LMFDB - Embedded newform 2352.4.a.cm.1.3

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			1.3
Root			$-6.46638$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} +0.726342 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +0.726342 q^{5} +9.00000 q^{9} +64.4895 q^{11} +71.8475 q^{13} -2.17903 q^{15} +48.9034 q^{17} -34.3968 q^{19} +0.903350 q^{23} -124.472 q^{25} -27.0000 q^{27} +226.686 q^{29} +275.795 q^{31} -193.468 q^{33} +295.209 q^{37} -215.543 q^{39} +186.604 q^{41} +455.317 q^{43} +6.53708 q^{45} -282.334 q^{47} -146.710 q^{51} +356.214 q^{53} +46.8414 q^{55} +103.191 q^{57} -729.370 q^{59} +274.353 q^{61} +52.1859 q^{65} -193.272 q^{67} -2.71005 q^{69} -40.5277 q^{71} -206.472 q^{73} +373.417 q^{75} -937.741 q^{79} +81.0000 q^{81} -911.607 q^{83} +35.5206 q^{85} -680.059 q^{87} -949.975 q^{89} -827.385 q^{93} -24.9839 q^{95} +39.4687 q^{97} +580.405 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.726342	0.0649660	0.0324830	−	0.999472i	$-0.489659\pi$
$5$	0.726342	0.0649660	0.0324830	+	0.999472i	$0.489659\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	64.4895	1.76766	0.883832	−	0.467804i	$-0.154955\pi$
$11$	64.4895	1.76766	0.883832	+	0.467804i	$0.154955\pi$
$12$	0	0
$13$	71.8475	1.53284	0.766420	−	0.642340i	$-0.222036\pi$
$13$	71.8475	1.53284	0.766420	+	0.642340i	$0.222036\pi$
$14$	0	0
$15$	−2.17903	−0.0375081
$16$	0	0
$17$	48.9034	0.697694	0.348847	−	0.937180i	$-0.386573\pi$
$17$	48.9034	0.697694	0.348847	+	0.937180i	$0.386573\pi$
$18$	0	0
$19$	−34.3968	−0.415325	−0.207663	−	0.978201i	$-0.566586\pi$
$19$	−34.3968	−0.415325	−0.207663	+	0.978201i	$0.566586\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.903350	0.00818963	0.00409482	−	0.999992i	$-0.498697\pi$
$23$	0.903350	0.00818963	0.00409482	+	0.999992i	$0.498697\pi$
$24$	0	0
$25$	−124.472	−0.995779
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	226.686	1.45154	0.725769	−	0.687939i	$-0.241484\pi$
$29$	226.686	1.45154	0.725769	+	0.687939i	$0.241484\pi$
$30$	0	0
$31$	275.795	1.59788	0.798940	−	0.601411i	$-0.205395\pi$
$31$	275.795	1.59788	0.798940	+	0.601411i	$0.205395\pi$
$32$	0	0
$33$	−193.468	−1.02056
$34$	0	0
$35$	0	0
$36$	0	0
$37$	295.209	1.31168	0.655839	−	0.754901i	$-0.272315\pi$
$37$	295.209	1.31168	0.655839	+	0.754901i	$0.272315\pi$
$38$	0	0
$39$	−215.543	−0.884985
$40$	0	0
$41$	186.604	0.710798	0.355399	−	0.934715i	$-0.384345\pi$
$41$	186.604	0.710798	0.355399	+	0.934715i	$0.384345\pi$
$42$	0	0
$43$	455.317	1.61477	0.807386	−	0.590023i	$-0.200881\pi$
$43$	455.317	1.61477	0.807386	+	0.590023i	$0.200881\pi$
$44$	0	0
$45$	6.53708	0.0216553
$46$	0	0
$47$	−282.334	−0.876228	−0.438114	−	0.898920i	$-0.644353\pi$
$47$	−282.334	−0.876228	−0.438114	+	0.898920i	$0.644353\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−146.710	−0.402814
$52$	0	0
$53$	356.214	0.923204	0.461602	−	0.887087i	$-0.347275\pi$
$53$	356.214	0.923204	0.461602	+	0.887087i	$0.347275\pi$
$54$	0	0
$55$	46.8414	0.114838
$56$	0	0
$57$	103.191	0.239788
$58$	0	0
$59$	−729.370	−1.60942	−0.804711	−	0.593667i	$-0.797680\pi$
$59$	−729.370	−1.60942	−0.804711	+	0.593667i	$0.797680\pi$
$60$	0	0
$61$	274.353	0.575856	0.287928	−	0.957652i	$-0.407034\pi$
$61$	274.353	0.575856	0.287928	+	0.957652i	$0.407034\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	52.1859	0.0995825
$66$	0	0
$67$	−193.272	−0.352417	−0.176209	−	0.984353i	$-0.556383\pi$
$67$	−193.272	−0.352417	−0.176209	+	0.984353i	$0.556383\pi$
$68$	0	0
$69$	−2.71005	−0.00472829
$70$	0	0
$71$	−40.5277	−0.0677429	−0.0338715	−	0.999426i	$-0.510784\pi$
$71$	−40.5277	−0.0677429	−0.0338715	+	0.999426i	$0.510784\pi$
$72$	0	0
$73$	−206.472	−0.331038	−0.165519	−	0.986207i	$-0.552930\pi$
$73$	−206.472	−0.331038	−0.165519	+	0.986207i	$0.552930\pi$
$74$	0	0
$75$	373.417	0.574914
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−937.741	−1.33549	−0.667747	−	0.744388i	$-0.732741\pi$
$79$	−937.741	−1.33549	−0.667747	+	0.744388i	$0.732741\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−911.607	−1.20556	−0.602782	−	0.797906i	$-0.705941\pi$
$83$	−911.607	−1.20556	−0.602782	+	0.797906i	$0.705941\pi$
$84$	0	0
$85$	35.5206	0.0453264
$86$	0	0
$87$	−680.059	−0.838045
$88$	0	0
$89$	−949.975	−1.13143	−0.565715	−	0.824601i	$-0.691400\pi$
$89$	−949.975	−1.13143	−0.565715	+	0.824601i	$0.691400\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−827.385	−0.922536
$94$	0	0
$95$	−24.9839	−0.0269820
$96$	0	0
$97$	39.4687	0.0413138	0.0206569	−	0.999787i	$-0.493424\pi$
$97$	39.4687	0.0413138	0.0206569	+	0.999787i	$0.493424\pi$
$98$	0	0
$99$	580.405	0.589221
$100$	0	0
$101$	−316.874	−0.312180	−0.156090	−	0.987743i	$-0.549889\pi$
$101$	−316.874	−0.312180	−0.156090	+	0.987743i	$0.549889\pi$
$102$	0	0
$103$	322.634	0.308641	0.154321	−	0.988021i	$-0.450681\pi$
$103$	322.634	0.308641	0.154321	+	0.988021i	$0.450681\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−681.451	−0.615686	−0.307843	−	0.951437i	$-0.599607\pi$
$107$	−681.451	−0.615686	−0.307843	+	0.951437i	$0.599607\pi$
$108$	0	0
$109$	−455.967	−0.400677	−0.200338	−	0.979727i	$-0.564204\pi$
$109$	−455.967	−0.400677	−0.200338	+	0.979727i	$0.564204\pi$
$110$	0	0
$111$	−885.627	−0.757297
$112$	0	0
$113$	−796.025	−0.662688	−0.331344	−	0.943510i	$-0.607502\pi$
$113$	−796.025	−0.662688	−0.331344	+	0.943510i	$0.607502\pi$
$114$	0	0
$115$	0.656141	0.000532048 0
$116$	0	0
$117$	646.628	0.510947
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2827.89	2.12464
$122$	0	0
$123$	−559.813	−0.410379
$124$	0	0
$125$	−181.202	−0.129658
$126$	0	0
$127$	−2333.92	−1.63072	−0.815362	−	0.578952i	$-0.803462\pi$
$127$	−2333.92	−1.63072	−0.815362	+	0.578952i	$0.803462\pi$
$128$	0	0
$129$	−1365.95	−0.932289
$130$	0	0
$131$	1886.98	1.25852	0.629259	−	0.777195i	$-0.283358\pi$
$131$	1886.98	1.25852	0.629259	+	0.777195i	$0.283358\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−19.6112	−0.0125027
$136$	0	0
$137$	2947.26	1.83797	0.918983	−	0.394297i	$-0.129012\pi$
$137$	2947.26	1.83797	0.918983	+	0.394297i	$0.129012\pi$
$138$	0	0
$139$	−955.433	−0.583013	−0.291506	−	0.956569i	$-0.594156\pi$
$139$	−955.433	−0.583013	−0.291506	+	0.956569i	$0.594156\pi$
$140$	0	0
$141$	847.003	0.505890
$142$	0	0
$143$	4633.41	2.70955
$144$	0	0
$145$	164.652	0.0943006
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2183.74	1.20066	0.600332	−	0.799751i	$-0.295035\pi$
$149$	2183.74	1.20066	0.600332	+	0.799751i	$0.295035\pi$
$150$	0	0
$151$	−404.719	−0.218116	−0.109058	−	0.994035i	$-0.534783\pi$
$151$	−404.719	−0.218116	−0.109058	+	0.994035i	$0.534783\pi$
$152$	0	0
$153$	440.130	0.232565
$154$	0	0
$155$	200.322	0.103808
$156$	0	0
$157$	−929.582	−0.472540	−0.236270	−	0.971687i	$-0.575925\pi$
$157$	−929.582	−0.472540	−0.236270	+	0.971687i	$0.575925\pi$
$158$	0	0
$159$	−1068.64	−0.533012
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1425.68	0.685079	0.342540	−	0.939503i	$-0.388713\pi$
$163$	1425.68	0.685079	0.342540	+	0.939503i	$0.388713\pi$
$164$	0	0
$165$	−140.524	−0.0663018
$166$	0	0
$167$	−4185.15	−1.93926	−0.969631	−	0.244572i	$-0.921353\pi$
$167$	−4185.15	−1.93926	−0.969631	+	0.244572i	$0.921353\pi$
$168$	0	0
$169$	2965.07	1.34960
$170$	0	0
$171$	−309.572	−0.138442
$172$	0	0
$173$	2235.55	0.982462	0.491231	−	0.871029i	$-0.336547\pi$
$173$	2235.55	0.982462	0.491231	+	0.871029i	$0.336547\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2188.11	0.929200
$178$	0	0
$179$	941.479	0.393126	0.196563	−	0.980491i	$-0.437022\pi$
$179$	941.479	0.393126	0.196563	+	0.980491i	$0.437022\pi$
$180$	0	0
$181$	−467.540	−0.192000	−0.0960000	−	0.995381i	$-0.530605\pi$
$181$	−467.540	−0.192000	−0.0960000	+	0.995381i	$0.530605\pi$
$182$	0	0
$183$	−823.058	−0.332471
$184$	0	0
$185$	214.423	0.0852144
$186$	0	0
$187$	3153.75	1.23329
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−275.403	−0.104332	−0.0521660	−	0.998638i	$-0.516613\pi$
$191$	−275.403	−0.104332	−0.0521660	+	0.998638i	$0.516613\pi$
$192$	0	0
$193$	1640.30	0.611767	0.305884	−	0.952069i	$-0.401048\pi$
$193$	1640.30	0.611767	0.305884	+	0.952069i	$0.401048\pi$
$194$	0	0
$195$	−156.558	−0.0574940
$196$	0	0
$197$	1303.88	0.471560	0.235780	−	0.971806i	$-0.424236\pi$
$197$	1303.88	0.471560	0.235780	+	0.971806i	$0.424236\pi$
$198$	0	0
$199$	1327.36	0.472833	0.236417	−	0.971652i	$-0.424027\pi$
$199$	1327.36	0.472833	0.236417	+	0.971652i	$0.424027\pi$
$200$	0	0
$201$	579.817	0.203468
$202$	0	0
$203$	0	0
$204$	0	0
$205$	135.539	0.0461777
$206$	0	0
$207$	8.13015	0.00272988
$208$	0	0
$209$	−2218.23	−0.734155
$210$	0	0
$211$	4753.28	1.55085	0.775426	−	0.631439i	$-0.217535\pi$
$211$	4753.28	1.55085	0.775426	+	0.631439i	$0.217535\pi$
$212$	0	0
$213$	121.583	0.0391114
$214$	0	0
$215$	330.716	0.104905
$216$	0	0
$217$	0	0
$218$	0	0
$219$	619.417	0.191125
$220$	0	0
$221$	3513.58	1.06945
$222$	0	0
$223$	−513.149	−0.154094	−0.0770470	−	0.997027i	$-0.524549\pi$
$223$	−513.149	−0.154094	−0.0770470	+	0.997027i	$0.524549\pi$
$224$	0	0
$225$	−1120.25	−0.331926
$226$	0	0
$227$	−3309.34	−0.967616	−0.483808	−	0.875174i	$-0.660747\pi$
$227$	−3309.34	−0.967616	−0.483808	+	0.875174i	$0.660747\pi$
$228$	0	0
$229$	−5909.47	−1.70528	−0.852639	−	0.522501i	$-0.824999\pi$
$229$	−5909.47	−1.70528	−0.852639	+	0.522501i	$0.824999\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−353.572	−0.0994131	−0.0497065	−	0.998764i	$-0.515829\pi$
$233$	−353.572	−0.0994131	−0.0497065	+	0.998764i	$0.515829\pi$
$234$	0	0
$235$	−205.071	−0.0569250
$236$	0	0
$237$	2813.22	0.771048
$238$	0	0
$239$	1652.55	0.447259	0.223629	−	0.974674i	$-0.428209\pi$
$239$	1652.55	0.447259	0.223629	+	0.974674i	$0.428209\pi$
$240$	0	0
$241$	−3106.95	−0.830440	−0.415220	−	0.909721i	$-0.636295\pi$
$241$	−3106.95	−0.830440	−0.415220	+	0.909721i	$0.636295\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2471.33	−0.636627
$248$	0	0
$249$	2734.82	0.696033
$250$	0	0
$251$	−1771.77	−0.445551	−0.222776	−	0.974870i	$-0.571512\pi$
$251$	−1771.77	−0.445551	−0.222776	+	0.974870i	$0.571512\pi$
$252$	0	0
$253$	58.2566	0.0144765
$254$	0	0
$255$	−106.562	−0.0261692
$256$	0	0
$257$	1478.13	0.358766	0.179383	−	0.983779i	$-0.442590\pi$
$257$	1478.13	0.358766	0.179383	+	0.983779i	$0.442590\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2040.18	0.483846
$262$	0	0
$263$	1431.97	0.335739	0.167869	−	0.985809i	$-0.446311\pi$
$263$	1431.97	0.335739	0.167869	+	0.985809i	$0.446311\pi$
$264$	0	0
$265$	258.734	0.0599769
$266$	0	0
$267$	2849.93	0.653231
$268$	0	0
$269$	285.398	0.0646879	0.0323439	−	0.999477i	$-0.489703\pi$
$269$	285.398	0.0646879	0.0323439	+	0.999477i	$0.489703\pi$
$270$	0	0
$271$	5249.33	1.17666	0.588328	−	0.808622i	$-0.299786\pi$
$271$	5249.33	1.17666	0.588328	+	0.808622i	$0.299786\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8027.16	−1.76020
$276$	0	0
$277$	7147.87	1.55045	0.775224	−	0.631687i	$-0.217637\pi$
$277$	7147.87	1.55045	0.775224	+	0.631687i	$0.217637\pi$
$278$	0	0
$279$	2482.16	0.532627
$280$	0	0
$281$	4228.36	0.897661	0.448831	−	0.893617i	$-0.351841\pi$
$281$	4228.36	0.897661	0.448831	+	0.893617i	$0.351841\pi$
$282$	0	0
$283$	−8342.38	−1.75231	−0.876154	−	0.482031i	$-0.839899\pi$
$283$	−8342.38	−1.75231	−0.876154	+	0.482031i	$0.839899\pi$
$284$	0	0
$285$	74.9516	0.0155781
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2521.46	−0.513223
$290$	0	0
$291$	−118.406	−0.0238525
$292$	0	0
$293$	5038.84	1.00468	0.502342	−	0.864669i	$-0.332472\pi$
$293$	5038.84	1.00468	0.502342	+	0.864669i	$0.332472\pi$
$294$	0	0
$295$	−529.772	−0.104558
$296$	0	0
$297$	−1741.22	−0.340187
$298$	0	0
$299$	64.9035	0.0125534
$300$	0	0
$301$	0	0
$302$	0	0
$303$	950.623	0.180237
$304$	0	0
$305$	199.274	0.0374111
$306$	0	0
$307$	4869.67	0.905300	0.452650	−	0.891688i	$-0.350479\pi$
$307$	4869.67	0.905300	0.452650	+	0.891688i	$0.350479\pi$
$308$	0	0
$309$	−967.901	−0.178194
$310$	0	0
$311$	1452.84	0.264897	0.132449	−	0.991190i	$-0.457716\pi$
$311$	1452.84	0.264897	0.132449	+	0.991190i	$0.457716\pi$
$312$	0	0
$313$	5696.27	1.02867	0.514333	−	0.857591i	$-0.328040\pi$
$313$	5696.27	1.02867	0.514333	+	0.857591i	$0.328040\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3471.47	−0.615070	−0.307535	−	0.951537i	$-0.599504\pi$
$317$	−3471.47	−0.615070	−0.307535	+	0.951537i	$0.599504\pi$
$318$	0	0
$319$	14618.9	2.56583
$320$	0	0
$321$	2044.35	0.355466
$322$	0	0
$323$	−1682.12	−0.289770
$324$	0	0
$325$	−8943.03	−1.52637
$326$	0	0
$327$	1367.90	0.231331
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6254.38	−1.03859	−0.519293	−	0.854597i	$-0.673805\pi$
$331$	−6254.38	−1.03859	−0.519293	+	0.854597i	$0.673805\pi$
$332$	0	0
$333$	2656.88	0.437226
$334$	0	0
$335$	−140.382	−0.0228951
$336$	0	0
$337$	8006.96	1.29426	0.647132	−	0.762378i	$-0.275968\pi$
$337$	8006.96	1.29426	0.647132	+	0.762378i	$0.275968\pi$
$338$	0	0
$339$	2388.08	0.382603
$340$	0	0
$341$	17785.9	2.82451
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.96842	−0.000307178 0
$346$	0	0
$347$	−7635.59	−1.18127	−0.590634	−	0.806940i	$-0.701122\pi$
$347$	−7635.59	−1.18127	−0.590634	+	0.806940i	$0.701122\pi$
$348$	0	0
$349$	−10358.0	−1.58869	−0.794345	−	0.607468i	$-0.792185\pi$
$349$	−10358.0	−1.58869	−0.794345	+	0.607468i	$0.792185\pi$
$350$	0	0
$351$	−1939.88	−0.294995
$352$	0	0
$353$	3384.23	0.510268	0.255134	−	0.966906i	$-0.417880\pi$
$353$	3384.23	0.510268	0.255134	+	0.966906i	$0.417880\pi$
$354$	0	0
$355$	−29.4369	−0.00440099
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8194.61	1.20472	0.602361	−	0.798224i	$-0.294227\pi$
$359$	8194.61	1.20472	0.602361	+	0.798224i	$0.294227\pi$
$360$	0	0
$361$	−5675.86	−0.827505
$362$	0	0
$363$	−8483.67	−1.22666
$364$	0	0
$365$	−149.970	−0.0215062
$366$	0	0
$367$	−805.510	−0.114570	−0.0572851	−	0.998358i	$-0.518244\pi$
$367$	−805.510	−0.114570	−0.0572851	+	0.998358i	$0.518244\pi$
$368$	0	0
$369$	1679.44	0.236933
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7905.71	1.09743	0.548716	−	0.836009i	$-0.315117\pi$
$373$	7905.71	1.09743	0.548716	+	0.836009i	$0.315117\pi$
$374$	0	0
$375$	543.607	0.0748580
$376$	0	0
$377$	16286.8	2.22497
$378$	0	0
$379$	−3324.24	−0.450540	−0.225270	−	0.974296i	$-0.572326\pi$
$379$	−3324.24	−0.450540	−0.225270	+	0.974296i	$0.572326\pi$
$380$	0	0
$381$	7001.76	0.941499
$382$	0	0
$383$	8470.76	1.13012	0.565060	−	0.825050i	$-0.308853\pi$
$383$	8470.76	1.13012	0.565060	+	0.825050i	$0.308853\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4097.85	0.538258
$388$	0	0
$389$	−7908.40	−1.03078	−0.515388	−	0.856957i	$-0.672352\pi$
$389$	−7908.40	−1.03078	−0.515388	+	0.856957i	$0.672352\pi$
$390$	0	0
$391$	44.1769	0.00571386
$392$	0	0
$393$	−5660.93	−0.726606
$394$	0	0
$395$	−681.120	−0.0867617
$396$	0	0
$397$	−9781.58	−1.23658	−0.618292	−	0.785949i	$-0.712175\pi$
$397$	−9781.58	−1.23658	−0.618292	+	0.785949i	$0.712175\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−795.653	−0.0990849	−0.0495424	−	0.998772i	$-0.515776\pi$
$401$	−795.653	−0.0990849	−0.0495424	+	0.998772i	$0.515776\pi$
$402$	0	0
$403$	19815.2	2.44929
$404$	0	0
$405$	58.8337	0.00721844
$406$	0	0
$407$	19037.9	2.31860
$408$	0	0
$409$	8584.72	1.03787	0.518933	−	0.854815i	$-0.326329\pi$
$409$	8584.72	1.03787	0.518933	+	0.854815i	$0.326329\pi$
$410$	0	0
$411$	−8841.78	−1.06115
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−662.138	−0.0783207
$416$	0	0
$417$	2866.30	0.336603
$418$	0	0
$419$	7447.09	0.868292	0.434146	−	0.900843i	$-0.357050\pi$
$419$	7447.09	0.868292	0.434146	+	0.900843i	$0.357050\pi$
$420$	0	0
$421$	−4446.76	−0.514779	−0.257390	−	0.966308i	$-0.582862\pi$
$421$	−4446.76	−0.514779	−0.257390	+	0.966308i	$0.582862\pi$
$422$	0	0
$423$	−2541.01	−0.292076
$424$	0	0
$425$	−6087.12	−0.694750
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13900.2	−1.56436
$430$	0	0
$431$	8963.70	1.00178	0.500889	−	0.865512i	$-0.333006\pi$
$431$	8963.70	1.00178	0.500889	+	0.865512i	$0.333006\pi$
$432$	0	0
$433$	−16173.6	−1.79504	−0.897520	−	0.440973i	$-0.854633\pi$
$433$	−16173.6	−1.79504	−0.897520	+	0.440973i	$0.854633\pi$
$434$	0	0
$435$	−493.955	−0.0544445
$436$	0	0
$437$	−31.0724	−0.00340136
$438$	0	0
$439$	6298.43	0.684755	0.342378	−	0.939562i	$-0.388768\pi$
$439$	6298.43	0.684755	0.342378	+	0.939562i	$0.388768\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16784.7	1.80015	0.900074	−	0.435738i	$-0.143512\pi$
$443$	16784.7	1.80015	0.900074	+	0.435738i	$0.143512\pi$
$444$	0	0
$445$	−690.007	−0.0735044
$446$	0	0
$447$	−6551.22	−0.693203
$448$	0	0
$449$	2733.88	0.287349	0.143674	−	0.989625i	$-0.454108\pi$
$449$	2733.88	0.287349	0.143674	+	0.989625i	$0.454108\pi$
$450$	0	0
$451$	12034.0	1.25645
$452$	0	0
$453$	1214.16	0.125930
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9229.20	0.944691	0.472345	−	0.881414i	$-0.343408\pi$
$457$	9229.20	0.944691	0.472345	+	0.881414i	$0.343408\pi$
$458$	0	0
$459$	−1320.39	−0.134271
$460$	0	0
$461$	−19726.7	−1.99298	−0.996491	−	0.0837048i	$-0.973325\pi$
$461$	−19726.7	−1.99298	−0.996491	+	0.0837048i	$0.973325\pi$
$462$	0	0
$463$	−368.924	−0.0370310	−0.0185155	−	0.999829i	$-0.505894\pi$
$463$	−368.924	−0.0370310	−0.0185155	+	0.999829i	$0.505894\pi$
$464$	0	0
$465$	−600.965	−0.0599335
$466$	0	0
$467$	13309.6	1.31883	0.659414	−	0.751780i	$-0.270804\pi$
$467$	13309.6	1.31883	0.659414	+	0.751780i	$0.270804\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2788.75	0.272821
$472$	0	0
$473$	29363.2	2.85438
$474$	0	0
$475$	4281.46	0.413572
$476$	0	0
$477$	3205.93	0.307735
$478$	0	0
$479$	−11562.8	−1.10296	−0.551479	−	0.834189i	$-0.685936\pi$
$479$	−11562.8	−1.10296	−0.551479	+	0.834189i	$0.685936\pi$
$480$	0	0
$481$	21210.0	2.01059
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.6677	0.00268399
$486$	0	0
$487$	−2628.13	−0.244542	−0.122271	−	0.992497i	$-0.539018\pi$
$487$	−2628.13	−0.244542	−0.122271	+	0.992497i	$0.539018\pi$
$488$	0	0
$489$	−4277.04	−0.395531
$490$	0	0
$491$	12319.9	1.13236	0.566181	−	0.824281i	$-0.308420\pi$
$491$	12319.9	1.13236	0.566181	+	0.824281i	$0.308420\pi$
$492$	0	0
$493$	11085.7	1.01273
$494$	0	0
$495$	421.573	0.0382794
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3305.81	0.296570	0.148285	−	0.988945i	$-0.452625\pi$
$499$	3305.81	0.296570	0.148285	+	0.988945i	$0.452625\pi$
$500$	0	0
$501$	12555.5	1.11963
$502$	0	0
$503$	3072.72	0.272377	0.136189	−	0.990683i	$-0.456515\pi$
$503$	3072.72	0.272377	0.136189	+	0.990683i	$0.456515\pi$
$504$	0	0
$505$	−230.159	−0.0202811
$506$	0	0
$507$	−8895.20	−0.779190
$508$	0	0
$509$	13569.8	1.18167	0.590836	−	0.806791i	$-0.298798\pi$
$509$	13569.8	1.18167	0.590836	+	0.806791i	$0.298798\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	928.715	0.0799294
$514$	0	0
$515$	234.342	0.0200512
$516$	0	0
$517$	−18207.6	−1.54888
$518$	0	0
$519$	−6706.66	−0.567225
$520$	0	0
$521$	−10733.4	−0.902566	−0.451283	−	0.892381i	$-0.649033\pi$
$521$	−10733.4	−0.902566	−0.451283	+	0.892381i	$0.649033\pi$
$522$	0	0
$523$	−8348.15	−0.697971	−0.348986	−	0.937128i	$-0.613474\pi$
$523$	−8348.15	−0.697971	−0.348986	+	0.937128i	$0.613474\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13487.3	1.11483
$528$	0	0
$529$	−12166.2	−0.999933
$530$	0	0
$531$	−6564.33	−0.536474
$532$	0	0
$533$	13407.1	1.08954
$534$	0	0
$535$	−494.967	−0.0399987
$536$	0	0
$537$	−2824.44	−0.226971
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−23107.3	−1.83634	−0.918170	−	0.396186i	$-0.870334\pi$
$541$	−23107.3	−1.83634	−0.918170	+	0.396186i	$0.870334\pi$
$542$	0	0
$543$	1402.62	0.110851
$544$	0	0
$545$	−331.188	−0.0260304
$546$	0	0
$547$	−13935.2	−1.08926	−0.544630	−	0.838676i	$-0.683330\pi$
$547$	−13935.2	−1.08926	−0.544630	+	0.838676i	$0.683330\pi$
$548$	0	0
$549$	2469.17	0.191952
$550$	0	0
$551$	−7797.29	−0.602860
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−643.268	−0.0491986
$556$	0	0
$557$	−15047.5	−1.14467	−0.572337	−	0.820018i	$-0.693963\pi$
$557$	−15047.5	−1.14467	−0.572337	+	0.820018i	$0.693963\pi$
$558$	0	0
$559$	32713.4	2.47519
$560$	0	0
$561$	−9461.25	−0.712040
$562$	0	0
$563$	−15442.2	−1.15597	−0.577983	−	0.816049i	$-0.696160\pi$
$563$	−15442.2	−1.15597	−0.577983	+	0.816049i	$0.696160\pi$
$564$	0	0
$565$	−578.187	−0.0430522
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13362.1	−0.984481	−0.492240	−	0.870459i	$-0.663822\pi$
$569$	−13362.1	−0.984481	−0.492240	+	0.870459i	$0.663822\pi$
$570$	0	0
$571$	22185.3	1.62596	0.812981	−	0.582290i	$-0.197843\pi$
$571$	22185.3	1.62596	0.812981	+	0.582290i	$0.197843\pi$
$572$	0	0
$573$	826.208	0.0602362
$574$	0	0
$575$	−112.442	−0.00815507
$576$	0	0
$577$	−18951.4	−1.36734	−0.683671	−	0.729791i	$-0.739617\pi$
$577$	−18951.4	−1.36734	−0.683671	+	0.729791i	$0.739617\pi$
$578$	0	0
$579$	−4920.89	−0.353204
$580$	0	0
$581$	0	0
$582$	0	0
$583$	22972.1	1.63191
$584$	0	0
$585$	469.673	0.0331942
$586$	0	0
$587$	−19579.5	−1.37672	−0.688359	−	0.725371i	$-0.741668\pi$
$587$	−19579.5	−1.37672	−0.688359	+	0.725371i	$0.741668\pi$
$588$	0	0
$589$	−9486.48	−0.663640
$590$	0	0
$591$	−3911.63	−0.272255
$592$	0	0
$593$	5613.01	0.388699	0.194350	−	0.980932i	$-0.437740\pi$
$593$	5613.01	0.388699	0.194350	+	0.980932i	$0.437740\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3982.07	−0.272990
$598$	0	0
$599$	−21925.5	−1.49558	−0.747790	−	0.663936i	$-0.768885\pi$
$599$	−21925.5	−1.49558	−0.747790	+	0.663936i	$0.768885\pi$
$600$	0	0
$601$	−2067.51	−0.140326	−0.0701628	−	0.997536i	$-0.522352\pi$
$601$	−2067.51	−0.140326	−0.0701628	+	0.997536i	$0.522352\pi$
$602$	0	0
$603$	−1739.45	−0.117472
$604$	0	0
$605$	2054.02	0.138029
$606$	0	0
$607$	10167.2	0.679859	0.339930	−	0.940451i	$-0.389597\pi$
$607$	10167.2	0.679859	0.339930	+	0.940451i	$0.389597\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20285.0	−1.34312
$612$	0	0
$613$	−18186.1	−1.19825	−0.599127	−	0.800654i	$-0.704485\pi$
$613$	−18186.1	−1.19825	−0.599127	+	0.800654i	$0.704485\pi$
$614$	0	0
$615$	−406.616	−0.0266607
$616$	0	0
$617$	6584.41	0.429625	0.214812	−	0.976655i	$-0.431086\pi$
$617$	6584.41	0.429625	0.214812	+	0.976655i	$0.431086\pi$
$618$	0	0
$619$	7779.72	0.505159	0.252579	−	0.967576i	$-0.418721\pi$
$619$	7779.72	0.505159	0.252579	+	0.967576i	$0.418721\pi$
$620$	0	0
$621$	−24.3905	−0.00157610
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15427.4	0.987356
$626$	0	0
$627$	6654.70	0.423865
$628$	0	0
$629$	14436.7	0.915150
$630$	0	0
$631$	−3787.78	−0.238969	−0.119484	−	0.992836i	$-0.538124\pi$
$631$	−3787.78	−0.238969	−0.119484	+	0.992836i	$0.538124\pi$
$632$	0	0
$633$	−14259.9	−0.895384
$634$	0	0
$635$	−1695.22	−0.105942
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−364.749	−0.0225810
$640$	0	0
$641$	−19930.9	−1.22812	−0.614058	−	0.789261i	$-0.710464\pi$
$641$	−19930.9	−1.22812	−0.614058	+	0.789261i	$0.710464\pi$
$642$	0	0
$643$	−3185.18	−0.195352	−0.0976759	−	0.995218i	$-0.531141\pi$
$643$	−3185.18	−0.195352	−0.0976759	+	0.995218i	$0.531141\pi$
$644$	0	0
$645$	−992.148	−0.0605671
$646$	0	0
$647$	16943.0	1.02952	0.514759	−	0.857335i	$-0.327881\pi$
$647$	16943.0	1.02952	0.514759	+	0.857335i	$0.327881\pi$
$648$	0	0
$649$	−47036.7	−2.84492
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20078.8	−1.20328	−0.601640	−	0.798767i	$-0.705486\pi$
$653$	−20078.8	−1.20328	−0.601640	+	0.798767i	$0.705486\pi$
$654$	0	0
$655$	1370.59	0.0817609
$656$	0	0
$657$	−1858.25	−0.110346
$658$	0	0
$659$	9442.46	0.558158	0.279079	−	0.960268i	$-0.409971\pi$
$659$	9442.46	0.558158	0.279079	+	0.960268i	$0.409971\pi$
$660$	0	0
$661$	761.990	0.0448381	0.0224190	−	0.999749i	$-0.492863\pi$
$661$	761.990	0.0448381	0.0224190	+	0.999749i	$0.492863\pi$
$662$	0	0
$663$	−10540.8	−0.617449
$664$	0	0
$665$	0	0
$666$	0	0
$667$	204.777	0.0118876
$668$	0	0
$669$	1539.45	0.0889663
$670$	0	0
$671$	17692.8	1.01792
$672$	0	0
$673$	−16111.6	−0.922818	−0.461409	−	0.887188i	$-0.652656\pi$
$673$	−16111.6	−0.922818	−0.461409	+	0.887188i	$0.652656\pi$
$674$	0	0
$675$	3360.76	0.191638
$676$	0	0
$677$	31241.5	1.77357	0.886786	−	0.462180i	$-0.152933\pi$
$677$	31241.5	1.77357	0.886786	+	0.462180i	$0.152933\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9928.03	0.558653
$682$	0	0
$683$	30113.5	1.68706	0.843530	−	0.537082i	$-0.180473\pi$
$683$	30113.5	1.68706	0.843530	+	0.537082i	$0.180473\pi$
$684$	0	0
$685$	2140.72	0.119405
$686$	0	0
$687$	17728.4	0.984542
$688$	0	0
$689$	25593.1	1.41512
$690$	0	0
$691$	9750.85	0.536816	0.268408	−	0.963305i	$-0.413502\pi$
$691$	9750.85	0.536816	0.268408	+	0.963305i	$0.413502\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−693.971	−0.0378760
$696$	0	0
$697$	9125.58	0.495920
$698$	0	0
$699$	1060.71	0.0573962
$700$	0	0
$701$	11851.6	0.638559	0.319280	−	0.947661i	$-0.396559\pi$
$701$	11851.6	0.638559	0.319280	+	0.947661i	$0.396559\pi$
$702$	0	0
$703$	−10154.3	−0.544773
$704$	0	0
$705$	615.214	0.0328657
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1322.39	0.0700472	0.0350236	−	0.999386i	$-0.488849\pi$
$709$	1322.39	0.0700472	0.0350236	+	0.999386i	$0.488849\pi$
$710$	0	0
$711$	−8439.67	−0.445165
$712$	0	0
$713$	249.140	0.0130861
$714$	0	0
$715$	3365.44	0.176028
$716$	0	0
$717$	−4957.66	−0.258225
$718$	0	0
$719$	−7301.56	−0.378723	−0.189362	−	0.981907i	$-0.560642\pi$
$719$	−7301.56	−0.378723	−0.189362	+	0.981907i	$0.560642\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	9320.84	0.479455
$724$	0	0
$725$	−28216.2	−1.44541
$726$	0	0
$727$	6088.72	0.310616	0.155308	−	0.987866i	$-0.450363\pi$
$727$	6088.72	0.310616	0.155308	+	0.987866i	$0.450363\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	22266.5	1.12662
$732$	0	0
$733$	−26578.5	−1.33929	−0.669644	−	0.742683i	$-0.733553\pi$
$733$	−26578.5	−1.33929	−0.669644	+	0.742683i	$0.733553\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12464.0	−0.622955
$738$	0	0
$739$	−25219.8	−1.25538	−0.627689	−	0.778464i	$-0.715999\pi$
$739$	−25219.8	−1.25538	−0.627689	+	0.778464i	$0.715999\pi$
$740$	0	0
$741$	7413.98	0.367557
$742$	0	0
$743$	−2634.28	−0.130071	−0.0650353	−	0.997883i	$-0.520716\pi$
$743$	−2634.28	−0.130071	−0.0650353	+	0.997883i	$0.520716\pi$
$744$	0	0
$745$	1586.14	0.0780023
$746$	0	0
$747$	−8204.46	−0.401855
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−5767.29	−0.280228	−0.140114	−	0.990135i	$-0.544747\pi$
$751$	−5767.29	−0.280228	−0.140114	+	0.990135i	$0.544747\pi$
$752$	0	0
$753$	5315.32	0.257239
$754$	0	0
$755$	−293.965	−0.0141702
$756$	0	0
$757$	33378.7	1.60260	0.801302	−	0.598260i	$-0.204141\pi$
$757$	33378.7	1.60260	0.801302	+	0.598260i	$0.204141\pi$
$758$	0	0
$759$	−174.770	−0.00835802
$760$	0	0
$761$	3626.09	0.172727	0.0863637	−	0.996264i	$-0.472475\pi$
$761$	3626.09	0.172727	0.0863637	+	0.996264i	$0.472475\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	319.685	0.0151088
$766$	0	0
$767$	−52403.4	−2.46698
$768$	0	0
$769$	3426.15	0.160663	0.0803316	−	0.996768i	$-0.474402\pi$
$769$	3426.15	0.160663	0.0803316	+	0.996768i	$0.474402\pi$
$770$	0	0
$771$	−4434.38	−0.207134
$772$	0	0
$773$	13108.2	0.609920	0.304960	−	0.952365i	$-0.401357\pi$
$773$	13108.2	0.609920	0.304960	+	0.952365i	$0.401357\pi$
$774$	0	0
$775$	−34328.9	−1.59114
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6418.60	−0.295212
$780$	0	0
$781$	−2613.61	−0.119747
$782$	0	0
$783$	−6120.53	−0.279348
$784$	0	0
$785$	−675.194	−0.0306990
$786$	0	0
$787$	−39319.7	−1.78094	−0.890468	−	0.455045i	$-0.849623\pi$
$787$	−39319.7	−1.78094	−0.890468	+	0.455045i	$0.849623\pi$
$788$	0	0
$789$	−4295.92	−0.193839
$790$	0	0
$791$	0	0
$792$	0	0
$793$	19711.5	0.882696
$794$	0	0
$795$	−776.201	−0.0346277
$796$	0	0
$797$	−14597.8	−0.648785	−0.324393	−	0.945923i	$-0.605160\pi$
$797$	−14597.8	−0.648785	−0.324393	+	0.945923i	$0.605160\pi$
$798$	0	0
$799$	−13807.1	−0.611339
$800$	0	0
$801$	−8549.78	−0.377143
$802$	0	0
$803$	−13315.3	−0.585164
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−856.194	−0.0373475
$808$	0	0
$809$	26827.4	1.16589	0.582943	−	0.812513i	$-0.301901\pi$
$809$	26827.4	1.16589	0.582943	+	0.812513i	$0.301901\pi$
$810$	0	0
$811$	5177.09	0.224158	0.112079	−	0.993699i	$-0.464249\pi$
$811$	5177.09	0.224158	0.112079	+	0.993699i	$0.464249\pi$
$812$	0	0
$813$	−15748.0	−0.679343
$814$	0	0
$815$	1035.53	0.0445068
$816$	0	0
$817$	−15661.5	−0.670656
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20356.8	0.865355	0.432678	−	0.901549i	$-0.357569\pi$
$821$	20356.8	0.865355	0.432678	+	0.901549i	$0.357569\pi$
$822$	0	0
$823$	−57.5738	−0.00243851	−0.00121926	−	0.999999i	$-0.500388\pi$
$823$	−57.5738	−0.00243851	−0.00121926	+	0.999999i	$0.500388\pi$
$824$	0	0
$825$	24081.5	1.01625
$826$	0	0
$827$	12296.6	0.517044	0.258522	−	0.966005i	$-0.416765\pi$
$827$	12296.6	0.517044	0.258522	+	0.966005i	$0.416765\pi$
$828$	0	0
$829$	22871.7	0.958224	0.479112	−	0.877754i	$-0.340959\pi$
$829$	22871.7	0.958224	0.479112	+	0.877754i	$0.340959\pi$
$830$	0	0
$831$	−21443.6	−0.895151
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3039.85	−0.125986
$836$	0	0
$837$	−7446.47	−0.307512
$838$	0	0
$839$	−37571.3	−1.54601	−0.773007	−	0.634398i	$-0.781248\pi$
$839$	−37571.3	−1.54601	−0.773007	+	0.634398i	$0.781248\pi$
$840$	0	0
$841$	26997.6	1.10696
$842$	0	0
$843$	−12685.1	−0.518265
$844$	0	0
$845$	2153.65	0.0876780
$846$	0	0
$847$	0	0
$848$	0	0
$849$	25027.1	1.01170
$850$	0	0
$851$	266.677	0.0107422
$852$	0	0
$853$	−38354.5	−1.53955	−0.769774	−	0.638317i	$-0.779631\pi$
$853$	−38354.5	−1.53955	−0.769774	+	0.638317i	$0.779631\pi$
$854$	0	0
$855$	−224.855	−0.00899401
$856$	0	0
$857$	36545.7	1.45668	0.728341	−	0.685215i	$-0.240292\pi$
$857$	36545.7	1.45668	0.728341	+	0.685215i	$0.240292\pi$
$858$	0	0
$859$	18564.3	0.737374	0.368687	−	0.929554i	$-0.379807\pi$
$859$	18564.3	0.737374	0.368687	+	0.929554i	$0.379807\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37457.7	−1.47749	−0.738746	−	0.673984i	$-0.764582\pi$
$863$	−37457.7	−1.47749	−0.738746	+	0.673984i	$0.764582\pi$
$864$	0	0
$865$	1623.78	0.0638266
$866$	0	0
$867$	7564.39	0.296309
$868$	0	0
$869$	−60474.4	−2.36071
$870$	0	0
$871$	−13886.1	−0.540199
$872$	0	0
$873$	355.218	0.0137713
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27128.8	−1.04455	−0.522277	−	0.852776i	$-0.674917\pi$
$877$	−27128.8	−1.04455	−0.522277	+	0.852776i	$0.674917\pi$
$878$	0	0
$879$	−15116.5	−0.580055
$880$	0	0
$881$	23647.8	0.904329	0.452165	−	0.891935i	$-0.350652\pi$
$881$	23647.8	0.904329	0.452165	+	0.891935i	$0.350652\pi$
$882$	0	0
$883$	4488.47	0.171063	0.0855316	−	0.996335i	$-0.472741\pi$
$883$	4488.47	0.171063	0.0855316	+	0.996335i	$0.472741\pi$
$884$	0	0
$885$	1589.32	0.0603664
$886$	0	0
$887$	41026.8	1.55304	0.776520	−	0.630093i	$-0.216983\pi$
$887$	41026.8	1.55304	0.776520	+	0.630093i	$0.216983\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5223.65	0.196407
$892$	0	0
$893$	9711.41	0.363919
$894$	0	0
$895$	683.836	0.0255398
$896$	0	0
$897$	−194.710	−0.00724771
$898$	0	0
$899$	62519.0	2.31938
$900$	0	0
$901$	17420.1	0.644114
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−339.594	−0.0124735
$906$	0	0
$907$	736.305	0.0269555	0.0134777	−	0.999909i	$-0.495710\pi$
$907$	736.305	0.0269555	0.0134777	+	0.999909i	$0.495710\pi$
$908$	0	0
$909$	−2851.87	−0.104060
$910$	0	0
$911$	−1287.54	−0.0468256	−0.0234128	−	0.999726i	$-0.507453\pi$
$911$	−1287.54	−0.0468256	−0.0234128	+	0.999726i	$0.507453\pi$
$912$	0	0
$913$	−58789.0	−2.13103
$914$	0	0
$915$	−597.821	−0.0215993
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−34280.0	−1.23046	−0.615230	−	0.788348i	$-0.710937\pi$
$919$	−34280.0	−1.23046	−0.615230	+	0.788348i	$0.710937\pi$
$920$	0	0
$921$	−14609.0	−0.522675
$922$	0	0
$923$	−2911.81	−0.103839
$924$	0	0
$925$	−36745.4	−1.30614
$926$	0	0
$927$	2903.70	0.102880
$928$	0	0
$929$	−31475.8	−1.11161	−0.555806	−	0.831312i	$-0.687590\pi$
$929$	−31475.8	−1.11161	−0.555806	+	0.831312i	$0.687590\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−4358.52	−0.152939
$934$	0	0
$935$	2290.70	0.0801219
$936$	0	0
$937$	52560.6	1.83253	0.916264	−	0.400575i	$-0.131190\pi$
$937$	52560.6	1.83253	0.916264	+	0.400575i	$0.131190\pi$
$938$	0	0
$939$	−17088.8	−0.593900
$940$	0	0
$941$	15976.7	0.553482	0.276741	−	0.960945i	$-0.410746\pi$
$941$	15976.7	0.553482	0.276741	+	0.960945i	$0.410746\pi$
$942$	0	0
$943$	168.569	0.00582118
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8504.43	0.291824	0.145912	−	0.989298i	$-0.453388\pi$
$947$	8504.43	0.291824	0.145912	+	0.989298i	$0.453388\pi$
$948$	0	0
$949$	−14834.5	−0.507428
$950$	0	0
$951$	10414.4	0.355111
$952$	0	0
$953$	29417.5	0.999923	0.499961	−	0.866048i	$-0.333348\pi$
$953$	29417.5	0.999923	0.499961	+	0.866048i	$0.333348\pi$
$954$	0	0
$955$	−200.036	−0.00677804
$956$	0	0
$957$	−43856.6	−1.48138
$958$	0	0
$959$	0	0
$960$	0	0
$961$	46272.0	1.55322
$962$	0	0
$963$	−6133.06	−0.205229
$964$	0	0
$965$	1191.42	0.0397441
$966$	0	0
$967$	−3461.97	−0.115129	−0.0575643	−	0.998342i	$-0.518333\pi$
$967$	−3461.97	−0.115129	−0.0575643	+	0.998342i	$0.518333\pi$
$968$	0	0
$969$	5046.36	0.167299
$970$	0	0
$971$	43558.8	1.43962	0.719809	−	0.694172i	$-0.244229\pi$
$971$	43558.8	1.43962	0.719809	+	0.694172i	$0.244229\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	26829.1	0.881250
$976$	0	0
$977$	−5504.95	−0.180265	−0.0901325	−	0.995930i	$-0.528729\pi$
$977$	−5504.95	−0.180265	−0.0901325	+	0.995930i	$0.528729\pi$
$978$	0	0
$979$	−61263.4	−1.99999
$980$	0	0
$981$	−4103.71	−0.133559
$982$	0	0
$983$	30789.7	0.999023	0.499511	−	0.866307i	$-0.333513\pi$
$983$	30789.7	0.999023	0.499511	+	0.866307i	$0.333513\pi$
$984$	0	0
$985$	947.060	0.0306354
$986$	0	0
$987$	0	0
$988$	0	0
$989$	411.311	0.0132244
$990$	0	0
$991$	21629.1	0.693312	0.346656	−	0.937992i	$-0.387317\pi$
$991$	21629.1	0.693312	0.346656	+	0.937992i	$0.387317\pi$
$992$	0	0
$993$	18763.1	0.599627
$994$	0	0
$995$	964.114	0.0307181
$996$	0	0
$997$	−38195.8	−1.21331	−0.606656	−	0.794964i	$-0.707490\pi$
$997$	−38195.8	−1.21331	−0.606656	+	0.794964i	$0.707490\pi$
$998$	0	0
$999$	−7970.64	−0.252432

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cm.1.3		4
4.3	odd	2		1176.4.a.bd.1.3		4
7.2	even	3		336.4.q.m.193.2		8
7.4	even	3		336.4.q.m.289.2		8
7.6	odd	2		2352.4.a.cp.1.2		4
28.11	odd	6		168.4.q.f.121.2	yes	8
28.23	odd	6		168.4.q.f.25.2	✓	8
28.27	even	2		1176.4.a.ba.1.2		4
84.11	even	6		504.4.s.j.289.3		8
84.23	even	6		504.4.s.j.361.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.f.25.2	✓	8	28.23	odd	6
168.4.q.f.121.2	yes	8	28.11	odd	6
336.4.q.m.193.2		8	7.2	even	3
336.4.q.m.289.2		8	7.4	even	3
504.4.s.j.289.3		8	84.11	even	6
504.4.s.j.361.3		8	84.23	even	6
1176.4.a.ba.1.2		4	28.27	even	2
1176.4.a.bd.1.3		4	4.3	odd	2
2352.4.a.cm.1.3		4	1.1	even	1	trivial
2352.4.a.cp.1.2		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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +0.726342 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +0.726342 q^{5} +9.00000 q^{9} +64.4895 q^{11} +71.8475 q^{13} -2.17903 q^{15} +48.9034 q^{17} -34.3968 q^{19} +0.903350 q^{23} -124.472 q^{25} -27.0000 q^{27} +226.686 q^{29} +275.795 q^{31} -193.468 q^{33} +295.209 q^{37} -215.543 q^{39} +186.604 q^{41} +455.317 q^{43} +6.53708 q^{45} -282.334 q^{47} -146.710 q^{51} +356.214 q^{53} +46.8414 q^{55} +103.191 q^{57} -729.370 q^{59} +274.353 q^{61} +52.1859 q^{65} -193.272 q^{67} -2.71005 q^{69} -40.5277 q^{71} -206.472 q^{73} +373.417 q^{75} -937.741 q^{79} +81.0000 q^{81} -911.607 q^{83} +35.5206 q^{85} -680.059 q^{87} -949.975 q^{89} -827.385 q^{93} -24.9839 q^{95} +39.4687 q^{97} +580.405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 4 q^{5} + 36 q^{9}+O(q^{10}) \) 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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