LMFDB - Embedded newform 1100.3.e.c.549.8

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			549.8
Root			$-1.49129i$ of defining polynomial
Character	$\chi$	$=$	1100.549
Dual form			1100.3.e.c.549.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.638826i q^{3} +9.06537 q^{7} +8.59190 q^{9} +(10.0956 - 4.36788i) q^{11} +10.4302 q^{13} -3.39108 q^{17} +28.0187i q^{19} -5.79120i q^{21} +18.8226i q^{23} -11.2382i q^{27} -22.2253i q^{29} +0.00744511 q^{31} +(-2.79032 - 6.44935i) q^{33} -8.96796i q^{37} -6.66311i q^{39} +16.5975i q^{41} -81.5368 q^{43} -59.1685i q^{47} +33.1809 q^{49} +2.16631i q^{51} +60.1508i q^{53} +17.8991 q^{57} +81.5361 q^{59} +114.695i q^{61} +77.8888 q^{63} -99.1084i q^{67} +12.0244 q^{69} +70.3598 q^{71} -16.4827 q^{73} +(91.5206 - 39.5965i) q^{77} +118.929i q^{79} +70.1479 q^{81} -30.6387 q^{83} -14.1981 q^{87} +4.02316 q^{89} +94.5539 q^{91} -0.00475614i q^{93} -101.970i q^{97} +(86.7406 - 37.5284i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	0.638826i		−	0.212942i	−0.994316	−	0.106471i	$-0.966045\pi$
$3$		−	0.638826i		−	0.212942i	0.994316	−	0.106471i	$-0.0339552\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	9.06537			1.29505			0.647526	−	0.762043i	$-0.275804\pi$
$7$	9.06537			1.29505			0.647526	+	0.762043i	$0.275804\pi$
$8$		0			0
$9$	8.59190			0.954656
$10$		0			0
$11$	10.0956	−	4.36788i	0.917784	−	0.397080i
$11$	10.0956	−	4.36788i	0.917784	−	0.397080i
$12$		0			0
$13$	10.4302			0.802326			0.401163	−	0.916007i	$-0.368606\pi$
$13$	10.4302			0.802326			0.401163	+	0.916007i	$0.368606\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.39108			−0.199475			−0.0997376	−	0.995014i	$-0.531800\pi$
$17$	−3.39108			−0.199475			−0.0997376	+	0.995014i	$0.531800\pi$
$18$		0			0
$19$			28.0187i			1.47467i	0.675529	+	0.737333i	$0.263915\pi$
$19$			28.0187i			1.47467i	−0.675529	+	0.737333i	$0.736085\pi$
$20$		0			0
$21$		−	5.79120i		−	0.275771i
$22$		0			0
$23$			18.8226i			0.818375i	0.912450	+	0.409188i	$0.134188\pi$
$23$			18.8226i			0.818375i	−0.912450	+	0.409188i	$0.865812\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	11.2382i		−	0.416229i
$28$		0			0
$29$		−	22.2253i		−	0.766391i	−0.923667	−	0.383195i	$-0.874824\pi$
$29$		−	22.2253i		−	0.766391i	0.923667	−	0.383195i	$-0.125176\pi$
$30$		0			0
$31$	0.00744511			0.000240165			0.000120082	−	1.00000i	$-0.499962\pi$
$31$	0.00744511			0.000240165			0.000120082		1.00000i	$0.499962\pi$
$32$		0			0
$33$	−2.79032	−	6.44935i	−0.0845551	−	0.195435i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	8.96796i		−	0.242377i	−0.992629	−	0.121189i	$-0.961329\pi$
$37$		−	8.96796i		−	0.242377i	0.992629	−	0.121189i	$-0.0386706\pi$
$38$		0			0
$39$		−	6.66311i		−	0.170849i
$40$		0			0
$41$			16.5975i			0.404817i	0.979301	+	0.202408i	$0.0648768\pi$
$41$			16.5975i			0.404817i	−0.979301	+	0.202408i	$0.935123\pi$
$42$		0			0
$43$	−81.5368			−1.89621			−0.948103	−	0.317964i	$-0.897001\pi$
$43$	−81.5368			−1.89621			−0.948103	+	0.317964i	$0.897001\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	59.1685i		−	1.25890i	−0.777039	−	0.629452i	$-0.783279\pi$
$47$		−	59.1685i		−	1.25890i	0.777039	−	0.629452i	$-0.216721\pi$
$48$		0			0
$49$	33.1809			0.677162
$50$		0			0
$51$			2.16631i			0.0424767i
$52$		0			0
$53$			60.1508i			1.13492i	0.823401	+	0.567460i	$0.192074\pi$
$53$			60.1508i			1.13492i	−0.823401	+	0.567460i	$0.807926\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	17.8991			0.314019
$58$		0			0
$59$	81.5361			1.38197			0.690984	−	0.722870i	$-0.257177\pi$
$59$	81.5361			1.38197			0.690984	+	0.722870i	$0.257177\pi$
$60$		0			0
$61$			114.695i			1.88024i	0.340843	+	0.940120i	$0.389288\pi$
$61$			114.695i			1.88024i	−0.340843	+	0.940120i	$0.610712\pi$
$62$		0			0
$63$	77.8888			1.23633
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	99.1084i		−	1.47923i	−0.673030	−	0.739615i	$-0.735008\pi$
$67$		−	99.1084i		−	1.47923i	0.673030	−	0.739615i	$-0.264992\pi$
$68$		0			0
$69$	12.0244			0.174267
$70$		0			0
$71$	70.3598			0.990983			0.495491	−	0.868613i	$-0.334988\pi$
$71$	70.3598			0.990983			0.495491	+	0.868613i	$0.334988\pi$
$72$		0			0
$73$	−16.4827			−0.225791			−0.112895	−	0.993607i	$-0.536012\pi$
$73$	−16.4827			−0.225791			−0.112895	+	0.993607i	$0.536012\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	91.5206	−	39.5965i	1.18858	−	0.514240i
$78$		0			0
$79$			118.929i			1.50543i	0.658344	+	0.752717i	$0.271257\pi$
$79$			118.929i			1.50543i	−0.658344	+	0.752717i	$0.728743\pi$
$80$		0			0
$81$	70.1479			0.866023
$82$		0			0
$83$	−30.6387			−0.369141			−0.184570	−	0.982819i	$-0.559089\pi$
$83$	−30.6387			−0.369141			−0.184570	+	0.982819i	$0.559089\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−14.1981			−0.163197
$88$		0			0
$89$	4.02316			0.0452041			0.0226020	−	0.999745i	$-0.492805\pi$
$89$	4.02316			0.0452041			0.0226020	+	0.999745i	$0.492805\pi$
$90$		0			0
$91$	94.5539			1.03905
$92$		0			0
$93$		−	0.00475614i		−	5.11412e-5i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	101.970i		−	1.05124i	−0.850720	−	0.525619i	$-0.823834\pi$
$97$		−	101.970i		−	1.05124i	0.850720	−	0.525619i	$-0.176166\pi$
$98$		0			0
$99$	86.7406	−	37.5284i	0.876168	−	0.379075i
$100$		0			0
$101$		−	149.468i		−	1.47988i	−0.672672	−	0.739940i	$-0.734854\pi$
$101$		−	149.468i		−	1.47988i	0.672672	−	0.739940i	$-0.265146\pi$
$102$		0			0
$103$		−	74.9240i		−	0.727417i	−0.931513	−	0.363709i	$-0.881510\pi$
$103$		−	74.9240i		−	0.727417i	0.931513	−	0.363709i	$-0.118490\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	94.4230			0.882458			0.441229	−	0.897395i	$-0.354543\pi$
$107$	94.4230			0.882458			0.441229	+	0.897395i	$0.354543\pi$
$108$		0			0
$109$			30.1807i			0.276887i	0.990370	+	0.138444i	$0.0442100\pi$
$109$			30.1807i			0.276887i	−0.990370	+	0.138444i	$0.955790\pi$
$110$		0			0
$111$	−5.72897			−0.0516124
$112$		0			0
$113$			74.7503i			0.661507i	0.943717	+	0.330753i	$0.107303\pi$
$113$			74.7503i			0.661507i	−0.943717	+	0.330753i	$0.892697\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	89.6155			0.765945
$118$		0			0
$119$	−30.7414			−0.258331
$120$		0			0
$121$	82.8432	−	88.1930i	0.684655	−	0.728868i
$122$		0			0
$123$	10.6029			0.0862025
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−107.488			−0.846361			−0.423181	−	0.906045i	$-0.639086\pi$
$127$	−107.488			−0.846361			−0.423181	+	0.906045i	$0.639086\pi$
$128$		0			0
$129$			52.0879i			0.403782i
$130$		0			0
$131$			201.887i			1.54112i	0.637367	+	0.770560i	$0.280024\pi$
$131$			201.887i			1.54112i	−0.637367	+	0.770560i	$0.719976\pi$
$132$		0			0
$133$			254.000i			1.90977i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	136.064i		−	0.993166i	−0.867989	−	0.496583i	$-0.834588\pi$
$137$		−	136.064i		−	0.993166i	0.867989	−	0.496583i	$-0.165412\pi$
$138$		0			0
$139$			6.92439i			0.0498157i	0.999690	+	0.0249079i	$0.00792924\pi$
$139$			6.92439i			0.0498157i	−0.999690	+	0.0249079i	$0.992071\pi$
$140$		0			0
$141$	−37.7984			−0.268074
$142$		0			0
$143$	105.300	−	45.5580i	0.736362	−	0.318588i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	21.1968i		−	0.144196i
$148$		0			0
$149$		−	94.2807i		−	0.632756i	−0.948633	−	0.316378i	$-0.897533\pi$
$149$		−	94.2807i		−	0.632756i	0.948633	−	0.316378i	$-0.102467\pi$
$150$		0			0
$151$		−	152.478i		−	1.00979i	−0.863181	−	0.504894i	$-0.831532\pi$
$151$		−	152.478i		−	1.00979i	0.863181	−	0.504894i	$-0.168468\pi$
$152$		0			0
$153$	−29.1358			−0.190430
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	168.579i		−	1.07375i	−0.843661	−	0.536876i	$-0.819604\pi$
$157$		−	168.579i		−	1.07375i	0.843661	−	0.536876i	$-0.180396\pi$
$158$		0			0
$159$	38.4259			0.241672
$160$		0			0
$161$			170.634i			1.05984i
$162$		0			0
$163$		−	21.6126i		−	0.132592i	−0.997800	−	0.0662962i	$-0.978882\pi$
$163$		−	21.6126i		−	0.132592i	0.997800	−	0.0662962i	$-0.0211182\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	247.745			1.48350			0.741750	−	0.670676i	$-0.233996\pi$
$167$	247.745			1.48350			0.741750	+	0.670676i	$0.233996\pi$
$168$		0			0
$169$	−60.2103			−0.356274
$170$		0			0
$171$			240.734i			1.40780i
$172$		0			0
$173$	−208.255			−1.20379			−0.601894	−	0.798576i	$-0.705587\pi$
$173$	−208.255			−1.20379			−0.601894	+	0.798576i	$0.705587\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	52.0874i		−	0.294279i
$178$		0			0
$179$	147.992			0.826770			0.413385	−	0.910556i	$-0.364346\pi$
$179$	147.992			0.826770			0.413385	+	0.910556i	$0.364346\pi$
$180$		0			0
$181$	−11.0571			−0.0610887			−0.0305444	−	0.999533i	$-0.509724\pi$
$181$	−11.0571			−0.0610887			−0.0305444	+	0.999533i	$0.509724\pi$
$182$		0			0
$183$	73.2700			0.400382
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−34.2350	+	14.8118i	−0.183075	+	0.0792076i
$188$		0			0
$189$		−	101.878i		−	0.539038i
$190$		0			0
$191$	104.092			0.544985			0.272492	−	0.962158i	$-0.412152\pi$
$191$	104.092			0.544985			0.272492	+	0.962158i	$0.412152\pi$
$192$		0			0
$193$	−267.361			−1.38529			−0.692644	−	0.721280i	$-0.743554\pi$
$193$	−267.361			−1.38529			−0.692644	+	0.721280i	$0.743554\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−264.837			−1.34435			−0.672175	−	0.740392i	$-0.734640\pi$
$197$	−264.837			−1.34435			−0.672175	+	0.740392i	$0.734640\pi$
$198$		0			0
$199$	108.882			0.547145			0.273572	−	0.961851i	$-0.411795\pi$
$199$	108.882			0.547145			0.273572	+	0.961851i	$0.411795\pi$
$200$		0			0
$201$	−63.3131			−0.314990
$202$		0			0
$203$		−	201.481i		−	0.992516i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			161.722i			0.781266i
$208$		0			0
$209$	122.382	+	282.866i	0.585561	+	1.35343i
$210$		0			0
$211$		−	91.2096i		−	0.432273i	−0.976363	−	0.216136i	$-0.930654\pi$
$211$		−	91.2096i		−	0.432273i	0.976363	−	0.216136i	$-0.0693456\pi$
$212$		0			0
$213$		−	44.9477i		−	0.211022i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.0674927			0.000311026
$218$		0			0
$219$			10.5296i			0.0480803i
$220$		0			0
$221$	−35.3697			−0.160044
$222$		0			0
$223$		−	2.12864i		−	0.00954548i	−0.999989	−	0.00477274i	$-0.998481\pi$
$223$		−	2.12864i		−	0.00954548i	0.999989	−	0.00477274i	$-0.00151922\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−247.145			−1.08875			−0.544373	−	0.838843i	$-0.683232\pi$
$227$	−247.145			−1.08875			−0.544373	+	0.838843i	$0.683232\pi$
$228$		0			0
$229$	−307.707			−1.34370			−0.671849	−	0.740688i	$-0.734500\pi$
$229$	−307.707			−1.34370			−0.671849	+	0.740688i	$0.734500\pi$
$230$		0			0
$231$	−25.2953	−	58.4658i	−0.109503	−	0.253098i
$232$		0			0
$233$	274.895			1.17981			0.589903	−	0.807474i	$-0.299166\pi$
$233$	274.895			1.17981			0.589903	+	0.807474i	$0.299166\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	75.9752			0.320571
$238$		0			0
$239$		−	191.500i		−	0.801253i	−0.916241	−	0.400627i	$-0.868792\pi$
$239$		−	191.500i		−	0.801253i	0.916241	−	0.400627i	$-0.131208\pi$
$240$		0			0
$241$			111.228i			0.461525i	0.973010	+	0.230763i	$0.0741221\pi$
$241$			111.228i			0.461525i	−0.973010	+	0.230763i	$0.925878\pi$
$242$		0			0
$243$		−	145.956i		−	0.600641i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			292.241i			1.18316i
$248$		0			0
$249$			19.5728i			0.0786056i
$250$		0			0
$251$	−144.363			−0.575152			−0.287576	−	0.957758i	$-0.592849\pi$
$251$	−144.363			−0.575152			−0.287576	+	0.957758i	$0.592849\pi$
$252$		0			0
$253$	82.2150	+	190.026i	0.324961	+	0.751092i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			406.571i			1.58199i	0.611823	+	0.790994i	$0.290436\pi$
$257$			406.571i			1.58199i	−0.611823	+	0.790994i	$0.709564\pi$
$258$		0			0
$259$		−	81.2979i		−	0.313892i
$260$		0			0
$261$		−	190.958i		−	0.731639i
$262$		0			0
$263$	249.632			0.949171			0.474585	−	0.880209i	$-0.342598\pi$
$263$	249.632			0.949171			0.474585	+	0.880209i	$0.342598\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	2.57010i		−	0.00962585i
$268$		0			0
$269$	−183.714			−0.682952			−0.341476	−	0.939890i	$-0.610927\pi$
$269$	−183.714			−0.682952			−0.341476	+	0.939890i	$0.610927\pi$
$270$		0			0
$271$		−	236.527i		−	0.872792i	−0.899755	−	0.436396i	$-0.856255\pi$
$271$		−	236.527i		−	0.872792i	0.899755	−	0.436396i	$-0.143745\pi$
$272$		0			0
$273$		−	60.4035i		−	0.221258i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−22.7429			−0.0821043			−0.0410522	−	0.999157i	$-0.513071\pi$
$277$	−22.7429			−0.0821043			−0.0410522	+	0.999157i	$0.513071\pi$
$278$		0			0
$279$	0.0639677			0.000229275
$280$		0			0
$281$		−	316.963i		−	1.12798i	−0.825781	−	0.563991i	$-0.809265\pi$
$281$		−	316.963i		−	1.12798i	0.825781	−	0.563991i	$-0.190735\pi$
$282$		0			0
$283$	218.829			0.773246			0.386623	−	0.922238i	$-0.373642\pi$
$283$	218.829			0.773246			0.386623	+	0.922238i	$0.373642\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			150.462i			0.524259i
$288$		0			0
$289$	−277.501			−0.960210
$290$		0			0
$291$	−65.1412			−0.223853
$292$		0			0
$293$	−205.688			−0.702005			−0.351003	−	0.936374i	$-0.614159\pi$
$293$	−205.688			−0.702005			−0.351003	+	0.936374i	$0.614159\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−49.0870	−	113.456i	−0.165276	−	0.382008i
$298$		0			0
$299$			196.324i			0.656603i
$300$		0			0
$301$	−739.161			−2.45569
$302$		0			0
$303$	−95.4841			−0.315129
$304$		0			0
$305$		0			0
$306$		0			0
$307$	351.513			1.14499			0.572497	−	0.819907i	$-0.305975\pi$
$307$	351.513			1.14499			0.572497	+	0.819907i	$0.305975\pi$
$308$		0			0
$309$	−47.8634			−0.154898
$310$		0			0
$311$	395.742			1.27248			0.636241	−	0.771490i	$-0.280488\pi$
$311$	395.742			1.27248			0.636241	+	0.771490i	$0.280488\pi$
$312$		0			0
$313$			340.880i			1.08907i	0.838736	+	0.544537i	$0.183295\pi$
$313$			340.880i			1.08907i	−0.838736	+	0.544537i	$0.816705\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			613.787i			1.93624i	0.250493	+	0.968118i	$0.419407\pi$
$317$			613.787i			1.93624i	−0.250493	+	0.968118i	$0.580593\pi$
$318$		0			0
$319$	−97.0776	−	224.379i	−0.304319	−	0.703381i
$320$		0			0
$321$		−	60.3199i		−	0.187912i
$322$		0			0
$323$		−	95.0134i		−	0.294159i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	19.2802			0.0589609
$328$		0			0
$329$		−	536.385i		−	1.63035i
$330$		0			0
$331$	−2.96493			−0.00895750			−0.00447875	−	0.999990i	$-0.501426\pi$
$331$	−2.96493			−0.00895750			−0.00447875	+	0.999990i	$0.501426\pi$
$332$		0			0
$333$		−	77.0519i		−	0.231387i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−44.4138			−0.131792			−0.0658958	−	0.997827i	$-0.520991\pi$
$337$	−44.4138			−0.131792			−0.0658958	+	0.997827i	$0.520991\pi$
$338$		0			0
$339$	47.7525			0.140863
$340$		0			0
$341$	0.0751631	−	0.0325194i	0.000220420	−	9.53647e-5i
$342$		0			0
$343$	−143.406			−0.418093
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−481.360			−1.38720			−0.693602	−	0.720358i	$-0.743977\pi$
$347$	−481.360			−1.38720			−0.693602	+	0.720358i	$0.743977\pi$
$348$		0			0
$349$			75.1868i			0.215435i	0.994182	+	0.107717i	$0.0343542\pi$
$349$			75.1868i			0.215435i	−0.994182	+	0.107717i	$0.965646\pi$
$350$		0			0
$351$		−	117.217i		−	0.333951i
$352$		0			0
$353$		−	187.670i		−	0.531643i	−0.964022	−	0.265822i	$-0.914357\pi$
$353$		−	187.670i		−	0.531643i	0.964022	−	0.265822i	$-0.0856432\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			19.6384i			0.0550095i
$358$		0			0
$359$			402.781i			1.12195i	0.827832	+	0.560976i	$0.189574\pi$
$359$			402.781i			1.12195i	−0.827832	+	0.560976i	$0.810426\pi$
$360$		0			0
$361$	−424.046			−1.17464
$362$		0			0
$363$	−56.3400	−	52.9224i	−0.155207	−	0.145792i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			292.427i			0.796805i	0.917211	+	0.398402i	$0.130435\pi$
$367$			292.427i			0.796805i	−0.917211	+	0.398402i	$0.869565\pi$
$368$		0			0
$369$			142.604i			0.386460i
$370$		0			0
$371$			545.289i			1.46978i
$372$		0			0
$373$	−232.155			−0.622398			−0.311199	−	0.950345i	$-0.600731\pi$
$373$	−232.155			−0.622398			−0.311199	+	0.950345i	$0.600731\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	231.815i		−	0.614895i
$378$		0			0
$379$	−467.391			−1.23322			−0.616611	−	0.787268i	$-0.711495\pi$
$379$	−467.391			−1.23322			−0.616611	+	0.787268i	$0.711495\pi$
$380$		0			0
$381$			68.6661i			0.180226i
$382$		0			0
$383$		−	721.494i		−	1.88380i	−0.335899	−	0.941898i	$-0.609040\pi$
$383$		−	721.494i		−	1.88380i	0.335899	−	0.941898i	$-0.390960\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−700.556			−1.81022
$388$		0			0
$389$	−643.293			−1.65371			−0.826854	−	0.562416i	$-0.809872\pi$
$389$	−643.293			−1.65371			−0.826854	+	0.562416i	$0.809872\pi$
$390$		0			0
$391$		−	63.8290i		−	0.163245i
$392$		0			0
$393$	128.971			0.328170
$394$		0			0
$395$		0			0
$396$		0			0
$397$			151.518i			0.381658i	0.981623	+	0.190829i	$0.0611176\pi$
$397$			151.518i			0.381658i	−0.981623	+	0.190829i	$0.938882\pi$
$398$		0			0
$399$	162.262			0.406671
$400$		0			0
$401$	−314.057			−0.783184			−0.391592	−	0.920139i	$-0.628076\pi$
$401$	−314.057			−0.783184			−0.391592	+	0.920139i	$0.628076\pi$
$402$		0			0
$403$	0.0776543			0.000192690
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−39.1710	−	90.5372i	−0.0962433	−	0.222450i
$408$		0			0
$409$		−	691.237i		−	1.69007i	−0.534713	−	0.845034i	$-0.679580\pi$
$409$		−	691.237i		−	1.69007i	0.534713	−	0.845034i	$-0.320420\pi$
$410$		0			0
$411$	−86.9211			−0.211487
$412$		0			0
$413$	739.155			1.78972
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.42348			0.0106079
$418$		0			0
$419$	439.877			1.04983			0.524913	−	0.851156i	$-0.324098\pi$
$419$	439.877			1.04983			0.524913	+	0.851156i	$0.324098\pi$
$420$		0			0
$421$	−274.014			−0.650864			−0.325432	−	0.945565i	$-0.605510\pi$
$421$	−274.014			−0.650864			−0.325432	+	0.945565i	$0.605510\pi$
$422$		0			0
$423$		−	508.370i		−	1.20182i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			1039.75i			2.43501i
$428$		0			0
$429$	−29.1037	−	67.2682i	−0.0678407	−	0.156802i
$430$		0			0
$431$			419.940i			0.974340i	0.873307	+	0.487170i	$0.161971\pi$
$431$			419.940i			0.974340i	−0.873307	+	0.487170i	$0.838029\pi$
$432$		0			0
$433$			396.492i			0.915686i	0.889033	+	0.457843i	$0.151378\pi$
$433$			396.492i			0.915686i	−0.889033	+	0.457843i	$0.848622\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−527.385			−1.20683
$438$		0			0
$439$			120.848i			0.275279i	0.990482	+	0.137640i	$0.0439516\pi$
$439$			120.848i			0.275279i	−0.990482	+	0.137640i	$0.956048\pi$
$440$		0			0
$441$	285.087			0.646456
$442$		0			0
$443$		−	374.096i		−	0.844461i	−0.906488	−	0.422231i	$-0.861247\pi$
$443$		−	374.096i		−	0.844461i	0.906488	−	0.422231i	$-0.138753\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−60.2290			−0.134741
$448$		0			0
$449$	−282.403			−0.628959			−0.314480	−	0.949264i	$-0.601830\pi$
$449$	−282.403			−0.628959			−0.314480	+	0.949264i	$0.601830\pi$
$450$		0			0
$451$	72.4958	+	167.562i	0.160745	+	0.371534i
$452$		0			0
$453$	−97.4069			−0.215026
$454$		0			0
$455$		0			0
$456$		0			0
$457$	513.069			1.12269			0.561345	−	0.827582i	$-0.310284\pi$
$457$	513.069			1.12269			0.561345	+	0.827582i	$0.310284\pi$
$458$		0			0
$459$			38.1095i			0.0830272i
$460$		0			0
$461$		−	712.598i		−	1.54577i	−0.634549	−	0.772883i	$-0.718814\pi$
$461$		−	712.598i		−	1.54577i	0.634549	−	0.772883i	$-0.281186\pi$
$462$		0			0
$463$		−	129.526i		−	0.279754i	−0.990169	−	0.139877i	$-0.955329\pi$
$463$		−	129.526i		−	0.279754i	0.990169	−	0.139877i	$-0.0446708\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			397.672i			0.851547i	0.904830	+	0.425773i	$0.139998\pi$
$467$			397.672i			0.851547i	−0.904830	+	0.425773i	$0.860002\pi$
$468$		0			0
$469$		−	898.454i		−	1.91568i
$470$		0			0
$471$	−107.693			−0.228647
$472$		0			0
$473$	−823.165	+	356.143i	−1.74031	+	0.752946i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			516.809i			1.08346i
$478$		0			0
$479$			409.935i			0.855815i	0.903823	+	0.427907i	$0.140749\pi$
$479$			409.935i			0.855815i	−0.903823	+	0.427907i	$0.859251\pi$
$480$		0			0
$481$		−	93.5380i		−	0.194466i
$482$		0			0
$483$	109.006			0.225684
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	847.021i		−	1.73926i	−0.493702	−	0.869631i	$-0.664357\pi$
$487$		−	847.021i		−	1.73926i	0.493702	−	0.869631i	$-0.335643\pi$
$488$		0			0
$489$	−13.8067			−0.0282345
$490$		0			0
$491$		−	634.763i		−	1.29280i	−0.763000	−	0.646398i	$-0.776275\pi$
$491$		−	634.763i		−	1.29280i	0.763000	−	0.646398i	$-0.223725\pi$
$492$		0			0
$493$			75.3678i			0.152876i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	637.837			1.28337
$498$		0			0
$499$	−360.573			−0.722591			−0.361296	−	0.932451i	$-0.617665\pi$
$499$	−360.573			−0.722591			−0.361296	+	0.932451i	$0.617665\pi$
$500$		0			0
$501$		−	158.266i		−	0.315900i
$502$		0			0
$503$	239.586			0.476314			0.238157	−	0.971227i	$-0.423457\pi$
$503$	239.586			0.476314			0.238157	+	0.971227i	$0.423457\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			38.4639i			0.0758657i
$508$		0			0
$509$	92.3407			0.181416			0.0907080	−	0.995878i	$-0.471087\pi$
$509$	92.3407			0.181416			0.0907080	+	0.995878i	$0.471087\pi$
$510$		0			0
$511$	−149.422			−0.292411
$512$		0			0
$513$	314.879			0.613798
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−258.441	−	597.343i	−0.499886	−	1.15540i
$518$		0			0
$519$			133.039i			0.256337i
$520$		0			0
$521$	−439.053			−0.842712			−0.421356	−	0.906895i	$-0.638446\pi$
$521$	−439.053			−0.842712			−0.421356	+	0.906895i	$0.638446\pi$
$522$		0			0
$523$	495.619			0.947647			0.473823	−	0.880620i	$-0.342874\pi$
$523$	495.619			0.947647			0.473823	+	0.880620i	$0.342874\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.0252470			−4.79069e−5
$528$		0			0
$529$	174.709			0.330262
$530$		0			0
$531$	700.550			1.31930
$532$		0			0
$533$			173.116i			0.324795i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	94.5410i		−	0.176054i
$538$		0			0
$539$	334.982	−	144.930i	0.621488	−	0.268887i
$540$		0			0
$541$			572.642i			1.05849i	0.848470	+	0.529244i	$0.177524\pi$
$541$			572.642i			1.05849i	−0.848470	+	0.529244i	$0.822476\pi$
$542$		0			0
$543$			7.06354i			0.0130084i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	131.117			0.239702			0.119851	−	0.992792i	$-0.461758\pi$
$547$	131.117			0.239702			0.119851	+	0.992792i	$0.461758\pi$
$548$		0			0
$549$			985.445i			1.79498i
$550$		0			0
$551$	622.724			1.13017
$552$		0			0
$553$			1078.14i			1.94962i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	433.908			0.779008			0.389504	−	0.921025i	$-0.372646\pi$
$557$	433.908			0.779008			0.389504	+	0.921025i	$0.372646\pi$
$558$		0			0
$559$	−850.448			−1.52137
$560$		0			0
$561$	9.46218	+	21.8702i	0.0168666	+	0.0389844i
$562$		0			0
$563$	−405.058			−0.719464			−0.359732	−	0.933056i	$-0.617132\pi$
$563$	−405.058			−0.719464			−0.359732	+	0.933056i	$0.617132\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	635.916			1.12155
$568$		0			0
$569$			422.003i			0.741657i	0.928701	+	0.370829i	$0.120926\pi$
$569$			422.003i			0.741657i	−0.928701	+	0.370829i	$0.879074\pi$
$570$		0			0
$571$		−	395.151i		−	0.692034i	−0.938228	−	0.346017i	$-0.887534\pi$
$571$		−	395.151i		−	0.692034i	0.938228	−	0.346017i	$-0.112466\pi$
$572$		0			0
$573$		−	66.4968i		−	0.116050i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	728.545i		−	1.26264i	−0.775521	−	0.631322i	$-0.782513\pi$
$577$		−	728.545i		−	1.26264i	0.775521	−	0.631322i	$-0.217487\pi$
$578$		0			0
$579$			170.797i			0.294986i
$580$		0			0
$581$	−277.751			−0.478057
$582$		0			0
$583$	262.731	+	607.259i	0.450654	+	1.04161i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			292.478i			0.498259i	0.968470	+	0.249129i	$0.0801444\pi$
$587$			292.478i			0.498259i	−0.968470	+	0.249129i	$0.919856\pi$
$588$		0			0
$589$			0.208602i			0.000354163i
$590$		0			0
$591$			169.185i			0.286269i
$592$		0			0
$593$	189.061			0.318821			0.159410	−	0.987212i	$-0.449041\pi$
$593$	189.061			0.318821			0.159410	+	0.987212i	$0.449041\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	69.5566i		−	0.116510i
$598$		0			0
$599$	333.334			0.556485			0.278242	−	0.960511i	$-0.410248\pi$
$599$	333.334			0.556485			0.278242	+	0.960511i	$0.410248\pi$
$600$		0			0
$601$			473.020i			0.787055i	0.919313	+	0.393527i	$0.128745\pi$
$601$			473.020i			0.787055i	−0.919313	+	0.393527i	$0.871255\pi$
$602$		0			0
$603$		−	851.529i		−	1.41215i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−1062.30			−1.75008			−0.875039	−	0.484052i	$-0.839164\pi$
$607$	−1062.30			−1.75008			−0.875039	+	0.484052i	$0.839164\pi$
$608$		0			0
$609$	−128.711			−0.211349
$610$		0			0
$611$		−	617.141i		−	1.01005i
$612$		0			0
$613$	−1093.66			−1.78411			−0.892057	−	0.451923i	$-0.850738\pi$
$613$	−1093.66			−1.78411			−0.892057	+	0.451923i	$0.850738\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	793.012i		−	1.28527i	−0.766172	−	0.642636i	$-0.777841\pi$
$617$		−	793.012i		−	1.28527i	0.766172	−	0.642636i	$-0.222159\pi$
$618$		0			0
$619$	1156.20			1.86786			0.933928	−	0.357462i	$-0.116358\pi$
$619$	1156.20			1.86786			0.933928	+	0.357462i	$0.116358\pi$
$620$		0			0
$621$	211.532			0.340631
$622$		0			0
$623$	36.4715			0.0585417
$624$		0			0
$625$		0			0
$626$		0			0
$627$	180.702	−	78.1810i	0.288201	−	0.124691i
$628$		0			0
$629$			30.4111i			0.0483483i
$630$		0			0
$631$	−184.009			−0.291614			−0.145807	−	0.989313i	$-0.546578\pi$
$631$	−184.009			−0.291614			−0.145807	+	0.989313i	$0.546578\pi$
$632$		0			0
$633$	−58.2671			−0.0920491
$634$		0			0
$635$		0			0
$636$		0			0
$637$	346.085			0.543304
$638$		0			0
$639$	604.524			0.946047
$640$		0			0
$641$	1060.80			1.65492			0.827458	−	0.561528i	$-0.189786\pi$
$641$	1060.80			1.65492			0.827458	+	0.561528i	$0.189786\pi$
$642$		0			0
$643$			345.954i			0.538031i	0.963136	+	0.269015i	$0.0866982\pi$
$643$			345.954i			0.538031i	−0.963136	+	0.269015i	$0.913302\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			582.248i			0.899919i	0.893049	+	0.449959i	$0.148562\pi$
$647$			582.248i			0.899919i	−0.893049	+	0.449959i	$0.851438\pi$
$648$		0			0
$649$	823.158	−	356.140i	1.26835	−	0.548752i
$650$		0			0
$651$		−	0.0431161i		−	6.62306e-5i
$652$		0			0
$653$		−	860.284i		−	1.31743i	−0.752391	−	0.658717i	$-0.771100\pi$
$653$		−	860.284i		−	1.31743i	0.752391	−	0.658717i	$-0.228900\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−141.618			−0.215552
$658$		0			0
$659$		−	494.828i		−	0.750876i	−0.926847	−	0.375438i	$-0.877492\pi$
$659$		−	494.828i		−	0.750876i	0.926847	−	0.375438i	$-0.122508\pi$
$660$		0			0
$661$	−321.241			−0.485993			−0.242996	−	0.970027i	$-0.578130\pi$
$661$	−321.241			−0.485993			−0.242996	+	0.970027i	$0.578130\pi$
$662$		0			0
$663$			22.5951i			0.0340801i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	418.339			0.627195
$668$		0			0
$669$	−1.35983			−0.00203264
$670$		0			0
$671$	500.973	+	1157.91i	0.746606	+	1.72565i
$672$		0			0
$673$	−775.168			−1.15181			−0.575905	−	0.817517i	$-0.695350\pi$
$673$	−775.168			−1.15181			−0.575905	+	0.817517i	$0.695350\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−273.752			−0.404360			−0.202180	−	0.979348i	$-0.564803\pi$
$677$	−273.752			−0.404360			−0.202180	+	0.979348i	$0.564803\pi$
$678$		0			0
$679$		−	924.397i		−	1.36141i
$680$		0			0
$681$			157.883i			0.231840i
$682$		0			0
$683$			645.794i			0.945526i	0.881190	+	0.472763i	$0.156743\pi$
$683$			645.794i			0.945526i	−0.881190	+	0.472763i	$0.843257\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			196.571i			0.286130i
$688$		0			0
$689$			627.386i			0.910575i
$690$		0			0
$691$	−519.100			−0.751231			−0.375615	−	0.926776i	$-0.622569\pi$
$691$	−519.100			−0.751231			−0.375615	+	0.926776i	$0.622569\pi$
$692$		0			0
$693$	786.335	−	340.209i	1.13468	−	0.490922i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	56.2833i		−	0.0807508i
$698$		0			0
$699$		−	175.610i		−	0.251231i
$700$		0			0
$701$			450.375i			0.642475i	0.946999	+	0.321237i	$0.104099\pi$
$701$			450.375i			0.642475i	−0.946999	+	0.321237i	$0.895901\pi$
$702$		0			0
$703$	251.270			0.357426
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	1354.98i		−	1.91652i
$708$		0			0
$709$	−527.930			−0.744612			−0.372306	−	0.928110i	$-0.621433\pi$
$709$	−527.930			−0.744612			−0.372306	+	0.928110i	$0.621433\pi$
$710$		0			0
$711$			1021.83i			1.43717i
$712$		0			0
$713$			0.140137i			0.000196545i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−122.335			−0.170621
$718$		0			0
$719$	312.167			0.434168			0.217084	−	0.976153i	$-0.430345\pi$
$719$	312.167			0.434168			0.217084	+	0.976153i	$0.430345\pi$
$720$		0			0
$721$		−	679.213i		−	0.942043i
$722$		0			0
$723$	71.0551			0.0982782
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	1414.10i		−	1.94512i	−0.232662	−	0.972558i	$-0.574744\pi$
$727$		−	1414.10i		−	1.94512i	0.232662	−	0.972558i	$-0.425256\pi$
$728$		0			0
$729$	538.090			0.738121
$730$		0			0
$731$	276.498			0.378246
$732$		0			0
$733$	122.714			0.167413			0.0837067	−	0.996490i	$-0.473324\pi$
$733$	122.714			0.167413			0.0837067	+	0.996490i	$0.473324\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−432.894	−	1000.56i	−0.587373	−	1.35761i
$738$		0			0
$739$			1027.05i			1.38978i	0.719115	+	0.694891i	$0.244547\pi$
$739$			1027.05i			1.38978i	−0.719115	+	0.694891i	$0.755453\pi$
$740$		0			0
$741$	186.691			0.251945
$742$		0			0
$743$	−609.811			−0.820742			−0.410371	−	0.911919i	$-0.634601\pi$
$743$	−609.811			−0.820742			−0.410371	+	0.911919i	$0.634601\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−263.244			−0.352402
$748$		0			0
$749$	855.979			1.14283
$750$		0			0
$751$	232.075			0.309021			0.154511	−	0.987991i	$-0.450620\pi$
$751$	232.075			0.309021			0.154511	+	0.987991i	$0.450620\pi$
$752$		0			0
$753$			92.2229i			0.122474i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			929.780i			1.22824i	0.789211	+	0.614122i	$0.210490\pi$
$757$			929.780i			1.22824i	−0.789211	+	0.614122i	$0.789510\pi$
$758$		0			0
$759$	121.394	−	52.5211i	0.159939	−	0.0691978i
$760$		0			0
$761$			759.425i			0.997931i	0.866622	+	0.498965i	$0.166287\pi$
$761$			759.425i			0.997931i	−0.866622	+	0.498965i	$0.833713\pi$
$762$		0			0
$763$			273.599i			0.358583i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	850.441			1.10879
$768$		0			0
$769$			1242.48i			1.61571i	0.589384	+	0.807853i	$0.299370\pi$
$769$			1242.48i			1.61571i	−0.589384	+	0.807853i	$0.700630\pi$
$770$		0			0
$771$	259.728			0.336872
$772$		0			0
$773$			542.327i			0.701588i	0.936453	+	0.350794i	$0.114088\pi$
$773$			542.327i			0.701588i	−0.936453	+	0.350794i	$0.885912\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−51.9353			−0.0668407
$778$		0			0
$779$	−465.039			−0.596969
$780$		0			0
$781$	710.326	−	307.323i	0.909508	−	0.393500i
$782$		0			0
$783$	−249.772			−0.318994
$784$		0			0
$785$		0			0
$786$		0			0
$787$	1265.30			1.60775			0.803877	−	0.594796i	$-0.202767\pi$
$787$	1265.30			1.60775			0.803877	+	0.594796i	$0.202767\pi$
$788$		0			0
$789$		−	159.471i		−	0.202118i
$790$		0			0
$791$			677.639i			0.856686i
$792$		0			0
$793$			1196.29i			1.50857i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			3.70455i			0.00464812i	0.999997	+	0.00232406i	$0.000739772\pi$
$797$			3.70455i			0.00464812i	−0.999997	+	0.00232406i	$0.999260\pi$
$798$		0			0
$799$			200.645i			0.251120i
$800$		0			0
$801$	34.5666			0.0431543
$802$		0			0
$803$	−166.403	+	71.9945i	−0.207227	+	0.0896569i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			117.361i			0.145429i
$808$		0			0
$809$		−	387.788i		−	0.479343i	−0.970854	−	0.239671i	$-0.922960\pi$
$809$		−	387.788i		−	0.479343i	0.970854	−	0.239671i	$-0.0770397\pi$
$810$		0			0
$811$		−	653.974i		−	0.806379i	−0.915116	−	0.403190i	$-0.867901\pi$
$811$		−	653.974i		−	0.806379i	0.915116	−	0.403190i	$-0.132099\pi$
$812$		0			0
$813$	−151.100			−0.185854
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	2284.55i		−	2.79627i
$818$		0			0
$819$	812.398			0.991939
$820$		0			0
$821$		−	1276.47i		−	1.55477i	−0.629023	−	0.777387i	$-0.716545\pi$
$821$		−	1276.47i		−	1.55477i	0.629023	−	0.777387i	$-0.283455\pi$
$822$		0			0
$823$			946.469i			1.15002i	0.818145	+	0.575012i	$0.195003\pi$
$823$			946.469i			1.15002i	−0.818145	+	0.575012i	$0.804997\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	785.770			0.950146			0.475073	−	0.879946i	$-0.342422\pi$
$827$	785.770			0.950146			0.475073	+	0.879946i	$0.342422\pi$
$828$		0			0
$829$	−711.369			−0.858105			−0.429052	−	0.903280i	$-0.641152\pi$
$829$	−711.369			−0.858105			−0.429052	+	0.903280i	$0.641152\pi$
$830$		0			0
$831$			14.5288i			0.0174835i
$832$		0			0
$833$	−112.519			−0.135077
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	0.0836695i		−	9.99635e-5i
$838$		0			0
$839$	1072.89			1.27877			0.639386	−	0.768886i	$-0.279189\pi$
$839$	1072.89			1.27877			0.639386	+	0.768886i	$0.279189\pi$
$840$		0			0
$841$	347.035			0.412645
$842$		0			0
$843$	−202.484			−0.240195
$844$		0			0
$845$		0			0
$846$		0			0
$847$	751.004	−	799.502i	0.886664	−	0.943922i
$848$		0			0
$849$		−	139.793i		−	0.164657i
$850$		0			0
$851$	168.801			0.198356
$852$		0			0
$853$	−678.071			−0.794925			−0.397462	−	0.917618i	$-0.630109\pi$
$853$	−678.071			−0.794925			−0.397462	+	0.917618i	$0.630109\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	514.861			0.600771			0.300386	−	0.953818i	$-0.402885\pi$
$857$	514.861			0.600771			0.300386	+	0.953818i	$0.402885\pi$
$858$		0			0
$859$	−1048.57			−1.22069			−0.610346	−	0.792135i	$-0.708970\pi$
$859$	−1048.57			−1.22069			−0.610346	+	0.792135i	$0.708970\pi$
$860$		0			0
$861$	96.1193			0.111637
$862$		0			0
$863$		−	962.788i		−	1.11563i	−0.829966	−	0.557814i	$-0.811640\pi$
$863$		−	962.788i		−	1.11563i	0.829966	−	0.557814i	$-0.188360\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			177.275i			0.204469i
$868$		0			0
$869$	519.469	+	1200.67i	0.597778	+	1.38166i
$870$		0			0
$871$		−	1033.72i		−	1.18682i
$872$		0			0
$873$		−	876.117i		−	1.00357i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−297.443			−0.339160			−0.169580	−	0.985516i	$-0.554241\pi$
$877$	−297.443			−0.339160			−0.169580	+	0.985516i	$0.554241\pi$
$878$		0			0
$879$			131.399i			0.149487i
$880$		0			0
$881$	1259.46			1.42958			0.714789	−	0.699341i	$-0.246523\pi$
$881$	1259.46			1.42958			0.714789	+	0.699341i	$0.246523\pi$
$882$		0			0
$883$			645.648i			0.731199i	0.930772	+	0.365599i	$0.119136\pi$
$883$			645.648i			0.731199i	−0.930772	+	0.365599i	$0.880864\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−545.781			−0.615312			−0.307656	−	0.951498i	$-0.599545\pi$
$887$	−545.781			−0.615312			−0.307656	+	0.951498i	$0.599545\pi$
$888$		0			0
$889$	−974.417			−1.09608
$890$		0			0
$891$	708.186	−	306.398i	0.794822	−	0.343881i
$892$		0			0
$893$	1657.82			1.85646
$894$		0			0
$895$		0			0
$896$		0			0
$897$	125.417			0.139819
$898$		0			0
$899$		−	0.165470i		−	0.000184060i
$900$		0			0
$901$		−	203.976i		−	0.226388i
$902$		0			0
$903$			472.196i			0.522919i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	146.166i		−	0.161153i	−0.996748	−	0.0805766i	$-0.974324\pi$
$907$		−	146.166i		−	0.161153i	0.996748	−	0.0805766i	$-0.0256762\pi$
$908$		0			0
$909$		−	1284.21i		−	1.41278i
$910$		0			0
$911$	−1533.17			−1.68296			−0.841478	−	0.540292i	$-0.818314\pi$
$911$	−1533.17			−1.68296			−0.841478	+	0.540292i	$0.818314\pi$
$912$		0			0
$913$	−309.317	+	133.826i	−0.338791	+	0.146578i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1830.18i			1.99583i
$918$		0			0
$919$			402.371i			0.437836i	0.975743	+	0.218918i	$0.0702527\pi$
$919$			402.371i			0.437836i	−0.975743	+	0.218918i	$0.929747\pi$
$920$		0			0
$921$		−	224.556i		−	0.243817i
$922$		0			0
$923$	733.869			0.795091
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	643.739i		−	0.694433i
$928$		0			0
$929$	602.857			0.648932			0.324466	−	0.945897i	$-0.394815\pi$
$929$	602.857			0.648932			0.324466	+	0.945897i	$0.394815\pi$
$930$		0			0
$931$			929.685i			0.998587i
$932$		0			0
$933$		−	252.811i		−	0.270965i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	847.671			0.904665			0.452333	−	0.891849i	$-0.350592\pi$
$937$	847.671			0.904665			0.452333	+	0.891849i	$0.350592\pi$
$938$		0			0
$939$	217.763			0.231910
$940$		0			0
$941$			1337.24i			1.42109i	0.703653	+	0.710544i	$0.251551\pi$
$941$			1337.24i			1.42109i	−0.703653	+	0.710544i	$0.748449\pi$
$942$		0			0
$943$	−312.408			−0.331292
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1205.82i		−	1.27331i	−0.771151	−	0.636653i	$-0.780318\pi$
$947$		−	1205.82i		−	1.27331i	0.771151	−	0.636653i	$-0.219682\pi$
$948$		0			0
$949$	−171.918			−0.181158
$950$		0			0
$951$	392.103			0.412306
$952$		0			0
$953$	647.299			0.679223			0.339611	−	0.940566i	$-0.389704\pi$
$953$	647.299			0.679223			0.339611	+	0.940566i	$0.389704\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−143.339	+	62.0157i	−0.149779	+	0.0648022i
$958$		0			0
$959$		−	1233.47i		−	1.28620i
$960$		0			0
$961$	−961.000			−1.00000
$962$		0			0
$963$	811.273			0.842443
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−887.321			−0.917602			−0.458801	−	0.888539i	$-0.651721\pi$
$967$	−887.321			−0.917602			−0.458801	+	0.888539i	$0.651721\pi$
$968$		0			0
$969$	−60.6971			−0.0626389
$970$		0			0
$971$	1104.79			1.13778			0.568891	−	0.822413i	$-0.307373\pi$
$971$	1104.79			1.13778			0.568891	+	0.822413i	$0.307373\pi$
$972$		0			0
$973$			62.7721i			0.0645140i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1577.70i			1.61484i	0.589976	+	0.807420i	$0.299137\pi$
$977$			1577.70i			1.61484i	−0.589976	+	0.807420i	$0.700863\pi$
$978$		0			0
$979$	40.6163	−	17.5727i	0.0414876	−	0.0179496i
$980$		0			0
$981$			259.310i			0.264332i
$982$		0			0
$983$		−	860.925i		−	0.875814i	−0.899020	−	0.437907i	$-0.855720\pi$
$983$		−	860.925i		−	0.875814i	0.899020	−	0.437907i	$-0.144280\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−342.657			−0.347170
$988$		0			0
$989$		−	1534.74i		−	1.55181i
$990$		0			0
$991$	−620.455			−0.626090			−0.313045	−	0.949738i	$-0.601349\pi$
$991$	−620.455			−0.626090			−0.313045	+	0.949738i	$0.601349\pi$
$992$		0			0
$993$			1.89408i			0.00190743i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	148.972			0.149420			0.0747100	−	0.997205i	$-0.476197\pi$
$997$	148.972			0.149420			0.0747100	+	0.997205i	$0.476197\pi$
$998$		0			0
$999$	−100.784			−0.100884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.3.e.c.549.8		16
5.2	odd	4		1100.3.f.e.901.4	yes	8
5.3	odd	4		1100.3.f.c.901.5	✓	8
5.4	even	2	inner	1100.3.e.c.549.9		16
11.10	odd	2	inner	1100.3.e.c.549.7		16
55.32	even	4		1100.3.f.e.901.3	yes	8
55.43	even	4		1100.3.f.c.901.6	yes	8
55.54	odd	2	inner	1100.3.e.c.549.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.3.e.c.549.7		16	11.10	odd	2	inner
1100.3.e.c.549.8		16	1.1	even	1	trivial
1100.3.e.c.549.9		16	5.4	even	2	inner
1100.3.e.c.549.10		16	55.54	odd	2	inner
1100.3.f.c.901.5	✓	8	5.3	odd	4
1100.3.f.c.901.6	yes	8	55.43	even	4
1100.3.f.e.901.3	yes	8	55.32	even	4
1100.3.f.e.901.4	yes	8	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

\(f(q)\)	\(=\)	\(q-0.638826i q^{3} +9.06537 q^{7} +8.59190 q^{9} +(10.0956 - 4.36788i) q^{11} +10.4302 q^{13} -3.39108 q^{17} +28.0187i q^{19} -5.79120i q^{21} +18.8226i q^{23} -11.2382i q^{27} -22.2253i q^{29} +0.00744511 q^{31} +(-2.79032 - 6.44935i) q^{33} -8.96796i q^{37} -6.66311i q^{39} +16.5975i q^{41} -81.5368 q^{43} -59.1685i q^{47} +33.1809 q^{49} +2.16631i q^{51} +60.1508i q^{53} +17.8991 q^{57} +81.5361 q^{59} +114.695i q^{61} +77.8888 q^{63} -99.1084i q^{67} +12.0244 q^{69} +70.3598 q^{71} -16.4827 q^{73} +(91.5206 - 39.5965i) q^{77} +118.929i q^{79} +70.1479 q^{81} -30.6387 q^{83} -14.1981 q^{87} +4.02316 q^{89} +94.5539 q^{91} -0.00475614i q^{93} -101.970i q^{97} +(86.7406 - 37.5284i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9} + 6 q^{11} + 28 q^{31} - 28 q^{49} - 256 q^{59} + 352 q^{69} - 68 q^{71} - 256 q^{81} + 292 q^{89} + 228 q^{91} - 16 q^{99}+O(q^{100}) \) 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

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