LMFDB - Embedded newform 855.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, 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("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +1.41421 q^{7} -1.58579 q^{8} +O(q^{10})$	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +1.41421 q^{7} -1.58579 q^{8} +0.414214 q^{10} -6.24264 q^{11} -0.585786 q^{13} +0.585786 q^{14} +3.00000 q^{16} -6.82843 q^{17} -1.00000 q^{19} -1.82843 q^{20} -2.58579 q^{22} +3.65685 q^{23} +1.00000 q^{25} -0.242641 q^{26} -2.58579 q^{28} +1.41421 q^{29} -8.82843 q^{31} +4.41421 q^{32} -2.82843 q^{34} +1.41421 q^{35} -0.585786 q^{37} -0.414214 q^{38} -1.58579 q^{40} -8.24264 q^{41} +3.75736 q^{43} +11.4142 q^{44} +1.51472 q^{46} -3.65685 q^{47} -5.00000 q^{49} +0.414214 q^{50} +1.07107 q^{52} -8.00000 q^{53} -6.24264 q^{55} -2.24264 q^{56} +0.585786 q^{58} +4.48528 q^{59} -15.3137 q^{61} -3.65685 q^{62} -4.17157 q^{64} -0.585786 q^{65} +1.65685 q^{67} +12.4853 q^{68} +0.585786 q^{70} +5.17157 q^{71} +3.65685 q^{73} -0.242641 q^{74} +1.82843 q^{76} -8.82843 q^{77} +3.00000 q^{80} -3.41421 q^{82} -7.17157 q^{83} -6.82843 q^{85} +1.55635 q^{86} +9.89949 q^{88} +13.8995 q^{89} -0.828427 q^{91} -6.68629 q^{92} -1.51472 q^{94} -1.00000 q^{95} -18.2426 q^{97} -2.07107 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8} - 2 q^{10} - 4 q^{11} - 4 q^{13} + 4 q^{14} + 6 q^{16} - 8 q^{17} - 2 q^{19} + 2 q^{20} - 8 q^{22} - 4 q^{23} + 2 q^{25} + 8 q^{26} - 8 q^{28} - 12 q^{31} + 6 q^{32} - 4 q^{37} + 2 q^{38} - 6 q^{40} - 8 q^{41} + 16 q^{43} + 20 q^{44} + 20 q^{46} + 4 q^{47} - 10 q^{49} - 2 q^{50} - 12 q^{52} - 16 q^{53} - 4 q^{55} + 4 q^{56} + 4 q^{58} - 8 q^{59} - 8 q^{61} + 4 q^{62} - 14 q^{64} - 4 q^{65} - 8 q^{67} + 8 q^{68} + 4 q^{70} + 16 q^{71} - 4 q^{73} + 8 q^{74} - 2 q^{76} - 12 q^{77} + 6 q^{80} - 4 q^{82} - 20 q^{83} - 8 q^{85} - 28 q^{86} + 8 q^{89} + 4 q^{91} - 36 q^{92} - 20 q^{94} - 2 q^{95} - 28 q^{97} + 10 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8 - 2 * q^10 - 4 * q^11 - 4 * q^13 + 4 * q^14 + 6 * q^16 - 8 * q^17 - 2 * q^19 + 2 * q^20 - 8 * q^22 - 4 * q^23 + 2 * q^25 + 8 * q^26 - 8 * q^28 - 12 * q^31 + 6 * q^32 - 4 * q^37 + 2 * q^38 - 6 * q^40 - 8 * q^41 + 16 * q^43 + 20 * q^44 + 20 * q^46 + 4 * q^47 - 10 * q^49 - 2 * q^50 - 12 * q^52 - 16 * q^53 - 4 * q^55 + 4 * q^56 + 4 * q^58 - 8 * q^59 - 8 * q^61 + 4 * q^62 - 14 * q^64 - 4 * q^65 - 8 * q^67 + 8 * q^68 + 4 * q^70 + 16 * q^71 - 4 * q^73 + 8 * q^74 - 2 * q^76 - 12 * q^77 + 6 * q^80 - 4 * q^82 - 20 * q^83 - 8 * q^85 - 28 * q^86 + 8 * q^89 + 4 * q^91 - 36 * q^92 - 20 * q^94 - 2 * q^95 - 28 * q^97 + 10 * q^98

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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	−1.58579	−0.560660
$9$	0	0
$10$	0.414214	0.130986
$11$	−6.24264	−1.88223	−0.941113	−	0.338091i	$-0.890219\pi$
$11$	−6.24264	−1.88223	−0.941113	+	0.338091i	$0.890219\pi$
$12$	0	0
$13$	−0.585786	−0.162468	−0.0812340	−	0.996695i	$-0.525886\pi$
$13$	−0.585786	−0.162468	−0.0812340	+	0.996695i	$0.525886\pi$
$14$	0.585786	0.156558
$15$	0	0
$16$	3.00000	0.750000
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.82843	−0.408849
$21$	0	0
$22$	−2.58579	−0.551292
$23$	3.65685	0.762507	0.381253	−	0.924471i	$-0.375493\pi$
$23$	3.65685	0.762507	0.381253	+	0.924471i	$0.375493\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.242641	−0.0475858
$27$	0	0
$28$	−2.58579	−0.488668
$29$	1.41421	0.262613	0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421	0.262613	0.131306	+	0.991342i	$0.458083\pi$
$30$	0	0
$31$	−8.82843	−1.58563	−0.792816	−	0.609461i	$-0.791386\pi$
$31$	−8.82843	−1.58563	−0.792816	+	0.609461i	$0.791386\pi$
$32$	4.41421	0.780330
$33$	0	0
$34$	−2.82843	−0.485071
$35$	1.41421	0.239046
$36$	0	0
$37$	−0.585786	−0.0963027	−0.0481513	−	0.998840i	$-0.515333\pi$
$37$	−0.585786	−0.0963027	−0.0481513	+	0.998840i	$0.515333\pi$
$38$	−0.414214	−0.0671943
$39$	0	0
$40$	−1.58579	−0.250735
$41$	−8.24264	−1.28728	−0.643642	−	0.765327i	$-0.722577\pi$
$41$	−8.24264	−1.28728	−0.643642	+	0.765327i	$0.722577\pi$
$42$	0	0
$43$	3.75736	0.572992	0.286496	−	0.958081i	$-0.407509\pi$
$43$	3.75736	0.572992	0.286496	+	0.958081i	$0.407509\pi$
$44$	11.4142	1.72076
$45$	0	0
$46$	1.51472	0.223333
$47$	−3.65685	−0.533407	−0.266704	−	0.963779i	$-0.585934\pi$
$47$	−3.65685	−0.533407	−0.266704	+	0.963779i	$0.585934\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0.414214	0.0585786
$51$	0	0
$52$	1.07107	0.148530
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	−6.24264	−0.841757
$56$	−2.24264	−0.299685
$57$	0	0
$58$	0.585786	0.0769175
$59$	4.48528	0.583934	0.291967	−	0.956428i	$-0.405690\pi$
$59$	4.48528	0.583934	0.291967	+	0.956428i	$0.405690\pi$
$60$	0	0
$61$	−15.3137	−1.96072	−0.980360	−	0.197218i	$-0.936809\pi$
$61$	−15.3137	−1.96072	−0.980360	+	0.197218i	$0.936809\pi$
$62$	−3.65685	−0.464421
$63$	0	0
$64$	−4.17157	−0.521447
$65$	−0.585786	−0.0726579
$66$	0	0
$67$	1.65685	0.202417	0.101208	−	0.994865i	$-0.467729\pi$
$67$	1.65685	0.202417	0.101208	+	0.994865i	$0.467729\pi$
$68$	12.4853	1.51406
$69$	0	0
$70$	0.585786	0.0700149
$71$	5.17157	0.613753	0.306876	−	0.951749i	$-0.400716\pi$
$71$	5.17157	0.613753	0.306876	+	0.951749i	$0.400716\pi$
$72$	0	0
$73$	3.65685	0.428002	0.214001	−	0.976833i	$-0.431350\pi$
$73$	3.65685	0.428002	0.214001	+	0.976833i	$0.431350\pi$
$74$	−0.242641	−0.0282064
$75$	0	0
$76$	1.82843	0.209735
$77$	−8.82843	−1.00609
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−3.41421	−0.377037
$83$	−7.17157	−0.787182	−0.393591	−	0.919286i	$-0.628767\pi$
$83$	−7.17157	−0.787182	−0.393591	+	0.919286i	$0.628767\pi$
$84$	0	0
$85$	−6.82843	−0.740647
$86$	1.55635	0.167825
$87$	0	0
$88$	9.89949	1.05529
$89$	13.8995	1.47334	0.736672	−	0.676250i	$-0.236396\pi$
$89$	13.8995	1.47334	0.736672	+	0.676250i	$0.236396\pi$
$90$	0	0
$91$	−0.828427	−0.0868428
$92$	−6.68629	−0.697094
$93$	0	0
$94$	−1.51472	−0.156231
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−18.2426	−1.85226	−0.926130	−	0.377205i	$-0.876885\pi$
$97$	−18.2426	−1.85226	−0.926130	+	0.377205i	$0.876885\pi$
$98$	−2.07107	−0.209209
$99$	0	0
$100$	−1.82843	−0.182843
$101$	8.82843	0.878461	0.439231	−	0.898374i	$-0.355251\pi$
$101$	8.82843	0.878461	0.439231	+	0.898374i	$0.355251\pi$
$102$	0	0
$103$	15.3137	1.50890	0.754452	−	0.656355i	$-0.227903\pi$
$103$	15.3137	1.50890	0.754452	+	0.656355i	$0.227903\pi$
$104$	0.928932	0.0910893
$105$	0	0
$106$	−3.31371	−0.321856
$107$	3.31371	0.320348	0.160174	−	0.987089i	$-0.448794\pi$
$107$	3.31371	0.320348	0.160174	+	0.987089i	$0.448794\pi$
$108$	0	0
$109$	10.4853	1.00431	0.502154	−	0.864778i	$-0.332541\pi$
$109$	10.4853	1.00431	0.502154	+	0.864778i	$0.332541\pi$
$110$	−2.58579	−0.246545
$111$	0	0
$112$	4.24264	0.400892
$113$	18.1421	1.70667	0.853334	−	0.521364i	$-0.174577\pi$
$113$	18.1421	1.70667	0.853334	+	0.521364i	$0.174577\pi$
$114$	0	0
$115$	3.65685	0.341003
$116$	−2.58579	−0.240084
$117$	0	0
$118$	1.85786	0.171030
$119$	−9.65685	−0.885242
$120$	0	0
$121$	27.9706	2.54278
$122$	−6.34315	−0.574281
$123$	0	0
$124$	16.1421	1.44961
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.31371	−0.294044	−0.147022	−	0.989133i	$-0.546969\pi$
$127$	−3.31371	−0.294044	−0.147022	+	0.989133i	$0.546969\pi$
$128$	−10.5563	−0.933058
$129$	0	0
$130$	−0.242641	−0.0212810
$131$	11.4142	0.997264	0.498632	−	0.866814i	$-0.333836\pi$
$131$	11.4142	0.997264	0.498632	+	0.866814i	$0.333836\pi$
$132$	0	0
$133$	−1.41421	−0.122628
$134$	0.686292	0.0592866
$135$	0	0
$136$	10.8284	0.928530
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	14.1421	1.19952	0.599760	−	0.800180i	$-0.295263\pi$
$139$	14.1421	1.19952	0.599760	+	0.800180i	$0.295263\pi$
$140$	−2.58579	−0.218539
$141$	0	0
$142$	2.14214	0.179764
$143$	3.65685	0.305802
$144$	0	0
$145$	1.41421	0.117444
$146$	1.51472	0.125359
$147$	0	0
$148$	1.07107	0.0880412
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	10.4853	0.853280	0.426640	−	0.904422i	$-0.359697\pi$
$151$	10.4853	0.853280	0.426640	+	0.904422i	$0.359697\pi$
$152$	1.58579	0.128624
$153$	0	0
$154$	−3.65685	−0.294678
$155$	−8.82843	−0.709116
$156$	0	0
$157$	−6.48528	−0.517582	−0.258791	−	0.965933i	$-0.583324\pi$
$157$	−6.48528	−0.517582	−0.258791	+	0.965933i	$0.583324\pi$
$158$	0	0
$159$	0	0
$160$	4.41421	0.348974
$161$	5.17157	0.407577
$162$	0	0
$163$	2.10051	0.164524	0.0822621	−	0.996611i	$-0.473786\pi$
$163$	2.10051	0.164524	0.0822621	+	0.996611i	$0.473786\pi$
$164$	15.0711	1.17685
$165$	0	0
$166$	−2.97056	−0.230560
$167$	−5.31371	−0.411187	−0.205594	−	0.978637i	$-0.565912\pi$
$167$	−5.31371	−0.411187	−0.205594	+	0.978637i	$0.565912\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	−2.82843	−0.216930
$171$	0	0
$172$	−6.87006	−0.523837
$173$	−19.7990	−1.50529	−0.752645	−	0.658427i	$-0.771222\pi$
$173$	−19.7990	−1.50529	−0.752645	+	0.658427i	$0.771222\pi$
$174$	0	0
$175$	1.41421	0.106904
$176$	−18.7279	−1.41167
$177$	0	0
$178$	5.75736	0.431532
$179$	0.485281	0.0362716	0.0181358	−	0.999836i	$-0.494227\pi$
$179$	0.485281	0.0362716	0.0181358	+	0.999836i	$0.494227\pi$
$180$	0	0
$181$	−15.1716	−1.12769	−0.563847	−	0.825879i	$-0.690679\pi$
$181$	−15.1716	−1.12769	−0.563847	+	0.825879i	$0.690679\pi$
$182$	−0.343146	−0.0254357
$183$	0	0
$184$	−5.79899	−0.427507
$185$	−0.585786	−0.0430679
$186$	0	0
$187$	42.6274	3.11723
$188$	6.68629	0.487648
$189$	0	0
$190$	−0.414214	−0.0300502
$191$	−1.75736	−0.127158	−0.0635790	−	0.997977i	$-0.520251\pi$
$191$	−1.75736	−0.127158	−0.0635790	+	0.997977i	$0.520251\pi$
$192$	0	0
$193$	−9.07107	−0.652950	−0.326475	−	0.945206i	$-0.605861\pi$
$193$	−9.07107	−0.652950	−0.326475	+	0.945206i	$0.605861\pi$
$194$	−7.55635	−0.542514
$195$	0	0
$196$	9.14214	0.653010
$197$	−1.17157	−0.0834711	−0.0417356	−	0.999129i	$-0.513289\pi$
$197$	−1.17157	−0.0834711	−0.0417356	+	0.999129i	$0.513289\pi$
$198$	0	0
$199$	−10.1421	−0.718957	−0.359478	−	0.933153i	$-0.617045\pi$
$199$	−10.1421	−0.718957	−0.359478	+	0.933153i	$0.617045\pi$
$200$	−1.58579	−0.112132
$201$	0	0
$202$	3.65685	0.257295
$203$	2.00000	0.140372
$204$	0	0
$205$	−8.24264	−0.575691
$206$	6.34315	0.441948
$207$	0	0
$208$	−1.75736	−0.121851
$209$	6.24264	0.431812
$210$	0	0
$211$	−15.3137	−1.05424	−0.527120	−	0.849791i	$-0.676728\pi$
$211$	−15.3137	−1.05424	−0.527120	+	0.849791i	$0.676728\pi$
$212$	14.6274	1.00462
$213$	0	0
$214$	1.37258	0.0938278
$215$	3.75736	0.256250
$216$	0	0
$217$	−12.4853	−0.847556
$218$	4.34315	0.294155
$219$	0	0
$220$	11.4142	0.769546
$221$	4.00000	0.269069
$222$	0	0
$223$	−26.6274	−1.78310	−0.891552	−	0.452919i	$-0.850383\pi$
$223$	−26.6274	−1.78310	−0.891552	+	0.452919i	$0.850383\pi$
$224$	6.24264	0.417104
$225$	0	0
$226$	7.51472	0.499872
$227$	18.9706	1.25912	0.629560	−	0.776952i	$-0.283235\pi$
$227$	18.9706	1.25912	0.629560	+	0.776952i	$0.283235\pi$
$228$	0	0
$229$	22.6274	1.49526	0.747631	−	0.664114i	$-0.231191\pi$
$229$	22.6274	1.49526	0.747631	+	0.664114i	$0.231191\pi$
$230$	1.51472	0.0998776
$231$	0	0
$232$	−2.24264	−0.147237
$233$	11.6569	0.763666	0.381833	−	0.924231i	$-0.375293\pi$
$233$	11.6569	0.763666	0.381833	+	0.924231i	$0.375293\pi$
$234$	0	0
$235$	−3.65685	−0.238547
$236$	−8.20101	−0.533840
$237$	0	0
$238$	−4.00000	−0.259281
$239$	1.27208	0.0822839	0.0411419	−	0.999153i	$-0.486900\pi$
$239$	1.27208	0.0822839	0.0411419	+	0.999153i	$0.486900\pi$
$240$	0	0
$241$	−8.34315	−0.537429	−0.268715	−	0.963220i	$-0.586599\pi$
$241$	−8.34315	−0.537429	−0.268715	+	0.963220i	$0.586599\pi$
$242$	11.5858	0.744763
$243$	0	0
$244$	28.0000	1.79252
$245$	−5.00000	−0.319438
$246$	0	0
$247$	0.585786	0.0372727
$248$	14.0000	0.889001
$249$	0	0
$250$	0.414214	0.0261972
$251$	10.2426	0.646510	0.323255	−	0.946312i	$-0.395223\pi$
$251$	10.2426	0.646510	0.323255	+	0.946312i	$0.395223\pi$
$252$	0	0
$253$	−22.8284	−1.43521
$254$	−1.37258	−0.0861235
$255$	0	0
$256$	3.97056	0.248160
$257$	−12.4853	−0.778810	−0.389405	−	0.921067i	$-0.627319\pi$
$257$	−12.4853	−0.778810	−0.389405	+	0.921067i	$0.627319\pi$
$258$	0	0
$259$	−0.828427	−0.0514760
$260$	1.07107	0.0664248
$261$	0	0
$262$	4.72792	0.292092
$263$	−27.4558	−1.69300	−0.846500	−	0.532389i	$-0.821294\pi$
$263$	−27.4558	−1.69300	−0.846500	+	0.532389i	$0.821294\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	−0.585786	−0.0359169
$267$	0	0
$268$	−3.02944	−0.185052
$269$	−11.0711	−0.675015	−0.337507	−	0.941323i	$-0.609584\pi$
$269$	−11.0711	−0.675015	−0.337507	+	0.941323i	$0.609584\pi$
$270$	0	0
$271$	−5.17157	−0.314151	−0.157075	−	0.987587i	$-0.550207\pi$
$271$	−5.17157	−0.314151	−0.157075	+	0.987587i	$0.550207\pi$
$272$	−20.4853	−1.24210
$273$	0	0
$274$	−4.14214	−0.250236
$275$	−6.24264	−0.376445
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	5.85786	0.351331
$279$	0	0
$280$	−2.24264	−0.134023
$281$	17.4142	1.03884	0.519422	−	0.854518i	$-0.326147\pi$
$281$	17.4142	1.03884	0.519422	+	0.854518i	$0.326147\pi$
$282$	0	0
$283$	26.3848	1.56841	0.784206	−	0.620500i	$-0.213071\pi$
$283$	26.3848	1.56841	0.784206	+	0.620500i	$0.213071\pi$
$284$	−9.45584	−0.561101
$285$	0	0
$286$	1.51472	0.0895672
$287$	−11.6569	−0.688082
$288$	0	0
$289$	29.6274	1.74279
$290$	0.585786	0.0343986
$291$	0	0
$292$	−6.68629	−0.391286
$293$	28.4853	1.66413	0.832064	−	0.554680i	$-0.187159\pi$
$293$	28.4853	1.66413	0.832064	+	0.554680i	$0.187159\pi$
$294$	0	0
$295$	4.48528	0.261143
$296$	0.928932	0.0539931
$297$	0	0
$298$	2.48528	0.143968
$299$	−2.14214	−0.123883
$300$	0	0
$301$	5.31371	0.306277
$302$	4.34315	0.249920
$303$	0	0
$304$	−3.00000	−0.172062
$305$	−15.3137	−0.876860
$306$	0	0
$307$	26.8284	1.53118	0.765590	−	0.643329i	$-0.222447\pi$
$307$	26.8284	1.53118	0.765590	+	0.643329i	$0.222447\pi$
$308$	16.1421	0.919784
$309$	0	0
$310$	−3.65685	−0.207695
$311$	−2.24264	−0.127168	−0.0635842	−	0.997976i	$-0.520253\pi$
$311$	−2.24264	−0.127168	−0.0635842	+	0.997976i	$0.520253\pi$
$312$	0	0
$313$	−33.7990	−1.91043	−0.955216	−	0.295910i	$-0.904377\pi$
$313$	−33.7990	−1.91043	−0.955216	+	0.295910i	$0.904377\pi$
$314$	−2.68629	−0.151596
$315$	0	0
$316$	0	0
$317$	−18.6274	−1.04622	−0.523110	−	0.852265i	$-0.675228\pi$
$317$	−18.6274	−1.04622	−0.523110	+	0.852265i	$0.675228\pi$
$318$	0	0
$319$	−8.82843	−0.494297
$320$	−4.17157	−0.233198
$321$	0	0
$322$	2.14214	0.119377
$323$	6.82843	0.379944
$324$	0	0
$325$	−0.585786	−0.0324936
$326$	0.870058	0.0481880
$327$	0	0
$328$	13.0711	0.721729
$329$	−5.17157	−0.285118
$330$	0	0
$331$	−0.142136	−0.00781248	−0.00390624	−	0.999992i	$-0.501243\pi$
$331$	−0.142136	−0.00781248	−0.00390624	+	0.999992i	$0.501243\pi$
$332$	13.1127	0.719653
$333$	0	0
$334$	−2.20101	−0.120434
$335$	1.65685	0.0905236
$336$	0	0
$337$	−27.4142	−1.49335	−0.746674	−	0.665191i	$-0.768350\pi$
$337$	−27.4142	−1.49335	−0.746674	+	0.665191i	$0.768350\pi$
$338$	−5.24264	−0.285162
$339$	0	0
$340$	12.4853	0.677109
$341$	55.1127	2.98452
$342$	0	0
$343$	−16.9706	−0.916324
$344$	−5.95837	−0.321254
$345$	0	0
$346$	−8.20101	−0.440889
$347$	−23.4558	−1.25918	−0.629588	−	0.776929i	$-0.716776\pi$
$347$	−23.4558	−1.25918	−0.629588	+	0.776929i	$0.716776\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0.585786	0.0313116
$351$	0	0
$352$	−27.5563	−1.46876
$353$	7.65685	0.407533	0.203767	−	0.979019i	$-0.434682\pi$
$353$	7.65685	0.407533	0.203767	+	0.979019i	$0.434682\pi$
$354$	0	0
$355$	5.17157	0.274479
$356$	−25.4142	−1.34695
$357$	0	0
$358$	0.201010	0.0106237
$359$	28.8701	1.52370	0.761852	−	0.647752i	$-0.224290\pi$
$359$	28.8701	1.52370	0.761852	+	0.647752i	$0.224290\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−6.28427	−0.330294
$363$	0	0
$364$	1.51472	0.0793928
$365$	3.65685	0.191408
$366$	0	0
$367$	22.3848	1.16848	0.584238	−	0.811582i	$-0.301394\pi$
$367$	22.3848	1.16848	0.584238	+	0.811582i	$0.301394\pi$
$368$	10.9706	0.571880
$369$	0	0
$370$	−0.242641	−0.0126143
$371$	−11.3137	−0.587378
$372$	0	0
$373$	3.41421	0.176781	0.0883906	−	0.996086i	$-0.471828\pi$
$373$	3.41421	0.176781	0.0883906	+	0.996086i	$0.471828\pi$
$374$	17.6569	0.913014
$375$	0	0
$376$	5.79899	0.299060
$377$	−0.828427	−0.0426662
$378$	0	0
$379$	8.82843	0.453486	0.226743	−	0.973955i	$-0.427192\pi$
$379$	8.82843	0.453486	0.226743	+	0.973955i	$0.427192\pi$
$380$	1.82843	0.0937963
$381$	0	0
$382$	−0.727922	−0.0372437
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	0	0
$385$	−8.82843	−0.449938
$386$	−3.75736	−0.191245
$387$	0	0
$388$	33.3553	1.69336
$389$	2.97056	0.150614	0.0753068	−	0.997160i	$-0.476006\pi$
$389$	2.97056	0.150614	0.0753068	+	0.997160i	$0.476006\pi$
$390$	0	0
$391$	−24.9706	−1.26282
$392$	7.92893	0.400472
$393$	0	0
$394$	−0.485281	−0.0244481
$395$	0	0
$396$	0	0
$397$	−16.6274	−0.834506	−0.417253	−	0.908790i	$-0.637007\pi$
$397$	−16.6274	−0.834506	−0.417253	+	0.908790i	$0.637007\pi$
$398$	−4.20101	−0.210578
$399$	0	0
$400$	3.00000	0.150000
$401$	16.2426	0.811119	0.405559	−	0.914069i	$-0.367077\pi$
$401$	16.2426	0.811119	0.405559	+	0.914069i	$0.367077\pi$
$402$	0	0
$403$	5.17157	0.257614
$404$	−16.1421	−0.803101
$405$	0	0
$406$	0.828427	0.0411141
$407$	3.65685	0.181264
$408$	0	0
$409$	−7.17157	−0.354611	−0.177306	−	0.984156i	$-0.556738\pi$
$409$	−7.17157	−0.354611	−0.177306	+	0.984156i	$0.556738\pi$
$410$	−3.41421	−0.168616
$411$	0	0
$412$	−28.0000	−1.37946
$413$	6.34315	0.312126
$414$	0	0
$415$	−7.17157	−0.352039
$416$	−2.58579	−0.126779
$417$	0	0
$418$	2.58579	0.126475
$419$	−0.585786	−0.0286175	−0.0143088	−	0.999898i	$-0.504555\pi$
$419$	−0.585786	−0.0286175	−0.0143088	+	0.999898i	$0.504555\pi$
$420$	0	0
$421$	−13.3137	−0.648870	−0.324435	−	0.945908i	$-0.605174\pi$
$421$	−13.3137	−0.648870	−0.324435	+	0.945908i	$0.605174\pi$
$422$	−6.34315	−0.308780
$423$	0	0
$424$	12.6863	0.616101
$425$	−6.82843	−0.331227
$426$	0	0
$427$	−21.6569	−1.04805
$428$	−6.05887	−0.292867
$429$	0	0
$430$	1.55635	0.0750538
$431$	−31.1127	−1.49865	−0.749323	−	0.662205i	$-0.769621\pi$
$431$	−31.1127	−1.49865	−0.749323	+	0.662205i	$0.769621\pi$
$432$	0	0
$433$	−25.0711	−1.20484	−0.602419	−	0.798180i	$-0.705796\pi$
$433$	−25.0711	−1.20484	−0.602419	+	0.798180i	$0.705796\pi$
$434$	−5.17157	−0.248243
$435$	0	0
$436$	−19.1716	−0.918152
$437$	−3.65685	−0.174931
$438$	0	0
$439$	−10.3431	−0.493651	−0.246826	−	0.969060i	$-0.579388\pi$
$439$	−10.3431	−0.493651	−0.246826	+	0.969060i	$0.579388\pi$
$440$	9.89949	0.471940
$441$	0	0
$442$	1.65685	0.0788085
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	13.8995	0.658899
$446$	−11.0294	−0.522259
$447$	0	0
$448$	−5.89949	−0.278725
$449$	−26.8701	−1.26808	−0.634038	−	0.773302i	$-0.718604\pi$
$449$	−26.8701	−1.26808	−0.634038	+	0.773302i	$0.718604\pi$
$450$	0	0
$451$	51.4558	2.42296
$452$	−33.1716	−1.56026
$453$	0	0
$454$	7.85786	0.368788
$455$	−0.828427	−0.0388373
$456$	0	0
$457$	27.1716	1.27103	0.635516	−	0.772087i	$-0.280787\pi$
$457$	27.1716	1.27103	0.635516	+	0.772087i	$0.280787\pi$
$458$	9.37258	0.437952
$459$	0	0
$460$	−6.68629	−0.311750
$461$	−8.34315	−0.388579	−0.194290	−	0.980944i	$-0.562240\pi$
$461$	−8.34315	−0.388579	−0.194290	+	0.980944i	$0.562240\pi$
$462$	0	0
$463$	15.7574	0.732307	0.366153	−	0.930555i	$-0.380675\pi$
$463$	15.7574	0.732307	0.366153	+	0.930555i	$0.380675\pi$
$464$	4.24264	0.196960
$465$	0	0
$466$	4.82843	0.223673
$467$	−24.3431	−1.12647	−0.563233	−	0.826298i	$-0.690443\pi$
$467$	−24.3431	−1.12647	−0.563233	+	0.826298i	$0.690443\pi$
$468$	0	0
$469$	2.34315	0.108196
$470$	−1.51472	−0.0698688
$471$	0	0
$472$	−7.11270	−0.327388
$473$	−23.4558	−1.07850
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	17.6569	0.809301
$477$	0	0
$478$	0.526912	0.0241004
$479$	−2.92893	−0.133826	−0.0669132	−	0.997759i	$-0.521315\pi$
$479$	−2.92893	−0.133826	−0.0669132	+	0.997759i	$0.521315\pi$
$480$	0	0
$481$	0.343146	0.0156461
$482$	−3.45584	−0.157409
$483$	0	0
$484$	−51.1421	−2.32464
$485$	−18.2426	−0.828356
$486$	0	0
$487$	3.51472	0.159267	0.0796336	−	0.996824i	$-0.474625\pi$
$487$	3.51472	0.159267	0.0796336	+	0.996824i	$0.474625\pi$
$488$	24.2843	1.09930
$489$	0	0
$490$	−2.07107	−0.0935613
$491$	10.2426	0.462244	0.231122	−	0.972925i	$-0.425760\pi$
$491$	10.2426	0.462244	0.231122	+	0.972925i	$0.425760\pi$
$492$	0	0
$493$	−9.65685	−0.434923
$494$	0.242641	0.0109169
$495$	0	0
$496$	−26.4853	−1.18922
$497$	7.31371	0.328065
$498$	0	0
$499$	−10.8284	−0.484747	−0.242373	−	0.970183i	$-0.577926\pi$
$499$	−10.8284	−0.484747	−0.242373	+	0.970183i	$0.577926\pi$
$500$	−1.82843	−0.0817697
$501$	0	0
$502$	4.24264	0.189358
$503$	0.828427	0.0369377	0.0184689	−	0.999829i	$-0.494121\pi$
$503$	0.828427	0.0369377	0.0184689	+	0.999829i	$0.494121\pi$
$504$	0	0
$505$	8.82843	0.392860
$506$	−9.45584	−0.420364
$507$	0	0
$508$	6.05887	0.268819
$509$	−24.7279	−1.09605	−0.548023	−	0.836463i	$-0.684619\pi$
$509$	−24.7279	−1.09605	−0.548023	+	0.836463i	$0.684619\pi$
$510$	0	0
$511$	5.17157	0.228777
$512$	22.7574	1.00574
$513$	0	0
$514$	−5.17157	−0.228108
$515$	15.3137	0.674803
$516$	0	0
$517$	22.8284	1.00399
$518$	−0.343146	−0.0150770
$519$	0	0
$520$	0.928932	0.0407364
$521$	−16.2426	−0.711603	−0.355802	−	0.934562i	$-0.615792\pi$
$521$	−16.2426	−0.711603	−0.355802	+	0.934562i	$0.615792\pi$
$522$	0	0
$523$	−32.4853	−1.42048	−0.710241	−	0.703959i	$-0.751414\pi$
$523$	−32.4853	−1.42048	−0.710241	+	0.703959i	$0.751414\pi$
$524$	−20.8701	−0.911713
$525$	0	0
$526$	−11.3726	−0.495868
$527$	60.2843	2.62602
$528$	0	0
$529$	−9.62742	−0.418583
$530$	−3.31371	−0.143938
$531$	0	0
$532$	2.58579	0.112108
$533$	4.82843	0.209142
$534$	0	0
$535$	3.31371	0.143264
$536$	−2.62742	−0.113487
$537$	0	0
$538$	−4.58579	−0.197707
$539$	31.2132	1.34445
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−2.14214	−0.0920126
$543$	0	0
$544$	−30.1421	−1.29233
$545$	10.4853	0.449140
$546$	0	0
$547$	7.51472	0.321306	0.160653	−	0.987011i	$-0.448640\pi$
$547$	7.51472	0.321306	0.160653	+	0.987011i	$0.448640\pi$
$548$	18.2843	0.781065
$549$	0	0
$550$	−2.58579	−0.110258
$551$	−1.41421	−0.0602475
$552$	0	0
$553$	0	0
$554$	−9.11270	−0.387161
$555$	0	0
$556$	−25.8579	−1.09662
$557$	2.68629	0.113822	0.0569109	−	0.998379i	$-0.481875\pi$
$557$	2.68629	0.113822	0.0569109	+	0.998379i	$0.481875\pi$
$558$	0	0
$559$	−2.20101	−0.0930928
$560$	4.24264	0.179284
$561$	0	0
$562$	7.21320	0.304271
$563$	−26.2843	−1.10775	−0.553875	−	0.832600i	$-0.686851\pi$
$563$	−26.2843	−1.10775	−0.553875	+	0.832600i	$0.686851\pi$
$564$	0	0
$565$	18.1421	0.763245
$566$	10.9289	0.459377
$567$	0	0
$568$	−8.20101	−0.344107
$569$	−16.9289	−0.709698	−0.354849	−	0.934924i	$-0.615468\pi$
$569$	−16.9289	−0.709698	−0.354849	+	0.934924i	$0.615468\pi$
$570$	0	0
$571$	14.8284	0.620550	0.310275	−	0.950647i	$-0.399579\pi$
$571$	14.8284	0.620550	0.310275	+	0.950647i	$0.399579\pi$
$572$	−6.68629	−0.279568
$573$	0	0
$574$	−4.82843	−0.201535
$575$	3.65685	0.152501
$576$	0	0
$577$	−31.4558	−1.30952	−0.654762	−	0.755835i	$-0.727231\pi$
$577$	−31.4558	−1.30952	−0.654762	+	0.755835i	$0.727231\pi$
$578$	12.2721	0.510451
$579$	0	0
$580$	−2.58579	−0.107369
$581$	−10.1421	−0.420767
$582$	0	0
$583$	49.9411	2.06835
$584$	−5.79899	−0.239964
$585$	0	0
$586$	11.7990	0.487412
$587$	23.6569	0.976423	0.488211	−	0.872725i	$-0.337649\pi$
$587$	23.6569	0.976423	0.488211	+	0.872725i	$0.337649\pi$
$588$	0	0
$589$	8.82843	0.363769
$590$	1.85786	0.0764871
$591$	0	0
$592$	−1.75736	−0.0722270
$593$	8.62742	0.354286	0.177143	−	0.984185i	$-0.443315\pi$
$593$	8.62742	0.354286	0.177143	+	0.984185i	$0.443315\pi$
$594$	0	0
$595$	−9.65685	−0.395892
$596$	−10.9706	−0.449372
$597$	0	0
$598$	−0.887302	−0.0362845
$599$	36.9706	1.51058	0.755288	−	0.655393i	$-0.227497\pi$
$599$	36.9706	1.51058	0.755288	+	0.655393i	$0.227497\pi$
$600$	0	0
$601$	−20.1421	−0.821615	−0.410807	−	0.911722i	$-0.634753\pi$
$601$	−20.1421	−0.821615	−0.410807	+	0.911722i	$0.634753\pi$
$602$	2.20101	0.0897065
$603$	0	0
$604$	−19.1716	−0.780080
$605$	27.9706	1.13717
$606$	0	0
$607$	−0.485281	−0.0196970	−0.00984848	−	0.999952i	$-0.503135\pi$
$607$	−0.485281	−0.0196970	−0.00984848	+	0.999952i	$0.503135\pi$
$608$	−4.41421	−0.179020
$609$	0	0
$610$	−6.34315	−0.256826
$611$	2.14214	0.0866615
$612$	0	0
$613$	6.48528	0.261938	0.130969	−	0.991386i	$-0.458191\pi$
$613$	6.48528	0.261938	0.130969	+	0.991386i	$0.458191\pi$
$614$	11.1127	0.448472
$615$	0	0
$616$	14.0000	0.564076
$617$	−11.5147	−0.463565	−0.231783	−	0.972768i	$-0.574456\pi$
$617$	−11.5147	−0.463565	−0.231783	+	0.972768i	$0.574456\pi$
$618$	0	0
$619$	−3.51472	−0.141268	−0.0706342	−	0.997502i	$-0.522502\pi$
$619$	−3.51472	−0.141268	−0.0706342	+	0.997502i	$0.522502\pi$
$620$	16.1421	0.648284
$621$	0	0
$622$	−0.928932	−0.0372468
$623$	19.6569	0.787535
$624$	0	0
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	11.8579	0.473180
$629$	4.00000	0.159490
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	−7.71573	−0.306431
$635$	−3.31371	−0.131501
$636$	0	0
$637$	2.92893	0.116049
$638$	−3.65685	−0.144776
$639$	0	0
$640$	−10.5563	−0.417276
$641$	−39.3553	−1.55444	−0.777221	−	0.629227i	$-0.783371\pi$
$641$	−39.3553	−1.55444	−0.777221	+	0.629227i	$0.783371\pi$
$642$	0	0
$643$	5.61522	0.221443	0.110721	−	0.993851i	$-0.464684\pi$
$643$	5.61522	0.221443	0.110721	+	0.993851i	$0.464684\pi$
$644$	−9.45584	−0.372612
$645$	0	0
$646$	2.82843	0.111283
$647$	7.85786	0.308925	0.154462	−	0.987999i	$-0.450635\pi$
$647$	7.85786	0.308925	0.154462	+	0.987999i	$0.450635\pi$
$648$	0	0
$649$	−28.0000	−1.09910
$650$	−0.242641	−0.00951715
$651$	0	0
$652$	−3.84062	−0.150410
$653$	−37.1716	−1.45464	−0.727318	−	0.686301i	$-0.759233\pi$
$653$	−37.1716	−1.45464	−0.727318	+	0.686301i	$0.759233\pi$
$654$	0	0
$655$	11.4142	0.445990
$656$	−24.7279	−0.965463
$657$	0	0
$658$	−2.14214	−0.0835091
$659$	−26.6274	−1.03726	−0.518628	−	0.855000i	$-0.673557\pi$
$659$	−26.6274	−1.03726	−0.518628	+	0.855000i	$0.673557\pi$
$660$	0	0
$661$	−43.4558	−1.69024	−0.845118	−	0.534579i	$-0.820470\pi$
$661$	−43.4558	−1.69024	−0.845118	+	0.534579i	$0.820470\pi$
$662$	−0.0588745	−0.00228822
$663$	0	0
$664$	11.3726	0.441342
$665$	−1.41421	−0.0548408
$666$	0	0
$667$	5.17157	0.200244
$668$	9.71573	0.375913
$669$	0	0
$670$	0.686292	0.0265138
$671$	95.5980	3.69052
$672$	0	0
$673$	32.1838	1.24059	0.620297	−	0.784367i	$-0.287012\pi$
$673$	32.1838	1.24059	0.620297	+	0.784367i	$0.287012\pi$
$674$	−11.3553	−0.437391
$675$	0	0
$676$	23.1421	0.890082
$677$	16.9706	0.652232	0.326116	−	0.945330i	$-0.394260\pi$
$677$	16.9706	0.652232	0.326116	+	0.945330i	$0.394260\pi$
$678$	0	0
$679$	−25.7990	−0.990074
$680$	10.8284	0.415251
$681$	0	0
$682$	22.8284	0.874146
$683$	−29.6569	−1.13479	−0.567394	−	0.823446i	$-0.692048\pi$
$683$	−29.6569	−1.13479	−0.567394	+	0.823446i	$0.692048\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	−7.02944	−0.268385
$687$	0	0
$688$	11.2721	0.429744
$689$	4.68629	0.178533
$690$	0	0
$691$	41.1716	1.56624	0.783120	−	0.621870i	$-0.213627\pi$
$691$	41.1716	1.56624	0.783120	+	0.621870i	$0.213627\pi$
$692$	36.2010	1.37616
$693$	0	0
$694$	−9.71573	−0.368804
$695$	14.1421	0.536442
$696$	0	0
$697$	56.2843	2.13192
$698$	7.45584	0.282208
$699$	0	0
$700$	−2.58579	−0.0977335
$701$	−29.3137	−1.10716	−0.553582	−	0.832795i	$-0.686739\pi$
$701$	−29.3137	−1.10716	−0.553582	+	0.832795i	$0.686739\pi$
$702$	0	0
$703$	0.585786	0.0220934
$704$	26.0416	0.981481
$705$	0	0
$706$	3.17157	0.119364
$707$	12.4853	0.469557
$708$	0	0
$709$	−4.97056	−0.186673	−0.0933367	−	0.995635i	$-0.529753\pi$
$709$	−4.97056	−0.186673	−0.0933367	+	0.995635i	$0.529753\pi$
$710$	2.14214	0.0803929
$711$	0	0
$712$	−22.0416	−0.826045
$713$	−32.2843	−1.20906
$714$	0	0
$715$	3.65685	0.136759
$716$	−0.887302	−0.0331600
$717$	0	0
$718$	11.9584	0.446282
$719$	−13.0711	−0.487469	−0.243734	−	0.969842i	$-0.578372\pi$
$719$	−13.0711	−0.487469	−0.243734	+	0.969842i	$0.578372\pi$
$720$	0	0
$721$	21.6569	0.806543
$722$	0.414214	0.0154154
$723$	0	0
$724$	27.7401	1.03095
$725$	1.41421	0.0525226
$726$	0	0
$727$	−23.3553	−0.866202	−0.433101	−	0.901345i	$-0.642581\pi$
$727$	−23.3553	−0.866202	−0.433101	+	0.901345i	$0.642581\pi$
$728$	1.31371	0.0486893
$729$	0	0
$730$	1.51472	0.0560623
$731$	−25.6569	−0.948953
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	9.27208	0.342239
$735$	0	0
$736$	16.1421	0.595007
$737$	−10.3431	−0.380995
$738$	0	0
$739$	25.6569	0.943803	0.471901	−	0.881651i	$-0.343568\pi$
$739$	25.6569	0.943803	0.471901	+	0.881651i	$0.343568\pi$
$740$	1.07107	0.0393732
$741$	0	0
$742$	−4.68629	−0.172039
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	1.41421	0.0517780
$747$	0	0
$748$	−77.9411	−2.84981
$749$	4.68629	0.171233
$750$	0	0
$751$	4.14214	0.151149	0.0755743	−	0.997140i	$-0.475921\pi$
$751$	4.14214	0.151149	0.0755743	+	0.997140i	$0.475921\pi$
$752$	−10.9706	−0.400055
$753$	0	0
$754$	−0.343146	−0.0124966
$755$	10.4853	0.381598
$756$	0	0
$757$	52.4264	1.90547	0.952735	−	0.303802i	$-0.0982562\pi$
$757$	52.4264	1.90547	0.952735	+	0.303802i	$0.0982562\pi$
$758$	3.65685	0.132823
$759$	0	0
$760$	1.58579	0.0575225
$761$	23.9411	0.867865	0.433933	−	0.900945i	$-0.357126\pi$
$761$	23.9411	0.867865	0.433933	+	0.900945i	$0.357126\pi$
$762$	0	0
$763$	14.8284	0.536825
$764$	3.21320	0.116250
$765$	0	0
$766$	−11.5980	−0.419052
$767$	−2.62742	−0.0948705
$768$	0	0
$769$	9.02944	0.325610	0.162805	−	0.986658i	$-0.447946\pi$
$769$	9.02944	0.325610	0.162805	+	0.986658i	$0.447946\pi$
$770$	−3.65685	−0.131784
$771$	0	0
$772$	16.5858	0.596936
$773$	14.3431	0.515887	0.257944	−	0.966160i	$-0.416955\pi$
$773$	14.3431	0.515887	0.257944	+	0.966160i	$0.416955\pi$
$774$	0	0
$775$	−8.82843	−0.317126
$776$	28.9289	1.03849
$777$	0	0
$778$	1.23045	0.0441137
$779$	8.24264	0.295323
$780$	0	0
$781$	−32.2843	−1.15522
$782$	−10.3431	−0.369870
$783$	0	0
$784$	−15.0000	−0.535714
$785$	−6.48528	−0.231470
$786$	0	0
$787$	−37.1716	−1.32502	−0.662512	−	0.749052i	$-0.730510\pi$
$787$	−37.1716	−1.32502	−0.662512	+	0.749052i	$0.730510\pi$
$788$	2.14214	0.0763104
$789$	0	0
$790$	0	0
$791$	25.6569	0.912253
$792$	0	0
$793$	8.97056	0.318554
$794$	−6.88730	−0.244421
$795$	0	0
$796$	18.5442	0.657280
$797$	−14.1421	−0.500940	−0.250470	−	0.968124i	$-0.580585\pi$
$797$	−14.1421	−0.500940	−0.250470	+	0.968124i	$0.580585\pi$
$798$	0	0
$799$	24.9706	0.883395
$800$	4.41421	0.156066
$801$	0	0
$802$	6.72792	0.237571
$803$	−22.8284	−0.805598
$804$	0	0
$805$	5.17157	0.182274
$806$	2.14214	0.0754535
$807$	0	0
$808$	−14.0000	−0.492518
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	31.3137	1.09957	0.549787	−	0.835305i	$-0.314709\pi$
$811$	31.3137	1.09957	0.549787	+	0.835305i	$0.314709\pi$
$812$	−3.65685	−0.128330
$813$	0	0
$814$	1.51472	0.0530909
$815$	2.10051	0.0735775
$816$	0	0
$817$	−3.75736	−0.131453
$818$	−2.97056	−0.103863
$819$	0	0
$820$	15.0711	0.526305
$821$	2.48528	0.0867369	0.0433685	−	0.999059i	$-0.486191\pi$
$821$	2.48528	0.0867369	0.0433685	+	0.999059i	$0.486191\pi$
$822$	0	0
$823$	57.0122	1.98732	0.993660	−	0.112426i	$-0.0358622\pi$
$823$	57.0122	1.98732	0.993660	+	0.112426i	$0.0358622\pi$
$824$	−24.2843	−0.845983
$825$	0	0
$826$	2.62742	0.0914195
$827$	25.3137	0.880244	0.440122	−	0.897938i	$-0.354935\pi$
$827$	25.3137	0.880244	0.440122	+	0.897938i	$0.354935\pi$
$828$	0	0
$829$	29.5147	1.02509	0.512544	−	0.858661i	$-0.328703\pi$
$829$	29.5147	1.02509	0.512544	+	0.858661i	$0.328703\pi$
$830$	−2.97056	−0.103110
$831$	0	0
$832$	2.44365	0.0847183
$833$	34.1421	1.18295
$834$	0	0
$835$	−5.31371	−0.183888
$836$	−11.4142	−0.394769
$837$	0	0
$838$	−0.242641	−0.00838188
$839$	38.1421	1.31681	0.658406	−	0.752663i	$-0.271231\pi$
$839$	38.1421	1.31681	0.658406	+	0.752663i	$0.271231\pi$
$840$	0	0
$841$	−27.0000	−0.931034
$842$	−5.51472	−0.190050
$843$	0	0
$844$	28.0000	0.963800
$845$	−12.6569	−0.435409
$846$	0	0
$847$	39.5563	1.35917
$848$	−24.0000	−0.824163
$849$	0	0
$850$	−2.82843	−0.0970143
$851$	−2.14214	−0.0734315
$852$	0	0
$853$	−11.1716	−0.382507	−0.191254	−	0.981541i	$-0.561255\pi$
$853$	−11.1716	−0.382507	−0.191254	+	0.981541i	$0.561255\pi$
$854$	−8.97056	−0.306966
$855$	0	0
$856$	−5.25483	−0.179607
$857$	−35.3137	−1.20629	−0.603147	−	0.797630i	$-0.706087\pi$
$857$	−35.3137	−1.20629	−0.603147	+	0.797630i	$0.706087\pi$
$858$	0	0
$859$	17.9411	0.612143	0.306072	−	0.952008i	$-0.400985\pi$
$859$	17.9411	0.612143	0.306072	+	0.952008i	$0.400985\pi$
$860$	−6.87006	−0.234267
$861$	0	0
$862$	−12.8873	−0.438943
$863$	−31.3137	−1.06593	−0.532966	−	0.846137i	$-0.678922\pi$
$863$	−31.3137	−1.06593	−0.532966	+	0.846137i	$0.678922\pi$
$864$	0	0
$865$	−19.7990	−0.673186
$866$	−10.3848	−0.352889
$867$	0	0
$868$	22.8284	0.774847
$869$	0	0
$870$	0	0
$871$	−0.970563	−0.0328863
$872$	−16.6274	−0.563075
$873$	0	0
$874$	−1.51472	−0.0512361
$875$	1.41421	0.0478091
$876$	0	0
$877$	−49.0711	−1.65701	−0.828506	−	0.559980i	$-0.810809\pi$
$877$	−49.0711	−1.65701	−0.828506	+	0.559980i	$0.810809\pi$
$878$	−4.28427	−0.144587
$879$	0	0
$880$	−18.7279	−0.631318
$881$	−33.7990	−1.13872	−0.569358	−	0.822089i	$-0.692808\pi$
$881$	−33.7990	−1.13872	−0.569358	+	0.822089i	$0.692808\pi$
$882$	0	0
$883$	36.0416	1.21290	0.606449	−	0.795123i	$-0.292594\pi$
$883$	36.0416	1.21290	0.606449	+	0.795123i	$0.292594\pi$
$884$	−7.31371	−0.245987
$885$	0	0
$886$	−7.45584	−0.250484
$887$	41.9411	1.40825	0.704123	−	0.710078i	$-0.251341\pi$
$887$	41.9411	1.40825	0.704123	+	0.710078i	$0.251341\pi$
$888$	0	0
$889$	−4.68629	−0.157173
$890$	5.75736	0.192987
$891$	0	0
$892$	48.6863	1.63014
$893$	3.65685	0.122372
$894$	0	0
$895$	0.485281	0.0162212
$896$	−14.9289	−0.498741
$897$	0	0
$898$	−11.1299	−0.371411
$899$	−12.4853	−0.416407
$900$	0	0
$901$	54.6274	1.81990
$902$	21.3137	0.709669
$903$	0	0
$904$	−28.7696	−0.956861
$905$	−15.1716	−0.504320
$906$	0	0
$907$	10.1421	0.336764	0.168382	−	0.985722i	$-0.446146\pi$
$907$	10.1421	0.336764	0.168382	+	0.985722i	$0.446146\pi$
$908$	−34.6863	−1.15111
$909$	0	0
$910$	−0.343146	−0.0113752
$911$	−31.3137	−1.03747	−0.518735	−	0.854935i	$-0.673597\pi$
$911$	−31.3137	−1.03747	−0.518735	+	0.854935i	$0.673597\pi$
$912$	0	0
$913$	44.7696	1.48166
$914$	11.2548	0.372277
$915$	0	0
$916$	−41.3726	−1.36699
$917$	16.1421	0.533060
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	−5.79899	−0.191187
$921$	0	0
$922$	−3.45584	−0.113812
$923$	−3.02944	−0.0997151
$924$	0	0
$925$	−0.585786	−0.0192605
$926$	6.52691	0.214488
$927$	0	0
$928$	6.24264	0.204925
$929$	29.1127	0.955157	0.477578	−	0.878589i	$-0.341515\pi$
$929$	29.1127	0.955157	0.477578	+	0.878589i	$0.341515\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	−21.3137	−0.698154
$933$	0	0
$934$	−10.0833	−0.329934
$935$	42.6274	1.39407
$936$	0	0
$937$	9.79899	0.320119	0.160060	−	0.987107i	$-0.448831\pi$
$937$	9.79899	0.320119	0.160060	+	0.987107i	$0.448831\pi$
$938$	0.970563	0.0316900
$939$	0	0
$940$	6.68629	0.218083
$941$	30.8701	1.00634	0.503168	−	0.864189i	$-0.332168\pi$
$941$	30.8701	1.00634	0.503168	+	0.864189i	$0.332168\pi$
$942$	0	0
$943$	−30.1421	−0.981563
$944$	13.4558	0.437950
$945$	0	0
$946$	−9.71573	−0.315886
$947$	52.8284	1.71669	0.858347	−	0.513070i	$-0.171492\pi$
$947$	52.8284	1.71669	0.858347	+	0.513070i	$0.171492\pi$
$948$	0	0
$949$	−2.14214	−0.0695367
$950$	−0.414214	−0.0134389
$951$	0	0
$952$	15.3137	0.496320
$953$	2.54416	0.0824133	0.0412066	−	0.999151i	$-0.486880\pi$
$953$	2.54416	0.0824133	0.0412066	+	0.999151i	$0.486880\pi$
$954$	0	0
$955$	−1.75736	−0.0568668
$956$	−2.32590	−0.0752250
$957$	0	0
$958$	−1.21320	−0.0391968
$959$	−14.1421	−0.456673
$960$	0	0
$961$	46.9411	1.51423
$962$	0.142136	0.00458264
$963$	0	0
$964$	15.2548	0.491325
$965$	−9.07107	−0.292008
$966$	0	0
$967$	−40.0416	−1.28765	−0.643826	−	0.765172i	$-0.722654\pi$
$967$	−40.0416	−1.28765	−0.643826	+	0.765172i	$0.722654\pi$
$968$	−44.3553	−1.42563
$969$	0	0
$970$	−7.55635	−0.242620
$971$	−33.6569	−1.08010	−0.540050	−	0.841633i	$-0.681595\pi$
$971$	−33.6569	−1.08010	−0.540050	+	0.841633i	$0.681595\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	1.45584	0.0466483
$975$	0	0
$976$	−45.9411	−1.47054
$977$	24.2843	0.776923	0.388461	−	0.921465i	$-0.373007\pi$
$977$	24.2843	0.776923	0.388461	+	0.921465i	$0.373007\pi$
$978$	0	0
$979$	−86.7696	−2.77317
$980$	9.14214	0.292035
$981$	0	0
$982$	4.24264	0.135388
$983$	−20.6274	−0.657912	−0.328956	−	0.944345i	$-0.606697\pi$
$983$	−20.6274	−0.657912	−0.328956	+	0.944345i	$0.606697\pi$
$984$	0	0
$985$	−1.17157	−0.0373294
$986$	−4.00000	−0.127386
$987$	0	0
$988$	−1.07107	−0.0340752
$989$	13.7401	0.436910
$990$	0	0
$991$	13.6569	0.433824	0.216912	−	0.976191i	$-0.430401\pi$
$991$	13.6569	0.433824	0.216912	+	0.976191i	$0.430401\pi$
$992$	−38.9706	−1.23732
$993$	0	0
$994$	3.02944	0.0960879
$995$	−10.1421	−0.321527
$996$	0	0
$997$	58.0833	1.83952	0.919758	−	0.392487i	$-0.128385\pi$
$997$	58.0833	1.83952	0.919758	+	0.392487i	$0.128385\pi$
$998$	−4.48528	−0.141979
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.d.1.2		2
3.2	odd	2		285.2.a.g.1.1	✓	2
5.4	even	2		4275.2.a.y.1.1		2
12.11	even	2		4560.2.a.bf.1.1		2
15.2	even	4		1425.2.c.l.799.2		4
15.8	even	4		1425.2.c.l.799.3		4
15.14	odd	2		1425.2.a.k.1.2		2
57.56	even	2		5415.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.g.1.1	✓	2	3.2	odd	2
855.2.a.d.1.2		2	1.1	even	1	trivial
1425.2.a.k.1.2		2	15.14	odd	2
1425.2.c.l.799.2		4	15.2	even	4
1425.2.c.l.799.3		4	15.8	even	4
4275.2.a.y.1.1		2	5.4	even	2
4560.2.a.bf.1.1		2	12.11	even	2
5415.2.a.n.1.2		2	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +1.41421 q^{7} -1.58579 q^{8} +O(q^{10})\)	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +1.41421 q^{7} -1.58579 q^{8} +0.414214 q^{10} -6.24264 q^{11} -0.585786 q^{13} +0.585786 q^{14} +3.00000 q^{16} -6.82843 q^{17} -1.00000 q^{19} -1.82843 q^{20} -2.58579 q^{22} +3.65685 q^{23} +1.00000 q^{25} -0.242641 q^{26} -2.58579 q^{28} +1.41421 q^{29} -8.82843 q^{31} +4.41421 q^{32} -2.82843 q^{34} +1.41421 q^{35} -0.585786 q^{37} -0.414214 q^{38} -1.58579 q^{40} -8.24264 q^{41} +3.75736 q^{43} +11.4142 q^{44} +1.51472 q^{46} -3.65685 q^{47} -5.00000 q^{49} +0.414214 q^{50} +1.07107 q^{52} -8.00000 q^{53} -6.24264 q^{55} -2.24264 q^{56} +0.585786 q^{58} +4.48528 q^{59} -15.3137 q^{61} -3.65685 q^{62} -4.17157 q^{64} -0.585786 q^{65} +1.65685 q^{67} +12.4853 q^{68} +0.585786 q^{70} +5.17157 q^{71} +3.65685 q^{73} -0.242641 q^{74} +1.82843 q^{76} -8.82843 q^{77} +3.00000 q^{80} -3.41421 q^{82} -7.17157 q^{83} -6.82843 q^{85} +1.55635 q^{86} +9.89949 q^{88} +13.8995 q^{89} -0.828427 q^{91} -6.68629 q^{92} -1.51472 q^{94} -1.00000 q^{95} -18.2426 q^{97} -2.07107 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 6 q^{8} - 2 q^{10} - 4 q^{11} - 4 q^{13} + 4 q^{14} + 6 q^{16} - 8 q^{17} - 2 q^{19} + 2 q^{20} - 8 q^{22} - 4 q^{23} + 2 q^{25} + 8 q^{26} - 8 q^{28} - 12 q^{31} + 6 q^{32} - 4 q^{37} + 2 q^{38} - 6 q^{40} - 8 q^{41} + 16 q^{43} + 20 q^{44} + 20 q^{46} + 4 q^{47} - 10 q^{49} - 2 q^{50} - 12 q^{52} - 16 q^{53} - 4 q^{55} + 4 q^{56} + 4 q^{58} - 8 q^{59} - 8 q^{61} + 4 q^{62} - 14 q^{64} - 4 q^{65} - 8 q^{67} + 8 q^{68} + 4 q^{70} + 16 q^{71} - 4 q^{73} + 8 q^{74} - 2 q^{76} - 12 q^{77} + 6 q^{80} - 4 q^{82} - 20 q^{83} - 8 q^{85} - 28 q^{86} + 8 q^{89} + 4 q^{91} - 36 q^{92} - 20 q^{94} - 2 q^{95} - 28 q^{97} + 10 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 6 * q^8 - 2 * q^10 - 4 * q^11 - 4 * q^13 + 4 * q^14 + 6 * q^16 - 8 * q^17 - 2 * q^19 + 2 * q^20 - 8 * q^22 - 4 * q^23 + 2 * q^25 + 8 * q^26 - 8 * q^28 - 12 * q^31 + 6 * q^32 - 4 * q^37 + 2 * q^38 - 6 * q^40 - 8 * q^41 + 16 * q^43 + 20 * q^44 + 20 * q^46 + 4 * q^47 - 10 * q^49 - 2 * q^50 - 12 * q^52 - 16 * q^53 - 4 * q^55 + 4 * q^56 + 4 * q^58 - 8 * q^59 - 8 * q^61 + 4 * q^62 - 14 * q^64 - 4 * q^65 - 8 * q^67 + 8 * q^68 + 4 * q^70 + 16 * q^71 - 4 * q^73 + 8 * q^74 - 2 * q^76 - 12 * q^77 + 6 * q^80 - 4 * q^82 - 20 * q^83 - 8 * q^85 - 28 * q^86 + 8 * q^89 + 4 * q^91 - 36 * q^92 - 20 * q^94 - 2 * q^95 - 28 * q^97 + 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more