LMFDB - Embedded newform 1200.4.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, 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("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.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:	$70.8022920069$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	1200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -22.2094 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -22.2094 q^{7} +9.00000 q^{9} +1.79063 q^{11} +58.2094 q^{13} +18.9844 q^{17} -104.837 q^{19} +66.6281 q^{21} +49.6125 q^{23} -27.0000 q^{27} -293.466 q^{29} -64.4187 q^{31} -5.37188 q^{33} -19.8844 q^{37} -174.628 q^{39} -165.581 q^{41} +247.350 q^{43} -384.544 q^{47} +150.256 q^{49} -56.9531 q^{51} +463.528 q^{53} +314.512 q^{57} +73.7906 q^{59} -137.350 q^{61} -199.884 q^{63} -173.906 q^{67} -148.837 q^{69} +594.281 q^{71} +320.231 q^{73} -39.7687 q^{77} +770.469 q^{79} +81.0000 q^{81} -173.925 q^{83} +880.397 q^{87} +1019.02 q^{89} -1292.79 q^{91} +193.256 q^{93} +384.375 q^{97} +16.1156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9} + 42 q^{11} + 78 q^{13} + 102 q^{17} - 56 q^{19} + 18 q^{21} + 48 q^{23} - 54 q^{27} - 318 q^{29} - 52 q^{31} - 126 q^{33} + 306 q^{37} - 234 q^{39} - 408 q^{41} - 120 q^{43} - 180 q^{47} + 70 q^{49} - 306 q^{51} + 402 q^{53} + 168 q^{57} + 186 q^{59} + 340 q^{61} - 54 q^{63} - 732 q^{67} - 144 q^{69} + 36 q^{71} + 1332 q^{73} + 612 q^{77} - 380 q^{79} + 162 q^{81} + 984 q^{83} + 954 q^{87} + 1116 q^{89} - 972 q^{91} + 156 q^{93} - 768 q^{97} + 378 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9 + 42 * q^11 + 78 * q^13 + 102 * q^17 - 56 * q^19 + 18 * q^21 + 48 * q^23 - 54 * q^27 - 318 * q^29 - 52 * q^31 - 126 * q^33 + 306 * q^37 - 234 * q^39 - 408 * q^41 - 120 * q^43 - 180 * q^47 + 70 * q^49 - 306 * q^51 + 402 * q^53 + 168 * q^57 + 186 * q^59 + 340 * q^61 - 54 * q^63 - 732 * q^67 - 144 * q^69 + 36 * q^71 + 1332 * q^73 + 612 * q^77 - 380 * q^79 + 162 * q^81 + 984 * q^83 + 954 * q^87 + 1116 * q^89 - 972 * q^91 + 156 * q^93 - 768 * q^97 + 378 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−22.2094	−1.19919	−0.599597	−	0.800302i	$-0.704672\pi$
$7$	−22.2094	−1.19919	−0.599597	+	0.800302i	$0.704672\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	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	$0.286752\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.9844	0.270846	0.135423	−	0.990788i	$-0.456761\pi$
$17$	18.9844	0.270846	0.135423	+	0.990788i	$0.456761\pi$
$18$	0	0
$19$	−104.837	−1.26586	−0.632931	−	0.774208i	$-0.718148\pi$
$19$	−104.837	−1.26586	−0.632931	+	0.774208i	$0.718148\pi$
$20$	0	0
$21$	66.6281	0.692355
$22$	0	0
$23$	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
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−293.466	−1.87914	−0.939572	−	0.342350i	$-0.888777\pi$
$29$	−293.466	−1.87914	−0.939572	+	0.342350i	$0.888777\pi$
$30$	0	0
$31$	−64.4187	−0.373224	−0.186612	−	0.982434i	$-0.559751\pi$
$31$	−64.4187	−0.373224	−0.186612	+	0.982434i	$0.559751\pi$
$32$	0	0
$33$	−5.37188	−0.0283371
$34$	0	0
$35$	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	$0.514066\pi$
$38$	0	0
$39$	−174.628	−0.716997
$40$	0	0
$41$	−165.581	−0.630718	−0.315359	−	0.948972i	$-0.602125\pi$
$41$	−165.581	−0.630718	−0.315359	+	0.948972i	$0.602125\pi$
$42$	0	0
$43$	247.350	0.877221	0.438611	−	0.898677i	$-0.355471\pi$
$43$	247.350	0.877221	0.438611	+	0.898677i	$0.355471\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−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
$51$	−56.9531	−0.156373
$52$	0	0
$53$	463.528	1.20133	0.600665	−	0.799501i	$-0.294903\pi$
$53$	463.528	1.20133	0.600665	+	0.799501i	$0.294903\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	314.512	0.730846
$58$	0	0
$59$	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	$0.546044\pi$
$62$	0	0
$63$	−199.884	−0.399731
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−173.906	−0.317105	−0.158552	−	0.987351i	$-0.550683\pi$
$67$	−173.906	−0.317105	−0.158552	+	0.987351i	$0.550683\pi$
$68$	0	0
$69$	−148.837	−0.259680
$70$	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	$0.417360\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−39.7687	−0.0588580
$78$	0	0
$79$	770.469	1.09727	0.548636	−	0.836061i	$-0.315147\pi$
$79$	770.469	1.09727	0.548636	+	0.836061i	$0.315147\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−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
$86$	0	0
$87$	880.397	1.08492
$88$	0	0
$89$	1019.02	1.21367	0.606834	−	0.794829i	$-0.292439\pi$
$89$	1019.02	1.21367	0.606834	+	0.794829i	$0.292439\pi$
$90$	0	0
$91$	−1292.79	−1.48925
$92$	0	0
$93$	193.256	0.215481
$94$	0	0
$95$	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	$0.435525\pi$
$98$	0	0
$99$	16.1156	0.0163604
$100$	0	0
$101$	34.4906	0.0339796	0.0169898	−	0.999856i	$-0.494592\pi$
$101$	34.4906	0.0339796	0.0169898	+	0.999856i	$0.494592\pi$
$102$	0	0
$103$	1756.30	1.68013	0.840066	−	0.542484i	$-0.182516\pi$
$103$	1756.30	1.68013	0.840066	+	0.542484i	$0.182516\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	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
$111$	59.6531	0.0510092
$112$	0	0
$113$	−1582.25	−1.31721	−0.658607	−	0.752487i	$-0.728854\pi$
$113$	−1582.25	−1.31721	−0.658607	+	0.752487i	$0.728854\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	523.884	0.413958
$118$	0	0
$119$	−421.631	−0.324797
$120$	0	0
$121$	−1327.79	−0.997591
$122$	0	0
$123$	496.744	0.364145
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1197.14	0.836449	0.418225	−	0.908344i	$-0.362652\pi$
$127$	1197.14	0.836449	0.418225	+	0.908344i	$0.362652\pi$
$128$	0	0
$129$	−742.050	−0.506464
$130$	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
$134$	0	0
$135$	0	0
$136$	0	0
$137$	354.291	0.220942	0.110471	−	0.993879i	$-0.464764\pi$
$137$	354.291	0.220942	0.110471	+	0.993879i	$0.464764\pi$
$138$	0	0
$139$	−77.2562	−0.0471424	−0.0235712	−	0.999722i	$-0.507504\pi$
$139$	−77.2562	−0.0471424	−0.0235712	+	0.999722i	$0.507504\pi$
$140$	0	0
$141$	1153.63	0.689030
$142$	0	0
$143$	104.231	0.0609529
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−450.769	−0.252917
$148$	0	0
$149$	−1705.38	−0.937651	−0.468826	−	0.883291i	$-0.655323\pi$
$149$	−1705.38	−0.937651	−0.468826	+	0.883291i	$0.655323\pi$
$150$	0	0
$151$	−758.281	−0.408663	−0.204331	−	0.978902i	$-0.565502\pi$
$151$	−758.281	−0.408663	−0.204331	+	0.978902i	$0.565502\pi$
$152$	0	0
$153$	170.859	0.0902821
$154$	0	0
$155$	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	$0.648446\pi$
$158$	0	0
$159$	−1390.58	−0.693588
$160$	0	0
$161$	−1101.86	−0.539372
$162$	0	0
$163$	881.719	0.423690	0.211845	−	0.977303i	$-0.432053\pi$
$163$	881.719	0.423690	0.211845	+	0.977303i	$0.432053\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	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
$171$	−943.537	−0.421954
$172$	0	0
$173$	4125.91	1.81322	0.906610	−	0.421970i	$-0.138661\pi$
$173$	4125.91	1.81322	0.906610	+	0.421970i	$0.138661\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−221.372	−0.0940075
$178$	0	0
$179$	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	$0.254529\pi$
$182$	0	0
$183$	412.050	0.166446
$184$	0	0
$185$	0	0
$186$	0	0
$187$	33.9939	0.0132935
$188$	0	0
$189$	599.653	0.230785
$190$	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	$0.391534\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1678.19	0.606935	0.303467	−	0.952842i	$-0.401856\pi$
$197$	1678.19	0.606935	0.303467	+	0.952842i	$0.401856\pi$
$198$	0	0
$199$	−3108.23	−1.10722	−0.553610	−	0.832776i	$-0.686750\pi$
$199$	−3108.23	−1.10722	−0.553610	+	0.832776i	$0.686750\pi$
$200$	0	0
$201$	521.719	0.183081
$202$	0	0
$203$	6517.69	2.25346
$204$	0	0
$205$	0	0
$206$	0	0
$207$	446.512	0.149926
$208$	0	0
$209$	−187.725	−0.0621301
$210$	0	0
$211$	4473.27	1.45949	0.729745	−	0.683719i	$-0.239639\pi$
$211$	4473.27	1.45949	0.729745	+	0.683719i	$0.239639\pi$
$212$	0	0
$213$	−1782.84	−0.573514
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1430.70	0.447568
$218$	0	0
$219$	−960.694	−0.296428
$220$	0	0
$221$	1105.07	0.336357
$222$	0	0
$223$	1753.42	0.526535	0.263268	−	0.964723i	$-0.415200\pi$
$223$	1753.42	0.526535	0.263268	+	0.964723i	$0.415200\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	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
$231$	119.306	0.0339817
$232$	0	0
$233$	2295.01	0.645284	0.322642	−	0.946521i	$-0.395429\pi$
$233$	2295.01	0.645284	0.322642	+	0.946521i	$0.395429\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2311.41	−0.633510
$238$	0	0
$239$	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	$0.483731\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6102.52	−1.57204
$248$	0	0
$249$	521.775	0.132796
$250$	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
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−92.7843	−0.0225203	−0.0112602	−	0.999937i	$-0.503584\pi$
$257$	−92.7843	−0.0225203	−0.0112602	+	0.999937i	$0.503584\pi$
$258$	0	0
$259$	441.619	0.105949
$260$	0	0
$261$	−2641.19	−0.626382
$262$	0	0
$263$	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
$266$	0	0
$267$	−3057.07	−0.700711
$268$	0	0
$269$	7582.41	1.71862	0.859309	−	0.511458i	$-0.170894\pi$
$269$	7582.41	1.71862	0.859309	+	0.511458i	$0.170894\pi$
$270$	0	0
$271$	−7943.69	−1.78061	−0.890304	−	0.455366i	$-0.849508\pi$
$271$	−7943.69	−1.78061	−0.890304	+	0.455366i	$0.849508\pi$
$272$	0	0
$273$	3878.38	0.859818
$274$	0	0
$275$	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	$0.234830\pi$
$278$	0	0
$279$	−579.769	−0.124408
$280$	0	0
$281$	3315.86	0.703942	0.351971	−	0.936011i	$-0.385512\pi$
$281$	3315.86	0.703942	0.351971	+	0.936011i	$0.385512\pi$
$282$	0	0
$283$	−6602.76	−1.38690	−0.693451	−	0.720504i	$-0.743910\pi$
$283$	−6602.76	−1.38690	−0.693451	+	0.720504i	$0.743910\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3677.46	0.756353
$288$	0	0
$289$	−4552.59	−0.926642
$290$	0	0
$291$	−1153.12	−0.232293
$292$	0	0
$293$	5814.14	1.15927	0.579634	−	0.814877i	$-0.303195\pi$
$293$	5814.14	1.15927	0.579634	+	0.814877i	$0.303195\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−48.3469	−0.00944570
$298$	0	0
$299$	2887.91	0.558570
$300$	0	0
$301$	−5493.49	−1.05196
$302$	0	0
$303$	−103.472	−0.0196181
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8124.86	1.51046	0.755229	−	0.655462i	$-0.227526\pi$
$307$	8124.86	1.51046	0.755229	+	0.655462i	$0.227526\pi$
$308$	0	0
$309$	−5268.91	−0.970025
$310$	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	$0.563732\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10008.9	1.77336	0.886679	−	0.462386i	$-0.153007\pi$
$317$	10008.9	1.77336	0.886679	+	0.462386i	$0.153007\pi$
$318$	0	0
$319$	−525.488	−0.0922309
$320$	0	0
$321$	−4085.21	−0.710325
$322$	0	0
$323$	−1990.27	−0.342854
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−963.356	−0.162917
$328$	0	0
$329$	8540.47	1.43116
$330$	0	0
$331$	8695.94	1.44402	0.722012	−	0.691881i	$-0.243218\pi$
$331$	8695.94	1.44402	0.722012	+	0.691881i	$0.243218\pi$
$332$	0	0
$333$	−178.959	−0.0294502
$334$	0	0
$335$	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	$0.704088\pi$
$338$	0	0
$339$	4746.74	0.760494
$340$	0	0
$341$	−115.350	−0.0183183
$342$	0	0
$343$	4280.72	0.673869
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−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
$351$	−1571.65	−0.238999
$352$	0	0
$353$	−12163.0	−1.83392	−0.916959	−	0.398981i	$-0.869364\pi$
$353$	−12163.0	−1.83392	−0.916959	+	0.398981i	$0.869364\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1264.89	0.187522
$358$	0	0
$359$	−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
$362$	0	0
$363$	3983.38	0.575959
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6086.09	0.865644	0.432822	−	0.901479i	$-0.357518\pi$
$367$	6086.09	0.865644	0.432822	+	0.901479i	$0.357518\pi$
$368$	0	0
$369$	−1490.23	−0.210239
$370$	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	$0.762567\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−17082.4	−2.33366
$378$	0	0
$379$	−11655.2	−1.57964	−0.789822	−	0.613336i	$-0.789827\pi$
$379$	−11655.2	−1.57964	−0.789822	+	0.613336i	$0.789827\pi$
$380$	0	0
$381$	−3591.42	−0.482924
$382$	0	0
$383$	−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
$386$	0	0
$387$	2226.15	0.292407
$388$	0	0
$389$	6134.33	0.799545	0.399773	−	0.916614i	$-0.369089\pi$
$389$	6134.33	0.799545	0.399773	+	0.916614i	$0.369089\pi$
$390$	0	0
$391$	941.862	0.121821
$392$	0	0
$393$	−964.941	−0.123855
$394$	0	0
$395$	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	$0.288723\pi$
$398$	0	0
$399$	−6985.12	−0.876425
$400$	0	0
$401$	−1306.44	−0.162695	−0.0813474	−	0.996686i	$-0.525922\pi$
$401$	−1306.44	−0.162695	−0.0813474	+	0.996686i	$0.525922\pi$
$402$	0	0
$403$	−3749.77	−0.463498
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−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
$411$	−1062.87	−0.127561
$412$	0	0
$413$	−1638.84	−0.195260
$414$	0	0
$415$	0	0
$416$	0	0
$417$	231.769	0.0272177
$418$	0	0
$419$	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	$0.655742\pi$
$422$	0	0
$423$	−3460.89	−0.397812
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3050.46	0.345719
$428$	0	0
$429$	−312.694	−0.0351911
$430$	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	$0.387232\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5201.25	−0.569358
$438$	0	0
$439$	4230.97	0.459984	0.229992	−	0.973192i	$-0.426130\pi$
$439$	4230.97	0.459984	0.229992	+	0.973192i	$0.426130\pi$
$440$	0	0
$441$	1352.31	0.146022
$442$	0	0
$443$	−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
$446$	0	0
$447$	5116.13	0.541353
$448$	0	0
$449$	−9349.71	−0.982717	−0.491358	−	0.870957i	$-0.663499\pi$
$449$	−9349.71	−0.982717	−0.491358	+	0.870957i	$0.663499\pi$
$450$	0	0
$451$	−296.494	−0.0309565
$452$	0	0
$453$	2274.84	0.235941
$454$	0	0
$455$	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	$0.662505\pi$
$458$	0	0
$459$	−512.578	−0.0521244
$460$	0	0
$461$	6237.23	0.630145	0.315073	−	0.949068i	$-0.397971\pi$
$461$	6237.23	0.630145	0.315073	+	0.949068i	$0.397971\pi$
$462$	0	0
$463$	−6469.98	−0.649428	−0.324714	−	0.945812i	$-0.605268\pi$
$463$	−6469.98	−0.649428	−0.324714	+	0.945812i	$0.605268\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−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
$471$	5307.16	0.519195
$472$	0	0
$473$	442.912	0.0430552
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4171.75	0.400443
$478$	0	0
$479$	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
$482$	0	0
$483$	3305.59	0.311407
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12757.1	−1.18702	−0.593510	−	0.804827i	$-0.702258\pi$
$487$	−12757.1	−1.18702	−0.593510	+	0.804827i	$0.702258\pi$
$488$	0	0
$489$	−2645.16	−0.244618
$490$	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
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13198.6	−1.19122
$498$	0	0
$499$	−11372.3	−1.02023	−0.510113	−	0.860107i	$-0.670396\pi$
$499$	−11372.3	−1.02023	−0.510113	+	0.860107i	$0.670396\pi$
$500$	0	0
$501$	−650.700	−0.0580262
$502$	0	0
$503$	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
$506$	0	0
$507$	−3573.99	−0.313070
$508$	0	0
$509$	16256.7	1.41565	0.707825	−	0.706388i	$-0.249676\pi$
$509$	16256.7	1.41565	0.707825	+	0.706388i	$0.249676\pi$
$510$	0	0
$511$	−7112.14	−0.615699
$512$	0	0
$513$	2830.61	0.243615
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−688.574	−0.0585754
$518$	0	0
$519$	−12377.7	−1.04686
$520$	0	0
$521$	19748.4	1.66064	0.830320	−	0.557286i	$-0.188157\pi$
$521$	19748.4	1.66064	0.830320	+	0.557286i	$0.188157\pi$
$522$	0	0
$523$	7843.44	0.655774	0.327887	−	0.944717i	$-0.393663\pi$
$523$	7843.44	0.655774	0.327887	+	0.944717i	$0.393663\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1222.95	−0.101086
$528$	0	0
$529$	−9705.60	−0.797699
$530$	0	0
$531$	664.116	0.0542753
$532$	0	0
$533$	−9638.38	−0.783273
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9639.42	−0.774622
$538$	0	0
$539$	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	$0.405221\pi$
$542$	0	0
$543$	−10183.3	−0.804798
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3354.90	0.262240	0.131120	−	0.991367i	$-0.458143\pi$
$547$	3354.90	0.262240	0.131120	+	0.991367i	$0.458143\pi$
$548$	0	0
$549$	−1236.15	−0.0960976
$550$	0	0
$551$	30766.2	2.37874
$552$	0	0
$553$	−17111.6	−1.31584
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20771.8	1.58012	0.790061	−	0.613028i	$-0.210049\pi$
$557$	20771.8	1.58012	0.790061	+	0.613028i	$0.210049\pi$
$558$	0	0
$559$	14398.1	1.08940
$560$	0	0
$561$	−101.982	−0.00767500
$562$	0	0
$563$	−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
$566$	0	0
$567$	−1798.96	−0.133244
$568$	0	0
$569$	11549.5	0.850931	0.425466	−	0.904975i	$-0.360110\pi$
$569$	11549.5	0.850931	0.425466	+	0.904975i	$0.360110\pi$
$570$	0	0
$571$	−1482.54	−0.108655	−0.0543277	−	0.998523i	$-0.517302\pi$
$571$	−1482.54	−0.108655	−0.0543277	+	0.998523i	$0.517302\pi$
$572$	0	0
$573$	−10402.5	−0.758412
$574$	0	0
$575$	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	$0.685620\pi$
$578$	0	0
$579$	−5376.43	−0.385901
$580$	0	0
$581$	3862.76	0.275825
$582$	0	0
$583$	830.006	0.0589628
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−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
$591$	−5034.57	−0.350414
$592$	0	0
$593$	−11764.8	−0.814707	−0.407353	−	0.913271i	$-0.633548\pi$
$593$	−11764.8	−0.814707	−0.407353	+	0.913271i	$0.633548\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9324.69	0.639253
$598$	0	0
$599$	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	$0.533895\pi$
$602$	0	0
$603$	−1565.16	−0.105702
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−22700.8	−1.51795	−0.758975	−	0.651120i	$-0.774300\pi$
$607$	−22700.8	−1.51795	−0.758975	+	0.651120i	$0.774300\pi$
$608$	0	0
$609$	−19553.1	−1.30103
$610$	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	$0.0985325\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5566.87	−0.363231	−0.181616	−	0.983370i	$-0.558133\pi$
$617$	−5566.87	−0.363231	−0.181616	+	0.983370i	$0.558133\pi$
$618$	0	0
$619$	−4150.32	−0.269492	−0.134746	−	0.990880i	$-0.543022\pi$
$619$	−4150.32	−0.269492	−0.134746	+	0.990880i	$0.543022\pi$
$620$	0	0
$621$	−1339.54	−0.0865600
$622$	0	0
$623$	−22631.9	−1.45542
$624$	0	0
$625$	0	0
$626$	0	0
$627$	563.175	0.0358709
$628$	0	0
$629$	−377.492	−0.0239294
$630$	0	0
$631$	4090.09	0.258041	0.129021	−	0.991642i	$-0.458817\pi$
$631$	4090.09	0.258041	0.129021	+	0.991642i	$0.458817\pi$
$632$	0	0
$633$	−13419.8	−0.842637
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8746.32	0.544022
$638$	0	0
$639$	5348.53	0.331118
$640$	0	0
$641$	3909.35	0.240890	0.120445	−	0.992720i	$-0.461568\pi$
$641$	3909.35	0.240890	0.120445	+	0.992720i	$0.461568\pi$
$642$	0	0
$643$	−30539.5	−1.87303	−0.936516	−	0.350624i	$-0.885969\pi$
$643$	−30539.5	−1.87303	−0.936516	+	0.350624i	$0.885969\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	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
$651$	−4292.10	−0.258403
$652$	0	0
$653$	12777.6	0.765737	0.382869	−	0.923803i	$-0.374936\pi$
$653$	12777.6	0.765737	0.382869	+	0.923803i	$0.374936\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2882.08	0.171143
$658$	0	0
$659$	−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	$0.540958\pi$
$662$	0	0
$663$	−3315.21	−0.194196
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−14559.6	−0.845200
$668$	0	0
$669$	−5260.25	−0.303995
$670$	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	$0.575488\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28057.1	1.59279	0.796397	−	0.604774i	$-0.206737\pi$
$677$	28057.1	1.59279	0.796397	+	0.604774i	$0.206737\pi$
$678$	0	0
$679$	−8536.73	−0.482488
$680$	0	0
$681$	−2810.70	−0.158159
$682$	0	0
$683$	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
$686$	0	0
$687$	7746.17	0.430182
$688$	0	0
$689$	26981.7	1.49190
$690$	0	0
$691$	−12964.8	−0.713757	−0.356879	−	0.934151i	$-0.616159\pi$
$691$	−12964.8	−0.713757	−0.356879	+	0.934151i	$0.616159\pi$
$692$	0	0
$693$	−357.918	−0.0196193
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3143.46	−0.170828
$698$	0	0
$699$	−6885.03	−0.372555
$700$	0	0
$701$	−16162.1	−0.870806	−0.435403	−	0.900236i	$-0.643394\pi$
$701$	−16162.1	−0.870806	−0.435403	+	0.900236i	$0.643394\pi$
$702$	0	0
$703$	2084.63	0.111839
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−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
$711$	6934.22	0.365757
$712$	0	0
$713$	−3195.97	−0.167868
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6882.02	−0.358457
$718$	0	0
$719$	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
$722$	0	0
$723$	−1146.86	−0.0589934
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2525.52	0.128840	0.0644199	−	0.997923i	$-0.479480\pi$
$727$	2525.52	0.128840	0.0644199	+	0.997923i	$0.479480\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	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	$0.567882\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−311.401	−0.0155639
$738$	0	0
$739$	19689.1	0.980074	0.490037	−	0.871702i	$-0.336983\pi$
$739$	19689.1	0.980074	0.490037	+	0.871702i	$0.336983\pi$
$740$	0	0
$741$	18307.6	0.907619
$742$	0	0
$743$	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
$746$	0	0
$747$	−1565.32	−0.0766696
$748$	0	0
$749$	−30243.3	−1.47539
$750$	0	0
$751$	−34691.1	−1.68562	−0.842808	−	0.538215i	$-0.819099\pi$
$751$	−34691.1	−1.68562	−0.842808	+	0.538215i	$0.819099\pi$
$752$	0	0
$753$	−6779.95	−0.328121
$754$	0	0
$755$	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	$0.449198\pi$
$758$	0	0
$759$	−266.512	−0.0127454
$760$	0	0
$761$	−29368.7	−1.39897	−0.699483	−	0.714649i	$-0.746586\pi$
$761$	−29368.7	−1.39897	−0.699483	+	0.714649i	$0.746586\pi$
$762$	0	0
$763$	−7131.84	−0.338388
$764$	0	0
$765$	0	0
$766$	0	0
$767$	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
$771$	278.353	0.0130021
$772$	0	0
$773$	−28047.5	−1.30504	−0.652522	−	0.757770i	$-0.726289\pi$
$773$	−28047.5	−1.30504	−0.652522	+	0.757770i	$0.726289\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1324.86	−0.0611699
$778$	0	0
$779$	17359.1	0.798402
$780$	0	0
$781$	1064.14	0.0487552
$782$	0	0
$783$	7923.57	0.361642
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22172.1	1.00426	0.502128	−	0.864793i	$-0.332551\pi$
$787$	22172.1	1.00426	0.502128	+	0.864793i	$0.332551\pi$
$788$	0	0
$789$	−1704.94	−0.0769295
$790$	0	0
$791$	35140.7	1.57960
$792$	0	0
$793$	−7995.06	−0.358024
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24170.3	−1.07422	−0.537112	−	0.843511i	$-0.680485\pi$
$797$	−24170.3	−1.07422	−0.537112	+	0.843511i	$0.680485\pi$
$798$	0	0
$799$	−7300.32	−0.323238
$800$	0	0
$801$	9171.22	0.404556
$802$	0	0
$803$	573.415	0.0251997
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−22747.2	−0.992244
$808$	0	0
$809$	15304.2	0.665102	0.332551	−	0.943085i	$-0.392091\pi$
$809$	15304.2	0.665102	0.332551	+	0.943085i	$0.392091\pi$
$810$	0	0
$811$	27002.2	1.16914	0.584572	−	0.811342i	$-0.301262\pi$
$811$	27002.2	1.16914	0.584572	+	0.811342i	$0.301262\pi$
$812$	0	0
$813$	23831.1	1.02803
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−25931.5	−1.11044
$818$	0	0
$819$	−11635.1	−0.496416
$820$	0	0
$821$	25061.4	1.06535	0.532673	−	0.846321i	$-0.321188\pi$
$821$	25061.4	1.06535	0.532673	+	0.846321i	$0.321188\pi$
$822$	0	0
$823$	24896.4	1.05448	0.527238	−	0.849718i	$-0.323228\pi$
$823$	24896.4	1.05448	0.527238	+	0.849718i	$0.323228\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	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
$831$	−20469.0	−0.854467
$832$	0	0
$833$	2852.52	0.118648
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1739.31	0.0718270
$838$	0	0
$839$	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
$842$	0	0
$843$	−9947.59	−0.406421
$844$	0	0
$845$	0	0
$846$	0	0
$847$	29489.5	1.19630
$848$	0	0
$849$	19808.3	0.800728
$850$	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	$0.295906\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22045.2	0.878706	0.439353	−	0.898314i	$-0.355208\pi$
$857$	22045.2	0.878706	0.439353	+	0.898314i	$0.355208\pi$
$858$	0	0
$859$	−33609.5	−1.33497	−0.667487	−	0.744622i	$-0.732630\pi$
$859$	−33609.5	−1.33497	−0.667487	+	0.744622i	$0.732630\pi$
$860$	0	0
$861$	−11032.4	−0.436681
$862$	0	0
$863$	−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
$866$	0	0
$867$	13657.8	0.534997
$868$	0	0
$869$	1379.62	0.0538556
$870$	0	0
$871$	−10123.0	−0.393805
$872$	0	0
$873$	3459.37	0.134115
$874$	0	0
$875$	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	$0.421775\pi$
$878$	0	0
$879$	−17442.4	−0.669304
$880$	0	0
$881$	−6579.45	−0.251609	−0.125804	−	0.992055i	$-0.540151\pi$
$881$	−6579.45	−0.251609	−0.125804	+	0.992055i	$0.540151\pi$
$882$	0	0
$883$	50442.1	1.92244	0.961219	−	0.275786i	$-0.0889382\pi$
$883$	50442.1	1.92244	0.961219	+	0.275786i	$0.0889382\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−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
$891$	145.041	0.00545348
$892$	0	0
$893$	40314.6	1.51072
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8663.74	−0.322490
$898$	0	0
$899$	18904.7	0.701342
$900$	0	0
$901$	8799.79	0.325376
$902$	0	0
$903$	16480.5	0.607348
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−43679.9	−1.59908	−0.799541	−	0.600612i	$-0.794924\pi$
$907$	−43679.9	−1.59908	−0.799541	+	0.600612i	$0.794924\pi$
$908$	0	0
$909$	310.415	0.0113265
$910$	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
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7143.58	−0.257254
$918$	0	0
$919$	−11451.9	−0.411059	−0.205530	−	0.978651i	$-0.565892\pi$
$919$	−11451.9	−0.411059	−0.205530	+	0.978651i	$0.565892\pi$
$920$	0	0
$921$	−24374.6	−0.872063
$922$	0	0
$923$	34592.7	1.23362
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15806.7	0.560044
$928$	0	0
$929$	−27701.8	−0.978326	−0.489163	−	0.872192i	$-0.662698\pi$
$929$	−27701.8	−0.978326	−0.489163	+	0.872192i	$0.662698\pi$
$930$	0	0
$931$	−15752.5	−0.554529
$932$	0	0
$933$	22008.8	0.772277
$934$	0	0
$935$	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	$0.532674\pi$
$938$	0	0
$939$	6607.97	0.229652
$940$	0	0
$941$	−28786.0	−0.997234	−0.498617	−	0.866823i	$-0.666159\pi$
$941$	−28786.0	−0.997234	−0.498617	+	0.866823i	$0.666159\pi$
$942$	0	0
$943$	−8214.90	−0.283684
$944$	0	0
$945$	0	0
$946$	0	0
$947$	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
$951$	−30026.6	−1.02385
$952$	0	0
$953$	31929.4	1.08530	0.542651	−	0.839958i	$-0.317420\pi$
$953$	31929.4	1.08530	0.542651	+	0.839958i	$0.317420\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1576.46	0.0532495
$958$	0	0
$959$	−7868.57	−0.264952
$960$	0	0
$961$	−25641.2	−0.860704
$962$	0	0
$963$	12255.6	0.410106
$964$	0	0
$965$	0	0
$966$	0	0
$967$	10897.1	0.362385	0.181193	−	0.983448i	$-0.442004\pi$
$967$	10897.1	0.362385	0.181193	+	0.983448i	$0.442004\pi$
$968$	0	0
$969$	5970.82	0.197947
$970$	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
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37607.6	−1.23150	−0.615749	−	0.787943i	$-0.711146\pi$
$977$	−37607.6	−1.23150	−0.615749	+	0.787943i	$0.711146\pi$
$978$	0	0
$979$	1824.69	0.0595684
$980$	0	0
$981$	2890.07	0.0940599
$982$	0	0
$983$	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
$986$	0	0
$987$	−25621.4	−0.826281
$988$	0	0
$989$	12271.6	0.394556
$990$	0	0
$991$	41686.5	1.33624	0.668120	−	0.744053i	$-0.267099\pi$
$991$	41686.5	1.33624	0.668120	+	0.744053i	$0.267099\pi$
$992$	0	0
$993$	−26087.8	−0.833708
$994$	0	0
$995$	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	$0.367456\pi$
$998$	0	0
$999$	536.878	0.0170031

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.b.a.4.2	✓	4	20.7	even	4
15.4.b.a.4.3	yes	4	20.3	even	4
45.4.b.b.19.2		4	60.23	odd	4
45.4.b.b.19.3		4	60.47	odd	4
75.4.a.c.1.2		2	20.19	odd	2
75.4.a.f.1.1		2	4.3	odd	2
225.4.a.i.1.2		2	12.11	even	2
225.4.a.o.1.1		2	60.59	even	2
240.4.f.f.49.1		4	5.3	odd	4
240.4.f.f.49.3		4	5.2	odd	4
720.4.f.j.289.3		4	15.2	even	4
720.4.f.j.289.4		4	15.8	even	4
960.4.f.p.769.2		4	40.37	odd	4
960.4.f.p.769.4		4	40.13	odd	4
960.4.f.q.769.2		4	40.3	even	4
960.4.f.q.769.4		4	40.27	even	4
1200.4.a.bn.1.1		2	1.1	even	1	trivial
1200.4.a.bt.1.2		2	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -22.2094 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -22.2094 q^{7} +9.00000 q^{9} +1.79063 q^{11} +58.2094 q^{13} +18.9844 q^{17} -104.837 q^{19} +66.6281 q^{21} +49.6125 q^{23} -27.0000 q^{27} -293.466 q^{29} -64.4187 q^{31} -5.37188 q^{33} -19.8844 q^{37} -174.628 q^{39} -165.581 q^{41} +247.350 q^{43} -384.544 q^{47} +150.256 q^{49} -56.9531 q^{51} +463.528 q^{53} +314.512 q^{57} +73.7906 q^{59} -137.350 q^{61} -199.884 q^{63} -173.906 q^{67} -148.837 q^{69} +594.281 q^{71} +320.231 q^{73} -39.7687 q^{77} +770.469 q^{79} +81.0000 q^{81} -173.925 q^{83} +880.397 q^{87} +1019.02 q^{89} -1292.79 q^{91} +193.256 q^{93} +384.375 q^{97} +16.1156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 6 q^{7} + 18 q^{9} + 42 q^{11} + 78 q^{13} + 102 q^{17} - 56 q^{19} + 18 q^{21} + 48 q^{23} - 54 q^{27} - 318 q^{29} - 52 q^{31} - 126 q^{33} + 306 q^{37} - 234 q^{39} - 408 q^{41} - 120 q^{43} - 180 q^{47} + 70 q^{49} - 306 q^{51} + 402 q^{53} + 168 q^{57} + 186 q^{59} + 340 q^{61} - 54 q^{63} - 732 q^{67} - 144 q^{69} + 36 q^{71} + 1332 q^{73} + 612 q^{77} - 380 q^{79} + 162 q^{81} + 984 q^{83} + 954 q^{87} + 1116 q^{89} - 972 q^{91} + 156 q^{93} - 768 q^{97} + 378 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 6 * q^7 + 18 * q^9 + 42 * q^11 + 78 * q^13 + 102 * q^17 - 56 * q^19 + 18 * q^21 + 48 * q^23 - 54 * q^27 - 318 * q^29 - 52 * q^31 - 126 * q^33 + 306 * q^37 - 234 * q^39 - 408 * q^41 - 120 * q^43 - 180 * q^47 + 70 * q^49 - 306 * q^51 + 402 * q^53 + 168 * q^57 + 186 * q^59 + 340 * q^61 - 54 * q^63 - 732 * q^67 - 144 * q^69 + 36 * q^71 + 1332 * q^73 + 612 * q^77 - 380 * q^79 + 162 * q^81 + 984 * q^83 + 954 * q^87 + 1116 * q^89 - 972 * q^91 + 156 * q^93 - 768 * q^97 + 378 * q^99

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