LMFDB - Embedded newform 560.6.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [560,6,Mod(1,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("560.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	560.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-5,0,50,0,-98,0,91] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$89.8149390953$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1129}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 282 $ x^2 - x - 282
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-16.3003$ of defining polynomial
Character	$\chi$	$=$	560.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+14.3003 q^{3} +25.0000 q^{5} -49.0000 q^{7} -38.5015 q^{9} -425.904 q^{11} +399.303 q^{13} +357.507 q^{15} +1751.52 q^{17} -2874.23 q^{19} -700.715 q^{21} +2313.43 q^{23} +625.000 q^{25} -4025.56 q^{27} -2127.93 q^{29} +10262.5 q^{31} -6090.55 q^{33} -1225.00 q^{35} -7266.05 q^{37} +5710.16 q^{39} -5893.44 q^{41} -20157.7 q^{43} -962.537 q^{45} -20056.5 q^{47} +2401.00 q^{49} +25047.2 q^{51} -33954.9 q^{53} -10647.6 q^{55} -41102.4 q^{57} +4319.34 q^{59} -12253.6 q^{61} +1886.57 q^{63} +9982.58 q^{65} +17533.8 q^{67} +33082.7 q^{69} -1658.18 q^{71} -8246.91 q^{73} +8937.69 q^{75} +20869.3 q^{77} +9168.61 q^{79} -48210.8 q^{81} -95203.2 q^{83} +43788.0 q^{85} -30430.0 q^{87} -14441.8 q^{89} -19565.9 q^{91} +146757. q^{93} -71855.8 q^{95} +62132.4 q^{97} +16397.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{3} + 50 q^{5} - 98 q^{7} + 91 q^{9} - 415 q^{11} + 429 q^{13} - 125 q^{15} + 1319 q^{17} - 1918 q^{19} + 245 q^{21} + 1334 q^{23} + 1250 q^{25} - 1835 q^{27} - 1131 q^{29} + 5472 q^{31} - 6301 q^{33}+ \cdots + 17810 q^{99}+O(q^{100}) $ 2 * q - 5 * q^3 + 50 * q^5 - 98 * q^7 + 91 * q^9 - 415 * q^11 + 429 * q^13 - 125 * q^15 + 1319 * q^17 - 1918 * q^19 + 245 * q^21 + 1334 * q^23 + 1250 * q^25 - 1835 * q^27 - 1131 * q^29 + 5472 * q^31 - 6301 * q^33 - 2450 * q^35 - 9156 * q^37 + 5137 * q^39 - 7822 * q^41 - 2078 * q^43 + 2275 * q^45 + 8507 * q^47 + 4802 * q^49 + 33395 * q^51 - 34242 * q^53 - 10375 * q^55 - 59558 * q^57 - 6952 * q^59 - 45138 * q^61 - 4459 * q^63 + 10725 * q^65 + 54556 * q^67 + 51986 * q^69 + 62272 * q^71 + 40896 * q^73 - 3125 * q^75 + 20335 * q^77 - 62069 * q^79 - 121958 * q^81 - 189600 * q^83 + 32975 * q^85 - 49671 * q^87 + 64190 * q^89 - 21021 * q^91 + 239216 * q^93 - 47950 * q^95 + 155547 * q^97 + 17810 * 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
$3$	14.3003	0.917365	0.458682	−	0.888600i	$-0.348322\pi$
$3$	14.3003	0.917365	0.458682	+	0.888600i	$0.348322\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−38.5015	−0.158442
$10$	0	0
$11$	−425.904	−1.06128	−0.530640	−	0.847597i	$-0.678048\pi$
$11$	−425.904	−1.06128	−0.530640	+	0.847597i	$0.678048\pi$
$12$	0	0
$13$	399.303	0.655307	0.327653	−	0.944798i	$-0.393742\pi$
$13$	399.303	0.655307	0.327653	+	0.944798i	$0.393742\pi$
$14$	0	0
$15$	357.507	0.410258
$16$	0	0
$17$	1751.52	1.46992	0.734958	−	0.678112i	$-0.237202\pi$
$17$	1751.52	1.46992	0.734958	+	0.678112i	$0.237202\pi$
$18$	0	0
$19$	−2874.23	−1.82658	−0.913289	−	0.407313i	$-0.866466\pi$
$19$	−2874.23	−1.82658	−0.913289	+	0.407313i	$0.866466\pi$
$20$	0	0
$21$	−700.715	−0.346731
$22$	0	0
$23$	2313.43	0.911878	0.455939	−	0.890011i	$-0.349304\pi$
$23$	2313.43	0.911878	0.455939	+	0.890011i	$0.349304\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−4025.56	−1.06271
$28$	0	0
$29$	−2127.93	−0.469853	−0.234926	−	0.972013i	$-0.575485\pi$
$29$	−2127.93	−0.469853	−0.234926	+	0.972013i	$0.575485\pi$
$30$	0	0
$31$	10262.5	1.91801	0.959004	−	0.283393i	$-0.0914601\pi$
$31$	10262.5	1.91801	0.959004	+	0.283393i	$0.0914601\pi$
$32$	0	0
$33$	−6090.55	−0.973580
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	−7266.05	−0.872557	−0.436279	−	0.899812i	$-0.643704\pi$
$37$	−7266.05	−0.872557	−0.436279	+	0.899812i	$0.643704\pi$
$38$	0	0
$39$	5710.16	0.601155
$40$	0	0
$41$	−5893.44	−0.547531	−0.273766	−	0.961796i	$-0.588269\pi$
$41$	−5893.44	−0.547531	−0.273766	+	0.961796i	$0.588269\pi$
$42$	0	0
$43$	−20157.7	−1.66253	−0.831267	−	0.555873i	$-0.812384\pi$
$43$	−20157.7	−1.66253	−0.831267	+	0.555873i	$0.812384\pi$
$44$	0	0
$45$	−962.537	−0.0708576
$46$	0	0
$47$	−20056.5	−1.32438	−0.662188	−	0.749338i	$-0.730372\pi$
$47$	−20056.5	−1.32438	−0.662188	+	0.749338i	$0.730372\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	25047.2	1.34845
$52$	0	0
$53$	−33954.9	−1.66040	−0.830200	−	0.557466i	$-0.811774\pi$
$53$	−33954.9	−1.66040	−0.830200	+	0.557466i	$0.811774\pi$
$54$	0	0
$55$	−10647.6	−0.474619
$56$	0	0
$57$	−41102.4	−1.67564
$58$	0	0
$59$	4319.34	0.161543	0.0807713	−	0.996733i	$-0.474262\pi$
$59$	4319.34	0.161543	0.0807713	+	0.996733i	$0.474262\pi$
$60$	0	0
$61$	−12253.6	−0.421638	−0.210819	−	0.977525i	$-0.567613\pi$
$61$	−12253.6	−0.421638	−0.210819	+	0.977525i	$0.567613\pi$
$62$	0	0
$63$	1886.57	0.0598856
$64$	0	0
$65$	9982.58	0.293062
$66$	0	0
$67$	17533.8	0.477188	0.238594	−	0.971119i	$-0.423313\pi$
$67$	17533.8	0.477188	0.238594	+	0.971119i	$0.423313\pi$
$68$	0	0
$69$	33082.7	0.836524
$70$	0	0
$71$	−1658.18	−0.0390379	−0.0195189	−	0.999809i	$-0.506213\pi$
$71$	−1658.18	−0.0390379	−0.0195189	+	0.999809i	$0.506213\pi$
$72$	0	0
$73$	−8246.91	−0.181127	−0.0905637	−	0.995891i	$-0.528867\pi$
$73$	−8246.91	−0.181127	−0.0905637	+	0.995891i	$0.528867\pi$
$74$	0	0
$75$	8937.69	0.183473
$76$	0	0
$77$	20869.3	0.401126
$78$	0	0
$79$	9168.61	0.165286	0.0826430	−	0.996579i	$-0.473664\pi$
$79$	9168.61	0.165286	0.0826430	+	0.996579i	$0.473664\pi$
$80$	0	0
$81$	−48210.8	−0.816454
$82$	0	0
$83$	−95203.2	−1.51690	−0.758449	−	0.651732i	$-0.774043\pi$
$83$	−95203.2	−1.51690	−0.758449	+	0.651732i	$0.774043\pi$
$84$	0	0
$85$	43788.0	0.657367
$86$	0	0
$87$	−30430.0	−0.431026
$88$	0	0
$89$	−14441.8	−0.193262	−0.0966311	−	0.995320i	$-0.530807\pi$
$89$	−14441.8	−0.193262	−0.0966311	+	0.995320i	$0.530807\pi$
$90$	0	0
$91$	−19565.9	−0.247683
$92$	0	0
$93$	146757.	1.75951
$94$	0	0
$95$	−71855.8	−0.816870
$96$	0	0
$97$	62132.4	0.670485	0.335242	−	0.942132i	$-0.391182\pi$
$97$	62132.4	0.670485	0.335242	+	0.942132i	$0.391182\pi$
$98$	0	0
$99$	16397.9	0.168152
$100$	0	0
$101$	−108138.	−1.05481	−0.527404	−	0.849615i	$-0.676835\pi$
$101$	−108138.	−1.05481	−0.527404	+	0.849615i	$0.676835\pi$
$102$	0	0
$103$	−138034.	−1.28201	−0.641005	−	0.767536i	$-0.721482\pi$
$103$	−138034.	−1.28201	−0.641005	+	0.767536i	$0.721482\pi$
$104$	0	0
$105$	−17517.9	−0.155063
$106$	0	0
$107$	−6189.61	−0.0522641	−0.0261321	−	0.999658i	$-0.508319\pi$
$107$	−6189.61	−0.0522641	−0.0261321	+	0.999658i	$0.508319\pi$
$108$	0	0
$109$	−68652.9	−0.553468	−0.276734	−	0.960947i	$-0.589252\pi$
$109$	−68652.9	−0.553468	−0.276734	+	0.960947i	$0.589252\pi$
$110$	0	0
$111$	−103907.	−0.800453
$112$	0	0
$113$	−62835.9	−0.462926	−0.231463	−	0.972844i	$-0.574351\pi$
$113$	−62835.9	−0.462926	−0.231463	+	0.972844i	$0.574351\pi$
$114$	0	0
$115$	57835.7	0.407804
$116$	0	0
$117$	−15373.8	−0.103828
$118$	0	0
$119$	−85824.4	−0.555576
$120$	0	0
$121$	20343.1	0.126315
$122$	0	0
$123$	−84277.9	−0.502286
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	198049.	1.08959	0.544794	−	0.838570i	$-0.316608\pi$
$127$	198049.	1.08959	0.544794	+	0.838570i	$0.316608\pi$
$128$	0	0
$129$	−288262.	−1.52515
$130$	0	0
$131$	285132.	1.45167	0.725835	−	0.687868i	$-0.241453\pi$
$131$	285132.	1.45167	0.725835	+	0.687868i	$0.241453\pi$
$132$	0	0
$133$	140837.	0.690381
$134$	0	0
$135$	−100639.	−0.475260
$136$	0	0
$137$	406979.	1.85255	0.926277	−	0.376844i	$-0.122991\pi$
$137$	406979.	1.85255	0.926277	+	0.376844i	$0.122991\pi$
$138$	0	0
$139$	−13530.4	−0.0593985	−0.0296992	−	0.999559i	$-0.509455\pi$
$139$	−13530.4	−0.0593985	−0.0296992	+	0.999559i	$0.509455\pi$
$140$	0	0
$141$	−286814.	−1.21494
$142$	0	0
$143$	−170065.	−0.695464
$144$	0	0
$145$	−53198.2	−0.210125
$146$	0	0
$147$	34335.0	0.131052
$148$	0	0
$149$	77129.8	0.284614	0.142307	−	0.989823i	$-0.454548\pi$
$149$	77129.8	0.284614	0.142307	+	0.989823i	$0.454548\pi$
$150$	0	0
$151$	420464.	1.50067	0.750336	−	0.661056i	$-0.229892\pi$
$151$	420464.	1.50067	0.750336	+	0.661056i	$0.229892\pi$
$152$	0	0
$153$	−67436.1	−0.232897
$154$	0	0
$155$	256563.	0.857759
$156$	0	0
$157$	−481908.	−1.56032	−0.780162	−	0.625577i	$-0.784864\pi$
$157$	−481908.	−1.56032	−0.780162	+	0.625577i	$0.784864\pi$
$158$	0	0
$159$	−485565.	−1.52319
$160$	0	0
$161$	−113358.	−0.344657
$162$	0	0
$163$	−282039.	−0.831458	−0.415729	−	0.909488i	$-0.636474\pi$
$163$	−282039.	−0.831458	−0.415729	+	0.909488i	$0.636474\pi$
$164$	0	0
$165$	−152264.	−0.435398
$166$	0	0
$167$	−131971.	−0.366173	−0.183086	−	0.983097i	$-0.558609\pi$
$167$	−131971.	−0.366173	−0.183086	+	0.983097i	$0.558609\pi$
$168$	0	0
$169$	−211850.	−0.570573
$170$	0	0
$171$	110662.	0.289407
$172$	0	0
$173$	55501.5	0.140990	0.0704952	−	0.997512i	$-0.477542\pi$
$173$	55501.5	0.140990	0.0704952	+	0.997512i	$0.477542\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	61767.8	0.148193
$178$	0	0
$179$	−20768.3	−0.0484471	−0.0242236	−	0.999707i	$-0.507711\pi$
$179$	−20768.3	−0.0484471	−0.0242236	+	0.999707i	$0.507711\pi$
$180$	0	0
$181$	−255866.	−0.580520	−0.290260	−	0.956948i	$-0.593742\pi$
$181$	−255866.	−0.580520	−0.290260	+	0.956948i	$0.593742\pi$
$182$	0	0
$183$	−175230.	−0.386796
$184$	0	0
$185$	−181651.	−0.390219
$186$	0	0
$187$	−745979.	−1.55999
$188$	0	0
$189$	197252.	0.401668
$190$	0	0
$191$	−649260.	−1.28776	−0.643881	−	0.765126i	$-0.722677\pi$
$191$	−649260.	−1.28776	−0.643881	+	0.765126i	$0.722677\pi$
$192$	0	0
$193$	−768642.	−1.48536	−0.742678	−	0.669649i	$-0.766445\pi$
$193$	−768642.	−1.48536	−0.742678	+	0.669649i	$0.766445\pi$
$194$	0	0
$195$	142754.	0.268845
$196$	0	0
$197$	−166160.	−0.305044	−0.152522	−	0.988300i	$-0.548739\pi$
$197$	−166160.	−0.305044	−0.152522	+	0.988300i	$0.548739\pi$
$198$	0	0
$199$	74756.1	0.133818	0.0669089	−	0.997759i	$-0.478686\pi$
$199$	74756.1	0.133818	0.0669089	+	0.997759i	$0.478686\pi$
$200$	0	0
$201$	250739.	0.437756
$202$	0	0
$203$	104268.	0.177588
$204$	0	0
$205$	−147336.	−0.244863
$206$	0	0
$207$	−89070.5	−0.144480
$208$	0	0
$209$	1.22415e6	1.93851
$210$	0	0
$211$	1.08646e6	1.67999	0.839997	−	0.542591i	$-0.182557\pi$
$211$	1.08646e6	1.67999	0.839997	+	0.542591i	$0.182557\pi$
$212$	0	0
$213$	−23712.5	−0.0358120
$214$	0	0
$215$	−503943.	−0.743508
$216$	0	0
$217$	−502864.	−0.724939
$218$	0	0
$219$	−117933.	−0.166160
$220$	0	0
$221$	699387.	0.963246
$222$	0	0
$223$	430809.	0.580126	0.290063	−	0.957008i	$-0.406324\pi$
$223$	430809.	0.580126	0.290063	+	0.957008i	$0.406324\pi$
$224$	0	0
$225$	−24063.4	−0.0316885
$226$	0	0
$227$	228724.	0.294610	0.147305	−	0.989091i	$-0.452940\pi$
$227$	228724.	0.294610	0.147305	+	0.989091i	$0.452940\pi$
$228$	0	0
$229$	−1.01797e6	−1.28276	−0.641378	−	0.767225i	$-0.721637\pi$
$229$	−1.01797e6	−1.28276	−0.641378	+	0.767225i	$0.721637\pi$
$230$	0	0
$231$	298437.	0.367979
$232$	0	0
$233$	−236710.	−0.285645	−0.142822	−	0.989748i	$-0.545618\pi$
$233$	−236710.	−0.285645	−0.142822	+	0.989748i	$0.545618\pi$
$234$	0	0
$235$	−501413.	−0.592279
$236$	0	0
$237$	131114.	0.151627
$238$	0	0
$239$	−1.73994e6	−1.97034	−0.985169	−	0.171587i	$-0.945111\pi$
$239$	−1.73994e6	−1.97034	−0.985169	+	0.171587i	$0.945111\pi$
$240$	0	0
$241$	−332840.	−0.369142	−0.184571	−	0.982819i	$-0.559090\pi$
$241$	−332840.	−0.369142	−0.184571	+	0.982819i	$0.559090\pi$
$242$	0	0
$243$	288781.	0.313728
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−1.14769e6	−1.19697
$248$	0	0
$249$	−1.36143e6	−1.39155
$250$	0	0
$251$	−1.88585e6	−1.88939	−0.944696	−	0.327948i	$-0.893643\pi$
$251$	−1.88585e6	−1.88939	−0.944696	+	0.327948i	$0.893643\pi$
$252$	0	0
$253$	−985298.	−0.967757
$254$	0	0
$255$	626181.	0.603045
$256$	0	0
$257$	432608.	0.408566	0.204283	−	0.978912i	$-0.434514\pi$
$257$	432608.	0.408566	0.204283	+	0.978912i	$0.434514\pi$
$258$	0	0
$259$	356036.	0.329796
$260$	0	0
$261$	81928.4	0.0744446
$262$	0	0
$263$	1.20388e6	1.07323	0.536614	−	0.843828i	$-0.319703\pi$
$263$	1.20388e6	1.07323	0.536614	+	0.843828i	$0.319703\pi$
$264$	0	0
$265$	−848872.	−0.742554
$266$	0	0
$267$	−206522.	−0.177292
$268$	0	0
$269$	548133.	0.461855	0.230927	−	0.972971i	$-0.425824\pi$
$269$	548133.	0.461855	0.230927	+	0.972971i	$0.425824\pi$
$270$	0	0
$271$	906590.	0.749873	0.374936	−	0.927051i	$-0.377665\pi$
$271$	906590.	0.749873	0.374936	+	0.927051i	$0.377665\pi$
$272$	0	0
$273$	−279798.	−0.227215
$274$	0	0
$275$	−266190.	−0.212256
$276$	0	0
$277$	1.35338e6	1.05979	0.529895	−	0.848064i	$-0.322231\pi$
$277$	1.35338e6	1.05979	0.529895	+	0.848064i	$0.322231\pi$
$278$	0	0
$279$	−395123.	−0.303894
$280$	0	0
$281$	752309.	0.568369	0.284185	−	0.958770i	$-0.408277\pi$
$281$	752309.	0.568369	0.284185	+	0.958770i	$0.408277\pi$
$282$	0	0
$283$	820710.	0.609149	0.304575	−	0.952489i	$-0.401486\pi$
$283$	820710.	0.609149	0.304575	+	0.952489i	$0.401486\pi$
$284$	0	0
$285$	−1.02756e6	−0.749368
$286$	0	0
$287$	288778.	0.206947
$288$	0	0
$289$	1.64796e6	1.16065
$290$	0	0
$291$	888512.	0.615079
$292$	0	0
$293$	−1.92239e6	−1.30819	−0.654097	−	0.756411i	$-0.726951\pi$
$293$	−1.92239e6	−1.30819	−0.654097	+	0.756411i	$0.726951\pi$
$294$	0	0
$295$	107983.	0.0722441
$296$	0	0
$297$	1.71450e6	1.12784
$298$	0	0
$299$	923760.	0.597559
$300$	0	0
$301$	987729.	0.628379
$302$	0	0
$303$	−1.54640e6	−0.967643
$304$	0	0
$305$	−306340.	−0.188562
$306$	0	0
$307$	−223229.	−0.135177	−0.0675887	−	0.997713i	$-0.521531\pi$
$307$	−223229.	−0.135177	−0.0675887	+	0.997713i	$0.521531\pi$
$308$	0	0
$309$	−1.97392e6	−1.17607
$310$	0	0
$311$	−2.29818e6	−1.34736	−0.673678	−	0.739025i	$-0.735286\pi$
$311$	−2.29818e6	−1.34736	−0.673678	+	0.739025i	$0.735286\pi$
$312$	0	0
$313$	−1.62881e6	−0.939744	−0.469872	−	0.882735i	$-0.655700\pi$
$313$	−1.62881e6	−0.939744	−0.469872	+	0.882735i	$0.655700\pi$
$314$	0	0
$315$	47164.3	0.0267816
$316$	0	0
$317$	1.13620e6	0.635049	0.317524	−	0.948250i	$-0.397149\pi$
$317$	1.13620e6	0.635049	0.317524	+	0.948250i	$0.397149\pi$
$318$	0	0
$319$	906293.	0.498645
$320$	0	0
$321$	−88513.3	−0.0479453
$322$	0	0
$323$	−5.03428e6	−2.68492
$324$	0	0
$325$	249565.	0.131061
$326$	0	0
$327$	−981757.	−0.507732
$328$	0	0
$329$	982770.	0.500567
$330$	0	0
$331$	3.77549e6	1.89410	0.947051	−	0.321084i	$-0.104047\pi$
$331$	3.77549e6	1.89410	0.947051	+	0.321084i	$0.104047\pi$
$332$	0	0
$333$	279754.	0.138250
$334$	0	0
$335$	438346.	0.213405
$336$	0	0
$337$	3.74913e6	1.79827	0.899136	−	0.437669i	$-0.144196\pi$
$337$	3.74913e6	1.79827	0.899136	+	0.437669i	$0.144196\pi$
$338$	0	0
$339$	−898571.	−0.424672
$340$	0	0
$341$	−4.37085e6	−2.03554
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	827068.	0.374105
$346$	0	0
$347$	1.14843e6	0.512012	0.256006	−	0.966675i	$-0.417593\pi$
$347$	1.14843e6	0.512012	0.256006	+	0.966675i	$0.417593\pi$
$348$	0	0
$349$	4.03275e6	1.77230	0.886151	−	0.463396i	$-0.153369\pi$
$349$	4.03275e6	1.77230	0.886151	+	0.463396i	$0.153369\pi$
$350$	0	0
$351$	−1.60742e6	−0.696403
$352$	0	0
$353$	652660.	0.278773	0.139386	−	0.990238i	$-0.455487\pi$
$353$	652660.	0.278773	0.139386	+	0.990238i	$0.455487\pi$
$354$	0	0
$355$	−41454.5	−0.0174583
$356$	0	0
$357$	−1.22732e6	−0.509666
$358$	0	0
$359$	2.84604e6	1.16548	0.582741	−	0.812658i	$-0.301980\pi$
$359$	2.84604e6	1.16548	0.582741	+	0.812658i	$0.301980\pi$
$360$	0	0
$361$	5.78512e6	2.33639
$362$	0	0
$363$	290912.	0.115877
$364$	0	0
$365$	−206173.	−0.0810026
$366$	0	0
$367$	2.65232e6	1.02792	0.513962	−	0.857813i	$-0.328177\pi$
$367$	2.65232e6	1.02792	0.513962	+	0.857813i	$0.328177\pi$
$368$	0	0
$369$	226906.	0.0867521
$370$	0	0
$371$	1.66379e6	0.627572
$372$	0	0
$373$	2.83283e6	1.05426	0.527132	−	0.849784i	$-0.323267\pi$
$373$	2.83283e6	1.05426	0.527132	+	0.849784i	$0.323267\pi$
$374$	0	0
$375$	223442.	0.0820516
$376$	0	0
$377$	−849688.	−0.307898
$378$	0	0
$379$	1.20678e6	0.431551	0.215775	−	0.976443i	$-0.430772\pi$
$379$	1.20678e6	0.431551	0.215775	+	0.976443i	$0.430772\pi$
$380$	0	0
$381$	2.83215e6	0.999550
$382$	0	0
$383$	2.22793e6	0.776076	0.388038	−	0.921643i	$-0.373153\pi$
$383$	2.22793e6	0.776076	0.388038	+	0.921643i	$0.373153\pi$
$384$	0	0
$385$	521732.	0.179389
$386$	0	0
$387$	776103.	0.263416
$388$	0	0
$389$	−3.74004e6	−1.25315	−0.626574	−	0.779362i	$-0.715543\pi$
$389$	−3.74004e6	−1.25315	−0.626574	+	0.779362i	$0.715543\pi$
$390$	0	0
$391$	4.05202e6	1.34038
$392$	0	0
$393$	4.07748e6	1.33171
$394$	0	0
$395$	229215.	0.0739181
$396$	0	0
$397$	−2.90335e6	−0.924533	−0.462267	−	0.886741i	$-0.652964\pi$
$397$	−2.90335e6	−0.924533	−0.462267	+	0.886741i	$0.652964\pi$
$398$	0	0
$399$	2.01402e6	0.633331
$400$	0	0
$401$	1.43777e6	0.446506	0.223253	−	0.974761i	$-0.428332\pi$
$401$	1.43777e6	0.446506	0.223253	+	0.974761i	$0.428332\pi$
$402$	0	0
$403$	4.09786e6	1.25688
$404$	0	0
$405$	−1.20527e6	−0.365129
$406$	0	0
$407$	3.09464e6	0.926027
$408$	0	0
$409$	−4.78825e6	−1.41536	−0.707682	−	0.706531i	$-0.750259\pi$
$409$	−4.78825e6	−1.41536	−0.707682	+	0.706531i	$0.750259\pi$
$410$	0	0
$411$	5.81992e6	1.69947
$412$	0	0
$413$	−211648.	−0.0610574
$414$	0	0
$415$	−2.38008e6	−0.678377
$416$	0	0
$417$	−193489.	−0.0544900
$418$	0	0
$419$	2.08411e6	0.579943	0.289972	−	0.957035i	$-0.406354\pi$
$419$	2.08411e6	0.579943	0.289972	+	0.957035i	$0.406354\pi$
$420$	0	0
$421$	−1.00260e6	−0.275690	−0.137845	−	0.990454i	$-0.544018\pi$
$421$	−1.00260e6	−0.275690	−0.137845	+	0.990454i	$0.544018\pi$
$422$	0	0
$423$	772206.	0.209837
$424$	0	0
$425$	1.09470e6	0.293983
$426$	0	0
$427$	600427.	0.159364
$428$	0	0
$429$	−2.43198e6	−0.637994
$430$	0	0
$431$	−1.45620e6	−0.377595	−0.188798	−	0.982016i	$-0.560459\pi$
$431$	−1.45620e6	−0.377595	−0.188798	+	0.982016i	$0.560459\pi$
$432$	0	0
$433$	−1.10726e6	−0.283812	−0.141906	−	0.989880i	$-0.545323\pi$
$433$	−1.10726e6	−0.283812	−0.141906	+	0.989880i	$0.545323\pi$
$434$	0	0
$435$	−760750.	−0.192761
$436$	0	0
$437$	−6.64934e6	−1.66562
$438$	0	0
$439$	−4.15410e6	−1.02876	−0.514382	−	0.857561i	$-0.671979\pi$
$439$	−4.15410e6	−1.02876	−0.514382	+	0.857561i	$0.671979\pi$
$440$	0	0
$441$	−92442.1	−0.0226346
$442$	0	0
$443$	7.20799e6	1.74504	0.872519	−	0.488581i	$-0.162485\pi$
$443$	7.20799e6	1.74504	0.872519	+	0.488581i	$0.162485\pi$
$444$	0	0
$445$	−361046.	−0.0864295
$446$	0	0
$447$	1.10298e6	0.261095
$448$	0	0
$449$	5.58367e6	1.30709	0.653543	−	0.756890i	$-0.273282\pi$
$449$	5.58367e6	1.30709	0.653543	+	0.756890i	$0.273282\pi$
$450$	0	0
$451$	2.51004e6	0.581084
$452$	0	0
$453$	6.01275e6	1.37666
$454$	0	0
$455$	−489147.	−0.110767
$456$	0	0
$457$	−3.54344e6	−0.793661	−0.396831	−	0.917892i	$-0.629890\pi$
$457$	−3.54344e6	−0.793661	−0.396831	+	0.917892i	$0.629890\pi$
$458$	0	0
$459$	−7.05084e6	−1.56210
$460$	0	0
$461$	5.88069e6	1.28877	0.644386	−	0.764700i	$-0.277113\pi$
$461$	5.88069e6	1.28877	0.644386	+	0.764700i	$0.277113\pi$
$462$	0	0
$463$	−6.65045e6	−1.44178	−0.720890	−	0.693050i	$-0.756267\pi$
$463$	−6.65045e6	−1.44178	−0.720890	+	0.693050i	$0.756267\pi$
$464$	0	0
$465$	3.66893e6	0.786878
$466$	0	0
$467$	−4.34743e6	−0.922445	−0.461222	−	0.887285i	$-0.652589\pi$
$467$	−4.34743e6	−0.922445	−0.461222	+	0.887285i	$0.652589\pi$
$468$	0	0
$469$	−859158.	−0.180360
$470$	0	0
$471$	−6.89143e6	−1.43139
$472$	0	0
$473$	8.58526e6	1.76441
$474$	0	0
$475$	−1.79640e6	−0.365316
$476$	0	0
$477$	1.30731e6	0.263078
$478$	0	0
$479$	929680.	0.185138	0.0925688	−	0.995706i	$-0.470492\pi$
$479$	929680.	0.185138	0.0925688	+	0.995706i	$0.470492\pi$
$480$	0	0
$481$	−2.90136e6	−0.571792
$482$	0	0
$483$	−1.62105e6	−0.316176
$484$	0	0
$485$	1.55331e6	0.299850
$486$	0	0
$487$	5.44887e6	1.04108	0.520540	−	0.853837i	$-0.325731\pi$
$487$	5.44887e6	1.04108	0.520540	+	0.853837i	$0.325731\pi$
$488$	0	0
$489$	−4.03324e6	−0.762750
$490$	0	0
$491$	670070.	0.125434	0.0627172	−	0.998031i	$-0.480023\pi$
$491$	670070.	0.125434	0.0627172	+	0.998031i	$0.480023\pi$
$492$	0	0
$493$	−3.72711e6	−0.690644
$494$	0	0
$495$	409948.	0.0751997
$496$	0	0
$497$	81250.9	0.0147549
$498$	0	0
$499$	−6.58017e6	−1.18300	−0.591501	−	0.806304i	$-0.701465\pi$
$499$	−6.58017e6	−1.18300	−0.591501	+	0.806304i	$0.701465\pi$
$500$	0	0
$501$	−1.88722e6	−0.335914
$502$	0	0
$503$	1.47714e6	0.260317	0.130159	−	0.991493i	$-0.458451\pi$
$503$	1.47714e6	0.260317	0.130159	+	0.991493i	$0.458451\pi$
$504$	0	0
$505$	−2.70344e6	−0.471724
$506$	0	0
$507$	−3.02952e6	−0.523424
$508$	0	0
$509$	5.16474e6	0.883597	0.441799	−	0.897114i	$-0.354341\pi$
$509$	5.16474e6	0.883597	0.441799	+	0.897114i	$0.354341\pi$
$510$	0	0
$511$	404099.	0.0684597
$512$	0	0
$513$	1.15704e7	1.94113
$514$	0	0
$515$	−3.45084e6	−0.573333
$516$	0	0
$517$	8.54215e6	1.40553
$518$	0	0
$519$	793688.	0.129340
$520$	0	0
$521$	294103.	0.0474684	0.0237342	−	0.999718i	$-0.492444\pi$
$521$	294103.	0.0474684	0.0237342	+	0.999718i	$0.492444\pi$
$522$	0	0
$523$	−1.39098e6	−0.222365	−0.111182	−	0.993800i	$-0.535464\pi$
$523$	−1.39098e6	−0.222365	−0.111182	+	0.993800i	$0.535464\pi$
$524$	0	0
$525$	−437947.	−0.0693462
$526$	0	0
$527$	1.79750e7	2.81931
$528$	0	0
$529$	−1.08439e6	−0.168479
$530$	0	0
$531$	−166301.	−0.0255952
$532$	0	0
$533$	−2.35327e6	−0.358801
$534$	0	0
$535$	−154740.	−0.0233732
$536$	0	0
$537$	−296993.	−0.0444437
$538$	0	0
$539$	−1.02260e6	−0.151611
$540$	0	0
$541$	2.36315e6	0.347134	0.173567	−	0.984822i	$-0.444471\pi$
$541$	2.36315e6	0.347134	0.173567	+	0.984822i	$0.444471\pi$
$542$	0	0
$543$	−3.65897e6	−0.532548
$544$	0	0
$545$	−1.71632e6	−0.247519
$546$	0	0
$547$	−4.52298e6	−0.646333	−0.323167	−	0.946342i	$-0.604747\pi$
$547$	−4.52298e6	−0.646333	−0.323167	+	0.946342i	$0.604747\pi$
$548$	0	0
$549$	471782.	0.0668053
$550$	0	0
$551$	6.11616e6	0.858223
$552$	0	0
$553$	−449262.	−0.0624722
$554$	0	0
$555$	−2.59767e6	−0.357973
$556$	0	0
$557$	−8.40670e6	−1.14812	−0.574060	−	0.818813i	$-0.694633\pi$
$557$	−8.40670e6	−1.14812	−0.574060	+	0.818813i	$0.694633\pi$
$558$	0	0
$559$	−8.04905e6	−1.08947
$560$	0	0
$561$	−1.06677e7	−1.43108
$562$	0	0
$563$	2.22527e6	0.295877	0.147938	−	0.988997i	$-0.452736\pi$
$563$	2.22527e6	0.295877	0.147938	+	0.988997i	$0.452736\pi$
$564$	0	0
$565$	−1.57090e6	−0.207027
$566$	0	0
$567$	2.36233e6	0.308590
$568$	0	0
$569$	1.58379e6	0.205078	0.102539	−	0.994729i	$-0.467303\pi$
$569$	1.58379e6	0.205078	0.102539	+	0.994729i	$0.467303\pi$
$570$	0	0
$571$	−1.25695e7	−1.61335	−0.806676	−	0.590994i	$-0.798736\pi$
$571$	−1.25695e7	−1.61335	−0.806676	+	0.590994i	$0.798736\pi$
$572$	0	0
$573$	−9.28462e6	−1.18135
$574$	0	0
$575$	1.44589e6	0.182376
$576$	0	0
$577$	1.37636e7	1.72104	0.860521	−	0.509414i	$-0.170138\pi$
$577$	1.37636e7	1.72104	0.860521	+	0.509414i	$0.170138\pi$
$578$	0	0
$579$	−1.09918e7	−1.36261
$580$	0	0
$581$	4.66496e6	0.573334
$582$	0	0
$583$	1.44615e7	1.76215
$584$	0	0
$585$	−384344.	−0.0464334
$586$	0	0
$587$	−1.43475e6	−0.171862	−0.0859311	−	0.996301i	$-0.527386\pi$
$587$	−1.43475e6	−0.171862	−0.0859311	+	0.996301i	$0.527386\pi$
$588$	0	0
$589$	−2.94969e7	−3.50339
$590$	0	0
$591$	−2.37614e6	−0.279836
$592$	0	0
$593$	2.58486e6	0.301856	0.150928	−	0.988545i	$-0.451774\pi$
$593$	2.58486e6	0.301856	0.150928	+	0.988545i	$0.451774\pi$
$594$	0	0
$595$	−2.14561e6	−0.248461
$596$	0	0
$597$	1.06903e6	0.122760
$598$	0	0
$599$	5.51209e6	0.627696	0.313848	−	0.949473i	$-0.398382\pi$
$599$	5.51209e6	0.627696	0.313848	+	0.949473i	$0.398382\pi$
$600$	0	0
$601$	6.92526e6	0.782077	0.391039	−	0.920374i	$-0.372116\pi$
$601$	6.92526e6	0.782077	0.391039	+	0.920374i	$0.372116\pi$
$602$	0	0
$603$	−675078.	−0.0756068
$604$	0	0
$605$	508578.	0.0564896
$606$	0	0
$607$	−1.19865e7	−1.32045	−0.660225	−	0.751067i	$-0.729539\pi$
$607$	−1.19865e7	−1.32045	−0.660225	+	0.751067i	$0.729539\pi$
$608$	0	0
$609$	1.49107e6	0.162913
$610$	0	0
$611$	−8.00864e6	−0.867872
$612$	0	0
$613$	−4.84046e6	−0.520278	−0.260139	−	0.965571i	$-0.583768\pi$
$613$	−4.84046e6	−0.520278	−0.260139	+	0.965571i	$0.583768\pi$
$614$	0	0
$615$	−2.10695e6	−0.224629
$616$	0	0
$617$	−9.94864e6	−1.05208	−0.526042	−	0.850458i	$-0.676325\pi$
$617$	−9.94864e6	−1.05208	−0.526042	+	0.850458i	$0.676325\pi$
$618$	0	0
$619$	1.38913e7	1.45719	0.728597	−	0.684943i	$-0.240173\pi$
$619$	1.38913e7	1.45719	0.728597	+	0.684943i	$0.240173\pi$
$620$	0	0
$621$	−9.31284e6	−0.969065
$622$	0	0
$623$	707649.	0.0730463
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	1.75057e7	1.77832
$628$	0	0
$629$	−1.27266e7	−1.28259
$630$	0	0
$631$	7.29060e6	0.728937	0.364469	−	0.931216i	$-0.381251\pi$
$631$	7.29060e6	0.728937	0.364469	+	0.931216i	$0.381251\pi$
$632$	0	0
$633$	1.55367e7	1.54117
$634$	0	0
$635$	4.95121e6	0.487279
$636$	0	0
$637$	958727.	0.0936152
$638$	0	0
$639$	63842.4	0.00618525
$640$	0	0
$641$	1.58175e7	1.52052	0.760259	−	0.649620i	$-0.225072\pi$
$641$	1.58175e7	1.52052	0.760259	+	0.649620i	$0.225072\pi$
$642$	0	0
$643$	7.35926e6	0.701951	0.350975	−	0.936385i	$-0.385850\pi$
$643$	7.35926e6	0.701951	0.350975	+	0.936385i	$0.385850\pi$
$644$	0	0
$645$	−7.20654e6	−0.682068
$646$	0	0
$647$	−1.08644e7	−1.02034	−0.510172	−	0.860072i	$-0.670418\pi$
$647$	−1.08644e7	−1.02034	−0.510172	+	0.860072i	$0.670418\pi$
$648$	0	0
$649$	−1.83962e6	−0.171442
$650$	0	0
$651$	−7.19111e6	−0.665033
$652$	0	0
$653$	−2.85006e6	−0.261560	−0.130780	−	0.991411i	$-0.541748\pi$
$653$	−2.85006e6	−0.261560	−0.130780	+	0.991411i	$0.541748\pi$
$654$	0	0
$655$	7.12831e6	0.649207
$656$	0	0
$657$	317518.	0.0286982
$658$	0	0
$659$	214583.	0.0192479	0.00962393	−	0.999954i	$-0.496937\pi$
$659$	214583.	0.0192479	0.00962393	+	0.999954i	$0.496937\pi$
$660$	0	0
$661$	2.26579e6	0.201704	0.100852	−	0.994901i	$-0.467843\pi$
$661$	2.26579e6	0.201704	0.100852	+	0.994901i	$0.467843\pi$
$662$	0	0
$663$	1.00014e7	0.883648
$664$	0	0
$665$	3.52094e6	0.308748
$666$	0	0
$667$	−4.92281e6	−0.428448
$668$	0	0
$669$	6.16070e6	0.532187
$670$	0	0
$671$	5.21886e6	0.447476
$672$	0	0
$673$	−9.52991e6	−0.811057	−0.405528	−	0.914082i	$-0.632912\pi$
$673$	−9.52991e6	−0.811057	−0.405528	+	0.914082i	$0.632912\pi$
$674$	0	0
$675$	−2.51597e6	−0.212543
$676$	0	0
$677$	2.14827e7	1.80143	0.900714	−	0.434412i	$-0.143044\pi$
$677$	2.14827e7	1.80143	0.900714	+	0.434412i	$0.143044\pi$
$678$	0	0
$679$	−3.04449e6	−0.253419
$680$	0	0
$681$	3.27082e6	0.270264
$682$	0	0
$683$	2.89672e6	0.237605	0.118802	−	0.992918i	$-0.462095\pi$
$683$	2.89672e6	0.237605	0.118802	+	0.992918i	$0.462095\pi$
$684$	0	0
$685$	1.01745e7	0.828487
$686$	0	0
$687$	−1.45572e7	−1.17676
$688$	0	0
$689$	−1.35583e7	−1.08807
$690$	0	0
$691$	2.09710e6	0.167080	0.0835399	−	0.996504i	$-0.473377\pi$
$691$	2.09710e6	0.167080	0.0835399	+	0.996504i	$0.473377\pi$
$692$	0	0
$693$	−803499.	−0.0635553
$694$	0	0
$695$	−338261.	−0.0265638
$696$	0	0
$697$	−1.03225e7	−0.804825
$698$	0	0
$699$	−3.38502e6	−0.262041
$700$	0	0
$701$	6.59970e6	0.507258	0.253629	−	0.967302i	$-0.418376\pi$
$701$	6.59970e6	0.507258	0.253629	+	0.967302i	$0.418376\pi$
$702$	0	0
$703$	2.08843e7	1.59379
$704$	0	0
$705$	−7.17036e6	−0.543336
$706$	0	0
$707$	5.29874e6	0.398680
$708$	0	0
$709$	−2.22011e7	−1.65866	−0.829332	−	0.558756i	$-0.811279\pi$
$709$	−2.22011e7	−1.65866	−0.829332	+	0.558756i	$0.811279\pi$
$710$	0	0
$711$	−353005.	−0.0261883
$712$	0	0
$713$	2.37416e7	1.74899
$714$	0	0
$715$	−4.25162e6	−0.311021
$716$	0	0
$717$	−2.48817e7	−1.80752
$718$	0	0
$719$	−1.87540e7	−1.35292	−0.676460	−	0.736479i	$-0.736487\pi$
$719$	−1.87540e7	−1.35292	−0.676460	+	0.736479i	$0.736487\pi$
$720$	0	0
$721$	6.76365e6	0.484555
$722$	0	0
$723$	−4.75972e6	−0.338638
$724$	0	0
$725$	−1.32995e6	−0.0939706
$726$	0	0
$727$	−1.38593e7	−0.972537	−0.486269	−	0.873809i	$-0.661642\pi$
$727$	−1.38593e7	−0.972537	−0.486269	+	0.873809i	$0.661642\pi$
$728$	0	0
$729$	1.58449e7	1.10426
$730$	0	0
$731$	−3.53067e7	−2.44379
$732$	0	0
$733$	1.08280e7	0.744372	0.372186	−	0.928158i	$-0.378608\pi$
$733$	1.08280e7	0.744372	0.372186	+	0.928158i	$0.378608\pi$
$734$	0	0
$735$	858375.	0.0586083
$736$	0	0
$737$	−7.46772e6	−0.506430
$738$	0	0
$739$	906866.	0.0610846	0.0305423	−	0.999533i	$-0.490277\pi$
$739$	906866.	0.0610846	0.0305423	+	0.999533i	$0.490277\pi$
$740$	0	0
$741$	−1.64123e7	−1.09806
$742$	0	0
$743$	1.76804e7	1.17495	0.587475	−	0.809242i	$-0.300122\pi$
$743$	1.76804e7	1.17495	0.587475	+	0.809242i	$0.300122\pi$
$744$	0	0
$745$	1.92825e6	0.127283
$746$	0	0
$747$	3.66547e6	0.240341
$748$	0	0
$749$	303291.	0.0197540
$750$	0	0
$751$	−1.96975e7	−1.27441	−0.637207	−	0.770692i	$-0.719911\pi$
$751$	−1.96975e7	−1.27441	−0.637207	+	0.770692i	$0.719911\pi$
$752$	0	0
$753$	−2.69682e7	−1.73326
$754$	0	0
$755$	1.05116e7	0.671121
$756$	0	0
$757$	−1.88790e7	−1.19740	−0.598699	−	0.800974i	$-0.704315\pi$
$757$	−1.88790e7	−1.19740	−0.598699	+	0.800974i	$0.704315\pi$
$758$	0	0
$759$	−1.40901e7	−0.887786
$760$	0	0
$761$	9.74849e6	0.610205	0.305102	−	0.952320i	$-0.401309\pi$
$761$	9.74849e6	0.610205	0.305102	+	0.952320i	$0.401309\pi$
$762$	0	0
$763$	3.36399e6	0.209191
$764$	0	0
$765$	−1.68590e6	−0.104155
$766$	0	0
$767$	1.72473e6	0.105860
$768$	0	0
$769$	6.11920e6	0.373146	0.186573	−	0.982441i	$-0.440262\pi$
$769$	6.11920e6	0.373146	0.186573	+	0.982441i	$0.440262\pi$
$770$	0	0
$771$	6.18642e6	0.374804
$772$	0	0
$773$	1.47893e6	0.0890223	0.0445111	−	0.999009i	$-0.485827\pi$
$773$	1.47893e6	0.0890223	0.0445111	+	0.999009i	$0.485827\pi$
$774$	0	0
$775$	6.41408e6	0.383602
$776$	0	0
$777$	5.09143e6	0.302543
$778$	0	0
$779$	1.69391e7	1.00011
$780$	0	0
$781$	706226.	0.0414301
$782$	0	0
$783$	8.56609e6	0.499319
$784$	0	0
$785$	−1.20477e7	−0.697799
$786$	0	0
$787$	−2.39360e7	−1.37758	−0.688788	−	0.724963i	$-0.741857\pi$
$787$	−2.39360e7	−1.37758	−0.688788	+	0.724963i	$0.741857\pi$
$788$	0	0
$789$	1.72158e7	0.984542
$790$	0	0
$791$	3.07896e6	0.174970
$792$	0	0
$793$	−4.89291e6	−0.276302
$794$	0	0
$795$	−1.21391e7	−0.681192
$796$	0	0
$797$	6.21796e6	0.346738	0.173369	−	0.984857i	$-0.444535\pi$
$797$	6.21796e6	0.346738	0.173369	+	0.984857i	$0.444535\pi$
$798$	0	0
$799$	−3.51294e7	−1.94672
$800$	0	0
$801$	556032.	0.0306209
$802$	0	0
$803$	3.51239e6	0.192227
$804$	0	0
$805$	−2.83395e6	−0.154135
$806$	0	0
$807$	7.83847e6	0.423689
$808$	0	0
$809$	−4.98100e6	−0.267575	−0.133787	−	0.991010i	$-0.542714\pi$
$809$	−4.98100e6	−0.267575	−0.133787	+	0.991010i	$0.542714\pi$
$810$	0	0
$811$	3.09722e7	1.65356	0.826780	−	0.562525i	$-0.190170\pi$
$811$	3.09722e7	1.65356	0.826780	+	0.562525i	$0.190170\pi$
$812$	0	0
$813$	1.29645e7	0.687907
$814$	0	0
$815$	−7.05098e6	−0.371839
$816$	0	0
$817$	5.79381e7	3.03675
$818$	0	0
$819$	753315.	0.0392434
$820$	0	0
$821$	−2.87667e7	−1.48947	−0.744734	−	0.667361i	$-0.767424\pi$
$821$	−2.87667e7	−1.48947	−0.744734	+	0.667361i	$0.767424\pi$
$822$	0	0
$823$	−3.55713e7	−1.83063	−0.915313	−	0.402743i	$-0.868057\pi$
$823$	−3.55713e7	−1.83063	−0.915313	+	0.402743i	$0.868057\pi$
$824$	0	0
$825$	−3.80660e6	−0.194716
$826$	0	0
$827$	−3.01542e7	−1.53315	−0.766575	−	0.642155i	$-0.778040\pi$
$827$	−3.01542e7	−1.53315	−0.766575	+	0.642155i	$0.778040\pi$
$828$	0	0
$829$	−3.11697e7	−1.57524	−0.787618	−	0.616164i	$-0.788686\pi$
$829$	−3.11697e7	−1.57524	−0.787618	+	0.616164i	$0.788686\pi$
$830$	0	0
$831$	1.93537e7	0.972213
$832$	0	0
$833$	4.20540e6	0.209988
$834$	0	0
$835$	−3.29926e6	−0.163757
$836$	0	0
$837$	−4.13124e7	−2.03829
$838$	0	0
$839$	−3.63050e7	−1.78058	−0.890291	−	0.455393i	$-0.849499\pi$
$839$	−3.63050e7	−1.78058	−0.890291	+	0.455393i	$0.849499\pi$
$840$	0	0
$841$	−1.59831e7	−0.779238
$842$	0	0
$843$	1.07582e7	0.521402
$844$	0	0
$845$	−5.29625e6	−0.255168
$846$	0	0
$847$	−996812.	−0.0477425
$848$	0	0
$849$	1.17364e7	0.558812
$850$	0	0
$851$	−1.68095e7	−0.795666
$852$	0	0
$853$	1.00191e7	0.471472	0.235736	−	0.971817i	$-0.424250\pi$
$853$	1.00191e7	0.471472	0.235736	+	0.971817i	$0.424250\pi$
$854$	0	0
$855$	2.76656e6	0.129427
$856$	0	0
$857$	−1.17397e7	−0.546014	−0.273007	−	0.962012i	$-0.588018\pi$
$857$	−1.17397e7	−0.546014	−0.273007	+	0.962012i	$0.588018\pi$
$858$	0	0
$859$	1.69498e7	0.783756	0.391878	−	0.920017i	$-0.371825\pi$
$859$	1.69498e7	0.783756	0.391878	+	0.920017i	$0.371825\pi$
$860$	0	0
$861$	4.12962e6	0.189846
$862$	0	0
$863$	2.11590e7	0.967091	0.483546	−	0.875319i	$-0.339349\pi$
$863$	2.11590e7	0.967091	0.483546	+	0.875319i	$0.339349\pi$
$864$	0	0
$865$	1.38754e6	0.0630528
$866$	0	0
$867$	2.35664e7	1.06474
$868$	0	0
$869$	−3.90495e6	−0.175415
$870$	0	0
$871$	7.00131e6	0.312705
$872$	0	0
$873$	−2.39219e6	−0.106233
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	1.01611e7	0.446109	0.223055	−	0.974806i	$-0.428397\pi$
$877$	1.01611e7	0.446109	0.223055	+	0.974806i	$0.428397\pi$
$878$	0	0
$879$	−2.74907e7	−1.20009
$880$	0	0
$881$	−3.50223e7	−1.52021	−0.760106	−	0.649799i	$-0.774853\pi$
$881$	−3.50223e7	−1.52021	−0.760106	+	0.649799i	$0.774853\pi$
$882$	0	0
$883$	2.15600e7	0.930566	0.465283	−	0.885162i	$-0.345953\pi$
$883$	2.15600e7	0.930566	0.465283	+	0.885162i	$0.345953\pi$
$884$	0	0
$885$	1.54420e6	0.0662741
$886$	0	0
$887$	1.29121e7	0.551048	0.275524	−	0.961294i	$-0.411149\pi$
$887$	1.29121e7	0.551048	0.275524	+	0.961294i	$0.411149\pi$
$888$	0	0
$889$	−9.70438e6	−0.411826
$890$	0	0
$891$	2.05332e7	0.866486
$892$	0	0
$893$	5.76472e7	2.41907
$894$	0	0
$895$	−519207.	−0.0216662
$896$	0	0
$897$	1.32100e7	0.548180
$898$	0	0
$899$	−2.18379e7	−0.901181
$900$	0	0
$901$	−5.94727e7	−2.44065
$902$	0	0
$903$	1.41248e7	0.576453
$904$	0	0
$905$	−6.39666e6	−0.259616
$906$	0	0
$907$	−2.63522e7	−1.06365	−0.531825	−	0.846854i	$-0.678494\pi$
$907$	−2.63522e7	−1.06365	−0.531825	+	0.846854i	$0.678494\pi$
$908$	0	0
$909$	4.16346e6	0.167126
$910$	0	0
$911$	4.82227e6	0.192511	0.0962555	−	0.995357i	$-0.469313\pi$
$911$	4.82227e6	0.192511	0.0962555	+	0.995357i	$0.469313\pi$
$912$	0	0
$913$	4.05474e7	1.60985
$914$	0	0
$915$	−4.38076e6	−0.172980
$916$	0	0
$917$	−1.39715e7	−0.548680
$918$	0	0
$919$	121562.	0.00474798	0.00237399	−	0.999997i	$-0.499244\pi$
$919$	121562.	0.00474798	0.00237399	+	0.999997i	$0.499244\pi$
$920$	0	0
$921$	−3.19224e6	−0.124007
$922$	0	0
$923$	−662117.	−0.0255818
$924$	0	0
$925$	−4.54128e6	−0.174511
$926$	0	0
$927$	5.31450e6	0.203125
$928$	0	0
$929$	1.74856e7	0.664725	0.332362	−	0.943152i	$-0.392154\pi$
$929$	1.74856e7	0.664725	0.332362	+	0.943152i	$0.392154\pi$
$930$	0	0
$931$	−6.90104e6	−0.260940
$932$	0	0
$933$	−3.28646e7	−1.23602
$934$	0	0
$935$	−1.86495e7	−0.697650
$936$	0	0
$937$	−1.97882e7	−0.736304	−0.368152	−	0.929766i	$-0.620009\pi$
$937$	−1.97882e7	−0.736304	−0.368152	+	0.929766i	$0.620009\pi$
$938$	0	0
$939$	−2.32925e7	−0.862088
$940$	0	0
$941$	2.05948e7	0.758199	0.379100	−	0.925356i	$-0.376234\pi$
$941$	2.05948e7	0.758199	0.379100	+	0.925356i	$0.376234\pi$
$942$	0	0
$943$	−1.36340e7	−0.499282
$944$	0	0
$945$	4.93130e6	0.179631
$946$	0	0
$947$	−6.90803e6	−0.250311	−0.125155	−	0.992137i	$-0.539943\pi$
$947$	−6.90803e6	−0.250311	−0.125155	+	0.992137i	$0.539943\pi$
$948$	0	0
$949$	−3.29302e6	−0.118694
$950$	0	0
$951$	1.62480e7	0.582571
$952$	0	0
$953$	3.72040e7	1.32696	0.663480	−	0.748194i	$-0.269079\pi$
$953$	3.72040e7	1.32696	0.663480	+	0.748194i	$0.269079\pi$
$954$	0	0
$955$	−1.62315e7	−0.575905
$956$	0	0
$957$	1.29603e7	0.457440
$958$	0	0
$959$	−1.99420e7	−0.700199
$960$	0	0
$961$	7.66904e7	2.67875
$962$	0	0
$963$	238309.	0.00828085
$964$	0	0
$965$	−1.92160e7	−0.664272
$966$	0	0
$967$	−1.88878e7	−0.649555	−0.324778	−	0.945790i	$-0.605289\pi$
$967$	−1.88878e7	−0.649555	−0.324778	+	0.945790i	$0.605289\pi$
$968$	0	0
$969$	−7.19916e7	−2.46305
$970$	0	0
$971$	4.20566e7	1.43148	0.715741	−	0.698366i	$-0.246089\pi$
$971$	4.20566e7	1.43148	0.715741	+	0.698366i	$0.246089\pi$
$972$	0	0
$973$	662992.	0.0224505
$974$	0	0
$975$	3.56885e6	0.120231
$976$	0	0
$977$	4.61174e7	1.54571	0.772855	−	0.634583i	$-0.218828\pi$
$977$	4.61174e7	1.54571	0.772855	+	0.634583i	$0.218828\pi$
$978$	0	0
$979$	6.15083e6	0.205105
$980$	0	0
$981$	2.64324e6	0.0876928
$982$	0	0
$983$	−3.25827e7	−1.07548	−0.537741	−	0.843110i	$-0.680722\pi$
$983$	−3.25827e7	−1.07548	−0.537741	+	0.843110i	$0.680722\pi$
$984$	0	0
$985$	−4.15401e6	−0.136420
$986$	0	0
$987$	1.40539e7	0.459202
$988$	0	0
$989$	−4.66335e7	−1.51603
$990$	0	0
$991$	−1.68209e7	−0.544084	−0.272042	−	0.962285i	$-0.587699\pi$
$991$	−1.68209e7	−0.544084	−0.272042	+	0.962285i	$0.587699\pi$
$992$	0	0
$993$	5.39906e7	1.73758
$994$	0	0
$995$	1.86890e6	0.0598451
$996$	0	0
$997$	7.67044e6	0.244389	0.122195	−	0.992506i	$-0.461007\pi$
$997$	7.67044e6	0.244389	0.122195	+	0.992506i	$0.461007\pi$
$998$	0	0
$999$	2.92499e7	0.927279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.a.k.1.2		2
4.3	odd	2		70.6.a.h.1.1	✓	2
12.11	even	2		630.6.a.s.1.1		2
20.3	even	4		350.6.c.k.99.1		4
20.7	even	4		350.6.c.k.99.4		4
20.19	odd	2		350.6.a.p.1.2		2
28.27	even	2		490.6.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.h.1.1	✓	2	4.3	odd	2
350.6.a.p.1.2		2	20.19	odd	2
350.6.c.k.99.1		4	20.3	even	4
350.6.c.k.99.4		4	20.7	even	4
490.6.a.u.1.2		2	28.27	even	2
560.6.a.k.1.2		2	1.1	even	1	trivial
630.6.a.s.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q+14.3003 q^{3} +25.0000 q^{5} -49.0000 q^{7} -38.5015 q^{9} -425.904 q^{11} +399.303 q^{13} +357.507 q^{15} +1751.52 q^{17} -2874.23 q^{19} -700.715 q^{21} +2313.43 q^{23} +625.000 q^{25} -4025.56 q^{27} -2127.93 q^{29} +10262.5 q^{31} -6090.55 q^{33} -1225.00 q^{35} -7266.05 q^{37} +5710.16 q^{39} -5893.44 q^{41} -20157.7 q^{43} -962.537 q^{45} -20056.5 q^{47} +2401.00 q^{49} +25047.2 q^{51} -33954.9 q^{53} -10647.6 q^{55} -41102.4 q^{57} +4319.34 q^{59} -12253.6 q^{61} +1886.57 q^{63} +9982.58 q^{65} +17533.8 q^{67} +33082.7 q^{69} -1658.18 q^{71} -8246.91 q^{73} +8937.69 q^{75} +20869.3 q^{77} +9168.61 q^{79} -48210.8 q^{81} -95203.2 q^{83} +43788.0 q^{85} -30430.0 q^{87} -14441.8 q^{89} -19565.9 q^{91} +146757. q^{93} -71855.8 q^{95} +62132.4 q^{97} +16397.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{3} + 50 q^{5} - 98 q^{7} + 91 q^{9} - 415 q^{11} + 429 q^{13} - 125 q^{15} + 1319 q^{17} - 1918 q^{19} + 245 q^{21} + 1334 q^{23} + 1250 q^{25} - 1835 q^{27} - 1131 q^{29} + 5472 q^{31} - 6301 q^{33}+ \cdots + 17810 q^{99}+O(q^{100}) \) 2 * q - 5 * q^3 + 50 * q^5 - 98 * q^7 + 91 * q^9 - 415 * q^11 + 429 * q^13 - 125 * q^15 + 1319 * q^17 - 1918 * q^19 + 245 * q^21 + 1334 * q^23 + 1250 * q^25 - 1835 * q^27 - 1131 * q^29 + 5472 * q^31 - 6301 * q^33 - 2450 * q^35 - 9156 * q^37 + 5137 * q^39 - 7822 * q^41 - 2078 * q^43 + 2275 * q^45 + 8507 * q^47 + 4802 * q^49 + 33395 * q^51 - 34242 * q^53 - 10375 * q^55 - 59558 * q^57 - 6952 * q^59 - 45138 * q^61 - 4459 * q^63 + 10725 * q^65 + 54556 * q^67 + 51986 * q^69 + 62272 * q^71 + 40896 * q^73 - 3125 * q^75 + 20335 * q^77 - 62069 * q^79 - 121958 * q^81 - 189600 * q^83 + 32975 * q^85 - 49671 * q^87 + 64190 * q^89 - 21021 * q^91 + 239216 * q^93 - 47950 * q^95 + 155547 * q^97 + 17810 * q^99

Introduction

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