LMFDB - Embedded newform 1805.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1805.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1805 = 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1805.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:	$14.4129975648$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.90211$ of defining polynomial
Character	$\chi$	$=$	1805.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.726543 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -2.47214 q^{9} +O(q^{10})$	$q+0.726543 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -2.47214 q^{9} +0.854102 q^{11} -1.45309 q^{12} -0.726543 q^{13} -0.726543 q^{15} +4.00000 q^{16} +3.85410 q^{17} +2.00000 q^{20} +1.17557 q^{21} -3.47214 q^{23} +1.00000 q^{25} -3.97574 q^{27} -3.23607 q^{28} +2.35114 q^{29} -6.88191 q^{31} +0.620541 q^{33} -1.61803 q^{35} +4.94427 q^{36} +7.77997 q^{37} -0.527864 q^{39} -5.98385 q^{41} -9.61803 q^{43} -1.70820 q^{44} +2.47214 q^{45} -8.70820 q^{47} +2.90617 q^{48} -4.38197 q^{49} +2.80017 q^{51} +1.45309 q^{52} -12.5882 q^{53} -0.854102 q^{55} +2.62866 q^{59} +1.45309 q^{60} -2.76393 q^{61} -4.00000 q^{63} -8.00000 q^{64} +0.726543 q^{65} +3.35520 q^{67} -7.70820 q^{68} -2.52265 q^{69} +13.0373 q^{71} -11.0000 q^{73} +0.726543 q^{75} +1.38197 q^{77} -16.2865 q^{79} -4.00000 q^{80} +4.52786 q^{81} -2.70820 q^{83} -2.35114 q^{84} -3.85410 q^{85} +1.70820 q^{87} +13.3148 q^{89} -1.17557 q^{91} +6.94427 q^{92} -5.00000 q^{93} +7.77997 q^{97} -2.11146 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 - 4 * q^5 + 2 * q^7 + 8 * q^9	$ 4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9} - 10 q^{11} + 16 q^{16} + 2 q^{17} + 8 q^{20} + 4 q^{23} + 4 q^{25} - 4 q^{28} - 2 q^{35} - 16 q^{36} - 20 q^{39} - 34 q^{43} + 20 q^{44} - 8 q^{45} - 8 q^{47} - 22 q^{49} + 10 q^{55} - 20 q^{61} - 16 q^{63} - 32 q^{64} - 4 q^{68} - 44 q^{73} + 10 q^{77} - 16 q^{80} + 36 q^{81} + 16 q^{83} - 2 q^{85} - 20 q^{87} - 8 q^{92} - 20 q^{93} - 80 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^5 + 2 * q^7 + 8 * q^9 - 10 * q^11 + 16 * q^16 + 2 * q^17 + 8 * q^20 + 4 * q^23 + 4 * q^25 - 4 * q^28 - 2 * q^35 - 16 * q^36 - 20 * q^39 - 34 * q^43 + 20 * q^44 - 8 * q^45 - 8 * q^47 - 22 * q^49 + 10 * q^55 - 20 * q^61 - 16 * q^63 - 32 * q^64 - 4 * q^68 - 44 * q^73 + 10 * q^77 - 16 * q^80 + 36 * q^81 + 16 * q^83 - 2 * q^85 - 20 * q^87 - 8 * q^92 - 20 * q^93 - 80 * 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0.726543	0.419470	0.209735	−	0.977758i	$-0.432740\pi$
$3$	0.726543	0.419470	0.209735	+	0.977758i	$0.432740\pi$
$4$	−2.00000	−1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.61803	0.611559	0.305780	−	0.952102i	$-0.401083\pi$
$7$	1.61803	0.611559	0.305780	+	0.952102i	$0.401083\pi$
$8$	0	0
$9$	−2.47214	−0.824045
$10$	0	0
$11$	0.854102	0.257521	0.128761	−	0.991676i	$-0.458900\pi$
$11$	0.854102	0.257521	0.128761	+	0.991676i	$0.458900\pi$
$12$	−1.45309	−0.419470
$13$	−0.726543	−0.201507	−0.100753	−	0.994911i	$-0.532125\pi$
$13$	−0.726543	−0.201507	−0.100753	+	0.994911i	$0.532125\pi$
$14$	0	0
$15$	−0.726543	−0.187592
$16$	4.00000	1.00000
$17$	3.85410	0.934757	0.467379	−	0.884057i	$-0.345199\pi$
$17$	3.85410	0.934757	0.467379	+	0.884057i	$0.345199\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	2.00000	0.447214
$21$	1.17557	0.256531
$22$	0	0
$23$	−3.47214	−0.723990	−0.361995	−	0.932180i	$-0.617904\pi$
$23$	−3.47214	−0.723990	−0.361995	+	0.932180i	$0.617904\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.97574	−0.765131
$28$	−3.23607	−0.611559
$29$	2.35114	0.436596	0.218298	−	0.975882i	$-0.429950\pi$
$29$	2.35114	0.436596	0.218298	+	0.975882i	$0.429950\pi$
$30$	0	0
$31$	−6.88191	−1.23603	−0.618014	−	0.786167i	$-0.712062\pi$
$31$	−6.88191	−1.23603	−0.618014	+	0.786167i	$0.712062\pi$
$32$	0	0
$33$	0.620541	0.108022
$34$	0	0
$35$	−1.61803	−0.273498
$36$	4.94427	0.824045
$37$	7.77997	1.27902	0.639509	−	0.768783i	$-0.279138\pi$
$37$	7.77997	1.27902	0.639509	+	0.768783i	$0.279138\pi$
$38$	0	0
$39$	−0.527864	−0.0845259
$40$	0	0
$41$	−5.98385	−0.934521	−0.467260	−	0.884120i	$-0.654759\pi$
$41$	−5.98385	−0.934521	−0.467260	+	0.884120i	$0.654759\pi$
$42$	0	0
$43$	−9.61803	−1.46674	−0.733368	−	0.679832i	$-0.762053\pi$
$43$	−9.61803	−1.46674	−0.733368	+	0.679832i	$0.762053\pi$
$44$	−1.70820	−0.257521
$45$	2.47214	0.368524
$46$	0	0
$47$	−8.70820	−1.27022	−0.635111	−	0.772421i	$-0.719046\pi$
$47$	−8.70820	−1.27022	−0.635111	+	0.772421i	$0.719046\pi$
$48$	2.90617	0.419470
$49$	−4.38197	−0.625995
$50$	0	0
$51$	2.80017	0.392102
$52$	1.45309	0.201507
$53$	−12.5882	−1.72913	−0.864564	−	0.502522i	$-0.832406\pi$
$53$	−12.5882	−1.72913	−0.864564	+	0.502522i	$0.832406\pi$
$54$	0	0
$55$	−0.854102	−0.115167
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.62866	0.342222	0.171111	−	0.985252i	$-0.445264\pi$
$59$	2.62866	0.342222	0.171111	+	0.985252i	$0.445264\pi$
$60$	1.45309	0.187592
$61$	−2.76393	−0.353885	−0.176943	−	0.984221i	$-0.556621\pi$
$61$	−2.76393	−0.353885	−0.176943	+	0.984221i	$0.556621\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	−8.00000	−1.00000
$65$	0.726543	0.0901165
$66$	0	0
$67$	3.35520	0.409903	0.204951	−	0.978772i	$-0.434296\pi$
$67$	3.35520	0.409903	0.204951	+	0.978772i	$0.434296\pi$
$68$	−7.70820	−0.934757
$69$	−2.52265	−0.303692
$70$	0	0
$71$	13.0373	1.54724	0.773620	−	0.633650i	$-0.218444\pi$
$71$	13.0373	1.54724	0.773620	+	0.633650i	$0.218444\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	0.726543	0.0838939
$76$	0	0
$77$	1.38197	0.157490
$78$	0	0
$79$	−16.2865	−1.83237	−0.916186	−	0.400754i	$-0.868748\pi$
$79$	−16.2865	−1.83237	−0.916186	+	0.400754i	$0.868748\pi$
$80$	−4.00000	−0.447214
$81$	4.52786	0.503096
$82$	0	0
$83$	−2.70820	−0.297264	−0.148632	−	0.988893i	$-0.547487\pi$
$83$	−2.70820	−0.297264	−0.148632	+	0.988893i	$0.547487\pi$
$84$	−2.35114	−0.256531
$85$	−3.85410	−0.418036
$86$	0	0
$87$	1.70820	0.183139
$88$	0	0
$89$	13.3148	1.41137	0.705683	−	0.708528i	$-0.250640\pi$
$89$	13.3148	1.41137	0.705683	+	0.708528i	$0.250640\pi$
$90$	0	0
$91$	−1.17557	−0.123233
$92$	6.94427	0.723990
$93$	−5.00000	−0.518476
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.77997	0.789936	0.394968	−	0.918695i	$-0.370756\pi$
$97$	7.77997	0.789936	0.394968	+	0.918695i	$0.370756\pi$
$98$	0	0
$99$	−2.11146	−0.212209
$100$	−2.00000	−0.200000
$101$	−13.0902	−1.30252	−0.651260	−	0.758854i	$-0.725759\pi$
$101$	−13.0902	−1.30252	−0.651260	+	0.758854i	$0.725759\pi$
$102$	0	0
$103$	10.8576	1.06984	0.534918	−	0.844904i	$-0.320343\pi$
$103$	10.8576	1.06984	0.534918	+	0.844904i	$0.320343\pi$
$104$	0	0
$105$	−1.17557	−0.114724
$106$	0	0
$107$	−17.2905	−1.67154	−0.835769	−	0.549081i	$-0.814978\pi$
$107$	−17.2905	−1.67154	−0.835769	+	0.549081i	$0.814978\pi$
$108$	7.95148	0.765131
$109$	13.5923	1.30191	0.650953	−	0.759118i	$-0.274369\pi$
$109$	13.5923	1.30191	0.650953	+	0.759118i	$0.274369\pi$
$110$	0	0
$111$	5.65248	0.536509
$112$	6.47214	0.611559
$113$	−8.78402	−0.826331	−0.413166	−	0.910656i	$-0.635577\pi$
$113$	−8.78402	−0.826331	−0.413166	+	0.910656i	$0.635577\pi$
$114$	0	0
$115$	3.47214	0.323778
$116$	−4.70228	−0.436596
$117$	1.79611	0.166051
$118$	0	0
$119$	6.23607	0.571659
$120$	0	0
$121$	−10.2705	−0.933683
$122$	0	0
$123$	−4.34752	−0.392003
$124$	13.7638	1.23603
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−18.0171	−1.59876	−0.799378	−	0.600828i	$-0.794838\pi$
$127$	−18.0171	−1.59876	−0.799378	+	0.600828i	$0.794838\pi$
$128$	0	0
$129$	−6.98791	−0.615251
$130$	0	0
$131$	0.763932	0.0667451	0.0333725	−	0.999443i	$-0.489375\pi$
$131$	0.763932	0.0667451	0.0333725	+	0.999443i	$0.489375\pi$
$132$	−1.24108	−0.108022
$133$	0	0
$134$	0	0
$135$	3.97574	0.342177
$136$	0	0
$137$	−5.56231	−0.475220	−0.237610	−	0.971361i	$-0.576364\pi$
$137$	−5.56231	−0.475220	−0.237610	+	0.971361i	$0.576364\pi$
$138$	0	0
$139$	−2.61803	−0.222059	−0.111029	−	0.993817i	$-0.535415\pi$
$139$	−2.61803	−0.222059	−0.111029	+	0.993817i	$0.535415\pi$
$140$	3.23607	0.273498
$141$	−6.32688	−0.532819
$142$	0	0
$143$	−0.620541	−0.0518923
$144$	−9.88854	−0.824045
$145$	−2.35114	−0.195252
$146$	0	0
$147$	−3.18368	−0.262586
$148$	−15.5599	−1.27902
$149$	−8.23607	−0.674725	−0.337362	−	0.941375i	$-0.609535\pi$
$149$	−8.23607	−0.674725	−0.337362	+	0.941375i	$0.609535\pi$
$150$	0	0
$151$	7.33094	0.596583	0.298292	−	0.954475i	$-0.403583\pi$
$151$	7.33094	0.596583	0.298292	+	0.954475i	$0.403583\pi$
$152$	0	0
$153$	−9.52786	−0.770282
$154$	0	0
$155$	6.88191	0.552768
$156$	1.05573	0.0845259
$157$	−14.0344	−1.12007	−0.560035	−	0.828469i	$-0.689212\pi$
$157$	−14.0344	−1.12007	−0.560035	+	0.828469i	$0.689212\pi$
$158$	0	0
$159$	−9.14590	−0.725317
$160$	0	0
$161$	−5.61803	−0.442763
$162$	0	0
$163$	−0.909830	−0.0712634	−0.0356317	−	0.999365i	$-0.511344\pi$
$163$	−0.909830	−0.0712634	−0.0356317	+	0.999365i	$0.511344\pi$
$164$	11.9677	0.934521
$165$	−0.620541	−0.0483091
$166$	0	0
$167$	17.2905	1.33798	0.668991	−	0.743271i	$-0.266727\pi$
$167$	17.2905	1.33798	0.668991	+	0.743271i	$0.266727\pi$
$168$	0	0
$169$	−12.4721	−0.959395
$170$	0	0
$171$	0	0
$172$	19.2361	1.46674
$173$	6.60440	0.502123	0.251061	−	0.967971i	$-0.419220\pi$
$173$	6.60440	0.502123	0.251061	+	0.967971i	$0.419220\pi$
$174$	0	0
$175$	1.61803	0.122312
$176$	3.41641	0.257521
$177$	1.90983	0.143552
$178$	0	0
$179$	4.53077	0.338646	0.169323	−	0.985561i	$-0.445842\pi$
$179$	4.53077	0.338646	0.169323	+	0.985561i	$0.445842\pi$
$180$	−4.94427	−0.368524
$181$	7.43694	0.552783	0.276392	−	0.961045i	$-0.410861\pi$
$181$	7.43694	0.552783	0.276392	+	0.961045i	$0.410861\pi$
$182$	0	0
$183$	−2.00811	−0.148444
$184$	0	0
$185$	−7.77997	−0.571994
$186$	0	0
$187$	3.29180	0.240720
$188$	17.4164	1.27022
$189$	−6.43288	−0.467923
$190$	0	0
$191$	−5.09017	−0.368312	−0.184156	−	0.982897i	$-0.558955\pi$
$191$	−5.09017	−0.368312	−0.184156	+	0.982897i	$0.558955\pi$
$192$	−5.81234	−0.419470
$193$	−3.91023	−0.281464	−0.140732	−	0.990048i	$-0.544946\pi$
$193$	−3.91023	−0.281464	−0.140732	+	0.990048i	$0.544946\pi$
$194$	0	0
$195$	0.527864	0.0378011
$196$	8.76393	0.625995
$197$	22.6525	1.61392	0.806961	−	0.590605i	$-0.201111\pi$
$197$	22.6525	1.61392	0.806961	+	0.590605i	$0.201111\pi$
$198$	0	0
$199$	12.5623	0.890518	0.445259	−	0.895402i	$-0.353112\pi$
$199$	12.5623	0.890518	0.445259	+	0.895402i	$0.353112\pi$
$200$	0	0
$201$	2.43769	0.171942
$202$	0	0
$203$	3.80423	0.267004
$204$	−5.60034	−0.392102
$205$	5.98385	0.417930
$206$	0	0
$207$	8.58359	0.596601
$208$	−2.90617	−0.201507
$209$	0	0
$210$	0	0
$211$	−2.80017	−0.192772	−0.0963858	−	0.995344i	$-0.530728\pi$
$211$	−2.80017	−0.192772	−0.0963858	+	0.995344i	$0.530728\pi$
$212$	25.1765	1.72913
$213$	9.47214	0.649020
$214$	0	0
$215$	9.61803	0.655944
$216$	0	0
$217$	−11.1352	−0.755904
$218$	0	0
$219$	−7.99197	−0.540047
$220$	1.70820	0.115167
$221$	−2.80017	−0.188360
$222$	0	0
$223$	−5.15131	−0.344957	−0.172479	−	0.985013i	$-0.555178\pi$
$223$	−5.15131	−0.344957	−0.172479	+	0.985013i	$0.555178\pi$
$224$	0	0
$225$	−2.47214	−0.164809
$226$	0	0
$227$	26.9726	1.79023	0.895117	−	0.445830i	$-0.147092\pi$
$227$	26.9726	1.79023	0.895117	+	0.445830i	$0.147092\pi$
$228$	0	0
$229$	−4.85410	−0.320768	−0.160384	−	0.987055i	$-0.551273\pi$
$229$	−4.85410	−0.320768	−0.160384	+	0.987055i	$0.551273\pi$
$230$	0	0
$231$	1.00406	0.0660621
$232$	0	0
$233$	12.9443	0.848007	0.424004	−	0.905660i	$-0.360624\pi$
$233$	12.9443	0.848007	0.424004	+	0.905660i	$0.360624\pi$
$234$	0	0
$235$	8.70820	0.568061
$236$	−5.25731	−0.342222
$237$	−11.8328	−0.768624
$238$	0	0
$239$	3.94427	0.255134	0.127567	−	0.991830i	$-0.459283\pi$
$239$	3.94427	0.255134	0.127567	+	0.991830i	$0.459283\pi$
$240$	−2.90617	−0.187592
$241$	−20.3682	−1.31203	−0.656016	−	0.754747i	$-0.727760\pi$
$241$	−20.3682	−1.31203	−0.656016	+	0.754747i	$0.727760\pi$
$242$	0	0
$243$	15.2169	0.976165
$244$	5.52786	0.353885
$245$	4.38197	0.279954
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.96763	−0.124693
$250$	0	0
$251$	6.14590	0.387926	0.193963	−	0.981009i	$-0.437866\pi$
$251$	6.14590	0.387926	0.193963	+	0.981009i	$0.437866\pi$
$252$	8.00000	0.503953
$253$	−2.96556	−0.186443
$254$	0	0
$255$	−2.80017	−0.175353
$256$	16.0000	1.00000
$257$	−29.6013	−1.84648	−0.923238	−	0.384228i	$-0.874468\pi$
$257$	−29.6013	−1.84648	−0.923238	+	0.384228i	$0.874468\pi$
$258$	0	0
$259$	12.5882	0.782196
$260$	−1.45309	−0.0901165
$261$	−5.81234	−0.359775
$262$	0	0
$263$	10.9098	0.672729	0.336364	−	0.941732i	$-0.390803\pi$
$263$	10.9098	0.672729	0.336364	+	0.941732i	$0.390803\pi$
$264$	0	0
$265$	12.5882	0.773290
$266$	0	0
$267$	9.67376	0.592025
$268$	−6.71040	−0.409903
$269$	−19.0866	−1.16373	−0.581867	−	0.813284i	$-0.697677\pi$
$269$	−19.0866	−1.16373	−0.581867	+	0.813284i	$0.697677\pi$
$270$	0	0
$271$	−18.4164	−1.11872	−0.559359	−	0.828926i	$-0.688952\pi$
$271$	−18.4164	−1.11872	−0.559359	+	0.828926i	$0.688952\pi$
$272$	15.4164	0.934757
$273$	−0.854102	−0.0516926
$274$	0	0
$275$	0.854102	0.0515043
$276$	5.04531	0.303692
$277$	25.3607	1.52378	0.761888	−	0.647709i	$-0.224273\pi$
$277$	25.3607	1.52378	0.761888	+	0.647709i	$0.224273\pi$
$278$	0	0
$279$	17.0130	1.01854
$280$	0	0
$281$	4.42477	0.263959	0.131980	−	0.991252i	$-0.457867\pi$
$281$	4.42477	0.263959	0.131980	+	0.991252i	$0.457867\pi$
$282$	0	0
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	−26.0746	−1.54724
$285$	0	0
$286$	0	0
$287$	−9.68208	−0.571515
$288$	0	0
$289$	−2.14590	−0.126229
$290$	0	0
$291$	5.65248	0.331354
$292$	22.0000	1.28745
$293$	6.04937	0.353408	0.176704	−	0.984264i	$-0.443457\pi$
$293$	6.04937	0.353408	0.176704	+	0.984264i	$0.443457\pi$
$294$	0	0
$295$	−2.62866	−0.153046
$296$	0	0
$297$	−3.39569	−0.197038
$298$	0	0
$299$	2.52265	0.145889
$300$	−1.45309	−0.0838939
$301$	−15.5623	−0.896996
$302$	0	0
$303$	−9.51057	−0.546368
$304$	0	0
$305$	2.76393	0.158262
$306$	0	0
$307$	24.0009	1.36981	0.684903	−	0.728635i	$-0.259845\pi$
$307$	24.0009	1.36981	0.684903	+	0.728635i	$0.259845\pi$
$308$	−2.76393	−0.157490
$309$	7.88854	0.448764
$310$	0	0
$311$	2.03444	0.115363	0.0576813	−	0.998335i	$-0.481629\pi$
$311$	2.03444	0.115363	0.0576813	+	0.998335i	$0.481629\pi$
$312$	0	0
$313$	29.3262	1.65762	0.828808	−	0.559532i	$-0.189019\pi$
$313$	29.3262	1.65762	0.828808	+	0.559532i	$0.189019\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	32.5729	1.83237
$317$	−8.12299	−0.456233	−0.228116	−	0.973634i	$-0.573257\pi$
$317$	−8.12299	−0.456233	−0.228116	+	0.973634i	$0.573257\pi$
$318$	0	0
$319$	2.00811	0.112433
$320$	8.00000	0.447214
$321$	−12.5623	−0.701160
$322$	0	0
$323$	0	0
$324$	−9.05573	−0.503096
$325$	−0.726543	−0.0403013
$326$	0	0
$327$	9.87539	0.546110
$328$	0	0
$329$	−14.0902	−0.776816
$330$	0	0
$331$	17.1845	0.944547	0.472274	−	0.881452i	$-0.343433\pi$
$331$	17.1845	0.944547	0.472274	+	0.881452i	$0.343433\pi$
$332$	5.41641	0.297264
$333$	−19.2331	−1.05397
$334$	0	0
$335$	−3.35520	−0.183314
$336$	4.70228	0.256531
$337$	−13.0373	−0.710186	−0.355093	−	0.934831i	$-0.615551\pi$
$337$	−13.0373	−0.710186	−0.355093	+	0.934831i	$0.615551\pi$
$338$	0	0
$339$	−6.38197	−0.346621
$340$	7.70820	0.418036
$341$	−5.87785	−0.318304
$342$	0	0
$343$	−18.4164	−0.994393
$344$	0	0
$345$	2.52265	0.135815
$346$	0	0
$347$	−11.2918	−0.606175	−0.303088	−	0.952963i	$-0.598018\pi$
$347$	−11.2918	−0.606175	−0.303088	+	0.952963i	$0.598018\pi$
$348$	−3.41641	−0.183139
$349$	−30.1246	−1.61253	−0.806267	−	0.591552i	$-0.798515\pi$
$349$	−30.1246	−1.61253	−0.806267	+	0.591552i	$0.798515\pi$
$350$	0	0
$351$	2.88854	0.154179
$352$	0	0
$353$	17.6180	0.937713	0.468857	−	0.883274i	$-0.344666\pi$
$353$	17.6180	0.937713	0.468857	+	0.883274i	$0.344666\pi$
$354$	0	0
$355$	−13.0373	−0.691947
$356$	−26.6296	−1.41137
$357$	4.53077	0.239794
$358$	0	0
$359$	−26.5623	−1.40190	−0.700952	−	0.713208i	$-0.747241\pi$
$359$	−26.5623	−1.40190	−0.700952	+	0.713208i	$0.747241\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−7.46196	−0.391651
$364$	2.35114	0.123233
$365$	11.0000	0.575766
$366$	0	0
$367$	17.6525	0.921452	0.460726	−	0.887542i	$-0.347589\pi$
$367$	17.6525	0.921452	0.460726	+	0.887542i	$0.347589\pi$
$368$	−13.8885	−0.723990
$369$	14.7929	0.770088
$370$	0	0
$371$	−20.3682	−1.05746
$372$	10.0000	0.518476
$373$	30.9888	1.60454	0.802271	−	0.596961i	$-0.203625\pi$
$373$	30.9888	1.60454	0.802271	+	0.596961i	$0.203625\pi$
$374$	0	0
$375$	−0.726543	−0.0375185
$376$	0	0
$377$	−1.70820	−0.0879770
$378$	0	0
$379$	−7.26543	−0.373200	−0.186600	−	0.982436i	$-0.559747\pi$
$379$	−7.26543	−0.373200	−0.186600	+	0.982436i	$0.559747\pi$
$380$	0	0
$381$	−13.0902	−0.670630
$382$	0	0
$383$	−20.9888	−1.07248	−0.536238	−	0.844067i	$-0.680155\pi$
$383$	−20.9888	−1.07248	−0.536238	+	0.844067i	$0.680155\pi$
$384$	0	0
$385$	−1.38197	−0.0704315
$386$	0	0
$387$	23.7771	1.20866
$388$	−15.5599	−0.789936
$389$	27.5623	1.39746	0.698732	−	0.715383i	$-0.253748\pi$
$389$	27.5623	1.39746	0.698732	+	0.715383i	$0.253748\pi$
$390$	0	0
$391$	−13.3820	−0.676755
$392$	0	0
$393$	0.555029	0.0279975
$394$	0	0
$395$	16.2865	0.819461
$396$	4.22291	0.212209
$397$	−7.52786	−0.377813	−0.188906	−	0.981995i	$-0.560494\pi$
$397$	−7.52786	−0.377813	−0.188906	+	0.981995i	$0.560494\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	20.8172	1.03956	0.519782	−	0.854299i	$-0.326013\pi$
$401$	20.8172	1.03956	0.519782	+	0.854299i	$0.326013\pi$
$402$	0	0
$403$	5.00000	0.249068
$404$	26.1803	1.30252
$405$	−4.52786	−0.224991
$406$	0	0
$407$	6.64488	0.329375
$408$	0	0
$409$	2.62866	0.129979	0.0649893	−	0.997886i	$-0.479299\pi$
$409$	2.62866	0.129979	0.0649893	+	0.997886i	$0.479299\pi$
$410$	0	0
$411$	−4.04125	−0.199340
$412$	−21.7153	−1.06984
$413$	4.25325	0.209289
$414$	0	0
$415$	2.70820	0.132941
$416$	0	0
$417$	−1.90211	−0.0931469
$418$	0	0
$419$	−36.7082	−1.79331	−0.896657	−	0.442727i	$-0.854011\pi$
$419$	−36.7082	−1.79331	−0.896657	+	0.442727i	$0.854011\pi$
$420$	2.35114	0.114724
$421$	2.52265	0.122947	0.0614733	−	0.998109i	$-0.480420\pi$
$421$	2.52265	0.122947	0.0614733	+	0.998109i	$0.480420\pi$
$422$	0	0
$423$	21.5279	1.04672
$424$	0	0
$425$	3.85410	0.186951
$426$	0	0
$427$	−4.47214	−0.216422
$428$	34.5811	1.67154
$429$	−0.450850	−0.0217672
$430$	0	0
$431$	30.6053	1.47421	0.737103	−	0.675780i	$-0.236193\pi$
$431$	30.6053	1.47421	0.737103	+	0.675780i	$0.236193\pi$
$432$	−15.9030	−0.765131
$433$	22.5478	1.08358	0.541790	−	0.840514i	$-0.317747\pi$
$433$	22.5478	1.08358	0.541790	+	0.840514i	$0.317747\pi$
$434$	0	0
$435$	−1.70820	−0.0819021
$436$	−27.1846	−1.30191
$437$	0	0
$438$	0	0
$439$	−28.5972	−1.36487	−0.682435	−	0.730946i	$-0.739079\pi$
$439$	−28.5972	−1.36487	−0.682435	+	0.730946i	$0.739079\pi$
$440$	0	0
$441$	10.8328	0.515848
$442$	0	0
$443$	0.0557281	0.00264772	0.00132386	−	0.999999i	$-0.499579\pi$
$443$	0.0557281	0.00264772	0.00132386	+	0.999999i	$0.499579\pi$
$444$	−11.3050	−0.536509
$445$	−13.3148	−0.631182
$446$	0	0
$447$	−5.98385	−0.283027
$448$	−12.9443	−0.611559
$449$	39.1118	1.84580	0.922901	−	0.385038i	$-0.125812\pi$
$449$	39.1118	1.84580	0.922901	+	0.385038i	$0.125812\pi$
$450$	0	0
$451$	−5.11082	−0.240659
$452$	17.5680	0.826331
$453$	5.32624	0.250248
$454$	0	0
$455$	1.17557	0.0551116
$456$	0	0
$457$	18.6525	0.872526	0.436263	−	0.899819i	$-0.356302\pi$
$457$	18.6525	0.872526	0.436263	+	0.899819i	$0.356302\pi$
$458$	0	0
$459$	−15.3229	−0.715212
$460$	−6.94427	−0.323778
$461$	20.4164	0.950887	0.475443	−	0.879746i	$-0.342288\pi$
$461$	20.4164	0.950887	0.475443	+	0.879746i	$0.342288\pi$
$462$	0	0
$463$	−30.7082	−1.42713	−0.713566	−	0.700588i	$-0.752921\pi$
$463$	−30.7082	−1.42713	−0.713566	+	0.700588i	$0.752921\pi$
$464$	9.40456	0.436596
$465$	5.00000	0.231869
$466$	0	0
$467$	−6.94427	−0.321343	−0.160671	−	0.987008i	$-0.551366\pi$
$467$	−6.94427	−0.321343	−0.160671	+	0.987008i	$0.551366\pi$
$468$	−3.59222	−0.166051
$469$	5.42882	0.250680
$470$	0	0
$471$	−10.1966	−0.469835
$472$	0	0
$473$	−8.21478	−0.377716
$474$	0	0
$475$	0	0
$476$	−12.4721	−0.571659
$477$	31.1199	1.42488
$478$	0	0
$479$	31.8541	1.45545	0.727726	−	0.685868i	$-0.240577\pi$
$479$	31.8541	1.45545	0.727726	+	0.685868i	$0.240577\pi$
$480$	0	0
$481$	−5.65248	−0.257731
$482$	0	0
$483$	−4.08174	−0.185726
$484$	20.5410	0.933683
$485$	−7.77997	−0.353270
$486$	0	0
$487$	1.79611	0.0813896	0.0406948	−	0.999172i	$-0.487043\pi$
$487$	1.79611	0.0813896	0.0406948	+	0.999172i	$0.487043\pi$
$488$	0	0
$489$	−0.661030	−0.0298928
$490$	0	0
$491$	−28.3820	−1.28086	−0.640430	−	0.768016i	$-0.721244\pi$
$491$	−28.3820	−1.28086	−0.640430	+	0.768016i	$0.721244\pi$
$492$	8.69505	0.392003
$493$	9.06154	0.408111
$494$	0	0
$495$	2.11146	0.0949029
$496$	−27.5276	−1.23603
$497$	21.0948	0.946229
$498$	0	0
$499$	−24.1246	−1.07997	−0.539983	−	0.841676i	$-0.681569\pi$
$499$	−24.1246	−1.07997	−0.539983	+	0.841676i	$0.681569\pi$
$500$	2.00000	0.0894427
$501$	12.5623	0.561242
$502$	0	0
$503$	31.8541	1.42030	0.710152	−	0.704048i	$-0.248626\pi$
$503$	31.8541	1.42030	0.710152	+	0.704048i	$0.248626\pi$
$504$	0	0
$505$	13.0902	0.582505
$506$	0	0
$507$	−9.06154	−0.402437
$508$	36.0341	1.59876
$509$	−32.1239	−1.42387	−0.711934	−	0.702247i	$-0.752180\pi$
$509$	−32.1239	−1.42387	−0.711934	+	0.702247i	$0.752180\pi$
$510$	0	0
$511$	−17.7984	−0.787354
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.8576	−0.478445
$516$	13.9758	0.615251
$517$	−7.43769	−0.327109
$518$	0	0
$519$	4.79837	0.210625
$520$	0	0
$521$	4.76779	0.208881	0.104440	−	0.994531i	$-0.466695\pi$
$521$	4.76779	0.208881	0.104440	+	0.994531i	$0.466695\pi$
$522$	0	0
$523$	−11.9677	−0.523311	−0.261656	−	0.965161i	$-0.584268\pi$
$523$	−11.9677	−0.523311	−0.261656	+	0.965161i	$0.584268\pi$
$524$	−1.52786	−0.0667451
$525$	1.17557	0.0513061
$526$	0	0
$527$	−26.5236	−1.15539
$528$	2.48217	0.108022
$529$	−10.9443	−0.475838
$530$	0	0
$531$	−6.49839	−0.282006
$532$	0	0
$533$	4.34752	0.188312
$534$	0	0
$535$	17.2905	0.747535
$536$	0	0
$537$	3.29180	0.142051
$538$	0	0
$539$	−3.74265	−0.161207
$540$	−7.95148	−0.342177
$541$	−11.0557	−0.475323	−0.237661	−	0.971348i	$-0.576381\pi$
$541$	−11.0557	−0.475323	−0.237661	+	0.971348i	$0.576381\pi$
$542$	0	0
$543$	5.40325	0.231876
$544$	0	0
$545$	−13.5923	−0.582230
$546$	0	0
$547$	9.61657	0.411175	0.205587	−	0.978639i	$-0.434090\pi$
$547$	9.61657	0.411175	0.205587	+	0.978639i	$0.434090\pi$
$548$	11.1246	0.475220
$549$	6.83282	0.291617
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−26.3521	−1.12060
$554$	0	0
$555$	−5.65248	−0.239934
$556$	5.23607	0.222059
$557$	10.3607	0.438996	0.219498	−	0.975613i	$-0.429558\pi$
$557$	10.3607	0.438996	0.219498	+	0.975613i	$0.429558\pi$
$558$	0	0
$559$	6.98791	0.295557
$560$	−6.47214	−0.273498
$561$	2.39163	0.100975
$562$	0	0
$563$	5.98385	0.252189	0.126095	−	0.992018i	$-0.459756\pi$
$563$	5.98385	0.252189	0.126095	+	0.992018i	$0.459756\pi$
$564$	12.6538	0.532819
$565$	8.78402	0.369547
$566$	0	0
$567$	7.32624	0.307673
$568$	0	0
$569$	−17.3560	−0.727603	−0.363802	−	0.931476i	$-0.618521\pi$
$569$	−17.3560	−0.727603	−0.363802	+	0.931476i	$0.618521\pi$
$570$	0	0
$571$	28.5410	1.19440	0.597202	−	0.802091i	$-0.296279\pi$
$571$	28.5410	1.19440	0.597202	+	0.802091i	$0.296279\pi$
$572$	1.24108	0.0518923
$573$	−3.69822	−0.154496
$574$	0	0
$575$	−3.47214	−0.144798
$576$	19.7771	0.824045
$577$	28.1246	1.17084	0.585421	−	0.810729i	$-0.300929\pi$
$577$	28.1246	1.17084	0.585421	+	0.810729i	$0.300929\pi$
$578$	0	0
$579$	−2.84095	−0.118066
$580$	4.70228	0.195252
$581$	−4.38197	−0.181795
$582$	0	0
$583$	−10.7516	−0.445288
$584$	0	0
$585$	−1.79611	−0.0742601
$586$	0	0
$587$	15.8885	0.655790	0.327895	−	0.944714i	$-0.393661\pi$
$587$	15.8885	0.655790	0.327895	+	0.944714i	$0.393661\pi$
$588$	6.36737	0.262586
$589$	0	0
$590$	0	0
$591$	16.4580	0.676991
$592$	31.1199	1.27902
$593$	27.2148	1.11758	0.558789	−	0.829310i	$-0.311266\pi$
$593$	27.2148	1.11758	0.558789	+	0.829310i	$0.311266\pi$
$594$	0	0
$595$	−6.23607	−0.255654
$596$	16.4721	0.674725
$597$	9.12705	0.373545
$598$	0	0
$599$	1.55909	0.0637025	0.0318513	−	0.999493i	$-0.489860\pi$
$599$	1.55909	0.0637025	0.0318513	+	0.999493i	$0.489860\pi$
$600$	0	0
$601$	1.24108	0.0506248	0.0253124	−	0.999680i	$-0.491942\pi$
$601$	1.24108	0.0506248	0.0253124	+	0.999680i	$0.491942\pi$
$602$	0	0
$603$	−8.29451	−0.337778
$604$	−14.6619	−0.596583
$605$	10.2705	0.417556
$606$	0	0
$607$	26.2866	1.06694	0.533469	−	0.845819i	$-0.320888\pi$
$607$	26.2866	1.06694	0.533469	+	0.845819i	$0.320888\pi$
$608$	0	0
$609$	2.76393	0.112000
$610$	0	0
$611$	6.32688	0.255958
$612$	19.0557	0.770282
$613$	10.2705	0.414822	0.207411	−	0.978254i	$-0.433496\pi$
$613$	10.2705	0.414822	0.207411	+	0.978254i	$0.433496\pi$
$614$	0	0
$615$	4.34752	0.175309
$616$	0	0
$617$	−34.8885	−1.40456	−0.702280	−	0.711901i	$-0.747835\pi$
$617$	−34.8885	−1.40456	−0.702280	+	0.711901i	$0.747835\pi$
$618$	0	0
$619$	−32.0902	−1.28981	−0.644906	−	0.764262i	$-0.723104\pi$
$619$	−32.0902	−1.28981	−0.644906	+	0.764262i	$0.723104\pi$
$620$	−13.7638	−0.552768
$621$	13.8043	0.553948
$622$	0	0
$623$	21.5438	0.863134
$624$	−2.11146	−0.0845259
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	28.0689	1.12007
$629$	29.9848	1.19557
$630$	0	0
$631$	−15.5279	−0.618155	−0.309077	−	0.951037i	$-0.600020\pi$
$631$	−15.5279	−0.618155	−0.309077	+	0.951037i	$0.600020\pi$
$632$	0	0
$633$	−2.03444	−0.0808618
$634$	0	0
$635$	18.0171	0.714986
$636$	18.2918	0.725317
$637$	3.18368	0.126142
$638$	0	0
$639$	−32.2299	−1.27500
$640$	0	0
$641$	14.3188	0.565561	0.282780	−	0.959185i	$-0.408743\pi$
$641$	14.3188	0.565561	0.282780	+	0.959185i	$0.408743\pi$
$642$	0	0
$643$	−2.94427	−0.116111	−0.0580554	−	0.998313i	$-0.518490\pi$
$643$	−2.94427	−0.116111	−0.0580554	+	0.998313i	$0.518490\pi$
$644$	11.2361	0.442763
$645$	6.98791	0.275149
$646$	0	0
$647$	0.639320	0.0251343	0.0125671	−	0.999921i	$-0.496000\pi$
$647$	0.639320	0.0251343	0.0125671	+	0.999921i	$0.496000\pi$
$648$	0	0
$649$	2.24514	0.0881294
$650$	0	0
$651$	−8.09017	−0.317079
$652$	1.81966	0.0712634
$653$	6.32624	0.247565	0.123782	−	0.992309i	$-0.460498\pi$
$653$	6.32624	0.247565	0.123782	+	0.992309i	$0.460498\pi$
$654$	0	0
$655$	−0.763932	−0.0298493
$656$	−23.9354	−0.934521
$657$	27.1935	1.06092
$658$	0	0
$659$	−5.36331	−0.208925	−0.104462	−	0.994529i	$-0.533312\pi$
$659$	−5.36331	−0.208925	−0.104462	+	0.994529i	$0.533312\pi$
$660$	1.24108	0.0483091
$661$	−11.2412	−0.437231	−0.218615	−	0.975811i	$-0.570154\pi$
$661$	−11.2412	−0.437231	−0.218615	+	0.975811i	$0.570154\pi$
$662$	0	0
$663$	−2.03444	−0.0790112
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.16348	−0.316091
$668$	−34.5811	−1.33798
$669$	−3.74265	−0.144699
$670$	0	0
$671$	−2.36068	−0.0911330
$672$	0	0
$673$	−37.2097	−1.43433	−0.717165	−	0.696904i	$-0.754560\pi$
$673$	−37.2097	−1.43433	−0.717165	+	0.696904i	$0.754560\pi$
$674$	0	0
$675$	−3.97574	−0.153026
$676$	24.9443	0.959395
$677$	−44.8182	−1.72250	−0.861251	−	0.508180i	$-0.830319\pi$
$677$	−44.8182	−1.72250	−0.861251	+	0.508180i	$0.830319\pi$
$678$	0	0
$679$	12.5882	0.483093
$680$	0	0
$681$	19.5967	0.750949
$682$	0	0
$683$	3.42071	0.130890	0.0654449	−	0.997856i	$-0.479153\pi$
$683$	3.42071	0.130890	0.0654449	+	0.997856i	$0.479153\pi$
$684$	0	0
$685$	5.56231	0.212525
$686$	0	0
$687$	−3.52671	−0.134552
$688$	−38.4721	−1.46674
$689$	9.14590	0.348431
$690$	0	0
$691$	−11.0557	−0.420580	−0.210290	−	0.977639i	$-0.567441\pi$
$691$	−11.0557	−0.420580	−0.210290	+	0.977639i	$0.567441\pi$
$692$	−13.2088	−0.502123
$693$	−3.41641	−0.129779
$694$	0	0
$695$	2.61803	0.0993077
$696$	0	0
$697$	−23.0624	−0.873550
$698$	0	0
$699$	9.40456	0.355713
$700$	−3.23607	−0.122312
$701$	25.0000	0.944237	0.472118	−	0.881535i	$-0.343489\pi$
$701$	25.0000	0.944237	0.472118	+	0.881535i	$0.343489\pi$
$702$	0	0
$703$	0	0
$704$	−6.83282	−0.257521
$705$	6.32688	0.238284
$706$	0	0
$707$	−21.1803	−0.796569
$708$	−3.81966	−0.143552
$709$	−34.4721	−1.29463	−0.647314	−	0.762223i	$-0.724108\pi$
$709$	−34.4721	−1.29463	−0.647314	+	0.762223i	$0.724108\pi$
$710$	0	0
$711$	40.2624	1.50996
$712$	0	0
$713$	23.8949	0.894872
$714$	0	0
$715$	0.620541	0.0232069
$716$	−9.06154	−0.338646
$717$	2.86568	0.107021
$718$	0	0
$719$	3.76393	0.140371	0.0701855	−	0.997534i	$-0.477641\pi$
$719$	3.76393	0.140371	0.0701855	+	0.997534i	$0.477641\pi$
$720$	9.88854	0.368524
$721$	17.5680	0.654268
$722$	0	0
$723$	−14.7984	−0.550357
$724$	−14.8739	−0.552783
$725$	2.35114	0.0873192
$726$	0	0
$727$	−37.0689	−1.37481	−0.687404	−	0.726275i	$-0.741250\pi$
$727$	−37.0689	−1.37481	−0.687404	+	0.726275i	$0.741250\pi$
$728$	0	0
$729$	−2.52786	−0.0936246
$730$	0	0
$731$	−37.0689	−1.37104
$732$	4.01623	0.148444
$733$	−1.20163	−0.0443831	−0.0221915	−	0.999754i	$-0.507064\pi$
$733$	−1.20163	−0.0443831	−0.0221915	+	0.999754i	$0.507064\pi$
$734$	0	0
$735$	3.18368	0.117432
$736$	0	0
$737$	2.86568	0.105559
$738$	0	0
$739$	−11.5836	−0.426109	−0.213055	−	0.977040i	$-0.568341\pi$
$739$	−11.5836	−0.426109	−0.213055	+	0.977040i	$0.568341\pi$
$740$	15.5599	0.571994
$741$	0	0
$742$	0	0
$743$	36.6952	1.34622	0.673108	−	0.739544i	$-0.264959\pi$
$743$	36.6952	1.34622	0.673108	+	0.739544i	$0.264959\pi$
$744$	0	0
$745$	8.23607	0.301746
$746$	0	0
$747$	6.69505	0.244959
$748$	−6.58359	−0.240720
$749$	−27.9767	−1.02225
$750$	0	0
$751$	27.6992	1.01076	0.505378	−	0.862898i	$-0.331353\pi$
$751$	27.6992	1.01076	0.505378	+	0.862898i	$0.331353\pi$
$752$	−34.8328	−1.27022
$753$	4.46526	0.162723
$754$	0	0
$755$	−7.33094	−0.266800
$756$	12.8658	0.467923
$757$	−18.7771	−0.682465	−0.341232	−	0.939979i	$-0.610844\pi$
$757$	−18.7771	−0.682465	−0.341232	+	0.939979i	$0.610844\pi$
$758$	0	0
$759$	−2.15460	−0.0782072
$760$	0	0
$761$	19.0689	0.691246	0.345623	−	0.938373i	$-0.387668\pi$
$761$	19.0689	0.691246	0.345623	+	0.938373i	$0.387668\pi$
$762$	0	0
$763$	21.9928	0.796193
$764$	10.1803	0.368312
$765$	9.52786	0.344481
$766$	0	0
$767$	−1.90983	−0.0689600
$768$	11.6247	0.419470
$769$	−26.3607	−0.950590	−0.475295	−	0.879826i	$-0.657659\pi$
$769$	−26.3607	−0.950590	−0.475295	+	0.879826i	$0.657659\pi$
$770$	0	0
$771$	−21.5066	−0.774540
$772$	7.82045	0.281464
$773$	7.95148	0.285995	0.142997	−	0.989723i	$-0.454326\pi$
$773$	7.95148	0.285995	0.142997	+	0.989723i	$0.454326\pi$
$774$	0	0
$775$	−6.88191	−0.247205
$776$	0	0
$777$	9.14590	0.328107
$778$	0	0
$779$	0	0
$780$	−1.05573	−0.0378011
$781$	11.1352	0.398447
$782$	0	0
$783$	−9.34752	−0.334053
$784$	−17.5279	−0.625995
$785$	14.0344	0.500911
$786$	0	0
$787$	−44.1976	−1.57548	−0.787738	−	0.616011i	$-0.788748\pi$
$787$	−44.1976	−1.57548	−0.787738	+	0.616011i	$0.788748\pi$
$788$	−45.3050	−1.61392
$789$	7.92646	0.282189
$790$	0	0
$791$	−14.2128	−0.505351
$792$	0	0
$793$	2.00811	0.0713102
$794$	0	0
$795$	9.14590	0.324372
$796$	−25.1246	−0.890518
$797$	15.9434	0.564746	0.282373	−	0.959305i	$-0.408878\pi$
$797$	15.9434	0.564746	0.282373	+	0.959305i	$0.408878\pi$
$798$	0	0
$799$	−33.5623	−1.18735
$800$	0	0
$801$	−32.9160	−1.16303
$802$	0	0
$803$	−9.39512	−0.331547
$804$	−4.87539	−0.171942
$805$	5.61803	0.198010
$806$	0	0
$807$	−13.8673	−0.488151
$808$	0	0
$809$	37.7639	1.32771	0.663855	−	0.747862i	$-0.268919\pi$
$809$	37.7639	1.32771	0.663855	+	0.747862i	$0.268919\pi$
$810$	0	0
$811$	−9.16754	−0.321916	−0.160958	−	0.986961i	$-0.551458\pi$
$811$	−9.16754	−0.321916	−0.160958	+	0.986961i	$0.551458\pi$
$812$	−7.60845	−0.267004
$813$	−13.3803	−0.469268
$814$	0	0
$815$	0.909830	0.0318700
$816$	11.2007	0.392102
$817$	0	0
$818$	0	0
$819$	2.90617	0.101550
$820$	−11.9677	−0.417930
$821$	−31.5967	−1.10273	−0.551367	−	0.834263i	$-0.685894\pi$
$821$	−31.5967	−1.10273	−0.551367	+	0.834263i	$0.685894\pi$
$822$	0	0
$823$	−16.8885	−0.588698	−0.294349	−	0.955698i	$-0.595103\pi$
$823$	−16.8885	−0.588698	−0.294349	+	0.955698i	$0.595103\pi$
$824$	0	0
$825$	0.620541	0.0216045
$826$	0	0
$827$	−10.7921	−0.375279	−0.187640	−	0.982238i	$-0.560084\pi$
$827$	−10.7921	−0.375279	−0.187640	+	0.982238i	$0.560084\pi$
$828$	−17.1672	−0.596601
$829$	11.9272	0.414249	0.207125	−	0.978315i	$-0.433589\pi$
$829$	11.9272	0.414249	0.207125	+	0.978315i	$0.433589\pi$
$830$	0	0
$831$	18.4256	0.639177
$832$	5.81234	0.201507
$833$	−16.8885	−0.585153
$834$	0	0
$835$	−17.2905	−0.598363
$836$	0	0
$837$	27.3607	0.945723
$838$	0	0
$839$	−18.8496	−0.650761	−0.325381	−	0.945583i	$-0.605492\pi$
$839$	−18.8496	−0.650761	−0.325381	+	0.945583i	$0.605492\pi$
$840$	0	0
$841$	−23.4721	−0.809384
$842$	0	0
$843$	3.21478	0.110723
$844$	5.60034	0.192772
$845$	12.4721	0.429055
$846$	0	0
$847$	−16.6180	−0.571002
$848$	−50.3530	−1.72913
$849$	−17.4370	−0.598437
$850$	0	0
$851$	−27.0131	−0.925997
$852$	−18.9443	−0.649020
$853$	3.27051	0.111980	0.0559901	−	0.998431i	$-0.482168\pi$
$853$	3.27051	0.111980	0.0559901	+	0.998431i	$0.482168\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.237026	−0.00809664	−0.00404832	−	0.999992i	$-0.501289\pi$
$857$	−0.237026	−0.00809664	−0.00404832	+	0.999992i	$0.501289\pi$
$858$	0	0
$859$	−46.8328	−1.59792	−0.798958	−	0.601387i	$-0.794615\pi$
$859$	−46.8328	−1.59792	−0.798958	+	0.601387i	$0.794615\pi$
$860$	−19.2361	−0.655944
$861$	−7.03444	−0.239733
$862$	0	0
$863$	44.3691	1.51034	0.755172	−	0.655527i	$-0.227554\pi$
$863$	44.3691	1.51034	0.755172	+	0.655527i	$0.227554\pi$
$864$	0	0
$865$	−6.60440	−0.224556
$866$	0	0
$867$	−1.55909	−0.0529493
$868$	22.2703	0.755904
$869$	−13.9103	−0.471875
$870$	0	0
$871$	−2.43769	−0.0825981
$872$	0	0
$873$	−19.2331	−0.650943
$874$	0	0
$875$	−1.61803	−0.0546995
$876$	15.9839	0.540047
$877$	−29.4953	−0.995984	−0.497992	−	0.867182i	$-0.665929\pi$
$877$	−29.4953	−0.995984	−0.497992	+	0.867182i	$0.665929\pi$
$878$	0	0
$879$	4.39512	0.148244
$880$	−3.41641	−0.115167
$881$	−19.5967	−0.660231	−0.330116	−	0.943941i	$-0.607088\pi$
$881$	−19.5967	−0.660231	−0.330116	+	0.943941i	$0.607088\pi$
$882$	0	0
$883$	−20.9098	−0.703672	−0.351836	−	0.936062i	$-0.614442\pi$
$883$	−20.9098	−0.703672	−0.351836	+	0.936062i	$0.614442\pi$
$884$	5.60034	0.188360
$885$	−1.90983	−0.0641982
$886$	0	0
$887$	49.4800	1.66137	0.830687	−	0.556739i	$-0.187948\pi$
$887$	49.4800	1.66137	0.830687	+	0.556739i	$0.187948\pi$
$888$	0	0
$889$	−29.1522	−0.977735
$890$	0	0
$891$	3.86726	0.129558
$892$	10.3026	0.344957
$893$	0	0
$894$	0	0
$895$	−4.53077	−0.151447
$896$	0	0
$897$	1.83282	0.0611959
$898$	0	0
$899$	−16.1803	−0.539645
$900$	4.94427	0.164809
$901$	−48.5164	−1.61632
$902$	0	0
$903$	−11.3067	−0.376263
$904$	0	0
$905$	−7.43694	−0.247212
$906$	0	0
$907$	51.4881	1.70963	0.854817	−	0.518930i	$-0.173669\pi$
$907$	51.4881	1.70963	0.854817	+	0.518930i	$0.173669\pi$
$908$	−53.9452	−1.79023
$909$	32.3607	1.07334
$910$	0	0
$911$	−1.69011	−0.0559959	−0.0279979	−	0.999608i	$-0.508913\pi$
$911$	−1.69011	−0.0559959	−0.0279979	+	0.999608i	$0.508913\pi$
$912$	0	0
$913$	−2.31308	−0.0765519
$914$	0	0
$915$	2.00811	0.0663862
$916$	9.70820	0.320768
$917$	1.23607	0.0408186
$918$	0	0
$919$	−39.5967	−1.30618	−0.653088	−	0.757282i	$-0.726527\pi$
$919$	−39.5967	−1.30618	−0.653088	+	0.757282i	$0.726527\pi$
$920$	0	0
$921$	17.4377	0.574592
$922$	0	0
$923$	−9.47214	−0.311779
$924$	−2.00811	−0.0660621
$925$	7.77997	0.255804
$926$	0	0
$927$	−26.8416	−0.881593
$928$	0	0
$929$	43.5410	1.42853	0.714267	−	0.699873i	$-0.246760\pi$
$929$	43.5410	1.42853	0.714267	+	0.699873i	$0.246760\pi$
$930$	0	0
$931$	0	0
$932$	−25.8885	−0.848007
$933$	1.47811	0.0483911
$934$	0	0
$935$	−3.29180	−0.107653
$936$	0	0
$937$	−39.5623	−1.29244	−0.646222	−	0.763149i	$-0.723652\pi$
$937$	−39.5623	−1.29244	−0.646222	+	0.763149i	$0.723652\pi$
$938$	0	0
$939$	21.3068	0.695320
$940$	−17.4164	−0.568061
$941$	49.7980	1.62337	0.811684	−	0.584097i	$-0.198551\pi$
$941$	49.7980	1.62337	0.811684	+	0.584097i	$0.198551\pi$
$942$	0	0
$943$	20.7768	0.676584
$944$	10.5146	0.342222
$945$	6.43288	0.209262
$946$	0	0
$947$	49.1033	1.59564	0.797822	−	0.602893i	$-0.205986\pi$
$947$	49.1033	1.59564	0.797822	+	0.602893i	$0.205986\pi$
$948$	23.6656	0.768624
$949$	7.99197	0.259430
$950$	0	0
$951$	−5.90170	−0.191376
$952$	0	0
$953$	−30.2218	−0.978980	−0.489490	−	0.872009i	$-0.662817\pi$
$953$	−30.2218	−0.978980	−0.489490	+	0.872009i	$0.662817\pi$
$954$	0	0
$955$	5.09017	0.164714
$956$	−7.88854	−0.255134
$957$	1.45898	0.0471621
$958$	0	0
$959$	−9.00000	−0.290625
$960$	5.81234	0.187592
$961$	16.3607	0.527764
$962$	0	0
$963$	42.7445	1.37742
$964$	40.7364	1.31203
$965$	3.91023	0.125875
$966$	0	0
$967$	16.0689	0.516740	0.258370	−	0.966046i	$-0.416815\pi$
$967$	16.0689	0.516740	0.258370	+	0.966046i	$0.416815\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−58.1985	−1.86768	−0.933839	−	0.357694i	$-0.883563\pi$
$971$	−58.1985	−1.86768	−0.933839	+	0.357694i	$0.883563\pi$
$972$	−30.4338	−0.976165
$973$	−4.23607	−0.135802
$974$	0	0
$975$	−0.527864	−0.0169052
$976$	−11.0557	−0.353885
$977$	19.1271	0.611931	0.305966	−	0.952043i	$-0.401021\pi$
$977$	19.1271	0.611931	0.305966	+	0.952043i	$0.401021\pi$
$978$	0	0
$979$	11.3722	0.363457
$980$	−8.76393	−0.279954
$981$	−33.6020	−1.07283
$982$	0	0
$983$	−24.6215	−0.785303	−0.392651	−	0.919687i	$-0.628442\pi$
$983$	−24.6215	−0.785303	−0.392651	+	0.919687i	$0.628442\pi$
$984$	0	0
$985$	−22.6525	−0.721768
$986$	0	0
$987$	−10.2371	−0.325851
$988$	0	0
$989$	33.3951	1.06190
$990$	0	0
$991$	−28.3602	−0.900891	−0.450445	−	0.892804i	$-0.648735\pi$
$991$	−28.3602	−0.900891	−0.450445	+	0.892804i	$0.648735\pi$
$992$	0	0
$993$	12.4853	0.396209
$994$	0	0
$995$	−12.5623	−0.398252
$996$	3.93525	0.124693
$997$	4.76393	0.150875	0.0754376	−	0.997151i	$-0.475965\pi$
$997$	4.76393	0.150875	0.0754376	+	0.997151i	$0.475965\pi$
$998$	0	0
$999$	−30.9311	−0.978617

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1805.2.a.k.1.3	yes	4
5.4	even	2		9025.2.a.bi.1.2		4
19.18	odd	2	inner	1805.2.a.k.1.2	✓	4
95.94	odd	2		9025.2.a.bi.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.k.1.2	✓	4	19.18	odd	2	inner
1805.2.a.k.1.3	yes	4	1.1	even	1	trivial
9025.2.a.bi.1.2		4	5.4	even	2
9025.2.a.bi.1.3		4	95.94	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1805 = 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1805.a (trivial)

\(f(q)\)	\(=\)	\(q+0.726543 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -2.47214 q^{9} +O(q^{10})\)	\(q+0.726543 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.61803 q^{7} -2.47214 q^{9} +0.854102 q^{11} -1.45309 q^{12} -0.726543 q^{13} -0.726543 q^{15} +4.00000 q^{16} +3.85410 q^{17} +2.00000 q^{20} +1.17557 q^{21} -3.47214 q^{23} +1.00000 q^{25} -3.97574 q^{27} -3.23607 q^{28} +2.35114 q^{29} -6.88191 q^{31} +0.620541 q^{33} -1.61803 q^{35} +4.94427 q^{36} +7.77997 q^{37} -0.527864 q^{39} -5.98385 q^{41} -9.61803 q^{43} -1.70820 q^{44} +2.47214 q^{45} -8.70820 q^{47} +2.90617 q^{48} -4.38197 q^{49} +2.80017 q^{51} +1.45309 q^{52} -12.5882 q^{53} -0.854102 q^{55} +2.62866 q^{59} +1.45309 q^{60} -2.76393 q^{61} -4.00000 q^{63} -8.00000 q^{64} +0.726543 q^{65} +3.35520 q^{67} -7.70820 q^{68} -2.52265 q^{69} +13.0373 q^{71} -11.0000 q^{73} +0.726543 q^{75} +1.38197 q^{77} -16.2865 q^{79} -4.00000 q^{80} +4.52786 q^{81} -2.70820 q^{83} -2.35114 q^{84} -3.85410 q^{85} +1.70820 q^{87} +13.3148 q^{89} -1.17557 q^{91} +6.94427 q^{92} -5.00000 q^{93} +7.77997 q^{97} -2.11146 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 - 4 * q^5 + 2 * q^7 + 8 * q^9	\( 4 q - 8 q^{4} - 4 q^{5} + 2 q^{7} + 8 q^{9} - 10 q^{11} + 16 q^{16} + 2 q^{17} + 8 q^{20} + 4 q^{23} + 4 q^{25} - 4 q^{28} - 2 q^{35} - 16 q^{36} - 20 q^{39} - 34 q^{43} + 20 q^{44} - 8 q^{45} - 8 q^{47} - 22 q^{49} + 10 q^{55} - 20 q^{61} - 16 q^{63} - 32 q^{64} - 4 q^{68} - 44 q^{73} + 10 q^{77} - 16 q^{80} + 36 q^{81} + 16 q^{83} - 2 q^{85} - 20 q^{87} - 8 q^{92} - 20 q^{93} - 80 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^5 + 2 * q^7 + 8 * q^9 - 10 * q^11 + 16 * q^16 + 2 * q^17 + 8 * q^20 + 4 * q^23 + 4 * q^25 - 4 * q^28 - 2 * q^35 - 16 * q^36 - 20 * q^39 - 34 * q^43 + 20 * q^44 - 8 * q^45 - 8 * q^47 - 22 * q^49 + 10 * q^55 - 20 * q^61 - 16 * q^63 - 32 * q^64 - 4 * q^68 - 44 * q^73 + 10 * q^77 - 16 * q^80 + 36 * q^81 + 16 * q^83 - 2 * q^85 - 20 * q^87 - 8 * q^92 - 20 * q^93 - 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more