LMFDB - Embedded newform 1620.4.i.u.1081.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1620 = 2^{2} \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$ , degree $2$ , not 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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} + 91x^{4} - 294x^{3} + 8292x^{2} - 17280x + 36864$ x^6 - x^5 + 91x^4 - 294x^3 + 8292x^2 - 17280x + 36864
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}\cdot 3^{3}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.2
Root			$4.38373 + 7.59284i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.4.i.u.541.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 - 4.33013i) q^{5} +(-0.474792 - 0.822364i) q^{7} +(31.3276 + 54.2610i) q^{11} +(-3.47479 + 6.01852i) q^{13} +103.411 q^{17} +73.8064 q^{19} +(43.0795 - 74.6159i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-27.7812 - 48.1184i) q^{29} +(-99.9323 + 173.088i) q^{31} -4.74792 q^{35} -18.9070 q^{37} +(-28.6512 + 49.6254i) q^{41} +(-148.016 - 256.371i) q^{43} +(28.7692 + 49.8298i) q^{47} +(171.049 - 296.266i) q^{49} -672.005 q^{53} +313.276 q^{55} +(-273.322 + 473.408i) q^{59} +(395.780 + 685.510i) q^{61} +(17.3740 + 30.0926i) q^{65} +(-315.780 + 546.947i) q^{67} -102.942 q^{71} -200.652 q^{73} +(29.7482 - 51.5253i) q^{77} +(332.966 + 576.714i) q^{79} +(673.703 + 1166.89i) q^{83} +(258.528 - 447.783i) q^{85} +12.6866 q^{89} +6.59922 q^{91} +(184.516 - 319.591i) q^{95} +(318.689 + 551.986i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 15 q^{5} - 15 q^{7} + 24 q^{11} - 33 q^{13} - 84 q^{17} + 42 q^{19} - 33 q^{23} - 75 q^{25} + 222 q^{29} - 132 q^{31} - 150 q^{35} + 348 q^{37} - 99 q^{41} + 120 q^{43} + 537 q^{47} - 492 q^{49} - 534 q^{53}+ \cdots + 2556 q^{97}+O(q^{100})$ 6 * q + 15 * q^5 - 15 * q^7 + 24 * q^11 - 33 * q^13 - 84 * q^17 + 42 * q^19 - 33 * q^23 - 75 * q^25 + 222 * q^29 - 132 * q^31 - 150 * q^35 + 348 * q^37 - 99 * q^41 + 120 * q^43 + 537 * q^47 - 492 * q^49 - 534 * q^53 + 240 * q^55 - 225 * q^59 + 480 * q^61 + 165 * q^65 - 12 * q^67 - 1140 * q^71 + 2124 * q^73 + 312 * q^77 + 1026 * q^79 + 702 * q^83 - 210 * q^85 - 2280 * q^89 + 3222 * q^91 + 105 * q^95 + 2556 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$ .

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	−0.474792	−	0.822364i	−0.0256364	−	0.0444035i	0.852923	−	0.522037i	$-0.174828\pi$
$7$	−0.474792	−	0.822364i	−0.0256364	−	0.0444035i	−0.878559	+	0.477634i	$0.841495\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	31.3276	+	54.2610i	0.858693	+	1.48730i	0.873176	+	0.487404i	$0.162056\pi$
$11$	31.3276	+	54.2610i	0.858693	+	1.48730i	−0.0144837	+	0.999895i	$0.504610\pi$
$12$		0			0
$13$	−3.47479	+	6.01852i	−0.0741334	+	0.128403i	−0.900709	−	0.434423i	$-0.856952\pi$
$13$	−3.47479	+	6.01852i	−0.0741334	+	0.128403i	0.826576	+	0.562826i	$0.190286\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	103.411			1.47535			0.737673	−	0.675158i	$-0.235925\pi$
$17$	103.411			1.47535			0.737673	+	0.675158i	$0.235925\pi$
$18$		0			0
$19$	73.8064			0.891176			0.445588	−	0.895238i	$-0.352995\pi$
$19$	73.8064			0.891176			0.445588	+	0.895238i	$0.352995\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	43.0795	−	74.6159i	0.390552	−	0.676456i	−0.601970	−	0.798519i	$-0.705617\pi$
$23$	43.0795	−	74.6159i	0.390552	−	0.676456i	0.992522	+	0.122062i	$0.0389507\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−27.7812	−	48.1184i	−0.177891	−	0.308116i	0.763267	−	0.646083i	$-0.223594\pi$
$29$	−27.7812	−	48.1184i	−0.177891	−	0.308116i	−0.941158	+	0.337967i	$0.890261\pi$
$30$		0			0
$31$	−99.9323	+	173.088i	−0.578980	+	1.00282i	0.416617	+	0.909082i	$0.363216\pi$
$31$	−99.9323	+	173.088i	−0.578980	+	1.00282i	−0.995597	+	0.0937403i	$0.970118\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.74792			−0.0229299
$36$		0			0
$37$	−18.9070			−0.0840078			−0.0420039	−	0.999117i	$-0.513374\pi$
$37$	−18.9070			−0.0840078			−0.0420039	+	0.999117i	$0.513374\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−28.6512	+	49.6254i	−0.109136	+	0.189029i	−0.915420	−	0.402499i	$-0.868142\pi$
$41$	−28.6512	+	49.6254i	−0.109136	+	0.189029i	0.806285	+	0.591528i	$0.201475\pi$
$42$		0			0
$43$	−148.016	−	256.371i	−0.524935	−	0.909214i	−0.999578	−	0.0290360i	$-0.990756\pi$
$43$	−148.016	−	256.371i	−0.524935	−	0.909214i	0.474643	−	0.880178i	$-0.342577\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	28.7692	+	49.8298i	0.0892856	+	0.154647i	0.907209	−	0.420679i	$-0.138208\pi$
$47$	28.7692	+	49.8298i	0.0892856	+	0.154647i	−0.817924	+	0.575327i	$0.804875\pi$
$48$		0			0
$49$	171.049	−	296.266i	0.498686	−	0.863749i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−672.005			−1.74164			−0.870821	−	0.491600i	$-0.836412\pi$
$53$	−672.005			−1.74164			−0.870821	+	0.491600i	$0.836412\pi$
$54$		0			0
$55$	313.276			0.768038
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−273.322	+	473.408i	−0.603110	+	1.04462i	0.389237	+	0.921138i	$0.372739\pi$
$59$	−273.322	+	473.408i	−0.603110	+	1.04462i	−0.992347	+	0.123480i	$0.960595\pi$
$60$		0			0
$61$	395.780	+	685.510i	0.830728	+	1.43886i	0.897462	+	0.441092i	$0.145409\pi$
$61$	395.780	+	685.510i	0.830728	+	1.43886i	−0.0667342	+	0.997771i	$0.521258\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	17.3740	+	30.0926i	0.0331535	+	0.0574235i
$66$		0			0
$67$	−315.780	+	546.947i	−0.575801	+	0.997316i	0.420153	+	0.907453i	$0.361976\pi$
$67$	−315.780	+	546.947i	−0.575801	+	0.997316i	−0.995954	+	0.0898632i	$0.971357\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−102.942			−0.172069			−0.0860346	−	0.996292i	$-0.527420\pi$
$71$	−102.942			−0.172069			−0.0860346	+	0.996292i	$0.527420\pi$
$72$		0			0
$73$	−200.652			−0.321707			−0.160853	−	0.986978i	$-0.551425\pi$
$73$	−200.652			−0.321707			−0.160853	+	0.986978i	$0.551425\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	29.7482	−	51.5253i	0.0440275	−	0.0762579i
$78$		0			0
$79$	332.966	+	576.714i	0.474197	+	0.821333i	0.999564	−	0.0295426i	$-0.00940508\pi$
$79$	332.966	+	576.714i	0.474197	+	0.821333i	−0.525366	+	0.850876i	$0.676072\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	673.703	+	1166.89i	0.890946	+	1.54316i	0.838743	+	0.544527i	$0.183291\pi$
$83$	673.703	+	1166.89i	0.890946	+	1.54316i	0.0522027	+	0.998637i	$0.483376\pi$
$84$		0			0
$85$	258.528	−	447.783i	0.329897	−	0.571399i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.6866			0.0151099			0.00755496	−	0.999971i	$-0.497595\pi$
$89$	12.6866			0.0151099			0.00755496	+	0.999971i	$0.497595\pi$
$90$		0			0
$91$	6.59922			0.00760204
$92$		0			0
$93$		0			0
$94$		0			0
$95$	184.516	−	319.591i	0.199273	−	0.345151i
$96$		0			0
$97$	318.689	+	551.986i	0.333588	+	0.577791i	0.983213	−	0.182464i	$-0.0584073\pi$
$97$	318.689	+	551.986i	0.333588	+	0.577791i	−0.649625	+	0.760255i	$0.725074\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−368.452	−	638.177i	−0.362993	−	0.628723i	0.625459	−	0.780257i	$-0.284912\pi$
$101$	−368.452	−	638.177i	−0.362993	−	0.628723i	−0.988452	+	0.151535i	$0.951579\pi$
$102$		0			0
$103$	698.320	−	1209.53i	0.668034	−	1.15707i	−0.310419	−	0.950600i	$-0.600469\pi$
$103$	698.320	−	1209.53i	0.668034	−	1.15707i	0.978453	−	0.206470i	$-0.0661974\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	181.225			0.163735			0.0818677	−	0.996643i	$-0.473911\pi$
$107$	181.225			0.163735			0.0818677	+	0.996643i	$0.473911\pi$
$108$		0			0
$109$	1495.62			1.31426			0.657132	−	0.753775i	$-0.271769\pi$
$109$	1495.62			1.31426			0.657132	+	0.753775i	$0.271769\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1119.25	−	1938.60i	0.931771	−	1.61387i	0.151477	−	0.988461i	$-0.451597\pi$
$113$	1119.25	−	1938.60i	0.931771	−	1.61387i	0.780294	−	0.625413i	$-0.215070\pi$
$114$		0			0
$115$	−215.398	−	373.080i	−0.174660	−	0.302521i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−49.0988	−	85.0416i	−0.0378225	−	0.0655105i
$120$		0			0
$121$	−1297.33	+	2247.05i	−0.974706	+	1.68824i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	1163.69			0.813075			0.406537	−	0.913634i	$-0.366736\pi$
$127$	1163.69			0.813075			0.406537	+	0.913634i	$0.366736\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−49.4973	+	85.7319i	−0.0330122	+	0.0571789i	−0.882059	−	0.471138i	$-0.843843\pi$
$131$	−49.4973	+	85.7319i	−0.0330122	+	0.0571789i	0.849047	+	0.528317i	$0.177177\pi$
$132$		0			0
$133$	−35.0427	−	60.6957i	−0.0228465	−	0.0395713i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	87.4537	+	151.474i	0.0545378	+	0.0944622i	0.892005	−	0.452025i	$-0.149298\pi$
$137$	87.4537	+	151.474i	0.0545378	+	0.0944622i	−0.837468	+	0.546487i	$0.815965\pi$
$138$		0			0
$139$	−490.757	+	850.016i	−0.299464	+	0.518686i	−0.976013	−	0.217711i	$-0.930141\pi$
$139$	−490.757	+	850.016i	−0.299464	+	0.518686i	0.676550	+	0.736397i	$0.263474\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−435.427			−0.254631
$144$		0			0
$145$	−277.812			−0.159111
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1153.47	−	1997.86i	0.634200	−	1.09847i	−0.352484	−	0.935818i	$-0.614663\pi$
$149$	1153.47	−	1997.86i	0.634200	−	1.09847i	0.986684	−	0.162649i	$-0.0520037\pi$
$150$		0			0
$151$	990.901	+	1716.29i	0.534029	+	0.924965i	0.999210	+	0.0397495i	$0.0126560\pi$
$151$	990.901	+	1716.29i	0.534029	+	0.924965i	−0.465181	+	0.885216i	$0.654011\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	499.662	+	865.439i	0.258928	+	0.448476i
$156$		0			0
$157$	216.573	−	375.116i	0.110092	−	0.190685i	−0.805715	−	0.592303i	$-0.798219\pi$
$157$	216.573	−	375.116i	0.110092	−	0.190685i	0.915807	+	0.401618i	$0.131552\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−81.8153			−0.0400494
$162$		0			0
$163$	3447.72			1.65673			0.828363	−	0.560192i	$-0.189273\pi$
$163$	3447.72			1.65673			0.828363	+	0.560192i	$0.189273\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	527.445	−	913.561i	0.244401	−	0.423314i	−0.717562	−	0.696494i	$-0.754742\pi$
$167$	527.445	−	913.561i	0.244401	−	0.423314i	0.961963	+	0.273180i	$0.0880755\pi$
$168$		0			0
$169$	1074.35	+	1860.83i	0.489008	+	0.846988i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	316.072	+	547.453i	0.138905	+	0.240590i	0.927082	−	0.374858i	$-0.122309\pi$
$173$	316.072	+	547.453i	0.138905	+	0.240590i	−0.788178	+	0.615448i	$0.788975\pi$
$174$		0			0
$175$	−11.8698	+	20.5591i	−0.00512727	+	0.00888070i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1329.33			−0.555078			−0.277539	−	0.960714i	$-0.589519\pi$
$179$	−1329.33			−0.555078			−0.277539	+	0.960714i	$0.589519\pi$
$180$		0			0
$181$	1468.52			0.603062			0.301531	−	0.953456i	$-0.402502\pi$
$181$	1468.52			0.603062			0.301531	+	0.953456i	$0.402502\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−47.2674	+	81.8696i	−0.0187847	+	0.0325361i
$186$		0			0
$187$	3239.62	+	5611.19i	1.26687	+	2.19428i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	241.195	+	417.762i	0.0913732	+	0.158263i	0.908089	−	0.418777i	$-0.137541\pi$
$191$	241.195	+	417.762i	0.0913732	+	0.158263i	−0.816716	+	0.577040i	$0.804208\pi$
$192$		0			0
$193$	171.677	−	297.353i	0.0640287	−	0.110901i	−0.832234	−	0.554425i	$-0.812938\pi$
$193$	171.677	−	297.353i	0.0640287	−	0.110901i	0.896263	+	0.443524i	$0.146272\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4507.54			−1.63020			−0.815099	−	0.579321i	$-0.803318\pi$
$197$	−4507.54			−1.63020			−0.815099	+	0.579321i	$0.803318\pi$
$198$		0			0
$199$	−1750.03			−0.623397			−0.311699	−	0.950181i	$-0.600898\pi$
$199$	−1750.03			−0.623397			−0.311699	+	0.950181i	$0.600898\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−26.3806	+	45.6925i	−0.00912096	+	0.0157980i
$204$		0			0
$205$	143.256	+	248.127i	0.0488071	+	0.0845363i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2312.18	+	4004.81i	0.765247	+	1.32545i
$210$		0			0
$211$	2600.02	−	4503.38i	0.848308	−	1.46931i	−0.0344082	−	0.999408i	$-0.510955\pi$
$211$	2600.02	−	4503.38i	0.848308	−	1.46931i	0.882717	−	0.469906i	$-0.155712\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1480.16			−0.469516
$216$		0			0
$217$	189.788			0.0593717
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−359.332	+	622.382i	−0.109372	+	0.189439i
$222$		0			0
$223$	1452.74	+	2516.21i	0.436244	+	0.755597i	0.997396	−	0.0721155i	$-0.0229750\pi$
$223$	1452.74	+	2516.21i	0.436244	+	0.755597i	−0.561152	+	0.827713i	$0.689642\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	867.010	+	1501.71i	0.253504	+	0.439082i	0.964488	−	0.264126i	$-0.0850835\pi$
$227$	867.010	+	1501.71i	0.253504	+	0.439082i	−0.710984	+	0.703208i	$0.751750\pi$
$228$		0			0
$229$	−1736.48	+	3007.68i	−0.501092	+	0.867917i	0.498907	+	0.866656i	$0.333735\pi$
$229$	−1736.48	+	3007.68i	−0.501092	+	0.867917i	−0.999999	+	0.00126168i	$0.999598\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	595.216			0.167356			0.0836779	−	0.996493i	$-0.473333\pi$
$233$	595.216			0.167356			0.0836779	+	0.996493i	$0.473333\pi$
$234$		0			0
$235$	287.692			0.0798595
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1089.58	−	1887.21i	0.294892	−	0.510768i	−0.680068	−	0.733149i	$-0.738050\pi$
$239$	1089.58	−	1887.21i	0.294892	−	0.510768i	0.974960	+	0.222381i	$0.0713830\pi$
$240$		0			0
$241$	2181.08	+	3777.74i	0.582970	+	1.00973i	0.995125	+	0.0986197i	$0.0314427\pi$
$241$	2181.08	+	3777.74i	0.582970	+	1.00973i	−0.412155	+	0.911114i	$0.635224\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−855.246	−	1481.33i	−0.223019	−	0.386280i
$246$		0			0
$247$	−256.462	+	444.205i	−0.0660659	+	0.114430i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2738.00			−0.688531			−0.344266	−	0.938872i	$-0.611872\pi$
$251$	−2738.00			−0.688531			−0.344266	+	0.938872i	$0.611872\pi$
$252$		0			0
$253$	5398.31			1.34146
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−997.546	+	1727.80i	−0.242122	+	0.419367i	−0.961318	−	0.275440i	$-0.911177\pi$
$257$	−997.546	+	1727.80i	−0.242122	+	0.419367i	0.719197	+	0.694806i	$0.244510\pi$
$258$		0			0
$259$	8.97689	+	15.5484i	0.00215365	+	0.00373024i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1276.78	−	2211.45i	−0.299352	−	0.518493i	0.676636	−	0.736318i	$-0.263437\pi$
$263$	−1276.78	−	2211.45i	−0.299352	−	0.518493i	−0.975988	+	0.217825i	$0.930104\pi$
$264$		0			0
$265$	−1680.01	+	2909.87i	−0.389443	+	0.674535i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4534.66			−1.02782			−0.513909	−	0.857845i	$-0.671803\pi$
$269$	−4534.66			−1.02782			−0.513909	+	0.857845i	$0.671803\pi$
$270$		0			0
$271$	4390.49			0.984145			0.492073	−	0.870554i	$-0.336239\pi$
$271$	4390.49			0.984145			0.492073	+	0.870554i	$0.336239\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	783.189	−	1356.52i	0.171739	−	0.297460i
$276$		0			0
$277$	−3363.39	−	5825.57i	−0.729555	−	1.26363i	−0.957072	−	0.289852i	$-0.906394\pi$
$277$	−3363.39	−	5825.57i	−0.729555	−	1.26363i	0.227517	−	0.973774i	$-0.426939\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3619.36	+	6268.92i	0.768374	+	1.33086i	0.938444	+	0.345431i	$0.112267\pi$
$281$	3619.36	+	6268.92i	0.768374	+	1.33086i	−0.170070	+	0.985432i	$0.554399\pi$
$282$		0			0
$283$	−3726.68	+	6454.79i	−0.782784	+	1.35582i	0.147530	+	0.989058i	$0.452868\pi$
$283$	−3726.68	+	6454.79i	−0.782784	+	1.35582i	−0.930314	+	0.366764i	$0.880466\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	54.4135			0.0111914
$288$		0			0
$289$	5780.86			1.17665
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1532.37	−	2654.14i	0.305535	−	0.529203i	−0.671845	−	0.740692i	$-0.734498\pi$
$293$	1532.37	−	2654.14i	0.305535	−	0.529203i	0.977380	+	0.211489i	$0.0678312\pi$
$294$		0			0
$295$	1366.61	+	2367.04i	0.269719	+	0.467167i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	299.385	+	518.550i	0.0579059	+	0.100296i
$300$		0			0
$301$	−140.554	+	243.446i	−0.0269149	+	0.0466179i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3957.80			0.743026
$306$		0			0
$307$	7427.82			1.38087			0.690437	−	0.723393i	$-0.257418\pi$
$307$	7427.82			1.38087			0.690437	+	0.723393i	$0.257418\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3787.39	−	6559.96i	0.690557	−	1.19608i	−0.281098	−	0.959679i	$-0.590699\pi$
$311$	3787.39	−	6559.96i	0.690557	−	1.19608i	0.971655	−	0.236402i	$-0.0759681\pi$
$312$		0			0
$313$	4314.30	+	7472.59i	0.779101	+	1.34944i	0.932460	+	0.361273i	$0.117658\pi$
$313$	4314.30	+	7472.59i	0.779101	+	1.34944i	−0.153359	+	0.988171i	$0.549009\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1473.04	+	2551.38i	0.260992	+	0.452051i	0.966506	−	0.256645i	$-0.0826172\pi$
$317$	1473.04	+	2551.38i	0.260992	+	0.452051i	−0.705514	+	0.708696i	$0.749284\pi$
$318$		0			0
$319$	1740.63	−	3014.87i	0.305507	−	0.529154i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7632.40			1.31479
$324$		0			0
$325$	173.740			0.0296534
$326$		0			0
$327$		0			0
$328$		0			0
$329$	27.3188	−	47.3176i	0.00457792	−	0.00792918i
$330$		0			0
$331$	−1568.72	−	2717.11i	−0.260498	−	0.451196i	0.705876	−	0.708335i	$-0.250553\pi$
$331$	−1568.72	−	2717.11i	−0.260498	−	0.451196i	−0.966374	+	0.257139i	$0.917220\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1578.90	+	2734.73i	0.257506	+	0.446013i
$336$		0			0
$337$	1857.87	−	3217.93i	0.300311	−	0.520153i	−0.675896	−	0.736997i	$-0.736243\pi$
$337$	1857.87	−	3217.93i	0.300311	−	0.520153i	0.976206	+	0.216844i	$0.0695763\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12522.5			−1.98866
$342$		0			0
$343$	−650.559			−0.102411
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3030.01	−	5248.13i	0.468759	−	0.811915i	−0.530603	−	0.847620i	$-0.678035\pi$
$347$	3030.01	−	5248.13i	0.468759	−	0.811915i	0.999362	+	0.0357056i	$0.0113679\pi$
$348$		0			0
$349$	2817.33	+	4879.75i	0.432115	+	0.748444i	0.997055	−	0.0766872i	$-0.0244343\pi$
$349$	2817.33	+	4879.75i	0.432115	+	0.748444i	−0.564941	+	0.825132i	$0.691101\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2278.94	+	3947.24i	0.343614	+	0.595158i	0.985101	−	0.171977i	$-0.0550154\pi$
$353$	2278.94	+	3947.24i	0.343614	+	0.595158i	−0.641487	+	0.767134i	$0.721682\pi$
$354$		0			0
$355$	−257.354	+	445.750i	−0.0384758	+	0.0666421i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3206.54			0.471406			0.235703	−	0.971825i	$-0.424261\pi$
$359$	3206.54			0.471406			0.235703	+	0.971825i	$0.424261\pi$
$360$		0			0
$361$	−1411.62			−0.205805
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−501.631	+	868.850i	−0.0719358	+	0.124596i
$366$		0			0
$367$	4350.11	+	7534.61i	0.618730	+	1.07167i	0.989718	+	0.143033i	$0.0456857\pi$
$367$	4350.11	+	7534.61i	0.618730	+	1.07167i	−0.370988	+	0.928638i	$0.620981\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	319.063	+	552.633i	0.0446494	+	0.0773350i
$372$		0			0
$373$	−6361.23	+	11018.0i	−0.883035	+	1.52946i	−0.0350846	+	0.999384i	$0.511170\pi$
$373$	−6361.23	+	11018.0i	−0.883035	+	1.52946i	−0.847950	+	0.530076i	$0.822163\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	386.135			0.0527506
$378$		0			0
$379$	1339.34			0.181523			0.0907613	−	0.995873i	$-0.471070\pi$
$379$	1339.34			0.181523			0.0907613	+	0.995873i	$0.471070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2364.75	+	4095.87i	−0.315491	+	0.546447i	−0.979542	−	0.201241i	$-0.935503\pi$
$383$	−2364.75	+	4095.87i	−0.315491	+	0.546447i	0.664051	+	0.747688i	$0.268836\pi$
$384$		0			0
$385$	−148.741	−	257.627i	−0.0196897	−	0.0341036i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2419.19	−	4190.16i	−0.315316	−	0.546142i	0.664189	−	0.747565i	$-0.268777\pi$
$389$	−2419.19	−	4190.16i	−0.315316	−	0.546142i	−0.979505	+	0.201422i	$0.935444\pi$
$390$		0			0
$391$	4454.90	−	7716.12i	0.576200	−	0.998007i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3329.66			0.424135
$396$		0			0
$397$	−7768.23			−0.982056			−0.491028	−	0.871144i	$-0.663379\pi$
$397$	−7768.23			−0.982056			−0.491028	+	0.871144i	$0.663379\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2603.53	+	4509.45i	−0.324225	+	0.561575i	−0.981355	−	0.192203i	$-0.938437\pi$
$401$	−2603.53	+	4509.45i	−0.324225	+	0.561575i	0.657130	+	0.753777i	$0.271770\pi$
$402$		0			0
$403$	−694.488	−	1202.89i	−0.0858435	−	0.148685i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−592.310	−	1025.91i	−0.0721369	−	0.124945i
$408$		0			0
$409$	3915.71	−	6782.22i	0.473398	−	0.819949i	−0.526139	−	0.850399i	$-0.676361\pi$
$409$	3915.71	−	6782.22i	0.473398	−	0.819949i	0.999536	+	0.0304502i	$0.00969409\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	519.085			0.0618462
$414$		0			0
$415$	6737.03			0.796886
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5850.12	+	10132.7i	−0.682093	+	1.18142i	0.292248	+	0.956343i	$0.405597\pi$
$419$	−5850.12	+	10132.7i	−0.682093	+	1.18142i	−0.974341	+	0.225077i	$0.927737\pi$
$420$		0			0
$421$	1618.00	+	2802.45i	0.187307	+	0.324425i	0.944352	−	0.328938i	$-0.106691\pi$
$421$	1618.00	+	2802.45i	0.187307	+	0.324425i	−0.757044	+	0.653363i	$0.773357\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1292.64	−	2238.92i	−0.147535	−	0.255537i
$426$		0			0
$427$	375.826	−	650.950i	0.0425937	−	0.0737744i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7828.48			−0.874907			−0.437453	−	0.899241i	$-0.644119\pi$
$431$	−7828.48			−0.874907			−0.437453	+	0.899241i	$0.644119\pi$
$432$		0			0
$433$	−11016.3			−1.22265			−0.611327	−	0.791378i	$-0.709364\pi$
$433$	−11016.3			−1.22265			−0.611327	+	0.791378i	$0.709364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3179.54	−	5507.13i	0.348051	−	0.602842i
$438$		0			0
$439$	324.458	+	561.977i	0.0352745	+	0.0610973i	0.883124	−	0.469140i	$-0.155436\pi$
$439$	324.458	+	561.977i	0.0352745	+	0.0610973i	−0.847849	+	0.530237i	$0.822103\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	457.211	+	791.913i	0.0490356	+	0.0849322i	0.889501	−	0.456932i	$-0.151052\pi$
$443$	457.211	+	791.913i	0.0490356	+	0.0849322i	−0.840466	+	0.541865i	$0.817719\pi$
$444$		0			0
$445$	31.7166	−	54.9348i	0.00337868	−	0.00585204i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9663.37			−1.01569			−0.507843	−	0.861450i	$-0.669557\pi$
$449$	−9663.37			−1.01569			−0.507843	+	0.861450i	$0.669557\pi$
$450$		0			0
$451$	−3590.30			−0.374857
$452$		0			0
$453$		0			0
$454$		0			0
$455$	16.4980	−	28.5754i	0.00169987	−	0.00294426i
$456$		0			0
$457$	−4472.92	−	7747.32i	−0.457843	−	0.793007i	0.541004	−	0.841020i	$-0.318044\pi$
$457$	−4472.92	−	7747.32i	−0.457843	−	0.793007i	−0.998847	+	0.0480128i	$0.984711\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3194.05	−	5532.26i	−0.322694	−	0.558922i	0.658349	−	0.752713i	$-0.271255\pi$
$461$	−3194.05	−	5532.26i	−0.322694	−	0.558922i	−0.981043	+	0.193791i	$0.937922\pi$
$462$		0			0
$463$	94.8824	−	164.341i	0.00952389	−	0.0164959i	−0.861224	−	0.508225i	$-0.830302\pi$
$463$	94.8824	−	164.341i	0.00952389	−	0.0164959i	0.870748	+	0.491729i	$0.163635\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17087.3			−1.69317			−0.846583	−	0.532257i	$-0.821344\pi$
$467$	−17087.3			−1.69317			−0.846583	+	0.532257i	$0.821344\pi$
$468$		0			0
$469$	599.719			0.0590458
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9273.96	−	16063.0i	0.901516	−	1.56147i
$474$		0			0
$475$	−922.580	−	1597.96i	−0.0891176	−	0.154356i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3874.09	−	6710.13i	−0.369545	−	0.640070i	0.619950	−	0.784641i	$-0.287153\pi$
$479$	−3874.09	−	6710.13i	−0.369545	−	0.640070i	−0.989494	+	0.144572i	$0.953820\pi$
$480$		0			0
$481$	65.6978	−	113.792i	0.00622778	−	0.0107868i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3186.89			0.298370
$486$		0			0
$487$	5330.93			0.496032			0.248016	−	0.968756i	$-0.420221\pi$
$487$	5330.93			0.496032			0.248016	+	0.968756i	$0.420221\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7248.89	+	12555.4i	−0.666269	+	1.15401i	0.312671	+	0.949861i	$0.398776\pi$
$491$	−7248.89	+	12555.4i	−0.666269	+	1.15401i	−0.978940	+	0.204150i	$0.934557\pi$
$492$		0			0
$493$	−2872.88	−	4975.98i	−0.262451	−	0.454578i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	48.8758	+	84.6554i	0.00441123	+	0.00764047i
$498$		0			0
$499$	1531.42	−	2652.50i	0.137386	−	0.237960i	−0.789120	−	0.614239i	$-0.789463\pi$
$499$	1531.42	−	2652.50i	0.137386	−	0.237960i	0.926506	+	0.376279i	$0.122797\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10683.8			0.947055			0.473527	−	0.880779i	$-0.342980\pi$
$503$	10683.8			0.947055			0.473527	+	0.880779i	$0.342980\pi$
$504$		0			0
$505$	−3684.52			−0.324671
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7187.19	−	12448.6i	0.625868	−	1.08403i	−0.362505	−	0.931982i	$-0.618078\pi$
$509$	7187.19	−	12448.6i	0.625868	−	1.08403i	0.988372	−	0.152053i	$-0.0485883\pi$
$510$		0			0
$511$	95.2682	+	165.009i	0.00824739	+	0.0142849i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3491.60	−	6047.63i	−0.298754	−	0.517457i
$516$		0			0
$517$	−1802.54	+	3122.09i	−0.153338	+	0.265589i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9386.82			0.789336			0.394668	−	0.918824i	$-0.370860\pi$
$521$	9386.82			0.789336			0.394668	+	0.918824i	$0.370860\pi$
$522$		0			0
$523$	15352.7			1.28361			0.641805	−	0.766868i	$-0.278186\pi$
$523$	15352.7			1.28361			0.641805	+	0.766868i	$0.278186\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10334.1	+	17899.2i	−0.854196	+	1.47951i
$528$		0			0
$529$	2371.81	+	4108.09i	0.194938	+	0.337642i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−199.114	−	344.876i	−0.0161812	−	0.0280267i
$534$		0			0
$535$	453.063	−	784.728i	0.0366124	−	0.0634145i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	21434.2			1.71287
$540$		0			0
$541$	−23730.6			−1.88588			−0.942938	−	0.332968i	$-0.891950\pi$
$541$	−23730.6			−1.88588			−0.942938	+	0.332968i	$0.891950\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3739.06	−	6476.25i	0.293879	−	0.509013i
$546$		0			0
$547$	6570.32	+	11380.1i	0.513577	+	0.889541i	0.999876	+	0.0157490i	$0.00501327\pi$
$547$	6570.32	+	11380.1i	0.513577	+	0.889541i	−0.486299	+	0.873792i	$0.661653\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2050.43	−	3551.45i	−0.158532	−	0.274586i
$552$		0			0
$553$	316.179	−	547.638i	0.0243134	−	0.0421120i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19848.3			1.50987			0.754935	−	0.655800i	$-0.227668\pi$
$557$	19848.3			1.50987			0.754935	+	0.655800i	$0.227668\pi$
$558$		0			0
$559$	2057.30			0.155661
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9300.96	−	16109.7i	0.696250	−	1.20594i	−0.273508	−	0.961870i	$-0.588184\pi$
$563$	9300.96	−	16109.7i	0.696250	−	1.20594i	0.969758	−	0.244070i	$-0.0784827\pi$
$564$		0			0
$565$	−5596.24	−	9692.98i	−0.416700	−	0.721746i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9689.01	+	16781.9i	0.713857	+	1.23644i	0.963399	+	0.268072i	$0.0863867\pi$
$569$	9689.01	+	16781.9i	0.713857	+	1.23644i	−0.249542	+	0.968364i	$0.580280\pi$
$570$		0			0
$571$	−2341.16	+	4055.02i	−0.171584	+	0.297193i	−0.938974	−	0.343988i	$-0.888222\pi$
$571$	−2341.16	+	4055.02i	−0.171584	+	0.297193i	0.767390	+	0.641181i	$0.221555\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2153.98			−0.156221
$576$		0			0
$577$	−20543.6			−1.48222			−0.741111	−	0.671382i	$-0.765701\pi$
$577$	−20543.6			−1.48222			−0.741111	+	0.671382i	$0.765701\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	639.738	−	1108.06i	0.0456812	−	0.0791222i
$582$		0			0
$583$	−21052.3	−	36463.6i	−1.49554	−	2.59034i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1849.17	−	3202.86i	−0.130023	−	0.225206i	0.793662	−	0.608359i	$-0.208172\pi$
$587$	−1849.17	−	3202.86i	−0.130023	−	0.225206i	−0.923685	+	0.383152i	$0.874838\pi$
$588$		0			0
$589$	−7375.64	+	12775.0i	−0.515973	+	0.893692i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9685.31			0.670705			0.335353	−	0.942093i	$-0.391145\pi$
$593$	9685.31			0.670705			0.335353	+	0.942093i	$0.391145\pi$
$594$		0			0
$595$	−490.988			−0.0338295
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7004.86	+	12132.8i	−0.477814	+	0.827599i	−0.999677	−	0.0254309i	$-0.991904\pi$
$599$	−7004.86	+	12132.8i	−0.477814	+	0.827599i	0.521862	+	0.853030i	$0.325238\pi$
$600$		0			0
$601$	1621.78	+	2809.01i	0.110073	+	0.190652i	0.915800	−	0.401636i	$-0.131558\pi$
$601$	1621.78	+	2809.01i	0.110073	+	0.190652i	−0.805726	+	0.592288i	$0.798225\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6486.67	+	11235.2i	0.435902	+	0.755004i
$606$		0			0
$607$	5863.32	−	10155.6i	0.392068	−	0.679081i	−0.600655	−	0.799509i	$-0.705093\pi$
$607$	5863.32	−	10155.6i	0.392068	−	0.679081i	0.992722	+	0.120428i	$0.0384266\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−399.868			−0.0264762
$612$		0			0
$613$	20343.9			1.34043			0.670215	−	0.742167i	$-0.266202\pi$
$613$	20343.9			1.34043			0.670215	+	0.742167i	$0.266202\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10946.7	+	18960.3i	−0.714260	+	1.23714i	0.248984	+	0.968508i	$0.419903\pi$
$617$	−10946.7	+	18960.3i	−0.714260	+	1.23714i	−0.963244	+	0.268627i	$0.913430\pi$
$618$		0			0
$619$	−14883.5	−	25779.0i	−0.966426	−	1.67390i	−0.705733	−	0.708478i	$-0.749382\pi$
$619$	−14883.5	−	25779.0i	−0.966426	−	1.67390i	−0.260693	−	0.965422i	$-0.583951\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.02352	−	10.4330i	−0.000387363	−	0.000670933i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1955.19			−0.123941
$630$		0			0
$631$	−6594.67			−0.416054			−0.208027	−	0.978123i	$-0.566704\pi$
$631$	−6594.67			−0.416054			−0.208027	+	0.978123i	$0.566704\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2909.22	−	5038.91i	0.181809	−	0.314902i
$636$		0			0
$637$	1188.72	+	2058.92i	0.0739385	+	0.128065i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8080.71	−	13996.2i	−0.497923	−	0.862429i	0.502074	−	0.864825i	$-0.332571\pi$
$641$	−8080.71	−	13996.2i	−0.497923	−	0.862429i	−0.999997	+	0.00239620i	$0.999237\pi$
$642$		0			0
$643$	7938.09	−	13749.2i	0.486855	−	0.843258i	−0.513031	−	0.858370i	$-0.671477\pi$
$643$	7938.09	−	13749.2i	0.486855	−	0.843258i	0.999886	+	0.0151126i	$0.00481067\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−624.258			−0.0379322			−0.0189661	−	0.999820i	$-0.506037\pi$
$647$	−624.258			−0.0379322			−0.0189661	+	0.999820i	$0.506037\pi$
$648$		0			0
$649$	−34250.1			−2.07155
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5073.19	+	8787.03i	−0.304027	+	0.526589i	−0.977044	−	0.213037i	$-0.931664\pi$
$653$	−5073.19	+	8787.03i	−0.304027	+	0.526589i	0.673018	+	0.739626i	$0.264998\pi$
$654$		0			0
$655$	247.487	+	428.659i	0.0147635	+	0.0255712i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12025.2	−	20828.2i	−0.710826	−	1.23119i	−0.964548	−	0.263909i	$-0.914988\pi$
$659$	−12025.2	−	20828.2i	−0.710826	−	1.23119i	0.253722	−	0.967277i	$-0.418345\pi$
$660$		0			0
$661$	493.002	−	853.904i	0.0290099	−	0.0502466i	−0.851156	−	0.524913i	$-0.824098\pi$
$661$	493.002	−	853.904i	0.0290099	−	0.0502466i	0.880166	+	0.474666i	$0.157431\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−350.427			−0.0204345
$666$		0			0
$667$	−4787.20			−0.277903
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24797.6	+	42950.8i	−1.42668	+	2.47108i
$672$		0			0
$673$	−15027.8	−	26028.8i	−0.860739	−	1.49084i	−0.871217	−	0.490899i	$-0.836669\pi$
$673$	−15027.8	−	26028.8i	−0.860739	−	1.49084i	0.0104776	−	0.999945i	$-0.496665\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11185.3	−	19373.5i	−0.634986	−	1.09983i	−0.986518	−	0.163652i	$-0.947672\pi$
$677$	−11185.3	−	19373.5i	−0.634986	−	1.09983i	0.351532	−	0.936176i	$-0.385661\pi$
$678$		0			0
$679$	302.622	−	524.158i	0.0171040	−	0.0296249i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	27040.9			1.51492			0.757460	−	0.652881i	$-0.226440\pi$
$683$	27040.9			1.51492			0.757460	+	0.652881i	$0.226440\pi$
$684$		0			0
$685$	874.537			0.0487801
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2335.08	−	4044.47i	0.129114	−	0.223632i
$690$		0			0
$691$	11300.7	+	19573.5i	0.622143	+	1.07758i	0.989086	+	0.147340i	$0.0470710\pi$
$691$	11300.7	+	19573.5i	0.622143	+	1.07758i	−0.366943	+	0.930243i	$0.619596\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2453.78	+	4250.08i	0.133924	+	0.231964i
$696$		0			0
$697$	−2962.86	+	5131.82i	−0.161013	+	0.278883i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20238.5			−1.09044			−0.545219	−	0.838294i	$-0.683554\pi$
$701$	−20238.5			−1.09044			−0.545219	+	0.838294i	$0.683554\pi$
$702$		0			0
$703$	−1395.46			−0.0748658
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−349.876	+	606.003i	−0.0186117	+	0.0322363i
$708$		0			0
$709$	−3979.88	−	6893.36i	−0.210815	−	0.365142i	0.741155	−	0.671334i	$-0.234278\pi$
$709$	−3979.88	−	6893.36i	−0.210815	−	0.365142i	−0.951970	+	0.306192i	$0.900945\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8610.07	+	14913.1i	0.452244	+	0.783309i
$714$		0			0
$715$	−1088.57	+	1885.46i	−0.0569373	+	0.0986182i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−22376.4			−1.16064			−0.580320	−	0.814389i	$-0.697072\pi$
$719$	−22376.4			−1.16064			−0.580320	+	0.814389i	$0.697072\pi$
$720$		0			0
$721$	−1326.23			−0.0685039
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−694.530	+	1202.96i	−0.0355782	+	0.0616232i
$726$		0			0
$727$	−6639.11	−	11499.3i	−0.338694	−	0.586636i	0.645493	−	0.763766i	$-0.276652\pi$
$727$	−6639.11	−	11499.3i	−0.338694	−	0.586636i	−0.984187	+	0.177130i	$0.943319\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−15306.5	−	26511.6i	−0.774461	−	1.34141i
$732$		0			0
$733$	5863.33	−	10155.6i	0.295453	−	0.511740i	−0.679637	−	0.733548i	$-0.737863\pi$
$733$	5863.33	−	10155.6i	0.295453	−	0.511740i	0.975090	+	0.221809i	$0.0711961\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−39570.5			−1.97774
$738$		0			0
$739$	−10822.8			−0.538733			−0.269366	−	0.963038i	$-0.586814\pi$
$739$	−10822.8			−0.538733			−0.269366	+	0.963038i	$0.586814\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7554.29	−	13084.4i	0.373002	−	0.646058i	−0.617024	−	0.786944i	$-0.711662\pi$
$743$	7554.29	−	13084.4i	0.373002	−	0.646058i	0.990026	+	0.140886i	$0.0449953\pi$
$744$		0			0
$745$	−5767.34	−	9989.32i	−0.283623	−	0.491249i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−86.0442	−	149.033i	−0.00419758	−	0.00727042i
$750$		0			0
$751$	1964.75	−	3403.05i	0.0954659	−	0.165352i	−0.814337	−	0.580392i	$-0.802899\pi$
$751$	1964.75	−	3403.05i	0.0954659	−	0.165352i	0.909803	+	0.415040i	$0.136233\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9909.01			0.477650
$756$		0			0
$757$	−32082.2			−1.54035			−0.770176	−	0.637831i	$-0.779832\pi$
$757$	−32082.2			−1.54035			−0.770176	+	0.637831i	$0.779832\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−14674.1	+	25416.2i	−0.698994	+	1.21069i	0.269822	+	0.962910i	$0.413035\pi$
$761$	−14674.1	+	25416.2i	−0.698994	+	1.21069i	−0.968816	+	0.247782i	$0.920298\pi$
$762$		0			0
$763$	−710.111	−	1229.95i	−0.0336930	−	0.0583579i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1899.48	−	3289.99i	−0.0894212	−	0.154882i
$768$		0			0
$769$	−7775.42	+	13467.4i	−0.364615	+	0.631532i	−0.988714	−	0.149813i	$-0.952133\pi$
$769$	−7775.42	+	13467.4i	−0.364615	+	0.631532i	0.624099	+	0.781345i	$0.285466\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	29954.5			1.39378			0.696888	−	0.717180i	$-0.254567\pi$
$773$	29954.5			1.39378			0.696888	+	0.717180i	$0.254567\pi$
$774$		0			0
$775$	4996.62			0.231592
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2114.65	+	3662.67i	−0.0972594	+	0.168458i
$780$		0			0
$781$	−3224.91	−	5585.71i	−0.147755	−	0.255918i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1082.87	−	1875.58i	−0.0492346	−	0.0852768i
$786$		0			0
$787$	7801.11	−	13511.9i	0.353341	−	0.612005i	−0.633491	−	0.773750i	$-0.718379\pi$
$787$	7801.11	−	13511.9i	0.353341	−	0.612005i	0.986833	+	0.161745i	$0.0517121\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2125.64			−0.0955488
$792$		0			0
$793$	−5501.01			−0.246339
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15314.5	+	26525.4i	−0.680635	+	1.17889i	0.294152	+	0.955759i	$0.404963\pi$
$797$	−15314.5	+	26525.4i	−0.680635	+	1.17889i	−0.974787	+	0.223136i	$0.928371\pi$
$798$		0			0
$799$	2975.06	+	5152.95i	0.131727	+	0.228158i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6285.95	−	10887.6i	−0.276247	−	0.478474i
$804$		0			0
$805$	−204.538	+	354.271i	−0.00895531	+	0.0155111i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	19959.7			0.867425			0.433713	−	0.901051i	$-0.357203\pi$
$809$	19959.7			0.867425			0.433713	+	0.901051i	$0.357203\pi$
$810$		0			0
$811$	3768.40			0.163165			0.0815823	−	0.996667i	$-0.474003\pi$
$811$	3768.40			0.163165			0.0815823	+	0.996667i	$0.474003\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8619.30	−	14929.1i	0.370455	−	0.641647i
$816$		0			0
$817$	−10924.5	−	18921.8i	−0.467810	−	0.810270i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	14643.8	+	25363.8i	0.622500	+	1.07820i	0.989019	+	0.147791i	$0.0472162\pi$
$821$	14643.8	+	25363.8i	0.622500	+	1.07820i	−0.366519	+	0.930411i	$0.619451\pi$
$822$		0			0
$823$	530.591	−	919.010i	0.0224729	−	0.0389243i	−0.854570	−	0.519336i	$-0.826179\pi$
$823$	530.591	−	919.010i	0.0224729	−	0.0389243i	0.877043	+	0.480412i	$0.159513\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	43016.7			1.80875			0.904375	−	0.426739i	$-0.140338\pi$
$827$	43016.7			1.80875			0.904375	+	0.426739i	$0.140338\pi$
$828$		0			0
$829$	−27947.5			−1.17088			−0.585439	−	0.810716i	$-0.699078\pi$
$829$	−27947.5			−1.17088			−0.585439	+	0.810716i	$0.699078\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	17688.4	−	30637.2i	0.735734	−	1.27433i
$834$		0			0
$835$	−2637.22	−	4567.81i	−0.109299	−	0.189312i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6547.78	+	11341.1i	0.269433	+	0.466672i	0.968716	−	0.248174i	$-0.0798303\pi$
$839$	6547.78	+	11341.1i	0.269433	+	0.466672i	−0.699282	+	0.714846i	$0.746497\pi$
$840$		0			0
$841$	10650.9	−	18447.9i	0.436710	−	0.756403i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10743.5			0.437382
$846$		0			0
$847$	2463.86			0.0999517
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−814.504	+	1410.76i	−0.0328094	+	0.0568276i
$852$		0			0
$853$	−16112.7	−	27908.1i	−0.646764	−	1.12023i	−0.983891	−	0.178769i	$-0.942789\pi$
$853$	−16112.7	−	27908.1i	−0.646764	−	1.12023i	0.337127	−	0.941459i	$-0.390545\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2625.97	−	4548.32i	−0.104669	−	0.181293i	0.808934	−	0.587900i	$-0.200045\pi$
$857$	−2625.97	−	4548.32i	−0.104669	−	0.181293i	−0.913603	+	0.406607i	$0.866712\pi$
$858$		0			0
$859$	17564.8	−	30423.1i	0.697675	−	1.20841i	−0.271595	−	0.962412i	$-0.587551\pi$
$859$	17564.8	−	30423.1i	0.697675	−	1.20841i	0.969270	−	0.245998i	$-0.0791156\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	47402.3			1.86975			0.934874	−	0.354979i	$-0.115512\pi$
$863$	47402.3			1.86975			0.934874	+	0.354979i	$0.115512\pi$
$864$		0			0
$865$	3160.72			0.124240
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−20862.0	+	36134.1i	−0.814379	+	1.41055i
$870$		0			0
$871$	−2194.54	−	3801.05i	−0.0853721	−	0.147869i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	59.3490	+	102.796i	0.00229299	+	0.00397157i
$876$		0			0
$877$	−11863.6	+	20548.4i	−0.456792	+	0.791187i	−0.998789	−	0.0491932i	$-0.984335\pi$
$877$	−11863.6	+	20548.4i	−0.456792	+	0.791187i	0.541997	+	0.840380i	$0.317668\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−43690.9			−1.67081			−0.835406	−	0.549634i	$-0.814767\pi$
$881$	−43690.9			−1.67081			−0.835406	+	0.549634i	$0.814767\pi$
$882$		0			0
$883$	7852.57			0.299275			0.149638	−	0.988741i	$-0.452189\pi$
$883$	7852.57			0.299275			0.149638	+	0.988741i	$0.452189\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8007.15	−	13868.8i	0.303105	−	0.524993i	−0.673733	−	0.738975i	$-0.735310\pi$
$887$	8007.15	−	13868.8i	0.303105	−	0.524993i	0.976838	+	0.213982i	$0.0686435\pi$
$888$		0			0
$889$	−552.509	−	956.974i	−0.0208443	−	0.0361034i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2123.35	+	3677.75i	0.0795692	+	0.137818i
$894$		0			0
$895$	−3323.33	+	5756.18i	−0.124119	+	0.214981i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11105.0			0.411981
$900$		0			0
$901$	−69492.8			−2.56952
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3671.30	−	6358.88i	0.134849	−	0.233565i
$906$		0			0
$907$	−12936.7	−	22407.0i	−0.473600	−	0.820299i	0.525943	−	0.850520i	$-0.323712\pi$
$907$	−12936.7	−	22407.0i	−0.473600	−	0.820299i	−0.999543	+	0.0302205i	$0.990379\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−11646.5	−	20172.4i	−0.423564	−	0.733634i	0.572721	−	0.819750i	$-0.305888\pi$
$911$	−11646.5	−	20172.4i	−0.423564	−	0.733634i	−0.996285	+	0.0861158i	$0.972554\pi$
$912$		0			0
$913$	−42211.0	+	73111.5i	−1.53010	+	2.65021i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	94.0038			0.00338525
$918$		0			0
$919$	42422.5			1.52273			0.761365	−	0.648323i	$-0.224529\pi$
$919$	42422.5			1.52273			0.761365	+	0.648323i	$0.224529\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	357.700	−	619.555i	0.0127561	−	0.0220942i
$924$		0			0
$925$	236.337	+	409.348i	0.00840078	+	0.0145506i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17298.0	−	29961.0i	−0.610903	−	1.05811i	−0.991089	−	0.133205i	$-0.957473\pi$
$929$	−17298.0	−	29961.0i	−0.610903	−	1.05811i	0.380186	−	0.924910i	$-0.375860\pi$
$930$		0			0
$931$	12624.5	−	21866.3i	0.444417	−	0.769752i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	32396.2			1.13312
$936$		0			0
$937$	17173.8			0.598767			0.299383	−	0.954133i	$-0.403219\pi$
$937$	17173.8			0.598767			0.299383	+	0.954133i	$0.403219\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	16746.7	−	29006.2i	0.580158	−	1.00486i	−0.415303	−	0.909683i	$-0.636324\pi$
$941$	16746.7	−	29006.2i	0.580158	−	1.00486i	0.995460	−	0.0951790i	$-0.0303423\pi$
$942$		0			0
$943$	2468.56	+	4275.68i	0.0852466	+	0.147651i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−10738.5	−	18599.6i	−0.368484	−	0.638233i	0.620845	−	0.783934i	$-0.286790\pi$
$947$	−10738.5	−	18599.6i	−0.368484	−	0.638233i	−0.989329	+	0.145700i	$0.953456\pi$
$948$		0			0
$949$	697.225	−	1207.63i	0.0238492	−	0.0413080i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	19415.1			0.659934			0.329967	−	0.943992i	$-0.392962\pi$
$953$	19415.1			0.659934			0.329967	+	0.943992i	$0.392962\pi$
$954$		0			0
$955$	2411.95			0.0817266
$956$		0			0
$957$		0			0
$958$		0			0
$959$	83.0447	−	143.838i	0.00279630	−	0.00484334i
$960$		0			0
$961$	−5077.43	−	8794.37i	−0.170435	−	0.295202i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−858.383	−	1486.76i	−0.0286345	−	0.0495965i
$966$		0			0
$967$	27465.9	−	47572.4i	0.913386	−	1.58203i	0.104139	−	0.994563i	$-0.466791\pi$
$967$	27465.9	−	47572.4i	0.913386	−	1.58203i	0.809247	−	0.587469i	$-0.199875\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.202715			6.69972e−6			3.34986e−6	−	1.00000i	$-0.499999\pi$
$971$	0.202715			6.69972e−6			3.34986e−6		1.00000i	$0.499999\pi$
$972$		0			0
$973$	932.030			0.0307086
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11436.7	−	19809.0i	0.374508	−	0.648666i	−0.615746	−	0.787945i	$-0.711145\pi$
$977$	11436.7	−	19809.0i	0.374508	−	0.648666i	0.990253	+	0.139279i	$0.0444784\pi$
$978$		0			0
$979$	397.442	+	688.390i	0.0129748	+	0.0224730i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13129.7	−	22741.4i	−0.426016	−	0.737881i	0.570499	−	0.821299i	$-0.306750\pi$
$983$	−13129.7	−	22741.4i	−0.426016	−	0.737881i	−0.996515	+	0.0834171i	$0.973417\pi$
$984$		0			0
$985$	−11268.9	+	19518.2i	−0.364524	+	0.631373i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−25505.8			−0.820058
$990$		0			0
$991$	−40250.1			−1.29020			−0.645099	−	0.764099i	$-0.723184\pi$
$991$	−40250.1			−1.29020			−0.645099	+	0.764099i	$0.723184\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4375.06	+	7577.83i	−0.139396	+	0.241441i
$996$		0			0
$997$	29543.6	+	51171.0i	0.938469	+	1.62548i	0.768327	+	0.640057i	$0.221089\pi$
$997$	29543.6	+	51171.0i	0.938469	+	1.62548i	0.170142	+	0.985420i	$0.445577\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.u.1081.2		6
3.2	odd	2		1620.4.i.s.1081.2		6
9.2	odd	6		1620.4.i.s.541.2		6
9.4	even	3		1620.4.a.d.1.2	✓	3
9.5	odd	6		1620.4.a.f.1.2	yes	3
9.7	even	3	inner	1620.4.i.u.541.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.d.1.2	✓	3	9.4	even	3
1620.4.a.f.1.2	yes	3	9.5	odd	6
1620.4.i.s.541.2		6	9.2	odd	6
1620.4.i.s.1081.2		6	3.2	odd	2
1620.4.i.u.541.2		6	9.7	even	3	inner
1620.4.i.u.1081.2		6	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

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	1620.4.i.u.1081.2

Level	$1620$
Weight	$4$
Character	1620.1081
Analytic conductor	$95.583$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$