LMFDB - Embedded newform 2205.4.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.70156 q^{2} +14.1047 q^{4} -5.00000 q^{5} -28.7016 q^{8} +O(q^{10})$	$q-4.70156 q^{2} +14.1047 q^{4} -5.00000 q^{5} -28.7016 q^{8} +23.5078 q^{10} -24.5969 q^{11} +35.0156 q^{13} +22.1047 q^{16} -18.4187 q^{17} +67.4031 q^{19} -70.5234 q^{20} +115.644 q^{22} +145.675 q^{23} +25.0000 q^{25} -164.628 q^{26} -214.419 q^{29} +88.6594 q^{31} +125.686 q^{32} +86.5969 q^{34} +162.125 q^{37} -316.900 q^{38} +143.508 q^{40} -337.769 q^{41} +122.156 q^{43} -346.931 q^{44} -684.900 q^{46} +354.219 q^{47} -117.539 q^{50} +493.884 q^{52} -676.691 q^{53} +122.984 q^{55} +1008.10 q^{58} +501.319 q^{59} +708.931 q^{61} -416.837 q^{62} -767.758 q^{64} -175.078 q^{65} -907.956 q^{67} -259.791 q^{68} -430.334 q^{71} -41.3406 q^{73} -762.241 q^{74} +950.700 q^{76} +890.388 q^{79} -110.523 q^{80} +1588.04 q^{82} -1057.15 q^{83} +92.0937 q^{85} -574.325 q^{86} +705.969 q^{88} +1473.72 q^{89} +2054.70 q^{92} -1665.38 q^{94} -337.016 q^{95} -555.034 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 9 q^{4} - 10 q^{5} - 51 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 + 9 * q^4 - 10 * q^5 - 51 * q^8	$ 2 q - 3 q^{2} + 9 q^{4} - 10 q^{5} - 51 q^{8} + 15 q^{10} - 62 q^{11} + 6 q^{13} + 25 q^{16} + 40 q^{17} + 122 q^{19} - 45 q^{20} + 52 q^{22} - 16 q^{23} + 50 q^{25} - 214 q^{26} - 352 q^{29} - 66 q^{31} + 309 q^{32} + 186 q^{34} - 188 q^{37} - 224 q^{38} + 255 q^{40} + 16 q^{41} - 396 q^{43} - 156 q^{44} - 960 q^{46} - 188 q^{47} - 75 q^{50} + 642 q^{52} - 982 q^{53} + 310 q^{55} + 774 q^{58} + 516 q^{59} + 880 q^{61} - 680 q^{62} - 479 q^{64} - 30 q^{65} - 356 q^{67} - 558 q^{68} - 310 q^{71} - 326 q^{73} - 1358 q^{74} + 672 q^{76} + 1832 q^{79} - 125 q^{80} + 2190 q^{82} - 680 q^{83} - 200 q^{85} - 1456 q^{86} + 1540 q^{88} + 796 q^{89} + 2880 q^{92} - 2588 q^{94} - 610 q^{95} + 670 q^{97}+O(q^{100}) $ 2 * q - 3 * q^2 + 9 * q^4 - 10 * q^5 - 51 * q^8 + 15 * q^10 - 62 * q^11 + 6 * q^13 + 25 * q^16 + 40 * q^17 + 122 * q^19 - 45 * q^20 + 52 * q^22 - 16 * q^23 + 50 * q^25 - 214 * q^26 - 352 * q^29 - 66 * q^31 + 309 * q^32 + 186 * q^34 - 188 * q^37 - 224 * q^38 + 255 * q^40 + 16 * q^41 - 396 * q^43 - 156 * q^44 - 960 * q^46 - 188 * q^47 - 75 * q^50 + 642 * q^52 - 982 * q^53 + 310 * q^55 + 774 * q^58 + 516 * q^59 + 880 * q^61 - 680 * q^62 - 479 * q^64 - 30 * q^65 - 356 * q^67 - 558 * q^68 - 310 * q^71 - 326 * q^73 - 1358 * q^74 + 672 * q^76 + 1832 * q^79 - 125 * q^80 + 2190 * q^82 - 680 * q^83 - 200 * q^85 - 1456 * q^86 + 1540 * q^88 + 796 * q^89 + 2880 * q^92 - 2588 * q^94 - 610 * q^95 + 670 * 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 $

$2$	−4.70156	−1.66225	−0.831127	−	0.556083i	$-0.812304\pi$
$2$	−4.70156	−1.66225	−0.831127	+	0.556083i	$0.812304\pi$
$3$	0	0
$3$	0	0
$4$	14.1047	1.76309
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−28.7016	−1.26844
$9$	0	0
$10$	23.5078	0.743382
$11$	−24.5969	−0.674203	−0.337102	−	0.941468i	$-0.609447\pi$
$11$	−24.5969	−0.674203	−0.337102	+	0.941468i	$0.609447\pi$
$12$	0	0
$13$	35.0156	0.747045	0.373523	−	0.927621i	$-0.378150\pi$
$13$	35.0156	0.747045	0.373523	+	0.927621i	$0.378150\pi$
$14$	0	0
$15$	0	0
$16$	22.1047	0.345386
$17$	−18.4187	−0.262777	−0.131388	−	0.991331i	$-0.541943\pi$
$17$	−18.4187	−0.262777	−0.131388	+	0.991331i	$0.541943\pi$
$18$	0	0
$19$	67.4031	0.813860	0.406930	−	0.913459i	$-0.366599\pi$
$19$	67.4031	0.813860	0.406930	+	0.913459i	$0.366599\pi$
$20$	−70.5234	−0.788476
$21$	0	0
$22$	115.644	1.12070
$23$	145.675	1.32067	0.660333	−	0.750973i	$-0.270415\pi$
$23$	145.675	1.32067	0.660333	+	0.750973i	$0.270415\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−164.628	−1.24178
$27$	0	0
$28$	0	0
$29$	−214.419	−1.37298	−0.686492	−	0.727137i	$-0.740851\pi$
$29$	−214.419	−1.37298	−0.686492	+	0.727137i	$0.740851\pi$
$30$	0	0
$31$	88.6594	0.513667	0.256834	−	0.966456i	$-0.417321\pi$
$31$	88.6594	0.513667	0.256834	+	0.966456i	$0.417321\pi$
$32$	125.686	0.694323
$33$	0	0
$34$	86.5969	0.436801
$35$	0	0
$36$	0	0
$37$	162.125	0.720356	0.360178	−	0.932884i	$-0.382716\pi$
$37$	162.125	0.720356	0.360178	+	0.932884i	$0.382716\pi$
$38$	−316.900	−1.35284
$39$	0	0
$40$	143.508	0.567264
$41$	−337.769	−1.28660	−0.643300	−	0.765614i	$-0.722435\pi$
$41$	−337.769	−1.28660	−0.643300	+	0.765614i	$0.722435\pi$
$42$	0	0
$43$	122.156	0.433224	0.216612	−	0.976258i	$-0.430499\pi$
$43$	122.156	0.433224	0.216612	+	0.976258i	$0.430499\pi$
$44$	−346.931	−1.18868
$45$	0	0
$46$	−684.900	−2.19528
$47$	354.219	1.09932	0.549661	−	0.835388i	$-0.314757\pi$
$47$	354.219	1.09932	0.549661	+	0.835388i	$0.314757\pi$
$48$	0	0
$49$	0	0
$50$	−117.539	−0.332451
$51$	0	0
$52$	493.884	1.31710
$53$	−676.691	−1.75378	−0.876892	−	0.480687i	$-0.840387\pi$
$53$	−676.691	−1.75378	−0.876892	+	0.480687i	$0.840387\pi$
$54$	0	0
$55$	122.984	0.301513
$56$	0	0
$57$	0	0
$58$	1008.10	2.28225
$59$	501.319	1.10621	0.553103	−	0.833113i	$-0.313444\pi$
$59$	501.319	1.10621	0.553103	+	0.833113i	$0.313444\pi$
$60$	0	0
$61$	708.931	1.48802	0.744011	−	0.668167i	$-0.232921\pi$
$61$	708.931	1.48802	0.744011	+	0.668167i	$0.232921\pi$
$62$	−416.837	−0.853845
$63$	0	0
$64$	−767.758	−1.49953
$65$	−175.078	−0.334089
$66$	0	0
$67$	−907.956	−1.65559	−0.827795	−	0.561031i	$-0.810405\pi$
$67$	−907.956	−1.65559	−0.827795	+	0.561031i	$0.810405\pi$
$68$	−259.791	−0.463298
$69$	0	0
$70$	0	0
$71$	−430.334	−0.719314	−0.359657	−	0.933085i	$-0.617106\pi$
$71$	−430.334	−0.719314	−0.359657	+	0.933085i	$0.617106\pi$
$72$	0	0
$73$	−41.3406	−0.0662816	−0.0331408	−	0.999451i	$-0.510551\pi$
$73$	−41.3406	−0.0662816	−0.0331408	+	0.999451i	$0.510551\pi$
$74$	−762.241	−1.19741
$75$	0	0
$76$	950.700	1.43490
$77$	0	0
$78$	0	0
$79$	890.388	1.26806	0.634028	−	0.773310i	$-0.281400\pi$
$79$	890.388	1.26806	0.634028	+	0.773310i	$0.281400\pi$
$80$	−110.523	−0.154461
$81$	0	0
$82$	1588.04	2.13866
$83$	−1057.15	−1.39804	−0.699020	−	0.715102i	$-0.746380\pi$
$83$	−1057.15	−1.39804	−0.699020	+	0.715102i	$0.746380\pi$
$84$	0	0
$85$	92.0937	0.117517
$86$	−574.325	−0.720129
$87$	0	0
$88$	705.969	0.855188
$89$	1473.72	1.75522	0.877610	−	0.479376i	$-0.159137\pi$
$89$	1473.72	1.75522	0.877610	+	0.479376i	$0.159137\pi$
$90$	0	0
$91$	0	0
$92$	2054.70	2.32845
$93$	0	0
$94$	−1665.38	−1.82735
$95$	−337.016	−0.363969
$96$	0	0
$97$	−555.034	−0.580981	−0.290491	−	0.956878i	$-0.593819\pi$
$97$	−555.034	−0.580981	−0.290491	+	0.956878i	$0.593819\pi$
$98$	0	0
$99$	0	0
$100$	352.617	0.352617
$101$	1890.14	1.86214	0.931071	−	0.364838i	$-0.118876\pi$
$101$	1890.14	1.86214	0.931071	+	0.364838i	$0.118876\pi$
$102$	0	0
$103$	−662.700	−0.633959	−0.316979	−	0.948432i	$-0.602669\pi$
$103$	−662.700	−0.633959	−0.316979	+	0.948432i	$0.602669\pi$
$104$	−1005.00	−0.947583
$105$	0	0
$106$	3181.50	2.91523
$107$	−1614.53	−1.45872	−0.729358	−	0.684132i	$-0.760181\pi$
$107$	−1614.53	−1.45872	−0.729358	+	0.684132i	$0.760181\pi$
$108$	0	0
$109$	217.206	0.190868	0.0954339	−	0.995436i	$-0.469576\pi$
$109$	217.206	0.190868	0.0954339	+	0.995436i	$0.469576\pi$
$110$	−578.219	−0.501191
$111$	0	0
$112$	0	0
$113$	1658.20	1.38044	0.690221	−	0.723598i	$-0.257513\pi$
$113$	1658.20	1.38044	0.690221	+	0.723598i	$0.257513\pi$
$114$	0	0
$115$	−728.375	−0.590620
$116$	−3024.31	−2.42069
$117$	0	0
$118$	−2356.98	−1.83879
$119$	0	0
$120$	0	0
$121$	−725.994	−0.545450
$122$	−3333.08	−2.47347
$123$	0	0
$124$	1250.51	0.905640
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1108.81	−0.774734	−0.387367	−	0.921926i	$-0.626615\pi$
$127$	−1108.81	−0.774734	−0.387367	+	0.921926i	$0.626615\pi$
$128$	2604.17	1.79827
$129$	0	0
$130$	823.141	0.555340
$131$	185.488	0.123711	0.0618554	−	0.998085i	$-0.480298\pi$
$131$	185.488	0.123711	0.0618554	+	0.998085i	$0.480298\pi$
$132$	0	0
$133$	0	0
$134$	4268.81	2.75201
$135$	0	0
$136$	528.647	0.333317
$137$	37.9907	0.0236917	0.0118458	−	0.999930i	$-0.496229\pi$
$137$	37.9907	0.0236917	0.0118458	+	0.999930i	$0.496229\pi$
$138$	0	0
$139$	−183.609	−0.112040	−0.0560199	−	0.998430i	$-0.517841\pi$
$139$	−183.609	−0.112040	−0.0560199	+	0.998430i	$0.517841\pi$
$140$	0	0
$141$	0	0
$142$	2023.24	1.19568
$143$	−861.275	−0.503660
$144$	0	0
$145$	1072.09	0.614018
$146$	194.366	0.110177
$147$	0	0
$148$	2286.72	1.27005
$149$	1383.34	0.760587	0.380293	−	0.924866i	$-0.375823\pi$
$149$	1383.34	0.760587	0.380293	+	0.924866i	$0.375823\pi$
$150$	0	0
$151$	765.256	0.412422	0.206211	−	0.978508i	$-0.433887\pi$
$151$	765.256	0.412422	0.206211	+	0.978508i	$0.433887\pi$
$152$	−1934.57	−1.03233
$153$	0	0
$154$	0	0
$155$	−443.297	−0.229719
$156$	0	0
$157$	2366.76	1.20311	0.601554	−	0.798832i	$-0.294548\pi$
$157$	2366.76	1.20311	0.601554	+	0.798832i	$0.294548\pi$
$158$	−4186.21	−2.10783
$159$	0	0
$160$	−628.430	−0.310511
$161$	0	0
$162$	0	0
$163$	−3137.69	−1.50775	−0.753875	−	0.657018i	$-0.771817\pi$
$163$	−3137.69	−1.50775	−0.753875	+	0.657018i	$0.771817\pi$
$164$	−4764.12	−2.26839
$165$	0	0
$166$	4970.26	2.32390
$167$	146.469	0.0678688	0.0339344	−	0.999424i	$-0.489196\pi$
$167$	146.469	0.0678688	0.0339344	+	0.999424i	$0.489196\pi$
$168$	0	0
$169$	−970.906	−0.441924
$170$	−432.984	−0.195343
$171$	0	0
$172$	1722.98	0.763812
$173$	−1424.12	−0.625860	−0.312930	−	0.949776i	$-0.601311\pi$
$173$	−1424.12	−0.625860	−0.312930	+	0.949776i	$0.601311\pi$
$174$	0	0
$175$	0	0
$176$	−543.706	−0.232860
$177$	0	0
$178$	−6928.81	−2.91762
$179$	−1244.70	−0.519737	−0.259869	−	0.965644i	$-0.583679\pi$
$179$	−1244.70	−0.519737	−0.259869	+	0.965644i	$0.583679\pi$
$180$	0	0
$181$	3879.09	1.59299	0.796493	−	0.604648i	$-0.206686\pi$
$181$	3879.09	1.59299	0.796493	+	0.604648i	$0.206686\pi$
$182$	0	0
$183$	0	0
$184$	−4181.10	−1.67519
$185$	−810.625	−0.322153
$186$	0	0
$187$	453.044	0.177165
$188$	4996.14	1.93820
$189$	0	0
$190$	1584.50	0.605009
$191$	−1574.90	−0.596628	−0.298314	−	0.954468i	$-0.596424\pi$
$191$	−1574.90	−0.596628	−0.298314	+	0.954468i	$0.596424\pi$
$192$	0	0
$193$	−4775.67	−1.78114	−0.890572	−	0.454843i	$-0.849695\pi$
$193$	−4775.67	−1.78114	−0.890572	+	0.454843i	$0.849695\pi$
$194$	2609.53	0.965738
$195$	0	0
$196$	0	0
$197$	2803.58	1.01394	0.506971	−	0.861963i	$-0.330765\pi$
$197$	2803.58	1.01394	0.506971	+	0.861963i	$0.330765\pi$
$198$	0	0
$199$	−4102.92	−1.46155	−0.730774	−	0.682620i	$-0.760841\pi$
$199$	−4102.92	−1.46155	−0.730774	+	0.682620i	$0.760841\pi$
$200$	−717.539	−0.253688
$201$	0	0
$202$	−8886.63	−3.09535
$203$	0	0
$204$	0	0
$205$	1688.84	0.575385
$206$	3115.72	1.05380
$207$	0	0
$208$	774.009	0.258019
$209$	−1657.91	−0.548707
$210$	0	0
$211$	−823.512	−0.268687	−0.134343	−	0.990935i	$-0.542893\pi$
$211$	−823.512	−0.268687	−0.134343	+	0.990935i	$0.542893\pi$
$212$	−9544.51	−3.09207
$213$	0	0
$214$	7590.82	2.42476
$215$	−610.781	−0.193744
$216$	0	0
$217$	0	0
$218$	−1021.21	−0.317271
$219$	0	0
$220$	1734.66	0.531593
$221$	−644.944	−0.196306
$222$	0	0
$223$	−817.194	−0.245396	−0.122698	−	0.992444i	$-0.539155\pi$
$223$	−817.194	−0.245396	−0.122698	+	0.992444i	$0.539155\pi$
$224$	0	0
$225$	0	0
$226$	−7796.12	−2.29465
$227$	3655.85	1.06893	0.534465	−	0.845190i	$-0.320513\pi$
$227$	3655.85	1.06893	0.534465	+	0.845190i	$0.320513\pi$
$228$	0	0
$229$	−939.393	−0.271078	−0.135539	−	0.990772i	$-0.543277\pi$
$229$	−939.393	−0.271078	−0.135539	+	0.990772i	$0.543277\pi$
$230$	3424.50	0.981760
$231$	0	0
$232$	6154.15	1.74155
$233$	7.64701	0.00215010	0.00107505	−	0.999999i	$-0.499658\pi$
$233$	7.64701	0.00215010	0.00107505	+	0.999999i	$0.499658\pi$
$234$	0	0
$235$	−1771.09	−0.491631
$236$	7070.94	1.95034
$237$	0	0
$238$	0	0
$239$	889.115	0.240636	0.120318	−	0.992735i	$-0.461609\pi$
$239$	889.115	0.240636	0.120318	+	0.992735i	$0.461609\pi$
$240$	0	0
$241$	−2140.23	−0.572051	−0.286026	−	0.958222i	$-0.592334\pi$
$241$	−2140.23	−0.572051	−0.286026	+	0.958222i	$0.592334\pi$
$242$	3413.30	0.906676
$243$	0	0
$244$	9999.25	2.62351
$245$	0	0
$246$	0	0
$247$	2360.16	0.607990
$248$	−2544.66	−0.651557
$249$	0	0
$250$	587.695	0.148676
$251$	−6749.81	−1.69739	−0.848693	−	0.528886i	$-0.822610\pi$
$251$	−6749.81	−1.69739	−0.848693	+	0.528886i	$0.822610\pi$
$252$	0	0
$253$	−3583.15	−0.890398
$254$	5213.15	1.28780
$255$	0	0
$256$	−6101.62	−1.48965
$257$	3068.64	0.744811	0.372405	−	0.928070i	$-0.378533\pi$
$257$	3068.64	0.744811	0.372405	+	0.928070i	$0.378533\pi$
$258$	0	0
$259$	0	0
$260$	−2469.42	−0.589027
$261$	0	0
$262$	−872.081	−0.205639
$263$	4674.12	1.09589	0.547944	−	0.836515i	$-0.315411\pi$
$263$	4674.12	1.09589	0.547944	+	0.836515i	$0.315411\pi$
$264$	0	0
$265$	3383.45	0.784316
$266$	0	0
$267$	0	0
$268$	−12806.4	−2.91895
$269$	2417.38	0.547919	0.273960	−	0.961741i	$-0.411667\pi$
$269$	2417.38	0.547919	0.273960	+	0.961741i	$0.411667\pi$
$270$	0	0
$271$	−7724.30	−1.73143	−0.865715	−	0.500537i	$-0.833136\pi$
$271$	−7724.30	−1.73143	−0.865715	+	0.500537i	$0.833136\pi$
$272$	−407.141	−0.0907593
$273$	0	0
$274$	−178.616	−0.0393816
$275$	−614.922	−0.134841
$276$	0	0
$277$	−4576.17	−0.992620	−0.496310	−	0.868145i	$-0.665312\pi$
$277$	−4576.17	−0.992620	−0.496310	+	0.868145i	$0.665312\pi$
$278$	863.250	0.186239
$279$	0	0
$280$	0	0
$281$	1358.56	0.288415	0.144208	−	0.989547i	$-0.453937\pi$
$281$	1358.56	0.288415	0.144208	+	0.989547i	$0.453937\pi$
$282$	0	0
$283$	−3885.04	−0.816048	−0.408024	−	0.912971i	$-0.633782\pi$
$283$	−3885.04	−0.816048	−0.408024	+	0.912971i	$0.633782\pi$
$284$	−6069.73	−1.26821
$285$	0	0
$286$	4049.34	0.837211
$287$	0	0
$288$	0	0
$289$	−4573.75	−0.930948
$290$	−5040.52	−1.02065
$291$	0	0
$292$	−583.097	−0.116860
$293$	−4033.91	−0.804312	−0.402156	−	0.915571i	$-0.631739\pi$
$293$	−4033.91	−0.804312	−0.402156	+	0.915571i	$0.631739\pi$
$294$	0	0
$295$	−2506.59	−0.494710
$296$	−4653.24	−0.913730
$297$	0	0
$298$	−6503.85	−1.26429
$299$	5100.90	0.986598
$300$	0	0
$301$	0	0
$302$	−3597.90	−0.685549
$303$	0	0
$304$	1489.92	0.281096
$305$	−3544.66	−0.665464
$306$	0	0
$307$	4620.36	0.858950	0.429475	−	0.903079i	$-0.358699\pi$
$307$	4620.36	0.858950	0.429475	+	0.903079i	$0.358699\pi$
$308$	0	0
$309$	0	0
$310$	2084.19	0.381851
$311$	6675.89	1.21722	0.608609	−	0.793470i	$-0.291728\pi$
$311$	6675.89	1.21722	0.608609	+	0.793470i	$0.291728\pi$
$312$	0	0
$313$	−2836.78	−0.512283	−0.256141	−	0.966639i	$-0.582451\pi$
$313$	−2836.78	−0.512283	−0.256141	+	0.966639i	$0.582451\pi$
$314$	−11127.5	−1.99987
$315$	0	0
$316$	12558.6	2.23569
$317$	−4010.63	−0.710597	−0.355299	−	0.934753i	$-0.615621\pi$
$317$	−4010.63	−0.710597	−0.355299	+	0.934753i	$0.615621\pi$
$318$	0	0
$319$	5274.03	0.925671
$320$	3838.79	0.670609
$321$	0	0
$322$	0	0
$323$	−1241.48	−0.213863
$324$	0	0
$325$	875.391	0.149409
$326$	14752.1	2.50626
$327$	0	0
$328$	9694.49	1.63198
$329$	0	0
$330$	0	0
$331$	11087.5	1.84117	0.920583	−	0.390546i	$-0.127714\pi$
$331$	11087.5	1.84117	0.920583	+	0.390546i	$0.127714\pi$
$332$	−14910.8	−2.46486
$333$	0	0
$334$	−688.631	−0.112815
$335$	4539.78	0.740402
$336$	0	0
$337$	12118.7	1.95890	0.979450	−	0.201689i	$-0.0646431\pi$
$337$	12118.7	1.95890	0.979450	+	0.201689i	$0.0646431\pi$
$338$	4564.78	0.734589
$339$	0	0
$340$	1298.95	0.207193
$341$	−2180.74	−0.346316
$342$	0	0
$343$	0	0
$344$	−3506.07	−0.549520
$345$	0	0
$346$	6695.58	1.04034
$347$	6361.22	0.984116	0.492058	−	0.870562i	$-0.336245\pi$
$347$	6361.22	0.984116	0.492058	+	0.870562i	$0.336245\pi$
$348$	0	0
$349$	3115.18	0.477799	0.238899	−	0.971044i	$-0.423213\pi$
$349$	3115.18	0.477799	0.238899	+	0.971044i	$0.423213\pi$
$350$	0	0
$351$	0	0
$352$	−3091.48	−0.468115
$353$	−11927.4	−1.79839	−0.899194	−	0.437550i	$-0.855846\pi$
$353$	−11927.4	−1.79839	−0.899194	+	0.437550i	$0.855846\pi$
$354$	0	0
$355$	2151.67	0.321687
$356$	20786.4	3.09460
$357$	0	0
$358$	5852.02	0.863935
$359$	6143.95	0.903245	0.451623	−	0.892209i	$-0.350845\pi$
$359$	6143.95	0.903245	0.451623	+	0.892209i	$0.350845\pi$
$360$	0	0
$361$	−2315.82	−0.337632
$362$	−18237.8	−2.64794
$363$	0	0
$364$	0	0
$365$	206.703	0.0296420
$366$	0	0
$367$	1927.67	0.274178	0.137089	−	0.990559i	$-0.456225\pi$
$367$	1927.67	0.274178	0.137089	+	0.990559i	$0.456225\pi$
$368$	3220.10	0.456139
$369$	0	0
$370$	3811.20	0.535500
$371$	0	0
$372$	0	0
$373$	10452.0	1.45090	0.725449	−	0.688276i	$-0.241632\pi$
$373$	10452.0	1.45090	0.725449	+	0.688276i	$0.241632\pi$
$374$	−2130.01	−0.294493
$375$	0	0
$376$	−10166.6	−1.39443
$377$	−7508.01	−1.02568
$378$	0	0
$379$	7066.43	0.957726	0.478863	−	0.877890i	$-0.341049\pi$
$379$	7066.43	0.957726	0.478863	+	0.877890i	$0.341049\pi$
$380$	−4753.50	−0.641709
$381$	0	0
$382$	7404.51	0.991747
$383$	7168.04	0.956318	0.478159	−	0.878273i	$-0.341304\pi$
$383$	7168.04	0.956318	0.478159	+	0.878273i	$0.341304\pi$
$384$	0	0
$385$	0	0
$386$	22453.1	2.96071
$387$	0	0
$388$	−7828.58	−1.02432
$389$	7414.06	0.966344	0.483172	−	0.875525i	$-0.339485\pi$
$389$	7414.06	0.966344	0.483172	+	0.875525i	$0.339485\pi$
$390$	0	0
$391$	−2683.15	−0.347040
$392$	0	0
$393$	0	0
$394$	−13181.2	−1.68543
$395$	−4451.94	−0.567092
$396$	0	0
$397$	8936.01	1.12969	0.564843	−	0.825198i	$-0.308937\pi$
$397$	8936.01	1.12969	0.564843	+	0.825198i	$0.308937\pi$
$398$	19290.1	2.42946
$399$	0	0
$400$	552.617	0.0690771
$401$	−1782.91	−0.222031	−0.111015	−	0.993819i	$-0.535410\pi$
$401$	−1782.91	−0.222031	−0.111015	+	0.993819i	$0.535410\pi$
$402$	0	0
$403$	3104.46	0.383733
$404$	26659.9	3.28312
$405$	0	0
$406$	0	0
$407$	−3987.77	−0.485667
$408$	0	0
$409$	8759.92	1.05905	0.529524	−	0.848295i	$-0.322371\pi$
$409$	8759.92	1.05905	0.529524	+	0.848295i	$0.322371\pi$
$410$	−7940.20	−0.956436
$411$	0	0
$412$	−9347.17	−1.11772
$413$	0	0
$414$	0	0
$415$	5285.75	0.625222
$416$	4400.97	0.518691
$417$	0	0
$418$	7794.75	0.912090
$419$	−3212.74	−0.374588	−0.187294	−	0.982304i	$-0.559972\pi$
$419$	−3212.74	−0.374588	−0.187294	+	0.982304i	$0.559972\pi$
$420$	0	0
$421$	15757.8	1.82420	0.912101	−	0.409965i	$-0.134459\pi$
$421$	15757.8	1.82420	0.912101	+	0.409965i	$0.134459\pi$
$422$	3871.79	0.446626
$423$	0	0
$424$	19422.1	2.22457
$425$	−460.469	−0.0525553
$426$	0	0
$427$	0	0
$428$	−22772.5	−2.57184
$429$	0	0
$430$	2871.63	0.322051
$431$	405.917	0.0453650	0.0226825	−	0.999743i	$-0.492779\pi$
$431$	405.917	0.0453650	0.0226825	+	0.999743i	$0.492779\pi$
$432$	0	0
$433$	7845.25	0.870713	0.435357	−	0.900258i	$-0.356622\pi$
$433$	7845.25	0.870713	0.435357	+	0.900258i	$0.356622\pi$
$434$	0	0
$435$	0	0
$436$	3063.63	0.336516
$437$	9818.95	1.07484
$438$	0	0
$439$	−423.029	−0.0459911	−0.0229955	−	0.999736i	$-0.507320\pi$
$439$	−423.029	−0.0459911	−0.0229955	+	0.999736i	$0.507320\pi$
$440$	−3529.84	−0.382452
$441$	0	0
$442$	3032.24	0.326310
$443$	16058.7	1.72229	0.861143	−	0.508362i	$-0.169749\pi$
$443$	16058.7	1.72229	0.861143	+	0.508362i	$0.169749\pi$
$444$	0	0
$445$	−7368.62	−0.784958
$446$	3842.09	0.407911
$447$	0	0
$448$	0	0
$449$	−2186.75	−0.229842	−0.114921	−	0.993375i	$-0.536662\pi$
$449$	−2186.75	−0.229842	−0.114921	+	0.993375i	$0.536662\pi$
$450$	0	0
$451$	8308.05	0.867430
$452$	23388.3	2.43384
$453$	0	0
$454$	−17188.2	−1.77683
$455$	0	0
$456$	0	0
$457$	−5799.22	−0.593602	−0.296801	−	0.954939i	$-0.595920\pi$
$457$	−5799.22	−0.593602	−0.296801	+	0.954939i	$0.595920\pi$
$458$	4416.62	0.450600
$459$	0	0
$460$	−10273.5	−1.04131
$461$	9873.35	0.997500	0.498750	−	0.866746i	$-0.333793\pi$
$461$	9873.35	0.997500	0.498750	+	0.866746i	$0.333793\pi$
$462$	0	0
$463$	−6181.84	−0.620506	−0.310253	−	0.950654i	$-0.600414\pi$
$463$	−6181.84	−0.620506	−0.310253	+	0.950654i	$0.600414\pi$
$464$	−4739.66	−0.474209
$465$	0	0
$466$	−35.9529	−0.00357400
$467$	6145.50	0.608950	0.304475	−	0.952520i	$-0.401519\pi$
$467$	6145.50	0.608950	0.304475	+	0.952520i	$0.401519\pi$
$468$	0	0
$469$	0	0
$470$	8326.91	0.817216
$471$	0	0
$472$	−14388.6	−1.40316
$473$	−3004.66	−0.292081
$474$	0	0
$475$	1685.08	0.162772
$476$	0	0
$477$	0	0
$478$	−4180.23	−0.399999
$479$	10879.4	1.03777	0.518887	−	0.854843i	$-0.326347\pi$
$479$	10879.4	1.03777	0.518887	+	0.854843i	$0.326347\pi$
$480$	0	0
$481$	5676.91	0.538139
$482$	10062.4	0.950894
$483$	0	0
$484$	−10239.9	−0.961675
$485$	2775.17	0.259823
$486$	0	0
$487$	8087.51	0.752526	0.376263	−	0.926513i	$-0.377209\pi$
$487$	8087.51	0.752526	0.376263	+	0.926513i	$0.377209\pi$
$488$	−20347.4	−1.88747
$489$	0	0
$490$	0	0
$491$	6959.90	0.639707	0.319853	−	0.947467i	$-0.396366\pi$
$491$	6959.90	0.639707	0.319853	+	0.947467i	$0.396366\pi$
$492$	0	0
$493$	3949.32	0.360788
$494$	−11096.4	−1.01063
$495$	0	0
$496$	1959.79	0.177413
$497$	0	0
$498$	0	0
$499$	18632.0	1.67151	0.835756	−	0.549101i	$-0.185030\pi$
$499$	18632.0	1.67151	0.835756	+	0.549101i	$0.185030\pi$
$500$	−1763.09	−0.157695
$501$	0	0
$502$	31734.6	2.82149
$503$	4627.62	0.410209	0.205105	−	0.978740i	$-0.434247\pi$
$503$	4627.62	0.410209	0.205105	+	0.978740i	$0.434247\pi$
$504$	0	0
$505$	−9450.72	−0.832775
$506$	16846.4	1.48007
$507$	0	0
$508$	−15639.4	−1.36592
$509$	−11351.8	−0.988528	−0.494264	−	0.869312i	$-0.664562\pi$
$509$	−11351.8	−0.988528	−0.494264	+	0.869312i	$0.664562\pi$
$510$	0	0
$511$	0	0
$512$	7853.76	0.677911
$513$	0	0
$514$	−14427.4	−1.23806
$515$	3313.50	0.283515
$516$	0	0
$517$	−8712.67	−0.741166
$518$	0	0
$519$	0	0
$520$	5025.02	0.423772
$521$	19096.1	1.60579	0.802893	−	0.596123i	$-0.203293\pi$
$521$	19096.1	1.60579	0.802893	+	0.596123i	$0.203293\pi$
$522$	0	0
$523$	3145.11	0.262956	0.131478	−	0.991319i	$-0.458028\pi$
$523$	3145.11	0.262956	0.131478	+	0.991319i	$0.458028\pi$
$524$	2616.24	0.218113
$525$	0	0
$526$	−21975.7	−1.82164
$527$	−1632.99	−0.134980
$528$	0	0
$529$	9054.20	0.744160
$530$	−15907.5	−1.30373
$531$	0	0
$532$	0	0
$533$	−11827.2	−0.961148
$534$	0	0
$535$	8072.66	0.652358
$536$	26059.8	2.10002
$537$	0	0
$538$	−11365.5	−0.910781
$539$	0	0
$540$	0	0
$541$	8776.12	0.697440	0.348720	−	0.937227i	$-0.386616\pi$
$541$	8776.12	0.697440	0.348720	+	0.937227i	$0.386616\pi$
$542$	36316.3	2.87808
$543$	0	0
$544$	−2314.98	−0.182452
$545$	−1086.03	−0.0853587
$546$	0	0
$547$	−13695.1	−1.07049	−0.535247	−	0.844696i	$-0.679781\pi$
$547$	−13695.1	−1.07049	−0.535247	+	0.844696i	$0.679781\pi$
$548$	535.847	0.0417705
$549$	0	0
$550$	2891.09	0.224139
$551$	−14452.5	−1.11742
$552$	0	0
$553$	0	0
$554$	21515.2	1.64999
$555$	0	0
$556$	−2589.75	−0.197536
$557$	−7850.44	−0.597188	−0.298594	−	0.954380i	$-0.596518\pi$
$557$	−7850.44	−0.597188	−0.298594	+	0.954380i	$0.596518\pi$
$558$	0	0
$559$	4277.38	0.323638
$560$	0	0
$561$	0	0
$562$	−6387.33	−0.479419
$563$	−4948.81	−0.370457	−0.185229	−	0.982695i	$-0.559303\pi$
$563$	−4948.81	−0.370457	−0.185229	+	0.982695i	$0.559303\pi$
$564$	0	0
$565$	−8290.98	−0.617353
$566$	18265.7	1.35648
$567$	0	0
$568$	12351.3	0.912408
$569$	8115.76	0.597945	0.298972	−	0.954262i	$-0.403356\pi$
$569$	8115.76	0.597945	0.298972	+	0.954262i	$0.403356\pi$
$570$	0	0
$571$	5656.42	0.414560	0.207280	−	0.978282i	$-0.433539\pi$
$571$	5656.42	0.414560	0.207280	+	0.978282i	$0.433539\pi$
$572$	−12148.0	−0.887996
$573$	0	0
$574$	0	0
$575$	3641.87	0.264133
$576$	0	0
$577$	9536.77	0.688078	0.344039	−	0.938955i	$-0.388205\pi$
$577$	9536.77	0.688078	0.344039	+	0.938955i	$0.388205\pi$
$578$	21503.8	1.54747
$579$	0	0
$580$	15121.5	1.08257
$581$	0	0
$582$	0	0
$583$	16644.5	1.18241
$584$	1186.54	0.0840743
$585$	0	0
$586$	18965.7	1.33697
$587$	−13089.6	−0.920383	−0.460191	−	0.887820i	$-0.652219\pi$
$587$	−13089.6	−0.920383	−0.460191	+	0.887820i	$0.652219\pi$
$588$	0	0
$589$	5975.92	0.418053
$590$	11784.9	0.822334
$591$	0	0
$592$	3583.72	0.248801
$593$	4281.96	0.296524	0.148262	−	0.988948i	$-0.452632\pi$
$593$	4281.96	0.296524	0.148262	+	0.988948i	$0.452632\pi$
$594$	0	0
$595$	0	0
$596$	19511.5	1.34098
$597$	0	0
$598$	−23982.2	−1.63997
$599$	−3699.92	−0.252378	−0.126189	−	0.992006i	$-0.540275\pi$
$599$	−3699.92	−0.252378	−0.126189	+	0.992006i	$0.540275\pi$
$600$	0	0
$601$	17286.1	1.17323	0.586616	−	0.809865i	$-0.300460\pi$
$601$	17286.1	1.17323	0.586616	+	0.809865i	$0.300460\pi$
$602$	0	0
$603$	0	0
$604$	10793.7	0.727135
$605$	3629.97	0.243933
$606$	0	0
$607$	−14456.7	−0.966689	−0.483344	−	0.875430i	$-0.660578\pi$
$607$	−14456.7	−0.966689	−0.483344	+	0.875430i	$0.660578\pi$
$608$	8471.63	0.565082
$609$	0	0
$610$	16665.4	1.10617
$611$	12403.2	0.821243
$612$	0	0
$613$	17981.9	1.18480	0.592400	−	0.805644i	$-0.298181\pi$
$613$	17981.9	1.18480	0.592400	+	0.805644i	$0.298181\pi$
$614$	−21722.9	−1.42779
$615$	0	0
$616$	0	0
$617$	−19614.7	−1.27983	−0.639916	−	0.768445i	$-0.721031\pi$
$617$	−19614.7	−1.27983	−0.639916	+	0.768445i	$0.721031\pi$
$618$	0	0
$619$	10462.9	0.679385	0.339692	−	0.940537i	$-0.389677\pi$
$619$	10462.9	0.679385	0.339692	+	0.940537i	$0.389677\pi$
$620$	−6252.56	−0.405014
$621$	0	0
$622$	−31387.1	−2.02332
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	13337.3	0.851544
$627$	0	0
$628$	33382.4	2.12118
$629$	−2986.14	−0.189293
$630$	0	0
$631$	24481.9	1.54454	0.772272	−	0.635292i	$-0.219120\pi$
$631$	24481.9	1.54454	0.772272	+	0.635292i	$0.219120\pi$
$632$	−25555.5	−1.60846
$633$	0	0
$634$	18856.2	1.18119
$635$	5544.06	0.346471
$636$	0	0
$637$	0	0
$638$	−24796.2	−1.53870
$639$	0	0
$640$	−13020.9	−0.804211
$641$	1109.39	0.0683595	0.0341797	−	0.999416i	$-0.489118\pi$
$641$	1109.39	0.0683595	0.0341797	+	0.999416i	$0.489118\pi$
$642$	0	0
$643$	−30112.5	−1.84684	−0.923422	−	0.383787i	$-0.874620\pi$
$643$	−30112.5	−1.84684	−0.923422	+	0.383787i	$0.874620\pi$
$644$	0	0
$645$	0	0
$646$	5836.90	0.355495
$647$	−4260.27	−0.258869	−0.129435	−	0.991588i	$-0.541316\pi$
$647$	−4260.27	−0.258869	−0.129435	+	0.991588i	$0.541316\pi$
$648$	0	0
$649$	−12330.9	−0.745808
$650$	−4115.70	−0.248356
$651$	0	0
$652$	−44256.2	−2.65829
$653$	10576.8	0.633844	0.316922	−	0.948452i	$-0.397351\pi$
$653$	10576.8	0.633844	0.316922	+	0.948452i	$0.397351\pi$
$654$	0	0
$655$	−927.438	−0.0553252
$656$	−7466.27	−0.444373
$657$	0	0
$658$	0	0
$659$	−3394.70	−0.200666	−0.100333	−	0.994954i	$-0.531991\pi$
$659$	−3394.70	−0.200666	−0.100333	+	0.994954i	$0.531991\pi$
$660$	0	0
$661$	33174.4	1.95210	0.976048	−	0.217554i	$-0.0698079\pi$
$661$	33174.4	1.95210	0.976048	+	0.217554i	$0.0698079\pi$
$662$	−52128.7	−3.06048
$663$	0	0
$664$	30341.9	1.77333
$665$	0	0
$666$	0	0
$667$	−31235.4	−1.81326
$668$	2065.89	0.119658
$669$	0	0
$670$	−21344.1	−1.23074
$671$	−17437.5	−1.00323
$672$	0	0
$673$	753.881	0.0431797	0.0215899	−	0.999767i	$-0.493127\pi$
$673$	753.881	0.0431797	0.0215899	+	0.999767i	$0.493127\pi$
$674$	−56976.9	−3.25619
$675$	0	0
$676$	−13694.3	−0.779149
$677$	15668.8	0.889511	0.444756	−	0.895652i	$-0.353291\pi$
$677$	15668.8	0.889511	0.444756	+	0.895652i	$0.353291\pi$
$678$	0	0
$679$	0	0
$680$	−2643.23	−0.149064
$681$	0	0
$682$	10252.9	0.575665
$683$	11557.4	0.647485	0.323742	−	0.946145i	$-0.395059\pi$
$683$	11557.4	0.647485	0.323742	+	0.946145i	$0.395059\pi$
$684$	0	0
$685$	−189.953	−0.0105952
$686$	0	0
$687$	0	0
$688$	2700.22	0.149630
$689$	−23694.7	−1.31016
$690$	0	0
$691$	18503.1	1.01866	0.509328	−	0.860572i	$-0.329894\pi$
$691$	18503.1	1.01866	0.509328	+	0.860572i	$0.329894\pi$
$692$	−20086.8	−1.10344
$693$	0	0
$694$	−29907.7	−1.63585
$695$	918.046	0.0501057
$696$	0	0
$697$	6221.28	0.338088
$698$	−14646.2	−0.794223
$699$	0	0
$700$	0	0
$701$	−22580.4	−1.21662	−0.608311	−	0.793699i	$-0.708153\pi$
$701$	−22580.4	−1.21662	−0.608311	+	0.793699i	$0.708153\pi$
$702$	0	0
$703$	10927.7	0.586269
$704$	18884.4	1.01099
$705$	0	0
$706$	56077.4	2.98938
$707$	0	0
$708$	0	0
$709$	−27426.6	−1.45279	−0.726394	−	0.687278i	$-0.758805\pi$
$709$	−27426.6	−1.45279	−0.726394	+	0.687278i	$0.758805\pi$
$710$	−10116.2	−0.534725
$711$	0	0
$712$	−42298.2	−2.22639
$713$	12915.5	0.678383
$714$	0	0
$715$	4306.37	0.225244
$716$	−17556.1	−0.916342
$717$	0	0
$718$	−28886.1	−1.50142
$719$	19383.0	1.00538	0.502688	−	0.864468i	$-0.332344\pi$
$719$	19383.0	1.00538	0.502688	+	0.864468i	$0.332344\pi$
$720$	0	0
$721$	0	0
$722$	10888.0	0.561230
$723$	0	0
$724$	54713.3	2.80857
$725$	−5360.47	−0.274597
$726$	0	0
$727$	12317.3	0.628368	0.314184	−	0.949362i	$-0.398269\pi$
$727$	12317.3	0.628368	0.314184	+	0.949362i	$0.398269\pi$
$728$	0	0
$729$	0	0
$730$	−971.828	−0.0492726
$731$	−2249.96	−0.113841
$732$	0	0
$733$	−1234.02	−0.0621822	−0.0310911	−	0.999517i	$-0.509898\pi$
$733$	−1234.02	−0.0621822	−0.0310911	+	0.999517i	$0.509898\pi$
$734$	−9063.05	−0.455754
$735$	0	0
$736$	18309.3	0.916970
$737$	22332.9	1.11620
$738$	0	0
$739$	−15257.3	−0.759473	−0.379736	−	0.925095i	$-0.623985\pi$
$739$	−15257.3	−0.759473	−0.379736	+	0.925095i	$0.623985\pi$
$740$	−11433.6	−0.567984
$741$	0	0
$742$	0	0
$743$	35565.1	1.75606	0.878032	−	0.478602i	$-0.158856\pi$
$743$	35565.1	1.75606	0.878032	+	0.478602i	$0.158856\pi$
$744$	0	0
$745$	−6916.69	−0.340145
$746$	−49140.8	−2.41176
$747$	0	0
$748$	6390.04	0.312357
$749$	0	0
$750$	0	0
$751$	14266.7	0.693209	0.346605	−	0.938011i	$-0.387335\pi$
$751$	14266.7	0.693209	0.346605	+	0.938011i	$0.387335\pi$
$752$	7829.89	0.379690
$753$	0	0
$754$	35299.4	1.70494
$755$	−3826.28	−0.184441
$756$	0	0
$757$	−15927.9	−0.764744	−0.382372	−	0.924009i	$-0.624893\pi$
$757$	−15927.9	−0.764744	−0.382372	+	0.924009i	$0.624893\pi$
$758$	−33223.3	−1.59198
$759$	0	0
$760$	9672.87	0.461674
$761$	−2566.48	−0.122253	−0.0611266	−	0.998130i	$-0.519469\pi$
$761$	−2566.48	−0.122253	−0.0611266	+	0.998130i	$0.519469\pi$
$762$	0	0
$763$	0	0
$764$	−22213.5	−1.05191
$765$	0	0
$766$	−33701.0	−1.58964
$767$	17554.0	0.826386
$768$	0	0
$769$	−14433.1	−0.676816	−0.338408	−	0.940999i	$-0.609888\pi$
$769$	−14433.1	−0.676816	−0.338408	+	0.940999i	$0.609888\pi$
$770$	0	0
$771$	0	0
$772$	−67359.4	−3.14031
$773$	−29443.2	−1.36999	−0.684993	−	0.728550i	$-0.740195\pi$
$773$	−29443.2	−1.36999	−0.684993	+	0.728550i	$0.740195\pi$
$774$	0	0
$775$	2216.48	0.102733
$776$	15930.4	0.736941
$777$	0	0
$778$	−34857.7	−1.60631
$779$	−22766.7	−1.04711
$780$	0	0
$781$	10584.9	0.484964
$782$	12615.0	0.576869
$783$	0	0
$784$	0	0
$785$	−11833.8	−0.538046
$786$	0	0
$787$	−26390.6	−1.19533	−0.597664	−	0.801747i	$-0.703904\pi$
$787$	−26390.6	−1.19533	−0.597664	+	0.801747i	$0.703904\pi$
$788$	39543.6	1.78767
$789$	0	0
$790$	20931.1	0.942650
$791$	0	0
$792$	0	0
$793$	24823.7	1.11162
$794$	−42013.2	−1.87783
$795$	0	0
$796$	−57870.3	−2.57683
$797$	−3738.33	−0.166146	−0.0830730	−	0.996543i	$-0.526473\pi$
$797$	−3738.33	−0.166146	−0.0830730	+	0.996543i	$0.526473\pi$
$798$	0	0
$799$	−6524.26	−0.288876
$800$	3142.15	0.138865
$801$	0	0
$802$	8382.48	0.369072
$803$	1016.85	0.0446873
$804$	0	0
$805$	0	0
$806$	−14595.8	−0.637861
$807$	0	0
$808$	−54250.1	−2.36202
$809$	−43204.1	−1.87760	−0.938798	−	0.344468i	$-0.888059\pi$
$809$	−43204.1	−1.87760	−0.938798	+	0.344468i	$0.888059\pi$
$810$	0	0
$811$	30192.4	1.30727	0.653637	−	0.756809i	$-0.273242\pi$
$811$	30192.4	1.30727	0.653637	+	0.756809i	$0.273242\pi$
$812$	0	0
$813$	0	0
$814$	18748.7	0.807301
$815$	15688.5	0.674286
$816$	0	0
$817$	8233.71	0.352584
$818$	−41185.3	−1.76041
$819$	0	0
$820$	23820.6	1.01445
$821$	40274.7	1.71206	0.856028	−	0.516929i	$-0.172925\pi$
$821$	40274.7	1.71206	0.856028	+	0.516929i	$0.172925\pi$
$822$	0	0
$823$	25184.2	1.06667	0.533334	−	0.845905i	$-0.320939\pi$
$823$	25184.2	1.06667	0.533334	+	0.845905i	$0.320939\pi$
$824$	19020.5	0.804140
$825$	0	0
$826$	0	0
$827$	38941.7	1.63741	0.818703	−	0.574218i	$-0.194694\pi$
$827$	38941.7	1.63741	0.818703	+	0.574218i	$0.194694\pi$
$828$	0	0
$829$	8327.05	0.348867	0.174433	−	0.984669i	$-0.444191\pi$
$829$	8327.05	0.348867	0.174433	+	0.984669i	$0.444191\pi$
$830$	−24851.3	−1.03928
$831$	0	0
$832$	−26883.5	−1.12021
$833$	0	0
$834$	0	0
$835$	−732.343	−0.0303518
$836$	−23384.2	−0.967418
$837$	0	0
$838$	15104.9	0.622660
$839$	8784.41	0.361468	0.180734	−	0.983532i	$-0.442153\pi$
$839$	8784.41	0.361468	0.180734	+	0.983532i	$0.442153\pi$
$840$	0	0
$841$	21586.4	0.885087
$842$	−74086.4	−3.03229
$843$	0	0
$844$	−11615.4	−0.473718
$845$	4854.53	0.197634
$846$	0	0
$847$	0	0
$848$	−14958.0	−0.605732
$849$	0	0
$850$	2164.92	0.0873602
$851$	23617.6	0.951350
$852$	0	0
$853$	9076.15	0.364316	0.182158	−	0.983269i	$-0.441692\pi$
$853$	9076.15	0.364316	0.182158	+	0.983269i	$0.441692\pi$
$854$	0	0
$855$	0	0
$856$	46339.6	1.85030
$857$	36396.7	1.45074	0.725372	−	0.688357i	$-0.241668\pi$
$857$	36396.7	1.45074	0.725372	+	0.688357i	$0.241668\pi$
$858$	0	0
$859$	−8915.27	−0.354115	−0.177058	−	0.984200i	$-0.556658\pi$
$859$	−8915.27	−0.354115	−0.177058	+	0.984200i	$0.556658\pi$
$860$	−8614.88	−0.341587
$861$	0	0
$862$	−1908.44	−0.0754081
$863$	6148.26	0.242514	0.121257	−	0.992621i	$-0.461308\pi$
$863$	6148.26	0.242514	0.121257	+	0.992621i	$0.461308\pi$
$864$	0	0
$865$	7120.59	0.279893
$866$	−36884.9	−1.44735
$867$	0	0
$868$	0	0
$869$	−21900.8	−0.854928
$870$	0	0
$871$	−31792.6	−1.23680
$872$	−6234.16	−0.242105
$873$	0	0
$874$	−46164.4	−1.78665
$875$	0	0
$876$	0	0
$877$	−14287.0	−0.550101	−0.275050	−	0.961430i	$-0.588694\pi$
$877$	−14287.0	−0.550101	−0.275050	+	0.961430i	$0.588694\pi$
$878$	1988.90	0.0764488
$879$	0	0
$880$	2718.53	0.104138
$881$	−13315.9	−0.509221	−0.254610	−	0.967044i	$-0.581947\pi$
$881$	−13315.9	−0.509221	−0.254610	+	0.967044i	$0.581947\pi$
$882$	0	0
$883$	−5271.78	−0.200917	−0.100458	−	0.994941i	$-0.532031\pi$
$883$	−5271.78	−0.200917	−0.100458	+	0.994941i	$0.532031\pi$
$884$	−9096.73	−0.346104
$885$	0	0
$886$	−75501.1	−2.86288
$887$	2606.07	0.0986507	0.0493253	−	0.998783i	$-0.484293\pi$
$887$	2606.07	0.0986507	0.0493253	+	0.998783i	$0.484293\pi$
$888$	0	0
$889$	0	0
$890$	34644.0	1.30480
$891$	0	0
$892$	−11526.3	−0.432654
$893$	23875.4	0.894694
$894$	0	0
$895$	6223.48	0.232434
$896$	0	0
$897$	0	0
$898$	10281.1	0.382056
$899$	−19010.2	−0.705258
$900$	0	0
$901$	12463.8	0.460854
$902$	−39060.8	−1.44189
$903$	0	0
$904$	−47592.8	−1.75101
$905$	−19395.4	−0.712405
$906$	0	0
$907$	18610.6	0.681317	0.340659	−	0.940187i	$-0.389350\pi$
$907$	18610.6	0.681317	0.340659	+	0.940187i	$0.389350\pi$
$908$	51564.6	1.88462
$909$	0	0
$910$	0	0
$911$	−41091.7	−1.49443	−0.747216	−	0.664581i	$-0.768610\pi$
$911$	−41091.7	−1.49443	−0.747216	+	0.664581i	$0.768610\pi$
$912$	0	0
$913$	26002.6	0.942563
$914$	27265.4	0.986716
$915$	0	0
$916$	−13249.8	−0.477934
$917$	0	0
$918$	0	0
$919$	38891.3	1.39598	0.697990	−	0.716107i	$-0.254078\pi$
$919$	38891.3	1.39598	0.697990	+	0.716107i	$0.254078\pi$
$920$	20905.5	0.749167
$921$	0	0
$922$	−46420.2	−1.65810
$923$	−15068.4	−0.537360
$924$	0	0
$925$	4053.12	0.144071
$926$	29064.3	1.03144
$927$	0	0
$928$	−26949.4	−0.953295
$929$	18699.4	0.660396	0.330198	−	0.943912i	$-0.392885\pi$
$929$	18699.4	0.660396	0.330198	+	0.943912i	$0.392885\pi$
$930$	0	0
$931$	0	0
$932$	107.859	0.00379080
$933$	0	0
$934$	−28893.4	−1.01223
$935$	−2265.22	−0.0792305
$936$	0	0
$937$	−21509.6	−0.749933	−0.374967	−	0.927038i	$-0.622346\pi$
$937$	−21509.6	−0.749933	−0.374967	+	0.927038i	$0.622346\pi$
$938$	0	0
$939$	0	0
$940$	−24980.7	−0.866788
$941$	−11241.7	−0.389448	−0.194724	−	0.980858i	$-0.562381\pi$
$941$	−11241.7	−0.389448	−0.194724	+	0.980858i	$0.562381\pi$
$942$	0	0
$943$	−49204.5	−1.69917
$944$	11081.5	0.382068
$945$	0	0
$946$	14126.6	0.485513
$947$	36556.3	1.25441	0.627203	−	0.778856i	$-0.284200\pi$
$947$	36556.3	1.25441	0.627203	+	0.778856i	$0.284200\pi$
$948$	0	0
$949$	−1447.57	−0.0495153
$950$	−7922.50	−0.270568
$951$	0	0
$952$	0	0
$953$	36633.4	1.24520	0.622598	−	0.782542i	$-0.286077\pi$
$953$	36633.4	1.24520	0.622598	+	0.782542i	$0.286077\pi$
$954$	0	0
$955$	7874.52	0.266820
$956$	12540.7	0.424263
$957$	0	0
$958$	−51150.3	−1.72504
$959$	0	0
$960$	0	0
$961$	−21930.5	−0.736146
$962$	−26690.3	−0.894523
$963$	0	0
$964$	−30187.3	−1.00858
$965$	23878.4	0.796551
$966$	0	0
$967$	−35515.8	−1.18109	−0.590544	−	0.807006i	$-0.701087\pi$
$967$	−35515.8	−1.18109	−0.590544	+	0.807006i	$0.701087\pi$
$968$	20837.2	0.691871
$969$	0	0
$970$	−13047.6	−0.431891
$971$	−39661.0	−1.31080	−0.655398	−	0.755283i	$-0.727499\pi$
$971$	−39661.0	−1.31080	−0.655398	+	0.755283i	$0.727499\pi$
$972$	0	0
$973$	0	0
$974$	−38023.9	−1.25089
$975$	0	0
$976$	15670.7	0.513942
$977$	−50325.3	−1.64795	−0.823977	−	0.566624i	$-0.808249\pi$
$977$	−50325.3	−1.64795	−0.823977	+	0.566624i	$0.808249\pi$
$978$	0	0
$979$	−36249.0	−1.18337
$980$	0	0
$981$	0	0
$982$	−32722.4	−1.06335
$983$	51189.0	1.66091	0.830456	−	0.557084i	$-0.188080\pi$
$983$	51189.0	1.66091	0.830456	+	0.557084i	$0.188080\pi$
$984$	0	0
$985$	−14017.9	−0.453449
$986$	−18568.0	−0.599721
$987$	0	0
$988$	33289.3	1.07194
$989$	17795.1	0.572145
$990$	0	0
$991$	−55137.3	−1.76740	−0.883700	−	0.468054i	$-0.844955\pi$
$991$	−55137.3	−1.76740	−0.883700	+	0.468054i	$0.844955\pi$
$992$	11143.2	0.356651
$993$	0	0
$994$	0	0
$995$	20514.6	0.653624
$996$	0	0
$997$	−41606.5	−1.32166	−0.660828	−	0.750537i	$-0.729795\pi$
$997$	−41606.5	−1.32166	−0.660828	+	0.750537i	$0.729795\pi$
$998$	−87599.6	−2.77848
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.v.1.1		2
3.2	odd	2		735.4.a.q.1.2		2
7.6	odd	2		315.4.a.g.1.1		2
21.20	even	2		105.4.a.g.1.2	✓	2
35.34	odd	2		1575.4.a.y.1.2		2
84.83	odd	2		1680.4.a.y.1.2		2
105.62	odd	4		525.4.d.j.274.4		4
105.83	odd	4		525.4.d.j.274.1		4
105.104	even	2		525.4.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.g.1.2	✓	2	21.20	even	2
315.4.a.g.1.1		2	7.6	odd	2
525.4.a.i.1.1		2	105.104	even	2
525.4.d.j.274.1		4	105.83	odd	4
525.4.d.j.274.4		4	105.62	odd	4
735.4.a.q.1.2		2	3.2	odd	2
1575.4.a.y.1.2		2	35.34	odd	2
1680.4.a.y.1.2		2	84.83	odd	2
2205.4.a.v.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-4.70156 q^{2} +14.1047 q^{4} -5.00000 q^{5} -28.7016 q^{8} +O(q^{10})\)	\(q-4.70156 q^{2} +14.1047 q^{4} -5.00000 q^{5} -28.7016 q^{8} +23.5078 q^{10} -24.5969 q^{11} +35.0156 q^{13} +22.1047 q^{16} -18.4187 q^{17} +67.4031 q^{19} -70.5234 q^{20} +115.644 q^{22} +145.675 q^{23} +25.0000 q^{25} -164.628 q^{26} -214.419 q^{29} +88.6594 q^{31} +125.686 q^{32} +86.5969 q^{34} +162.125 q^{37} -316.900 q^{38} +143.508 q^{40} -337.769 q^{41} +122.156 q^{43} -346.931 q^{44} -684.900 q^{46} +354.219 q^{47} -117.539 q^{50} +493.884 q^{52} -676.691 q^{53} +122.984 q^{55} +1008.10 q^{58} +501.319 q^{59} +708.931 q^{61} -416.837 q^{62} -767.758 q^{64} -175.078 q^{65} -907.956 q^{67} -259.791 q^{68} -430.334 q^{71} -41.3406 q^{73} -762.241 q^{74} +950.700 q^{76} +890.388 q^{79} -110.523 q^{80} +1588.04 q^{82} -1057.15 q^{83} +92.0937 q^{85} -574.325 q^{86} +705.969 q^{88} +1473.72 q^{89} +2054.70 q^{92} -1665.38 q^{94} -337.016 q^{95} -555.034 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 9 q^{4} - 10 q^{5} - 51 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 + 9 * q^4 - 10 * q^5 - 51 * q^8	\( 2 q - 3 q^{2} + 9 q^{4} - 10 q^{5} - 51 q^{8} + 15 q^{10} - 62 q^{11} + 6 q^{13} + 25 q^{16} + 40 q^{17} + 122 q^{19} - 45 q^{20} + 52 q^{22} - 16 q^{23} + 50 q^{25} - 214 q^{26} - 352 q^{29} - 66 q^{31} + 309 q^{32} + 186 q^{34} - 188 q^{37} - 224 q^{38} + 255 q^{40} + 16 q^{41} - 396 q^{43} - 156 q^{44} - 960 q^{46} - 188 q^{47} - 75 q^{50} + 642 q^{52} - 982 q^{53} + 310 q^{55} + 774 q^{58} + 516 q^{59} + 880 q^{61} - 680 q^{62} - 479 q^{64} - 30 q^{65} - 356 q^{67} - 558 q^{68} - 310 q^{71} - 326 q^{73} - 1358 q^{74} + 672 q^{76} + 1832 q^{79} - 125 q^{80} + 2190 q^{82} - 680 q^{83} - 200 q^{85} - 1456 q^{86} + 1540 q^{88} + 796 q^{89} + 2880 q^{92} - 2588 q^{94} - 610 q^{95} + 670 q^{97}+O(q^{100}) \) 2 * q - 3 * q^2 + 9 * q^4 - 10 * q^5 - 51 * q^8 + 15 * q^10 - 62 * q^11 + 6 * q^13 + 25 * q^16 + 40 * q^17 + 122 * q^19 - 45 * q^20 + 52 * q^22 - 16 * q^23 + 50 * q^25 - 214 * q^26 - 352 * q^29 - 66 * q^31 + 309 * q^32 + 186 * q^34 - 188 * q^37 - 224 * q^38 + 255 * q^40 + 16 * q^41 - 396 * q^43 - 156 * q^44 - 960 * q^46 - 188 * q^47 - 75 * q^50 + 642 * q^52 - 982 * q^53 + 310 * q^55 + 774 * q^58 + 516 * q^59 + 880 * q^61 - 680 * q^62 - 479 * q^64 - 30 * q^65 - 356 * q^67 - 558 * q^68 - 310 * q^71 - 326 * q^73 - 1358 * q^74 + 672 * q^76 + 1832 * q^79 - 125 * q^80 + 2190 * q^82 - 680 * q^83 - 200 * q^85 - 1456 * q^86 + 1540 * q^88 + 796 * q^89 + 2880 * q^92 - 2588 * q^94 - 610 * q^95 + 670 * q^97

Introduction

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