LMFDB - Embedded newform 252.6.k.d.37.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,6,Mod(37,252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("252.37"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$252 = 2^{2} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	252.k (of order $3$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$40.4167225929$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{109})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + 28x^{2} + 27x + 729$ x^4 - x^3 + 28x^2 + 27x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$2^{2}\cdot 3$
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.2
Root			$2.86008 - 4.95380i$ of defining polynomial
Character	$\chi$	$=$	252.37
Dual form			252.6.k.d.109.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(41.8209 - 72.4360i) q^{5} +(28.0000 + 126.582i) q^{7} +(274.623 + 475.661i) q^{11} -823.135 q^{13} +(72.7837 + 126.065i) q^{17} +(-884.019 + 1531.17i) q^{19} +(-761.363 + 1318.72i) q^{23} +(-1935.48 - 3352.35i) q^{25} +741.943 q^{29} +(1602.09 + 2774.90i) q^{31} +(10340.1 + 3265.57i) q^{35} +(1774.77 - 3074.00i) q^{37} +6461.29 q^{41} +6716.00 q^{43} +(-9588.40 + 16607.6i) q^{47} +(-15239.0 + 7088.59i) q^{49} +(-10639.3 - 18427.8i) q^{53} +45940.0 q^{55} +(18264.3 + 31634.8i) q^{59} +(-21716.9 + 37614.7i) q^{61} +(-34424.3 + 59624.6i) q^{65} +(-2514.07 - 4354.50i) q^{67} +6311.32 q^{71} +(21299.0 + 36890.9i) q^{73} +(-52520.7 + 48080.9i) q^{77} +(-47534.6 + 82332.3i) q^{79} +34978.2 q^{83} +12175.5 q^{85} +(50276.7 - 87081.8i) q^{89} +(-23047.8 - 104194. i) q^{91} +(73940.9 + 128069. i) q^{95} +76794.5 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 42 q^{5} + 112 q^{7} + 660 q^{11} - 1288 q^{13} - 210 q^{17} - 3724 q^{19} + 24 q^{23} - 2480 q^{25} - 11064 q^{29} - 2800 q^{31} + 28644 q^{35} + 13238 q^{37} - 8232 q^{41} + 26864 q^{43} + 8064 q^{47}+ \cdots + 16520 q^{97}+O(q^{100})$ 4 * q + 42 * q^5 + 112 * q^7 + 660 * q^11 - 1288 * q^13 - 210 * q^17 - 3724 * q^19 + 24 * q^23 - 2480 * q^25 - 11064 * q^29 - 2800 * q^31 + 28644 * q^35 + 13238 * q^37 - 8232 * q^41 + 26864 * q^43 + 8064 * q^47 - 60956 * q^49 - 53958 * q^53 + 82656 * q^55 + 36036 * q^59 - 83986 * q^61 - 76308 * q^65 + 2660 * q^67 - 143136 * q^71 - 31318 * q^73 - 77658 * q^77 - 51136 * q^79 - 12432 * q^83 + 53964 * q^85 + 166278 * q^89 - 36064 * q^91 + 66432 * q^95 + 16520 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$ .

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	41.8209	−	72.4360i	0.748115	−	1.29577i	−0.200610	−	0.979671i	$-0.564292\pi$
$5$	41.8209	−	72.4360i	0.748115	−	1.29577i	0.948725	−	0.316103i	$-0.102374\pi$
$6$		0			0
$7$	28.0000	+	126.582i	0.215980	+	0.976398i
$7$	28.0000	+	126.582i	0.215980	+	0.976398i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	274.623	+	475.661i	0.684314	+	1.18527i	0.973652	+	0.228040i	$0.0732316\pi$
$11$	274.623	+	475.661i	0.684314	+	1.18527i	−0.289338	+	0.957227i	$0.593435\pi$
$12$		0			0
$13$	−823.135			−1.35087			−0.675433	−	0.737421i	$-0.736043\pi$
$13$	−823.135			−1.35087			−0.675433	+	0.737421i	$0.736043\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	72.7837	+	126.065i	0.0610818	+	0.105797i	0.894949	−	0.446168i	$-0.147212\pi$
$17$	72.7837	+	126.065i	0.0610818	+	0.105797i	−0.833867	+	0.551965i	$0.813878\pi$
$18$		0			0
$19$	−884.019	+	1531.17i	−0.561794	+	0.973056i	0.435546	+	0.900167i	$0.356555\pi$
$19$	−884.019	+	1531.17i	−0.561794	+	0.973056i	−0.997340	+	0.0728898i	$0.976778\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−761.363	+	1318.72i	−0.300104	+	0.519796i	−0.976159	−	0.217055i	$-0.930355\pi$
$23$	−761.363	+	1318.72i	−0.300104	+	0.519796i	0.676055	+	0.736851i	$0.263688\pi$
$24$		0			0
$25$	−1935.48	−	3352.35i	−0.619353	−	1.07275i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	741.943			0.163823			0.0819116	−	0.996640i	$-0.473897\pi$
$29$	741.943			0.163823			0.0819116	+	0.996640i	$0.473897\pi$
$30$		0			0
$31$	1602.09	+	2774.90i	0.299421	+	0.518612i	0.976004	−	0.217755i	$-0.0698732\pi$
$31$	1602.09	+	2774.90i	0.299421	+	0.518612i	−0.676583	+	0.736367i	$0.736540\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10340.1	+	3265.57i	1.42677	+	0.450597i
$36$		0			0
$37$	1774.77	−	3074.00i	0.213127	−	0.369147i	−0.739564	−	0.673086i	$-0.764968\pi$
$37$	1774.77	−	3074.00i	0.213127	−	0.369147i	0.952692	+	0.303939i	$0.0983018\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6461.29			0.600288			0.300144	−	0.953894i	$-0.402965\pi$
$41$	6461.29			0.600288			0.300144	+	0.953894i	$0.402965\pi$
$42$		0			0
$43$	6716.00			0.553910			0.276955	−	0.960883i	$-0.410675\pi$
$43$	6716.00			0.553910			0.276955	+	0.960883i	$0.410675\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9588.40	+	16607.6i	−0.633143	+	1.09664i	0.353763	+	0.935335i	$0.384902\pi$
$47$	−9588.40	+	16607.6i	−0.633143	+	1.09664i	−0.986905	+	0.161300i	$0.948431\pi$
$48$		0			0
$49$	−15239.0	+	7088.59i	−0.906706	+	0.421764i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10639.3	−	18427.8i	−0.520263	−	0.901123i	−0.999722	−	0.0235583i	$-0.992500\pi$
$53$	−10639.3	−	18427.8i	−0.520263	−	0.901123i	0.479459	−	0.877564i	$-0.340833\pi$
$54$		0			0
$55$	45940.0			2.04778
$56$		0			0
$57$		0			0
$58$		0			0
$59$	18264.3	+	31634.8i	0.683083	+	1.18314i	0.974035	+	0.226397i	$0.0726948\pi$
$59$	18264.3	+	31634.8i	0.683083	+	1.18314i	−0.290952	+	0.956738i	$0.593972\pi$
$60$		0			0
$61$	−21716.9	+	37614.7i	−0.747262	+	1.29430i	0.201869	+	0.979413i	$0.435299\pi$
$61$	−21716.9	+	37614.7i	−0.747262	+	1.29430i	−0.949130	+	0.314883i	$0.898035\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−34424.3	+	59624.6i	−1.01060	+	1.75042i
$66$		0			0
$67$	−2514.07	−	4354.50i	−0.0684212	−	0.118509i	0.829785	−	0.558083i	$-0.188463\pi$
$67$	−2514.07	−	4354.50i	−0.0684212	−	0.118509i	−0.898207	+	0.439574i	$0.855130\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6311.32			0.148585			0.0742923	−	0.997237i	$-0.476330\pi$
$71$	6311.32			0.148585			0.0742923	+	0.997237i	$0.476330\pi$
$72$		0			0
$73$	21299.0	+	36890.9i	0.467790	+	0.810236i	0.999323	−	0.0368015i	$-0.0117169\pi$
$73$	21299.0	+	36890.9i	0.467790	+	0.810236i	−0.531532	+	0.847038i	$0.678384\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−52520.7	+	48080.9i	−1.00949	+	0.924156i
$78$		0			0
$79$	−47534.6	+	82332.3i	−0.856923	+	1.48423i	0.0179267	+	0.999839i	$0.494293\pi$
$79$	−47534.6	+	82332.3i	−0.856923	+	1.48423i	−0.874850	+	0.484395i	$0.839040\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	34978.2			0.557318			0.278659	−	0.960390i	$-0.410110\pi$
$83$	34978.2			0.557318			0.278659	+	0.960390i	$0.410110\pi$
$84$		0			0
$85$	12175.5			0.182785
$86$		0			0
$87$		0			0
$88$		0			0
$89$	50276.7	−	87081.8i	0.672809	−	1.16534i	−0.304295	−	0.952578i	$-0.598421\pi$
$89$	50276.7	−	87081.8i	0.672809	−	1.16534i	0.977104	−	0.212762i	$-0.0682458\pi$
$90$		0			0
$91$	−23047.8	−	104194.i	−0.291760	−	1.31898i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	73940.9	+	128069.i	0.840574	+	1.45592i
$96$		0			0
$97$	76794.5			0.828707			0.414353	−	0.910116i	$-0.364008\pi$
$97$	76794.5			0.828707			0.414353	+	0.910116i	$0.364008\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	57020.2	+	98761.9i	0.556193	+	0.963355i	0.997810	+	0.0661508i	$0.0210718\pi$
$101$	57020.2	+	98761.9i	0.556193	+	0.963355i	−0.441617	+	0.897204i	$0.645595\pi$
$102$		0			0
$103$	61818.5	−	107073.i	0.574150	−	0.994457i	−0.421983	−	0.906604i	$-0.638666\pi$
$103$	61818.5	−	107073.i	0.574150	−	0.994457i	0.996133	−	0.0878533i	$-0.0280007\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11670.1	+	20213.2i	−0.0985404	+	0.170677i	−0.911081	−	0.412228i	$-0.864751\pi$
$107$	−11670.1	+	20213.2i	−0.0985404	+	0.170677i	0.812540	+	0.582905i	$0.198084\pi$
$108$		0			0
$109$	13601.6	+	23558.6i	0.109654	+	0.189926i	0.915630	−	0.402022i	$-0.131692\pi$
$109$	13601.6	+	23558.6i	0.109654	+	0.189926i	−0.805976	+	0.591948i	$0.798359\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	157394.			1.15955			0.579777	−	0.814775i	$-0.303140\pi$
$113$	157394.			1.15955			0.579777	+	0.814775i	$0.303140\pi$
$114$		0			0
$115$	63681.8	+	110300.i	0.449025	+	0.777734i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−13919.6	+	12742.9i	−0.0901073	+	0.0824901i
$120$		0			0
$121$	−70310.3	+	121781.i	−0.436572	+	0.756165i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−62393.2			−0.357160
$126$		0			0
$127$	−253263.			−1.39336			−0.696679	−	0.717383i	$-0.745340\pi$
$127$	−253263.			−1.39336			−0.696679	+	0.717383i	$0.745340\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−57772.6	+	100065.i	−0.294133	+	0.509453i	−0.974783	−	0.223156i	$-0.928364\pi$
$131$	−57772.6	+	100065.i	−0.294133	+	0.509453i	0.680650	+	0.732609i	$0.261698\pi$
$132$		0			0
$133$	−218570.	−	69028.2i	−1.07143	−	0.338374i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−103904.	−	179967.i	−0.472966	−	0.819202i	0.526555	−	0.850141i	$-0.323483\pi$
$137$	−103904.	−	179967.i	−0.472966	−	0.819202i	−0.999521	+	0.0309393i	$0.990150\pi$
$138$		0			0
$139$	−28993.1			−0.127279			−0.0636395	−	0.997973i	$-0.520271\pi$
$139$	−28993.1			−0.127279			−0.0636395	+	0.997973i	$0.520271\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−226052.	−	391533.i	−0.924417	−	1.60114i
$144$		0			0
$145$	31028.7	−	53743.4i	0.122559	−	0.212278i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−165490.	+	286637.i	−0.610668	+	1.05771i	0.380460	+	0.924798i	$0.375766\pi$
$149$	−165490.	+	286637.i	−0.610668	+	1.05771i	−0.991128	+	0.132911i	$0.957567\pi$
$150$		0			0
$151$	−54428.3	−	94272.6i	−0.194260	−	0.336468i	0.752398	−	0.658709i	$-0.228897\pi$
$151$	−54428.3	−	94272.6i	−0.194260	−	0.336468i	−0.946658	+	0.322241i	$0.895564\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	268003.			0.896005
$156$		0			0
$157$	248873.	+	431060.i	0.805801	+	1.39569i	0.915749	+	0.401752i	$0.131598\pi$
$157$	248873.	+	431060.i	0.805801	+	1.39569i	−0.109947	+	0.993937i	$0.535068\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−188244.	−	59450.7i	−0.572344	−	0.180756i
$162$		0			0
$163$	179551.	−	310992.i	0.529321	−	0.916811i	−0.470094	−	0.882616i	$-0.655780\pi$
$163$	179551.	−	310992.i	0.529321	−	0.916811i	0.999415	−	0.0341950i	$-0.0108867\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	50645.2			0.140523			0.0702614	−	0.997529i	$-0.477617\pi$
$167$	50645.2			0.140523			0.0702614	+	0.997529i	$0.477617\pi$
$168$		0			0
$169$	306258.			0.824841
$170$		0			0
$171$		0			0
$172$		0			0
$173$	151595.	−	262569.i	0.385095	−	0.667005i	−0.606687	−	0.794941i	$-0.707502\pi$
$173$	151595.	−	262569.i	0.385095	−	0.667005i	0.991782	+	0.127936i	$0.0408352\pi$
$174$		0			0
$175$	370153.	−	338862.i	0.913664	−	0.836428i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−341272.	−	591100.i	−0.796100	−	1.37889i	−0.922138	−	0.386860i	$-0.873560\pi$
$179$	−341272.	−	591100.i	−0.796100	−	1.37889i	0.126039	−	0.992025i	$-0.459774\pi$
$180$		0			0
$181$	182455.			0.413961			0.206980	−	0.978345i	$-0.433636\pi$
$181$	182455.			0.413961			0.206980	+	0.978345i	$0.433636\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−148445.	−	257115.i	−0.318888	−	0.552329i
$186$		0			0
$187$	−39976.2	+	69240.8i	−0.0835983	+	0.144796i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	341772.	−	591966.i	0.677879	−	1.17412i	−0.297739	−	0.954647i	$-0.596232\pi$
$191$	341772.	−	591966.i	0.677879	−	1.17412i	0.975618	−	0.219474i	$-0.0704342\pi$
$192$		0			0
$193$	−473251.	−	819696.i	−0.914532	−	1.58402i	−0.807586	−	0.589750i	$-0.799226\pi$
$193$	−473251.	−	819696.i	−0.914532	−	1.58402i	−0.106946	−	0.994265i	$-0.534107\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	184209.			0.338178			0.169089	−	0.985601i	$-0.445918\pi$
$197$	184209.			0.338178			0.169089	+	0.985601i	$0.445918\pi$
$198$		0			0
$199$	−34103.3	−	59068.6i	−0.0610469	−	0.105736i	0.833887	−	0.551936i	$-0.186111\pi$
$199$	−34103.3	−	59068.6i	−0.0610469	−	0.105736i	−0.894934	+	0.446199i	$0.852777\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	20774.4	+	93916.6i	0.0353825	+	0.159957i
$204$		0			0
$205$	270217.	−	468030.i	0.449085	−	0.777837i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−971088.			−1.53778
$210$		0			0
$211$	−1.19270e6			−1.84427			−0.922134	−	0.386871i	$-0.873556\pi$
$211$	−1.19270e6			−1.84427			−0.922134	+	0.386871i	$0.873556\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	280869.	−	486480.i	0.414389	−	0.717743i
$216$		0			0
$217$	−306394.	+	280493.i	−0.441703	+	0.404364i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−59910.8	−	103769.i	−0.0825134	−	0.142917i
$222$		0			0
$223$	200502.			0.269995			0.134998	−	0.990846i	$-0.456897\pi$
$223$	200502.			0.269995			0.134998	+	0.990846i	$0.456897\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	145384.	+	251813.i	0.187264	+	0.324350i	0.944337	−	0.328980i	$-0.106705\pi$
$227$	145384.	+	251813.i	0.187264	+	0.324350i	−0.757073	+	0.653330i	$0.773371\pi$
$228$		0			0
$229$	−472471.	+	818344.i	−0.595369	+	1.03121i	0.398125	+	0.917331i	$0.369661\pi$
$229$	−472471.	+	818344.i	−0.595369	+	1.03121i	−0.993495	+	0.113879i	$0.963672\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	108192.	−	187394.i	0.130558	−	0.226134i	−0.793334	−	0.608787i	$-0.791656\pi$
$233$	108192.	−	187394.i	0.130558	−	0.226134i	0.923892	+	0.382653i	$0.124990\pi$
$234$		0			0
$235$	801991.	+	1.38909e6i	0.947327	+	1.64082i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	668807.			0.757366			0.378683	−	0.925526i	$-0.376377\pi$
$239$	668807.			0.757366			0.378683	+	0.925526i	$0.376377\pi$
$240$		0			0
$241$	−578006.	−	1.00114e6i	−0.641047	−	1.11033i	−0.985199	−	0.171413i	$-0.945167\pi$
$241$	−578006.	−	1.00114e6i	−0.641047	−	1.11033i	0.344152	−	0.938914i	$-0.388166\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−123840.	+	1.40030e6i	−0.131809	+	1.49041i
$246$		0			0
$247$	727666.	−	1.26036e6i	0.758909	−	1.31447i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	77574.9			0.0777207			0.0388604	−	0.999245i	$-0.487627\pi$
$251$	77574.9			0.0777207			0.0388604	+	0.999245i	$0.487627\pi$
$252$		0			0
$253$	−836351.			−0.821462
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−140101.	+	242662.i	−0.132315	+	0.229176i	−0.924569	−	0.381016i	$-0.875574\pi$
$257$	−140101.	+	242662.i	−0.132315	+	0.229176i	0.792254	+	0.610192i	$0.208908\pi$
$258$		0			0
$259$	438807.	+	138583.i	0.406466	+	0.128369i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−134983.	−	233797.i	−0.120334	−	0.208425i	0.799565	−	0.600579i	$-0.205063\pi$
$263$	−134983.	−	233797.i	−0.120334	−	0.208425i	−0.919900	+	0.392154i	$0.871730\pi$
$264$		0			0
$265$	−1.77978e6			−1.55687
$266$		0			0
$267$		0			0
$268$		0			0
$269$	445133.	+	770994.i	0.375068	+	0.649636i	0.990337	−	0.138681i	$-0.0442862\pi$
$269$	445133.	+	770994.i	0.375068	+	0.649636i	−0.615270	+	0.788317i	$0.710953\pi$
$270$		0			0
$271$	−192140.	+	332796.i	−0.158926	+	0.275267i	−0.934482	−	0.356012i	$-0.884136\pi$
$271$	−192140.	+	332796.i	−0.158926	+	0.275267i	0.775556	+	0.631279i	$0.217470\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.06305e6	−	1.84126e6i	0.847664	−	1.46820i
$276$		0			0
$277$	545967.	+	945642.i	0.427530	+	0.740504i	0.996653	−	0.0817484i	$-0.0260504\pi$
$277$	545967.	+	945642.i	0.427530	+	0.740504i	−0.569123	+	0.822253i	$0.692717\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−771640.			−0.582974			−0.291487	−	0.956575i	$-0.594150\pi$
$281$	−771640.			−0.582974			−0.291487	+	0.956575i	$0.594150\pi$
$282$		0			0
$283$	270902.	+	469217.i	0.201070	+	0.348263i	0.948873	−	0.315657i	$-0.102225\pi$
$283$	270902.	+	469217.i	0.201070	+	0.348263i	−0.747804	+	0.663920i	$0.768892\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	180916.	+	817883.i	0.129650	+	0.586120i
$288$		0			0
$289$	699334.	−	1.21128e6i	0.492538	−	0.853101i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.69804e6			−1.15552			−0.577762	−	0.816205i	$-0.696074\pi$
$293$	−1.69804e6			−1.15552			−0.577762	+	0.816205i	$0.696074\pi$
$294$		0			0
$295$	3.05532e6			2.04410
$296$		0			0
$297$		0			0
$298$		0			0
$299$	626704.	−	1.08548e6i	0.405401	−	0.702175i
$300$		0			0
$301$	188048.	+	850125.i	0.119633	+	0.540837i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.81644e6	+	3.14617e6i	1.11808	+	1.93656i
$306$		0			0
$307$	−288376.			−0.174627			−0.0873137	−	0.996181i	$-0.527828\pi$
$307$	−288376.			−0.174627			−0.0873137	+	0.996181i	$0.527828\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	173801.	+	301032.i	0.101894	+	0.176486i	0.912465	−	0.409154i	$-0.134176\pi$
$311$	173801.	+	301032.i	0.101894	+	0.176486i	−0.810571	+	0.585641i	$0.800843\pi$
$312$		0			0
$313$	385201.	−	667188.i	0.222242	−	0.384935i	−0.733246	−	0.679963i	$-0.761996\pi$
$313$	385201.	−	667188.i	0.222242	−	0.384935i	0.955489	+	0.295028i	$0.0953290\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.00876e6	+	1.74722e6i	−0.563819	+	0.976564i	0.433339	+	0.901231i	$0.357335\pi$
$317$	−1.00876e6	+	1.74722e6i	−0.563819	+	0.976564i	−0.997158	+	0.0753328i	$0.975998\pi$
$318$		0			0
$319$	203755.	+	352914.i	0.112107	+	0.194174i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−257369.			−0.137262
$324$		0			0
$325$	1.59316e6	+	2.75943e6i	0.836664	+	1.44914i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.37070e6	−	748706.i	−1.20750	−	0.381348i
$330$		0			0
$331$	−212966.	+	368868.i	−0.106842	+	0.185055i	−0.914489	−	0.404610i	$-0.867407\pi$
$331$	−212966.	+	368868.i	−0.106842	+	0.185055i	0.807648	+	0.589666i	$0.200740\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−420563.			−0.204748
$336$		0			0
$337$	1.15035e6			0.551767			0.275884	−	0.961191i	$-0.411030\pi$
$337$	1.15035e6			0.551767			0.275884	+	0.961191i	$0.411030\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−879941.	+	1.52410e6i	−0.409796	+	0.709787i
$342$		0			0
$343$	−1.32398e6	−	1.73050e6i	−0.607640	−	0.794213i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−156568.	−	271184.i	−0.0698039	−	0.120904i	0.829011	−	0.559232i	$-0.188904\pi$
$347$	−156568.	−	271184.i	−0.0698039	−	0.120904i	−0.898815	+	0.438328i	$0.855571\pi$
$348$		0			0
$349$	−3.01459e6			−1.32484			−0.662422	−	0.749131i	$-0.730471\pi$
$349$	−3.01459e6			−1.32484			−0.662422	+	0.749131i	$0.730471\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.19230e6	+	3.79717e6i	0.936404	+	1.62190i	0.772111	+	0.635487i	$0.219201\pi$
$353$	2.19230e6	+	3.79717e6i	0.936404	+	1.62190i	0.164292	+	0.986412i	$0.447466\pi$
$354$		0			0
$355$	263945.	−	457166.i	0.111158	−	0.192532i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−96327.8	+	166845.i	−0.0394471	+	0.0683244i	−0.885075	−	0.465449i	$-0.845893\pi$
$359$	−96327.8	+	166845.i	−0.0394471	+	0.0683244i	0.845628	+	0.533773i	$0.179226\pi$
$360$		0			0
$361$	−324928.	−	562792.i	−0.131226	−	0.227290i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.56297e6			1.39984
$366$		0			0
$367$	1.16991e6	+	2.02634e6i	0.453405	+	0.785320i	0.998595	−	0.0529924i	$-0.0168759\pi$
$367$	1.16991e6	+	2.02634e6i	0.453405	+	0.785320i	−0.545190	+	0.838312i	$0.683543\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.03473e6	−	1.86272e6i	0.767488	−	0.702608i
$372$		0			0
$373$	−1.96217e6	+	3.39857e6i	−0.730237	+	1.26481i	0.226545	+	0.974001i	$0.427257\pi$
$373$	−1.96217e6	+	3.39857e6i	−0.730237	+	1.26481i	−0.956782	+	0.290806i	$0.906077\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−610719.			−0.221303
$378$		0			0
$379$	−3.26556e6			−1.16778			−0.583889	−	0.811834i	$-0.698470\pi$
$379$	−3.26556e6			−1.16778			−0.583889	+	0.811834i	$0.698470\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.21488e6	+	3.83628e6i	−0.771529	+	1.33633i	0.165195	+	0.986261i	$0.447175\pi$
$383$	−2.21488e6	+	3.83628e6i	−0.771529	+	1.33633i	−0.936725	+	0.350067i	$0.886159\pi$
$384$		0			0
$385$	1.28632e6	+	5.81517e6i	0.442280	+	1.99945i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.55733e6	−	4.42943e6i	−0.856867	−	1.48414i	−0.874902	−	0.484300i	$-0.839074\pi$
$389$	−2.55733e6	−	4.42943e6i	−0.856867	−	1.48414i	0.0180345	−	0.999837i	$-0.494259\pi$
$390$		0			0
$391$	−221659.			−0.0733236
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.97588e6	+	6.88642e6i	1.28215	+	2.22076i
$396$		0			0
$397$	770944.	−	1.33531e6i	0.245497	−	0.425214i	−0.716774	−	0.697306i	$-0.754382\pi$
$397$	770944.	−	1.33531e6i	0.245497	−	0.425214i	0.962271	+	0.272092i	$0.0877154\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.72114e6	−	4.71315e6i	0.845064	−	1.46369i	−0.0405025	−	0.999179i	$-0.512896\pi$
$401$	2.72114e6	−	4.71315e6i	0.845064	−	1.46369i	0.885566	−	0.464514i	$-0.153771\pi$
$402$		0			0
$403$	−1.31873e6	−	2.28411e6i	−0.404478	−	0.700576i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.94958e6			0.583384
$408$		0			0
$409$	−2.59241e6	−	4.49019e6i	−0.766294	−	1.32726i	−0.939560	−	0.342385i	$-0.888765\pi$
$409$	−2.59241e6	−	4.49019e6i	−0.766294	−	1.32726i	0.173265	−	0.984875i	$-0.444568\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.49299e6	+	3.19771e6i	−1.00768	+	0.922494i
$414$		0			0
$415$	1.46282e6	−	2.53368e6i	0.416938	−	0.722157i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.17093e6			0.604102			0.302051	−	0.953292i	$-0.402329\pi$
$419$	2.17093e6			0.604102			0.302051	+	0.953292i	$0.402329\pi$
$420$		0			0
$421$	6.61967e6			1.82025			0.910125	−	0.414334i	$-0.135986\pi$
$421$	6.61967e6			1.82025			0.910125	+	0.414334i	$0.135986\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	281743.	−	487992.i	0.0756624	−	0.131051i
$426$		0			0
$427$	−5.36942e6	−	1.69575e6i	−1.42514	−	0.450083i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.44992e6	−	2.51134e6i	−0.375968	−	0.651196i	0.614503	−	0.788914i	$-0.289357\pi$
$431$	−1.44992e6	−	2.51134e6i	−0.375968	−	0.651196i	−0.990471	+	0.137718i	$0.956023\pi$
$432$		0			0
$433$	4.71803e6			1.20932			0.604659	−	0.796484i	$-0.293309\pi$
$433$	4.71803e6			1.20932			0.604659	+	0.796484i	$0.293309\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.34612e6	−	2.33154e6i	−0.337194	−	0.584036i
$438$		0			0
$439$	1.73850e6	−	3.01117e6i	0.430540	−	0.745718i	−0.566379	−	0.824145i	$-0.691656\pi$
$439$	1.73850e6	−	3.01117e6i	0.430540	−	0.745718i	0.996920	+	0.0784267i	$0.0249896\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.64829e6	−	6.31902e6i	0.883243	−	1.52982i	0.0355278	−	0.999369i	$-0.488689\pi$
$443$	3.64829e6	−	6.31902e6i	0.883243	−	1.52982i	0.847715	−	0.530452i	$-0.177978\pi$
$444$		0			0
$445$	−4.20524e6	−	7.28368e6i	−1.00668	−	1.74362i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3.65867e6			−0.856462			−0.428231	−	0.903669i	$-0.640863\pi$
$449$	−3.65867e6			−0.856462			−0.428231	+	0.903669i	$0.640863\pi$
$450$		0			0
$451$	1.77442e6	+	3.07339e6i	0.410786	+	0.711501i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.51127e6	−	2.68800e6i	−1.92737	−	0.608697i
$456$		0			0
$457$	−3.22031e6	+	5.57774e6i	−0.721286	+	1.24930i	0.239199	+	0.970971i	$0.423115\pi$
$457$	−3.22031e6	+	5.57774e6i	−0.721286	+	1.24930i	−0.960485	+	0.278333i	$0.910218\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.47898e6			−0.981583			−0.490791	−	0.871277i	$-0.663292\pi$
$461$	−4.47898e6			−0.981583			−0.490791	+	0.871277i	$0.663292\pi$
$462$		0			0
$463$	−2.11214e6			−0.457899			−0.228949	−	0.973438i	$-0.573529\pi$
$463$	−2.11214e6			−0.457899			−0.228949	+	0.973438i	$0.573529\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.92919e6	−	6.80556e6i	0.833702	−	1.44401i	−0.0613806	−	0.998114i	$-0.519550\pi$
$467$	3.92919e6	−	6.80556e6i	0.833702	−	1.44401i	0.895083	−	0.445900i	$-0.147116\pi$
$468$		0			0
$469$	480808.	−	440162.i	0.100934	−	0.0924019i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.84437e6	+	3.19454e6i	0.379049	+	0.656532i
$474$		0			0
$475$	6.84400e6			1.39180
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.24373e6	+	7.35035e6i	0.845101	+	1.46376i	0.885533	+	0.464576i	$0.153793\pi$
$479$	4.24373e6	+	7.35035e6i	0.845101	+	1.46376i	−0.0404324	+	0.999182i	$0.512874\pi$
$480$		0			0
$481$	−1.46088e6	+	2.53032e6i	−0.287907	+	0.498669i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.21162e6	−	5.56269e6i	0.619968	−	1.07382i
$486$		0			0
$487$	−1.37808e6	−	2.38691e6i	−0.263301	−	0.456051i	0.703816	−	0.710382i	$-0.251478\pi$
$487$	−1.37808e6	−	2.38691e6i	−0.263301	−	0.456051i	−0.967117	+	0.254331i	$0.918145\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.45533e6			−0.459628			−0.229814	−	0.973235i	$-0.573812\pi$
$491$	−2.45533e6			−0.459628			−0.229814	+	0.973235i	$0.573812\pi$
$492$		0			0
$493$	54001.3	+	93533.1i	0.0100066	+	0.0173320i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	176717.	+	798899.i	0.0320913	+	0.145078i
$498$		0			0
$499$	2.22925e6	−	3.86117e6i	0.400781	−	0.694172i	−0.593040	−	0.805173i	$-0.702072\pi$
$499$	2.22925e6	−	3.86117e6i	0.400781	−	0.694172i	0.993820	+	0.111001i	$0.0354056\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.64080e6			−0.465389			−0.232694	−	0.972550i	$-0.574754\pi$
$503$	−2.64080e6			−0.465389			−0.232694	+	0.972550i	$0.574754\pi$
$504$		0			0
$505$	9.53856e6			1.66439
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.35520e6	−	4.07933e6i	0.402934	−	0.697902i	−0.591145	−	0.806566i	$-0.701324\pi$
$509$	2.35520e6	−	4.07933e6i	0.402934	−	0.697902i	0.994079	+	0.108663i	$0.0346571\pi$
$510$		0			0
$511$	−4.07335e6	+	3.72901e6i	−0.690080	+	0.631744i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.17061e6	−	8.95576e6i	−0.859061	−	1.48794i
$516$		0			0
$517$	−1.05328e7			−1.73307
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.40134e6	−	2.42720e6i	−0.226178	−	0.391752i	0.730494	−	0.682919i	$-0.239290\pi$
$521$	−1.40134e6	−	2.42720e6i	−0.226178	−	0.391752i	−0.956672	+	0.291167i	$0.905956\pi$
$522$		0			0
$523$	1.49878e6	−	2.59596e6i	0.239598	−	0.414995i	−0.721001	−	0.692934i	$-0.756318\pi$
$523$	1.49878e6	−	2.59596e6i	0.239598	−	0.414995i	0.960599	+	0.277939i	$0.0896512\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−233212.	+	403934.i	−0.0365783	+	0.0633555i
$528$		0			0
$529$	2.05883e6	+	3.56599e6i	0.319875	+	0.554040i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5.31851e6			−0.810909
$534$		0			0
$535$	976107.	+	1.69067e6i	0.147439	+	0.255372i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7.55675e6	−	5.30191e6i	−1.12037	−	0.786069i
$540$		0			0
$541$	−1.65252e6	+	2.86225e6i	−0.242747	+	0.420450i	−0.961496	−	0.274820i	$-0.911382\pi$
$541$	−1.65252e6	+	2.86225e6i	−0.242747	+	0.420450i	0.718749	+	0.695270i	$0.244715\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2.27532e6			0.328135
$546$		0			0
$547$	3.51435e6			0.502201			0.251100	−	0.967961i	$-0.419208\pi$
$547$	3.51435e6			0.502201			0.251100	+	0.967961i	$0.419208\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−655891.	+	1.13604e6i	−0.0920350	+	0.159409i
$552$		0			0
$553$	−1.17528e7	−	3.71172e6i	−1.63428	−	0.516133i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.99093e6	−	5.18045e6i	−0.408478	−	0.707505i	0.586241	−	0.810137i	$-0.300607\pi$
$557$	−2.99093e6	−	5.18045e6i	−0.408478	−	0.707505i	−0.994719	+	0.102632i	$0.967274\pi$
$558$		0			0
$559$	−5.52817e6			−0.748259
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.36921e6	+	9.29974e6i	0.713903	+	1.23652i	0.963381	+	0.268136i	$0.0864075\pi$
$563$	5.36921e6	+	9.29974e6i	0.713903	+	1.23652i	−0.249478	+	0.968380i	$0.580259\pi$
$564$		0			0
$565$	6.58235e6	−	1.14010e7i	0.867481	−	1.50252i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.29774e6	+	5.71186e6i	−0.427008	+	0.739600i	−0.996606	−	0.0823241i	$-0.973766\pi$
$569$	−3.29774e6	+	5.71186e6i	−0.427008	+	0.739600i	0.569598	+	0.821924i	$0.307099\pi$
$570$		0			0
$571$	1.30789e6	+	2.26533e6i	0.167873	+	0.290764i	0.937672	−	0.347522i	$-0.112977\pi$
$571$	1.30789e6	+	2.26533e6i	0.167873	+	0.290764i	−0.769799	+	0.638286i	$0.779644\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.89440e6			0.743482
$576$		0			0
$577$	619490.	+	1.07299e6i	0.0774630	+	0.134170i	0.902155	−	0.431413i	$-0.141985\pi$
$577$	619490.	+	1.07299e6i	0.0774630	+	0.134170i	−0.824692	+	0.565583i	$0.808651\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	979391.	+	4.42761e6i	0.120369	+	0.544164i
$582$		0			0
$583$	5.84360e6	−	1.01214e7i	0.712047	−	1.23330i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.02896e7			−1.23254			−0.616270	−	0.787535i	$-0.711357\pi$
$587$	−1.02896e7			−1.23254			−0.616270	+	0.787535i	$0.711357\pi$
$588$		0			0
$589$	−5.66510e6			−0.672852
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7.65889e6	−	1.32656e7i	0.894394	−	1.54914i	0.0598415	−	0.998208i	$-0.480940\pi$
$593$	7.65889e6	−	1.32656e7i	0.894394	−	1.54914i	0.834553	−	0.550928i	$-0.185726\pi$
$594$		0			0
$595$	340915.	+	1.54120e6i	0.0394778	+	0.178471i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.34748e6	+	2.33391e6i	0.153446	+	0.265776i	0.932492	−	0.361190i	$-0.117630\pi$
$599$	1.34748e6	+	2.33391e6i	0.153446	+	0.265776i	−0.779046	+	0.626967i	$0.784296\pi$
$600$		0			0
$601$	1.54704e6			0.174709			0.0873547	−	0.996177i	$-0.472159\pi$
$601$	1.54704e6			0.174709			0.0873547	+	0.996177i	$0.472159\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.88088e6	+	1.01860e7i	0.653212	+	1.13140i
$606$		0			0
$607$	−1.89569e6	+	3.28343e6i	−0.208831	+	0.361706i	−0.951347	−	0.308123i	$-0.900299\pi$
$607$	−1.89569e6	+	3.28343e6i	−0.208831	+	0.361706i	0.742516	+	0.669829i	$0.233633\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.89255e6	−	1.36703e7i	0.855291	−	1.48141i
$612$		0			0
$613$	−3.09603e6	−	5.36249e6i	−0.332778	−	0.576388i	0.650277	−	0.759697i	$-0.274653\pi$
$613$	−3.09603e6	−	5.36249e6i	−0.332778	−	0.576388i	−0.983055	+	0.183308i	$0.941319\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.59349e7			1.68514			0.842571	−	0.538585i	$-0.181041\pi$
$617$	1.59349e7			1.68514			0.842571	+	0.538585i	$0.181041\pi$
$618$		0			0
$619$	5.78660e6	+	1.00227e7i	0.607011	+	1.05137i	0.991730	+	0.128341i	$0.0409651\pi$
$619$	5.78660e6	+	1.00227e7i	0.607011	+	1.05137i	−0.384719	+	0.923034i	$0.625702\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.24307e7	+	3.92584e6i	1.28315	+	0.405240i
$624$		0			0
$625$	3.43903e6	−	5.95657e6i	0.352156	−	0.609953i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	516699.			0.0520728
$630$		0			0
$631$	900933.			0.0900781			0.0450390	−	0.998985i	$-0.485659\pi$
$631$	900933.			0.0900781			0.0450390	+	0.998985i	$0.485659\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.05917e7	+	1.83454e7i	−1.04239	+	1.80548i
$636$		0			0
$637$	1.25437e7	−	5.83487e6i	1.22484	−	0.569747i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−424385.	−	735057.i	−0.0407958	−	0.0706604i	0.844907	−	0.534914i	$-0.179656\pi$
$641$	−424385.	−	735057.i	−0.0407958	−	0.0706604i	−0.885702	+	0.464254i	$0.846323\pi$
$642$		0			0
$643$	−1.37977e7			−1.31607			−0.658036	−	0.752986i	$-0.728613\pi$
$643$	−1.37977e7			−1.31607			−0.658036	+	0.752986i	$0.728613\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.82675e6	−	3.16402e6i	−0.171561	−	0.297152i	0.767405	−	0.641163i	$-0.221548\pi$
$647$	−1.82675e6	−	3.16402e6i	−0.171561	−	0.297152i	−0.938966	+	0.344011i	$0.888214\pi$
$648$		0			0
$649$	−1.00316e7	+	1.73753e7i	−0.934887	+	1.61927i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.89558e6	−	1.02115e7i	0.541058	−	0.937140i	−0.457786	−	0.889063i	$-0.651357\pi$
$653$	5.89558e6	−	1.02115e7i	0.541058	−	0.937140i	0.998844	−	0.0480774i	$-0.0153094\pi$
$654$		0			0
$655$	4.83221e6	+	8.36963e6i	0.440091	+	0.762260i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.15208e7			1.03340			0.516700	−	0.856166i	$-0.327160\pi$
$659$	1.15208e7			1.03340			0.516700	+	0.856166i	$0.327160\pi$
$660$		0			0
$661$	−2.68607e6	−	4.65241e6i	−0.239119	−	0.414166i	0.721343	−	0.692578i	$-0.243525\pi$
$661$	−2.68607e6	−	4.65241e6i	−0.239119	−	0.414166i	−0.960462	+	0.278412i	$0.910192\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.41409e7	+	1.29455e7i	−1.24001	+	1.13518i
$666$		0			0
$667$	−564888.	+	978414.i	−0.0491640	+	0.0851546i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.38558e7			−2.04545
$672$		0			0
$673$	1.48042e7			1.25993			0.629966	−	0.776623i	$-0.283069\pi$
$673$	1.48042e7			1.25993			0.629966	+	0.776623i	$0.283069\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.95353e6	+	5.11567e6i	−0.247668	+	0.428974i	−0.962878	−	0.269935i	$-0.912998\pi$
$677$	−2.95353e6	+	5.11567e6i	−0.247668	+	0.428974i	0.715210	+	0.698909i	$0.246331\pi$
$678$		0			0
$679$	2.15025e6	+	9.72080e6i	0.178984	+	0.809148i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.10080e7	+	1.90663e7i	0.902932	+	1.56392i	0.823670	+	0.567070i	$0.191923\pi$
$683$	1.10080e7	+	1.90663e7i	0.902932	+	1.56392i	0.0792618	+	0.996854i	$0.474744\pi$
$684$		0			0
$685$	−1.73814e7			−1.41533
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.75757e6	+	1.51686e7i	0.702806	+	1.21730i
$690$		0			0
$691$	−6.53254e6	+	1.13147e7i	−0.520460	+	0.901463i	0.479257	+	0.877674i	$0.340906\pi$
$691$	−6.53254e6	+	1.13147e7i	−0.520460	+	0.901463i	−0.999717	+	0.0237881i	$0.992427\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.21252e6	+	2.10014e6i	−0.0952194	+	0.164925i
$696$		0			0
$697$	470276.	+	814543.i	0.0366667	+	0.0635085i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.20043e7			−0.922658			−0.461329	−	0.887229i	$-0.652627\pi$
$701$	−1.20043e7			−0.922658			−0.461329	+	0.887229i	$0.652627\pi$
$702$		0			0
$703$	3.13787e6	+	5.43495e6i	0.239467	+	0.414770i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.09049e7	+	9.98307e6i	−0.820491	+	0.751131i
$708$		0			0
$709$	8.23972e6	−	1.42716e7i	0.615597	−	1.06625i	−0.374682	−	0.927153i	$-0.622248\pi$
$709$	8.23972e6	−	1.42716e7i	0.615597	−	1.06625i	0.990279	−	0.139093i	$-0.0444185\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.87908e6			−0.359430
$714$		0			0
$715$	−3.78148e7			−2.76628
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.66177e6	+	6.34237e6i	−0.264161	+	0.457540i	−0.967343	−	0.253469i	$-0.918428\pi$
$719$	−3.66177e6	+	6.34237e6i	−0.264161	+	0.457540i	0.703183	+	0.711009i	$0.251762\pi$
$720$		0			0
$721$	1.52844e7	+	4.82707e6i	1.09499	+	0.345816i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.43601e6	−	2.48725e6i	−0.101464	−	0.175742i
$726$		0			0
$727$	1.83777e7			1.28960			0.644798	−	0.764353i	$-0.276941\pi$
$727$	1.83777e7			1.28960			0.644798	+	0.764353i	$0.276941\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	488815.	+	846653.i	0.0338338	+	0.0586019i
$732$		0			0
$733$	5.92488e6	−	1.02622e7i	0.407305	−	0.705472i	−0.587282	−	0.809382i	$-0.699802\pi$
$733$	5.92488e6	−	1.02622e7i	0.407305	−	0.705472i	0.994587	+	0.103910i	$0.0331354\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.38085e6	−	2.39170e6i	0.0936432	−	0.162195i
$738$		0			0
$739$	1.04756e7	+	1.81442e7i	0.705613	+	1.22216i	0.966470	+	0.256780i	$0.0826616\pi$
$739$	1.04756e7	+	1.81442e7i	0.705613	+	1.22216i	−0.260857	+	0.965377i	$0.584005\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.42524e7			0.947141			0.473570	−	0.880756i	$-0.342965\pi$
$743$	1.42524e7			0.947141			0.473570	+	0.880756i	$0.342965\pi$
$744$		0			0
$745$	1.38419e7	+	2.39748e7i	0.913701	+	1.58258i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.88539e6	−	911253.i	−0.187931	−	0.0593519i
$750$		0			0
$751$	−659846.	+	1.14289e6i	−0.0426916	+	0.0739441i	−0.886582	−	0.462572i	$-0.846927\pi$
$751$	−659846.	+	1.14289e6i	−0.0426916	+	0.0739441i	0.843890	+	0.536516i	$0.180260\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9.10497e6			−0.581315
$756$		0			0
$757$	102138.			0.00647812			0.00323906	−	0.999995i	$-0.498969\pi$
$757$	102138.			0.00647812			0.00323906	+	0.999995i	$0.498969\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.09596e7	−	1.89826e7i	0.686016	−	1.18821i	−0.287100	−	0.957901i	$-0.592691\pi$
$761$	1.09596e7	−	1.89826e7i	0.686016	−	1.18821i	0.973116	−	0.230314i	$-0.0739754\pi$
$762$		0			0
$763$	−2.60126e6	+	2.38136e6i	−0.161760	+	0.148086i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.50340e7	−	2.60397e7i	−0.922755	−	1.59826i
$768$		0			0
$769$	6.44098e6			0.392768			0.196384	−	0.980527i	$-0.437080\pi$
$769$	6.44098e6			0.392768			0.196384	+	0.980527i	$0.437080\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.45580e6	+	2.52152e6i	0.0876302	+	0.151780i	0.906509	−	0.422187i	$-0.138737\pi$
$773$	1.45580e6	+	2.52152e6i	0.0876302	+	0.151780i	−0.818879	+	0.573966i	$0.805404\pi$
$774$		0			0
$775$	6.20161e6	−	1.07415e7i	0.370894	−	0.642408i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.71190e6	+	9.89330e6i	−0.337238	+	0.584114i
$780$		0			0
$781$	1.73323e6	+	3.00205e6i	0.101679	+	0.176113i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.16323e7			2.41133
$786$		0			0
$787$	−1.27290e7	−	2.20472e7i	−0.732582	−	1.26887i	−0.955776	−	0.294095i	$-0.904982\pi$
$787$	−1.27290e7	−	2.20472e7i	−0.732582	−	1.26887i	0.223195	−	0.974774i	$-0.428351\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4.40702e6	+	1.99232e7i	0.250440	+	1.13219i
$792$		0			0
$793$	1.78759e7	−	3.09620e7i	1.00945	−	1.74842i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.04344e7			1.13950			0.569752	−	0.821817i	$-0.307039\pi$
$797$	2.04344e7			1.13950			0.569752	+	0.821817i	$0.307039\pi$
$798$		0			0
$799$	−2.79152e6			−0.154694
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.16984e7	+	2.02622e7i	−0.640231	+	1.10891i
$804$		0			0
$805$	−1.21789e7	+	1.11494e7i	−0.662398	+	0.606402i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.41416e7	+	2.44939e7i	0.759672	+	1.31579i	0.943018	+	0.332743i	$0.107974\pi$
$809$	1.41416e7	+	2.44939e7i	0.759672	+	1.31579i	−0.183345	+	0.983049i	$0.558693\pi$
$810$		0			0
$811$	1.58852e7			0.848087			0.424043	−	0.905642i	$-0.360610\pi$
$811$	1.58852e7			0.848087			0.424043	+	0.905642i	$0.360610\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.50180e7	−	2.60119e7i	−0.791987	−	1.37176i
$816$		0			0
$817$	−5.93707e6	+	1.02833e7i	−0.311184	+	0.538986i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.72198e6	−	1.33749e7i	0.399826	−	0.692519i	−0.593878	−	0.804555i	$-0.702404\pi$
$821$	7.72198e6	−	1.33749e7i	0.399826	−	0.692519i	0.993704	+	0.112036i	$0.0357372\pi$
$822$		0			0
$823$	−1.48943e7	−	2.57977e7i	−0.766516	−	1.32764i	−0.939441	−	0.342710i	$-0.888655\pi$
$823$	−1.48943e7	−	2.57977e7i	−0.766516	−	1.32764i	0.172926	−	0.984935i	$-0.444678\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.66831e7			−0.848230			−0.424115	−	0.905608i	$-0.639415\pi$
$827$	−1.66831e7			−0.848230			−0.424115	+	0.905608i	$0.639415\pi$
$828$		0			0
$829$	1.10568e7	+	1.91510e7i	0.558785	+	0.967844i	0.997598	+	0.0692656i	$0.0220656\pi$
$829$	1.10568e7	+	1.91510e7i	0.558785	+	0.967844i	−0.438813	+	0.898578i	$0.644601\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.00277e6	−	1.40517e6i	−0.100004	−	0.0701644i
$834$		0			0
$835$	2.11803e6	−	3.66853e6i	0.105127	−	0.182086i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−539166.			−0.0264434			−0.0132217	−	0.999913i	$-0.504209\pi$
$839$	−539166.			−0.0264434			−0.0132217	+	0.999913i	$0.504209\pi$
$840$		0			0
$841$	−1.99607e7			−0.973162
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.28080e7	−	2.21841e7i	0.617076	−	1.06881i
$846$		0			0
$847$	−1.73840e7	−	5.49015e6i	−0.832608	−	0.262952i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.70249e6	+	4.68086e6i	0.127921	+	0.221565i
$852$		0			0
$853$	2.55452e6			0.120209			0.0601044	−	0.998192i	$-0.480857\pi$
$853$	2.55452e6			0.120209			0.0601044	+	0.998192i	$0.480857\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	120039.	+	207913.i	0.00558303	+	0.00967008i	0.868803	−	0.495157i	$-0.164890\pi$
$857$	120039.	+	207913.i	0.00558303	+	0.00967008i	−0.863220	+	0.504827i	$0.831556\pi$
$858$		0			0
$859$	−3.33955e6	+	5.78428e6i	−0.154421	+	0.267464i	−0.932848	−	0.360270i	$-0.882684\pi$
$859$	−3.33955e6	+	5.78428e6i	−0.154421	+	0.267464i	0.778427	+	0.627735i	$0.216018\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.02530e7	+	1.77587e7i	−0.468622	+	0.811678i	−0.999357	−	0.0358604i	$-0.988583\pi$
$863$	−1.02530e7	+	1.77587e7i	−0.468622	+	0.811678i	0.530734	+	0.847538i	$0.321916\pi$
$864$		0			0
$865$	−1.26796e7	−	2.19618e7i	−0.576192	−	0.997993i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.22164e7			−2.34562
$870$		0			0
$871$	2.06942e6	+	3.58434e6i	0.0924280	+	0.160090i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.74701e6	−	7.89786e6i	−0.0771392	−	0.348730i
$876$		0			0
$877$	−1.26387e7	+	2.18909e7i	−0.554888	+	0.961093i	0.443025	+	0.896509i	$0.353905\pi$
$877$	−1.26387e7	+	2.18909e7i	−0.554888	+	0.961093i	−0.997912	+	0.0645840i	$0.979428\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.45932e7			1.06752			0.533759	−	0.845637i	$-0.320779\pi$
$881$	2.45932e7			1.06752			0.533759	+	0.845637i	$0.320779\pi$
$882$		0			0
$883$	2.78301e7			1.20120			0.600598	−	0.799551i	$-0.294929\pi$
$883$	2.78301e7			1.20120			0.600598	+	0.799551i	$0.294929\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.74928e7	−	3.02984e7i	0.746534	−	1.29304i	−0.202940	−	0.979191i	$-0.565050\pi$
$887$	1.74928e7	−	3.02984e7i	0.746534	−	1.29304i	0.949474	−	0.313844i	$-0.101617\pi$
$888$		0			0
$889$	−7.09137e6	−	3.20586e7i	−0.300937	−	1.36047i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.69526e7	−	2.93629e7i	−0.711392	−	1.23217i
$894$		0			0
$895$	−5.70892e7			−2.38230
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.18866e6	+	2.05882e6i	0.0490521	+	0.0849607i
$900$		0			0
$901$	1.54873e6	−	2.68249e6i	0.0635572	−	0.110084i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.63044e6	−	1.32163e7i	0.309691	−	0.536400i
$906$		0			0
$907$	2.03300e7	+	3.52127e7i	0.820578	+	1.42128i	0.905252	+	0.424874i	$0.139682\pi$
$907$	2.03300e7	+	3.52127e7i	0.820578	+	1.42128i	−0.0846741	+	0.996409i	$0.526985\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.87014e7			1.14579			0.572897	−	0.819627i	$-0.305820\pi$
$911$	2.87014e7			1.14579			0.572897	+	0.819627i	$0.305820\pi$
$912$		0			0
$913$	9.60584e6	+	1.66378e7i	0.381380	+	0.660570i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.42841e7	−	4.51115e6i	−0.560956	−	0.177159i
$918$		0			0
$919$	−1.47415e7	+	2.55330e7i	−0.575773	+	0.997269i	0.420184	+	0.907439i	$0.361966\pi$
$919$	−1.47415e7	+	2.55330e7i	−0.575773	+	0.997269i	−0.995957	+	0.0898297i	$0.971368\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.19506e6			−0.200718
$924$		0			0
$925$	−1.37402e7			−0.528004
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7.94112e6	+	1.37544e7i	−0.301886	+	0.522881i	−0.976563	−	0.215231i	$-0.930949\pi$
$929$	−7.94112e6	+	1.37544e7i	−0.301886	+	0.522881i	0.674677	+	0.738113i	$0.264283\pi$
$930$		0			0
$931$	2.61776e6	−	2.95999e7i	0.0989817	−	1.11922i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.34368e6	+	5.79143e6i	0.125082	+	0.216649i
$936$		0			0
$937$	1.04452e7			0.388656			0.194328	−	0.980937i	$-0.437747\pi$
$937$	1.04452e7			0.388656			0.194328	+	0.980937i	$0.437747\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.34241e7	−	2.32513e7i	−0.494211	−	0.855998i	0.505767	−	0.862670i	$-0.331209\pi$
$941$	−1.34241e7	−	2.32513e7i	−0.494211	−	0.855998i	−0.999978	+	0.00667223i	$0.997876\pi$
$942$		0			0
$943$	−4.91938e6	+	8.52062e6i	−0.180149	+	0.312027i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.62047e6	+	1.31990e7i	−0.276126	+	0.478264i	−0.970418	−	0.241429i	$-0.922384\pi$
$947$	−7.62047e6	+	1.31990e7i	−0.276126	+	0.478264i	0.694293	+	0.719693i	$0.255717\pi$
$948$		0			0
$949$	−1.75319e7	−	3.03662e7i	−0.631922	−	1.09452i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.29112e7			0.460504			0.230252	−	0.973131i	$-0.426045\pi$
$953$	1.29112e7			0.460504			0.230252	+	0.973131i	$0.426045\pi$
$954$		0			0
$955$	−2.85864e7	−	4.95131e7i	−1.01426	−	1.75676i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.98712e7	−	1.81914e7i	0.697716	−	0.638734i
$960$		0			0
$961$	9.18121e6	−	1.59023e7i	0.320694	−	0.555459i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.91672e7			−2.73670
$966$		0			0
$967$	4.60406e7			1.58334			0.791671	−	0.610948i	$-0.209211\pi$
$967$	4.60406e7			1.58334			0.791671	+	0.610948i	$0.209211\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	9.36580e6	−	1.62220e7i	0.318784	−	0.552150i	−0.661451	−	0.749989i	$-0.730059\pi$
$971$	9.36580e6	−	1.62220e7i	0.318784	−	0.552150i	0.980235	+	0.197839i	$0.0633922\pi$
$972$		0			0
$973$	−811806.	−	3.67000e6i	−0.0274897	−	0.124275i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.44602e7	−	2.50458e7i	−0.484660	−	0.839456i	0.515184	−	0.857079i	$-0.327723\pi$
$977$	−1.44602e7	−	2.50458e7i	−0.484660	−	0.839456i	−0.999845	+	0.0176231i	$0.994390\pi$
$978$		0			0
$979$	5.52286e7			1.84165
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.08834e7	+	1.88505e7i	0.359235	+	0.622214i	0.987833	−	0.155517i	$-0.0497042\pi$
$983$	1.08834e7	+	1.88505e7i	0.359235	+	0.622214i	−0.628598	+	0.777730i	$0.716371\pi$
$984$		0			0
$985$	7.70379e6	−	1.33433e7i	0.252996	−	0.438202i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5.11331e6	+	8.85651e6i	−0.166231	+	0.287920i
$990$		0			0
$991$	9.39333e6	+	1.62697e7i	0.303833	+	0.526255i	0.977001	−	0.213235i	$-0.0684001\pi$
$991$	9.39333e6	+	1.62697e7i	0.303833	+	0.526255i	−0.673168	+	0.739490i	$0.735067\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.70492e6			−0.182680
$996$		0			0
$997$	7.38382e6	+	1.27891e7i	0.235257	+	0.407477i	0.959347	−	0.282228i	$-0.0910734\pi$
$997$	7.38382e6	+	1.27891e7i	0.235257	+	0.407477i	−0.724090	+	0.689705i	$0.757740\pi$
$998$		0			0
$999$		0			0

Display

a_p

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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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.6.k.d.37.2		4
3.2	odd	2		28.6.e.b.9.1	✓	4
7.4	even	3	inner	252.6.k.d.109.2		4
12.11	even	2		112.6.i.e.65.2		4
21.2	odd	6		196.6.a.j.1.2		2
21.5	even	6		196.6.a.h.1.1		2
21.11	odd	6		28.6.e.b.25.1	yes	4
21.17	even	6		196.6.e.k.165.2		4
21.20	even	2		196.6.e.k.177.2		4
84.11	even	6		112.6.i.e.81.2		4
84.23	even	6		784.6.a.o.1.1		2
84.47	odd	6		784.6.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.e.b.9.1	✓	4	3.2	odd	2
28.6.e.b.25.1	yes	4	21.11	odd	6
112.6.i.e.65.2		4	12.11	even	2
112.6.i.e.81.2		4	84.11	even	6
196.6.a.h.1.1		2	21.5	even	6
196.6.a.j.1.2		2	21.2	odd	6
196.6.e.k.165.2		4	21.17	even	6
196.6.e.k.177.2		4	21.20	even	2
252.6.k.d.37.2		4	1.1	even	1	trivial
252.6.k.d.109.2		4	7.4	even	3	inner
784.6.a.o.1.1		2	84.23	even	6
784.6.a.bd.1.2		2	84.47	odd	6

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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Newspace parameters

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$q$ -expansion

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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	252.6.k.d.37.2

Level	$252$
Weight	$6$
Character	252.37
Analytic conductor	$40.417$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$