LMFDB - Embedded newform 280.6.a.i.1.1

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			1.1
Root			$-26.7452$ 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)

$f(q)$	$=$	$q-27.7452 q^{3} -25.0000 q^{5} -49.0000 q^{7} +526.795 q^{9} +O(q^{10})$	$q-27.7452 q^{3} -25.0000 q^{5} -49.0000 q^{7} +526.795 q^{9} +15.5398 q^{11} -603.520 q^{13} +693.629 q^{15} -2242.22 q^{17} -3094.03 q^{19} +1359.51 q^{21} -3530.37 q^{23} +625.000 q^{25} -7873.93 q^{27} +2013.08 q^{29} -2924.24 q^{31} -431.155 q^{33} +1225.00 q^{35} +11955.1 q^{37} +16744.8 q^{39} +2905.39 q^{41} -19224.3 q^{43} -13169.9 q^{45} +459.421 q^{47} +2401.00 q^{49} +62210.9 q^{51} -26542.2 q^{53} -388.495 q^{55} +85844.4 q^{57} -23574.5 q^{59} -18235.1 q^{61} -25812.9 q^{63} +15088.0 q^{65} -39512.0 q^{67} +97950.8 q^{69} +3573.82 q^{71} +71844.7 q^{73} -17340.7 q^{75} -761.451 q^{77} +67361.0 q^{79} +90452.4 q^{81} +13524.1 q^{83} +56055.6 q^{85} -55853.4 q^{87} +144109. q^{89} +29572.5 q^{91} +81133.5 q^{93} +77350.7 q^{95} +80839.7 q^{97} +8186.29 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−27.7452	−1.77985	−0.889927	−	0.456103i	$-0.849245\pi$
$3$	−27.7452	−1.77985	−0.889927	+	0.456103i	$0.849245\pi$
$4$	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
$9$	526.795	2.16788
$10$	0	0
$11$	15.5398	0.0387226	0.0193613	−	0.999813i	$-0.493837\pi$
$11$	15.5398	0.0387226	0.0193613	+	0.999813i	$0.493837\pi$
$12$	0	0
$13$	−603.520	−0.990451	−0.495226	−	0.868764i	$-0.664915\pi$
$13$	−603.520	−0.990451	−0.495226	+	0.868764i	$0.664915\pi$
$14$	0	0
$15$	693.629	0.795975
$16$	0	0
$17$	−2242.22	−1.88173	−0.940864	−	0.338786i	$-0.889984\pi$
$17$	−2242.22	−1.88173	−0.940864	+	0.338786i	$0.889984\pi$
$18$	0	0
$19$	−3094.03	−1.96626	−0.983129	−	0.182915i	$-0.941447\pi$
$19$	−3094.03	−1.96626	−0.983129	+	0.182915i	$0.941447\pi$
$20$	0	0
$21$	1359.51	0.672721
$22$	0	0
$23$	−3530.37	−1.39156	−0.695778	−	0.718257i	$-0.744940\pi$
$23$	−3530.37	−1.39156	−0.695778	+	0.718257i	$0.744940\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−7873.93	−2.07865
$28$	0	0
$29$	2013.08	0.444495	0.222248	−	0.974990i	$-0.428661\pi$
$29$	2013.08	0.444495	0.222248	+	0.974990i	$0.428661\pi$
$30$	0	0
$31$	−2924.24	−0.546523	−0.273261	−	0.961940i	$-0.588102\pi$
$31$	−2924.24	−0.546523	−0.273261	+	0.961940i	$0.588102\pi$
$32$	0	0
$33$	−431.155	−0.0689205
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	11955.1	1.43565	0.717825	−	0.696224i	$-0.245138\pi$
$37$	11955.1	1.43565	0.717825	+	0.696224i	$0.245138\pi$
$38$	0	0
$39$	16744.8	1.76286
$40$	0	0
$41$	2905.39	0.269926	0.134963	−	0.990851i	$-0.456908\pi$
$41$	2905.39	0.269926	0.134963	+	0.990851i	$0.456908\pi$
$42$	0	0
$43$	−19224.3	−1.58555	−0.792775	−	0.609515i	$-0.791364\pi$
$43$	−19224.3	−1.58555	−0.792775	+	0.609515i	$0.791364\pi$
$44$	0	0
$45$	−13169.9	−0.969505
$46$	0	0
$47$	459.421	0.0303365	0.0151683	−	0.999885i	$-0.495172\pi$
$47$	459.421	0.0303365	0.0151683	+	0.999885i	$0.495172\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	62210.9	3.34920
$52$	0	0
$53$	−26542.2	−1.29792	−0.648959	−	0.760824i	$-0.724795\pi$
$53$	−26542.2	−1.29792	−0.648959	+	0.760824i	$0.724795\pi$
$54$	0	0
$55$	−388.495	−0.0173173
$56$	0	0
$57$	85844.4	3.49965
$58$	0	0
$59$	−23574.5	−0.881681	−0.440841	−	0.897585i	$-0.645320\pi$
$59$	−23574.5	−0.881681	−0.440841	+	0.897585i	$0.645320\pi$
$60$	0	0
$61$	−18235.1	−0.627456	−0.313728	−	0.949513i	$-0.601578\pi$
$61$	−18235.1	−0.627456	−0.313728	+	0.949513i	$0.601578\pi$
$62$	0	0
$63$	−25812.9	−0.819381
$64$	0	0
$65$	15088.0	0.442943
$66$	0	0
$67$	−39512.0	−1.07533	−0.537666	−	0.843158i	$-0.680694\pi$
$67$	−39512.0	−1.07533	−0.537666	+	0.843158i	$0.680694\pi$
$68$	0	0
$69$	97950.8	2.47677
$70$	0	0
$71$	3573.82	0.0841369	0.0420685	−	0.999115i	$-0.486605\pi$
$71$	3573.82	0.0841369	0.0420685	+	0.999115i	$0.486605\pi$
$72$	0	0
$73$	71844.7	1.57793	0.788965	−	0.614439i	$-0.210617\pi$
$73$	71844.7	1.57793	0.788965	+	0.614439i	$0.210617\pi$
$74$	0	0
$75$	−17340.7	−0.355971
$76$	0	0
$77$	−761.451	−0.0146358
$78$	0	0
$79$	67361.0	1.21434	0.607171	−	0.794571i	$-0.292304\pi$
$79$	67361.0	1.21434	0.607171	+	0.794571i	$0.292304\pi$
$80$	0	0
$81$	90452.4	1.53182
$82$	0	0
$83$	13524.1	0.215482	0.107741	−	0.994179i	$-0.465638\pi$
$83$	13524.1	0.215482	0.107741	+	0.994179i	$0.465638\pi$
$84$	0	0
$85$	56055.6	0.841534
$86$	0	0
$87$	−55853.4	−0.791136
$88$	0	0
$89$	144109.	1.92849	0.964245	−	0.265013i	$-0.0853762\pi$
$89$	144109.	1.92849	0.964245	+	0.265013i	$0.0853762\pi$
$90$	0	0
$91$	29572.5	0.374355
$92$	0	0
$93$	81133.5	0.972731
$94$	0	0
$95$	77350.7	0.879337
$96$	0	0
$97$	80839.7	0.872360	0.436180	−	0.899860i	$-0.356331\pi$
$97$	80839.7	0.872360	0.436180	+	0.899860i	$0.356331\pi$
$98$	0	0
$99$	8186.29	0.0839459
$100$	0	0
$101$	−129532.	−1.26350	−0.631749	−	0.775173i	$-0.717663\pi$
$101$	−129532.	−1.26350	−0.631749	+	0.775173i	$0.717663\pi$
$102$	0	0
$103$	23515.1	0.218400	0.109200	−	0.994020i	$-0.465171\pi$
$103$	23515.1	0.218400	0.109200	+	0.994020i	$0.465171\pi$
$104$	0	0
$105$	−33987.8	−0.300850
$106$	0	0
$107$	22478.3	0.189804	0.0949018	−	0.995487i	$-0.469746\pi$
$107$	22478.3	0.189804	0.0949018	+	0.995487i	$0.469746\pi$
$108$	0	0
$109$	−156309.	−1.26013	−0.630067	−	0.776541i	$-0.716973\pi$
$109$	−156309.	−1.26013	−0.630067	+	0.776541i	$0.716973\pi$
$110$	0	0
$111$	−331696.	−2.55525
$112$	0	0
$113$	−97473.5	−0.718109	−0.359055	−	0.933317i	$-0.616901\pi$
$113$	−97473.5	−0.718109	−0.359055	+	0.933317i	$0.616901\pi$
$114$	0	0
$115$	88259.3	0.622323
$116$	0	0
$117$	−317931.	−2.14718
$118$	0	0
$119$	109869.	0.711226
$120$	0	0
$121$	−160810.	−0.998501
$122$	0	0
$123$	−80610.5	−0.480429
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	49466.4	0.272145	0.136073	−	0.990699i	$-0.456552\pi$
$127$	49466.4	0.272145	0.136073	+	0.990699i	$0.456552\pi$
$128$	0	0
$129$	533382.	2.82205
$130$	0	0
$131$	151749.	0.772586	0.386293	−	0.922376i	$-0.373755\pi$
$131$	151749.	0.772586	0.386293	+	0.922376i	$0.373755\pi$
$132$	0	0
$133$	151607.	0.743175
$134$	0	0
$135$	196848.	0.929602
$136$	0	0
$137$	−210460.	−0.958008	−0.479004	−	0.877813i	$-0.659002\pi$
$137$	−210460.	−0.958008	−0.479004	+	0.877813i	$0.659002\pi$
$138$	0	0
$139$	−394132.	−1.73023	−0.865117	−	0.501570i	$-0.832756\pi$
$139$	−394132.	−1.73023	−0.865117	+	0.501570i	$0.832756\pi$
$140$	0	0
$141$	−12746.7	−0.0539946
$142$	0	0
$143$	−9378.59	−0.0383528
$144$	0	0
$145$	−50327.1	−0.198784
$146$	0	0
$147$	−66616.2	−0.254265
$148$	0	0
$149$	510392.	1.88338	0.941690	−	0.336482i	$-0.109237\pi$
$149$	510392.	1.88338	0.941690	+	0.336482i	$0.109237\pi$
$150$	0	0
$151$	264504.	0.944039	0.472019	−	0.881588i	$-0.343525\pi$
$151$	264504.	0.944039	0.472019	+	0.881588i	$0.343525\pi$
$152$	0	0
$153$	−1.18119e6	−4.07936
$154$	0	0
$155$	73105.9	0.244412
$156$	0	0
$157$	333310.	1.07919	0.539597	−	0.841923i	$-0.318577\pi$
$157$	333310.	1.07919	0.539597	+	0.841923i	$0.318577\pi$
$158$	0	0
$159$	736418.	2.31010
$160$	0	0
$161$	172988.	0.525959
$162$	0	0
$163$	−324233.	−0.955846	−0.477923	−	0.878402i	$-0.658610\pi$
$163$	−324233.	−0.955846	−0.477923	+	0.878402i	$0.658610\pi$
$164$	0	0
$165$	10778.9	0.0308222
$166$	0	0
$167$	−312966.	−0.868372	−0.434186	−	0.900823i	$-0.642964\pi$
$167$	−312966.	−0.868372	−0.434186	+	0.900823i	$0.642964\pi$
$168$	0	0
$169$	−7056.99	−0.0190065
$170$	0	0
$171$	−1.62992e6	−4.26261
$172$	0	0
$173$	8203.32	0.0208389	0.0104194	−	0.999946i	$-0.496683\pi$
$173$	8203.32	0.0208389	0.0104194	+	0.999946i	$0.496683\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	654077.	1.56926
$178$	0	0
$179$	694532.	1.62017	0.810083	−	0.586315i	$-0.199422\pi$
$179$	694532.	1.62017	0.810083	+	0.586315i	$0.199422\pi$
$180$	0	0
$181$	100483.	0.227979	0.113990	−	0.993482i	$-0.463637\pi$
$181$	100483.	0.227979	0.113990	+	0.993482i	$0.463637\pi$
$182$	0	0
$183$	505935.	1.11678
$184$	0	0
$185$	−298877.	−0.642042
$186$	0	0
$187$	−34843.8	−0.0728653
$188$	0	0
$189$	385823.	0.785657
$190$	0	0
$191$	−333988.	−0.662442	−0.331221	−	0.943553i	$-0.607461\pi$
$191$	−333988.	−0.662442	−0.331221	+	0.943553i	$0.607461\pi$
$192$	0	0
$193$	−332090.	−0.641745	−0.320872	−	0.947122i	$-0.603976\pi$
$193$	−332090.	−0.641745	−0.320872	+	0.947122i	$0.603976\pi$
$194$	0	0
$195$	−418619.	−0.788374
$196$	0	0
$197$	143897.	0.264171	0.132085	−	0.991238i	$-0.457833\pi$
$197$	143897.	0.264171	0.132085	+	0.991238i	$0.457833\pi$
$198$	0	0
$199$	469576.	0.840569	0.420285	−	0.907392i	$-0.361930\pi$
$199$	469576.	0.840569	0.420285	+	0.907392i	$0.361930\pi$
$200$	0	0
$201$	1.09627e6	1.91393
$202$	0	0
$203$	−98641.1	−0.168003
$204$	0	0
$205$	−72634.7	−0.120715
$206$	0	0
$207$	−1.85978e6	−3.01673
$208$	0	0
$209$	−48080.7	−0.0761386
$210$	0	0
$211$	126139.	0.195048	0.0975242	−	0.995233i	$-0.468908\pi$
$211$	126139.	0.195048	0.0975242	+	0.995233i	$0.468908\pi$
$212$	0	0
$213$	−99156.2	−0.149751
$214$	0	0
$215$	480608.	0.709079
$216$	0	0
$217$	143288.	0.206566
$218$	0	0
$219$	−1.99334e6	−2.80848
$220$	0	0
$221$	1.35323e6	1.86376
$222$	0	0
$223$	−130611.	−0.175881	−0.0879403	−	0.996126i	$-0.528028\pi$
$223$	−130611.	−0.175881	−0.0879403	+	0.996126i	$0.528028\pi$
$224$	0	0
$225$	329247.	0.433576
$226$	0	0
$227$	−407980.	−0.525502	−0.262751	−	0.964864i	$-0.584630\pi$
$227$	−407980.	−0.525502	−0.262751	+	0.964864i	$0.584630\pi$
$228$	0	0
$229$	−699994.	−0.882075	−0.441038	−	0.897489i	$-0.645389\pi$
$229$	−699994.	−0.882075	−0.441038	+	0.897489i	$0.645389\pi$
$230$	0	0
$231$	21126.6	0.0260495
$232$	0	0
$233$	16292.3	0.0196604	0.00983021	−	0.999952i	$-0.496871\pi$
$233$	16292.3	0.0196604	0.00983021	+	0.999952i	$0.496871\pi$
$234$	0	0
$235$	−11485.5	−0.0135669
$236$	0	0
$237$	−1.86894e6	−2.16135
$238$	0	0
$239$	1.28857e6	1.45919	0.729595	−	0.683879i	$-0.239708\pi$
$239$	1.28857e6	1.45919	0.729595	+	0.683879i	$0.239708\pi$
$240$	0	0
$241$	−125785.	−0.139503	−0.0697517	−	0.997564i	$-0.522221\pi$
$241$	−125785.	−0.139503	−0.0697517	+	0.997564i	$0.522221\pi$
$242$	0	0
$243$	−596253.	−0.647762
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	1.86731e6	1.94748
$248$	0	0
$249$	−375227.	−0.383527
$250$	0	0
$251$	−611234.	−0.612383	−0.306192	−	0.951970i	$-0.599055\pi$
$251$	−611234.	−0.612383	−0.306192	+	0.951970i	$0.599055\pi$
$252$	0	0
$253$	−54861.3	−0.0538847
$254$	0	0
$255$	−1.55527e6	−1.49781
$256$	0	0
$257$	370771.	0.350165	0.175083	−	0.984554i	$-0.443981\pi$
$257$	370771.	0.350165	0.175083	+	0.984554i	$0.443981\pi$
$258$	0	0
$259$	−585799.	−0.542624
$260$	0	0
$261$	1.06048e6	0.963612
$262$	0	0
$263$	−756765.	−0.674639	−0.337320	−	0.941390i	$-0.609520\pi$
$263$	−756765.	−0.674639	−0.337320	+	0.941390i	$0.609520\pi$
$264$	0	0
$265$	663555.	0.580446
$266$	0	0
$267$	−3.99834e6	−3.43243
$268$	0	0
$269$	−949722.	−0.800232	−0.400116	−	0.916465i	$-0.631030\pi$
$269$	−949722.	−0.800232	−0.400116	+	0.916465i	$0.631030\pi$
$270$	0	0
$271$	−1.27199e6	−1.05211	−0.526053	−	0.850452i	$-0.676329\pi$
$271$	−1.27199e6	−1.05211	−0.526053	+	0.850452i	$0.676329\pi$
$272$	0	0
$273$	−820493.	−0.666298
$274$	0	0
$275$	9712.39	0.00774452
$276$	0	0
$277$	−1.48800e6	−1.16521	−0.582606	−	0.812755i	$-0.697967\pi$
$277$	−1.48800e6	−1.16521	−0.582606	+	0.812755i	$0.697967\pi$
$278$	0	0
$279$	−1.54047e6	−1.18480
$280$	0	0
$281$	449345.	0.339480	0.169740	−	0.985489i	$-0.445707\pi$
$281$	449345.	0.339480	0.169740	+	0.985489i	$0.445707\pi$
$282$	0	0
$283$	1.04984e6	0.779217	0.389609	−	0.920981i	$-0.372610\pi$
$283$	1.04984e6	0.779217	0.389609	+	0.920981i	$0.372610\pi$
$284$	0	0
$285$	−2.14611e6	−1.56509
$286$	0	0
$287$	−142364.	−0.102022
$288$	0	0
$289$	3.60771e6	2.54090
$290$	0	0
$291$	−2.24291e6	−1.55267
$292$	0	0
$293$	−187253.	−0.127427	−0.0637133	−	0.997968i	$-0.520294\pi$
$293$	−187253.	−0.127427	−0.0637133	+	0.997968i	$0.520294\pi$
$294$	0	0
$295$	589361.	0.394300
$296$	0	0
$297$	−122359.	−0.0804908
$298$	0	0
$299$	2.13065e6	1.37827
$300$	0	0
$301$	941991.	0.599281
$302$	0	0
$303$	3.59390e6	2.24884
$304$	0	0
$305$	455877.	0.280607
$306$	0	0
$307$	−259697.	−0.157261	−0.0786306	−	0.996904i	$-0.525055\pi$
$307$	−259697.	−0.157261	−0.0786306	+	0.996904i	$0.525055\pi$
$308$	0	0
$309$	−652429.	−0.388720
$310$	0	0
$311$	−1.73231e6	−1.01560	−0.507802	−	0.861474i	$-0.669542\pi$
$311$	−1.73231e6	−1.01560	−0.507802	+	0.861474i	$0.669542\pi$
$312$	0	0
$313$	1.19739e6	0.690836	0.345418	−	0.938449i	$-0.387737\pi$
$313$	1.19739e6	0.690836	0.345418	+	0.938449i	$0.387737\pi$
$314$	0	0
$315$	645323.	0.366438
$316$	0	0
$317$	1.85807e6	1.03852	0.519259	−	0.854617i	$-0.326208\pi$
$317$	1.85807e6	1.03852	0.519259	+	0.854617i	$0.326208\pi$
$318$	0	0
$319$	31283.0	0.0172120
$320$	0	0
$321$	−623665.	−0.337822
$322$	0	0
$323$	6.93751e6	3.69996
$324$	0	0
$325$	−377200.	−0.198090
$326$	0	0
$327$	4.33681e6	2.24286
$328$	0	0
$329$	−22511.6	−0.0114661
$330$	0	0
$331$	−1.22596e6	−0.615047	−0.307523	−	0.951541i	$-0.599500\pi$
$331$	−1.22596e6	−0.615047	−0.307523	+	0.951541i	$0.599500\pi$
$332$	0	0
$333$	6.29787e6	3.11231
$334$	0	0
$335$	987801.	0.480903
$336$	0	0
$337$	−2.26758e6	−1.08765	−0.543824	−	0.839199i	$-0.683024\pi$
$337$	−2.26758e6	−1.08765	−0.543824	+	0.839199i	$0.683024\pi$
$338$	0	0
$339$	2.70442e6	1.27813
$340$	0	0
$341$	−45442.1	−0.0211628
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−2.44877e6	−1.10764
$346$	0	0
$347$	669021.	0.298274	0.149137	−	0.988817i	$-0.452350\pi$
$347$	669021.	0.298274	0.149137	+	0.988817i	$0.452350\pi$
$348$	0	0
$349$	−3.87549e6	−1.70319	−0.851595	−	0.524200i	$-0.824364\pi$
$349$	−3.87549e6	−1.70319	−0.851595	+	0.524200i	$0.824364\pi$
$350$	0	0
$351$	4.75207e6	2.05880
$352$	0	0
$353$	−2.58079e6	−1.10234	−0.551171	−	0.834392i	$-0.685819\pi$
$353$	−2.58079e6	−1.10234	−0.551171	+	0.834392i	$0.685819\pi$
$354$	0	0
$355$	−89345.5	−0.0376272
$356$	0	0
$357$	−3.04833e6	−1.26588
$358$	0	0
$359$	3.07484e6	1.25918	0.629588	−	0.776929i	$-0.283224\pi$
$359$	3.07484e6	1.25918	0.629588	+	0.776929i	$0.283224\pi$
$360$	0	0
$361$	7.09692e6	2.86617
$362$	0	0
$363$	4.46169e6	1.77718
$364$	0	0
$365$	−1.79612e6	−0.705671
$366$	0	0
$367$	10739.6	0.00416219	0.00208110	−	0.999998i	$-0.499338\pi$
$367$	10739.6	0.00416219	0.00208110	+	0.999998i	$0.499338\pi$
$368$	0	0
$369$	1.53054e6	0.585167
$370$	0	0
$371$	1.30057e6	0.490567
$372$	0	0
$373$	2.28210e6	0.849303	0.424651	−	0.905357i	$-0.360397\pi$
$373$	2.28210e6	0.849303	0.424651	+	0.905357i	$0.360397\pi$
$374$	0	0
$375$	433518.	0.159195
$376$	0	0
$377$	−1.21494e6	−0.440251
$378$	0	0
$379$	−3.03460e6	−1.08518	−0.542592	−	0.839996i	$-0.682557\pi$
$379$	−3.03460e6	−1.08518	−0.542592	+	0.839996i	$0.682557\pi$
$380$	0	0
$381$	−1.37245e6	−0.484379
$382$	0	0
$383$	3.61160e6	1.25806	0.629031	−	0.777380i	$-0.283452\pi$
$383$	3.61160e6	1.25806	0.629031	+	0.777380i	$0.283452\pi$
$384$	0	0
$385$	19036.3	0.00654531
$386$	0	0
$387$	−1.01273e7	−3.43728
$388$	0	0
$389$	−3.84566e6	−1.28854	−0.644269	−	0.764799i	$-0.722838\pi$
$389$	−3.84566e6	−1.28854	−0.644269	+	0.764799i	$0.722838\pi$
$390$	0	0
$391$	7.91588e6	2.61853
$392$	0	0
$393$	−4.21030e6	−1.37509
$394$	0	0
$395$	−1.68403e6	−0.543070
$396$	0	0
$397$	−1.17695e6	−0.374785	−0.187393	−	0.982285i	$-0.560004\pi$
$397$	−1.17695e6	−0.374785	−0.187393	+	0.982285i	$0.560004\pi$
$398$	0	0
$399$	−4.20637e6	−1.32274
$400$	0	0
$401$	−4.09062e6	−1.27036	−0.635182	−	0.772363i	$-0.719075\pi$
$401$	−4.09062e6	−1.27036	−0.635182	+	0.772363i	$0.719075\pi$
$402$	0	0
$403$	1.76483e6	0.541304
$404$	0	0
$405$	−2.26131e6	−0.685051
$406$	0	0
$407$	185780.	0.0555920
$408$	0	0
$409$	2.09415e6	0.619013	0.309506	−	0.950897i	$-0.399836\pi$
$409$	2.09415e6	0.619013	0.309506	+	0.950897i	$0.399836\pi$
$410$	0	0
$411$	5.83926e6	1.70511
$412$	0	0
$413$	1.15515e6	0.333244
$414$	0	0
$415$	−338101.	−0.0963666
$416$	0	0
$417$	1.09353e7	3.07956
$418$	0	0
$419$	5.24417e6	1.45929	0.729645	−	0.683826i	$-0.239685\pi$
$419$	5.24417e6	1.45929	0.729645	+	0.683826i	$0.239685\pi$
$420$	0	0
$421$	−3.98861e6	−1.09677	−0.548386	−	0.836225i	$-0.684758\pi$
$421$	−3.98861e6	−1.09677	−0.548386	+	0.836225i	$0.684758\pi$
$422$	0	0
$423$	242020.	0.0657660
$424$	0	0
$425$	−1.40139e6	−0.376345
$426$	0	0
$427$	893519.	0.237156
$428$	0	0
$429$	260211.	0.0682624
$430$	0	0
$431$	577280.	0.149690	0.0748451	−	0.997195i	$-0.476154\pi$
$431$	577280.	0.149690	0.0748451	+	0.997195i	$0.476154\pi$
$432$	0	0
$433$	263712.	0.0675943	0.0337972	−	0.999429i	$-0.489240\pi$
$433$	263712.	0.0675943	0.0337972	+	0.999429i	$0.489240\pi$
$434$	0	0
$435$	1.39633e6	0.353807
$436$	0	0
$437$	1.09231e7	2.73616
$438$	0	0
$439$	−868392.	−0.215058	−0.107529	−	0.994202i	$-0.534294\pi$
$439$	−868392.	−0.215058	−0.107529	+	0.994202i	$0.534294\pi$
$440$	0	0
$441$	1.26483e6	0.309697
$442$	0	0
$443$	−4.95264e6	−1.19902	−0.599511	−	0.800367i	$-0.704638\pi$
$443$	−4.95264e6	−1.19902	−0.599511	+	0.800367i	$0.704638\pi$
$444$	0	0
$445$	−3.60274e6	−0.862447
$446$	0	0
$447$	−1.41609e7	−3.35214
$448$	0	0
$449$	−5.41763e6	−1.26822	−0.634109	−	0.773244i	$-0.718633\pi$
$449$	−5.41763e6	−1.26822	−0.634109	+	0.773244i	$0.718633\pi$
$450$	0	0
$451$	45149.2	0.0104522
$452$	0	0
$453$	−7.33871e6	−1.68025
$454$	0	0
$455$	−739312.	−0.167417
$456$	0	0
$457$	−7.88054e6	−1.76508	−0.882542	−	0.470234i	$-0.844170\pi$
$457$	−7.88054e6	−1.76508	−0.882542	+	0.470234i	$0.844170\pi$
$458$	0	0
$459$	1.76551e7	3.91146
$460$	0	0
$461$	3.99457e6	0.875423	0.437711	−	0.899116i	$-0.355789\pi$
$461$	3.99457e6	0.875423	0.437711	+	0.899116i	$0.355789\pi$
$462$	0	0
$463$	−8.00166e6	−1.73471	−0.867356	−	0.497688i	$-0.834183\pi$
$463$	−8.00166e6	−1.73471	−0.867356	+	0.497688i	$0.834183\pi$
$464$	0	0
$465$	−2.02834e6	−0.435018
$466$	0	0
$467$	4.48324e6	0.951260	0.475630	−	0.879645i	$-0.342220\pi$
$467$	4.48324e6	0.951260	0.475630	+	0.879645i	$0.342220\pi$
$468$	0	0
$469$	1.93609e6	0.406437
$470$	0	0
$471$	−9.24775e6	−1.92081
$472$	0	0
$473$	−298742.	−0.0613965
$474$	0	0
$475$	−1.93377e6	−0.393251
$476$	0	0
$477$	−1.39823e7	−2.81373
$478$	0	0
$479$	−4.55585e6	−0.907258	−0.453629	−	0.891191i	$-0.649871\pi$
$479$	−4.55585e6	−0.907258	−0.453629	+	0.891191i	$0.649871\pi$
$480$	0	0
$481$	−7.21513e6	−1.42194
$482$	0	0
$483$	−4.79959e6	−0.936130
$484$	0	0
$485$	−2.02099e6	−0.390131
$486$	0	0
$487$	−924758.	−0.176687	−0.0883437	−	0.996090i	$-0.528157\pi$
$487$	−924758.	−0.176687	−0.0883437	+	0.996090i	$0.528157\pi$
$488$	0	0
$489$	8.99590e6	1.70127
$490$	0	0
$491$	2.83860e6	0.531374	0.265687	−	0.964059i	$-0.414401\pi$
$491$	2.83860e6	0.531374	0.265687	+	0.964059i	$0.414401\pi$
$492$	0	0
$493$	−4.51379e6	−0.836419
$494$	0	0
$495$	−204657.	−0.0375417
$496$	0	0
$497$	−175117.	−0.0318008
$498$	0	0
$499$	381260.	0.0685441	0.0342720	−	0.999413i	$-0.489089\pi$
$499$	381260.	0.0685441	0.0342720	+	0.999413i	$0.489089\pi$
$500$	0	0
$501$	8.68329e6	1.54557
$502$	0	0
$503$	4.72055e6	0.831904	0.415952	−	0.909387i	$-0.363448\pi$
$503$	4.72055e6	0.831904	0.415952	+	0.909387i	$0.363448\pi$
$504$	0	0
$505$	3.23831e6	0.565054
$506$	0	0
$507$	195797.	0.0338288
$508$	0	0
$509$	−231549.	−0.0396139	−0.0198070	−	0.999804i	$-0.506305\pi$
$509$	−231549.	−0.0396139	−0.0198070	+	0.999804i	$0.506305\pi$
$510$	0	0
$511$	−3.52039e6	−0.596401
$512$	0	0
$513$	2.43622e7	4.08717
$514$	0	0
$515$	−587877.	−0.0976715
$516$	0	0
$517$	7139.32	0.00117471
$518$	0	0
$519$	−227602.	−0.0370901
$520$	0	0
$521$	3.36903e6	0.543765	0.271882	−	0.962330i	$-0.412354\pi$
$521$	3.36903e6	0.543765	0.271882	+	0.962330i	$0.412354\pi$
$522$	0	0
$523$	2.87220e6	0.459156	0.229578	−	0.973290i	$-0.426265\pi$
$523$	2.87220e6	0.459156	0.229578	+	0.973290i	$0.426265\pi$
$524$	0	0
$525$	849696.	0.134544
$526$	0	0
$527$	6.55679e6	1.02841
$528$	0	0
$529$	6.02718e6	0.936430
$530$	0	0
$531$	−1.24189e7	−1.91138
$532$	0	0
$533$	−1.75346e6	−0.267348
$534$	0	0
$535$	−561958.	−0.0848827
$536$	0	0
$537$	−1.92699e7	−2.88366
$538$	0	0
$539$	37311.1	0.00553180
$540$	0	0
$541$	−9.78651e6	−1.43759	−0.718794	−	0.695223i	$-0.755306\pi$
$541$	−9.78651e6	−1.43759	−0.718794	+	0.695223i	$0.755306\pi$
$542$	0	0
$543$	−2.78792e6	−0.405770
$544$	0	0
$545$	3.90772e6	0.563549
$546$	0	0
$547$	2.76714e6	0.395423	0.197712	−	0.980260i	$-0.436649\pi$
$547$	2.76714e6	0.395423	0.197712	+	0.980260i	$0.436649\pi$
$548$	0	0
$549$	−9.60614e6	−1.36025
$550$	0	0
$551$	−6.22854e6	−0.873992
$552$	0	0
$553$	−3.30069e6	−0.458978
$554$	0	0
$555$	8.29240e6	1.14274
$556$	0	0
$557$	640592.	0.0874870	0.0437435	−	0.999043i	$-0.486072\pi$
$557$	640592.	0.0874870	0.0437435	+	0.999043i	$0.486072\pi$
$558$	0	0
$559$	1.16023e7	1.57041
$560$	0	0
$561$	966746.	0.129690
$562$	0	0
$563$	−4.56278e6	−0.606678	−0.303339	−	0.952883i	$-0.598101\pi$
$563$	−4.56278e6	−0.606678	−0.303339	+	0.952883i	$0.598101\pi$
$564$	0	0
$565$	2.43684e6	0.321148
$566$	0	0
$567$	−4.43217e6	−0.578973
$568$	0	0
$569$	6.43354e6	0.833046	0.416523	−	0.909125i	$-0.363249\pi$
$569$	6.43354e6	0.833046	0.416523	+	0.909125i	$0.363249\pi$
$570$	0	0
$571$	9.59413e6	1.23145	0.615723	−	0.787963i	$-0.288864\pi$
$571$	9.59413e6	1.23145	0.615723	+	0.787963i	$0.288864\pi$
$572$	0	0
$573$	9.26656e6	1.17905
$574$	0	0
$575$	−2.20648e6	−0.278311
$576$	0	0
$577$	−9.68049e6	−1.21048	−0.605241	−	0.796043i	$-0.706923\pi$
$577$	−9.68049e6	−1.21048	−0.605241	+	0.796043i	$0.706923\pi$
$578$	0	0
$579$	9.21389e6	1.14221
$580$	0	0
$581$	−662678.	−0.0814446
$582$	0	0
$583$	−412461.	−0.0502587
$584$	0	0
$585$	7.94827e6	0.960247
$586$	0	0
$587$	−1.40974e7	−1.68867	−0.844336	−	0.535814i	$-0.820005\pi$
$587$	−1.40974e7	−1.68867	−0.844336	+	0.535814i	$0.820005\pi$
$588$	0	0
$589$	9.04768e6	1.07460
$590$	0	0
$591$	−3.99243e6	−0.470185
$592$	0	0
$593$	1.10161e7	1.28644	0.643220	−	0.765682i	$-0.277598\pi$
$593$	1.10161e7	1.28644	0.643220	+	0.765682i	$0.277598\pi$
$594$	0	0
$595$	−2.74672e6	−0.318070
$596$	0	0
$597$	−1.30285e7	−1.49609
$598$	0	0
$599$	1.21863e6	0.138772	0.0693862	−	0.997590i	$-0.477896\pi$
$599$	1.21863e6	0.138772	0.0693862	+	0.997590i	$0.477896\pi$
$600$	0	0
$601$	−1.41559e7	−1.59864	−0.799319	−	0.600907i	$-0.794806\pi$
$601$	−1.41559e7	−1.59864	−0.799319	+	0.600907i	$0.794806\pi$
$602$	0	0
$603$	−2.08147e7	−2.33119
$604$	0	0
$605$	4.02024e6	0.446543
$606$	0	0
$607$	5.72231e6	0.630375	0.315188	−	0.949029i	$-0.397933\pi$
$607$	5.72231e6	0.630375	0.315188	+	0.949029i	$0.397933\pi$
$608$	0	0
$609$	2.73682e6	0.299021
$610$	0	0
$611$	−277270.	−0.0300469
$612$	0	0
$613$	−9.70043e6	−1.04265	−0.521326	−	0.853357i	$-0.674563\pi$
$613$	−9.70043e6	−1.04265	−0.521326	+	0.853357i	$0.674563\pi$
$614$	0	0
$615$	2.01526e6	0.214854
$616$	0	0
$617$	−1.11750e7	−1.18178	−0.590890	−	0.806752i	$-0.701223\pi$
$617$	−1.11750e7	−1.18178	−0.590890	+	0.806752i	$0.701223\pi$
$618$	0	0
$619$	1.18302e7	1.24098	0.620491	−	0.784213i	$-0.286933\pi$
$619$	1.18302e7	1.24098	0.620491	+	0.784213i	$0.286933\pi$
$620$	0	0
$621$	2.77979e7	2.89256
$622$	0	0
$623$	−7.06136e6	−0.728901
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	1.33401e6	0.135515
$628$	0	0
$629$	−2.68060e7	−2.70150
$630$	0	0
$631$	−1.77594e7	−1.77564	−0.887822	−	0.460187i	$-0.847782\pi$
$631$	−1.77594e7	−1.77564	−0.887822	+	0.460187i	$0.847782\pi$
$632$	0	0
$633$	−3.49974e6	−0.347158
$634$	0	0
$635$	−1.23666e6	−0.121707
$636$	0	0
$637$	−1.44905e6	−0.141493
$638$	0	0
$639$	1.88267e6	0.182399
$640$	0	0
$641$	2.63776e6	0.253566	0.126783	−	0.991930i	$-0.459535\pi$
$641$	2.63776e6	0.253566	0.126783	+	0.991930i	$0.459535\pi$
$642$	0	0
$643$	−6.98017e6	−0.665792	−0.332896	−	0.942964i	$-0.608026\pi$
$643$	−6.98017e6	−0.665792	−0.332896	+	0.942964i	$0.608026\pi$
$644$	0	0
$645$	−1.33345e7	−1.26206
$646$	0	0
$647$	−4.45907e6	−0.418777	−0.209389	−	0.977832i	$-0.567147\pi$
$647$	−4.45907e6	−0.418777	−0.209389	+	0.977832i	$0.567147\pi$
$648$	0	0
$649$	−366343.	−0.0341410
$650$	0	0
$651$	−3.97554e6	−0.367658
$652$	0	0
$653$	1.14971e7	1.05513	0.527566	−	0.849514i	$-0.323105\pi$
$653$	1.14971e7	1.05513	0.527566	+	0.849514i	$0.323105\pi$
$654$	0	0
$655$	−3.79372e6	−0.345511
$656$	0	0
$657$	3.78474e7	3.42076
$658$	0	0
$659$	1.80060e7	1.61511	0.807557	−	0.589789i	$-0.200789\pi$
$659$	1.80060e7	1.61511	0.807557	+	0.589789i	$0.200789\pi$
$660$	0	0
$661$	−1.99624e7	−1.77709	−0.888546	−	0.458788i	$-0.848284\pi$
$661$	−1.99624e7	−1.77709	−0.888546	+	0.458788i	$0.848284\pi$
$662$	0	0
$663$	−3.75455e7	−3.31722
$664$	0	0
$665$	−3.79019e6	−0.332358
$666$	0	0
$667$	−7.10694e6	−0.618540
$668$	0	0
$669$	3.62383e6	0.313042
$670$	0	0
$671$	−283370.	−0.0242967
$672$	0	0
$673$	1.45909e7	1.24178	0.620891	−	0.783897i	$-0.286771\pi$
$673$	1.45909e7	1.24178	0.620891	+	0.783897i	$0.286771\pi$
$674$	0	0
$675$	−4.92121e6	−0.415731
$676$	0	0
$677$	254237.	0.0213191	0.0106595	−	0.999943i	$-0.496607\pi$
$677$	254237.	0.0213191	0.0106595	+	0.999943i	$0.496607\pi$
$678$	0	0
$679$	−3.96115e6	−0.329721
$680$	0	0
$681$	1.13195e7	0.935316
$682$	0	0
$683$	1.44796e7	1.18770	0.593848	−	0.804578i	$-0.297608\pi$
$683$	1.44796e7	1.18770	0.593848	+	0.804578i	$0.297608\pi$
$684$	0	0
$685$	5.26151e6	0.428434
$686$	0	0
$687$	1.94215e7	1.56996
$688$	0	0
$689$	1.60187e7	1.28552
$690$	0	0
$691$	−4.35540e6	−0.347003	−0.173501	−	0.984834i	$-0.555508\pi$
$691$	−4.35540e6	−0.347003	−0.173501	+	0.984834i	$0.555508\pi$
$692$	0	0
$693$	−401128.	−0.0317286
$694$	0	0
$695$	9.85330e6	0.773784
$696$	0	0
$697$	−6.51453e6	−0.507927
$698$	0	0
$699$	−452033.	−0.0349927
$700$	0	0
$701$	−1.40619e7	−1.08081	−0.540403	−	0.841406i	$-0.681728\pi$
$701$	−1.40619e7	−1.08081	−0.540403	+	0.841406i	$0.681728\pi$
$702$	0	0
$703$	−3.69894e7	−2.82286
$704$	0	0
$705$	318668.	0.0241471
$706$	0	0
$707$	6.34708e6	0.477558
$708$	0	0
$709$	−8.88588e6	−0.663872	−0.331936	−	0.943302i	$-0.607702\pi$
$709$	−8.88588e6	−0.663872	−0.331936	+	0.943302i	$0.607702\pi$
$710$	0	0
$711$	3.54854e7	2.63255
$712$	0	0
$713$	1.03236e7	0.760518
$714$	0	0
$715$	234465.	0.0171519
$716$	0	0
$717$	−3.57515e7	−2.59715
$718$	0	0
$719$	1.90540e7	1.37456	0.687282	−	0.726391i	$-0.258804\pi$
$719$	1.90540e7	1.37456	0.687282	+	0.726391i	$0.258804\pi$
$720$	0	0
$721$	−1.15224e6	−0.0825475
$722$	0	0
$723$	3.48992e6	0.248296
$724$	0	0
$725$	1.25818e6	0.0888990
$726$	0	0
$727$	−5.69409e6	−0.399566	−0.199783	−	0.979840i	$-0.564024\pi$
$727$	−5.69409e6	−0.399566	−0.199783	+	0.979840i	$0.564024\pi$
$728$	0	0
$729$	−5.43679e6	−0.378899
$730$	0	0
$731$	4.31052e7	2.98357
$732$	0	0
$733$	−4.37552e6	−0.300795	−0.150397	−	0.988626i	$-0.548055\pi$
$733$	−4.37552e6	−0.300795	−0.150397	+	0.988626i	$0.548055\pi$
$734$	0	0
$735$	1.66540e6	0.113711
$736$	0	0
$737$	−614010.	−0.0416396
$738$	0	0
$739$	9.82284e6	0.661647	0.330823	−	0.943693i	$-0.392674\pi$
$739$	9.82284e6	0.661647	0.330823	+	0.943693i	$0.392674\pi$
$740$	0	0
$741$	−5.18088e7	−3.46623
$742$	0	0
$743$	8.59501e6	0.571181	0.285591	−	0.958352i	$-0.407810\pi$
$743$	8.59501e6	0.571181	0.285591	+	0.958352i	$0.407810\pi$
$744$	0	0
$745$	−1.27598e7	−0.842273
$746$	0	0
$747$	7.12440e6	0.467139
$748$	0	0
$749$	−1.10144e6	−0.0717390
$750$	0	0
$751$	−2.14533e7	−1.38801	−0.694006	−	0.719969i	$-0.744156\pi$
$751$	−2.14533e7	−1.38801	−0.694006	+	0.719969i	$0.744156\pi$
$752$	0	0
$753$	1.69588e7	1.08995
$754$	0	0
$755$	−6.61260e6	−0.422187
$756$	0	0
$757$	2.22538e7	1.41144	0.705722	−	0.708489i	$-0.250623\pi$
$757$	2.22538e7	1.41144	0.705722	+	0.708489i	$0.250623\pi$
$758$	0	0
$759$	1.52214e6	0.0959068
$760$	0	0
$761$	1.83061e7	1.14587	0.572933	−	0.819602i	$-0.305805\pi$
$761$	1.83061e7	1.14587	0.572933	+	0.819602i	$0.305805\pi$
$762$	0	0
$763$	7.65913e6	0.476286
$764$	0	0
$765$	2.95298e7	1.82434
$766$	0	0
$767$	1.42276e7	0.873262
$768$	0	0
$769$	−2.14181e7	−1.30607	−0.653034	−	0.757329i	$-0.726504\pi$
$769$	−2.14181e7	−1.30607	−0.653034	+	0.757329i	$0.726504\pi$
$770$	0	0
$771$	−1.02871e7	−0.623243
$772$	0	0
$773$	−9.62690e6	−0.579479	−0.289739	−	0.957106i	$-0.593569\pi$
$773$	−9.62690e6	−0.579479	−0.289739	+	0.957106i	$0.593569\pi$
$774$	0	0
$775$	−1.82765e6	−0.109305
$776$	0	0
$777$	1.62531e7	0.965792
$778$	0	0
$779$	−8.98936e6	−0.530744
$780$	0	0
$781$	55536.5	0.00325800
$782$	0	0
$783$	−1.58509e7	−0.923951
$784$	0	0
$785$	−8.33276e6	−0.482630
$786$	0	0
$787$	−1.97448e7	−1.13636	−0.568181	−	0.822904i	$-0.692353\pi$
$787$	−1.97448e7	−1.13636	−0.568181	+	0.822904i	$0.692353\pi$
$788$	0	0
$789$	2.09966e7	1.20076
$790$	0	0
$791$	4.77620e6	0.271420
$792$	0	0
$793$	1.10052e7	0.621464
$794$	0	0
$795$	−1.84104e7	−1.03311
$796$	0	0
$797$	1.95132e7	1.08813	0.544067	−	0.839042i	$-0.316884\pi$
$797$	1.95132e7	1.08813	0.544067	+	0.839042i	$0.316884\pi$
$798$	0	0
$799$	−1.03012e6	−0.0570851
$800$	0	0
$801$	7.59161e7	4.18073
$802$	0	0
$803$	1.11645e6	0.0611015
$804$	0	0
$805$	−4.32471e6	−0.235216
$806$	0	0
$807$	2.63502e7	1.42430
$808$	0	0
$809$	5.61647e6	0.301712	0.150856	−	0.988556i	$-0.451797\pi$
$809$	5.61647e6	0.301712	0.150856	+	0.988556i	$0.451797\pi$
$810$	0	0
$811$	−7.46677e6	−0.398640	−0.199320	−	0.979934i	$-0.563873\pi$
$811$	−7.46677e6	−0.398640	−0.199320	+	0.979934i	$0.563873\pi$
$812$	0	0
$813$	3.52915e7	1.87260
$814$	0	0
$815$	8.10582e6	0.427468
$816$	0	0
$817$	5.94806e7	3.11760
$818$	0	0
$819$	1.55786e7	0.811557
$820$	0	0
$821$	−5.16909e6	−0.267643	−0.133822	−	0.991005i	$-0.542725\pi$
$821$	−5.16909e6	−0.267643	−0.133822	+	0.991005i	$0.542725\pi$
$822$	0	0
$823$	−3.05051e7	−1.56990	−0.784951	−	0.619558i	$-0.787312\pi$
$823$	−3.05051e7	−1.56990	−0.784951	+	0.619558i	$0.787312\pi$
$824$	0	0
$825$	−269472.	−0.0137841
$826$	0	0
$827$	2.16192e7	1.09920	0.549599	−	0.835429i	$-0.314781\pi$
$827$	2.16192e7	1.09920	0.549599	+	0.835429i	$0.314781\pi$
$828$	0	0
$829$	−8.85564e6	−0.447542	−0.223771	−	0.974642i	$-0.571837\pi$
$829$	−8.85564e6	−0.447542	−0.223771	+	0.974642i	$0.571837\pi$
$830$	0	0
$831$	4.12849e7	2.07391
$832$	0	0
$833$	−5.38358e6	−0.268818
$834$	0	0
$835$	7.82414e6	0.388348
$836$	0	0
$837$	2.30252e7	1.13603
$838$	0	0
$839$	−5.78319e6	−0.283637	−0.141818	−	0.989893i	$-0.545295\pi$
$839$	−5.78319e6	−0.283637	−0.141818	+	0.989893i	$0.545295\pi$
$840$	0	0
$841$	−1.64586e7	−0.802424
$842$	0	0
$843$	−1.24672e7	−0.604225
$844$	0	0
$845$	176425.	0.00849997
$846$	0	0
$847$	7.87967e6	0.377398
$848$	0	0
$849$	−2.91281e7	−1.38689
$850$	0	0
$851$	−4.22059e7	−1.99779
$852$	0	0
$853$	1.02000e7	0.479985	0.239992	−	0.970775i	$-0.422855\pi$
$853$	1.02000e7	0.479985	0.239992	+	0.970775i	$0.422855\pi$
$854$	0	0
$855$	4.07479e7	1.90630
$856$	0	0
$857$	2.20918e7	1.02750	0.513748	−	0.857941i	$-0.328257\pi$
$857$	2.20918e7	1.02750	0.513748	+	0.857941i	$0.328257\pi$
$858$	0	0
$859$	−1.82725e7	−0.844921	−0.422460	−	0.906381i	$-0.638833\pi$
$859$	−1.82725e7	−0.844921	−0.422460	+	0.906381i	$0.638833\pi$
$860$	0	0
$861$	3.94991e6	0.181585
$862$	0	0
$863$	3.69356e7	1.68818	0.844090	−	0.536202i	$-0.180141\pi$
$863$	3.69356e7	1.68818	0.844090	+	0.536202i	$0.180141\pi$
$864$	0	0
$865$	−205083.	−0.00931943
$866$	0	0
$867$	−1.00097e8	−4.52243
$868$	0	0
$869$	1.04678e6	0.0470224
$870$	0	0
$871$	2.38463e7	1.06506
$872$	0	0
$873$	4.25859e7	1.89117
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	9.20727e6	0.404233	0.202117	−	0.979361i	$-0.435218\pi$
$877$	9.20727e6	0.404233	0.202117	+	0.979361i	$0.435218\pi$
$878$	0	0
$879$	5.19537e6	0.226801
$880$	0	0
$881$	−7.44673e6	−0.323240	−0.161620	−	0.986853i	$-0.551672\pi$
$881$	−7.44673e6	−0.323240	−0.161620	+	0.986853i	$0.551672\pi$
$882$	0	0
$883$	2.26696e7	0.978459	0.489230	−	0.872155i	$-0.337278\pi$
$883$	2.26696e7	0.978459	0.489230	+	0.872155i	$0.337278\pi$
$884$	0	0
$885$	−1.63519e7	−0.701796
$886$	0	0
$887$	−2.48453e7	−1.06032	−0.530158	−	0.847899i	$-0.677867\pi$
$887$	−2.48453e7	−1.06032	−0.530158	+	0.847899i	$0.677867\pi$
$888$	0	0
$889$	−2.42385e6	−0.102861
$890$	0	0
$891$	1.40561e6	0.0593160
$892$	0	0
$893$	−1.42146e6	−0.0596495
$894$	0	0
$895$	−1.73633e7	−0.724560
$896$	0	0
$897$	−5.91152e7	−2.45312
$898$	0	0
$899$	−5.88674e6	−0.242927
$900$	0	0
$901$	5.95135e7	2.44233
$902$	0	0
$903$	−2.61357e7	−1.06663
$904$	0	0
$905$	−2.51207e6	−0.101956
$906$	0	0
$907$	2.10526e7	0.849744	0.424872	−	0.905253i	$-0.360319\pi$
$907$	2.10526e7	0.849744	0.424872	+	0.905253i	$0.360319\pi$
$908$	0	0
$909$	−6.82369e7	−2.73911
$910$	0	0
$911$	9.90759e6	0.395523	0.197762	−	0.980250i	$-0.436633\pi$
$911$	9.90759e6	0.395523	0.197762	+	0.980250i	$0.436633\pi$
$912$	0	0
$913$	210161.	0.00834403
$914$	0	0
$915$	−1.26484e7	−0.499439
$916$	0	0
$917$	−7.43569e6	−0.292010
$918$	0	0
$919$	−4.73407e7	−1.84904	−0.924520	−	0.381134i	$-0.875534\pi$
$919$	−4.73407e7	−1.84904	−0.924520	+	0.381134i	$0.875534\pi$
$920$	0	0
$921$	7.20535e6	0.279902
$922$	0	0
$923$	−2.15687e6	−0.0833335
$924$	0	0
$925$	7.47193e6	0.287130
$926$	0	0
$927$	1.23876e7	0.473465
$928$	0	0
$929$	1.35426e6	0.0514830	0.0257415	−	0.999669i	$-0.491805\pi$
$929$	1.35426e6	0.0514830	0.0257415	+	0.999669i	$0.491805\pi$
$930$	0	0
$931$	−7.42876e6	−0.280894
$932$	0	0
$933$	4.80632e7	1.80763
$934$	0	0
$935$	871094.	0.0325864
$936$	0	0
$937$	2.84275e7	1.05776	0.528882	−	0.848695i	$-0.322611\pi$
$937$	2.84275e7	1.05776	0.528882	+	0.848695i	$0.322611\pi$
$938$	0	0
$939$	−3.32218e7	−1.22959
$940$	0	0
$941$	−2.69890e7	−0.993603	−0.496802	−	0.867864i	$-0.665492\pi$
$941$	−2.69890e7	−0.993603	−0.496802	+	0.867864i	$0.665492\pi$
$942$	0	0
$943$	−1.02571e7	−0.375617
$944$	0	0
$945$	−9.64556e6	−0.351357
$946$	0	0
$947$	−1.74341e7	−0.631718	−0.315859	−	0.948806i	$-0.602293\pi$
$947$	−1.74341e7	−0.631718	−0.315859	+	0.948806i	$0.602293\pi$
$948$	0	0
$949$	−4.33597e7	−1.56286
$950$	0	0
$951$	−5.15524e7	−1.84841
$952$	0	0
$953$	3.26242e6	0.116361	0.0581806	−	0.998306i	$-0.481470\pi$
$953$	3.26242e6	0.116361	0.0581806	+	0.998306i	$0.481470\pi$
$954$	0	0
$955$	8.34970e6	0.296253
$956$	0	0
$957$	−867951.	−0.0306348
$958$	0	0
$959$	1.03126e7	0.362093
$960$	0	0
$961$	−2.00780e7	−0.701313
$962$	0	0
$963$	1.18415e7	0.411471
$964$	0	0
$965$	8.30224e6	0.286997
$966$	0	0
$967$	4.34272e7	1.49347	0.746733	−	0.665124i	$-0.231621\pi$
$967$	4.34272e7	1.49347	0.746733	+	0.665124i	$0.231621\pi$
$968$	0	0
$969$	−1.92482e8	−6.58539
$970$	0	0
$971$	−9.78787e6	−0.333150	−0.166575	−	0.986029i	$-0.553271\pi$
$971$	−9.78787e6	−0.333150	−0.166575	+	0.986029i	$0.553271\pi$
$972$	0	0
$973$	1.93125e7	0.653967
$974$	0	0
$975$	1.04655e7	0.352572
$976$	0	0
$977$	4.64804e6	0.155788	0.0778938	−	0.996962i	$-0.475180\pi$
$977$	4.64804e6	0.155788	0.0778938	+	0.996962i	$0.475180\pi$
$978$	0	0
$979$	2.23943e6	0.0746761
$980$	0	0
$981$	−8.23426e7	−2.73182
$982$	0	0
$983$	−1.66362e7	−0.549124	−0.274562	−	0.961569i	$-0.588533\pi$
$983$	−1.66362e7	−0.549124	−0.274562	+	0.961569i	$0.588533\pi$
$984$	0	0
$985$	−3.59741e6	−0.118141
$986$	0	0
$987$	624589.	0.0204080
$988$	0	0
$989$	6.78690e7	2.20638
$990$	0	0
$991$	−5.08150e7	−1.64364	−0.821822	−	0.569744i	$-0.807042\pi$
$991$	−5.08150e7	−1.64364	−0.821822	+	0.569744i	$0.807042\pi$
$992$	0	0
$993$	3.40146e7	1.09469
$994$	0	0
$995$	−1.17394e7	−0.375914
$996$	0	0
$997$	1.31935e7	0.420361	0.210180	−	0.977663i	$-0.432595\pi$
$997$	1.31935e7	0.420361	0.210180	+	0.977663i	$0.432595\pi$
$998$	0	0
$999$	−9.41335e7	−2.98422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.i.1.1	✓	5
4.3	odd	2		560.6.a.z.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.i.1.1	✓	5	1.1	even	1	trivial
560.6.a.z.1.5		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)

\(f(q)\)	\(=\)	\(q-27.7452 q^{3} -25.0000 q^{5} -49.0000 q^{7} +526.795 q^{9} +O(q^{10})\)	\(q-27.7452 q^{3} -25.0000 q^{5} -49.0000 q^{7} +526.795 q^{9} +15.5398 q^{11} -603.520 q^{13} +693.629 q^{15} -2242.22 q^{17} -3094.03 q^{19} +1359.51 q^{21} -3530.37 q^{23} +625.000 q^{25} -7873.93 q^{27} +2013.08 q^{29} -2924.24 q^{31} -431.155 q^{33} +1225.00 q^{35} +11955.1 q^{37} +16744.8 q^{39} +2905.39 q^{41} -19224.3 q^{43} -13169.9 q^{45} +459.421 q^{47} +2401.00 q^{49} +62210.9 q^{51} -26542.2 q^{53} -388.495 q^{55} +85844.4 q^{57} -23574.5 q^{59} -18235.1 q^{61} -25812.9 q^{63} +15088.0 q^{65} -39512.0 q^{67} +97950.8 q^{69} +3573.82 q^{71} +71844.7 q^{73} -17340.7 q^{75} -761.451 q^{77} +67361.0 q^{79} +90452.4 q^{81} +13524.1 q^{83} +56055.6 q^{85} -55853.4 q^{87} +144109. q^{89} +29572.5 q^{91} +81133.5 q^{93} +77350.7 q^{95} +80839.7 q^{97} +8186.29 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} - 125 q^{5} - 245 q^{7} + 372 q^{9}+O(q^{10}) \) 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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