LMFDB - Embedded newform 1120.2.g.b.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(449,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, 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("1120.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.g (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:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 $ x^10 + 13x^8 + 56x^6 + 97x^4 + 61x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.84576i$ of defining polynomial
Character	$\chi$	$=$	1120.449
Dual form			1120.2.g.b.449.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.25260i q^{3} +(-1.49436 + 1.66340i) q^{5} -1.00000i q^{7} -7.57939 q^{9} +O(q^{10})$	$q-3.25260i q^{3} +(-1.49436 + 1.66340i) q^{5} -1.00000i q^{7} -7.57939 q^{9} -2.88786 q^{11} +6.94412i q^{13} +(5.41035 + 4.86056i) q^{15} -0.549797i q^{17} +4.98872 q^{19} -3.25260 q^{21} +2.33807i q^{23} +(-0.533766 - 4.97143i) q^{25} +14.8949i q^{27} +5.14173 q^{29} -5.40560 q^{31} +9.39306i q^{33} +(1.66340 + 1.49436i) q^{35} +8.31551i q^{37} +22.5864 q^{39} -2.87785 q^{41} +7.75318i q^{43} +(11.3263 - 12.6075i) q^{45} -8.24132i q^{47} -1.00000 q^{49} -1.78827 q^{51} -1.81032i q^{53} +(4.31551 - 4.80366i) q^{55} -16.2263i q^{57} -9.04790 q^{59} -2.11087 q^{61} +7.57939i q^{63} +(-11.5508 - 10.3770i) q^{65} +13.1588i q^{67} +7.60479 q^{69} -3.28928 q^{71} +2.18968i q^{73} +(-16.1701 + 1.73613i) q^{75} +2.88786i q^{77} +2.04460 q^{79} +25.7089 q^{81} -3.38141i q^{83} +(0.914530 + 0.821596i) q^{85} -16.7240i q^{87} +3.24798 q^{89} +6.94412 q^{91} +17.5822i q^{93} +(-7.45496 + 8.29822i) q^{95} -5.22593i q^{97} +21.8882 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 14 q^{9}+O(q^{10}) $ 10 * q - 2 * q^5 - 14 * q^9	$ 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65} - 8 q^{69} - 8 q^{71} - 64 q^{75} + 64 q^{79} + 2 q^{81} - 12 q^{85} - 4 q^{89} - 60 q^{95} + 80 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 14 * q^9 - 8 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 6 * q^25 + 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 + 10 * q^45 - 10 * q^49 - 24 * q^51 - 16 * q^55 + 32 * q^59 - 20 * q^61 - 8 * q^65 - 8 * q^69 - 8 * q^71 - 64 * q^75 + 64 * q^79 + 2 * q^81 - 12 * q^85 - 4 * q^89 - 60 * q^95 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	3.25260i		−	1.87789i	−0.344070	−	0.938944i	$-0.611806\pi$
$3$		−	3.25260i		−	1.87789i	0.344070	−	0.938944i	$-0.388194\pi$
$4$		0			0
$5$	−1.49436	+	1.66340i	−0.668299	+	0.743893i
$5$	−1.49436	+	1.66340i	−0.668299	+	0.743893i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−7.57939			−2.52646
$10$		0			0
$11$	−2.88786			−0.870724			−0.435362	−	0.900256i	$-0.643380\pi$
$11$	−2.88786			−0.870724			−0.435362	+	0.900256i	$0.643380\pi$
$12$		0			0
$13$			6.94412i			1.92595i	0.269587	+	0.962976i	$0.413113\pi$
$13$			6.94412i			1.92595i	−0.269587	+	0.962976i	$0.586887\pi$
$14$		0			0
$15$	5.41035	+	4.86056i	1.39695	+	1.25499i
$16$		0			0
$17$		−	0.549797i		−	0.133345i	−0.997775	−	0.0666727i	$-0.978762\pi$
$17$		−	0.549797i		−	0.133345i	0.997775	−	0.0666727i	$-0.0212383\pi$
$18$		0			0
$19$	4.98872			1.14449			0.572246	−	0.820082i	$-0.306072\pi$
$19$	4.98872			1.14449			0.572246	+	0.820082i	$0.306072\pi$
$20$		0			0
$21$	−3.25260			−0.709775
$22$		0			0
$23$			2.33807i			0.487521i	0.969836	+	0.243760i	$0.0783810\pi$
$23$			2.33807i			0.487521i	−0.969836	+	0.243760i	$0.921619\pi$
$24$		0			0
$25$	−0.533766	−	4.97143i	−0.106753	−	0.994286i
$26$		0			0
$27$			14.8949i			2.86652i
$28$		0			0
$29$	5.14173			0.954794			0.477397	−	0.878688i	$-0.341580\pi$
$29$	5.14173			0.954794			0.477397	+	0.878688i	$0.341580\pi$
$30$		0			0
$31$	−5.40560			−0.970874			−0.485437	−	0.874272i	$-0.661340\pi$
$31$	−5.40560			−0.970874			−0.485437	+	0.874272i	$0.661340\pi$
$32$		0			0
$33$			9.39306i			1.63512i
$34$		0			0
$35$	1.66340	+	1.49436i	0.281165	+	0.252593i
$36$		0			0
$37$			8.31551i			1.36706i	0.729921	+	0.683531i	$0.239557\pi$
$37$			8.31551i			1.36706i	−0.729921	+	0.683531i	$0.760443\pi$
$38$		0			0
$39$	22.5864			3.61672
$40$		0			0
$41$	−2.87785			−0.449445			−0.224722	−	0.974423i	$-0.572148\pi$
$41$	−2.87785			−0.449445			−0.224722	+	0.974423i	$0.572148\pi$
$42$		0			0
$43$			7.75318i			1.18235i	0.806544	+	0.591174i	$0.201335\pi$
$43$			7.75318i			1.18235i	−0.806544	+	0.591174i	$0.798665\pi$
$44$		0			0
$45$	11.3263	−	12.6075i	1.68843	−	1.87942i
$46$		0			0
$47$		−	8.24132i		−	1.20212i	−0.799204	−	0.601060i	$-0.794745\pi$
$47$		−	8.24132i		−	1.20212i	0.799204	−	0.601060i	$-0.205255\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−1.78827			−0.250408
$52$		0			0
$53$		−	1.81032i		−	0.248667i	−0.992241	−	0.124333i	$-0.960321\pi$
$53$		−	1.81032i		−	0.248667i	0.992241	−	0.124333i	$-0.0396792\pi$
$54$		0			0
$55$	4.31551	−	4.80366i	0.581904	−	0.647725i
$56$		0			0
$57$		−	16.2263i		−	2.14923i
$58$		0			0
$59$	−9.04790			−1.17794			−0.588968	−	0.808156i	$-0.700466\pi$
$59$	−9.04790			−1.17794			−0.588968	+	0.808156i	$0.700466\pi$
$60$		0			0
$61$	−2.11087			−0.270269			−0.135135	−	0.990827i	$-0.543147\pi$
$61$	−2.11087			−0.270269			−0.135135	+	0.990827i	$0.543147\pi$
$62$		0			0
$63$			7.57939i			0.954913i
$64$		0			0
$65$	−11.5508	−	10.3770i	−1.43270	−	1.28711i
$66$		0			0
$67$			13.1588i			1.60760i	0.594900	+	0.803800i	$0.297192\pi$
$67$			13.1588i			1.60760i	−0.594900	+	0.803800i	$0.702808\pi$
$68$		0			0
$69$	7.60479			0.915509
$70$		0			0
$71$	−3.28928			−0.390365			−0.195183	−	0.980767i	$-0.562530\pi$
$71$	−3.28928			−0.390365			−0.195183	+	0.980767i	$0.562530\pi$
$72$		0			0
$73$			2.18968i			0.256283i	0.991756	+	0.128141i	$0.0409011\pi$
$73$			2.18968i			0.256283i	−0.991756	+	0.128141i	$0.959099\pi$
$74$		0			0
$75$	−16.1701	+	1.73613i	−1.86716	+	0.200471i
$76$		0			0
$77$			2.88786i			0.329103i
$78$		0			0
$79$	2.04460			0.230036			0.115018	−	0.993363i	$-0.463307\pi$
$79$	2.04460			0.230036			0.115018	+	0.993363i	$0.463307\pi$
$80$		0			0
$81$	25.7089			2.85655
$82$		0			0
$83$		−	3.38141i		−	0.371157i	−0.982629	−	0.185579i	$-0.940584\pi$
$83$		−	3.38141i		−	0.371157i	0.982629	−	0.185579i	$-0.0594160\pi$
$84$		0			0
$85$	0.914530	+	0.821596i	0.0991947	+	0.0891146i
$86$		0			0
$87$		−	16.7240i		−	1.79300i
$88$		0			0
$89$	3.24798			0.344285			0.172143	−	0.985072i	$-0.444931\pi$
$89$	3.24798			0.344285			0.172143	+	0.985072i	$0.444931\pi$
$90$		0			0
$91$	6.94412			0.727942
$92$		0			0
$93$			17.5822i			1.82319i
$94$		0			0
$95$	−7.45496	+	8.29822i	−0.764862	+	0.851379i
$96$		0			0
$97$		−	5.22593i		−	0.530613i	−0.964164	−	0.265306i	$-0.914527\pi$
$97$		−	5.22593i		−	0.530613i	0.964164	−	0.265306i	$-0.0854731\pi$
$98$		0			0
$99$	21.8882			2.19985
$100$		0			0
$101$	−4.24594			−0.422486			−0.211243	−	0.977434i	$-0.567751\pi$
$101$	−4.24594			−0.422486			−0.211243	+	0.977434i	$0.567751\pi$
$102$		0			0
$103$			12.9175i			1.27279i	0.771361	+	0.636397i	$0.219576\pi$
$103$			12.9175i			1.27279i	−0.771361	+	0.636397i	$0.780424\pi$
$104$		0			0
$105$	4.86056	−	5.41035i	0.474342	−	0.527996i
$106$		0			0
$107$		−	9.68652i		−	0.936431i	−0.883614	−	0.468216i	$-0.844897\pi$
$107$		−	9.68652i		−	0.936431i	0.883614	−	0.468216i	$-0.155103\pi$
$108$		0			0
$109$	−0.465592			−0.0445956			−0.0222978	−	0.999751i	$-0.507098\pi$
$109$	−0.465592			−0.0445956			−0.0222978	+	0.999751i	$0.507098\pi$
$110$		0			0
$111$	27.0470			2.56719
$112$		0			0
$113$			17.3259i			1.62988i	0.579543	+	0.814942i	$0.303231\pi$
$113$			17.3259i			1.62988i	−0.579543	+	0.814942i	$0.696769\pi$
$114$		0			0
$115$	−3.88913	−	3.49392i	−0.362663	−	0.325809i
$116$		0			0
$117$		−	52.6322i		−	4.86585i
$118$		0			0
$119$	−0.549797			−0.0503998
$120$		0			0
$121$	−2.66024			−0.241840
$122$		0			0
$123$			9.36049i			0.844007i
$124$		0			0
$125$	9.06709	+	6.54125i	0.810985	+	0.585067i
$126$		0			0
$127$			18.2038i			1.61532i	0.589647	+	0.807661i	$0.299267\pi$
$127$			18.2038i			1.61532i	−0.589647	+	0.807661i	$0.700733\pi$
$128$		0			0
$129$	25.2180			2.22032
$130$		0			0
$131$	13.7974			1.20548			0.602742	−	0.797936i	$-0.294075\pi$
$131$	13.7974			1.20548			0.602742	+	0.797936i	$0.294075\pi$
$132$		0			0
$133$		−	4.98872i		−	0.432577i
$134$		0			0
$135$	−24.7761	−	22.2584i	−2.13239	−	1.91570i
$136$		0			0
$137$			17.8349i			1.52374i	0.647731	+	0.761869i	$0.275718\pi$
$137$			17.8349i			1.52374i	−0.647731	+	0.761869i	$0.724282\pi$
$138$		0			0
$139$	−1.26965			−0.107690			−0.0538450	−	0.998549i	$-0.517148\pi$
$139$	−1.26965			−0.107690			−0.0538450	+	0.998549i	$0.517148\pi$
$140$		0			0
$141$	−26.8057			−2.25745
$142$		0			0
$143$		−	20.0537i		−	1.67697i
$144$		0			0
$145$	−7.68360	+	8.55272i	−0.638088	+	0.710265i
$146$		0			0
$147$			3.25260i			0.268270i
$148$		0			0
$149$	−11.3072			−0.926319			−0.463159	−	0.886275i	$-0.653284\pi$
$149$	−11.3072			−0.926319			−0.463159	+	0.886275i	$0.653284\pi$
$150$		0			0
$151$	1.54144			0.125441			0.0627205	−	0.998031i	$-0.480022\pi$
$151$	1.54144			0.125441			0.0627205	+	0.998031i	$0.480022\pi$
$152$		0			0
$153$			4.16713i			0.336892i
$154$		0			0
$155$	8.07792	−	8.99165i	0.648834	−	0.722226i
$156$		0			0
$157$			13.0133i			1.03858i	0.854600	+	0.519288i	$0.173803\pi$
$157$			13.0133i			1.03858i	−0.854600	+	0.519288i	$0.826197\pi$
$158$		0			0
$159$	−5.88824			−0.466968
$160$		0			0
$161$	2.33807			0.184265
$162$		0			0
$163$		−	16.2609i		−	1.27365i	−0.771008	−	0.636826i	$-0.780247\pi$
$163$		−	16.2609i		−	1.27365i	0.771008	−	0.636826i	$-0.219753\pi$
$164$		0			0
$165$	−15.6244	−	14.0366i	−1.21636	−	1.09275i
$166$		0			0
$167$		−	14.6732i		−	1.13544i	−0.823221	−	0.567722i	$-0.807825\pi$
$167$		−	14.6732i		−	1.13544i	0.823221	−	0.567722i	$-0.192175\pi$
$168$		0			0
$169$	−35.2208			−2.70929
$170$		0			0
$171$	−37.8115			−2.89152
$172$		0			0
$173$			3.86708i			0.294009i	0.989136	+	0.147004i	$0.0469631\pi$
$173$			3.86708i			0.294009i	−0.989136	+	0.147004i	$0.953037\pi$
$174$		0			0
$175$	−4.97143	+	0.533766i	−0.375805	+	0.0403489i
$176$		0			0
$177$			29.4292i			2.21203i
$178$		0			0
$179$	−3.95489			−0.295603			−0.147801	−	0.989017i	$-0.547220\pi$
$179$	−3.95489			−0.295603			−0.147801	+	0.989017i	$0.547220\pi$
$180$		0			0
$181$	−15.1804			−1.12835			−0.564177	−	0.825654i	$-0.690806\pi$
$181$	−15.1804			−1.12835			−0.564177	+	0.825654i	$0.690806\pi$
$182$		0			0
$183$			6.86581i			0.507536i
$184$		0			0
$185$	−13.8320	−	12.4264i	−1.01695	−	0.913606i
$186$		0			0
$187$			1.58774i			0.116107i
$188$		0			0
$189$	14.8949			1.08344
$190$		0			0
$191$	−4.89990			−0.354544			−0.177272	−	0.984162i	$-0.556727\pi$
$191$	−4.89990			−0.354544			−0.177272	+	0.984162i	$0.556727\pi$
$192$		0			0
$193$		−	4.78408i		−	0.344366i	−0.985065	−	0.172183i	$-0.944918\pi$
$193$		−	4.78408i		−	0.344366i	0.985065	−	0.172183i	$-0.0550820\pi$
$194$		0			0
$195$	−33.7523	+	37.5701i	−2.41705	+	2.69045i
$196$		0			0
$197$			16.4280i			1.17045i	0.810872	+	0.585224i	$0.198993\pi$
$197$			16.4280i			1.17045i	−0.810872	+	0.585224i	$0.801007\pi$
$198$		0			0
$199$	7.12215			0.504876			0.252438	−	0.967613i	$-0.418768\pi$
$199$	7.12215			0.504876			0.252438	+	0.967613i	$0.418768\pi$
$200$		0			0
$201$	42.8002			3.01889
$202$		0			0
$203$		−	5.14173i		−	0.360878i
$204$		0			0
$205$	4.30055	−	4.78700i	0.300364	−	0.334339i
$206$		0			0
$207$		−	17.7211i		−	1.23170i
$208$		0			0
$209$	−14.4068			−0.996536
$210$		0			0
$211$	−24.4793			−1.68523			−0.842613	−	0.538519i	$-0.818984\pi$
$211$	−24.4793			−1.68523			−0.842613	+	0.538519i	$0.818984\pi$
$212$		0			0
$213$			10.6987i			0.733062i
$214$		0			0
$215$	−12.8966	−	11.5860i	−0.879540	−	0.790162i
$216$		0			0
$217$			5.40560i			0.366956i
$218$		0			0
$219$	7.12215			0.481270
$220$		0			0
$221$	3.81786			0.256817
$222$		0			0
$223$		−	8.09878i		−	0.542335i	−0.962532	−	0.271167i	$-0.912590\pi$
$223$		−	8.09878i		−	0.542335i	0.962532	−	0.271167i	$-0.0874096\pi$
$224$		0			0
$225$	4.04562	+	37.6804i	0.269708	+	2.51203i
$226$		0			0
$227$		−	10.0546i		−	0.667345i	−0.942689	−	0.333672i	$-0.891712\pi$
$227$		−	10.0546i		−	0.667345i	0.942689	−	0.333672i	$-0.108288\pi$
$228$		0			0
$229$	−1.58312			−0.104616			−0.0523079	−	0.998631i	$-0.516658\pi$
$229$	−1.58312			−0.104616			−0.0523079	+	0.998631i	$0.516658\pi$
$230$		0			0
$231$	9.39306			0.618018
$232$		0			0
$233$			0.0691827i			0.00453231i	0.999997	+	0.00226615i	$0.000721340\pi$
$233$			0.0691827i			0.00453231i	−0.999997	+	0.00226615i	$0.999279\pi$
$234$		0			0
$235$	13.7086	+	12.3155i	0.894249	+	0.803376i
$236$		0			0
$237$		−	6.65027i		−	0.431982i
$238$		0			0
$239$	20.8212			1.34681			0.673406	−	0.739273i	$-0.264831\pi$
$239$	20.8212			1.34681			0.673406	+	0.739273i	$0.264831\pi$
$240$		0			0
$241$	−6.20172			−0.399488			−0.199744	−	0.979848i	$-0.564011\pi$
$241$	−6.20172			−0.399488			−0.199744	+	0.979848i	$0.564011\pi$
$242$		0			0
$243$		−	38.9361i		−	2.49775i
$244$		0			0
$245$	1.49436	−	1.66340i	0.0954713	−	0.106270i
$246$		0			0
$247$			34.6423i			2.20424i
$248$		0			0
$249$	−10.9983			−0.696992
$250$		0			0
$251$	−15.9924			−1.00943			−0.504716	−	0.863285i	$-0.668403\pi$
$251$	−15.9924			−1.00943			−0.504716	+	0.863285i	$0.668403\pi$
$252$		0			0
$253$		−	6.75202i		−	0.424496i
$254$		0			0
$255$	2.67232	−	2.97460i	0.167347	−	0.186277i
$256$		0			0
$257$		−	22.4413i		−	1.39985i	−0.714215	−	0.699926i	$-0.753216\pi$
$257$		−	22.4413i		−	1.39985i	0.714215	−	0.699926i	$-0.246784\pi$
$258$		0			0
$259$	8.31551			0.516701
$260$		0			0
$261$	−38.9711			−2.41225
$262$		0			0
$263$		−	5.30716i		−	0.327254i	−0.986522	−	0.163627i	$-0.947681\pi$
$263$		−	5.30716i		−	0.327254i	0.986522	−	0.163627i	$-0.0523193\pi$
$264$		0			0
$265$	3.01128	+	2.70527i	0.184981	+	0.166184i
$266$		0			0
$267$		−	10.5644i		−	0.646529i
$268$		0			0
$269$	−25.2030			−1.53665			−0.768327	−	0.640058i	$-0.778910\pi$
$269$	−25.2030			−1.53665			−0.768327	+	0.640058i	$0.778910\pi$
$270$		0			0
$271$	−6.68945			−0.406355			−0.203178	−	0.979142i	$-0.565127\pi$
$271$	−6.68945			−0.406355			−0.203178	+	0.979142i	$0.565127\pi$
$272$		0			0
$273$		−	22.5864i		−	1.36699i
$274$		0			0
$275$	1.54144	+	14.3568i	0.0929526	+	0.865748i
$276$		0			0
$277$			19.2001i			1.15362i	0.816878	+	0.576810i	$0.195703\pi$
$277$			19.2001i			1.15362i	−0.816878	+	0.576810i	$0.804297\pi$
$278$		0			0
$279$	40.9711			2.45288
$280$		0			0
$281$	9.89122			0.590060			0.295030	−	0.955488i	$-0.404670\pi$
$281$	9.89122			0.590060			0.295030	+	0.955488i	$0.404670\pi$
$282$		0			0
$283$		−	4.96915i		−	0.295385i	−0.989033	−	0.147693i	$-0.952815\pi$
$283$		−	4.96915i		−	0.295385i	0.989033	−	0.147693i	$-0.0471846\pi$
$284$		0			0
$285$	26.9908	+	24.2480i	1.59879	+	1.43633i
$286$		0			0
$287$			2.87785i			0.169874i
$288$		0			0
$289$	16.6977			0.982219
$290$		0			0
$291$	−16.9978			−0.996431
$292$		0			0
$293$		−	2.47071i		−	0.144340i	−0.997392	−	0.0721702i	$-0.977008\pi$
$293$		−	2.47071i		−	0.144340i	0.997392	−	0.0721702i	$-0.0229925\pi$
$294$		0			0
$295$	13.5208	−	15.0502i	0.787214	−	0.876259i
$296$		0			0
$297$		−	43.0144i		−	2.49595i
$298$		0			0
$299$	−16.2358			−0.938941
$300$		0			0
$301$	7.75318			0.446886
$302$		0			0
$303$			13.8103i			0.793382i
$304$		0			0
$305$	3.15441	−	3.51121i	0.180621	−	0.201051i
$306$		0			0
$307$		−	15.4292i		−	0.880593i	−0.897852	−	0.440296i	$-0.854873\pi$
$307$		−	15.4292i		−	0.880593i	0.897852	−	0.440296i	$-0.145127\pi$
$308$		0			0
$309$	42.0153			2.39017
$310$		0			0
$311$	−24.8169			−1.40724			−0.703619	−	0.710578i	$-0.748434\pi$
$311$	−24.8169			−1.40724			−0.703619	+	0.710578i	$0.748434\pi$
$312$		0			0
$313$		−	2.00873i		−	0.113540i	−0.998387	−	0.0567700i	$-0.981920\pi$
$313$		−	2.00873i		−	0.113540i	0.998387	−	0.0567700i	$-0.0180802\pi$
$314$		0			0
$315$	−12.6075	−	11.3263i	−0.710353	−	0.638167i
$316$		0			0
$317$			11.2442i			0.631535i	0.948837	+	0.315768i	$0.102262\pi$
$317$			11.2442i			0.631535i	−0.948837	+	0.315768i	$0.897738\pi$
$318$		0			0
$319$	−14.8486			−0.831362
$320$		0			0
$321$	−31.5064			−1.75851
$322$		0			0
$323$		−	2.74279i		−	0.152613i
$324$		0			0
$325$	34.5222	−	3.70654i	1.91495	−	0.205602i
$326$		0			0
$327$			1.51438i			0.0837456i
$328$		0			0
$329$	−8.24132			−0.454359
$330$		0			0
$331$	17.7990			0.978323			0.489162	−	0.872193i	$-0.337303\pi$
$331$	17.7990			0.978323			0.489162	+	0.872193i	$0.337303\pi$
$332$		0			0
$333$		−	63.0265i		−	3.45383i
$334$		0			0
$335$	−21.8882	−	19.6640i	−1.19588	−	1.07436i
$336$		0			0
$337$		−	9.47429i		−	0.516097i	−0.966132	−	0.258049i	$-0.916921\pi$
$337$		−	9.47429i		−	0.516097i	0.966132	−	0.258049i	$-0.0830795\pi$
$338$		0			0
$339$	56.3542			3.06074
$340$		0			0
$341$	15.6106			0.845363
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	−11.3643	+	12.6498i	−0.611834	+	0.681041i
$346$		0			0
$347$			0.0984387i			0.00528447i	0.999997	+	0.00264223i	$0.000841050\pi$
$347$			0.0984387i			0.00528447i	−0.999997	+	0.00264223i	$0.999159\pi$
$348$		0			0
$349$	−3.00875			−0.161055			−0.0805273	−	0.996752i	$-0.525660\pi$
$349$	−3.00875			−0.161055			−0.0805273	+	0.996752i	$0.525660\pi$
$350$		0			0
$351$	−103.432			−5.52079
$352$		0			0
$353$		−	24.7556i		−	1.31761i	−0.752315	−	0.658803i	$-0.771063\pi$
$353$		−	24.7556i		−	1.31761i	0.752315	−	0.658803i	$-0.228937\pi$
$354$		0			0
$355$	4.91537	−	5.47136i	0.260881	−	0.290390i
$356$		0			0
$357$			1.78827i			0.0946452i
$358$		0			0
$359$	−28.5167			−1.50506			−0.752528	−	0.658560i	$-0.771166\pi$
$359$	−28.5167			−1.50506			−0.752528	+	0.658560i	$0.771166\pi$
$360$		0			0
$361$	5.88736			0.309861
$362$		0			0
$363$			8.65269i			0.454149i
$364$		0			0
$365$	−3.64230	−	3.27217i	−0.190647	−	0.171273i
$366$		0			0
$367$		−	0.444045i		−	0.0231790i	−0.999933	−	0.0115895i	$-0.996311\pi$
$367$		−	0.444045i		−	0.0231790i	0.999933	−	0.0115895i	$-0.00368913\pi$
$368$		0			0
$369$	21.8124			1.13551
$370$		0			0
$371$	−1.81032			−0.0939871
$372$		0			0
$373$		−	5.16929i		−	0.267656i	−0.991005	−	0.133828i	$-0.957273\pi$
$373$		−	5.16929i		−	0.267656i	0.991005	−	0.133828i	$-0.0427269\pi$
$374$		0			0
$375$	21.2760	−	29.4916i	1.09869	−	1.52294i
$376$		0			0
$377$			35.7048i			1.83889i
$378$		0			0
$379$	29.8632			1.53397			0.766984	−	0.641667i	$-0.221757\pi$
$379$	29.8632			1.53397			0.766984	+	0.641667i	$0.221757\pi$
$380$		0			0
$381$	59.2095			3.03339
$382$		0			0
$383$			26.3117i			1.34447i	0.740340	+	0.672233i	$0.234664\pi$
$383$			26.3117i			1.34447i	−0.740340	+	0.672233i	$0.765336\pi$
$384$		0			0
$385$	−4.80366	−	4.31551i	−0.244817	−	0.219939i
$386$		0			0
$387$		−	58.7643i		−	2.98716i
$388$		0			0
$389$	28.6632			1.45328			0.726640	−	0.687019i	$-0.241081\pi$
$389$	28.6632			1.45328			0.726640	+	0.687019i	$0.241081\pi$
$390$		0			0
$391$	1.28546			0.0650087
$392$		0			0
$393$		−	44.8774i		−	2.26376i
$394$		0			0
$395$	−3.05538	+	3.40098i	−0.153733	+	0.171122i
$396$		0			0
$397$			5.18842i			0.260399i	0.991488	+	0.130200i	$0.0415618\pi$
$397$			5.18842i			0.260399i	−0.991488	+	0.130200i	$0.958438\pi$
$398$		0			0
$399$	−16.2263			−0.812331
$400$		0			0
$401$	4.94367			0.246875			0.123438	−	0.992352i	$-0.460608\pi$
$401$	4.94367			0.246875			0.123438	+	0.992352i	$0.460608\pi$
$402$		0			0
$403$		−	37.5371i		−	1.86986i
$404$		0			0
$405$	−38.4185	+	42.7641i	−1.90903	+	2.12497i
$406$		0			0
$407$		−	24.0141i		−	1.19033i
$408$		0			0
$409$	24.3534			1.20420			0.602100	−	0.798421i	$-0.294331\pi$
$409$	24.3534			1.20420			0.602100	+	0.798421i	$0.294331\pi$
$410$		0			0
$411$	58.0098			2.86141
$412$		0			0
$413$			9.04790i			0.445218i
$414$		0			0
$415$	5.62461	+	5.05304i	0.276101	+	0.248044i
$416$		0			0
$417$			4.12965i			0.202230i
$418$		0			0
$419$	−23.5172			−1.14889			−0.574446	−	0.818543i	$-0.694782\pi$
$419$	−23.5172			−1.14889			−0.574446	+	0.818543i	$0.694782\pi$
$420$		0			0
$421$	25.8853			1.26157			0.630785	−	0.775957i	$-0.282733\pi$
$421$	25.8853			1.26157			0.630785	+	0.775957i	$0.282733\pi$
$422$		0			0
$423$			62.4642i			3.03711i
$424$		0			0
$425$	−2.73328	+	0.293463i	−0.132583	+	0.0142351i
$426$		0			0
$427$			2.11087i			0.102152i
$428$		0			0
$429$	−65.2265			−3.14917
$430$		0			0
$431$	−3.00038			−0.144523			−0.0722615	−	0.997386i	$-0.523022\pi$
$431$	−3.00038			−0.144523			−0.0722615	+	0.997386i	$0.523022\pi$
$432$		0			0
$433$		−	34.8640i		−	1.67546i	−0.546086	−	0.837729i	$-0.683883\pi$
$433$		−	34.8640i		−	1.67546i	0.546086	−	0.837729i	$-0.316117\pi$
$434$		0			0
$435$	27.8186	+	24.9916i	1.33380	+	1.19826i
$436$		0			0
$437$			11.6640i			0.557963i
$438$		0			0
$439$	23.5314			1.12309			0.561547	−	0.827445i	$-0.310206\pi$
$439$	23.5314			1.12309			0.561547	+	0.827445i	$0.310206\pi$
$440$		0			0
$441$	7.57939			0.360923
$442$		0			0
$443$			20.8761i			0.991852i	0.868365	+	0.495926i	$0.165171\pi$
$443$			20.8761i			0.991852i	−0.868365	+	0.495926i	$0.834829\pi$
$444$		0			0
$445$	−4.85366	+	5.40268i	−0.230085	+	0.256111i
$446$		0			0
$447$			36.7776i			1.73952i
$448$		0			0
$449$	13.6977			0.646436			0.323218	−	0.946325i	$-0.395235\pi$
$449$	13.6977			0.646436			0.323218	+	0.946325i	$0.395235\pi$
$450$		0			0
$451$	8.31085			0.391342
$452$		0			0
$453$		−	5.01370i		−	0.235564i
$454$		0			0
$455$	−10.3770	+	11.5508i	−0.486483	+	0.541511i
$456$		0			0
$457$			35.4159i			1.65668i	0.560223	+	0.828342i	$0.310716\pi$
$457$			35.4159i			1.65668i	−0.560223	+	0.828342i	$0.689284\pi$
$458$		0			0
$459$	8.18918			0.382238
$460$		0			0
$461$	6.01582			0.280185			0.140092	−	0.990138i	$-0.455260\pi$
$461$	6.01582			0.280185			0.140092	+	0.990138i	$0.455260\pi$
$462$		0			0
$463$			22.4759i			1.04455i	0.852779	+	0.522273i	$0.174916\pi$
$463$			22.4759i			1.04455i	−0.852779	+	0.522273i	$0.825084\pi$
$464$		0			0
$465$	−29.2462	−	26.2742i	−1.35626	−	1.21844i
$466$		0			0
$467$			34.3196i			1.58812i	0.607837	+	0.794062i	$0.292037\pi$
$467$			34.3196i			1.58812i	−0.607837	+	0.794062i	$0.707963\pi$
$468$		0			0
$469$	13.1588			0.607616
$470$		0			0
$471$	42.3271			1.95033
$472$		0			0
$473$		−	22.3901i		−	1.02950i
$474$		0			0
$475$	−2.66281	−	24.8011i	−0.122178	−	1.13795i
$476$		0			0
$477$			13.7211i			0.628247i
$478$		0			0
$479$	28.5059			1.30247			0.651235	−	0.758876i	$-0.274251\pi$
$479$	28.5059			1.30247			0.651235	+	0.758876i	$0.274251\pi$
$480$		0			0
$481$	−57.7439			−2.63290
$482$		0			0
$483$		−	7.60479i		−	0.346030i
$484$		0			0
$485$	8.69279	+	7.80943i	0.394719	+	0.354608i
$486$		0			0
$487$			2.04791i			0.0927998i	0.998923	+	0.0463999i	$0.0147748\pi$
$487$			2.04791i			0.0927998i	−0.998923	+	0.0463999i	$0.985225\pi$
$488$		0			0
$489$	−52.8901			−2.39178
$490$		0			0
$491$	−26.1659			−1.18085			−0.590424	−	0.807093i	$-0.701040\pi$
$491$	−26.1659			−1.18085			−0.590424	+	0.807093i	$0.701040\pi$
$492$		0			0
$493$		−	2.82691i		−	0.127318i
$494$		0			0
$495$	−32.7089	+	36.4088i	−1.47016	+	1.63645i
$496$		0			0
$497$			3.28928i			0.147544i
$498$		0			0
$499$	27.9282			1.25024			0.625118	−	0.780530i	$-0.285051\pi$
$499$	27.9282			1.25024			0.625118	+	0.780530i	$0.285051\pi$
$500$		0			0
$501$	−47.7259			−2.13223
$502$		0			0
$503$		−	29.9952i		−	1.33742i	−0.743522	−	0.668711i	$-0.766846\pi$
$503$		−	29.9952i		−	1.33742i	0.743522	−	0.668711i	$-0.233154\pi$
$504$		0			0
$505$	6.34496	−	7.06267i	0.282347	−	0.314285i
$506$		0			0
$507$			114.559i			5.08775i
$508$		0			0
$509$	1.67818			0.0743839			0.0371920	−	0.999308i	$-0.488159\pi$
$509$	1.67818			0.0743839			0.0371920	+	0.999308i	$0.488159\pi$
$510$		0			0
$511$	2.18968			0.0968658
$512$		0			0
$513$			74.3065i			3.28071i
$514$		0			0
$515$	−21.4868	−	19.3033i	−0.946823	−	0.850607i
$516$		0			0
$517$			23.7998i			1.04671i
$518$		0			0
$519$	12.5780			0.552115
$520$		0			0
$521$	−20.4801			−0.897250			−0.448625	−	0.893720i	$-0.648086\pi$
$521$	−20.4801			−0.897250			−0.448625	+	0.893720i	$0.648086\pi$
$522$		0			0
$523$			26.2659i			1.14853i	0.818671	+	0.574263i	$0.194711\pi$
$523$			26.2659i			1.14853i	−0.818671	+	0.574263i	$0.805289\pi$
$524$		0			0
$525$	1.73613	+	16.1701i	0.0757708	+	0.705719i
$526$		0			0
$527$			2.97198i			0.129462i
$528$		0			0
$529$	17.5334			0.762324
$530$		0			0
$531$	68.5776			2.97601
$532$		0			0
$533$		−	19.9841i		−	0.865610i
$534$		0			0
$535$	16.1125	+	14.4752i	0.696605	+	0.625816i
$536$		0			0
$537$			12.8637i			0.555109i
$538$		0			0
$539$	2.88786			0.124389
$540$		0			0
$541$	2.94671			0.126689			0.0633445	−	0.997992i	$-0.479823\pi$
$541$	2.94671			0.126689			0.0633445	+	0.997992i	$0.479823\pi$
$542$		0			0
$543$			49.3758i			2.11892i
$544$		0			0
$545$	0.695763	−	0.774464i	0.0298032	−	0.0331744i
$546$		0			0
$547$		−	27.4113i		−	1.17202i	−0.810303	−	0.586011i	$-0.800697\pi$
$547$		−	27.4113i		−	1.17202i	0.810303	−	0.586011i	$-0.199303\pi$
$548$		0			0
$549$	15.9991			0.682825
$550$		0			0
$551$	25.6506			1.09275
$552$		0			0
$553$		−	2.04460i		−	0.0869454i
$554$		0			0
$555$	−40.4180	+	44.9899i	−1.71565	+	1.90971i
$556$		0			0
$557$		−	24.8481i		−	1.05285i	−0.850222	−	0.526424i	$-0.823532\pi$
$557$		−	24.8481i		−	1.05285i	0.850222	−	0.526424i	$-0.176468\pi$
$558$		0			0
$559$	−53.8390			−2.27715
$560$		0			0
$561$	5.16428			0.218036
$562$		0			0
$563$		−	20.3969i		−	0.859625i	−0.902918	−	0.429812i	$-0.858580\pi$
$563$		−	20.3969i		−	0.859625i	0.902918	−	0.429812i	$-0.141420\pi$
$564$		0			0
$565$	−28.8198	−	25.8912i	−1.21246	−	1.08925i
$566$		0			0
$567$		−	25.7089i		−	1.07967i
$568$		0			0
$569$	−1.13166			−0.0474415			−0.0237208	−	0.999719i	$-0.507551\pi$
$569$	−1.13166			−0.0474415			−0.0237208	+	0.999719i	$0.507551\pi$
$570$		0			0
$571$	−38.4752			−1.61014			−0.805068	−	0.593183i	$-0.797871\pi$
$571$	−38.4752			−1.61014			−0.805068	+	0.593183i	$0.797871\pi$
$572$		0			0
$573$			15.9374i			0.665795i
$574$		0			0
$575$	11.6235	−	1.24798i	0.484735	−	0.0520444i
$576$		0			0
$577$		−	6.91322i		−	0.287801i	−0.989592	−	0.143901i	$-0.954035\pi$
$577$		−	6.91322i		−	0.287801i	0.989592	−	0.143901i	$-0.0459646\pi$
$578$		0			0
$579$	−15.5607			−0.646680
$580$		0			0
$581$	−3.38141			−0.140284
$582$		0			0
$583$			5.22796i			0.216520i
$584$		0			0
$585$	87.5481	+	78.6515i	3.61967	+	3.25184i
$586$		0			0
$587$		−	18.1908i		−	0.750816i	−0.926860	−	0.375408i	$-0.877503\pi$
$587$		−	18.1908i		−	0.750816i	0.926860	−	0.375408i	$-0.122497\pi$
$588$		0			0
$589$	−26.9670			−1.11116
$590$		0			0
$591$	53.4337			2.19797
$592$		0			0
$593$		−	22.4130i		−	0.920390i	−0.887818	−	0.460195i	$-0.847780\pi$
$593$		−	22.4130i		−	0.920390i	0.887818	−	0.460195i	$-0.152220\pi$
$594$		0			0
$595$	0.821596	−	0.914530i	0.0336822	−	0.0374921i
$596$		0			0
$597$		−	23.1655i		−	0.948100i
$598$		0			0
$599$	−1.30435			−0.0532941			−0.0266471	−	0.999645i	$-0.508483\pi$
$599$	−1.30435			−0.0532941			−0.0266471	+	0.999645i	$0.508483\pi$
$600$		0			0
$601$	12.8453			0.523971			0.261985	−	0.965072i	$-0.415623\pi$
$601$	12.8453			0.523971			0.261985	+	0.965072i	$0.415623\pi$
$602$		0			0
$603$		−	99.7354i		−	4.06154i
$604$		0			0
$605$	3.97536	−	4.42503i	0.161621	−	0.179903i
$606$		0			0
$607$		−	34.6132i		−	1.40491i	−0.711730	−	0.702453i	$-0.752088\pi$
$607$		−	34.6132i		−	1.40491i	0.711730	−	0.702453i	$-0.247912\pi$
$608$		0			0
$609$	−16.7240			−0.677689
$610$		0			0
$611$	57.2287			2.31523
$612$		0			0
$613$			5.53685i			0.223631i	0.993729	+	0.111816i	$0.0356666\pi$
$613$			5.53685i			0.223631i	−0.993729	+	0.111816i	$0.964333\pi$
$614$		0			0
$615$	−15.5702	−	13.9880i	−0.627851	−	0.564049i
$616$		0			0
$617$			1.35593i			0.0545876i	0.999627	+	0.0272938i	$0.00868897\pi$
$617$			1.35593i			0.0545876i	−0.999627	+	0.0272938i	$0.991311\pi$
$618$		0			0
$619$	0.0467467			0.00187891			0.000939455	−	1.00000i	$-0.499701\pi$
$619$	0.0467467			0.00187891			0.000939455		1.00000i	$0.499701\pi$
$620$		0			0
$621$	−34.8253			−1.39749
$622$		0			0
$623$		−	3.24798i		−	0.130128i
$624$		0			0
$625$	−24.4302	+	5.30716i	−0.977207	+	0.212286i
$626$		0			0
$627$			46.8594i			1.87138i
$628$		0			0
$629$	4.57185			0.182292
$630$		0			0
$631$	−6.71412			−0.267285			−0.133642	−	0.991030i	$-0.542667\pi$
$631$	−6.71412			−0.267285			−0.133642	+	0.991030i	$0.542667\pi$
$632$		0			0
$633$			79.6214i			3.16467i
$634$		0			0
$635$	−30.2800	−	27.2030i	−1.20163	−	1.07952i
$636$		0			0
$637$		−	6.94412i		−	0.275136i
$638$		0			0
$639$	24.9307			0.986243
$640$		0			0
$641$	−14.1605			−0.559306			−0.279653	−	0.960101i	$-0.590219\pi$
$641$	−14.1605			−0.559306			−0.279653	+	0.960101i	$0.590219\pi$
$642$		0			0
$643$			17.6511i			0.696092i	0.937477	+	0.348046i	$0.113155\pi$
$643$			17.6511i			0.696092i	−0.937477	+	0.348046i	$0.886845\pi$
$644$		0			0
$645$	−37.6847	+	41.9474i	−1.48384	+	1.65168i
$646$		0			0
$647$		−	3.14217i		−	0.123532i	−0.998091	−	0.0617658i	$-0.980327\pi$
$647$		−	3.14217i		−	0.123532i	0.998091	−	0.0617658i	$-0.0196732\pi$
$648$		0			0
$649$	26.1291			1.02566
$650$		0			0
$651$	17.5822			0.689102
$652$		0			0
$653$		−	4.85733i		−	0.190082i	−0.995473	−	0.0950411i	$-0.969702\pi$
$653$		−	4.85733i		−	0.190082i	0.995473	−	0.0950411i	$-0.0302982\pi$
$654$		0			0
$655$	−20.6183	+	22.9505i	−0.805623	+	0.896751i
$656$		0			0
$657$		−	16.5964i		−	0.647489i
$658$		0			0
$659$	−37.2495			−1.45104			−0.725518	−	0.688204i	$-0.758400\pi$
$659$	−37.2495			−1.45104			−0.725518	+	0.688204i	$0.758400\pi$
$660$		0			0
$661$	26.9869			1.04967			0.524835	−	0.851204i	$-0.324127\pi$
$661$	26.9869			1.04967			0.524835	+	0.851204i	$0.324127\pi$
$662$		0			0
$663$		−	12.4180i		−	0.482273i
$664$		0			0
$665$	8.29822	+	7.45496i	0.321791	+	0.289091i
$666$		0			0
$667$			12.0217i			0.465482i
$668$		0			0
$669$	−26.3421			−1.01844
$670$		0			0
$671$	6.09591			0.235330
$672$		0			0
$673$		−	22.2835i		−	0.858964i	−0.903075	−	0.429482i	$-0.858696\pi$
$673$		−	22.2835i		−	0.858964i	0.903075	−	0.429482i	$-0.141304\pi$
$674$		0			0
$675$	74.0489	−	7.95039i	2.85014	−	0.306011i
$676$		0			0
$677$			26.7857i			1.02946i	0.857353	+	0.514730i	$0.172108\pi$
$677$			26.7857i			1.02946i	−0.857353	+	0.514730i	$0.827892\pi$
$678$		0			0
$679$	−5.22593			−0.200553
$680$		0			0
$681$	−32.7034			−1.25320
$682$		0			0
$683$		−	45.4881i		−	1.74055i	−0.492563	−	0.870277i	$-0.663940\pi$
$683$		−	45.4881i		−	1.74055i	0.492563	−	0.870277i	$-0.336060\pi$
$684$		0			0
$685$	−29.6665	−	26.6518i	−1.13350	−	1.01831i
$686$		0			0
$687$			5.14927i			0.196457i
$688$		0			0
$689$	12.5711			0.478920
$690$		0			0
$691$	17.8566			0.679296			0.339648	−	0.940553i	$-0.389692\pi$
$691$	17.8566			0.679296			0.339648	+	0.940553i	$0.389692\pi$
$692$		0			0
$693$		−	21.8882i		−	0.831465i
$694$		0			0
$695$	1.89731	−	2.11192i	0.0719691	−	0.0801098i
$696$		0			0
$697$			1.58224i			0.0599314i
$698$		0			0
$699$	0.225023			0.00851117
$700$		0			0
$701$	−21.4385			−0.809721			−0.404860	−	0.914378i	$-0.632680\pi$
$701$	−21.4385			−0.809721			−0.404860	+	0.914378i	$0.632680\pi$
$702$		0			0
$703$			41.4838i			1.56459i
$704$		0			0
$705$	40.0574	−	44.5885i	1.50865	−	1.67930i
$706$		0			0
$707$			4.24594i			0.159685i
$708$		0			0
$709$	36.2421			1.36110			0.680550	−	0.732702i	$-0.261741\pi$
$709$	36.2421			1.36110			0.680550	+	0.732702i	$0.261741\pi$
$710$		0			0
$711$	−15.4968			−0.581177
$712$		0			0
$713$		−	12.6387i		−	0.473321i
$714$		0			0
$715$	33.3572	+	29.9674i	1.24749	+	1.12072i
$716$		0			0
$717$		−	67.7230i		−	2.52916i
$718$		0			0
$719$	4.91195			0.183185			0.0915924	−	0.995797i	$-0.470804\pi$
$719$	4.91195			0.183185			0.0915924	+	0.995797i	$0.470804\pi$
$720$		0			0
$721$	12.9175			0.481071
$722$		0			0
$723$			20.1717i			0.750193i
$724$		0			0
$725$	−2.74448	−	25.5617i	−0.101927	−	0.949338i
$726$		0			0
$727$		−	4.47341i		−	0.165910i	−0.996553	−	0.0829548i	$-0.973564\pi$
$727$		−	4.47341i		−	0.165910i	0.996553	−	0.0829548i	$-0.0264357\pi$
$728$		0			0
$729$	−49.5167			−1.83395
$730$		0			0
$731$	4.26268			0.157661
$732$		0			0
$733$		−	30.0754i		−	1.11086i	−0.831563	−	0.555430i	$-0.812554\pi$
$733$		−	30.0754i		−	1.11086i	0.831563	−	0.555430i	$-0.187446\pi$
$734$		0			0
$735$	−5.41035	−	4.86056i	−0.199564	−	0.179284i
$736$		0			0
$737$		−	38.0008i		−	1.39978i
$738$		0			0
$739$	43.0252			1.58271			0.791355	−	0.611357i	$-0.209376\pi$
$739$	43.0252			1.58271			0.791355	+	0.611357i	$0.209376\pi$
$740$		0			0
$741$	112.677			4.13931
$742$		0			0
$743$			10.0854i			0.369997i	0.982739	+	0.184999i	$0.0592281\pi$
$743$			10.0854i			0.369997i	−0.982739	+	0.184999i	$0.940772\pi$
$744$		0			0
$745$	16.8970	−	18.8083i	0.619058	−	0.689082i
$746$		0			0
$747$			25.6290i			0.937715i
$748$		0			0
$749$	−9.68652			−0.353938
$750$		0			0
$751$	−13.2964			−0.485192			−0.242596	−	0.970127i	$-0.577999\pi$
$751$	−13.2964			−0.485192			−0.242596	+	0.970127i	$0.577999\pi$
$752$		0			0
$753$			52.0169i			1.89560i
$754$		0			0
$755$	−2.30348	+	2.56403i	−0.0838321	+	0.0933146i
$756$		0			0
$757$			49.0007i			1.78096i	0.455023	+	0.890480i	$0.349631\pi$
$757$			49.0007i			1.78096i	−0.455023	+	0.890480i	$0.650369\pi$
$758$		0			0
$759$	−21.9616			−0.797155
$760$		0			0
$761$	46.9511			1.70198			0.850988	−	0.525185i	$-0.176004\pi$
$761$	46.9511			1.70198			0.850988	+	0.525185i	$0.176004\pi$
$762$		0			0
$763$			0.465592i			0.0168556i
$764$		0			0
$765$	−6.93158	−	6.22720i	−0.250612	−	0.225145i
$766$		0			0
$767$		−	62.8297i		−	2.26865i
$768$		0			0
$769$	47.5364			1.71420			0.857102	−	0.515146i	$-0.172262\pi$
$769$	47.5364			1.71420			0.857102	+	0.515146i	$0.172262\pi$
$770$		0			0
$771$	−72.9927			−2.62877
$772$		0			0
$773$			41.3651i			1.48780i	0.668292	+	0.743899i	$0.267026\pi$
$773$			41.3651i			1.48780i	−0.668292	+	0.743899i	$0.732974\pi$
$774$		0			0
$775$	2.88533	+	26.8735i	0.103644	+	0.965326i
$776$		0			0
$777$		−	27.0470i		−	0.970306i
$778$		0			0
$779$	−14.3568			−0.514386
$780$		0			0
$781$	9.49898			0.339900
$782$		0			0
$783$			76.5855i			2.73694i
$784$		0			0
$785$	−21.6463	−	19.4466i	−0.772589	−	0.694079i
$786$		0			0
$787$		−	26.2363i		−	0.935222i	−0.883934	−	0.467611i	$-0.845115\pi$
$787$		−	26.2363i		−	0.935222i	0.883934	−	0.467611i	$-0.154885\pi$
$788$		0			0
$789$	−17.2621			−0.614545
$790$		0			0
$791$	17.3259			0.616038
$792$		0			0
$793$		−	14.6581i		−	0.520526i
$794$		0			0
$795$	8.79916	−	9.79447i	0.312074	−	0.347374i
$796$		0			0
$797$			10.0462i			0.355856i	0.984043	+	0.177928i	$0.0569395\pi$
$797$			10.0462i			0.355856i	−0.984043	+	0.177928i	$0.943061\pi$
$798$		0			0
$799$	−4.53106			−0.160297
$800$		0			0
$801$	−24.6177			−0.869824
$802$		0			0
$803$		−	6.32350i		−	0.223151i
$804$		0			0
$805$	−3.49392	+	3.88913i	−0.123144	+	0.137074i
$806$		0			0
$807$			81.9752i			2.88566i
$808$		0			0
$809$	48.6717			1.71120			0.855602	−	0.517634i	$-0.173187\pi$
$809$	48.6717			1.71120			0.855602	+	0.517634i	$0.173187\pi$
$810$		0			0
$811$	16.4792			0.578664			0.289332	−	0.957229i	$-0.406567\pi$
$811$	16.4792			0.578664			0.289332	+	0.957229i	$0.406567\pi$
$812$		0			0
$813$			21.7581i			0.763090i
$814$		0			0
$815$	27.0483	+	24.2997i	0.947461	+	0.851180i
$816$		0			0
$817$			38.6784i			1.35319i
$818$		0			0
$819$	−52.6322			−1.83912
$820$		0			0
$821$	12.0845			0.421750			0.210875	−	0.977513i	$-0.432369\pi$
$821$	12.0845			0.421750			0.210875	+	0.977513i	$0.432369\pi$
$822$		0			0
$823$		−	19.0037i		−	0.662427i	−0.943556	−	0.331213i	$-0.892542\pi$
$823$		−	19.0037i		−	0.662427i	0.943556	−	0.331213i	$-0.107458\pi$
$824$		0			0
$825$	46.6969	−	5.01370i	1.62578	−	0.174555i
$826$		0			0
$827$			55.9890i			1.94693i	0.228840	+	0.973464i	$0.426507\pi$
$827$			55.9890i			1.94693i	−0.228840	+	0.973464i	$0.573493\pi$
$828$		0			0
$829$	−50.1242			−1.74088			−0.870442	−	0.492271i	$-0.836167\pi$
$829$	−50.1242			−1.74088			−0.870442	+	0.492271i	$0.836167\pi$
$830$		0			0
$831$	62.4501			2.16637
$832$		0			0
$833$			0.549797i			0.0190494i
$834$		0			0
$835$	24.4073	+	21.9270i	0.844648	+	0.758815i
$836$		0			0
$837$		−	80.5159i		−	2.78304i
$838$		0			0
$839$	−35.4505			−1.22389			−0.611943	−	0.790902i	$-0.709612\pi$
$839$	−35.4505			−1.22389			−0.611943	+	0.790902i	$0.709612\pi$
$840$		0			0
$841$	−2.56266			−0.0883675
$842$		0			0
$843$		−	32.1721i		−	1.10807i
$844$		0			0
$845$	52.6326	−	58.5861i	1.81062	−	2.01542i
$846$		0			0
$847$			2.66024i			0.0914070i
$848$		0			0
$849$	−16.1626			−0.554700
$850$		0			0
$851$	−19.4422			−0.666471
$852$		0			0
$853$			20.7224i			0.709521i	0.934957	+	0.354760i	$0.115438\pi$
$853$			20.7224i			0.709521i	−0.934957	+	0.354760i	$0.884562\pi$
$854$		0			0
$855$	56.5040	−	62.8954i	1.93240	−	2.15098i
$856$		0			0
$857$			22.8881i			0.781843i	0.920424	+	0.390922i	$0.127844\pi$
$857$			22.8881i			0.781843i	−0.920424	+	0.390922i	$0.872156\pi$
$858$		0			0
$859$	24.8011			0.846202			0.423101	−	0.906083i	$-0.360942\pi$
$859$	24.8011			0.846202			0.423101	+	0.906083i	$0.360942\pi$
$860$		0			0
$861$	9.36049			0.319005
$862$		0			0
$863$		−	42.3316i		−	1.44099i	−0.693463	−	0.720493i	$-0.743916\pi$
$863$		−	42.3316i		−	1.44099i	0.693463	−	0.720493i	$-0.256084\pi$
$864$		0			0
$865$	−6.43248	−	5.77881i	−0.218711	−	0.196486i
$866$		0			0
$867$		−	54.3110i		−	1.84450i
$868$		0			0
$869$	−5.90454			−0.200298
$870$		0			0
$871$	−91.3761			−3.09616
$872$		0			0
$873$			39.6094i			1.34057i
$874$		0			0
$875$	6.54125	−	9.06709i	0.221135	−	0.306524i
$876$		0			0
$877$		−	40.7056i		−	1.37453i	−0.726407	−	0.687265i	$-0.758811\pi$
$877$		−	40.7056i		−	1.37453i	0.726407	−	0.687265i	$-0.241189\pi$
$878$		0			0
$879$	−8.03623			−0.271055
$880$		0			0
$881$	−26.1484			−0.880962			−0.440481	−	0.897762i	$-0.645192\pi$
$881$	−26.1484			−0.880962			−0.440481	+	0.897762i	$0.645192\pi$
$882$		0			0
$883$			44.6977i			1.50420i	0.659051	+	0.752098i	$0.270958\pi$
$883$			44.6977i			1.50420i	−0.659051	+	0.752098i	$0.729042\pi$
$884$		0			0
$885$	−48.9524	−	43.9778i	−1.64552	−	1.47830i
$886$		0			0
$887$			3.43894i			0.115468i	0.998332	+	0.0577342i	$0.0183876\pi$
$887$			3.43894i			0.115468i	−0.998332	+	0.0577342i	$0.981612\pi$
$888$		0			0
$889$	18.2038			0.610534
$890$		0			0
$891$	−74.2439			−2.48727
$892$		0			0
$893$		−	41.1137i		−	1.37582i
$894$		0			0
$895$	5.91004	−	6.57855i	0.197551	−	0.219897i
$896$		0			0
$897$			52.8086i			1.76323i
$898$		0			0
$899$	−27.7941			−0.926985
$900$		0			0
$901$	−0.995309			−0.0331585
$902$		0			0
$903$		−	25.2180i		−	0.839201i
$904$		0			0
$905$	22.6851	−	25.2511i	0.754077	−	0.839374i
$906$		0			0
$907$		−	46.5643i		−	1.54614i	−0.634320	−	0.773071i	$-0.718720\pi$
$907$		−	46.5643i		−	1.54614i	0.634320	−	0.773071i	$-0.281280\pi$
$908$		0			0
$909$	32.1816			1.06740
$910$		0			0
$911$	9.64904			0.319687			0.159843	−	0.987142i	$-0.448901\pi$
$911$	9.64904			0.319687			0.159843	+	0.987142i	$0.448901\pi$
$912$		0			0
$913$			9.76504i			0.323176i
$914$		0			0
$915$	−11.4206	−	10.2600i	−0.377552	−	0.339185i
$916$		0			0
$917$		−	13.7974i		−	0.455630i
$918$		0			0
$919$	−2.84743			−0.0939279			−0.0469640	−	0.998897i	$-0.514955\pi$
$919$	−2.84743			−0.0939279			−0.0469640	+	0.998897i	$0.514955\pi$
$920$		0			0
$921$	−50.1851			−1.65365
$922$		0			0
$923$		−	22.8411i		−	0.751825i
$924$		0			0
$925$	41.3400	−	4.43854i	1.35925	−	0.145938i
$926$		0			0
$927$		−	97.9064i		−	3.21567i
$928$		0			0
$929$	33.9290			1.11318			0.556588	−	0.830789i	$-0.312110\pi$
$929$	33.9290			1.11318			0.556588	+	0.830789i	$0.312110\pi$
$930$		0			0
$931$	−4.98872			−0.163499
$932$		0			0
$933$			80.7194i			2.64263i
$934$		0			0
$935$	−2.64104	−	2.37266i	−0.0863712	−	0.0775942i
$936$		0			0
$937$		−	48.7997i		−	1.59422i	−0.603837	−	0.797108i	$-0.706362\pi$
$937$		−	48.7997i		−	1.59422i	0.603837	−	0.797108i	$-0.293638\pi$
$938$		0			0
$939$	−6.53359			−0.213216
$940$		0			0
$941$	32.5802			1.06208			0.531042	−	0.847346i	$-0.321801\pi$
$941$	32.5802			1.06208			0.531042	+	0.847346i	$0.321801\pi$
$942$		0			0
$943$		−	6.72861i		−	0.219114i
$944$		0			0
$945$	−22.2584	+	24.7761i	−0.724065	+	0.805967i
$946$		0			0
$947$		−	13.2672i		−	0.431127i	−0.976490	−	0.215564i	$-0.930841\pi$
$947$		−	13.2672i		−	0.431127i	0.976490	−	0.215564i	$-0.0691589\pi$
$948$		0			0
$949$	−15.2054			−0.493588
$950$		0			0
$951$	36.5727			1.18595
$952$		0			0
$953$			17.4038i			0.563765i	0.959449	+	0.281883i	$0.0909588\pi$
$953$			17.4038i			0.563765i	−0.959449	+	0.281883i	$0.909041\pi$
$954$		0			0
$955$	7.32223	−	8.15047i	0.236942	−	0.263743i
$956$		0			0
$957$			48.2965i			1.56120i
$958$		0			0
$959$	17.8349			0.575919
$960$		0			0
$961$	−1.77950			−0.0574031
$962$		0			0
$963$			73.4179i			2.36586i
$964$		0			0
$965$	7.95782	+	7.14915i	0.256171	+	0.230139i
$966$		0			0
$967$			11.8019i			0.379525i	0.981830	+	0.189762i	$0.0607718\pi$
$967$			11.8019i			0.379525i	−0.981830	+	0.189762i	$0.939228\pi$
$968$		0			0
$969$	−8.92118			−0.286590
$970$		0			0
$971$	−50.5207			−1.62129			−0.810643	−	0.585541i	$-0.800882\pi$
$971$	−50.5207			−1.62129			−0.810643	+	0.585541i	$0.800882\pi$
$972$		0			0
$973$			1.26965i			0.0407030i
$974$		0			0
$975$	−12.0559	−	112.287i	−0.386097	−	3.59605i
$976$		0			0
$977$			0.572854i			0.0183272i	0.999958	+	0.00916360i	$0.00291691\pi$
$977$			0.572854i			0.0183272i	−0.999958	+	0.00916360i	$0.997083\pi$
$978$		0			0
$979$	−9.37973			−0.299777
$980$		0			0
$981$	3.52890			0.112669
$982$		0			0
$983$			26.3915i			0.841757i	0.907117	+	0.420878i	$0.138278\pi$
$983$			26.3915i			0.841757i	−0.907117	+	0.420878i	$0.861722\pi$
$984$		0			0
$985$	−27.3263	−	24.5494i	−0.870688	−	0.782209i
$986$		0			0
$987$			26.8057i			0.853235i
$988$		0			0
$989$	−18.1274			−0.576419
$990$		0			0
$991$	−47.2933			−1.50232			−0.751161	−	0.660119i	$-0.770506\pi$
$991$	−47.2933			−1.50232			−0.751161	+	0.660119i	$0.770506\pi$
$992$		0			0
$993$		−	57.8931i		−	1.83718i
$994$		0			0
$995$	−10.6431	+	11.8469i	−0.337408	+	0.375573i
$996$		0			0
$997$		−	36.4981i		−	1.15591i	−0.816070	−	0.577953i	$-0.803852\pi$
$997$		−	36.4981i		−	1.15591i	0.816070	−	0.577953i	$-0.196148\pi$
$998$		0			0
$999$	−123.859			−3.91872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.g.b.449.1	✓	10
4.3	odd	2		1120.2.g.c.449.10	yes	10
5.2	odd	4		5600.2.a.bu.1.1		5
5.3	odd	4		5600.2.a.bw.1.5		5
5.4	even	2	inner	1120.2.g.b.449.10	yes	10
8.3	odd	2		2240.2.g.n.449.1		10
8.5	even	2		2240.2.g.o.449.10		10
20.3	even	4		5600.2.a.bv.1.1		5
20.7	even	4		5600.2.a.bx.1.5		5
20.19	odd	2		1120.2.g.c.449.1	yes	10
40.19	odd	2		2240.2.g.n.449.10		10
40.29	even	2		2240.2.g.o.449.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.b.449.1	✓	10	1.1	even	1	trivial
1120.2.g.b.449.10	yes	10	5.4	even	2	inner
1120.2.g.c.449.1	yes	10	20.19	odd	2
1120.2.g.c.449.10	yes	10	4.3	odd	2
2240.2.g.n.449.1		10	8.3	odd	2
2240.2.g.n.449.10		10	40.19	odd	2
2240.2.g.o.449.1		10	40.29	even	2
2240.2.g.o.449.10		10	8.5	even	2
5600.2.a.bu.1.1		5	5.2	odd	4
5600.2.a.bv.1.1		5	20.3	even	4
5600.2.a.bw.1.5		5	5.3	odd	4
5600.2.a.bx.1.5		5	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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.25260i q^{3} +(-1.49436 + 1.66340i) q^{5} -1.00000i q^{7} -7.57939 q^{9} +O(q^{10})\)	\(q-3.25260i q^{3} +(-1.49436 + 1.66340i) q^{5} -1.00000i q^{7} -7.57939 q^{9} -2.88786 q^{11} +6.94412i q^{13} +(5.41035 + 4.86056i) q^{15} -0.549797i q^{17} +4.98872 q^{19} -3.25260 q^{21} +2.33807i q^{23} +(-0.533766 - 4.97143i) q^{25} +14.8949i q^{27} +5.14173 q^{29} -5.40560 q^{31} +9.39306i q^{33} +(1.66340 + 1.49436i) q^{35} +8.31551i q^{37} +22.5864 q^{39} -2.87785 q^{41} +7.75318i q^{43} +(11.3263 - 12.6075i) q^{45} -8.24132i q^{47} -1.00000 q^{49} -1.78827 q^{51} -1.81032i q^{53} +(4.31551 - 4.80366i) q^{55} -16.2263i q^{57} -9.04790 q^{59} -2.11087 q^{61} +7.57939i q^{63} +(-11.5508 - 10.3770i) q^{65} +13.1588i q^{67} +7.60479 q^{69} -3.28928 q^{71} +2.18968i q^{73} +(-16.1701 + 1.73613i) q^{75} +2.88786i q^{77} +2.04460 q^{79} +25.7089 q^{81} -3.38141i q^{83} +(0.914530 + 0.821596i) q^{85} -16.7240i q^{87} +3.24798 q^{89} +6.94412 q^{91} +17.5822i q^{93} +(-7.45496 + 8.29822i) q^{95} -5.22593i q^{97} +21.8882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 14 q^{9}+O(q^{10}) \) 10 * q - 2 * q^5 - 14 * q^9	\( 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65} - 8 q^{69} - 8 q^{71} - 64 q^{75} + 64 q^{79} + 2 q^{81} - 12 q^{85} - 4 q^{89} - 60 q^{95} + 80 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 14 * q^9 - 8 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 6 * q^25 + 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 + 10 * q^45 - 10 * q^49 - 24 * q^51 - 16 * q^55 + 32 * q^59 - 20 * q^61 - 8 * q^65 - 8 * q^69 - 8 * q^71 - 64 * q^75 + 64 * q^79 + 2 * q^81 - 12 * q^85 - 4 * q^89 - 60 * q^95 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more