LMFDB - Embedded newform 5445.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
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			1.1
Root			$-0.477260$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.77222 q^{2} +5.68522 q^{4} +1.00000 q^{5} -2.27759 q^{7} -10.2163 q^{8} +O(q^{10})$	$q-2.77222 q^{2} +5.68522 q^{4} +1.00000 q^{5} -2.27759 q^{7} -10.2163 q^{8} -2.77222 q^{10} -0.435737 q^{13} +6.31399 q^{14} +16.9513 q^{16} -5.00000 q^{17} +4.69596 q^{19} +5.68522 q^{20} +0.845811 q^{23} +1.00000 q^{25} +1.20796 q^{26} -12.9486 q^{28} +2.65711 q^{29} -4.66785 q^{31} -26.5602 q^{32} +13.8611 q^{34} -2.27759 q^{35} -8.86239 q^{37} -13.0182 q^{38} -10.2163 q^{40} +4.29417 q^{41} +7.00317 q^{43} -2.34478 q^{46} -0.468179 q^{47} -1.81258 q^{49} -2.77222 q^{50} -2.47726 q^{52} +10.8386 q^{53} +23.2684 q^{56} -7.36611 q^{58} -3.93598 q^{59} +2.96549 q^{61} +12.9403 q^{62} +39.7283 q^{64} -0.435737 q^{65} -2.47048 q^{67} -28.4261 q^{68} +6.31399 q^{70} +11.3140 q^{71} +8.42910 q^{73} +24.5685 q^{74} +26.6975 q^{76} -10.8707 q^{79} +16.9513 q^{80} -11.9044 q^{82} +5.92050 q^{83} -5.00000 q^{85} -19.4143 q^{86} -5.89958 q^{89} +0.992430 q^{91} +4.80862 q^{92} +1.29790 q^{94} +4.69596 q^{95} -8.64803 q^{97} +5.02487 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{2} + 9 q^{4} + 4 q^{5} - 2 q^{7} - 15 q^{8}+O(q^{10}) $ 4 * q - 5 * q^2 + 9 * q^4 + 4 * q^5 - 2 * q^7 - 15 * q^8	$ 4 q - 5 q^{2} + 9 q^{4} + 4 q^{5} - 2 q^{7} - 15 q^{8} - 5 q^{10} + 3 q^{13} + 5 q^{14} + 15 q^{16} - 20 q^{17} + 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} + 3 q^{28} - 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} - 2 q^{35} - 7 q^{37} - q^{38} - 15 q^{40} - 20 q^{41} - 2 q^{43} + 7 q^{46} + 20 q^{47} + 8 q^{49} - 5 q^{50} - 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} - 7 q^{61} + 12 q^{62} + 49 q^{64} + 3 q^{65} - 13 q^{67} - 45 q^{68} + 5 q^{70} + 25 q^{71} + 23 q^{73} + 7 q^{74} - 7 q^{76} + 15 q^{80} + 11 q^{82} - 33 q^{83} - 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} - 17 q^{94} + 3 q^{95} - 25 q^{98}+O(q^{100}) $ 4 * q - 5 * q^2 + 9 * q^4 + 4 * q^5 - 2 * q^7 - 15 * q^8 - 5 * q^10 + 3 * q^13 + 5 * q^14 + 15 * q^16 - 20 * q^17 + 3 * q^19 + 9 * q^20 + 5 * q^23 + 4 * q^25 - 6 * q^26 + 3 * q^28 - 5 * q^29 - q^31 - 30 * q^32 + 25 * q^34 - 2 * q^35 - 7 * q^37 - q^38 - 15 * q^40 - 20 * q^41 - 2 * q^43 + 7 * q^46 + 20 * q^47 + 8 * q^49 - 5 * q^50 - 7 * q^52 - 6 * q^53 - 10 * q^56 - 21 * q^58 + 5 * q^59 - 7 * q^61 + 12 * q^62 + 49 * q^64 + 3 * q^65 - 13 * q^67 - 45 * q^68 + 5 * q^70 + 25 * q^71 + 23 * q^73 + 7 * q^74 - 7 * q^76 + 15 * q^80 + 11 * q^82 - 33 * q^83 - 20 * q^85 + 12 * q^86 - 16 * q^89 - 24 * q^91 - 17 * q^94 + 3 * q^95 - 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 $

$2$	−2.77222	−1.96026	−0.980129	−	0.198362i	$-0.936438\pi$
$2$	−2.77222	−1.96026	−0.980129	+	0.198362i	$0.936438\pi$
$3$	0	0
$3$	0	0
$4$	5.68522	2.84261
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.27759	−0.860849	−0.430424	−	0.902627i	$-0.641636\pi$
$7$	−2.27759	−0.860849	−0.430424	+	0.902627i	$0.641636\pi$
$8$	−10.2163	−3.61199
$9$	0	0
$10$	−2.77222	−0.876654
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.435737	−0.120852	−0.0604258	−	0.998173i	$-0.519246\pi$
$13$	−0.435737	−0.120852	−0.0604258	+	0.998173i	$0.519246\pi$
$14$	6.31399	1.68748
$15$	0	0
$16$	16.9513	4.23782
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	4.69596	1.07733	0.538663	−	0.842521i	$-0.318930\pi$
$19$	4.69596	1.07733	0.538663	+	0.842521i	$0.318930\pi$
$20$	5.68522	1.27125
$21$	0	0
$22$	0	0
$23$	0.845811	0.176364	0.0881819	−	0.996104i	$-0.471894\pi$
$23$	0.845811	0.176364	0.0881819	+	0.996104i	$0.471894\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.20796	0.236900
$27$	0	0
$28$	−12.9486	−2.44706
$29$	2.65711	0.493413	0.246707	−	0.969090i	$-0.420652\pi$
$29$	2.65711	0.493413	0.246707	+	0.969090i	$0.420652\pi$
$30$	0	0
$31$	−4.66785	−0.838370	−0.419185	−	0.907901i	$-0.637684\pi$
$31$	−4.66785	−0.838370	−0.419185	+	0.907901i	$0.637684\pi$
$32$	−26.5602	−4.69523
$33$	0	0
$34$	13.8611	2.37716
$35$	−2.27759	−0.384983
$36$	0	0
$37$	−8.86239	−1.45697	−0.728484	−	0.685063i	$-0.759775\pi$
$37$	−8.86239	−1.45697	−0.728484	+	0.685063i	$0.759775\pi$
$38$	−13.0182	−2.11184
$39$	0	0
$40$	−10.2163	−1.61533
$41$	4.29417	0.670637	0.335319	−	0.942105i	$-0.391156\pi$
$41$	4.29417	0.670637	0.335319	+	0.942105i	$0.391156\pi$
$42$	0	0
$43$	7.00317	1.06797	0.533986	−	0.845493i	$-0.320693\pi$
$43$	7.00317	1.06797	0.533986	+	0.845493i	$0.320693\pi$
$44$	0	0
$45$	0	0
$46$	−2.34478	−0.345718
$47$	−0.468179	−0.0682909	−0.0341455	−	0.999417i	$-0.510871\pi$
$47$	−0.468179	−0.0682909	−0.0341455	+	0.999417i	$0.510871\pi$
$48$	0	0
$49$	−1.81258	−0.258940
$50$	−2.77222	−0.392052
$51$	0	0
$52$	−2.47726	−0.343534
$53$	10.8386	1.48880	0.744399	−	0.667735i	$-0.232736\pi$
$53$	10.8386	1.48880	0.744399	+	0.667735i	$0.232736\pi$
$54$	0	0
$55$	0	0
$56$	23.2684	3.10938
$57$	0	0
$58$	−7.36611	−0.967217
$59$	−3.93598	−0.512421	−0.256211	−	0.966621i	$-0.582474\pi$
$59$	−3.93598	−0.512421	−0.256211	+	0.966621i	$0.582474\pi$
$60$	0	0
$61$	2.96549	0.379692	0.189846	−	0.981814i	$-0.439201\pi$
$61$	2.96549	0.379692	0.189846	+	0.981814i	$0.439201\pi$
$62$	12.9403	1.64342
$63$	0	0
$64$	39.7283	4.96604
$65$	−0.435737	−0.0540465
$66$	0	0
$67$	−2.47048	−0.301817	−0.150909	−	0.988548i	$-0.548220\pi$
$67$	−2.47048	−0.301817	−0.150909	+	0.988548i	$0.548220\pi$
$68$	−28.4261	−3.44717
$69$	0	0
$70$	6.31399	0.754666
$71$	11.3140	1.34272	0.671362	−	0.741130i	$-0.265710\pi$
$71$	11.3140	1.34272	0.671362	+	0.741130i	$0.265710\pi$
$72$	0	0
$73$	8.42910	0.986552	0.493276	−	0.869873i	$-0.335799\pi$
$73$	8.42910	0.986552	0.493276	+	0.869873i	$0.335799\pi$
$74$	24.5685	2.85603
$75$	0	0
$76$	26.6975	3.06242
$77$	0	0
$78$	0	0
$79$	−10.8707	−1.22305	−0.611524	−	0.791226i	$-0.709443\pi$
$79$	−10.8707	−1.22305	−0.611524	+	0.791226i	$0.709443\pi$
$80$	16.9513	1.89521
$81$	0	0
$82$	−11.9044	−1.31462
$83$	5.92050	0.649859	0.324930	−	0.945738i	$-0.394659\pi$
$83$	5.92050	0.649859	0.324930	+	0.945738i	$0.394659\pi$
$84$	0	0
$85$	−5.00000	−0.542326
$86$	−19.4143	−2.09350
$87$	0	0
$88$	0	0
$89$	−5.89958	−0.625354	−0.312677	−	0.949859i	$-0.601226\pi$
$89$	−5.89958	−0.625354	−0.312677	+	0.949859i	$0.601226\pi$
$90$	0	0
$91$	0.992430	0.104035
$92$	4.80862	0.501333
$93$	0	0
$94$	1.29790	0.133868
$95$	4.69596	0.481795
$96$	0	0
$97$	−8.64803	−0.878074	−0.439037	−	0.898469i	$-0.644680\pi$
$97$	−8.64803	−0.878074	−0.439037	+	0.898469i	$0.644680\pi$
$98$	5.02487	0.507589
$99$	0	0
$100$	5.68522	0.568522
$101$	5.68126	0.565307	0.282653	−	0.959222i	$-0.408785\pi$
$101$	5.68126	0.565307	0.282653	+	0.959222i	$0.408785\pi$
$102$	0	0
$103$	−1.29613	−0.127711	−0.0638557	−	0.997959i	$-0.520340\pi$
$103$	−1.29613	−0.127711	−0.0638557	+	0.997959i	$0.520340\pi$
$104$	4.45160	0.436515
$105$	0	0
$106$	−30.0471	−2.91843
$107$	12.2008	1.17949	0.589746	−	0.807589i	$-0.299228\pi$
$107$	12.2008	1.17949	0.589746	+	0.807589i	$0.299228\pi$
$108$	0	0
$109$	−12.1644	−1.16514	−0.582568	−	0.812782i	$-0.697952\pi$
$109$	−12.1644	−1.16514	−0.582568	+	0.812782i	$0.697952\pi$
$110$	0	0
$111$	0	0
$112$	−38.6081	−3.64812
$113$	−0.142052	−0.0133632	−0.00668158	−	0.999978i	$-0.502127\pi$
$113$	−0.142052	−0.0133632	−0.00668158	+	0.999978i	$0.502127\pi$
$114$	0	0
$115$	0.845811	0.0788723
$116$	15.1063	1.40258
$117$	0	0
$118$	10.9114	1.00448
$119$	11.3880	1.04393
$120$	0	0
$121$	0	0
$122$	−8.22100	−0.744294
$123$	0	0
$124$	−26.5377	−2.38316
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.503713	−0.0446973	−0.0223486	−	0.999750i	$-0.507114\pi$
$127$	−0.503713	−0.0446973	−0.0223486	+	0.999750i	$0.507114\pi$
$128$	−57.0153	−5.03949
$129$	0	0
$130$	1.20796	0.105945
$131$	−19.1098	−1.66963	−0.834816	−	0.550529i	$-0.814426\pi$
$131$	−19.1098	−1.66963	−0.834816	+	0.550529i	$0.814426\pi$
$132$	0	0
$133$	−10.6955	−0.927415
$134$	6.84872	0.591640
$135$	0	0
$136$	51.0813	4.38018
$137$	−12.3448	−1.05469	−0.527343	−	0.849653i	$-0.676812\pi$
$137$	−12.3448	−1.05469	−0.527343	+	0.849653i	$0.676812\pi$
$138$	0	0
$139$	−7.73236	−0.655850	−0.327925	−	0.944704i	$-0.606349\pi$
$139$	−7.73236	−0.655850	−0.327925	+	0.944704i	$0.606349\pi$
$140$	−12.9486	−1.09436
$141$	0	0
$142$	−31.3649	−2.63208
$143$	0	0
$144$	0	0
$145$	2.65711	0.220661
$146$	−23.3673	−1.93390
$147$	0	0
$148$	−50.3847	−4.14159
$149$	−7.60895	−0.623350	−0.311675	−	0.950189i	$-0.600890\pi$
$149$	−7.60895	−0.623350	−0.311675	+	0.950189i	$0.600890\pi$
$150$	0	0
$151$	−2.67425	−0.217627	−0.108814	−	0.994062i	$-0.534705\pi$
$151$	−2.67425	−0.217627	−0.108814	+	0.994062i	$0.534705\pi$
$152$	−47.9751	−3.89129
$153$	0	0
$154$	0	0
$155$	−4.66785	−0.374931
$156$	0	0
$157$	20.2000	1.61213	0.806067	−	0.591825i	$-0.201592\pi$
$157$	20.2000	1.61213	0.806067	+	0.591825i	$0.201592\pi$
$158$	30.1360	2.39749
$159$	0	0
$160$	−26.5602	−2.09977
$161$	−1.92641	−0.151823
$162$	0	0
$163$	−9.79582	−0.767268	−0.383634	−	0.923485i	$-0.625328\pi$
$163$	−9.79582	−0.767268	−0.383634	+	0.923485i	$0.625328\pi$
$164$	24.4133	1.90636
$165$	0	0
$166$	−16.4129	−1.27389
$167$	−25.5776	−1.97925	−0.989627	−	0.143658i	$-0.954114\pi$
$167$	−25.5776	−1.97925	−0.989627	+	0.143658i	$0.954114\pi$
$168$	0	0
$169$	−12.8101	−0.985395
$170$	13.8611	1.06310
$171$	0	0
$172$	39.8145	3.03583
$173$	6.75406	0.513502	0.256751	−	0.966478i	$-0.417348\pi$
$173$	6.75406	0.513502	0.256751	+	0.966478i	$0.417348\pi$
$174$	0	0
$175$	−2.27759	−0.172170
$176$	0	0
$177$	0	0
$178$	16.3550	1.22586
$179$	−6.52195	−0.487473	−0.243737	−	0.969841i	$-0.578373\pi$
$179$	−6.52195	−0.487473	−0.243737	+	0.969841i	$0.578373\pi$
$180$	0	0
$181$	13.7522	1.02219	0.511095	−	0.859524i	$-0.329240\pi$
$181$	13.7522	1.02219	0.511095	+	0.859524i	$0.329240\pi$
$182$	−2.75124	−0.203935
$183$	0	0
$184$	−8.64102	−0.637024
$185$	−8.86239	−0.651576
$186$	0	0
$187$	0	0
$188$	−2.66170	−0.194124
$189$	0	0
$190$	−13.0182	−0.944442
$191$	19.4360	1.40634	0.703171	−	0.711020i	$-0.251767\pi$
$191$	19.4360	1.40634	0.703171	+	0.711020i	$0.251767\pi$
$192$	0	0
$193$	22.6124	1.62767	0.813836	−	0.581094i	$-0.197375\pi$
$193$	22.6124	1.62767	0.813836	+	0.581094i	$0.197375\pi$
$194$	23.9743	1.72125
$195$	0	0
$196$	−10.3049	−0.736065
$197$	−23.0300	−1.64082	−0.820410	−	0.571776i	$-0.806255\pi$
$197$	−23.0300	−1.64082	−0.820410	+	0.571776i	$0.806255\pi$
$198$	0	0
$199$	−19.6216	−1.39094	−0.695468	−	0.718557i	$-0.744803\pi$
$199$	−19.6216	−1.39094	−0.695468	+	0.718557i	$0.744803\pi$
$200$	−10.2163	−0.722398
$201$	0	0
$202$	−15.7497	−1.10815
$203$	−6.05181	−0.424754
$204$	0	0
$205$	4.29417	0.299918
$206$	3.59316	0.250347
$207$	0	0
$208$	−7.38630	−0.512148
$209$	0	0
$210$	0	0
$211$	0.0499252	0.00343699	0.00171850	−	0.999999i	$-0.499453\pi$
$211$	0.0499252	0.00343699	0.00171850	+	0.999999i	$0.499453\pi$
$212$	61.6199	4.23207
$213$	0	0
$214$	−33.8232	−2.31211
$215$	7.00317	0.477612
$216$	0	0
$217$	10.6314	0.721710
$218$	33.7223	2.28397
$219$	0	0
$220$	0	0
$221$	2.17868	0.146554
$222$	0	0
$223$	0.754962	0.0505560	0.0252780	−	0.999680i	$-0.491953\pi$
$223$	0.754962	0.0505560	0.0252780	+	0.999680i	$0.491953\pi$
$224$	60.4934	4.04188
$225$	0	0
$226$	0.393801	0.0261952
$227$	−8.28399	−0.549828	−0.274914	−	0.961469i	$-0.588649\pi$
$227$	−8.28399	−0.549828	−0.274914	+	0.961469i	$0.588649\pi$
$228$	0	0
$229$	−2.87622	−0.190066	−0.0950330	−	0.995474i	$-0.530296\pi$
$229$	−2.87622	−0.190066	−0.0950330	+	0.995474i	$0.530296\pi$
$230$	−2.34478	−0.154610
$231$	0	0
$232$	−27.1457	−1.78220
$233$	5.17868	0.339267	0.169633	−	0.985507i	$-0.445742\pi$
$233$	5.17868	0.339267	0.169633	+	0.985507i	$0.445742\pi$
$234$	0	0
$235$	−0.468179	−0.0305406
$236$	−22.3769	−1.45661
$237$	0	0
$238$	−31.5700	−2.04638
$239$	−10.8328	−0.700714	−0.350357	−	0.936616i	$-0.613940\pi$
$239$	−10.8328	−0.700714	−0.350357	+	0.936616i	$0.613940\pi$
$240$	0	0
$241$	−2.96526	−0.191009	−0.0955045	−	0.995429i	$-0.530446\pi$
$241$	−2.96526	−0.191009	−0.0955045	+	0.995429i	$0.530446\pi$
$242$	0	0
$243$	0	0
$244$	16.8595	1.07932
$245$	−1.81258	−0.115801
$246$	0	0
$247$	−2.04620	−0.130197
$248$	47.6879	3.02818
$249$	0	0
$250$	−2.77222	−0.175331
$251$	17.6144	1.11181	0.555906	−	0.831245i	$-0.312371\pi$
$251$	17.6144	1.11181	0.555906	+	0.831245i	$0.312371\pi$
$252$	0	0
$253$	0	0
$254$	1.39640	0.0876182
$255$	0	0
$256$	78.6025	4.91266
$257$	−20.9647	−1.30774	−0.653871	−	0.756606i	$-0.726856\pi$
$257$	−20.9647	−1.30774	−0.653871	+	0.756606i	$0.726856\pi$
$258$	0	0
$259$	20.1849	1.25423
$260$	−2.47726	−0.153633
$261$	0	0
$262$	52.9766	3.27291
$263$	−8.43471	−0.520107	−0.260053	−	0.965594i	$-0.583740\pi$
$263$	−8.43471	−0.520107	−0.260053	+	0.965594i	$0.583740\pi$
$264$	0	0
$265$	10.8386	0.665811
$266$	29.6502	1.81797
$267$	0	0
$268$	−14.0452	−0.857949
$269$	−9.64346	−0.587972	−0.293986	−	0.955810i	$-0.594982\pi$
$269$	−9.64346	−0.587972	−0.293986	+	0.955810i	$0.594982\pi$
$270$	0	0
$271$	10.2278	0.621293	0.310647	−	0.950525i	$-0.399454\pi$
$271$	10.2278	0.621293	0.310647	+	0.950525i	$0.399454\pi$
$272$	−84.7564	−5.13911
$273$	0	0
$274$	34.2225	2.06746
$275$	0	0
$276$	0	0
$277$	−5.83903	−0.350833	−0.175417	−	0.984494i	$-0.556127\pi$
$277$	−5.83903	−0.350833	−0.175417	+	0.984494i	$0.556127\pi$
$278$	21.4358	1.28563
$279$	0	0
$280$	23.2684	1.39056
$281$	−14.4139	−0.859858	−0.429929	−	0.902863i	$-0.641461\pi$
$281$	−14.4139	−0.859858	−0.429929	+	0.902863i	$0.641461\pi$
$282$	0	0
$283$	−10.6334	−0.632093	−0.316046	−	0.948744i	$-0.602356\pi$
$283$	−10.6334	−0.632093	−0.316046	+	0.948744i	$0.602356\pi$
$284$	64.3225	3.81684
$285$	0	0
$286$	0	0
$287$	−9.78037	−0.577317
$288$	0	0
$289$	8.00000	0.470588
$290$	−7.36611	−0.432553
$291$	0	0
$292$	47.9213	2.80438
$293$	8.39576	0.490485	0.245243	−	0.969462i	$-0.421132\pi$
$293$	8.39576	0.490485	0.245243	+	0.969462i	$0.421132\pi$
$294$	0	0
$295$	−3.93598	−0.229162
$296$	90.5404	5.26256
$297$	0	0
$298$	21.0937	1.22193
$299$	−0.368551	−0.0213139
$300$	0	0
$301$	−15.9504	−0.919363
$302$	7.41362	0.426606
$303$	0	0
$304$	79.6025	4.56552
$305$	2.96549	0.169803
$306$	0	0
$307$	4.51902	0.257914	0.128957	−	0.991650i	$-0.458837\pi$
$307$	4.51902	0.257914	0.128957	+	0.991650i	$0.458837\pi$
$308$	0	0
$309$	0	0
$310$	12.9403	0.734961
$311$	−23.9970	−1.36074	−0.680372	−	0.732867i	$-0.738182\pi$
$311$	−23.9970	−1.36074	−0.680372	+	0.732867i	$0.738182\pi$
$312$	0	0
$313$	−25.8686	−1.46218	−0.731090	−	0.682281i	$-0.760988\pi$
$313$	−25.8686	−1.46218	−0.731090	+	0.682281i	$0.760988\pi$
$314$	−55.9988	−3.16020
$315$	0	0
$316$	−61.8022	−3.47665
$317$	−2.90319	−0.163060	−0.0815298	−	0.996671i	$-0.525981\pi$
$317$	−2.90319	−0.163060	−0.0815298	+	0.996671i	$0.525981\pi$
$318$	0	0
$319$	0	0
$320$	39.7283	2.22088
$321$	0	0
$322$	5.34044	0.297611
$323$	−23.4798	−1.30645
$324$	0	0
$325$	−0.435737	−0.0241703
$326$	27.1562	1.50404
$327$	0	0
$328$	−43.8703	−2.42233
$329$	1.06632	0.0587881
$330$	0	0
$331$	10.9837	0.603720	0.301860	−	0.953352i	$-0.402392\pi$
$331$	10.9837	0.603720	0.301860	+	0.953352i	$0.402392\pi$
$332$	33.6593	1.84730
$333$	0	0
$334$	70.9068	3.87985
$335$	−2.47048	−0.134977
$336$	0	0
$337$	12.8462	0.699775	0.349887	−	0.936792i	$-0.386220\pi$
$337$	12.8462	0.699775	0.349887	+	0.936792i	$0.386220\pi$
$338$	35.5125	1.93163
$339$	0	0
$340$	−28.4261	−1.54162
$341$	0	0
$342$	0	0
$343$	20.0715	1.08376
$344$	−71.5461	−3.85751
$345$	0	0
$346$	−18.7238	−1.00660
$347$	1.88971	0.101445	0.0507225	−	0.998713i	$-0.483848\pi$
$347$	1.88971	0.101445	0.0507225	+	0.998713i	$0.483848\pi$
$348$	0	0
$349$	23.7787	1.27284	0.636422	−	0.771341i	$-0.280414\pi$
$349$	23.7787	1.27284	0.636422	+	0.771341i	$0.280414\pi$
$350$	6.31399	0.337497
$351$	0	0
$352$	0	0
$353$	20.2294	1.07670	0.538352	−	0.842720i	$-0.319047\pi$
$353$	20.2294	1.07670	0.538352	+	0.842720i	$0.319047\pi$
$354$	0	0
$355$	11.3140	0.600484
$356$	−33.5404	−1.77764
$357$	0	0
$358$	18.0803	0.955573
$359$	18.5897	0.981129	0.490565	−	0.871405i	$-0.336791\pi$
$359$	18.5897	0.981129	0.490565	+	0.871405i	$0.336791\pi$
$360$	0	0
$361$	3.05200	0.160632
$362$	−38.1241	−2.00376
$363$	0	0
$364$	5.64219	0.295731
$365$	8.42910	0.441199
$366$	0	0
$367$	5.90811	0.308401	0.154200	−	0.988040i	$-0.450720\pi$
$367$	5.90811	0.308401	0.154200	+	0.988040i	$0.450720\pi$
$368$	14.3376	0.747398
$369$	0	0
$370$	24.5685	1.27726
$371$	−24.6859	−1.28163
$372$	0	0
$373$	−1.66992	−0.0864650	−0.0432325	−	0.999065i	$-0.513766\pi$
$373$	−1.66992	−0.0864650	−0.0432325	+	0.999065i	$0.513766\pi$
$374$	0	0
$375$	0	0
$376$	4.78303	0.246666
$377$	−1.15780	−0.0596298
$378$	0	0
$379$	8.43518	0.433286	0.216643	−	0.976251i	$-0.430489\pi$
$379$	8.43518	0.433286	0.216643	+	0.976251i	$0.430489\pi$
$380$	26.6975	1.36956
$381$	0	0
$382$	−53.8810	−2.75679
$383$	−21.8815	−1.11809	−0.559047	−	0.829136i	$-0.688833\pi$
$383$	−21.8815	−1.11809	−0.559047	+	0.829136i	$0.688833\pi$
$384$	0	0
$385$	0	0
$386$	−62.6865	−3.19066
$387$	0	0
$388$	−49.1660	−2.49602
$389$	0.392092	0.0198799	0.00993993	−	0.999951i	$-0.496836\pi$
$389$	0.392092	0.0198799	0.00993993	+	0.999951i	$0.496836\pi$
$390$	0	0
$391$	−4.22906	−0.213873
$392$	18.5178	0.935288
$393$	0	0
$394$	63.8443	3.21643
$395$	−10.8707	−0.546963
$396$	0	0
$397$	35.7823	1.79586	0.897932	−	0.440134i	$-0.145069\pi$
$397$	35.7823	1.79586	0.897932	+	0.440134i	$0.145069\pi$
$398$	54.3954	2.72659
$399$	0	0
$400$	16.9513	0.847564
$401$	−30.4165	−1.51893	−0.759463	−	0.650551i	$-0.774538\pi$
$401$	−30.4165	−1.51893	−0.759463	+	0.650551i	$0.774538\pi$
$402$	0	0
$403$	2.03395	0.101318
$404$	32.2992	1.60695
$405$	0	0
$406$	16.7770	0.832627
$407$	0	0
$408$	0	0
$409$	−16.2697	−0.804486	−0.402243	−	0.915533i	$-0.631769\pi$
$409$	−16.2697	−0.804486	−0.402243	+	0.915533i	$0.631769\pi$
$410$	−11.9044	−0.587917
$411$	0	0
$412$	−7.36878	−0.363034
$413$	8.96456	0.441117
$414$	0	0
$415$	5.92050	0.290626
$416$	11.5733	0.567427
$417$	0	0
$418$	0	0
$419$	40.0703	1.95756	0.978781	−	0.204910i	$-0.0656903\pi$
$419$	40.0703	1.95756	0.978781	+	0.204910i	$0.0656903\pi$
$420$	0	0
$421$	−19.3943	−0.945219	−0.472609	−	0.881272i	$-0.656688\pi$
$421$	−19.3943	−0.945219	−0.472609	+	0.881272i	$0.656688\pi$
$422$	−0.138404	−0.00673739
$423$	0	0
$424$	−110.730	−5.37753
$425$	−5.00000	−0.242536
$426$	0	0
$427$	−6.75417	−0.326857
$428$	69.3640	3.35284
$429$	0	0
$430$	−19.4143	−0.936243
$431$	−33.9444	−1.63505	−0.817523	−	0.575896i	$-0.804653\pi$
$431$	−33.9444	−1.63505	−0.817523	+	0.575896i	$0.804653\pi$
$432$	0	0
$433$	−13.1155	−0.630290	−0.315145	−	0.949044i	$-0.602053\pi$
$433$	−13.1155	−0.630290	−0.315145	+	0.949044i	$0.602053\pi$
$434$	−29.4727	−1.41474
$435$	0	0
$436$	−69.1571	−3.31202
$437$	3.97189	0.190001
$438$	0	0
$439$	−9.64731	−0.460441	−0.230220	−	0.973138i	$-0.573945\pi$
$439$	−9.64731	−0.460441	−0.230220	+	0.973138i	$0.573945\pi$
$440$	0	0
$441$	0	0
$442$	−6.03980	−0.287284
$443$	8.91410	0.423521	0.211761	−	0.977322i	$-0.432080\pi$
$443$	8.91410	0.423521	0.211761	+	0.977322i	$0.432080\pi$
$444$	0	0
$445$	−5.89958	−0.279667
$446$	−2.09292	−0.0991028
$447$	0	0
$448$	−90.4849	−4.27501
$449$	12.4368	0.586931	0.293465	−	0.955970i	$-0.405191\pi$
$449$	12.4368	0.586931	0.293465	+	0.955970i	$0.405191\pi$
$450$	0	0
$451$	0	0
$452$	−0.807599	−0.0379863
$453$	0	0
$454$	22.9651	1.07780
$455$	0.992430	0.0465259
$456$	0	0
$457$	−38.1583	−1.78497	−0.892484	−	0.451078i	$-0.851040\pi$
$457$	−38.1583	−1.78497	−0.892484	+	0.451078i	$0.851040\pi$
$458$	7.97353	0.372578
$459$	0	0
$460$	4.80862	0.224203
$461$	−34.3847	−1.60145	−0.800726	−	0.599030i	$-0.795553\pi$
$461$	−34.3847	−1.60145	−0.800726	+	0.599030i	$0.795553\pi$
$462$	0	0
$463$	−40.2561	−1.87086	−0.935430	−	0.353511i	$-0.884988\pi$
$463$	−40.2561	−1.87086	−0.935430	+	0.353511i	$0.884988\pi$
$464$	45.0415	2.09100
$465$	0	0
$466$	−14.3565	−0.665051
$467$	−15.1161	−0.699490	−0.349745	−	0.936845i	$-0.613732\pi$
$467$	−15.1161	−0.699490	−0.349745	+	0.936845i	$0.613732\pi$
$468$	0	0
$469$	5.62674	0.259819
$470$	1.29790	0.0598675
$471$	0	0
$472$	40.2110	1.85086
$473$	0	0
$474$	0	0
$475$	4.69596	0.215465
$476$	64.7430	2.96749
$477$	0	0
$478$	30.0309	1.37358
$479$	28.8045	1.31611	0.658055	−	0.752970i	$-0.271379\pi$
$479$	28.8045	1.31611	0.658055	+	0.752970i	$0.271379\pi$
$480$	0	0
$481$	3.86167	0.176077
$482$	8.22035	0.374427
$483$	0	0
$484$	0	0
$485$	−8.64803	−0.392687
$486$	0	0
$487$	28.8525	1.30743	0.653716	−	0.756740i	$-0.273209\pi$
$487$	28.8525	1.30743	0.653716	+	0.756740i	$0.273209\pi$
$488$	−30.2962	−1.37144
$489$	0	0
$490$	5.02487	0.227001
$491$	14.9367	0.674084	0.337042	−	0.941490i	$-0.390574\pi$
$491$	14.9367	0.674084	0.337042	+	0.941490i	$0.390574\pi$
$492$	0	0
$493$	−13.2856	−0.598351
$494$	5.67253	0.255219
$495$	0	0
$496$	−79.1260	−3.55286
$497$	−25.7686	−1.15588
$498$	0	0
$499$	22.1990	0.993764	0.496882	−	0.867818i	$-0.334478\pi$
$499$	22.1990	0.993764	0.496882	+	0.867818i	$0.334478\pi$
$500$	5.68522	0.254251
$501$	0	0
$502$	−48.8311	−2.17944
$503$	−0.155356	−0.00692698	−0.00346349	−	0.999994i	$-0.501102\pi$
$503$	−0.155356	−0.00692698	−0.00346349	+	0.999994i	$0.501102\pi$
$504$	0	0
$505$	5.68126	0.252813
$506$	0	0
$507$	0	0
$508$	−2.86372	−0.127057
$509$	18.6375	0.826095	0.413047	−	0.910710i	$-0.364464\pi$
$509$	18.6375	0.826095	0.413047	+	0.910710i	$0.364464\pi$
$510$	0	0
$511$	−19.1980	−0.849272
$512$	−103.873	−4.59058
$513$	0	0
$514$	58.1188	2.56351
$515$	−1.29613	−0.0571143
$516$	0	0
$517$	0	0
$518$	−55.9571	−2.45861
$519$	0	0
$520$	4.45160	0.195215
$521$	−31.3888	−1.37517	−0.687585	−	0.726104i	$-0.741329\pi$
$521$	−31.3888	−1.37517	−0.687585	+	0.726104i	$0.741329\pi$
$522$	0	0
$523$	−23.9841	−1.04875	−0.524377	−	0.851486i	$-0.675702\pi$
$523$	−23.9841	−1.04875	−0.524377	+	0.851486i	$0.675702\pi$
$524$	−108.643	−4.74611
$525$	0	0
$526$	23.3829	1.01954
$527$	23.3392	1.01667
$528$	0	0
$529$	−22.2846	−0.968896
$530$	−30.0471	−1.30516
$531$	0	0
$532$	−60.8061	−2.63628
$533$	−1.87113	−0.0810476
$534$	0	0
$535$	12.2008	0.527485
$536$	25.2390	1.09016
$537$	0	0
$538$	26.7338	1.15258
$539$	0	0
$540$	0	0
$541$	9.97322	0.428782	0.214391	−	0.976748i	$-0.431223\pi$
$541$	9.97322	0.428782	0.214391	+	0.976748i	$0.431223\pi$
$542$	−28.3537	−1.21789
$543$	0	0
$544$	132.801	5.69380
$545$	−12.1644	−0.521064
$546$	0	0
$547$	−40.0167	−1.71099	−0.855496	−	0.517809i	$-0.826748\pi$
$547$	−40.0167	−1.71099	−0.855496	+	0.517809i	$0.826748\pi$
$548$	−70.1828	−2.99806
$549$	0	0
$550$	0	0
$551$	12.4777	0.531567
$552$	0	0
$553$	24.7590	1.05286
$554$	16.1871	0.687724
$555$	0	0
$556$	−43.9601	−1.86433
$557$	7.76046	0.328821	0.164411	−	0.986392i	$-0.447428\pi$
$557$	7.76046	0.328821	0.164411	+	0.986392i	$0.447428\pi$
$558$	0	0
$559$	−3.05154	−0.129066
$560$	−38.6081	−1.63149
$561$	0	0
$562$	39.9584	1.68554
$563$	−18.4355	−0.776964	−0.388482	−	0.921456i	$-0.627000\pi$
$563$	−18.4355	−0.776964	−0.388482	+	0.921456i	$0.627000\pi$
$564$	0	0
$565$	−0.142052	−0.00597619
$566$	29.4783	1.23906
$567$	0	0
$568$	−115.587	−4.84990
$569$	−11.0239	−0.462148	−0.231074	−	0.972936i	$-0.574224\pi$
$569$	−11.0239	−0.462148	−0.231074	+	0.972936i	$0.574224\pi$
$570$	0	0
$571$	−38.1338	−1.59585	−0.797926	−	0.602756i	$-0.794069\pi$
$571$	−38.1338	−1.59585	−0.797926	+	0.602756i	$0.794069\pi$
$572$	0	0
$573$	0	0
$574$	27.1134	1.13169
$575$	0.845811	0.0352728
$576$	0	0
$577$	11.0122	0.458446	0.229223	−	0.973374i	$-0.426382\pi$
$577$	11.0122	0.458446	0.229223	+	0.973374i	$0.426382\pi$
$578$	−22.1778	−0.922474
$579$	0	0
$580$	15.1063	0.627253
$581$	−13.4845	−0.559430
$582$	0	0
$583$	0	0
$584$	−86.1138	−3.56341
$585$	0	0
$586$	−23.2749	−0.961478
$587$	−11.8440	−0.488853	−0.244427	−	0.969668i	$-0.578600\pi$
$587$	−11.8440	−0.488853	−0.244427	+	0.969668i	$0.578600\pi$
$588$	0	0
$589$	−21.9200	−0.903198
$590$	10.9114	0.449216
$591$	0	0
$592$	−150.229	−6.17437
$593$	−9.25062	−0.379877	−0.189939	−	0.981796i	$-0.560829\pi$
$593$	−9.25062	−0.379877	−0.189939	+	0.981796i	$0.560829\pi$
$594$	0	0
$595$	11.3880	0.466861
$596$	−43.2586	−1.77194
$597$	0	0
$598$	1.02171	0.0417807
$599$	−23.6834	−0.967678	−0.483839	−	0.875157i	$-0.660758\pi$
$599$	−23.6834	−0.967678	−0.483839	+	0.875157i	$0.660758\pi$
$600$	0	0
$601$	36.2731	1.47961	0.739806	−	0.672821i	$-0.234917\pi$
$601$	36.2731	1.47961	0.739806	+	0.672821i	$0.234917\pi$
$602$	44.2179	1.80219
$603$	0	0
$604$	−15.2037	−0.618630
$605$	0	0
$606$	0	0
$607$	−24.5068	−0.994699	−0.497350	−	0.867550i	$-0.665693\pi$
$607$	−24.5068	−0.994699	−0.497350	+	0.867550i	$0.665693\pi$
$608$	−124.726	−5.05830
$609$	0	0
$610$	−8.22100	−0.332858
$611$	0.204003	0.00825307
$612$	0	0
$613$	4.64498	0.187609	0.0938044	−	0.995591i	$-0.470097\pi$
$613$	4.64498	0.187609	0.0938044	+	0.995591i	$0.470097\pi$
$614$	−12.5277	−0.505578
$615$	0	0
$616$	0	0
$617$	−41.7419	−1.68047	−0.840233	−	0.542226i	$-0.817582\pi$
$617$	−41.7419	−1.68047	−0.840233	+	0.542226i	$0.817582\pi$
$618$	0	0
$619$	−44.2019	−1.77663	−0.888313	−	0.459239i	$-0.848122\pi$
$619$	−44.2019	−1.77663	−0.888313	+	0.459239i	$0.848122\pi$
$620$	−26.5377	−1.06578
$621$	0	0
$622$	66.5250	2.66741
$623$	13.4368	0.538335
$624$	0	0
$625$	1.00000	0.0400000
$626$	71.7136	2.86625
$627$	0	0
$628$	114.841	4.58267
$629$	44.3120	1.76683
$630$	0	0
$631$	−16.1161	−0.641572	−0.320786	−	0.947152i	$-0.603947\pi$
$631$	−16.1161	−0.641572	−0.320786	+	0.947152i	$0.603947\pi$
$632$	111.058	4.41764
$633$	0	0
$634$	8.04830	0.319639
$635$	−0.503713	−0.0199892
$636$	0	0
$637$	0.789807	0.0312933
$638$	0	0
$639$	0	0
$640$	−57.0153	−2.25373
$641$	9.20985	0.363767	0.181884	−	0.983320i	$-0.441781\pi$
$641$	9.20985	0.363767	0.181884	+	0.983320i	$0.441781\pi$
$642$	0	0
$643$	4.01530	0.158348	0.0791741	−	0.996861i	$-0.474772\pi$
$643$	4.01530	0.158348	0.0791741	+	0.996861i	$0.474772\pi$
$644$	−10.9521	−0.431572
$645$	0	0
$646$	65.0912	2.56098
$647$	−14.2693	−0.560984	−0.280492	−	0.959856i	$-0.590498\pi$
$647$	−14.2693	−0.560984	−0.280492	+	0.959856i	$0.590498\pi$
$648$	0	0
$649$	0	0
$650$	1.20796	0.0473801
$651$	0	0
$652$	−55.6914	−2.18104
$653$	6.52172	0.255215	0.127607	−	0.991825i	$-0.459270\pi$
$653$	6.52172	0.255215	0.127607	+	0.991825i	$0.459270\pi$
$654$	0	0
$655$	−19.1098	−0.746682
$656$	72.7917	2.84204
$657$	0	0
$658$	−2.95608	−0.115240
$659$	−6.74928	−0.262915	−0.131457	−	0.991322i	$-0.541966\pi$
$659$	−6.74928	−0.262915	−0.131457	+	0.991322i	$0.541966\pi$
$660$	0	0
$661$	9.65248	0.375438	0.187719	−	0.982223i	$-0.439891\pi$
$661$	9.65248	0.375438	0.187719	+	0.982223i	$0.439891\pi$
$662$	−30.4493	−1.18345
$663$	0	0
$664$	−60.4853	−2.34728
$665$	−10.6955	−0.414752
$666$	0	0
$667$	2.24741	0.0870202
$668$	−145.414	−5.62625
$669$	0	0
$670$	6.84872	0.264589
$671$	0	0
$672$	0	0
$673$	−28.3594	−1.09318	−0.546588	−	0.837402i	$-0.684074\pi$
$673$	−28.3594	−1.09318	−0.546588	+	0.837402i	$0.684074\pi$
$674$	−35.6124	−1.37174
$675$	0	0
$676$	−72.8284	−2.80109
$677$	5.08697	0.195508	0.0977540	−	0.995211i	$-0.468834\pi$
$677$	5.08697	0.195508	0.0977540	+	0.995211i	$0.468834\pi$
$678$	0	0
$679$	19.6967	0.755889
$680$	51.0813	1.95888
$681$	0	0
$682$	0	0
$683$	−17.3612	−0.664310	−0.332155	−	0.943225i	$-0.607776\pi$
$683$	−17.3612	−0.664310	−0.332155	+	0.943225i	$0.607776\pi$
$684$	0	0
$685$	−12.3448	−0.471670
$686$	−55.6425	−2.12444
$687$	0	0
$688$	118.713	4.52588
$689$	−4.72279	−0.179924
$690$	0	0
$691$	−42.9791	−1.63500	−0.817500	−	0.575928i	$-0.804641\pi$
$691$	−42.9791	−1.63500	−0.817500	+	0.575928i	$0.804641\pi$
$692$	38.3983	1.45969
$693$	0	0
$694$	−5.23870	−0.198858
$695$	−7.73236	−0.293305
$696$	0	0
$697$	−21.4709	−0.813267
$698$	−65.9198	−2.49510
$699$	0	0
$700$	−12.9486	−0.489411
$701$	14.3882	0.543436	0.271718	−	0.962377i	$-0.412408\pi$
$701$	14.3882	0.543436	0.271718	+	0.962377i	$0.412408\pi$
$702$	0	0
$703$	−41.6174	−1.56963
$704$	0	0
$705$	0	0
$706$	−56.0805	−2.11062
$707$	−12.9396	−0.486644
$708$	0	0
$709$	51.1751	1.92192	0.960961	−	0.276682i	$-0.0892349\pi$
$709$	51.1751	1.92192	0.960961	+	0.276682i	$0.0892349\pi$
$710$	−31.3649	−1.17710
$711$	0	0
$712$	60.2716	2.25877
$713$	−3.94812	−0.147858
$714$	0	0
$715$	0	0
$716$	−37.0787	−1.38570
$717$	0	0
$718$	−51.5349	−1.92327
$719$	−52.0371	−1.94066	−0.970328	−	0.241793i	$-0.922265\pi$
$719$	−52.0371	−1.94066	−0.970328	+	0.241793i	$0.922265\pi$
$720$	0	0
$721$	2.95205	0.109940
$722$	−8.46084	−0.314880
$723$	0	0
$724$	78.1841	2.90569
$725$	2.65711	0.0986826
$726$	0	0
$727$	−18.2951	−0.678528	−0.339264	−	0.940691i	$-0.610178\pi$
$727$	−18.2951	−0.678528	−0.339264	+	0.940691i	$0.610178\pi$
$728$	−10.1389	−0.375773
$729$	0	0
$730$	−23.3673	−0.864864
$731$	−35.0158	−1.29511
$732$	0	0
$733$	37.5153	1.38566	0.692830	−	0.721101i	$-0.256364\pi$
$733$	37.5153	1.38566	0.692830	+	0.721101i	$0.256364\pi$
$734$	−16.3786	−0.604545
$735$	0	0
$736$	−22.4649	−0.828069
$737$	0	0
$738$	0	0
$739$	−27.3138	−1.00475	−0.502377	−	0.864649i	$-0.667541\pi$
$739$	−27.3138	−1.00475	−0.502377	+	0.864649i	$0.667541\pi$
$740$	−50.3847	−1.85218
$741$	0	0
$742$	68.4349	2.51233
$743$	−18.8234	−0.690562	−0.345281	−	0.938499i	$-0.612216\pi$
$743$	−18.8234	−0.690562	−0.345281	+	0.938499i	$0.612216\pi$
$744$	0	0
$745$	−7.60895	−0.278770
$746$	4.62938	0.169494
$747$	0	0
$748$	0	0
$749$	−27.7884	−1.01536
$750$	0	0
$751$	−25.1894	−0.919174	−0.459587	−	0.888133i	$-0.652003\pi$
$751$	−25.1894	−0.919174	−0.459587	+	0.888133i	$0.652003\pi$
$752$	−7.93624	−0.289405
$753$	0	0
$754$	3.20968	0.116890
$755$	−2.67425	−0.0973259
$756$	0	0
$757$	45.7939	1.66441	0.832204	−	0.554470i	$-0.187079\pi$
$757$	45.7939	1.66441	0.832204	+	0.554470i	$0.187079\pi$
$758$	−23.3842	−0.849352
$759$	0	0
$760$	−47.9751	−1.74024
$761$	−8.01793	−0.290650	−0.145325	−	0.989384i	$-0.546423\pi$
$761$	−8.01793	−0.290650	−0.145325	+	0.989384i	$0.546423\pi$
$762$	0	0
$763$	27.7055	1.00300
$764$	110.498	3.99768
$765$	0	0
$766$	60.6605	2.19175
$767$	1.71505	0.0619269
$768$	0	0
$769$	40.0439	1.44402	0.722010	−	0.691883i	$-0.243218\pi$
$769$	40.0439	1.44402	0.722010	+	0.691883i	$0.243218\pi$
$770$	0	0
$771$	0	0
$772$	128.556	4.62684
$773$	23.8250	0.856925	0.428462	−	0.903560i	$-0.359055\pi$
$773$	23.8250	0.856925	0.428462	+	0.903560i	$0.359055\pi$
$774$	0	0
$775$	−4.66785	−0.167674
$776$	88.3504	3.17160
$777$	0	0
$778$	−1.08697	−0.0389697
$779$	20.1652	0.722495
$780$	0	0
$781$	0	0
$782$	11.7239	0.419245
$783$	0	0
$784$	−30.7255	−1.09734
$785$	20.2000	0.720968
$786$	0	0
$787$	−9.48879	−0.338239	−0.169119	−	0.985596i	$-0.554092\pi$
$787$	−9.48879	−0.338239	−0.169119	+	0.985596i	$0.554092\pi$
$788$	−130.931	−4.66421
$789$	0	0
$790$	30.1360	1.07219
$791$	0.323537	0.0115037
$792$	0	0
$793$	−1.29217	−0.0458864
$794$	−99.1966	−3.52036
$795$	0	0
$796$	−111.553	−3.95389
$797$	23.7639	0.841759	0.420880	−	0.907117i	$-0.361721\pi$
$797$	23.7639	0.841759	0.420880	+	0.907117i	$0.361721\pi$
$798$	0	0
$799$	2.34090	0.0828149
$800$	−26.5602	−0.939046
$801$	0	0
$802$	84.3212	2.97749
$803$	0	0
$804$	0	0
$805$	−1.92641	−0.0678971
$806$	−5.63857	−0.198610
$807$	0	0
$808$	−58.0412	−2.04188
$809$	18.4128	0.647361	0.323680	−	0.946167i	$-0.395080\pi$
$809$	18.4128	0.647361	0.323680	+	0.946167i	$0.395080\pi$
$810$	0	0
$811$	23.6451	0.830293	0.415147	−	0.909755i	$-0.363730\pi$
$811$	23.6451	0.830293	0.415147	+	0.909755i	$0.363730\pi$
$812$	−34.4059	−1.20741
$813$	0	0
$814$	0	0
$815$	−9.79582	−0.343133
$816$	0	0
$817$	32.8866	1.15056
$818$	45.1033	1.57700
$819$	0	0
$820$	24.4133	0.852550
$821$	−26.3153	−0.918409	−0.459204	−	0.888331i	$-0.651865\pi$
$821$	−26.3153	−0.918409	−0.459204	+	0.888331i	$0.651865\pi$
$822$	0	0
$823$	40.4318	1.40936	0.704681	−	0.709524i	$-0.251090\pi$
$823$	40.4318	1.40936	0.704681	+	0.709524i	$0.251090\pi$
$824$	13.2416	0.461293
$825$	0	0
$826$	−24.8517	−0.864703
$827$	−16.5927	−0.576984	−0.288492	−	0.957482i	$-0.593154\pi$
$827$	−16.5927	−0.576984	−0.288492	+	0.957482i	$0.593154\pi$
$828$	0	0
$829$	−34.3073	−1.19154	−0.595770	−	0.803155i	$-0.703153\pi$
$829$	−34.3073	−1.19154	−0.595770	+	0.803155i	$0.703153\pi$
$830$	−16.4129	−0.569701
$831$	0	0
$832$	−17.3111	−0.600154
$833$	9.06289	0.314011
$834$	0	0
$835$	−25.5776	−0.885150
$836$	0	0
$837$	0	0
$838$	−111.084	−3.83732
$839$	3.38811	0.116971	0.0584853	−	0.998288i	$-0.481373\pi$
$839$	3.38811	0.116971	0.0584853	+	0.998288i	$0.481373\pi$
$840$	0	0
$841$	−21.9398	−0.756543
$842$	53.7652	1.85287
$843$	0	0
$844$	0.283836	0.00977003
$845$	−12.8101	−0.440682
$846$	0	0
$847$	0	0
$848$	183.729	6.30926
$849$	0	0
$850$	13.8611	0.475432
$851$	−7.49591	−0.256956
$852$	0	0
$853$	16.4060	0.561732	0.280866	−	0.959747i	$-0.409378\pi$
$853$	16.4060	0.561732	0.280866	+	0.959747i	$0.409378\pi$
$854$	18.7241	0.640725
$855$	0	0
$856$	−124.646	−4.26032
$857$	3.37817	0.115396	0.0576981	−	0.998334i	$-0.481624\pi$
$857$	3.37817	0.115396	0.0576981	+	0.998334i	$0.481624\pi$
$858$	0	0
$859$	−2.32376	−0.0792855	−0.0396428	−	0.999214i	$-0.512622\pi$
$859$	−2.32376	−0.0792855	−0.0396428	+	0.999214i	$0.512622\pi$
$860$	39.8145	1.35766
$861$	0	0
$862$	94.1015	3.20511
$863$	20.2652	0.689837	0.344918	−	0.938633i	$-0.387907\pi$
$863$	20.2652	0.689837	0.344918	+	0.938633i	$0.387907\pi$
$864$	0	0
$865$	6.75406	0.229645
$866$	36.3591	1.23553
$867$	0	0
$868$	60.4421	2.05154
$869$	0	0
$870$	0	0
$871$	1.07648	0.0364751
$872$	124.274	4.20846
$873$	0	0
$874$	−11.0110	−0.372452
$875$	−2.27759	−0.0769966
$876$	0	0
$877$	−16.6208	−0.561244	−0.280622	−	0.959818i	$-0.590541\pi$
$877$	−16.6208	−0.561244	−0.280622	+	0.959818i	$0.590541\pi$
$878$	26.7445	0.902583
$879$	0	0
$880$	0	0
$881$	29.8895	1.00700	0.503502	−	0.863994i	$-0.332045\pi$
$881$	29.8895	1.00700	0.503502	+	0.863994i	$0.332045\pi$
$882$	0	0
$883$	14.0135	0.471593	0.235796	−	0.971803i	$-0.424230\pi$
$883$	14.0135	0.471593	0.235796	+	0.971803i	$0.424230\pi$
$884$	12.3863	0.416596
$885$	0	0
$886$	−24.7119	−0.830211
$887$	−28.6081	−0.960567	−0.480283	−	0.877113i	$-0.659466\pi$
$887$	−28.6081	−0.960567	−0.480283	+	0.877113i	$0.659466\pi$
$888$	0	0
$889$	1.14725	0.0384776
$890$	16.3550	0.548219
$891$	0	0
$892$	4.29213	0.143711
$893$	−2.19855	−0.0735716
$894$	0	0
$895$	−6.52195	−0.218005
$896$	129.858	4.33824
$897$	0	0
$898$	−34.4777	−1.15054
$899$	−12.4030	−0.413663
$900$	0	0
$901$	−54.1931	−1.80543
$902$	0	0
$903$	0	0
$904$	1.45124	0.0482676
$905$	13.7522	0.457138
$906$	0	0
$907$	26.9693	0.895499	0.447750	−	0.894159i	$-0.352226\pi$
$907$	26.9693	0.895499	0.447750	+	0.894159i	$0.352226\pi$
$908$	−47.0963	−1.56295
$909$	0	0
$910$	−2.75124	−0.0912027
$911$	−7.13660	−0.236446	−0.118223	−	0.992987i	$-0.537720\pi$
$911$	−7.13660	−0.236446	−0.118223	+	0.992987i	$0.537720\pi$
$912$	0	0
$913$	0	0
$914$	105.783	3.49900
$915$	0	0
$916$	−16.3519	−0.540284
$917$	43.5243	1.43730
$918$	0	0
$919$	−6.89424	−0.227420	−0.113710	−	0.993514i	$-0.536273\pi$
$919$	−6.89424	−0.227420	−0.113710	+	0.993514i	$0.536273\pi$
$920$	−8.64102	−0.284886
$921$	0	0
$922$	95.3219	3.13926
$923$	−4.92992	−0.162270
$924$	0	0
$925$	−8.86239	−0.291394
$926$	111.599	3.66737
$927$	0	0
$928$	−70.5735	−2.31669
$929$	36.3816	1.19364	0.596821	−	0.802375i	$-0.296430\pi$
$929$	36.3816	1.19364	0.596821	+	0.802375i	$0.296430\pi$
$930$	0	0
$931$	−8.51179	−0.278963
$932$	29.4420	0.964403
$933$	0	0
$934$	41.9052	1.37118
$935$	0	0
$936$	0	0
$937$	52.6934	1.72142	0.860710	−	0.509096i	$-0.170020\pi$
$937$	52.6934	1.72142	0.860710	+	0.509096i	$0.170020\pi$
$938$	−15.5986	−0.509312
$939$	0	0
$940$	−2.66170	−0.0868151
$941$	20.2984	0.661709	0.330854	−	0.943682i	$-0.392663\pi$
$941$	20.2984	0.661709	0.330854	+	0.943682i	$0.392663\pi$
$942$	0	0
$943$	3.63206	0.118276
$944$	−66.7199	−2.17155
$945$	0	0
$946$	0	0
$947$	−14.5680	−0.473395	−0.236698	−	0.971583i	$-0.576065\pi$
$947$	−14.5680	−0.473395	−0.236698	+	0.971583i	$0.576065\pi$
$948$	0	0
$949$	−3.67287	−0.119226
$950$	−13.0182	−0.422367
$951$	0	0
$952$	−116.342	−3.77067
$953$	39.5054	1.27970	0.639852	−	0.768498i	$-0.278996\pi$
$953$	39.5054	1.27970	0.639852	+	0.768498i	$0.278996\pi$
$954$	0	0
$955$	19.4360	0.628936
$956$	−61.5867	−1.99186
$957$	0	0
$958$	−79.8524	−2.57991
$959$	28.1164	0.907924
$960$	0	0
$961$	−9.21120	−0.297135
$962$	−10.7054	−0.345156
$963$	0	0
$964$	−16.8581	−0.542964
$965$	22.6124	0.727917
$966$	0	0
$967$	−44.8051	−1.44083	−0.720417	−	0.693541i	$-0.756050\pi$
$967$	−44.8051	−1.44083	−0.720417	+	0.693541i	$0.756050\pi$
$968$	0	0
$969$	0	0
$970$	23.9743	0.769767
$971$	−18.1454	−0.582313	−0.291156	−	0.956675i	$-0.594040\pi$
$971$	−18.1454	−0.582313	−0.291156	+	0.956675i	$0.594040\pi$
$972$	0	0
$973$	17.6111	0.564587
$974$	−79.9856	−2.56290
$975$	0	0
$976$	50.2689	1.60907
$977$	−51.9967	−1.66352	−0.831761	−	0.555134i	$-0.812667\pi$
$977$	−51.9967	−1.66352	−0.831761	+	0.555134i	$0.812667\pi$
$978$	0	0
$979$	0	0
$980$	−10.3049	−0.329178
$981$	0	0
$982$	−41.4079	−1.32138
$983$	−44.3322	−1.41398	−0.706989	−	0.707224i	$-0.749947\pi$
$983$	−44.3322	−1.41398	−0.706989	+	0.707224i	$0.749947\pi$
$984$	0	0
$985$	−23.0300	−0.733797
$986$	36.8305	1.17292
$987$	0	0
$988$	−11.6331	−0.370098
$989$	5.92336	0.188352
$990$	0	0
$991$	10.4084	0.330635	0.165317	−	0.986240i	$-0.447135\pi$
$991$	10.4084	0.330635	0.165317	+	0.986240i	$0.447135\pi$
$992$	123.979	3.93634
$993$	0	0
$994$	71.4364	2.26583
$995$	−19.6216	−0.622046
$996$	0	0
$997$	6.52202	0.206554	0.103277	−	0.994653i	$-0.467067\pi$
$997$	6.52202	0.206554	0.103277	+	0.994653i	$0.467067\pi$
$998$	−61.5406	−1.94803
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.be.1.1		4
3.2	odd	2		1815.2.a.x.1.4		4
11.5	even	5		495.2.n.d.91.2		8
11.9	even	5		495.2.n.d.136.2		8
11.10	odd	2		5445.2.a.bv.1.4		4
15.14	odd	2		9075.2.a.cl.1.1		4
33.5	odd	10		165.2.m.a.91.1	✓	8
33.20	odd	10		165.2.m.a.136.1	yes	8
33.32	even	2		1815.2.a.o.1.1		4
165.38	even	20		825.2.bx.h.124.4		16
165.53	even	20		825.2.bx.h.499.1		16
165.104	odd	10		825.2.n.k.751.2		8
165.119	odd	10		825.2.n.k.301.2		8
165.137	even	20		825.2.bx.h.124.1		16
165.152	even	20		825.2.bx.h.499.4		16
165.164	even	2		9075.2.a.dj.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.a.91.1	✓	8	33.5	odd	10
165.2.m.a.136.1	yes	8	33.20	odd	10
495.2.n.d.91.2		8	11.5	even	5
495.2.n.d.136.2		8	11.9	even	5
825.2.n.k.301.2		8	165.119	odd	10
825.2.n.k.751.2		8	165.104	odd	10
825.2.bx.h.124.1		16	165.137	even	20
825.2.bx.h.124.4		16	165.38	even	20
825.2.bx.h.499.1		16	165.53	even	20
825.2.bx.h.499.4		16	165.152	even	20
1815.2.a.o.1.1		4	33.32	even	2
1815.2.a.x.1.4		4	3.2	odd	2
5445.2.a.be.1.1		4	1.1	even	1	trivial
5445.2.a.bv.1.4		4	11.10	odd	2
9075.2.a.cl.1.1		4	15.14	odd	2
9075.2.a.dj.1.4		4	165.164	even	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.77222 q^{2} +5.68522 q^{4} +1.00000 q^{5} -2.27759 q^{7} -10.2163 q^{8} +O(q^{10})\)	\(q-2.77222 q^{2} +5.68522 q^{4} +1.00000 q^{5} -2.27759 q^{7} -10.2163 q^{8} -2.77222 q^{10} -0.435737 q^{13} +6.31399 q^{14} +16.9513 q^{16} -5.00000 q^{17} +4.69596 q^{19} +5.68522 q^{20} +0.845811 q^{23} +1.00000 q^{25} +1.20796 q^{26} -12.9486 q^{28} +2.65711 q^{29} -4.66785 q^{31} -26.5602 q^{32} +13.8611 q^{34} -2.27759 q^{35} -8.86239 q^{37} -13.0182 q^{38} -10.2163 q^{40} +4.29417 q^{41} +7.00317 q^{43} -2.34478 q^{46} -0.468179 q^{47} -1.81258 q^{49} -2.77222 q^{50} -2.47726 q^{52} +10.8386 q^{53} +23.2684 q^{56} -7.36611 q^{58} -3.93598 q^{59} +2.96549 q^{61} +12.9403 q^{62} +39.7283 q^{64} -0.435737 q^{65} -2.47048 q^{67} -28.4261 q^{68} +6.31399 q^{70} +11.3140 q^{71} +8.42910 q^{73} +24.5685 q^{74} +26.6975 q^{76} -10.8707 q^{79} +16.9513 q^{80} -11.9044 q^{82} +5.92050 q^{83} -5.00000 q^{85} -19.4143 q^{86} -5.89958 q^{89} +0.992430 q^{91} +4.80862 q^{92} +1.29790 q^{94} +4.69596 q^{95} -8.64803 q^{97} +5.02487 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} + 9 q^{4} + 4 q^{5} - 2 q^{7} - 15 q^{8}+O(q^{10}) \) 4 * q - 5 * q^2 + 9 * q^4 + 4 * q^5 - 2 * q^7 - 15 * q^8	\( 4 q - 5 q^{2} + 9 q^{4} + 4 q^{5} - 2 q^{7} - 15 q^{8} - 5 q^{10} + 3 q^{13} + 5 q^{14} + 15 q^{16} - 20 q^{17} + 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} + 3 q^{28} - 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} - 2 q^{35} - 7 q^{37} - q^{38} - 15 q^{40} - 20 q^{41} - 2 q^{43} + 7 q^{46} + 20 q^{47} + 8 q^{49} - 5 q^{50} - 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} - 7 q^{61} + 12 q^{62} + 49 q^{64} + 3 q^{65} - 13 q^{67} - 45 q^{68} + 5 q^{70} + 25 q^{71} + 23 q^{73} + 7 q^{74} - 7 q^{76} + 15 q^{80} + 11 q^{82} - 33 q^{83} - 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} - 17 q^{94} + 3 q^{95} - 25 q^{98}+O(q^{100}) \) 4 * q - 5 * q^2 + 9 * q^4 + 4 * q^5 - 2 * q^7 - 15 * q^8 - 5 * q^10 + 3 * q^13 + 5 * q^14 + 15 * q^16 - 20 * q^17 + 3 * q^19 + 9 * q^20 + 5 * q^23 + 4 * q^25 - 6 * q^26 + 3 * q^28 - 5 * q^29 - q^31 - 30 * q^32 + 25 * q^34 - 2 * q^35 - 7 * q^37 - q^38 - 15 * q^40 - 20 * q^41 - 2 * q^43 + 7 * q^46 + 20 * q^47 + 8 * q^49 - 5 * q^50 - 7 * q^52 - 6 * q^53 - 10 * q^56 - 21 * q^58 + 5 * q^59 - 7 * q^61 + 12 * q^62 + 49 * q^64 + 3 * q^65 - 13 * q^67 - 45 * q^68 + 5 * q^70 + 25 * q^71 + 23 * q^73 + 7 * q^74 - 7 * q^76 + 15 * q^80 + 11 * q^82 - 33 * q^83 - 20 * q^85 + 12 * q^86 - 16 * q^89 - 24 * q^91 - 17 * q^94 + 3 * q^95 - 25 * q^98

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