LMFDB - Embedded newform 260.6.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,6,Mod(1,260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("260.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 260 = 2^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	260.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:	$41.6997931514$
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} - 2x^{3} - 335x^{2} - 928x + 4264 $ x^4 - 2x^3 - 335x^2 - 928*x + 4264
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.44457$ of defining polynomial
Character	$\chi$	$=$	260.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.2247 q^{3} +25.0000 q^{5} -113.468 q^{7} -93.5569 q^{9} +O(q^{10})$	$q+12.2247 q^{3} +25.0000 q^{5} -113.468 q^{7} -93.5569 q^{9} +286.946 q^{11} -169.000 q^{13} +305.617 q^{15} +27.0836 q^{17} -1400.34 q^{19} -1387.12 q^{21} -1877.84 q^{23} +625.000 q^{25} -4114.30 q^{27} -5309.41 q^{29} -9.68719 q^{31} +3507.82 q^{33} -2836.71 q^{35} -578.181 q^{37} -2065.97 q^{39} -909.694 q^{41} -2990.34 q^{43} -2338.92 q^{45} +21981.2 q^{47} -3931.94 q^{49} +331.088 q^{51} -23669.2 q^{53} +7173.64 q^{55} -17118.7 q^{57} -46895.4 q^{59} +36538.9 q^{61} +10615.7 q^{63} -4225.00 q^{65} +30806.6 q^{67} -22956.0 q^{69} -72500.7 q^{71} -81602.9 q^{73} +7640.43 q^{75} -32559.2 q^{77} +41698.9 q^{79} -27561.8 q^{81} -17708.6 q^{83} +677.089 q^{85} -64905.9 q^{87} +35240.9 q^{89} +19176.1 q^{91} -118.423 q^{93} -35008.4 q^{95} +88496.5 q^{97} -26845.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{3} + 100 q^{5} - 24 q^{7} - 84 q^{9}+O(q^{10}) $ 4 * q - 20 * q^3 + 100 * q^5 - 24 * q^7 - 84 * q^9	$ 4 q - 20 q^{3} + 100 q^{5} - 24 q^{7} - 84 q^{9} - 12 q^{11} - 676 q^{13} - 500 q^{15} - 416 q^{17} - 644 q^{19} - 376 q^{21} + 1748 q^{23} + 2500 q^{25} - 1664 q^{27} - 4080 q^{29} - 1500 q^{31} + 2096 q^{33} - 600 q^{35} - 3848 q^{37} + 3380 q^{39} - 17968 q^{41} + 3020 q^{43} - 2100 q^{45} - 12328 q^{47} - 28108 q^{49} - 15704 q^{51} - 52672 q^{53} - 300 q^{55} - 61696 q^{57} - 58988 q^{59} - 56736 q^{61} - 17768 q^{63} - 16900 q^{65} - 56480 q^{67} - 120392 q^{69} - 68260 q^{71} - 113000 q^{73} - 12500 q^{75} - 126600 q^{77} - 43640 q^{79} - 135684 q^{81} - 52248 q^{83} - 10400 q^{85} - 7760 q^{87} - 114376 q^{89} + 4056 q^{91} + 17680 q^{93} - 16100 q^{95} - 21432 q^{97} + 44092 q^{99}+O(q^{100}) $ 4 * q - 20 * q^3 + 100 * q^5 - 24 * q^7 - 84 * q^9 - 12 * q^11 - 676 * q^13 - 500 * q^15 - 416 * q^17 - 644 * q^19 - 376 * q^21 + 1748 * q^23 + 2500 * q^25 - 1664 * q^27 - 4080 * q^29 - 1500 * q^31 + 2096 * q^33 - 600 * q^35 - 3848 * q^37 + 3380 * q^39 - 17968 * q^41 + 3020 * q^43 - 2100 * q^45 - 12328 * q^47 - 28108 * q^49 - 15704 * q^51 - 52672 * q^53 - 300 * q^55 - 61696 * q^57 - 58988 * q^59 - 56736 * q^61 - 17768 * q^63 - 16900 * q^65 - 56480 * q^67 - 120392 * q^69 - 68260 * q^71 - 113000 * q^73 - 12500 * q^75 - 126600 * q^77 - 43640 * q^79 - 135684 * q^81 - 52248 * q^83 - 10400 * q^85 - 7760 * q^87 - 114376 * q^89 + 4056 * q^91 + 17680 * q^93 - 16100 * q^95 - 21432 * q^97 + 44092 * 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
$3$	12.2247	0.784214	0.392107	−	0.919920i	$-0.371746\pi$
$3$	12.2247	0.784214	0.392107	+	0.919920i	$0.371746\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−113.468	−0.875245	−0.437622	−	0.899159i	$-0.644179\pi$
$7$	−113.468	−0.875245	−0.437622	+	0.899159i	$0.644179\pi$
$8$	0	0
$9$	−93.5569	−0.385008
$10$	0	0
$11$	286.946	0.715019	0.357510	−	0.933909i	$-0.383626\pi$
$11$	286.946	0.715019	0.357510	+	0.933909i	$0.383626\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	305.617	0.350711
$16$	0	0
$17$	27.0836	0.0227292	0.0113646	−	0.999935i	$-0.496382\pi$
$17$	27.0836	0.0227292	0.0113646	+	0.999935i	$0.496382\pi$
$18$	0	0
$19$	−1400.34	−0.889914	−0.444957	−	0.895552i	$-0.646781\pi$
$19$	−1400.34	−0.889914	−0.444957	+	0.895552i	$0.646781\pi$
$20$	0	0
$21$	−1387.12	−0.686380
$22$	0	0
$23$	−1877.84	−0.740183	−0.370091	−	0.928995i	$-0.620674\pi$
$23$	−1877.84	−0.740183	−0.370091	+	0.928995i	$0.620674\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−4114.30	−1.08614
$28$	0	0
$29$	−5309.41	−1.17233	−0.586167	−	0.810190i	$-0.699364\pi$
$29$	−5309.41	−1.17233	−0.586167	+	0.810190i	$0.699364\pi$
$30$	0	0
$31$	−9.68719	−0.00181048	−0.000905240	−	1.00000i	$-0.500288\pi$
$31$	−9.68719	−0.00181048	−0.000905240		1.00000i	$0.500288\pi$
$32$	0	0
$33$	3507.82	0.560728
$34$	0	0
$35$	−2836.71	−0.391421
$36$	0	0
$37$	−578.181	−0.0694320	−0.0347160	−	0.999397i	$-0.511053\pi$
$37$	−578.181	−0.0694320	−0.0347160	+	0.999397i	$0.511053\pi$
$38$	0	0
$39$	−2065.97	−0.217502
$40$	0	0
$41$	−909.694	−0.0845154	−0.0422577	−	0.999107i	$-0.513455\pi$
$41$	−909.694	−0.0845154	−0.0422577	+	0.999107i	$0.513455\pi$
$42$	0	0
$43$	−2990.34	−0.246632	−0.123316	−	0.992367i	$-0.539353\pi$
$43$	−2990.34	−0.246632	−0.123316	+	0.992367i	$0.539353\pi$
$44$	0	0
$45$	−2338.92	−0.172181
$46$	0	0
$47$	21981.2	1.45147	0.725734	−	0.687975i	$-0.241500\pi$
$47$	21981.2	1.45147	0.725734	+	0.687975i	$0.241500\pi$
$48$	0	0
$49$	−3931.94	−0.233947
$50$	0	0
$51$	331.088	0.0178245
$52$	0	0
$53$	−23669.2	−1.15743	−0.578714	−	0.815531i	$-0.696445\pi$
$53$	−23669.2	−1.15743	−0.578714	+	0.815531i	$0.696445\pi$
$54$	0	0
$55$	7173.64	0.319766
$56$	0	0
$57$	−17118.7	−0.697883
$58$	0	0
$59$	−46895.4	−1.75388	−0.876940	−	0.480600i	$-0.840419\pi$
$59$	−46895.4	−1.75388	−0.876940	+	0.480600i	$0.840419\pi$
$60$	0	0
$61$	36538.9	1.25728	0.628639	−	0.777698i	$-0.283612\pi$
$61$	36538.9	1.25728	0.628639	+	0.777698i	$0.283612\pi$
$62$	0	0
$63$	10615.7	0.336976
$64$	0	0
$65$	−4225.00	−0.124035
$66$	0	0
$67$	30806.6	0.838410	0.419205	−	0.907892i	$-0.362309\pi$
$67$	30806.6	0.838410	0.419205	+	0.907892i	$0.362309\pi$
$68$	0	0
$69$	−22956.0	−0.580462
$70$	0	0
$71$	−72500.7	−1.70685	−0.853427	−	0.521213i	$-0.825480\pi$
$71$	−72500.7	−1.70685	−0.853427	+	0.521213i	$0.825480\pi$
$72$	0	0
$73$	−81602.9	−1.79225	−0.896125	−	0.443802i	$-0.853629\pi$
$73$	−81602.9	−1.79225	−0.896125	+	0.443802i	$0.853629\pi$
$74$	0	0
$75$	7640.43	0.156843
$76$	0	0
$77$	−32559.2	−0.625817
$78$	0	0
$79$	41698.9	0.751722	0.375861	−	0.926676i	$-0.377347\pi$
$79$	41698.9	0.751722	0.375861	+	0.926676i	$0.377347\pi$
$80$	0	0
$81$	−27561.8	−0.466761
$82$	0	0
$83$	−17708.6	−0.282156	−0.141078	−	0.989998i	$-0.545057\pi$
$83$	−17708.6	−0.282156	−0.141078	+	0.989998i	$0.545057\pi$
$84$	0	0
$85$	677.089	0.0101648
$86$	0	0
$87$	−64905.9	−0.919361
$88$	0	0
$89$	35240.9	0.471597	0.235799	−	0.971802i	$-0.424229\pi$
$89$	35240.9	0.471597	0.235799	+	0.971802i	$0.424229\pi$
$90$	0	0
$91$	19176.1	0.242749
$92$	0	0
$93$	−118.423	−0.00141980
$94$	0	0
$95$	−35008.4	−0.397982
$96$	0	0
$97$	88496.5	0.954986	0.477493	−	0.878636i	$-0.341546\pi$
$97$	88496.5	0.954986	0.477493	+	0.878636i	$0.341546\pi$
$98$	0	0
$99$	−26845.7	−0.275288
$100$	0	0
$101$	107065.	1.04435	0.522175	−	0.852838i	$-0.325121\pi$
$101$	107065.	1.04435	0.522175	+	0.852838i	$0.325121\pi$
$102$	0	0
$103$	−170687.	−1.58528	−0.792641	−	0.609689i	$-0.791294\pi$
$103$	−170687.	−1.58528	−0.792641	+	0.609689i	$0.791294\pi$
$104$	0	0
$105$	−34677.9	−0.306958
$106$	0	0
$107$	−85959.1	−0.725826	−0.362913	−	0.931823i	$-0.618218\pi$
$107$	−85959.1	−0.725826	−0.362913	+	0.931823i	$0.618218\pi$
$108$	0	0
$109$	−123344.	−0.994375	−0.497187	−	0.867643i	$-0.665634\pi$
$109$	−123344.	−0.994375	−0.497187	+	0.867643i	$0.665634\pi$
$110$	0	0
$111$	−7068.08	−0.0544495
$112$	0	0
$113$	−43120.4	−0.317677	−0.158839	−	0.987305i	$-0.550775\pi$
$113$	−43120.4	−0.317677	−0.158839	+	0.987305i	$0.550775\pi$
$114$	0	0
$115$	−46946.0	−0.331020
$116$	0	0
$117$	15811.1	0.106782
$118$	0	0
$119$	−3073.13	−0.0198936
$120$	0	0
$121$	−78713.2	−0.488747
$122$	0	0
$123$	−11120.7	−0.0662782
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	786.478	0.00432690	0.00216345	−	0.999998i	$-0.499311\pi$
$127$	786.478	0.00432690	0.00216345	+	0.999998i	$0.499311\pi$
$128$	0	0
$129$	−36556.0	−0.193412
$130$	0	0
$131$	−75752.6	−0.385673	−0.192836	−	0.981231i	$-0.561769\pi$
$131$	−75752.6	−0.385673	−0.192836	+	0.981231i	$0.561769\pi$
$132$	0	0
$133$	158894.	0.778892
$134$	0	0
$135$	−102858.	−0.485738
$136$	0	0
$137$	−142333.	−0.647892	−0.323946	−	0.946076i	$-0.605010\pi$
$137$	−142333.	−0.647892	−0.323946	+	0.946076i	$0.605010\pi$
$138$	0	0
$139$	184905.	0.811728	0.405864	−	0.913933i	$-0.366971\pi$
$139$	184905.	0.811728	0.405864	+	0.913933i	$0.366971\pi$
$140$	0	0
$141$	268714.	1.13826
$142$	0	0
$143$	−48493.8	−0.198311
$144$	0	0
$145$	−132735.	−0.524284
$146$	0	0
$147$	−48066.8	−0.183464
$148$	0	0
$149$	−60807.5	−0.224384	−0.112192	−	0.993687i	$-0.535787\pi$
$149$	−60807.5	−0.224384	−0.112192	+	0.993687i	$0.535787\pi$
$150$	0	0
$151$	86947.9	0.310325	0.155163	−	0.987889i	$-0.450410\pi$
$151$	86947.9	0.310325	0.155163	+	0.987889i	$0.450410\pi$
$152$	0	0
$153$	−2533.85	−0.00875090
$154$	0	0
$155$	−242.180	−0.000809671 0
$156$	0	0
$157$	435822.	1.41111	0.705554	−	0.708656i	$-0.250698\pi$
$157$	435822.	1.41111	0.705554	+	0.708656i	$0.250698\pi$
$158$	0	0
$159$	−289349.	−0.907671
$160$	0	0
$161$	213075.	0.647841
$162$	0	0
$163$	489082.	1.44183	0.720913	−	0.693026i	$-0.243723\pi$
$163$	489082.	1.44183	0.720913	+	0.693026i	$0.243723\pi$
$164$	0	0
$165$	87695.5	0.250765
$166$	0	0
$167$	408070.	1.13225	0.566127	−	0.824318i	$-0.308441\pi$
$167$	408070.	1.13225	0.566127	+	0.824318i	$0.308441\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	131011.	0.342624
$172$	0	0
$173$	−165606.	−0.420690	−0.210345	−	0.977627i	$-0.567459\pi$
$173$	−165606.	−0.420690	−0.210345	+	0.977627i	$0.567459\pi$
$174$	0	0
$175$	−70917.7	−0.175049
$176$	0	0
$177$	−573281.	−1.37542
$178$	0	0
$179$	465517.	1.08593	0.542967	−	0.839754i	$-0.317301\pi$
$179$	465517.	1.08593	0.542967	+	0.839754i	$0.317301\pi$
$180$	0	0
$181$	111786.	0.253624	0.126812	−	0.991927i	$-0.459526\pi$
$181$	111786.	0.253624	0.126812	+	0.991927i	$0.459526\pi$
$182$	0	0
$183$	446677.	0.985975
$184$	0	0
$185$	−14454.5	−0.0310509
$186$	0	0
$187$	7771.51	0.0162518
$188$	0	0
$189$	466843.	0.950641
$190$	0	0
$191$	821476.	1.62934	0.814669	−	0.579926i	$-0.196918\pi$
$191$	821476.	1.62934	0.814669	+	0.579926i	$0.196918\pi$
$192$	0	0
$193$	−159551.	−0.308324	−0.154162	−	0.988046i	$-0.549268\pi$
$193$	−159551.	−0.308324	−0.154162	+	0.988046i	$0.549268\pi$
$194$	0	0
$195$	−51649.3	−0.0972698
$196$	0	0
$197$	558853.	1.02596	0.512982	−	0.858399i	$-0.328541\pi$
$197$	558853.	1.02596	0.512982	+	0.858399i	$0.328541\pi$
$198$	0	0
$199$	61616.3	0.110297	0.0551484	−	0.998478i	$-0.482437\pi$
$199$	61616.3	0.110297	0.0551484	+	0.998478i	$0.482437\pi$
$200$	0	0
$201$	376601.	0.657493
$202$	0	0
$203$	602450.	1.02608
$204$	0	0
$205$	−22742.3	−0.0377964
$206$	0	0
$207$	175685.	0.284976
$208$	0	0
$209$	−401820.	−0.636306
$210$	0	0
$211$	1.00941e6	1.56085	0.780425	−	0.625249i	$-0.215003\pi$
$211$	1.00941e6	1.56085	0.780425	+	0.625249i	$0.215003\pi$
$212$	0	0
$213$	−886299.	−1.33854
$214$	0	0
$215$	−74758.5	−0.110297
$216$	0	0
$217$	1099.19	0.00158461
$218$	0	0
$219$	−997571.	−1.40551
$220$	0	0
$221$	−4577.12	−0.00630393
$222$	0	0
$223$	567430.	0.764101	0.382050	−	0.924142i	$-0.375218\pi$
$223$	567430.	0.764101	0.382050	+	0.924142i	$0.375218\pi$
$224$	0	0
$225$	−58473.1	−0.0770016
$226$	0	0
$227$	186823.	0.240639	0.120319	−	0.992735i	$-0.461608\pi$
$227$	186823.	0.240639	0.120319	+	0.992735i	$0.461608\pi$
$228$	0	0
$229$	−1.15069e6	−1.45001	−0.725005	−	0.688744i	$-0.758162\pi$
$229$	−1.15069e6	−1.45001	−0.725005	+	0.688744i	$0.758162\pi$
$230$	0	0
$231$	−398027.	−0.490775
$232$	0	0
$233$	−38942.4	−0.0469930	−0.0234965	−	0.999724i	$-0.507480\pi$
$233$	−38942.4	−0.0469930	−0.0234965	+	0.999724i	$0.507480\pi$
$234$	0	0
$235$	549531.	0.649116
$236$	0	0
$237$	509756.	0.589511
$238$	0	0
$239$	−284879.	−0.322600	−0.161300	−	0.986905i	$-0.551569\pi$
$239$	−284879.	−0.322600	−0.161300	+	0.986905i	$0.551569\pi$
$240$	0	0
$241$	−1.41307e6	−1.56719	−0.783594	−	0.621273i	$-0.786616\pi$
$241$	−1.41307e6	−1.56719	−0.783594	+	0.621273i	$0.786616\pi$
$242$	0	0
$243$	662842.	0.720102
$244$	0	0
$245$	−98298.5	−0.104624
$246$	0	0
$247$	236657.	0.246818
$248$	0	0
$249$	−216483.	−0.221271
$250$	0	0
$251$	−1.10531e6	−1.10739	−0.553694	−	0.832720i	$-0.686782\pi$
$251$	−1.10531e6	−1.10739	−0.553694	+	0.832720i	$0.686782\pi$
$252$	0	0
$253$	−538838.	−0.529245
$254$	0	0
$255$	8277.20	0.00797137
$256$	0	0
$257$	111150.	0.104973	0.0524864	−	0.998622i	$-0.483285\pi$
$257$	111150.	0.104973	0.0524864	+	0.998622i	$0.483285\pi$
$258$	0	0
$259$	65605.2	0.0607700
$260$	0	0
$261$	496732.	0.451358
$262$	0	0
$263$	1.48222e6	1.32137	0.660683	−	0.750665i	$-0.270267\pi$
$263$	1.48222e6	1.32137	0.660683	+	0.750665i	$0.270267\pi$
$264$	0	0
$265$	−591730.	−0.517617
$266$	0	0
$267$	430809.	0.369833
$268$	0	0
$269$	−3984.78	−0.00335755	−0.00167878	−	0.999999i	$-0.500534\pi$
$269$	−3984.78	−0.00335755	−0.00167878	+	0.999999i	$0.500534\pi$
$270$	0	0
$271$	−614876.	−0.508586	−0.254293	−	0.967127i	$-0.581843\pi$
$271$	−614876.	−0.508586	−0.254293	+	0.967127i	$0.581843\pi$
$272$	0	0
$273$	234422.	0.190367
$274$	0	0
$275$	179341.	0.143004
$276$	0	0
$277$	−326483.	−0.255659	−0.127830	−	0.991796i	$-0.540801\pi$
$277$	−326483.	−0.255659	−0.127830	+	0.991796i	$0.540801\pi$
$278$	0	0
$279$	906.304	0.000697049 0
$280$	0	0
$281$	−562943.	−0.425304	−0.212652	−	0.977128i	$-0.568210\pi$
$281$	−562943.	−0.425304	−0.212652	+	0.977128i	$0.568210\pi$
$282$	0	0
$283$	1.11323e6	0.826264	0.413132	−	0.910671i	$-0.364435\pi$
$283$	1.11323e6	0.826264	0.413132	+	0.910671i	$0.364435\pi$
$284$	0	0
$285$	−427967.	−0.312103
$286$	0	0
$287$	103221.	0.0739716
$288$	0	0
$289$	−1.41912e6	−0.999483
$290$	0	0
$291$	1.08184e6	0.748914
$292$	0	0
$293$	1.47580e6	1.00429	0.502144	−	0.864784i	$-0.332545\pi$
$293$	1.47580e6	1.00429	0.502144	+	0.864784i	$0.332545\pi$
$294$	0	0
$295$	−1.17238e6	−0.784359
$296$	0	0
$297$	−1.18058e6	−0.776613
$298$	0	0
$299$	317355.	0.205290
$300$	0	0
$301$	339309.	0.215863
$302$	0	0
$303$	1.30884e6	0.818994
$304$	0	0
$305$	913473.	0.562271
$306$	0	0
$307$	578749.	0.350465	0.175232	−	0.984527i	$-0.443932\pi$
$307$	578749.	0.350465	0.175232	+	0.984527i	$0.443932\pi$
$308$	0	0
$309$	−2.08659e6	−1.24320
$310$	0	0
$311$	589037.	0.345336	0.172668	−	0.984980i	$-0.444761\pi$
$311$	589037.	0.345336	0.172668	+	0.984980i	$0.444761\pi$
$312$	0	0
$313$	−469092.	−0.270643	−0.135322	−	0.990802i	$-0.543207\pi$
$313$	−469092.	−0.270643	−0.135322	+	0.990802i	$0.543207\pi$
$314$	0	0
$315$	265394.	0.150700
$316$	0	0
$317$	−1.54670e6	−0.864484	−0.432242	−	0.901758i	$-0.642277\pi$
$317$	−1.54670e6	−0.864484	−0.432242	+	0.901758i	$0.642277\pi$
$318$	0	0
$319$	−1.52351e6	−0.838241
$320$	0	0
$321$	−1.05082e6	−0.569203
$322$	0	0
$323$	−37926.1	−0.0202270
$324$	0	0
$325$	−105625.	−0.0554700
$326$	0	0
$327$	−1.50784e6	−0.779803
$328$	0	0
$329$	−2.49417e6	−1.27039
$330$	0	0
$331$	−537750.	−0.269781	−0.134890	−	0.990861i	$-0.543068\pi$
$331$	−537750.	−0.269781	−0.134890	+	0.990861i	$0.543068\pi$
$332$	0	0
$333$	54092.8	0.0267318
$334$	0	0
$335$	770165.	0.374948
$336$	0	0
$337$	1.25740e6	0.603115	0.301558	−	0.953448i	$-0.402493\pi$
$337$	1.25740e6	0.603115	0.301558	+	0.953448i	$0.402493\pi$
$338$	0	0
$339$	−527133.	−0.249127
$340$	0	0
$341$	−2779.70	−0.00129453
$342$	0	0
$343$	2.35321e6	1.08001
$344$	0	0
$345$	−573901.	−0.259591
$346$	0	0
$347$	−2.40945e6	−1.07422	−0.537111	−	0.843512i	$-0.680484\pi$
$347$	−2.40945e6	−1.07422	−0.537111	+	0.843512i	$0.680484\pi$
$348$	0	0
$349$	−2.14151e6	−0.941143	−0.470571	−	0.882362i	$-0.655952\pi$
$349$	−2.14151e6	−0.941143	−0.470571	+	0.882362i	$0.655952\pi$
$350$	0	0
$351$	695317.	0.301242
$352$	0	0
$353$	3.15733e6	1.34860	0.674299	−	0.738458i	$-0.264446\pi$
$353$	3.15733e6	1.34860	0.674299	+	0.738458i	$0.264446\pi$
$354$	0	0
$355$	−1.81252e6	−0.763328
$356$	0	0
$357$	−37568.0	−0.0156008
$358$	0	0
$359$	−1.39999e6	−0.573308	−0.286654	−	0.958034i	$-0.592543\pi$
$359$	−1.39999e6	−0.573308	−0.286654	+	0.958034i	$0.592543\pi$
$360$	0	0
$361$	−515161.	−0.208053
$362$	0	0
$363$	−962245.	−0.383283
$364$	0	0
$365$	−2.04007e6	−0.801518
$366$	0	0
$367$	−462495.	−0.179243	−0.0896214	−	0.995976i	$-0.528566\pi$
$367$	−462495.	−0.179243	−0.0896214	+	0.995976i	$0.528566\pi$
$368$	0	0
$369$	85108.1	0.0325391
$370$	0	0
$371$	2.68570e6	1.01303
$372$	0	0
$373$	1.92966e6	0.718138	0.359069	−	0.933311i	$-0.383094\pi$
$373$	1.92966e6	0.718138	0.359069	+	0.933311i	$0.383094\pi$
$374$	0	0
$375$	191011.	0.0701423
$376$	0	0
$377$	897290.	0.325147
$378$	0	0
$379$	−860927.	−0.307870	−0.153935	−	0.988081i	$-0.549195\pi$
$379$	−860927.	−0.307870	−0.153935	+	0.988081i	$0.549195\pi$
$380$	0	0
$381$	9614.45	0.00339322
$382$	0	0
$383$	1.86918e6	0.651109	0.325555	−	0.945523i	$-0.394449\pi$
$383$	1.86918e6	0.651109	0.325555	+	0.945523i	$0.394449\pi$
$384$	0	0
$385$	−813981.	−0.279874
$386$	0	0
$387$	279767.	0.0949552
$388$	0	0
$389$	−3.71296e6	−1.24407	−0.622036	−	0.782988i	$-0.713694\pi$
$389$	−3.71296e6	−1.24407	−0.622036	+	0.782988i	$0.713694\pi$
$390$	0	0
$391$	−50858.6	−0.0168237
$392$	0	0
$393$	−926052.	−0.302450
$394$	0	0
$395$	1.04247e6	0.336180
$396$	0	0
$397$	−4.73148e6	−1.50668	−0.753340	−	0.657632i	$-0.771558\pi$
$397$	−4.73148e6	−1.50668	−0.753340	+	0.657632i	$0.771558\pi$
$398$	0	0
$399$	1.94243e6	0.610819
$400$	0	0
$401$	−4.76111e6	−1.47859	−0.739295	−	0.673382i	$-0.764841\pi$
$401$	−4.76111e6	−1.47859	−0.739295	+	0.673382i	$0.764841\pi$
$402$	0	0
$403$	1637.14	0.000502137 0
$404$	0	0
$405$	−689045.	−0.208742
$406$	0	0
$407$	−165906.	−0.0496452
$408$	0	0
$409$	−3.34970e6	−0.990144	−0.495072	−	0.868852i	$-0.664858\pi$
$409$	−3.34970e6	−0.990144	−0.495072	+	0.868852i	$0.664858\pi$
$410$	0	0
$411$	−1.73997e6	−0.508086
$412$	0	0
$413$	5.32114e6	1.53507
$414$	0	0
$415$	−442716.	−0.126184
$416$	0	0
$417$	2.26040e6	0.636569
$418$	0	0
$419$	921763.	0.256498	0.128249	−	0.991742i	$-0.459064\pi$
$419$	921763.	0.256498	0.128249	+	0.991742i	$0.459064\pi$
$420$	0	0
$421$	287143.	0.0789574	0.0394787	−	0.999220i	$-0.487430\pi$
$421$	287143.	0.0789574	0.0394787	+	0.999220i	$0.487430\pi$
$422$	0	0
$423$	−2.05650e6	−0.558827
$424$	0	0
$425$	16927.2	0.00454583
$426$	0	0
$427$	−4.14601e6	−1.10043
$428$	0	0
$429$	−592822.	−0.155518
$430$	0	0
$431$	1.56463e6	0.405714	0.202857	−	0.979208i	$-0.434977\pi$
$431$	1.56463e6	0.405714	0.202857	+	0.979208i	$0.434977\pi$
$432$	0	0
$433$	−2.92795e6	−0.750487	−0.375244	−	0.926926i	$-0.622441\pi$
$433$	−2.92795e6	−0.750487	−0.375244	+	0.926926i	$0.622441\pi$
$434$	0	0
$435$	−1.62265e6	−0.411151
$436$	0	0
$437$	2.62961e6	0.658699
$438$	0	0
$439$	4.28834e6	1.06201	0.531004	−	0.847369i	$-0.321815\pi$
$439$	4.28834e6	1.06201	0.531004	+	0.847369i	$0.321815\pi$
$440$	0	0
$441$	367860.	0.0900713
$442$	0	0
$443$	3.41036e6	0.825640	0.412820	−	0.910813i	$-0.364544\pi$
$443$	3.41036e6	0.825640	0.412820	+	0.910813i	$0.364544\pi$
$444$	0	0
$445$	881021.	0.210905
$446$	0	0
$447$	−743353.	−0.175965
$448$	0	0
$449$	−4.44696e6	−1.04099	−0.520496	−	0.853864i	$-0.674253\pi$
$449$	−4.44696e6	−1.04099	−0.520496	+	0.853864i	$0.674253\pi$
$450$	0	0
$451$	−261033.	−0.0604301
$452$	0	0
$453$	1.06291e6	0.243361
$454$	0	0
$455$	479404.	0.108561
$456$	0	0
$457$	2.27279e6	0.509060	0.254530	−	0.967065i	$-0.418079\pi$
$457$	2.27279e6	0.509060	0.254530	+	0.967065i	$0.418079\pi$
$458$	0	0
$459$	−111430.	−0.0246871
$460$	0	0
$461$	6.64978e6	1.45732	0.728660	−	0.684876i	$-0.240144\pi$
$461$	6.64978e6	1.45732	0.728660	+	0.684876i	$0.240144\pi$
$462$	0	0
$463$	2.10458e6	0.456260	0.228130	−	0.973631i	$-0.426739\pi$
$463$	2.10458e6	0.456260	0.228130	+	0.973631i	$0.426739\pi$
$464$	0	0
$465$	−2960.57	−0.000634956 0
$466$	0	0
$467$	2.30580e6	0.489248	0.244624	−	0.969618i	$-0.421336\pi$
$467$	2.30580e6	0.489248	0.244624	+	0.969618i	$0.421336\pi$
$468$	0	0
$469$	−3.49557e6	−0.733814
$470$	0	0
$471$	5.32779e6	1.10661
$472$	0	0
$473$	−858064.	−0.176347
$474$	0	0
$475$	−875209.	−0.177983
$476$	0	0
$477$	2.21442e6	0.445619
$478$	0	0
$479$	−9.22828e6	−1.83773	−0.918866	−	0.394570i	$-0.870894\pi$
$479$	−9.22828e6	−1.83773	−0.918866	+	0.394570i	$0.870894\pi$
$480$	0	0
$481$	97712.6	0.0192570
$482$	0	0
$483$	2.60478e6	0.508046
$484$	0	0
$485$	2.21241e6	0.427083
$486$	0	0
$487$	−51888.8	−0.00991406	−0.00495703	−	0.999988i	$-0.501578\pi$
$487$	−51888.8	−0.00991406	−0.00495703	+	0.999988i	$0.501578\pi$
$488$	0	0
$489$	5.97888e6	1.13070
$490$	0	0
$491$	2.45778e6	0.460086	0.230043	−	0.973180i	$-0.426113\pi$
$491$	2.45778e6	0.460086	0.230043	+	0.973180i	$0.426113\pi$
$492$	0	0
$493$	−143798.	−0.0266462
$494$	0	0
$495$	−671143.	−0.123113
$496$	0	0
$497$	8.22653e6	1.49391
$498$	0	0
$499$	−2.11011e6	−0.379362	−0.189681	−	0.981846i	$-0.560745\pi$
$499$	−2.11011e6	−0.379362	−0.189681	+	0.981846i	$0.560745\pi$
$500$	0	0
$501$	4.98853e6	0.887930
$502$	0	0
$503$	1.57748e6	0.277999	0.139000	−	0.990292i	$-0.455611\pi$
$503$	1.57748e6	0.277999	0.139000	+	0.990292i	$0.455611\pi$
$504$	0	0
$505$	2.67664e6	0.467047
$506$	0	0
$507$	349149.	0.0603242
$508$	0	0
$509$	6.26403e6	1.07167	0.535833	−	0.844324i	$-0.319998\pi$
$509$	6.26403e6	1.07167	0.535833	+	0.844324i	$0.319998\pi$
$510$	0	0
$511$	9.25935e6	1.56866
$512$	0	0
$513$	5.76141e6	0.966574
$514$	0	0
$515$	−4.26717e6	−0.708960
$516$	0	0
$517$	6.30742e6	1.03783
$518$	0	0
$519$	−2.02449e6	−0.329911
$520$	0	0
$521$	−5.03713e6	−0.812997	−0.406499	−	0.913651i	$-0.633250\pi$
$521$	−5.03713e6	−0.812997	−0.406499	+	0.913651i	$0.633250\pi$
$522$	0	0
$523$	−1.74854e6	−0.279525	−0.139763	−	0.990185i	$-0.544634\pi$
$523$	−1.74854e6	−0.279525	−0.139763	+	0.990185i	$0.544634\pi$
$524$	0	0
$525$	−866947.	−0.137276
$526$	0	0
$527$	−262.364	−4.11507e−5 0
$528$	0	0
$529$	−2.91006e6	−0.452129
$530$	0	0
$531$	4.38738e6	0.675257
$532$	0	0
$533$	153738.	0.0234403
$534$	0	0
$535$	−2.14898e6	−0.324599
$536$	0	0
$537$	5.69080e6	0.851604
$538$	0	0
$539$	−1.12825e6	−0.167276
$540$	0	0
$541$	−5.86702e6	−0.861836	−0.430918	−	0.902391i	$-0.641810\pi$
$541$	−5.86702e6	−0.861836	−0.430918	+	0.902391i	$0.641810\pi$
$542$	0	0
$543$	1.36655e6	0.198895
$544$	0	0
$545$	−3.08359e6	−0.444698
$546$	0	0
$547$	8.69078e6	1.24191	0.620955	−	0.783846i	$-0.286745\pi$
$547$	8.69078e6	1.24191	0.620955	+	0.783846i	$0.286745\pi$
$548$	0	0
$549$	−3.41847e6	−0.484062
$550$	0	0
$551$	7.43495e6	1.04328
$552$	0	0
$553$	−4.73151e6	−0.657940
$554$	0	0
$555$	−176702.	−0.0243506
$556$	0	0
$557$	6.88157e6	0.939831	0.469915	−	0.882711i	$-0.344284\pi$
$557$	6.88157e6	0.939831	0.469915	+	0.882711i	$0.344284\pi$
$558$	0	0
$559$	505367.	0.0684034
$560$	0	0
$561$	95004.3	0.0127449
$562$	0	0
$563$	1.17681e7	1.56472	0.782358	−	0.622829i	$-0.214017\pi$
$563$	1.17681e7	1.56472	0.782358	+	0.622829i	$0.214017\pi$
$564$	0	0
$565$	−1.07801e6	−0.142070
$566$	0	0
$567$	3.12739e6	0.408530
$568$	0	0
$569$	−1.10766e7	−1.43425	−0.717124	−	0.696946i	$-0.754542\pi$
$569$	−1.10766e7	−1.43425	−0.717124	+	0.696946i	$0.754542\pi$
$570$	0	0
$571$	−1.14920e7	−1.47505	−0.737524	−	0.675321i	$-0.764005\pi$
$571$	−1.14920e7	−1.47505	−0.737524	+	0.675321i	$0.764005\pi$
$572$	0	0
$573$	1.00423e7	1.27775
$574$	0	0
$575$	−1.17365e6	−0.148037
$576$	0	0
$577$	−9.07057e6	−1.13421	−0.567107	−	0.823644i	$-0.691937\pi$
$577$	−9.07057e6	−1.13421	−0.567107	+	0.823644i	$0.691937\pi$
$578$	0	0
$579$	−1.95047e6	−0.241792
$580$	0	0
$581$	2.00937e6	0.246956
$582$	0	0
$583$	−6.79177e6	−0.827583
$584$	0	0
$585$	395278.	0.0477543
$586$	0	0
$587$	−1.11097e7	−1.33078	−0.665392	−	0.746494i	$-0.731735\pi$
$587$	−1.11097e7	−1.33078	−0.665392	+	0.746494i	$0.731735\pi$
$588$	0	0
$589$	13565.3	0.00161117
$590$	0	0
$591$	6.83181e6	0.804576
$592$	0	0
$593$	−4.96834e6	−0.580196	−0.290098	−	0.956997i	$-0.593688\pi$
$593$	−4.96834e6	−0.580196	−0.290098	+	0.956997i	$0.593688\pi$
$594$	0	0
$595$	−76828.1	−0.00889668
$596$	0	0
$597$	753240.	0.0864963
$598$	0	0
$599$	6.86030e6	0.781225	0.390613	−	0.920555i	$-0.372263\pi$
$599$	6.86030e6	0.781225	0.390613	+	0.920555i	$0.372263\pi$
$600$	0	0
$601$	1.51402e7	1.70980	0.854900	−	0.518793i	$-0.173619\pi$
$601$	1.51402e7	1.70980	0.854900	+	0.518793i	$0.173619\pi$
$602$	0	0
$603$	−2.88217e6	−0.322794
$604$	0	0
$605$	−1.96783e6	−0.218574
$606$	0	0
$607$	−238891.	−0.0263165	−0.0131582	−	0.999913i	$-0.504189\pi$
$607$	−238891.	−0.0263165	−0.0131582	+	0.999913i	$0.504189\pi$
$608$	0	0
$609$	7.36476e6	0.804666
$610$	0	0
$611$	−3.71483e6	−0.402565
$612$	0	0
$613$	−1.14348e7	−1.22907	−0.614536	−	0.788889i	$-0.710657\pi$
$613$	−1.14348e7	−1.22907	−0.614536	+	0.788889i	$0.710657\pi$
$614$	0	0
$615$	−278018.	−0.0296405
$616$	0	0
$617$	−4.90217e6	−0.518413	−0.259206	−	0.965822i	$-0.583461\pi$
$617$	−4.90217e6	−0.518413	−0.259206	+	0.965822i	$0.583461\pi$
$618$	0	0
$619$	−3.15524e6	−0.330983	−0.165492	−	0.986211i	$-0.552921\pi$
$619$	−3.15524e6	−0.330983	−0.165492	+	0.986211i	$0.552921\pi$
$620$	0	0
$621$	7.72601e6	0.803945
$622$	0	0
$623$	−3.99872e6	−0.412763
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−4.91213e6	−0.499000
$628$	0	0
$629$	−15659.2	−0.00157813
$630$	0	0
$631$	−5.07621e6	−0.507535	−0.253768	−	0.967265i	$-0.581670\pi$
$631$	−5.07621e6	−0.507535	−0.253768	+	0.967265i	$0.581670\pi$
$632$	0	0
$633$	1.23397e7	1.22404
$634$	0	0
$635$	19662.0	0.00193505
$636$	0	0
$637$	664498.	0.0648851
$638$	0	0
$639$	6.78294e6	0.657152
$640$	0	0
$641$	−1.13280e7	−1.08895	−0.544475	−	0.838777i	$-0.683271\pi$
$641$	−1.13280e7	−1.08895	−0.544475	+	0.838777i	$0.683271\pi$
$642$	0	0
$643$	−1.54746e7	−1.47602	−0.738008	−	0.674792i	$-0.764233\pi$
$643$	−1.54746e7	−1.47602	−0.738008	+	0.674792i	$0.764233\pi$
$644$	0	0
$645$	−913899.	−0.0864966
$646$	0	0
$647$	4.12040e6	0.386971	0.193486	−	0.981103i	$-0.438021\pi$
$647$	4.12040e6	0.386971	0.193486	+	0.981103i	$0.438021\pi$
$648$	0	0
$649$	−1.34564e7	−1.25406
$650$	0	0
$651$	13437.3	0.00124268
$652$	0	0
$653$	6.22702e6	0.571475	0.285737	−	0.958308i	$-0.407761\pi$
$653$	6.22702e6	0.571475	0.285737	+	0.958308i	$0.407761\pi$
$654$	0	0
$655$	−1.89381e6	−0.172478
$656$	0	0
$657$	7.63451e6	0.690030
$658$	0	0
$659$	3.85134e6	0.345460	0.172730	−	0.984969i	$-0.444741\pi$
$659$	3.85134e6	0.345460	0.172730	+	0.984969i	$0.444741\pi$
$660$	0	0
$661$	−1.00528e7	−0.894915	−0.447458	−	0.894305i	$-0.647670\pi$
$661$	−1.00528e7	−0.894915	−0.447458	+	0.894305i	$0.647670\pi$
$662$	0	0
$663$	−55953.9	−0.00494364
$664$	0	0
$665$	3.97234e6	0.348331
$666$	0	0
$667$	9.97022e6	0.867741
$668$	0	0
$669$	6.93666e6	0.599219
$670$	0	0
$671$	1.04847e7	0.898978
$672$	0	0
$673$	−1.64753e7	−1.40216	−0.701078	−	0.713085i	$-0.747298\pi$
$673$	−1.64753e7	−1.40216	−0.701078	+	0.713085i	$0.747298\pi$
$674$	0	0
$675$	−2.57144e6	−0.217229
$676$	0	0
$677$	−1.56813e7	−1.31495	−0.657477	−	0.753474i	$-0.728376\pi$
$677$	−1.56813e7	−1.31495	−0.657477	+	0.753474i	$0.728376\pi$
$678$	0	0
$679$	−1.00416e7	−0.835846
$680$	0	0
$681$	2.28385e6	0.188713
$682$	0	0
$683$	−1.38324e6	−0.113461	−0.0567305	−	0.998390i	$-0.518068\pi$
$683$	−1.38324e6	−0.113461	−0.0567305	+	0.998390i	$0.518068\pi$
$684$	0	0
$685$	−3.55831e6	−0.289746
$686$	0	0
$687$	−1.40669e7	−1.13712
$688$	0	0
$689$	4.00009e6	0.321013
$690$	0	0
$691$	−3.50867e6	−0.279542	−0.139771	−	0.990184i	$-0.544637\pi$
$691$	−3.50867e6	−0.279542	−0.139771	+	0.990184i	$0.544637\pi$
$692$	0	0
$693$	3.04614e6	0.240944
$694$	0	0
$695$	4.62262e6	0.363016
$696$	0	0
$697$	−24637.7	−0.00192096
$698$	0	0
$699$	−476059.	−0.0368526
$700$	0	0
$701$	−1.36167e7	−1.04659	−0.523294	−	0.852152i	$-0.675297\pi$
$701$	−1.36167e7	−1.04659	−0.523294	+	0.852152i	$0.675297\pi$
$702$	0	0
$703$	809647.	0.0617885
$704$	0	0
$705$	6.71785e6	0.509046
$706$	0	0
$707$	−1.21485e7	−0.914062
$708$	0	0
$709$	−1.63025e7	−1.21797	−0.608986	−	0.793181i	$-0.708424\pi$
$709$	−1.63025e7	−1.21797	−0.608986	+	0.793181i	$0.708424\pi$
$710$	0	0
$711$	−3.90122e6	−0.289419
$712$	0	0
$713$	18191.0	0.00134009
$714$	0	0
$715$	−1.21234e6	−0.0886872
$716$	0	0
$717$	−3.48255e6	−0.252988
$718$	0	0
$719$	2.23923e7	1.61539	0.807693	−	0.589603i	$-0.200716\pi$
$719$	2.23923e7	1.61539	0.807693	+	0.589603i	$0.200716\pi$
$720$	0	0
$721$	1.93675e7	1.38751
$722$	0	0
$723$	−1.72744e7	−1.22901
$724$	0	0
$725$	−3.31838e6	−0.234467
$726$	0	0
$727$	−2.57850e7	−1.80939	−0.904693	−	0.426065i	$-0.859900\pi$
$727$	−2.57850e7	−1.80939	−0.904693	+	0.426065i	$0.859900\pi$
$728$	0	0
$729$	1.48005e7	1.03148
$730$	0	0
$731$	−80989.0	−0.00560573
$732$	0	0
$733$	−358431.	−0.0246403	−0.0123201	−	0.999924i	$-0.503922\pi$
$733$	−358431.	−0.0246403	−0.0123201	+	0.999924i	$0.503922\pi$
$734$	0	0
$735$	−1.20167e6	−0.0820477
$736$	0	0
$737$	8.83981e6	0.599479
$738$	0	0
$739$	−2.47885e6	−0.166970	−0.0834851	−	0.996509i	$-0.526605\pi$
$739$	−2.47885e6	−0.166970	−0.0834851	+	0.996509i	$0.526605\pi$
$740$	0	0
$741$	2.89305e6	0.193558
$742$	0	0
$743$	−5.92881e6	−0.394000	−0.197000	−	0.980404i	$-0.563120\pi$
$743$	−5.92881e6	−0.394000	−0.197000	+	0.980404i	$0.563120\pi$
$744$	0	0
$745$	−1.52019e6	−0.100347
$746$	0	0
$747$	1.65676e6	0.108632
$748$	0	0
$749$	9.75364e6	0.635275
$750$	0	0
$751$	−9.56161e6	−0.618630	−0.309315	−	0.950960i	$-0.600100\pi$
$751$	−9.56161e6	−0.618630	−0.309315	+	0.950960i	$0.600100\pi$
$752$	0	0
$753$	−1.35121e7	−0.868429
$754$	0	0
$755$	2.17370e6	0.138782
$756$	0	0
$757$	−1.71151e7	−1.08553	−0.542763	−	0.839886i	$-0.682622\pi$
$757$	−1.71151e7	−1.08553	−0.542763	+	0.839886i	$0.682622\pi$
$758$	0	0
$759$	−6.58713e6	−0.415042
$760$	0	0
$761$	−2.59574e7	−1.62480	−0.812400	−	0.583100i	$-0.801840\pi$
$761$	−2.59574e7	−1.62480	−0.812400	+	0.583100i	$0.801840\pi$
$762$	0	0
$763$	1.39956e7	0.870321
$764$	0	0
$765$	−63346.3	−0.00391352
$766$	0	0
$767$	7.92532e6	0.486439
$768$	0	0
$769$	−1.44637e7	−0.881991	−0.440995	−	0.897509i	$-0.645375\pi$
$769$	−1.44637e7	−0.881991	−0.440995	+	0.897509i	$0.645375\pi$
$770$	0	0
$771$	1.35877e6	0.0823211
$772$	0	0
$773$	3.11310e7	1.87389	0.936946	−	0.349473i	$-0.113640\pi$
$773$	3.11310e7	1.87389	0.936946	+	0.349473i	$0.113640\pi$
$774$	0	0
$775$	−6054.50	−0.000362096 0
$776$	0	0
$777$	802004.	0.0476567
$778$	0	0
$779$	1.27388e6	0.0752114
$780$	0	0
$781$	−2.08038e7	−1.22043
$782$	0	0
$783$	2.18445e7	1.27332
$784$	0	0
$785$	1.08956e7	0.631066
$786$	0	0
$787$	−2.91122e7	−1.67547	−0.837737	−	0.546074i	$-0.816122\pi$
$787$	−2.91122e7	−1.67547	−0.837737	+	0.546074i	$0.816122\pi$
$788$	0	0
$789$	1.81197e7	1.03623
$790$	0	0
$791$	4.89280e6	0.278045
$792$	0	0
$793$	−6.17508e6	−0.348706
$794$	0	0
$795$	−7.23372e6	−0.405923
$796$	0	0
$797$	2.95059e7	1.64537	0.822684	−	0.568499i	$-0.192476\pi$
$797$	2.95059e7	1.64537	0.822684	+	0.568499i	$0.192476\pi$
$798$	0	0
$799$	595330.	0.0329907
$800$	0	0
$801$	−3.29702e6	−0.181569
$802$	0	0
$803$	−2.34156e7	−1.28149
$804$	0	0
$805$	5.32688e6	0.289723
$806$	0	0
$807$	−48712.7	−0.00263304
$808$	0	0
$809$	213894.	0.0114902	0.00574510	−	0.999983i	$-0.498171\pi$
$809$	213894.	0.0114902	0.00574510	+	0.999983i	$0.498171\pi$
$810$	0	0
$811$	−1.50737e7	−0.804761	−0.402380	−	0.915473i	$-0.631817\pi$
$811$	−1.50737e7	−0.804761	−0.402380	+	0.915473i	$0.631817\pi$
$812$	0	0
$813$	−7.51667e6	−0.398841
$814$	0	0
$815$	1.22270e7	0.644804
$816$	0	0
$817$	4.18748e6	0.219481
$818$	0	0
$819$	−1.79406e6	−0.0934603
$820$	0	0
$821$	−3.12224e6	−0.161662	−0.0808311	−	0.996728i	$-0.525757\pi$
$821$	−3.12224e6	−0.161662	−0.0808311	+	0.996728i	$0.525757\pi$
$822$	0	0
$823$	9.14671e6	0.470723	0.235362	−	0.971908i	$-0.424373\pi$
$823$	9.14671e6	0.470723	0.235362	+	0.971908i	$0.424373\pi$
$824$	0	0
$825$	2.19239e6	0.112146
$826$	0	0
$827$	2.03955e7	1.03698	0.518491	−	0.855083i	$-0.326494\pi$
$827$	2.03955e7	1.03698	0.518491	+	0.855083i	$0.326494\pi$
$828$	0	0
$829$	3.29182e7	1.66360	0.831802	−	0.555072i	$-0.187309\pi$
$829$	3.29182e7	1.66360	0.831802	+	0.555072i	$0.187309\pi$
$830$	0	0
$831$	−3.99116e6	−0.200492
$832$	0	0
$833$	−106491.	−0.00531741
$834$	0	0
$835$	1.02018e7	0.506359
$836$	0	0
$837$	39856.1	0.00196644
$838$	0	0
$839$	−1.21243e7	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$839$	−1.21243e7	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$840$	0	0
$841$	7.67868e6	0.374366
$842$	0	0
$843$	−6.88181e6	−0.333529
$844$	0	0
$845$	714025.	0.0344010
$846$	0	0
$847$	8.93146e6	0.427774
$848$	0	0
$849$	1.36089e7	0.647968
$850$	0	0
$851$	1.08573e6	0.0513923
$852$	0	0
$853$	1.17119e7	0.551132	0.275566	−	0.961282i	$-0.411135\pi$
$853$	1.17119e7	0.551132	0.275566	+	0.961282i	$0.411135\pi$
$854$	0	0
$855$	3.27527e6	0.153226
$856$	0	0
$857$	2.02412e7	0.941422	0.470711	−	0.882287i	$-0.343997\pi$
$857$	2.02412e7	0.941422	0.470711	+	0.882287i	$0.343997\pi$
$858$	0	0
$859$	−4.17853e7	−1.93215	−0.966076	−	0.258259i	$-0.916851\pi$
$859$	−4.17853e7	−1.93215	−0.966076	+	0.258259i	$0.916851\pi$
$860$	0	0
$861$	1.26185e6	0.0580096
$862$	0	0
$863$	−2.67589e7	−1.22304	−0.611520	−	0.791229i	$-0.709442\pi$
$863$	−2.67589e7	−1.22304	−0.611520	+	0.791229i	$0.709442\pi$
$864$	0	0
$865$	−4.14016e6	−0.188138
$866$	0	0
$867$	−1.73483e7	−0.783809
$868$	0	0
$869$	1.19653e7	0.537495
$870$	0	0
$871$	−5.20631e6	−0.232533
$872$	0	0
$873$	−8.27946e6	−0.367677
$874$	0	0
$875$	−1.77294e6	−0.0782843
$876$	0	0
$877$	2.48901e7	1.09277	0.546384	−	0.837535i	$-0.316004\pi$
$877$	2.48901e7	1.09277	0.546384	+	0.837535i	$0.316004\pi$
$878$	0	0
$879$	1.80412e7	0.787578
$880$	0	0
$881$	−2.02286e7	−0.878063	−0.439032	−	0.898472i	$-0.644678\pi$
$881$	−2.02286e7	−0.878063	−0.439032	+	0.898472i	$0.644678\pi$
$882$	0	0
$883$	−1.25103e7	−0.539965	−0.269982	−	0.962865i	$-0.587018\pi$
$883$	−1.25103e7	−0.539965	−0.269982	+	0.962865i	$0.587018\pi$
$884$	0	0
$885$	−1.43320e7	−0.615106
$886$	0	0
$887$	−1.10737e7	−0.472588	−0.236294	−	0.971682i	$-0.575933\pi$
$887$	−1.10737e7	−0.472588	−0.236294	+	0.971682i	$0.575933\pi$
$888$	0	0
$889$	−89240.3	−0.00378710
$890$	0	0
$891$	−7.90873e6	−0.333743
$892$	0	0
$893$	−3.07811e7	−1.29168
$894$	0	0
$895$	1.16379e7	0.485644
$896$	0	0
$897$	3.87957e6	0.160991
$898$	0	0
$899$	51433.3	0.00212249
$900$	0	0
$901$	−641046.	−0.0263074
$902$	0	0
$903$	4.14794e6	0.169283
$904$	0	0
$905$	2.79464e6	0.113424
$906$	0	0
$907$	−3.33261e7	−1.34514	−0.672568	−	0.740035i	$-0.734809\pi$
$907$	−3.33261e7	−1.34514	−0.672568	+	0.740035i	$0.734809\pi$
$908$	0	0
$909$	−1.00167e7	−0.402083
$910$	0	0
$911$	−2.09229e7	−0.835267	−0.417633	−	0.908616i	$-0.637140\pi$
$911$	−2.09229e7	−0.835267	−0.417633	+	0.908616i	$0.637140\pi$
$912$	0	0
$913$	−5.08141e6	−0.201747
$914$	0	0
$915$	1.11669e7	0.440941
$916$	0	0
$917$	8.59552e6	0.337558
$918$	0	0
$919$	1.45678e7	0.568990	0.284495	−	0.958678i	$-0.408174\pi$
$919$	1.45678e7	0.568990	0.284495	+	0.958678i	$0.408174\pi$
$920$	0	0
$921$	7.07503e6	0.274840
$922$	0	0
$923$	1.22526e7	0.473396
$924$	0	0
$925$	−361363.	−0.0138864
$926$	0	0
$927$	1.59689e7	0.610346
$928$	0	0
$929$	4.92771e7	1.87329	0.936647	−	0.350274i	$-0.113911\pi$
$929$	4.92771e7	1.87329	0.936647	+	0.350274i	$0.113911\pi$
$930$	0	0
$931$	5.50603e6	0.208192
$932$	0	0
$933$	7.20079e6	0.270817
$934$	0	0
$935$	194288.	0.00726802
$936$	0	0
$937$	2.73238e7	1.01670	0.508349	−	0.861151i	$-0.330256\pi$
$937$	2.73238e7	1.01670	0.508349	+	0.861151i	$0.330256\pi$
$938$	0	0
$939$	−5.73450e6	−0.212242
$940$	0	0
$941$	1.52874e7	0.562809	0.281404	−	0.959589i	$-0.409200\pi$
$941$	1.52874e7	0.562809	0.281404	+	0.959589i	$0.409200\pi$
$942$	0	0
$943$	1.70826e6	0.0625568
$944$	0	0
$945$	1.16711e7	0.425140
$946$	0	0
$947$	−3.31946e7	−1.20280	−0.601398	−	0.798949i	$-0.705389\pi$
$947$	−3.31946e7	−1.20280	−0.601398	+	0.798949i	$0.705389\pi$
$948$	0	0
$949$	1.37909e7	0.497081
$950$	0	0
$951$	−1.89079e7	−0.677941
$952$	0	0
$953$	4.44705e7	1.58613	0.793067	−	0.609134i	$-0.208483\pi$
$953$	4.44705e7	1.58613	0.793067	+	0.609134i	$0.208483\pi$
$954$	0	0
$955$	2.05369e7	0.728662
$956$	0	0
$957$	−1.86245e7	−0.657361
$958$	0	0
$959$	1.61502e7	0.567064
$960$	0	0
$961$	−2.86291e7	−0.999997
$962$	0	0
$963$	8.04207e6	0.279449
$964$	0	0
$965$	−3.98878e6	−0.137887
$966$	0	0
$967$	−1.61648e7	−0.555910	−0.277955	−	0.960594i	$-0.589657\pi$
$967$	−1.61648e7	−0.555910	−0.277955	+	0.960594i	$0.589657\pi$
$968$	0	0
$969$	−463634.	−0.0158623
$970$	0	0
$971$	1.11145e7	0.378304	0.189152	−	0.981948i	$-0.439426\pi$
$971$	1.11145e7	0.378304	0.189152	+	0.981948i	$0.439426\pi$
$972$	0	0
$973$	−2.09808e7	−0.710461
$974$	0	0
$975$	−1.29123e6	−0.0435004
$976$	0	0
$977$	2.15544e7	0.722438	0.361219	−	0.932481i	$-0.382361\pi$
$977$	2.15544e7	0.722438	0.361219	+	0.932481i	$0.382361\pi$
$978$	0	0
$979$	1.01122e7	0.337201
$980$	0	0
$981$	1.15396e7	0.382842
$982$	0	0
$983$	2.08098e7	0.686886	0.343443	−	0.939174i	$-0.388407\pi$
$983$	2.08098e7	0.686886	0.343443	+	0.939174i	$0.388407\pi$
$984$	0	0
$985$	1.39713e7	0.458825
$986$	0	0
$987$	−3.04905e7	−0.996258
$988$	0	0
$989$	5.61538e6	0.182553
$990$	0	0
$991$	4.88426e7	1.57984	0.789922	−	0.613207i	$-0.210121\pi$
$991$	4.88426e7	1.57984	0.789922	+	0.613207i	$0.210121\pi$
$992$	0	0
$993$	−6.57383e6	−0.211566
$994$	0	0
$995$	1.54041e6	0.0493262
$996$	0	0
$997$	3.53695e7	1.12691	0.563457	−	0.826145i	$-0.309471\pi$
$997$	3.53695e7	1.12691	0.563457	+	0.826145i	$0.309471\pi$
$998$	0	0
$999$	2.37881e6	0.0754130

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	260.6.a.d.1.4	✓	4
4.3	odd	2		1040.6.a.p.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.6.a.d.1.4	✓	4	1.1	even	1	trivial
1040.6.a.p.1.1		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:	\( N \)	\(=\)	\( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	260.a (trivial)

\(f(q)\)	\(=\)	\(q+12.2247 q^{3} +25.0000 q^{5} -113.468 q^{7} -93.5569 q^{9} +O(q^{10})\)	\(q+12.2247 q^{3} +25.0000 q^{5} -113.468 q^{7} -93.5569 q^{9} +286.946 q^{11} -169.000 q^{13} +305.617 q^{15} +27.0836 q^{17} -1400.34 q^{19} -1387.12 q^{21} -1877.84 q^{23} +625.000 q^{25} -4114.30 q^{27} -5309.41 q^{29} -9.68719 q^{31} +3507.82 q^{33} -2836.71 q^{35} -578.181 q^{37} -2065.97 q^{39} -909.694 q^{41} -2990.34 q^{43} -2338.92 q^{45} +21981.2 q^{47} -3931.94 q^{49} +331.088 q^{51} -23669.2 q^{53} +7173.64 q^{55} -17118.7 q^{57} -46895.4 q^{59} +36538.9 q^{61} +10615.7 q^{63} -4225.00 q^{65} +30806.6 q^{67} -22956.0 q^{69} -72500.7 q^{71} -81602.9 q^{73} +7640.43 q^{75} -32559.2 q^{77} +41698.9 q^{79} -27561.8 q^{81} -17708.6 q^{83} +677.089 q^{85} -64905.9 q^{87} +35240.9 q^{89} +19176.1 q^{91} -118.423 q^{93} -35008.4 q^{95} +88496.5 q^{97} -26845.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{3} + 100 q^{5} - 24 q^{7} - 84 q^{9}+O(q^{10}) \) 4 * q - 20 * q^3 + 100 * q^5 - 24 * q^7 - 84 * q^9	\( 4 q - 20 q^{3} + 100 q^{5} - 24 q^{7} - 84 q^{9} - 12 q^{11} - 676 q^{13} - 500 q^{15} - 416 q^{17} - 644 q^{19} - 376 q^{21} + 1748 q^{23} + 2500 q^{25} - 1664 q^{27} - 4080 q^{29} - 1500 q^{31} + 2096 q^{33} - 600 q^{35} - 3848 q^{37} + 3380 q^{39} - 17968 q^{41} + 3020 q^{43} - 2100 q^{45} - 12328 q^{47} - 28108 q^{49} - 15704 q^{51} - 52672 q^{53} - 300 q^{55} - 61696 q^{57} - 58988 q^{59} - 56736 q^{61} - 17768 q^{63} - 16900 q^{65} - 56480 q^{67} - 120392 q^{69} - 68260 q^{71} - 113000 q^{73} - 12500 q^{75} - 126600 q^{77} - 43640 q^{79} - 135684 q^{81} - 52248 q^{83} - 10400 q^{85} - 7760 q^{87} - 114376 q^{89} + 4056 q^{91} + 17680 q^{93} - 16100 q^{95} - 21432 q^{97} + 44092 q^{99}+O(q^{100}) \) 4 * q - 20 * q^3 + 100 * q^5 - 24 * q^7 - 84 * q^9 - 12 * q^11 - 676 * q^13 - 500 * q^15 - 416 * q^17 - 644 * q^19 - 376 * q^21 + 1748 * q^23 + 2500 * q^25 - 1664 * q^27 - 4080 * q^29 - 1500 * q^31 + 2096 * q^33 - 600 * q^35 - 3848 * q^37 + 3380 * q^39 - 17968 * q^41 + 3020 * q^43 - 2100 * q^45 - 12328 * q^47 - 28108 * q^49 - 15704 * q^51 - 52672 * q^53 - 300 * q^55 - 61696 * q^57 - 58988 * q^59 - 56736 * q^61 - 17768 * q^63 - 16900 * q^65 - 56480 * q^67 - 120392 * q^69 - 68260 * q^71 - 113000 * q^73 - 12500 * q^75 - 126600 * q^77 - 43640 * q^79 - 135684 * q^81 - 52248 * q^83 - 10400 * q^85 - 7760 * q^87 - 114376 * q^89 + 4056 * q^91 + 17680 * q^93 - 16100 * q^95 - 21432 * q^97 + 44092 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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