LMFDB - Embedded newform 1100.3.e.b.549.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,3,Mod(549,1100)]

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, 1]))

N = Newforms(chi, 3, names="a")

//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);

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

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} + 85x^{14} + 2456x^{12} + 32605x^{10} + 215801x^{8} + 712960x^{6} + 1098976x^{4} + 633600x^{2} + 92416 $ x^16 + 85x^14 + 2456x^12 + 32605x^10 + 215801x^8 + 712960x^6 + 1098976x^4 + 633600*x^2 + 92416
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{25}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			549.12
Root			$-1.73790i$ of defining polynomial
Character	$\chi$	$=$	1100.549
Dual form			1100.3.e.b.549.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.41553i q^{3} +0.558647 q^{7} -2.66584 q^{9} +O(q^{10})$	$q+3.41553i q^{3} +0.558647 q^{7} -2.66584 q^{9} +(2.82054 + 10.6322i) q^{11} -18.1072 q^{13} +32.0402 q^{17} +15.7408i q^{19} +1.90807i q^{21} -31.7468i q^{23} +21.6345i q^{27} +48.3396i q^{29} -43.3557 q^{31} +(-36.3147 + 9.63363i) q^{33} +21.4366i q^{37} -61.8458i q^{39} +38.9133i q^{41} -23.3044 q^{43} -75.2975i q^{47} -48.6879 q^{49} +109.434i q^{51} +8.43533i q^{53} -53.7631 q^{57} +29.6094 q^{59} +64.1200i q^{61} -1.48926 q^{63} +18.8211i q^{67} +108.432 q^{69} +94.8085 q^{71} +0.945835 q^{73} +(1.57568 + 5.93967i) q^{77} -73.2825i q^{79} -97.8859 q^{81} -161.659 q^{83} -165.105 q^{87} -91.5556 q^{89} -10.1155 q^{91} -148.083i q^{93} -33.1491i q^{97} +(-7.51910 - 28.3438i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 56 q^{9}+O(q^{10}) $ 16 * q - 56 * q^9	$ 16 q - 56 q^{9} + 48 q^{11} + 40 q^{31} + 488 q^{49} + 32 q^{59} + 112 q^{69} - 440 q^{71} - 448 q^{81} - 440 q^{89} - 144 q^{91} + 80 q^{99}+O(q^{100}) $ 16 * q - 56 * q^9 + 48 * q^11 + 40 * q^31 + 488 * q^49 + 32 * q^59 + 112 * q^69 - 440 * q^71 - 448 * q^81 - 440 * q^89 - 144 * q^91 + 80 * q^99

Display 100 coefficients 10 coefficients

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
$3$			3.41553i			1.13851i	0.822161	+	0.569255i	$0.192768\pi$
$3$			3.41553i			1.13851i	−0.822161	+	0.569255i	$0.807232\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.558647			0.0798067			0.0399033	−	0.999204i	$-0.487295\pi$
$7$	0.558647			0.0798067			0.0399033	+	0.999204i	$0.487295\pi$
$8$		0			0
$9$	−2.66584			−0.296204
$10$		0			0
$11$	2.82054	+	10.6322i	0.256413	+	0.966567i
$11$	2.82054	+	10.6322i	0.256413	+	0.966567i
$12$		0			0
$13$	−18.1072			−1.39286			−0.696432	−	0.717623i	$-0.745230\pi$
$13$	−18.1072			−1.39286			−0.696432	+	0.717623i	$0.745230\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	32.0402			1.88472			0.942359	−	0.334605i	$-0.108603\pi$
$17$	32.0402			1.88472			0.942359	+	0.334605i	$0.108603\pi$
$18$		0			0
$19$			15.7408i			0.828462i	0.910172	+	0.414231i	$0.135949\pi$
$19$			15.7408i			0.828462i	−0.910172	+	0.414231i	$0.864051\pi$
$20$		0			0
$21$			1.90807i			0.0908607i
$22$		0			0
$23$		−	31.7468i		−	1.38030i	−0.723667	−	0.690149i	$-0.757545\pi$
$23$		−	31.7468i		−	1.38030i	0.723667	−	0.690149i	$-0.242455\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			21.6345i			0.801278i
$28$		0			0
$29$			48.3396i			1.66688i	0.552608	+	0.833442i	$0.313633\pi$
$29$			48.3396i			1.66688i	−0.552608	+	0.833442i	$0.686367\pi$
$30$		0			0
$31$	−43.3557			−1.39857			−0.699286	−	0.714842i	$-0.746498\pi$
$31$	−43.3557			−1.39857			−0.699286	+	0.714842i	$0.746498\pi$
$32$		0			0
$33$	−36.3147	+	9.63363i	−1.10045	+	0.291928i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			21.4366i			0.579367i	0.957123	+	0.289683i	$0.0935500\pi$
$37$			21.4366i			0.579367i	−0.957123	+	0.289683i	$0.906450\pi$
$38$		0			0
$39$		−	61.8458i		−	1.58579i
$40$		0			0
$41$			38.9133i			0.949106i	0.880227	+	0.474553i	$0.157390\pi$
$41$			38.9133i			0.949106i	−0.880227	+	0.474553i	$0.842610\pi$
$42$		0			0
$43$	−23.3044			−0.541964			−0.270982	−	0.962584i	$-0.587348\pi$
$43$	−23.3044			−0.541964			−0.270982	+	0.962584i	$0.587348\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	75.2975i		−	1.60207i	−0.598614	−	0.801037i	$-0.704282\pi$
$47$		−	75.2975i		−	1.60207i	0.598614	−	0.801037i	$-0.295718\pi$
$48$		0			0
$49$	−48.6879			−0.993631
$50$		0			0
$51$			109.434i			2.14577i
$52$		0			0
$53$			8.43533i			0.159157i	0.996829	+	0.0795786i	$0.0253575\pi$
$53$			8.43533i			0.159157i	−0.996829	+	0.0795786i	$0.974643\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−53.7631			−0.943212
$58$		0			0
$59$	29.6094			0.501854			0.250927	−	0.968006i	$-0.419265\pi$
$59$	29.6094			0.501854			0.250927	+	0.968006i	$0.419265\pi$
$60$		0			0
$61$			64.1200i			1.05115i	0.850748	+	0.525573i	$0.176149\pi$
$61$			64.1200i			1.05115i	−0.850748	+	0.525573i	$0.823851\pi$
$62$		0			0
$63$	−1.48926			−0.0236391
$64$		0			0
$65$		0			0
$66$		0			0
$67$			18.8211i			0.280912i	0.990087	+	0.140456i	$0.0448569\pi$
$67$			18.8211i			0.280912i	−0.990087	+	0.140456i	$0.955143\pi$
$68$		0			0
$69$	108.432			1.57148
$70$		0			0
$71$	94.8085			1.33533			0.667666	−	0.744461i	$-0.267294\pi$
$71$	94.8085			1.33533			0.667666	+	0.744461i	$0.267294\pi$
$72$		0			0
$73$	0.945835			0.0129566			0.00647832	−	0.999979i	$-0.497938\pi$
$73$	0.945835			0.0129566			0.00647832	+	0.999979i	$0.497938\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.57568	+	5.93967i	0.0204634	+	0.0771385i
$78$		0			0
$79$		−	73.2825i		−	0.927626i	−0.885933	−	0.463813i	$-0.846481\pi$
$79$		−	73.2825i		−	0.927626i	0.885933	−	0.463813i	$-0.153519\pi$
$80$		0			0
$81$	−97.8859			−1.20847
$82$		0			0
$83$	−161.659			−1.94770			−0.973850	−	0.227192i	$-0.927045\pi$
$83$	−161.659			−1.94770			−0.973850	+	0.227192i	$0.927045\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−165.105			−1.89776
$88$		0			0
$89$	−91.5556			−1.02871			−0.514357	−	0.857576i	$-0.671969\pi$
$89$	−91.5556			−1.02871			−0.514357	+	0.857576i	$0.671969\pi$
$90$		0			0
$91$	−10.1155			−0.111160
$92$		0			0
$93$		−	148.083i		−	1.59229i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	33.1491i		−	0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$97$		−	33.1491i		−	0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$98$		0			0
$99$	−7.51910	−	28.3438i	−0.0759505	−	0.286301i
$100$		0			0
$101$		−	42.0879i		−	0.416712i	−0.978053	−	0.208356i	$-0.933189\pi$
$101$		−	42.0879i		−	0.416712i	0.978053	−	0.208356i	$-0.0668112\pi$
$102$		0			0
$103$			75.9577i			0.737453i	0.929538	+	0.368726i	$0.120206\pi$
$103$			75.9577i			0.737453i	−0.929538	+	0.368726i	$0.879794\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−36.5864			−0.341929			−0.170965	−	0.985277i	$-0.554688\pi$
$107$	−36.5864			−0.341929			−0.170965	+	0.985277i	$0.554688\pi$
$108$		0			0
$109$		−	21.7920i		−	0.199927i	−0.994991	−	0.0999633i	$-0.968127\pi$
$109$		−	21.7920i		−	0.199927i	0.994991	−	0.0999633i	$-0.0318725\pi$
$110$		0			0
$111$	−73.2172			−0.659615
$112$		0			0
$113$			47.1444i			0.417207i	0.978000	+	0.208604i	$0.0668919\pi$
$113$			47.1444i			0.417207i	−0.978000	+	0.208604i	$0.933108\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	48.2710			0.412572
$118$		0			0
$119$	17.8991			0.150413
$120$		0			0
$121$	−105.089	+	59.9773i	−0.868505	+	0.495680i
$122$		0			0
$123$	−132.910			−1.08057
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−157.516			−1.24028			−0.620141	−	0.784490i	$-0.712925\pi$
$127$	−157.516			−1.24028			−0.620141	+	0.784490i	$0.712925\pi$
$128$		0			0
$129$		−	79.5970i		−	0.617031i
$130$		0			0
$131$		−	106.106i		−	0.809966i	−0.914324	−	0.404983i	$-0.867277\pi$
$131$		−	106.106i		−	0.809966i	0.914324	−	0.404983i	$-0.132723\pi$
$132$		0			0
$133$			8.79353i			0.0661168i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			87.9617i			0.642057i	0.947070	+	0.321028i	$0.104028\pi$
$137$			87.9617i			0.642057i	−0.947070	+	0.321028i	$0.895972\pi$
$138$		0			0
$139$			24.4159i			0.175654i	0.996136	+	0.0878268i	$0.0279922\pi$
$139$			24.4159i			0.175654i	−0.996136	+	0.0878268i	$0.972008\pi$
$140$		0			0
$141$	257.181			1.82398
$142$		0			0
$143$	−51.0722	−	192.521i	−0.357148	−	1.34630i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	166.295i		−	1.13126i
$148$		0			0
$149$			186.092i			1.24894i	0.781050	+	0.624469i	$0.214685\pi$
$149$			186.092i			1.24894i	−0.781050	+	0.624469i	$0.785315\pi$
$150$		0			0
$151$			5.26044i			0.0348374i	0.999848	+	0.0174187i	$0.00554482\pi$
$151$			5.26044i			0.0348374i	−0.999848	+	0.0174187i	$0.994455\pi$
$152$		0			0
$153$	−85.4140			−0.558261
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	220.052i		−	1.40161i	−0.713354	−	0.700804i	$-0.752825\pi$
$157$		−	220.052i		−	1.40161i	0.713354	−	0.700804i	$-0.247175\pi$
$158$		0			0
$159$	−28.8111			−0.181202
$160$		0			0
$161$		−	17.7353i		−	0.110157i
$162$		0			0
$163$			11.9134i			0.0730881i	0.999332	+	0.0365441i	$0.0116349\pi$
$163$			11.9134i			0.0730881i	−0.999332	+	0.0365441i	$0.988365\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.5984			0.0634633			0.0317317	−	0.999496i	$-0.489898\pi$
$167$	10.5984			0.0634633			0.0317317	+	0.999496i	$0.489898\pi$
$168$		0			0
$169$	158.872			0.940071
$170$		0			0
$171$		−	41.9624i		−	0.245394i
$172$		0			0
$173$	−181.818			−1.05097			−0.525485	−	0.850803i	$-0.676116\pi$
$173$	−181.818			−1.05097			−0.525485	+	0.850803i	$0.676116\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			101.132i			0.571366i
$178$		0			0
$179$	−72.6441			−0.405833			−0.202916	−	0.979196i	$-0.565042\pi$
$179$	−72.6441			−0.405833			−0.202916	+	0.979196i	$0.565042\pi$
$180$		0			0
$181$	325.085			1.79605			0.898024	−	0.439946i	$-0.145003\pi$
$181$	325.085			1.79605			0.898024	+	0.439946i	$0.145003\pi$
$182$		0			0
$183$	−219.004			−1.19674
$184$		0			0
$185$		0			0
$186$		0			0
$187$	90.3706	+	340.659i	0.483265	+	1.82171i
$188$		0			0
$189$			12.0860i			0.0639473i
$190$		0			0
$191$	−199.153			−1.04268			−0.521342	−	0.853348i	$-0.674568\pi$
$191$	−199.153			−1.04268			−0.521342	+	0.853348i	$0.674568\pi$
$192$		0			0
$193$	284.097			1.47201			0.736003	−	0.676978i	$-0.236711\pi$
$193$	284.097			1.47201			0.736003	+	0.676978i	$0.236711\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	230.008			1.16755			0.583777	−	0.811914i	$-0.301574\pi$
$197$	230.008			1.16755			0.583777	+	0.811914i	$0.301574\pi$
$198$		0			0
$199$	123.328			0.619740			0.309870	−	0.950779i	$-0.399714\pi$
$199$	123.328			0.619740			0.309870	+	0.950779i	$0.399714\pi$
$200$		0			0
$201$	−64.2841			−0.319822
$202$		0			0
$203$			27.0048i			0.133028i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			84.6320i			0.408850i
$208$		0			0
$209$	−167.360	+	44.3975i	−0.800764	+	0.212428i
$210$		0			0
$211$			214.886i			1.01842i	0.860643	+	0.509209i	$0.170062\pi$
$211$			214.886i			1.01842i	−0.860643	+	0.509209i	$0.829938\pi$
$212$		0			0
$213$			323.821i			1.52029i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−24.2205			−0.111615
$218$		0			0
$219$			3.23053i			0.0147513i
$220$		0			0
$221$	−580.159			−2.62516
$222$		0			0
$223$		−	59.8548i		−	0.268407i	−0.990954	−	0.134204i	$-0.957152\pi$
$223$		−	59.8548i		−	0.268407i	0.990954	−	0.134204i	$-0.0428476\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−203.787			−0.897740			−0.448870	−	0.893597i	$-0.648173\pi$
$227$	−203.787			−0.897740			−0.448870	+	0.893597i	$0.648173\pi$
$228$		0			0
$229$	−265.375			−1.15884			−0.579421	−	0.815028i	$-0.696721\pi$
$229$	−265.375			−1.15884			−0.579421	+	0.815028i	$0.696721\pi$
$230$		0			0
$231$	−20.2871	+	5.38180i	−0.0878230	+	0.0232978i
$232$		0			0
$233$	313.018			1.34343			0.671713	−	0.740812i	$-0.265559\pi$
$233$	313.018			1.34343			0.671713	+	0.740812i	$0.265559\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	250.298			1.05611
$238$		0			0
$239$			159.796i			0.668604i	0.942466	+	0.334302i	$0.108501\pi$
$239$			159.796i			0.668604i	−0.942466	+	0.334302i	$0.891499\pi$
$240$		0			0
$241$			467.884i			1.94143i	0.240242	+	0.970713i	$0.422773\pi$
$241$			467.884i			1.94143i	−0.240242	+	0.970713i	$0.577227\pi$
$242$		0			0
$243$		−	139.621i		−	0.574574i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	285.022i		−	1.15393i
$248$		0			0
$249$		−	552.151i		−	2.21748i
$250$		0			0
$251$	312.122			1.24351			0.621757	−	0.783210i	$-0.286419\pi$
$251$	312.122			1.24351			0.621757	+	0.783210i	$0.286419\pi$
$252$		0			0
$253$	337.540	−	89.5432i	1.33415	−	0.353926i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			139.735i			0.543714i	0.962338	+	0.271857i	$0.0876378\pi$
$257$			139.735i			0.543714i	−0.962338	+	0.271857i	$0.912362\pi$
$258$		0			0
$259$			11.9755i			0.0462373i
$260$		0			0
$261$		−	128.866i		−	0.493738i
$262$		0			0
$263$	181.115			0.688651			0.344326	−	0.938850i	$-0.388108\pi$
$263$	181.115			0.688651			0.344326	+	0.938850i	$0.388108\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	312.711i		−	1.17120i
$268$		0			0
$269$	96.4486			0.358545			0.179273	−	0.983799i	$-0.442626\pi$
$269$	96.4486			0.358545			0.179273	+	0.983799i	$0.442626\pi$
$270$		0			0
$271$		−	296.758i		−	1.09505i	−0.836790	−	0.547525i	$-0.815570\pi$
$271$		−	296.758i		−	1.09505i	0.836790	−	0.547525i	$-0.184430\pi$
$272$		0			0
$273$		−	34.5499i		−	0.126557i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	165.044			0.595826			0.297913	−	0.954593i	$-0.403709\pi$
$277$	165.044			0.595826			0.297913	+	0.954593i	$0.403709\pi$
$278$		0			0
$279$	115.579			0.414263
$280$		0			0
$281$			229.230i			0.815765i	0.913034	+	0.407883i	$0.133733\pi$
$281$			229.230i			0.815765i	−0.913034	+	0.407883i	$0.866267\pi$
$282$		0			0
$283$	133.408			0.471407			0.235703	−	0.971825i	$-0.424261\pi$
$283$	133.408			0.471407			0.235703	+	0.971825i	$0.424261\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			21.7388i			0.0757449i
$288$		0			0
$289$	737.574			2.55216
$290$		0			0
$291$	113.222			0.389078
$292$		0			0
$293$	141.969			0.484535			0.242268	−	0.970209i	$-0.422109\pi$
$293$	141.969			0.484535			0.242268	+	0.970209i	$0.422109\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−230.023	+	61.0210i	−0.774489	+	0.205458i
$298$		0			0
$299$			574.848i			1.92257i
$300$		0			0
$301$	−13.0189			−0.0432523
$302$		0			0
$303$	143.752			0.474430
$304$		0			0
$305$		0			0
$306$		0			0
$307$	405.167			1.31976			0.659881	−	0.751370i	$-0.270607\pi$
$307$	405.167			1.31976			0.659881	+	0.751370i	$0.270607\pi$
$308$		0			0
$309$	−259.436			−0.839597
$310$		0			0
$311$	518.227			1.66633			0.833163	−	0.553028i	$-0.186528\pi$
$311$	518.227			1.66633			0.833163	+	0.553028i	$0.186528\pi$
$312$		0			0
$313$			177.327i			0.566539i	0.959040	+	0.283270i	$0.0914191\pi$
$313$			177.327i			0.566539i	−0.959040	+	0.283270i	$0.908581\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			509.414i			1.60698i	0.595315	+	0.803492i	$0.297027\pi$
$317$			509.414i			1.60698i	−0.595315	+	0.803492i	$0.702973\pi$
$318$		0			0
$319$	−513.958	+	136.344i	−1.61115	+	0.427410i
$320$		0			0
$321$		−	124.962i		−	0.389290i
$322$		0			0
$323$			504.337i			1.56142i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	74.4312			0.227618
$328$		0			0
$329$		−	42.0647i		−	0.127856i
$330$		0			0
$331$	−360.510			−1.08915			−0.544577	−	0.838711i	$-0.683310\pi$
$331$	−360.510			−1.08915			−0.544577	+	0.838711i	$0.683310\pi$
$332$		0			0
$333$		−	57.1464i		−	0.171611i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	419.994			1.24627			0.623136	−	0.782113i	$-0.285858\pi$
$337$	419.994			1.24627			0.623136	+	0.782113i	$0.285858\pi$
$338$		0			0
$339$	−161.023			−0.474995
$340$		0			0
$341$	−122.286	−	460.968i	−0.358611	−	1.35181i
$342$		0			0
$343$	−54.5730			−0.159105
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−278.399			−0.802303			−0.401151	−	0.916012i	$-0.631390\pi$
$347$	−278.399			−0.802303			−0.401151	+	0.916012i	$0.631390\pi$
$348$		0			0
$349$		−	426.489i		−	1.22203i	−0.791618	−	0.611016i	$-0.790761\pi$
$349$		−	426.489i		−	1.22203i	0.791618	−	0.611016i	$-0.209239\pi$
$350$		0			0
$351$		−	391.741i		−	1.11607i
$352$		0			0
$353$			114.688i			0.324894i	0.986717	+	0.162447i	$0.0519387\pi$
$353$			114.688i			0.324894i	−0.986717	+	0.162447i	$0.948061\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			61.1351i			0.171247i
$358$		0			0
$359$			545.543i			1.51962i	0.650147	+	0.759809i	$0.274707\pi$
$359$			545.543i			1.51962i	−0.650147	+	0.759809i	$0.725293\pi$
$360$		0			0
$361$	113.228			0.313651
$362$		0			0
$363$	−204.854	−	358.935i	−0.564337	−	0.988802i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			265.619i			0.723759i	0.932225	+	0.361879i	$0.117865\pi$
$367$			265.619i			0.723759i	−0.932225	+	0.361879i	$0.882135\pi$
$368$		0			0
$369$		−	103.737i		−	0.281129i
$370$		0			0
$371$			4.71237i			0.0127018i
$372$		0			0
$373$	202.250			0.542225			0.271113	−	0.962548i	$-0.412608\pi$
$373$	202.250			0.542225			0.271113	+	0.962548i	$0.412608\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	875.297i		−	2.32174i
$378$		0			0
$379$	−114.650			−0.302506			−0.151253	−	0.988495i	$-0.548331\pi$
$379$	−114.650			−0.302506			−0.151253	+	0.988495i	$0.548331\pi$
$380$		0			0
$381$		−	538.000i		−	1.41207i
$382$		0			0
$383$			373.418i			0.974982i	0.873128	+	0.487491i	$0.162088\pi$
$383$			373.418i			0.974982i	−0.873128	+	0.487491i	$0.837912\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	62.1259			0.160532
$388$		0			0
$389$	−241.139			−0.619896			−0.309948	−	0.950754i	$-0.600312\pi$
$389$	−241.139			−0.619896			−0.309948	+	0.950754i	$0.600312\pi$
$390$		0			0
$391$		−	1017.18i		−	2.60147i
$392$		0			0
$393$	362.406			0.922154
$394$		0			0
$395$		0			0
$396$		0			0
$397$			467.321i			1.17713i	0.808449	+	0.588566i	$0.200307\pi$
$397$			467.321i			1.17713i	−0.808449	+	0.588566i	$0.799693\pi$
$398$		0			0
$399$	−30.0346			−0.0752746
$400$		0			0
$401$	680.945			1.69812			0.849059	−	0.528298i	$-0.177170\pi$
$401$	680.945			1.69812			0.849059	+	0.528298i	$0.177170\pi$
$402$		0			0
$403$	785.052			1.94802
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−227.919	+	60.4627i	−0.559997	+	0.148557i
$408$		0			0
$409$		−	599.500i		−	1.46577i	−0.680353	−	0.732885i	$-0.738173\pi$
$409$		−	599.500i		−	1.46577i	0.680353	−	0.732885i	$-0.261827\pi$
$410$		0			0
$411$	−300.436			−0.730988
$412$		0			0
$413$	16.5412			0.0400513
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−83.3931			−0.199983
$418$		0			0
$419$	256.808			0.612906			0.306453	−	0.951886i	$-0.400858\pi$
$419$	256.808			0.612906			0.306453	+	0.951886i	$0.400858\pi$
$420$		0			0
$421$	23.4654			0.0557373			0.0278686	−	0.999612i	$-0.491128\pi$
$421$	23.4654			0.0557373			0.0278686	+	0.999612i	$0.491128\pi$
$422$		0			0
$423$			200.731i			0.474541i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			35.8204i			0.0838885i
$428$		0			0
$429$	657.559	−	174.438i	1.53277	−	0.406616i
$430$		0			0
$431$		−	228.370i		−	0.529860i	−0.964268	−	0.264930i	$-0.914651\pi$
$431$		−	228.370i		−	0.529860i	0.964268	−	0.264930i	$-0.0853488\pi$
$432$		0			0
$433$		−	564.969i		−	1.30478i	−0.757884	−	0.652389i	$-0.773767\pi$
$433$		−	564.969i		−	1.30478i	0.757884	−	0.652389i	$-0.226233\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	499.720			1.14352
$438$		0			0
$439$			500.617i			1.14036i	0.821521	+	0.570178i	$0.193126\pi$
$439$			500.617i			1.14036i	−0.821521	+	0.570178i	$0.806874\pi$
$440$		0			0
$441$	129.794			0.294318
$442$		0			0
$443$		−	365.948i		−	0.826069i	−0.910715	−	0.413034i	$-0.864469\pi$
$443$		−	365.948i		−	0.826069i	0.910715	−	0.413034i	$-0.135531\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−635.602			−1.42193
$448$		0			0
$449$	316.528			0.704963			0.352481	−	0.935819i	$-0.385338\pi$
$449$	316.528			0.704963			0.352481	+	0.935819i	$0.385338\pi$
$450$		0			0
$451$	−413.736	+	109.757i	−0.917375	+	0.243363i
$452$		0			0
$453$	−17.9672			−0.0396627
$454$		0			0
$455$		0			0
$456$		0			0
$457$	291.964			0.638870			0.319435	−	0.947608i	$-0.396507\pi$
$457$	291.964			0.638870			0.319435	+	0.947608i	$0.396507\pi$
$458$		0			0
$459$			693.174i			1.51018i
$460$		0			0
$461$		−	431.260i		−	0.935489i	−0.883864	−	0.467744i	$-0.845067\pi$
$461$		−	431.260i		−	0.935489i	0.883864	−	0.467744i	$-0.154933\pi$
$462$		0			0
$463$		−	264.022i		−	0.570243i	−0.958491	−	0.285121i	$-0.907966\pi$
$463$		−	264.022i		−	0.570243i	0.958491	−	0.285121i	$-0.0920339\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	477.602i		−	1.02270i	−0.859372	−	0.511351i	$-0.829145\pi$
$467$		−	477.602i		−	1.02270i	0.859372	−	0.511351i	$-0.170855\pi$
$468$		0			0
$469$			10.5144i			0.0224187i
$470$		0			0
$471$	751.595			1.59574
$472$		0			0
$473$	−65.7311	−	247.778i	−0.138966	−	0.523844i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	22.4872i		−	0.0471430i
$478$		0			0
$479$		−	279.214i		−	0.582909i	−0.956585	−	0.291455i	$-0.905861\pi$
$479$		−	279.214i		−	0.582909i	0.956585	−	0.291455i	$-0.0941393\pi$
$480$		0			0
$481$		−	388.157i		−	0.806979i
$482$		0			0
$483$	60.5753			0.125415
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	279.655i		−	0.574241i	−0.957894	−	0.287121i	$-0.907302\pi$
$487$		−	279.655i		−	0.574241i	0.957894	−	0.287121i	$-0.0926980\pi$
$488$		0			0
$489$	−40.6904			−0.0832115
$490$		0			0
$491$			442.142i			0.900493i	0.892904	+	0.450246i	$0.148664\pi$
$491$			442.142i			0.900493i	−0.892904	+	0.450246i	$0.851336\pi$
$492$		0			0
$493$			1548.81i			3.14160i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	52.9645			0.106568
$498$		0			0
$499$	−430.853			−0.863432			−0.431716	−	0.902010i	$-0.642092\pi$
$499$	−430.853			−0.863432			−0.431716	+	0.902010i	$0.642092\pi$
$500$		0			0
$501$			36.1991i			0.0722536i
$502$		0			0
$503$	293.745			0.583985			0.291993	−	0.956421i	$-0.405682\pi$
$503$	293.745			0.583985			0.291993	+	0.956421i	$0.405682\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			542.632i			1.07028i
$508$		0			0
$509$	−276.608			−0.543434			−0.271717	−	0.962377i	$-0.587592\pi$
$509$	−276.608			−0.543434			−0.271717	+	0.962377i	$0.587592\pi$
$510$		0			0
$511$	0.528388			0.00103403
$512$		0			0
$513$	−340.544			−0.663828
$514$		0			0
$515$		0			0
$516$		0			0
$517$	800.581	−	212.380i	1.54851	−	0.410792i
$518$		0			0
$519$		−	621.004i		−	1.19654i
$520$		0			0
$521$	571.383			1.09670			0.548352	−	0.836248i	$-0.315255\pi$
$521$	571.383			1.09670			0.548352	+	0.836248i	$0.315255\pi$
$522$		0			0
$523$	−255.631			−0.488778			−0.244389	−	0.969677i	$-0.578587\pi$
$523$	−255.631			−0.488778			−0.244389	+	0.969677i	$0.578587\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1389.13			−2.63591
$528$		0			0
$529$	−478.862			−0.905222
$530$		0			0
$531$	−78.9339			−0.148651
$532$		0			0
$533$		−	704.613i		−	1.32198i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	248.118i		−	0.462045i
$538$		0			0
$539$	−137.326	−	517.662i	−0.254780	−	0.960411i
$540$		0			0
$541$		−	214.227i		−	0.395983i	−0.980204	−	0.197991i	$-0.936558\pi$
$541$		−	214.227i		−	0.395983i	0.980204	−	0.197991i	$-0.0634418\pi$
$542$		0			0
$543$			1110.34i			2.04482i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−552.014			−1.00917			−0.504584	−	0.863363i	$-0.668354\pi$
$547$	−552.014			−1.00917			−0.504584	+	0.863363i	$0.668354\pi$
$548$		0			0
$549$		−	170.933i		−	0.311354i
$550$		0			0
$551$	−760.903			−1.38095
$552$		0			0
$553$		−	40.9390i		−	0.0740308i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−159.593			−0.286523			−0.143261	−	0.989685i	$-0.545759\pi$
$557$	−159.593			−0.286523			−0.143261	+	0.989685i	$0.545759\pi$
$558$		0			0
$559$	421.979			0.754882
$560$		0			0
$561$	−1163.53	+	308.663i	−2.07403	+	0.550202i
$562$		0			0
$563$	603.391			1.07174			0.535871	−	0.844300i	$-0.319983\pi$
$563$	603.391			1.07174			0.535871	+	0.844300i	$0.319983\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−54.6836			−0.0964437
$568$		0			0
$569$		−	228.804i		−	0.402115i	−0.979579	−	0.201058i	$-0.935562\pi$
$569$		−	228.804i		−	0.402115i	0.979579	−	0.201058i	$-0.0644379\pi$
$570$		0			0
$571$			682.868i			1.19592i	0.801527	+	0.597958i	$0.204021\pi$
$571$			682.868i			1.19592i	−0.801527	+	0.597958i	$0.795979\pi$
$572$		0			0
$573$		−	680.212i		−	1.18711i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			486.375i			0.842937i	0.906843	+	0.421468i	$0.138485\pi$
$577$			486.375i			0.842937i	−0.906843	+	0.421468i	$0.861515\pi$
$578$		0			0
$579$			970.342i			1.67589i
$580$		0			0
$581$	−90.3103			−0.155439
$582$		0			0
$583$	−89.6864	+	23.7922i	−0.153836	+	0.0408099i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			411.640i			0.701260i	0.936514	+	0.350630i	$0.114033\pi$
$587$			411.640i			0.701260i	−0.936514	+	0.350630i	$0.885967\pi$
$588$		0			0
$589$		−	682.452i		−	1.15866i
$590$		0			0
$591$			785.599i			1.32927i
$592$		0			0
$593$	−495.443			−0.835486			−0.417743	−	0.908565i	$-0.637179\pi$
$593$	−495.443			−0.835486			−0.417743	+	0.908565i	$0.637179\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			421.231i			0.705580i
$598$		0			0
$599$	482.567			0.805621			0.402811	−	0.915283i	$-0.368033\pi$
$599$	482.567			0.805621			0.402811	+	0.915283i	$0.368033\pi$
$600$		0			0
$601$			1171.94i			1.94999i	0.222237	+	0.974993i	$0.428664\pi$
$601$			1171.94i			1.94999i	−0.222237	+	0.974993i	$0.571336\pi$
$602$		0			0
$603$		−	50.1741i		−	0.0832075i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−22.6334			−0.0372873			−0.0186437	−	0.999826i	$-0.505935\pi$
$607$	−22.6334			−0.0372873			−0.0186437	+	0.999826i	$0.505935\pi$
$608$		0			0
$609$	−92.2355			−0.151454
$610$		0			0
$611$			1363.43i			2.23147i
$612$		0			0
$613$	257.299			0.419738			0.209869	−	0.977730i	$-0.432696\pi$
$613$	257.299			0.419738			0.209869	+	0.977730i	$0.432696\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	250.845i		−	0.406555i	−0.979121	−	0.203278i	$-0.934841\pi$
$617$		−	250.845i		−	0.406555i	0.979121	−	0.203278i	$-0.0651594\pi$
$618$		0			0
$619$	454.630			0.734459			0.367230	−	0.930130i	$-0.380306\pi$
$619$	454.630			0.734459			0.367230	+	0.930130i	$0.380306\pi$
$620$		0			0
$621$	686.828			1.10600
$622$		0			0
$623$	−51.1472			−0.0820982
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−151.641	−	571.622i	−0.241851	−	0.911678i
$628$		0			0
$629$			686.832i			1.09194i
$630$		0			0
$631$	1097.79			1.73977			0.869883	−	0.493257i	$-0.164194\pi$
$631$	1097.79			1.73977			0.869883	+	0.493257i	$0.164194\pi$
$632$		0			0
$633$	−733.950			−1.15948
$634$		0			0
$635$		0			0
$636$		0			0
$637$	881.604			1.38399
$638$		0			0
$639$	−252.744			−0.395531
$640$		0			0
$641$	−931.874			−1.45378			−0.726891	−	0.686753i	$-0.759035\pi$
$641$	−931.874			−1.45378			−0.726891	+	0.686753i	$0.759035\pi$
$642$		0			0
$643$			462.937i			0.719963i	0.932959	+	0.359982i	$0.117217\pi$
$643$			462.937i			0.719963i	−0.932959	+	0.359982i	$0.882783\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	337.698i		−	0.521944i	−0.965346	−	0.260972i	$-0.915957\pi$
$647$		−	337.698i		−	0.521944i	0.965346	−	0.260972i	$-0.0840430\pi$
$648$		0			0
$649$	83.5144	+	314.814i	0.128682	+	0.485076i
$650$		0			0
$651$		−	82.7259i		−	0.127075i
$652$		0			0
$653$		−	237.239i		−	0.363307i	−0.983363	−	0.181653i	$-0.941855\pi$
$653$		−	237.239i		−	0.363307i	0.983363	−	0.181653i	$-0.0581449\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−2.52144			−0.00383782
$658$		0			0
$659$		−	482.840i		−	0.732686i	−0.930480	−	0.366343i	$-0.880610\pi$
$659$		−	482.840i		−	0.732686i	0.930480	−	0.366343i	$-0.119390\pi$
$660$		0			0
$661$	−649.514			−0.982624			−0.491312	−	0.870984i	$-0.663483\pi$
$661$	−649.514			−0.982624			−0.491312	+	0.870984i	$0.663483\pi$
$662$		0			0
$663$		−	1981.55i		−	2.98876i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1534.63			2.30079
$668$		0			0
$669$	204.436			0.305584
$670$		0			0
$671$	−681.739	+	180.853i	−1.01600	+	0.269527i
$672$		0			0
$673$	−1077.34			−1.60080			−0.800400	−	0.599467i	$-0.795379\pi$
$673$	−1077.34			−1.60080			−0.800400	+	0.599467i	$0.795379\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	82.8635			0.122398			0.0611990	−	0.998126i	$-0.480508\pi$
$677$	82.8635			0.122398			0.0611990	+	0.998126i	$0.480508\pi$
$678$		0			0
$679$		−	18.5186i		−	0.0272734i
$680$		0			0
$681$		−	696.041i		−	1.02209i
$682$		0			0
$683$		−	388.525i		−	0.568851i	−0.958698	−	0.284426i	$-0.908197\pi$
$683$		−	388.525i		−	0.568851i	0.958698	−	0.284426i	$-0.0918029\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	906.396i		−	1.31935i
$688$		0			0
$689$		−	152.740i		−	0.221684i
$690$		0			0
$691$	764.841			1.10686			0.553430	−	0.832896i	$-0.313319\pi$
$691$	764.841			1.10686			0.553430	+	0.832896i	$0.313319\pi$
$692$		0			0
$693$	−4.20052	−	15.8342i	−0.00606136	−	0.0228488i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			1246.79i			1.78880i
$698$		0			0
$699$			1069.12i			1.52950i
$700$		0			0
$701$			229.490i			0.327375i	0.986512	+	0.163688i	$0.0523389\pi$
$701$			229.490i			0.327375i	−0.986512	+	0.163688i	$0.947661\pi$
$702$		0			0
$703$	−337.428			−0.479983
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	23.5122i		−	0.0332564i
$708$		0			0
$709$	197.960			0.279210			0.139605	−	0.990207i	$-0.455417\pi$
$709$	197.960			0.279210			0.139605	+	0.990207i	$0.455417\pi$
$710$		0			0
$711$			195.359i			0.274767i
$712$		0			0
$713$			1376.41i			1.93045i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−545.789			−0.761213
$718$		0			0
$719$	−955.392			−1.32878			−0.664390	−	0.747386i	$-0.731308\pi$
$719$	−955.392			−1.32878			−0.664390	+	0.747386i	$0.731308\pi$
$720$		0			0
$721$			42.4335i			0.0588537i
$722$		0			0
$723$	−1598.07			−2.21033
$724$		0			0
$725$		0			0
$726$		0			0
$727$			149.231i			0.205270i	0.994719	+	0.102635i	$0.0327273\pi$
$727$			149.231i			0.205270i	−0.994719	+	0.102635i	$0.967273\pi$
$728$		0			0
$729$	−404.092			−0.554310
$730$		0			0
$731$	−746.679			−1.02145
$732$		0			0
$733$	−884.031			−1.20604			−0.603022	−	0.797724i	$-0.706037\pi$
$733$	−884.031			−1.20604			−0.603022	+	0.797724i	$0.706037\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−200.111	+	53.0857i	−0.271521	+	0.0720295i
$738$		0			0
$739$			185.448i			0.250944i	0.992097	+	0.125472i	$0.0400445\pi$
$739$			185.448i			0.250944i	−0.992097	+	0.125472i	$0.959955\pi$
$740$		0			0
$741$	973.501			1.31377
$742$		0			0
$743$	−114.936			−0.154691			−0.0773457	−	0.997004i	$-0.524645\pi$
$743$	−114.936			−0.154691			−0.0773457	+	0.997004i	$0.524645\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	430.957			0.576917
$748$		0			0
$749$	−20.4389			−0.0272882
$750$		0			0
$751$	−440.004			−0.585891			−0.292945	−	0.956129i	$-0.594635\pi$
$751$	−440.004			−0.585891			−0.292945	+	0.956129i	$0.594635\pi$
$752$		0			0
$753$			1066.06i			1.41575i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	434.602i		−	0.574112i	−0.957914	−	0.287056i	$-0.907323\pi$
$757$		−	434.602i		−	0.574112i	0.957914	−	0.287056i	$-0.0926765\pi$
$758$		0			0
$759$	305.837	+	1152.88i	0.402948	+	1.51894i
$760$		0			0
$761$			1172.82i			1.54116i	0.637346	+	0.770578i	$0.280032\pi$
$761$			1172.82i			1.54116i	−0.637346	+	0.770578i	$0.719968\pi$
$762$		0			0
$763$		−	12.1740i		−	0.0159555i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−536.144			−0.699015
$768$		0			0
$769$		−	947.915i		−	1.23266i	−0.787488	−	0.616330i	$-0.788619\pi$
$769$		−	947.915i		−	1.23266i	0.787488	−	0.616330i	$-0.211381\pi$
$770$		0			0
$771$	−477.268			−0.619024
$772$		0			0
$773$		−	1281.52i		−	1.65785i	−0.559359	−	0.828925i	$-0.688953\pi$
$773$		−	1281.52i		−	1.65785i	0.559359	−	0.828925i	$-0.311047\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−40.9026			−0.0526416
$778$		0			0
$779$	−612.526			−0.786298
$780$		0			0
$781$	267.411	+	1008.03i	0.342396	+	1.29069i
$782$		0			0
$783$	−1045.80			−1.33564
$784$		0			0
$785$		0			0
$786$		0			0
$787$	707.698			0.899235			0.449617	−	0.893221i	$-0.351560\pi$
$787$	707.698			0.899235			0.449617	+	0.893221i	$0.351560\pi$
$788$		0			0
$789$			618.604i			0.784036i
$790$		0			0
$791$			26.3371i			0.0332959i
$792$		0			0
$793$		−	1161.04i		−	1.46411i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			241.596i			0.303132i	0.988447	+	0.151566i	$0.0484316\pi$
$797$			241.596i			0.303132i	−0.988447	+	0.151566i	$0.951568\pi$
$798$		0			0
$799$		−	2412.55i		−	3.01946i
$800$		0			0
$801$	244.072			0.304710
$802$		0			0
$803$	2.66777	+	10.0563i	0.00332225	+	0.0125235i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			329.423i			0.408207i
$808$		0			0
$809$			1063.62i			1.31473i	0.753571	+	0.657366i	$0.228330\pi$
$809$			1063.62i			1.31473i	−0.753571	+	0.657366i	$0.771670\pi$
$810$		0			0
$811$		−	976.484i		−	1.20405i	−0.798477	−	0.602025i	$-0.794361\pi$
$811$		−	976.484i		−	1.20405i	0.798477	−	0.602025i	$-0.205639\pi$
$812$		0			0
$813$	1013.59			1.24672
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	366.830i		−	0.448996i
$818$		0			0
$819$	26.9664			0.0329260
$820$		0			0
$821$		−	189.653i		−	0.231003i	−0.993307	−	0.115501i	$-0.963153\pi$
$821$		−	189.653i		−	0.231003i	0.993307	−	0.115501i	$-0.0368475\pi$
$822$		0			0
$823$			155.966i			0.189510i	0.995501	+	0.0947548i	$0.0302067\pi$
$823$			155.966i			0.189510i	−0.995501	+	0.0947548i	$0.969793\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1612.24			−1.94950			−0.974752	−	0.223292i	$-0.928320\pi$
$827$	−1612.24			−1.94950			−0.974752	+	0.223292i	$0.928320\pi$
$828$		0			0
$829$	555.833			0.670486			0.335243	−	0.942132i	$-0.391182\pi$
$829$	555.833			0.670486			0.335243	+	0.942132i	$0.391182\pi$
$830$		0			0
$831$			563.712i			0.678354i
$832$		0			0
$833$	−1559.97			−1.87271
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	937.980i		−	1.12064i
$838$		0			0
$839$	−622.115			−0.741496			−0.370748	−	0.928733i	$-0.620899\pi$
$839$	−622.115			−0.741496			−0.370748	+	0.928733i	$0.620899\pi$
$840$		0			0
$841$	−1495.72			−1.77850
$842$		0			0
$843$	−782.942			−0.928757
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−58.7077	+	33.5061i	−0.0693125	+	0.0395586i
$848$		0			0
$849$			455.659i			0.536701i
$850$		0			0
$851$	680.543			0.799698
$852$		0			0
$853$	1205.37			1.41309			0.706545	−	0.707668i	$-0.250253\pi$
$853$	1205.37			1.41309			0.706545	+	0.707668i	$0.250253\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−884.739			−1.03237			−0.516184	−	0.856478i	$-0.672648\pi$
$857$	−884.739			−1.03237			−0.516184	+	0.856478i	$0.672648\pi$
$858$		0			0
$859$	1362.42			1.58605			0.793024	−	0.609191i	$-0.208506\pi$
$859$	1362.42			1.58605			0.793024	+	0.609191i	$0.208506\pi$
$860$		0			0
$861$	−74.2495			−0.0862364
$862$		0			0
$863$			1254.23i			1.45333i	0.686990	+	0.726667i	$0.258932\pi$
$863$			1254.23i			1.45333i	−0.686990	+	0.726667i	$0.741068\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			2519.20i			2.90566i
$868$		0			0
$869$	779.157	−	206.696i	0.896613	−	0.237855i
$870$		0			0
$871$		−	340.799i		−	0.391273i
$872$		0			0
$873$			88.3701i			0.101226i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−86.1504			−0.0982331			−0.0491165	−	0.998793i	$-0.515641\pi$
$877$	−86.1504			−0.0982331			−0.0491165	+	0.998793i	$0.515641\pi$
$878$		0			0
$879$			484.899i			0.551648i
$880$		0			0
$881$	1011.19			1.14778			0.573888	−	0.818934i	$-0.305434\pi$
$881$	1011.19			1.14778			0.573888	+	0.818934i	$0.305434\pi$
$882$		0			0
$883$		−	442.707i		−	0.501367i	−0.968069	−	0.250683i	$-0.919345\pi$
$883$		−	442.707i		−	0.501367i	0.968069	−	0.250683i	$-0.0806553\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−706.419			−0.796414			−0.398207	−	0.917296i	$-0.630367\pi$
$887$	−706.419			−0.796414			−0.398207	+	0.917296i	$0.630367\pi$
$888$		0			0
$889$	−87.9957			−0.0989828
$890$		0			0
$891$	−276.091	−	1040.75i	−0.309866	−	1.16807i
$892$		0			0
$893$	1185.24			1.32726
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1963.41			−2.18886
$898$		0			0
$899$		−	2095.80i		−	2.33126i
$900$		0			0
$901$			270.270i			0.299966i
$902$		0			0
$903$		−	44.4666i		−	0.0492432i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	963.841i		−	1.06267i	−0.847162	−	0.531335i	$-0.821691\pi$
$907$		−	963.841i		−	1.06267i	0.847162	−	0.531335i	$-0.178309\pi$
$908$		0			0
$909$			112.199i			0.123432i
$910$		0			0
$911$	859.938			0.943949			0.471975	−	0.881612i	$-0.343541\pi$
$911$	859.938			0.943949			0.471975	+	0.881612i	$0.343541\pi$
$912$		0			0
$913$	−455.966	−	1718.80i	−0.499415	−	1.88258i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	59.2755i		−	0.0646407i
$918$		0			0
$919$		−	884.874i		−	0.962866i	−0.876483	−	0.481433i	$-0.840116\pi$
$919$		−	884.874i		−	0.962866i	0.876483	−	0.481433i	$-0.159884\pi$
$920$		0			0
$921$			1383.86i			1.50256i
$922$		0			0
$923$	−1716.72			−1.85994
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	202.491i		−	0.218437i
$928$		0			0
$929$	537.098			0.578146			0.289073	−	0.957307i	$-0.406653\pi$
$929$	537.098			0.578146			0.289073	+	0.957307i	$0.406653\pi$
$930$		0			0
$931$		−	766.385i		−	0.823185i
$932$		0			0
$933$			1770.02i			1.89713i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1151.99			1.22945			0.614724	−	0.788742i	$-0.289267\pi$
$937$	1151.99			1.22945			0.614724	+	0.788742i	$0.289267\pi$
$938$		0			0
$939$	−605.665			−0.645010
$940$		0			0
$941$		−	1358.32i		−	1.44349i	−0.692161	−	0.721743i	$-0.743341\pi$
$941$		−	1358.32i		−	1.44349i	0.692161	−	0.721743i	$-0.256659\pi$
$942$		0			0
$943$	1235.38			1.31005
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1834.97i		−	1.93767i	−0.247710	−	0.968834i	$-0.579678\pi$
$947$		−	1834.97i		−	1.93767i	0.247710	−	0.968834i	$-0.420322\pi$
$948$		0			0
$949$	−17.1265			−0.0180469
$950$		0			0
$951$	−1739.92			−1.82957
$952$		0			0
$953$	249.548			0.261856			0.130928	−	0.991392i	$-0.458204\pi$
$953$	249.548			0.261856			0.130928	+	0.991392i	$0.458204\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−465.686	−	1755.44i	−0.486610	−	1.83432i
$958$		0			0
$959$			49.1395i			0.0512404i
$960$		0			0
$961$	918.718			0.956002
$962$		0			0
$963$	97.5336			0.101281
$964$		0			0
$965$		0			0
$966$		0			0
$967$	903.841			0.934686			0.467343	−	0.884076i	$-0.345211\pi$
$967$	903.841			0.934686			0.467343	+	0.884076i	$0.345211\pi$
$968$		0			0
$969$	−1722.58			−1.77769
$970$		0			0
$971$	−500.888			−0.515847			−0.257924	−	0.966165i	$-0.583038\pi$
$971$	−500.888			−0.515847			−0.257924	+	0.966165i	$0.583038\pi$
$972$		0			0
$973$			13.6398i			0.0140183i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	549.099i		−	0.562026i	−0.959704	−	0.281013i	$-0.909330\pi$
$977$		−	549.099i		−	0.562026i	0.959704	−	0.281013i	$-0.0906703\pi$
$978$		0			0
$979$	−258.236	−	973.441i	−0.263775	−	0.994322i
$980$		0			0
$981$			58.0940i			0.0592191i
$982$		0			0
$983$			245.607i			0.249855i	0.992166	+	0.124927i	$0.0398698\pi$
$983$			245.607i			0.249855i	−0.992166	+	0.124927i	$0.960130\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	143.673			0.145566
$988$		0			0
$989$			739.842i			0.748071i
$990$		0			0
$991$	1154.02			1.16450			0.582251	−	0.813009i	$-0.302172\pi$
$991$	1154.02			1.16450			0.582251	+	0.813009i	$0.302172\pi$
$992$		0			0
$993$		−	1231.33i		−	1.24001i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1220.05			−1.22372			−0.611861	−	0.790966i	$-0.709579\pi$
$997$	−1220.05			−1.22372			−0.611861	+	0.790966i	$0.709579\pi$
$998$		0			0
$999$	−463.770			−0.464234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.3.e.b.549.12		16
5.2	odd	4		1100.3.f.f.901.6		8
5.3	odd	4		220.3.f.a.21.3	✓	8
5.4	even	2	inner	1100.3.e.b.549.5		16
11.10	odd	2	inner	1100.3.e.b.549.11		16
15.8	even	4		1980.3.b.a.901.2		8
20.3	even	4		880.3.j.b.241.6		8
55.32	even	4		1100.3.f.f.901.5		8
55.43	even	4		220.3.f.a.21.4	yes	8
55.54	odd	2	inner	1100.3.e.b.549.6		16
165.98	odd	4		1980.3.b.a.901.3		8
220.43	odd	4		880.3.j.b.241.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.3.f.a.21.3	✓	8	5.3	odd	4
220.3.f.a.21.4	yes	8	55.43	even	4
880.3.j.b.241.5		8	220.43	odd	4
880.3.j.b.241.6		8	20.3	even	4
1100.3.e.b.549.5		16	5.4	even	2	inner
1100.3.e.b.549.6		16	55.54	odd	2	inner
1100.3.e.b.549.11		16	11.10	odd	2	inner
1100.3.e.b.549.12		16	1.1	even	1	trivial
1100.3.f.f.901.5		8	55.32	even	4
1100.3.f.f.901.6		8	5.2	odd	4
1980.3.b.a.901.2		8	15.8	even	4
1980.3.b.a.901.3		8	165.98	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.41553i q^{3} +0.558647 q^{7} -2.66584 q^{9} +O(q^{10})\)	\(q+3.41553i q^{3} +0.558647 q^{7} -2.66584 q^{9} +(2.82054 + 10.6322i) q^{11} -18.1072 q^{13} +32.0402 q^{17} +15.7408i q^{19} +1.90807i q^{21} -31.7468i q^{23} +21.6345i q^{27} +48.3396i q^{29} -43.3557 q^{31} +(-36.3147 + 9.63363i) q^{33} +21.4366i q^{37} -61.8458i q^{39} +38.9133i q^{41} -23.3044 q^{43} -75.2975i q^{47} -48.6879 q^{49} +109.434i q^{51} +8.43533i q^{53} -53.7631 q^{57} +29.6094 q^{59} +64.1200i q^{61} -1.48926 q^{63} +18.8211i q^{67} +108.432 q^{69} +94.8085 q^{71} +0.945835 q^{73} +(1.57568 + 5.93967i) q^{77} -73.2825i q^{79} -97.8859 q^{81} -161.659 q^{83} -165.105 q^{87} -91.5556 q^{89} -10.1155 q^{91} -148.083i q^{93} -33.1491i q^{97} +(-7.51910 - 28.3438i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 56 q^{9}+O(q^{10}) \) 16 * q - 56 * q^9	\( 16 q - 56 q^{9} + 48 q^{11} + 40 q^{31} + 488 q^{49} + 32 q^{59} + 112 q^{69} - 440 q^{71} - 448 q^{81} - 440 q^{89} - 144 q^{91} + 80 q^{99}+O(q^{100}) \) 16 * q - 56 * q^9 + 48 * q^11 + 40 * q^31 + 488 * q^49 + 32 * q^59 + 112 * q^69 - 440 * q^71 - 448 * q^81 - 440 * q^89 - 144 * q^91 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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