LMFDB - Embedded newform 1472.4.a.bg.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,4,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1472.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.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:	$86.8508115285$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 $ x^8 - 4x^7 - 137x^6 + 344x^5 + 6175x^4 - 7924x^3 - 89643x^2 + 45072*x + 51084
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 736)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.557812$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.55781 q^{3} +15.7437 q^{5} -25.1863 q^{7} -20.4576 q^{9} -15.8480 q^{11} -15.4997 q^{13} -40.2693 q^{15} +56.7767 q^{17} +107.137 q^{19} +64.4219 q^{21} -23.0000 q^{23} +122.863 q^{25} +121.388 q^{27} +267.565 q^{29} +35.0813 q^{31} +40.5361 q^{33} -396.525 q^{35} -84.9213 q^{37} +39.6452 q^{39} +296.807 q^{41} -353.475 q^{43} -322.078 q^{45} -86.6482 q^{47} +291.352 q^{49} -145.224 q^{51} +126.747 q^{53} -249.505 q^{55} -274.036 q^{57} -853.472 q^{59} -647.571 q^{61} +515.252 q^{63} -244.021 q^{65} -603.269 q^{67} +58.8297 q^{69} -467.385 q^{71} +301.578 q^{73} -314.260 q^{75} +399.152 q^{77} +766.955 q^{79} +241.868 q^{81} -660.843 q^{83} +893.873 q^{85} -684.381 q^{87} -1101.08 q^{89} +390.380 q^{91} -89.7314 q^{93} +1686.73 q^{95} -1494.52 q^{97} +324.211 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) $ 8 * q - 12 * q^3 + 12 * q^5 - 14 * q^7 + 90 * q^9 - 88 * q^11 + 30 * q^13 + 30 * q^15 + 58 * q^17 - 190 * q^19 + 66 * q^21 - 184 * q^23 + 28 * q^25 - 432 * q^27 - 190 * q^29 - 60 * q^31 + 346 * q^33 - 192 * q^35 + 156 * q^37 + 156 * q^39 + 282 * q^41 - 810 * q^43 - 358 * q^45 + 564 * q^47 + 752 * q^49 - 1614 * q^51 + 230 * q^53 + 1292 * q^55 + 796 * q^57 - 1916 * q^59 - 22 * q^61 + 1036 * q^63 + 222 * q^65 - 2292 * q^67 + 276 * q^69 + 2376 * q^71 - 630 * q^73 - 3952 * q^75 + 948 * q^77 + 892 * q^79 - 1044 * q^81 - 2800 * q^83 - 288 * q^85 + 2756 * q^87 - 308 * q^89 - 4610 * q^91 + 982 * q^93 + 1300 * q^95 + 854 * q^97 - 5986 * 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
$3$	−2.55781	−0.492251	−0.246126	−	0.969238i	$-0.579158\pi$
$3$	−2.55781	−0.492251	−0.246126	+	0.969238i	$0.579158\pi$
$4$	0	0
$5$	15.7437	1.40816	0.704078	−	0.710123i	$-0.251360\pi$
$5$	15.7437	1.40816	0.704078	+	0.710123i	$0.251360\pi$
$6$	0	0
$7$	−25.1863	−1.35993	−0.679967	−	0.733242i	$-0.738006\pi$
$7$	−25.1863	−1.35993	−0.679967	+	0.733242i	$0.738006\pi$
$8$	0	0
$9$	−20.4576	−0.757689
$10$	0	0
$11$	−15.8480	−0.434394	−0.217197	−	0.976128i	$-0.569691\pi$
$11$	−15.8480	−0.434394	−0.217197	+	0.976128i	$0.569691\pi$
$12$	0	0
$13$	−15.4997	−0.330679	−0.165340	−	0.986237i	$-0.552872\pi$
$13$	−15.4997	−0.330679	−0.165340	+	0.986237i	$0.552872\pi$
$14$	0	0
$15$	−40.2693	−0.693167
$16$	0	0
$17$	56.7767	0.810021	0.405011	−	0.914312i	$-0.367268\pi$
$17$	56.7767	0.810021	0.405011	+	0.914312i	$0.367268\pi$
$18$	0	0
$19$	107.137	1.29362	0.646812	−	0.762649i	$-0.276102\pi$
$19$	107.137	1.29362	0.646812	+	0.762649i	$0.276102\pi$
$20$	0	0
$21$	64.4219	0.669430
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	122.863	0.982904
$26$	0	0
$27$	121.388	0.865224
$28$	0	0
$29$	267.565	1.71330	0.856648	−	0.515901i	$-0.172543\pi$
$29$	267.565	1.71330	0.856648	+	0.515901i	$0.172543\pi$
$30$	0	0
$31$	35.0813	0.203251	0.101626	−	0.994823i	$-0.467596\pi$
$31$	35.0813	0.203251	0.101626	+	0.994823i	$0.467596\pi$
$32$	0	0
$33$	40.5361	0.213831
$34$	0	0
$35$	−396.525	−1.91500
$36$	0	0
$37$	−84.9213	−0.377324	−0.188662	−	0.982042i	$-0.560415\pi$
$37$	−84.9213	−0.377324	−0.188662	+	0.982042i	$0.560415\pi$
$38$	0	0
$39$	39.6452	0.162777
$40$	0	0
$41$	296.807	1.13057	0.565286	−	0.824895i	$-0.308766\pi$
$41$	296.807	1.13057	0.565286	+	0.824895i	$0.308766\pi$
$42$	0	0
$43$	−353.475	−1.25359	−0.626795	−	0.779184i	$-0.715634\pi$
$43$	−353.475	−1.25359	−0.626795	+	0.779184i	$0.715634\pi$
$44$	0	0
$45$	−322.078	−1.06694
$46$	0	0
$47$	−86.6482	−0.268913	−0.134457	−	0.990919i	$-0.542929\pi$
$47$	−86.6482	−0.268913	−0.134457	+	0.990919i	$0.542929\pi$
$48$	0	0
$49$	291.352	0.849423
$50$	0	0
$51$	−145.224	−0.398734
$52$	0	0
$53$	126.747	0.328491	0.164246	−	0.986419i	$-0.447481\pi$
$53$	126.747	0.328491	0.164246	+	0.986419i	$0.447481\pi$
$54$	0	0
$55$	−249.505	−0.611695
$56$	0	0
$57$	−274.036	−0.636788
$58$	0	0
$59$	−853.472	−1.88326	−0.941632	−	0.336644i	$-0.890708\pi$
$59$	−853.472	−1.88326	−0.941632	+	0.336644i	$0.890708\pi$
$60$	0	0
$61$	−647.571	−1.35923	−0.679614	−	0.733570i	$-0.737853\pi$
$61$	−647.571	−1.35923	−0.679614	+	0.733570i	$0.737853\pi$
$62$	0	0
$63$	515.252	1.03041
$64$	0	0
$65$	−244.021	−0.465648
$66$	0	0
$67$	−603.269	−1.10002	−0.550008	−	0.835160i	$-0.685375\pi$
$67$	−603.269	−1.10002	−0.550008	+	0.835160i	$0.685375\pi$
$68$	0	0
$69$	58.8297	0.102641
$70$	0	0
$71$	−467.385	−0.781245	−0.390623	−	0.920551i	$-0.627740\pi$
$71$	−467.385	−0.781245	−0.390623	+	0.920551i	$0.627740\pi$
$72$	0	0
$73$	301.578	0.483522	0.241761	−	0.970336i	$-0.422275\pi$
$73$	301.578	0.483522	0.241761	+	0.970336i	$0.422275\pi$
$74$	0	0
$75$	−314.260	−0.483836
$76$	0	0
$77$	399.152	0.590748
$78$	0	0
$79$	766.955	1.09227	0.546134	−	0.837698i	$-0.316099\pi$
$79$	766.955	1.09227	0.546134	+	0.837698i	$0.316099\pi$
$80$	0	0
$81$	241.868	0.331781
$82$	0	0
$83$	−660.843	−0.873939	−0.436969	−	0.899476i	$-0.643948\pi$
$83$	−660.843	−0.873939	−0.436969	+	0.899476i	$0.643948\pi$
$84$	0	0
$85$	893.873	1.14064
$86$	0	0
$87$	−684.381	−0.843372
$88$	0	0
$89$	−1101.08	−1.31140	−0.655700	−	0.755021i	$-0.727626\pi$
$89$	−1101.08	−1.31140	−0.655700	+	0.755021i	$0.727626\pi$
$90$	0	0
$91$	390.380	0.449702
$92$	0	0
$93$	−89.7314	−0.100051
$94$	0	0
$95$	1686.73	1.82163
$96$	0	0
$97$	−1494.52	−1.56438	−0.782192	−	0.623037i	$-0.785899\pi$
$97$	−1494.52	−1.56438	−0.782192	+	0.623037i	$0.785899\pi$
$98$	0	0
$99$	324.211	0.329136
$100$	0	0
$101$	1184.71	1.16716	0.583578	−	0.812057i	$-0.301652\pi$
$101$	1184.71	1.16716	0.583578	+	0.812057i	$0.301652\pi$
$102$	0	0
$103$	374.047	0.357825	0.178912	−	0.983865i	$-0.442742\pi$
$103$	374.047	0.357825	0.178912	+	0.983865i	$0.442742\pi$
$104$	0	0
$105$	1014.24	0.942661
$106$	0	0
$107$	−2052.88	−1.85476	−0.927380	−	0.374122i	$-0.877944\pi$
$107$	−2052.88	−1.85476	−0.927380	+	0.374122i	$0.877944\pi$
$108$	0	0
$109$	−547.325	−0.480956	−0.240478	−	0.970655i	$-0.577304\pi$
$109$	−547.325	−0.480956	−0.240478	+	0.970655i	$0.577304\pi$
$110$	0	0
$111$	217.213	0.185738
$112$	0	0
$113$	250.013	0.208135	0.104067	−	0.994570i	$-0.466814\pi$
$113$	250.013	0.208135	0.104067	+	0.994570i	$0.466814\pi$
$114$	0	0
$115$	−362.104	−0.293621
$116$	0	0
$117$	317.086	0.250552
$118$	0	0
$119$	−1430.00	−1.10158
$120$	0	0
$121$	−1079.84	−0.811302
$122$	0	0
$123$	−759.176	−0.556525
$124$	0	0
$125$	−33.6445	−0.0240741
$126$	0	0
$127$	654.293	0.457158	0.228579	−	0.973525i	$-0.426592\pi$
$127$	654.293	0.457158	0.228579	+	0.973525i	$0.426592\pi$
$128$	0	0
$129$	904.122	0.617081
$130$	0	0
$131$	1679.16	1.11992	0.559958	−	0.828521i	$-0.310817\pi$
$131$	1679.16	1.11992	0.559958	+	0.828521i	$0.310817\pi$
$132$	0	0
$133$	−2698.38	−1.75924
$134$	0	0
$135$	1911.09	1.21837
$136$	0	0
$137$	−1344.51	−0.838463	−0.419232	−	0.907879i	$-0.637700\pi$
$137$	−1344.51	−0.838463	−0.419232	+	0.907879i	$0.637700\pi$
$138$	0	0
$139$	1729.97	1.05564	0.527821	−	0.849356i	$-0.323009\pi$
$139$	1729.97	1.05564	0.527821	+	0.849356i	$0.323009\pi$
$140$	0	0
$141$	221.630	0.132373
$142$	0	0
$143$	245.638	0.143645
$144$	0	0
$145$	4212.46	2.41259
$146$	0	0
$147$	−745.224	−0.418129
$148$	0	0
$149$	−272.695	−0.149933	−0.0749667	−	0.997186i	$-0.523885\pi$
$149$	−272.695	−0.149933	−0.0749667	+	0.997186i	$0.523885\pi$
$150$	0	0
$151$	−1696.91	−0.914519	−0.457259	−	0.889333i	$-0.651169\pi$
$151$	−1696.91	−0.914519	−0.457259	+	0.889333i	$0.651169\pi$
$152$	0	0
$153$	−1161.51	−0.613744
$154$	0	0
$155$	552.308	0.286209
$156$	0	0
$157$	2111.93	1.07357	0.536784	−	0.843720i	$-0.319639\pi$
$157$	2111.93	1.07357	0.536784	+	0.843720i	$0.319639\pi$
$158$	0	0
$159$	−324.195	−0.161700
$160$	0	0
$161$	579.286	0.283566
$162$	0	0
$163$	−726.275	−0.348995	−0.174498	−	0.984658i	$-0.555830\pi$
$163$	−726.275	−0.348995	−0.174498	+	0.984658i	$0.555830\pi$
$164$	0	0
$165$	638.187	0.301108
$166$	0	0
$167$	1282.83	0.594423	0.297212	−	0.954812i	$-0.403943\pi$
$167$	1282.83	0.594423	0.297212	+	0.954812i	$0.403943\pi$
$168$	0	0
$169$	−1956.76	−0.890651
$170$	0	0
$171$	−2191.76	−0.980165
$172$	0	0
$173$	−1437.63	−0.631797	−0.315899	−	0.948793i	$-0.602306\pi$
$173$	−1437.63	−0.631797	−0.315899	+	0.948793i	$0.602306\pi$
$174$	0	0
$175$	−3094.47	−1.33669
$176$	0	0
$177$	2183.02	0.927039
$178$	0	0
$179$	−4243.83	−1.77206	−0.886031	−	0.463627i	$-0.846548\pi$
$179$	−4243.83	−1.77206	−0.886031	+	0.463627i	$0.846548\pi$
$180$	0	0
$181$	−693.537	−0.284808	−0.142404	−	0.989809i	$-0.545483\pi$
$181$	−693.537	−0.284808	−0.142404	+	0.989809i	$0.545483\pi$
$182$	0	0
$183$	1656.36	0.669082
$184$	0	0
$185$	−1336.97	−0.531331
$186$	0	0
$187$	−899.794	−0.351869
$188$	0	0
$189$	−3057.31	−1.17665
$190$	0	0
$191$	1692.66	0.641239	0.320620	−	0.947208i	$-0.396109\pi$
$191$	1692.66	0.641239	0.320620	+	0.947208i	$0.396109\pi$
$192$	0	0
$193$	1028.12	0.383447	0.191724	−	0.981449i	$-0.438592\pi$
$193$	1028.12	0.383447	0.191724	+	0.981449i	$0.438592\pi$
$194$	0	0
$195$	624.161	0.229216
$196$	0	0
$197$	548.484	0.198365	0.0991823	−	0.995069i	$-0.468377\pi$
$197$	548.484	0.198365	0.0991823	+	0.995069i	$0.468377\pi$
$198$	0	0
$199$	−2784.57	−0.991925	−0.495962	−	0.868344i	$-0.665185\pi$
$199$	−2784.57	−0.991925	−0.495962	+	0.868344i	$0.665185\pi$
$200$	0	0
$201$	1543.05	0.541484
$202$	0	0
$203$	−6738.99	−2.32997
$204$	0	0
$205$	4672.83	1.59202
$206$	0	0
$207$	470.525	0.157989
$208$	0	0
$209$	−1697.90	−0.561943
$210$	0	0
$211$	3344.97	1.09136	0.545681	−	0.837993i	$-0.316271\pi$
$211$	3344.97	1.09136	0.545681	+	0.837993i	$0.316271\pi$
$212$	0	0
$213$	1195.48	0.384569
$214$	0	0
$215$	−5564.98	−1.76525
$216$	0	0
$217$	−883.569	−0.276408
$218$	0	0
$219$	−771.381	−0.238014
$220$	0	0
$221$	−880.019	−0.267857
$222$	0	0
$223$	−3449.19	−1.03576	−0.517881	−	0.855453i	$-0.673279\pi$
$223$	−3449.19	−1.03576	−0.517881	+	0.855453i	$0.673279\pi$
$224$	0	0
$225$	−2513.48	−0.744735
$226$	0	0
$227$	−2897.08	−0.847075	−0.423538	−	0.905879i	$-0.639212\pi$
$227$	−2897.08	−0.847075	−0.423538	+	0.905879i	$0.639212\pi$
$228$	0	0
$229$	461.258	0.133104	0.0665519	−	0.997783i	$-0.478800\pi$
$229$	461.258	0.133104	0.0665519	+	0.997783i	$0.478800\pi$
$230$	0	0
$231$	−1020.96	−0.290796
$232$	0	0
$233$	−889.861	−0.250200	−0.125100	−	0.992144i	$-0.539925\pi$
$233$	−889.861	−0.250200	−0.125100	+	0.992144i	$0.539925\pi$
$234$	0	0
$235$	−1364.16	−0.378672
$236$	0	0
$237$	−1961.73	−0.537670
$238$	0	0
$239$	4239.79	1.14749	0.573744	−	0.819035i	$-0.305491\pi$
$239$	4239.79	1.14749	0.573744	+	0.819035i	$0.305491\pi$
$240$	0	0
$241$	4493.26	1.20098	0.600491	−	0.799632i	$-0.294972\pi$
$241$	4493.26	1.20098	0.600491	+	0.799632i	$0.294972\pi$
$242$	0	0
$243$	−3896.12	−1.02854
$244$	0	0
$245$	4586.95	1.19612
$246$	0	0
$247$	−1660.58	−0.427775
$248$	0	0
$249$	1690.31	0.430197
$250$	0	0
$251$	−5012.13	−1.26041	−0.630205	−	0.776429i	$-0.717029\pi$
$251$	−5012.13	−1.26041	−0.630205	+	0.776429i	$0.717029\pi$
$252$	0	0
$253$	364.503	0.0905775
$254$	0	0
$255$	−2286.36	−0.561480
$256$	0	0
$257$	−3557.61	−0.863493	−0.431746	−	0.901995i	$-0.642102\pi$
$257$	−3557.61	−0.863493	−0.431746	+	0.901995i	$0.642102\pi$
$258$	0	0
$259$	2138.86	0.513136
$260$	0	0
$261$	−5473.74	−1.29815
$262$	0	0
$263$	−6499.08	−1.52377	−0.761883	−	0.647715i	$-0.775725\pi$
$263$	−6499.08	−1.52377	−0.761883	+	0.647715i	$0.775725\pi$
$264$	0	0
$265$	1995.46	0.462567
$266$	0	0
$267$	2816.37	0.645538
$268$	0	0
$269$	−1314.12	−0.297856	−0.148928	−	0.988848i	$-0.547582\pi$
$269$	−1314.12	−0.297856	−0.148928	+	0.988848i	$0.547582\pi$
$270$	0	0
$271$	2987.30	0.669614	0.334807	−	0.942287i	$-0.391329\pi$
$271$	2987.30	0.669614	0.334807	+	0.942287i	$0.391329\pi$
$272$	0	0
$273$	−998.518	−0.221367
$274$	0	0
$275$	−1947.13	−0.426968
$276$	0	0
$277$	−3036.46	−0.658641	−0.329320	−	0.944218i	$-0.606820\pi$
$277$	−3036.46	−0.658641	−0.329320	+	0.944218i	$0.606820\pi$
$278$	0	0
$279$	−717.679	−0.154001
$280$	0	0
$281$	2686.34	0.570298	0.285149	−	0.958483i	$-0.407957\pi$
$281$	2686.34	0.570298	0.285149	+	0.958483i	$0.407957\pi$
$282$	0	0
$283$	2167.83	0.455349	0.227675	−	0.973737i	$-0.426888\pi$
$283$	2167.83	0.455349	0.227675	+	0.973737i	$0.426888\pi$
$284$	0	0
$285$	−4314.33	−0.896697
$286$	0	0
$287$	−7475.48	−1.53750
$288$	0	0
$289$	−1689.41	−0.343865
$290$	0	0
$291$	3822.70	0.770070
$292$	0	0
$293$	−7312.10	−1.45794	−0.728972	−	0.684543i	$-0.760002\pi$
$293$	−7312.10	−1.45794	−0.728972	+	0.684543i	$0.760002\pi$
$294$	0	0
$295$	−13436.8	−2.65193
$296$	0	0
$297$	−1923.75	−0.375849
$298$	0	0
$299$	356.492	0.0689514
$300$	0	0
$301$	8902.73	1.70480
$302$	0	0
$303$	−3030.26	−0.574534
$304$	0	0
$305$	−10195.1	−1.91401
$306$	0	0
$307$	4204.98	0.781729	0.390864	−	0.920448i	$-0.372176\pi$
$307$	4204.98	0.781729	0.390864	+	0.920448i	$0.372176\pi$
$308$	0	0
$309$	−956.742	−0.176140
$310$	0	0
$311$	−6334.77	−1.15502	−0.577511	−	0.816383i	$-0.695976\pi$
$311$	−6334.77	−1.15502	−0.577511	+	0.816383i	$0.695976\pi$
$312$	0	0
$313$	−3328.27	−0.601038	−0.300519	−	0.953776i	$-0.597160\pi$
$313$	−3328.27	−0.601038	−0.300519	+	0.953776i	$0.597160\pi$
$314$	0	0
$315$	8111.96	1.45097
$316$	0	0
$317$	−8132.70	−1.44094	−0.720470	−	0.693487i	$-0.756074\pi$
$317$	−8132.70	−1.44094	−0.720470	+	0.693487i	$0.756074\pi$
$318$	0	0
$319$	−4240.36	−0.744246
$320$	0	0
$321$	5250.88	0.913007
$322$	0	0
$323$	6082.87	1.04786
$324$	0	0
$325$	−1904.33	−0.325026
$326$	0	0
$327$	1399.95	0.236751
$328$	0	0
$329$	2182.35	0.365705
$330$	0	0
$331$	−6300.52	−1.04625	−0.523124	−	0.852257i	$-0.675234\pi$
$331$	−6300.52	−1.04625	−0.523124	+	0.852257i	$0.675234\pi$
$332$	0	0
$333$	1737.29	0.285894
$334$	0	0
$335$	−9497.66	−1.54899
$336$	0	0
$337$	−8473.63	−1.36970	−0.684848	−	0.728686i	$-0.740132\pi$
$337$	−8473.63	−1.36970	−0.684848	+	0.728686i	$0.740132\pi$
$338$	0	0
$339$	−639.486	−0.102455
$340$	0	0
$341$	−555.967	−0.0882911
$342$	0	0
$343$	1300.82	0.204775
$344$	0	0
$345$	926.195	0.144535
$346$	0	0
$347$	2859.61	0.442398	0.221199	−	0.975229i	$-0.429003\pi$
$347$	2859.61	0.442398	0.221199	+	0.975229i	$0.429003\pi$
$348$	0	0
$349$	10651.8	1.63375	0.816876	−	0.576814i	$-0.195704\pi$
$349$	10651.8	1.63375	0.816876	+	0.576814i	$0.195704\pi$
$350$	0	0
$351$	−1881.47	−0.286112
$352$	0	0
$353$	7723.04	1.16446	0.582232	−	0.813022i	$-0.302179\pi$
$353$	7723.04	1.16446	0.582232	+	0.813022i	$0.302179\pi$
$354$	0	0
$355$	−7358.36	−1.10012
$356$	0	0
$357$	3657.66	0.542252
$358$	0	0
$359$	5468.59	0.803959	0.401979	−	0.915649i	$-0.368322\pi$
$359$	5468.59	0.803959	0.401979	+	0.915649i	$0.368322\pi$
$360$	0	0
$361$	4619.29	0.673464
$362$	0	0
$363$	2762.03	0.399364
$364$	0	0
$365$	4747.95	0.680874
$366$	0	0
$367$	13353.5	1.89931	0.949656	−	0.313295i	$-0.101433\pi$
$367$	13353.5	1.89931	0.949656	+	0.313295i	$0.101433\pi$
$368$	0	0
$369$	−6071.95	−0.856621
$370$	0	0
$371$	−3192.29	−0.446727
$372$	0	0
$373$	−4189.05	−0.581503	−0.290752	−	0.956799i	$-0.593905\pi$
$373$	−4189.05	−0.581503	−0.290752	+	0.956799i	$0.593905\pi$
$374$	0	0
$375$	86.0564	0.0118505
$376$	0	0
$377$	−4147.17	−0.566552
$378$	0	0
$379$	10371.6	1.40568	0.702839	−	0.711349i	$-0.251915\pi$
$379$	10371.6	1.40568	0.702839	+	0.711349i	$0.251915\pi$
$380$	0	0
$381$	−1673.56	−0.225037
$382$	0	0
$383$	6828.72	0.911047	0.455524	−	0.890224i	$-0.349452\pi$
$383$	6828.72	0.911047	0.455524	+	0.890224i	$0.349452\pi$
$384$	0	0
$385$	6284.12	0.831865
$386$	0	0
$387$	7231.24	0.949831
$388$	0	0
$389$	1413.57	0.184244	0.0921221	−	0.995748i	$-0.470635\pi$
$389$	1413.57	0.184244	0.0921221	+	0.995748i	$0.470635\pi$
$390$	0	0
$391$	−1305.86	−0.168901
$392$	0	0
$393$	−4294.98	−0.551280
$394$	0	0
$395$	12074.7	1.53808
$396$	0	0
$397$	12730.3	1.60936	0.804679	−	0.593711i	$-0.202338\pi$
$397$	12730.3	1.60936	0.804679	+	0.593711i	$0.202338\pi$
$398$	0	0
$399$	6901.96	0.865990
$400$	0	0
$401$	11371.4	1.41611	0.708056	−	0.706157i	$-0.249573\pi$
$401$	11371.4	1.41611	0.708056	+	0.706157i	$0.249573\pi$
$402$	0	0
$403$	−543.748	−0.0672110
$404$	0	0
$405$	3807.89	0.467199
$406$	0	0
$407$	1345.83	0.163907
$408$	0	0
$409$	9102.21	1.10043	0.550215	−	0.835023i	$-0.314546\pi$
$409$	9102.21	1.10043	0.550215	+	0.835023i	$0.314546\pi$
$410$	0	0
$411$	3439.01	0.412735
$412$	0	0
$413$	21495.8	2.56112
$414$	0	0
$415$	−10404.1	−1.23064
$416$	0	0
$417$	−4424.94	−0.519641
$418$	0	0
$419$	−15235.3	−1.77635	−0.888176	−	0.459504i	$-0.848027\pi$
$419$	−15235.3	−1.77635	−0.888176	+	0.459504i	$0.848027\pi$
$420$	0	0
$421$	−7711.38	−0.892707	−0.446354	−	0.894857i	$-0.647278\pi$
$421$	−7711.38	−0.892707	−0.446354	+	0.894857i	$0.647278\pi$
$422$	0	0
$423$	1772.61	0.203753
$424$	0	0
$425$	6975.75	0.796173
$426$	0	0
$427$	16309.9	1.84846
$428$	0	0
$429$	−628.296	−0.0707096
$430$	0	0
$431$	2248.78	0.251322	0.125661	−	0.992073i	$-0.459895\pi$
$431$	2248.78	0.251322	0.125661	+	0.992073i	$0.459895\pi$
$432$	0	0
$433$	7405.23	0.821877	0.410938	−	0.911663i	$-0.365201\pi$
$433$	7405.23	0.821877	0.410938	+	0.911663i	$0.365201\pi$
$434$	0	0
$435$	−10774.7	−1.18760
$436$	0	0
$437$	−2464.15	−0.269739
$438$	0	0
$439$	6226.42	0.676927	0.338463	−	0.940980i	$-0.390093\pi$
$439$	6226.42	0.676927	0.338463	+	0.940980i	$0.390093\pi$
$440$	0	0
$441$	−5960.36	−0.643598
$442$	0	0
$443$	−15367.3	−1.64813	−0.824064	−	0.566497i	$-0.808298\pi$
$443$	−15367.3	−1.64813	−0.824064	+	0.566497i	$0.808298\pi$
$444$	0	0
$445$	−17335.1	−1.84666
$446$	0	0
$447$	697.503	0.0738049
$448$	0	0
$449$	−16205.8	−1.70334	−0.851668	−	0.524081i	$-0.824409\pi$
$449$	−16205.8	−1.70334	−0.851668	+	0.524081i	$0.824409\pi$
$450$	0	0
$451$	−4703.78	−0.491114
$452$	0	0
$453$	4340.37	0.450173
$454$	0	0
$455$	6146.01	0.633251
$456$	0	0
$457$	−15393.8	−1.57569	−0.787846	−	0.615872i	$-0.788804\pi$
$457$	−15393.8	−1.57569	−0.787846	+	0.615872i	$0.788804\pi$
$458$	0	0
$459$	6891.98	0.700850
$460$	0	0
$461$	−3078.57	−0.311026	−0.155513	−	0.987834i	$-0.549703\pi$
$461$	−3078.57	−0.311026	−0.155513	+	0.987834i	$0.549703\pi$
$462$	0	0
$463$	12389.8	1.24364	0.621820	−	0.783160i	$-0.286394\pi$
$463$	12389.8	1.24364	0.621820	+	0.783160i	$0.286394\pi$
$464$	0	0
$465$	−1412.70	−0.140887
$466$	0	0
$467$	9711.19	0.962271	0.481135	−	0.876646i	$-0.340225\pi$
$467$	9711.19	0.962271	0.481135	+	0.876646i	$0.340225\pi$
$468$	0	0
$469$	15194.1	1.49595
$470$	0	0
$471$	−5401.91	−0.528465
$472$	0	0
$473$	5601.85	0.544552
$474$	0	0
$475$	13163.1	1.27151
$476$	0	0
$477$	−2592.94	−0.248894
$478$	0	0
$479$	−11418.2	−1.08917	−0.544583	−	0.838707i	$-0.683312\pi$
$479$	−11418.2	−1.08917	−0.544583	+	0.838707i	$0.683312\pi$
$480$	0	0
$481$	1316.25	0.124773
$482$	0	0
$483$	−1481.70	−0.139586
$484$	0	0
$485$	−23529.2	−2.20290
$486$	0	0
$487$	8515.00	0.792303	0.396151	−	0.918185i	$-0.370346\pi$
$487$	8515.00	0.792303	0.396151	+	0.918185i	$0.370346\pi$
$488$	0	0
$489$	1857.68	0.171793
$490$	0	0
$491$	−5868.20	−0.539365	−0.269683	−	0.962949i	$-0.586919\pi$
$491$	−5868.20	−0.539365	−0.269683	+	0.962949i	$0.586919\pi$
$492$	0	0
$493$	15191.5	1.38781
$494$	0	0
$495$	5104.27	0.463474
$496$	0	0
$497$	11771.7	1.06244
$498$	0	0
$499$	−8500.38	−0.762584	−0.381292	−	0.924455i	$-0.624521\pi$
$499$	−8500.38	−0.762584	−0.381292	+	0.924455i	$0.624521\pi$
$500$	0	0
$501$	−3281.25	−0.292606
$502$	0	0
$503$	12383.7	1.09774	0.548869	−	0.835909i	$-0.315059\pi$
$503$	12383.7	1.09774	0.548869	+	0.835909i	$0.315059\pi$
$504$	0	0
$505$	18651.6	1.64354
$506$	0	0
$507$	5005.03	0.438424
$508$	0	0
$509$	−7608.92	−0.662593	−0.331296	−	0.943527i	$-0.607486\pi$
$509$	−7608.92	−0.662593	−0.331296	+	0.943527i	$0.607486\pi$
$510$	0	0
$511$	−7595.66	−0.657558
$512$	0	0
$513$	13005.1	1.11928
$514$	0	0
$515$	5888.87	0.503873
$516$	0	0
$517$	1373.20	0.116814
$518$	0	0
$519$	3677.19	0.311003
$520$	0	0
$521$	19155.0	1.61074	0.805372	−	0.592770i	$-0.201966\pi$
$521$	19155.0	1.61074	0.805372	+	0.592770i	$0.201966\pi$
$522$	0	0
$523$	2705.89	0.226234	0.113117	−	0.993582i	$-0.463917\pi$
$523$	2705.89	0.226234	0.113117	+	0.993582i	$0.463917\pi$
$524$	0	0
$525$	7915.07	0.657985
$526$	0	0
$527$	1991.80	0.164638
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	17460.0	1.42693
$532$	0	0
$533$	−4600.40	−0.373857
$534$	0	0
$535$	−32319.8	−2.61179
$536$	0	0
$537$	10854.9	0.872299
$538$	0	0
$539$	−4617.33	−0.368984
$540$	0	0
$541$	−14435.0	−1.14715	−0.573575	−	0.819153i	$-0.694444\pi$
$541$	−14435.0	−1.14715	−0.573575	+	0.819153i	$0.694444\pi$
$542$	0	0
$543$	1773.94	0.140197
$544$	0	0
$545$	−8616.90	−0.677262
$546$	0	0
$547$	−6800.62	−0.531578	−0.265789	−	0.964031i	$-0.585633\pi$
$547$	−6800.62	−0.531578	−0.265789	+	0.964031i	$0.585633\pi$
$548$	0	0
$549$	13247.7	1.02987
$550$	0	0
$551$	28666.1	2.21636
$552$	0	0
$553$	−19316.8	−1.48541
$554$	0	0
$555$	3419.73	0.261548
$556$	0	0
$557$	−19804.9	−1.50657	−0.753285	−	0.657694i	$-0.771532\pi$
$557$	−19804.9	−1.50657	−0.753285	+	0.657694i	$0.771532\pi$
$558$	0	0
$559$	5478.74	0.414536
$560$	0	0
$561$	2301.50	0.173208
$562$	0	0
$563$	−13125.5	−0.982546	−0.491273	−	0.871006i	$-0.663468\pi$
$563$	−13125.5	−0.982546	−0.491273	+	0.871006i	$0.663468\pi$
$564$	0	0
$565$	3936.12	0.293087
$566$	0	0
$567$	−6091.78	−0.451200
$568$	0	0
$569$	−14961.4	−1.10231	−0.551155	−	0.834403i	$-0.685813\pi$
$569$	−14961.4	−1.10231	−0.551155	+	0.834403i	$0.685813\pi$
$570$	0	0
$571$	−6809.52	−0.499071	−0.249536	−	0.968366i	$-0.580278\pi$
$571$	−6809.52	−0.499071	−0.249536	+	0.968366i	$0.580278\pi$
$572$	0	0
$573$	−4329.51	−0.315651
$574$	0	0
$575$	−2825.85	−0.204950
$576$	0	0
$577$	−21373.0	−1.54206	−0.771032	−	0.636796i	$-0.780259\pi$
$577$	−21373.0	−1.54206	−0.771032	+	0.636796i	$0.780259\pi$
$578$	0	0
$579$	−2629.73	−0.188752
$580$	0	0
$581$	16644.2	1.18850
$582$	0	0
$583$	−2008.68	−0.142695
$584$	0	0
$585$	4992.09	0.352816
$586$	0	0
$587$	1796.18	0.126297	0.0631483	−	0.998004i	$-0.479886\pi$
$587$	1796.18	0.126297	0.0631483	+	0.998004i	$0.479886\pi$
$588$	0	0
$589$	3758.50	0.262931
$590$	0	0
$591$	−1402.92	−0.0976452
$592$	0	0
$593$	14393.9	0.996773	0.498386	−	0.866955i	$-0.333926\pi$
$593$	14393.9	0.996773	0.498386	+	0.866955i	$0.333926\pi$
$594$	0	0
$595$	−22513.4	−1.55119
$596$	0	0
$597$	7122.41	0.488276
$598$	0	0
$599$	−6674.39	−0.455272	−0.227636	−	0.973746i	$-0.573100\pi$
$599$	−6674.39	−0.455272	−0.227636	+	0.973746i	$0.573100\pi$
$600$	0	0
$601$	−10008.1	−0.679263	−0.339631	−	0.940559i	$-0.610302\pi$
$601$	−10008.1	−0.679263	−0.339631	+	0.940559i	$0.610302\pi$
$602$	0	0
$603$	12341.4	0.833469
$604$	0	0
$605$	−17000.7	−1.14244
$606$	0	0
$607$	25442.5	1.70128	0.850642	−	0.525745i	$-0.176213\pi$
$607$	25442.5	1.70128	0.850642	+	0.525745i	$0.176213\pi$
$608$	0	0
$609$	17237.1	1.14693
$610$	0	0
$611$	1343.02	0.0889241
$612$	0	0
$613$	−7269.39	−0.478969	−0.239484	−	0.970900i	$-0.576978\pi$
$613$	−7269.39	−0.478969	−0.239484	+	0.970900i	$0.576978\pi$
$614$	0	0
$615$	−11952.2	−0.783674
$616$	0	0
$617$	−13634.2	−0.889614	−0.444807	−	0.895626i	$-0.646728\pi$
$617$	−13634.2	−0.889614	−0.444807	+	0.895626i	$0.646728\pi$
$618$	0	0
$619$	16180.2	1.05063	0.525313	−	0.850909i	$-0.323948\pi$
$619$	16180.2	1.05063	0.525313	+	0.850909i	$0.323948\pi$
$620$	0	0
$621$	−2791.92	−0.180412
$622$	0	0
$623$	27732.3	1.78342
$624$	0	0
$625$	−15887.6	−1.01680
$626$	0	0
$627$	4342.91	0.276617
$628$	0	0
$629$	−4821.55	−0.305640
$630$	0	0
$631$	4454.90	0.281057	0.140528	−	0.990077i	$-0.455120\pi$
$631$	4454.90	0.281057	0.140528	+	0.990077i	$0.455120\pi$
$632$	0	0
$633$	−8555.81	−0.537224
$634$	0	0
$635$	10301.0	0.643750
$636$	0	0
$637$	−4515.86	−0.280887
$638$	0	0
$639$	9561.58	0.591941
$640$	0	0
$641$	−14638.1	−0.901980	−0.450990	−	0.892529i	$-0.648929\pi$
$641$	−14638.1	−0.901980	−0.450990	+	0.892529i	$0.648929\pi$
$642$	0	0
$643$	−18574.8	−1.13922	−0.569610	−	0.821915i	$-0.692906\pi$
$643$	−18574.8	−1.13922	−0.569610	+	0.821915i	$0.692906\pi$
$644$	0	0
$645$	14234.2	0.868947
$646$	0	0
$647$	−9846.84	−0.598330	−0.299165	−	0.954201i	$-0.596708\pi$
$647$	−9846.84	−0.598330	−0.299165	+	0.954201i	$0.596708\pi$
$648$	0	0
$649$	13525.8	0.818079
$650$	0	0
$651$	2260.00	0.136062
$652$	0	0
$653$	23223.5	1.39174	0.695869	−	0.718169i	$-0.255020\pi$
$653$	23223.5	1.39174	0.695869	+	0.718169i	$0.255020\pi$
$654$	0	0
$655$	26436.1	1.57702
$656$	0	0
$657$	−6169.57	−0.366359
$658$	0	0
$659$	−28368.7	−1.67692	−0.838459	−	0.544965i	$-0.816543\pi$
$659$	−28368.7	−1.67692	−0.838459	+	0.544965i	$0.816543\pi$
$660$	0	0
$661$	−15263.8	−0.898173	−0.449087	−	0.893488i	$-0.648251\pi$
$661$	−15263.8	−0.898173	−0.449087	+	0.893488i	$0.648251\pi$
$662$	0	0
$663$	2250.92	0.131853
$664$	0	0
$665$	−42482.5	−2.47729
$666$	0	0
$667$	−6154.00	−0.357247
$668$	0	0
$669$	8822.38	0.509855
$670$	0	0
$671$	10262.7	0.590441
$672$	0	0
$673$	−11576.8	−0.663082	−0.331541	−	0.943441i	$-0.607569\pi$
$673$	−11576.8	−0.663082	−0.331541	+	0.943441i	$0.607569\pi$
$674$	0	0
$675$	14914.0	0.850432
$676$	0	0
$677$	8245.18	0.468077	0.234039	−	0.972227i	$-0.424806\pi$
$677$	8245.18	0.468077	0.234039	+	0.972227i	$0.424806\pi$
$678$	0	0
$679$	37641.5	2.12746
$680$	0	0
$681$	7410.19	0.416974
$682$	0	0
$683$	34421.5	1.92841	0.964203	−	0.265167i	$-0.0854269\pi$
$683$	34421.5	1.92841	0.964203	+	0.265167i	$0.0854269\pi$
$684$	0	0
$685$	−21167.6	−1.18069
$686$	0	0
$687$	−1179.81	−0.0655205
$688$	0	0
$689$	−1964.54	−0.108625
$690$	0	0
$691$	−28258.4	−1.55571	−0.777857	−	0.628441i	$-0.783693\pi$
$691$	−28258.4	−1.55571	−0.777857	+	0.628441i	$0.783693\pi$
$692$	0	0
$693$	−8165.69	−0.447603
$694$	0	0
$695$	27236.1	1.48651
$696$	0	0
$697$	16851.7	0.915787
$698$	0	0
$699$	2276.10	0.123161
$700$	0	0
$701$	−19513.9	−1.05140	−0.525700	−	0.850670i	$-0.676196\pi$
$701$	−19513.9	−1.05140	−0.525700	+	0.850670i	$0.676196\pi$
$702$	0	0
$703$	−9098.20	−0.488115
$704$	0	0
$705$	3489.26	0.186402
$706$	0	0
$707$	−29838.4	−1.58725
$708$	0	0
$709$	−12679.3	−0.671624	−0.335812	−	0.941929i	$-0.609011\pi$
$709$	−12679.3	−0.671624	−0.335812	+	0.941929i	$0.609011\pi$
$710$	0	0
$711$	−15690.0	−0.827599
$712$	0	0
$713$	−806.870	−0.0423808
$714$	0	0
$715$	3867.24	0.202275
$716$	0	0
$717$	−10844.6	−0.564852
$718$	0	0
$719$	25606.3	1.32817	0.664085	−	0.747657i	$-0.268821\pi$
$719$	25606.3	1.32817	0.664085	+	0.747657i	$0.268821\pi$
$720$	0	0
$721$	−9420.87	−0.486618
$722$	0	0
$723$	−11492.9	−0.591184
$724$	0	0
$725$	32873.8	1.68401
$726$	0	0
$727$	5687.29	0.290138	0.145069	−	0.989422i	$-0.453660\pi$
$727$	5687.29	0.290138	0.145069	+	0.989422i	$0.453660\pi$
$728$	0	0
$729$	3435.10	0.174521
$730$	0	0
$731$	−20069.1	−1.01543
$732$	0	0
$733$	−26753.0	−1.34808	−0.674040	−	0.738694i	$-0.735443\pi$
$733$	−26753.0	−1.34808	−0.674040	+	0.738694i	$0.735443\pi$
$734$	0	0
$735$	−11732.6	−0.588791
$736$	0	0
$737$	9560.58	0.477840
$738$	0	0
$739$	18859.2	0.938763	0.469381	−	0.882996i	$-0.344477\pi$
$739$	18859.2	0.938763	0.469381	+	0.882996i	$0.344477\pi$
$740$	0	0
$741$	4247.46	0.210573
$742$	0	0
$743$	−19561.0	−0.965848	−0.482924	−	0.875662i	$-0.660425\pi$
$743$	−19561.0	−0.965848	−0.482924	+	0.875662i	$0.660425\pi$
$744$	0	0
$745$	−4293.22	−0.211130
$746$	0	0
$747$	13519.3	0.662173
$748$	0	0
$749$	51704.5	2.52235
$750$	0	0
$751$	14946.6	0.726242	0.363121	−	0.931742i	$-0.381711\pi$
$751$	14946.6	0.726242	0.363121	+	0.931742i	$0.381711\pi$
$752$	0	0
$753$	12820.1	0.620438
$754$	0	0
$755$	−26715.5	−1.28778
$756$	0	0
$757$	−13741.0	−0.659743	−0.329871	−	0.944026i	$-0.607005\pi$
$757$	−13741.0	−0.659743	−0.329871	+	0.944026i	$0.607005\pi$
$758$	0	0
$759$	−932.330	−0.0445869
$760$	0	0
$761$	−25846.0	−1.23116	−0.615582	−	0.788073i	$-0.711079\pi$
$761$	−25846.0	−1.23116	−0.615582	+	0.788073i	$0.711079\pi$
$762$	0	0
$763$	13785.1	0.654069
$764$	0	0
$765$	−18286.5	−0.864247
$766$	0	0
$767$	13228.5	0.622757
$768$	0	0
$769$	16125.9	0.756195	0.378097	−	0.925766i	$-0.376578\pi$
$769$	16125.9	0.756195	0.378097	+	0.925766i	$0.376578\pi$
$770$	0	0
$771$	9099.70	0.425055
$772$	0	0
$773$	−8845.77	−0.411592	−0.205796	−	0.978595i	$-0.565978\pi$
$773$	−8845.77	−0.411592	−0.205796	+	0.978595i	$0.565978\pi$
$774$	0	0
$775$	4310.19	0.199776
$776$	0	0
$777$	−5470.80	−0.252592
$778$	0	0
$779$	31798.9	1.46253
$780$	0	0
$781$	7407.10	0.339369
$782$	0	0
$783$	32479.1	1.48239
$784$	0	0
$785$	33249.5	1.51175
$786$	0	0
$787$	15915.8	0.720885	0.360443	−	0.932781i	$-0.382626\pi$
$787$	15915.8	0.720885	0.360443	+	0.932781i	$0.382626\pi$
$788$	0	0
$789$	16623.4	0.750076
$790$	0	0
$791$	−6296.91	−0.283050
$792$	0	0
$793$	10037.1	0.449469
$794$	0	0
$795$	−5104.02	−0.227699
$796$	0	0
$797$	16815.4	0.747342	0.373671	−	0.927561i	$-0.378099\pi$
$797$	16815.4	0.747342	0.373671	+	0.927561i	$0.378099\pi$
$798$	0	0
$799$	−4919.59	−0.217826
$800$	0	0
$801$	22525.5	0.993633
$802$	0	0
$803$	−4779.40	−0.210039
$804$	0	0
$805$	9120.08	0.399305
$806$	0	0
$807$	3361.27	0.146620
$808$	0	0
$809$	1908.47	0.0829395	0.0414698	−	0.999140i	$-0.486796\pi$
$809$	1908.47	0.0829395	0.0414698	+	0.999140i	$0.486796\pi$
$810$	0	0
$811$	−18224.0	−0.789066	−0.394533	−	0.918882i	$-0.629094\pi$
$811$	−18224.0	−0.789066	−0.394533	+	0.918882i	$0.629094\pi$
$812$	0	0
$813$	−7640.95	−0.329618
$814$	0	0
$815$	−11434.2	−0.491440
$816$	0	0
$817$	−37870.1	−1.62167
$818$	0	0
$819$	−7986.23	−0.340734
$820$	0	0
$821$	16782.4	0.713410	0.356705	−	0.934217i	$-0.383900\pi$
$821$	16782.4	0.713410	0.356705	+	0.934217i	$0.383900\pi$
$822$	0	0
$823$	−35573.1	−1.50668	−0.753341	−	0.657630i	$-0.771559\pi$
$823$	−35573.1	−1.50668	−0.753341	+	0.657630i	$0.771559\pi$
$824$	0	0
$825$	4980.38	0.210175
$826$	0	0
$827$	23590.3	0.991918	0.495959	−	0.868346i	$-0.334817\pi$
$827$	23590.3	0.991918	0.495959	+	0.868346i	$0.334817\pi$
$828$	0	0
$829$	20356.1	0.852829	0.426414	−	0.904528i	$-0.359777\pi$
$829$	20356.1	0.852829	0.426414	+	0.904528i	$0.359777\pi$
$830$	0	0
$831$	7766.70	0.324217
$832$	0	0
$833$	16542.0	0.688050
$834$	0	0
$835$	20196.5	0.837041
$836$	0	0
$837$	4258.43	0.175858
$838$	0	0
$839$	43791.0	1.80195	0.900974	−	0.433874i	$-0.142854\pi$
$839$	43791.0	1.80195	0.900974	+	0.433874i	$0.142854\pi$
$840$	0	0
$841$	47202.1	1.93538
$842$	0	0
$843$	−6871.16	−0.280730
$844$	0	0
$845$	−30806.6	−1.25418
$846$	0	0
$847$	27197.3	1.10332
$848$	0	0
$849$	−5544.89	−0.224146
$850$	0	0
$851$	1953.19	0.0786775
$852$	0	0
$853$	−4492.42	−0.180325	−0.0901627	−	0.995927i	$-0.528739\pi$
$853$	−4492.42	−0.180325	−0.0901627	+	0.995927i	$0.528739\pi$
$854$	0	0
$855$	−34506.3	−1.38022
$856$	0	0
$857$	33204.5	1.32351	0.661753	−	0.749722i	$-0.269813\pi$
$857$	33204.5	1.32351	0.661753	+	0.749722i	$0.269813\pi$
$858$	0	0
$859$	7947.14	0.315661	0.157831	−	0.987466i	$-0.449550\pi$
$859$	7947.14	0.315661	0.157831	+	0.987466i	$0.449550\pi$
$860$	0	0
$861$	19120.9	0.756838
$862$	0	0
$863$	10840.6	0.427599	0.213799	−	0.976878i	$-0.431416\pi$
$863$	10840.6	0.427599	0.213799	+	0.976878i	$0.431416\pi$
$864$	0	0
$865$	−22633.6	−0.889669
$866$	0	0
$867$	4321.20	0.169268
$868$	0	0
$869$	−12154.7	−0.474475
$870$	0	0
$871$	9350.46	0.363752
$872$	0	0
$873$	30574.3	1.18532
$874$	0	0
$875$	847.383	0.0327392
$876$	0	0
$877$	−11412.4	−0.439419	−0.219709	−	0.975565i	$-0.570511\pi$
$877$	−11412.4	−0.439419	−0.219709	+	0.975565i	$0.570511\pi$
$878$	0	0
$879$	18703.0	0.717675
$880$	0	0
$881$	2502.70	0.0957073	0.0478537	−	0.998854i	$-0.484762\pi$
$881$	2502.70	0.0957073	0.0478537	+	0.998854i	$0.484762\pi$
$882$	0	0
$883$	52137.0	1.98703	0.993517	−	0.113684i	$-0.0362651\pi$
$883$	52137.0	1.98703	0.993517	+	0.113684i	$0.0362651\pi$
$884$	0	0
$885$	34368.7	1.30542
$886$	0	0
$887$	14793.6	0.560002	0.280001	−	0.960000i	$-0.409665\pi$
$887$	14793.6	0.560002	0.280001	+	0.960000i	$0.409665\pi$
$888$	0	0
$889$	−16479.2	−0.621705
$890$	0	0
$891$	−3833.12	−0.144124
$892$	0	0
$893$	−9283.20	−0.347873
$894$	0	0
$895$	−66813.5	−2.49534
$896$	0	0
$897$	−911.840	−0.0339414
$898$	0	0
$899$	9386.53	0.348229
$900$	0	0
$901$	7196.27	0.266085
$902$	0	0
$903$	−22771.5	−0.839190
$904$	0	0
$905$	−10918.8	−0.401054
$906$	0	0
$907$	−13661.8	−0.500146	−0.250073	−	0.968227i	$-0.580455\pi$
$907$	−13661.8	−0.500146	−0.250073	+	0.968227i	$0.580455\pi$
$908$	0	0
$909$	−24236.2	−0.884340
$910$	0	0
$911$	−6644.55	−0.241651	−0.120825	−	0.992674i	$-0.538554\pi$
$911$	−6644.55	−0.241651	−0.120825	+	0.992674i	$0.538554\pi$
$912$	0	0
$913$	10473.0	0.379634
$914$	0	0
$915$	26077.3	0.942172
$916$	0	0
$917$	−42291.9	−1.52301
$918$	0	0
$919$	−9464.13	−0.339709	−0.169855	−	0.985469i	$-0.554330\pi$
$919$	−9464.13	−0.339709	−0.169855	+	0.985469i	$0.554330\pi$
$920$	0	0
$921$	−10755.5	−0.384807
$922$	0	0
$923$	7244.31	0.258342
$924$	0	0
$925$	−10433.7	−0.370873
$926$	0	0
$927$	−7652.10	−0.271120
$928$	0	0
$929$	−16562.5	−0.584927	−0.292464	−	0.956277i	$-0.594475\pi$
$929$	−16562.5	−0.584927	−0.292464	+	0.956277i	$0.594475\pi$
$930$	0	0
$931$	31214.5	1.09883
$932$	0	0
$933$	16203.2	0.568561
$934$	0	0
$935$	−14166.1	−0.495486
$936$	0	0
$937$	−38227.3	−1.33280	−0.666398	−	0.745596i	$-0.732165\pi$
$937$	−38227.3	−1.33280	−0.666398	+	0.745596i	$0.732165\pi$
$938$	0	0
$939$	8513.09	0.295862
$940$	0	0
$941$	47320.8	1.63933	0.819667	−	0.572840i	$-0.194158\pi$
$941$	47320.8	1.63933	0.819667	+	0.572840i	$0.194158\pi$
$942$	0	0
$943$	−6826.56	−0.235740
$944$	0	0
$945$	−48133.3	−1.65691
$946$	0	0
$947$	−7613.32	−0.261246	−0.130623	−	0.991432i	$-0.541698\pi$
$947$	−7613.32	−0.261246	−0.130623	+	0.991432i	$0.541698\pi$
$948$	0	0
$949$	−4674.36	−0.159891
$950$	0	0
$951$	20801.9	0.709304
$952$	0	0
$953$	−44717.8	−1.51999	−0.759995	−	0.649928i	$-0.774799\pi$
$953$	−44717.8	−1.51999	−0.759995	+	0.649928i	$0.774799\pi$
$954$	0	0
$955$	26648.7	0.902965
$956$	0	0
$957$	10846.0	0.366356
$958$	0	0
$959$	33863.4	1.14026
$960$	0	0
$961$	−28560.3	−0.958689
$962$	0	0
$963$	41997.0	1.40533
$964$	0	0
$965$	16186.3	0.539954
$966$	0	0
$967$	−47052.4	−1.56474	−0.782370	−	0.622813i	$-0.785990\pi$
$967$	−47052.4	−1.56474	−0.782370	+	0.622813i	$0.785990\pi$
$968$	0	0
$969$	−15558.8	−0.515812
$970$	0	0
$971$	40080.3	1.32465	0.662326	−	0.749215i	$-0.269569\pi$
$971$	40080.3	1.32465	0.662326	+	0.749215i	$0.269569\pi$
$972$	0	0
$973$	−43571.6	−1.43560
$974$	0	0
$975$	4870.93	0.159994
$976$	0	0
$977$	56773.8	1.85911	0.929557	−	0.368679i	$-0.120190\pi$
$977$	56773.8	1.85911	0.929557	+	0.368679i	$0.120190\pi$
$978$	0	0
$979$	17449.9	0.569665
$980$	0	0
$981$	11197.0	0.364415
$982$	0	0
$983$	33919.1	1.10056	0.550281	−	0.834980i	$-0.314521\pi$
$983$	33919.1	1.10056	0.550281	+	0.834980i	$0.314521\pi$
$984$	0	0
$985$	8635.14	0.279328
$986$	0	0
$987$	−5582.04	−0.180019
$988$	0	0
$989$	8129.91	0.261392
$990$	0	0
$991$	−5183.79	−0.166164	−0.0830820	−	0.996543i	$-0.526476\pi$
$991$	−5183.79	−0.166164	−0.0830820	+	0.996543i	$0.526476\pi$
$992$	0	0
$993$	16115.6	0.515017
$994$	0	0
$995$	−43839.3	−1.39678
$996$	0	0
$997$	−41570.9	−1.32053	−0.660263	−	0.751034i	$-0.729555\pi$
$997$	−41570.9	−1.32053	−0.660263	+	0.751034i	$0.729555\pi$
$998$	0	0
$999$	−10308.4	−0.326470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bg.1.4		8
4.3	odd	2		1472.4.a.bh.1.5		8
8.3	odd	2		736.4.a.e.1.4	✓	8
8.5	even	2		736.4.a.f.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.e.1.4	✓	8	8.3	odd	2
736.4.a.f.1.5	yes	8	8.5	even	2
1472.4.a.bg.1.4		8	1.1	even	1	trivial
1472.4.a.bh.1.5		8	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q-2.55781 q^{3} +15.7437 q^{5} -25.1863 q^{7} -20.4576 q^{9} -15.8480 q^{11} -15.4997 q^{13} -40.2693 q^{15} +56.7767 q^{17} +107.137 q^{19} +64.4219 q^{21} -23.0000 q^{23} +122.863 q^{25} +121.388 q^{27} +267.565 q^{29} +35.0813 q^{31} +40.5361 q^{33} -396.525 q^{35} -84.9213 q^{37} +39.6452 q^{39} +296.807 q^{41} -353.475 q^{43} -322.078 q^{45} -86.6482 q^{47} +291.352 q^{49} -145.224 q^{51} +126.747 q^{53} -249.505 q^{55} -274.036 q^{57} -853.472 q^{59} -647.571 q^{61} +515.252 q^{63} -244.021 q^{65} -603.269 q^{67} +58.8297 q^{69} -467.385 q^{71} +301.578 q^{73} -314.260 q^{75} +399.152 q^{77} +766.955 q^{79} +241.868 q^{81} -660.843 q^{83} +893.873 q^{85} -684.381 q^{87} -1101.08 q^{89} +390.380 q^{91} -89.7314 q^{93} +1686.73 q^{95} -1494.52 q^{97} +324.211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 8 * q - 12 * q^3 + 12 * q^5 - 14 * q^7 + 90 * q^9 - 88 * q^11 + 30 * q^13 + 30 * q^15 + 58 * q^17 - 190 * q^19 + 66 * q^21 - 184 * q^23 + 28 * q^25 - 432 * q^27 - 190 * q^29 - 60 * q^31 + 346 * q^33 - 192 * q^35 + 156 * q^37 + 156 * q^39 + 282 * q^41 - 810 * q^43 - 358 * q^45 + 564 * q^47 + 752 * q^49 - 1614 * q^51 + 230 * q^53 + 1292 * q^55 + 796 * q^57 - 1916 * q^59 - 22 * q^61 + 1036 * q^63 + 222 * q^65 - 2292 * q^67 + 276 * q^69 + 2376 * q^71 - 630 * q^73 - 3952 * q^75 + 948 * q^77 + 892 * q^79 - 1044 * q^81 - 2800 * q^83 - 288 * q^85 + 2756 * q^87 - 308 * q^89 - 4610 * q^91 + 982 * q^93 + 1300 * q^95 + 854 * q^97 - 5986 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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