LMFDB - Embedded newform 4275.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.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:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} -2.23607 q^{7} -2.23607 q^{8} +4.00000 q^{11} +2.47214 q^{13} -3.61803 q^{14} -4.85410 q^{16} -3.23607 q^{17} -1.00000 q^{19} +6.47214 q^{22} -1.23607 q^{23} +4.00000 q^{26} -1.38197 q^{28} +1.47214 q^{29} -1.52786 q^{31} -3.38197 q^{32} -5.23607 q^{34} -7.23607 q^{37} -1.61803 q^{38} +5.00000 q^{41} -4.00000 q^{43} +2.47214 q^{44} -2.00000 q^{46} -8.47214 q^{47} -2.00000 q^{49} +1.52786 q^{52} -5.00000 q^{53} +5.00000 q^{56} +2.38197 q^{58} +1.29180 q^{59} -9.94427 q^{61} -2.47214 q^{62} +4.23607 q^{64} -6.94427 q^{67} -2.00000 q^{68} +13.1803 q^{71} +5.47214 q^{73} -11.7082 q^{74} -0.618034 q^{76} -8.94427 q^{77} -15.7082 q^{79} +8.09017 q^{82} +10.9443 q^{83} -6.47214 q^{86} -8.94427 q^{88} -7.94427 q^{89} -5.52786 q^{91} -0.763932 q^{92} -13.7082 q^{94} -11.7082 q^{97} -3.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 8 q^{11} - 4 q^{13} - 5 q^{14} - 3 q^{16} - 2 q^{17} - 2 q^{19} + 4 q^{22} + 2 q^{23} + 8 q^{26} - 5 q^{28} - 6 q^{29} - 12 q^{31} - 9 q^{32} - 6 q^{34} - 10 q^{37} - q^{38} + 10 q^{41}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 8 * q^11 - 4 * q^13 - 5 * q^14 - 3 * q^16 - 2 * q^17 - 2 * q^19 + 4 * q^22 + 2 * q^23 + 8 * q^26 - 5 * q^28 - 6 * q^29 - 12 * q^31 - 9 * q^32 - 6 * q^34 - 10 * q^37 - q^38 + 10 * q^41 - 8 * q^43 - 4 * q^44 - 4 * q^46 - 8 * q^47 - 4 * q^49 + 12 * q^52 - 10 * q^53 + 10 * q^56 + 7 * q^58 + 16 * q^59 - 2 * q^61 + 4 * q^62 + 4 * q^64 + 4 * q^67 - 4 * q^68 + 4 * q^71 + 2 * q^73 - 10 * q^74 + q^76 - 18 * q^79 + 5 * q^82 + 4 * q^83 - 4 * q^86 + 2 * q^89 - 20 * q^91 - 6 * q^92 - 14 * q^94 - 10 * q^97 - 2 * 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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.23607	−0.845154	−0.422577	−	0.906327i	$-0.638874\pi$
$7$	−2.23607	−0.845154	−0.422577	+	0.906327i	$0.638874\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	2.47214	0.685647	0.342824	−	0.939400i	$-0.388617\pi$
$13$	2.47214	0.685647	0.342824	+	0.939400i	$0.388617\pi$
$14$	−3.61803	−0.966960
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−3.23607	−0.784862	−0.392431	−	0.919781i	$-0.628366\pi$
$17$	−3.23607	−0.784862	−0.392431	+	0.919781i	$0.628366\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	6.47214	1.37986
$23$	−1.23607	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$24$	0	0
$25$	0	0
$26$	4.00000	0.784465
$27$	0	0
$28$	−1.38197	−0.261167
$29$	1.47214	0.273369	0.136684	−	0.990615i	$-0.456355\pi$
$29$	1.47214	0.273369	0.136684	+	0.990615i	$0.456355\pi$
$30$	0	0
$31$	−1.52786	−0.274412	−0.137206	−	0.990543i	$-0.543812\pi$
$31$	−1.52786	−0.274412	−0.137206	+	0.990543i	$0.543812\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	−5.23607	−0.897978
$35$	0	0
$36$	0	0
$37$	−7.23607	−1.18960	−0.594801	−	0.803873i	$-0.702769\pi$
$37$	−7.23607	−1.18960	−0.594801	+	0.803873i	$0.702769\pi$
$38$	−1.61803	−0.262480
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	2.47214	0.372689
$45$	0	0
$46$	−2.00000	−0.294884
$47$	−8.47214	−1.23579	−0.617894	−	0.786261i	$-0.712014\pi$
$47$	−8.47214	−1.23579	−0.617894	+	0.786261i	$0.712014\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	1.52786	0.211877
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	0	0
$55$	0	0
$56$	5.00000	0.668153
$57$	0	0
$58$	2.38197	0.312767
$59$	1.29180	0.168178	0.0840888	−	0.996458i	$-0.473202\pi$
$59$	1.29180	0.168178	0.0840888	+	0.996458i	$0.473202\pi$
$60$	0	0
$61$	−9.94427	−1.27323	−0.636617	−	0.771180i	$-0.719667\pi$
$61$	−9.94427	−1.27323	−0.636617	+	0.771180i	$0.719667\pi$
$62$	−2.47214	−0.313962
$63$	0	0
$64$	4.23607	0.529508
$65$	0	0
$66$	0	0
$67$	−6.94427	−0.848378	−0.424189	−	0.905574i	$-0.639441\pi$
$67$	−6.94427	−0.848378	−0.424189	+	0.905574i	$0.639441\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	13.1803	1.56422	0.782109	−	0.623141i	$-0.214144\pi$
$71$	13.1803	1.56422	0.782109	+	0.623141i	$0.214144\pi$
$72$	0	0
$73$	5.47214	0.640465	0.320233	−	0.947339i	$-0.396239\pi$
$73$	5.47214	0.640465	0.320233	+	0.947339i	$0.396239\pi$
$74$	−11.7082	−1.36105
$75$	0	0
$76$	−0.618034	−0.0708934
$77$	−8.94427	−1.01929
$78$	0	0
$79$	−15.7082	−1.76731	−0.883656	−	0.468138i	$-0.844925\pi$
$79$	−15.7082	−1.76731	−0.883656	+	0.468138i	$0.844925\pi$
$80$	0	0
$81$	0	0
$82$	8.09017	0.893410
$83$	10.9443	1.20129	0.600645	−	0.799516i	$-0.294911\pi$
$83$	10.9443	1.20129	0.600645	+	0.799516i	$0.294911\pi$
$84$	0	0
$85$	0	0
$86$	−6.47214	−0.697908
$87$	0	0
$88$	−8.94427	−0.953463
$89$	−7.94427	−0.842091	−0.421046	−	0.907039i	$-0.638337\pi$
$89$	−7.94427	−0.842091	−0.421046	+	0.907039i	$0.638337\pi$
$90$	0	0
$91$	−5.52786	−0.579478
$92$	−0.763932	−0.0796454
$93$	0	0
$94$	−13.7082	−1.41389
$95$	0	0
$96$	0	0
$97$	−11.7082	−1.18879	−0.594394	−	0.804174i	$-0.702608\pi$
$97$	−11.7082	−1.18879	−0.594394	+	0.804174i	$0.702608\pi$
$98$	−3.23607	−0.326892
$99$	0	0
$100$	0	0
$101$	−8.18034	−0.813974	−0.406987	−	0.913434i	$-0.633421\pi$
$101$	−8.18034	−0.813974	−0.406987	+	0.913434i	$0.633421\pi$
$102$	0	0
$103$	0.944272	0.0930419	0.0465209	−	0.998917i	$-0.485187\pi$
$103$	0.944272	0.0930419	0.0465209	+	0.998917i	$0.485187\pi$
$104$	−5.52786	−0.542052
$105$	0	0
$106$	−8.09017	−0.785787
$107$	−11.7639	−1.13726	−0.568631	−	0.822593i	$-0.692527\pi$
$107$	−11.7639	−1.13726	−0.568631	+	0.822593i	$0.692527\pi$
$108$	0	0
$109$	5.70820	0.546747	0.273373	−	0.961908i	$-0.411861\pi$
$109$	5.70820	0.546747	0.273373	+	0.961908i	$0.411861\pi$
$110$	0	0
$111$	0	0
$112$	10.8541	1.02562
$113$	−3.52786	−0.331874	−0.165937	−	0.986136i	$-0.553065\pi$
$113$	−3.52786	−0.331874	−0.165937	+	0.986136i	$0.553065\pi$
$114$	0	0
$115$	0	0
$116$	0.909830	0.0844756
$117$	0	0
$118$	2.09017	0.192416
$119$	7.23607	0.663329
$120$	0	0
$121$	5.00000	0.454545
$122$	−16.0902	−1.45674
$123$	0	0
$124$	−0.944272	−0.0847981
$125$	0	0
$126$	0	0
$127$	−15.2361	−1.35198	−0.675991	−	0.736910i	$-0.736284\pi$
$127$	−15.2361	−1.35198	−0.675991	+	0.736910i	$0.736284\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	0	0
$131$	−10.9443	−0.956205	−0.478103	−	0.878304i	$-0.658675\pi$
$131$	−10.9443	−0.956205	−0.478103	+	0.878304i	$0.658675\pi$
$132$	0	0
$133$	2.23607	0.193892
$134$	−11.2361	−0.970648
$135$	0	0
$136$	7.23607	0.620488
$137$	17.7082	1.51291	0.756457	−	0.654043i	$-0.226929\pi$
$137$	17.7082	1.51291	0.756457	+	0.654043i	$0.226929\pi$
$138$	0	0
$139$	−20.2361	−1.71640	−0.858200	−	0.513315i	$-0.828417\pi$
$139$	−20.2361	−1.71640	−0.858200	+	0.513315i	$0.828417\pi$
$140$	0	0
$141$	0	0
$142$	21.3262	1.78966
$143$	9.88854	0.826922
$144$	0	0
$145$	0	0
$146$	8.85410	0.732771
$147$	0	0
$148$	−4.47214	−0.367607
$149$	−7.41641	−0.607576	−0.303788	−	0.952740i	$-0.598251\pi$
$149$	−7.41641	−0.607576	−0.303788	+	0.952740i	$0.598251\pi$
$150$	0	0
$151$	−1.70820	−0.139012	−0.0695058	−	0.997582i	$-0.522142\pi$
$151$	−1.70820	−0.139012	−0.0695058	+	0.997582i	$0.522142\pi$
$152$	2.23607	0.181369
$153$	0	0
$154$	−14.4721	−1.16620
$155$	0	0
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	−25.4164	−2.02202
$159$	0	0
$160$	0	0
$161$	2.76393	0.217828
$162$	0	0
$163$	−11.1803	−0.875712	−0.437856	−	0.899045i	$-0.644262\pi$
$163$	−11.1803	−0.875712	−0.437856	+	0.899045i	$0.644262\pi$
$164$	3.09017	0.241302
$165$	0	0
$166$	17.7082	1.37442
$167$	24.5967	1.90335	0.951677	−	0.307102i	$-0.0993591\pi$
$167$	24.5967	1.90335	0.951677	+	0.307102i	$0.0993591\pi$
$168$	0	0
$169$	−6.88854	−0.529888
$170$	0	0
$171$	0	0
$172$	−2.47214	−0.188499
$173$	−4.52786	−0.344247	−0.172124	−	0.985075i	$-0.555063\pi$
$173$	−4.52786	−0.344247	−0.172124	+	0.985075i	$0.555063\pi$
$174$	0	0
$175$	0	0
$176$	−19.4164	−1.46357
$177$	0	0
$178$	−12.8541	−0.963456
$179$	18.1246	1.35470	0.677349	−	0.735662i	$-0.263129\pi$
$179$	18.1246	1.35470	0.677349	+	0.735662i	$0.263129\pi$
$180$	0	0
$181$	1.52786	0.113565	0.0567826	−	0.998387i	$-0.481916\pi$
$181$	1.52786	0.113565	0.0567826	+	0.998387i	$0.481916\pi$
$182$	−8.94427	−0.662994
$183$	0	0
$184$	2.76393	0.203760
$185$	0	0
$186$	0	0
$187$	−12.9443	−0.946579
$188$	−5.23607	−0.381880
$189$	0	0
$190$	0	0
$191$	19.4164	1.40492	0.702461	−	0.711722i	$-0.252085\pi$
$191$	19.4164	1.40492	0.702461	+	0.711722i	$0.252085\pi$
$192$	0	0
$193$	−1.23607	−0.0889741	−0.0444871	−	0.999010i	$-0.514165\pi$
$193$	−1.23607	−0.0889741	−0.0444871	+	0.999010i	$0.514165\pi$
$194$	−18.9443	−1.36012
$195$	0	0
$196$	−1.23607	−0.0882906
$197$	8.65248	0.616463	0.308232	−	0.951311i	$-0.400263\pi$
$197$	8.65248	0.616463	0.308232	+	0.951311i	$0.400263\pi$
$198$	0	0
$199$	6.23607	0.442063	0.221032	−	0.975267i	$-0.429058\pi$
$199$	6.23607	0.442063	0.221032	+	0.975267i	$0.429058\pi$
$200$	0	0
$201$	0	0
$202$	−13.2361	−0.931286
$203$	−3.29180	−0.231039
$204$	0	0
$205$	0	0
$206$	1.52786	0.106451
$207$	0	0
$208$	−12.0000	−0.832050
$209$	−4.00000	−0.276686
$210$	0	0
$211$	28.1803	1.94001	0.970007	−	0.243076i	$-0.0781564\pi$
$211$	28.1803	1.94001	0.970007	+	0.243076i	$0.0781564\pi$
$212$	−3.09017	−0.212234
$213$	0	0
$214$	−19.0344	−1.30117
$215$	0	0
$216$	0	0
$217$	3.41641	0.231921
$218$	9.23607	0.625545
$219$	0	0
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−4.76393	−0.319016	−0.159508	−	0.987197i	$-0.550991\pi$
$223$	−4.76393	−0.319016	−0.159508	+	0.987197i	$0.550991\pi$
$224$	7.56231	0.505278
$225$	0	0
$226$	−5.70820	−0.379704
$227$	−12.1246	−0.804739	−0.402369	−	0.915477i	$-0.631813\pi$
$227$	−12.1246	−0.804739	−0.402369	+	0.915477i	$0.631813\pi$
$228$	0	0
$229$	4.47214	0.295527	0.147764	−	0.989023i	$-0.452793\pi$
$229$	4.47214	0.295527	0.147764	+	0.989023i	$0.452793\pi$
$230$	0	0
$231$	0	0
$232$	−3.29180	−0.216117
$233$	−14.9443	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$233$	−14.9443	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$234$	0	0
$235$	0	0
$236$	0.798374	0.0519697
$237$	0	0
$238$	11.7082	0.758930
$239$	−13.7082	−0.886710	−0.443355	−	0.896346i	$-0.646212\pi$
$239$	−13.7082	−0.886710	−0.443355	+	0.896346i	$0.646212\pi$
$240$	0	0
$241$	6.18034	0.398111	0.199055	−	0.979988i	$-0.436213\pi$
$241$	6.18034	0.398111	0.199055	+	0.979988i	$0.436213\pi$
$242$	8.09017	0.520056
$243$	0	0
$244$	−6.14590	−0.393451
$245$	0	0
$246$	0	0
$247$	−2.47214	−0.157298
$248$	3.41641	0.216942
$249$	0	0
$250$	0	0
$251$	9.70820	0.612776	0.306388	−	0.951907i	$-0.400879\pi$
$251$	9.70820	0.612776	0.306388	+	0.951907i	$0.400879\pi$
$252$	0	0
$253$	−4.94427	−0.310844
$254$	−24.6525	−1.54683
$255$	0	0
$256$	13.5623	0.847644
$257$	−13.0000	−0.810918	−0.405459	−	0.914113i	$-0.632888\pi$
$257$	−13.0000	−0.810918	−0.405459	+	0.914113i	$0.632888\pi$
$258$	0	0
$259$	16.1803	1.00540
$260$	0	0
$261$	0	0
$262$	−17.7082	−1.09402
$263$	−23.7082	−1.46191	−0.730955	−	0.682425i	$-0.760925\pi$
$263$	−23.7082	−1.46191	−0.730955	+	0.682425i	$0.760925\pi$
$264$	0	0
$265$	0	0
$266$	3.61803	0.221836
$267$	0	0
$268$	−4.29180	−0.262163
$269$	−11.5279	−0.702866	−0.351433	−	0.936213i	$-0.614306\pi$
$269$	−11.5279	−0.702866	−0.351433	+	0.936213i	$0.614306\pi$
$270$	0	0
$271$	−19.1803	−1.16512	−0.582561	−	0.812787i	$-0.697949\pi$
$271$	−19.1803	−1.16512	−0.582561	+	0.812787i	$0.697949\pi$
$272$	15.7082	0.952450
$273$	0	0
$274$	28.6525	1.73096
$275$	0	0
$276$	0	0
$277$	−32.4164	−1.94771	−0.973857	−	0.227164i	$-0.927055\pi$
$277$	−32.4164	−1.94771	−0.973857	+	0.227164i	$0.927055\pi$
$278$	−32.7426	−1.96377
$279$	0	0
$280$	0	0
$281$	27.8885	1.66369	0.831846	−	0.555007i	$-0.187285\pi$
$281$	27.8885	1.66369	0.831846	+	0.555007i	$0.187285\pi$
$282$	0	0
$283$	25.8885	1.53891	0.769457	−	0.638699i	$-0.220527\pi$
$283$	25.8885	1.53891	0.769457	+	0.638699i	$0.220527\pi$
$284$	8.14590	0.483370
$285$	0	0
$286$	16.0000	0.946100
$287$	−11.1803	−0.659955
$288$	0	0
$289$	−6.52786	−0.383992
$290$	0	0
$291$	0	0
$292$	3.38197	0.197915
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	0	0
$296$	16.1803	0.940463
$297$	0	0
$298$	−12.0000	−0.695141
$299$	−3.05573	−0.176717
$300$	0	0
$301$	8.94427	0.515539
$302$	−2.76393	−0.159046
$303$	0	0
$304$	4.85410	0.278402
$305$	0	0
$306$	0	0
$307$	20.9443	1.19535	0.597676	−	0.801737i	$-0.296091\pi$
$307$	20.9443	1.19535	0.597676	+	0.801737i	$0.296091\pi$
$308$	−5.52786	−0.314979
$309$	0	0
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	13.0557	0.737953	0.368977	−	0.929439i	$-0.379708\pi$
$313$	13.0557	0.737953	0.368977	+	0.929439i	$0.379708\pi$
$314$	−11.3262	−0.639177
$315$	0	0
$316$	−9.70820	−0.546129
$317$	9.94427	0.558526	0.279263	−	0.960215i	$-0.409910\pi$
$317$	9.94427	0.558526	0.279263	+	0.960215i	$0.409910\pi$
$318$	0	0
$319$	5.88854	0.329695
$320$	0	0
$321$	0	0
$322$	4.47214	0.249222
$323$	3.23607	0.180060
$324$	0	0
$325$	0	0
$326$	−18.0902	−1.00192
$327$	0	0
$328$	−11.1803	−0.617331
$329$	18.9443	1.04443
$330$	0	0
$331$	18.3607	1.00919	0.504597	−	0.863355i	$-0.331641\pi$
$331$	18.3607	1.00919	0.504597	+	0.863355i	$0.331641\pi$
$332$	6.76393	0.371219
$333$	0	0
$334$	39.7984	2.17767
$335$	0	0
$336$	0	0
$337$	−0.472136	−0.0257189	−0.0128594	−	0.999917i	$-0.504093\pi$
$337$	−0.472136	−0.0257189	−0.0128594	+	0.999917i	$0.504093\pi$
$338$	−11.1459	−0.606257
$339$	0	0
$340$	0	0
$341$	−6.11146	−0.330954
$342$	0	0
$343$	20.1246	1.08663
$344$	8.94427	0.482243
$345$	0	0
$346$	−7.32624	−0.393861
$347$	11.7082	0.628529	0.314265	−	0.949335i	$-0.398242\pi$
$347$	11.7082	0.628529	0.314265	+	0.949335i	$0.398242\pi$
$348$	0	0
$349$	35.8328	1.91809	0.959043	−	0.283259i	$-0.0914157\pi$
$349$	35.8328	1.91809	0.959043	+	0.283259i	$0.0914157\pi$
$350$	0	0
$351$	0	0
$352$	−13.5279	−0.721038
$353$	−17.5279	−0.932914	−0.466457	−	0.884544i	$-0.654470\pi$
$353$	−17.5279	−0.932914	−0.466457	+	0.884544i	$0.654470\pi$
$354$	0	0
$355$	0	0
$356$	−4.90983	−0.260220
$357$	0	0
$358$	29.3262	1.54994
$359$	20.7639	1.09588	0.547939	−	0.836518i	$-0.315413\pi$
$359$	20.7639	1.09588	0.547939	+	0.836518i	$0.315413\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.47214	0.129933
$363$	0	0
$364$	−3.41641	−0.179068
$365$	0	0
$366$	0	0
$367$	29.3050	1.52971	0.764853	−	0.644205i	$-0.222812\pi$
$367$	29.3050	1.52971	0.764853	+	0.644205i	$0.222812\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	0	0
$371$	11.1803	0.580454
$372$	0	0
$373$	10.1803	0.527118	0.263559	−	0.964643i	$-0.415104\pi$
$373$	10.1803	0.527118	0.263559	+	0.964643i	$0.415104\pi$
$374$	−20.9443	−1.08300
$375$	0	0
$376$	18.9443	0.976976
$377$	3.63932	0.187435
$378$	0	0
$379$	11.2361	0.577158	0.288579	−	0.957456i	$-0.406817\pi$
$379$	11.2361	0.577158	0.288579	+	0.957456i	$0.406817\pi$
$380$	0	0
$381$	0	0
$382$	31.4164	1.60740
$383$	−24.1246	−1.23271	−0.616355	−	0.787468i	$-0.711391\pi$
$383$	−24.1246	−1.23271	−0.616355	+	0.787468i	$0.711391\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−7.23607	−0.367356
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	4.47214	0.225877
$393$	0	0
$394$	14.0000	0.705310
$395$	0	0
$396$	0	0
$397$	22.3607	1.12225	0.561125	−	0.827731i	$-0.310369\pi$
$397$	22.3607	1.12225	0.561125	+	0.827731i	$0.310369\pi$
$398$	10.0902	0.505775
$399$	0	0
$400$	0	0
$401$	−32.8328	−1.63959	−0.819796	−	0.572655i	$-0.805913\pi$
$401$	−32.8328	−1.63959	−0.819796	+	0.572655i	$0.805913\pi$
$402$	0	0
$403$	−3.77709	−0.188150
$404$	−5.05573	−0.251532
$405$	0	0
$406$	−5.32624	−0.264337
$407$	−28.9443	−1.43471
$408$	0	0
$409$	13.2361	0.654481	0.327241	−	0.944941i	$-0.393881\pi$
$409$	13.2361	0.654481	0.327241	+	0.944941i	$0.393881\pi$
$410$	0	0
$411$	0	0
$412$	0.583592	0.0287515
$413$	−2.88854	−0.142136
$414$	0	0
$415$	0	0
$416$	−8.36068	−0.409916
$417$	0	0
$418$	−6.47214	−0.316563
$419$	23.3050	1.13852	0.569261	−	0.822157i	$-0.307230\pi$
$419$	23.3050	1.13852	0.569261	+	0.822157i	$0.307230\pi$
$420$	0	0
$421$	20.4721	0.997751	0.498875	−	0.866674i	$-0.333747\pi$
$421$	20.4721	0.997751	0.498875	+	0.866674i	$0.333747\pi$
$422$	45.5967	2.21961
$423$	0	0
$424$	11.1803	0.542965
$425$	0	0
$426$	0	0
$427$	22.2361	1.07608
$428$	−7.27051	−0.351433
$429$	0	0
$430$	0	0
$431$	−17.1803	−0.827548	−0.413774	−	0.910380i	$-0.635790\pi$
$431$	−17.1803	−0.827548	−0.413774	+	0.910380i	$0.635790\pi$
$432$	0	0
$433$	28.4721	1.36828	0.684142	−	0.729349i	$-0.260177\pi$
$433$	28.4721	1.36828	0.684142	+	0.729349i	$0.260177\pi$
$434$	5.52786	0.265346
$435$	0	0
$436$	3.52786	0.168954
$437$	1.23607	0.0591292
$438$	0	0
$439$	−21.2361	−1.01354	−0.506771	−	0.862081i	$-0.669161\pi$
$439$	−21.2361	−1.01354	−0.506771	+	0.862081i	$0.669161\pi$
$440$	0	0
$441$	0	0
$442$	−12.9443	−0.615696
$443$	−13.4164	−0.637433	−0.318716	−	0.947850i	$-0.603252\pi$
$443$	−13.4164	−0.637433	−0.318716	+	0.947850i	$0.603252\pi$
$444$	0	0
$445$	0	0
$446$	−7.70820	−0.364994
$447$	0	0
$448$	−9.47214	−0.447516
$449$	16.5279	0.779998	0.389999	−	0.920815i	$-0.372475\pi$
$449$	16.5279	0.779998	0.389999	+	0.920815i	$0.372475\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	−2.18034	−0.102555
$453$	0	0
$454$	−19.6180	−0.920720
$455$	0	0
$456$	0	0
$457$	26.8885	1.25779	0.628897	−	0.777489i	$-0.283507\pi$
$457$	26.8885	1.25779	0.628897	+	0.777489i	$0.283507\pi$
$458$	7.23607	0.338119
$459$	0	0
$460$	0	0
$461$	24.1803	1.12619	0.563095	−	0.826392i	$-0.309610\pi$
$461$	24.1803	1.12619	0.563095	+	0.826392i	$0.309610\pi$
$462$	0	0
$463$	−35.4164	−1.64594	−0.822970	−	0.568085i	$-0.807685\pi$
$463$	−35.4164	−1.64594	−0.822970	+	0.568085i	$0.807685\pi$
$464$	−7.14590	−0.331740
$465$	0	0
$466$	−24.1803	−1.12013
$467$	−3.81966	−0.176753	−0.0883764	−	0.996087i	$-0.528168\pi$
$467$	−3.81966	−0.176753	−0.0883764	+	0.996087i	$0.528168\pi$
$468$	0	0
$469$	15.5279	0.717010
$470$	0	0
$471$	0	0
$472$	−2.88854	−0.132956
$473$	−16.0000	−0.735681
$474$	0	0
$475$	0	0
$476$	4.47214	0.204980
$477$	0	0
$478$	−22.1803	−1.01451
$479$	43.2361	1.97551	0.987753	−	0.156025i	$-0.0498679\pi$
$479$	43.2361	1.97551	0.987753	+	0.156025i	$0.0498679\pi$
$480$	0	0
$481$	−17.8885	−0.815647
$482$	10.0000	0.455488
$483$	0	0
$484$	3.09017	0.140462
$485$	0	0
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	22.2361	1.00658
$489$	0	0
$490$	0	0
$491$	21.7082	0.979678	0.489839	−	0.871813i	$-0.337056\pi$
$491$	21.7082	0.979678	0.489839	+	0.871813i	$0.337056\pi$
$492$	0	0
$493$	−4.76393	−0.214557
$494$	−4.00000	−0.179969
$495$	0	0
$496$	7.41641	0.333007
$497$	−29.4721	−1.32201
$498$	0	0
$499$	−8.81966	−0.394822	−0.197411	−	0.980321i	$-0.563253\pi$
$499$	−8.81966	−0.394822	−0.197411	+	0.980321i	$0.563253\pi$
$500$	0	0
$501$	0	0
$502$	15.7082	0.701091
$503$	−18.4721	−0.823632	−0.411816	−	0.911267i	$-0.635105\pi$
$503$	−18.4721	−0.823632	−0.411816	+	0.911267i	$0.635105\pi$
$504$	0	0
$505$	0	0
$506$	−8.00000	−0.355643
$507$	0	0
$508$	−9.41641	−0.417786
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	−12.2361	−0.541292
$512$	−5.29180	−0.233867
$513$	0	0
$514$	−21.0344	−0.927789
$515$	0	0
$516$	0	0
$517$	−33.8885	−1.49042
$518$	26.1803	1.15030
$519$	0	0
$520$	0	0
$521$	−17.8328	−0.781270	−0.390635	−	0.920546i	$-0.627745\pi$
$521$	−17.8328	−0.781270	−0.390635	+	0.920546i	$0.627745\pi$
$522$	0	0
$523$	−0.875388	−0.0382781	−0.0191390	−	0.999817i	$-0.506093\pi$
$523$	−0.875388	−0.0382781	−0.0191390	+	0.999817i	$0.506093\pi$
$524$	−6.76393	−0.295484
$525$	0	0
$526$	−38.3607	−1.67261
$527$	4.94427	0.215376
$528$	0	0
$529$	−21.4721	−0.933571
$530$	0	0
$531$	0	0
$532$	1.38197	0.0599158
$533$	12.3607	0.535400
$534$	0	0
$535$	0	0
$536$	15.5279	0.670702
$537$	0	0
$538$	−18.6525	−0.804165
$539$	−8.00000	−0.344584
$540$	0	0
$541$	−28.8328	−1.23962	−0.619810	−	0.784752i	$-0.712790\pi$
$541$	−28.8328	−1.23962	−0.619810	+	0.784752i	$0.712790\pi$
$542$	−31.0344	−1.33304
$543$	0	0
$544$	10.9443	0.469232
$545$	0	0
$546$	0	0
$547$	−5.70820	−0.244065	−0.122033	−	0.992526i	$-0.538941\pi$
$547$	−5.70820	−0.244065	−0.122033	+	0.992526i	$0.538941\pi$
$548$	10.9443	0.467516
$549$	0	0
$550$	0	0
$551$	−1.47214	−0.0627151
$552$	0	0
$553$	35.1246	1.49365
$554$	−52.4508	−2.22842
$555$	0	0
$556$	−12.5066	−0.530397
$557$	1.88854	0.0800202	0.0400101	−	0.999199i	$-0.487261\pi$
$557$	1.88854	0.0800202	0.0400101	+	0.999199i	$0.487261\pi$
$558$	0	0
$559$	−9.88854	−0.418241
$560$	0	0
$561$	0	0
$562$	45.1246	1.90347
$563$	−7.65248	−0.322513	−0.161257	−	0.986912i	$-0.551555\pi$
$563$	−7.65248	−0.322513	−0.161257	+	0.986912i	$0.551555\pi$
$564$	0	0
$565$	0	0
$566$	41.8885	1.76071
$567$	0	0
$568$	−29.4721	−1.23662
$569$	−14.8885	−0.624160	−0.312080	−	0.950056i	$-0.601026\pi$
$569$	−14.8885	−0.624160	−0.312080	+	0.950056i	$0.601026\pi$
$570$	0	0
$571$	15.6525	0.655036	0.327518	−	0.944845i	$-0.393788\pi$
$571$	15.6525	0.655036	0.327518	+	0.944845i	$0.393788\pi$
$572$	6.11146	0.255533
$573$	0	0
$574$	−18.0902	−0.755069
$575$	0	0
$576$	0	0
$577$	−25.4164	−1.05810	−0.529049	−	0.848591i	$-0.677451\pi$
$577$	−25.4164	−1.05810	−0.529049	+	0.848591i	$0.677451\pi$
$578$	−10.5623	−0.439334
$579$	0	0
$580$	0	0
$581$	−24.4721	−1.01528
$582$	0	0
$583$	−20.0000	−0.828315
$584$	−12.2361	−0.506332
$585$	0	0
$586$	3.23607	0.133681
$587$	−18.6525	−0.769870	−0.384935	−	0.922944i	$-0.625776\pi$
$587$	−18.6525	−0.769870	−0.384935	+	0.922944i	$0.625776\pi$
$588$	0	0
$589$	1.52786	0.0629545
$590$	0	0
$591$	0	0
$592$	35.1246	1.44361
$593$	21.8885	0.898855	0.449427	−	0.893317i	$-0.351628\pi$
$593$	21.8885	0.898855	0.449427	+	0.893317i	$0.351628\pi$
$594$	0	0
$595$	0	0
$596$	−4.58359	−0.187751
$597$	0	0
$598$	−4.94427	−0.202186
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	35.7771	1.45938	0.729689	−	0.683779i	$-0.239665\pi$
$601$	35.7771	1.45938	0.729689	+	0.683779i	$0.239665\pi$
$602$	14.4721	0.589840
$603$	0	0
$604$	−1.05573	−0.0429570
$605$	0	0
$606$	0	0
$607$	−21.1246	−0.857422	−0.428711	−	0.903442i	$-0.641032\pi$
$607$	−21.1246	−0.857422	−0.428711	+	0.903442i	$0.641032\pi$
$608$	3.38197	0.137157
$609$	0	0
$610$	0	0
$611$	−20.9443	−0.847315
$612$	0	0
$613$	−23.0000	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$613$	−23.0000	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$614$	33.8885	1.36763
$615$	0	0
$616$	20.0000	0.805823
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	17.2918	0.695016	0.347508	−	0.937677i	$-0.387028\pi$
$619$	17.2918	0.695016	0.347508	+	0.937677i	$0.387028\pi$
$620$	0	0
$621$	0	0
$622$	−6.47214	−0.259509
$623$	17.7639	0.711697
$624$	0	0
$625$	0	0
$626$	21.1246	0.844309
$627$	0	0
$628$	−4.32624	−0.172636
$629$	23.4164	0.933673
$630$	0	0
$631$	−16.5836	−0.660182	−0.330091	−	0.943949i	$-0.607080\pi$
$631$	−16.5836	−0.660182	−0.330091	+	0.943949i	$0.607080\pi$
$632$	35.1246	1.39718
$633$	0	0
$634$	16.0902	0.639022
$635$	0	0
$636$	0	0
$637$	−4.94427	−0.195899
$638$	9.52786	0.377212
$639$	0	0
$640$	0	0
$641$	10.9443	0.432273	0.216136	−	0.976363i	$-0.430654\pi$
$641$	10.9443	0.432273	0.216136	+	0.976363i	$0.430654\pi$
$642$	0	0
$643$	−5.18034	−0.204293	−0.102146	−	0.994769i	$-0.532571\pi$
$643$	−5.18034	−0.204293	−0.102146	+	0.994769i	$0.532571\pi$
$644$	1.70820	0.0673127
$645$	0	0
$646$	5.23607	0.206010
$647$	−45.3050	−1.78112	−0.890561	−	0.454864i	$-0.849688\pi$
$647$	−45.3050	−1.78112	−0.890561	+	0.454864i	$0.849688\pi$
$648$	0	0
$649$	5.16718	0.202830
$650$	0	0
$651$	0	0
$652$	−6.90983	−0.270610
$653$	−9.88854	−0.386969	−0.193484	−	0.981103i	$-0.561979\pi$
$653$	−9.88854	−0.386969	−0.193484	+	0.981103i	$0.561979\pi$
$654$	0	0
$655$	0	0
$656$	−24.2705	−0.947604
$657$	0	0
$658$	30.6525	1.19496
$659$	−7.05573	−0.274852	−0.137426	−	0.990512i	$-0.543883\pi$
$659$	−7.05573	−0.274852	−0.137426	+	0.990512i	$0.543883\pi$
$660$	0	0
$661$	17.1246	0.666070	0.333035	−	0.942914i	$-0.391927\pi$
$661$	17.1246	0.666070	0.333035	+	0.942914i	$0.391927\pi$
$662$	29.7082	1.15464
$663$	0	0
$664$	−24.4721	−0.949703
$665$	0	0
$666$	0	0
$667$	−1.81966	−0.0704575
$668$	15.2016	0.588169
$669$	0	0
$670$	0	0
$671$	−39.7771	−1.53558
$672$	0	0
$673$	−46.2492	−1.78278	−0.891388	−	0.453240i	$-0.850268\pi$
$673$	−46.2492	−1.78278	−0.891388	+	0.453240i	$0.850268\pi$
$674$	−0.763932	−0.0294256
$675$	0	0
$676$	−4.25735	−0.163744
$677$	−47.8328	−1.83836	−0.919182	−	0.393833i	$-0.871149\pi$
$677$	−47.8328	−1.83836	−0.919182	+	0.393833i	$0.871149\pi$
$678$	0	0
$679$	26.1803	1.00471
$680$	0	0
$681$	0	0
$682$	−9.88854	−0.378652
$683$	39.6525	1.51726	0.758630	−	0.651522i	$-0.225869\pi$
$683$	39.6525	1.51726	0.758630	+	0.651522i	$0.225869\pi$
$684$	0	0
$685$	0	0
$686$	32.5623	1.24323
$687$	0	0
$688$	19.4164	0.740244
$689$	−12.3607	−0.470904
$690$	0	0
$691$	−24.9443	−0.948925	−0.474462	−	0.880276i	$-0.657358\pi$
$691$	−24.9443	−0.948925	−0.474462	+	0.880276i	$0.657358\pi$
$692$	−2.79837	−0.106378
$693$	0	0
$694$	18.9443	0.719115
$695$	0	0
$696$	0	0
$697$	−16.1803	−0.612874
$698$	57.9787	2.19453
$699$	0	0
$700$	0	0
$701$	−48.6525	−1.83758	−0.918789	−	0.394748i	$-0.870832\pi$
$701$	−48.6525	−1.83758	−0.918789	+	0.394748i	$0.870832\pi$
$702$	0	0
$703$	7.23607	0.272913
$704$	16.9443	0.638611
$705$	0	0
$706$	−28.3607	−1.06737
$707$	18.2918	0.687934
$708$	0	0
$709$	−13.5836	−0.510143	−0.255071	−	0.966922i	$-0.582099\pi$
$709$	−13.5836	−0.510143	−0.255071	+	0.966922i	$0.582099\pi$
$710$	0	0
$711$	0	0
$712$	17.7639	0.665731
$713$	1.88854	0.0707265
$714$	0	0
$715$	0	0
$716$	11.2016	0.418624
$717$	0	0
$718$	33.5967	1.25382
$719$	46.4721	1.73312	0.866559	−	0.499074i	$-0.166327\pi$
$719$	46.4721	1.73312	0.866559	+	0.499074i	$0.166327\pi$
$720$	0	0
$721$	−2.11146	−0.0786347
$722$	1.61803	0.0602170
$723$	0	0
$724$	0.944272	0.0350936
$725$	0	0
$726$	0	0
$727$	20.2361	0.750514	0.375257	−	0.926921i	$-0.377554\pi$
$727$	20.2361	0.750514	0.375257	+	0.926921i	$0.377554\pi$
$728$	12.3607	0.458117
$729$	0	0
$730$	0	0
$731$	12.9443	0.478761
$732$	0	0
$733$	−14.4164	−0.532482	−0.266241	−	0.963906i	$-0.585782\pi$
$733$	−14.4164	−0.532482	−0.266241	+	0.963906i	$0.585782\pi$
$734$	47.4164	1.75017
$735$	0	0
$736$	4.18034	0.154089
$737$	−27.7771	−1.02318
$738$	0	0
$739$	44.5967	1.64052	0.820259	−	0.571992i	$-0.193829\pi$
$739$	44.5967	1.64052	0.820259	+	0.571992i	$0.193829\pi$
$740$	0	0
$741$	0	0
$742$	18.0902	0.664111
$743$	1.87539	0.0688013	0.0344007	−	0.999408i	$-0.489048\pi$
$743$	1.87539	0.0688013	0.0344007	+	0.999408i	$0.489048\pi$
$744$	0	0
$745$	0	0
$746$	16.4721	0.603088
$747$	0	0
$748$	−8.00000	−0.292509
$749$	26.3050	0.961162
$750$	0	0
$751$	−11.5279	−0.420658	−0.210329	−	0.977631i	$-0.567453\pi$
$751$	−11.5279	−0.420658	−0.210329	+	0.977631i	$0.567453\pi$
$752$	41.1246	1.49966
$753$	0	0
$754$	5.88854	0.214448
$755$	0	0
$756$	0	0
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	18.1803	0.660340
$759$	0	0
$760$	0	0
$761$	−15.1246	−0.548267	−0.274133	−	0.961692i	$-0.588391\pi$
$761$	−15.1246	−0.548267	−0.274133	+	0.961692i	$0.588391\pi$
$762$	0	0
$763$	−12.7639	−0.462085
$764$	12.0000	0.434145
$765$	0	0
$766$	−39.0344	−1.41037
$767$	3.19350	0.115310
$768$	0	0
$769$	−32.4164	−1.16897	−0.584483	−	0.811406i	$-0.698703\pi$
$769$	−32.4164	−1.16897	−0.584483	+	0.811406i	$0.698703\pi$
$770$	0	0
$771$	0	0
$772$	−0.763932	−0.0274945
$773$	13.9443	0.501541	0.250770	−	0.968047i	$-0.419316\pi$
$773$	13.9443	0.501541	0.250770	+	0.968047i	$0.419316\pi$
$774$	0	0
$775$	0	0
$776$	26.1803	0.939819
$777$	0	0
$778$	−12.9443	−0.464075
$779$	−5.00000	−0.179144
$780$	0	0
$781$	52.7214	1.88652
$782$	6.47214	0.231443
$783$	0	0
$784$	9.70820	0.346722
$785$	0	0
$786$	0	0
$787$	50.6525	1.80557	0.902783	−	0.430097i	$-0.141520\pi$
$787$	50.6525	1.80557	0.902783	+	0.430097i	$0.141520\pi$
$788$	5.34752	0.190498
$789$	0	0
$790$	0	0
$791$	7.88854	0.280484
$792$	0	0
$793$	−24.5836	−0.872989
$794$	36.1803	1.28399
$795$	0	0
$796$	3.85410	0.136605
$797$	13.3607	0.473260	0.236630	−	0.971600i	$-0.423957\pi$
$797$	13.3607	0.473260	0.236630	+	0.971600i	$0.423957\pi$
$798$	0	0
$799$	27.4164	0.969923
$800$	0	0
$801$	0	0
$802$	−53.1246	−1.87590
$803$	21.8885	0.772430
$804$	0	0
$805$	0	0
$806$	−6.11146	−0.215267
$807$	0	0
$808$	18.2918	0.643503
$809$	−36.8328	−1.29497	−0.647486	−	0.762077i	$-0.724180\pi$
$809$	−36.8328	−1.29497	−0.647486	+	0.762077i	$0.724180\pi$
$810$	0	0
$811$	−42.7214	−1.50015	−0.750075	−	0.661353i	$-0.769983\pi$
$811$	−42.7214	−1.50015	−0.750075	+	0.661353i	$0.769983\pi$
$812$	−2.03444	−0.0713949
$813$	0	0
$814$	−46.8328	−1.64149
$815$	0	0
$816$	0	0
$817$	4.00000	0.139942
$818$	21.4164	0.748807
$819$	0	0
$820$	0	0
$821$	35.1246	1.22586	0.612929	−	0.790138i	$-0.289991\pi$
$821$	35.1246	1.22586	0.612929	+	0.790138i	$0.289991\pi$
$822$	0	0
$823$	10.8197	0.377150	0.188575	−	0.982059i	$-0.439613\pi$
$823$	10.8197	0.377150	0.188575	+	0.982059i	$0.439613\pi$
$824$	−2.11146	−0.0735561
$825$	0	0
$826$	−4.67376	−0.162621
$827$	−19.0557	−0.662633	−0.331316	−	0.943520i	$-0.607493\pi$
$827$	−19.0557	−0.662633	−0.331316	+	0.943520i	$0.607493\pi$
$828$	0	0
$829$	−51.6656	−1.79442	−0.897211	−	0.441603i	$-0.854410\pi$
$829$	−51.6656	−1.79442	−0.897211	+	0.441603i	$0.854410\pi$
$830$	0	0
$831$	0	0
$832$	10.4721	0.363056
$833$	6.47214	0.224246
$834$	0	0
$835$	0	0
$836$	−2.47214	−0.0855006
$837$	0	0
$838$	37.7082	1.30261
$839$	33.7639	1.16566	0.582830	−	0.812594i	$-0.301945\pi$
$839$	33.7639	1.16566	0.582830	+	0.812594i	$0.301945\pi$
$840$	0	0
$841$	−26.8328	−0.925270
$842$	33.1246	1.14155
$843$	0	0
$844$	17.4164	0.599497
$845$	0	0
$846$	0	0
$847$	−11.1803	−0.384161
$848$	24.2705	0.833453
$849$	0	0
$850$	0	0
$851$	8.94427	0.306606
$852$	0	0
$853$	1.11146	0.0380555	0.0190278	−	0.999819i	$-0.493943\pi$
$853$	1.11146	0.0380555	0.0190278	+	0.999819i	$0.493943\pi$
$854$	35.9787	1.23117
$855$	0	0
$856$	26.3050	0.899085
$857$	−38.7771	−1.32460	−0.662300	−	0.749239i	$-0.730420\pi$
$857$	−38.7771	−1.32460	−0.662300	+	0.749239i	$0.730420\pi$
$858$	0	0
$859$	−34.7082	−1.18423	−0.592114	−	0.805854i	$-0.701707\pi$
$859$	−34.7082	−1.18423	−0.592114	+	0.805854i	$0.701707\pi$
$860$	0	0
$861$	0	0
$862$	−27.7984	−0.946816
$863$	57.1803	1.94644	0.973221	−	0.229873i	$-0.0738310\pi$
$863$	57.1803	1.94644	0.973221	+	0.229873i	$0.0738310\pi$
$864$	0	0
$865$	0	0
$866$	46.0689	1.56548
$867$	0	0
$868$	2.11146	0.0716675
$869$	−62.8328	−2.13146
$870$	0	0
$871$	−17.1672	−0.581688
$872$	−12.7639	−0.432241
$873$	0	0
$874$	2.00000	0.0676510
$875$	0	0
$876$	0	0
$877$	27.0557	0.913607	0.456804	−	0.889568i	$-0.348994\pi$
$877$	27.0557	0.913607	0.456804	+	0.889568i	$0.348994\pi$
$878$	−34.3607	−1.15962
$879$	0	0
$880$	0	0
$881$	−16.9443	−0.570867	−0.285434	−	0.958399i	$-0.592138\pi$
$881$	−16.9443	−0.570867	−0.285434	+	0.958399i	$0.592138\pi$
$882$	0	0
$883$	−14.3475	−0.482833	−0.241416	−	0.970422i	$-0.577612\pi$
$883$	−14.3475	−0.482833	−0.241416	+	0.970422i	$0.577612\pi$
$884$	−4.94427	−0.166294
$885$	0	0
$886$	−21.7082	−0.729301
$887$	34.8328	1.16957	0.584786	−	0.811188i	$-0.301179\pi$
$887$	34.8328	1.16957	0.584786	+	0.811188i	$0.301179\pi$
$888$	0	0
$889$	34.0689	1.14263
$890$	0	0
$891$	0	0
$892$	−2.94427	−0.0985815
$893$	8.47214	0.283509
$894$	0	0
$895$	0	0
$896$	−30.4508	−1.01729
$897$	0	0
$898$	26.7426	0.892414
$899$	−2.24922	−0.0750158
$900$	0	0
$901$	16.1803	0.539045
$902$	32.3607	1.07749
$903$	0	0
$904$	7.88854	0.262369
$905$	0	0
$906$	0	0
$907$	52.1803	1.73262	0.866310	−	0.499507i	$-0.166485\pi$
$907$	52.1803	1.73262	0.866310	+	0.499507i	$0.166485\pi$
$908$	−7.49342	−0.248678
$909$	0	0
$910$	0	0
$911$	−16.2361	−0.537925	−0.268962	−	0.963151i	$-0.586681\pi$
$911$	−16.2361	−0.537925	−0.268962	+	0.963151i	$0.586681\pi$
$912$	0	0
$913$	43.7771	1.44881
$914$	43.5066	1.43907
$915$	0	0
$916$	2.76393	0.0913229
$917$	24.4721	0.808141
$918$	0	0
$919$	54.7082	1.80466	0.902329	−	0.431049i	$-0.141856\pi$
$919$	54.7082	1.80466	0.902329	+	0.431049i	$0.141856\pi$
$920$	0	0
$921$	0	0
$922$	39.1246	1.28850
$923$	32.5836	1.07250
$924$	0	0
$925$	0	0
$926$	−57.3050	−1.88316
$927$	0	0
$928$	−4.97871	−0.163434
$929$	1.59675	0.0523876	0.0261938	−	0.999657i	$-0.491661\pi$
$929$	1.59675	0.0523876	0.0261938	+	0.999657i	$0.491661\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	−9.23607	−0.302537
$933$	0	0
$934$	−6.18034	−0.202227
$935$	0	0
$936$	0	0
$937$	−22.0557	−0.720529	−0.360265	−	0.932850i	$-0.617314\pi$
$937$	−22.0557	−0.720529	−0.360265	+	0.932850i	$0.617314\pi$
$938$	25.1246	0.820348
$939$	0	0
$940$	0	0
$941$	−27.5279	−0.897383	−0.448691	−	0.893687i	$-0.648110\pi$
$941$	−27.5279	−0.897383	−0.448691	+	0.893687i	$0.648110\pi$
$942$	0	0
$943$	−6.18034	−0.201260
$944$	−6.27051	−0.204088
$945$	0	0
$946$	−25.8885	−0.841709
$947$	−19.0557	−0.619228	−0.309614	−	0.950862i	$-0.600200\pi$
$947$	−19.0557	−0.619228	−0.309614	+	0.950862i	$0.600200\pi$
$948$	0	0
$949$	13.5279	0.439133
$950$	0	0
$951$	0	0
$952$	−16.1803	−0.524408
$953$	−27.2492	−0.882689	−0.441344	−	0.897338i	$-0.645498\pi$
$953$	−27.2492	−0.882689	−0.441344	+	0.897338i	$0.645498\pi$
$954$	0	0
$955$	0	0
$956$	−8.47214	−0.274008
$957$	0	0
$958$	69.9574	2.26022
$959$	−39.5967	−1.27865
$960$	0	0
$961$	−28.6656	−0.924698
$962$	−28.9443	−0.933201
$963$	0	0
$964$	3.81966	0.123023
$965$	0	0
$966$	0	0
$967$	26.5967	0.855294	0.427647	−	0.903946i	$-0.359343\pi$
$967$	26.5967	0.855294	0.427647	+	0.903946i	$0.359343\pi$
$968$	−11.1803	−0.359350
$969$	0	0
$970$	0	0
$971$	37.0689	1.18960	0.594799	−	0.803875i	$-0.297232\pi$
$971$	37.0689	1.18960	0.594799	+	0.803875i	$0.297232\pi$
$972$	0	0
$973$	45.2492	1.45062
$974$	−38.8328	−1.24428
$975$	0	0
$976$	48.2705	1.54510
$977$	22.9443	0.734052	0.367026	−	0.930211i	$-0.380376\pi$
$977$	22.9443	0.734052	0.367026	+	0.930211i	$0.380376\pi$
$978$	0	0
$979$	−31.7771	−1.01560
$980$	0	0
$981$	0	0
$982$	35.1246	1.12087
$983$	−5.30495	−0.169202	−0.0846008	−	0.996415i	$-0.526962\pi$
$983$	−5.30495	−0.169202	−0.0846008	+	0.996415i	$0.526962\pi$
$984$	0	0
$985$	0	0
$986$	−7.70820	−0.245479
$987$	0	0
$988$	−1.52786	−0.0486078
$989$	4.94427	0.157219
$990$	0	0
$991$	−54.9443	−1.74536	−0.872681	−	0.488290i	$-0.837621\pi$
$991$	−54.9443	−1.74536	−0.872681	+	0.488290i	$0.837621\pi$
$992$	5.16718	0.164058
$993$	0	0
$994$	−47.6869	−1.51254
$995$	0	0
$996$	0	0
$997$	−3.52786	−0.111729	−0.0558643	−	0.998438i	$-0.517791\pi$
$997$	−3.52786	−0.111729	−0.0558643	+	0.998438i	$0.517791\pi$
$998$	−14.2705	−0.451725
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.v.1.2		2
3.2	odd	2		1425.2.a.n.1.1	✓	2
5.4	even	2		4275.2.a.s.1.1		2
15.2	even	4		1425.2.c.m.799.1		4
15.8	even	4		1425.2.c.m.799.4		4
15.14	odd	2		1425.2.a.q.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.n.1.1	✓	2	3.2	odd	2
1425.2.a.q.1.2	yes	2	15.14	odd	2
1425.2.c.m.799.1		4	15.2	even	4
1425.2.c.m.799.4		4	15.8	even	4
4275.2.a.s.1.1		2	5.4	even	2
4275.2.a.v.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} -2.23607 q^{7} -2.23607 q^{8} +4.00000 q^{11} +2.47214 q^{13} -3.61803 q^{14} -4.85410 q^{16} -3.23607 q^{17} -1.00000 q^{19} +6.47214 q^{22} -1.23607 q^{23} +4.00000 q^{26} -1.38197 q^{28} +1.47214 q^{29} -1.52786 q^{31} -3.38197 q^{32} -5.23607 q^{34} -7.23607 q^{37} -1.61803 q^{38} +5.00000 q^{41} -4.00000 q^{43} +2.47214 q^{44} -2.00000 q^{46} -8.47214 q^{47} -2.00000 q^{49} +1.52786 q^{52} -5.00000 q^{53} +5.00000 q^{56} +2.38197 q^{58} +1.29180 q^{59} -9.94427 q^{61} -2.47214 q^{62} +4.23607 q^{64} -6.94427 q^{67} -2.00000 q^{68} +13.1803 q^{71} +5.47214 q^{73} -11.7082 q^{74} -0.618034 q^{76} -8.94427 q^{77} -15.7082 q^{79} +8.09017 q^{82} +10.9443 q^{83} -6.47214 q^{86} -8.94427 q^{88} -7.94427 q^{89} -5.52786 q^{91} -0.763932 q^{92} -13.7082 q^{94} -11.7082 q^{97} -3.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 8 q^{11} - 4 q^{13} - 5 q^{14} - 3 q^{16} - 2 q^{17} - 2 q^{19} + 4 q^{22} + 2 q^{23} + 8 q^{26} - 5 q^{28} - 6 q^{29} - 12 q^{31} - 9 q^{32} - 6 q^{34} - 10 q^{37} - q^{38} + 10 q^{41}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 8 * q^11 - 4 * q^13 - 5 * q^14 - 3 * q^16 - 2 * q^17 - 2 * q^19 + 4 * q^22 + 2 * q^23 + 8 * q^26 - 5 * q^28 - 6 * q^29 - 12 * q^31 - 9 * q^32 - 6 * q^34 - 10 * q^37 - q^38 + 10 * q^41 - 8 * q^43 - 4 * q^44 - 4 * q^46 - 8 * q^47 - 4 * q^49 + 12 * q^52 - 10 * q^53 + 10 * q^56 + 7 * q^58 + 16 * q^59 - 2 * q^61 + 4 * q^62 + 4 * q^64 + 4 * q^67 - 4 * q^68 + 4 * q^71 + 2 * q^73 - 10 * q^74 + q^76 - 18 * q^79 + 5 * q^82 + 4 * q^83 - 4 * q^86 + 2 * q^89 - 20 * q^91 - 6 * q^92 - 14 * q^94 - 10 * q^97 - 2 * q^98

Introduction

L-functions

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