LMFDB - Embedded newform 1176.4.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,4,Mod(1,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1176.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1176.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$69.3862461668$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{113}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 28 $ x^2 - x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.81507$ of defining polynomial
Character	$\chi$	$=$	1176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +7.63015 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +7.63015 q^{5} +9.00000 q^{9} -30.8904 q^{11} +42.5206 q^{13} -22.8904 q^{15} -4.36985 q^{17} -27.2603 q^{19} -56.6713 q^{23} -66.7809 q^{25} -27.0000 q^{27} +219.343 q^{29} -172.301 q^{31} +92.6713 q^{33} -31.3426 q^{37} -127.562 q^{39} -274.754 q^{41} +188.000 q^{43} +68.6713 q^{45} -50.0823 q^{47} +13.1096 q^{51} +282.904 q^{53} -235.699 q^{55} +81.7809 q^{57} +64.1647 q^{59} +97.1780 q^{61} +324.438 q^{65} -716.466 q^{67} +170.014 q^{69} +141.986 q^{71} -641.343 q^{73} +200.343 q^{75} +391.590 q^{79} +81.0000 q^{81} -952.165 q^{83} -33.3426 q^{85} -658.028 q^{87} +360.205 q^{89} +516.904 q^{93} -208.000 q^{95} +58.5485 q^{97} -278.014 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 6 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 6 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} - 6 q^{5} + 18 q^{9} + 2 q^{11} + 18 q^{15} - 30 q^{17} - 12 q^{19} + 78 q^{23} - 6 q^{25} - 54 q^{27} + 56 q^{29} - 132 q^{31} - 6 q^{33} + 320 q^{37} - 18 q^{41} + 376 q^{43} - 54 q^{45} + 240 q^{47} + 90 q^{51} - 72 q^{53} - 684 q^{55} + 36 q^{57} - 552 q^{59} + 492 q^{61} + 904 q^{65} - 540 q^{67} - 234 q^{69} + 858 q^{71} - 900 q^{73} + 18 q^{75} - 620 q^{79} + 162 q^{81} - 1224 q^{83} + 316 q^{85} - 168 q^{87} + 1422 q^{89} + 396 q^{93} - 416 q^{95} - 1116 q^{97} + 18 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 6 * q^5 + 18 * q^9 + 2 * q^11 + 18 * q^15 - 30 * q^17 - 12 * q^19 + 78 * q^23 - 6 * q^25 - 54 * q^27 + 56 * q^29 - 132 * q^31 - 6 * q^33 + 320 * q^37 - 18 * q^41 + 376 * q^43 - 54 * q^45 + 240 * q^47 + 90 * q^51 - 72 * q^53 - 684 * q^55 + 36 * q^57 - 552 * q^59 + 492 * q^61 + 904 * q^65 - 540 * q^67 - 234 * q^69 + 858 * q^71 - 900 * q^73 + 18 * q^75 - 620 * q^79 + 162 * q^81 - 1224 * q^83 + 316 * q^85 - 168 * q^87 + 1422 * q^89 + 396 * q^93 - 416 * q^95 - 1116 * q^97 + 18 * 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
$5$	7.63015	0.682461	0.341230	−	0.939980i	$-0.389156\pi$
$5$	7.63015	0.682461	0.341230	+	0.939980i	$0.389156\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−30.8904	−0.846711	−0.423355	−	0.905964i	$-0.639148\pi$
$11$	−30.8904	−0.846711	−0.423355	+	0.905964i	$0.639148\pi$
$12$	0	0
$13$	42.5206	0.907161	0.453580	−	0.891215i	$-0.350147\pi$
$13$	42.5206	0.907161	0.453580	+	0.891215i	$0.350147\pi$
$14$	0	0
$15$	−22.8904	−0.394019
$16$	0	0
$17$	−4.36985	−0.0623438	−0.0311719	−	0.999514i	$-0.509924\pi$
$17$	−4.36985	−0.0623438	−0.0311719	+	0.999514i	$0.509924\pi$
$18$	0	0
$19$	−27.2603	−0.329155	−0.164577	−	0.986364i	$-0.552626\pi$
$19$	−27.2603	−0.329155	−0.164577	+	0.986364i	$0.552626\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−56.6713	−0.513773	−0.256887	−	0.966442i	$-0.582697\pi$
$23$	−56.6713	−0.513773	−0.256887	+	0.966442i	$0.582697\pi$
$24$	0	0
$25$	−66.7809	−0.534247
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	219.343	1.40451	0.702257	−	0.711924i	$-0.252176\pi$
$29$	219.343	1.40451	0.702257	+	0.711924i	$0.252176\pi$
$30$	0	0
$31$	−172.301	−0.998266	−0.499133	−	0.866525i	$-0.666348\pi$
$31$	−172.301	−0.998266	−0.499133	+	0.866525i	$0.666348\pi$
$32$	0	0
$33$	92.6713	0.488849
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−31.3426	−0.139262	−0.0696310	−	0.997573i	$-0.522182\pi$
$37$	−31.3426	−0.139262	−0.0696310	+	0.997573i	$0.522182\pi$
$38$	0	0
$39$	−127.562	−0.523749
$40$	0	0
$41$	−274.754	−1.04657	−0.523284	−	0.852158i	$-0.675293\pi$
$41$	−274.754	−1.04657	−0.523284	+	0.852158i	$0.675293\pi$
$42$	0	0
$43$	188.000	0.666738	0.333369	−	0.942796i	$-0.391815\pi$
$43$	188.000	0.666738	0.333369	+	0.942796i	$0.391815\pi$
$44$	0	0
$45$	68.6713	0.227487
$46$	0	0
$47$	−50.0823	−0.155431	−0.0777155	−	0.996976i	$-0.524763\pi$
$47$	−50.0823	−0.155431	−0.0777155	+	0.996976i	$0.524763\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	13.1096	0.0359942
$52$	0	0
$53$	282.904	0.733206	0.366603	−	0.930378i	$-0.380521\pi$
$53$	282.904	0.733206	0.366603	+	0.930378i	$0.380521\pi$
$54$	0	0
$55$	−235.699	−0.577847
$56$	0	0
$57$	81.7809	0.190038
$58$	0	0
$59$	64.1647	0.141585	0.0707926	−	0.997491i	$-0.477447\pi$
$59$	64.1647	0.141585	0.0707926	+	0.997491i	$0.477447\pi$
$60$	0	0
$61$	97.1780	0.203973	0.101987	−	0.994786i	$-0.467480\pi$
$61$	97.1780	0.203973	0.101987	+	0.994786i	$0.467480\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	324.438	0.619102
$66$	0	0
$67$	−716.466	−1.30642	−0.653211	−	0.757176i	$-0.726579\pi$
$67$	−716.466	−1.30642	−0.653211	+	0.757176i	$0.726579\pi$
$68$	0	0
$69$	170.014	0.296627
$70$	0	0
$71$	141.986	0.237333	0.118667	−	0.992934i	$-0.462138\pi$
$71$	141.986	0.237333	0.118667	+	0.992934i	$0.462138\pi$
$72$	0	0
$73$	−641.343	−1.02827	−0.514133	−	0.857710i	$-0.671886\pi$
$73$	−641.343	−1.02827	−0.514133	+	0.857710i	$0.671886\pi$
$74$	0	0
$75$	200.343	0.308448
$76$	0	0
$77$	0	0
$78$	0	0
$79$	391.590	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$79$	391.590	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−952.165	−1.25920	−0.629600	−	0.776919i	$-0.716781\pi$
$83$	−952.165	−1.25920	−0.629600	+	0.776919i	$0.716781\pi$
$84$	0	0
$85$	−33.3426	−0.0425472
$86$	0	0
$87$	−658.028	−0.810896
$88$	0	0
$89$	360.205	0.429008	0.214504	−	0.976723i	$-0.431187\pi$
$89$	360.205	0.429008	0.214504	+	0.976723i	$0.431187\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	516.904	0.576349
$94$	0	0
$95$	−208.000	−0.224635
$96$	0	0
$97$	58.5485	0.0612855	0.0306428	−	0.999530i	$-0.490245\pi$
$97$	58.5485	0.0612855	0.0306428	+	0.999530i	$0.490245\pi$
$98$	0	0
$99$	−278.014	−0.282237
$100$	0	0
$101$	−1676.67	−1.65183	−0.825916	−	0.563793i	$-0.809341\pi$
$101$	−1676.67	−1.65183	−0.825916	+	0.563793i	$0.809341\pi$
$102$	0	0
$103$	1244.96	1.19097	0.595483	−	0.803368i	$-0.296961\pi$
$103$	1244.96	1.19097	0.595483	+	0.803368i	$0.296961\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	50.9183	0.0460043	0.0230021	−	0.999735i	$-0.492678\pi$
$107$	50.9183	0.0460043	0.0230021	+	0.999735i	$0.492678\pi$
$108$	0	0
$109$	1504.30	1.32189	0.660945	−	0.750434i	$-0.270156\pi$
$109$	1504.30	1.32189	0.660945	+	0.750434i	$0.270156\pi$
$110$	0	0
$111$	94.0279	0.0804030
$112$	0	0
$113$	−526.000	−0.437893	−0.218947	−	0.975737i	$-0.570262\pi$
$113$	−526.000	−0.437893	−0.218947	+	0.975737i	$0.570262\pi$
$114$	0	0
$115$	−432.410	−0.350630
$116$	0	0
$117$	382.685	0.302387
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−376.781	−0.283081
$122$	0	0
$123$	824.261	0.604237
$124$	0	0
$125$	−1463.32	−1.04706
$126$	0	0
$127$	481.343	0.336317	0.168158	−	0.985760i	$-0.446218\pi$
$127$	481.343	0.336317	0.168158	+	0.985760i	$0.446218\pi$
$128$	0	0
$129$	−564.000	−0.384941
$130$	0	0
$131$	−1710.47	−1.14080	−0.570398	−	0.821369i	$-0.693211\pi$
$131$	−1710.47	−1.14080	−0.570398	+	0.821369i	$0.693211\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−206.014	−0.131340
$136$	0	0
$137$	−182.494	−0.113807	−0.0569033	−	0.998380i	$-0.518123\pi$
$137$	−182.494	−0.113807	−0.0569033	+	0.998380i	$0.518123\pi$
$138$	0	0
$139$	−2930.85	−1.78843	−0.894214	−	0.447640i	$-0.852265\pi$
$139$	−2930.85	−1.78843	−0.894214	+	0.447640i	$0.852265\pi$
$140$	0	0
$141$	150.247	0.0897382
$142$	0	0
$143$	−1313.48	−0.768103
$144$	0	0
$145$	1673.62	0.958526
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1655.40	0.910171	0.455086	−	0.890448i	$-0.349609\pi$
$149$	1655.40	0.910171	0.455086	+	0.890448i	$0.349609\pi$
$150$	0	0
$151$	−649.753	−0.350173	−0.175087	−	0.984553i	$-0.556021\pi$
$151$	−649.753	−0.350173	−0.175087	+	0.984553i	$0.556021\pi$
$152$	0	0
$153$	−39.3287	−0.0207813
$154$	0	0
$155$	−1314.69	−0.681278
$156$	0	0
$157$	−3378.11	−1.71721	−0.858607	−	0.512635i	$-0.828670\pi$
$157$	−3378.11	−1.71721	−0.858607	+	0.512635i	$0.828670\pi$
$158$	0	0
$159$	−848.713	−0.423317
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3836.52	−1.84356	−0.921778	−	0.387718i	$-0.873263\pi$
$163$	−3836.52	−1.84356	−0.921778	+	0.387718i	$0.873263\pi$
$164$	0	0
$165$	707.096	0.333620
$166$	0	0
$167$	−3344.60	−1.54978	−0.774890	−	0.632097i	$-0.782195\pi$
$167$	−3344.60	−1.54978	−0.774890	+	0.632097i	$0.782195\pi$
$168$	0	0
$169$	−389.000	−0.177060
$170$	0	0
$171$	−245.343	−0.109718
$172$	0	0
$173$	2941.74	1.29281	0.646406	−	0.762994i	$-0.276271\pi$
$173$	2941.74	1.29281	0.646406	+	0.762994i	$0.276271\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−192.494	−0.0817443
$178$	0	0
$179$	1119.77	0.467572	0.233786	−	0.972288i	$-0.424889\pi$
$179$	1119.77	0.467572	0.233786	+	0.972288i	$0.424889\pi$
$180$	0	0
$181$	220.932	0.0907280	0.0453640	−	0.998971i	$-0.485555\pi$
$181$	220.932	0.0907280	0.0453640	+	0.998971i	$0.485555\pi$
$182$	0	0
$183$	−291.534	−0.117764
$184$	0	0
$185$	−239.149	−0.0950409
$186$	0	0
$187$	134.987	0.0527872
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1129.19	−0.427778	−0.213889	−	0.976858i	$-0.568613\pi$
$191$	−1129.19	−0.427778	−0.213889	+	0.976858i	$0.568613\pi$
$192$	0	0
$193$	−4340.00	−1.61865	−0.809327	−	0.587358i	$-0.800168\pi$
$193$	−4340.00	−1.61865	−0.809327	+	0.587358i	$0.800168\pi$
$194$	0	0
$195$	−973.315	−0.357439
$196$	0	0
$197$	−2891.40	−1.04570	−0.522852	−	0.852424i	$-0.675132\pi$
$197$	−2891.40	−1.04570	−0.522852	+	0.852424i	$0.675132\pi$
$198$	0	0
$199$	−1337.75	−0.476537	−0.238268	−	0.971199i	$-0.576580\pi$
$199$	−1337.75	−0.476537	−0.238268	+	0.971199i	$0.576580\pi$
$200$	0	0
$201$	2149.40	0.754263
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2096.41	−0.714242
$206$	0	0
$207$	−510.042	−0.171258
$208$	0	0
$209$	842.082	0.278699
$210$	0	0
$211$	745.426	0.243210	0.121605	−	0.992579i	$-0.461196\pi$
$211$	745.426	0.243210	0.121605	+	0.992579i	$0.461196\pi$
$212$	0	0
$213$	−425.958	−0.137024
$214$	0	0
$215$	1434.47	0.455023
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1924.03	0.593670
$220$	0	0
$221$	−185.809	−0.0565559
$222$	0	0
$223$	−3172.99	−0.952820	−0.476410	−	0.879223i	$-0.658062\pi$
$223$	−3172.99	−0.952820	−0.476410	+	0.879223i	$0.658062\pi$
$224$	0	0
$225$	−601.028	−0.178082
$226$	0	0
$227$	−5304.66	−1.55103	−0.775513	−	0.631332i	$-0.782509\pi$
$227$	−5304.66	−1.55103	−0.775513	+	0.631332i	$0.782509\pi$
$228$	0	0
$229$	541.642	0.156300	0.0781500	−	0.996942i	$-0.475099\pi$
$229$	541.642	0.156300	0.0781500	+	0.996942i	$0.475099\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2048.41	−0.575949	−0.287974	−	0.957638i	$-0.592982\pi$
$233$	−2048.41	−0.575949	−0.287974	+	0.957638i	$0.592982\pi$
$234$	0	0
$235$	−382.136	−0.106076
$236$	0	0
$237$	−1174.77	−0.321981
$238$	0	0
$239$	3540.81	0.958310	0.479155	−	0.877730i	$-0.340943\pi$
$239$	3540.81	0.958310	0.479155	+	0.877730i	$0.340943\pi$
$240$	0	0
$241$	−5192.91	−1.38799	−0.693993	−	0.719982i	$-0.744150\pi$
$241$	−5192.91	−1.38799	−0.693993	+	0.719982i	$0.744150\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1159.12	−0.298596
$248$	0	0
$249$	2856.49	0.727000
$250$	0	0
$251$	−3333.97	−0.838400	−0.419200	−	0.907894i	$-0.637689\pi$
$251$	−3333.97	−0.838400	−0.419200	+	0.907894i	$0.637689\pi$
$252$	0	0
$253$	1750.60	0.435017
$254$	0	0
$255$	100.028	0.0245647
$256$	0	0
$257$	3788.75	0.919595	0.459798	−	0.888024i	$-0.347922\pi$
$257$	3788.75	0.919595	0.459798	+	0.888024i	$0.347922\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1974.08	0.468171
$262$	0	0
$263$	4236.75	0.993344	0.496672	−	0.867938i	$-0.334555\pi$
$263$	4236.75	0.993344	0.496672	+	0.867938i	$0.334555\pi$
$264$	0	0
$265$	2158.60	0.500384
$266$	0	0
$267$	−1080.62	−0.247688
$268$	0	0
$269$	−3167.44	−0.717926	−0.358963	−	0.933352i	$-0.616870\pi$
$269$	−3167.44	−0.717926	−0.358963	+	0.933352i	$0.616870\pi$
$270$	0	0
$271$	−4253.84	−0.953514	−0.476757	−	0.879035i	$-0.658188\pi$
$271$	−4253.84	−0.953514	−0.476757	+	0.879035i	$0.658188\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2062.89	0.452353
$276$	0	0
$277$	−2103.67	−0.456308	−0.228154	−	0.973625i	$-0.573269\pi$
$277$	−2103.67	−0.456308	−0.228154	+	0.973625i	$0.573269\pi$
$278$	0	0
$279$	−1550.71	−0.332755
$280$	0	0
$281$	1518.99	0.322474	0.161237	−	0.986916i	$-0.448452\pi$
$281$	1518.99	0.322474	0.161237	+	0.986916i	$0.448452\pi$
$282$	0	0
$283$	4359.65	0.915739	0.457870	−	0.889019i	$-0.348613\pi$
$283$	4359.65	0.915739	0.457870	+	0.889019i	$0.348613\pi$
$284$	0	0
$285$	624.000	0.129693
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4893.90	−0.996113
$290$	0	0
$291$	−175.645	−0.0353832
$292$	0	0
$293$	7845.69	1.56433	0.782167	−	0.623068i	$-0.214114\pi$
$293$	7845.69	1.56433	0.782167	+	0.623068i	$0.214114\pi$
$294$	0	0
$295$	489.586	0.0966264
$296$	0	0
$297$	834.042	0.162950
$298$	0	0
$299$	−2409.70	−0.466075
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5030.01	0.953686
$304$	0	0
$305$	741.482	0.139204
$306$	0	0
$307$	−7944.25	−1.47688	−0.738440	−	0.674319i	$-0.764437\pi$
$307$	−7944.25	−1.47688	−0.738440	+	0.674319i	$0.764437\pi$
$308$	0	0
$309$	−3734.88	−0.687605
$310$	0	0
$311$	5958.69	1.08645	0.543225	−	0.839587i	$-0.317203\pi$
$311$	5958.69	1.08645	0.543225	+	0.839587i	$0.317203\pi$
$312$	0	0
$313$	−361.976	−0.0653677	−0.0326839	−	0.999466i	$-0.510405\pi$
$313$	−361.976	−0.0653677	−0.0326839	+	0.999466i	$0.510405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8210.31	1.45469	0.727345	−	0.686272i	$-0.240754\pi$
$317$	8210.31	1.45469	0.727345	+	0.686272i	$0.240754\pi$
$318$	0	0
$319$	−6775.59	−1.18922
$320$	0	0
$321$	−152.755	−0.0265606
$322$	0	0
$323$	119.123	0.0205208
$324$	0	0
$325$	−2839.56	−0.484648
$326$	0	0
$327$	−4512.91	−0.763194
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1330.96	−0.221015	−0.110508	−	0.993875i	$-0.535248\pi$
$331$	−1330.96	−0.221015	−0.110508	+	0.993875i	$0.535248\pi$
$332$	0	0
$333$	−282.084	−0.0464207
$334$	0	0
$335$	−5466.74	−0.891582
$336$	0	0
$337$	10144.0	1.63970	0.819848	−	0.572581i	$-0.194058\pi$
$337$	10144.0	1.63970	0.819848	+	0.572581i	$0.194058\pi$
$338$	0	0
$339$	1578.00	0.252818
$340$	0	0
$341$	5322.47	0.845243
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1297.23	0.202436
$346$	0	0
$347$	8030.95	1.24243	0.621216	−	0.783640i	$-0.286639\pi$
$347$	8030.95	1.24243	0.621216	+	0.783640i	$0.286639\pi$
$348$	0	0
$349$	2491.48	0.382137	0.191068	−	0.981577i	$-0.438805\pi$
$349$	2491.48	0.382137	0.191068	+	0.981577i	$0.438805\pi$
$350$	0	0
$351$	−1148.06	−0.174583
$352$	0	0
$353$	−3380.65	−0.509727	−0.254864	−	0.966977i	$-0.582031\pi$
$353$	−3380.65	−0.509727	−0.254864	+	0.966977i	$0.582031\pi$
$354$	0	0
$355$	1083.37	0.161971
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12659.6	1.86114	0.930571	−	0.366111i	$-0.119311\pi$
$359$	12659.6	1.86114	0.930571	+	0.366111i	$0.119311\pi$
$360$	0	0
$361$	−6115.88	−0.891657
$362$	0	0
$363$	1130.34	0.163437
$364$	0	0
$365$	−4893.54	−0.701752
$366$	0	0
$367$	3820.27	0.543369	0.271685	−	0.962386i	$-0.412419\pi$
$367$	3820.27	0.543369	0.271685	+	0.962386i	$0.412419\pi$
$368$	0	0
$369$	−2472.78	−0.348856
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4186.65	−0.581171	−0.290585	−	0.956849i	$-0.593850\pi$
$373$	−4186.65	−0.581171	−0.290585	+	0.956849i	$0.593850\pi$
$374$	0	0
$375$	4389.95	0.604523
$376$	0	0
$377$	9326.58	1.27412
$378$	0	0
$379$	−7653.27	−1.03726	−0.518630	−	0.854998i	$-0.673558\pi$
$379$	−7653.27	−1.03726	−0.518630	+	0.854998i	$0.673558\pi$
$380$	0	0
$381$	−1444.03	−0.194173
$382$	0	0
$383$	−5774.85	−0.770447	−0.385223	−	0.922823i	$-0.625876\pi$
$383$	−5774.85	−0.770447	−0.385223	+	0.922823i	$0.625876\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1692.00	0.222246
$388$	0	0
$389$	−1429.42	−0.186310	−0.0931550	−	0.995652i	$-0.529695\pi$
$389$	−1429.42	−0.186310	−0.0931550	+	0.995652i	$0.529695\pi$
$390$	0	0
$391$	247.645	0.0320306
$392$	0	0
$393$	5131.40	0.658639
$394$	0	0
$395$	2987.89	0.380600
$396$	0	0
$397$	9709.57	1.22748	0.613739	−	0.789509i	$-0.289665\pi$
$397$	9709.57	1.22748	0.613739	+	0.789509i	$0.289665\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	534.654	0.0665819	0.0332909	−	0.999446i	$-0.489401\pi$
$401$	534.654	0.0665819	0.0332909	+	0.999446i	$0.489401\pi$
$402$	0	0
$403$	−7326.36	−0.905588
$404$	0	0
$405$	618.042	0.0758290
$406$	0	0
$407$	968.187	0.117915
$408$	0	0
$409$	13964.5	1.68826	0.844131	−	0.536138i	$-0.180117\pi$
$409$	13964.5	1.68826	0.844131	+	0.536138i	$0.180117\pi$
$410$	0	0
$411$	547.482	0.0657063
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7265.16	−0.859355
$416$	0	0
$417$	8792.55	1.03255
$418$	0	0
$419$	11475.1	1.33793	0.668967	−	0.743292i	$-0.266737\pi$
$419$	11475.1	1.33793	0.668967	+	0.743292i	$0.266737\pi$
$420$	0	0
$421$	12518.8	1.44924	0.724619	−	0.689150i	$-0.242016\pi$
$421$	12518.8	1.44924	0.724619	+	0.689150i	$0.242016\pi$
$422$	0	0
$423$	−450.741	−0.0518103
$424$	0	0
$425$	291.823	0.0333070
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3940.44	0.443464
$430$	0	0
$431$	−385.528	−0.0430863	−0.0215432	−	0.999768i	$-0.506858\pi$
$431$	−385.528	−0.0430863	−0.0215432	+	0.999768i	$0.506858\pi$
$432$	0	0
$433$	15640.4	1.73586	0.867932	−	0.496683i	$-0.165449\pi$
$433$	15640.4	1.73586	0.867932	+	0.496683i	$0.165449\pi$
$434$	0	0
$435$	−5020.85	−0.553405
$436$	0	0
$437$	1544.88	0.169111
$438$	0	0
$439$	−9921.04	−1.07860	−0.539300	−	0.842114i	$-0.681311\pi$
$439$	−9921.04	−1.07860	−0.539300	+	0.842114i	$0.681311\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5660.07	−0.607039	−0.303519	−	0.952825i	$-0.598162\pi$
$443$	−5660.07	−0.607039	−0.303519	+	0.952825i	$0.598162\pi$
$444$	0	0
$445$	2748.42	0.292781
$446$	0	0
$447$	−4966.20	−0.525488
$448$	0	0
$449$	−3111.47	−0.327037	−0.163518	−	0.986540i	$-0.552284\pi$
$449$	−3111.47	−0.327037	−0.163518	+	0.986540i	$0.552284\pi$
$450$	0	0
$451$	8487.26	0.886141
$452$	0	0
$453$	1949.26	0.202173
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16305.0	−1.66896	−0.834479	−	0.551039i	$-0.814231\pi$
$457$	−16305.0	−1.66896	−0.834479	+	0.551039i	$0.814231\pi$
$458$	0	0
$459$	117.986	0.0119981
$460$	0	0
$461$	−1726.54	−0.174432	−0.0872159	−	0.996189i	$-0.527797\pi$
$461$	−1726.54	−0.174432	−0.0872159	+	0.996189i	$0.527797\pi$
$462$	0	0
$463$	15118.3	1.51751	0.758757	−	0.651374i	$-0.225807\pi$
$463$	15118.3	1.51751	0.758757	+	0.651374i	$0.225807\pi$
$464$	0	0
$465$	3944.06	0.393336
$466$	0	0
$467$	−8736.49	−0.865689	−0.432844	−	0.901469i	$-0.642490\pi$
$467$	−8736.49	−0.865689	−0.432844	+	0.901469i	$0.642490\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	10134.3	0.991434
$472$	0	0
$473$	−5807.40	−0.564534
$474$	0	0
$475$	1820.47	0.175850
$476$	0	0
$477$	2546.14	0.244402
$478$	0	0
$479$	13085.3	1.24819	0.624096	−	0.781348i	$-0.285468\pi$
$479$	13085.3	1.24819	0.624096	+	0.781348i	$0.285468\pi$
$480$	0	0
$481$	−1332.71	−0.126333
$482$	0	0
$483$	0	0
$484$	0	0
$485$	446.733	0.0418250
$486$	0	0
$487$	−13941.5	−1.29723	−0.648615	−	0.761116i	$-0.724652\pi$
$487$	−13941.5	−1.29723	−0.648615	+	0.761116i	$0.724652\pi$
$488$	0	0
$489$	11509.6	1.06438
$490$	0	0
$491$	−6613.66	−0.607882	−0.303941	−	0.952691i	$-0.598303\pi$
$491$	−6613.66	−0.607882	−0.303941	+	0.952691i	$0.598303\pi$
$492$	0	0
$493$	−958.495	−0.0875628
$494$	0	0
$495$	−2121.29	−0.192616
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−11315.0	−1.01508	−0.507542	−	0.861627i	$-0.669446\pi$
$499$	−11315.0	−1.01508	−0.507542	+	0.861627i	$0.669446\pi$
$500$	0	0
$501$	10033.8	0.894765
$502$	0	0
$503$	−3026.86	−0.268312	−0.134156	−	0.990960i	$-0.542832\pi$
$503$	−3026.86	−0.268312	−0.134156	+	0.990960i	$0.542832\pi$
$504$	0	0
$505$	−12793.2	−1.12731
$506$	0	0
$507$	1167.00	0.102225
$508$	0	0
$509$	−1890.35	−0.164613	−0.0823066	−	0.996607i	$-0.526229\pi$
$509$	−1890.35	−0.164613	−0.0823066	+	0.996607i	$0.526229\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	736.028	0.0633459
$514$	0	0
$515$	9499.23	0.812788
$516$	0	0
$517$	1547.07	0.131605
$518$	0	0
$519$	−8825.22	−0.746405
$520$	0	0
$521$	14498.9	1.21921	0.609603	−	0.792707i	$-0.291329\pi$
$521$	14498.9	1.21921	0.609603	+	0.792707i	$0.291329\pi$
$522$	0	0
$523$	−20164.2	−1.68589	−0.842945	−	0.538000i	$-0.819180\pi$
$523$	−20164.2	−1.68589	−0.842945	+	0.538000i	$0.819180\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	752.932	0.0622358
$528$	0	0
$529$	−8955.36	−0.736037
$530$	0	0
$531$	577.482	0.0471951
$532$	0	0
$533$	−11682.7	−0.949406
$534$	0	0
$535$	388.514	0.0313961
$536$	0	0
$537$	−3359.30	−0.269953
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22800.2	−1.81194	−0.905969	−	0.423344i	$-0.860856\pi$
$541$	−22800.2	−1.81194	−0.905969	+	0.423344i	$0.860856\pi$
$542$	0	0
$543$	−662.797	−0.0523818
$544$	0	0
$545$	11478.0	0.902139
$546$	0	0
$547$	4682.34	0.366001	0.183000	−	0.983113i	$-0.441419\pi$
$547$	4682.34	0.366001	0.183000	+	0.983113i	$0.441419\pi$
$548$	0	0
$549$	874.602	0.0679911
$550$	0	0
$551$	−5979.34	−0.462302
$552$	0	0
$553$	0	0
$554$	0	0
$555$	717.446	0.0548719
$556$	0	0
$557$	−5113.81	−0.389011	−0.194506	−	0.980901i	$-0.562310\pi$
$557$	−5113.81	−0.389011	−0.194506	+	0.980901i	$0.562310\pi$
$558$	0	0
$559$	7993.87	0.604838
$560$	0	0
$561$	−404.960	−0.0304767
$562$	0	0
$563$	17129.9	1.28231	0.641155	−	0.767412i	$-0.278456\pi$
$563$	17129.9	1.28231	0.641155	+	0.767412i	$0.278456\pi$
$564$	0	0
$565$	−4013.46	−0.298845
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20904.4	1.54017	0.770087	−	0.637939i	$-0.220213\pi$
$569$	20904.4	1.54017	0.770087	+	0.637939i	$0.220213\pi$
$570$	0	0
$571$	−8893.18	−0.651782	−0.325891	−	0.945407i	$-0.605664\pi$
$571$	−8893.18	−0.651782	−0.325891	+	0.945407i	$0.605664\pi$
$572$	0	0
$573$	3387.58	0.246978
$574$	0	0
$575$	3784.56	0.274482
$576$	0	0
$577$	9556.10	0.689473	0.344736	−	0.938700i	$-0.387968\pi$
$577$	9556.10	0.689473	0.344736	+	0.938700i	$0.387968\pi$
$578$	0	0
$579$	13020.0	0.934531
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−8739.04	−0.620813
$584$	0	0
$585$	2919.94	0.206367
$586$	0	0
$587$	−4694.14	−0.330065	−0.165032	−	0.986288i	$-0.552773\pi$
$587$	−4694.14	−0.330065	−0.165032	+	0.986288i	$0.552773\pi$
$588$	0	0
$589$	4696.99	0.328584
$590$	0	0
$591$	8674.20	0.603737
$592$	0	0
$593$	−10021.3	−0.693973	−0.346986	−	0.937870i	$-0.612795\pi$
$593$	−10021.3	−0.693973	−0.346986	+	0.937870i	$0.612795\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4013.26	0.275129
$598$	0	0
$599$	18938.3	1.29182	0.645909	−	0.763414i	$-0.276478\pi$
$599$	18938.3	1.29182	0.645909	+	0.763414i	$0.276478\pi$
$600$	0	0
$601$	17991.2	1.22109	0.610546	−	0.791981i	$-0.290950\pi$
$601$	17991.2	1.22109	0.610546	+	0.791981i	$0.290950\pi$
$602$	0	0
$603$	−6448.20	−0.435474
$604$	0	0
$605$	−2874.89	−0.193192
$606$	0	0
$607$	17372.0	1.16163	0.580814	−	0.814036i	$-0.302734\pi$
$607$	17372.0	1.16163	0.580814	+	0.814036i	$0.302734\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2129.53	−0.141001
$612$	0	0
$613$	−655.976	−0.0432212	−0.0216106	−	0.999766i	$-0.506879\pi$
$613$	−655.976	−0.0432212	−0.0216106	+	0.999766i	$0.506879\pi$
$614$	0	0
$615$	6289.23	0.412368
$616$	0	0
$617$	−13669.6	−0.891922	−0.445961	−	0.895052i	$-0.647138\pi$
$617$	−13669.6	−0.891922	−0.445961	+	0.895052i	$0.647138\pi$
$618$	0	0
$619$	−719.461	−0.0467166	−0.0233583	−	0.999727i	$-0.507436\pi$
$619$	−719.461	−0.0467166	−0.0233583	+	0.999727i	$0.507436\pi$
$620$	0	0
$621$	1530.13	0.0988757
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−2817.71	−0.180333
$626$	0	0
$627$	−2526.25	−0.160907
$628$	0	0
$629$	136.963	0.00868213
$630$	0	0
$631$	−1430.12	−0.0902250	−0.0451125	−	0.998982i	$-0.514365\pi$
$631$	−1430.12	−0.0902250	−0.0451125	+	0.998982i	$0.514365\pi$
$632$	0	0
$633$	−2236.28	−0.140417
$634$	0	0
$635$	3672.71	0.229523
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1277.87	0.0791110
$640$	0	0
$641$	12079.4	0.744316	0.372158	−	0.928169i	$-0.378618\pi$
$641$	12079.4	0.744316	0.372158	+	0.928169i	$0.378618\pi$
$642$	0	0
$643$	5406.82	0.331608	0.165804	−	0.986159i	$-0.446978\pi$
$643$	5406.82	0.331608	0.165804	+	0.986159i	$0.446978\pi$
$644$	0	0
$645$	−4303.40	−0.262707
$646$	0	0
$647$	4465.81	0.271359	0.135679	−	0.990753i	$-0.456678\pi$
$647$	4465.81	0.271359	0.135679	+	0.990753i	$0.456678\pi$
$648$	0	0
$649$	−1982.07	−0.119882
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14362.6	0.860725	0.430362	−	0.902656i	$-0.358386\pi$
$653$	14362.6	0.860725	0.430362	+	0.902656i	$0.358386\pi$
$654$	0	0
$655$	−13051.1	−0.778549
$656$	0	0
$657$	−5772.08	−0.342756
$658$	0	0
$659$	16057.0	0.949151	0.474575	−	0.880215i	$-0.342602\pi$
$659$	16057.0	0.949151	0.474575	+	0.880215i	$0.342602\pi$
$660$	0	0
$661$	7872.21	0.463227	0.231614	−	0.972808i	$-0.425599\pi$
$661$	7872.21	0.463227	0.231614	+	0.972808i	$0.425599\pi$
$662$	0	0
$663$	557.426	0.0326526
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12430.4	−0.721602
$668$	0	0
$669$	9518.96	0.550111
$670$	0	0
$671$	−3001.87	−0.172706
$672$	0	0
$673$	4037.09	0.231231	0.115615	−	0.993294i	$-0.463116\pi$
$673$	4037.09	0.231231	0.115615	+	0.993294i	$0.463116\pi$
$674$	0	0
$675$	1803.08	0.102816
$676$	0	0
$677$	−2755.36	−0.156421	−0.0782107	−	0.996937i	$-0.524921\pi$
$677$	−2755.36	−0.156421	−0.0782107	+	0.996937i	$0.524921\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	15914.0	0.895485
$682$	0	0
$683$	35181.3	1.97097	0.985487	−	0.169749i	$-0.0542955\pi$
$683$	35181.3	1.97097	0.985487	+	0.169749i	$0.0542955\pi$
$684$	0	0
$685$	−1392.46	−0.0776686
$686$	0	0
$687$	−1624.92	−0.0902398
$688$	0	0
$689$	12029.3	0.665135
$690$	0	0
$691$	−17204.7	−0.947173	−0.473587	−	0.880747i	$-0.657041\pi$
$691$	−17204.7	−0.947173	−0.473587	+	0.880747i	$0.657041\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−22362.8	−1.22053
$696$	0	0
$697$	1200.63	0.0652471
$698$	0	0
$699$	6145.24	0.332524
$700$	0	0
$701$	−3882.63	−0.209194	−0.104597	−	0.994515i	$-0.533355\pi$
$701$	−3882.63	−0.209194	−0.104597	+	0.994515i	$0.533355\pi$
$702$	0	0
$703$	854.409	0.0458388
$704$	0	0
$705$	1146.41	0.0612428
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−907.203	−0.0480546	−0.0240273	−	0.999711i	$-0.507649\pi$
$709$	−907.203	−0.0480546	−0.0240273	+	0.999711i	$0.507649\pi$
$710$	0	0
$711$	3524.31	0.185896
$712$	0	0
$713$	9764.55	0.512883
$714$	0	0
$715$	−10022.0	−0.524200
$716$	0	0
$717$	−10622.4	−0.553280
$718$	0	0
$719$	−3524.99	−0.182837	−0.0914186	−	0.995813i	$-0.529140\pi$
$719$	−3524.99	−0.182837	−0.0914186	+	0.995813i	$0.529140\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	15578.7	0.801354
$724$	0	0
$725$	−14647.9	−0.750357
$726$	0	0
$727$	30342.1	1.54790	0.773951	−	0.633245i	$-0.218277\pi$
$727$	30342.1	1.54790	0.773951	+	0.633245i	$0.218277\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−821.533	−0.0415670
$732$	0	0
$733$	21254.2	1.07100	0.535498	−	0.844536i	$-0.320124\pi$
$733$	21254.2	1.07100	0.535498	+	0.844536i	$0.320124\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22132.0	1.10616
$738$	0	0
$739$	−23838.2	−1.18660	−0.593302	−	0.804980i	$-0.702176\pi$
$739$	−23838.2	−1.18660	−0.593302	+	0.804980i	$0.702176\pi$
$740$	0	0
$741$	3477.37	0.172395
$742$	0	0
$743$	23378.5	1.15434	0.577169	−	0.816625i	$-0.304157\pi$
$743$	23378.5	1.15434	0.577169	+	0.816625i	$0.304157\pi$
$744$	0	0
$745$	12630.9	0.621156
$746$	0	0
$747$	−8569.48	−0.419734
$748$	0	0
$749$	0	0
$750$	0	0
$751$	13961.8	0.678392	0.339196	−	0.940716i	$-0.389845\pi$
$751$	13961.8	0.678392	0.339196	+	0.940716i	$0.389845\pi$
$752$	0	0
$753$	10001.9	0.484050
$754$	0	0
$755$	−4957.71	−0.238980
$756$	0	0
$757$	19381.1	0.930541	0.465270	−	0.885169i	$-0.345957\pi$
$757$	19381.1	0.930541	0.465270	+	0.885169i	$0.345957\pi$
$758$	0	0
$759$	−5251.80	−0.251157
$760$	0	0
$761$	−22136.5	−1.05446	−0.527231	−	0.849722i	$-0.676770\pi$
$761$	−22136.5	−1.05446	−0.527231	+	0.849722i	$0.676770\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−300.084	−0.0141824
$766$	0	0
$767$	2728.32	0.128441
$768$	0	0
$769$	−32106.6	−1.50558	−0.752792	−	0.658259i	$-0.771293\pi$
$769$	−32106.6	−1.50558	−0.752792	+	0.658259i	$0.771293\pi$
$770$	0	0
$771$	−11366.3	−0.530929
$772$	0	0
$773$	−14991.2	−0.697536	−0.348768	−	0.937209i	$-0.613400\pi$
$773$	−14991.2	−0.697536	−0.348768	+	0.937209i	$0.613400\pi$
$774$	0	0
$775$	11506.4	0.533321
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7489.86	0.344483
$780$	0	0
$781$	−4386.01	−0.200952
$782$	0	0
$783$	−5922.25	−0.270299
$784$	0	0
$785$	−25775.5	−1.17193
$786$	0	0
$787$	356.725	0.0161574	0.00807869	−	0.999967i	$-0.497428\pi$
$787$	356.725	0.0161574	0.00807869	+	0.999967i	$0.497428\pi$
$788$	0	0
$789$	−12710.3	−0.573507
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4132.06	0.185036
$794$	0	0
$795$	−6475.80	−0.288897
$796$	0	0
$797$	−43780.4	−1.94577	−0.972887	−	0.231282i	$-0.925708\pi$
$797$	−43780.4	−1.94577	−0.972887	+	0.231282i	$0.925708\pi$
$798$	0	0
$799$	218.852	0.00969017
$800$	0	0
$801$	3241.85	0.143003
$802$	0	0
$803$	19811.4	0.870644
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9502.31	0.414495
$808$	0	0
$809$	−33995.6	−1.47741	−0.738703	−	0.674031i	$-0.764561\pi$
$809$	−33995.6	−1.47741	−0.738703	+	0.674031i	$0.764561\pi$
$810$	0	0
$811$	−21542.4	−0.932746	−0.466373	−	0.884588i	$-0.654440\pi$
$811$	−21542.4	−0.932746	−0.466373	+	0.884588i	$0.654440\pi$
$812$	0	0
$813$	12761.5	0.550511
$814$	0	0
$815$	−29273.2	−1.25815
$816$	0	0
$817$	−5124.93	−0.219460
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.0244	0.00140385	0.000701924	−	1.00000i	$-0.499777\pi$
$821$	33.0244	0.00140385	0.000701924		1.00000i	$0.499777\pi$
$822$	0	0
$823$	−12081.7	−0.511714	−0.255857	−	0.966715i	$-0.582358\pi$
$823$	−12081.7	−0.511714	−0.255857	+	0.966715i	$0.582358\pi$
$824$	0	0
$825$	−6188.67	−0.261166
$826$	0	0
$827$	1047.62	0.0440500	0.0220250	−	0.999757i	$-0.492989\pi$
$827$	1047.62	0.0440500	0.0220250	+	0.999757i	$0.492989\pi$
$828$	0	0
$829$	−32458.6	−1.35987	−0.679937	−	0.733271i	$-0.737993\pi$
$829$	−32458.6	−1.35987	−0.679937	+	0.733271i	$0.737993\pi$
$830$	0	0
$831$	6311.01	0.263449
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−25519.8	−1.05766
$836$	0	0
$837$	4652.14	0.192116
$838$	0	0
$839$	24735.2	1.01783	0.508913	−	0.860818i	$-0.330048\pi$
$839$	24735.2	1.01783	0.508913	+	0.860818i	$0.330048\pi$
$840$	0	0
$841$	23722.2	0.972659
$842$	0	0
$843$	−4556.96	−0.186181
$844$	0	0
$845$	−2968.13	−0.120836
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−13078.9	−0.528702
$850$	0	0
$851$	1776.23	0.0715491
$852$	0	0
$853$	−42545.4	−1.70777	−0.853883	−	0.520464i	$-0.825759\pi$
$853$	−42545.4	−1.70777	−0.853883	+	0.520464i	$0.825759\pi$
$854$	0	0
$855$	−1872.00	−0.0748784
$856$	0	0
$857$	30570.7	1.21853	0.609263	−	0.792968i	$-0.291465\pi$
$857$	30570.7	1.21853	0.609263	+	0.792968i	$0.291465\pi$
$858$	0	0
$859$	11588.6	0.460302	0.230151	−	0.973155i	$-0.426078\pi$
$859$	11588.6	0.460302	0.230151	+	0.973155i	$0.426078\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10547.9	0.416054	0.208027	−	0.978123i	$-0.433296\pi$
$863$	10547.9	0.416054	0.208027	+	0.978123i	$0.433296\pi$
$864$	0	0
$865$	22445.9	0.882293
$866$	0	0
$867$	14681.7	0.575106
$868$	0	0
$869$	−12096.4	−0.472200
$870$	0	0
$871$	−30464.6	−1.18513
$872$	0	0
$873$	526.936	0.0204285
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18678.9	−0.719203	−0.359602	−	0.933106i	$-0.617087\pi$
$877$	−18678.9	−0.719203	−0.359602	+	0.933106i	$0.617087\pi$
$878$	0	0
$879$	−23537.1	−0.903169
$880$	0	0
$881$	19726.8	0.754386	0.377193	−	0.926135i	$-0.376889\pi$
$881$	19726.8	0.754386	0.377193	+	0.926135i	$0.376889\pi$
$882$	0	0
$883$	27468.4	1.04687	0.523435	−	0.852066i	$-0.324650\pi$
$883$	27468.4	1.04687	0.523435	+	0.852066i	$0.324650\pi$
$884$	0	0
$885$	−1468.76	−0.0557873
$886$	0	0
$887$	13657.6	0.516997	0.258499	−	0.966012i	$-0.416772\pi$
$887$	13657.6	0.516997	0.258499	+	0.966012i	$0.416772\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2502.13	−0.0940790
$892$	0	0
$893$	1365.26	0.0511609
$894$	0	0
$895$	8543.99	0.319099
$896$	0	0
$897$	7229.09	0.269088
$898$	0	0
$899$	−37793.1	−1.40208
$900$	0	0
$901$	−1236.25	−0.0457109
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1685.75	0.0619183
$906$	0	0
$907$	−13066.6	−0.478356	−0.239178	−	0.970976i	$-0.576878\pi$
$907$	−13066.6	−0.478356	−0.239178	+	0.970976i	$0.576878\pi$
$908$	0	0
$909$	−15090.0	−0.550611
$910$	0	0
$911$	42088.8	1.53070	0.765348	−	0.643617i	$-0.222567\pi$
$911$	42088.8	1.53070	0.765348	+	0.643617i	$0.222567\pi$
$912$	0	0
$913$	29412.8	1.06618
$914$	0	0
$915$	−2224.45	−0.0803693
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4464.29	−0.160243	−0.0801216	−	0.996785i	$-0.525531\pi$
$919$	−4464.29	−0.160243	−0.0801216	+	0.996785i	$0.525531\pi$
$920$	0	0
$921$	23832.8	0.852677
$922$	0	0
$923$	6037.33	0.215299
$924$	0	0
$925$	2093.09	0.0744003
$926$	0	0
$927$	11204.6	0.396989
$928$	0	0
$929$	5537.87	0.195578	0.0977888	−	0.995207i	$-0.468823\pi$
$929$	5537.87	0.195578	0.0977888	+	0.995207i	$0.468823\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−17876.1	−0.627263
$934$	0	0
$935$	1029.97	0.0360252
$936$	0	0
$937$	14375.3	0.501194	0.250597	−	0.968091i	$-0.419373\pi$
$937$	14375.3	0.501194	0.250597	+	0.968091i	$0.419373\pi$
$938$	0	0
$939$	1085.93	0.0377401
$940$	0	0
$941$	−54815.5	−1.89897	−0.949487	−	0.313806i	$-0.898396\pi$
$941$	−54815.5	−1.89897	−0.949487	+	0.313806i	$0.898396\pi$
$942$	0	0
$943$	15570.6	0.537699
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24132.1	0.828077	0.414039	−	0.910259i	$-0.364118\pi$
$947$	24132.1	0.828077	0.414039	+	0.910259i	$0.364118\pi$
$948$	0	0
$949$	−27270.3	−0.932803
$950$	0	0
$951$	−24630.9	−0.839866
$952$	0	0
$953$	−13438.4	−0.456782	−0.228391	−	0.973569i	$-0.573346\pi$
$953$	−13438.4	−0.456782	−0.228391	+	0.973569i	$0.573346\pi$
$954$	0	0
$955$	−8615.91	−0.291942
$956$	0	0
$957$	20326.8	0.686595
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−103.208	−0.00346439
$962$	0	0
$963$	458.265	0.0153348
$964$	0	0
$965$	−33114.9	−1.10467
$966$	0	0
$967$	−43033.7	−1.43110	−0.715549	−	0.698562i	$-0.753823\pi$
$967$	−43033.7	−1.43110	−0.715549	+	0.698562i	$0.753823\pi$
$968$	0	0
$969$	−357.370	−0.0118477
$970$	0	0
$971$	56125.9	1.85496	0.927481	−	0.373871i	$-0.121970\pi$
$971$	56125.9	1.85496	0.927481	+	0.373871i	$0.121970\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	8518.69	0.279812
$976$	0	0
$977$	−12329.5	−0.403742	−0.201871	−	0.979412i	$-0.564702\pi$
$977$	−12329.5	−0.403742	−0.201871	+	0.979412i	$0.564702\pi$
$978$	0	0
$979$	−11126.9	−0.363245
$980$	0	0
$981$	13538.7	0.440630
$982$	0	0
$983$	−16982.8	−0.551036	−0.275518	−	0.961296i	$-0.588849\pi$
$983$	−16982.8	−0.551036	−0.275518	+	0.961296i	$0.588849\pi$
$984$	0	0
$985$	−22061.8	−0.713652
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10654.2	−0.342552
$990$	0	0
$991$	26256.6	0.841644	0.420822	−	0.907143i	$-0.361742\pi$
$991$	26256.6	0.841644	0.420822	+	0.907143i	$0.361742\pi$
$992$	0	0
$993$	3992.87	0.127603
$994$	0	0
$995$	−10207.3	−0.325218
$996$	0	0
$997$	−51375.2	−1.63196	−0.815982	−	0.578077i	$-0.803803\pi$
$997$	−51375.2	−1.63196	−0.815982	+	0.578077i	$0.803803\pi$
$998$	0	0
$999$	846.251	0.0268010

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.4.a.q.1.2	✓	2
4.3	odd	2		2352.4.a.by.1.2		2
7.6	odd	2		1176.4.a.v.1.1	yes	2
28.27	even	2		2352.4.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.q.1.2	✓	2	1.1	even	1	trivial
1176.4.a.v.1.1	yes	2	7.6	odd	2
2352.4.a.bs.1.1		2	28.27	even	2
2352.4.a.by.1.2		2	4.3	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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1176.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +7.63015 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +7.63015 q^{5} +9.00000 q^{9} -30.8904 q^{11} +42.5206 q^{13} -22.8904 q^{15} -4.36985 q^{17} -27.2603 q^{19} -56.6713 q^{23} -66.7809 q^{25} -27.0000 q^{27} +219.343 q^{29} -172.301 q^{31} +92.6713 q^{33} -31.3426 q^{37} -127.562 q^{39} -274.754 q^{41} +188.000 q^{43} +68.6713 q^{45} -50.0823 q^{47} +13.1096 q^{51} +282.904 q^{53} -235.699 q^{55} +81.7809 q^{57} +64.1647 q^{59} +97.1780 q^{61} +324.438 q^{65} -716.466 q^{67} +170.014 q^{69} +141.986 q^{71} -641.343 q^{73} +200.343 q^{75} +391.590 q^{79} +81.0000 q^{81} -952.165 q^{83} -33.3426 q^{85} -658.028 q^{87} +360.205 q^{89} +516.904 q^{93} -208.000 q^{95} +58.5485 q^{97} -278.014 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 6 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 6 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} - 6 q^{5} + 18 q^{9} + 2 q^{11} + 18 q^{15} - 30 q^{17} - 12 q^{19} + 78 q^{23} - 6 q^{25} - 54 q^{27} + 56 q^{29} - 132 q^{31} - 6 q^{33} + 320 q^{37} - 18 q^{41} + 376 q^{43} - 54 q^{45} + 240 q^{47} + 90 q^{51} - 72 q^{53} - 684 q^{55} + 36 q^{57} - 552 q^{59} + 492 q^{61} + 904 q^{65} - 540 q^{67} - 234 q^{69} + 858 q^{71} - 900 q^{73} + 18 q^{75} - 620 q^{79} + 162 q^{81} - 1224 q^{83} + 316 q^{85} - 168 q^{87} + 1422 q^{89} + 396 q^{93} - 416 q^{95} - 1116 q^{97} + 18 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 6 * q^5 + 18 * q^9 + 2 * q^11 + 18 * q^15 - 30 * q^17 - 12 * q^19 + 78 * q^23 - 6 * q^25 - 54 * q^27 + 56 * q^29 - 132 * q^31 - 6 * q^33 + 320 * q^37 - 18 * q^41 + 376 * q^43 - 54 * q^45 + 240 * q^47 + 90 * q^51 - 72 * q^53 - 684 * q^55 + 36 * q^57 - 552 * q^59 + 492 * q^61 + 904 * q^65 - 540 * q^67 - 234 * q^69 + 858 * q^71 - 900 * q^73 + 18 * q^75 - 620 * q^79 + 162 * q^81 - 1224 * q^83 + 316 * q^85 - 168 * q^87 + 1422 * q^89 + 396 * q^93 - 416 * q^95 - 1116 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

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