LMFDB - Embedded newform 2352.4.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 84)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +12.8248 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +12.8248 q^{5} +9.00000 q^{9} +36.8248 q^{11} -87.1238 q^{13} -38.4743 q^{15} -102.598 q^{17} +95.8248 q^{19} +96.0000 q^{23} +39.4743 q^{25} -27.0000 q^{27} -212.021 q^{29} -159.248 q^{31} -110.474 q^{33} +128.670 q^{37} +261.371 q^{39} +298.042 q^{41} +33.3297 q^{43} +115.423 q^{45} +271.196 q^{47} +307.794 q^{51} +448.268 q^{53} +472.268 q^{55} -287.474 q^{57} -668.474 q^{59} +243.691 q^{61} -1117.34 q^{65} +335.577 q^{67} -288.000 q^{69} +339.608 q^{71} +918.320 q^{73} -118.423 q^{75} +136.299 q^{79} +81.0000 q^{81} +287.464 q^{83} -1315.79 q^{85} +636.062 q^{87} -161.855 q^{89} +477.743 q^{93} +1228.93 q^{95} -182.680 q^{97} +331.423 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 3 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 3 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} + 3 q^{5} + 18 q^{9} + 51 q^{11} - 61 q^{13} - 9 q^{15} - 24 q^{17} + 169 q^{19} + 192 q^{23} + 11 q^{25} - 54 q^{27} - 39 q^{29} - 92 q^{31} - 153 q^{33} - 173 q^{37} + 183 q^{39} - 174 q^{41} + 497 q^{43} + 27 q^{45} + 180 q^{47} + 72 q^{51} + 285 q^{53} + 333 q^{55} - 507 q^{57} - 1269 q^{59} - 328 q^{61} - 1374 q^{65} + 875 q^{67} - 576 q^{69} + 1404 q^{71} + 1361 q^{73} - 33 q^{75} + 182 q^{79} + 162 q^{81} - 399 q^{83} - 2088 q^{85} + 117 q^{87} - 822 q^{89} + 276 q^{93} + 510 q^{95} - 841 q^{97} + 459 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 3 * q^5 + 18 * q^9 + 51 * q^11 - 61 * q^13 - 9 * q^15 - 24 * q^17 + 169 * q^19 + 192 * q^23 + 11 * q^25 - 54 * q^27 - 39 * q^29 - 92 * q^31 - 153 * q^33 - 173 * q^37 + 183 * q^39 - 174 * q^41 + 497 * q^43 + 27 * q^45 + 180 * q^47 + 72 * q^51 + 285 * q^53 + 333 * q^55 - 507 * q^57 - 1269 * q^59 - 328 * q^61 - 1374 * q^65 + 875 * q^67 - 576 * q^69 + 1404 * q^71 + 1361 * q^73 - 33 * q^75 + 182 * q^79 + 162 * q^81 - 399 * q^83 - 2088 * q^85 + 117 * q^87 - 822 * q^89 + 276 * q^93 + 510 * q^95 - 841 * q^97 + 459 * q^99

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	12.8248	1.14708	0.573540	−	0.819177i	$-0.305570\pi$
$5$	12.8248	1.14708	0.573540	+	0.819177i	$0.305570\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	36.8248	1.00937	0.504685	−	0.863303i	$-0.331608\pi$
$11$	36.8248	1.00937	0.504685	+	0.863303i	$0.331608\pi$
$12$	0	0
$13$	−87.1238	−1.85875	−0.929376	−	0.369134i	$-0.879654\pi$
$13$	−87.1238	−1.85875	−0.929376	+	0.369134i	$0.879654\pi$
$14$	0	0
$15$	−38.4743	−0.662267
$16$	0	0
$17$	−102.598	−1.46375	−0.731873	−	0.681441i	$-0.761353\pi$
$17$	−102.598	−1.46375	−0.731873	+	0.681441i	$0.761353\pi$
$18$	0	0
$19$	95.8248	1.15704	0.578519	−	0.815669i	$-0.303631\pi$
$19$	95.8248	1.15704	0.578519	+	0.815669i	$0.303631\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	0	0
$25$	39.4743	0.315794
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−212.021	−1.35763	−0.678815	−	0.734309i	$-0.737506\pi$
$29$	−212.021	−1.35763	−0.678815	+	0.734309i	$0.737506\pi$
$30$	0	0
$31$	−159.248	−0.922635	−0.461318	−	0.887235i	$-0.652623\pi$
$31$	−159.248	−0.922635	−0.461318	+	0.887235i	$0.652623\pi$
$32$	0	0
$33$	−110.474	−0.582761
$34$	0	0
$35$	0	0
$36$	0	0
$37$	128.670	0.571710	0.285855	−	0.958273i	$-0.407722\pi$
$37$	128.670	0.571710	0.285855	+	0.958273i	$0.407722\pi$
$38$	0	0
$39$	261.371	1.07315
$40$	0	0
$41$	298.042	1.13527	0.567637	−	0.823279i	$-0.307858\pi$
$41$	298.042	1.13527	0.567637	+	0.823279i	$0.307858\pi$
$42$	0	0
$43$	33.3297	0.118203	0.0591016	−	0.998252i	$-0.481176\pi$
$43$	33.3297	0.118203	0.0591016	+	0.998252i	$0.481176\pi$
$44$	0	0
$45$	115.423	0.382360
$46$	0	0
$47$	271.196	0.841660	0.420830	−	0.907140i	$-0.361739\pi$
$47$	271.196	0.841660	0.420830	+	0.907140i	$0.361739\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	307.794	0.845094
$52$	0	0
$53$	448.268	1.16178	0.580890	−	0.813982i	$-0.302704\pi$
$53$	448.268	1.16178	0.580890	+	0.813982i	$0.302704\pi$
$54$	0	0
$55$	472.268	1.15783
$56$	0	0
$57$	−287.474	−0.668016
$58$	0	0
$59$	−668.474	−1.47505	−0.737525	−	0.675320i	$-0.764006\pi$
$59$	−668.474	−1.47505	−0.737525	+	0.675320i	$0.764006\pi$
$60$	0	0
$61$	243.691	0.511499	0.255750	−	0.966743i	$-0.417678\pi$
$61$	243.691	0.511499	0.255750	+	0.966743i	$0.417678\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1117.34	−2.13214
$66$	0	0
$67$	335.577	0.611900	0.305950	−	0.952048i	$-0.401026\pi$
$67$	335.577	0.611900	0.305950	+	0.952048i	$0.401026\pi$
$68$	0	0
$69$	−288.000	−0.502480
$70$	0	0
$71$	339.608	0.567663	0.283831	−	0.958874i	$-0.408394\pi$
$71$	339.608	0.567663	0.283831	+	0.958874i	$0.408394\pi$
$72$	0	0
$73$	918.320	1.47235	0.736173	−	0.676794i	$-0.236631\pi$
$73$	918.320	1.47235	0.736173	+	0.676794i	$0.236631\pi$
$74$	0	0
$75$	−118.423	−0.182324
$76$	0	0
$77$	0	0
$78$	0	0
$79$	136.299	0.194112	0.0970559	−	0.995279i	$-0.469057\pi$
$79$	136.299	0.194112	0.0970559	+	0.995279i	$0.469057\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	287.464	0.380160	0.190080	−	0.981769i	$-0.439125\pi$
$83$	287.464	0.380160	0.190080	+	0.981769i	$0.439125\pi$
$84$	0	0
$85$	−1315.79	−1.67903
$86$	0	0
$87$	636.062	0.783828
$88$	0	0
$89$	−161.855	−0.192771	−0.0963856	−	0.995344i	$-0.530728\pi$
$89$	−161.855	−0.192771	−0.0963856	+	0.995344i	$0.530728\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	477.743	0.532684
$94$	0	0
$95$	1228.93	1.32721
$96$	0	0
$97$	−182.680	−0.191220	−0.0956101	−	0.995419i	$-0.530480\pi$
$97$	−182.680	−0.191220	−0.0956101	+	0.995419i	$0.530480\pi$
$98$	0	0
$99$	331.423	0.336457
$100$	0	0
$101$	1532.10	1.50941	0.754703	−	0.656067i	$-0.227781\pi$
$101$	1532.10	1.50941	0.754703	+	0.656067i	$0.227781\pi$
$102$	0	0
$103$	487.908	0.466747	0.233374	−	0.972387i	$-0.425023\pi$
$103$	487.908	0.466747	0.233374	+	0.972387i	$0.425023\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−492.351	−0.444836	−0.222418	−	0.974951i	$-0.571395\pi$
$107$	−492.351	−0.444836	−0.222418	+	0.974951i	$0.571395\pi$
$108$	0	0
$109$	848.072	0.745235	0.372617	−	0.927985i	$-0.378460\pi$
$109$	848.072	0.745235	0.372617	+	0.927985i	$0.378460\pi$
$110$	0	0
$111$	−386.011	−0.330077
$112$	0	0
$113$	−736.350	−0.613009	−0.306505	−	0.951869i	$-0.599159\pi$
$113$	−736.350	−0.613009	−0.306505	+	0.951869i	$0.599159\pi$
$114$	0	0
$115$	1231.18	0.998328
$116$	0	0
$117$	−784.114	−0.619584
$118$	0	0
$119$	0	0
$120$	0	0
$121$	25.0623	0.0188297
$122$	0	0
$123$	−894.125	−0.655451
$124$	0	0
$125$	−1096.85	−0.784839
$126$	0	0
$127$	2511.37	1.75471	0.877355	−	0.479841i	$-0.159306\pi$
$127$	2511.37	1.75471	0.877355	+	0.479841i	$0.159306\pi$
$128$	0	0
$129$	−99.9892	−0.0682446
$130$	0	0
$131$	−679.423	−0.453141	−0.226570	−	0.973995i	$-0.572751\pi$
$131$	−679.423	−0.453141	−0.226570	+	0.973995i	$0.572751\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−346.268	−0.220756
$136$	0	0
$137$	−164.537	−0.102608	−0.0513040	−	0.998683i	$-0.516338\pi$
$137$	−164.537	−0.102608	−0.0513040	+	0.998683i	$0.516338\pi$
$138$	0	0
$139$	−521.991	−0.318523	−0.159261	−	0.987236i	$-0.550911\pi$
$139$	−521.991	−0.318523	−0.159261	+	0.987236i	$0.550911\pi$
$140$	0	0
$141$	−813.588	−0.485932
$142$	0	0
$143$	−3208.31	−1.87617
$144$	0	0
$145$	−2719.11	−1.55731
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2412.12	−1.32623	−0.663117	−	0.748516i	$-0.730767\pi$
$149$	−2412.12	−1.32623	−0.663117	+	0.748516i	$0.730767\pi$
$150$	0	0
$151$	1574.58	0.848591	0.424296	−	0.905524i	$-0.360522\pi$
$151$	1574.58	0.848591	0.424296	+	0.905524i	$0.360522\pi$
$152$	0	0
$153$	−923.382	−0.487915
$154$	0	0
$155$	−2042.31	−1.05834
$156$	0	0
$157$	−2078.74	−1.05670	−0.528349	−	0.849027i	$-0.677189\pi$
$157$	−2078.74	−1.05670	−0.528349	+	0.849027i	$0.677189\pi$
$158$	0	0
$159$	−1344.80	−0.670754
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3179.43	1.52780	0.763901	−	0.645333i	$-0.223281\pi$
$163$	3179.43	1.52780	0.763901	+	0.645333i	$0.223281\pi$
$164$	0	0
$165$	−1416.80	−0.668473
$166$	0	0
$167$	−2979.28	−1.38050	−0.690250	−	0.723571i	$-0.742500\pi$
$167$	−2979.28	−1.38050	−0.690250	+	0.723571i	$0.742500\pi$
$168$	0	0
$169$	5393.55	2.45496
$170$	0	0
$171$	862.423	0.385679
$172$	0	0
$173$	−16.3704	−0.00719431	−0.00359716	−	0.999994i	$-0.501145\pi$
$173$	−16.3704	−0.00719431	−0.00359716	+	0.999994i	$0.501145\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2005.42	0.851620
$178$	0	0
$179$	−2698.29	−1.12670	−0.563351	−	0.826218i	$-0.690488\pi$
$179$	−2698.29	−1.12670	−0.563351	+	0.826218i	$0.690488\pi$
$180$	0	0
$181$	−31.3297	−0.0128659	−0.00643293	−	0.999979i	$-0.502048\pi$
$181$	−31.3297	−0.0128659	−0.00643293	+	0.999979i	$0.502048\pi$
$182$	0	0
$183$	−731.073	−0.295314
$184$	0	0
$185$	1650.16	0.655797
$186$	0	0
$187$	−3778.15	−1.47746
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1545.17	0.585366	0.292683	−	0.956210i	$-0.405452\pi$
$191$	1545.17	0.585366	0.292683	+	0.956210i	$0.405452\pi$
$192$	0	0
$193$	−1830.20	−0.682593	−0.341297	−	0.939956i	$-0.610866\pi$
$193$	−1830.20	−0.682593	−0.341297	+	0.939956i	$0.610866\pi$
$194$	0	0
$195$	3352.02	1.23099
$196$	0	0
$197$	4728.45	1.71009	0.855047	−	0.518551i	$-0.173528\pi$
$197$	4728.45	1.71009	0.855047	+	0.518551i	$0.173528\pi$
$198$	0	0
$199$	−328.249	−0.116930	−0.0584648	−	0.998289i	$-0.518621\pi$
$199$	−328.249	−0.116930	−0.0584648	+	0.998289i	$0.518621\pi$
$200$	0	0
$201$	−1006.73	−0.353280
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3822.31	1.30225
$206$	0	0
$207$	864.000	0.290107
$208$	0	0
$209$	3528.72	1.16788
$210$	0	0
$211$	4935.76	1.61039	0.805193	−	0.593013i	$-0.202062\pi$
$211$	4935.76	1.61039	0.805193	+	0.593013i	$0.202062\pi$
$212$	0	0
$213$	−1018.82	−0.327740
$214$	0	0
$215$	427.445	0.135589
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2754.96	−0.850059
$220$	0	0
$221$	8938.72	2.72074
$222$	0	0
$223$	3446.00	1.03480	0.517402	−	0.855742i	$-0.326899\pi$
$223$	3446.00	1.03480	0.517402	+	0.855742i	$0.326899\pi$
$224$	0	0
$225$	355.268	0.105265
$226$	0	0
$227$	5724.75	1.67385	0.836927	−	0.547315i	$-0.184350\pi$
$227$	5724.75	1.67385	0.836927	+	0.547315i	$0.184350\pi$
$228$	0	0
$229$	−3017.15	−0.870649	−0.435325	−	0.900274i	$-0.643366\pi$
$229$	−3017.15	−0.870649	−0.435325	+	0.900274i	$0.643366\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	381.713	0.107325	0.0536627	−	0.998559i	$-0.482910\pi$
$233$	381.713	0.107325	0.0536627	+	0.998559i	$0.482910\pi$
$234$	0	0
$235$	3478.02	0.965452
$236$	0	0
$237$	−408.897	−0.112071
$238$	0	0
$239$	1377.38	0.372785	0.186392	−	0.982475i	$-0.440320\pi$
$239$	1377.38	0.372785	0.186392	+	0.982475i	$0.440320\pi$
$240$	0	0
$241$	−5806.72	−1.55205	−0.776025	−	0.630702i	$-0.782767\pi$
$241$	−5806.72	−1.55205	−0.776025	+	0.630702i	$0.782767\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8348.61	−2.15065
$248$	0	0
$249$	−862.393	−0.219486
$250$	0	0
$251$	−4348.52	−1.09353	−0.546765	−	0.837286i	$-0.684141\pi$
$251$	−4348.52	−1.09353	−0.546765	+	0.837286i	$0.684141\pi$
$252$	0	0
$253$	3535.18	0.878477
$254$	0	0
$255$	3947.38	0.969391
$256$	0	0
$257$	4691.11	1.13861	0.569307	−	0.822125i	$-0.307212\pi$
$257$	4691.11	1.13861	0.569307	+	0.822125i	$0.307212\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1908.19	−0.452543
$262$	0	0
$263$	7581.61	1.77758	0.888788	−	0.458319i	$-0.151548\pi$
$263$	7581.61	1.77758	0.888788	+	0.458319i	$0.151548\pi$
$264$	0	0
$265$	5748.93	1.33266
$266$	0	0
$267$	485.566	0.111297
$268$	0	0
$269$	3601.90	0.816400	0.408200	−	0.912893i	$-0.366157\pi$
$269$	3601.90	0.816400	0.408200	+	0.912893i	$0.366157\pi$
$270$	0	0
$271$	−3836.27	−0.859914	−0.429957	−	0.902849i	$-0.641471\pi$
$271$	−3836.27	−0.859914	−0.429957	+	0.902849i	$0.641471\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1453.63	0.318753
$276$	0	0
$277$	5403.04	1.17198	0.585988	−	0.810320i	$-0.300706\pi$
$277$	5403.04	1.17198	0.585988	+	0.810320i	$0.300706\pi$
$278$	0	0
$279$	−1433.23	−0.307545
$280$	0	0
$281$	150.842	0.0320230	0.0160115	−	0.999872i	$-0.494903\pi$
$281$	150.842	0.0320230	0.0160115	+	0.999872i	$0.494903\pi$
$282$	0	0
$283$	1817.14	0.381689	0.190844	−	0.981620i	$-0.438877\pi$
$283$	1817.14	0.381689	0.190844	+	0.981620i	$0.438877\pi$
$284$	0	0
$285$	−3686.79	−0.766268
$286$	0	0
$287$	0	0
$288$	0	0
$289$	5613.35	1.14255
$290$	0	0
$291$	548.041	0.110401
$292$	0	0
$293$	2817.59	0.561792	0.280896	−	0.959738i	$-0.409368\pi$
$293$	2817.59	0.561792	0.280896	+	0.959738i	$0.409368\pi$
$294$	0	0
$295$	−8573.02	−1.69200
$296$	0	0
$297$	−994.268	−0.194254
$298$	0	0
$299$	−8363.88	−1.61771
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4596.31	−0.871456
$304$	0	0
$305$	3125.28	0.586731
$306$	0	0
$307$	8589.21	1.59678	0.798391	−	0.602139i	$-0.205685\pi$
$307$	8589.21	1.59678	0.798391	+	0.602139i	$0.205685\pi$
$308$	0	0
$309$	−1463.72	−0.269477
$310$	0	0
$311$	−5998.29	−1.09367	−0.546836	−	0.837240i	$-0.684168\pi$
$311$	−5998.29	−1.09367	−0.546836	+	0.837240i	$0.684168\pi$
$312$	0	0
$313$	4963.27	0.896297	0.448148	−	0.893959i	$-0.352084\pi$
$313$	4963.27	0.896297	0.448148	+	0.893959i	$0.352084\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3904.60	−0.691812	−0.345906	−	0.938269i	$-0.612428\pi$
$317$	−3904.60	−0.691812	−0.345906	+	0.938269i	$0.612428\pi$
$318$	0	0
$319$	−7807.61	−1.37035
$320$	0	0
$321$	1477.05	0.256826
$322$	0	0
$323$	−9831.43	−1.69361
$324$	0	0
$325$	−3439.15	−0.586983
$326$	0	0
$327$	−2544.22	−0.430262
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2449.73	0.406795	0.203397	−	0.979096i	$-0.434802\pi$
$331$	2449.73	0.406795	0.203397	+	0.979096i	$0.434802\pi$
$332$	0	0
$333$	1158.03	0.190570
$334$	0	0
$335$	4303.69	0.701898
$336$	0	0
$337$	−1770.59	−0.286203	−0.143101	−	0.989708i	$-0.545707\pi$
$337$	−1770.59	−0.286203	−0.143101	+	0.989708i	$0.545707\pi$
$338$	0	0
$339$	2209.05	0.353921
$340$	0	0
$341$	−5864.25	−0.931281
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3693.53	−0.576385
$346$	0	0
$347$	4034.72	0.624194	0.312097	−	0.950050i	$-0.398969\pi$
$347$	4034.72	0.624194	0.312097	+	0.950050i	$0.398969\pi$
$348$	0	0
$349$	−6791.53	−1.04167	−0.520834	−	0.853658i	$-0.674379\pi$
$349$	−6791.53	−1.04167	−0.520834	+	0.853658i	$0.674379\pi$
$350$	0	0
$351$	2352.34	0.357717
$352$	0	0
$353$	10156.8	1.53142	0.765712	−	0.643184i	$-0.222387\pi$
$353$	10156.8	1.53142	0.765712	+	0.643184i	$0.222387\pi$
$354$	0	0
$355$	4355.39	0.651155
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12842.7	1.88806	0.944029	−	0.329861i	$-0.107002\pi$
$359$	12842.7	1.88806	0.944029	+	0.329861i	$0.107002\pi$
$360$	0	0
$361$	2323.38	0.338735
$362$	0	0
$363$	−75.1870	−0.0108713
$364$	0	0
$365$	11777.2	1.68890
$366$	0	0
$367$	−1914.82	−0.272350	−0.136175	−	0.990685i	$-0.543481\pi$
$367$	−1914.82	−0.272350	−0.136175	+	0.990685i	$0.543481\pi$
$368$	0	0
$369$	2682.37	0.378425
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5714.86	0.793309	0.396654	−	0.917968i	$-0.370171\pi$
$373$	5714.86	0.793309	0.396654	+	0.917968i	$0.370171\pi$
$374$	0	0
$375$	3290.54	0.453127
$376$	0	0
$377$	18472.0	2.52350
$378$	0	0
$379$	11570.3	1.56815	0.784075	−	0.620666i	$-0.213138\pi$
$379$	11570.3	1.56815	0.784075	+	0.620666i	$0.213138\pi$
$380$	0	0
$381$	−7534.12	−1.01308
$382$	0	0
$383$	6118.02	0.816231	0.408115	−	0.912930i	$-0.366186\pi$
$383$	6118.02	0.816231	0.408115	+	0.912930i	$0.366186\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	299.967	0.0394010
$388$	0	0
$389$	−3258.46	−0.424705	−0.212353	−	0.977193i	$-0.568113\pi$
$389$	−3258.46	−0.424705	−0.212353	+	0.977193i	$0.568113\pi$
$390$	0	0
$391$	−9849.41	−1.27393
$392$	0	0
$393$	2038.27	0.261621
$394$	0	0
$395$	1748.00	0.222662
$396$	0	0
$397$	−723.926	−0.0915184	−0.0457592	−	0.998952i	$-0.514571\pi$
$397$	−723.926	−0.0915184	−0.0457592	+	0.998952i	$0.514571\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11029.8	1.37357	0.686787	−	0.726859i	$-0.259021\pi$
$401$	11029.8	1.37357	0.686787	+	0.726859i	$0.259021\pi$
$402$	0	0
$403$	13874.2	1.71495
$404$	0	0
$405$	1038.80	0.127453
$406$	0	0
$407$	4738.25	0.577067
$408$	0	0
$409$	−2683.56	−0.324434	−0.162217	−	0.986755i	$-0.551864\pi$
$409$	−2683.56	−0.324434	−0.162217	+	0.986755i	$0.551864\pi$
$410$	0	0
$411$	493.610	0.0592408
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3686.66	0.436075
$416$	0	0
$417$	1565.97	0.183899
$418$	0	0
$419$	10024.9	1.16885	0.584427	−	0.811446i	$-0.301319\pi$
$419$	10024.9	1.16885	0.584427	+	0.811446i	$0.301319\pi$
$420$	0	0
$421$	−5560.68	−0.643731	−0.321866	−	0.946785i	$-0.604310\pi$
$421$	−5560.68	−0.643731	−0.321866	+	0.946785i	$0.604310\pi$
$422$	0	0
$423$	2440.76	0.280553
$424$	0	0
$425$	−4049.98	−0.462242
$426$	0	0
$427$	0	0
$428$	0	0
$429$	9624.93	1.08321
$430$	0	0
$431$	−11526.2	−1.28816	−0.644081	−	0.764957i	$-0.722760\pi$
$431$	−11526.2	−1.28816	−0.644081	+	0.764957i	$0.722760\pi$
$432$	0	0
$433$	−2228.79	−0.247365	−0.123683	−	0.992322i	$-0.539470\pi$
$433$	−2228.79	−0.247365	−0.123683	+	0.992322i	$0.539470\pi$
$434$	0	0
$435$	8157.34	0.899114
$436$	0	0
$437$	9199.18	1.00699
$438$	0	0
$439$	4609.26	0.501112	0.250556	−	0.968102i	$-0.419387\pi$
$439$	4609.26	0.501112	0.250556	+	0.968102i	$0.419387\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1063.32	−0.114041	−0.0570203	−	0.998373i	$-0.518160\pi$
$443$	−1063.32	−0.114041	−0.0570203	+	0.998373i	$0.518160\pi$
$444$	0	0
$445$	−2075.76	−0.221124
$446$	0	0
$447$	7236.37	0.765702
$448$	0	0
$449$	−12265.9	−1.28923	−0.644613	−	0.764509i	$-0.722982\pi$
$449$	−12265.9	−1.28923	−0.644613	+	0.764509i	$0.722982\pi$
$450$	0	0
$451$	10975.3	1.14591
$452$	0	0
$453$	−4723.73	−0.489934
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17791.2	−1.82109	−0.910543	−	0.413415i	$-0.864336\pi$
$457$	−17791.2	−1.82109	−0.910543	+	0.413415i	$0.864336\pi$
$458$	0	0
$459$	2770.15	0.281698
$460$	0	0
$461$	−15368.9	−1.55272	−0.776358	−	0.630293i	$-0.782935\pi$
$461$	−15368.9	−1.55272	−0.776358	+	0.630293i	$0.782935\pi$
$462$	0	0
$463$	4104.98	0.412040	0.206020	−	0.978548i	$-0.433949\pi$
$463$	4104.98	0.412040	0.206020	+	0.978548i	$0.433949\pi$
$464$	0	0
$465$	6126.93	0.611031
$466$	0	0
$467$	3807.36	0.377267	0.188634	−	0.982048i	$-0.439594\pi$
$467$	3807.36	0.377267	0.188634	+	0.982048i	$0.439594\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6236.23	0.610085
$472$	0	0
$473$	1227.36	0.119311
$474$	0	0
$475$	3782.61	0.365385
$476$	0	0
$477$	4034.41	0.387260
$478$	0	0
$479$	9874.65	0.941930	0.470965	−	0.882152i	$-0.343906\pi$
$479$	9874.65	0.941930	0.470965	+	0.882152i	$0.343906\pi$
$480$	0	0
$481$	−11210.2	−1.06267
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2342.83	−0.219345
$486$	0	0
$487$	6763.72	0.629350	0.314675	−	0.949199i	$-0.398104\pi$
$487$	6763.72	0.629350	0.314675	+	0.949199i	$0.398104\pi$
$488$	0	0
$489$	−9538.28	−0.882077
$490$	0	0
$491$	5574.29	0.512351	0.256175	−	0.966630i	$-0.417538\pi$
$491$	5574.29	0.512351	0.256175	+	0.966630i	$0.417538\pi$
$492$	0	0
$493$	21752.9	1.98722
$494$	0	0
$495$	4250.41	0.385943
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−5787.46	−0.519203	−0.259601	−	0.965716i	$-0.583591\pi$
$499$	−5787.46	−0.519203	−0.259601	+	0.965716i	$0.583591\pi$
$500$	0	0
$501$	8937.84	0.797032
$502$	0	0
$503$	8296.10	0.735397	0.367699	−	0.929945i	$-0.380146\pi$
$503$	8296.10	0.735397	0.367699	+	0.929945i	$0.380146\pi$
$504$	0	0
$505$	19648.8	1.73141
$506$	0	0
$507$	−16180.6	−1.41737
$508$	0	0
$509$	9760.38	0.849943	0.424972	−	0.905207i	$-0.360284\pi$
$509$	9760.38	0.849943	0.424972	+	0.905207i	$0.360284\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2587.27	−0.222672
$514$	0	0
$515$	6257.30	0.535397
$516$	0	0
$517$	9986.73	0.849547
$518$	0	0
$519$	49.1111	0.00415364
$520$	0	0
$521$	6537.98	0.549778	0.274889	−	0.961476i	$-0.411359\pi$
$521$	6537.98	0.549778	0.274889	+	0.961476i	$0.411359\pi$
$522$	0	0
$523$	−10343.8	−0.864826	−0.432413	−	0.901676i	$-0.642338\pi$
$523$	−10343.8	−0.864826	−0.432413	+	0.901676i	$0.642338\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16338.5	1.35050
$528$	0	0
$529$	−2951.00	−0.242541
$530$	0	0
$531$	−6016.27	−0.491683
$532$	0	0
$533$	−25966.5	−2.11020
$534$	0	0
$535$	−6314.28	−0.510262
$536$	0	0
$537$	8094.87	0.650502
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6843.19	−0.543829	−0.271915	−	0.962321i	$-0.587657\pi$
$541$	−6843.19	−0.543829	−0.271915	+	0.962321i	$0.587657\pi$
$542$	0	0
$543$	93.9892	0.00742810
$544$	0	0
$545$	10876.3	0.854844
$546$	0	0
$547$	18402.1	1.43842	0.719211	−	0.694791i	$-0.244503\pi$
$547$	18402.1	1.43842	0.719211	+	0.694791i	$0.244503\pi$
$548$	0	0
$549$	2193.22	0.170500
$550$	0	0
$551$	−20316.8	−1.57083
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4950.49	−0.378625
$556$	0	0
$557$	646.478	0.0491780	0.0245890	−	0.999698i	$-0.492172\pi$
$557$	646.478	0.0491780	0.0245890	+	0.999698i	$0.492172\pi$
$558$	0	0
$559$	−2903.81	−0.219710
$560$	0	0
$561$	11334.4	0.853013
$562$	0	0
$563$	6174.80	0.462232	0.231116	−	0.972926i	$-0.425762\pi$
$563$	6174.80	0.462232	0.231116	+	0.972926i	$0.425762\pi$
$564$	0	0
$565$	−9443.51	−0.703171
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6837.17	−0.503742	−0.251871	−	0.967761i	$-0.581046\pi$
$569$	−6837.17	−0.503742	−0.251871	+	0.967761i	$0.581046\pi$
$570$	0	0
$571$	5885.54	0.431353	0.215676	−	0.976465i	$-0.430804\pi$
$571$	5885.54	0.431353	0.215676	+	0.976465i	$0.430804\pi$
$572$	0	0
$573$	−4635.52	−0.337961
$574$	0	0
$575$	3789.53	0.274842
$576$	0	0
$577$	12003.2	0.866033	0.433017	−	0.901386i	$-0.357449\pi$
$577$	12003.2	0.866033	0.433017	+	0.901386i	$0.357449\pi$
$578$	0	0
$579$	5490.59	0.394095
$580$	0	0
$581$	0	0
$582$	0	0
$583$	16507.4	1.17267
$584$	0	0
$585$	−10056.1	−0.710713
$586$	0	0
$587$	−12719.4	−0.894358	−0.447179	−	0.894445i	$-0.647571\pi$
$587$	−12719.4	−0.894358	−0.447179	+	0.894445i	$0.647571\pi$
$588$	0	0
$589$	−15259.9	−1.06752
$590$	0	0
$591$	−14185.4	−0.987323
$592$	0	0
$593$	−19160.4	−1.32685	−0.663426	−	0.748242i	$-0.730898\pi$
$593$	−19160.4	−1.32685	−0.663426	+	0.748242i	$0.730898\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	984.748	0.0675093
$598$	0	0
$599$	−1067.97	−0.0728479	−0.0364240	−	0.999336i	$-0.511597\pi$
$599$	−1067.97	−0.0728479	−0.0364240	+	0.999336i	$0.511597\pi$
$600$	0	0
$601$	−7554.96	−0.512768	−0.256384	−	0.966575i	$-0.582531\pi$
$601$	−7554.96	−0.512768	−0.256384	+	0.966575i	$0.582531\pi$
$602$	0	0
$603$	3020.20	0.203967
$604$	0	0
$605$	321.418	0.0215992
$606$	0	0
$607$	−11777.4	−0.787529	−0.393765	−	0.919211i	$-0.628828\pi$
$607$	−11777.4	−0.787529	−0.393765	+	0.919211i	$0.628828\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−23627.6	−1.56444
$612$	0	0
$613$	26852.9	1.76930	0.884649	−	0.466258i	$-0.154398\pi$
$613$	26852.9	1.76930	0.884649	+	0.466258i	$0.154398\pi$
$614$	0	0
$615$	−11466.9	−0.751855
$616$	0	0
$617$	−6816.72	−0.444782	−0.222391	−	0.974958i	$-0.571386\pi$
$617$	−6816.72	−0.444782	−0.222391	+	0.974958i	$0.571386\pi$
$618$	0	0
$619$	−16713.5	−1.08525	−0.542627	−	0.839974i	$-0.682570\pi$
$619$	−16713.5	−1.08525	−0.542627	+	0.839974i	$0.682570\pi$
$620$	0	0
$621$	−2592.00	−0.167493
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19001.1	−1.21607
$626$	0	0
$627$	−10586.2	−0.674276
$628$	0	0
$629$	−13201.3	−0.836838
$630$	0	0
$631$	−592.225	−0.0373631	−0.0186815	−	0.999825i	$-0.505947\pi$
$631$	−592.225	−0.0373631	−0.0186815	+	0.999825i	$0.505947\pi$
$632$	0	0
$633$	−14807.3	−0.929757
$634$	0	0
$635$	32207.7	2.01279
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3056.47	0.189221
$640$	0	0
$641$	6963.91	0.429108	0.214554	−	0.976712i	$-0.431170\pi$
$641$	6963.91	0.429108	0.214554	+	0.976712i	$0.431170\pi$
$642$	0	0
$643$	5466.06	0.335242	0.167621	−	0.985852i	$-0.446392\pi$
$643$	5466.06	0.335242	0.167621	+	0.985852i	$0.446392\pi$
$644$	0	0
$645$	−1282.34	−0.0782821
$646$	0	0
$647$	−1237.27	−0.0751808	−0.0375904	−	0.999293i	$-0.511968\pi$
$647$	−1237.27	−0.0751808	−0.0375904	+	0.999293i	$0.511968\pi$
$648$	0	0
$649$	−24616.4	−1.48887
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27151.8	1.62716	0.813578	−	0.581455i	$-0.197516\pi$
$653$	27151.8	1.62716	0.813578	+	0.581455i	$0.197516\pi$
$654$	0	0
$655$	−8713.43	−0.519789
$656$	0	0
$657$	8264.88	0.490782
$658$	0	0
$659$	−5900.66	−0.348797	−0.174398	−	0.984675i	$-0.555798\pi$
$659$	−5900.66	−0.348797	−0.174398	+	0.984675i	$0.555798\pi$
$660$	0	0
$661$	−3618.98	−0.212953	−0.106477	−	0.994315i	$-0.533957\pi$
$661$	−3618.98	−0.212953	−0.106477	+	0.994315i	$0.533957\pi$
$662$	0	0
$663$	−26816.2	−1.57082
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20354.0	−1.18157
$668$	0	0
$669$	−10338.0	−0.597445
$670$	0	0
$671$	8973.86	0.516292
$672$	0	0
$673$	−13952.5	−0.799151	−0.399576	−	0.916700i	$-0.630842\pi$
$673$	−13952.5	−0.799151	−0.399576	+	0.916700i	$0.630842\pi$
$674$	0	0
$675$	−1065.80	−0.0607746
$676$	0	0
$677$	28918.9	1.64172	0.820859	−	0.571131i	$-0.193495\pi$
$677$	28918.9	1.64172	0.820859	+	0.571131i	$0.193495\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−17174.2	−0.966400
$682$	0	0
$683$	11130.1	0.623547	0.311773	−	0.950156i	$-0.399077\pi$
$683$	11130.1	0.623547	0.311773	+	0.950156i	$0.399077\pi$
$684$	0	0
$685$	−2110.14	−0.117700
$686$	0	0
$687$	9051.44	0.502670
$688$	0	0
$689$	−39054.8	−2.15946
$690$	0	0
$691$	−14025.3	−0.772137	−0.386069	−	0.922470i	$-0.626167\pi$
$691$	−14025.3	−0.772137	−0.386069	+	0.922470i	$0.626167\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6694.40	−0.365372
$696$	0	0
$697$	−30578.5	−1.66175
$698$	0	0
$699$	−1145.14	−0.0619644
$700$	0	0
$701$	30366.7	1.63614	0.818070	−	0.575119i	$-0.195044\pi$
$701$	30366.7	1.63614	0.818070	+	0.575119i	$0.195044\pi$
$702$	0	0
$703$	12329.8	0.661490
$704$	0	0
$705$	−10434.1	−0.557404
$706$	0	0
$707$	0	0
$708$	0	0
$709$	16891.0	0.894716	0.447358	−	0.894355i	$-0.352365\pi$
$709$	16891.0	0.894716	0.447358	+	0.894355i	$0.352365\pi$
$710$	0	0
$711$	1226.69	0.0647040
$712$	0	0
$713$	−15287.8	−0.802989
$714$	0	0
$715$	−41145.8	−2.15212
$716$	0	0
$717$	−4132.15	−0.215227
$718$	0	0
$719$	11007.2	0.570932	0.285466	−	0.958389i	$-0.407852\pi$
$719$	11007.2	0.570932	0.285466	+	0.958389i	$0.407852\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	17420.2	0.896076
$724$	0	0
$725$	−8369.36	−0.428731
$726$	0	0
$727$	−14618.0	−0.745737	−0.372869	−	0.927884i	$-0.621626\pi$
$727$	−14618.0	−0.745737	−0.372869	+	0.927884i	$0.621626\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−3419.56	−0.173019
$732$	0	0
$733$	28522.2	1.43723	0.718617	−	0.695406i	$-0.244775\pi$
$733$	28522.2	1.43723	0.718617	+	0.695406i	$0.244775\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12357.5	0.617634
$738$	0	0
$739$	−20239.5	−1.00747	−0.503735	−	0.863858i	$-0.668041\pi$
$739$	−20239.5	−1.00747	−0.503735	+	0.863858i	$0.668041\pi$
$740$	0	0
$741$	25045.8	1.24168
$742$	0	0
$743$	−13977.6	−0.690159	−0.345079	−	0.938573i	$-0.612148\pi$
$743$	−13977.6	−0.690159	−0.345079	+	0.938573i	$0.612148\pi$
$744$	0	0
$745$	−30934.9	−1.52130
$746$	0	0
$747$	2587.18	0.126720
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−15027.8	−0.730190	−0.365095	−	0.930970i	$-0.618963\pi$
$751$	−15027.8	−0.730190	−0.365095	+	0.930970i	$0.618963\pi$
$752$	0	0
$753$	13045.6	0.631350
$754$	0	0
$755$	20193.6	0.973403
$756$	0	0
$757$	20769.4	0.997196	0.498598	−	0.866833i	$-0.333848\pi$
$757$	20769.4	0.997196	0.498598	+	0.866833i	$0.333848\pi$
$758$	0	0
$759$	−10605.5	−0.507189
$760$	0	0
$761$	−11211.1	−0.534039	−0.267019	−	0.963691i	$-0.586039\pi$
$761$	−11211.1	−0.534039	−0.267019	+	0.963691i	$0.586039\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−11842.1	−0.559678
$766$	0	0
$767$	58240.0	2.74175
$768$	0	0
$769$	−4305.86	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$769$	−4305.86	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$770$	0	0
$771$	−14073.3	−0.657379
$772$	0	0
$773$	−16640.5	−0.774277	−0.387139	−	0.922022i	$-0.626536\pi$
$773$	−16640.5	−0.774277	−0.387139	+	0.922022i	$0.626536\pi$
$774$	0	0
$775$	−6286.18	−0.291363
$776$	0	0
$777$	0	0
$778$	0	0
$779$	28559.8	1.31356
$780$	0	0
$781$	12506.0	0.572982
$782$	0	0
$783$	5724.56	0.261276
$784$	0	0
$785$	−26659.4	−1.21212
$786$	0	0
$787$	−9767.05	−0.442386	−0.221193	−	0.975230i	$-0.570995\pi$
$787$	−9767.05	−0.442386	−0.221193	+	0.975230i	$0.570995\pi$
$788$	0	0
$789$	−22744.8	−1.02628
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−21231.3	−0.950750
$794$	0	0
$795$	−17246.8	−0.769410
$796$	0	0
$797$	−23118.5	−1.02748	−0.513738	−	0.857947i	$-0.671740\pi$
$797$	−23118.5	−1.02748	−0.513738	+	0.857947i	$0.671740\pi$
$798$	0	0
$799$	−27824.2	−1.23198
$800$	0	0
$801$	−1456.70	−0.0642571
$802$	0	0
$803$	33816.9	1.48614
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10805.7	−0.471349
$808$	0	0
$809$	10460.7	0.454607	0.227304	−	0.973824i	$-0.427009\pi$
$809$	10460.7	0.454607	0.227304	+	0.973824i	$0.427009\pi$
$810$	0	0
$811$	−9167.55	−0.396937	−0.198469	−	0.980107i	$-0.563597\pi$
$811$	−9167.55	−0.396937	−0.198469	+	0.980107i	$0.563597\pi$
$812$	0	0
$813$	11508.8	0.496472
$814$	0	0
$815$	40775.3	1.75251
$816$	0	0
$817$	3193.81	0.136765
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28827.3	1.22543	0.612717	−	0.790303i	$-0.290077\pi$
$821$	28827.3	1.22543	0.612717	+	0.790303i	$0.290077\pi$
$822$	0	0
$823$	−42794.3	−1.81254	−0.906268	−	0.422704i	$-0.861081\pi$
$823$	−42794.3	−1.81254	−0.906268	+	0.422704i	$0.861081\pi$
$824$	0	0
$825$	−4360.89	−0.184032
$826$	0	0
$827$	6028.87	0.253500	0.126750	−	0.991935i	$-0.459545\pi$
$827$	6028.87	0.253500	0.126750	+	0.991935i	$0.459545\pi$
$828$	0	0
$829$	19407.4	0.813085	0.406542	−	0.913632i	$-0.366734\pi$
$829$	19407.4	0.813085	0.406542	+	0.913632i	$0.366734\pi$
$830$	0	0
$831$	−16209.1	−0.676641
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−38208.5	−1.58355
$836$	0	0
$837$	4299.68	0.177561
$838$	0	0
$839$	−10599.5	−0.436156	−0.218078	−	0.975931i	$-0.569979\pi$
$839$	−10599.5	−0.436156	−0.218078	+	0.975931i	$0.569979\pi$
$840$	0	0
$841$	20563.8	0.843159
$842$	0	0
$843$	−452.526	−0.0184885
$844$	0	0
$845$	69170.9	2.81604
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−5451.43	−0.220368
$850$	0	0
$851$	12352.3	0.497571
$852$	0	0
$853$	34766.1	1.39551	0.697754	−	0.716338i	$-0.254183\pi$
$853$	34766.1	1.39551	0.697754	+	0.716338i	$0.254183\pi$
$854$	0	0
$855$	11060.4	0.442405
$856$	0	0
$857$	−30004.0	−1.19594	−0.597968	−	0.801520i	$-0.704025\pi$
$857$	−30004.0	−1.19594	−0.597968	+	0.801520i	$0.704025\pi$
$858$	0	0
$859$	−26130.6	−1.03791	−0.518955	−	0.854802i	$-0.673679\pi$
$859$	−26130.6	−1.03791	−0.518955	+	0.854802i	$0.673679\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44668.3	−1.76191	−0.880955	−	0.473201i	$-0.843099\pi$
$863$	−44668.3	−1.76191	−0.880955	+	0.473201i	$0.843099\pi$
$864$	0	0
$865$	−209.946	−0.00825245
$866$	0	0
$867$	−16840.1	−0.659652
$868$	0	0
$869$	5019.18	0.195931
$870$	0	0
$871$	−29236.7	−1.13737
$872$	0	0
$873$	−1644.12	−0.0637401
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3678.56	0.141637	0.0708187	−	0.997489i	$-0.477439\pi$
$877$	3678.56	0.141637	0.0708187	+	0.997489i	$0.477439\pi$
$878$	0	0
$879$	−8452.76	−0.324351
$880$	0	0
$881$	33443.6	1.27894	0.639468	−	0.768817i	$-0.279154\pi$
$881$	33443.6	1.27894	0.639468	+	0.768817i	$0.279154\pi$
$882$	0	0
$883$	21095.4	0.803983	0.401991	−	0.915644i	$-0.368318\pi$
$883$	21095.4	0.803983	0.401991	+	0.915644i	$0.368318\pi$
$884$	0	0
$885$	25719.0	0.976877
$886$	0	0
$887$	14453.3	0.547119	0.273560	−	0.961855i	$-0.411799\pi$
$887$	14453.3	0.547119	0.273560	+	0.961855i	$0.411799\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2982.80	0.112152
$892$	0	0
$893$	25987.3	0.973832
$894$	0	0
$895$	−34604.9	−1.29242
$896$	0	0
$897$	25091.6	0.933986
$898$	0	0
$899$	33763.8	1.25260
$900$	0	0
$901$	−45991.4	−1.70055
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−401.796	−0.0147582
$906$	0	0
$907$	5204.09	0.190517	0.0952584	−	0.995453i	$-0.469632\pi$
$907$	5204.09	0.190517	0.0952584	+	0.995453i	$0.469632\pi$
$908$	0	0
$909$	13788.9	0.503135
$910$	0	0
$911$	−31584.8	−1.14868	−0.574342	−	0.818616i	$-0.694742\pi$
$911$	−31584.8	−1.14868	−0.574342	+	0.818616i	$0.694742\pi$
$912$	0	0
$913$	10585.8	0.383723
$914$	0	0
$915$	−9375.83	−0.338749
$916$	0	0
$917$	0	0
$918$	0	0
$919$	46630.6	1.67378	0.836889	−	0.547373i	$-0.184372\pi$
$919$	46630.6	1.67378	0.836889	+	0.547373i	$0.184372\pi$
$920$	0	0
$921$	−25767.6	−0.921902
$922$	0	0
$923$	−29587.9	−1.05514
$924$	0	0
$925$	5079.16	0.180543
$926$	0	0
$927$	4391.17	0.155582
$928$	0	0
$929$	38492.6	1.35942	0.679711	−	0.733480i	$-0.262105\pi$
$929$	38492.6	1.35942	0.679711	+	0.733480i	$0.262105\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	17994.9	0.631432
$934$	0	0
$935$	−48453.8	−1.69477
$936$	0	0
$937$	−39893.8	−1.39090	−0.695450	−	0.718574i	$-0.744795\pi$
$937$	−39893.8	−1.39090	−0.695450	+	0.718574i	$0.744795\pi$
$938$	0	0
$939$	−14889.8	−0.517477
$940$	0	0
$941$	32630.7	1.13043	0.565213	−	0.824945i	$-0.308794\pi$
$941$	32630.7	1.13043	0.565213	+	0.824945i	$0.308794\pi$
$942$	0	0
$943$	28612.0	0.988054
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30029.3	−1.03043	−0.515217	−	0.857060i	$-0.672289\pi$
$947$	−30029.3	−1.03043	−0.515217	+	0.857060i	$0.672289\pi$
$948$	0	0
$949$	−80007.5	−2.73673
$950$	0	0
$951$	11713.8	0.399418
$952$	0	0
$953$	−34963.6	−1.18844	−0.594220	−	0.804303i	$-0.702539\pi$
$953$	−34963.6	−1.18844	−0.594220	+	0.804303i	$0.702539\pi$
$954$	0	0
$955$	19816.5	0.671462
$956$	0	0
$957$	23422.8	0.791173
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−4431.23	−0.148744
$962$	0	0
$963$	−4431.16	−0.148279
$964$	0	0
$965$	−23471.8	−0.782990
$966$	0	0
$967$	−20520.5	−0.682414	−0.341207	−	0.939988i	$-0.610836\pi$
$967$	−20520.5	−0.682414	−0.341207	+	0.939988i	$0.610836\pi$
$968$	0	0
$969$	29494.3	0.977805
$970$	0	0
$971$	39775.3	1.31457	0.657286	−	0.753641i	$-0.271704\pi$
$971$	39775.3	1.31457	0.657286	+	0.753641i	$0.271704\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	10317.4	0.338895
$976$	0	0
$977$	−7172.04	−0.234856	−0.117428	−	0.993081i	$-0.537465\pi$
$977$	−7172.04	−0.234856	−0.117428	+	0.993081i	$0.537465\pi$
$978$	0	0
$979$	−5960.29	−0.194578
$980$	0	0
$981$	7632.65	0.248412
$982$	0	0
$983$	48348.3	1.56874	0.784369	−	0.620294i	$-0.212987\pi$
$983$	48348.3	1.56874	0.784369	+	0.620294i	$0.212987\pi$
$984$	0	0
$985$	60641.2	1.96161
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3199.65	0.102875
$990$	0	0
$991$	−36944.6	−1.18424	−0.592121	−	0.805849i	$-0.701709\pi$
$991$	−36944.6	−1.18424	−0.592121	+	0.805849i	$0.701709\pi$
$992$	0	0
$993$	−7349.18	−0.234863
$994$	0	0
$995$	−4209.72	−0.134128
$996$	0	0
$997$	−56589.6	−1.79760	−0.898802	−	0.438355i	$-0.855561\pi$
$997$	−56589.6	−1.79760	−0.898802	+	0.438355i	$0.855561\pi$
$998$	0	0
$999$	−3474.10	−0.110026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bp.1.2		2
4.3	odd	2		588.4.a.h.1.2		2
7.3	odd	6		336.4.q.h.289.2		4
7.5	odd	6		336.4.q.h.193.2		4
7.6	odd	2		2352.4.a.cb.1.1		2
12.11	even	2		1764.4.a.p.1.1		2
28.3	even	6		84.4.i.b.37.2	yes	4
28.11	odd	6		588.4.i.i.373.1		4
28.19	even	6		84.4.i.b.25.2	✓	4
28.23	odd	6		588.4.i.i.361.1		4
28.27	even	2		588.4.a.g.1.1		2
84.11	even	6		1764.4.k.z.1549.2		4
84.23	even	6		1764.4.k.z.361.2		4
84.47	odd	6		252.4.k.d.109.1		4
84.59	odd	6		252.4.k.d.37.1		4
84.83	odd	2		1764.4.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.i.b.25.2	✓	4	28.19	even	6
84.4.i.b.37.2	yes	4	28.3	even	6
252.4.k.d.37.1		4	84.59	odd	6
252.4.k.d.109.1		4	84.47	odd	6
336.4.q.h.193.2		4	7.5	odd	6
336.4.q.h.289.2		4	7.3	odd	6
588.4.a.g.1.1		2	28.27	even	2
588.4.a.h.1.2		2	4.3	odd	2
588.4.i.i.361.1		4	28.23	odd	6
588.4.i.i.373.1		4	28.11	odd	6
1764.4.a.p.1.1		2	12.11	even	2
1764.4.a.x.1.2		2	84.83	odd	2
1764.4.k.z.361.2		4	84.23	even	6
1764.4.k.z.1549.2		4	84.11	even	6
2352.4.a.bp.1.2		2	1.1	even	1	trivial
2352.4.a.cb.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +12.8248 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +12.8248 q^{5} +9.00000 q^{9} +36.8248 q^{11} -87.1238 q^{13} -38.4743 q^{15} -102.598 q^{17} +95.8248 q^{19} +96.0000 q^{23} +39.4743 q^{25} -27.0000 q^{27} -212.021 q^{29} -159.248 q^{31} -110.474 q^{33} +128.670 q^{37} +261.371 q^{39} +298.042 q^{41} +33.3297 q^{43} +115.423 q^{45} +271.196 q^{47} +307.794 q^{51} +448.268 q^{53} +472.268 q^{55} -287.474 q^{57} -668.474 q^{59} +243.691 q^{61} -1117.34 q^{65} +335.577 q^{67} -288.000 q^{69} +339.608 q^{71} +918.320 q^{73} -118.423 q^{75} +136.299 q^{79} +81.0000 q^{81} +287.464 q^{83} -1315.79 q^{85} +636.062 q^{87} -161.855 q^{89} +477.743 q^{93} +1228.93 q^{95} -182.680 q^{97} +331.423 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 3 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 3 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} + 3 q^{5} + 18 q^{9} + 51 q^{11} - 61 q^{13} - 9 q^{15} - 24 q^{17} + 169 q^{19} + 192 q^{23} + 11 q^{25} - 54 q^{27} - 39 q^{29} - 92 q^{31} - 153 q^{33} - 173 q^{37} + 183 q^{39} - 174 q^{41} + 497 q^{43} + 27 q^{45} + 180 q^{47} + 72 q^{51} + 285 q^{53} + 333 q^{55} - 507 q^{57} - 1269 q^{59} - 328 q^{61} - 1374 q^{65} + 875 q^{67} - 576 q^{69} + 1404 q^{71} + 1361 q^{73} - 33 q^{75} + 182 q^{79} + 162 q^{81} - 399 q^{83} - 2088 q^{85} + 117 q^{87} - 822 q^{89} + 276 q^{93} + 510 q^{95} - 841 q^{97} + 459 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 3 * q^5 + 18 * q^9 + 51 * q^11 - 61 * q^13 - 9 * q^15 - 24 * q^17 + 169 * q^19 + 192 * q^23 + 11 * q^25 - 54 * q^27 - 39 * q^29 - 92 * q^31 - 153 * q^33 - 173 * q^37 + 183 * q^39 - 174 * q^41 + 497 * q^43 + 27 * q^45 + 180 * q^47 + 72 * q^51 + 285 * q^53 + 333 * q^55 - 507 * q^57 - 1269 * q^59 - 328 * q^61 - 1374 * q^65 + 875 * q^67 - 576 * q^69 + 1404 * q^71 + 1361 * q^73 - 33 * q^75 + 182 * q^79 + 162 * q^81 - 399 * q^83 - 2088 * q^85 + 117 * q^87 - 822 * q^89 + 276 * q^93 + 510 * q^95 - 841 * q^97 + 459 * q^99

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