LMFDB - Embedded newform 1805.2.b.h.1084.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1084,1805)]

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([1, 0]))

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

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

chi := DirichletCharacter("1805.1084");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1805 = 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1805.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$14.4129975648$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.280944640000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 80x^{4} + 128x^{2} + 19 $ x^8 + 16x^6 + 80x^4 + 128*x^2 + 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1084.5
Root			$0.406045i$ of defining polynomial
Character	$\chi$	$=$	1805.1084
Dual form			1805.2.b.h.1084.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.406045i q^{2} +3.27727i q^{3} +1.83513 q^{4} +2.23607 q^{5} -1.33072 q^{6} +1.55723i q^{8} -7.74048 q^{9} +O(q^{10})$	$q+0.406045i q^{2} +3.27727i q^{3} +1.83513 q^{4} +2.23607 q^{5} -1.33072 q^{6} +1.55723i q^{8} -7.74048 q^{9} +0.907944i q^{10} -2.92978 q^{11} +6.01420i q^{12} +6.51610i q^{13} +7.32819i q^{15} +3.03795 q^{16} -3.14298i q^{18} +4.10347 q^{20} -1.18962i q^{22} -5.10347 q^{24} +5.00000 q^{25} -2.64583 q^{26} -15.5358i q^{27} -2.97557 q^{30} +4.34801i q^{32} -9.60166i q^{33} -14.2048 q^{36} -8.01591i q^{37} -21.3550 q^{39} +3.48208i q^{40} -5.37651 q^{44} -17.3082 q^{45} +9.95617i q^{48} +7.00000 q^{49} +2.03022i q^{50} +11.9579i q^{52} +5.09315i q^{53} +6.30824 q^{54} -6.55118 q^{55} +13.4482i q^{60} -1.11908 q^{61} +4.31041 q^{64} +14.5704i q^{65} +3.89870 q^{66} -8.95237i q^{67} -12.0537i q^{72} +3.25482 q^{74} +16.3863i q^{75} -8.67109i q^{78} +6.79306 q^{80} +27.6936 q^{81} -4.56235i q^{88} -7.02792i q^{90} -14.2496 q^{96} +11.6477i q^{97} +2.84231i q^{98} +22.6779 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} - 24 q^{9}+O(q^{10}) $ 8 * q - 16 * q^4 - 24 * q^9	$ 8 q - 16 q^{4} - 24 q^{9} + 32 q^{16} - 8 q^{24} + 40 q^{25} + 24 q^{26} - 40 q^{30} - 8 q^{36} - 72 q^{44} + 56 q^{49} - 88 q^{54} - 64 q^{64} + 104 q^{66} + 120 q^{80} + 72 q^{81} - 120 q^{96} - 32 q^{99}+O(q^{100}) $ 8 * q - 16 * q^4 - 24 * q^9 + 32 * q^16 - 8 * q^24 + 40 * q^25 + 24 * q^26 - 40 * q^30 - 8 * q^36 - 72 * q^44 + 56 * q^49 - 88 * q^54 - 64 * q^64 + 104 * q^66 + 120 * q^80 + 72 * q^81 - 120 * q^96 - 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$.

$n$	$362$	$1446$
$\chi(n)$	$-1$	$1$

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 $

$2$	0.406045i	0.287117i	0.989642	+	0.143559i	$0.0458545\pi$
$2$	0.406045i	0.287117i	−0.989642	+	0.143559i	$0.954145\pi$
$3$	3.27727i	1.89213i	0.323976	+	0.946065i	$0.394980\pi$
$3$	3.27727i	1.89213i	−0.323976	+	0.946065i	$0.605020\pi$
$4$	1.83513	0.917564
$5$	2.23607	1.00000
$5$	2.23607	1.00000
$6$	−1.33072	−0.543263
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.55723i	0.550565i
$9$	−7.74048	−2.58016
$10$	0.907944i	0.287117i
$11$	−2.92978	−0.883361	−0.441680	−	0.897172i	$-0.645618\pi$
$11$	−2.92978	−0.883361	−0.441680	+	0.897172i	$0.645618\pi$
$12$	6.01420i	1.73615i
$13$	6.51610i	1.80724i	0.428333	+	0.903621i	$0.359101\pi$
$13$	6.51610i	1.80724i	−0.428333	+	0.903621i	$0.640899\pi$
$14$	0	0
$15$	7.32819i	1.89213i
$16$	3.03795	0.759487
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 3.14298i	− 0.740808i
$19$	0	0
$19$	0	0
$20$	4.10347	0.917564
$21$	0	0
$22$	− 1.18962i	− 0.253628i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−5.10347	−1.04174
$25$	5.00000	1.00000
$26$	−2.64583	−0.518890
$27$	− 15.5358i	− 2.98987i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−2.97557	−0.543263
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	4.34801i	0.768627i
$33$	− 9.60166i	− 1.67143i
$34$	0	0
$35$	0	0
$36$	−14.2048	−2.36746
$37$	− 8.01591i	− 1.31781i	−0.752227	−	0.658904i	$-0.771020\pi$
$37$	− 8.01591i	− 1.31781i	0.752227	−	0.658904i	$-0.228980\pi$
$38$	0	0
$39$	−21.3550	−3.41954
$40$	3.48208i	0.550565i
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−5.37651	−0.810540
$45$	−17.3082	−2.58016
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	9.95617i	1.43705i
$49$	7.00000	1.00000
$50$	2.03022i	0.287117i
$51$	0	0
$52$	11.9579i	1.65826i
$53$	5.09315i	0.699599i	0.936825	+	0.349799i	$0.113750\pi$
$53$	5.09315i	0.699599i	−0.936825	+	0.349799i	$0.886250\pi$
$54$	6.30824	0.858442
$55$	−6.55118	−0.883361
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	13.4482i	1.73615i
$61$	−1.11908	−0.143283	−0.0716414	−	0.997430i	$-0.522824\pi$
$61$	−1.11908	−0.143283	−0.0716414	+	0.997430i	$0.522824\pi$
$62$	0	0
$63$	0	0
$64$	4.31041	0.538801
$65$	14.5704i	1.80724i
$66$	3.89870	0.479897
$67$	− 8.95237i	− 1.09371i	−0.837229	−	0.546853i	$-0.815826\pi$
$67$	− 8.95237i	− 1.09371i	0.837229	−	0.546853i	$-0.184174\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 12.0537i	− 1.42055i
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	3.25482	0.378365
$75$	16.3863i	1.89213i
$76$	0	0
$77$	0	0
$78$	− 8.67109i	− 0.981807i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	6.79306	0.759487
$81$	27.6936	3.07706
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	− 4.56235i	− 0.486348i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	− 7.02792i	− 0.740808i
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	−14.2496	−1.45434
$97$	11.6477i	1.18264i	0.806436	+	0.591322i	$0.201394\pi$
$97$	11.6477i	1.18264i	−0.806436	+	0.591322i	$0.798606\pi$
$98$	2.84231i	0.287117i
$99$	22.6779	2.27921
$100$	9.17564	0.917564
$101$	−20.0810	−1.99813	−0.999067	−	0.0431977i	$-0.986245\pi$
$101$	−20.0810	−1.99813	−0.999067	+	0.0431977i	$0.986245\pi$
$102$	0	0
$103$	4.07983i	0.401998i	0.979591	+	0.200999i	$0.0644188\pi$
$103$	4.07983i	0.401998i	−0.979591	+	0.200999i	$0.935581\pi$
$104$	−10.1471	−0.995004
$105$	0	0
$106$	−2.06805	−0.200867
$107$	− 20.3604i	− 1.96831i	−0.177300	−	0.984157i	$-0.556736\pi$
$107$	− 20.3604i	− 1.96831i	0.177300	−	0.984157i	$-0.443264\pi$
$108$	− 28.5102i	− 2.74340i
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	− 2.66007i	− 0.253628i
$111$	26.2703	2.49347
$112$	0	0
$113$	21.1725i	1.99174i	0.0907914	+	0.995870i	$0.471060\pi$
$113$	21.1725i	1.99174i	−0.0907914	+	0.995870i	$0.528940\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 50.4377i	− 4.66297i
$118$	0	0
$119$	0	0
$120$	−11.4117	−1.04174
$121$	−2.41641	−0.219673
$122$	− 0.454395i	− 0.0411390i
$123$	0	0
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	6.20003i	0.550163i	0.961421	+	0.275082i	$0.0887049\pi$
$127$	6.20003i	0.550163i	−0.961421	+	0.275082i	$0.911295\pi$
$128$	10.4462i	0.923326i
$129$	0	0
$130$	−5.91625	−0.518890
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	− 17.6203i	− 1.53365i
$133$	0	0
$134$	3.63506	0.314022
$135$	− 34.7391i	− 2.98987i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	21.8917	1.85683	0.928414	−	0.371546i	$-0.121172\pi$
$139$	21.8917	1.85683	0.928414	+	0.371546i	$0.121172\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 19.0907i	− 1.59645i
$144$	−23.5152	−1.95960
$145$	0	0
$146$	0	0
$147$	22.9409i	1.89213i
$148$	− 14.7102i	− 1.20917i
$149$	8.36188	0.685032	0.342516	−	0.939512i	$-0.388721\pi$
$149$	8.36188	0.685032	0.342516	+	0.939512i	$0.388721\pi$
$150$	−6.65359	−0.543263
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−39.1892	−3.13764
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−16.6916	−1.32373
$160$	9.72245i	0.768627i
$161$	0	0
$162$	11.2448i	0.883477i
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	− 21.4700i	− 1.67143i
$166$	0	0
$167$	− 19.3091i	− 1.49418i	−0.664721	−	0.747091i	$-0.731450\pi$
$167$	− 19.3091i	− 1.49418i	0.664721	−	0.747091i	$-0.268550\pi$
$168$	0	0
$169$	−29.4596	−2.26612
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.7967i	1.73320i	0.499005	+	0.866599i	$0.333699\pi$
$173$	22.7967i	1.73320i	−0.499005	+	0.866599i	$0.666301\pi$
$174$	0	0
$175$	0	0
$176$	−8.90051	−0.670901
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−31.7628	−2.36746
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	− 3.66751i	− 0.271110i
$184$	0	0
$185$	− 17.9241i	− 1.31781i
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.94427	−0.647185	−0.323592	−	0.946197i	$-0.604891\pi$
$191$	−8.94427	−0.647185	−0.323592	+	0.946197i	$0.604891\pi$
$192$	14.1264i	1.01948i
$193$	27.6795i	1.99242i	0.0870089	+	0.996208i	$0.472269\pi$
$193$	27.6795i	1.99242i	−0.0870089	+	0.996208i	$0.527731\pi$
$194$	−4.72948	−0.339557
$195$	−47.7512	−3.41954
$196$	12.8459	0.917564
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	9.20823i	0.654401i
$199$	26.8328	1.90213	0.951064	−	0.308994i	$-0.0999924\pi$
$199$	26.8328	1.90213	0.951064	+	0.308994i	$0.0999924\pi$
$200$	7.78617i	0.550565i
$201$	29.3393	2.06944
$202$	− 8.15378i	− 0.573698i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−1.65660	−0.115420
$207$	0	0
$208$	19.7956i	1.37258i
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	9.34659i	0.641926i
$213$	0	0
$214$	8.26723	0.565136
$215$	0	0
$216$	24.1929	1.64612
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−12.0223	−0.810540
$221$	0	0
$222$	10.6669i	0.715916i
$223$	− 25.8636i	− 1.73196i	−0.500082	−	0.865978i	$-0.666697\pi$
$223$	− 25.8636i	− 1.73196i	0.500082	−	0.865978i	$-0.333303\pi$
$224$	0	0
$225$	−38.7024	−2.58016
$226$	−8.59698	−0.571862
$227$	− 23.6088i	− 1.56697i	−0.621412	−	0.783484i	$-0.713441\pi$
$227$	− 23.6088i	− 1.56697i	0.621412	−	0.783484i	$-0.286559\pi$
$228$	0	0
$229$	29.5619	1.95351	0.976754	−	0.214362i	$-0.0687674\pi$
$229$	29.5619	1.95351	0.976754	+	0.214362i	$0.0687674\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	20.4800	1.33882
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	22.2627i	1.43705i
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	− 0.981170i	− 0.0630720i
$243$	44.1518i	2.83234i
$244$	−2.05365	−0.131471
$245$	15.6525	1.00000
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	4.53972i	0.287117i
$251$	17.8885	1.12911	0.564557	−	0.825394i	$-0.309047\pi$
$251$	17.8885	1.12911	0.564557	+	0.825394i	$0.309047\pi$
$252$	0	0
$253$	0	0
$254$	−2.51749	−0.157961
$255$	0	0
$256$	4.37918	0.273699
$257$	− 24.7568i	− 1.54428i	−0.635450	−	0.772142i	$-0.719185\pi$
$257$	− 24.7568i	− 1.54428i	0.635450	−	0.772142i	$-0.280815\pi$
$258$	0	0
$259$	0	0
$260$	26.7386i	1.65826i
$261$	0	0
$262$	4.87254i	0.301026i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	14.9520	0.920234
$265$	11.3886i	0.699599i
$266$	0	0
$267$	0	0
$268$	− 16.4287i	− 1.00355i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	14.1056	0.858442
$271$	24.1298	1.46578	0.732892	−	0.680345i	$-0.238170\pi$
$271$	24.1298	1.46578	0.732892	+	0.680345i	$0.238170\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−14.6489	−0.883361
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	8.88901i	0.533127i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	7.75169	0.458367
$287$	0	0
$288$	− 33.6557i	− 1.98318i
$289$	17.0000	1.00000
$290$	0	0
$291$	−38.1726	−2.23772
$292$	0	0
$293$	− 34.2340i	− 1.99997i	−0.00505234	−	0.999987i	$-0.501608\pi$
$293$	− 34.2340i	− 1.99997i	0.00505234	−	0.999987i	$-0.498392\pi$
$294$	−9.31502	−0.543263
$295$	0	0
$296$	12.4826	0.725539
$297$	45.5165i	2.64113i
$298$	3.39530i	0.196684i
$299$	0	0
$300$	30.0710i	1.73615i
$301$	0	0
$302$	0	0
$303$	− 65.8108i	− 3.78073i
$304$	0	0
$305$	−2.50233	−0.143283
$306$	0	0
$307$	35.0168i	1.99851i	0.0385528	+	0.999257i	$0.487725\pi$
$307$	35.0168i	1.99851i	−0.0385528	+	0.999257i	$0.512275\pi$
$308$	0	0
$309$	−13.3707	−0.760633
$310$	0	0
$311$	31.3726	1.77898	0.889490	−	0.456955i	$-0.151060\pi$
$311$	31.3726	1.77898	0.889490	+	0.456955i	$0.151060\pi$
$312$	− 33.2547i	− 1.88268i
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.2999i	0.915496i	0.889082	+	0.457748i	$0.151344\pi$
$317$	16.2999i	0.915496i	−0.889082	+	0.457748i	$0.848656\pi$
$318$	− 6.77755i	− 0.380066i
$319$	0	0
$320$	9.63837	0.538801
$321$	66.7265	3.72431
$322$	0	0
$323$	0	0
$324$	50.8212	2.82340
$325$	32.5805i	1.80724i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	8.71777	0.479897
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	62.0470i	3.40015i
$334$	7.84036	0.429005
$335$	− 20.0181i	− 1.09371i
$336$	0	0
$337$	1.64356i	0.0895307i	0.998998	+	0.0447653i	$0.0142540\pi$
$337$	1.64356i	0.0895307i	−0.998998	+	0.0447653i	$0.985746\pi$
$338$	− 11.9619i	− 0.650642i
$339$	−69.3879	−3.76863
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−9.25647	−0.497631
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−13.4164	−0.718164	−0.359082	−	0.933306i	$-0.616910\pi$
$349$	−13.4164	−0.718164	−0.359082	+	0.933306i	$0.616910\pi$
$350$	0	0
$351$	101.233	5.40341
$352$	− 12.7387i	− 0.678975i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.5131	1.34653	0.673265	−	0.739401i	$-0.264891\pi$
$359$	25.5131	1.34653	0.673265	+	0.739401i	$0.264891\pi$
$360$	− 26.9530i	− 1.42055i
$361$	0	0
$362$	0	0
$363$	− 7.91921i	− 0.415651i
$364$	0	0
$365$	0	0
$366$	1.48917	0.0778403
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	7.27800	0.378365
$371$	0	0
$372$	0	0
$373$	31.3113i	1.62124i	0.585575	+	0.810619i	$0.300869\pi$
$373$	31.3113i	1.62124i	−0.585575	+	0.810619i	$0.699131\pi$
$374$	0	0
$375$	36.6410i	1.89213i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−20.3191	−1.04098
$382$	− 3.63178i	− 0.185818i
$383$	25.2329i	1.28934i	0.764460	+	0.644671i	$0.223006\pi$
$383$	25.2329i	1.28934i	−0.764460	+	0.644671i	$0.776994\pi$
$384$	−34.2351	−1.74705
$385$	0	0
$386$	−11.2391	−0.572056
$387$	0	0
$388$	21.3750i	1.08515i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	− 19.3891i	− 0.981807i
$391$	0	0
$392$	10.9006i	0.550565i
$393$	39.3272i	1.98379i
$394$	0	0
$395$	0	0
$396$	41.6168	2.09132
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	10.8953i	0.546133i
$399$	0	0
$400$	15.1897	0.759487
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	11.9131i	0.594170i
$403$	0	0
$404$	−36.8512	−1.83341
$405$	61.9247	3.07706
$406$	0	0
$407$	23.4848i	1.16410i
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	7.48701i	0.368859i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−28.3321	−1.38909
$417$	71.7449i	3.51336i
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	0	0
$424$	−7.93123	−0.385175
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 37.3639i	− 1.80605i
$429$	62.5654	3.02069
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 47.1970i	− 2.27077i
$433$	− 37.4530i	− 1.79988i	−0.436015	−	0.899939i	$-0.643611\pi$
$433$	− 37.4530i	− 1.79988i	0.436015	−	0.899939i	$-0.356389\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	− 10.2017i	− 0.486348i
$441$	−54.1833	−2.58016
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	48.2093	2.28791
$445$	0	0
$446$	10.5018	0.497274
$447$	27.4041i	1.29617i
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	− 15.7149i	− 0.740808i
$451$	0	0
$452$	38.8542i	1.82755i
$453$	0	0
$454$	9.58621	0.449903
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	12.0035i	0.560886i
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	− 92.5597i	− 4.27857i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 39.4235i	− 1.80508i
$478$	− 9.74507i	− 0.445729i
$479$	−43.0918	−1.96891	−0.984456	−	0.175630i	$-0.943804\pi$
$479$	−43.0918	−1.96891	−0.984456	+	0.175630i	$0.943804\pi$
$480$	−31.8631	−1.45434
$481$	52.2325	2.38160
$482$	0	0
$483$	0	0
$484$	−4.43442	−0.201564
$485$	26.0450i	1.18264i
$486$	−17.9276	−0.813213
$487$	− 30.1442i	− 1.36597i	−0.730434	−	0.682983i	$-0.760682\pi$
$487$	− 30.1442i	− 1.36597i	0.730434	−	0.682983i	$-0.239318\pi$
$488$	− 1.74266i	− 0.0788866i
$489$	0	0
$490$	6.35561i	0.287117i
$491$	−35.7771	−1.61460	−0.807299	−	0.590143i	$-0.799071\pi$
$491$	−35.7771	−1.61460	−0.807299	+	0.590143i	$0.799071\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	50.7093	2.27921
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.31303	0.193078	0.0965389	−	0.995329i	$-0.469223\pi$
$499$	4.31303	0.193078	0.0965389	+	0.995329i	$0.469223\pi$
$500$	20.5174	0.917564
$501$	63.2811	2.82719
$502$	7.26355i	0.324188i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−44.9025	−1.99813
$506$	0	0
$507$	− 96.5469i	− 4.28780i
$508$	11.3778i	0.504810i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	22.6706i	1.00191i
$513$	0	0
$514$	10.0524	0.443390
$515$	9.12278i	0.401998i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−74.7108	−3.27944
$520$	−22.6896	−0.995004
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	− 22.2319i	− 0.972131i	−0.873922	−	0.486066i	$-0.838432\pi$
$523$	− 22.2319i	− 0.972131i	0.873922	−	0.486066i	$-0.161568\pi$
$524$	22.0215	0.962015
$525$	0	0
$526$	0	0
$527$	0	0
$528$	− 29.1694i	− 1.26943i
$529$	23.0000	1.00000
$530$	−4.62430	−0.200867
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 45.5272i	− 1.96831i
$536$	13.9409	0.602157
$537$	0	0
$538$	0	0
$539$	−20.5084	−0.883361
$540$	− 63.7507i	− 2.74340i
$541$	27.3238	1.17474	0.587371	−	0.809318i	$-0.300163\pi$
$541$	27.3238	1.17474	0.587371	+	0.809318i	$0.300163\pi$
$542$	9.79780i	0.420851i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.56825i	0.109810i	0.998492	+	0.0549052i	$0.0174857\pi$
$547$	2.56825i	0.109810i	−0.998492	+	0.0549052i	$0.982514\pi$
$548$	0	0
$549$	8.66218	0.369693
$550$	− 5.94810i	− 0.253628i
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	58.7421	2.49347
$556$	40.1740	1.70376
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.7864i	1.21320i	0.795007	+	0.606601i	$0.207467\pi$
$563$	28.7864i	1.21320i	−0.795007	+	0.606601i	$0.792533\pi$
$564$	0	0
$565$	47.3431i	1.99174i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	39.4704	1.65178	0.825891	−	0.563829i	$-0.190672\pi$
$571$	39.4704	1.65178	0.825891	+	0.563829i	$0.190672\pi$
$572$	− 35.0339i	− 1.46484i
$573$	− 29.3128i	− 1.22456i
$574$	0	0
$575$	0	0
$576$	−33.3646	−1.39019
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	6.90276i	0.287117i
$579$	−90.7132	−3.76991
$580$	0	0
$581$	0	0
$582$	− 15.4998i	− 0.642486i
$583$	− 14.9218i	− 0.617998i
$584$	0	0
$585$	− 112.782i	− 4.66297i
$586$	13.9006	0.574227
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	42.0994i	1.73615i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	− 24.3519i	− 1.00086i
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	−18.4817	−0.758314
$595$	0	0
$596$	15.3451	0.628561
$597$	87.9383i	3.59907i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	−25.5174	−1.04174
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	69.2956i	2.82194i
$604$	0	0
$605$	−5.40325	−0.219673
$606$	26.7221	1.08551
$607$	− 13.8249i	− 0.561136i	−0.959834	−	0.280568i	$-0.909477\pi$
$607$	− 13.8249i	− 0.561136i	0.959834	−	0.280568i	$-0.0905228\pi$
$608$	0	0
$609$	0	0
$610$	− 1.01606i	− 0.0411390i
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−14.2184	−0.573807
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	− 5.42910i	− 0.218391i
$619$	−34.9941	−1.40653	−0.703265	−	0.710928i	$-0.748275\pi$
$619$	−34.9941	−1.40653	−0.703265	+	0.710928i	$0.748275\pi$
$620$	0	0
$621$	0	0
$622$	12.7387i	0.510775i
$623$	0	0
$624$	−64.8754	−2.59709
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−11.0275	−0.438997	−0.219499	−	0.975613i	$-0.570442\pi$
$631$	−11.0275	−0.438997	−0.219499	+	0.975613i	$0.570442\pi$
$632$	0	0
$633$	0	0
$634$	−6.61851	−0.262855
$635$	13.8637i	0.550163i
$636$	−30.6313	−1.21461
$637$	45.6127i	1.80724i
$638$	0	0
$639$	0	0
$640$	23.3585i	0.923326i
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	27.0939i	1.06931i
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	43.1254i	1.69412i
$649$	0	0
$650$	−13.2291	−0.518890
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	26.8328	1.04844
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	− 39.4001i	− 1.53365i
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−25.1939	−0.976242
$667$	0	0
$668$	− 35.4347i	− 1.37101i
$669$	84.7620	3.27709
$670$	8.12825	0.314022
$671$	3.27864	0.126571
$672$	0	0
$673$	− 50.4853i	− 1.94606i	−0.230671	−	0.973032i	$-0.574092\pi$
$673$	− 50.4853i	− 1.94606i	0.230671	−	0.973032i	$-0.425908\pi$
$674$	−0.667361	−0.0257058
$675$	− 77.6791i	− 2.98987i
$676$	−54.0621	−2.07931
$677$	− 44.4204i	− 1.70721i	−0.520918	−	0.853607i	$-0.674410\pi$
$677$	− 44.4204i	− 1.70721i	0.520918	−	0.853607i	$-0.325590\pi$
$678$	− 28.1746i	− 1.08204i
$679$	0	0
$680$	0	0
$681$	77.3722	2.96491
$682$	0	0
$683$	− 10.5408i	− 0.403333i	−0.979454	−	0.201667i	$-0.935364\pi$
$683$	− 10.5408i	− 0.403333i	0.979454	−	0.201667i	$-0.0646357\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	96.8824i	3.69629i
$688$	0	0
$689$	−33.1875	−1.26434
$690$	0	0
$691$	−52.5727	−1.99996	−0.999980	−	0.00630823i	$-0.997992\pi$
$691$	−52.5727	−1.99996	−0.999980	+	0.00630823i	$0.997992\pi$
$692$	41.8348i	1.59032i
$693$	0	0
$694$	0	0
$695$	48.9513	1.85683
$696$	0	0
$697$	0	0
$698$	− 5.44766i	− 0.206197i
$699$	0	0
$700$	0	0
$701$	−48.5239	−1.83272	−0.916360	−	0.400354i	$-0.868887\pi$
$701$	−48.5239	−1.83272	−0.916360	+	0.400354i	$0.868887\pi$
$702$	41.1051i	1.55141i
$703$	0	0
$704$	−12.6285	−0.475956
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	40.2492	1.51159	0.755796	−	0.654808i	$-0.227250\pi$
$709$	40.2492	1.51159	0.755796	+	0.654808i	$0.227250\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 42.6882i	− 1.59645i
$716$	0	0
$717$	− 78.6544i	− 2.93740i
$718$	10.3595i	0.386612i
$719$	−13.7940	−0.514429	−0.257214	−	0.966354i	$-0.582805\pi$
$719$	−13.7940	−0.514429	−0.257214	+	0.966354i	$0.582805\pi$
$720$	−52.5815	−1.95960
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	3.21556	0.119340
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−61.6165	−2.28209
$730$	0	0
$731$	0	0
$732$	− 6.73035i	− 0.248761i
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	51.2973i	1.89213i
$736$	0	0
$737$	26.2284i	0.966137i
$738$	0	0
$739$	−53.6656	−1.97412	−0.987061	−	0.160345i	$-0.948739\pi$
$739$	−53.6656	−1.97412	−0.987061	+	0.160345i	$0.948739\pi$
$740$	− 32.8931i	− 1.20917i
$741$	0	0
$742$	0	0
$743$	− 39.6817i	− 1.45578i	−0.685693	−	0.727890i	$-0.740501\pi$
$743$	− 39.6817i	− 1.45578i	0.685693	−	0.727890i	$-0.259499\pi$
$744$	0	0
$745$	18.6977	0.685032
$746$	−12.7138	−0.465485
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−14.8779	−0.543263
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	58.6255i	2.13643i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−54.3834	−1.97140	−0.985699	−	0.168518i	$-0.946102\pi$
$761$	−54.3834	−1.97140	−0.985699	+	0.168518i	$0.946102\pi$
$762$	− 8.25048i	− 0.298883i
$763$	0	0
$764$	−16.4139	−0.593833
$765$	0	0
$766$	−10.2457	−0.370192
$767$	0	0
$768$	14.3517i	0.517874i
$769$	47.1406	1.69993	0.849967	−	0.526836i	$-0.176622\pi$
$769$	47.1406	1.69993	0.849967	+	0.526836i	$0.176622\pi$
$770$	0	0
$771$	81.1345	2.92199
$772$	50.7954i	1.82817i
$773$	− 13.0516i	− 0.469433i	−0.972064	−	0.234717i	$-0.924584\pi$
$773$	− 13.0516i	− 0.469433i	0.972064	−	0.234717i	$-0.0754162\pi$
$774$	0	0
$775$	0	0
$776$	−18.1382	−0.651122
$777$	0	0
$778$	− 2.43627i	− 0.0873445i
$779$	0	0
$780$	−87.6296	−3.13764
$781$	0	0
$782$	0	0
$783$	0	0
$784$	21.2656	0.759487
$785$	0	0
$786$	−15.9686	−0.569581
$787$	17.0954i	0.609383i	0.952451	+	0.304692i	$0.0985534\pi$
$787$	17.0954i	0.609383i	−0.952451	+	0.304692i	$0.901447\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	35.3147i	1.25485i
$793$	− 7.29201i	− 0.258947i
$794$	0	0
$795$	−37.3236	−1.32373
$796$	49.2416	1.74532
$797$	13.8614i	0.490997i	0.969397	+	0.245499i	$0.0789517\pi$
$797$	13.8614i	0.490997i	−0.969397	+	0.245499i	$0.921048\pi$
$798$	0	0
$799$	0	0
$800$	21.7401i	0.768627i
$801$	0	0
$802$	0	0
$803$	0	0
$804$	53.8414	1.89884
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 31.2708i	− 1.10010i
$809$	22.3607	0.786160	0.393080	−	0.919504i	$-0.371410\pi$
$809$	22.3607	0.786160	0.393080	+	0.919504i	$0.371410\pi$
$810$	25.1442i	0.883477i
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	79.0799i	2.77345i
$814$	−9.53590	−0.334233
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	−6.35325	−0.221326
$825$	− 48.0083i	− 1.67143i
$826$	0	0
$827$	48.4500i	1.68477i	0.538875	+	0.842386i	$0.318849\pi$
$827$	48.4500i	1.68477i	−0.538875	+	0.842386i	$0.681151\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	28.0871i	0.973744i
$833$	0	0
$834$	−29.1316	−1.00875
$835$	− 43.1764i	− 1.49418i
$836$	0	0
$837$	0	0
$838$	− 14.6176i	− 0.504957i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−65.8736	−2.26612
$846$	0	0
$847$	0	0
$848$	15.4727i	0.531336i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	31.7059	1.08369
$857$	33.5250i	1.14519i	0.819837	+	0.572597i	$0.194064\pi$
$857$	33.5250i	1.14519i	−0.819837	+	0.572597i	$0.805936\pi$
$858$	25.4044i	0.867290i
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 58.6363i	− 1.99600i	−0.0631923	−	0.998001i	$-0.520128\pi$
$863$	− 58.6363i	− 1.99600i	0.0631923	−	0.998001i	$-0.479872\pi$
$864$	67.5499	2.29809
$865$	50.9749i	1.73320i
$866$	15.2076	0.516776
$867$	55.7135i	1.89213i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	58.3346	1.97659
$872$	0	0
$873$	− 90.1587i	− 3.05141i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.752363i	0.0254055i	0.999919	+	0.0127028i	$0.00404352\pi$
$877$	0.752363i	0.0254055i	−0.999919	+	0.0127028i	$0.995956\pi$
$878$	0	0
$879$	112.194	3.78421
$880$	−19.9021	−0.670901
$881$	9.21678	0.310521	0.155261	−	0.987874i	$-0.450378\pi$
$881$	9.21678	0.310521	0.155261	+	0.987874i	$0.450378\pi$
$882$	− 22.0009i	− 0.740808i
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 28.4813i	− 0.956308i	−0.878276	−	0.478154i	$-0.841306\pi$
$887$	− 28.4813i	− 0.956308i	0.878276	−	0.478154i	$-0.158694\pi$
$888$	40.9090i	1.37282i
$889$	0	0
$890$	0	0
$891$	−81.1360	−2.71816
$892$	− 47.4631i	− 1.58918i
$893$	0	0
$894$	−11.1273	−0.372153
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−71.0238	−2.36746
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−32.9705	−1.09658
$905$	0	0
$906$	0	0
$907$	− 52.9215i	− 1.75723i	−0.477531	−	0.878615i	$-0.658468\pi$
$907$	− 52.9215i	− 1.75723i	0.477531	−	0.878615i	$-0.341532\pi$
$908$	− 43.3251i	− 1.43779i
$909$	155.436	5.15550
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 8.20080i	− 0.271110i
$916$	54.2499	1.79247
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−114.759	−3.78145
$922$	7.30881i	0.240703i
$923$	0	0
$924$	0	0
$925$	− 40.0796i	− 1.31781i
$926$	0	0
$927$	− 31.5799i	− 1.03722i
$928$	0	0
$929$	31.3050	1.02708	0.513541	−	0.858065i	$-0.328333\pi$
$929$	31.3050	1.02708	0.513541	+	0.858065i	$0.328333\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	102.817i	3.36606i
$934$	0	0
$935$	0	0
$936$	78.5434	2.56727
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−53.4193	−1.73224
$952$	0	0
$953$	5.80217i	0.187951i	0.995575	+	0.0939754i	$0.0299575\pi$
$953$	5.80217i	0.187951i	−0.995575	+	0.0939754i	$0.970043\pi$
$954$	16.0077	0.518268
$955$	−20.0000	−0.647185
$956$	−44.0431	−1.42445
$957$	0	0
$958$	− 17.4972i	− 0.565308i
$959$	0	0
$960$	31.5875i	1.01948i
$961$	−31.0000	−1.00000
$962$	21.2087i	0.683797i
$963$	157.599i	5.07856i
$964$	0	0
$965$	61.8933i	1.99242i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	− 3.76291i	− 0.120945i
$969$	0	0
$970$	−10.5754	−0.339557
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	81.0242i	2.59885i
$973$	0	0
$974$	12.2399	0.392192
$975$	−106.775	−3.41954
$976$	−3.39969	−0.108822
$977$	− 60.2691i	− 1.92818i	−0.265577	−	0.964090i	$-0.585562\pi$
$977$	− 60.2691i	− 1.92818i	0.265577	−	0.964090i	$-0.414438\pi$
$978$	0	0
$979$	0	0
$980$	28.7243	0.917564
$981$	0	0
$982$	− 14.5271i	− 0.463578i
$983$	− 62.7054i	− 1.99999i	−0.00306979	−	0.999995i	$-0.500977\pi$
$983$	− 62.7054i	− 1.99999i	0.00306979	−	0.999995i	$-0.499023\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	20.5902i	0.654401i
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	60.0000	1.90213
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	1.75128i	0.0554359i
$999$	−124.534	−3.94007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1805.2.b.h.1084.5	yes	8
5.2	odd	4		9025.2.a.cb.1.4		8
5.3	odd	4		9025.2.a.cb.1.5		8
5.4	even	2	inner	1805.2.b.h.1084.4	✓	8
19.18	odd	2	inner	1805.2.b.h.1084.4	✓	8
95.18	even	4		9025.2.a.cb.1.4		8
95.37	even	4		9025.2.a.cb.1.5		8
95.94	odd	2	CM	1805.2.b.h.1084.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.h.1084.4	✓	8	5.4	even	2	inner
1805.2.b.h.1084.4	✓	8	19.18	odd	2	inner
1805.2.b.h.1084.5	yes	8	1.1	even	1	trivial
1805.2.b.h.1084.5	yes	8	95.94	odd	2	CM
9025.2.a.cb.1.4		8	5.2	odd	4
9025.2.a.cb.1.4		8	95.18	even	4
9025.2.a.cb.1.5		8	5.3	odd	4
9025.2.a.cb.1.5		8	95.37	even	4

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.406045i q^{2} +3.27727i q^{3} +1.83513 q^{4} +2.23607 q^{5} -1.33072 q^{6} +1.55723i q^{8} -7.74048 q^{9} +O(q^{10})\)	\(q+0.406045i q^{2} +3.27727i q^{3} +1.83513 q^{4} +2.23607 q^{5} -1.33072 q^{6} +1.55723i q^{8} -7.74048 q^{9} +0.907944i q^{10} -2.92978 q^{11} +6.01420i q^{12} +6.51610i q^{13} +7.32819i q^{15} +3.03795 q^{16} -3.14298i q^{18} +4.10347 q^{20} -1.18962i q^{22} -5.10347 q^{24} +5.00000 q^{25} -2.64583 q^{26} -15.5358i q^{27} -2.97557 q^{30} +4.34801i q^{32} -9.60166i q^{33} -14.2048 q^{36} -8.01591i q^{37} -21.3550 q^{39} +3.48208i q^{40} -5.37651 q^{44} -17.3082 q^{45} +9.95617i q^{48} +7.00000 q^{49} +2.03022i q^{50} +11.9579i q^{52} +5.09315i q^{53} +6.30824 q^{54} -6.55118 q^{55} +13.4482i q^{60} -1.11908 q^{61} +4.31041 q^{64} +14.5704i q^{65} +3.89870 q^{66} -8.95237i q^{67} -12.0537i q^{72} +3.25482 q^{74} +16.3863i q^{75} -8.67109i q^{78} +6.79306 q^{80} +27.6936 q^{81} -4.56235i q^{88} -7.02792i q^{90} -14.2496 q^{96} +11.6477i q^{97} +2.84231i q^{98} +22.6779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \) 8 * q - 16 * q^4 - 24 * q^9	\( 8 q - 16 q^{4} - 24 q^{9} + 32 q^{16} - 8 q^{24} + 40 q^{25} + 24 q^{26} - 40 q^{30} - 8 q^{36} - 72 q^{44} + 56 q^{49} - 88 q^{54} - 64 q^{64} + 104 q^{66} + 120 q^{80} + 72 q^{81} - 120 q^{96} - 32 q^{99}+O(q^{100}) \) 8 * q - 16 * q^4 - 24 * q^9 + 32 * q^16 - 8 * q^24 + 40 * q^25 + 24 * q^26 - 40 * q^30 - 8 * q^36 - 72 * q^44 + 56 * q^49 - 88 * q^54 - 64 * q^64 + 104 * q^66 + 120 * q^80 + 72 * q^81 - 120 * q^96 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more