LMFDB - Embedded newform 2352.3.m.l.1471.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(1471,2352)]

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([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2352.1471");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.m (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.2
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2352.1471
Dual form			2352.3.m.l.1471.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +7.41421 q^{5} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +7.41421 q^{5} -3.00000 q^{9} -20.1903i q^{11} +21.8995 q^{13} -12.8418i q^{15} +13.0711 q^{17} -14.6969i q^{19} -40.9749i q^{23} +29.9706 q^{25} +5.19615i q^{27} +32.4853 q^{29} +51.0190i q^{31} -34.9706 q^{33} -33.9411 q^{37} -37.9310i q^{39} -8.10051 q^{41} +69.7744i q^{43} -22.2426 q^{45} -28.5533i q^{47} -22.6398i q^{51} -72.9117 q^{53} -149.695i q^{55} -25.4558 q^{57} -4.40661i q^{59} -33.9828 q^{61} +162.368 q^{65} +30.5826i q^{67} -70.9706 q^{69} +27.6108i q^{71} +61.4975 q^{73} -51.9105i q^{75} -90.5590i q^{79} +9.00000 q^{81} -5.24714i q^{83} +96.9117 q^{85} -56.2662i q^{87} -56.8701 q^{89} +88.3675 q^{93} -108.966i q^{95} +123.556 q^{97} +60.5708i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 24 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 24 * q^5 - 12 * q^9	$ 4 q + 24 q^{5} - 12 q^{9} + 48 q^{13} + 24 q^{17} + 52 q^{25} + 96 q^{29} - 72 q^{33} - 72 q^{41} - 72 q^{45} - 88 q^{53} + 96 q^{61} + 344 q^{65} - 216 q^{69} + 48 q^{73} + 36 q^{81} + 184 q^{85} - 120 q^{89} + 48 q^{93} + 432 q^{97}+O(q^{100}) $ 4 * q + 24 * q^5 - 12 * q^9 + 48 * q^13 + 24 * q^17 + 52 * q^25 + 96 * q^29 - 72 * q^33 - 72 * q^41 - 72 * q^45 - 88 * q^53 + 96 * q^61 + 344 * q^65 - 216 * q^69 + 48 * q^73 + 36 * q^81 + 184 * q^85 - 120 * q^89 + 48 * q^93 + 432 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	7.41421	1.48284	0.741421	−	0.671040i	$-0.234152\pi$
$5$	7.41421	1.48284	0.741421	+	0.671040i	$0.234152\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 20.1903i	− 1.83548i	−0.397183	−	0.917739i	$-0.630012\pi$
$11$	− 20.1903i	− 1.83548i	0.397183	−	0.917739i	$-0.369988\pi$
$12$	0	0
$13$	21.8995	1.68458	0.842288	−	0.539027i	$-0.181208\pi$
$13$	21.8995	1.68458	0.842288	+	0.539027i	$0.181208\pi$
$14$	0	0
$15$	− 12.8418i	− 0.856120i
$16$	0	0
$17$	13.0711	0.768886	0.384443	−	0.923149i	$-0.374393\pi$
$17$	13.0711	0.768886	0.384443	+	0.923149i	$0.374393\pi$
$18$	0	0
$19$	− 14.6969i	− 0.773523i	−0.922180	−	0.386762i	$-0.873594\pi$
$19$	− 14.6969i	− 0.773523i	0.922180	−	0.386762i	$-0.126406\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 40.9749i	− 1.78152i	−0.454478	−	0.890758i	$-0.650174\pi$
$23$	− 40.9749i	− 1.78152i	0.454478	−	0.890758i	$-0.349826\pi$
$24$	0	0
$25$	29.9706	1.19882
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	32.4853	1.12018	0.560091	−	0.828431i	$-0.310766\pi$
$29$	32.4853	1.12018	0.560091	+	0.828431i	$0.310766\pi$
$30$	0	0
$31$	51.0190i	1.64577i	0.568204	+	0.822887i	$0.307638\pi$
$31$	51.0190i	1.64577i	−0.568204	+	0.822887i	$0.692362\pi$
$32$	0	0
$33$	−34.9706	−1.05971
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−33.9411	−0.917328	−0.458664	−	0.888610i	$-0.651672\pi$
$37$	−33.9411	−0.917328	−0.458664	+	0.888610i	$0.651672\pi$
$38$	0	0
$39$	− 37.9310i	− 0.972591i
$40$	0	0
$41$	−8.10051	−0.197573	−0.0987866	−	0.995109i	$-0.531496\pi$
$41$	−8.10051	−0.197573	−0.0987866	+	0.995109i	$0.531496\pi$
$42$	0	0
$43$	69.7744i	1.62266i	0.584588	+	0.811330i	$0.301256\pi$
$43$	69.7744i	1.62266i	−0.584588	+	0.811330i	$0.698744\pi$
$44$	0	0
$45$	−22.2426	−0.494281
$46$	0	0
$47$	− 28.5533i	− 0.607518i	−0.952749	−	0.303759i	$-0.901758\pi$
$47$	− 28.5533i	− 0.607518i	0.952749	−	0.303759i	$-0.0982418\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 22.6398i	− 0.443917i
$52$	0	0
$53$	−72.9117	−1.37569	−0.687846	−	0.725857i	$-0.741444\pi$
$53$	−72.9117	−1.37569	−0.687846	+	0.725857i	$0.741444\pi$
$54$	0	0
$55$	− 149.695i	− 2.72173i
$56$	0	0
$57$	−25.4558	−0.446594
$58$	0	0
$59$	− 4.40661i	− 0.0746883i	−0.999302	−	0.0373441i	$-0.988110\pi$
$59$	− 4.40661i	− 0.0746883i	0.999302	−	0.0373441i	$-0.0118898\pi$
$60$	0	0
$61$	−33.9828	−0.557094	−0.278547	−	0.960423i	$-0.589853\pi$
$61$	−33.9828	−0.557094	−0.278547	+	0.960423i	$0.589853\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	162.368	2.49796
$66$	0	0
$67$	30.5826i	0.456456i	0.973608	+	0.228228i	$0.0732932\pi$
$67$	30.5826i	0.456456i	−0.973608	+	0.228228i	$0.926707\pi$
$68$	0	0
$69$	−70.9706	−1.02856
$70$	0	0
$71$	27.6108i	0.388885i	0.980914	+	0.194443i	$0.0622898\pi$
$71$	27.6108i	0.388885i	−0.980914	+	0.194443i	$0.937710\pi$
$72$	0	0
$73$	61.4975	0.842431	0.421216	−	0.906961i	$-0.361604\pi$
$73$	61.4975	0.842431	0.421216	+	0.906961i	$0.361604\pi$
$74$	0	0
$75$	− 51.9105i	− 0.692140i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 90.5590i	− 1.14632i	−0.819445	−	0.573158i	$-0.805718\pi$
$79$	− 90.5590i	− 1.14632i	0.819445	−	0.573158i	$-0.194282\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 5.24714i	− 0.0632185i	−0.999500	−	0.0316093i	$-0.989937\pi$
$83$	− 5.24714i	− 0.0632185i	0.999500	−	0.0316093i	$-0.0100632\pi$
$84$	0	0
$85$	96.9117	1.14014
$86$	0	0
$87$	− 56.2662i	− 0.646737i
$88$	0	0
$89$	−56.8701	−0.638989	−0.319495	−	0.947588i	$-0.603513\pi$
$89$	−56.8701	−0.638989	−0.319495	+	0.947588i	$0.603513\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	88.3675	0.950189
$94$	0	0
$95$	− 108.966i	− 1.14701i
$96$	0	0
$97$	123.556	1.27378	0.636888	−	0.770956i	$-0.280221\pi$
$97$	123.556	1.27378	0.636888	+	0.770956i	$0.280221\pi$
$98$	0	0
$99$	60.5708i	0.611826i
$100$	0	0
$101$	−70.3259	−0.696296	−0.348148	−	0.937440i	$-0.613189\pi$
$101$	−70.3259	−0.696296	−0.348148	+	0.937440i	$0.613189\pi$
$102$	0	0
$103$	109.807i	1.06609i	0.846088	+	0.533043i	$0.178951\pi$
$103$	109.807i	1.06609i	−0.846088	+	0.533043i	$0.821049\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.1471i	0.141561i	0.997492	+	0.0707807i	$0.0225491\pi$
$107$	15.1471i	0.141561i	−0.997492	+	0.0707807i	$0.977451\pi$
$108$	0	0
$109$	72.8528	0.668374	0.334187	−	0.942507i	$-0.391538\pi$
$109$	72.8528	0.668374	0.334187	+	0.942507i	$0.391538\pi$
$110$	0	0
$111$	58.7878i	0.529619i
$112$	0	0
$113$	−87.8234	−0.777198	−0.388599	−	0.921407i	$-0.627041\pi$
$113$	−87.8234	−0.777198	−0.388599	+	0.921407i	$0.627041\pi$
$114$	0	0
$115$	− 303.796i	− 2.64171i
$116$	0	0
$117$	−65.6985	−0.561526
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−286.647	−2.36898
$122$	0	0
$123$	14.0305i	0.114069i
$124$	0	0
$125$	36.8528	0.294823
$126$	0	0
$127$	104.211i	0.820563i	0.911959	+	0.410281i	$0.134570\pi$
$127$	104.211i	0.820563i	−0.911959	+	0.410281i	$0.865430\pi$
$128$	0	0
$129$	120.853	0.936844
$130$	0	0
$131$	− 91.7477i	− 0.700364i	−0.936682	−	0.350182i	$-0.886120\pi$
$131$	− 91.7477i	− 0.700364i	0.936682	−	0.350182i	$-0.113880\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	38.5254i	0.285373i
$136$	0	0
$137$	186.426	1.36078	0.680388	−	0.732852i	$-0.261811\pi$
$137$	186.426	1.36078	0.680388	+	0.732852i	$0.261811\pi$
$138$	0	0
$139$	141.722i	1.01958i	0.860298	+	0.509792i	$0.170278\pi$
$139$	141.722i	1.01958i	−0.860298	+	0.509792i	$0.829722\pi$
$140$	0	0
$141$	−49.4558	−0.350751
$142$	0	0
$143$	− 442.157i	− 3.09200i
$144$	0	0
$145$	240.853	1.66105
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−48.9117	−0.328266	−0.164133	−	0.986438i	$-0.552483\pi$
$149$	−48.9117	−0.328266	−0.164133	+	0.986438i	$0.552483\pi$
$150$	0	0
$151$	− 214.366i	− 1.41965i	−0.704381	−	0.709823i	$-0.748775\pi$
$151$	− 214.366i	− 1.41965i	0.704381	−	0.709823i	$-0.251225\pi$
$152$	0	0
$153$	−39.2132	−0.256295
$154$	0	0
$155$	378.266i	2.44043i
$156$	0	0
$157$	55.0711	0.350771	0.175386	−	0.984500i	$-0.443883\pi$
$157$	55.0711	0.350771	0.175386	+	0.984500i	$0.443883\pi$
$158$	0	0
$159$	126.287i	0.794256i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	128.274i	0.786955i	0.919334	+	0.393478i	$0.128728\pi$
$163$	128.274i	0.786955i	−0.919334	+	0.393478i	$0.871272\pi$
$164$	0	0
$165$	−259.279	−1.57139
$166$	0	0
$167$	− 239.762i	− 1.43570i	−0.696199	−	0.717849i	$-0.745127\pi$
$167$	− 239.762i	− 1.43570i	0.696199	−	0.717849i	$-0.254873\pi$
$168$	0	0
$169$	310.588	1.83780
$170$	0	0
$171$	44.0908i	0.257841i
$172$	0	0
$173$	−15.7330	−0.0909420	−0.0454710	−	0.998966i	$-0.514479\pi$
$173$	−15.7330	−0.0909420	−0.0454710	+	0.998966i	$0.514479\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.63247	−0.0431213
$178$	0	0
$179$	95.9081i	0.535800i	0.963447	+	0.267900i	$0.0863296\pi$
$179$	95.9081i	0.535800i	−0.963447	+	0.267900i	$0.913670\pi$
$180$	0	0
$181$	221.012	1.22106	0.610531	−	0.791992i	$-0.290956\pi$
$181$	221.012	1.22106	0.610531	+	0.791992i	$0.290956\pi$
$182$	0	0
$183$	58.8599i	0.321639i
$184$	0	0
$185$	−251.647	−1.36025
$186$	0	0
$187$	− 263.908i	− 1.41127i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	51.9615i	0.272050i	0.990705	+	0.136025i	$0.0434327\pi$
$191$	51.9615i	0.272050i	−0.990705	+	0.136025i	$0.956567\pi$
$192$	0	0
$193$	−187.823	−0.973178	−0.486589	−	0.873631i	$-0.661759\pi$
$193$	−187.823	−0.973178	−0.486589	+	0.873631i	$0.661759\pi$
$194$	0	0
$195$	− 281.229i	− 1.44220i
$196$	0	0
$197$	−47.0883	−0.239027	−0.119513	−	0.992833i	$-0.538133\pi$
$197$	−47.0883	−0.239027	−0.119513	+	0.992833i	$0.538133\pi$
$198$	0	0
$199$	− 252.777i	− 1.27024i	−0.772414	−	0.635119i	$-0.780951\pi$
$199$	− 252.777i	− 1.27024i	0.772414	−	0.635119i	$-0.219049\pi$
$200$	0	0
$201$	52.9706	0.263535
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−60.0589	−0.292970
$206$	0	0
$207$	122.925i	0.593839i
$208$	0	0
$209$	−296.735	−1.41978
$210$	0	0
$211$	350.061i	1.65906i	0.558465	+	0.829528i	$0.311390\pi$
$211$	350.061i	1.65906i	−0.558465	+	0.829528i	$0.688610\pi$
$212$	0	0
$213$	47.8234	0.224523
$214$	0	0
$215$	517.322i	2.40615i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 106.517i	− 0.486378i
$220$	0	0
$221$	286.250	1.29525
$222$	0	0
$223$	17.2185i	0.0772132i	0.999254	+	0.0386066i	$0.0122919\pi$
$223$	17.2185i	0.0772132i	−0.999254	+	0.0386066i	$0.987708\pi$
$224$	0	0
$225$	−89.9117	−0.399608
$226$	0	0
$227$	61.1054i	0.269187i	0.990901	+	0.134593i	$0.0429728\pi$
$227$	61.1054i	0.269187i	−0.990901	+	0.134593i	$0.957027\pi$
$228$	0	0
$229$	2.10051	0.00917251	0.00458626	−	0.999989i	$-0.498540\pi$
$229$	2.10051	0.00917251	0.00458626	+	0.999989i	$0.498540\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	425.220	1.82498	0.912490	−	0.409099i	$-0.134157\pi$
$233$	425.220	1.82498	0.912490	+	0.409099i	$0.134157\pi$
$234$	0	0
$235$	− 211.701i	− 0.900854i
$236$	0	0
$237$	−156.853	−0.661826
$238$	0	0
$239$	230.991i	0.966488i	0.875486	+	0.483244i	$0.160542\pi$
$239$	230.991i	0.966488i	−0.875486	+	0.483244i	$0.839458\pi$
$240$	0	0
$241$	53.1787	0.220659	0.110329	−	0.993895i	$-0.464809\pi$
$241$	53.1787	0.220659	0.110329	+	0.993895i	$0.464809\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 321.856i	− 1.30306i
$248$	0	0
$249$	−9.08831	−0.0364992
$250$	0	0
$251$	− 269.359i	− 1.07314i	−0.843854	−	0.536572i	$-0.819719\pi$
$251$	− 269.359i	− 1.07314i	0.843854	−	0.536572i	$-0.180281\pi$
$252$	0	0
$253$	−827.294	−3.26993
$254$	0	0
$255$	− 167.856i	− 0.658259i
$256$	0	0
$257$	116.517	0.453373	0.226686	−	0.973968i	$-0.427211\pi$
$257$	116.517	0.453373	0.226686	+	0.973968i	$0.427211\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−97.4558	−0.373394
$262$	0	0
$263$	− 164.782i	− 0.626549i	−0.949663	−	0.313274i	$-0.898574\pi$
$263$	− 164.782i	− 0.626549i	0.949663	−	0.313274i	$-0.101426\pi$
$264$	0	0
$265$	−540.583	−2.03994
$266$	0	0
$267$	98.5018i	0.368921i
$268$	0	0
$269$	−482.590	−1.79401	−0.897007	−	0.442016i	$-0.854264\pi$
$269$	−482.590	−1.79401	−0.897007	+	0.442016i	$0.854264\pi$
$270$	0	0
$271$	− 225.701i	− 0.832846i	−0.909171	−	0.416423i	$-0.863284\pi$
$271$	− 225.701i	− 0.832846i	0.909171	−	0.416423i	$-0.136716\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 605.114i	− 2.20041i
$276$	0	0
$277$	−76.1766	−0.275006	−0.137503	−	0.990501i	$-0.543908\pi$
$277$	−76.1766	−0.275006	−0.137503	+	0.990501i	$0.543908\pi$
$278$	0	0
$279$	− 153.057i	− 0.548592i
$280$	0	0
$281$	−86.3087	−0.307148	−0.153574	−	0.988137i	$-0.549078\pi$
$281$	−86.3087	−0.307148	−0.153574	+	0.988137i	$0.549078\pi$
$282$	0	0
$283$	− 285.101i	− 1.00742i	−0.863872	−	0.503712i	$-0.831967\pi$
$283$	− 285.101i	− 1.00742i	0.863872	−	0.503712i	$-0.168033\pi$
$284$	0	0
$285$	−188.735	−0.662228
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−118.147	−0.408814
$290$	0	0
$291$	− 214.006i	− 0.735415i
$292$	0	0
$293$	454.056	1.54968	0.774839	−	0.632158i	$-0.217831\pi$
$293$	454.056	1.54968	0.774839	+	0.632158i	$0.217831\pi$
$294$	0	0
$295$	− 32.6715i	− 0.110751i
$296$	0	0
$297$	104.912	0.353238
$298$	0	0
$299$	− 897.329i	− 3.00110i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	121.808i	0.402007i
$304$	0	0
$305$	−251.955	−0.826083
$306$	0	0
$307$	− 300.230i	− 0.977949i	−0.872298	−	0.488975i	$-0.837371\pi$
$307$	− 300.230i	− 0.977949i	0.872298	−	0.488975i	$-0.162629\pi$
$308$	0	0
$309$	190.191	0.615505
$310$	0	0
$311$	253.210i	0.814180i	0.913388	+	0.407090i	$0.133457\pi$
$311$	253.210i	0.814180i	−0.913388	+	0.407090i	$0.866543\pi$
$312$	0	0
$313$	−333.296	−1.06484	−0.532422	−	0.846479i	$-0.678718\pi$
$313$	−333.296	−1.06484	−0.532422	+	0.846479i	$0.678718\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	109.265	0.344684	0.172342	−	0.985037i	$-0.444867\pi$
$317$	109.265	0.344684	0.172342	+	0.985037i	$0.444867\pi$
$318$	0	0
$319$	− 655.886i	− 2.05607i
$320$	0	0
$321$	26.2355	0.0817305
$322$	0	0
$323$	− 192.105i	− 0.594751i
$324$	0	0
$325$	656.340	2.01951
$326$	0	0
$327$	− 126.185i	− 0.385886i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 345.018i	− 1.04235i	−0.853450	−	0.521175i	$-0.825494\pi$
$331$	− 345.018i	− 1.04235i	0.853450	−	0.521175i	$-0.174506\pi$
$332$	0	0
$333$	101.823	0.305776
$334$	0	0
$335$	226.746i	0.676853i
$336$	0	0
$337$	374.558	1.11145	0.555725	−	0.831366i	$-0.312441\pi$
$337$	374.558	1.11145	0.555725	+	0.831366i	$0.312441\pi$
$338$	0	0
$339$	152.115i	0.448715i
$340$	0	0
$341$	1030.09	3.02078
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−526.191	−1.52519
$346$	0	0
$347$	− 333.148i	− 0.960081i	−0.877246	−	0.480040i	$-0.840622\pi$
$347$	− 333.148i	− 0.960081i	0.877246	−	0.480040i	$-0.159378\pi$
$348$	0	0
$349$	−119.958	−0.343720	−0.171860	−	0.985121i	$-0.554978\pi$
$349$	−119.958	−0.343720	−0.171860	+	0.985121i	$0.554978\pi$
$350$	0	0
$351$	113.793i	0.324197i
$352$	0	0
$353$	−82.6589	−0.234161	−0.117081	−	0.993122i	$-0.537354\pi$
$353$	−82.6589	−0.234161	−0.117081	+	0.993122i	$0.537354\pi$
$354$	0	0
$355$	204.713i	0.576655i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 22.5676i	− 0.0628625i	−0.999506	−	0.0314313i	$-0.989993\pi$
$359$	− 22.5676i	− 0.0628625i	0.999506	−	0.0314313i	$-0.0100065\pi$
$360$	0	0
$361$	145.000	0.401662
$362$	0	0
$363$	496.487i	1.36773i
$364$	0	0
$365$	455.955	1.24919
$366$	0	0
$367$	583.242i	1.58922i	0.607123	+	0.794608i	$0.292324\pi$
$367$	583.242i	1.58922i	−0.607123	+	0.794608i	$0.707676\pi$
$368$	0	0
$369$	24.3015	0.0658578
$370$	0	0
$371$	0	0
$372$	0	0
$373$	87.4701	0.234504	0.117252	−	0.993102i	$-0.462591\pi$
$373$	87.4701	0.234504	0.117252	+	0.993102i	$0.462591\pi$
$374$	0	0
$375$	− 63.8309i	− 0.170216i
$376$	0	0
$377$	711.411	1.88703
$378$	0	0
$379$	− 122.330i	− 0.322771i	−0.986891	−	0.161386i	$-0.948404\pi$
$379$	− 122.330i	− 0.322771i	0.986891	−	0.161386i	$-0.0515963\pi$
$380$	0	0
$381$	180.500	0.473752
$382$	0	0
$383$	− 13.4485i	− 0.0351136i	−0.999846	−	0.0175568i	$-0.994411\pi$
$383$	− 13.4485i	− 0.0351136i	0.999846	−	0.0175568i	$-0.00558879\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 209.323i	− 0.540887i
$388$	0	0
$389$	−79.2792	−0.203803	−0.101901	−	0.994795i	$-0.532493\pi$
$389$	−79.2792	−0.203803	−0.101901	+	0.994795i	$0.532493\pi$
$390$	0	0
$391$	− 535.585i	− 1.36978i
$392$	0	0
$393$	−158.912	−0.404355
$394$	0	0
$395$	− 671.424i	− 1.69981i
$396$	0	0
$397$	−374.600	−0.943577	−0.471789	−	0.881712i	$-0.656391\pi$
$397$	−374.600	−0.943577	−0.471789	+	0.881712i	$0.656391\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	60.1035	0.149884	0.0749420	−	0.997188i	$-0.476123\pi$
$401$	60.1035	0.149884	0.0749420	+	0.997188i	$0.476123\pi$
$402$	0	0
$403$	1117.29i	2.77243i
$404$	0	0
$405$	66.7279	0.164760
$406$	0	0
$407$	685.280i	1.68374i
$408$	0	0
$409$	−801.296	−1.95916	−0.979580	−	0.201055i	$-0.935563\pi$
$409$	−801.296	−1.95916	−0.979580	+	0.201055i	$0.935563\pi$
$410$	0	0
$411$	− 322.900i	− 0.785645i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 38.9034i	− 0.0937432i
$416$	0	0
$417$	245.470	0.588657
$418$	0	0
$419$	− 171.957i	− 0.410398i	−0.978720	−	0.205199i	$-0.934216\pi$
$419$	− 171.957i	− 0.410398i	0.978720	−	0.205199i	$-0.0657841\pi$
$420$	0	0
$421$	−369.294	−0.877182	−0.438591	−	0.898687i	$-0.644522\pi$
$421$	−369.294	−0.877182	−0.438591	+	0.898687i	$0.644522\pi$
$422$	0	0
$423$	85.6600i	0.202506i
$424$	0	0
$425$	391.747	0.921758
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−765.838	−1.78517
$430$	0	0
$431$	− 415.675i	− 0.964442i	−0.876049	−	0.482221i	$-0.839830\pi$
$431$	− 415.675i	− 0.964442i	0.876049	−	0.482221i	$-0.160170\pi$
$432$	0	0
$433$	−484.409	−1.11873	−0.559364	−	0.828922i	$-0.688955\pi$
$433$	−484.409	−1.11873	−0.559364	+	0.828922i	$0.688955\pi$
$434$	0	0
$435$	− 417.169i	− 0.959010i
$436$	0	0
$437$	−602.205	−1.37804
$438$	0	0
$439$	543.762i	1.23864i	0.785140	+	0.619319i	$0.212591\pi$
$439$	543.762i	1.23864i	−0.785140	+	0.619319i	$0.787409\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	418.952i	0.945716i	0.881139	+	0.472858i	$0.156778\pi$
$443$	418.952i	0.945716i	−0.881139	+	0.472858i	$0.843222\pi$
$444$	0	0
$445$	−421.647	−0.947521
$446$	0	0
$447$	84.7175i	0.189525i
$448$	0	0
$449$	697.647	1.55378	0.776889	−	0.629637i	$-0.216796\pi$
$449$	697.647	1.55378	0.776889	+	0.629637i	$0.216796\pi$
$450$	0	0
$451$	163.551i	0.362642i
$452$	0	0
$453$	−371.294	−0.819632
$454$	0	0
$455$	0	0
$456$	0	0
$457$	129.647	0.283691	0.141845	−	0.989889i	$-0.454696\pi$
$457$	129.647	0.283691	0.141845	+	0.989889i	$0.454696\pi$
$458$	0	0
$459$	67.9193i	0.147972i
$460$	0	0
$461$	185.002	0.401306	0.200653	−	0.979662i	$-0.435694\pi$
$461$	185.002	0.401306	0.200653	+	0.979662i	$0.435694\pi$
$462$	0	0
$463$	632.436i	1.36595i	0.730441	+	0.682976i	$0.239315\pi$
$463$	632.436i	1.36595i	−0.730441	+	0.682976i	$0.760685\pi$
$464$	0	0
$465$	655.176	1.40898
$466$	0	0
$467$	716.992i	1.53531i	0.640861	+	0.767657i	$0.278578\pi$
$467$	716.992i	1.53531i	−0.640861	+	0.767657i	$0.721422\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 95.3859i	− 0.202518i
$472$	0	0
$473$	1408.76	2.97836
$474$	0	0
$475$	− 440.476i	− 0.927317i
$476$	0	0
$477$	218.735	0.458564
$478$	0	0
$479$	466.915i	0.974771i	0.873187	+	0.487385i	$0.162049\pi$
$479$	466.915i	0.974771i	−0.873187	+	0.487385i	$0.837951\pi$
$480$	0	0
$481$	−743.294	−1.54531
$482$	0	0
$483$	0	0
$484$	0	0
$485$	916.073	1.88881
$486$	0	0
$487$	358.093i	0.735304i	0.929963	+	0.367652i	$0.119838\pi$
$487$	358.093i	0.735304i	−0.929963	+	0.367652i	$0.880162\pi$
$488$	0	0
$489$	222.177	0.454349
$490$	0	0
$491$	− 908.850i	− 1.85102i	−0.378724	−	0.925509i	$-0.623637\pi$
$491$	− 908.850i	− 1.85102i	0.378724	−	0.925509i	$-0.376363\pi$
$492$	0	0
$493$	424.617	0.861293
$494$	0	0
$495$	449.085i	0.907242i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	678.760i	1.36024i	0.733101	+	0.680120i	$0.238072\pi$
$499$	678.760i	1.36024i	−0.733101	+	0.680120i	$0.761928\pi$
$500$	0	0
$501$	−415.279	−0.828901
$502$	0	0
$503$	154.102i	0.306365i	0.988198	+	0.153182i	$0.0489522\pi$
$503$	154.102i	0.306365i	−0.988198	+	0.153182i	$0.951048\pi$
$504$	0	0
$505$	−521.411	−1.03250
$506$	0	0
$507$	− 537.954i	− 1.06105i
$508$	0	0
$509$	−474.561	−0.932341	−0.466170	−	0.884695i	$-0.654367\pi$
$509$	−474.561	−0.932341	−0.466170	+	0.884695i	$0.654367\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	76.3675	0.148865
$514$	0	0
$515$	814.131i	1.58084i
$516$	0	0
$517$	−576.500	−1.11509
$518$	0	0
$519$	27.2503i	0.0525054i
$520$	0	0
$521$	527.282	1.01206	0.506029	−	0.862516i	$-0.331113\pi$
$521$	527.282	1.01206	0.506029	+	0.862516i	$0.331113\pi$
$522$	0	0
$523$	77.4834i	0.148152i	0.997253	+	0.0740759i	$0.0236007\pi$
$523$	77.4834i	0.148152i	−0.997253	+	0.0740759i	$0.976399\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	666.873i	1.26541i
$528$	0	0
$529$	−1149.94	−2.17380
$530$	0	0
$531$	13.2198i	0.0248961i
$532$	0	0
$533$	−177.397	−0.332827
$534$	0	0
$535$	112.304i	0.209913i
$536$	0	0
$537$	166.118	0.309344
$538$	0	0
$539$	0	0
$540$	0	0
$541$	986.764	1.82396	0.911981	−	0.410232i	$-0.134552\pi$
$541$	986.764	1.82396	0.911981	+	0.410232i	$0.134552\pi$
$542$	0	0
$543$	− 382.804i	− 0.704980i
$544$	0	0
$545$	540.146	0.991094
$546$	0	0
$547$	− 152.301i	− 0.278430i	−0.990262	−	0.139215i	$-0.955542\pi$
$547$	− 152.301i	− 0.278430i	0.990262	−	0.139215i	$-0.0444579\pi$
$548$	0	0
$549$	101.948	0.185698
$550$	0	0
$551$	− 477.434i	− 0.866487i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	435.865i	0.785342i
$556$	0	0
$557$	−4.20519	−0.00754972	−0.00377486	−	0.999993i	$-0.501202\pi$
$557$	−4.20519	−0.00754972	−0.00377486	+	0.999993i	$0.501202\pi$
$558$	0	0
$559$	1528.02i	2.73350i
$560$	0	0
$561$	−457.103	−0.814800
$562$	0	0
$563$	− 564.571i	− 1.00279i	−0.865218	−	0.501395i	$-0.832820\pi$
$563$	− 564.571i	− 1.00279i	0.865218	−	0.501395i	$-0.167180\pi$
$564$	0	0
$565$	−651.141	−1.15246
$566$	0	0
$567$	0	0
$568$	0	0
$569$	818.662	1.43877	0.719387	−	0.694610i	$-0.244423\pi$
$569$	818.662	1.43877	0.719387	+	0.694610i	$0.244423\pi$
$570$	0	0
$571$	472.968i	0.828315i	0.910205	+	0.414157i	$0.135924\pi$
$571$	472.968i	0.828315i	−0.910205	+	0.414157i	$0.864076\pi$
$572$	0	0
$573$	90.0000	0.157068
$574$	0	0
$575$	− 1228.04i	− 2.13572i
$576$	0	0
$577$	950.060	1.64655	0.823276	−	0.567642i	$-0.192144\pi$
$577$	950.060	1.64655	0.823276	+	0.567642i	$0.192144\pi$
$578$	0	0
$579$	325.320i	0.561865i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1472.11i	2.52505i
$584$	0	0
$585$	−487.103	−0.832654
$586$	0	0
$587$	1123.87i	1.91460i	0.289095	+	0.957300i	$0.406646\pi$
$587$	1123.87i	1.91460i	−0.289095	+	0.957300i	$0.593354\pi$
$588$	0	0
$589$	749.823	1.27304
$590$	0	0
$591$	81.5593i	0.138002i
$592$	0	0
$593$	562.159	0.947992	0.473996	−	0.880527i	$-0.342811\pi$
$593$	562.159	0.947992	0.473996	+	0.880527i	$0.342811\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−437.823	−0.733372
$598$	0	0
$599$	394.025i	0.657804i	0.944364	+	0.328902i	$0.106679\pi$
$599$	394.025i	0.657804i	−0.944364	+	0.328902i	$0.893321\pi$
$600$	0	0
$601$	−824.943	−1.37262	−0.686309	−	0.727310i	$-0.740770\pi$
$601$	−824.943	−1.37262	−0.686309	+	0.727310i	$0.740770\pi$
$602$	0	0
$603$	− 91.7477i	− 0.152152i
$604$	0	0
$605$	−2125.26	−3.51283
$606$	0	0
$607$	− 380.031i	− 0.626081i	−0.949740	−	0.313041i	$-0.898652\pi$
$607$	− 380.031i	− 0.626081i	0.949740	−	0.313041i	$-0.101348\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 625.304i	− 1.02341i
$612$	0	0
$613$	53.4701	0.0872270	0.0436135	−	0.999048i	$-0.486113\pi$
$613$	53.4701	0.0872270	0.0436135	+	0.999048i	$0.486113\pi$
$614$	0	0
$615$	104.025i	0.169146i
$616$	0	0
$617$	−304.514	−0.493539	−0.246770	−	0.969074i	$-0.579369\pi$
$617$	−304.514	−0.493539	−0.246770	+	0.969074i	$0.579369\pi$
$618$	0	0
$619$	− 523.843i	− 0.846272i	−0.906066	−	0.423136i	$-0.860929\pi$
$619$	− 523.843i	− 0.846272i	0.906066	−	0.423136i	$-0.139071\pi$
$620$	0	0
$621$	212.912	0.342853
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−476.029	−0.761647
$626$	0	0
$627$	513.960i	0.819713i
$628$	0	0
$629$	−443.647	−0.705321
$630$	0	0
$631$	− 214.943i	− 0.340639i	−0.985389	−	0.170320i	$-0.945520\pi$
$631$	− 214.943i	− 0.340639i	0.985389	−	0.170320i	$-0.0544800\pi$
$632$	0	0
$633$	606.323	0.957856
$634$	0	0
$635$	772.646i	1.21677i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 82.8325i	− 0.129628i
$640$	0	0
$641$	1216.51	1.89784	0.948919	−	0.315520i	$-0.102179\pi$
$641$	1216.51	1.89784	0.948919	+	0.315520i	$0.102179\pi$
$642$	0	0
$643$	522.341i	0.812350i	0.913795	+	0.406175i	$0.133138\pi$
$643$	522.341i	0.812350i	−0.913795	+	0.406175i	$0.866862\pi$
$644$	0	0
$645$	896.029	1.38919
$646$	0	0
$647$	1072.42i	1.65753i	0.559600	+	0.828763i	$0.310955\pi$
$647$	1072.42i	1.65753i	−0.559600	+	0.828763i	$0.689045\pi$
$648$	0	0
$649$	−88.9706	−0.137089
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−672.043	−1.02916	−0.514581	−	0.857442i	$-0.672053\pi$
$653$	−672.043	−1.02916	−0.514581	+	0.857442i	$0.672053\pi$
$654$	0	0
$655$	− 680.237i	− 1.03853i
$656$	0	0
$657$	−184.492	−0.280810
$658$	0	0
$659$	329.005i	0.499249i	0.968343	+	0.249625i	$0.0803072\pi$
$659$	329.005i	0.499249i	−0.968343	+	0.249625i	$0.919693\pi$
$660$	0	0
$661$	845.658	1.27936	0.639681	−	0.768641i	$-0.279067\pi$
$661$	845.658	1.27936	0.639681	+	0.768641i	$0.279067\pi$
$662$	0	0
$663$	− 495.799i	− 0.747812i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1331.08i	− 1.99562i
$668$	0	0
$669$	29.8234	0.0445790
$670$	0	0
$671$	686.121i	1.02253i
$672$	0	0
$673$	−214.648	−0.318941	−0.159471	−	0.987203i	$-0.550979\pi$
$673$	−214.648	−0.318941	−0.159471	+	0.987203i	$0.550979\pi$
$674$	0	0
$675$	155.732i	0.230713i
$676$	0	0
$677$	−234.145	−0.345857	−0.172928	−	0.984934i	$-0.555323\pi$
$677$	−234.145	−0.345857	−0.172928	+	0.984934i	$0.555323\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	105.838	0.155415
$682$	0	0
$683$	738.396i	1.08111i	0.841310	+	0.540553i	$0.181785\pi$
$683$	738.396i	1.08111i	−0.841310	+	0.540553i	$0.818215\pi$
$684$	0	0
$685$	1382.21	2.01782
$686$	0	0
$687$	− 3.63818i	− 0.00529575i
$688$	0	0
$689$	−1596.73	−2.31746
$690$	0	0
$691$	− 732.505i	− 1.06006i	−0.847978	−	0.530032i	$-0.822180\pi$
$691$	− 732.505i	− 1.06006i	0.847978	−	0.530032i	$-0.177820\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1050.76i	1.51188i
$696$	0	0
$697$	−105.882	−0.151911
$698$	0	0
$699$	− 736.503i	− 1.05365i
$700$	0	0
$701$	538.690	0.768460	0.384230	−	0.923237i	$-0.374467\pi$
$701$	538.690	0.768460	0.384230	+	0.923237i	$0.374467\pi$
$702$	0	0
$703$	498.831i	0.709574i
$704$	0	0
$705$	−366.676	−0.520108
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1277.97	−1.80250	−0.901248	−	0.433304i	$-0.857348\pi$
$709$	−1277.97	−1.80250	−0.901248	+	0.433304i	$0.857348\pi$
$710$	0	0
$711$	271.677i	0.382106i
$712$	0	0
$713$	2090.50	2.93197
$714$	0	0
$715$	− 3278.24i	− 4.58496i
$716$	0	0
$717$	400.087	0.558002
$718$	0	0
$719$	615.541i	0.856107i	0.903753	+	0.428053i	$0.140801\pi$
$719$	615.541i	0.856107i	−0.903753	+	0.428053i	$0.859199\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 92.1082i	− 0.127397i
$724$	0	0
$725$	973.602	1.34290
$726$	0	0
$727$	273.129i	0.375694i	0.982198	+	0.187847i	$0.0601509\pi$
$727$	273.129i	0.375694i	−0.982198	+	0.187847i	$0.939849\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	912.026i	1.24764i
$732$	0	0
$733$	1013.51	1.38269	0.691345	−	0.722525i	$-0.257018\pi$
$733$	1013.51	1.38269	0.691345	+	0.722525i	$0.257018\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	617.470	0.837816
$738$	0	0
$739$	− 152.048i	− 0.205748i	−0.994694	−	0.102874i	$-0.967196\pi$
$739$	− 152.048i	− 0.205748i	0.994694	−	0.102874i	$-0.0328038\pi$
$740$	0	0
$741$	−557.470	−0.752321
$742$	0	0
$743$	601.582i	0.809667i	0.914390	+	0.404833i	$0.132670\pi$
$743$	601.582i	0.809667i	−0.914390	+	0.404833i	$0.867330\pi$
$744$	0	0
$745$	−362.642	−0.486767
$746$	0	0
$747$	15.7414i	0.0210728i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	739.602i	0.984822i	0.870363	+	0.492411i	$0.163884\pi$
$751$	739.602i	0.984822i	−0.870363	+	0.492411i	$0.836116\pi$
$752$	0	0
$753$	−466.544	−0.619581
$754$	0	0
$755$	− 1589.36i	− 2.10511i
$756$	0	0
$757$	−14.9117	−0.0196984	−0.00984920	−	0.999951i	$-0.503135\pi$
$757$	−14.9117	−0.0196984	−0.00984920	+	0.999951i	$0.503135\pi$
$758$	0	0
$759$	1432.91i	1.88790i
$760$	0	0
$761$	−361.341	−0.474824	−0.237412	−	0.971409i	$-0.576299\pi$
$761$	−361.341	−0.474824	−0.237412	+	0.971409i	$0.576299\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−290.735	−0.380046
$766$	0	0
$767$	− 96.5025i	− 0.125818i
$768$	0	0
$769$	−137.762	−0.179144	−0.0895719	−	0.995980i	$-0.528550\pi$
$769$	−137.762	−0.179144	−0.0895719	+	0.995980i	$0.528550\pi$
$770$	0	0
$771$	− 201.813i	− 0.261755i
$772$	0	0
$773$	512.974	0.663614	0.331807	−	0.943347i	$-0.392342\pi$
$773$	512.974	0.663614	0.331807	+	0.943347i	$0.392342\pi$
$774$	0	0
$775$	1529.07i	1.97299i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	119.053i	0.152828i
$780$	0	0
$781$	557.470	0.713790
$782$	0	0
$783$	168.798i	0.215579i
$784$	0	0
$785$	408.309	0.520138
$786$	0	0
$787$	1531.59i	1.94611i	0.230569	+	0.973056i	$0.425941\pi$
$787$	1531.59i	1.94611i	−0.230569	+	0.973056i	$0.574059\pi$
$788$	0	0
$789$	−285.411	−0.361738
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−744.205	−0.938468
$794$	0	0
$795$	936.317i	1.17776i
$796$	0	0
$797$	−1319.33	−1.65536	−0.827682	−	0.561197i	$-0.810341\pi$
$797$	−1319.33	−1.65536	−0.827682	+	0.561197i	$0.810341\pi$
$798$	0	0
$799$	− 373.223i	− 0.467112i
$800$	0	0
$801$	170.610	0.212996
$802$	0	0
$803$	− 1241.65i	− 1.54626i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	835.870i	1.03577i
$808$	0	0
$809$	−1487.53	−1.83872	−0.919362	−	0.393413i	$-0.871294\pi$
$809$	−1487.53	−1.83872	−0.919362	+	0.393413i	$0.871294\pi$
$810$	0	0
$811$	50.1785i	0.0618724i	0.999521	+	0.0309362i	$0.00984886\pi$
$811$	50.1785i	0.0618724i	−0.999521	+	0.0309362i	$0.990151\pi$
$812$	0	0
$813$	−390.926	−0.480844
$814$	0	0
$815$	951.049i	1.16693i
$816$	0	0
$817$	1025.47	1.25517
$818$	0	0
$819$	0	0
$820$	0	0
$821$	798.029	0.972020	0.486010	−	0.873953i	$-0.338452\pi$
$821$	798.029	0.972020	0.486010	+	0.873953i	$0.338452\pi$
$822$	0	0
$823$	351.861i	0.427535i	0.976885	+	0.213767i	$0.0685735\pi$
$823$	351.861i	0.427535i	−0.976885	+	0.213767i	$0.931427\pi$
$824$	0	0
$825$	−1048.09	−1.27041
$826$	0	0
$827$	− 1240.79i	− 1.50035i	−0.661237	−	0.750177i	$-0.729968\pi$
$827$	− 1240.79i	− 1.50035i	0.661237	−	0.750177i	$-0.270032\pi$
$828$	0	0
$829$	978.281	1.18007	0.590037	−	0.807376i	$-0.299113\pi$
$829$	978.281	1.18007	0.590037	+	0.807376i	$0.299113\pi$
$830$	0	0
$831$	131.942i	0.158775i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 1777.64i	− 2.12891i
$836$	0	0
$837$	−265.103	−0.316730
$838$	0	0
$839$	− 263.296i	− 0.313822i	−0.987613	−	0.156911i	$-0.949846\pi$
$839$	− 263.296i	− 0.313822i	0.987613	−	0.156911i	$-0.0501535\pi$
$840$	0	0
$841$	214.294	0.254808
$842$	0	0
$843$	149.491i	0.177332i
$844$	0	0
$845$	2302.76	2.72517
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−493.809	−0.581636
$850$	0	0
$851$	1390.73i	1.63423i
$852$	0	0
$853$	591.806	0.693794	0.346897	−	0.937903i	$-0.387235\pi$
$853$	591.806	0.693794	0.346897	+	0.937903i	$0.387235\pi$
$854$	0	0
$855$	326.899i	0.382338i
$856$	0	0
$857$	−201.869	−0.235553	−0.117777	−	0.993040i	$-0.537577\pi$
$857$	−201.869	−0.235553	−0.117777	+	0.993040i	$0.537577\pi$
$858$	0	0
$859$	− 822.596i	− 0.957620i	−0.877918	−	0.478810i	$-0.841068\pi$
$859$	− 822.596i	− 0.957620i	0.877918	−	0.478810i	$-0.158932\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1464.70i	− 1.69722i	−0.529017	−	0.848611i	$-0.677439\pi$
$863$	− 1464.70i	− 1.69722i	0.529017	−	0.848611i	$-0.322561\pi$
$864$	0	0
$865$	−116.648	−0.134853
$866$	0	0
$867$	204.637i	0.236029i
$868$	0	0
$869$	−1828.41	−2.10404
$870$	0	0
$871$	669.743i	0.768935i
$872$	0	0
$873$	−370.669	−0.424592
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−217.352	−0.247836	−0.123918	−	0.992292i	$-0.539546\pi$
$877$	−217.352	−0.247836	−0.123918	+	0.992292i	$0.539546\pi$
$878$	0	0
$879$	− 786.448i	− 0.894708i
$880$	0	0
$881$	−99.8995	−0.113393	−0.0566966	−	0.998391i	$-0.518057\pi$
$881$	−99.8995	−0.113393	−0.0566966	+	0.998391i	$0.518057\pi$
$882$	0	0
$883$	86.1277i	0.0975398i	0.998810	+	0.0487699i	$0.0155301\pi$
$883$	86.1277i	0.0975398i	−0.998810	+	0.0487699i	$0.984470\pi$
$884$	0	0
$885$	−56.5887	−0.0639421
$886$	0	0
$887$	81.0247i	0.0913469i	0.998956	+	0.0456735i	$0.0145434\pi$
$887$	81.0247i	0.0913469i	−0.998956	+	0.0456735i	$0.985457\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 181.712i	− 0.203942i
$892$	0	0
$893$	−419.647	−0.469929
$894$	0	0
$895$	711.083i	0.794507i
$896$	0	0
$897$	−1554.22	−1.73269
$898$	0	0
$899$	1657.37i	1.84357i
$900$	0	0
$901$	−953.034	−1.05775
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1638.63	1.81064
$906$	0	0
$907$	− 1289.83i	− 1.42209i	−0.703147	−	0.711044i	$-0.748222\pi$
$907$	− 1289.83i	− 1.42209i	0.703147	−	0.711044i	$-0.251778\pi$
$908$	0	0
$909$	210.978	0.232099
$910$	0	0
$911$	− 495.859i	− 0.544302i	−0.962255	−	0.272151i	$-0.912265\pi$
$911$	− 495.859i	− 0.544302i	0.962255	−	0.272151i	$-0.0877350\pi$
$912$	0	0
$913$	−105.941	−0.116036
$914$	0	0
$915$	436.400i	0.476939i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1323.91i	1.44060i	0.693662	+	0.720301i	$0.255996\pi$
$919$	1323.91i	1.44060i	−0.693662	+	0.720301i	$0.744004\pi$
$920$	0	0
$921$	−520.014	−0.564619
$922$	0	0
$923$	604.663i	0.655107i
$924$	0	0
$925$	−1017.23	−1.09971
$926$	0	0
$927$	− 329.420i	− 0.355362i
$928$	0	0
$929$	−439.664	−0.473266	−0.236633	−	0.971599i	$-0.576044\pi$
$929$	−439.664	−0.473266	−0.236633	+	0.971599i	$0.576044\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	438.573	0.470067
$934$	0	0
$935$	− 1956.67i	− 2.09270i
$936$	0	0
$937$	−30.0315	−0.0320507	−0.0160254	−	0.999872i	$-0.505101\pi$
$937$	−30.0315	−0.0320507	−0.0160254	+	0.999872i	$0.505101\pi$
$938$	0	0
$939$	577.286i	0.614789i
$940$	0	0
$941$	−981.266	−1.04279	−0.521395	−	0.853315i	$-0.674588\pi$
$941$	−981.266	−1.04279	−0.521395	+	0.853315i	$0.674588\pi$
$942$	0	0
$943$	331.917i	0.351980i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	633.642i	0.669105i	0.942377	+	0.334552i	$0.108585\pi$
$947$	633.642i	0.669105i	−0.942377	+	0.334552i	$0.891415\pi$
$948$	0	0
$949$	1346.76	1.41914
$950$	0	0
$951$	− 189.252i	− 0.199004i
$952$	0	0
$953$	1058.76	1.11098	0.555490	−	0.831523i	$-0.312531\pi$
$953$	1058.76	1.11098	0.555490	+	0.831523i	$0.312531\pi$
$954$	0	0
$955$	385.254i	0.403407i
$956$	0	0
$957$	−1136.03	−1.18707
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1641.94	−1.70857
$962$	0	0
$963$	− 45.4412i	− 0.0471871i
$964$	0	0
$965$	−1392.56	−1.44307
$966$	0	0
$967$	309.103i	0.319652i	0.987145	+	0.159826i	$0.0510933\pi$
$967$	309.103i	0.319652i	−0.987145	+	0.159826i	$0.948907\pi$
$968$	0	0
$969$	−332.735	−0.343380
$970$	0	0
$971$	− 1197.76i	− 1.23354i	−0.787145	−	0.616768i	$-0.788442\pi$
$971$	− 1197.76i	− 1.23354i	0.787145	−	0.616768i	$-0.211558\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 1136.81i	− 1.16596i
$976$	0	0
$977$	837.131	0.856838	0.428419	−	0.903580i	$-0.359071\pi$
$977$	837.131	0.856838	0.428419	+	0.903580i	$0.359071\pi$
$978$	0	0
$979$	1148.22i	1.17285i
$980$	0	0
$981$	−218.558	−0.222791
$982$	0	0
$983$	512.687i	0.521553i	0.965399	+	0.260777i	$0.0839786\pi$
$983$	512.687i	0.521553i	−0.965399	+	0.260777i	$0.916021\pi$
$984$	0	0
$985$	−349.123	−0.354439
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2859.00	2.89080
$990$	0	0
$991$	984.874i	0.993818i	0.867803	+	0.496909i	$0.165532\pi$
$991$	984.874i	0.993818i	−0.867803	+	0.496909i	$0.834468\pi$
$992$	0	0
$993$	−597.588	−0.601800
$994$	0	0
$995$	− 1874.15i	− 1.88356i
$996$	0	0
$997$	−880.929	−0.883580	−0.441790	−	0.897119i	$-0.645656\pi$
$997$	−880.929	−0.883580	−0.441790	+	0.897119i	$0.645656\pi$
$998$	0	0
$999$	− 176.363i	− 0.176540i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.m.l.1471.2	yes	4
4.3	odd	2	inner	2352.3.m.l.1471.4	yes	4
7.6	odd	2		2352.3.m.d.1471.3	yes	4
28.27	even	2		2352.3.m.d.1471.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.3.m.d.1471.1	✓	4	28.27	even	2
2352.3.m.d.1471.3	yes	4	7.6	odd	2
2352.3.m.l.1471.2	yes	4	1.1	even	1	trivial
2352.3.m.l.1471.4	yes	4	4.3	odd	2	inner

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +7.41421 q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +7.41421 q^{5} -3.00000 q^{9} -20.1903i q^{11} +21.8995 q^{13} -12.8418i q^{15} +13.0711 q^{17} -14.6969i q^{19} -40.9749i q^{23} +29.9706 q^{25} +5.19615i q^{27} +32.4853 q^{29} +51.0190i q^{31} -34.9706 q^{33} -33.9411 q^{37} -37.9310i q^{39} -8.10051 q^{41} +69.7744i q^{43} -22.2426 q^{45} -28.5533i q^{47} -22.6398i q^{51} -72.9117 q^{53} -149.695i q^{55} -25.4558 q^{57} -4.40661i q^{59} -33.9828 q^{61} +162.368 q^{65} +30.5826i q^{67} -70.9706 q^{69} +27.6108i q^{71} +61.4975 q^{73} -51.9105i q^{75} -90.5590i q^{79} +9.00000 q^{81} -5.24714i q^{83} +96.9117 q^{85} -56.2662i q^{87} -56.8701 q^{89} +88.3675 q^{93} -108.966i q^{95} +123.556 q^{97} +60.5708i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 24 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 24 * q^5 - 12 * q^9	\( 4 q + 24 q^{5} - 12 q^{9} + 48 q^{13} + 24 q^{17} + 52 q^{25} + 96 q^{29} - 72 q^{33} - 72 q^{41} - 72 q^{45} - 88 q^{53} + 96 q^{61} + 344 q^{65} - 216 q^{69} + 48 q^{73} + 36 q^{81} + 184 q^{85} - 120 q^{89} + 48 q^{93} + 432 q^{97}+O(q^{100}) \) 4 * q + 24 * q^5 - 12 * q^9 + 48 * q^13 + 24 * q^17 + 52 * q^25 + 96 * q^29 - 72 * q^33 - 72 * q^41 - 72 * q^45 - 88 * q^53 + 96 * q^61 + 344 * q^65 - 216 * q^69 + 48 * q^73 + 36 * q^81 + 184 * q^85 - 120 * q^89 + 48 * q^93 + 432 * q^97

Introduction

L-functions

Modular forms

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Fields

Representations

Groups

Database

Properties

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