LMFDB - Embedded newform 2400.4.a.bf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.23109.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 37x + 10 $ x^3 - x^2 - 37*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.74427$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +6.81960 q^{7} +9.00000 q^{9} -25.7967 q^{11} -32.1575 q^{13} +9.16680 q^{17} +99.2784 q^{19} -20.4588 q^{21} -61.2597 q^{23} -27.0000 q^{27} +17.1991 q^{29} -74.9635 q^{31} +77.3900 q^{33} +55.9740 q^{37} +96.4724 q^{39} -54.1960 q^{41} -33.9813 q^{43} -152.955 q^{47} -296.493 q^{49} -27.5004 q^{51} -26.7289 q^{53} -297.835 q^{57} +567.984 q^{59} -249.963 q^{61} +61.3764 q^{63} +987.608 q^{67} +183.779 q^{69} -705.600 q^{71} +278.004 q^{73} -175.923 q^{77} +1137.94 q^{79} +81.0000 q^{81} -1058.03 q^{83} -51.5972 q^{87} +4.63145 q^{89} -219.301 q^{91} +224.890 q^{93} -1459.17 q^{97} -232.170 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 32 q^{7} + 27 q^{9} + 48 q^{11} - 76 q^{13} - 16 q^{17} + 88 q^{19} + 96 q^{21} - 20 q^{23} - 81 q^{27} + 58 q^{29} - 56 q^{31} - 144 q^{33} - 436 q^{37} + 228 q^{39} + 362 q^{41} - 148 q^{43}+ \cdots + 432 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 32 * q^7 + 27 * q^9 + 48 * q^11 - 76 * q^13 - 16 * q^17 + 88 * q^19 + 96 * q^21 - 20 * q^23 - 81 * q^27 + 58 * q^29 - 56 * q^31 - 144 * q^33 - 436 * q^37 + 228 * q^39 + 362 * q^41 - 148 * q^43 - 52 * q^47 - 109 * q^49 + 48 * q^51 - 940 * q^53 - 264 * q^57 + 816 * q^59 - 842 * q^61 - 288 * q^63 + 780 * q^67 + 60 * q^69 + 304 * q^71 - 512 * q^73 - 1536 * q^77 - 440 * q^79 + 243 * q^81 + 1212 * q^83 - 174 * q^87 - 590 * q^89 + 944 * q^91 + 168 * q^93 - 1752 * q^97 + 432 * q^99

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	0
$5$	0	0
$6$	0	0
$7$	6.81960	0.368224	0.184112	−	0.982905i	$-0.441059\pi$
$7$	6.81960	0.368224	0.184112	+	0.982905i	$0.441059\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−25.7967	−0.707090	−0.353545	−	0.935418i	$-0.615024\pi$
$11$	−25.7967	−0.707090	−0.353545	+	0.935418i	$0.615024\pi$
$12$	0	0
$13$	−32.1575	−0.686067	−0.343034	−	0.939323i	$-0.611455\pi$
$13$	−32.1575	−0.686067	−0.343034	+	0.939323i	$0.611455\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	9.16680	0.130781	0.0653904	−	0.997860i	$-0.479171\pi$
$17$	9.16680	0.130781	0.0653904	+	0.997860i	$0.479171\pi$
$18$	0	0
$19$	99.2784	1.19874	0.599369	−	0.800473i	$-0.295418\pi$
$19$	99.2784	1.19874	0.599369	+	0.800473i	$0.295418\pi$
$20$	0	0
$21$	−20.4588	−0.212594
$22$	0	0
$23$	−61.2597	−0.555371	−0.277686	−	0.960672i	$-0.589567\pi$
$23$	−61.2597	−0.555371	−0.277686	+	0.960672i	$0.589567\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	17.1991	0.110131	0.0550653	−	0.998483i	$-0.482463\pi$
$29$	17.1991	0.110131	0.0550653	+	0.998483i	$0.482463\pi$
$30$	0	0
$31$	−74.9635	−0.434317	−0.217159	−	0.976136i	$-0.569679\pi$
$31$	−74.9635	−0.434317	−0.217159	+	0.976136i	$0.569679\pi$
$32$	0	0
$33$	77.3900	0.408238
$34$	0	0
$35$	0	0
$36$	0	0
$37$	55.9740	0.248705	0.124352	−	0.992238i	$-0.460315\pi$
$37$	55.9740	0.248705	0.124352	+	0.992238i	$0.460315\pi$
$38$	0	0
$39$	96.4724	0.396101
$40$	0	0
$41$	−54.1960	−0.206439	−0.103219	−	0.994659i	$-0.532914\pi$
$41$	−54.1960	−0.206439	−0.103219	+	0.994659i	$0.532914\pi$
$42$	0	0
$43$	−33.9813	−0.120514	−0.0602570	−	0.998183i	$-0.519192\pi$
$43$	−33.9813	−0.120514	−0.0602570	+	0.998183i	$0.519192\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−152.955	−0.474697	−0.237349	−	0.971425i	$-0.576278\pi$
$47$	−152.955	−0.474697	−0.237349	+	0.971425i	$0.576278\pi$
$48$	0	0
$49$	−296.493	−0.864411
$50$	0	0
$51$	−27.5004	−0.0755064
$52$	0	0
$53$	−26.7289	−0.0692736	−0.0346368	−	0.999400i	$-0.511027\pi$
$53$	−26.7289	−0.0692736	−0.0346368	+	0.999400i	$0.511027\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−297.835	−0.692092
$58$	0	0
$59$	567.984	1.25331	0.626655	−	0.779297i	$-0.284424\pi$
$59$	567.984	1.25331	0.626655	+	0.779297i	$0.284424\pi$
$60$	0	0
$61$	−249.963	−0.524663	−0.262331	−	0.964978i	$-0.584491\pi$
$61$	−249.963	−0.524663	−0.262331	+	0.964978i	$0.584491\pi$
$62$	0	0
$63$	61.3764	0.122741
$64$	0	0
$65$	0	0
$66$	0	0
$67$	987.608	1.80083	0.900415	−	0.435033i	$-0.143263\pi$
$67$	987.608	1.80083	0.900415	+	0.435033i	$0.143263\pi$
$68$	0	0
$69$	183.779	0.320644
$70$	0	0
$71$	−705.600	−1.17943	−0.589713	−	0.807613i	$-0.700759\pi$
$71$	−705.600	−1.17943	−0.589713	+	0.807613i	$0.700759\pi$
$72$	0	0
$73$	278.004	0.445725	0.222862	−	0.974850i	$-0.428460\pi$
$73$	278.004	0.445725	0.222862	+	0.974850i	$0.428460\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−175.923	−0.260367
$78$	0	0
$79$	1137.94	1.62061	0.810303	−	0.586010i	$-0.199302\pi$
$79$	1137.94	1.62061	0.810303	+	0.586010i	$0.199302\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1058.03	−1.39921	−0.699603	−	0.714532i	$-0.746640\pi$
$83$	−1058.03	−1.39921	−0.699603	+	0.714532i	$0.746640\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−51.5972	−0.0635839
$88$	0	0
$89$	4.63145	0.00551610	0.00275805	−	0.999996i	$-0.499122\pi$
$89$	4.63145	0.00551610	0.00275805	+	0.999996i	$0.499122\pi$
$90$	0	0
$91$	−219.301	−0.252626
$92$	0	0
$93$	224.890	0.250753
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1459.17	−1.52739	−0.763694	−	0.645578i	$-0.776617\pi$
$97$	−1459.17	−1.52739	−0.763694	+	0.645578i	$0.776617\pi$
$98$	0	0
$99$	−232.170	−0.235697
$100$	0	0
$101$	1423.35	1.40227	0.701133	−	0.713031i	$-0.252678\pi$
$101$	1423.35	1.40227	0.701133	+	0.713031i	$0.252678\pi$
$102$	0	0
$103$	620.400	0.593493	0.296747	−	0.954956i	$-0.404098\pi$
$103$	620.400	0.593493	0.296747	+	0.954956i	$0.404098\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	765.835	0.691926	0.345963	−	0.938248i	$-0.387552\pi$
$107$	765.835	0.691926	0.345963	+	0.938248i	$0.387552\pi$
$108$	0	0
$109$	1192.78	1.04814	0.524071	−	0.851675i	$-0.324413\pi$
$109$	1192.78	1.04814	0.524071	+	0.851675i	$0.324413\pi$
$110$	0	0
$111$	−167.922	−0.143590
$112$	0	0
$113$	−2049.68	−1.70635	−0.853177	−	0.521621i	$-0.825327\pi$
$113$	−2049.68	−1.70635	−0.853177	+	0.521621i	$0.825327\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−289.417	−0.228689
$118$	0	0
$119$	62.5139	0.0481566
$120$	0	0
$121$	−665.532	−0.500024
$122$	0	0
$123$	162.588	0.119188
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1721.60	−1.20289	−0.601445	−	0.798914i	$-0.705408\pi$
$127$	−1721.60	−1.20289	−0.601445	+	0.798914i	$0.705408\pi$
$128$	0	0
$129$	101.944	0.0695788
$130$	0	0
$131$	2064.73	1.37707	0.688535	−	0.725203i	$-0.258254\pi$
$131$	2064.73	1.37707	0.688535	+	0.725203i	$0.258254\pi$
$132$	0	0
$133$	677.039	0.441404
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2096.85	1.30764	0.653818	−	0.756652i	$-0.273166\pi$
$137$	2096.85	1.30764	0.653818	+	0.756652i	$0.273166\pi$
$138$	0	0
$139$	−453.572	−0.276773	−0.138386	−	0.990378i	$-0.544192\pi$
$139$	−453.572	−0.276773	−0.138386	+	0.990378i	$0.544192\pi$
$140$	0	0
$141$	458.865	0.274067
$142$	0	0
$143$	829.555	0.485111
$144$	0	0
$145$	0	0
$146$	0	0
$147$	889.479	0.499068
$148$	0	0
$149$	−1373.69	−0.755281	−0.377641	−	0.925952i	$-0.623265\pi$
$149$	−1373.69	−0.755281	−0.377641	+	0.925952i	$0.623265\pi$
$150$	0	0
$151$	2477.83	1.33538	0.667692	−	0.744437i	$-0.267282\pi$
$151$	2477.83	1.33538	0.667692	+	0.744437i	$0.267282\pi$
$152$	0	0
$153$	82.5012	0.0435936
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1670.28	−0.849065	−0.424532	−	0.905413i	$-0.639561\pi$
$157$	−1670.28	−0.849065	−0.424532	+	0.905413i	$0.639561\pi$
$158$	0	0
$159$	80.1868	0.0399951
$160$	0	0
$161$	−417.767	−0.204501
$162$	0	0
$163$	−1097.50	−0.527378	−0.263689	−	0.964608i	$-0.584939\pi$
$163$	−1097.50	−0.527378	−0.263689	+	0.964608i	$0.584939\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2332.34	−1.08073	−0.540365	−	0.841430i	$-0.681714\pi$
$167$	−2332.34	−1.08073	−0.540365	+	0.841430i	$0.681714\pi$
$168$	0	0
$169$	−1162.90	−0.529311
$170$	0	0
$171$	893.506	0.399579
$172$	0	0
$173$	−4112.25	−1.80722	−0.903608	−	0.428360i	$-0.859092\pi$
$173$	−4112.25	−1.80722	−0.903608	+	0.428360i	$0.859092\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1703.95	−0.723598
$178$	0	0
$179$	−1156.81	−0.483040	−0.241520	−	0.970396i	$-0.577646\pi$
$179$	−1156.81	−0.483040	−0.241520	+	0.970396i	$0.577646\pi$
$180$	0	0
$181$	−1211.29	−0.497426	−0.248713	−	0.968577i	$-0.580008\pi$
$181$	−1211.29	−0.497426	−0.248713	+	0.968577i	$0.580008\pi$
$182$	0	0
$183$	749.888	0.302914
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−236.473	−0.0924738
$188$	0	0
$189$	−184.129	−0.0708647
$190$	0	0
$191$	4192.97	1.58844	0.794221	−	0.607629i	$-0.207879\pi$
$191$	4192.97	1.58844	0.794221	+	0.607629i	$0.207879\pi$
$192$	0	0
$193$	2388.88	0.890960	0.445480	−	0.895292i	$-0.353033\pi$
$193$	2388.88	0.890960	0.445480	+	0.895292i	$0.353033\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	346.366	0.125267	0.0626334	−	0.998037i	$-0.480050\pi$
$197$	346.366	0.125267	0.0626334	+	0.998037i	$0.480050\pi$
$198$	0	0
$199$	1089.75	0.388192	0.194096	−	0.980983i	$-0.437823\pi$
$199$	1089.75	0.388192	0.194096	+	0.980983i	$0.437823\pi$
$200$	0	0
$201$	−2962.82	−1.03971
$202$	0	0
$203$	117.291	0.0405527
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−551.338	−0.185124
$208$	0	0
$209$	−2561.05	−0.847616
$210$	0	0
$211$	5613.32	1.83146	0.915728	−	0.401799i	$-0.131615\pi$
$211$	5613.32	1.83146	0.915728	+	0.401799i	$0.131615\pi$
$212$	0	0
$213$	2116.80	0.680942
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−511.221	−0.159926
$218$	0	0
$219$	−834.012	−0.257339
$220$	0	0
$221$	−294.781	−0.0897245
$222$	0	0
$223$	14.7386	0.00442588	0.00221294	−	0.999998i	$-0.499296\pi$
$223$	14.7386	0.00442588	0.00221294	+	0.999998i	$0.499296\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3008.40	0.879622	0.439811	−	0.898090i	$-0.355045\pi$
$227$	3008.40	0.879622	0.439811	+	0.898090i	$0.355045\pi$
$228$	0	0
$229$	3704.88	1.06911	0.534554	−	0.845135i	$-0.320480\pi$
$229$	3704.88	1.06911	0.534554	+	0.845135i	$0.320480\pi$
$230$	0	0
$231$	527.769	0.150323
$232$	0	0
$233$	−305.357	−0.0858566	−0.0429283	−	0.999078i	$-0.513669\pi$
$233$	−305.357	−0.0858566	−0.0429283	+	0.999078i	$0.513669\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3413.81	−0.935658
$238$	0	0
$239$	−254.763	−0.0689510	−0.0344755	−	0.999406i	$-0.510976\pi$
$239$	−254.763	−0.0689510	−0.0344755	+	0.999406i	$0.510976\pi$
$240$	0	0
$241$	4032.38	1.07780	0.538898	−	0.842371i	$-0.318841\pi$
$241$	4032.38	1.07780	0.538898	+	0.842371i	$0.318841\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3192.54	−0.822415
$248$	0	0
$249$	3174.09	0.807832
$250$	0	0
$251$	6009.93	1.51133	0.755664	−	0.654960i	$-0.227314\pi$
$251$	6009.93	1.51133	0.755664	+	0.654960i	$0.227314\pi$
$252$	0	0
$253$	1580.30	0.392697
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3638.54	0.883135	0.441567	−	0.897228i	$-0.354423\pi$
$257$	3638.54	0.883135	0.441567	+	0.897228i	$0.354423\pi$
$258$	0	0
$259$	381.720	0.0915789
$260$	0	0
$261$	154.792	0.0367102
$262$	0	0
$263$	4946.05	1.15964	0.579822	−	0.814743i	$-0.303122\pi$
$263$	4946.05	1.15964	0.579822	+	0.814743i	$0.303122\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−13.8944	−0.00318472
$268$	0	0
$269$	3081.01	0.698336	0.349168	−	0.937060i	$-0.386464\pi$
$269$	3081.01	0.698336	0.349168	+	0.937060i	$0.386464\pi$
$270$	0	0
$271$	7914.77	1.77413	0.887063	−	0.461649i	$-0.152742\pi$
$271$	7914.77	1.77413	0.887063	+	0.461649i	$0.152742\pi$
$272$	0	0
$273$	657.903	0.145854
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3559.58	0.772110	0.386055	−	0.922476i	$-0.373837\pi$
$277$	3559.58	0.772110	0.386055	+	0.922476i	$0.373837\pi$
$278$	0	0
$279$	−674.671	−0.144772
$280$	0	0
$281$	2025.25	0.429952	0.214976	−	0.976619i	$-0.431033\pi$
$281$	2025.25	0.429952	0.214976	+	0.976619i	$0.431033\pi$
$282$	0	0
$283$	3311.43	0.695563	0.347782	−	0.937576i	$-0.386935\pi$
$283$	3311.43	0.695563	0.347782	+	0.937576i	$0.386935\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−369.595	−0.0760157
$288$	0	0
$289$	−4828.97	−0.982896
$290$	0	0
$291$	4377.52	0.881838
$292$	0	0
$293$	−1735.70	−0.346077	−0.173039	−	0.984915i	$-0.555359\pi$
$293$	−1735.70	−0.346077	−0.173039	+	0.984915i	$0.555359\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	696.510	0.136079
$298$	0	0
$299$	1969.96	0.381022
$300$	0	0
$301$	−231.739	−0.0443761
$302$	0	0
$303$	−4270.06	−0.809598
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5444.54	1.01217	0.506085	−	0.862483i	$-0.331092\pi$
$307$	5444.54	1.01217	0.506085	+	0.862483i	$0.331092\pi$
$308$	0	0
$309$	−1861.20	−0.342654
$310$	0	0
$311$	2873.36	0.523901	0.261950	−	0.965081i	$-0.415634\pi$
$311$	2873.36	0.523901	0.261950	+	0.965081i	$0.415634\pi$
$312$	0	0
$313$	6010.68	1.08544	0.542722	−	0.839913i	$-0.317394\pi$
$313$	6010.68	1.08544	0.542722	+	0.839913i	$0.317394\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7883.11	1.39672	0.698359	−	0.715748i	$-0.253914\pi$
$317$	7883.11	1.39672	0.698359	+	0.715748i	$0.253914\pi$
$318$	0	0
$319$	−443.679	−0.0778722
$320$	0	0
$321$	−2297.51	−0.399484
$322$	0	0
$323$	910.065	0.156772
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3578.33	−0.605145
$328$	0	0
$329$	−1043.09	−0.174795
$330$	0	0
$331$	1636.58	0.271766	0.135883	−	0.990725i	$-0.456613\pi$
$331$	1636.58	0.271766	0.135883	+	0.990725i	$0.456613\pi$
$332$	0	0
$333$	503.766	0.0829015
$334$	0	0
$335$	0	0
$336$	0	0
$337$	287.115	0.0464099	0.0232050	−	0.999731i	$-0.492613\pi$
$337$	287.115	0.0464099	0.0232050	+	0.999731i	$0.492613\pi$
$338$	0	0
$339$	6149.05	0.985164
$340$	0	0
$341$	1933.81	0.307101
$342$	0	0
$343$	−4361.09	−0.686521
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3067.93	−0.474626	−0.237313	−	0.971433i	$-0.576267\pi$
$347$	−3067.93	−0.474626	−0.237313	+	0.971433i	$0.576267\pi$
$348$	0	0
$349$	6754.03	1.03592	0.517959	−	0.855406i	$-0.326692\pi$
$349$	6754.03	1.03592	0.517959	+	0.855406i	$0.326692\pi$
$350$	0	0
$351$	868.252	0.132034
$352$	0	0
$353$	−8318.47	−1.25424	−0.627121	−	0.778922i	$-0.715767\pi$
$353$	−8318.47	−1.25424	−0.627121	+	0.778922i	$0.715767\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−187.542	−0.0278032
$358$	0	0
$359$	7980.40	1.17323	0.586615	−	0.809866i	$-0.300460\pi$
$359$	7980.40	1.17323	0.586615	+	0.809866i	$0.300460\pi$
$360$	0	0
$361$	2997.20	0.436973
$362$	0	0
$363$	1996.60	0.288689
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6471.22	0.920422	0.460211	−	0.887810i	$-0.347774\pi$
$367$	6471.22	0.920422	0.460211	+	0.887810i	$0.347774\pi$
$368$	0	0
$369$	−487.764	−0.0688129
$370$	0	0
$371$	−182.281	−0.0255082
$372$	0	0
$373$	7441.94	1.03305	0.516527	−	0.856271i	$-0.327225\pi$
$373$	7441.94	1.03305	0.516527	+	0.856271i	$0.327225\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−553.078	−0.0755570
$378$	0	0
$379$	10962.3	1.48573	0.742867	−	0.669439i	$-0.233466\pi$
$379$	10962.3	1.48573	0.742867	+	0.669439i	$0.233466\pi$
$380$	0	0
$381$	5164.79	0.694489
$382$	0	0
$383$	−14038.1	−1.87288	−0.936441	−	0.350826i	$-0.885901\pi$
$383$	−14038.1	−1.87288	−0.936441	+	0.350826i	$0.885901\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−305.832	−0.0401714
$388$	0	0
$389$	−10282.9	−1.34027	−0.670135	−	0.742239i	$-0.733764\pi$
$389$	−10282.9	−1.34027	−0.670135	+	0.742239i	$0.733764\pi$
$390$	0	0
$391$	−561.556	−0.0726319
$392$	0	0
$393$	−6194.19	−0.795052
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6787.17	0.858032	0.429016	−	0.903297i	$-0.358860\pi$
$397$	6787.17	0.858032	0.429016	+	0.903297i	$0.358860\pi$
$398$	0	0
$399$	−2031.12	−0.254845
$400$	0	0
$401$	−1939.49	−0.241530	−0.120765	−	0.992681i	$-0.538535\pi$
$401$	−1939.49	−0.241530	−0.120765	+	0.992681i	$0.538535\pi$
$402$	0	0
$403$	2410.64	0.297971
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1443.94	−0.175856
$408$	0	0
$409$	−5528.51	−0.668379	−0.334190	−	0.942506i	$-0.608463\pi$
$409$	−5528.51	−0.668379	−0.334190	+	0.942506i	$0.608463\pi$
$410$	0	0
$411$	−6290.56	−0.754964
$412$	0	0
$413$	3873.42	0.461498
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1360.71	0.159795
$418$	0	0
$419$	6818.79	0.795035	0.397518	−	0.917595i	$-0.369872\pi$
$419$	6818.79	0.795035	0.397518	+	0.917595i	$0.369872\pi$
$420$	0	0
$421$	−15473.8	−1.79132	−0.895659	−	0.444741i	$-0.853296\pi$
$421$	−15473.8	−1.79132	−0.895659	+	0.444741i	$0.853296\pi$
$422$	0	0
$423$	−1376.59	−0.158232
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1704.65	−0.193193
$428$	0	0
$429$	−2488.67	−0.280079
$430$	0	0
$431$	11445.5	1.27914	0.639569	−	0.768734i	$-0.279113\pi$
$431$	11445.5	1.27914	0.639569	+	0.768734i	$0.279113\pi$
$432$	0	0
$433$	8471.01	0.940164	0.470082	−	0.882623i	$-0.344224\pi$
$433$	8471.01	0.940164	0.470082	+	0.882623i	$0.344224\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6081.77	−0.665745
$438$	0	0
$439$	2237.01	0.243205	0.121602	−	0.992579i	$-0.461197\pi$
$439$	2237.01	0.243205	0.121602	+	0.992579i	$0.461197\pi$
$440$	0	0
$441$	−2668.44	−0.288137
$442$	0	0
$443$	14329.1	1.53679	0.768394	−	0.639977i	$-0.221056\pi$
$443$	14329.1	1.53679	0.768394	+	0.639977i	$0.221056\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4121.06	0.436062
$448$	0	0
$449$	15604.9	1.64018	0.820089	−	0.572237i	$-0.193924\pi$
$449$	15604.9	1.64018	0.820089	+	0.572237i	$0.193924\pi$
$450$	0	0
$451$	1398.08	0.145971
$452$	0	0
$453$	−7433.50	−0.770985
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1300.53	−0.133121	−0.0665605	−	0.997782i	$-0.521203\pi$
$457$	−1300.53	−0.133121	−0.0665605	+	0.997782i	$0.521203\pi$
$458$	0	0
$459$	−247.504	−0.0251688
$460$	0	0
$461$	4748.14	0.479703	0.239851	−	0.970810i	$-0.422901\pi$
$461$	4748.14	0.479703	0.239851	+	0.970810i	$0.422901\pi$
$462$	0	0
$463$	−13565.9	−1.36168	−0.680842	−	0.732430i	$-0.738386\pi$
$463$	−13565.9	−1.36168	−0.680842	+	0.732430i	$0.738386\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11178.2	1.10764	0.553819	−	0.832637i	$-0.313170\pi$
$467$	11178.2	1.10764	0.553819	+	0.832637i	$0.313170\pi$
$468$	0	0
$469$	6735.09	0.663108
$470$	0	0
$471$	5010.85	0.490208
$472$	0	0
$473$	876.605	0.0852143
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−240.560	−0.0230912
$478$	0	0
$479$	5562.99	0.530646	0.265323	−	0.964160i	$-0.414521\pi$
$479$	5562.99	0.530646	0.265323	+	0.964160i	$0.414521\pi$
$480$	0	0
$481$	−1799.98	−0.170628
$482$	0	0
$483$	1253.30	0.118069
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−16716.3	−1.55542	−0.777710	−	0.628623i	$-0.783619\pi$
$487$	−16716.3	−1.55542	−0.777710	+	0.628623i	$0.783619\pi$
$488$	0	0
$489$	3292.49	0.304482
$490$	0	0
$491$	3320.29	0.305178	0.152589	−	0.988290i	$-0.451239\pi$
$491$	3320.29	0.305178	0.152589	+	0.988290i	$0.451239\pi$
$492$	0	0
$493$	157.660	0.0144030
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4811.91	−0.434293
$498$	0	0
$499$	−1649.53	−0.147982	−0.0739910	−	0.997259i	$-0.523574\pi$
$499$	−1649.53	−0.147982	−0.0739910	+	0.997259i	$0.523574\pi$
$500$	0	0
$501$	6997.03	0.623960
$502$	0	0
$503$	−4173.99	−0.369998	−0.184999	−	0.982739i	$-0.559228\pi$
$503$	−4173.99	−0.369998	−0.184999	+	0.982739i	$0.559228\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3488.69	0.305598
$508$	0	0
$509$	−2945.71	−0.256515	−0.128258	−	0.991741i	$-0.540938\pi$
$509$	−2945.71	−0.256515	−0.128258	+	0.991741i	$0.540938\pi$
$510$	0	0
$511$	1895.88	0.164126
$512$	0	0
$513$	−2680.52	−0.230697
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3945.73	0.335654
$518$	0	0
$519$	12336.7	1.04340
$520$	0	0
$521$	−9799.89	−0.824071	−0.412035	−	0.911168i	$-0.635182\pi$
$521$	−9799.89	−0.824071	−0.412035	+	0.911168i	$0.635182\pi$
$522$	0	0
$523$	14268.0	1.19292	0.596458	−	0.802644i	$-0.296574\pi$
$523$	14268.0	1.19292	0.596458	+	0.802644i	$0.296574\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−687.175	−0.0568004
$528$	0	0
$529$	−8414.24	−0.691563
$530$	0	0
$531$	5111.86	0.417770
$532$	0	0
$533$	1742.81	0.141631
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3470.44	0.278884
$538$	0	0
$539$	7648.53	0.611216
$540$	0	0
$541$	6611.27	0.525399	0.262700	−	0.964878i	$-0.415387\pi$
$541$	6611.27	0.525399	0.262700	+	0.964878i	$0.415387\pi$
$542$	0	0
$543$	3633.86	0.287189
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16141.3	1.26171	0.630853	−	0.775903i	$-0.282705\pi$
$547$	16141.3	1.26171	0.630853	+	0.775903i	$0.282705\pi$
$548$	0	0
$549$	−2249.66	−0.174888
$550$	0	0
$551$	1707.50	0.132018
$552$	0	0
$553$	7760.28	0.596746
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16680.5	−1.26889	−0.634447	−	0.772966i	$-0.718772\pi$
$557$	−16680.5	−1.26889	−0.634447	+	0.772966i	$0.718772\pi$
$558$	0	0
$559$	1092.75	0.0826808
$560$	0	0
$561$	709.419	0.0533898
$562$	0	0
$563$	−15190.8	−1.13715	−0.568577	−	0.822630i	$-0.692506\pi$
$563$	−15190.8	−1.13715	−0.568577	+	0.822630i	$0.692506\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	552.388	0.0409138
$568$	0	0
$569$	7785.68	0.573625	0.286812	−	0.957987i	$-0.407404\pi$
$569$	7785.68	0.573625	0.286812	+	0.957987i	$0.407404\pi$
$570$	0	0
$571$	10274.5	0.753023	0.376511	−	0.926412i	$-0.377124\pi$
$571$	10274.5	0.753023	0.376511	+	0.926412i	$0.377124\pi$
$572$	0	0
$573$	−12578.9	−0.917087
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18429.3	1.32968	0.664838	−	0.746988i	$-0.268501\pi$
$577$	18429.3	1.32968	0.664838	+	0.746988i	$0.268501\pi$
$578$	0	0
$579$	−7166.63	−0.514396
$580$	0	0
$581$	−7215.35	−0.515221
$582$	0	0
$583$	689.517	0.0489827
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5225.99	0.367461	0.183731	−	0.982977i	$-0.441183\pi$
$587$	5225.99	0.367461	0.183731	+	0.982977i	$0.441183\pi$
$588$	0	0
$589$	−7442.25	−0.520633
$590$	0	0
$591$	−1039.10	−0.0723228
$592$	0	0
$593$	−602.367	−0.0417138	−0.0208569	−	0.999782i	$-0.506639\pi$
$593$	−602.367	−0.0417138	−0.0208569	+	0.999782i	$0.506639\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3269.24	−0.224123
$598$	0	0
$599$	−7714.91	−0.526248	−0.263124	−	0.964762i	$-0.584753\pi$
$599$	−7714.91	−0.526248	−0.263124	+	0.964762i	$0.584753\pi$
$600$	0	0
$601$	16822.2	1.14175	0.570874	−	0.821038i	$-0.306605\pi$
$601$	16822.2	1.14175	0.570874	+	0.821038i	$0.306605\pi$
$602$	0	0
$603$	8888.47	0.600276
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8251.11	−0.551733	−0.275867	−	0.961196i	$-0.588965\pi$
$607$	−8251.11	−0.551733	−0.275867	+	0.961196i	$0.588965\pi$
$608$	0	0
$609$	−351.872	−0.0234131
$610$	0	0
$611$	4918.64	0.325674
$612$	0	0
$613$	−14423.3	−0.950330	−0.475165	−	0.879897i	$-0.657612\pi$
$613$	−14423.3	−0.950330	−0.475165	+	0.879897i	$0.657612\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17060.0	−1.11314	−0.556571	−	0.830800i	$-0.687883\pi$
$617$	−17060.0	−1.11314	−0.556571	+	0.830800i	$0.687883\pi$
$618$	0	0
$619$	−4179.63	−0.271395	−0.135697	−	0.990750i	$-0.543327\pi$
$619$	−4179.63	−0.271395	−0.135697	+	0.990750i	$0.543327\pi$
$620$	0	0
$621$	1654.01	0.106881
$622$	0	0
$623$	31.5846	0.00203116
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7683.16	0.489371
$628$	0	0
$629$	513.102	0.0325258
$630$	0	0
$631$	10401.2	0.656208	0.328104	−	0.944642i	$-0.393590\pi$
$631$	10401.2	0.656208	0.328104	+	0.944642i	$0.393590\pi$
$632$	0	0
$633$	−16840.0	−1.05739
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9534.47	0.593044
$638$	0	0
$639$	−6350.40	−0.393142
$640$	0	0
$641$	9629.38	0.593350	0.296675	−	0.954978i	$-0.404122\pi$
$641$	9629.38	0.593350	0.296675	+	0.954978i	$0.404122\pi$
$642$	0	0
$643$	−554.156	−0.0339872	−0.0169936	−	0.999856i	$-0.505409\pi$
$643$	−554.156	−0.0339872	−0.0169936	+	0.999856i	$0.505409\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29213.5	−1.77512	−0.887559	−	0.460694i	$-0.847600\pi$
$647$	−29213.5	−1.77512	−0.887559	+	0.460694i	$0.847600\pi$
$648$	0	0
$649$	−14652.1	−0.886202
$650$	0	0
$651$	1533.66	0.0923333
$652$	0	0
$653$	8519.64	0.510565	0.255283	−	0.966866i	$-0.417831\pi$
$653$	8519.64	0.510565	0.255283	+	0.966866i	$0.417831\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2502.04	0.148575
$658$	0	0
$659$	−13709.4	−0.810382	−0.405191	−	0.914232i	$-0.632795\pi$
$659$	−13709.4	−0.810382	−0.405191	+	0.914232i	$0.632795\pi$
$660$	0	0
$661$	−3470.85	−0.204237	−0.102118	−	0.994772i	$-0.532562\pi$
$661$	−3470.85	−0.204237	−0.102118	+	0.994772i	$0.532562\pi$
$662$	0	0
$663$	884.343	0.0518025
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1053.61	−0.0611633
$668$	0	0
$669$	−44.2158	−0.00255528
$670$	0	0
$671$	6448.20	0.370984
$672$	0	0
$673$	13746.2	0.787336	0.393668	−	0.919253i	$-0.371206\pi$
$673$	13746.2	0.787336	0.393668	+	0.919253i	$0.371206\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21098.5	−1.19776	−0.598878	−	0.800840i	$-0.704386\pi$
$677$	−21098.5	−1.19776	−0.598878	+	0.800840i	$0.704386\pi$
$678$	0	0
$679$	−9950.99	−0.562421
$680$	0	0
$681$	−9025.19	−0.507850
$682$	0	0
$683$	16080.3	0.900872	0.450436	−	0.892809i	$-0.351269\pi$
$683$	16080.3	0.900872	0.450436	+	0.892809i	$0.351269\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−11114.7	−0.617249
$688$	0	0
$689$	859.535	0.0475264
$690$	0	0
$691$	14310.6	0.787844	0.393922	−	0.919144i	$-0.371118\pi$
$691$	14310.6	0.787844	0.393922	+	0.919144i	$0.371118\pi$
$692$	0	0
$693$	−1583.31	−0.0867891
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−496.804	−0.0269983
$698$	0	0
$699$	916.071	0.0495694
$700$	0	0
$701$	−9834.76	−0.529891	−0.264946	−	0.964263i	$-0.585354\pi$
$701$	−9834.76	−0.529891	−0.264946	+	0.964263i	$0.585354\pi$
$702$	0	0
$703$	5557.01	0.298132
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9706.69	0.516347
$708$	0	0
$709$	10055.7	0.532650	0.266325	−	0.963883i	$-0.414191\pi$
$709$	10055.7	0.532650	0.266325	+	0.963883i	$0.414191\pi$
$710$	0	0
$711$	10241.4	0.540202
$712$	0	0
$713$	4592.24	0.241207
$714$	0	0
$715$	0	0
$716$	0	0
$717$	764.290	0.0398089
$718$	0	0
$719$	−22769.9	−1.18105	−0.590524	−	0.807020i	$-0.701079\pi$
$719$	−22769.9	−1.18105	−0.590524	+	0.807020i	$0.701079\pi$
$720$	0	0
$721$	4230.88	0.218538
$722$	0	0
$723$	−12097.2	−0.622265
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−23421.7	−1.19486	−0.597429	−	0.801922i	$-0.703811\pi$
$727$	−23421.7	−1.19486	−0.597429	+	0.801922i	$0.703811\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−311.500	−0.0157609
$732$	0	0
$733$	38228.0	1.92631	0.963153	−	0.268953i	$-0.0866776\pi$
$733$	38228.0	1.92631	0.963153	+	0.268953i	$0.0866776\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25477.0	−1.27335
$738$	0	0
$739$	−21579.3	−1.07416	−0.537082	−	0.843530i	$-0.680473\pi$
$739$	−21579.3	−1.07416	−0.537082	+	0.843530i	$0.680473\pi$
$740$	0	0
$741$	9577.63	0.474822
$742$	0	0
$743$	−894.461	−0.0441650	−0.0220825	−	0.999756i	$-0.507030\pi$
$743$	−894.461	−0.0441650	−0.0220825	+	0.999756i	$0.507030\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9522.28	−0.466402
$748$	0	0
$749$	5222.69	0.254784
$750$	0	0
$751$	−13206.6	−0.641700	−0.320850	−	0.947130i	$-0.603968\pi$
$751$	−13206.6	−0.641700	−0.320850	+	0.947130i	$0.603968\pi$
$752$	0	0
$753$	−18029.8	−0.872566
$754$	0	0
$755$	0	0
$756$	0	0
$757$	16163.8	0.776066	0.388033	−	0.921645i	$-0.373155\pi$
$757$	16163.8	0.776066	0.388033	+	0.921645i	$0.373155\pi$
$758$	0	0
$759$	−4740.89	−0.226724
$760$	0	0
$761$	−2172.37	−0.103480	−0.0517400	−	0.998661i	$-0.516477\pi$
$761$	−2172.37	−0.103480	−0.0517400	+	0.998661i	$0.516477\pi$
$762$	0	0
$763$	8134.27	0.385951
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18264.9	−0.859855
$768$	0	0
$769$	35993.9	1.68787	0.843935	−	0.536446i	$-0.180233\pi$
$769$	35993.9	1.68787	0.843935	+	0.536446i	$0.180233\pi$
$770$	0	0
$771$	−10915.6	−0.509878
$772$	0	0
$773$	−12656.4	−0.588900	−0.294450	−	0.955667i	$-0.595136\pi$
$773$	−12656.4	−0.588900	−0.294450	+	0.955667i	$0.595136\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1145.16	−0.0528731
$778$	0	0
$779$	−5380.49	−0.247466
$780$	0	0
$781$	18202.1	0.833960
$782$	0	0
$783$	−464.375	−0.0211946
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36055.0	1.63306	0.816532	−	0.577300i	$-0.195894\pi$
$787$	36055.0	1.63306	0.816532	+	0.577300i	$0.195894\pi$
$788$	0	0
$789$	−14838.1	−0.669521
$790$	0	0
$791$	−13978.0	−0.628320
$792$	0	0
$793$	8038.17	0.359954
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5228.07	−0.232356	−0.116178	−	0.993228i	$-0.537064\pi$
$797$	−5228.07	−0.232356	−0.116178	+	0.993228i	$0.537064\pi$
$798$	0	0
$799$	−1402.11	−0.0620813
$800$	0	0
$801$	41.6831	0.00183870
$802$	0	0
$803$	−7171.58	−0.315167
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9243.03	−0.403185
$808$	0	0
$809$	19721.0	0.857051	0.428525	−	0.903530i	$-0.359033\pi$
$809$	19721.0	0.857051	0.428525	+	0.903530i	$0.359033\pi$
$810$	0	0
$811$	−16305.2	−0.705984	−0.352992	−	0.935626i	$-0.614836\pi$
$811$	−16305.2	−0.705984	−0.352992	+	0.935626i	$0.614836\pi$
$812$	0	0
$813$	−23744.3	−1.02429
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3373.61	−0.144465
$818$	0	0
$819$	−1973.71	−0.0842088
$820$	0	0
$821$	−42456.1	−1.80478	−0.902392	−	0.430916i	$-0.858190\pi$
$821$	−42456.1	−1.80478	−0.902392	+	0.430916i	$0.858190\pi$
$822$	0	0
$823$	18745.3	0.793948	0.396974	−	0.917830i	$-0.370060\pi$
$823$	18745.3	0.793948	0.396974	+	0.917830i	$0.370060\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31886.6	1.34076	0.670379	−	0.742019i	$-0.266132\pi$
$827$	31886.6	1.34076	0.670379	+	0.742019i	$0.266132\pi$
$828$	0	0
$829$	−16850.5	−0.705962	−0.352981	−	0.935631i	$-0.614832\pi$
$829$	−16850.5	−0.705962	−0.352981	+	0.935631i	$0.614832\pi$
$830$	0	0
$831$	−10678.7	−0.445778
$832$	0	0
$833$	−2717.89	−0.113048
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2024.01	0.0835844
$838$	0	0
$839$	4400.11	0.181059	0.0905296	−	0.995894i	$-0.471144\pi$
$839$	4400.11	0.181059	0.0905296	+	0.995894i	$0.471144\pi$
$840$	0	0
$841$	−24093.2	−0.987871
$842$	0	0
$843$	−6075.76	−0.248233
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4538.66	−0.184121
$848$	0	0
$849$	−9934.30	−0.401584
$850$	0	0
$851$	−3428.95	−0.138123
$852$	0	0
$853$	37149.8	1.49119	0.745596	−	0.666398i	$-0.232165\pi$
$853$	37149.8	1.49119	0.745596	+	0.666398i	$0.232165\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1242.70	0.0495332	0.0247666	−	0.999693i	$-0.492116\pi$
$857$	1242.70	0.0495332	0.0247666	+	0.999693i	$0.492116\pi$
$858$	0	0
$859$	19076.5	0.757721	0.378860	−	0.925454i	$-0.376316\pi$
$859$	19076.5	0.757721	0.378860	+	0.925454i	$0.376316\pi$
$860$	0	0
$861$	1108.79	0.0438877
$862$	0	0
$863$	3400.46	0.134129	0.0670644	−	0.997749i	$-0.478637\pi$
$863$	3400.46	0.134129	0.0670644	+	0.997749i	$0.478637\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14486.9	0.567475
$868$	0	0
$869$	−29355.0	−1.14591
$870$	0	0
$871$	−31759.0	−1.23549
$872$	0	0
$873$	−13132.6	−0.509130
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−24717.7	−0.951719	−0.475860	−	0.879521i	$-0.657863\pi$
$877$	−24717.7	−0.951719	−0.475860	+	0.879521i	$0.657863\pi$
$878$	0	0
$879$	5207.10	0.199808
$880$	0	0
$881$	45748.4	1.74949	0.874746	−	0.484582i	$-0.161028\pi$
$881$	45748.4	1.74949	0.874746	+	0.484582i	$0.161028\pi$
$882$	0	0
$883$	−48052.1	−1.83135	−0.915676	−	0.401918i	$-0.868344\pi$
$883$	−48052.1	−1.83135	−0.915676	+	0.401918i	$0.868344\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29794.4	−1.12785	−0.563923	−	0.825828i	$-0.690708\pi$
$887$	−29794.4	−1.12785	−0.563923	+	0.825828i	$0.690708\pi$
$888$	0	0
$889$	−11740.6	−0.442933
$890$	0	0
$891$	−2089.53	−0.0785655
$892$	0	0
$893$	−15185.1	−0.569038
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−5909.87	−0.219983
$898$	0	0
$899$	−1289.30	−0.0478316
$900$	0	0
$901$	−245.019	−0.00905967
$902$	0	0
$903$	695.217	0.0256206
$904$	0	0
$905$	0	0
$906$	0	0
$907$	43880.6	1.60643	0.803214	−	0.595690i	$-0.203121\pi$
$907$	43880.6	1.60643	0.803214	+	0.595690i	$0.203121\pi$
$908$	0	0
$909$	12810.2	0.467422
$910$	0	0
$911$	−43185.1	−1.57057	−0.785284	−	0.619136i	$-0.787483\pi$
$911$	−43185.1	−1.57057	−0.785284	+	0.619136i	$0.787483\pi$
$912$	0	0
$913$	27293.7	0.989364
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14080.6	0.507070
$918$	0	0
$919$	14712.4	0.528093	0.264047	−	0.964510i	$-0.414943\pi$
$919$	14712.4	0.528093	0.264047	+	0.964510i	$0.414943\pi$
$920$	0	0
$921$	−16333.6	−0.584377
$922$	0	0
$923$	22690.3	0.809166
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5583.60	0.197831
$928$	0	0
$929$	−10082.0	−0.356061	−0.178031	−	0.984025i	$-0.556973\pi$
$929$	−10082.0	−0.356061	−0.178031	+	0.984025i	$0.556973\pi$
$930$	0	0
$931$	−29435.4	−1.03620
$932$	0	0
$933$	−8620.07	−0.302474
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26665.2	0.929686	0.464843	−	0.885393i	$-0.346111\pi$
$937$	26665.2	0.929686	0.464843	+	0.885393i	$0.346111\pi$
$938$	0	0
$939$	−18032.0	−0.626681
$940$	0	0
$941$	−14181.1	−0.491277	−0.245639	−	0.969361i	$-0.578998\pi$
$941$	−14181.1	−0.491277	−0.245639	+	0.969361i	$0.578998\pi$
$942$	0	0
$943$	3320.03	0.114650
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33047.2	1.13399	0.566995	−	0.823721i	$-0.308106\pi$
$947$	33047.2	1.13399	0.566995	+	0.823721i	$0.308106\pi$
$948$	0	0
$949$	−8939.90	−0.305797
$950$	0	0
$951$	−23649.3	−0.806395
$952$	0	0
$953$	−30618.0	−1.04073	−0.520365	−	0.853944i	$-0.674204\pi$
$953$	−30618.0	−1.04073	−0.520365	+	0.853944i	$0.674204\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1331.04	0.0449595
$958$	0	0
$959$	14299.7	0.481503
$960$	0	0
$961$	−24171.5	−0.811368
$962$	0	0
$963$	6892.52	0.230642
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−31993.1	−1.06394	−0.531969	−	0.846764i	$-0.678548\pi$
$967$	−31993.1	−1.06394	−0.531969	+	0.846764i	$0.678548\pi$
$968$	0	0
$969$	−2730.20	−0.0905124
$970$	0	0
$971$	−39141.8	−1.29364	−0.646818	−	0.762644i	$-0.723901\pi$
$971$	−39141.8	−1.29364	−0.646818	+	0.762644i	$0.723901\pi$
$972$	0	0
$973$	−3093.18	−0.101914
$974$	0	0
$975$	0	0
$976$	0	0
$977$	246.261	0.00806407	0.00403204	−	0.999992i	$-0.498717\pi$
$977$	246.261	0.00806407	0.00403204	+	0.999992i	$0.498717\pi$
$978$	0	0
$979$	−119.476	−0.00390038
$980$	0	0
$981$	10735.0	0.349380
$982$	0	0
$983$	−602.142	−0.0195375	−0.00976874	−	0.999952i	$-0.503110\pi$
$983$	−602.142	−0.0195375	−0.00976874	+	0.999952i	$0.503110\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3129.27	0.100918
$988$	0	0
$989$	2081.69	0.0669300
$990$	0	0
$991$	−15764.3	−0.505318	−0.252659	−	0.967555i	$-0.581305\pi$
$991$	−15764.3	−0.505318	−0.252659	+	0.967555i	$0.581305\pi$
$992$	0	0
$993$	−4909.73	−0.156904
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33598.4	1.06728	0.533638	−	0.845713i	$-0.320825\pi$
$997$	33598.4	1.06728	0.533638	+	0.845713i	$0.320825\pi$
$998$	0	0
$999$	−1511.30	−0.0478632

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bf.1.3		3
4.3	odd	2		2400.4.a.bw.1.1		3
5.2	odd	4		480.4.f.e.289.6	yes	6
5.3	odd	4		480.4.f.e.289.3	yes	6
5.4	even	2		2400.4.a.bx.1.1		3
15.2	even	4		1440.4.f.i.289.1		6
15.8	even	4		1440.4.f.i.289.2		6
20.3	even	4		480.4.f.d.289.6	yes	6
20.7	even	4		480.4.f.d.289.3	✓	6
20.19	odd	2		2400.4.a.be.1.3		3
40.3	even	4		960.4.f.s.769.1		6
40.13	odd	4		960.4.f.r.769.4		6
40.27	even	4		960.4.f.s.769.4		6
40.37	odd	4		960.4.f.r.769.1		6
60.23	odd	4		1440.4.f.j.289.2		6
60.47	odd	4		1440.4.f.j.289.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.f.d.289.3	✓	6	20.7	even	4
480.4.f.d.289.6	yes	6	20.3	even	4
480.4.f.e.289.3	yes	6	5.3	odd	4
480.4.f.e.289.6	yes	6	5.2	odd	4
960.4.f.r.769.1		6	40.37	odd	4
960.4.f.r.769.4		6	40.13	odd	4
960.4.f.s.769.1		6	40.3	even	4
960.4.f.s.769.4		6	40.27	even	4
1440.4.f.i.289.1		6	15.2	even	4
1440.4.f.i.289.2		6	15.8	even	4
1440.4.f.j.289.1		6	60.47	odd	4
1440.4.f.j.289.2		6	60.23	odd	4
2400.4.a.be.1.3		3	20.19	odd	2
2400.4.a.bf.1.3		3	1.1	even	1	trivial
2400.4.a.bw.1.1		3	4.3	odd	2
2400.4.a.bx.1.1		3	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +6.81960 q^{7} +9.00000 q^{9} -25.7967 q^{11} -32.1575 q^{13} +9.16680 q^{17} +99.2784 q^{19} -20.4588 q^{21} -61.2597 q^{23} -27.0000 q^{27} +17.1991 q^{29} -74.9635 q^{31} +77.3900 q^{33} +55.9740 q^{37} +96.4724 q^{39} -54.1960 q^{41} -33.9813 q^{43} -152.955 q^{47} -296.493 q^{49} -27.5004 q^{51} -26.7289 q^{53} -297.835 q^{57} +567.984 q^{59} -249.963 q^{61} +61.3764 q^{63} +987.608 q^{67} +183.779 q^{69} -705.600 q^{71} +278.004 q^{73} -175.923 q^{77} +1137.94 q^{79} +81.0000 q^{81} -1058.03 q^{83} -51.5972 q^{87} +4.63145 q^{89} -219.301 q^{91} +224.890 q^{93} -1459.17 q^{97} -232.170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 32 q^{7} + 27 q^{9} + 48 q^{11} - 76 q^{13} - 16 q^{17} + 88 q^{19} + 96 q^{21} - 20 q^{23} - 81 q^{27} + 58 q^{29} - 56 q^{31} - 144 q^{33} - 436 q^{37} + 228 q^{39} + 362 q^{41} - 148 q^{43}+ \cdots + 432 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 32 * q^7 + 27 * q^9 + 48 * q^11 - 76 * q^13 - 16 * q^17 + 88 * q^19 + 96 * q^21 - 20 * q^23 - 81 * q^27 + 58 * q^29 - 56 * q^31 - 144 * q^33 - 436 * q^37 + 228 * q^39 + 362 * q^41 - 148 * q^43 - 52 * q^47 - 109 * q^49 + 48 * q^51 - 940 * q^53 - 264 * q^57 + 816 * q^59 - 842 * q^61 - 288 * q^63 + 780 * q^67 + 60 * q^69 + 304 * q^71 - 512 * q^73 - 1536 * q^77 - 440 * q^79 + 243 * q^81 + 1212 * q^83 - 174 * q^87 - 590 * q^89 + 944 * q^91 + 168 * q^93 - 1752 * q^97 + 432 * q^99

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