LMFDB - Embedded newform 280.6.a.c.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:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.72015$ 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)

$f(q)$	$=$	$q-19.8806 q^{3} +25.0000 q^{5} -49.0000 q^{7} +152.239 q^{9} +O(q^{10})$	$q-19.8806 q^{3} +25.0000 q^{5} -49.0000 q^{7} +152.239 q^{9} +248.851 q^{11} -483.074 q^{13} -497.015 q^{15} -287.462 q^{17} +1718.82 q^{19} +974.150 q^{21} +1888.17 q^{23} +625.000 q^{25} +1804.39 q^{27} -459.984 q^{29} +6156.94 q^{31} -4947.31 q^{33} -1225.00 q^{35} -6887.51 q^{37} +9603.82 q^{39} -7399.96 q^{41} -15389.4 q^{43} +3805.97 q^{45} +8133.41 q^{47} +2401.00 q^{49} +5714.93 q^{51} -1218.64 q^{53} +6221.28 q^{55} -34171.2 q^{57} +1697.06 q^{59} -20630.3 q^{61} -7459.70 q^{63} -12076.9 q^{65} -44473.7 q^{67} -37537.9 q^{69} -31669.1 q^{71} -64792.8 q^{73} -12425.4 q^{75} -12193.7 q^{77} -62862.2 q^{79} -72866.4 q^{81} +18292.4 q^{83} -7186.56 q^{85} +9144.76 q^{87} +1721.31 q^{89} +23670.6 q^{91} -122404. q^{93} +42970.5 q^{95} -30216.1 q^{97} +37884.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 50 q^{5} - 98 q^{7} + 388 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 50 * q^5 - 98 * q^7 + 388 * q^9	$ 2 q + 2 q^{3} + 50 q^{5} - 98 q^{7} + 388 q^{9} - 254 q^{11} - 1342 q^{13} + 50 q^{15} - 1786 q^{17} + 1976 q^{19} - 98 q^{21} - 2112 q^{23} + 1250 q^{25} + 1646 q^{27} - 4762 q^{29} + 13692 q^{31} - 15950 q^{33} - 2450 q^{35} + 2136 q^{37} - 9190 q^{39} - 6740 q^{41} - 26728 q^{43} + 9700 q^{45} - 6326 q^{47} + 4802 q^{49} - 27074 q^{51} + 24624 q^{53} - 6350 q^{55} - 28544 q^{57} + 51336 q^{59} + 4468 q^{61} - 19012 q^{63} - 33550 q^{65} - 39168 q^{67} - 125064 q^{69} + 41232 q^{71} - 36124 q^{73} + 1250 q^{75} + 12446 q^{77} - 140842 q^{79} - 133622 q^{81} - 57712 q^{83} - 44650 q^{85} - 84986 q^{87} - 20236 q^{89} + 65758 q^{91} + 42468 q^{93} + 49400 q^{95} - 183586 q^{97} - 80668 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 50 * q^5 - 98 * q^7 + 388 * q^9 - 254 * q^11 - 1342 * q^13 + 50 * q^15 - 1786 * q^17 + 1976 * q^19 - 98 * q^21 - 2112 * q^23 + 1250 * q^25 + 1646 * q^27 - 4762 * q^29 + 13692 * q^31 - 15950 * q^33 - 2450 * q^35 + 2136 * q^37 - 9190 * q^39 - 6740 * q^41 - 26728 * q^43 + 9700 * q^45 - 6326 * q^47 + 4802 * q^49 - 27074 * q^51 + 24624 * q^53 - 6350 * q^55 - 28544 * q^57 + 51336 * q^59 + 4468 * q^61 - 19012 * q^63 - 33550 * q^65 - 39168 * q^67 - 125064 * q^69 + 41232 * q^71 - 36124 * q^73 + 1250 * q^75 + 12446 * q^77 - 140842 * q^79 - 133622 * q^81 - 57712 * q^83 - 44650 * q^85 - 84986 * q^87 - 20236 * q^89 + 65758 * q^91 + 42468 * q^93 + 49400 * q^95 - 183586 * q^97 - 80668 * 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
$3$	−19.8806	−1.27534	−0.637671	−	0.770309i	$-0.720102\pi$
$3$	−19.8806	−1.27534	−0.637671	+	0.770309i	$0.720102\pi$
$4$	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
$9$	152.239	0.626497
$10$	0	0
$11$	248.851	0.620094	0.310047	−	0.950721i	$-0.399655\pi$
$11$	248.851	0.620094	0.310047	+	0.950721i	$0.399655\pi$
$12$	0	0
$13$	−483.074	−0.792786	−0.396393	−	0.918081i	$-0.629738\pi$
$13$	−483.074	−0.792786	−0.396393	+	0.918081i	$0.629738\pi$
$14$	0	0
$15$	−497.015	−0.570350
$16$	0	0
$17$	−287.462	−0.241245	−0.120623	−	0.992698i	$-0.538489\pi$
$17$	−287.462	−0.241245	−0.120623	+	0.992698i	$0.538489\pi$
$18$	0	0
$19$	1718.82	1.09231	0.546156	−	0.837683i	$-0.316091\pi$
$19$	1718.82	1.09231	0.546156	+	0.837683i	$0.316091\pi$
$20$	0	0
$21$	974.150	0.482034
$22$	0	0
$23$	1888.17	0.744253	0.372127	−	0.928182i	$-0.378629\pi$
$23$	1888.17	0.744253	0.372127	+	0.928182i	$0.378629\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	1804.39	0.476344
$28$	0	0
$29$	−459.984	−0.101566	−0.0507829	−	0.998710i	$-0.516172\pi$
$29$	−459.984	−0.101566	−0.0507829	+	0.998710i	$0.516172\pi$
$30$	0	0
$31$	6156.94	1.15070	0.575348	−	0.817909i	$-0.304867\pi$
$31$	6156.94	1.15070	0.575348	+	0.817909i	$0.304867\pi$
$32$	0	0
$33$	−4947.31	−0.790832
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	−6887.51	−0.827100	−0.413550	−	0.910481i	$-0.635711\pi$
$37$	−6887.51	−0.827100	−0.413550	+	0.910481i	$0.635711\pi$
$38$	0	0
$39$	9603.82	1.01107
$40$	0	0
$41$	−7399.96	−0.687495	−0.343748	−	0.939062i	$-0.611696\pi$
$41$	−7399.96	−0.687495	−0.343748	+	0.939062i	$0.611696\pi$
$42$	0	0
$43$	−15389.4	−1.26926	−0.634631	−	0.772816i	$-0.718848\pi$
$43$	−15389.4	−1.26926	−0.634631	+	0.772816i	$0.718848\pi$
$44$	0	0
$45$	3805.97	0.280178
$46$	0	0
$47$	8133.41	0.537067	0.268533	−	0.963270i	$-0.413461\pi$
$47$	8133.41	0.537067	0.268533	+	0.963270i	$0.413461\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	5714.93	0.307670
$52$	0	0
$53$	−1218.64	−0.0595916	−0.0297958	−	0.999556i	$-0.509486\pi$
$53$	−1218.64	−0.0595916	−0.0297958	+	0.999556i	$0.509486\pi$
$54$	0	0
$55$	6221.28	0.277315
$56$	0	0
$57$	−34171.2	−1.39307
$58$	0	0
$59$	1697.06	0.0634697	0.0317348	−	0.999496i	$-0.489897\pi$
$59$	1697.06	0.0634697	0.0317348	+	0.999496i	$0.489897\pi$
$60$	0	0
$61$	−20630.3	−0.709872	−0.354936	−	0.934891i	$-0.615497\pi$
$61$	−20630.3	−0.709872	−0.354936	+	0.934891i	$0.615497\pi$
$62$	0	0
$63$	−7459.70	−0.236794
$64$	0	0
$65$	−12076.9	−0.354544
$66$	0	0
$67$	−44473.7	−1.21036	−0.605182	−	0.796087i	$-0.706900\pi$
$67$	−44473.7	−1.21036	−0.605182	+	0.796087i	$0.706900\pi$
$68$	0	0
$69$	−37537.9	−0.949177
$70$	0	0
$71$	−31669.1	−0.745571	−0.372786	−	0.927917i	$-0.621597\pi$
$71$	−31669.1	−0.745571	−0.372786	+	0.927917i	$0.621597\pi$
$72$	0	0
$73$	−64792.8	−1.42305	−0.711524	−	0.702662i	$-0.751995\pi$
$73$	−64792.8	−1.42305	−0.711524	+	0.702662i	$0.751995\pi$
$74$	0	0
$75$	−12425.4	−0.255068
$76$	0	0
$77$	−12193.7	−0.234374
$78$	0	0
$79$	−62862.2	−1.13324	−0.566620	−	0.823979i	$-0.691749\pi$
$79$	−62862.2	−1.13324	−0.566620	+	0.823979i	$0.691749\pi$
$80$	0	0
$81$	−72866.4	−1.23400
$82$	0	0
$83$	18292.4	0.291458	0.145729	−	0.989325i	$-0.453447\pi$
$83$	18292.4	0.291458	0.145729	+	0.989325i	$0.453447\pi$
$84$	0	0
$85$	−7186.56	−0.107888
$86$	0	0
$87$	9144.76	0.129531
$88$	0	0
$89$	1721.31	0.0230347	0.0115174	−	0.999934i	$-0.496334\pi$
$89$	1721.31	0.0230347	0.0115174	+	0.999934i	$0.496334\pi$
$90$	0	0
$91$	23670.6	0.299645
$92$	0	0
$93$	−122404.	−1.46753
$94$	0	0
$95$	42970.5	0.488497
$96$	0	0
$97$	−30216.1	−0.326068	−0.163034	−	0.986620i	$-0.552128\pi$
$97$	−30216.1	−0.326068	−0.163034	+	0.986620i	$0.552128\pi$
$98$	0	0
$99$	37884.8	0.388487
$100$	0	0
$101$	−14037.7	−0.136928	−0.0684641	−	0.997654i	$-0.521810\pi$
$101$	−14037.7	−0.136928	−0.0684641	+	0.997654i	$0.521810\pi$
$102$	0	0
$103$	−110492.	−1.02621	−0.513106	−	0.858325i	$-0.671505\pi$
$103$	−110492.	−1.02621	−0.513106	+	0.858325i	$0.671505\pi$
$104$	0	0
$105$	24353.8	0.215572
$106$	0	0
$107$	35526.7	0.299982	0.149991	−	0.988687i	$-0.452076\pi$
$107$	35526.7	0.299982	0.149991	+	0.988687i	$0.452076\pi$
$108$	0	0
$109$	138261.	1.11464	0.557319	−	0.830299i	$-0.311830\pi$
$109$	138261.	1.11464	0.557319	+	0.830299i	$0.311830\pi$
$110$	0	0
$111$	136928.	1.05484
$112$	0	0
$113$	−9323.26	−0.0686866	−0.0343433	−	0.999410i	$-0.510934\pi$
$113$	−9323.26	−0.0686866	−0.0343433	+	0.999410i	$0.510934\pi$
$114$	0	0
$115$	47204.2	0.332840
$116$	0	0
$117$	−73542.7	−0.496678
$118$	0	0
$119$	14085.6	0.0911821
$120$	0	0
$121$	−99124.2	−0.615483
$122$	0	0
$123$	147116.	0.876791
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−165251.	−0.909148	−0.454574	−	0.890709i	$-0.650208\pi$
$127$	−165251.	−0.909148	−0.454574	+	0.890709i	$0.650208\pi$
$128$	0	0
$129$	305951.	1.61874
$130$	0	0
$131$	−334329.	−1.70214	−0.851072	−	0.525050i	$-0.824047\pi$
$131$	−334329.	−1.70214	−0.851072	+	0.525050i	$0.824047\pi$
$132$	0	0
$133$	−84222.3	−0.412855
$134$	0	0
$135$	45109.7	0.213028
$136$	0	0
$137$	161876.	0.736852	0.368426	−	0.929657i	$-0.379897\pi$
$137$	161876.	0.736852	0.368426	+	0.929657i	$0.379897\pi$
$138$	0	0
$139$	402730.	1.76798	0.883990	−	0.467506i	$-0.154847\pi$
$139$	402730.	1.76798	0.883990	+	0.467506i	$0.154847\pi$
$140$	0	0
$141$	−161697.	−0.684943
$142$	0	0
$143$	−120214.	−0.491602
$144$	0	0
$145$	−11499.6	−0.0454216
$146$	0	0
$147$	−47733.4	−0.182192
$148$	0	0
$149$	112226.	0.414123	0.207062	−	0.978328i	$-0.433610\pi$
$149$	112226.	0.414123	0.207062	+	0.978328i	$0.433610\pi$
$150$	0	0
$151$	−199582.	−0.712327	−0.356163	−	0.934424i	$-0.615915\pi$
$151$	−199582.	−0.712327	−0.356163	+	0.934424i	$0.615915\pi$
$152$	0	0
$153$	−43762.9	−0.151139
$154$	0	0
$155$	153923.	0.514607
$156$	0	0
$157$	−199948.	−0.647393	−0.323696	−	0.946161i	$-0.604926\pi$
$157$	−199948.	−0.647393	−0.323696	+	0.946161i	$0.604926\pi$
$158$	0	0
$159$	24227.3	0.0759996
$160$	0	0
$161$	−92520.2	−0.281301
$162$	0	0
$163$	23155.0	0.0682614	0.0341307	−	0.999417i	$-0.489134\pi$
$163$	23155.0	0.0682614	0.0341307	+	0.999417i	$0.489134\pi$
$164$	0	0
$165$	−123683.	−0.353671
$166$	0	0
$167$	95639.4	0.265366	0.132683	−	0.991159i	$-0.457641\pi$
$167$	95639.4	0.265366	0.132683	+	0.991159i	$0.457641\pi$
$168$	0	0
$169$	−137932.	−0.371491
$170$	0	0
$171$	261671.	0.684330
$172$	0	0
$173$	−465433.	−1.18234	−0.591169	−	0.806548i	$-0.701333\pi$
$173$	−465433.	−1.18234	−0.591169	+	0.806548i	$0.701333\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	−33738.5	−0.0809455
$178$	0	0
$179$	658851.	1.53693	0.768466	−	0.639891i	$-0.221020\pi$
$179$	658851.	1.53693	0.768466	+	0.639891i	$0.221020\pi$
$180$	0	0
$181$	610326.	1.38473	0.692365	−	0.721547i	$-0.256569\pi$
$181$	610326.	1.38473	0.692365	+	0.721547i	$0.256569\pi$
$182$	0	0
$183$	410142.	0.905330
$184$	0	0
$185$	−172188.	−0.369890
$186$	0	0
$187$	−71535.3	−0.149595
$188$	0	0
$189$	−88415.1	−0.180041
$190$	0	0
$191$	−574926.	−1.14032	−0.570162	−	0.821532i	$-0.693120\pi$
$191$	−574926.	−1.14032	−0.570162	+	0.821532i	$0.693120\pi$
$192$	0	0
$193$	−549314.	−1.06152	−0.530760	−	0.847522i	$-0.678093\pi$
$193$	−549314.	−1.06152	−0.530760	+	0.847522i	$0.678093\pi$
$194$	0	0
$195$	240095.	0.452165
$196$	0	0
$197$	−748127.	−1.37344	−0.686720	−	0.726922i	$-0.740950\pi$
$197$	−748127.	−1.37344	−0.686720	+	0.726922i	$0.740950\pi$
$198$	0	0
$199$	525282.	0.940286	0.470143	−	0.882590i	$-0.344202\pi$
$199$	525282.	0.940286	0.470143	+	0.882590i	$0.344202\pi$
$200$	0	0
$201$	884164.	1.54363
$202$	0	0
$203$	22539.2	0.0383883
$204$	0	0
$205$	−184999.	−0.307457
$206$	0	0
$207$	287452.	0.466272
$208$	0	0
$209$	427730.	0.677337
$210$	0	0
$211$	−57400.7	−0.0887588	−0.0443794	−	0.999015i	$-0.514131\pi$
$211$	−57400.7	−0.0887588	−0.0443794	+	0.999015i	$0.514131\pi$
$212$	0	0
$213$	629600.	0.950858
$214$	0	0
$215$	−384735.	−0.567631
$216$	0	0
$217$	−301690.	−0.434922
$218$	0	0
$219$	1.28812e6	1.81487
$220$	0	0
$221$	138866.	0.191256
$222$	0	0
$223$	587256.	0.790798	0.395399	−	0.918510i	$-0.370606\pi$
$223$	587256.	0.790798	0.395399	+	0.918510i	$0.370606\pi$
$224$	0	0
$225$	95149.2	0.125299
$226$	0	0
$227$	390860.	0.503450	0.251725	−	0.967799i	$-0.419002\pi$
$227$	390860.	0.503450	0.251725	+	0.967799i	$0.419002\pi$
$228$	0	0
$229$	27202.7	0.0342786	0.0171393	−	0.999853i	$-0.494544\pi$
$229$	27202.7	0.0342786	0.0171393	+	0.999853i	$0.494544\pi$
$230$	0	0
$231$	242418.	0.298906
$232$	0	0
$233$	−1.22835e6	−1.48228	−0.741141	−	0.671349i	$-0.765715\pi$
$233$	−1.22835e6	−1.48228	−0.741141	+	0.671349i	$0.765715\pi$
$234$	0	0
$235$	203335.	0.240183
$236$	0	0
$237$	1.24974e6	1.44527
$238$	0	0
$239$	−743865.	−0.842363	−0.421182	−	0.906976i	$-0.638384\pi$
$239$	−743865.	−0.842363	−0.421182	+	0.906976i	$0.638384\pi$
$240$	0	0
$241$	−106938.	−0.118601	−0.0593005	−	0.998240i	$-0.518887\pi$
$241$	−106938.	−0.118601	−0.0593005	+	0.998240i	$0.518887\pi$
$242$	0	0
$243$	1.01016e6	1.09743
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−830319.	−0.865969
$248$	0	0
$249$	−363665.	−0.371709
$250$	0	0
$251$	1.43872e6	1.44143	0.720713	−	0.693234i	$-0.243815\pi$
$251$	1.43872e6	1.44143	0.720713	+	0.693234i	$0.243815\pi$
$252$	0	0
$253$	469872.	0.461507
$254$	0	0
$255$	142873.	0.137594
$256$	0	0
$257$	−75732.5	−0.0715236	−0.0357618	−	0.999360i	$-0.511386\pi$
$257$	−75732.5	−0.0715236	−0.0357618	+	0.999360i	$0.511386\pi$
$258$	0	0
$259$	337488.	0.312615
$260$	0	0
$261$	−70027.3	−0.0636306
$262$	0	0
$263$	685453.	0.611066	0.305533	−	0.952181i	$-0.401165\pi$
$263$	685453.	0.611066	0.305533	+	0.952181i	$0.401165\pi$
$264$	0	0
$265$	−30465.9	−0.0266502
$266$	0	0
$267$	−34220.6	−0.0293772
$268$	0	0
$269$	272852.	0.229904	0.114952	−	0.993371i	$-0.463329\pi$
$269$	272852.	0.229904	0.114952	+	0.993371i	$0.463329\pi$
$270$	0	0
$271$	−444009.	−0.367256	−0.183628	−	0.982996i	$-0.558784\pi$
$271$	−444009.	−0.367256	−0.183628	+	0.982996i	$0.558784\pi$
$272$	0	0
$273$	−470587.	−0.382150
$274$	0	0
$275$	155532.	0.124019
$276$	0	0
$277$	−1.65455e6	−1.29563	−0.647815	−	0.761798i	$-0.724317\pi$
$277$	−1.65455e6	−1.29563	−0.647815	+	0.761798i	$0.724317\pi$
$278$	0	0
$279$	937325.	0.720908
$280$	0	0
$281$	543803.	0.410843	0.205421	−	0.978674i	$-0.434143\pi$
$281$	543803.	0.410843	0.205421	+	0.978674i	$0.434143\pi$
$282$	0	0
$283$	2.33944e6	1.73638	0.868191	−	0.496231i	$-0.165283\pi$
$283$	2.33944e6	1.73638	0.868191	+	0.496231i	$0.165283\pi$
$284$	0	0
$285$	−854281.	−0.623001
$286$	0	0
$287$	362598.	0.259849
$288$	0	0
$289$	−1.33722e6	−0.941801
$290$	0	0
$291$	600714.	0.415849
$292$	0	0
$293$	1.75951e6	1.19735	0.598676	−	0.800991i	$-0.295694\pi$
$293$	1.75951e6	1.19735	0.598676	+	0.800991i	$0.295694\pi$
$294$	0	0
$295$	42426.4	0.0283845
$296$	0	0
$297$	449024.	0.295378
$298$	0	0
$299$	−912125.	−0.590033
$300$	0	0
$301$	754082.	0.479736
$302$	0	0
$303$	279078.	0.174630
$304$	0	0
$305$	−515757.	−0.317465
$306$	0	0
$307$	−2.69824e6	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$307$	−2.69824e6	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$308$	0	0
$309$	2.19665e6	1.30877
$310$	0	0
$311$	−2.76317e6	−1.61997	−0.809983	−	0.586453i	$-0.800524\pi$
$311$	−2.76317e6	−1.61997	−0.809983	+	0.586453i	$0.800524\pi$
$312$	0	0
$313$	529371.	0.305421	0.152711	−	0.988271i	$-0.451200\pi$
$313$	529371.	0.305421	0.152711	+	0.988271i	$0.451200\pi$
$314$	0	0
$315$	−186492.	−0.105897
$316$	0	0
$317$	−1.47473e6	−0.824258	−0.412129	−	0.911126i	$-0.635215\pi$
$317$	−1.47473e6	−0.824258	−0.412129	+	0.911126i	$0.635215\pi$
$318$	0	0
$319$	−114467.	−0.0629804
$320$	0	0
$321$	−706292.	−0.382580
$322$	0	0
$323$	−494096.	−0.263515
$324$	0	0
$325$	−301922.	−0.158557
$326$	0	0
$327$	−2.74871e6	−1.42154
$328$	0	0
$329$	−398537.	−0.202992
$330$	0	0
$331$	−1.22757e6	−0.615854	−0.307927	−	0.951410i	$-0.599635\pi$
$331$	−1.22757e6	−0.615854	−0.307927	+	0.951410i	$0.599635\pi$
$332$	0	0
$333$	−1.04855e6	−0.518176
$334$	0	0
$335$	−1.11184e6	−0.541291
$336$	0	0
$337$	−548187.	−0.262938	−0.131469	−	0.991320i	$-0.541969\pi$
$337$	−548187.	−0.262938	−0.131469	+	0.991320i	$0.541969\pi$
$338$	0	0
$339$	185352.	0.0875989
$340$	0	0
$341$	1.53216e6	0.713540
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−938448.	−0.424485
$346$	0	0
$347$	−4.03128e6	−1.79729	−0.898646	−	0.438674i	$-0.855448\pi$
$347$	−4.03128e6	−1.79729	−0.898646	+	0.438674i	$0.855448\pi$
$348$	0	0
$349$	210957.	0.0927108	0.0463554	−	0.998925i	$-0.485239\pi$
$349$	210957.	0.0927108	0.0463554	+	0.998925i	$0.485239\pi$
$350$	0	0
$351$	−871654.	−0.377639
$352$	0	0
$353$	−2.44145e6	−1.04282	−0.521412	−	0.853305i	$-0.674595\pi$
$353$	−2.44145e6	−1.04282	−0.521412	+	0.853305i	$0.674595\pi$
$354$	0	0
$355$	−791726.	−0.333430
$356$	0	0
$357$	−280031.	−0.116288
$358$	0	0
$359$	−1.61757e6	−0.662411	−0.331205	−	0.943559i	$-0.607455\pi$
$359$	−1.61757e6	−0.662411	−0.331205	+	0.943559i	$0.607455\pi$
$360$	0	0
$361$	478248.	0.193146
$362$	0	0
$363$	1.97065e6	0.784951
$364$	0	0
$365$	−1.61982e6	−0.636407
$366$	0	0
$367$	−847657.	−0.328515	−0.164257	−	0.986418i	$-0.552523\pi$
$367$	−847657.	−0.328515	−0.164257	+	0.986418i	$0.552523\pi$
$368$	0	0
$369$	−1.12656e6	−0.430714
$370$	0	0
$371$	59713.2	0.0225235
$372$	0	0
$373$	3.55105e6	1.32155	0.660777	−	0.750583i	$-0.270227\pi$
$373$	3.55105e6	1.32155	0.660777	+	0.750583i	$0.270227\pi$
$374$	0	0
$375$	−310635.	−0.114070
$376$	0	0
$377$	222206.	0.0805199
$378$	0	0
$379$	−2.22663e6	−0.796253	−0.398126	−	0.917331i	$-0.630339\pi$
$379$	−2.22663e6	−0.796253	−0.398126	+	0.917331i	$0.630339\pi$
$380$	0	0
$381$	3.28529e6	1.15947
$382$	0	0
$383$	−2.48191e6	−0.864549	−0.432275	−	0.901742i	$-0.642289\pi$
$383$	−2.48191e6	−0.864549	−0.432275	+	0.901742i	$0.642289\pi$
$384$	0	0
$385$	−304843.	−0.104815
$386$	0	0
$387$	−2.34287e6	−0.795189
$388$	0	0
$389$	−3.87685e6	−1.29899	−0.649493	−	0.760367i	$-0.725019\pi$
$389$	−3.87685e6	−1.29899	−0.649493	+	0.760367i	$0.725019\pi$
$390$	0	0
$391$	−542777.	−0.179547
$392$	0	0
$393$	6.64667e6	2.17081
$394$	0	0
$395$	−1.57156e6	−0.506800
$396$	0	0
$397$	3.85287e6	1.22690	0.613448	−	0.789735i	$-0.289782\pi$
$397$	3.85287e6	1.22690	0.613448	+	0.789735i	$0.289782\pi$
$398$	0	0
$399$	1.67439e6	0.526532
$400$	0	0
$401$	−2.26149e6	−0.702318	−0.351159	−	0.936316i	$-0.614212\pi$
$401$	−2.26149e6	−0.702318	−0.351159	+	0.936316i	$0.614212\pi$
$402$	0	0
$403$	−2.97426e6	−0.912255
$404$	0	0
$405$	−1.82166e6	−0.551861
$406$	0	0
$407$	−1.71396e6	−0.512880
$408$	0	0
$409$	3.52554e6	1.04212	0.521060	−	0.853520i	$-0.325537\pi$
$409$	3.52554e6	1.04212	0.521060	+	0.853520i	$0.325537\pi$
$410$	0	0
$411$	−3.21819e6	−0.939738
$412$	0	0
$413$	−83155.8	−0.0239893
$414$	0	0
$415$	457311.	0.130344
$416$	0	0
$417$	−8.00653e6	−2.25478
$418$	0	0
$419$	−1.10653e6	−0.307912	−0.153956	−	0.988078i	$-0.549201\pi$
$419$	−1.10653e6	−0.307912	−0.153956	+	0.988078i	$0.549201\pi$
$420$	0	0
$421$	6.39886e6	1.75953	0.879767	−	0.475405i	$-0.157699\pi$
$421$	6.39886e6	1.75953	0.879767	+	0.475405i	$0.157699\pi$
$422$	0	0
$423$	1.23822e6	0.336471
$424$	0	0
$425$	−179664.	−0.0482490
$426$	0	0
$427$	1.01088e6	0.268307
$428$	0	0
$429$	2.38992e6	0.626960
$430$	0	0
$431$	−1.59172e6	−0.412736	−0.206368	−	0.978474i	$-0.566164\pi$
$431$	−1.59172e6	−0.412736	−0.206368	+	0.978474i	$0.566164\pi$
$432$	0	0
$433$	−1.30012e6	−0.333244	−0.166622	−	0.986021i	$-0.553286\pi$
$433$	−1.30012e6	−0.333244	−0.166622	+	0.986021i	$0.553286\pi$
$434$	0	0
$435$	228619.	0.0579281
$436$	0	0
$437$	3.24542e6	0.812957
$438$	0	0
$439$	360890.	0.0893744	0.0446872	−	0.999001i	$-0.485771\pi$
$439$	360890.	0.0893744	0.0446872	+	0.999001i	$0.485771\pi$
$440$	0	0
$441$	365525.	0.0894996
$442$	0	0
$443$	7.25504e6	1.75643	0.878215	−	0.478267i	$-0.158735\pi$
$443$	7.25504e6	1.75643	0.878215	+	0.478267i	$0.158735\pi$
$444$	0	0
$445$	43032.7	0.0103015
$446$	0	0
$447$	−2.23113e6	−0.528149
$448$	0	0
$449$	3.04408e6	0.712591	0.356296	−	0.934373i	$-0.384040\pi$
$449$	3.04408e6	0.712591	0.356296	+	0.934373i	$0.384040\pi$
$450$	0	0
$451$	−1.84149e6	−0.426312
$452$	0	0
$453$	3.96781e6	0.908460
$454$	0	0
$455$	591766.	0.134005
$456$	0	0
$457$	500891.	0.112190	0.0560948	−	0.998425i	$-0.482135\pi$
$457$	500891.	0.112190	0.0560948	+	0.998425i	$0.482135\pi$
$458$	0	0
$459$	−518694.	−0.114916
$460$	0	0
$461$	1.38482e6	0.303488	0.151744	−	0.988420i	$-0.451511\pi$
$461$	1.38482e6	0.303488	0.151744	+	0.988420i	$0.451511\pi$
$462$	0	0
$463$	−839131.	−0.181919	−0.0909594	−	0.995855i	$-0.528993\pi$
$463$	−839131.	−0.181919	−0.0909594	+	0.995855i	$0.528993\pi$
$464$	0	0
$465$	−3.06009e6	−0.656300
$466$	0	0
$467$	−7.14636e6	−1.51633	−0.758164	−	0.652064i	$-0.773903\pi$
$467$	−7.14636e6	−1.51633	−0.758164	+	0.652064i	$0.773903\pi$
$468$	0	0
$469$	2.17921e6	0.457475
$470$	0	0
$471$	3.97509e6	0.825647
$472$	0	0
$473$	−3.82967e6	−0.787062
$474$	0	0
$475$	1.07426e6	0.218462
$476$	0	0
$477$	−185524.	−0.0373339
$478$	0	0
$479$	6.81773e6	1.35769	0.678845	−	0.734281i	$-0.262481\pi$
$479$	6.81773e6	1.35769	0.678845	+	0.734281i	$0.262481\pi$
$480$	0	0
$481$	3.32718e6	0.655713
$482$	0	0
$483$	1.83936e6	0.358755
$484$	0	0
$485$	−755402.	−0.145822
$486$	0	0
$487$	1.32728e6	0.253594	0.126797	−	0.991929i	$-0.459530\pi$
$487$	1.32728e6	0.253594	0.126797	+	0.991929i	$0.459530\pi$
$488$	0	0
$489$	−460335.	−0.0870567
$490$	0	0
$491$	−622174.	−0.116468	−0.0582342	−	0.998303i	$-0.518547\pi$
$491$	−622174.	−0.116468	−0.0582342	+	0.998303i	$0.518547\pi$
$492$	0	0
$493$	132228.	0.0245022
$494$	0	0
$495$	947119.	0.173737
$496$	0	0
$497$	1.55178e6	0.281799
$498$	0	0
$499$	9.40560e6	1.69097	0.845483	−	0.534002i	$-0.179312\pi$
$499$	9.40560e6	1.69097	0.845483	+	0.534002i	$0.179312\pi$
$500$	0	0
$501$	−1.90137e6	−0.338433
$502$	0	0
$503$	1.17009e6	0.206206	0.103103	−	0.994671i	$-0.467123\pi$
$503$	1.17009e6	0.206206	0.103103	+	0.994671i	$0.467123\pi$
$504$	0	0
$505$	−350943.	−0.0612361
$506$	0	0
$507$	2.74217e6	0.473778
$508$	0	0
$509$	2.75154e6	0.470739	0.235370	−	0.971906i	$-0.424370\pi$
$509$	2.75154e6	0.470739	0.235370	+	0.971906i	$0.424370\pi$
$510$	0	0
$511$	3.17485e6	0.537862
$512$	0	0
$513$	3.10142e6	0.520316
$514$	0	0
$515$	−2.76230e6	−0.458936
$516$	0	0
$517$	2.02401e6	0.333032
$518$	0	0
$519$	9.25309e6	1.50788
$520$	0	0
$521$	4.20968e6	0.679445	0.339723	−	0.940526i	$-0.389667\pi$
$521$	4.20968e6	0.679445	0.339723	+	0.940526i	$0.389667\pi$
$522$	0	0
$523$	−9.91592e6	−1.58518	−0.792591	−	0.609754i	$-0.791268\pi$
$523$	−9.91592e6	−1.58518	−0.792591	+	0.609754i	$0.791268\pi$
$524$	0	0
$525$	608844.	0.0964068
$526$	0	0
$527$	−1.76989e6	−0.277600
$528$	0	0
$529$	−2.87117e6	−0.446087
$530$	0	0
$531$	258358.	0.0397636
$532$	0	0
$533$	3.57473e6	0.545036
$534$	0	0
$535$	888167.	0.134156
$536$	0	0
$537$	−1.30984e7	−1.96011
$538$	0	0
$539$	597491.	0.0885849
$540$	0	0
$541$	−9.62766e6	−1.41425	−0.707127	−	0.707086i	$-0.750009\pi$
$541$	−9.62766e6	−1.41425	−0.707127	+	0.707086i	$0.750009\pi$
$542$	0	0
$543$	−1.21336e7	−1.76600
$544$	0	0
$545$	3.45653e6	0.498481
$546$	0	0
$547$	−1.37462e6	−0.196433	−0.0982166	−	0.995165i	$-0.531314\pi$
$547$	−1.37462e6	−0.196433	−0.0982166	+	0.995165i	$0.531314\pi$
$548$	0	0
$549$	−3.14073e6	−0.444733
$550$	0	0
$551$	−790630.	−0.110942
$552$	0	0
$553$	3.08025e6	0.428325
$554$	0	0
$555$	3.42320e6	0.471737
$556$	0	0
$557$	−2.33516e6	−0.318918	−0.159459	−	0.987205i	$-0.550975\pi$
$557$	−2.33516e6	−0.318918	−0.159459	+	0.987205i	$0.550975\pi$
$558$	0	0
$559$	7.43424e6	1.00625
$560$	0	0
$561$	1.42217e6	0.190784
$562$	0	0
$563$	−4.53506e6	−0.602992	−0.301496	−	0.953467i	$-0.597486\pi$
$563$	−4.53506e6	−0.602992	−0.301496	+	0.953467i	$0.597486\pi$
$564$	0	0
$565$	−233082.	−0.0307176
$566$	0	0
$567$	3.57045e6	0.466408
$568$	0	0
$569$	−19774.7	−0.00256053	−0.00128027	−	0.999999i	$-0.500408\pi$
$569$	−19774.7	−0.00256053	−0.00128027	+	0.999999i	$0.500408\pi$
$570$	0	0
$571$	−4.41002e6	−0.566045	−0.283022	−	0.959113i	$-0.591337\pi$
$571$	−4.41002e6	−0.566045	−0.283022	+	0.959113i	$0.591337\pi$
$572$	0	0
$573$	1.14299e7	1.45430
$574$	0	0
$575$	1.18010e6	0.148851
$576$	0	0
$577$	−6.54585e6	−0.818514	−0.409257	−	0.912419i	$-0.634212\pi$
$577$	−6.54585e6	−0.818514	−0.409257	+	0.912419i	$0.634212\pi$
$578$	0	0
$579$	1.09207e7	1.35380
$580$	0	0
$581$	−896329.	−0.110161
$582$	0	0
$583$	−303259.	−0.0369524
$584$	0	0
$585$	−1.83857e6	−0.222121
$586$	0	0
$587$	−1.20472e7	−1.44308	−0.721538	−	0.692375i	$-0.756564\pi$
$587$	−1.20472e7	−1.44308	−0.721538	+	0.692375i	$0.756564\pi$
$588$	0	0
$589$	1.05827e7	1.25692
$590$	0	0
$591$	1.48732e7	1.75161
$592$	0	0
$593$	−3.23357e6	−0.377611	−0.188806	−	0.982014i	$-0.560462\pi$
$593$	−3.23357e6	−0.377611	−0.188806	+	0.982014i	$0.560462\pi$
$594$	0	0
$595$	352141.	0.0407779
$596$	0	0
$597$	−1.04429e7	−1.19919
$598$	0	0
$599$	2.78022e6	0.316601	0.158300	−	0.987391i	$-0.449399\pi$
$599$	2.78022e6	0.316601	0.158300	+	0.987391i	$0.449399\pi$
$600$	0	0
$601$	−1.37257e7	−1.55006	−0.775029	−	0.631926i	$-0.782265\pi$
$601$	−1.37257e7	−1.55006	−0.775029	+	0.631926i	$0.782265\pi$
$602$	0	0
$603$	−6.77062e6	−0.758290
$604$	0	0
$605$	−2.47810e6	−0.275252
$606$	0	0
$607$	1.73396e7	1.91014	0.955072	−	0.296373i	$-0.0957772\pi$
$607$	1.73396e7	1.91014	0.955072	+	0.296373i	$0.0957772\pi$
$608$	0	0
$609$	−448093.	−0.0489581
$610$	0	0
$611$	−3.92904e6	−0.425779
$612$	0	0
$613$	1.29861e7	1.39581	0.697907	−	0.716188i	$-0.254115\pi$
$613$	1.29861e7	1.39581	0.697907	+	0.716188i	$0.254115\pi$
$614$	0	0
$615$	3.67789e6	0.392113
$616$	0	0
$617$	6.35198e6	0.671733	0.335866	−	0.941910i	$-0.390971\pi$
$617$	6.35198e6	0.671733	0.335866	+	0.941910i	$0.390971\pi$
$618$	0	0
$619$	1.29004e7	1.35325	0.676623	−	0.736330i	$-0.263443\pi$
$619$	1.29004e7	1.35325	0.676623	+	0.736330i	$0.263443\pi$
$620$	0	0
$621$	3.40699e6	0.354521
$622$	0	0
$623$	−84344.1	−0.00870632
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−8.50354e6	−0.863836
$628$	0	0
$629$	1.97990e6	0.199534
$630$	0	0
$631$	5.77912e6	0.577814	0.288907	−	0.957357i	$-0.406708\pi$
$631$	5.77912e6	0.577814	0.288907	+	0.957357i	$0.406708\pi$
$632$	0	0
$633$	1.14116e6	0.113198
$634$	0	0
$635$	−4.13127e6	−0.406583
$636$	0	0
$637$	−1.15986e6	−0.113255
$638$	0	0
$639$	−4.82126e6	−0.467098
$640$	0	0
$641$	−3.17406e6	−0.305120	−0.152560	−	0.988294i	$-0.548752\pi$
$641$	−3.17406e6	−0.305120	−0.152560	+	0.988294i	$0.548752\pi$
$642$	0	0
$643$	8.68677e6	0.828573	0.414287	−	0.910146i	$-0.364031\pi$
$643$	8.68677e6	0.828573	0.414287	+	0.910146i	$0.364031\pi$
$644$	0	0
$645$	7.64878e6	0.723924
$646$	0	0
$647$	3.27771e6	0.307829	0.153915	−	0.988084i	$-0.450812\pi$
$647$	3.27771e6	0.307829	0.153915	+	0.988084i	$0.450812\pi$
$648$	0	0
$649$	422314.	0.0393572
$650$	0	0
$651$	5.99778e6	0.554675
$652$	0	0
$653$	7.12705e6	0.654074	0.327037	−	0.945012i	$-0.393950\pi$
$653$	7.12705e6	0.654074	0.327037	+	0.945012i	$0.393950\pi$
$654$	0	0
$655$	−8.35823e6	−0.761222
$656$	0	0
$657$	−9.86398e6	−0.891536
$658$	0	0
$659$	3.43541e6	0.308152	0.154076	−	0.988059i	$-0.450760\pi$
$659$	3.43541e6	0.308152	0.154076	+	0.988059i	$0.450760\pi$
$660$	0	0
$661$	1.99759e7	1.77829	0.889147	−	0.457622i	$-0.151299\pi$
$661$	1.99759e7	1.77829	0.889147	+	0.457622i	$0.151299\pi$
$662$	0	0
$663$	−2.76073e6	−0.243916
$664$	0	0
$665$	−2.10556e6	−0.184634
$666$	0	0
$667$	−868526.	−0.0755906
$668$	0	0
$669$	−1.16750e7	−1.00854
$670$	0	0
$671$	−5.13386e6	−0.440188
$672$	0	0
$673$	9.83922e6	0.837381	0.418691	−	0.908129i	$-0.362489\pi$
$673$	9.83922e6	0.837381	0.418691	+	0.908129i	$0.362489\pi$
$674$	0	0
$675$	1.12774e6	0.0952688
$676$	0	0
$677$	−2.20208e6	−0.184655	−0.0923276	−	0.995729i	$-0.529431\pi$
$677$	−2.20208e6	−0.184655	−0.0923276	+	0.995729i	$0.529431\pi$
$678$	0	0
$679$	1.48059e6	0.123242
$680$	0	0
$681$	−7.77054e6	−0.642071
$682$	0	0
$683$	−1.52153e7	−1.24804	−0.624020	−	0.781409i	$-0.714502\pi$
$683$	−1.52153e7	−1.24804	−0.624020	+	0.781409i	$0.714502\pi$
$684$	0	0
$685$	4.04689e6	0.329530
$686$	0	0
$687$	−540806.	−0.0437169
$688$	0	0
$689$	588693.	0.0472433
$690$	0	0
$691$	−8.31991e6	−0.662862	−0.331431	−	0.943479i	$-0.607531\pi$
$691$	−8.31991e6	−0.662862	−0.331431	+	0.943479i	$0.607531\pi$
$692$	0	0
$693$	−1.85635e6	−0.146834
$694$	0	0
$695$	1.00683e7	0.790665
$696$	0	0
$697$	2.12721e6	0.165855
$698$	0	0
$699$	2.44203e7	1.89042
$700$	0	0
$701$	−1.02331e7	−0.786526	−0.393263	−	0.919426i	$-0.628654\pi$
$701$	−1.02331e7	−0.786526	−0.393263	+	0.919426i	$0.628654\pi$
$702$	0	0
$703$	−1.18384e7	−0.903452
$704$	0	0
$705$	−4.04243e6	−0.306316
$706$	0	0
$707$	687848.	0.0517540
$708$	0	0
$709$	3.53959e6	0.264446	0.132223	−	0.991220i	$-0.457788\pi$
$709$	3.53959e6	0.264446	0.132223	+	0.991220i	$0.457788\pi$
$710$	0	0
$711$	−9.57007e6	−0.709972
$712$	0	0
$713$	1.16253e7	0.856409
$714$	0	0
$715$	−3.00534e6	−0.219851
$716$	0	0
$717$	1.47885e7	1.07430
$718$	0	0
$719$	−4.03606e6	−0.291162	−0.145581	−	0.989346i	$-0.546505\pi$
$719$	−4.03606e6	−0.291162	−0.145581	+	0.989346i	$0.546505\pi$
$720$	0	0
$721$	5.41410e6	0.387872
$722$	0	0
$723$	2.12599e6	0.151257
$724$	0	0
$725$	−287490.	−0.0203132
$726$	0	0
$727$	2.23385e7	1.56754	0.783768	−	0.621054i	$-0.213295\pi$
$727$	2.23385e7	1.56754	0.783768	+	0.621054i	$0.213295\pi$
$728$	0	0
$729$	−2.37611e6	−0.165595
$730$	0	0
$731$	4.42388e6	0.306203
$732$	0	0
$733$	−2.52573e7	−1.73631	−0.868153	−	0.496296i	$-0.834693\pi$
$733$	−2.52573e7	−1.73631	−0.868153	+	0.496296i	$0.834693\pi$
$734$	0	0
$735$	−1.19333e6	−0.0814786
$736$	0	0
$737$	−1.10673e7	−0.750540
$738$	0	0
$739$	1.60773e7	1.08294	0.541468	−	0.840721i	$-0.317869\pi$
$739$	1.60773e7	1.08294	0.541468	+	0.840721i	$0.317869\pi$
$740$	0	0
$741$	1.65072e7	1.10441
$742$	0	0
$743$	−2.50696e7	−1.66600	−0.833002	−	0.553271i	$-0.813380\pi$
$743$	−2.50696e7	−1.66600	−0.833002	+	0.553271i	$0.813380\pi$
$744$	0	0
$745$	2.80566e6	0.185202
$746$	0	0
$747$	2.78482e6	0.182598
$748$	0	0
$749$	−1.74081e6	−0.113383
$750$	0	0
$751$	−2.32004e7	−1.50105	−0.750527	−	0.660840i	$-0.770200\pi$
$751$	−2.32004e7	−1.50105	−0.750527	+	0.660840i	$0.770200\pi$
$752$	0	0
$753$	−2.86027e7	−1.83831
$754$	0	0
$755$	−4.98955e6	−0.318562
$756$	0	0
$757$	1.16202e7	0.737010	0.368505	−	0.929626i	$-0.379870\pi$
$757$	1.16202e7	0.737010	0.368505	+	0.929626i	$0.379870\pi$
$758$	0	0
$759$	−9.34135e6	−0.588579
$760$	0	0
$761$	−2.35563e7	−1.47450	−0.737250	−	0.675620i	$-0.763876\pi$
$761$	−2.35563e7	−1.47450	−0.737250	+	0.675620i	$0.763876\pi$
$762$	0	0
$763$	−6.77479e6	−0.421293
$764$	0	0
$765$	−1.09407e6	−0.0675916
$766$	0	0
$767$	−819805.	−0.0503178
$768$	0	0
$769$	−2.93169e7	−1.78773	−0.893866	−	0.448333i	$-0.852018\pi$
$769$	−2.93169e7	−1.78773	−0.893866	+	0.448333i	$0.852018\pi$
$770$	0	0
$771$	1.50561e6	0.0912170
$772$	0	0
$773$	2.50967e6	0.151066	0.0755332	−	0.997143i	$-0.475934\pi$
$773$	2.50967e6	0.151066	0.0755332	+	0.997143i	$0.475934\pi$
$774$	0	0
$775$	3.84809e6	0.230139
$776$	0	0
$777$	−6.70947e6	−0.398690
$778$	0	0
$779$	−1.27192e7	−0.750959
$780$	0	0
$781$	−7.88088e6	−0.462325
$782$	0	0
$783$	−829989.	−0.0483802
$784$	0	0
$785$	−4.99870e6	−0.289523
$786$	0	0
$787$	−1.03559e7	−0.596008	−0.298004	−	0.954565i	$-0.596321\pi$
$787$	−1.03559e7	−0.596008	−0.298004	+	0.954565i	$0.596321\pi$
$788$	0	0
$789$	−1.36272e7	−0.779318
$790$	0	0
$791$	456840.	0.0259611
$792$	0	0
$793$	9.96596e6	0.562777
$794$	0	0
$795$	605681.	0.0339881
$796$	0	0
$797$	1.35622e7	0.756285	0.378143	−	0.925747i	$-0.376563\pi$
$797$	1.35622e7	0.756285	0.378143	+	0.925747i	$0.376563\pi$
$798$	0	0
$799$	−2.33805e6	−0.129565
$800$	0	0
$801$	262050.	0.0144312
$802$	0	0
$803$	−1.61238e7	−0.882424
$804$	0	0
$805$	−2.31300e6	−0.125802
$806$	0	0
$807$	−5.42447e6	−0.293206
$808$	0	0
$809$	−3.57146e7	−1.91856	−0.959279	−	0.282460i	$-0.908850\pi$
$809$	−3.57146e7	−1.91856	−0.959279	+	0.282460i	$0.908850\pi$
$810$	0	0
$811$	2.92173e7	1.55987	0.779934	−	0.625861i	$-0.215252\pi$
$811$	2.92173e7	1.55987	0.779934	+	0.625861i	$0.215252\pi$
$812$	0	0
$813$	8.82718e6	0.468377
$814$	0	0
$815$	578874.	0.0305274
$816$	0	0
$817$	−2.64517e7	−1.38643
$818$	0	0
$819$	3.60359e6	0.187727
$820$	0	0
$821$	−2.04749e7	−1.06014	−0.530070	−	0.847954i	$-0.677834\pi$
$821$	−2.04749e7	−1.06014	−0.530070	+	0.847954i	$0.677834\pi$
$822$	0	0
$823$	515564.	0.0265328	0.0132664	−	0.999912i	$-0.495777\pi$
$823$	515564.	0.0265328	0.0132664	+	0.999912i	$0.495777\pi$
$824$	0	0
$825$	−3.09207e6	−0.158166
$826$	0	0
$827$	−2.81057e7	−1.42899	−0.714497	−	0.699638i	$-0.753344\pi$
$827$	−2.81057e7	−1.42899	−0.714497	+	0.699638i	$0.753344\pi$
$828$	0	0
$829$	−1.80636e7	−0.912887	−0.456443	−	0.889752i	$-0.650877\pi$
$829$	−1.80636e7	−0.912887	−0.456443	+	0.889752i	$0.650877\pi$
$830$	0	0
$831$	3.28935e7	1.65237
$832$	0	0
$833$	−690197.	−0.0344636
$834$	0	0
$835$	2.39098e6	0.118675
$836$	0	0
$837$	1.11095e7	0.548127
$838$	0	0
$839$	7.37319e6	0.361618	0.180809	−	0.983518i	$-0.442128\pi$
$839$	7.37319e6	0.361618	0.180809	+	0.983518i	$0.442128\pi$
$840$	0	0
$841$	−2.02996e7	−0.989684
$842$	0	0
$843$	−1.08111e7	−0.523965
$844$	0	0
$845$	−3.44830e6	−0.166136
$846$	0	0
$847$	4.85708e6	0.232631
$848$	0	0
$849$	−4.65094e7	−2.21448
$850$	0	0
$851$	−1.30048e7	−0.615572
$852$	0	0
$853$	2.97937e7	1.40201	0.701007	−	0.713155i	$-0.252734\pi$
$853$	2.97937e7	1.40201	0.701007	+	0.713155i	$0.252734\pi$
$854$	0	0
$855$	6.54178e6	0.306042
$856$	0	0
$857$	1.59561e7	0.742120	0.371060	−	0.928609i	$-0.378994\pi$
$857$	1.59561e7	0.742120	0.371060	+	0.928609i	$0.378994\pi$
$858$	0	0
$859$	2.86950e7	1.32685	0.663427	−	0.748241i	$-0.269102\pi$
$859$	2.86950e7	1.32685	0.663427	+	0.748241i	$0.269102\pi$
$860$	0	0
$861$	−7.20867e6	−0.331396
$862$	0	0
$863$	1.81195e7	0.828169	0.414084	−	0.910239i	$-0.364102\pi$
$863$	1.81195e7	0.828169	0.414084	+	0.910239i	$0.364102\pi$
$864$	0	0
$865$	−1.16358e7	−0.528758
$866$	0	0
$867$	2.65848e7	1.20112
$868$	0	0
$869$	−1.56433e7	−0.702716
$870$	0	0
$871$	2.14841e7	0.959559
$872$	0	0
$873$	−4.60006e6	−0.204281
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	−2.26847e7	−0.995942	−0.497971	−	0.867194i	$-0.665921\pi$
$877$	−2.26847e7	−0.995942	−0.497971	+	0.867194i	$0.665921\pi$
$878$	0	0
$879$	−3.49801e7	−1.52703
$880$	0	0
$881$	−1.85071e7	−0.803339	−0.401670	−	0.915785i	$-0.631570\pi$
$881$	−1.85071e7	−0.803339	−0.401670	+	0.915785i	$0.631570\pi$
$882$	0	0
$883$	2.56694e7	1.10794	0.553968	−	0.832538i	$-0.313113\pi$
$883$	2.56694e7	1.10794	0.553968	+	0.832538i	$0.313113\pi$
$884$	0	0
$885$	−843463.	−0.0361999
$886$	0	0
$887$	1.63268e7	0.696774	0.348387	−	0.937351i	$-0.386730\pi$
$887$	1.63268e7	0.696774	0.348387	+	0.937351i	$0.386730\pi$
$888$	0	0
$889$	8.09729e6	0.343626
$890$	0	0
$891$	−1.81329e7	−0.765195
$892$	0	0
$893$	1.39799e7	0.586644
$894$	0	0
$895$	1.64713e7	0.687337
$896$	0	0
$897$	1.81336e7	0.752494
$898$	0	0
$899$	−2.83209e6	−0.116871
$900$	0	0
$901$	350312.	0.0143762
$902$	0	0
$903$	−1.49916e7	−0.611827
$904$	0	0
$905$	1.52581e7	0.619270
$906$	0	0
$907$	2.58505e7	1.04340	0.521700	−	0.853129i	$-0.325298\pi$
$907$	2.58505e7	1.04340	0.521700	+	0.853129i	$0.325298\pi$
$908$	0	0
$909$	−2.13708e6	−0.0857851
$910$	0	0
$911$	−3.39056e7	−1.35355	−0.676777	−	0.736188i	$-0.736624\pi$
$911$	−3.39056e7	−1.35355	−0.676777	+	0.736188i	$0.736624\pi$
$912$	0	0
$913$	4.55209e6	0.180731
$914$	0	0
$915$	1.02536e7	0.404876
$916$	0	0
$917$	1.63821e7	0.643350
$918$	0	0
$919$	−7.93650e6	−0.309985	−0.154992	−	0.987916i	$-0.549535\pi$
$919$	−7.93650e6	−0.309985	−0.154992	+	0.987916i	$0.549535\pi$
$920$	0	0
$921$	5.36427e7	2.08383
$922$	0	0
$923$	1.52985e7	0.591078
$924$	0	0
$925$	−4.30470e6	−0.165420
$926$	0	0
$927$	−1.68211e7	−0.642919
$928$	0	0
$929$	−3.57036e7	−1.35729	−0.678646	−	0.734466i	$-0.737433\pi$
$929$	−3.57036e7	−1.35729	−0.678646	+	0.734466i	$0.737433\pi$
$930$	0	0
$931$	4.12689e6	0.156045
$932$	0	0
$933$	5.49334e7	2.06601
$934$	0	0
$935$	−1.78838e6	−0.0669008
$936$	0	0
$937$	1.66283e7	0.618728	0.309364	−	0.950944i	$-0.399884\pi$
$937$	1.66283e7	0.618728	0.309364	+	0.950944i	$0.399884\pi$
$938$	0	0
$939$	−1.05242e7	−0.389516
$940$	0	0
$941$	2.02482e7	0.745440	0.372720	−	0.927944i	$-0.378425\pi$
$941$	2.02482e7	0.745440	0.372720	+	0.927944i	$0.378425\pi$
$942$	0	0
$943$	−1.39724e7	−0.511670
$944$	0	0
$945$	−2.21038e6	−0.0805168
$946$	0	0
$947$	−579062.	−0.0209822	−0.0104911	−	0.999945i	$-0.503339\pi$
$947$	−579062.	−0.0209822	−0.0104911	+	0.999945i	$0.503339\pi$
$948$	0	0
$949$	3.12998e7	1.12817
$950$	0	0
$951$	2.93184e7	1.05121
$952$	0	0
$953$	4.33037e7	1.54452	0.772259	−	0.635307i	$-0.219127\pi$
$953$	4.33037e7	1.54452	0.772259	+	0.635307i	$0.219127\pi$
$954$	0	0
$955$	−1.43731e7	−0.509969
$956$	0	0
$957$	2.27568e6	0.0803215
$958$	0	0
$959$	−7.93191e6	−0.278504
$960$	0	0
$961$	9.27876e6	0.324102
$962$	0	0
$963$	5.40854e6	0.187938
$964$	0	0
$965$	−1.37329e7	−0.474726
$966$	0	0
$967$	2.28301e7	0.785129	0.392564	−	0.919725i	$-0.371588\pi$
$967$	2.28301e7	0.785129	0.392564	+	0.919725i	$0.371588\pi$
$968$	0	0
$969$	9.82294e6	0.336072
$970$	0	0
$971$	2.47290e7	0.841701	0.420851	−	0.907130i	$-0.361732\pi$
$971$	2.47290e7	0.841701	0.420851	+	0.907130i	$0.361732\pi$
$972$	0	0
$973$	−1.97338e7	−0.668234
$974$	0	0
$975$	6.00239e6	0.202215
$976$	0	0
$977$	2.72538e6	0.0913462	0.0456731	−	0.998956i	$-0.485457\pi$
$977$	2.72538e6	0.0913462	0.0456731	+	0.998956i	$0.485457\pi$
$978$	0	0
$979$	428349.	0.0142837
$980$	0	0
$981$	2.10487e7	0.698317
$982$	0	0
$983$	−2.59662e7	−0.857088	−0.428544	−	0.903521i	$-0.640973\pi$
$983$	−2.59662e7	−0.857088	−0.428544	+	0.903521i	$0.640973\pi$
$984$	0	0
$985$	−1.87032e7	−0.614221
$986$	0	0
$987$	7.92316e6	0.258884
$988$	0	0
$989$	−2.90578e7	−0.944652
$990$	0	0
$991$	−7.15802e6	−0.231531	−0.115765	−	0.993277i	$-0.536932\pi$
$991$	−7.15802e6	−0.231531	−0.115765	+	0.993277i	$0.536932\pi$
$992$	0	0
$993$	2.44049e7	0.785425
$994$	0	0
$995$	1.31321e7	0.420509
$996$	0	0
$997$	6.17496e7	1.96741	0.983707	−	0.179777i	$-0.0575375\pi$
$997$	6.17496e7	1.96741	0.983707	+	0.179777i	$0.0575375\pi$
$998$	0	0
$999$	−1.24278e7	−0.393984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.c.1.1	✓	2
4.3	odd	2		560.6.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.c.1.1	✓	2	1.1	even	1	trivial
560.6.a.n.1.2		2	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)

\(f(q)\)	\(=\)	\(q-19.8806 q^{3} +25.0000 q^{5} -49.0000 q^{7} +152.239 q^{9} +O(q^{10})\)	\(q-19.8806 q^{3} +25.0000 q^{5} -49.0000 q^{7} +152.239 q^{9} +248.851 q^{11} -483.074 q^{13} -497.015 q^{15} -287.462 q^{17} +1718.82 q^{19} +974.150 q^{21} +1888.17 q^{23} +625.000 q^{25} +1804.39 q^{27} -459.984 q^{29} +6156.94 q^{31} -4947.31 q^{33} -1225.00 q^{35} -6887.51 q^{37} +9603.82 q^{39} -7399.96 q^{41} -15389.4 q^{43} +3805.97 q^{45} +8133.41 q^{47} +2401.00 q^{49} +5714.93 q^{51} -1218.64 q^{53} +6221.28 q^{55} -34171.2 q^{57} +1697.06 q^{59} -20630.3 q^{61} -7459.70 q^{63} -12076.9 q^{65} -44473.7 q^{67} -37537.9 q^{69} -31669.1 q^{71} -64792.8 q^{73} -12425.4 q^{75} -12193.7 q^{77} -62862.2 q^{79} -72866.4 q^{81} +18292.4 q^{83} -7186.56 q^{85} +9144.76 q^{87} +1721.31 q^{89} +23670.6 q^{91} -122404. q^{93} +42970.5 q^{95} -30216.1 q^{97} +37884.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 50 q^{5} - 98 q^{7} + 388 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 50 * q^5 - 98 * q^7 + 388 * q^9	\( 2 q + 2 q^{3} + 50 q^{5} - 98 q^{7} + 388 q^{9} - 254 q^{11} - 1342 q^{13} + 50 q^{15} - 1786 q^{17} + 1976 q^{19} - 98 q^{21} - 2112 q^{23} + 1250 q^{25} + 1646 q^{27} - 4762 q^{29} + 13692 q^{31} - 15950 q^{33} - 2450 q^{35} + 2136 q^{37} - 9190 q^{39} - 6740 q^{41} - 26728 q^{43} + 9700 q^{45} - 6326 q^{47} + 4802 q^{49} - 27074 q^{51} + 24624 q^{53} - 6350 q^{55} - 28544 q^{57} + 51336 q^{59} + 4468 q^{61} - 19012 q^{63} - 33550 q^{65} - 39168 q^{67} - 125064 q^{69} + 41232 q^{71} - 36124 q^{73} + 1250 q^{75} + 12446 q^{77} - 140842 q^{79} - 133622 q^{81} - 57712 q^{83} - 44650 q^{85} - 84986 q^{87} - 20236 q^{89} + 65758 q^{91} + 42468 q^{93} + 49400 q^{95} - 183586 q^{97} - 80668 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 50 * q^5 - 98 * q^7 + 388 * q^9 - 254 * q^11 - 1342 * q^13 + 50 * q^15 - 1786 * q^17 + 1976 * q^19 - 98 * q^21 - 2112 * q^23 + 1250 * q^25 + 1646 * q^27 - 4762 * q^29 + 13692 * q^31 - 15950 * q^33 - 2450 * q^35 + 2136 * q^37 - 9190 * q^39 - 6740 * q^41 - 26728 * q^43 + 9700 * q^45 - 6326 * q^47 + 4802 * q^49 - 27074 * q^51 + 24624 * q^53 - 6350 * q^55 - 28544 * q^57 + 51336 * q^59 + 4468 * q^61 - 19012 * q^63 - 33550 * q^65 - 39168 * q^67 - 125064 * q^69 + 41232 * q^71 - 36124 * q^73 + 1250 * q^75 + 12446 * q^77 - 140842 * q^79 - 133622 * q^81 - 57712 * q^83 - 44650 * q^85 - 84986 * q^87 - 20236 * q^89 + 65758 * q^91 + 42468 * q^93 + 49400 * q^95 - 183586 * q^97 - 80668 * q^99

Introduction

L-functions

Modular forms

Varieties

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