LMFDB - Embedded newform 4900.2.e.r.2549.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(2549,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4900.2549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (of order $2$, degree $1$, not 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:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 980)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.r.2549.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214i q^{3} +2.82843 q^{9} +O(q^{10})$	$q-0.414214i q^{3} +2.82843 q^{9} -3.82843 q^{11} +3.58579i q^{13} -6.41421i q^{17} -3.65685 q^{19} +0.585786i q^{23} -2.41421i q^{27} -6.65685 q^{29} +4.58579 q^{31} +1.58579i q^{33} +3.41421i q^{37} +1.48528 q^{39} +0.585786 q^{41} +11.6569i q^{43} +8.89949i q^{47} -2.65685 q^{51} -3.75736i q^{53} +1.51472i q^{57} -3.41421 q^{59} +5.17157 q^{61} +11.0711i q^{67} +0.242641 q^{69} +6.48528 q^{71} +5.17157i q^{73} -13.1421 q^{79} +7.48528 q^{81} -8.00000i q^{83} +2.75736i q^{87} -16.9706 q^{89} -1.89949i q^{93} -15.7279i q^{97} -10.8284 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{11} + 8 q^{19} - 4 q^{29} + 24 q^{31} - 28 q^{39} + 8 q^{41} + 12 q^{51} - 8 q^{59} + 32 q^{61} - 16 q^{69} - 8 q^{71} + 4 q^{79} - 4 q^{81} - 32 q^{99}+O(q^{100}) $ 4 * q - 4 * q^11 + 8 * q^19 - 4 * q^29 + 24 * q^31 - 28 * q^39 + 8 * q^41 + 12 * q^51 - 8 * q^59 + 32 * q^61 - 16 * q^69 - 8 * q^71 + 4 * q^79 - 4 * q^81 - 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 0.414214i	− 0.239146i	−0.992825	−	0.119573i	$-0.961847\pi$
$3$	− 0.414214i	− 0.239146i	0.992825	−	0.119573i	$-0.0381526\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	−3.82843	−1.15431	−0.577157	−	0.816633i	$-0.695838\pi$
$11$	−3.82843	−1.15431	−0.577157	+	0.816633i	$0.695838\pi$
$12$	0	0
$13$	3.58579i	0.994518i	0.867602	+	0.497259i	$0.165660\pi$
$13$	3.58579i	0.994518i	−0.867602	+	0.497259i	$0.834340\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.41421i	− 1.55568i	−0.628465	−	0.777838i	$-0.716317\pi$
$17$	− 6.41421i	− 1.55568i	0.628465	−	0.777838i	$-0.283683\pi$
$18$	0	0
$19$	−3.65685	−0.838940	−0.419470	−	0.907769i	$-0.637784\pi$
$19$	−3.65685	−0.838940	−0.419470	+	0.907769i	$0.637784\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.585786i	0.122145i	0.998133	+	0.0610725i	$0.0194521\pi$
$23$	0.585786i	0.122145i	−0.998133	+	0.0610725i	$0.980548\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.41421i	− 0.464616i
$28$	0	0
$29$	−6.65685	−1.23615	−0.618073	−	0.786120i	$-0.712087\pi$
$29$	−6.65685	−1.23615	−0.618073	+	0.786120i	$0.712087\pi$
$30$	0	0
$31$	4.58579	0.823632	0.411816	−	0.911267i	$-0.364895\pi$
$31$	4.58579	0.823632	0.411816	+	0.911267i	$0.364895\pi$
$32$	0	0
$33$	1.58579i	0.276050i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.41421i	0.561293i	0.959811	+	0.280647i	$0.0905489\pi$
$37$	3.41421i	0.561293i	−0.959811	+	0.280647i	$0.909451\pi$
$38$	0	0
$39$	1.48528	0.237835
$40$	0	0
$41$	0.585786	0.0914845	0.0457422	−	0.998953i	$-0.485435\pi$
$41$	0.585786	0.0914845	0.0457422	+	0.998953i	$0.485435\pi$
$42$	0	0
$43$	11.6569i	1.77765i	0.458243	+	0.888827i	$0.348479\pi$
$43$	11.6569i	1.77765i	−0.458243	+	0.888827i	$0.651521\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.89949i	1.29812i	0.760735	+	0.649062i	$0.224839\pi$
$47$	8.89949i	1.29812i	−0.760735	+	0.649062i	$0.775161\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.65685	−0.372034
$52$	0	0
$53$	− 3.75736i	− 0.516113i	−0.966130	−	0.258056i	$-0.916918\pi$
$53$	− 3.75736i	− 0.516113i	0.966130	−	0.258056i	$-0.0830821\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.51472i	0.200629i
$58$	0	0
$59$	−3.41421	−0.444493	−0.222246	−	0.974991i	$-0.571339\pi$
$59$	−3.41421	−0.444493	−0.222246	+	0.974991i	$0.571339\pi$
$60$	0	0
$61$	5.17157	0.662152	0.331076	−	0.943604i	$-0.392588\pi$
$61$	5.17157	0.662152	0.331076	+	0.943604i	$0.392588\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.0711i	1.35255i	0.736651	+	0.676273i	$0.236406\pi$
$67$	11.0711i	1.35255i	−0.736651	+	0.676273i	$0.763594\pi$
$68$	0	0
$69$	0.242641	0.0292105
$70$	0	0
$71$	6.48528	0.769661	0.384831	−	0.922987i	$-0.374260\pi$
$71$	6.48528	0.769661	0.384831	+	0.922987i	$0.374260\pi$
$72$	0	0
$73$	5.17157i	0.605287i	0.953104	+	0.302643i	$0.0978691\pi$
$73$	5.17157i	0.605287i	−0.953104	+	0.302643i	$0.902131\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.1421	−1.47861	−0.739303	−	0.673373i	$-0.764845\pi$
$79$	−13.1421	−1.47861	−0.739303	+	0.673373i	$0.764845\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.75736i	0.295620i
$88$	0	0
$89$	−16.9706	−1.79888	−0.899438	−	0.437048i	$-0.856024\pi$
$89$	−16.9706	−1.79888	−0.899438	+	0.437048i	$0.856024\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 1.89949i	− 0.196968i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 15.7279i	− 1.59693i	−0.602042	−	0.798464i	$-0.705646\pi$
$97$	− 15.7279i	− 1.59693i	0.602042	−	0.798464i	$-0.294354\pi$
$98$	0	0
$99$	−10.8284	−1.08830
$100$	0	0
$101$	−4.82843	−0.480446	−0.240223	−	0.970718i	$-0.577221\pi$
$101$	−4.82843	−0.480446	−0.240223	+	0.970718i	$0.577221\pi$
$102$	0	0
$103$	1.58579i	0.156252i	0.996943	+	0.0781261i	$0.0248937\pi$
$103$	1.58579i	0.156252i	−0.996943	+	0.0781261i	$0.975106\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.8284i	1.62687i	0.581659	+	0.813433i	$0.302404\pi$
$107$	16.8284i	1.62687i	−0.581659	+	0.813433i	$0.697596\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	1.41421	0.134231
$112$	0	0
$113$	− 5.07107i	− 0.477046i	−0.971137	−	0.238523i	$-0.923337\pi$
$113$	− 5.07107i	− 0.477046i	0.971137	−	0.238523i	$-0.0766632\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.1421i	0.937641i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.65685	0.332441
$122$	0	0
$123$	− 0.242641i	− 0.0218782i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	21.8995i	1.94327i	0.236494	+	0.971633i	$0.424002\pi$
$127$	21.8995i	1.94327i	−0.236494	+	0.971633i	$0.575998\pi$
$128$	0	0
$129$	4.82843	0.425119
$130$	0	0
$131$	−11.7574	−1.02725	−0.513623	−	0.858016i	$-0.671697\pi$
$131$	−11.7574	−1.02725	−0.513623	+	0.858016i	$0.671697\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.9706i	− 1.10815i	−0.832467	−	0.554075i	$-0.813072\pi$
$137$	− 12.9706i	− 1.10815i	0.832467	−	0.554075i	$-0.186928\pi$
$138$	0	0
$139$	13.8995	1.17894	0.589470	−	0.807790i	$-0.299337\pi$
$139$	13.8995	1.17894	0.589470	+	0.807790i	$0.299337\pi$
$140$	0	0
$141$	3.68629	0.310442
$142$	0	0
$143$	− 13.7279i	− 1.14799i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.51472	−0.287937	−0.143968	−	0.989582i	$-0.545986\pi$
$149$	−3.51472	−0.287937	−0.143968	+	0.989582i	$0.545986\pi$
$150$	0	0
$151$	−11.8284	−0.962584	−0.481292	−	0.876560i	$-0.659832\pi$
$151$	−11.8284	−0.962584	−0.481292	+	0.876560i	$0.659832\pi$
$152$	0	0
$153$	− 18.1421i	− 1.46670i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.4853i	− 0.836817i	−0.908259	−	0.418408i	$-0.862588\pi$
$157$	− 10.4853i	− 0.836817i	0.908259	−	0.418408i	$-0.137412\pi$
$158$	0	0
$159$	−1.55635	−0.123427
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.5563i	1.68842i	0.536010	+	0.844212i	$0.319931\pi$
$163$	21.5563i	1.68842i	−0.536010	+	0.844212i	$0.680069\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2.41421i	− 0.186817i	−0.995628	−	0.0934087i	$-0.970224\pi$
$167$	− 2.41421i	− 0.186817i	0.995628	−	0.0934087i	$-0.0297763\pi$
$168$	0	0
$169$	0.142136	0.0109335
$170$	0	0
$171$	−10.3431	−0.790960
$172$	0	0
$173$	18.5563i	1.41081i	0.708803	+	0.705407i	$0.249236\pi$
$173$	18.5563i	1.41081i	−0.708803	+	0.705407i	$0.750764\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.41421i	0.106299i
$178$	0	0
$179$	−6.48528	−0.484733	−0.242366	−	0.970185i	$-0.577924\pi$
$179$	−6.48528	−0.484733	−0.242366	+	0.970185i	$0.577924\pi$
$180$	0	0
$181$	1.75736	0.130623	0.0653117	−	0.997865i	$-0.479196\pi$
$181$	1.75736	0.130623	0.0653117	+	0.997865i	$0.479196\pi$
$182$	0	0
$183$	− 2.14214i	− 0.158351i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.5563i	1.79574i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.6569	1.20525	0.602624	−	0.798025i	$-0.294122\pi$
$191$	16.6569	1.20525	0.602624	+	0.798025i	$0.294122\pi$
$192$	0	0
$193$	5.65685i	0.407189i	0.979055	+	0.203595i	$0.0652625\pi$
$193$	5.65685i	0.407189i	−0.979055	+	0.203595i	$0.934738\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	27.5563i	1.96331i	0.190669	+	0.981654i	$0.438934\pi$
$197$	27.5563i	1.96331i	−0.190669	+	0.981654i	$0.561066\pi$
$198$	0	0
$199$	26.7279	1.89469	0.947346	−	0.320212i	$-0.103754\pi$
$199$	26.7279	1.89469	0.947346	+	0.320212i	$0.103754\pi$
$200$	0	0
$201$	4.58579	0.323456
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.65685i	0.115159i
$208$	0	0
$209$	14.0000	0.968400
$210$	0	0
$211$	−24.3137	−1.67382	−0.836912	−	0.547337i	$-0.815642\pi$
$211$	−24.3137	−1.67382	−0.836912	+	0.547337i	$0.815642\pi$
$212$	0	0
$213$	− 2.68629i	− 0.184062i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.14214	0.144752
$220$	0	0
$221$	23.0000	1.54715
$222$	0	0
$223$	24.0711i	1.61192i	0.591971	+	0.805959i	$0.298350\pi$
$223$	24.0711i	1.61192i	−0.591971	+	0.805959i	$0.701650\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 7.72792i	− 0.512920i	−0.966555	−	0.256460i	$-0.917444\pi$
$227$	− 7.72792i	− 0.512920i	0.966555	−	0.256460i	$-0.0825562\pi$
$228$	0	0
$229$	−23.8995	−1.57932	−0.789662	−	0.613543i	$-0.789744\pi$
$229$	−23.8995	−1.57932	−0.789662	+	0.613543i	$0.789744\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 9.17157i	− 0.600850i	−0.953805	−	0.300425i	$-0.902872\pi$
$233$	− 9.17157i	− 0.600850i	0.953805	−	0.300425i	$-0.0971285\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.44365i	0.353603i
$238$	0	0
$239$	−14.1716	−0.916683	−0.458341	−	0.888776i	$-0.651556\pi$
$239$	−14.1716	−0.916683	−0.458341	+	0.888776i	$0.651556\pi$
$240$	0	0
$241$	−3.55635	−0.229085	−0.114542	−	0.993418i	$-0.536540\pi$
$241$	−3.55635	−0.229085	−0.114542	+	0.993418i	$0.536540\pi$
$242$	0	0
$243$	− 10.3431i	− 0.663513i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 13.1127i	− 0.834341i
$248$	0	0
$249$	−3.31371	−0.209998
$250$	0	0
$251$	−27.0711	−1.70871	−0.854355	−	0.519689i	$-0.826048\pi$
$251$	−27.0711	−1.70871	−0.854355	+	0.519689i	$0.826048\pi$
$252$	0	0
$253$	− 2.24264i	− 0.140994i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25.7990i	1.60930i	0.593752	+	0.804648i	$0.297646\pi$
$257$	25.7990i	1.60930i	−0.593752	+	0.804648i	$0.702354\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−18.8284	−1.16545
$262$	0	0
$263$	10.0000i	0.616626i	0.951285	+	0.308313i	$0.0997645\pi$
$263$	10.0000i	0.616626i	−0.951285	+	0.308313i	$0.900236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.02944i	0.430195i
$268$	0	0
$269$	16.7279	1.01992	0.509960	−	0.860198i	$-0.329660\pi$
$269$	16.7279	1.01992	0.509960	+	0.860198i	$0.329660\pi$
$270$	0	0
$271$	28.9706	1.75984	0.879918	−	0.475125i	$-0.157597\pi$
$271$	28.9706	1.75984	0.879918	+	0.475125i	$0.157597\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.41421i	0.445477i	0.974878	+	0.222738i	$0.0714996\pi$
$277$	7.41421i	0.445477i	−0.974878	+	0.222738i	$0.928500\pi$
$278$	0	0
$279$	12.9706	0.776527
$280$	0	0
$281$	−1.00000	−0.0596550	−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000	−0.0596550	−0.0298275	+	0.999555i	$0.509496\pi$
$282$	0	0
$283$	− 18.7574i	− 1.11501i	−0.830174	−	0.557505i	$-0.811759\pi$
$283$	− 18.7574i	− 1.11501i	0.830174	−	0.557505i	$-0.188241\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−24.1421	−1.42013
$290$	0	0
$291$	−6.51472	−0.381900
$292$	0	0
$293$	− 14.4142i	− 0.842087i	−0.907040	−	0.421044i	$-0.861664\pi$
$293$	− 14.4142i	− 0.842087i	0.907040	−	0.421044i	$-0.138336\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	9.24264i	0.536312i
$298$	0	0
$299$	−2.10051	−0.121475
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.00000i	0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3.58579i	− 0.204652i	−0.994751	−	0.102326i	$-0.967372\pi$
$307$	− 3.58579i	− 0.204652i	0.994751	−	0.102326i	$-0.0326284\pi$
$308$	0	0
$309$	0.656854	0.0373671
$310$	0	0
$311$	6.97056	0.395264	0.197632	−	0.980276i	$-0.436675\pi$
$311$	6.97056	0.395264	0.197632	+	0.980276i	$0.436675\pi$
$312$	0	0
$313$	16.5563i	0.935820i	0.883776	+	0.467910i	$0.154993\pi$
$313$	16.5563i	0.935820i	−0.883776	+	0.467910i	$0.845007\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.9706i	0.840831i	0.907332	+	0.420415i	$0.138116\pi$
$317$	14.9706i	0.840831i	−0.907332	+	0.420415i	$0.861884\pi$
$318$	0	0
$319$	25.4853	1.42690
$320$	0	0
$321$	6.97056	0.389059
$322$	0	0
$323$	23.4558i	1.30512i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.72792i	0.206155i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.82843	0.265394	0.132697	−	0.991157i	$-0.457636\pi$
$331$	4.82843	0.265394	0.132697	+	0.991157i	$0.457636\pi$
$332$	0	0
$333$	9.65685i	0.529192i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.7279i	1.02017i	0.860123	+	0.510087i	$0.170387\pi$
$337$	18.7279i	1.02017i	−0.860123	+	0.510087i	$0.829613\pi$
$338$	0	0
$339$	−2.10051	−0.114084
$340$	0	0
$341$	−17.5563	−0.950730
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.41421i	0.398016i	0.979998	+	0.199008i	$0.0637719\pi$
$347$	7.41421i	0.398016i	−0.979998	+	0.199008i	$0.936228\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	8.65685	0.462069
$352$	0	0
$353$	− 2.07107i	− 0.110232i	−0.998480	−	0.0551159i	$-0.982447\pi$
$353$	− 2.07107i	− 0.110232i	0.998480	−	0.0551159i	$-0.0175528\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.3137	−0.597115	−0.298557	−	0.954392i	$-0.596505\pi$
$359$	−11.3137	−0.597115	−0.298557	+	0.954392i	$0.596505\pi$
$360$	0	0
$361$	−5.62742	−0.296180
$362$	0	0
$363$	− 1.51472i	− 0.0795021i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 19.7279i	− 1.02979i	−0.857254	−	0.514895i	$-0.827831\pi$
$367$	− 19.7279i	− 1.02979i	0.857254	−	0.514895i	$-0.172169\pi$
$368$	0	0
$369$	1.65685	0.0862524
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.4853i	0.853576i	0.904352	+	0.426788i	$0.140355\pi$
$373$	16.4853i	0.853576i	−0.904352	+	0.426788i	$0.859645\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 23.8701i	− 1.22937i
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	0	0
$381$	9.07107	0.464725
$382$	0	0
$383$	− 3.51472i	− 0.179594i	−0.995960	−	0.0897969i	$-0.971378\pi$
$383$	− 3.51472i	− 0.179594i	0.995960	−	0.0897969i	$-0.0286218\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	32.9706i	1.67599i
$388$	0	0
$389$	−13.1421	−0.666333	−0.333166	−	0.942868i	$-0.608117\pi$
$389$	−13.1421	−0.666333	−0.333166	+	0.942868i	$0.608117\pi$
$390$	0	0
$391$	3.75736	0.190018
$392$	0	0
$393$	4.87006i	0.245662i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 14.4142i	− 0.723429i	−0.932289	−	0.361714i	$-0.882192\pi$
$397$	− 14.4142i	− 0.723429i	0.932289	−	0.361714i	$-0.117808\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.514719	0.0257038	0.0128519	−	0.999917i	$-0.495909\pi$
$401$	0.514719	0.0257038	0.0128519	+	0.999917i	$0.495909\pi$
$402$	0	0
$403$	16.4437i	0.819117i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 13.0711i	− 0.647909i
$408$	0	0
$409$	−32.8284	−1.62326	−0.811631	−	0.584171i	$-0.801420\pi$
$409$	−32.8284	−1.62326	−0.811631	+	0.584171i	$0.801420\pi$
$410$	0	0
$411$	−5.37258	−0.265010
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5.75736i	− 0.281939i
$418$	0	0
$419$	9.55635	0.466858	0.233429	−	0.972374i	$-0.425005\pi$
$419$	9.55635	0.466858	0.233429	+	0.972374i	$0.425005\pi$
$420$	0	0
$421$	20.3137	0.990030	0.495015	−	0.868885i	$-0.335163\pi$
$421$	20.3137	0.990030	0.495015	+	0.868885i	$0.335163\pi$
$422$	0	0
$423$	25.1716i	1.22388i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.68629	−0.274537
$430$	0	0
$431$	−9.82843	−0.473419	−0.236709	−	0.971581i	$-0.576069\pi$
$431$	−9.82843	−0.473419	−0.236709	+	0.971581i	$0.576069\pi$
$432$	0	0
$433$	− 22.2843i	− 1.07091i	−0.844563	−	0.535457i	$-0.820139\pi$
$433$	− 22.2843i	− 1.07091i	0.844563	−	0.535457i	$-0.179861\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.14214i	− 0.102472i
$438$	0	0
$439$	−24.2426	−1.15704	−0.578519	−	0.815669i	$-0.696369\pi$
$439$	−24.2426	−1.15704	−0.578519	+	0.815669i	$0.696369\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 35.4558i	− 1.68456i	−0.539042	−	0.842279i	$-0.681214\pi$
$443$	− 35.4558i	− 1.68456i	0.539042	−	0.842279i	$-0.318786\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.45584i	0.0688591i
$448$	0	0
$449$	−23.8284	−1.12453	−0.562267	−	0.826956i	$-0.690070\pi$
$449$	−23.8284	−1.12453	−0.562267	+	0.826956i	$0.690070\pi$
$450$	0	0
$451$	−2.24264	−0.105602
$452$	0	0
$453$	4.89949i	0.230198i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.0711i	0.704995i	0.935813	+	0.352497i	$0.114667\pi$
$457$	15.0711i	0.704995i	−0.935813	+	0.352497i	$0.885333\pi$
$458$	0	0
$459$	−15.4853	−0.722791
$460$	0	0
$461$	22.3431	1.04062	0.520312	−	0.853976i	$-0.325816\pi$
$461$	22.3431	1.04062	0.520312	+	0.853976i	$0.325816\pi$
$462$	0	0
$463$	− 13.4558i	− 0.625346i	−0.949861	−	0.312673i	$-0.898776\pi$
$463$	− 13.4558i	− 0.625346i	0.949861	−	0.312673i	$-0.101224\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 0.899495i	− 0.0416237i	−0.999783	−	0.0208118i	$-0.993375\pi$
$467$	− 0.899495i	− 0.0416237i	0.999783	−	0.0208118i	$-0.00662509\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4.34315	−0.200122
$472$	0	0
$473$	− 44.6274i	− 2.05197i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.6274i	− 0.486596i
$478$	0	0
$479$	−9.41421	−0.430146	−0.215073	−	0.976598i	$-0.568999\pi$
$479$	−9.41421	−0.430146	−0.215073	+	0.976598i	$0.568999\pi$
$480$	0	0
$481$	−12.2426	−0.558216
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.41421i	0.335970i	0.985790	+	0.167985i	$0.0537260\pi$
$487$	7.41421i	0.335970i	−0.985790	+	0.167985i	$0.946274\pi$
$488$	0	0
$489$	8.92893	0.403780
$490$	0	0
$491$	17.2843	0.780028	0.390014	−	0.920809i	$-0.372470\pi$
$491$	17.2843	0.780028	0.390014	+	0.920809i	$0.372470\pi$
$492$	0	0
$493$	42.6985i	1.92304i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	13.6274	0.610047	0.305023	−	0.952345i	$-0.401336\pi$
$499$	13.6274	0.610047	0.305023	+	0.952345i	$0.401336\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	39.0416i	1.74078i	0.492363	+	0.870390i	$0.336133\pi$
$503$	39.0416i	1.74078i	−0.492363	+	0.870390i	$0.663867\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.0588745i	− 0.00261471i
$508$	0	0
$509$	35.0711	1.55450	0.777249	−	0.629193i	$-0.216615\pi$
$509$	35.0711	1.55450	0.777249	+	0.629193i	$0.216615\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	8.82843i	0.389785i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 34.0711i	− 1.49844i
$518$	0	0
$519$	7.68629	0.337391
$520$	0	0
$521$	6.68629	0.292932	0.146466	−	0.989216i	$-0.453210\pi$
$521$	6.68629	0.292932	0.146466	+	0.989216i	$0.453210\pi$
$522$	0	0
$523$	1.85786i	0.0812387i	0.999175	+	0.0406194i	$0.0129331\pi$
$523$	1.85786i	0.0812387i	−0.999175	+	0.0406194i	$0.987067\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 29.4142i	− 1.28130i
$528$	0	0
$529$	22.6569	0.985081
$530$	0	0
$531$	−9.65685	−0.419072
$532$	0	0
$533$	2.10051i	0.0909830i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.68629i	0.115922i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	15.9706	0.686628	0.343314	−	0.939221i	$-0.388450\pi$
$541$	15.9706	0.686628	0.343314	+	0.939221i	$0.388450\pi$
$542$	0	0
$543$	− 0.727922i	− 0.0312381i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 7.51472i	− 0.321306i	−0.987011	−	0.160653i	$-0.948640\pi$
$547$	− 7.51472i	− 0.321306i	0.987011	−	0.160653i	$-0.0513600\pi$
$548$	0	0
$549$	14.6274	0.624283
$550$	0	0
$551$	24.3431	1.03705
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.79899i	0.330454i	0.986256	+	0.165227i	$0.0528356\pi$
$557$	7.79899i	0.330454i	−0.986256	+	0.165227i	$0.947164\pi$
$558$	0	0
$559$	−41.7990	−1.76791
$560$	0	0
$561$	10.1716	0.429444
$562$	0	0
$563$	− 20.6274i	− 0.869342i	−0.900589	−	0.434671i	$-0.856865\pi$
$563$	− 20.6274i	− 0.869342i	0.900589	−	0.434671i	$-0.143135\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.7990	1.33308	0.666542	−	0.745468i	$-0.267774\pi$
$569$	31.7990	1.33308	0.666542	+	0.745468i	$0.267774\pi$
$570$	0	0
$571$	4.82843	0.202063	0.101032	−	0.994883i	$-0.467786\pi$
$571$	4.82843	0.202063	0.101032	+	0.994883i	$0.467786\pi$
$572$	0	0
$573$	− 6.89949i	− 0.288231i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.0711i	0.585786i	0.956145	+	0.292893i	$0.0946180\pi$
$577$	14.0711i	0.585786i	−0.956145	+	0.292893i	$0.905382\pi$
$578$	0	0
$579$	2.34315	0.0973778
$580$	0	0
$581$	0	0
$582$	0	0
$583$	14.3848i	0.595757i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.8284i	1.27243i	0.771514	+	0.636213i	$0.219500\pi$
$587$	30.8284i	1.27243i	−0.771514	+	0.636213i	$0.780500\pi$
$588$	0	0
$589$	−16.7696	−0.690977
$590$	0	0
$591$	11.4142	0.469518
$592$	0	0
$593$	17.7279i	0.727999i	0.931399	+	0.363999i	$0.118589\pi$
$593$	17.7279i	0.727999i	−0.931399	+	0.363999i	$0.881411\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11.0711i	− 0.453109i
$598$	0	0
$599$	12.7990	0.522953	0.261476	−	0.965210i	$-0.415791\pi$
$599$	12.7990	0.522953	0.261476	+	0.965210i	$0.415791\pi$
$600$	0	0
$601$	39.6569	1.61764	0.808818	−	0.588058i	$-0.200108\pi$
$601$	39.6569	1.61764	0.808818	+	0.588058i	$0.200108\pi$
$602$	0	0
$603$	31.3137i	1.27519i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 13.9289i	− 0.565358i	−0.959215	−	0.282679i	$-0.908777\pi$
$607$	− 13.9289i	− 0.565358i	0.959215	−	0.282679i	$-0.0912231\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.9117	−1.29101
$612$	0	0
$613$	− 41.3137i	− 1.66864i	−0.551277	−	0.834322i	$-0.685859\pi$
$613$	− 41.3137i	− 1.66864i	0.551277	−	0.834322i	$-0.314141\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 40.8701i	− 1.64537i	−0.568500	−	0.822683i	$-0.692476\pi$
$617$	− 40.8701i	− 1.64537i	0.568500	−	0.822683i	$-0.307524\pi$
$618$	0	0
$619$	−10.8701	−0.436905	−0.218452	−	0.975848i	$-0.570101\pi$
$619$	−10.8701	−0.436905	−0.218452	+	0.975848i	$0.570101\pi$
$620$	0	0
$621$	1.41421	0.0567504
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 5.79899i	− 0.231589i
$628$	0	0
$629$	21.8995	0.873190
$630$	0	0
$631$	−39.4853	−1.57188	−0.785942	−	0.618300i	$-0.787822\pi$
$631$	−39.4853	−1.57188	−0.785942	+	0.618300i	$0.787822\pi$
$632$	0	0
$633$	10.0711i	0.400289i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	18.3431	0.725644
$640$	0	0
$641$	39.3137	1.55280	0.776399	−	0.630242i	$-0.217044\pi$
$641$	39.3137	1.55280	0.776399	+	0.630242i	$0.217044\pi$
$642$	0	0
$643$	4.21320i	0.166153i	0.996543	+	0.0830763i	$0.0264745\pi$
$643$	4.21320i	0.166153i	−0.996543	+	0.0830763i	$0.973525\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 35.1127i	− 1.38042i	−0.723608	−	0.690211i	$-0.757518\pi$
$647$	− 35.1127i	− 1.38042i	0.723608	−	0.690211i	$-0.242482\pi$
$648$	0	0
$649$	13.0711	0.513084
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 26.6863i	− 1.04432i	−0.852849	−	0.522158i	$-0.825127\pi$
$653$	− 26.6863i	− 1.04432i	0.852849	−	0.522158i	$-0.174873\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.6274i	0.570670i
$658$	0	0
$659$	20.6569	0.804677	0.402338	−	0.915491i	$-0.368198\pi$
$659$	20.6569	0.804677	0.402338	+	0.915491i	$0.368198\pi$
$660$	0	0
$661$	−34.1421	−1.32798	−0.663988	−	0.747744i	$-0.731137\pi$
$661$	−34.1421	−1.32798	−0.663988	+	0.747744i	$0.731137\pi$
$662$	0	0
$663$	− 9.52691i	− 0.369995i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.89949i	− 0.150989i
$668$	0	0
$669$	9.97056	0.385484
$670$	0	0
$671$	−19.7990	−0.764332
$672$	0	0
$673$	− 27.5147i	− 1.06061i	−0.847806	−	0.530307i	$-0.822077\pi$
$673$	− 27.5147i	− 1.06061i	0.847806	−	0.530307i	$-0.177923\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.2721i	0.394788i	0.980324	+	0.197394i	$0.0632478\pi$
$677$	10.2721i	0.394788i	−0.980324	+	0.197394i	$0.936752\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.20101	−0.122663
$682$	0	0
$683$	35.1127i	1.34355i	0.740755	+	0.671775i	$0.234468\pi$
$683$	35.1127i	1.34355i	−0.740755	+	0.671775i	$0.765532\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.89949i	0.377689i
$688$	0	0
$689$	13.4731	0.513284
$690$	0	0
$691$	3.51472	0.133706	0.0668531	−	0.997763i	$-0.478704\pi$
$691$	3.51472	0.133706	0.0668531	+	0.997763i	$0.478704\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3.75736i	− 0.142320i
$698$	0	0
$699$	−3.79899	−0.143691
$700$	0	0
$701$	0.514719	0.0194407	0.00972033	−	0.999953i	$-0.496906\pi$
$701$	0.514719	0.0194407	0.00972033	+	0.999953i	$0.496906\pi$
$702$	0	0
$703$	− 12.4853i	− 0.470891i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−3.62742	−0.136231	−0.0681153	−	0.997677i	$-0.521699\pi$
$709$	−3.62742	−0.136231	−0.0681153	+	0.997677i	$0.521699\pi$
$710$	0	0
$711$	−37.1716	−1.39404
$712$	0	0
$713$	2.68629i	0.100602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.87006i	0.219221i
$718$	0	0
$719$	−15.7574	−0.587650	−0.293825	−	0.955859i	$-0.594928\pi$
$719$	−15.7574	−0.587650	−0.293825	+	0.955859i	$0.594928\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.47309i	0.0547847i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.0000i	0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$	18.0000i	0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$	0	0
$729$	18.1716	0.673021
$730$	0	0
$731$	74.7696	2.76545
$732$	0	0
$733$	− 30.6985i	− 1.13387i	−0.823761	−	0.566937i	$-0.808128\pi$
$733$	− 30.6985i	− 1.13387i	0.823761	−	0.566937i	$-0.191872\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 42.3848i	− 1.56126i
$738$	0	0
$739$	−20.6569	−0.759875	−0.379937	−	0.925012i	$-0.624054\pi$
$739$	−20.6569	−0.759875	−0.379937	+	0.925012i	$0.624054\pi$
$740$	0	0
$741$	−5.43146	−0.199530
$742$	0	0
$743$	− 8.92893i	− 0.327571i	−0.986496	−	0.163785i	$-0.947630\pi$
$743$	− 8.92893i	− 0.327571i	0.986496	−	0.163785i	$-0.0523705\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 22.6274i	− 0.827894i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.14214	0.0416771	0.0208386	−	0.999783i	$-0.493366\pi$
$751$	1.14214	0.0416771	0.0208386	+	0.999783i	$0.493366\pi$
$752$	0	0
$753$	11.2132i	0.408632i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 37.1716i	− 1.35102i	−0.737349	−	0.675512i	$-0.763923\pi$
$757$	− 37.1716i	− 1.35102i	0.737349	−	0.675512i	$-0.236077\pi$
$758$	0	0
$759$	−0.928932	−0.0337181
$760$	0	0
$761$	−18.8701	−0.684039	−0.342020	−	0.939693i	$-0.611111\pi$
$761$	−18.8701	−0.684039	−0.342020	+	0.939693i	$0.611111\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 12.2426i	− 0.442056i
$768$	0	0
$769$	−28.1421	−1.01483	−0.507416	−	0.861701i	$-0.669399\pi$
$769$	−28.1421	−1.01483	−0.507416	+	0.861701i	$0.669399\pi$
$770$	0	0
$771$	10.6863	0.384857
$772$	0	0
$773$	− 43.7279i	− 1.57278i	−0.617728	−	0.786392i	$-0.711947\pi$
$773$	− 43.7279i	− 1.57278i	0.617728	−	0.786392i	$-0.288053\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.14214	−0.0767500
$780$	0	0
$781$	−24.8284	−0.888431
$782$	0	0
$783$	16.0711i	0.574333i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	43.7279i	1.55873i	0.626569	+	0.779366i	$0.284459\pi$
$787$	43.7279i	1.55873i	−0.626569	+	0.779366i	$0.715541\pi$
$788$	0	0
$789$	4.14214	0.147464
$790$	0	0
$791$	0	0
$792$	0	0
$793$	18.5442i	0.658522i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.3848i	1.25339i	0.779263	+	0.626697i	$0.215593\pi$
$797$	35.3848i	1.25339i	−0.779263	+	0.626697i	$0.784407\pi$
$798$	0	0
$799$	57.0833	2.01946
$800$	0	0
$801$	−48.0000	−1.69600
$802$	0	0
$803$	− 19.7990i	− 0.698691i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6.92893i	− 0.243910i
$808$	0	0
$809$	−6.02944	−0.211984	−0.105992	−	0.994367i	$-0.533802\pi$
$809$	−6.02944	−0.211984	−0.105992	+	0.994367i	$0.533802\pi$
$810$	0	0
$811$	−19.5563	−0.686716	−0.343358	−	0.939205i	$-0.611564\pi$
$811$	−19.5563	−0.686716	−0.343358	+	0.939205i	$0.611564\pi$
$812$	0	0
$813$	− 12.0000i	− 0.420858i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 42.6274i	− 1.49134i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.8284	1.66922	0.834612	−	0.550839i	$-0.185692\pi$
$821$	47.8284	1.66922	0.834612	+	0.550839i	$0.185692\pi$
$822$	0	0
$823$	36.5269i	1.27325i	0.771174	+	0.636624i	$0.219670\pi$
$823$	36.5269i	1.27325i	−0.771174	+	0.636624i	$0.780330\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.0416i	0.349182i	0.984641	+	0.174591i	$0.0558603\pi$
$827$	10.0416i	0.349182i	−0.984641	+	0.174591i	$0.944140\pi$
$828$	0	0
$829$	−22.7279	−0.789373	−0.394687	−	0.918816i	$-0.629147\pi$
$829$	−22.7279	−0.789373	−0.394687	+	0.918816i	$0.629147\pi$
$830$	0	0
$831$	3.07107	0.106534
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 11.0711i	− 0.382672i
$838$	0	0
$839$	−22.3848	−0.772808	−0.386404	−	0.922330i	$-0.626283\pi$
$839$	−22.3848	−0.772808	−0.386404	+	0.922330i	$0.626283\pi$
$840$	0	0
$841$	15.3137	0.528059
$842$	0	0
$843$	0.414214i	0.0142663i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7.76955	−0.266650
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	0	0
$853$	− 30.2843i	− 1.03691i	−0.855104	−	0.518457i	$-0.826507\pi$
$853$	− 30.2843i	− 1.03691i	0.855104	−	0.518457i	$-0.173493\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.8284i	0.711486i	0.934584	+	0.355743i	$0.115772\pi$
$857$	20.8284i	0.711486i	−0.934584	+	0.355743i	$0.884228\pi$
$858$	0	0
$859$	45.4558	1.55093	0.775467	−	0.631388i	$-0.217515\pi$
$859$	45.4558	1.55093	0.775467	+	0.631388i	$0.217515\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 19.7574i	− 0.672548i	−0.941764	−	0.336274i	$-0.890833\pi$
$863$	− 19.7574i	− 0.672548i	0.941764	−	0.336274i	$-0.109167\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10.0000i	0.339618i
$868$	0	0
$869$	50.3137	1.70678
$870$	0	0
$871$	−39.6985	−1.34513
$872$	0	0
$873$	− 44.4853i	− 1.50560i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 10.6863i	− 0.360850i	−0.983589	−	0.180425i	$-0.942253\pi$
$877$	− 10.6863i	− 0.360850i	0.983589	−	0.180425i	$-0.0577474\pi$
$878$	0	0
$879$	−5.97056	−0.201382
$880$	0	0
$881$	−56.2843	−1.89627	−0.948133	−	0.317875i	$-0.897031\pi$
$881$	−56.2843	−1.89627	−0.948133	+	0.317875i	$0.897031\pi$
$882$	0	0
$883$	− 7.11270i	− 0.239361i	−0.992812	−	0.119681i	$-0.961813\pi$
$883$	− 7.11270i	− 0.239361i	0.992812	−	0.119681i	$-0.0381871\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 57.5980i	− 1.93395i	−0.254870	−	0.966975i	$-0.582033\pi$
$887$	− 57.5980i	− 1.93395i	0.254870	−	0.966975i	$-0.417967\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−28.6569	−0.960041
$892$	0	0
$893$	− 32.5442i	− 1.08905i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.870058i	0.0290504i
$898$	0	0
$899$	−30.5269	−1.01813
$900$	0	0
$901$	−24.1005	−0.802904
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 22.5269i	− 0.747994i	−0.927430	−	0.373997i	$-0.877987\pi$
$907$	− 22.5269i	− 0.747994i	0.927430	−	0.373997i	$-0.122013\pi$
$908$	0	0
$909$	−13.6569	−0.452969
$910$	0	0
$911$	−55.5980	−1.84204	−0.921022	−	0.389511i	$-0.872644\pi$
$911$	−55.5980	−1.84204	−0.921022	+	0.389511i	$0.872644\pi$
$912$	0	0
$913$	30.6274i	1.01362i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	18.5147	0.610744	0.305372	−	0.952233i	$-0.401219\pi$
$919$	18.5147	0.610744	0.305372	+	0.952233i	$0.401219\pi$
$920$	0	0
$921$	−1.48528	−0.0489417
$922$	0	0
$923$	23.2548i	0.765442i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.48528i	0.147316i
$928$	0	0
$929$	21.2721	0.697914	0.348957	−	0.937139i	$-0.386536\pi$
$929$	21.2721	0.697914	0.348957	+	0.937139i	$0.386536\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 2.88730i	− 0.0945260i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.72792i	0.252460i	0.992001	+	0.126230i	$0.0402878\pi$
$937$	7.72792i	0.252460i	−0.992001	+	0.126230i	$0.959712\pi$
$938$	0	0
$939$	6.85786	0.223798
$940$	0	0
$941$	−4.00000	−0.130396	−0.0651981	−	0.997872i	$-0.520768\pi$
$941$	−4.00000	−0.130396	−0.0651981	+	0.997872i	$0.520768\pi$
$942$	0	0
$943$	0.343146i	0.0111744i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.0416i	1.04121i	0.853797	+	0.520607i	$0.174294\pi$
$947$	32.0416i	1.04121i	−0.853797	+	0.520607i	$0.825706\pi$
$948$	0	0
$949$	−18.5442	−0.601969
$950$	0	0
$951$	6.20101	0.201082
$952$	0	0
$953$	− 38.8701i	− 1.25912i	−0.776950	−	0.629562i	$-0.783234\pi$
$953$	− 38.8701i	− 1.25912i	0.776950	−	0.629562i	$-0.216766\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 10.5563i	− 0.341238i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−9.97056	−0.321631
$962$	0	0
$963$	47.5980i	1.53382i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.45584i	0.239764i	0.992788	+	0.119882i	$0.0382516\pi$
$967$	7.45584i	0.239764i	−0.992788	+	0.119882i	$0.961748\pi$
$968$	0	0
$969$	9.71573	0.312114
$970$	0	0
$971$	−22.5269	−0.722923	−0.361462	−	0.932387i	$-0.617722\pi$
$971$	−22.5269	−0.722923	−0.361462	+	0.932387i	$0.617722\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.14214i	0.260490i	0.991482	+	0.130245i	$0.0415764\pi$
$977$	8.14214i	0.260490i	−0.991482	+	0.130245i	$0.958424\pi$
$978$	0	0
$979$	64.9706	2.07647
$980$	0	0
$981$	−25.4558	−0.812743
$982$	0	0
$983$	− 42.0122i	− 1.33998i	−0.742370	−	0.669990i	$-0.766298\pi$
$983$	− 42.0122i	− 1.33998i	0.742370	−	0.669990i	$-0.233702\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.82843	−0.217131
$990$	0	0
$991$	25.3137	0.804116	0.402058	−	0.915614i	$-0.368295\pi$
$991$	25.3137	0.804116	0.402058	+	0.915614i	$0.368295\pi$
$992$	0	0
$993$	− 2.00000i	− 0.0634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	50.8406i	1.61014i	0.593181	+	0.805069i	$0.297872\pi$
$997$	50.8406i	1.61014i	−0.593181	+	0.805069i	$0.702128\pi$
$998$	0	0
$999$	8.24264	0.260786

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.r.2549.2		4
5.2	odd	4		980.2.a.k.1.1	yes	2
5.3	odd	4		4900.2.a.x.1.2		2
5.4	even	2	inner	4900.2.e.r.2549.3		4
7.6	odd	2		4900.2.e.q.2549.3		4
15.2	even	4		8820.2.a.bl.1.2		2
20.7	even	4		3920.2.a.bo.1.2		2
35.2	odd	12		980.2.i.k.361.2		4
35.12	even	12		980.2.i.l.361.1		4
35.13	even	4		4900.2.a.z.1.1		2
35.17	even	12		980.2.i.l.961.1		4
35.27	even	4		980.2.a.j.1.2	✓	2
35.32	odd	12		980.2.i.k.961.2		4
35.34	odd	2		4900.2.e.q.2549.2		4
105.62	odd	4		8820.2.a.bg.1.2		2
140.27	odd	4		3920.2.a.bx.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.a.j.1.2	✓	2	35.27	even	4
980.2.a.k.1.1	yes	2	5.2	odd	4
980.2.i.k.361.2		4	35.2	odd	12
980.2.i.k.961.2		4	35.32	odd	12
980.2.i.l.361.1		4	35.12	even	12
980.2.i.l.961.1		4	35.17	even	12
3920.2.a.bo.1.2		2	20.7	even	4
3920.2.a.bx.1.1		2	140.27	odd	4
4900.2.a.x.1.2		2	5.3	odd	4
4900.2.a.z.1.1		2	35.13	even	4
4900.2.e.q.2549.2		4	35.34	odd	2
4900.2.e.q.2549.3		4	7.6	odd	2
4900.2.e.r.2549.2		4	1.1	even	1	trivial
4900.2.e.r.2549.3		4	5.4	even	2	inner
8820.2.a.bg.1.2		2	105.62	odd	4
8820.2.a.bl.1.2		2	15.2	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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.414214i q^{3} +2.82843 q^{9} +O(q^{10})\)	\(q-0.414214i q^{3} +2.82843 q^{9} -3.82843 q^{11} +3.58579i q^{13} -6.41421i q^{17} -3.65685 q^{19} +0.585786i q^{23} -2.41421i q^{27} -6.65685 q^{29} +4.58579 q^{31} +1.58579i q^{33} +3.41421i q^{37} +1.48528 q^{39} +0.585786 q^{41} +11.6569i q^{43} +8.89949i q^{47} -2.65685 q^{51} -3.75736i q^{53} +1.51472i q^{57} -3.41421 q^{59} +5.17157 q^{61} +11.0711i q^{67} +0.242641 q^{69} +6.48528 q^{71} +5.17157i q^{73} -13.1421 q^{79} +7.48528 q^{81} -8.00000i q^{83} +2.75736i q^{87} -16.9706 q^{89} -1.89949i q^{93} -15.7279i q^{97} -10.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{11} + 8 q^{19} - 4 q^{29} + 24 q^{31} - 28 q^{39} + 8 q^{41} + 12 q^{51} - 8 q^{59} + 32 q^{61} - 16 q^{69} - 8 q^{71} + 4 q^{79} - 4 q^{81} - 32 q^{99}+O(q^{100}) \) 4 * q - 4 * q^11 + 8 * q^19 - 4 * q^29 + 24 * q^31 - 28 * q^39 + 8 * q^41 + 12 * q^51 - 8 * q^59 + 32 * q^61 - 16 * q^69 - 8 * q^71 + 4 * q^79 - 4 * q^81 - 32 * q^99

Introduction

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