LMFDB - Embedded newform 4900.2.e.p.2549.3

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:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 196)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.p.2549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{3} -5.00000 q^{9} +4.00000 q^{11} -4.24264i q^{13} +1.41421i q^{17} -2.82843 q^{19} +4.00000i q^{23} -5.65685i q^{27} -8.00000 q^{29} +11.3137i q^{33} -8.00000i q^{37} +12.0000 q^{39} -7.07107 q^{41} +4.00000i q^{43} +5.65685i q^{47} -4.00000 q^{51} -10.0000i q^{53} -8.00000i q^{57} -14.1421 q^{59} -7.07107 q^{61} -11.3137 q^{69} +7.07107i q^{73} -8.00000 q^{79} +1.00000 q^{81} +14.1421i q^{83} -22.6274i q^{87} -7.07107 q^{89} -1.41421i q^{97} -20.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{9} + 16 q^{11} - 32 q^{29} + 48 q^{39} - 16 q^{51} - 32 q^{79} + 4 q^{81} - 80 q^{99}+O(q^{100}) $ 4 * q - 20 * q^9 + 16 * q^11 - 32 * q^29 + 48 * q^39 - 16 * q^51 - 32 * q^79 + 4 * q^81 - 80 * q^99

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
$3$	2.82843i	1.63299i	0.577350	+	0.816497i	$0.304087\pi$
$3$	2.82843i	1.63299i	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−5.00000	−1.66667
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	− 4.24264i	− 1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$	− 4.24264i	− 1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.41421i	0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$	1.41421i	0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	11.3137i	1.96946i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	0	0
$39$	12.0000	1.92154
$40$	0	0
$41$	−7.07107	−1.10432	−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107	−1.10432	−0.552158	+	0.833740i	$0.686195\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.65685i	0.825137i	0.910927	+	0.412568i	$0.135368\pi$
$47$	5.65685i	0.825137i	−0.910927	+	0.412568i	$0.864632\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8.00000i	− 1.05963i
$58$	0	0
$59$	−14.1421	−1.84115	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	−14.1421	−1.84115	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	−7.07107	−0.905357	−0.452679	−	0.891674i	$-0.649532\pi$
$61$	−7.07107	−0.905357	−0.452679	+	0.891674i	$0.649532\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−11.3137	−1.36201
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	7.07107i	0.827606i	0.910366	+	0.413803i	$0.135800\pi$
$73$	7.07107i	0.827606i	−0.910366	+	0.413803i	$0.864200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.1421i	1.55230i	0.630548	+	0.776151i	$0.282830\pi$
$83$	14.1421i	1.55230i	−0.630548	+	0.776151i	$0.717170\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 22.6274i	− 2.42591i
$88$	0	0
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.41421i	− 0.143592i	−0.997419	−	0.0717958i	$-0.977127\pi$
$97$	− 1.41421i	− 0.143592i	0.997419	−	0.0717958i	$-0.0228730\pi$
$98$	0	0
$99$	−20.0000	−2.01008
$100$	0	0
$101$	−12.7279	−1.26648	−0.633238	−	0.773957i	$-0.718274\pi$
$101$	−12.7279	−1.26648	−0.633238	+	0.773957i	$0.718274\pi$
$102$	0	0
$103$	11.3137i	1.11477i	0.830253	+	0.557386i	$0.188196\pi$
$103$	11.3137i	1.11477i	−0.830253	+	0.557386i	$0.811804\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	22.6274	2.14770
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	21.2132i	1.96116i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	− 20.0000i	− 1.80334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 20.0000i	− 1.77471i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	− 20.0000i	− 1.77471i	0.461084	−	0.887357i	$-0.347461\pi$
$128$	0	0
$129$	−11.3137	−0.996116
$130$	0	0
$131$	−8.48528	−0.741362	−0.370681	−	0.928760i	$-0.620876\pi$
$131$	−8.48528	−0.741362	−0.370681	+	0.928760i	$0.620876\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−2.82843	−0.239904	−0.119952	−	0.992780i	$-0.538274\pi$
$139$	−2.82843	−0.239904	−0.119952	+	0.992780i	$0.538274\pi$
$140$	0	0
$141$	−16.0000	−1.34744
$142$	0	0
$143$	− 16.9706i	− 1.41915i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	− 7.07107i	− 0.571662i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.07107i	0.564333i	0.959366	+	0.282166i	$0.0910530\pi$
$157$	7.07107i	0.564333i	−0.959366	+	0.282166i	$0.908947\pi$
$158$	0	0
$159$	28.2843	2.24309
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.65685i	− 0.437741i	−0.975754	−	0.218870i	$-0.929763\pi$
$167$	− 5.65685i	− 0.437741i	0.975754	−	0.218870i	$-0.0702371\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	14.1421	1.08148
$172$	0	0
$173$	4.24264i	0.322562i	0.986909	+	0.161281i	$0.0515625\pi$
$173$	4.24264i	0.322562i	−0.986909	+	0.161281i	$0.948437\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 40.0000i	− 3.00658i
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−21.2132	−1.57676	−0.788382	−	0.615185i	$-0.789081\pi$
$181$	−21.2132	−1.57676	−0.788382	+	0.615185i	$0.789081\pi$
$182$	0	0
$183$	− 20.0000i	− 1.47844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.65685i	0.413670i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 10.0000i	− 0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$	− 10.0000i	− 0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 20.0000i	− 1.39010i
$208$	0	0
$209$	−11.3137	−0.782586
$210$	0	0
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−20.0000	−1.35147
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	− 16.9706i	− 1.13643i	−0.822879	−	0.568216i	$-0.807634\pi$
$223$	− 16.9706i	− 1.13643i	0.822879	−	0.568216i	$-0.192366\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.48528i	0.563188i	0.959534	+	0.281594i	$0.0908631\pi$
$227$	8.48528i	0.563188i	−0.959534	+	0.281594i	$0.909137\pi$
$228$	0	0
$229$	21.2132	1.40181	0.700904	−	0.713256i	$-0.252780\pi$
$229$	21.2132	1.40181	0.700904	+	0.713256i	$0.252780\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 22.6274i	− 1.46981i
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	12.7279	0.819878	0.409939	−	0.912113i	$-0.365550\pi$
$241$	12.7279	0.819878	0.409939	+	0.912113i	$0.365550\pi$
$242$	0	0
$243$	− 14.1421i	− 0.907218i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000i	0.763542i
$248$	0	0
$249$	−40.0000	−2.53490
$250$	0	0
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.2132i	1.32324i	0.749838	+	0.661622i	$0.230131\pi$
$257$	21.2132i	1.32324i	−0.749838	+	0.661622i	$0.769869\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	40.0000	2.47594
$262$	0	0
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 20.0000i	− 1.22398i
$268$	0	0
$269$	18.3848	1.12094	0.560470	−	0.828175i	$-0.310621\pi$
$269$	18.3848	1.12094	0.560470	+	0.828175i	$0.310621\pi$
$270$	0	0
$271$	28.2843	1.71815	0.859074	−	0.511852i	$-0.171040\pi$
$271$	28.2843	1.71815	0.859074	+	0.511852i	$0.171040\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	2.82843i	0.168133i	0.996460	+	0.0840663i	$0.0267907\pi$
$283$	2.82843i	0.168133i	−0.996460	+	0.0840663i	$0.973209\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	4.00000	0.234484
$292$	0	0
$293$	− 32.5269i	− 1.90024i	−0.311881	−	0.950121i	$-0.600959\pi$
$293$	− 32.5269i	− 1.90024i	0.311881	−	0.950121i	$-0.399041\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 22.6274i	− 1.31298i
$298$	0	0
$299$	16.9706	0.981433
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 36.0000i	− 2.06815i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 19.7990i	− 1.12999i	−0.825095	−	0.564994i	$-0.808878\pi$
$307$	− 19.7990i	− 1.12999i	0.825095	−	0.564994i	$-0.191122\pi$
$308$	0	0
$309$	−32.0000	−1.82042
$310$	0	0
$311$	−22.6274	−1.28308	−0.641542	−	0.767088i	$-0.721705\pi$
$311$	−22.6274	−1.28308	−0.641542	+	0.767088i	$0.721705\pi$
$312$	0	0
$313$	− 4.24264i	− 0.239808i	−0.992785	−	0.119904i	$-0.961741\pi$
$313$	− 4.24264i	− 0.239808i	0.992785	−	0.119904i	$-0.0382587\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	0	0
$319$	−32.0000	−1.79166
$320$	0	0
$321$	−22.6274	−1.26294
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	22.6274i	1.25130i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	40.0000i	2.19199i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.00000i	0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$	8.00000i	0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	0	0
$339$	16.9706	0.921714
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	4.24264	0.227103	0.113552	−	0.993532i	$-0.463777\pi$
$349$	4.24264	0.227103	0.113552	+	0.993532i	$0.463777\pi$
$350$	0	0
$351$	−24.0000	−1.28103
$352$	0	0
$353$	− 9.89949i	− 0.526897i	−0.964673	−	0.263448i	$-0.915140\pi$
$353$	− 9.89949i	− 0.526897i	0.964673	−	0.263448i	$-0.0848599\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	14.1421i	0.742270i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.65685i	0.295285i	0.989041	+	0.147643i	$0.0471686\pi$
$367$	5.65685i	0.295285i	−0.989041	+	0.147643i	$0.952831\pi$
$368$	0	0
$369$	35.3553	1.84053
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	33.9411i	1.74806i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	56.5685	2.89809
$382$	0	0
$383$	5.65685i	0.289052i	0.989501	+	0.144526i	$0.0461657\pi$
$383$	5.65685i	0.289052i	−0.989501	+	0.144526i	$0.953834\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 20.0000i	− 1.01666i
$388$	0	0
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	− 24.0000i	− 1.21064i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.5563i	0.780751i	0.920656	+	0.390375i	$0.127655\pi$
$397$	15.5563i	0.780751i	−0.920656	+	0.390375i	$0.872345\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 32.0000i	− 1.58618i
$408$	0	0
$409$	−38.1838	−1.88807	−0.944033	−	0.329851i	$-0.893001\pi$
$409$	−38.1838	−1.88807	−0.944033	+	0.329851i	$0.893001\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8.00000i	− 0.391762i
$418$	0	0
$419$	−14.1421	−0.690889	−0.345444	−	0.938439i	$-0.612272\pi$
$419$	−14.1421	−0.690889	−0.345444	+	0.938439i	$0.612272\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	− 28.2843i	− 1.37523i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	48.0000	2.31746
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	21.2132i	1.01944i	0.860340	+	0.509721i	$0.170251\pi$
$433$	21.2132i	1.01944i	−0.860340	+	0.509721i	$0.829749\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 11.3137i	− 0.541208i
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.0000i	0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$	16.0000i	0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 28.2843i	− 1.33780i
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−28.2843	−1.33185
$452$	0	0
$453$	− 11.3137i	− 0.531564i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.0000i	1.40334i	0.712502	+	0.701670i	$0.247562\pi$
$457$	30.0000i	1.40334i	−0.712502	+	0.701670i	$0.752438\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	7.07107	0.329332	0.164666	−	0.986349i	$-0.447345\pi$
$461$	7.07107	0.329332	0.164666	+	0.986349i	$0.447345\pi$
$462$	0	0
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 19.7990i	− 0.916188i	−0.888904	−	0.458094i	$-0.848532\pi$
$467$	− 19.7990i	− 0.916188i	0.888904	−	0.458094i	$-0.151468\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	50.0000i	2.28934i
$478$	0	0
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	0	0
$481$	−33.9411	−1.54758
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 12.0000i	− 0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$	− 12.0000i	− 0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	0	0
$489$	11.3137	0.511624
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	− 11.3137i	− 0.509544i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	39.5980i	1.76559i	0.469762	+	0.882793i	$0.344340\pi$
$503$	39.5980i	1.76559i	−0.469762	+	0.882793i	$0.655660\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 14.1421i	− 0.628074i
$508$	0	0
$509$	18.3848	0.814891	0.407445	−	0.913230i	$-0.366420\pi$
$509$	18.3848	0.814891	0.407445	+	0.913230i	$0.366420\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	16.0000i	0.706417i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	22.6274i	0.995153i
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−41.0122	−1.79678	−0.898388	−	0.439202i	$-0.855261\pi$
$521$	−41.0122	−1.79678	−0.898388	+	0.439202i	$0.855261\pi$
$522$	0	0
$523$	− 42.4264i	− 1.85518i	−0.373603	−	0.927589i	$-0.621878\pi$
$523$	− 42.4264i	− 1.85518i	0.373603	−	0.927589i	$-0.378122\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	70.7107	3.06858
$532$	0	0
$533$	30.0000i	1.29944i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	− 60.0000i	− 2.57485i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	0	0
$549$	35.3553	1.50893
$550$	0	0
$551$	22.6274	0.963960
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	16.9706	0.717778
$560$	0	0
$561$	−16.0000	−0.675521
$562$	0	0
$563$	− 14.1421i	− 0.596020i	−0.954563	−	0.298010i	$-0.903677\pi$
$563$	− 14.1421i	− 0.596020i	0.954563	−	0.298010i	$-0.0963229\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−40.0000	−1.67689	−0.838444	−	0.544988i	$-0.816534\pi$
$569$	−40.0000	−1.67689	−0.838444	+	0.544988i	$0.816534\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	− 45.2548i	− 1.89055i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.7279i	0.529870i	0.964266	+	0.264935i	$0.0853506\pi$
$577$	12.7279i	0.529870i	−0.964266	+	0.264935i	$0.914649\pi$
$578$	0	0
$579$	−28.2843	−1.17545
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 40.0000i	− 1.65663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.4558i	1.05068i	0.850894	+	0.525338i	$0.176061\pi$
$587$	25.4558i	1.05068i	−0.850894	+	0.525338i	$0.823939\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	28.2843	1.16346
$592$	0	0
$593$	9.89949i	0.406524i	0.979124	+	0.203262i	$0.0651542\pi$
$593$	9.89949i	0.406524i	−0.979124	+	0.203262i	$0.934846\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	−29.6985	−1.21143	−0.605713	−	0.795683i	$-0.707112\pi$
$601$	−29.6985	−1.21143	−0.605713	+	0.795683i	$0.707112\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 33.9411i	− 1.37763i	−0.724938	−	0.688814i	$-0.758132\pi$
$607$	− 33.9411i	− 1.37763i	0.724938	−	0.688814i	$-0.241868\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	0	0
$613$	− 24.0000i	− 0.969351i	−0.874694	−	0.484675i	$-0.838938\pi$
$613$	− 24.0000i	− 0.969351i	0.874694	−	0.484675i	$-0.161062\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 8.00000i	− 0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$	− 8.00000i	− 0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$	0	0
$619$	31.1127	1.25052	0.625262	−	0.780415i	$-0.284992\pi$
$619$	31.1127	1.25052	0.625262	+	0.780415i	$0.284992\pi$
$620$	0	0
$621$	22.6274	0.908007
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 32.0000i	− 1.27796i
$628$	0	0
$629$	11.3137	0.451107
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	67.8823i	2.69808i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.00000	0.315981	0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000	0.315981	0.157991	+	0.987441i	$0.449498\pi$
$642$	0	0
$643$	− 2.82843i	− 0.111542i	−0.998444	−	0.0557711i	$-0.982238\pi$
$643$	− 2.82843i	− 0.111542i	0.998444	−	0.0557711i	$-0.0177617\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−56.5685	−2.22051
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.0000i	0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$	24.0000i	0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 35.3553i	− 1.37934i
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−21.2132	−0.825098	−0.412549	−	0.910935i	$-0.635361\pi$
$661$	−21.2132	−0.825098	−0.412549	+	0.910935i	$0.635361\pi$
$662$	0	0
$663$	16.9706i	0.659082i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 32.0000i	− 1.23904i
$668$	0	0
$669$	48.0000	1.85579
$670$	0	0
$671$	−28.2843	−1.09190
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.7279i	0.489174i	0.969627	+	0.244587i	$0.0786523\pi$
$677$	12.7279i	0.489174i	−0.969627	+	0.244587i	$0.921348\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	60.0000i	2.28914i
$688$	0	0
$689$	−42.4264	−1.61632
$690$	0	0
$691$	−42.4264	−1.61398	−0.806988	−	0.590567i	$-0.798904\pi$
$691$	−42.4264	−1.61398	−0.806988	+	0.590567i	$0.798904\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 10.0000i	− 0.378777i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	22.6274i	0.853409i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	0	0
$711$	40.0000	1.50012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	33.9411i	1.26755i
$718$	0	0
$719$	−39.5980	−1.47676	−0.738378	−	0.674387i	$-0.764408\pi$
$719$	−39.5980	−1.47676	−0.738378	+	0.674387i	$0.764408\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	36.0000i	1.33885i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.2843i	1.04901i	0.851409	+	0.524503i	$0.175749\pi$
$727$	28.2843i	1.04901i	−0.851409	+	0.524503i	$0.824251\pi$
$728$	0	0
$729$	43.0000	1.59259
$730$	0	0
$731$	−5.65685	−0.209226
$732$	0	0
$733$	38.1838i	1.41035i	0.709034	+	0.705175i	$0.249131\pi$
$733$	38.1838i	1.41035i	−0.709034	+	0.705175i	$0.750869\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	−33.9411	−1.24686
$742$	0	0
$743$	20.0000i	0.733729i	0.930274	+	0.366864i	$0.119569\pi$
$743$	20.0000i	0.733729i	−0.930274	+	0.366864i	$0.880431\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 70.7107i	− 2.58717i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	56.0000i	2.04075i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 40.0000i	− 1.45382i	−0.686730	−	0.726912i	$-0.740955\pi$
$757$	− 40.0000i	− 1.45382i	0.686730	−	0.726912i	$-0.259045\pi$
$758$	0	0
$759$	−45.2548	−1.64265
$760$	0	0
$761$	41.0122	1.48669	0.743345	−	0.668908i	$-0.233238\pi$
$761$	41.0122	1.48669	0.743345	+	0.668908i	$0.233238\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	60.0000i	2.16647i
$768$	0	0
$769$	−46.6690	−1.68293	−0.841464	−	0.540312i	$-0.818306\pi$
$769$	−46.6690	−1.68293	−0.841464	+	0.540312i	$0.818306\pi$
$770$	0	0
$771$	−60.0000	−2.16085
$772$	0	0
$773$	− 24.0416i	− 0.864717i	−0.901702	−	0.432359i	$-0.857681\pi$
$773$	− 24.0416i	− 0.864717i	0.901702	−	0.432359i	$-0.142319\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.0000	0.716574
$780$	0	0
$781$	0	0
$782$	0	0
$783$	45.2548i	1.61728i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	48.0833i	1.71398i	0.515330	+	0.856992i	$0.327669\pi$
$787$	48.0833i	1.71398i	−0.515330	+	0.856992i	$0.672331\pi$
$788$	0	0
$789$	−67.8823	−2.41667
$790$	0	0
$791$	0	0
$792$	0	0
$793$	30.0000i	1.06533i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 29.6985i	− 1.05197i	−0.850493	−	0.525987i	$-0.823696\pi$
$797$	− 29.6985i	− 1.05197i	0.850493	−	0.525987i	$-0.176304\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	35.3553	1.24922
$802$	0	0
$803$	28.2843i	0.998130i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	52.0000i	1.83049i
$808$	0	0
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	8.48528	0.297959	0.148979	−	0.988840i	$-0.452401\pi$
$811$	8.48528	0.297959	0.148979	+	0.988840i	$0.452401\pi$
$812$	0	0
$813$	80.0000i	2.80572i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 11.3137i	− 0.395817i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	− 40.0000i	− 1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$	− 40.0000i	− 1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	48.0000i	1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$	48.0000i	1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$	0	0
$829$	7.07107	0.245588	0.122794	−	0.992432i	$-0.460815\pi$
$829$	7.07107	0.245588	0.122794	+	0.992432i	$0.460815\pi$
$830$	0	0
$831$	−62.2254	−2.15858
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	11.3137	0.390593	0.195296	−	0.980744i	$-0.437433\pi$
$839$	11.3137	0.390593	0.195296	+	0.980744i	$0.437433\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	45.2548i	1.55866i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	32.0000	1.09695
$852$	0	0
$853$	− 12.7279i	− 0.435796i	−0.975972	−	0.217898i	$-0.930080\pi$
$853$	− 12.7279i	− 0.435796i	0.975972	−	0.217898i	$-0.0699200\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.07107i	0.241543i	0.992680	+	0.120772i	$0.0385368\pi$
$857$	7.07107i	0.241543i	−0.992680	+	0.120772i	$0.961463\pi$
$858$	0	0
$859$	−14.1421	−0.482523	−0.241262	−	0.970460i	$-0.577561\pi$
$859$	−14.1421	−0.482523	−0.241262	+	0.970460i	$0.577561\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.0000i	1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$863$	40.0000i	1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	42.4264i	1.44088i
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	0	0
$872$	0	0
$873$	7.07107i	0.239319i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.00000i	0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$	8.00000i	0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$	0	0
$879$	92.0000	3.10308
$880$	0	0
$881$	−21.2132	−0.714691	−0.357345	−	0.933972i	$-0.616318\pi$
$881$	−21.2132	−0.714691	−0.357345	+	0.933972i	$0.616318\pi$
$882$	0	0
$883$	40.0000i	1.34611i	0.739594	+	0.673054i	$0.235018\pi$
$883$	40.0000i	1.34611i	−0.739594	+	0.673054i	$0.764982\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.2843i	0.949693i	0.880069	+	0.474846i	$0.157496\pi$
$887$	28.2843i	0.949693i	−0.880069	+	0.474846i	$0.842504\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	− 16.0000i	− 0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	48.0000i	1.60267i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	14.1421	0.471143
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 8.00000i	− 0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$	− 8.00000i	− 0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$	0	0
$909$	63.6396	2.11079
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	56.5685i	1.87215i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	56.0000	1.84526
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 56.5685i	− 1.85795i
$928$	0	0
$929$	−35.3553	−1.15997	−0.579986	−	0.814627i	$-0.696942\pi$
$929$	−35.3553	−1.15997	−0.579986	+	0.814627i	$0.696942\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 64.0000i	− 2.09527i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15.5563i	0.508204i	0.967177	+	0.254102i	$0.0817799\pi$
$937$	15.5563i	0.508204i	−0.967177	+	0.254102i	$0.918220\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	0	0
$941$	−7.07107	−0.230510	−0.115255	−	0.993336i	$-0.536769\pi$
$941$	−7.07107	−0.230510	−0.115255	+	0.993336i	$0.536769\pi$
$942$	0	0
$943$	− 28.2843i	− 0.921063i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	60.0000i	1.94974i	0.222779	+	0.974869i	$0.428487\pi$
$947$	60.0000i	1.94974i	−0.222779	+	0.974869i	$0.571513\pi$
$948$	0	0
$949$	30.0000	0.973841
$950$	0	0
$951$	5.65685	0.183436
$952$	0	0
$953$	− 10.0000i	− 0.323932i	−0.986796	−	0.161966i	$-0.948217\pi$
$953$	− 10.0000i	− 0.323932i	0.986796	−	0.161966i	$-0.0517835\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 90.5097i	− 2.92576i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 40.0000i	− 1.28898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.0000i	0.385894i	0.981209	+	0.192947i	$0.0618045\pi$
$967$	12.0000i	0.385894i	−0.981209	+	0.192947i	$0.938195\pi$
$968$	0	0
$969$	11.3137	0.363449
$970$	0	0
$971$	14.1421	0.453843	0.226921	−	0.973913i	$-0.427134\pi$
$971$	14.1421	0.453843	0.226921	+	0.973913i	$0.427134\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.0000i	1.53566i	0.640656	+	0.767828i	$0.278662\pi$
$977$	48.0000i	1.53566i	−0.640656	+	0.767828i	$0.721338\pi$
$978$	0	0
$979$	−28.2843	−0.903969
$980$	0	0
$981$	−40.0000	−1.27710
$982$	0	0
$983$	− 45.2548i	− 1.44341i	−0.692203	−	0.721703i	$-0.743360\pi$
$983$	− 45.2548i	− 1.44341i	0.692203	−	0.721703i	$-0.256640\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	− 56.5685i	− 1.79515i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.41421i	0.0447886i	0.999749	+	0.0223943i	$0.00712892\pi$
$997$	1.41421i	0.0447886i	−0.999749	+	0.0223943i	$0.992871\pi$
$998$	0	0
$999$	−45.2548	−1.43180

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.p.2549.3		4
5.2	odd	4		4900.2.a.y.1.2		2
5.3	odd	4		196.2.a.c.1.1	✓	2
5.4	even	2	inner	4900.2.e.p.2549.1		4
7.6	odd	2	inner	4900.2.e.p.2549.2		4
15.8	even	4		1764.2.a.l.1.1		2
20.3	even	4		784.2.a.m.1.2		2
35.3	even	12		196.2.e.b.177.1		4
35.13	even	4		196.2.a.c.1.2	yes	2
35.18	odd	12		196.2.e.b.177.2		4
35.23	odd	12		196.2.e.b.165.2		4
35.27	even	4		4900.2.a.y.1.1		2
35.33	even	12		196.2.e.b.165.1		4
35.34	odd	2	inner	4900.2.e.p.2549.4		4
40.3	even	4		3136.2.a.bs.1.1		2
40.13	odd	4		3136.2.a.br.1.2		2
60.23	odd	4		7056.2.a.cr.1.1		2
105.23	even	12		1764.2.k.l.361.2		4
105.38	odd	12		1764.2.k.l.1549.1		4
105.53	even	12		1764.2.k.l.1549.2		4
105.68	odd	12		1764.2.k.l.361.1		4
105.83	odd	4		1764.2.a.l.1.2		2
140.3	odd	12		784.2.i.l.177.2		4
140.23	even	12		784.2.i.l.753.1		4
140.83	odd	4		784.2.a.m.1.1		2
140.103	odd	12		784.2.i.l.753.2		4
140.123	even	12		784.2.i.l.177.1		4
280.13	even	4		3136.2.a.br.1.1		2
280.83	odd	4		3136.2.a.bs.1.2		2
420.83	even	4		7056.2.a.cr.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.a.c.1.1	✓	2	5.3	odd	4
196.2.a.c.1.2	yes	2	35.13	even	4
196.2.e.b.165.1		4	35.33	even	12
196.2.e.b.165.2		4	35.23	odd	12
196.2.e.b.177.1		4	35.3	even	12
196.2.e.b.177.2		4	35.18	odd	12
784.2.a.m.1.1		2	140.83	odd	4
784.2.a.m.1.2		2	20.3	even	4
784.2.i.l.177.1		4	140.123	even	12
784.2.i.l.177.2		4	140.3	odd	12
784.2.i.l.753.1		4	140.23	even	12
784.2.i.l.753.2		4	140.103	odd	12
1764.2.a.l.1.1		2	15.8	even	4
1764.2.a.l.1.2		2	105.83	odd	4
1764.2.k.l.361.1		4	105.68	odd	12
1764.2.k.l.361.2		4	105.23	even	12
1764.2.k.l.1549.1		4	105.38	odd	12
1764.2.k.l.1549.2		4	105.53	even	12
3136.2.a.br.1.1		2	280.13	even	4
3136.2.a.br.1.2		2	40.13	odd	4
3136.2.a.bs.1.1		2	40.3	even	4
3136.2.a.bs.1.2		2	280.83	odd	4
4900.2.a.y.1.1		2	35.27	even	4
4900.2.a.y.1.2		2	5.2	odd	4
4900.2.e.p.2549.1		4	5.4	even	2	inner
4900.2.e.p.2549.2		4	7.6	odd	2	inner
4900.2.e.p.2549.3		4	1.1	even	1	trivial
4900.2.e.p.2549.4		4	35.34	odd	2	inner
7056.2.a.cr.1.1		2	60.23	odd	4
7056.2.a.cr.1.2		2	420.83	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)

\(f(q)\)	\(=\)	\(q+2.82843i q^{3} -5.00000 q^{9} +4.00000 q^{11} -4.24264i q^{13} +1.41421i q^{17} -2.82843 q^{19} +4.00000i q^{23} -5.65685i q^{27} -8.00000 q^{29} +11.3137i q^{33} -8.00000i q^{37} +12.0000 q^{39} -7.07107 q^{41} +4.00000i q^{43} +5.65685i q^{47} -4.00000 q^{51} -10.0000i q^{53} -8.00000i q^{57} -14.1421 q^{59} -7.07107 q^{61} -11.3137 q^{69} +7.07107i q^{73} -8.00000 q^{79} +1.00000 q^{81} +14.1421i q^{83} -22.6274i q^{87} -7.07107 q^{89} -1.41421i q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{9} + 16 q^{11} - 32 q^{29} + 48 q^{39} - 16 q^{51} - 32 q^{79} + 4 q^{81} - 80 q^{99}+O(q^{100}) \) 4 * q - 20 * q^9 + 16 * q^11 - 32 * q^29 + 48 * q^39 - 16 * q^51 - 32 * q^79 + 4 * q^81 - 80 * q^99

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