LMFDB - Embedded newform 2352.4.a.ca.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(\sqrt{1345}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 336 $ x^2 - x - 336
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-17.8371$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +15.8371 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +15.8371 q^{5} +9.00000 q^{9} -51.8371 q^{11} -38.8371 q^{13} +47.5114 q^{15} -27.3485 q^{17} +76.5114 q^{19} -147.348 q^{23} +125.814 q^{25} +27.0000 q^{27} +240.208 q^{29} -296.674 q^{31} -155.511 q^{33} -161.534 q^{37} -116.511 q^{39} +102.977 q^{41} +328.557 q^{43} +142.534 q^{45} +67.9546 q^{47} -82.0454 q^{51} -66.4886 q^{53} -820.951 q^{55} +229.534 q^{57} -461.928 q^{59} -185.348 q^{61} -615.068 q^{65} -545.208 q^{67} -442.045 q^{69} +130.742 q^{71} -181.299 q^{73} +377.443 q^{75} +409.697 q^{79} +81.0000 q^{81} +347.928 q^{83} -433.121 q^{85} +720.625 q^{87} -1157.16 q^{89} -890.023 q^{93} +1211.72 q^{95} -1618.30 q^{97} -466.534 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 5 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 5 * q^5 + 18 * q^9	$ 2 q + 6 q^{3} - 5 q^{5} + 18 q^{9} - 67 q^{11} - 41 q^{13} - 15 q^{15} + 92 q^{17} + 43 q^{19} - 148 q^{23} + 435 q^{25} + 54 q^{27} + 77 q^{29} - 520 q^{31} - 201 q^{33} + 7 q^{37} - 123 q^{39} + 426 q^{41} + 107 q^{43} - 45 q^{45} + 576 q^{47} + 276 q^{51} - 243 q^{53} - 505 q^{55} + 129 q^{57} - 7 q^{59} - 224 q^{61} - 570 q^{65} - 687 q^{67} - 444 q^{69} - 472 q^{71} + 921 q^{73} + 1305 q^{75} + 526 q^{79} + 162 q^{81} - 221 q^{83} - 2920 q^{85} + 231 q^{87} - 774 q^{89} - 1560 q^{93} + 1910 q^{95} - 1953 q^{97} - 603 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 5 * q^5 + 18 * q^9 - 67 * q^11 - 41 * q^13 - 15 * q^15 + 92 * q^17 + 43 * q^19 - 148 * q^23 + 435 * q^25 + 54 * q^27 + 77 * q^29 - 520 * q^31 - 201 * q^33 + 7 * q^37 - 123 * q^39 + 426 * q^41 + 107 * q^43 - 45 * q^45 + 576 * q^47 + 276 * q^51 - 243 * q^53 - 505 * q^55 + 129 * q^57 - 7 * q^59 - 224 * q^61 - 570 * q^65 - 687 * q^67 - 444 * q^69 - 472 * q^71 + 921 * q^73 + 1305 * q^75 + 526 * q^79 + 162 * q^81 - 221 * q^83 - 2920 * q^85 + 231 * q^87 - 774 * q^89 - 1560 * q^93 + 1910 * q^95 - 1953 * q^97 - 603 * 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
$5$	15.8371	1.41652	0.708258	−	0.705954i	$-0.249482\pi$
$5$	15.8371	1.41652	0.708258	+	0.705954i	$0.249482\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−51.8371	−1.42086	−0.710431	−	0.703767i	$-0.751500\pi$
$11$	−51.8371	−1.42086	−0.710431	+	0.703767i	$0.751500\pi$
$12$	0	0
$13$	−38.8371	−0.828575	−0.414288	−	0.910146i	$-0.635969\pi$
$13$	−38.8371	−0.828575	−0.414288	+	0.910146i	$0.635969\pi$
$14$	0	0
$15$	47.5114	0.817825
$16$	0	0
$17$	−27.3485	−0.390175	−0.195088	−	0.980786i	$-0.562499\pi$
$17$	−27.3485	−0.390175	−0.195088	+	0.980786i	$0.562499\pi$
$18$	0	0
$19$	76.5114	0.923837	0.461919	−	0.886922i	$-0.347161\pi$
$19$	76.5114	0.923837	0.461919	+	0.886922i	$0.347161\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−147.348	−1.33584	−0.667919	−	0.744234i	$-0.732815\pi$
$23$	−147.348	−1.33584	−0.667919	+	0.744234i	$0.732815\pi$
$24$	0	0
$25$	125.814	1.00652
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	240.208	1.53812	0.769061	−	0.639175i	$-0.220724\pi$
$29$	240.208	1.53812	0.769061	+	0.639175i	$0.220724\pi$
$30$	0	0
$31$	−296.674	−1.71885	−0.859424	−	0.511264i	$-0.829177\pi$
$31$	−296.674	−1.71885	−0.859424	+	0.511264i	$0.829177\pi$
$32$	0	0
$33$	−155.511	−0.820335
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−161.534	−0.717731	−0.358865	−	0.933389i	$-0.616836\pi$
$37$	−161.534	−0.717731	−0.358865	+	0.933389i	$0.616836\pi$
$38$	0	0
$39$	−116.511	−0.478378
$40$	0	0
$41$	102.977	0.392252	0.196126	−	0.980579i	$-0.437164\pi$
$41$	102.977	0.392252	0.196126	+	0.980579i	$0.437164\pi$
$42$	0	0
$43$	328.557	1.16522	0.582610	−	0.812752i	$-0.302032\pi$
$43$	328.557	1.16522	0.582610	+	0.812752i	$0.302032\pi$
$44$	0	0
$45$	142.534	0.472172
$46$	0	0
$47$	67.9546	0.210898	0.105449	−	0.994425i	$-0.466372\pi$
$47$	67.9546	0.210898	0.105449	+	0.994425i	$0.466372\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−82.0454	−0.225268
$52$	0	0
$53$	−66.4886	−0.172319	−0.0861596	−	0.996281i	$-0.527459\pi$
$53$	−66.4886	−0.172319	−0.0861596	+	0.996281i	$0.527459\pi$
$54$	0	0
$55$	−820.951	−2.01267
$56$	0	0
$57$	229.534	0.533378
$58$	0	0
$59$	−461.928	−1.01929	−0.509643	−	0.860386i	$-0.670223\pi$
$59$	−461.928	−1.01929	−0.509643	+	0.860386i	$0.670223\pi$
$60$	0	0
$61$	−185.348	−0.389040	−0.194520	−	0.980899i	$-0.562315\pi$
$61$	−185.348	−0.389040	−0.194520	+	0.980899i	$0.562315\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−615.068	−1.17369
$66$	0	0
$67$	−545.208	−0.994146	−0.497073	−	0.867709i	$-0.665592\pi$
$67$	−545.208	−0.994146	−0.497073	+	0.867709i	$0.665592\pi$
$68$	0	0
$69$	−442.045	−0.771247
$70$	0	0
$71$	130.742	0.218539	0.109270	−	0.994012i	$-0.465149\pi$
$71$	130.742	0.218539	0.109270	+	0.994012i	$0.465149\pi$
$72$	0	0
$73$	−181.299	−0.290678	−0.145339	−	0.989382i	$-0.546427\pi$
$73$	−181.299	−0.290678	−0.145339	+	0.989382i	$0.546427\pi$
$74$	0	0
$75$	377.443	0.581112
$76$	0	0
$77$	0	0
$78$	0	0
$79$	409.697	0.583475	0.291737	−	0.956498i	$-0.405767\pi$
$79$	409.697	0.583475	0.291737	+	0.956498i	$0.405767\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	347.928	0.460121	0.230061	−	0.973176i	$-0.426108\pi$
$83$	347.928	0.460121	0.230061	+	0.973176i	$0.426108\pi$
$84$	0	0
$85$	−433.121	−0.552689
$86$	0	0
$87$	720.625	0.888036
$88$	0	0
$89$	−1157.16	−1.37819	−0.689093	−	0.724673i	$-0.741991\pi$
$89$	−1157.16	−1.37819	−0.689093	+	0.724673i	$0.741991\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−890.023	−0.992377
$94$	0	0
$95$	1211.72	1.30863
$96$	0	0
$97$	−1618.30	−1.69395	−0.846976	−	0.531631i	$-0.821579\pi$
$97$	−1618.30	−1.69395	−0.846976	+	0.531631i	$0.821579\pi$
$98$	0	0
$99$	−466.534	−0.473621
$100$	0	0
$101$	−718.742	−0.708094	−0.354047	−	0.935228i	$-0.615195\pi$
$101$	−718.742	−0.708094	−0.354047	+	0.935228i	$0.615195\pi$
$102$	0	0
$103$	−1611.58	−1.54169	−0.770843	−	0.637025i	$-0.780165\pi$
$103$	−1611.58	−1.54169	−0.770843	+	0.637025i	$0.780165\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−934.670	−0.844467	−0.422234	−	0.906487i	$-0.638754\pi$
$107$	−934.670	−0.844467	−0.422234	+	0.906487i	$0.638754\pi$
$108$	0	0
$109$	−1197.02	−1.05187	−0.525934	−	0.850525i	$-0.676284\pi$
$109$	−1197.02	−1.05187	−0.525934	+	0.850525i	$0.676284\pi$
$110$	0	0
$111$	−484.602	−0.414382
$112$	0	0
$113$	−2384.64	−1.98521	−0.992604	−	0.121400i	$-0.961262\pi$
$113$	−2384.64	−1.98521	−0.992604	+	0.121400i	$0.961262\pi$
$114$	0	0
$115$	−2333.58	−1.89224
$116$	0	0
$117$	−349.534	−0.276192
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1356.09	1.01885
$122$	0	0
$123$	308.932	0.226467
$124$	0	0
$125$	12.8977	0.00922883
$126$	0	0
$127$	2673.92	1.86829	0.934143	−	0.356898i	$-0.116166\pi$
$127$	2673.92	1.86829	0.934143	+	0.356898i	$0.116166\pi$
$128$	0	0
$129$	985.670	0.672740
$130$	0	0
$131$	−38.8598	−0.0259176	−0.0129588	−	0.999916i	$-0.504125\pi$
$131$	−38.8598	−0.0259176	−0.0129588	+	0.999916i	$0.504125\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	427.602	0.272608
$136$	0	0
$137$	−768.144	−0.479029	−0.239514	−	0.970893i	$-0.576988\pi$
$137$	−768.144	−0.479029	−0.239514	+	0.970893i	$0.576988\pi$
$138$	0	0
$139$	1052.55	0.642274	0.321137	−	0.947033i	$-0.395935\pi$
$139$	1052.55	0.642274	0.321137	+	0.947033i	$0.395935\pi$
$140$	0	0
$141$	203.864	0.121762
$142$	0	0
$143$	2013.20	1.17729
$144$	0	0
$145$	3804.21	2.17877
$146$	0	0
$147$	0	0
$148$	0	0
$149$	360.977	0.198473	0.0992363	−	0.995064i	$-0.468360\pi$
$149$	360.977	0.198473	0.0992363	+	0.995064i	$0.468360\pi$
$150$	0	0
$151$	−1548.39	−0.834478	−0.417239	−	0.908797i	$-0.637002\pi$
$151$	−1548.39	−0.834478	−0.417239	+	0.908797i	$0.637002\pi$
$152$	0	0
$153$	−246.136	−0.130058
$154$	0	0
$155$	−4698.47	−2.43477
$156$	0	0
$157$	967.068	0.491595	0.245798	−	0.969321i	$-0.420950\pi$
$157$	967.068	0.491595	0.245798	+	0.969321i	$0.420950\pi$
$158$	0	0
$159$	−199.466	−0.0994885
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1326.50	0.637420	0.318710	−	0.947852i	$-0.396750\pi$
$163$	1326.50	0.637420	0.318710	+	0.947852i	$0.396750\pi$
$164$	0	0
$165$	−2462.85	−1.16202
$166$	0	0
$167$	1416.70	0.656451	0.328225	−	0.944599i	$-0.393549\pi$
$167$	1416.70	0.656451	0.328225	+	0.944599i	$0.393549\pi$
$168$	0	0
$169$	−688.678	−0.313463
$170$	0	0
$171$	688.602	0.307946
$172$	0	0
$173$	1036.60	0.455556	0.227778	−	0.973713i	$-0.426854\pi$
$173$	1036.60	0.455556	0.227778	+	0.973713i	$0.426854\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1385.78	−0.588485
$178$	0	0
$179$	767.432	0.320450	0.160225	−	0.987081i	$-0.448778\pi$
$179$	767.432	0.320450	0.160225	+	0.987081i	$0.448778\pi$
$180$	0	0
$181$	3957.71	1.62527	0.812636	−	0.582772i	$-0.198032\pi$
$181$	3957.71	1.62527	0.812636	+	0.582772i	$0.198032\pi$
$182$	0	0
$183$	−556.045	−0.224612
$184$	0	0
$185$	−2558.23	−1.01668
$186$	0	0
$187$	1417.67	0.554385
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1805.30	0.683909	0.341954	−	0.939717i	$-0.388911\pi$
$191$	1805.30	0.683909	0.341954	+	0.939717i	$0.388911\pi$
$192$	0	0
$193$	3370.84	1.25719	0.628597	−	0.777731i	$-0.283630\pi$
$193$	3370.84	1.25719	0.628597	+	0.777731i	$0.283630\pi$
$194$	0	0
$195$	−1845.20	−0.677630
$196$	0	0
$197$	−4612.31	−1.66809	−0.834044	−	0.551697i	$-0.813980\pi$
$197$	−4612.31	−1.66809	−0.834044	+	0.551697i	$0.813980\pi$
$198$	0	0
$199$	−2229.86	−0.794326	−0.397163	−	0.917748i	$-0.630005\pi$
$199$	−2229.86	−0.794326	−0.397163	+	0.917748i	$0.630005\pi$
$200$	0	0
$201$	−1635.62	−0.573971
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1630.86	0.555631
$206$	0	0
$207$	−1326.14	−0.445279
$208$	0	0
$209$	−3966.13	−1.31265
$210$	0	0
$211$	−912.614	−0.297758	−0.148879	−	0.988855i	$-0.547566\pi$
$211$	−912.614	−0.297758	−0.148879	+	0.988855i	$0.547566\pi$
$212$	0	0
$213$	392.227	0.126174
$214$	0	0
$215$	5203.39	1.65055
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−543.898	−0.167823
$220$	0	0
$221$	1062.14	0.323290
$222$	0	0
$223$	−4319.47	−1.29710	−0.648549	−	0.761173i	$-0.724624\pi$
$223$	−4319.47	−1.29710	−0.648549	+	0.761173i	$0.724624\pi$
$224$	0	0
$225$	1132.33	0.335505
$226$	0	0
$227$	2061.28	0.602697	0.301349	−	0.953514i	$-0.402563\pi$
$227$	2061.28	0.602697	0.301349	+	0.953514i	$0.402563\pi$
$228$	0	0
$229$	3474.63	1.00266	0.501332	−	0.865255i	$-0.332843\pi$
$229$	3474.63	1.00266	0.501332	+	0.865255i	$0.332843\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	776.099	0.218214	0.109107	−	0.994030i	$-0.465201\pi$
$233$	776.099	0.218214	0.109107	+	0.994030i	$0.465201\pi$
$234$	0	0
$235$	1076.20	0.298740
$236$	0	0
$237$	1229.09	0.336869
$238$	0	0
$239$	−2006.80	−0.543133	−0.271567	−	0.962420i	$-0.587542\pi$
$239$	−2006.80	−0.543133	−0.271567	+	0.962420i	$0.587542\pi$
$240$	0	0
$241$	−805.648	−0.215337	−0.107669	−	0.994187i	$-0.534339\pi$
$241$	−805.648	−0.215337	−0.107669	+	0.994187i	$0.534339\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2971.48	−0.765469
$248$	0	0
$249$	1043.78	0.265651
$250$	0	0
$251$	1421.78	0.357539	0.178769	−	0.983891i	$-0.442788\pi$
$251$	1421.78	0.357539	0.178769	+	0.983891i	$0.442788\pi$
$252$	0	0
$253$	7638.12	1.89804
$254$	0	0
$255$	−1299.36	−0.319095
$256$	0	0
$257$	1465.82	0.355779	0.177890	−	0.984050i	$-0.443073\pi$
$257$	1465.82	0.355779	0.177890	+	0.984050i	$0.443073\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2161.87	0.512708
$262$	0	0
$263$	−6991.39	−1.63919	−0.819596	−	0.572942i	$-0.805802\pi$
$263$	−6991.39	−1.63919	−0.819596	+	0.572942i	$0.805802\pi$
$264$	0	0
$265$	−1052.99	−0.244093
$266$	0	0
$267$	−3471.48	−0.795696
$268$	0	0
$269$	−808.958	−0.183357	−0.0916786	−	0.995789i	$-0.529223\pi$
$269$	−808.958	−0.183357	−0.0916786	+	0.995789i	$0.529223\pi$
$270$	0	0
$271$	6661.77	1.49326	0.746630	−	0.665239i	$-0.231670\pi$
$271$	6661.77	1.49326	0.746630	+	0.665239i	$0.231670\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6521.86	−1.43012
$276$	0	0
$277$	−7531.47	−1.63365	−0.816827	−	0.576883i	$-0.804269\pi$
$277$	−7531.47	−1.63365	−0.816827	+	0.576883i	$0.804269\pi$
$278$	0	0
$279$	−2670.07	−0.572949
$280$	0	0
$281$	1690.19	0.358819	0.179410	−	0.983774i	$-0.442581\pi$
$281$	1690.19	0.358819	0.179410	+	0.983774i	$0.442581\pi$
$282$	0	0
$283$	3178.23	0.667584	0.333792	−	0.942647i	$-0.391672\pi$
$283$	3178.23	0.667584	0.333792	+	0.942647i	$0.391672\pi$
$284$	0	0
$285$	3635.16	0.755538
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4165.06	−0.847763
$290$	0	0
$291$	−4854.90	−0.978004
$292$	0	0
$293$	2176.53	0.433974	0.216987	−	0.976174i	$-0.430377\pi$
$293$	2176.53	0.433974	0.216987	+	0.976174i	$0.430377\pi$
$294$	0	0
$295$	−7315.61	−1.44383
$296$	0	0
$297$	−1399.60	−0.273445
$298$	0	0
$299$	5722.59	1.10684
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2156.23	−0.408819
$304$	0	0
$305$	−2935.39	−0.551081
$306$	0	0
$307$	623.504	0.115913	0.0579564	−	0.998319i	$-0.481542\pi$
$307$	623.504	0.115913	0.0579564	+	0.998319i	$0.481542\pi$
$308$	0	0
$309$	−4834.74	−0.890093
$310$	0	0
$311$	467.992	0.0853293	0.0426647	−	0.999089i	$-0.486415\pi$
$311$	467.992	0.0853293	0.0426647	+	0.999089i	$0.486415\pi$
$312$	0	0
$313$	−3612.81	−0.652422	−0.326211	−	0.945297i	$-0.605772\pi$
$313$	−3612.81	−0.652422	−0.326211	+	0.945297i	$0.605772\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4531.74	0.802927	0.401463	−	0.915875i	$-0.368502\pi$
$317$	4531.74	0.802927	0.401463	+	0.915875i	$0.368502\pi$
$318$	0	0
$319$	−12451.7	−2.18546
$320$	0	0
$321$	−2804.01	−0.487553
$322$	0	0
$323$	−2092.47	−0.360459
$324$	0	0
$325$	−4886.27	−0.833974
$326$	0	0
$327$	−3591.06	−0.607296
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1237.06	0.205422	0.102711	−	0.994711i	$-0.467248\pi$
$331$	1237.06	0.205422	0.102711	+	0.994711i	$0.467248\pi$
$332$	0	0
$333$	−1453.81	−0.239244
$334$	0	0
$335$	−8634.53	−1.40822
$336$	0	0
$337$	−1867.83	−0.301921	−0.150960	−	0.988540i	$-0.548237\pi$
$337$	−1867.83	−0.301921	−0.150960	+	0.988540i	$0.548237\pi$
$338$	0	0
$339$	−7153.93	−1.14616
$340$	0	0
$341$	15378.7	2.44224
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7000.73	−1.09248
$346$	0	0
$347$	63.3637	0.00980271	0.00490136	−	0.999988i	$-0.498440\pi$
$347$	63.3637	0.00980271	0.00490136	+	0.999988i	$0.498440\pi$
$348$	0	0
$349$	−1223.79	−0.187702	−0.0938508	−	0.995586i	$-0.529918\pi$
$349$	−1223.79	−0.187702	−0.0938508	+	0.995586i	$0.529918\pi$
$350$	0	0
$351$	−1048.60	−0.159459
$352$	0	0
$353$	4515.61	0.680855	0.340428	−	0.940271i	$-0.389428\pi$
$353$	4515.61	0.680855	0.340428	+	0.940271i	$0.389428\pi$
$354$	0	0
$355$	2070.58	0.309564
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2228.49	−0.327619	−0.163810	−	0.986492i	$-0.552378\pi$
$359$	−2228.49	−0.327619	−0.163810	+	0.986492i	$0.552378\pi$
$360$	0	0
$361$	−1005.01	−0.146524
$362$	0	0
$363$	4068.26	0.588232
$364$	0	0
$365$	−2871.26	−0.411749
$366$	0	0
$367$	−1437.34	−0.204438	−0.102219	−	0.994762i	$-0.532594\pi$
$367$	−1437.34	−0.204438	−0.102219	+	0.994762i	$0.532594\pi$
$368$	0	0
$369$	926.795	0.130751
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−12237.4	−1.69874	−0.849370	−	0.527798i	$-0.823018\pi$
$373$	−12237.4	−1.69874	−0.849370	+	0.527798i	$0.823018\pi$
$374$	0	0
$375$	38.6931	0.00532827
$376$	0	0
$377$	−9329.00	−1.27445
$378$	0	0
$379$	10647.0	1.44301	0.721503	−	0.692411i	$-0.243451\pi$
$379$	10647.0	1.44301	0.721503	+	0.692411i	$0.243451\pi$
$380$	0	0
$381$	8021.77	1.07866
$382$	0	0
$383$	−6714.81	−0.895851	−0.447925	−	0.894071i	$-0.647837\pi$
$383$	−6714.81	−0.895851	−0.447925	+	0.894071i	$0.647837\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2957.01	0.388407
$388$	0	0
$389$	−10653.1	−1.38852	−0.694258	−	0.719727i	$-0.744267\pi$
$389$	−10653.1	−1.38852	−0.694258	+	0.719727i	$0.744267\pi$
$390$	0	0
$391$	4029.76	0.521211
$392$	0	0
$393$	−116.580	−0.0149635
$394$	0	0
$395$	6488.42	0.826501
$396$	0	0
$397$	−3221.04	−0.407203	−0.203601	−	0.979054i	$-0.565265\pi$
$397$	−3221.04	−0.407203	−0.203601	+	0.979054i	$0.565265\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12485.0	1.55479	0.777395	−	0.629012i	$-0.216540\pi$
$401$	12485.0	1.55479	0.777395	+	0.629012i	$0.216540\pi$
$402$	0	0
$403$	11522.0	1.42419
$404$	0	0
$405$	1282.81	0.157391
$406$	0	0
$407$	8373.46	1.01980
$408$	0	0
$409$	−7037.39	−0.850798	−0.425399	−	0.905006i	$-0.639866\pi$
$409$	−7037.39	−0.850798	−0.425399	+	0.905006i	$0.639866\pi$
$410$	0	0
$411$	−2304.43	−0.276567
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5510.18	0.651769
$416$	0	0
$417$	3157.65	0.370817
$418$	0	0
$419$	−1549.66	−0.180682	−0.0903410	−	0.995911i	$-0.528796\pi$
$419$	−1549.66	−0.180682	−0.0903410	+	0.995911i	$0.528796\pi$
$420$	0	0
$421$	5531.63	0.640369	0.320184	−	0.947355i	$-0.396255\pi$
$421$	5531.63	0.640369	0.320184	+	0.947355i	$0.396255\pi$
$422$	0	0
$423$	611.591	0.0702992
$424$	0	0
$425$	−3440.83	−0.392717
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6039.61	0.679709
$430$	0	0
$431$	2029.93	0.226864	0.113432	−	0.993546i	$-0.463816\pi$
$431$	2029.93	0.226864	0.113432	+	0.993546i	$0.463816\pi$
$432$	0	0
$433$	327.739	0.0363744	0.0181872	−	0.999835i	$-0.494211\pi$
$433$	327.739	0.0363744	0.0181872	+	0.999835i	$0.494211\pi$
$434$	0	0
$435$	11412.6	1.25792
$436$	0	0
$437$	−11273.8	−1.23410
$438$	0	0
$439$	7908.68	0.859819	0.429910	−	0.902872i	$-0.358545\pi$
$439$	7908.68	0.859819	0.429910	+	0.902872i	$0.358545\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2920.82	0.313256	0.156628	−	0.987658i	$-0.449938\pi$
$443$	2920.82	0.313256	0.156628	+	0.987658i	$0.449938\pi$
$444$	0	0
$445$	−18326.1	−1.95222
$446$	0	0
$447$	1082.93	0.114588
$448$	0	0
$449$	−10240.2	−1.07631	−0.538156	−	0.842845i	$-0.680879\pi$
$449$	−10240.2	−1.07631	−0.538156	+	0.842845i	$0.680879\pi$
$450$	0	0
$451$	−5338.05	−0.557336
$452$	0	0
$453$	−4645.17	−0.481786
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5892.63	0.603163	0.301582	−	0.953440i	$-0.402485\pi$
$457$	5892.63	0.603163	0.301582	+	0.953440i	$0.402485\pi$
$458$	0	0
$459$	−738.409	−0.0750893
$460$	0	0
$461$	12643.4	1.27735	0.638677	−	0.769475i	$-0.279482\pi$
$461$	12643.4	1.27735	0.638677	+	0.769475i	$0.279482\pi$
$462$	0	0
$463$	−15093.2	−1.51499	−0.757494	−	0.652842i	$-0.773577\pi$
$463$	−15093.2	−1.51499	−0.757494	+	0.652842i	$0.773577\pi$
$464$	0	0
$465$	−14095.4	−1.40572
$466$	0	0
$467$	−2820.23	−0.279454	−0.139727	−	0.990190i	$-0.544622\pi$
$467$	−2820.23	−0.279454	−0.139727	+	0.990190i	$0.544622\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2901.20	0.283823
$472$	0	0
$473$	−17031.4	−1.65562
$474$	0	0
$475$	9626.23	0.929856
$476$	0	0
$477$	−598.398	−0.0574397
$478$	0	0
$479$	−16448.5	−1.56900	−0.784500	−	0.620129i	$-0.787080\pi$
$479$	−16448.5	−1.56900	−0.784500	+	0.620129i	$0.787080\pi$
$480$	0	0
$481$	6273.52	0.594694
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−25629.2	−2.39951
$486$	0	0
$487$	−6331.07	−0.589093	−0.294546	−	0.955637i	$-0.595169\pi$
$487$	−6331.07	−0.589093	−0.294546	+	0.955637i	$0.595169\pi$
$488$	0	0
$489$	3979.50	0.368015
$490$	0	0
$491$	9286.90	0.853588	0.426794	−	0.904349i	$-0.359643\pi$
$491$	9286.90	0.853588	0.426794	+	0.904349i	$0.359643\pi$
$492$	0	0
$493$	−6569.33	−0.600138
$494$	0	0
$495$	−7388.56	−0.670891
$496$	0	0
$497$	0	0
$498$	0	0
$499$	243.451	0.0218404	0.0109202	−	0.999940i	$-0.496524\pi$
$499$	243.451	0.0218404	0.0109202	+	0.999940i	$0.496524\pi$
$500$	0	0
$501$	4250.09	0.379002
$502$	0	0
$503$	8499.30	0.753409	0.376705	−	0.926333i	$-0.377057\pi$
$503$	8499.30	0.753409	0.376705	+	0.926333i	$0.377057\pi$
$504$	0	0
$505$	−11382.8	−1.00303
$506$	0	0
$507$	−2066.03	−0.180978
$508$	0	0
$509$	7683.10	0.669052	0.334526	−	0.942387i	$-0.391424\pi$
$509$	7683.10	0.669052	0.334526	+	0.942387i	$0.391424\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2065.81	0.177793
$514$	0	0
$515$	−25522.8	−2.18382
$516$	0	0
$517$	−3522.57	−0.299656
$518$	0	0
$519$	3109.80	0.263015
$520$	0	0
$521$	21530.6	1.81051	0.905253	−	0.424874i	$-0.139681\pi$
$521$	21530.6	1.81051	0.905253	+	0.424874i	$0.139681\pi$
$522$	0	0
$523$	−16847.1	−1.40855	−0.704274	−	0.709929i	$-0.748727\pi$
$523$	−16847.1	−1.40855	−0.704274	+	0.709929i	$0.748727\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8113.59	0.670652
$528$	0	0
$529$	9544.58	0.784464
$530$	0	0
$531$	−4157.35	−0.339762
$532$	0	0
$533$	−3999.34	−0.325011
$534$	0	0
$535$	−14802.5	−1.19620
$536$	0	0
$537$	2302.30	0.185012
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17440.0	1.38596	0.692981	−	0.720955i	$-0.256297\pi$
$541$	17440.0	1.38596	0.692981	+	0.720955i	$0.256297\pi$
$542$	0	0
$543$	11873.1	0.938351
$544$	0	0
$545$	−18957.3	−1.48999
$546$	0	0
$547$	−11520.7	−0.900530	−0.450265	−	0.892895i	$-0.648671\pi$
$547$	−11520.7	−0.900530	−0.450265	+	0.892895i	$0.648671\pi$
$548$	0	0
$549$	−1668.14	−0.129680
$550$	0	0
$551$	18378.7	1.42098
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7674.70	−0.586978
$556$	0	0
$557$	11493.0	0.874282	0.437141	−	0.899393i	$-0.355991\pi$
$557$	11493.0	0.874282	0.437141	+	0.899393i	$0.355991\pi$
$558$	0	0
$559$	−12760.2	−0.965472
$560$	0	0
$561$	4253.00	0.320075
$562$	0	0
$563$	18111.3	1.35577	0.677886	−	0.735167i	$-0.262896\pi$
$563$	18111.3	1.35577	0.677886	+	0.735167i	$0.262896\pi$
$564$	0	0
$565$	−37765.9	−2.81208
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4417.61	0.325476	0.162738	−	0.986669i	$-0.447967\pi$
$569$	4417.61	0.325476	0.162738	+	0.986669i	$0.447967\pi$
$570$	0	0
$571$	−13219.7	−0.968878	−0.484439	−	0.874825i	$-0.660976\pi$
$571$	−13219.7	−0.968878	−0.484439	+	0.874825i	$0.660976\pi$
$572$	0	0
$573$	5415.89	0.394855
$574$	0	0
$575$	−18538.6	−1.34454
$576$	0	0
$577$	17496.4	1.26236	0.631182	−	0.775634i	$-0.282570\pi$
$577$	17496.4	1.26236	0.631182	+	0.775634i	$0.282570\pi$
$578$	0	0
$579$	10112.5	0.725841
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3446.58	0.244842
$584$	0	0
$585$	−5535.61	−0.391230
$586$	0	0
$587$	−4280.53	−0.300982	−0.150491	−	0.988611i	$-0.548085\pi$
$587$	−4280.53	−0.300982	−0.150491	+	0.988611i	$0.548085\pi$
$588$	0	0
$589$	−22699.0	−1.58794
$590$	0	0
$591$	−13836.9	−0.963072
$592$	0	0
$593$	1590.93	0.110172	0.0550858	−	0.998482i	$-0.482457\pi$
$593$	1590.93	0.110172	0.0550858	+	0.998482i	$0.482457\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6689.59	−0.458604
$598$	0	0
$599$	13922.8	0.949703	0.474851	−	0.880066i	$-0.342502\pi$
$599$	13922.8	0.949703	0.474851	+	0.880066i	$0.342502\pi$
$600$	0	0
$601$	−12559.7	−0.852446	−0.426223	−	0.904618i	$-0.640156\pi$
$601$	−12559.7	−0.852446	−0.426223	+	0.904618i	$0.640156\pi$
$602$	0	0
$603$	−4906.87	−0.331382
$604$	0	0
$605$	21476.5	1.44321
$606$	0	0
$607$	7678.37	0.513436	0.256718	−	0.966486i	$-0.417359\pi$
$607$	7678.37	0.513436	0.256718	+	0.966486i	$0.417359\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2639.16	−0.174745
$612$	0	0
$613$	6158.37	0.405766	0.202883	−	0.979203i	$-0.434969\pi$
$613$	6158.37	0.405766	0.202883	+	0.979203i	$0.434969\pi$
$614$	0	0
$615$	4892.59	0.320794
$616$	0	0
$617$	8813.12	0.575045	0.287523	−	0.957774i	$-0.407168\pi$
$617$	8813.12	0.575045	0.287523	+	0.957774i	$0.407168\pi$
$618$	0	0
$619$	23189.9	1.50579	0.752894	−	0.658142i	$-0.228658\pi$
$619$	23189.9	1.50579	0.752894	+	0.658142i	$0.228658\pi$
$620$	0	0
$621$	−3978.41	−0.257082
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−15522.5	−0.993442
$626$	0	0
$627$	−11898.4	−0.757856
$628$	0	0
$629$	4417.71	0.280041
$630$	0	0
$631$	−7936.94	−0.500736	−0.250368	−	0.968151i	$-0.580552\pi$
$631$	−7936.94	−0.500736	−0.250368	+	0.968151i	$0.580552\pi$
$632$	0	0
$633$	−2737.84	−0.171911
$634$	0	0
$635$	42347.3	2.64646
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1176.68	0.0728463
$640$	0	0
$641$	32114.6	1.97886	0.989432	−	0.144996i	$-0.0463169\pi$
$641$	32114.6	1.97886	0.989432	+	0.144996i	$0.0463169\pi$
$642$	0	0
$643$	−24786.7	−1.52021	−0.760104	−	0.649802i	$-0.774852\pi$
$643$	−24786.7	−1.52021	−0.760104	+	0.649802i	$0.774852\pi$
$644$	0	0
$645$	15610.2	0.952946
$646$	0	0
$647$	7545.59	0.458497	0.229249	−	0.973368i	$-0.426373\pi$
$647$	7545.59	0.458497	0.229249	+	0.973368i	$0.426373\pi$
$648$	0	0
$649$	23945.0	1.44827
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4888.99	−0.292988	−0.146494	−	0.989212i	$-0.546799\pi$
$653$	−4888.99	−0.292988	−0.146494	+	0.989212i	$0.546799\pi$
$654$	0	0
$655$	−615.428	−0.0367126
$656$	0	0
$657$	−1631.69	−0.0968926
$658$	0	0
$659$	−25895.9	−1.53075	−0.765374	−	0.643586i	$-0.777446\pi$
$659$	−25895.9	−1.53075	−0.765374	+	0.643586i	$0.777446\pi$
$660$	0	0
$661$	−8183.37	−0.481537	−0.240769	−	0.970583i	$-0.577399\pi$
$661$	−8183.37	−0.481537	−0.240769	+	0.970583i	$0.577399\pi$
$662$	0	0
$663$	3186.41	0.186651
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−35394.3	−2.05468
$668$	0	0
$669$	−12958.4	−0.748880
$670$	0	0
$671$	9607.93	0.552772
$672$	0	0
$673$	−4635.02	−0.265478	−0.132739	−	0.991151i	$-0.542377\pi$
$673$	−4635.02	−0.265478	−0.132739	+	0.991151i	$0.542377\pi$
$674$	0	0
$675$	3396.99	0.193704
$676$	0	0
$677$	24385.8	1.38437	0.692187	−	0.721718i	$-0.256647\pi$
$677$	24385.8	1.38437	0.692187	+	0.721718i	$0.256647\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6183.85	0.347967
$682$	0	0
$683$	18393.4	1.03046	0.515230	−	0.857052i	$-0.327706\pi$
$683$	18393.4	1.03046	0.515230	+	0.857052i	$0.327706\pi$
$684$	0	0
$685$	−12165.2	−0.678552
$686$	0	0
$687$	10423.9	0.578889
$688$	0	0
$689$	2582.23	0.142779
$690$	0	0
$691$	−14898.9	−0.820231	−0.410116	−	0.912034i	$-0.634512\pi$
$691$	−14898.9	−0.820231	−0.410116	+	0.912034i	$0.634512\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16669.3	0.909791
$696$	0	0
$697$	−2816.27	−0.153047
$698$	0	0
$699$	2328.30	0.125986
$700$	0	0
$701$	−5725.70	−0.308497	−0.154249	−	0.988032i	$-0.549296\pi$
$701$	−5725.70	−0.308497	−0.154249	+	0.988032i	$0.549296\pi$
$702$	0	0
$703$	−12359.2	−0.663067
$704$	0	0
$705$	3228.61	0.172477
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−23456.8	−1.24251	−0.621255	−	0.783609i	$-0.713377\pi$
$709$	−23456.8	−1.24251	−0.621255	+	0.783609i	$0.713377\pi$
$710$	0	0
$711$	3687.27	0.194492
$712$	0	0
$713$	43714.5	2.29610
$714$	0	0
$715$	31883.4	1.66765
$716$	0	0
$717$	−6020.39	−0.313578
$718$	0	0
$719$	−8459.00	−0.438759	−0.219379	−	0.975640i	$-0.570403\pi$
$719$	−8459.00	−0.438759	−0.219379	+	0.975640i	$0.570403\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2416.94	−0.124325
$724$	0	0
$725$	30221.7	1.54814
$726$	0	0
$727$	−11822.2	−0.603111	−0.301555	−	0.953449i	$-0.597506\pi$
$727$	−11822.2	−0.603111	−0.301555	+	0.953449i	$0.597506\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−8985.53	−0.454640
$732$	0	0
$733$	−5028.95	−0.253409	−0.126704	−	0.991941i	$-0.540440\pi$
$733$	−5028.95	−0.253409	−0.126704	+	0.991941i	$0.540440\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28262.0	1.41254
$738$	0	0
$739$	17743.9	0.883247	0.441624	−	0.897200i	$-0.354403\pi$
$739$	17743.9	0.883247	0.441624	+	0.897200i	$0.354403\pi$
$740$	0	0
$741$	−8914.44	−0.441944
$742$	0	0
$743$	13202.3	0.651877	0.325938	−	0.945391i	$-0.394320\pi$
$743$	13202.3	0.651877	0.325938	+	0.945391i	$0.394320\pi$
$744$	0	0
$745$	5716.84	0.281139
$746$	0	0
$747$	3131.35	0.153374
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−15601.0	−0.758040	−0.379020	−	0.925388i	$-0.623739\pi$
$751$	−15601.0	−0.758040	−0.379020	+	0.925388i	$0.623739\pi$
$752$	0	0
$753$	4265.35	0.206425
$754$	0	0
$755$	−24522.0	−1.18205
$756$	0	0
$757$	2948.08	0.141545	0.0707725	−	0.997492i	$-0.477454\pi$
$757$	2948.08	0.141545	0.0707725	+	0.997492i	$0.477454\pi$
$758$	0	0
$759$	22914.4	1.09583
$760$	0	0
$761$	−1697.03	−0.0808375	−0.0404187	−	0.999183i	$-0.512869\pi$
$761$	−1697.03	−0.0808375	−0.0404187	+	0.999183i	$0.512869\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3898.09	−0.184230
$766$	0	0
$767$	17940.0	0.844556
$768$	0	0
$769$	96.7799	0.00453833	0.00226916	−	0.999997i	$-0.499278\pi$
$769$	96.7799	0.00453833	0.00226916	+	0.999997i	$0.499278\pi$
$770$	0	0
$771$	4397.45	0.205409
$772$	0	0
$773$	−36326.8	−1.69028	−0.845138	−	0.534548i	$-0.820482\pi$
$773$	−36326.8	−1.69028	−0.845138	+	0.534548i	$0.820482\pi$
$774$	0	0
$775$	−37325.9	−1.73005
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7878.93	0.362377
$780$	0	0
$781$	−6777.31	−0.310514
$782$	0	0
$783$	6485.62	0.296012
$784$	0	0
$785$	15315.6	0.696352
$786$	0	0
$787$	7096.46	0.321425	0.160713	−	0.987001i	$-0.448621\pi$
$787$	7096.46	0.321425	0.160713	+	0.987001i	$0.448621\pi$
$788$	0	0
$789$	−20974.2	−0.946388
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7198.40	0.322349
$794$	0	0
$795$	−3158.97	−0.140927
$796$	0	0
$797$	−40289.6	−1.79063	−0.895314	−	0.445436i	$-0.853049\pi$
$797$	−40289.6	−1.79063	−0.895314	+	0.445436i	$0.853049\pi$
$798$	0	0
$799$	−1858.45	−0.0822871
$800$	0	0
$801$	−10414.4	−0.459396
$802$	0	0
$803$	9398.03	0.413013
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2426.88	−0.105861
$808$	0	0
$809$	−11566.0	−0.502642	−0.251321	−	0.967904i	$-0.580865\pi$
$809$	−11566.0	−0.502642	−0.251321	+	0.967904i	$0.580865\pi$
$810$	0	0
$811$	18014.2	0.779981	0.389991	−	0.920819i	$-0.372478\pi$
$811$	18014.2	0.779981	0.389991	+	0.920819i	$0.372478\pi$
$812$	0	0
$813$	19985.3	0.862134
$814$	0	0
$815$	21007.9	0.902915
$816$	0	0
$817$	25138.3	1.07647
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16792.4	−0.713837	−0.356918	−	0.934136i	$-0.616173\pi$
$821$	−16792.4	−0.713837	−0.356918	+	0.934136i	$0.616173\pi$
$822$	0	0
$823$	8819.03	0.373526	0.186763	−	0.982405i	$-0.440200\pi$
$823$	8819.03	0.373526	0.186763	+	0.982405i	$0.440200\pi$
$824$	0	0
$825$	−19565.6	−0.825680
$826$	0	0
$827$	−8250.13	−0.346898	−0.173449	−	0.984843i	$-0.555491\pi$
$827$	−8250.13	−0.346898	−0.173449	+	0.984843i	$0.555491\pi$
$828$	0	0
$829$	21550.5	0.902871	0.451435	−	0.892304i	$-0.350912\pi$
$829$	21550.5	0.902871	0.451435	+	0.892304i	$0.350912\pi$
$830$	0	0
$831$	−22594.4	−0.943190
$832$	0	0
$833$	0	0
$834$	0	0
$835$	22436.4	0.929873
$836$	0	0
$837$	−8010.20	−0.330792
$838$	0	0
$839$	30130.4	1.23983	0.619916	−	0.784669i	$-0.287167\pi$
$839$	30130.4	1.23983	0.619916	+	0.784669i	$0.287167\pi$
$840$	0	0
$841$	33311.0	1.36582
$842$	0	0
$843$	5070.57	0.207164
$844$	0	0
$845$	−10906.7	−0.444025
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9534.69	0.385430
$850$	0	0
$851$	23801.8	0.958772
$852$	0	0
$853$	40738.6	1.63525	0.817623	−	0.575754i	$-0.195291\pi$
$853$	40738.6	1.63525	0.817623	+	0.575754i	$0.195291\pi$
$854$	0	0
$855$	10905.5	0.436210
$856$	0	0
$857$	36508.2	1.45519	0.727594	−	0.686008i	$-0.240638\pi$
$857$	36508.2	1.45519	0.727594	+	0.686008i	$0.240638\pi$
$858$	0	0
$859$	−21980.8	−0.873081	−0.436541	−	0.899685i	$-0.643796\pi$
$859$	−21980.8	−0.873081	−0.436541	+	0.899685i	$0.643796\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23426.1	0.924024	0.462012	−	0.886874i	$-0.347128\pi$
$863$	23426.1	0.924024	0.462012	+	0.886874i	$0.347128\pi$
$864$	0	0
$865$	16416.7	0.645301
$866$	0	0
$867$	−12495.2	−0.489456
$868$	0	0
$869$	−21237.5	−0.829037
$870$	0	0
$871$	21174.3	0.823725
$872$	0	0
$873$	−14564.7	−0.564651
$874$	0	0
$875$	0	0
$876$	0	0
$877$	307.401	0.0118360	0.00591802	−	0.999982i	$-0.498116\pi$
$877$	307.401	0.0118360	0.00591802	+	0.999982i	$0.498116\pi$
$878$	0	0
$879$	6529.60	0.250555
$880$	0	0
$881$	19941.7	0.762605	0.381302	−	0.924450i	$-0.375476\pi$
$881$	19941.7	0.762605	0.381302	+	0.924450i	$0.375476\pi$
$882$	0	0
$883$	37524.1	1.43011	0.715056	−	0.699068i	$-0.246401\pi$
$883$	37524.1	1.43011	0.715056	+	0.699068i	$0.246401\pi$
$884$	0	0
$885$	−21946.8	−0.833598
$886$	0	0
$887$	−2880.20	−0.109028	−0.0545140	−	0.998513i	$-0.517361\pi$
$887$	−2880.20	−0.109028	−0.0545140	+	0.998513i	$0.517361\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4198.81	−0.157874
$892$	0	0
$893$	5199.30	0.194835
$894$	0	0
$895$	12153.9	0.453922
$896$	0	0
$897$	17167.8	0.639036
$898$	0	0
$899$	−71263.6	−2.64380
$900$	0	0
$901$	1818.36	0.0672347
$902$	0	0
$903$	0	0
$904$	0	0
$905$	62678.7	2.30222
$906$	0	0
$907$	−18319.2	−0.670651	−0.335326	−	0.942102i	$-0.608846\pi$
$907$	−18319.2	−0.670651	−0.335326	+	0.942102i	$0.608846\pi$
$908$	0	0
$909$	−6468.68	−0.236031
$910$	0	0
$911$	−46150.7	−1.67842	−0.839210	−	0.543807i	$-0.816982\pi$
$911$	−46150.7	−1.67842	−0.839210	+	0.543807i	$0.816982\pi$
$912$	0	0
$913$	−18035.6	−0.653769
$914$	0	0
$915$	−8806.16	−0.318167
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−47264.3	−1.69652	−0.848261	−	0.529578i	$-0.822350\pi$
$919$	−47264.3	−1.69652	−0.848261	+	0.529578i	$0.822350\pi$
$920$	0	0
$921$	1870.51	0.0669223
$922$	0	0
$923$	−5077.66	−0.181076
$924$	0	0
$925$	−20323.3	−0.722407
$926$	0	0
$927$	−14504.2	−0.513895
$928$	0	0
$929$	53271.5	1.88136	0.940678	−	0.339301i	$-0.110190\pi$
$929$	53271.5	1.88136	0.940678	+	0.339301i	$0.110190\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1403.98	0.0492649
$934$	0	0
$935$	22451.8	0.785295
$936$	0	0
$937$	−17197.8	−0.599602	−0.299801	−	0.954002i	$-0.596920\pi$
$937$	−17197.8	−0.599602	−0.299801	+	0.954002i	$0.596920\pi$
$938$	0	0
$939$	−10838.4	−0.376676
$940$	0	0
$941$	−28834.9	−0.998926	−0.499463	−	0.866335i	$-0.666469\pi$
$941$	−28834.9	−0.998926	−0.499463	+	0.866335i	$0.666469\pi$
$942$	0	0
$943$	−15173.5	−0.523986
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51841.2	1.77890	0.889448	−	0.457037i	$-0.151089\pi$
$947$	51841.2	1.77890	0.889448	+	0.457037i	$0.151089\pi$
$948$	0	0
$949$	7041.14	0.240848
$950$	0	0
$951$	13595.2	0.463570
$952$	0	0
$953$	−5887.31	−0.200114	−0.100057	−	0.994982i	$-0.531903\pi$
$953$	−5887.31	−0.200114	−0.100057	+	0.994982i	$0.531903\pi$
$954$	0	0
$955$	28590.7	0.968767
$956$	0	0
$957$	−37355.1	−1.26178
$958$	0	0
$959$	0	0
$960$	0	0
$961$	58224.6	1.95444
$962$	0	0
$963$	−8412.03	−0.281489
$964$	0	0
$965$	53384.4	1.78083
$966$	0	0
$967$	36620.0	1.21781	0.608904	−	0.793244i	$-0.291609\pi$
$967$	36620.0	1.21781	0.608904	+	0.793244i	$0.291609\pi$
$968$	0	0
$969$	−6277.41	−0.208111
$970$	0	0
$971$	−42364.9	−1.40016	−0.700079	−	0.714065i	$-0.746852\pi$
$971$	−42364.9	−1.40016	−0.700079	+	0.714065i	$0.746852\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−14658.8	−0.481495
$976$	0	0
$977$	5022.32	0.164461	0.0822304	−	0.996613i	$-0.473796\pi$
$977$	5022.32	0.164461	0.0822304	+	0.996613i	$0.473796\pi$
$978$	0	0
$979$	59983.8	1.95821
$980$	0	0
$981$	−10773.2	−0.350623
$982$	0	0
$983$	28292.8	0.918005	0.459003	−	0.888435i	$-0.348207\pi$
$983$	28292.8	0.918005	0.459003	+	0.888435i	$0.348207\pi$
$984$	0	0
$985$	−73045.7	−2.36287
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48412.3	−1.55655
$990$	0	0
$991$	−36401.1	−1.16682	−0.583410	−	0.812178i	$-0.698282\pi$
$991$	−36401.1	−1.16682	−0.583410	+	0.812178i	$0.698282\pi$
$992$	0	0
$993$	3711.17	0.118601
$994$	0	0
$995$	−35314.6	−1.12517
$996$	0	0
$997$	2357.19	0.0748777	0.0374389	−	0.999299i	$-0.488080\pi$
$997$	2357.19	0.0748777	0.0374389	+	0.999299i	$0.488080\pi$
$998$	0	0
$999$	−4361.42	−0.138127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.ca.1.2		2
4.3	odd	2		294.4.a.m.1.2		2
7.3	odd	6		336.4.q.j.289.2		4
7.5	odd	6		336.4.q.j.193.2		4
7.6	odd	2		2352.4.a.bq.1.1		2
12.11	even	2		882.4.a.z.1.1		2
28.3	even	6		42.4.e.c.37.2	yes	4
28.11	odd	6		294.4.e.l.79.1		4
28.19	even	6		42.4.e.c.25.2	✓	4
28.23	odd	6		294.4.e.l.67.1		4
28.27	even	2		294.4.a.n.1.1		2
84.11	even	6		882.4.g.bf.667.2		4
84.23	even	6		882.4.g.bf.361.2		4
84.47	odd	6		126.4.g.g.109.1		4
84.59	odd	6		126.4.g.g.37.1		4
84.83	odd	2		882.4.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.e.c.25.2	✓	4	28.19	even	6
42.4.e.c.37.2	yes	4	28.3	even	6
126.4.g.g.37.1		4	84.59	odd	6
126.4.g.g.109.1		4	84.47	odd	6
294.4.a.m.1.2		2	4.3	odd	2
294.4.a.n.1.1		2	28.27	even	2
294.4.e.l.67.1		4	28.23	odd	6
294.4.e.l.79.1		4	28.11	odd	6
336.4.q.j.193.2		4	7.5	odd	6
336.4.q.j.289.2		4	7.3	odd	6
882.4.a.v.1.2		2	84.83	odd	2
882.4.a.z.1.1		2	12.11	even	2
882.4.g.bf.361.2		4	84.23	even	6
882.4.g.bf.667.2		4	84.11	even	6
2352.4.a.bq.1.1		2	7.6	odd	2
2352.4.a.ca.1.2		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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +15.8371 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +15.8371 q^{5} +9.00000 q^{9} -51.8371 q^{11} -38.8371 q^{13} +47.5114 q^{15} -27.3485 q^{17} +76.5114 q^{19} -147.348 q^{23} +125.814 q^{25} +27.0000 q^{27} +240.208 q^{29} -296.674 q^{31} -155.511 q^{33} -161.534 q^{37} -116.511 q^{39} +102.977 q^{41} +328.557 q^{43} +142.534 q^{45} +67.9546 q^{47} -82.0454 q^{51} -66.4886 q^{53} -820.951 q^{55} +229.534 q^{57} -461.928 q^{59} -185.348 q^{61} -615.068 q^{65} -545.208 q^{67} -442.045 q^{69} +130.742 q^{71} -181.299 q^{73} +377.443 q^{75} +409.697 q^{79} +81.0000 q^{81} +347.928 q^{83} -433.121 q^{85} +720.625 q^{87} -1157.16 q^{89} -890.023 q^{93} +1211.72 q^{95} -1618.30 q^{97} -466.534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 5 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 5 * q^5 + 18 * q^9	\( 2 q + 6 q^{3} - 5 q^{5} + 18 q^{9} - 67 q^{11} - 41 q^{13} - 15 q^{15} + 92 q^{17} + 43 q^{19} - 148 q^{23} + 435 q^{25} + 54 q^{27} + 77 q^{29} - 520 q^{31} - 201 q^{33} + 7 q^{37} - 123 q^{39} + 426 q^{41} + 107 q^{43} - 45 q^{45} + 576 q^{47} + 276 q^{51} - 243 q^{53} - 505 q^{55} + 129 q^{57} - 7 q^{59} - 224 q^{61} - 570 q^{65} - 687 q^{67} - 444 q^{69} - 472 q^{71} + 921 q^{73} + 1305 q^{75} + 526 q^{79} + 162 q^{81} - 221 q^{83} - 2920 q^{85} + 231 q^{87} - 774 q^{89} - 1560 q^{93} + 1910 q^{95} - 1953 q^{97} - 603 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 5 * q^5 + 18 * q^9 - 67 * q^11 - 41 * q^13 - 15 * q^15 + 92 * q^17 + 43 * q^19 - 148 * q^23 + 435 * q^25 + 54 * q^27 + 77 * q^29 - 520 * q^31 - 201 * q^33 + 7 * q^37 - 123 * q^39 + 426 * q^41 + 107 * q^43 - 45 * q^45 + 576 * q^47 + 276 * q^51 - 243 * q^53 - 505 * q^55 + 129 * q^57 - 7 * q^59 - 224 * q^61 - 570 * q^65 - 687 * q^67 - 444 * q^69 - 472 * q^71 + 921 * q^73 + 1305 * q^75 + 526 * q^79 + 162 * q^81 - 221 * q^83 - 2920 * q^85 + 231 * q^87 - 774 * q^89 - 1560 * q^93 + 1910 * q^95 - 1953 * q^97 - 603 * q^99

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