LMFDB - Embedded newform 1872.4.a.bo.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,4,Mod(1,1872)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, 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("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1872.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:	$110.451575531$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1520092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 40x^{2} - 9x + 81 $ x^4 - x^3 - 40x^2 - 9x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$6.81072$ of defining polynomial
Character	$\chi$	$=$	1872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.0859 q^{5} -24.2643 q^{7} +O(q^{10})$	$q+14.0859 q^{5} -24.2643 q^{7} -3.10739 q^{11} -13.0000 q^{13} +43.9142 q^{17} +85.8504 q^{19} -203.829 q^{23} +73.4140 q^{25} -31.0739 q^{29} +135.736 q^{31} -341.786 q^{35} +290.229 q^{37} -148.731 q^{41} -281.057 q^{43} +225.991 q^{47} +245.758 q^{49} +172.755 q^{53} -43.7706 q^{55} +41.2175 q^{59} +499.815 q^{61} -183.117 q^{65} -503.506 q^{67} -946.442 q^{71} -1115.97 q^{73} +75.3989 q^{77} -674.803 q^{79} -59.4512 q^{83} +618.574 q^{85} -1218.41 q^{89} +315.436 q^{91} +1209.28 q^{95} -879.746 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{7}+O(q^{10}) $ 4 * q - 36 * q^7	$ 4 q - 36 q^{7} - 52 q^{13} - 84 q^{19} + 660 q^{25} + 604 q^{31} + 184 q^{37} - 880 q^{43} - 116 q^{49} - 1152 q^{55} + 656 q^{61} - 3052 q^{67} - 312 q^{73} + 720 q^{79} + 32 q^{85} + 468 q^{91} - 344 q^{97}+O(q^{100}) $ 4 * q - 36 * q^7 - 52 * q^13 - 84 * q^19 + 660 * q^25 + 604 * q^31 + 184 * q^37 - 880 * q^43 - 116 * q^49 - 1152 * q^55 + 656 * q^61 - 3052 * q^67 - 312 * q^73 + 720 * q^79 + 32 * q^85 + 468 * q^91 - 344 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	14.0859	1.25989	0.629943	−	0.776642i	$-0.283078\pi$
$5$	14.0859	1.25989	0.629943	+	0.776642i	$0.283078\pi$
$6$	0	0
$7$	−24.2643	−1.31015	−0.655076	−	0.755563i	$-0.727363\pi$
$7$	−24.2643	−1.31015	−0.655076	+	0.755563i	$0.727363\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.10739	−0.0851741	−0.0425870	−	0.999093i	$-0.513560\pi$
$11$	−3.10739	−0.0851741	−0.0425870	+	0.999093i	$0.513560\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	43.9142	0.626515	0.313258	−	0.949668i	$-0.398580\pi$
$17$	43.9142	0.626515	0.313258	+	0.949668i	$0.398580\pi$
$18$	0	0
$19$	85.8504	1.03660	0.518301	−	0.855198i	$-0.326565\pi$
$19$	85.8504	1.03660	0.518301	+	0.855198i	$0.326565\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−203.829	−1.84788	−0.923940	−	0.382537i	$-0.875050\pi$
$23$	−203.829	−1.84788	−0.923940	+	0.382537i	$0.875050\pi$
$24$	0	0
$25$	73.4140	0.587312
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−31.0739	−0.198975	−0.0994877	−	0.995039i	$-0.531720\pi$
$29$	−31.0739	−0.198975	−0.0994877	+	0.995039i	$0.531720\pi$
$30$	0	0
$31$	135.736	0.786414	0.393207	−	0.919450i	$-0.371366\pi$
$31$	135.736	0.786414	0.393207	+	0.919450i	$0.371366\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−341.786	−1.65064
$36$	0	0
$37$	290.229	1.28955	0.644776	−	0.764372i	$-0.276951\pi$
$37$	290.229	1.28955	0.644776	+	0.764372i	$0.276951\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−148.731	−0.566532	−0.283266	−	0.959041i	$-0.591418\pi$
$41$	−148.731	−0.566532	−0.283266	+	0.959041i	$0.591418\pi$
$42$	0	0
$43$	−281.057	−0.996764	−0.498382	−	0.866958i	$-0.666072\pi$
$43$	−281.057	−0.996764	−0.498382	+	0.866958i	$0.666072\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	225.991	0.701366	0.350683	−	0.936494i	$-0.385949\pi$
$47$	225.991	0.701366	0.350683	+	0.936494i	$0.385949\pi$
$48$	0	0
$49$	245.758	0.716496
$50$	0	0
$51$	0	0
$52$	0	0
$53$	172.755	0.447730	0.223865	−	0.974620i	$-0.428132\pi$
$53$	172.755	0.447730	0.223865	+	0.974620i	$0.428132\pi$
$54$	0	0
$55$	−43.7706	−0.107310
$56$	0	0
$57$	0	0
$58$	0	0
$59$	41.2175	0.0909503	0.0454751	−	0.998965i	$-0.485520\pi$
$59$	41.2175	0.0909503	0.0454751	+	0.998965i	$0.485520\pi$
$60$	0	0
$61$	499.815	1.04910	0.524548	−	0.851381i	$-0.324234\pi$
$61$	499.815	1.04910	0.524548	+	0.851381i	$0.324234\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−183.117	−0.349429
$66$	0	0
$67$	−503.506	−0.918106	−0.459053	−	0.888409i	$-0.651811\pi$
$67$	−503.506	−0.918106	−0.459053	+	0.888409i	$0.651811\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−946.442	−1.58200	−0.791000	−	0.611816i	$-0.790439\pi$
$71$	−946.442	−1.58200	−0.791000	+	0.611816i	$0.790439\pi$
$72$	0	0
$73$	−1115.97	−1.78925	−0.894623	−	0.446821i	$-0.852556\pi$
$73$	−1115.97	−1.78925	−0.894623	+	0.446821i	$0.852556\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	75.3989	0.111591
$78$	0	0
$79$	−674.803	−0.961029	−0.480514	−	0.876987i	$-0.659550\pi$
$79$	−674.803	−0.961029	−0.480514	+	0.876987i	$0.659550\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−59.4512	−0.0786219	−0.0393109	−	0.999227i	$-0.512516\pi$
$83$	−59.4512	−0.0786219	−0.0393109	+	0.999227i	$0.512516\pi$
$84$	0	0
$85$	618.574	0.789338
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1218.41	−1.45114	−0.725571	−	0.688147i	$-0.758424\pi$
$89$	−1218.41	−1.45114	−0.725571	+	0.688147i	$0.758424\pi$
$90$	0	0
$91$	315.436	0.363371
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1209.28	1.30600
$96$	0	0
$97$	−879.746	−0.920872	−0.460436	−	0.887693i	$-0.652307\pi$
$97$	−879.746	−0.920872	−0.460436	+	0.887693i	$0.652307\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−255.628	−0.251841	−0.125920	−	0.992040i	$-0.540188\pi$
$101$	−255.628	−0.251841	−0.125920	+	0.992040i	$0.540188\pi$
$102$	0	0
$103$	−176.369	−0.168720	−0.0843600	−	0.996435i	$-0.526885\pi$
$103$	−176.369	−0.168720	−0.0843600	+	0.996435i	$0.526885\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	816.164	0.737397	0.368699	−	0.929549i	$-0.379803\pi$
$107$	816.164	0.737397	0.368699	+	0.929549i	$0.379803\pi$
$108$	0	0
$109$	174.369	0.153225	0.0766125	−	0.997061i	$-0.475590\pi$
$109$	174.369	0.153225	0.0766125	+	0.997061i	$0.475590\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−199.667	−0.166222	−0.0831112	−	0.996540i	$-0.526486\pi$
$113$	−199.667	−0.166222	−0.0831112	+	0.996540i	$0.526486\pi$
$114$	0	0
$115$	−2871.12	−2.32812
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1065.55	−0.820830
$120$	0	0
$121$	−1321.34	−0.992745
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−726.638	−0.519940
$126$	0	0
$127$	−407.840	−0.284961	−0.142480	−	0.989798i	$-0.545508\pi$
$127$	−407.840	−0.284961	−0.142480	+	0.989798i	$0.545508\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2177.50	1.45229	0.726143	−	0.687544i	$-0.241311\pi$
$131$	2177.50	1.45229	0.726143	+	0.687544i	$0.241311\pi$
$132$	0	0
$133$	−2083.10	−1.35810
$134$	0	0
$135$	0	0
$136$	0	0
$137$	723.750	0.451344	0.225672	−	0.974203i	$-0.427542\pi$
$137$	723.750	0.451344	0.225672	+	0.974203i	$0.427542\pi$
$138$	0	0
$139$	1520.85	0.928033	0.464017	−	0.885826i	$-0.346408\pi$
$139$	1520.85	0.928033	0.464017	+	0.885826i	$0.346408\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	40.3961	0.0236230
$144$	0	0
$145$	−437.706	−0.250686
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2027.51	−1.11477	−0.557384	−	0.830255i	$-0.688195\pi$
$149$	−2027.51	−1.11477	−0.557384	+	0.830255i	$0.688195\pi$
$150$	0	0
$151$	11.8952	0.00641071	0.00320535	−	0.999995i	$-0.498980\pi$
$151$	11.8952	0.00641071	0.00320535	+	0.999995i	$0.498980\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1911.97	0.990792
$156$	0	0
$157$	1369.10	0.695963	0.347982	−	0.937501i	$-0.386867\pi$
$157$	1369.10	0.695963	0.347982	+	0.937501i	$0.386867\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4945.77	2.42100
$162$	0	0
$163$	−1650.54	−0.793130	−0.396565	−	0.918007i	$-0.629798\pi$
$163$	−1650.54	−0.793130	−0.396565	+	0.918007i	$0.629798\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−872.686	−0.404374	−0.202187	−	0.979347i	$-0.564805\pi$
$167$	−872.686	−0.404374	−0.202187	+	0.979347i	$0.564805\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	732.880	0.322080	0.161040	−	0.986948i	$-0.448515\pi$
$173$	732.880	0.322080	0.161040	+	0.986948i	$0.448515\pi$
$174$	0	0
$175$	−1781.34	−0.769467
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1530.81	0.639207	0.319604	−	0.947551i	$-0.396450\pi$
$179$	1530.81	0.639207	0.319604	+	0.947551i	$0.396450\pi$
$180$	0	0
$181$	−1198.21	−0.492056	−0.246028	−	0.969263i	$-0.579126\pi$
$181$	−1198.21	−0.492056	−0.246028	+	0.969263i	$0.579126\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4088.16	1.62469
$186$	0	0
$187$	−136.459	−0.0533629
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2465.44	−0.933994	−0.466997	−	0.884259i	$-0.654664\pi$
$191$	−2465.44	−0.933994	−0.466997	+	0.884259i	$0.654664\pi$
$192$	0	0
$193$	−4588.52	−1.71134	−0.855671	−	0.517520i	$-0.826855\pi$
$193$	−4588.52	−1.71134	−0.855671	+	0.517520i	$0.826855\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3862.82	1.39703	0.698514	−	0.715596i	$-0.253845\pi$
$197$	3862.82	1.39703	0.698514	+	0.715596i	$0.253845\pi$
$198$	0	0
$199$	−1434.59	−0.511033	−0.255517	−	0.966805i	$-0.582246\pi$
$199$	−1434.59	−0.511033	−0.255517	+	0.966805i	$0.582246\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	753.989	0.260688
$204$	0	0
$205$	−2095.01	−0.713766
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−266.771	−0.0882915
$210$	0	0
$211$	1577.24	0.514604	0.257302	−	0.966331i	$-0.417166\pi$
$211$	1577.24	0.514604	0.257302	+	0.966331i	$0.417166\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3958.96	−1.25581
$216$	0	0
$217$	−3293.54	−1.03032
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−570.885	−0.173764
$222$	0	0
$223$	3123.69	0.938016	0.469008	−	0.883194i	$-0.344612\pi$
$223$	3123.69	0.938016	0.469008	+	0.883194i	$0.344612\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3006.22	−0.878987	−0.439493	−	0.898246i	$-0.644842\pi$
$227$	−3006.22	−0.878987	−0.439493	+	0.898246i	$0.644842\pi$
$228$	0	0
$229$	−2344.43	−0.676526	−0.338263	−	0.941052i	$-0.609839\pi$
$229$	−2344.43	−0.676526	−0.338263	+	0.941052i	$0.609839\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5913.50	−1.66269	−0.831344	−	0.555759i	$-0.812428\pi$
$233$	−5913.50	−1.66269	−0.831344	+	0.555759i	$0.812428\pi$
$234$	0	0
$235$	3183.30	0.883641
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−250.440	−0.0677807	−0.0338904	−	0.999426i	$-0.510790\pi$
$239$	−250.440	−0.0677807	−0.0338904	+	0.999426i	$0.510790\pi$
$240$	0	0
$241$	1112.43	0.297337	0.148668	−	0.988887i	$-0.452501\pi$
$241$	1112.43	0.297337	0.148668	+	0.988887i	$0.452501\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3461.74	0.902703
$246$	0	0
$247$	−1116.05	−0.287501
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3496.19	0.879194	0.439597	−	0.898195i	$-0.355121\pi$
$251$	3496.19	0.879194	0.439597	+	0.898195i	$0.355121\pi$
$252$	0	0
$253$	633.376	0.157391
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4644.20	−1.12723	−0.563614	−	0.826039i	$-0.690589\pi$
$257$	−4644.20	−1.12723	−0.563614	+	0.826039i	$0.690589\pi$
$258$	0	0
$259$	−7042.22	−1.68951
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4930.41	−1.15598	−0.577989	−	0.816045i	$-0.696162\pi$
$263$	−4930.41	−1.15598	−0.577989	+	0.816045i	$0.696162\pi$
$264$	0	0
$265$	2433.42	0.564089
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1276.44	0.289316	0.144658	−	0.989482i	$-0.453792\pi$
$269$	1276.44	0.289316	0.144658	+	0.989482i	$0.453792\pi$
$270$	0	0
$271$	2519.91	0.564848	0.282424	−	0.959290i	$-0.408861\pi$
$271$	2519.91	0.564848	0.282424	+	0.959290i	$0.408861\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−228.126	−0.0500237
$276$	0	0
$277$	−2462.00	−0.534033	−0.267017	−	0.963692i	$-0.586038\pi$
$277$	−2462.00	−0.534033	−0.267017	+	0.963692i	$0.586038\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5058.09	−1.07381	−0.536904	−	0.843643i	$-0.680406\pi$
$281$	−5058.09	−1.07381	−0.536904	+	0.843643i	$0.680406\pi$
$282$	0	0
$283$	−3078.55	−0.646647	−0.323323	−	0.946289i	$-0.604800\pi$
$283$	−3078.55	−0.646647	−0.323323	+	0.946289i	$0.604800\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3608.85	0.742243
$288$	0	0
$289$	−2984.54	−0.607478
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8626.31	−1.71998	−0.859990	−	0.510310i	$-0.829531\pi$
$293$	−8626.31	−1.71998	−0.859990	+	0.510310i	$0.829531\pi$
$294$	0	0
$295$	580.588	0.114587
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2649.77	0.512510
$300$	0	0
$301$	6819.67	1.30591
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7040.37	1.32174
$306$	0	0
$307$	−545.591	−0.101428	−0.0507142	−	0.998713i	$-0.516150\pi$
$307$	−545.591	−0.101428	−0.0507142	+	0.998713i	$0.516150\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3024.25	0.551413	0.275707	−	0.961242i	$-0.411088\pi$
$311$	3024.25	0.551413	0.275707	+	0.961242i	$0.411088\pi$
$312$	0	0
$313$	−10082.4	−1.82074	−0.910370	−	0.413796i	$-0.864203\pi$
$313$	−10082.4	−1.82074	−0.910370	+	0.413796i	$0.864203\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8498.67	1.50578	0.752891	−	0.658145i	$-0.228659\pi$
$317$	8498.67	1.50578	0.752891	+	0.658145i	$0.228659\pi$
$318$	0	0
$319$	96.5590	0.0169475
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3770.05	0.649447
$324$	0	0
$325$	−954.382	−0.162891
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5483.53	−0.918896
$330$	0	0
$331$	−9969.94	−1.65558	−0.827791	−	0.561037i	$-0.810403\pi$
$331$	−9969.94	−1.65558	−0.827791	+	0.561037i	$0.810403\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7092.36	−1.15671
$336$	0	0
$337$	−3231.96	−0.522422	−0.261211	−	0.965282i	$-0.584122\pi$
$337$	−3231.96	−0.522422	−0.261211	+	0.965282i	$0.584122\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−421.784	−0.0669821
$342$	0	0
$343$	2359.51	0.371433
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5989.75	0.926647	0.463323	−	0.886189i	$-0.346657\pi$
$347$	5989.75	0.926647	0.463323	+	0.886189i	$0.346657\pi$
$348$	0	0
$349$	−9974.91	−1.52993	−0.764964	−	0.644074i	$-0.777243\pi$
$349$	−9974.91	−1.52993	−0.764964	+	0.644074i	$0.777243\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10834.0	−1.63352	−0.816761	−	0.576976i	$-0.804233\pi$
$353$	−10834.0	−1.63352	−0.816761	+	0.576976i	$0.804233\pi$
$354$	0	0
$355$	−13331.5	−1.99314
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4315.76	0.634477	0.317239	−	0.948346i	$-0.397244\pi$
$359$	4315.76	0.634477	0.317239	+	0.948346i	$0.397244\pi$
$360$	0	0
$361$	511.285	0.0745422
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15719.6	−2.25425
$366$	0	0
$367$	−10505.3	−1.49420	−0.747101	−	0.664711i	$-0.768555\pi$
$367$	−10505.3	−1.49420	−0.747101	+	0.664711i	$0.768555\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4191.78	−0.586594
$372$	0	0
$373$	5869.44	0.814767	0.407384	−	0.913257i	$-0.366441\pi$
$373$	5869.44	0.814767	0.407384	+	0.913257i	$0.366441\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	403.961	0.0551858
$378$	0	0
$379$	6525.75	0.884447	0.442223	−	0.896905i	$-0.354190\pi$
$379$	6525.75	0.884447	0.442223	+	0.896905i	$0.354190\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7042.66	−0.939590	−0.469795	−	0.882775i	$-0.655672\pi$
$383$	−7042.66	−0.939590	−0.469795	+	0.882775i	$0.655672\pi$
$384$	0	0
$385$	1062.06	0.140592
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10339.4	1.34763	0.673815	−	0.738900i	$-0.264655\pi$
$389$	10339.4	1.34763	0.673815	+	0.738900i	$0.264655\pi$
$390$	0	0
$391$	−8950.98	−1.15773
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9505.24	−1.21079
$396$	0	0
$397$	−7465.86	−0.943831	−0.471915	−	0.881644i	$-0.656437\pi$
$397$	−7465.86	−0.943831	−0.471915	+	0.881644i	$0.656437\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3224.37	0.401539	0.200770	−	0.979638i	$-0.435656\pi$
$401$	3224.37	0.401539	0.200770	+	0.979638i	$0.435656\pi$
$402$	0	0
$403$	−1764.56	−0.218112
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−901.857	−0.109836
$408$	0	0
$409$	12391.9	1.49814	0.749072	−	0.662489i	$-0.230500\pi$
$409$	12391.9	1.49814	0.749072	+	0.662489i	$0.230500\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1000.12	−0.119159
$414$	0	0
$415$	−837.426	−0.0990546
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13464.4	−1.56988	−0.784938	−	0.619574i	$-0.787306\pi$
$419$	−13464.4	−1.56988	−0.784938	+	0.619574i	$0.787306\pi$
$420$	0	0
$421$	−6265.49	−0.725324	−0.362662	−	0.931921i	$-0.618132\pi$
$421$	−6265.49	−0.725324	−0.362662	+	0.931921i	$0.618132\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3223.92	0.367960
$426$	0	0
$427$	−12127.7	−1.37447
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9499.09	1.06161	0.530806	−	0.847493i	$-0.321889\pi$
$431$	9499.09	1.06161	0.530806	+	0.847493i	$0.321889\pi$
$432$	0	0
$433$	5002.67	0.555226	0.277613	−	0.960693i	$-0.410457\pi$
$433$	5002.67	0.555226	0.277613	+	0.960693i	$0.410457\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17498.8	−1.91551
$438$	0	0
$439$	−13664.9	−1.48563	−0.742816	−	0.669496i	$-0.766510\pi$
$439$	−13664.9	−1.48563	−0.742816	+	0.669496i	$0.766510\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10780.6	1.15621	0.578105	−	0.815962i	$-0.303792\pi$
$443$	10780.6	1.15621	0.578105	+	0.815962i	$0.303792\pi$
$444$	0	0
$445$	−17162.5	−1.82827
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11959.1	1.25699	0.628493	−	0.777815i	$-0.283672\pi$
$449$	11959.1	1.25699	0.628493	+	0.777815i	$0.283672\pi$
$450$	0	0
$451$	462.165	0.0482539
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4443.22	0.457805
$456$	0	0
$457$	−2120.66	−0.217069	−0.108534	−	0.994093i	$-0.534616\pi$
$457$	−2120.66	−0.217069	−0.108534	+	0.994093i	$0.534616\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15462.9	1.56221	0.781103	−	0.624402i	$-0.214657\pi$
$461$	15462.9	1.56221	0.781103	+	0.624402i	$0.214657\pi$
$462$	0	0
$463$	10395.4	1.04344	0.521722	−	0.853116i	$-0.325290\pi$
$463$	10395.4	1.04344	0.521722	+	0.853116i	$0.325290\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11845.8	1.17379	0.586893	−	0.809664i	$-0.300351\pi$
$467$	11845.8	1.17379	0.586893	+	0.809664i	$0.300351\pi$
$468$	0	0
$469$	12217.2	1.20286
$470$	0	0
$471$	0	0
$472$	0	0
$473$	873.356	0.0848984
$474$	0	0
$475$	6302.62	0.608808
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12879.9	−1.22860	−0.614299	−	0.789074i	$-0.710561\pi$
$479$	−12879.9	−1.22860	−0.614299	+	0.789074i	$0.710561\pi$
$480$	0	0
$481$	−3772.98	−0.357657
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12392.1	−1.16019
$486$	0	0
$487$	4946.81	0.460290	0.230145	−	0.973156i	$-0.426080\pi$
$487$	4946.81	0.460290	0.230145	+	0.973156i	$0.426080\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14390.3	−1.32266	−0.661330	−	0.750095i	$-0.730008\pi$
$491$	−14390.3	−1.32266	−0.661330	+	0.750095i	$0.730008\pi$
$492$	0	0
$493$	−1364.59	−0.124661
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22964.8	2.07266
$498$	0	0
$499$	1427.01	0.128019	0.0640096	−	0.997949i	$-0.479611\pi$
$499$	1427.01	0.128019	0.0640096	+	0.997949i	$0.479611\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18033.6	1.59857	0.799283	−	0.600955i	$-0.205213\pi$
$503$	18033.6	1.59857	0.799283	+	0.600955i	$0.205213\pi$
$504$	0	0
$505$	−3600.76	−0.317290
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7512.52	0.654198	0.327099	−	0.944990i	$-0.393929\pi$
$509$	7512.52	0.654198	0.327099	+	0.944990i	$0.393929\pi$
$510$	0	0
$511$	27078.4	2.34418
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2484.33	−0.212568
$516$	0	0
$517$	−702.244	−0.0597382
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21355.8	1.79580	0.897901	−	0.440198i	$-0.145092\pi$
$521$	21355.8	1.79580	0.897901	+	0.440198i	$0.145092\pi$
$522$	0	0
$523$	−3086.57	−0.258062	−0.129031	−	0.991641i	$-0.541187\pi$
$523$	−3086.57	−0.258062	−0.129031	+	0.991641i	$0.541187\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5960.73	0.492701
$528$	0	0
$529$	29379.2	2.41466
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1933.50	0.157128
$534$	0	0
$535$	11496.4	0.929036
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−763.667	−0.0610269
$540$	0	0
$541$	−2029.46	−0.161282	−0.0806408	−	0.996743i	$-0.525697\pi$
$541$	−2029.46	−0.161282	−0.0806408	+	0.996743i	$0.525697\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2456.16	0.193046
$546$	0	0
$547$	−22144.8	−1.73098	−0.865488	−	0.500929i	$-0.832992\pi$
$547$	−22144.8	−1.73098	−0.865488	+	0.500929i	$0.832992\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2667.71	−0.206258
$552$	0	0
$553$	16373.6	1.25909
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23136.5	1.76001	0.880004	−	0.474967i	$-0.157540\pi$
$557$	23136.5	1.76001	0.880004	+	0.474967i	$0.157540\pi$
$558$	0	0
$559$	3653.75	0.276453
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3978.29	0.297806	0.148903	−	0.988852i	$-0.452426\pi$
$563$	3978.29	0.297806	0.148903	+	0.988852i	$0.452426\pi$
$564$	0	0
$565$	−2812.51	−0.209421
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25871.5	−1.90613	−0.953065	−	0.302764i	$-0.902090\pi$
$569$	−25871.5	−1.90613	−0.953065	+	0.302764i	$0.902090\pi$
$570$	0	0
$571$	7241.31	0.530717	0.265358	−	0.964150i	$-0.414510\pi$
$571$	7241.31	0.530717	0.265358	+	0.964150i	$0.414510\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−14963.9	−1.08528
$576$	0	0
$577$	11537.4	0.832426	0.416213	−	0.909267i	$-0.363357\pi$
$577$	11537.4	0.832426	0.416213	+	0.909267i	$0.363357\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1442.54	0.103007
$582$	0	0
$583$	−536.817	−0.0381350
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16195.2	−1.13875	−0.569376	−	0.822078i	$-0.692815\pi$
$587$	−16195.2	−1.13875	−0.569376	+	0.822078i	$0.692815\pi$
$588$	0	0
$589$	11653.0	0.815198
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14885.6	−1.03083	−0.515413	−	0.856942i	$-0.672362\pi$
$593$	−14885.6	−1.03083	−0.515413	+	0.856942i	$0.672362\pi$
$594$	0	0
$595$	−15009.3	−1.03415
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21516.2	1.46766	0.733829	−	0.679334i	$-0.237731\pi$
$599$	21516.2	1.46766	0.733829	+	0.679334i	$0.237731\pi$
$600$	0	0
$601$	−675.727	−0.0458627	−0.0229313	−	0.999737i	$-0.507300\pi$
$601$	−675.727	−0.0458627	−0.0229313	+	0.999737i	$0.507300\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−18612.4	−1.25075
$606$	0	0
$607$	12166.0	0.813512	0.406756	−	0.913537i	$-0.366660\pi$
$607$	12166.0	0.813512	0.406756	+	0.913537i	$0.366660\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2937.89	−0.194524
$612$	0	0
$613$	−22799.0	−1.50219	−0.751095	−	0.660194i	$-0.770474\pi$
$613$	−22799.0	−1.50219	−0.751095	+	0.660194i	$0.770474\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1635.30	−0.106701	−0.0533506	−	0.998576i	$-0.516990\pi$
$617$	−1635.30	−0.106701	−0.0533506	+	0.998576i	$0.516990\pi$
$618$	0	0
$619$	4435.20	0.287990	0.143995	−	0.989578i	$-0.454005\pi$
$619$	4435.20	0.287990	0.143995	+	0.989578i	$0.454005\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	29564.0	1.90122
$624$	0	0
$625$	−19412.1	−1.24238
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12745.2	0.807924
$630$	0	0
$631$	17645.7	1.11325	0.556626	−	0.830763i	$-0.312096\pi$
$631$	17645.7	1.11325	0.556626	+	0.830763i	$0.312096\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5744.82	−0.359018
$636$	0	0
$637$	−3194.85	−0.198720
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1482.41	0.0913440	0.0456720	−	0.998956i	$-0.485457\pi$
$641$	1482.41	0.0913440	0.0456720	+	0.998956i	$0.485457\pi$
$642$	0	0
$643$	−9629.03	−0.590563	−0.295281	−	0.955410i	$-0.595413\pi$
$643$	−9629.03	−0.590563	−0.295281	+	0.955410i	$0.595413\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15991.6	−0.971709	−0.485854	−	0.874040i	$-0.661491\pi$
$647$	−15991.6	−0.971709	−0.485854	+	0.874040i	$0.661491\pi$
$648$	0	0
$649$	−128.079	−0.00774660
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20321.2	1.21781	0.608907	−	0.793242i	$-0.291608\pi$
$653$	20321.2	1.21781	0.608907	+	0.793242i	$0.291608\pi$
$654$	0	0
$655$	30672.2	1.82971
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21597.1	1.27664	0.638318	−	0.769773i	$-0.279631\pi$
$659$	21597.1	1.27664	0.638318	+	0.769773i	$0.279631\pi$
$660$	0	0
$661$	513.172	0.0301968	0.0150984	−	0.999886i	$-0.495194\pi$
$661$	513.172	0.0301968	0.0150984	+	0.999886i	$0.495194\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−29342.5	−1.71106
$666$	0	0
$667$	6333.76	0.367683
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1553.12	−0.0893557
$672$	0	0
$673$	11983.8	0.686389	0.343195	−	0.939264i	$-0.388491\pi$
$673$	11983.8	0.686389	0.343195	+	0.939264i	$0.388491\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21876.5	−1.24192	−0.620960	−	0.783842i	$-0.713257\pi$
$677$	−21876.5	−1.24192	−0.620960	+	0.783842i	$0.713257\pi$
$678$	0	0
$679$	21346.4	1.20648
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2551.17	−0.142925	−0.0714625	−	0.997443i	$-0.522767\pi$
$683$	−2551.17	−0.142925	−0.0714625	+	0.997443i	$0.522767\pi$
$684$	0	0
$685$	10194.7	0.568642
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2245.81	−0.124178
$690$	0	0
$691$	15321.2	0.843481	0.421741	−	0.906717i	$-0.361419\pi$
$691$	15321.2	0.843481	0.421741	+	0.906717i	$0.361419\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21422.6	1.16922
$696$	0	0
$697$	−6531.39	−0.354941
$698$	0	0
$699$	0	0
$700$	0	0
$701$	558.289	0.0300803	0.0150401	−	0.999887i	$-0.495212\pi$
$701$	558.289	0.0300803	0.0150401	+	0.999887i	$0.495212\pi$
$702$	0	0
$703$	24916.3	1.33675
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6202.64	0.329949
$708$	0	0
$709$	−7058.31	−0.373879	−0.186940	−	0.982371i	$-0.559857\pi$
$709$	−7058.31	−0.373879	−0.186940	+	0.982371i	$0.559857\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−27666.8	−1.45320
$714$	0	0
$715$	569.018	0.0297623
$716$	0	0
$717$	0	0
$718$	0	0
$719$	14673.0	0.761073	0.380536	−	0.924766i	$-0.375739\pi$
$719$	14673.0	0.761073	0.380536	+	0.924766i	$0.375739\pi$
$720$	0	0
$721$	4279.48	0.221049
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2281.26	−0.116861
$726$	0	0
$727$	−35867.8	−1.82980	−0.914899	−	0.403684i	$-0.867730\pi$
$727$	−35867.8	−1.82980	−0.914899	+	0.403684i	$0.867730\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12342.4	−0.624488
$732$	0	0
$733$	27817.7	1.40173	0.700865	−	0.713294i	$-0.252797\pi$
$733$	27817.7	1.40173	0.700865	+	0.713294i	$0.252797\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1564.59	0.0781988
$738$	0	0
$739$	7162.20	0.356517	0.178258	−	0.983984i	$-0.442954\pi$
$739$	7162.20	0.356517	0.178258	+	0.983984i	$0.442954\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12529.8	−0.618671	−0.309336	−	0.950953i	$-0.600107\pi$
$743$	−12529.8	−0.618671	−0.309336	+	0.950953i	$0.600107\pi$
$744$	0	0
$745$	−28559.5	−1.40448
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−19803.7	−0.966102
$750$	0	0
$751$	−8282.62	−0.402446	−0.201223	−	0.979545i	$-0.564492\pi$
$751$	−8282.62	−0.402446	−0.201223	+	0.979545i	$0.564492\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	167.555	0.00807676
$756$	0	0
$757$	−22044.3	−1.05840	−0.529202	−	0.848496i	$-0.677509\pi$
$757$	−22044.3	−1.05840	−0.529202	+	0.848496i	$0.677509\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5710.64	0.272025	0.136012	−	0.990707i	$-0.456571\pi$
$761$	5710.64	0.272025	0.136012	+	0.990707i	$0.456571\pi$
$762$	0	0
$763$	−4230.95	−0.200748
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−535.828	−0.0252251
$768$	0	0
$769$	16851.6	0.790228	0.395114	−	0.918632i	$-0.370705\pi$
$769$	16851.6	0.790228	0.395114	+	0.918632i	$0.370705\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	593.103	0.0275970	0.0137985	−	0.999905i	$-0.495608\pi$
$773$	593.103	0.0275970	0.0137985	+	0.999905i	$0.495608\pi$
$774$	0	0
$775$	9964.89	0.461870
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12768.6	−0.587268
$780$	0	0
$781$	2940.97	0.134745
$782$	0	0
$783$	0	0
$784$	0	0
$785$	19285.1	0.876834
$786$	0	0
$787$	−18564.0	−0.840834	−0.420417	−	0.907331i	$-0.638116\pi$
$787$	−18564.0	−0.840834	−0.420417	+	0.907331i	$0.638116\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4844.80	0.217777
$792$	0	0
$793$	−6497.60	−0.290967
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5227.16	0.232316	0.116158	−	0.993231i	$-0.462942\pi$
$797$	5227.16	0.232316	0.116158	+	0.993231i	$0.462942\pi$
$798$	0	0
$799$	9924.23	0.439417
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3467.77	0.152397
$804$	0	0
$805$	69665.9	3.05019
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8907.33	−0.387101	−0.193551	−	0.981090i	$-0.562000\pi$
$809$	−8907.33	−0.387101	−0.193551	+	0.981090i	$0.562000\pi$
$810$	0	0
$811$	15352.3	0.664725	0.332363	−	0.943152i	$-0.392154\pi$
$811$	15352.3	0.664725	0.332363	+	0.943152i	$0.392154\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23249.4	−0.999253
$816$	0	0
$817$	−24128.9	−1.03325
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12869.6	−0.547080	−0.273540	−	0.961861i	$-0.588195\pi$
$821$	−12869.6	−0.547080	−0.273540	+	0.961861i	$0.588195\pi$
$822$	0	0
$823$	402.065	0.0170293	0.00851464	−	0.999964i	$-0.497290\pi$
$823$	402.065	0.0170293	0.00851464	+	0.999964i	$0.497290\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−42772.7	−1.79849	−0.899245	−	0.437446i	$-0.855883\pi$
$827$	−42772.7	−1.79849	−0.899245	+	0.437446i	$0.855883\pi$
$828$	0	0
$829$	−22933.5	−0.960814	−0.480407	−	0.877046i	$-0.659511\pi$
$829$	−22933.5	−0.960814	−0.480407	+	0.877046i	$0.659511\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10792.3	0.448896
$834$	0	0
$835$	−12292.6	−0.509465
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42510.1	−1.74924	−0.874619	−	0.484811i	$-0.838888\pi$
$839$	−42510.1	−1.74924	−0.874619	+	0.484811i	$0.838888\pi$
$840$	0	0
$841$	−23423.4	−0.960409
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2380.53	0.0969143
$846$	0	0
$847$	32061.5	1.30065
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−59157.1	−2.38294
$852$	0	0
$853$	40464.6	1.62425	0.812123	−	0.583487i	$-0.198312\pi$
$853$	40464.6	1.62425	0.812123	+	0.583487i	$0.198312\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38964.3	1.55308	0.776542	−	0.630065i	$-0.216972\pi$
$857$	38964.3	1.55308	0.776542	+	0.630065i	$0.216972\pi$
$858$	0	0
$859$	6379.47	0.253393	0.126697	−	0.991942i	$-0.459563\pi$
$859$	6379.47	0.253393	0.126697	+	0.991942i	$0.459563\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33744.0	1.33101	0.665504	−	0.746395i	$-0.268217\pi$
$863$	33744.0	1.33101	0.665504	+	0.746395i	$0.268217\pi$
$864$	0	0
$865$	10323.3	0.405784
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2096.88	0.0818547
$870$	0	0
$871$	6545.58	0.254637
$872$	0	0
$873$	0	0
$874$	0	0
$875$	17631.4	0.681200
$876$	0	0
$877$	42898.6	1.65175	0.825874	−	0.563854i	$-0.190682\pi$
$877$	42898.6	1.65175	0.825874	+	0.563854i	$0.190682\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1750.90	0.0669572	0.0334786	−	0.999439i	$-0.489341\pi$
$881$	1750.90	0.0669572	0.0334786	+	0.999439i	$0.489341\pi$
$882$	0	0
$883$	−33196.7	−1.26518	−0.632591	−	0.774486i	$-0.718009\pi$
$883$	−33196.7	−1.26518	−0.632591	+	0.774486i	$0.718009\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29998.9	1.13559	0.567794	−	0.823171i	$-0.307797\pi$
$887$	29998.9	1.13559	0.567794	+	0.823171i	$0.307797\pi$
$888$	0	0
$889$	9895.98	0.373341
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19401.4	0.727037
$894$	0	0
$895$	21562.9	0.805328
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4217.84	−0.156477
$900$	0	0
$901$	7586.39	0.280510
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16877.9	−0.619935
$906$	0	0
$907$	−24629.3	−0.901657	−0.450829	−	0.892611i	$-0.648871\pi$
$907$	−24629.3	−0.901657	−0.450829	+	0.892611i	$0.648871\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7338.72	−0.266896	−0.133448	−	0.991056i	$-0.542605\pi$
$911$	−7338.72	−0.266896	−0.133448	+	0.991056i	$0.542605\pi$
$912$	0	0
$913$	184.738	0.00669654
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−52835.7	−1.90271
$918$	0	0
$919$	1449.13	0.0520155	0.0260078	−	0.999662i	$-0.491721\pi$
$919$	1449.13	0.0520155	0.0260078	+	0.999662i	$0.491721\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12303.7	0.438768
$924$	0	0
$925$	21306.9	0.757369
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6478.07	−0.228782	−0.114391	−	0.993436i	$-0.536492\pi$
$929$	−6478.07	−0.228782	−0.114391	+	0.993436i	$0.536492\pi$
$930$	0	0
$931$	21098.4	0.742720
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1922.15	−0.0672311
$936$	0	0
$937$	4679.24	0.163142	0.0815710	−	0.996668i	$-0.474006\pi$
$937$	4679.24	0.163142	0.0815710	+	0.996668i	$0.474006\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	47081.7	1.63105	0.815525	−	0.578722i	$-0.196448\pi$
$941$	47081.7	1.63105	0.815525	+	0.578722i	$0.196448\pi$
$942$	0	0
$943$	30315.6	1.04688
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4264.62	−0.146338	−0.0731688	−	0.997320i	$-0.523311\pi$
$947$	−4264.62	−0.146338	−0.0731688	+	0.997320i	$0.523311\pi$
$948$	0	0
$949$	14507.7	0.496248
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24368.1	0.828292	0.414146	−	0.910211i	$-0.364080\pi$
$953$	24368.1	0.828292	0.414146	+	0.910211i	$0.364080\pi$
$954$	0	0
$955$	−34728.0	−1.17673
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17561.3	−0.591329
$960$	0	0
$961$	−11366.8	−0.381552
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−64633.7	−2.15610
$966$	0	0
$967$	−11355.1	−0.377616	−0.188808	−	0.982014i	$-0.560462\pi$
$967$	−11355.1	−0.377616	−0.188808	+	0.982014i	$0.560462\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3024.06	−0.0999451	−0.0499726	−	0.998751i	$-0.515913\pi$
$971$	−3024.06	−0.0999451	−0.0499726	+	0.998751i	$0.515913\pi$
$972$	0	0
$973$	−36902.4	−1.21586
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41218.9	1.34975	0.674877	−	0.737930i	$-0.264197\pi$
$977$	41218.9	1.34975	0.674877	+	0.737930i	$0.264197\pi$
$978$	0	0
$979$	3786.09	0.123600
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21579.3	0.700177	0.350088	−	0.936717i	$-0.386152\pi$
$983$	21579.3	0.700177	0.350088	+	0.936717i	$0.386152\pi$
$984$	0	0
$985$	54411.5	1.76010
$986$	0	0
$987$	0	0
$988$	0	0
$989$	57287.6	1.84190
$990$	0	0
$991$	34613.5	1.10952	0.554760	−	0.832010i	$-0.312810\pi$
$991$	34613.5	1.10952	0.554760	+	0.832010i	$0.312810\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−20207.6	−0.643843
$996$	0	0
$997$	20841.5	0.662043	0.331021	−	0.943623i	$-0.392607\pi$
$997$	20841.5	0.662043	0.331021	+	0.943623i	$0.392607\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.bo.1.3		4
3.2	odd	2	inner	1872.4.a.bo.1.2		4
4.3	odd	2		117.4.a.g.1.1	✓	4
12.11	even	2		117.4.a.g.1.4	yes	4
52.51	odd	2		1521.4.a.ba.1.4		4
156.155	even	2		1521.4.a.ba.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.a.g.1.1	✓	4	4.3	odd	2
117.4.a.g.1.4	yes	4	12.11	even	2
1521.4.a.ba.1.1		4	156.155	even	2
1521.4.a.ba.1.4		4	52.51	odd	2
1872.4.a.bo.1.2		4	3.2	odd	2	inner
1872.4.a.bo.1.3		4	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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q+14.0859 q^{5} -24.2643 q^{7} +O(q^{10})\)	\(q+14.0859 q^{5} -24.2643 q^{7} -3.10739 q^{11} -13.0000 q^{13} +43.9142 q^{17} +85.8504 q^{19} -203.829 q^{23} +73.4140 q^{25} -31.0739 q^{29} +135.736 q^{31} -341.786 q^{35} +290.229 q^{37} -148.731 q^{41} -281.057 q^{43} +225.991 q^{47} +245.758 q^{49} +172.755 q^{53} -43.7706 q^{55} +41.2175 q^{59} +499.815 q^{61} -183.117 q^{65} -503.506 q^{67} -946.442 q^{71} -1115.97 q^{73} +75.3989 q^{77} -674.803 q^{79} -59.4512 q^{83} +618.574 q^{85} -1218.41 q^{89} +315.436 q^{91} +1209.28 q^{95} -879.746 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{7}+O(q^{10}) \) 4 * q - 36 * q^7	\( 4 q - 36 q^{7} - 52 q^{13} - 84 q^{19} + 660 q^{25} + 604 q^{31} + 184 q^{37} - 880 q^{43} - 116 q^{49} - 1152 q^{55} + 656 q^{61} - 3052 q^{67} - 312 q^{73} + 720 q^{79} + 32 q^{85} + 468 q^{91} - 344 q^{97}+O(q^{100}) \) 4 * q - 36 * q^7 - 52 * q^13 - 84 * q^19 + 660 * q^25 + 604 * q^31 + 184 * q^37 - 880 * q^43 - 116 * q^49 - 1152 * q^55 + 656 * q^61 - 3052 * q^67 - 312 * q^73 + 720 * q^79 + 32 * q^85 + 468 * q^91 - 344 * q^97

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