LMFDB - Embedded newform 1472.4.a.bg.1.8

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			1.8
Root			$9.42564$ 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)

$f(q)$	$=$	$q+7.42564 q^{3} -4.36302 q^{5} +9.40153 q^{7} +28.1401 q^{9} -67.5361 q^{11} +33.5730 q^{13} -32.3982 q^{15} +24.0641 q^{17} -71.4057 q^{19} +69.8123 q^{21} -23.0000 q^{23} -105.964 q^{25} +8.46606 q^{27} +96.4102 q^{29} +63.1309 q^{31} -501.498 q^{33} -41.0191 q^{35} -84.0774 q^{37} +249.301 q^{39} -144.498 q^{41} +75.4198 q^{43} -122.776 q^{45} -215.075 q^{47} -254.611 q^{49} +178.691 q^{51} +55.0233 q^{53} +294.661 q^{55} -530.233 q^{57} -139.853 q^{59} -772.119 q^{61} +264.560 q^{63} -146.480 q^{65} -871.806 q^{67} -170.790 q^{69} -148.801 q^{71} +745.997 q^{73} -786.851 q^{75} -634.942 q^{77} +461.010 q^{79} -696.917 q^{81} +345.272 q^{83} -104.992 q^{85} +715.907 q^{87} +1276.79 q^{89} +315.637 q^{91} +468.787 q^{93} +311.545 q^{95} -1024.86 q^{97} -1900.47 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	7.42564	1.42906	0.714532	−	0.699602i	$-0.246639\pi$
$3$	7.42564	1.42906	0.714532	+	0.699602i	$0.246639\pi$
$4$	0	0
$5$	−4.36302	−0.390241	−0.195120	−	0.980779i	$-0.562510\pi$
$5$	−4.36302	−0.390241	−0.195120	+	0.980779i	$0.562510\pi$
$6$	0	0
$7$	9.40153	0.507635	0.253817	−	0.967252i	$-0.418314\pi$
$7$	9.40153	0.507635	0.253817	+	0.967252i	$0.418314\pi$
$8$	0	0
$9$	28.1401	1.04223
$10$	0	0
$11$	−67.5361	−1.85117	−0.925586	−	0.378538i	$-0.876427\pi$
$11$	−67.5361	−1.85117	−0.925586	+	0.378538i	$0.876427\pi$
$12$	0	0
$13$	33.5730	0.716267	0.358133	−	0.933670i	$-0.383413\pi$
$13$	33.5730	0.716267	0.358133	+	0.933670i	$0.383413\pi$
$14$	0	0
$15$	−32.3982	−0.557679
$16$	0	0
$17$	24.0641	0.343317	0.171659	−	0.985156i	$-0.445087\pi$
$17$	24.0641	0.343317	0.171659	+	0.985156i	$0.445087\pi$
$18$	0	0
$19$	−71.4057	−0.862189	−0.431094	−	0.902307i	$-0.641872\pi$
$19$	−71.4057	−0.862189	−0.431094	+	0.902307i	$0.641872\pi$
$20$	0	0
$21$	69.8123	0.725443
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−105.964	−0.847712
$26$	0	0
$27$	8.46606	0.0603442
$28$	0	0
$29$	96.4102	0.617342	0.308671	−	0.951169i	$-0.400116\pi$
$29$	96.4102	0.617342	0.308671	+	0.951169i	$0.400116\pi$
$30$	0	0
$31$	63.1309	0.365763	0.182881	−	0.983135i	$-0.441458\pi$
$31$	63.1309	0.365763	0.182881	+	0.983135i	$0.441458\pi$
$32$	0	0
$33$	−501.498	−2.64544
$34$	0	0
$35$	−41.0191	−0.198100
$36$	0	0
$37$	−84.0774	−0.373574	−0.186787	−	0.982400i	$-0.559807\pi$
$37$	−84.0774	−0.373574	−0.186787	+	0.982400i	$0.559807\pi$
$38$	0	0
$39$	249.301	1.02359
$40$	0	0
$41$	−144.498	−0.550411	−0.275206	−	0.961385i	$-0.588746\pi$
$41$	−144.498	−0.550411	−0.275206	+	0.961385i	$0.588746\pi$
$42$	0	0
$43$	75.4198	0.267475	0.133737	−	0.991017i	$-0.457302\pi$
$43$	75.4198	0.267475	0.133737	+	0.991017i	$0.457302\pi$
$44$	0	0
$45$	−122.776	−0.406719
$46$	0	0
$47$	−215.075	−0.667486	−0.333743	−	0.942664i	$-0.608312\pi$
$47$	−215.075	−0.667486	−0.333743	+	0.942664i	$0.608312\pi$
$48$	0	0
$49$	−254.611	−0.742307
$50$	0	0
$51$	178.691	0.490623
$52$	0	0
$53$	55.0233	0.142604	0.0713021	−	0.997455i	$-0.477285\pi$
$53$	55.0233	0.142604	0.0713021	+	0.997455i	$0.477285\pi$
$54$	0	0
$55$	294.661	0.722402
$56$	0	0
$57$	−530.233	−1.23212
$58$	0	0
$59$	−139.853	−0.308598	−0.154299	−	0.988024i	$-0.549312\pi$
$59$	−139.853	−0.308598	−0.154299	+	0.988024i	$0.549312\pi$
$60$	0	0
$61$	−772.119	−1.62065	−0.810325	−	0.585981i	$-0.800709\pi$
$61$	−772.119	−1.62065	−0.810325	+	0.585981i	$0.800709\pi$
$62$	0	0
$63$	264.560	0.529070
$64$	0	0
$65$	−146.480	−0.279516
$66$	0	0
$67$	−871.806	−1.58967	−0.794836	−	0.606824i	$-0.792443\pi$
$67$	−871.806	−1.58967	−0.794836	+	0.606824i	$0.792443\pi$
$68$	0	0
$69$	−170.790	−0.297981
$70$	0	0
$71$	−148.801	−0.248725	−0.124362	−	0.992237i	$-0.539689\pi$
$71$	−148.801	−0.248725	−0.124362	+	0.992237i	$0.539689\pi$
$72$	0	0
$73$	745.997	1.19606	0.598030	−	0.801474i	$-0.295950\pi$
$73$	745.997	1.19606	0.598030	+	0.801474i	$0.295950\pi$
$74$	0	0
$75$	−786.851	−1.21144
$76$	0	0
$77$	−634.942	−0.939719
$78$	0	0
$79$	461.010	0.656552	0.328276	−	0.944582i	$-0.393532\pi$
$79$	461.010	0.656552	0.328276	+	0.944582i	$0.393532\pi$
$80$	0	0
$81$	−696.917	−0.955991
$82$	0	0
$83$	345.272	0.456609	0.228305	−	0.973590i	$-0.426682\pi$
$83$	345.272	0.456609	0.228305	+	0.973590i	$0.426682\pi$
$84$	0	0
$85$	−104.992	−0.133976
$86$	0	0
$87$	715.907	0.882222
$88$	0	0
$89$	1276.79	1.52067	0.760337	−	0.649529i	$-0.225034\pi$
$89$	1276.79	1.52067	0.760337	+	0.649529i	$0.225034\pi$
$90$	0	0
$91$	315.637	0.363602
$92$	0	0
$93$	468.787	0.522699
$94$	0	0
$95$	311.545	0.336461
$96$	0	0
$97$	−1024.86	−1.07277	−0.536386	−	0.843973i	$-0.680211\pi$
$97$	−1024.86	−1.07277	−0.536386	+	0.843973i	$0.680211\pi$
$98$	0	0
$99$	−1900.47	−1.92934
$100$	0	0
$101$	−126.508	−0.124634	−0.0623169	−	0.998056i	$-0.519849\pi$
$101$	−126.508	−0.124634	−0.0623169	+	0.998056i	$0.519849\pi$
$102$	0	0
$103$	−258.017	−0.246827	−0.123413	−	0.992355i	$-0.539384\pi$
$103$	−258.017	−0.246827	−0.123413	+	0.992355i	$0.539384\pi$
$104$	0	0
$105$	−304.593	−0.283097
$106$	0	0
$107$	−1017.46	−0.919265	−0.459633	−	0.888109i	$-0.652019\pi$
$107$	−1017.46	−0.919265	−0.459633	+	0.888109i	$0.652019\pi$
$108$	0	0
$109$	−1861.29	−1.63559	−0.817797	−	0.575507i	$-0.804805\pi$
$109$	−1861.29	−1.63559	−0.817797	+	0.575507i	$0.804805\pi$
$110$	0	0
$111$	−624.329	−0.533862
$112$	0	0
$113$	−49.7040	−0.0413784	−0.0206892	−	0.999786i	$-0.506586\pi$
$113$	−49.7040	−0.0413784	−0.0206892	+	0.999786i	$0.506586\pi$
$114$	0	0
$115$	100.350	0.0813708
$116$	0	0
$117$	944.748	0.746512
$118$	0	0
$119$	226.239	0.174280
$120$	0	0
$121$	3230.12	2.42684
$122$	0	0
$123$	−1072.99	−0.786573
$124$	0	0
$125$	1007.70	0.721052
$126$	0	0
$127$	−1755.96	−1.22690	−0.613449	−	0.789734i	$-0.710218\pi$
$127$	−1755.96	−1.22690	−0.613449	+	0.789734i	$0.710218\pi$
$128$	0	0
$129$	560.040	0.382239
$130$	0	0
$131$	−377.103	−0.251509	−0.125754	−	0.992061i	$-0.540135\pi$
$131$	−377.103	−0.251509	−0.125754	+	0.992061i	$0.540135\pi$
$132$	0	0
$133$	−671.322	−0.437677
$134$	0	0
$135$	−36.9376	−0.0235488
$136$	0	0
$137$	376.686	0.234908	0.117454	−	0.993078i	$-0.462527\pi$
$137$	376.686	0.234908	0.117454	+	0.993078i	$0.462527\pi$
$138$	0	0
$139$	−2297.98	−1.40224	−0.701122	−	0.713041i	$-0.747317\pi$
$139$	−2297.98	−1.40224	−0.701122	+	0.713041i	$0.747317\pi$
$140$	0	0
$141$	−1597.07	−0.953881
$142$	0	0
$143$	−2267.39	−1.32593
$144$	0	0
$145$	−420.640	−0.240912
$146$	0	0
$147$	−1890.65	−1.06080
$148$	0	0
$149$	−1191.27	−0.654987	−0.327493	−	0.944853i	$-0.606204\pi$
$149$	−1191.27	−0.654987	−0.327493	+	0.944853i	$0.606204\pi$
$150$	0	0
$151$	2536.35	1.36692	0.683462	−	0.729986i	$-0.260473\pi$
$151$	2536.35	1.36692	0.683462	+	0.729986i	$0.260473\pi$
$152$	0	0
$153$	677.166	0.357814
$154$	0	0
$155$	−275.442	−0.142736
$156$	0	0
$157$	−1112.51	−0.565529	−0.282764	−	0.959189i	$-0.591251\pi$
$157$	−1112.51	−0.565529	−0.282764	+	0.959189i	$0.591251\pi$
$158$	0	0
$159$	408.583	0.203791
$160$	0	0
$161$	−216.235	−0.105849
$162$	0	0
$163$	−1481.99	−0.712135	−0.356068	−	0.934460i	$-0.615883\pi$
$163$	−1481.99	−0.712135	−0.356068	+	0.934460i	$0.615883\pi$
$164$	0	0
$165$	2188.05	1.03236
$166$	0	0
$167$	3939.40	1.82539	0.912695	−	0.408642i	$-0.133997\pi$
$167$	3939.40	1.82539	0.912695	+	0.408642i	$0.133997\pi$
$168$	0	0
$169$	−1069.85	−0.486962
$170$	0	0
$171$	−2009.36	−0.898596
$172$	0	0
$173$	1332.84	0.585744	0.292872	−	0.956152i	$-0.405389\pi$
$173$	1332.84	0.585744	0.292872	+	0.956152i	$0.405389\pi$
$174$	0	0
$175$	−996.224	−0.430328
$176$	0	0
$177$	−1038.50	−0.441006
$178$	0	0
$179$	−603.340	−0.251931	−0.125966	−	0.992035i	$-0.540203\pi$
$179$	−603.340	−0.251931	−0.125966	+	0.992035i	$0.540203\pi$
$180$	0	0
$181$	4132.64	1.69711	0.848554	−	0.529109i	$-0.177474\pi$
$181$	4132.64	1.69711	0.848554	+	0.529109i	$0.177474\pi$
$182$	0	0
$183$	−5733.47	−2.31601
$184$	0	0
$185$	366.832	0.145784
$186$	0	0
$187$	−1625.19	−0.635539
$188$	0	0
$189$	79.5939	0.0306328
$190$	0	0
$191$	2364.21	0.895647	0.447823	−	0.894122i	$-0.352199\pi$
$191$	2364.21	0.895647	0.447823	+	0.894122i	$0.352199\pi$
$192$	0	0
$193$	−3292.16	−1.22785	−0.613925	−	0.789365i	$-0.710410\pi$
$193$	−3292.16	−1.22785	−0.613925	+	0.789365i	$0.710410\pi$
$194$	0	0
$195$	−1087.71	−0.399447
$196$	0	0
$197$	3375.84	1.22091	0.610453	−	0.792052i	$-0.290987\pi$
$197$	3375.84	1.22091	0.610453	+	0.792052i	$0.290987\pi$
$198$	0	0
$199$	−4578.09	−1.63082	−0.815408	−	0.578886i	$-0.803488\pi$
$199$	−4578.09	−1.63082	−0.815408	+	0.578886i	$0.803488\pi$
$200$	0	0
$201$	−6473.71	−2.27174
$202$	0	0
$203$	906.403	0.313384
$204$	0	0
$205$	630.450	0.214793
$206$	0	0
$207$	−647.223	−0.217319
$208$	0	0
$209$	4822.46	1.59606
$210$	0	0
$211$	2634.81	0.859659	0.429829	−	0.902910i	$-0.358574\pi$
$211$	2634.81	0.859659	0.429829	+	0.902910i	$0.358574\pi$
$212$	0	0
$213$	−1104.94	−0.355444
$214$	0	0
$215$	−329.058	−0.104380
$216$	0	0
$217$	593.527	0.185674
$218$	0	0
$219$	5539.50	1.70925
$220$	0	0
$221$	807.902	0.245907
$222$	0	0
$223$	551.110	0.165493	0.0827467	−	0.996571i	$-0.473631\pi$
$223$	551.110	0.165493	0.0827467	+	0.996571i	$0.473631\pi$
$224$	0	0
$225$	−2981.84	−0.883508
$226$	0	0
$227$	−1519.50	−0.444286	−0.222143	−	0.975014i	$-0.571305\pi$
$227$	−1519.50	−0.444286	−0.222143	+	0.975014i	$0.571305\pi$
$228$	0	0
$229$	−3162.04	−0.912461	−0.456231	−	0.889862i	$-0.650801\pi$
$229$	−3162.04	−0.912461	−0.456231	+	0.889862i	$0.650801\pi$
$230$	0	0
$231$	−4714.85	−1.34292
$232$	0	0
$233$	4659.33	1.31005	0.655027	−	0.755605i	$-0.272657\pi$
$233$	4659.33	1.31005	0.655027	+	0.755605i	$0.272657\pi$
$234$	0	0
$235$	938.376	0.260480
$236$	0	0
$237$	3423.29	0.938256
$238$	0	0
$239$	1993.51	0.539536	0.269768	−	0.962925i	$-0.413053\pi$
$239$	1993.51	0.539536	0.269768	+	0.962925i	$0.413053\pi$
$240$	0	0
$241$	1772.18	0.473677	0.236838	−	0.971549i	$-0.423889\pi$
$241$	1772.18	0.473677	0.236838	+	0.971549i	$0.423889\pi$
$242$	0	0
$243$	−5403.64	−1.42652
$244$	0	0
$245$	1110.87	0.289678
$246$	0	0
$247$	−2397.30	−0.617557
$248$	0	0
$249$	2563.87	0.652524
$250$	0	0
$251$	−1484.40	−0.373285	−0.186643	−	0.982428i	$-0.559761\pi$
$251$	−1484.40	−0.373285	−0.186643	+	0.982428i	$0.559761\pi$
$252$	0	0
$253$	1553.33	0.385996
$254$	0	0
$255$	−779.633	−0.191461
$256$	0	0
$257$	2791.47	0.677537	0.338769	−	0.940870i	$-0.389990\pi$
$257$	2791.47	0.677537	0.338769	+	0.940870i	$0.389990\pi$
$258$	0	0
$259$	−790.456	−0.189639
$260$	0	0
$261$	2712.99	0.643410
$262$	0	0
$263$	−6898.74	−1.61747	−0.808735	−	0.588173i	$-0.799847\pi$
$263$	−6898.74	−1.61747	−0.808735	+	0.588173i	$0.799847\pi$
$264$	0	0
$265$	−240.068	−0.0556500
$266$	0	0
$267$	9481.01	2.17314
$268$	0	0
$269$	−2861.59	−0.648604	−0.324302	−	0.945954i	$-0.605129\pi$
$269$	−2861.59	−0.648604	−0.324302	+	0.945954i	$0.605129\pi$
$270$	0	0
$271$	2226.19	0.499009	0.249504	−	0.968374i	$-0.419732\pi$
$271$	2226.19	0.499009	0.249504	+	0.968374i	$0.419732\pi$
$272$	0	0
$273$	2343.81	0.519611
$274$	0	0
$275$	7156.39	1.56926
$276$	0	0
$277$	5283.50	1.14605	0.573023	−	0.819539i	$-0.305770\pi$
$277$	5283.50	1.14605	0.573023	+	0.819539i	$0.305770\pi$
$278$	0	0
$279$	1776.51	0.381208
$280$	0	0
$281$	7449.27	1.58145	0.790723	−	0.612174i	$-0.209705\pi$
$281$	7449.27	1.58145	0.790723	+	0.612174i	$0.209705\pi$
$282$	0	0
$283$	−5206.96	−1.09372	−0.546858	−	0.837225i	$-0.684176\pi$
$283$	−5206.96	−1.09372	−0.546858	+	0.837225i	$0.684176\pi$
$284$	0	0
$285$	2313.42	0.480825
$286$	0	0
$287$	−1358.51	−0.279408
$288$	0	0
$289$	−4333.92	−0.882133
$290$	0	0
$291$	−7610.24	−1.53306
$292$	0	0
$293$	3588.15	0.715433	0.357717	−	0.933830i	$-0.383555\pi$
$293$	3588.15	0.715433	0.357717	+	0.933830i	$0.383555\pi$
$294$	0	0
$295$	610.180	0.120427
$296$	0	0
$297$	−571.764	−0.111708
$298$	0	0
$299$	−772.179	−0.149352
$300$	0	0
$301$	709.062	0.135779
$302$	0	0
$303$	−939.402	−0.178110
$304$	0	0
$305$	3368.77	0.632443
$306$	0	0
$307$	−1222.71	−0.227309	−0.113654	−	0.993520i	$-0.536256\pi$
$307$	−1222.71	−0.227309	−0.113654	+	0.993520i	$0.536256\pi$
$308$	0	0
$309$	−1915.94	−0.352731
$310$	0	0
$311$	158.959	0.0289831	0.0144916	−	0.999895i	$-0.495387\pi$
$311$	158.959	0.0289831	0.0144916	+	0.999895i	$0.495387\pi$
$312$	0	0
$313$	1532.53	0.276754	0.138377	−	0.990380i	$-0.455811\pi$
$313$	1532.53	0.276754	0.138377	+	0.990380i	$0.455811\pi$
$314$	0	0
$315$	−1154.28	−0.206465
$316$	0	0
$317$	2444.39	0.433093	0.216546	−	0.976272i	$-0.430521\pi$
$317$	2444.39	0.433093	0.216546	+	0.976272i	$0.430521\pi$
$318$	0	0
$319$	−6511.16	−1.14281
$320$	0	0
$321$	−7555.27	−1.31369
$322$	0	0
$323$	−1718.31	−0.296004
$324$	0	0
$325$	−3557.53	−0.607188
$326$	0	0
$327$	−13821.3	−2.33737
$328$	0	0
$329$	−2022.03	−0.338839
$330$	0	0
$331$	3229.27	0.536245	0.268122	−	0.963385i	$-0.413597\pi$
$331$	3229.27	0.536245	0.268122	+	0.963385i	$0.413597\pi$
$332$	0	0
$333$	−2365.95	−0.389349
$334$	0	0
$335$	3803.71	0.620354
$336$	0	0
$337$	2563.42	0.414357	0.207178	−	0.978303i	$-0.433572\pi$
$337$	2563.42	0.414357	0.207178	+	0.978303i	$0.433572\pi$
$338$	0	0
$339$	−369.084	−0.0591325
$340$	0	0
$341$	−4263.61	−0.677090
$342$	0	0
$343$	−5618.46	−0.884456
$344$	0	0
$345$	745.159	0.116284
$346$	0	0
$347$	−5755.34	−0.890383	−0.445192	−	0.895435i	$-0.646864\pi$
$347$	−5755.34	−0.890383	−0.445192	+	0.895435i	$0.646864\pi$
$348$	0	0
$349$	−10312.7	−1.58174	−0.790868	−	0.611987i	$-0.790370\pi$
$349$	−10312.7	−1.58174	−0.790868	+	0.611987i	$0.790370\pi$
$350$	0	0
$351$	284.231	0.0432226
$352$	0	0
$353$	5515.25	0.831578	0.415789	−	0.909461i	$-0.363506\pi$
$353$	5515.25	0.831578	0.415789	+	0.909461i	$0.363506\pi$
$354$	0	0
$355$	649.223	0.0970625
$356$	0	0
$357$	1679.97	0.249057
$358$	0	0
$359$	563.557	0.0828507	0.0414254	−	0.999142i	$-0.486810\pi$
$359$	563.557	0.0828507	0.0414254	+	0.999142i	$0.486810\pi$
$360$	0	0
$361$	−1760.23	−0.256631
$362$	0	0
$363$	23985.7	3.46811
$364$	0	0
$365$	−3254.80	−0.466751
$366$	0	0
$367$	−7705.08	−1.09592	−0.547959	−	0.836505i	$-0.684595\pi$
$367$	−7705.08	−1.09592	−0.547959	+	0.836505i	$0.684595\pi$
$368$	0	0
$369$	−4066.20	−0.573653
$370$	0	0
$371$	517.303	0.0723909
$372$	0	0
$373$	2346.53	0.325734	0.162867	−	0.986648i	$-0.447926\pi$
$373$	2346.53	0.325734	0.162867	+	0.986648i	$0.447926\pi$
$374$	0	0
$375$	7482.83	1.03043
$376$	0	0
$377$	3236.78	0.442182
$378$	0	0
$379$	−3390.67	−0.459544	−0.229772	−	0.973244i	$-0.573798\pi$
$379$	−3390.67	−0.459544	−0.229772	+	0.973244i	$0.573798\pi$
$380$	0	0
$381$	−13039.1	−1.75332
$382$	0	0
$383$	−837.622	−0.111751	−0.0558753	−	0.998438i	$-0.517795\pi$
$383$	−837.622	−0.111751	−0.0558753	+	0.998438i	$0.517795\pi$
$384$	0	0
$385$	2770.27	0.366717
$386$	0	0
$387$	2122.32	0.278769
$388$	0	0
$389$	−12669.0	−1.65126	−0.825632	−	0.564209i	$-0.809181\pi$
$389$	−12669.0	−1.65126	−0.825632	+	0.564209i	$0.809181\pi$
$390$	0	0
$391$	−553.474	−0.0715866
$392$	0	0
$393$	−2800.23	−0.359422
$394$	0	0
$395$	−2011.40	−0.256213
$396$	0	0
$397$	−8731.37	−1.10382	−0.551908	−	0.833905i	$-0.686100\pi$
$397$	−8731.37	−1.10382	−0.551908	+	0.833905i	$0.686100\pi$
$398$	0	0
$399$	−4985.00	−0.625469
$400$	0	0
$401$	−8265.25	−1.02929	−0.514647	−	0.857402i	$-0.672077\pi$
$401$	−8265.25	−1.02929	−0.514647	+	0.857402i	$0.672077\pi$
$402$	0	0
$403$	2119.49	0.261984
$404$	0	0
$405$	3040.66	0.373066
$406$	0	0
$407$	5678.26	0.691550
$408$	0	0
$409$	15088.7	1.82418	0.912091	−	0.409988i	$-0.134467\pi$
$409$	15088.7	1.82418	0.912091	+	0.409988i	$0.134467\pi$
$410$	0	0
$411$	2797.13	0.335699
$412$	0	0
$413$	−1314.83	−0.156655
$414$	0	0
$415$	−1506.43	−0.178187
$416$	0	0
$417$	−17064.0	−2.00390
$418$	0	0
$419$	−6923.33	−0.807224	−0.403612	−	0.914930i	$-0.632245\pi$
$419$	−6923.33	−0.807224	−0.403612	+	0.914930i	$0.632245\pi$
$420$	0	0
$421$	3156.36	0.365396	0.182698	−	0.983169i	$-0.441517\pi$
$421$	3156.36	0.365396	0.182698	+	0.983169i	$0.441517\pi$
$422$	0	0
$423$	−6052.23	−0.695672
$424$	0	0
$425$	−2549.93	−0.291034
$426$	0	0
$427$	−7259.09	−0.822698
$428$	0	0
$429$	−16836.8	−1.89484
$430$	0	0
$431$	−6355.74	−0.710314	−0.355157	−	0.934807i	$-0.615573\pi$
$431$	−6355.74	−0.710314	−0.355157	+	0.934807i	$0.615573\pi$
$432$	0	0
$433$	−5151.76	−0.571773	−0.285886	−	0.958264i	$-0.592288\pi$
$433$	−5151.76	−0.571773	−0.285886	+	0.958264i	$0.592288\pi$
$434$	0	0
$435$	−3123.52	−0.344279
$436$	0	0
$437$	1642.33	0.179779
$438$	0	0
$439$	−11082.9	−1.20492	−0.602458	−	0.798150i	$-0.705812\pi$
$439$	−11082.9	−1.20492	−0.602458	+	0.798150i	$0.705812\pi$
$440$	0	0
$441$	−7164.79	−0.773652
$442$	0	0
$443$	13618.8	1.46060	0.730301	−	0.683126i	$-0.239380\pi$
$443$	13618.8	1.46060	0.730301	+	0.683126i	$0.239380\pi$
$444$	0	0
$445$	−5570.68	−0.593428
$446$	0	0
$447$	−8845.98	−0.936019
$448$	0	0
$449$	−13134.4	−1.38051	−0.690257	−	0.723564i	$-0.742503\pi$
$449$	−13134.4	−1.38051	−0.690257	+	0.723564i	$0.742503\pi$
$450$	0	0
$451$	9758.85	1.01891
$452$	0	0
$453$	18834.1	1.95342
$454$	0	0
$455$	−1377.13	−0.141892
$456$	0	0
$457$	12711.1	1.30109	0.650546	−	0.759467i	$-0.274540\pi$
$457$	12711.1	1.30109	0.650546	+	0.759467i	$0.274540\pi$
$458$	0	0
$459$	203.728	0.0207172
$460$	0	0
$461$	−3915.10	−0.395541	−0.197771	−	0.980248i	$-0.563370\pi$
$461$	−3915.10	−0.395541	−0.197771	+	0.980248i	$0.563370\pi$
$462$	0	0
$463$	16351.0	1.64124	0.820619	−	0.571475i	$-0.193629\pi$
$463$	16351.0	1.64124	0.820619	+	0.571475i	$0.193629\pi$
$464$	0	0
$465$	−2045.33	−0.203978
$466$	0	0
$467$	−7265.82	−0.719961	−0.359981	−	0.932960i	$-0.617217\pi$
$467$	−7265.82	−0.719961	−0.359981	+	0.932960i	$0.617217\pi$
$468$	0	0
$469$	−8196.30	−0.806973
$470$	0	0
$471$	−8261.10	−0.808177
$472$	0	0
$473$	−5093.56	−0.495142
$474$	0	0
$475$	7566.43	0.730888
$476$	0	0
$477$	1548.36	0.148626
$478$	0	0
$479$	19081.1	1.82012	0.910060	−	0.414476i	$-0.136035\pi$
$479$	19081.1	1.82012	0.910060	+	0.414476i	$0.136035\pi$
$480$	0	0
$481$	−2822.73	−0.267579
$482$	0	0
$483$	−1605.68	−0.151265
$484$	0	0
$485$	4471.49	0.418639
$486$	0	0
$487$	12350.2	1.14916	0.574580	−	0.818448i	$-0.305165\pi$
$487$	12350.2	1.14916	0.574580	+	0.818448i	$0.305165\pi$
$488$	0	0
$489$	−11004.7	−1.01769
$490$	0	0
$491$	−11944.1	−1.09782	−0.548910	−	0.835881i	$-0.684957\pi$
$491$	−11944.1	−1.09782	−0.548910	+	0.835881i	$0.684957\pi$
$492$	0	0
$493$	2320.02	0.211944
$494$	0	0
$495$	8291.80	0.752907
$496$	0	0
$497$	−1398.96	−0.126261
$498$	0	0
$499$	−1772.57	−0.159020	−0.0795101	−	0.996834i	$-0.525336\pi$
$499$	−1772.57	−0.159020	−0.0795101	+	0.996834i	$0.525336\pi$
$500$	0	0
$501$	29252.6	2.60860
$502$	0	0
$503$	5517.89	0.489126	0.244563	−	0.969633i	$-0.421355\pi$
$503$	5517.89	0.489126	0.244563	+	0.969633i	$0.421355\pi$
$504$	0	0
$505$	551.957	0.0486371
$506$	0	0
$507$	−7944.36	−0.695900
$508$	0	0
$509$	21201.4	1.84624	0.923120	−	0.384512i	$-0.125630\pi$
$509$	21201.4	1.84624	0.923120	+	0.384512i	$0.125630\pi$
$510$	0	0
$511$	7013.51	0.607161
$512$	0	0
$513$	−604.525	−0.0520281
$514$	0	0
$515$	1125.73	0.0963218
$516$	0	0
$517$	14525.3	1.23563
$518$	0	0
$519$	9897.17	0.837067
$520$	0	0
$521$	−9203.33	−0.773906	−0.386953	−	0.922099i	$-0.626472\pi$
$521$	−9203.33	−0.773906	−0.386953	+	0.922099i	$0.626472\pi$
$522$	0	0
$523$	21900.1	1.83103	0.915513	−	0.402289i	$-0.131785\pi$
$523$	21900.1	1.83103	0.915513	+	0.402289i	$0.131785\pi$
$524$	0	0
$525$	−7397.60	−0.614967
$526$	0	0
$527$	1519.19	0.125573
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−3935.47	−0.321629
$532$	0	0
$533$	−4851.24	−0.394241
$534$	0	0
$535$	4439.19	0.358735
$536$	0	0
$537$	−4480.18	−0.360026
$538$	0	0
$539$	17195.4	1.37414
$540$	0	0
$541$	3490.20	0.277367	0.138684	−	0.990337i	$-0.455713\pi$
$541$	3490.20	0.277367	0.138684	+	0.990337i	$0.455713\pi$
$542$	0	0
$543$	30687.5	2.42528
$544$	0	0
$545$	8120.87	0.638275
$546$	0	0
$547$	16506.0	1.29021	0.645106	−	0.764093i	$-0.276813\pi$
$547$	16506.0	1.29021	0.645106	+	0.764093i	$0.276813\pi$
$548$	0	0
$549$	−21727.5	−1.68908
$550$	0	0
$551$	−6884.23	−0.532265
$552$	0	0
$553$	4334.19	0.333289
$554$	0	0
$555$	2723.96	0.208335
$556$	0	0
$557$	10101.1	0.768400	0.384200	−	0.923250i	$-0.374477\pi$
$557$	10101.1	0.768400	0.384200	+	0.923250i	$0.374477\pi$
$558$	0	0
$559$	2532.07	0.191583
$560$	0	0
$561$	−12068.1	−0.908227
$562$	0	0
$563$	−1303.81	−0.0976005	−0.0488003	−	0.998809i	$-0.515540\pi$
$563$	−1303.81	−0.0976005	−0.0488003	+	0.998809i	$0.515540\pi$
$564$	0	0
$565$	216.860	0.0161475
$566$	0	0
$567$	−6552.09	−0.485294
$568$	0	0
$569$	1993.03	0.146840	0.0734201	−	0.997301i	$-0.476609\pi$
$569$	1993.03	0.146840	0.0734201	+	0.997301i	$0.476609\pi$
$570$	0	0
$571$	−1776.69	−0.130214	−0.0651071	−	0.997878i	$-0.520739\pi$
$571$	−1776.69	−0.130214	−0.0651071	+	0.997878i	$0.520739\pi$
$572$	0	0
$573$	17555.8	1.27994
$574$	0	0
$575$	2437.17	0.176760
$576$	0	0
$577$	9542.03	0.688457	0.344229	−	0.938886i	$-0.388140\pi$
$577$	9542.03	0.688457	0.344229	+	0.938886i	$0.388140\pi$
$578$	0	0
$579$	−24446.4	−1.75468
$580$	0	0
$581$	3246.09	0.231791
$582$	0	0
$583$	−3716.05	−0.263985
$584$	0	0
$585$	−4121.95	−0.291319
$586$	0	0
$587$	−8518.89	−0.598999	−0.299499	−	0.954096i	$-0.596820\pi$
$587$	−8518.89	−0.598999	−0.299499	+	0.954096i	$0.596820\pi$
$588$	0	0
$589$	−4507.91	−0.315357
$590$	0	0
$591$	25067.8	1.74475
$592$	0	0
$593$	11295.1	0.782181	0.391090	−	0.920352i	$-0.372098\pi$
$593$	11295.1	0.782181	0.391090	+	0.920352i	$0.372098\pi$
$594$	0	0
$595$	−987.086	−0.0680110
$596$	0	0
$597$	−33995.3	−2.33054
$598$	0	0
$599$	7880.18	0.537522	0.268761	−	0.963207i	$-0.413386\pi$
$599$	7880.18	0.537522	0.268761	+	0.963207i	$0.413386\pi$
$600$	0	0
$601$	21198.1	1.43875	0.719375	−	0.694622i	$-0.244428\pi$
$601$	21198.1	1.43875	0.719375	+	0.694622i	$0.244428\pi$
$602$	0	0
$603$	−24532.7	−1.65680
$604$	0	0
$605$	−14093.1	−0.947050
$606$	0	0
$607$	−13439.0	−0.898638	−0.449319	−	0.893371i	$-0.648333\pi$
$607$	−13439.0	−0.898638	−0.449319	+	0.893371i	$0.648333\pi$
$608$	0	0
$609$	6730.62	0.447846
$610$	0	0
$611$	−7220.70	−0.478098
$612$	0	0
$613$	24209.4	1.59512	0.797561	−	0.603238i	$-0.206123\pi$
$613$	24209.4	1.59512	0.797561	+	0.603238i	$0.206123\pi$
$614$	0	0
$615$	4681.49	0.306953
$616$	0	0
$617$	−21713.2	−1.41676	−0.708379	−	0.705832i	$-0.750573\pi$
$617$	−21713.2	−1.41676	−0.708379	+	0.705832i	$0.750573\pi$
$618$	0	0
$619$	−15809.5	−1.02656	−0.513279	−	0.858222i	$-0.671569\pi$
$619$	−15809.5	−1.02656	−0.513279	+	0.858222i	$0.671569\pi$
$620$	0	0
$621$	−194.719	−0.0125826
$622$	0	0
$623$	12003.8	0.771947
$624$	0	0
$625$	8848.88	0.566328
$626$	0	0
$627$	35809.8	2.28087
$628$	0	0
$629$	−2023.25	−0.128254
$630$	0	0
$631$	−29674.4	−1.87214	−0.936068	−	0.351818i	$-0.885564\pi$
$631$	−29674.4	−1.87214	−0.936068	+	0.351818i	$0.885564\pi$
$632$	0	0
$633$	19565.2	1.22851
$634$	0	0
$635$	7661.28	0.478785
$636$	0	0
$637$	−8548.06	−0.531690
$638$	0	0
$639$	−4187.28	−0.259227
$640$	0	0
$641$	−6257.34	−0.385569	−0.192785	−	0.981241i	$-0.561752\pi$
$641$	−6257.34	−0.385569	−0.192785	+	0.981241i	$0.561752\pi$
$642$	0	0
$643$	−1466.42	−0.0899377	−0.0449688	−	0.998988i	$-0.514319\pi$
$643$	−1466.42	−0.0899377	−0.0449688	+	0.998988i	$0.514319\pi$
$644$	0	0
$645$	−2443.47	−0.149165
$646$	0	0
$647$	25202.9	1.53142	0.765711	−	0.643185i	$-0.222387\pi$
$647$	25202.9	1.53142	0.765711	+	0.643185i	$0.222387\pi$
$648$	0	0
$649$	9445.10	0.571267
$650$	0	0
$651$	4407.32	0.265340
$652$	0	0
$653$	27800.2	1.66601	0.833006	−	0.553263i	$-0.186618\pi$
$653$	27800.2	1.66601	0.833006	+	0.553263i	$0.186618\pi$
$654$	0	0
$655$	1645.31	0.0981489
$656$	0	0
$657$	20992.4	1.24656
$658$	0	0
$659$	−1065.63	−0.0629912	−0.0314956	−	0.999504i	$-0.510027\pi$
$659$	−1065.63	−0.0629912	−0.0314956	+	0.999504i	$0.510027\pi$
$660$	0	0
$661$	14149.4	0.832599	0.416300	−	0.909227i	$-0.363327\pi$
$661$	14149.4	0.832599	0.416300	+	0.909227i	$0.363327\pi$
$662$	0	0
$663$	5999.19	0.351417
$664$	0	0
$665$	2928.99	0.170799
$666$	0	0
$667$	−2217.43	−0.128725
$668$	0	0
$669$	4092.34	0.236501
$670$	0	0
$671$	52145.9	3.00010
$672$	0	0
$673$	1643.03	0.0941073	0.0470536	−	0.998892i	$-0.485017\pi$
$673$	1643.03	0.0941073	0.0470536	+	0.998892i	$0.485017\pi$
$674$	0	0
$675$	−897.098	−0.0511545
$676$	0	0
$677$	−30414.5	−1.72662	−0.863311	−	0.504672i	$-0.831613\pi$
$677$	−30414.5	−1.72662	−0.863311	+	0.504672i	$0.831613\pi$
$678$	0	0
$679$	−9635.25	−0.544576
$680$	0	0
$681$	−11283.3	−0.634914
$682$	0	0
$683$	−18182.8	−1.01866	−0.509331	−	0.860571i	$-0.670107\pi$
$683$	−18182.8	−1.01866	−0.509331	+	0.860571i	$0.670107\pi$
$684$	0	0
$685$	−1643.49	−0.0916708
$686$	0	0
$687$	−23480.2	−1.30397
$688$	0	0
$689$	1847.29	0.102143
$690$	0	0
$691$	−15330.4	−0.843989	−0.421995	−	0.906598i	$-0.638670\pi$
$691$	−15330.4	−0.843989	−0.421995	+	0.906598i	$0.638670\pi$
$692$	0	0
$693$	−17867.3	−0.979400
$694$	0	0
$695$	10026.1	0.547212
$696$	0	0
$697$	−3477.22	−0.188966
$698$	0	0
$699$	34598.5	1.87215
$700$	0	0
$701$	1245.98	0.0671328	0.0335664	−	0.999436i	$-0.489313\pi$
$701$	1245.98	0.0671328	0.0335664	+	0.999436i	$0.489313\pi$
$702$	0	0
$703$	6003.61	0.322092
$704$	0	0
$705$	6968.04	0.372243
$706$	0	0
$707$	−1189.37	−0.0632684
$708$	0	0
$709$	10282.8	0.544683	0.272341	−	0.962201i	$-0.412202\pi$
$709$	10282.8	0.544683	0.272341	+	0.962201i	$0.412202\pi$
$710$	0	0
$711$	12972.9	0.684276
$712$	0	0
$713$	−1452.01	−0.0762668
$714$	0	0
$715$	9892.66	0.517433
$716$	0	0
$717$	14803.1	0.771032
$718$	0	0
$719$	14800.1	0.767666	0.383833	−	0.923403i	$-0.374604\pi$
$719$	14800.1	0.767666	0.383833	+	0.923403i	$0.374604\pi$
$720$	0	0
$721$	−2425.75	−0.125298
$722$	0	0
$723$	13159.6	0.676915
$724$	0	0
$725$	−10216.0	−0.523328
$726$	0	0
$727$	24168.5	1.23296	0.616478	−	0.787372i	$-0.288559\pi$
$727$	24168.5	1.23296	0.616478	+	0.787372i	$0.288559\pi$
$728$	0	0
$729$	−21308.7	−1.08259
$730$	0	0
$731$	1814.91	0.0918287
$732$	0	0
$733$	37052.5	1.86707	0.933537	−	0.358481i	$-0.116705\pi$
$733$	37052.5	1.86707	0.933537	+	0.358481i	$0.116705\pi$
$734$	0	0
$735$	8248.95	0.413969
$736$	0	0
$737$	58878.3	2.94276
$738$	0	0
$739$	2133.67	0.106209	0.0531045	−	0.998589i	$-0.483088\pi$
$739$	2133.67	0.106209	0.0531045	+	0.998589i	$0.483088\pi$
$740$	0	0
$741$	−17801.5	−0.882529
$742$	0	0
$743$	26389.0	1.30299	0.651494	−	0.758654i	$-0.274143\pi$
$743$	26389.0	1.30299	0.651494	+	0.758654i	$0.274143\pi$
$744$	0	0
$745$	5197.56	0.255602
$746$	0	0
$747$	9716.00	0.475890
$748$	0	0
$749$	−9565.66	−0.466651
$750$	0	0
$751$	−10857.9	−0.527579	−0.263789	−	0.964580i	$-0.584972\pi$
$751$	−10857.9	−0.527579	−0.263789	+	0.964580i	$0.584972\pi$
$752$	0	0
$753$	−11022.6	−0.533449
$754$	0	0
$755$	−11066.2	−0.533430
$756$	0	0
$757$	−4597.95	−0.220760	−0.110380	−	0.993889i	$-0.535207\pi$
$757$	−4597.95	−0.220760	−0.110380	+	0.993889i	$0.535207\pi$
$758$	0	0
$759$	11534.5	0.551613
$760$	0	0
$761$	1093.90	0.0521077	0.0260538	−	0.999661i	$-0.491706\pi$
$761$	1093.90	0.0521077	0.0260538	+	0.999661i	$0.491706\pi$
$762$	0	0
$763$	−17499.0	−0.830284
$764$	0	0
$765$	−2954.49	−0.139634
$766$	0	0
$767$	−4695.27	−0.221038
$768$	0	0
$769$	32217.7	1.51079	0.755397	−	0.655267i	$-0.227444\pi$
$769$	32217.7	1.51079	0.755397	+	0.655267i	$0.227444\pi$
$770$	0	0
$771$	20728.4	0.968244
$772$	0	0
$773$	2841.25	0.132203	0.0661014	−	0.997813i	$-0.478944\pi$
$773$	2841.25	0.132203	0.0661014	+	0.997813i	$0.478944\pi$
$774$	0	0
$775$	−6689.61	−0.310062
$776$	0	0
$777$	−5869.64	−0.271007
$778$	0	0
$779$	10318.0	0.474558
$780$	0	0
$781$	10049.4	0.460432
$782$	0	0
$783$	816.214	0.0372530
$784$	0	0
$785$	4853.91	0.220692
$786$	0	0
$787$	−14015.7	−0.634822	−0.317411	−	0.948288i	$-0.602814\pi$
$787$	−14015.7	−0.634822	−0.317411	+	0.948288i	$0.602814\pi$
$788$	0	0
$789$	−51227.6	−2.31147
$790$	0	0
$791$	−467.294	−0.0210051
$792$	0	0
$793$	−25922.3	−1.16082
$794$	0	0
$795$	−1782.66	−0.0795274
$796$	0	0
$797$	33612.0	1.49385	0.746925	−	0.664908i	$-0.231529\pi$
$797$	33612.0	1.49385	0.746925	+	0.664908i	$0.231529\pi$
$798$	0	0
$799$	−5175.57	−0.229160
$800$	0	0
$801$	35929.1	1.58489
$802$	0	0
$803$	−50381.7	−2.21411
$804$	0	0
$805$	943.439	0.0413066
$806$	0	0
$807$	−21249.2	−0.926897
$808$	0	0
$809$	−377.089	−0.0163878	−0.00819390	−	0.999966i	$-0.502608\pi$
$809$	−377.089	−0.0163878	−0.00819390	+	0.999966i	$0.502608\pi$
$810$	0	0
$811$	24291.0	1.05175	0.525877	−	0.850561i	$-0.323737\pi$
$811$	24291.0	1.05175	0.525877	+	0.850561i	$0.323737\pi$
$812$	0	0
$813$	16530.9	0.713116
$814$	0	0
$815$	6465.93	0.277904
$816$	0	0
$817$	−5385.40	−0.230614
$818$	0	0
$819$	8882.07	0.378956
$820$	0	0
$821$	−5538.66	−0.235445	−0.117723	−	0.993047i	$-0.537559\pi$
$821$	−5538.66	−0.235445	−0.117723	+	0.993047i	$0.537559\pi$
$822$	0	0
$823$	−41353.5	−1.75151	−0.875756	−	0.482755i	$-0.839636\pi$
$823$	−41353.5	−1.75151	−0.875756	+	0.482755i	$0.839636\pi$
$824$	0	0
$825$	53140.8	2.24258
$826$	0	0
$827$	−3467.59	−0.145804	−0.0729019	−	0.997339i	$-0.523226\pi$
$827$	−3467.59	−0.145804	−0.0729019	+	0.997339i	$0.523226\pi$
$828$	0	0
$829$	5524.89	0.231468	0.115734	−	0.993280i	$-0.463078\pi$
$829$	5524.89	0.231468	0.115734	+	0.993280i	$0.463078\pi$
$830$	0	0
$831$	39233.4	1.63778
$832$	0	0
$833$	−6126.98	−0.254847
$834$	0	0
$835$	−17187.7	−0.712341
$836$	0	0
$837$	534.470	0.0220717
$838$	0	0
$839$	−42684.4	−1.75641	−0.878206	−	0.478283i	$-0.841259\pi$
$839$	−42684.4	−1.75641	−0.878206	+	0.478283i	$0.841259\pi$
$840$	0	0
$841$	−15094.1	−0.618889
$842$	0	0
$843$	55315.6	2.25999
$844$	0	0
$845$	4667.80	0.190032
$846$	0	0
$847$	30368.1	1.23195
$848$	0	0
$849$	−38665.0	−1.56299
$850$	0	0
$851$	1933.78	0.0778956
$852$	0	0
$853$	39711.0	1.59400	0.796998	−	0.603981i	$-0.206420\pi$
$853$	39711.0	1.59400	0.796998	+	0.603981i	$0.206420\pi$
$854$	0	0
$855$	8766.90	0.350669
$856$	0	0
$857$	17088.5	0.681136	0.340568	−	0.940220i	$-0.389381\pi$
$857$	17088.5	0.681136	0.340568	+	0.940220i	$0.389381\pi$
$858$	0	0
$859$	34573.9	1.37328	0.686639	−	0.726999i	$-0.259085\pi$
$859$	34573.9	1.37328	0.686639	+	0.726999i	$0.259085\pi$
$860$	0	0
$861$	−10087.8	−0.399292
$862$	0	0
$863$	34786.0	1.37211	0.686054	−	0.727551i	$-0.259342\pi$
$863$	34786.0	1.37211	0.686054	+	0.727551i	$0.259342\pi$
$864$	0	0
$865$	−5815.20	−0.228581
$866$	0	0
$867$	−32182.1	−1.26063
$868$	0	0
$869$	−31134.8	−1.21539
$870$	0	0
$871$	−29269.1	−1.13863
$872$	0	0
$873$	−28839.7	−1.11807
$874$	0	0
$875$	9473.93	0.366031
$876$	0	0
$877$	10814.7	0.416406	0.208203	−	0.978086i	$-0.433239\pi$
$877$	10814.7	0.416406	0.208203	+	0.978086i	$0.433239\pi$
$878$	0	0
$879$	26644.3	1.02240
$880$	0	0
$881$	10411.1	0.398135	0.199068	−	0.979986i	$-0.436209\pi$
$881$	10411.1	0.398135	0.199068	+	0.979986i	$0.436209\pi$
$882$	0	0
$883$	−28277.5	−1.07771	−0.538853	−	0.842400i	$-0.681142\pi$
$883$	−28277.5	−1.07771	−0.538853	+	0.842400i	$0.681142\pi$
$884$	0	0
$885$	4530.98	0.172099
$886$	0	0
$887$	−16488.4	−0.624154	−0.312077	−	0.950057i	$-0.601025\pi$
$887$	−16488.4	−0.624154	−0.312077	+	0.950057i	$0.601025\pi$
$888$	0	0
$889$	−16508.7	−0.622816
$890$	0	0
$891$	47067.0	1.76970
$892$	0	0
$893$	15357.6	0.575499
$894$	0	0
$895$	2632.38	0.0983139
$896$	0	0
$897$	−5733.92	−0.213434
$898$	0	0
$899$	6086.46	0.225801
$900$	0	0
$901$	1324.08	0.0489585
$902$	0	0
$903$	5265.23	0.194038
$904$	0	0
$905$	−18030.8	−0.662280
$906$	0	0
$907$	42168.8	1.54376	0.771880	−	0.635768i	$-0.219316\pi$
$907$	42168.8	1.54376	0.771880	+	0.635768i	$0.219316\pi$
$908$	0	0
$909$	−3559.95	−0.129897
$910$	0	0
$911$	−20373.7	−0.740954	−0.370477	−	0.928842i	$-0.620806\pi$
$911$	−20373.7	−0.740954	−0.370477	+	0.928842i	$0.620806\pi$
$912$	0	0
$913$	−23318.3	−0.845262
$914$	0	0
$915$	25015.3	0.903803
$916$	0	0
$917$	−3545.34	−0.127675
$918$	0	0
$919$	−36823.5	−1.32176	−0.660879	−	0.750492i	$-0.729817\pi$
$919$	−36823.5	−1.32176	−0.660879	+	0.750492i	$0.729817\pi$
$920$	0	0
$921$	−9079.41	−0.324839
$922$	0	0
$923$	−4995.70	−0.178153
$924$	0	0
$925$	8909.19	0.316683
$926$	0	0
$927$	−7260.62	−0.257249
$928$	0	0
$929$	10251.0	0.362028	0.181014	−	0.983480i	$-0.442062\pi$
$929$	10251.0	0.362028	0.181014	+	0.983480i	$0.442062\pi$
$930$	0	0
$931$	18180.7	0.640009
$932$	0	0
$933$	1180.37	0.0414187
$934$	0	0
$935$	7090.75	0.248013
$936$	0	0
$937$	−28272.1	−0.985708	−0.492854	−	0.870112i	$-0.664046\pi$
$937$	−28272.1	−0.985708	−0.492854	+	0.870112i	$0.664046\pi$
$938$	0	0
$939$	11380.0	0.395499
$940$	0	0
$941$	43672.5	1.51295	0.756473	−	0.654025i	$-0.226921\pi$
$941$	43672.5	1.51295	0.756473	+	0.654025i	$0.226921\pi$
$942$	0	0
$943$	3323.46	0.114769
$944$	0	0
$945$	−347.270	−0.0119542
$946$	0	0
$947$	−43064.2	−1.47772	−0.738858	−	0.673861i	$-0.764635\pi$
$947$	−43064.2	−1.47772	−0.738858	+	0.673861i	$0.764635\pi$
$948$	0	0
$949$	25045.3	0.856698
$950$	0	0
$951$	18151.1	0.618918
$952$	0	0
$953$	−11297.0	−0.383994	−0.191997	−	0.981396i	$-0.561496\pi$
$953$	−11297.0	−0.383994	−0.191997	+	0.981396i	$0.561496\pi$
$954$	0	0
$955$	−10315.1	−0.349518
$956$	0	0
$957$	−48349.5	−1.63314
$958$	0	0
$959$	3541.42	0.119248
$960$	0	0
$961$	−25805.5	−0.866218
$962$	0	0
$963$	−28631.4	−0.958082
$964$	0	0
$965$	14363.8	0.479157
$966$	0	0
$967$	−25167.0	−0.836937	−0.418468	−	0.908231i	$-0.637433\pi$
$967$	−25167.0	−0.836937	−0.418468	+	0.908231i	$0.637433\pi$
$968$	0	0
$969$	−12759.6	−0.423009
$970$	0	0
$971$	2387.35	0.0789018	0.0394509	−	0.999222i	$-0.487439\pi$
$971$	2387.35	0.0789018	0.0394509	+	0.999222i	$0.487439\pi$
$972$	0	0
$973$	−21604.5	−0.711828
$974$	0	0
$975$	−26416.9	−0.867711
$976$	0	0
$977$	−50653.2	−1.65869	−0.829345	−	0.558736i	$-0.811286\pi$
$977$	−50653.2	−1.65869	−0.829345	+	0.558736i	$0.811286\pi$
$978$	0	0
$979$	−86229.7	−2.81503
$980$	0	0
$981$	−52377.0	−1.70466
$982$	0	0
$983$	−3927.47	−0.127433	−0.0637166	−	0.997968i	$-0.520295\pi$
$983$	−3927.47	−0.127433	−0.0637166	+	0.997968i	$0.520295\pi$
$984$	0	0
$985$	−14728.9	−0.476447
$986$	0	0
$987$	−15014.9	−0.484223
$988$	0	0
$989$	−1734.66	−0.0557723
$990$	0	0
$991$	−41415.4	−1.32755	−0.663776	−	0.747931i	$-0.731047\pi$
$991$	−41415.4	−1.32755	−0.663776	+	0.747931i	$0.731047\pi$
$992$	0	0
$993$	23979.4	0.766328
$994$	0	0
$995$	19974.3	0.636411
$996$	0	0
$997$	−54644.4	−1.73581	−0.867907	−	0.496727i	$-0.834535\pi$
$997$	−54644.4	−1.73581	−0.867907	+	0.496727i	$0.834535\pi$
$998$	0	0
$999$	−711.805	−0.0225430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bg.1.8		8
4.3	odd	2		1472.4.a.bh.1.1		8
8.3	odd	2		736.4.a.e.1.8	✓	8
8.5	even	2		736.4.a.f.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.e.1.8	✓	8	8.3	odd	2
736.4.a.f.1.1	yes	8	8.5	even	2
1472.4.a.bg.1.8		8	1.1	even	1	trivial
1472.4.a.bh.1.1		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)

\(f(q)\)	\(=\)	\(q+7.42564 q^{3} -4.36302 q^{5} +9.40153 q^{7} +28.1401 q^{9} -67.5361 q^{11} +33.5730 q^{13} -32.3982 q^{15} +24.0641 q^{17} -71.4057 q^{19} +69.8123 q^{21} -23.0000 q^{23} -105.964 q^{25} +8.46606 q^{27} +96.4102 q^{29} +63.1309 q^{31} -501.498 q^{33} -41.0191 q^{35} -84.0774 q^{37} +249.301 q^{39} -144.498 q^{41} +75.4198 q^{43} -122.776 q^{45} -215.075 q^{47} -254.611 q^{49} +178.691 q^{51} +55.0233 q^{53} +294.661 q^{55} -530.233 q^{57} -139.853 q^{59} -772.119 q^{61} +264.560 q^{63} -146.480 q^{65} -871.806 q^{67} -170.790 q^{69} -148.801 q^{71} +745.997 q^{73} -786.851 q^{75} -634.942 q^{77} +461.010 q^{79} -696.917 q^{81} +345.272 q^{83} -104.992 q^{85} +715.907 q^{87} +1276.79 q^{89} +315.637 q^{91} +468.787 q^{93} +311.545 q^{95} -1024.86 q^{97} -1900.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} + 12 q^{5} - 14 q^{7} + 90 q^{9} - 88 q^{11} + 30 q^{13} + 30 q^{15} + 58 q^{17} - 190 q^{19} + 66 q^{21} - 184 q^{23} + 28 q^{25} - 432 q^{27} - 190 q^{29} - 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 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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