LMFDB - Embedded newform 2450.2.c.n.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(99,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.c (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:	$19.5633484952$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 490)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2450.99
Dual form			2450.2.c.n.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +2.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +8.00000i q^{17} -1.00000i q^{18} +6.00000 q^{19} -4.00000i q^{22} +4.00000i q^{23} -2.00000 q^{24} +2.00000 q^{26} -4.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +8.00000i q^{33} -8.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +6.00000i q^{38} -4.00000 q^{39} +4.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} +4.00000i q^{47} -2.00000i q^{48} +16.0000 q^{51} +2.00000i q^{52} -10.0000i q^{53} +4.00000 q^{54} -12.0000i q^{57} +6.00000i q^{58} +14.0000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -8.00000 q^{66} -4.00000i q^{67} -8.00000i q^{68} +8.00000 q^{69} +12.0000 q^{71} +1.00000i q^{72} -4.00000i q^{73} +10.0000 q^{74} -6.00000 q^{76} -4.00000i q^{78} -4.00000 q^{79} -11.0000 q^{81} +4.00000i q^{82} +2.00000i q^{83} +4.00000 q^{86} -12.0000i q^{87} +4.00000i q^{88} -8.00000 q^{89} -4.00000i q^{92} -8.00000i q^{93} -4.00000 q^{94} +2.00000 q^{96} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} + 12 q^{19} - 4 q^{24} + 4 q^{26} + 12 q^{29} + 8 q^{31} - 16 q^{34} + 2 q^{36} - 8 q^{39} + 8 q^{41} + 8 q^{44} - 8 q^{46} + 32 q^{51} + 8 q^{54} + 28 q^{59} - 20 q^{61} - 2 q^{64} - 16 q^{66} + 16 q^{69} + 24 q^{71} + 20 q^{74} - 12 q^{76} - 8 q^{79} - 22 q^{81} + 8 q^{86} - 16 q^{89} - 8 q^{94} + 4 q^{96} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^16 + 12 * q^19 - 4 * q^24 + 4 * q^26 + 12 * q^29 + 8 * q^31 - 16 * q^34 + 2 * q^36 - 8 * q^39 + 8 * q^41 + 8 * q^44 - 8 * q^46 + 32 * q^51 + 8 * q^54 + 28 * q^59 - 20 * q^61 - 2 * q^64 - 16 * q^66 + 16 * q^69 + 24 * q^71 + 20 * q^74 - 12 * q^76 - 8 * q^79 - 22 * q^81 + 8 * q^86 - 16 * q^89 - 8 * q^94 + 4 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$.

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

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	2.00000i	0.577350i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	8.00000i	1.94029i	0.242536	+	0.970143i	$0.422021\pi$
$17$	8.00000i	1.94029i	−0.242536	+	0.970143i	$0.577979\pi$
$18$	− 1.00000i	− 0.235702i
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	2.00000	0.392232
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000i	0.176777i
$33$	8.00000i	1.39262i
$34$	−8.00000	−1.37199
$35$	0	0
$36$	1.00000	0.166667
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	6.00000i	0.973329i
$39$	−4.00000	−0.640513
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	− 2.00000i	− 0.288675i
$49$	0	0
$50$	0	0
$51$	16.0000	2.24045
$52$	2.00000i	0.277350i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	0	0
$57$	− 12.0000i	− 1.58944i
$58$	6.00000i	0.787839i
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−8.00000	−0.984732
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 8.00000i	− 0.970143i
$69$	8.00000	0.963087
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	1.00000i	0.117851i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	− 4.00000i	− 0.452911i
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	4.00000i	0.441726i
$83$	2.00000i	0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$	2.00000i	0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	− 12.0000i	− 1.28654i
$88$	4.00000i	0.426401i
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	0	0
$92$	− 4.00000i	− 0.417029i
$93$	− 8.00000i	− 0.829561i
$94$	−4.00000	−0.412568
$95$	0	0
$96$	2.00000	0.204124
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	16.0000i	1.58424i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	10.0000	0.971286
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	4.00000i	0.384900i
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−20.0000	−1.89832
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	12.0000	1.12390
$115$	0	0
$116$	−6.00000	−0.557086
$117$	2.00000i	0.184900i
$118$	14.0000i	1.28880i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	− 10.0000i	− 0.905357i
$123$	− 8.00000i	− 0.721336i
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	− 12.0000i	− 1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$	− 12.0000i	− 1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	− 8.00000i	− 0.696311i
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	8.00000	0.685994
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	8.00000i	0.681005i
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	12.0000i	1.00702i
$143$	8.00000i	0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	10.0000i	0.821995i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	− 6.00000i	− 0.486664i
$153$	− 8.00000i	− 0.646762i
$154$	0	0
$155$	0	0
$156$	4.00000	0.320256
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	− 4.00000i	− 0.318223i
$159$	−20.0000	−1.58610
$160$	0	0
$161$	0	0
$162$	− 11.0000i	− 0.864242i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	−2.00000	−0.155230
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	4.00000i	0.304997i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	−4.00000	−0.301511
$177$	− 28.0000i	− 2.10461i
$178$	− 8.00000i	− 0.599625i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	0	0
$183$	20.0000i	1.47844i
$184$	4.00000	0.294884
$185$	0	0
$186$	8.00000	0.586588
$187$	− 32.0000i	− 2.34007i
$188$	− 4.00000i	− 0.291730i
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	2.00000i	0.144338i
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	4.00000i	0.284268i
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	− 2.00000i	− 0.140720i
$203$	0	0
$204$	−16.0000	−1.12022
$205$	0	0
$206$	4.00000	0.278693
$207$	− 4.00000i	− 0.278019i
$208$	− 2.00000i	− 0.138675i
$209$	−24.0000	−1.66011
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	10.0000i	0.686803i
$213$	− 24.0000i	− 1.64445i
$214$	12.0000	0.820303
$215$	0	0
$216$	−4.00000	−0.272166
$217$	0	0
$218$	− 10.0000i	− 0.677285i
$219$	−8.00000	−0.540590
$220$	0	0
$221$	16.0000	1.07628
$222$	− 20.0000i	− 1.34231i
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	0	0
$225$	0	0
$226$	−2.00000	−0.133038
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	12.0000i	0.794719i
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 22.0000i	− 1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	− 22.0000i	− 1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−14.0000	−0.911322
$237$	8.00000i	0.519656i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	5.00000i	0.321412i
$243$	10.0000i	0.641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	8.00000	0.510061
$247$	− 12.0000i	− 0.763542i
$248$	− 4.00000i	− 0.254000i
$249$	4.00000	0.253490
$250$	0	0
$251$	22.0000	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	− 8.00000i	− 0.498058i
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	18.0000i	1.11204i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	8.00000	0.492366
$265$	0	0
$266$	0	0
$267$	16.0000i	0.979184i
$268$	4.00000i	0.244339i
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	8.00000i	0.485071i
$273$	0	0
$274$	2.00000	0.120824
$275$	0	0
$276$	−8.00000	−0.481543
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	10.0000i	0.599760i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	8.00000i	0.476393i
$283$	26.0000i	1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$	26.0000i	1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−8.00000	−0.473050
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	−47.0000	−2.76471
$290$	0	0
$291$	0	0
$292$	4.00000i	0.234082i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	−10.0000	−0.581238
$297$	16.0000i	0.928414i
$298$	10.0000i	0.579284i
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4.00000i	0.229794i
$304$	6.00000	0.344124
$305$	0	0
$306$	8.00000	0.457330
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	4.00000i	0.226455i
$313$	8.00000i	0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$	8.00000i	0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	4.00000	0.225018
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	− 20.0000i	− 1.12154i
$319$	−24.0000	−1.34374
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	48.0000i	2.67079i
$324$	11.0000	0.611111
$325$	0	0
$326$	−4.00000	−0.221540
$327$	20.0000i	1.10600i
$328$	− 4.00000i	− 0.220863i
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	− 2.00000i	− 0.109764i
$333$	10.0000i	0.547997i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	9.00000i	0.489535i
$339$	4.00000	0.217250
$340$	0	0
$341$	−16.0000	−0.866449
$342$	− 6.00000i	− 0.324443i
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	18.0000	0.967686
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	12.0000i	0.643268i
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	− 4.00000i	− 0.213201i
$353$	12.0000i	0.638696i	0.947638	+	0.319348i	$0.103464\pi$
$353$	12.0000i	0.638696i	−0.947638	+	0.319348i	$0.896536\pi$
$354$	28.0000	1.48818
$355$	0	0
$356$	8.00000	0.423999
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	26.0000i	1.36653i
$363$	− 10.0000i	− 0.524864i
$364$	0	0
$365$	0	0
$366$	−20.0000	−1.04542
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	4.00000i	0.208514i
$369$	−4.00000	−0.208232
$370$	0	0
$371$	0	0
$372$	8.00000i	0.414781i
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	32.0000	1.65468
$375$	0	0
$376$	4.00000	0.206284
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	−24.0000	−1.22956
$382$	12.0000i	0.613973i
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	−18.0000	−0.916176
$387$	4.00000i	0.203331i
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−32.0000	−1.61831
$392$	0	0
$393$	− 36.0000i	− 1.81596i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	−4.00000	−0.201008
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	− 4.00000i	− 0.200502i
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	− 8.00000i	− 0.399004i
$403$	− 8.00000i	− 0.398508i
$404$	2.00000	0.0995037
$405$	0	0
$406$	0	0
$407$	40.0000i	1.98273i
$408$	− 16.0000i	− 0.792118i
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	4.00000i	0.197066i
$413$	0	0
$414$	4.00000	0.196589
$415$	0	0
$416$	2.00000	0.0980581
$417$	− 20.0000i	− 0.979404i
$418$	− 24.0000i	− 1.17388i
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	20.0000i	0.973585i
$423$	− 4.00000i	− 0.194487i
$424$	−10.0000	−0.485643
$425$	0	0
$426$	24.0000	1.16280
$427$	0	0
$428$	12.0000i	0.580042i
$429$	16.0000	0.772487
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 4.00000i	− 0.192450i
$433$	40.0000i	1.92228i	0.276066	+	0.961139i	$0.410969\pi$
$433$	40.0000i	1.92228i	−0.276066	+	0.961139i	$0.589031\pi$
$434$	0	0
$435$	0	0
$436$	10.0000	0.478913
$437$	24.0000i	1.14808i
$438$	− 8.00000i	− 0.382255i
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	0	0
$442$	16.0000i	0.761042i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	20.0000	0.949158
$445$	0	0
$446$	−8.00000	−0.378811
$447$	− 20.0000i	− 0.945968i
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	− 2.00000i	− 0.0940721i
$453$	0	0
$454$	18.0000	0.844782
$455$	0	0
$456$	−12.0000	−0.561951
$457$	6.00000i	0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$	6.00000i	0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	14.0000i	0.654177i
$459$	32.0000	1.49363
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	22.0000	1.01913
$467$	− 10.0000i	− 0.462745i	−0.972865	−	0.231372i	$-0.925678\pi$
$467$	− 10.0000i	− 0.462745i	0.972865	−	0.231372i	$-0.0743216\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	4.00000	0.184310
$472$	− 14.0000i	− 0.644402i
$473$	16.0000i	0.735681i
$474$	−8.00000	−0.367452
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	8.00000i	0.365911i
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	4.00000i	0.182195i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−10.0000	−0.453609
$487$	− 44.0000i	− 1.99383i	−0.0784867	−	0.996915i	$-0.525009\pi$
$487$	− 44.0000i	− 1.99383i	0.0784867	−	0.996915i	$-0.474991\pi$
$488$	10.0000i	0.452679i
$489$	8.00000	0.361773
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	8.00000i	0.360668i
$493$	48.0000i	2.16181i
$494$	12.0000	0.539906
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	4.00000i	0.179244i
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	22.0000i	0.981908i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	0	0
$505$	0	0
$506$	16.0000	0.711287
$507$	− 18.0000i	− 0.799408i
$508$	12.0000i	0.532414i
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	− 24.0000i	− 1.05963i
$514$	−12.0000	−0.529297
$515$	0	0
$516$	8.00000	0.352180
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	−36.0000	−1.58022
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	− 6.00000i	− 0.262613i
$523$	34.0000i	1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$	34.0000i	1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	0	0
$527$	32.0000i	1.39394i
$528$	8.00000i	0.348155i
$529$	7.00000	0.304348
$530$	0	0
$531$	−14.0000	−0.607548
$532$	0	0
$533$	− 8.00000i	− 0.346518i
$534$	−16.0000	−0.692388
$535$	0	0
$536$	−4.00000	−0.172774
$537$	8.00000i	0.345225i
$538$	− 26.0000i	− 1.12094i
$539$	0	0
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	16.0000i	0.687259i
$543$	− 52.0000i	− 2.23153i
$544$	−8.00000	−0.342997
$545$	0	0
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	2.00000i	0.0854358i
$549$	10.0000	0.426790
$550$	0	0
$551$	36.0000	1.53365
$552$	− 8.00000i	− 0.340503i
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−10.0000	−0.424094
$557$	− 6.00000i	− 0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$	− 6.00000i	− 0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$	− 4.00000i	− 0.169334i
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−64.0000	−2.70208
$562$	− 10.0000i	− 0.421825i
$563$	− 18.0000i	− 0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$	− 18.0000i	− 0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−26.0000	−1.09286
$567$	0	0
$568$	− 12.0000i	− 0.503509i
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	− 8.00000i	− 0.334497i
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 20.0000i	− 0.832611i	−0.909225	−	0.416305i	$-0.863325\pi$
$577$	− 20.0000i	− 0.832611i	0.909225	−	0.416305i	$-0.136675\pi$
$578$	− 47.0000i	− 1.95494i
$579$	36.0000	1.49611
$580$	0	0
$581$	0	0
$582$	0	0
$583$	40.0000i	1.65663i
$584$	−4.00000	−0.165521
$585$	0	0
$586$	6.00000	0.247858
$587$	2.00000i	0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$	2.00000i	0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	36.0000	1.48084
$592$	− 10.0000i	− 0.410997i
$593$	20.0000i	0.821302i	0.911793	+	0.410651i	$0.134698\pi$
$593$	20.0000i	0.821302i	−0.911793	+	0.410651i	$0.865302\pi$
$594$	−16.0000	−0.656488
$595$	0	0
$596$	−10.0000	−0.409616
$597$	8.00000i	0.327418i
$598$	8.00000i	0.327144i
$599$	28.0000	1.14405	0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000	1.14405	0.572024	+	0.820237i	$0.306158\pi$
$600$	0	0
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	0	0
$606$	−4.00000	−0.162489
$607$	− 16.0000i	− 0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$	− 16.0000i	− 0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$	6.00000i	0.243332i
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	8.00000i	0.323381i
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	− 8.00000i	− 0.321807i
$619$	6.00000	0.241160	0.120580	−	0.992704i	$-0.461525\pi$
$619$	6.00000	0.241160	0.120580	+	0.992704i	$0.461525\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	16.0000i	0.641542i
$623$	0	0
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−8.00000	−0.319744
$627$	48.0000i	1.91694i
$628$	− 2.00000i	− 0.0798087i
$629$	80.0000	3.18981
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	4.00000i	0.159111i
$633$	− 40.0000i	− 1.58986i
$634$	−18.0000	−0.714871
$635$	0	0
$636$	20.0000	0.793052
$637$	0	0
$638$	− 24.0000i	− 0.950169i
$639$	−12.0000	−0.474713
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	− 24.0000i	− 0.947204i
$643$	− 18.0000i	− 0.709851i	−0.934895	−	0.354925i	$-0.884506\pi$
$643$	− 18.0000i	− 0.709851i	0.934895	−	0.354925i	$-0.115494\pi$
$644$	0	0
$645$	0	0
$646$	−48.0000	−1.88853
$647$	28.0000i	1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$647$	28.0000i	1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$648$	11.0000i	0.432121i
$649$	−56.0000	−2.19819
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	−20.0000	−0.782062
$655$	0	0
$656$	4.00000	0.156174
$657$	4.00000i	0.156055i
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	− 28.0000i	− 1.08825i
$663$	− 32.0000i	− 1.24278i
$664$	2.00000	0.0776151
$665$	0	0
$666$	−10.0000	−0.387492
$667$	24.0000i	0.929284i
$668$	− 12.0000i	− 0.464294i
$669$	16.0000	0.618596
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	− 22.0000i	− 0.848038i	−0.905653	−	0.424019i	$-0.860619\pi$
$673$	− 22.0000i	− 0.848038i	0.905653	−	0.424019i	$-0.139381\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 46.0000i	− 1.76792i	−0.467559	−	0.883962i	$-0.654866\pi$
$677$	− 46.0000i	− 1.76792i	0.467559	−	0.883962i	$-0.345134\pi$
$678$	4.00000i	0.153619i
$679$	0	0
$680$	0	0
$681$	−36.0000	−1.37952
$682$	− 16.0000i	− 0.612672i
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	6.00000	0.229416
$685$	0	0
$686$	0	0
$687$	− 28.0000i	− 1.06827i
$688$	− 4.00000i	− 0.152499i
$689$	−20.0000	−0.761939
$690$	0	0
$691$	−46.0000	−1.74992	−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000	−1.74992	−0.874961	+	0.484193i	$0.839113\pi$
$692$	18.0000i	0.684257i
$693$	0	0
$694$	4.00000	0.151838
$695$	0	0
$696$	−12.0000	−0.454859
$697$	32.0000i	1.21209i
$698$	− 10.0000i	− 0.378506i
$699$	−44.0000	−1.66423
$700$	0	0
$701$	−38.0000	−1.43524	−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000	−1.43524	−0.717620	+	0.696435i	$0.754769\pi$
$702$	− 8.00000i	− 0.301941i
$703$	− 60.0000i	− 2.26294i
$704$	4.00000	0.150756
$705$	0	0
$706$	−12.0000	−0.451626
$707$	0	0
$708$	28.0000i	1.05230i
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	8.00000i	0.299813i
$713$	16.0000i	0.599205i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	− 16.0000i	− 0.597531i
$718$	8.00000i	0.298557i
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	0	0
$722$	17.0000i	0.632674i
$723$	− 8.00000i	− 0.297523i
$724$	−26.0000	−0.966282
$725$	0	0
$726$	10.0000	0.371135
$727$	20.0000i	0.741759i	0.928681	+	0.370879i	$0.120944\pi$
$727$	20.0000i	0.741759i	−0.928681	+	0.370879i	$0.879056\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	32.0000	1.18356
$732$	− 20.0000i	− 0.739221i
$733$	− 30.0000i	− 1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$733$	− 30.0000i	− 1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−4.00000	−0.147442
$737$	16.0000i	0.589368i
$738$	− 4.00000i	− 0.147242i
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	−24.0000	−0.881662
$742$	0	0
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	−6.00000	−0.219676
$747$	− 2.00000i	− 0.0731762i
$748$	32.0000i	1.17004i
$749$	0	0
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	4.00000i	0.145865i
$753$	− 44.0000i	− 1.60345i
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	4.00000i	0.145287i
$759$	−32.0000	−1.16153
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	− 24.0000i	− 0.869428i
$763$	0	0
$764$	−12.0000	−0.434145
$765$	0	0
$766$	20.0000	0.722629
$767$	− 28.0000i	− 1.01102i
$768$	− 2.00000i	− 0.0721688i
$769$	−28.0000	−1.00971	−0.504853	−	0.863205i	$-0.668453\pi$
$769$	−28.0000	−1.00971	−0.504853	+	0.863205i	$0.668453\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	− 18.0000i	− 0.647834i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	0	0
$777$	0	0
$778$	− 18.0000i	− 0.645331i
$779$	24.0000	0.859889
$780$	0	0
$781$	−48.0000	−1.71758
$782$	− 32.0000i	− 1.14432i
$783$	− 24.0000i	− 0.857690i
$784$	0	0
$785$	0	0
$786$	36.0000	1.28408
$787$	6.00000i	0.213877i	0.994266	+	0.106938i	$0.0341048\pi$
$787$	6.00000i	0.213877i	−0.994266	+	0.106938i	$0.965895\pi$
$788$	− 18.0000i	− 0.641223i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	− 4.00000i	− 0.142134i
$793$	20.0000i	0.710221i
$794$	−26.0000	−0.922705
$795$	0	0
$796$	4.00000	0.141776
$797$	2.00000i	0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$	2.00000i	0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	0	0
$801$	8.00000	0.282666
$802$	− 14.0000i	− 0.494357i
$803$	16.0000i	0.564628i
$804$	8.00000	0.282138
$805$	0	0
$806$	8.00000	0.281788
$807$	52.0000i	1.83049i
$808$	2.00000i	0.0703598i
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	− 32.0000i	− 1.12229i
$814$	−40.0000	−1.40200
$815$	0	0
$816$	16.0000	0.560112
$817$	− 24.0000i	− 0.839654i
$818$	20.0000i	0.699284i
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	− 4.00000i	− 0.139516i
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	44.0000i	1.53003i	0.644013	+	0.765015i	$0.277268\pi$
$827$	44.0000i	1.53003i	−0.644013	+	0.765015i	$0.722732\pi$
$828$	4.00000i	0.139010i
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	2.00000i	0.0693375i
$833$	0	0
$834$	20.0000	0.692543
$835$	0	0
$836$	24.0000	0.830057
$837$	− 16.0000i	− 0.553041i
$838$	− 6.00000i	− 0.207267i
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 34.0000i	− 1.17172i
$843$	20.0000i	0.688837i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	4.00000	0.137523
$847$	0	0
$848$	− 10.0000i	− 0.343401i
$849$	52.0000	1.78464
$850$	0	0
$851$	40.0000	1.37118
$852$	24.0000i	0.822226i
$853$	− 50.0000i	− 1.71197i	−0.517003	−	0.855984i	$-0.672952\pi$
$853$	− 50.0000i	− 1.71197i	0.517003	−	0.855984i	$-0.327048\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 24.0000i	− 0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$	− 24.0000i	− 0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$	16.0000i	0.546231i
$859$	−38.0000	−1.29654	−0.648272	−	0.761409i	$-0.724508\pi$
$859$	−38.0000	−1.29654	−0.648272	+	0.761409i	$0.724508\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	−40.0000	−1.35926
$867$	94.0000i	3.19241i
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	−8.00000	−0.271070
$872$	10.0000i	0.338643i
$873$	0	0
$874$	−24.0000	−0.811812
$875$	0	0
$876$	8.00000	0.270295
$877$	− 42.0000i	− 1.41824i	−0.705088	−	0.709120i	$-0.749093\pi$
$877$	− 42.0000i	− 1.41824i	0.705088	−	0.709120i	$-0.250907\pi$
$878$	32.0000i	1.07995i
$879$	−12.0000	−0.404750
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	0	0
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	−16.0000	−0.538138
$885$	0	0
$886$	4.00000	0.134383
$887$	4.00000i	0.134307i	0.997743	+	0.0671534i	$0.0213917\pi$
$887$	4.00000i	0.134307i	−0.997743	+	0.0671534i	$0.978608\pi$
$888$	20.0000i	0.671156i
$889$	0	0
$890$	0	0
$891$	44.0000	1.47406
$892$	− 8.00000i	− 0.267860i
$893$	24.0000i	0.803129i
$894$	20.0000	0.668900
$895$	0	0
$896$	0	0
$897$	− 16.0000i	− 0.534224i
$898$	− 18.0000i	− 0.600668i
$899$	24.0000	0.800445
$900$	0	0
$901$	80.0000	2.66519
$902$	− 16.0000i	− 0.532742i
$903$	0	0
$904$	2.00000	0.0665190
$905$	0	0
$906$	0	0
$907$	4.00000i	0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$	4.00000i	0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	18.0000i	0.597351i
$909$	2.00000	0.0663358
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	− 12.0000i	− 0.397360i
$913$	− 8.00000i	− 0.264761i
$914$	−6.00000	−0.198462
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	32.0000i	1.05616i
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	− 6.00000i	− 0.197599i
$923$	− 24.0000i	− 0.789970i
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	4.00000i	0.131377i
$928$	6.00000i	0.196960i
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	0	0
$932$	22.0000i	0.720634i
$933$	− 32.0000i	− 1.04763i
$934$	10.0000	0.327210
$935$	0	0
$936$	2.00000	0.0653720
$937$	12.0000i	0.392023i	0.980602	+	0.196011i	$0.0627990\pi$
$937$	12.0000i	0.392023i	−0.980602	+	0.196011i	$0.937201\pi$
$938$	0	0
$939$	16.0000	0.522140
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	4.00000i	0.130327i
$943$	16.0000i	0.521032i
$944$	14.0000	0.455661
$945$	0	0
$946$	−16.0000	−0.520205
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	− 8.00000i	− 0.259828i
$949$	−8.00000	−0.259691
$950$	0	0
$951$	36.0000	1.16738
$952$	0	0
$953$	− 2.00000i	− 0.0647864i	−0.999475	−	0.0323932i	$-0.989687\pi$
$953$	− 2.00000i	− 0.0647864i	0.999475	−	0.0323932i	$-0.0103129\pi$
$954$	−10.0000	−0.323762
$955$	0	0
$956$	−8.00000	−0.258738
$957$	48.0000i	1.55162i
$958$	− 4.00000i	− 0.129234i
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 20.0000i	− 0.644826i
$963$	12.0000i	0.386695i
$964$	−4.00000	−0.128831
$965$	0	0
$966$	0	0
$967$	− 52.0000i	− 1.67221i	−0.548572	−	0.836104i	$-0.684828\pi$
$967$	− 52.0000i	− 1.67221i	0.548572	−	0.836104i	$-0.315172\pi$
$968$	− 5.00000i	− 0.160706i
$969$	96.0000	3.08396
$970$	0	0
$971$	−62.0000	−1.98967	−0.994837	−	0.101482i	$-0.967641\pi$
$971$	−62.0000	−1.98967	−0.994837	+	0.101482i	$0.967641\pi$
$972$	− 10.0000i	− 0.320750i
$973$	0	0
$974$	44.0000	1.40985
$975$	0	0
$976$	−10.0000	−0.320092
$977$	54.0000i	1.72761i	0.503824	+	0.863807i	$0.331926\pi$
$977$	54.0000i	1.72761i	−0.503824	+	0.863807i	$0.668074\pi$
$978$	8.00000i	0.255812i
$979$	32.0000	1.02272
$980$	0	0
$981$	10.0000	0.319275
$982$	− 12.0000i	− 0.382935i
$983$	− 20.0000i	− 0.637901i	−0.947771	−	0.318950i	$-0.896670\pi$
$983$	− 20.0000i	− 0.637901i	0.947771	−	0.318950i	$-0.103330\pi$
$984$	−8.00000	−0.255031
$985$	0	0
$986$	−48.0000	−1.52863
$987$	0	0
$988$	12.0000i	0.381771i
$989$	16.0000	0.508770
$990$	0	0
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	4.00000i	0.127000i
$993$	56.0000i	1.77711i
$994$	0	0
$995$	0	0
$996$	−4.00000	−0.126745
$997$	− 42.0000i	− 1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$	− 42.0000i	− 1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$	− 28.0000i	− 0.886325i
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.c.n.99.2		2
5.2	odd	4		2450.2.a.d.1.1		1
5.3	odd	4		490.2.a.i.1.1	yes	1
5.4	even	2	inner	2450.2.c.n.99.1		2
7.6	odd	2		2450.2.c.b.99.2		2
15.8	even	4		4410.2.a.i.1.1		1
20.3	even	4		3920.2.a.j.1.1		1
35.3	even	12		490.2.e.e.471.1		2
35.13	even	4		490.2.a.f.1.1	✓	1
35.18	odd	12		490.2.e.b.471.1		2
35.23	odd	12		490.2.e.b.361.1		2
35.27	even	4		2450.2.a.n.1.1		1
35.33	even	12		490.2.e.e.361.1		2
35.34	odd	2		2450.2.c.b.99.1		2
105.83	odd	4		4410.2.a.s.1.1		1
140.83	odd	4		3920.2.a.bg.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.a.f.1.1	✓	1	35.13	even	4
490.2.a.i.1.1	yes	1	5.3	odd	4
490.2.e.b.361.1		2	35.23	odd	12
490.2.e.b.471.1		2	35.18	odd	12
490.2.e.e.361.1		2	35.33	even	12
490.2.e.e.471.1		2	35.3	even	12
2450.2.a.d.1.1		1	5.2	odd	4
2450.2.a.n.1.1		1	35.27	even	4
2450.2.c.b.99.1		2	35.34	odd	2
2450.2.c.b.99.2		2	7.6	odd	2
2450.2.c.n.99.1		2	5.4	even	2	inner
2450.2.c.n.99.2		2	1.1	even	1	trivial
3920.2.a.j.1.1		1	20.3	even	4
3920.2.a.bg.1.1		1	140.83	odd	4
4410.2.a.i.1.1		1	15.8	even	4
4410.2.a.s.1.1		1	105.83	odd	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +2.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} +8.00000i q^{17} -1.00000i q^{18} +6.00000 q^{19} -4.00000i q^{22} +4.00000i q^{23} -2.00000 q^{24} +2.00000 q^{26} -4.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +8.00000i q^{33} -8.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} +6.00000i q^{38} -4.00000 q^{39} +4.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} +4.00000i q^{47} -2.00000i q^{48} +16.0000 q^{51} +2.00000i q^{52} -10.0000i q^{53} +4.00000 q^{54} -12.0000i q^{57} +6.00000i q^{58} +14.0000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -8.00000 q^{66} -4.00000i q^{67} -8.00000i q^{68} +8.00000 q^{69} +12.0000 q^{71} +1.00000i q^{72} -4.00000i q^{73} +10.0000 q^{74} -6.00000 q^{76} -4.00000i q^{78} -4.00000 q^{79} -11.0000 q^{81} +4.00000i q^{82} +2.00000i q^{83} +4.00000 q^{86} -12.0000i q^{87} +4.00000i q^{88} -8.00000 q^{89} -4.00000i q^{92} -8.00000i q^{93} -4.00000 q^{94} +2.00000 q^{96} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} + 12 q^{19} - 4 q^{24} + 4 q^{26} + 12 q^{29} + 8 q^{31} - 16 q^{34} + 2 q^{36} - 8 q^{39} + 8 q^{41} + 8 q^{44} - 8 q^{46} + 32 q^{51} + 8 q^{54} + 28 q^{59} - 20 q^{61} - 2 q^{64} - 16 q^{66} + 16 q^{69} + 24 q^{71} + 20 q^{74} - 12 q^{76} - 8 q^{79} - 22 q^{81} + 8 q^{86} - 16 q^{89} - 8 q^{94} + 4 q^{96} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^16 + 12 * q^19 - 4 * q^24 + 4 * q^26 + 12 * q^29 + 8 * q^31 - 16 * q^34 + 2 * q^36 - 8 * q^39 + 8 * q^41 + 8 * q^44 - 8 * q^46 + 32 * q^51 + 8 * q^54 + 28 * q^59 - 20 * q^61 - 2 * q^64 - 16 * q^66 + 16 * q^69 + 24 * q^71 + 20 * q^74 - 12 * q^76 - 8 * q^79 - 22 * q^81 + 8 * q^86 - 16 * q^89 - 8 * q^94 + 4 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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