LMFDB - Embedded newform 1100.3.e.c.549.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1100,3,Mod(549,1100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1100.549"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1100.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$29.9728290796$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 174 x^{14} + 10969 x^{12} + 318076 x^{10} + 4442560 x^{8} + 28982576 x^{6} + 77210944 x^{4} + \cdots + 26790976 $ x^16 + 174x^14 + 10969x^12 + 318076x^10 + 4442560x^8 + 28982576x^6 + 77210944x^4 + 82074624*x^2 + 26790976
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			549.15
Root			$-3.45778i$ of defining polynomial
Character	$\chi$	$=$	1100.549
Dual form			1100.3.e.c.549.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.69333i q^{3} -7.54848 q^{7} -13.0273 q^{9} +(5.78376 + 9.35672i) q^{11} +0.829047 q^{13} -29.8374 q^{17} +36.2155i q^{19} -35.4275i q^{21} -29.7480i q^{23} -18.9015i q^{27} -10.0356i q^{29} +34.6221 q^{31} +(-43.9141 + 27.1451i) q^{33} -61.9103i q^{37} +3.89099i q^{39} +11.1392i q^{41} -39.4907 q^{43} +6.35892i q^{47} +7.97952 q^{49} -140.037i q^{51} -56.5691i q^{53} -169.971 q^{57} -70.4682 q^{59} +8.41148i q^{61} +98.3364 q^{63} -18.7532i q^{67} +139.617 q^{69} -3.79141 q^{71} +70.9217 q^{73} +(-43.6586 - 70.6290i) q^{77} -127.952i q^{79} -28.5349 q^{81} +100.235 q^{83} +47.1004 q^{87} +66.0818 q^{89} -6.25804 q^{91} +162.493i q^{93} -1.65712i q^{97} +(-75.3468 - 121.893i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9} + 6 q^{11} + 28 q^{31} - 28 q^{49} - 256 q^{59} + 352 q^{69} - 68 q^{71} - 256 q^{81} + 292 q^{89} + 228 q^{91} - 16 q^{99}+O(q^{100}) $ 16 * q - 32 * q^9 + 6 * q^11 + 28 * q^31 - 28 * q^49 - 256 * q^59 + 352 * q^69 - 68 * q^71 - 256 * q^81 + 292 * q^89 + 228 * q^91 - 16 * 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
$3$			4.69333i			1.56444i	0.623001	+	0.782221i	$0.285913\pi$
$3$			4.69333i			1.56444i	−0.623001	+	0.782221i	$0.714087\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−7.54848			−1.07835			−0.539177	−	0.842192i	$-0.681265\pi$
$7$	−7.54848			−1.07835			−0.539177	+	0.842192i	$0.681265\pi$
$8$		0			0
$9$	−13.0273			−1.44748
$10$		0			0
$11$	5.78376	+	9.35672i	0.525796	+	0.850611i
$11$	5.78376	+	9.35672i	0.525796	+	0.850611i
$12$		0			0
$13$	0.829047			0.0637728			0.0318864	−	0.999491i	$-0.489849\pi$
$13$	0.829047			0.0637728			0.0318864	+	0.999491i	$0.489849\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−29.8374			−1.75514			−0.877570	−	0.479449i	$-0.840837\pi$
$17$	−29.8374			−1.75514			−0.877570	+	0.479449i	$0.840837\pi$
$18$		0			0
$19$			36.2155i			1.90608i	0.302846	+	0.953040i	$0.402063\pi$
$19$			36.2155i			1.90608i	−0.302846	+	0.953040i	$0.597937\pi$
$20$		0			0
$21$		−	35.4275i		−	1.68702i
$22$		0			0
$23$		−	29.7480i		−	1.29339i	−0.762749	−	0.646695i	$-0.776151\pi$
$23$		−	29.7480i		−	1.29339i	0.762749	−	0.646695i	$-0.223849\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	18.9015i		−	0.700056i
$28$		0			0
$29$		−	10.0356i		−	0.346056i	−0.984917	−	0.173028i	$-0.944645\pi$
$29$		−	10.0356i		−	0.346056i	0.984917	−	0.173028i	$-0.0553551\pi$
$30$		0			0
$31$	34.6221			1.11684			0.558422	−	0.829557i	$-0.311407\pi$
$31$	34.6221			1.11684			0.558422	+	0.829557i	$0.311407\pi$
$32$		0			0
$33$	−43.9141	+	27.1451i	−1.33073	+	0.822578i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	61.9103i		−	1.67325i	−0.547775	−	0.836626i	$-0.684525\pi$
$37$		−	61.9103i		−	1.67325i	0.547775	−	0.836626i	$-0.315475\pi$
$38$		0			0
$39$			3.89099i			0.0997689i
$40$		0			0
$41$			11.1392i			0.271687i	0.990730	+	0.135844i	$0.0433745\pi$
$41$			11.1392i			0.271687i	−0.990730	+	0.135844i	$0.956626\pi$
$42$		0			0
$43$	−39.4907			−0.918388			−0.459194	−	0.888336i	$-0.651862\pi$
$43$	−39.4907			−0.918388			−0.459194	+	0.888336i	$0.651862\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.35892i			0.135296i	0.997709	+	0.0676481i	$0.0215495\pi$
$47$			6.35892i			0.135296i	−0.997709	+	0.0676481i	$0.978450\pi$
$48$		0			0
$49$	7.97952			0.162847
$50$		0			0
$51$		−	140.037i		−	2.74582i
$52$		0			0
$53$		−	56.5691i		−	1.06734i	−0.845693	−	0.533670i	$-0.820812\pi$
$53$		−	56.5691i		−	1.06734i	0.845693	−	0.533670i	$-0.179188\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−169.971			−2.98195
$58$		0			0
$59$	−70.4682			−1.19438			−0.597188	−	0.802101i	$-0.703715\pi$
$59$	−70.4682			−1.19438			−0.597188	+	0.802101i	$0.703715\pi$
$60$		0			0
$61$			8.41148i			0.137893i	0.997620	+	0.0689466i	$0.0219638\pi$
$61$			8.41148i			0.137893i	−0.997620	+	0.0689466i	$0.978036\pi$
$62$		0			0
$63$	98.3364			1.56090
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	18.7532i		−	0.279899i	−0.990159	−	0.139950i	$-0.955306\pi$
$67$		−	18.7532i		−	0.279899i	0.990159	−	0.139950i	$-0.0446940\pi$
$68$		0			0
$69$	139.617			2.02343
$70$		0			0
$71$	−3.79141			−0.0534001			−0.0267001	−	0.999643i	$-0.508500\pi$
$71$	−3.79141			−0.0534001			−0.0267001	+	0.999643i	$0.508500\pi$
$72$		0			0
$73$	70.9217			0.971531			0.485765	−	0.874089i	$-0.338541\pi$
$73$	70.9217			0.971531			0.485765	+	0.874089i	$0.338541\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−43.6586	−	70.6290i	−0.566994	−	0.917259i
$78$		0			0
$79$		−	127.952i		−	1.61965i	−0.586671	−	0.809825i	$-0.699562\pi$
$79$		−	127.952i		−	1.61965i	0.586671	−	0.809825i	$-0.300438\pi$
$80$		0			0
$81$	−28.5349			−0.352283
$82$		0			0
$83$	100.235			1.20765			0.603826	−	0.797117i	$-0.293642\pi$
$83$	100.235			1.20765			0.603826	+	0.797117i	$0.293642\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	47.1004			0.541384
$88$		0			0
$89$	66.0818			0.742493			0.371246	−	0.928534i	$-0.378931\pi$
$89$	66.0818			0.742493			0.371246	+	0.928534i	$0.378931\pi$
$90$		0			0
$91$	−6.25804			−0.0687697
$92$		0			0
$93$			162.493i			1.74724i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1.65712i		−	0.0170838i	−0.999964	−	0.00854188i	$-0.997281\pi$
$97$		−	1.65712i		−	0.0170838i	0.999964	−	0.00854188i	$-0.00271900\pi$
$98$		0			0
$99$	−75.3468	−	121.893i	−0.761079	−	1.23124i
$100$		0			0
$101$			75.2305i			0.744857i	0.928061	+	0.372428i	$0.121475\pi$
$101$			75.2305i			0.744857i	−0.928061	+	0.372428i	$0.878525\pi$
$102$		0			0
$103$			135.557i			1.31609i	0.752980	+	0.658043i	$0.228615\pi$
$103$			135.557i			1.31609i	−0.752980	+	0.658043i	$0.771385\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−80.0123			−0.747779			−0.373889	−	0.927473i	$-0.621976\pi$
$107$	−80.0123			−0.747779			−0.373889	+	0.927473i	$0.621976\pi$
$108$		0			0
$109$		−	85.3259i		−	0.782806i	−0.920219	−	0.391403i	$-0.871990\pi$
$109$		−	85.3259i		−	0.782806i	0.920219	−	0.391403i	$-0.128010\pi$
$110$		0			0
$111$	290.565			2.61770
$112$		0			0
$113$			139.235i			1.23217i	0.787680	+	0.616084i	$0.211282\pi$
$113$			139.235i			1.23217i	−0.787680	+	0.616084i	$0.788718\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−10.8003			−0.0923098
$118$		0			0
$119$	225.227			1.89266
$120$		0			0
$121$	−54.0963	+	108.234i	−0.447077	+	0.894495i
$122$		0			0
$123$	−52.2798			−0.425039
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−19.3840			−0.152630			−0.0763148	−	0.997084i	$-0.524315\pi$
$127$	−19.3840			−0.152630			−0.0763148	+	0.997084i	$0.524315\pi$
$128$		0			0
$129$		−	185.343i		−	1.43676i
$130$		0			0
$131$		−	206.006i		−	1.57257i	−0.617866	−	0.786284i	$-0.712002\pi$
$131$		−	206.006i		−	1.57257i	0.617866	−	0.786284i	$-0.287998\pi$
$132$		0			0
$133$		−	273.372i		−	2.05543i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			99.8842i			0.729082i	0.931187	+	0.364541i	$0.118774\pi$
$137$			99.8842i			0.729082i	−0.931187	+	0.364541i	$0.881226\pi$
$138$		0			0
$139$		−	111.069i		−	0.799059i	−0.916720	−	0.399530i	$-0.869173\pi$
$139$		−	111.069i		−	0.799059i	0.916720	−	0.399530i	$-0.130827\pi$
$140$		0			0
$141$	−29.8445			−0.211663
$142$		0			0
$143$	4.79500	+	7.75716i	0.0335315	+	0.0542458i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			37.4505i			0.254765i
$148$		0			0
$149$		−	90.7317i		−	0.608937i	−0.952522	−	0.304469i	$-0.901521\pi$
$149$		−	90.7317i		−	0.608937i	0.952522	−	0.304469i	$-0.0984789\pi$
$150$		0			0
$151$		−	50.8689i		−	0.336880i	−0.985712	−	0.168440i	$-0.946127\pi$
$151$		−	50.8689i		−	0.336880i	0.985712	−	0.168440i	$-0.0538729\pi$
$152$		0			0
$153$	388.701			2.54053
$154$		0			0
$155$		0			0
$156$		0			0
$157$			14.6351i			0.0932170i	0.998913	+	0.0466085i	$0.0148413\pi$
$157$			14.6351i			0.0932170i	−0.998913	+	0.0466085i	$0.985159\pi$
$158$		0			0
$159$	265.497			1.66979
$160$		0			0
$161$			224.552i			1.39473i
$162$		0			0
$163$		−	33.1487i		−	0.203366i	−0.994817	−	0.101683i	$-0.967577\pi$
$163$		−	33.1487i		−	0.203366i	0.994817	−	0.101683i	$-0.0324228\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	236.993			1.41912			0.709558	−	0.704647i	$-0.248894\pi$
$167$	236.993			1.41912			0.709558	+	0.704647i	$0.248894\pi$
$168$		0			0
$169$	−168.313			−0.995933
$170$		0			0
$171$		−	471.791i		−	2.75901i
$172$		0			0
$173$	−273.880			−1.58312			−0.791561	−	0.611090i	$-0.790731\pi$
$173$	−273.880			−1.58312			−0.791561	+	0.611090i	$0.790731\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	330.730i		−	1.86853i
$178$		0			0
$179$	309.422			1.72861			0.864307	−	0.502964i	$-0.167757\pi$
$179$	309.422			1.72861			0.864307	+	0.502964i	$0.167757\pi$
$180$		0			0
$181$	−259.071			−1.43133			−0.715667	−	0.698442i	$-0.753877\pi$
$181$	−259.071			−1.43133			−0.715667	+	0.698442i	$0.753877\pi$
$182$		0			0
$183$	−39.4778			−0.215726
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−172.572	−	279.180i	−0.922846	−	1.49294i
$188$		0			0
$189$			142.678i			0.754908i
$190$		0			0
$191$	−124.976			−0.654323			−0.327162	−	0.944968i	$-0.606092\pi$
$191$	−124.976			−0.654323			−0.327162	+	0.944968i	$0.606092\pi$
$192$		0			0
$193$	95.0964			0.492728			0.246364	−	0.969177i	$-0.420764\pi$
$193$	95.0964			0.492728			0.246364	+	0.969177i	$0.420764\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−286.580			−1.45472			−0.727361	−	0.686255i	$-0.759253\pi$
$197$	−286.580			−1.45472			−0.727361	+	0.686255i	$0.759253\pi$
$198$		0			0
$199$	−382.548			−1.92235			−0.961176	−	0.275936i	$-0.911012\pi$
$199$	−382.548			−1.92235			−0.961176	+	0.275936i	$0.911012\pi$
$200$		0			0
$201$	88.0151			0.437886
$202$		0			0
$203$			75.7536i			0.373171i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			387.536i			1.87215i
$208$		0			0
$209$	−338.858	+	209.462i	−1.62133	+	1.00221i
$210$		0			0
$211$			234.103i			1.10949i	0.832019	+	0.554747i	$0.187185\pi$
$211$			234.103i			1.10949i	−0.832019	+	0.554747i	$0.812815\pi$
$212$		0			0
$213$		−	17.7943i		−	0.0835414i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−261.344			−1.20435
$218$		0			0
$219$			332.859i			1.51990i
$220$		0			0
$221$	−24.7366			−0.111930
$222$		0			0
$223$			152.390i			0.683365i	0.939815	+	0.341682i	$0.110997\pi$
$223$			152.390i			0.683365i	−0.939815	+	0.341682i	$0.889003\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−70.2453			−0.309451			−0.154725	−	0.987958i	$-0.549449\pi$
$227$	−70.2453			−0.309451			−0.154725	+	0.987958i	$0.549449\pi$
$228$		0			0
$229$	−166.019			−0.724973			−0.362486	−	0.931989i	$-0.618072\pi$
$229$	−166.019			−0.724973			−0.362486	+	0.931989i	$0.618072\pi$
$230$		0			0
$231$	331.485	−	204.904i	1.43500	−	0.887030i
$232$		0			0
$233$	−234.357			−1.00583			−0.502913	−	0.864337i	$-0.667738\pi$
$233$	−234.357			−1.00583			−0.502913	+	0.864337i	$0.667738\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	600.522			2.53385
$238$		0			0
$239$		−	155.065i		−	0.648808i	−0.945919	−	0.324404i	$-0.894836\pi$
$239$		−	155.065i		−	0.648808i	0.945919	−	0.324404i	$-0.105164\pi$
$240$		0			0
$241$			276.909i			1.14900i	0.818505	+	0.574500i	$0.194803\pi$
$241$			276.909i			1.14900i	−0.818505	+	0.574500i	$0.805197\pi$
$242$		0			0
$243$		−	304.037i		−	1.25118i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			30.0243i			0.121556i
$248$		0			0
$249$			470.436i			1.88930i
$250$		0			0
$251$	−5.23446			−0.0208544			−0.0104272	−	0.999946i	$-0.503319\pi$
$251$	−5.23446			−0.0208544			−0.0104272	+	0.999946i	$0.503319\pi$
$252$		0			0
$253$	278.343	−	172.055i	1.10017	−	0.680059i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	222.515i		−	0.865818i	−0.901438	−	0.432909i	$-0.857487\pi$
$257$		−	222.515i		−	0.865818i	0.901438	−	0.432909i	$-0.142513\pi$
$258$		0			0
$259$			467.328i			1.80436i
$260$		0			0
$261$			130.737i			0.500909i
$262$		0			0
$263$	−401.324			−1.52595			−0.762973	−	0.646431i	$-0.776261\pi$
$263$	−401.324			−1.52595			−0.762973	+	0.646431i	$0.776261\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			310.144i			1.16159i
$268$		0			0
$269$	77.6187			0.288545			0.144273	−	0.989538i	$-0.453916\pi$
$269$	77.6187			0.288545			0.144273	+	0.989538i	$0.453916\pi$
$270$		0			0
$271$			192.430i			0.710072i	0.934853	+	0.355036i	$0.115531\pi$
$271$			192.430i			0.710072i	−0.934853	+	0.355036i	$0.884469\pi$
$272$		0			0
$273$		−	29.3710i		−	0.107586i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−299.161			−1.08000			−0.540002	−	0.841664i	$-0.681576\pi$
$277$	−299.161			−1.08000			−0.540002	+	0.841664i	$0.681576\pi$
$278$		0			0
$279$	−451.034			−1.61661
$280$		0			0
$281$		−	101.945i		−	0.362795i	−0.983410	−	0.181398i	$-0.941938\pi$
$281$		−	101.945i		−	0.362795i	0.983410	−	0.181398i	$-0.0580621\pi$
$282$		0			0
$283$	−381.022			−1.34637			−0.673183	−	0.739476i	$-0.735073\pi$
$283$	−381.022			−1.34637			−0.673183	+	0.739476i	$0.735073\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	84.0838i		−	0.292975i
$288$		0			0
$289$	601.269			2.08052
$290$		0			0
$291$	7.77743			0.0267265
$292$		0			0
$293$	−250.254			−0.854110			−0.427055	−	0.904226i	$-0.640449\pi$
$293$	−250.254			−0.854110			−0.427055	+	0.904226i	$0.640449\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	176.856	−	109.322i	0.595475	−	0.368086i
$298$		0			0
$299$		−	24.6624i		−	0.0824831i
$300$		0			0
$301$	298.095			0.990347
$302$		0			0
$303$	−353.081			−1.16529
$304$		0			0
$305$		0			0
$306$		0			0
$307$	558.527			1.81931			0.909653	−	0.415369i	$-0.136348\pi$
$307$	558.527			1.81931			0.909653	+	0.415369i	$0.136348\pi$
$308$		0			0
$309$	−636.213			−2.05894
$310$		0			0
$311$	−476.323			−1.53159			−0.765793	−	0.643087i	$-0.777653\pi$
$311$	−476.323			−1.53159			−0.765793	+	0.643087i	$0.777653\pi$
$312$		0			0
$313$			232.746i			0.743598i	0.928313	+	0.371799i	$0.121259\pi$
$313$			232.746i			0.743598i	−0.928313	+	0.371799i	$0.878741\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			323.974i			1.02200i	0.859581	+	0.511000i	$0.170725\pi$
$317$			323.974i			1.02200i	−0.859581	+	0.511000i	$0.829275\pi$
$318$		0			0
$319$	93.9004	−	58.0436i	0.294359	−	0.181955i
$320$		0			0
$321$		−	375.524i		−	1.16986i
$322$		0			0
$323$		−	1080.58i		−	3.34544i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	400.462			1.22466
$328$		0			0
$329$		−	48.0002i		−	0.145897i
$330$		0			0
$331$	196.429			0.593441			0.296721	−	0.954964i	$-0.404107\pi$
$331$	196.429			0.593441			0.296721	+	0.954964i	$0.404107\pi$
$332$		0			0
$333$			806.525i			2.42200i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	69.8212			0.207185			0.103592	−	0.994620i	$-0.466966\pi$
$337$	69.8212			0.207185			0.103592	+	0.994620i	$0.466966\pi$
$338$		0			0
$339$	−653.476			−1.92766
$340$		0			0
$341$	200.246	+	323.950i	0.587232	+	0.949999i
$342$		0			0
$343$	309.642			0.902747
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−303.143			−0.873611			−0.436805	−	0.899556i	$-0.643890\pi$
$347$	−303.143			−0.873611			−0.436805	+	0.899556i	$0.643890\pi$
$348$		0			0
$349$			544.750i			1.56089i	0.625226	+	0.780444i	$0.285007\pi$
$349$			544.750i			1.56089i	−0.625226	+	0.780444i	$0.714993\pi$
$350$		0			0
$351$		−	15.6702i		−	0.0446445i
$352$		0			0
$353$			132.757i			0.376083i	0.982161	+	0.188042i	$0.0602140\pi$
$353$			132.757i			0.376083i	−0.982161	+	0.188042i	$0.939786\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			1057.06i			2.96096i
$358$		0			0
$359$		−	575.713i		−	1.60366i	−0.597553	−	0.801829i	$-0.703860\pi$
$359$		−	575.713i		−	1.60366i	0.597553	−	0.801829i	$-0.296140\pi$
$360$		0			0
$361$	−950.563			−2.63314
$362$		0			0
$363$	−507.977	−	253.892i	−1.39939	−	0.699426i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			548.913i			1.49568i	0.663881	+	0.747838i	$0.268908\pi$
$367$			548.913i			1.49568i	−0.663881	+	0.747838i	$0.731092\pi$
$368$		0			0
$369$		−	145.113i		−	0.393262i
$370$		0			0
$371$			427.010i			1.15097i
$372$		0			0
$373$	240.103			0.643707			0.321853	−	0.946790i	$-0.395694\pi$
$373$	240.103			0.643707			0.321853	+	0.946790i	$0.395694\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	8.32000i		−	0.0220690i
$378$		0			0
$379$	700.496			1.84827			0.924137	−	0.382061i	$-0.124786\pi$
$379$	700.496			1.84827			0.924137	+	0.382061i	$0.124786\pi$
$380$		0			0
$381$		−	90.9752i		−	0.238780i
$382$		0			0
$383$		−	60.8936i		−	0.158991i	−0.996835	−	0.0794955i	$-0.974669\pi$
$383$		−	60.8936i		−	0.158991i	0.996835	−	0.0794955i	$-0.0253309\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	514.458			1.32935
$388$		0			0
$389$	−198.172			−0.509439			−0.254719	−	0.967015i	$-0.581983\pi$
$389$	−198.172			−0.509439			−0.254719	+	0.967015i	$0.581983\pi$
$390$		0			0
$391$			887.601i			2.27008i
$392$		0			0
$393$	966.855			2.46019
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	728.154i		−	1.83414i	−0.398724	−	0.917071i	$-0.630547\pi$
$397$		−	728.154i		−	1.83414i	0.398724	−	0.917071i	$-0.369453\pi$
$398$		0			0
$399$	1283.02			3.21560
$400$		0			0
$401$	12.2406			0.0305253			0.0152627	−	0.999884i	$-0.495142\pi$
$401$	12.2406			0.0305253			0.0152627	+	0.999884i	$0.495142\pi$
$402$		0			0
$403$	28.7034			0.0712242
$404$		0			0
$405$		0			0
$406$		0			0
$407$	579.277	−	358.074i	1.42329	−	0.879789i
$408$		0			0
$409$			384.072i			0.939050i	0.882919	+	0.469525i	$0.155575\pi$
$409$			384.072i			0.939050i	−0.882919	+	0.469525i	$0.844425\pi$
$410$		0			0
$411$	−468.789			−1.14061
$412$		0			0
$413$	531.927			1.28796
$414$		0			0
$415$		0			0
$416$		0			0
$417$	521.284			1.25008
$418$		0			0
$419$	−579.481			−1.38301			−0.691505	−	0.722372i	$-0.743052\pi$
$419$	−579.481			−1.38301			−0.691505	+	0.722372i	$0.743052\pi$
$420$		0			0
$421$	−218.272			−0.518460			−0.259230	−	0.965816i	$-0.583469\pi$
$421$	−218.272			−0.518460			−0.259230	+	0.965816i	$0.583469\pi$
$422$		0			0
$423$		−	82.8397i		−	0.195838i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	63.4939i		−	0.148698i
$428$		0			0
$429$	−36.4069	+	22.5045i	−0.0848645	+	0.0524581i
$430$		0			0
$431$			262.440i			0.608910i	0.952527	+	0.304455i	$0.0984743\pi$
$431$			262.440i			0.608910i	−0.952527	+	0.304455i	$0.901526\pi$
$432$		0			0
$433$			48.5618i			0.112152i	0.998427	+	0.0560759i	$0.0178589\pi$
$433$			48.5618i			0.112152i	−0.998427	+	0.0560759i	$0.982141\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1077.34			2.46530
$438$		0			0
$439$		−	401.532i		−	0.914652i	−0.889299	−	0.457326i	$-0.848807\pi$
$439$		−	401.532i		−	0.914652i	0.889299	−	0.457326i	$-0.151193\pi$
$440$		0			0
$441$	−103.952			−0.235718
$442$		0			0
$443$			263.646i			0.595138i	0.954700	+	0.297569i	$0.0961759\pi$
$443$			263.646i			0.595138i	−0.954700	+	0.297569i	$0.903824\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	425.833			0.952647
$448$		0			0
$449$	−138.896			−0.309345			−0.154673	−	0.987966i	$-0.549432\pi$
$449$	−138.896			−0.309345			−0.154673	+	0.987966i	$0.549432\pi$
$450$		0			0
$451$	−104.226	+	64.4263i	−0.231100	+	0.142852i
$452$		0			0
$453$	238.744			0.527029
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−618.063			−1.35244			−0.676218	−	0.736701i	$-0.736382\pi$
$457$	−618.063			−1.35244			−0.676218	+	0.736701i	$0.736382\pi$
$458$		0			0
$459$			563.971i			1.22870i
$460$		0			0
$461$		−	17.2274i		−	0.0373697i	−0.999825	−	0.0186848i	$-0.994052\pi$
$461$		−	17.2274i		−	0.0373697i	0.999825	−	0.0186848i	$-0.00594791\pi$
$462$		0			0
$463$			721.198i			1.55766i	0.627233	+	0.778831i	$0.284187\pi$
$463$			721.198i			1.55766i	−0.627233	+	0.778831i	$0.715813\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			205.184i			0.439366i	0.975571	+	0.219683i	$0.0705023\pi$
$467$			205.184i			0.439366i	−0.975571	+	0.219683i	$0.929498\pi$
$468$		0			0
$469$			141.558i			0.301830i
$470$		0			0
$471$	−68.6872			−0.145833
$472$		0			0
$473$	−228.404	−	369.503i	−0.482885	−	0.781191i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			736.943i			1.54495i
$478$		0			0
$479$		−	858.946i		−	1.79321i	−0.442834	−	0.896604i	$-0.646027\pi$
$479$		−	858.946i		−	1.79321i	0.442834	−	0.896604i	$-0.353973\pi$
$480$		0			0
$481$		−	51.3265i		−	0.106708i
$482$		0			0
$483$	−1053.89			−2.18198
$484$		0			0
$485$		0			0
$486$		0			0
$487$			483.597i			0.993013i	0.868033	+	0.496506i	$0.165384\pi$
$487$			483.597i			0.993013i	−0.868033	+	0.496506i	$0.834616\pi$
$488$		0			0
$489$	155.578			0.318155
$490$		0			0
$491$		−	817.941i		−	1.66587i	−0.553373	−	0.832934i	$-0.686660\pi$
$491$		−	817.941i		−	1.66587i	0.553373	−	0.832934i	$-0.313340\pi$
$492$		0			0
$493$			299.437i			0.607376i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	28.6194			0.0575842
$498$		0			0
$499$	−376.487			−0.754483			−0.377242	−	0.926115i	$-0.623127\pi$
$499$	−376.487			−0.754483			−0.377242	+	0.926115i	$0.623127\pi$
$500$		0			0
$501$			1112.28i			2.22013i
$502$		0			0
$503$	4.61169			0.00916837			0.00458418	−	0.999989i	$-0.498541\pi$
$503$	4.61169			0.00916837			0.00458418	+	0.999989i	$0.498541\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	789.946i		−	1.55808i
$508$		0			0
$509$	525.844			1.03309			0.516546	−	0.856259i	$-0.327217\pi$
$509$	525.844			1.03309			0.516546	+	0.856259i	$0.327217\pi$
$510$		0			0
$511$	−535.351			−1.04765
$512$		0			0
$513$	684.527			1.33436
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−59.4986	+	36.7784i	−0.115084	+	0.0711382i
$518$		0			0
$519$		−	1285.41i		−	2.47670i
$520$		0			0
$521$	762.348			1.46324			0.731620	−	0.681712i	$-0.238764\pi$
$521$	762.348			1.46324			0.731620	+	0.681712i	$0.238764\pi$
$522$		0			0
$523$	−558.597			−1.06806			−0.534032	−	0.845464i	$-0.679324\pi$
$523$	−558.597			−1.06806			−0.534032	+	0.845464i	$0.679324\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1033.03			−1.96022
$528$		0			0
$529$	−355.941			−0.672856
$530$		0			0
$531$	918.011			1.72883
$532$		0			0
$533$			9.23489i			0.0173263i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			1452.22i			2.70432i
$538$		0			0
$539$	46.1516	+	74.6621i	0.0856245	+	0.138520i
$540$		0			0
$541$		−	75.8646i		−	0.140230i	−0.997539	−	0.0701152i	$-0.977663\pi$
$541$		−	75.8646i		−	0.140230i	0.997539	−	0.0701152i	$-0.0223367\pi$
$542$		0			0
$543$		−	1215.91i		−	2.23924i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−782.767			−1.43102			−0.715509	−	0.698603i	$-0.753805\pi$
$547$	−782.767			−1.43102			−0.715509	+	0.698603i	$0.753805\pi$
$548$		0			0
$549$		−	109.579i		−	0.199598i
$550$		0			0
$551$	363.445			0.659610
$552$		0			0
$553$			965.846i			1.74656i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	271.032			0.486592			0.243296	−	0.969952i	$-0.421771\pi$
$557$	271.032			0.486592			0.243296	+	0.969952i	$0.421771\pi$
$558$		0			0
$559$	−32.7396			−0.0585682
$560$		0			0
$561$	1310.28	−	809.937i	2.33562	−	1.44374i
$562$		0			0
$563$	−345.532			−0.613734			−0.306867	−	0.951752i	$-0.599281\pi$
$563$	−345.532			−0.613734			−0.306867	+	0.951752i	$0.599281\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	215.395			0.379886
$568$		0			0
$569$		−	474.767i		−	0.834389i	−0.908817	−	0.417195i	$-0.863013\pi$
$569$		−	474.767i		−	0.834389i	0.908817	−	0.417195i	$-0.136987\pi$
$570$		0			0
$571$			786.977i			1.37824i	0.724645	+	0.689122i	$0.242004\pi$
$571$			786.977i			1.37824i	−0.724645	+	0.689122i	$0.757996\pi$
$572$		0			0
$573$		−	586.552i		−	1.02365i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	58.9156i		−	0.102107i	−0.998696	−	0.0510534i	$-0.983742\pi$
$577$		−	58.9156i		−	0.102107i	0.998696	−	0.0510534i	$-0.0162579\pi$
$578$		0			0
$579$			446.319i			0.770844i
$580$		0			0
$581$	−756.622			−1.30228
$582$		0			0
$583$	529.301	−	327.182i	0.907891	−	0.561204i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			246.978i			0.420746i	0.977621	+	0.210373i	$0.0674678\pi$
$587$			246.978i			0.420746i	−0.977621	+	0.210373i	$0.932532\pi$
$588$		0			0
$589$			1253.86i			2.12879i
$590$		0			0
$591$		−	1345.01i		−	2.27583i
$592$		0			0
$593$	−230.921			−0.389411			−0.194705	−	0.980862i	$-0.562375\pi$
$593$	−230.921			−0.389411			−0.194705	+	0.980862i	$0.562375\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	1795.42i		−	3.00741i
$598$		0			0
$599$	633.026			1.05681			0.528403	−	0.848994i	$-0.322791\pi$
$599$	633.026			1.05681			0.528403	+	0.848994i	$0.322791\pi$
$600$		0			0
$601$		−	120.426i		−	0.200376i	−0.994969	−	0.100188i	$-0.968056\pi$
$601$		−	120.426i		−	0.200376i	0.994969	−	0.100188i	$-0.0319444\pi$
$602$		0			0
$603$			244.304i			0.405148i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−380.480			−0.626821			−0.313410	−	0.949618i	$-0.601471\pi$
$607$	−380.480			−0.626821			−0.313410	+	0.949618i	$0.601471\pi$
$608$		0			0
$609$	−355.537			−0.583804
$610$		0			0
$611$			5.27184i			0.00862822i
$612$		0			0
$613$	489.507			0.798543			0.399272	−	0.916833i	$-0.369263\pi$
$613$	489.507			0.798543			0.399272	+	0.916833i	$0.369263\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	825.996i		−	1.33873i	−0.742934	−	0.669365i	$-0.766566\pi$
$617$		−	825.996i		−	1.33873i	0.742934	−	0.669365i	$-0.233434\pi$
$618$		0			0
$619$	−391.081			−0.631795			−0.315897	−	0.948793i	$-0.602306\pi$
$619$	−391.081			−0.631795			−0.315897	+	0.948793i	$0.602306\pi$
$620$		0			0
$621$	−562.281			−0.905444
$622$		0			0
$623$	−498.817			−0.800670
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−983.072	−	1590.37i	−1.56790	−	2.53648i
$628$		0			0
$629$			1847.24i			2.93679i
$630$		0			0
$631$	−806.374			−1.27793			−0.638965	−	0.769236i	$-0.720637\pi$
$631$	−806.374			−1.27793			−0.638965	+	0.769236i	$0.720637\pi$
$632$		0			0
$633$	−1098.72			−1.73574
$634$		0			0
$635$		0			0
$636$		0			0
$637$	6.61539			0.0103852
$638$		0			0
$639$	49.3919			0.0772955
$640$		0			0
$641$	238.517			0.372101			0.186051	−	0.982540i	$-0.440431\pi$
$641$	238.517			0.372101			0.186051	+	0.982540i	$0.440431\pi$
$642$		0			0
$643$		−	55.8795i		−	0.0869044i	−0.999056	−	0.0434522i	$-0.986164\pi$
$643$		−	55.8795i		−	0.0869044i	0.999056	−	0.0434522i	$-0.0138356\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			773.182i			1.19503i	0.801859	+	0.597513i	$0.203845\pi$
$647$			773.182i			1.19503i	−0.801859	+	0.597513i	$0.796155\pi$
$648$		0			0
$649$	−407.571	−	659.351i	−0.627998	−	1.01595i
$650$		0			0
$651$		−	1226.57i		−	1.88414i
$652$		0			0
$653$			605.962i			0.927967i	0.885844	+	0.463983i	$0.153580\pi$
$653$			605.962i			0.927967i	−0.885844	+	0.463983i	$0.846420\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−923.920			−1.40627
$658$		0			0
$659$			515.938i			0.782910i	0.920197	+	0.391455i	$0.128028\pi$
$659$			515.938i			0.782910i	−0.920197	+	0.391455i	$0.871972\pi$
$660$		0			0
$661$	154.673			0.233998			0.116999	−	0.993132i	$-0.462672\pi$
$661$	154.673			0.233998			0.116999	+	0.993132i	$0.462672\pi$
$662$		0			0
$663$		−	116.097i		−	0.175108i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−298.539			−0.447585
$668$		0			0
$669$	−715.218			−1.06908
$670$		0			0
$671$	−78.7039	+	48.6500i	−0.117293	+	0.0725037i
$672$		0			0
$673$	594.884			0.883929			0.441965	−	0.897033i	$-0.354282\pi$
$673$	594.884			0.883929			0.441965	+	0.897033i	$0.354282\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1067.80			−1.57726			−0.788629	−	0.614869i	$-0.789209\pi$
$677$	−1067.80			−1.57726			−0.788629	+	0.614869i	$0.789209\pi$
$678$		0			0
$679$			12.5088i			0.0184223i
$680$		0			0
$681$		−	329.684i		−	0.484118i
$682$		0			0
$683$		−	406.047i		−	0.594505i	−0.954799	−	0.297253i	$-0.903930\pi$
$683$		−	406.047i		−	0.594505i	0.954799	−	0.297253i	$-0.0960704\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	779.180i		−	1.13418i
$688$		0			0
$689$		−	46.8984i		−	0.0680673i
$690$		0			0
$691$	−76.5068			−0.110719			−0.0553595	−	0.998466i	$-0.517630\pi$
$691$	−76.5068			−0.110719			−0.0553595	+	0.998466i	$0.517630\pi$
$692$		0			0
$693$	568.754	+	920.106i	0.820712	+	1.32771i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	332.364i		−	0.476849i
$698$		0			0
$699$		−	1099.92i		−	1.57356i
$700$		0			0
$701$		−	526.761i		−	0.751443i	−0.926733	−	0.375721i	$-0.877395\pi$
$701$		−	526.761i		−	0.751443i	0.926733	−	0.375721i	$-0.122605\pi$
$702$		0			0
$703$	2242.11			3.18935
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	567.876i		−	0.803219i
$708$		0			0
$709$	−746.180			−1.05244			−0.526220	−	0.850348i	$-0.676391\pi$
$709$	−746.180			−1.05244			−0.526220	+	0.850348i	$0.676391\pi$
$710$		0			0
$711$			1666.88i			2.34441i
$712$		0			0
$713$		−	1029.94i		−	1.44451i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	727.772			1.01502
$718$		0			0
$719$	421.903			0.586792			0.293396	−	0.955991i	$-0.405215\pi$
$719$	421.903			0.586792			0.293396	+	0.955991i	$0.405215\pi$
$720$		0			0
$721$		−	1023.25i		−	1.41921i
$722$		0			0
$723$	−1299.62			−1.79754
$724$		0			0
$725$		0			0
$726$		0			0
$727$			343.364i			0.472303i	0.971716	+	0.236152i	$0.0758862\pi$
$727$			343.364i			0.472303i	−0.971716	+	0.236152i	$0.924114\pi$
$728$		0			0
$729$	1170.13			1.60512
$730$		0			0
$731$	1178.30			1.61190
$732$		0			0
$733$	−494.615			−0.674782			−0.337391	−	0.941365i	$-0.609544\pi$
$733$	−494.615			−0.674782			−0.337391	+	0.941365i	$0.609544\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	175.469	−	108.464i	0.238085	−	0.147170i
$738$		0			0
$739$			589.547i			0.797763i	0.917003	+	0.398881i	$0.130602\pi$
$739$			589.547i			0.797763i	−0.917003	+	0.398881i	$0.869398\pi$
$740$		0			0
$741$	−140.914			−0.190167
$742$		0			0
$743$	59.7902			0.0804714			0.0402357	−	0.999190i	$-0.487189\pi$
$743$	59.7902			0.0804714			0.0402357	+	0.999190i	$0.487189\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1305.79			−1.74805
$748$		0			0
$749$	603.971			0.806370
$750$		0			0
$751$	847.143			1.12802			0.564010	−	0.825768i	$-0.309258\pi$
$751$	847.143			1.12802			0.564010	+	0.825768i	$0.309258\pi$
$752$		0			0
$753$		−	24.5670i		−	0.0326255i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	218.552i		−	0.288708i	−0.989526	−	0.144354i	$-0.953890\pi$
$757$		−	218.552i		−	0.288708i	0.989526	−	0.144354i	$-0.0461104\pi$
$758$		0			0
$759$	807.510	+	1306.36i	1.06391	+	1.72115i
$760$		0			0
$761$		−	6.43775i		−	0.00845959i	−0.999991	−	0.00422979i	$-0.998654\pi$
$761$		−	6.43775i		−	0.00845959i	0.999991	−	0.00422979i	$-0.00134639\pi$
$762$		0			0
$763$			644.081i			0.844142i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−58.4214			−0.0761687
$768$		0			0
$769$		−	255.543i		−	0.332305i	−0.986100	−	0.166153i	$-0.946866\pi$
$769$		−	255.543i		−	0.332305i	0.986100	−	0.166153i	$-0.0531345\pi$
$770$		0			0
$771$	1044.34			1.35452
$772$		0			0
$773$		−	634.112i		−	0.820326i	−0.912012	−	0.410163i	$-0.865472\pi$
$773$		−	634.112i		−	0.820326i	0.912012	−	0.410163i	$-0.134528\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−2193.33			−2.82281
$778$		0			0
$779$	−403.411			−0.517857
$780$		0			0
$781$	−21.9286	−	35.4751i	−0.0280776	−	0.0454227i
$782$		0			0
$783$	−189.688			−0.242258
$784$		0			0
$785$		0			0
$786$		0			0
$787$	390.112			0.495695			0.247848	−	0.968799i	$-0.420277\pi$
$787$	390.112			0.495695			0.247848	+	0.968799i	$0.420277\pi$
$788$		0			0
$789$		−	1883.54i		−	2.38725i
$790$		0			0
$791$		−	1051.01i		−	1.32871i
$792$		0			0
$793$			6.97351i			0.00879384i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			729.894i			0.915801i	0.889003	+	0.457901i	$0.151398\pi$
$797$			729.894i			0.915801i	−0.889003	+	0.457901i	$0.848602\pi$
$798$		0			0
$799$		−	189.734i		−	0.237464i
$800$		0			0
$801$	−860.869			−1.07474
$802$		0			0
$803$	410.194	+	663.595i	0.510827	+	0.826394i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			364.290i			0.451413i
$808$		0			0
$809$		−	1360.50i		−	1.68171i	−0.541260	−	0.840855i	$-0.682053\pi$
$809$		−	1360.50i		−	1.68171i	0.541260	−	0.840855i	$-0.317947\pi$
$810$		0			0
$811$		−	265.266i		−	0.327085i	−0.986536	−	0.163542i	$-0.947708\pi$
$811$		−	265.266i		−	0.327085i	0.986536	−	0.163542i	$-0.0522920\pi$
$812$		0			0
$813$	−903.135			−1.11087
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	1430.17i		−	1.75052i
$818$		0			0
$819$	81.5255			0.0995427
$820$		0			0
$821$		−	236.318i		−	0.287842i	−0.989589	−	0.143921i	$-0.954029\pi$
$821$		−	236.318i		−	0.287842i	0.989589	−	0.143921i	$-0.0459711\pi$
$822$		0			0
$823$		−	820.686i		−	0.997188i	−0.866836	−	0.498594i	$-0.833850\pi$
$823$		−	820.686i		−	0.997188i	0.866836	−	0.498594i	$-0.166150\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	405.095			0.489837			0.244918	−	0.969544i	$-0.421239\pi$
$827$	405.095			0.489837			0.244918	+	0.969544i	$0.421239\pi$
$828$		0			0
$829$	−1103.14			−1.33068			−0.665341	−	0.746539i	$-0.731714\pi$
$829$	−1103.14			−1.33068			−0.665341	+	0.746539i	$0.731714\pi$
$830$		0			0
$831$		−	1404.06i		−	1.68960i
$832$		0			0
$833$	−238.088			−0.285820
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	654.410i		−	0.781852i
$838$		0			0
$839$	720.265			0.858480			0.429240	−	0.903190i	$-0.358781\pi$
$839$	720.265			0.858480			0.429240	+	0.903190i	$0.358781\pi$
$840$		0			0
$841$	740.286			0.880245
$842$		0			0
$843$	478.463			0.567572
$844$		0			0
$845$		0			0
$846$		0			0
$847$	408.345	−	817.002i	0.482107	−	0.964583i
$848$		0			0
$849$		−	1788.26i		−	2.10631i
$850$		0			0
$851$	−1841.70			−2.16417
$852$		0			0
$853$	1350.75			1.58353			0.791766	−	0.610824i	$-0.209162\pi$
$853$	1350.75			1.58353			0.791766	+	0.610824i	$0.209162\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−726.116			−0.847276			−0.423638	−	0.905832i	$-0.639247\pi$
$857$	−726.116			−0.847276			−0.423638	+	0.905832i	$0.639247\pi$
$858$		0			0
$859$	955.715			1.11259			0.556295	−	0.830985i	$-0.312222\pi$
$859$	955.715			1.11259			0.556295	+	0.830985i	$0.312222\pi$
$860$		0			0
$861$	394.633			0.458342
$862$		0			0
$863$			1154.66i			1.33796i	0.743279	+	0.668982i	$0.233270\pi$
$863$			1154.66i			1.33796i	−0.743279	+	0.668982i	$0.766730\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			2821.95i			3.25485i
$868$		0			0
$869$	1197.21	−	740.045i	1.37769	−	0.851606i
$870$		0			0
$871$		−	15.5473i		−	0.0178500i
$872$		0			0
$873$			21.5879i			0.0247284i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	112.344			0.128101			0.0640503	−	0.997947i	$-0.479598\pi$
$877$	112.344			0.128101			0.0640503	+	0.997947i	$0.479598\pi$
$878$		0			0
$879$		−	1174.52i		−	1.33621i
$880$		0			0
$881$	653.270			0.741509			0.370755	−	0.928731i	$-0.379099\pi$
$881$	653.270			0.741509			0.370755	+	0.928731i	$0.379099\pi$
$882$		0			0
$883$			702.158i			0.795196i	0.917560	+	0.397598i	$0.130156\pi$
$883$			702.158i			0.795196i	−0.917560	+	0.397598i	$0.869844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1046.64			1.17997			0.589987	−	0.807413i	$-0.299133\pi$
$887$	1046.64			1.17997			0.589987	+	0.807413i	$0.299133\pi$
$888$		0			0
$889$	146.319			0.164589
$890$		0			0
$891$	−165.039	−	266.993i	−0.185229	−	0.299656i
$892$		0			0
$893$	−230.292			−0.257885
$894$		0			0
$895$		0			0
$896$		0			0
$897$	115.749			0.129040
$898$		0			0
$899$		−	347.455i		−	0.386490i
$900$		0			0
$901$			1687.87i			1.87333i
$902$		0			0
$903$			1399.05i			1.54934i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			214.057i			0.236006i	0.993013	+	0.118003i	$0.0376493\pi$
$907$			214.057i			0.236006i	−0.993013	+	0.118003i	$0.962351\pi$
$908$		0			0
$909$		−	980.052i		−	1.07816i
$910$		0			0
$911$	829.046			0.910039			0.455020	−	0.890481i	$-0.349632\pi$
$911$	829.046			0.910039			0.455020	+	0.890481i	$0.349632\pi$
$912$		0			0
$913$	579.735	+	937.871i	0.634978	+	1.02724i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1555.03i			1.69578i
$918$		0			0
$919$			970.008i			1.05550i	0.849399	+	0.527752i	$0.176965\pi$
$919$			970.008i			1.05550i	−0.849399	+	0.527752i	$0.823035\pi$
$920$		0			0
$921$			2621.35i			2.84620i
$922$		0			0
$923$	−3.14325			−0.00340548
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	1765.94i		−	1.90501i
$928$		0			0
$929$	−1530.72			−1.64771			−0.823853	−	0.566803i	$-0.808180\pi$
$929$	−1530.72			−1.64771			−0.823853	+	0.566803i	$0.808180\pi$
$930$		0			0
$931$			288.982i			0.310400i
$932$		0			0
$933$		−	2235.54i		−	2.39608i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−798.725			−0.852428			−0.426214	−	0.904622i	$-0.640153\pi$
$937$	−798.725			−0.852428			−0.426214	+	0.904622i	$0.640153\pi$
$938$		0			0
$939$	−1092.35			−1.16332
$940$		0			0
$941$		−	661.163i		−	0.702618i	−0.936260	−	0.351309i	$-0.885737\pi$
$941$		−	661.163i		−	0.702618i	0.936260	−	0.351309i	$-0.114263\pi$
$942$		0			0
$943$	331.368			0.351397
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1191.07i			1.25773i	0.777514	+	0.628865i	$0.216480\pi$
$947$			1191.07i			1.25773i	−0.777514	+	0.628865i	$0.783520\pi$
$948$		0			0
$949$	58.7974			0.0619573
$950$		0			0
$951$	−1520.52			−1.59886
$952$		0			0
$953$	368.044			0.386195			0.193098	−	0.981180i	$-0.438147\pi$
$953$	368.044			0.386195			0.193098	+	0.981180i	$0.438147\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	272.417	+	440.705i	0.284658	+	0.460507i
$958$		0			0
$959$		−	753.974i		−	0.786208i
$960$		0			0
$961$	237.693			0.247339
$962$		0			0
$963$	1042.35			1.08239
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1224.13			1.26590			0.632951	−	0.774192i	$-0.281843\pi$
$967$	1224.13			1.26590			0.632951	+	0.774192i	$0.281843\pi$
$968$		0			0
$969$	5071.50			5.23374
$970$		0			0
$971$	1370.18			1.41110			0.705551	−	0.708659i	$-0.250699\pi$
$971$	1370.18			1.41110			0.705551	+	0.708659i	$0.250699\pi$
$972$		0			0
$973$			838.404i			0.861669i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1150.60i		−	1.17768i	−0.808248	−	0.588842i	$-0.799584\pi$
$977$		−	1150.60i		−	1.17768i	0.808248	−	0.588842i	$-0.200416\pi$
$978$		0			0
$979$	382.201	+	618.309i	0.390400	+	0.631572i
$980$		0			0
$981$			1111.57i			1.13310i
$982$		0			0
$983$			597.726i			0.608063i	0.952662	+	0.304032i	$0.0983329\pi$
$983$			597.726i			0.608063i	−0.952662	+	0.304032i	$0.901667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	225.280			0.228248
$988$		0			0
$989$			1174.77i			1.18783i
$990$		0			0
$991$	894.339			0.902461			0.451230	−	0.892407i	$-0.350985\pi$
$991$	894.339			0.902461			0.451230	+	0.892407i	$0.350985\pi$
$992$		0			0
$993$			921.906i			0.928405i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−136.661			−0.137072			−0.0685359	−	0.997649i	$-0.521833\pi$
$997$	−136.661			−0.137072			−0.0685359	+	0.997649i	$0.521833\pi$
$998$		0			0
$999$	−1170.20			−1.17137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.3.e.c.549.15		16
5.2	odd	4		1100.3.f.e.901.7	yes	8
5.3	odd	4		1100.3.f.c.901.2	yes	8
5.4	even	2	inner	1100.3.e.c.549.2		16
11.10	odd	2	inner	1100.3.e.c.549.16		16
55.32	even	4		1100.3.f.e.901.8	yes	8
55.43	even	4		1100.3.f.c.901.1	✓	8
55.54	odd	2	inner	1100.3.e.c.549.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.3.e.c.549.1		16	55.54	odd	2	inner
1100.3.e.c.549.2		16	5.4	even	2	inner
1100.3.e.c.549.15		16	1.1	even	1	trivial
1100.3.e.c.549.16		16	11.10	odd	2	inner
1100.3.f.c.901.1	✓	8	55.43	even	4
1100.3.f.c.901.2	yes	8	5.3	odd	4
1100.3.f.e.901.7	yes	8	5.2	odd	4
1100.3.f.e.901.8	yes	8	55.32	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1100.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.69333i q^{3} -7.54848 q^{7} -13.0273 q^{9} +(5.78376 + 9.35672i) q^{11} +0.829047 q^{13} -29.8374 q^{17} +36.2155i q^{19} -35.4275i q^{21} -29.7480i q^{23} -18.9015i q^{27} -10.0356i q^{29} +34.6221 q^{31} +(-43.9141 + 27.1451i) q^{33} -61.9103i q^{37} +3.89099i q^{39} +11.1392i q^{41} -39.4907 q^{43} +6.35892i q^{47} +7.97952 q^{49} -140.037i q^{51} -56.5691i q^{53} -169.971 q^{57} -70.4682 q^{59} +8.41148i q^{61} +98.3364 q^{63} -18.7532i q^{67} +139.617 q^{69} -3.79141 q^{71} +70.9217 q^{73} +(-43.6586 - 70.6290i) q^{77} -127.952i q^{79} -28.5349 q^{81} +100.235 q^{83} +47.1004 q^{87} +66.0818 q^{89} -6.25804 q^{91} +162.493i q^{93} -1.65712i q^{97} +(-75.3468 - 121.893i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9} + 6 q^{11} + 28 q^{31} - 28 q^{49} - 256 q^{59} + 352 q^{69} - 68 q^{71} - 256 q^{81} + 292 q^{89} + 228 q^{91} - 16 q^{99}+O(q^{100}) \) 16 * q - 32 * q^9 + 6 * q^11 + 28 * q^31 - 28 * q^49 - 256 * q^59 + 352 * q^69 - 68 * q^71 - 256 * q^81 + 292 * q^89 + 228 * q^91 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more