LMFDB - Embedded newform 448.6.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("448.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$448 = 2^{6} \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	448.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:	$71.8519512762$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
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			$4.27492$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-25.6495 q^{3} -28.7492 q^{5} -49.0000 q^{7} +414.897 q^{9} -270.090 q^{11} -300.640 q^{13} +737.402 q^{15} +613.106 q^{17} -1700.95 q^{19} +1256.83 q^{21} -3188.15 q^{23} -2298.49 q^{25} -4409.07 q^{27} -4299.28 q^{29} -2028.46 q^{31} +6927.67 q^{33} +1408.71 q^{35} -5154.46 q^{37} +7711.26 q^{39} -7146.21 q^{41} -19584.3 q^{43} -11927.9 q^{45} -19998.4 q^{47} +2401.00 q^{49} -15725.9 q^{51} -3948.82 q^{53} +7764.86 q^{55} +43628.5 q^{57} -29707.6 q^{59} +50519.3 q^{61} -20330.0 q^{63} +8643.14 q^{65} +5053.56 q^{67} +81774.5 q^{69} -32853.3 q^{71} -11115.0 q^{73} +58955.0 q^{75} +13234.4 q^{77} -81889.4 q^{79} +12270.6 q^{81} +118234. q^{83} -17626.3 q^{85} +110274. q^{87} -41695.4 q^{89} +14731.3 q^{91} +52028.9 q^{93} +48900.9 q^{95} +43682.8 q^{97} -112059. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 6 q^{3} + 18 q^{5} - 98 q^{7} + 558 q^{9} + 396 q^{11} + 350 q^{13} + 1656 q^{15} + 1800 q^{17} - 3266 q^{19} + 294 q^{21} - 2088 q^{23} - 3238 q^{25} - 6372 q^{27} - 6696 q^{29} + 20 q^{31} + 20016 q^{33}+ \cdots - 16740 q^{99}+O(q^{100})$ 2 * q - 6 * q^3 + 18 * q^5 - 98 * q^7 + 558 * q^9 + 396 * q^11 + 350 * q^13 + 1656 * q^15 + 1800 * q^17 - 3266 * q^19 + 294 * q^21 - 2088 * q^23 - 3238 * q^25 - 6372 * q^27 - 6696 * q^29 + 20 * q^31 + 20016 * q^33 - 882 * q^35 - 6232 * q^37 + 20496 * q^39 - 6048 * q^41 - 3020 * q^43 - 5238 * q^45 - 11700 * q^47 + 4802 * q^49 + 7596 * q^51 - 9468 * q^53 + 38904 * q^55 + 12876 * q^57 - 43938 * q^59 + 64754 * q^61 - 27342 * q^63 + 39060 * q^65 + 24784 * q^67 + 103392 * q^69 - 97416 * q^71 + 17452 * q^73 + 40494 * q^75 - 19404 * q^77 - 51256 * q^79 - 61074 * q^81 + 117558 * q^83 + 37860 * q^85 + 63180 * q^87 + 84276 * q^89 - 17150 * q^91 + 92280 * q^93 - 24264 * q^95 + 20776 * q^97 - 16740 * 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
$3$	−25.6495	−1.64542	−0.822708	−	0.568464i	$-0.807538\pi$
$3$	−25.6495	−1.64542	−0.822708	+	0.568464i	$0.807538\pi$
$4$	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
$9$	414.897	1.70740
$10$	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	$-0.579344\pi$
$13$	−300.640	−0.493387	−0.246694	+	0.969094i	$0.579344\pi$
$14$	0	0
$15$	737.402	0.846206
$16$	0	0
$17$	613.106	0.514533	0.257267	−	0.966340i	$-0.417178\pi$
$17$	613.106	0.514533	0.257267	+	0.966340i	$0.417178\pi$
$18$	0	0
$19$	−1700.95	−1.08095	−0.540477	−	0.841359i	$-0.681756\pi$
$19$	−1700.95	−1.08095	−0.540477	+	0.841359i	$0.681756\pi$
$20$	0	0
$21$	1256.83	0.621909
$22$	0	0
$23$	−3188.15	−1.25667	−0.628333	−	0.777945i	$-0.716262\pi$
$23$	−3188.15	−1.25667	−0.628333	+	0.777945i	$0.716262\pi$
$24$	0	0
$25$	−2298.49	−0.735515
$26$	0	0
$27$	−4409.07	−1.16396
$28$	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
$33$	6927.67	1.10739
$34$	0	0
$35$	1408.71	0.194380
$36$	0	0
$37$	−5154.46	−0.618983	−0.309491	−	0.950902i	$-0.600159\pi$
$37$	−5154.46	−0.618983	−0.309491	+	0.950902i	$0.600159\pi$
$38$	0	0
$39$	7711.26	0.811827
$40$	0	0
$41$	−7146.21	−0.663921	−0.331960	−	0.943293i	$-0.607710\pi$
$41$	−7146.21	−0.663921	−0.331960	+	0.943293i	$0.607710\pi$
$42$	0	0
$43$	−19584.3	−1.61524	−0.807620	−	0.589703i	$-0.799245\pi$
$43$	−19584.3	−1.61524	−0.807620	+	0.589703i	$0.799245\pi$
$44$	0	0
$45$	−11927.9	−0.878081
$46$	0	0
$47$	−19998.4	−1.32054	−0.660268	−	0.751030i	$-0.729557\pi$
$47$	−19998.4	−1.32054	−0.660268	+	0.751030i	$0.729557\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−15725.9	−0.846621
$52$	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
$57$	43628.5	1.77862
$58$	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	$-0.164657\pi$
$61$	50519.3	1.73833	0.869165	+	0.494522i	$0.164657\pi$
$62$	0	0
$63$	−20330.0	−0.645335
$64$	0	0
$65$	8643.14	0.253740
$66$	0	0
$67$	5053.56	0.137534	0.0687671	−	0.997633i	$-0.478093\pi$
$67$	5053.56	0.137534	0.0687671	+	0.997633i	$0.478093\pi$
$68$	0	0
$69$	81774.5	2.06774
$70$	0	0
$71$	−32853.3	−0.773453	−0.386726	−	0.922195i	$-0.626394\pi$
$71$	−32853.3	−0.773453	−0.386726	+	0.922195i	$0.626394\pi$
$72$	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
$75$	58955.0	1.21023
$76$	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
$81$	12270.6	0.207803
$82$	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
$87$	110274.	1.56198
$88$	0	0
$89$	−41695.4	−0.557972	−0.278986	−	0.960295i	$-0.589998\pi$
$89$	−41695.4	−0.557972	−0.278986	+	0.960295i	$0.589998\pi$
$90$	0	0
$91$	14731.3	0.186483
$92$	0	0
$93$	52028.9	0.623788
$94$	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
$99$	−112059.	−1.14911
$100$	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
$105$	−36132.7	−0.319836
$106$	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	$-0.387292\pi$
$109$	86017.6	0.693459	0.346730	+	0.937965i	$0.387292\pi$
$110$	0	0
$111$	132209.	1.01848
$112$	0	0
$113$	137568.	1.01349	0.506745	−	0.862096i	$-0.330848\pi$
$113$	137568.	1.01349	0.506745	+	0.862096i	$0.330848\pi$
$114$	0	0
$115$	91656.8	0.646279
$116$	0	0
$117$	−124734.	−0.842407
$118$	0	0
$119$	−30042.2	−0.194475
$120$	0	0
$121$	−88102.5	−0.547047
$122$	0	0
$123$	183297.	1.09243
$124$	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
$129$	502328.	2.65774
$130$	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
$135$	126757.	0.598602
$136$	0	0
$137$	−1989.94	−0.00905813	−0.00452907	−	0.999990i	$-0.501442\pi$
$137$	−1989.94	−0.00905813	−0.00452907	+	0.999990i	$0.501442\pi$
$138$	0	0
$139$	366409.	1.60853	0.804264	−	0.594272i	$-0.202560\pi$
$139$	366409.	1.60853	0.804264	+	0.594272i	$0.202560\pi$
$140$	0	0
$141$	512949.	2.17283
$142$	0	0
$143$	81199.7	0.332058
$144$	0	0
$145$	123601.	0.488204
$146$	0	0
$147$	−61584.5	−0.235059
$148$	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
$153$	254376.	0.878512
$154$	0	0
$155$	58316.4	0.194967
$156$	0	0
$157$	89794.6	0.290738	0.145369	−	0.989378i	$-0.453563\pi$
$157$	89794.6	0.290738	0.145369	+	0.989378i	$0.453563\pi$
$158$	0	0
$159$	101285.	0.317726
$160$	0	0
$161$	156219.	0.474975
$162$	0	0
$163$	−481230.	−1.41868	−0.709339	−	0.704867i	$-0.751006\pi$
$163$	−481230.	−1.41868	−0.709339	+	0.704867i	$0.751006\pi$
$164$	0	0
$165$	−199165.	−0.569512
$166$	0	0
$167$	86572.7	0.240209	0.120105	−	0.992761i	$-0.461677\pi$
$167$	86572.7	0.240209	0.120105	+	0.992761i	$0.461677\pi$
$168$	0	0
$169$	−280909.	−0.756569
$170$	0	0
$171$	−705718.	−1.84562
$172$	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
$177$	761985.	1.82816
$178$	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	$-0.602507\pi$
$181$	−278996.	−0.632996	−0.316498	+	0.948593i	$0.602507\pi$
$182$	0	0
$183$	−1.29579e6	−2.86028
$184$	0	0
$185$	148186.	0.318331
$186$	0	0
$187$	−165594.	−0.346290
$188$	0	0
$189$	216045.	0.439935
$190$	0	0
$191$	445132.	0.882888	0.441444	−	0.897289i	$-0.354466\pi$
$191$	445132.	0.882888	0.441444	+	0.897289i	$0.354466\pi$
$192$	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
$195$	−221692.	−0.417507
$196$	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
$201$	−129621.	−0.226301
$202$	0	0
$203$	210665.	0.358799
$204$	0	0
$205$	205448.	0.341442
$206$	0	0
$207$	−1.32276e6	−2.14562
$208$	0	0
$209$	459409.	0.727501
$210$	0	0
$211$	750147.	1.15995	0.579976	−	0.814633i	$-0.303062\pi$
$211$	750147.	1.15995	0.579976	+	0.814633i	$0.303062\pi$
$212$	0	0
$213$	842672.	1.27265
$214$	0	0
$215$	563033.	0.830687
$216$	0	0
$217$	99394.3	0.143289
$218$	0	0
$219$	285093.	0.401677
$220$	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
$225$	−953635.	−1.25582
$226$	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	$-0.724892\pi$
$229$	−1.03036e6	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$230$	0	0
$231$	−339456.	−0.418556
$232$	0	0
$233$	−119211.	−0.143856	−0.0719278	−	0.997410i	$-0.522915\pi$
$233$	−119211.	−0.143856	−0.0719278	+	0.997410i	$0.522915\pi$
$234$	0	0
$235$	574937.	0.679127
$236$	0	0
$237$	2.10042e6	2.42905
$238$	0	0
$239$	254090.	0.287735	0.143868	−	0.989597i	$-0.454046\pi$
$239$	254090.	0.287735	0.143868	+	0.989597i	$0.454046\pi$
$240$	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
$243$	756671.	0.822037
$244$	0	0
$245$	−69026.8	−0.0734687
$246$	0	0
$247$	511372.	0.533329
$248$	0	0
$249$	−3.03264e6	−3.09972
$250$	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
$255$	452106.	0.435401
$256$	0	0
$257$	726996.	0.686593	0.343296	−	0.939227i	$-0.388456\pi$
$257$	726996.	0.686593	0.343296	+	0.939227i	$0.388456\pi$
$258$	0	0
$259$	252568.	0.233953
$260$	0	0
$261$	−1.78376e6	−1.62082
$262$	0	0
$263$	225880.	0.201367	0.100684	−	0.994918i	$-0.467897\pi$
$263$	225880.	0.201367	0.100684	+	0.994918i	$0.467897\pi$
$264$	0	0
$265$	113525.	0.0993065
$266$	0	0
$267$	1.06947e6	0.918097
$268$	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
$273$	−377852.	−0.306842
$274$	0	0
$275$	620797.	0.495015
$276$	0	0
$277$	−2.23055e6	−1.74668	−0.873338	−	0.487115i	$-0.838049\pi$
$277$	−2.23055e6	−1.74668	−0.873338	+	0.487115i	$0.838049\pi$
$278$	0	0
$279$	−841600.	−0.647285
$280$	0	0
$281$	1.67140e6	1.26274	0.631371	−	0.775481i	$-0.282493\pi$
$281$	1.67140e6	1.26274	0.631371	+	0.775481i	$0.282493\pi$
$282$	0	0
$283$	−396152.	−0.294033	−0.147016	−	0.989134i	$-0.546967\pi$
$283$	−396152.	−0.294033	−0.147016	+	0.989134i	$0.546967\pi$
$284$	0	0
$285$	−1.25428e6	−0.914710
$286$	0	0
$287$	350164.	0.250938
$288$	0	0
$289$	−1.04396e6	−0.735256
$290$	0	0
$291$	−1.12044e6	−0.775634
$292$	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
$297$	1.19085e6	0.783365
$298$	0	0
$299$	958485.	0.620022
$300$	0	0
$301$	959631.	0.610503
$302$	0	0
$303$	657860.	0.411649
$304$	0	0
$305$	−1.45239e6	−0.893990
$306$	0	0
$307$	1.83295e6	1.10995	0.554976	−	0.831866i	$-0.312727\pi$
$307$	1.83295e6	1.10995	0.554976	+	0.831866i	$0.312727\pi$
$308$	0	0
$309$	−367302.	−0.218840
$310$	0	0
$311$	2.29685e6	1.34658	0.673289	−	0.739379i	$-0.264881\pi$
$311$	2.29685e6	1.34658	0.673289	+	0.739379i	$0.264881\pi$
$312$	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
$315$	584469.	0.331883
$316$	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
$321$	−441217.	−0.238996
$322$	0	0
$323$	−1.04286e6	−0.556187
$324$	0	0
$325$	691016.	0.362894
$326$	0	0
$327$	−2.20631e6	−1.14103
$328$	0	0
$329$	979921.	0.499116
$330$	0	0
$331$	966164.	0.484709	0.242354	−	0.970188i	$-0.422080\pi$
$331$	966164.	0.484709	0.242354	+	0.970188i	$0.422080\pi$
$332$	0	0
$333$	−2.13857e6	−1.05685
$334$	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
$339$	−3.52854e6	−1.66761
$340$	0	0
$341$	547865.	0.255145
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−2.35095e6	−1.06340
$346$	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	$-0.490197\pi$
$349$	140128.	0.0615830	0.0307915	+	0.999526i	$0.490197\pi$
$350$	0	0
$351$	1.32554e6	0.574283
$352$	0	0
$353$	3.48141e6	1.48703	0.743514	−	0.668721i	$-0.233158\pi$
$353$	3.48141e6	1.48703	0.743514	+	0.668721i	$0.233158\pi$
$354$	0	0
$355$	944507.	0.397772
$356$	0	0
$357$	770568.	0.319993
$358$	0	0
$359$	−1.75285e6	−0.717810	−0.358905	−	0.933374i	$-0.616850\pi$
$359$	−1.75285e6	−0.717810	−0.358905	+	0.933374i	$0.616850\pi$
$360$	0	0
$361$	417127.	0.168461
$362$	0	0
$363$	2.25979e6	0.900121
$364$	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
$369$	−2.96494e6	−1.13357
$370$	0	0
$371$	193492.	0.0729841
$372$	0	0
$373$	4.16212e6	1.54897	0.774485	−	0.632592i	$-0.218009\pi$
$373$	4.16212e6	1.54897	0.774485	+	0.632592i	$0.218009\pi$
$374$	0	0
$375$	−3.99929e6	−1.46860
$376$	0	0
$377$	1.29253e6	0.468369
$378$	0	0
$379$	618163.	0.221057	0.110529	−	0.993873i	$-0.464746\pi$
$379$	618163.	0.221057	0.110529	+	0.993873i	$0.464746\pi$
$380$	0	0
$381$	−1.81001e6	−0.638805
$382$	0	0
$383$	4.11163e6	1.43225	0.716123	−	0.697974i	$-0.245915\pi$
$383$	4.11163e6	1.43225	0.716123	+	0.697974i	$0.245915\pi$
$384$	0	0
$385$	−380478.	−0.130821
$386$	0	0
$387$	−8.12547e6	−2.75785
$388$	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
$393$	4.45564e6	1.45522
$394$	0	0
$395$	2.35425e6	0.759207
$396$	0	0
$397$	−5.07349e6	−1.61559	−0.807794	−	0.589465i	$-0.799339\pi$
$397$	−5.07349e6	−1.61559	−0.807794	+	0.589465i	$0.799339\pi$
$398$	0	0
$399$	−2.13780e6	−0.672255
$400$	0	0
$401$	−1.48056e6	−0.459795	−0.229898	−	0.973215i	$-0.573839\pi$
$401$	−1.48056e6	−0.459795	−0.229898	+	0.973215i	$0.573839\pi$
$402$	0	0
$403$	609834.	0.187046
$404$	0	0
$405$	−352768.	−0.106869
$406$	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
$411$	51041.0	0.0149044
$412$	0	0
$413$	1.45567e6	0.419941
$414$	0	0
$415$	−3.39913e6	−0.968829
$416$	0	0
$417$	−9.39820e6	−2.64670
$418$	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	$-0.437832\pi$
$421$	1.41151e6	0.388132	0.194066	+	0.980988i	$0.437832\pi$
$422$	0	0
$423$	−8.29727e6	−2.25468
$424$	0	0
$425$	−1.40922e6	−0.378447
$426$	0	0
$427$	−2.47544e6	−0.657027
$428$	0	0
$429$	−2.08273e6	−0.546374
$430$	0	0
$431$	−1.07640e6	−0.279113	−0.139557	−	0.990214i	$-0.544568\pi$
$431$	−1.07640e6	−0.279113	−0.139557	+	0.990214i	$0.544568\pi$
$432$	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
$435$	−3.17030e6	−0.803298
$436$	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
$441$	996168.	0.243914
$442$	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
$447$	3.60936e6	0.854401
$448$	0	0
$449$	−6.96544e6	−1.63054	−0.815272	−	0.579078i	$-0.803413\pi$
$449$	−6.96544e6	−1.63054	−0.815272	+	0.579078i	$0.803413\pi$
$450$	0	0
$451$	1.93012e6	0.446830
$452$	0	0
$453$	1.28413e6	0.294011
$454$	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
$459$	−2.70323e6	−0.598896
$460$	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
$465$	−1.49579e6	−0.320802
$466$	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
$471$	−2.30319e6	−0.478384
$472$	0	0
$473$	5.28952e6	1.08708
$474$	0	0
$475$	3.90960e6	0.795058
$476$	0	0
$477$	−1.63835e6	−0.329694
$478$	0	0
$479$	2.41693e6	0.481311	0.240655	−	0.970611i	$-0.422638\pi$
$479$	2.41693e6	0.481311	0.240655	+	0.970611i	$0.422638\pi$
$480$	0	0
$481$	1.54963e6	0.305398
$482$	0	0
$483$	−4.00695e6	−0.781531
$484$	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
$489$	1.23433e7	2.33432
$490$	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
$495$	3.22162e6	0.590964
$496$	0	0
$497$	1.60981e6	0.292338
$498$	0	0
$499$	−3.29606e6	−0.592576	−0.296288	−	0.955099i	$-0.595749\pi$
$499$	−3.29606e6	−0.592576	−0.296288	+	0.955099i	$0.595749\pi$
$500$	0	0
$501$	−2.22055e6	−0.395244
$502$	0	0
$503$	1.06512e7	1.87706	0.938528	−	0.345204i	$-0.112190\pi$
$503$	1.06512e7	1.87706	0.938528	+	0.345204i	$0.112190\pi$
$504$	0	0
$505$	737361.	0.128662
$506$	0	0
$507$	7.20517e6	1.24487
$508$	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
$513$	7.49961e6	1.25819
$514$	0	0
$515$	−411689.	−0.0683992
$516$	0	0
$517$	5.40136e6	0.888744
$518$	0	0
$519$	−1.49120e6	−0.243006
$520$	0	0
$521$	4.97077e6	0.802286	0.401143	−	0.916015i	$-0.368613\pi$
$521$	4.97077e6	0.802286	0.401143	+	0.916015i	$0.368613\pi$
$522$	0	0
$523$	2.41579e6	0.386193	0.193096	−	0.981180i	$-0.438147\pi$
$523$	2.41579e6	0.386193	0.193096	+	0.981180i	$0.438147\pi$
$524$	0	0
$525$	−2.88880e6	−0.457424
$526$	0	0
$527$	−1.24366e6	−0.195063
$528$	0	0
$529$	3.72798e6	0.579207
$530$	0	0
$531$	−1.23256e7	−1.89702
$532$	0	0
$533$	2.14843e6	0.327570
$534$	0	0
$535$	−494537.	−0.0746989
$536$	0	0
$537$	5.37050e6	0.803672
$538$	0	0
$539$	−648485.	−0.0961454
$540$	0	0
$541$	−472165.	−0.0693587	−0.0346794	−	0.999398i	$-0.511041\pi$
$541$	−472165.	−0.0693587	−0.0346794	+	0.999398i	$0.511041\pi$
$542$	0	0
$543$	7.15610e6	1.04154
$544$	0	0
$545$	−2.47293e6	−0.356633
$546$	0	0
$547$	7.63716e6	1.09135	0.545675	−	0.837997i	$-0.316273\pi$
$547$	7.63716e6	1.09135	0.545675	+	0.837997i	$0.316273\pi$
$548$	0	0
$549$	2.09603e7	2.96802
$550$	0	0
$551$	7.31285e6	1.02614
$552$	0	0
$553$	4.01258e6	0.557970
$554$	0	0
$555$	−3.80091e6	−0.523787
$556$	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
$561$	4.24740e6	0.569791
$562$	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
$567$	−601258.	−0.0785422
$568$	0	0
$569$	−1.13325e7	−1.46739	−0.733696	−	0.679478i	$-0.762206\pi$
$569$	−1.13325e7	−1.46739	−0.733696	+	0.679478i	$0.762206\pi$
$570$	0	0
$571$	−843773.	−0.108302	−0.0541509	−	0.998533i	$-0.517245\pi$
$571$	−843773.	−0.108302	−0.0541509	+	0.998533i	$0.517245\pi$
$572$	0	0
$573$	−1.14174e7	−1.45272
$574$	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
$579$	1.86423e7	2.31102
$580$	0	0
$581$	−5.79346e6	−0.712029
$582$	0	0
$583$	1.06653e6	0.129958
$584$	0	0
$585$	3.58601e6	0.433234
$586$	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
$591$	−9.35942e6	−1.10225
$592$	0	0
$593$	8.00167e6	0.934424	0.467212	−	0.884145i	$-0.345258\pi$
$593$	8.00167e6	0.934424	0.467212	+	0.884145i	$0.345258\pi$
$594$	0	0
$595$	863689.	0.100015
$596$	0	0
$597$	7.42058e6	0.852123
$598$	0	0
$599$	−1.45899e7	−1.66144	−0.830719	−	0.556692i	$-0.812070\pi$
$599$	−1.45899e7	−1.66144	−0.830719	+	0.556692i	$0.812070\pi$
$600$	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
$603$	2.09671e6	0.234825
$604$	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
$609$	−5.40344e6	−0.590374
$610$	0	0
$611$	6.01231e6	0.651536
$612$	0	0
$613$	−2.35101e6	−0.252699	−0.126350	−	0.991986i	$-0.540326\pi$
$613$	−2.35101e6	−0.252699	−0.126350	+	0.991986i	$0.540326\pi$
$614$	0	0
$615$	−5.26963e6	−0.561814
$616$	0	0
$617$	9.63523e6	1.01894	0.509470	−	0.860488i	$-0.329841\pi$
$617$	9.63523e6	1.01894	0.509470	+	0.860488i	$0.329841\pi$
$618$	0	0
$619$	−4.86148e6	−0.509967	−0.254983	−	0.966945i	$-0.582070\pi$
$619$	−4.86148e6	−0.509967	−0.254983	+	0.966945i	$0.582070\pi$
$620$	0	0
$621$	1.40568e7	1.46271
$622$	0	0
$623$	2.04307e6	0.210894
$624$	0	0
$625$	2.70017e6	0.276498
$626$	0	0
$627$	−1.17836e7	−1.19704
$628$	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
$633$	−1.92409e7	−1.90860
$634$	0	0
$635$	−2.02874e6	−0.199661
$636$	0	0
$637$	−721836.	−0.0704839
$638$	0	0
$639$	−1.36308e7	−1.32059
$640$	0	0
$641$	1.44525e7	1.38930	0.694651	−	0.719347i	$-0.255559\pi$
$641$	1.44525e7	1.38930	0.694651	+	0.719347i	$0.255559\pi$
$642$	0	0
$643$	−1.54720e7	−1.47577	−0.737886	−	0.674926i	$-0.764176\pi$
$643$	−1.54720e7	−1.47577	−0.737886	+	0.674926i	$0.764176\pi$
$644$	0	0
$645$	−1.44415e7	−1.36683
$646$	0	0
$647$	−1.66647e7	−1.56508	−0.782540	−	0.622601i	$-0.786076\pi$
$647$	−1.66647e7	−1.56508	−0.782540	+	0.622601i	$0.786076\pi$
$648$	0	0
$649$	8.02371e6	0.747762
$650$	0	0
$651$	−2.54941e6	−0.235770
$652$	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
$657$	−4.61157e6	−0.416807
$658$	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	$-0.659632\pi$
$661$	−1.08005e7	−0.961478	−0.480739	+	0.876864i	$0.659632\pi$
$662$	0	0
$663$	4.72782e6	0.417712
$664$	0	0
$665$	−2.39614e6	−0.210116
$666$	0	0
$667$	1.37068e7	1.19294
$668$	0	0
$669$	1.37070e7	1.18407
$670$	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
$675$	1.01342e7	0.856110
$676$	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
$681$	1.05323e7	0.870272
$682$	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
$687$	2.64283e7	2.13637
$688$	0	0
$689$	1.18717e6	0.0952720
$690$	0	0
$691$	7.11964e6	0.567235	0.283617	−	0.958938i	$-0.408465\pi$
$691$	7.11964e6	0.567235	0.283617	+	0.958938i	$0.408465\pi$
$692$	0	0
$693$	5.49091e6	0.434322
$694$	0	0
$695$	−1.05339e7	−0.827235
$696$	0	0
$697$	−4.38139e6	−0.341609
$698$	0	0
$699$	3.05770e6	0.236702
$700$	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
$705$	−1.47469e7	−1.11745
$706$	0	0
$707$	1.25676e6	0.0945589
$708$	0	0
$709$	8.84454e6	0.660784	0.330392	−	0.943844i	$-0.392819\pi$
$709$	8.84454e6	0.660784	0.330392	+	0.943844i	$0.392819\pi$
$710$	0	0
$711$	−3.39757e7	−2.52054
$712$	0	0
$713$	6.46703e6	0.476410
$714$	0	0
$715$	−2.33442e6	−0.170771
$716$	0	0
$717$	−6.51728e6	−0.473444
$718$	0	0
$719$	−6.58086e6	−0.474745	−0.237373	−	0.971419i	$-0.576286\pi$
$719$	−6.58086e6	−0.474745	−0.237373	+	0.971419i	$0.576286\pi$
$720$	0	0
$721$	−701682.	−0.0502692
$722$	0	0
$723$	−3.62301e7	−2.57765
$724$	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
$729$	−2.23900e7	−1.56040
$730$	0	0
$731$	−1.20073e7	−0.831095
$732$	0	0
$733$	2.78330e6	0.191337	0.0956687	−	0.995413i	$-0.469501\pi$
$733$	2.78330e6	0.191337	0.0956687	+	0.995413i	$0.469501\pi$
$734$	0	0
$735$	1.77050e6	0.120887
$736$	0	0
$737$	−1.36491e6	−0.0925629
$738$	0	0
$739$	−2.48970e7	−1.67701	−0.838505	−	0.544894i	$-0.816570\pi$
$739$	−2.48970e7	−1.67701	−0.838505	+	0.544894i	$0.816570\pi$
$740$	0	0
$741$	−1.31164e7	−0.877548
$742$	0	0
$743$	3.86085e6	0.256573	0.128286	−	0.991737i	$-0.459052\pi$
$743$	3.86085e6	0.256573	0.128286	+	0.991737i	$0.459052\pi$
$744$	0	0
$745$	4.04554e6	0.267046
$746$	0	0
$747$	4.90549e7	3.21648
$748$	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
$753$	4.29737e7	2.76195
$754$	0	0
$755$	1.43932e6	0.0918943
$756$	0	0
$757$	−2.17782e7	−1.38128	−0.690642	−	0.723197i	$-0.742672\pi$
$757$	−2.17782e7	−1.38128	−0.690642	+	0.723197i	$0.742672\pi$
$758$	0	0
$759$	−2.20865e7	−1.39162
$760$	0	0
$761$	−2.57074e7	−1.60915	−0.804575	−	0.593851i	$-0.797607\pi$
$761$	−2.57074e7	−1.60915	−0.804575	+	0.593851i	$0.797607\pi$
$762$	0	0
$763$	−4.21486e6	−0.262103
$764$	0	0
$765$	−7.31310e6	−0.451802
$766$	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
$771$	−1.86471e7	−1.12973
$772$	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
$777$	−6.47825e6	−0.384951
$778$	0	0
$779$	1.21553e7	0.717667
$780$	0	0
$781$	8.87335e6	0.520547
$782$	0	0
$783$	1.89558e7	1.10494
$784$	0	0
$785$	−2.58152e6	−0.149521
$786$	0	0
$787$	−2.07672e6	−0.119520	−0.0597602	−	0.998213i	$-0.519034\pi$
$787$	−2.07672e6	−0.119520	−0.0597602	+	0.998213i	$0.519034\pi$
$788$	0	0
$789$	−5.79372e6	−0.331333
$790$	0	0
$791$	−6.74081e6	−0.383064
$792$	0	0
$793$	−1.51881e7	−0.857670
$794$	0	0
$795$	−2.91187e6	−0.163401
$796$	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
$801$	−1.72993e7	−0.952679
$802$	0	0
$803$	3.00204e6	0.164296
$804$	0	0
$805$	−4.49118e6	−0.244270
$806$	0	0
$807$	4.63043e7	2.50286
$808$	0	0
$809$	1.96864e7	1.05754	0.528769	−	0.848766i	$-0.322654\pi$
$809$	1.96864e7	1.05754	0.528769	+	0.848766i	$0.322654\pi$
$810$	0	0
$811$	8.50101e6	0.453856	0.226928	−	0.973912i	$-0.427132\pi$
$811$	8.50101e6	0.453856	0.226928	+	0.973912i	$0.427132\pi$
$812$	0	0
$813$	−4.39580e7	−2.33245
$814$	0	0
$815$	1.38350e7	0.729599
$816$	0	0
$817$	3.33119e7	1.74600
$818$	0	0
$819$	6.11199e6	0.318400
$820$	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
$825$	−1.59231e7	−0.814505
$826$	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	$-0.454482\pi$
$829$	5.63984e6	0.285023	0.142512	+	0.989793i	$0.454482\pi$
$830$	0	0
$831$	5.72125e7	2.87401
$832$	0	0
$833$	1.47207e6	0.0735048
$834$	0	0
$835$	−2.48889e6	−0.123535
$836$	0	0
$837$	8.94361e6	0.441265
$838$	0	0
$839$	1.16351e7	0.570642	0.285321	−	0.958432i	$-0.407900\pi$
$839$	1.16351e7	0.570642	0.285321	+	0.958432i	$0.407900\pi$
$840$	0	0
$841$	−2.02735e6	−0.0988413
$842$	0	0
$843$	−4.28706e7	−2.07774
$844$	0	0
$845$	8.07590e6	0.389089
$846$	0	0
$847$	4.31702e6	0.206764
$848$	0	0
$849$	1.01611e7	0.483806
$850$	0	0
$851$	1.64332e7	0.777854
$852$	0	0
$853$	−2.85205e7	−1.34210	−0.671049	−	0.741413i	$-0.734156\pi$
$853$	−2.85205e7	−1.34210	−0.671049	+	0.741413i	$0.734156\pi$
$854$	0	0
$855$	2.02888e7	0.949165
$856$	0	0
$857$	−9.95725e6	−0.463113	−0.231557	−	0.972821i	$-0.574382\pi$
$857$	−9.95725e6	−0.463113	−0.231557	+	0.972821i	$0.574382\pi$
$858$	0	0
$859$	−1.49322e7	−0.690463	−0.345232	−	0.938517i	$-0.612200\pi$
$859$	−1.49322e7	−0.690463	−0.345232	+	0.938517i	$0.612200\pi$
$860$	0	0
$861$	−8.98154e6	−0.412898
$862$	0	0
$863$	−3.84933e7	−1.75937	−0.879687	−	0.475553i	$-0.842248\pi$
$863$	−3.84933e7	−1.75937	−0.879687	+	0.475553i	$0.842248\pi$
$864$	0	0
$865$	−1.67140e6	−0.0759523
$866$	0	0
$867$	2.67770e7	1.20980
$868$	0	0
$869$	2.21175e7	0.993542
$870$	0	0
$871$	−1.51930e6	−0.0678576
$872$	0	0
$873$	1.81239e7	0.804850
$874$	0	0
$875$	−7.64011e6	−0.337349
$876$	0	0
$877$	−9.40311e6	−0.412831	−0.206416	−	0.978464i	$-0.566180\pi$
$877$	−9.40311e6	−0.412831	−0.206416	+	0.978464i	$0.566180\pi$
$878$	0	0
$879$	−2.38403e7	−1.04073
$880$	0	0
$881$	1.10395e6	0.0479194	0.0239597	−	0.999713i	$-0.492373\pi$
$881$	1.10395e6	0.0479194	0.0239597	+	0.999713i	$0.492373\pi$
$882$	0	0
$883$	8.06579e6	0.348133	0.174067	−	0.984734i	$-0.444309\pi$
$883$	8.06579e6	0.348133	0.174067	+	0.984734i	$0.444309\pi$
$884$	0	0
$885$	−2.19064e7	−0.940185
$886$	0	0
$887$	1.49902e7	0.639732	0.319866	−	0.947463i	$-0.396362\pi$
$887$	1.49902e7	0.639732	0.319866	+	0.947463i	$0.396362\pi$
$888$	0	0
$889$	−3.45779e6	−0.146738
$890$	0	0
$891$	−3.31415e6	−0.139855
$892$	0	0
$893$	3.40162e7	1.42744
$894$	0	0
$895$	6.01950e6	0.251190
$896$	0	0
$897$	−2.45847e7	−1.02019
$898$	0	0
$899$	8.72090e6	0.359883
$900$	0	0
$901$	−2.42104e6	−0.0993553
$902$	0	0
$903$	−2.46141e7	−1.00453
$904$	0	0
$905$	8.02089e6	0.325538
$906$	0	0
$907$	4.12622e6	0.166546	0.0832730	−	0.996527i	$-0.473463\pi$
$907$	4.12622e6	0.166546	0.0832730	+	0.996527i	$0.473463\pi$
$908$	0	0
$909$	−1.06413e7	−0.427155
$910$	0	0
$911$	−4.04272e7	−1.61391	−0.806953	−	0.590616i	$-0.798885\pi$
$911$	−4.04272e7	−1.61391	−0.806953	+	0.590616i	$0.798885\pi$
$912$	0	0
$913$	−3.19338e7	−1.26787
$914$	0	0
$915$	3.72530e7	1.47099
$916$	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
$921$	−4.70142e7	−1.82633
$922$	0	0
$923$	9.87702e6	0.381612
$924$	0	0
$925$	1.18474e7	0.455271
$926$	0	0
$927$	5.94134e6	0.227083
$928$	0	0
$929$	1.06843e7	0.406169	0.203085	−	0.979161i	$-0.434903\pi$
$929$	1.06843e7	0.406169	0.203085	+	0.979161i	$0.434903\pi$
$930$	0	0
$931$	−4.08398e6	−0.154422
$932$	0	0
$933$	−5.89130e7	−2.21568
$934$	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
$939$	8.78419e7	3.25116
$940$	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
$945$	−6.21110e6	−0.226250
$946$	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
$951$	7.54879e7	2.70661
$952$	0	0
$953$	−3.07901e7	−1.09819	−0.549096	−	0.835759i	$-0.685028\pi$
$953$	−3.07901e7	−1.09819	−0.549096	+	0.835759i	$0.685028\pi$
$954$	0	0
$955$	−1.27972e7	−0.454053
$956$	0	0
$957$	−2.97840e7	−1.05124
$958$	0	0
$959$	97507.1	0.00342365
$960$	0	0
$961$	−2.45145e7	−0.856278
$962$	0	0
$963$	7.13697e6	0.247998
$964$	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
$969$	2.67489e7	0.915159
$970$	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
$975$	−1.77242e7	−0.597111
$976$	0	0
$977$	7.48673e6	0.250932	0.125466	−	0.992098i	$-0.459957\pi$
$977$	7.48673e6	0.250932	0.125466	+	0.992098i	$0.459957\pi$
$978$	0	0
$979$	1.12615e7	0.375525
$980$	0	0
$981$	3.56884e7	1.18401
$982$	0	0
$983$	−1.79815e7	−0.593528	−0.296764	−	0.954951i	$-0.595907\pi$
$983$	−1.79815e7	−0.593528	−0.296764	+	0.954951i	$0.595907\pi$
$984$	0	0
$985$	−1.04905e7	−0.344512
$986$	0	0
$987$	−2.51345e7	−0.821253
$988$	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
$993$	−2.47816e7	−0.797548
$994$	0	0
$995$	8.31734e6	0.266334
$996$	0	0
$997$	−4.87422e7	−1.55298	−0.776492	−	0.630128i	$-0.783003\pi$
$997$	−4.87422e7	−1.55298	−0.776492	+	0.630128i	$0.783003\pi$
$998$	0	0
$999$	2.27264e7	0.720471

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.u.1.1		2
4.3	odd	2		448.6.a.w.1.2		2
8.3	odd	2		7.6.a.b.1.2	✓	2
8.5	even	2		112.6.a.h.1.2		2
24.5	odd	2		1008.6.a.bq.1.1		2
24.11	even	2		63.6.a.f.1.1		2
40.3	even	4		175.6.b.c.99.1		4
40.19	odd	2		175.6.a.c.1.1		2
40.27	even	4		175.6.b.c.99.4		4
56.3	even	6		49.6.c.d.30.1		4
56.11	odd	6		49.6.c.e.30.1		4
56.13	odd	2		784.6.a.v.1.1		2
56.19	even	6		49.6.c.d.18.1		4
56.27	even	2		49.6.a.f.1.2		2
56.51	odd	6		49.6.c.e.18.1		4
88.43	even	2		847.6.a.c.1.1		2
168.83	odd	2		441.6.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.b.1.2	✓	2	8.3	odd	2
49.6.a.f.1.2		2	56.27	even	2
49.6.c.d.18.1		4	56.19	even	6
49.6.c.d.30.1		4	56.3	even	6
49.6.c.e.18.1		4	56.51	odd	6
49.6.c.e.30.1		4	56.11	odd	6
63.6.a.f.1.1		2	24.11	even	2
112.6.a.h.1.2		2	8.5	even	2
175.6.a.c.1.1		2	40.19	odd	2
175.6.b.c.99.1		4	40.3	even	4
175.6.b.c.99.4		4	40.27	even	4
441.6.a.l.1.1		2	168.83	odd	2
448.6.a.u.1.1		2	1.1	even	1	trivial
448.6.a.w.1.2		2	4.3	odd	2
784.6.a.v.1.1		2	56.13	odd	2
847.6.a.c.1.1		2	88.43	even	2
1008.6.a.bq.1.1		2	24.5	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	448.6.a.u.1.1

Level	$448$
Weight	$6$
Character	448.1
Self dual	yes
Analytic conductor	$71.852$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$