LMFDB - Embedded newform 9075.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.a (trivial)

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:	yes
Analytic conductor:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1815)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -2.23607 q^{6} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	$q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -2.23607 q^{6} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{12} -4.47214 q^{13} -1.00000 q^{16} -4.47214 q^{17} +2.23607 q^{18} +4.00000 q^{23} -2.23607 q^{24} -10.0000 q^{26} -1.00000 q^{27} +8.94427 q^{29} -6.70820 q^{32} -10.0000 q^{34} +3.00000 q^{36} +8.00000 q^{37} +4.47214 q^{39} -8.94427 q^{41} +8.94427 q^{43} +8.94427 q^{46} +12.0000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +4.47214 q^{51} -13.4164 q^{52} +4.00000 q^{53} -2.23607 q^{54} +20.0000 q^{58} -13.0000 q^{64} +12.0000 q^{67} -13.4164 q^{68} -4.00000 q^{69} +12.0000 q^{71} +2.23607 q^{72} +13.4164 q^{73} +17.8885 q^{74} +10.0000 q^{78} +1.00000 q^{81} -20.0000 q^{82} +20.0000 q^{86} -8.94427 q^{87} +6.00000 q^{89} +12.0000 q^{92} +26.8328 q^{94} +6.70820 q^{96} +8.00000 q^{97} -15.6525 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9	$ 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} - 6 q^{12} - 2 q^{16} + 8 q^{23} - 20 q^{26} - 2 q^{27} - 20 q^{34} + 6 q^{36} + 16 q^{37} + 24 q^{47} + 2 q^{48} - 14 q^{49} + 8 q^{53} + 40 q^{58} - 26 q^{64} + 24 q^{67} - 8 q^{69} + 24 q^{71} + 20 q^{78} + 2 q^{81} - 40 q^{82} + 40 q^{86} + 12 q^{89} + 24 q^{92} + 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9 - 6 * q^12 - 2 * q^16 + 8 * q^23 - 20 * q^26 - 2 * q^27 - 20 * q^34 + 6 * q^36 + 16 * q^37 + 24 * q^47 + 2 * q^48 - 14 * q^49 + 8 * q^53 + 40 * q^58 - 26 * q^64 + 24 * q^67 - 8 * q^69 + 24 * q^71 + 20 * q^78 + 2 * q^81 - 40 * q^82 + 40 * q^86 + 12 * q^89 + 24 * q^92 + 16 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.00000	1.50000
$5$	0	0
$5$	0	0
$6$	−2.23607	−0.912871
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−3.00000	−0.866025
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	2.23607	0.527046
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	−10.0000	−1.96116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.94427	1.66091	0.830455	−	0.557086i	$-0.188081\pi$
$29$	8.94427	1.66091	0.830455	+	0.557086i	$0.188081\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	−10.0000	−1.71499
$35$	0	0
$36$	3.00000	0.500000
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	4.47214	0.716115
$40$	0	0
$41$	−8.94427	−1.39686	−0.698430	−	0.715678i	$-0.746118\pi$
$41$	−8.94427	−1.39686	−0.698430	+	0.715678i	$0.746118\pi$
$42$	0	0
$43$	8.94427	1.36399	0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427	1.36399	0.681994	+	0.731357i	$0.261113\pi$
$44$	0	0
$45$	0	0
$46$	8.94427	1.31876
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	0	0
$51$	4.47214	0.626224
$52$	−13.4164	−1.86052
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−2.23607	−0.304290
$55$	0	0
$56$	0	0
$57$	0	0
$58$	20.0000	2.62613
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	−13.0000	−1.62500
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−13.4164	−1.62698
$69$	−4.00000	−0.481543
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	2.23607	0.263523
$73$	13.4164	1.57027	0.785136	−	0.619324i	$-0.212593\pi$
$73$	13.4164	1.57027	0.785136	+	0.619324i	$0.212593\pi$
$74$	17.8885	2.07950
$75$	0	0
$76$	0	0
$77$	0	0
$78$	10.0000	1.13228
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−20.0000	−2.20863
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	20.0000	2.15666
$87$	−8.94427	−0.958927
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	12.0000	1.25109
$93$	0	0
$94$	26.8328	2.76759
$95$	0	0
$96$	6.70820	0.684653
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−15.6525	−1.58114
$99$	0	0
$100$	0	0
$101$	−8.94427	−0.889988	−0.444994	−	0.895533i	$-0.646794\pi$
$101$	−8.94427	−0.889988	−0.444994	+	0.895533i	$0.646794\pi$
$102$	10.0000	0.990148
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−10.0000	−0.980581
$105$	0	0
$106$	8.94427	0.868744
$107$	17.8885	1.72935	0.864675	−	0.502331i	$-0.167524\pi$
$107$	17.8885	1.72935	0.864675	+	0.502331i	$0.167524\pi$
$108$	−3.00000	−0.288675
$109$	−17.8885	−1.71341	−0.856706	−	0.515805i	$-0.827493\pi$
$109$	−17.8885	−1.71341	−0.856706	+	0.515805i	$0.827493\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	0	0
$116$	26.8328	2.49136
$117$	−4.47214	−0.413449
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	8.94427	0.806478
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−15.6525	−1.38350
$129$	−8.94427	−0.787499
$130$	0	0
$131$	−17.8885	−1.56293	−0.781465	−	0.623949i	$-0.785527\pi$
$131$	−17.8885	−1.56293	−0.781465	+	0.623949i	$0.785527\pi$
$132$	0	0
$133$	0	0
$134$	26.8328	2.31800
$135$	0	0
$136$	−10.0000	−0.857493
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−8.94427	−0.761387
$139$	17.8885	1.51729	0.758643	−	0.651506i	$-0.225863\pi$
$139$	17.8885	1.51729	0.758643	+	0.651506i	$0.225863\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	26.8328	2.25176
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	30.0000	2.48282
$147$	7.00000	0.577350
$148$	24.0000	1.97279
$149$	8.94427	0.732743	0.366372	−	0.930469i	$-0.380600\pi$
$149$	8.94427	0.732743	0.366372	+	0.930469i	$0.380600\pi$
$150$	0	0
$151$	17.8885	1.45575	0.727875	−	0.685710i	$-0.240508\pi$
$151$	17.8885	1.45575	0.727875	+	0.685710i	$0.240508\pi$
$152$	0	0
$153$	−4.47214	−0.361551
$154$	0	0
$155$	0	0
$156$	13.4164	1.07417
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	0	0
$162$	2.23607	0.175682
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−26.8328	−2.09529
$165$	0	0
$166$	0	0
$167$	−8.94427	−0.692129	−0.346064	−	0.938211i	$-0.612482\pi$
$167$	−8.94427	−0.692129	−0.346064	+	0.938211i	$0.612482\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	26.8328	2.04598
$173$	−13.4164	−1.02003	−0.510015	−	0.860165i	$-0.670360\pi$
$173$	−13.4164	−1.02003	−0.510015	+	0.860165i	$0.670360\pi$
$174$	−20.0000	−1.51620
$175$	0	0
$176$	0	0
$177$	0	0
$178$	13.4164	1.00560
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	8.94427	0.659380
$185$	0	0
$186$	0	0
$187$	0	0
$188$	36.0000	2.62557
$189$	0	0
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	13.0000	0.938194
$193$	−4.47214	−0.321911	−0.160956	−	0.986962i	$-0.551458\pi$
$193$	−4.47214	−0.321911	−0.160956	+	0.986962i	$0.551458\pi$
$194$	17.8885	1.28432
$195$	0	0
$196$	−21.0000	−1.50000
$197$	4.47214	0.318626	0.159313	−	0.987228i	$-0.449072\pi$
$197$	4.47214	0.318626	0.159313	+	0.987228i	$0.449072\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	−20.0000	−1.40720
$203$	0	0
$204$	13.4164	0.939336
$205$	0	0
$206$	8.94427	0.623177
$207$	4.00000	0.278019
$208$	4.47214	0.310087
$209$	0	0
$210$	0	0
$211$	−17.8885	−1.23150	−0.615749	−	0.787942i	$-0.711146\pi$
$211$	−17.8885	−1.23150	−0.615749	+	0.787942i	$0.711146\pi$
$212$	12.0000	0.824163
$213$	−12.0000	−0.822226
$214$	40.0000	2.73434
$215$	0	0
$216$	−2.23607	−0.152145
$217$	0	0
$218$	−40.0000	−2.70914
$219$	−13.4164	−0.906597
$220$	0	0
$221$	20.0000	1.34535
$222$	−17.8885	−1.20060
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	0	0
$226$	8.94427	0.594964
$227$	−17.8885	−1.18730	−0.593652	−	0.804722i	$-0.702314\pi$
$227$	−17.8885	−1.18730	−0.593652	+	0.804722i	$0.702314\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	20.0000	1.31306
$233$	22.3607	1.46490	0.732448	−	0.680823i	$-0.238378\pi$
$233$	22.3607	1.46490	0.732448	+	0.680823i	$0.238378\pi$
$234$	−10.0000	−0.653720
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.8885	1.15711	0.578557	−	0.815642i	$-0.303616\pi$
$239$	17.8885	1.15711	0.578557	+	0.815642i	$0.303616\pi$
$240$	0	0
$241$	−17.8885	−1.15230	−0.576151	−	0.817343i	$-0.695446\pi$
$241$	−17.8885	−1.15230	−0.576151	+	0.817343i	$0.695446\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	20.0000	1.27515
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−9.00000	−0.562500
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	−20.0000	−1.24515
$259$	0	0
$260$	0	0
$261$	8.94427	0.553637
$262$	−40.0000	−2.47121
$263$	−8.94427	−0.551527	−0.275764	−	0.961225i	$-0.588931\pi$
$263$	−8.94427	−0.551527	−0.275764	+	0.961225i	$0.588931\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	36.0000	2.19905
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	17.8885	1.08665	0.543326	−	0.839522i	$-0.317165\pi$
$271$	17.8885	1.08665	0.543326	+	0.839522i	$0.317165\pi$
$272$	4.47214	0.271163
$273$	0	0
$274$	−26.8328	−1.62103
$275$	0	0
$276$	−12.0000	−0.722315
$277$	13.4164	0.806114	0.403057	−	0.915175i	$-0.367948\pi$
$277$	13.4164	0.806114	0.403057	+	0.915175i	$0.367948\pi$
$278$	40.0000	2.39904
$279$	0	0
$280$	0	0
$281$	−8.94427	−0.533571	−0.266785	−	0.963756i	$-0.585961\pi$
$281$	−8.94427	−0.533571	−0.266785	+	0.963756i	$0.585961\pi$
$282$	−26.8328	−1.59787
$283$	−8.94427	−0.531682	−0.265841	−	0.964017i	$-0.585650\pi$
$283$	−8.94427	−0.531682	−0.265841	+	0.964017i	$0.585650\pi$
$284$	36.0000	2.13621
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−6.70820	−0.395285
$289$	3.00000	0.176471
$290$	0	0
$291$	−8.00000	−0.468968
$292$	40.2492	2.35541
$293$	−4.47214	−0.261265	−0.130632	−	0.991431i	$-0.541701\pi$
$293$	−4.47214	−0.261265	−0.130632	+	0.991431i	$0.541701\pi$
$294$	15.6525	0.912871
$295$	0	0
$296$	17.8885	1.03975
$297$	0	0
$298$	20.0000	1.15857
$299$	−17.8885	−1.03452
$300$	0	0
$301$	0	0
$302$	40.0000	2.30174
$303$	8.94427	0.513835
$304$	0	0
$305$	0	0
$306$	−10.0000	−0.571662
$307$	8.94427	0.510477	0.255238	−	0.966878i	$-0.417846\pi$
$307$	8.94427	0.510477	0.255238	+	0.966878i	$0.417846\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	10.0000	0.566139
$313$	16.0000	0.904373	0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000	0.904373	0.452187	+	0.891923i	$0.350644\pi$
$314$	17.8885	1.00951
$315$	0	0
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	−8.94427	−0.501570
$319$	0	0
$320$	0	0
$321$	−17.8885	−0.998441
$322$	0	0
$323$	0	0
$324$	3.00000	0.166667
$325$	0	0
$326$	−8.94427	−0.495377
$327$	17.8885	0.989239
$328$	−20.0000	−1.10432
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	−20.0000	−1.09435
$335$	0	0
$336$	0	0
$337$	31.3050	1.70529	0.852645	−	0.522491i	$-0.174997\pi$
$337$	31.3050	1.70529	0.852645	+	0.522491i	$0.174997\pi$
$338$	15.6525	0.851382
$339$	−4.00000	−0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	20.0000	1.07833
$345$	0	0
$346$	−30.0000	−1.61281
$347$	−17.8885	−0.960307	−0.480154	−	0.877184i	$-0.659419\pi$
$347$	−17.8885	−0.960307	−0.480154	+	0.877184i	$0.659419\pi$
$348$	−26.8328	−1.43839
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	0	0
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	0	0
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	35.7771	1.89088
$359$	17.8885	0.944121	0.472061	−	0.881566i	$-0.343510\pi$
$359$	17.8885	0.944121	0.472061	+	0.881566i	$0.343510\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−22.3607	−1.17525
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	−4.00000	−0.208514
$369$	−8.94427	−0.465620
$370$	0	0
$371$	0	0
$372$	0	0
$373$	22.3607	1.15779	0.578896	−	0.815401i	$-0.303484\pi$
$373$	22.3607	1.15779	0.578896	+	0.815401i	$0.303484\pi$
$374$	0	0
$375$	0	0
$376$	26.8328	1.38380
$377$	−40.0000	−2.06010
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	−44.7214	−2.28814
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	15.6525	0.798762
$385$	0	0
$386$	−10.0000	−0.508987
$387$	8.94427	0.454663
$388$	24.0000	1.21842
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−17.8885	−0.904663
$392$	−15.6525	−0.790569
$393$	17.8885	0.902358
$394$	10.0000	0.503793
$395$	0	0
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−26.8328	−1.33830
$403$	0	0
$404$	−26.8328	−1.33498
$405$	0	0
$406$	0	0
$407$	0	0
$408$	10.0000	0.495074
$409$	17.8885	0.884532	0.442266	−	0.896884i	$-0.354175\pi$
$409$	17.8885	0.884532	0.442266	+	0.896884i	$0.354175\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	12.0000	0.591198
$413$	0	0
$414$	8.94427	0.439587
$415$	0	0
$416$	30.0000	1.47087
$417$	−17.8885	−0.876006
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−40.0000	−1.94717
$423$	12.0000	0.583460
$424$	8.94427	0.434372
$425$	0	0
$426$	−26.8328	−1.30005
$427$	0	0
$428$	53.6656	2.59403
$429$	0	0
$430$	0	0
$431$	17.8885	0.861661	0.430830	−	0.902433i	$-0.358221\pi$
$431$	17.8885	0.861661	0.430830	+	0.902433i	$0.358221\pi$
$432$	1.00000	0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	0	0
$436$	−53.6656	−2.57012
$437$	0	0
$438$	−30.0000	−1.43346
$439$	17.8885	0.853774	0.426887	−	0.904305i	$-0.359610\pi$
$439$	17.8885	0.853774	0.426887	+	0.904305i	$0.359610\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	44.7214	2.12718
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−24.0000	−1.13899
$445$	0	0
$446$	−8.94427	−0.423524
$447$	−8.94427	−0.423050
$448$	0	0
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	0	0
$452$	12.0000	0.564433
$453$	−17.8885	−0.840477
$454$	−40.0000	−1.87729
$455$	0	0
$456$	0	0
$457$	−13.4164	−0.627593	−0.313797	−	0.949490i	$-0.601601\pi$
$457$	−13.4164	−0.627593	−0.313797	+	0.949490i	$0.601601\pi$
$458$	22.3607	1.04485
$459$	4.47214	0.208741
$460$	0	0
$461$	−26.8328	−1.24973	−0.624864	−	0.780733i	$-0.714846\pi$
$461$	−26.8328	−1.24973	−0.624864	+	0.780733i	$0.714846\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−8.94427	−0.415227
$465$	0	0
$466$	50.0000	2.31621
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	−13.4164	−0.620174
$469$	0	0
$470$	0	0
$471$	−8.00000	−0.368621
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.00000	0.183147
$478$	40.0000	1.82956
$479$	17.8885	0.817348	0.408674	−	0.912680i	$-0.365991\pi$
$479$	17.8885	0.817348	0.408674	+	0.912680i	$0.365991\pi$
$480$	0	0
$481$	−35.7771	−1.63129
$482$	−40.0000	−1.82195
$483$	0	0
$484$	0	0
$485$	0	0
$486$	−2.23607	−0.101430
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	35.7771	1.61460	0.807299	−	0.590143i	$-0.200929\pi$
$491$	35.7771	1.61460	0.807299	+	0.590143i	$0.200929\pi$
$492$	26.8328	1.20972
$493$	−40.0000	−1.80151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	8.94427	0.399601
$502$	17.8885	0.798405
$503$	26.8328	1.19642	0.598208	−	0.801341i	$-0.295880\pi$
$503$	26.8328	1.19642	0.598208	+	0.801341i	$0.295880\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7.00000	−0.310881
$508$	0	0
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	0	0
$512$	11.1803	0.494106
$513$	0	0
$514$	26.8328	1.18354
$515$	0	0
$516$	−26.8328	−1.18125
$517$	0	0
$518$	0	0
$519$	13.4164	0.588915
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	20.0000	0.875376
$523$	−8.94427	−0.391106	−0.195553	−	0.980693i	$-0.562650\pi$
$523$	−8.94427	−0.391106	−0.195553	+	0.980693i	$0.562650\pi$
$524$	−53.6656	−2.34439
$525$	0	0
$526$	−20.0000	−0.872041
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	40.0000	1.73259
$534$	−13.4164	−0.580585
$535$	0	0
$536$	26.8328	1.15900
$537$	−16.0000	−0.690451
$538$	−22.3607	−0.964037
$539$	0	0
$540$	0	0
$541$	−17.8885	−0.769089	−0.384544	−	0.923107i	$-0.625641\pi$
$541$	−17.8885	−0.769089	−0.384544	+	0.923107i	$0.625641\pi$
$542$	40.0000	1.71815
$543$	10.0000	0.429141
$544$	30.0000	1.28624
$545$	0	0
$546$	0	0
$547$	−44.7214	−1.91215	−0.956074	−	0.293127i	$-0.905304\pi$
$547$	−44.7214	−1.91215	−0.956074	+	0.293127i	$0.905304\pi$
$548$	−36.0000	−1.53784
$549$	0	0
$550$	0	0
$551$	0	0
$552$	−8.94427	−0.380693
$553$	0	0
$554$	30.0000	1.27458
$555$	0	0
$556$	53.6656	2.27593
$557$	−13.4164	−0.568471	−0.284236	−	0.958754i	$-0.591740\pi$
$557$	−13.4164	−0.568471	−0.284236	+	0.958754i	$0.591740\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	0	0
$562$	−20.0000	−0.843649
$563$	−17.8885	−0.753912	−0.376956	−	0.926231i	$-0.623029\pi$
$563$	−17.8885	−0.753912	−0.376956	+	0.926231i	$0.623029\pi$
$564$	−36.0000	−1.51587
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	26.8328	1.12588
$569$	−8.94427	−0.374963	−0.187482	−	0.982268i	$-0.560033\pi$
$569$	−8.94427	−0.374963	−0.187482	+	0.982268i	$0.560033\pi$
$570$	0	0
$571$	35.7771	1.49722	0.748612	−	0.663008i	$-0.230720\pi$
$571$	35.7771	1.49722	0.748612	+	0.663008i	$0.230720\pi$
$572$	0	0
$573$	20.0000	0.835512
$574$	0	0
$575$	0	0
$576$	−13.0000	−0.541667
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	6.70820	0.279024
$579$	4.47214	0.185856
$580$	0	0
$581$	0	0
$582$	−17.8885	−0.741504
$583$	0	0
$584$	30.0000	1.24141
$585$	0	0
$586$	−10.0000	−0.413096
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	21.0000	0.866025
$589$	0	0
$590$	0	0
$591$	−4.47214	−0.183959
$592$	−8.00000	−0.328798
$593$	−4.47214	−0.183649	−0.0918243	−	0.995775i	$-0.529270\pi$
$593$	−4.47214	−0.183649	−0.0918243	+	0.995775i	$0.529270\pi$
$594$	0	0
$595$	0	0
$596$	26.8328	1.09911
$597$	0	0
$598$	−40.0000	−1.63572
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	35.7771	1.45938	0.729689	−	0.683779i	$-0.239665\pi$
$601$	35.7771	1.45938	0.729689	+	0.683779i	$0.239665\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	53.6656	2.18362
$605$	0	0
$606$	20.0000	0.812444
$607$	17.8885	0.726074	0.363037	−	0.931775i	$-0.381740\pi$
$607$	17.8885	0.726074	0.363037	+	0.931775i	$0.381740\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−53.6656	−2.17108
$612$	−13.4164	−0.542326
$613$	−13.4164	−0.541884	−0.270942	−	0.962596i	$-0.587335\pi$
$613$	−13.4164	−0.541884	−0.270942	+	0.962596i	$0.587335\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	−8.94427	−0.359791
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	−26.8328	−1.07590
$623$	0	0
$624$	−4.47214	−0.179029
$625$	0	0
$626$	35.7771	1.42994
$627$	0	0
$628$	24.0000	0.957704
$629$	−35.7771	−1.42653
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	17.8885	0.711006
$634$	26.8328	1.06567
$635$	0	0
$636$	−12.0000	−0.475831
$637$	31.3050	1.24035
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−40.0000	−1.57867
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	2.23607	0.0878410
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−12.0000	−0.469956
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	40.0000	1.56412
$655$	0	0
$656$	8.94427	0.349215
$657$	13.4164	0.523424
$658$	0	0
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	−44.7214	−1.73814
$663$	−20.0000	−0.776736
$664$	0	0
$665$	0	0
$666$	17.8885	0.693167
$667$	35.7771	1.38529
$668$	−26.8328	−1.03819
$669$	4.00000	0.154649
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.47214	−0.172388	−0.0861941	−	0.996278i	$-0.527471\pi$
$673$	−4.47214	−0.172388	−0.0861941	+	0.996278i	$0.527471\pi$
$674$	70.0000	2.69630
$675$	0	0
$676$	21.0000	0.807692
$677$	31.3050	1.20315	0.601574	−	0.798817i	$-0.294541\pi$
$677$	31.3050	1.20315	0.601574	+	0.798817i	$0.294541\pi$
$678$	−8.94427	−0.343503
$679$	0	0
$680$	0	0
$681$	17.8885	0.685490
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.0000	−0.381524
$688$	−8.94427	−0.340997
$689$	−17.8885	−0.681499
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−40.2492	−1.53005
$693$	0	0
$694$	−40.0000	−1.51838
$695$	0	0
$696$	−20.0000	−0.758098
$697$	40.0000	1.51511
$698$	0	0
$699$	−22.3607	−0.845759
$700$	0	0
$701$	−8.94427	−0.337820	−0.168910	−	0.985631i	$-0.554025\pi$
$701$	−8.94427	−0.337820	−0.168910	+	0.985631i	$0.554025\pi$
$702$	10.0000	0.377426
$703$	0	0
$704$	0	0
$705$	0	0
$706$	8.94427	0.336622
$707$	0	0
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	0	0
$712$	13.4164	0.502801
$713$	0	0
$714$	0	0
$715$	0	0
$716$	48.0000	1.79384
$717$	−17.8885	−0.668060
$718$	40.0000	1.49279
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	−42.4853	−1.58114
$723$	17.8885	0.665282
$724$	−30.0000	−1.11494
$725$	0	0
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.0000	−1.47945
$732$	0	0
$733$	−31.3050	−1.15627	−0.578137	−	0.815939i	$-0.696220\pi$
$733$	−31.3050	−1.15627	−0.578137	+	0.815939i	$0.696220\pi$
$734$	−62.6099	−2.31097
$735$	0	0
$736$	−26.8328	−0.989071
$737$	0	0
$738$	−20.0000	−0.736210
$739$	−17.8885	−0.658041	−0.329020	−	0.944323i	$-0.606718\pi$
$739$	−17.8885	−0.658041	−0.329020	+	0.944323i	$0.606718\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.8328	−0.984401	−0.492200	−	0.870482i	$-0.663807\pi$
$743$	−26.8328	−0.984401	−0.492200	+	0.870482i	$0.663807\pi$
$744$	0	0
$745$	0	0
$746$	50.0000	1.83063
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	−12.0000	−0.437595
$753$	−8.00000	−0.291536
$754$	−89.4427	−3.25731
$755$	0	0
$756$	0	0
$757$	48.0000	1.74459	0.872295	−	0.488980i	$-0.162631\pi$
$757$	48.0000	1.74459	0.872295	+	0.488980i	$0.162631\pi$
$758$	−44.7214	−1.62435
$759$	0	0
$760$	0	0
$761$	−8.94427	−0.324230	−0.162115	−	0.986772i	$-0.551831\pi$
$761$	−8.94427	−0.324230	−0.162115	+	0.986772i	$0.551831\pi$
$762$	0	0
$763$	0	0
$764$	−60.0000	−2.17072
$765$	0	0
$766$	−8.94427	−0.323170
$767$	0	0
$768$	9.00000	0.324760
$769$	−35.7771	−1.29015	−0.645077	−	0.764117i	$-0.723175\pi$
$769$	−35.7771	−1.29015	−0.645077	+	0.764117i	$0.723175\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	−13.4164	−0.482867
$773$	36.0000	1.29483	0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000	1.29483	0.647415	+	0.762138i	$0.275850\pi$
$774$	20.0000	0.718885
$775$	0	0
$776$	17.8885	0.642161
$777$	0	0
$778$	−67.0820	−2.40501
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−40.0000	−1.43040
$783$	−8.94427	−0.319642
$784$	7.00000	0.250000
$785$	0	0
$786$	40.0000	1.42675
$787$	26.8328	0.956487	0.478243	−	0.878227i	$-0.341274\pi$
$787$	26.8328	0.956487	0.478243	+	0.878227i	$0.341274\pi$
$788$	13.4164	0.477940
$789$	8.94427	0.318425
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	17.8885	0.634841
$795$	0	0
$796$	0	0
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	−53.6656	−1.89855
$800$	0	0
$801$	6.00000	0.212000
$802$	67.0820	2.36875
$803$	0	0
$804$	−36.0000	−1.26962
$805$	0	0
$806$	0	0
$807$	10.0000	0.352017
$808$	−20.0000	−0.703598
$809$	−26.8328	−0.943392	−0.471696	−	0.881761i	$-0.656358\pi$
$809$	−26.8328	−0.943392	−0.471696	+	0.881761i	$0.656358\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	−17.8885	−0.627379
$814$	0	0
$815$	0	0
$816$	−4.47214	−0.156556
$817$	0	0
$818$	40.0000	1.39857
$819$	0	0
$820$	0	0
$821$	8.94427	0.312157	0.156079	−	0.987745i	$-0.450115\pi$
$821$	8.94427	0.312157	0.156079	+	0.987745i	$0.450115\pi$
$822$	26.8328	0.935902
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	8.94427	0.311588
$825$	0	0
$826$	0	0
$827$	−17.8885	−0.622046	−0.311023	−	0.950402i	$-0.600672\pi$
$827$	−17.8885	−0.622046	−0.311023	+	0.950402i	$0.600672\pi$
$828$	12.0000	0.417029
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	−13.4164	−0.465410
$832$	58.1378	2.01556
$833$	31.3050	1.08465
$834$	−40.0000	−1.38509
$835$	0	0
$836$	0	0
$837$	0	0
$838$	53.6656	1.85385
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	51.0000	1.75862
$842$	−22.3607	−0.770600
$843$	8.94427	0.308057
$844$	−53.6656	−1.84725
$845$	0	0
$846$	26.8328	0.922531
$847$	0	0
$848$	−4.00000	−0.137361
$849$	8.94427	0.306967
$850$	0	0
$851$	32.0000	1.09695
$852$	−36.0000	−1.23334
$853$	−49.1935	−1.68435	−0.842177	−	0.539202i	$-0.818726\pi$
$853$	−49.1935	−1.68435	−0.842177	+	0.539202i	$0.818726\pi$
$854$	0	0
$855$	0	0
$856$	40.0000	1.36717
$857$	13.4164	0.458296	0.229148	−	0.973392i	$-0.426406\pi$
$857$	13.4164	0.458296	0.229148	+	0.973392i	$0.426406\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	0	0
$862$	40.0000	1.36241
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	6.70820	0.228218
$865$	0	0
$866$	35.7771	1.21575
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−53.6656	−1.81839
$872$	−40.0000	−1.35457
$873$	8.00000	0.270759
$874$	0	0
$875$	0	0
$876$	−40.2492	−1.35990
$877$	−31.3050	−1.05709	−0.528547	−	0.848904i	$-0.677263\pi$
$877$	−31.3050	−1.05709	−0.528547	+	0.848904i	$0.677263\pi$
$878$	40.0000	1.34993
$879$	4.47214	0.150841
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	−15.6525	−0.527046
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	60.0000	2.01802
$885$	0	0
$886$	−80.4984	−2.70440
$887$	44.7214	1.50160	0.750798	−	0.660532i	$-0.229669\pi$
$887$	44.7214	1.50160	0.750798	+	0.660532i	$0.229669\pi$
$888$	−17.8885	−0.600300
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−12.0000	−0.401790
$893$	0	0
$894$	−20.0000	−0.668900
$895$	0	0
$896$	0	0
$897$	17.8885	0.597281
$898$	76.0263	2.53703
$899$	0	0
$900$	0	0
$901$	−17.8885	−0.595954
$902$	0	0
$903$	0	0
$904$	8.94427	0.297482
$905$	0	0
$906$	−40.0000	−1.32891
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−53.6656	−1.78096
$909$	−8.94427	−0.296663
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	0	0
$914$	−30.0000	−0.992312
$915$	0	0
$916$	30.0000	0.991228
$917$	0	0
$918$	10.0000	0.330049
$919$	−53.6656	−1.77027	−0.885133	−	0.465338i	$-0.845933\pi$
$919$	−53.6656	−1.77027	−0.885133	+	0.465338i	$0.845933\pi$
$920$	0	0
$921$	−8.94427	−0.294724
$922$	−60.0000	−1.97599
$923$	−53.6656	−1.76643
$924$	0	0
$925$	0	0
$926$	−8.94427	−0.293927
$927$	4.00000	0.131377
$928$	−60.0000	−1.96960
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	0	0
$932$	67.0820	2.19735
$933$	12.0000	0.392862
$934$	62.6099	2.04866
$935$	0	0
$936$	−10.0000	−0.326860
$937$	22.3607	0.730492	0.365246	−	0.930911i	$-0.380985\pi$
$937$	22.3607	0.730492	0.365246	+	0.930911i	$0.380985\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	0	0
$941$	−8.94427	−0.291575	−0.145787	−	0.989316i	$-0.546572\pi$
$941$	−8.94427	−0.291575	−0.145787	+	0.989316i	$0.546572\pi$
$942$	−17.8885	−0.582840
$943$	−35.7771	−1.16506
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	−13.4164	−0.434600	−0.217300	−	0.976105i	$-0.569725\pi$
$953$	−13.4164	−0.434600	−0.217300	+	0.976105i	$0.569725\pi$
$954$	8.94427	0.289581
$955$	0	0
$956$	53.6656	1.73567
$957$	0	0
$958$	40.0000	1.29234
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−80.0000	−2.57930
$963$	17.8885	0.576450
$964$	−53.6656	−1.72845
$965$	0	0
$966$	0	0
$967$	17.8885	0.575257	0.287628	−	0.957742i	$-0.407133\pi$
$967$	17.8885	0.575257	0.287628	+	0.957742i	$0.407133\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	−3.00000	−0.0962250
$973$	0	0
$974$	−62.6099	−2.00615
$975$	0	0
$976$	0	0
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	8.94427	0.286006
$979$	0	0
$980$	0	0
$981$	−17.8885	−0.571137
$982$	80.0000	2.55290
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	20.0000	0.637577
$985$	0	0
$986$	−89.4427	−2.84844
$987$	0	0
$988$	0	0
$989$	35.7771	1.13765
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−13.4164	−0.424902	−0.212451	−	0.977172i	$-0.568145\pi$
$997$	−13.4164	−0.424902	−0.212451	+	0.977172i	$0.568145\pi$
$998$	44.7214	1.41563
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.bj.1.2		2
5.2	odd	4		1815.2.c.c.364.4	yes	4
5.3	odd	4		1815.2.c.c.364.1	✓	4
5.4	even	2		9075.2.a.bq.1.1		2
11.10	odd	2	inner	9075.2.a.bj.1.1		2
55.32	even	4		1815.2.c.c.364.2	yes	4
55.43	even	4		1815.2.c.c.364.3	yes	4
55.54	odd	2		9075.2.a.bq.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.c.c.364.1	✓	4	5.3	odd	4
1815.2.c.c.364.2	yes	4	55.32	even	4
1815.2.c.c.364.3	yes	4	55.43	even	4
1815.2.c.c.364.4	yes	4	5.2	odd	4
9075.2.a.bj.1.1		2	11.10	odd	2	inner
9075.2.a.bj.1.2		2	1.1	even	1	trivial
9075.2.a.bq.1.1		2	5.4	even	2
9075.2.a.bq.1.2		2	55.54	odd	2

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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -2.23607 q^{6} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -2.23607 q^{6} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{12} -4.47214 q^{13} -1.00000 q^{16} -4.47214 q^{17} +2.23607 q^{18} +4.00000 q^{23} -2.23607 q^{24} -10.0000 q^{26} -1.00000 q^{27} +8.94427 q^{29} -6.70820 q^{32} -10.0000 q^{34} +3.00000 q^{36} +8.00000 q^{37} +4.47214 q^{39} -8.94427 q^{41} +8.94427 q^{43} +8.94427 q^{46} +12.0000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +4.47214 q^{51} -13.4164 q^{52} +4.00000 q^{53} -2.23607 q^{54} +20.0000 q^{58} -13.0000 q^{64} +12.0000 q^{67} -13.4164 q^{68} -4.00000 q^{69} +12.0000 q^{71} +2.23607 q^{72} +13.4164 q^{73} +17.8885 q^{74} +10.0000 q^{78} +1.00000 q^{81} -20.0000 q^{82} +20.0000 q^{86} -8.94427 q^{87} +6.00000 q^{89} +12.0000 q^{92} +26.8328 q^{94} +6.70820 q^{96} +8.00000 q^{97} -15.6525 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} - 6 q^{12} - 2 q^{16} + 8 q^{23} - 20 q^{26} - 2 q^{27} - 20 q^{34} + 6 q^{36} + 16 q^{37} + 24 q^{47} + 2 q^{48} - 14 q^{49} + 8 q^{53} + 40 q^{58} - 26 q^{64} + 24 q^{67} - 8 q^{69} + 24 q^{71} + 20 q^{78} + 2 q^{81} - 40 q^{82} + 40 q^{86} + 12 q^{89} + 24 q^{92} + 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9 - 6 * q^12 - 2 * q^16 + 8 * q^23 - 20 * q^26 - 2 * q^27 - 20 * q^34 + 6 * q^36 + 16 * q^37 + 24 * q^47 + 2 * q^48 - 14 * q^49 + 8 * q^53 + 40 * q^58 - 26 * q^64 + 24 * q^67 - 8 * q^69 + 24 * q^71 + 20 * q^78 + 2 * q^81 - 40 * q^82 + 40 * q^86 + 12 * q^89 + 24 * q^92 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more