LMFDB - Embedded newform 588.6.f.c.293.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,Mod(293,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("588.293");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$588 = 2^{2} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	588.f (of order $2$ , degree $1$ , minimal)

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:	no
Analytic conductor:	$94.3056860500$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			293.20
Character	$\chi$	$=$	588.293
Dual form			588.6.f.c.293.19

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.39153 + 15.5262i) q^{3} -69.8984 q^{5} +(-239.127 - 43.2103i) q^{9} +465.771i q^{11} +285.857i q^{13} +(97.2655 - 1085.26i) q^{15} +898.214 q^{17} +2096.30i q^{19} +4901.65i q^{23} +1760.78 q^{25} +(1003.65 - 3652.62i) q^{27} +5878.95i q^{29} +3854.71i q^{31} +(-7231.66 - 648.133i) q^{33} -3618.34 q^{37} +(-4438.28 - 397.778i) q^{39} +17142.4 q^{41} -18567.3 q^{43} +(16714.6 + 3020.33i) q^{45} +2422.15 q^{47} +(-1249.89 + 13945.9i) q^{51} +7618.31i q^{53} -32556.6i q^{55} +(-32547.6 - 2917.06i) q^{57} -10644.1 q^{59} +15624.4i q^{61} -19980.9i q^{65} -2973.20 q^{67} +(-76104.2 - 6820.79i) q^{69} +16268.3i q^{71} +14677.7i q^{73} +(-2450.17 + 27338.3i) q^{75} -86199.9 q^{79} +(55314.7 + 20665.5i) q^{81} +2747.03 q^{83} -62783.7 q^{85} +(-91277.9 - 8180.72i) q^{87} +101804. q^{89} +(-59849.1 - 5363.94i) q^{93} -146528. i q^{95} -23406.7i q^{97} +(20126.1 - 111378. i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$40 q + 440 q^{9} - 1224 q^{15} + 19896 q^{25} + 26224 q^{37} - 17384 q^{39} - 21584 q^{43} - 52776 q^{51} + 34880 q^{57} + 89776 q^{67} - 288240 q^{79} + 72504 q^{81} - 50640 q^{85} + 263376 q^{93} - 87056 q^{99}+O(q^{100})$ 40 * q + 440 * q^9 - 1224 * q^15 + 19896 * q^25 + 26224 * q^37 - 17384 * q^39 - 21584 * q^43 - 52776 * q^51 + 34880 * q^57 + 89776 * q^67 - 288240 * q^79 + 72504 * q^81 - 50640 * q^85 + 263376 * q^93 - 87056 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$ .

$n$	$197$	$295$	$493$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−1.39153	+	15.5262i	−0.0892665	+	0.996008i
$3$	−1.39153	+	15.5262i	−0.0892665	+	0.996008i
$4$		0			0
$5$	−69.8984			−1.25038			−0.625190	−	0.780473i	$-0.714978\pi$
$5$	−69.8984			−1.25038			−0.625190	+	0.780473i	$0.714978\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−239.127	−	43.2103i	−0.984063	−	0.177820i
$10$		0			0
$11$			465.771i			1.16062i	0.814395	+	0.580310i	$0.197069\pi$
$11$			465.771i			1.16062i	−0.814395	+	0.580310i	$0.802931\pi$
$12$		0			0
$13$			285.857i			0.469127i	0.972101	+	0.234563i	$0.0753660\pi$
$13$			285.857i			0.469127i	−0.972101	+	0.234563i	$0.924634\pi$
$14$		0			0
$15$	97.2655	−	1085.26i	0.111617	−	1.24539i
$16$		0			0
$17$	898.214			0.753803			0.376901	−	0.926253i	$-0.376990\pi$
$17$	898.214			0.753803			0.376901	+	0.926253i	$0.376990\pi$
$18$		0			0
$19$			2096.30i			1.33220i	0.745863	+	0.666100i	$0.232037\pi$
$19$			2096.30i			1.33220i	−0.745863	+	0.666100i	$0.767963\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4901.65i			1.93207i	0.258409	+	0.966036i	$0.416802\pi$
$23$			4901.65i			1.93207i	−0.258409	+	0.966036i	$0.583198\pi$
$24$		0			0
$25$	1760.78			0.563450
$26$		0			0
$27$	1003.65	−	3652.62i	0.264954	−	0.964261i
$28$		0			0
$29$			5878.95i			1.29809i	0.760750	+	0.649045i	$0.224831\pi$
$29$			5878.95i			1.29809i	−0.760750	+	0.649045i	$0.775169\pi$
$30$		0			0
$31$			3854.71i			0.720423i	0.932871	+	0.360211i	$0.117295\pi$
$31$			3854.71i			0.720423i	−0.932871	+	0.360211i	$0.882705\pi$
$32$		0			0
$33$	−7231.66	−	648.133i	−1.15599	−	0.103605i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3618.34			−0.434515			−0.217257	−	0.976114i	$-0.569711\pi$
$37$	−3618.34			−0.434515			−0.217257	+	0.976114i	$0.569711\pi$
$38$		0			0
$39$	−4438.28	−	397.778i	−0.467254	−	0.0418773i
$40$		0			0
$41$	17142.4			1.59262			0.796308	−	0.604891i	$-0.206783\pi$
$41$	17142.4			1.59262			0.796308	+	0.604891i	$0.206783\pi$
$42$		0			0
$43$	−18567.3			−1.53136			−0.765680	−	0.643221i	$-0.777598\pi$
$43$	−18567.3			−1.53136			−0.765680	+	0.643221i	$0.777598\pi$
$44$		0			0
$45$	16714.6	+	3020.33i	1.23045	+	0.222343i
$46$		0			0
$47$	2422.15			0.159940			0.0799698	−	0.996797i	$-0.474518\pi$
$47$	2422.15			0.159940			0.0799698	+	0.996797i	$0.474518\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−1249.89	+	13945.9i	−0.0672894	+	0.750793i
$52$		0			0
$53$			7618.31i			0.372537i	0.982499	+	0.186268i	$0.0596394\pi$
$53$			7618.31i			0.372537i	−0.982499	+	0.186268i	$0.940361\pi$
$54$		0			0
$55$		−	32556.6i		−	1.45122i
$56$		0			0
$57$	−32547.6	−	2917.06i	−1.32688	−	0.118921i
$58$		0			0
$59$	−10644.1			−0.398089			−0.199044	−	0.979990i	$-0.563784\pi$
$59$	−10644.1			−0.398089			−0.199044	+	0.979990i	$0.563784\pi$
$60$		0			0
$61$			15624.4i			0.537624i	0.963193	+	0.268812i	$0.0866310\pi$
$61$			15624.4i			0.537624i	−0.963193	+	0.268812i	$0.913369\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	19980.9i		−	0.586586i
$66$		0			0
$67$	−2973.20			−0.0809165			−0.0404582	−	0.999181i	$-0.512882\pi$
$67$	−2973.20			−0.0809165			−0.0404582	+	0.999181i	$0.512882\pi$
$68$		0			0
$69$	−76104.2	−	6820.79i	−1.92436	−	0.172469i
$70$		0			0
$71$			16268.3i			0.382997i	0.981493	+	0.191498i	$0.0613347\pi$
$71$			16268.3i			0.382997i	−0.981493	+	0.191498i	$0.938665\pi$
$72$		0			0
$73$			14677.7i			0.322368i	0.986924	+	0.161184i	$0.0515312\pi$
$73$			14677.7i			0.322368i	−0.986924	+	0.161184i	$0.948469\pi$
$74$		0			0
$75$	−2450.17	+	27338.3i	−0.0502972	+	0.561200i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−86199.9			−1.55396			−0.776978	−	0.629527i	$-0.783249\pi$
$79$	−86199.9			−1.55396			−0.776978	+	0.629527i	$0.783249\pi$
$80$		0			0
$81$	55314.7	+	20665.5i	0.936760	+	0.349973i
$82$		0			0
$83$	2747.03			0.0437691			0.0218846	−	0.999761i	$-0.493033\pi$
$83$	2747.03			0.0437691			0.0218846	+	0.999761i	$0.493033\pi$
$84$		0			0
$85$	−62783.7			−0.942540
$86$		0			0
$87$	−91277.9	−	8180.72i	−1.29291	−	0.115876i
$88$		0			0
$89$	101804.			1.36235			0.681173	−	0.732122i	$-0.261470\pi$
$89$	101804.			1.36235			0.681173	+	0.732122i	$0.261470\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−59849.1	−	5363.94i	−0.717547	−	0.0643097i
$94$		0			0
$95$		−	146528.i		−	1.66576i
$96$		0			0
$97$		−	23406.7i		−	0.252587i	−0.991993	−	0.126294i	$-0.959692\pi$
$97$		−	23406.7i		−	0.252587i	0.991993	−	0.126294i	$-0.0403082\pi$
$98$		0			0
$99$	20126.1	−	111378.i	0.206382	−	1.14212i
$100$		0			0
$101$	62202.4			0.606741			0.303371	−	0.952873i	$-0.401888\pi$
$101$	62202.4			0.606741			0.303371	+	0.952873i	$0.401888\pi$
$102$		0			0
$103$		−	38730.7i		−	0.359718i	−0.983692	−	0.179859i	$-0.942436\pi$
$103$		−	38730.7i		−	0.359718i	0.983692	−	0.179859i	$-0.0575641\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	188678.i		−	1.59317i	−0.604525	−	0.796586i	$-0.706637\pi$
$107$		−	188678.i		−	1.59317i	0.604525	−	0.796586i	$-0.293363\pi$
$108$		0			0
$109$	158294.			1.27614			0.638071	−	0.769978i	$-0.279733\pi$
$109$	158294.			1.27614			0.638071	+	0.769978i	$0.279733\pi$
$110$		0			0
$111$	5035.02	−	56179.1i	0.0387877	−	0.432780i
$112$		0			0
$113$			168309.i			1.23997i	0.784615	+	0.619984i	$0.212861\pi$
$113$			168309.i			1.23997i	−0.784615	+	0.619984i	$0.787139\pi$
$114$		0			0
$115$		−	342618.i		−	2.41582i
$116$		0			0
$117$	12352.0	−	68356.2i	0.0834203	−	0.461650i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−55891.3			−0.347041
$122$		0			0
$123$	−23854.1	+	266156.i	−0.142167	+	1.58626i
$124$		0			0
$125$	95356.7			0.545854
$126$		0			0
$127$	−120698.			−0.664036			−0.332018	−	0.943273i	$-0.607730\pi$
$127$	−120698.			−0.664036			−0.332018	+	0.943273i	$0.607730\pi$
$128$		0			0
$129$	25836.9	−	288280.i	0.136699	−	1.52525i
$130$		0			0
$131$	−242876.			−1.23653			−0.618267	−	0.785968i	$-0.712165\pi$
$131$	−242876.			−1.23653			−0.618267	+	0.785968i	$0.712165\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−70153.2	+	255312.i	−0.331294	+	1.20569i
$136$		0			0
$137$		−	208676.i		−	0.949887i	−0.880016	−	0.474944i	$-0.842468\pi$
$137$		−	208676.i		−	0.949887i	0.880016	−	0.474944i	$-0.157532\pi$
$138$		0			0
$139$			202179.i			0.887563i	0.896135	+	0.443782i	$0.146363\pi$
$139$			202179.i			0.887563i	−0.896135	+	0.443782i	$0.853637\pi$
$140$		0			0
$141$	−3370.48	+	37606.8i	−0.0142772	+	0.159301i
$142$		0			0
$143$	−133144.			−0.544478
$144$		0			0
$145$		−	410929.i		−	1.62310i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			501543.i			1.85073i	0.379081	+	0.925364i	$0.376240\pi$
$149$			501543.i			1.85073i	−0.379081	+	0.925364i	$0.623760\pi$
$150$		0			0
$151$	352128.			1.25678			0.628389	−	0.777899i	$-0.283715\pi$
$151$	352128.			1.25678			0.628389	+	0.777899i	$0.283715\pi$
$152$		0			0
$153$	−214788.	−	38812.1i	−0.741789	−	0.134041i
$154$		0			0
$155$		−	269438.i		−	0.900802i
$156$		0			0
$157$		−	547248.i		−	1.77188i	−0.463797	−	0.885941i	$-0.653513\pi$
$157$		−	547248.i		−	1.77188i	0.463797	−	0.885941i	$-0.346487\pi$
$158$		0			0
$159$	−118284.	−	10601.1i	−0.371050	−	0.0332551i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−629293.			−1.85517			−0.927585	−	0.373611i	$-0.878119\pi$
$163$	−629293.			−1.85517			−0.927585	+	0.373611i	$0.878119\pi$
$164$		0			0
$165$	505481.	+	45303.4i	1.44542	+	0.129545i
$166$		0			0
$167$	588586.			1.63312			0.816562	−	0.577258i	$-0.195877\pi$
$167$	588586.			1.63312			0.816562	+	0.577258i	$0.195877\pi$
$168$		0			0
$169$	289579.			0.779920
$170$		0			0
$171$	90581.8	−	501282.i	0.236892	−	1.31097i
$172$		0			0
$173$	−161240.			−0.409598			−0.204799	−	0.978804i	$-0.565654\pi$
$173$	−161240.			−0.409598			−0.204799	+	0.978804i	$0.565654\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	14811.6	−	165263.i	0.0355360	−	0.396499i
$178$		0			0
$179$			330630.i			0.771275i	0.922650	+	0.385638i	$0.126018\pi$
$179$			330630.i			0.771275i	−0.922650	+	0.385638i	$0.873982\pi$
$180$		0			0
$181$		−	421555.i		−	0.956439i	−0.878240	−	0.478220i	$-0.841282\pi$
$181$		−	421555.i		−	0.956439i	0.878240	−	0.478220i	$-0.158718\pi$
$182$		0			0
$183$	−242588.	−	21741.8i	−0.535477	−	0.0479918i
$184$		0			0
$185$	252916.			0.543309
$186$		0			0
$187$			418362.i			0.874879i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	398002.i		−	0.789408i	−0.918808	−	0.394704i	$-0.870847\pi$
$191$		−	398002.i		−	0.789408i	0.918808	−	0.394704i	$-0.129153\pi$
$192$		0			0
$193$	519175.			1.00328			0.501638	−	0.865078i	$-0.332731\pi$
$193$	519175.			1.00328			0.501638	+	0.865078i	$0.332731\pi$
$194$		0			0
$195$	310228.	+	27804.0i	0.584245	+	0.0523626i
$196$		0			0
$197$			580266.i			1.06527i	0.846344	+	0.532637i	$0.178799\pi$
$197$			580266.i			1.06527i	−0.846344	+	0.532637i	$0.821201\pi$
$198$		0			0
$199$		−	844112.i		−	1.51101i	−0.655142	−	0.755505i	$-0.727391\pi$
$199$		−	844112.i		−	1.51101i	0.655142	−	0.755505i	$-0.272609\pi$
$200$		0			0
$201$	4137.29	−	46162.5i	0.00722313	−	0.0805934i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.19822e6			−1.99138
$206$		0			0
$207$	211802.	−	1.17212e6i	0.343562	−	1.90128i
$208$		0			0
$209$	−976395.			−1.54618
$210$		0			0
$211$	−1.04382e6			−1.61406			−0.807032	−	0.590508i	$-0.798927\pi$
$211$	−1.04382e6			−1.61406			−0.807032	+	0.590508i	$0.798927\pi$
$212$		0			0
$213$	−252585.	−	22637.7i	−0.381468	−	0.0341888i
$214$		0			0
$215$	1.29782e6			1.91478
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−227890.	−	20424.5i	−0.321081	−	0.0287767i
$220$		0			0
$221$			256761.i			0.353629i
$222$		0			0
$223$			185286.i			0.249506i	0.992188	+	0.124753i	$0.0398138\pi$
$223$			185286.i			0.249506i	−0.992188	+	0.124753i	$0.960186\pi$
$224$		0			0
$225$	−421051.	−	76083.9i	−0.554470	−	0.100193i
$226$		0			0
$227$	−877852.			−1.13072			−0.565362	−	0.824843i	$-0.691264\pi$
$227$	−877852.			−1.13072			−0.565362	+	0.824843i	$0.691264\pi$
$228$		0			0
$229$		−	1.22612e6i		−	1.54505i	−0.634983	−	0.772526i	$-0.718993\pi$
$229$		−	1.22612e6i		−	1.54505i	0.634983	−	0.772526i	$-0.281007\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	75659.1i		−	0.0913001i	−0.998957	−	0.0456501i	$-0.985464\pi$
$233$		−	75659.1i		−	0.0913001i	0.998957	−	0.0456501i	$-0.0145359\pi$
$234$		0			0
$235$	−169304.			−0.199985
$236$		0			0
$237$	119950.	−	1.33836e6i	0.138716	−	1.54775i
$238$		0			0
$239$		−	658418.i		−	0.745601i	−0.927911	−	0.372801i	$-0.878397\pi$
$239$		−	658418.i		−	0.745601i	0.927911	−	0.372801i	$-0.121603\pi$
$240$		0			0
$241$			567326.i			0.629202i	0.949224	+	0.314601i	$0.101871\pi$
$241$			567326.i			0.629202i	−0.949224	+	0.314601i	$0.898129\pi$
$242$		0			0
$243$	−397830.	+	830072.i	−0.432197	+	0.901779i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−599241.			−0.624970
$248$		0			0
$249$	−3822.57	+	42651.0i	−0.00390712	+	0.0435944i
$250$		0			0
$251$	−1.83665e6			−1.84010			−0.920050	−	0.391800i	$-0.871852\pi$
$251$	−1.83665e6			−1.84010			−0.920050	+	0.391800i	$0.871852\pi$
$252$		0			0
$253$	−2.28305e6			−2.24240
$254$		0			0
$255$	87365.3	−	974794.i	0.0841373	−	0.938777i
$256$		0			0
$257$	−902411.			−0.852259			−0.426129	−	0.904662i	$-0.640123\pi$
$257$	−902411.			−0.852259			−0.426129	+	0.904662i	$0.640123\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	254031.	−	1.40582e6i	0.230827	−	1.27740i
$262$		0			0
$263$			933055.i			0.831798i	0.909411	+	0.415899i	$0.136533\pi$
$263$			933055.i			0.831798i	−0.909411	+	0.415899i	$0.863467\pi$
$264$		0			0
$265$		−	532508.i		−	0.465813i
$266$		0			0
$267$	−141662.	+	1.58062e6i	−0.121612	+	1.35691i
$268$		0			0
$269$	−533435.			−0.449470			−0.224735	−	0.974420i	$-0.572152\pi$
$269$	−533435.			−0.449470			−0.224735	+	0.974420i	$0.572152\pi$
$270$		0			0
$271$			798687.i			0.660622i	0.943872	+	0.330311i	$0.107154\pi$
$271$			798687.i			0.660622i	−0.943872	+	0.330311i	$0.892846\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			820120.i			0.653951i
$276$		0			0
$277$	1.18032e6			0.924271			0.462136	−	0.886809i	$-0.347083\pi$
$277$	1.18032e6			0.924271			0.462136	+	0.886809i	$0.347083\pi$
$278$		0			0
$279$	166563.	−	921766.i	0.128106	−	0.708941i
$280$		0			0
$281$		−	1.56254e6i		−	1.18050i	−0.807221	−	0.590250i	$-0.799029\pi$
$281$		−	1.56254e6i		−	1.18050i	0.807221	−	0.590250i	$-0.200971\pi$
$282$		0			0
$283$			1.08464e6i			0.805044i	0.915410	+	0.402522i	$0.131866\pi$
$283$			1.08464e6i			0.805044i	−0.915410	+	0.402522i	$0.868134\pi$
$284$		0			0
$285$	2.27502e6	+	203898.i	1.65911	+	0.148696i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−613068.			−0.431782
$290$		0			0
$291$	363418.	+	32571.1i	0.251579	+	0.0225476i
$292$		0			0
$293$	1.39243e6			0.947552			0.473776	−	0.880645i	$-0.342891\pi$
$293$	1.39243e6			0.947552			0.473776	+	0.880645i	$0.342891\pi$
$294$		0			0
$295$	744007.			0.497762
$296$		0			0
$297$	1.70128e6	+	467469.i	1.11914	+	0.307512i
$298$		0			0
$299$	−1.40117e6			−0.906386
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−86556.3	+	965768.i	−0.0541617	+	0.604319i
$304$		0			0
$305$		−	1.09212e6i		−	0.672234i
$306$		0			0
$307$			1.19400e6i			0.723034i	0.932366	+	0.361517i	$0.117741\pi$
$307$			1.19400e6i			0.723034i	−0.932366	+	0.361517i	$0.882259\pi$
$308$		0			0
$309$	601341.	+	53894.8i	0.358282	+	0.0321108i
$310$		0			0
$311$	2.62556e6			1.53929			0.769646	−	0.638470i	$-0.220433\pi$
$311$	2.62556e6			1.53929			0.769646	+	0.638470i	$0.220433\pi$
$312$		0			0
$313$		−	887758.i		−	0.512193i	−0.966651	−	0.256097i	$-0.917563\pi$
$313$		−	887758.i		−	0.512193i	0.966651	−	0.256097i	$-0.0824365\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			1.81415e6i			1.01397i	0.861955	+	0.506986i	$0.169240\pi$
$317$			1.81415e6i			1.01397i	−0.861955	+	0.506986i	$0.830760\pi$
$318$		0			0
$319$	−2.73824e6			−1.50659
$320$		0			0
$321$	2.92946e6	+	262551.i	1.58681	+	0.142217i
$322$		0			0
$323$			1.88293e6i			1.00422i
$324$		0			0
$325$			503331.i			0.264329i
$326$		0			0
$327$	−220271.	+	2.45771e6i	−0.113917	+	1.27105i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.55409e6			−1.28135			−0.640673	−	0.767814i	$-0.721345\pi$
$331$	−2.55409e6			−1.28135			−0.640673	+	0.767814i	$0.721345\pi$
$332$		0			0
$333$	865243.	+	156350.i	0.427590	+	0.0772656i
$334$		0			0
$335$	207822.			0.101176
$336$		0			0
$337$	1.68914e6			0.810196			0.405098	−	0.914273i	$-0.367237\pi$
$337$	1.68914e6			0.810196			0.405098	+	0.914273i	$0.367237\pi$
$338$		0			0
$339$	−2.61320e6	−	234206.i	−1.23502	−	0.110688i
$340$		0			0
$341$	−1.79541e6			−0.836138
$342$		0			0
$343$		0			0
$344$		0			0
$345$	5.31956e6	+	476762.i	2.40618	+	0.215652i
$346$		0			0
$347$			169943.i			0.0757668i	0.999282	+	0.0378834i	$0.0120615\pi$
$347$			169943.i			0.0757668i	−0.999282	+	0.0378834i	$0.987938\pi$
$348$		0			0
$349$			2.36988e6i			1.04151i	0.853706	+	0.520755i	$0.174350\pi$
$349$			2.36988e6i			1.04151i	−0.853706	+	0.520755i	$0.825650\pi$
$350$		0			0
$351$	1.04412e6	+	286899.i	0.452361	+	0.124297i
$352$		0			0
$353$	−2.02957e6			−0.866897			−0.433448	−	0.901178i	$-0.642703\pi$
$353$	−2.02957e6			−0.866897			−0.433448	+	0.901178i	$0.642703\pi$
$354$		0			0
$355$		−	1.13712e6i		−	0.478892i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	352368.i		−	0.144298i	−0.997394	−	0.0721489i	$-0.977014\pi$
$359$		−	352368.i		−	0.144298i	0.997394	−	0.0721489i	$-0.0229857\pi$
$360$		0			0
$361$	−1.91837e6			−0.774755
$362$		0			0
$363$	77774.3	−	867781.i	0.0309791	−	0.345655i
$364$		0			0
$365$		−	1.02595e6i		−	0.403082i
$366$		0			0
$367$			1.80805e6i			0.700723i	0.936615	+	0.350362i	$0.113941\pi$
$367$			1.80805e6i			0.700723i	−0.936615	+	0.350362i	$0.886059\pi$
$368$		0			0
$369$	−4.09921e6	−	740728.i	−1.56723	−	0.283200i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	3.63970e6			1.35454			0.677272	−	0.735733i	$-0.263162\pi$
$373$	3.63970e6			1.35454			0.677272	+	0.735733i	$0.263162\pi$
$374$		0			0
$375$	−132692.	+	1.48053e6i	−0.0487265	+	0.543675i
$376$		0			0
$377$	−1.68054e6			−0.608968
$378$		0			0
$379$	4.81458e6			1.72171			0.860856	−	0.508849i	$-0.169929\pi$
$379$	4.81458e6			1.72171			0.860856	+	0.508849i	$0.169929\pi$
$380$		0			0
$381$	167955.	−	1.87399e6i	0.0592762	−	0.661385i
$382$		0			0
$383$	3.70544e6			1.29075			0.645376	−	0.763865i	$-0.276701\pi$
$383$	3.70544e6			1.29075			0.645376	+	0.763865i	$0.276701\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.43995e6	+	802299.i	1.50696	+	0.272307i
$388$		0			0
$389$			1.09914e6i			0.368282i	0.982900	+	0.184141i	$0.0589503\pi$
$389$			1.09914e6i			0.368282i	−0.982900	+	0.184141i	$0.941050\pi$
$390$		0			0
$391$			4.40274e6i			1.45640i
$392$		0			0
$393$	337969.	−	3.77095e6i	0.110381	−	1.23160i
$394$		0			0
$395$	6.02523e6			1.94304
$396$		0			0
$397$		−	5.54292e6i		−	1.76507i	−0.470246	−	0.882535i	$-0.655835\pi$
$397$		−	5.54292e6i		−	1.76507i	0.470246	−	0.882535i	$-0.344165\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	3.50177e6i		−	1.08749i	−0.839250	−	0.543746i	$-0.817005\pi$
$401$		−	3.50177e6i		−	1.08749i	0.839250	−	0.543746i	$-0.182995\pi$
$402$		0			0
$403$	−1.10189e6			−0.337970
$404$		0			0
$405$	−3.86641e6	−	1.44449e6i	−1.17131	−	0.437599i
$406$		0			0
$407$		−	1.68532e6i		−	0.504307i
$408$		0			0
$409$			4.96495e6i			1.46760i	0.679367	+	0.733799i	$0.262254\pi$
$409$			4.96495e6i			1.46760i	−0.679367	+	0.733799i	$0.737746\pi$
$410$		0			0
$411$	3.23996e6	+	290379.i	0.946095	+	0.0847932i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−192013.			−0.0547281
$416$		0			0
$417$	−3.13908e6	−	281338.i	−0.884020	−	0.0792297i
$418$		0			0
$419$	2.93856e6			0.817711			0.408855	−	0.912599i	$-0.365928\pi$
$419$	2.93856e6			0.817711			0.408855	+	0.912599i	$0.365928\pi$
$420$		0			0
$421$	−90444.5			−0.0248701			−0.0124350	−	0.999923i	$-0.503958\pi$
$421$	−90444.5			−0.0248701			−0.0124350	+	0.999923i	$0.503958\pi$
$422$		0			0
$423$	−579201.	−	104662.i	−0.157391	−	0.0284405i
$424$		0			0
$425$	1.58156e6			0.424730
$426$		0			0
$427$		0			0
$428$		0			0
$429$	185273.	−	2.06722e6i	0.0486037	−	0.542305i
$430$		0			0
$431$			295281.i			0.0765672i	0.999267	+	0.0382836i	$0.0121890\pi$
$431$			295281.i			0.0765672i	−0.999267	+	0.0382836i	$0.987811\pi$
$432$		0			0
$433$			7.22505e6i			1.85191i	0.377628	+	0.925957i	$0.376740\pi$
$433$			7.22505e6i			1.85191i	−0.377628	+	0.925957i	$0.623260\pi$
$434$		0			0
$435$	6.38017e6	+	571819.i	1.61663	+	0.144889i
$436$		0			0
$437$	−1.02753e7			−2.57390
$438$		0			0
$439$			1.59769e6i			0.395668i	0.980236	+	0.197834i	$0.0633906\pi$
$439$			1.59769e6i			0.395668i	−0.980236	+	0.197834i	$0.936609\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	5.69173e6i		−	1.37795i	−0.724783	−	0.688977i	$-0.758060\pi$
$443$		−	5.69173e6i		−	1.37795i	0.724783	−	0.688977i	$-0.241940\pi$
$444$		0			0
$445$	−7.11590e6			−1.70345
$446$		0			0
$447$	−7.78707e6	−	697911.i	−1.84334	−	0.165208i
$448$		0			0
$449$			7.13493e6i			1.67022i	0.550082	+	0.835111i	$0.314597\pi$
$449$			7.13493e6i			1.67022i	−0.550082	+	0.835111i	$0.685403\pi$
$450$		0			0
$451$			7.98441e6i			1.84842i
$452$		0			0
$453$	−489997.	+	5.46723e6i	−0.112188	+	1.25176i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.94203e6			0.434976			0.217488	−	0.976063i	$-0.430214\pi$
$457$	1.94203e6			0.434976			0.217488	+	0.976063i	$0.430214\pi$
$458$		0			0
$459$	901489.	−	3.28083e6i	0.199723	−	0.726862i
$460$		0			0
$461$	6.10948e6			1.33891			0.669456	−	0.742852i	$-0.266527\pi$
$461$	6.10948e6			1.33891			0.669456	+	0.742852i	$0.266527\pi$
$462$		0			0
$463$	4.08866e6			0.886398			0.443199	−	0.896423i	$-0.353844\pi$
$463$	4.08866e6			0.886398			0.443199	+	0.896423i	$0.353844\pi$
$464$		0			0
$465$	4.18335e6	+	374930.i	0.897206	+	0.0804115i
$466$		0			0
$467$	−444247.			−0.0942611			−0.0471306	−	0.998889i	$-0.515008\pi$
$467$	−444247.			−0.0942611			−0.0471306	+	0.998889i	$0.515008\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	8.49669e6	+	761511.i	1.76481	+	0.158170i
$472$		0			0
$473$		−	8.64810e6i		−	1.77733i
$474$		0			0
$475$			3.69112e6i			0.750627i
$476$		0			0
$477$	329190.	−	1.82175e6i	0.0662446	−	0.366600i
$478$		0			0
$479$	−1.50806e6			−0.300317			−0.150159	−	0.988662i	$-0.547978\pi$
$479$	−1.50806e6			−0.300317			−0.150159	+	0.988662i	$0.547978\pi$
$480$		0			0
$481$		−	1.03433e6i		−	0.203843i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			1.63609e6i			0.315830i
$486$		0			0
$487$	2.85810e6			0.546078			0.273039	−	0.962003i	$-0.411971\pi$
$487$	2.85810e6			0.546078			0.273039	+	0.962003i	$0.411971\pi$
$488$		0			0
$489$	875679.	−	9.77054e6i	0.165605	−	1.84776i
$490$		0			0
$491$		−	4.95391e6i		−	0.927352i	−0.886005	−	0.463676i	$-0.846530\pi$
$491$		−	4.95391e6i		−	0.927352i	0.886005	−	0.463676i	$-0.153470\pi$
$492$		0			0
$493$			5.28056e6i			0.978503i
$494$		0			0
$495$	−1.40678e6	+	7.78517e6i	−0.258056	+	1.42809i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−7.98813e6			−1.43613			−0.718065	−	0.695976i	$-0.754972\pi$
$499$	−7.98813e6			−1.43613			−0.718065	+	0.695976i	$0.754972\pi$
$500$		0			0
$501$	−819034.	+	9.13853e6i	−0.145783	+	1.62660i
$502$		0			0
$503$	2.57392e6			0.453602			0.226801	−	0.973941i	$-0.427173\pi$
$503$	2.57392e6			0.453602			0.226801	+	0.973941i	$0.427173\pi$
$504$		0			0
$505$	−4.34784e6			−0.758657
$506$		0			0
$507$	−402957.	+	4.49607e6i	−0.0696208	+	0.776807i
$508$		0			0
$509$	9.29786e6			1.59070			0.795350	−	0.606150i	$-0.207287\pi$
$509$	9.29786e6			1.59070			0.795350	+	0.606150i	$0.207287\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	7.65697e6	+	2.10394e6i	1.28459	+	0.352972i
$514$		0			0
$515$			2.70721e6i			0.449784i
$516$		0			0
$517$			1.12816e6i			0.185629i
$518$		0			0
$519$	224370.	−	2.50345e6i	0.0365634	−	0.407963i
$520$		0			0
$521$	1.09866e7			1.77325			0.886624	−	0.462490i	$-0.153044\pi$
$521$	1.09866e7			1.77325			0.886624	+	0.462490i	$0.153044\pi$
$522$		0			0
$523$		−	278694.i		−	0.0445527i	−0.999752	−	0.0222763i	$-0.992909\pi$
$523$		−	278694.i		−	0.0445527i	0.999752	−	0.0222763i	$-0.00709137\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			3.46235e6i			0.543057i
$528$		0			0
$529$	−1.75899e7			−2.73290
$530$		0			0
$531$	2.54530e6	+	459936.i	0.391744	+	0.0707883i
$532$		0			0
$533$			4.90026e6i			0.747139i
$534$		0			0
$535$			1.31883e7i			1.99207i
$536$		0			0
$537$	−5.13343e6	−	460080.i	−0.768196	−	0.0688491i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.24150e6			−0.182370			−0.0911849	−	0.995834i	$-0.529065\pi$
$541$	−1.24150e6			−0.182370			−0.0911849	+	0.995834i	$0.529065\pi$
$542$		0			0
$543$	6.54515e6	+	586605.i	0.952621	+	0.0853780i
$544$		0			0
$545$	−1.10645e7			−1.59566
$546$		0			0
$547$	−5.28542e6			−0.755286			−0.377643	−	0.925951i	$-0.623265\pi$
$547$	−5.28542e6			−0.755286			−0.377643	+	0.925951i	$0.623265\pi$
$548$		0			0
$549$	675135.	−	3.73622e6i	0.0956004	−	0.529056i
$550$		0			0
$551$	−1.23240e7			−1.72931
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−351939.	+	3.92683e6i	−0.0484993	+	0.541140i
$556$		0			0
$557$			7.68636e6i			1.04974i	0.851182	+	0.524871i	$0.175887\pi$
$557$			7.68636e6i			1.04974i	−0.851182	+	0.524871i	$0.824113\pi$
$558$		0			0
$559$		−	5.30759e6i		−	0.718402i
$560$		0			0
$561$	−6.49558e6	−	582162.i	−0.871386	−	0.0780974i
$562$		0			0
$563$	4.56529e6			0.607012			0.303506	−	0.952830i	$-0.401843\pi$
$563$	4.56529e6			0.607012			0.303506	+	0.952830i	$0.401843\pi$
$564$		0			0
$565$		−	1.17645e7i		−	1.55043i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			4.68731e6i			0.606936i	0.952842	+	0.303468i	$0.0981446\pi$
$569$			4.68731e6i			0.606936i	−0.952842	+	0.303468i	$0.901855\pi$
$570$		0			0
$571$	5.13561e6			0.659177			0.329589	−	0.944125i	$-0.393090\pi$
$571$	5.13561e6			0.659177			0.329589	+	0.944125i	$0.393090\pi$
$572$		0			0
$573$	6.17947e6	+	553831.i	0.786257	+	0.0704678i
$574$		0			0
$575$			8.63073e6i			1.08862i
$576$		0			0
$577$		−	667514.i		−	0.0834682i	−0.999129	−	0.0417341i	$-0.986712\pi$
$577$		−	667514.i		−	0.0834682i	0.999129	−	0.0417341i	$-0.0132882\pi$
$578$		0			0
$579$	−722447.	+	8.06083e6i	−0.0895590	+	0.999271i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−3.54839e6			−0.432374
$584$		0			0
$585$	−863382.	+	4.77798e6i	−0.104307	+	0.577238i
$586$		0			0
$587$	9.60922e6			1.15105			0.575523	−	0.817785i	$-0.304798\pi$
$587$	9.60922e6			1.15105			0.575523	+	0.817785i	$0.304798\pi$
$588$		0			0
$589$	−8.08062e6			−0.959747
$590$		0			0
$591$	−9.00934e6	−	807456.i	−1.06102	−	0.0950934i
$592$		0			0
$593$	−1.00253e7			−1.17074			−0.585371	−	0.810766i	$-0.699051\pi$
$593$	−1.00253e7			−1.17074			−0.585371	+	0.810766i	$0.699051\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.31059e7	+	1.17461e6i	1.50498	+	0.134883i
$598$		0			0
$599$		−	4.96759e6i		−	0.565690i	−0.959166	−	0.282845i	$-0.908722\pi$
$599$		−	4.96759e6i		−	0.565690i	0.959166	−	0.282845i	$-0.0912782\pi$
$600$		0			0
$601$			2.91804e6i			0.329538i	0.986332	+	0.164769i	$0.0526878\pi$
$601$			2.91804e6i			0.329538i	−0.986332	+	0.164769i	$0.947312\pi$
$602$		0			0
$603$	710973.	+	128473.i	0.0796269	+	0.0143886i
$604$		0			0
$605$	3.90671e6			0.433933
$606$		0			0
$607$			5.53707e6i			0.609969i	0.952357	+	0.304985i	$0.0986513\pi$
$607$			5.53707e6i			0.609969i	−0.952357	+	0.304985i	$0.901349\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			692387.i			0.0750319i
$612$		0			0
$613$	−5.39229e6			−0.579591			−0.289796	−	0.957089i	$-0.593587\pi$
$613$	−5.39229e6			−0.579591			−0.289796	+	0.957089i	$0.593587\pi$
$614$		0			0
$615$	1.66736e6	−	1.86039e7i	0.177763	−	1.98343i
$616$		0			0
$617$		−	1.18727e7i		−	1.25556i	−0.778392	−	0.627778i	$-0.783965\pi$
$617$		−	1.18727e7i		−	1.25556i	0.778392	−	0.627778i	$-0.216035\pi$
$618$		0			0
$619$			6.71087e6i			0.703967i	0.936006	+	0.351983i	$0.114493\pi$
$619$			6.71087e6i			0.703967i	−0.936006	+	0.351983i	$0.885507\pi$
$620$		0			0
$621$	1.79039e7	+	4.91953e6i	1.86302	+	0.511911i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.21677e7			−1.24597
$626$		0			0
$627$	1.35868e6	−	1.51597e7i	0.138022	−	1.54001i
$628$		0			0
$629$	−3.25004e6			−0.327539
$630$		0			0
$631$	1.01133e7			1.01116			0.505580	−	0.862780i	$-0.331279\pi$
$631$	1.01133e7			1.01116			0.505580	+	0.862780i	$0.331279\pi$
$632$		0			0
$633$	1.45251e6	−	1.62066e7i	0.144082	−	1.60762i
$634$		0			0
$635$	8.43661e6			0.830297
$636$		0			0
$637$		0			0
$638$		0			0
$639$	702957.	−	3.89019e6i	0.0681046	−	0.376893i
$640$		0			0
$641$		−	8.43591e6i		−	0.810937i	−0.914109	−	0.405468i	$-0.867108\pi$
$641$		−	8.43591e6i		−	0.810937i	0.914109	−	0.405468i	$-0.132892\pi$
$642$		0			0
$643$			1.51051e7i			1.44077i	0.693573	+	0.720386i	$0.256035\pi$
$643$			1.51051e7i			1.44077i	−0.693573	+	0.720386i	$0.743965\pi$
$644$		0			0
$645$	−1.80596e6	+	2.01503e7i	−0.170926	+	1.90714i
$646$		0			0
$647$	−1.08522e7			−1.01919			−0.509597	−	0.860413i	$-0.670205\pi$
$647$	−1.08522e7			−1.01919			−0.509597	+	0.860413i	$0.670205\pi$
$648$		0			0
$649$		−	4.95772e6i		−	0.462030i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			590041.i			0.0541501i	0.999633	+	0.0270750i	$0.00861931\pi$
$653$			590041.i			0.0541501i	−0.999633	+	0.0270750i	$0.991381\pi$
$654$		0			0
$655$	1.69766e7			1.54614
$656$		0			0
$657$	634229.	−	3.50984e6i	0.0573235	−	0.317230i
$658$		0			0
$659$		−	1.38679e7i		−	1.24394i	−0.783042	−	0.621969i	$-0.786333\pi$
$659$		−	1.38679e7i		−	1.24394i	0.783042	−	0.621969i	$-0.213667\pi$
$660$		0			0
$661$		−	1.62028e7i		−	1.44240i	−0.692727	−	0.721200i	$-0.743591\pi$
$661$		−	1.62028e7i		−	1.44240i	0.692727	−	0.721200i	$-0.256409\pi$
$662$		0			0
$663$	−3.98652e6	−	357290.i	−0.352217	−	0.0315672i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.88166e7			−2.50800
$668$		0			0
$669$	−2.87679e6	−	257831.i	−0.248510	−	0.0222725i
$670$		0			0
$671$	−7.27738e6			−0.623977
$672$		0			0
$673$	−1.09799e7			−0.934456			−0.467228	−	0.884137i	$-0.654747\pi$
$673$	−1.09799e7			−0.934456			−0.467228	+	0.884137i	$0.654747\pi$
$674$		0			0
$675$	1.76720e6	−	6.43145e6i	0.149288	−	0.543312i
$676$		0			0
$677$	−190993.			−0.0160157			−0.00800785	−	0.999968i	$-0.502549\pi$
$677$	−190993.			−0.0160157			−0.00800785	+	0.999968i	$0.502549\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.22156e6	−	1.36297e7i	0.100936	−	1.12621i
$682$		0			0
$683$			1.35316e7i			1.10993i	0.831873	+	0.554967i	$0.187269\pi$
$683$			1.35316e7i			1.10993i	−0.831873	+	0.554967i	$0.812731\pi$
$684$		0			0
$685$			1.45861e7i			1.18772i
$686$		0			0
$687$	1.90370e7	+	1.70618e6i	1.53888	+	0.137922i
$688$		0			0
$689$	−2.17775e6			−0.174767
$690$		0			0
$691$		−	9.88908e6i		−	0.787881i	−0.919136	−	0.393941i	$-0.871112\pi$
$691$		−	9.88908e6i		−	0.787881i	0.919136	−	0.393941i	$-0.128888\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	1.41320e7i		−	1.10979i
$696$		0			0
$697$	1.53975e7			1.20052
$698$		0			0
$699$	1.17470e6	+	105282.i	0.0909357	+	0.00815005i
$700$		0			0
$701$			1.46554e6i			0.112643i	0.998413	+	0.0563213i	$0.0179371\pi$
$701$			1.46554e6i			0.112643i	−0.998413	+	0.0563213i	$0.982063\pi$
$702$		0			0
$703$		−	7.58512e6i		−	0.578861i
$704$		0			0
$705$	235591.	−	2.62865e6i	0.0178520	−	0.199187i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.66337e6			0.124272			0.0621361	−	0.998068i	$-0.480209\pi$
$709$	1.66337e6			0.124272			0.0621361	+	0.998068i	$0.480209\pi$
$710$		0			0
$711$	2.06127e7	+	3.72473e6i	1.52919	+	0.276325i
$712$		0			0
$713$	−1.88945e7			−1.39191
$714$		0			0
$715$	9.30652e6			0.680805
$716$		0			0
$717$	1.02227e7	+	916207.i	0.742625	+	0.0665573i
$718$		0			0
$719$	7.28664e6			0.525660			0.262830	−	0.964842i	$-0.415344\pi$
$719$	7.28664e6			0.525660			0.262830	+	0.964842i	$0.415344\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−8.80843e6	−	789449.i	−0.626690	−	0.0561667i
$724$		0			0
$725$			1.03515e7i			0.731408i
$726$		0			0
$727$		−	2.37534e6i		−	0.166682i	−0.996521	−	0.0833411i	$-0.973441\pi$
$727$		−	2.37534e6i		−	0.166682i	0.996521	−	0.0833411i	$-0.0265591\pi$
$728$		0			0
$729$	−1.23343e7	−	7.33187e6i	−0.859598	−	0.510970i
$730$		0			0
$731$	−1.66774e7			−1.15434
$732$		0			0
$733$		−	2.12018e7i		−	1.45751i	−0.684774	−	0.728755i	$-0.740099\pi$
$733$		−	2.12018e7i		−	1.45751i	0.684774	−	0.728755i	$-0.259901\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.38483e6i		−	0.0939133i
$738$		0			0
$739$	8248.53			0.000555604			0.000277802	−	1.00000i	$-0.499912\pi$
$739$	8248.53			0.000555604			0.000277802		1.00000i	$0.499912\pi$
$740$		0			0
$741$	833861.	−	9.30395e6i	0.0557889	−	0.622475i
$742$		0			0
$743$		−	2.10286e7i		−	1.39746i	−0.715386	−	0.698730i	$-0.753749\pi$
$743$		−	2.10286e7i		−	1.39746i	0.715386	−	0.698730i	$-0.246251\pi$
$744$		0			0
$745$		−	3.50570e7i		−	2.31411i
$746$		0			0
$747$	−656890.	−	118700.i	−0.0430716	−	0.00778305i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.24333e7			−0.804426			−0.402213	−	0.915546i	$-0.631759\pi$
$751$	−1.24333e7			−0.804426			−0.402213	+	0.915546i	$0.631759\pi$
$752$		0			0
$753$	2.55575e6	−	2.85162e7i	0.164259	−	1.83275i
$754$		0			0
$755$	−2.46132e7			−1.57145
$756$		0			0
$757$	−1.56314e7			−0.991421			−0.495710	−	0.868488i	$-0.665092\pi$
$757$	−1.56314e7			−0.991421			−0.495710	+	0.868488i	$0.665092\pi$
$758$		0			0
$759$	3.17692e6	−	3.54471e7i	0.200172	−	2.23345i
$760$		0			0
$761$	3.54527e6			0.221915			0.110958	−	0.993825i	$-0.464608\pi$
$761$	3.54527e6			0.221915			0.110958	+	0.993825i	$0.464608\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.50133e7	+	2.71291e6i	0.927518	+	0.167603i
$766$		0			0
$767$		−	3.04269e6i		−	0.186754i
$768$		0			0
$769$		−	4.01821e6i		−	0.245028i	−0.992467	−	0.122514i	$-0.960904\pi$
$769$		−	4.01821e6i		−	0.245028i	0.992467	−	0.122514i	$-0.0390957\pi$
$770$		0			0
$771$	1.25573e6	−	1.40110e7i	0.0760782	−	0.848857i
$772$		0			0
$773$	−2.96681e7			−1.78584			−0.892918	−	0.450219i	$-0.851346\pi$
$773$	−2.96681e7			−1.78584			−0.892918	+	0.450219i	$0.851346\pi$
$774$		0			0
$775$			6.78729e6i			0.405922i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			3.59355e7i			2.12168i
$780$		0			0
$781$	−7.57728e6			−0.444514
$782$		0			0
$783$	2.14735e7	+	5.90038e6i	1.25170	+	0.343934i
$784$		0			0
$785$			3.82517e7i			2.21553i
$786$		0			0
$787$		−	6.48183e6i		−	0.373045i	−0.982451	−	0.186522i	$-0.940278\pi$
$787$		−	6.48183e6i		−	0.373045i	0.982451	−	0.186522i	$-0.0597217\pi$
$788$		0			0
$789$	−1.44868e7	−	1.29837e6i	−0.828477	−	0.0742517i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−4.46634e6			−0.252214
$794$		0			0
$795$	8.26783e6	+	740999.i	0.463953	+	0.0415815i
$796$		0			0
$797$	3.52053e7			1.96319			0.981594	−	0.190979i	$-0.0611663\pi$
$797$	3.52053e7			1.96319			0.981594	+	0.190979i	$0.0611663\pi$
$798$		0			0
$799$	2.17561e6			0.120563
$800$		0			0
$801$	−2.43440e7	−	4.39896e6i	−1.34064	−	0.242253i
$802$		0			0
$803$	−6.83645e6			−0.374147
$804$		0			0
$805$		0			0
$806$		0			0
$807$	742290.	−	8.28224e6i	0.0401227	−	0.447676i
$808$		0			0
$809$			1.62772e7i			0.874397i	0.899365	+	0.437198i	$0.144029\pi$
$809$			1.62772e7i			0.874397i	−0.899365	+	0.437198i	$0.855971\pi$
$810$		0			0
$811$		−	823682.i		−	0.0439752i	−0.999758	−	0.0219876i	$-0.993001\pi$
$811$		−	823682.i		−	0.0439752i	0.999758	−	0.0219876i	$-0.00699943\pi$
$812$		0			0
$813$	−1.24006e7	−	1.11139e6i	−0.657985	−	0.0589715i
$814$		0			0
$815$	4.39865e7			2.31967
$816$		0			0
$817$		−	3.89226e7i		−	2.04008i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			4.33763e6i			0.224592i	0.993675	+	0.112296i	$0.0358205\pi$
$821$			4.33763e6i			0.224592i	−0.993675	+	0.112296i	$0.964179\pi$
$822$		0			0
$823$	1.01908e7			0.524454			0.262227	−	0.965006i	$-0.415543\pi$
$823$	1.01908e7			0.524454			0.262227	+	0.965006i	$0.415543\pi$
$824$		0			0
$825$	−1.27334e7	−	1.14122e6i	−0.651341	−	0.0583760i
$826$		0			0
$827$			1.90780e7i			0.969996i	0.874515	+	0.484998i	$0.161180\pi$
$827$			1.90780e7i			0.969996i	−0.874515	+	0.484998i	$0.838820\pi$
$828$		0			0
$829$		−	5.64200e6i		−	0.285133i	−0.989785	−	0.142566i	$-0.954465\pi$
$829$		−	5.64200e6i		−	0.285133i	0.989785	−	0.142566i	$-0.0455354\pi$
$830$		0			0
$831$	−1.64244e6	+	1.83259e7i	−0.0825065	+	0.920581i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−4.11412e7			−2.04202
$836$		0			0
$837$	1.40798e7	+	3.86876e6i	0.694675	+	0.190879i
$838$		0			0
$839$	1.11858e7			0.548608			0.274304	−	0.961643i	$-0.411552\pi$
$839$	1.11858e7			0.548608			0.274304	+	0.961643i	$0.411552\pi$
$840$		0			0
$841$	−1.40509e7			−0.685037
$842$		0			0
$843$	2.42604e7	+	2.17432e6i	1.17579	+	0.105379i
$844$		0			0
$845$	−2.02411e7			−0.975196
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.68404e7	−	1.50931e6i	−0.801830	−	0.0718635i
$850$		0			0
$851$		−	1.77358e7i		−	0.839514i
$852$		0			0
$853$		−	8.52648e6i		−	0.401233i	−0.979670	−	0.200617i	$-0.935705\pi$
$853$		−	8.52648e6i		−	0.401233i	0.979670	−	0.200617i	$-0.0642946\pi$
$854$		0			0
$855$	−6.33152e6	+	3.50388e7i	−0.296205	+	1.63921i
$856$		0			0
$857$	2.71350e7			1.26205			0.631027	−	0.775761i	$-0.282634\pi$
$857$	2.71350e7			1.26205			0.631027	+	0.775761i	$0.282634\pi$
$858$		0			0
$859$			2.17238e7i			1.00451i	0.864720	+	0.502254i	$0.167496\pi$
$859$			2.17238e7i			1.00451i	−0.864720	+	0.502254i	$0.832504\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.77634e7i		−	0.811895i	−0.913896	−	0.405948i	$-0.866942\pi$
$863$		−	1.77634e7i		−	0.811895i	0.913896	−	0.405948i	$-0.133058\pi$
$864$		0			0
$865$	1.12704e7			0.512153
$866$		0			0
$867$	853101.	−	9.51863e6i	0.0385437	−	0.430058i
$868$		0			0
$869$		−	4.01494e7i		−	1.80355i
$870$		0			0
$871$		−	849909.i		−	0.0379601i
$872$		0			0
$873$	−1.01141e6	+	5.59719e6i	−0.0449152	+	0.248562i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−621725.			−0.0272960			−0.0136480	−	0.999907i	$-0.504344\pi$
$877$	−621725.			−0.0272960			−0.0136480	+	0.999907i	$0.504344\pi$
$878$		0			0
$879$	−1.93760e6	+	2.16191e7i	−0.0845847	+	0.943769i
$880$		0			0
$881$	3.18993e7			1.38465			0.692327	−	0.721584i	$-0.256586\pi$
$881$	3.18993e7			1.38465			0.692327	+	0.721584i	$0.256586\pi$
$882$		0			0
$883$	−1.60937e7			−0.694633			−0.347316	−	0.937748i	$-0.612907\pi$
$883$	−1.60937e7			−0.694633			−0.347316	+	0.937748i	$0.612907\pi$
$884$		0			0
$885$	−1.03531e6	+	1.15516e7i	−0.0444335	+	0.495775i
$886$		0			0
$887$	2.19434e7			0.936471			0.468235	−	0.883604i	$-0.344890\pi$
$887$	2.19434e7			0.936471			0.468235	+	0.883604i	$0.344890\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−9.62541e6	+	2.57640e7i	−0.406186	+	1.08722i
$892$		0			0
$893$			5.07754e6i			0.213071i
$894$		0			0
$895$		−	2.31105e7i		−	0.964387i
$896$		0			0
$897$	1.94977e6	−	2.17549e7i	0.0809100	−	0.902768i
$898$		0			0
$899$	−2.26616e7			−0.935173
$900$		0			0
$901$			6.84288e6i			0.280819i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			2.94660e7i			1.19591i
$906$		0			0
$907$	2.41122e7			0.973238			0.486619	−	0.873614i	$-0.338230\pi$
$907$	2.41122e7			0.973238			0.486619	+	0.873614i	$0.338230\pi$
$908$		0			0
$909$	−1.48743e7	−	2.68779e6i	−0.597071	−	0.107891i
$910$		0			0
$911$			1.16994e7i			0.467056i	0.972350	+	0.233528i	$0.0750271\pi$
$911$			1.16994e7i			0.467056i	−0.972350	+	0.233528i	$0.924973\pi$
$912$		0			0
$913$			1.27949e6i			0.0507994i
$914$		0			0
$915$	1.69565e7	+	1.51971e6i	0.669550	+	0.0600080i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.35972e7			0.531080			0.265540	−	0.964100i	$-0.414450\pi$
$919$	1.35972e7			0.531080			0.265540	+	0.964100i	$0.414450\pi$
$920$		0			0
$921$	−1.85383e7	−	1.66148e6i	−0.720147	−	0.0645427i
$922$		0			0
$923$	−4.65039e6			−0.179674
$924$		0			0
$925$	−6.37110e6			−0.244827
$926$		0			0
$927$	−1.67357e6	+	9.26156e6i	−0.0639651	+	0.353985i
$928$		0			0
$929$	−1.13421e7			−0.431177			−0.215589	−	0.976484i	$-0.569167\pi$
$929$	−1.13421e7			−0.431177			−0.215589	+	0.976484i	$0.569167\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−3.65354e6	+	4.07651e7i	−0.137407	+	1.53315i
$934$		0			0
$935$		−	2.92428e7i		−	1.09393i
$936$		0			0
$937$		−	2.45048e7i		−	0.911806i	−0.890029	−	0.455903i	$-0.849316\pi$
$937$		−	2.45048e7i		−	0.911806i	0.890029	−	0.455903i	$-0.150684\pi$
$938$		0			0
$939$	1.37835e7	+	1.23534e6i	0.510149	+	0.0457217i
$940$		0			0
$941$	5.41453e6			0.199336			0.0996682	−	0.995021i	$-0.468222\pi$
$941$	5.41453e6			0.199336			0.0996682	+	0.995021i	$0.468222\pi$
$942$		0			0
$943$			8.40260e7i			3.07705i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	3.00820e7i		−	1.09001i	−0.838432	−	0.545006i	$-0.816527\pi$
$947$		−	3.00820e7i		−	1.09001i	0.838432	−	0.545006i	$-0.183473\pi$
$948$		0			0
$949$	−4.19573e6			−0.151231
$950$		0			0
$951$	−2.81669e7	−	2.52444e6i	−1.00992	−	0.0905137i
$952$		0			0
$953$		−	2.03196e7i		−	0.724742i	−0.932034	−	0.362371i	$-0.881967\pi$
$953$		−	2.03196e7i		−	0.724742i	0.932034	−	0.362371i	$-0.118033\pi$
$954$		0			0
$955$			2.78197e7i			0.987060i
$956$		0			0
$957$	3.81034e6	−	4.25146e7i	0.134488	−	1.50058i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.37704e7			0.480991
$962$		0			0
$963$	−8.15286e6	+	4.51182e7i	−0.283299	+	1.56778i
$964$		0			0
$965$	−3.62895e7			−1.25448
$966$		0			0
$967$	−2.69494e7			−0.926792			−0.463396	−	0.886151i	$-0.653369\pi$
$967$	−2.69494e7			−0.926792			−0.463396	+	0.886151i	$0.653369\pi$
$968$		0			0
$969$	−2.92347e7	−	2.62014e6i	−1.00021	−	0.0896428i
$970$		0			0
$971$	−3.42537e7			−1.16590			−0.582948	−	0.812510i	$-0.698101\pi$
$971$	−3.42537e7			−1.16590			−0.582948	+	0.812510i	$0.698101\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−7.81483e6	−	700399.i	−0.263274	−	0.0235958i
$976$		0			0
$977$		−	2.50876e7i		−	0.840858i	−0.907325	−	0.420429i	$-0.861880\pi$
$977$		−	2.50876e7i		−	0.840858i	0.907325	−	0.420429i	$-0.138120\pi$
$978$		0			0
$979$			4.74171e7i			1.58117i
$980$		0			0
$981$	−3.78525e7	−	6.83995e6i	−1.25580	−	0.226924i
$982$		0			0
$983$	−2.36116e7			−0.779367			−0.389684	−	0.920949i	$-0.627416\pi$
$983$	−2.36116e7			−0.779367			−0.389684	+	0.920949i	$0.627416\pi$
$984$		0			0
$985$		−	4.05596e7i		−	1.33200i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	9.10105e7i		−	2.95870i
$990$		0			0
$991$	−2.17500e7			−0.703519			−0.351760	−	0.936090i	$-0.614417\pi$
$991$	−2.17500e7			−0.703519			−0.351760	+	0.936090i	$0.614417\pi$
$992$		0			0
$993$	3.55409e6	−	3.96554e7i	0.114381	−	1.27623i
$994$		0			0
$995$			5.90021e7i			1.88934i
$996$		0			0
$997$			4.35621e7i			1.38794i	0.720004	+	0.693970i	$0.244140\pi$
$997$			4.35621e7i			1.38794i	−0.720004	+	0.693970i	$0.755860\pi$
$998$		0			0
$999$	−3.63153e6	+	1.32164e7i	−0.115127	+	0.418986i

Display

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.6.f.c.293.20	yes	40
3.2	odd	2	inner	588.6.f.c.293.22	yes	40
7.6	odd	2	inner	588.6.f.c.293.21	yes	40
21.20	even	2	inner	588.6.f.c.293.19	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.6.f.c.293.19	✓	40	21.20	even	2	inner
588.6.f.c.293.20	yes	40	1.1	even	1	trivial
588.6.f.c.293.21	yes	40	7.6	odd	2	inner
588.6.f.c.293.22	yes	40	3.2	odd	2	inner

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	588.6.f.c.293.20

Level	$588$
Weight	$6$
Character	588.293
Analytic conductor	$94.306$
Analytic rank	$0$
Dimension	$40$
Inner twists	$4$