LMFDB - Embedded newform 920.2.e.c.369.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(369,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("920.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.e (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:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} + \cdots + 1024 $ x^16 - 2x^15 + 2x^14 + 6x^13 + 100x^12 - 196x^11 + 210x^10 + 702x^9 + 1572x^8 - 1594x^7 + 2464x^6 + 9568x^5 + 15457x^4 + 4336x^3 + 128x^2 - 512*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.2
Root			$1.65386 - 1.65386i$ of defining polynomial
Character	$\chi$	$=$	920.369
Dual form			920.2.e.c.369.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.89996i q^{3} +(-1.50491 - 1.65386i) q^{5} +0.580879i q^{7} -5.40975 q^{9} +O(q^{10})$	$q-2.89996i q^{3} +(-1.50491 - 1.65386i) q^{5} +0.580879i q^{7} -5.40975 q^{9} -0.0809094 q^{11} -3.02066i q^{13} +(-4.79611 + 4.36418i) q^{15} -0.280529i q^{17} -4.72683 q^{19} +1.68452 q^{21} +1.00000i q^{23} +(-0.470482 + 4.97782i) q^{25} +6.98817i q^{27} -1.38057 q^{29} -5.70770 q^{31} +0.234634i q^{33} +(0.960690 - 0.874171i) q^{35} -2.61536i q^{37} -8.75977 q^{39} +5.31570 q^{41} -7.30240i q^{43} +(8.14119 + 8.94695i) q^{45} +12.2117i q^{47} +6.66258 q^{49} -0.813523 q^{51} +6.64032i q^{53} +(0.121761 + 0.133812i) q^{55} +13.7076i q^{57} -3.70211 q^{59} +11.5113 q^{61} -3.14241i q^{63} +(-4.99573 + 4.54582i) q^{65} -5.99193i q^{67} +2.89996 q^{69} -13.8324 q^{71} -1.14465i q^{73} +(14.4354 + 1.36438i) q^{75} -0.0469985i q^{77} +1.68167 q^{79} +4.03614 q^{81} -9.86877i q^{83} +(-0.463955 + 0.422172i) q^{85} +4.00360i q^{87} -6.69235 q^{89} +1.75463 q^{91} +16.5521i q^{93} +(7.11347 + 7.81751i) q^{95} -12.8190i q^{97} +0.437699 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} - 22 q^{9}+O(q^{10}) $ 16 * q - 2 * q^5 - 22 * q^9	$ 16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99}+O(q^{100}) $ 16 * q - 2 * q^5 - 22 * q^9 + 14 * q^11 + 6 * q^15 - 22 * q^19 + 12 * q^25 - 44 * q^29 + 18 * q^31 + 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 - 38 * q^51 + 30 * q^55 - 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 + 30 * q^71 + 56 * q^75 + 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 - 70 * q^91 + 38 * q^95 - 122 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/920\mathbb{Z}\right)^\times$.

$n$	$231$	$281$	$461$	$737$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	2.89996i		−	1.67429i	−0.546980	−	0.837145i	$-0.684223\pi$
$3$		−	2.89996i		−	1.67429i	0.546980	−	0.837145i	$-0.315777\pi$
$4$		0			0
$5$	−1.50491	−	1.65386i	−0.673017	−	0.739627i
$5$	−1.50491	−	1.65386i	−0.673017	−	0.739627i
$6$		0			0
$7$			0.580879i			0.219552i	0.993956	+	0.109776i	$0.0350133\pi$
$7$			0.580879i			0.219552i	−0.993956	+	0.109776i	$0.964987\pi$
$8$		0			0
$9$	−5.40975			−1.80325
$10$		0			0
$11$	−0.0809094			−0.0243951			−0.0121975	−	0.999926i	$-0.503883\pi$
$11$	−0.0809094			−0.0243951			−0.0121975	+	0.999926i	$0.503883\pi$
$12$		0			0
$13$		−	3.02066i		−	0.837779i	−0.908037	−	0.418890i	$-0.862419\pi$
$13$		−	3.02066i		−	0.837779i	0.908037	−	0.418890i	$-0.137581\pi$
$14$		0			0
$15$	−4.79611	+	4.36418i	−1.23835	+	1.12683i
$16$		0			0
$17$		−	0.280529i		−	0.0680384i	−0.999421	−	0.0340192i	$-0.989169\pi$
$17$		−	0.280529i		−	0.0680384i	0.999421	−	0.0340192i	$-0.0108307\pi$
$18$		0			0
$19$	−4.72683			−1.08441			−0.542205	−	0.840246i	$-0.682410\pi$
$19$	−4.72683			−1.08441			−0.542205	+	0.840246i	$0.682410\pi$
$20$		0			0
$21$	1.68452			0.367593
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−0.470482	+	4.97782i	−0.0940965	+	0.995563i
$26$		0			0
$27$			6.98817i			1.34487i
$28$		0			0
$29$	−1.38057			−0.256366			−0.128183	−	0.991751i	$-0.540914\pi$
$29$	−1.38057			−0.256366			−0.128183	+	0.991751i	$0.540914\pi$
$30$		0			0
$31$	−5.70770			−1.02513			−0.512567	−	0.858647i	$-0.671305\pi$
$31$	−5.70770			−1.02513			−0.512567	+	0.858647i	$0.671305\pi$
$32$		0			0
$33$			0.234634i			0.0408445i
$34$		0			0
$35$	0.960690	−	0.874171i	0.162386	−	0.147762i
$36$		0			0
$37$		−	2.61536i		−	0.429962i	−0.976618	−	0.214981i	$-0.931031\pi$
$37$		−	2.61536i		−	0.429962i	0.976618	−	0.214981i	$-0.0689690\pi$
$38$		0			0
$39$	−8.75977			−1.40269
$40$		0			0
$41$	5.31570			0.830172			0.415086	−	0.909782i	$-0.363751\pi$
$41$	5.31570			0.830172			0.415086	+	0.909782i	$0.363751\pi$
$42$		0			0
$43$		−	7.30240i		−	1.11361i	−0.830645	−	0.556803i	$-0.812028\pi$
$43$		−	7.30240i		−	1.11361i	0.830645	−	0.556803i	$-0.187972\pi$
$44$		0			0
$45$	8.14119	+	8.94695i	1.21362	+	1.33373i
$46$		0			0
$47$			12.2117i			1.78125i	0.454736	+	0.890626i	$0.349734\pi$
$47$			12.2117i			1.78125i	−0.454736	+	0.890626i	$0.650266\pi$
$48$		0			0
$49$	6.66258			0.951797
$50$		0			0
$51$	−0.813523			−0.113916
$52$		0			0
$53$			6.64032i			0.912117i	0.889950	+	0.456059i	$0.150739\pi$
$53$			6.64032i			0.912117i	−0.889950	+	0.456059i	$0.849261\pi$
$54$		0			0
$55$	0.121761	+	0.133812i	0.0164183	+	0.0180433i
$56$		0			0
$57$			13.7076i			1.81562i
$58$		0			0
$59$	−3.70211			−0.481974			−0.240987	−	0.970528i	$-0.577471\pi$
$59$	−3.70211			−0.481974			−0.240987	+	0.970528i	$0.577471\pi$
$60$		0			0
$61$	11.5113			1.47387			0.736936	−	0.675962i	$-0.236272\pi$
$61$	11.5113			1.47387			0.736936	+	0.675962i	$0.236272\pi$
$62$		0			0
$63$		−	3.14241i		−	0.395906i
$64$		0			0
$65$	−4.99573	+	4.54582i	−0.619644	+	0.563840i
$66$		0			0
$67$		−	5.99193i		−	0.732031i	−0.930609	−	0.366015i	$-0.880722\pi$
$67$		−	5.99193i		−	0.732031i	0.930609	−	0.366015i	$-0.119278\pi$
$68$		0			0
$69$	2.89996			0.349114
$70$		0			0
$71$	−13.8324			−1.64160			−0.820801	−	0.571214i	$-0.806473\pi$
$71$	−13.8324			−1.64160			−0.820801	+	0.571214i	$0.806473\pi$
$72$		0			0
$73$		−	1.14465i		−	0.133971i	−0.997754	−	0.0669857i	$-0.978662\pi$
$73$		−	1.14465i		−	0.133971i	0.997754	−	0.0669857i	$-0.0213382\pi$
$74$		0			0
$75$	14.4354	+	1.36438i	1.66686	+	0.157545i
$76$		0			0
$77$		−	0.0469985i		−	0.00535598i
$78$		0			0
$79$	1.68167			0.189203			0.0946015	−	0.995515i	$-0.469842\pi$
$79$	1.68167			0.189203			0.0946015	+	0.995515i	$0.469842\pi$
$80$		0			0
$81$	4.03614			0.448460
$82$		0			0
$83$		−	9.86877i		−	1.08324i	−0.840624	−	0.541619i	$-0.817811\pi$
$83$		−	9.86877i		−	1.08324i	0.840624	−	0.541619i	$-0.182189\pi$
$84$		0			0
$85$	−0.463955	+	0.422172i	−0.0503230	+	0.0457910i
$86$		0			0
$87$			4.00360i			0.429231i
$88$		0			0
$89$	−6.69235			−0.709388			−0.354694	−	0.934982i	$-0.615415\pi$
$89$	−6.69235			−0.709388			−0.354694	+	0.934982i	$0.615415\pi$
$90$		0			0
$91$	1.75463			0.183936
$92$		0			0
$93$			16.5521i			1.71637i
$94$		0			0
$95$	7.11347	+	7.81751i	0.729826	+	0.802059i
$96$		0			0
$97$		−	12.8190i		−	1.30158i	−0.759260	−	0.650788i	$-0.774439\pi$
$97$		−	12.8190i		−	1.30158i	0.759260	−	0.650788i	$-0.225561\pi$
$98$		0			0
$99$	0.437699			0.0439904
$100$		0			0
$101$	−1.01958			−0.101452			−0.0507262	−	0.998713i	$-0.516154\pi$
$101$	−1.01958			−0.101452			−0.0507262	+	0.998713i	$0.516154\pi$
$102$		0			0
$103$		−	2.36255i		−	0.232789i	−0.993203	−	0.116394i	$-0.962866\pi$
$103$		−	2.36255i		−	0.232789i	0.993203	−	0.116394i	$-0.0371336\pi$
$104$		0			0
$105$	−2.53506	−	2.78596i	−0.247396	−	0.271882i
$106$		0			0
$107$			5.58207i			0.539639i	0.962911	+	0.269820i	$0.0869641\pi$
$107$			5.58207i			0.539639i	−0.962911	+	0.269820i	$0.913036\pi$
$108$		0			0
$109$	−6.09973			−0.584248			−0.292124	−	0.956380i	$-0.594362\pi$
$109$	−6.09973			−0.584248			−0.292124	+	0.956380i	$0.594362\pi$
$110$		0			0
$111$	−7.58443			−0.719882
$112$		0			0
$113$		−	17.3938i		−	1.63627i	−0.575023	−	0.818137i	$-0.695007\pi$
$113$		−	17.3938i		−	1.63627i	0.575023	−	0.818137i	$-0.304993\pi$
$114$		0			0
$115$	1.65386	−	1.50491i	0.154223	−	0.140334i
$116$		0			0
$117$			16.3410i			1.51073i
$118$		0			0
$119$	0.162954			0.0149379
$120$		0			0
$121$	−10.9935			−0.999405
$122$		0			0
$123$		−	15.4153i		−	1.38995i
$124$		0			0
$125$	8.94063	−	6.71306i	0.799674	−	0.600434i
$126$		0			0
$127$		−	11.9365i		−	1.05919i	−0.848250	−	0.529597i	$-0.822343\pi$
$127$		−	11.9365i		−	1.05919i	0.848250	−	0.529597i	$-0.177657\pi$
$128$		0			0
$129$	−21.1767			−1.86450
$130$		0			0
$131$	−9.84214			−0.859912			−0.429956	−	0.902850i	$-0.641471\pi$
$131$	−9.84214			−0.859912			−0.429956	+	0.902850i	$0.641471\pi$
$132$		0			0
$133$		−	2.74572i		−	0.238084i
$134$		0			0
$135$	11.5574	−	10.5166i	0.994705	−	0.905123i
$136$		0			0
$137$		−	2.93256i		−	0.250545i	−0.992122	−	0.125273i	$-0.960019\pi$
$137$		−	2.93256i		−	0.250545i	0.992122	−	0.125273i	$-0.0399805\pi$
$138$		0			0
$139$	−4.96075			−0.420766			−0.210383	−	0.977619i	$-0.567471\pi$
$139$	−4.96075			−0.420766			−0.210383	+	0.977619i	$0.567471\pi$
$140$		0			0
$141$	35.4133			2.98233
$142$		0			0
$143$			0.244399i			0.0204377i
$144$		0			0
$145$	2.07764	+	2.28327i	0.172539	+	0.189615i
$146$		0			0
$147$		−	19.3212i		−	1.59359i
$148$		0			0
$149$	−18.0375			−1.47769			−0.738846	−	0.673874i	$-0.764629\pi$
$149$	−18.0375			−1.47769			−0.738846	+	0.673874i	$0.764629\pi$
$150$		0			0
$151$	−18.1026			−1.47317			−0.736585	−	0.676345i	$-0.763563\pi$
$151$	−18.1026			−1.47317			−0.736585	+	0.676345i	$0.763563\pi$
$152$		0			0
$153$			1.51759i			0.122690i
$154$		0			0
$155$	8.58958	+	9.43972i	0.689932	+	0.758216i
$156$		0			0
$157$		−	15.2698i		−	1.21867i	−0.792915	−	0.609333i	$-0.791437\pi$
$157$		−	15.2698i		−	1.21867i	0.792915	−	0.609333i	$-0.208563\pi$
$158$		0			0
$159$	19.2566			1.52715
$160$		0			0
$161$	−0.580879			−0.0457797
$162$		0			0
$163$		−	16.1488i		−	1.26487i	−0.774613	−	0.632435i	$-0.782055\pi$
$163$		−	16.1488i		−	1.26487i	0.774613	−	0.632435i	$-0.217945\pi$
$164$		0			0
$165$	0.388050	−	0.353103i	0.0302097	−	0.0274890i
$166$		0			0
$167$		−	6.77915i		−	0.524586i	−0.964988	−	0.262293i	$-0.915521\pi$
$167$		−	6.77915i		−	0.524586i	0.964988	−	0.262293i	$-0.0844787\pi$
$168$		0			0
$169$	3.87563			0.298126
$170$		0			0
$171$	25.5710			1.95546
$172$		0			0
$173$			21.9428i			1.66828i	0.551553	+	0.834140i	$0.314035\pi$
$173$			21.9428i			1.66828i	−0.551553	+	0.834140i	$0.685965\pi$
$174$		0			0
$175$	−2.89151	−	0.273293i	−0.218577	−	0.0206590i
$176$		0			0
$177$			10.7360i			0.806964i
$178$		0			0
$179$	−6.76302			−0.505492			−0.252746	−	0.967533i	$-0.581334\pi$
$179$	−6.76302			−0.505492			−0.252746	+	0.967533i	$0.581334\pi$
$180$		0			0
$181$	12.1572			0.903641			0.451820	−	0.892109i	$-0.350775\pi$
$181$	12.1572			0.903641			0.451820	+	0.892109i	$0.350775\pi$
$182$		0			0
$183$		−	33.3823i		−	2.46769i
$184$		0			0
$185$	−4.32543	+	3.93589i	−0.318012	+	0.289372i
$186$		0			0
$187$			0.0226975i			0.00165980i
$188$		0			0
$189$	−4.05928			−0.295269
$190$		0			0
$191$	8.87427			0.642119			0.321060	−	0.947059i	$-0.395961\pi$
$191$	8.87427			0.642119			0.321060	+	0.947059i	$0.395961\pi$
$192$		0			0
$193$		−	11.7026i		−	0.842374i	−0.906974	−	0.421187i	$-0.861614\pi$
$193$		−	11.7026i		−	0.842374i	0.906974	−	0.421187i	$-0.138386\pi$
$194$		0			0
$195$	13.1827	+	14.4874i	0.944032	+	1.03746i
$196$		0			0
$197$			0.330763i			0.0235659i	0.999931	+	0.0117829i	$0.00375071\pi$
$197$			0.330763i			0.0235659i	−0.999931	+	0.0117829i	$0.996249\pi$
$198$		0			0
$199$	−9.72600			−0.689458			−0.344729	−	0.938702i	$-0.612029\pi$
$199$	−9.72600			−0.689458			−0.344729	+	0.938702i	$0.612029\pi$
$200$		0			0
$201$	−17.3763			−1.22563
$202$		0			0
$203$		−	0.801945i		−	0.0562855i
$204$		0			0
$205$	−7.99965	−	8.79140i	−0.558720	−	0.614018i
$206$		0			0
$207$		−	5.40975i		−	0.376004i
$208$		0			0
$209$	0.382445			0.0264543
$210$		0			0
$211$	−6.63709			−0.456916			−0.228458	−	0.973554i	$-0.573368\pi$
$211$	−6.63709			−0.456916			−0.228458	+	0.973554i	$0.573368\pi$
$212$		0			0
$213$			40.1133i			2.74852i
$214$		0			0
$215$	−12.0771	+	10.9895i	−0.823653	+	0.749476i
$216$		0			0
$217$		−	3.31548i		−	0.225070i
$218$		0			0
$219$	−3.31944			−0.224307
$220$		0			0
$221$	−0.847383			−0.0570012
$222$		0			0
$223$			27.4323i			1.83700i	0.395420	+	0.918500i	$0.370599\pi$
$223$			27.4323i			1.83700i	−0.395420	+	0.918500i	$0.629401\pi$
$224$		0			0
$225$	2.54519	−	26.9287i	0.169679	−	1.79525i
$226$		0			0
$227$		−	20.6195i		−	1.36856i	−0.729218	−	0.684281i	$-0.760116\pi$
$227$		−	20.6195i		−	1.36856i	0.729218	−	0.684281i	$-0.239884\pi$
$228$		0			0
$229$	22.2949			1.47329			0.736644	−	0.676280i	$-0.236409\pi$
$229$	22.2949			1.47329			0.736644	+	0.676280i	$0.236409\pi$
$230$		0			0
$231$	−0.136294			−0.00896747
$232$		0			0
$233$		−	25.5331i		−	1.67273i	−0.548173	−	0.836365i	$-0.684676\pi$
$233$		−	25.5331i		−	1.67273i	0.548173	−	0.836365i	$-0.315324\pi$
$234$		0			0
$235$	20.1963	−	18.3775i	1.31746	−	1.19881i
$236$		0			0
$237$		−	4.87678i		−	0.316781i
$238$		0			0
$239$	−21.6017			−1.39730			−0.698649	−	0.715465i	$-0.746215\pi$
$239$	−21.6017			−1.39730			−0.698649	+	0.715465i	$0.746215\pi$
$240$		0			0
$241$	21.6850			1.39685			0.698426	−	0.715682i	$-0.253884\pi$
$241$	21.6850			1.39685			0.698426	+	0.715682i	$0.253884\pi$
$242$		0			0
$243$			9.25988i			0.594021i
$244$		0			0
$245$	−10.0266	−	11.0190i	−0.640576	−	0.703975i
$246$		0			0
$247$			14.2781i			0.908496i
$248$		0			0
$249$	−28.6190			−1.81366
$250$		0			0
$251$	26.5991			1.67892			0.839461	−	0.543420i	$-0.182871\pi$
$251$	26.5991			1.67892			0.839461	+	0.543420i	$0.182871\pi$
$252$		0			0
$253$		−	0.0809094i		−	0.00508673i
$254$		0			0
$255$	1.22428	+	1.34545i	0.0766674	+	0.0842554i
$256$		0			0
$257$			0.156392i			0.00975548i	0.999988	+	0.00487774i	$0.00155264\pi$
$257$			0.156392i			0.00975548i	−0.999988	+	0.00487774i	$0.998447\pi$
$258$		0			0
$259$	1.51921			0.0943989
$260$		0			0
$261$	7.46855			0.462292
$262$		0			0
$263$			24.1557i			1.48951i	0.667340	+	0.744753i	$0.267433\pi$
$263$			24.1557i			1.48951i	−0.667340	+	0.744753i	$0.732567\pi$
$264$		0			0
$265$	10.9821	−	9.99309i	0.674627	−	0.613870i
$266$		0			0
$267$			19.4075i			1.18772i
$268$		0			0
$269$	−0.270387			−0.0164858			−0.00824289	−	0.999966i	$-0.502624\pi$
$269$	−0.270387			−0.0164858			−0.00824289	+	0.999966i	$0.502624\pi$
$270$		0			0
$271$	10.0041			0.607704			0.303852	−	0.952719i	$-0.401727\pi$
$271$	10.0041			0.607704			0.303852	+	0.952719i	$0.401727\pi$
$272$		0			0
$273$		−	5.08837i		−	0.307962i
$274$		0			0
$275$	0.0380664	−	0.402752i	0.00229549	−	0.0242869i
$276$		0			0
$277$			12.0842i			0.726070i	0.931776	+	0.363035i	$0.118259\pi$
$277$			12.0842i			0.726070i	−0.931776	+	0.363035i	$0.881741\pi$
$278$		0			0
$279$	30.8772			1.84857
$280$		0			0
$281$	22.2912			1.32978			0.664890	−	0.746941i	$-0.268478\pi$
$281$	22.2912			1.32978			0.664890	+	0.746941i	$0.268478\pi$
$282$		0			0
$283$			10.5506i			0.627165i	0.949561	+	0.313583i	$0.101529\pi$
$283$			10.5506i			0.627165i	−0.949561	+	0.313583i	$0.898471\pi$
$284$		0			0
$285$	22.6704	−	20.6287i	1.34288	−	1.22194i
$286$		0			0
$287$			3.08777i			0.182266i
$288$		0			0
$289$	16.9213			0.995371
$290$		0			0
$291$	−37.1746			−2.17922
$292$		0			0
$293$		−	8.49506i		−	0.496287i	−0.968723	−	0.248143i	$-0.920180\pi$
$293$		−	8.49506i		−	0.496287i	0.968723	−	0.248143i	$-0.0798204\pi$
$294$		0			0
$295$	5.57135	+	6.12276i	0.324376	+	0.356481i
$296$		0			0
$297$		−	0.565408i		−	0.0328083i
$298$		0			0
$299$	3.02066			0.174689
$300$		0			0
$301$	4.24181			0.244494
$302$		0			0
$303$			2.95675i			0.169861i
$304$		0			0
$305$	−17.3235	−	19.0381i	−0.991941	−	1.09012i
$306$		0			0
$307$		−	16.1607i		−	0.922339i	−0.887312	−	0.461170i	$-0.847430\pi$
$307$		−	16.1607i		−	0.922339i	0.887312	−	0.461170i	$-0.152570\pi$
$308$		0			0
$309$	−6.85128			−0.389756
$310$		0			0
$311$	33.3027			1.88842			0.944211	−	0.329340i	$-0.106826\pi$
$311$	33.3027			1.88842			0.944211	+	0.329340i	$0.106826\pi$
$312$		0			0
$313$			8.59232i			0.485666i	0.970068	+	0.242833i	$0.0780768\pi$
$313$			8.59232i			0.485666i	−0.970068	+	0.242833i	$0.921923\pi$
$314$		0			0
$315$	−5.19709	+	4.72905i	−0.292823	+	0.266452i
$316$		0			0
$317$		−	32.9109i		−	1.84846i	−0.381837	−	0.924230i	$-0.624709\pi$
$317$		−	32.9109i		−	1.84846i	0.381837	−	0.924230i	$-0.375291\pi$
$318$		0			0
$319$	0.111701			0.00625407
$320$		0			0
$321$	16.1878			0.903513
$322$		0			0
$323$			1.32602i			0.0737815i
$324$		0			0
$325$	15.0363	+	1.42117i	0.834062	+	0.0788321i
$326$		0			0
$327$			17.6889i			0.978201i
$328$		0			0
$329$	−7.09349			−0.391077
$330$		0			0
$331$	8.19245			0.450298			0.225149	−	0.974324i	$-0.427713\pi$
$331$	8.19245			0.450298			0.225149	+	0.974324i	$0.427713\pi$
$332$		0			0
$333$			14.1484i			0.775330i
$334$		0			0
$335$	−9.90979	+	9.01733i	−0.541430	+	0.492669i
$336$		0			0
$337$		−	25.7372i		−	1.40199i	−0.713165	−	0.700996i	$-0.752739\pi$
$337$		−	25.7372i		−	1.40199i	0.713165	−	0.700996i	$-0.247261\pi$
$338$		0			0
$339$	−50.4414			−2.73960
$340$		0			0
$341$	0.461806			0.0250082
$342$		0			0
$343$			7.93630i			0.428520i
$344$		0			0
$345$	−4.36418	−	4.79611i	−0.234959	−	0.258214i
$346$		0			0
$347$			12.1539i			0.652455i	0.945291	+	0.326228i	$0.105778\pi$
$347$			12.1539i			0.652455i	−0.945291	+	0.326228i	$0.894222\pi$
$348$		0			0
$349$	−4.51456			−0.241659			−0.120829	−	0.992673i	$-0.538555\pi$
$349$	−4.51456			−0.241659			−0.120829	+	0.992673i	$0.538555\pi$
$350$		0			0
$351$	21.1089			1.12671
$352$		0			0
$353$			29.7767i			1.58485i	0.609968	+	0.792426i	$0.291182\pi$
$353$			29.7767i			1.58485i	−0.609968	+	0.792426i	$0.708818\pi$
$354$		0			0
$355$	20.8165	+	22.8768i	1.10483	+	1.21417i
$356$		0			0
$357$		−	0.472558i		−	0.0250104i
$358$		0			0
$359$	4.22436			0.222953			0.111477	−	0.993767i	$-0.464442\pi$
$359$	4.22436			0.222953			0.111477	+	0.993767i	$0.464442\pi$
$360$		0			0
$361$	3.34296			0.175945
$362$		0			0
$363$			31.8805i			1.67329i
$364$		0			0
$365$	−1.89309	+	1.72260i	−0.0990889	+	0.0901650i
$366$		0			0
$367$		−	31.9394i		−	1.66722i	−0.552351	−	0.833611i	$-0.686269\pi$
$367$		−	31.9394i		−	1.66722i	0.552351	−	0.833611i	$-0.313731\pi$
$368$		0			0
$369$	−28.7566			−1.49701
$370$		0			0
$371$	−3.85722			−0.200257
$372$		0			0
$373$			13.3524i			0.691361i	0.938352	+	0.345681i	$0.112352\pi$
$373$			13.3524i			0.691361i	−0.938352	+	0.345681i	$0.887648\pi$
$374$		0			0
$375$	−19.4676	−	25.9274i	−1.00530	−	1.33889i
$376$		0			0
$377$			4.17024i			0.214778i
$378$		0			0
$379$	−20.0481			−1.02980			−0.514901	−	0.857250i	$-0.672171\pi$
$379$	−20.0481			−1.02980			−0.514901	+	0.857250i	$0.672171\pi$
$380$		0			0
$381$	−34.6153			−1.77340
$382$		0			0
$383$			12.7850i			0.653281i	0.945149	+	0.326640i	$0.105917\pi$
$383$			12.7850i			0.653281i	−0.945149	+	0.326640i	$0.894083\pi$
$384$		0			0
$385$	−0.0777288	+	0.0707286i	−0.00396143	+	0.00360466i
$386$		0			0
$387$			39.5042i			2.00811i
$388$		0			0
$389$	−5.69492			−0.288744			−0.144372	−	0.989523i	$-0.546116\pi$
$389$	−5.69492			−0.288744			−0.144372	+	0.989523i	$0.546116\pi$
$390$		0			0
$391$	0.280529			0.0141870
$392$		0			0
$393$			28.5418i			1.43974i
$394$		0			0
$395$	−2.53077	−	2.78125i	−0.127337	−	0.139940i
$396$		0			0
$397$		−	11.2231i		−	0.563269i	−0.959522	−	0.281635i	$-0.909123\pi$
$397$		−	11.2231i		−	0.563269i	0.959522	−	0.281635i	$-0.0908766\pi$
$398$		0			0
$399$	−7.96246			−0.398622
$400$		0			0
$401$	−27.1979			−1.35820			−0.679098	−	0.734047i	$-0.737629\pi$
$401$	−27.1979			−1.35820			−0.679098	+	0.734047i	$0.737629\pi$
$402$		0			0
$403$			17.2410i			0.858836i
$404$		0			0
$405$	−6.07403	−	6.67519i	−0.301821	−	0.331693i
$406$		0			0
$407$			0.211607i			0.0104890i
$408$		0			0
$409$	28.2012			1.39446			0.697230	−	0.716848i	$-0.254416\pi$
$409$	28.2012			1.39446			0.697230	+	0.716848i	$0.254416\pi$
$410$		0			0
$411$	−8.50428			−0.419485
$412$		0			0
$413$		−	2.15048i		−	0.105818i
$414$		0			0
$415$	−16.3215	+	14.8516i	−0.801192	+	0.729038i
$416$		0			0
$417$			14.3860i			0.704484i
$418$		0			0
$419$	−18.5731			−0.907356			−0.453678	−	0.891166i	$-0.649888\pi$
$419$	−18.5731			−0.907356			−0.453678	+	0.891166i	$0.649888\pi$
$420$		0			0
$421$	−16.6531			−0.811622			−0.405811	−	0.913957i	$-0.633011\pi$
$421$	−16.6531			−0.811622			−0.405811	+	0.913957i	$0.633011\pi$
$422$		0			0
$423$		−	66.0620i		−	3.21204i
$424$		0			0
$425$	1.39642	+	0.131984i	0.0677365	+	0.00640217i
$426$		0			0
$427$			6.68668i			0.323591i
$428$		0			0
$429$	0.708748			0.0342187
$430$		0			0
$431$	−0.446399			−0.0215023			−0.0107511	−	0.999942i	$-0.503422\pi$
$431$	−0.446399			−0.0215023			−0.0107511	+	0.999942i	$0.503422\pi$
$432$		0			0
$433$		−	1.87406i		−	0.0900616i	−0.998986	−	0.0450308i	$-0.985661\pi$
$433$		−	1.87406i		−	0.0900616i	0.998986	−	0.0450308i	$-0.0143386\pi$
$434$		0			0
$435$	6.62138	−	6.02507i	0.317471	−	0.288880i
$436$		0			0
$437$		−	4.72683i		−	0.226115i
$438$		0			0
$439$	13.6607			0.651991			0.325995	−	0.945371i	$-0.394301\pi$
$439$	13.6607			0.651991			0.325995	+	0.945371i	$0.394301\pi$
$440$		0			0
$441$	−36.0429			−1.71633
$442$		0			0
$443$			21.6380i			1.02805i	0.857774	+	0.514027i	$0.171847\pi$
$443$			21.6380i			1.02805i	−0.857774	+	0.514027i	$0.828153\pi$
$444$		0			0
$445$	10.0714	+	11.0682i	0.477430	+	0.524683i
$446$		0			0
$447$			52.3080i			2.47409i
$448$		0			0
$449$	22.3547			1.05499			0.527493	−	0.849560i	$-0.323132\pi$
$449$	22.3547			1.05499			0.527493	+	0.849560i	$0.323132\pi$
$450$		0			0
$451$	−0.430090			−0.0202521
$452$		0			0
$453$			52.4968i			2.46652i
$454$		0			0
$455$	−2.64057	−	2.90191i	−0.123792	−	0.136044i
$456$		0			0
$457$		−	13.8614i		−	0.648408i	−0.945987	−	0.324204i	$-0.894904\pi$
$457$		−	13.8614i		−	0.648408i	0.945987	−	0.324204i	$-0.105096\pi$
$458$		0			0
$459$	1.96039			0.0915030
$460$		0			0
$461$	−26.4880			−1.23367			−0.616834	−	0.787093i	$-0.711585\pi$
$461$	−26.4880			−1.23367			−0.616834	+	0.787093i	$0.711585\pi$
$462$		0			0
$463$		−	31.3980i		−	1.45919i	−0.683881	−	0.729594i	$-0.739709\pi$
$463$		−	31.3980i		−	1.45919i	0.683881	−	0.729594i	$-0.260291\pi$
$464$		0			0
$465$	27.3748	−	24.9094i	1.26947	−	1.15515i
$466$		0			0
$467$			3.27210i			0.151415i	0.997130	+	0.0757074i	$0.0241215\pi$
$467$			3.27210i			0.151415i	−0.997130	+	0.0757074i	$0.975879\pi$
$468$		0			0
$469$	3.48058			0.160718
$470$		0			0
$471$	−44.2819			−2.04040
$472$		0			0
$473$			0.590833i			0.0271665i
$474$		0			0
$475$	2.22389	−	23.5293i	0.102039	−	1.07960i
$476$		0			0
$477$		−	35.9224i		−	1.64478i
$478$		0			0
$479$	30.4865			1.39296			0.696482	−	0.717574i	$-0.254747\pi$
$479$	30.4865			1.39296			0.696482	+	0.717574i	$0.254747\pi$
$480$		0			0
$481$	−7.90010			−0.360214
$482$		0			0
$483$			1.68452i			0.0766485i
$484$		0			0
$485$	−21.2008	+	19.2915i	−0.962680	+	0.875982i
$486$		0			0
$487$		−	12.8997i		−	0.584542i	−0.956336	−	0.292271i	$-0.905589\pi$
$487$		−	12.8997i		−	0.584542i	0.956336	−	0.292271i	$-0.0944109\pi$
$488$		0			0
$489$	−46.8308			−2.11776
$490$		0			0
$491$	1.53828			0.0694217			0.0347109	−	0.999397i	$-0.488949\pi$
$491$	1.53828			0.0694217			0.0347109	+	0.999397i	$0.488949\pi$
$492$		0			0
$493$			0.387291i			0.0174427i
$494$		0			0
$495$	−0.658699	−	0.723892i	−0.0296063	−	0.0325365i
$496$		0			0
$497$		−	8.03494i		−	0.360416i
$498$		0			0
$499$	−18.0529			−0.808160			−0.404080	−	0.914724i	$-0.632408\pi$
$499$	−18.0529			−0.808160			−0.404080	+	0.914724i	$0.632408\pi$
$500$		0			0
$501$	−19.6592			−0.878310
$502$		0			0
$503$		−	7.35252i		−	0.327832i	−0.986474	−	0.163916i	$-0.947587\pi$
$503$		−	7.35252i		−	0.327832i	0.986474	−	0.163916i	$-0.0524127\pi$
$504$		0			0
$505$	1.53439	+	1.68625i	0.0682792	+	0.0750370i
$506$		0			0
$507$		−	11.2392i		−	0.499149i
$508$		0			0
$509$	29.3564			1.30120			0.650600	−	0.759421i	$-0.274517\pi$
$509$	29.3564			1.30120			0.650600	+	0.759421i	$0.274517\pi$
$510$		0			0
$511$	0.664904			0.0294136
$512$		0			0
$513$		−	33.0319i		−	1.45839i
$514$		0			0
$515$	−3.90731	+	3.55542i	−0.172177	+	0.156671i
$516$		0			0
$517$		−	0.988037i		−	0.0434538i
$518$		0			0
$519$	63.6331			2.79318
$520$		0			0
$521$	−38.1800			−1.67270			−0.836349	−	0.548198i	$-0.815314\pi$
$521$	−38.1800			−1.67270			−0.836349	+	0.548198i	$0.815314\pi$
$522$		0			0
$523$		−	37.7527i		−	1.65081i	−0.564540	−	0.825405i	$-0.690947\pi$
$523$		−	37.7527i		−	1.65081i	0.564540	−	0.825405i	$-0.309053\pi$
$524$		0			0
$525$	−0.792538	+	8.38524i	−0.0345892	+	0.365962i
$526$		0			0
$527$			1.60118i			0.0697484i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	20.0275			0.869119
$532$		0			0
$533$		−	16.0569i		−	0.695501i
$534$		0			0
$535$	9.23194	−	8.40052i	0.399132	−	0.363186i
$536$		0			0
$537$			19.6125i			0.846341i
$538$		0			0
$539$	−0.539065			−0.0232192
$540$		0			0
$541$	−6.30694			−0.271157			−0.135578	−	0.990767i	$-0.543289\pi$
$541$	−6.30694			−0.271157			−0.135578	+	0.990767i	$0.543289\pi$
$542$		0			0
$543$		−	35.2555i		−	1.51296i
$544$		0			0
$545$	9.17955	+	10.0881i	0.393209	+	0.432126i
$546$		0			0
$547$			40.9574i			1.75121i	0.483027	+	0.875606i	$0.339537\pi$
$547$			40.9574i			1.75121i	−0.483027	+	0.875606i	$0.660463\pi$
$548$		0			0
$549$	−62.2733			−2.65776
$550$		0			0
$551$	6.52574			0.278006
$552$		0			0
$553$			0.976848i			0.0415398i
$554$		0			0
$555$	11.4139	+	12.5436i	0.484493	+	0.532444i
$556$		0			0
$557$		−	30.6143i		−	1.29717i	−0.761142	−	0.648586i	$-0.775361\pi$
$557$		−	30.6143i		−	1.29717i	0.761142	−	0.648586i	$-0.224639\pi$
$558$		0			0
$559$	−22.0581			−0.932956
$560$		0			0
$561$	0.0658217			0.00277899
$562$		0			0
$563$			19.8221i			0.835400i	0.908585	+	0.417700i	$0.137164\pi$
$563$			19.8221i			0.835400i	−0.908585	+	0.417700i	$0.862836\pi$
$564$		0			0
$565$	−28.7669	+	26.1762i	−1.21023	+	1.10124i
$566$		0			0
$567$			2.34451i			0.0984600i
$568$		0			0
$569$	−34.8830			−1.46237			−0.731185	−	0.682179i	$-0.761032\pi$
$569$	−34.8830			−1.46237			−0.731185	+	0.682179i	$0.761032\pi$
$570$		0			0
$571$	10.5730			0.442466			0.221233	−	0.975221i	$-0.428992\pi$
$571$	10.5730			0.442466			0.221233	+	0.975221i	$0.428992\pi$
$572$		0			0
$573$		−	25.7350i		−	1.07509i
$574$		0			0
$575$	−4.97782	−	0.470482i	−0.207589	−	0.0196205i
$576$		0			0
$577$			12.7839i			0.532201i	0.963945	+	0.266101i	$0.0857354\pi$
$577$			12.7839i			0.532201i	−0.963945	+	0.266101i	$0.914265\pi$
$578$		0			0
$579$	−33.9371			−1.41038
$580$		0			0
$581$	5.73256			0.237827
$582$		0			0
$583$		−	0.537264i		−	0.0222512i
$584$		0			0
$585$	27.0257	−	24.5918i	1.11737	−	1.01674i
$586$		0			0
$587$			10.4015i			0.429317i	0.976689	+	0.214658i	$0.0688638\pi$
$587$			10.4015i			0.429317i	−0.976689	+	0.214658i	$0.931136\pi$
$588$		0			0
$589$	26.9794			1.11166
$590$		0			0
$591$	0.959198			0.0394561
$592$		0			0
$593$			25.7468i			1.05729i	0.848842	+	0.528647i	$0.177300\pi$
$593$			25.7468i			1.05729i	−0.848842	+	0.528647i	$0.822700\pi$
$594$		0			0
$595$	−0.245231	−	0.269502i	−0.0100535	−	0.0110485i
$596$		0			0
$597$			28.2050i			1.15435i
$598$		0			0
$599$	−21.2360			−0.867680			−0.433840	−	0.900990i	$-0.642842\pi$
$599$	−21.2360			−0.867680			−0.433840	+	0.900990i	$0.642842\pi$
$600$		0			0
$601$	−10.9088			−0.444980			−0.222490	−	0.974935i	$-0.571418\pi$
$601$	−10.9088			−0.444980			−0.222490	+	0.974935i	$0.571418\pi$
$602$		0			0
$603$			32.4148i			1.32003i
$604$		0			0
$605$	16.5442	+	18.1816i	0.672616	+	0.739187i
$606$		0			0
$607$			6.15713i			0.249910i	0.992162	+	0.124955i	$0.0398787\pi$
$607$			6.15713i			0.249910i	−0.992162	+	0.124955i	$0.960121\pi$
$608$		0			0
$609$	−2.32561			−0.0942383
$610$		0			0
$611$	36.8872			1.49230
$612$		0			0
$613$			20.3588i			0.822283i	0.911572	+	0.411142i	$0.134870\pi$
$613$			20.3588i			0.822283i	−0.911572	+	0.411142i	$0.865130\pi$
$614$		0			0
$615$	−25.4947	+	23.1986i	−1.02804	+	0.935460i
$616$		0			0
$617$			13.9290i			0.560759i	0.959889	+	0.280380i	$0.0904604\pi$
$617$			13.9290i			0.560759i	−0.959889	+	0.280380i	$0.909540\pi$
$618$		0			0
$619$	45.4617			1.82726			0.913629	−	0.406548i	$-0.133268\pi$
$619$	45.4617			1.82726			0.913629	+	0.406548i	$0.133268\pi$
$620$		0			0
$621$	−6.98817			−0.280426
$622$		0			0
$623$		−	3.88745i		−	0.155747i
$624$		0			0
$625$	−24.5573	−	4.68395i	−0.982292	−	0.187358i
$626$		0			0
$627$		−	1.10907i		−	0.0442922i
$628$		0			0
$629$	−0.733685			−0.0292540
$630$		0			0
$631$	39.7205			1.58125			0.790624	−	0.612303i	$-0.209757\pi$
$631$	39.7205			1.58125			0.790624	+	0.612303i	$0.209757\pi$
$632$		0			0
$633$			19.2473i			0.765011i
$634$		0			0
$635$	−19.7413	+	17.9634i	−0.783408	+	0.712855i
$636$		0			0
$637$		−	20.1254i		−	0.797396i
$638$		0			0
$639$	74.8297			2.96022
$640$		0			0
$641$	19.4730			0.769137			0.384568	−	0.923096i	$-0.374350\pi$
$641$	19.4730			0.769137			0.384568	+	0.923096i	$0.374350\pi$
$642$		0			0
$643$		−	36.3269i		−	1.43259i	−0.697795	−	0.716297i	$-0.745836\pi$
$643$		−	36.3269i		−	1.43259i	0.697795	−	0.716297i	$-0.254164\pi$
$644$		0			0
$645$	31.8690	+	35.0231i	1.25484	+	1.37904i
$646$		0			0
$647$		−	4.18108i		−	0.164375i	−0.996617	−	0.0821875i	$-0.973809\pi$
$647$		−	4.18108i		−	0.164375i	0.996617	−	0.0821875i	$-0.0261906\pi$
$648$		0			0
$649$	0.299535			0.0117578
$650$		0			0
$651$	−9.61475			−0.376832
$652$		0			0
$653$			30.5378i			1.19504i	0.801855	+	0.597519i	$0.203847\pi$
$653$			30.5378i			1.19504i	−0.801855	+	0.597519i	$0.796153\pi$
$654$		0			0
$655$	14.8116	+	16.2775i	0.578735	+	0.636014i
$656$		0			0
$657$			6.19228i			0.241584i
$658$		0			0
$659$	39.7062			1.54673			0.773366	−	0.633960i	$-0.218571\pi$
$659$	39.7062			1.54673			0.773366	+	0.633960i	$0.218571\pi$
$660$		0			0
$661$	4.53963			0.176571			0.0882857	−	0.996095i	$-0.471861\pi$
$661$	4.53963			0.176571			0.0882857	+	0.996095i	$0.471861\pi$
$662$		0			0
$663$			2.45737i			0.0954365i
$664$		0			0
$665$	−4.54102	+	4.13206i	−0.176093	+	0.160234i
$666$		0			0
$667$		−	1.38057i		−	0.0534560i
$668$		0			0
$669$	79.5524			3.07567
$670$		0			0
$671$	−0.931373			−0.0359553
$672$		0			0
$673$			28.2447i			1.08875i	0.838841	+	0.544377i	$0.183234\pi$
$673$			28.2447i			1.08875i	−0.838841	+	0.544377i	$0.816766\pi$
$674$		0			0
$675$	−34.7858	−	3.28781i	−1.33891	−	0.126548i
$676$		0			0
$677$			10.0502i			0.386260i	0.981173	+	0.193130i	$0.0618639\pi$
$677$			10.0502i			0.386260i	−0.981173	+	0.193130i	$0.938136\pi$
$678$		0			0
$679$	7.44630			0.285763
$680$		0			0
$681$	−59.7956			−2.29137
$682$		0			0
$683$		−	13.1814i		−	0.504372i	−0.967679	−	0.252186i	$-0.918851\pi$
$683$		−	13.1814i		−	0.504372i	0.967679	−	0.252186i	$-0.0811495\pi$
$684$		0			0
$685$	−4.85003	+	4.41324i	−0.185310	+	0.168621i
$686$		0			0
$687$		−	64.6542i		−	2.46671i
$688$		0			0
$689$	20.0581			0.764153
$690$		0			0
$691$	48.3908			1.84087			0.920436	−	0.390894i	$-0.127834\pi$
$691$	48.3908			1.84087			0.920436	+	0.390894i	$0.127834\pi$
$692$		0			0
$693$			0.254250i			0.00965817i
$694$		0			0
$695$	7.46550	+	8.20437i	0.283182	+	0.311210i
$696$		0			0
$697$		−	1.49121i		−	0.0564836i
$698$		0			0
$699$	−74.0449			−2.80064
$700$		0			0
$701$	34.4310			1.30044			0.650220	−	0.759746i	$-0.274677\pi$
$701$	34.4310			1.30044			0.650220	+	0.759746i	$0.274677\pi$
$702$		0			0
$703$			12.3624i			0.466256i
$704$		0			0
$705$	−53.2938	−	58.5685i	−2.00716	−	2.20582i
$706$		0			0
$707$		−	0.592255i		−	0.0222740i
$708$		0			0
$709$	−13.7221			−0.515346			−0.257673	−	0.966232i	$-0.582956\pi$
$709$	−13.7221			−0.515346			−0.257673	+	0.966232i	$0.582956\pi$
$710$		0			0
$711$	−9.09743			−0.341180
$712$		0			0
$713$		−	5.70770i		−	0.213755i
$714$		0			0
$715$	0.404202	−	0.367799i	0.0151163	−	0.0137549i
$716$		0			0
$717$			62.6440i			2.33948i
$718$		0			0
$719$	34.7900			1.29745			0.648724	−	0.761024i	$-0.275303\pi$
$719$	34.7900			1.29745			0.648724	+	0.761024i	$0.275303\pi$
$720$		0			0
$721$	1.37235			0.0511091
$722$		0			0
$723$		−	62.8855i		−	2.33874i
$724$		0			0
$725$	0.649535	−	6.87224i	0.0241231	−	0.255228i
$726$		0			0
$727$			3.35114i			0.124287i	0.998067	+	0.0621435i	$0.0197937\pi$
$727$			3.35114i			0.124287i	−0.998067	+	0.0621435i	$0.980206\pi$
$728$		0			0
$729$	38.9617			1.44302
$730$		0			0
$731$	−2.04854			−0.0757680
$732$		0			0
$733$			3.20440i			0.118357i	0.998247	+	0.0591786i	$0.0188482\pi$
$733$			3.20440i			0.118357i	−0.998247	+	0.0591786i	$0.981152\pi$
$734$		0			0
$735$	−31.9545	+	29.0767i	−1.17866	+	1.07251i
$736$		0			0
$737$			0.484803i			0.0178580i
$738$		0			0
$739$	−38.3376			−1.41027			−0.705136	−	0.709072i	$-0.749114\pi$
$739$	−38.3376			−1.41027			−0.705136	+	0.709072i	$0.749114\pi$
$740$		0			0
$741$	41.4060			1.52109
$742$		0			0
$743$			21.5137i			0.789261i	0.918840	+	0.394631i	$0.129127\pi$
$743$			21.5137i			0.789261i	−0.918840	+	0.394631i	$0.870873\pi$
$744$		0			0
$745$	27.1449	+	29.8315i	0.994512	+	1.09294i
$746$		0			0
$747$			53.3876i			1.95335i
$748$		0			0
$749$	−3.24251			−0.118479
$750$		0			0
$751$	−29.9353			−1.09236			−0.546178	−	0.837669i	$-0.683918\pi$
$751$	−29.9353			−1.09236			−0.546178	+	0.837669i	$0.683918\pi$
$752$		0			0
$753$		−	77.1363i		−	2.81100i
$754$		0			0
$755$	27.2429	+	29.9391i	0.991469	+	1.08960i
$756$		0			0
$757$		−	23.2518i		−	0.845102i	−0.906339	−	0.422551i	$-0.861135\pi$
$757$		−	23.2518i		−	0.845102i	0.906339	−	0.422551i	$-0.138865\pi$
$758$		0			0
$759$	−0.234634			−0.00851666
$760$		0			0
$761$	40.7791			1.47824			0.739120	−	0.673574i	$-0.235242\pi$
$761$	40.7791			1.47824			0.739120	+	0.673574i	$0.235242\pi$
$762$		0			0
$763$		−	3.54320i		−	0.128272i
$764$		0			0
$765$	2.50988	−	2.28384i	0.0907450	−	0.0825726i
$766$		0			0
$767$			11.1828i			0.403788i
$768$		0			0
$769$	1.08570			0.0391515			0.0195757	−	0.999808i	$-0.493768\pi$
$769$	1.08570			0.0391515			0.0195757	+	0.999808i	$0.493768\pi$
$770$		0			0
$771$	0.453531			0.0163335
$772$		0			0
$773$		−	37.7659i		−	1.35835i	−0.733978	−	0.679173i	$-0.762338\pi$
$773$		−	37.7659i		−	1.35835i	0.733978	−	0.679173i	$-0.237662\pi$
$774$		0			0
$775$	2.68537	−	28.4119i	0.0964614	−	1.02058i
$776$		0			0
$777$		−	4.40563i		−	0.158051i
$778$		0			0
$779$	−25.1264			−0.900247
$780$		0			0
$781$	1.11917			0.0400470
$782$		0			0
$783$		−	9.64767i		−	0.344780i
$784$		0			0
$785$	−25.2541	+	22.9798i	−0.901358	+	0.820183i
$786$		0			0
$787$		−	38.0489i		−	1.35630i	−0.734925	−	0.678149i	$-0.762782\pi$
$787$		−	38.0489i		−	1.35630i	0.734925	−	0.678149i	$-0.237218\pi$
$788$		0			0
$789$	70.0506			2.49387
$790$		0			0
$791$	10.1037			0.359246
$792$		0			0
$793$		−	34.7717i		−	1.23478i
$794$		0			0
$795$	−28.9795	−	31.8477i	−1.02780	−	1.12952i
$796$		0			0
$797$			12.8798i			0.456227i	0.973634	+	0.228114i	$0.0732558\pi$
$797$			12.8798i			0.456227i	−0.973634	+	0.228114i	$0.926744\pi$
$798$		0			0
$799$	3.42573			0.121194
$800$		0			0
$801$	36.2040			1.27920
$802$		0			0
$803$			0.0926131i			0.00326825i
$804$		0			0
$805$	0.874171	+	0.960690i	0.0308105	+	0.0338599i
$806$		0			0
$807$			0.784110i			0.0276020i
$808$		0			0
$809$	0.0775700			0.00272722			0.00136361	−	0.999999i	$-0.499566\pi$
$809$	0.0775700			0.00272722			0.00136361	+	0.999999i	$0.499566\pi$
$810$		0			0
$811$	−0.0163427			−0.000573869			−0.000286934	−	1.00000i	$-0.500091\pi$
$811$	−0.0163427			−0.000573869			−0.000286934		1.00000i	$0.500091\pi$
$812$		0			0
$813$		−	29.0114i		−	1.01747i
$814$		0			0
$815$	−26.7078	+	24.3025i	−0.935533	+	0.851279i
$816$		0			0
$817$			34.5173i			1.20761i
$818$		0			0
$819$	−9.49214			−0.331682
$820$		0			0
$821$	6.85832			0.239357			0.119678	−	0.992813i	$-0.461814\pi$
$821$	6.85832			0.239357			0.119678	+	0.992813i	$0.461814\pi$
$822$		0			0
$823$		−	24.4740i		−	0.853109i	−0.904462	−	0.426554i	$-0.859727\pi$
$823$		−	24.4740i		−	0.853109i	0.904462	−	0.426554i	$-0.140273\pi$
$824$		0			0
$825$	−1.16796	−	0.110391i	−0.0406633	−	0.00384332i
$826$		0			0
$827$		−	11.5556i		−	0.401828i	−0.979609	−	0.200914i	$-0.935609\pi$
$827$		−	11.5556i		−	0.401828i	0.979609	−	0.200914i	$-0.0643912\pi$
$828$		0			0
$829$	42.7496			1.48475			0.742377	−	0.669983i	$-0.233698\pi$
$829$	42.7496			1.48475			0.742377	+	0.669983i	$0.233698\pi$
$830$		0			0
$831$	35.0437			1.21565
$832$		0			0
$833$		−	1.86905i		−	0.0647587i
$834$		0			0
$835$	−11.2117	+	10.2020i	−0.387998	+	0.353055i
$836$		0			0
$837$		−	39.8864i		−	1.37867i
$838$		0			0
$839$	−52.4396			−1.81042			−0.905209	−	0.424968i	$-0.860286\pi$
$839$	−52.4396			−1.81042			−0.905209	+	0.424968i	$0.860286\pi$
$840$		0			0
$841$	−27.0940			−0.934277
$842$		0			0
$843$		−	64.6434i		−	2.22644i
$844$		0			0
$845$	−5.83249	−	6.40974i	−0.200644	−	0.220502i
$846$		0			0
$847$		−	6.38586i		−	0.219421i
$848$		0			0
$849$	30.5961			1.05006
$850$		0			0
$851$	2.61536			0.0896534
$852$		0			0
$853$			23.6690i			0.810410i	0.914226	+	0.405205i	$0.132800\pi$
$853$			23.6690i			0.810410i	−0.914226	+	0.405205i	$0.867200\pi$
$854$		0			0
$855$	−38.4821	−	42.2907i	−1.31606	−	1.44631i
$856$		0			0
$857$		−	23.1587i		−	0.791088i	−0.918447	−	0.395544i	$-0.870556\pi$
$857$		−	23.1587i		−	0.791088i	0.918447	−	0.395544i	$-0.129444\pi$
$858$		0			0
$859$	11.6044			0.395939			0.197969	−	0.980208i	$-0.436565\pi$
$859$	11.6044			0.395939			0.197969	+	0.980208i	$0.436565\pi$
$860$		0			0
$861$	8.95441			0.305166
$862$		0			0
$863$		−	2.71131i		−	0.0922940i	−0.998935	−	0.0461470i	$-0.985306\pi$
$863$		−	2.71131i		−	0.0922940i	0.998935	−	0.0461470i	$-0.0146943\pi$
$864$		0			0
$865$	36.2902	−	33.0220i	1.23390	−	1.12278i
$866$		0			0
$867$		−	49.0710i		−	1.66654i
$868$		0			0
$869$	−0.136063			−0.00461562
$870$		0			0
$871$	−18.0996			−0.613280
$872$		0			0
$873$			69.3477i			2.34706i
$874$		0			0
$875$	3.89947	+	5.19342i	0.131826	+	0.175570i
$876$		0			0
$877$			48.0862i			1.62375i	0.583828	+	0.811877i	$0.301554\pi$
$877$			48.0862i			1.62375i	−0.583828	+	0.811877i	$0.698446\pi$
$878$		0			0
$879$	−24.6353			−0.830928
$880$		0			0
$881$	2.70507			0.0911361			0.0455680	−	0.998961i	$-0.485490\pi$
$881$	2.70507			0.0911361			0.0455680	+	0.998961i	$0.485490\pi$
$882$		0			0
$883$		−	13.5827i		−	0.457094i	−0.973533	−	0.228547i	$-0.926602\pi$
$883$		−	13.5827i		−	0.457094i	0.973533	−	0.228547i	$-0.0733975\pi$
$884$		0			0
$885$	17.7557	−	16.1567i	0.596853	−	0.543101i
$886$		0			0
$887$		−	3.42335i		−	0.114945i	−0.998347	−	0.0574724i	$-0.981696\pi$
$887$		−	3.42335i		−	0.114945i	0.998347	−	0.0574724i	$-0.0183041\pi$
$888$		0			0
$889$	6.93366			0.232547
$890$		0			0
$891$	−0.326561			−0.0109402
$892$		0			0
$893$		−	57.7225i		−	1.93161i
$894$		0			0
$895$	10.1778	+	11.1851i	0.340205	+	0.373876i
$896$		0			0
$897$		−	8.75977i		−	0.292480i
$898$		0			0
$899$	7.87990			0.262809
$900$		0			0
$901$	1.86280			0.0620590
$902$		0			0
$903$		−	12.3011i		−	0.409354i
$904$		0			0
$905$	−18.2956	−	20.1063i	−0.608165	−	0.668357i
$906$		0			0
$907$			9.90426i			0.328865i	0.986388	+	0.164433i	$0.0525793\pi$
$907$			9.90426i			0.328865i	−0.986388	+	0.164433i	$0.947421\pi$
$908$		0			0
$909$	5.51570			0.182944
$910$		0			0
$911$	−1.95943			−0.0649187			−0.0324593	−	0.999473i	$-0.510334\pi$
$911$	−1.95943			−0.0649187			−0.0324593	+	0.999473i	$0.510334\pi$
$912$		0			0
$913$			0.798476i			0.0264257i
$914$		0			0
$915$	−55.2096	+	50.2374i	−1.82517	+	1.66080i
$916$		0			0
$917$		−	5.71709i		−	0.188795i
$918$		0			0
$919$	55.8522			1.84239			0.921197	−	0.389096i	$-0.127213\pi$
$919$	55.8522			1.84239			0.921197	+	0.389096i	$0.127213\pi$
$920$		0			0
$921$	−46.8653			−1.54426
$922$		0			0
$923$			41.7829i			1.37530i
$924$		0			0
$925$	13.0188	+	1.23048i	0.428055	+	0.0404580i
$926$		0			0
$927$			12.7808i			0.419776i
$928$		0			0
$929$	−17.0522			−0.559464			−0.279732	−	0.960078i	$-0.590246\pi$
$929$	−17.0522			−0.559464			−0.279732	+	0.960078i	$0.590246\pi$
$930$		0			0
$931$	−31.4929			−1.03214
$932$		0			0
$933$		−	96.5764i		−	3.16177i
$934$		0			0
$935$	0.0375383	−	0.0341577i	0.00122764	−	0.00111708i
$936$		0			0
$937$		−	30.9074i		−	1.00970i	−0.863207	−	0.504850i	$-0.831548\pi$
$937$		−	30.9074i		−	1.00970i	0.863207	−	0.504850i	$-0.168452\pi$
$938$		0			0
$939$	24.9173			0.813147
$940$		0			0
$941$	−41.4508			−1.35126			−0.675629	−	0.737242i	$-0.736128\pi$
$941$	−41.4508			−1.35126			−0.675629	+	0.737242i	$0.736128\pi$
$942$		0			0
$943$			5.31570i			0.173103i
$944$		0			0
$945$	6.10885	+	6.71346i	0.198721	+	0.218389i
$946$		0			0
$947$		−	34.4896i		−	1.12076i	−0.828236	−	0.560380i	$-0.810655\pi$
$947$		−	34.4896i		−	1.12076i	0.828236	−	0.560380i	$-0.189345\pi$
$948$		0			0
$949$	−3.45760			−0.112239
$950$		0			0
$951$	−95.4401			−3.09486
$952$		0			0
$953$			28.8622i			0.934939i	0.884009	+	0.467470i	$0.154834\pi$
$953$			28.8622i			0.934939i	−0.884009	+	0.467470i	$0.845166\pi$
$954$		0			0
$955$	−13.3550	−	14.6768i	−0.432157	−	0.474929i
$956$		0			0
$957$		−	0.323929i		−	0.0104711i
$958$		0			0
$959$	1.70346			0.0550076
$960$		0			0
$961$	1.57784			0.0508982
$962$		0			0
$963$		−	30.1976i		−	0.973104i
$964$		0			0
$965$	−19.3545	+	17.6114i	−0.623042	+	0.566932i
$966$		0			0
$967$		−	6.47559i		−	0.208241i	−0.994565	−	0.104121i	$-0.966797\pi$
$967$		−	6.47559i		−	0.208241i	0.994565	−	0.104121i	$-0.0332028\pi$
$968$		0			0
$969$	3.84539			0.123532
$970$		0			0
$971$	33.9296			1.08885			0.544426	−	0.838809i	$-0.316747\pi$
$971$	33.9296			1.08885			0.544426	+	0.838809i	$0.316747\pi$
$972$		0			0
$973$		−	2.88160i		−	0.0923797i
$974$		0			0
$975$	4.12132	−	43.6045i	0.131988	−	1.39646i
$976$		0			0
$977$		−	2.90638i		−	0.0929832i	−0.998919	−	0.0464916i	$-0.985196\pi$
$977$		−	2.90638i		−	0.0929832i	0.998919	−	0.0464916i	$-0.0148041\pi$
$978$		0			0
$979$	0.541474			0.0173056
$980$		0			0
$981$	32.9980			1.05354
$982$		0			0
$983$		−	8.66470i		−	0.276361i	−0.990407	−	0.138180i	$-0.955875\pi$
$983$		−	8.66470i		−	0.276361i	0.990407	−	0.138180i	$-0.0441254\pi$
$984$		0			0
$985$	0.547034	−	0.497769i	0.0174300	−	0.0158602i
$986$		0			0
$987$			20.5708i			0.654776i
$988$		0			0
$989$	7.30240			0.232203
$990$		0			0
$991$	−32.4587			−1.03108			−0.515542	−	0.856864i	$-0.672410\pi$
$991$	−32.4587			−1.03108			−0.515542	+	0.856864i	$0.672410\pi$
$992$		0			0
$993$		−	23.7577i		−	0.753929i
$994$		0			0
$995$	14.6368	+	16.0854i	0.464017	+	0.509942i
$996$		0			0
$997$		−	61.7623i		−	1.95603i	−0.208534	−	0.978015i	$-0.566869\pi$
$997$		−	61.7623i		−	1.95603i	0.208534	−	0.978015i	$-0.433131\pi$
$998$		0			0
$999$	18.2766			0.578245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.e.c.369.2	✓	16
4.3	odd	2		1840.2.e.h.369.15		16
5.2	odd	4		4600.2.a.bj.1.2		8
5.3	odd	4		4600.2.a.bk.1.7		8
5.4	even	2	inner	920.2.e.c.369.15	yes	16
20.3	even	4		9200.2.a.dd.1.2		8
20.7	even	4		9200.2.a.de.1.7		8
20.19	odd	2		1840.2.e.h.369.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.2	✓	16	1.1	even	1	trivial
920.2.e.c.369.15	yes	16	5.4	even	2	inner
1840.2.e.h.369.2		16	20.19	odd	2
1840.2.e.h.369.15		16	4.3	odd	2
4600.2.a.bj.1.2		8	5.2	odd	4
4600.2.a.bk.1.7		8	5.3	odd	4
9200.2.a.dd.1.2		8	20.3	even	4
9200.2.a.de.1.7		8	20.7	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.89996i q^{3} +(-1.50491 - 1.65386i) q^{5} +0.580879i q^{7} -5.40975 q^{9} +O(q^{10})\)	\(q-2.89996i q^{3} +(-1.50491 - 1.65386i) q^{5} +0.580879i q^{7} -5.40975 q^{9} -0.0809094 q^{11} -3.02066i q^{13} +(-4.79611 + 4.36418i) q^{15} -0.280529i q^{17} -4.72683 q^{19} +1.68452 q^{21} +1.00000i q^{23} +(-0.470482 + 4.97782i) q^{25} +6.98817i q^{27} -1.38057 q^{29} -5.70770 q^{31} +0.234634i q^{33} +(0.960690 - 0.874171i) q^{35} -2.61536i q^{37} -8.75977 q^{39} +5.31570 q^{41} -7.30240i q^{43} +(8.14119 + 8.94695i) q^{45} +12.2117i q^{47} +6.66258 q^{49} -0.813523 q^{51} +6.64032i q^{53} +(0.121761 + 0.133812i) q^{55} +13.7076i q^{57} -3.70211 q^{59} +11.5113 q^{61} -3.14241i q^{63} +(-4.99573 + 4.54582i) q^{65} -5.99193i q^{67} +2.89996 q^{69} -13.8324 q^{71} -1.14465i q^{73} +(14.4354 + 1.36438i) q^{75} -0.0469985i q^{77} +1.68167 q^{79} +4.03614 q^{81} -9.86877i q^{83} +(-0.463955 + 0.422172i) q^{85} +4.00360i q^{87} -6.69235 q^{89} +1.75463 q^{91} +16.5521i q^{93} +(7.11347 + 7.81751i) q^{95} -12.8190i q^{97} +0.437699 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} - 22 q^{9}+O(q^{10}) \) 16 * q - 2 * q^5 - 22 * q^9	\( 16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99}+O(q^{100}) \) 16 * q - 2 * q^5 - 22 * q^9 + 14 * q^11 + 6 * q^15 - 22 * q^19 + 12 * q^25 - 44 * q^29 + 18 * q^31 + 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 - 38 * q^51 + 30 * q^55 - 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 + 30 * q^71 + 56 * q^75 + 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 - 70 * q^91 + 38 * q^95 - 122 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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