LMFDB - Embedded newform 648.4.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(1,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, 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("648.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.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:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 13 $ x^4 - 8*x^2 + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.50597$ of defining polynomial
Character	$\chi$	$=$	648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+17.9847 q^{5} +7.28309 q^{7} +O(q^{10})$	$q+17.9847 q^{5} +7.28309 q^{7} -4.61468 q^{11} +29.8602 q^{13} -67.9721 q^{17} +111.462 q^{19} +218.506 q^{23} +198.450 q^{25} -34.2713 q^{29} -77.7046 q^{31} +130.984 q^{35} -347.228 q^{37} -234.546 q^{41} +53.3798 q^{43} +385.402 q^{47} -289.957 q^{49} +461.661 q^{53} -82.9938 q^{55} -7.16328 q^{59} +416.109 q^{61} +537.027 q^{65} -869.274 q^{67} +585.943 q^{71} -733.324 q^{73} -33.6092 q^{77} +1170.90 q^{79} +67.4364 q^{83} -1222.46 q^{85} +965.886 q^{89} +217.475 q^{91} +2004.62 q^{95} +1.43299 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5}+O(q^{10}) $ 4 * q + 8 * q^5	$ 4 q + 8 q^{5} + 8 q^{11} + 4 q^{13} + 16 q^{17} + 80 q^{19} + 200 q^{23} + 8 q^{25} + 216 q^{29} + 80 q^{31} + 408 q^{35} - 276 q^{37} + 384 q^{41} + 160 q^{43} + 768 q^{47} - 268 q^{49} + 944 q^{53} + 304 q^{55} + 992 q^{59} - 548 q^{61} + 1328 q^{65} + 464 q^{67} + 1720 q^{71} - 764 q^{73} + 1728 q^{77} + 688 q^{79} + 2128 q^{83} - 1324 q^{85} + 2112 q^{89} + 1776 q^{91} + 2056 q^{95} - 1816 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 8 * q^11 + 4 * q^13 + 16 * q^17 + 80 * q^19 + 200 * q^23 + 8 * q^25 + 216 * q^29 + 80 * q^31 + 408 * q^35 - 276 * q^37 + 384 * q^41 + 160 * q^43 + 768 * q^47 - 268 * q^49 + 944 * q^53 + 304 * q^55 + 992 * q^59 - 548 * q^61 + 1328 * q^65 + 464 * q^67 + 1720 * q^71 - 764 * q^73 + 1728 * q^77 + 688 * q^79 + 2128 * q^83 - 1324 * q^85 + 2112 * q^89 + 1776 * q^91 + 2056 * q^95 - 1816 * 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	17.9847	1.60860	0.804301	−	0.594222i	$-0.202540\pi$
$5$	17.9847	1.60860	0.804301	+	0.594222i	$0.202540\pi$
$6$	0	0
$7$	7.28309	0.393250	0.196625	−	0.980479i	$-0.437002\pi$
$7$	7.28309	0.393250	0.196625	+	0.980479i	$0.437002\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.61468	−0.126489	−0.0632445	−	0.997998i	$-0.520145\pi$
$11$	−4.61468	−0.126489	−0.0632445	+	0.997998i	$0.520145\pi$
$12$	0	0
$13$	29.8602	0.637056	0.318528	−	0.947913i	$-0.396811\pi$
$13$	29.8602	0.637056	0.318528	+	0.947913i	$0.396811\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−67.9721	−0.969744	−0.484872	−	0.874585i	$-0.661134\pi$
$17$	−67.9721	−0.969744	−0.484872	+	0.874585i	$0.661134\pi$
$18$	0	0
$19$	111.462	1.34585	0.672926	−	0.739710i	$-0.265037\pi$
$19$	111.462	1.34585	0.672926	+	0.739710i	$0.265037\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	218.506	1.98094	0.990472	−	0.137711i	$-0.0439746\pi$
$23$	218.506	1.98094	0.990472	+	0.137711i	$0.0439746\pi$
$24$	0	0
$25$	198.450	1.58760
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−34.2713	−0.219449	−0.109725	−	0.993962i	$-0.534997\pi$
$29$	−34.2713	−0.219449	−0.109725	+	0.993962i	$0.534997\pi$
$30$	0	0
$31$	−77.7046	−0.450199	−0.225099	−	0.974336i	$-0.572271\pi$
$31$	−77.7046	−0.450199	−0.225099	+	0.974336i	$0.572271\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	130.984	0.632583
$36$	0	0
$37$	−347.228	−1.54281	−0.771404	−	0.636346i	$-0.780445\pi$
$37$	−347.228	−1.54281	−0.771404	+	0.636346i	$0.780445\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−234.546	−0.893413	−0.446707	−	0.894680i	$-0.647403\pi$
$41$	−234.546	−0.893413	−0.446707	+	0.894680i	$0.647403\pi$
$42$	0	0
$43$	53.3798	0.189310	0.0946552	−	0.995510i	$-0.469825\pi$
$43$	53.3798	0.189310	0.0946552	+	0.995510i	$0.469825\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	385.402	1.19610	0.598049	−	0.801460i	$-0.295943\pi$
$47$	385.402	1.19610	0.598049	+	0.801460i	$0.295943\pi$
$48$	0	0
$49$	−289.957	−0.845354
$50$	0	0
$51$	0	0
$52$	0	0
$53$	461.661	1.19649	0.598245	−	0.801313i	$-0.295865\pi$
$53$	461.661	1.19649	0.598245	+	0.801313i	$0.295865\pi$
$54$	0	0
$55$	−82.9938	−0.203471
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.16328	−0.0158064	−0.00790322	−	0.999969i	$-0.502516\pi$
$59$	−7.16328	−0.0158064	−0.00790322	+	0.999969i	$0.502516\pi$
$60$	0	0
$61$	416.109	0.873398	0.436699	−	0.899608i	$-0.356147\pi$
$61$	416.109	0.873398	0.436699	+	0.899608i	$0.356147\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	537.027	1.02477
$66$	0	0
$67$	−869.274	−1.58506	−0.792528	−	0.609835i	$-0.791236\pi$
$67$	−869.274	−1.58506	−0.792528	+	0.609835i	$0.791236\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	585.943	0.979417	0.489709	−	0.871886i	$-0.337103\pi$
$71$	585.943	0.979417	0.489709	+	0.871886i	$0.337103\pi$
$72$	0	0
$73$	−733.324	−1.17574	−0.587870	−	0.808955i	$-0.700034\pi$
$73$	−733.324	−1.17574	−0.587870	+	0.808955i	$0.700034\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−33.6092	−0.0497418
$78$	0	0
$79$	1170.90	1.66754	0.833772	−	0.552108i	$-0.186177\pi$
$79$	1170.90	1.66754	0.833772	+	0.552108i	$0.186177\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	67.4364	0.0891820	0.0445910	−	0.999005i	$-0.485802\pi$
$83$	67.4364	0.0891820	0.0445910	+	0.999005i	$0.485802\pi$
$84$	0	0
$85$	−1222.46	−1.55993
$86$	0	0
$87$	0	0
$88$	0	0
$89$	965.886	1.15038	0.575189	−	0.818020i	$-0.304928\pi$
$89$	965.886	1.15038	0.575189	+	0.818020i	$0.304928\pi$
$90$	0	0
$91$	217.475	0.250522
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2004.62	2.16494
$96$	0	0
$97$	1.43299	0.00149998	0.000749988	−	1.00000i	$-0.499761\pi$
$97$	1.43299	0.00149998	0.000749988		1.00000i	$0.499761\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	896.177	0.882901	0.441450	−	0.897286i	$-0.354464\pi$
$101$	896.177	0.882901	0.441450	+	0.897286i	$0.354464\pi$
$102$	0	0
$103$	1940.81	1.85664	0.928318	−	0.371788i	$-0.121255\pi$
$103$	1940.81	1.85664	0.928318	+	0.371788i	$0.121255\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1888.04	−1.70583	−0.852915	−	0.522050i	$-0.825168\pi$
$107$	−1888.04	−1.70583	−0.852915	+	0.522050i	$0.825168\pi$
$108$	0	0
$109$	1201.72	1.05600	0.528001	−	0.849244i	$-0.322942\pi$
$109$	1201.72	1.05600	0.528001	+	0.849244i	$0.322942\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1522.42	1.26741	0.633706	−	0.773574i	$-0.281533\pi$
$113$	1522.42	1.26741	0.633706	+	0.773574i	$0.281533\pi$
$114$	0	0
$115$	3929.78	3.18655
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−495.047	−0.381352
$120$	0	0
$121$	−1309.70	−0.984001
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1320.98	0.945214
$126$	0	0
$127$	−2300.50	−1.60738	−0.803688	−	0.595051i	$-0.797132\pi$
$127$	−2300.50	−1.60738	−0.803688	+	0.595051i	$0.797132\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1038.33	0.692517	0.346258	−	0.938139i	$-0.387452\pi$
$131$	1038.33	0.692517	0.346258	+	0.938139i	$0.387452\pi$
$132$	0	0
$133$	811.790	0.529256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1089.53	0.679448	0.339724	−	0.940525i	$-0.389666\pi$
$137$	1089.53	0.679448	0.339724	+	0.940525i	$0.389666\pi$
$138$	0	0
$139$	−1919.33	−1.17119	−0.585594	−	0.810605i	$-0.699139\pi$
$139$	−1919.33	−1.17119	−0.585594	+	0.810605i	$0.699139\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−137.795	−0.0805807
$144$	0	0
$145$	−616.360	−0.353006
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2131.77	−1.17209	−0.586044	−	0.810279i	$-0.699316\pi$
$149$	−2131.77	−1.17209	−0.586044	+	0.810279i	$0.699316\pi$
$150$	0	0
$151$	−1165.86	−0.628323	−0.314161	−	0.949370i	$-0.601723\pi$
$151$	−1165.86	−0.628323	−0.314161	+	0.949370i	$0.601723\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1397.50	−0.724191
$156$	0	0
$157$	36.7762	0.0186947	0.00934733	−	0.999956i	$-0.497025\pi$
$157$	36.7762	0.0186947	0.00934733	+	0.999956i	$0.497025\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1591.40	0.779007
$162$	0	0
$163$	3208.90	1.54197	0.770983	−	0.636856i	$-0.219765\pi$
$163$	3208.90	1.54197	0.770983	+	0.636856i	$0.219765\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1796.09	0.832248	0.416124	−	0.909308i	$-0.363388\pi$
$167$	1796.09	0.832248	0.416124	+	0.909308i	$0.363388\pi$
$168$	0	0
$169$	−1305.37	−0.594159
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1694.71	0.744778	0.372389	−	0.928077i	$-0.378539\pi$
$173$	1694.71	0.744778	0.372389	+	0.928077i	$0.378539\pi$
$174$	0	0
$175$	1445.33	0.624324
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−327.128	−0.136596	−0.0682981	−	0.997665i	$-0.521757\pi$
$179$	−327.128	−0.136596	−0.0682981	+	0.997665i	$0.521757\pi$
$180$	0	0
$181$	−4449.32	−1.82716	−0.913578	−	0.406663i	$-0.866692\pi$
$181$	−4449.32	−1.82716	−0.913578	+	0.406663i	$0.866692\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6244.79	−2.48176
$186$	0	0
$187$	313.670	0.122662
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2078.96	0.787583	0.393792	−	0.919200i	$-0.371163\pi$
$191$	2078.96	0.787583	0.393792	+	0.919200i	$0.371163\pi$
$192$	0	0
$193$	−2887.17	−1.07680	−0.538401	−	0.842688i	$-0.680972\pi$
$193$	−2887.17	−1.07680	−0.538401	+	0.842688i	$0.680972\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2679.74	0.969154	0.484577	−	0.874749i	$-0.338973\pi$
$197$	2679.74	0.969154	0.484577	+	0.874749i	$0.338973\pi$
$198$	0	0
$199$	341.646	0.121702	0.0608508	−	0.998147i	$-0.480619\pi$
$199$	341.646	0.121702	0.0608508	+	0.998147i	$0.480619\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−249.601	−0.0862984
$204$	0	0
$205$	−4218.25	−1.43715
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−514.363	−0.170236
$210$	0	0
$211$	1575.78	0.514128	0.257064	−	0.966394i	$-0.417245\pi$
$211$	1575.78	0.514128	0.257064	+	0.966394i	$0.417245\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	960.021	0.304525
$216$	0	0
$217$	−565.930	−0.177041
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2029.66	−0.617782
$222$	0	0
$223$	−2982.80	−0.895708	−0.447854	−	0.894107i	$-0.647812\pi$
$223$	−2982.80	−0.895708	−0.447854	+	0.894107i	$0.647812\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2139.81	−0.625656	−0.312828	−	0.949810i	$-0.601276\pi$
$227$	−2139.81	−0.625656	−0.312828	+	0.949810i	$0.601276\pi$
$228$	0	0
$229$	619.401	0.178739	0.0893694	−	0.995999i	$-0.471515\pi$
$229$	619.401	0.178739	0.0893694	+	0.995999i	$0.471515\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4272.01	−1.20115	−0.600577	−	0.799567i	$-0.705062\pi$
$233$	−4272.01	−1.20115	−0.600577	+	0.799567i	$0.705062\pi$
$234$	0	0
$235$	6931.34	1.92405
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6252.62	1.69225	0.846126	−	0.532982i	$-0.178929\pi$
$239$	6252.62	1.69225	0.846126	+	0.532982i	$0.178929\pi$
$240$	0	0
$241$	250.780	0.0670296	0.0335148	−	0.999438i	$-0.489330\pi$
$241$	250.780	0.0670296	0.0335148	+	0.999438i	$0.489330\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5214.79	−1.35984
$246$	0	0
$247$	3328.29	0.857384
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7146.55	−1.79716	−0.898579	−	0.438812i	$-0.855399\pi$
$251$	−7146.55	−1.79716	−0.898579	+	0.438812i	$0.855399\pi$
$252$	0	0
$253$	−1008.34	−0.250568
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3027.64	−0.734859	−0.367429	−	0.930051i	$-0.619762\pi$
$257$	−3027.64	−0.734859	−0.367429	+	0.930051i	$0.619762\pi$
$258$	0	0
$259$	−2528.89	−0.606709
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5218.96	−1.22363	−0.611816	−	0.791000i	$-0.709561\pi$
$263$	−5218.96	−1.22363	−0.611816	+	0.791000i	$0.709561\pi$
$264$	0	0
$265$	8302.84	1.92468
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6038.18	−1.36860	−0.684302	−	0.729199i	$-0.739893\pi$
$269$	−6038.18	−1.36860	−0.684302	+	0.729199i	$0.739893\pi$
$270$	0	0
$271$	3726.70	0.835354	0.417677	−	0.908596i	$-0.362844\pi$
$271$	3726.70	0.835354	0.417677	+	0.908596i	$0.362844\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−915.784	−0.200814
$276$	0	0
$277$	−572.095	−0.124093	−0.0620466	−	0.998073i	$-0.519763\pi$
$277$	−572.095	−0.124093	−0.0620466	+	0.998073i	$0.519763\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2047.98	−0.434777	−0.217389	−	0.976085i	$-0.569754\pi$
$281$	−2047.98	−0.434777	−0.217389	+	0.976085i	$0.569754\pi$
$282$	0	0
$283$	4793.97	1.00697	0.503484	−	0.864005i	$-0.332051\pi$
$283$	4793.97	1.00697	0.503484	+	0.864005i	$0.332051\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1708.22	−0.351335
$288$	0	0
$289$	−292.797	−0.0595963
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−69.4967	−0.0138568	−0.00692840	−	0.999976i	$-0.502205\pi$
$293$	−69.4967	−0.0138568	−0.00692840	+	0.999976i	$0.502205\pi$
$294$	0	0
$295$	−128.830	−0.0254263
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6524.65	1.26197
$300$	0	0
$301$	388.770	0.0744463
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7483.60	1.40495
$306$	0	0
$307$	665.210	0.123666	0.0618331	−	0.998087i	$-0.480305\pi$
$307$	665.210	0.123666	0.0618331	+	0.998087i	$0.480305\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	252.564	0.0460501	0.0230251	−	0.999735i	$-0.492670\pi$
$311$	252.564	0.0460501	0.0230251	+	0.999735i	$0.492670\pi$
$312$	0	0
$313$	−8826.49	−1.59394	−0.796969	−	0.604020i	$-0.793565\pi$
$313$	−8826.49	−1.59394	−0.796969	+	0.604020i	$0.793565\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−917.091	−0.162489	−0.0812444	−	0.996694i	$-0.525889\pi$
$317$	−917.091	−0.162489	−0.0812444	+	0.996694i	$0.525889\pi$
$318$	0	0
$319$	158.151	0.0277579
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7576.32	−1.30513
$324$	0	0
$325$	5925.76	1.01139
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2806.92	0.470365
$330$	0	0
$331$	−1662.81	−0.276122	−0.138061	−	0.990424i	$-0.544087\pi$
$331$	−1662.81	−0.276122	−0.138061	+	0.990424i	$0.544087\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15633.7	−2.54972
$336$	0	0
$337$	−2403.96	−0.388582	−0.194291	−	0.980944i	$-0.562241\pi$
$337$	−2403.96	−0.388582	−0.194291	+	0.980944i	$0.562241\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	358.582	0.0569452
$342$	0	0
$343$	−4609.88	−0.725686
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7871.63	−1.21778	−0.608892	−	0.793253i	$-0.708386\pi$
$347$	−7871.63	−1.21778	−0.608892	+	0.793253i	$0.708386\pi$
$348$	0	0
$349$	−11474.1	−1.75987	−0.879934	−	0.475096i	$-0.842413\pi$
$349$	−11474.1	−1.75987	−0.879934	+	0.475096i	$0.842413\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4021.19	0.606306	0.303153	−	0.952942i	$-0.401961\pi$
$353$	4021.19	0.606306	0.303153	+	0.952942i	$0.401961\pi$
$354$	0	0
$355$	10538.0	1.57549
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1662.93	−0.244473	−0.122237	−	0.992501i	$-0.539007\pi$
$359$	−1662.93	−0.244473	−0.122237	+	0.992501i	$0.539007\pi$
$360$	0	0
$361$	5564.84	0.811319
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−13188.6	−1.89130
$366$	0	0
$367$	−2938.64	−0.417972	−0.208986	−	0.977919i	$-0.567016\pi$
$367$	−2938.64	−0.417972	−0.208986	+	0.977919i	$0.567016\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3362.32	0.470520
$372$	0	0
$373$	−7619.53	−1.05771	−0.528853	−	0.848713i	$-0.677378\pi$
$373$	−7619.53	−1.05771	−0.528853	+	0.848713i	$0.677378\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1023.35	−0.139802
$378$	0	0
$379$	−13118.3	−1.77795	−0.888975	−	0.457955i	$-0.848582\pi$
$379$	−13118.3	−1.77795	−0.888975	+	0.457955i	$0.848582\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8602.70	−1.14772	−0.573861	−	0.818953i	$-0.694555\pi$
$383$	−8602.70	−1.14772	−0.573861	+	0.818953i	$0.694555\pi$
$384$	0	0
$385$	−604.451	−0.0800148
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12921.6	1.68419	0.842095	−	0.539329i	$-0.181322\pi$
$389$	12921.6	1.68419	0.842095	+	0.539329i	$0.181322\pi$
$390$	0	0
$391$	−14852.3	−1.92101
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21058.2	2.68242
$396$	0	0
$397$	5511.90	0.696812	0.348406	−	0.937344i	$-0.386723\pi$
$397$	5511.90	0.696812	0.348406	+	0.937344i	$0.386723\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2233.31	0.278120	0.139060	−	0.990284i	$-0.455592\pi$
$401$	2233.31	0.278120	0.139060	+	0.990284i	$0.455592\pi$
$402$	0	0
$403$	−2320.28	−0.286802
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1602.35	0.195148
$408$	0	0
$409$	2225.31	0.269033	0.134517	−	0.990911i	$-0.457052\pi$
$409$	2225.31	0.269033	0.134517	+	0.990911i	$0.457052\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−52.1708	−0.00621588
$414$	0	0
$415$	1212.82	0.143458
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5421.12	−0.632074	−0.316037	−	0.948747i	$-0.602352\pi$
$419$	−5421.12	−0.632074	−0.316037	+	0.948747i	$0.602352\pi$
$420$	0	0
$421$	12820.4	1.48415	0.742075	−	0.670317i	$-0.233842\pi$
$421$	12820.4	1.48415	0.742075	+	0.670317i	$0.233842\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−13489.1	−1.53957
$426$	0	0
$427$	3030.56	0.343464
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10070.7	−1.12549	−0.562747	−	0.826629i	$-0.690255\pi$
$431$	−10070.7	−1.12549	−0.562747	+	0.826629i	$0.690255\pi$
$432$	0	0
$433$	−5708.09	−0.633518	−0.316759	−	0.948506i	$-0.602595\pi$
$433$	−5708.09	−0.633518	−0.316759	+	0.948506i	$0.602595\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24355.2	2.66606
$438$	0	0
$439$	−2979.59	−0.323936	−0.161968	−	0.986796i	$-0.551784\pi$
$439$	−2979.59	−0.323936	−0.161968	+	0.986796i	$0.551784\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10492.4	1.12530	0.562649	−	0.826696i	$-0.309782\pi$
$443$	10492.4	1.12530	0.562649	+	0.826696i	$0.309782\pi$
$444$	0	0
$445$	17371.2	1.85050
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4846.95	−0.509447	−0.254724	−	0.967014i	$-0.581985\pi$
$449$	−4846.95	−0.509447	−0.254724	+	0.967014i	$0.581985\pi$
$450$	0	0
$451$	1082.36	0.113007
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3911.22	0.402991
$456$	0	0
$457$	5141.73	0.526302	0.263151	−	0.964755i	$-0.415238\pi$
$457$	5141.73	0.526302	0.263151	+	0.964755i	$0.415238\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3992.66	−0.403377	−0.201689	−	0.979450i	$-0.564643\pi$
$461$	−3992.66	−0.403377	−0.201689	+	0.979450i	$0.564643\pi$
$462$	0	0
$463$	−9507.06	−0.954277	−0.477139	−	0.878828i	$-0.658326\pi$
$463$	−9507.06	−0.954277	−0.477139	+	0.878828i	$0.658326\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1373.54	−0.136102	−0.0680511	−	0.997682i	$-0.521678\pi$
$467$	−1373.54	−0.136102	−0.0680511	+	0.997682i	$0.521678\pi$
$468$	0	0
$469$	−6331.00	−0.623323
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−246.331	−0.0239457
$474$	0	0
$475$	22119.7	2.13668
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17107.8	−1.63189	−0.815943	−	0.578133i	$-0.803782\pi$
$479$	−17107.8	−1.63189	−0.815943	+	0.578133i	$0.803782\pi$
$480$	0	0
$481$	−10368.3	−0.982855
$482$	0	0
$483$	0	0
$484$	0	0
$485$	25.7718	0.00241286
$486$	0	0
$487$	−4627.18	−0.430549	−0.215275	−	0.976554i	$-0.569065\pi$
$487$	−4627.18	−0.430549	−0.215275	+	0.976554i	$0.569065\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9161.66	0.842078	0.421039	−	0.907043i	$-0.361666\pi$
$491$	9161.66	0.842078	0.421039	+	0.907043i	$0.361666\pi$
$492$	0	0
$493$	2329.49	0.212810
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4267.48	0.385156
$498$	0	0
$499$	8299.08	0.744524	0.372262	−	0.928128i	$-0.378582\pi$
$499$	8299.08	0.744524	0.372262	+	0.928128i	$0.378582\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9568.57	0.848194	0.424097	−	0.905617i	$-0.360592\pi$
$503$	9568.57	0.848194	0.424097	+	0.905617i	$0.360592\pi$
$504$	0	0
$505$	16117.5	1.42024
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8749.02	0.761874	0.380937	−	0.924601i	$-0.375602\pi$
$509$	8749.02	0.761874	0.380937	+	0.924601i	$0.375602\pi$
$510$	0	0
$511$	−5340.87	−0.462360
$512$	0	0
$513$	0	0
$514$	0	0
$515$	34904.9	2.98659
$516$	0	0
$517$	−1778.51	−0.151293
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4974.67	−0.418319	−0.209160	−	0.977882i	$-0.567073\pi$
$521$	−4974.67	−0.418319	−0.209160	+	0.977882i	$0.567073\pi$
$522$	0	0
$523$	10144.9	0.848196	0.424098	−	0.905616i	$-0.360591\pi$
$523$	10144.9	0.848196	0.424098	+	0.905616i	$0.360591\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5281.74	0.436578
$528$	0	0
$529$	35578.0	2.92414
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7003.60	−0.569155
$534$	0	0
$535$	−33955.9	−2.74400
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1338.06	0.106928
$540$	0	0
$541$	−9130.53	−0.725604	−0.362802	−	0.931866i	$-0.618180\pi$
$541$	−9130.53	−0.725604	−0.362802	+	0.931866i	$0.618180\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	21612.7	1.69869
$546$	0	0
$547$	−3754.31	−0.293460	−0.146730	−	0.989177i	$-0.546875\pi$
$547$	−3754.31	−0.293460	−0.146730	+	0.989177i	$0.546875\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3819.96	−0.295346
$552$	0	0
$553$	8527.74	0.655762
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9750.59	−0.741734	−0.370867	−	0.928686i	$-0.620939\pi$
$557$	−9750.59	−0.741734	−0.370867	+	0.928686i	$0.620939\pi$
$558$	0	0
$559$	1593.93	0.120601
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2886.34	−0.216065	−0.108033	−	0.994147i	$-0.534455\pi$
$563$	−2886.34	−0.216065	−0.108033	+	0.994147i	$0.534455\pi$
$564$	0	0
$565$	27380.3	2.03876
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7508.07	0.553171	0.276586	−	0.960989i	$-0.410797\pi$
$569$	7508.07	0.553171	0.276586	+	0.960989i	$0.410797\pi$
$570$	0	0
$571$	9160.32	0.671361	0.335681	−	0.941976i	$-0.391034\pi$
$571$	9160.32	0.671361	0.335681	+	0.941976i	$0.391034\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	43362.6	3.14495
$576$	0	0
$577$	−21578.4	−1.55688	−0.778441	−	0.627718i	$-0.783989\pi$
$577$	−21578.4	−1.55688	−0.778441	+	0.627718i	$0.783989\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	491.145	0.0350708
$582$	0	0
$583$	−2130.42	−0.151343
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22754.1	1.59994	0.799968	−	0.600042i	$-0.204850\pi$
$587$	22754.1	1.59994	0.799968	+	0.600042i	$0.204850\pi$
$588$	0	0
$589$	−8661.13	−0.605901
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11462.9	−0.793801	−0.396901	−	0.917862i	$-0.629914\pi$
$593$	−11462.9	−0.793801	−0.396901	+	0.917862i	$0.629914\pi$
$594$	0	0
$595$	−8903.28	−0.613443
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6000.18	−0.409283	−0.204642	−	0.978837i	$-0.565603\pi$
$599$	−6000.18	−0.409283	−0.204642	+	0.978837i	$0.565603\pi$
$600$	0	0
$601$	−12874.8	−0.873835	−0.436917	−	0.899502i	$-0.643930\pi$
$601$	−12874.8	−0.873835	−0.436917	+	0.899502i	$0.643930\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−23554.7	−1.58287
$606$	0	0
$607$	−23646.1	−1.58116	−0.790580	−	0.612359i	$-0.790221\pi$
$607$	−23646.1	−1.58116	−0.790580	+	0.612359i	$0.790221\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11508.2	0.761982
$612$	0	0
$613$	1047.98	0.0690500	0.0345250	−	0.999404i	$-0.489008\pi$
$613$	1047.98	0.0690500	0.0345250	+	0.999404i	$0.489008\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1998.69	−0.130412	−0.0652059	−	0.997872i	$-0.520770\pi$
$617$	−1998.69	−0.130412	−0.0652059	+	0.997872i	$0.520770\pi$
$618$	0	0
$619$	−15246.5	−0.989997	−0.494999	−	0.868894i	$-0.664831\pi$
$619$	−15246.5	−0.989997	−0.494999	+	0.868894i	$0.664831\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7034.64	0.452386
$624$	0	0
$625$	−1048.85	−0.0671267
$626$	0	0
$627$	0	0
$628$	0	0
$629$	23601.8	1.49613
$630$	0	0
$631$	−13870.7	−0.875094	−0.437547	−	0.899195i	$-0.644153\pi$
$631$	−13870.7	−0.875094	−0.437547	+	0.899195i	$0.644153\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−41373.9	−2.58563
$636$	0	0
$637$	−8658.17	−0.538539
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19759.3	1.21755	0.608773	−	0.793345i	$-0.291662\pi$
$641$	19759.3	1.21755	0.608773	+	0.793345i	$0.291662\pi$
$642$	0	0
$643$	−11694.7	−0.717252	−0.358626	−	0.933481i	$-0.616755\pi$
$643$	−11694.7	−0.717252	−0.358626	+	0.933481i	$0.616755\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3214.68	0.195335	0.0976676	−	0.995219i	$-0.468862\pi$
$647$	3214.68	0.195335	0.0976676	+	0.995219i	$0.468862\pi$
$648$	0	0
$649$	33.0563	0.00199934
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9247.49	0.554184	0.277092	−	0.960843i	$-0.410629\pi$
$653$	9247.49	0.554184	0.277092	+	0.960843i	$0.410629\pi$
$654$	0	0
$655$	18674.1	1.11398
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7982.01	0.471829	0.235914	−	0.971774i	$-0.424192\pi$
$659$	7982.01	0.471829	0.235914	+	0.971774i	$0.424192\pi$
$660$	0	0
$661$	−17033.5	−1.00231	−0.501154	−	0.865358i	$-0.667091\pi$
$661$	−17033.5	−1.00231	−0.501154	+	0.865358i	$0.667091\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	14599.8	0.851363
$666$	0	0
$667$	−7488.50	−0.434717
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1920.21	−0.110475
$672$	0	0
$673$	18527.4	1.06119	0.530594	−	0.847626i	$-0.321969\pi$
$673$	18527.4	1.06119	0.530594	+	0.847626i	$0.321969\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32963.3	1.87132	0.935659	−	0.352905i	$-0.114806\pi$
$677$	32963.3	1.87132	0.935659	+	0.352905i	$0.114806\pi$
$678$	0	0
$679$	10.4366	0.000589865 0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−9336.54	−0.523064	−0.261532	−	0.965195i	$-0.584228\pi$
$683$	−9336.54	−0.523064	−0.261532	+	0.965195i	$0.584228\pi$
$684$	0	0
$685$	19594.8	1.09296
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13785.3	0.762232
$690$	0	0
$691$	28848.3	1.58819	0.794096	−	0.607793i	$-0.207945\pi$
$691$	28848.3	1.58819	0.794096	+	0.607793i	$0.207945\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−34518.5	−1.88397
$696$	0	0
$697$	15942.6	0.866382
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−28870.8	−1.55554	−0.777772	−	0.628547i	$-0.783650\pi$
$701$	−28870.8	−1.55554	−0.777772	+	0.628547i	$0.783650\pi$
$702$	0	0
$703$	−38702.8	−2.07639
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6526.94	0.347201
$708$	0	0
$709$	15233.6	0.806927	0.403463	−	0.914996i	$-0.367806\pi$
$709$	15233.6	0.806927	0.403463	+	0.914996i	$0.367806\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−16979.0	−0.891819
$714$	0	0
$715$	−2478.21	−0.129622
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9946.12	−0.515894	−0.257947	−	0.966159i	$-0.583046\pi$
$719$	−9946.12	−0.515894	−0.257947	+	0.966159i	$0.583046\pi$
$720$	0	0
$721$	14135.1	0.730122
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6801.14	−0.348397
$726$	0	0
$727$	−30407.4	−1.55124	−0.775619	−	0.631202i	$-0.782562\pi$
$727$	−30407.4	−1.55124	−0.775619	+	0.631202i	$0.782562\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3628.34	−0.183583
$732$	0	0
$733$	15075.8	0.759670	0.379835	−	0.925054i	$-0.375981\pi$
$733$	15075.8	0.759670	0.379835	+	0.925054i	$0.375981\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4011.43	0.200492
$738$	0	0
$739$	19253.6	0.958399	0.479200	−	0.877706i	$-0.340927\pi$
$739$	19253.6	0.958399	0.479200	+	0.877706i	$0.340927\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15733.0	0.776835	0.388417	−	0.921484i	$-0.373022\pi$
$743$	15733.0	0.776835	0.388417	+	0.921484i	$0.373022\pi$
$744$	0	0
$745$	−38339.2	−1.88542
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13750.8	−0.670818
$750$	0	0
$751$	13538.6	0.657832	0.328916	−	0.944359i	$-0.393317\pi$
$751$	13538.6	0.657832	0.328916	+	0.944359i	$0.393317\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−20967.7	−1.01072
$756$	0	0
$757$	25434.5	1.22118	0.610590	−	0.791947i	$-0.290932\pi$
$757$	25434.5	1.22118	0.610590	+	0.791947i	$0.290932\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13679.5	0.651620	0.325810	−	0.945435i	$-0.394363\pi$
$761$	13679.5	0.651620	0.325810	+	0.945435i	$0.394363\pi$
$762$	0	0
$763$	8752.26	0.415273
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−213.897	−0.0100696
$768$	0	0
$769$	−9198.81	−0.431362	−0.215681	−	0.976464i	$-0.569197\pi$
$769$	−9198.81	−0.431362	−0.215681	+	0.976464i	$0.569197\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19726.5	0.917868	0.458934	−	0.888470i	$-0.348231\pi$
$773$	19726.5	0.917868	0.458934	+	0.888470i	$0.348231\pi$
$774$	0	0
$775$	−15420.5	−0.714735
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−26143.0	−1.20240
$780$	0	0
$781$	−2703.94	−0.123886
$782$	0	0
$783$	0	0
$784$	0	0
$785$	661.410	0.0300723
$786$	0	0
$787$	15430.4	0.698901	0.349451	−	0.936955i	$-0.386368\pi$
$787$	15430.4	0.698901	0.349451	+	0.936955i	$0.386368\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11087.9	0.498409
$792$	0	0
$793$	12425.1	0.556404
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15181.2	0.674712	0.337356	−	0.941377i	$-0.390467\pi$
$797$	15181.2	0.674712	0.337356	+	0.941377i	$0.390467\pi$
$798$	0	0
$799$	−26196.5	−1.15991
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3384.06	0.148718
$804$	0	0
$805$	28620.9	1.25311
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38480.1	−1.67230	−0.836149	−	0.548503i	$-0.815198\pi$
$809$	−38480.1	−1.67230	−0.836149	+	0.548503i	$0.815198\pi$
$810$	0	0
$811$	32207.1	1.39451	0.697253	−	0.716825i	$-0.254405\pi$
$811$	32207.1	1.39451	0.697253	+	0.716825i	$0.254405\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	57711.1	2.48041
$816$	0	0
$817$	5949.84	0.254784
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5861.45	−0.249167	−0.124583	−	0.992209i	$-0.539759\pi$
$821$	−5861.45	−0.249167	−0.124583	+	0.992209i	$0.539759\pi$
$822$	0	0
$823$	34051.2	1.44222	0.721112	−	0.692819i	$-0.243631\pi$
$823$	34051.2	1.44222	0.721112	+	0.692819i	$0.243631\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17214.4	−0.723824	−0.361912	−	0.932212i	$-0.617876\pi$
$827$	−17214.4	−0.723824	−0.361912	+	0.932212i	$0.617876\pi$
$828$	0	0
$829$	−5144.24	−0.215521	−0.107760	−	0.994177i	$-0.534368\pi$
$829$	−5144.24	−0.215521	−0.107760	+	0.994177i	$0.534368\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	19709.0	0.819778
$834$	0	0
$835$	32302.1	1.33876
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−15437.1	−0.635218	−0.317609	−	0.948222i	$-0.602880\pi$
$839$	−15437.1	−0.635218	−0.317609	+	0.948222i	$0.602880\pi$
$840$	0	0
$841$	−23214.5	−0.951842
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−23476.7	−0.955765
$846$	0	0
$847$	−9538.70	−0.386958
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−75871.5	−3.05622
$852$	0	0
$853$	−19811.6	−0.795234	−0.397617	−	0.917551i	$-0.630163\pi$
$853$	−19811.6	−0.795234	−0.397617	+	0.917551i	$0.630163\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37919.6	−1.51145	−0.755723	−	0.654891i	$-0.772714\pi$
$857$	−37919.6	−1.51145	−0.755723	+	0.654891i	$0.772714\pi$
$858$	0	0
$859$	8600.38	0.341608	0.170804	−	0.985305i	$-0.445364\pi$
$859$	8600.38	0.341608	0.170804	+	0.985305i	$0.445364\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16423.6	−0.647816	−0.323908	−	0.946089i	$-0.604997\pi$
$863$	−16423.6	−0.647816	−0.323908	+	0.946089i	$0.604997\pi$
$864$	0	0
$865$	30478.9	1.19805
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5403.31	−0.210926
$870$	0	0
$871$	−25956.7	−1.00977
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9620.80	0.371705
$876$	0	0
$877$	23079.8	0.888655	0.444328	−	0.895864i	$-0.353443\pi$
$877$	23079.8	0.888655	0.444328	+	0.895864i	$0.353443\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22691.9	0.867775	0.433887	−	0.900967i	$-0.357142\pi$
$881$	22691.9	0.867775	0.433887	+	0.900967i	$0.357142\pi$
$882$	0	0
$883$	19739.1	0.752291	0.376146	−	0.926561i	$-0.377249\pi$
$883$	19739.1	0.752291	0.376146	+	0.926561i	$0.377249\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7738.38	−0.292931	−0.146465	−	0.989216i	$-0.546790\pi$
$887$	−7738.38	−0.292931	−0.146465	+	0.989216i	$0.546790\pi$
$888$	0	0
$889$	−16754.8	−0.632100
$890$	0	0
$891$	0	0
$892$	0	0
$893$	42957.7	1.60977
$894$	0	0
$895$	−5883.31	−0.219729
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2663.04	0.0987958
$900$	0	0
$901$	−31380.1	−1.16029
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−80019.7	−2.93917
$906$	0	0
$907$	−19862.2	−0.727139	−0.363569	−	0.931567i	$-0.618442\pi$
$907$	−19862.2	−0.727139	−0.363569	+	0.931567i	$0.618442\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−50188.4	−1.82527	−0.912633	−	0.408780i	$-0.865954\pi$
$911$	−50188.4	−1.82527	−0.912633	+	0.408780i	$0.865954\pi$
$912$	0	0
$913$	−311.198	−0.0112805
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7562.28	0.272332
$918$	0	0
$919$	5586.87	0.200537	0.100269	−	0.994960i	$-0.468030\pi$
$919$	5586.87	0.200537	0.100269	+	0.994960i	$0.468030\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	17496.4	0.623944
$924$	0	0
$925$	−68907.3	−2.44936
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10242.2	0.361717	0.180858	−	0.983509i	$-0.442112\pi$
$929$	10242.2	0.361717	0.180858	+	0.983509i	$0.442112\pi$
$930$	0	0
$931$	−32319.2	−1.13772
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5641.26	0.197314
$936$	0	0
$937$	−52148.1	−1.81815	−0.909074	−	0.416634i	$-0.863210\pi$
$937$	−52148.1	−1.81815	−0.909074	+	0.416634i	$0.863210\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−44437.5	−1.53945	−0.769724	−	0.638377i	$-0.779606\pi$
$941$	−44437.5	−1.53945	−0.769724	+	0.638377i	$0.779606\pi$
$942$	0	0
$943$	−51249.8	−1.76980
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10115.3	0.347100	0.173550	−	0.984825i	$-0.444476\pi$
$947$	10115.3	0.347100	0.173550	+	0.984825i	$0.444476\pi$
$948$	0	0
$949$	−21897.2	−0.749013
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14693.2	0.499433	0.249716	−	0.968319i	$-0.419663\pi$
$953$	14693.2	0.499433	0.249716	+	0.968319i	$0.419663\pi$
$954$	0	0
$955$	37389.5	1.26691
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7935.11	0.267193
$960$	0	0
$961$	−23753.0	−0.797321
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−51924.9	−1.73215
$966$	0	0
$967$	55659.8	1.85098	0.925491	−	0.378769i	$-0.123653\pi$
$967$	55659.8	1.85098	0.925491	+	0.378769i	$0.123653\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−52103.0	−1.72200	−0.861001	−	0.508604i	$-0.830162\pi$
$971$	−52103.0	−1.72200	−0.861001	+	0.508604i	$0.830162\pi$
$972$	0	0
$973$	−13978.6	−0.460569
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−46135.2	−1.51074	−0.755372	−	0.655296i	$-0.772544\pi$
$977$	−46135.2	−1.51074	−0.755372	+	0.655296i	$0.772544\pi$
$978$	0	0
$979$	−4457.26	−0.145510
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1617.78	−0.0524915	−0.0262458	−	0.999656i	$-0.508355\pi$
$983$	−1617.78	−0.0524915	−0.0262458	+	0.999656i	$0.508355\pi$
$984$	0	0
$985$	48194.3	1.55898
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11663.8	0.375013
$990$	0	0
$991$	6422.76	0.205879	0.102939	−	0.994688i	$-0.467175\pi$
$991$	6422.76	0.205879	0.102939	+	0.994688i	$0.467175\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6144.40	0.195770
$996$	0	0
$997$	39064.7	1.24091	0.620457	−	0.784240i	$-0.286947\pi$
$997$	39064.7	1.24091	0.620457	+	0.784240i	$0.286947\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.a.j.1.4	yes	4
3.2	odd	2		648.4.a.g.1.1	✓	4
4.3	odd	2		1296.4.a.bb.1.4		4
9.2	odd	6		648.4.i.v.433.4		8
9.4	even	3		648.4.i.u.217.1		8
9.5	odd	6		648.4.i.v.217.4		8
9.7	even	3		648.4.i.u.433.1		8
12.11	even	2		1296.4.a.x.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.g.1.1	✓	4	3.2	odd	2
648.4.a.j.1.4	yes	4	1.1	even	1	trivial
648.4.i.u.217.1		8	9.4	even	3
648.4.i.u.433.1		8	9.7	even	3
648.4.i.v.217.4		8	9.5	odd	6
648.4.i.v.433.4		8	9.2	odd	6
1296.4.a.x.1.1		4	12.11	even	2
1296.4.a.bb.1.4		4	4.3	odd	2

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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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q+17.9847 q^{5} +7.28309 q^{7} +O(q^{10})\)	\(q+17.9847 q^{5} +7.28309 q^{7} -4.61468 q^{11} +29.8602 q^{13} -67.9721 q^{17} +111.462 q^{19} +218.506 q^{23} +198.450 q^{25} -34.2713 q^{29} -77.7046 q^{31} +130.984 q^{35} -347.228 q^{37} -234.546 q^{41} +53.3798 q^{43} +385.402 q^{47} -289.957 q^{49} +461.661 q^{53} -82.9938 q^{55} -7.16328 q^{59} +416.109 q^{61} +537.027 q^{65} -869.274 q^{67} +585.943 q^{71} -733.324 q^{73} -33.6092 q^{77} +1170.90 q^{79} +67.4364 q^{83} -1222.46 q^{85} +965.886 q^{89} +217.475 q^{91} +2004.62 q^{95} +1.43299 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5}+O(q^{10}) \) 4 * q + 8 * q^5	\( 4 q + 8 q^{5} + 8 q^{11} + 4 q^{13} + 16 q^{17} + 80 q^{19} + 200 q^{23} + 8 q^{25} + 216 q^{29} + 80 q^{31} + 408 q^{35} - 276 q^{37} + 384 q^{41} + 160 q^{43} + 768 q^{47} - 268 q^{49} + 944 q^{53} + 304 q^{55} + 992 q^{59} - 548 q^{61} + 1328 q^{65} + 464 q^{67} + 1720 q^{71} - 764 q^{73} + 1728 q^{77} + 688 q^{79} + 2128 q^{83} - 1324 q^{85} + 2112 q^{89} + 1776 q^{91} + 2056 q^{95} - 1816 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 8 * q^11 + 4 * q^13 + 16 * q^17 + 80 * q^19 + 200 * q^23 + 8 * q^25 + 216 * q^29 + 80 * q^31 + 408 * q^35 - 276 * q^37 + 384 * q^41 + 160 * q^43 + 768 * q^47 - 268 * q^49 + 944 * q^53 + 304 * q^55 + 992 * q^59 - 548 * q^61 + 1328 * q^65 + 464 * q^67 + 1720 * q^71 - 764 * q^73 + 1728 * q^77 + 688 * q^79 + 2128 * q^83 - 1324 * q^85 + 2112 * q^89 + 1776 * q^91 + 2056 * q^95 - 1816 * q^97

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