LMFDB - Embedded newform 280.6.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,6,Mod(1,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, 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("280.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	280.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:	$44.9074695476$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 791x^{3} + 280x^{2} + 24832x + 39040 $ x^5 - 2x^4 - 791x^3 + 280x^2 + 24832x + 39040
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.39132$ of defining polynomial
Character	$\chi$	$=$	280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.39132 q^{3} -25.0000 q^{5} -49.0000 q^{7} -213.934 q^{9} +O(q^{10})$	$q-5.39132 q^{3} -25.0000 q^{5} -49.0000 q^{7} -213.934 q^{9} -284.992 q^{11} -915.021 q^{13} +134.783 q^{15} +1008.73 q^{17} +2995.84 q^{19} +264.175 q^{21} -3921.15 q^{23} +625.000 q^{25} +2463.47 q^{27} -581.388 q^{29} -7689.26 q^{31} +1536.48 q^{33} +1225.00 q^{35} -7076.68 q^{37} +4933.17 q^{39} +14359.4 q^{41} +9125.17 q^{43} +5348.34 q^{45} +11266.7 q^{47} +2401.00 q^{49} -5438.39 q^{51} +22913.2 q^{53} +7124.81 q^{55} -16151.5 q^{57} -32940.2 q^{59} -4442.75 q^{61} +10482.8 q^{63} +22875.5 q^{65} +25402.7 q^{67} +21140.2 q^{69} +21764.9 q^{71} +80087.7 q^{73} -3369.57 q^{75} +13964.6 q^{77} +78210.5 q^{79} +38704.5 q^{81} -90710.5 q^{83} -25218.3 q^{85} +3134.45 q^{87} +25177.3 q^{89} +44836.1 q^{91} +41455.3 q^{93} -74896.0 q^{95} -50387.6 q^{97} +60969.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 - 125 * q^5 - 245 * q^7 + 372 * q^9	$ 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9} - 263 q^{11} - 729 q^{13} + 75 q^{15} - 1003 q^{17} - 2506 q^{19} + 147 q^{21} - 1066 q^{23} + 3125 q^{25} + 615 q^{27} + 3489 q^{29} + 7880 q^{31} + 4863 q^{33} + 6125 q^{35} + 13118 q^{37} + 27189 q^{39} + 23972 q^{41} + 3978 q^{43} - 9300 q^{45} + 9057 q^{47} + 12005 q^{49} + 96639 q^{51} + 2128 q^{53} + 6575 q^{55} + 61674 q^{57} - 15512 q^{59} + 4560 q^{61} - 18228 q^{63} + 18225 q^{65} + 7780 q^{67} + 126474 q^{69} + 32752 q^{71} + 189498 q^{73} - 1875 q^{75} + 12887 q^{77} + 42055 q^{79} + 294645 q^{81} + 58420 q^{83} + 25075 q^{85} - 765 q^{87} + 231324 q^{89} + 35721 q^{91} + 395736 q^{93} + 62650 q^{95} + 247569 q^{97} + 30606 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 - 125 * q^5 - 245 * q^7 + 372 * q^9 - 263 * q^11 - 729 * q^13 + 75 * q^15 - 1003 * q^17 - 2506 * q^19 + 147 * q^21 - 1066 * q^23 + 3125 * q^25 + 615 * q^27 + 3489 * q^29 + 7880 * q^31 + 4863 * q^33 + 6125 * q^35 + 13118 * q^37 + 27189 * q^39 + 23972 * q^41 + 3978 * q^43 - 9300 * q^45 + 9057 * q^47 + 12005 * q^49 + 96639 * q^51 + 2128 * q^53 + 6575 * q^55 + 61674 * q^57 - 15512 * q^59 + 4560 * q^61 - 18228 * q^63 + 18225 * q^65 + 7780 * q^67 + 126474 * q^69 + 32752 * q^71 + 189498 * q^73 - 1875 * q^75 + 12887 * q^77 + 42055 * q^79 + 294645 * q^81 + 58420 * q^83 + 25075 * q^85 - 765 * q^87 + 231324 * q^89 + 35721 * q^91 + 395736 * q^93 + 62650 * q^95 + 247569 * q^97 + 30606 * q^99

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
$3$	−5.39132	−0.345853	−0.172927	−	0.984935i	$-0.555322\pi$
$3$	−5.39132	−0.345853	−0.172927	+	0.984935i	$0.555322\pi$
$4$	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
$9$	−213.934	−0.880386
$10$	0	0
$11$	−284.992	−0.710152	−0.355076	−	0.934837i	$-0.615545\pi$
$11$	−284.992	−0.710152	−0.355076	+	0.934837i	$0.615545\pi$
$12$	0	0
$13$	−915.021	−1.50166	−0.750832	−	0.660493i	$-0.770347\pi$
$13$	−915.021	−1.50166	−0.750832	+	0.660493i	$0.770347\pi$
$14$	0	0
$15$	134.783	0.154670
$16$	0	0
$17$	1008.73	0.846551	0.423276	−	0.906001i	$-0.360880\pi$
$17$	1008.73	0.846551	0.423276	+	0.906001i	$0.360880\pi$
$18$	0	0
$19$	2995.84	1.90386	0.951929	−	0.306319i	$-0.0990975\pi$
$19$	2995.84	1.90386	0.951929	+	0.306319i	$0.0990975\pi$
$20$	0	0
$21$	264.175	0.130720
$22$	0	0
$23$	−3921.15	−1.54559	−0.772794	−	0.634657i	$-0.781142\pi$
$23$	−3921.15	−1.54559	−0.772794	+	0.634657i	$0.781142\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	2463.47	0.650337
$28$	0	0
$29$	−581.388	−0.128372	−0.0641861	−	0.997938i	$-0.520445\pi$
$29$	−581.388	−0.128372	−0.0641861	+	0.997938i	$0.520445\pi$
$30$	0	0
$31$	−7689.26	−1.43708	−0.718539	−	0.695486i	$-0.755189\pi$
$31$	−7689.26	−1.43708	−0.718539	+	0.695486i	$0.755189\pi$
$32$	0	0
$33$	1536.48	0.245608
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	−7076.68	−0.849816	−0.424908	−	0.905237i	$-0.639694\pi$
$37$	−7076.68	−0.849816	−0.424908	+	0.905237i	$0.639694\pi$
$38$	0	0
$39$	4933.17	0.519355
$40$	0	0
$41$	14359.4	1.33406	0.667032	−	0.745029i	$-0.267564\pi$
$41$	14359.4	1.33406	0.667032	+	0.745029i	$0.267564\pi$
$42$	0	0
$43$	9125.17	0.752610	0.376305	−	0.926496i	$-0.377195\pi$
$43$	9125.17	0.752610	0.376305	+	0.926496i	$0.377195\pi$
$44$	0	0
$45$	5348.34	0.393720
$46$	0	0
$47$	11266.7	0.743961	0.371981	−	0.928241i	$-0.378679\pi$
$47$	11266.7	0.743961	0.371981	+	0.928241i	$0.378679\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−5438.39	−0.292782
$52$	0	0
$53$	22913.2	1.12046	0.560229	−	0.828338i	$-0.310713\pi$
$53$	22913.2	1.12046	0.560229	+	0.828338i	$0.310713\pi$
$54$	0	0
$55$	7124.81	0.317590
$56$	0	0
$57$	−16151.5	−0.658455
$58$	0	0
$59$	−32940.2	−1.23196	−0.615979	−	0.787763i	$-0.711239\pi$
$59$	−32940.2	−1.23196	−0.615979	+	0.787763i	$0.711239\pi$
$60$	0	0
$61$	−4442.75	−0.152872	−0.0764359	−	0.997074i	$-0.524354\pi$
$61$	−4442.75	−0.152872	−0.0764359	+	0.997074i	$0.524354\pi$
$62$	0	0
$63$	10482.8	0.332754
$64$	0	0
$65$	22875.5	0.671565
$66$	0	0
$67$	25402.7	0.691342	0.345671	−	0.938356i	$-0.387651\pi$
$67$	25402.7	0.691342	0.345671	+	0.938356i	$0.387651\pi$
$68$	0	0
$69$	21140.2	0.534547
$70$	0	0
$71$	21764.9	0.512402	0.256201	−	0.966624i	$-0.417529\pi$
$71$	21764.9	0.512402	0.256201	+	0.966624i	$0.417529\pi$
$72$	0	0
$73$	80087.7	1.75897	0.879485	−	0.475926i	$-0.157887\pi$
$73$	80087.7	1.75897	0.879485	+	0.475926i	$0.157887\pi$
$74$	0	0
$75$	−3369.57	−0.0691706
$76$	0	0
$77$	13964.6	0.268412
$78$	0	0
$79$	78210.5	1.40993	0.704965	−	0.709243i	$-0.250963\pi$
$79$	78210.5	1.40993	0.704965	+	0.709243i	$0.250963\pi$
$80$	0	0
$81$	38704.5	0.655464
$82$	0	0
$83$	−90710.5	−1.44531	−0.722657	−	0.691207i	$-0.757079\pi$
$83$	−90710.5	−1.44531	−0.722657	+	0.691207i	$0.757079\pi$
$84$	0	0
$85$	−25218.3	−0.378589
$86$	0	0
$87$	3134.45	0.0443979
$88$	0	0
$89$	25177.3	0.336926	0.168463	−	0.985708i	$-0.446120\pi$
$89$	25177.3	0.336926	0.168463	+	0.985708i	$0.446120\pi$
$90$	0	0
$91$	44836.1	0.567576
$92$	0	0
$93$	41455.3	0.497018
$94$	0	0
$95$	−74896.0	−0.851431
$96$	0	0
$97$	−50387.6	−0.543743	−0.271872	−	0.962334i	$-0.587643\pi$
$97$	−50387.6	−0.543743	−0.271872	+	0.962334i	$0.587643\pi$
$98$	0	0
$99$	60969.5	0.625208
$100$	0	0
$101$	154075.	1.50289	0.751446	−	0.659795i	$-0.229357\pi$
$101$	154075.	1.50289	0.751446	+	0.659795i	$0.229357\pi$
$102$	0	0
$103$	−29489.1	−0.273885	−0.136942	−	0.990579i	$-0.543728\pi$
$103$	−29489.1	−0.273885	−0.136942	+	0.990579i	$0.543728\pi$
$104$	0	0
$105$	−6604.36	−0.0584599
$106$	0	0
$107$	7103.73	0.0599828	0.0299914	−	0.999550i	$-0.490452\pi$
$107$	7103.73	0.0599828	0.0299914	+	0.999550i	$0.490452\pi$
$108$	0	0
$109$	102173.	0.823703	0.411852	−	0.911251i	$-0.364882\pi$
$109$	102173.	0.823703	0.411852	+	0.911251i	$0.364882\pi$
$110$	0	0
$111$	38152.6	0.293912
$112$	0	0
$113$	33687.9	0.248186	0.124093	−	0.992271i	$-0.460398\pi$
$113$	33687.9	0.248186	0.124093	+	0.992271i	$0.460398\pi$
$114$	0	0
$115$	98028.8	0.691208
$116$	0	0
$117$	195754.	1.32204
$118$	0	0
$119$	−49427.8	−0.319966
$120$	0	0
$121$	−79830.4	−0.495684
$122$	0	0
$123$	−77416.1	−0.461391
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−126810.	−0.697662	−0.348831	−	0.937186i	$-0.613421\pi$
$127$	−126810.	−0.697662	−0.348831	+	0.937186i	$0.613421\pi$
$128$	0	0
$129$	−49196.7	−0.260292
$130$	0	0
$131$	−105884.	−0.539079	−0.269540	−	0.962989i	$-0.586872\pi$
$131$	−105884.	−0.539079	−0.269540	+	0.962989i	$0.586872\pi$
$132$	0	0
$133$	−146796.	−0.719591
$134$	0	0
$135$	−61586.9	−0.290840
$136$	0	0
$137$	308062.	1.40229	0.701144	−	0.713020i	$-0.252673\pi$
$137$	308062.	1.40229	0.701144	+	0.713020i	$0.252673\pi$
$138$	0	0
$139$	154365.	0.677658	0.338829	−	0.940848i	$-0.389969\pi$
$139$	154365.	0.677658	0.338829	+	0.940848i	$0.389969\pi$
$140$	0	0
$141$	−60742.1	−0.257301
$142$	0	0
$143$	260774.	1.06641
$144$	0	0
$145$	14534.7	0.0574098
$146$	0	0
$147$	−12944.6	−0.0494076
$148$	0	0
$149$	322762.	1.19101	0.595507	−	0.803350i	$-0.296951\pi$
$149$	322762.	1.19101	0.595507	+	0.803350i	$0.296951\pi$
$150$	0	0
$151$	−459185.	−1.63887	−0.819437	−	0.573169i	$-0.805714\pi$
$151$	−459185.	−1.63887	−0.819437	+	0.573169i	$0.805714\pi$
$152$	0	0
$153$	−215802.	−0.745292
$154$	0	0
$155$	192232.	0.642681
$156$	0	0
$157$	−604025.	−1.95571	−0.977857	−	0.209272i	$-0.932891\pi$
$157$	−604025.	−1.95571	−0.977857	+	0.209272i	$0.932891\pi$
$158$	0	0
$159$	−123532.	−0.387514
$160$	0	0
$161$	192136.	0.584178
$162$	0	0
$163$	−50783.9	−0.149712	−0.0748561	−	0.997194i	$-0.523850\pi$
$163$	−50783.9	−0.149712	−0.0748561	+	0.997194i	$0.523850\pi$
$164$	0	0
$165$	−38412.1	−0.109839
$166$	0	0
$167$	−417011.	−1.15706	−0.578530	−	0.815661i	$-0.696373\pi$
$167$	−417011.	−1.15706	−0.578530	+	0.815661i	$0.696373\pi$
$168$	0	0
$169$	465971.	1.25500
$170$	0	0
$171$	−640911.	−1.67613
$172$	0	0
$173$	−370255.	−0.940559	−0.470279	−	0.882518i	$-0.655847\pi$
$173$	−370255.	−0.940559	−0.470279	+	0.882518i	$0.655847\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	177591.	0.426076
$178$	0	0
$179$	4700.87	0.0109659	0.00548297	−	0.999985i	$-0.498255\pi$
$179$	4700.87	0.0109659	0.00548297	+	0.999985i	$0.498255\pi$
$180$	0	0
$181$	607645.	1.37865	0.689324	−	0.724453i	$-0.257908\pi$
$181$	607645.	1.37865	0.689324	+	0.724453i	$0.257908\pi$
$182$	0	0
$183$	23952.3	0.0528712
$184$	0	0
$185$	176917.	0.380049
$186$	0	0
$187$	−287481.	−0.601180
$188$	0	0
$189$	−120710.	−0.245804
$190$	0	0
$191$	654723.	1.29860	0.649298	−	0.760534i	$-0.275063\pi$
$191$	654723.	1.29860	0.649298	+	0.760534i	$0.275063\pi$
$192$	0	0
$193$	345897.	0.668426	0.334213	−	0.942498i	$-0.391529\pi$
$193$	345897.	0.668426	0.334213	+	0.942498i	$0.391529\pi$
$194$	0	0
$195$	−123329.	−0.232263
$196$	0	0
$197$	−509172.	−0.934757	−0.467378	−	0.884057i	$-0.654801\pi$
$197$	−509172.	−0.934757	−0.467378	+	0.884057i	$0.654801\pi$
$198$	0	0
$199$	−467290.	−0.836477	−0.418239	−	0.908337i	$-0.637352\pi$
$199$	−467290.	−0.836477	−0.418239	+	0.908337i	$0.637352\pi$
$200$	0	0
$201$	−136954.	−0.239103
$202$	0	0
$203$	28488.0	0.0485201
$204$	0	0
$205$	−358985.	−0.596612
$206$	0	0
$207$	838866.	1.36071
$208$	0	0
$209$	−853791.	−1.35203
$210$	0	0
$211$	−613859.	−0.949210	−0.474605	−	0.880199i	$-0.657409\pi$
$211$	−613859.	−0.949210	−0.474605	+	0.880199i	$0.657409\pi$
$212$	0	0
$213$	−117341.	−0.177216
$214$	0	0
$215$	−228129.	−0.336577
$216$	0	0
$217$	376774.	0.543165
$218$	0	0
$219$	−431778.	−0.608345
$220$	0	0
$221$	−923011.	−1.27124
$222$	0	0
$223$	−286299.	−0.385529	−0.192765	−	0.981245i	$-0.561745\pi$
$223$	−286299.	−0.385529	−0.192765	+	0.981245i	$0.561745\pi$
$224$	0	0
$225$	−133709.	−0.176077
$226$	0	0
$227$	−920931.	−1.18621	−0.593106	−	0.805124i	$-0.702099\pi$
$227$	−920931.	−1.18621	−0.593106	+	0.805124i	$0.702099\pi$
$228$	0	0
$229$	1.27010e6	1.60048	0.800240	−	0.599680i	$-0.204705\pi$
$229$	1.27010e6	1.60048	0.800240	+	0.599680i	$0.204705\pi$
$230$	0	0
$231$	−75287.7	−0.0928313
$232$	0	0
$233$	1.29745e6	1.56567	0.782835	−	0.622229i	$-0.213773\pi$
$233$	1.29745e6	1.56567	0.782835	+	0.622229i	$0.213773\pi$
$234$	0	0
$235$	−281666.	−0.332709
$236$	0	0
$237$	−421658.	−0.487628
$238$	0	0
$239$	−624070.	−0.706706	−0.353353	−	0.935490i	$-0.614959\pi$
$239$	−624070.	−0.706706	−0.353353	+	0.935490i	$0.614959\pi$
$240$	0	0
$241$	970316.	1.07614	0.538072	−	0.842899i	$-0.319153\pi$
$241$	970316.	1.07614	0.538072	+	0.842899i	$0.319153\pi$
$242$	0	0
$243$	−807293.	−0.877032
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	−2.74126e6	−2.85896
$248$	0	0
$249$	489049.	0.499867
$250$	0	0
$251$	−1.09826e6	−1.10032	−0.550161	−	0.835059i	$-0.685434\pi$
$251$	−1.09826e6	−1.10032	−0.550161	+	0.835059i	$0.685434\pi$
$252$	0	0
$253$	1.11750e6	1.09760
$254$	0	0
$255$	135960.	0.130936
$256$	0	0
$257$	436773.	0.412500	0.206250	−	0.978499i	$-0.433874\pi$
$257$	436773.	0.412500	0.206250	+	0.978499i	$0.433874\pi$
$258$	0	0
$259$	346757.	0.321200
$260$	0	0
$261$	124378.	0.113017
$262$	0	0
$263$	1.11673e6	0.995540	0.497770	−	0.867309i	$-0.334152\pi$
$263$	1.11673e6	0.995540	0.497770	+	0.867309i	$0.334152\pi$
$264$	0	0
$265$	−572829.	−0.501084
$266$	0	0
$267$	−135739.	−0.116527
$268$	0	0
$269$	100175.	0.0844073	0.0422037	−	0.999109i	$-0.486562\pi$
$269$	100175.	0.0844073	0.0422037	+	0.999109i	$0.486562\pi$
$270$	0	0
$271$	1.97803e6	1.63610	0.818051	−	0.575146i	$-0.195055\pi$
$271$	1.97803e6	1.63610	0.818051	+	0.575146i	$0.195055\pi$
$272$	0	0
$273$	−241725.	−0.196298
$274$	0	0
$275$	−178120.	−0.142030
$276$	0	0
$277$	−2.18213e6	−1.70876	−0.854381	−	0.519646i	$-0.826064\pi$
$277$	−2.18213e6	−1.70876	−0.854381	+	0.519646i	$0.826064\pi$
$278$	0	0
$279$	1.64499e6	1.26518
$280$	0	0
$281$	−1.97760e6	−1.49408	−0.747039	−	0.664780i	$-0.768525\pi$
$281$	−1.97760e6	−1.49408	−0.747039	+	0.664780i	$0.768525\pi$
$282$	0	0
$283$	−1.53027e6	−1.13580	−0.567900	−	0.823097i	$-0.692244\pi$
$283$	−1.53027e6	−1.13580	−0.567900	+	0.823097i	$0.692244\pi$
$284$	0	0
$285$	403788.	0.294470
$286$	0	0
$287$	−703611.	−0.504229
$288$	0	0
$289$	−402318.	−0.283351
$290$	0	0
$291$	271655.	0.188055
$292$	0	0
$293$	−1.01426e6	−0.690205	−0.345103	−	0.938565i	$-0.612156\pi$
$293$	−1.01426e6	−0.690205	−0.345103	+	0.938565i	$0.612156\pi$
$294$	0	0
$295$	823504.	0.550948
$296$	0	0
$297$	−702071.	−0.461839
$298$	0	0
$299$	3.58794e6	2.32096
$300$	0	0
$301$	−447133.	−0.284460
$302$	0	0
$303$	−830665.	−0.519780
$304$	0	0
$305$	111069.	0.0683664
$306$	0	0
$307$	−436639.	−0.264409	−0.132205	−	0.991222i	$-0.542206\pi$
$307$	−436639.	−0.264409	−0.132205	+	0.991222i	$0.542206\pi$
$308$	0	0
$309$	158985.	0.0947239
$310$	0	0
$311$	1.94043e6	1.13762	0.568809	−	0.822469i	$-0.307404\pi$
$311$	1.94043e6	1.13762	0.568809	+	0.822469i	$0.307404\pi$
$312$	0	0
$313$	−1.11477e6	−0.643166	−0.321583	−	0.946881i	$-0.604215\pi$
$313$	−1.11477e6	−0.643166	−0.321583	+	0.946881i	$0.604215\pi$
$314$	0	0
$315$	−262069.	−0.148812
$316$	0	0
$317$	2.82079e6	1.57660	0.788302	−	0.615288i	$-0.210960\pi$
$317$	2.82079e6	1.57660	0.788302	+	0.615288i	$0.210960\pi$
$318$	0	0
$319$	165691.	0.0911638
$320$	0	0
$321$	−38298.5	−0.0207453
$322$	0	0
$323$	3.02200e6	1.61171
$324$	0	0
$325$	−571888.	−0.300333
$326$	0	0
$327$	−550848.	−0.284880
$328$	0	0
$329$	−552066.	−0.281191
$330$	0	0
$331$	2.95822e6	1.48409	0.742045	−	0.670350i	$-0.233856\pi$
$331$	2.95822e6	1.48409	0.742045	+	0.670350i	$0.233856\pi$
$332$	0	0
$333$	1.51394e6	0.748166
$334$	0	0
$335$	−635068.	−0.309178
$336$	0	0
$337$	−1.01511e6	−0.486896	−0.243448	−	0.969914i	$-0.578279\pi$
$337$	−1.01511e6	−0.486896	−0.243448	+	0.969914i	$0.578279\pi$
$338$	0	0
$339$	−181622.	−0.0858361
$340$	0	0
$341$	2.19138e6	1.02054
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−528504.	−0.239057
$346$	0	0
$347$	723269.	0.322460	0.161230	−	0.986917i	$-0.448454\pi$
$347$	723269.	0.322460	0.161230	+	0.986917i	$0.448454\pi$
$348$	0	0
$349$	1.45403e6	0.639014	0.319507	−	0.947584i	$-0.396483\pi$
$349$	1.45403e6	0.639014	0.319507	+	0.947584i	$0.396483\pi$
$350$	0	0
$351$	−2.25413e6	−0.976588
$352$	0	0
$353$	2.92578e6	1.24970	0.624849	−	0.780746i	$-0.285161\pi$
$353$	2.92578e6	1.24970	0.624849	+	0.780746i	$0.285161\pi$
$354$	0	0
$355$	−544122.	−0.229153
$356$	0	0
$357$	266481.	0.110661
$358$	0	0
$359$	2.61108e6	1.06926	0.534631	−	0.845085i	$-0.320450\pi$
$359$	2.61108e6	1.06926	0.534631	+	0.845085i	$0.320450\pi$
$360$	0	0
$361$	6.49895e6	2.62467
$362$	0	0
$363$	430391.	0.171434
$364$	0	0
$365$	−2.00219e6	−0.786635
$366$	0	0
$367$	1.84927e6	0.716697	0.358349	−	0.933588i	$-0.383340\pi$
$367$	1.84927e6	0.716697	0.358349	+	0.933588i	$0.383340\pi$
$368$	0	0
$369$	−3.07196e6	−1.17449
$370$	0	0
$371$	−1.12275e6	−0.423493
$372$	0	0
$373$	−2.95585e6	−1.10004	−0.550022	−	0.835150i	$-0.685381\pi$
$373$	−2.95585e6	−1.10004	−0.550022	+	0.835150i	$0.685381\pi$
$374$	0	0
$375$	84239.3	0.0309340
$376$	0	0
$377$	531982.	0.192772
$378$	0	0
$379$	3.31928e6	1.18699	0.593494	−	0.804839i	$-0.297748\pi$
$379$	3.31928e6	1.18699	0.593494	+	0.804839i	$0.297748\pi$
$380$	0	0
$381$	683675.	0.241289
$382$	0	0
$383$	4.01556e6	1.39878	0.699390	−	0.714740i	$-0.253455\pi$
$383$	4.01556e6	1.39878	0.699390	+	0.714740i	$0.253455\pi$
$384$	0	0
$385$	−349116.	−0.120038
$386$	0	0
$387$	−1.95218e6	−0.662587
$388$	0	0
$389$	2.74600e6	0.920081	0.460041	−	0.887898i	$-0.347835\pi$
$389$	2.74600e6	0.920081	0.460041	+	0.887898i	$0.347835\pi$
$390$	0	0
$391$	−3.95539e6	−1.30842
$392$	0	0
$393$	570855.	0.186442
$394$	0	0
$395$	−1.95526e6	−0.630539
$396$	0	0
$397$	−3.44483e6	−1.09696	−0.548481	−	0.836163i	$-0.684794\pi$
$397$	−3.44483e6	−1.09696	−0.548481	+	0.836163i	$0.684794\pi$
$398$	0	0
$399$	791424.	0.248873
$400$	0	0
$401$	−212934.	−0.0661278	−0.0330639	−	0.999453i	$-0.510526\pi$
$401$	−212934.	−0.0661278	−0.0330639	+	0.999453i	$0.510526\pi$
$402$	0	0
$403$	7.03584e6	2.15801
$404$	0	0
$405$	−967613.	−0.293133
$406$	0	0
$407$	2.01680e6	0.603499
$408$	0	0
$409$	−2.94632e6	−0.870906	−0.435453	−	0.900211i	$-0.643412\pi$
$409$	−2.94632e6	−0.870906	−0.435453	+	0.900211i	$0.643412\pi$
$410$	0	0
$411$	−1.66086e6	−0.484986
$412$	0	0
$413$	1.61407e6	0.465636
$414$	0	0
$415$	2.26776e6	0.646364
$416$	0	0
$417$	−832229.	−0.234370
$418$	0	0
$419$	−371737.	−0.103443	−0.0517215	−	0.998662i	$-0.516471\pi$
$419$	−371737.	−0.103443	−0.0517215	+	0.998662i	$0.516471\pi$
$420$	0	0
$421$	−2.00473e6	−0.551254	−0.275627	−	0.961265i	$-0.588885\pi$
$421$	−2.00473e6	−0.551254	−0.275627	+	0.961265i	$0.588885\pi$
$422$	0	0
$423$	−2.41032e6	−0.654973
$424$	0	0
$425$	630457.	0.169310
$426$	0	0
$427$	217695.	0.0577801
$428$	0	0
$429$	−1.40592e6	−0.368821
$430$	0	0
$431$	1.66230e6	0.431038	0.215519	−	0.976500i	$-0.430856\pi$
$431$	1.66230e6	0.431038	0.215519	+	0.976500i	$0.430856\pi$
$432$	0	0
$433$	5.19092e6	1.33053	0.665265	−	0.746607i	$-0.268318\pi$
$433$	5.19092e6	1.33053	0.665265	+	0.746607i	$0.268318\pi$
$434$	0	0
$435$	−78361.1	−0.0198554
$436$	0	0
$437$	−1.17471e7	−2.94258
$438$	0	0
$439$	4.69324e6	1.16228	0.581141	−	0.813803i	$-0.302606\pi$
$439$	4.69324e6	1.16228	0.581141	+	0.813803i	$0.302606\pi$
$440$	0	0
$441$	−513655.	−0.125769
$442$	0	0
$443$	6.01295e6	1.45572	0.727861	−	0.685724i	$-0.240515\pi$
$443$	6.01295e6	1.45572	0.727861	+	0.685724i	$0.240515\pi$
$444$	0	0
$445$	−629433.	−0.150678
$446$	0	0
$447$	−1.74011e6	−0.411916
$448$	0	0
$449$	3.87731e6	0.907641	0.453821	−	0.891093i	$-0.350061\pi$
$449$	3.87731e6	0.907641	0.453821	+	0.891093i	$0.350061\pi$
$450$	0	0
$451$	−4.09232e6	−0.947389
$452$	0	0
$453$	2.47561e6	0.566810
$454$	0	0
$455$	−1.12090e6	−0.253828
$456$	0	0
$457$	−790341.	−0.177021	−0.0885104	−	0.996075i	$-0.528211\pi$
$457$	−790341.	−0.177021	−0.0885104	+	0.996075i	$0.528211\pi$
$458$	0	0
$459$	2.48498e6	0.550544
$460$	0	0
$461$	−6.10556e6	−1.33805	−0.669026	−	0.743239i	$-0.733289\pi$
$461$	−6.10556e6	−1.33805	−0.669026	+	0.743239i	$0.733289\pi$
$462$	0	0
$463$	−7.36944e6	−1.59765	−0.798825	−	0.601563i	$-0.794545\pi$
$463$	−7.36944e6	−1.59765	−0.798825	+	0.601563i	$0.794545\pi$
$464$	0	0
$465$	−1.03638e6	−0.222273
$466$	0	0
$467$	−3.09713e6	−0.657155	−0.328577	−	0.944477i	$-0.606569\pi$
$467$	−3.09713e6	−0.657155	−0.328577	+	0.944477i	$0.606569\pi$
$468$	0	0
$469$	−1.24473e6	−0.261303
$470$	0	0
$471$	3.25649e6	0.676390
$472$	0	0
$473$	−2.60060e6	−0.534468
$474$	0	0
$475$	1.87240e6	0.380772
$476$	0	0
$477$	−4.90190e6	−0.986435
$478$	0	0
$479$	6.64063e6	1.32242	0.661212	−	0.750199i	$-0.270042\pi$
$479$	6.64063e6	1.32242	0.661212	+	0.750199i	$0.270042\pi$
$480$	0	0
$481$	6.47531e6	1.27614
$482$	0	0
$483$	−1.03587e6	−0.202040
$484$	0	0
$485$	1.25969e6	0.243169
$486$	0	0
$487$	832952.	0.159147	0.0795734	−	0.996829i	$-0.474644\pi$
$487$	832952.	0.159147	0.0795734	+	0.996829i	$0.474644\pi$
$488$	0	0
$489$	273792.	0.0517785
$490$	0	0
$491$	3.80367e6	0.712031	0.356016	−	0.934480i	$-0.384135\pi$
$491$	3.80367e6	0.712031	0.356016	+	0.934480i	$0.384135\pi$
$492$	0	0
$493$	−586464.	−0.108674
$494$	0	0
$495$	−1.52424e6	−0.279601
$496$	0	0
$497$	−1.06648e6	−0.193670
$498$	0	0
$499$	944675.	0.169836	0.0849182	−	0.996388i	$-0.472937\pi$
$499$	944675.	0.169836	0.0849182	+	0.996388i	$0.472937\pi$
$500$	0	0
$501$	2.24824e6	0.400173
$502$	0	0
$503$	−1.72555e6	−0.304094	−0.152047	−	0.988373i	$-0.548587\pi$
$503$	−1.72555e6	−0.304094	−0.152047	+	0.988373i	$0.548587\pi$
$504$	0	0
$505$	−3.85186e6	−0.672113
$506$	0	0
$507$	−2.51220e6	−0.434044
$508$	0	0
$509$	−449727.	−0.0769405	−0.0384702	−	0.999260i	$-0.512248\pi$
$509$	−449727.	−0.0769405	−0.0384702	+	0.999260i	$0.512248\pi$
$510$	0	0
$511$	−3.92430e6	−0.664828
$512$	0	0
$513$	7.38017e6	1.23815
$514$	0	0
$515$	737227.	0.122485
$516$	0	0
$517$	−3.21091e6	−0.528326
$518$	0	0
$519$	1.99616e6	0.325295
$520$	0	0
$521$	789610.	0.127444	0.0637218	−	0.997968i	$-0.479703\pi$
$521$	789610.	0.127444	0.0637218	+	0.997968i	$0.479703\pi$
$522$	0	0
$523$	−1.29322e6	−0.206737	−0.103369	−	0.994643i	$-0.532962\pi$
$523$	−1.29322e6	−0.206737	−0.103369	+	0.994643i	$0.532962\pi$
$524$	0	0
$525$	165109.	0.0261440
$526$	0	0
$527$	−7.75640e6	−1.21656
$528$	0	0
$529$	8.93908e6	1.38884
$530$	0	0
$531$	7.04701e6	1.08460
$532$	0	0
$533$	−1.31392e7	−2.00332
$534$	0	0
$535$	−177593.	−0.0268251
$536$	0	0
$537$	−25343.9	−0.00379260
$538$	0	0
$539$	−684267.	−0.101450
$540$	0	0
$541$	−1.07872e7	−1.58458	−0.792292	−	0.610142i	$-0.791112\pi$
$541$	−1.07872e7	−1.58458	−0.792292	+	0.610142i	$0.791112\pi$
$542$	0	0
$543$	−3.27601e6	−0.476810
$544$	0	0
$545$	−2.55433e6	−0.368371
$546$	0	0
$547$	−7.96568e6	−1.13829	−0.569147	−	0.822236i	$-0.692727\pi$
$547$	−7.96568e6	−1.13829	−0.569147	+	0.822236i	$0.692727\pi$
$548$	0	0
$549$	950454.	0.134586
$550$	0	0
$551$	−1.74174e6	−0.244402
$552$	0	0
$553$	−3.83231e6	−0.532903
$554$	0	0
$555$	−953815.	−0.131441
$556$	0	0
$557$	6.32381e6	0.863657	0.431828	−	0.901956i	$-0.357869\pi$
$557$	6.32381e6	0.863657	0.431828	+	0.901956i	$0.357869\pi$
$558$	0	0
$559$	−8.34973e6	−1.13017
$560$	0	0
$561$	1.54990e6	0.207920
$562$	0	0
$563$	−4.18746e6	−0.556775	−0.278387	−	0.960469i	$-0.589800\pi$
$563$	−4.18746e6	−0.556775	−0.278387	+	0.960469i	$0.589800\pi$
$564$	0	0
$565$	−842198.	−0.110992
$566$	0	0
$567$	−1.89652e6	−0.247742
$568$	0	0
$569$	9.26185e6	1.19927	0.599635	−	0.800274i	$-0.295312\pi$
$569$	9.26185e6	1.19927	0.599635	+	0.800274i	$0.295312\pi$
$570$	0	0
$571$	3.55620e6	0.456453	0.228227	−	0.973608i	$-0.426707\pi$
$571$	3.55620e6	0.456453	0.228227	+	0.973608i	$0.426707\pi$
$572$	0	0
$573$	−3.52982e6	−0.449123
$574$	0	0
$575$	−2.45072e6	−0.309118
$576$	0	0
$577$	−3.76112e6	−0.470303	−0.235151	−	0.971959i	$-0.575559\pi$
$577$	−3.76112e6	−0.470303	−0.235151	+	0.971959i	$0.575559\pi$
$578$	0	0
$579$	−1.86484e6	−0.231177
$580$	0	0
$581$	4.44482e6	0.546278
$582$	0	0
$583$	−6.53008e6	−0.795696
$584$	0	0
$585$	−4.89385e6	−0.591236
$586$	0	0
$587$	−4.88290e6	−0.584901	−0.292450	−	0.956281i	$-0.594471\pi$
$587$	−4.88290e6	−0.584901	−0.292450	+	0.956281i	$0.594471\pi$
$588$	0	0
$589$	−2.30358e7	−2.73599
$590$	0	0
$591$	2.74511e6	0.323289
$592$	0	0
$593$	1.03655e7	1.21047	0.605233	−	0.796049i	$-0.293080\pi$
$593$	1.03655e7	1.21047	0.605233	+	0.796049i	$0.293080\pi$
$594$	0	0
$595$	1.23570e6	0.143093
$596$	0	0
$597$	2.51931e6	0.289298
$598$	0	0
$599$	1.52765e7	1.73963	0.869815	−	0.493378i	$-0.164238\pi$
$599$	1.52765e7	1.73963	0.869815	+	0.493378i	$0.164238\pi$
$600$	0	0
$601$	−8.53778e6	−0.964181	−0.482091	−	0.876121i	$-0.660122\pi$
$601$	−8.53778e6	−0.964181	−0.482091	+	0.876121i	$0.660122\pi$
$602$	0	0
$603$	−5.43450e6	−0.608648
$604$	0	0
$605$	1.99576e6	0.221677
$606$	0	0
$607$	−88828.1	−0.00978540	−0.00489270	−	0.999988i	$-0.501557\pi$
$607$	−88828.1	−0.00978540	−0.00489270	+	0.999988i	$0.501557\pi$
$608$	0	0
$609$	−153588.	−0.0167808
$610$	0	0
$611$	−1.03092e7	−1.11718
$612$	0	0
$613$	1.10099e7	1.18340	0.591699	−	0.806159i	$-0.298458\pi$
$613$	1.10099e7	1.18340	0.591699	+	0.806159i	$0.298458\pi$
$614$	0	0
$615$	1.93540e6	0.206340
$616$	0	0
$617$	6.32250e6	0.668614	0.334307	−	0.942464i	$-0.391498\pi$
$617$	6.32250e6	0.668614	0.334307	+	0.942464i	$0.391498\pi$
$618$	0	0
$619$	5.53925e6	0.581065	0.290532	−	0.956865i	$-0.406168\pi$
$619$	5.53925e6	0.581065	0.290532	+	0.956865i	$0.406168\pi$
$620$	0	0
$621$	−9.65965e6	−1.00515
$622$	0	0
$623$	−1.23369e6	−0.127346
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	4.60306e6	0.467603
$628$	0	0
$629$	−7.13847e6	−0.719413
$630$	0	0
$631$	7.42573e6	0.742448	0.371224	−	0.928543i	$-0.378938\pi$
$631$	7.42573e6	0.742448	0.371224	+	0.928543i	$0.378938\pi$
$632$	0	0
$633$	3.30951e6	0.328287
$634$	0	0
$635$	3.17026e6	0.312004
$636$	0	0
$637$	−2.19697e6	−0.214523
$638$	0	0
$639$	−4.65624e6	−0.451111
$640$	0	0
$641$	1.55803e7	1.49772	0.748861	−	0.662727i	$-0.230601\pi$
$641$	1.55803e7	1.49772	0.748861	+	0.662727i	$0.230601\pi$
$642$	0	0
$643$	−1.23417e7	−1.17719	−0.588595	−	0.808428i	$-0.700319\pi$
$643$	−1.23417e7	−1.17719	−0.588595	+	0.808428i	$0.700319\pi$
$644$	0	0
$645$	1.22992e6	0.116406
$646$	0	0
$647$	95855.2	0.00900233	0.00450117	−	0.999990i	$-0.498567\pi$
$647$	95855.2	0.00900233	0.00450117	+	0.999990i	$0.498567\pi$
$648$	0	0
$649$	9.38769e6	0.874877
$650$	0	0
$651$	−2.03131e6	−0.187855
$652$	0	0
$653$	1.93372e7	1.77464	0.887320	−	0.461154i	$-0.152564\pi$
$653$	1.93372e7	1.77464	0.887320	+	0.461154i	$0.152564\pi$
$654$	0	0
$655$	2.64710e6	0.241084
$656$	0	0
$657$	−1.71334e7	−1.54857
$658$	0	0
$659$	1.21349e7	1.08848	0.544241	−	0.838929i	$-0.316818\pi$
$659$	1.21349e7	1.08848	0.544241	+	0.838929i	$0.316818\pi$
$660$	0	0
$661$	−1.07221e7	−0.954504	−0.477252	−	0.878767i	$-0.658367\pi$
$661$	−1.07221e7	−0.954504	−0.477252	+	0.878767i	$0.658367\pi$
$662$	0	0
$663$	4.97625e6	0.439661
$664$	0	0
$665$	3.66990e6	0.321811
$666$	0	0
$667$	2.27971e6	0.198411
$668$	0	0
$669$	1.54353e6	0.133337
$670$	0	0
$671$	1.26615e6	0.108562
$672$	0	0
$673$	3.15152e6	0.268215	0.134107	−	0.990967i	$-0.457183\pi$
$673$	3.15152e6	0.268215	0.134107	+	0.990967i	$0.457183\pi$
$674$	0	0
$675$	1.53967e6	0.130067
$676$	0	0
$677$	2.98568e6	0.250364	0.125182	−	0.992134i	$-0.460049\pi$
$677$	2.98568e6	0.250364	0.125182	+	0.992134i	$0.460049\pi$
$678$	0	0
$679$	2.46899e6	0.205516
$680$	0	0
$681$	4.96503e6	0.410255
$682$	0	0
$683$	−6.64218e6	−0.544828	−0.272414	−	0.962180i	$-0.587822\pi$
$683$	−6.64218e6	−0.544828	−0.272414	+	0.962180i	$0.587822\pi$
$684$	0	0
$685$	−7.70156e6	−0.627122
$686$	0	0
$687$	−6.84753e6	−0.553531
$688$	0	0
$689$	−2.09660e7	−1.68255
$690$	0	0
$691$	−1.47761e7	−1.17724	−0.588622	−	0.808409i	$-0.700329\pi$
$691$	−1.47761e7	−1.17724	−0.588622	+	0.808409i	$0.700329\pi$
$692$	0	0
$693$	−2.98750e6	−0.236306
$694$	0	0
$695$	−3.85912e6	−0.303058
$696$	0	0
$697$	1.44848e7	1.12935
$698$	0	0
$699$	−6.99495e6	−0.541492
$700$	0	0
$701$	1.33502e7	1.02611	0.513053	−	0.858357i	$-0.328514\pi$
$701$	1.33502e7	1.02611	0.513053	+	0.858357i	$0.328514\pi$
$702$	0	0
$703$	−2.12006e7	−1.61793
$704$	0	0
$705$	1.51855e6	0.115069
$706$	0	0
$707$	−7.54966e6	−0.568040
$708$	0	0
$709$	−5.09670e6	−0.380780	−0.190390	−	0.981709i	$-0.560975\pi$
$709$	−5.09670e6	−0.380780	−0.190390	+	0.981709i	$0.560975\pi$
$710$	0	0
$711$	−1.67319e7	−1.24128
$712$	0	0
$713$	3.01508e7	2.22113
$714$	0	0
$715$	−6.51935e6	−0.476913
$716$	0	0
$717$	3.36456e6	0.244416
$718$	0	0
$719$	1.23855e7	0.893494	0.446747	−	0.894660i	$-0.352582\pi$
$719$	1.23855e7	0.893494	0.446747	+	0.894660i	$0.352582\pi$
$720$	0	0
$721$	1.44496e6	0.103519
$722$	0	0
$723$	−5.23128e6	−0.372188
$724$	0	0
$725$	−363367.	−0.0256744
$726$	0	0
$727$	−2.10781e7	−1.47909	−0.739547	−	0.673104i	$-0.764960\pi$
$727$	−2.10781e7	−1.47909	−0.739547	+	0.673104i	$0.764960\pi$
$728$	0	0
$729$	−5.05283e6	−0.352140
$730$	0	0
$731$	9.20485e6	0.637123
$732$	0	0
$733$	1.56667e7	1.07700	0.538501	−	0.842625i	$-0.318991\pi$
$733$	1.56667e7	1.07700	0.538501	+	0.842625i	$0.318991\pi$
$734$	0	0
$735$	323614.	0.0220957
$736$	0	0
$737$	−7.23958e6	−0.490958
$738$	0	0
$739$	1.25419e7	0.844796	0.422398	−	0.906411i	$-0.361189\pi$
$739$	1.25419e7	0.844796	0.422398	+	0.906411i	$0.361189\pi$
$740$	0	0
$741$	1.47790e7	0.988779
$742$	0	0
$743$	2.15075e7	1.42928	0.714642	−	0.699490i	$-0.246590\pi$
$743$	2.15075e7	1.42928	0.714642	+	0.699490i	$0.246590\pi$
$744$	0	0
$745$	−8.06905e6	−0.532637
$746$	0	0
$747$	1.94060e7	1.27243
$748$	0	0
$749$	−348083.	−0.0226714
$750$	0	0
$751$	−761944.	−0.0492973	−0.0246487	−	0.999696i	$-0.507847\pi$
$751$	−761944.	−0.0492973	−0.0246487	+	0.999696i	$0.507847\pi$
$752$	0	0
$753$	5.92105e6	0.380550
$754$	0	0
$755$	1.14796e7	0.732927
$756$	0	0
$757$	7.51459e6	0.476612	0.238306	−	0.971190i	$-0.423408\pi$
$757$	7.51459e6	0.476612	0.238306	+	0.971190i	$0.423408\pi$
$758$	0	0
$759$	−6.02479e6	−0.379610
$760$	0	0
$761$	2.68820e7	1.68267	0.841336	−	0.540513i	$-0.181770\pi$
$761$	2.68820e7	1.68267	0.841336	+	0.540513i	$0.181770\pi$
$762$	0	0
$763$	−5.00649e6	−0.311331
$764$	0	0
$765$	5.39504e6	0.333305
$766$	0	0
$767$	3.01410e7	1.84999
$768$	0	0
$769$	2.75789e7	1.68175	0.840875	−	0.541230i	$-0.182041\pi$
$769$	2.75789e7	1.68175	0.840875	+	0.541230i	$0.182041\pi$
$770$	0	0
$771$	−2.35478e6	−0.142664
$772$	0	0
$773$	−2.81806e7	−1.69629	−0.848147	−	0.529760i	$-0.822282\pi$
$773$	−2.81806e7	−1.69629	−0.848147	+	0.529760i	$0.822282\pi$
$774$	0	0
$775$	−4.80579e6	−0.287416
$776$	0	0
$777$	−1.86948e6	−0.111088
$778$	0	0
$779$	4.30185e7	2.53987
$780$	0	0
$781$	−6.20283e6	−0.363883
$782$	0	0
$783$	−1.43223e6	−0.0834852
$784$	0	0
$785$	1.51006e7	0.874622
$786$	0	0
$787$	7.25427e6	0.417501	0.208750	−	0.977969i	$-0.433060\pi$
$787$	7.25427e6	0.417501	0.208750	+	0.977969i	$0.433060\pi$
$788$	0	0
$789$	−6.02065e6	−0.344311
$790$	0	0
$791$	−1.65071e6	−0.0938056
$792$	0	0
$793$	4.06521e6	0.229562
$794$	0	0
$795$	3.08831e6	0.173302
$796$	0	0
$797$	1.41658e7	0.789944	0.394972	−	0.918693i	$-0.370754\pi$
$797$	1.41658e7	0.789944	0.394972	+	0.918693i	$0.370754\pi$
$798$	0	0
$799$	1.13650e7	0.629801
$800$	0	0
$801$	−5.38627e6	−0.296625
$802$	0	0
$803$	−2.28244e7	−1.24914
$804$	0	0
$805$	−4.80341e6	−0.261252
$806$	0	0
$807$	−540077.	−0.0291925
$808$	0	0
$809$	−1.24133e7	−0.666833	−0.333416	−	0.942780i	$-0.608201\pi$
$809$	−1.24133e7	−0.666833	−0.333416	+	0.942780i	$0.608201\pi$
$810$	0	0
$811$	−2.81050e7	−1.50048	−0.750242	−	0.661163i	$-0.770063\pi$
$811$	−2.81050e7	−1.50048	−0.750242	+	0.661163i	$0.770063\pi$
$812$	0	0
$813$	−1.06642e7	−0.565851
$814$	0	0
$815$	1.26960e6	0.0669534
$816$	0	0
$817$	2.73375e7	1.43286
$818$	0	0
$819$	−9.59194e6	−0.499686
$820$	0	0
$821$	−865419.	−0.0448093	−0.0224047	−	0.999749i	$-0.507132\pi$
$821$	−865419.	−0.0448093	−0.0224047	+	0.999749i	$0.507132\pi$
$822$	0	0
$823$	1.80157e6	0.0927151	0.0463576	−	0.998925i	$-0.485239\pi$
$823$	1.80157e6	0.0927151	0.0463576	+	0.998925i	$0.485239\pi$
$824$	0	0
$825$	960303.	0.0491217
$826$	0	0
$827$	−2.51361e7	−1.27801	−0.639005	−	0.769203i	$-0.720654\pi$
$827$	−2.51361e7	−1.27801	−0.639005	+	0.769203i	$0.720654\pi$
$828$	0	0
$829$	3.08361e6	0.155838	0.0779188	−	0.996960i	$-0.475172\pi$
$829$	3.08361e6	0.155838	0.0779188	+	0.996960i	$0.475172\pi$
$830$	0	0
$831$	1.17646e7	0.590981
$832$	0	0
$833$	2.42196e6	0.120936
$834$	0	0
$835$	1.04253e7	0.517453
$836$	0	0
$837$	−1.89423e7	−0.934586
$838$	0	0
$839$	−3.78148e7	−1.85463	−0.927314	−	0.374284i	$-0.877888\pi$
$839$	−3.78148e7	−1.85463	−0.927314	+	0.374284i	$0.877888\pi$
$840$	0	0
$841$	−2.01731e7	−0.983521
$842$	0	0
$843$	1.06619e7	0.516732
$844$	0	0
$845$	−1.16493e7	−0.561251
$846$	0	0
$847$	3.91169e6	0.187351
$848$	0	0
$849$	8.25018e6	0.392820
$850$	0	0
$851$	2.77487e7	1.31347
$852$	0	0
$853$	−175925.	−0.00827856	−0.00413928	−	0.999991i	$-0.501318\pi$
$853$	−175925.	−0.00827856	−0.00413928	+	0.999991i	$0.501318\pi$
$854$	0	0
$855$	1.60228e7	0.749588
$856$	0	0
$857$	3.92402e7	1.82507	0.912535	−	0.408998i	$-0.134122\pi$
$857$	3.92402e7	1.82507	0.912535	+	0.408998i	$0.134122\pi$
$858$	0	0
$859$	−1.05079e7	−0.485883	−0.242941	−	0.970041i	$-0.578112\pi$
$859$	−1.05079e7	−0.485883	−0.242941	+	0.970041i	$0.578112\pi$
$860$	0	0
$861$	3.79339e6	0.174389
$862$	0	0
$863$	−1.99897e6	−0.0913649	−0.0456824	−	0.998956i	$-0.514546\pi$
$863$	−1.99897e6	−0.0913649	−0.0456824	+	0.998956i	$0.514546\pi$
$864$	0	0
$865$	9.25638e6	0.420631
$866$	0	0
$867$	2.16902e6	0.0979978
$868$	0	0
$869$	−2.22894e7	−1.00126
$870$	0	0
$871$	−2.32440e7	−1.03816
$872$	0	0
$873$	1.07796e7	0.478704
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	3.36354e7	1.47672	0.738360	−	0.674407i	$-0.235601\pi$
$877$	3.36354e7	1.47672	0.738360	+	0.674407i	$0.235601\pi$
$878$	0	0
$879$	5.46817e6	0.238710
$880$	0	0
$881$	2.48413e7	1.07829	0.539144	−	0.842214i	$-0.318748\pi$
$881$	2.48413e7	1.07829	0.539144	+	0.842214i	$0.318748\pi$
$882$	0	0
$883$	2.11044e7	0.910900	0.455450	−	0.890261i	$-0.349478\pi$
$883$	2.11044e7	0.910900	0.455450	+	0.890261i	$0.349478\pi$
$884$	0	0
$885$	−4.43977e6	−0.190547
$886$	0	0
$887$	2.53051e7	1.07994	0.539969	−	0.841685i	$-0.318436\pi$
$887$	2.53051e7	1.07994	0.539969	+	0.841685i	$0.318436\pi$
$888$	0	0
$889$	6.21371e6	0.263692
$890$	0	0
$891$	−1.10305e7	−0.465479
$892$	0	0
$893$	3.37531e7	1.41640
$894$	0	0
$895$	−117522.	−0.00490412
$896$	0	0
$897$	−1.93437e7	−0.802710
$898$	0	0
$899$	4.47044e6	0.184481
$900$	0	0
$901$	2.31132e7	0.948525
$902$	0	0
$903$	2.41064e6	0.0983813
$904$	0	0
$905$	−1.51911e7	−0.616550
$906$	0	0
$907$	−2.58521e7	−1.04346	−0.521732	−	0.853109i	$-0.674714\pi$
$907$	−2.58521e7	−1.04346	−0.521732	+	0.853109i	$0.674714\pi$
$908$	0	0
$909$	−3.29617e7	−1.32312
$910$	0	0
$911$	−2.15237e7	−0.859251	−0.429626	−	0.903007i	$-0.641355\pi$
$911$	−2.15237e7	−0.859251	−0.429626	+	0.903007i	$0.641355\pi$
$912$	0	0
$913$	2.58518e7	1.02639
$914$	0	0
$915$	−598807.	−0.0236447
$916$	0	0
$917$	5.18832e6	0.203753
$918$	0	0
$919$	−2.74997e6	−0.107408	−0.0537042	−	0.998557i	$-0.517103\pi$
$919$	−2.74997e6	−0.107408	−0.0537042	+	0.998557i	$0.517103\pi$
$920$	0	0
$921$	2.35406e6	0.0914468
$922$	0	0
$923$	−1.99153e7	−0.769455
$924$	0	0
$925$	−4.42292e6	−0.169963
$926$	0	0
$927$	6.30871e6	0.241124
$928$	0	0
$929$	−2.77814e7	−1.05612	−0.528062	−	0.849206i	$-0.677081\pi$
$929$	−2.77814e7	−1.05612	−0.528062	+	0.849206i	$0.677081\pi$
$930$	0	0
$931$	7.19301e6	0.271980
$932$	0	0
$933$	−1.04615e7	−0.393449
$934$	0	0
$935$	7.18702e6	0.268856
$936$	0	0
$937$	−6.77223e6	−0.251990	−0.125995	−	0.992031i	$-0.540212\pi$
$937$	−6.77223e6	−0.251990	−0.125995	+	0.992031i	$0.540212\pi$
$938$	0	0
$939$	6.01006e6	0.222441
$940$	0	0
$941$	−2.58486e6	−0.0951620	−0.0475810	−	0.998867i	$-0.515151\pi$
$941$	−2.58486e6	−0.0951620	−0.0475810	+	0.998867i	$0.515151\pi$
$942$	0	0
$943$	−5.63054e7	−2.06192
$944$	0	0
$945$	3.01776e6	0.109927
$946$	0	0
$947$	1.58435e7	0.574086	0.287043	−	0.957918i	$-0.407328\pi$
$947$	1.58435e7	0.574086	0.287043	+	0.957918i	$0.407328\pi$
$948$	0	0
$949$	−7.32819e7	−2.64138
$950$	0	0
$951$	−1.52078e7	−0.545274
$952$	0	0
$953$	−1.62541e7	−0.579735	−0.289867	−	0.957067i	$-0.593611\pi$
$953$	−1.62541e7	−0.579735	−0.289867	+	0.957067i	$0.593611\pi$
$954$	0	0
$955$	−1.63681e7	−0.580750
$956$	0	0
$957$	−893293.	−0.0315293
$958$	0	0
$959$	−1.50950e7	−0.530015
$960$	0	0
$961$	3.04956e7	1.06519
$962$	0	0
$963$	−1.51973e6	−0.0528080
$964$	0	0
$965$	−8.64742e6	−0.298929
$966$	0	0
$967$	4.11445e7	1.41496	0.707482	−	0.706732i	$-0.249831\pi$
$967$	4.11445e7	1.41496	0.707482	+	0.706732i	$0.249831\pi$
$968$	0	0
$969$	−1.62925e7	−0.557416
$970$	0	0
$971$	2.84531e7	0.968460	0.484230	−	0.874941i	$-0.339100\pi$
$971$	2.84531e7	0.968460	0.484230	+	0.874941i	$0.339100\pi$
$972$	0	0
$973$	−7.56387e6	−0.256131
$974$	0	0
$975$	3.08323e6	0.103871
$976$	0	0
$977$	1.18848e7	0.398341	0.199171	−	0.979965i	$-0.436175\pi$
$977$	1.18848e7	0.398341	0.199171	+	0.979965i	$0.436175\pi$
$978$	0	0
$979$	−7.17534e6	−0.239269
$980$	0	0
$981$	−2.18583e7	−0.725177
$982$	0	0
$983$	−3.96769e6	−0.130965	−0.0654823	−	0.997854i	$-0.520859\pi$
$983$	−3.96769e6	−0.130965	−0.0654823	+	0.997854i	$0.520859\pi$
$984$	0	0
$985$	1.27293e7	0.418036
$986$	0	0
$987$	2.97636e6	0.0972507
$988$	0	0
$989$	−3.57812e7	−1.16323
$990$	0	0
$991$	4.35666e7	1.40919	0.704596	−	0.709609i	$-0.251128\pi$
$991$	4.35666e7	1.40919	0.704596	+	0.709609i	$0.251128\pi$
$992$	0	0
$993$	−1.59487e7	−0.513277
$994$	0	0
$995$	1.16823e7	0.374084
$996$	0	0
$997$	−9.29888e6	−0.296273	−0.148137	−	0.988967i	$-0.547328\pi$
$997$	−9.29888e6	−0.296273	−0.148137	+	0.988967i	$0.547328\pi$
$998$	0	0
$999$	−1.74332e7	−0.552667

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.i.1.2	✓	5
4.3	odd	2		560.6.a.z.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.i.1.2	✓	5	1.1	even	1	trivial
560.6.a.z.1.4		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	280.a (trivial)

\(f(q)\)	\(=\)	\(q-5.39132 q^{3} -25.0000 q^{5} -49.0000 q^{7} -213.934 q^{9} +O(q^{10})\)	\(q-5.39132 q^{3} -25.0000 q^{5} -49.0000 q^{7} -213.934 q^{9} -284.992 q^{11} -915.021 q^{13} +134.783 q^{15} +1008.73 q^{17} +2995.84 q^{19} +264.175 q^{21} -3921.15 q^{23} +625.000 q^{25} +2463.47 q^{27} -581.388 q^{29} -7689.26 q^{31} +1536.48 q^{33} +1225.00 q^{35} -7076.68 q^{37} +4933.17 q^{39} +14359.4 q^{41} +9125.17 q^{43} +5348.34 q^{45} +11266.7 q^{47} +2401.00 q^{49} -5438.39 q^{51} +22913.2 q^{53} +7124.81 q^{55} -16151.5 q^{57} -32940.2 q^{59} -4442.75 q^{61} +10482.8 q^{63} +22875.5 q^{65} +25402.7 q^{67} +21140.2 q^{69} +21764.9 q^{71} +80087.7 q^{73} -3369.57 q^{75} +13964.6 q^{77} +78210.5 q^{79} +38704.5 q^{81} -90710.5 q^{83} -25218.3 q^{85} +3134.45 q^{87} +25177.3 q^{89} +44836.1 q^{91} +41455.3 q^{93} -74896.0 q^{95} -50387.6 q^{97} +60969.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 - 125 * q^5 - 245 * q^7 + 372 * q^9	\( 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9} - 263 q^{11} - 729 q^{13} + 75 q^{15} - 1003 q^{17} - 2506 q^{19} + 147 q^{21} - 1066 q^{23} + 3125 q^{25} + 615 q^{27} + 3489 q^{29} + 7880 q^{31} + 4863 q^{33} + 6125 q^{35} + 13118 q^{37} + 27189 q^{39} + 23972 q^{41} + 3978 q^{43} - 9300 q^{45} + 9057 q^{47} + 12005 q^{49} + 96639 q^{51} + 2128 q^{53} + 6575 q^{55} + 61674 q^{57} - 15512 q^{59} + 4560 q^{61} - 18228 q^{63} + 18225 q^{65} + 7780 q^{67} + 126474 q^{69} + 32752 q^{71} + 189498 q^{73} - 1875 q^{75} + 12887 q^{77} + 42055 q^{79} + 294645 q^{81} + 58420 q^{83} + 25075 q^{85} - 765 q^{87} + 231324 q^{89} + 35721 q^{91} + 395736 q^{93} + 62650 q^{95} + 247569 q^{97} + 30606 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 - 125 * q^5 - 245 * q^7 + 372 * q^9 - 263 * q^11 - 729 * q^13 + 75 * q^15 - 1003 * q^17 - 2506 * q^19 + 147 * q^21 - 1066 * q^23 + 3125 * q^25 + 615 * q^27 + 3489 * q^29 + 7880 * q^31 + 4863 * q^33 + 6125 * q^35 + 13118 * q^37 + 27189 * q^39 + 23972 * q^41 + 3978 * q^43 - 9300 * q^45 + 9057 * q^47 + 12005 * q^49 + 96639 * q^51 + 2128 * q^53 + 6575 * q^55 + 61674 * q^57 - 15512 * q^59 + 4560 * q^61 - 18228 * q^63 + 18225 * q^65 + 7780 * q^67 + 126474 * q^69 + 32752 * q^71 + 189498 * q^73 - 1875 * q^75 + 12887 * q^77 + 42055 * q^79 + 294645 * q^81 + 58420 * q^83 + 25075 * q^85 - 765 * q^87 + 231324 * q^89 + 35721 * q^91 + 395736 * q^93 + 62650 * q^95 + 247569 * q^97 + 30606 * q^99

Introduction

L-functions

Modular forms

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