LMFDB - Embedded newform 1764.4.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(1,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, 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("1764.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.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:	$104.079369250$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1764.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.89949 q^{5} +O(q^{10})$	$q-9.89949 q^{5} +28.0000 q^{11} -4.24264 q^{13} +49.4975 q^{17} +5.65685 q^{19} -112.000 q^{23} -27.0000 q^{25} -154.000 q^{29} +33.9411 q^{31} -20.0000 q^{37} +168.291 q^{41} -76.0000 q^{43} +435.578 q^{47} +532.000 q^{53} -277.186 q^{55} +316.784 q^{59} +168.291 q^{61} +42.0000 q^{65} -372.000 q^{67} +168.000 q^{71} +253.144 q^{73} -64.0000 q^{79} -673.166 q^{83} -490.000 q^{85} +425.678 q^{89} -56.0000 q^{95} -1059.25 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 56 q^{11} - 224 q^{23} - 54 q^{25} - 308 q^{29} - 40 q^{37} - 152 q^{43} + 1064 q^{53} + 84 q^{65} - 744 q^{67} + 336 q^{71} - 128 q^{79} - 980 q^{85} - 112 q^{95}+O(q^{100}) $ 2 * q + 56 * q^11 - 224 * q^23 - 54 * q^25 - 308 * q^29 - 40 * q^37 - 152 * q^43 + 1064 * q^53 + 84 * q^65 - 744 * q^67 + 336 * q^71 - 128 * q^79 - 980 * q^85 - 112 * q^95

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
$5$	−9.89949	−0.885438	−0.442719	−	0.896660i	$-0.645986\pi$
$5$	−9.89949	−0.885438	−0.442719	+	0.896660i	$0.645986\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	28.0000	0.767483	0.383742	−	0.923440i	$-0.374635\pi$
$11$	28.0000	0.767483	0.383742	+	0.923440i	$0.374635\pi$
$12$	0	0
$13$	−4.24264	−0.0905151	−0.0452576	−	0.998975i	$-0.514411\pi$
$13$	−4.24264	−0.0905151	−0.0452576	+	0.998975i	$0.514411\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	49.4975	0.706171	0.353085	−	0.935591i	$-0.385133\pi$
$17$	49.4975	0.706171	0.353085	+	0.935591i	$0.385133\pi$
$18$	0	0
$19$	5.65685	0.0683038	0.0341519	−	0.999417i	$-0.489127\pi$
$19$	5.65685	0.0683038	0.0341519	+	0.999417i	$0.489127\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−112.000	−1.01537	−0.507687	−	0.861541i	$-0.669499\pi$
$23$	−112.000	−1.01537	−0.507687	+	0.861541i	$0.669499\pi$
$24$	0	0
$25$	−27.0000	−0.216000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−154.000	−0.986106	−0.493053	−	0.869999i	$-0.664119\pi$
$29$	−154.000	−0.986106	−0.493053	+	0.869999i	$0.664119\pi$
$30$	0	0
$31$	33.9411	0.196645	0.0983227	−	0.995155i	$-0.468652\pi$
$31$	33.9411	0.196645	0.0983227	+	0.995155i	$0.468652\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−20.0000	−0.0888643	−0.0444322	−	0.999012i	$-0.514148\pi$
$37$	−20.0000	−0.0888643	−0.0444322	+	0.999012i	$0.514148\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	168.291	0.641042	0.320521	−	0.947241i	$-0.396142\pi$
$41$	168.291	0.641042	0.320521	+	0.947241i	$0.396142\pi$
$42$	0	0
$43$	−76.0000	−0.269532	−0.134766	−	0.990877i	$-0.543028\pi$
$43$	−76.0000	−0.269532	−0.134766	+	0.990877i	$0.543028\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	435.578	1.35182	0.675910	−	0.736984i	$-0.263751\pi$
$47$	435.578	1.35182	0.675910	+	0.736984i	$0.263751\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	532.000	1.37879	0.689395	−	0.724386i	$-0.257877\pi$
$53$	532.000	1.37879	0.689395	+	0.724386i	$0.257877\pi$
$54$	0	0
$55$	−277.186	−0.679559
$56$	0	0
$57$	0	0
$58$	0	0
$59$	316.784	0.699013	0.349506	−	0.936934i	$-0.386349\pi$
$59$	316.784	0.699013	0.349506	+	0.936934i	$0.386349\pi$
$60$	0	0
$61$	168.291	0.353238	0.176619	−	0.984279i	$-0.443484\pi$
$61$	168.291	0.353238	0.176619	+	0.984279i	$0.443484\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	42.0000	0.0801455
$66$	0	0
$67$	−372.000	−0.678314	−0.339157	−	0.940730i	$-0.610142\pi$
$67$	−372.000	−0.678314	−0.339157	+	0.940730i	$0.610142\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	168.000	0.280816	0.140408	−	0.990094i	$-0.455159\pi$
$71$	168.000	0.280816	0.140408	+	0.990094i	$0.455159\pi$
$72$	0	0
$73$	253.144	0.405867	0.202933	−	0.979193i	$-0.434952\pi$
$73$	253.144	0.405867	0.202933	+	0.979193i	$0.434952\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−64.0000	−0.0911464	−0.0455732	−	0.998961i	$-0.514511\pi$
$79$	−64.0000	−0.0911464	−0.0455732	+	0.998961i	$0.514511\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−673.166	−0.890235	−0.445118	−	0.895472i	$-0.646838\pi$
$83$	−673.166	−0.890235	−0.445118	+	0.895472i	$0.646838\pi$
$84$	0	0
$85$	−490.000	−0.625270
$86$	0	0
$87$	0	0
$88$	0	0
$89$	425.678	0.506987	0.253493	−	0.967337i	$-0.418420\pi$
$89$	425.678	0.506987	0.253493	+	0.967337i	$0.418420\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−56.0000	−0.0604787
$96$	0	0
$97$	−1059.25	−1.10876	−0.554382	−	0.832262i	$-0.687045\pi$
$97$	−1059.25	−1.10876	−0.554382	+	0.832262i	$0.687045\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−564.271	−0.555912	−0.277956	−	0.960594i	$-0.589657\pi$
$101$	−564.271	−0.555912	−0.277956	+	0.960594i	$0.589657\pi$
$102$	0	0
$103$	469.519	0.449156	0.224578	−	0.974456i	$-0.427900\pi$
$103$	469.519	0.449156	0.224578	+	0.974456i	$0.427900\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−700.000	−0.632444	−0.316222	−	0.948685i	$-0.602415\pi$
$107$	−700.000	−0.632444	−0.316222	+	0.948685i	$0.602415\pi$
$108$	0	0
$109$	−1364.00	−1.19860	−0.599300	−	0.800524i	$-0.704555\pi$
$109$	−1364.00	−1.19860	−0.599300	+	0.800524i	$0.704555\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	1108.74	0.899051
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−547.000	−0.410969
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1504.72	1.07669
$126$	0	0
$127$	−1016.00	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−1016.00	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2494.67	−1.66382	−0.831911	−	0.554910i	$-0.812753\pi$
$131$	−2494.67	−1.66382	−0.831911	+	0.554910i	$0.812753\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1974.00	−1.23102	−0.615512	−	0.788128i	$-0.711051\pi$
$137$	−1974.00	−1.23102	−0.615512	+	0.788128i	$0.711051\pi$
$138$	0	0
$139$	−2539.93	−1.54988	−0.774942	−	0.632032i	$-0.782221\pi$
$139$	−2539.93	−1.54988	−0.774942	+	0.632032i	$0.782221\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−118.794	−0.0694689
$144$	0	0
$145$	1524.52	0.873136
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2716.00	1.49331	0.746656	−	0.665211i	$-0.231658\pi$
$149$	2716.00	1.49331	0.746656	+	0.665211i	$0.231658\pi$
$150$	0	0
$151$	−2368.00	−1.27619	−0.638096	−	0.769956i	$-0.720278\pi$
$151$	−2368.00	−1.27619	−0.638096	+	0.769956i	$0.720278\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−336.000	−0.174117
$156$	0	0
$157$	1376.03	0.699485	0.349742	−	0.936846i	$-0.386269\pi$
$157$	1376.03	0.699485	0.349742	+	0.936846i	$0.386269\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−68.0000	−0.0326759	−0.0163379	−	0.999867i	$-0.505201\pi$
$163$	−68.0000	−0.0326759	−0.0163379	+	0.999867i	$0.505201\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1742.31	−0.807330	−0.403665	−	0.914907i	$-0.632264\pi$
$167$	−1742.31	−0.807330	−0.403665	+	0.914907i	$0.632264\pi$
$168$	0	0
$169$	−2179.00	−0.991807
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−861.256	−0.378498	−0.189249	−	0.981929i	$-0.560605\pi$
$173$	−861.256	−0.378498	−0.189249	+	0.981929i	$0.560605\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1092.00	−0.455977	−0.227989	−	0.973664i	$-0.573215\pi$
$179$	−1092.00	−0.455977	−0.227989	+	0.973664i	$0.573215\pi$
$180$	0	0
$181$	−3915.96	−1.60813	−0.804063	−	0.594544i	$-0.797333\pi$
$181$	−3915.96	−1.60813	−0.804063	+	0.594544i	$0.797333\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	197.990	0.0786838
$186$	0	0
$187$	1385.93	0.541974
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1288.00	−0.487939	−0.243970	−	0.969783i	$-0.578450\pi$
$191$	−1288.00	−0.487939	−0.243970	+	0.969783i	$0.578450\pi$
$192$	0	0
$193$	−4402.00	−1.64178	−0.820888	−	0.571089i	$-0.806521\pi$
$193$	−4402.00	−1.64178	−0.820888	+	0.571089i	$0.806521\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1484.00	0.536704	0.268352	−	0.963321i	$-0.413521\pi$
$197$	1484.00	0.536704	0.268352	+	0.963321i	$0.413521\pi$
$198$	0	0
$199$	3207.44	1.14256	0.571279	−	0.820756i	$-0.306447\pi$
$199$	3207.44	1.14256	0.571279	+	0.820756i	$0.306447\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1666.00	−0.567602
$206$	0	0
$207$	0	0
$208$	0	0
$209$	158.392	0.0524220
$210$	0	0
$211$	−3180.00	−1.03754	−0.518768	−	0.854915i	$-0.673609\pi$
$211$	−3180.00	−1.03754	−0.518768	+	0.854915i	$0.673609\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	752.362	0.238654
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−210.000	−0.0639191
$222$	0	0
$223$	−6058.49	−1.81931	−0.909656	−	0.415363i	$-0.863655\pi$
$223$	−6058.49	−1.81931	−0.909656	+	0.415363i	$0.863655\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4870.55	−1.42410	−0.712048	−	0.702131i	$-0.752232\pi$
$227$	−4870.55	−1.42410	−0.712048	+	0.702131i	$0.752232\pi$
$228$	0	0
$229$	2736.50	0.789665	0.394832	−	0.918753i	$-0.370803\pi$
$229$	2736.50	0.789665	0.394832	+	0.918753i	$0.370803\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−42.0000	−0.0118091	−0.00590453	−	0.999983i	$-0.501879\pi$
$233$	−42.0000	−0.0118091	−0.00590453	+	0.999983i	$0.501879\pi$
$234$	0	0
$235$	−4312.00	−1.19695
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3696.00	1.00031	0.500156	−	0.865936i	$-0.333276\pi$
$239$	3696.00	1.00031	0.500156	+	0.865936i	$0.333276\pi$
$240$	0	0
$241$	1930.40	0.515967	0.257984	−	0.966149i	$-0.416942\pi$
$241$	1930.40	0.515967	0.257984	+	0.966149i	$0.416942\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−24.0000	−0.00618252
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−910.754	−0.229029	−0.114514	−	0.993422i	$-0.536531\pi$
$251$	−910.754	−0.229029	−0.114514	+	0.993422i	$0.536531\pi$
$252$	0	0
$253$	−3136.00	−0.779283
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4603.27	1.11729	0.558646	−	0.829407i	$-0.311321\pi$
$257$	4603.27	1.11729	0.558646	+	0.829407i	$0.311321\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6384.00	1.49678	0.748392	−	0.663256i	$-0.230826\pi$
$263$	6384.00	1.49678	0.748392	+	0.663256i	$0.230826\pi$
$264$	0	0
$265$	−5266.53	−1.22083
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6187.18	1.40238	0.701188	−	0.712976i	$-0.252653\pi$
$269$	6187.18	1.40238	0.701188	+	0.712976i	$0.252653\pi$
$270$	0	0
$271$	3128.24	0.701207	0.350603	−	0.936524i	$-0.385977\pi$
$271$	3128.24	0.701207	0.350603	+	0.936524i	$0.385977\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−756.000	−0.165776
$276$	0	0
$277$	−7022.00	−1.52314	−0.761572	−	0.648080i	$-0.775572\pi$
$277$	−7022.00	−1.52314	−0.761572	+	0.648080i	$0.775572\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8666.00	−1.83975	−0.919876	−	0.392210i	$-0.871711\pi$
$281$	−8666.00	−1.83975	−0.919876	+	0.392210i	$0.871711\pi$
$282$	0	0
$283$	6460.13	1.35694	0.678471	−	0.734627i	$-0.262643\pi$
$283$	6460.13	1.35694	0.678471	+	0.734627i	$0.262643\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2463.00	−0.501323
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1732.41	0.345422	0.172711	−	0.984973i	$-0.444747\pi$
$293$	1732.41	0.345422	0.172711	+	0.984973i	$0.444747\pi$
$294$	0	0
$295$	−3136.00	−0.618932
$296$	0	0
$297$	0	0
$298$	0	0
$299$	475.176	0.0919068
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1666.00	−0.312770
$306$	0	0
$307$	−8083.64	−1.50279	−0.751397	−	0.659850i	$-0.770620\pi$
$307$	−8083.64	−1.50279	−0.751397	+	0.659850i	$0.770620\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7167.23	1.30681	0.653403	−	0.757010i	$-0.273341\pi$
$311$	7167.23	1.30681	0.653403	+	0.757010i	$0.273341\pi$
$312$	0	0
$313$	3983.84	0.719425	0.359712	−	0.933063i	$-0.382875\pi$
$313$	3983.84	0.719425	0.359712	+	0.933063i	$0.382875\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	364.000	0.0644930	0.0322465	−	0.999480i	$-0.489734\pi$
$317$	364.000	0.0644930	0.0322465	+	0.999480i	$0.489734\pi$
$318$	0	0
$319$	−4312.00	−0.756820
$320$	0	0
$321$	0	0
$322$	0	0
$323$	280.000	0.0482341
$324$	0	0
$325$	114.551	0.0195513
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	820.000	0.136167	0.0680835	−	0.997680i	$-0.478312\pi$
$331$	820.000	0.136167	0.0680835	+	0.997680i	$0.478312\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3682.61	0.600605
$336$	0	0
$337$	−5200.00	−0.840540	−0.420270	−	0.907399i	$-0.638065\pi$
$337$	−5200.00	−0.840540	−0.420270	+	0.907399i	$0.638065\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	950.352	0.150922
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4620.00	0.714739	0.357370	−	0.933963i	$-0.383674\pi$
$347$	4620.00	0.714739	0.357370	+	0.933963i	$0.383674\pi$
$348$	0	0
$349$	−4246.88	−0.651377	−0.325688	−	0.945477i	$-0.605596\pi$
$349$	−4246.88	−0.651377	−0.325688	+	0.945477i	$0.605596\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6899.95	−1.04036	−0.520180	−	0.854057i	$-0.674135\pi$
$353$	−6899.95	−1.04036	−0.520180	+	0.854057i	$0.674135\pi$
$354$	0	0
$355$	−1663.12	−0.248645
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8120.00	−1.19375	−0.596876	−	0.802333i	$-0.703592\pi$
$359$	−8120.00	−1.19375	−0.596876	+	0.802333i	$0.703592\pi$
$360$	0	0
$361$	−6827.00	−0.995335
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2506.00	−0.359370
$366$	0	0
$367$	6409.22	0.911603	0.455802	−	0.890081i	$-0.349353\pi$
$367$	6409.22	0.911603	0.455802	+	0.890081i	$0.349353\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5006.00	−0.694908	−0.347454	−	0.937697i	$-0.612954\pi$
$373$	−5006.00	−0.694908	−0.347454	+	0.937697i	$0.612954\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	653.367	0.0892575
$378$	0	0
$379$	−860.000	−0.116557	−0.0582787	−	0.998300i	$-0.518561\pi$
$379$	−860.000	−0.116557	−0.0582787	+	0.998300i	$0.518561\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1623.52	−0.216600	−0.108300	−	0.994118i	$-0.534541\pi$
$383$	−1623.52	−0.216600	−0.108300	+	0.994118i	$0.534541\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4382.00	0.571147	0.285574	−	0.958357i	$-0.407816\pi$
$389$	4382.00	0.571147	0.285574	+	0.958357i	$0.407816\pi$
$390$	0	0
$391$	−5543.72	−0.717028
$392$	0	0
$393$	0	0
$394$	0	0
$395$	633.568	0.0807044
$396$	0	0
$397$	6984.80	0.883015	0.441508	−	0.897257i	$-0.354444\pi$
$397$	6984.80	0.883015	0.441508	+	0.897257i	$0.354444\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3570.00	−0.444582	−0.222291	−	0.974980i	$-0.571353\pi$
$401$	−3570.00	−0.444582	−0.222291	+	0.974980i	$0.571353\pi$
$402$	0	0
$403$	−144.000	−0.0177994
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−560.000	−0.0682019
$408$	0	0
$409$	−2490.43	−0.301085	−0.150543	−	0.988604i	$-0.548102\pi$
$409$	−2490.43	−0.301085	−0.150543	+	0.988604i	$0.548102\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6664.00	0.788248
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3247.03	−0.378587	−0.189294	−	0.981921i	$-0.560620\pi$
$419$	−3247.03	−0.378587	−0.189294	+	0.981921i	$0.560620\pi$
$420$	0	0
$421$	−3102.00	−0.359103	−0.179551	−	0.983749i	$-0.557465\pi$
$421$	−3102.00	−0.359103	−0.179551	+	0.983749i	$0.557465\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1336.43	−0.152533
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13440.0	−1.50205	−0.751023	−	0.660276i	$-0.770439\pi$
$431$	−13440.0	−1.50205	−0.751023	+	0.660276i	$0.770439\pi$
$432$	0	0
$433$	−1251.58	−0.138908	−0.0694539	−	0.997585i	$-0.522126\pi$
$433$	−1251.58	−0.138908	−0.0694539	+	0.997585i	$0.522126\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−633.568	−0.0693539
$438$	0	0
$439$	−2138.29	−0.232472	−0.116236	−	0.993222i	$-0.537083\pi$
$439$	−2138.29	−0.232472	−0.116236	+	0.993222i	$0.537083\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14588.0	1.56455	0.782276	−	0.622932i	$-0.214059\pi$
$443$	14588.0	1.56455	0.782276	+	0.622932i	$0.214059\pi$
$444$	0	0
$445$	−4214.00	−0.448905
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17640.0	1.85408	0.927041	−	0.374959i	$-0.122343\pi$
$449$	17640.0	1.85408	0.927041	+	0.374959i	$0.122343\pi$
$450$	0	0
$451$	4712.16	0.491989
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6390.00	−0.654074	−0.327037	−	0.945012i	$-0.606050\pi$
$457$	−6390.00	−0.654074	−0.327037	+	0.945012i	$0.606050\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14720.5	−1.48721	−0.743606	−	0.668619i	$-0.766886\pi$
$461$	−14720.5	−1.48721	−0.743606	+	0.668619i	$0.766886\pi$
$462$	0	0
$463$	160.000	0.0160601	0.00803005	−	0.999968i	$-0.497444\pi$
$463$	160.000	0.0160601	0.00803005	+	0.999968i	$0.497444\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−514.774	−0.0510083	−0.0255042	−	0.999675i	$-0.508119\pi$
$467$	−514.774	−0.0510083	−0.0255042	+	0.999675i	$0.508119\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2128.00	−0.206862
$474$	0	0
$475$	−152.735	−0.0147536
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4078.59	−0.389051	−0.194526	−	0.980897i	$-0.562317\pi$
$479$	−4078.59	−0.389051	−0.194526	+	0.980897i	$0.562317\pi$
$480$	0	0
$481$	84.8528	0.00804357
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10486.0	0.981742
$486$	0	0
$487$	7176.00	0.667712	0.333856	−	0.942624i	$-0.391650\pi$
$487$	7176.00	0.667712	0.333856	+	0.942624i	$0.391650\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5460.00	−0.501846	−0.250923	−	0.968007i	$-0.580734\pi$
$491$	−5460.00	−0.501846	−0.250923	+	0.968007i	$0.580734\pi$
$492$	0	0
$493$	−7622.61	−0.696359
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3956.00	0.354900	0.177450	−	0.984130i	$-0.443215\pi$
$499$	3956.00	0.354900	0.177450	+	0.984130i	$0.443215\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10968.6	0.972301	0.486151	−	0.873875i	$-0.338401\pi$
$503$	10968.6	0.972301	0.486151	+	0.873875i	$0.338401\pi$
$504$	0	0
$505$	5586.00	0.492225
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5929.80	−0.516373	−0.258186	−	0.966095i	$-0.583125\pi$
$509$	−5929.80	−0.516373	−0.258186	+	0.966095i	$0.583125\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4648.00	−0.397700
$516$	0	0
$517$	12196.2	1.03750
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3573.72	−0.300513	−0.150257	−	0.988647i	$-0.548010\pi$
$521$	−3573.72	−0.300513	−0.150257	+	0.988647i	$0.548010\pi$
$522$	0	0
$523$	−5260.87	−0.439851	−0.219925	−	0.975517i	$-0.570581\pi$
$523$	−5260.87	−0.439851	−0.219925	+	0.975517i	$0.570581\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1680.00	0.138865
$528$	0	0
$529$	377.000	0.0309855
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−714.000	−0.0580240
$534$	0	0
$535$	6929.65	0.559990
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4922.00	−0.391152	−0.195576	−	0.980689i	$-0.562658\pi$
$541$	−4922.00	−0.391152	−0.195576	+	0.980689i	$0.562658\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13502.9	1.06129
$546$	0	0
$547$	3796.00	0.296719	0.148359	−	0.988934i	$-0.452601\pi$
$547$	3796.00	0.296719	0.148359	+	0.988934i	$0.452601\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−871.156	−0.0673548
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3948.00	−0.300327	−0.150163	−	0.988661i	$-0.547980\pi$
$557$	−3948.00	−0.300327	−0.150163	+	0.988661i	$0.547980\pi$
$558$	0	0
$559$	322.441	0.0243968
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12988.1	−0.972264	−0.486132	−	0.873885i	$-0.661593\pi$
$563$	−12988.1	−0.972264	−0.486132	+	0.873885i	$0.661593\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10262.0	−0.756073	−0.378036	−	0.925791i	$-0.623401\pi$
$569$	−10262.0	−0.756073	−0.378036	+	0.925791i	$0.623401\pi$
$570$	0	0
$571$	12172.0	0.892088	0.446044	−	0.895011i	$-0.352832\pi$
$571$	12172.0	0.892088	0.446044	+	0.895011i	$0.352832\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3024.00	0.219321
$576$	0	0
$577$	−20309.5	−1.46533	−0.732666	−	0.680589i	$-0.761724\pi$
$577$	−20309.5	−1.46533	−0.732666	+	0.680589i	$0.761724\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	14896.0	1.05820
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15562.0	−1.09423	−0.547115	−	0.837058i	$-0.684274\pi$
$587$	−15562.0	−1.09423	−0.547115	+	0.837058i	$0.684274\pi$
$588$	0	0
$589$	192.000	0.0134316
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16146.1	−1.11811	−0.559056	−	0.829130i	$-0.688836\pi$
$593$	−16146.1	−1.11811	−0.559056	+	0.829130i	$0.688836\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14728.0	−1.00462	−0.502312	−	0.864686i	$-0.667517\pi$
$599$	−14728.0	−1.00462	−0.502312	+	0.864686i	$0.667517\pi$
$600$	0	0
$601$	19783.4	1.34273	0.671367	−	0.741125i	$-0.265707\pi$
$601$	19783.4	1.34273	0.671367	+	0.741125i	$0.265707\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5415.02	0.363888
$606$	0	0
$607$	−15052.9	−1.00655	−0.503277	−	0.864125i	$-0.667872\pi$
$607$	−15052.9	−1.00655	−0.503277	+	0.864125i	$0.667872\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1848.00	−0.122360
$612$	0	0
$613$	9060.00	0.596949	0.298475	−	0.954418i	$-0.403522\pi$
$613$	9060.00	0.596949	0.298475	+	0.954418i	$0.403522\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2170.00	0.141590	0.0707949	−	0.997491i	$-0.477446\pi$
$617$	2170.00	0.141590	0.0707949	+	0.997491i	$0.477446\pi$
$618$	0	0
$619$	5351.38	0.347480	0.173740	−	0.984792i	$-0.444415\pi$
$619$	5351.38	0.347480	0.173740	+	0.984792i	$0.444415\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11521.0	−0.737344
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−989.949	−0.0627534
$630$	0	0
$631$	21296.0	1.34355	0.671775	−	0.740755i	$-0.265532\pi$
$631$	21296.0	1.34355	0.671775	+	0.740755i	$0.265532\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10057.9	0.628559
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17150.0	1.05676	0.528381	−	0.849007i	$-0.322799\pi$
$641$	17150.0	1.05676	0.528381	+	0.849007i	$0.322799\pi$
$642$	0	0
$643$	15092.5	0.925645	0.462822	−	0.886451i	$-0.346837\pi$
$643$	15092.5	0.925645	0.462822	+	0.886451i	$0.346837\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27560.2	1.67466	0.837328	−	0.546700i	$-0.184116\pi$
$647$	27560.2	1.67466	0.837328	+	0.546700i	$0.184116\pi$
$648$	0	0
$649$	8869.95	0.536481
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22890.0	−1.37175	−0.685877	−	0.727718i	$-0.740581\pi$
$653$	−22890.0	−1.37175	−0.685877	+	0.727718i	$0.740581\pi$
$654$	0	0
$655$	24696.0	1.47321
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29484.0	−1.74284	−0.871422	−	0.490535i	$-0.836801\pi$
$659$	−29484.0	−1.74284	−0.871422	+	0.490535i	$0.836801\pi$
$660$	0	0
$661$	−23695.1	−1.39430	−0.697152	−	0.716924i	$-0.745550\pi$
$661$	−23695.1	−1.39430	−0.697152	+	0.716924i	$0.745550\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	17248.0	1.00127
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4712.16	0.271104
$672$	0	0
$673$	5832.00	0.334037	0.167019	−	0.985954i	$-0.446586\pi$
$673$	5832.00	0.334037	0.167019	+	0.985954i	$0.446586\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7394.92	−0.419808	−0.209904	−	0.977722i	$-0.567315\pi$
$677$	−7394.92	−0.419808	−0.209904	+	0.977722i	$0.567315\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27132.0	1.52003	0.760013	−	0.649908i	$-0.225193\pi$
$683$	27132.0	1.52003	0.760013	+	0.649908i	$0.225193\pi$
$684$	0	0
$685$	19541.6	1.08999
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2257.08	−0.124801
$690$	0	0
$691$	−22723.6	−1.25101	−0.625504	−	0.780221i	$-0.715106\pi$
$691$	−22723.6	−1.25101	−0.625504	+	0.780221i	$0.715106\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	25144.0	1.37233
$696$	0	0
$697$	8330.00	0.452685
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7770.00	0.418643	0.209322	−	0.977847i	$-0.432874\pi$
$701$	7770.00	0.418643	0.209322	+	0.977847i	$0.432874\pi$
$702$	0	0
$703$	−113.137	−0.00606977
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10796.0	−0.571865	−0.285933	−	0.958250i	$-0.592303\pi$
$709$	−10796.0	−0.571865	−0.285933	+	0.958250i	$0.592303\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3801.41	−0.199669
$714$	0	0
$715$	1176.00	0.0615104
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24194.4	1.25493	0.627467	−	0.778643i	$-0.284092\pi$
$719$	24194.4	1.25493	0.627467	+	0.778643i	$0.284092\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4158.00	0.212999
$726$	0	0
$727$	14973.7	0.763884	0.381942	−	0.924186i	$-0.375255\pi$
$727$	14973.7	0.763884	0.381942	+	0.924186i	$0.375255\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3761.81	−0.190336
$732$	0	0
$733$	−13487.4	−0.679627	−0.339814	−	0.940493i	$-0.610364\pi$
$733$	−13487.4	−0.679627	−0.339814	+	0.940493i	$0.610364\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10416.0	−0.520595
$738$	0	0
$739$	31380.0	1.56202	0.781009	−	0.624519i	$-0.214705\pi$
$739$	31380.0	1.56202	0.781009	+	0.624519i	$0.214705\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23128.0	1.14197	0.570985	−	0.820960i	$-0.306561\pi$
$743$	23128.0	1.14197	0.570985	+	0.820960i	$0.306561\pi$
$744$	0	0
$745$	−26887.0	−1.32223
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	13448.0	0.653428	0.326714	−	0.945123i	$-0.394059\pi$
$751$	13448.0	0.653428	0.326714	+	0.945123i	$0.394059\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	23442.0	1.12999
$756$	0	0
$757$	21740.0	1.04380	0.521898	−	0.853008i	$-0.325224\pi$
$757$	21740.0	1.04380	0.521898	+	0.853008i	$0.325224\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30243.0	1.44061	0.720306	−	0.693656i	$-0.244001\pi$
$761$	30243.0	1.44061	0.720306	+	0.693656i	$0.244001\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1344.00	−0.0632712
$768$	0	0
$769$	38663.2	1.81304	0.906522	−	0.422160i	$-0.138728\pi$
$769$	38663.2	1.81304	0.906522	+	0.422160i	$0.138728\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3553.92	−0.165363	−0.0826815	−	0.996576i	$-0.526348\pi$
$773$	−3553.92	−0.165363	−0.0826815	+	0.996576i	$0.526348\pi$
$774$	0	0
$775$	−916.410	−0.0424754
$776$	0	0
$777$	0	0
$778$	0	0
$779$	952.000	0.0437855
$780$	0	0
$781$	4704.00	0.215522
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13622.0	−0.619350
$786$	0	0
$787$	13265.3	0.600836	0.300418	−	0.953808i	$-0.402874\pi$
$787$	13265.3	0.600836	0.300418	+	0.953808i	$0.402874\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−714.000	−0.0319734
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13394.0	−0.595283	−0.297641	−	0.954678i	$-0.596200\pi$
$797$	−13394.0	−0.595283	−0.297641	+	0.954678i	$0.596200\pi$
$798$	0	0
$799$	21560.0	0.954616
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7088.04	0.311496
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3416.00	0.148455	0.0742275	−	0.997241i	$-0.476351\pi$
$809$	3416.00	0.148455	0.0742275	+	0.997241i	$0.476351\pi$
$810$	0	0
$811$	12117.0	0.524642	0.262321	−	0.964981i	$-0.415512\pi$
$811$	12117.0	0.524642	0.262321	+	0.964981i	$0.415512\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	673.166	0.0289325
$816$	0	0
$817$	−429.921	−0.0184101
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3444.00	−0.146402	−0.0732012	−	0.997317i	$-0.523322\pi$
$821$	−3444.00	−0.146402	−0.0732012	+	0.997317i	$0.523322\pi$
$822$	0	0
$823$	33448.0	1.41668	0.708338	−	0.705874i	$-0.249445\pi$
$823$	33448.0	1.41668	0.708338	+	0.705874i	$0.249445\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19236.0	−0.808829	−0.404414	−	0.914576i	$-0.632525\pi$
$827$	−19236.0	−0.808829	−0.404414	+	0.914576i	$0.632525\pi$
$828$	0	0
$829$	−40128.3	−1.68120	−0.840599	−	0.541657i	$-0.817797\pi$
$829$	−40128.3	−1.68120	−0.840599	+	0.541657i	$0.817797\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	17248.0	0.714840
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10057.9	0.413870	0.206935	−	0.978355i	$-0.433651\pi$
$839$	10057.9	0.413870	0.206935	+	0.978355i	$0.433651\pi$
$840$	0	0
$841$	−673.000	−0.0275944
$842$	0	0
$843$	0	0
$844$	0	0
$845$	21571.0	0.878183
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2240.00	0.0902306
$852$	0	0
$853$	29880.9	1.19942	0.599709	−	0.800218i	$-0.295283\pi$
$853$	29880.9	1.19942	0.599709	+	0.800218i	$0.295283\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10523.2	0.419445	0.209723	−	0.977761i	$-0.432744\pi$
$857$	10523.2	0.419445	0.209723	+	0.977761i	$0.432744\pi$
$858$	0	0
$859$	−42567.8	−1.69080	−0.845399	−	0.534135i	$-0.820637\pi$
$859$	−42567.8	−1.69080	−0.845399	+	0.534135i	$0.820637\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−23632.0	−0.932147	−0.466073	−	0.884746i	$-0.654332\pi$
$863$	−23632.0	−0.932147	−0.466073	+	0.884746i	$0.654332\pi$
$864$	0	0
$865$	8526.00	0.335136
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1792.00	−0.0699533
$870$	0	0
$871$	1578.26	0.0613977
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−24820.0	−0.955658	−0.477829	−	0.878453i	$-0.658576\pi$
$877$	−24820.0	−0.955658	−0.477829	+	0.878453i	$0.658576\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−40776.0	−1.55934	−0.779670	−	0.626190i	$-0.784613\pi$
$881$	−40776.0	−1.55934	−0.779670	+	0.626190i	$0.784613\pi$
$882$	0	0
$883$	26972.0	1.02795	0.513975	−	0.857805i	$-0.328172\pi$
$883$	26972.0	1.02795	0.513975	+	0.857805i	$0.328172\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38568.4	−1.45998	−0.729989	−	0.683458i	$-0.760475\pi$
$887$	−38568.4	−1.45998	−0.729989	+	0.683458i	$0.760475\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2464.00	0.0923344
$894$	0	0
$895$	10810.2	0.403739
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5226.93	−0.193913
$900$	0	0
$901$	26332.7	0.973660
$902$	0	0
$903$	0	0
$904$	0	0
$905$	38766.0	1.42390
$906$	0	0
$907$	28708.0	1.05097	0.525487	−	0.850802i	$-0.323883\pi$
$907$	28708.0	1.05097	0.525487	+	0.850802i	$0.323883\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	46592.0	1.69447	0.847235	−	0.531219i	$-0.178266\pi$
$911$	46592.0	1.69447	0.847235	+	0.531219i	$0.178266\pi$
$912$	0	0
$913$	−18848.6	−0.683241
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	38536.0	1.38323	0.691613	−	0.722268i	$-0.256900\pi$
$919$	38536.0	1.38323	0.691613	+	0.722268i	$0.256900\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−712.764	−0.0254181
$924$	0	0
$925$	540.000	0.0191947
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−38617.9	−1.36385	−0.681923	−	0.731424i	$-0.738856\pi$
$929$	−38617.9	−1.36385	−0.681923	+	0.731424i	$0.738856\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13720.0	−0.479884
$936$	0	0
$937$	15902.8	0.554453	0.277227	−	0.960805i	$-0.410585\pi$
$937$	15902.8	0.554453	0.277227	+	0.960805i	$0.410585\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4227.08	−0.146439	−0.0732195	−	0.997316i	$-0.523327\pi$
$941$	−4227.08	−0.146439	−0.0732195	+	0.997316i	$0.523327\pi$
$942$	0	0
$943$	−18848.6	−0.650897
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13356.0	−0.458302	−0.229151	−	0.973391i	$-0.573595\pi$
$947$	−13356.0	−0.458302	−0.229151	+	0.973391i	$0.573595\pi$
$948$	0	0
$949$	−1074.00	−0.0367371
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−7112.00	−0.241742	−0.120871	−	0.992668i	$-0.538569\pi$
$953$	−7112.00	−0.241742	−0.120871	+	0.992668i	$0.538569\pi$
$954$	0	0
$955$	12750.5	0.432040
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28639.0	−0.961331
$962$	0	0
$963$	0	0
$964$	0	0
$965$	43577.6	1.45369
$966$	0	0
$967$	−624.000	−0.0207513	−0.0103756	−	0.999946i	$-0.503303\pi$
$967$	−624.000	−0.0207513	−0.0103756	+	0.999946i	$0.503303\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−34133.5	−1.12811	−0.564055	−	0.825737i	$-0.690759\pi$
$971$	−34133.5	−1.12811	−0.564055	+	0.825737i	$0.690759\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19614.0	−0.642280	−0.321140	−	0.947032i	$-0.604066\pi$
$977$	−19614.0	−0.642280	−0.321140	+	0.947032i	$0.604066\pi$
$978$	0	0
$979$	11919.0	0.389104
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1029.55	−0.0334054	−0.0167027	−	0.999861i	$-0.505317\pi$
$983$	−1029.55	−0.0334054	−0.0167027	+	0.999861i	$0.505317\pi$
$984$	0	0
$985$	−14690.9	−0.475218
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8512.00	0.273676
$990$	0	0
$991$	25208.0	0.808031	0.404015	−	0.914752i	$-0.367614\pi$
$991$	25208.0	0.808031	0.404015	+	0.914752i	$0.367614\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−31752.0	−1.01166
$996$	0	0
$997$	−31761.8	−1.00893	−0.504467	−	0.863431i	$-0.668311\pi$
$997$	−31761.8	−1.00893	−0.504467	+	0.863431i	$0.668311\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.a.w.1.1	yes	2
3.2	odd	2		1764.4.a.q.1.2	yes	2
7.2	even	3		1764.4.k.r.361.2		4
7.3	odd	6		1764.4.k.r.1549.1		4
7.4	even	3		1764.4.k.r.1549.2		4
7.5	odd	6		1764.4.k.r.361.1		4
7.6	odd	2	inner	1764.4.a.w.1.2	yes	2
21.2	odd	6		1764.4.k.y.361.1		4
21.5	even	6		1764.4.k.y.361.2		4
21.11	odd	6		1764.4.k.y.1549.1		4
21.17	even	6		1764.4.k.y.1549.2		4
21.20	even	2		1764.4.a.q.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.4.a.q.1.1	✓	2	21.20	even	2
1764.4.a.q.1.2	yes	2	3.2	odd	2
1764.4.a.w.1.1	yes	2	1.1	even	1	trivial
1764.4.a.w.1.2	yes	2	7.6	odd	2	inner
1764.4.k.r.361.1		4	7.5	odd	6
1764.4.k.r.361.2		4	7.2	even	3
1764.4.k.r.1549.1		4	7.3	odd	6
1764.4.k.r.1549.2		4	7.4	even	3
1764.4.k.y.361.1		4	21.2	odd	6
1764.4.k.y.361.2		4	21.5	even	6
1764.4.k.y.1549.1		4	21.11	odd	6
1764.4.k.y.1549.2		4	21.17	even	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.a (trivial)

\(f(q)\)	\(=\)	\(q-9.89949 q^{5} +O(q^{10})\)	\(q-9.89949 q^{5} +28.0000 q^{11} -4.24264 q^{13} +49.4975 q^{17} +5.65685 q^{19} -112.000 q^{23} -27.0000 q^{25} -154.000 q^{29} +33.9411 q^{31} -20.0000 q^{37} +168.291 q^{41} -76.0000 q^{43} +435.578 q^{47} +532.000 q^{53} -277.186 q^{55} +316.784 q^{59} +168.291 q^{61} +42.0000 q^{65} -372.000 q^{67} +168.000 q^{71} +253.144 q^{73} -64.0000 q^{79} -673.166 q^{83} -490.000 q^{85} +425.678 q^{89} -56.0000 q^{95} -1059.25 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 56 q^{11} - 224 q^{23} - 54 q^{25} - 308 q^{29} - 40 q^{37} - 152 q^{43} + 1064 q^{53} + 84 q^{65} - 744 q^{67} + 336 q^{71} - 128 q^{79} - 980 q^{85} - 112 q^{95}+O(q^{100}) \) 2 * q + 56 * q^11 - 224 * q^23 - 54 * q^25 - 308 * q^29 - 40 * q^37 - 152 * q^43 + 1064 * q^53 + 84 * q^65 - 744 * q^67 + 336 * q^71 - 128 * q^79 - 980 * q^85 - 112 * q^95

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