LMFDB - Embedded newform 2352.4.a.bn.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:	$1$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-3.00000 q^{3} +3.89949 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +3.89949 q^{5} +9.00000 q^{9} +61.3970 q^{11} -53.6985 q^{13} -11.6985 q^{15} -32.1005 q^{17} -55.7990 q^{19} +94.6030 q^{23} -109.794 q^{25} -27.0000 q^{27} +138.191 q^{29} +132.603 q^{31} -184.191 q^{33} +149.206 q^{37} +161.095 q^{39} -427.497 q^{41} -437.588 q^{43} +35.0955 q^{45} +57.0051 q^{47} +96.3015 q^{51} -263.588 q^{53} +239.417 q^{55} +167.397 q^{57} +451.799 q^{59} -579.307 q^{61} -209.397 q^{65} -309.588 q^{67} -283.809 q^{69} +1058.98 q^{71} -1193.66 q^{73} +329.382 q^{75} -1319.56 q^{79} +81.0000 q^{81} +1190.33 q^{83} -125.176 q^{85} -414.573 q^{87} +233.085 q^{89} -397.809 q^{93} -217.588 q^{95} +1609.44 q^{97} +552.573 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 12 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} - 12 q^{5} + 18 q^{9} + 4 q^{11} - 48 q^{13} + 36 q^{15} - 84 q^{17} - 72 q^{19} + 308 q^{23} + 18 q^{25} - 54 q^{27} - 80 q^{29} + 384 q^{31} - 12 q^{33} + 536 q^{37} + 144 q^{39} - 756 q^{41} - 400 q^{43} - 108 q^{45} + 312 q^{47} + 252 q^{51} - 52 q^{53} + 1152 q^{55} + 216 q^{57} + 864 q^{59} - 1416 q^{61} - 300 q^{65} - 144 q^{67} - 924 q^{69} + 1524 q^{71} - 744 q^{73} - 54 q^{75} - 976 q^{79} + 162 q^{81} - 312 q^{83} + 700 q^{85} + 240 q^{87} - 108 q^{89} - 1152 q^{93} + 40 q^{95} + 744 q^{97} + 36 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 12 * q^5 + 18 * q^9 + 4 * q^11 - 48 * q^13 + 36 * q^15 - 84 * q^17 - 72 * q^19 + 308 * q^23 + 18 * q^25 - 54 * q^27 - 80 * q^29 + 384 * q^31 - 12 * q^33 + 536 * q^37 + 144 * q^39 - 756 * q^41 - 400 * q^43 - 108 * q^45 + 312 * q^47 + 252 * q^51 - 52 * q^53 + 1152 * q^55 + 216 * q^57 + 864 * q^59 - 1416 * q^61 - 300 * q^65 - 144 * q^67 - 924 * q^69 + 1524 * q^71 - 744 * q^73 - 54 * q^75 - 976 * q^79 + 162 * q^81 - 312 * q^83 + 700 * q^85 + 240 * q^87 - 108 * q^89 - 1152 * q^93 + 40 * q^95 + 744 * q^97 + 36 * 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
$5$	3.89949	0.348781	0.174391	−	0.984677i	$-0.444204\pi$
$5$	3.89949	0.348781	0.174391	+	0.984677i	$0.444204\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	61.3970	1.68290	0.841449	−	0.540336i	$-0.181703\pi$
$11$	61.3970	1.68290	0.841449	+	0.540336i	$0.181703\pi$
$12$	0	0
$13$	−53.6985	−1.14564	−0.572818	−	0.819682i	$-0.694150\pi$
$13$	−53.6985	−1.14564	−0.572818	+	0.819682i	$0.694150\pi$
$14$	0	0
$15$	−11.6985	−0.201369
$16$	0	0
$17$	−32.1005	−0.457972	−0.228986	−	0.973430i	$-0.573541\pi$
$17$	−32.1005	−0.457972	−0.228986	+	0.973430i	$0.573541\pi$
$18$	0	0
$19$	−55.7990	−0.673746	−0.336873	−	0.941550i	$-0.609369\pi$
$19$	−55.7990	−0.673746	−0.336873	+	0.941550i	$0.609369\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	94.6030	0.857656	0.428828	−	0.903386i	$-0.358927\pi$
$23$	94.6030	0.857656	0.428828	+	0.903386i	$0.358927\pi$
$24$	0	0
$25$	−109.794	−0.878352
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	138.191	0.884876	0.442438	−	0.896799i	$-0.354114\pi$
$29$	138.191	0.884876	0.442438	+	0.896799i	$0.354114\pi$
$30$	0	0
$31$	132.603	0.768265	0.384132	−	0.923278i	$-0.374501\pi$
$31$	132.603	0.768265	0.384132	+	0.923278i	$0.374501\pi$
$32$	0	0
$33$	−184.191	−0.971622
$34$	0	0
$35$	0	0
$36$	0	0
$37$	149.206	0.662955	0.331477	−	0.943463i	$-0.392453\pi$
$37$	149.206	0.662955	0.331477	+	0.943463i	$0.392453\pi$
$38$	0	0
$39$	161.095	0.661434
$40$	0	0
$41$	−427.497	−1.62839	−0.814194	−	0.580593i	$-0.802821\pi$
$41$	−427.497	−1.62839	−0.814194	+	0.580593i	$0.802821\pi$
$42$	0	0
$43$	−437.588	−1.55190	−0.775948	−	0.630797i	$-0.782728\pi$
$43$	−437.588	−1.55190	−0.775948	+	0.630797i	$0.782728\pi$
$44$	0	0
$45$	35.0955	0.116260
$46$	0	0
$47$	57.0051	0.176916	0.0884579	−	0.996080i	$-0.471806\pi$
$47$	57.0051	0.176916	0.0884579	+	0.996080i	$0.471806\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	96.3015	0.264410
$52$	0	0
$53$	−263.588	−0.683143	−0.341572	−	0.939856i	$-0.610959\pi$
$53$	−263.588	−0.683143	−0.341572	+	0.939856i	$0.610959\pi$
$54$	0	0
$55$	239.417	0.586964
$56$	0	0
$57$	167.397	0.388987
$58$	0	0
$59$	451.799	0.996936	0.498468	−	0.866908i	$-0.333896\pi$
$59$	451.799	0.996936	0.498468	+	0.866908i	$0.333896\pi$
$60$	0	0
$61$	−579.307	−1.21594	−0.607972	−	0.793958i	$-0.708017\pi$
$61$	−579.307	−1.21594	−0.607972	+	0.793958i	$0.708017\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−209.397	−0.399577
$66$	0	0
$67$	−309.588	−0.564510	−0.282255	−	0.959339i	$-0.591082\pi$
$67$	−309.588	−0.564510	−0.282255	+	0.959339i	$0.591082\pi$
$68$	0	0
$69$	−283.809	−0.495168
$70$	0	0
$71$	1058.98	1.77012	0.885059	−	0.465479i	$-0.154118\pi$
$71$	1058.98	1.77012	0.885059	+	0.465479i	$0.154118\pi$
$72$	0	0
$73$	−1193.66	−1.91380	−0.956898	−	0.290424i	$-0.906204\pi$
$73$	−1193.66	−1.91380	−0.956898	+	0.290424i	$0.906204\pi$
$74$	0	0
$75$	329.382	0.507116
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1319.56	−1.87926	−0.939632	−	0.342187i	$-0.888832\pi$
$79$	−1319.56	−1.87926	−0.939632	+	0.342187i	$0.888832\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1190.33	1.57417	0.787083	−	0.616847i	$-0.211590\pi$
$83$	1190.33	1.57417	0.787083	+	0.616847i	$0.211590\pi$
$84$	0	0
$85$	−125.176	−0.159732
$86$	0	0
$87$	−414.573	−0.510883
$88$	0	0
$89$	233.085	0.277607	0.138803	−	0.990320i	$-0.455674\pi$
$89$	233.085	0.277607	0.138803	+	0.990320i	$0.455674\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−397.809	−0.443558
$94$	0	0
$95$	−217.588	−0.234990
$96$	0	0
$97$	1609.44	1.68468	0.842338	−	0.538950i	$-0.181179\pi$
$97$	1609.44	1.68468	0.842338	+	0.538950i	$0.181179\pi$
$98$	0	0
$99$	552.573	0.560966
$100$	0	0
$101$	1479.26	1.45734	0.728671	−	0.684864i	$-0.240139\pi$
$101$	1479.26	1.45734	0.728671	+	0.684864i	$0.240139\pi$
$102$	0	0
$103$	−1145.35	−1.09567	−0.547837	−	0.836585i	$-0.684548\pi$
$103$	−1145.35	−1.09567	−0.547837	+	0.836585i	$0.684548\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−436.955	−0.394785	−0.197392	−	0.980325i	$-0.563247\pi$
$107$	−436.955	−0.394785	−0.197392	+	0.980325i	$0.563247\pi$
$108$	0	0
$109$	−166.352	−0.146180	−0.0730898	−	0.997325i	$-0.523286\pi$
$109$	−166.352	−0.146180	−0.0730898	+	0.997325i	$0.523286\pi$
$110$	0	0
$111$	−447.618	−0.382757
$112$	0	0
$113$	490.824	0.408609	0.204305	−	0.978907i	$-0.434507\pi$
$113$	490.824	0.408609	0.204305	+	0.978907i	$0.434507\pi$
$114$	0	0
$115$	368.904	0.299135
$116$	0	0
$117$	−483.286	−0.381879
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2438.59	1.83215
$122$	0	0
$123$	1282.49	0.940150
$124$	0	0
$125$	−915.578	−0.655134
$126$	0	0
$127$	2616.70	1.82831	0.914153	−	0.405369i	$-0.132857\pi$
$127$	2616.70	1.82831	0.914153	+	0.405369i	$0.132857\pi$
$128$	0	0
$129$	1312.76	0.895988
$130$	0	0
$131$	177.588	0.118442	0.0592211	−	0.998245i	$-0.481138\pi$
$131$	177.588	0.118442	0.0592211	+	0.998245i	$0.481138\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−105.286	−0.0671230
$136$	0	0
$137$	27.0152	0.0168472	0.00842358	−	0.999965i	$-0.497319\pi$
$137$	27.0152	0.0168472	0.00842358	+	0.999965i	$0.497319\pi$
$138$	0	0
$139$	−922.754	−0.563071	−0.281536	−	0.959551i	$-0.590844\pi$
$139$	−922.754	−0.563071	−0.281536	+	0.959551i	$0.590844\pi$
$140$	0	0
$141$	−171.015	−0.102142
$142$	0	0
$143$	−3296.92	−1.92799
$144$	0	0
$145$	538.875	0.308628
$146$	0	0
$147$	0	0
$148$	0	0
$149$	746.703	0.410552	0.205276	−	0.978704i	$-0.434191\pi$
$149$	746.703	0.410552	0.205276	+	0.978704i	$0.434191\pi$
$150$	0	0
$151$	−2073.15	−1.11729	−0.558643	−	0.829408i	$-0.688678\pi$
$151$	−2073.15	−1.11729	−0.558643	+	0.829408i	$0.688678\pi$
$152$	0	0
$153$	−288.905	−0.152657
$154$	0	0
$155$	517.085	0.267956
$156$	0	0
$157$	−1566.22	−0.796166	−0.398083	−	0.917349i	$-0.630324\pi$
$157$	−1566.22	−0.796166	−0.398083	+	0.917349i	$0.630324\pi$
$158$	0	0
$159$	790.764	0.394413
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−98.7333	−0.0474441	−0.0237221	−	0.999719i	$-0.507552\pi$
$163$	−98.7333	−0.0474441	−0.0237221	+	0.999719i	$0.507552\pi$
$164$	0	0
$165$	−718.252	−0.338884
$166$	0	0
$167$	−2231.36	−1.03394	−0.516969	−	0.856004i	$-0.672940\pi$
$167$	−2231.36	−1.03394	−0.516969	+	0.856004i	$0.672940\pi$
$168$	0	0
$169$	686.527	0.312484
$170$	0	0
$171$	−502.191	−0.224582
$172$	0	0
$173$	−2100.84	−0.923261	−0.461631	−	0.887072i	$-0.652735\pi$
$173$	−2100.84	−0.923261	−0.461631	+	0.887072i	$0.652735\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1355.40	−0.575581
$178$	0	0
$179$	−1722.54	−0.719267	−0.359634	−	0.933094i	$-0.617098\pi$
$179$	−1722.54	−0.719267	−0.359634	+	0.933094i	$0.617098\pi$
$180$	0	0
$181$	−1655.00	−0.679644	−0.339822	−	0.940490i	$-0.610367\pi$
$181$	−1655.00	−0.679644	−0.339822	+	0.940490i	$0.610367\pi$
$182$	0	0
$183$	1737.92	0.702026
$184$	0	0
$185$	581.828	0.231226
$186$	0	0
$187$	−1970.87	−0.770720
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1007.69	0.381747	0.190874	−	0.981615i	$-0.438868\pi$
$191$	1007.69	0.381747	0.190874	+	0.981615i	$0.438868\pi$
$192$	0	0
$193$	−7.64849	−0.00285259	−0.00142630	−	0.999999i	$-0.500454\pi$
$193$	−7.64849	−0.00285259	−0.00142630	+	0.999999i	$0.500454\pi$
$194$	0	0
$195$	628.191	0.230696
$196$	0	0
$197$	2689.88	0.972822	0.486411	−	0.873730i	$-0.338306\pi$
$197$	2689.88	0.972822	0.486411	+	0.873730i	$0.338306\pi$
$198$	0	0
$199$	867.497	0.309021	0.154511	−	0.987991i	$-0.450620\pi$
$199$	867.497	0.309021	0.154511	+	0.987991i	$0.450620\pi$
$200$	0	0
$201$	928.764	0.325920
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1667.02	−0.567951
$206$	0	0
$207$	851.427	0.285885
$208$	0	0
$209$	−3425.89	−1.13385
$210$	0	0
$211$	−162.030	−0.0528655	−0.0264328	−	0.999651i	$-0.508415\pi$
$211$	−162.030	−0.0528655	−0.0264328	+	0.999651i	$0.508415\pi$
$212$	0	0
$213$	−3176.95	−1.02198
$214$	0	0
$215$	−1706.37	−0.541272
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3580.97	1.10493
$220$	0	0
$221$	1723.75	0.524669
$222$	0	0
$223$	−4577.85	−1.37469	−0.687344	−	0.726332i	$-0.741223\pi$
$223$	−4577.85	−1.37469	−0.687344	+	0.726332i	$0.741223\pi$
$224$	0	0
$225$	−988.145	−0.292784
$226$	0	0
$227$	−2218.19	−0.648575	−0.324287	−	0.945959i	$-0.605125\pi$
$227$	−2218.19	−0.648575	−0.324287	+	0.945959i	$0.605125\pi$
$228$	0	0
$229$	−785.217	−0.226588	−0.113294	−	0.993562i	$-0.536140\pi$
$229$	−785.217	−0.226588	−0.113294	+	0.993562i	$0.536140\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5369.12	−1.50963	−0.754813	−	0.655940i	$-0.772273\pi$
$233$	−5369.12	−1.50963	−0.754813	+	0.655940i	$0.772273\pi$
$234$	0	0
$235$	222.291	0.0617049
$236$	0	0
$237$	3958.67	1.08499
$238$	0	0
$239$	−3713.28	−1.00499	−0.502493	−	0.864581i	$-0.667584\pi$
$239$	−3713.28	−1.00499	−0.502493	+	0.864581i	$0.667584\pi$
$240$	0	0
$241$	−6998.62	−1.87063	−0.935313	−	0.353821i	$-0.884882\pi$
$241$	−6998.62	−1.87063	−0.935313	+	0.353821i	$0.884882\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2996.32	0.771868
$248$	0	0
$249$	−3570.99	−0.908846
$250$	0	0
$251$	3722.75	0.936168	0.468084	−	0.883684i	$-0.344945\pi$
$251$	3722.75	0.936168	0.468084	+	0.883684i	$0.344945\pi$
$252$	0	0
$253$	5808.34	1.44335
$254$	0	0
$255$	375.527	0.0922213
$256$	0	0
$257$	−1230.15	−0.298578	−0.149289	−	0.988794i	$-0.547699\pi$
$257$	−1230.15	−0.298578	−0.149289	+	0.988794i	$0.547699\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1243.72	0.294959
$262$	0	0
$263$	−2388.63	−0.560036	−0.280018	−	0.959995i	$-0.590340\pi$
$263$	−2388.63	−0.560036	−0.280018	+	0.959995i	$0.590340\pi$
$264$	0	0
$265$	−1027.86	−0.238268
$266$	0	0
$267$	−699.256	−0.160276
$268$	0	0
$269$	−6702.31	−1.51913	−0.759567	−	0.650429i	$-0.774589\pi$
$269$	−6702.31	−1.51913	−0.759567	+	0.650429i	$0.774589\pi$
$270$	0	0
$271$	4950.37	1.10964	0.554822	−	0.831969i	$-0.312786\pi$
$271$	4950.37	1.10964	0.554822	+	0.831969i	$0.312786\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6741.02	−1.47818
$276$	0	0
$277$	−3705.18	−0.803691	−0.401846	−	0.915707i	$-0.631631\pi$
$277$	−3705.18	−0.803691	−0.401846	+	0.915707i	$0.631631\pi$
$278$	0	0
$279$	1193.43	0.256088
$280$	0	0
$281$	9324.74	1.97960	0.989800	−	0.142465i	$-0.0455029\pi$
$281$	9324.74	1.97960	0.989800	+	0.142465i	$0.0455029\pi$
$282$	0	0
$283$	−5569.49	−1.16986	−0.584932	−	0.811082i	$-0.698879\pi$
$283$	−5569.49	−1.16986	−0.584932	+	0.811082i	$0.698879\pi$
$284$	0	0
$285$	652.764	0.135672
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3882.56	−0.790262
$290$	0	0
$291$	−4828.31	−0.972648
$292$	0	0
$293$	1665.31	0.332042	0.166021	−	0.986122i	$-0.446908\pi$
$293$	1665.31	0.332042	0.166021	+	0.986122i	$0.446908\pi$
$294$	0	0
$295$	1761.79	0.347713
$296$	0	0
$297$	−1657.72	−0.323874
$298$	0	0
$299$	−5080.04	−0.982563
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4437.77	−0.841396
$304$	0	0
$305$	−2259.00	−0.424099
$306$	0	0
$307$	−5303.32	−0.985916	−0.492958	−	0.870053i	$-0.664084\pi$
$307$	−5303.32	−0.985916	−0.492958	+	0.870053i	$0.664084\pi$
$308$	0	0
$309$	3436.04	0.632587
$310$	0	0
$311$	1125.99	0.205302	0.102651	−	0.994717i	$-0.467267\pi$
$311$	1125.99	0.205302	0.102651	+	0.994717i	$0.467267\pi$
$312$	0	0
$313$	8299.51	1.49877	0.749386	−	0.662133i	$-0.230349\pi$
$313$	8299.51	1.49877	0.749386	+	0.662133i	$0.230349\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4278.76	−0.758105	−0.379053	−	0.925375i	$-0.623750\pi$
$317$	−4278.76	−0.758105	−0.379053	+	0.925375i	$0.623750\pi$
$318$	0	0
$319$	8484.50	1.48916
$320$	0	0
$321$	1310.86	0.227929
$322$	0	0
$323$	1791.18	0.308556
$324$	0	0
$325$	5895.77	1.00627
$326$	0	0
$327$	499.055	0.0843969
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1707.12	0.283479	0.141739	−	0.989904i	$-0.454730\pi$
$331$	1707.12	0.283479	0.141739	+	0.989904i	$0.454730\pi$
$332$	0	0
$333$	1342.85	0.220985
$334$	0	0
$335$	−1207.24	−0.196891
$336$	0	0
$337$	1710.67	0.276517	0.138259	−	0.990396i	$-0.455849\pi$
$337$	1710.67	0.276517	0.138259	+	0.990396i	$0.455849\pi$
$338$	0	0
$339$	−1472.47	−0.235911
$340$	0	0
$341$	8141.42	1.29291
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1106.71	−0.172705
$346$	0	0
$347$	−8910.30	−1.37847	−0.689236	−	0.724537i	$-0.742054\pi$
$347$	−8910.30	−1.37847	−0.689236	+	0.724537i	$0.742054\pi$
$348$	0	0
$349$	−5378.68	−0.824969	−0.412485	−	0.910965i	$-0.635339\pi$
$349$	−5378.68	−0.824969	−0.412485	+	0.910965i	$0.635339\pi$
$350$	0	0
$351$	1449.86	0.220478
$352$	0	0
$353$	4252.56	0.641193	0.320596	−	0.947216i	$-0.396117\pi$
$353$	4252.56	0.641193	0.320596	+	0.947216i	$0.396117\pi$
$354$	0	0
$355$	4129.51	0.617384
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4903.89	−0.720940	−0.360470	−	0.932771i	$-0.617384\pi$
$359$	−4903.89	−0.720940	−0.360470	+	0.932771i	$0.617384\pi$
$360$	0	0
$361$	−3745.47	−0.546067
$362$	0	0
$363$	−7315.76	−1.05779
$364$	0	0
$365$	−4654.66	−0.667497
$366$	0	0
$367$	−4041.57	−0.574845	−0.287423	−	0.957804i	$-0.592798\pi$
$367$	−4041.57	−0.574845	−0.287423	+	0.957804i	$0.592798\pi$
$368$	0	0
$369$	−3847.48	−0.542796
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7451.35	−1.03436	−0.517180	−	0.855877i	$-0.673018\pi$
$373$	−7451.35	−1.03436	−0.517180	+	0.855877i	$0.673018\pi$
$374$	0	0
$375$	2746.73	0.378242
$376$	0	0
$377$	−7420.64	−1.01375
$378$	0	0
$379$	12564.4	1.70288	0.851438	−	0.524456i	$-0.175731\pi$
$379$	12564.4	1.70288	0.851438	+	0.524456i	$0.175731\pi$
$380$	0	0
$381$	−7850.11	−1.05557
$382$	0	0
$383$	−4289.93	−0.572337	−0.286169	−	0.958179i	$-0.592382\pi$
$383$	−4289.93	−0.572337	−0.286169	+	0.958179i	$0.592382\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3938.29	−0.517299
$388$	0	0
$389$	5662.77	0.738082	0.369041	−	0.929413i	$-0.379686\pi$
$389$	5662.77	0.738082	0.369041	+	0.929413i	$0.379686\pi$
$390$	0	0
$391$	−3036.81	−0.392782
$392$	0	0
$393$	−532.764	−0.0683826
$394$	0	0
$395$	−5145.61	−0.655452
$396$	0	0
$397$	−14561.4	−1.84084	−0.920421	−	0.390928i	$-0.872154\pi$
$397$	−14561.4	−1.84084	−0.920421	+	0.390928i	$0.872154\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3742.19	−0.466025	−0.233013	−	0.972474i	$-0.574858\pi$
$401$	−3742.19	−0.466025	−0.233013	+	0.972474i	$0.574858\pi$
$402$	0	0
$403$	−7120.58	−0.880152
$404$	0	0
$405$	315.859	0.0387535
$406$	0	0
$407$	9160.80	1.11569
$408$	0	0
$409$	3517.17	0.425215	0.212608	−	0.977138i	$-0.431804\pi$
$409$	3517.17	0.425215	0.212608	+	0.977138i	$0.431804\pi$
$410$	0	0
$411$	−81.0455	−0.00972671
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4641.69	0.549040
$416$	0	0
$417$	2768.26	0.325089
$418$	0	0
$419$	−7579.52	−0.883732	−0.441866	−	0.897081i	$-0.645683\pi$
$419$	−7579.52	−0.883732	−0.441866	+	0.897081i	$0.645683\pi$
$420$	0	0
$421$	−4980.87	−0.576610	−0.288305	−	0.957539i	$-0.593092\pi$
$421$	−4980.87	−0.576610	−0.288305	+	0.957539i	$0.593092\pi$
$422$	0	0
$423$	513.045	0.0589719
$424$	0	0
$425$	3524.44	0.402260
$426$	0	0
$427$	0	0
$428$	0	0
$429$	9890.77	1.11313
$430$	0	0
$431$	−14203.3	−1.58736	−0.793678	−	0.608339i	$-0.791836\pi$
$431$	−14203.3	−1.58736	−0.793678	+	0.608339i	$0.791836\pi$
$432$	0	0
$433$	−3874.82	−0.430051	−0.215026	−	0.976608i	$-0.568983\pi$
$433$	−3874.82	−0.430051	−0.215026	+	0.976608i	$0.568983\pi$
$434$	0	0
$435$	−1616.62	−0.178187
$436$	0	0
$437$	−5278.75	−0.577842
$438$	0	0
$439$	7763.82	0.844070	0.422035	−	0.906579i	$-0.361316\pi$
$439$	7763.82	0.844070	0.422035	+	0.906579i	$0.361316\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9662.24	−1.03627	−0.518134	−	0.855299i	$-0.673373\pi$
$443$	−9662.24	−1.03627	−0.518134	+	0.855299i	$0.673373\pi$
$444$	0	0
$445$	908.915	0.0968241
$446$	0	0
$447$	−2240.11	−0.237032
$448$	0	0
$449$	−10942.6	−1.15014	−0.575069	−	0.818105i	$-0.695025\pi$
$449$	−10942.6	−1.15014	−0.575069	+	0.818105i	$0.695025\pi$
$450$	0	0
$451$	−26247.0	−2.74041
$452$	0	0
$453$	6219.44	0.645065
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13618.3	1.39396	0.696979	−	0.717091i	$-0.254527\pi$
$457$	13618.3	1.39396	0.696979	+	0.717091i	$0.254527\pi$
$458$	0	0
$459$	866.714	0.0881367
$460$	0	0
$461$	−11955.8	−1.20789	−0.603947	−	0.797025i	$-0.706406\pi$
$461$	−11955.8	−1.20789	−0.603947	+	0.797025i	$0.706406\pi$
$462$	0	0
$463$	−648.503	−0.0650939	−0.0325470	−	0.999470i	$-0.510362\pi$
$463$	−648.503	−0.0650939	−0.0325470	+	0.999470i	$0.510362\pi$
$464$	0	0
$465$	−1551.25	−0.154705
$466$	0	0
$467$	2784.74	0.275937	0.137969	−	0.990437i	$-0.455943\pi$
$467$	2784.74	0.275937	0.137969	+	0.990437i	$0.455943\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4698.66	0.459667
$472$	0	0
$473$	−26866.6	−2.61168
$474$	0	0
$475$	6126.39	0.591785
$476$	0	0
$477$	−2372.29	−0.227714
$478$	0	0
$479$	11113.4	1.06009	0.530046	−	0.847969i	$-0.322175\pi$
$479$	11113.4	1.06009	0.530046	+	0.847969i	$0.322175\pi$
$480$	0	0
$481$	−8012.14	−0.759505
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6275.99	0.587584
$486$	0	0
$487$	3786.27	0.352305	0.176152	−	0.984363i	$-0.443635\pi$
$487$	3786.27	0.352305	0.176152	+	0.984363i	$0.443635\pi$
$488$	0	0
$489$	296.200	0.0273919
$490$	0	0
$491$	9582.12	0.880723	0.440361	−	0.897821i	$-0.354850\pi$
$491$	9582.12	0.880723	0.440361	+	0.897821i	$0.354850\pi$
$492$	0	0
$493$	−4436.00	−0.405248
$494$	0	0
$495$	2154.75	0.195655
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−5581.55	−0.500730	−0.250365	−	0.968152i	$-0.580551\pi$
$499$	−5581.55	−0.500730	−0.250365	+	0.968152i	$0.580551\pi$
$500$	0	0
$501$	6694.07	0.596944
$502$	0	0
$503$	−14116.3	−1.25132	−0.625661	−	0.780095i	$-0.715171\pi$
$503$	−14116.3	−1.25132	−0.625661	+	0.780095i	$0.715171\pi$
$504$	0	0
$505$	5768.35	0.508294
$506$	0	0
$507$	−2059.58	−0.180413
$508$	0	0
$509$	−16787.6	−1.46188	−0.730941	−	0.682441i	$-0.760919\pi$
$509$	−16787.6	−1.46188	−0.730941	+	0.682441i	$0.760919\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1506.57	0.129662
$514$	0	0
$515$	−4466.27	−0.382150
$516$	0	0
$517$	3499.94	0.297731
$518$	0	0
$519$	6302.53	0.533045
$520$	0	0
$521$	5598.61	0.470786	0.235393	−	0.971900i	$-0.424362\pi$
$521$	5598.61	0.470786	0.235393	+	0.971900i	$0.424362\pi$
$522$	0	0
$523$	−13270.7	−1.10954	−0.554769	−	0.832004i	$-0.687193\pi$
$523$	−13270.7	−1.10954	−0.554769	+	0.832004i	$0.687193\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4256.62	−0.351843
$528$	0	0
$529$	−3217.27	−0.264426
$530$	0	0
$531$	4066.19	0.332312
$532$	0	0
$533$	22956.0	1.86554
$534$	0	0
$535$	−1703.90	−0.137694
$536$	0	0
$537$	5167.63	0.415269
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−23017.5	−1.82921	−0.914603	−	0.404353i	$-0.867497\pi$
$541$	−23017.5	−1.82921	−0.914603	+	0.404353i	$0.867497\pi$
$542$	0	0
$543$	4965.01	0.392393
$544$	0	0
$545$	−648.687	−0.0509848
$546$	0	0
$547$	−4475.84	−0.349859	−0.174930	−	0.984581i	$-0.555970\pi$
$547$	−4475.84	−0.349859	−0.174930	+	0.984581i	$0.555970\pi$
$548$	0	0
$549$	−5213.76	−0.405315
$550$	0	0
$551$	−7710.91	−0.596181
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1745.48	−0.133499
$556$	0	0
$557$	1541.70	0.117278	0.0586390	−	0.998279i	$-0.481324\pi$
$557$	1541.70	0.117278	0.0586390	+	0.998279i	$0.481324\pi$
$558$	0	0
$559$	23497.8	1.77791
$560$	0	0
$561$	5912.62	0.444975
$562$	0	0
$563$	13079.7	0.979115	0.489557	−	0.871971i	$-0.337158\pi$
$563$	13079.7	0.979115	0.489557	+	0.871971i	$0.337158\pi$
$564$	0	0
$565$	1913.97	0.142515
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11411.2	0.840740	0.420370	−	0.907353i	$-0.361900\pi$
$569$	11411.2	0.840740	0.420370	+	0.907353i	$0.361900\pi$
$570$	0	0
$571$	−2311.74	−0.169428	−0.0847139	−	0.996405i	$-0.526998\pi$
$571$	−2311.74	−0.169428	−0.0847139	+	0.996405i	$0.526998\pi$
$572$	0	0
$573$	−3023.06	−0.220402
$574$	0	0
$575$	−10386.8	−0.753324
$576$	0	0
$577$	−25097.0	−1.81075	−0.905373	−	0.424617i	$-0.860409\pi$
$577$	−25097.0	−1.81075	−0.905373	+	0.424617i	$0.860409\pi$
$578$	0	0
$579$	22.9455	0.00164694
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−16183.5	−1.14966
$584$	0	0
$585$	−1884.57	−0.133192
$586$	0	0
$587$	19789.1	1.39145	0.695726	−	0.718307i	$-0.255083\pi$
$587$	19789.1	1.39145	0.695726	+	0.718307i	$0.255083\pi$
$588$	0	0
$589$	−7399.12	−0.517615
$590$	0	0
$591$	−8069.64	−0.561659
$592$	0	0
$593$	2774.13	0.192108	0.0960540	−	0.995376i	$-0.469378\pi$
$593$	2774.13	0.192108	0.0960540	+	0.995376i	$0.469378\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2602.49	−0.178414
$598$	0	0
$599$	26190.3	1.78649	0.893245	−	0.449570i	$-0.148423\pi$
$599$	26190.3	1.78649	0.893245	+	0.449570i	$0.148423\pi$
$600$	0	0
$601$	11038.6	0.749211	0.374605	−	0.927184i	$-0.377778\pi$
$601$	11038.6	0.749211	0.374605	+	0.927184i	$0.377778\pi$
$602$	0	0
$603$	−2786.29	−0.188170
$604$	0	0
$605$	9509.26	0.639019
$606$	0	0
$607$	−1854.39	−0.123999	−0.0619996	−	0.998076i	$-0.519748\pi$
$607$	−1854.39	−0.123999	−0.0619996	+	0.998076i	$0.519748\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3061.08	−0.202681
$612$	0	0
$613$	−14856.8	−0.978894	−0.489447	−	0.872033i	$-0.662801\pi$
$613$	−14856.8	−0.978894	−0.489447	+	0.872033i	$0.662801\pi$
$614$	0	0
$615$	5001.07	0.327907
$616$	0	0
$617$	7600.00	0.495890	0.247945	−	0.968774i	$-0.420245\pi$
$617$	7600.00	0.495890	0.247945	+	0.968774i	$0.420245\pi$
$618$	0	0
$619$	22685.3	1.47302	0.736509	−	0.676427i	$-0.236473\pi$
$619$	22685.3	1.47302	0.736509	+	0.676427i	$0.236473\pi$
$620$	0	0
$621$	−2554.28	−0.165056
$622$	0	0
$623$	0	0
$624$	0	0
$625$	10154.0	0.649853
$626$	0	0
$627$	10277.7	0.654626
$628$	0	0
$629$	−4789.59	−0.303614
$630$	0	0
$631$	12024.1	0.758595	0.379297	−	0.925275i	$-0.376166\pi$
$631$	12024.1	0.758595	0.379297	+	0.925275i	$0.376166\pi$
$632$	0	0
$633$	486.091	0.0305219
$634$	0	0
$635$	10203.8	0.637679
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9530.86	0.590039
$640$	0	0
$641$	11320.6	0.697559	0.348779	−	0.937205i	$-0.386596\pi$
$641$	11320.6	0.697559	0.348779	+	0.937205i	$0.386596\pi$
$642$	0	0
$643$	16843.6	1.03304	0.516521	−	0.856275i	$-0.327227\pi$
$643$	16843.6	1.03304	0.516521	+	0.856275i	$0.327227\pi$
$644$	0	0
$645$	5119.12	0.312504
$646$	0	0
$647$	7719.32	0.469054	0.234527	−	0.972110i	$-0.424646\pi$
$647$	7719.32	0.469054	0.234527	+	0.972110i	$0.424646\pi$
$648$	0	0
$649$	27739.1	1.67774
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30803.2	−1.84597	−0.922987	−	0.384832i	$-0.874259\pi$
$653$	−30803.2	−1.84597	−0.922987	+	0.384832i	$0.874259\pi$
$654$	0	0
$655$	692.503	0.0413104
$656$	0	0
$657$	−10742.9	−0.637932
$658$	0	0
$659$	−9760.68	−0.576968	−0.288484	−	0.957485i	$-0.593151\pi$
$659$	−9760.68	−0.576968	−0.288484	+	0.957485i	$0.593151\pi$
$660$	0	0
$661$	−18071.1	−1.06336	−0.531682	−	0.846944i	$-0.678440\pi$
$661$	−18071.1	−1.06336	−0.531682	+	0.846944i	$0.678440\pi$
$662$	0	0
$663$	−5171.25	−0.302918
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13073.3	0.758920
$668$	0	0
$669$	13733.5	0.793676
$670$	0	0
$671$	−35567.7	−2.04631
$672$	0	0
$673$	26591.1	1.52305	0.761524	−	0.648137i	$-0.224451\pi$
$673$	26591.1	1.52305	0.761524	+	0.648137i	$0.224451\pi$
$674$	0	0
$675$	2964.44	0.169039
$676$	0	0
$677$	−33080.3	−1.87796	−0.938979	−	0.343975i	$-0.888227\pi$
$677$	−33080.3	−1.87796	−0.938979	+	0.343975i	$0.888227\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6654.57	0.374455
$682$	0	0
$683$	10466.1	0.586344	0.293172	−	0.956060i	$-0.405289\pi$
$683$	10466.1	0.586344	0.293172	+	0.956060i	$0.405289\pi$
$684$	0	0
$685$	105.345	0.00587597
$686$	0	0
$687$	2355.65	0.130820
$688$	0	0
$689$	14154.3	0.782634
$690$	0	0
$691$	13269.9	0.730551	0.365275	−	0.930900i	$-0.380975\pi$
$691$	13269.9	0.730551	0.365275	+	0.930900i	$0.380975\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3598.27	−0.196389
$696$	0	0
$697$	13722.9	0.745755
$698$	0	0
$699$	16107.4	0.871583
$700$	0	0
$701$	−15169.9	−0.817344	−0.408672	−	0.912681i	$-0.634008\pi$
$701$	−15169.9	−0.817344	−0.408672	+	0.912681i	$0.634008\pi$
$702$	0	0
$703$	−8325.55	−0.446663
$704$	0	0
$705$	−666.873	−0.0356254
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26317.7	1.39405	0.697026	−	0.717046i	$-0.254506\pi$
$709$	26317.7	1.39405	0.697026	+	0.717046i	$0.254506\pi$
$710$	0	0
$711$	−11876.0	−0.626421
$712$	0	0
$713$	12544.6	0.658907
$714$	0	0
$715$	−12856.3	−0.672447
$716$	0	0
$717$	11139.8	0.580229
$718$	0	0
$719$	23013.1	1.19366	0.596831	−	0.802367i	$-0.296426\pi$
$719$	23013.1	1.19366	0.596831	+	0.802367i	$0.296426\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	20995.9	1.08001
$724$	0	0
$725$	−15172.5	−0.777232
$726$	0	0
$727$	−16265.4	−0.829780	−0.414890	−	0.909872i	$-0.636180\pi$
$727$	−16265.4	−0.829780	−0.414890	+	0.909872i	$0.636180\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	14046.8	0.710724
$732$	0	0
$733$	866.444	0.0436601	0.0218300	−	0.999762i	$-0.493051\pi$
$733$	866.444	0.0436601	0.0218300	+	0.999762i	$0.493051\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19007.8	−0.950013
$738$	0	0
$739$	−10990.1	−0.547058	−0.273529	−	0.961864i	$-0.588191\pi$
$739$	−10990.1	−0.547058	−0.273529	+	0.961864i	$0.588191\pi$
$740$	0	0
$741$	−8988.96	−0.445638
$742$	0	0
$743$	22416.4	1.10683	0.553417	−	0.832905i	$-0.313324\pi$
$743$	22416.4	1.10683	0.553417	+	0.832905i	$0.313324\pi$
$744$	0	0
$745$	2911.76	0.143193
$746$	0	0
$747$	10713.0	0.524722
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19717.9	0.958080	0.479040	−	0.877793i	$-0.340985\pi$
$751$	19717.9	0.958080	0.479040	+	0.877793i	$0.340985\pi$
$752$	0	0
$753$	−11168.3	−0.540497
$754$	0	0
$755$	−8084.22	−0.389689
$756$	0	0
$757$	839.321	0.0402981	0.0201490	−	0.999797i	$-0.493586\pi$
$757$	839.321	0.0402981	0.0201490	+	0.999797i	$0.493586\pi$
$758$	0	0
$759$	−17425.0	−0.833318
$760$	0	0
$761$	−18364.5	−0.874786	−0.437393	−	0.899270i	$-0.644098\pi$
$761$	−18364.5	−0.874786	−0.437393	+	0.899270i	$0.644098\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1126.58	−0.0532440
$766$	0	0
$767$	−24260.9	−1.14213
$768$	0	0
$769$	14890.8	0.698277	0.349138	−	0.937071i	$-0.386474\pi$
$769$	14890.8	0.698277	0.349138	+	0.937071i	$0.386474\pi$
$770$	0	0
$771$	3690.45	0.172384
$772$	0	0
$773$	16606.2	0.772683	0.386341	−	0.922356i	$-0.373739\pi$
$773$	16606.2	0.772683	0.386341	+	0.922356i	$0.373739\pi$
$774$	0	0
$775$	−14559.0	−0.674807
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23853.9	1.09712
$780$	0	0
$781$	65018.5	2.97893
$782$	0	0
$783$	−3731.15	−0.170294
$784$	0	0
$785$	−6107.47	−0.277688
$786$	0	0
$787$	3156.20	0.142956	0.0714781	−	0.997442i	$-0.477228\pi$
$787$	3156.20	0.142956	0.0714781	+	0.997442i	$0.477228\pi$
$788$	0	0
$789$	7165.90	0.323337
$790$	0	0
$791$	0	0
$792$	0	0
$793$	31107.9	1.39303
$794$	0	0
$795$	3083.58	0.137564
$796$	0	0
$797$	13514.8	0.600652	0.300326	−	0.953837i	$-0.402904\pi$
$797$	13514.8	0.600652	0.300326	+	0.953837i	$0.402904\pi$
$798$	0	0
$799$	−1829.89	−0.0810224
$800$	0	0
$801$	2097.77	0.0925356
$802$	0	0
$803$	−73287.0	−3.22072
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20106.9	0.877073
$808$	0	0
$809$	−26758.1	−1.16287	−0.581437	−	0.813591i	$-0.697509\pi$
$809$	−26758.1	−1.16287	−0.581437	+	0.813591i	$0.697509\pi$
$810$	0	0
$811$	−15920.7	−0.689338	−0.344669	−	0.938724i	$-0.612009\pi$
$811$	−15920.7	−0.689338	−0.344669	+	0.938724i	$0.612009\pi$
$812$	0	0
$813$	−14851.1	−0.640653
$814$	0	0
$815$	−385.010	−0.0165476
$816$	0	0
$817$	24417.0	1.04558
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19306.2	−0.820696	−0.410348	−	0.911929i	$-0.634593\pi$
$821$	−19306.2	−0.820696	−0.410348	+	0.911929i	$0.634593\pi$
$822$	0	0
$823$	−791.000	−0.0335025	−0.0167512	−	0.999860i	$-0.505332\pi$
$823$	−791.000	−0.0335025	−0.0167512	+	0.999860i	$0.505332\pi$
$824$	0	0
$825$	20223.0	0.853426
$826$	0	0
$827$	−29537.6	−1.24199	−0.620993	−	0.783816i	$-0.713270\pi$
$827$	−29537.6	−1.24199	−0.620993	+	0.783816i	$0.713270\pi$
$828$	0	0
$829$	5766.52	0.241592	0.120796	−	0.992677i	$-0.461455\pi$
$829$	5766.52	0.241592	0.120796	+	0.992677i	$0.461455\pi$
$830$	0	0
$831$	11115.5	0.464011
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8701.16	−0.360618
$836$	0	0
$837$	−3580.28	−0.147853
$838$	0	0
$839$	29726.4	1.22321	0.611603	−	0.791165i	$-0.290525\pi$
$839$	29726.4	1.22321	0.611603	+	0.791165i	$0.290525\pi$
$840$	0	0
$841$	−5292.27	−0.216994
$842$	0	0
$843$	−27974.2	−1.14292
$844$	0	0
$845$	2677.11	0.108989
$846$	0	0
$847$	0	0
$848$	0	0
$849$	16708.5	0.675421
$850$	0	0
$851$	14115.3	0.568587
$852$	0	0
$853$	17829.6	0.715677	0.357838	−	0.933784i	$-0.383514\pi$
$853$	17829.6	0.715677	0.357838	+	0.933784i	$0.383514\pi$
$854$	0	0
$855$	−1958.29	−0.0783300
$856$	0	0
$857$	39682.4	1.58171	0.790856	−	0.612003i	$-0.209636\pi$
$857$	39682.4	1.58171	0.790856	+	0.612003i	$0.209636\pi$
$858$	0	0
$859$	−2195.13	−0.0871909	−0.0435955	−	0.999049i	$-0.513881\pi$
$859$	−2195.13	−0.0871909	−0.0435955	+	0.999049i	$0.513881\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31917.1	−1.25894	−0.629472	−	0.777023i	$-0.716729\pi$
$863$	−31917.1	−1.25894	−0.629472	+	0.777023i	$0.716729\pi$
$864$	0	0
$865$	−8192.23	−0.322016
$866$	0	0
$867$	11647.7	0.456258
$868$	0	0
$869$	−81016.8	−3.16261
$870$	0	0
$871$	16624.4	0.646724
$872$	0	0
$873$	14484.9	0.561559
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28842.9	1.11055	0.555276	−	0.831666i	$-0.312613\pi$
$877$	28842.9	1.11055	0.555276	+	0.831666i	$0.312613\pi$
$878$	0	0
$879$	−4995.92	−0.191704
$880$	0	0
$881$	15350.2	0.587015	0.293508	−	0.955957i	$-0.405177\pi$
$881$	15350.2	0.587015	0.293508	+	0.955957i	$0.405177\pi$
$882$	0	0
$883$	−6089.64	−0.232087	−0.116043	−	0.993244i	$-0.537021\pi$
$883$	−6089.64	−0.232087	−0.116043	+	0.993244i	$0.537021\pi$
$884$	0	0
$885$	−5285.36	−0.200752
$886$	0	0
$887$	−384.441	−0.0145527	−0.00727637	−	0.999974i	$-0.502316\pi$
$887$	−384.441	−0.0145527	−0.00727637	+	0.999974i	$0.502316\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4973.15	0.186989
$892$	0	0
$893$	−3180.82	−0.119196
$894$	0	0
$895$	−6717.05	−0.250867
$896$	0	0
$897$	15240.1	0.567283
$898$	0	0
$899$	18324.5	0.679819
$900$	0	0
$901$	8461.30	0.312860
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6453.68	−0.237047
$906$	0	0
$907$	7267.93	0.266073	0.133036	−	0.991111i	$-0.457527\pi$
$907$	7267.93	0.266073	0.133036	+	0.991111i	$0.457527\pi$
$908$	0	0
$909$	13313.3	0.485780
$910$	0	0
$911$	8535.12	0.310408	0.155204	−	0.987882i	$-0.450397\pi$
$911$	8535.12	0.310408	0.155204	+	0.987882i	$0.450397\pi$
$912$	0	0
$913$	73082.7	2.64916
$914$	0	0
$915$	6777.01	0.244854
$916$	0	0
$917$	0	0
$918$	0	0
$919$	10851.7	0.389516	0.194758	−	0.980851i	$-0.437608\pi$
$919$	10851.7	0.389516	0.194758	+	0.980851i	$0.437608\pi$
$920$	0	0
$921$	15909.9	0.569219
$922$	0	0
$923$	−56865.9	−2.02791
$924$	0	0
$925$	−16381.9	−0.582307
$926$	0	0
$927$	−10308.1	−0.365224
$928$	0	0
$929$	1560.64	0.0551161	0.0275581	−	0.999620i	$-0.491227\pi$
$929$	1560.64	0.0551161	0.0275581	+	0.999620i	$0.491227\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−3377.97	−0.118531
$934$	0	0
$935$	−7685.41	−0.268813
$936$	0	0
$937$	−11978.4	−0.417627	−0.208813	−	0.977956i	$-0.566960\pi$
$937$	−11978.4	−0.417627	−0.208813	+	0.977956i	$0.566960\pi$
$938$	0	0
$939$	−24898.5	−0.865317
$940$	0	0
$941$	−24597.7	−0.852137	−0.426068	−	0.904691i	$-0.640102\pi$
$941$	−24597.7	−0.852137	−0.426068	+	0.904691i	$0.640102\pi$
$942$	0	0
$943$	−40442.6	−1.39660
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10834.6	0.371783	0.185891	−	0.982570i	$-0.440483\pi$
$947$	10834.6	0.371783	0.185891	+	0.982570i	$0.440483\pi$
$948$	0	0
$949$	64097.6	2.19252
$950$	0	0
$951$	12836.3	0.437692
$952$	0	0
$953$	701.418	0.0238417	0.0119209	−	0.999929i	$-0.496205\pi$
$953$	701.418	0.0238417	0.0119209	+	0.999929i	$0.496205\pi$
$954$	0	0
$955$	3929.47	0.133146
$956$	0	0
$957$	−25453.5	−0.859765
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−12207.4	−0.409769
$962$	0	0
$963$	−3932.59	−0.131595
$964$	0	0
$965$	−29.8252	−0.000994931 0
$966$	0	0
$967$	−42402.5	−1.41011	−0.705053	−	0.709154i	$-0.749077\pi$
$967$	−42402.5	−1.41011	−0.705053	+	0.709154i	$0.749077\pi$
$968$	0	0
$969$	−5373.53	−0.178145
$970$	0	0
$971$	15463.4	0.511064	0.255532	−	0.966801i	$-0.417749\pi$
$971$	15463.4	0.511064	0.255532	+	0.966801i	$0.417749\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−17687.3	−0.580971
$976$	0	0
$977$	−34484.1	−1.12922	−0.564609	−	0.825359i	$-0.690973\pi$
$977$	−34484.1	−1.12922	−0.564609	+	0.825359i	$0.690973\pi$
$978$	0	0
$979$	14310.7	0.467184
$980$	0	0
$981$	−1497.16	−0.0487266
$982$	0	0
$983$	7335.36	0.238008	0.119004	−	0.992894i	$-0.462030\pi$
$983$	7335.36	0.238008	0.119004	+	0.992894i	$0.462030\pi$
$984$	0	0
$985$	10489.2	0.339302
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−41397.1	−1.33099
$990$	0	0
$991$	10123.0	0.324487	0.162243	−	0.986751i	$-0.448127\pi$
$991$	10123.0	0.324487	0.162243	+	0.986751i	$0.448127\pi$
$992$	0	0
$993$	−5121.35	−0.163667
$994$	0	0
$995$	3382.80	0.107781
$996$	0	0
$997$	−56669.2	−1.80013	−0.900066	−	0.435755i	$-0.856482\pi$
$997$	−56669.2	−1.80013	−0.900066	+	0.435755i	$0.856482\pi$
$998$	0	0
$999$	−4028.56	−0.127586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bn.1.2		2
4.3	odd	2		294.4.a.k.1.2	yes	2
7.6	odd	2		2352.4.a.cd.1.1		2
12.11	even	2		882.4.a.bi.1.1		2
28.3	even	6		294.4.e.o.79.2		4
28.11	odd	6		294.4.e.n.79.1		4
28.19	even	6		294.4.e.o.67.2		4
28.23	odd	6		294.4.e.n.67.1		4
28.27	even	2		294.4.a.j.1.1	✓	2
84.11	even	6		882.4.g.y.667.2		4
84.23	even	6		882.4.g.y.361.2		4
84.47	odd	6		882.4.g.bd.361.1		4
84.59	odd	6		882.4.g.bd.667.1		4
84.83	odd	2		882.4.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.4.a.j.1.1	✓	2	28.27	even	2
294.4.a.k.1.2	yes	2	4.3	odd	2
294.4.e.n.67.1		4	28.23	odd	6
294.4.e.n.79.1		4	28.11	odd	6
294.4.e.o.67.2		4	28.19	even	6
294.4.e.o.79.2		4	28.3	even	6
882.4.a.bc.1.2		2	84.83	odd	2
882.4.a.bi.1.1		2	12.11	even	2
882.4.g.y.361.2		4	84.23	even	6
882.4.g.y.667.2		4	84.11	even	6
882.4.g.bd.361.1		4	84.47	odd	6
882.4.g.bd.667.1		4	84.59	odd	6
2352.4.a.bn.1.2		2	1.1	even	1	trivial
2352.4.a.cd.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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +3.89949 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +3.89949 q^{5} +9.00000 q^{9} +61.3970 q^{11} -53.6985 q^{13} -11.6985 q^{15} -32.1005 q^{17} -55.7990 q^{19} +94.6030 q^{23} -109.794 q^{25} -27.0000 q^{27} +138.191 q^{29} +132.603 q^{31} -184.191 q^{33} +149.206 q^{37} +161.095 q^{39} -427.497 q^{41} -437.588 q^{43} +35.0955 q^{45} +57.0051 q^{47} +96.3015 q^{51} -263.588 q^{53} +239.417 q^{55} +167.397 q^{57} +451.799 q^{59} -579.307 q^{61} -209.397 q^{65} -309.588 q^{67} -283.809 q^{69} +1058.98 q^{71} -1193.66 q^{73} +329.382 q^{75} -1319.56 q^{79} +81.0000 q^{81} +1190.33 q^{83} -125.176 q^{85} -414.573 q^{87} +233.085 q^{89} -397.809 q^{93} -217.588 q^{95} +1609.44 q^{97} +552.573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 12 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} - 12 q^{5} + 18 q^{9} + 4 q^{11} - 48 q^{13} + 36 q^{15} - 84 q^{17} - 72 q^{19} + 308 q^{23} + 18 q^{25} - 54 q^{27} - 80 q^{29} + 384 q^{31} - 12 q^{33} + 536 q^{37} + 144 q^{39} - 756 q^{41} - 400 q^{43} - 108 q^{45} + 312 q^{47} + 252 q^{51} - 52 q^{53} + 1152 q^{55} + 216 q^{57} + 864 q^{59} - 1416 q^{61} - 300 q^{65} - 144 q^{67} - 924 q^{69} + 1524 q^{71} - 744 q^{73} - 54 q^{75} - 976 q^{79} + 162 q^{81} - 312 q^{83} + 700 q^{85} + 240 q^{87} - 108 q^{89} - 1152 q^{93} + 40 q^{95} + 744 q^{97} + 36 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 12 * q^5 + 18 * q^9 + 4 * q^11 - 48 * q^13 + 36 * q^15 - 84 * q^17 - 72 * q^19 + 308 * q^23 + 18 * q^25 - 54 * q^27 - 80 * q^29 + 384 * q^31 - 12 * q^33 + 536 * q^37 + 144 * q^39 - 756 * q^41 - 400 * q^43 - 108 * q^45 + 312 * q^47 + 252 * q^51 - 52 * q^53 + 1152 * q^55 + 216 * q^57 + 864 * q^59 - 1416 * q^61 - 300 * q^65 - 144 * q^67 - 924 * q^69 + 1524 * q^71 - 744 * q^73 - 54 * q^75 - 976 * q^79 + 162 * q^81 - 312 * q^83 + 700 * q^85 + 240 * q^87 - 108 * q^89 - 1152 * q^93 + 40 * q^95 + 744 * q^97 + 36 * q^99

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