LMFDB - Embedded newform 225.4.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,4,Mod(1,225)] 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("225.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-3,0,9,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.2754297513$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70156 q^{2} -5.10469 q^{4} +22.2094 q^{7} -22.2984 q^{8} +1.79063 q^{11} +58.2094 q^{13} +37.7906 q^{14} +2.89531 q^{16} -18.9844 q^{17} +104.837 q^{19} +3.04686 q^{22} +49.6125 q^{23} +99.0469 q^{26} -113.372 q^{28} +293.466 q^{29} +64.4187 q^{31} +183.314 q^{32} -32.3031 q^{34} -19.8844 q^{37} +178.388 q^{38} +165.581 q^{41} -247.350 q^{43} -9.14059 q^{44} +84.4187 q^{46} -384.544 q^{47} +150.256 q^{49} -297.141 q^{52} -463.528 q^{53} -495.234 q^{56} +499.350 q^{58} +73.7906 q^{59} -137.350 q^{61} +109.612 q^{62} +288.758 q^{64} +173.906 q^{67} +96.9093 q^{68} +594.281 q^{71} +320.231 q^{73} -33.8345 q^{74} -535.163 q^{76} +39.7687 q^{77} -770.469 q^{79} +281.747 q^{82} -173.925 q^{83} -420.881 q^{86} -39.9282 q^{88} -1019.02 q^{89} +1292.79 q^{91} -253.256 q^{92} -654.325 q^{94} +384.375 q^{97} +255.670 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 9 q^{4} + 6 q^{7} - 51 q^{8} + 42 q^{11} + 78 q^{13} + 114 q^{14} + 25 q^{16} - 102 q^{17} + 56 q^{19} - 186 q^{22} + 48 q^{23} + 6 q^{26} - 342 q^{28} + 318 q^{29} + 52 q^{31} + 309 q^{32}+ \cdots + 633 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 9 * q^4 + 6 * q^7 - 51 * q^8 + 42 * q^11 + 78 * q^13 + 114 * q^14 + 25 * q^16 - 102 * q^17 + 56 * q^19 - 186 * q^22 + 48 * q^23 + 6 * q^26 - 342 * q^28 + 318 * q^29 + 52 * q^31 + 309 * q^32 + 358 * q^34 + 306 * q^37 + 408 * q^38 + 408 * q^41 + 120 * q^43 + 558 * q^44 + 92 * q^46 - 180 * q^47 + 70 * q^49 - 18 * q^52 - 402 * q^53 - 30 * q^56 + 384 * q^58 + 186 * q^59 + 340 * q^61 + 168 * q^62 - 479 * q^64 + 732 * q^67 - 1074 * q^68 + 36 * q^71 + 1332 * q^73 - 1566 * q^74 - 1224 * q^76 - 612 * q^77 + 380 * q^79 - 858 * q^82 + 984 * q^83 - 2148 * q^86 - 1194 * q^88 - 1116 * q^89 + 972 * q^91 - 276 * q^92 - 1616 * q^94 - 768 * q^97 + 633 * q^98

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 $

$2$	1.70156	0.601593	0.300797	−	0.953688i	$-0.402747\pi$
$2$	1.70156	0.601593	0.300797	+	0.953688i	$0.402747\pi$
$3$	0	0
$3$	0	0
$4$	−5.10469	−0.638086
$5$	0	0
$5$	0	0
$6$	0	0
$7$	22.2094	1.19919	0.599597	−	0.800302i	$-0.295328\pi$
$7$	22.2094	1.19919	0.599597	+	0.800302i	$0.295328\pi$
$8$	−22.2984	−0.985461
$9$	0	0
$10$	0	0
$11$	1.79063	0.0490813	0.0245407	−	0.999699i	$-0.492188\pi$
$11$	1.79063	0.0490813	0.0245407	+	0.999699i	$0.492188\pi$
$12$	0	0
$13$	58.2094	1.24188	0.620938	−	0.783860i	$-0.286752\pi$
$13$	58.2094	1.24188	0.620938	+	0.783860i	$0.286752\pi$
$14$	37.7906	0.721426
$15$	0	0
$16$	2.89531	0.0452393
$17$	−18.9844	−0.270846	−0.135423	−	0.990788i	$-0.543239\pi$
$17$	−18.9844	−0.270846	−0.135423	+	0.990788i	$0.543239\pi$
$18$	0	0
$19$	104.837	1.26586	0.632931	−	0.774208i	$-0.281852\pi$
$19$	104.837	1.26586	0.632931	+	0.774208i	$0.281852\pi$
$20$	0	0
$21$	0	0
$22$	3.04686	0.0295270
$23$	49.6125	0.449779	0.224890	−	0.974384i	$-0.427798\pi$
$23$	49.6125	0.449779	0.224890	+	0.974384i	$0.427798\pi$
$24$	0	0
$25$	0	0
$26$	99.0469	0.747103
$27$	0	0
$28$	−113.372	−0.765188
$29$	293.466	1.87914	0.939572	−	0.342350i	$-0.111223\pi$
$29$	293.466	1.87914	0.939572	+	0.342350i	$0.111223\pi$
$30$	0	0
$31$	64.4187	0.373224	0.186612	−	0.982434i	$-0.440249\pi$
$31$	64.4187	0.373224	0.186612	+	0.982434i	$0.440249\pi$
$32$	183.314	1.01268
$33$	0	0
$34$	−32.3031	−0.162939
$35$	0	0
$36$	0	0
$37$	−19.8844	−0.0883505	−0.0441752	−	0.999024i	$-0.514066\pi$
$37$	−19.8844	−0.0883505	−0.0441752	+	0.999024i	$0.514066\pi$
$38$	178.388	0.761534
$39$	0	0
$40$	0	0
$41$	165.581	0.630718	0.315359	−	0.948972i	$-0.397875\pi$
$41$	165.581	0.630718	0.315359	+	0.948972i	$0.397875\pi$
$42$	0	0
$43$	−247.350	−0.877221	−0.438611	−	0.898677i	$-0.644529\pi$
$43$	−247.350	−0.877221	−0.438611	+	0.898677i	$0.644529\pi$
$44$	−9.14059	−0.0313181
$45$	0	0
$46$	84.4187	0.270584
$47$	−384.544	−1.19344	−0.596718	−	0.802451i	$-0.703529\pi$
$47$	−384.544	−1.19344	−0.596718	+	0.802451i	$0.703529\pi$
$48$	0	0
$49$	150.256	0.438065
$50$	0	0
$51$	0	0
$52$	−297.141	−0.792423
$53$	−463.528	−1.20133	−0.600665	−	0.799501i	$-0.705097\pi$
$53$	−463.528	−1.20133	−0.600665	+	0.799501i	$0.705097\pi$
$54$	0	0
$55$	0	0
$56$	−495.234	−1.18176
$57$	0	0
$58$	499.350	1.13048
$59$	73.7906	0.162826	0.0814129	−	0.996680i	$-0.474057\pi$
$59$	73.7906	0.162826	0.0814129	+	0.996680i	$0.474057\pi$
$60$	0	0
$61$	−137.350	−0.288293	−0.144146	−	0.989556i	$-0.546044\pi$
$61$	−137.350	−0.288293	−0.144146	+	0.989556i	$0.546044\pi$
$62$	109.612	0.224529
$63$	0	0
$64$	288.758	0.563980
$65$	0	0
$66$	0	0
$67$	173.906	0.317105	0.158552	−	0.987351i	$-0.449317\pi$
$67$	173.906	0.317105	0.158552	+	0.987351i	$0.449317\pi$
$68$	96.9093	0.172823
$69$	0	0
$70$	0	0
$71$	594.281	0.993355	0.496677	−	0.867935i	$-0.334553\pi$
$71$	594.281	0.993355	0.496677	+	0.867935i	$0.334553\pi$
$72$	0	0
$73$	320.231	0.513428	0.256714	−	0.966487i	$-0.417360\pi$
$73$	320.231	0.513428	0.256714	+	0.966487i	$0.417360\pi$
$74$	−33.8345	−0.0531510
$75$	0	0
$76$	−535.163	−0.807728
$77$	39.7687	0.0588580
$78$	0	0
$79$	−770.469	−1.09727	−0.548636	−	0.836061i	$-0.684853\pi$
$79$	−770.469	−1.09727	−0.548636	+	0.836061i	$0.684853\pi$
$80$	0	0
$81$	0	0
$82$	281.747	0.379436
$83$	−173.925	−0.230009	−0.115004	−	0.993365i	$-0.536688\pi$
$83$	−173.925	−0.230009	−0.115004	+	0.993365i	$0.536688\pi$
$84$	0	0
$85$	0	0
$86$	−420.881	−0.527730
$87$	0	0
$88$	−39.9282	−0.0483677
$89$	−1019.02	−1.21367	−0.606834	−	0.794829i	$-0.707561\pi$
$89$	−1019.02	−1.21367	−0.606834	+	0.794829i	$0.707561\pi$
$90$	0	0
$91$	1292.79	1.48925
$92$	−253.256	−0.286998
$93$	0	0
$94$	−654.325	−0.717962
$95$	0	0
$96$	0	0
$97$	384.375	0.402344	0.201172	−	0.979556i	$-0.435525\pi$
$97$	384.375	0.402344	0.201172	+	0.979556i	$0.435525\pi$
$98$	255.670	0.263537
$99$	0	0
$100$	0	0
$101$	−34.4906	−0.0339796	−0.0169898	−	0.999856i	$-0.505408\pi$
$101$	−34.4906	−0.0339796	−0.0169898	+	0.999856i	$0.505408\pi$
$102$	0	0
$103$	−1756.30	−1.68013	−0.840066	−	0.542484i	$-0.817484\pi$
$103$	−1756.30	−1.68013	−0.840066	+	0.542484i	$0.817484\pi$
$104$	−1297.98	−1.22382
$105$	0	0
$106$	−788.722	−0.722712
$107$	1361.74	1.23032	0.615159	−	0.788403i	$-0.289092\pi$
$107$	1361.74	1.23032	0.615159	+	0.788403i	$0.289092\pi$
$108$	0	0
$109$	321.119	0.282180	0.141090	−	0.989997i	$-0.454939\pi$
$109$	321.119	0.282180	0.141090	+	0.989997i	$0.454939\pi$
$110$	0	0
$111$	0	0
$112$	64.3031	0.0542506
$113$	1582.25	1.31721	0.658607	−	0.752487i	$-0.271146\pi$
$113$	1582.25	1.31721	0.658607	+	0.752487i	$0.271146\pi$
$114$	0	0
$115$	0	0
$116$	−1498.05	−1.19906
$117$	0	0
$118$	125.559	0.0979549
$119$	−421.631	−0.324797
$120$	0	0
$121$	−1327.79	−0.997591
$122$	−233.709	−0.173435
$123$	0	0
$124$	−328.837	−0.238149
$125$	0	0
$126$	0	0
$127$	−1197.14	−0.836449	−0.418225	−	0.908344i	$-0.637348\pi$
$127$	−1197.14	−0.836449	−0.418225	+	0.908344i	$0.637348\pi$
$128$	−975.173	−0.673390
$129$	0	0
$130$	0	0
$131$	321.647	0.214522	0.107261	−	0.994231i	$-0.465792\pi$
$131$	321.647	0.214522	0.107261	+	0.994231i	$0.465792\pi$
$132$	0	0
$133$	2328.37	1.51801
$134$	295.912	0.190768
$135$	0	0
$136$	423.322	0.266909
$137$	−354.291	−0.220942	−0.110471	−	0.993879i	$-0.535236\pi$
$137$	−354.291	−0.220942	−0.110471	+	0.993879i	$0.535236\pi$
$138$	0	0
$139$	77.2562	0.0471424	0.0235712	−	0.999722i	$-0.492496\pi$
$139$	77.2562	0.0471424	0.0235712	+	0.999722i	$0.492496\pi$
$140$	0	0
$141$	0	0
$142$	1011.21	0.597595
$143$	104.231	0.0609529
$144$	0	0
$145$	0	0
$146$	544.893	0.308875
$147$	0	0
$148$	101.503	0.0563752
$149$	1705.38	0.937651	0.468826	−	0.883291i	$-0.344677\pi$
$149$	1705.38	0.937651	0.468826	+	0.883291i	$0.344677\pi$
$150$	0	0
$151$	758.281	0.408663	0.204331	−	0.978902i	$-0.434498\pi$
$151$	758.281	0.408663	0.204331	+	0.978902i	$0.434498\pi$
$152$	−2337.71	−1.24746
$153$	0	0
$154$	67.6689	0.0354086
$155$	0	0
$156$	0	0
$157$	−1769.05	−0.899273	−0.449636	−	0.893212i	$-0.648446\pi$
$157$	−1769.05	−0.899273	−0.449636	+	0.893212i	$0.648446\pi$
$158$	−1311.00	−0.660111
$159$	0	0
$160$	0	0
$161$	1101.86	0.539372
$162$	0	0
$163$	−881.719	−0.423690	−0.211845	−	0.977303i	$-0.567947\pi$
$163$	−881.719	−0.423690	−0.211845	+	0.977303i	$0.567947\pi$
$164$	−845.240	−0.402452
$165$	0	0
$166$	−295.944	−0.138372
$167$	216.900	0.100504	0.0502522	−	0.998737i	$-0.483997\pi$
$167$	216.900	0.100504	0.0502522	+	0.998737i	$0.483997\pi$
$168$	0	0
$169$	1191.33	0.542254
$170$	0	0
$171$	0	0
$172$	1262.64	0.559742
$173$	−4125.91	−1.81322	−0.906610	−	0.421970i	$-0.861339\pi$
$173$	−4125.91	−1.81322	−0.906610	+	0.421970i	$0.861339\pi$
$174$	0	0
$175$	0	0
$176$	5.18443	0.00222040
$177$	0	0
$178$	−1733.93	−0.730134
$179$	3213.14	1.34168	0.670842	−	0.741600i	$-0.265933\pi$
$179$	3213.14	1.34168	0.670842	+	0.741600i	$0.265933\pi$
$180$	0	0
$181$	3394.42	1.39395	0.696976	−	0.717095i	$-0.254529\pi$
$181$	3394.42	1.39395	0.696976	+	0.717095i	$0.254529\pi$
$182$	2199.77	0.895921
$183$	0	0
$184$	−1106.28	−0.443240
$185$	0	0
$186$	0	0
$187$	−33.9939	−0.0132935
$188$	1962.98	0.761514
$189$	0	0
$190$	0	0
$191$	3467.49	1.31361	0.656804	−	0.754062i	$-0.271908\pi$
$191$	3467.49	1.31361	0.656804	+	0.754062i	$0.271908\pi$
$192$	0	0
$193$	1792.14	0.668401	0.334200	−	0.942502i	$-0.391534\pi$
$193$	1792.14	0.668401	0.334200	+	0.942502i	$0.391534\pi$
$194$	654.038	0.242047
$195$	0	0
$196$	−767.011	−0.279523
$197$	−1678.19	−0.606935	−0.303467	−	0.952842i	$-0.598144\pi$
$197$	−1678.19	−0.606935	−0.303467	+	0.952842i	$0.598144\pi$
$198$	0	0
$199$	3108.23	1.10722	0.553610	−	0.832776i	$-0.313250\pi$
$199$	3108.23	1.10722	0.553610	+	0.832776i	$0.313250\pi$
$200$	0	0
$201$	0	0
$202$	−58.6878	−0.0204419
$203$	6517.69	2.25346
$204$	0	0
$205$	0	0
$206$	−2988.46	−1.01076
$207$	0	0
$208$	168.534	0.0561815
$209$	187.725	0.0621301
$210$	0	0
$211$	−4473.27	−1.45949	−0.729745	−	0.683719i	$-0.760361\pi$
$211$	−4473.27	−1.45949	−0.729745	+	0.683719i	$0.760361\pi$
$212$	2366.17	0.766551
$213$	0	0
$214$	2317.08	0.740151
$215$	0	0
$216$	0	0
$217$	1430.70	0.447568
$218$	546.403	0.169757
$219$	0	0
$220$	0	0
$221$	−1105.07	−0.336357
$222$	0	0
$223$	−1753.42	−0.526535	−0.263268	−	0.964723i	$-0.584800\pi$
$223$	−1753.42	−0.526535	−0.263268	+	0.964723i	$0.584800\pi$
$224$	4071.29	1.21440
$225$	0	0
$226$	2692.29	0.792427
$227$	936.900	0.273939	0.136970	−	0.990575i	$-0.456264\pi$
$227$	936.900	0.273939	0.136970	+	0.990575i	$0.456264\pi$
$228$	0	0
$229$	−2582.06	−0.745096	−0.372548	−	0.928013i	$-0.621516\pi$
$229$	−2582.06	−0.745096	−0.372548	+	0.928013i	$0.621516\pi$
$230$	0	0
$231$	0	0
$232$	−6543.82	−1.85182
$233$	−2295.01	−0.645284	−0.322642	−	0.946521i	$-0.604571\pi$
$233$	−2295.01	−0.645284	−0.322642	+	0.946521i	$0.604571\pi$
$234$	0	0
$235$	0	0
$236$	−376.678	−0.103897
$237$	0	0
$238$	−717.432	−0.195396
$239$	2294.01	0.620866	0.310433	−	0.950595i	$-0.399526\pi$
$239$	2294.01	0.620866	0.310433	+	0.950595i	$0.399526\pi$
$240$	0	0
$241$	382.287	0.102180	0.0510898	−	0.998694i	$-0.483731\pi$
$241$	382.287	0.102180	0.0510898	+	0.998694i	$0.483731\pi$
$242$	−2259.32	−0.600144
$243$	0	0
$244$	701.128	0.183956
$245$	0	0
$246$	0	0
$247$	6102.52	1.57204
$248$	−1436.44	−0.367798
$249$	0	0
$250$	0	0
$251$	2259.98	0.568322	0.284161	−	0.958777i	$-0.408285\pi$
$251$	2259.98	0.568322	0.284161	+	0.958777i	$0.408285\pi$
$252$	0	0
$253$	88.8375	0.0220758
$254$	−2037.01	−0.503202
$255$	0	0
$256$	−3969.38	−0.969087
$257$	92.7843	0.0225203	0.0112602	−	0.999937i	$-0.496416\pi$
$257$	92.7843	0.0225203	0.0112602	+	0.999937i	$0.496416\pi$
$258$	0	0
$259$	−441.619	−0.105949
$260$	0	0
$261$	0	0
$262$	547.302	0.129055
$263$	568.312	0.133246	0.0666229	−	0.997778i	$-0.478778\pi$
$263$	568.312	0.133246	0.0666229	+	0.997778i	$0.478778\pi$
$264$	0	0
$265$	0	0
$266$	3961.87	0.913226
$267$	0	0
$268$	−887.737	−0.202340
$269$	−7582.41	−1.71862	−0.859309	−	0.511458i	$-0.829106\pi$
$269$	−7582.41	−1.71862	−0.859309	+	0.511458i	$0.829106\pi$
$270$	0	0
$271$	7943.69	1.78061	0.890304	−	0.455366i	$-0.150492\pi$
$271$	7943.69	1.78061	0.890304	+	0.455366i	$0.150492\pi$
$272$	−54.9657	−0.0122529
$273$	0	0
$274$	−602.847	−0.132917
$275$	0	0
$276$	0	0
$277$	6823.00	1.47998	0.739990	−	0.672618i	$-0.234830\pi$
$277$	6823.00	1.47998	0.739990	+	0.672618i	$0.234830\pi$
$278$	131.456	0.0283605
$279$	0	0
$280$	0	0
$281$	−3315.86	−0.703942	−0.351971	−	0.936011i	$-0.614488\pi$
$281$	−3315.86	−0.703942	−0.351971	+	0.936011i	$0.614488\pi$
$282$	0	0
$283$	6602.76	1.38690	0.693451	−	0.720504i	$-0.256090\pi$
$283$	6602.76	1.38690	0.693451	+	0.720504i	$0.256090\pi$
$284$	−3033.62	−0.633846
$285$	0	0
$286$	177.356	0.0366688
$287$	3677.46	0.756353
$288$	0	0
$289$	−4552.59	−0.926642
$290$	0	0
$291$	0	0
$292$	−1634.68	−0.327611
$293$	−5814.14	−1.15927	−0.579634	−	0.814877i	$-0.696805\pi$
$293$	−5814.14	−1.15927	−0.579634	+	0.814877i	$0.696805\pi$
$294$	0	0
$295$	0	0
$296$	443.390	0.0870660
$297$	0	0
$298$	2901.81	0.564084
$299$	2887.91	0.558570
$300$	0	0
$301$	−5493.49	−1.05196
$302$	1290.26	0.245849
$303$	0	0
$304$	303.537	0.0572667
$305$	0	0
$306$	0	0
$307$	−8124.86	−1.51046	−0.755229	−	0.655462i	$-0.772474\pi$
$307$	−8124.86	−1.51046	−0.755229	+	0.655462i	$0.772474\pi$
$308$	−203.007	−0.0375564
$309$	0	0
$310$	0	0
$311$	−7336.26	−1.33762	−0.668812	−	0.743432i	$-0.733197\pi$
$311$	−7336.26	−1.33762	−0.668812	+	0.743432i	$0.733197\pi$
$312$	0	0
$313$	−2202.66	−0.397768	−0.198884	−	0.980023i	$-0.563732\pi$
$313$	−2202.66	−0.397768	−0.198884	+	0.980023i	$0.563732\pi$
$314$	−3010.15	−0.540996
$315$	0	0
$316$	3933.00	0.700154
$317$	−10008.9	−1.77336	−0.886679	−	0.462386i	$-0.846993\pi$
$317$	−10008.9	−1.77336	−0.886679	+	0.462386i	$0.846993\pi$
$318$	0	0
$319$	525.488	0.0922309
$320$	0	0
$321$	0	0
$322$	1874.89	0.324483
$323$	−1990.27	−0.342854
$324$	0	0
$325$	0	0
$326$	−1500.30	−0.254889
$327$	0	0
$328$	−3692.20	−0.621548
$329$	−8540.47	−1.43116
$330$	0	0
$331$	−8695.94	−1.44402	−0.722012	−	0.691881i	$-0.756782\pi$
$331$	−8695.94	−1.44402	−0.722012	+	0.691881i	$0.756782\pi$
$332$	887.832	0.146765
$333$	0	0
$334$	369.069	0.0604627
$335$	0	0
$336$	0	0
$337$	−7400.61	−1.19625	−0.598126	−	0.801402i	$-0.704088\pi$
$337$	−7400.61	−1.19625	−0.598126	+	0.801402i	$0.704088\pi$
$338$	2027.12	0.326216
$339$	0	0
$340$	0	0
$341$	115.350	0.0183183
$342$	0	0
$343$	−4280.72	−0.673869
$344$	5515.52	0.864467
$345$	0	0
$346$	−7020.49	−1.09082
$347$	−7841.44	−1.21311	−0.606557	−	0.795040i	$-0.707450\pi$
$347$	−7841.44	−1.21311	−0.606557	+	0.795040i	$0.707450\pi$
$348$	0	0
$349$	−4961.26	−0.760946	−0.380473	−	0.924792i	$-0.624239\pi$
$349$	−4961.26	−0.760946	−0.380473	+	0.924792i	$0.624239\pi$
$350$	0	0
$351$	0	0
$352$	328.247	0.0497035
$353$	12163.0	1.83392	0.916959	−	0.398981i	$-0.130636\pi$
$353$	12163.0	1.83392	0.916959	+	0.398981i	$0.130636\pi$
$354$	0	0
$355$	0	0
$356$	5201.80	0.774424
$357$	0	0
$358$	5467.36	0.807148
$359$	−5193.79	−0.763559	−0.381779	−	0.924253i	$-0.624689\pi$
$359$	−5193.79	−0.763559	−0.381779	+	0.924253i	$0.624689\pi$
$360$	0	0
$361$	4131.90	0.602406
$362$	5775.81	0.838591
$363$	0	0
$364$	−6599.31	−0.950268
$365$	0	0
$366$	0	0
$367$	−6086.09	−0.865644	−0.432822	−	0.901479i	$-0.642482\pi$
$367$	−6086.09	−0.865644	−0.432822	+	0.901479i	$0.642482\pi$
$368$	143.644	0.0203477
$369$	0	0
$370$	0	0
$371$	−10294.7	−1.44063
$372$	0	0
$373$	−10581.9	−1.46893	−0.734466	−	0.678646i	$-0.762567\pi$
$373$	−10581.9	−1.46893	−0.734466	+	0.678646i	$0.762567\pi$
$374$	−57.8428	−0.00799727
$375$	0	0
$376$	8574.72	1.17608
$377$	17082.4	2.33366
$378$	0	0
$379$	11655.2	1.57964	0.789822	−	0.613336i	$-0.210173\pi$
$379$	11655.2	1.57964	0.789822	+	0.613336i	$0.210173\pi$
$380$	0	0
$381$	0	0
$382$	5900.16	0.790257
$383$	−6364.97	−0.849177	−0.424588	−	0.905387i	$-0.639581\pi$
$383$	−6364.97	−0.849177	−0.424588	+	0.905387i	$0.639581\pi$
$384$	0	0
$385$	0	0
$386$	3049.44	0.402105
$387$	0	0
$388$	−1962.11	−0.256730
$389$	−6134.33	−0.799545	−0.399773	−	0.916614i	$-0.630911\pi$
$389$	−6134.33	−0.799545	−0.399773	+	0.916614i	$0.630911\pi$
$390$	0	0
$391$	−941.862	−0.121821
$392$	−3350.48	−0.431696
$393$	0	0
$394$	−2855.55	−0.365128
$395$	0	0
$396$	0	0
$397$	9746.46	1.23214	0.616072	−	0.787690i	$-0.288723\pi$
$397$	9746.46	1.23214	0.616072	+	0.787690i	$0.288723\pi$
$398$	5288.85	0.666096
$399$	0	0
$400$	0	0
$401$	1306.44	0.162695	0.0813474	−	0.996686i	$-0.474078\pi$
$401$	1306.44	0.162695	0.0813474	+	0.996686i	$0.474078\pi$
$402$	0	0
$403$	3749.77	0.463498
$404$	176.063	0.0216819
$405$	0	0
$406$	11090.2	1.35566
$407$	−35.6055	−0.00433636
$408$	0	0
$409$	−3876.93	−0.468709	−0.234354	−	0.972151i	$-0.575298\pi$
$409$	−3876.93	−0.468709	−0.234354	+	0.972151i	$0.575298\pi$
$410$	0	0
$411$	0	0
$412$	8965.38	1.07207
$413$	1638.84	0.195260
$414$	0	0
$415$	0	0
$416$	10670.6	1.25762
$417$	0	0
$418$	319.426	0.0373771
$419$	16022.5	1.86814	0.934071	−	0.357088i	$-0.116230\pi$
$419$	16022.5	1.86814	0.934071	+	0.357088i	$0.116230\pi$
$420$	0	0
$421$	−8119.73	−0.939980	−0.469990	−	0.882672i	$-0.655742\pi$
$421$	−8119.73	−0.939980	−0.469990	+	0.882672i	$0.655742\pi$
$422$	−7611.54	−0.878019
$423$	0	0
$424$	10336.0	1.18386
$425$	0	0
$426$	0	0
$427$	−3050.46	−0.345719
$428$	−6951.24	−0.785049
$429$	0	0
$430$	0	0
$431$	5713.99	0.638592	0.319296	−	0.947655i	$-0.396554\pi$
$431$	5713.99	0.638592	0.319296	+	0.947655i	$0.396554\pi$
$432$	0	0
$433$	6251.34	0.693811	0.346906	−	0.937900i	$-0.387232\pi$
$433$	6251.34	0.693811	0.346906	+	0.937900i	$0.387232\pi$
$434$	2434.42	0.269254
$435$	0	0
$436$	−1639.21	−0.180055
$437$	5201.25	0.569358
$438$	0	0
$439$	−4230.97	−0.459984	−0.229992	−	0.973192i	$-0.573870\pi$
$439$	−4230.97	−0.459984	−0.229992	+	0.973192i	$0.573870\pi$
$440$	0	0
$441$	0	0
$442$	−1880.34	−0.202350
$443$	−6314.29	−0.677203	−0.338601	−	0.940930i	$-0.609954\pi$
$443$	−6314.29	−0.677203	−0.338601	+	0.940930i	$0.609954\pi$
$444$	0	0
$445$	0	0
$446$	−2983.55	−0.316760
$447$	0	0
$448$	6413.13	0.676321
$449$	9349.71	0.982717	0.491358	−	0.870957i	$-0.336501\pi$
$449$	9349.71	0.982717	0.491358	+	0.870957i	$0.336501\pi$
$450$	0	0
$451$	296.494	0.0309565
$452$	−8076.87	−0.840496
$453$	0	0
$454$	1594.19	0.164800
$455$	0	0
$456$	0	0
$457$	−9547.46	−0.977268	−0.488634	−	0.872489i	$-0.662505\pi$
$457$	−9547.46	−0.977268	−0.488634	+	0.872489i	$0.662505\pi$
$458$	−4393.53	−0.448245
$459$	0	0
$460$	0	0
$461$	−6237.23	−0.630145	−0.315073	−	0.949068i	$-0.602029\pi$
$461$	−6237.23	−0.630145	−0.315073	+	0.949068i	$0.602029\pi$
$462$	0	0
$463$	6469.98	0.649428	0.324714	−	0.945812i	$-0.394732\pi$
$463$	6469.98	0.649428	0.324714	+	0.945812i	$0.394732\pi$
$464$	849.675	0.0850111
$465$	0	0
$466$	−3905.10	−0.388198
$467$	−7206.64	−0.714097	−0.357049	−	0.934086i	$-0.616217\pi$
$467$	−7206.64	−0.714097	−0.357049	+	0.934086i	$0.616217\pi$
$468$	0	0
$469$	3862.35	0.380270
$470$	0	0
$471$	0	0
$472$	−1645.42	−0.160458
$473$	−442.912	−0.0430552
$474$	0	0
$475$	0	0
$476$	2152.29	0.207248
$477$	0	0
$478$	3903.39	0.373509
$479$	10851.8	1.03514	0.517571	−	0.855640i	$-0.326836\pi$
$479$	10851.8	1.03514	0.517571	+	0.855640i	$0.326836\pi$
$480$	0	0
$481$	−1157.46	−0.109720
$482$	650.485	0.0614705
$483$	0	0
$484$	6777.97	0.636549
$485$	0	0
$486$	0	0
$487$	12757.1	1.18702	0.593510	−	0.804827i	$-0.297742\pi$
$487$	12757.1	1.18702	0.593510	+	0.804827i	$0.297742\pi$
$488$	3062.69	0.284101
$489$	0	0
$490$	0	0
$491$	7016.52	0.644911	0.322455	−	0.946585i	$-0.395492\pi$
$491$	7016.52	0.644911	0.322455	+	0.946585i	$0.395492\pi$
$492$	0	0
$493$	−5571.26	−0.508960
$494$	10383.8	0.945729
$495$	0	0
$496$	186.512	0.0168844
$497$	13198.6	1.19122
$498$	0	0
$499$	11372.3	1.02023	0.510113	−	0.860107i	$-0.329604\pi$
$499$	11372.3	1.02023	0.510113	+	0.860107i	$0.329604\pi$
$500$	0	0
$501$	0	0
$502$	3845.50	0.341899
$503$	5587.37	0.495285	0.247643	−	0.968851i	$-0.420344\pi$
$503$	5587.37	0.495285	0.247643	+	0.968851i	$0.420344\pi$
$504$	0	0
$505$	0	0
$506$	151.163	0.0132806
$507$	0	0
$508$	6111.03	0.533726
$509$	−16256.7	−1.41565	−0.707825	−	0.706388i	$-0.750324\pi$
$509$	−16256.7	−1.41565	−0.707825	+	0.706388i	$0.750324\pi$
$510$	0	0
$511$	7112.14	0.615699
$512$	1047.24	0.0903943
$513$	0	0
$514$	157.878	0.0135481
$515$	0	0
$516$	0	0
$517$	−688.574	−0.0585754
$518$	−751.442	−0.0637384
$519$	0	0
$520$	0	0
$521$	−19748.4	−1.66064	−0.830320	−	0.557286i	$-0.811843\pi$
$521$	−19748.4	−1.66064	−0.830320	+	0.557286i	$0.811843\pi$
$522$	0	0
$523$	−7843.44	−0.655774	−0.327887	−	0.944717i	$-0.606337\pi$
$523$	−7843.44	−0.655774	−0.327887	+	0.944717i	$0.606337\pi$
$524$	−1641.91	−0.136884
$525$	0	0
$526$	967.019	0.0801597
$527$	−1222.95	−0.101086
$528$	0	0
$529$	−9705.60	−0.797699
$530$	0	0
$531$	0	0
$532$	−11885.6	−0.968622
$533$	9638.38	0.783273
$534$	0	0
$535$	0	0
$536$	−3877.84	−0.312495
$537$	0	0
$538$	−12902.0	−1.03391
$539$	269.053	0.0215008
$540$	0	0
$541$	7383.29	0.586751	0.293376	−	0.955997i	$-0.405221\pi$
$541$	7383.29	0.586751	0.293376	+	0.955997i	$0.405221\pi$
$542$	13516.7	1.07120
$543$	0	0
$544$	−3480.10	−0.274280
$545$	0	0
$546$	0	0
$547$	−3354.90	−0.262240	−0.131120	−	0.991367i	$-0.541857\pi$
$547$	−3354.90	−0.262240	−0.131120	+	0.991367i	$0.541857\pi$
$548$	1808.54	0.140980
$549$	0	0
$550$	0	0
$551$	30766.2	2.37874
$552$	0	0
$553$	−17111.6	−1.31584
$554$	11609.8	0.890346
$555$	0	0
$556$	−394.369	−0.0300809
$557$	−20771.8	−1.58012	−0.790061	−	0.613028i	$-0.789951\pi$
$557$	−20771.8	−1.58012	−0.790061	+	0.613028i	$0.789951\pi$
$558$	0	0
$559$	−14398.1	−1.08940
$560$	0	0
$561$	0	0
$562$	−5642.15	−0.423487
$563$	−7194.86	−0.538592	−0.269296	−	0.963057i	$-0.586791\pi$
$563$	−7194.86	−0.538592	−0.269296	+	0.963057i	$0.586791\pi$
$564$	0	0
$565$	0	0
$566$	11235.0	0.834350
$567$	0	0
$568$	−13251.5	−0.978913
$569$	−11549.5	−0.850931	−0.425466	−	0.904975i	$-0.639890\pi$
$569$	−11549.5	−0.850931	−0.425466	+	0.904975i	$0.639890\pi$
$570$	0	0
$571$	1482.54	0.108655	0.0543277	−	0.998523i	$-0.482698\pi$
$571$	1482.54	0.108655	0.0543277	+	0.998523i	$0.482698\pi$
$572$	−532.068	−0.0388932
$573$	0	0
$574$	6257.42	0.455017
$575$	0	0
$576$	0	0
$577$	−15264.0	−1.10130	−0.550649	−	0.834737i	$-0.685620\pi$
$577$	−15264.0	−1.10130	−0.550649	+	0.834737i	$0.685620\pi$
$578$	−7746.52	−0.557462
$579$	0	0
$580$	0	0
$581$	−3862.76	−0.275825
$582$	0	0
$583$	−830.006	−0.0589628
$584$	−7140.66	−0.505963
$585$	0	0
$586$	−9893.12	−0.697408
$587$	−1736.89	−0.122128	−0.0610639	−	0.998134i	$-0.519449\pi$
$587$	−1736.89	−0.122128	−0.0610639	+	0.998134i	$0.519449\pi$
$588$	0	0
$589$	6753.50	0.472450
$590$	0	0
$591$	0	0
$592$	−57.5714	−0.00399691
$593$	11764.8	0.814707	0.407353	−	0.913271i	$-0.366452\pi$
$593$	11764.8	0.814707	0.407353	+	0.913271i	$0.366452\pi$
$594$	0	0
$595$	0	0
$596$	−8705.42	−0.598302
$597$	0	0
$598$	4913.96	0.336032
$599$	9451.99	0.644737	0.322369	−	0.946614i	$-0.395521\pi$
$599$	9451.99	0.644737	0.322369	+	0.946614i	$0.395521\pi$
$600$	0	0
$601$	−3131.93	−0.212569	−0.106285	−	0.994336i	$-0.533895\pi$
$601$	−3131.93	−0.212569	−0.106285	+	0.994336i	$0.533895\pi$
$602$	−9347.51	−0.632851
$603$	0	0
$604$	−3870.79	−0.260762
$605$	0	0
$606$	0	0
$607$	22700.8	1.51795	0.758975	−	0.651120i	$-0.225700\pi$
$607$	22700.8	1.51795	0.758975	+	0.651120i	$0.225700\pi$
$608$	19218.2	1.28191
$609$	0	0
$610$	0	0
$611$	−22384.0	−1.48210
$612$	0	0
$613$	28911.6	1.90494	0.952471	−	0.304629i	$-0.0985325\pi$
$613$	28911.6	1.90494	0.952471	+	0.304629i	$0.0985325\pi$
$614$	−13825.0	−0.908681
$615$	0	0
$616$	−886.780	−0.0580023
$617$	5566.87	0.363231	0.181616	−	0.983370i	$-0.441867\pi$
$617$	5566.87	0.363231	0.181616	+	0.983370i	$0.441867\pi$
$618$	0	0
$619$	4150.32	0.269492	0.134746	−	0.990880i	$-0.456978\pi$
$619$	4150.32	0.269492	0.134746	+	0.990880i	$0.456978\pi$
$620$	0	0
$621$	0	0
$622$	−12483.1	−0.804705
$623$	−22631.9	−1.45542
$624$	0	0
$625$	0	0
$626$	−3747.96	−0.239295
$627$	0	0
$628$	9030.46	0.573813
$629$	377.492	0.0239294
$630$	0	0
$631$	−4090.09	−0.258041	−0.129021	−	0.991642i	$-0.541183\pi$
$631$	−4090.09	−0.258041	−0.129021	+	0.991642i	$0.541183\pi$
$632$	17180.2	1.08132
$633$	0	0
$634$	−17030.7	−1.06684
$635$	0	0
$636$	0	0
$637$	8746.32	0.544022
$638$	894.150	0.0554855
$639$	0	0
$640$	0	0
$641$	−3909.35	−0.240890	−0.120445	−	0.992720i	$-0.538432\pi$
$641$	−3909.35	−0.240890	−0.120445	+	0.992720i	$0.538432\pi$
$642$	0	0
$643$	30539.5	1.87303	0.936516	−	0.350624i	$-0.114031\pi$
$643$	30539.5	1.87303	0.936516	+	0.350624i	$0.114031\pi$
$644$	−5624.66	−0.344166
$645$	0	0
$646$	−3386.58	−0.206259
$647$	12707.7	0.772167	0.386083	−	0.922464i	$-0.373828\pi$
$647$	12707.7	0.772167	0.386083	+	0.922464i	$0.373828\pi$
$648$	0	0
$649$	132.132	0.00799170
$650$	0	0
$651$	0	0
$652$	4500.90	0.270351
$653$	−12777.6	−0.765737	−0.382869	−	0.923803i	$-0.625064\pi$
$653$	−12777.6	−0.765737	−0.382869	+	0.923803i	$0.625064\pi$
$654$	0	0
$655$	0	0
$656$	479.410	0.0285332
$657$	0	0
$658$	−14532.1	−0.860976
$659$	−23563.5	−1.39287	−0.696435	−	0.717620i	$-0.745232\pi$
$659$	−23563.5	−1.39287	−0.696435	+	0.717620i	$0.745232\pi$
$660$	0	0
$661$	−4361.31	−0.256634	−0.128317	−	0.991733i	$-0.540958\pi$
$661$	−4361.31	−0.256634	−0.128317	+	0.991733i	$0.540958\pi$
$662$	−14796.7	−0.868715
$663$	0	0
$664$	3878.25	0.226665
$665$	0	0
$666$	0	0
$667$	14559.6	0.845200
$668$	−1107.21	−0.0641304
$669$	0	0
$670$	0	0
$671$	−245.943	−0.0141498
$672$	0	0
$673$	−8203.52	−0.469870	−0.234935	−	0.972011i	$-0.575488\pi$
$673$	−8203.52	−0.469870	−0.234935	+	0.972011i	$0.575488\pi$
$674$	−12592.6	−0.719657
$675$	0	0
$676$	−6081.37	−0.346004
$677$	−28057.1	−1.59279	−0.796397	−	0.604774i	$-0.793263\pi$
$677$	−28057.1	−1.59279	−0.796397	+	0.604774i	$0.793263\pi$
$678$	0	0
$679$	8536.73	0.482488
$680$	0	0
$681$	0	0
$682$	196.275	0.0110202
$683$	3344.62	0.187377	0.0936885	−	0.995602i	$-0.470134\pi$
$683$	3344.62	0.187377	0.0936885	+	0.995602i	$0.470134\pi$
$684$	0	0
$685$	0	0
$686$	−7283.91	−0.405395
$687$	0	0
$688$	−716.156	−0.0396849
$689$	−26981.7	−1.49190
$690$	0	0
$691$	12964.8	0.713757	0.356879	−	0.934151i	$-0.383841\pi$
$691$	12964.8	0.713757	0.356879	+	0.934151i	$0.383841\pi$
$692$	21061.5	1.15699
$693$	0	0
$694$	−13342.7	−0.729801
$695$	0	0
$696$	0	0
$697$	−3143.46	−0.170828
$698$	−8441.90	−0.457780
$699$	0	0
$700$	0	0
$701$	16162.1	0.870806	0.435403	−	0.900236i	$-0.356606\pi$
$701$	16162.1	0.870806	0.435403	+	0.900236i	$0.356606\pi$
$702$	0	0
$703$	−2084.63	−0.111839
$704$	517.058	0.0276809
$705$	0	0
$706$	20696.2	1.10327
$707$	−766.014	−0.0407481
$708$	0	0
$709$	−14244.4	−0.754529	−0.377265	−	0.926105i	$-0.623135\pi$
$709$	−14244.4	−0.754529	−0.377265	+	0.926105i	$0.623135\pi$
$710$	0	0
$711$	0	0
$712$	22722.7	1.19602
$713$	3195.97	0.167868
$714$	0	0
$715$	0	0
$716$	−16402.1	−0.856109
$717$	0	0
$718$	−8837.55	−0.459352
$719$	27638.5	1.43358	0.716790	−	0.697289i	$-0.245611\pi$
$719$	27638.5	1.43358	0.716790	+	0.697289i	$0.245611\pi$
$720$	0	0
$721$	−39006.4	−2.01480
$722$	7030.68	0.362403
$723$	0	0
$724$	−17327.4	−0.889460
$725$	0	0
$726$	0	0
$727$	−2525.52	−0.128840	−0.0644199	−	0.997923i	$-0.520520\pi$
$727$	−2525.52	−0.128840	−0.0644199	+	0.997923i	$0.520520\pi$
$728$	−28827.3	−1.46760
$729$	0	0
$730$	0	0
$731$	4695.79	0.237592
$732$	0	0
$733$	−8400.27	−0.423289	−0.211645	−	0.977347i	$-0.567882\pi$
$733$	−8400.27	−0.423289	−0.211645	+	0.977347i	$0.567882\pi$
$734$	−10355.9	−0.520765
$735$	0	0
$736$	9094.67	0.455481
$737$	311.401	0.0155639
$738$	0	0
$739$	−19689.1	−0.980074	−0.490037	−	0.871702i	$-0.663017\pi$
$739$	−19689.1	−0.980074	−0.490037	+	0.871702i	$0.663017\pi$
$740$	0	0
$741$	0	0
$742$	−17517.0	−0.866671
$743$	22526.6	1.11227	0.556137	−	0.831091i	$-0.312283\pi$
$743$	22526.6	1.11227	0.556137	+	0.831091i	$0.312283\pi$
$744$	0	0
$745$	0	0
$746$	−18005.8	−0.883699
$747$	0	0
$748$	173.528	0.00848239
$749$	30243.3	1.47539
$750$	0	0
$751$	34691.1	1.68562	0.842808	−	0.538215i	$-0.180901\pi$
$751$	34691.1	1.68562	0.842808	+	0.538215i	$0.180901\pi$
$752$	−1113.37	−0.0539902
$753$	0	0
$754$	29066.8	1.40392
$755$	0	0
$756$	0	0
$757$	6619.98	0.317843	0.158922	−	0.987291i	$-0.449198\pi$
$757$	6619.98	0.317843	0.158922	+	0.987291i	$0.449198\pi$
$758$	19832.0	0.950303
$759$	0	0
$760$	0	0
$761$	29368.7	1.39897	0.699483	−	0.714649i	$-0.253414\pi$
$761$	29368.7	1.39897	0.699483	+	0.714649i	$0.253414\pi$
$762$	0	0
$763$	7131.84	0.338388
$764$	−17700.5	−0.838194
$765$	0	0
$766$	−10830.4	−0.510859
$767$	4295.31	0.202209
$768$	0	0
$769$	32677.4	1.53235	0.766174	−	0.642633i	$-0.222158\pi$
$769$	32677.4	1.53235	0.766174	+	0.642633i	$0.222158\pi$
$770$	0	0
$771$	0	0
$772$	−9148.33	−0.426497
$773$	28047.5	1.30504	0.652522	−	0.757770i	$-0.273711\pi$
$773$	28047.5	1.30504	0.652522	+	0.757770i	$0.273711\pi$
$774$	0	0
$775$	0	0
$776$	−8570.96	−0.396494
$777$	0	0
$778$	−10438.0	−0.481001
$779$	17359.1	0.798402
$780$	0	0
$781$	1064.14	0.0487552
$782$	−1602.64	−0.0732867
$783$	0	0
$784$	435.039	0.0198177
$785$	0	0
$786$	0	0
$787$	−22172.1	−1.00426	−0.502128	−	0.864793i	$-0.667449\pi$
$787$	−22172.1	−1.00426	−0.502128	+	0.864793i	$0.667449\pi$
$788$	8566.64	0.387276
$789$	0	0
$790$	0	0
$791$	35140.7	1.57960
$792$	0	0
$793$	−7995.06	−0.358024
$794$	16584.2	0.741249
$795$	0	0
$796$	−15866.5	−0.706501
$797$	24170.3	1.07422	0.537112	−	0.843511i	$-0.319515\pi$
$797$	24170.3	1.07422	0.537112	+	0.843511i	$0.319515\pi$
$798$	0	0
$799$	7300.32	0.323238
$800$	0	0
$801$	0	0
$802$	2222.99	0.0978761
$803$	573.415	0.0251997
$804$	0	0
$805$	0	0
$806$	6380.47	0.278837
$807$	0	0
$808$	769.085	0.0334856
$809$	−15304.2	−0.665102	−0.332551	−	0.943085i	$-0.607909\pi$
$809$	−15304.2	−0.665102	−0.332551	+	0.943085i	$0.607909\pi$
$810$	0	0
$811$	−27002.2	−1.16914	−0.584572	−	0.811342i	$-0.698738\pi$
$811$	−27002.2	−1.16914	−0.584572	+	0.811342i	$0.698738\pi$
$812$	−33270.7	−1.43790
$813$	0	0
$814$	−60.5849	−0.00260872
$815$	0	0
$816$	0	0
$817$	−25931.5	−1.11044
$818$	−6596.84	−0.281972
$819$	0	0
$820$	0	0
$821$	−25061.4	−1.06535	−0.532673	−	0.846321i	$-0.678812\pi$
$821$	−25061.4	−1.06535	−0.532673	+	0.846321i	$0.678812\pi$
$822$	0	0
$823$	−24896.4	−1.05448	−0.527238	−	0.849718i	$-0.676772\pi$
$823$	−24896.4	−1.05448	−0.527238	+	0.849718i	$0.676772\pi$
$824$	39162.8	1.65571
$825$	0	0
$826$	2788.59	0.117467
$827$	20063.2	0.843612	0.421806	−	0.906686i	$-0.361396\pi$
$827$	20063.2	0.843612	0.421806	+	0.906686i	$0.361396\pi$
$828$	0	0
$829$	−13884.2	−0.581687	−0.290844	−	0.956771i	$-0.593936\pi$
$829$	−13884.2	−0.581687	−0.290844	+	0.956771i	$0.593936\pi$
$830$	0	0
$831$	0	0
$832$	16808.4	0.700393
$833$	−2852.52	−0.118648
$834$	0	0
$835$	0	0
$836$	−958.277	−0.0396444
$837$	0	0
$838$	27263.3	1.12386
$839$	13678.1	0.562838	0.281419	−	0.959585i	$-0.409195\pi$
$839$	13678.1	0.562838	0.281419	+	0.959585i	$0.409195\pi$
$840$	0	0
$841$	61733.1	2.53118
$842$	−13816.2	−0.565485
$843$	0	0
$844$	22834.6	0.931280
$845$	0	0
$846$	0	0
$847$	−29489.5	−1.19630
$848$	−1342.06	−0.0543473
$849$	0	0
$850$	0	0
$851$	−986.512	−0.0397382
$852$	0	0
$853$	29802.9	1.19629	0.598143	−	0.801390i	$-0.295906\pi$
$853$	29802.9	1.19629	0.598143	+	0.801390i	$0.295906\pi$
$854$	−5190.54	−0.207982
$855$	0	0
$856$	−30364.6	−1.21243
$857$	−22045.2	−0.878706	−0.439353	−	0.898314i	$-0.644792\pi$
$857$	−22045.2	−0.878706	−0.439353	+	0.898314i	$0.644792\pi$
$858$	0	0
$859$	33609.5	1.33497	0.667487	−	0.744622i	$-0.267370\pi$
$859$	33609.5	1.33497	0.667487	+	0.744622i	$0.267370\pi$
$860$	0	0
$861$	0	0
$862$	9722.70	0.384172
$863$	−33775.6	−1.33226	−0.666128	−	0.745838i	$-0.732049\pi$
$863$	−33775.6	−1.33226	−0.666128	+	0.745838i	$0.732049\pi$
$864$	0	0
$865$	0	0
$866$	10637.0	0.417392
$867$	0	0
$868$	−7303.27	−0.285587
$869$	−1379.62	−0.0538556
$870$	0	0
$871$	10123.0	0.393805
$872$	−7160.44	−0.278077
$873$	0	0
$874$	8850.25	0.342522
$875$	0	0
$876$	0	0
$877$	12637.0	0.486570	0.243285	−	0.969955i	$-0.421775\pi$
$877$	12637.0	0.486570	0.243285	+	0.969955i	$0.421775\pi$
$878$	−7199.25	−0.276723
$879$	0	0
$880$	0	0
$881$	6579.45	0.251609	0.125804	−	0.992055i	$-0.459849\pi$
$881$	6579.45	0.251609	0.125804	+	0.992055i	$0.459849\pi$
$882$	0	0
$883$	−50442.1	−1.92244	−0.961219	−	0.275786i	$-0.911062\pi$
$883$	−50442.1	−1.92244	−0.961219	+	0.275786i	$0.911062\pi$
$884$	5641.03	0.214625
$885$	0	0
$886$	−10744.2	−0.407401
$887$	−984.823	−0.0372797	−0.0186399	−	0.999826i	$-0.505934\pi$
$887$	−984.823	−0.0372797	−0.0186399	+	0.999826i	$0.505934\pi$
$888$	0	0
$889$	−26587.7	−1.00306
$890$	0	0
$891$	0	0
$892$	8950.64	0.335975
$893$	−40314.6	−1.51072
$894$	0	0
$895$	0	0
$896$	−21658.0	−0.807525
$897$	0	0
$898$	15909.1	0.591196
$899$	18904.7	0.701342
$900$	0	0
$901$	8799.79	0.325376
$902$	504.503	0.0186232
$903$	0	0
$904$	−35281.6	−1.29806
$905$	0	0
$906$	0	0
$907$	43679.9	1.59908	0.799541	−	0.600612i	$-0.205076\pi$
$907$	43679.9	1.59908	0.799541	+	0.600612i	$0.205076\pi$
$908$	−4782.58	−0.174797
$909$	0	0
$910$	0	0
$911$	10364.3	0.376930	0.188465	−	0.982080i	$-0.439649\pi$
$911$	10364.3	0.376930	0.188465	+	0.982080i	$0.439649\pi$
$912$	0	0
$913$	−311.435	−0.0112891
$914$	−16245.6	−0.587918
$915$	0	0
$916$	13180.6	0.475435
$917$	7143.58	0.257254
$918$	0	0
$919$	11451.9	0.411059	0.205530	−	0.978651i	$-0.434108\pi$
$919$	11451.9	0.411059	0.205530	+	0.978651i	$0.434108\pi$
$920$	0	0
$921$	0	0
$922$	−10613.0	−0.379091
$923$	34592.7	1.23362
$924$	0	0
$925$	0	0
$926$	11009.1	0.390692
$927$	0	0
$928$	53796.4	1.90297
$929$	27701.8	0.978326	0.489163	−	0.872192i	$-0.337302\pi$
$929$	27701.8	0.978326	0.489163	+	0.872192i	$0.337302\pi$
$930$	0	0
$931$	15752.5	0.554529
$932$	11715.3	0.411746
$933$	0	0
$934$	−12262.5	−0.429596
$935$	0	0
$936$	0	0
$937$	−5878.01	−0.204937	−0.102469	−	0.994736i	$-0.532674\pi$
$937$	−5878.01	−0.204937	−0.102469	+	0.994736i	$0.532674\pi$
$938$	6572.03	0.228768
$939$	0	0
$940$	0	0
$941$	28786.0	0.997234	0.498617	−	0.866823i	$-0.333841\pi$
$941$	28786.0	0.997234	0.498617	+	0.866823i	$0.333841\pi$
$942$	0	0
$943$	8214.90	0.283684
$944$	213.647	0.00736612
$945$	0	0
$946$	−753.642	−0.0259017
$947$	1695.04	0.0581641	0.0290821	−	0.999577i	$-0.490742\pi$
$947$	1695.04	0.0581641	0.0290821	+	0.999577i	$0.490742\pi$
$948$	0	0
$949$	18640.5	0.637613
$950$	0	0
$951$	0	0
$952$	9401.72	0.320075
$953$	−31929.4	−1.08530	−0.542651	−	0.839958i	$-0.682580\pi$
$953$	−31929.4	−1.08530	−0.542651	+	0.839958i	$0.682580\pi$
$954$	0	0
$955$	0	0
$956$	−11710.2	−0.396166
$957$	0	0
$958$	18465.1	0.622735
$959$	−7868.57	−0.264952
$960$	0	0
$961$	−25641.2	−0.860704
$962$	−1969.48	−0.0660069
$963$	0	0
$964$	−1951.46	−0.0651994
$965$	0	0
$966$	0	0
$967$	−10897.1	−0.362385	−0.181193	−	0.983448i	$-0.557996\pi$
$967$	−10897.1	−0.362385	−0.181193	+	0.983448i	$0.557996\pi$
$968$	29607.7	0.983087
$969$	0	0
$970$	0	0
$971$	−7041.97	−0.232737	−0.116368	−	0.993206i	$-0.537125\pi$
$971$	−7041.97	−0.232737	−0.116368	+	0.993206i	$0.537125\pi$
$972$	0	0
$973$	1715.81	0.0565328
$974$	21707.0	0.714103
$975$	0	0
$976$	−397.671	−0.0130422
$977$	37607.6	1.23150	0.615749	−	0.787943i	$-0.288854\pi$
$977$	37607.6	1.23150	0.615749	+	0.787943i	$0.288854\pi$
$978$	0	0
$979$	−1824.69	−0.0595684
$980$	0	0
$981$	0	0
$982$	11939.0	0.387974
$983$	25297.7	0.820826	0.410413	−	0.911900i	$-0.365385\pi$
$983$	25297.7	0.820826	0.410413	+	0.911900i	$0.365385\pi$
$984$	0	0
$985$	0	0
$986$	−9479.85	−0.306187
$987$	0	0
$988$	−31151.5	−1.00310
$989$	−12271.6	−0.394556
$990$	0	0
$991$	−41686.5	−1.33624	−0.668120	−	0.744053i	$-0.732901\pi$
$991$	−41686.5	−1.33624	−0.668120	+	0.744053i	$0.732901\pi$
$992$	11808.9	0.377955
$993$	0	0
$994$	22458.3	0.716633
$995$	0	0
$996$	0	0
$997$	25465.9	0.808939	0.404470	−	0.914551i	$-0.367456\pi$
$997$	25465.9	0.808939	0.404470	+	0.914551i	$0.367456\pi$
$998$	19350.6	0.613761
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.a.i.1.2		2
3.2	odd	2		75.4.a.f.1.1		2
5.2	odd	4		45.4.b.b.19.3		4
5.3	odd	4		45.4.b.b.19.2		4
5.4	even	2		225.4.a.o.1.1		2
12.11	even	2		1200.4.a.bn.1.1		2
15.2	even	4		15.4.b.a.4.2	✓	4
15.8	even	4		15.4.b.a.4.3	yes	4
15.14	odd	2		75.4.a.c.1.2		2
20.3	even	4		720.4.f.j.289.4		4
20.7	even	4		720.4.f.j.289.3		4
60.23	odd	4		240.4.f.f.49.1		4
60.47	odd	4		240.4.f.f.49.3		4
60.59	even	2		1200.4.a.bt.1.2		2
120.53	even	4		960.4.f.q.769.2		4
120.77	even	4		960.4.f.q.769.4		4
120.83	odd	4		960.4.f.p.769.4		4
120.107	odd	4		960.4.f.p.769.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.b.a.4.2	✓	4	15.2	even	4
15.4.b.a.4.3	yes	4	15.8	even	4
45.4.b.b.19.2		4	5.3	odd	4
45.4.b.b.19.3		4	5.2	odd	4
75.4.a.c.1.2		2	15.14	odd	2
75.4.a.f.1.1		2	3.2	odd	2
225.4.a.i.1.2		2	1.1	even	1	trivial
225.4.a.o.1.1		2	5.4	even	2
240.4.f.f.49.1		4	60.23	odd	4
240.4.f.f.49.3		4	60.47	odd	4
720.4.f.j.289.3		4	20.7	even	4
720.4.f.j.289.4		4	20.3	even	4
960.4.f.p.769.2		4	120.107	odd	4
960.4.f.p.769.4		4	120.83	odd	4
960.4.f.q.769.2		4	120.53	even	4
960.4.f.q.769.4		4	120.77	even	4
1200.4.a.bn.1.1		2	12.11	even	2
1200.4.a.bt.1.2		2	60.59	even	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70156 q^{2} -5.10469 q^{4} +22.2094 q^{7} -22.2984 q^{8} +1.79063 q^{11} +58.2094 q^{13} +37.7906 q^{14} +2.89531 q^{16} -18.9844 q^{17} +104.837 q^{19} +3.04686 q^{22} +49.6125 q^{23} +99.0469 q^{26} -113.372 q^{28} +293.466 q^{29} +64.4187 q^{31} +183.314 q^{32} -32.3031 q^{34} -19.8844 q^{37} +178.388 q^{38} +165.581 q^{41} -247.350 q^{43} -9.14059 q^{44} +84.4187 q^{46} -384.544 q^{47} +150.256 q^{49} -297.141 q^{52} -463.528 q^{53} -495.234 q^{56} +499.350 q^{58} +73.7906 q^{59} -137.350 q^{61} +109.612 q^{62} +288.758 q^{64} +173.906 q^{67} +96.9093 q^{68} +594.281 q^{71} +320.231 q^{73} -33.8345 q^{74} -535.163 q^{76} +39.7687 q^{77} -770.469 q^{79} +281.747 q^{82} -173.925 q^{83} -420.881 q^{86} -39.9282 q^{88} -1019.02 q^{89} +1292.79 q^{91} -253.256 q^{92} -654.325 q^{94} +384.375 q^{97} +255.670 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 9 q^{4} + 6 q^{7} - 51 q^{8} + 42 q^{11} + 78 q^{13} + 114 q^{14} + 25 q^{16} - 102 q^{17} + 56 q^{19} - 186 q^{22} + 48 q^{23} + 6 q^{26} - 342 q^{28} + 318 q^{29} + 52 q^{31} + 309 q^{32}+ \cdots + 633 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 9 * q^4 + 6 * q^7 - 51 * q^8 + 42 * q^11 + 78 * q^13 + 114 * q^14 + 25 * q^16 - 102 * q^17 + 56 * q^19 - 186 * q^22 + 48 * q^23 + 6 * q^26 - 342 * q^28 + 318 * q^29 + 52 * q^31 + 309 * q^32 + 358 * q^34 + 306 * q^37 + 408 * q^38 + 408 * q^41 + 120 * q^43 + 558 * q^44 + 92 * q^46 - 180 * q^47 + 70 * q^49 - 18 * q^52 - 402 * q^53 - 30 * q^56 + 384 * q^58 + 186 * q^59 + 340 * q^61 + 168 * q^62 - 479 * q^64 + 732 * q^67 - 1074 * q^68 + 36 * q^71 + 1332 * q^73 - 1566 * q^74 - 1224 * q^76 - 612 * q^77 + 380 * q^79 - 858 * q^82 + 984 * q^83 - 2148 * q^86 - 1194 * q^88 - 1116 * q^89 + 972 * q^91 - 276 * q^92 - 1616 * q^94 - 768 * q^97 + 633 * q^98

Introduction

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