LMFDB - Embedded newform 1008.6.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,6,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1008.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1008.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:	$161.666890371$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	1008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-28.7492 q^{5} -49.0000 q^{7} +O(q^{10})$	$q-28.7492 q^{5} -49.0000 q^{7} -270.090 q^{11} +300.640 q^{13} -613.106 q^{17} +1700.95 q^{19} +3188.15 q^{23} -2298.49 q^{25} -4299.28 q^{29} -2028.46 q^{31} +1408.71 q^{35} +5154.46 q^{37} +7146.21 q^{41} +19584.3 q^{43} +19998.4 q^{47} +2401.00 q^{49} -3948.82 q^{53} +7764.86 q^{55} -29707.6 q^{59} -50519.3 q^{61} -8643.14 q^{65} -5053.56 q^{67} +32853.3 q^{71} -11115.0 q^{73} +13234.4 q^{77} -81889.4 q^{79} +118234. q^{83} +17626.3 q^{85} +41695.4 q^{89} -14731.3 q^{91} -48900.9 q^{95} +43682.8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{5} - 98 q^{7}+O(q^{10}) $ 2 * q + 18 * q^5 - 98 * q^7	$ 2 q + 18 q^{5} - 98 q^{7} + 396 q^{11} - 350 q^{13} - 1800 q^{17} + 3266 q^{19} + 2088 q^{23} - 3238 q^{25} - 6696 q^{29} + 20 q^{31} - 882 q^{35} + 6232 q^{37} + 6048 q^{41} + 3020 q^{43} + 11700 q^{47} + 4802 q^{49} - 9468 q^{53} + 38904 q^{55} - 43938 q^{59} - 64754 q^{61} - 39060 q^{65} - 24784 q^{67} + 97416 q^{71} + 17452 q^{73} - 19404 q^{77} - 51256 q^{79} + 117558 q^{83} - 37860 q^{85} - 84276 q^{89} + 17150 q^{91} + 24264 q^{95} + 20776 q^{97}+O(q^{100}) $ 2 * q + 18 * q^5 - 98 * q^7 + 396 * q^11 - 350 * q^13 - 1800 * q^17 + 3266 * q^19 + 2088 * q^23 - 3238 * q^25 - 6696 * q^29 + 20 * q^31 - 882 * q^35 + 6232 * q^37 + 6048 * q^41 + 3020 * q^43 + 11700 * q^47 + 4802 * q^49 - 9468 * q^53 + 38904 * q^55 - 43938 * q^59 - 64754 * q^61 - 39060 * q^65 - 24784 * q^67 + 97416 * q^71 + 17452 * q^73 - 19404 * q^77 - 51256 * q^79 + 117558 * q^83 - 37860 * q^85 - 84276 * q^89 + 17150 * q^91 + 24264 * q^95 + 20776 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−28.7492	−0.514281	−0.257140	−	0.966374i	$-0.582780\pi$
$5$	−28.7492	−0.514281	−0.257140	+	0.966374i	$0.582780\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−270.090	−0.673018	−0.336509	−	0.941680i	$-0.609246\pi$
$11$	−270.090	−0.673018	−0.336509	+	0.941680i	$0.609246\pi$
$12$	0	0
$13$	300.640	0.493387	0.246694	−	0.969094i	$-0.420656\pi$
$13$	300.640	0.493387	0.246694	+	0.969094i	$0.420656\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−613.106	−0.514533	−0.257267	−	0.966340i	$-0.582822\pi$
$17$	−613.106	−0.514533	−0.257267	+	0.966340i	$0.582822\pi$
$18$	0	0
$19$	1700.95	1.08095	0.540477	−	0.841359i	$-0.318244\pi$
$19$	1700.95	1.08095	0.540477	+	0.841359i	$0.318244\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3188.15	1.25667	0.628333	−	0.777945i	$-0.283738\pi$
$23$	3188.15	1.25667	0.628333	+	0.777945i	$0.283738\pi$
$24$	0	0
$25$	−2298.49	−0.735515
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4299.28	−0.949294	−0.474647	−	0.880176i	$-0.657424\pi$
$29$	−4299.28	−0.949294	−0.474647	+	0.880176i	$0.657424\pi$
$30$	0	0
$31$	−2028.46	−0.379106	−0.189553	−	0.981870i	$-0.560704\pi$
$31$	−2028.46	−0.379106	−0.189553	+	0.981870i	$0.560704\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1408.71	0.194380
$36$	0	0
$37$	5154.46	0.618983	0.309491	−	0.950902i	$-0.399841\pi$
$37$	5154.46	0.618983	0.309491	+	0.950902i	$0.399841\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7146.21	0.663921	0.331960	−	0.943293i	$-0.392290\pi$
$41$	7146.21	0.663921	0.331960	+	0.943293i	$0.392290\pi$
$42$	0	0
$43$	19584.3	1.61524	0.807620	−	0.589703i	$-0.200755\pi$
$43$	19584.3	1.61524	0.807620	+	0.589703i	$0.200755\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	19998.4	1.32054	0.660268	−	0.751030i	$-0.270443\pi$
$47$	19998.4	1.32054	0.660268	+	0.751030i	$0.270443\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3948.82	−0.193098	−0.0965489	−	0.995328i	$-0.530780\pi$
$53$	−3948.82	−0.193098	−0.0965489	+	0.995328i	$0.530780\pi$
$54$	0	0
$55$	7764.86	0.346120
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−29707.6	−1.11106	−0.555530	−	0.831497i	$-0.687484\pi$
$59$	−29707.6	−1.11106	−0.555530	+	0.831497i	$0.687484\pi$
$60$	0	0
$61$	−50519.3	−1.73833	−0.869165	−	0.494522i	$-0.835343\pi$
$61$	−50519.3	−1.73833	−0.869165	+	0.494522i	$0.835343\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8643.14	−0.253740
$66$	0	0
$67$	−5053.56	−0.137534	−0.0687671	−	0.997633i	$-0.521907\pi$
$67$	−5053.56	−0.137534	−0.0687671	+	0.997633i	$0.521907\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	32853.3	0.773453	0.386726	−	0.922195i	$-0.373606\pi$
$71$	32853.3	0.773453	0.386726	+	0.922195i	$0.373606\pi$
$72$	0	0
$73$	−11115.0	−0.244119	−0.122059	−	0.992523i	$-0.538950\pi$
$73$	−11115.0	−0.244119	−0.122059	+	0.992523i	$0.538950\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13234.4	0.254377
$78$	0	0
$79$	−81889.4	−1.47625	−0.738125	−	0.674664i	$-0.764288\pi$
$79$	−81889.4	−1.47625	−0.738125	+	0.674664i	$0.764288\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	118234.	1.88385	0.941926	−	0.335819i	$-0.109013\pi$
$83$	118234.	1.88385	0.941926	+	0.335819i	$0.109013\pi$
$84$	0	0
$85$	17626.3	0.264615
$86$	0	0
$87$	0	0
$88$	0	0
$89$	41695.4	0.557972	0.278986	−	0.960295i	$-0.410002\pi$
$89$	41695.4	0.557972	0.278986	+	0.960295i	$0.410002\pi$
$90$	0	0
$91$	−14731.3	−0.186483
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−48900.9	−0.555914
$96$	0	0
$97$	43682.8	0.471391	0.235695	−	0.971827i	$-0.424263\pi$
$97$	43682.8	0.471391	0.235695	+	0.971827i	$0.424263\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−25648.1	−0.250179	−0.125090	−	0.992145i	$-0.539922\pi$
$101$	−25648.1	−0.250179	−0.125090	+	0.992145i	$0.539922\pi$
$102$	0	0
$103$	14320.0	0.133000	0.0664999	−	0.997786i	$-0.478817\pi$
$103$	14320.0	0.133000	0.0664999	+	0.997786i	$0.478817\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17201.8	0.145249	0.0726247	−	0.997359i	$-0.476862\pi$
$107$	17201.8	0.145249	0.0726247	+	0.997359i	$0.476862\pi$
$108$	0	0
$109$	−86017.6	−0.693459	−0.346730	−	0.937965i	$-0.612708\pi$
$109$	−86017.6	−0.693459	−0.346730	+	0.937965i	$0.612708\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−137568.	−1.01349	−0.506745	−	0.862096i	$-0.669152\pi$
$113$	−137568.	−1.01349	−0.506745	+	0.862096i	$0.669152\pi$
$114$	0	0
$115$	−91656.8	−0.646279
$116$	0	0
$117$	0	0
$118$	0	0
$119$	30042.2	0.194475
$120$	0	0
$121$	−88102.5	−0.547047
$122$	0	0
$123$	0	0
$124$	0	0
$125$	155921.	0.892542
$126$	0	0
$127$	70567.1	0.388233	0.194117	−	0.980978i	$-0.437816\pi$
$127$	70567.1	0.388233	0.194117	+	0.980978i	$0.437816\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−173712.	−0.884408	−0.442204	−	0.896914i	$-0.645803\pi$
$131$	−173712.	−0.884408	−0.442204	+	0.896914i	$0.645803\pi$
$132$	0	0
$133$	−83346.5	−0.408562
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1989.94	0.00905813	0.00452907	−	0.999990i	$-0.498558\pi$
$137$	1989.94	0.00905813	0.00452907	+	0.999990i	$0.498558\pi$
$138$	0	0
$139$	−366409.	−1.60853	−0.804264	−	0.594272i	$-0.797440\pi$
$139$	−366409.	−1.60853	−0.804264	+	0.594272i	$0.797440\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−81199.7	−0.332058
$144$	0	0
$145$	123601.	0.488204
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−140719.	−0.519261	−0.259631	−	0.965708i	$-0.583601\pi$
$149$	−140719.	−0.519261	−0.259631	+	0.965708i	$0.583601\pi$
$150$	0	0
$151$	−50064.6	−0.178685	−0.0893425	−	0.996001i	$-0.528477\pi$
$151$	−50064.6	−0.178685	−0.0893425	+	0.996001i	$0.528477\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	58316.4	0.194967
$156$	0	0
$157$	−89794.6	−0.290738	−0.145369	−	0.989378i	$-0.546437\pi$
$157$	−89794.6	−0.290738	−0.145369	+	0.989378i	$0.546437\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−156219.	−0.474975
$162$	0	0
$163$	481230.	1.41868	0.709339	−	0.704867i	$-0.248994\pi$
$163$	481230.	1.41868	0.709339	+	0.704867i	$0.248994\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−86572.7	−0.240209	−0.120105	−	0.992761i	$-0.538323\pi$
$167$	−86572.7	−0.240209	−0.120105	+	0.992761i	$0.538323\pi$
$168$	0	0
$169$	−280909.	−0.756569
$170$	0	0
$171$	0	0
$172$	0	0
$173$	58137.4	0.147686	0.0738432	−	0.997270i	$-0.476474\pi$
$173$	58137.4	0.147686	0.0738432	+	0.997270i	$0.476474\pi$
$174$	0	0
$175$	112626.	0.277999
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−209380.	−0.488431	−0.244215	−	0.969721i	$-0.578530\pi$
$179$	−209380.	−0.488431	−0.244215	+	0.969721i	$0.578530\pi$
$180$	0	0
$181$	278996.	0.632996	0.316498	−	0.948593i	$-0.397493\pi$
$181$	278996.	0.632996	0.316498	+	0.948593i	$0.397493\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−148186.	−0.318331
$186$	0	0
$187$	165594.	0.346290
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−445132.	−0.882888	−0.441444	−	0.897289i	$-0.645534\pi$
$191$	−445132.	−0.882888	−0.441444	+	0.897289i	$0.645534\pi$
$192$	0	0
$193$	−726811.	−1.40452	−0.702260	−	0.711920i	$-0.747826\pi$
$193$	−726811.	−1.40452	−0.702260	+	0.711920i	$0.747826\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	364897.	0.669892	0.334946	−	0.942237i	$-0.391282\pi$
$197$	364897.	0.669892	0.334946	+	0.942237i	$0.391282\pi$
$198$	0	0
$199$	−289307.	−0.517877	−0.258938	−	0.965894i	$-0.583373\pi$
$199$	−289307.	−0.517877	−0.258938	+	0.965894i	$0.583373\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	210665.	0.358799
$204$	0	0
$205$	−205448.	−0.341442
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−459409.	−0.727501
$210$	0	0
$211$	−750147.	−1.15995	−0.579976	−	0.814633i	$-0.696938\pi$
$211$	−750147.	−1.15995	−0.579976	+	0.814633i	$0.696938\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−563033.	−0.830687
$216$	0	0
$217$	99394.3	0.143289
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−184324.	−0.253864
$222$	0	0
$223$	−534398.	−0.719619	−0.359810	−	0.933026i	$-0.617158\pi$
$223$	−534398.	−0.719619	−0.359810	+	0.933026i	$0.617158\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−410624.	−0.528907	−0.264453	−	0.964398i	$-0.585192\pi$
$227$	−410624.	−0.528907	−0.264453	+	0.964398i	$0.585192\pi$
$228$	0	0
$229$	1.03036e6	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$229$	1.03036e6	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	119211.	0.143856	0.0719278	−	0.997410i	$-0.477085\pi$
$233$	119211.	0.143856	0.0719278	+	0.997410i	$0.477085\pi$
$234$	0	0
$235$	−574937.	−0.679127
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−254090.	−0.287735	−0.143868	−	0.989597i	$-0.545954\pi$
$239$	−254090.	−0.287735	−0.143868	+	0.989597i	$0.545954\pi$
$240$	0	0
$241$	1.41251e6	1.56656	0.783282	−	0.621667i	$-0.213544\pi$
$241$	1.41251e6	1.56656	0.783282	+	0.621667i	$0.213544\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−69026.8	−0.0734687
$246$	0	0
$247$	511372.	0.533329
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.67542e6	−1.67857	−0.839286	−	0.543690i	$-0.817027\pi$
$251$	−1.67542e6	−1.67857	−0.839286	+	0.543690i	$0.817027\pi$
$252$	0	0
$253$	−861087.	−0.845758
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−726996.	−0.686593	−0.343296	−	0.939227i	$-0.611544\pi$
$257$	−726996.	−0.686593	−0.343296	+	0.939227i	$0.611544\pi$
$258$	0	0
$259$	−252568.	−0.233953
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−225880.	−0.201367	−0.100684	−	0.994918i	$-0.532103\pi$
$263$	−225880.	−0.201367	−0.100684	+	0.994918i	$0.532103\pi$
$264$	0	0
$265$	113525.	0.0993065
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.80527e6	−1.52111	−0.760557	−	0.649272i	$-0.775074\pi$
$269$	−1.80527e6	−1.52111	−0.760557	+	0.649272i	$0.775074\pi$
$270$	0	0
$271$	1.71380e6	1.41754	0.708771	−	0.705439i	$-0.249250\pi$
$271$	1.71380e6	1.41754	0.708771	+	0.705439i	$0.249250\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	620797.	0.495015
$276$	0	0
$277$	2.23055e6	1.74668	0.873338	−	0.487115i	$-0.161951\pi$
$277$	2.23055e6	1.74668	0.873338	+	0.487115i	$0.161951\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.67140e6	−1.26274	−0.631371	−	0.775481i	$-0.717507\pi$
$281$	−1.67140e6	−1.26274	−0.631371	+	0.775481i	$0.717507\pi$
$282$	0	0
$283$	396152.	0.294033	0.147016	−	0.989134i	$-0.453033\pi$
$283$	396152.	0.294033	0.147016	+	0.989134i	$0.453033\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−350164.	−0.250938
$288$	0	0
$289$	−1.04396e6	−0.735256
$290$	0	0
$291$	0	0
$292$	0	0
$293$	929465.	0.632505	0.316252	−	0.948675i	$-0.397575\pi$
$293$	929465.	0.632505	0.316252	+	0.948675i	$0.397575\pi$
$294$	0	0
$295$	854068.	0.571397
$296$	0	0
$297$	0	0
$298$	0	0
$299$	958485.	0.620022
$300$	0	0
$301$	−959631.	−0.610503
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.45239e6	0.893990
$306$	0	0
$307$	−1.83295e6	−1.10995	−0.554976	−	0.831866i	$-0.687273\pi$
$307$	−1.83295e6	−1.10995	−0.554976	+	0.831866i	$0.687273\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.29685e6	−1.34658	−0.673289	−	0.739379i	$-0.735119\pi$
$311$	−2.29685e6	−1.34658	−0.673289	+	0.739379i	$0.735119\pi$
$312$	0	0
$313$	−3.42470e6	−1.97589	−0.987943	−	0.154817i	$-0.950521\pi$
$313$	−3.42470e6	−1.97589	−0.987943	+	0.154817i	$0.950521\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.94305e6	−1.64494	−0.822470	−	0.568808i	$-0.807405\pi$
$317$	−2.94305e6	−1.64494	−0.822470	+	0.568808i	$0.807405\pi$
$318$	0	0
$319$	1.16119e6	0.638891
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.04286e6	−0.556187
$324$	0	0
$325$	−691016.	−0.362894
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−979921.	−0.499116
$330$	0	0
$331$	−966164.	−0.484709	−0.242354	−	0.970188i	$-0.577920\pi$
$331$	−966164.	−0.484709	−0.242354	+	0.970188i	$0.577920\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	145286.	0.0707312
$336$	0	0
$337$	136417.	0.0654327	0.0327163	−	0.999465i	$-0.489584\pi$
$337$	136417.	0.0654327	0.0327163	+	0.999465i	$0.489584\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	547865.	0.255145
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	355408.	0.158454	0.0792270	−	0.996857i	$-0.474755\pi$
$347$	355408.	0.158454	0.0792270	+	0.996857i	$0.474755\pi$
$348$	0	0
$349$	−140128.	−0.0615830	−0.0307915	−	0.999526i	$-0.509803\pi$
$349$	−140128.	−0.0615830	−0.0307915	+	0.999526i	$0.509803\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.48141e6	−1.48703	−0.743514	−	0.668721i	$-0.766842\pi$
$353$	−3.48141e6	−1.48703	−0.743514	+	0.668721i	$0.766842\pi$
$354$	0	0
$355$	−944507.	−0.397772
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.75285e6	0.717810	0.358905	−	0.933374i	$-0.383150\pi$
$359$	1.75285e6	0.717810	0.358905	+	0.933374i	$0.383150\pi$
$360$	0	0
$361$	417127.	0.168461
$362$	0	0
$363$	0	0
$364$	0	0
$365$	319546.	0.125546
$366$	0	0
$367$	1.76939e6	0.685738	0.342869	−	0.939383i	$-0.388601\pi$
$367$	1.76939e6	0.685738	0.342869	+	0.939383i	$0.388601\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	193492.	0.0729841
$372$	0	0
$373$	−4.16212e6	−1.54897	−0.774485	−	0.632592i	$-0.781991\pi$
$373$	−4.16212e6	−1.54897	−0.774485	+	0.632592i	$0.781991\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.29253e6	−0.468369
$378$	0	0
$379$	−618163.	−0.221057	−0.110529	−	0.993873i	$-0.535254\pi$
$379$	−618163.	−0.221057	−0.110529	+	0.993873i	$0.535254\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.11163e6	−1.43225	−0.716123	−	0.697974i	$-0.754085\pi$
$383$	−4.11163e6	−1.43225	−0.716123	+	0.697974i	$0.754085\pi$
$384$	0	0
$385$	−380478.	−0.130821
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.62076e6	−1.54824	−0.774122	−	0.633037i	$-0.781808\pi$
$389$	−4.62076e6	−1.54824	−0.774122	+	0.633037i	$0.781808\pi$
$390$	0	0
$391$	−1.95468e6	−0.646596
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.35425e6	0.759207
$396$	0	0
$397$	5.07349e6	1.61559	0.807794	−	0.589465i	$-0.200661\pi$
$397$	5.07349e6	1.61559	0.807794	+	0.589465i	$0.200661\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.48056e6	0.459795	0.229898	−	0.973215i	$-0.426161\pi$
$401$	1.48056e6	0.459795	0.229898	+	0.973215i	$0.426161\pi$
$402$	0	0
$403$	−609834.	−0.187046
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.39217e6	−0.416586
$408$	0	0
$409$	−4.53379e6	−1.34015	−0.670075	−	0.742294i	$-0.733738\pi$
$409$	−4.53379e6	−1.34015	−0.670075	+	0.742294i	$0.733738\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.45567e6	0.419941
$414$	0	0
$415$	−3.39913e6	−0.968829
$416$	0	0
$417$	0	0
$418$	0	0
$419$	111026.	0.0308952	0.0154476	−	0.999881i	$-0.495083\pi$
$419$	111026.	0.0308952	0.0154476	+	0.999881i	$0.495083\pi$
$420$	0	0
$421$	−1.41151e6	−0.388132	−0.194066	−	0.980988i	$-0.562168\pi$
$421$	−1.41151e6	−0.388132	−0.194066	+	0.980988i	$0.562168\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.40922e6	0.378447
$426$	0	0
$427$	2.47544e6	0.657027
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.07640e6	0.279113	0.139557	−	0.990214i	$-0.455432\pi$
$431$	1.07640e6	0.279113	0.139557	+	0.990214i	$0.455432\pi$
$432$	0	0
$433$	−310172.	−0.0795029	−0.0397515	−	0.999210i	$-0.512657\pi$
$433$	−310172.	−0.0795029	−0.0397515	+	0.999210i	$0.512657\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.42288e6	1.35840
$438$	0	0
$439$	−5.67650e6	−1.40579	−0.702893	−	0.711296i	$-0.748109\pi$
$439$	−5.67650e6	−1.40579	−0.702893	+	0.711296i	$0.748109\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.05966e6	0.982834	0.491417	−	0.870924i	$-0.336479\pi$
$443$	4.05966e6	0.982834	0.491417	+	0.870924i	$0.336479\pi$
$444$	0	0
$445$	−1.19871e6	−0.286955
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.96544e6	1.63054	0.815272	−	0.579078i	$-0.196587\pi$
$449$	6.96544e6	1.63054	0.815272	+	0.579078i	$0.196587\pi$
$450$	0	0
$451$	−1.93012e6	−0.446830
$452$	0	0
$453$	0	0
$454$	0	0
$455$	423514.	0.0959045
$456$	0	0
$457$	1.79523e6	0.402096	0.201048	−	0.979581i	$-0.435565\pi$
$457$	1.79523e6	0.402096	0.201048	+	0.979581i	$0.435565\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.11294e6	0.463058	0.231529	−	0.972828i	$-0.425627\pi$
$461$	2.11294e6	0.463058	0.231529	+	0.972828i	$0.425627\pi$
$462$	0	0
$463$	−1.26223e6	−0.273643	−0.136822	−	0.990596i	$-0.543689\pi$
$463$	−1.26223e6	−0.273643	−0.136822	+	0.990596i	$0.543689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.58926e6	−0.761576	−0.380788	−	0.924662i	$-0.624347\pi$
$467$	−3.58926e6	−0.761576	−0.380788	+	0.924662i	$0.624347\pi$
$468$	0	0
$469$	247624.	0.0519830
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.28952e6	−1.08708
$474$	0	0
$475$	−3.90960e6	−0.795058
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.41693e6	−0.481311	−0.240655	−	0.970611i	$-0.577362\pi$
$479$	−2.41693e6	−0.481311	−0.240655	+	0.970611i	$0.577362\pi$
$480$	0	0
$481$	1.54963e6	0.305398
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.25584e6	−0.242427
$486$	0	0
$487$	5.19403e6	0.992388	0.496194	−	0.868212i	$-0.334730\pi$
$487$	5.19403e6	0.992388	0.496194	+	0.868212i	$0.334730\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.38961e6	1.00891	0.504456	−	0.863437i	$-0.331693\pi$
$491$	5.38961e6	1.00891	0.504456	+	0.863437i	$0.331693\pi$
$492$	0	0
$493$	2.63592e6	0.488443
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.60981e6	−0.292338
$498$	0	0
$499$	3.29606e6	0.592576	0.296288	−	0.955099i	$-0.404251\pi$
$499$	3.29606e6	0.592576	0.296288	+	0.955099i	$0.404251\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.06512e7	−1.87706	−0.938528	−	0.345204i	$-0.887810\pi$
$503$	−1.06512e7	−1.87706	−0.938528	+	0.345204i	$0.887810\pi$
$504$	0	0
$505$	737361.	0.128662
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.74268e6	0.469225	0.234612	−	0.972089i	$-0.424618\pi$
$509$	2.74268e6	0.469225	0.234612	+	0.972089i	$0.424618\pi$
$510$	0	0
$511$	544633.	0.0922682
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−411689.	−0.0683992
$516$	0	0
$517$	−5.40136e6	−0.888744
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.97077e6	−0.802286	−0.401143	−	0.916015i	$-0.631387\pi$
$521$	−4.97077e6	−0.802286	−0.401143	+	0.916015i	$0.631387\pi$
$522$	0	0
$523$	−2.41579e6	−0.386193	−0.193096	−	0.981180i	$-0.561853\pi$
$523$	−2.41579e6	−0.386193	−0.193096	+	0.981180i	$0.561853\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.24366e6	0.195063
$528$	0	0
$529$	3.72798e6	0.579207
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.14843e6	0.327570
$534$	0	0
$535$	−494537.	−0.0746989
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−648485.	−0.0961454
$540$	0	0
$541$	472165.	0.0693587	0.0346794	−	0.999398i	$-0.488959\pi$
$541$	472165.	0.0693587	0.0346794	+	0.999398i	$0.488959\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.47293e6	0.356633
$546$	0	0
$547$	−7.63716e6	−1.09135	−0.545675	−	0.837997i	$-0.683727\pi$
$547$	−7.63716e6	−1.09135	−0.545675	+	0.837997i	$0.683727\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.31285e6	−1.02614
$552$	0	0
$553$	4.01258e6	0.557970
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.48807e6	0.612946	0.306473	−	0.951879i	$-0.400851\pi$
$557$	4.48807e6	0.612946	0.306473	+	0.951879i	$0.400851\pi$
$558$	0	0
$559$	5.88782e6	0.796938
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.16500e6	−0.287864	−0.143932	−	0.989588i	$-0.545975\pi$
$563$	−2.16500e6	−0.287864	−0.143932	+	0.989588i	$0.545975\pi$
$564$	0	0
$565$	3.95495e6	0.521219
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.13325e7	1.46739	0.733696	−	0.679478i	$-0.237794\pi$
$569$	1.13325e7	1.46739	0.733696	+	0.679478i	$0.237794\pi$
$570$	0	0
$571$	843773.	0.108302	0.0541509	−	0.998533i	$-0.482755\pi$
$571$	843773.	0.108302	0.0541509	+	0.998533i	$0.482755\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.32792e6	−0.924296
$576$	0	0
$577$	−2.23784e6	−0.279827	−0.139914	−	0.990164i	$-0.544682\pi$
$577$	−2.23784e6	−0.279827	−0.139914	+	0.990164i	$0.544682\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.79346e6	−0.712029
$582$	0	0
$583$	1.06653e6	0.129958
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.21190e7	1.45168	0.725839	−	0.687864i	$-0.241452\pi$
$587$	1.21190e7	1.45168	0.725839	+	0.687864i	$0.241452\pi$
$588$	0	0
$589$	−3.45030e6	−0.409797
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.00167e6	−0.934424	−0.467212	−	0.884145i	$-0.654742\pi$
$593$	−8.00167e6	−0.934424	−0.467212	+	0.884145i	$0.654742\pi$
$594$	0	0
$595$	−863689.	−0.100015
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.45899e7	1.66144	0.830719	−	0.556692i	$-0.187930\pi$
$599$	1.45899e7	1.66144	0.830719	+	0.556692i	$0.187930\pi$
$600$	0	0
$601$	−8.67178e6	−0.979314	−0.489657	−	0.871915i	$-0.662878\pi$
$601$	−8.67178e6	−0.979314	−0.489657	+	0.871915i	$0.662878\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.53287e6	0.281336
$606$	0	0
$607$	1.33059e7	1.46580	0.732898	−	0.680339i	$-0.238167\pi$
$607$	1.33059e7	1.46580	0.732898	+	0.680339i	$0.238167\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.01231e6	0.651536
$612$	0	0
$613$	2.35101e6	0.252699	0.126350	−	0.991986i	$-0.459674\pi$
$613$	2.35101e6	0.252699	0.126350	+	0.991986i	$0.459674\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.63523e6	−1.01894	−0.509470	−	0.860488i	$-0.670159\pi$
$617$	−9.63523e6	−1.01894	−0.509470	+	0.860488i	$0.670159\pi$
$618$	0	0
$619$	4.86148e6	0.509967	0.254983	−	0.966945i	$-0.417930\pi$
$619$	4.86148e6	0.509967	0.254983	+	0.966945i	$0.417930\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.04307e6	−0.210894
$624$	0	0
$625$	2.70017e6	0.276498
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.16023e6	−0.318487
$630$	0	0
$631$	6.59770e6	0.659659	0.329829	−	0.944041i	$-0.393009\pi$
$631$	6.59770e6	0.659659	0.329829	+	0.944041i	$0.393009\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.02874e6	−0.199661
$636$	0	0
$637$	721836.	0.0704839
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.44525e7	−1.38930	−0.694651	−	0.719347i	$-0.744441\pi$
$641$	−1.44525e7	−1.38930	−0.694651	+	0.719347i	$0.744441\pi$
$642$	0	0
$643$	1.54720e7	1.47577	0.737886	−	0.674926i	$-0.235824\pi$
$643$	1.54720e7	1.47577	0.737886	+	0.674926i	$0.235824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.66647e7	1.56508	0.782540	−	0.622601i	$-0.213924\pi$
$647$	1.66647e7	1.56508	0.782540	+	0.622601i	$0.213924\pi$
$648$	0	0
$649$	8.02371e6	0.747762
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.33451e7	1.22472	0.612361	−	0.790578i	$-0.290220\pi$
$653$	1.33451e7	1.22472	0.612361	+	0.790578i	$0.290220\pi$
$654$	0	0
$655$	4.99409e6	0.454834
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4.00667e6	−0.359393	−0.179697	−	0.983722i	$-0.557512\pi$
$659$	−4.00667e6	−0.359393	−0.179697	+	0.983722i	$0.557512\pi$
$660$	0	0
$661$	1.08005e7	0.961478	0.480739	−	0.876864i	$-0.340368\pi$
$661$	1.08005e7	0.961478	0.480739	+	0.876864i	$0.340368\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.39614e6	0.210116
$666$	0	0
$667$	−1.37068e7	−1.19294
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.36447e7	1.16993
$672$	0	0
$673$	1.09119e7	0.928676	0.464338	−	0.885658i	$-0.346292\pi$
$673$	1.09119e7	0.928676	0.464338	+	0.885658i	$0.346292\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.35765e7	1.13846	0.569229	−	0.822179i	$-0.307242\pi$
$677$	1.35765e7	1.13846	0.569229	+	0.822179i	$0.307242\pi$
$678$	0	0
$679$	−2.14046e6	−0.178169
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.26726e7	−1.03948	−0.519738	−	0.854326i	$-0.673970\pi$
$683$	−1.26726e7	−1.03948	−0.519738	+	0.854326i	$0.673970\pi$
$684$	0	0
$685$	−57209.2	−0.00465843
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.18717e6	−0.0952720
$690$	0	0
$691$	−7.11964e6	−0.567235	−0.283617	−	0.958938i	$-0.591535\pi$
$691$	−7.11964e6	−0.567235	−0.283617	+	0.958938i	$0.591535\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.05339e7	0.827235
$696$	0	0
$697$	−4.38139e6	−0.341609
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.00155e7	0.769803	0.384902	−	0.922958i	$-0.374235\pi$
$701$	1.00155e7	0.769803	0.384902	+	0.922958i	$0.374235\pi$
$702$	0	0
$703$	8.76746e6	0.669092
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.25676e6	0.0945589
$708$	0	0
$709$	−8.84454e6	−0.660784	−0.330392	−	0.943844i	$-0.607181\pi$
$709$	−8.84454e6	−0.660784	−0.330392	+	0.943844i	$0.607181\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.46703e6	−0.476410
$714$	0	0
$715$	2.33442e6	0.170771
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.58086e6	0.474745	0.237373	−	0.971419i	$-0.423714\pi$
$719$	6.58086e6	0.474745	0.237373	+	0.971419i	$0.423714\pi$
$720$	0	0
$721$	−701682.	−0.0502692
$722$	0	0
$723$	0	0
$724$	0	0
$725$	9.88183e6	0.698220
$726$	0	0
$727$	−1.88401e7	−1.32205	−0.661023	−	0.750365i	$-0.729878\pi$
$727$	−1.88401e7	−1.32205	−0.661023	+	0.750365i	$0.729878\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.20073e7	−0.831095
$732$	0	0
$733$	−2.78330e6	−0.191337	−0.0956687	−	0.995413i	$-0.530499\pi$
$733$	−2.78330e6	−0.191337	−0.0956687	+	0.995413i	$0.530499\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.36491e6	0.0925629
$738$	0	0
$739$	2.48970e7	1.67701	0.838505	−	0.544894i	$-0.183430\pi$
$739$	2.48970e7	1.67701	0.838505	+	0.544894i	$0.183430\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.86085e6	−0.256573	−0.128286	−	0.991737i	$-0.540948\pi$
$743$	−3.86085e6	−0.256573	−0.128286	+	0.991737i	$0.540948\pi$
$744$	0	0
$745$	4.04554e6	0.267046
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−842888.	−0.0548991
$750$	0	0
$751$	−6.72737e6	−0.435257	−0.217628	−	0.976032i	$-0.569832\pi$
$751$	−6.72737e6	−0.435257	−0.217628	+	0.976032i	$0.569832\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.43932e6	0.0918943
$756$	0	0
$757$	2.17782e7	1.38128	0.690642	−	0.723197i	$-0.257328\pi$
$757$	2.17782e7	1.38128	0.690642	+	0.723197i	$0.257328\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.57074e7	1.60915	0.804575	−	0.593851i	$-0.202393\pi$
$761$	2.57074e7	1.60915	0.804575	+	0.593851i	$0.202393\pi$
$762$	0	0
$763$	4.21486e6	0.262103
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.93127e6	−0.548182
$768$	0	0
$769$	−1.34375e7	−0.819413	−0.409706	−	0.912217i	$-0.634369\pi$
$769$	−1.34375e7	−0.819413	−0.409706	+	0.912217i	$0.634369\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.05572e7	−1.83935	−0.919674	−	0.392682i	$-0.871547\pi$
$773$	−3.05572e7	−1.83935	−0.919674	+	0.392682i	$0.871547\pi$
$774$	0	0
$775$	4.66237e6	0.278839
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.21553e7	0.717667
$780$	0	0
$781$	−8.87335e6	−0.520547
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.58152e6	0.149521
$786$	0	0
$787$	2.07672e6	0.119520	0.0597602	−	0.998213i	$-0.480966\pi$
$787$	2.07672e6	0.119520	0.0597602	+	0.998213i	$0.480966\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.74081e6	0.383064
$792$	0	0
$793$	−1.51881e7	−0.857670
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.98563e6	0.333783	0.166892	−	0.985975i	$-0.446627\pi$
$797$	5.98563e6	0.333783	0.166892	+	0.985975i	$0.446627\pi$
$798$	0	0
$799$	−1.22611e7	−0.679460
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.00204e6	0.164296
$804$	0	0
$805$	4.49118e6	0.244270
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.96864e7	−1.05754	−0.528769	−	0.848766i	$-0.677346\pi$
$809$	−1.96864e7	−1.05754	−0.528769	+	0.848766i	$0.677346\pi$
$810$	0	0
$811$	−8.50101e6	−0.453856	−0.226928	−	0.973912i	$-0.572868\pi$
$811$	−8.50101e6	−0.453856	−0.226928	+	0.973912i	$0.572868\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.38350e7	−0.729599
$816$	0	0
$817$	3.33119e7	1.74600
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.36199e6	0.0705204	0.0352602	−	0.999378i	$-0.488774\pi$
$821$	1.36199e6	0.0705204	0.0352602	+	0.999378i	$0.488774\pi$
$822$	0	0
$823$	1.35934e6	0.0699566	0.0349783	−	0.999388i	$-0.488864\pi$
$823$	1.35934e6	0.0699566	0.0349783	+	0.999388i	$0.488864\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.00727e7	0.512132	0.256066	−	0.966659i	$-0.417574\pi$
$827$	1.00727e7	0.512132	0.256066	+	0.966659i	$0.417574\pi$
$828$	0	0
$829$	−5.63984e6	−0.285023	−0.142512	−	0.989793i	$-0.545518\pi$
$829$	−5.63984e6	−0.285023	−0.142512	+	0.989793i	$0.545518\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.47207e6	−0.0735048
$834$	0	0
$835$	2.48889e6	0.123535
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.16351e7	−0.570642	−0.285321	−	0.958432i	$-0.592100\pi$
$839$	−1.16351e7	−0.570642	−0.285321	+	0.958432i	$0.592100\pi$
$840$	0	0
$841$	−2.02735e6	−0.0988413
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.07590e6	0.389089
$846$	0	0
$847$	4.31702e6	0.206764
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.64332e7	0.777854
$852$	0	0
$853$	2.85205e7	1.34210	0.671049	−	0.741413i	$-0.265844\pi$
$853$	2.85205e7	1.34210	0.671049	+	0.741413i	$0.265844\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.95725e6	0.463113	0.231557	−	0.972821i	$-0.425618\pi$
$857$	9.95725e6	0.463113	0.231557	+	0.972821i	$0.425618\pi$
$858$	0	0
$859$	1.49322e7	0.690463	0.345232	−	0.938517i	$-0.387800\pi$
$859$	1.49322e7	0.690463	0.345232	+	0.938517i	$0.387800\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.84933e7	1.75937	0.879687	−	0.475553i	$-0.157752\pi$
$863$	3.84933e7	1.75937	0.879687	+	0.475553i	$0.157752\pi$
$864$	0	0
$865$	−1.67140e6	−0.0759523
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.21175e7	0.993542
$870$	0	0
$871$	−1.51930e6	−0.0678576
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7.64011e6	−0.337349
$876$	0	0
$877$	9.40311e6	0.412831	0.206416	−	0.978464i	$-0.433820\pi$
$877$	9.40311e6	0.412831	0.206416	+	0.978464i	$0.433820\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.10395e6	−0.0479194	−0.0239597	−	0.999713i	$-0.507627\pi$
$881$	−1.10395e6	−0.0479194	−0.0239597	+	0.999713i	$0.507627\pi$
$882$	0	0
$883$	−8.06579e6	−0.348133	−0.174067	−	0.984734i	$-0.555691\pi$
$883$	−8.06579e6	−0.348133	−0.174067	+	0.984734i	$0.555691\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.49902e7	−0.639732	−0.319866	−	0.947463i	$-0.603638\pi$
$887$	−1.49902e7	−0.639732	−0.319866	+	0.947463i	$0.603638\pi$
$888$	0	0
$889$	−3.45779e6	−0.146738
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.40162e7	1.42744
$894$	0	0
$895$	6.01950e6	0.251190
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.72090e6	0.359883
$900$	0	0
$901$	2.42104e6	0.0993553
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.02089e6	−0.325538
$906$	0	0
$907$	−4.12622e6	−0.166546	−0.0832730	−	0.996527i	$-0.526537\pi$
$907$	−4.12622e6	−0.166546	−0.0832730	+	0.996527i	$0.526537\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.04272e7	1.61391	0.806953	−	0.590616i	$-0.201115\pi$
$911$	4.04272e7	1.61391	0.806953	+	0.590616i	$0.201115\pi$
$912$	0	0
$913$	−3.19338e7	−1.26787
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.51191e6	0.334275
$918$	0	0
$919$	−2.18546e7	−0.853600	−0.426800	−	0.904346i	$-0.640359\pi$
$919$	−2.18546e7	−0.853600	−0.426800	+	0.904346i	$0.640359\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.87702e6	0.381612
$924$	0	0
$925$	−1.18474e7	−0.455271
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.06843e7	−0.406169	−0.203085	−	0.979161i	$-0.565097\pi$
$929$	−1.06843e7	−0.406169	−0.203085	+	0.979161i	$0.565097\pi$
$930$	0	0
$931$	4.08398e6	0.154422
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.76068e6	−0.178090
$936$	0	0
$937$	−3.99105e7	−1.48504	−0.742521	−	0.669823i	$-0.766370\pi$
$937$	−3.99105e7	−1.48504	−0.742521	+	0.669823i	$0.766370\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.32350e6	−0.0487248	−0.0243624	−	0.999703i	$-0.507756\pi$
$941$	−1.32350e6	−0.0487248	−0.0243624	+	0.999703i	$0.507756\pi$
$942$	0	0
$943$	2.27832e7	0.834326
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.76322e7	−1.00124	−0.500622	−	0.865666i	$-0.666895\pi$
$947$	−2.76322e7	−1.00124	−0.500622	+	0.865666i	$0.666895\pi$
$948$	0	0
$949$	−3.34160e6	−0.120445
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.07901e7	1.09819	0.549096	−	0.835759i	$-0.314972\pi$
$953$	3.07901e7	1.09819	0.549096	+	0.835759i	$0.314972\pi$
$954$	0	0
$955$	1.27972e7	0.454053
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−97507.1	−0.00342365
$960$	0	0
$961$	−2.45145e7	−0.856278
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.08952e7	0.722318
$966$	0	0
$967$	−2.92557e6	−0.100611	−0.0503055	−	0.998734i	$-0.516019\pi$
$967$	−2.92557e6	−0.100611	−0.0503055	+	0.998734i	$0.516019\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.78109e6	0.0946601	0.0473301	−	0.998879i	$-0.484929\pi$
$971$	2.78109e6	0.0946601	0.0473301	+	0.998879i	$0.484929\pi$
$972$	0	0
$973$	1.79540e7	0.607967
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.48673e6	−0.250932	−0.125466	−	0.992098i	$-0.540043\pi$
$977$	−7.48673e6	−0.250932	−0.125466	+	0.992098i	$0.540043\pi$
$978$	0	0
$979$	−1.12615e7	−0.375525
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.79815e7	0.593528	0.296764	−	0.954951i	$-0.404093\pi$
$983$	1.79815e7	0.593528	0.296764	+	0.954951i	$0.404093\pi$
$984$	0	0
$985$	−1.04905e7	−0.344512
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.24378e7	2.02982
$990$	0	0
$991$	−3.72778e7	−1.20578	−0.602888	−	0.797826i	$-0.705983\pi$
$991$	−3.72778e7	−1.20578	−0.602888	+	0.797826i	$0.705983\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.31734e6	0.266334
$996$	0	0
$997$	4.87422e7	1.55298	0.776492	−	0.630128i	$-0.216997\pi$
$997$	4.87422e7	1.55298	0.776492	+	0.630128i	$0.216997\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.6.a.bq.1.1		2
3.2	odd	2		112.6.a.h.1.2		2
4.3	odd	2		63.6.a.f.1.1		2
12.11	even	2		7.6.a.b.1.2	✓	2
21.20	even	2		784.6.a.v.1.1		2
24.5	odd	2		448.6.a.u.1.1		2
24.11	even	2		448.6.a.w.1.2		2
28.27	even	2		441.6.a.l.1.1		2
60.23	odd	4		175.6.b.c.99.1		4
60.47	odd	4		175.6.b.c.99.4		4
60.59	even	2		175.6.a.c.1.1		2
84.11	even	6		49.6.c.e.30.1		4
84.23	even	6		49.6.c.e.18.1		4
84.47	odd	6		49.6.c.d.18.1		4
84.59	odd	6		49.6.c.d.30.1		4
84.83	odd	2		49.6.a.f.1.2		2
132.131	odd	2		847.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.b.1.2	✓	2	12.11	even	2
49.6.a.f.1.2		2	84.83	odd	2
49.6.c.d.18.1		4	84.47	odd	6
49.6.c.d.30.1		4	84.59	odd	6
49.6.c.e.18.1		4	84.23	even	6
49.6.c.e.30.1		4	84.11	even	6
63.6.a.f.1.1		2	4.3	odd	2
112.6.a.h.1.2		2	3.2	odd	2
175.6.a.c.1.1		2	60.59	even	2
175.6.b.c.99.1		4	60.23	odd	4
175.6.b.c.99.4		4	60.47	odd	4
441.6.a.l.1.1		2	28.27	even	2
448.6.a.u.1.1		2	24.5	odd	2
448.6.a.w.1.2		2	24.11	even	2
784.6.a.v.1.1		2	21.20	even	2
847.6.a.c.1.1		2	132.131	odd	2
1008.6.a.bq.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

\(f(q)\)	\(=\)	\(q-28.7492 q^{5} -49.0000 q^{7} +O(q^{10})\)	\(q-28.7492 q^{5} -49.0000 q^{7} -270.090 q^{11} +300.640 q^{13} -613.106 q^{17} +1700.95 q^{19} +3188.15 q^{23} -2298.49 q^{25} -4299.28 q^{29} -2028.46 q^{31} +1408.71 q^{35} +5154.46 q^{37} +7146.21 q^{41} +19584.3 q^{43} +19998.4 q^{47} +2401.00 q^{49} -3948.82 q^{53} +7764.86 q^{55} -29707.6 q^{59} -50519.3 q^{61} -8643.14 q^{65} -5053.56 q^{67} +32853.3 q^{71} -11115.0 q^{73} +13234.4 q^{77} -81889.4 q^{79} +118234. q^{83} +17626.3 q^{85} +41695.4 q^{89} -14731.3 q^{91} -48900.9 q^{95} +43682.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{5} - 98 q^{7}+O(q^{10}) \) 2 * q + 18 * q^5 - 98 * q^7	\( 2 q + 18 q^{5} - 98 q^{7} + 396 q^{11} - 350 q^{13} - 1800 q^{17} + 3266 q^{19} + 2088 q^{23} - 3238 q^{25} - 6696 q^{29} + 20 q^{31} - 882 q^{35} + 6232 q^{37} + 6048 q^{41} + 3020 q^{43} + 11700 q^{47} + 4802 q^{49} - 9468 q^{53} + 38904 q^{55} - 43938 q^{59} - 64754 q^{61} - 39060 q^{65} - 24784 q^{67} + 97416 q^{71} + 17452 q^{73} - 19404 q^{77} - 51256 q^{79} + 117558 q^{83} - 37860 q^{85} - 84276 q^{89} + 17150 q^{91} + 24264 q^{95} + 20776 q^{97}+O(q^{100}) \) 2 * q + 18 * q^5 - 98 * q^7 + 396 * q^11 - 350 * q^13 - 1800 * q^17 + 3266 * q^19 + 2088 * q^23 - 3238 * q^25 - 6696 * q^29 + 20 * q^31 - 882 * q^35 + 6232 * q^37 + 6048 * q^41 + 3020 * q^43 + 11700 * q^47 + 4802 * q^49 - 9468 * q^53 + 38904 * q^55 - 43938 * q^59 - 64754 * q^61 - 39060 * q^65 - 24784 * q^67 + 97416 * q^71 + 17452 * q^73 - 19404 * q^77 - 51256 * q^79 + 117558 * q^83 - 37860 * q^85 - 84276 * q^89 + 17150 * q^91 + 24264 * q^95 + 20776 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more