LMFDB - Embedded newform 1100.4.b.j.749.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,4,Mod(749,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1100.749");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1100.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:	$64.9021010063$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 187x^{8} + 12537x^{6} + 358849x^{4} + 3893659x^{2} + 7371225 $ x^10 + 187x^8 + 12537x^6 + 358849x^4 + 3893659x^2 + 7371225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.8
Root			$4.57551i$ of defining polynomial
Character	$\chi$	$=$	1100.749
Dual form			1100.4.b.j.749.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.57551i q^{3} +20.7640i q^{7} -4.08629 q^{9} +11.0000 q^{11} +37.3507i q^{13} -32.4508i q^{17} -71.7593 q^{19} -115.770 q^{21} +86.2767i q^{23} +127.756i q^{27} +19.4826 q^{29} -113.280 q^{31} +61.3306i q^{33} +36.6594i q^{37} -208.249 q^{39} +152.019 q^{41} +75.7880i q^{43} +216.190i q^{47} -88.1454 q^{49} +180.930 q^{51} +291.842i q^{53} -400.095i q^{57} -673.318 q^{59} +267.522 q^{61} -84.8479i q^{63} -682.646i q^{67} -481.036 q^{69} +410.937 q^{71} -137.968i q^{73} +228.404i q^{77} -918.618 q^{79} -822.632 q^{81} +959.791i q^{83} +108.626i q^{87} -622.193 q^{89} -775.552 q^{91} -631.594i q^{93} -1390.64i q^{97} -44.9492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 118 q^{9} + 110 q^{11} - 6 q^{19} + 316 q^{21} - 314 q^{29} + 114 q^{31} - 1036 q^{39} + 148 q^{41} - 1054 q^{49} - 2292 q^{51} - 1384 q^{59} - 1996 q^{61} - 2492 q^{69} - 3290 q^{71} - 2640 q^{79}+ \cdots - 1298 q^{99}+O(q^{100}) $ 10 * q - 118 * q^9 + 110 * q^11 - 6 * q^19 + 316 * q^21 - 314 * q^29 + 114 * q^31 - 1036 * q^39 + 148 * q^41 - 1054 * q^49 - 2292 * q^51 - 1384 * q^59 - 1996 * q^61 - 2492 * q^69 - 3290 * q^71 - 2640 * q^79 - 1550 * q^81 - 2482 * q^89 - 2432 * q^91 - 1298 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	5.57551i	1.07301i	0.843898	+	0.536504i	$0.180255\pi$
$3$	5.57551i	1.07301i	−0.843898	+	0.536504i	$0.819745\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	20.7640i	1.12115i	0.828103	+	0.560576i	$0.189420\pi$
$7$	20.7640i	1.12115i	−0.828103	+	0.560576i	$0.810580\pi$
$8$	0	0
$9$	−4.08629	−0.151344
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	37.3507i	0.796864i	0.917198	+	0.398432i	$0.130446\pi$
$13$	37.3507i	0.796864i	−0.917198	+	0.398432i	$0.869554\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 32.4508i	− 0.462969i	−0.972839	−	0.231485i	$-0.925642\pi$
$17$	− 32.4508i	− 0.462969i	0.972839	−	0.231485i	$-0.0743583\pi$
$18$	0	0
$19$	−71.7593	−0.866459	−0.433229	−	0.901284i	$-0.642626\pi$
$19$	−71.7593	−0.866459	−0.433229	+	0.901284i	$0.642626\pi$
$20$	0	0
$21$	−115.770	−1.20300
$22$	0	0
$23$	86.2767i	0.782171i	0.920354	+	0.391086i	$0.127900\pi$
$23$	86.2767i	0.782171i	−0.920354	+	0.391086i	$0.872100\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	127.756i	0.910614i
$28$	0	0
$29$	19.4826	0.124753	0.0623764	−	0.998053i	$-0.480132\pi$
$29$	19.4826	0.124753	0.0623764	+	0.998053i	$0.480132\pi$
$30$	0	0
$31$	−113.280	−0.656313	−0.328156	−	0.944623i	$-0.606427\pi$
$31$	−113.280	−0.656313	−0.328156	+	0.944623i	$0.606427\pi$
$32$	0	0
$33$	61.3306i	0.323524i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	36.6594i	0.162886i	0.996678	+	0.0814429i	$0.0259528\pi$
$37$	36.6594i	0.162886i	−0.996678	+	0.0814429i	$0.974047\pi$
$38$	0	0
$39$	−208.249	−0.855041
$40$	0	0
$41$	152.019	0.579058	0.289529	−	0.957169i	$-0.406501\pi$
$41$	152.019	0.579058	0.289529	+	0.957169i	$0.406501\pi$
$42$	0	0
$43$	75.7880i	0.268781i	0.990928	+	0.134390i	$0.0429076\pi$
$43$	75.7880i	0.268781i	−0.990928	+	0.134390i	$0.957092\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	216.190i	0.670947i	0.942050	+	0.335474i	$0.108896\pi$
$47$	216.190i	0.670947i	−0.942050	+	0.335474i	$0.891104\pi$
$48$	0	0
$49$	−88.1454	−0.256984
$50$	0	0
$51$	180.930	0.496769
$52$	0	0
$53$	291.842i	0.756370i	0.925730	+	0.378185i	$0.123452\pi$
$53$	291.842i	0.756370i	−0.925730	+	0.378185i	$0.876548\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 400.095i	− 0.929716i
$58$	0	0
$59$	−673.318	−1.48574	−0.742869	−	0.669437i	$-0.766535\pi$
$59$	−673.318	−1.48574	−0.742869	+	0.669437i	$0.766535\pi$
$60$	0	0
$61$	267.522	0.561520	0.280760	−	0.959778i	$-0.409413\pi$
$61$	267.522	0.561520	0.280760	+	0.959778i	$0.409413\pi$
$62$	0	0
$63$	− 84.8479i	− 0.169680i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 682.646i	− 1.24475i	−0.782718	−	0.622377i	$-0.786167\pi$
$67$	− 682.646i	− 1.24475i	0.782718	−	0.622377i	$-0.213833\pi$
$68$	0	0
$69$	−481.036	−0.839275
$70$	0	0
$71$	410.937	0.686890	0.343445	−	0.939173i	$-0.388406\pi$
$71$	410.937	0.686890	0.343445	+	0.939173i	$0.388406\pi$
$72$	0	0
$73$	− 137.968i	− 0.221205i	−0.993865	−	0.110603i	$-0.964722\pi$
$73$	− 137.968i	− 0.221205i	0.993865	−	0.110603i	$-0.0352781\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	228.404i	0.338040i
$78$	0	0
$79$	−918.618	−1.30826	−0.654130	−	0.756382i	$-0.726965\pi$
$79$	−918.618	−1.30826	−0.654130	+	0.756382i	$0.726965\pi$
$80$	0	0
$81$	−822.632	−1.12844
$82$	0	0
$83$	959.791i	1.26929i	0.772805	+	0.634643i	$0.218853\pi$
$83$	959.791i	1.26929i	−0.772805	+	0.634643i	$0.781147\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	108.626i	0.133861i
$88$	0	0
$89$	−622.193	−0.741038	−0.370519	−	0.928825i	$-0.620820\pi$
$89$	−622.193	−0.741038	−0.370519	+	0.928825i	$0.620820\pi$
$90$	0	0
$91$	−775.552	−0.893406
$92$	0	0
$93$	− 631.594i	− 0.704228i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1390.64i	− 1.45565i	−0.685765	−	0.727823i	$-0.740532\pi$
$97$	− 1390.64i	− 1.45565i	0.685765	−	0.727823i	$-0.259468\pi$
$98$	0	0
$99$	−44.9492	−0.0456320
$100$	0	0
$101$	94.0599	0.0926664	0.0463332	−	0.998926i	$-0.485246\pi$
$101$	94.0599	0.0926664	0.0463332	+	0.998926i	$0.485246\pi$
$102$	0	0
$103$	1032.75i	0.987963i	0.869472	+	0.493982i	$0.164459\pi$
$103$	1032.75i	0.987963i	−0.869472	+	0.493982i	$0.835541\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 743.781i	− 0.672000i	−0.941862	−	0.336000i	$-0.890926\pi$
$107$	− 743.781i	− 0.672000i	0.941862	−	0.336000i	$-0.109074\pi$
$108$	0	0
$109$	668.924	0.587811	0.293905	−	0.955835i	$-0.405045\pi$
$109$	668.924	0.587811	0.293905	+	0.955835i	$0.405045\pi$
$110$	0	0
$111$	−204.395	−0.174778
$112$	0	0
$113$	80.1193i	0.0666990i	0.999444	+	0.0333495i	$0.0106174\pi$
$113$	80.1193i	0.0666990i	−0.999444	+	0.0333495i	$0.989383\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 152.626i	− 0.120601i
$118$	0	0
$119$	673.810	0.519059
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	847.583i	0.621333i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1126.07i	− 0.786788i	−0.919370	−	0.393394i	$-0.871301\pi$
$127$	− 1126.07i	− 0.786788i	0.919370	−	0.393394i	$-0.128699\pi$
$128$	0	0
$129$	−422.557	−0.288403
$130$	0	0
$131$	−2360.03	−1.57402	−0.787011	−	0.616940i	$-0.788372\pi$
$131$	−2360.03	−1.57402	−0.787011	+	0.616940i	$0.788372\pi$
$132$	0	0
$133$	− 1490.01i	− 0.971433i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1823.51i	− 1.13718i	−0.822622	−	0.568588i	$-0.807490\pi$
$137$	− 1823.51i	− 1.13718i	0.822622	−	0.568588i	$-0.192510\pi$
$138$	0	0
$139$	1826.28	1.11441	0.557206	−	0.830374i	$-0.311873\pi$
$139$	1826.28	1.11441	0.557206	+	0.830374i	$0.311873\pi$
$140$	0	0
$141$	−1205.37	−0.719931
$142$	0	0
$143$	410.858i	0.240264i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 491.455i	− 0.275745i
$148$	0	0
$149$	−287.973	−0.158333	−0.0791666	−	0.996861i	$-0.525226\pi$
$149$	−287.973	−0.158333	−0.0791666	+	0.996861i	$0.525226\pi$
$150$	0	0
$151$	−2881.09	−1.55272	−0.776358	−	0.630292i	$-0.782935\pi$
$151$	−2881.09	−1.55272	−0.776358	+	0.630292i	$0.782935\pi$
$152$	0	0
$153$	132.603i	0.0700677i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2405.81i	− 1.22296i	−0.791260	−	0.611480i	$-0.790575\pi$
$157$	− 2405.81i	− 1.22296i	0.791260	−	0.611480i	$-0.209425\pi$
$158$	0	0
$159$	−1627.17	−0.811590
$160$	0	0
$161$	−1791.45	−0.876933
$162$	0	0
$163$	1160.53i	0.557667i	0.960339	+	0.278834i	$0.0899478\pi$
$163$	1160.53i	0.557667i	−0.960339	+	0.278834i	$0.910052\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	468.123i	0.216913i	0.994101	+	0.108456i	$0.0345908\pi$
$167$	468.123i	0.216913i	−0.994101	+	0.108456i	$0.965409\pi$
$168$	0	0
$169$	801.922	0.365008
$170$	0	0
$171$	293.229	0.131133
$172$	0	0
$173$	− 56.1596i	− 0.0246806i	−0.999924	−	0.0123403i	$-0.996072\pi$
$173$	− 56.1596i	− 0.0246806i	0.999924	−	0.0123403i	$-0.00392813\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3754.09i	− 1.59421i
$178$	0	0
$179$	−1508.71	−0.629978	−0.314989	−	0.949095i	$-0.602001\pi$
$179$	−1508.71	−0.629978	−0.314989	+	0.949095i	$0.602001\pi$
$180$	0	0
$181$	438.185	0.179945	0.0899725	−	0.995944i	$-0.471322\pi$
$181$	438.185	0.179945	0.0899725	+	0.995944i	$0.471322\pi$
$182$	0	0
$183$	1491.57i	0.602515i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 356.959i	− 0.139591i
$188$	0	0
$189$	−2652.72	−1.02094
$190$	0	0
$191$	−3870.97	−1.46646	−0.733229	−	0.679982i	$-0.761988\pi$
$191$	−3870.97	−1.46646	−0.733229	+	0.679982i	$0.761988\pi$
$192$	0	0
$193$	− 4160.69i	− 1.55178i	−0.630871	−	0.775888i	$-0.717302\pi$
$193$	− 4160.69i	− 1.55178i	0.630871	−	0.775888i	$-0.282698\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 419.528i	− 0.151727i	−0.997118	−	0.0758633i	$-0.975829\pi$
$197$	− 419.528i	− 0.151727i	0.997118	−	0.0758633i	$-0.0241713\pi$
$198$	0	0
$199$	2278.61	0.811689	0.405845	−	0.913942i	$-0.366977\pi$
$199$	2278.61	0.811689	0.405845	+	0.913942i	$0.366977\pi$
$200$	0	0
$201$	3806.10	1.33563
$202$	0	0
$203$	404.538i	0.139867i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 352.552i	− 0.118377i
$208$	0	0
$209$	−789.352	−0.261247
$210$	0	0
$211$	757.812	0.247251	0.123625	−	0.992329i	$-0.460548\pi$
$211$	757.812	0.247251	0.123625	+	0.992329i	$0.460548\pi$
$212$	0	0
$213$	2291.18i	0.737038i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2352.15i	− 0.735827i
$218$	0	0
$219$	769.243	0.237355
$220$	0	0
$221$	1212.06	0.368924
$222$	0	0
$223$	− 2132.39i	− 0.640337i	−0.947361	−	0.320169i	$-0.896260\pi$
$223$	− 2132.39i	− 0.640337i	0.947361	−	0.320169i	$-0.103740\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3797.04i	− 1.11021i	−0.831779	−	0.555107i	$-0.812677\pi$
$227$	− 3797.04i	− 1.11021i	0.831779	−	0.555107i	$-0.187323\pi$
$228$	0	0
$229$	−1965.66	−0.567226	−0.283613	−	0.958939i	$-0.591533\pi$
$229$	−1965.66	−0.567226	−0.283613	+	0.958939i	$0.591533\pi$
$230$	0	0
$231$	−1273.47	−0.362720
$232$	0	0
$233$	638.658i	0.179570i	0.995961	+	0.0897851i	$0.0286180\pi$
$233$	638.658i	0.179570i	−0.995961	+	0.0897851i	$0.971382\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 5121.76i	− 1.40377i
$238$	0	0
$239$	4736.52	1.28193	0.640963	−	0.767572i	$-0.278535\pi$
$239$	4736.52	1.28193	0.640963	+	0.767572i	$0.278535\pi$
$240$	0	0
$241$	−2834.52	−0.757625	−0.378812	−	0.925473i	$-0.623667\pi$
$241$	−2834.52	−0.757625	−0.378812	+	0.925473i	$0.623667\pi$
$242$	0	0
$243$	− 1137.19i	− 0.300209i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2680.26i	− 0.690450i
$248$	0	0
$249$	−5351.32	−1.36195
$250$	0	0
$251$	3290.48	0.827464	0.413732	−	0.910399i	$-0.364225\pi$
$251$	3290.48	0.827464	0.413732	+	0.910399i	$0.364225\pi$
$252$	0	0
$253$	949.044i	0.235833i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2904.96i	0.705084i	0.935796	+	0.352542i	$0.114683\pi$
$257$	2904.96i	0.705084i	−0.935796	+	0.352542i	$0.885317\pi$
$258$	0	0
$259$	−761.198	−0.182620
$260$	0	0
$261$	−79.6117	−0.0188806
$262$	0	0
$263$	4947.48i	1.15998i	0.814624	+	0.579989i	$0.196943\pi$
$263$	4947.48i	1.15998i	−0.814624	+	0.579989i	$0.803057\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3469.04i	− 0.795139i
$268$	0	0
$269$	−6891.83	−1.56209	−0.781046	−	0.624474i	$-0.785313\pi$
$269$	−6891.83	−1.56209	−0.781046	+	0.624474i	$0.785313\pi$
$270$	0	0
$271$	−3875.28	−0.868660	−0.434330	−	0.900754i	$-0.643015\pi$
$271$	−3875.28	−0.868660	−0.434330	+	0.900754i	$0.643015\pi$
$272$	0	0
$273$	− 4324.10i	− 0.958631i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6658.20i	1.44423i	0.691771	+	0.722117i	$0.256831\pi$
$277$	6658.20i	1.44423i	−0.691771	+	0.722117i	$0.743169\pi$
$278$	0	0
$279$	462.895	0.0993291
$280$	0	0
$281$	−2564.20	−0.544369	−0.272184	−	0.962245i	$-0.587746\pi$
$281$	−2564.20	−0.544369	−0.272184	+	0.962245i	$0.587746\pi$
$282$	0	0
$283$	5367.22i	1.12738i	0.825987	+	0.563689i	$0.190618\pi$
$283$	5367.22i	1.12738i	−0.825987	+	0.563689i	$0.809382\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3156.53i	0.649212i
$288$	0	0
$289$	3859.94	0.785659
$290$	0	0
$291$	7753.50	1.56192
$292$	0	0
$293$	− 3189.28i	− 0.635903i	−0.948107	−	0.317952i	$-0.897005\pi$
$293$	− 3189.28i	− 0.635903i	0.948107	−	0.317952i	$-0.102995\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1405.31i	0.274560i
$298$	0	0
$299$	−3222.50	−0.623284
$300$	0	0
$301$	−1573.67	−0.301344
$302$	0	0
$303$	524.431i	0.0994317i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5223.47i	0.971072i	0.874217	+	0.485536i	$0.161376\pi$
$307$	5223.47i	0.971072i	−0.874217	+	0.485536i	$0.838624\pi$
$308$	0	0
$309$	−5758.13	−1.06009
$310$	0	0
$311$	−778.172	−0.141885	−0.0709423	−	0.997480i	$-0.522601\pi$
$311$	−778.172	−0.141885	−0.0709423	+	0.997480i	$0.522601\pi$
$312$	0	0
$313$	5903.88i	1.06616i	0.846066	+	0.533078i	$0.178965\pi$
$313$	5903.88i	1.06616i	−0.846066	+	0.533078i	$0.821035\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8015.21i	1.42012i	0.704140	+	0.710062i	$0.251333\pi$
$317$	8015.21i	1.42012i	−0.704140	+	0.710062i	$0.748667\pi$
$318$	0	0
$319$	214.309	0.0376144
$320$	0	0
$321$	4146.96	0.721061
$322$	0	0
$323$	2328.65i	0.401144i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3729.59i	0.630725i
$328$	0	0
$329$	−4488.97	−0.752234
$330$	0	0
$331$	1945.09	0.322996	0.161498	−	0.986873i	$-0.448367\pi$
$331$	1945.09	0.322996	0.161498	+	0.986873i	$0.448367\pi$
$332$	0	0
$333$	− 149.801i	− 0.0246518i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3833.73i	0.619693i	0.950787	+	0.309847i	$0.100278\pi$
$337$	3833.73i	0.619693i	−0.950787	+	0.309847i	$0.899722\pi$
$338$	0	0
$339$	−446.706	−0.0715685
$340$	0	0
$341$	−1246.08	−0.197886
$342$	0	0
$343$	5291.81i	0.833035i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3031.17i	− 0.468938i	−0.972124	−	0.234469i	$-0.924665\pi$
$347$	− 3031.17i	− 0.468938i	0.972124	−	0.234469i	$-0.0753351\pi$
$348$	0	0
$349$	9140.37	1.40193	0.700964	−	0.713197i	$-0.252754\pi$
$349$	9140.37	1.40193	0.700964	+	0.713197i	$0.252754\pi$
$350$	0	0
$351$	−4771.77	−0.725635
$352$	0	0
$353$	352.940i	0.0532156i	0.999646	+	0.0266078i	$0.00847052\pi$
$353$	352.940i	0.0532156i	−0.999646	+	0.0266078i	$0.991529\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3756.83i	0.556954i
$358$	0	0
$359$	9308.76	1.36852	0.684258	−	0.729240i	$-0.260126\pi$
$359$	9308.76	1.36852	0.684258	+	0.729240i	$0.260126\pi$
$360$	0	0
$361$	−1709.60	−0.249249
$362$	0	0
$363$	674.636i	0.0975461i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	730.769i	0.103940i	0.998649	+	0.0519698i	$0.0165499\pi$
$367$	730.769i	0.103940i	−0.998649	+	0.0519698i	$0.983450\pi$
$368$	0	0
$369$	−621.193	−0.0876370
$370$	0	0
$371$	−6059.82	−0.848006
$372$	0	0
$373$	3886.41i	0.539493i	0.962931	+	0.269746i	$0.0869399\pi$
$373$	3886.41i	0.539493i	−0.962931	+	0.269746i	$0.913060\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	727.690i	0.0994111i
$378$	0	0
$379$	−9925.81	−1.34526	−0.672631	−	0.739978i	$-0.734836\pi$
$379$	−9925.81	−1.34526	−0.672631	+	0.739978i	$0.734836\pi$
$380$	0	0
$381$	6278.38	0.844229
$382$	0	0
$383$	4658.20i	0.621469i	0.950497	+	0.310735i	$0.100575\pi$
$383$	4658.20i	0.621469i	−0.950497	+	0.310735i	$0.899425\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 309.692i	− 0.0406784i
$388$	0	0
$389$	−9334.00	−1.21659	−0.608294	−	0.793712i	$-0.708146\pi$
$389$	−9334.00	−1.21659	−0.608294	+	0.793712i	$0.708146\pi$
$390$	0	0
$391$	2799.75	0.362121
$392$	0	0
$393$	− 13158.4i	− 1.68894i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 3045.11i	− 0.384961i	−0.981301	−	0.192481i	$-0.938347\pi$
$397$	− 3045.11i	− 0.384961i	0.981301	−	0.192481i	$-0.0616532\pi$
$398$	0	0
$399$	8307.58	1.04235
$400$	0	0
$401$	8566.65	1.06683	0.533414	−	0.845854i	$-0.320909\pi$
$401$	8566.65	1.06683	0.533414	+	0.845854i	$0.320909\pi$
$402$	0	0
$403$	− 4231.09i	− 0.522992i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	403.254i	0.0491119i
$408$	0	0
$409$	2673.21	0.323183	0.161592	−	0.986858i	$-0.448337\pi$
$409$	2673.21	0.323183	0.161592	+	0.986858i	$0.448337\pi$
$410$	0	0
$411$	10167.0	1.22020
$412$	0	0
$413$	− 13980.8i	− 1.66574i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10182.5i	1.19577i
$418$	0	0
$419$	−585.643	−0.0682829	−0.0341415	−	0.999417i	$-0.510870\pi$
$419$	−585.643	−0.0682829	−0.0341415	+	0.999417i	$0.510870\pi$
$420$	0	0
$421$	−2506.29	−0.290141	−0.145070	−	0.989421i	$-0.546341\pi$
$421$	−2506.29	−0.290141	−0.145070	+	0.989421i	$0.546341\pi$
$422$	0	0
$423$	− 883.414i	− 0.101544i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5554.85i	0.629550i
$428$	0	0
$429$	−2290.74	−0.257804
$430$	0	0
$431$	4307.54	0.481408	0.240704	−	0.970599i	$-0.422622\pi$
$431$	4307.54	0.481408	0.240704	+	0.970599i	$0.422622\pi$
$432$	0	0
$433$	7599.55i	0.843443i	0.906725	+	0.421722i	$0.138574\pi$
$433$	7599.55i	0.843443i	−0.906725	+	0.421722i	$0.861426\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6191.16i	− 0.677719i
$438$	0	0
$439$	14291.1	1.55371	0.776855	−	0.629680i	$-0.216814\pi$
$439$	14291.1	1.55371	0.776855	+	0.629680i	$0.216814\pi$
$440$	0	0
$441$	360.188	0.0388930
$442$	0	0
$443$	− 8334.33i	− 0.893851i	−0.894571	−	0.446925i	$-0.852519\pi$
$443$	− 8334.33i	− 0.893851i	0.894571	−	0.446925i	$-0.147481\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1605.59i	− 0.169893i
$448$	0	0
$449$	3682.55	0.387061	0.193530	−	0.981094i	$-0.438006\pi$
$449$	3682.55	0.387061	0.193530	+	0.981094i	$0.438006\pi$
$450$	0	0
$451$	1672.21	0.174592
$452$	0	0
$453$	− 16063.6i	− 1.66607i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1348.62i	0.138044i	0.997615	+	0.0690218i	$0.0219878\pi$
$457$	1348.62i	0.138044i	−0.997615	+	0.0690218i	$0.978012\pi$
$458$	0	0
$459$	4145.77	0.421586
$460$	0	0
$461$	2058.01	0.207920	0.103960	−	0.994582i	$-0.466849\pi$
$461$	2058.01	0.207920	0.103960	+	0.994582i	$0.466849\pi$
$462$	0	0
$463$	8850.65i	0.888390i	0.895930	+	0.444195i	$0.146510\pi$
$463$	8850.65i	0.888390i	−0.895930	+	0.444195i	$0.853490\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19339.3i	1.91631i	0.286252	+	0.958154i	$0.407591\pi$
$467$	19339.3i	1.91631i	−0.286252	+	0.958154i	$0.592409\pi$
$468$	0	0
$469$	14174.5	1.39556
$470$	0	0
$471$	13413.6	1.31224
$472$	0	0
$473$	833.668i	0.0810404i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1192.55i	− 0.114472i
$478$	0	0
$479$	19045.4	1.81671	0.908356	−	0.418198i	$-0.137338\pi$
$479$	19045.4	1.81671	0.908356	+	0.418198i	$0.137338\pi$
$480$	0	0
$481$	−1369.26	−0.129798
$482$	0	0
$483$	− 9988.26i	− 0.940956i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 7083.95i	− 0.659147i	−0.944130	−	0.329573i	$-0.893095\pi$
$487$	− 7083.95i	− 0.659147i	0.944130	−	0.329573i	$-0.106905\pi$
$488$	0	0
$489$	−6470.55	−0.598381
$490$	0	0
$491$	11531.5	1.05990	0.529949	−	0.848030i	$-0.322211\pi$
$491$	11531.5	1.05990	0.529949	+	0.848030i	$0.322211\pi$
$492$	0	0
$493$	− 632.227i	− 0.0577567i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8532.71i	0.770109i
$498$	0	0
$499$	−8817.35	−0.791019	−0.395510	−	0.918462i	$-0.629432\pi$
$499$	−8817.35	−0.791019	−0.395510	+	0.918462i	$0.629432\pi$
$500$	0	0
$501$	−2610.03	−0.232749
$502$	0	0
$503$	− 10596.4i	− 0.939308i	−0.882851	−	0.469654i	$-0.844379\pi$
$503$	− 10596.4i	− 0.939308i	0.882851	−	0.469654i	$-0.155621\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4471.12i	0.391656i
$508$	0	0
$509$	18347.7	1.59773	0.798867	−	0.601508i	$-0.205433\pi$
$509$	18347.7	1.59773	0.798867	+	0.601508i	$0.205433\pi$
$510$	0	0
$511$	2864.78	0.248005
$512$	0	0
$513$	− 9167.65i	− 0.789009i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2378.09i	0.202298i
$518$	0	0
$519$	313.119	0.0264824
$520$	0	0
$521$	−18494.2	−1.55518	−0.777588	−	0.628775i	$-0.783557\pi$
$521$	−18494.2	−1.55518	−0.777588	+	0.628775i	$0.783557\pi$
$522$	0	0
$523$	15387.2i	1.28649i	0.765661	+	0.643244i	$0.222412\pi$
$523$	15387.2i	1.28649i	−0.765661	+	0.643244i	$0.777588\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3676.03i	0.303853i
$528$	0	0
$529$	4723.33	0.388208
$530$	0	0
$531$	2751.37	0.224858
$532$	0	0
$533$	5678.02i	0.461430i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 8411.81i	− 0.675971i
$538$	0	0
$539$	−969.599	−0.0774835
$540$	0	0
$541$	5424.06	0.431051	0.215525	−	0.976498i	$-0.430854\pi$
$541$	5424.06	0.431051	0.215525	+	0.976498i	$0.430854\pi$
$542$	0	0
$543$	2443.10i	0.193082i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 10544.7i	− 0.824241i	−0.911129	−	0.412120i	$-0.864788\pi$
$547$	− 10544.7i	− 0.824241i	0.911129	−	0.412120i	$-0.135212\pi$
$548$	0	0
$549$	−1093.17	−0.0849828
$550$	0	0
$551$	−1398.06	−0.108093
$552$	0	0
$553$	− 19074.2i	− 1.46676i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15977.6i	1.21542i	0.794158	+	0.607712i	$0.207912\pi$
$557$	15977.6i	1.21542i	−0.794158	+	0.607712i	$0.792088\pi$
$558$	0	0
$559$	−2830.74	−0.214182
$560$	0	0
$561$	1990.23	0.149782
$562$	0	0
$563$	17264.6i	1.29239i	0.763172	+	0.646196i	$0.223641\pi$
$563$	17264.6i	1.29239i	−0.763172	+	0.646196i	$0.776359\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 17081.2i	− 1.26515i
$568$	0	0
$569$	11626.6	0.856613	0.428306	−	0.903634i	$-0.359110\pi$
$569$	11626.6	0.856613	0.428306	+	0.903634i	$0.359110\pi$
$570$	0	0
$571$	26947.0	1.97495	0.987474	−	0.157781i	$-0.0504339\pi$
$571$	26947.0	1.97495	0.987474	+	0.157781i	$0.0504339\pi$
$572$	0	0
$573$	− 21582.6i	− 1.57352i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23399.2i	1.68825i	0.536143	+	0.844127i	$0.319881\pi$
$577$	23399.2i	1.68825i	−0.536143	+	0.844127i	$0.680119\pi$
$578$	0	0
$579$	23197.9	1.66507
$580$	0	0
$581$	−19929.1	−1.42306
$582$	0	0
$583$	3210.26i	0.228054i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4653.92i	− 0.327236i	−0.986524	−	0.163618i	$-0.947684\pi$
$587$	− 4653.92i	− 0.327236i	0.986524	−	0.163618i	$-0.0523165\pi$
$588$	0	0
$589$	8128.90	0.568668
$590$	0	0
$591$	2339.08	0.162804
$592$	0	0
$593$	− 8192.48i	− 0.567327i	−0.958924	−	0.283663i	$-0.908450\pi$
$593$	− 8192.48i	− 0.567327i	0.958924	−	0.283663i	$-0.0915498\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12704.4i	0.870948i
$598$	0	0
$599$	−6761.31	−0.461201	−0.230601	−	0.973048i	$-0.574069\pi$
$599$	−6761.31	−0.461201	−0.230601	+	0.973048i	$0.574069\pi$
$600$	0	0
$601$	−15266.1	−1.03613	−0.518067	−	0.855340i	$-0.673348\pi$
$601$	−15266.1	−1.03613	−0.518067	+	0.855340i	$0.673348\pi$
$602$	0	0
$603$	2789.49i	0.188386i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	25104.0i	1.67865i	0.543629	+	0.839326i	$0.317050\pi$
$607$	25104.0i	1.67865i	−0.543629	+	0.839326i	$0.682950\pi$
$608$	0	0
$609$	−2255.50	−0.150078
$610$	0	0
$611$	−8074.85	−0.534654
$612$	0	0
$613$	− 8848.85i	− 0.583037i	−0.956565	−	0.291519i	$-0.905840\pi$
$613$	− 8848.85i	− 0.583037i	0.956565	−	0.291519i	$-0.0941605\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3042.72i	− 0.198534i	−0.995061	−	0.0992668i	$-0.968350\pi$
$617$	− 3042.72i	− 0.198534i	0.995061	−	0.0992668i	$-0.0316497\pi$
$618$	0	0
$619$	3391.98	0.220251	0.110125	−	0.993918i	$-0.464875\pi$
$619$	3391.98	0.220251	0.110125	+	0.993918i	$0.464875\pi$
$620$	0	0
$621$	−11022.3	−0.712256
$622$	0	0
$623$	− 12919.2i	− 0.830816i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4401.04i	− 0.280320i
$628$	0	0
$629$	1189.63	0.0754111
$630$	0	0
$631$	5791.13	0.365358	0.182679	−	0.983173i	$-0.441523\pi$
$631$	5791.13	0.365358	0.182679	+	0.983173i	$0.441523\pi$
$632$	0	0
$633$	4225.19i	0.265302i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3292.30i	− 0.204781i
$638$	0	0
$639$	−1679.21	−0.103957
$640$	0	0
$641$	−25616.3	−1.57844	−0.789221	−	0.614110i	$-0.789515\pi$
$641$	−25616.3	−1.57844	−0.789221	+	0.614110i	$0.789515\pi$
$642$	0	0
$643$	16568.5i	1.01617i	0.861306	+	0.508086i	$0.169647\pi$
$643$	16568.5i	1.01617i	−0.861306	+	0.508086i	$0.830353\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8340.07i	0.506773i	0.967365	+	0.253386i	$0.0815444\pi$
$647$	8340.07i	0.506773i	−0.967365	+	0.253386i	$0.918456\pi$
$648$	0	0
$649$	−7406.50	−0.447967
$650$	0	0
$651$	13114.4	0.789547
$652$	0	0
$653$	− 8570.19i	− 0.513595i	−0.966465	−	0.256798i	$-0.917333\pi$
$653$	− 8570.19i	− 0.513595i	0.966465	−	0.256798i	$-0.0826673\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	563.779i	0.0334781i
$658$	0	0
$659$	3184.90	0.188264	0.0941320	−	0.995560i	$-0.469992\pi$
$659$	3184.90	0.188264	0.0941320	+	0.995560i	$0.469992\pi$
$660$	0	0
$661$	−5043.56	−0.296780	−0.148390	−	0.988929i	$-0.547409\pi$
$661$	−5043.56	−0.296780	−0.148390	+	0.988929i	$0.547409\pi$
$662$	0	0
$663$	6757.86i	0.395858i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1680.90i	0.0975781i
$668$	0	0
$669$	11889.1	0.687087
$670$	0	0
$671$	2942.75	0.169305
$672$	0	0
$673$	17554.6i	1.00547i	0.864440	+	0.502735i	$0.167673\pi$
$673$	17554.6i	1.00547i	−0.864440	+	0.502735i	$0.832327\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2667.67i	− 0.151443i	−0.997129	−	0.0757214i	$-0.975874\pi$
$677$	− 2667.67i	− 0.151443i	0.997129	−	0.0757214i	$-0.0241260\pi$
$678$	0	0
$679$	28875.2	1.63200
$680$	0	0
$681$	21170.4	1.19127
$682$	0	0
$683$	23516.3i	1.31746i	0.752378	+	0.658732i	$0.228907\pi$
$683$	23516.3i	1.31746i	−0.752378	+	0.658732i	$0.771093\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 10959.6i	− 0.608637i
$688$	0	0
$689$	−10900.5	−0.602724
$690$	0	0
$691$	32061.4	1.76508	0.882542	−	0.470234i	$-0.155831\pi$
$691$	32061.4	1.76508	0.882542	+	0.470234i	$0.155831\pi$
$692$	0	0
$693$	− 933.327i	− 0.0511604i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4933.14i	− 0.268086i
$698$	0	0
$699$	−3560.84	−0.192680
$700$	0	0
$701$	−4590.65	−0.247341	−0.123671	−	0.992323i	$-0.539467\pi$
$701$	−4590.65	−0.247341	−0.123671	+	0.992323i	$0.539467\pi$
$702$	0	0
$703$	− 2630.66i	− 0.141134i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1953.06i	0.103893i
$708$	0	0
$709$	−24836.3	−1.31558	−0.657791	−	0.753201i	$-0.728509\pi$
$709$	−24836.3	−1.31558	−0.657791	+	0.753201i	$0.728509\pi$
$710$	0	0
$711$	3753.74	0.197998
$712$	0	0
$713$	− 9773.43i	− 0.513349i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	26408.5i	1.37552i
$718$	0	0
$719$	−33950.9	−1.76099	−0.880497	−	0.474051i	$-0.842791\pi$
$719$	−33950.9	−1.76099	−0.880497	+	0.474051i	$0.842791\pi$
$720$	0	0
$721$	−21444.1	−1.10766
$722$	0	0
$723$	− 15803.9i	− 0.812937i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 16954.1i	− 0.864913i	−0.901655	−	0.432457i	$-0.857647\pi$
$727$	− 16954.1i	− 0.864913i	0.901655	−	0.432457i	$-0.142353\pi$
$728$	0	0
$729$	−15870.6	−0.806312
$730$	0	0
$731$	2459.38	0.124437
$732$	0	0
$733$	23218.8i	1.16999i	0.811036	+	0.584996i	$0.198904\pi$
$733$	23218.8i	1.16999i	−0.811036	+	0.584996i	$0.801096\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7509.11i	− 0.375307i
$738$	0	0
$739$	−33365.8	−1.66086	−0.830432	−	0.557119i	$-0.811907\pi$
$739$	−33365.8	−1.66086	−0.830432	+	0.557119i	$0.811907\pi$
$740$	0	0
$741$	14943.8	0.740857
$742$	0	0
$743$	− 13612.4i	− 0.672128i	−0.941839	−	0.336064i	$-0.890904\pi$
$743$	− 13612.4i	− 0.672128i	0.941839	−	0.336064i	$-0.109096\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3921.99i	− 0.192099i
$748$	0	0
$749$	15443.9	0.753415
$750$	0	0
$751$	18882.2	0.917470	0.458735	−	0.888573i	$-0.348303\pi$
$751$	18882.2	0.917470	0.458735	+	0.888573i	$0.348303\pi$
$752$	0	0
$753$	18346.1i	0.887875i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	31967.8i	1.53486i	0.641134	+	0.767429i	$0.278464\pi$
$757$	31967.8i	1.53486i	−0.641134	+	0.767429i	$0.721536\pi$
$758$	0	0
$759$	−5291.40	−0.253051
$760$	0	0
$761$	20438.6	0.973588	0.486794	−	0.873517i	$-0.338166\pi$
$761$	20438.6	0.973588	0.486794	+	0.873517i	$0.338166\pi$
$762$	0	0
$763$	13889.6i	0.659025i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 25148.9i	− 1.18393i
$768$	0	0
$769$	−26825.0	−1.25791	−0.628956	−	0.777441i	$-0.716517\pi$
$769$	−26825.0	−1.25791	−0.628956	+	0.777441i	$0.716517\pi$
$770$	0	0
$771$	−16196.6	−0.756560
$772$	0	0
$773$	1020.11i	0.0474655i	0.999718	+	0.0237327i	$0.00755508\pi$
$773$	1020.11i	0.0474655i	−0.999718	+	0.0237327i	$0.992445\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4244.07i	− 0.195952i
$778$	0	0
$779$	−10908.8	−0.501729
$780$	0	0
$781$	4520.30	0.207105
$782$	0	0
$783$	2489.01i	0.113602i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 23236.5i	− 1.05247i	−0.850339	−	0.526235i	$-0.823603\pi$
$787$	− 23236.5i	− 1.05247i	0.850339	−	0.526235i	$-0.176397\pi$
$788$	0	0
$789$	−27584.7	−1.24467
$790$	0	0
$791$	−1663.60	−0.0747798
$792$	0	0
$793$	9992.16i	0.447455i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20354.2i	0.904619i	0.891861	+	0.452310i	$0.149400\pi$
$797$	20354.2i	0.904619i	−0.891861	+	0.452310i	$0.850600\pi$
$798$	0	0
$799$	7015.53	0.310628
$800$	0	0
$801$	2542.46	0.112152
$802$	0	0
$803$	− 1517.65i	− 0.0666958i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 38425.5i	− 1.67613i
$808$	0	0
$809$	−14015.2	−0.609084	−0.304542	−	0.952499i	$-0.598503\pi$
$809$	−14015.2	−0.609084	−0.304542	+	0.952499i	$0.598503\pi$
$810$	0	0
$811$	18874.0	0.817207	0.408604	−	0.912712i	$-0.366016\pi$
$811$	18874.0	0.817207	0.408604	+	0.912712i	$0.366016\pi$
$812$	0	0
$813$	− 21606.7i	− 0.932078i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5438.50i	− 0.232887i
$818$	0	0
$819$	3169.13	0.135212
$820$	0	0
$821$	2299.94	0.0977690	0.0488845	−	0.998804i	$-0.484433\pi$
$821$	2299.94	0.0977690	0.0488845	+	0.998804i	$0.484433\pi$
$822$	0	0
$823$	− 11685.4i	− 0.494931i	−0.968897	−	0.247466i	$-0.920402\pi$
$823$	− 11685.4i	− 0.494931i	0.968897	−	0.247466i	$-0.0795977\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38022.6i	1.59876i	0.600825	+	0.799380i	$0.294839\pi$
$827$	38022.6i	1.59876i	−0.600825	+	0.799380i	$0.705161\pi$
$828$	0	0
$829$	−2077.11	−0.0870219	−0.0435110	−	0.999053i	$-0.513854\pi$
$829$	−2077.11	−0.0870219	−0.0435110	+	0.999053i	$0.513854\pi$
$830$	0	0
$831$	−37122.9	−1.54967
$832$	0	0
$833$	2860.39i	0.118976i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 14472.2i	− 0.597647i
$838$	0	0
$839$	26534.0	1.09184	0.545921	−	0.837837i	$-0.316180\pi$
$839$	26534.0	1.09184	0.545921	+	0.837837i	$0.316180\pi$
$840$	0	0
$841$	−24009.4	−0.984437
$842$	0	0
$843$	− 14296.7i	− 0.584111i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2512.45i	0.101923i
$848$	0	0
$849$	−29925.0	−1.20968
$850$	0	0
$851$	−3162.86	−0.127405
$852$	0	0
$853$	24845.2i	0.997283i	0.866808	+	0.498641i	$0.166168\pi$
$853$	24845.2i	0.997283i	−0.866808	+	0.498641i	$0.833832\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18710.0i	− 0.745767i	−0.927878	−	0.372883i	$-0.878369\pi$
$857$	− 18710.0i	− 0.745767i	0.927878	−	0.372883i	$-0.121631\pi$
$858$	0	0
$859$	4277.39	0.169898	0.0849492	−	0.996385i	$-0.472927\pi$
$859$	4277.39	0.169898	0.0849492	+	0.996385i	$0.472927\pi$
$860$	0	0
$861$	−17599.2	−0.696609
$862$	0	0
$863$	17711.3i	0.698611i	0.937009	+	0.349306i	$0.113582\pi$
$863$	17711.3i	0.698611i	−0.937009	+	0.349306i	$0.886418\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	21521.2i	0.843018i
$868$	0	0
$869$	−10104.8	−0.394455
$870$	0	0
$871$	25497.3	0.991899
$872$	0	0
$873$	5682.54i	0.220303i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	789.730i	0.0304074i	0.999884	+	0.0152037i	$0.00483967\pi$
$877$	789.730i	0.0304074i	−0.999884	+	0.0152037i	$0.995160\pi$
$878$	0	0
$879$	17781.8	0.682329
$880$	0	0
$881$	−34340.8	−1.31325	−0.656624	−	0.754218i	$-0.728016\pi$
$881$	−34340.8	−1.31325	−0.656624	+	0.754218i	$0.728016\pi$
$882$	0	0
$883$	43311.3i	1.65067i	0.564644	+	0.825335i	$0.309014\pi$
$883$	43311.3i	1.65067i	−0.564644	+	0.825335i	$0.690986\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 39802.8i	− 1.50670i	−0.657617	−	0.753352i	$-0.728435\pi$
$887$	− 39802.8i	− 1.50670i	0.657617	−	0.753352i	$-0.271565\pi$
$888$	0	0
$889$	23381.7	0.882110
$890$	0	0
$891$	−9048.95	−0.340237
$892$	0	0
$893$	− 15513.6i	− 0.581348i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 17967.1i	− 0.668788i
$898$	0	0
$899$	−2206.99	−0.0818769
$900$	0	0
$901$	9470.52	0.350176
$902$	0	0
$903$	− 8773.99i	− 0.323344i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 28916.2i	− 1.05859i	−0.848436	−	0.529297i	$-0.822456\pi$
$907$	− 28916.2i	− 1.05859i	0.848436	−	0.529297i	$-0.177544\pi$
$908$	0	0
$909$	−384.356	−0.0140245
$910$	0	0
$911$	−3182.17	−0.115730	−0.0578651	−	0.998324i	$-0.518429\pi$
$911$	−3182.17	−0.115730	−0.0578651	+	0.998324i	$0.518429\pi$
$912$	0	0
$913$	10557.7i	0.382704i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 49003.8i	− 1.76472i
$918$	0	0
$919$	−6946.10	−0.249326	−0.124663	−	0.992199i	$-0.539785\pi$
$919$	−6946.10	−0.249326	−0.124663	+	0.992199i	$0.539785\pi$
$920$	0	0
$921$	−29123.5	−1.04197
$922$	0	0
$923$	15348.8i	0.547358i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4220.13i	− 0.149522i
$928$	0	0
$929$	−44519.8	−1.57228	−0.786139	−	0.618050i	$-0.787923\pi$
$929$	−44519.8	−1.57228	−0.786139	+	0.618050i	$0.787923\pi$
$930$	0	0
$931$	6325.25	0.222666
$932$	0	0
$933$	− 4338.70i	− 0.152243i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 7802.11i	− 0.272021i	−0.990707	−	0.136011i	$-0.956572\pi$
$937$	− 7802.11i	− 0.272021i	0.990707	−	0.136011i	$-0.0434281\pi$
$938$	0	0
$939$	−32917.1	−1.14399
$940$	0	0
$941$	−24835.3	−0.860369	−0.430184	−	0.902741i	$-0.641551\pi$
$941$	−24835.3	−0.860369	−0.430184	+	0.902741i	$0.641551\pi$
$942$	0	0
$943$	13115.7i	0.452922i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26433.4i	0.907045i	0.891245	+	0.453522i	$0.149833\pi$
$947$	26433.4i	0.907045i	−0.891245	+	0.453522i	$0.850167\pi$
$948$	0	0
$949$	5153.22	0.176270
$950$	0	0
$951$	−44688.9	−1.52380
$952$	0	0
$953$	− 40389.2i	− 1.37286i	−0.727197	−	0.686429i	$-0.759177\pi$
$953$	− 40389.2i	− 1.37286i	0.727197	−	0.686429i	$-0.240823\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1194.88i	0.0403605i
$958$	0	0
$959$	37863.5	1.27495
$960$	0	0
$961$	−16958.6	−0.569254
$962$	0	0
$963$	3039.31i	0.101703i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 20108.7i	− 0.668719i	−0.942446	−	0.334359i	$-0.891480\pi$
$967$	− 20108.7i	− 0.668719i	0.942446	−	0.334359i	$-0.108520\pi$
$968$	0	0
$969$	−12983.4	−0.430430
$970$	0	0
$971$	−6054.35	−0.200096	−0.100048	−	0.994983i	$-0.531900\pi$
$971$	−6054.35	−0.200096	−0.100048	+	0.994983i	$0.531900\pi$
$972$	0	0
$973$	37921.0i	1.24943i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4573.59i	0.149767i	0.997192	+	0.0748833i	$0.0238584\pi$
$977$	4573.59i	0.149767i	−0.997192	+	0.0748833i	$0.976142\pi$
$978$	0	0
$979$	−6844.12	−0.223431
$980$	0	0
$981$	−2733.42	−0.0889617
$982$	0	0
$983$	27498.3i	0.892226i	0.894977	+	0.446113i	$0.147192\pi$
$983$	27498.3i	0.892226i	−0.894977	+	0.446113i	$0.852808\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 25028.3i	− 0.807153i
$988$	0	0
$989$	−6538.74	−0.210232
$990$	0	0
$991$	−22415.6	−0.718521	−0.359261	−	0.933237i	$-0.616971\pi$
$991$	−22415.6	−0.718521	−0.359261	+	0.933237i	$0.616971\pi$
$992$	0	0
$993$	10844.8i	0.346577i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	27064.0i	0.859706i	0.902899	+	0.429853i	$0.141435\pi$
$997$	27064.0i	0.859706i	−0.902899	+	0.429853i	$0.858565\pi$
$998$	0	0
$999$	−4683.45	−0.148326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.4.b.j.749.8		10
5.2	odd	4		1100.4.a.m.1.4	yes	5
5.3	odd	4		1100.4.a.j.1.2	✓	5
5.4	even	2	inner	1100.4.b.j.749.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.4.a.j.1.2	✓	5	5.3	odd	4
1100.4.a.m.1.4	yes	5	5.2	odd	4
1100.4.b.j.749.3		10	5.4	even	2	inner
1100.4.b.j.749.8		10	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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1100.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.57551i q^{3} +20.7640i q^{7} -4.08629 q^{9} +11.0000 q^{11} +37.3507i q^{13} -32.4508i q^{17} -71.7593 q^{19} -115.770 q^{21} +86.2767i q^{23} +127.756i q^{27} +19.4826 q^{29} -113.280 q^{31} +61.3306i q^{33} +36.6594i q^{37} -208.249 q^{39} +152.019 q^{41} +75.7880i q^{43} +216.190i q^{47} -88.1454 q^{49} +180.930 q^{51} +291.842i q^{53} -400.095i q^{57} -673.318 q^{59} +267.522 q^{61} -84.8479i q^{63} -682.646i q^{67} -481.036 q^{69} +410.937 q^{71} -137.968i q^{73} +228.404i q^{77} -918.618 q^{79} -822.632 q^{81} +959.791i q^{83} +108.626i q^{87} -622.193 q^{89} -775.552 q^{91} -631.594i q^{93} -1390.64i q^{97} -44.9492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 118 q^{9} + 110 q^{11} - 6 q^{19} + 316 q^{21} - 314 q^{29} + 114 q^{31} - 1036 q^{39} + 148 q^{41} - 1054 q^{49} - 2292 q^{51} - 1384 q^{59} - 1996 q^{61} - 2492 q^{69} - 3290 q^{71} - 2640 q^{79}+ \cdots - 1298 q^{99}+O(q^{100}) \) 10 * q - 118 * q^9 + 110 * q^11 - 6 * q^19 + 316 * q^21 - 314 * q^29 + 114 * q^31 - 1036 * q^39 + 148 * q^41 - 1054 * q^49 - 2292 * q^51 - 1384 * q^59 - 1996 * q^61 - 2492 * q^69 - 3290 * q^71 - 2640 * q^79 - 1550 * q^81 - 2482 * q^89 - 2432 * q^91 - 1298 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more