−0.361831	+	0.932244i	$0.617848\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	101.872	0.0527906
$156$	0	0
$157$	−2495.82	−1.26871	−0.634357	−	0.773041i	$-0.718735\pi$
$157$	−2495.82	−1.26871	−0.634357	+	0.773041i	$0.718735\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2012.55	−0.985164
$162$	0	0
$163$	−1961.58	−0.942595	−0.471297	−	0.881974i	$-0.656214\pi$
$163$	−1961.58	−0.942595	−0.471297	+	0.881974i	$0.656214\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2179.24	1.00979	0.504894	−	0.863182i	$-0.331532\pi$
$167$	2179.24	1.00979	0.504894	+	0.863182i	$0.331532\pi$
$168$	0	0
$169$	−1794.94	−0.816995
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3111.45	−1.36739	−0.683697	−	0.729766i	$-0.739629\pi$
$173$	−3111.45	−1.36739	−0.683697	+	0.729766i	$0.739629\pi$
$174$	0	0
$175$	−444.713	−0.192098
$176$	0	0
$177$	0	0
$178$	0	0
$179$	810.106	0.338269	0.169135	−	0.985593i	$-0.445903\pi$
$179$	810.106	0.338269	0.169135	+	0.985593i	$0.445903\pi$
$180$	0	0
$181$	−3356.23	−1.37827	−0.689134	−	0.724634i	$-0.742009\pi$
$181$	−3356.23	−1.37827	−0.689134	+	0.724634i	$0.742009\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4266.22	1.69545
$186$	0	0
$187$	−55.1500	−0.0215667
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1338.41	0.507038	0.253519	−	0.967330i	$-0.418412\pi$
$191$	1338.41	0.507038	0.253519	+	0.967330i	$0.418412\pi$
$192$	0	0
$193$	−227.465	−0.0848358	−0.0424179	−	0.999100i	$-0.513506\pi$
$193$	−227.465	−0.0848358	−0.0424179	+	0.999100i	$0.513506\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−815.549	−0.294952	−0.147476	−	0.989066i	$-0.547115\pi$
$197$	−815.549	−0.294952	−0.147476	+	0.989066i	$0.547115\pi$
$198$	0	0
$199$	−1866.90	−0.665030	−0.332515	−	0.943098i	$-0.607897\pi$
$199$	−1866.90	−0.665030	−0.332515	+	0.943098i	$0.607897\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7475.19	−2.58451
$204$	0	0
$205$	−2317.54	−0.789581
$206$	0	0
$207$	0	0
$208$	0	0
$209$	186.039	0.0615721
$210$	0	0
$211$	1102.88	0.359836	0.179918	−	0.983682i	$-0.442417\pi$
$211$	1102.88	0.359836	0.179918	+	0.983682i	$0.442417\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	885.008	0.280731
$216$	0	0
$217$	−223.297	−0.0698542
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−340.875	−0.103754
$222$	0	0
$223$	568.848	0.170820	0.0854100	−	0.996346i	$-0.472780\pi$
$223$	568.848	0.170820	0.0854100	+	0.996346i	$0.472780\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2106.99	0.616061	0.308030	−	0.951377i	$-0.400330\pi$
$227$	2106.99	0.616061	0.308030	+	0.951377i	$0.400330\pi$
$228$	0	0
$229$	4336.30	1.25131	0.625656	−	0.780099i	$-0.284831\pi$
$229$	4336.30	1.25131	0.625656	+	0.780099i	$0.284831\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4517.39	1.27014	0.635072	−	0.772453i	$-0.280970\pi$
$233$	4517.39	1.27014	0.635072	+	0.772453i	$0.280970\pi$
$234$	0	0
$235$	281.405	0.0781143
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5300.88	1.43467	0.717333	−	0.696731i	$-0.245363\pi$
$239$	5300.88	1.43467	0.717333	+	0.696731i	$0.245363\pi$
$240$	0	0
$241$	−1368.82	−0.365864	−0.182932	−	0.983126i	$-0.558559\pi$
$241$	−1368.82	−0.365864	−0.182932	+	0.983126i	$0.558559\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4044.40	1.05464
$246$	0	0
$247$	1149.88	0.296216
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5547.63	−1.39507	−0.697536	−	0.716549i	$-0.745720\pi$
$251$	−5547.63	−1.39507	−0.697536	+	0.716549i	$0.745720\pi$
$252$	0	0
$253$	−249.939	−0.0621088
$254$	0	0
$255$	0	0
$256$	0	0
$257$	193.949	0.0470748	0.0235374	−	0.999723i	$-0.492507\pi$
$257$	193.949	0.0470748	0.0235374	+	0.999723i	$0.492507\pi$
$258$	0	0
$259$	−9351.29	−2.24348
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1345.63	−0.315494	−0.157747	−	0.987480i	$-0.550423\pi$
$263$	−1345.63	−0.315494	−0.157747	+	0.987480i	$0.550423\pi$
$264$	0	0
$265$	−1243.34	−0.288218
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3083.04	0.698797	0.349398	−	0.936974i	$-0.386386\pi$
$269$	3083.04	0.698797	0.349398	+	0.936974i	$0.386386\pi$
$270$	0	0
$271$	422.163	0.0946294	0.0473147	−	0.998880i	$-0.484934\pi$
$271$	422.163	0.0946294	0.0473147	+	0.998880i	$0.484934\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−55.2288	−0.0121106
$276$	0	0
$277$	−8260.00	−1.79168	−0.895840	−	0.444377i	$-0.853425\pi$
$277$	−8260.00	−1.79168	−0.895840	+	0.444377i	$0.853425\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3321.91	−0.705226	−0.352613	−	0.935769i	$-0.614707\pi$
$281$	−3321.91	−0.705226	−0.352613	+	0.935769i	$0.614707\pi$
$282$	0	0
$283$	7954.43	1.67082	0.835409	−	0.549629i	$-0.185231\pi$
$283$	7954.43	1.67082	0.835409	+	0.549629i	$0.185231\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5079.91	1.04480
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1171.99	0.233681	0.116841	−	0.993151i	$-0.462723\pi$
$293$	1171.99	0.233681	0.116841	+	0.993151i	$0.462723\pi$
$294$	0	0
$295$	2971.76	0.586517
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1544.84	−0.298798
$300$	0	0
$301$	−1939.88	−0.371472
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4414.01	−0.828675
$306$	0	0
$307$	−865.763	−0.160950	−0.0804751	−	0.996757i	$-0.525644\pi$
$307$	−865.763	−0.160950	−0.0804751	+	0.996757i	$0.525644\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6994.83	1.27537	0.637685	−	0.770297i	$-0.279892\pi$
$311$	6994.83	1.27537	0.637685	+	0.770297i	$0.279892\pi$
$312$	0	0
$313$	3442.33	0.621635	0.310818	−	0.950470i	$-0.399397\pi$
$313$	3442.33	0.621635	0.310818	+	0.950470i	$0.399397\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2066.15	−0.366078	−0.183039	−	0.983106i	$-0.558593\pi$
$317$	−2066.15	−0.366078	−0.183039	+	0.983106i	$0.558593\pi$
$318$	0	0
$319$	−928.344	−0.162938
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−974.892	−0.167939
$324$	0	0
$325$	−341.362	−0.0582627
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−616.823	−0.103363
$330$	0	0
$331$	−9027.44	−1.49907	−0.749536	−	0.661964i	$-0.769723\pi$
$331$	−9027.44	−1.49907	−0.749536	+	0.661964i	$0.769723\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11198.2	−1.82633
$336$	0	0
$337$	204.309	0.0330250	0.0165125	−	0.999864i	$-0.494744\pi$
$337$	204.309	0.0330250	0.0165125	+	0.999864i	$0.494744\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−27.7312	−0.00440390
$342$	0	0
$343$	94.8397	0.0149296
$344$	0	0
$345$	0	0
$346$	0	0
$347$	143.063	0.0221326	0.0110663	−	0.999939i	$-0.496477\pi$
$347$	143.063	0.0221326	0.0110663	+	0.999939i	$0.496477\pi$
$348$	0	0
$349$	−3998.42	−0.613268	−0.306634	−	0.951828i	$-0.599203\pi$
$349$	−3998.42	−0.613268	−0.306634	+	0.951828i	$0.599203\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5809.57	0.875956	0.437978	−	0.898986i	$-0.355695\pi$
$353$	5809.57	0.875956	0.437978	+	0.898986i	$0.355695\pi$
$354$	0	0
$355$	−6199.39	−0.926844
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4895.37	−0.719687	−0.359844	−	0.933013i	$-0.617170\pi$
$359$	−4895.37	−0.719687	−0.359844	+	0.933013i	$0.617170\pi$
$360$	0	0
$361$	−3570.37	−0.520538
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4156.08	0.595998
$366$	0	0
$367$	−528.151	−0.0751206	−0.0375603	−	0.999294i	$-0.511959\pi$
$367$	−528.151	−0.0751206	−0.0375603	+	0.999294i	$0.511959\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2725.33	0.381380
$372$	0	0
$373$	−10113.5	−1.40390	−0.701950	−	0.712226i	$-0.747687\pi$
$373$	−10113.5	−1.40390	−0.701950	+	0.712226i	$0.747687\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5737.98	−0.783875
$378$	0	0
$379$	−729.385	−0.0988548	−0.0494274	−	0.998778i	$-0.515740\pi$
$379$	−729.385	−0.0988548	−0.0494274	+	0.998778i	$0.515740\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1608.08	−0.214540	−0.107270	−	0.994230i	$-0.534211\pi$
$383$	−1608.08	−0.214540	−0.107270	+	0.994230i	$0.534211\pi$
$384$	0	0
$385$	1009.92	0.133689
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9824.09	−1.28047	−0.640233	−	0.768181i	$-0.721162\pi$
$389$	−9824.09	−1.28047	−0.640233	+	0.768181i	$0.721162\pi$
$390$	0	0
$391$	1309.74	0.169403
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11367.1	1.44796
$396$	0	0
$397$	2876.88	0.363694	0.181847	−	0.983327i	$-0.441792\pi$
$397$	2876.88	0.363694	0.181847	+	0.983327i	$0.441792\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6515.91	0.811444	0.405722	−	0.913996i	$-0.367020\pi$
$401$	6515.91	0.811444	0.405722	+	0.913996i	$0.367020\pi$
$402$	0	0
$403$	−171.403	−0.0211866
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1161.34	−0.141438
$408$	0	0
$409$	−8870.10	−1.07237	−0.536183	−	0.844101i	$-0.680134\pi$
$409$	−8870.10	−1.07237	−0.536183	+	0.844101i	$0.680134\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6513.92	−0.776099
$414$	0	0
$415$	−16854.5	−1.99363
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1009.53	−0.117706	−0.0588531	−	0.998267i	$-0.518744\pi$
$419$	−1009.53	−0.117706	−0.0588531	+	0.998267i	$0.518744\pi$
$420$	0	0
$421$	−3253.60	−0.376652	−0.188326	−	0.982107i	$-0.560306\pi$
$421$	−3253.60	−0.376652	−0.188326	+	0.982107i	$0.560306\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	289.413	0.0330320
$426$	0	0
$427$	9675.25	1.09653
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2352.51	−0.262915	−0.131457	−	0.991322i	$-0.541966\pi$
$431$	−2352.51	−0.262915	−0.131457	+	0.991322i	$0.541966\pi$
$432$	0	0
$433$	−5860.51	−0.650434	−0.325217	−	0.945639i	$-0.605437\pi$
$433$	−5860.51	−0.650434	−0.325217	+	0.945639i	$0.605437\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4418.20	−0.483641
$438$	0	0
$439$	−2894.17	−0.314650	−0.157325	−	0.987547i	$-0.550287\pi$
$439$	−2894.17	−0.314650	−0.157325	+	0.987547i	$0.550287\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8256.85	−0.885541	−0.442771	−	0.896635i	$-0.646004\pi$
$443$	−8256.85	−0.885541	−0.442771	+	0.896635i	$0.646004\pi$
$444$	0	0
$445$	5793.42	0.617156
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15487.1	−1.62779	−0.813897	−	0.581009i	$-0.802658\pi$
$449$	−15487.1	−1.62779	−0.813897	+	0.581009i	$0.802658\pi$
$450$	0	0
$451$	630.874	0.0658685
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6242.19	0.643162
$456$	0	0
$457$	−16055.6	−1.64343	−0.821716	−	0.569897i	$-0.806983\pi$
$457$	−16055.6	−1.64343	−0.821716	+	0.569897i	$0.806983\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14064.0	−1.42088	−0.710440	−	0.703758i	$-0.751504\pi$
$461$	−14064.0	−1.42088	−0.710440	+	0.703758i	$0.751504\pi$
$462$	0	0
$463$	8071.30	0.810162	0.405081	−	0.914281i	$-0.367243\pi$
$463$	8071.30	0.810162	0.405081	+	0.914281i	$0.367243\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8582.41	0.850421	0.425211	−	0.905094i	$-0.360200\pi$
$467$	8582.41	0.850421	0.425211	+	0.905094i	$0.360200\pi$
$468$	0	0
$469$	24545.7	2.41667
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−240.914	−0.0234191
$474$	0	0
$475$	−976.286	−0.0943054
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6320.96	−0.602948	−0.301474	−	0.953474i	$-0.597479\pi$
$479$	−6320.96	−0.602948	−0.301474	+	0.953474i	$0.597479\pi$
$480$	0	0
$481$	−7178.07	−0.680441
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8166.77	−0.764606
$486$	0	0
$487$	−7336.47	−0.682643	−0.341321	−	0.939947i	$-0.610874\pi$
$487$	−7336.47	−0.682643	−0.341321	+	0.939947i	$0.610874\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6672.53	0.613294	0.306647	−	0.951823i	$-0.400793\pi$
$491$	6672.53	0.613294	0.306647	+	0.951823i	$0.400793\pi$
$492$	0	0
$493$	4864.76	0.444417
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13588.7	1.22643
$498$	0	0
$499$	−17920.9	−1.60772	−0.803858	−	0.594821i	$-0.797223\pi$
$499$	−17920.9	−1.60772	−0.803858	+	0.594821i	$0.797223\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11325.3	−1.00392	−0.501959	−	0.864891i	$-0.667387\pi$
$503$	−11325.3	−1.00392	−0.501959	+	0.864891i	$0.667387\pi$
$504$	0	0
$505$	−10305.6	−0.908108
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8313.78	0.723972	0.361986	−	0.932184i	$-0.382099\pi$
$509$	8313.78	0.723972	0.361986	+	0.932184i	$0.382099\pi$
$510$	0	0
$511$	−9109.87	−0.788644
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−22415.5	−1.91795
$516$	0	0
$517$	−76.6033	−0.00651646
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5121.64	−0.430677	−0.215339	−	0.976539i	$-0.569086\pi$
$521$	−5121.64	−0.430677	−0.215339	+	0.976539i	$0.569086\pi$
$522$	0	0
$523$	−13378.5	−1.11855	−0.559275	−	0.828982i	$-0.688920\pi$
$523$	−13378.5	−1.11855	−0.559275	+	0.828982i	$0.688920\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	145.319	0.0120117
$528$	0	0
$529$	−6231.26	−0.512144
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3899.35	0.316885
$534$	0	0
$535$	−391.068	−0.0316025
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1100.95	−0.0879804
$540$	0	0
$541$	9906.81	0.787296	0.393648	−	0.919261i	$-0.371213\pi$
$541$	9906.81	0.787296	0.393648	+	0.919261i	$0.371213\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6301.05	0.495243
$546$	0	0
$547$	−16399.6	−1.28189	−0.640947	−	0.767585i	$-0.721458\pi$
$547$	−16399.6	−1.28189	−0.640947	+	0.767585i	$0.721458\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16410.4	−1.26880
$552$	0	0
$553$	−24916.1	−1.91598
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22044.3	−1.67692	−0.838461	−	0.544962i	$-0.816544\pi$
$557$	−22044.3	−1.67692	−0.838461	+	0.544962i	$0.816544\pi$
$558$	0	0
$559$	−1489.06	−0.112666
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12048.8	0.901947	0.450973	−	0.892537i	$-0.351077\pi$
$563$	12048.8	0.901947	0.450973	+	0.892537i	$0.351077\pi$
$564$	0	0
$565$	−4942.04	−0.367988
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23785.4	1.75243	0.876217	−	0.481916i	$-0.160059\pi$
$569$	23785.4	1.75243	0.876217	+	0.481916i	$0.160059\pi$
$570$	0	0
$571$	10878.3	0.797271	0.398635	−	0.917110i	$-0.369484\pi$
$571$	10878.3	0.797271	0.398635	+	0.917110i	$0.369484\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1311.62	0.0951274
$576$	0	0
$577$	6315.86	0.455689	0.227845	−	0.973698i	$-0.426832\pi$
$577$	6315.86	0.455689	0.227845	+	0.973698i	$0.426832\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	36944.1	2.63804
$582$	0	0
$583$	338.459	0.0240438
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18192.1	1.27916	0.639581	−	0.768724i	$-0.279108\pi$
$587$	18192.1	1.27916	0.639581	+	0.768724i	$0.279108\pi$
$588$	0	0
$589$	−490.207	−0.0342931
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9828.72	0.680636	0.340318	−	0.940310i	$-0.389465\pi$
$593$	9828.72	0.680636	0.340318	+	0.940310i	$0.389465\pi$
$594$	0	0
$595$	−5292.25	−0.364640
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4662.57	0.318043	0.159021	−	0.987275i	$-0.449166\pi$
$599$	4662.57	0.318043	0.159021	+	0.987275i	$0.449166\pi$
$600$	0	0
$601$	21658.6	1.47000	0.735001	−	0.678066i	$-0.237182\pi$
$601$	21658.6	1.47000	0.735001	+	0.678066i	$0.237182\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−15736.6	−1.05750
$606$	0	0
$607$	−25764.7	−1.72283	−0.861415	−	0.507902i	$-0.830421\pi$
$607$	−25764.7	−1.72283	−0.861415	+	0.507902i	$0.830421\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−473.475	−0.0313498
$612$	0	0
$613$	16018.1	1.05541	0.527705	−	0.849428i	$-0.323053\pi$
$613$	16018.1	1.05541	0.527705	+	0.849428i	$0.323053\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22250.3	−1.45180	−0.725902	−	0.687798i	$-0.758578\pi$
$617$	−22250.3	−1.45180	−0.725902	+	0.687798i	$0.758578\pi$
$618$	0	0
$619$	3765.95	0.244534	0.122267	−	0.992497i	$-0.460984\pi$
$619$	3765.95	0.244534	0.122267	+	0.992497i	$0.460984\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12698.8	−0.816642
$624$	0	0
$625$	−17463.2	−1.11765
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6085.70	0.385775
$630$	0	0
$631$	20806.5	1.31267	0.656334	−	0.754470i	$-0.272106\pi$
$631$	20806.5	1.31267	0.656334	+	0.754470i	$0.272106\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7103.88	−0.443951
$636$	0	0
$637$	−6804.85	−0.423262
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2439.58	−0.150324	−0.0751620	−	0.997171i	$-0.523947\pi$
$641$	−2439.58	−0.150324	−0.0751620	+	0.997171i	$0.523947\pi$
$642$	0	0
$643$	19320.1	1.18493	0.592466	−	0.805595i	$-0.298154\pi$
$643$	19320.1	1.18493	0.592466	+	0.805595i	$0.298154\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14067.1	−0.854766	−0.427383	−	0.904071i	$-0.640564\pi$
$647$	−14067.1	−0.854766	−0.427383	+	0.904071i	$0.640564\pi$
$648$	0	0
$649$	−808.963	−0.0489285
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15893.7	−0.952478	−0.476239	−	0.879316i	$-0.658000\pi$
$653$	−15893.7	−0.952478	−0.476239	+	0.879316i	$0.658000\pi$
$654$	0	0
$655$	1451.77	0.0866033
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9653.54	−0.570635	−0.285318	−	0.958433i	$-0.592099\pi$
$659$	−9653.54	−0.570635	−0.285318	+	0.958433i	$0.592099\pi$
$660$	0	0
$661$	5389.06	0.317111	0.158555	−	0.987350i	$-0.449316\pi$
$661$	5389.06	0.317111	0.158555	+	0.987350i	$0.449316\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	17852.5	1.04104
$666$	0	0
$667$	22047.0	1.27986
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1201.57	0.0691297
$672$	0	0
$673$	−3032.18	−0.173673	−0.0868366	−	0.996223i	$-0.527676\pi$
$673$	−3032.18	−0.173673	−0.0868366	+	0.996223i	$0.527676\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22029.2	1.25059	0.625295	−	0.780388i	$-0.284979\pi$
$677$	22029.2	1.25059	0.625295	+	0.780388i	$0.284979\pi$
$678$	0	0
$679$	17901.1	1.01175
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−9040.72	−0.506491	−0.253246	−	0.967402i	$-0.581498\pi$
$683$	−9040.72	−0.506491	−0.253246	+	0.967402i	$0.581498\pi$
$684$	0	0
$685$	10694.3	0.596506
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2091.97	0.115672
$690$	0	0
$691$	22863.5	1.25871	0.629355	−	0.777118i	$-0.283319\pi$
$691$	22863.5	1.25871	0.629355	+	0.777118i	$0.283319\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	25188.7	1.37477
$696$	0	0
$697$	−3305.94	−0.179658
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1753.00	−0.0944507	−0.0472253	−	0.998884i	$-0.515038\pi$
$701$	−1753.00	−0.0944507	−0.0472253	+	0.998884i	$0.515038\pi$
$702$	0	0
$703$	−20529.1	−1.10138
$704$	0	0
$705$	0	0
$706$	0	0
$707$	22589.3	1.20164
$708$	0	0
$709$	11547.0	0.611645	0.305823	−	0.952089i	$-0.401069\pi$
$709$	11547.0	0.611645	0.305823	+	0.952089i	$0.401069\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	658.582	0.0345920
$714$	0	0
$715$	775.218	0.0405476
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10289.8	−0.533720	−0.266860	−	0.963735i	$-0.585986\pi$
$719$	−10289.8	−0.533720	−0.266860	+	0.963735i	$0.585986\pi$
$720$	0	0
$721$	49133.4	2.53790
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4871.72	0.249560
$726$	0	0
$727$	−2950.10	−0.150499	−0.0752497	−	0.997165i	$-0.523975\pi$
$727$	−2950.10	−0.150499	−0.0752497	+	0.997165i	$0.523975\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1262.45	0.0638762
$732$	0	0
$733$	24348.2	1.22691	0.613453	−	0.789731i	$-0.289780\pi$
$733$	24348.2	1.22691	0.613453	+	0.789731i	$0.289780\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3048.33	0.152357
$738$	0	0
$739$	29233.5	1.45517	0.727585	−	0.686017i	$-0.240642\pi$
$739$	29233.5	1.45517	0.727585	+	0.686017i	$0.240642\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15340.6	0.757457	0.378729	−	0.925508i	$-0.376361\pi$
$743$	15340.6	0.757457	0.378729	+	0.925508i	$0.376361\pi$
$744$	0	0
$745$	−30755.9	−1.51250
$746$	0	0
$747$	0	0
$748$	0	0
$749$	857.197	0.0418175
$750$	0	0
$751$	−39862.6	−1.93689	−0.968446	−	0.249223i	$-0.919825\pi$
$751$	−39862.6	−1.93689	−0.968446	+	0.249223i	$0.919825\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16002.3	−0.771369
$756$	0	0
$757$	26375.1	1.26634	0.633169	−	0.774013i	$-0.281754\pi$
$757$	26375.1	1.26634	0.633169	+	0.774013i	$0.281754\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7848.63	−0.373867	−0.186933	−	0.982373i	$-0.559855\pi$
$761$	−7848.63	−0.373867	−0.186933	+	0.982373i	$0.559855\pi$
$762$	0	0
$763$	−13811.5	−0.655322
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5000.10	−0.235389
$768$	0	0
$769$	31818.9	1.49209	0.746046	−	0.665895i	$-0.231950\pi$
$769$	31818.9	1.49209	0.746046	+	0.665895i	$0.231950\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29559.8	1.37541	0.687706	−	0.725990i	$-0.258618\pi$
$773$	29559.8	1.37541	0.687706	+	0.725990i	$0.258618\pi$
$774$	0	0
$775$	145.527	0.00674512
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11152.0	0.512917
$780$	0	0
$781$	1687.58	0.0773193
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−29743.6	−1.35235
$786$	0	0
$787$	28038.7	1.26998	0.634989	−	0.772521i	$-0.281005\pi$
$787$	28038.7	1.26998	0.634989	+	0.772521i	$0.281005\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10832.6	0.486934
$792$	0	0
$793$	7426.75	0.332574
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5320.45	−0.236462	−0.118231	−	0.992986i	$-0.537722\pi$
$797$	−5320.45	−0.236462	−0.118231	+	0.992986i	$0.537722\pi$
$798$	0	0
$799$	401.421	0.0177738
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1131.35	−0.0497194
$804$	0	0
$805$	−23984.4	−1.05011
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−30934.0	−1.34435	−0.672177	−	0.740390i	$-0.734641\pi$
$809$	−30934.0	−1.34435	−0.672177	+	0.740390i	$0.734641\pi$
$810$	0	0
$811$	40364.5	1.74771	0.873854	−	0.486189i	$-0.161613\pi$
$811$	40364.5	1.74771	0.873854	+	0.486189i	$0.161613\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23376.9	−1.00473
$816$	0	0
$817$	−4258.66	−0.182364
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19799.7	−0.841672	−0.420836	−	0.907137i	$-0.638263\pi$
$821$	−19799.7	−0.841672	−0.420836	+	0.907137i	$0.638263\pi$
$822$	0	0
$823$	−18756.4	−0.794419	−0.397210	−	0.917728i	$-0.630021\pi$
$823$	−18756.4	−0.794419	−0.397210	+	0.917728i	$0.630021\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20958.0	0.881234	0.440617	−	0.897695i	$-0.354760\pi$
$827$	20958.0	0.881234	0.440617	+	0.897695i	$0.354760\pi$
$828$	0	0
$829$	−31320.3	−1.31218	−0.656091	−	0.754682i	$-0.727791\pi$
$829$	−31320.3	−1.31218	−0.656091	+	0.754682i	$0.727791\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5769.28	0.239968
$834$	0	0
$835$	25970.8	1.07636
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30290.6	−1.24642	−0.623212	−	0.782053i	$-0.714173\pi$
$839$	−30290.6	−1.24642	−0.623212	+	0.782053i	$0.714173\pi$
$840$	0	0
$841$	57499.9	2.35762
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−21391.0	−0.870855
$846$	0	0
$847$	34493.7	1.39931
$848$	0	0
$849$	0	0
$850$	0	0
$851$	27580.3	1.11098
$852$	0	0
$853$	−21111.8	−0.847425	−0.423712	−	0.905797i	$-0.639273\pi$
$853$	−21111.8	−0.847425	−0.423712	+	0.905797i	$0.639273\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39983.0	1.59369	0.796845	−	0.604184i	$-0.206501\pi$
$857$	39983.0	1.59369	0.796845	+	0.604184i	$0.206501\pi$
$858$	0	0
$859$	39503.3	1.56907	0.784537	−	0.620082i	$-0.212901\pi$
$859$	39503.3	1.56907	0.784537	+	0.620082i	$0.212901\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26019.9	1.02634	0.513168	−	0.858288i	$-0.328472\pi$
$863$	26019.9	1.02634	0.513168	+	0.858288i	$0.328472\pi$
$864$	0	0
$865$	−37080.4	−1.45754
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3094.33	−0.120791
$870$	0	0
$871$	18841.4	0.732968
$872$	0	0
$873$	0	0
$874$	0	0
$875$	33613.8	1.29869
$876$	0	0
$877$	15038.3	0.579027	0.289514	−	0.957174i	$-0.406506\pi$
$877$	15038.3	0.579027	0.289514	+	0.957174i	$0.406506\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18334.3	0.701133	0.350567	−	0.936538i	$-0.385989\pi$
$881$	18334.3	0.701133	0.350567	+	0.936538i	$0.385989\pi$
$882$	0	0
$883$	−26659.5	−1.01604	−0.508020	−	0.861345i	$-0.669622\pi$
$883$	−26659.5	−1.01604	−0.508020	+	0.861345i	$0.669622\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11473.7	0.434327	0.217163	−	0.976135i	$-0.430320\pi$
$887$	11473.7	0.434327	0.217163	+	0.976135i	$0.430320\pi$
$888$	0	0
$889$	15571.3	0.587450
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1354.12	−0.0507436
$894$	0	0
$895$	9654.36	0.360569
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2446.16	0.0907498
$900$	0	0
$901$	−1773.61	−0.0655799
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−39997.5	−1.46913
$906$	0	0
$907$	20361.6	0.745421	0.372710	−	0.927948i	$-0.378429\pi$
$907$	20361.6	0.745421	0.372710	+	0.927948i	$0.378429\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19261.9	−0.700523	−0.350262	−	0.936652i	$-0.613907\pi$
$911$	−19261.9	−0.700523	−0.350262	+	0.936652i	$0.613907\pi$
$912$	0	0
$913$	4588.09	0.166313
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3182.18	−0.114596
$918$	0	0
$919$	21191.8	0.760666	0.380333	−	0.924850i	$-0.375809\pi$
$919$	21191.8	0.760666	0.380333	+	0.924850i	$0.375809\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10430.7	0.371973
$924$	0	0
$925$	6094.40	0.216630
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−40815.0	−1.44144	−0.720719	−	0.693227i	$-0.756188\pi$
$929$	−40815.0	−1.44144	−0.720719	+	0.693227i	$0.756188\pi$
$930$	0	0
$931$	−19461.7	−0.685102
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−657.244	−0.0229884
$936$	0	0
$937$	−38439.1	−1.34018	−0.670092	−	0.742278i	$-0.733745\pi$
$937$	−38439.1	−1.34018	−0.670092	+	0.742278i	$0.733745\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2244.08	0.0777415	0.0388708	−	0.999244i	$-0.487624\pi$
$941$	2244.08	0.0777415	0.0388708	+	0.999244i	$0.487624\pi$
$942$	0	0
$943$	−14982.5	−0.517388
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42289.0	−1.45112	−0.725559	−	0.688160i	$-0.758419\pi$
$947$	−42289.0	−1.45112	−0.725559	+	0.688160i	$0.758419\pi$
$948$	0	0
$949$	−6992.76	−0.239193
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−37426.2	−1.27214	−0.636072	−	0.771629i	$-0.719442\pi$
$953$	−37426.2	−1.27214	−0.636072	+	0.771629i	$0.719442\pi$
$954$	0	0
$955$	15950.4	0.540464
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−23441.2	−0.789317
$960$	0	0
$961$	−29717.9	−0.997547
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2710.80	−0.0904286
$966$	0	0
$967$	−1088.56	−0.0362003	−0.0181001	−	0.999836i	$-0.505762\pi$
$967$	−1088.56	−0.0362003	−0.0181001	+	0.999836i	$0.505762\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39506.5	1.30569	0.652845	−	0.757491i	$-0.273575\pi$
$971$	39506.5	1.30569	0.652845	+	0.757491i	$0.273575\pi$
$972$	0	0
$973$	−55212.1	−1.81914
$974$	0	0
$975$	0	0
$976$	0	0
$977$	43326.8	1.41878	0.709389	−	0.704817i	$-0.248971\pi$
$977$	43326.8	1.41878	0.709389	+	0.704817i	$0.248971\pi$
$978$	0	0
$979$	−1577.07	−0.0514845
$980$	0	0
$981$	0	0
$982$	0	0
$983$	10664.1	0.346014	0.173007	−	0.984921i	$-0.444652\pi$
$983$	10664.1	0.346014	0.173007	+	0.984921i	$0.444652\pi$
$984$	0	0
$985$	−9719.22	−0.314396
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5721.42	0.183954
$990$	0	0
$991$	−15461.4	−0.495609	−0.247804	−	0.968810i	$-0.579709\pi$
$991$	−15461.4	−0.495609	−0.247804	+	0.968810i	$0.579709\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−22248.6	−0.708871
$996$	0	0
$997$	46061.1	1.46316	0.731579	−	0.681756i	$-0.238784\pi$
$997$	46061.1	1.46316	0.731579	+	0.681756i	$0.238784\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.4.a.bi.1.3		3
3.2	odd	2		272.4.a.h.1.3		3
4.3	odd	2		153.4.a.g.1.1		3
12.11	even	2		17.4.a.b.1.3	✓	3
24.5	odd	2		1088.4.a.x.1.1		3
24.11	even	2		1088.4.a.v.1.3		3
60.23	odd	4		425.4.b.f.324.2		6
60.47	odd	4		425.4.b.f.324.5		6
60.59	even	2		425.4.a.g.1.1		3
84.83	odd	2		833.4.a.d.1.3		3
132.131	odd	2		2057.4.a.e.1.1		3
204.47	even	4		289.4.b.b.288.1		6
204.191	even	4		289.4.b.b.288.2		6
204.203	even	2		289.4.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.a.b.1.3	✓	3	12.11	even	2
153.4.a.g.1.1		3	4.3	odd	2
272.4.a.h.1.3		3	3.2	odd	2
289.4.a.b.1.3		3	204.203	even	2
289.4.b.b.288.1		6	204.47	even	4
289.4.b.b.288.2		6	204.191	even	4
425.4.a.g.1.1		3	60.59	even	2
425.4.b.f.324.2		6	60.23	odd	4
425.4.b.f.324.5		6	60.47	odd	4
833.4.a.d.1.3		3	84.83	odd	2
1088.4.a.v.1.3		3	24.11	even	2
1088.4.a.x.1.1		3	24.5	odd	2
2057.4.a.e.1.1		3	132.131	odd	2
2448.4.a.bi.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2448.a (trivial)

\(f(q)\)	\(=\)	\(q+11.9174 q^{5} -26.1222 q^{7} +O(q^{10})\)	\(q+11.9174 q^{5} -26.1222 q^{7} -3.24412 q^{11} -20.0515 q^{13} +17.0000 q^{17} -57.3466 q^{19} +77.0438 q^{23} +17.0243 q^{25} +286.162 q^{29} +8.54816 q^{31} -311.309 q^{35} +357.982 q^{37} -194.467 q^{41} +74.2619 q^{43} +23.6130 q^{47} +339.369 q^{49} -104.330 q^{53} -38.6614 q^{55} +249.363 q^{59} -370.384 q^{61} -238.961 q^{65} -939.650 q^{67} -520.197 q^{71} +348.741 q^{73} +84.7434 q^{77} +953.827 q^{79} -1414.28 q^{83} +202.596 q^{85} +486.132 q^{89} +523.788 q^{91} -683.422 q^{95} -685.281 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 8 q^{5} - 22 q^{7}+O(q^{10}) \) 3 * q + 8 * q^5 - 22 * q^7	\( 3 q + 8 q^{5} - 22 q^{7} - 28 q^{11} + 30 q^{13} + 51 q^{17} - 80 q^{19} + 142 q^{23} - 223 q^{25} + 456 q^{29} - 230 q^{31} - 332 q^{35} + 356 q^{37} + 294 q^{41} - 556 q^{43} + 640 q^{47} - 269 q^{49} - 302 q^{53} - 76 q^{55} + 636 q^{59} - 84 q^{61} - 408 q^{65} - 1008 q^{67} - 402 q^{71} + 838 q^{73} + 504 q^{77} + 594 q^{79} - 2396 q^{83} + 136 q^{85} + 170 q^{89} + 1016 q^{91} - 472 q^{95} - 270 q^{97}+O(q^{100}) \) 3 * q + 8 * q^5 - 22 * q^7 - 28 * q^11 + 30 * q^13 + 51 * q^17 - 80 * q^19 + 142 * q^23 - 223 * q^25 + 456 * q^29 - 230 * q^31 - 332 * q^35 + 356 * q^37 + 294 * q^41 - 556 * q^43 + 640 * q^47 - 269 * q^49 - 302 * q^53 - 76 * q^55 + 636 * q^59 - 84 * q^61 - 408 * q^65 - 1008 * q^67 - 402 * q^71 + 838 * q^73 + 504 * q^77 + 594 * q^79 - 2396 * q^83 + 136 * q^85 + 170 * q^89 + 1016 * q^91 - 472 * q^95 - 270 * q^97

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