LMFDB - Embedded newform 9075.2.a.cl.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.737640$ 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)

$f(q)$	$=$	$q-2.45589 q^{2} -1.00000 q^{3} +4.03138 q^{4} +2.45589 q^{6} -3.28684 q^{7} -4.98884 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.45589 q^{2} -1.00000 q^{3} +4.03138 q^{4} +2.45589 q^{6} -3.28684 q^{7} -4.98884 q^{8} +1.00000 q^{9} -4.03138 q^{12} +0.313133 q^{13} +8.07211 q^{14} +4.18926 q^{16} -5.00000 q^{17} -2.45589 q^{18} -7.45408 q^{19} +3.28684 q^{21} -1.07392 q^{23} +4.98884 q^{24} -0.769020 q^{26} -1.00000 q^{27} -13.2505 q^{28} -5.03647 q^{29} +3.44899 q^{31} -0.310680 q^{32} +12.2794 q^{34} +4.03138 q^{36} -2.63428 q^{37} +18.3064 q^{38} -0.313133 q^{39} +10.8472 q^{41} -8.07211 q^{42} +5.51468 q^{43} +2.63743 q^{46} +11.9982 q^{47} -4.18926 q^{48} +3.80333 q^{49} +5.00000 q^{51} +1.26236 q^{52} -4.93543 q^{53} +2.45589 q^{54} +16.3975 q^{56} +7.45408 q^{57} +12.3690 q^{58} -9.16409 q^{59} +9.18431 q^{61} -8.47033 q^{62} -3.28684 q^{63} -7.61553 q^{64} +15.2739 q^{67} -20.1569 q^{68} +1.07392 q^{69} +3.07211 q^{71} -4.98884 q^{72} +8.65269 q^{73} +6.46950 q^{74} -30.0502 q^{76} +0.769020 q^{78} +5.41446 q^{79} +1.00000 q^{81} -26.6395 q^{82} -16.2454 q^{83} +13.2505 q^{84} -13.5434 q^{86} +5.03647 q^{87} +1.62118 q^{89} -1.02922 q^{91} -4.32938 q^{92} -3.44899 q^{93} -29.4662 q^{94} +0.310680 q^{96} -0.224082 q^{97} -9.34054 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 5 * q^2 - 4 * q^3 + 9 * q^4 + 5 * q^6 + 2 * q^7 - 15 * q^8 + 4 * q^9	$ 4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9} - 9 q^{12} - 3 q^{13} - 5 q^{14} + 15 q^{16} - 20 q^{17} - 5 q^{18} + 3 q^{19} - 2 q^{21} + 5 q^{23} + 15 q^{24} + 6 q^{26} - 4 q^{27} - 3 q^{28} + 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} + 9 q^{36} + 7 q^{37} - q^{38} + 3 q^{39} + 20 q^{41} + 5 q^{42} + 2 q^{43} + 7 q^{46} + 20 q^{47} - 15 q^{48} + 8 q^{49} + 20 q^{51} + 7 q^{52} - 6 q^{53} + 5 q^{54} + 10 q^{56} - 3 q^{57} + 21 q^{58} - 5 q^{59} - 7 q^{61} + 12 q^{62} + 2 q^{63} + 49 q^{64} + 13 q^{67} - 45 q^{68} - 5 q^{69} - 25 q^{71} - 15 q^{72} - 23 q^{73} - 7 q^{74} - 7 q^{76} - 6 q^{78} + 4 q^{81} - 11 q^{82} - 33 q^{83} + 3 q^{84} - 12 q^{86} - 5 q^{87} + 16 q^{89} - 24 q^{91} + q^{93} - 17 q^{94} + 30 q^{96} - 25 q^{98}+O(q^{100}) $ 4 * q - 5 * q^2 - 4 * q^3 + 9 * q^4 + 5 * q^6 + 2 * q^7 - 15 * q^8 + 4 * q^9 - 9 * q^12 - 3 * q^13 - 5 * q^14 + 15 * q^16 - 20 * q^17 - 5 * q^18 + 3 * q^19 - 2 * q^21 + 5 * q^23 + 15 * q^24 + 6 * q^26 - 4 * q^27 - 3 * q^28 + 5 * q^29 - q^31 - 30 * q^32 + 25 * q^34 + 9 * q^36 + 7 * q^37 - q^38 + 3 * q^39 + 20 * q^41 + 5 * q^42 + 2 * q^43 + 7 * q^46 + 20 * q^47 - 15 * q^48 + 8 * q^49 + 20 * q^51 + 7 * q^52 - 6 * q^53 + 5 * q^54 + 10 * q^56 - 3 * q^57 + 21 * q^58 - 5 * q^59 - 7 * q^61 + 12 * q^62 + 2 * q^63 + 49 * q^64 + 13 * q^67 - 45 * q^68 - 5 * q^69 - 25 * q^71 - 15 * q^72 - 23 * q^73 - 7 * q^74 - 7 * q^76 - 6 * q^78 + 4 * q^81 - 11 * q^82 - 33 * q^83 + 3 * q^84 - 12 * q^86 - 5 * q^87 + 16 * q^89 - 24 * q^91 + q^93 - 17 * q^94 + 30 * q^96 - 25 * q^98

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 $

$2$	−2.45589	−1.73657	−0.868287	−	0.496062i	$-0.834779\pi$
$2$	−2.45589	−1.73657	−0.868287	+	0.496062i	$0.834779\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.03138	2.01569
$5$	0	0
$5$	0	0
$6$	2.45589	1.00261
$7$	−3.28684	−1.24231	−0.621155	−	0.783688i	$-0.713336\pi$
$7$	−3.28684	−1.24231	−0.621155	+	0.783688i	$0.713336\pi$
$8$	−4.98884	−1.76382
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−4.03138	−1.16376
$13$	0.313133	0.0868476	0.0434238	−	0.999057i	$-0.486173\pi$
$13$	0.313133	0.0868476	0.0434238	+	0.999057i	$0.486173\pi$
$14$	8.07211	2.15736
$15$	0	0
$16$	4.18926	1.04732
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	−2.45589	−0.578858
$19$	−7.45408	−1.71008	−0.855041	−	0.518560i	$-0.826468\pi$
$19$	−7.45408	−1.71008	−0.855041	+	0.518560i	$0.826468\pi$
$20$	0	0
$21$	3.28684	0.717248
$22$	0	0
$23$	−1.07392	−0.223928	−0.111964	−	0.993712i	$-0.535714\pi$
$23$	−1.07392	−0.223928	−0.111964	+	0.993712i	$0.535714\pi$
$24$	4.98884	1.01834
$25$	0	0
$26$	−0.769020	−0.150817
$27$	−1.00000	−0.192450
$28$	−13.2505	−2.50411
$29$	−5.03647	−0.935249	−0.467624	−	0.883927i	$-0.654890\pi$
$29$	−5.03647	−0.935249	−0.467624	+	0.883927i	$0.654890\pi$
$30$	0	0
$31$	3.44899	0.619457	0.309728	−	0.950825i	$-0.399762\pi$
$31$	3.44899	0.619457	0.309728	+	0.950825i	$0.399762\pi$
$32$	−0.310680	−0.0549210
$33$	0	0
$34$	12.2794	2.10591
$35$	0	0
$36$	4.03138	0.671897
$37$	−2.63428	−0.433073	−0.216537	−	0.976274i	$-0.569476\pi$
$37$	−2.63428	−0.433073	−0.216537	+	0.976274i	$0.569476\pi$
$38$	18.3064	2.96969
$39$	−0.313133	−0.0501415
$40$	0	0
$41$	10.8472	1.69405	0.847024	−	0.531554i	$-0.178392\pi$
$41$	10.8472	1.69405	0.847024	+	0.531554i	$0.178392\pi$
$42$	−8.07211	−1.24555
$43$	5.51468	0.840980	0.420490	−	0.907297i	$-0.361858\pi$
$43$	5.51468	0.840980	0.420490	+	0.907297i	$0.361858\pi$
$44$	0	0
$45$	0	0
$46$	2.63743	0.388868
$47$	11.9982	1.75012	0.875058	−	0.484018i	$-0.160823\pi$
$47$	11.9982	1.75012	0.875058	+	0.484018i	$0.160823\pi$
$48$	−4.18926	−0.604668
$49$	3.80333	0.543333
$50$	0	0
$51$	5.00000	0.700140
$52$	1.26236	0.175058
$53$	−4.93543	−0.677934	−0.338967	−	0.940798i	$-0.610077\pi$
$53$	−4.93543	−0.677934	−0.338967	+	0.940798i	$0.610077\pi$
$54$	2.45589	0.334204
$55$	0	0
$56$	16.3975	2.19121
$57$	7.45408	0.987317
$58$	12.3690	1.62413
$59$	−9.16409	−1.19306	−0.596531	−	0.802590i	$-0.703455\pi$
$59$	−9.16409	−1.19306	−0.596531	+	0.802590i	$0.703455\pi$
$60$	0	0
$61$	9.18431	1.17593	0.587965	−	0.808886i	$-0.299929\pi$
$61$	9.18431	1.17593	0.587965	+	0.808886i	$0.299929\pi$
$62$	−8.47033	−1.07573
$63$	−3.28684	−0.414103
$64$	−7.61553	−0.951942
$65$	0	0
$66$	0	0
$67$	15.2739	1.86600	0.933000	−	0.359876i	$-0.117181\pi$
$67$	15.2739	1.86600	0.933000	+	0.359876i	$0.117181\pi$
$68$	−20.1569	−2.44438
$69$	1.07392	0.129285
$70$	0	0
$71$	3.07211	0.364593	0.182296	−	0.983244i	$-0.441647\pi$
$71$	3.07211	0.364593	0.182296	+	0.983244i	$0.441647\pi$
$72$	−4.98884	−0.587940
$73$	8.65269	1.01272	0.506361	−	0.862322i	$-0.330991\pi$
$73$	8.65269	1.01272	0.506361	+	0.862322i	$0.330991\pi$
$74$	6.46950	0.752064
$75$	0	0
$76$	−30.0502	−3.44700
$77$	0	0
$78$	0.769020	0.0870744
$79$	5.41446	0.609175	0.304587	−	0.952484i	$-0.401481\pi$
$79$	5.41446	0.609175	0.304587	+	0.952484i	$0.401481\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−26.6395	−2.94184
$83$	−16.2454	−1.78317	−0.891583	−	0.452857i	$-0.850405\pi$
$83$	−16.2454	−1.78317	−0.891583	+	0.452857i	$0.850405\pi$
$84$	13.2505	1.44575
$85$	0	0
$86$	−13.5434	−1.46042
$87$	5.03647	0.539966
$88$	0	0
$89$	1.62118	0.171845	0.0859223	−	0.996302i	$-0.472616\pi$
$89$	1.62118	0.171845	0.0859223	+	0.996302i	$0.472616\pi$
$90$	0	0
$91$	−1.02922	−0.107892
$92$	−4.32938	−0.451369
$93$	−3.44899	−0.357643
$94$	−29.4662	−3.03921
$95$	0	0
$96$	0.310680	0.0317086
$97$	−0.224082	−0.0227521	−0.0113760	−	0.999935i	$-0.503621\pi$
$97$	−0.224082	−0.0227521	−0.0113760	+	0.999935i	$0.503621\pi$
$98$	−9.34054	−0.943537
$99$	0	0
$100$	0	0
$101$	−0.505326	−0.0502818	−0.0251409	−	0.999684i	$-0.508003\pi$
$101$	−0.505326	−0.0502818	−0.0251409	+	0.999684i	$0.508003\pi$
$102$	−12.2794	−1.21585
$103$	6.40197	0.630805	0.315402	−	0.948958i	$-0.397861\pi$
$103$	6.40197	0.630805	0.315402	+	0.948958i	$0.397861\pi$
$104$	−1.56217	−0.153184
$105$	0	0
$106$	12.1209	1.17728
$107$	−2.09249	−0.202289	−0.101144	−	0.994872i	$-0.532250\pi$
$107$	−2.09249	−0.202289	−0.101144	+	0.994872i	$0.532250\pi$
$108$	−4.03138	−0.387920
$109$	−6.69278	−0.641052	−0.320526	−	0.947240i	$-0.603860\pi$
$109$	−6.69278	−0.641052	−0.320526	+	0.947240i	$0.603860\pi$
$110$	0	0
$111$	2.63428	0.250035
$112$	−13.7694	−1.30109
$113$	10.7941	1.01542	0.507712	−	0.861527i	$-0.330491\pi$
$113$	10.7941	1.01542	0.507712	+	0.861527i	$0.330491\pi$
$114$	−18.3064	−1.71455
$115$	0	0
$116$	−20.3039	−1.88517
$117$	0.313133	0.0289492
$118$	22.5060	2.07184
$119$	16.4342	1.50652
$120$	0	0
$121$	0	0
$122$	−22.5556	−2.04209
$123$	−10.8472	−0.978059
$124$	13.9042	1.24863
$125$	0	0
$126$	8.07211	0.719121
$127$	17.0033	1.50880	0.754398	−	0.656417i	$-0.227929\pi$
$127$	17.0033	1.50880	0.754398	+	0.656417i	$0.227929\pi$
$128$	19.3242	1.70804
$129$	−5.51468	−0.485540
$130$	0	0
$131$	−0.0430508	−0.00376136	−0.00188068	−	0.999998i	$-0.500599\pi$
$131$	−0.0430508	−0.00376136	−0.00188068	+	0.999998i	$0.500599\pi$
$132$	0	0
$133$	24.5004	2.12445
$134$	−37.5109	−3.24045
$135$	0	0
$136$	24.9442	2.13895
$137$	−7.36257	−0.629027	−0.314513	−	0.949253i	$-0.601841\pi$
$137$	−7.36257	−0.629027	−0.314513	+	0.949253i	$0.601841\pi$
$138$	−2.63743	−0.224513
$139$	13.2393	1.12295	0.561473	−	0.827495i	$-0.310235\pi$
$139$	13.2393	1.12295	0.561473	+	0.827495i	$0.310235\pi$
$140$	0	0
$141$	−11.9982	−1.01043
$142$	−7.54476	−0.633142
$143$	0	0
$144$	4.18926	0.349105
$145$	0	0
$146$	−21.2500	−1.75867
$147$	−3.80333	−0.313693
$148$	−10.6198	−0.872942
$149$	−5.87858	−0.481592	−0.240796	−	0.970576i	$-0.577409\pi$
$149$	−5.87858	−0.481592	−0.240796	+	0.970576i	$0.577409\pi$
$150$	0	0
$151$	7.62821	0.620775	0.310387	−	0.950610i	$-0.399541\pi$
$151$	7.62821	0.620775	0.310387	+	0.950610i	$0.399541\pi$
$152$	37.1872	3.01628
$153$	−5.00000	−0.404226
$154$	0	0
$155$	0	0
$156$	−1.26236	−0.101070
$157$	10.1332	0.808719	0.404360	−	0.914600i	$-0.367494\pi$
$157$	10.1332	0.808719	0.404360	+	0.914600i	$0.367494\pi$
$158$	−13.2973	−1.05788
$159$	4.93543	0.391405
$160$	0	0
$161$	3.52981	0.278188
$162$	−2.45589	−0.192953
$163$	5.02906	0.393906	0.196953	−	0.980413i	$-0.436895\pi$
$163$	5.02906	0.393906	0.196953	+	0.980413i	$0.436895\pi$
$164$	43.7292	3.41468
$165$	0	0
$166$	39.8969	3.09660
$167$	−5.79105	−0.448125	−0.224062	−	0.974575i	$-0.571932\pi$
$167$	−5.79105	−0.448125	−0.224062	+	0.974575i	$0.571932\pi$
$168$	−16.3975	−1.26510
$169$	−12.9019	−0.992457
$170$	0	0
$171$	−7.45408	−0.570028
$172$	22.2318	1.69516
$173$	−16.0652	−1.22142	−0.610708	−	0.791856i	$-0.709115\pi$
$173$	−16.0652	−1.22142	−0.610708	+	0.791856i	$0.709115\pi$
$174$	−12.3690	−0.937691
$175$	0	0
$176$	0	0
$177$	9.16409	0.688815
$178$	−3.98143	−0.298421
$179$	−8.30309	−0.620602	−0.310301	−	0.950638i	$-0.600430\pi$
$179$	−8.30309	−0.620602	−0.310301	+	0.950638i	$0.600430\pi$
$180$	0	0
$181$	−6.46425	−0.480484	−0.240242	−	0.970713i	$-0.577227\pi$
$181$	−6.46425	−0.480484	−0.240242	+	0.970713i	$0.577227\pi$
$182$	2.52765	0.187362
$183$	−9.18431	−0.678924
$184$	5.35762	0.394969
$185$	0	0
$186$	8.47033	0.621074
$187$	0	0
$188$	48.3693	3.52769
$189$	3.28684	0.239083
$190$	0	0
$191$	15.3693	1.11208	0.556041	−	0.831155i	$-0.312320\pi$
$191$	15.3693	1.11208	0.556041	+	0.831155i	$0.312320\pi$
$192$	7.61553	0.549604
$193$	−15.7518	−1.13384	−0.566919	−	0.823773i	$-0.691865\pi$
$193$	−15.7518	−1.13384	−0.566919	+	0.823773i	$0.691865\pi$
$194$	0.550320	0.0395107
$195$	0	0
$196$	15.3327	1.09519
$197$	−16.3940	−1.16802	−0.584010	−	0.811746i	$-0.698517\pi$
$197$	−16.3940	−1.16802	−0.584010	+	0.811746i	$0.698517\pi$
$198$	0	0
$199$	6.96500	0.493736	0.246868	−	0.969049i	$-0.420599\pi$
$199$	6.96500	0.493736	0.246868	+	0.969049i	$0.420599\pi$
$200$	0	0
$201$	−15.2739	−1.07734
$202$	1.24102	0.0873180
$203$	16.5541	1.16187
$204$	20.1569	1.41127
$205$	0	0
$206$	−15.7225	−1.09544
$207$	−1.07392	−0.0746427
$208$	1.31180	0.0909569
$209$	0	0
$210$	0	0
$211$	−19.9531	−1.37363	−0.686814	−	0.726833i	$-0.740992\pi$
$211$	−19.9531	−1.37363	−0.686814	+	0.726833i	$0.740992\pi$
$212$	−19.8966	−1.36650
$213$	−3.07211	−0.210498
$214$	5.13892	0.351289
$215$	0	0
$216$	4.98884	0.339447
$217$	−11.3363	−0.769557
$218$	16.4367	1.11323
$219$	−8.65269	−0.584695
$220$	0	0
$221$	−1.56567	−0.105318
$222$	−6.46950	−0.434204
$223$	20.1466	1.34912	0.674559	−	0.738221i	$-0.264334\pi$
$223$	20.1466	1.34912	0.674559	+	0.738221i	$0.264334\pi$
$224$	1.02116	0.0682289
$225$	0	0
$226$	−26.5091	−1.76336
$227$	−0.533937	−0.0354386	−0.0177193	−	0.999843i	$-0.505641\pi$
$227$	−0.533937	−0.0354386	−0.0177193	+	0.999843i	$0.505641\pi$
$228$	30.0502	1.99012
$229$	−22.1931	−1.46656	−0.733279	−	0.679928i	$-0.762011\pi$
$229$	−22.1931	−1.46656	−0.733279	+	0.679928i	$0.762011\pi$
$230$	0	0
$231$	0	0
$232$	25.1261	1.64961
$233$	4.56567	0.299107	0.149553	−	0.988754i	$-0.452216\pi$
$233$	4.56567	0.299107	0.149553	+	0.988754i	$0.452216\pi$
$234$	−0.769020	−0.0502724
$235$	0	0
$236$	−36.9439	−2.40485
$237$	−5.41446	−0.351707
$238$	−40.3606	−2.61619
$239$	−5.86053	−0.379086	−0.189543	−	0.981872i	$-0.560701\pi$
$239$	−5.86053	−0.379086	−0.189543	+	0.981872i	$0.560701\pi$
$240$	0	0
$241$	9.96074	0.641628	0.320814	−	0.947142i	$-0.396044\pi$
$241$	9.96074	0.641628	0.320814	+	0.947142i	$0.396044\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	37.0254	2.37031
$245$	0	0
$246$	26.6395	1.69847
$247$	−2.33412	−0.148517
$248$	−17.2065	−1.09261
$249$	16.2454	1.02951
$250$	0	0
$251$	−16.8788	−1.06538	−0.532690	−	0.846310i	$-0.678819\pi$
$251$	−16.8788	−1.06538	−0.532690	+	0.846310i	$0.678819\pi$
$252$	−13.2505	−0.834704
$253$	0	0
$254$	−41.7581	−2.62014
$255$	0	0
$256$	−32.2271	−2.01419
$257$	−11.1436	−0.695117	−0.347559	−	0.937658i	$-0.612989\pi$
$257$	−11.1436	−0.695117	−0.347559	+	0.937658i	$0.612989\pi$
$258$	13.5434	0.843177
$259$	8.65847	0.538011
$260$	0	0
$261$	−5.03647	−0.311750
$262$	0.105728	0.00653189
$263$	26.8726	1.65704	0.828519	−	0.559961i	$-0.189184\pi$
$263$	26.8726	1.65704	0.828519	+	0.559961i	$0.189184\pi$
$264$	0	0
$265$	0	0
$266$	−60.1701	−3.68927
$267$	−1.62118	−0.0992145
$268$	61.5748	3.76128
$269$	−10.0629	−0.613545	−0.306773	−	0.951783i	$-0.599249\pi$
$269$	−10.0629	−0.613545	−0.306773	+	0.951783i	$0.599249\pi$
$270$	0	0
$271$	10.5441	0.640509	0.320255	−	0.947331i	$-0.396232\pi$
$271$	10.5441	0.640509	0.320255	+	0.947331i	$0.396232\pi$
$272$	−20.9463	−1.27006
$273$	1.02922	0.0622912
$274$	18.0816	1.09235
$275$	0	0
$276$	4.32938	0.260598
$277$	17.9376	1.07777	0.538883	−	0.842381i	$-0.318847\pi$
$277$	17.9376	1.07777	0.538883	+	0.842381i	$0.318847\pi$
$278$	−32.5143	−1.95008
$279$	3.44899	0.206486
$280$	0	0
$281$	7.41103	0.442105	0.221052	−	0.975262i	$-0.429051\pi$
$281$	7.41103	0.442105	0.221052	+	0.975262i	$0.429051\pi$
$282$	29.4662	1.75469
$283$	−5.38684	−0.320214	−0.160107	−	0.987100i	$-0.551184\pi$
$283$	−5.38684	−0.320214	−0.160107	+	0.987100i	$0.551184\pi$
$284$	12.3848	0.734905
$285$	0	0
$286$	0	0
$287$	−35.6530	−2.10453
$288$	−0.310680	−0.0183070
$289$	8.00000	0.470588
$290$	0	0
$291$	0.224082	0.0131359
$292$	34.8823	2.04133
$293$	1.74006	0.101655	0.0508277	−	0.998707i	$-0.483814\pi$
$293$	1.74006	0.101655	0.0508277	+	0.998707i	$0.483814\pi$
$294$	9.34054	0.544752
$295$	0	0
$296$	13.1420	0.763864
$297$	0	0
$298$	14.4371	0.836321
$299$	−0.336280	−0.0194476
$300$	0	0
$301$	−18.1259	−1.04476
$302$	−18.7340	−1.07802
$303$	0.505326	0.0290302
$304$	−31.2271	−1.79100
$305$	0	0
$306$	12.2794	0.701969
$307$	−21.3566	−1.21889	−0.609444	−	0.792829i	$-0.708607\pi$
$307$	−21.3566	−1.21889	−0.609444	+	0.792829i	$0.708607\pi$
$308$	0	0
$309$	−6.40197	−0.364195
$310$	0	0
$311$	32.8096	1.86046	0.930231	−	0.366975i	$-0.119607\pi$
$311$	32.8096	1.86046	0.930231	+	0.366975i	$0.119607\pi$
$312$	1.56217	0.0884406
$313$	3.45852	0.195487	0.0977436	−	0.995212i	$-0.468837\pi$
$313$	3.45852	0.195487	0.0977436	+	0.995212i	$0.468837\pi$
$314$	−24.8860	−1.40440
$315$	0	0
$316$	21.8278	1.22791
$317$	2.87566	0.161513	0.0807565	−	0.996734i	$-0.474266\pi$
$317$	2.87566	0.161513	0.0807565	+	0.996734i	$0.474266\pi$
$318$	−12.1209	−0.679704
$319$	0	0
$320$	0	0
$321$	2.09249	0.116791
$322$	−8.66881	−0.483094
$323$	37.2704	2.07378
$324$	4.03138	0.223966
$325$	0	0
$326$	−12.3508	−0.684048
$327$	6.69278	0.370112
$328$	−54.1150	−2.98800
$329$	−39.4362	−2.17419
$330$	0	0
$331$	−14.1221	−0.776219	−0.388109	−	0.921613i	$-0.626872\pi$
$331$	−14.1221	−0.776219	−0.388109	+	0.921613i	$0.626872\pi$
$332$	−65.4915	−3.59431
$333$	−2.63428	−0.144358
$334$	14.2222	0.778202
$335$	0	0
$336$	13.7694	0.751185
$337$	−15.9490	−0.868796	−0.434398	−	0.900721i	$-0.643039\pi$
$337$	−15.9490	−0.868796	−0.434398	+	0.900721i	$0.643039\pi$
$338$	31.6857	1.72348
$339$	−10.7941	−0.586256
$340$	0	0
$341$	0	0
$342$	18.3064	0.989895
$343$	10.5070	0.567322
$344$	−27.5118	−1.48334
$345$	0	0
$346$	39.4543	2.12108
$347$	−29.6801	−1.59331	−0.796656	−	0.604433i	$-0.793400\pi$
$347$	−29.6801	−1.59331	−0.796656	+	0.604433i	$0.793400\pi$
$348$	20.3039	1.08840
$349$	31.6937	1.69653	0.848263	−	0.529574i	$-0.177648\pi$
$349$	31.6937	1.69653	0.848263	+	0.529574i	$0.177648\pi$
$350$	0	0
$351$	−0.313133	−0.0167138
$352$	0	0
$353$	−1.20189	−0.0639703	−0.0319852	−	0.999488i	$-0.510183\pi$
$353$	−1.20189	−0.0639703	−0.0319852	+	0.999488i	$0.510183\pi$
$354$	−22.5060	−1.19618
$355$	0	0
$356$	6.53559	0.346385
$357$	−16.4342	−0.869791
$358$	20.3915	1.07772
$359$	−11.6591	−0.615343	−0.307671	−	0.951493i	$-0.599550\pi$
$359$	−11.6591	−0.615343	−0.307671	+	0.951493i	$0.599550\pi$
$360$	0	0
$361$	36.5633	1.92438
$362$	15.8755	0.834396
$363$	0	0
$364$	−4.14918	−0.217476
$365$	0	0
$366$	22.5556	1.17900
$367$	−15.9860	−0.834465	−0.417232	−	0.908800i	$-0.637000\pi$
$367$	−15.9860	−0.834465	−0.417232	+	0.908800i	$0.637000\pi$
$368$	−4.49894	−0.234523
$369$	10.8472	0.564683
$370$	0	0
$371$	16.2220	0.842203
$372$	−13.9042	−0.720898
$373$	−0.321975	−0.0166712	−0.00833561	−	0.999965i	$-0.502653\pi$
$373$	−0.321975	−0.0166712	−0.00833561	+	0.999965i	$0.502653\pi$
$374$	0	0
$375$	0	0
$376$	−59.8570	−3.08689
$377$	−1.57709	−0.0812241
$378$	−8.07211	−0.415185
$379$	11.4174	0.586475	0.293237	−	0.956040i	$-0.405267\pi$
$379$	11.4174	0.586475	0.293237	+	0.956040i	$0.405267\pi$
$380$	0	0
$381$	−17.0033	−0.871104
$382$	−37.7452	−1.93121
$383$	−28.3673	−1.44950	−0.724750	−	0.689012i	$-0.758045\pi$
$383$	−28.3673	−1.44950	−0.724750	+	0.689012i	$0.758045\pi$
$384$	−19.3242	−0.986136
$385$	0	0
$386$	38.6846	1.96899
$387$	5.51468	0.280327
$388$	−0.903359	−0.0458611
$389$	15.1802	0.769666	0.384833	−	0.922986i	$-0.374259\pi$
$389$	15.1802	0.769666	0.384833	+	0.922986i	$0.374259\pi$
$390$	0	0
$391$	5.36960	0.271553
$392$	−18.9742	−0.958341
$393$	0.0430508	0.00217163
$394$	40.2617	2.02835
$395$	0	0
$396$	0	0
$397$	−5.22461	−0.262216	−0.131108	−	0.991368i	$-0.541853\pi$
$397$	−5.22461	−0.262216	−0.131108	+	0.991368i	$0.541853\pi$
$398$	−17.1053	−0.857409
$399$	−24.5004	−1.22655
$400$	0	0
$401$	14.0007	0.699160	0.349580	−	0.936907i	$-0.386324\pi$
$401$	14.0007	0.699160	0.349580	+	0.936907i	$0.386324\pi$
$402$	37.5109	1.87087
$403$	1.07999	0.0537983
$404$	−2.03716	−0.101352
$405$	0	0
$406$	−40.6549	−2.01767
$407$	0	0
$408$	−24.9442	−1.23492
$409$	33.3112	1.64713	0.823567	−	0.567218i	$-0.191980\pi$
$409$	33.3112	1.64713	0.823567	+	0.567218i	$0.191980\pi$
$410$	0	0
$411$	7.36257	0.363169
$412$	25.8088	1.27151
$413$	30.1209	1.48215
$414$	2.63743	0.129623
$415$	0	0
$416$	−0.0972843	−0.00476975
$417$	−13.2393	−0.648334
$418$	0	0
$419$	−5.28460	−0.258170	−0.129085	−	0.991634i	$-0.541204\pi$
$419$	−5.28460	−0.258170	−0.129085	+	0.991634i	$0.541204\pi$
$420$	0	0
$421$	−30.7810	−1.50017	−0.750087	−	0.661340i	$-0.769988\pi$
$421$	−30.7810	−1.50017	−0.750087	+	0.661340i	$0.769988\pi$
$422$	49.0026	2.38541
$423$	11.9982	0.583372
$424$	24.6221	1.19575
$425$	0	0
$426$	7.54476	0.365545
$427$	−30.1874	−1.46087
$428$	−8.43562	−0.407751
$429$	0	0
$430$	0	0
$431$	12.3506	0.594910	0.297455	−	0.954736i	$-0.403862\pi$
$431$	12.3506	0.594910	0.297455	+	0.954736i	$0.403862\pi$
$432$	−4.18926	−0.201556
$433$	−1.41287	−0.0678983	−0.0339491	−	0.999424i	$-0.510808\pi$
$433$	−1.41287	−0.0678983	−0.0339491	+	0.999424i	$0.510808\pi$
$434$	27.8406	1.33639
$435$	0	0
$436$	−26.9811	−1.29216
$437$	8.00509	0.382935
$438$	21.2500	1.01537
$439$	−7.58532	−0.362028	−0.181014	−	0.983481i	$-0.557938\pi$
$439$	−7.58532	−0.362028	−0.181014	+	0.983481i	$0.557938\pi$
$440$	0	0
$441$	3.80333	0.181111
$442$	3.84510	0.182893
$443$	−11.0662	−0.525771	−0.262885	−	0.964827i	$-0.584674\pi$
$443$	−11.0662	−0.525771	−0.262885	+	0.964827i	$0.584674\pi$
$444$	10.6198	0.503993
$445$	0	0
$446$	−49.4779	−2.34284
$447$	5.87858	0.278047
$448$	25.0311	1.18261
$449$	6.32856	0.298663	0.149332	−	0.988787i	$-0.452288\pi$
$449$	6.32856	0.298663	0.149332	+	0.988787i	$0.452288\pi$
$450$	0	0
$451$	0	0
$452$	43.5152	2.04678
$453$	−7.62821	−0.358405
$454$	1.31129	0.0615418
$455$	0	0
$456$	−37.1872	−1.74145
$457$	0.189579	0.00886814	0.00443407	−	0.999990i	$-0.498589\pi$
$457$	0.189579	0.00886814	0.00443407	+	0.999990i	$0.498589\pi$
$458$	54.5036	2.54679
$459$	5.00000	0.233380
$460$	0	0
$461$	−26.6198	−1.23981	−0.619904	−	0.784678i	$-0.712828\pi$
$461$	−26.6198	−1.23981	−0.619904	+	0.784678i	$0.712828\pi$
$462$	0	0
$463$	−20.9935	−0.975652	−0.487826	−	0.872941i	$-0.662210\pi$
$463$	−20.9935	−0.975652	−0.487826	+	0.872941i	$0.662210\pi$
$464$	−21.0991	−0.979501
$465$	0	0
$466$	−11.2128	−0.519421
$467$	−7.89989	−0.365563	−0.182782	−	0.983154i	$-0.558510\pi$
$467$	−7.89989	−0.365563	−0.182782	+	0.983154i	$0.558510\pi$
$468$	1.26236	0.0583526
$469$	−50.2028	−2.31815
$470$	0	0
$471$	−10.1332	−0.466914
$472$	45.7182	2.10435
$473$	0	0
$474$	13.2973	0.610766
$475$	0	0
$476$	66.2525	3.03668
$477$	−4.93543	−0.225978
$478$	14.3928	0.658311
$479$	39.9728	1.82640	0.913201	−	0.407509i	$-0.133602\pi$
$479$	39.9728	1.82640	0.913201	+	0.407509i	$0.133602\pi$
$480$	0	0
$481$	−0.824882	−0.0376114
$482$	−24.4624	−1.11423
$483$	−3.52981	−0.160612
$484$	0	0
$485$	0	0
$486$	2.45589	0.111401
$487$	9.93556	0.450223	0.225112	−	0.974333i	$-0.427725\pi$
$487$	9.93556	0.450223	0.225112	+	0.974333i	$0.427725\pi$
$488$	−45.8190	−2.07413
$489$	−5.02906	−0.227422
$490$	0	0
$491$	4.97349	0.224451	0.112225	−	0.993683i	$-0.464202\pi$
$491$	4.97349	0.224451	0.112225	+	0.993683i	$0.464202\pi$
$492$	−43.7292	−1.97146
$493$	25.1823	1.13416
$494$	5.73234	0.257910
$495$	0	0
$496$	14.4487	0.648767
$497$	−10.0975	−0.452937
$498$	−39.8969	−1.78782
$499$	43.7757	1.95967	0.979834	−	0.199812i	$-0.0640332\pi$
$499$	43.7757	1.95967	0.979834	+	0.199812i	$0.0640332\pi$
$500$	0	0
$501$	5.79105	0.258725
$502$	41.4524	1.85011
$503$	−6.28236	−0.280117	−0.140058	−	0.990143i	$-0.544729\pi$
$503$	−6.28236	−0.280117	−0.140058	+	0.990143i	$0.544729\pi$
$504$	16.3975	0.730404
$505$	0	0
$506$	0	0
$507$	12.9019	0.572996
$508$	68.5467	3.04127
$509$	24.8381	1.10093	0.550465	−	0.834858i	$-0.314450\pi$
$509$	24.8381	1.10093	0.550465	+	0.834858i	$0.314450\pi$
$510$	0	0
$511$	−28.4400	−1.25811
$512$	40.4976	1.78976
$513$	7.45408	0.329106
$514$	27.3674	1.20712
$515$	0	0
$516$	−22.2318	−0.978699
$517$	0	0
$518$	−21.2642	−0.934296
$519$	16.0652	0.705185
$520$	0	0
$521$	−6.94869	−0.304428	−0.152214	−	0.988348i	$-0.548640\pi$
$521$	−6.94869	−0.304428	−0.152214	+	0.988348i	$0.548640\pi$
$522$	12.3690	0.541376
$523$	26.7510	1.16974	0.584869	−	0.811128i	$-0.301146\pi$
$523$	26.7510	1.16974	0.584869	+	0.811128i	$0.301146\pi$
$524$	−0.173554	−0.00758175
$525$	0	0
$526$	−65.9962	−2.87757
$527$	−17.2449	−0.751202
$528$	0	0
$529$	−21.8467	−0.949856
$530$	0	0
$531$	−9.16409	−0.397688
$532$	98.7703	4.28224
$533$	3.39662	0.147124
$534$	3.98143	0.172293
$535$	0	0
$536$	−76.1989	−3.29129
$537$	8.30309	0.358305
$538$	24.7133	1.06547
$539$	0	0
$540$	0	0
$541$	14.5084	0.623767	0.311883	−	0.950120i	$-0.399040\pi$
$541$	14.5084	0.623767	0.311883	+	0.950120i	$0.399040\pi$
$542$	−25.8951	−1.11229
$543$	6.46425	0.277408
$544$	1.55340	0.0666015
$545$	0	0
$546$	−2.52765	−0.108173
$547$	26.7346	1.14309	0.571543	−	0.820572i	$-0.306345\pi$
$547$	26.7346	1.14309	0.571543	+	0.820572i	$0.306345\pi$
$548$	−29.6813	−1.26792
$549$	9.18431	0.391977
$550$	0	0
$551$	37.5422	1.59935
$552$	−5.35762	−0.228035
$553$	−17.7965	−0.756784
$554$	−44.0527	−1.87162
$555$	0	0
$556$	53.3728	2.26351
$557$	−17.2444	−0.730670	−0.365335	−	0.930876i	$-0.619046\pi$
$557$	−17.2444	−0.730670	−0.365335	+	0.930876i	$0.619046\pi$
$558$	−8.47033	−0.358578
$559$	1.72683	0.0730371
$560$	0	0
$561$	0	0
$562$	−18.2006	−0.767748
$563$	0.831914	0.0350610	0.0175305	−	0.999846i	$-0.494420\pi$
$563$	0.831914	0.0350610	0.0175305	+	0.999846i	$0.494420\pi$
$564$	−48.3693	−2.03671
$565$	0	0
$566$	13.2295	0.556076
$567$	−3.28684	−0.138034
$568$	−15.3263	−0.643076
$569$	11.5961	0.486132	0.243066	−	0.970010i	$-0.421847\pi$
$569$	11.5961	0.486132	0.243066	+	0.970010i	$0.421847\pi$
$570$	0	0
$571$	21.8414	0.914034	0.457017	−	0.889458i	$-0.348918\pi$
$571$	21.8414	0.914034	0.457017	+	0.889458i	$0.348918\pi$
$572$	0	0
$573$	−15.3693	−0.642061
$574$	87.5598	3.65468
$575$	0	0
$576$	−7.61553	−0.317314
$577$	−9.74587	−0.405726	−0.202863	−	0.979207i	$-0.565025\pi$
$577$	−9.74587	−0.405726	−0.202863	+	0.979207i	$0.565025\pi$
$578$	−19.6471	−0.817211
$579$	15.7518	0.654622
$580$	0	0
$581$	53.3961	2.21524
$582$	−0.550320	−0.0228115
$583$	0	0
$584$	−43.1669	−1.78626
$585$	0	0
$586$	−4.27339	−0.176532
$587$	−22.9441	−0.947005	−0.473502	−	0.880793i	$-0.657010\pi$
$587$	−22.9441	−0.947005	−0.473502	+	0.880793i	$0.657010\pi$
$588$	−15.3327	−0.632308
$589$	−25.7090	−1.05932
$590$	0	0
$591$	16.3940	0.674357
$592$	−11.0357	−0.453565
$593$	−28.7819	−1.18193	−0.590965	−	0.806697i	$-0.701253\pi$
$593$	−28.7819	−1.18193	−0.590965	+	0.806697i	$0.701253\pi$
$594$	0	0
$595$	0	0
$596$	−23.6988	−0.970741
$597$	−6.96500	−0.285059
$598$	0.825867	0.0337722
$599$	−29.1951	−1.19288	−0.596440	−	0.802657i	$-0.703419\pi$
$599$	−29.1951	−1.19288	−0.596440	+	0.802657i	$0.703419\pi$
$600$	0	0
$601$	−6.68087	−0.272518	−0.136259	−	0.990673i	$-0.543508\pi$
$601$	−6.68087	−0.272518	−0.136259	+	0.990673i	$0.543508\pi$
$602$	44.5151	1.81430
$603$	15.2739	0.622000
$604$	30.7522	1.25129
$605$	0	0
$606$	−1.24102	−0.0504131
$607$	42.6108	1.72952	0.864759	−	0.502188i	$-0.167471\pi$
$607$	42.6108	1.72952	0.864759	+	0.502188i	$0.167471\pi$
$608$	2.31583	0.0939194
$609$	−16.5541	−0.670805
$610$	0	0
$611$	3.75703	0.151993
$612$	−20.1569	−0.814794
$613$	5.83156	0.235535	0.117767	−	0.993041i	$-0.462426\pi$
$613$	5.83156	0.235535	0.117767	+	0.993041i	$0.462426\pi$
$614$	52.4495	2.11669
$615$	0	0
$616$	0	0
$617$	33.6386	1.35424	0.677119	−	0.735874i	$-0.263228\pi$
$617$	33.6386	1.35424	0.677119	+	0.735874i	$0.263228\pi$
$618$	15.7225	0.632452
$619$	42.5616	1.71070	0.855349	−	0.518053i	$-0.173343\pi$
$619$	42.5616	1.71070	0.855349	+	0.518053i	$0.173343\pi$
$620$	0	0
$621$	1.07392	0.0430950
$622$	−80.5766	−3.23083
$623$	−5.32856	−0.213484
$624$	−1.31180	−0.0525140
$625$	0	0
$626$	−8.49374	−0.339478
$627$	0	0
$628$	40.8509	1.63013
$629$	13.1714	0.525179
$630$	0	0
$631$	−8.89989	−0.354299	−0.177149	−	0.984184i	$-0.556688\pi$
$631$	−8.89989	−0.354299	−0.177149	+	0.984184i	$0.556688\pi$
$632$	−27.0119	−1.07448
$633$	19.9531	0.793065
$634$	−7.06228	−0.280479
$635$	0	0
$636$	19.8966	0.788951
$637$	1.19095	0.0471871
$638$	0	0
$639$	3.07211	0.121531
$640$	0	0
$641$	−6.16806	−0.243624	−0.121812	−	0.992553i	$-0.538870\pi$
$641$	−6.16806	−0.243624	−0.121812	+	0.992553i	$0.538870\pi$
$642$	−5.13892	−0.202817
$643$	−4.35335	−0.171680	−0.0858398	−	0.996309i	$-0.527357\pi$
$643$	−4.35335	−0.171680	−0.0858398	+	0.996309i	$0.527357\pi$
$644$	14.2300	0.560740
$645$	0	0
$646$	−91.5318	−3.60127
$647$	−13.4933	−0.530478	−0.265239	−	0.964183i	$-0.585451\pi$
$647$	−13.4933	−0.530478	−0.265239	+	0.964183i	$0.585451\pi$
$648$	−4.98884	−0.195980
$649$	0	0
$650$	0	0
$651$	11.3363	0.444304
$652$	20.2741	0.793993
$653$	−27.4481	−1.07413	−0.537064	−	0.843541i	$-0.680467\pi$
$653$	−27.4481	−1.07413	−0.537064	+	0.843541i	$0.680467\pi$
$654$	−16.4367	−0.642726
$655$	0	0
$656$	45.4418	1.77420
$657$	8.65269	0.337574
$658$	96.8507	3.77563
$659$	−18.7768	−0.731441	−0.365721	−	0.930725i	$-0.619177\pi$
$659$	−18.7768	−0.731441	−0.365721	+	0.930725i	$0.619177\pi$
$660$	0	0
$661$	−21.6525	−0.842184	−0.421092	−	0.907018i	$-0.638353\pi$
$661$	−21.6525	−0.842184	−0.421092	+	0.907018i	$0.638353\pi$
$662$	34.6822	1.34796
$663$	1.56567	0.0608055
$664$	81.0458	3.14519
$665$	0	0
$666$	6.46950	0.250688
$667$	5.40877	0.209428
$668$	−23.3459	−0.903281
$669$	−20.1466	−0.778914
$670$	0	0
$671$	0	0
$672$	−1.02116	−0.0393919
$673$	−23.3021	−0.898232	−0.449116	−	0.893474i	$-0.648261\pi$
$673$	−23.3021	−0.898232	−0.449116	+	0.893474i	$0.648261\pi$
$674$	39.1689	1.50873
$675$	0	0
$676$	−52.0127	−2.00049
$677$	−33.2808	−1.27909	−0.639543	−	0.768756i	$-0.720876\pi$
$677$	−33.2808	−1.27909	−0.639543	+	0.768756i	$0.720876\pi$
$678$	26.5091	1.01808
$679$	0.736522	0.0282651
$680$	0	0
$681$	0.533937	0.0204605
$682$	0	0
$683$	−16.9244	−0.647593	−0.323796	−	0.946127i	$-0.604959\pi$
$683$	−16.9244	−0.647593	−0.323796	+	0.946127i	$0.604959\pi$
$684$	−30.0502	−1.14900
$685$	0	0
$686$	−25.8039	−0.985197
$687$	22.1931	0.846718
$688$	23.1024	0.880772
$689$	−1.54545	−0.0588769
$690$	0	0
$691$	−48.4335	−1.84250	−0.921249	−	0.388973i	$-0.872830\pi$
$691$	−48.4335	−1.84250	−0.921249	+	0.388973i	$0.872830\pi$
$692$	−64.7650	−2.46200
$693$	0	0
$694$	72.8910	2.76690
$695$	0	0
$696$	−25.1261	−0.952403
$697$	−54.2360	−2.05434
$698$	−77.8362	−2.94614
$699$	−4.56567	−0.172689
$700$	0	0
$701$	−45.4161	−1.71534	−0.857672	−	0.514197i	$-0.828090\pi$
$701$	−45.4161	−1.71534	−0.857672	+	0.514197i	$0.828090\pi$
$702$	0.769020	0.0290248
$703$	19.6361	0.740591
$704$	0	0
$705$	0	0
$706$	2.95171	0.111089
$707$	1.66093	0.0624655
$708$	36.9439	1.38844
$709$	−1.76497	−0.0662848	−0.0331424	−	0.999451i	$-0.510551\pi$
$709$	−1.76497	−0.0662848	−0.0331424	+	0.999451i	$0.510551\pi$
$710$	0	0
$711$	5.41446	0.203058
$712$	−8.08780	−0.303103
$713$	−3.70394	−0.138714
$714$	40.3606	1.51046
$715$	0	0
$716$	−33.4729	−1.25094
$717$	5.86053	0.218865
$718$	28.6334	1.06859
$719$	−3.29998	−0.123069	−0.0615343	−	0.998105i	$-0.519599\pi$
$719$	−3.29998	−0.123069	−0.0615343	+	0.998105i	$0.519599\pi$
$720$	0	0
$721$	−21.0423	−0.783655
$722$	−89.7952	−3.34183
$723$	−9.96074	−0.370444
$724$	−26.0599	−0.968507
$725$	0	0
$726$	0	0
$727$	−11.7838	−0.437037	−0.218519	−	0.975833i	$-0.570122\pi$
$727$	−11.7838	−0.437037	−0.218519	+	0.975833i	$0.570122\pi$
$728$	5.13461	0.190301
$729$	1.00000	0.0370370
$730$	0	0
$731$	−27.5734	−1.01984
$732$	−37.0254	−1.36850
$733$	−5.73108	−0.211682	−0.105841	−	0.994383i	$-0.533754\pi$
$733$	−5.73108	−0.211682	−0.105841	+	0.994383i	$0.533754\pi$
$734$	39.2599	1.44911
$735$	0	0
$736$	0.333646	0.0122983
$737$	0	0
$738$	−26.6395	−0.980614
$739$	−21.0551	−0.774524	−0.387262	−	0.921970i	$-0.626579\pi$
$739$	−21.0551	−0.774524	−0.387262	+	0.921970i	$0.626579\pi$
$740$	0	0
$741$	2.33412	0.0857461
$742$	−39.8393	−1.46255
$743$	−13.1283	−0.481630	−0.240815	−	0.970571i	$-0.577415\pi$
$743$	−13.1283	−0.481630	−0.240815	+	0.970571i	$0.577415\pi$
$744$	17.2065	0.630819
$745$	0	0
$746$	0.790734	0.0289508
$747$	−16.2454	−0.594389
$748$	0	0
$749$	6.87768	0.251305
$750$	0	0
$751$	25.6251	0.935073	0.467537	−	0.883974i	$-0.345142\pi$
$751$	25.6251	0.935073	0.467537	+	0.883974i	$0.345142\pi$
$752$	50.2636	1.83292
$753$	16.8788	0.615098
$754$	3.87315	0.141052
$755$	0	0
$756$	13.2505	0.481916
$757$	−31.1970	−1.13387	−0.566936	−	0.823762i	$-0.691871\pi$
$757$	−31.1970	−1.13387	−0.566936	+	0.823762i	$0.691871\pi$
$758$	−28.0400	−1.01846
$759$	0	0
$760$	0	0
$761$	11.3761	0.412382	0.206191	−	0.978512i	$-0.433893\pi$
$761$	11.3761	0.412382	0.206191	+	0.978512i	$0.433893\pi$
$762$	41.7581	1.51274
$763$	21.9981	0.796385
$764$	61.9594	2.24161
$765$	0	0
$766$	69.6668	2.51717
$767$	−2.86958	−0.103615
$768$	32.2271	1.16290
$769$	10.3938	0.374811	0.187405	−	0.982283i	$-0.439992\pi$
$769$	10.3938	0.374811	0.187405	+	0.982283i	$0.439992\pi$
$770$	0	0
$771$	11.1436	0.401326
$772$	−63.5014	−2.28547
$773$	−14.0348	−0.504796	−0.252398	−	0.967623i	$-0.581219\pi$
$773$	−14.0348	−0.504796	−0.252398	+	0.967623i	$0.581219\pi$
$774$	−13.5434	−0.486808
$775$	0	0
$776$	1.11791	0.0401306
$777$	−8.65847	−0.310621
$778$	−37.2808	−1.33658
$779$	−80.8559	−2.89696
$780$	0	0
$781$	0	0
$782$	−13.1871	−0.471571
$783$	5.03647	0.179989
$784$	15.9331	0.569041
$785$	0	0
$786$	−0.105728	−0.00377119
$787$	13.8176	0.492545	0.246273	−	0.969201i	$-0.420794\pi$
$787$	13.8176	0.492545	0.246273	+	0.969201i	$0.420794\pi$
$788$	−66.0902	−2.35437
$789$	−26.8726	−0.956691
$790$	0	0
$791$	−35.4785	−1.26147
$792$	0	0
$793$	2.87591	0.102127
$794$	12.8311	0.455357
$795$	0	0
$796$	28.0786	0.995218
$797$	5.38594	0.190780	0.0953898	−	0.995440i	$-0.469590\pi$
$797$	5.38594	0.190780	0.0953898	+	0.995440i	$0.469590\pi$
$798$	60.1701	2.13000
$799$	−59.9910	−2.12233
$800$	0	0
$801$	1.62118	0.0572815
$802$	−34.3840	−1.21414
$803$	0	0
$804$	−61.5748	−2.17157
$805$	0	0
$806$	−2.65234	−0.0934248
$807$	10.0629	0.354231
$808$	2.52099	0.0886880
$809$	23.7748	0.835876	0.417938	−	0.908476i	$-0.362753\pi$
$809$	23.7748	0.835876	0.417938	+	0.908476i	$0.362753\pi$
$810$	0	0
$811$	9.46335	0.332303	0.166152	−	0.986100i	$-0.446866\pi$
$811$	9.46335	0.332303	0.166152	+	0.986100i	$0.446866\pi$
$812$	66.7358	2.34197
$813$	−10.5441	−0.369798
$814$	0	0
$815$	0	0
$816$	20.9463	0.733268
$817$	−41.1068	−1.43815
$818$	−81.8086	−2.86037
$819$	−1.02922	−0.0359639
$820$	0	0
$821$	−10.4189	−0.363622	−0.181811	−	0.983334i	$-0.558196\pi$
$821$	−10.4189	−0.363622	−0.181811	+	0.983334i	$0.558196\pi$
$822$	−18.0816	−0.630670
$823$	−24.3540	−0.848928	−0.424464	−	0.905445i	$-0.639537\pi$
$823$	−24.3540	−0.848928	−0.424464	+	0.905445i	$0.639537\pi$
$824$	−31.9384	−1.11263
$825$	0	0
$826$	−73.9736	−2.57387
$827$	22.0006	0.765037	0.382519	−	0.923948i	$-0.375057\pi$
$827$	22.0006	0.765037	0.382519	+	0.923948i	$0.375057\pi$
$828$	−4.32938	−0.150456
$829$	3.03289	0.105337	0.0526683	−	0.998612i	$-0.483227\pi$
$829$	3.03289	0.105337	0.0526683	+	0.998612i	$0.483227\pi$
$830$	0	0
$831$	−17.9376	−0.622248
$832$	−2.38468	−0.0826738
$833$	−19.0166	−0.658888
$834$	32.5143	1.12588
$835$	0	0
$836$	0	0
$837$	−3.44899	−0.119214
$838$	12.9784	0.448331
$839$	−48.5383	−1.67573	−0.837864	−	0.545879i	$-0.816196\pi$
$839$	−48.5383	−1.67573	−0.837864	+	0.545879i	$0.816196\pi$
$840$	0	0
$841$	−3.63399	−0.125310
$842$	75.5946	2.60516
$843$	−7.41103	−0.255249
$844$	−80.4386	−2.76881
$845$	0	0
$846$	−29.4662	−1.01307
$847$	0	0
$848$	−20.6758	−0.710011
$849$	5.38684	0.184876
$850$	0	0
$851$	2.82901	0.0969773
$852$	−12.3848	−0.424298
$853$	11.7632	0.402766	0.201383	−	0.979513i	$-0.435456\pi$
$853$	11.7632	0.402766	0.201383	+	0.979513i	$0.435456\pi$
$854$	74.1368	2.53691
$855$	0	0
$856$	10.4391	0.356801
$857$	−1.61311	−0.0551026	−0.0275513	−	0.999620i	$-0.508771\pi$
$857$	−1.61311	−0.0551026	−0.0275513	+	0.999620i	$0.508771\pi$
$858$	0	0
$859$	47.3263	1.61475	0.807376	−	0.590038i	$-0.200887\pi$
$859$	47.3263	1.61475	0.807376	+	0.590038i	$0.200887\pi$
$860$	0	0
$861$	35.6530	1.21505
$862$	−30.3318	−1.03310
$863$	9.97233	0.339462	0.169731	−	0.985490i	$-0.445710\pi$
$863$	9.97233	0.339462	0.169731	+	0.985490i	$0.445710\pi$
$864$	0.310680	0.0105695
$865$	0	0
$866$	3.46985	0.117910
$867$	−8.00000	−0.271694
$868$	−45.7009	−1.55119
$869$	0	0
$870$	0	0
$871$	4.78276	0.162058
$872$	33.3892	1.13070
$873$	−0.224082	−0.00758402
$874$	−19.6596	−0.664996
$875$	0	0
$876$	−34.8823	−1.17856
$877$	−26.0057	−0.878151	−0.439076	−	0.898450i	$-0.644694\pi$
$877$	−26.0057	−0.878151	−0.439076	+	0.898450i	$0.644694\pi$
$878$	18.6287	0.628688
$879$	−1.74006	−0.0586907
$880$	0	0
$881$	10.4081	0.350657	0.175329	−	0.984510i	$-0.443901\pi$
$881$	10.4081	0.350657	0.175329	+	0.984510i	$0.443901\pi$
$882$	−9.34054	−0.314512
$883$	53.7283	1.80810	0.904051	−	0.427424i	$-0.140579\pi$
$883$	53.7283	1.80810	0.904051	+	0.427424i	$0.140579\pi$
$884$	−6.31180	−0.212289
$885$	0	0
$886$	27.1773	0.913040
$887$	40.6246	1.36404	0.682021	−	0.731333i	$-0.261101\pi$
$887$	40.6246	1.36404	0.682021	+	0.731333i	$0.261101\pi$
$888$	−13.1420	−0.441017
$889$	−55.8871	−1.87439
$890$	0	0
$891$	0	0
$892$	81.2188	2.71941
$893$	−89.4354	−2.99284
$894$	−14.4371	−0.482850
$895$	0	0
$896$	−63.5157	−2.12191
$897$	0.336280	0.0112281
$898$	−15.5422	−0.518651
$899$	−17.3707	−0.579346
$900$	0	0
$901$	24.6772	0.822115
$902$	0	0
$903$	18.1259	0.603191
$904$	−53.8501	−1.79103
$905$	0	0
$906$	18.7340	0.622396
$907$	19.4070	0.644398	0.322199	−	0.946672i	$-0.395578\pi$
$907$	19.4070	0.644398	0.322199	+	0.946672i	$0.395578\pi$
$908$	−2.15250	−0.0714333
$909$	−0.505326	−0.0167606
$910$	0	0
$911$	10.7208	0.355195	0.177597	−	0.984103i	$-0.443168\pi$
$911$	10.7208	0.355195	0.177597	+	0.984103i	$0.443168\pi$
$912$	31.2271	1.03403
$913$	0	0
$914$	−0.465585	−0.0154002
$915$	0	0
$916$	−89.4686	−2.95613
$917$	0.141501	0.00467278
$918$	−12.2794	−0.405282
$919$	−23.1310	−0.763021	−0.381511	−	0.924364i	$-0.624596\pi$
$919$	−23.1310	−0.763021	−0.381511	+	0.924364i	$0.624596\pi$
$920$	0	0
$921$	21.3566	0.703725
$922$	65.3752	2.15302
$923$	0.961981	0.0316640
$924$	0	0
$925$	0	0
$926$	51.5577	1.69429
$927$	6.40197	0.210268
$928$	1.56473	0.0513648
$929$	26.2273	0.860489	0.430245	−	0.902712i	$-0.358427\pi$
$929$	26.2273	0.860489	0.430245	+	0.902712i	$0.358427\pi$
$930$	0	0
$931$	−28.3503	−0.929144
$932$	18.4059	0.602907
$933$	−32.8096	−1.07414
$934$	19.4012	0.634828
$935$	0	0
$936$	−1.56217	−0.0510612
$937$	−23.4011	−0.764480	−0.382240	−	0.924063i	$-0.624847\pi$
$937$	−23.4011	−0.764480	−0.382240	+	0.924063i	$0.624847\pi$
$938$	123.292	4.02564
$939$	−3.45852	−0.112865
$940$	0	0
$941$	−10.9687	−0.357570	−0.178785	−	0.983888i	$-0.557217\pi$
$941$	−10.9687	−0.357570	−0.178785	+	0.983888i	$0.557217\pi$
$942$	24.8860	0.810831
$943$	−11.6490	−0.379345
$944$	−38.3908	−1.24951
$945$	0	0
$946$	0	0
$947$	13.3652	0.434310	0.217155	−	0.976137i	$-0.430322\pi$
$947$	13.3652	0.434310	0.217155	+	0.976137i	$0.430322\pi$
$948$	−21.8278	−0.708933
$949$	2.70945	0.0879524
$950$	0	0
$951$	−2.87566	−0.0932495
$952$	−81.9876	−2.65723
$953$	21.8242	0.706956	0.353478	−	0.935443i	$-0.384999\pi$
$953$	21.8242	0.706956	0.353478	+	0.935443i	$0.384999\pi$
$954$	12.1209	0.392427
$955$	0	0
$956$	−23.6260	−0.764120
$957$	0	0
$958$	−98.1686	−3.17168
$959$	24.1996	0.781446
$960$	0	0
$961$	−19.1045	−0.616273
$962$	2.02582	0.0653150
$963$	−2.09249	−0.0674295
$964$	40.1555	1.29332
$965$	0	0
$966$	8.66881	0.278914
$967$	−16.6600	−0.535750	−0.267875	−	0.963454i	$-0.586321\pi$
$967$	−16.6600	−0.535750	−0.267875	+	0.963454i	$0.586321\pi$
$968$	0	0
$969$	−37.2704	−1.19730
$970$	0	0
$971$	11.1032	0.356320	0.178160	−	0.984002i	$-0.442986\pi$
$971$	11.1032	0.356320	0.178160	+	0.984002i	$0.442986\pi$
$972$	−4.03138	−0.129307
$973$	−43.5156	−1.39505
$974$	−24.4006	−0.781846
$975$	0	0
$976$	38.4755	1.23157
$977$	−18.8144	−0.601926	−0.300963	−	0.953636i	$-0.597308\pi$
$977$	−18.8144	−0.601926	−0.300963	+	0.953636i	$0.597308\pi$
$978$	12.3508	0.394935
$979$	0	0
$980$	0	0
$981$	−6.69278	−0.213684
$982$	−12.2143	−0.389775
$983$	1.37848	0.0439667	0.0219833	−	0.999758i	$-0.493002\pi$
$983$	1.37848	0.0439667	0.0219833	+	0.999758i	$0.493002\pi$
$984$	54.1150	1.72512
$985$	0	0
$986$	−61.8450	−1.96955
$987$	39.4362	1.25527
$988$	−9.40973	−0.299363
$989$	−5.92233	−0.188319
$990$	0	0
$991$	−46.3186	−1.47136	−0.735680	−	0.677329i	$-0.763137\pi$
$991$	−46.3186	−1.47136	−0.735680	+	0.677329i	$0.763137\pi$
$992$	−1.07153	−0.0340212
$993$	14.1221	0.448150
$994$	24.7984	0.786558
$995$	0	0
$996$	65.4915	2.07518
$997$	−14.5470	−0.460709	−0.230355	−	0.973107i	$-0.573989\pi$
$997$	−14.5470	−0.460709	−0.230355	+	0.973107i	$0.573989\pi$
$998$	−107.508	−3.40311
$999$	2.63428	0.0833450

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cl.1.2		4
5.4	even	2		1815.2.a.x.1.3		4
11.3	even	5		825.2.n.k.526.1		8
11.4	even	5		825.2.n.k.676.1		8
11.10	odd	2		9075.2.a.dj.1.3		4
15.14	odd	2		5445.2.a.be.1.2		4
55.3	odd	20		825.2.bx.h.724.4		16
55.4	even	10		165.2.m.a.16.2	✓	8
55.14	even	10		165.2.m.a.31.2	yes	8
55.37	odd	20		825.2.bx.h.49.4		16
55.47	odd	20		825.2.bx.h.724.1		16
55.48	odd	20		825.2.bx.h.49.1		16
55.54	odd	2		1815.2.a.o.1.2		4
165.14	odd	10		495.2.n.d.361.1		8
165.59	odd	10		495.2.n.d.181.1		8
165.164	even	2		5445.2.a.bv.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.a.16.2	✓	8	55.4	even	10
165.2.m.a.31.2	yes	8	55.14	even	10
495.2.n.d.181.1		8	165.59	odd	10
495.2.n.d.361.1		8	165.14	odd	10
825.2.n.k.526.1		8	11.3	even	5
825.2.n.k.676.1		8	11.4	even	5
825.2.bx.h.49.1		16	55.48	odd	20
825.2.bx.h.49.4		16	55.37	odd	20
825.2.bx.h.724.1		16	55.47	odd	20
825.2.bx.h.724.4		16	55.3	odd	20
1815.2.a.o.1.2		4	55.54	odd	2
1815.2.a.x.1.3		4	5.4	even	2
5445.2.a.be.1.2		4	15.14	odd	2
5445.2.a.bv.1.3		4	165.164	even	2
9075.2.a.cl.1.2		4	1.1	even	1	trivial
9075.2.a.dj.1.3		4	11.10	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)

\(f(q)\)	\(=\)	\(q-2.45589 q^{2} -1.00000 q^{3} +4.03138 q^{4} +2.45589 q^{6} -3.28684 q^{7} -4.98884 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.45589 q^{2} -1.00000 q^{3} +4.03138 q^{4} +2.45589 q^{6} -3.28684 q^{7} -4.98884 q^{8} +1.00000 q^{9} -4.03138 q^{12} +0.313133 q^{13} +8.07211 q^{14} +4.18926 q^{16} -5.00000 q^{17} -2.45589 q^{18} -7.45408 q^{19} +3.28684 q^{21} -1.07392 q^{23} +4.98884 q^{24} -0.769020 q^{26} -1.00000 q^{27} -13.2505 q^{28} -5.03647 q^{29} +3.44899 q^{31} -0.310680 q^{32} +12.2794 q^{34} +4.03138 q^{36} -2.63428 q^{37} +18.3064 q^{38} -0.313133 q^{39} +10.8472 q^{41} -8.07211 q^{42} +5.51468 q^{43} +2.63743 q^{46} +11.9982 q^{47} -4.18926 q^{48} +3.80333 q^{49} +5.00000 q^{51} +1.26236 q^{52} -4.93543 q^{53} +2.45589 q^{54} +16.3975 q^{56} +7.45408 q^{57} +12.3690 q^{58} -9.16409 q^{59} +9.18431 q^{61} -8.47033 q^{62} -3.28684 q^{63} -7.61553 q^{64} +15.2739 q^{67} -20.1569 q^{68} +1.07392 q^{69} +3.07211 q^{71} -4.98884 q^{72} +8.65269 q^{73} +6.46950 q^{74} -30.0502 q^{76} +0.769020 q^{78} +5.41446 q^{79} +1.00000 q^{81} -26.6395 q^{82} -16.2454 q^{83} +13.2505 q^{84} -13.5434 q^{86} +5.03647 q^{87} +1.62118 q^{89} -1.02922 q^{91} -4.32938 q^{92} -3.44899 q^{93} -29.4662 q^{94} +0.310680 q^{96} -0.224082 q^{97} -9.34054 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 5 * q^2 - 4 * q^3 + 9 * q^4 + 5 * q^6 + 2 * q^7 - 15 * q^8 + 4 * q^9	\( 4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9} - 9 q^{12} - 3 q^{13} - 5 q^{14} + 15 q^{16} - 20 q^{17} - 5 q^{18} + 3 q^{19} - 2 q^{21} + 5 q^{23} + 15 q^{24} + 6 q^{26} - 4 q^{27} - 3 q^{28} + 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} + 9 q^{36} + 7 q^{37} - q^{38} + 3 q^{39} + 20 q^{41} + 5 q^{42} + 2 q^{43} + 7 q^{46} + 20 q^{47} - 15 q^{48} + 8 q^{49} + 20 q^{51} + 7 q^{52} - 6 q^{53} + 5 q^{54} + 10 q^{56} - 3 q^{57} + 21 q^{58} - 5 q^{59} - 7 q^{61} + 12 q^{62} + 2 q^{63} + 49 q^{64} + 13 q^{67} - 45 q^{68} - 5 q^{69} - 25 q^{71} - 15 q^{72} - 23 q^{73} - 7 q^{74} - 7 q^{76} - 6 q^{78} + 4 q^{81} - 11 q^{82} - 33 q^{83} + 3 q^{84} - 12 q^{86} - 5 q^{87} + 16 q^{89} - 24 q^{91} + q^{93} - 17 q^{94} + 30 q^{96} - 25 q^{98}+O(q^{100}) \) 4 * q - 5 * q^2 - 4 * q^3 + 9 * q^4 + 5 * q^6 + 2 * q^7 - 15 * q^8 + 4 * q^9 - 9 * q^12 - 3 * q^13 - 5 * q^14 + 15 * q^16 - 20 * q^17 - 5 * q^18 + 3 * q^19 - 2 * q^21 + 5 * q^23 + 15 * q^24 + 6 * q^26 - 4 * q^27 - 3 * q^28 + 5 * q^29 - q^31 - 30 * q^32 + 25 * q^34 + 9 * q^36 + 7 * q^37 - q^38 + 3 * q^39 + 20 * q^41 + 5 * q^42 + 2 * q^43 + 7 * q^46 + 20 * q^47 - 15 * q^48 + 8 * q^49 + 20 * q^51 + 7 * q^52 - 6 * q^53 + 5 * q^54 + 10 * q^56 - 3 * q^57 + 21 * q^58 - 5 * q^59 - 7 * q^61 + 12 * q^62 + 2 * q^63 + 49 * q^64 + 13 * q^67 - 45 * q^68 - 5 * q^69 - 25 * q^71 - 15 * q^72 - 23 * q^73 - 7 * q^74 - 7 * q^76 - 6 * q^78 + 4 * q^81 - 11 * q^82 - 33 * q^83 + 3 * q^84 - 12 * q^86 - 5 * q^87 + 16 * q^89 - 24 * q^91 + q^93 - 17 * q^94 + 30 * q^96 - 25 * q^98

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