LMFDB - Embedded newform 2205.2.d.m.1324.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,2,Mod(1324,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2205.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.d (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:	$17.6070136457$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.309760000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 $ x^8 - 4x^7 + 8x^6 - 2x^5 - x^4 - 2x^3 + 18x^2 + 6x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 735)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1324.2
Root			$1.74861 + 1.74861i$ of defining polynomial
Character	$\chi$	$=$	2205.1324
Dual form			2205.2.d.m.1324.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49721i q^{2} -4.23607 q^{4} +(1.54336 - 1.61803i) q^{5} +5.58394i q^{8} +O(q^{10})$	\(q-2.49721i q^{2} -4.23607 q^{4} +(1.54336 - 1.61803i) q^{5} +5.58394i q^{8} +(-4.04057 - 3.85410i) q^{10} +4.47214 q^{11} +5.23607i q^{13} +5.47214 q^{16} +0.763932i q^{17} +8.08115 q^{19} +(-6.53779 + 6.85410i) q^{20} -11.1679i q^{22} +3.08672i q^{23} +(-0.236068 - 4.99442i) q^{25} +13.0756 q^{26} +2.00000 q^{29} +1.90770 q^{31} -2.49721i q^{32} +1.90770 q^{34} -6.17345i q^{37} -20.1803i q^{38} +(9.03500 + 8.61803i) q^{40} +6.90212 q^{41} +9.98885i q^{43} -18.9443 q^{44} +7.70820 q^{46} -4.94427i q^{47} +(-12.4721 + 0.589512i) q^{50} -22.1803i q^{52} +1.90770i q^{53} +(6.90212 - 7.23607i) q^{55} -4.99442i q^{58} +9.98885 q^{59} -3.81540 q^{61} -4.76393i q^{62} +4.70820 q^{64} +(8.47214 + 8.08115i) q^{65} +6.17345i q^{67} -3.23607i q^{68} -0.472136 q^{71} +2.76393i q^{73} -15.4164 q^{74} -34.2323 q^{76} -12.9443 q^{79} +(8.44549 - 8.85410i) q^{80} -17.2361i q^{82} -6.47214i q^{83} +(1.23607 + 1.17902i) q^{85} +24.9443 q^{86} +24.9721i q^{88} +9.26017 q^{89} -13.0756i q^{92} -12.3469 q^{94} +(12.4721 - 13.0756i) q^{95} -10.1803i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4}+O(q^{10}) $ 8 * q - 16 * q^4	$ 8 q - 16 q^{4} + 8 q^{16} + 16 q^{25} + 16 q^{29} - 80 q^{44} + 8 q^{46} - 64 q^{50} - 16 q^{64} + 32 q^{65} + 32 q^{71} - 16 q^{74} - 32 q^{79} - 8 q^{85} + 128 q^{86} + 64 q^{95}+O(q^{100}) $ 8 * q - 16 * q^4 + 8 * q^16 + 16 * q^25 + 16 * q^29 - 80 * q^44 + 8 * q^46 - 64 * q^50 - 16 * q^64 + 32 * q^65 + 32 * q^71 - 16 * q^74 - 32 * q^79 - 8 * q^85 + 128 * q^86 + 64 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times$.

$n$	$442$	$1081$	$1226$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	2.49721i		−	1.76580i	−0.469565	−	0.882898i	$-0.655589\pi$
$2$		−	2.49721i		−	1.76580i	0.469565	−	0.882898i	$-0.344411\pi$
$3$		0			0
$3$		0			0
$4$	−4.23607			−2.11803
$5$	1.54336	−	1.61803i	0.690212	−	0.723607i
$5$	1.54336	−	1.61803i	0.690212	−	0.723607i
$6$		0			0
$7$		0			0
$7$		0			0
$8$			5.58394i			1.97422i
$9$		0			0
$10$	−4.04057	−	3.85410i	−1.27774	−	1.21877i
$11$	4.47214			1.34840			0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214			1.34840			0.674200	+	0.738549i	$0.264489\pi$
$12$		0			0
$13$			5.23607i			1.45222i	0.687576	+	0.726112i	$0.258675\pi$
$13$			5.23607i			1.45222i	−0.687576	+	0.726112i	$0.741325\pi$
$14$		0			0
$15$		0			0
$16$	5.47214			1.36803
$17$			0.763932i			0.185281i	0.995700	+	0.0926404i	$0.0295307\pi$
$17$			0.763932i			0.185281i	−0.995700	+	0.0926404i	$0.970469\pi$
$18$		0			0
$19$	8.08115			1.85394			0.926971	−	0.375132i	$-0.122403\pi$
$19$	8.08115			1.85394			0.926971	+	0.375132i	$0.122403\pi$
$20$	−6.53779	+	6.85410i	−1.46189	+	1.53262i
$21$		0			0
$22$		−	11.1679i		−	2.38100i
$23$			3.08672i			0.643626i	0.946803	+	0.321813i	$0.104292\pi$
$23$			3.08672i			0.643626i	−0.946803	+	0.321813i	$0.895708\pi$
$24$		0			0
$25$	−0.236068	−	4.99442i	−0.0472136	−	0.998885i
$26$	13.0756			2.56433
$27$		0			0
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	1.90770			0.342633			0.171317	−	0.985216i	$-0.445198\pi$
$31$	1.90770			0.342633			0.171317	+	0.985216i	$0.445198\pi$
$32$		−	2.49721i		−	0.441449i
$33$		0			0
$34$	1.90770			0.327168
$35$		0			0
$36$		0			0
$37$		−	6.17345i		−	1.01491i	−0.861679	−	0.507454i	$-0.830587\pi$
$37$		−	6.17345i		−	1.01491i	0.861679	−	0.507454i	$-0.169413\pi$
$38$		−	20.1803i		−	3.27368i
$39$		0			0
$40$	9.03500	+	8.61803i	1.42856	+	1.36263i
$41$	6.90212			1.07793			0.538965	−	0.842328i	$-0.318815\pi$
$41$	6.90212			1.07793			0.538965	+	0.842328i	$0.318815\pi$
$42$		0			0
$43$			9.98885i			1.52329i	0.647997	+	0.761643i	$0.275607\pi$
$43$			9.98885i			1.52329i	−0.647997	+	0.761643i	$0.724393\pi$
$44$	−18.9443			−2.85596
$45$		0			0
$46$	7.70820			1.13651
$47$		−	4.94427i		−	0.721196i	−0.932721	−	0.360598i	$-0.882573\pi$
$47$		−	4.94427i		−	0.721196i	0.932721	−	0.360598i	$-0.117427\pi$
$48$		0			0
$49$		0			0
$50$	−12.4721	+	0.589512i	−1.76383	+	0.0833696i
$51$		0			0
$52$		−	22.1803i		−	3.07586i
$53$			1.90770i			0.262043i	0.991380	+	0.131021i	$0.0418257\pi$
$53$			1.90770i			0.262043i	−0.991380	+	0.131021i	$0.958174\pi$
$54$		0			0
$55$	6.90212	−	7.23607i	0.930682	−	0.975711i
$56$		0			0
$57$		0			0
$58$		−	4.99442i		−	0.655800i
$59$	9.98885			1.30044			0.650219	−	0.759747i	$-0.274677\pi$
$59$	9.98885			1.30044			0.650219	+	0.759747i	$0.274677\pi$
$60$		0			0
$61$	−3.81540			−0.488512			−0.244256	−	0.969711i	$-0.578544\pi$
$61$	−3.81540			−0.488512			−0.244256	+	0.969711i	$0.578544\pi$
$62$		−	4.76393i		−	0.605020i
$63$		0			0
$64$	4.70820			0.588525
$65$	8.47214	+	8.08115i	1.05084	+	1.00234i
$66$		0			0
$67$			6.17345i			0.754207i	0.926171	+	0.377103i	$0.123080\pi$
$67$			6.17345i			0.754207i	−0.926171	+	0.377103i	$0.876920\pi$
$68$		−	3.23607i		−	0.392431i
$69$		0			0
$70$		0			0
$71$	−0.472136			−0.0560322			−0.0280161	−	0.999607i	$-0.508919\pi$
$71$	−0.472136			−0.0560322			−0.0280161	+	0.999607i	$0.508919\pi$
$72$		0			0
$73$			2.76393i			0.323494i	0.986832	+	0.161747i	$0.0517128\pi$
$73$			2.76393i			0.323494i	−0.986832	+	0.161747i	$0.948287\pi$
$74$	−15.4164			−1.79212
$75$		0			0
$76$	−34.2323			−3.92671
$77$		0			0
$78$		0			0
$79$	−12.9443			−1.45634			−0.728172	−	0.685394i	$-0.759630\pi$
$79$	−12.9443			−1.45634			−0.728172	+	0.685394i	$0.759630\pi$
$80$	8.44549	−	8.85410i	0.944234	−	0.989919i
$81$		0			0
$82$		−	17.2361i		−	1.90341i
$83$		−	6.47214i		−	0.710409i	−0.934789	−	0.355205i	$-0.884411\pi$
$83$		−	6.47214i		−	0.710409i	0.934789	−	0.355205i	$-0.115589\pi$
$84$		0			0
$85$	1.23607	+	1.17902i	0.134070	+	0.127883i
$86$	24.9443			2.68981
$87$		0			0
$88$			24.9721i			2.66204i
$89$	9.26017			0.981576			0.490788	−	0.871279i	$-0.336709\pi$
$89$	9.26017			0.981576			0.490788	+	0.871279i	$0.336709\pi$
$90$		0			0
$91$		0			0
$92$		−	13.0756i		−	1.36322i
$93$		0			0
$94$	−12.3469			−1.27349
$95$	12.4721	−	13.0756i	1.27961	−	1.34153i
$96$		0			0
$97$		−	10.1803i		−	1.03366i	−0.856089	−	0.516828i	$-0.827113\pi$
$97$		−	10.1803i		−	1.03366i	0.856089	−	0.516828i	$-0.172887\pi$
$98$		0			0
$99$		0			0
$100$	1.00000	+	21.1567i	0.100000	+	2.11567i
$101$	−13.0756			−1.30107			−0.650534	−	0.759477i	$-0.725455\pi$
$101$	−13.0756			−1.30107			−0.650534	+	0.759477i	$0.725455\pi$
$102$		0			0
$103$			1.52786i			0.150545i	0.997163	+	0.0752725i	$0.0239827\pi$
$103$			1.52786i			0.150545i	−0.997163	+	0.0752725i	$0.976017\pi$
$104$	−29.2379			−2.86701
$105$		0			0
$106$	4.76393			0.462714
$107$			6.90212i			0.667254i	0.942705	+	0.333627i	$0.108273\pi$
$107$			6.90212i			0.667254i	−0.942705	+	0.333627i	$0.891727\pi$
$108$		0			0
$109$	−12.4721			−1.19461			−0.597307	−	0.802013i	$-0.703763\pi$
$109$	−12.4721			−1.19461			−0.597307	+	0.802013i	$0.703763\pi$
$110$	−18.0700	−	17.2361i	−1.72291	−	1.64339i
$111$		0			0
$112$		0			0
$113$		−	4.26575i		−	0.401288i	−0.979664	−	0.200644i	$-0.935697\pi$
$113$		−	4.26575i		−	0.401288i	0.979664	−	0.200644i	$-0.0643034\pi$
$114$		0			0
$115$	4.99442	+	4.76393i	0.465732	+	0.444239i
$116$	−8.47214			−0.786618
$117$		0			0
$118$		−	24.9443i		−	2.29631i
$119$		0			0
$120$		0			0
$121$	9.00000			0.818182
$122$			9.52786i			0.862612i
$123$		0			0
$124$	−8.08115			−0.725709
$125$	−8.44549	−	7.32624i	−0.755387	−	0.655279i
$126$		0			0
$127$		−	3.81540i		−	0.338562i	−0.985568	−	0.169281i	$-0.945855\pi$
$127$		−	3.81540i		−	0.338562i	0.985568	−	0.169281i	$-0.0541445\pi$
$128$		−	16.7518i		−	1.48066i
$129$		0			0
$130$	20.1803	−	21.1567i	1.76993	−	1.85557i
$131$	9.98885			0.872730			0.436365	−	0.899770i	$-0.356266\pi$
$131$	9.98885			0.872730			0.436365	+	0.899770i	$0.356266\pi$
$132$		0			0
$133$		0			0
$134$	15.4164			1.33177
$135$		0			0
$136$	−4.26575			−0.365785
$137$			8.08115i			0.690419i	0.938526	+	0.345210i	$0.112192\pi$
$137$			8.08115i			0.690419i	−0.938526	+	0.345210i	$0.887808\pi$
$138$		0			0
$139$	−14.2546			−1.20906			−0.604530	−	0.796583i	$-0.706639\pi$
$139$	−14.2546			−1.20906			−0.604530	+	0.796583i	$0.706639\pi$
$140$		0			0
$141$		0			0
$142$			1.17902i			0.0989415i
$143$			23.4164i			1.95818i
$144$		0			0
$145$	3.08672	−	3.23607i	0.256338	−	0.268741i
$146$	6.90212			0.571224
$147$		0			0
$148$			26.1511i			2.14961i
$149$	−18.9443			−1.55198			−0.775988	−	0.630748i	$-0.782748\pi$
$149$	−18.9443			−1.55198			−0.775988	+	0.630748i	$0.782748\pi$
$150$		0			0
$151$	−8.94427			−0.727875			−0.363937	−	0.931423i	$-0.618568\pi$
$151$	−8.94427			−0.727875			−0.363937	+	0.931423i	$0.618568\pi$
$152$			45.1246i			3.66009i
$153$		0			0
$154$		0			0
$155$	2.94427	−	3.08672i	0.236490	−	0.247932i
$156$		0			0
$157$			8.65248i			0.690543i	0.938503	+	0.345271i	$0.112213\pi$
$157$			8.65248i			0.690543i	−0.938503	+	0.345271i	$0.887787\pi$
$158$			32.3246i			2.57161i
$159$		0			0
$160$	−4.04057	−	3.85410i	−0.319435	−	0.304694i
$161$		0			0
$162$		0			0
$163$		−	3.81540i		−	0.298845i	−0.988773	−	0.149423i	$-0.952259\pi$
$163$		−	3.81540i		−	0.298845i	0.988773	−	0.149423i	$-0.0477415\pi$
$164$	−29.2379			−2.28309
$165$		0			0
$166$	−16.1623			−1.25444
$167$		−	15.4164i		−	1.19296i	−0.802629	−	0.596479i	$-0.796566\pi$
$167$		−	15.4164i		−	1.19296i	0.802629	−	0.596479i	$-0.203434\pi$
$168$		0			0
$169$	−14.4164			−1.10895
$170$	2.94427	−	3.08672i	0.225815	−	0.236741i
$171$		0			0
$172$		−	42.3134i		−	3.22637i
$173$		−	19.2361i		−	1.46249i	−0.682114	−	0.731246i	$-0.738939\pi$
$173$		−	19.2361i		−	1.46249i	0.682114	−	0.731246i	$-0.261061\pi$
$174$		0			0
$175$		0			0
$176$	24.4721			1.84466
$177$		0			0
$178$		−	23.1246i		−	1.73326i
$179$	−7.52786			−0.562659			−0.281329	−	0.959611i	$-0.590775\pi$
$179$	−7.52786			−0.562659			−0.281329	+	0.959611i	$0.590775\pi$
$180$		0			0
$181$	9.98885			0.742465			0.371233	−	0.928540i	$-0.378935\pi$
$181$	9.98885			0.742465			0.371233	+	0.928540i	$0.378935\pi$
$182$		0			0
$183$		0			0
$184$	−17.2361			−1.27066
$185$	−9.98885	−	9.52786i	−0.734395	−	0.700502i
$186$		0			0
$187$			3.41641i			0.249832i
$188$			20.9443i			1.52752i
$189$		0			0
$190$	−32.6525	−	31.1456i	−2.36886	−	2.25954i
$191$	3.52786			0.255267			0.127634	−	0.991821i	$-0.459262\pi$
$191$	3.52786			0.255267			0.127634	+	0.991821i	$0.459262\pi$
$192$		0			0
$193$			13.8042i			0.993652i	0.867850	+	0.496826i	$0.165501\pi$
$193$			13.8042i			0.993652i	−0.867850	+	0.496826i	$0.834499\pi$
$194$	−25.4225			−1.82523
$195$		0			0
$196$		0			0
$197$			1.90770i			0.135918i	0.997688	+	0.0679590i	$0.0216487\pi$
$197$			1.90770i			0.135918i	−0.997688	+	0.0679590i	$0.978351\pi$
$198$		0			0
$199$	−11.8965			−0.843324			−0.421662	−	0.906753i	$-0.638553\pi$
$199$	−11.8965			−0.843324			−0.421662	+	0.906753i	$0.638553\pi$
$200$	27.8885	−	1.31819i	1.97202	−	0.0932100i
$201$		0			0
$202$			32.6525i			2.29742i
$203$		0			0
$204$		0			0
$205$	10.6525	−	11.1679i	0.744001	−	0.779998i
$206$	3.81540			0.265832
$207$		0			0
$208$			28.6525i			1.98669i
$209$	36.1400			2.49986
$210$		0			0
$211$	−12.9443			−0.891120			−0.445560	−	0.895252i	$-0.646995\pi$
$211$	−12.9443			−0.891120			−0.445560	+	0.895252i	$0.646995\pi$
$212$		−	8.08115i		−	0.555016i
$213$		0			0
$214$	17.2361			1.17823
$215$	16.1623	+	15.4164i	1.10226	+	1.05139i
$216$		0			0
$217$		0			0
$218$			31.1456i			2.10944i
$219$		0			0
$220$	−29.2379	+	30.6525i	−1.97122	+	2.06659i
$221$	−4.00000			−0.269069
$222$		0			0
$223$		−	17.8885i		−	1.19791i	−0.800784	−	0.598953i	$-0.795584\pi$
$223$		−	17.8885i		−	1.19791i	0.800784	−	0.598953i	$-0.204416\pi$
$224$		0			0
$225$		0			0
$226$	−10.6525			−0.708592
$227$		−	1.52786i		−	0.101408i	−0.998714	−	0.0507039i	$-0.983854\pi$
$227$		−	1.52786i		−	0.101408i	0.998714	−	0.0507039i	$-0.0161465\pi$
$228$		0			0
$229$	22.3357			1.47599			0.737994	−	0.674808i	$-0.235773\pi$
$229$	22.3357			1.47599			0.737994	+	0.674808i	$0.235773\pi$
$230$	11.8965	−	12.4721i	0.784435	−	0.822388i
$231$		0			0
$232$			11.1679i			0.733207i
$233$		−	11.8965i		−	0.779369i	−0.920948	−	0.389684i	$-0.872584\pi$
$233$		−	11.8965i		−	0.779369i	0.920948	−	0.389684i	$-0.127416\pi$
$234$		0			0
$235$	−8.00000	−	7.63080i	−0.521862	−	0.497779i
$236$	−42.3134			−2.75437
$237$		0			0
$238$		0			0
$239$	−7.52786			−0.486937			−0.243469	−	0.969909i	$-0.578285\pi$
$239$	−7.52786			−0.486937			−0.243469	+	0.969909i	$0.578285\pi$
$240$		0			0
$241$	6.17345			0.397667			0.198833	−	0.980033i	$-0.436285\pi$
$241$	6.17345			0.397667			0.198833	+	0.980033i	$0.436285\pi$
$242$		−	22.4749i		−	1.44474i
$243$		0			0
$244$	16.1623			1.03468
$245$		0			0
$246$		0			0
$247$			42.3134i			2.69234i
$248$			10.6525i			0.676433i
$249$		0			0
$250$	−18.2952	+	21.0902i	−1.15709	+	1.33386i
$251$	2.35805			0.148839			0.0744193	−	0.997227i	$-0.476290\pi$
$251$	2.35805			0.148839			0.0744193	+	0.997227i	$0.476290\pi$
$252$		0			0
$253$			13.8042i			0.867866i
$254$	−9.52786			−0.597831
$255$		0			0
$256$	−32.4164			−2.02603
$257$			21.7082i			1.35412i	0.735928	+	0.677060i	$0.236746\pi$
$257$			21.7082i			1.35412i	−0.735928	+	0.677060i	$0.763254\pi$
$258$		0			0
$259$		0			0
$260$	−35.8885	−	34.2323i	−2.22571	−	2.12300i
$261$		0			0
$262$		−	24.9443i		−	1.54106i
$263$		−	15.4336i		−	0.951678i	−0.879532	−	0.475839i	$-0.842145\pi$
$263$		−	15.4336i		−	0.951678i	0.879532	−	0.475839i	$-0.157855\pi$
$264$		0			0
$265$	3.08672	+	2.94427i	0.189616	+	0.180865i
$266$		0			0
$267$		0			0
$268$		−	26.1511i		−	1.59744i
$269$	6.90212			0.420830			0.210415	−	0.977612i	$-0.432518\pi$
$269$	6.90212			0.420830			0.210415	+	0.977612i	$0.432518\pi$
$270$		0			0
$271$	18.0700			1.09767			0.548837	−	0.835929i	$-0.315071\pi$
$271$	18.0700			1.09767			0.548837	+	0.835929i	$0.315071\pi$
$272$			4.18034i			0.253470i
$273$		0			0
$274$	20.1803			1.21914
$275$	−1.05573	−	22.3357i	−0.0636628	−	1.34690i
$276$		0			0
$277$			9.98885i			0.600172i	0.953912	+	0.300086i	$0.0970153\pi$
$277$			9.98885i			0.600172i	−0.953912	+	0.300086i	$0.902985\pi$
$278$			35.5967i			2.13495i
$279$		0			0
$280$		0			0
$281$	22.0000			1.31241			0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000			1.31241			0.656205	+	0.754583i	$0.272161\pi$
$282$		0			0
$283$			12.3607i			0.734766i	0.930070	+	0.367383i	$0.119746\pi$
$283$			12.3607i			0.734766i	−0.930070	+	0.367383i	$0.880254\pi$
$284$	2.00000			0.118678
$285$		0			0
$286$	58.4757			3.45774
$287$		0			0
$288$		0			0
$289$	16.4164			0.965671
$290$	−8.08115	−	7.70820i	−0.474541	−	0.452641i
$291$		0			0
$292$		−	11.7082i		−	0.685171i
$293$		−	20.1803i		−	1.17895i	−0.807787	−	0.589474i	$-0.799335\pi$
$293$		−	20.1803i		−	1.17895i	0.807787	−	0.589474i	$-0.200665\pi$
$294$		0			0
$295$	15.4164	−	16.1623i	0.897578	−	0.941005i
$296$	34.4721			2.00365
$297$		0			0
$298$			47.3079i			2.74047i
$299$	−16.1623			−0.934690
$300$		0			0
$301$		0			0
$302$			22.3357i			1.28528i
$303$		0			0
$304$	44.2211			2.53626
$305$	−5.88854	+	6.17345i	−0.337177	+	0.353491i
$306$		0			0
$307$		−	2.47214i		−	0.141092i	−0.997509	−	0.0705461i	$-0.977526\pi$
$307$		−	2.47214i		−	0.141092i	0.997509	−	0.0705461i	$-0.0224742\pi$
$308$		0			0
$309$		0			0
$310$	−7.70820	−	7.35247i	−0.437797	−	0.417592i
$311$	−3.81540			−0.216352			−0.108176	−	0.994132i	$-0.534501\pi$
$311$	−3.81540			−0.216352			−0.108176	+	0.994132i	$0.534501\pi$
$312$		0			0
$313$			24.6525i			1.39344i	0.717343	+	0.696720i	$0.245358\pi$
$313$			24.6525i			1.39344i	−0.717343	+	0.696720i	$0.754642\pi$
$314$	21.6071			1.21936
$315$		0			0
$316$	54.8328			3.08459
$317$			10.4392i			0.586324i	0.956063	+	0.293162i	$0.0947075\pi$
$317$			10.4392i			0.586324i	−0.956063	+	0.293162i	$0.905293\pi$
$318$		0			0
$319$	8.94427			0.500783
$320$	7.26646	−	7.61803i	0.406208	−	0.425861i
$321$		0			0
$322$		0			0
$323$			6.17345i			0.343500i
$324$		0			0
$325$	26.1511	−	1.23607i	1.45060	−	0.0685647i
$326$	−9.52786			−0.527700
$327$		0			0
$328$			38.5410i			2.12807i
$329$		0			0
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$			27.4164i			1.50467i
$333$		0			0
$334$	−38.4980			−2.10652
$335$	9.98885	+	9.52786i	0.545749	+	0.520563i
$336$		0			0
$337$		−	3.81540i		−	0.207838i	−0.994586	−	0.103919i	$-0.966862\pi$
$337$		−	3.81540i		−	0.207838i	0.994586	−	0.103919i	$-0.0331383\pi$
$338$			36.0008i			1.95819i
$339$		0			0
$340$	−5.23607	−	4.99442i	−0.283966	−	0.270861i
$341$	8.53149			0.462006
$342$		0			0
$343$		0			0
$344$	−55.7771			−3.00730
$345$		0			0
$346$	−48.0365			−2.58246
$347$		−	25.4225i		−	1.36475i	−0.731003	−	0.682375i	$-0.760947\pi$
$347$		−	25.4225i		−	1.36475i	0.731003	−	0.682375i	$-0.239053\pi$
$348$		0			0
$349$	8.53149			0.456680			0.228340	−	0.973581i	$-0.426670\pi$
$349$	8.53149			0.456680			0.228340	+	0.973581i	$0.426670\pi$
$350$		0			0
$351$		0			0
$352$		−	11.1679i		−	0.595250i
$353$			3.23607i			0.172239i	0.996285	+	0.0861193i	$0.0274466\pi$
$353$			3.23607i			0.172239i	−0.996285	+	0.0861193i	$0.972553\pi$
$354$		0			0
$355$	−0.728677	+	0.763932i	−0.0386741	+	0.0405453i
$356$	−39.2267			−2.07901
$357$		0			0
$358$			18.7987i			0.993541i
$359$	20.4721			1.08048			0.540239	−	0.841512i	$-0.318334\pi$
$359$	20.4721			1.08048			0.540239	+	0.841512i	$0.318334\pi$
$360$		0			0
$361$	46.3050			2.43710
$362$		−	24.9443i		−	1.31104i
$363$		0			0
$364$		0			0
$365$	4.47214	+	4.26575i	0.234082	+	0.223279i
$366$		0			0
$367$			16.0000i			0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$			16.0000i			0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$			16.8910i			0.880503i
$369$		0			0
$370$	−23.7931	+	24.9443i	−1.23694	+	1.29679i
$371$		0			0
$372$		0			0
$373$		−	32.3246i		−	1.67370i	−0.547429	−	0.836852i	$-0.684393\pi$
$373$		−	32.3246i		−	1.67370i	0.547429	−	0.836852i	$-0.315607\pi$
$374$	8.53149			0.441153
$375$		0			0
$376$	27.6085			1.42380
$377$			10.4721i			0.539342i
$378$		0			0
$379$	−3.05573			−0.156962			−0.0784811	−	0.996916i	$-0.525007\pi$
$379$	−3.05573			−0.156962			−0.0784811	+	0.996916i	$0.525007\pi$
$380$	−52.8328	+	55.3890i	−2.71027	+	2.84140i
$381$		0			0
$382$		−	8.80982i		−	0.450750i
$383$		−	10.4721i		−	0.535101i	−0.963544	−	0.267551i	$-0.913786\pi$
$383$		−	10.4721i		−	0.535101i	0.963544	−	0.267551i	$-0.0862142\pi$
$384$		0			0
$385$		0			0
$386$	34.4721			1.75459
$387$		0			0
$388$			43.1246i			2.18932i
$389$	−11.8885			−0.602773			−0.301387	−	0.953502i	$-0.597449\pi$
$389$	−11.8885			−0.602773			−0.301387	+	0.953502i	$0.597449\pi$
$390$		0			0
$391$	−2.35805			−0.119252
$392$		0			0
$393$		0			0
$394$	4.76393			0.240003
$395$	−19.9777	+	20.9443i	−1.00519	+	1.05382i
$396$		0			0
$397$			9.23607i			0.463545i	0.972770	+	0.231772i	$0.0744525\pi$
$397$			9.23607i			0.463545i	−0.972770	+	0.231772i	$0.925548\pi$
$398$			29.7082i			1.48914i
$399$		0			0
$400$	−1.29180	−	27.3302i	−0.0645898	−	1.36651i
$401$	−36.8328			−1.83934			−0.919672	−	0.392689i	$-0.871545\pi$
$401$	−36.8328			−1.83934			−0.919672	+	0.392689i	$0.871545\pi$
$402$		0			0
$403$			9.98885i			0.497580i
$404$	55.3890			2.75571
$405$		0			0
$406$		0			0
$407$		−	27.6085i		−	1.36850i
$408$		0			0
$409$	−33.7819			−1.67041			−0.835205	−	0.549939i	$-0.814651\pi$
$409$	−33.7819			−1.67041			−0.835205	+	0.549939i	$0.814651\pi$
$410$	−27.8885	−	26.6015i	−1.37732	−	1.31375i
$411$		0			0
$412$		−	6.47214i		−	0.318859i
$413$		0			0
$414$		0			0
$415$	−10.4721	−	9.98885i	−0.514057	−	0.490333i
$416$	13.0756			0.641083
$417$		0			0
$418$		−	90.2492i		−	4.41423i
$419$	1.45735			0.0711964			0.0355982	−	0.999366i	$-0.488666\pi$
$419$	1.45735			0.0711964			0.0355982	+	0.999366i	$0.488666\pi$
$420$		0			0
$421$	16.4721			0.802803			0.401401	−	0.915902i	$-0.368523\pi$
$421$	16.4721			0.802803			0.401401	+	0.915902i	$0.368523\pi$
$422$			32.3246i			1.57354i
$423$		0			0
$424$	−10.6525			−0.517330
$425$	3.81540	−	0.180340i	0.185074	−	0.00874777i
$426$		0			0
$427$		0			0
$428$		−	29.2379i		−	1.41327i
$429$		0			0
$430$	38.4980	−	40.3607i	1.85654	−	1.94636i
$431$	4.47214			0.215415			0.107708	−	0.994183i	$-0.465649\pi$
$431$	4.47214			0.215415			0.107708	+	0.994183i	$0.465649\pi$
$432$		0			0
$433$			35.7082i			1.71603i	0.513627	+	0.858013i	$0.328301\pi$
$433$			35.7082i			1.71603i	−0.513627	+	0.858013i	$0.671699\pi$
$434$		0			0
$435$		0			0
$436$	52.8328			2.53023
$437$			24.9443i			1.19325i
$438$		0			0
$439$	−24.2434			−1.15708			−0.578538	−	0.815655i	$-0.696377\pi$
$439$	−24.2434			−1.15708			−0.578538	+	0.815655i	$0.696377\pi$
$440$	40.4057	+	38.5410i	1.92627	+	1.83737i
$441$		0			0
$442$			9.98885i			0.475121i
$443$		−	31.5959i		−	1.50117i	−0.660775	−	0.750584i	$-0.729772\pi$
$443$		−	31.5959i		−	1.50117i	0.660775	−	0.750584i	$-0.270228\pi$
$444$		0			0
$445$	14.2918	−	14.9833i	0.677496	−	0.710275i
$446$	−44.6715			−2.11526
$447$		0			0
$448$		0			0
$449$	−17.0557			−0.804910			−0.402455	−	0.915440i	$-0.631843\pi$
$449$	−17.0557			−0.804910			−0.402455	+	0.915440i	$0.631843\pi$
$450$		0			0
$451$	30.8672			1.45348
$452$			18.0700i			0.849941i
$453$		0			0
$454$	−3.81540			−0.179066
$455$		0			0
$456$		0			0
$457$			23.7931i			1.11299i	0.830850	+	0.556497i	$0.187855\pi$
$457$			23.7931i			1.11299i	−0.830850	+	0.556497i	$0.812145\pi$
$458$		−	55.7771i		−	2.60629i
$459$		0			0
$460$	−21.1567	−	20.1803i	−0.986437	−	0.940913i
$461$	−36.8687			−1.71715			−0.858573	−	0.512692i	$-0.828648\pi$
$461$	−36.8687			−1.71715			−0.858573	+	0.512692i	$0.828648\pi$
$462$		0			0
$463$		−	28.5092i		−	1.32493i	−0.749091	−	0.662467i	$-0.769509\pi$
$463$		−	28.5092i		−	1.32493i	0.749091	−	0.662467i	$-0.230491\pi$
$464$	10.9443			0.508075
$465$		0			0
$466$	−29.7082			−1.37621
$467$		−	27.4164i		−	1.26868i	−0.773054	−	0.634340i	$-0.781272\pi$
$467$		−	27.4164i		−	1.26868i	0.773054	−	0.634340i	$-0.218728\pi$
$468$		0			0
$469$		0			0
$470$	−19.0557	+	19.9777i	−0.878975	+	0.921502i
$471$		0			0
$472$			55.7771i			2.56735i
$473$			44.6715i			2.05400i
$474$		0			0
$475$	−1.90770	−	40.3607i	−0.0875313	−	1.85187i
$476$		0			0
$477$		0			0
$478$			18.7987i			0.859831i
$479$	28.5092			1.30262			0.651309	−	0.758813i	$-0.274220\pi$
$479$	28.5092			1.30262			0.651309	+	0.758813i	$0.274220\pi$
$480$		0			0
$481$	32.3246			1.47387
$482$		−	15.4164i		−	0.702198i
$483$		0			0
$484$	−38.1246			−1.73294
$485$	−16.4721	−	15.7119i	−0.747961	−	0.713443i
$486$		0			0
$487$			9.98885i			0.452638i	0.974053	+	0.226319i	$0.0726692\pi$
$487$			9.98885i			0.452638i	−0.974053	+	0.226319i	$0.927331\pi$
$488$		−	21.3050i		−	0.964430i
$489$		0			0
$490$		0			0
$491$	−18.3607			−0.828606			−0.414303	−	0.910139i	$-0.635975\pi$
$491$	−18.3607			−0.828606			−0.414303	+	0.910139i	$0.635975\pi$
$492$		0			0
$493$			1.52786i			0.0688115i
$494$	105.666			4.75412
$495$		0			0
$496$	10.4392			0.468734
$497$		0			0
$498$		0			0
$499$	−8.00000			−0.358129			−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000			−0.358129			−0.179065	+	0.983837i	$0.557307\pi$
$500$	35.7757	+	31.0344i	1.59994	+	1.38790i
$501$		0			0
$502$		−	5.88854i		−	0.262819i
$503$		−	29.5279i		−	1.31658i	−0.752763	−	0.658291i	$-0.771280\pi$
$503$		−	29.5279i		−	1.31658i	0.752763	−	0.658291i	$-0.228720\pi$
$504$		0			0
$505$	−20.1803	+	21.1567i	−0.898013	+	0.941462i
$506$	34.4721			1.53247
$507$		0			0
$508$			16.1623i			0.717086i
$509$	13.0756			0.579565			0.289782	−	0.957093i	$-0.406417\pi$
$509$	13.0756			0.579565			0.289782	+	0.957093i	$0.406417\pi$
$510$		0			0
$511$		0			0
$512$			47.4470i			2.09688i
$513$		0			0
$514$	54.2100			2.39110
$515$	2.47214	+	2.35805i	0.108935	+	0.103908i
$516$		0			0
$517$		−	22.1115i		−	0.972461i
$518$		0			0
$519$		0			0
$520$	−45.1246	+	47.3079i	−1.97885	+	2.07459i
$521$	39.2267			1.71855			0.859277	−	0.511511i	$-0.170914\pi$
$521$	39.2267			1.71855			0.859277	+	0.511511i	$0.170914\pi$
$522$		0			0
$523$			15.0557i			0.658341i	0.944270	+	0.329171i	$0.106769\pi$
$523$			15.0557i			0.658341i	−0.944270	+	0.329171i	$0.893231\pi$
$524$	−42.3134			−1.84847
$525$		0			0
$526$	−38.5410			−1.68047
$527$			1.45735i			0.0634833i
$528$		0			0
$529$	13.4721			0.585745
$530$	7.35247	−	7.70820i	0.319371	−	0.334823i
$531$		0			0
$532$		0			0
$533$			36.1400i			1.56540i
$534$		0			0
$535$	11.1679	+	10.6525i	0.482829	+	0.460547i
$536$	−34.4721			−1.48897
$537$		0			0
$538$		−	17.2361i		−	0.743100i
$539$		0			0
$540$		0			0
$541$	−24.4721			−1.05214			−0.526070	−	0.850441i	$-0.676335\pi$
$541$	−24.4721			−1.05214			−0.526070	+	0.850441i	$0.676335\pi$
$542$		−	45.1246i		−	1.93827i
$543$		0			0
$544$	1.90770			0.0817920
$545$	−19.2490	+	20.1803i	−0.824537	+	0.864431i
$546$		0			0
$547$		−	38.4980i		−	1.64606i	−0.568000	−	0.823029i	$-0.692283\pi$
$547$		−	38.4980i		−	1.64606i	0.568000	−	0.823029i	$-0.307717\pi$
$548$		−	34.2323i		−	1.46233i
$549$		0			0
$550$	−55.7771	+	2.63638i	−2.37834	+	0.112415i
$551$	16.1623			0.688537
$552$		0			0
$553$		0			0
$554$	24.9443			1.05978
$555$		0			0
$556$	60.3834			2.56083
$557$			30.4169i			1.28881i	0.764686	+	0.644403i	$0.222894\pi$
$557$			30.4169i			1.28881i	−0.764686	+	0.644403i	$0.777106\pi$
$558$		0			0
$559$	−52.3023			−2.21215
$560$		0			0
$561$		0			0
$562$		−	54.9387i		−	2.31745i
$563$		−	13.8885i		−	0.585332i	−0.956215	−	0.292666i	$-0.905458\pi$
$563$		−	13.8885i		−	0.585332i	0.956215	−	0.292666i	$-0.0945425\pi$
$564$		0			0
$565$	−6.90212	−	6.58359i	−0.290375	−	0.276974i
$566$	30.8672			1.29745
$567$		0			0
$568$		−	2.63638i		−	0.110620i
$569$	36.8328			1.54411			0.772056	−	0.635555i	$-0.219228\pi$
$569$	36.8328			1.54411			0.772056	+	0.635555i	$0.219228\pi$
$570$		0			0
$571$	−42.8328			−1.79250			−0.896249	−	0.443552i	$-0.853718\pi$
$571$	−42.8328			−1.79250			−0.896249	+	0.443552i	$0.853718\pi$
$572$		−	99.1935i		−	4.14749i
$573$		0			0
$574$		0			0
$575$	15.4164	−	0.728677i	0.642909	−	0.0303879i
$576$		0			0
$577$		−	16.6525i		−	0.693252i	−0.938003	−	0.346626i	$-0.887327\pi$
$577$		−	16.6525i		−	0.693252i	0.938003	−	0.346626i	$-0.112673\pi$
$578$		−	40.9953i		−	1.70518i
$579$		0			0
$580$	−13.0756	+	13.7082i	−0.542934	+	0.569202i
$581$		0			0
$582$		0			0
$583$			8.53149i			0.353338i
$584$	−15.4336			−0.638648
$585$		0			0
$586$	−50.3946			−2.08178
$587$			8.94427i			0.369170i	0.982817	+	0.184585i	$0.0590940\pi$
$587$			8.94427i			0.369170i	−0.982817	+	0.184585i	$0.940906\pi$
$588$		0			0
$589$	15.4164			0.635222
$590$	−40.3607	−	38.4980i	−1.66162	−	1.58494i
$591$		0			0
$592$		−	33.7819i		−	1.38843i
$593$			17.7082i			0.727189i	0.931557	+	0.363594i	$0.118451\pi$
$593$			17.7082i			0.727189i	−0.931557	+	0.363594i	$0.881549\pi$
$594$		0			0
$595$		0			0
$596$	80.2492			3.28714
$597$		0			0
$598$			40.3607i			1.65047i
$599$	9.41641			0.384744			0.192372	−	0.981322i	$-0.438382\pi$
$599$	9.41641			0.384744			0.192372	+	0.981322i	$0.438382\pi$
$600$		0			0
$601$	12.3469			0.503640			0.251820	−	0.967774i	$-0.418971\pi$
$601$	12.3469			0.503640			0.251820	+	0.967774i	$0.418971\pi$
$602$		0			0
$603$		0			0
$604$	37.8885			1.54166
$605$	13.8903	−	14.5623i	0.564719	−	0.592042i
$606$		0			0
$607$		−	38.8328i		−	1.57618i	−0.615563	−	0.788088i	$-0.711071\pi$
$607$		−	38.8328i		−	1.57618i	0.615563	−	0.788088i	$-0.288929\pi$
$608$		−	20.1803i		−	0.818421i
$609$		0			0
$610$	15.4164	+	14.7049i	0.624192	+	0.595386i
$611$	25.8885			1.04734
$612$		0			0
$613$		−	44.6715i		−	1.80426i	−0.431460	−	0.902132i	$-0.642001\pi$
$613$		−	44.6715i		−	1.80426i	0.431460	−	0.902132i	$-0.357999\pi$
$614$	−6.17345			−0.249140
$615$		0			0
$616$		0			0
$617$		−	20.4280i		−	0.822402i	−0.911545	−	0.411201i	$-0.865109\pi$
$617$		−	20.4280i		−	0.822402i	0.911545	−	0.411201i	$-0.134891\pi$
$618$		0			0
$619$	22.7861			0.915850			0.457925	−	0.888991i	$-0.348593\pi$
$619$	22.7861			0.915850			0.457925	+	0.888991i	$0.348593\pi$
$620$	−12.4721	+	13.0756i	−0.500893	+	0.525128i
$621$		0			0
$622$			9.52786i			0.382033i
$623$		0			0
$624$		0			0
$625$	−24.8885	+	2.35805i	−0.995542	+	0.0943219i
$626$	61.5625			2.46053
$627$		0			0
$628$		−	36.6525i		−	1.46259i
$629$	4.71609			0.188043
$630$		0			0
$631$	27.0557			1.07707			0.538536	−	0.842603i	$-0.318978\pi$
$631$	27.0557			1.07707			0.538536	+	0.842603i	$0.318978\pi$
$632$		−	72.2800i		−	2.87514i
$633$		0			0
$634$	26.0689			1.03533
$635$	−6.17345	−	5.88854i	−0.244986	−	0.233680i
$636$		0			0
$637$		0			0
$638$		−	22.3357i		−	0.884281i
$639$		0			0
$640$	−27.1050	−	25.8541i	−1.07142	−	1.02197i
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$			28.3607i			1.11844i	0.829021	+	0.559218i	$0.188899\pi$
$643$			28.3607i			1.11844i	−0.829021	+	0.559218i	$0.811101\pi$
$644$		0			0
$645$		0			0
$646$	15.4164			0.606550
$647$		−	47.4164i		−	1.86413i	−0.362289	−	0.932066i	$-0.618005\pi$
$647$		−	47.4164i		−	1.86413i	0.362289	−	0.932066i	$-0.381995\pi$
$648$		0			0
$649$	44.6715			1.75351
$650$	−3.08672	−	65.3050i	−0.121071	−	2.56147i
$651$		0			0
$652$			16.1623i			0.632964i
$653$			42.7638i			1.67348i	0.547603	+	0.836738i	$0.315540\pi$
$653$			42.7638i			1.67348i	−0.547603	+	0.836738i	$0.684460\pi$
$654$		0			0
$655$	15.4164	−	16.1623i	0.602369	−	0.631513i
$656$	37.7694			1.47465
$657$		0			0
$658$		0			0
$659$	−8.47214			−0.330028			−0.165014	−	0.986291i	$-0.552767\pi$
$659$	−8.47214			−0.330028			−0.165014	+	0.986291i	$0.552767\pi$
$660$		0			0
$661$	3.81540			0.148402			0.0742009	−	0.997243i	$-0.476359\pi$
$661$	3.81540			0.148402			0.0742009	+	0.997243i	$0.476359\pi$
$662$		−	19.9777i		−	0.776455i
$663$		0			0
$664$	36.1400			1.40250
$665$		0			0
$666$		0			0
$667$			6.17345i			0.239037i
$668$			65.3050i			2.52672i
$669$		0			0
$670$	23.7931	−	24.9443i	0.919208	−	0.963681i
$671$	−17.0630			−0.658709
$672$		0			0
$673$			30.8672i			1.18984i	0.803783	+	0.594922i	$0.202817\pi$
$673$			30.8672i			1.18984i	−0.803783	+	0.594922i	$0.797183\pi$
$674$	−9.52786			−0.367000
$675$		0			0
$676$	61.0689			2.34880
$677$			31.5967i			1.21436i	0.794564	+	0.607181i	$0.207700\pi$
$677$			31.5967i			1.21436i	−0.794564	+	0.607181i	$0.792300\pi$
$678$		0			0
$679$		0			0
$680$	−6.58359	+	6.90212i	−0.252469	+	0.264684i
$681$		0			0
$682$		−	21.3050i		−	0.815809i
$683$		−	26.8798i		−	1.02853i	−0.857632	−	0.514264i	$-0.828065\pi$
$683$		−	26.8798i		−	1.02853i	0.857632	−	0.514264i	$-0.171935\pi$
$684$		0			0
$685$	13.0756	+	12.4721i	0.499592	+	0.476536i
$686$		0			0
$687$		0			0
$688$			54.6603i			2.08391i
$689$	−9.98885			−0.380545
$690$		0			0
$691$	−34.2323			−1.30226			−0.651129	−	0.758967i	$-0.725704\pi$
$691$	−34.2323			−1.30226			−0.651129	+	0.758967i	$0.725704\pi$
$692$			81.4853i			3.09761i
$693$		0			0
$694$	−63.4853			−2.40987
$695$	−22.0000	+	23.0644i	−0.834508	+	0.874883i
$696$		0			0
$697$			5.27275i			0.199720i
$698$		−	21.3050i		−	0.806404i
$699$		0			0
$700$		0			0
$701$	−8.83282			−0.333611			−0.166805	−	0.985990i	$-0.553345\pi$
$701$	−8.83282			−0.333611			−0.166805	+	0.985990i	$0.553345\pi$
$702$		0			0
$703$		−	49.8885i		−	1.88158i
$704$	21.0557			0.793568
$705$		0			0
$706$	8.08115			0.304138
$707$		0			0
$708$		0			0
$709$	34.0000			1.27690			0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000			1.27690			0.638448	+	0.769665i	$0.279577\pi$
$710$	1.90770	+	1.81966i	0.0715947	+	0.0682906i
$711$		0			0
$712$			51.7082i			1.93785i
$713$			5.88854i			0.220528i
$714$		0			0
$715$	37.8885	+	36.1400i	1.41695	+	1.35156i
$716$	31.8885			1.19173
$717$		0			0
$718$		−	51.1233i		−	1.90790i
$719$	−23.7931			−0.887333			−0.443666	−	0.896192i	$-0.646322\pi$
$719$	−23.7931			−0.887333			−0.443666	+	0.896192i	$0.646322\pi$
$720$		0			0
$721$		0			0
$722$		−	115.633i		−	4.30343i
$723$		0			0
$724$	−42.3134			−1.57257
$725$	−0.472136	−	9.98885i	−0.0175347	−	0.370977i
$726$		0			0
$727$			1.52786i			0.0566653i	0.999599	+	0.0283327i	$0.00901978\pi$
$727$			1.52786i			0.0566653i	−0.999599	+	0.0283327i	$0.990980\pi$
$728$		0			0
$729$		0			0
$730$	10.6525	−	11.1679i	0.394266	−	0.413341i
$731$	−7.63080			−0.282235
$732$		0			0
$733$			34.1803i			1.26248i	0.775588	+	0.631240i	$0.217454\pi$
$733$			34.1803i			1.26248i	−0.775588	+	0.631240i	$0.782546\pi$
$734$	39.9554			1.47478
$735$		0			0
$736$	7.70820			0.284128
$737$			27.6085i			1.01697i
$738$		0			0
$739$	−24.9443			−0.917590			−0.458795	−	0.888542i	$-0.651719\pi$
$739$	−24.9443			−0.917590			−0.458795	+	0.888542i	$0.651719\pi$
$740$	42.3134	+	40.3607i	1.55547	+	1.48369i
$741$		0			0
$742$		0			0
$743$			3.08672i			0.113241i	0.998396	+	0.0566205i	$0.0180325\pi$
$743$			3.08672i			0.113241i	−0.998396	+	0.0566205i	$0.981968\pi$
$744$		0			0
$745$	−29.2379	+	30.6525i	−1.07119	+	1.12302i
$746$	−80.7214			−2.95542
$747$		0			0
$748$		−	14.4721i		−	0.529154i
$749$		0			0
$750$		0			0
$751$	−10.8328			−0.395295			−0.197648	−	0.980273i	$-0.563330\pi$
$751$	−10.8328			−0.395295			−0.197648	+	0.980273i	$0.563330\pi$
$752$		−	27.0557i		−	0.986621i
$753$		0			0
$754$	26.1511			0.952368
$755$	−13.8042	+	14.4721i	−0.502388	+	0.526695i
$756$		0			0
$757$		−	28.5092i		−	1.03618i	−0.855325	−	0.518092i	$-0.826642\pi$
$757$		−	28.5092i		−	1.03618i	0.855325	−	0.518092i	$-0.173358\pi$
$758$			7.63080i			0.277163i
$759$		0			0
$760$	73.0132	+	69.6436i	2.64847	+	2.52624i
$761$	4.54408			0.164723			0.0823613	−	0.996603i	$-0.473754\pi$
$761$	4.54408			0.164723			0.0823613	+	0.996603i	$0.473754\pi$
$762$		0			0
$763$		0			0
$764$	−14.9443			−0.540665
$765$		0			0
$766$	−26.1511			−0.944879
$767$			52.3023i			1.88853i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		−	58.4757i		−	2.10459i
$773$		−	14.6525i		−	0.527013i	−0.964658	−	0.263506i	$-0.915121\pi$
$773$		−	14.6525i		−	0.527013i	0.964658	−	0.263506i	$-0.0848790\pi$
$774$		0			0
$775$	−0.450347	−	9.52786i	−0.0161769	−	0.342251i
$776$	56.8464			2.04067
$777$		0			0
$778$			29.6882i			1.06437i
$779$	55.7771			1.99842
$780$		0			0
$781$	−2.11146			−0.0755538
$782$			5.88854i			0.210574i
$783$		0			0
$784$		0			0
$785$	14.0000	+	13.3539i	0.499681	+	0.476621i
$786$		0			0
$787$			40.9443i			1.45951i	0.683711	+	0.729753i	$0.260365\pi$
$787$			40.9443i			1.45951i	−0.683711	+	0.729753i	$0.739635\pi$
$788$		−	8.08115i		−	0.287879i
$789$		0			0
$790$	52.3023	+	49.8885i	1.86083	+	1.77495i
$791$		0			0
$792$		0			0
$793$		−	19.9777i		−	0.709429i
$794$	23.0644			0.818526
$795$		0			0
$796$	50.3946			1.78619
$797$			27.2361i			0.964751i	0.875965	+	0.482376i	$0.160226\pi$
$797$			27.2361i			0.964751i	−0.875965	+	0.482376i	$0.839774\pi$
$798$		0			0
$799$	3.77709			0.133624
$800$	−12.4721	+	0.589512i	−0.440957	+	0.0208424i
$801$		0			0
$802$			91.9794i			3.24790i
$803$			12.3607i			0.436199i
$804$		0			0
$805$		0			0
$806$	24.9443			0.878625
$807$		0			0
$808$		−	73.0132i		−	2.56859i
$809$	−30.9443			−1.08794			−0.543971	−	0.839104i	$-0.683080\pi$
$809$	−30.9443			−1.08794			−0.543971	+	0.839104i	$0.683080\pi$
$810$		0			0
$811$	−30.4169			−1.06808			−0.534041	−	0.845459i	$-0.679327\pi$
$811$	−30.4169			−1.06808			−0.534041	+	0.845459i	$0.679327\pi$
$812$		0			0
$813$		0			0
$814$	−68.9443			−2.41650
$815$	−6.17345	−	5.88854i	−0.216246	−	0.206267i
$816$		0			0
$817$			80.7214i			2.82408i
$818$			84.3607i			2.94960i
$819$		0			0
$820$	−45.1246	+	47.3079i	−1.57582	+	1.65206i
$821$	−7.88854			−0.275312			−0.137656	−	0.990480i	$-0.543957\pi$
$821$	−7.88854			−0.275312			−0.137656	+	0.990480i	$0.543957\pi$
$822$		0			0
$823$		−	11.4462i		−	0.398990i	−0.979899	−	0.199495i	$-0.936070\pi$
$823$		−	11.4462i		−	0.398990i	0.979899	−	0.199495i	$-0.0639301\pi$
$824$	−8.53149			−0.297209
$825$		0			0
$826$		0			0
$827$			46.8575i			1.62940i	0.579886	+	0.814698i	$0.303097\pi$
$827$			46.8575i			1.62940i	−0.579886	+	0.814698i	$0.696903\pi$
$828$		0			0
$829$	17.6196			0.611956			0.305978	−	0.952039i	$-0.401017\pi$
$829$	17.6196			0.611956			0.305978	+	0.952039i	$0.401017\pi$
$830$	−24.9443	+	26.1511i	−0.865828	+	0.907719i
$831$		0			0
$832$			24.6525i			0.854671i
$833$		0			0
$834$		0			0
$835$	−24.9443	−	23.7931i	−0.863232	−	0.823394i
$836$	−153.091			−5.29478
$837$		0			0
$838$		−	3.63932i		−	0.125718i
$839$	−52.3023			−1.80568			−0.902838	−	0.429981i	$-0.858520\pi$
$839$	−52.3023			−1.80568			−0.902838	+	0.429981i	$0.858520\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		−	41.1344i		−	1.41759i
$843$		0			0
$844$	54.8328			1.88742
$845$	−22.2497	+	23.3262i	−0.765414	+	0.802447i
$846$		0			0
$847$		0			0
$848$			10.4392i			0.358483i
$849$		0			0
$850$	−0.450347	−	9.52786i	−0.0154468	−	0.326803i
$851$	19.0557			0.653222
$852$		0			0
$853$			4.87539i			0.166930i	0.996511	+	0.0834651i	$0.0265987\pi$
$853$			4.87539i			0.166930i	−0.996511	+	0.0834651i	$0.973401\pi$
$854$		0			0
$855$		0			0
$856$	−38.5410			−1.31730
$857$			45.1246i			1.54143i	0.637182	+	0.770714i	$0.280100\pi$
$857$			45.1246i			1.54143i	−0.637182	+	0.770714i	$0.719900\pi$
$858$		0			0
$859$	−0.450347			−0.0153656			−0.00768282	−	0.999970i	$-0.502446\pi$
$859$	−0.450347			−0.0153656			−0.00768282	+	0.999970i	$0.502446\pi$
$860$	−68.4646	−	65.3050i	−2.33462	−	2.22688i
$861$		0			0
$862$		−	11.1679i		−	0.380379i
$863$			10.7175i			0.364829i	0.983222	+	0.182414i	$0.0583912\pi$
$863$			10.7175i			0.364829i	−0.983222	+	0.182414i	$0.941609\pi$
$864$		0			0
$865$	−31.1246	−	29.6882i	−1.05827	−	1.00943i
$866$	89.1710			3.03015
$867$		0			0
$868$		0			0
$869$	−57.8885			−1.96373
$870$		0			0
$871$	−32.3246			−1.09528
$872$		−	69.6436i		−	2.35843i
$873$		0			0
$874$	62.2911			2.10703
$875$		0			0
$876$		0			0
$877$			15.2616i			0.515348i	0.966232	+	0.257674i	$0.0829560\pi$
$877$			15.2616i			0.515348i	−0.966232	+	0.257674i	$0.917044\pi$
$878$			60.5410i			2.04316i
$879$		0			0
$880$	37.7694	−	39.5967i	1.27320	−	1.33481i
$881$	41.5848			1.40103			0.700513	−	0.713640i	$-0.252955\pi$
$881$	41.5848			1.40103			0.700513	+	0.713640i	$0.252955\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	16.9443			0.569898
$885$		0			0
$886$	−78.9017			−2.65075
$887$			35.7771i			1.20128i	0.799521	+	0.600639i	$0.205087\pi$
$887$			35.7771i			1.20128i	−0.799521	+	0.600639i	$0.794913\pi$
$888$		0			0
$889$		0			0
$890$	−37.4164	−	35.6896i	−1.25420	−	1.19632i
$891$		0			0
$892$			75.7771i			2.53720i
$893$		−	39.9554i		−	1.33706i
$894$		0			0
$895$	−11.6182	+	12.1803i	−0.388354	+	0.407144i
$896$		0			0
$897$		0			0
$898$			42.5918i			1.42131i
$899$	3.81540			0.127251
$900$		0			0
$901$	−1.45735			−0.0485515
$902$		−	77.0820i		−	2.56655i
$903$		0			0
$904$	23.8197			0.792230
$905$	15.4164	−	16.1623i	0.512459	−	0.537253i
$906$		0			0
$907$			39.9554i			1.32670i	0.748311	+	0.663349i	$0.230865\pi$
$907$			39.9554i			1.32670i	−0.748311	+	0.663349i	$0.769135\pi$
$908$			6.47214i			0.214785i
$909$		0			0
$910$		0			0
$911$	−25.4164			−0.842083			−0.421042	−	0.907041i	$-0.638335\pi$
$911$	−25.4164			−0.842083			−0.421042	+	0.907041i	$0.638335\pi$
$912$		0			0
$913$		−	28.9443i		−	0.957916i
$914$	59.4164			1.96532
$915$		0			0
$916$	−94.6157			−3.12619
$917$		0			0
$918$		0			0
$919$	−4.00000			−0.131948			−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000			−0.131948			−0.0659739	+	0.997821i	$0.521015\pi$
$920$	−26.6015	+	27.8885i	−0.877025	+	0.919458i
$921$		0			0
$922$			92.0689i			3.03213i
$923$		−	2.47214i		−	0.0813713i
$924$		0			0
$925$	−30.8328	+	1.45735i	−1.01378	+	0.0479175i
$926$	−71.1935			−2.33956
$927$		0			0
$928$		−	4.99442i		−	0.163950i
$929$	3.08672			0.101272			0.0506361	−	0.998717i	$-0.483875\pi$
$929$	3.08672			0.101272			0.0506361	+	0.998717i	$0.483875\pi$
$930$		0			0
$931$		0			0
$932$			50.3946i			1.65073i
$933$		0			0
$934$	−68.4646			−2.24023
$935$	5.52786	+	5.27275i	0.180780	+	0.172437i
$936$		0			0
$937$			26.5410i			0.867057i	0.901140	+	0.433529i	$0.142732\pi$
$937$			26.5410i			0.867057i	−0.901140	+	0.433529i	$0.857268\pi$
$938$		0			0
$939$		0			0
$940$	33.8885	+	32.3246i	1.10532	+	1.05431i
$941$	−35.4113			−1.15438			−0.577188	−	0.816611i	$-0.695850\pi$
$941$	−35.4113			−1.15438			−0.577188	+	0.816611i	$0.695850\pi$
$942$		0			0
$943$			21.3050i			0.693785i
$944$	54.6603			1.77904
$945$		0			0
$946$	111.554			3.62694
$947$		−	14.5329i		−	0.472257i	−0.971722	−	0.236128i	$-0.924121\pi$
$947$		−	14.5329i		−	0.472257i	0.971722	−	0.236128i	$-0.0758786\pi$
$948$		0			0
$949$	−14.4721			−0.469785
$950$	−100.789	+	4.76393i	−3.27003	+	0.154562i
$951$		0			0
$952$		0			0
$953$		−	51.8519i		−	1.67965i	−0.542858	−	0.839825i	$-0.682658\pi$
$953$		−	51.8519i		−	1.67965i	0.542858	−	0.839825i	$-0.317342\pi$
$954$		0			0
$955$	5.44477	−	5.70820i	0.176189	−	0.184713i
$956$	31.8885			1.03135
$957$		0			0
$958$		−	71.1935i		−	2.30016i
$959$		0			0
$960$		0			0
$961$	−27.3607			−0.882603
$962$		−	80.7214i		−	2.60256i
$963$		0			0
$964$	−26.1511			−0.842272
$965$	22.3357	+	21.3050i	0.719013	+	0.685831i
$966$		0			0
$967$			26.1511i			0.840964i	0.907301	+	0.420482i	$0.138139\pi$
$967$			26.1511i			0.840964i	−0.907301	+	0.420482i	$0.861861\pi$
$968$			50.2554i			1.61527i
$969$		0			0
$970$	−39.2361	+	41.1344i	−1.25979	+	1.32075i
$971$	−58.4757			−1.87658			−0.938288	−	0.345855i	$-0.887589\pi$
$971$	−58.4757			−1.87658			−0.938288	+	0.345855i	$0.887589\pi$
$972$		0			0
$973$		0			0
$974$	24.9443			0.799266
$975$		0			0
$976$	−20.8784			−0.668301
$977$			24.2434i			0.775616i	0.921740	+	0.387808i	$0.126768\pi$
$977$			24.2434i			0.775616i	−0.921740	+	0.387808i	$0.873232\pi$
$978$		0			0
$979$	41.4127			1.32356
$980$		0			0
$981$		0			0
$982$			45.8505i			1.46315i
$983$			38.8328i			1.23857i	0.785165	+	0.619287i	$0.212578\pi$
$983$			38.8328i			1.23857i	−0.785165	+	0.619287i	$0.787422\pi$
$984$		0			0
$985$	3.08672	+	2.94427i	0.0983512	+	0.0938123i
$986$	3.81540			0.121507
$987$		0			0
$988$		−	179.243i		−	5.70247i
$989$	−30.8328			−0.980427
$990$		0			0
$991$	36.0000			1.14358			0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000			1.14358			0.571789	+	0.820401i	$0.306250\pi$
$992$		−	4.76393i		−	0.151255i
$993$		0			0
$994$		0			0
$995$	−18.3607	+	19.2490i	−0.582073	+	0.610235i
$996$		0			0
$997$		−	27.7082i		−	0.877528i	−0.898602	−	0.438764i	$-0.855416\pi$
$997$		−	27.7082i		−	0.877528i	0.898602	−	0.438764i	$-0.144584\pi$
$998$			19.9777i			0.632383i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.d.m.1324.2		8
3.2	odd	2		735.2.d.c.589.8	yes	8
5.4	even	2	inner	2205.2.d.m.1324.8		8
7.6	odd	2	inner	2205.2.d.m.1324.1		8
15.2	even	4		3675.2.a.bv.1.1		4
15.8	even	4		3675.2.a.bt.1.4		4
15.14	odd	2		735.2.d.c.589.1	✓	8
21.2	odd	6		735.2.q.h.214.7		16
21.5	even	6		735.2.q.h.214.8		16
21.11	odd	6		735.2.q.h.79.2		16
21.17	even	6		735.2.q.h.79.1		16
21.20	even	2		735.2.d.c.589.7	yes	8
35.34	odd	2	inner	2205.2.d.m.1324.7		8
105.44	odd	6		735.2.q.h.214.2		16
105.59	even	6		735.2.q.h.79.8		16
105.62	odd	4		3675.2.a.bt.1.1		4
105.74	odd	6		735.2.q.h.79.7		16
105.83	odd	4		3675.2.a.bv.1.4		4
105.89	even	6		735.2.q.h.214.1		16
105.104	even	2		735.2.d.c.589.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.d.c.589.1	✓	8	15.14	odd	2
735.2.d.c.589.2	yes	8	105.104	even	2
735.2.d.c.589.7	yes	8	21.20	even	2
735.2.d.c.589.8	yes	8	3.2	odd	2
735.2.q.h.79.1		16	21.17	even	6
735.2.q.h.79.2		16	21.11	odd	6
735.2.q.h.79.7		16	105.74	odd	6
735.2.q.h.79.8		16	105.59	even	6
735.2.q.h.214.1		16	105.89	even	6
735.2.q.h.214.2		16	105.44	odd	6
735.2.q.h.214.7		16	21.2	odd	6
735.2.q.h.214.8		16	21.5	even	6
2205.2.d.m.1324.1		8	7.6	odd	2	inner
2205.2.d.m.1324.2		8	1.1	even	1	trivial
2205.2.d.m.1324.7		8	35.34	odd	2	inner
2205.2.d.m.1324.8		8	5.4	even	2	inner
3675.2.a.bt.1.1		4	105.62	odd	4
3675.2.a.bt.1.4		4	15.8	even	4
3675.2.a.bv.1.1		4	15.2	even	4
3675.2.a.bv.1.4		4	105.83	odd	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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.49721i q^{2} -4.23607 q^{4} +(1.54336 - 1.61803i) q^{5} +5.58394i q^{8} +O(q^{10})\)	\(q-2.49721i q^{2} -4.23607 q^{4} +(1.54336 - 1.61803i) q^{5} +5.58394i q^{8} +(-4.04057 - 3.85410i) q^{10} +4.47214 q^{11} +5.23607i q^{13} +5.47214 q^{16} +0.763932i q^{17} +8.08115 q^{19} +(-6.53779 + 6.85410i) q^{20} -11.1679i q^{22} +3.08672i q^{23} +(-0.236068 - 4.99442i) q^{25} +13.0756 q^{26} +2.00000 q^{29} +1.90770 q^{31} -2.49721i q^{32} +1.90770 q^{34} -6.17345i q^{37} -20.1803i q^{38} +(9.03500 + 8.61803i) q^{40} +6.90212 q^{41} +9.98885i q^{43} -18.9443 q^{44} +7.70820 q^{46} -4.94427i q^{47} +(-12.4721 + 0.589512i) q^{50} -22.1803i q^{52} +1.90770i q^{53} +(6.90212 - 7.23607i) q^{55} -4.99442i q^{58} +9.98885 q^{59} -3.81540 q^{61} -4.76393i q^{62} +4.70820 q^{64} +(8.47214 + 8.08115i) q^{65} +6.17345i q^{67} -3.23607i q^{68} -0.472136 q^{71} +2.76393i q^{73} -15.4164 q^{74} -34.2323 q^{76} -12.9443 q^{79} +(8.44549 - 8.85410i) q^{80} -17.2361i q^{82} -6.47214i q^{83} +(1.23607 + 1.17902i) q^{85} +24.9443 q^{86} +24.9721i q^{88} +9.26017 q^{89} -13.0756i q^{92} -12.3469 q^{94} +(12.4721 - 13.0756i) q^{95} -10.1803i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \) 8 * q - 16 * q^4	\( 8 q - 16 q^{4} + 8 q^{16} + 16 q^{25} + 16 q^{29} - 80 q^{44} + 8 q^{46} - 64 q^{50} - 16 q^{64} + 32 q^{65} + 32 q^{71} - 16 q^{74} - 32 q^{79} - 8 q^{85} + 128 q^{86} + 64 q^{95}+O(q^{100}) \) 8 * q - 16 * q^4 + 8 * q^16 + 16 * q^25 + 16 * q^29 - 80 * q^44 + 8 * q^46 - 64 * q^50 - 16 * q^64 + 32 * q^65 + 32 * q^71 - 16 * q^74 - 32 * q^79 - 8 * q^85 + 128 * q^86 + 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more