LMFDB - Embedded newform 4400.2.b.y.4049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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, 1, 0]))

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (of order $2$, degree $1$, 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:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.4
Root			$2.30278i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.y.4049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278i q^{3} -0.697224i q^{7} -2.30278 q^{9} +1.00000 q^{11} -5.00000i q^{13} +6.90833i q^{17} -1.00000 q^{19} +1.60555 q^{21} -7.30278i q^{23} +1.60555i q^{27} -0.908327 q^{29} -10.2111 q^{31} +2.30278i q^{33} +2.39445i q^{37} +11.5139 q^{39} -5.60555 q^{41} +7.21110i q^{43} +3.00000i q^{47} +6.51388 q^{49} -15.9083 q^{51} +1.30278i q^{53} -2.30278i q^{57} -14.2111 q^{59} -7.90833 q^{61} +1.60555i q^{63} +4.00000i q^{67} +16.8167 q^{69} +2.60555 q^{71} +7.90833i q^{73} -0.697224i q^{77} -10.9083 q^{79} -10.6056 q^{81} -3.51388i q^{83} -2.09167i q^{87} -1.69722 q^{89} -3.48612 q^{91} -23.5139i q^{93} +15.3028i q^{97} -2.30278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - 2 * q^9 + 4 * q^11 - 4 * q^19 - 8 * q^21 + 18 * q^29 - 12 * q^31 + 10 * q^39 - 8 * q^41 - 10 * q^49 - 42 * q^51 - 28 * q^59 - 10 * q^61 + 24 * q^69 - 4 * q^71 - 22 * q^79 - 28 * q^81 - 14 * q^89 - 50 * q^91 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	2.30278i	1.32951i	0.747062	+	0.664754i	$0.231464\pi$
$3$	2.30278i	1.32951i	−0.747062	+	0.664754i	$0.768536\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.697224i	− 0.263526i	−0.991281	−	0.131763i	$-0.957936\pi$
$7$	− 0.697224i	− 0.263526i	0.991281	−	0.131763i	$-0.0420638\pi$
$8$	0	0
$9$	−2.30278	−0.767592
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 5.00000i	− 1.38675i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$	− 5.00000i	− 1.38675i	0.720577	−	0.693375i	$-0.243877\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.90833i	1.67552i	0.546042	+	0.837758i	$0.316134\pi$
$17$	6.90833i	1.67552i	−0.546042	+	0.837758i	$0.683866\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	1.60555	0.350360
$22$	0	0
$23$	− 7.30278i	− 1.52273i	−0.648321	−	0.761367i	$-0.724529\pi$
$23$	− 7.30278i	− 1.52273i	0.648321	−	0.761367i	$-0.275471\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.60555i	0.308988i
$28$	0	0
$29$	−0.908327	−0.168672	−0.0843360	−	0.996437i	$-0.526877\pi$
$29$	−0.908327	−0.168672	−0.0843360	+	0.996437i	$0.526877\pi$
$30$	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
$33$	2.30278i	0.400862i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.39445i	0.393645i	0.980439	+	0.196822i	$0.0630623\pi$
$37$	2.39445i	0.393645i	−0.980439	+	0.196822i	$0.936938\pi$
$38$	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
$43$	7.21110i	1.09968i	0.835269	+	0.549841i	$0.185312\pi$
$43$	7.21110i	1.09968i	−0.835269	+	0.549841i	$0.814688\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	0	0
$49$	6.51388	0.930554
$50$	0	0
$51$	−15.9083	−2.22761
$52$	0	0
$53$	1.30278i	0.178950i	0.995989	+	0.0894750i	$0.0285189\pi$
$53$	1.30278i	0.178950i	−0.995989	+	0.0894750i	$0.971481\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.30278i	− 0.305010i
$58$	0	0
$59$	−14.2111	−1.85013	−0.925064	−	0.379811i	$-0.875989\pi$
$59$	−14.2111	−1.85013	−0.925064	+	0.379811i	$0.875989\pi$
$60$	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
$63$	1.60555i	0.202280i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	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
$73$	7.90833i	0.925600i	0.886463	+	0.462800i	$0.153155\pi$
$73$	7.90833i	0.925600i	−0.886463	+	0.462800i	$0.846845\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.697224i	− 0.0794561i
$78$	0	0
$79$	−10.9083	−1.22728	−0.613641	−	0.789585i	$-0.710296\pi$
$79$	−10.9083	−1.22728	−0.613641	+	0.789585i	$0.710296\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	− 3.51388i	− 0.385698i	−0.981228	−	0.192849i	$-0.938227\pi$
$83$	− 3.51388i	− 0.385698i	0.981228	−	0.192849i	$-0.0617728\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.09167i	− 0.224251i
$88$	0	0
$89$	−1.69722	−0.179905	−0.0899527	−	0.995946i	$-0.528672\pi$
$89$	−1.69722	−0.179905	−0.0899527	+	0.995946i	$0.528672\pi$
$90$	0	0
$91$	−3.48612	−0.365445
$92$	0	0
$93$	− 23.5139i	− 2.43828i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	15.3028i	1.55376i	0.629648	+	0.776881i	$0.283199\pi$
$97$	15.3028i	1.55376i	−0.629648	+	0.776881i	$0.716801\pi$
$98$	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
$103$	2.90833i	0.286566i	0.989682	+	0.143283i	$0.0457659\pi$
$103$	2.90833i	0.286566i	−0.989682	+	0.143283i	$0.954234\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	−11.5139	−1.10283	−0.551415	−	0.834231i	$-0.685912\pi$
$109$	−11.5139	−1.10283	−0.551415	+	0.834231i	$0.685912\pi$
$110$	0	0
$111$	−5.51388	−0.523354
$112$	0	0
$113$	10.8167i	1.01755i	0.860901	+	0.508773i	$0.169901\pi$
$113$	10.8167i	1.01755i	−0.860901	+	0.508773i	$0.830099\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11.5139i	1.06446i
$118$	0	0
$119$	4.81665	0.441542
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 12.9083i	− 1.16390i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.11943i	− 0.720483i	−0.932859	−	0.360241i	$-0.882694\pi$
$127$	− 8.11943i	− 0.720483i	0.932859	−	0.360241i	$-0.117306\pi$
$128$	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
$133$	0.697224i	0.0604570i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.9083i	− 1.10283i	−0.834230	−	0.551416i	$-0.814088\pi$
$137$	− 12.9083i	− 1.10283i	0.834230	−	0.551416i	$-0.185912\pi$
$138$	0	0
$139$	−6.21110	−0.526819	−0.263409	−	0.964684i	$-0.584847\pi$
$139$	−6.21110	−0.526819	−0.263409	+	0.964684i	$0.584847\pi$
$140$	0	0
$141$	−6.90833	−0.581786
$142$	0	0
$143$	− 5.00000i	− 0.418121i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.0000i	1.23718i
$148$	0	0
$149$	−17.2111	−1.40999	−0.704994	−	0.709213i	$-0.749050\pi$
$149$	−17.2111	−1.40999	−0.704994	+	0.709213i	$0.749050\pi$
$150$	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
$153$	− 15.9083i	− 1.28611i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.2111i	1.53321i	0.642117	+	0.766606i	$0.278056\pi$
$157$	19.2111i	1.53321i	−0.642117	+	0.766606i	$0.721944\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−5.09167	−0.401280
$162$	0	0
$163$	9.30278i	0.728650i	0.931272	+	0.364325i	$0.118700\pi$
$163$	9.30278i	0.728650i	−0.931272	+	0.364325i	$0.881300\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 13.4222i	− 1.03864i	−0.854579	−	0.519321i	$-0.826185\pi$
$167$	− 13.4222i	− 1.03864i	0.854579	−	0.519321i	$-0.173815\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.30278	0.176098
$172$	0	0
$173$	4.81665i	0.366203i	0.983094	+	0.183102i	$0.0586138\pi$
$173$	4.81665i	0.366203i	−0.983094	+	0.183102i	$0.941386\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 32.7250i	− 2.45976i
$178$	0	0
$179$	12.5139	0.935331	0.467666	−	0.883905i	$-0.345095\pi$
$179$	12.5139	0.935331	0.467666	+	0.883905i	$0.345095\pi$
$180$	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
$183$	− 18.2111i	− 1.34620i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.90833i	0.505187i
$188$	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
$193$	− 13.2111i	− 0.950956i	−0.879728	−	0.475478i	$-0.842275\pi$
$193$	− 13.2111i	− 0.950956i	0.879728	−	0.475478i	$-0.157725\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.3028i	0.947784i	0.880583	+	0.473892i	$0.157151\pi$
$197$	13.3028i	0.947784i	−0.880583	+	0.473892i	$0.842849\pi$
$198$	0	0
$199$	−6.48612	−0.459789	−0.229894	−	0.973216i	$-0.573838\pi$
$199$	−6.48612	−0.459789	−0.229894	+	0.973216i	$0.573838\pi$
$200$	0	0
$201$	−9.21110	−0.649701
$202$	0	0
$203$	0.633308i	0.0444495i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	16.8167i	1.16884i
$208$	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
$213$	6.00000i	0.411113i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.11943i	0.483298i
$218$	0	0
$219$	−18.2111	−1.23059
$220$	0	0
$221$	34.5416	2.32352
$222$	0	0
$223$	− 22.6333i	− 1.51564i	−0.652465	−	0.757819i	$-0.726265\pi$
$223$	− 22.6333i	− 1.51564i	0.652465	−	0.757819i	$-0.273735\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.69722i	− 0.112649i	−0.998413	−	0.0563244i	$-0.982062\pi$
$227$	− 1.69722i	− 0.112649i	0.998413	−	0.0563244i	$-0.0179381\pi$
$228$	0	0
$229$	18.7250	1.23738	0.618691	−	0.785635i	$-0.287663\pi$
$229$	18.7250	1.23738	0.618691	+	0.785635i	$0.287663\pi$
$230$	0	0
$231$	1.60555	0.105638
$232$	0	0
$233$	− 15.9083i	− 1.04219i	−0.853499	−	0.521095i	$-0.825524\pi$
$233$	− 15.9083i	− 1.04219i	0.853499	−	0.521095i	$-0.174476\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 25.1194i	− 1.63168i
$238$	0	0
$239$	21.1194	1.36610	0.683051	−	0.730371i	$-0.260653\pi$
$239$	21.1194	1.36610	0.683051	+	0.730371i	$0.260653\pi$
$240$	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
$243$	− 19.6056i	− 1.25770i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.00000i	0.318142i
$248$	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
$253$	− 7.30278i	− 0.459122i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	1.66947	0.103736
$260$	0	0
$261$	2.09167	0.129471
$262$	0	0
$263$	− 22.8167i	− 1.40694i	−0.710727	−	0.703468i	$-0.751634\pi$
$263$	− 22.8167i	− 1.40694i	0.710727	−	0.703468i	$-0.248366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3.90833i	− 0.239186i
$268$	0	0
$269$	−8.72498	−0.531971	−0.265986	−	0.963977i	$-0.585697\pi$
$269$	−8.72498	−0.531971	−0.265986	+	0.963977i	$0.585697\pi$
$270$	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
$273$	− 8.02776i	− 0.485862i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.3944i	0.864879i	0.901663	+	0.432439i	$0.142347\pi$
$277$	14.3944i	0.864879i	−0.901663	+	0.432439i	$0.857653\pi$
$278$	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
$283$	6.30278i	0.374661i	0.982297	+	0.187331i	$0.0599836\pi$
$283$	6.30278i	0.374661i	−0.982297	+	0.187331i	$0.940016\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.90833i	0.230701i
$288$	0	0
$289$	−30.7250	−1.80735
$290$	0	0
$291$	−35.2389	−2.06574
$292$	0	0
$293$	− 0.788897i	− 0.0460879i	−0.999734	−	0.0230439i	$-0.992664\pi$
$293$	− 0.788897i	− 0.0460879i	0.999734	−	0.0230439i	$-0.00733576\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.60555i	0.0931635i
$298$	0	0
$299$	−36.5139	−2.11165
$300$	0	0
$301$	5.02776	0.289795
$302$	0	0
$303$	− 1.18335i	− 0.0679815i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.9083i	0.965009i	0.875893	+	0.482505i	$0.160273\pi$
$307$	16.9083i	0.965009i	−0.875893	+	0.482505i	$0.839727\pi$
$308$	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
$313$	− 0.183346i	− 0.0103633i	−0.999987	−	0.00518167i	$-0.998351\pi$
$313$	− 0.183346i	− 0.0103633i	0.999987	−	0.00518167i	$-0.00164938\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.908327i	− 0.0510167i	−0.999675	−	0.0255084i	$-0.991880\pi$
$317$	− 0.908327i	− 0.0510167i	0.999675	−	0.0255084i	$-0.00812044\pi$
$318$	0	0
$319$	−0.908327	−0.0508565
$320$	0	0
$321$	−6.90833	−0.385585
$322$	0	0
$323$	− 6.90833i	− 0.384390i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 26.5139i	− 1.46622i
$328$	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
$333$	− 5.51388i	− 0.302159i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 30.8444i	− 1.68020i	−0.542430	−	0.840101i	$-0.682496\pi$
$337$	− 30.8444i	− 1.68020i	0.542430	−	0.840101i	$-0.317504\pi$
$338$	0	0
$339$	−24.9083	−1.35283
$340$	0	0
$341$	−10.2111	−0.552962
$342$	0	0
$343$	− 9.42221i	− 0.508751i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.5139i	0.671780i	0.941901	+	0.335890i	$0.109037\pi$
$347$	12.5139i	0.671780i	−0.941901	+	0.335890i	$0.890963\pi$
$348$	0	0
$349$	5.18335	0.277458	0.138729	−	0.990330i	$-0.455698\pi$
$349$	5.18335	0.277458	0.138729	+	0.990330i	$0.455698\pi$
$350$	0	0
$351$	8.02776	0.428490
$352$	0	0
$353$	− 18.6333i	− 0.991751i	−0.868394	−	0.495875i	$-0.834847\pi$
$353$	− 18.6333i	− 0.991751i	0.868394	−	0.495875i	$-0.165153\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	11.0917i	0.587034i
$358$	0	0
$359$	0.788897	0.0416364	0.0208182	−	0.999783i	$-0.493373\pi$
$359$	0.788897	0.0416364	0.0208182	+	0.999783i	$0.493373\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	2.30278i	0.120864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	20.6972i	1.08039i	0.841541	+	0.540193i	$0.181649\pi$
$367$	20.6972i	1.08039i	−0.841541	+	0.540193i	$0.818351\pi$
$368$	0	0
$369$	12.9083	0.671981
$370$	0	0
$371$	0.908327	0.0471580
$372$	0	0
$373$	− 27.4222i	− 1.41987i	−0.704268	−	0.709934i	$-0.748725\pi$
$373$	− 27.4222i	− 1.41987i	0.704268	−	0.709934i	$-0.251275\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.54163i	0.233906i
$378$	0	0
$379$	3.18335	0.163518	0.0817588	−	0.996652i	$-0.473946\pi$
$379$	3.18335	0.163518	0.0817588	+	0.996652i	$0.473946\pi$
$380$	0	0
$381$	18.6972	0.957888
$382$	0	0
$383$	21.6333i	1.10541i	0.833377	+	0.552705i	$0.186404\pi$
$383$	21.6333i	1.10541i	−0.833377	+	0.552705i	$0.813596\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 16.6056i	− 0.844108i
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	50.4500	2.55136
$392$	0	0
$393$	22.8167i	1.15095i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	21.6972i	1.08895i	0.838776	+	0.544476i	$0.183272\pi$
$397$	21.6972i	1.08895i	−0.838776	+	0.544476i	$0.816728\pi$
$398$	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
$403$	51.0555i	2.54326i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.39445i	0.118688i
$408$	0	0
$409$	6.21110	0.307119	0.153560	−	0.988139i	$-0.450926\pi$
$409$	6.21110	0.307119	0.153560	+	0.988139i	$0.450926\pi$
$410$	0	0
$411$	29.7250	1.46623
$412$	0	0
$413$	9.90833i	0.487557i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 14.3028i	− 0.700410i
$418$	0	0
$419$	6.39445	0.312389	0.156195	−	0.987726i	$-0.450077\pi$
$419$	6.39445	0.312389	0.156195	+	0.987726i	$0.450077\pi$
$420$	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
$423$	− 6.90833i	− 0.335894i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.51388i	0.266835i
$428$	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
$433$	− 5.00000i	− 0.240285i	−0.992757	−	0.120142i	$-0.961665\pi$
$433$	− 5.00000i	− 0.240285i	0.992757	−	0.120142i	$-0.0383351\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.30278i	0.349339i
$438$	0	0
$439$	24.3028	1.15991	0.579954	−	0.814649i	$-0.303070\pi$
$439$	24.3028	1.15991	0.579954	+	0.814649i	$0.303070\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	0	0
$443$	− 8.60555i	− 0.408862i	−0.978881	−	0.204431i	$-0.934466\pi$
$443$	− 8.60555i	− 0.408862i	0.978881	−	0.204431i	$-0.0655344\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 39.6333i	− 1.87459i
$448$	0	0
$449$	−23.4861	−1.10838	−0.554189	−	0.832391i	$-0.686972\pi$
$449$	−23.4861	−1.10838	−0.554189	+	0.832391i	$0.686972\pi$
$450$	0	0
$451$	−5.60555	−0.263955
$452$	0	0
$453$	− 1.88057i	− 0.0883569i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 20.6972i	− 0.968175i	−0.875020	−	0.484088i	$-0.839152\pi$
$457$	− 20.6972i	− 0.968175i	0.875020	−	0.484088i	$-0.160848\pi$
$458$	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
$463$	11.7889i	0.547877i	0.961747	+	0.273938i	$0.0883264\pi$
$463$	11.7889i	0.547877i	−0.961747	+	0.273938i	$0.911674\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.6333i	0.862247i	0.902293	+	0.431123i	$0.141883\pi$
$467$	18.6333i	0.862247i	−0.902293	+	0.431123i	$0.858117\pi$
$468$	0	0
$469$	2.78890	0.128779
$470$	0	0
$471$	−44.2389	−2.03842
$472$	0	0
$473$	7.21110i	0.331567i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.00000i	− 0.137361i
$478$	0	0
$479$	−34.8167	−1.59081	−0.795407	−	0.606076i	$-0.792743\pi$
$479$	−34.8167	−1.59081	−0.795407	+	0.606076i	$0.792743\pi$
$480$	0	0
$481$	11.9722	0.545887
$482$	0	0
$483$	− 11.7250i	− 0.533505i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.21110i	− 0.190823i	−0.995438	−	0.0954116i	$-0.969583\pi$
$487$	− 4.21110i	− 0.190823i	0.995438	−	0.0954116i	$-0.0304167\pi$
$488$	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
$493$	− 6.27502i	− 0.282613i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.81665i	− 0.0814881i
$498$	0	0
$499$	−3.48612	−0.156060	−0.0780301	−	0.996951i	$-0.524863\pi$
$499$	−3.48612	−0.156060	−0.0780301	+	0.996951i	$0.524863\pi$
$500$	0	0
$501$	30.9083	1.38088
$502$	0	0
$503$	9.39445i	0.418878i	0.977822	+	0.209439i	$0.0671637\pi$
$503$	9.39445i	0.418878i	−0.977822	+	0.209439i	$0.932836\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 27.6333i	− 1.22724i
$508$	0	0
$509$	22.6972	1.00604	0.503018	−	0.864276i	$-0.332223\pi$
$509$	22.6972	1.00604	0.503018	+	0.864276i	$0.332223\pi$
$510$	0	0
$511$	5.51388	0.243920
$512$	0	0
$513$	− 1.60555i	− 0.0708868i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.00000i	0.131940i
$518$	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
$523$	− 32.4222i	− 1.41772i	−0.705347	−	0.708862i	$-0.749209\pi$
$523$	− 32.4222i	− 1.41772i	0.705347	−	0.708862i	$-0.250791\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 70.5416i	− 3.07284i
$528$	0	0
$529$	−30.3305	−1.31872
$530$	0	0
$531$	32.7250	1.42014
$532$	0	0
$533$	28.0278i	1.21402i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	28.8167i	1.24353i
$538$	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
$543$	− 45.8444i	− 1.96737i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.11943i	0.304405i	0.988349	+	0.152202i	$0.0486366\pi$
$547$	7.11943i	0.304405i	−0.988349	+	0.152202i	$0.951363\pi$
$548$	0	0
$549$	18.2111	0.777231
$550$	0	0
$551$	0.908327	0.0386960
$552$	0	0
$553$	7.60555i	0.323421i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.4222i	0.822945i	0.911422	+	0.411473i	$0.134985\pi$
$557$	19.4222i	0.822945i	−0.911422	+	0.411473i	$0.865015\pi$
$558$	0	0
$559$	36.0555	1.52499
$560$	0	0
$561$	−15.9083	−0.671650
$562$	0	0
$563$	8.09167i	0.341023i	0.985356	+	0.170512i	$0.0545421\pi$
$563$	8.09167i	0.341023i	−0.985356	+	0.170512i	$0.945458\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.39445i	0.310538i
$568$	0	0
$569$	46.1472	1.93459	0.967295	−	0.253653i	$-0.0816321\pi$
$569$	46.1472	1.93459	0.967295	+	0.253653i	$0.0816321\pi$
$570$	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
$573$	− 23.7250i	− 0.991125i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	44.3583i	1.84666i	0.384008	+	0.923330i	$0.374544\pi$
$577$	44.3583i	1.84666i	−0.384008	+	0.923330i	$0.625456\pi$
$578$	0	0
$579$	30.4222	1.26430
$580$	0	0
$581$	−2.44996	−0.101642
$582$	0	0
$583$	1.30278i	0.0539555i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 16.5416i	− 0.682746i	−0.939928	−	0.341373i	$-0.889108\pi$
$587$	− 16.5416i	− 0.682746i	0.939928	−	0.341373i	$-0.110892\pi$
$588$	0	0
$589$	10.2111	0.420741
$590$	0	0
$591$	−30.6333	−1.26009
$592$	0	0
$593$	6.39445i	0.262589i	0.991343	+	0.131294i	$0.0419133\pi$
$593$	6.39445i	0.262589i	−0.991343	+	0.131294i	$0.958087\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 14.9361i	− 0.611293i
$598$	0	0
$599$	−24.9083	−1.01773	−0.508863	−	0.860847i	$-0.669934\pi$
$599$	−24.9083	−1.01773	−0.508863	+	0.860847i	$0.669934\pi$
$600$	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
$603$	− 9.21110i	− 0.375105i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7.21110i	− 0.292690i	−0.989234	−	0.146345i	$-0.953249\pi$
$607$	− 7.21110i	− 0.292690i	0.989234	−	0.146345i	$-0.0467509\pi$
$608$	0	0
$609$	−1.45837	−0.0590959
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	15.8806i	0.641410i	0.947179	+	0.320705i	$0.103920\pi$
$613$	15.8806i	0.641410i	−0.947179	+	0.320705i	$0.896080\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.39445i	0.136655i	0.997663	+	0.0683277i	$0.0217663\pi$
$617$	3.39445i	0.136655i	−0.997663	+	0.0683277i	$0.978234\pi$
$618$	0	0
$619$	−11.4222	−0.459097	−0.229549	−	0.973297i	$-0.573725\pi$
$619$	−11.4222	−0.459097	−0.229549	+	0.973297i	$0.573725\pi$
$620$	0	0
$621$	11.7250	0.470507
$622$	0	0
$623$	1.18335i	0.0474098i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.30278i	− 0.0919640i
$628$	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
$633$	58.1194i	2.31004i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 32.5694i	− 1.29045i
$638$	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
$643$	− 22.0000i	− 0.867595i	−0.901010	−	0.433798i	$-0.857173\pi$
$643$	− 22.0000i	− 0.867595i	0.901010	−	0.433798i	$-0.142827\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.2389i	1.30675i	0.757033	+	0.653377i	$0.226648\pi$
$647$	33.2389i	1.30675i	−0.757033	+	0.653377i	$0.773352\pi$
$648$	0	0
$649$	−14.2111	−0.557835
$650$	0	0
$651$	−16.3944	−0.642549
$652$	0	0
$653$	− 6.11943i	− 0.239472i	−0.992806	−	0.119736i	$-0.961795\pi$
$653$	− 6.11943i	− 0.239472i	0.992806	−	0.119736i	$-0.0382048\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 18.2111i	− 0.710483i
$658$	0	0
$659$	−30.9083	−1.20402	−0.602009	−	0.798489i	$-0.705633\pi$
$659$	−30.9083	−1.20402	−0.602009	+	0.798489i	$0.705633\pi$
$660$	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
$663$	79.5416i	3.08914i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.63331i	0.256843i
$668$	0	0
$669$	52.1194	2.01505
$670$	0	0
$671$	−7.90833	−0.305298
$672$	0	0
$673$	− 30.0278i	− 1.15748i	−0.815510	−	0.578742i	$-0.803544\pi$
$673$	− 30.0278i	− 1.15748i	0.815510	−	0.578742i	$-0.196456\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.2389i	0.931575i	0.884897	+	0.465788i	$0.154229\pi$
$677$	24.2389i	0.931575i	−0.884897	+	0.465788i	$0.845771\pi$
$678$	0	0
$679$	10.6695	0.409457
$680$	0	0
$681$	3.90833	0.149767
$682$	0	0
$683$	47.8444i	1.83072i	0.402642	+	0.915358i	$0.368092\pi$
$683$	47.8444i	1.83072i	−0.402642	+	0.915358i	$0.631908\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	43.1194i	1.64511i
$688$	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
$693$	1.60555i	0.0609898i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 38.7250i	− 1.46681i
$698$	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
$703$	− 2.39445i	− 0.0903083i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.358288i	0.0134748i
$708$	0	0
$709$	31.6333	1.18801	0.594007	−	0.804460i	$-0.297545\pi$
$709$	31.6333	1.18801	0.594007	+	0.804460i	$0.297545\pi$
$710$	0	0
$711$	25.1194	0.942052
$712$	0	0
$713$	74.5694i	2.79265i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	48.6333i	1.81624i
$718$	0	0
$719$	7.18335	0.267894	0.133947	−	0.990989i	$-0.457235\pi$
$719$	7.18335	0.267894	0.133947	+	0.990989i	$0.457235\pi$
$720$	0	0
$721$	2.02776	0.0755176
$722$	0	0
$723$	50.5139i	1.87863i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	39.3305i	1.45869i	0.684147	+	0.729344i	$0.260175\pi$
$727$	39.3305i	1.45869i	−0.684147	+	0.729344i	$0.739825\pi$
$728$	0	0
$729$	13.3305	0.493723
$730$	0	0
$731$	−49.8167	−1.84254
$732$	0	0
$733$	− 19.6056i	− 0.724148i	−0.932149	−	0.362074i	$-0.882069\pi$
$733$	− 19.6056i	− 0.724148i	0.932149	−	0.362074i	$-0.117931\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000i	0.147342i
$738$	0	0
$739$	35.1194	1.29189	0.645945	−	0.763384i	$-0.276464\pi$
$739$	35.1194	1.29189	0.645945	+	0.763384i	$0.276464\pi$
$740$	0	0
$741$	−11.5139	−0.422973
$742$	0	0
$743$	40.6972i	1.49304i	0.665365	+	0.746518i	$0.268276\pi$
$743$	40.6972i	1.49304i	−0.665365	+	0.746518i	$0.731724\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.09167i	0.296059i
$748$	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
$753$	− 15.9083i	− 0.579732i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	49.0555i	1.78295i	0.453067	+	0.891476i	$0.350330\pi$
$757$	49.0555i	1.78295i	−0.453067	+	0.891476i	$0.649670\pi$
$758$	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
$763$	8.02776i	0.290624i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	71.0555i	2.56567i
$768$	0	0
$769$	5.18335	0.186916	0.0934581	−	0.995623i	$-0.470208\pi$
$769$	5.18335	0.186916	0.0934581	+	0.995623i	$0.470208\pi$
$770$	0	0
$771$	41.4500	1.49278
$772$	0	0
$773$	− 3.11943i	− 0.112198i	−0.998425	−	0.0560990i	$-0.982134\pi$
$773$	− 3.11943i	− 0.112198i	0.998425	−	0.0560990i	$-0.0178663\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.84441i	0.137917i
$778$	0	0
$779$	5.60555	0.200840
$780$	0	0
$781$	2.60555	0.0932340
$782$	0	0
$783$	− 1.45837i	− 0.0521177i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 10.2111i	− 0.363986i	−0.983300	−	0.181993i	$-0.941745\pi$
$787$	− 10.2111i	− 0.363986i	0.983300	−	0.181993i	$-0.0582549\pi$
$788$	0	0
$789$	52.5416	1.87053
$790$	0	0
$791$	7.54163	0.268150
$792$	0	0
$793$	39.5416i	1.40416i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.51388i	0.124468i	0.998062	+	0.0622340i	$0.0198225\pi$
$797$	3.51388i	0.124468i	−0.998062	+	0.0622340i	$0.980178\pi$
$798$	0	0
$799$	−20.7250	−0.733197
$800$	0	0
$801$	3.90833	0.138094
$802$	0	0
$803$	7.90833i	0.279079i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 20.0917i	− 0.707260i
$808$	0	0
$809$	−39.6333	−1.39343	−0.696716	−	0.717347i	$-0.745356\pi$
$809$	−39.6333	−1.39343	−0.696716	+	0.717347i	$0.745356\pi$
$810$	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
$813$	0.486122i	0.0170490i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.21110i	− 0.252285i
$818$	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
$823$	18.4222i	0.642158i	0.947052	+	0.321079i	$0.104045\pi$
$823$	18.4222i	0.642158i	−0.947052	+	0.321079i	$0.895955\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13.8167i	− 0.480452i	−0.970717	−	0.240226i	$-0.922778\pi$
$827$	− 13.8167i	− 0.480452i	0.970717	−	0.240226i	$-0.0772216\pi$
$828$	0	0
$829$	−29.7527	−1.03336	−0.516678	−	0.856180i	$-0.672831\pi$
$829$	−29.7527	−1.03336	−0.516678	+	0.856180i	$0.672831\pi$
$830$	0	0
$831$	−33.1472	−1.14986
$832$	0	0
$833$	45.0000i	1.55916i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 16.3944i	− 0.566675i
$838$	0	0
$839$	−9.11943	−0.314838	−0.157419	−	0.987532i	$-0.550317\pi$
$839$	−9.11943	−0.314838	−0.157419	+	0.987532i	$0.550317\pi$
$840$	0	0
$841$	−28.1749	−0.971550
$842$	0	0
$843$	− 2.72498i	− 0.0938533i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.697224i	− 0.0239569i
$848$	0	0
$849$	−14.5139	−0.498115
$850$	0	0
$851$	17.4861	0.599417
$852$	0	0
$853$	12.7250i	0.435695i	0.975983	+	0.217848i	$0.0699035\pi$
$853$	12.7250i	0.435695i	−0.975983	+	0.217848i	$0.930096\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.00000i	− 0.102478i	−0.998686	−	0.0512390i	$-0.983683\pi$
$857$	− 3.00000i	− 0.102478i	0.998686	−	0.0512390i	$-0.0163170\pi$
$858$	0	0
$859$	41.3944	1.41236	0.706180	−	0.708032i	$-0.250417\pi$
$859$	41.3944	1.41236	0.706180	+	0.708032i	$0.250417\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	0	0
$863$	− 12.3944i	− 0.421912i	−0.977496	−	0.210956i	$-0.932342\pi$
$863$	− 12.3944i	− 0.421912i	0.977496	−	0.210956i	$-0.0676577\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 70.7527i	− 2.40289i
$868$	0	0
$869$	−10.9083	−0.370040
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	− 35.2389i	− 1.19265i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.0000i	0.675352i	0.941262	+	0.337676i	$0.109641\pi$
$877$	20.0000i	0.675352i	−0.941262	+	0.337676i	$0.890359\pi$
$878$	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
$883$	52.4500i	1.76508i	0.470236	+	0.882541i	$0.344169\pi$
$883$	52.4500i	1.76508i	−0.470236	+	0.882541i	$0.655831\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.23886i	0.108750i	0.998521	+	0.0543751i	$0.0173167\pi$
$887$	3.23886i	0.108750i	−0.998521	+	0.0543751i	$0.982683\pi$
$888$	0	0
$889$	−5.66106	−0.189866
$890$	0	0
$891$	−10.6056	−0.355299
$892$	0	0
$893$	− 3.00000i	− 0.100391i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 84.0833i	− 2.80746i
$898$	0	0
$899$	9.27502	0.309339
$900$	0	0
$901$	−9.00000	−0.299833
$902$	0	0
$903$	11.5778i	0.385285i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.00000i	0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$	4.00000i	0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	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
$913$	− 3.51388i	− 0.116292i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6.90833i	− 0.228133i
$918$	0	0
$919$	26.7889	0.883684	0.441842	−	0.897093i	$-0.354325\pi$
$919$	26.7889	0.883684	0.441842	+	0.897093i	$0.354325\pi$
$920$	0	0
$921$	−38.9361	−1.28299
$922$	0	0
$923$	− 13.0278i	− 0.428814i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6.69722i	− 0.219966i
$928$	0	0
$929$	−53.6056	−1.75874	−0.879371	−	0.476138i	$-0.842036\pi$
$929$	−53.6056	−1.75874	−0.879371	+	0.476138i	$0.842036\pi$
$930$	0	0
$931$	−6.51388	−0.213484
$932$	0	0
$933$	− 11.0917i	− 0.363125i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 9.21110i	− 0.300914i	−0.988617	−	0.150457i	$-0.951926\pi$
$937$	− 9.21110i	− 0.300914i	0.988617	−	0.150457i	$-0.0480744\pi$
$938$	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
$943$	40.9361i	1.33306i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 6.63331i	− 0.215554i	−0.994175	−	0.107777i	$-0.965627\pi$
$947$	− 6.63331i	− 0.215554i	0.994175	−	0.107777i	$-0.0343732\pi$
$948$	0	0
$949$	39.5416	1.28358
$950$	0	0
$951$	2.09167	0.0678271
$952$	0	0
$953$	− 37.2666i	− 1.20718i	−0.797293	−	0.603592i	$-0.793736\pi$
$953$	− 37.2666i	− 1.20718i	0.797293	−	0.603592i	$-0.206264\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.09167i	− 0.0676142i
$958$	0	0
$959$	−9.00000	−0.290625
$960$	0	0
$961$	73.2666	2.36344
$962$	0	0
$963$	− 6.90833i	− 0.222618i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 14.9083i	− 0.479419i	−0.970845	−	0.239710i	$-0.922948\pi$
$967$	− 14.9083i	− 0.479419i	0.970845	−	0.239710i	$-0.0770523\pi$
$968$	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
$973$	4.33053i	0.138830i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 52.0278i	− 1.66452i	−0.554389	−	0.832258i	$-0.687048\pi$
$977$	− 52.0278i	− 1.66452i	0.554389	−	0.832258i	$-0.312952\pi$
$978$	0	0
$979$	−1.69722	−0.0542435
$980$	0	0
$981$	26.5139	0.846523
$982$	0	0
$983$	8.84441i	0.282093i	0.990003	+	0.141046i	$0.0450467\pi$
$983$	8.84441i	0.282093i	−0.990003	+	0.141046i	$0.954953\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.81665i	0.153316i
$988$	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
$993$	49.7527i	1.57886i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.7250i	1.47979i	0.672720	+	0.739897i	$0.265126\pi$
$997$	46.7250i	1.47979i	−0.672720	+	0.739897i	$0.734874\pi$
$998$	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.b.y.4049.4		4
4.3	odd	2		275.2.b.c.199.2		4
5.2	odd	4		4400.2.a.bs.1.2		2
5.3	odd	4		4400.2.a.bh.1.1		2
5.4	even	2	inner	4400.2.b.y.4049.1		4
12.11	even	2		2475.2.c.k.199.3		4
20.3	even	4		275.2.a.f.1.1	yes	2
20.7	even	4		275.2.a.e.1.2	✓	2
20.19	odd	2		275.2.b.c.199.3		4
60.23	odd	4		2475.2.a.o.1.2		2
60.47	odd	4		2475.2.a.t.1.1		2
60.59	even	2		2475.2.c.k.199.2		4
220.43	odd	4		3025.2.a.h.1.2		2
220.87	odd	4		3025.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.2	✓	2	20.7	even	4
275.2.a.f.1.1	yes	2	20.3	even	4
275.2.b.c.199.2		4	4.3	odd	2
275.2.b.c.199.3		4	20.19	odd	2
2475.2.a.o.1.2		2	60.23	odd	4
2475.2.a.t.1.1		2	60.47	odd	4
2475.2.c.k.199.2		4	60.59	even	2
2475.2.c.k.199.3		4	12.11	even	2
3025.2.a.h.1.2		2	220.43	odd	4
3025.2.a.n.1.1		2	220.87	odd	4
4400.2.a.bh.1.1		2	5.3	odd	4
4400.2.a.bs.1.2		2	5.2	odd	4
4400.2.b.y.4049.1		4	5.4	even	2	inner
4400.2.b.y.4049.4		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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.30278i q^{3} -0.697224i q^{7} -2.30278 q^{9} +1.00000 q^{11} -5.00000i q^{13} +6.90833i q^{17} -1.00000 q^{19} +1.60555 q^{21} -7.30278i q^{23} +1.60555i q^{27} -0.908327 q^{29} -10.2111 q^{31} +2.30278i q^{33} +2.39445i q^{37} +11.5139 q^{39} -5.60555 q^{41} +7.21110i q^{43} +3.00000i q^{47} +6.51388 q^{49} -15.9083 q^{51} +1.30278i q^{53} -2.30278i q^{57} -14.2111 q^{59} -7.90833 q^{61} +1.60555i q^{63} +4.00000i q^{67} +16.8167 q^{69} +2.60555 q^{71} +7.90833i q^{73} -0.697224i q^{77} -10.9083 q^{79} -10.6056 q^{81} -3.51388i q^{83} -2.09167i q^{87} -1.69722 q^{89} -3.48612 q^{91} -23.5139i q^{93} +15.3028i q^{97} -2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - 2 * q^9 + 4 * q^11 - 4 * q^19 - 8 * q^21 + 18 * q^29 - 12 * q^31 + 10 * q^39 - 8 * q^41 - 10 * q^49 - 42 * q^51 - 28 * q^59 - 10 * q^61 + 24 * q^69 - 4 * q^71 - 22 * q^79 - 28 * q^81 - 14 * q^89 - 50 * q^91 - 2 * q^99

Introduction

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Modular forms

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