LMFDB - Embedded newform 4400.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(1,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{3} +0.697224 q^{7} +2.30278 q^{9} +1.00000 q^{11} -5.00000 q^{13} -6.90833 q^{17} +1.00000 q^{19} +1.60555 q^{21} -7.30278 q^{23} -1.60555 q^{27} +0.908327 q^{29} -10.2111 q^{31} +2.30278 q^{33} -2.39445 q^{37} -11.5139 q^{39} -5.60555 q^{41} +7.21110 q^{43} -3.00000 q^{47} -6.51388 q^{49} -15.9083 q^{51} +1.30278 q^{53} +2.30278 q^{57} +14.2111 q^{59} -7.90833 q^{61} +1.60555 q^{63} -4.00000 q^{67} -16.8167 q^{69} +2.60555 q^{71} +7.90833 q^{73} +0.697224 q^{77} +10.9083 q^{79} -10.6056 q^{81} -3.51388 q^{83} +2.09167 q^{87} +1.69722 q^{89} -3.48612 q^{91} -23.5139 q^{93} -15.3028 q^{97} +2.30278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 5 q^{7} + q^{9} + 2 q^{11} - 10 q^{13} - 3 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} + 4 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} - 12 q^{37} - 5 q^{39} - 4 q^{41} - 6 q^{47} + 5 q^{49} - 21 q^{51}+ \cdots + q^{99}+O(q^{100}) $ 2 * q + q^3 + 5 * q^7 + q^9 + 2 * q^11 - 10 * q^13 - 3 * q^17 + 2 * q^19 - 4 * q^21 - 11 * q^23 + 4 * q^27 - 9 * q^29 - 6 * q^31 + q^33 - 12 * q^37 - 5 * q^39 - 4 * q^41 - 6 * q^47 + 5 * q^49 - 21 * q^51 - q^53 + q^57 + 14 * q^59 - 5 * q^61 - 4 * q^63 - 8 * q^67 - 12 * q^69 - 2 * q^71 + 5 * q^73 + 5 * q^77 + 11 * q^79 - 14 * q^81 + 11 * q^83 + 15 * q^87 + 7 * q^89 - 25 * q^91 - 29 * q^93 - 27 * q^97 + q^99

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
$3$	2.30278	1.32951	0.664754	−	0.747062i	$-0.268536\pi$
$3$	2.30278	1.32951	0.664754	+	0.747062i	$0.268536\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.697224	0.263526	0.131763	−	0.991281i	$-0.457936\pi$
$7$	0.697224	0.263526	0.131763	+	0.991281i	$0.457936\pi$
$8$	0	0
$9$	2.30278	0.767592
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.90833	−1.67552	−0.837758	−	0.546042i	$-0.816134\pi$
$17$	−6.90833	−1.67552	−0.837758	+	0.546042i	$0.816134\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	1.60555	0.350360
$22$	0	0
$23$	−7.30278	−1.52273	−0.761367	−	0.648321i	$-0.775471\pi$
$23$	−7.30278	−1.52273	−0.761367	+	0.648321i	$0.775471\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.60555	−0.308988
$28$	0	0
$29$	0.908327	0.168672	0.0843360	−	0.996437i	$-0.473123\pi$
$29$	0.908327	0.168672	0.0843360	+	0.996437i	$0.473123\pi$
$30$	0	0
$31$	−10.2111	−1.83397	−0.916984	−	0.398924i	$-0.869384\pi$
$31$	−10.2111	−1.83397	−0.916984	+	0.398924i	$0.869384\pi$
$32$	0	0
$33$	2.30278	0.400862
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.39445	−0.393645	−0.196822	−	0.980439i	$-0.563062\pi$
$37$	−2.39445	−0.393645	−0.196822	+	0.980439i	$0.563062\pi$
$38$	0	0
$39$	−11.5139	−1.84370
$40$	0	0
$41$	−5.60555	−0.875440	−0.437720	−	0.899111i	$-0.644214\pi$
$41$	−5.60555	−0.875440	−0.437720	+	0.899111i	$0.644214\pi$
$42$	0	0
$43$	7.21110	1.09968	0.549841	−	0.835269i	$-0.314688\pi$
$43$	7.21110	1.09968	0.549841	+	0.835269i	$0.314688\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	−6.51388	−0.930554
$50$	0	0
$51$	−15.9083	−2.22761
$52$	0	0
$53$	1.30278	0.178950	0.0894750	−	0.995989i	$-0.471481\pi$
$53$	1.30278	0.178950	0.0894750	+	0.995989i	$0.471481\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.30278	0.305010
$58$	0	0
$59$	14.2111	1.85013	0.925064	−	0.379811i	$-0.124011\pi$
$59$	14.2111	1.85013	0.925064	+	0.379811i	$0.124011\pi$
$60$	0	0
$61$	−7.90833	−1.01256	−0.506279	−	0.862370i	$-0.668979\pi$
$61$	−7.90833	−1.01256	−0.506279	+	0.862370i	$0.668979\pi$
$62$	0	0
$63$	1.60555	0.202280
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−16.8167	−2.02449
$70$	0	0
$71$	2.60555	0.309222	0.154611	−	0.987975i	$-0.450588\pi$
$71$	2.60555	0.309222	0.154611	+	0.987975i	$0.450588\pi$
$72$	0	0
$73$	7.90833	0.925600	0.462800	−	0.886463i	$-0.346845\pi$
$73$	7.90833	0.925600	0.462800	+	0.886463i	$0.346845\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.697224	0.0794561
$78$	0	0
$79$	10.9083	1.22728	0.613641	−	0.789585i	$-0.289704\pi$
$79$	10.9083	1.22728	0.613641	+	0.789585i	$0.289704\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	−3.51388	−0.385698	−0.192849	−	0.981228i	$-0.561773\pi$
$83$	−3.51388	−0.385698	−0.192849	+	0.981228i	$0.561773\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.09167	0.224251
$88$	0	0
$89$	1.69722	0.179905	0.0899527	−	0.995946i	$-0.471328\pi$
$89$	1.69722	0.179905	0.0899527	+	0.995946i	$0.471328\pi$
$90$	0	0
$91$	−3.48612	−0.365445
$92$	0	0
$93$	−23.5139	−2.43828
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.3028	−1.55376	−0.776881	−	0.629648i	$-0.783199\pi$
$97$	−15.3028	−1.55376	−0.776881	+	0.629648i	$0.783199\pi$
$98$	0	0
$99$	2.30278	0.231438
$100$	0	0
$101$	−0.513878	−0.0511328	−0.0255664	−	0.999673i	$-0.508139\pi$
$101$	−0.513878	−0.0511328	−0.0255664	+	0.999673i	$0.508139\pi$
$102$	0	0
$103$	2.90833	0.286566	0.143283	−	0.989682i	$-0.454234\pi$
$103$	2.90833	0.286566	0.143283	+	0.989682i	$0.454234\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	11.5139	1.10283	0.551415	−	0.834231i	$-0.314088\pi$
$109$	11.5139	1.10283	0.551415	+	0.834231i	$0.314088\pi$
$110$	0	0
$111$	−5.51388	−0.523354
$112$	0	0
$113$	10.8167	1.01755	0.508773	−	0.860901i	$-0.330099\pi$
$113$	10.8167	1.01755	0.508773	+	0.860901i	$0.330099\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−11.5139	−1.06446
$118$	0	0
$119$	−4.81665	−0.441542
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.9083	−1.16390
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.11943	0.720483	0.360241	−	0.932859i	$-0.382694\pi$
$127$	8.11943	0.720483	0.360241	+	0.932859i	$0.382694\pi$
$128$	0	0
$129$	16.6056	1.46204
$130$	0	0
$131$	9.90833	0.865695	0.432847	−	0.901467i	$-0.357509\pi$
$131$	9.90833	0.865695	0.432847	+	0.901467i	$0.357509\pi$
$132$	0	0
$133$	0.697224	0.0604570
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.9083	1.10283	0.551416	−	0.834230i	$-0.314088\pi$
$137$	12.9083	1.10283	0.551416	+	0.834230i	$0.314088\pi$
$138$	0	0
$139$	6.21110	0.526819	0.263409	−	0.964684i	$-0.415153\pi$
$139$	6.21110	0.526819	0.263409	+	0.964684i	$0.415153\pi$
$140$	0	0
$141$	−6.90833	−0.581786
$142$	0	0
$143$	−5.00000	−0.418121
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−15.0000	−1.23718
$148$	0	0
$149$	17.2111	1.40999	0.704994	−	0.709213i	$-0.250950\pi$
$149$	17.2111	1.40999	0.704994	+	0.709213i	$0.250950\pi$
$150$	0	0
$151$	−0.816654	−0.0664583	−0.0332292	−	0.999448i	$-0.510579\pi$
$151$	−0.816654	−0.0664583	−0.0332292	+	0.999448i	$0.510579\pi$
$152$	0	0
$153$	−15.9083	−1.28611
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−19.2111	−1.53321	−0.766606	−	0.642117i	$-0.778056\pi$
$157$	−19.2111	−1.53321	−0.766606	+	0.642117i	$0.778056\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	−5.09167	−0.401280
$162$	0	0
$163$	9.30278	0.728650	0.364325	−	0.931272i	$-0.381300\pi$
$163$	9.30278	0.728650	0.364325	+	0.931272i	$0.381300\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.4222	1.03864	0.519321	−	0.854579i	$-0.326185\pi$
$167$	13.4222	1.03864	0.519321	+	0.854579i	$0.326185\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	2.30278	0.176098
$172$	0	0
$173$	4.81665	0.366203	0.183102	−	0.983094i	$-0.441386\pi$
$173$	4.81665	0.366203	0.183102	+	0.983094i	$0.441386\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	32.7250	2.45976
$178$	0	0
$179$	−12.5139	−0.935331	−0.467666	−	0.883905i	$-0.654905\pi$
$179$	−12.5139	−0.935331	−0.467666	+	0.883905i	$0.654905\pi$
$180$	0	0
$181$	−19.9083	−1.47977	−0.739887	−	0.672731i	$-0.765121\pi$
$181$	−19.9083	−1.47977	−0.739887	+	0.672731i	$0.765121\pi$
$182$	0	0
$183$	−18.2111	−1.34620
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.90833	−0.505187
$188$	0	0
$189$	−1.11943	−0.0814265
$190$	0	0
$191$	−10.3028	−0.745483	−0.372741	−	0.927935i	$-0.621582\pi$
$191$	−10.3028	−0.745483	−0.372741	+	0.927935i	$0.621582\pi$
$192$	0	0
$193$	−13.2111	−0.950956	−0.475478	−	0.879728i	$-0.657725\pi$
$193$	−13.2111	−0.950956	−0.475478	+	0.879728i	$0.657725\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.3028	−0.947784	−0.473892	−	0.880583i	$-0.657151\pi$
$197$	−13.3028	−0.947784	−0.473892	+	0.880583i	$0.657151\pi$
$198$	0	0
$199$	6.48612	0.459789	0.229894	−	0.973216i	$-0.426162\pi$
$199$	6.48612	0.459789	0.229894	+	0.973216i	$0.426162\pi$
$200$	0	0
$201$	−9.21110	−0.649701
$202$	0	0
$203$	0.633308	0.0444495
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−16.8167	−1.16884
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	25.2389	1.73751	0.868757	−	0.495238i	$-0.164919\pi$
$211$	25.2389	1.73751	0.868757	+	0.495238i	$0.164919\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7.11943	−0.483298
$218$	0	0
$219$	18.2111	1.23059
$220$	0	0
$221$	34.5416	2.32352
$222$	0	0
$223$	−22.6333	−1.51564	−0.757819	−	0.652465i	$-0.773735\pi$
$223$	−22.6333	−1.51564	−0.757819	+	0.652465i	$0.773735\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.69722	0.112649	0.0563244	−	0.998413i	$-0.482062\pi$
$227$	1.69722	0.112649	0.0563244	+	0.998413i	$0.482062\pi$
$228$	0	0
$229$	−18.7250	−1.23738	−0.618691	−	0.785635i	$-0.712337\pi$
$229$	−18.7250	−1.23738	−0.618691	+	0.785635i	$0.712337\pi$
$230$	0	0
$231$	1.60555	0.105638
$232$	0	0
$233$	−15.9083	−1.04219	−0.521095	−	0.853499i	$-0.674476\pi$
$233$	−15.9083	−1.04219	−0.521095	+	0.853499i	$0.674476\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	25.1194	1.63168
$238$	0	0
$239$	−21.1194	−1.36610	−0.683051	−	0.730371i	$-0.739347\pi$
$239$	−21.1194	−1.36610	−0.683051	+	0.730371i	$0.739347\pi$
$240$	0	0
$241$	21.9361	1.41303	0.706514	−	0.707699i	$-0.250267\pi$
$241$	21.9361	1.41303	0.706514	+	0.707699i	$0.250267\pi$
$242$	0	0
$243$	−19.6056	−1.25770
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.00000	−0.318142
$248$	0	0
$249$	−8.09167	−0.512789
$250$	0	0
$251$	−6.90833	−0.436050	−0.218025	−	0.975943i	$-0.569961\pi$
$251$	−6.90833	−0.436050	−0.218025	+	0.975943i	$0.569961\pi$
$252$	0	0
$253$	−7.30278	−0.459122
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−1.66947	−0.103736
$260$	0	0
$261$	2.09167	0.129471
$262$	0	0
$263$	−22.8167	−1.40694	−0.703468	−	0.710727i	$-0.748366\pi$
$263$	−22.8167	−1.40694	−0.703468	+	0.710727i	$0.748366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.90833	0.239186
$268$	0	0
$269$	8.72498	0.531971	0.265986	−	0.963977i	$-0.414303\pi$
$269$	8.72498	0.531971	0.265986	+	0.963977i	$0.414303\pi$
$270$	0	0
$271$	0.211103	0.0128236	0.00641178	−	0.999979i	$-0.497959\pi$
$271$	0.211103	0.0128236	0.00641178	+	0.999979i	$0.497959\pi$
$272$	0	0
$273$	−8.02776	−0.485862
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.3944	−0.864879	−0.432439	−	0.901663i	$-0.642347\pi$
$277$	−14.3944	−0.864879	−0.432439	+	0.901663i	$0.642347\pi$
$278$	0	0
$279$	−23.5139	−1.40774
$280$	0	0
$281$	−1.18335	−0.0705925	−0.0352963	−	0.999377i	$-0.511237\pi$
$281$	−1.18335	−0.0705925	−0.0352963	+	0.999377i	$0.511237\pi$
$282$	0	0
$283$	6.30278	0.374661	0.187331	−	0.982297i	$-0.440016\pi$
$283$	6.30278	0.374661	0.187331	+	0.982297i	$0.440016\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.90833	−0.230701
$288$	0	0
$289$	30.7250	1.80735
$290$	0	0
$291$	−35.2389	−2.06574
$292$	0	0
$293$	−0.788897	−0.0460879	−0.0230439	−	0.999734i	$-0.507336\pi$
$293$	−0.788897	−0.0460879	−0.0230439	+	0.999734i	$0.507336\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.60555	−0.0931635
$298$	0	0
$299$	36.5139	2.11165
$300$	0	0
$301$	5.02776	0.289795
$302$	0	0
$303$	−1.18335	−0.0679815
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−16.9083	−0.965009	−0.482505	−	0.875893i	$-0.660273\pi$
$307$	−16.9083	−0.965009	−0.482505	+	0.875893i	$0.660273\pi$
$308$	0	0
$309$	6.69722	0.380992
$310$	0	0
$311$	−4.81665	−0.273127	−0.136564	−	0.990631i	$-0.543606\pi$
$311$	−4.81665	−0.273127	−0.136564	+	0.990631i	$0.543606\pi$
$312$	0	0
$313$	−0.183346	−0.0103633	−0.00518167	−	0.999987i	$-0.501649\pi$
$313$	−0.183346	−0.0103633	−0.00518167	+	0.999987i	$0.501649\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.908327	0.0510167	0.0255084	−	0.999675i	$-0.491880\pi$
$317$	0.908327	0.0510167	0.0255084	+	0.999675i	$0.491880\pi$
$318$	0	0
$319$	0.908327	0.0508565
$320$	0	0
$321$	−6.90833	−0.385585
$322$	0	0
$323$	−6.90833	−0.384390
$324$	0	0
$325$	0	0
$326$	0	0
$327$	26.5139	1.46622
$328$	0	0
$329$	−2.09167	−0.115318
$330$	0	0
$331$	21.6056	1.18755	0.593774	−	0.804632i	$-0.297637\pi$
$331$	21.6056	1.18755	0.593774	+	0.804632i	$0.297637\pi$
$332$	0	0
$333$	−5.51388	−0.302159
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.8444	1.68020	0.840101	−	0.542430i	$-0.182496\pi$
$337$	30.8444	1.68020	0.840101	+	0.542430i	$0.182496\pi$
$338$	0	0
$339$	24.9083	1.35283
$340$	0	0
$341$	−10.2111	−0.552962
$342$	0	0
$343$	−9.42221	−0.508751
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.5139	−0.671780	−0.335890	−	0.941901i	$-0.609037\pi$
$347$	−12.5139	−0.671780	−0.335890	+	0.941901i	$0.609037\pi$
$348$	0	0
$349$	−5.18335	−0.277458	−0.138729	−	0.990330i	$-0.544302\pi$
$349$	−5.18335	−0.277458	−0.138729	+	0.990330i	$0.544302\pi$
$350$	0	0
$351$	8.02776	0.428490
$352$	0	0
$353$	−18.6333	−0.991751	−0.495875	−	0.868394i	$-0.665153\pi$
$353$	−18.6333	−0.991751	−0.495875	+	0.868394i	$0.665153\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−11.0917	−0.587034
$358$	0	0
$359$	−0.788897	−0.0416364	−0.0208182	−	0.999783i	$-0.506627\pi$
$359$	−0.788897	−0.0416364	−0.0208182	+	0.999783i	$0.506627\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	2.30278	0.120864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−20.6972	−1.08039	−0.540193	−	0.841541i	$-0.681649\pi$
$367$	−20.6972	−1.08039	−0.540193	+	0.841541i	$0.681649\pi$
$368$	0	0
$369$	−12.9083	−0.671981
$370$	0	0
$371$	0.908327	0.0471580
$372$	0	0
$373$	−27.4222	−1.41987	−0.709934	−	0.704268i	$-0.751275\pi$
$373$	−27.4222	−1.41987	−0.709934	+	0.704268i	$0.751275\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.54163	−0.233906
$378$	0	0
$379$	−3.18335	−0.163518	−0.0817588	−	0.996652i	$-0.526054\pi$
$379$	−3.18335	−0.163518	−0.0817588	+	0.996652i	$0.526054\pi$
$380$	0	0
$381$	18.6972	0.957888
$382$	0	0
$383$	21.6333	1.10541	0.552705	−	0.833377i	$-0.313596\pi$
$383$	21.6333	1.10541	0.552705	+	0.833377i	$0.313596\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.6056	0.844108
$388$	0	0
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	50.4500	2.55136
$392$	0	0
$393$	22.8167	1.15095
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−21.6972	−1.08895	−0.544476	−	0.838776i	$-0.683272\pi$
$397$	−21.6972	−1.08895	−0.544476	+	0.838776i	$0.683272\pi$
$398$	0	0
$399$	1.60555	0.0803781
$400$	0	0
$401$	−12.7889	−0.638647	−0.319324	−	0.947646i	$-0.603456\pi$
$401$	−12.7889	−0.638647	−0.319324	+	0.947646i	$0.603456\pi$
$402$	0	0
$403$	51.0555	2.54326
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.39445	−0.118688
$408$	0	0
$409$	−6.21110	−0.307119	−0.153560	−	0.988139i	$-0.549074\pi$
$409$	−6.21110	−0.307119	−0.153560	+	0.988139i	$0.549074\pi$
$410$	0	0
$411$	29.7250	1.46623
$412$	0	0
$413$	9.90833	0.487557
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14.3028	0.700410
$418$	0	0
$419$	−6.39445	−0.312389	−0.156195	−	0.987726i	$-0.549923\pi$
$419$	−6.39445	−0.312389	−0.156195	+	0.987726i	$0.549923\pi$
$420$	0	0
$421$	0.697224	0.0339806	0.0169903	−	0.999856i	$-0.494592\pi$
$421$	0.697224	0.0339806	0.0169903	+	0.999856i	$0.494592\pi$
$422$	0	0
$423$	−6.90833	−0.335894
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.51388	−0.266835
$428$	0	0
$429$	−11.5139	−0.555895
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	0	0
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.30278	−0.349339
$438$	0	0
$439$	−24.3028	−1.15991	−0.579954	−	0.814649i	$-0.696930\pi$
$439$	−24.3028	−1.15991	−0.579954	+	0.814649i	$0.696930\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	0	0
$443$	−8.60555	−0.408862	−0.204431	−	0.978881i	$-0.565534\pi$
$443$	−8.60555	−0.408862	−0.204431	+	0.978881i	$0.565534\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	39.6333	1.87459
$448$	0	0
$449$	23.4861	1.10838	0.554189	−	0.832391i	$-0.313028\pi$
$449$	23.4861	1.10838	0.554189	+	0.832391i	$0.313028\pi$
$450$	0	0
$451$	−5.60555	−0.263955
$452$	0	0
$453$	−1.88057	−0.0883569
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.6972	0.968175	0.484088	−	0.875020i	$-0.339152\pi$
$457$	20.6972	0.968175	0.484088	+	0.875020i	$0.339152\pi$
$458$	0	0
$459$	11.0917	0.517715
$460$	0	0
$461$	32.2111	1.50022	0.750110	−	0.661313i	$-0.230000\pi$
$461$	32.2111	1.50022	0.750110	+	0.661313i	$0.230000\pi$
$462$	0	0
$463$	11.7889	0.547877	0.273938	−	0.961747i	$-0.411674\pi$
$463$	11.7889	0.547877	0.273938	+	0.961747i	$0.411674\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.6333	−0.862247	−0.431123	−	0.902293i	$-0.641883\pi$
$467$	−18.6333	−0.862247	−0.431123	+	0.902293i	$0.641883\pi$
$468$	0	0
$469$	−2.78890	−0.128779
$470$	0	0
$471$	−44.2389	−2.03842
$472$	0	0
$473$	7.21110	0.331567
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	34.8167	1.59081	0.795407	−	0.606076i	$-0.207257\pi$
$479$	34.8167	1.59081	0.795407	+	0.606076i	$0.207257\pi$
$480$	0	0
$481$	11.9722	0.545887
$482$	0	0
$483$	−11.7250	−0.533505
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.21110	0.190823	0.0954116	−	0.995438i	$-0.469583\pi$
$487$	4.21110	0.190823	0.0954116	+	0.995438i	$0.469583\pi$
$488$	0	0
$489$	21.4222	0.968746
$490$	0	0
$491$	9.78890	0.441767	0.220883	−	0.975300i	$-0.429106\pi$
$491$	9.78890	0.441767	0.220883	+	0.975300i	$0.429106\pi$
$492$	0	0
$493$	−6.27502	−0.282613
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.81665	0.0814881
$498$	0	0
$499$	3.48612	0.156060	0.0780301	−	0.996951i	$-0.475137\pi$
$499$	3.48612	0.156060	0.0780301	+	0.996951i	$0.475137\pi$
$500$	0	0
$501$	30.9083	1.38088
$502$	0	0
$503$	9.39445	0.418878	0.209439	−	0.977822i	$-0.432836\pi$
$503$	9.39445	0.418878	0.209439	+	0.977822i	$0.432836\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	27.6333	1.22724
$508$	0	0
$509$	−22.6972	−1.00604	−0.503018	−	0.864276i	$-0.667777\pi$
$509$	−22.6972	−1.00604	−0.503018	+	0.864276i	$0.667777\pi$
$510$	0	0
$511$	5.51388	0.243920
$512$	0	0
$513$	−1.60555	−0.0708868
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.00000	−0.131940
$518$	0	0
$519$	11.0917	0.486870
$520$	0	0
$521$	−41.4500	−1.81596	−0.907978	−	0.419018i	$-0.862374\pi$
$521$	−41.4500	−1.81596	−0.907978	+	0.419018i	$0.862374\pi$
$522$	0	0
$523$	−32.4222	−1.41772	−0.708862	−	0.705347i	$-0.750791\pi$
$523$	−32.4222	−1.41772	−0.708862	+	0.705347i	$0.750791\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	70.5416	3.07284
$528$	0	0
$529$	30.3305	1.31872
$530$	0	0
$531$	32.7250	1.42014
$532$	0	0
$533$	28.0278	1.21402
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−28.8167	−1.24353
$538$	0	0
$539$	−6.51388	−0.280573
$540$	0	0
$541$	25.7250	1.10600	0.553002	−	0.833180i	$-0.313482\pi$
$541$	25.7250	1.10600	0.553002	+	0.833180i	$0.313482\pi$
$542$	0	0
$543$	−45.8444	−1.96737
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.11943	−0.304405	−0.152202	−	0.988349i	$-0.548637\pi$
$547$	−7.11943	−0.304405	−0.152202	+	0.988349i	$0.548637\pi$
$548$	0	0
$549$	−18.2111	−0.777231
$550$	0	0
$551$	0.908327	0.0386960
$552$	0	0
$553$	7.60555	0.323421
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.4222	−0.822945	−0.411473	−	0.911422i	$-0.634985\pi$
$557$	−19.4222	−0.822945	−0.411473	+	0.911422i	$0.634985\pi$
$558$	0	0
$559$	−36.0555	−1.52499
$560$	0	0
$561$	−15.9083	−0.671650
$562$	0	0
$563$	8.09167	0.341023	0.170512	−	0.985356i	$-0.445458\pi$
$563$	8.09167	0.341023	0.170512	+	0.985356i	$0.445458\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.39445	−0.310538
$568$	0	0
$569$	−46.1472	−1.93459	−0.967295	−	0.253653i	$-0.918368\pi$
$569$	−46.1472	−1.93459	−0.967295	+	0.253653i	$0.918368\pi$
$570$	0	0
$571$	−22.3305	−0.934504	−0.467252	−	0.884124i	$-0.654756\pi$
$571$	−22.3305	−0.934504	−0.467252	+	0.884124i	$0.654756\pi$
$572$	0	0
$573$	−23.7250	−0.991125
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−44.3583	−1.84666	−0.923330	−	0.384008i	$-0.874544\pi$
$577$	−44.3583	−1.84666	−0.923330	+	0.384008i	$0.874544\pi$
$578$	0	0
$579$	−30.4222	−1.26430
$580$	0	0
$581$	−2.44996	−0.101642
$582$	0	0
$583$	1.30278	0.0539555
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.5416	0.682746	0.341373	−	0.939928i	$-0.389108\pi$
$587$	16.5416	0.682746	0.341373	+	0.939928i	$0.389108\pi$
$588$	0	0
$589$	−10.2111	−0.420741
$590$	0	0
$591$	−30.6333	−1.26009
$592$	0	0
$593$	6.39445	0.262589	0.131294	−	0.991343i	$-0.458087\pi$
$593$	6.39445	0.262589	0.131294	+	0.991343i	$0.458087\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.9361	0.611293
$598$	0	0
$599$	24.9083	1.01773	0.508863	−	0.860847i	$-0.330066\pi$
$599$	24.9083	1.01773	0.508863	+	0.860847i	$0.330066\pi$
$600$	0	0
$601$	−1.90833	−0.0778423	−0.0389211	−	0.999242i	$-0.512392\pi$
$601$	−1.90833	−0.0778423	−0.0389211	+	0.999242i	$0.512392\pi$
$602$	0	0
$603$	−9.21110	−0.375105
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.21110	0.292690	0.146345	−	0.989234i	$-0.453249\pi$
$607$	7.21110	0.292690	0.146345	+	0.989234i	$0.453249\pi$
$608$	0	0
$609$	1.45837	0.0590959
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	15.8806	0.641410	0.320705	−	0.947179i	$-0.396080\pi$
$613$	15.8806	0.641410	0.320705	+	0.947179i	$0.396080\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.39445	−0.136655	−0.0683277	−	0.997663i	$-0.521766\pi$
$617$	−3.39445	−0.136655	−0.0683277	+	0.997663i	$0.521766\pi$
$618$	0	0
$619$	11.4222	0.459097	0.229549	−	0.973297i	$-0.426275\pi$
$619$	11.4222	0.459097	0.229549	+	0.973297i	$0.426275\pi$
$620$	0	0
$621$	11.7250	0.470507
$622$	0	0
$623$	1.18335	0.0474098
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.30278	0.0919640
$628$	0	0
$629$	16.5416	0.659558
$630$	0	0
$631$	−6.93608	−0.276121	−0.138061	−	0.990424i	$-0.544087\pi$
$631$	−6.93608	−0.276121	−0.138061	+	0.990424i	$0.544087\pi$
$632$	0	0
$633$	58.1194	2.31004
$634$	0	0
$635$	0	0
$636$	0	0
$637$	32.5694	1.29045
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	27.7889	1.09760	0.548798	−	0.835955i	$-0.315086\pi$
$641$	27.7889	1.09760	0.548798	+	0.835955i	$0.315086\pi$
$642$	0	0
$643$	−22.0000	−0.867595	−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000	−0.867595	−0.433798	+	0.901010i	$0.642827\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.2389	−1.30675	−0.653377	−	0.757033i	$-0.726648\pi$
$647$	−33.2389	−1.30675	−0.653377	+	0.757033i	$0.726648\pi$
$648$	0	0
$649$	14.2111	0.557835
$650$	0	0
$651$	−16.3944	−0.642549
$652$	0	0
$653$	−6.11943	−0.239472	−0.119736	−	0.992806i	$-0.538205\pi$
$653$	−6.11943	−0.239472	−0.119736	+	0.992806i	$0.538205\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	18.2111	0.710483
$658$	0	0
$659$	30.9083	1.20402	0.602009	−	0.798489i	$-0.294367\pi$
$659$	30.9083	1.20402	0.602009	+	0.798489i	$0.294367\pi$
$660$	0	0
$661$	−8.81665	−0.342928	−0.171464	−	0.985190i	$-0.554850\pi$
$661$	−8.81665	−0.342928	−0.171464	+	0.985190i	$0.554850\pi$
$662$	0	0
$663$	79.5416	3.08914
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.63331	−0.256843
$668$	0	0
$669$	−52.1194	−2.01505
$670$	0	0
$671$	−7.90833	−0.305298
$672$	0	0
$673$	−30.0278	−1.15748	−0.578742	−	0.815510i	$-0.696456\pi$
$673$	−30.0278	−1.15748	−0.578742	+	0.815510i	$0.696456\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24.2389	−0.931575	−0.465788	−	0.884897i	$-0.654229\pi$
$677$	−24.2389	−0.931575	−0.465788	+	0.884897i	$0.654229\pi$
$678$	0	0
$679$	−10.6695	−0.409457
$680$	0	0
$681$	3.90833	0.149767
$682$	0	0
$683$	47.8444	1.83072	0.915358	−	0.402642i	$-0.131908\pi$
$683$	47.8444	1.83072	0.915358	+	0.402642i	$0.131908\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−43.1194	−1.64511
$688$	0	0
$689$	−6.51388	−0.248159
$690$	0	0
$691$	−27.5416	−1.04773	−0.523867	−	0.851800i	$-0.675511\pi$
$691$	−27.5416	−1.04773	−0.523867	+	0.851800i	$0.675511\pi$
$692$	0	0
$693$	1.60555	0.0609898
$694$	0	0
$695$	0	0
$696$	0	0
$697$	38.7250	1.46681
$698$	0	0
$699$	−36.6333	−1.38560
$700$	0	0
$701$	−41.2111	−1.55652	−0.778261	−	0.627941i	$-0.783898\pi$
$701$	−41.2111	−1.55652	−0.778261	+	0.627941i	$0.783898\pi$
$702$	0	0
$703$	−2.39445	−0.0903083
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.358288	−0.0134748
$708$	0	0
$709$	−31.6333	−1.18801	−0.594007	−	0.804460i	$-0.702455\pi$
$709$	−31.6333	−1.18801	−0.594007	+	0.804460i	$0.702455\pi$
$710$	0	0
$711$	25.1194	0.942052
$712$	0	0
$713$	74.5694	2.79265
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−48.6333	−1.81624
$718$	0	0
$719$	−7.18335	−0.267894	−0.133947	−	0.990989i	$-0.542765\pi$
$719$	−7.18335	−0.267894	−0.133947	+	0.990989i	$0.542765\pi$
$720$	0	0
$721$	2.02776	0.0755176
$722$	0	0
$723$	50.5139	1.87863
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−39.3305	−1.45869	−0.729344	−	0.684147i	$-0.760175\pi$
$727$	−39.3305	−1.45869	−0.729344	+	0.684147i	$0.760175\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	−49.8167	−1.84254
$732$	0	0
$733$	−19.6056	−0.724148	−0.362074	−	0.932149i	$-0.617931\pi$
$733$	−19.6056	−0.724148	−0.362074	+	0.932149i	$0.617931\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−35.1194	−1.29189	−0.645945	−	0.763384i	$-0.723536\pi$
$739$	−35.1194	−1.29189	−0.645945	+	0.763384i	$0.723536\pi$
$740$	0	0
$741$	−11.5139	−0.422973
$742$	0	0
$743$	40.6972	1.49304	0.746518	−	0.665365i	$-0.231724\pi$
$743$	40.6972	1.49304	0.746518	+	0.665365i	$0.231724\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.09167	−0.296059
$748$	0	0
$749$	−2.09167	−0.0764281
$750$	0	0
$751$	45.3305	1.65413	0.827067	−	0.562103i	$-0.190008\pi$
$751$	45.3305	1.65413	0.827067	+	0.562103i	$0.190008\pi$
$752$	0	0
$753$	−15.9083	−0.579732
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−49.0555	−1.78295	−0.891476	−	0.453067i	$-0.850330\pi$
$757$	−49.0555	−1.78295	−0.891476	+	0.453067i	$0.850330\pi$
$758$	0	0
$759$	−16.8167	−0.610406
$760$	0	0
$761$	−13.5778	−0.492195	−0.246097	−	0.969245i	$-0.579148\pi$
$761$	−13.5778	−0.492195	−0.246097	+	0.969245i	$0.579148\pi$
$762$	0	0
$763$	8.02776	0.290624
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−71.0555	−2.56567
$768$	0	0
$769$	−5.18335	−0.186916	−0.0934581	−	0.995623i	$-0.529792\pi$
$769$	−5.18335	−0.186916	−0.0934581	+	0.995623i	$0.529792\pi$
$770$	0	0
$771$	41.4500	1.49278
$772$	0	0
$773$	−3.11943	−0.112198	−0.0560990	−	0.998425i	$-0.517866\pi$
$773$	−3.11943	−0.112198	−0.0560990	+	0.998425i	$0.517866\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.84441	−0.137917
$778$	0	0
$779$	−5.60555	−0.200840
$780$	0	0
$781$	2.60555	0.0932340
$782$	0	0
$783$	−1.45837	−0.0521177
$784$	0	0
$785$	0	0
$786$	0	0
$787$	10.2111	0.363986	0.181993	−	0.983300i	$-0.441745\pi$
$787$	10.2111	0.363986	0.181993	+	0.983300i	$0.441745\pi$
$788$	0	0
$789$	−52.5416	−1.87053
$790$	0	0
$791$	7.54163	0.268150
$792$	0	0
$793$	39.5416	1.40416
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.51388	−0.124468	−0.0622340	−	0.998062i	$-0.519822\pi$
$797$	−3.51388	−0.124468	−0.0622340	+	0.998062i	$0.519822\pi$
$798$	0	0
$799$	20.7250	0.733197
$800$	0	0
$801$	3.90833	0.138094
$802$	0	0
$803$	7.90833	0.279079
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.0917	0.707260
$808$	0	0
$809$	39.6333	1.39343	0.696716	−	0.717347i	$-0.254644\pi$
$809$	39.6333	1.39343	0.696716	+	0.717347i	$0.254644\pi$
$810$	0	0
$811$	−38.8722	−1.36499	−0.682493	−	0.730892i	$-0.739104\pi$
$811$	−38.8722	−1.36499	−0.682493	+	0.730892i	$0.739104\pi$
$812$	0	0
$813$	0.486122	0.0170490
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.21110	0.252285
$818$	0	0
$819$	−8.02776	−0.280513
$820$	0	0
$821$	12.0000	0.418803	0.209401	−	0.977830i	$-0.432848\pi$
$821$	12.0000	0.418803	0.209401	+	0.977830i	$0.432848\pi$
$822$	0	0
$823$	18.4222	0.642158	0.321079	−	0.947052i	$-0.395955\pi$
$823$	18.4222	0.642158	0.321079	+	0.947052i	$0.395955\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.8167	0.480452	0.240226	−	0.970717i	$-0.422778\pi$
$827$	13.8167	0.480452	0.240226	+	0.970717i	$0.422778\pi$
$828$	0	0
$829$	29.7527	1.03336	0.516678	−	0.856180i	$-0.327169\pi$
$829$	29.7527	1.03336	0.516678	+	0.856180i	$0.327169\pi$
$830$	0	0
$831$	−33.1472	−1.14986
$832$	0	0
$833$	45.0000	1.55916
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.3944	0.566675
$838$	0	0
$839$	9.11943	0.314838	0.157419	−	0.987532i	$-0.449683\pi$
$839$	9.11943	0.314838	0.157419	+	0.987532i	$0.449683\pi$
$840$	0	0
$841$	−28.1749	−0.971550
$842$	0	0
$843$	−2.72498	−0.0938533
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.697224	0.0239569
$848$	0	0
$849$	14.5139	0.498115
$850$	0	0
$851$	17.4861	0.599417
$852$	0	0
$853$	12.7250	0.435695	0.217848	−	0.975983i	$-0.430096\pi$
$853$	12.7250	0.435695	0.217848	+	0.975983i	$0.430096\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	−41.3944	−1.41236	−0.706180	−	0.708032i	$-0.749583\pi$
$859$	−41.3944	−1.41236	−0.706180	+	0.708032i	$0.749583\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	0	0
$863$	−12.3944	−0.421912	−0.210956	−	0.977496i	$-0.567658\pi$
$863$	−12.3944	−0.421912	−0.210956	+	0.977496i	$0.567658\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	70.7527	2.40289
$868$	0	0
$869$	10.9083	0.370040
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	−35.2389	−1.19265
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−20.0000	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$878$	0	0
$879$	−1.81665	−0.0612742
$880$	0	0
$881$	19.5416	0.658374	0.329187	−	0.944265i	$-0.393225\pi$
$881$	19.5416	0.658374	0.329187	+	0.944265i	$0.393225\pi$
$882$	0	0
$883$	52.4500	1.76508	0.882541	−	0.470236i	$-0.155831\pi$
$883$	52.4500	1.76508	0.882541	+	0.470236i	$0.155831\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.23886	−0.108750	−0.0543751	−	0.998521i	$-0.517317\pi$
$887$	−3.23886	−0.108750	−0.0543751	+	0.998521i	$0.517317\pi$
$888$	0	0
$889$	5.66106	0.189866
$890$	0	0
$891$	−10.6056	−0.355299
$892$	0	0
$893$	−3.00000	−0.100391
$894$	0	0
$895$	0	0
$896$	0	0
$897$	84.0833	2.80746
$898$	0	0
$899$	−9.27502	−0.309339
$900$	0	0
$901$	−9.00000	−0.299833
$902$	0	0
$903$	11.5778	0.385285
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	−1.18335	−0.0392491
$910$	0	0
$911$	24.7889	0.821293	0.410646	−	0.911795i	$-0.365303\pi$
$911$	24.7889	0.821293	0.410646	+	0.911795i	$0.365303\pi$
$912$	0	0
$913$	−3.51388	−0.116292
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.90833	0.228133
$918$	0	0
$919$	−26.7889	−0.883684	−0.441842	−	0.897093i	$-0.645675\pi$
$919$	−26.7889	−0.883684	−0.441842	+	0.897093i	$0.645675\pi$
$920$	0	0
$921$	−38.9361	−1.28299
$922$	0	0
$923$	−13.0278	−0.428814
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.69722	0.219966
$928$	0	0
$929$	53.6056	1.75874	0.879371	−	0.476138i	$-0.157964\pi$
$929$	53.6056	1.75874	0.879371	+	0.476138i	$0.157964\pi$
$930$	0	0
$931$	−6.51388	−0.213484
$932$	0	0
$933$	−11.0917	−0.363125
$934$	0	0
$935$	0	0
$936$	0	0
$937$	9.21110	0.300914	0.150457	−	0.988617i	$-0.451926\pi$
$937$	9.21110	0.300914	0.150457	+	0.988617i	$0.451926\pi$
$938$	0	0
$939$	−0.422205	−0.0137781
$940$	0	0
$941$	59.6056	1.94309	0.971543	−	0.236864i	$-0.0761197\pi$
$941$	59.6056	1.94309	0.971543	+	0.236864i	$0.0761197\pi$
$942$	0	0
$943$	40.9361	1.33306
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.63331	0.215554	0.107777	−	0.994175i	$-0.465627\pi$
$947$	6.63331	0.215554	0.107777	+	0.994175i	$0.465627\pi$
$948$	0	0
$949$	−39.5416	−1.28358
$950$	0	0
$951$	2.09167	0.0678271
$952$	0	0
$953$	−37.2666	−1.20718	−0.603592	−	0.797293i	$-0.706264\pi$
$953$	−37.2666	−1.20718	−0.603592	+	0.797293i	$0.706264\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.09167	0.0676142
$958$	0	0
$959$	9.00000	0.290625
$960$	0	0
$961$	73.2666	2.36344
$962$	0	0
$963$	−6.90833	−0.222618
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14.9083	0.479419	0.239710	−	0.970845i	$-0.422948\pi$
$967$	14.9083	0.479419	0.239710	+	0.970845i	$0.422948\pi$
$968$	0	0
$969$	−15.9083	−0.511049
$970$	0	0
$971$	−45.3583	−1.45562	−0.727808	−	0.685781i	$-0.759461\pi$
$971$	−45.3583	−1.45562	−0.727808	+	0.685781i	$0.759461\pi$
$972$	0	0
$973$	4.33053	0.138830
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.0278	1.66452	0.832258	−	0.554389i	$-0.187048\pi$
$977$	52.0278	1.66452	0.832258	+	0.554389i	$0.187048\pi$
$978$	0	0
$979$	1.69722	0.0542435
$980$	0	0
$981$	26.5139	0.846523
$982$	0	0
$983$	8.84441	0.282093	0.141046	−	0.990003i	$-0.454953\pi$
$983$	8.84441	0.282093	0.141046	+	0.990003i	$0.454953\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.81665	−0.153316
$988$	0	0
$989$	−52.6611	−1.67452
$990$	0	0
$991$	16.9083	0.537111	0.268555	−	0.963264i	$-0.413454\pi$
$991$	16.9083	0.537111	0.268555	+	0.963264i	$0.413454\pi$
$992$	0	0
$993$	49.7527	1.57886
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−46.7250	−1.47979	−0.739897	−	0.672720i	$-0.765126\pi$
$997$	−46.7250	−1.47979	−0.739897	+	0.672720i	$0.765126\pi$
$998$	0	0
$999$	3.84441	0.121632

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bs.1.2		2
4.3	odd	2		275.2.a.e.1.2	✓	2
5.2	odd	4		4400.2.b.y.4049.1		4
5.3	odd	4		4400.2.b.y.4049.4		4
5.4	even	2		4400.2.a.bh.1.1		2
12.11	even	2		2475.2.a.t.1.1		2
20.3	even	4		275.2.b.c.199.2		4
20.7	even	4		275.2.b.c.199.3		4
20.19	odd	2		275.2.a.f.1.1	yes	2
44.43	even	2		3025.2.a.n.1.1		2
60.23	odd	4		2475.2.c.k.199.3		4
60.47	odd	4		2475.2.c.k.199.2		4
60.59	even	2		2475.2.a.o.1.2		2
220.219	even	2		3025.2.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.2	✓	2	4.3	odd	2
275.2.a.f.1.1	yes	2	20.19	odd	2
275.2.b.c.199.2		4	20.3	even	4
275.2.b.c.199.3		4	20.7	even	4
2475.2.a.o.1.2		2	60.59	even	2
2475.2.a.t.1.1		2	12.11	even	2
2475.2.c.k.199.2		4	60.47	odd	4
2475.2.c.k.199.3		4	60.23	odd	4
3025.2.a.h.1.2		2	220.219	even	2
3025.2.a.n.1.1		2	44.43	even	2
4400.2.a.bh.1.1		2	5.4	even	2
4400.2.a.bs.1.2		2	1.1	even	1	trivial
4400.2.b.y.4049.1		4	5.2	odd	4
4400.2.b.y.4049.4		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{3} +0.697224 q^{7} +2.30278 q^{9} +1.00000 q^{11} -5.00000 q^{13} -6.90833 q^{17} +1.00000 q^{19} +1.60555 q^{21} -7.30278 q^{23} -1.60555 q^{27} +0.908327 q^{29} -10.2111 q^{31} +2.30278 q^{33} -2.39445 q^{37} -11.5139 q^{39} -5.60555 q^{41} +7.21110 q^{43} -3.00000 q^{47} -6.51388 q^{49} -15.9083 q^{51} +1.30278 q^{53} +2.30278 q^{57} +14.2111 q^{59} -7.90833 q^{61} +1.60555 q^{63} -4.00000 q^{67} -16.8167 q^{69} +2.60555 q^{71} +7.90833 q^{73} +0.697224 q^{77} +10.9083 q^{79} -10.6056 q^{81} -3.51388 q^{83} +2.09167 q^{87} +1.69722 q^{89} -3.48612 q^{91} -23.5139 q^{93} -15.3028 q^{97} +2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 5 q^{7} + q^{9} + 2 q^{11} - 10 q^{13} - 3 q^{17} + 2 q^{19} - 4 q^{21} - 11 q^{23} + 4 q^{27} - 9 q^{29} - 6 q^{31} + q^{33} - 12 q^{37} - 5 q^{39} - 4 q^{41} - 6 q^{47} + 5 q^{49} - 21 q^{51}+ \cdots + q^{99}+O(q^{100}) \) 2 * q + q^3 + 5 * q^7 + q^9 + 2 * q^11 - 10 * q^13 - 3 * q^17 + 2 * q^19 - 4 * q^21 - 11 * q^23 + 4 * q^27 - 9 * q^29 - 6 * q^31 + q^33 - 12 * q^37 - 5 * q^39 - 4 * q^41 - 6 * q^47 + 5 * q^49 - 21 * q^51 - q^53 + q^57 + 14 * q^59 - 5 * q^61 - 4 * q^63 - 8 * q^67 - 12 * q^69 - 2 * q^71 + 5 * q^73 + 5 * q^77 + 11 * q^79 - 14 * q^81 + 11 * q^83 + 15 * q^87 + 7 * q^89 - 25 * q^91 - 29 * q^93 - 27 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more