LMFDB - Embedded newform 5520.2.be.a.1471.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1471,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))

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

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

chi := DirichletCharacter("5520.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.be (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:	$44.0774219157$
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} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} + \cdots + 64 $ x^16 - 4x^15 + 8x^14 + 28x^13 + 373x^12 - 920x^11 + 1088x^10 - 168x^9 + 16460x^8 - 45408x^7 + 62624x^6 - 18048x^5 + 2160x^4 - 1664x^3 + 6272x^2 - 896*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.1
Root			$0.0727486 - 0.0727486i$ of defining polynomial
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.a.1471.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -4.59925 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.00000i q^{5} -4.59925 q^{7} -1.00000 q^{9} +4.88207 q^{11} +1.41451 q^{13} +1.00000 q^{15} -4.35085i q^{17} -1.34013 q^{19} +4.59925i q^{21} +(0.751600 + 4.73657i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -8.47505 q^{29} +4.83021i q^{31} -4.88207i q^{33} -4.59925i q^{35} -2.55810i q^{37} -1.41451i q^{39} -1.94751 q^{41} +9.76592 q^{43} -1.00000i q^{45} +5.05444i q^{47} +14.1531 q^{49} -4.35085 q^{51} -11.0842i q^{53} +4.88207i q^{55} +1.34013i q^{57} +1.36588i q^{59} -9.11481i q^{61} +4.59925 q^{63} +1.41451i q^{65} +8.47015 q^{67} +(4.73657 - 0.751600i) q^{69} -0.721095i q^{71} +2.29450 q^{73} +1.00000i q^{75} -22.4538 q^{77} -15.2680 q^{79} +1.00000 q^{81} +13.3145 q^{83} +4.35085 q^{85} +8.47505i q^{87} +8.99269i q^{89} -6.50567 q^{91} +4.83021 q^{93} -1.34013i q^{95} +1.26729i q^{97} -4.88207 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{7} - 16 q^{9}+O(q^{10}) $ 16 * q - 8 * q^7 - 16 * q^9	$ 16 q - 8 q^{7} - 16 q^{9} + 8 q^{11} + 8 q^{13} + 16 q^{15} + 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} - 4 q^{51} + 8 q^{63} - 16 q^{67} + 40 q^{73} + 24 q^{77} + 32 q^{79} + 16 q^{81} + 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) $ 16 * q - 8 * q^7 - 16 * q^9 + 8 * q^11 + 8 * q^13 + 16 * q^15 + 12 * q^23 - 16 * q^25 - 4 * q^29 + 4 * q^41 + 20 * q^49 - 4 * q^51 + 8 * q^63 - 16 * q^67 + 40 * q^73 + 24 * q^77 + 32 * q^79 + 16 * q^81 + 4 * q^85 - 48 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1841$	$4417$	$4831$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−4.59925			−1.73835			−0.869176	−	0.494502i	$-0.835350\pi$
$7$	−4.59925			−1.73835			−0.869176	+	0.494502i	$0.835350\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	4.88207			1.47200			0.735999	−	0.676982i	$-0.236713\pi$
$11$	4.88207			1.47200			0.735999	+	0.676982i	$0.236713\pi$
$12$		0			0
$13$	1.41451			0.392314			0.196157	−	0.980573i	$-0.437154\pi$
$13$	1.41451			0.392314			0.196157	+	0.980573i	$0.437154\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$		−	4.35085i		−	1.05524i	−0.849482	−	0.527618i	$-0.823085\pi$
$17$		−	4.35085i		−	1.05524i	0.849482	−	0.527618i	$-0.176915\pi$
$18$		0			0
$19$	−1.34013			−0.307448			−0.153724	−	0.988114i	$-0.549127\pi$
$19$	−1.34013			−0.307448			−0.153724	+	0.988114i	$0.549127\pi$
$20$		0			0
$21$			4.59925i			1.00364i
$22$		0			0
$23$	0.751600	+	4.73657i	0.156719	+	0.987643i
$23$	0.751600	+	4.73657i	0.156719	+	0.987643i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−8.47505			−1.57378			−0.786888	−	0.617095i	$-0.788309\pi$
$29$	−8.47505			−1.57378			−0.786888	+	0.617095i	$0.788309\pi$
$30$		0			0
$31$			4.83021i			0.867531i	0.901026	+	0.433766i	$0.142815\pi$
$31$			4.83021i			0.867531i	−0.901026	+	0.433766i	$0.857185\pi$
$32$		0			0
$33$		−	4.88207i		−	0.849859i
$34$		0			0
$35$		−	4.59925i		−	0.777415i
$36$		0			0
$37$		−	2.55810i		−	0.420549i	−0.977642	−	0.210274i	$-0.932564\pi$
$37$		−	2.55810i		−	0.420549i	0.977642	−	0.210274i	$-0.0674357\pi$
$38$		0			0
$39$		−	1.41451i		−	0.226502i
$40$		0			0
$41$	−1.94751			−0.304150			−0.152075	−	0.988369i	$-0.548596\pi$
$41$	−1.94751			−0.304150			−0.152075	+	0.988369i	$0.548596\pi$
$42$		0			0
$43$	9.76592			1.48929			0.744644	−	0.667461i	$-0.232619\pi$
$43$	9.76592			1.48929			0.744644	+	0.667461i	$0.232619\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$			5.05444i			0.737266i	0.929575	+	0.368633i	$0.120174\pi$
$47$			5.05444i			0.737266i	−0.929575	+	0.368633i	$0.879826\pi$
$48$		0			0
$49$	14.1531			2.02187
$50$		0			0
$51$	−4.35085			−0.609241
$52$		0			0
$53$		−	11.0842i		−	1.52253i	−0.648441	−	0.761265i	$-0.724578\pi$
$53$		−	11.0842i		−	1.52253i	0.648441	−	0.761265i	$-0.275422\pi$
$54$		0			0
$55$			4.88207i			0.658298i
$56$		0			0
$57$			1.34013i			0.177505i
$58$		0			0
$59$			1.36588i			0.177822i	0.996040	+	0.0889111i	$0.0283387\pi$
$59$			1.36588i			0.177822i	−0.996040	+	0.0889111i	$0.971661\pi$
$60$		0			0
$61$		−	9.11481i		−	1.16703i	−0.812102	−	0.583516i	$-0.801676\pi$
$61$		−	9.11481i		−	1.16703i	0.812102	−	0.583516i	$-0.198324\pi$
$62$		0			0
$63$	4.59925			0.579451
$64$		0			0
$65$			1.41451i			0.175448i
$66$		0			0
$67$	8.47015			1.03479			0.517397	−	0.855746i	$-0.326901\pi$
$67$	8.47015			1.03479			0.517397	+	0.855746i	$0.326901\pi$
$68$		0			0
$69$	4.73657	−	0.751600i	0.570216	−	0.0904820i
$70$		0			0
$71$		−	0.721095i		−	0.0855783i	−0.999084	−	0.0427891i	$-0.986376\pi$
$71$		−	0.721095i		−	0.0855783i	0.999084	−	0.0427891i	$-0.0136244\pi$
$72$		0			0
$73$	2.29450			0.268551			0.134275	−	0.990944i	$-0.457129\pi$
$73$	2.29450			0.268551			0.134275	+	0.990944i	$0.457129\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	−22.4538			−2.55885
$78$		0			0
$79$	−15.2680			−1.71778			−0.858890	−	0.512160i	$-0.828846\pi$
$79$	−15.2680			−1.71778			−0.858890	+	0.512160i	$0.828846\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	13.3145			1.46146			0.730729	−	0.682667i	$-0.239180\pi$
$83$	13.3145			1.46146			0.730729	+	0.682667i	$0.239180\pi$
$84$		0			0
$85$	4.35085			0.471916
$86$		0			0
$87$			8.47505i			0.908620i
$88$		0			0
$89$			8.99269i			0.953223i	0.879114	+	0.476612i	$0.158135\pi$
$89$			8.99269i			0.953223i	−0.879114	+	0.476612i	$0.841865\pi$
$90$		0			0
$91$	−6.50567			−0.681980
$92$		0			0
$93$	4.83021			0.500869
$94$		0			0
$95$		−	1.34013i		−	0.137495i
$96$		0			0
$97$			1.26729i			0.128674i	0.997928	+	0.0643370i	$0.0204932\pi$
$97$			1.26729i			0.128674i	−0.997928	+	0.0643370i	$0.979507\pi$
$98$		0			0
$99$	−4.88207			−0.490666
$100$		0			0
$101$	−14.3919			−1.43205			−0.716025	−	0.698075i	$-0.754040\pi$
$101$	−14.3919			−1.43205			−0.716025	+	0.698075i	$0.754040\pi$
$102$		0			0
$103$	17.0569			1.68066			0.840331	−	0.542074i	$-0.182361\pi$
$103$	17.0569			1.68066			0.840331	+	0.542074i	$0.182361\pi$
$104$		0			0
$105$	−4.59925			−0.448841
$106$		0			0
$107$	−3.92835			−0.379768			−0.189884	−	0.981807i	$-0.560811\pi$
$107$	−3.92835			−0.379768			−0.189884	+	0.981807i	$0.560811\pi$
$108$		0			0
$109$			5.40581i			0.517783i	0.965906	+	0.258891i	$0.0833571\pi$
$109$			5.40581i			0.517783i	−0.965906	+	0.258891i	$0.916643\pi$
$110$		0			0
$111$	−2.55810			−0.242804
$112$		0			0
$113$		−	13.5574i		−	1.27537i	−0.770296	−	0.637686i	$-0.779892\pi$
$113$		−	13.5574i		−	1.27537i	0.770296	−	0.637686i	$-0.220108\pi$
$114$		0			0
$115$	−4.73657	+	0.751600i	−0.441687	+	0.0700870i
$116$		0			0
$117$	−1.41451			−0.130771
$118$		0			0
$119$			20.0106i			1.83437i
$120$		0			0
$121$	12.8346			1.16678
$122$		0			0
$123$			1.94751i			0.175601i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	13.2910i		−	1.17938i	−0.807628	−	0.589692i	$-0.799249\pi$
$127$		−	13.2910i		−	1.17938i	0.807628	−	0.589692i	$-0.200751\pi$
$128$		0			0
$129$		−	9.76592i		−	0.859841i
$130$		0			0
$131$		−	8.00063i		−	0.699019i	−0.936933	−	0.349509i	$-0.886348\pi$
$131$		−	8.00063i		−	0.699019i	0.936933	−	0.349509i	$-0.113652\pi$
$132$		0			0
$133$	6.16361			0.534453
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$			1.42388i			0.121650i	0.998148	+	0.0608251i	$0.0193732\pi$
$137$			1.42388i			0.121650i	−0.998148	+	0.0608251i	$0.980627\pi$
$138$		0			0
$139$		−	18.1054i		−	1.53568i	−0.640641	−	0.767841i	$-0.721331\pi$
$139$		−	18.1054i		−	1.53568i	0.640641	−	0.767841i	$-0.278669\pi$
$140$		0			0
$141$	5.05444			0.425661
$142$		0			0
$143$	6.90572			0.577485
$144$		0			0
$145$		−	8.47505i		−	0.703814i
$146$		0			0
$147$		−	14.1531i		−	1.16733i
$148$		0			0
$149$		−	13.4277i		−	1.10004i	−0.835151	−	0.550021i	$-0.814620\pi$
$149$		−	13.4277i		−	1.10004i	0.835151	−	0.550021i	$-0.185380\pi$
$150$		0			0
$151$			11.2003i			0.911466i	0.890116	+	0.455733i	$0.150623\pi$
$151$			11.2003i			0.911466i	−0.890116	+	0.455733i	$0.849377\pi$
$152$		0			0
$153$			4.35085i			0.351745i
$154$		0			0
$155$	−4.83021			−0.387972
$156$		0			0
$157$		−	18.4353i		−	1.47129i	−0.677365	−	0.735647i	$-0.736878\pi$
$157$		−	18.4353i		−	1.47129i	0.677365	−	0.735647i	$-0.263122\pi$
$158$		0			0
$159$	−11.0842			−0.879033
$160$		0			0
$161$	−3.45679	−	21.7847i	−0.272434	−	1.71687i
$162$		0			0
$163$			12.1912i			0.954888i	0.878662	+	0.477444i	$0.158437\pi$
$163$			12.1912i			0.954888i	−0.878662	+	0.477444i	$0.841563\pi$
$164$		0			0
$165$	4.88207			0.380068
$166$		0			0
$167$			3.44145i			0.266307i	0.991095	+	0.133154i	$0.0425104\pi$
$167$			3.44145i			0.266307i	−0.991095	+	0.133154i	$0.957490\pi$
$168$		0			0
$169$	−10.9992			−0.846090
$170$		0			0
$171$	1.34013			0.102483
$172$		0			0
$173$	21.8497			1.66120			0.830602	−	0.556867i	$-0.187997\pi$
$173$	21.8497			1.66120			0.830602	+	0.556867i	$0.187997\pi$
$174$		0			0
$175$	4.59925			0.347671
$176$		0			0
$177$	1.36588			0.102666
$178$		0			0
$179$		−	23.5630i		−	1.76118i	−0.473876	−	0.880591i	$-0.657146\pi$
$179$		−	23.5630i		−	1.76118i	0.473876	−	0.880591i	$-0.342854\pi$
$180$		0			0
$181$		−	3.96386i		−	0.294631i	−0.989090	−	0.147316i	$-0.952937\pi$
$181$		−	3.96386i		−	0.294631i	0.989090	−	0.147316i	$-0.0470633\pi$
$182$		0			0
$183$	−9.11481			−0.673786
$184$		0			0
$185$	2.55810			0.188075
$186$		0			0
$187$		−	21.2411i		−	1.55331i
$188$		0			0
$189$		−	4.59925i		−	0.334546i
$190$		0			0
$191$	13.0157			0.941782			0.470891	−	0.882191i	$-0.343933\pi$
$191$	13.0157			0.941782			0.470891	+	0.882191i	$0.343933\pi$
$192$		0			0
$193$	−7.98629			−0.574866			−0.287433	−	0.957801i	$-0.592802\pi$
$193$	−7.98629			−0.574866			−0.287433	+	0.957801i	$0.592802\pi$
$194$		0			0
$195$	1.41451			0.101295
$196$		0			0
$197$	22.2055			1.58208			0.791039	−	0.611765i	$-0.209540\pi$
$197$	22.2055			1.58208			0.791039	+	0.611765i	$0.209540\pi$
$198$		0			0
$199$	0.0306913			0.00217565			0.00108783	−	0.999999i	$-0.499654\pi$
$199$	0.0306913			0.00217565			0.00108783	+	0.999999i	$0.499654\pi$
$200$		0			0
$201$		−	8.47015i		−	0.597438i
$202$		0			0
$203$	38.9789			2.73578
$204$		0			0
$205$		−	1.94751i		−	0.136020i
$206$		0			0
$207$	−0.751600	−	4.73657i	−0.0522398	−	0.329214i
$208$		0			0
$209$	−6.54263			−0.452563
$210$		0			0
$211$		−	10.9831i		−	0.756110i	−0.925783	−	0.378055i	$-0.876593\pi$
$211$		−	10.9831i		−	0.756110i	0.925783	−	0.378055i	$-0.123407\pi$
$212$		0			0
$213$	−0.721095			−0.0494086
$214$		0			0
$215$			9.76592i			0.666030i
$216$		0			0
$217$		−	22.2153i		−	1.50808i
$218$		0			0
$219$		−	2.29450i		−	0.155048i
$220$		0			0
$221$		−	6.15431i		−	0.413983i
$222$		0			0
$223$		−	27.7153i		−	1.85595i	−0.372637	−	0.927977i	$-0.621546\pi$
$223$		−	27.7153i		−	1.85595i	0.372637	−	0.927977i	$-0.378454\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	1.03113			0.0684385			0.0342192	−	0.999414i	$-0.489106\pi$
$227$	1.03113			0.0684385			0.0342192	+	0.999414i	$0.489106\pi$
$228$		0			0
$229$		−	8.25485i		−	0.545496i	−0.962086	−	0.272748i	$-0.912067\pi$
$229$		−	8.25485i		−	0.545496i	0.962086	−	0.272748i	$-0.0879325\pi$
$230$		0			0
$231$			22.4538i			1.47735i
$232$		0			0
$233$	9.59119			0.628340			0.314170	−	0.949367i	$-0.398274\pi$
$233$	9.59119			0.628340			0.314170	+	0.949367i	$0.398274\pi$
$234$		0			0
$235$	−5.05444			−0.329716
$236$		0			0
$237$			15.2680i			0.991761i
$238$		0			0
$239$		−	21.5163i		−	1.39177i	−0.718153	−	0.695885i	$-0.755012\pi$
$239$		−	21.5163i		−	1.39177i	0.718153	−	0.695885i	$-0.244988\pi$
$240$		0			0
$241$			8.52582i			0.549196i	0.961559	+	0.274598i	$0.0885448\pi$
$241$			8.52582i			0.549196i	−0.961559	+	0.274598i	$0.911455\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			14.1531i			0.904208i
$246$		0			0
$247$	−1.89563			−0.120616
$248$		0			0
$249$		−	13.3145i		−	0.843773i
$250$		0			0
$251$	−11.6159			−0.733189			−0.366595	−	0.930381i	$-0.619476\pi$
$251$	−11.6159			−0.733189			−0.366595	+	0.930381i	$0.619476\pi$
$252$		0			0
$253$	3.66936	+	23.1243i	0.230691	+	1.45381i
$254$		0			0
$255$		−	4.35085i		−	0.272461i
$256$		0			0
$257$	5.71183			0.356294			0.178147	−	0.984004i	$-0.442990\pi$
$257$	5.71183			0.356294			0.178147	+	0.984004i	$0.442990\pi$
$258$		0			0
$259$			11.7653i			0.731062i
$260$		0			0
$261$	8.47505			0.524592
$262$		0			0
$263$	−7.96311			−0.491027			−0.245513	−	0.969393i	$-0.578957\pi$
$263$	−7.96311			−0.491027			−0.245513	+	0.969393i	$0.578957\pi$
$264$		0			0
$265$	11.0842			0.680896
$266$		0			0
$267$	8.99269			0.550344
$268$		0			0
$269$	2.35108			0.143348			0.0716740	−	0.997428i	$-0.477166\pi$
$269$	2.35108			0.143348			0.0716740	+	0.997428i	$0.477166\pi$
$270$		0			0
$271$		−	25.2408i		−	1.53327i	−0.642085	−	0.766634i	$-0.721930\pi$
$271$		−	25.2408i		−	1.53327i	0.642085	−	0.766634i	$-0.278070\pi$
$272$		0			0
$273$			6.50567i			0.393741i
$274$		0			0
$275$	−4.88207			−0.294400
$276$		0			0
$277$	13.6182			0.818237			0.409118	−	0.912481i	$-0.365836\pi$
$277$	13.6182			0.818237			0.409118	+	0.912481i	$0.365836\pi$
$278$		0			0
$279$		−	4.83021i		−	0.289177i
$280$		0			0
$281$		−	28.3953i		−	1.69392i	−0.531654	−	0.846962i	$-0.678429\pi$
$281$		−	28.3953i		−	1.69392i	0.531654	−	0.846962i	$-0.321571\pi$
$282$		0			0
$283$	−8.70296			−0.517338			−0.258669	−	0.965966i	$-0.583284\pi$
$283$	−8.70296			−0.517338			−0.258669	+	0.965966i	$0.583284\pi$
$284$		0			0
$285$	−1.34013			−0.0793827
$286$		0			0
$287$	8.95709			0.528720
$288$		0			0
$289$	−1.92988			−0.113523
$290$		0			0
$291$	1.26729			0.0742899
$292$		0			0
$293$		−	13.6653i		−	0.798333i	−0.916879	−	0.399166i	$-0.869300\pi$
$293$		−	13.6653i		−	0.798333i	0.916879	−	0.399166i	$-0.130700\pi$
$294$		0			0
$295$	−1.36588			−0.0795245
$296$		0			0
$297$			4.88207i			0.283286i
$298$		0			0
$299$	1.06314	+	6.69991i	0.0614831	+	0.387466i
$300$		0			0
$301$	−44.9159			−2.58891
$302$		0			0
$303$			14.3919i			0.826794i
$304$		0			0
$305$	9.11481			0.521913
$306$		0			0
$307$		−	6.87777i		−	0.392535i	−0.980550	−	0.196268i	$-0.937118\pi$
$307$		−	6.87777i		−	0.392535i	0.980550	−	0.196268i	$-0.0628821\pi$
$308$		0			0
$309$		−	17.0569i		−	0.970331i
$310$		0			0
$311$			13.4738i			0.764028i	0.924157	+	0.382014i	$0.124769\pi$
$311$			13.4738i			0.764028i	−0.924157	+	0.382014i	$0.875231\pi$
$312$		0			0
$313$			2.61198i			0.147638i	0.997272	+	0.0738189i	$0.0235187\pi$
$313$			2.61198i			0.147638i	−0.997272	+	0.0738189i	$0.976481\pi$
$314$		0			0
$315$			4.59925i			0.259138i
$316$		0			0
$317$	−15.2649			−0.857361			−0.428680	−	0.903456i	$-0.641021\pi$
$317$	−15.2649			−0.857361			−0.428680	+	0.903456i	$0.641021\pi$
$318$		0			0
$319$	−41.3757			−2.31660
$320$		0			0
$321$			3.92835i			0.219259i
$322$		0			0
$323$			5.83072i			0.324430i
$324$		0			0
$325$	−1.41451			−0.0784627
$326$		0			0
$327$	5.40581			0.298942
$328$		0			0
$329$		−	23.2466i		−	1.28163i
$330$		0			0
$331$			24.1660i			1.32829i	0.747606	+	0.664143i	$0.231203\pi$
$331$			24.1660i			1.32829i	−0.747606	+	0.664143i	$0.768797\pi$
$332$		0			0
$333$			2.55810i			0.140183i
$334$		0			0
$335$			8.47015i			0.462774i
$336$		0			0
$337$		−	28.9408i		−	1.57651i	−0.615350	−	0.788254i	$-0.710985\pi$
$337$		−	28.9408i		−	1.57651i	0.615350	−	0.788254i	$-0.289015\pi$
$338$		0			0
$339$	−13.5574			−0.736336
$340$		0			0
$341$			23.5814i			1.27700i
$342$		0			0
$343$	−32.8989			−1.77637
$344$		0			0
$345$	0.751600	+	4.73657i	0.0404648	+	0.255008i
$346$		0			0
$347$		−	10.1722i		−	0.546071i	−0.962004	−	0.273036i	$-0.911972\pi$
$347$		−	10.1722i		−	0.546071i	0.962004	−	0.273036i	$-0.0880277\pi$
$348$		0			0
$349$	2.12044			0.113505			0.0567524	−	0.998388i	$-0.481925\pi$
$349$	2.12044			0.113505			0.0567524	+	0.998388i	$0.481925\pi$
$350$		0			0
$351$			1.41451i			0.0755008i
$352$		0			0
$353$	7.05780			0.375649			0.187824	−	0.982203i	$-0.439856\pi$
$353$	7.05780			0.375649			0.187824	+	0.982203i	$0.439856\pi$
$354$		0			0
$355$	0.721095			0.0382718
$356$		0			0
$357$	20.0106			1.05908
$358$		0			0
$359$	−19.6238			−1.03571			−0.517853	−	0.855470i	$-0.673268\pi$
$359$	−19.6238			−1.03571			−0.517853	+	0.855470i	$0.673268\pi$
$360$		0			0
$361$	−17.2040			−0.905476
$362$		0			0
$363$		−	12.8346i		−	0.673641i
$364$		0			0
$365$			2.29450i			0.120100i
$366$		0			0
$367$	−0.388755			−0.0202928			−0.0101464	−	0.999949i	$-0.503230\pi$
$367$	−0.388755			−0.0202928			−0.0101464	+	0.999949i	$0.503230\pi$
$368$		0			0
$369$	1.94751			0.101383
$370$		0			0
$371$			50.9790i			2.64670i
$372$		0			0
$373$		−	6.08246i		−	0.314938i	−0.987524	−	0.157469i	$-0.949667\pi$
$373$		−	6.08246i		−	0.314938i	0.987524	−	0.157469i	$-0.0503334\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	−11.9880			−0.617414
$378$		0			0
$379$	3.12546			0.160544			0.0802720	−	0.996773i	$-0.474421\pi$
$379$	3.12546			0.160544			0.0802720	+	0.996773i	$0.474421\pi$
$380$		0			0
$381$	−13.2910			−0.680918
$382$		0			0
$383$	3.85924			0.197198			0.0985989	−	0.995127i	$-0.468564\pi$
$383$	3.85924			0.197198			0.0985989	+	0.995127i	$0.468564\pi$
$384$		0			0
$385$		−	22.4538i		−	1.14435i
$386$		0			0
$387$	−9.76592			−0.496430
$388$		0			0
$389$			20.7792i			1.05355i	0.850006	+	0.526773i	$0.176598\pi$
$389$			20.7792i			1.05355i	−0.850006	+	0.526773i	$0.823402\pi$
$390$		0			0
$391$	20.6081	−	3.27010i	1.04220	−	0.165376i
$392$		0			0
$393$	−8.00063			−0.403579
$394$		0			0
$395$		−	15.2680i		−	0.768215i
$396$		0			0
$397$	−19.0440			−0.955791			−0.477896	−	0.878417i	$-0.658600\pi$
$397$	−19.0440			−0.955791			−0.477896	+	0.878417i	$0.658600\pi$
$398$		0			0
$399$		−	6.16361i		−	0.308566i
$400$		0			0
$401$			5.95750i			0.297503i	0.988875	+	0.148752i	$0.0475255\pi$
$401$			5.95750i			0.297503i	−0.988875	+	0.148752i	$0.952475\pi$
$402$		0			0
$403$			6.83237i			0.340344i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$		−	12.4888i		−	0.619047i
$408$		0			0
$409$	15.8548			0.783968			0.391984	−	0.919972i	$-0.371789\pi$
$409$	15.8548			0.783968			0.391984	+	0.919972i	$0.371789\pi$
$410$		0			0
$411$	1.42388			0.0702348
$412$		0			0
$413$		−	6.28201i		−	0.309118i
$414$		0			0
$415$			13.3145i			0.653584i
$416$		0			0
$417$	−18.1054			−0.886626
$418$		0			0
$419$	35.4203			1.73040			0.865198	−	0.501431i	$-0.167193\pi$
$419$	35.4203			1.73040			0.865198	+	0.501431i	$0.167193\pi$
$420$		0			0
$421$		−	10.6010i		−	0.516661i	−0.966057	−	0.258331i	$-0.916828\pi$
$421$		−	10.6010i		−	0.516661i	0.966057	−	0.258331i	$-0.0831724\pi$
$422$		0			0
$423$		−	5.05444i		−	0.245755i
$424$		0			0
$425$			4.35085i			0.211047i
$426$		0			0
$427$			41.9213i			2.02871i
$428$		0			0
$429$		−	6.90572i		−	0.333411i
$430$		0			0
$431$	33.4942			1.61336			0.806679	−	0.590989i	$-0.201263\pi$
$431$	33.4942			1.61336			0.806679	+	0.590989i	$0.201263\pi$
$432$		0			0
$433$		−	11.1564i		−	0.536143i	−0.963399	−	0.268071i	$-0.913614\pi$
$433$		−	11.1564i		−	0.536143i	0.963399	−	0.268071i	$-0.0863863\pi$
$434$		0			0
$435$	−8.47505			−0.406347
$436$		0			0
$437$	−1.00724	−	6.34764i	−0.0481830	−	0.303649i
$438$		0			0
$439$			26.7286i			1.27568i	0.770167	+	0.637842i	$0.220173\pi$
$439$			26.7286i			1.27568i	−0.770167	+	0.637842i	$0.779827\pi$
$440$		0			0
$441$	−14.1531			−0.673957
$442$		0			0
$443$			8.02805i			0.381424i	0.981646	+	0.190712i	$0.0610797\pi$
$443$			8.02805i			0.381424i	−0.981646	+	0.190712i	$0.938920\pi$
$444$		0			0
$445$	−8.99269			−0.426294
$446$		0			0
$447$	−13.4277			−0.635109
$448$		0			0
$449$	41.8897			1.97690			0.988449	−	0.151557i	$-0.0484287\pi$
$449$	41.8897			1.97690			0.988449	+	0.151557i	$0.0484287\pi$
$450$		0			0
$451$	−9.50788			−0.447709
$452$		0			0
$453$	11.2003			0.526235
$454$		0			0
$455$		−	6.50567i		−	0.304991i
$456$		0			0
$457$		−	36.0975i		−	1.68857i	−0.535894	−	0.844285i	$-0.680025\pi$
$457$		−	36.0975i		−	1.68857i	0.535894	−	0.844285i	$-0.319975\pi$
$458$		0			0
$459$	4.35085			0.203080
$460$		0			0
$461$	27.2086			1.26723			0.633616	−	0.773648i	$-0.281570\pi$
$461$	27.2086			1.26723			0.633616	+	0.773648i	$0.281570\pi$
$462$		0			0
$463$			21.3134i			0.990519i	0.868745	+	0.495259i	$0.164927\pi$
$463$			21.3134i			0.990519i	−0.868745	+	0.495259i	$0.835073\pi$
$464$		0			0
$465$			4.83021i			0.223996i
$466$		0			0
$467$	−11.2590			−0.521004			−0.260502	−	0.965473i	$-0.583888\pi$
$467$	−11.2590			−0.521004			−0.260502	+	0.965473i	$0.583888\pi$
$468$		0			0
$469$	−38.9563			−1.79884
$470$		0			0
$471$	−18.4353			−0.849452
$472$		0			0
$473$	47.6779			2.19223
$474$		0			0
$475$	1.34013			0.0614896
$476$		0			0
$477$			11.0842i			0.507510i
$478$		0			0
$479$	−22.5960			−1.03244			−0.516219	−	0.856457i	$-0.672661\pi$
$479$	−22.5960			−1.03244			−0.516219	+	0.856457i	$0.672661\pi$
$480$		0			0
$481$		−	3.61845i		−	0.164987i
$482$		0			0
$483$	−21.7847	+	3.45679i	−0.991237	+	0.157290i
$484$		0			0
$485$	−1.26729			−0.0575447
$486$		0			0
$487$			14.2500i			0.645728i	0.946445	+	0.322864i	$0.104646\pi$
$487$			14.2500i			0.645728i	−0.946445	+	0.322864i	$0.895354\pi$
$488$		0			0
$489$	12.1912			0.551305
$490$		0			0
$491$			28.6030i			1.29084i	0.763830	+	0.645418i	$0.223317\pi$
$491$			28.6030i			1.29084i	−0.763830	+	0.645418i	$0.776683\pi$
$492$		0			0
$493$			36.8736i			1.66071i
$494$		0			0
$495$		−	4.88207i		−	0.219433i
$496$		0			0
$497$			3.31650i			0.148765i
$498$		0			0
$499$			17.5242i			0.784490i	0.919861	+	0.392245i	$0.128302\pi$
$499$			17.5242i			0.784490i	−0.919861	+	0.392245i	$0.871698\pi$
$500$		0			0
$501$	3.44145			0.153753
$502$		0			0
$503$	−16.4284			−0.732508			−0.366254	−	0.930515i	$-0.619360\pi$
$503$	−16.4284			−0.732508			−0.366254	+	0.930515i	$0.619360\pi$
$504$		0			0
$505$		−	14.3919i		−	0.640432i
$506$		0			0
$507$			10.9992i			0.488490i
$508$		0			0
$509$	11.6812			0.517762			0.258881	−	0.965909i	$-0.416646\pi$
$509$	11.6812			0.517762			0.258881	+	0.965909i	$0.416646\pi$
$510$		0			0
$511$	−10.5530			−0.466836
$512$		0			0
$513$		−	1.34013i		−	0.0591684i
$514$		0			0
$515$			17.0569i			0.751615i
$516$		0			0
$517$			24.6761i			1.08526i
$518$		0			0
$519$		−	21.8497i		−	0.959096i
$520$		0			0
$521$		−	22.9944i		−	1.00740i	−0.863877	−	0.503702i	$-0.831971\pi$
$521$		−	22.9944i		−	1.00740i	0.863877	−	0.503702i	$-0.168029\pi$
$522$		0			0
$523$	−16.7340			−0.731727			−0.365864	−	0.930668i	$-0.619226\pi$
$523$	−16.7340			−0.731727			−0.365864	+	0.930668i	$0.619226\pi$
$524$		0			0
$525$		−	4.59925i		−	0.200728i
$526$		0			0
$527$	21.0155			0.915450
$528$		0			0
$529$	−21.8702	+	7.12001i	−0.950878	+	0.309566i
$530$		0			0
$531$		−	1.36588i		−	0.0592741i
$532$		0			0
$533$	−2.75477			−0.119322
$534$		0			0
$535$		−	3.92835i		−	0.169838i
$536$		0			0
$537$	−23.5630			−1.01682
$538$		0			0
$539$	69.0964			2.97619
$540$		0			0
$541$	30.2431			1.30025			0.650126	−	0.759826i	$-0.274716\pi$
$541$	30.2431			1.30025			0.650126	+	0.759826i	$0.274716\pi$
$542$		0			0
$543$	−3.96386			−0.170105
$544$		0			0
$545$	−5.40581			−0.231559
$546$		0			0
$547$			39.8256i			1.70282i	0.524502	+	0.851409i	$0.324251\pi$
$547$			39.8256i			1.70282i	−0.524502	+	0.851409i	$0.675749\pi$
$548$		0			0
$549$			9.11481i			0.389011i
$550$		0			0
$551$	11.3577			0.483854
$552$		0			0
$553$	70.2212			2.98611
$554$		0			0
$555$		−	2.55810i		−	0.108585i
$556$		0			0
$557$			9.14289i			0.387397i	0.981061	+	0.193698i	$0.0620483\pi$
$557$			9.14289i			0.387397i	−0.981061	+	0.193698i	$0.937952\pi$
$558$		0			0
$559$	13.8140			0.584268
$560$		0			0
$561$	−21.2411			−0.896801
$562$		0			0
$563$	−5.43927			−0.229238			−0.114619	−	0.993410i	$-0.536565\pi$
$563$	−5.43927			−0.229238			−0.114619	+	0.993410i	$0.536565\pi$
$564$		0			0
$565$	13.5574			0.570364
$566$		0			0
$567$	−4.59925			−0.193150
$568$		0			0
$569$			15.3131i			0.641960i	0.947086	+	0.320980i	$0.104012\pi$
$569$			15.3131i			0.641960i	−0.947086	+	0.320980i	$0.895988\pi$
$570$		0			0
$571$	−28.7285			−1.20225			−0.601126	−	0.799154i	$-0.705281\pi$
$571$	−28.7285			−1.20225			−0.601126	+	0.799154i	$0.705281\pi$
$572$		0			0
$573$		−	13.0157i		−	0.543738i
$574$		0			0
$575$	−0.751600	−	4.73657i	−0.0313439	−	0.197529i
$576$		0			0
$577$	−45.7627			−1.90513			−0.952563	−	0.304340i	$-0.901564\pi$
$577$	−45.7627			−1.90513			−0.952563	+	0.304340i	$0.901564\pi$
$578$		0			0
$579$			7.98629i			0.331899i
$580$		0			0
$581$	−61.2368			−2.54053
$582$		0			0
$583$		−	54.1138i		−	2.24116i
$584$		0			0
$585$		−	1.41451i		−	0.0584827i
$586$		0			0
$587$		−	20.5254i		−	0.847176i	−0.905855	−	0.423588i	$-0.860770\pi$
$587$		−	20.5254i		−	0.847176i	0.905855	−	0.423588i	$-0.139230\pi$
$588$		0			0
$589$		−	6.47313i		−	0.266721i
$590$		0			0
$591$		−	22.2055i		−	0.913413i
$592$		0			0
$593$	42.3259			1.73812			0.869058	−	0.494710i	$-0.164726\pi$
$593$	42.3259			1.73812			0.869058	+	0.494710i	$0.164726\pi$
$594$		0			0
$595$	−20.0106			−0.820356
$596$		0			0
$597$		−	0.0306913i		−	0.00125611i
$598$		0			0
$599$			8.21773i			0.335767i	0.985807	+	0.167884i	$0.0536933\pi$
$599$			8.21773i			0.335767i	−0.985807	+	0.167884i	$0.946307\pi$
$600$		0			0
$601$	−21.9012			−0.893370			−0.446685	−	0.894691i	$-0.647396\pi$
$601$	−21.9012			−0.893370			−0.446685	+	0.894691i	$0.647396\pi$
$602$		0			0
$603$	−8.47015			−0.344931
$604$		0			0
$605$			12.8346i			0.521800i
$606$		0			0
$607$		−	41.6105i		−	1.68892i	−0.535620	−	0.844459i	$-0.679922\pi$
$607$		−	41.6105i		−	1.68892i	0.535620	−	0.844459i	$-0.320078\pi$
$608$		0			0
$609$		−	38.9789i		−	1.57950i
$610$		0			0
$611$			7.14955i			0.289240i
$612$		0			0
$613$		−	35.6614i		−	1.44035i	−0.693792	−	0.720175i	$-0.744061\pi$
$613$		−	35.6614i		−	1.44035i	0.693792	−	0.720175i	$-0.255939\pi$
$614$		0			0
$615$	−1.94751			−0.0785312
$616$		0			0
$617$			33.4403i			1.34626i	0.739526	+	0.673128i	$0.235050\pi$
$617$			33.4403i			1.34626i	−0.739526	+	0.673128i	$0.764950\pi$
$618$		0			0
$619$	7.06018			0.283773			0.141886	−	0.989883i	$-0.454683\pi$
$619$	7.06018			0.283773			0.141886	+	0.989883i	$0.454683\pi$
$620$		0			0
$621$	−4.73657	+	0.751600i	−0.190072	+	0.0301607i
$622$		0			0
$623$		−	41.3596i		−	1.65704i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			6.54263i			0.261287i
$628$		0			0
$629$	−11.1299			−0.443778
$630$		0			0
$631$	22.1241			0.880748			0.440374	−	0.897814i	$-0.354846\pi$
$631$	22.1241			0.880748			0.440374	+	0.897814i	$0.354846\pi$
$632$		0			0
$633$	−10.9831			−0.436540
$634$		0			0
$635$	13.2910			0.527437
$636$		0			0
$637$	20.0196			0.793207
$638$		0			0
$639$			0.721095i			0.0285261i
$640$		0			0
$641$		−	28.6565i		−	1.13187i	−0.824451	−	0.565933i	$-0.808516\pi$
$641$		−	28.6565i		−	1.13187i	0.824451	−	0.565933i	$-0.191484\pi$
$642$		0			0
$643$	−35.1114			−1.38466			−0.692329	−	0.721582i	$-0.743415\pi$
$643$	−35.1114			−1.38466			−0.692329	+	0.721582i	$0.743415\pi$
$644$		0			0
$645$	9.76592			0.384533
$646$		0			0
$647$		−	27.4563i		−	1.07942i	−0.841852	−	0.539709i	$-0.818534\pi$
$647$		−	27.4563i		−	1.07942i	0.841852	−	0.539709i	$-0.181466\pi$
$648$		0			0
$649$			6.66831i			0.261754i
$650$		0			0
$651$	−22.2153			−0.870688
$652$		0			0
$653$	−25.9667			−1.01615			−0.508077	−	0.861311i	$-0.669644\pi$
$653$	−25.9667			−1.01615			−0.508077	+	0.861311i	$0.669644\pi$
$654$		0			0
$655$	8.00063			0.312611
$656$		0			0
$657$	−2.29450			−0.0895169
$658$		0			0
$659$	23.5211			0.916253			0.458127	−	0.888887i	$-0.348521\pi$
$659$	23.5211			0.916253			0.458127	+	0.888887i	$0.348521\pi$
$660$		0			0
$661$			41.0127i			1.59521i	0.603181	+	0.797605i	$0.293900\pi$
$661$			41.0127i			1.59521i	−0.603181	+	0.797605i	$0.706100\pi$
$662$		0			0
$663$	−6.15431			−0.239013
$664$		0			0
$665$			6.16361i			0.239015i
$666$		0			0
$667$	−6.36984	−	40.1427i	−0.246641	−	1.55433i
$668$		0			0
$669$	−27.7153			−1.07154
$670$		0			0
$671$		−	44.4991i		−	1.71787i
$672$		0			0
$673$	4.56660			0.176030			0.0880148	−	0.996119i	$-0.471948\pi$
$673$	4.56660			0.176030			0.0880148	+	0.996119i	$0.471948\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$		−	41.1466i		−	1.58139i	−0.612210	−	0.790695i	$-0.709719\pi$
$677$		−	41.1466i		−	1.58139i	0.612210	−	0.790695i	$-0.290281\pi$
$678$		0			0
$679$		−	5.82859i		−	0.223681i
$680$		0			0
$681$		−	1.03113i		−	0.0395130i
$682$		0			0
$683$			35.5129i			1.35886i	0.733739	+	0.679432i	$0.237774\pi$
$683$			35.5129i			1.35886i	−0.733739	+	0.679432i	$0.762226\pi$
$684$		0			0
$685$	−1.42388			−0.0544036
$686$		0			0
$687$	−8.25485			−0.314942
$688$		0			0
$689$		−	15.6787i		−	0.597310i
$690$		0			0
$691$		−	31.3788i		−	1.19370i	−0.802351	−	0.596852i	$-0.796418\pi$
$691$		−	31.3788i		−	1.19370i	0.802351	−	0.596852i	$-0.203582\pi$
$692$		0			0
$693$	22.4538			0.852951
$694$		0			0
$695$	18.1054			0.686777
$696$		0			0
$697$			8.47333i			0.320950i
$698$		0			0
$699$		−	9.59119i		−	0.362772i
$700$		0			0
$701$			23.9305i			0.903843i	0.892058	+	0.451921i	$0.149261\pi$
$701$			23.9305i			0.903843i	−0.892058	+	0.451921i	$0.850739\pi$
$702$		0			0
$703$			3.42819i			0.129297i
$704$		0			0
$705$			5.05444i			0.190361i
$706$		0			0
$707$	66.1920			2.48941
$708$		0			0
$709$			40.1421i			1.50757i	0.657122	+	0.753784i	$0.271774\pi$
$709$			40.1421i			1.50757i	−0.657122	+	0.753784i	$0.728226\pi$
$710$		0			0
$711$	15.2680			0.572593
$712$		0			0
$713$	−22.8786	+	3.63038i	−0.856811	+	0.135959i
$714$		0			0
$715$			6.90572i			0.258259i
$716$		0			0
$717$	−21.5163			−0.803539
$718$		0			0
$719$		−	9.22449i		−	0.344015i	−0.985096	−	0.172008i	$-0.944975\pi$
$719$		−	9.22449i		−	0.344015i	0.985096	−	0.172008i	$-0.0550254\pi$
$720$		0			0
$721$	−78.4487			−2.92158
$722$		0			0
$723$	8.52582			0.317079
$724$		0			0
$725$	8.47505			0.314755
$726$		0			0
$727$	18.4434			0.684027			0.342014	−	0.939695i	$-0.388891\pi$
$727$	18.4434			0.684027			0.342014	+	0.939695i	$0.388891\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	42.4900i		−	1.57155i
$732$		0			0
$733$			51.1562i			1.88950i	0.327797	+	0.944748i	$0.393694\pi$
$733$			51.1562i			1.88950i	−0.327797	+	0.944748i	$0.606306\pi$
$734$		0			0
$735$	14.1531			0.522045
$736$		0			0
$737$	41.3518			1.52321
$738$		0			0
$739$			17.9784i			0.661346i	0.943745	+	0.330673i	$0.107276\pi$
$739$			17.9784i			0.661346i	−0.943745	+	0.330673i	$0.892724\pi$
$740$		0			0
$741$			1.89563i			0.0696377i
$742$		0			0
$743$	−45.8180			−1.68090			−0.840449	−	0.541890i	$-0.817709\pi$
$743$	−45.8180			−1.68090			−0.840449	+	0.541890i	$0.817709\pi$
$744$		0			0
$745$	13.4277			0.491954
$746$		0			0
$747$	−13.3145			−0.487153
$748$		0			0
$749$	18.0675			0.660171
$750$		0			0
$751$	−5.29283			−0.193138			−0.0965691	−	0.995326i	$-0.530787\pi$
$751$	−5.29283			−0.193138			−0.0965691	+	0.995326i	$0.530787\pi$
$752$		0			0
$753$			11.6159i			0.423307i
$754$		0			0
$755$	−11.2003			−0.407620
$756$		0			0
$757$		−	24.6833i		−	0.897130i	−0.893750	−	0.448565i	$-0.851935\pi$
$757$		−	24.6833i		−	0.897130i	0.893750	−	0.448565i	$-0.148065\pi$
$758$		0			0
$759$	23.1243	−	3.66936i	0.839357	−	0.133189i
$760$		0			0
$761$	−19.9805			−0.724293			−0.362147	−	0.932121i	$-0.617956\pi$
$761$	−19.9805			−0.724293			−0.362147	+	0.932121i	$0.617956\pi$
$762$		0			0
$763$		−	24.8627i		−	0.900089i
$764$		0			0
$765$	−4.35085			−0.157305
$766$		0			0
$767$			1.93204i			0.0697621i
$768$		0			0
$769$			4.23081i			0.152567i	0.997086	+	0.0762834i	$0.0243054\pi$
$769$			4.23081i			0.152567i	−0.997086	+	0.0762834i	$0.975695\pi$
$770$		0			0
$771$		−	5.71183i		−	0.205706i
$772$		0			0
$773$		−	4.37581i		−	0.157387i	−0.996899	−	0.0786934i	$-0.974925\pi$
$773$		−	4.37581i		−	0.157387i	0.996899	−	0.0786934i	$-0.0250748\pi$
$774$		0			0
$775$		−	4.83021i		−	0.173506i
$776$		0			0
$777$	11.7653			0.422079
$778$		0			0
$779$	2.60993			0.0935103
$780$		0			0
$781$		−	3.52044i		−	0.125971i
$782$		0			0
$783$		−	8.47505i		−	0.302873i
$784$		0			0
$785$	18.4353			0.657983
$786$		0			0
$787$	−5.64990			−0.201397			−0.100699	−	0.994917i	$-0.532108\pi$
$787$	−5.64990			−0.201397			−0.100699	+	0.994917i	$0.532108\pi$
$788$		0			0
$789$			7.96311i			0.283494i
$790$		0			0
$791$			62.3538i			2.21705i
$792$		0			0
$793$		−	12.8930i		−	0.457843i
$794$		0			0
$795$		−	11.0842i		−	0.393116i
$796$		0			0
$797$			33.8817i			1.20015i	0.799943	+	0.600076i	$0.204863\pi$
$797$			33.8817i			1.20015i	−0.799943	+	0.600076i	$0.795137\pi$
$798$		0			0
$799$	21.9911			0.777990
$800$		0			0
$801$		−	8.99269i		−	0.317741i
$802$		0			0
$803$	11.2019			0.395306
$804$		0			0
$805$	21.7847	−	3.45679i	0.767809	−	0.121836i
$806$		0			0
$807$		−	2.35108i		−	0.0827620i
$808$		0			0
$809$	−9.74040			−0.342454			−0.171227	−	0.985232i	$-0.554773\pi$
$809$	−9.74040			−0.342454			−0.171227	+	0.985232i	$0.554773\pi$
$810$		0			0
$811$			2.50103i			0.0878230i	0.999035	+	0.0439115i	$0.0139820\pi$
$811$			2.50103i			0.0878230i	−0.999035	+	0.0439115i	$0.986018\pi$
$812$		0			0
$813$	−25.2408			−0.885232
$814$		0			0
$815$	−12.1912			−0.427039
$816$		0			0
$817$	−13.0876			−0.457879
$818$		0			0
$819$	6.50567			0.227327
$820$		0			0
$821$	24.1942			0.844382			0.422191	−	0.906507i	$-0.361261\pi$
$821$	24.1942			0.844382			0.422191	+	0.906507i	$0.361261\pi$
$822$		0			0
$823$			33.5620i			1.16990i	0.811070	+	0.584950i	$0.198886\pi$
$823$			33.5620i			1.16990i	−0.811070	+	0.584950i	$0.801114\pi$
$824$		0			0
$825$			4.88207i			0.169972i
$826$		0			0
$827$	34.5490			1.20139			0.600693	−	0.799480i	$-0.294891\pi$
$827$	34.5490			1.20139			0.600693	+	0.799480i	$0.294891\pi$
$828$		0			0
$829$	−1.62965			−0.0566000			−0.0283000	−	0.999599i	$-0.509009\pi$
$829$	−1.62965			−0.0566000			−0.0283000	+	0.999599i	$0.509009\pi$
$830$		0			0
$831$		−	13.6182i		−	0.472409i
$832$		0			0
$833$		−	61.5780i		−	2.13355i
$834$		0			0
$835$	−3.44145			−0.119096
$836$		0			0
$837$	−4.83021			−0.166956
$838$		0			0
$839$	36.7302			1.26807			0.634034	−	0.773305i	$-0.281398\pi$
$839$	36.7302			1.26807			0.634034	+	0.773305i	$0.281398\pi$
$840$		0			0
$841$	42.8264			1.47677
$842$		0			0
$843$	−28.3953			−0.977987
$844$		0			0
$845$		−	10.9992i		−	0.378383i
$846$		0			0
$847$	−59.0294			−2.02828
$848$		0			0
$849$			8.70296i			0.298685i
$850$		0			0
$851$	12.1166	−	1.92267i	0.415352	−	0.0659081i
$852$		0			0
$853$	1.46669			0.0502186			0.0251093	−	0.999685i	$-0.492007\pi$
$853$	1.46669			0.0502186			0.0251093	+	0.999685i	$0.492007\pi$
$854$		0			0
$855$			1.34013i			0.0458316i
$856$		0			0
$857$	−16.8164			−0.574437			−0.287219	−	0.957865i	$-0.592731\pi$
$857$	−16.8164			−0.574437			−0.287219	+	0.957865i	$0.592731\pi$
$858$		0			0
$859$		−	47.4832i		−	1.62011i	−0.586357	−	0.810053i	$-0.699439\pi$
$859$		−	47.4832i		−	1.62011i	0.586357	−	0.810053i	$-0.300561\pi$
$860$		0			0
$861$		−	8.95709i		−	0.305257i
$862$		0			0
$863$		−	15.5777i		−	0.530273i	−0.964211	−	0.265136i	$-0.914583\pi$
$863$		−	15.5777i		−	0.530273i	0.964211	−	0.265136i	$-0.0854169\pi$
$864$		0			0
$865$			21.8497i			0.742913i
$866$		0			0
$867$			1.92988i			0.0655423i
$868$		0			0
$869$	−74.5392			−2.52857
$870$		0			0
$871$	11.9811			0.405963
$872$		0			0
$873$		−	1.26729i		−	0.0428913i
$874$		0			0
$875$			4.59925i			0.155483i
$876$		0			0
$877$	54.1273			1.82775			0.913875	−	0.405995i	$-0.133075\pi$
$877$	54.1273			1.82775			0.913875	+	0.405995i	$0.133075\pi$
$878$		0			0
$879$	−13.6653			−0.460918
$880$		0			0
$881$			9.45016i			0.318384i	0.987248	+	0.159192i	$0.0508889\pi$
$881$			9.45016i			0.318384i	−0.987248	+	0.159192i	$0.949111\pi$
$882$		0			0
$883$		−	40.8216i		−	1.37376i	−0.726772	−	0.686879i	$-0.758980\pi$
$883$		−	40.8216i		−	1.37376i	0.726772	−	0.686879i	$-0.241020\pi$
$884$		0			0
$885$			1.36588i			0.0459135i
$886$		0			0
$887$		−	1.39569i		−	0.0468625i	−0.999725	−	0.0234313i	$-0.992541\pi$
$887$		−	1.39569i		−	0.0468625i	0.999725	−	0.0234313i	$-0.00745909\pi$
$888$		0			0
$889$			61.1286i			2.05019i
$890$		0			0
$891$	4.88207			0.163555
$892$		0			0
$893$		−	6.77363i		−	0.226671i
$894$		0			0
$895$	23.5630			0.787625
$896$		0			0
$897$	6.69991	−	1.06314i	0.223704	−	0.0354973i
$898$		0			0
$899$		−	40.9363i		−	1.36530i
$900$		0			0
$901$	−48.2256			−1.60663
$902$		0			0
$903$			44.9159i			1.49471i
$904$		0			0
$905$	3.96386			0.131763
$906$		0			0
$907$	−6.66378			−0.221267			−0.110634	−	0.993861i	$-0.535288\pi$
$907$	−6.66378			−0.221267			−0.110634	+	0.993861i	$0.535288\pi$
$908$		0			0
$909$	14.3919			0.477350
$910$		0			0
$911$	21.8114			0.722644			0.361322	−	0.932441i	$-0.382326\pi$
$911$	21.8114			0.722644			0.361322	+	0.932441i	$0.382326\pi$
$912$		0			0
$913$	65.0024			2.15126
$914$		0			0
$915$		−	9.11481i		−	0.301326i
$916$		0			0
$917$			36.7969i			1.21514i
$918$		0			0
$919$	21.6832			0.715263			0.357632	−	0.933863i	$-0.383584\pi$
$919$	21.6832			0.715263			0.357632	+	0.933863i	$0.383584\pi$
$920$		0			0
$921$	−6.87777			−0.226630
$922$		0			0
$923$		−	1.01999i		−	0.0335735i
$924$		0			0
$925$			2.55810i			0.0841097i
$926$		0			0
$927$	−17.0569			−0.560221
$928$		0			0
$929$	−38.0711			−1.24907			−0.624535	−	0.780997i	$-0.714712\pi$
$929$	−38.0711			−1.24907			−0.624535	+	0.780997i	$0.714712\pi$
$930$		0			0
$931$	−18.9670			−0.621620
$932$		0			0
$933$	13.4738			0.441112
$934$		0			0
$935$	21.2411			0.694659
$936$		0			0
$937$		−	11.8843i		−	0.388244i	−0.980977	−	0.194122i	$-0.937814\pi$
$937$		−	11.8843i		−	0.388244i	0.980977	−	0.194122i	$-0.0621857\pi$
$938$		0			0
$939$	2.61198			0.0852387
$940$		0			0
$941$			54.4909i			1.77635i	0.459503	+	0.888176i	$0.348028\pi$
$941$			54.4909i			1.77635i	−0.459503	+	0.888176i	$0.651972\pi$
$942$		0			0
$943$	−1.46375	−	9.22452i	−0.0476662	−	0.300392i
$944$		0			0
$945$	4.59925			0.149614
$946$		0			0
$947$			44.3947i			1.44263i	0.692605	+	0.721317i	$0.256463\pi$
$947$			44.3947i			1.44263i	−0.692605	+	0.721317i	$0.743537\pi$
$948$		0			0
$949$	3.24558			0.105356
$950$		0			0
$951$			15.2649i			0.494998i
$952$		0			0
$953$			49.4810i			1.60285i	0.598098	+	0.801423i	$0.295923\pi$
$953$			49.4810i			1.60285i	−0.598098	+	0.801423i	$0.704077\pi$
$954$		0			0
$955$			13.0157i			0.421178i
$956$		0			0
$957$			41.3757i			1.33749i
$958$		0			0
$959$		−	6.54877i		−	0.211471i
$960$		0			0
$961$	7.66907			0.247389
$962$		0			0
$963$	3.92835			0.126589
$964$		0			0
$965$		−	7.98629i		−	0.257088i
$966$		0			0
$967$		−	45.3379i		−	1.45797i	−0.684530	−	0.728984i	$-0.739993\pi$
$967$		−	45.3379i		−	1.45797i	0.684530	−	0.728984i	$-0.260007\pi$
$968$		0			0
$969$	5.83072			0.187310
$970$		0			0
$971$	24.5821			0.788877			0.394438	−	0.918922i	$-0.370939\pi$
$971$	24.5821			0.788877			0.394438	+	0.918922i	$0.370939\pi$
$972$		0			0
$973$			83.2713i			2.66956i
$974$		0			0
$975$			1.41451i			0.0453005i
$976$		0			0
$977$			48.9650i			1.56653i	0.621688	+	0.783265i	$0.286447\pi$
$977$			48.9650i			1.56653i	−0.621688	+	0.783265i	$0.713553\pi$
$978$		0			0
$979$			43.9029i			1.40314i
$980$		0			0
$981$		−	5.40581i		−	0.172594i
$982$		0			0
$983$	7.75279			0.247276			0.123638	−	0.992327i	$-0.460544\pi$
$983$	7.75279			0.247276			0.123638	+	0.992327i	$0.460544\pi$
$984$		0			0
$985$			22.2055i			0.707527i
$986$		0			0
$987$	−23.2466			−0.739949
$988$		0			0
$989$	7.34006	+	46.2570i	0.233400	+	1.47089i
$990$		0			0
$991$			38.3077i			1.21688i	0.793599	+	0.608441i	$0.208205\pi$
$991$			38.3077i			1.21688i	−0.793599	+	0.608441i	$0.791795\pi$
$992$		0			0
$993$	24.1660			0.766886
$994$		0			0
$995$			0.0306913i			0.000972981i
$996$		0			0
$997$	40.6501			1.28740			0.643700	−	0.765278i	$-0.277398\pi$
$997$	40.6501			1.28740			0.643700	+	0.765278i	$0.277398\pi$
$998$		0			0
$999$	2.55810			0.0809346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.a.1471.1	✓	16
4.3	odd	2		5520.2.be.b.1471.16	yes	16
23.22	odd	2		5520.2.be.b.1471.8	yes	16
92.91	even	2	inner	5520.2.be.a.1471.9	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.a.1471.1	✓	16	1.1	even	1	trivial
5520.2.be.a.1471.9	yes	16	92.91	even	2	inner
5520.2.be.b.1471.8	yes	16	23.22	odd	2
5520.2.be.b.1471.16	yes	16	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.be (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -4.59925 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.00000i q^{5} -4.59925 q^{7} -1.00000 q^{9} +4.88207 q^{11} +1.41451 q^{13} +1.00000 q^{15} -4.35085i q^{17} -1.34013 q^{19} +4.59925i q^{21} +(0.751600 + 4.73657i) q^{23} -1.00000 q^{25} +1.00000i q^{27} -8.47505 q^{29} +4.83021i q^{31} -4.88207i q^{33} -4.59925i q^{35} -2.55810i q^{37} -1.41451i q^{39} -1.94751 q^{41} +9.76592 q^{43} -1.00000i q^{45} +5.05444i q^{47} +14.1531 q^{49} -4.35085 q^{51} -11.0842i q^{53} +4.88207i q^{55} +1.34013i q^{57} +1.36588i q^{59} -9.11481i q^{61} +4.59925 q^{63} +1.41451i q^{65} +8.47015 q^{67} +(4.73657 - 0.751600i) q^{69} -0.721095i q^{71} +2.29450 q^{73} +1.00000i q^{75} -22.4538 q^{77} -15.2680 q^{79} +1.00000 q^{81} +13.3145 q^{83} +4.35085 q^{85} +8.47505i q^{87} +8.99269i q^{89} -6.50567 q^{91} +4.83021 q^{93} -1.34013i q^{95} +1.26729i q^{97} -4.88207 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{7} - 16 q^{9}+O(q^{10}) \) 16 * q - 8 * q^7 - 16 * q^9	\( 16 q - 8 q^{7} - 16 q^{9} + 8 q^{11} + 8 q^{13} + 16 q^{15} + 12 q^{23} - 16 q^{25} - 4 q^{29} + 4 q^{41} + 20 q^{49} - 4 q^{51} + 8 q^{63} - 16 q^{67} + 40 q^{73} + 24 q^{77} + 32 q^{79} + 16 q^{81} + 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) \) 16 * q - 8 * q^7 - 16 * q^9 + 8 * q^11 + 8 * q^13 + 16 * q^15 + 12 * q^23 - 16 * q^25 - 4 * q^29 + 4 * q^41 + 20 * q^49 - 4 * q^51 + 8 * q^63 - 16 * q^67 + 40 * q^73 + 24 * q^77 + 32 * q^79 + 16 * q^81 + 4 * q^85 - 48 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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