LMFDB - Embedded newform 2352.4.a.bl.1.1

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:	$1$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -19.8995 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -19.8995 q^{5} +9.00000 q^{9} -23.9411 q^{11} -87.3553 q^{13} +59.6985 q^{15} +5.63961 q^{17} +64.8873 q^{19} +25.5980 q^{23} +270.990 q^{25} -27.0000 q^{27} +60.3188 q^{29} -122.711 q^{31} +71.8234 q^{33} -56.1177 q^{37} +262.066 q^{39} +299.713 q^{41} +501.421 q^{43} -179.095 q^{45} +305.553 q^{47} -16.9188 q^{51} -375.117 q^{53} +476.416 q^{55} -194.662 q^{57} -627.612 q^{59} -3.75736 q^{61} +1738.33 q^{65} +813.048 q^{67} -76.7939 q^{69} -165.902 q^{71} -619.100 q^{73} -812.970 q^{75} +138.246 q^{79} +81.0000 q^{81} +621.137 q^{83} -112.225 q^{85} -180.956 q^{87} -285.418 q^{89} +368.132 q^{93} -1291.22 q^{95} +603.114 q^{97} -215.470 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 20 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 20 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} - 20 q^{5} + 18 q^{9} + 20 q^{11} - 104 q^{13} + 60 q^{15} - 116 q^{17} + 192 q^{19} - 28 q^{23} + 146 q^{25} - 54 q^{27} + 296 q^{29} - 104 q^{31} - 60 q^{33} - 248 q^{37} + 312 q^{39} - 20 q^{41} + 720 q^{43} - 180 q^{45} - 96 q^{47} + 348 q^{51} + 268 q^{53} + 472 q^{55} - 576 q^{57} - 616 q^{59} - 16 q^{61} + 1740 q^{65} + 144 q^{67} + 84 q^{69} - 988 q^{71} - 104 q^{73} - 438 q^{75} + 944 q^{79} + 162 q^{81} + 1016 q^{83} - 100 q^{85} - 888 q^{87} + 388 q^{89} + 312 q^{93} - 1304 q^{95} - 488 q^{97} + 180 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 20 * q^5 + 18 * q^9 + 20 * q^11 - 104 * q^13 + 60 * q^15 - 116 * q^17 + 192 * q^19 - 28 * q^23 + 146 * q^25 - 54 * q^27 + 296 * q^29 - 104 * q^31 - 60 * q^33 - 248 * q^37 + 312 * q^39 - 20 * q^41 + 720 * q^43 - 180 * q^45 - 96 * q^47 + 348 * q^51 + 268 * q^53 + 472 * q^55 - 576 * q^57 - 616 * q^59 - 16 * q^61 + 1740 * q^65 + 144 * q^67 + 84 * q^69 - 988 * q^71 - 104 * q^73 - 438 * q^75 + 944 * q^79 + 162 * q^81 + 1016 * q^83 - 100 * q^85 - 888 * q^87 + 388 * q^89 + 312 * q^93 - 1304 * q^95 - 488 * q^97 + 180 * 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
$5$	−19.8995	−1.77986	−0.889932	−	0.456092i	$-0.849249\pi$
$5$	−19.8995	−1.77986	−0.889932	+	0.456092i	$0.849249\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−23.9411	−0.656229	−0.328115	−	0.944638i	$-0.606413\pi$
$11$	−23.9411	−0.656229	−0.328115	+	0.944638i	$0.606413\pi$
$12$	0	0
$13$	−87.3553	−1.86369	−0.931847	−	0.362852i	$-0.881803\pi$
$13$	−87.3553	−1.86369	−0.931847	+	0.362852i	$0.881803\pi$
$14$	0	0
$15$	59.6985	1.02761
$16$	0	0
$17$	5.63961	0.0804592	0.0402296	−	0.999190i	$-0.487191\pi$
$17$	5.63961	0.0804592	0.0402296	+	0.999190i	$0.487191\pi$
$18$	0	0
$19$	64.8873	0.783483	0.391741	−	0.920075i	$-0.371873\pi$
$19$	64.8873	0.783483	0.391741	+	0.920075i	$0.371873\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	25.5980	0.232067	0.116034	−	0.993245i	$-0.462982\pi$
$23$	25.5980	0.232067	0.116034	+	0.993245i	$0.462982\pi$
$24$	0	0
$25$	270.990	2.16792
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	60.3188	0.386238	0.193119	−	0.981175i	$-0.438140\pi$
$29$	60.3188	0.386238	0.193119	+	0.981175i	$0.438140\pi$
$30$	0	0
$31$	−122.711	−0.710951	−0.355476	−	0.934686i	$-0.615681\pi$
$31$	−122.711	−0.710951	−0.355476	+	0.934686i	$0.615681\pi$
$32$	0	0
$33$	71.8234	0.378874
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−56.1177	−0.249343	−0.124672	−	0.992198i	$-0.539788\pi$
$37$	−56.1177	−0.249343	−0.124672	+	0.992198i	$0.539788\pi$
$38$	0	0
$39$	262.066	1.07600
$40$	0	0
$41$	299.713	1.14164	0.570820	−	0.821075i	$-0.306625\pi$
$41$	299.713	1.14164	0.570820	+	0.821075i	$0.306625\pi$
$42$	0	0
$43$	501.421	1.77828	0.889140	−	0.457635i	$-0.151303\pi$
$43$	501.421	1.77828	0.889140	+	0.457635i	$0.151303\pi$
$44$	0	0
$45$	−179.095	−0.593288
$46$	0	0
$47$	305.553	0.948288	0.474144	−	0.880447i	$-0.342758\pi$
$47$	305.553	0.948288	0.474144	+	0.880447i	$0.342758\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−16.9188	−0.0464531
$52$	0	0
$53$	−375.117	−0.972194	−0.486097	−	0.873905i	$-0.661580\pi$
$53$	−375.117	−0.972194	−0.486097	+	0.873905i	$0.661580\pi$
$54$	0	0
$55$	476.416	1.16800
$56$	0	0
$57$	−194.662	−0.452344
$58$	0	0
$59$	−627.612	−1.38488	−0.692442	−	0.721474i	$-0.743465\pi$
$59$	−627.612	−1.38488	−0.692442	+	0.721474i	$0.743465\pi$
$60$	0	0
$61$	−3.75736	−0.00788657	−0.00394328	−	0.999992i	$-0.501255\pi$
$61$	−3.75736	−0.00788657	−0.00394328	+	0.999992i	$0.501255\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1738.33	3.31712
$66$	0	0
$67$	813.048	1.48253	0.741266	−	0.671212i	$-0.234226\pi$
$67$	813.048	1.48253	0.741266	+	0.671212i	$0.234226\pi$
$68$	0	0
$69$	−76.7939	−0.133984
$70$	0	0
$71$	−165.902	−0.277310	−0.138655	−	0.990341i	$-0.544278\pi$
$71$	−165.902	−0.277310	−0.138655	+	0.990341i	$0.544278\pi$
$72$	0	0
$73$	−619.100	−0.992605	−0.496302	−	0.868150i	$-0.665309\pi$
$73$	−619.100	−0.992605	−0.496302	+	0.868150i	$0.665309\pi$
$74$	0	0
$75$	−812.970	−1.25165
$76$	0	0
$77$	0	0
$78$	0	0
$79$	138.246	0.196884	0.0984421	−	0.995143i	$-0.468614\pi$
$79$	138.246	0.196884	0.0984421	+	0.995143i	$0.468614\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	621.137	0.821430	0.410715	−	0.911764i	$-0.365279\pi$
$83$	621.137	0.821430	0.410715	+	0.911764i	$0.365279\pi$
$84$	0	0
$85$	−112.225	−0.143207
$86$	0	0
$87$	−180.956	−0.222995
$88$	0	0
$89$	−285.418	−0.339936	−0.169968	−	0.985450i	$-0.554366\pi$
$89$	−285.418	−0.339936	−0.169968	+	0.985450i	$0.554366\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	368.132	0.410468
$94$	0	0
$95$	−1291.22	−1.39449
$96$	0	0
$97$	603.114	0.631309	0.315654	−	0.948874i	$-0.397776\pi$
$97$	603.114	0.631309	0.315654	+	0.948874i	$0.397776\pi$
$98$	0	0
$99$	−215.470	−0.218743
$100$	0	0
$101$	−457.209	−0.450436	−0.225218	−	0.974308i	$-0.572309\pi$
$101$	−457.209	−0.450436	−0.225218	+	0.974308i	$0.572309\pi$
$102$	0	0
$103$	786.045	0.751954	0.375977	−	0.926629i	$-0.377307\pi$
$103$	786.045	0.751954	0.375977	+	0.926629i	$0.377307\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	196.461	0.177501	0.0887504	−	0.996054i	$-0.471713\pi$
$107$	196.461	0.177501	0.0887504	+	0.996054i	$0.471713\pi$
$108$	0	0
$109$	−306.343	−0.269196	−0.134598	−	0.990900i	$-0.542974\pi$
$109$	−306.343	−0.269196	−0.134598	+	0.990900i	$0.542974\pi$
$110$	0	0
$111$	168.353	0.143958
$112$	0	0
$113$	1997.63	1.66302	0.831508	−	0.555512i	$-0.187478\pi$
$113$	1997.63	1.66302	0.831508	+	0.555512i	$0.187478\pi$
$114$	0	0
$115$	−509.387	−0.413048
$116$	0	0
$117$	−786.198	−0.621231
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−757.823	−0.569363
$122$	0	0
$123$	−899.138	−0.659127
$124$	0	0
$125$	−2905.13	−2.07874
$126$	0	0
$127$	2311.40	1.61499	0.807494	−	0.589875i	$-0.200823\pi$
$127$	2311.40	1.61499	0.807494	+	0.589875i	$0.200823\pi$
$128$	0	0
$129$	−1504.26	−1.02669
$130$	0	0
$131$	−155.018	−0.103389	−0.0516945	−	0.998663i	$-0.516462\pi$
$131$	−155.018	−0.103389	−0.0516945	+	0.998663i	$0.516462\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	537.286	0.342535
$136$	0	0
$137$	516.936	0.322371	0.161186	−	0.986924i	$-0.448468\pi$
$137$	516.936	0.322371	0.161186	+	0.986924i	$0.448468\pi$
$138$	0	0
$139$	958.067	0.584620	0.292310	−	0.956324i	$-0.405576\pi$
$139$	958.067	0.584620	0.292310	+	0.956324i	$0.405576\pi$
$140$	0	0
$141$	−916.660	−0.547494
$142$	0	0
$143$	2091.39	1.22301
$144$	0	0
$145$	−1200.31	−0.687452
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1770.63	−0.973526	−0.486763	−	0.873534i	$-0.661822\pi$
$149$	−1770.63	−0.973526	−0.486763	+	0.873534i	$0.661822\pi$
$150$	0	0
$151$	2540.24	1.36902	0.684508	−	0.729005i	$-0.260017\pi$
$151$	2540.24	1.36902	0.684508	+	0.729005i	$0.260017\pi$
$152$	0	0
$153$	50.7565	0.0268197
$154$	0	0
$155$	2441.88	1.26540
$156$	0	0
$157$	1083.34	0.550702	0.275351	−	0.961344i	$-0.411206\pi$
$157$	1083.34	0.550702	0.275351	+	0.961344i	$0.411206\pi$
$158$	0	0
$159$	1125.35	0.561296
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2968.72	−1.42655	−0.713277	−	0.700882i	$-0.752790\pi$
$163$	−2968.72	−1.42655	−0.713277	+	0.700882i	$0.752790\pi$
$164$	0	0
$165$	−1429.25	−0.674345
$166$	0	0
$167$	−2091.53	−0.969149	−0.484574	−	0.874750i	$-0.661025\pi$
$167$	−2091.53	−0.969149	−0.484574	+	0.874750i	$0.661025\pi$
$168$	0	0
$169$	5433.96	2.47335
$170$	0	0
$171$	583.986	0.261161
$172$	0	0
$173$	470.148	0.206617	0.103308	−	0.994649i	$-0.467057\pi$
$173$	470.148	0.206617	0.103308	+	0.994649i	$0.467057\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1882.84	0.799563
$178$	0	0
$179$	−1056.46	−0.441137	−0.220569	−	0.975371i	$-0.570791\pi$
$179$	−1056.46	−0.441137	−0.220569	+	0.975371i	$0.570791\pi$
$180$	0	0
$181$	406.470	0.166921	0.0834605	−	0.996511i	$-0.473403\pi$
$181$	406.470	0.166921	0.0834605	+	0.996511i	$0.473403\pi$
$182$	0	0
$183$	11.2721	0.00455331
$184$	0	0
$185$	1116.71	0.443797
$186$	0	0
$187$	−135.019	−0.0527997
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−179.410	−0.0679666	−0.0339833	−	0.999422i	$-0.510819\pi$
$191$	−179.410	−0.0679666	−0.0339833	+	0.999422i	$0.510819\pi$
$192$	0	0
$193$	−2388.37	−0.890769	−0.445385	−	0.895339i	$-0.646933\pi$
$193$	−2388.37	−0.890769	−0.445385	+	0.895339i	$0.646933\pi$
$194$	0	0
$195$	−5214.98	−1.91514
$196$	0	0
$197$	−2665.35	−0.963952	−0.481976	−	0.876184i	$-0.660081\pi$
$197$	−2665.35	−0.963952	−0.481976	+	0.876184i	$0.660081\pi$
$198$	0	0
$199$	−1342.31	−0.478159	−0.239079	−	0.971000i	$-0.576846\pi$
$199$	−1342.31	−0.478159	−0.239079	+	0.971000i	$0.576846\pi$
$200$	0	0
$201$	−2439.14	−0.855940
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−5964.13	−2.03197
$206$	0	0
$207$	230.382	0.0773558
$208$	0	0
$209$	−1553.48	−0.514144
$210$	0	0
$211$	−628.442	−0.205042	−0.102521	−	0.994731i	$-0.532691\pi$
$211$	−628.442	−0.205042	−0.102521	+	0.994731i	$0.532691\pi$
$212$	0	0
$213$	497.707	0.160105
$214$	0	0
$215$	−9978.03	−3.16510
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1857.30	0.573081
$220$	0	0
$221$	−492.650	−0.149951
$222$	0	0
$223$	−969.970	−0.291273	−0.145637	−	0.989338i	$-0.546523\pi$
$223$	−969.970	−0.291273	−0.145637	+	0.989338i	$0.546523\pi$
$224$	0	0
$225$	2438.91	0.722640
$226$	0	0
$227$	−4748.64	−1.38845	−0.694225	−	0.719758i	$-0.744253\pi$
$227$	−4748.64	−1.38845	−0.694225	+	0.719758i	$0.744253\pi$
$228$	0	0
$229$	4801.99	1.38570	0.692848	−	0.721083i	$-0.256356\pi$
$229$	4801.99	1.38570	0.692848	+	0.721083i	$0.256356\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3155.29	−0.887166	−0.443583	−	0.896233i	$-0.646293\pi$
$233$	−3155.29	−0.887166	−0.443583	+	0.896233i	$0.646293\pi$
$234$	0	0
$235$	−6080.36	−1.68782
$236$	0	0
$237$	−414.737	−0.113671
$238$	0	0
$239$	−4241.93	−1.14806	−0.574032	−	0.818833i	$-0.694622\pi$
$239$	−4241.93	−1.14806	−0.574032	+	0.818833i	$0.694622\pi$
$240$	0	0
$241$	4342.99	1.16081	0.580407	−	0.814326i	$-0.302893\pi$
$241$	4342.99	1.16081	0.580407	+	0.814326i	$0.302893\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5668.25	−1.46017
$248$	0	0
$249$	−1863.41	−0.474253
$250$	0	0
$251$	3003.01	0.755172	0.377586	−	0.925974i	$-0.376754\pi$
$251$	3003.01	0.755172	0.377586	+	0.925974i	$0.376754\pi$
$252$	0	0
$253$	−612.844	−0.152289
$254$	0	0
$255$	336.676	0.0826803
$256$	0	0
$257$	−4468.84	−1.08466	−0.542332	−	0.840164i	$-0.682459\pi$
$257$	−4468.84	−1.08466	−0.542332	+	0.840164i	$0.682459\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	542.869	0.128746
$262$	0	0
$263$	−6160.13	−1.44430	−0.722148	−	0.691738i	$-0.756845\pi$
$263$	−6160.13	−1.44430	−0.722148	+	0.691738i	$0.756845\pi$
$264$	0	0
$265$	7464.64	1.73037
$266$	0	0
$267$	856.255	0.196262
$268$	0	0
$269$	−4988.90	−1.13078	−0.565388	−	0.824825i	$-0.691274\pi$
$269$	−4988.90	−1.13078	−0.565388	+	0.824825i	$0.691274\pi$
$270$	0	0
$271$	4433.73	0.993837	0.496918	−	0.867797i	$-0.334465\pi$
$271$	4433.73	0.993837	0.496918	+	0.867797i	$0.334465\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6487.80	−1.42265
$276$	0	0
$277$	1112.37	0.241284	0.120642	−	0.992696i	$-0.461505\pi$
$277$	1112.37	0.241284	0.120642	+	0.992696i	$0.461505\pi$
$278$	0	0
$279$	−1104.40	−0.236984
$280$	0	0
$281$	2813.22	0.597233	0.298616	−	0.954373i	$-0.403475\pi$
$281$	2813.22	0.597233	0.298616	+	0.954373i	$0.403475\pi$
$282$	0	0
$283$	3147.54	0.661137	0.330569	−	0.943782i	$-0.392759\pi$
$283$	3147.54	0.661137	0.330569	+	0.943782i	$0.392759\pi$
$284$	0	0
$285$	3873.67	0.805111
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4881.19	−0.993526
$290$	0	0
$291$	−1809.34	−0.364486
$292$	0	0
$293$	−9143.04	−1.82301	−0.911505	−	0.411289i	$-0.865079\pi$
$293$	−9143.04	−1.82301	−0.911505	+	0.411289i	$0.865079\pi$
$294$	0	0
$295$	12489.2	2.46491
$296$	0	0
$297$	646.410	0.126291
$298$	0	0
$299$	−2236.12	−0.432502
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1371.63	0.260059
$304$	0	0
$305$	74.7696	0.0140370
$306$	0	0
$307$	−4648.90	−0.864257	−0.432129	−	0.901812i	$-0.642237\pi$
$307$	−4648.90	−0.864257	−0.432129	+	0.901812i	$0.642237\pi$
$308$	0	0
$309$	−2358.13	−0.434141
$310$	0	0
$311$	−6417.18	−1.17005	−0.585024	−	0.811016i	$-0.698915\pi$
$311$	−6417.18	−1.17005	−0.585024	+	0.811016i	$0.698915\pi$
$312$	0	0
$313$	−5868.22	−1.05972	−0.529858	−	0.848086i	$-0.677755\pi$
$313$	−5868.22	−1.05972	−0.529858	+	0.848086i	$0.677755\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3974.52	−0.704200	−0.352100	−	0.935962i	$-0.614532\pi$
$317$	−3974.52	−0.704200	−0.352100	+	0.935962i	$0.614532\pi$
$318$	0	0
$319$	−1444.10	−0.253461
$320$	0	0
$321$	−589.383	−0.102480
$322$	0	0
$323$	365.939	0.0630384
$324$	0	0
$325$	−23672.4	−4.04034
$326$	0	0
$327$	919.029	0.155420
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8912.82	1.48004	0.740020	−	0.672585i	$-0.234816\pi$
$331$	8912.82	1.48004	0.740020	+	0.672585i	$0.234816\pi$
$332$	0	0
$333$	−505.060	−0.0831144
$334$	0	0
$335$	−16179.2	−2.63871
$336$	0	0
$337$	3977.06	0.642862	0.321431	−	0.946933i	$-0.395836\pi$
$337$	3977.06	0.642862	0.321431	+	0.946933i	$0.395836\pi$
$338$	0	0
$339$	−5992.88	−0.960143
$340$	0	0
$341$	2937.83	0.466547
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1528.16	0.238474
$346$	0	0
$347$	−6826.43	−1.05609	−0.528043	−	0.849218i	$-0.677074\pi$
$347$	−6826.43	−1.05609	−0.528043	+	0.849218i	$0.677074\pi$
$348$	0	0
$349$	807.342	0.123828	0.0619141	−	0.998081i	$-0.480280\pi$
$349$	807.342	0.123828	0.0619141	+	0.998081i	$0.480280\pi$
$350$	0	0
$351$	2358.59	0.358668
$352$	0	0
$353$	7919.20	1.19404	0.597020	−	0.802226i	$-0.296351\pi$
$353$	7919.20	1.19404	0.597020	+	0.802226i	$0.296351\pi$
$354$	0	0
$355$	3301.38	0.493574
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8819.21	1.29655	0.648273	−	0.761408i	$-0.275491\pi$
$359$	8819.21	1.29655	0.648273	+	0.761408i	$0.275491\pi$
$360$	0	0
$361$	−2648.64	−0.386155
$362$	0	0
$363$	2273.47	0.328722
$364$	0	0
$365$	12319.8	1.76670
$366$	0	0
$367$	−11161.8	−1.58758	−0.793788	−	0.608194i	$-0.791894\pi$
$367$	−11161.8	−1.58758	−0.793788	+	0.608194i	$0.791894\pi$
$368$	0	0
$369$	2697.41	0.380547
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2727.86	0.378668	0.189334	−	0.981913i	$-0.439367\pi$
$373$	2727.86	0.378668	0.189334	+	0.981913i	$0.439367\pi$
$374$	0	0
$375$	8715.38	1.20016
$376$	0	0
$377$	−5269.17	−0.719830
$378$	0	0
$379$	−4086.49	−0.553849	−0.276924	−	0.960892i	$-0.589315\pi$
$379$	−4086.49	−0.553849	−0.276924	+	0.960892i	$0.589315\pi$
$380$	0	0
$381$	−6934.20	−0.932414
$382$	0	0
$383$	13032.9	1.73878	0.869389	−	0.494129i	$-0.164513\pi$
$383$	13032.9	1.73878	0.869389	+	0.494129i	$0.164513\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4512.79	0.592760
$388$	0	0
$389$	−194.991	−0.0254151	−0.0127075	−	0.999919i	$-0.504045\pi$
$389$	−194.991	−0.0254151	−0.0127075	+	0.999919i	$0.504045\pi$
$390$	0	0
$391$	144.363	0.0186719
$392$	0	0
$393$	465.053	0.0596916
$394$	0	0
$395$	−2751.02	−0.350427
$396$	0	0
$397$	14183.0	1.79300	0.896501	−	0.443042i	$-0.146101\pi$
$397$	14183.0	1.79300	0.896501	+	0.443042i	$0.146101\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10005.0	1.24596	0.622978	−	0.782239i	$-0.285923\pi$
$401$	10005.0	1.24596	0.622978	+	0.782239i	$0.285923\pi$
$402$	0	0
$403$	10719.4	1.32500
$404$	0	0
$405$	−1611.86	−0.197763
$406$	0	0
$407$	1343.52	0.163626
$408$	0	0
$409$	−4634.93	−0.560349	−0.280174	−	0.959949i	$-0.590392\pi$
$409$	−4634.93	−0.560349	−0.280174	+	0.959949i	$0.590392\pi$
$410$	0	0
$411$	−1550.81	−0.186121
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12360.3	−1.46203
$416$	0	0
$417$	−2874.20	−0.337531
$418$	0	0
$419$	4998.31	0.582777	0.291388	−	0.956605i	$-0.405883\pi$
$419$	4998.31	0.582777	0.291388	+	0.956605i	$0.405883\pi$
$420$	0	0
$421$	−704.160	−0.0815170	−0.0407585	−	0.999169i	$-0.512977\pi$
$421$	−704.160	−0.0815170	−0.0407585	+	0.999169i	$0.512977\pi$
$422$	0	0
$423$	2749.98	0.316096
$424$	0	0
$425$	1528.28	0.174429
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6274.16	−0.706105
$430$	0	0
$431$	10332.8	1.15479	0.577393	−	0.816466i	$-0.304070\pi$
$431$	10332.8	1.15479	0.577393	+	0.816466i	$0.304070\pi$
$432$	0	0
$433$	−11106.8	−1.23270	−0.616348	−	0.787474i	$-0.711389\pi$
$433$	−11106.8	−1.23270	−0.616348	+	0.787474i	$0.711389\pi$
$434$	0	0
$435$	3600.94	0.396901
$436$	0	0
$437$	1660.98	0.181821
$438$	0	0
$439$	−7299.28	−0.793566	−0.396783	−	0.917912i	$-0.629874\pi$
$439$	−7299.28	−0.793566	−0.396783	+	0.917912i	$0.629874\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16089.7	−1.72560	−0.862802	−	0.505542i	$-0.831293\pi$
$443$	−16089.7	−1.72560	−0.862802	+	0.505542i	$0.831293\pi$
$444$	0	0
$445$	5679.68	0.605040
$446$	0	0
$447$	5311.88	0.562065
$448$	0	0
$449$	13561.7	1.42543	0.712715	−	0.701454i	$-0.247466\pi$
$449$	13561.7	1.42543	0.712715	+	0.701454i	$0.247466\pi$
$450$	0	0
$451$	−7175.46	−0.749178
$452$	0	0
$453$	−7620.71	−0.790402
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10848.6	1.11045	0.555224	−	0.831701i	$-0.312633\pi$
$457$	10848.6	1.11045	0.555224	+	0.831701i	$0.312633\pi$
$458$	0	0
$459$	−152.269	−0.0154844
$460$	0	0
$461$	1758.69	0.177679	0.0888397	−	0.996046i	$-0.471684\pi$
$461$	1758.69	0.177679	0.0888397	+	0.996046i	$0.471684\pi$
$462$	0	0
$463$	5411.95	0.543228	0.271614	−	0.962406i	$-0.412443\pi$
$463$	5411.95	0.543228	0.271614	+	0.962406i	$0.412443\pi$
$464$	0	0
$465$	−7325.64	−0.730577
$466$	0	0
$467$	8111.34	0.803744	0.401872	−	0.915696i	$-0.368360\pi$
$467$	8111.34	0.803744	0.401872	+	0.915696i	$0.368360\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3250.03	−0.317948
$472$	0	0
$473$	−12004.6	−1.16696
$474$	0	0
$475$	17583.8	1.69853
$476$	0	0
$477$	−3376.05	−0.324065
$478$	0	0
$479$	2095.76	0.199912	0.0999559	−	0.994992i	$-0.468130\pi$
$479$	2095.76	0.199912	0.0999559	+	0.994992i	$0.468130\pi$
$480$	0	0
$481$	4902.18	0.464699
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12001.7	−1.12364
$486$	0	0
$487$	−9610.08	−0.894197	−0.447099	−	0.894485i	$-0.647543\pi$
$487$	−9610.08	−0.894197	−0.447099	+	0.894485i	$0.647543\pi$
$488$	0	0
$489$	8906.17	0.823622
$490$	0	0
$491$	11717.3	1.07698	0.538488	−	0.842633i	$-0.318996\pi$
$491$	11717.3	1.07698	0.538488	+	0.842633i	$0.318996\pi$
$492$	0	0
$493$	340.174	0.0310764
$494$	0	0
$495$	4287.75	0.389333
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9195.19	0.824916	0.412458	−	0.910977i	$-0.364670\pi$
$499$	9195.19	0.824916	0.412458	+	0.910977i	$0.364670\pi$
$500$	0	0
$501$	6274.60	0.559538
$502$	0	0
$503$	16118.8	1.42883	0.714414	−	0.699724i	$-0.246693\pi$
$503$	16118.8	1.42883	0.714414	+	0.699724i	$0.246693\pi$
$504$	0	0
$505$	9098.23	0.801715
$506$	0	0
$507$	−16301.9	−1.42799
$508$	0	0
$509$	−4918.78	−0.428333	−0.214166	−	0.976797i	$-0.568703\pi$
$509$	−4918.78	−0.428333	−0.214166	+	0.976797i	$0.568703\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1751.96	−0.150781
$514$	0	0
$515$	−15641.9	−1.33838
$516$	0	0
$517$	−7315.29	−0.622294
$518$	0	0
$519$	−1410.44	−0.119290
$520$	0	0
$521$	−13963.4	−1.17418	−0.587089	−	0.809522i	$-0.699726\pi$
$521$	−13963.4	−1.17418	−0.587089	+	0.809522i	$0.699726\pi$
$522$	0	0
$523$	−13755.3	−1.15005	−0.575024	−	0.818136i	$-0.695007\pi$
$523$	−13755.3	−1.15005	−0.575024	+	0.818136i	$0.695007\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−692.040	−0.0572026
$528$	0	0
$529$	−11511.7	−0.946145
$530$	0	0
$531$	−5648.51	−0.461628
$532$	0	0
$533$	−26181.5	−2.12767
$534$	0	0
$535$	−3909.47	−0.315928
$536$	0	0
$537$	3169.38	0.254691
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14462.7	1.14935	0.574676	−	0.818381i	$-0.305128\pi$
$541$	14462.7	1.14935	0.574676	+	0.818381i	$0.305128\pi$
$542$	0	0
$543$	−1219.41	−0.0963719
$544$	0	0
$545$	6096.07	0.479132
$546$	0	0
$547$	−13682.5	−1.06951	−0.534755	−	0.845007i	$-0.679596\pi$
$547$	−13682.5	−1.06951	−0.534755	+	0.845007i	$0.679596\pi$
$548$	0	0
$549$	−33.8162	−0.00262886
$550$	0	0
$551$	3913.92	0.302611
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3350.14	−0.256227
$556$	0	0
$557$	−7663.13	−0.582939	−0.291470	−	0.956580i	$-0.594144\pi$
$557$	−7663.13	−0.582939	−0.291470	+	0.956580i	$0.594144\pi$
$558$	0	0
$559$	−43801.8	−3.31417
$560$	0	0
$561$	405.056	0.0304839
$562$	0	0
$563$	17470.5	1.30780	0.653902	−	0.756580i	$-0.273131\pi$
$563$	17470.5	1.30780	0.653902	+	0.756580i	$0.273131\pi$
$564$	0	0
$565$	−39751.8	−2.95995
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13873.9	−1.02219	−0.511094	−	0.859525i	$-0.670760\pi$
$569$	−13873.9	−1.02219	−0.511094	+	0.859525i	$0.670760\pi$
$570$	0	0
$571$	3777.52	0.276855	0.138428	−	0.990373i	$-0.455795\pi$
$571$	3777.52	0.276855	0.138428	+	0.990373i	$0.455795\pi$
$572$	0	0
$573$	538.229	0.0392405
$574$	0	0
$575$	6936.79	0.503103
$576$	0	0
$577$	−13880.5	−1.00148	−0.500738	−	0.865599i	$-0.666938\pi$
$577$	−13880.5	−1.00148	−0.500738	+	0.865599i	$0.666938\pi$
$578$	0	0
$579$	7165.10	0.514286
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8980.72	0.637982
$584$	0	0
$585$	15644.9	1.10571
$586$	0	0
$587$	−2395.61	−0.168445	−0.0842227	−	0.996447i	$-0.526841\pi$
$587$	−2395.61	−0.168445	−0.0842227	+	0.996447i	$0.526841\pi$
$588$	0	0
$589$	−7962.36	−0.557018
$590$	0	0
$591$	7996.06	0.556538
$592$	0	0
$593$	−6603.50	−0.457290	−0.228645	−	0.973510i	$-0.573430\pi$
$593$	−6603.50	−0.457290	−0.228645	+	0.973510i	$0.573430\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4026.92	0.276065
$598$	0	0
$599$	−17252.1	−1.17680	−0.588399	−	0.808571i	$-0.700241\pi$
$599$	−17252.1	−1.17680	−0.588399	+	0.808571i	$0.700241\pi$
$600$	0	0
$601$	−12833.1	−0.871005	−0.435503	−	0.900187i	$-0.643429\pi$
$601$	−12833.1	−0.871005	−0.435503	+	0.900187i	$0.643429\pi$
$602$	0	0
$603$	7317.43	0.494177
$604$	0	0
$605$	15080.3	1.01339
$606$	0	0
$607$	8620.16	0.576411	0.288206	−	0.957569i	$-0.406941\pi$
$607$	8620.16	0.576411	0.288206	+	0.957569i	$0.406941\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−26691.7	−1.76732
$612$	0	0
$613$	5136.73	0.338451	0.169226	−	0.985577i	$-0.445873\pi$
$613$	5136.73	0.338451	0.169226	+	0.985577i	$0.445873\pi$
$614$	0	0
$615$	17892.4	1.17316
$616$	0	0
$617$	−1759.82	−0.114826	−0.0574131	−	0.998351i	$-0.518285\pi$
$617$	−1759.82	−0.114826	−0.0574131	+	0.998351i	$0.518285\pi$
$618$	0	0
$619$	−3560.24	−0.231176	−0.115588	−	0.993297i	$-0.536875\pi$
$619$	−3560.24	−0.231176	−0.115588	+	0.993297i	$0.536875\pi$
$620$	0	0
$621$	−691.145	−0.0446614
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23936.8	1.53195
$626$	0	0
$627$	4660.43	0.296841
$628$	0	0
$629$	−316.482	−0.0200620
$630$	0	0
$631$	27321.4	1.72369	0.861845	−	0.507172i	$-0.169309\pi$
$631$	27321.4	1.72369	0.861845	+	0.507172i	$0.169309\pi$
$632$	0	0
$633$	1885.33	0.118381
$634$	0	0
$635$	−45995.7	−2.87446
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1493.12	−0.0924366
$640$	0	0
$641$	−21927.0	−1.35112	−0.675558	−	0.737307i	$-0.736097\pi$
$641$	−21927.0	−1.35112	−0.675558	+	0.737307i	$0.736097\pi$
$642$	0	0
$643$	−5826.04	−0.357320	−0.178660	−	0.983911i	$-0.557176\pi$
$643$	−5826.04	−0.357320	−0.178660	+	0.983911i	$0.557176\pi$
$644$	0	0
$645$	29934.1	1.82737
$646$	0	0
$647$	24210.7	1.47113	0.735565	−	0.677454i	$-0.236917\pi$
$647$	24210.7	1.47113	0.735565	+	0.677454i	$0.236917\pi$
$648$	0	0
$649$	15025.7	0.908801
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25623.3	−1.53556	−0.767778	−	0.640716i	$-0.778637\pi$
$653$	−25623.3	−1.53556	−0.767778	+	0.640716i	$0.778637\pi$
$654$	0	0
$655$	3084.77	0.184018
$656$	0	0
$657$	−5571.90	−0.330868
$658$	0	0
$659$	23273.7	1.37574	0.687871	−	0.725833i	$-0.258545\pi$
$659$	23273.7	1.37574	0.687871	+	0.725833i	$0.258545\pi$
$660$	0	0
$661$	20036.5	1.17902	0.589508	−	0.807763i	$-0.299322\pi$
$661$	20036.5	1.17902	0.589508	+	0.807763i	$0.299322\pi$
$662$	0	0
$663$	1477.95	0.0865744
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1544.04	0.0896333
$668$	0	0
$669$	2909.91	0.168167
$670$	0	0
$671$	89.9554	0.00517540
$672$	0	0
$673$	−18127.8	−1.03830	−0.519149	−	0.854684i	$-0.673751\pi$
$673$	−18127.8	−1.03830	−0.519149	+	0.854684i	$0.673751\pi$
$674$	0	0
$675$	−7316.73	−0.417216
$676$	0	0
$677$	−13815.5	−0.784301	−0.392150	−	0.919901i	$-0.628269\pi$
$677$	−13815.5	−0.784301	−0.392150	+	0.919901i	$0.628269\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	14245.9	0.801622
$682$	0	0
$683$	4604.23	0.257944	0.128972	−	0.991648i	$-0.458832\pi$
$683$	4604.23	0.257944	0.128972	+	0.991648i	$0.458832\pi$
$684$	0	0
$685$	−10286.8	−0.573777
$686$	0	0
$687$	−14406.0	−0.800032
$688$	0	0
$689$	32768.5	1.81187
$690$	0	0
$691$	17913.7	0.986205	0.493103	−	0.869971i	$-0.335863\pi$
$691$	17913.7	0.986205	0.493103	+	0.869971i	$0.335863\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19065.1	−1.04054
$696$	0	0
$697$	1690.26	0.0918555
$698$	0	0
$699$	9465.86	0.512206
$700$	0	0
$701$	11303.7	0.609035	0.304518	−	0.952507i	$-0.401505\pi$
$701$	11303.7	0.609035	0.304518	+	0.952507i	$0.401505\pi$
$702$	0	0
$703$	−3641.33	−0.195356
$704$	0	0
$705$	18241.1	0.974466
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−16046.3	−0.849973	−0.424987	−	0.905200i	$-0.639721\pi$
$709$	−16046.3	−0.849973	−0.424987	+	0.905200i	$0.639721\pi$
$710$	0	0
$711$	1244.21	0.0656280
$712$	0	0
$713$	−3141.15	−0.164989
$714$	0	0
$715$	−41617.5	−2.17679
$716$	0	0
$717$	12725.8	0.662836
$718$	0	0
$719$	−25190.5	−1.30660	−0.653300	−	0.757099i	$-0.726616\pi$
$719$	−25190.5	−1.30660	−0.653300	+	0.757099i	$0.726616\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−13029.0	−0.670197
$724$	0	0
$725$	16345.8	0.837334
$726$	0	0
$727$	11277.2	0.575307	0.287653	−	0.957735i	$-0.407125\pi$
$727$	11277.2	0.575307	0.287653	+	0.957735i	$0.407125\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	2827.82	0.143079
$732$	0	0
$733$	−23720.0	−1.19525	−0.597626	−	0.801775i	$-0.703889\pi$
$733$	−23720.0	−1.19525	−0.597626	+	0.801775i	$0.703889\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19465.3	−0.972880
$738$	0	0
$739$	−8124.72	−0.404428	−0.202214	−	0.979341i	$-0.564814\pi$
$739$	−8124.72	−0.404428	−0.202214	+	0.979341i	$0.564814\pi$
$740$	0	0
$741$	17004.8	0.843030
$742$	0	0
$743$	−20955.3	−1.03469	−0.517346	−	0.855777i	$-0.673080\pi$
$743$	−20955.3	−1.03469	−0.517346	+	0.855777i	$0.673080\pi$
$744$	0	0
$745$	35234.6	1.73274
$746$	0	0
$747$	5590.23	0.273810
$748$	0	0
$749$	0	0
$750$	0	0
$751$	38208.1	1.85650	0.928251	−	0.371954i	$-0.121312\pi$
$751$	38208.1	1.85650	0.928251	+	0.371954i	$0.121312\pi$
$752$	0	0
$753$	−9009.02	−0.435999
$754$	0	0
$755$	−50549.4	−2.43666
$756$	0	0
$757$	30958.1	1.48638	0.743191	−	0.669079i	$-0.233311\pi$
$757$	30958.1	1.48638	0.743191	+	0.669079i	$0.233311\pi$
$758$	0	0
$759$	1838.53	0.0879243
$760$	0	0
$761$	40049.4	1.90774	0.953871	−	0.300218i	$-0.0970594\pi$
$761$	40049.4	1.90774	0.953871	+	0.300218i	$0.0970594\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1010.03	−0.0477355
$766$	0	0
$767$	54825.3	2.58100
$768$	0	0
$769$	8002.01	0.375240	0.187620	−	0.982242i	$-0.439923\pi$
$769$	8002.01	0.375240	0.187620	+	0.982242i	$0.439923\pi$
$770$	0	0
$771$	13406.5	0.626231
$772$	0	0
$773$	13933.4	0.648316	0.324158	−	0.946003i	$-0.394919\pi$
$773$	13933.4	0.648316	0.324158	+	0.946003i	$0.394919\pi$
$774$	0	0
$775$	−33253.4	−1.54128
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19447.6	0.894456
$780$	0	0
$781$	3971.89	0.181979
$782$	0	0
$783$	−1628.61	−0.0743316
$784$	0	0
$785$	−21558.0	−0.980175
$786$	0	0
$787$	37581.7	1.70222	0.851108	−	0.524991i	$-0.175931\pi$
$787$	37581.7	1.70222	0.851108	+	0.524991i	$0.175931\pi$
$788$	0	0
$789$	18480.4	0.833865
$790$	0	0
$791$	0	0
$792$	0	0
$793$	328.225	0.0146981
$794$	0	0
$795$	−22393.9	−0.999032
$796$	0	0
$797$	9458.78	0.420385	0.210193	−	0.977660i	$-0.432591\pi$
$797$	9458.78	0.420385	0.210193	+	0.977660i	$0.432591\pi$
$798$	0	0
$799$	1723.20	0.0762985
$800$	0	0
$801$	−2568.77	−0.113312
$802$	0	0
$803$	14821.9	0.651376
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14966.7	0.652854
$808$	0	0
$809$	−1909.94	−0.0830034	−0.0415017	−	0.999138i	$-0.513214\pi$
$809$	−1909.94	−0.0830034	−0.0415017	+	0.999138i	$0.513214\pi$
$810$	0	0
$811$	43110.6	1.86661	0.933303	−	0.359091i	$-0.116913\pi$
$811$	43110.6	1.86661	0.933303	+	0.359091i	$0.116913\pi$
$812$	0	0
$813$	−13301.2	−0.573792
$814$	0	0
$815$	59076.1	2.53907
$816$	0	0
$817$	32535.9	1.39325
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4026.56	0.171167	0.0855834	−	0.996331i	$-0.472725\pi$
$821$	4026.56	0.171167	0.0855834	+	0.996331i	$0.472725\pi$
$822$	0	0
$823$	−39668.1	−1.68012	−0.840062	−	0.542491i	$-0.817481\pi$
$823$	−39668.1	−1.68012	−0.840062	+	0.542491i	$0.817481\pi$
$824$	0	0
$825$	19463.4	0.821368
$826$	0	0
$827$	−30137.7	−1.26722	−0.633611	−	0.773652i	$-0.718428\pi$
$827$	−30137.7	−1.26722	−0.633611	+	0.773652i	$0.718428\pi$
$828$	0	0
$829$	−23278.0	−0.975245	−0.487622	−	0.873055i	$-0.662136\pi$
$829$	−23278.0	−0.975245	−0.487622	+	0.873055i	$0.662136\pi$
$830$	0	0
$831$	−3337.10	−0.139306
$832$	0	0
$833$	0	0
$834$	0	0
$835$	41620.5	1.72495
$836$	0	0
$837$	3313.19	0.136823
$838$	0	0
$839$	−9494.43	−0.390684	−0.195342	−	0.980735i	$-0.562582\pi$
$839$	−9494.43	−0.390684	−0.195342	+	0.980735i	$0.562582\pi$
$840$	0	0
$841$	−20750.6	−0.850820
$842$	0	0
$843$	−8439.65	−0.344813
$844$	0	0
$845$	−108133.	−4.40223
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9442.62	−0.381708
$850$	0	0
$851$	−1436.50	−0.0578644
$852$	0	0
$853$	12692.4	0.509471	0.254736	−	0.967011i	$-0.418012\pi$
$853$	12692.4	0.509471	0.254736	+	0.967011i	$0.418012\pi$
$854$	0	0
$855$	−11621.0	−0.464831
$856$	0	0
$857$	−22206.0	−0.885114	−0.442557	−	0.896740i	$-0.645929\pi$
$857$	−22206.0	−0.885114	−0.442557	+	0.896740i	$0.645929\pi$
$858$	0	0
$859$	19820.5	0.787271	0.393636	−	0.919267i	$-0.371217\pi$
$859$	19820.5	0.787271	0.393636	+	0.919267i	$0.371217\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36413.8	−1.43631	−0.718157	−	0.695881i	$-0.755014\pi$
$863$	−36413.8	−1.43631	−0.718157	+	0.695881i	$0.755014\pi$
$864$	0	0
$865$	−9355.70	−0.367749
$866$	0	0
$867$	14643.6	0.573613
$868$	0	0
$869$	−3309.76	−0.129201
$870$	0	0
$871$	−71024.1	−2.76298
$872$	0	0
$873$	5428.03	0.210436
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19442.6	0.748609	0.374305	−	0.927306i	$-0.377881\pi$
$877$	19442.6	0.748609	0.374305	+	0.927306i	$0.377881\pi$
$878$	0	0
$879$	27429.1	1.05252
$880$	0	0
$881$	25184.2	0.963082	0.481541	−	0.876423i	$-0.340077\pi$
$881$	25184.2	0.963082	0.481541	+	0.876423i	$0.340077\pi$
$882$	0	0
$883$	4050.03	0.154354	0.0771769	−	0.997017i	$-0.475409\pi$
$883$	4050.03	0.154354	0.0771769	+	0.997017i	$0.475409\pi$
$884$	0	0
$885$	−37467.5	−1.42311
$886$	0	0
$887$	41604.2	1.57489	0.787447	−	0.616383i	$-0.211402\pi$
$887$	41604.2	1.57489	0.787447	+	0.616383i	$0.211402\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1939.23	−0.0729144
$892$	0	0
$893$	19826.5	0.742967
$894$	0	0
$895$	21023.0	0.785165
$896$	0	0
$897$	6708.36	0.249705
$898$	0	0
$899$	−7401.76	−0.274597
$900$	0	0
$901$	−2115.51	−0.0782219
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8088.56	−0.297097
$906$	0	0
$907$	3572.60	0.130790	0.0653949	−	0.997859i	$-0.479169\pi$
$907$	3572.60	0.130790	0.0653949	+	0.997859i	$0.479169\pi$
$908$	0	0
$909$	−4114.88	−0.150145
$910$	0	0
$911$	−29457.7	−1.07133	−0.535663	−	0.844432i	$-0.679938\pi$
$911$	−29457.7	−1.07133	−0.535663	+	0.844432i	$0.679938\pi$
$912$	0	0
$913$	−14870.7	−0.539046
$914$	0	0
$915$	−224.309	−0.00810428
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3310.65	0.118834	0.0594168	−	0.998233i	$-0.481076\pi$
$919$	3310.65	0.118834	0.0594168	+	0.998233i	$0.481076\pi$
$920$	0	0
$921$	13946.7	0.498979
$922$	0	0
$923$	14492.5	0.516820
$924$	0	0
$925$	−15207.3	−0.540556
$926$	0	0
$927$	7074.40	0.250651
$928$	0	0
$929$	31467.5	1.11132	0.555660	−	0.831410i	$-0.312466\pi$
$929$	31467.5	1.11132	0.555660	+	0.831410i	$0.312466\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	19251.5	0.675527
$934$	0	0
$935$	2686.80	0.0939763
$936$	0	0
$937$	17363.4	0.605375	0.302688	−	0.953090i	$-0.402116\pi$
$937$	17363.4	0.605375	0.302688	+	0.953090i	$0.402116\pi$
$938$	0	0
$939$	17604.6	0.611828
$940$	0	0
$941$	5547.77	0.192192	0.0960958	−	0.995372i	$-0.469364\pi$
$941$	5547.77	0.192192	0.0960958	+	0.995372i	$0.469364\pi$
$942$	0	0
$943$	7672.04	0.264937
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37960.3	1.30258	0.651290	−	0.758829i	$-0.274228\pi$
$947$	37960.3	1.30258	0.651290	+	0.758829i	$0.274228\pi$
$948$	0	0
$949$	54081.7	1.84991
$950$	0	0
$951$	11923.6	0.406570
$952$	0	0
$953$	−10019.3	−0.340563	−0.170282	−	0.985395i	$-0.554468\pi$
$953$	−10019.3	−0.340563	−0.170282	+	0.985395i	$0.554468\pi$
$954$	0	0
$955$	3570.16	0.120971
$956$	0	0
$957$	4332.30	0.146336
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−14733.1	−0.494548
$962$	0	0
$963$	1768.15	0.0591670
$964$	0	0
$965$	47527.3	1.58545
$966$	0	0
$967$	−27834.4	−0.925641	−0.462820	−	0.886452i	$-0.653163\pi$
$967$	−27834.4	−0.925641	−0.462820	+	0.886452i	$0.653163\pi$
$968$	0	0
$969$	−1097.82	−0.0363952
$970$	0	0
$971$	−18275.3	−0.604000	−0.302000	−	0.953308i	$-0.597654\pi$
$971$	−18275.3	−0.604000	−0.302000	+	0.953308i	$0.597654\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	71017.2	2.33269
$976$	0	0
$977$	−43028.9	−1.40903	−0.704513	−	0.709691i	$-0.748834\pi$
$977$	−43028.9	−1.40903	−0.704513	+	0.709691i	$0.748834\pi$
$978$	0	0
$979$	6833.24	0.223076
$980$	0	0
$981$	−2757.09	−0.0897320
$982$	0	0
$983$	30559.2	0.991544	0.495772	−	0.868453i	$-0.334885\pi$
$983$	30559.2	0.991544	0.495772	+	0.868453i	$0.334885\pi$
$984$	0	0
$985$	53039.2	1.71571
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12835.4	0.412681
$990$	0	0
$991$	44945.9	1.44072	0.720360	−	0.693600i	$-0.243976\pi$
$991$	44945.9	1.44072	0.720360	+	0.693600i	$0.243976\pi$
$992$	0	0
$993$	−26738.5	−0.854501
$994$	0	0
$995$	26711.2	0.851058
$996$	0	0
$997$	−29006.1	−0.921397	−0.460698	−	0.887557i	$-0.652401\pi$
$997$	−29006.1	−0.921397	−0.460698	+	0.887557i	$0.652401\pi$
$998$	0	0
$999$	1515.18	0.0479861

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bl.1.1		2
4.3	odd	2		147.4.a.k.1.2	yes	2
7.6	odd	2		2352.4.a.cf.1.2		2
12.11	even	2		441.4.a.o.1.1		2
28.3	even	6		147.4.e.k.79.1		4
28.11	odd	6		147.4.e.j.79.1		4
28.19	even	6		147.4.e.k.67.1		4
28.23	odd	6		147.4.e.j.67.1		4
28.27	even	2		147.4.a.j.1.2	✓	2
84.11	even	6		441.4.e.u.226.2		4
84.23	even	6		441.4.e.u.361.2		4
84.47	odd	6		441.4.e.v.361.2		4
84.59	odd	6		441.4.e.v.226.2		4
84.83	odd	2		441.4.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.a.j.1.2	✓	2	28.27	even	2
147.4.a.k.1.2	yes	2	4.3	odd	2
147.4.e.j.67.1		4	28.23	odd	6
147.4.e.j.79.1		4	28.11	odd	6
147.4.e.k.67.1		4	28.19	even	6
147.4.e.k.79.1		4	28.3	even	6
441.4.a.n.1.1		2	84.83	odd	2
441.4.a.o.1.1		2	12.11	even	2
441.4.e.u.226.2		4	84.11	even	6
441.4.e.u.361.2		4	84.23	even	6
441.4.e.v.226.2		4	84.59	odd	6
441.4.e.v.361.2		4	84.47	odd	6
2352.4.a.bl.1.1		2	1.1	even	1	trivial
2352.4.a.cf.1.2		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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -19.8995 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -19.8995 q^{5} +9.00000 q^{9} -23.9411 q^{11} -87.3553 q^{13} +59.6985 q^{15} +5.63961 q^{17} +64.8873 q^{19} +25.5980 q^{23} +270.990 q^{25} -27.0000 q^{27} +60.3188 q^{29} -122.711 q^{31} +71.8234 q^{33} -56.1177 q^{37} +262.066 q^{39} +299.713 q^{41} +501.421 q^{43} -179.095 q^{45} +305.553 q^{47} -16.9188 q^{51} -375.117 q^{53} +476.416 q^{55} -194.662 q^{57} -627.612 q^{59} -3.75736 q^{61} +1738.33 q^{65} +813.048 q^{67} -76.7939 q^{69} -165.902 q^{71} -619.100 q^{73} -812.970 q^{75} +138.246 q^{79} +81.0000 q^{81} +621.137 q^{83} -112.225 q^{85} -180.956 q^{87} -285.418 q^{89} +368.132 q^{93} -1291.22 q^{95} +603.114 q^{97} -215.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 20 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 20 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} - 20 q^{5} + 18 q^{9} + 20 q^{11} - 104 q^{13} + 60 q^{15} - 116 q^{17} + 192 q^{19} - 28 q^{23} + 146 q^{25} - 54 q^{27} + 296 q^{29} - 104 q^{31} - 60 q^{33} - 248 q^{37} + 312 q^{39} - 20 q^{41} + 720 q^{43} - 180 q^{45} - 96 q^{47} + 348 q^{51} + 268 q^{53} + 472 q^{55} - 576 q^{57} - 616 q^{59} - 16 q^{61} + 1740 q^{65} + 144 q^{67} + 84 q^{69} - 988 q^{71} - 104 q^{73} - 438 q^{75} + 944 q^{79} + 162 q^{81} + 1016 q^{83} - 100 q^{85} - 888 q^{87} + 388 q^{89} + 312 q^{93} - 1304 q^{95} - 488 q^{97} + 180 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 20 * q^5 + 18 * q^9 + 20 * q^11 - 104 * q^13 + 60 * q^15 - 116 * q^17 + 192 * q^19 - 28 * q^23 + 146 * q^25 - 54 * q^27 + 296 * q^29 - 104 * q^31 - 60 * q^33 - 248 * q^37 + 312 * q^39 - 20 * q^41 + 720 * q^43 - 180 * q^45 - 96 * q^47 + 348 * q^51 + 268 * q^53 + 472 * q^55 - 576 * q^57 - 616 * q^59 - 16 * q^61 + 1740 * q^65 + 144 * q^67 + 84 * q^69 - 988 * q^71 - 104 * q^73 - 438 * q^75 + 944 * q^79 + 162 * q^81 + 1016 * q^83 - 100 * q^85 - 888 * q^87 + 388 * q^89 + 312 * q^93 - 1304 * q^95 - 488 * q^97 + 180 * q^99

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