LMFDB - Embedded newform 1800.3.v.k.1657.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,3,Mod(793,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1800.793");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1800.v (of order $4$, degree $2$, 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:	$49.0464475849$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1657.2
Root			$3.70156i$ of defining polynomial
Character	$\chi$	$=$	1800.1657
Dual form			1800.3.v.k.793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.298438 - 0.298438i) q^{7} +O(q^{10})$	$q+(-0.298438 - 0.298438i) q^{7} -11.4031 q^{11} +(1.59688 - 1.59688i) q^{13} +(10.4031 + 10.4031i) q^{17} +2.80625i q^{19} +(19.1047 - 19.1047i) q^{23} -5.61250i q^{29} -15.4031 q^{31} +(40.6125 + 40.6125i) q^{37} -70.6281 q^{41} +(-20.2984 + 20.2984i) q^{43} +(-42.5078 - 42.5078i) q^{47} -48.8219i q^{49} +(-3.00000 + 3.00000i) q^{53} -66.8062i q^{59} -44.5969 q^{61} +(-28.2984 - 28.2984i) q^{67} -32.5969 q^{71} +(-62.6125 + 62.6125i) q^{73} +(3.40312 + 3.40312i) q^{77} +118.450i q^{79} +(-11.2828 + 11.2828i) q^{83} +11.2250i q^{89} -0.953136 q^{91} +(-49.0000 - 49.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{7}+O(q^{10}) $ 4 * q - 14 * q^7	$ 4 q - 14 q^{7} - 20 q^{11} + 32 q^{13} + 16 q^{17} + 38 q^{23} - 36 q^{31} + 60 q^{37} - 52 q^{41} - 94 q^{43} - 106 q^{47} - 12 q^{53} - 204 q^{61} - 126 q^{67} - 156 q^{71} - 148 q^{73} - 12 q^{77} - 186 q^{83} - 388 q^{91} - 196 q^{97}+O(q^{100}) $ 4 * q - 14 * q^7 - 20 * q^11 + 32 * q^13 + 16 * q^17 + 38 * q^23 - 36 * q^31 + 60 * q^37 - 52 * q^41 - 94 * q^43 - 106 * q^47 - 12 * q^53 - 204 * q^61 - 126 * q^67 - 156 * q^71 - 148 * q^73 - 12 * q^77 - 186 * q^83 - 388 * q^91 - 196 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.298438	−	0.298438i	−0.0426340	−	0.0426340i	0.685468	−	0.728102i	$-0.259597\pi$
$7$	−0.298438	−	0.298438i	−0.0426340	−	0.0426340i	−0.728102	+	0.685468i	$0.759597\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−11.4031			−1.03665			−0.518324	−	0.855184i	$-0.673444\pi$
$11$	−11.4031			−1.03665			−0.518324	+	0.855184i	$0.673444\pi$
$12$		0			0
$13$	1.59688	−	1.59688i	0.122837	−	0.122837i	−0.643016	−	0.765853i	$-0.722317\pi$
$13$	1.59688	−	1.59688i	0.122837	−	0.122837i	0.765853	+	0.643016i	$0.222317\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	10.4031	+	10.4031i	0.611948	+	0.611948i	0.943453	−	0.331505i	$-0.107556\pi$
$17$	10.4031	+	10.4031i	0.611948	+	0.611948i	−0.331505	+	0.943453i	$0.607556\pi$
$18$		0			0
$19$			2.80625i			0.147697i	0.997269	+	0.0738486i	$0.0235282\pi$
$19$			2.80625i			0.147697i	−0.997269	+	0.0738486i	$0.976472\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	19.1047	−	19.1047i	0.830639	−	0.830639i	−0.156966	−	0.987604i	$-0.550171\pi$
$23$	19.1047	−	19.1047i	0.830639	−	0.830639i	0.987604	+	0.156966i	$0.0501712\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	5.61250i		−	0.193534i	−0.995307	−	0.0967672i	$-0.969150\pi$
$29$		−	5.61250i		−	0.193534i	0.995307	−	0.0967672i	$-0.0308502\pi$
$30$		0			0
$31$	−15.4031			−0.496875			−0.248437	−	0.968648i	$-0.579917\pi$
$31$	−15.4031			−0.496875			−0.248437	+	0.968648i	$0.579917\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	40.6125	+	40.6125i	1.09764	+	1.09764i	0.994687	+	0.102948i	$0.0328276\pi$
$37$	40.6125	+	40.6125i	1.09764	+	1.09764i	0.102948	+	0.994687i	$0.467172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−70.6281			−1.72264			−0.861319	−	0.508065i	$-0.830361\pi$
$41$	−70.6281			−1.72264			−0.861319	+	0.508065i	$0.830361\pi$
$42$		0			0
$43$	−20.2984	+	20.2984i	−0.472057	+	0.472057i	−0.902580	−	0.430523i	$-0.858329\pi$
$43$	−20.2984	+	20.2984i	−0.472057	+	0.472057i	0.430523	+	0.902580i	$0.358329\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−42.5078	−	42.5078i	−0.904422	−	0.904422i	0.0913934	−	0.995815i	$-0.470868\pi$
$47$	−42.5078	−	42.5078i	−0.904422	−	0.904422i	−0.995815	+	0.0913934i	$0.970868\pi$
$48$		0			0
$49$		−	48.8219i		−	0.996365i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.00000	+	3.00000i	−0.0566038	+	0.0566038i	−0.734842	−	0.678238i	$-0.762744\pi$
$53$	−3.00000	+	3.00000i	−0.0566038	+	0.0566038i	0.678238	+	0.734842i	$0.262744\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	66.8062i		−	1.13231i	−0.824299	−	0.566155i	$-0.808430\pi$
$59$		−	66.8062i		−	1.13231i	0.824299	−	0.566155i	$-0.191570\pi$
$60$		0			0
$61$	−44.5969			−0.731096			−0.365548	−	0.930792i	$-0.619118\pi$
$61$	−44.5969			−0.731096			−0.365548	+	0.930792i	$0.619118\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−28.2984	−	28.2984i	−0.422365	−	0.422365i	0.463652	−	0.886017i	$-0.346539\pi$
$67$	−28.2984	−	28.2984i	−0.422365	−	0.422365i	−0.886017	+	0.463652i	$0.846539\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−32.5969			−0.459111			−0.229555	−	0.973296i	$-0.573727\pi$
$71$	−32.5969			−0.459111			−0.229555	+	0.973296i	$0.573727\pi$
$72$		0			0
$73$	−62.6125	+	62.6125i	−0.857705	+	0.857705i	−0.991067	−	0.133362i	$-0.957423\pi$
$73$	−62.6125	+	62.6125i	−0.857705	+	0.857705i	0.133362	+	0.991067i	$0.457423\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.40312	+	3.40312i	0.0441964	+	0.0441964i
$78$		0			0
$79$			118.450i			1.49937i	0.661797	+	0.749683i	$0.269794\pi$
$79$			118.450i			1.49937i	−0.661797	+	0.749683i	$0.730206\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−11.2828	+	11.2828i	−0.135938	+	0.135938i	−0.771801	−	0.635864i	$-0.780644\pi$
$83$	−11.2828	+	11.2828i	−0.135938	+	0.135938i	0.635864	+	0.771801i	$0.280644\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			11.2250i			0.126124i	0.998010	+	0.0630618i	$0.0200865\pi$
$89$			11.2250i			0.126124i	−0.998010	+	0.0630618i	$0.979913\pi$
$90$		0			0
$91$	−0.953136			−0.0104740
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−49.0000	−	49.0000i	−0.505155	−	0.505155i	0.407881	−	0.913035i	$-0.366268\pi$
$97$	−49.0000	−	49.0000i	−0.505155	−	0.505155i	−0.913035	+	0.407881i	$0.866268\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	68.0312			0.673577			0.336788	−	0.941580i	$-0.390659\pi$
$101$	68.0312			0.673577			0.336788	+	0.941580i	$0.390659\pi$
$102$		0			0
$103$	−29.4922	+	29.4922i	−0.286332	+	0.286332i	−0.835628	−	0.549296i	$-0.814896\pi$
$103$	−29.4922	+	29.4922i	−0.286332	+	0.286332i	0.549296	+	0.835628i	$0.314896\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−92.5391	−	92.5391i	−0.864851	−	0.864851i	0.127046	−	0.991897i	$-0.459450\pi$
$107$	−92.5391	−	92.5391i	−0.864851	−	0.864851i	−0.991897	+	0.127046i	$0.959450\pi$
$108$		0			0
$109$		−	173.047i		−	1.58759i	−0.608188	−	0.793793i	$-0.708103\pi$
$109$		−	173.047i		−	1.58759i	0.608188	−	0.793793i	$-0.291897\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−80.8219	+	80.8219i	−0.715238	+	0.715238i	−0.967626	−	0.252388i	$-0.918784\pi$
$113$	−80.8219	+	80.8219i	−0.715238	+	0.715238i	0.252388	+	0.967626i	$0.418784\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	6.20937i		−	0.0521796i
$120$		0			0
$121$	9.03124			0.0746384
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−8.29844	−	8.29844i	−0.0653420	−	0.0653420i	0.673681	−	0.739023i	$-0.264712\pi$
$127$	−8.29844	−	8.29844i	−0.0653420	−	0.0653420i	−0.739023	+	0.673681i	$0.764712\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	107.884			0.823545			0.411772	−	0.911287i	$-0.364910\pi$
$131$	107.884			0.823545			0.411772	+	0.911287i	$0.364910\pi$
$132$		0			0
$133$	0.837491	−	0.837491i	0.00629692	−	0.00629692i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	35.3875	+	35.3875i	0.258303	+	0.258303i	0.824364	−	0.566061i	$-0.191533\pi$
$137$	35.3875	+	35.3875i	0.258303	+	0.258303i	−0.566061	+	0.824364i	$0.691533\pi$
$138$		0			0
$139$		−	221.194i		−	1.59132i	−0.605742	−	0.795661i	$-0.707124\pi$
$139$		−	221.194i		−	1.59132i	0.605742	−	0.795661i	$-0.292876\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−18.2094	+	18.2094i	−0.127338	+	0.127338i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			95.8531i			0.643309i	0.946857	+	0.321655i	$0.104239\pi$
$149$			95.8531i			0.643309i	−0.946857	+	0.321655i	$0.895761\pi$
$150$		0			0
$151$	−279.528			−1.85118			−0.925590	−	0.378528i	$-0.876430\pi$
$151$	−279.528			−1.85118			−0.925590	+	0.378528i	$0.876430\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−122.466	−	122.466i	−0.780036	−	0.780036i	0.199801	−	0.979837i	$-0.435970\pi$
$157$	−122.466	−	122.466i	−0.780036	−	0.780036i	−0.979837	+	0.199801i	$0.935970\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−11.4031			−0.0708269
$162$		0			0
$163$	−125.973	+	125.973i	−0.772843	+	0.772843i	−0.978603	−	0.205760i	$-0.934034\pi$
$163$	−125.973	+	125.973i	−0.772843	+	0.772843i	0.205760	+	0.978603i	$0.434034\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−81.6703	−	81.6703i	−0.489044	−	0.489044i	0.418961	−	0.908004i	$-0.362395\pi$
$167$	−81.6703	−	81.6703i	−0.489044	−	0.489044i	−0.908004	+	0.418961i	$0.862395\pi$
$168$		0			0
$169$			163.900i			0.969822i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−117.388	+	117.388i	−0.678540	+	0.678540i	−0.959670	−	0.281129i	$-0.909291\pi$
$173$	−117.388	+	117.388i	−0.678540	+	0.678540i	0.281129	+	0.959670i	$0.409291\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	104.419i		−	0.583345i	−0.956518	−	0.291672i	$-0.905788\pi$
$179$		−	104.419i		−	0.583345i	0.956518	−	0.291672i	$-0.0942117\pi$
$180$		0			0
$181$	−139.612			−0.771340			−0.385670	−	0.922637i	$-0.626030\pi$
$181$	−139.612			−0.771340			−0.385670	+	0.922637i	$0.626030\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−118.628	−	118.628i	−0.634375	−	0.634375i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−317.497			−1.66229			−0.831144	−	0.556058i	$-0.812313\pi$
$191$	−317.497			−1.66229			−0.831144	+	0.556058i	$0.812313\pi$
$192$		0			0
$193$	−15.1781	+	15.1781i	−0.0786432	+	0.0786432i	−0.745334	−	0.666691i	$-0.767710\pi$
$193$	−15.1781	+	15.1781i	−0.0786432	+	0.0786432i	0.666691	+	0.745334i	$0.267710\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	221.837	+	221.837i	1.12608	+	1.12608i	0.990809	+	0.135270i	$0.0431901\pi$
$197$	221.837	+	221.837i	1.12608	+	1.12608i	0.135270	+	0.990809i	$0.456810\pi$
$198$		0			0
$199$		−	210.512i		−	1.05785i	−0.848668	−	0.528926i	$-0.822595\pi$
$199$		−	210.512i		−	1.05785i	0.848668	−	0.528926i	$-0.177405\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.67498	+	1.67498i	−0.00825114	+	0.00825114i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	32.0000i		−	0.153110i
$210$		0			0
$211$	−217.372			−1.03020			−0.515099	−	0.857131i	$-0.672245\pi$
$211$	−217.372			−1.03020			−0.515099	+	0.857131i	$0.672245\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.59688	+	4.59688i	0.0211838	+	0.0211838i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	33.2250			0.150339
$222$		0			0
$223$	228.602	−	228.602i	1.02512	−	1.02512i	0.0254427	−	0.999676i	$-0.491900\pi$
$223$	228.602	−	228.602i	1.02512	−	1.02512i	0.999676	−	0.0254427i	$-0.00809954\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	123.167	+	123.167i	0.542587	+	0.542587i	0.924286	−	0.381700i	$-0.124661\pi$
$227$	123.167	+	123.167i	0.542587	+	0.542587i	−0.381700	+	0.924286i	$0.624661\pi$
$228$		0			0
$229$		−	60.0625i		−	0.262282i	−0.991364	−	0.131141i	$-0.958136\pi$
$229$		−	60.0625i		−	0.262282i	0.991364	−	0.131141i	$-0.0418640\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−99.2094	+	99.2094i	−0.425791	+	0.425791i	−0.887192	−	0.461401i	$-0.847347\pi$
$233$	−99.2094	+	99.2094i	−0.425791	+	0.425791i	0.461401	+	0.887192i	$0.347347\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	229.612i		−	0.960722i	−0.877071	−	0.480361i	$-0.840506\pi$
$239$		−	229.612i		−	0.960722i	0.877071	−	0.480361i	$-0.159494\pi$
$240$		0			0
$241$	0.178130			0.000739130			0.000369565	−	1.00000i	$-0.499882\pi$
$241$	0.178130			0.000739130			0.000369565		1.00000i	$0.499882\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	4.48123	+	4.48123i	0.0181426	+	0.0181426i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−263.466			−1.04966			−0.524832	−	0.851206i	$-0.675872\pi$
$251$	−263.466			−1.04966			−0.524832	+	0.851206i	$0.675872\pi$
$252$		0			0
$253$	−217.853	+	217.853i	−0.861079	+	0.861079i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−19.9219	−	19.9219i	−0.0775171	−	0.0775171i	0.667285	−	0.744802i	$-0.267456\pi$
$257$	−19.9219	−	19.9219i	−0.0775171	−	0.0775171i	−0.744802	+	0.667285i	$0.767456\pi$
$258$		0			0
$259$		−	24.2406i		−	0.0935931i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	220.361	−	220.361i	0.837874	−	0.837874i	−0.150705	−	0.988579i	$-0.548154\pi$
$263$	220.361	−	220.361i	0.837874	−	0.837874i	0.988579	+	0.150705i	$0.0481542\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			210.953i			0.784212i	0.919920	+	0.392106i	$0.128253\pi$
$269$			210.953i			0.784212i	−0.919920	+	0.392106i	$0.871747\pi$
$270$		0			0
$271$	−373.016			−1.37644			−0.688221	−	0.725501i	$-0.741608\pi$
$271$	−373.016			−1.37644			−0.688221	+	0.725501i	$0.741608\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−22.4031	−	22.4031i	−0.0808777	−	0.0808777i	0.665511	−	0.746388i	$-0.268214\pi$
$277$	−22.4031	−	22.4031i	−0.0808777	−	0.0808777i	−0.746388	+	0.665511i	$0.768214\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	153.372			0.545807			0.272904	−	0.962041i	$-0.412016\pi$
$281$	153.372			0.545807			0.272904	+	0.962041i	$0.412016\pi$
$282$		0			0
$283$	−55.9422	+	55.9422i	−0.197676	+	0.197676i	−0.799003	−	0.601327i	$-0.794639\pi$
$283$	−55.9422	+	55.9422i	−0.197676	+	0.197676i	0.601327	+	0.799003i	$0.294639\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	21.0781	+	21.0781i	0.0734429	+	0.0734429i
$288$		0			0
$289$		−	72.5500i		−	0.251038i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	237.691	−	237.691i	0.811231	−	0.811231i	−0.173588	−	0.984818i	$-0.555536\pi$
$293$	237.691	−	237.691i	0.811231	−	0.811231i	0.984818	+	0.173588i	$0.0555360\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	61.0156i		−	0.204066i
$300$		0			0
$301$	12.1156			0.0402513
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	115.702	+	115.702i	0.376878	+	0.376878i	0.869975	−	0.493097i	$-0.164135\pi$
$307$	115.702	+	115.702i	0.376878	+	0.376878i	−0.493097	+	0.869975i	$0.664135\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−57.3094			−0.184275			−0.0921373	−	0.995746i	$-0.529370\pi$
$311$	−57.3094			−0.184275			−0.0921373	+	0.995746i	$0.529370\pi$
$312$		0			0
$313$	251.334	−	251.334i	0.802985	−	0.802985i	−0.180576	−	0.983561i	$-0.557796\pi$
$313$	251.334	−	251.334i	0.802985	−	0.802985i	0.983561	+	0.180576i	$0.0577962\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−36.1093	−	36.1093i	−0.113910	−	0.113910i	0.647854	−	0.761764i	$-0.275666\pi$
$317$	−36.1093	−	36.1093i	−0.113910	−	0.113910i	−0.761764	+	0.647854i	$0.775666\pi$
$318$		0			0
$319$			64.0000i			0.200627i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−29.1938	+	29.1938i	−0.0903831	+	0.0903831i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$			25.3719i			0.0771182i
$330$		0			0
$331$	333.078			1.00628			0.503139	−	0.864205i	$-0.332178\pi$
$331$	333.078			1.00628			0.503139	+	0.864205i	$0.332178\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	368.675	+	368.675i	1.09399	+	1.09399i	0.995098	+	0.0988930i	$0.0315301\pi$
$337$	368.675	+	368.675i	1.09399	+	1.09399i	0.0988930	+	0.995098i	$0.468470\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	175.644			0.515084
$342$		0			0
$343$	−29.1938	+	29.1938i	−0.0851130	+	0.0851130i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−160.183	−	160.183i	−0.461622	−	0.461622i	0.437565	−	0.899187i	$-0.355841\pi$
$347$	−160.183	−	160.183i	−0.461622	−	0.461622i	−0.899187	+	0.437565i	$0.855841\pi$
$348$		0			0
$349$			424.962i			1.21766i	0.793302	+	0.608829i	$0.208360\pi$
$349$			424.962i			1.21766i	−0.793302	+	0.608829i	$0.791640\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−267.900	+	267.900i	−0.758923	+	0.758923i	−0.976126	−	0.217203i	$-0.930307\pi$
$353$	−267.900	+	267.900i	−0.758923	+	0.758923i	0.217203	+	0.976126i	$0.430307\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			545.675i			1.51999i	0.649931	+	0.759993i	$0.274798\pi$
$359$			545.675i			1.51999i	−0.649931	+	0.759993i	$0.725202\pi$
$360$		0			0
$361$	353.125			0.978186
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−355.523	−	355.523i	−0.968729	−	0.968729i	0.0307970	−	0.999526i	$-0.490195\pi$
$367$	−355.523	−	355.523i	−0.968729	−	0.968729i	−0.999526	+	0.0307970i	$0.990195\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.79063			0.00482649
$372$		0			0
$373$	−304.350	+	304.350i	−0.815952	+	0.815952i	−0.985519	−	0.169567i	$-0.945763\pi$
$373$	−304.350	+	304.350i	−0.815952	+	0.815952i	0.169567	+	0.985519i	$0.445763\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.96246	−	8.96246i	−0.0237731	−	0.0237731i
$378$		0			0
$379$		−	550.156i		−	1.45160i	−0.687906	−	0.725800i	$-0.741470\pi$
$379$		−	550.156i		−	1.45160i	0.687906	−	0.725800i	$-0.258530\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−215.702	+	215.702i	−0.563189	+	0.563189i	−0.930212	−	0.367023i	$-0.880377\pi$
$383$	−215.702	+	215.702i	−0.563189	+	0.563189i	0.367023	+	0.930212i	$0.380377\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	308.628i		−	0.793388i	−0.917951	−	0.396694i	$-0.870157\pi$
$389$		−	308.628i		−	0.793388i	0.917951	−	0.396694i	$-0.129843\pi$
$390$		0			0
$391$	397.497			1.01662
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	22.2250	+	22.2250i	0.0559824	+	0.0559824i	0.734544	−	0.678561i	$-0.237396\pi$
$397$	22.2250	+	22.2250i	0.0559824	+	0.0559824i	−0.678561	+	0.734544i	$0.737396\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−91.8000			−0.228928			−0.114464	−	0.993427i	$-0.536515\pi$
$401$	−91.8000			−0.228928			−0.114464	+	0.993427i	$0.536515\pi$
$402$		0			0
$403$	−24.5969	+	24.5969i	−0.0610344	+	0.0610344i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−463.109	−	463.109i	−1.13786	−	1.13786i
$408$		0			0
$409$			342.597i			0.837645i	0.908068	+	0.418823i	$0.137557\pi$
$409$			342.597i			0.837645i	−0.908068	+	0.418823i	$0.862443\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−19.9375	+	19.9375i	−0.0482749	+	0.0482749i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	588.356i		−	1.40419i	−0.712083	−	0.702096i	$-0.752248\pi$
$419$		−	588.356i		−	1.40419i	0.712083	−	0.702096i	$-0.247752\pi$
$420$		0			0
$421$	−52.7218			−0.125230			−0.0626150	−	0.998038i	$-0.519944\pi$
$421$	−52.7218			−0.125230			−0.0626150	+	0.998038i	$0.519944\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	13.3094	+	13.3094i	0.0311695	+	0.0311695i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−342.209			−0.793989			−0.396995	−	0.917821i	$-0.629947\pi$
$431$	−342.209			−0.793989			−0.396995	+	0.917821i	$0.629947\pi$
$432$		0			0
$433$	−4.34996	+	4.34996i	−0.0100461	+	0.0100461i	−0.712112	−	0.702066i	$-0.752261\pi$
$433$	−4.34996	+	4.34996i	−0.0100461	+	0.0100461i	0.702066	+	0.712112i	$0.252261\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	53.6125	+	53.6125i	0.122683	+	0.122683i
$438$		0			0
$439$			593.925i			1.35290i	0.736487	+	0.676452i	$0.236483\pi$
$439$			593.925i			1.35290i	−0.736487	+	0.676452i	$0.763517\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	449.492	−	449.492i	1.01466	−	1.01466i	0.0147640	−	0.999891i	$-0.495300\pi$
$443$	449.492	−	449.492i	1.01466	−	1.01466i	0.999891	−	0.0147640i	$-0.00469971\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	366.428i		−	0.816098i	−0.912960	−	0.408049i	$-0.866209\pi$
$449$		−	366.428i		−	0.816098i	0.912960	−	0.408049i	$-0.133791\pi$
$450$		0			0
$451$	805.381			1.78577
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	147.209	+	147.209i	0.322121	+	0.322121i	0.849580	−	0.527459i	$-0.176855\pi$
$457$	147.209	+	147.209i	0.322121	+	0.322121i	−0.527459	+	0.849580i	$0.676855\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	804.281			1.74464			0.872322	−	0.488931i	$-0.162613\pi$
$461$	804.281			1.74464			0.872322	+	0.488931i	$0.162613\pi$
$462$		0			0
$463$	185.670	−	185.670i	0.401016	−	0.401016i	−0.477575	−	0.878591i	$-0.658484\pi$
$463$	185.670	−	185.670i	0.401016	−	0.401016i	0.878591	+	0.477575i	$0.158484\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−135.345	−	135.345i	−0.289819	−	0.289819i	0.547190	−	0.837009i	$-0.315698\pi$
$467$	−135.345	−	135.345i	−0.289819	−	0.289819i	−0.837009	+	0.547190i	$0.815698\pi$
$468$		0			0
$469$			16.8907i			0.0360142i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	231.466	−	231.466i	0.489356	−	0.489356i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$			629.862i			1.31495i	0.753475	+	0.657476i	$0.228376\pi$
$479$			629.862i			1.31495i	−0.753475	+	0.657476i	$0.771624\pi$
$480$		0			0
$481$	129.706			0.269660
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	174.864	+	174.864i	0.359064	+	0.359064i	0.863468	−	0.504404i	$-0.168288\pi$
$487$	174.864	+	174.864i	0.359064	+	0.359064i	−0.504404	+	0.863468i	$0.668288\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	555.047			1.13044			0.565221	−	0.824940i	$-0.308791\pi$
$491$	555.047			1.13044			0.565221	+	0.824940i	$0.308791\pi$
$492$		0			0
$493$	58.3875	−	58.3875i	0.118433	−	0.118433i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.72814	+	9.72814i	0.0195737	+	0.0195737i
$498$		0			0
$499$			391.831i			0.785233i	0.919702	+	0.392616i	$0.128430\pi$
$499$			391.831i			0.785233i	−0.919702	+	0.392616i	$0.871570\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−103.283	+	103.283i	−0.205334	+	0.205334i	−0.802281	−	0.596947i	$-0.796380\pi$
$503$	−103.283	+	103.283i	−0.205334	+	0.205334i	0.596947	+	0.802281i	$0.296380\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$			349.737i			0.687107i	0.939133	+	0.343554i	$0.111631\pi$
$509$			349.737i			0.687107i	−0.939133	+	0.343554i	$0.888369\pi$
$510$		0			0
$511$	37.3719			0.0731348
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	484.722	+	484.722i	0.937566	+	0.937566i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	72.0937			0.138376			0.0691878	−	0.997604i	$-0.477959\pi$
$521$	72.0937			0.138376			0.0691878	+	0.997604i	$0.477959\pi$
$522$		0			0
$523$	383.345	−	383.345i	0.732974	−	0.732974i	−0.238234	−	0.971208i	$-0.576569\pi$
$523$	383.345	−	383.345i	0.732974	−	0.732974i	0.971208	+	0.238234i	$0.0765685\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−160.241	−	160.241i	−0.304062	−	0.304062i
$528$		0			0
$529$		−	200.978i		−	0.379921i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−112.784	+	112.784i	−0.211603	+	0.211603i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			556.722i			1.03288i
$540$		0			0
$541$	43.9688			0.0812731			0.0406366	−	0.999174i	$-0.487061\pi$
$541$	43.9688			0.0812731			0.0406366	+	0.999174i	$0.487061\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−111.942	−	111.942i	−0.204647	−	0.204647i	0.597340	−	0.801988i	$-0.296224\pi$
$547$	−111.942	−	111.942i	−0.204647	−	0.204647i	−0.801988	+	0.597340i	$0.796224\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	15.7501			0.0285845
$552$		0			0
$553$	35.3500	−	35.3500i	0.0639240	−	0.0639240i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	275.575	+	275.575i	0.494749	+	0.494749i	0.909799	−	0.415050i	$-0.136236\pi$
$557$	275.575	+	275.575i	0.494749	+	0.494749i	−0.415050	+	0.909799i	$0.636236\pi$
$558$		0			0
$559$			64.8282i			0.115972i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	240.236	−	240.236i	0.426707	−	0.426707i	−0.460798	−	0.887505i	$-0.652437\pi$
$563$	240.236	−	240.236i	0.426707	−	0.426707i	0.887505	+	0.460798i	$0.152437\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	108.753i		−	0.191130i	−0.995423	−	0.0955651i	$-0.969534\pi$
$569$		−	108.753i		−	0.191130i	0.995423	−	0.0955651i	$-0.0304658\pi$
$570$		0			0
$571$	−80.0718			−0.140231			−0.0701154	−	0.997539i	$-0.522337\pi$
$571$	−80.0718			−0.140231			−0.0701154	+	0.997539i	$0.522337\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	484.172	+	484.172i	0.839119	+	0.839119i	0.988743	−	0.149624i	$-0.0478062\pi$
$577$	484.172	+	484.172i	0.839119	+	0.839119i	−0.149624	+	0.988743i	$0.547806\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.73444			0.0115911
$582$		0			0
$583$	34.2094	−	34.2094i	0.0586782	−	0.0586782i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	342.517	+	342.517i	0.583504	+	0.583504i	0.935865	−	0.352360i	$-0.114621\pi$
$587$	342.517	+	342.517i	0.583504	+	0.583504i	−0.352360	+	0.935865i	$0.614621\pi$
$588$		0			0
$589$		−	43.2250i		−	0.0733871i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	39.4500	−	39.4500i	0.0665261	−	0.0665261i	−0.673061	−	0.739587i	$-0.735021\pi$
$593$	39.4500	−	39.4500i	0.0665261	−	0.0665261i	0.739587	+	0.673061i	$0.235021\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	589.987i		−	0.984954i	−0.870325	−	0.492477i	$-0.836092\pi$
$599$		−	589.987i		−	0.984954i	0.870325	−	0.492477i	$-0.163908\pi$
$600$		0			0
$601$	806.628			1.34214			0.671072	−	0.741393i	$-0.265834\pi$
$601$	806.628			1.34214			0.671072	+	0.741393i	$0.265834\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−391.167	−	391.167i	−0.644427	−	0.644427i	0.307214	−	0.951641i	$-0.400603\pi$
$607$	−391.167	−	391.167i	−0.644427	−	0.644427i	−0.951641	+	0.307214i	$0.900603\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−135.759			−0.222192
$612$		0			0
$613$	111.209	−	111.209i	0.181418	−	0.181418i	−0.610555	−	0.791974i	$-0.709054\pi$
$613$	111.209	−	111.209i	0.181418	−	0.181418i	0.791974	+	0.610555i	$0.209054\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	286.466	+	286.466i	0.464288	+	0.464288i	0.900058	−	0.435770i	$-0.143524\pi$
$617$	286.466	+	286.466i	0.464288	+	0.464288i	−0.435770	+	0.900058i	$0.643524\pi$
$618$		0			0
$619$			233.006i			0.376424i	0.982128	+	0.188212i	$0.0602692\pi$
$619$			233.006i			0.376424i	−0.982128	+	0.188212i	$0.939731\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.34996	−	3.34996i	0.00537715	−	0.00537715i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$			844.994i			1.34339i
$630$		0			0
$631$	79.7594			0.126402			0.0632008	−	0.998001i	$-0.479869\pi$
$631$	79.7594			0.126402			0.0632008	+	0.998001i	$0.479869\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−77.9625	−	77.9625i	−0.122390	−	0.122390i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−736.178			−1.14848			−0.574242	−	0.818686i	$-0.694703\pi$
$641$	−736.178			−1.14848			−0.574242	+	0.818686i	$0.694703\pi$
$642$		0			0
$643$	580.245	−	580.245i	0.902403	−	0.902403i	−0.0932404	−	0.995644i	$-0.529723\pi$
$643$	580.245	−	580.245i	0.902403	−	0.902403i	0.995644	+	0.0932404i	$0.0297225\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	711.586	+	711.586i	1.09982	+	1.09982i	0.994431	+	0.105393i	$0.0336101\pi$
$647$	711.586	+	711.586i	1.09982	+	1.09982i	0.105393	+	0.994431i	$0.466390\pi$
$648$		0			0
$649$			761.800i			1.17381i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	521.187	−	521.187i	0.798143	−	0.798143i	−0.184659	−	0.982803i	$-0.559118\pi$
$653$	521.187	−	521.187i	0.798143	−	0.798143i	0.982803	+	0.184659i	$0.0591182\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	444.606i		−	0.674668i	−0.941385	−	0.337334i	$-0.890475\pi$
$659$		−	444.606i		−	0.674668i	0.941385	−	0.337334i	$-0.109525\pi$
$660$		0			0
$661$	−479.947			−0.726092			−0.363046	−	0.931771i	$-0.618263\pi$
$661$	−479.947			−0.726092			−0.363046	+	0.931771i	$0.618263\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−107.225	−	107.225i	−0.160757	−	0.160757i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	508.544			0.757889
$672$		0			0
$673$	440.550	−	440.550i	0.654606	−	0.654606i	−0.299492	−	0.954099i	$-0.596817\pi$
$673$	440.550	−	440.550i	0.654606	−	0.654606i	0.954099	+	0.299492i	$0.0968173\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	494.087	+	494.087i	0.729819	+	0.729819i	0.970584	−	0.240765i	$-0.0773982\pi$
$677$	494.087	+	494.087i	0.729819	+	0.729819i	−0.240765	+	0.970584i	$0.577398\pi$
$678$		0			0
$679$			29.2469i			0.0430735i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	362.748	−	362.748i	0.531110	−	0.531110i	−0.389792	−	0.920903i	$-0.627453\pi$
$683$	362.748	−	362.748i	0.531110	−	0.531110i	0.920903	+	0.389792i	$0.127453\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$			9.58125i			0.0139060i
$690$		0			0
$691$	−228.597			−0.330820			−0.165410	−	0.986225i	$-0.552895\pi$
$691$	−228.597			−0.330820			−0.165410	+	0.986225i	$0.552895\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−734.753	−	734.753i	−1.05417	−	1.05417i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−264.178			−0.376859			−0.188429	−	0.982087i	$-0.560340\pi$
$701$	−264.178			−0.376859			−0.188429	+	0.982087i	$0.560340\pi$
$702$		0			0
$703$	−113.969	+	113.969i	−0.162118	+	0.162118i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−20.3031	−	20.3031i	−0.0287173	−	0.0287173i
$708$		0			0
$709$		−	1122.76i		−	1.58359i	−0.610790	−	0.791793i	$-0.709148\pi$
$709$		−	1122.76i		−	1.58359i	0.610790	−	0.791793i	$-0.290852\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−294.272	+	294.272i	−0.412724	+	0.412724i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			320.588i			0.445880i	0.974832	+	0.222940i	$0.0715654\pi$
$719$			320.588i			0.445880i	−0.974832	+	0.222940i	$0.928435\pi$
$720$		0			0
$721$	17.6032			0.0244149
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	521.502	+	521.502i	0.717334	+	0.717334i	0.968058	−	0.250725i	$-0.0806689\pi$
$727$	521.502	+	521.502i	0.717334	+	0.717334i	−0.250725	+	0.968058i	$0.580669\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−422.334			−0.577749
$732$		0			0
$733$	680.737	−	680.737i	0.928700	−	0.928700i	−0.0689216	−	0.997622i	$-0.521956\pi$
$733$	680.737	−	680.737i	0.928700	−	0.928700i	0.997622	+	0.0689216i	$0.0219558\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	322.691	+	322.691i	0.437843	+	0.437843i
$738$		0			0
$739$			408.669i			0.553002i	0.961014	+	0.276501i	$0.0891750\pi$
$739$			408.669i			0.553002i	−0.961014	+	0.276501i	$0.910825\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−819.301	+	819.301i	−1.10269	+	1.10269i	−0.108609	+	0.994085i	$0.534640\pi$
$743$	−819.301	+	819.301i	−1.10269	+	1.10269i	−0.994085	+	0.108609i	$0.965360\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			55.2343i			0.0737441i
$750$		0			0
$751$	48.5969			0.0647096			0.0323548	−	0.999476i	$-0.489699\pi$
$751$	48.5969			0.0647096			0.0323548	+	0.999476i	$0.489699\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−481.628	−	481.628i	−0.636233	−	0.636233i	0.313391	−	0.949624i	$-0.398535\pi$
$757$	−481.628	−	481.628i	−0.636233	−	0.636233i	−0.949624	+	0.313391i	$0.898535\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	588.325			0.773095			0.386547	−	0.922270i	$-0.373668\pi$
$761$	588.325			0.773095			0.386547	+	0.922270i	$0.373668\pi$
$762$		0			0
$763$	−51.6437	+	51.6437i	−0.0676851	+	0.0676851i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−106.681	−	106.681i	−0.139089	−	0.139089i
$768$		0			0
$769$			65.6750i			0.0854031i	0.999088	+	0.0427015i	$0.0135965\pi$
$769$			65.6750i			0.0854031i	−0.999088	+	0.0427015i	$0.986404\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−703.459	+	703.459i	−0.910038	+	0.910038i	−0.996275	−	0.0862368i	$-0.972516\pi$
$773$	−703.459	+	703.459i	−0.910038	+	0.910038i	0.0862368	+	0.996275i	$0.472516\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	198.200i		−	0.254429i
$780$		0			0
$781$	371.706			0.475936
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	1012.79	+	1012.79i	1.28690	+	1.28690i	0.936661	+	0.350237i	$0.113899\pi$
$787$	1012.79	+	1012.79i	1.28690	+	1.28690i	0.350237	+	0.936661i	$0.386101\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	48.2406			0.0609869
$792$		0			0
$793$	−71.2157	+	71.2157i	−0.0898054	+	0.0898054i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−996.109	−	996.109i	−1.24982	−	1.24982i	−0.955797	−	0.294026i	$-0.905005\pi$
$797$	−996.109	−	996.109i	−1.24982	−	1.24982i	−0.294026	−	0.955797i	$-0.594995\pi$
$798$		0			0
$799$		−	884.428i		−	1.10692i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	713.978	−	713.978i	0.889138	−	0.889138i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1105.92i			1.36703i	0.729938	+	0.683514i	$0.239549\pi$
$809$			1105.92i			1.36703i	−0.729938	+	0.683514i	$0.760451\pi$
$810$		0			0
$811$	−697.034			−0.859475			−0.429738	−	0.902954i	$-0.641394\pi$
$811$	−697.034			−0.859475			−0.429738	+	0.902954i	$0.641394\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−56.9625	−	56.9625i	−0.0697215	−	0.0697215i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1091.30			1.32923			0.664614	−	0.747187i	$-0.268596\pi$
$821$	1091.30			1.32923			0.664614	+	0.747187i	$0.268596\pi$
$822$		0			0
$823$	−1147.48	+	1147.48i	−1.39426	+	1.39426i	−0.578782	+	0.815482i	$0.696472\pi$
$823$	−1147.48	+	1147.48i	−1.39426	+	1.39426i	−0.815482	+	0.578782i	$0.803528\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	673.492	+	673.492i	0.814380	+	0.814380i	0.985287	−	0.170907i	$-0.0546698\pi$
$827$	673.492	+	673.492i	0.814380	+	0.814380i	−0.170907	+	0.985287i	$0.554670\pi$
$828$		0			0
$829$			586.197i			0.707113i	0.935413	+	0.353557i	$0.115028\pi$
$829$			586.197i			0.707113i	−0.935413	+	0.353557i	$0.884972\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	507.900	−	507.900i	0.609724	−	0.609724i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			306.512i			0.365331i	0.983175	+	0.182665i	$0.0584725\pi$
$839$			306.512i			0.365331i	−0.983175	+	0.182665i	$0.941528\pi$
$840$		0			0
$841$	809.500			0.962544
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−2.69526	−	2.69526i	−0.00318213	−	0.00318213i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1551.78			1.82348
$852$		0			0
$853$	−4.14059	+	4.14059i	−0.00485415	+	0.00485415i	−0.709530	−	0.704676i	$-0.751093\pi$
$853$	−4.14059	+	4.14059i	−0.00485415	+	0.00485415i	0.704676	+	0.709530i	$0.251093\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−644.737	−	644.737i	−0.752319	−	0.752319i	0.222592	−	0.974912i	$-0.428548\pi$
$857$	−644.737	−	644.737i	−0.752319	−	0.752319i	−0.974912	+	0.222592i	$0.928548\pi$
$858$		0			0
$859$		−	170.094i		−	0.198014i	−0.995087	−	0.0990068i	$-0.968433\pi$
$859$		−	170.094i		−	0.198014i	0.995087	−	0.0990068i	$-0.0315666\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−412.183	+	412.183i	−0.477616	+	0.477616i	−0.904369	−	0.426752i	$-0.859658\pi$
$863$	−412.183	+	412.183i	−0.477616	+	0.477616i	0.426752	+	0.904369i	$0.359658\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1350.70i		−	1.55432i
$870$		0			0
$871$	−90.3782			−0.103764
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−882.841	−	882.841i	−1.00666	−	1.00666i	−0.999978	−	0.00668198i	$-0.997873\pi$
$877$	−882.841	−	882.841i	−1.00666	−	1.00666i	−0.00668198	−	0.999978i	$-0.502127\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−685.078			−0.777614			−0.388807	−	0.921319i	$-0.627113\pi$
$881$	−685.078			−0.777614			−0.388807	+	0.921319i	$0.627113\pi$
$882$		0			0
$883$	742.214	−	742.214i	0.840559	−	0.840559i	−0.148372	−	0.988932i	$-0.547403\pi$
$883$	742.214	−	742.214i	0.840559	−	0.840559i	0.988932	+	0.148372i	$0.0474033\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−882.464	−	882.464i	−0.994886	−	0.994886i	0.00510089	−	0.999987i	$-0.498376\pi$
$887$	−882.464	−	882.464i	−0.994886	−	0.994886i	−0.999987	+	0.00510089i	$0.998376\pi$
$888$		0			0
$889$			4.95314i			0.00557158i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	119.287	−	119.287i	0.133581	−	0.133581i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			86.4500i			0.0961624i
$900$		0			0
$901$	−62.4187			−0.0692772
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	457.439	+	457.439i	0.504343	+	0.504343i	0.912784	−	0.408442i	$-0.133928\pi$
$907$	457.439	+	457.439i	0.504343	+	0.504343i	−0.408442	+	0.912784i	$0.633928\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	591.903			0.649729			0.324864	−	0.945761i	$-0.394681\pi$
$911$	591.903			0.649729			0.324864	+	0.945761i	$0.394681\pi$
$912$		0			0
$913$	128.659	−	128.659i	0.140919	−	0.140919i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−32.1968	−	32.1968i	−0.0351110	−	0.0351110i
$918$		0			0
$919$			1391.66i			1.51432i	0.653228	+	0.757161i	$0.273414\pi$
$919$			1391.66i			1.51432i	−0.653228	+	0.757161i	$0.726586\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−52.0532	+	52.0532i	−0.0563956	+	0.0563956i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	81.5719i		−	0.0878062i	−0.999036	−	0.0439031i	$-0.986021\pi$
$929$		−	81.5719i		−	0.0878062i	0.999036	−	0.0439031i	$-0.0139793\pi$
$930$		0			0
$931$	137.006			0.147160
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1126.84	−	1126.84i	−1.20260	−	1.20260i	−0.973372	−	0.229233i	$-0.926378\pi$
$937$	−1126.84	−	1126.84i	−1.20260	−	1.20260i	−0.229233	−	0.973372i	$-0.573622\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−983.987			−1.04568			−0.522841	−	0.852430i	$-0.675128\pi$
$941$	−983.987			−1.04568			−0.522841	+	0.852430i	$0.675128\pi$
$942$		0			0
$943$	−1349.33	+	1349.33i	−1.43089	+	1.43089i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	301.973	+	301.973i	0.318874	+	0.318874i	0.848334	−	0.529461i	$-0.177606\pi$
$947$	301.973	+	301.973i	0.318874	+	0.318874i	−0.529461	+	0.848334i	$0.677606\pi$
$948$		0			0
$949$			199.969i			0.210715i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−531.459	+	531.459i	−0.557670	+	0.557670i	−0.928643	−	0.370974i	$-0.879024\pi$
$953$	−531.459	+	531.459i	−0.557670	+	0.557670i	0.370974	+	0.928643i	$0.379024\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	21.1219i		−	0.0220250i
$960$		0			0
$961$	−723.744			−0.753115
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−74.3922	−	74.3922i	−0.0769309	−	0.0769309i	0.667594	−	0.744525i	$-0.267324\pi$
$967$	−74.3922	−	74.3922i	−0.0769309	−	0.0769309i	−0.744525	+	0.667594i	$0.767324\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	169.959			0.175035			0.0875177	−	0.996163i	$-0.472107\pi$
$971$	169.959			0.175035			0.0875177	+	0.996163i	$0.472107\pi$
$972$		0			0
$973$	−66.0126	+	66.0126i	−0.0678444	+	0.0678444i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−287.000	−	287.000i	−0.293756	−	0.293756i	0.544806	−	0.838562i	$-0.316603\pi$
$977$	−287.000	−	287.000i	−0.293756	−	0.293756i	−0.838562	+	0.544806i	$0.816603\pi$
$978$		0			0
$979$		−	128.000i		−	0.130746i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−772.477	+	772.477i	−0.785836	+	0.785836i	−0.980809	−	0.194973i	$-0.937538\pi$
$983$	−772.477	+	772.477i	−0.785836	+	0.785836i	0.194973	+	0.980809i	$0.437538\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			775.591i			0.784217i
$990$		0			0
$991$	1541.28			1.55528			0.777641	−	0.628709i	$-0.216416\pi$
$991$	1541.28			1.55528			0.777641	+	0.628709i	$0.216416\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1248.55	+	1248.55i	1.25230	+	1.25230i	0.954685	+	0.297619i	$0.0961925\pi$
$997$	1248.55	+	1248.55i	1.25230	+	1.25230i	0.297619	+	0.954685i	$0.403808\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.3.v.k.1657.2		4
3.2	odd	2		200.3.l.e.57.2		4
5.2	odd	4		360.3.v.c.73.1		4
5.3	odd	4	inner	1800.3.v.k.793.2		4
5.4	even	2		360.3.v.c.217.1		4
12.11	even	2		400.3.p.i.257.1		4
15.2	even	4		40.3.l.b.33.1	yes	4
15.8	even	4		200.3.l.e.193.2		4
15.14	odd	2		40.3.l.b.17.1	✓	4
20.7	even	4		720.3.bh.l.433.1		4
20.19	odd	2		720.3.bh.l.577.1		4
60.23	odd	4		400.3.p.i.193.1		4
60.47	odd	4		80.3.p.d.33.2		4
60.59	even	2		80.3.p.d.17.2		4
120.29	odd	2		320.3.p.l.257.2		4
120.59	even	2		320.3.p.i.257.1		4
120.77	even	4		320.3.p.l.193.2		4
120.107	odd	4		320.3.p.i.193.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.l.b.17.1	✓	4	15.14	odd	2
40.3.l.b.33.1	yes	4	15.2	even	4
80.3.p.d.17.2		4	60.59	even	2
80.3.p.d.33.2		4	60.47	odd	4
200.3.l.e.57.2		4	3.2	odd	2
200.3.l.e.193.2		4	15.8	even	4
320.3.p.i.193.1		4	120.107	odd	4
320.3.p.i.257.1		4	120.59	even	2
320.3.p.l.193.2		4	120.77	even	4
320.3.p.l.257.2		4	120.29	odd	2
360.3.v.c.73.1		4	5.2	odd	4
360.3.v.c.217.1		4	5.4	even	2
400.3.p.i.193.1		4	60.23	odd	4
400.3.p.i.257.1		4	12.11	even	2
720.3.bh.l.433.1		4	20.7	even	4
720.3.bh.l.577.1		4	20.19	odd	2
1800.3.v.k.793.2		4	5.3	odd	4	inner
1800.3.v.k.1657.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1800.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.298438 - 0.298438i) q^{7} +O(q^{10})\)	\(q+(-0.298438 - 0.298438i) q^{7} -11.4031 q^{11} +(1.59688 - 1.59688i) q^{13} +(10.4031 + 10.4031i) q^{17} +2.80625i q^{19} +(19.1047 - 19.1047i) q^{23} -5.61250i q^{29} -15.4031 q^{31} +(40.6125 + 40.6125i) q^{37} -70.6281 q^{41} +(-20.2984 + 20.2984i) q^{43} +(-42.5078 - 42.5078i) q^{47} -48.8219i q^{49} +(-3.00000 + 3.00000i) q^{53} -66.8062i q^{59} -44.5969 q^{61} +(-28.2984 - 28.2984i) q^{67} -32.5969 q^{71} +(-62.6125 + 62.6125i) q^{73} +(3.40312 + 3.40312i) q^{77} +118.450i q^{79} +(-11.2828 + 11.2828i) q^{83} +11.2250i q^{89} -0.953136 q^{91} +(-49.0000 - 49.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{7}+O(q^{10}) \) 4 * q - 14 * q^7	\( 4 q - 14 q^{7} - 20 q^{11} + 32 q^{13} + 16 q^{17} + 38 q^{23} - 36 q^{31} + 60 q^{37} - 52 q^{41} - 94 q^{43} - 106 q^{47} - 12 q^{53} - 204 q^{61} - 126 q^{67} - 156 q^{71} - 148 q^{73} - 12 q^{77} - 186 q^{83} - 388 q^{91} - 196 q^{97}+O(q^{100}) \) 4 * q - 14 * q^7 - 20 * q^11 + 32 * q^13 + 16 * q^17 + 38 * q^23 - 36 * q^31 + 60 * q^37 - 52 * q^41 - 94 * q^43 - 106 * q^47 - 12 * q^53 - 204 * q^61 - 126 * q^67 - 156 * q^71 - 148 * q^73 - 12 * q^77 - 186 * q^83 - 388 * q^91 - 196 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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