LMFDB - Embedded newform 2304.4.a.bx.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$2304 = 2^{8} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2304.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:	$135.940400653$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{10}, \sqrt{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 16x^{2} + 9$ x^4 - 16*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{9}$
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.764069$ of defining polynomial
Character	$\chi$	$=$	2304.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.32456 q^{5} -10.0000 q^{7} -37.9473 q^{11} +59.3296 q^{13} -75.0467 q^{17} +118.659 q^{19} -150.093 q^{23} -85.0000 q^{25} -246.658 q^{29} +62.0000 q^{31} +63.2456 q^{35} -59.3296 q^{37} +375.233 q^{41} +118.659 q^{43} +450.280 q^{47} -243.000 q^{49} +132.816 q^{53} +240.000 q^{55} -733.648 q^{59} -533.966 q^{61} -375.233 q^{65} -711.955 q^{67} -30.0000 q^{73} +379.473 q^{77} +94.0000 q^{79} -670.403 q^{83} +474.637 q^{85} -750.467 q^{89} -593.296 q^{91} -750.467 q^{95} +130.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 40 q^{7} - 340 q^{25} + 248 q^{31} - 972 q^{49} + 960 q^{55} - 120 q^{73} + 376 q^{79} + 520 q^{97}+O(q^{100})$ 4 * q - 40 * q^7 - 340 * q^25 + 248 * q^31 - 972 * q^49 + 960 * q^55 - 120 * q^73 + 376 * q^79 + 520 * q^97

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
$5$	−6.32456	−0.565685	−0.282843	−	0.959166i	$-0.591277\pi$
$5$	−6.32456	−0.565685	−0.282843	+	0.959166i	$0.591277\pi$
$6$	0	0
$7$	−10.0000	−0.539949	−0.269975	−	0.962867i	$-0.587015\pi$
$7$	−10.0000	−0.539949	−0.269975	+	0.962867i	$0.587015\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−37.9473	−1.04014	−0.520071	−	0.854123i	$-0.674094\pi$
$11$	−37.9473	−1.04014	−0.520071	+	0.854123i	$0.674094\pi$
$12$	0	0
$13$	59.3296	1.26577	0.632887	−	0.774244i	$-0.281870\pi$
$13$	59.3296	1.26577	0.632887	+	0.774244i	$0.281870\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−75.0467	−1.07068	−0.535338	−	0.844638i	$-0.679816\pi$
$17$	−75.0467	−1.07068	−0.535338	+	0.844638i	$0.679816\pi$
$18$	0	0
$19$	118.659	1.43275	0.716376	−	0.697715i	$-0.245800\pi$
$19$	118.659	1.43275	0.716376	+	0.697715i	$0.245800\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−150.093	−1.36072	−0.680361	−	0.732877i	$-0.738177\pi$
$23$	−150.093	−1.36072	−0.680361	+	0.732877i	$0.738177\pi$
$24$	0	0
$25$	−85.0000	−0.680000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−246.658	−1.57942	−0.789710	−	0.613480i	$-0.789769\pi$
$29$	−246.658	−1.57942	−0.789710	+	0.613480i	$0.789769\pi$
$30$	0	0
$31$	62.0000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	62.0000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	63.2456	0.305441
$36$	0	0
$37$	−59.3296	−0.263614	−0.131807	−	0.991275i	$-0.542078\pi$
$37$	−59.3296	−0.263614	−0.131807	+	0.991275i	$0.542078\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	375.233	1.42931	0.714654	−	0.699479i	$-0.246584\pi$
$41$	375.233	1.42931	0.714654	+	0.699479i	$0.246584\pi$
$42$	0	0
$43$	118.659	0.420822	0.210411	−	0.977613i	$-0.432520\pi$
$43$	118.659	0.420822	0.210411	+	0.977613i	$0.432520\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	450.280	1.39745	0.698724	−	0.715391i	$-0.253751\pi$
$47$	450.280	1.39745	0.698724	+	0.715391i	$0.253751\pi$
$48$	0	0
$49$	−243.000	−0.708455
$50$	0	0
$51$	0	0
$52$	0	0
$53$	132.816	0.344220	0.172110	−	0.985078i	$-0.444942\pi$
$53$	132.816	0.344220	0.172110	+	0.985078i	$0.444942\pi$
$54$	0	0
$55$	240.000	0.588393
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−733.648	−1.61886	−0.809431	−	0.587215i	$-0.800224\pi$
$59$	−733.648	−1.61886	−0.809431	+	0.587215i	$0.800224\pi$
$60$	0	0
$61$	−533.966	−1.12078	−0.560388	−	0.828230i	$-0.689348\pi$
$61$	−533.966	−1.12078	−0.560388	+	0.828230i	$0.689348\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−375.233	−0.716030
$66$	0	0
$67$	−711.955	−1.29820	−0.649098	−	0.760705i	$-0.724854\pi$
$67$	−711.955	−1.29820	−0.649098	+	0.760705i	$0.724854\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−30.0000	−0.0480991	−0.0240496	−	0.999711i	$-0.507656\pi$
$73$	−30.0000	−0.0480991	−0.0240496	+	0.999711i	$0.507656\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	379.473	0.561623
$78$	0	0
$79$	94.0000	0.133871	0.0669356	−	0.997757i	$-0.478678\pi$
$79$	94.0000	0.133871	0.0669356	+	0.997757i	$0.478678\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−670.403	−0.886582	−0.443291	−	0.896378i	$-0.646189\pi$
$83$	−670.403	−0.886582	−0.443291	+	0.896378i	$0.646189\pi$
$84$	0	0
$85$	474.637	0.605666
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−750.467	−0.893812	−0.446906	−	0.894581i	$-0.647474\pi$
$89$	−750.467	−0.893812	−0.446906	+	0.894581i	$0.647474\pi$
$90$	0	0
$91$	−593.296	−0.683454
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−750.467	−0.810487
$96$	0	0
$97$	130.000	0.136077	0.0680387	−	0.997683i	$-0.478326\pi$
$97$	130.000	0.136077	0.0680387	+	0.997683i	$0.478326\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	562.885	0.554546	0.277273	−	0.960791i	$-0.410569\pi$
$101$	562.885	0.554546	0.277273	+	0.960791i	$0.410569\pi$
$102$	0	0
$103$	570.000	0.545279	0.272640	−	0.962116i	$-0.412103\pi$
$103$	570.000	0.545279	0.272640	+	0.962116i	$0.412103\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1290.21	1.16569	0.582847	−	0.812582i	$-0.301939\pi$
$107$	1290.21	1.16569	0.582847	+	0.812582i	$0.301939\pi$
$108$	0	0
$109$	1720.56	1.51192	0.755961	−	0.654616i	$-0.227170\pi$
$109$	1720.56	1.51192	0.755961	+	0.654616i	$0.227170\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1350.84	1.12457	0.562285	−	0.826944i	$-0.309923\pi$
$113$	1350.84	1.12457	0.562285	+	0.826944i	$0.309923\pi$
$114$	0	0
$115$	949.273	0.769741
$116$	0	0
$117$	0	0
$118$	0	0
$119$	750.467	0.578111
$120$	0	0
$121$	109.000	0.0818933
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1328.16	0.950352
$126$	0	0
$127$	2530.00	1.76773	0.883863	−	0.467746i	$-0.154934\pi$
$127$	2530.00	1.76773	0.883863	+	0.467746i	$0.154934\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−252.982	−0.168726	−0.0843632	−	0.996435i	$-0.526886\pi$
$131$	−252.982	−0.168726	−0.0843632	+	0.996435i	$0.526886\pi$
$132$	0	0
$133$	−1186.59	−0.773613
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−675.420	−0.421204	−0.210602	−	0.977572i	$-0.567542\pi$
$137$	−675.420	−0.421204	−0.210602	+	0.977572i	$0.567542\pi$
$138$	0	0
$139$	237.318	0.144814	0.0724068	−	0.997375i	$-0.476932\pi$
$139$	237.318	0.144814	0.0724068	+	0.997375i	$0.476932\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2251.40	−1.31658
$144$	0	0
$145$	1560.00	0.893455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2940.92	−1.61698	−0.808488	−	0.588513i	$-0.799714\pi$
$149$	−2940.92	−1.61698	−0.808488	+	0.588513i	$0.799714\pi$
$150$	0	0
$151$	2262.00	1.21907	0.609533	−	0.792761i	$-0.291357\pi$
$151$	2262.00	1.21907	0.609533	+	0.792761i	$0.291357\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−392.122	−0.203200
$156$	0	0
$157$	−1720.56	−0.874621	−0.437310	−	0.899311i	$-0.644069\pi$
$157$	−1720.56	−0.874621	−0.437310	+	0.899311i	$0.644069\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1500.93	0.734721
$162$	0	0
$163$	2017.21	0.969324	0.484662	−	0.874702i	$-0.338943\pi$
$163$	2017.21	0.969324	0.484662	+	0.874702i	$0.338943\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1050.65	0.486838	0.243419	−	0.969921i	$-0.421731\pi$
$167$	1050.65	0.486838	0.243419	+	0.969921i	$0.421731\pi$
$168$	0	0
$169$	1323.00	0.602185
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1423.02	−0.625379	−0.312690	−	0.949855i	$-0.601230\pi$
$173$	−1423.02	−0.625379	−0.312690	+	0.949855i	$0.601230\pi$
$174$	0	0
$175$	850.000	0.367165
$176$	0	0
$177$	0	0
$178$	0	0
$179$	126.491	0.0528178	0.0264089	−	0.999651i	$-0.491593\pi$
$179$	126.491	0.0528178	0.0264089	+	0.999651i	$0.491593\pi$
$180$	0	0
$181$	59.3296	0.0243643	0.0121821	−	0.999926i	$-0.496122\pi$
$181$	59.3296	0.0243643	0.0121821	+	0.999926i	$0.496122\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	375.233	0.149123
$186$	0	0
$187$	2847.82	1.11365
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3752.33	1.42151	0.710757	−	0.703437i	$-0.248352\pi$
$191$	3752.33	1.42151	0.710757	+	0.703437i	$0.248352\pi$
$192$	0	0
$193$	−1350.00	−0.503498	−0.251749	−	0.967793i	$-0.581006\pi$
$193$	−1350.00	−0.503498	−0.251749	+	0.967793i	$0.581006\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1448.32	0.523801	0.261900	−	0.965095i	$-0.415651\pi$
$197$	1448.32	0.523801	0.261900	+	0.965095i	$0.415651\pi$
$198$	0	0
$199$	3194.00	1.13777	0.568886	−	0.822416i	$-0.307375\pi$
$199$	3194.00	1.13777	0.568886	+	0.822416i	$0.307375\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2466.58	0.852807
$204$	0	0
$205$	−2373.18	−0.808538
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4502.80	−1.49026
$210$	0	0
$211$	−5458.32	−1.78088	−0.890442	−	0.455097i	$-0.849604\pi$
$211$	−5458.32	−1.78088	−0.890442	+	0.455097i	$0.849604\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−750.467	−0.238053
$216$	0	0
$217$	−620.000	−0.193955
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4452.49	−1.35523
$222$	0	0
$223$	5330.00	1.60055	0.800276	−	0.599632i	$-0.204686\pi$
$223$	5330.00	1.60055	0.800276	+	0.599632i	$0.204686\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4768.71	1.39432	0.697160	−	0.716915i	$-0.254447\pi$
$227$	4768.71	1.39432	0.697160	+	0.716915i	$0.254447\pi$
$228$	0	0
$229$	3619.10	1.04435	0.522177	−	0.852837i	$-0.325120\pi$
$229$	3619.10	1.04435	0.522177	+	0.852837i	$0.325120\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	150.093	0.0422015	0.0211007	−	0.999777i	$-0.493283\pi$
$233$	150.093	0.0422015	0.0211007	+	0.999777i	$0.493283\pi$
$234$	0	0
$235$	−2847.82	−0.790516
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2251.40	0.609334	0.304667	−	0.952459i	$-0.401455\pi$
$239$	2251.40	0.609334	0.304667	+	0.952459i	$0.401455\pi$
$240$	0	0
$241$	−1162.00	−0.310585	−0.155293	−	0.987869i	$-0.549632\pi$
$241$	−1162.00	−0.310585	−0.155293	+	0.987869i	$0.549632\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1536.87	0.400763
$246$	0	0
$247$	7040.00	1.81354
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−645.105	−0.162226	−0.0811128	−	0.996705i	$-0.525847\pi$
$251$	−645.105	−0.162226	−0.0811128	+	0.996705i	$0.525847\pi$
$252$	0	0
$253$	5695.64	1.41534
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1801.12	−0.437162	−0.218581	−	0.975819i	$-0.570143\pi$
$257$	−1801.12	−0.437162	−0.218581	+	0.975819i	$0.570143\pi$
$258$	0	0
$259$	593.296	0.142338
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6604.11	−1.54839	−0.774195	−	0.632947i	$-0.781845\pi$
$263$	−6604.11	−1.54839	−0.774195	+	0.632947i	$0.781845\pi$
$264$	0	0
$265$	−840.000	−0.194720
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6052.60	1.37187	0.685936	−	0.727662i	$-0.259393\pi$
$269$	6052.60	1.37187	0.685936	+	0.727662i	$0.259393\pi$
$270$	0	0
$271$	6402.00	1.43503	0.717516	−	0.696542i	$-0.245279\pi$
$271$	6402.00	1.43503	0.717516	+	0.696542i	$0.245279\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3225.52	0.707296
$276$	0	0
$277$	−2076.54	−0.450422	−0.225211	−	0.974310i	$-0.572307\pi$
$277$	−2076.54	−0.450422	−0.225211	+	0.974310i	$0.572307\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9005.60	1.91185	0.955923	−	0.293616i	$-0.0948587\pi$
$281$	9005.60	1.91185	0.955923	+	0.293616i	$0.0948587\pi$
$282$	0	0
$283$	−5221.00	−1.09667	−0.548333	−	0.836260i	$-0.684737\pi$
$283$	−5221.00	−1.09667	−0.548333	+	0.836260i	$0.684737\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3752.33	−0.771753
$288$	0	0
$289$	719.000	0.146346
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5293.65	1.05549	0.527745	−	0.849403i	$-0.323038\pi$
$293$	5293.65	1.05549	0.527745	+	0.849403i	$0.323038\pi$
$294$	0	0
$295$	4640.00	0.915767
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8904.97	−1.72237
$300$	0	0
$301$	−1186.59	−0.227223
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3377.10	0.634007
$306$	0	0
$307$	4983.69	0.926495	0.463247	−	0.886229i	$-0.346684\pi$
$307$	4983.69	0.926495	0.463247	+	0.886229i	$0.346684\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1500.93	−0.273666	−0.136833	−	0.990594i	$-0.543692\pi$
$311$	−1500.93	−0.273666	−0.136833	+	0.990594i	$0.543692\pi$
$312$	0	0
$313$	−6350.00	−1.14672	−0.573360	−	0.819304i	$-0.694360\pi$
$313$	−6350.00	−1.14672	−0.573360	+	0.819304i	$0.694360\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1739.25	−0.308158	−0.154079	−	0.988059i	$-0.549241\pi$
$317$	−1739.25	−0.308158	−0.154079	+	0.988059i	$0.549241\pi$
$318$	0	0
$319$	9360.00	1.64282
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8904.97	−1.53401
$324$	0	0
$325$	−5043.01	−0.860727
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4502.80	−0.754551
$330$	0	0
$331$	3322.46	0.551718	0.275859	−	0.961198i	$-0.411038\pi$
$331$	3322.46	0.551718	0.275859	+	0.961198i	$0.411038\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4502.80	0.734371
$336$	0	0
$337$	−1430.00	−0.231149	−0.115574	−	0.993299i	$-0.536871\pi$
$337$	−1430.00	−0.231149	−0.115574	+	0.993299i	$0.536871\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2352.73	−0.373630
$342$	0	0
$343$	5860.00	0.922479
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10106.6	1.56355	0.781776	−	0.623559i	$-0.214314\pi$
$347$	10106.6	1.56355	0.781776	+	0.623559i	$0.214314\pi$
$348$	0	0
$349$	1127.26	0.172897	0.0864484	−	0.996256i	$-0.472448\pi$
$349$	1127.26	0.172897	0.0864484	+	0.996256i	$0.472448\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1350.84	−0.203677	−0.101838	−	0.994801i	$-0.532472\pi$
$353$	−1350.84	−0.203677	−0.101838	+	0.994801i	$0.532472\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12007.5	1.76526	0.882632	−	0.470065i	$-0.155769\pi$
$359$	12007.5	1.76526	0.882632	+	0.470065i	$0.155769\pi$
$360$	0	0
$361$	7221.00	1.05278
$362$	0	0
$363$	0	0
$364$	0	0
$365$	189.737	0.0272090
$366$	0	0
$367$	5070.00	0.721122	0.360561	−	0.932736i	$-0.382585\pi$
$367$	5070.00	0.721122	0.360561	+	0.932736i	$0.382585\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1328.16	−0.185861
$372$	0	0
$373$	−2432.51	−0.337670	−0.168835	−	0.985644i	$-0.554000\pi$
$373$	−2432.51	−0.337670	−0.168835	+	0.985644i	$0.554000\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14634.1	−1.99919
$378$	0	0
$379$	7712.85	1.04534	0.522668	−	0.852536i	$-0.324937\pi$
$379$	7712.85	1.04534	0.522668	+	0.852536i	$0.324937\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5853.64	−0.780958	−0.390479	−	0.920612i	$-0.627691\pi$
$383$	−5853.64	−0.780958	−0.390479	+	0.920612i	$0.627691\pi$
$384$	0	0
$385$	−2400.00	−0.317702
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2156.67	−0.281099	−0.140550	−	0.990074i	$-0.544887\pi$
$389$	−2156.67	−0.281099	−0.140550	+	0.990074i	$0.544887\pi$
$390$	0	0
$391$	11264.0	1.45689
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−594.508	−0.0757290
$396$	0	0
$397$	−12162.6	−1.53759	−0.768793	−	0.639498i	$-0.779142\pi$
$397$	−12162.6	−1.53759	−0.768793	+	0.639498i	$0.779142\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3377.10	0.420559	0.210280	−	0.977641i	$-0.432563\pi$
$401$	3377.10	0.420559	0.210280	+	0.977641i	$0.432563\pi$
$402$	0	0
$403$	3678.43	0.454680
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2251.40	0.274196
$408$	0	0
$409$	−3526.00	−0.426282	−0.213141	−	0.977021i	$-0.568369\pi$
$409$	−3526.00	−0.426282	−0.213141	+	0.977021i	$0.568369\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7336.48	0.874104
$414$	0	0
$415$	4240.00	0.501526
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2466.58	−0.287590	−0.143795	−	0.989608i	$-0.545931\pi$
$419$	−2466.58	−0.287590	−0.143795	+	0.989608i	$0.545931\pi$
$420$	0	0
$421$	13586.5	1.57284	0.786418	−	0.617694i	$-0.211933\pi$
$421$	13586.5	1.57284	0.786418	+	0.617694i	$0.211933\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6378.97	0.728059
$426$	0	0
$427$	5339.66	0.605163
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6754.20	−0.754845	−0.377423	−	0.926041i	$-0.623190\pi$
$431$	−6754.20	−0.754845	−0.377423	+	0.926041i	$0.623190\pi$
$432$	0	0
$433$	7790.00	0.864581	0.432290	−	0.901734i	$-0.357706\pi$
$433$	7790.00	0.864581	0.432290	+	0.901734i	$0.357706\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17809.9	−1.94958
$438$	0	0
$439$	9354.00	1.01695	0.508476	−	0.861076i	$-0.330209\pi$
$439$	9354.00	1.01695	0.508476	+	0.861076i	$0.330209\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6488.99	−0.695940	−0.347970	−	0.937506i	$-0.613129\pi$
$443$	−6488.99	−0.695940	−0.347970	+	0.937506i	$0.613129\pi$
$444$	0	0
$445$	4746.37	0.505617
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10131.3	1.06487	0.532434	−	0.846472i	$-0.321278\pi$
$449$	10131.3	1.06487	0.532434	+	0.846472i	$0.321278\pi$
$450$	0	0
$451$	−14239.1	−1.48668
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3752.33	0.386620
$456$	0	0
$457$	12010.0	1.22933	0.614665	−	0.788788i	$-0.289291\pi$
$457$	12010.0	1.22933	0.614665	+	0.788788i	$0.289291\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9316.07	−0.941199	−0.470599	−	0.882347i	$-0.655962\pi$
$461$	−9316.07	−0.941199	−0.470599	+	0.882347i	$0.655962\pi$
$462$	0	0
$463$	14770.0	1.48255	0.741274	−	0.671202i	$-0.234222\pi$
$463$	14770.0	1.48255	0.741274	+	0.671202i	$0.234222\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8740.54	−0.866089	−0.433045	−	0.901372i	$-0.642561\pi$
$467$	−8740.54	−0.866089	−0.433045	+	0.901372i	$0.642561\pi$
$468$	0	0
$469$	7119.55	0.700960
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4502.80	−0.437714
$474$	0	0
$475$	−10086.0	−0.974271
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3001.87	0.286344	0.143172	−	0.989698i	$-0.454270\pi$
$479$	3001.87	0.286344	0.143172	+	0.989698i	$0.454270\pi$
$480$	0	0
$481$	−3520.00	−0.333676
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−822.192	−0.0769770
$486$	0	0
$487$	9910.00	0.922105	0.461052	−	0.887373i	$-0.347472\pi$
$487$	9910.00	0.922105	0.461052	+	0.887373i	$0.347472\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3389.96	−0.311582	−0.155791	−	0.987790i	$-0.549793\pi$
$491$	−3389.96	−0.311582	−0.155791	+	0.987790i	$0.549793\pi$
$492$	0	0
$493$	18510.8	1.69105
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4271.73	0.383224	0.191612	−	0.981471i	$-0.438628\pi$
$499$	4271.73	0.383224	0.191612	+	0.981471i	$0.438628\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	900.560	0.0798290	0.0399145	−	0.999203i	$-0.487291\pi$
$503$	900.560	0.0798290	0.0399145	+	0.999203i	$0.487291\pi$
$504$	0	0
$505$	−3560.00	−0.313699
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−853.815	−0.0743510	−0.0371755	−	0.999309i	$-0.511836\pi$
$509$	−853.815	−0.0743510	−0.0371755	+	0.999309i	$0.511836\pi$
$510$	0	0
$511$	300.000	0.0259711
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3605.00	−0.308457
$516$	0	0
$517$	−17086.9	−1.45354
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7879.90	0.662619	0.331310	−	0.943522i	$-0.392510\pi$
$521$	7879.90	0.662619	0.331310	+	0.943522i	$0.392510\pi$
$522$	0	0
$523$	−10323.3	−0.863114	−0.431557	−	0.902086i	$-0.642036\pi$
$523$	−10323.3	−0.863114	−0.431557	+	0.902086i	$0.642036\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4652.89	−0.384598
$528$	0	0
$529$	10361.0	0.851566
$530$	0	0
$531$	0	0
$532$	0	0
$533$	22262.4	1.80918
$534$	0	0
$535$	−8160.00	−0.659416
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9221.20	0.736893
$540$	0	0
$541$	−15603.7	−1.24003	−0.620014	−	0.784591i	$-0.712873\pi$
$541$	−15603.7	−1.24003	−0.620014	+	0.784591i	$0.712873\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10881.8	−0.855273
$546$	0	0
$547$	2254.52	0.176228	0.0881138	−	0.996110i	$-0.471916\pi$
$547$	2254.52	0.176228	0.0881138	+	0.996110i	$0.471916\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−29268.2	−2.26292
$552$	0	0
$553$	−940.000	−0.0722837
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−284.605	−0.0216501	−0.0108250	−	0.999941i	$-0.503446\pi$
$557$	−284.605	−0.0216501	−0.0108250	+	0.999941i	$0.503446\pi$
$558$	0	0
$559$	7040.00	0.532666
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18328.6	1.37204	0.686018	−	0.727584i	$-0.259357\pi$
$563$	18328.6	1.37204	0.686018	+	0.727584i	$0.259357\pi$
$564$	0	0
$565$	−8543.46	−0.636152
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7879.90	0.580567	0.290283	−	0.956941i	$-0.406250\pi$
$569$	7879.90	0.580567	0.290283	+	0.956941i	$0.406250\pi$
$570$	0	0
$571$	−8306.14	−0.608759	−0.304379	−	0.952551i	$-0.598449\pi$
$571$	−8306.14	−0.608759	−0.304379	+	0.952551i	$0.598449\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12757.9	0.925291
$576$	0	0
$577$	−6330.00	−0.456709	−0.228355	−	0.973578i	$-0.573335\pi$
$577$	−6330.00	−0.456709	−0.228355	+	0.973578i	$0.573335\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6704.03	0.478709
$582$	0	0
$583$	−5040.00	−0.358037
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1922.66	0.135191	0.0675953	−	0.997713i	$-0.478467\pi$
$587$	1922.66	0.135191	0.0675953	+	0.997713i	$0.478467\pi$
$588$	0	0
$589$	7356.87	0.514660
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−10356.4	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$593$	−10356.4	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$594$	0	0
$595$	−4746.37	−0.327029
$596$	0	0
$597$	0	0
$598$	0	0
$599$	750.467	0.0511907	0.0255954	−	0.999672i	$-0.491852\pi$
$599$	750.467	0.0511907	0.0255954	+	0.999672i	$0.491852\pi$
$600$	0	0
$601$	−18578.0	−1.26092	−0.630460	−	0.776222i	$-0.717134\pi$
$601$	−18578.0	−1.26092	−0.630460	+	0.776222i	$0.717134\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−689.377	−0.0463259
$606$	0	0
$607$	−8030.00	−0.536948	−0.268474	−	0.963287i	$-0.586519\pi$
$607$	−8030.00	−0.536948	−0.268474	+	0.963287i	$0.586519\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26714.9	1.76885
$612$	0	0
$613$	−3856.42	−0.254094	−0.127047	−	0.991897i	$-0.540550\pi$
$613$	−3856.42	−0.254094	−0.127047	+	0.991897i	$0.540550\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	300.187	0.0195868	0.00979340	−	0.999952i	$-0.496883\pi$
$617$	300.187	0.0195868	0.00979340	+	0.999952i	$0.496883\pi$
$618$	0	0
$619$	1423.91	0.0924584	0.0462292	−	0.998931i	$-0.485280\pi$
$619$	1423.91	0.0924584	0.0462292	+	0.998931i	$0.485280\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7504.67	0.482613
$624$	0	0
$625$	2225.00	0.142400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4452.49	0.282245
$630$	0	0
$631$	12902.0	0.813979	0.406989	−	0.913433i	$-0.366579\pi$
$631$	12902.0	0.813979	0.406989	+	0.913433i	$0.366579\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16001.1	−0.999977
$636$	0	0
$637$	−14417.1	−0.896744
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19887.4	−1.22543	−0.612717	−	0.790302i	$-0.709924\pi$
$641$	−19887.4	−1.22543	−0.612717	+	0.790302i	$0.709924\pi$
$642$	0	0
$643$	29783.5	1.82666	0.913332	−	0.407216i	$-0.133500\pi$
$643$	29783.5	1.82666	0.913332	+	0.407216i	$0.133500\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13958.7	0.848180	0.424090	−	0.905620i	$-0.360594\pi$
$647$	13958.7	0.848180	0.424090	+	0.905620i	$0.360594\pi$
$648$	0	0
$649$	27840.0	1.68385
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18461.4	1.10635	0.553177	−	0.833064i	$-0.313415\pi$
$653$	18461.4	1.10635	0.553177	+	0.833064i	$0.313415\pi$
$654$	0	0
$655$	1600.00	0.0954461
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5160.84	−0.305065	−0.152532	−	0.988298i	$-0.548743\pi$
$659$	−5160.84	−0.305065	−0.152532	+	0.988298i	$0.548743\pi$
$660$	0	0
$661$	13467.8	0.792492	0.396246	−	0.918144i	$-0.370313\pi$
$661$	13467.8	0.792492	0.396246	+	0.918144i	$0.370313\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7504.67	0.437622
$666$	0	0
$667$	37021.7	2.14915
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20262.6	1.16577
$672$	0	0
$673$	−15010.0	−0.859722	−0.429861	−	0.902895i	$-0.641437\pi$
$673$	−15010.0	−0.859722	−0.429861	+	0.902895i	$0.641437\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6280.28	−0.356530	−0.178265	−	0.983983i	$-0.557048\pi$
$677$	−6280.28	−0.356530	−0.178265	+	0.983983i	$0.557048\pi$
$678$	0	0
$679$	−1300.00	−0.0734748
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12510.0	0.700850	0.350425	−	0.936591i	$-0.386037\pi$
$683$	12510.0	0.700850	0.350425	+	0.936591i	$0.386037\pi$
$684$	0	0
$685$	4271.73	0.238269
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7879.90	0.435704
$690$	0	0
$691$	−28359.5	−1.56128	−0.780642	−	0.624978i	$-0.785108\pi$
$691$	−28359.5	−1.56128	−0.780642	+	0.624978i	$0.785108\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1500.93	−0.0819189
$696$	0	0
$697$	−28160.0	−1.53032
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19802.2	−1.06693	−0.533465	−	0.845822i	$-0.679110\pi$
$701$	−19802.2	−1.06693	−0.533465	+	0.845822i	$0.679110\pi$
$702$	0	0
$703$	−7040.00	−0.377694
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5628.85	−0.299427
$708$	0	0
$709$	12874.5	0.681964	0.340982	−	0.940070i	$-0.389240\pi$
$709$	12874.5	0.681964	0.340982	+	0.940070i	$0.389240\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9305.78	−0.488786
$714$	0	0
$715$	14239.1	0.744772
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1500.93	0.0778517	0.0389258	−	0.999242i	$-0.487606\pi$
$719$	1500.93	0.0778517	0.0389258	+	0.999242i	$0.487606\pi$
$720$	0	0
$721$	−5700.00	−0.294423
$722$	0	0
$723$	0	0
$724$	0	0
$725$	20965.9	1.07401
$726$	0	0
$727$	−23850.0	−1.21671	−0.608355	−	0.793665i	$-0.708170\pi$
$727$	−23850.0	−1.21671	−0.608355	+	0.793665i	$0.708170\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8904.97	−0.450564
$732$	0	0
$733$	2195.19	0.110616	0.0553079	−	0.998469i	$-0.482386\pi$
$733$	2195.19	0.110616	0.0553079	+	0.998469i	$0.482386\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27016.8	1.35031
$738$	0	0
$739$	−2610.50	−0.129944	−0.0649722	−	0.997887i	$-0.520696\pi$
$739$	−2610.50	−0.129944	−0.0649722	+	0.997887i	$0.520696\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2851.77	0.140809	0.0704047	−	0.997519i	$-0.477571\pi$
$743$	2851.77	0.140809	0.0704047	+	0.997519i	$0.477571\pi$
$744$	0	0
$745$	18600.0	0.914700
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−12902.1	−0.629416
$750$	0	0
$751$	−20578.0	−0.999869	−0.499935	−	0.866063i	$-0.666643\pi$
$751$	−20578.0	−0.999869	−0.499935	+	0.866063i	$0.666643\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−14306.1	−0.689608
$756$	0	0
$757$	20468.7	0.982758	0.491379	−	0.870946i	$-0.336493\pi$
$757$	20468.7	0.982758	0.491379	+	0.870946i	$0.336493\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22138.8	1.05457	0.527286	−	0.849688i	$-0.323210\pi$
$761$	22138.8	1.05457	0.527286	+	0.849688i	$0.323210\pi$
$762$	0	0
$763$	−17205.6	−0.816362
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−43527.1	−2.04911
$768$	0	0
$769$	−6854.00	−0.321406	−0.160703	−	0.987003i	$-0.551376\pi$
$769$	−6854.00	−0.321406	−0.160703	+	0.987003i	$0.551376\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−38902.3	−1.81012	−0.905058	−	0.425288i	$-0.860173\pi$
$773$	−38902.3	−1.81012	−0.905058	+	0.425288i	$0.860173\pi$
$774$	0	0
$775$	−5270.00	−0.244263
$776$	0	0
$777$	0	0
$778$	0	0
$779$	44524.9	2.04784
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10881.8	0.494760
$786$	0	0
$787$	13171.2	0.596571	0.298286	−	0.954477i	$-0.403585\pi$
$787$	13171.2	0.596571	0.298286	+	0.954477i	$0.403585\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13508.4	−0.607210
$792$	0	0
$793$	−31680.0	−1.41865
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13730.6	0.610242	0.305121	−	0.952314i	$-0.401303\pi$
$797$	13730.6	0.610242	0.305121	+	0.952314i	$0.401303\pi$
$798$	0	0
$799$	−33792.0	−1.49621
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1138.42	0.0500298
$804$	0	0
$805$	−9492.73	−0.415621
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−30393.9	−1.32088	−0.660440	−	0.750879i	$-0.729630\pi$
$809$	−30393.9	−1.32088	−0.660440	+	0.750879i	$0.729630\pi$
$810$	0	0
$811$	−27647.6	−1.19709	−0.598544	−	0.801090i	$-0.704254\pi$
$811$	−27647.6	−1.19709	−0.598544	+	0.801090i	$0.704254\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12757.9	−0.548332
$816$	0	0
$817$	14080.0	0.602934
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26303.8	−1.11816	−0.559080	−	0.829114i	$-0.688846\pi$
$821$	−26303.8	−1.11816	−0.559080	+	0.829114i	$0.688846\pi$
$822$	0	0
$823$	−22970.0	−0.972884	−0.486442	−	0.873713i	$-0.661706\pi$
$823$	−22970.0	−0.972884	−0.486442	+	0.873713i	$0.661706\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7665.36	−0.322310	−0.161155	−	0.986929i	$-0.551522\pi$
$827$	−7665.36	−0.322310	−0.161155	+	0.986929i	$0.551522\pi$
$828$	0	0
$829$	17383.6	0.728295	0.364147	−	0.931341i	$-0.381360\pi$
$829$	17383.6	0.728295	0.364147	+	0.931341i	$0.381360\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18236.3	0.758525
$834$	0	0
$835$	−6644.91	−0.275397
$836$	0	0
$837$	0	0
$838$	0	0
$839$	33020.5	1.35875	0.679377	−	0.733789i	$-0.262250\pi$
$839$	33020.5	1.35875	0.679377	+	0.733789i	$0.262250\pi$
$840$	0	0
$841$	36451.0	1.49457
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8367.39	−0.340647
$846$	0	0
$847$	−1090.00	−0.0442182
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8904.97	0.358706
$852$	0	0
$853$	−46099.1	−1.85041	−0.925207	−	0.379463i	$-0.876109\pi$
$853$	−46099.1	−1.85041	−0.925207	+	0.379463i	$0.876109\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44352.6	−1.76786	−0.883929	−	0.467620i	$-0.845111\pi$
$857$	−44352.6	−1.76786	−0.883929	+	0.467620i	$0.845111\pi$
$858$	0	0
$859$	−7712.85	−0.306355	−0.153177	−	0.988199i	$-0.548951\pi$
$859$	−7712.85	−0.306355	−0.153177	+	0.988199i	$0.548951\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21163.2	−0.834765	−0.417383	−	0.908731i	$-0.637052\pi$
$863$	−21163.2	−0.834765	−0.417383	+	0.908731i	$0.637052\pi$
$864$	0	0
$865$	9000.00	0.353768
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3567.05	−0.139245
$870$	0	0
$871$	−42240.0	−1.64322
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−13281.6	−0.513142
$876$	0	0
$877$	−17146.3	−0.660191	−0.330096	−	0.943947i	$-0.607081\pi$
$877$	−17146.3	−0.660191	−0.330096	+	0.943947i	$0.607081\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42026.1	−1.60715	−0.803573	−	0.595206i	$-0.797071\pi$
$881$	−42026.1	−1.60715	−0.803573	+	0.595206i	$0.797071\pi$
$882$	0	0
$883$	4865.03	0.185415	0.0927073	−	0.995693i	$-0.470448\pi$
$883$	4865.03	0.185415	0.0927073	+	0.995693i	$0.470448\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11707.3	0.443170	0.221585	−	0.975141i	$-0.428877\pi$
$887$	11707.3	0.443170	0.221585	+	0.975141i	$0.428877\pi$
$888$	0	0
$889$	−25300.0	−0.954482
$890$	0	0
$891$	0	0
$892$	0	0
$893$	53429.8	2.00220
$894$	0	0
$895$	−800.000	−0.0298783
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15292.8	−0.567344
$900$	0	0
$901$	−9967.37	−0.368547
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−375.233	−0.0137825
$906$	0	0
$907$	−12933.9	−0.473497	−0.236748	−	0.971571i	$-0.576082\pi$
$907$	−12933.9	−0.473497	−0.236748	+	0.971571i	$0.576082\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20262.6	0.736915	0.368458	−	0.929645i	$-0.379886\pi$
$911$	20262.6	0.736915	0.368458	+	0.929645i	$0.379886\pi$
$912$	0	0
$913$	25440.0	0.922170
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2529.82	0.0911037
$918$	0	0
$919$	−22746.0	−0.816454	−0.408227	−	0.912880i	$-0.633853\pi$
$919$	−22746.0	−0.816454	−0.408227	+	0.912880i	$0.633853\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	5043.01	0.179258
$926$	0	0
$927$	0	0
$928$	0	0
$929$	13883.6	0.490320	0.245160	−	0.969483i	$-0.421160\pi$
$929$	13883.6	0.490320	0.245160	+	0.969483i	$0.421160\pi$
$930$	0	0
$931$	−28834.2	−1.01504
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18011.2	−0.629978
$936$	0	0
$937$	27850.0	0.970992	0.485496	−	0.874239i	$-0.338639\pi$
$937$	27850.0	0.970992	0.485496	+	0.874239i	$0.338639\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−36789.9	−1.27451	−0.637257	−	0.770651i	$-0.719931\pi$
$941$	−36789.9	−1.27451	−0.637257	+	0.770651i	$0.719931\pi$
$942$	0	0
$943$	−56320.0	−1.94489
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42728.7	−1.46620	−0.733102	−	0.680118i	$-0.761928\pi$
$947$	−42728.7	−1.46620	−0.733102	+	0.680118i	$0.761928\pi$
$948$	0	0
$949$	−1779.89	−0.0608826
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24990.5	0.849447	0.424723	−	0.905323i	$-0.360371\pi$
$953$	24990.5	0.849447	0.424723	+	0.905323i	$0.360371\pi$
$954$	0	0
$955$	−23731.8	−0.804130
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6754.20	0.227429
$960$	0	0
$961$	−25947.0	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8538.15	0.284822
$966$	0	0
$967$	−10550.0	−0.350843	−0.175421	−	0.984493i	$-0.556129\pi$
$967$	−10550.0	−0.350843	−0.175421	+	0.984493i	$0.556129\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−53012.4	−1.75206	−0.876030	−	0.482257i	$-0.839817\pi$
$971$	−53012.4	−1.75206	−0.876030	+	0.482257i	$0.839817\pi$
$972$	0	0
$973$	−2373.18	−0.0781920
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34596.5	1.13290	0.566448	−	0.824097i	$-0.308317\pi$
$977$	34596.5	1.13290	0.566448	+	0.824097i	$0.308317\pi$
$978$	0	0
$979$	28478.2	0.929691
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33921.1	1.10063	0.550313	−	0.834959i	$-0.314509\pi$
$983$	33921.1	1.10063	0.550313	+	0.834959i	$0.314509\pi$
$984$	0	0
$985$	−9160.00	−0.296306
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−17809.9	−0.572622
$990$	0	0
$991$	−8818.00	−0.282657	−0.141328	−	0.989963i	$-0.545137\pi$
$991$	−8818.00	−0.282657	−0.141328	+	0.989963i	$0.545137\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−20200.6	−0.643621
$996$	0	0
$997$	25096.4	0.797203	0.398602	−	0.917124i	$-0.369496\pi$
$997$	25096.4	0.797203	0.398602	+	0.917124i	$0.369496\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.4.a.bx.1.2		4
3.2	odd	2	inner	2304.4.a.bx.1.4		4
4.3	odd	2		2304.4.a.cc.1.2		4
8.3	odd	2		2304.4.a.cc.1.3		4
8.5	even	2	inner	2304.4.a.bx.1.3		4
12.11	even	2		2304.4.a.cc.1.4		4
16.3	odd	4		288.4.d.c.145.4		4
16.5	even	4		72.4.d.c.37.1	✓	4
16.11	odd	4		288.4.d.c.145.1		4
16.13	even	4		72.4.d.c.37.2	yes	4
24.5	odd	2	inner	2304.4.a.bx.1.1		4
24.11	even	2		2304.4.a.cc.1.1		4
48.5	odd	4		72.4.d.c.37.4	yes	4
48.11	even	4		288.4.d.c.145.3		4
48.29	odd	4		72.4.d.c.37.3	yes	4
48.35	even	4		288.4.d.c.145.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.d.c.37.1	✓	4	16.5	even	4
72.4.d.c.37.2	yes	4	16.13	even	4
72.4.d.c.37.3	yes	4	48.29	odd	4
72.4.d.c.37.4	yes	4	48.5	odd	4
288.4.d.c.145.1		4	16.11	odd	4
288.4.d.c.145.2		4	48.35	even	4
288.4.d.c.145.3		4	48.11	even	4
288.4.d.c.145.4		4	16.3	odd	4
2304.4.a.bx.1.1		4	24.5	odd	2	inner
2304.4.a.bx.1.2		4	1.1	even	1	trivial
2304.4.a.bx.1.3		4	8.5	even	2	inner
2304.4.a.bx.1.4		4	3.2	odd	2	inner
2304.4.a.cc.1.1		4	24.11	even	2
2304.4.a.cc.1.2		4	4.3	odd	2
2304.4.a.cc.1.3		4	8.3	odd	2
2304.4.a.cc.1.4		4	12.11	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2304.4.a.bx.1.2

Level	$2304$
Weight	$4$
Character	2304.1
Self dual	yes
Analytic conductor	$135.940$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$