LMFDB - Embedded newform 2352.4.a.cf.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:	$0$
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)

$f(q)$	$=$	$q+3.00000 q^{3} +0.100505 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +0.100505 q^{5} +9.00000 q^{9} +43.9411 q^{11} +16.6447 q^{13} +0.301515 q^{15} +121.640 q^{17} -127.113 q^{19} -53.5980 q^{23} -124.990 q^{25} +27.0000 q^{27} +235.681 q^{29} -18.7107 q^{31} +131.823 q^{33} -191.882 q^{37} +49.9340 q^{39} +319.713 q^{41} +218.579 q^{43} +0.904546 q^{45} +401.553 q^{47} +364.919 q^{51} +643.117 q^{53} +4.41631 q^{55} -381.338 q^{57} -11.6123 q^{59} +12.2426 q^{61} +1.67287 q^{65} -669.048 q^{67} -160.794 q^{69} -822.098 q^{71} -515.100 q^{73} -374.970 q^{75} +805.754 q^{79} +81.0000 q^{81} -394.863 q^{83} +12.2254 q^{85} +707.044 q^{87} -673.418 q^{89} -56.1320 q^{93} -12.7755 q^{95} +1091.11 q^{97} +395.470 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.100505	0.00898945	0.00449472	−	0.999990i	$-0.498569\pi$
$5$	0.100505	0.00898945	0.00449472	+	0.999990i	$0.498569\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	43.9411	1.20443	0.602216	−	0.798333i	$-0.294285\pi$
$11$	43.9411	1.20443	0.602216	+	0.798333i	$0.294285\pi$
$12$	0	0
$13$	16.6447	0.355108	0.177554	−	0.984111i	$-0.443182\pi$
$13$	16.6447	0.355108	0.177554	+	0.984111i	$0.443182\pi$
$14$	0	0
$15$	0.301515	0.00519006
$16$	0	0
$17$	121.640	1.73541	0.867704	−	0.497081i	$-0.165595\pi$
$17$	121.640	1.73541	0.867704	+	0.497081i	$0.165595\pi$
$18$	0	0
$19$	−127.113	−1.53482	−0.767412	−	0.641154i	$-0.778456\pi$
$19$	−127.113	−1.53482	−0.767412	+	0.641154i	$0.778456\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−53.5980	−0.485911	−0.242955	−	0.970037i	$-0.578117\pi$
$23$	−53.5980	−0.485911	−0.242955	+	0.970037i	$0.578117\pi$
$24$	0	0
$25$	−124.990	−0.999919
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	235.681	1.50913	0.754567	−	0.656223i	$-0.227847\pi$
$29$	235.681	1.50913	0.754567	+	0.656223i	$0.227847\pi$
$30$	0	0
$31$	−18.7107	−0.108404	−0.0542022	−	0.998530i	$-0.517262\pi$
$31$	−18.7107	−0.108404	−0.0542022	+	0.998530i	$0.517262\pi$
$32$	0	0
$33$	131.823	0.695379
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−191.882	−0.852574	−0.426287	−	0.904588i	$-0.640179\pi$
$37$	−191.882	−0.852574	−0.426287	+	0.904588i	$0.640179\pi$
$38$	0	0
$39$	49.9340	0.205021
$40$	0	0
$41$	319.713	1.21782	0.608912	−	0.793238i	$-0.291606\pi$
$41$	319.713	1.21782	0.608912	+	0.793238i	$0.291606\pi$
$42$	0	0
$43$	218.579	0.775184	0.387592	−	0.921831i	$-0.373307\pi$
$43$	218.579	0.775184	0.387592	+	0.921831i	$0.373307\pi$
$44$	0	0
$45$	0.904546	0.00299648
$46$	0	0
$47$	401.553	1.24623	0.623113	−	0.782132i	$-0.285868\pi$
$47$	401.553	1.24623	0.623113	+	0.782132i	$0.285868\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	364.919	1.00194
$52$	0	0
$53$	643.117	1.66677	0.833386	−	0.552692i	$-0.186399\pi$
$53$	643.117	1.66677	0.833386	+	0.552692i	$0.186399\pi$
$54$	0	0
$55$	4.41631	0.0108272
$56$	0	0
$57$	−381.338	−0.886131
$58$	0	0
$59$	−11.6123	−0.0256235	−0.0128118	−	0.999918i	$-0.504078\pi$
$59$	−11.6123	−0.0256235	−0.0128118	+	0.999918i	$0.504078\pi$
$60$	0	0
$61$	12.2426	0.0256969	0.0128484	−	0.999917i	$-0.495910\pi$
$61$	12.2426	0.0256969	0.0128484	+	0.999917i	$0.495910\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.67287	0.00319222
$66$	0	0
$67$	−669.048	−1.21996	−0.609979	−	0.792417i	$-0.708822\pi$
$67$	−669.048	−1.21996	−0.609979	+	0.792417i	$0.708822\pi$
$68$	0	0
$69$	−160.794	−0.280541
$70$	0	0
$71$	−822.098	−1.37416	−0.687078	−	0.726584i	$-0.741107\pi$
$71$	−822.098	−1.37416	−0.687078	+	0.726584i	$0.741107\pi$
$72$	0	0
$73$	−515.100	−0.825861	−0.412930	−	0.910763i	$-0.635495\pi$
$73$	−515.100	−0.825861	−0.412930	+	0.910763i	$0.635495\pi$
$74$	0	0
$75$	−374.970	−0.577304
$76$	0	0
$77$	0	0
$78$	0	0
$79$	805.754	1.14752	0.573762	−	0.819022i	$-0.305483\pi$
$79$	805.754	1.14752	0.573762	+	0.819022i	$0.305483\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−394.863	−0.522191	−0.261095	−	0.965313i	$-0.584084\pi$
$83$	−394.863	−0.522191	−0.261095	+	0.965313i	$0.584084\pi$
$84$	0	0
$85$	12.2254	0.0156004
$86$	0	0
$87$	707.044	0.871299
$88$	0	0
$89$	−673.418	−0.802047	−0.401024	−	0.916068i	$-0.631345\pi$
$89$	−673.418	−0.802047	−0.401024	+	0.916068i	$0.631345\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−56.1320	−0.0625873
$94$	0	0
$95$	−12.7755	−0.0137972
$96$	0	0
$97$	1091.11	1.14212	0.571061	−	0.820908i	$-0.306532\pi$
$97$	1091.11	1.14212	0.571061	+	0.820908i	$0.306532\pi$
$98$	0	0
$99$	395.470	0.401477
$100$	0	0
$101$	1370.79	1.35048	0.675242	−	0.737597i	$-0.264039\pi$
$101$	1370.79	1.35048	0.675242	+	0.737597i	$0.264039\pi$
$102$	0	0
$103$	−1413.96	−1.35263	−0.676316	−	0.736611i	$-0.736425\pi$
$103$	−1413.96	−1.35263	−0.676316	+	0.736611i	$0.736425\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	343.539	0.310385	0.155192	−	0.987884i	$-0.450400\pi$
$107$	343.539	0.310385	0.155192	+	0.987884i	$0.450400\pi$
$108$	0	0
$109$	−317.657	−0.279138	−0.139569	−	0.990212i	$-0.544572\pi$
$109$	−317.657	−0.279138	−0.139569	+	0.990212i	$0.544572\pi$
$110$	0	0
$111$	−575.647	−0.492234
$112$	0	0
$113$	798.373	0.664643	0.332321	−	0.943166i	$-0.392168\pi$
$113$	798.373	0.664643	0.332321	+	0.943166i	$0.392168\pi$
$114$	0	0
$115$	−5.38687	−0.00436807
$116$	0	0
$117$	149.802	0.118369
$118$	0	0
$119$	0	0
$120$	0	0
$121$	599.823	0.450656
$122$	0	0
$123$	959.138	0.703110
$124$	0	0
$125$	−25.1253	−0.0179782
$126$	0	0
$127$	−1071.40	−0.748593	−0.374297	−	0.927309i	$-0.622116\pi$
$127$	−1071.40	−0.748593	−0.374297	+	0.927309i	$0.622116\pi$
$128$	0	0
$129$	655.736	0.447553
$130$	0	0
$131$	−2515.02	−1.67739	−0.838695	−	0.544601i	$-0.816681\pi$
$131$	−2515.02	−1.67739	−0.838695	+	0.544601i	$0.816681\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.71364	0.00173002
$136$	0	0
$137$	251.064	0.156568	0.0782841	−	0.996931i	$-0.475056\pi$
$137$	251.064	0.156568	0.0782841	+	0.996931i	$0.475056\pi$
$138$	0	0
$139$	886.067	0.540685	0.270343	−	0.962764i	$-0.412863\pi$
$139$	886.067	0.540685	0.270343	+	0.962764i	$0.412863\pi$
$140$	0	0
$141$	1204.66	0.719508
$142$	0	0
$143$	731.385	0.427703
$144$	0	0
$145$	23.6872	0.0135663
$146$	0	0
$147$	0	0
$148$	0	0
$149$	582.626	0.320339	0.160170	−	0.987090i	$-0.448796\pi$
$149$	582.626	0.320339	0.160170	+	0.987090i	$0.448796\pi$
$150$	0	0
$151$	2811.76	1.51535	0.757676	−	0.652631i	$-0.226335\pi$
$151$	2811.76	1.51535	0.757676	+	0.652631i	$0.226335\pi$
$152$	0	0
$153$	1094.76	0.578469
$154$	0	0
$155$	−1.88052	−0.000974496 0
$156$	0	0
$157$	1691.34	0.859770	0.429885	−	0.902884i	$-0.358554\pi$
$157$	1691.34	0.859770	0.429885	+	0.902884i	$0.358554\pi$
$158$	0	0
$159$	1929.35	0.962311
$160$	0	0
$161$	0	0
$162$	0	0
$163$	40.7232	0.0195686	0.00978432	−	0.999952i	$-0.496886\pi$
$163$	40.7232	0.0195686	0.00978432	+	0.999952i	$0.496886\pi$
$164$	0	0
$165$	13.2489	0.00625107
$166$	0	0
$167$	2900.47	1.34398	0.671990	−	0.740560i	$-0.265440\pi$
$167$	2900.47	1.34398	0.671990	+	0.740560i	$0.265440\pi$
$168$	0	0
$169$	−1919.96	−0.873899
$170$	0	0
$171$	−1144.01	−0.511608
$172$	0	0
$173$	2146.15	0.943171	0.471585	−	0.881820i	$-0.343682\pi$
$173$	2146.15	0.943171	0.471585	+	0.881820i	$0.343682\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−34.8368	−0.0147938
$178$	0	0
$179$	−1203.54	−0.502552	−0.251276	−	0.967916i	$-0.580850\pi$
$179$	−1203.54	−0.502552	−0.251276	+	0.967916i	$0.580850\pi$
$180$	0	0
$181$	2990.47	1.22807	0.614033	−	0.789280i	$-0.289546\pi$
$181$	2990.47	1.22807	0.614033	+	0.789280i	$0.289546\pi$
$182$	0	0
$183$	36.7279	0.0148361
$184$	0	0
$185$	−19.2851	−0.00766417
$186$	0	0
$187$	5344.98	2.09018
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2807.41	1.06354	0.531772	−	0.846887i	$-0.321526\pi$
$191$	2807.41	1.06354	0.531772	+	0.846887i	$0.321526\pi$
$192$	0	0
$193$	3336.37	1.24434	0.622169	−	0.782883i	$-0.286252\pi$
$193$	3336.37	1.24434	0.622169	+	0.782883i	$0.286252\pi$
$194$	0	0
$195$	5.01862	0.00184303
$196$	0	0
$197$	−4226.65	−1.52861	−0.764305	−	0.644855i	$-0.776918\pi$
$197$	−4226.65	−1.52861	−0.764305	+	0.644855i	$0.776918\pi$
$198$	0	0
$199$	4385.69	1.56228	0.781140	−	0.624356i	$-0.214639\pi$
$199$	4385.69	1.56228	0.781140	+	0.624356i	$0.214639\pi$
$200$	0	0
$201$	−2007.14	−0.704343
$202$	0	0
$203$	0	0
$204$	0	0
$205$	32.1328	0.0109476
$206$	0	0
$207$	−482.382	−0.161970
$208$	0	0
$209$	−5585.48	−1.84859
$210$	0	0
$211$	−2291.56	−0.747665	−0.373833	−	0.927496i	$-0.621957\pi$
$211$	−2291.56	−0.747665	−0.373833	+	0.927496i	$0.621957\pi$
$212$	0	0
$213$	−2466.29	−0.793369
$214$	0	0
$215$	21.9683	0.00696848
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1545.30	−0.476811
$220$	0	0
$221$	2024.65	0.616257
$222$	0	0
$223$	−217.970	−0.0654544	−0.0327272	−	0.999464i	$-0.510419\pi$
$223$	−217.970	−0.0654544	−0.0327272	+	0.999464i	$0.510419\pi$
$224$	0	0
$225$	−1124.91	−0.333306
$226$	0	0
$227$	1835.36	0.536639	0.268320	−	0.963330i	$-0.413532\pi$
$227$	1835.36	0.536639	0.268320	+	0.963330i	$0.413532\pi$
$228$	0	0
$229$	−2774.01	−0.800488	−0.400244	−	0.916409i	$-0.631075\pi$
$229$	−2774.01	−0.800488	−0.400244	+	0.916409i	$0.631075\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−988.712	−0.277994	−0.138997	−	0.990293i	$-0.544388\pi$
$233$	−988.712	−0.277994	−0.138997	+	0.990293i	$0.544388\pi$
$234$	0	0
$235$	40.3581	0.0112029
$236$	0	0
$237$	2417.26	0.662524
$238$	0	0
$239$	837.928	0.226783	0.113391	−	0.993550i	$-0.463829\pi$
$239$	837.928	0.226783	0.113391	+	0.993550i	$0.463829\pi$
$240$	0	0
$241$	3454.99	0.923466	0.461733	−	0.887019i	$-0.347228\pi$
$241$	3454.99	0.923466	0.461733	+	0.887019i	$0.347228\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2115.75	−0.545028
$248$	0	0
$249$	−1184.59	−0.301487
$250$	0	0
$251$	5635.01	1.41705	0.708523	−	0.705688i	$-0.249362\pi$
$251$	5635.01	1.41705	0.708523	+	0.705688i	$0.249362\pi$
$252$	0	0
$253$	−2355.16	−0.585246
$254$	0	0
$255$	36.6762	0.00900687
$256$	0	0
$257$	2271.16	0.551248	0.275624	−	0.961265i	$-0.411115\pi$
$257$	2271.16	0.551248	0.275624	+	0.961265i	$0.411115\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2121.13	0.503045
$262$	0	0
$263$	−163.867	−0.0384201	−0.0192101	−	0.999815i	$-0.506115\pi$
$263$	−163.867	−0.0384201	−0.0192101	+	0.999815i	$0.506115\pi$
$264$	0	0
$265$	64.6365	0.0149834
$266$	0	0
$267$	−2020.26	−0.463062
$268$	0	0
$269$	5167.10	1.17116	0.585582	−	0.810613i	$-0.300866\pi$
$269$	5167.10	1.17116	0.585582	+	0.810613i	$0.300866\pi$
$270$	0	0
$271$	−1622.27	−0.363638	−0.181819	−	0.983332i	$-0.558199\pi$
$271$	−1622.27	−0.363638	−0.181819	+	0.983332i	$0.558199\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5492.20	−1.20433
$276$	0	0
$277$	−4612.37	−1.00047	−0.500235	−	0.865890i	$-0.666753\pi$
$277$	−4612.37	−1.00047	−0.500235	+	0.865890i	$0.666753\pi$
$278$	0	0
$279$	−168.396	−0.0361348
$280$	0	0
$281$	−2125.22	−0.451174	−0.225587	−	0.974223i	$-0.572430\pi$
$281$	−2125.22	−0.451174	−0.225587	+	0.974223i	$0.572430\pi$
$282$	0	0
$283$	2571.54	0.540149	0.270075	−	0.962839i	$-0.412952\pi$
$283$	2571.54	0.540149	0.270075	+	0.962839i	$0.412952\pi$
$284$	0	0
$285$	−38.3264	−0.00796583
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9883.19	2.01164
$290$	0	0
$291$	3273.34	0.659404
$292$	0	0
$293$	3324.96	0.662957	0.331478	−	0.943463i	$-0.392453\pi$
$293$	3324.96	0.662957	0.331478	+	0.943463i	$0.392453\pi$
$294$	0	0
$295$	−1.16709	−0.000230341 0
$296$	0	0
$297$	1186.41	0.231793
$298$	0	0
$299$	−892.120	−0.172551
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4112.37	0.779702
$304$	0	0
$305$	1.23045	0.000231001 0
$306$	0	0
$307$	887.096	0.164916	0.0824580	−	0.996595i	$-0.473723\pi$
$307$	887.096	0.164916	0.0824580	+	0.996595i	$0.473723\pi$
$308$	0	0
$309$	−4241.87	−0.780943
$310$	0	0
$311$	4510.82	0.822460	0.411230	−	0.911532i	$-0.365099\pi$
$311$	4510.82	0.822460	0.411230	+	0.911532i	$0.365099\pi$
$312$	0	0
$313$	3715.78	0.671018	0.335509	−	0.942037i	$-0.391092\pi$
$313$	3715.78	0.671018	0.335509	+	0.942037i	$0.391092\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6954.52	1.23219	0.616096	−	0.787671i	$-0.288713\pi$
$317$	6954.52	1.23219	0.616096	+	0.787671i	$0.288713\pi$
$318$	0	0
$319$	10356.1	1.81765
$320$	0	0
$321$	1030.62	0.179201
$322$	0	0
$323$	−15461.9	−2.66355
$324$	0	0
$325$	−2080.41	−0.355079
$326$	0	0
$327$	−952.971	−0.161160
$328$	0	0
$329$	0	0
$330$	0	0
$331$	9863.18	1.63785	0.818926	−	0.573899i	$-0.194570\pi$
$331$	9863.18	1.63785	0.818926	+	0.573899i	$0.194570\pi$
$332$	0	0
$333$	−1726.94	−0.284191
$334$	0	0
$335$	−67.2427	−0.0109668
$336$	0	0
$337$	−5945.06	−0.960974	−0.480487	−	0.877002i	$-0.659540\pi$
$337$	−5945.06	−0.960974	−0.480487	+	0.877002i	$0.659540\pi$
$338$	0	0
$339$	2395.12	0.383732
$340$	0	0
$341$	−822.168	−0.130566
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−16.1606	−0.00252191
$346$	0	0
$347$	−1169.57	−0.180939	−0.0904697	−	0.995899i	$-0.528837\pi$
$347$	−1169.57	−0.180939	−0.0904697	+	0.995899i	$0.528837\pi$
$348$	0	0
$349$	−9176.66	−1.40749	−0.703747	−	0.710451i	$-0.748491\pi$
$349$	−9176.66	−1.40749	−0.703747	+	0.710451i	$0.748491\pi$
$350$	0	0
$351$	449.406	0.0683405
$352$	0	0
$353$	10587.2	1.59632	0.798158	−	0.602448i	$-0.205808\pi$
$353$	10587.2	1.59632	0.798158	+	0.602448i	$0.205808\pi$
$354$	0	0
$355$	−82.6250	−0.0123529
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8615.21	−1.26656	−0.633278	−	0.773924i	$-0.718291\pi$
$359$	−8615.21	−1.26656	−0.633278	+	0.773924i	$0.718291\pi$
$360$	0	0
$361$	9298.64	1.35568
$362$	0	0
$363$	1799.47	0.260186
$364$	0	0
$365$	−51.7701	−0.00742403
$366$	0	0
$367$	−8297.79	−1.18022	−0.590110	−	0.807323i	$-0.700916\pi$
$367$	−8297.79	−1.18022	−0.590110	+	0.807323i	$0.700916\pi$
$368$	0	0
$369$	2877.41	0.405941
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5123.86	−0.711269	−0.355634	−	0.934625i	$-0.615735\pi$
$373$	−5123.86	−0.711269	−0.355634	+	0.934625i	$0.615735\pi$
$374$	0	0
$375$	−75.3758	−0.0103797
$376$	0	0
$377$	3922.83	0.535905
$378$	0	0
$379$	1502.49	0.203635	0.101817	−	0.994803i	$-0.467534\pi$
$379$	1502.49	0.203635	0.101817	+	0.994803i	$0.467534\pi$
$380$	0	0
$381$	−3214.20	−0.432200
$382$	0	0
$383$	10872.9	1.45060	0.725301	−	0.688431i	$-0.241700\pi$
$383$	10872.9	1.45060	0.725301	+	0.688431i	$0.241700\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1967.21	0.258395
$388$	0	0
$389$	4618.99	0.602036	0.301018	−	0.953618i	$-0.402674\pi$
$389$	4618.99	0.602036	0.301018	+	0.953618i	$0.402674\pi$
$390$	0	0
$391$	−6519.64	−0.843254
$392$	0	0
$393$	−7545.05	−0.968442
$394$	0	0
$395$	80.9824	0.0103156
$396$	0	0
$397$	9606.95	1.21451	0.607253	−	0.794508i	$-0.292271\pi$
$397$	9606.95	1.21451	0.607253	+	0.794508i	$0.292271\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10501.0	−1.30772	−0.653862	−	0.756614i	$-0.726852\pi$
$401$	−10501.0	−1.30772	−0.653862	+	0.756614i	$0.726852\pi$
$402$	0	0
$403$	−311.433	−0.0384952
$404$	0	0
$405$	8.14091	0.000998827 0
$406$	0	0
$407$	−8431.52	−1.02687
$408$	0	0
$409$	−12066.9	−1.45885	−0.729427	−	0.684059i	$-0.760213\pi$
$409$	−12066.9	−1.45885	−0.729427	+	0.684059i	$0.760213\pi$
$410$	0	0
$411$	753.192	0.0903947
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−39.6857	−0.00469421
$416$	0	0
$417$	2658.20	0.312165
$418$	0	0
$419$	6366.31	0.742278	0.371139	−	0.928577i	$-0.378967\pi$
$419$	6366.31	0.742278	0.371139	+	0.928577i	$0.378967\pi$
$420$	0	0
$421$	−4731.84	−0.547781	−0.273890	−	0.961761i	$-0.588311\pi$
$421$	−4731.84	−0.547781	−0.273890	+	0.961761i	$0.588311\pi$
$422$	0	0
$423$	3613.98	0.415408
$424$	0	0
$425$	−15203.7	−1.73527
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2194.16	0.246934
$430$	0	0
$431$	−3752.78	−0.419409	−0.209704	−	0.977765i	$-0.567250\pi$
$431$	−3752.78	−0.419409	−0.209704	+	0.977765i	$0.567250\pi$
$432$	0	0
$433$	11709.2	1.29956	0.649780	−	0.760122i	$-0.274861\pi$
$433$	11709.2	1.29956	0.649780	+	0.760122i	$0.274861\pi$
$434$	0	0
$435$	71.0615	0.00783250
$436$	0	0
$437$	6812.98	0.745788
$438$	0	0
$439$	14924.7	1.62259	0.811296	−	0.584635i	$-0.198762\pi$
$439$	14924.7	1.62259	0.811296	+	0.584635i	$0.198762\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8517.66	0.913513	0.456756	−	0.889592i	$-0.349011\pi$
$443$	8517.66	0.913513	0.456756	+	0.889592i	$0.349011\pi$
$444$	0	0
$445$	−67.6820	−0.00720996
$446$	0	0
$447$	1747.88	0.184948
$448$	0	0
$449$	−5965.73	−0.627038	−0.313519	−	0.949582i	$-0.601508\pi$
$449$	−5965.73	−0.627038	−0.313519	+	0.949582i	$0.601508\pi$
$450$	0	0
$451$	14048.5	1.46678
$452$	0	0
$453$	8435.29	0.874889
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13860.6	−1.41875	−0.709376	−	0.704830i	$-0.751023\pi$
$457$	−13860.6	−1.41875	−0.709376	+	0.704830i	$0.751023\pi$
$458$	0	0
$459$	3284.27	0.333979
$460$	0	0
$461$	−149.312	−0.0150850	−0.00754249	−	0.999972i	$-0.502401\pi$
$461$	−149.312	−0.0150850	−0.00754249	+	0.999972i	$0.502401\pi$
$462$	0	0
$463$	−5403.95	−0.542425	−0.271213	−	0.962519i	$-0.587425\pi$
$463$	−5403.95	−0.542425	−0.271213	+	0.962519i	$0.587425\pi$
$464$	0	0
$465$	−5.64155	−0.000562625 0
$466$	0	0
$467$	−3704.66	−0.367090	−0.183545	−	0.983011i	$-0.558757\pi$
$467$	−3704.66	−0.367090	−0.183545	+	0.983011i	$0.558757\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5074.03	0.496389
$472$	0	0
$473$	9604.59	0.933657
$474$	0	0
$475$	15887.8	1.53470
$476$	0	0
$477$	5788.05	0.555591
$478$	0	0
$479$	10671.8	1.01796	0.508982	−	0.860777i	$-0.330022\pi$
$479$	10671.8	1.01796	0.508982	+	0.860777i	$0.330022\pi$
$480$	0	0
$481$	−3193.82	−0.302756
$482$	0	0
$483$	0	0
$484$	0	0
$485$	109.662	0.0102670
$486$	0	0
$487$	−5853.92	−0.544695	−0.272348	−	0.962199i	$-0.587800\pi$
$487$	−5853.92	−0.544695	−0.272348	+	0.962199i	$0.587800\pi$
$488$	0	0
$489$	122.170	0.0112980
$490$	0	0
$491$	−4065.31	−0.373656	−0.186828	−	0.982393i	$-0.559821\pi$
$491$	−4065.31	−0.373656	−0.186828	+	0.982393i	$0.559821\pi$
$492$	0	0
$493$	28668.2	2.61896
$494$	0	0
$495$	39.7468	0.00360906
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4811.19	−0.431620	−0.215810	−	0.976435i	$-0.569239\pi$
$499$	−4811.19	−0.431620	−0.215810	+	0.976435i	$0.569239\pi$
$500$	0	0
$501$	8701.40	0.775947
$502$	0	0
$503$	−17001.2	−1.50705	−0.753526	−	0.657418i	$-0.771649\pi$
$503$	−17001.2	−1.50705	−0.753526	+	0.657418i	$0.771649\pi$
$504$	0	0
$505$	137.771	0.0121401
$506$	0	0
$507$	−5759.87	−0.504546
$508$	0	0
$509$	13797.2	1.20148	0.600738	−	0.799446i	$-0.294874\pi$
$509$	13797.2	1.20148	0.600738	+	0.799446i	$0.294874\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3432.04	−0.295377
$514$	0	0
$515$	−142.110	−0.0121594
$516$	0	0
$517$	17644.7	1.50099
$518$	0	0
$519$	6438.44	0.544540
$520$	0	0
$521$	3936.61	0.331029	0.165515	−	0.986207i	$-0.447072\pi$
$521$	3936.61	0.331029	0.165515	+	0.986207i	$0.447072\pi$
$522$	0	0
$523$	−17459.3	−1.45973	−0.729866	−	0.683590i	$-0.760418\pi$
$523$	−17459.3	−1.45973	−0.729866	+	0.683590i	$0.760418\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2275.96	−0.188126
$528$	0	0
$529$	−9294.26	−0.763891
$530$	0	0
$531$	−104.510	−0.00854118
$532$	0	0
$533$	5321.51	0.432458
$534$	0	0
$535$	34.5274	0.00279019
$536$	0	0
$537$	−3610.62	−0.290148
$538$	0	0
$539$	0	0
$540$	0	0
$541$	19101.3	1.51798	0.758992	−	0.651100i	$-0.225692\pi$
$541$	19101.3	1.51798	0.758992	+	0.651100i	$0.225692\pi$
$542$	0	0
$543$	8971.41	0.709024
$544$	0	0
$545$	−31.9261	−0.00250929
$546$	0	0
$547$	−15413.5	−1.20481	−0.602407	−	0.798189i	$-0.705792\pi$
$547$	−15413.5	−1.20481	−0.602407	+	0.798189i	$0.705792\pi$
$548$	0	0
$549$	110.184	0.00856563
$550$	0	0
$551$	−29958.1	−2.31626
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−57.8554	−0.00442491
$556$	0	0
$557$	−20492.9	−1.55891	−0.779453	−	0.626460i	$-0.784503\pi$
$557$	−20492.9	−1.55891	−0.779453	+	0.626460i	$0.784503\pi$
$558$	0	0
$559$	3638.17	0.275274
$560$	0	0
$561$	16034.9	1.20677
$562$	0	0
$563$	7142.49	0.534671	0.267336	−	0.963603i	$-0.413857\pi$
$563$	7142.49	0.534671	0.267336	+	0.963603i	$0.413857\pi$
$564$	0	0
$565$	80.2406	0.00597477
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4097.91	0.301922	0.150961	−	0.988540i	$-0.451763\pi$
$569$	4097.91	0.301922	0.150961	+	0.988540i	$0.451763\pi$
$570$	0	0
$571$	2838.48	0.208033	0.104016	−	0.994576i	$-0.466831\pi$
$571$	2838.48	0.208033	0.104016	+	0.994576i	$0.466831\pi$
$572$	0	0
$573$	8422.23	0.614038
$574$	0	0
$575$	6699.21	0.485872
$576$	0	0
$577$	−15464.5	−1.11576	−0.557881	−	0.829921i	$-0.688385\pi$
$577$	−15464.5	−1.11576	−0.557881	+	0.829921i	$0.688385\pi$
$578$	0	0
$579$	10009.1	0.718418
$580$	0	0
$581$	0	0
$582$	0	0
$583$	28259.3	2.00751
$584$	0	0
$585$	15.0559	0.00106407
$586$	0	0
$587$	−14003.6	−0.984652	−0.492326	−	0.870411i	$-0.663853\pi$
$587$	−14003.6	−0.984652	−0.492326	+	0.870411i	$0.663853\pi$
$588$	0	0
$589$	2378.36	0.166382
$590$	0	0
$591$	−12679.9	−0.882543
$592$	0	0
$593$	6504.50	0.450435	0.225217	−	0.974309i	$-0.427691\pi$
$593$	6504.50	0.450435	0.225217	+	0.974309i	$0.427691\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13157.1	0.901982
$598$	0	0
$599$	12616.1	0.860567	0.430284	−	0.902694i	$-0.358414\pi$
$599$	12616.1	0.860567	0.430284	+	0.902694i	$0.358414\pi$
$600$	0	0
$601$	8270.87	0.561358	0.280679	−	0.959802i	$-0.409440\pi$
$601$	8270.87	0.561358	0.280679	+	0.959802i	$0.409440\pi$
$602$	0	0
$603$	−6021.43	−0.406653
$604$	0	0
$605$	60.2852	0.00405114
$606$	0	0
$607$	−3811.84	−0.254889	−0.127445	−	0.991846i	$-0.540677\pi$
$607$	−3811.84	−0.254889	−0.127445	+	0.991846i	$0.540677\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6683.72	0.442544
$612$	0	0
$613$	11359.3	0.748445	0.374222	−	0.927339i	$-0.377910\pi$
$613$	11359.3	0.748445	0.374222	+	0.927339i	$0.377910\pi$
$614$	0	0
$615$	96.3983	0.00632057
$616$	0	0
$617$	−18272.2	−1.19224	−0.596118	−	0.802896i	$-0.703291\pi$
$617$	−18272.2	−1.19224	−0.596118	+	0.802896i	$0.703291\pi$
$618$	0	0
$619$	−29600.2	−1.92203	−0.961013	−	0.276503i	$-0.910824\pi$
$619$	−29600.2	−1.92203	−0.961013	+	0.276503i	$0.910824\pi$
$620$	0	0
$621$	−1447.15	−0.0935136
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15621.2	0.999758
$626$	0	0
$627$	−16756.4	−1.06728
$628$	0	0
$629$	−23340.5	−1.47956
$630$	0	0
$631$	−7185.41	−0.453322	−0.226661	−	0.973974i	$-0.572781\pi$
$631$	−7185.41	−0.453322	−0.226661	+	0.973974i	$0.572781\pi$
$632$	0	0
$633$	−6874.67	−0.431665
$634$	0	0
$635$	−107.681	−0.00672944
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7398.88	−0.458052
$640$	0	0
$641$	−232.982	−0.0143561	−0.00717803	−	0.999974i	$-0.502285\pi$
$641$	−232.982	−0.0143561	−0.00717803	+	0.999974i	$0.502285\pi$
$642$	0	0
$643$	1837.96	0.112725	0.0563624	−	0.998410i	$-0.482050\pi$
$643$	1837.96	0.112725	0.0563624	+	0.998410i	$0.482050\pi$
$644$	0	0
$645$	65.9048	0.00402325
$646$	0	0
$647$	18594.7	1.12988	0.564941	−	0.825131i	$-0.308899\pi$
$647$	18594.7	1.12988	0.564941	+	0.825131i	$0.308899\pi$
$648$	0	0
$649$	−510.256	−0.0308618
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−28864.7	−1.72980	−0.864902	−	0.501940i	$-0.832620\pi$
$653$	−28864.7	−1.72980	−0.864902	+	0.501940i	$0.832620\pi$
$654$	0	0
$655$	−252.772	−0.0150788
$656$	0	0
$657$	−4635.90	−0.275287
$658$	0	0
$659$	29066.3	1.71815	0.859076	−	0.511847i	$-0.171039\pi$
$659$	29066.3	1.71815	0.859076	+	0.511847i	$0.171039\pi$
$660$	0	0
$661$	−3979.51	−0.234168	−0.117084	−	0.993122i	$-0.537355\pi$
$661$	−3979.51	−0.234168	−0.117084	+	0.993122i	$0.537355\pi$
$662$	0	0
$663$	6073.95	0.355796
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12632.0	−0.733305
$668$	0	0
$669$	−653.909	−0.0377901
$670$	0	0
$671$	537.955	0.0309501
$672$	0	0
$673$	−184.229	−0.0105520	−0.00527601	−	0.999986i	$-0.501679\pi$
$673$	−184.229	−0.0105520	−0.00527601	+	0.999986i	$0.501679\pi$
$674$	0	0
$675$	−3374.73	−0.192435
$676$	0	0
$677$	−16683.5	−0.947116	−0.473558	−	0.880763i	$-0.657031\pi$
$677$	−16683.5	−0.947116	−0.473558	+	0.880763i	$0.657031\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5506.08	0.309829
$682$	0	0
$683$	−17808.2	−0.997676	−0.498838	−	0.866695i	$-0.666240\pi$
$683$	−17808.2	−0.997676	−0.498838	+	0.866695i	$0.666240\pi$
$684$	0	0
$685$	25.2332	0.00140746
$686$	0	0
$687$	−8322.03	−0.462162
$688$	0	0
$689$	10704.5	0.591883
$690$	0	0
$691$	20145.7	1.10908	0.554542	−	0.832156i	$-0.312894\pi$
$691$	20145.7	1.10908	0.554542	+	0.832156i	$0.312894\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	89.0542	0.00486046
$696$	0	0
$697$	38889.7	2.11342
$698$	0	0
$699$	−2966.14	−0.160500
$700$	0	0
$701$	−2719.67	−0.146534	−0.0732672	−	0.997312i	$-0.523343\pi$
$701$	−2719.67	−0.146534	−0.0732672	+	0.997312i	$0.523343\pi$
$702$	0	0
$703$	24390.7	1.30855
$704$	0	0
$705$	121.074	0.00646798
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−625.708	−0.0331438	−0.0165719	−	0.999863i	$-0.505275\pi$
$709$	−625.708	−0.0331438	−0.0165719	+	0.999863i	$0.505275\pi$
$710$	0	0
$711$	7251.79	0.382508
$712$	0	0
$713$	1002.85	0.0526749
$714$	0	0
$715$	73.5079	0.00384481
$716$	0	0
$717$	2513.78	0.130933
$718$	0	0
$719$	9577.54	0.496776	0.248388	−	0.968661i	$-0.420099\pi$
$719$	9577.54	0.496776	0.248388	+	0.968661i	$0.420099\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	10365.0	0.533163
$724$	0	0
$725$	−29457.8	−1.50901
$726$	0	0
$727$	16741.2	0.854053	0.427027	−	0.904239i	$-0.359561\pi$
$727$	16741.2	0.854053	0.427027	+	0.904239i	$0.359561\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	26587.8	1.34526
$732$	0	0
$733$	−6496.04	−0.327335	−0.163668	−	0.986516i	$-0.552332\pi$
$733$	−6496.04	−0.327335	−0.163668	+	0.986516i	$0.552332\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−29398.7	−1.46936
$738$	0	0
$739$	−499.280	−0.0248529	−0.0124265	−	0.999923i	$-0.503956\pi$
$739$	−499.280	−0.0248529	−0.0124265	+	0.999923i	$0.503956\pi$
$740$	0	0
$741$	−6347.24	−0.314672
$742$	0	0
$743$	6367.30	0.314393	0.157196	−	0.987567i	$-0.449754\pi$
$743$	6367.30	0.314393	0.157196	+	0.987567i	$0.449754\pi$
$744$	0	0
$745$	58.5568	0.00287967
$746$	0	0
$747$	−3553.77	−0.174064
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−496.098	−0.0241050	−0.0120525	−	0.999927i	$-0.503837\pi$
$751$	−496.098	−0.0241050	−0.0120525	+	0.999927i	$0.503837\pi$
$752$	0	0
$753$	16905.0	0.818132
$754$	0	0
$755$	282.597	0.0136222
$756$	0	0
$757$	13025.9	0.625408	0.312704	−	0.949851i	$-0.398765\pi$
$757$	13025.9	0.625408	0.312704	+	0.949851i	$0.398765\pi$
$758$	0	0
$759$	−7065.47	−0.337892
$760$	0	0
$761$	−25474.6	−1.21347	−0.606736	−	0.794904i	$-0.707521\pi$
$761$	−25474.6	−1.21347	−0.606736	+	0.794904i	$0.707521\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	110.029	0.00520012
$766$	0	0
$767$	−193.282	−0.00909911
$768$	0	0
$769$	−29054.0	−1.36244	−0.681218	−	0.732080i	$-0.738550\pi$
$769$	−29054.0	−1.36244	−0.681218	+	0.732080i	$0.738550\pi$
$770$	0	0
$771$	6813.47	0.318263
$772$	0	0
$773$	1897.35	0.0882834	0.0441417	−	0.999025i	$-0.485945\pi$
$773$	1897.35	0.0882834	0.0441417	+	0.999025i	$0.485945\pi$
$774$	0	0
$775$	2338.65	0.108396
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−40639.6	−1.86914
$780$	0	0
$781$	−36123.9	−1.65508
$782$	0	0
$783$	6363.39	0.290433
$784$	0	0
$785$	169.989	0.00772886
$786$	0	0
$787$	−42650.3	−1.93179	−0.965895	−	0.258936i	$-0.916628\pi$
$787$	−42650.3	−1.93179	−0.965895	+	0.258936i	$0.916628\pi$
$788$	0	0
$789$	−491.602	−0.0221819
$790$	0	0
$791$	0	0
$792$	0	0
$793$	203.775	0.00912516
$794$	0	0
$795$	193.910	0.00865064
$796$	0	0
$797$	36822.8	1.63655	0.818275	−	0.574828i	$-0.194931\pi$
$797$	36822.8	1.63655	0.818275	+	0.574828i	$0.194931\pi$
$798$	0	0
$799$	48844.8	2.16271
$800$	0	0
$801$	−6060.77	−0.267349
$802$	0	0
$803$	−22634.1	−0.994693
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15501.3	0.676172
$808$	0	0
$809$	5081.94	0.220855	0.110427	−	0.993884i	$-0.464778\pi$
$809$	5081.94	0.220855	0.110427	+	0.993884i	$0.464778\pi$
$810$	0	0
$811$	−11873.4	−0.514097	−0.257048	−	0.966399i	$-0.582750\pi$
$811$	−11873.4	−0.514097	−0.257048	+	0.966399i	$0.582750\pi$
$812$	0	0
$813$	−4866.82	−0.209947
$814$	0	0
$815$	4.09289	0.000175911 0
$816$	0	0
$817$	−27784.1	−1.18977
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16969.4	0.721361	0.360681	−	0.932689i	$-0.382544\pi$
$821$	16969.4	0.721361	0.360681	+	0.932689i	$0.382544\pi$
$822$	0	0
$823$	−3995.94	−0.169246	−0.0846231	−	0.996413i	$-0.526969\pi$
$823$	−3995.94	−0.169246	−0.0846231	+	0.996413i	$0.526969\pi$
$824$	0	0
$825$	−16476.6	−0.695323
$826$	0	0
$827$	13589.7	0.571417	0.285708	−	0.958317i	$-0.407771\pi$
$827$	13589.7	0.571417	0.285708	+	0.958317i	$0.407771\pi$
$828$	0	0
$829$	−30646.0	−1.28393	−0.641966	−	0.766733i	$-0.721881\pi$
$829$	−30646.0	−1.28393	−0.641966	+	0.766733i	$0.721881\pi$
$830$	0	0
$831$	−13837.1	−0.577622
$832$	0	0
$833$	0	0
$834$	0	0
$835$	291.511	0.0120816
$836$	0	0
$837$	−505.188	−0.0208624
$838$	0	0
$839$	7497.57	0.308516	0.154258	−	0.988031i	$-0.450701\pi$
$839$	7497.57	0.308516	0.154258	+	0.988031i	$0.450701\pi$
$840$	0	0
$841$	31156.6	1.27749
$842$	0	0
$843$	−6375.65	−0.260485
$844$	0	0
$845$	−192.965	−0.00785586
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7714.62	0.311855
$850$	0	0
$851$	10284.5	0.414275
$852$	0	0
$853$	−10347.6	−0.415352	−0.207676	−	0.978198i	$-0.566590\pi$
$853$	−10347.6	−0.415352	−0.207676	+	0.978198i	$0.566590\pi$
$854$	0	0
$855$	−114.979	−0.00459907
$856$	0	0
$857$	1550.00	0.0617817	0.0308909	−	0.999523i	$-0.490166\pi$
$857$	1550.00	0.0617817	0.0308909	+	0.999523i	$0.490166\pi$
$858$	0	0
$859$	−15187.5	−0.603250	−0.301625	−	0.953427i	$-0.597529\pi$
$859$	−15187.5	−0.603250	−0.301625	+	0.953427i	$0.597529\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41550.2	−1.63892	−0.819459	−	0.573138i	$-0.805726\pi$
$863$	−41550.2	−1.63892	−0.819459	+	0.573138i	$0.805726\pi$
$864$	0	0
$865$	215.699	0.00847858
$866$	0	0
$867$	29649.6	1.16142
$868$	0	0
$869$	35405.8	1.38212
$870$	0	0
$871$	−11136.1	−0.433216
$872$	0	0
$873$	9820.03	0.380707
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27837.4	1.07184	0.535919	−	0.844269i	$-0.319965\pi$
$877$	27837.4	1.07184	0.535919	+	0.844269i	$0.319965\pi$
$878$	0	0
$879$	9974.89	0.382758
$880$	0	0
$881$	−2587.85	−0.0989635	−0.0494817	−	0.998775i	$-0.515757\pi$
$881$	−2587.85	−0.0989635	−0.0494817	+	0.998775i	$0.515757\pi$
$882$	0	0
$883$	16382.0	0.624346	0.312173	−	0.950025i	$-0.398943\pi$
$883$	16382.0	0.624346	0.312173	+	0.950025i	$0.398943\pi$
$884$	0	0
$885$	−3.50127	−0.000132988 0
$886$	0	0
$887$	22980.2	0.869896	0.434948	−	0.900456i	$-0.356767\pi$
$887$	22980.2	0.869896	0.434948	+	0.900456i	$0.356767\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3559.23	0.133826
$892$	0	0
$893$	−51042.5	−1.91274
$894$	0	0
$895$	−120.962	−0.00451766
$896$	0	0
$897$	−2676.36	−0.0996222
$898$	0	0
$899$	−4409.76	−0.163597
$900$	0	0
$901$	78228.5	2.89253
$902$	0	0
$903$	0	0
$904$	0	0
$905$	300.557	0.0110396
$906$	0	0
$907$	22715.4	0.831590	0.415795	−	0.909458i	$-0.363503\pi$
$907$	22715.4	0.831590	0.415795	+	0.909458i	$0.363503\pi$
$908$	0	0
$909$	12337.1	0.450161
$910$	0	0
$911$	−34922.3	−1.27006	−0.635031	−	0.772487i	$-0.719013\pi$
$911$	−34922.3	−1.27006	−0.635031	+	0.772487i	$0.719013\pi$
$912$	0	0
$913$	−17350.7	−0.628943
$914$	0	0
$915$	3.69134	0.000133368 0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−11702.6	−0.420059	−0.210030	−	0.977695i	$-0.567356\pi$
$919$	−11702.6	−0.420059	−0.210030	+	0.977695i	$0.567356\pi$
$920$	0	0
$921$	2661.29	0.0952144
$922$	0	0
$923$	−13683.5	−0.487973
$924$	0	0
$925$	23983.3	0.852505
$926$	0	0
$927$	−12725.6	−0.450878
$928$	0	0
$929$	−53096.5	−1.87518	−0.937588	−	0.347747i	$-0.886947\pi$
$929$	−53096.5	−1.87518	−0.937588	+	0.347747i	$0.886947\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	13532.5	0.474848
$934$	0	0
$935$	537.198	0.0187896
$936$	0	0
$937$	−39020.6	−1.36046	−0.680229	−	0.733000i	$-0.738120\pi$
$937$	−39020.6	−1.36046	−0.680229	+	0.733000i	$0.738120\pi$
$938$	0	0
$939$	11147.4	0.387412
$940$	0	0
$941$	30743.8	1.06506	0.532528	−	0.846412i	$-0.321242\pi$
$941$	30743.8	1.06506	0.532528	+	0.846412i	$0.321242\pi$
$942$	0	0
$943$	−17136.0	−0.591754
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16300.3	−0.559332	−0.279666	−	0.960097i	$-0.590224\pi$
$947$	−16300.3	−0.559332	−0.279666	+	0.960097i	$0.590224\pi$
$948$	0	0
$949$	−8573.66	−0.293269
$950$	0	0
$951$	20863.6	0.711406
$952$	0	0
$953$	−11512.7	−0.391325	−0.195663	−	0.980671i	$-0.562686\pi$
$953$	−11512.7	−0.391325	−0.195663	+	0.980671i	$0.562686\pi$
$954$	0	0
$955$	282.159	0.00956068
$956$	0	0
$957$	31068.3	1.04942
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29440.9	−0.988248
$962$	0	0
$963$	3091.85	0.103462
$964$	0	0
$965$	335.322	0.0111859
$966$	0	0
$967$	18178.4	0.604528	0.302264	−	0.953224i	$-0.402258\pi$
$967$	18178.4	0.604528	0.302264	+	0.953224i	$0.402258\pi$
$968$	0	0
$969$	−46385.8	−1.53780
$970$	0	0
$971$	28276.7	0.934543	0.467271	−	0.884114i	$-0.345237\pi$
$971$	28276.7	0.934543	0.467271	+	0.884114i	$0.345237\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−6241.24	−0.205005
$976$	0	0
$977$	−13947.1	−0.456710	−0.228355	−	0.973578i	$-0.573335\pi$
$977$	−13947.1	−0.456710	−0.228355	+	0.973578i	$0.573335\pi$
$978$	0	0
$979$	−29590.8	−0.966011
$980$	0	0
$981$	−2858.91	−0.0930459
$982$	0	0
$983$	−26576.8	−0.862327	−0.431164	−	0.902274i	$-0.641897\pi$
$983$	−26576.8	−0.862327	−0.431164	+	0.902274i	$0.641897\pi$
$984$	0	0
$985$	−424.799	−0.0137414
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11715.4	−0.376671
$990$	0	0
$991$	−16249.9	−0.520884	−0.260442	−	0.965490i	$-0.583868\pi$
$991$	−16249.9	−0.520884	−0.260442	+	0.965490i	$0.583868\pi$
$992$	0	0
$993$	29589.5	0.945615
$994$	0	0
$995$	440.784	0.0140440
$996$	0	0
$997$	−18814.1	−0.597642	−0.298821	−	0.954309i	$-0.596593\pi$
$997$	−18814.1	−0.597642	−0.298821	+	0.954309i	$0.596593\pi$
$998$	0	0
$999$	−5180.82	−0.164078

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cf.1.1		2
4.3	odd	2		147.4.a.j.1.1	✓	2
7.6	odd	2		2352.4.a.bl.1.2		2
12.11	even	2		441.4.a.n.1.2		2
28.3	even	6		147.4.e.j.79.2		4
28.11	odd	6		147.4.e.k.79.2		4
28.19	even	6		147.4.e.j.67.2		4
28.23	odd	6		147.4.e.k.67.2		4
28.27	even	2		147.4.a.k.1.1	yes	2
84.11	even	6		441.4.e.v.226.1		4
84.23	even	6		441.4.e.v.361.1		4
84.47	odd	6		441.4.e.u.361.1		4
84.59	odd	6		441.4.e.u.226.1		4
84.83	odd	2		441.4.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.a.j.1.1	✓	2	4.3	odd	2
147.4.a.k.1.1	yes	2	28.27	even	2
147.4.e.j.67.2		4	28.19	even	6
147.4.e.j.79.2		4	28.3	even	6
147.4.e.k.67.2		4	28.23	odd	6
147.4.e.k.79.2		4	28.11	odd	6
441.4.a.n.1.2		2	12.11	even	2
441.4.a.o.1.2		2	84.83	odd	2
441.4.e.u.226.1		4	84.59	odd	6
441.4.e.u.361.1		4	84.47	odd	6
441.4.e.v.226.1		4	84.11	even	6
441.4.e.v.361.1		4	84.23	even	6
2352.4.a.bl.1.2		2	7.6	odd	2
2352.4.a.cf.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +0.100505 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +0.100505 q^{5} +9.00000 q^{9} +43.9411 q^{11} +16.6447 q^{13} +0.301515 q^{15} +121.640 q^{17} -127.113 q^{19} -53.5980 q^{23} -124.990 q^{25} +27.0000 q^{27} +235.681 q^{29} -18.7107 q^{31} +131.823 q^{33} -191.882 q^{37} +49.9340 q^{39} +319.713 q^{41} +218.579 q^{43} +0.904546 q^{45} +401.553 q^{47} +364.919 q^{51} +643.117 q^{53} +4.41631 q^{55} -381.338 q^{57} -11.6123 q^{59} +12.2426 q^{61} +1.67287 q^{65} -669.048 q^{67} -160.794 q^{69} -822.098 q^{71} -515.100 q^{73} -374.970 q^{75} +805.754 q^{79} +81.0000 q^{81} -394.863 q^{83} +12.2254 q^{85} +707.044 q^{87} -673.418 q^{89} -56.1320 q^{93} -12.7755 q^{95} +1091.11 q^{97} +395.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 20 q^{5} + 18 q^{9}+O(q^{10}) \) 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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