LMFDB - Embedded newform 280.6.a.h.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:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 766x^{2} + 2548x + 104520 $ x^4 - x^3 - 766x^2 + 2548x + 104520
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-26.1636$ 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)

$f(q)$	$=$	$q-25.1636 q^{3} -25.0000 q^{5} -49.0000 q^{7} +390.209 q^{9} +O(q^{10})$	$q-25.1636 q^{3} -25.0000 q^{5} -49.0000 q^{7} +390.209 q^{9} -365.186 q^{11} +247.916 q^{13} +629.091 q^{15} +1639.32 q^{17} -361.233 q^{19} +1233.02 q^{21} +1415.45 q^{23} +625.000 q^{25} -3704.31 q^{27} +5204.44 q^{29} +5883.20 q^{31} +9189.40 q^{33} +1225.00 q^{35} -11213.6 q^{37} -6238.47 q^{39} -7249.75 q^{41} +742.037 q^{43} -9755.22 q^{45} -5166.17 q^{47} +2401.00 q^{49} -41251.2 q^{51} +16015.5 q^{53} +9129.64 q^{55} +9089.94 q^{57} -19888.8 q^{59} -26621.9 q^{61} -19120.2 q^{63} -6197.90 q^{65} -15591.5 q^{67} -35617.8 q^{69} -46315.6 q^{71} +90796.7 q^{73} -15727.3 q^{75} +17894.1 q^{77} -103600. q^{79} -1606.84 q^{81} -96467.3 q^{83} -40982.9 q^{85} -130963. q^{87} -77336.2 q^{89} -12147.9 q^{91} -148043. q^{93} +9030.83 q^{95} +147486. q^{97} -142499. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{3} - 100 q^{5} - 196 q^{7} + 567 q^{9}+O(q^{10}) $ 4 * q + 5 * q^3 - 100 * q^5 - 196 * q^7 + 567 * q^9	$ 4 q + 5 q^{3} - 100 q^{5} - 196 q^{7} + 567 q^{9} - 123 q^{11} + 789 q^{13} - 125 q^{15} + 1051 q^{17} + 958 q^{19} - 245 q^{21} - 530 q^{23} + 2500 q^{25} - 3169 q^{27} - 5541 q^{29} - 10440 q^{31} - 3737 q^{33} + 4900 q^{35} - 24048 q^{37} - 9131 q^{39} - 35414 q^{41} + 2174 q^{43} - 14175 q^{45} - 2287 q^{47} + 9604 q^{49} - 54017 q^{51} - 22238 q^{53} + 3075 q^{55} + 5890 q^{57} - 22656 q^{59} - 20250 q^{61} - 27783 q^{63} - 19725 q^{65} - 19456 q^{67} - 47006 q^{69} - 72288 q^{71} + 59464 q^{73} + 3125 q^{75} + 6027 q^{77} - 232001 q^{79} - 149604 q^{81} - 169620 q^{83} - 26275 q^{85} - 270309 q^{87} + 141434 q^{89} - 38661 q^{91} - 203808 q^{93} - 23950 q^{95} + 167159 q^{97} - 292198 q^{99}+O(q^{100}) $ 4 * q + 5 * q^3 - 100 * q^5 - 196 * q^7 + 567 * q^9 - 123 * q^11 + 789 * q^13 - 125 * q^15 + 1051 * q^17 + 958 * q^19 - 245 * q^21 - 530 * q^23 + 2500 * q^25 - 3169 * q^27 - 5541 * q^29 - 10440 * q^31 - 3737 * q^33 + 4900 * q^35 - 24048 * q^37 - 9131 * q^39 - 35414 * q^41 + 2174 * q^43 - 14175 * q^45 - 2287 * q^47 + 9604 * q^49 - 54017 * q^51 - 22238 * q^53 + 3075 * q^55 + 5890 * q^57 - 22656 * q^59 - 20250 * q^61 - 27783 * q^63 - 19725 * q^65 - 19456 * q^67 - 47006 * q^69 - 72288 * q^71 + 59464 * q^73 + 3125 * q^75 + 6027 * q^77 - 232001 * q^79 - 149604 * q^81 - 169620 * q^83 - 26275 * q^85 - 270309 * q^87 + 141434 * q^89 - 38661 * q^91 - 203808 * q^93 - 23950 * q^95 + 167159 * q^97 - 292198 * 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
$3$	−25.1636	−1.61425	−0.807124	−	0.590382i	$-0.798977\pi$
$3$	−25.1636	−1.61425	−0.807124	+	0.590382i	$0.798977\pi$
$4$	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
$9$	390.209	1.60580
$10$	0	0
$11$	−365.186	−0.909981	−0.454990	−	0.890496i	$-0.650357\pi$
$11$	−365.186	−0.909981	−0.454990	+	0.890496i	$0.650357\pi$
$12$	0	0
$13$	247.916	0.406861	0.203431	−	0.979089i	$-0.434791\pi$
$13$	247.916	0.406861	0.203431	+	0.979089i	$0.434791\pi$
$14$	0	0
$15$	629.091	0.721914
$16$	0	0
$17$	1639.32	1.37575	0.687877	−	0.725827i	$-0.258543\pi$
$17$	1639.32	1.37575	0.687877	+	0.725827i	$0.258543\pi$
$18$	0	0
$19$	−361.233	−0.229564	−0.114782	−	0.993391i	$-0.536617\pi$
$19$	−361.233	−0.229564	−0.114782	+	0.993391i	$0.536617\pi$
$20$	0	0
$21$	1233.02	0.610128
$22$	0	0
$23$	1415.45	0.557923	0.278961	−	0.960302i	$-0.410010\pi$
$23$	1415.45	0.557923	0.278961	+	0.960302i	$0.410010\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−3704.31	−0.977908
$28$	0	0
$29$	5204.44	1.14916	0.574578	−	0.818450i	$-0.305166\pi$
$29$	5204.44	1.14916	0.574578	+	0.818450i	$0.305166\pi$
$30$	0	0
$31$	5883.20	1.09954	0.549768	−	0.835317i	$-0.314716\pi$
$31$	5883.20	1.09954	0.549768	+	0.835317i	$0.314716\pi$
$32$	0	0
$33$	9189.40	1.46893
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	−11213.6	−1.34661	−0.673306	−	0.739364i	$-0.735126\pi$
$37$	−11213.6	−1.34661	−0.673306	+	0.739364i	$0.735126\pi$
$38$	0	0
$39$	−6238.47	−0.656775
$40$	0	0
$41$	−7249.75	−0.673540	−0.336770	−	0.941587i	$-0.609335\pi$
$41$	−7249.75	−0.673540	−0.336770	+	0.941587i	$0.609335\pi$
$42$	0	0
$43$	742.037	0.0612004	0.0306002	−	0.999532i	$-0.490258\pi$
$43$	742.037	0.0612004	0.0306002	+	0.999532i	$0.490258\pi$
$44$	0	0
$45$	−9755.22	−0.718134
$46$	0	0
$47$	−5166.17	−0.341133	−0.170567	−	0.985346i	$-0.554560\pi$
$47$	−5166.17	−0.341133	−0.170567	+	0.985346i	$0.554560\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−41251.2	−2.22081
$52$	0	0
$53$	16015.5	0.783162	0.391581	−	0.920144i	$-0.371928\pi$
$53$	16015.5	0.783162	0.391581	+	0.920144i	$0.371928\pi$
$54$	0	0
$55$	9129.64	0.406956
$56$	0	0
$57$	9089.94	0.370573
$58$	0	0
$59$	−19888.8	−0.743837	−0.371919	−	0.928265i	$-0.621300\pi$
$59$	−19888.8	−0.743837	−0.371919	+	0.928265i	$0.621300\pi$
$60$	0	0
$61$	−26621.9	−0.916040	−0.458020	−	0.888942i	$-0.651441\pi$
$61$	−26621.9	−0.916040	−0.458020	+	0.888942i	$0.651441\pi$
$62$	0	0
$63$	−19120.2	−0.606934
$64$	0	0
$65$	−6197.90	−0.181954
$66$	0	0
$67$	−15591.5	−0.424327	−0.212164	−	0.977234i	$-0.568051\pi$
$67$	−15591.5	−0.424327	−0.212164	+	0.977234i	$0.568051\pi$
$68$	0	0
$69$	−35617.8	−0.900626
$70$	0	0
$71$	−46315.6	−1.09039	−0.545194	−	0.838310i	$-0.683544\pi$
$71$	−46315.6	−1.09039	−0.545194	+	0.838310i	$0.683544\pi$
$72$	0	0
$73$	90796.7	1.99417	0.997087	−	0.0762736i	$-0.0243023\pi$
$73$	90796.7	1.99417	0.997087	+	0.0762736i	$0.0243023\pi$
$74$	0	0
$75$	−15727.3	−0.322850
$76$	0	0
$77$	17894.1	0.343940
$78$	0	0
$79$	−103600.	−1.86764	−0.933822	−	0.357738i	$-0.883548\pi$
$79$	−103600.	−1.86764	−0.933822	+	0.357738i	$0.883548\pi$
$80$	0	0
$81$	−1606.84	−0.0272119
$82$	0	0
$83$	−96467.3	−1.53704	−0.768520	−	0.639826i	$-0.779006\pi$
$83$	−96467.3	−1.53704	−0.768520	+	0.639826i	$0.779006\pi$
$84$	0	0
$85$	−40982.9	−0.615256
$86$	0	0
$87$	−130963.	−1.85502
$88$	0	0
$89$	−77336.2	−1.03492	−0.517461	−	0.855707i	$-0.673123\pi$
$89$	−77336.2	−1.03492	−0.517461	+	0.855707i	$0.673123\pi$
$90$	0	0
$91$	−12147.9	−0.153779
$92$	0	0
$93$	−148043.	−1.77492
$94$	0	0
$95$	9030.83	0.102664
$96$	0	0
$97$	147486.	1.59155	0.795776	−	0.605591i	$-0.207063\pi$
$97$	147486.	1.59155	0.795776	+	0.605591i	$0.207063\pi$
$98$	0	0
$99$	−142499.	−1.46124
$100$	0	0
$101$	144298.	1.40753	0.703764	−	0.710433i	$-0.251501\pi$
$101$	144298.	1.40753	0.703764	+	0.710433i	$0.251501\pi$
$102$	0	0
$103$	137831.	1.28013	0.640064	−	0.768322i	$-0.278908\pi$
$103$	137831.	1.28013	0.640064	+	0.768322i	$0.278908\pi$
$104$	0	0
$105$	−30825.5	−0.272858
$106$	0	0
$107$	27833.0	0.235018	0.117509	−	0.993072i	$-0.462509\pi$
$107$	27833.0	0.235018	0.117509	+	0.993072i	$0.462509\pi$
$108$	0	0
$109$	−71739.7	−0.578353	−0.289177	−	0.957276i	$-0.593381\pi$
$109$	−71739.7	−0.578353	−0.289177	+	0.957276i	$0.593381\pi$
$110$	0	0
$111$	282176.	2.17377
$112$	0	0
$113$	−68723.5	−0.506301	−0.253151	−	0.967427i	$-0.581467\pi$
$113$	−68723.5	−0.506301	−0.253151	+	0.967427i	$0.581467\pi$
$114$	0	0
$115$	−35386.2	−0.249511
$116$	0	0
$117$	96739.1	0.653337
$118$	0	0
$119$	−80326.6	−0.519986
$120$	0	0
$121$	−27690.4	−0.171935
$122$	0	0
$123$	182430.	1.08726
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	18556.3	0.102090	0.0510449	−	0.998696i	$-0.483745\pi$
$127$	18556.3	0.102090	0.0510449	+	0.998696i	$0.483745\pi$
$128$	0	0
$129$	−18672.3	−0.0987926
$130$	0	0
$131$	63743.4	0.324532	0.162266	−	0.986747i	$-0.448120\pi$
$131$	63743.4	0.324532	0.162266	+	0.986747i	$0.448120\pi$
$132$	0	0
$133$	17700.4	0.0867670
$134$	0	0
$135$	92607.7	0.437334
$136$	0	0
$137$	−4154.35	−0.0189104	−0.00945522	−	0.999955i	$-0.503010\pi$
$137$	−4154.35	−0.0189104	−0.00945522	+	0.999955i	$0.503010\pi$
$138$	0	0
$139$	−71572.6	−0.314203	−0.157101	−	0.987582i	$-0.550215\pi$
$139$	−71572.6	−0.314203	−0.157101	+	0.987582i	$0.550215\pi$
$140$	0	0
$141$	130000.	0.550673
$142$	0	0
$143$	−90535.4	−0.370236
$144$	0	0
$145$	−130111.	−0.513918
$146$	0	0
$147$	−60417.9	−0.230607
$148$	0	0
$149$	−113390.	−0.418417	−0.209209	−	0.977871i	$-0.567089\pi$
$149$	−113390.	−0.418417	−0.209209	+	0.977871i	$0.567089\pi$
$150$	0	0
$151$	−130434.	−0.465530	−0.232765	−	0.972533i	$-0.574777\pi$
$151$	−130434.	−0.465530	−0.232765	+	0.972533i	$0.574777\pi$
$152$	0	0
$153$	639676.	2.20918
$154$	0	0
$155$	−147080.	−0.491728
$156$	0	0
$157$	−162457.	−0.526006	−0.263003	−	0.964795i	$-0.584713\pi$
$157$	−162457.	−0.526006	−0.263003	+	0.964795i	$0.584713\pi$
$158$	0	0
$159$	−403009.	−1.26422
$160$	0	0
$161$	−69356.9	−0.210875
$162$	0	0
$163$	105629.	0.311397	0.155699	−	0.987805i	$-0.450237\pi$
$163$	105629.	0.311397	0.155699	+	0.987805i	$0.450237\pi$
$164$	0	0
$165$	−229735.	−0.656927
$166$	0	0
$167$	458132.	1.27116	0.635578	−	0.772036i	$-0.280762\pi$
$167$	458132.	1.27116	0.635578	+	0.772036i	$0.280762\pi$
$168$	0	0
$169$	−309831.	−0.834464
$170$	0	0
$171$	−140956.	−0.368633
$172$	0	0
$173$	−188044.	−0.477688	−0.238844	−	0.971058i	$-0.576768\pi$
$173$	−188044.	−0.477688	−0.238844	+	0.971058i	$0.576768\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	500474.	1.20074
$178$	0	0
$179$	13432.9	0.0313355	0.0156678	−	0.999877i	$-0.495013\pi$
$179$	13432.9	0.0313355	0.0156678	+	0.999877i	$0.495013\pi$
$180$	0	0
$181$	637767.	1.44699	0.723495	−	0.690329i	$-0.242534\pi$
$181$	637767.	1.44699	0.723495	+	0.690329i	$0.242534\pi$
$182$	0	0
$183$	669904.	1.47872
$184$	0	0
$185$	280341.	0.602223
$186$	0	0
$187$	−598655.	−1.25191
$188$	0	0
$189$	181511.	0.369614
$190$	0	0
$191$	−260266.	−0.516218	−0.258109	−	0.966116i	$-0.583099\pi$
$191$	−260266.	−0.516218	−0.258109	+	0.966116i	$0.583099\pi$
$192$	0	0
$193$	−200921.	−0.388269	−0.194134	−	0.980975i	$-0.562190\pi$
$193$	−200921.	−0.388269	−0.194134	+	0.980975i	$0.562190\pi$
$194$	0	0
$195$	155962.	0.293719
$196$	0	0
$197$	−17087.6	−0.0313700	−0.0156850	−	0.999877i	$-0.504993\pi$
$197$	−17087.6	−0.0313700	−0.0156850	+	0.999877i	$0.504993\pi$
$198$	0	0
$199$	74041.3	0.132538	0.0662692	−	0.997802i	$-0.478890\pi$
$199$	74041.3	0.132538	0.0662692	+	0.997802i	$0.478890\pi$
$200$	0	0
$201$	392339.	0.684970
$202$	0	0
$203$	−255018.	−0.434340
$204$	0	0
$205$	181244.	0.301216
$206$	0	0
$207$	552320.	0.895911
$208$	0	0
$209$	131917.	0.208899
$210$	0	0
$211$	−1.11383e6	−1.72232	−0.861159	−	0.508335i	$-0.830261\pi$
$211$	−1.11383e6	−1.72232	−0.861159	+	0.508335i	$0.830261\pi$
$212$	0	0
$213$	1.16547e6	1.76016
$214$	0	0
$215$	−18550.9	−0.0273696
$216$	0	0
$217$	−288277.	−0.415586
$218$	0	0
$219$	−2.28478e6	−3.21909
$220$	0	0
$221$	406413.	0.559741
$222$	0	0
$223$	1.18513e6	1.59589	0.797944	−	0.602732i	$-0.205921\pi$
$223$	1.18513e6	1.59589	0.797944	+	0.602732i	$0.205921\pi$
$224$	0	0
$225$	243880.	0.321159
$226$	0	0
$227$	−242599.	−0.312481	−0.156241	−	0.987719i	$-0.549938\pi$
$227$	−242599.	−0.312481	−0.156241	+	0.987719i	$0.549938\pi$
$228$	0	0
$229$	−1.15820e6	−1.45947	−0.729733	−	0.683732i	$-0.760356\pi$
$229$	−1.15820e6	−1.45947	−0.729733	+	0.683732i	$0.760356\pi$
$230$	0	0
$231$	−450281.	−0.555205
$232$	0	0
$233$	−1.21102e6	−1.46137	−0.730685	−	0.682714i	$-0.760799\pi$
$233$	−1.21102e6	−1.46137	−0.730685	+	0.682714i	$0.760799\pi$
$234$	0	0
$235$	129154.	0.152559
$236$	0	0
$237$	2.60697e6	3.01484
$238$	0	0
$239$	26252.2	0.0297283	0.0148642	−	0.999890i	$-0.495268\pi$
$239$	26252.2	0.0297283	0.0148642	+	0.999890i	$0.495268\pi$
$240$	0	0
$241$	1.73781e6	1.92735	0.963674	−	0.267080i	$-0.0860586\pi$
$241$	1.73781e6	1.92735	0.963674	+	0.267080i	$0.0860586\pi$
$242$	0	0
$243$	940581.	1.02183
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	−89555.6	−0.0934007
$248$	0	0
$249$	2.42747e6	2.48116
$250$	0	0
$251$	−975118.	−0.976951	−0.488476	−	0.872578i	$-0.662447\pi$
$251$	−975118.	−0.976951	−0.488476	+	0.872578i	$0.662447\pi$
$252$	0	0
$253$	−516901.	−0.507699
$254$	0	0
$255$	1.03128e6	0.993176
$256$	0	0
$257$	−495232.	−0.467709	−0.233854	−	0.972272i	$-0.575134\pi$
$257$	−495232.	−0.467709	−0.233854	+	0.972272i	$0.575134\pi$
$258$	0	0
$259$	549469.	0.508972
$260$	0	0
$261$	2.03082e6	1.84531
$262$	0	0
$263$	135868.	0.121123	0.0605614	−	0.998164i	$-0.480711\pi$
$263$	135868.	0.121123	0.0605614	+	0.998164i	$0.480711\pi$
$264$	0	0
$265$	−400388.	−0.350241
$266$	0	0
$267$	1.94606e6	1.67062
$268$	0	0
$269$	−622707.	−0.524690	−0.262345	−	0.964974i	$-0.584496\pi$
$269$	−622707.	−0.524690	−0.262345	+	0.964974i	$0.584496\pi$
$270$	0	0
$271$	291685.	0.241263	0.120631	−	0.992697i	$-0.461508\pi$
$271$	291685.	0.241263	0.120631	+	0.992697i	$0.461508\pi$
$272$	0	0
$273$	305685.	0.248238
$274$	0	0
$275$	−228241.	−0.181996
$276$	0	0
$277$	2.42715e6	1.90063	0.950316	−	0.311287i	$-0.100760\pi$
$277$	2.42715e6	1.90063	0.950316	+	0.311287i	$0.100760\pi$
$278$	0	0
$279$	2.29568e6	1.76563
$280$	0	0
$281$	−677951.	−0.512192	−0.256096	−	0.966651i	$-0.582436\pi$
$281$	−677951.	−0.512192	−0.256096	+	0.966651i	$0.582436\pi$
$282$	0	0
$283$	−819948.	−0.608583	−0.304292	−	0.952579i	$-0.598420\pi$
$283$	−819948.	−0.608583	−0.304292	+	0.952579i	$0.598420\pi$
$284$	0	0
$285$	−227249.	−0.165725
$286$	0	0
$287$	355238.	0.254574
$288$	0	0
$289$	1.26751e6	0.892699
$290$	0	0
$291$	−3.71128e6	−2.56916
$292$	0	0
$293$	−2.36322e6	−1.60818	−0.804092	−	0.594505i	$-0.797348\pi$
$293$	−2.36322e6	−1.60818	−0.804092	+	0.594505i	$0.797348\pi$
$294$	0	0
$295$	497219.	0.332654
$296$	0	0
$297$	1.35276e6	0.889877
$298$	0	0
$299$	350912.	0.226997
$300$	0	0
$301$	−36359.8	−0.0231316
$302$	0	0
$303$	−3.63107e6	−2.27210
$304$	0	0
$305$	665547.	0.409665
$306$	0	0
$307$	−155573.	−0.0942081	−0.0471041	−	0.998890i	$-0.514999\pi$
$307$	−155573.	−0.0942081	−0.0471041	+	0.998890i	$0.514999\pi$
$308$	0	0
$309$	−3.46833e6	−2.06644
$310$	0	0
$311$	1.94499e6	1.14029	0.570146	−	0.821543i	$-0.306887\pi$
$311$	1.94499e6	1.14029	0.570146	+	0.821543i	$0.306887\pi$
$312$	0	0
$313$	−2.78719e6	−1.60808	−0.804038	−	0.594578i	$-0.797319\pi$
$313$	−2.78719e6	−1.60808	−0.804038	+	0.594578i	$0.797319\pi$
$314$	0	0
$315$	478006.	0.271429
$316$	0	0
$317$	−2.73346e6	−1.52780	−0.763898	−	0.645337i	$-0.776717\pi$
$317$	−2.73346e6	−1.52780	−0.763898	+	0.645337i	$0.776717\pi$
$318$	0	0
$319$	−1.90059e6	−1.04571
$320$	0	0
$321$	−700380.	−0.379377
$322$	0	0
$323$	−592176.	−0.315824
$324$	0	0
$325$	154948.	0.0813723
$326$	0	0
$327$	1.80523e6	0.933605
$328$	0	0
$329$	253142.	0.128936
$330$	0	0
$331$	−1.02078e6	−0.512108	−0.256054	−	0.966662i	$-0.582423\pi$
$331$	−1.02078e6	−0.512108	−0.256054	+	0.966662i	$0.582423\pi$
$332$	0	0
$333$	−4.37566e6	−2.16239
$334$	0	0
$335$	389788.	0.189765
$336$	0	0
$337$	1.32493e6	0.635504	0.317752	−	0.948174i	$-0.397072\pi$
$337$	1.32493e6	0.635504	0.317752	+	0.948174i	$0.397072\pi$
$338$	0	0
$339$	1.72933e6	0.817296
$340$	0	0
$341$	−2.14846e6	−1.00056
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	890445.	0.402772
$346$	0	0
$347$	−3.26525e6	−1.45577	−0.727884	−	0.685700i	$-0.759496\pi$
$347$	−3.26525e6	−1.45577	−0.727884	+	0.685700i	$0.759496\pi$
$348$	0	0
$349$	−2.65657e6	−1.16750	−0.583752	−	0.811932i	$-0.698416\pi$
$349$	−2.65657e6	−1.16750	−0.583752	+	0.811932i	$0.698416\pi$
$350$	0	0
$351$	−918358.	−0.397873
$352$	0	0
$353$	−2.74789e6	−1.17371	−0.586857	−	0.809691i	$-0.699635\pi$
$353$	−2.74789e6	−1.17371	−0.586857	+	0.809691i	$0.699635\pi$
$354$	0	0
$355$	1.15789e6	0.487637
$356$	0	0
$357$	2.02131e6	0.839387
$358$	0	0
$359$	2.29396e6	0.939399	0.469700	−	0.882826i	$-0.344362\pi$
$359$	2.29396e6	0.939399	0.469700	+	0.882826i	$0.344362\pi$
$360$	0	0
$361$	−2.34561e6	−0.947300
$362$	0	0
$363$	696791.	0.277546
$364$	0	0
$365$	−2.26992e6	−0.891822
$366$	0	0
$367$	−3.75348e6	−1.45469	−0.727343	−	0.686274i	$-0.759245\pi$
$367$	−3.75348e6	−1.45469	−0.727343	+	0.686274i	$0.759245\pi$
$368$	0	0
$369$	−2.82892e6	−1.08157
$370$	0	0
$371$	−784761.	−0.296007
$372$	0	0
$373$	249641.	0.0929061	0.0464531	−	0.998920i	$-0.485208\pi$
$373$	249641.	0.0929061	0.0464531	+	0.998920i	$0.485208\pi$
$374$	0	0
$375$	393182.	0.144383
$376$	0	0
$377$	1.29026e6	0.467547
$378$	0	0
$379$	3.78417e6	1.35323	0.676617	−	0.736335i	$-0.263445\pi$
$379$	3.78417e6	1.35323	0.676617	+	0.736335i	$0.263445\pi$
$380$	0	0
$381$	−466945.	−0.164798
$382$	0	0
$383$	−175121.	−0.0610016	−0.0305008	−	0.999535i	$-0.509710\pi$
$383$	−175121.	−0.0610016	−0.0305008	+	0.999535i	$0.509710\pi$
$384$	0	0
$385$	−447353.	−0.153815
$386$	0	0
$387$	289549.	0.0982754
$388$	0	0
$389$	−4.93382e6	−1.65314	−0.826570	−	0.562834i	$-0.809711\pi$
$389$	−4.93382e6	−1.65314	−0.826570	+	0.562834i	$0.809711\pi$
$390$	0	0
$391$	2.32037e6	0.767564
$392$	0	0
$393$	−1.60402e6	−0.523874
$394$	0	0
$395$	2.59001e6	0.835236
$396$	0	0
$397$	−3.55839e6	−1.13312	−0.566562	−	0.824019i	$-0.691727\pi$
$397$	−3.55839e6	−1.13312	−0.566562	+	0.824019i	$0.691727\pi$
$398$	0	0
$399$	−445407.	−0.140064
$400$	0	0
$401$	1.56407e6	0.485730	0.242865	−	0.970060i	$-0.421913\pi$
$401$	1.56407e6	0.485730	0.242865	+	0.970060i	$0.421913\pi$
$402$	0	0
$403$	1.45854e6	0.447359
$404$	0	0
$405$	40170.9	0.0121695
$406$	0	0
$407$	4.09506e6	1.22539
$408$	0	0
$409$	−1.72024e6	−0.508488	−0.254244	−	0.967140i	$-0.581827\pi$
$409$	−1.72024e6	−0.508488	−0.254244	+	0.967140i	$0.581827\pi$
$410$	0	0
$411$	104539.	0.0305262
$412$	0	0
$413$	974550.	0.281144
$414$	0	0
$415$	2.41168e6	0.687385
$416$	0	0
$417$	1.80103e6	0.507201
$418$	0	0
$419$	2.74755e6	0.764558	0.382279	−	0.924047i	$-0.375139\pi$
$419$	2.74755e6	0.764558	0.382279	+	0.924047i	$0.375139\pi$
$420$	0	0
$421$	−3.21284e6	−0.883454	−0.441727	−	0.897150i	$-0.645634\pi$
$421$	−3.21284e6	−0.883454	−0.441727	+	0.897150i	$0.645634\pi$
$422$	0	0
$423$	−2.01588e6	−0.547791
$424$	0	0
$425$	1.02457e6	0.275151
$426$	0	0
$427$	1.30447e6	0.346230
$428$	0	0
$429$	2.27820e6	0.597653
$430$	0	0
$431$	−5.78011e6	−1.49880	−0.749399	−	0.662119i	$-0.769657\pi$
$431$	−5.78011e6	−1.49880	−0.749399	+	0.662119i	$0.769657\pi$
$432$	0	0
$433$	−4.11053e6	−1.05360	−0.526802	−	0.849988i	$-0.676609\pi$
$433$	−4.11053e6	−1.05360	−0.526802	+	0.849988i	$0.676609\pi$
$434$	0	0
$435$	3.27407e6	0.829592
$436$	0	0
$437$	−511306.	−0.128079
$438$	0	0
$439$	−6.74527e6	−1.67047	−0.835234	−	0.549895i	$-0.814668\pi$
$439$	−6.74527e6	−1.67047	−0.835234	+	0.549895i	$0.814668\pi$
$440$	0	0
$441$	936891.	0.229400
$442$	0	0
$443$	−4.05057e6	−0.980634	−0.490317	−	0.871544i	$-0.663119\pi$
$443$	−4.05057e6	−0.980634	−0.490317	+	0.871544i	$0.663119\pi$
$444$	0	0
$445$	1.93341e6	0.462831
$446$	0	0
$447$	2.85331e6	0.675429
$448$	0	0
$449$	6.36601e6	1.49022	0.745112	−	0.666939i	$-0.232396\pi$
$449$	6.36601e6	1.49022	0.745112	+	0.666939i	$0.232396\pi$
$450$	0	0
$451$	2.64751e6	0.612909
$452$	0	0
$453$	3.28219e6	0.751481
$454$	0	0
$455$	303697.	0.0687721
$456$	0	0
$457$	3.97423e6	0.890150	0.445075	−	0.895493i	$-0.353177\pi$
$457$	3.97423e6	0.890150	0.445075	+	0.895493i	$0.353177\pi$
$458$	0	0
$459$	−6.07254e6	−1.34536
$460$	0	0
$461$	102643.	0.0224945	0.0112473	−	0.999937i	$-0.496420\pi$
$461$	102643.	0.0224945	0.0112473	+	0.999937i	$0.496420\pi$
$462$	0	0
$463$	6.67133e6	1.44631	0.723153	−	0.690688i	$-0.242692\pi$
$463$	6.67133e6	1.44631	0.723153	+	0.690688i	$0.242692\pi$
$464$	0	0
$465$	3.70107e6	0.793771
$466$	0	0
$467$	7.73210e6	1.64061	0.820305	−	0.571926i	$-0.193803\pi$
$467$	7.73210e6	1.64061	0.820305	+	0.571926i	$0.193803\pi$
$468$	0	0
$469$	763984.	0.160381
$470$	0	0
$471$	4.08802e6	0.849104
$472$	0	0
$473$	−270981.	−0.0556912
$474$	0	0
$475$	−225771.	−0.0459128
$476$	0	0
$477$	6.24940e6	1.25760
$478$	0	0
$479$	−8.12611e6	−1.61824	−0.809122	−	0.587641i	$-0.800057\pi$
$479$	−8.12611e6	−1.61824	−0.809122	+	0.587641i	$0.800057\pi$
$480$	0	0
$481$	−2.78004e6	−0.547884
$482$	0	0
$483$	1.74527e6	0.340404
$484$	0	0
$485$	−3.68715e6	−0.711764
$486$	0	0
$487$	−3.13029e6	−0.598084	−0.299042	−	0.954240i	$-0.596667\pi$
$487$	−3.13029e6	−0.598084	−0.299042	+	0.954240i	$0.596667\pi$
$488$	0	0
$489$	−2.65801e6	−0.502673
$490$	0	0
$491$	6.69469e6	1.25322	0.626610	−	0.779333i	$-0.284442\pi$
$491$	6.69469e6	1.25322	0.626610	+	0.779333i	$0.284442\pi$
$492$	0	0
$493$	8.53173e6	1.58096
$494$	0	0
$495$	3.56247e6	0.653488
$496$	0	0
$497$	2.26946e6	0.412128
$498$	0	0
$499$	1.47889e6	0.265880	0.132940	−	0.991124i	$-0.457558\pi$
$499$	1.47889e6	0.265880	0.132940	+	0.991124i	$0.457558\pi$
$500$	0	0
$501$	−1.15283e7	−2.05196
$502$	0	0
$503$	−6.97626e6	−1.22943	−0.614713	−	0.788751i	$-0.710728\pi$
$503$	−6.97626e6	−1.22943	−0.614713	+	0.788751i	$0.710728\pi$
$504$	0	0
$505$	−3.60745e6	−0.629466
$506$	0	0
$507$	7.79647e6	1.34703
$508$	0	0
$509$	−4.42031e6	−0.756237	−0.378118	−	0.925757i	$-0.623429\pi$
$509$	−4.42031e6	−0.756237	−0.378118	+	0.925757i	$0.623429\pi$
$510$	0	0
$511$	−4.44904e6	−0.753727
$512$	0	0
$513$	1.33812e6	0.224492
$514$	0	0
$515$	−3.44577e6	−0.572490
$516$	0	0
$517$	1.88661e6	0.310424
$518$	0	0
$519$	4.73187e6	0.771106
$520$	0	0
$521$	−216174.	−0.0348906	−0.0174453	−	0.999848i	$-0.505553\pi$
$521$	−216174.	−0.0348906	−0.0174453	+	0.999848i	$0.505553\pi$
$522$	0	0
$523$	3.58796e6	0.573579	0.286790	−	0.957994i	$-0.407412\pi$
$523$	3.58796e6	0.573579	0.286790	+	0.957994i	$0.407412\pi$
$524$	0	0
$525$	770636.	0.122026
$526$	0	0
$527$	9.64444e6	1.51269
$528$	0	0
$529$	−4.43285e6	−0.688722
$530$	0	0
$531$	−7.76077e6	−1.19445
$532$	0	0
$533$	−1.79733e6	−0.274038
$534$	0	0
$535$	−695826.	−0.105103
$536$	0	0
$537$	−338020.	−0.0505833
$538$	0	0
$539$	−876811.	−0.129997
$540$	0	0
$541$	1.22564e7	1.80040	0.900198	−	0.435480i	$-0.143421\pi$
$541$	1.22564e7	1.80040	0.900198	+	0.435480i	$0.143421\pi$
$542$	0	0
$543$	−1.60485e7	−2.33580
$544$	0	0
$545$	1.79349e6	0.258647
$546$	0	0
$547$	5.27402e6	0.753656	0.376828	−	0.926283i	$-0.377015\pi$
$547$	5.27402e6	0.753656	0.376828	+	0.926283i	$0.377015\pi$
$548$	0	0
$549$	−1.03881e7	−1.47097
$550$	0	0
$551$	−1.88002e6	−0.263805
$552$	0	0
$553$	5.07642e6	0.705903
$554$	0	0
$555$	−7.05440e6	−0.972138
$556$	0	0
$557$	−2.69954e6	−0.368682	−0.184341	−	0.982862i	$-0.559015\pi$
$557$	−2.69954e6	−0.368682	−0.184341	+	0.982862i	$0.559015\pi$
$558$	0	0
$559$	183963.	0.0249001
$560$	0	0
$561$	1.50644e7	2.02089
$562$	0	0
$563$	5.15141e6	0.684944	0.342472	−	0.939528i	$-0.388736\pi$
$563$	5.15141e6	0.684944	0.342472	+	0.939528i	$0.388736\pi$
$564$	0	0
$565$	1.71809e6	0.226425
$566$	0	0
$567$	78735.1	0.0102851
$568$	0	0
$569$	−1.41211e7	−1.82848	−0.914238	−	0.405177i	$-0.867210\pi$
$569$	−1.41211e7	−1.82848	−0.914238	+	0.405177i	$0.867210\pi$
$570$	0	0
$571$	8.36776e6	1.07404	0.537018	−	0.843571i	$-0.319551\pi$
$571$	8.36776e6	1.07404	0.537018	+	0.843571i	$0.319551\pi$
$572$	0	0
$573$	6.54923e6	0.833305
$574$	0	0
$575$	884654.	0.111585
$576$	0	0
$577$	9.79283e6	1.22453	0.612264	−	0.790653i	$-0.290259\pi$
$577$	9.79283e6	1.22453	0.612264	+	0.790653i	$0.290259\pi$
$578$	0	0
$579$	5.05591e6	0.626762
$580$	0	0
$581$	4.72690e6	0.580946
$582$	0	0
$583$	−5.84864e6	−0.712662
$584$	0	0
$585$	−2.41848e6	−0.292181
$586$	0	0
$587$	−1.25486e7	−1.50315	−0.751574	−	0.659649i	$-0.770705\pi$
$587$	−1.25486e7	−1.50315	−0.751574	+	0.659649i	$0.770705\pi$
$588$	0	0
$589$	−2.12521e6	−0.252414
$590$	0	0
$591$	429985.	0.0506389
$592$	0	0
$593$	2.67414e6	0.312282	0.156141	−	0.987735i	$-0.450095\pi$
$593$	2.67414e6	0.312282	0.156141	+	0.987735i	$0.450095\pi$
$594$	0	0
$595$	2.00816e6	0.232545
$596$	0	0
$597$	−1.86315e6	−0.213950
$598$	0	0
$599$	278772.	0.0317455	0.0158727	−	0.999874i	$-0.494947\pi$
$599$	278772.	0.0317455	0.0158727	+	0.999874i	$0.494947\pi$
$600$	0	0
$601$	1.52568e7	1.72296	0.861482	−	0.507788i	$-0.169537\pi$
$601$	1.52568e7	1.72296	0.861482	+	0.507788i	$0.169537\pi$
$602$	0	0
$603$	−6.08394e6	−0.681384
$604$	0	0
$605$	692259.	0.0768919
$606$	0	0
$607$	8.48247e6	0.934438	0.467219	−	0.884142i	$-0.345256\pi$
$607$	8.48247e6	0.934438	0.467219	+	0.884142i	$0.345256\pi$
$608$	0	0
$609$	6.41717e6	0.701133
$610$	0	0
$611$	−1.28078e6	−0.138794
$612$	0	0
$613$	−4.98365e6	−0.535668	−0.267834	−	0.963465i	$-0.586308\pi$
$613$	−4.98365e6	−0.535668	−0.267834	+	0.963465i	$0.586308\pi$
$614$	0	0
$615$	−4.56076e6	−0.486238
$616$	0	0
$617$	−1.09072e7	−1.15345	−0.576726	−	0.816938i	$-0.695670\pi$
$617$	−1.09072e7	−1.15345	−0.576726	+	0.816938i	$0.695670\pi$
$618$	0	0
$619$	1.77881e7	1.86596	0.932982	−	0.359923i	$-0.117197\pi$
$619$	1.77881e7	1.86596	0.932982	+	0.359923i	$0.117197\pi$
$620$	0	0
$621$	−5.24325e6	−0.545597
$622$	0	0
$623$	3.78947e6	0.391164
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−3.31952e6	−0.337214
$628$	0	0
$629$	−1.83827e7	−1.85261
$630$	0	0
$631$	5.89959e6	0.589859	0.294930	−	0.955519i	$-0.404704\pi$
$631$	5.89959e6	0.589859	0.294930	+	0.955519i	$0.404704\pi$
$632$	0	0
$633$	2.80281e7	2.78025
$634$	0	0
$635$	−463908.	−0.0456560
$636$	0	0
$637$	595247.	0.0581230
$638$	0	0
$639$	−1.80727e7	−1.75094
$640$	0	0
$641$	9.03424e6	0.868454	0.434227	−	0.900804i	$-0.357022\pi$
$641$	9.03424e6	0.868454	0.434227	+	0.900804i	$0.357022\pi$
$642$	0	0
$643$	1.02806e6	0.0980600	0.0490300	−	0.998797i	$-0.484387\pi$
$643$	1.02806e6	0.0980600	0.0490300	+	0.998797i	$0.484387\pi$
$644$	0	0
$645$	466809.	0.0441814
$646$	0	0
$647$	−1.40636e6	−0.132080	−0.0660399	−	0.997817i	$-0.521036\pi$
$647$	−1.40636e6	−0.132080	−0.0660399	+	0.997817i	$0.521036\pi$
$648$	0	0
$649$	7.26310e6	0.676877
$650$	0	0
$651$	7.25410e6	0.670859
$652$	0	0
$653$	4.98822e6	0.457787	0.228893	−	0.973452i	$-0.426489\pi$
$653$	4.98822e6	0.457787	0.228893	+	0.973452i	$0.426489\pi$
$654$	0	0
$655$	−1.59358e6	−0.145135
$656$	0	0
$657$	3.54297e7	3.20224
$658$	0	0
$659$	−1.27209e7	−1.14105	−0.570525	−	0.821280i	$-0.693260\pi$
$659$	−1.27209e7	−1.14105	−0.570525	+	0.821280i	$0.693260\pi$
$660$	0	0
$661$	−2.12617e7	−1.89275	−0.946375	−	0.323069i	$-0.895285\pi$
$661$	−2.12617e7	−1.89275	−0.946375	+	0.323069i	$0.895285\pi$
$662$	0	0
$663$	−1.02268e7	−0.903561
$664$	0	0
$665$	−442511.	−0.0388034
$666$	0	0
$667$	7.36661e6	0.641140
$668$	0	0
$669$	−2.98221e7	−2.57616
$670$	0	0
$671$	9.72193e6	0.833578
$672$	0	0
$673$	1.10908e7	0.943901	0.471950	−	0.881625i	$-0.343550\pi$
$673$	1.10908e7	0.943901	0.471950	+	0.881625i	$0.343550\pi$
$674$	0	0
$675$	−2.31519e6	−0.195582
$676$	0	0
$677$	−1.79324e7	−1.50372	−0.751858	−	0.659325i	$-0.770842\pi$
$677$	−1.79324e7	−1.50372	−0.751858	+	0.659325i	$0.770842\pi$
$678$	0	0
$679$	−7.22681e6	−0.601550
$680$	0	0
$681$	6.10467e6	0.504422
$682$	0	0
$683$	2.17980e7	1.78799	0.893995	−	0.448076i	$-0.147891\pi$
$683$	2.17980e7	1.78799	0.893995	+	0.448076i	$0.147891\pi$
$684$	0	0
$685$	103859.	0.00845701
$686$	0	0
$687$	2.91445e7	2.35594
$688$	0	0
$689$	3.97051e6	0.318638
$690$	0	0
$691$	−5.20702e6	−0.414853	−0.207426	−	0.978251i	$-0.566509\pi$
$691$	−5.20702e6	−0.414853	−0.207426	+	0.978251i	$0.566509\pi$
$692$	0	0
$693$	6.98244e6	0.552298
$694$	0	0
$695$	1.78932e6	0.140516
$696$	0	0
$697$	−1.18847e7	−0.926626
$698$	0	0
$699$	3.04736e7	2.35901
$700$	0	0
$701$	−4.54067e6	−0.349000	−0.174500	−	0.984657i	$-0.555831\pi$
$701$	−4.54067e6	−0.349000	−0.174500	+	0.984657i	$0.555831\pi$
$702$	0	0
$703$	4.05074e6	0.309134
$704$	0	0
$705$	−3.24999e6	−0.246269
$706$	0	0
$707$	−7.07061e6	−0.531996
$708$	0	0
$709$	−648628.	−0.0484597	−0.0242298	−	0.999706i	$-0.507713\pi$
$709$	−648628.	−0.0484597	−0.0242298	+	0.999706i	$0.507713\pi$
$710$	0	0
$711$	−4.04258e7	−2.99906
$712$	0	0
$713$	8.32736e6	0.613456
$714$	0	0
$715$	2.26339e6	0.165575
$716$	0	0
$717$	−660600.	−0.0479889
$718$	0	0
$719$	3.40688e6	0.245773	0.122887	−	0.992421i	$-0.460785\pi$
$719$	3.40688e6	0.245773	0.122887	+	0.992421i	$0.460785\pi$
$720$	0	0
$721$	−6.75371e6	−0.483843
$722$	0	0
$723$	−4.37297e7	−3.11122
$724$	0	0
$725$	3.25278e6	0.229831
$726$	0	0
$727$	2.67141e6	0.187458	0.0937290	−	0.995598i	$-0.470121\pi$
$727$	2.67141e6	0.187458	0.0937290	+	0.995598i	$0.470121\pi$
$728$	0	0
$729$	−2.32780e7	−1.62228
$730$	0	0
$731$	1.21643e6	0.0841967
$732$	0	0
$733$	2.26541e7	1.55735	0.778675	−	0.627427i	$-0.215892\pi$
$733$	2.26541e7	1.55735	0.778675	+	0.627427i	$0.215892\pi$
$734$	0	0
$735$	1.51045e6	0.103131
$736$	0	0
$737$	5.69379e6	0.386130
$738$	0	0
$739$	2.62466e6	0.176792	0.0883958	−	0.996085i	$-0.471826\pi$
$739$	2.62466e6	0.176792	0.0883958	+	0.996085i	$0.471826\pi$
$740$	0	0
$741$	2.25354e6	0.150772
$742$	0	0
$743$	−2.22277e7	−1.47714	−0.738570	−	0.674177i	$-0.764499\pi$
$743$	−2.22277e7	−1.47714	−0.738570	+	0.674177i	$0.764499\pi$
$744$	0	0
$745$	2.83475e6	0.187122
$746$	0	0
$747$	−3.76424e7	−2.46817
$748$	0	0
$749$	−1.36382e6	−0.0888284
$750$	0	0
$751$	1.83146e7	1.18495	0.592473	−	0.805591i	$-0.298152\pi$
$751$	1.83146e7	1.18495	0.592473	+	0.805591i	$0.298152\pi$
$752$	0	0
$753$	2.45375e7	1.57704
$754$	0	0
$755$	3.26084e6	0.208191
$756$	0	0
$757$	−1.32037e7	−0.837447	−0.418724	−	0.908114i	$-0.637522\pi$
$757$	−1.32037e7	−0.837447	−0.418724	+	0.908114i	$0.637522\pi$
$758$	0	0
$759$	1.30071e7	0.819552
$760$	0	0
$761$	−2.84707e7	−1.78212	−0.891059	−	0.453888i	$-0.850037\pi$
$761$	−2.84707e7	−1.78212	−0.891059	+	0.453888i	$0.850037\pi$
$762$	0	0
$763$	3.51524e6	0.218597
$764$	0	0
$765$	−1.59919e7	−0.987976
$766$	0	0
$767$	−4.93075e6	−0.302639
$768$	0	0
$769$	1.28629e7	0.784373	0.392187	−	0.919886i	$-0.371719\pi$
$769$	1.28629e7	0.784373	0.392187	+	0.919886i	$0.371719\pi$
$770$	0	0
$771$	1.24618e7	0.754998
$772$	0	0
$773$	−2.65300e6	−0.159694	−0.0798470	−	0.996807i	$-0.525443\pi$
$773$	−2.65300e6	−0.159694	−0.0798470	+	0.996807i	$0.525443\pi$
$774$	0	0
$775$	3.67700e6	0.219907
$776$	0	0
$777$	−1.38266e7	−0.821606
$778$	0	0
$779$	2.61885e6	0.154621
$780$	0	0
$781$	1.69138e7	0.992232
$782$	0	0
$783$	−1.92789e7	−1.12377
$784$	0	0
$785$	4.06144e6	0.235237
$786$	0	0
$787$	−8.41414e6	−0.484254	−0.242127	−	0.970245i	$-0.577845\pi$
$787$	−8.41414e6	−0.484254	−0.242127	+	0.970245i	$0.577845\pi$
$788$	0	0
$789$	−3.41892e6	−0.195522
$790$	0	0
$791$	3.36745e6	0.191364
$792$	0	0
$793$	−6.60000e6	−0.372701
$794$	0	0
$795$	1.00752e7	0.565375
$796$	0	0
$797$	7.14739e6	0.398567	0.199284	−	0.979942i	$-0.436138\pi$
$797$	7.14739e6	0.398567	0.199284	+	0.979942i	$0.436138\pi$
$798$	0	0
$799$	−8.46899e6	−0.469315
$800$	0	0
$801$	−3.01773e7	−1.66188
$802$	0	0
$803$	−3.31577e7	−1.81466
$804$	0	0
$805$	1.73392e6	0.0943061
$806$	0	0
$807$	1.56696e7	0.846981
$808$	0	0
$809$	−3.20475e7	−1.72156	−0.860782	−	0.508973i	$-0.830025\pi$
$809$	−3.20475e7	−1.72156	−0.860782	+	0.508973i	$0.830025\pi$
$810$	0	0
$811$	2.53424e7	1.35299	0.676496	−	0.736446i	$-0.263498\pi$
$811$	2.53424e7	1.35299	0.676496	+	0.736446i	$0.263498\pi$
$812$	0	0
$813$	−7.33985e6	−0.389458
$814$	0	0
$815$	−2.64073e6	−0.139261
$816$	0	0
$817$	−268048.	−0.0140494
$818$	0	0
$819$	−4.74021e6	−0.246938
$820$	0	0
$821$	−5.48235e6	−0.283863	−0.141931	−	0.989876i	$-0.545331\pi$
$821$	−5.48235e6	−0.283863	−0.141931	+	0.989876i	$0.545331\pi$
$822$	0	0
$823$	−1.86354e7	−0.959048	−0.479524	−	0.877529i	$-0.659191\pi$
$823$	−1.86354e7	−0.959048	−0.479524	+	0.877529i	$0.659191\pi$
$824$	0	0
$825$	5.74338e6	0.293787
$826$	0	0
$827$	−2.98317e7	−1.51675	−0.758375	−	0.651818i	$-0.774006\pi$
$827$	−2.98317e7	−1.51675	−0.758375	+	0.651818i	$0.774006\pi$
$828$	0	0
$829$	4.73659e6	0.239375	0.119688	−	0.992812i	$-0.461811\pi$
$829$	4.73659e6	0.239375	0.119688	+	0.992812i	$0.461811\pi$
$830$	0	0
$831$	−6.10760e7	−3.06809
$832$	0	0
$833$	3.93600e6	0.196536
$834$	0	0
$835$	−1.14533e7	−0.568479
$836$	0	0
$837$	−2.17932e7	−1.07525
$838$	0	0
$839$	−908022.	−0.0445340	−0.0222670	−	0.999752i	$-0.507088\pi$
$839$	−908022.	−0.0445340	−0.0222670	+	0.999752i	$0.507088\pi$
$840$	0	0
$841$	6.57505e6	0.320560
$842$	0	0
$843$	1.70597e7	0.826805
$844$	0	0
$845$	7.74576e6	0.373184
$846$	0	0
$847$	1.35683e6	0.0649855
$848$	0	0
$849$	2.06329e7	0.982405
$850$	0	0
$851$	−1.58723e7	−0.751305
$852$	0	0
$853$	−3.26310e7	−1.53553	−0.767764	−	0.640733i	$-0.778631\pi$
$853$	−3.26310e7	−1.53553	−0.767764	+	0.640733i	$0.778631\pi$
$854$	0	0
$855$	3.52391e6	0.164858
$856$	0	0
$857$	−1.37646e7	−0.640196	−0.320098	−	0.947385i	$-0.603716\pi$
$857$	−1.37646e7	−0.640196	−0.320098	+	0.947385i	$0.603716\pi$
$858$	0	0
$859$	−2.37219e7	−1.09690	−0.548449	−	0.836184i	$-0.684782\pi$
$859$	−2.37219e7	−1.09690	−0.548449	+	0.836184i	$0.684782\pi$
$860$	0	0
$861$	−8.93908e6	−0.410946
$862$	0	0
$863$	−3.65712e7	−1.67152	−0.835761	−	0.549094i	$-0.814973\pi$
$863$	−3.65712e7	−1.67152	−0.835761	+	0.549094i	$0.814973\pi$
$864$	0	0
$865$	4.70110e6	0.213628
$866$	0	0
$867$	−3.18951e7	−1.44104
$868$	0	0
$869$	3.78334e7	1.69952
$870$	0	0
$871$	−3.86539e6	−0.172642
$872$	0	0
$873$	5.75503e7	2.55571
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	1.19092e7	0.522857	0.261429	−	0.965223i	$-0.415806\pi$
$877$	1.19092e7	0.522857	0.261429	+	0.965223i	$0.415806\pi$
$878$	0	0
$879$	5.94673e7	2.59601
$880$	0	0
$881$	3.34066e7	1.45008	0.725042	−	0.688705i	$-0.241820\pi$
$881$	3.34066e7	1.45008	0.725042	+	0.688705i	$0.241820\pi$
$882$	0	0
$883$	−1.18505e7	−0.511489	−0.255745	−	0.966744i	$-0.582321\pi$
$883$	−1.18505e7	−0.511489	−0.255745	+	0.966744i	$0.582321\pi$
$884$	0	0
$885$	−1.25118e7	−0.536986
$886$	0	0
$887$	−3.57684e7	−1.52648	−0.763238	−	0.646117i	$-0.776392\pi$
$887$	−3.57684e7	−1.52648	−0.763238	+	0.646117i	$0.776392\pi$
$888$	0	0
$889$	−909260.	−0.0385863
$890$	0	0
$891$	586794.	0.0247623
$892$	0	0
$893$	1.86619e6	0.0783119
$894$	0	0
$895$	−335822.	−0.0140137
$896$	0	0
$897$	−8.83022e6	−0.366430
$898$	0	0
$899$	3.06188e7	1.26354
$900$	0	0
$901$	2.62545e7	1.07744
$902$	0	0
$903$	914945.	0.0373401
$904$	0	0
$905$	−1.59442e7	−0.647114
$906$	0	0
$907$	−4.92818e7	−1.98916	−0.994578	−	0.103995i	$-0.966837\pi$
$907$	−4.92818e7	−1.98916	−0.994578	+	0.103995i	$0.966837\pi$
$908$	0	0
$909$	5.63064e7	2.26021
$910$	0	0
$911$	2.36406e7	0.943761	0.471881	−	0.881662i	$-0.343575\pi$
$911$	2.36406e7	0.943761	0.471881	+	0.881662i	$0.343575\pi$
$912$	0	0
$913$	3.52285e7	1.39868
$914$	0	0
$915$	−1.67476e7	−0.661302
$916$	0	0
$917$	−3.12343e6	−0.122661
$918$	0	0
$919$	−122689.	−0.00479201	−0.00239600	−	0.999997i	$-0.500763\pi$
$919$	−122689.	−0.00479201	−0.00239600	+	0.999997i	$0.500763\pi$
$920$	0	0
$921$	3.91478e6	0.152075
$922$	0	0
$923$	−1.14824e7	−0.443637
$924$	0	0
$925$	−7.00853e6	−0.269322
$926$	0	0
$927$	5.37828e7	2.05563
$928$	0	0
$929$	−693132.	−0.0263498	−0.0131749	−	0.999913i	$-0.504194\pi$
$929$	−693132.	−0.0263498	−0.0131749	+	0.999913i	$0.504194\pi$
$930$	0	0
$931$	−867321.	−0.0327949
$932$	0	0
$933$	−4.89430e7	−1.84071
$934$	0	0
$935$	1.49664e7	0.559871
$936$	0	0
$937$	5.00421e7	1.86203	0.931014	−	0.364983i	$-0.118925\pi$
$937$	5.00421e7	1.86203	0.931014	+	0.364983i	$0.118925\pi$
$938$	0	0
$939$	7.01360e7	2.59583
$940$	0	0
$941$	−2.34763e7	−0.864282	−0.432141	−	0.901806i	$-0.642242\pi$
$941$	−2.34763e7	−0.864282	−0.432141	+	0.901806i	$0.642242\pi$
$942$	0	0
$943$	−1.02616e7	−0.375783
$944$	0	0
$945$	−4.53778e6	−0.165297
$946$	0	0
$947$	2.45096e7	0.888097	0.444049	−	0.896003i	$-0.353542\pi$
$947$	2.45096e7	0.888097	0.444049	+	0.896003i	$0.353542\pi$
$948$	0	0
$949$	2.25100e7	0.811352
$950$	0	0
$951$	6.87839e7	2.46624
$952$	0	0
$953$	−4.15441e7	−1.48176	−0.740879	−	0.671638i	$-0.765591\pi$
$953$	−4.15441e7	−1.48176	−0.740879	+	0.671638i	$0.765591\pi$
$954$	0	0
$955$	6.50664e6	0.230860
$956$	0	0
$957$	4.78257e7	1.68804
$958$	0	0
$959$	203563.	0.00714748
$960$	0	0
$961$	5.98294e6	0.208981
$962$	0	0
$963$	1.08607e7	0.377391
$964$	0	0
$965$	5.02303e6	0.173639
$966$	0	0
$967$	−1.77043e7	−0.608852	−0.304426	−	0.952536i	$-0.598465\pi$
$967$	−1.77043e7	−0.608852	−0.304426	+	0.952536i	$0.598465\pi$
$968$	0	0
$969$	1.49013e7	0.509818
$970$	0	0
$971$	−3.22470e7	−1.09759	−0.548797	−	0.835956i	$-0.684914\pi$
$971$	−3.22470e7	−1.09759	−0.548797	+	0.835956i	$0.684914\pi$
$972$	0	0
$973$	3.50706e6	0.118757
$974$	0	0
$975$	−3.89905e6	−0.131355
$976$	0	0
$977$	−3.15932e7	−1.05890	−0.529452	−	0.848340i	$-0.677602\pi$
$977$	−3.15932e7	−1.05890	−0.529452	+	0.848340i	$0.677602\pi$
$978$	0	0
$979$	2.82421e7	0.941760
$980$	0	0
$981$	−2.79934e7	−0.928718
$982$	0	0
$983$	3.72273e7	1.22879	0.614395	−	0.788999i	$-0.289400\pi$
$983$	3.72273e7	1.22879	0.614395	+	0.788999i	$0.289400\pi$
$984$	0	0
$985$	427189.	0.0140291
$986$	0	0
$987$	−6.36998e6	−0.208135
$988$	0	0
$989$	1.05031e6	0.0341451
$990$	0	0
$991$	3.07570e7	0.994855	0.497427	−	0.867506i	$-0.334278\pi$
$991$	3.07570e7	0.994855	0.497427	+	0.867506i	$0.334278\pi$
$992$	0	0
$993$	2.56865e7	0.826670
$994$	0	0
$995$	−1.85103e6	−0.0592729
$996$	0	0
$997$	2.38815e7	0.760892	0.380446	−	0.924803i	$-0.375770\pi$
$997$	2.38815e7	0.760892	0.380446	+	0.924803i	$0.375770\pi$
$998$	0	0
$999$	4.15388e7	1.31686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.h.1.1	✓	4
4.3	odd	2		560.6.a.w.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.h.1.1	✓	4	1.1	even	1	trivial
560.6.a.w.1.4		4	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)

\(f(q)\)	\(=\)	\(q-25.1636 q^{3} -25.0000 q^{5} -49.0000 q^{7} +390.209 q^{9} +O(q^{10})\)	\(q-25.1636 q^{3} -25.0000 q^{5} -49.0000 q^{7} +390.209 q^{9} -365.186 q^{11} +247.916 q^{13} +629.091 q^{15} +1639.32 q^{17} -361.233 q^{19} +1233.02 q^{21} +1415.45 q^{23} +625.000 q^{25} -3704.31 q^{27} +5204.44 q^{29} +5883.20 q^{31} +9189.40 q^{33} +1225.00 q^{35} -11213.6 q^{37} -6238.47 q^{39} -7249.75 q^{41} +742.037 q^{43} -9755.22 q^{45} -5166.17 q^{47} +2401.00 q^{49} -41251.2 q^{51} +16015.5 q^{53} +9129.64 q^{55} +9089.94 q^{57} -19888.8 q^{59} -26621.9 q^{61} -19120.2 q^{63} -6197.90 q^{65} -15591.5 q^{67} -35617.8 q^{69} -46315.6 q^{71} +90796.7 q^{73} -15727.3 q^{75} +17894.1 q^{77} -103600. q^{79} -1606.84 q^{81} -96467.3 q^{83} -40982.9 q^{85} -130963. q^{87} -77336.2 q^{89} -12147.9 q^{91} -148043. q^{93} +9030.83 q^{95} +147486. q^{97} -142499. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{3} - 100 q^{5} - 196 q^{7} + 567 q^{9}+O(q^{10}) \) 4 * q + 5 * q^3 - 100 * q^5 - 196 * q^7 + 567 * q^9	\( 4 q + 5 q^{3} - 100 q^{5} - 196 q^{7} + 567 q^{9} - 123 q^{11} + 789 q^{13} - 125 q^{15} + 1051 q^{17} + 958 q^{19} - 245 q^{21} - 530 q^{23} + 2500 q^{25} - 3169 q^{27} - 5541 q^{29} - 10440 q^{31} - 3737 q^{33} + 4900 q^{35} - 24048 q^{37} - 9131 q^{39} - 35414 q^{41} + 2174 q^{43} - 14175 q^{45} - 2287 q^{47} + 9604 q^{49} - 54017 q^{51} - 22238 q^{53} + 3075 q^{55} + 5890 q^{57} - 22656 q^{59} - 20250 q^{61} - 27783 q^{63} - 19725 q^{65} - 19456 q^{67} - 47006 q^{69} - 72288 q^{71} + 59464 q^{73} + 3125 q^{75} + 6027 q^{77} - 232001 q^{79} - 149604 q^{81} - 169620 q^{83} - 26275 q^{85} - 270309 q^{87} + 141434 q^{89} - 38661 q^{91} - 203808 q^{93} - 23950 q^{95} + 167159 q^{97} - 292198 q^{99}+O(q^{100}) \) 4 * q + 5 * q^3 - 100 * q^5 - 196 * q^7 + 567 * q^9 - 123 * q^11 + 789 * q^13 - 125 * q^15 + 1051 * q^17 + 958 * q^19 - 245 * q^21 - 530 * q^23 + 2500 * q^25 - 3169 * q^27 - 5541 * q^29 - 10440 * q^31 - 3737 * q^33 + 4900 * q^35 - 24048 * q^37 - 9131 * q^39 - 35414 * q^41 + 2174 * q^43 - 14175 * q^45 - 2287 * q^47 + 9604 * q^49 - 54017 * q^51 - 22238 * q^53 + 3075 * q^55 + 5890 * q^57 - 22656 * q^59 - 20250 * q^61 - 27783 * q^63 - 19725 * q^65 - 19456 * q^67 - 47006 * q^69 - 72288 * q^71 + 59464 * q^73 + 3125 * q^75 + 6027 * q^77 - 232001 * q^79 - 149604 * q^81 - 169620 * q^83 - 26275 * q^85 - 270309 * q^87 + 141434 * q^89 - 38661 * q^91 - 203808 * q^93 - 23950 * q^95 + 167159 * q^97 - 292198 * q^99

Introduction

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Modular forms

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