LMFDB - Embedded newform 2025.4.a.bk.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2025,4,Mod(1,2025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2025.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,0,0,54,0,0,0,0,0,0,-90,0,0,-102] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$119.478867762$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 91 x^{14} + 3268 x^{12} - 59128 x^{10} + 571975 x^{8} - 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 $ x^16 - 91x^14 + 3268x^12 - 59128x^10 + 571975x^8 - 2881141x^6 + 6555196x^4 - 4069504*x^2 + 614656
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.46385$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46385 q^{2} -1.92944 q^{4} -19.2401 q^{7} +24.4647 q^{8} +39.8548 q^{11} +1.00910 q^{13} +47.4048 q^{14} -44.8417 q^{16} +52.6170 q^{17} -49.5371 q^{19} -98.1964 q^{22} +27.4019 q^{23} -2.48626 q^{26} +37.1228 q^{28} -254.953 q^{29} -168.656 q^{31} -85.2341 q^{32} -129.640 q^{34} -419.938 q^{37} +122.052 q^{38} +398.570 q^{41} +358.828 q^{43} -76.8977 q^{44} -67.5141 q^{46} +141.302 q^{47} +27.1829 q^{49} -1.94699 q^{52} +290.878 q^{53} -470.703 q^{56} +628.166 q^{58} +28.7319 q^{59} +732.727 q^{61} +415.544 q^{62} +568.737 q^{64} -176.546 q^{67} -101.522 q^{68} -802.814 q^{71} +512.820 q^{73} +1034.66 q^{74} +95.5791 q^{76} -766.813 q^{77} +612.348 q^{79} -982.017 q^{82} +80.8162 q^{83} -884.098 q^{86} +975.035 q^{88} +24.0097 q^{89} -19.4151 q^{91} -52.8704 q^{92} -348.147 q^{94} +1367.92 q^{97} -66.9747 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 54 q^{4} - 90 q^{11} - 102 q^{14} + 146 q^{16} + 4 q^{19} - 468 q^{26} - 516 q^{29} + 38 q^{31} + 212 q^{34} - 576 q^{41} - 1644 q^{44} - 290 q^{46} - 4 q^{49} - 2430 q^{56} - 2202 q^{59} + 20 q^{61}+ \cdots - 1078 q^{94}+O(q^{100}) $ 16 * q + 54 * q^4 - 90 * q^11 - 102 * q^14 + 146 * q^16 + 4 * q^19 - 468 * q^26 - 516 * q^29 + 38 * q^31 + 212 * q^34 - 576 * q^41 - 1644 * q^44 - 290 * q^46 - 4 * q^49 - 2430 * q^56 - 2202 * q^59 + 20 * q^61 - 322 * q^64 - 2952 * q^71 - 4080 * q^74 - 396 * q^76 - 218 * q^79 - 6108 * q^86 - 4074 * q^89 - 942 * q^91 - 1078 * q^94

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 $

$2$	−2.46385	−0.871102	−0.435551	−	0.900164i	$-0.643447\pi$
$2$	−2.46385	−0.871102	−0.435551	+	0.900164i	$0.643447\pi$
$3$	0	0
$3$	0	0
$4$	−1.92944	−0.241181
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−19.2401	−1.03887	−0.519435	−	0.854510i	$-0.673858\pi$
$7$	−19.2401	−1.03887	−0.519435	+	0.854510i	$0.673858\pi$
$8$	24.4647	1.08120
$9$	0	0
$10$	0	0
$11$	39.8548	1.09243	0.546213	−	0.837646i	$-0.316069\pi$
$11$	39.8548	1.09243	0.546213	+	0.837646i	$0.316069\pi$
$12$	0	0
$13$	1.00910	0.0215287	0.0107643	−	0.999942i	$-0.496574\pi$
$13$	1.00910	0.0215287	0.0107643	+	0.999942i	$0.496574\pi$
$14$	47.4048	0.904962
$15$	0	0
$16$	−44.8417	−0.700651
$17$	52.6170	0.750676	0.375338	−	0.926888i	$-0.377527\pi$
$17$	52.6170	0.750676	0.375338	+	0.926888i	$0.377527\pi$
$18$	0	0
$19$	−49.5371	−0.598136	−0.299068	−	0.954232i	$-0.596676\pi$
$19$	−49.5371	−0.598136	−0.299068	+	0.954232i	$0.596676\pi$
$20$	0	0
$21$	0	0
$22$	−98.1964	−0.951615
$23$	27.4019	0.248421	0.124210	−	0.992256i	$-0.460360\pi$
$23$	27.4019	0.248421	0.124210	+	0.992256i	$0.460360\pi$
$24$	0	0
$25$	0	0
$26$	−2.48626	−0.0187537
$27$	0	0
$28$	37.1228	0.250555
$29$	−254.953	−1.63254	−0.816269	−	0.577672i	$-0.803961\pi$
$29$	−254.953	−1.63254	−0.816269	+	0.577672i	$0.803961\pi$
$30$	0	0
$31$	−168.656	−0.977147	−0.488574	−	0.872523i	$-0.662483\pi$
$31$	−168.656	−0.977147	−0.488574	+	0.872523i	$0.662483\pi$
$32$	−85.2341	−0.470856
$33$	0	0
$34$	−129.640	−0.653916
$35$	0	0
$36$	0	0
$37$	−419.938	−1.86588	−0.932938	−	0.360038i	$-0.882764\pi$
$37$	−419.938	−1.86588	−0.932938	+	0.360038i	$0.882764\pi$
$38$	122.052	0.521038
$39$	0	0
$40$	0	0
$41$	398.570	1.51820	0.759100	−	0.650974i	$-0.225639\pi$
$41$	398.570	1.51820	0.759100	+	0.650974i	$0.225639\pi$
$42$	0	0
$43$	358.828	1.27258	0.636288	−	0.771452i	$-0.280469\pi$
$43$	358.828	1.27258	0.636288	+	0.771452i	$0.280469\pi$
$44$	−76.8977	−0.263472
$45$	0	0
$46$	−67.5141	−0.216400
$47$	141.302	0.438532	0.219266	−	0.975665i	$-0.429634\pi$
$47$	141.302	0.438532	0.219266	+	0.975665i	$0.429634\pi$
$48$	0	0
$49$	27.1829	0.0792505
$50$	0	0
$51$	0	0
$52$	−1.94699	−0.00519230
$53$	290.878	0.753872	0.376936	−	0.926239i	$-0.376978\pi$
$53$	290.878	0.753872	0.376936	+	0.926239i	$0.376978\pi$
$54$	0	0
$55$	0	0
$56$	−470.703	−1.12322
$57$	0	0
$58$	628.166	1.42211
$59$	28.7319	0.0633995	0.0316998	−	0.999497i	$-0.489908\pi$
$59$	28.7319	0.0633995	0.0316998	+	0.999497i	$0.489908\pi$
$60$	0	0
$61$	732.727	1.53797	0.768985	−	0.639267i	$-0.220762\pi$
$61$	732.727	1.53797	0.768985	+	0.639267i	$0.220762\pi$
$62$	415.544	0.851195
$63$	0	0
$64$	568.737	1.11082
$65$	0	0
$66$	0	0
$67$	−176.546	−0.321918	−0.160959	−	0.986961i	$-0.551459\pi$
$67$	−176.546	−0.321918	−0.160959	+	0.986961i	$0.551459\pi$
$68$	−101.522	−0.181049
$69$	0	0
$70$	0	0
$71$	−802.814	−1.34192	−0.670961	−	0.741493i	$-0.734118\pi$
$71$	−802.814	−1.34192	−0.670961	+	0.741493i	$0.734118\pi$
$72$	0	0
$73$	512.820	0.822205	0.411103	−	0.911589i	$-0.365144\pi$
$73$	512.820	0.822205	0.411103	+	0.911589i	$0.365144\pi$
$74$	1034.66	1.62537
$75$	0	0
$76$	95.5791	0.144259
$77$	−766.813	−1.13489
$78$	0	0
$79$	612.348	0.872082	0.436041	−	0.899927i	$-0.356380\pi$
$79$	612.348	0.872082	0.436041	+	0.899927i	$0.356380\pi$
$80$	0	0
$81$	0	0
$82$	−982.017	−1.32251
$83$	80.8162	0.106876	0.0534382	−	0.998571i	$-0.482982\pi$
$83$	80.8162	0.106876	0.0534382	+	0.998571i	$0.482982\pi$
$84$	0	0
$85$	0	0
$86$	−884.098	−1.10854
$87$	0	0
$88$	975.035	1.18113
$89$	24.0097	0.0285957	0.0142979	−	0.999898i	$-0.495449\pi$
$89$	24.0097	0.0285957	0.0142979	+	0.999898i	$0.495449\pi$
$90$	0	0
$91$	−19.4151	−0.0223655
$92$	−52.8704	−0.0599143
$93$	0	0
$94$	−348.147	−0.382006
$95$	0	0
$96$	0	0
$97$	1367.92	1.43187	0.715934	−	0.698168i	$-0.246001\pi$
$97$	1367.92	1.43187	0.715934	+	0.698168i	$0.246001\pi$
$98$	−66.9747	−0.0690353
$99$	0	0
$100$	0	0
$101$	568.413	0.559992	0.279996	−	0.960001i	$-0.409667\pi$
$101$	568.413	0.559992	0.279996	+	0.960001i	$0.409667\pi$
$102$	0	0
$103$	475.159	0.454551	0.227276	−	0.973830i	$-0.427018\pi$
$103$	475.159	0.454551	0.227276	+	0.973830i	$0.427018\pi$
$104$	24.6872	0.0232767
$105$	0	0
$106$	−716.680	−0.656700
$107$	1452.60	1.31242	0.656208	−	0.754580i	$-0.272159\pi$
$107$	1452.60	1.31242	0.656208	+	0.754580i	$0.272159\pi$
$108$	0	0
$109$	−1962.92	−1.72490	−0.862448	−	0.506146i	$-0.831070\pi$
$109$	−1962.92	−1.72490	−0.862448	+	0.506146i	$0.831070\pi$
$110$	0	0
$111$	0	0
$112$	862.760	0.727886
$113$	−978.286	−0.814419	−0.407210	−	0.913335i	$-0.633498\pi$
$113$	−978.286	−0.814419	−0.407210	+	0.913335i	$0.633498\pi$
$114$	0	0
$115$	0	0
$116$	491.918	0.393736
$117$	0	0
$118$	−70.7910	−0.0552275
$119$	−1012.36	−0.779855
$120$	0	0
$121$	257.409	0.193395
$122$	−1805.33	−1.33973
$123$	0	0
$124$	325.413	0.235669
$125$	0	0
$126$	0	0
$127$	865.941	0.605038	0.302519	−	0.953143i	$-0.402172\pi$
$127$	865.941	0.605038	0.302519	+	0.953143i	$0.402172\pi$
$128$	−719.411	−0.496778
$129$	0	0
$130$	0	0
$131$	−1468.96	−0.979720	−0.489860	−	0.871801i	$-0.662952\pi$
$131$	−1468.96	−0.979720	−0.489860	+	0.871801i	$0.662952\pi$
$132$	0	0
$133$	953.101	0.621386
$134$	434.983	0.280424
$135$	0	0
$136$	1287.26	0.811628
$137$	−71.5353	−0.0446108	−0.0223054	−	0.999751i	$-0.507101\pi$
$137$	−71.5353	−0.0446108	−0.0223054	+	0.999751i	$0.507101\pi$
$138$	0	0
$139$	229.065	0.139777	0.0698886	−	0.997555i	$-0.477736\pi$
$139$	229.065	0.139777	0.0698886	+	0.997555i	$0.477736\pi$
$140$	0	0
$141$	0	0
$142$	1978.01	1.16895
$143$	40.2174	0.0235185
$144$	0	0
$145$	0	0
$146$	−1263.51	−0.716225
$147$	0	0
$148$	810.247	0.450013
$149$	−835.144	−0.459179	−0.229589	−	0.973288i	$-0.573738\pi$
$149$	−835.144	−0.459179	−0.229589	+	0.973288i	$0.573738\pi$
$150$	0	0
$151$	−1866.58	−1.00596	−0.502982	−	0.864297i	$-0.667764\pi$
$151$	−1866.58	−1.00596	−0.502982	+	0.864297i	$0.667764\pi$
$152$	−1211.91	−0.646702
$153$	0	0
$154$	1889.31	0.988604
$155$	0	0
$156$	0	0
$157$	2299.91	1.16913	0.584563	−	0.811348i	$-0.301266\pi$
$157$	2299.91	1.16913	0.584563	+	0.811348i	$0.301266\pi$
$158$	−1508.73	−0.759673
$159$	0	0
$160$	0	0
$161$	−527.215	−0.258077
$162$	0	0
$163$	−44.9023	−0.0215768	−0.0107884	−	0.999942i	$-0.503434\pi$
$163$	−44.9023	−0.0215768	−0.0107884	+	0.999942i	$0.503434\pi$
$164$	−769.019	−0.366160
$165$	0	0
$166$	−199.119	−0.0931002
$167$	1176.56	0.545180	0.272590	−	0.962130i	$-0.412120\pi$
$167$	1176.56	0.545180	0.272590	+	0.962130i	$0.412120\pi$
$168$	0	0
$169$	−2195.98	−0.999537
$170$	0	0
$171$	0	0
$172$	−692.339	−0.306921
$173$	−1068.09	−0.469396	−0.234698	−	0.972068i	$-0.575410\pi$
$173$	−1068.09	−0.469396	−0.234698	+	0.972068i	$0.575410\pi$
$174$	0	0
$175$	0	0
$176$	−1787.16	−0.765410
$177$	0	0
$178$	−59.1562	−0.0249098
$179$	−2779.66	−1.16068	−0.580339	−	0.814375i	$-0.697080\pi$
$179$	−2779.66	−1.16068	−0.580339	+	0.814375i	$0.697080\pi$
$180$	0	0
$181$	−3568.66	−1.46550	−0.732752	−	0.680496i	$-0.761764\pi$
$181$	−3568.66	−1.46550	−0.732752	+	0.680496i	$0.761764\pi$
$182$	47.8360	0.0194826
$183$	0	0
$184$	670.377	0.268592
$185$	0	0
$186$	0	0
$187$	2097.04	0.820059
$188$	−272.634	−0.105765
$189$	0	0
$190$	0	0
$191$	−3338.62	−1.26478	−0.632392	−	0.774648i	$-0.717927\pi$
$191$	−3338.62	−1.26478	−0.632392	+	0.774648i	$0.717927\pi$
$192$	0	0
$193$	−1940.87	−0.723868	−0.361934	−	0.932204i	$-0.617883\pi$
$193$	−1940.87	−0.723868	−0.361934	+	0.932204i	$0.617883\pi$
$194$	−3370.35	−1.24730
$195$	0	0
$196$	−52.4480	−0.0191137
$197$	1035.75	0.374588	0.187294	−	0.982304i	$-0.440028\pi$
$197$	1035.75	0.374588	0.187294	+	0.982304i	$0.440028\pi$
$198$	0	0
$199$	1565.53	0.557675	0.278838	−	0.960338i	$-0.410051\pi$
$199$	1565.53	0.557675	0.278838	+	0.960338i	$0.410051\pi$
$200$	0	0
$201$	0	0
$202$	−1400.48	−0.487810
$203$	4905.33	1.69599
$204$	0	0
$205$	0	0
$206$	−1170.72	−0.395961
$207$	0	0
$208$	−45.2496	−0.0150841
$209$	−1974.29	−0.653420
$210$	0	0
$211$	3385.27	1.10451	0.552255	−	0.833675i	$-0.313767\pi$
$211$	3385.27	1.10451	0.552255	+	0.833675i	$0.313767\pi$
$212$	−561.234	−0.181819
$213$	0	0
$214$	−3579.00	−1.14325
$215$	0	0
$216$	0	0
$217$	3244.97	1.01513
$218$	4836.34	1.50256
$219$	0	0
$220$	0	0
$221$	53.0956	0.0161611
$222$	0	0
$223$	−2365.54	−0.710352	−0.355176	−	0.934799i	$-0.615579\pi$
$223$	−2365.54	−0.710352	−0.355176	+	0.934799i	$0.615579\pi$
$224$	1639.92	0.489158
$225$	0	0
$226$	2410.35	0.709443
$227$	976.230	0.285439	0.142720	−	0.989763i	$-0.454415\pi$
$227$	976.230	0.285439	0.142720	+	0.989763i	$0.454415\pi$
$228$	0	0
$229$	1595.28	0.460345	0.230173	−	0.973150i	$-0.426071\pi$
$229$	1595.28	0.460345	0.230173	+	0.973150i	$0.426071\pi$
$230$	0	0
$231$	0	0
$232$	−6237.34	−1.76509
$233$	943.571	0.265302	0.132651	−	0.991163i	$-0.457651\pi$
$233$	943.571	0.265302	0.132651	+	0.991163i	$0.457651\pi$
$234$	0	0
$235$	0	0
$236$	−55.4366	−0.0152907
$237$	0	0
$238$	2494.30	0.679334
$239$	2731.50	0.739271	0.369635	−	0.929177i	$-0.379483\pi$
$239$	2731.50	0.739271	0.369635	+	0.929177i	$0.379483\pi$
$240$	0	0
$241$	3417.23	0.913375	0.456687	−	0.889627i	$-0.349036\pi$
$241$	3417.23	0.913375	0.456687	+	0.889627i	$0.349036\pi$
$242$	−634.217	−0.168467
$243$	0	0
$244$	−1413.76	−0.370928
$245$	0	0
$246$	0	0
$247$	−49.9877	−0.0128771
$248$	−4126.12	−1.05649
$249$	0	0
$250$	0	0
$251$	−3164.50	−0.795782	−0.397891	−	0.917433i	$-0.630258\pi$
$251$	−3164.50	−0.795782	−0.397891	+	0.917433i	$0.630258\pi$
$252$	0	0
$253$	1092.10	0.271382
$254$	−2133.55	−0.527050
$255$	0	0
$256$	−2777.38	−0.678071
$257$	5549.43	1.34694	0.673471	−	0.739214i	$-0.264803\pi$
$257$	5549.43	1.34694	0.673471	+	0.739214i	$0.264803\pi$
$258$	0	0
$259$	8079.67	1.93840
$260$	0	0
$261$	0	0
$262$	3619.29	0.853436
$263$	−4137.64	−0.970106	−0.485053	−	0.874485i	$-0.661200\pi$
$263$	−4137.64	−0.970106	−0.485053	+	0.874485i	$0.661200\pi$
$264$	0	0
$265$	0	0
$266$	−2348.30	−0.541291
$267$	0	0
$268$	340.636	0.0776404
$269$	−1026.74	−0.232719	−0.116359	−	0.993207i	$-0.537122\pi$
$269$	−1026.74	−0.232719	−0.116359	+	0.993207i	$0.537122\pi$
$270$	0	0
$271$	3400.66	0.762270	0.381135	−	0.924519i	$-0.375533\pi$
$271$	3400.66	0.762270	0.381135	+	0.924519i	$0.375533\pi$
$272$	−2359.44	−0.525962
$273$	0	0
$274$	176.252	0.0388605
$275$	0	0
$276$	0	0
$277$	−3568.81	−0.774112	−0.387056	−	0.922056i	$-0.626508\pi$
$277$	−3568.81	−0.774112	−0.387056	+	0.922056i	$0.626508\pi$
$278$	−564.381	−0.121760
$279$	0	0
$280$	0	0
$281$	−6018.68	−1.27774	−0.638869	−	0.769316i	$-0.720597\pi$
$281$	−6018.68	−1.27774	−0.638869	+	0.769316i	$0.720597\pi$
$282$	0	0
$283$	−4578.17	−0.961638	−0.480819	−	0.876820i	$-0.659661\pi$
$283$	−4578.17	−0.961638	−0.480819	+	0.876820i	$0.659661\pi$
$284$	1548.98	0.323646
$285$	0	0
$286$	−99.0895	−0.0204870
$287$	−7668.55	−1.57721
$288$	0	0
$289$	−2144.45	−0.436485
$290$	0	0
$291$	0	0
$292$	−989.457	−0.198300
$293$	6479.73	1.29198	0.645990	−	0.763346i	$-0.276445\pi$
$293$	6479.73	1.29198	0.645990	+	0.763346i	$0.276445\pi$
$294$	0	0
$295$	0	0
$296$	−10273.6	−2.01738
$297$	0	0
$298$	2057.67	0.399992
$299$	27.6511	0.00534818
$300$	0	0
$301$	−6903.90	−1.32204
$302$	4598.98	0.876297
$303$	0	0
$304$	2221.33	0.419085
$305$	0	0
$306$	0	0
$307$	708.452	0.131705	0.0658526	−	0.997829i	$-0.479023\pi$
$307$	708.452	0.131705	0.0658526	+	0.997829i	$0.479023\pi$
$308$	1479.52	0.273713
$309$	0	0
$310$	0	0
$311$	−6234.64	−1.13676	−0.568382	−	0.822765i	$-0.692430\pi$
$311$	−6234.64	−1.13676	−0.568382	+	0.822765i	$0.692430\pi$
$312$	0	0
$313$	−2631.41	−0.475196	−0.237598	−	0.971364i	$-0.576360\pi$
$313$	−2631.41	−0.475196	−0.237598	+	0.971364i	$0.576360\pi$
$314$	−5666.63	−1.01843
$315$	0	0
$316$	−1181.49	−0.210329
$317$	−502.534	−0.0890383	−0.0445191	−	0.999009i	$-0.514176\pi$
$317$	−502.534	−0.0890383	−0.0445191	+	0.999009i	$0.514176\pi$
$318$	0	0
$319$	−10161.1	−1.78343
$320$	0	0
$321$	0	0
$322$	1298.98	0.224812
$323$	−2606.49	−0.449007
$324$	0	0
$325$	0	0
$326$	110.632	0.0187956
$327$	0	0
$328$	9750.89	1.64147
$329$	−2718.67	−0.455578
$330$	0	0
$331$	3087.14	0.512642	0.256321	−	0.966592i	$-0.417490\pi$
$331$	3087.14	0.512642	0.256321	+	0.966592i	$0.417490\pi$
$332$	−155.930	−0.0257765
$333$	0	0
$334$	−2898.87	−0.474908
$335$	0	0
$336$	0	0
$337$	−5985.80	−0.967559	−0.483780	−	0.875190i	$-0.660736\pi$
$337$	−5985.80	−0.967559	−0.483780	+	0.875190i	$0.660736\pi$
$338$	5410.57	0.870699
$339$	0	0
$340$	0	0
$341$	−6721.77	−1.06746
$342$	0	0
$343$	6076.36	0.956539
$344$	8778.60	1.37590
$345$	0	0
$346$	2631.61	0.408892
$347$	1006.65	0.155734	0.0778671	−	0.996964i	$-0.475189\pi$
$347$	1006.65	0.155734	0.0778671	+	0.996964i	$0.475189\pi$
$348$	0	0
$349$	−1949.36	−0.298987	−0.149494	−	0.988763i	$-0.547764\pi$
$349$	−1949.36	−0.298987	−0.149494	+	0.988763i	$0.547764\pi$
$350$	0	0
$351$	0	0
$352$	−3396.99	−0.514376
$353$	2591.73	0.390776	0.195388	−	0.980726i	$-0.437403\pi$
$353$	2591.73	0.390776	0.195388	+	0.980726i	$0.437403\pi$
$354$	0	0
$355$	0	0
$356$	−46.3253	−0.00689674
$357$	0	0
$358$	6848.65	1.01107
$359$	−605.954	−0.0890837	−0.0445418	−	0.999008i	$-0.514183\pi$
$359$	−605.954	−0.0890837	−0.0445418	+	0.999008i	$0.514183\pi$
$360$	0	0
$361$	−4405.08	−0.642233
$362$	8792.63	1.27660
$363$	0	0
$364$	37.4604	0.00539412
$365$	0	0
$366$	0	0
$367$	−9760.26	−1.38823	−0.694117	−	0.719863i	$-0.744205\pi$
$367$	−9760.26	−1.38823	−0.694117	+	0.719863i	$0.744205\pi$
$368$	−1228.75	−0.174056
$369$	0	0
$370$	0	0
$371$	−5596.54	−0.783175
$372$	0	0
$373$	−7823.62	−1.08604	−0.543018	−	0.839721i	$-0.682719\pi$
$373$	−7823.62	−1.08604	−0.543018	+	0.839721i	$0.682719\pi$
$374$	−5166.80	−0.714355
$375$	0	0
$376$	3456.90	0.474139
$377$	−257.272	−0.0351464
$378$	0	0
$379$	2731.46	0.370200	0.185100	−	0.982720i	$-0.440739\pi$
$379$	2731.46	0.370200	0.185100	+	0.982720i	$0.440739\pi$
$380$	0	0
$381$	0	0
$382$	8225.85	1.10176
$383$	−12599.9	−1.68100	−0.840500	−	0.541812i	$-0.817738\pi$
$383$	−12599.9	−1.68100	−0.840500	+	0.541812i	$0.817738\pi$
$384$	0	0
$385$	0	0
$386$	4782.00	0.630563
$387$	0	0
$388$	−2639.33	−0.345339
$389$	−4943.46	−0.644327	−0.322163	−	0.946684i	$-0.604410\pi$
$389$	−4943.46	−0.644327	−0.322163	+	0.946684i	$0.604410\pi$
$390$	0	0
$391$	1441.80	0.186484
$392$	665.021	0.0856853
$393$	0	0
$394$	−2551.92	−0.326305
$395$	0	0
$396$	0	0
$397$	7067.06	0.893415	0.446707	−	0.894680i	$-0.352597\pi$
$397$	7067.06	0.893415	0.446707	+	0.894680i	$0.352597\pi$
$398$	−3857.23	−0.485792
$399$	0	0
$400$	0	0
$401$	−917.584	−0.114269	−0.0571346	−	0.998366i	$-0.518196\pi$
$401$	−917.584	−0.114269	−0.0571346	+	0.998366i	$0.518196\pi$
$402$	0	0
$403$	−170.190	−0.0210367
$404$	−1096.72	−0.135059
$405$	0	0
$406$	−12086.0	−1.47738
$407$	−16736.6	−2.03833
$408$	0	0
$409$	14735.5	1.78148	0.890739	−	0.454515i	$-0.150187\pi$
$409$	14735.5	1.78148	0.890739	+	0.454515i	$0.150187\pi$
$410$	0	0
$411$	0	0
$412$	−916.792	−0.109629
$413$	−552.805	−0.0658638
$414$	0	0
$415$	0	0
$416$	−86.0094	−0.0101369
$417$	0	0
$418$	4864.36	0.569196
$419$	−9254.92	−1.07907	−0.539537	−	0.841962i	$-0.681401\pi$
$419$	−9254.92	−1.07907	−0.539537	+	0.841962i	$0.681401\pi$
$420$	0	0
$421$	−10202.0	−1.18103	−0.590514	−	0.807027i	$-0.701075\pi$
$421$	−10202.0	−1.18103	−0.590514	+	0.807027i	$0.701075\pi$
$422$	−8340.80	−0.962142
$423$	0	0
$424$	7116.24	0.815083
$425$	0	0
$426$	0	0
$427$	−14097.8	−1.59775
$428$	−2802.72	−0.316529
$429$	0	0
$430$	0	0
$431$	8057.20	0.900468	0.450234	−	0.892911i	$-0.351341\pi$
$431$	8057.20	0.900468	0.450234	+	0.892911i	$0.351341\pi$
$432$	0	0
$433$	−4306.26	−0.477935	−0.238967	−	0.971028i	$-0.576809\pi$
$433$	−4306.26	−0.477935	−0.238967	+	0.971028i	$0.576809\pi$
$434$	−7995.12	−0.884281
$435$	0	0
$436$	3787.34	0.416011
$437$	−1357.41	−0.148590
$438$	0	0
$439$	−4721.79	−0.513346	−0.256673	−	0.966498i	$-0.582626\pi$
$439$	−4721.79	−0.513346	−0.256673	+	0.966498i	$0.582626\pi$
$440$	0	0
$441$	0	0
$442$	−130.820	−0.0140779
$443$	−15354.4	−1.64675	−0.823373	−	0.567501i	$-0.807910\pi$
$443$	−15354.4	−1.64675	−0.823373	+	0.567501i	$0.807910\pi$
$444$	0	0
$445$	0	0
$446$	5828.35	0.618790
$447$	0	0
$448$	−10942.6	−1.15399
$449$	5448.64	0.572688	0.286344	−	0.958127i	$-0.407560\pi$
$449$	5448.64	0.572688	0.286344	+	0.958127i	$0.407560\pi$
$450$	0	0
$451$	15885.0	1.65852
$452$	1887.55	0.196422
$453$	0	0
$454$	−2405.28	−0.248647
$455$	0	0
$456$	0	0
$457$	52.7307	0.00539746	0.00269873	−	0.999996i	$-0.499141\pi$
$457$	52.7307	0.00539746	0.00269873	+	0.999996i	$0.499141\pi$
$458$	−3930.53	−0.401008
$459$	0	0
$460$	0	0
$461$	−3167.02	−0.319963	−0.159982	−	0.987120i	$-0.551143\pi$
$461$	−3167.02	−0.319963	−0.159982	+	0.987120i	$0.551143\pi$
$462$	0	0
$463$	−13092.4	−1.31416	−0.657081	−	0.753820i	$-0.728209\pi$
$463$	−13092.4	−1.31416	−0.657081	+	0.753820i	$0.728209\pi$
$464$	11432.5	1.14384
$465$	0	0
$466$	−2324.82	−0.231105
$467$	13225.4	1.31049	0.655246	−	0.755416i	$-0.272565\pi$
$467$	13225.4	1.31049	0.655246	+	0.755416i	$0.272565\pi$
$468$	0	0
$469$	3396.77	0.334431
$470$	0	0
$471$	0	0
$472$	702.915	0.0685473
$473$	14301.0	1.39020
$474$	0	0
$475$	0	0
$476$	1953.29	0.188086
$477$	0	0
$478$	−6729.99	−0.643981
$479$	−8558.59	−0.816393	−0.408196	−	0.912894i	$-0.633842\pi$
$479$	−8558.59	−0.816393	−0.408196	+	0.912894i	$0.633842\pi$
$480$	0	0
$481$	−423.758	−0.0401698
$482$	−8419.55	−0.795643
$483$	0	0
$484$	−496.656	−0.0466432
$485$	0	0
$486$	0	0
$487$	−19531.3	−1.81735	−0.908674	−	0.417506i	$-0.862904\pi$
$487$	−19531.3	−1.81735	−0.908674	+	0.417506i	$0.862904\pi$
$488$	17925.9	1.66285
$489$	0	0
$490$	0	0
$491$	−19488.2	−1.79122	−0.895612	−	0.444835i	$-0.853262\pi$
$491$	−19488.2	−1.79122	−0.895612	+	0.444835i	$0.853262\pi$
$492$	0	0
$493$	−13414.9	−1.22551
$494$	123.162	0.0112173
$495$	0	0
$496$	7562.83	0.684640
$497$	15446.2	1.39408
$498$	0	0
$499$	−339.530	−0.0304598	−0.0152299	−	0.999884i	$-0.504848\pi$
$499$	−339.530	−0.0304598	−0.0152299	+	0.999884i	$0.504848\pi$
$500$	0	0
$501$	0	0
$502$	7796.84	0.693208
$503$	5550.25	0.491995	0.245998	−	0.969270i	$-0.420884\pi$
$503$	5550.25	0.491995	0.245998	+	0.969270i	$0.420884\pi$
$504$	0	0
$505$	0	0
$506$	−2690.76	−0.236401
$507$	0	0
$508$	−1670.79	−0.145923
$509$	5648.99	0.491920	0.245960	−	0.969280i	$-0.420897\pi$
$509$	5648.99	0.491920	0.245960	+	0.969280i	$0.420897\pi$
$510$	0	0
$511$	−9866.72	−0.854164
$512$	12598.3	1.08745
$513$	0	0
$514$	−13673.0	−1.17332
$515$	0	0
$516$	0	0
$517$	5631.57	0.479064
$518$	−19907.1	−1.68855
$519$	0	0
$520$	0	0
$521$	−11159.0	−0.938359	−0.469180	−	0.883103i	$-0.655450\pi$
$521$	−11159.0	−0.938359	−0.469180	+	0.883103i	$0.655450\pi$
$522$	0	0
$523$	−9476.61	−0.792320	−0.396160	−	0.918182i	$-0.629657\pi$
$523$	−9476.61	−0.792320	−0.396160	+	0.918182i	$0.629657\pi$
$524$	2834.27	0.236289
$525$	0	0
$526$	10194.5	0.845062
$527$	−8874.19	−0.733521
$528$	0	0
$529$	−11416.1	−0.938287
$530$	0	0
$531$	0	0
$532$	−1838.95	−0.149866
$533$	402.196	0.0326849
$534$	0	0
$535$	0	0
$536$	−4319.14	−0.348057
$537$	0	0
$538$	2529.73	0.202722
$539$	1083.37	0.0865754
$540$	0	0
$541$	8285.40	0.658442	0.329221	−	0.944253i	$-0.393214\pi$
$541$	8285.40	0.658442	0.329221	+	0.944253i	$0.393214\pi$
$542$	−8378.71	−0.664015
$543$	0	0
$544$	−4484.76	−0.353461
$545$	0	0
$546$	0	0
$547$	6918.30	0.540777	0.270389	−	0.962751i	$-0.412848\pi$
$547$	6918.30	0.540777	0.270389	+	0.962751i	$0.412848\pi$
$548$	138.023	0.0107593
$549$	0	0
$550$	0	0
$551$	12629.6	0.976480
$552$	0	0
$553$	−11781.7	−0.905980
$554$	8793.01	0.674331
$555$	0	0
$556$	−441.968	−0.0337115
$557$	−15622.7	−1.18843	−0.594216	−	0.804305i	$-0.702538\pi$
$557$	−15622.7	−1.18843	−0.594216	+	0.804305i	$0.702538\pi$
$558$	0	0
$559$	362.092	0.0273969
$560$	0	0
$561$	0	0
$562$	14829.1	1.11304
$563$	8129.30	0.608542	0.304271	−	0.952585i	$-0.401587\pi$
$563$	8129.30	0.608542	0.304271	+	0.952585i	$0.401587\pi$
$564$	0	0
$565$	0	0
$566$	11279.9	0.837686
$567$	0	0
$568$	−19640.6	−1.45088
$569$	8175.67	0.602359	0.301179	−	0.953568i	$-0.402620\pi$
$569$	8175.67	0.602359	0.301179	+	0.953568i	$0.402620\pi$
$570$	0	0
$571$	7878.84	0.577442	0.288721	−	0.957413i	$-0.406770\pi$
$571$	7878.84	0.577442	0.288721	+	0.957413i	$0.406770\pi$
$572$	−77.5972	−0.00567220
$573$	0	0
$574$	18894.1	1.37391
$575$	0	0
$576$	0	0
$577$	−16673.4	−1.20299	−0.601493	−	0.798878i	$-0.705427\pi$
$577$	−16673.4	−1.20299	−0.601493	+	0.798878i	$0.705427\pi$
$578$	5283.60	0.380223
$579$	0	0
$580$	0	0
$581$	−1554.92	−0.111031
$582$	0	0
$583$	11592.9	0.823549
$584$	12546.0	0.888965
$585$	0	0
$586$	−15965.1	−1.12545
$587$	3012.18	0.211799	0.105900	−	0.994377i	$-0.466228\pi$
$587$	3012.18	0.211799	0.105900	+	0.994377i	$0.466228\pi$
$588$	0	0
$589$	8354.74	0.584467
$590$	0	0
$591$	0	0
$592$	18830.7	1.30733
$593$	7922.71	0.548645	0.274323	−	0.961638i	$-0.411546\pi$
$593$	7922.71	0.548645	0.274323	+	0.961638i	$0.411546\pi$
$594$	0	0
$595$	0	0
$596$	1611.36	0.110745
$597$	0	0
$598$	−68.1282	−0.00465881
$599$	−22578.5	−1.54012	−0.770061	−	0.637970i	$-0.779774\pi$
$599$	−22578.5	−1.54012	−0.770061	+	0.637970i	$0.779774\pi$
$600$	0	0
$601$	10618.7	0.720709	0.360354	−	0.932815i	$-0.382656\pi$
$601$	10618.7	0.720709	0.360354	+	0.932815i	$0.382656\pi$
$602$	17010.2	1.15163
$603$	0	0
$604$	3601.47	0.242619
$605$	0	0
$606$	0	0
$607$	−13590.6	−0.908771	−0.454385	−	0.890805i	$-0.650141\pi$
$607$	−13590.6	−0.908771	−0.454385	+	0.890805i	$0.650141\pi$
$608$	4222.25	0.281636
$609$	0	0
$610$	0	0
$611$	142.587	0.00944102
$612$	0	0
$613$	−5457.57	−0.359591	−0.179796	−	0.983704i	$-0.557544\pi$
$613$	−5457.57	−0.359591	−0.179796	+	0.983704i	$0.557544\pi$
$614$	−1745.52	−0.114729
$615$	0	0
$616$	−18759.8	−1.22704
$617$	−5972.10	−0.389672	−0.194836	−	0.980836i	$-0.562417\pi$
$617$	−5972.10	−0.389672	−0.194836	+	0.980836i	$0.562417\pi$
$618$	0	0
$619$	−839.633	−0.0545197	−0.0272598	−	0.999628i	$-0.508678\pi$
$619$	−839.633	−0.0545197	−0.0272598	+	0.999628i	$0.508678\pi$
$620$	0	0
$621$	0	0
$622$	15361.2	0.990238
$623$	−461.949	−0.0297072
$624$	0	0
$625$	0	0
$626$	6483.41	0.413944
$627$	0	0
$628$	−4437.55	−0.281970
$629$	−22095.9	−1.40067
$630$	0	0
$631$	−13699.4	−0.864284	−0.432142	−	0.901806i	$-0.642242\pi$
$631$	−13699.4	−0.864284	−0.432142	+	0.901806i	$0.642242\pi$
$632$	14980.9	0.942892
$633$	0	0
$634$	1238.17	0.0775615
$635$	0	0
$636$	0	0
$637$	27.4302	0.00170616
$638$	25035.5	1.55355
$639$	0	0
$640$	0	0
$641$	−7939.36	−0.489214	−0.244607	−	0.969622i	$-0.578659\pi$
$641$	−7939.36	−0.489214	−0.244607	+	0.969622i	$0.578659\pi$
$642$	0	0
$643$	7635.95	0.468324	0.234162	−	0.972198i	$-0.424765\pi$
$643$	7635.95	0.468324	0.234162	+	0.972198i	$0.424765\pi$
$644$	1017.23	0.0622432
$645$	0	0
$646$	6422.01	0.391131
$647$	22084.3	1.34192	0.670962	−	0.741491i	$-0.265881\pi$
$647$	22084.3	1.34192	0.670962	+	0.741491i	$0.265881\pi$
$648$	0	0
$649$	1145.10	0.0692593
$650$	0	0
$651$	0	0
$652$	86.6364	0.00520390
$653$	−6637.97	−0.397801	−0.198900	−	0.980020i	$-0.563737\pi$
$653$	−6637.97	−0.397801	−0.198900	+	0.980020i	$0.563737\pi$
$654$	0	0
$655$	0	0
$656$	−17872.6	−1.06373
$657$	0	0
$658$	6698.39	0.396855
$659$	23812.3	1.40758	0.703789	−	0.710409i	$-0.251490\pi$
$659$	23812.3	1.40758	0.703789	+	0.710409i	$0.251490\pi$
$660$	0	0
$661$	−999.824	−0.0588330	−0.0294165	−	0.999567i	$-0.509365\pi$
$661$	−999.824	−0.0588330	−0.0294165	+	0.999567i	$0.509365\pi$
$662$	−7606.24	−0.446563
$663$	0	0
$664$	1977.14	0.115554
$665$	0	0
$666$	0	0
$667$	−6986.19	−0.405557
$668$	−2270.11	−0.131487
$669$	0	0
$670$	0	0
$671$	29202.7	1.68012
$672$	0	0
$673$	−10622.1	−0.608396	−0.304198	−	0.952609i	$-0.598388\pi$
$673$	−10622.1	−0.608396	−0.304198	+	0.952609i	$0.598388\pi$
$674$	14748.1	0.842843
$675$	0	0
$676$	4237.02	0.241069
$677$	−24907.8	−1.41401	−0.707005	−	0.707209i	$-0.749954\pi$
$677$	−24907.8	−1.41401	−0.707005	+	0.707209i	$0.749954\pi$
$678$	0	0
$679$	−26319.0	−1.48753
$680$	0	0
$681$	0	0
$682$	16561.4	0.929868
$683$	−17115.6	−0.958874	−0.479437	−	0.877576i	$-0.659159\pi$
$683$	−17115.6	−0.958874	−0.479437	+	0.877576i	$0.659159\pi$
$684$	0	0
$685$	0	0
$686$	−14971.2	−0.833243
$687$	0	0
$688$	−16090.5	−0.891632
$689$	293.524	0.0162299
$690$	0	0
$691$	28899.8	1.59103	0.795514	−	0.605935i	$-0.207201\pi$
$691$	28899.8	1.59103	0.795514	+	0.605935i	$0.207201\pi$
$692$	2060.82	0.113209
$693$	0	0
$694$	−2480.23	−0.135660
$695$	0	0
$696$	0	0
$697$	20971.6	1.13968
$698$	4802.92	0.260449
$699$	0	0
$700$	0	0
$701$	576.680	0.0310712	0.0155356	−	0.999879i	$-0.495055\pi$
$701$	576.680	0.0310712	0.0155356	+	0.999879i	$0.495055\pi$
$702$	0	0
$703$	20802.5	1.11605
$704$	22666.9	1.21348
$705$	0	0
$706$	−6385.63	−0.340406
$707$	−10936.3	−0.581759
$708$	0	0
$709$	5717.55	0.302859	0.151430	−	0.988468i	$-0.451612\pi$
$709$	5717.55	0.302859	0.151430	+	0.988468i	$0.451612\pi$
$710$	0	0
$711$	0	0
$712$	587.389	0.0309176
$713$	−4621.50	−0.242744
$714$	0	0
$715$	0	0
$716$	5363.19	0.279933
$717$	0	0
$718$	1492.98	0.0776010
$719$	−14799.9	−0.767655	−0.383827	−	0.923405i	$-0.625394\pi$
$719$	−14799.9	−0.767655	−0.383827	+	0.923405i	$0.625394\pi$
$720$	0	0
$721$	−9142.12	−0.472219
$722$	10853.4	0.559451
$723$	0	0
$724$	6885.52	0.353451
$725$	0	0
$726$	0	0
$727$	−6410.40	−0.327027	−0.163513	−	0.986541i	$-0.552283\pi$
$727$	−6410.40	−0.327027	−0.163513	+	0.986541i	$0.552283\pi$
$728$	−474.985	−0.0241815
$729$	0	0
$730$	0	0
$731$	18880.5	0.955292
$732$	0	0
$733$	17869.2	0.900431	0.450215	−	0.892920i	$-0.351347\pi$
$733$	17869.2	0.900431	0.450215	+	0.892920i	$0.351347\pi$
$734$	24047.8	1.20929
$735$	0	0
$736$	−2335.57	−0.116971
$737$	−7036.21	−0.351672
$738$	0	0
$739$	11099.7	0.552516	0.276258	−	0.961084i	$-0.410906\pi$
$739$	11099.7	0.552516	0.276258	+	0.961084i	$0.410906\pi$
$740$	0	0
$741$	0	0
$742$	13789.0	0.682225
$743$	−10857.6	−0.536107	−0.268053	−	0.963404i	$-0.586380\pi$
$743$	−10857.6	−0.536107	−0.268053	+	0.963404i	$0.586380\pi$
$744$	0	0
$745$	0	0
$746$	19276.2	0.946049
$747$	0	0
$748$	−4046.13	−0.197782
$749$	−27948.3	−1.36343
$750$	0	0
$751$	−32233.9	−1.56622	−0.783111	−	0.621882i	$-0.786368\pi$
$751$	−32233.9	−1.56622	−0.783111	+	0.621882i	$0.786368\pi$
$752$	−6336.22	−0.307258
$753$	0	0
$754$	633.880	0.0306161
$755$	0	0
$756$	0	0
$757$	9410.28	0.451813	0.225906	−	0.974149i	$-0.427466\pi$
$757$	9410.28	0.451813	0.225906	+	0.974149i	$0.427466\pi$
$758$	−6729.91	−0.322482
$759$	0	0
$760$	0	0
$761$	34767.1	1.65612	0.828060	−	0.560640i	$-0.189445\pi$
$761$	34767.1	1.65612	0.828060	+	0.560640i	$0.189445\pi$
$762$	0	0
$763$	37766.8	1.79194
$764$	6441.68	0.305042
$765$	0	0
$766$	31044.2	1.46432
$767$	28.9932	0.00136491
$768$	0	0
$769$	−29086.5	−1.36396	−0.681980	−	0.731371i	$-0.738881\pi$
$769$	−29086.5	−1.36396	−0.681980	+	0.731371i	$0.738881\pi$
$770$	0	0
$771$	0	0
$772$	3744.79	0.174583
$773$	2942.77	0.136927	0.0684633	−	0.997654i	$-0.478190\pi$
$773$	2942.77	0.136927	0.0684633	+	0.997654i	$0.478190\pi$
$774$	0	0
$775$	0	0
$776$	33465.7	1.54813
$777$	0	0
$778$	12179.9	0.561275
$779$	−19744.0	−0.908091
$780$	0	0
$781$	−31996.0	−1.46595
$782$	−3552.39	−0.162446
$783$	0	0
$784$	−1218.93	−0.0555270
$785$	0	0
$786$	0	0
$787$	−33829.1	−1.53225	−0.766124	−	0.642693i	$-0.777817\pi$
$787$	−33829.1	−1.53225	−0.766124	+	0.642693i	$0.777817\pi$
$788$	−1998.41	−0.0903434
$789$	0	0
$790$	0	0
$791$	18822.4	0.846076
$792$	0	0
$793$	739.392	0.0331105
$794$	−17412.2	−0.778256
$795$	0	0
$796$	−3020.60	−0.134500
$797$	22904.7	1.01797	0.508987	−	0.860774i	$-0.330020\pi$
$797$	22904.7	1.01797	0.508987	+	0.860774i	$0.330020\pi$
$798$	0	0
$799$	7434.89	0.329196
$800$	0	0
$801$	0	0
$802$	2260.79	0.0995402
$803$	20438.3	0.898199
$804$	0	0
$805$	0	0
$806$	419.324	0.0183251
$807$	0	0
$808$	13906.0	0.605461
$809$	−30665.7	−1.33269	−0.666347	−	0.745642i	$-0.732143\pi$
$809$	−30665.7	−1.33269	−0.666347	+	0.745642i	$0.732143\pi$
$810$	0	0
$811$	−35135.2	−1.52129	−0.760644	−	0.649169i	$-0.775117\pi$
$811$	−35135.2	−1.52129	−0.760644	+	0.649169i	$0.775117\pi$
$812$	−9464.57	−0.409041
$813$	0	0
$814$	41236.4	1.77560
$815$	0	0
$816$	0	0
$817$	−17775.3	−0.761174
$818$	−36306.1	−1.55185
$819$	0	0
$820$	0	0
$821$	34437.0	1.46390	0.731948	−	0.681361i	$-0.238611\pi$
$821$	34437.0	1.46390	0.731948	+	0.681361i	$0.238611\pi$
$822$	0	0
$823$	42801.8	1.81285	0.906426	−	0.422365i	$-0.138800\pi$
$823$	42801.8	1.81285	0.906426	+	0.422365i	$0.138800\pi$
$824$	11624.6	0.491459
$825$	0	0
$826$	1362.03	0.0573742
$827$	−23589.1	−0.991868	−0.495934	−	0.868360i	$-0.665174\pi$
$827$	−23589.1	−0.991868	−0.495934	+	0.868360i	$0.665174\pi$
$828$	0	0
$829$	−26766.2	−1.12139	−0.560693	−	0.828024i	$-0.689465\pi$
$829$	−26766.2	−1.12139	−0.560693	+	0.828024i	$0.689465\pi$
$830$	0	0
$831$	0	0
$832$	573.911	0.0239144
$833$	1430.28	0.0594915
$834$	0	0
$835$	0	0
$836$	3809.29	0.157592
$837$	0	0
$838$	22802.7	0.939985
$839$	−30334.3	−1.24822	−0.624110	−	0.781336i	$-0.714538\pi$
$839$	−30334.3	−1.24822	−0.624110	+	0.781336i	$0.714538\pi$
$840$	0	0
$841$	40612.1	1.66518
$842$	25136.1	1.02880
$843$	0	0
$844$	−6531.69	−0.266386
$845$	0	0
$846$	0	0
$847$	−4952.58	−0.200912
$848$	−13043.5	−0.528201
$849$	0	0
$850$	0	0
$851$	−11507.1	−0.463523
$852$	0	0
$853$	6533.77	0.262265	0.131133	−	0.991365i	$-0.458139\pi$
$853$	6533.77	0.262265	0.131133	+	0.991365i	$0.458139\pi$
$854$	34734.8	1.39180
$855$	0	0
$856$	35537.5	1.41898
$857$	37553.0	1.49683	0.748417	−	0.663228i	$-0.230814\pi$
$857$	37553.0	1.49683	0.748417	+	0.663228i	$0.230814\pi$
$858$	0	0
$859$	36637.9	1.45526	0.727631	−	0.685969i	$-0.240621\pi$
$859$	36637.9	1.45526	0.727631	+	0.685969i	$0.240621\pi$
$860$	0	0
$861$	0	0
$862$	−19851.7	−0.784400
$863$	14614.7	0.576467	0.288233	−	0.957560i	$-0.406932\pi$
$863$	14614.7	0.576467	0.288233	+	0.957560i	$0.406932\pi$
$864$	0	0
$865$	0	0
$866$	10610.0	0.416330
$867$	0	0
$868$	−6260.99	−0.244829
$869$	24405.0	0.952686
$870$	0	0
$871$	−178.152	−0.00693047
$872$	−48022.2	−1.86495
$873$	0	0
$874$	3344.45	0.129437
$875$	0	0
$876$	0	0
$877$	1388.56	0.0534646	0.0267323	−	0.999643i	$-0.491490\pi$
$877$	1388.56	0.0534646	0.0267323	+	0.999643i	$0.491490\pi$
$878$	11633.8	0.447177
$879$	0	0
$880$	0	0
$881$	23664.2	0.904957	0.452478	−	0.891775i	$-0.350540\pi$
$881$	23664.2	0.904957	0.452478	+	0.891775i	$0.350540\pi$
$882$	0	0
$883$	49607.6	1.89063	0.945317	−	0.326152i	$-0.105752\pi$
$883$	49607.6	1.89063	0.945317	+	0.326152i	$0.105752\pi$
$884$	−102.445	−0.00389774
$885$	0	0
$886$	37830.9	1.43448
$887$	11477.0	0.434454	0.217227	−	0.976121i	$-0.430299\pi$
$887$	11477.0	0.434454	0.217227	+	0.976121i	$0.430299\pi$
$888$	0	0
$889$	−16660.8	−0.628556
$890$	0	0
$891$	0	0
$892$	4564.19	0.171323
$893$	−6999.69	−0.262302
$894$	0	0
$895$	0	0
$896$	13841.6	0.516087
$897$	0	0
$898$	−13424.6	−0.498870
$899$	42999.4	1.59523
$900$	0	0
$901$	15305.1	0.565914
$902$	−39138.1	−1.44474
$903$	0	0
$904$	−23933.4	−0.880547
$905$	0	0
$906$	0	0
$907$	−11733.8	−0.429563	−0.214782	−	0.976662i	$-0.568904\pi$
$907$	−11733.8	−0.429563	−0.214782	+	0.976662i	$0.568904\pi$
$908$	−1883.58	−0.0688424
$909$	0	0
$910$	0	0
$911$	−35013.4	−1.27338	−0.636688	−	0.771122i	$-0.719696\pi$
$911$	−35013.4	−1.27338	−0.636688	+	0.771122i	$0.719696\pi$
$912$	0	0
$913$	3220.92	0.116755
$914$	−129.920	−0.00470174
$915$	0	0
$916$	−3078.00	−0.111026
$917$	28262.9	1.01780
$918$	0	0
$919$	16026.6	0.575264	0.287632	−	0.957741i	$-0.407132\pi$
$919$	16026.6	0.575264	0.287632	+	0.957741i	$0.407132\pi$
$920$	0	0
$921$	0	0
$922$	7803.07	0.278721
$923$	−810.116	−0.0288898
$924$	0	0
$925$	0	0
$926$	32257.8	1.14477
$927$	0	0
$928$	21730.7	0.768691
$929$	25886.8	0.914229	0.457114	−	0.889408i	$-0.348883\pi$
$929$	25886.8	0.914229	0.457114	+	0.889408i	$0.348883\pi$
$930$	0	0
$931$	−1346.56	−0.0474026
$932$	−1820.57	−0.0639857
$933$	0	0
$934$	−32585.5	−1.14157
$935$	0	0
$936$	0	0
$937$	6331.15	0.220736	0.110368	−	0.993891i	$-0.464797\pi$
$937$	6331.15	0.220736	0.110368	+	0.993891i	$0.464797\pi$
$938$	−8369.13	−0.291324
$939$	0	0
$940$	0	0
$941$	−28754.7	−0.996149	−0.498075	−	0.867134i	$-0.665959\pi$
$941$	−28754.7	−0.996149	−0.498075	+	0.867134i	$0.665959\pi$
$942$	0	0
$943$	10921.6	0.377153
$944$	−1288.39	−0.0444210
$945$	0	0
$946$	−35235.6	−1.21100
$947$	−29240.7	−1.00337	−0.501687	−	0.865049i	$-0.667287\pi$
$947$	−29240.7	−1.00337	−0.501687	+	0.865049i	$0.667287\pi$
$948$	0	0
$949$	517.484	0.0177010
$950$	0	0
$951$	0	0
$952$	−24767.0	−0.843176
$953$	−28758.3	−0.977517	−0.488758	−	0.872419i	$-0.662550\pi$
$953$	−28758.3	−0.977517	−0.488758	+	0.872419i	$0.662550\pi$
$954$	0	0
$955$	0	0
$956$	−5270.27	−0.178298
$957$	0	0
$958$	21087.1	0.711161
$959$	1376.35	0.0463448
$960$	0	0
$961$	−1346.05	−0.0451831
$962$	1044.08	0.0349920
$963$	0	0
$964$	−6593.36	−0.220288
$965$	0	0
$966$	0	0
$967$	−44695.1	−1.48635	−0.743174	−	0.669099i	$-0.766680\pi$
$967$	−44695.1	−1.48635	−0.743174	+	0.669099i	$0.766680\pi$
$968$	6297.42	0.209098
$969$	0	0
$970$	0	0
$971$	−30462.4	−1.00678	−0.503391	−	0.864059i	$-0.667914\pi$
$971$	−30462.4	−1.00678	−0.503391	+	0.864059i	$0.667914\pi$
$972$	0	0
$973$	−4407.24	−0.145210
$974$	48122.2	1.58310
$975$	0	0
$976$	−32856.7	−1.07758
$977$	−11808.0	−0.386665	−0.193332	−	0.981133i	$-0.561930\pi$
$977$	−11808.0	−0.386665	−0.193332	+	0.981133i	$0.561930\pi$
$978$	0	0
$979$	956.902	0.0312387
$980$	0	0
$981$	0	0
$982$	48016.1	1.56034
$983$	−24702.9	−0.801526	−0.400763	−	0.916182i	$-0.631255\pi$
$983$	−24702.9	−0.801526	−0.400763	+	0.916182i	$0.631255\pi$
$984$	0	0
$985$	0	0
$986$	33052.2	1.06754
$987$	0	0
$988$	96.4485	0.00310570
$989$	9832.55	0.316134
$990$	0	0
$991$	14443.0	0.462965	0.231482	−	0.972839i	$-0.425642\pi$
$991$	14443.0	0.462965	0.231482	+	0.972839i	$0.425642\pi$
$992$	14375.3	0.460096
$993$	0	0
$994$	−38057.2	−1.21439
$995$	0	0
$996$	0	0
$997$	−47669.3	−1.51425	−0.757123	−	0.653273i	$-0.773396\pi$
$997$	−47669.3	−1.51425	−0.757123	+	0.653273i	$0.773396\pi$
$998$	836.551	0.0265336
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bk.1.6		16
3.2	odd	2		2025.4.a.bl.1.11		16
5.2	odd	4		405.4.b.e.244.6		16
5.3	odd	4		405.4.b.e.244.11		16
5.4	even	2	inner	2025.4.a.bk.1.11		16
9.4	even	3		225.4.e.g.151.11		32
9.7	even	3		225.4.e.g.76.11		32
15.2	even	4		405.4.b.f.244.11		16
15.8	even	4		405.4.b.f.244.6		16
15.14	odd	2		2025.4.a.bl.1.6		16
45.2	even	12		135.4.j.a.64.11		32
45.4	even	6		225.4.e.g.151.6		32
45.7	odd	12		45.4.j.a.4.6	✓	32
45.13	odd	12		45.4.j.a.34.6	yes	32
45.22	odd	12		45.4.j.a.34.11	yes	32
45.23	even	12		135.4.j.a.19.11		32
45.32	even	12		135.4.j.a.19.6		32
45.34	even	6		225.4.e.g.76.6		32
45.38	even	12		135.4.j.a.64.6		32
45.43	odd	12		45.4.j.a.4.11	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.6	✓	32	45.7	odd	12
45.4.j.a.4.11	yes	32	45.43	odd	12
45.4.j.a.34.6	yes	32	45.13	odd	12
45.4.j.a.34.11	yes	32	45.22	odd	12
135.4.j.a.19.6		32	45.32	even	12
135.4.j.a.19.11		32	45.23	even	12
135.4.j.a.64.6		32	45.38	even	12
135.4.j.a.64.11		32	45.2	even	12
225.4.e.g.76.6		32	45.34	even	6
225.4.e.g.76.11		32	9.7	even	3
225.4.e.g.151.6		32	45.4	even	6
225.4.e.g.151.11		32	9.4	even	3
405.4.b.e.244.6		16	5.2	odd	4
405.4.b.e.244.11		16	5.3	odd	4
405.4.b.f.244.6		16	15.8	even	4
405.4.b.f.244.11		16	15.2	even	4
2025.4.a.bk.1.6		16	1.1	even	1	trivial
2025.4.a.bk.1.11		16	5.4	even	2	inner
2025.4.a.bl.1.6		16	15.14	odd	2
2025.4.a.bl.1.11		16	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46385 q^{2} -1.92944 q^{4} -19.2401 q^{7} +24.4647 q^{8} +39.8548 q^{11} +1.00910 q^{13} +47.4048 q^{14} -44.8417 q^{16} +52.6170 q^{17} -49.5371 q^{19} -98.1964 q^{22} +27.4019 q^{23} -2.48626 q^{26} +37.1228 q^{28} -254.953 q^{29} -168.656 q^{31} -85.2341 q^{32} -129.640 q^{34} -419.938 q^{37} +122.052 q^{38} +398.570 q^{41} +358.828 q^{43} -76.8977 q^{44} -67.5141 q^{46} +141.302 q^{47} +27.1829 q^{49} -1.94699 q^{52} +290.878 q^{53} -470.703 q^{56} +628.166 q^{58} +28.7319 q^{59} +732.727 q^{61} +415.544 q^{62} +568.737 q^{64} -176.546 q^{67} -101.522 q^{68} -802.814 q^{71} +512.820 q^{73} +1034.66 q^{74} +95.5791 q^{76} -766.813 q^{77} +612.348 q^{79} -982.017 q^{82} +80.8162 q^{83} -884.098 q^{86} +975.035 q^{88} +24.0097 q^{89} -19.4151 q^{91} -52.8704 q^{92} -348.147 q^{94} +1367.92 q^{97} -66.9747 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 54 q^{4} - 90 q^{11} - 102 q^{14} + 146 q^{16} + 4 q^{19} - 468 q^{26} - 516 q^{29} + 38 q^{31} + 212 q^{34} - 576 q^{41} - 1644 q^{44} - 290 q^{46} - 4 q^{49} - 2430 q^{56} - 2202 q^{59} + 20 q^{61}+ \cdots - 1078 q^{94}+O(q^{100}) \) 16 * q + 54 * q^4 - 90 * q^11 - 102 * q^14 + 146 * q^16 + 4 * q^19 - 468 * q^26 - 516 * q^29 + 38 * q^31 + 212 * q^34 - 576 * q^41 - 1644 * q^44 - 290 * q^46 - 4 * q^49 - 2430 * q^56 - 2202 * q^59 + 20 * q^61 - 322 * q^64 - 2952 * q^71 - 4080 * q^74 - 396 * q^76 - 218 * q^79 - 6108 * q^86 - 4074 * q^89 - 942 * q^91 - 1078 * q^94

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