LMFDB - Embedded newform 2450.2.c.m.99.1

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 50)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} +2.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{18} +5.00000 q^{19} +3.00000i q^{22} -6.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} +5.00000i q^{27} -2.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} +3.00000 q^{34} -2.00000 q^{36} +2.00000i q^{37} -5.00000i q^{38} +4.00000 q^{39} +3.00000 q^{41} +4.00000i q^{43} +3.00000 q^{44} -6.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} -3.00000 q^{51} +4.00000i q^{52} -6.00000i q^{53} +5.00000 q^{54} +5.00000i q^{57} -2.00000 q^{61} +2.00000i q^{62} -1.00000 q^{64} -3.00000 q^{66} -13.0000i q^{67} -3.00000i q^{68} +6.00000 q^{69} +12.0000 q^{71} +2.00000i q^{72} +11.0000i q^{73} +2.00000 q^{74} -5.00000 q^{76} -4.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} -3.00000i q^{82} -9.00000i q^{83} +4.00000 q^{86} -3.00000i q^{88} +15.0000 q^{89} +6.00000i q^{92} -2.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} - 6 q^{11} + 2 q^{16} + 10 q^{19} - 2 q^{24} - 8 q^{26} - 4 q^{31} + 6 q^{34} - 4 q^{36} + 8 q^{39} + 6 q^{41} + 6 q^{44} - 12 q^{46} - 6 q^{51} + 10 q^{54} - 4 q^{61} - 2 q^{64} - 6 q^{66} + 12 q^{69} + 24 q^{71} + 4 q^{74} - 10 q^{76} + 20 q^{79} + 2 q^{81} + 8 q^{86} + 30 q^{89} - 24 q^{94} + 2 q^{96} - 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 - 6 * q^11 + 2 * q^16 + 10 * q^19 - 2 * q^24 - 8 * q^26 - 4 * q^31 + 6 * q^34 - 4 * q^36 + 8 * q^39 + 6 * q^41 + 6 * q^44 - 12 * q^46 - 6 * q^51 + 10 * q^54 - 4 * q^61 - 2 * q^64 - 6 * q^66 + 12 * q^69 + 24 * q^71 + 4 * q^74 - 10 * q^76 + 20 * q^79 + 2 * q^81 + 8 * q^86 + 30 * q^89 - 24 * q^94 + 2 * q^96 - 12 * 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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	2.00000	0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	− 2.00000i	− 0.471405i
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	0	0
$22$	3.00000i	0.639602i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−4.00000	−0.784465
$27$	5.00000i	0.962250i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 3.00000i	− 0.522233i
$34$	3.00000	0.514496
$35$	0	0
$36$	−2.00000	−0.333333
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 5.00000i	− 0.811107i
$39$	4.00000	0.640513
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−6.00000	−0.884652
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	1.00000i	0.144338i
$49$	0	0
$50$	0	0
$51$	−3.00000	−0.420084
$52$	4.00000i	0.554700i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	0	0
$57$	5.00000i	0.662266i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	2.00000i	0.254000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	− 13.0000i	− 1.58820i	−0.607785	−	0.794101i	$-0.707942\pi$
$67$	− 13.0000i	− 1.58820i	0.607785	−	0.794101i	$-0.292058\pi$
$68$	− 3.00000i	− 0.363803i
$69$	6.00000	0.722315
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	2.00000i	0.235702i
$73$	11.0000i	1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$	11.0000i	1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−5.00000	−0.573539
$77$	0	0
$78$	− 4.00000i	− 0.452911i
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 3.00000i	− 0.331295i
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	− 3.00000i	− 0.319801i
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	0	0
$92$	6.00000i	0.625543i
$93$	− 2.00000i	− 0.207390i
$94$	−12.0000	−1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	3.00000i	0.297044i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 3.00000i	− 0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$	− 3.00000i	− 0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$	− 5.00000i	− 0.481125i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	0	0
$117$	− 8.00000i	− 0.739600i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.00000i	0.181071i
$123$	3.00000i	0.270501i
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	1.00000i	0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	3.00000i	0.261116i
$133$	0	0
$134$	−13.0000	−1.12303
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	− 6.00000i	− 0.510754i
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	− 12.0000i	− 1.00702i
$143$	12.0000i	1.00349i
$144$	2.00000	0.166667
$145$	0	0
$146$	11.0000	0.910366
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	5.00000i	0.405554i
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	− 10.0000i	− 0.795557i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−9.00000	−0.698535
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	10.0000	0.764719
$172$	− 4.00000i	− 0.304997i
$173$	− 24.0000i	− 1.82469i	−0.409426	−	0.912343i	$-0.634271\pi$
$173$	− 24.0000i	− 1.82469i	0.409426	−	0.912343i	$-0.365729\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	− 15.0000i	− 1.12430i
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	− 2.00000i	− 0.147844i
$184$	6.00000	0.442326
$185$	0	0
$186$	−2.00000	−0.146647
$187$	− 9.00000i	− 0.658145i
$188$	12.0000i	0.875190i
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	19.0000i	1.36765i	0.729646	+	0.683825i	$0.239685\pi$
$193$	19.0000i	1.36765i	−0.729646	+	0.683825i	$0.760315\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	6.00000i	0.426401i
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	− 18.0000i	− 1.26648i
$203$	0	0
$204$	3.00000	0.210042
$205$	0	0
$206$	−4.00000	−0.278693
$207$	− 12.0000i	− 0.834058i
$208$	− 4.00000i	− 0.277350i
$209$	−15.0000	−1.03757
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	6.00000i	0.412082i
$213$	12.0000i	0.822226i
$214$	−3.00000	−0.205076
$215$	0	0
$216$	−5.00000	−0.340207
$217$	0	0
$218$	− 10.0000i	− 0.677285i
$219$	−11.0000	−0.743311
$220$	0	0
$221$	12.0000	0.807207
$222$	2.00000i	0.134231i
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	0	0
$225$	0	0
$226$	9.00000	0.598671
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	− 5.00000i	− 0.331133i
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−8.00000	−0.522976
$235$	0	0
$236$	0	0
$237$	10.0000i	0.649570i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	2.00000i	0.128565i
$243$	16.0000i	1.02640i
$244$	2.00000	0.128037
$245$	0	0
$246$	3.00000	0.191273
$247$	− 20.0000i	− 1.27257i
$248$	− 2.00000i	− 0.127000i
$249$	9.00000	0.570352
$250$	0	0
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	0	0
$253$	18.0000i	1.13165i
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	4.00000i	0.249029i
$259$	0	0
$260$	0	0
$261$	0	0
$262$	12.0000i	0.741362i
$263$	− 6.00000i	− 0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$	− 6.00000i	− 0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	0	0
$267$	15.0000i	0.917985i
$268$	13.0000i	0.794101i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	3.00000i	0.181902i
$273$	0	0
$274$	−3.00000	−0.181237
$275$	0	0
$276$	−6.00000	−0.361158
$277$	32.0000i	1.92269i	0.275340	+	0.961347i	$0.411209\pi$
$277$	32.0000i	1.92269i	−0.275340	+	0.961347i	$0.588791\pi$
$278$	− 5.00000i	− 0.299880i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	− 12.0000i	− 0.714590i
$283$	11.0000i	0.653882i	0.945045	+	0.326941i	$0.106018\pi$
$283$	11.0000i	0.653882i	−0.945045	+	0.326941i	$0.893982\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	12.0000	0.709575
$287$	0	0
$288$	− 2.00000i	− 0.117851i
$289$	8.00000	0.470588
$290$	0	0
$291$	2.00000	0.117242
$292$	− 11.0000i	− 0.643726i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	− 15.0000i	− 0.870388i
$298$	0	0
$299$	−24.0000	−1.38796
$300$	0	0
$301$	0	0
$302$	− 2.00000i	− 0.115087i
$303$	18.0000i	1.03407i
$304$	5.00000	0.286770
$305$	0	0
$306$	6.00000	0.342997
$307$	− 17.0000i	− 0.970241i	−0.874447	−	0.485121i	$-0.838776\pi$
$307$	− 17.0000i	− 0.970241i	0.874447	−	0.485121i	$-0.161224\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	4.00000i	0.226455i
$313$	26.0000i	1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$	26.0000i	1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−10.0000	−0.562544
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	− 6.00000i	− 0.336463i
$319$	0	0
$320$	0	0
$321$	3.00000	0.167444
$322$	0	0
$323$	15.0000i	0.834622i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−11.0000	−0.609234
$327$	10.0000i	0.553001i
$328$	3.00000i	0.165647i
$329$	0	0
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	9.00000i	0.493939i
$333$	4.00000i	0.219199i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	− 13.0000i	− 0.708155i	−0.935216	−	0.354078i	$-0.884795\pi$
$337$	− 13.0000i	− 0.708155i	0.935216	−	0.354078i	$-0.115205\pi$
$338$	3.00000i	0.163178i
$339$	−9.00000	−0.488813
$340$	0	0
$341$	6.00000	0.324918
$342$	− 10.0000i	− 0.540738i
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−24.0000	−1.29025
$347$	− 3.00000i	− 0.161048i	−0.996753	−	0.0805242i	$-0.974341\pi$
$347$	− 3.00000i	− 0.161048i	0.996753	−	0.0805242i	$-0.0256594\pi$
$348$	0	0
$349$	−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
$351$	20.0000	1.06752
$352$	3.00000i	0.159901i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	−15.0000	−0.794998
$357$	0	0
$358$	− 15.0000i	− 0.792775i
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	2.00000i	0.105118i
$363$	− 2.00000i	− 0.104973i
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	28.0000i	1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$	28.0000i	1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$	− 6.00000i	− 0.312772i
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	2.00000i	0.103695i
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	12.0000	0.618853
$377$	0	0
$378$	0	0
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	18.0000i	0.920960i
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	19.0000	0.967075
$387$	8.00000i	0.406663i
$388$	2.00000i	0.101535i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	− 12.0000i	− 0.605320i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	6.00000	0.301511
$397$	28.0000i	1.40528i	0.711546	+	0.702640i	$0.247995\pi$
$397$	28.0000i	1.40528i	−0.711546	+	0.702640i	$0.752005\pi$
$398$	− 20.0000i	− 1.00251i
$399$	0	0
$400$	0	0
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	− 13.0000i	− 0.648381i
$403$	8.00000i	0.398508i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	− 3.00000i	− 0.148522i
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	4.00000i	0.197066i
$413$	0	0
$414$	−12.0000	−0.589768
$415$	0	0
$416$	−4.00000	−0.196116
$417$	5.00000i	0.244851i
$418$	15.0000i	0.733674i
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	13.0000i	0.632830i
$423$	− 24.0000i	− 1.16692i
$424$	6.00000	0.291386
$425$	0	0
$426$	12.0000	0.581402
$427$	0	0
$428$	3.00000i	0.145010i
$429$	−12.0000	−0.579365
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	5.00000i	0.240563i
$433$	− 19.0000i	− 0.913082i	−0.889702	−	0.456541i	$-0.849088\pi$
$433$	− 19.0000i	− 0.913082i	0.889702	−	0.456541i	$-0.150912\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 30.0000i	− 1.43509i
$438$	11.0000i	0.525600i
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	9.00000i	0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$	9.00000i	0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	−4.00000	−0.189405
$447$	0	0
$448$	0	0
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	− 9.00000i	− 0.423324i
$453$	2.00000i	0.0939682i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−5.00000	−0.234146
$457$	17.0000i	0.795226i	0.917553	+	0.397613i	$0.130161\pi$
$457$	17.0000i	0.795226i	−0.917553	+	0.397613i	$0.869839\pi$
$458$	− 20.0000i	− 0.934539i
$459$	−15.0000	−0.700140
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	0	0
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	8.00000i	0.369800i
$469$	0	0
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	− 12.0000i	− 0.551761i
$474$	10.0000	0.459315
$475$	0	0
$476$	0	0
$477$	− 12.0000i	− 0.549442i
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	17.0000i	0.774329i
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	16.0000	0.725775
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	11.0000	0.497437
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	− 3.00000i	− 0.135250i
$493$	0	0
$494$	−20.0000	−0.899843
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	− 9.00000i	− 0.403300i
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	27.0000i	1.20507i
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	0	0
$505$	0	0
$506$	18.0000	0.800198
$507$	− 3.00000i	− 0.133235i
$508$	− 2.00000i	− 0.0887357i
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	25.0000i	1.10378i
$514$	18.0000	0.793946
$515$	0	0
$516$	4.00000	0.176090
$517$	36.0000i	1.58328i
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	41.0000i	1.79280i	0.443241	+	0.896402i	$0.353829\pi$
$523$	41.0000i	1.79280i	−0.443241	+	0.896402i	$0.646171\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−6.00000	−0.261612
$527$	− 6.00000i	− 0.261364i
$528$	− 3.00000i	− 0.130558i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	15.0000	0.649113
$535$	0	0
$536$	13.0000	0.561514
$537$	15.0000i	0.647298i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	2.00000i	0.0859074i
$543$	− 2.00000i	− 0.0858282i
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	17.0000i	0.726868i	0.931620	+	0.363434i	$0.118396\pi$
$547$	17.0000i	0.726868i	−0.931620	+	0.363434i	$0.881604\pi$
$548$	3.00000i	0.128154i
$549$	−4.00000	−0.170716
$550$	0	0
$551$	0	0
$552$	6.00000i	0.255377i
$553$	0	0
$554$	32.0000	1.35955
$555$	0	0
$556$	−5.00000	−0.212047
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	4.00000i	0.169334i
$559$	16.0000	0.676728
$560$	0	0
$561$	9.00000	0.379980
$562$	18.0000i	0.759284i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	11.0000	0.462364
$567$	0	0
$568$	12.0000i	0.503509i
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	− 12.0000i	− 0.501745i
$573$	− 18.0000i	− 0.751961i
$574$	0	0
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	13.0000i	0.541197i	0.962692	+	0.270599i	$0.0872216\pi$
$577$	13.0000i	0.541197i	−0.962692	+	0.270599i	$0.912778\pi$
$578$	− 8.00000i	− 0.332756i
$579$	−19.0000	−0.789613
$580$	0	0
$581$	0	0
$582$	− 2.00000i	− 0.0829027i
$583$	18.0000i	0.745484i
$584$	−11.0000	−0.455183
$585$	0	0
$586$	6.00000	0.247858
$587$	33.0000i	1.36206i	0.732257	+	0.681028i	$0.238467\pi$
$587$	33.0000i	1.36206i	−0.732257	+	0.681028i	$0.761533\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	18.0000	0.740421
$592$	2.00000i	0.0821995i
$593$	− 9.00000i	− 0.369586i	−0.982777	−	0.184793i	$-0.940839\pi$
$593$	− 9.00000i	− 0.369586i	0.982777	−	0.184793i	$-0.0591614\pi$
$594$	−15.0000	−0.615457
$595$	0	0
$596$	0	0
$597$	20.0000i	0.818546i
$598$	24.0000i	0.981433i
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	13.0000	0.530281	0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000	0.530281	0.265141	+	0.964210i	$0.414582\pi$
$602$	0	0
$603$	− 26.0000i	− 1.05880i
$604$	−2.00000	−0.0813788
$605$	0	0
$606$	18.0000	0.731200
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	− 5.00000i	− 0.202777i
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	− 6.00000i	− 0.242536i
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−17.0000	−0.686064
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	− 4.00000i	− 0.160904i
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	30.0000	1.20386
$622$	− 18.0000i	− 0.721734i
$623$	0	0
$624$	4.00000	0.160128
$625$	0	0
$626$	26.0000	1.03917
$627$	− 15.0000i	− 0.599042i
$628$	2.00000i	0.0798087i
$629$	−6.00000	−0.239236
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	10.0000i	0.397779i
$633$	− 13.0000i	− 0.516704i
$634$	12.0000	0.476581
$635$	0	0
$636$	−6.00000	−0.237915
$637$	0	0
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	− 3.00000i	− 0.118401i
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	0	0
$646$	15.0000	0.590167
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	11.0000i	0.430793i
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	3.00000	0.117130
$657$	22.0000i	0.858302i
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	− 17.0000i	− 0.660724i
$663$	12.0000i	0.466041i
$664$	9.00000	0.349268
$665$	0	0
$666$	4.00000	0.154997
$667$	0	0
$668$	12.0000i	0.464294i
$669$	4.00000	0.154649
$670$	0	0
$671$	6.00000	0.231627
$672$	0	0
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	3.00000	0.115385
$677$	48.0000i	1.84479i	0.386248	+	0.922395i	$0.373771\pi$
$677$	48.0000i	1.84479i	−0.386248	+	0.922395i	$0.626229\pi$
$678$	9.00000i	0.345643i
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	− 6.00000i	− 0.229752i
$683$	39.0000i	1.49229i	0.665782	+	0.746147i	$0.268098\pi$
$683$	39.0000i	1.49229i	−0.665782	+	0.746147i	$0.731902\pi$
$684$	−10.0000	−0.382360
$685$	0	0
$686$	0	0
$687$	20.0000i	0.763048i
$688$	4.00000i	0.152499i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−17.0000	−0.646710	−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000	−0.646710	−0.323355	+	0.946278i	$0.604811\pi$
$692$	24.0000i	0.912343i
$693$	0	0
$694$	−3.00000	−0.113878
$695$	0	0
$696$	0	0
$697$	9.00000i	0.340899i
$698$	10.0000i	0.378506i
$699$	6.00000	0.226941
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	− 20.0000i	− 0.754851i
$703$	10.0000i	0.377157i
$704$	3.00000	0.113067
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	15.0000i	0.562149i
$713$	12.0000i	0.449404i
$714$	0	0
$715$	0	0
$716$	−15.0000	−0.560576
$717$	0	0
$718$	30.0000i	1.11959i
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	0	0
$722$	− 6.00000i	− 0.223297i
$723$	− 17.0000i	− 0.632237i
$724$	2.00000	0.0743294
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−12.0000	−0.443836
$732$	2.00000i	0.0739221i
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	−6.00000	−0.221163
$737$	39.0000i	1.43658i
$738$	− 6.00000i	− 0.220863i
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	20.0000	0.734718
$742$	0	0
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	−26.0000	−0.951928
$747$	− 18.0000i	− 0.658586i
$748$	9.00000i	0.329073i
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	− 12.0000i	− 0.437595i
$753$	− 27.0000i	− 0.983935i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	32.0000i	1.16306i	0.813525	+	0.581530i	$0.197546\pi$
$757$	32.0000i	1.16306i	−0.813525	+	0.581530i	$0.802454\pi$
$758$	− 25.0000i	− 0.908041i
$759$	−18.0000	−0.653359
$760$	0	0
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	2.00000i	0.0724524i
$763$	0	0
$764$	18.0000	0.651217
$765$	0	0
$766$	6.00000	0.216789
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	− 19.0000i	− 0.683825i
$773$	− 24.0000i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$	− 24.0000i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	30.0000i	1.07555i
$779$	15.0000	0.537431
$780$	0	0
$781$	−36.0000	−1.28818
$782$	− 18.0000i	− 0.643679i
$783$	0	0
$784$	0	0
$785$	0	0
$786$	−12.0000	−0.428026
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	18.0000i	0.641223i
$789$	6.00000	0.213606
$790$	0	0
$791$	0	0
$792$	− 6.00000i	− 0.213201i
$793$	8.00000i	0.284088i
$794$	28.0000	0.993683
$795$	0	0
$796$	−20.0000	−0.708881
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	0	0
$801$	30.0000	1.06000
$802$	3.00000i	0.105934i
$803$	− 33.0000i	− 1.16454i
$804$	−13.0000	−0.458475
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	18.0000i	0.633238i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	− 2.00000i	− 0.0701431i
$814$	−6.00000	−0.210300
$815$	0	0
$816$	−3.00000	−0.105021
$817$	20.0000i	0.699711i
$818$	− 5.00000i	− 0.174821i
$819$	0	0
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	− 3.00000i	− 0.104637i
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	− 33.0000i	− 1.14752i	−0.819023	−	0.573761i	$-0.805484\pi$
$827$	− 33.0000i	− 1.14752i	0.819023	−	0.573761i	$-0.194516\pi$
$828$	12.0000i	0.417029i
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	−32.0000	−1.11007
$832$	4.00000i	0.138675i
$833$	0	0
$834$	5.00000	0.173136
$835$	0	0
$836$	15.0000	0.518786
$837$	− 10.0000i	− 0.345651i
$838$	15.0000i	0.518166i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	28.0000i	0.964944i
$843$	− 18.0000i	− 0.619953i
$844$	13.0000	0.447478
$845$	0	0
$846$	−24.0000	−0.825137
$847$	0	0
$848$	− 6.00000i	− 0.206041i
$849$	−11.0000	−0.377519
$850$	0	0
$851$	12.0000	0.411355
$852$	− 12.0000i	− 0.411113i
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	0	0
$855$	0	0
$856$	3.00000	0.102538
$857$	3.00000i	0.102478i	0.998686	+	0.0512390i	$0.0163170\pi$
$857$	3.00000i	0.102478i	−0.998686	+	0.0512390i	$0.983683\pi$
$858$	12.0000i	0.409673i
$859$	5.00000	0.170598	0.0852989	−	0.996355i	$-0.472815\pi$
$859$	5.00000	0.170598	0.0852989	+	0.996355i	$0.472815\pi$
$860$	0	0
$861$	0	0
$862$	18.0000i	0.613082i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−19.0000	−0.645646
$867$	8.00000i	0.271694i
$868$	0	0
$869$	−30.0000	−1.01768
$870$	0	0
$871$	−52.0000	−1.76195
$872$	10.0000i	0.338643i
$873$	− 4.00000i	− 0.135379i
$874$	−30.0000	−1.01477
$875$	0	0
$876$	11.0000	0.371656
$877$	− 28.0000i	− 0.945493i	−0.881199	−	0.472746i	$-0.843263\pi$
$877$	− 28.0000i	− 0.945493i	0.881199	−	0.472746i	$-0.156737\pi$
$878$	40.0000i	1.34993i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	− 41.0000i	− 1.37976i	−0.723924	−	0.689880i	$-0.757663\pi$
$883$	− 41.0000i	− 1.37976i	0.723924	−	0.689880i	$-0.242337\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	9.00000	0.302361
$887$	− 12.0000i	− 0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$	− 12.0000i	− 0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	0	0
$890$	0	0
$891$	−3.00000	−0.100504
$892$	4.00000i	0.133930i
$893$	− 60.0000i	− 2.00782i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 24.0000i	− 0.801337i
$898$	− 15.0000i	− 0.500556i
$899$	0	0
$900$	0	0
$901$	18.0000	0.599667
$902$	9.00000i	0.299667i
$903$	0	0
$904$	−9.00000	−0.299336
$905$	0	0
$906$	2.00000	0.0664455
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	12.0000i	0.398234i
$909$	36.0000	1.19404
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	5.00000i	0.165567i
$913$	27.0000i	0.893570i
$914$	17.0000	0.562310
$915$	0	0
$916$	−20.0000	−0.660819
$917$	0	0
$918$	15.0000i	0.495074i
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	17.0000	0.560169
$922$	12.0000i	0.395199i
$923$	− 48.0000i	− 1.57994i
$924$	0	0
$925$	0	0
$926$	4.00000	0.131448
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	6.00000i	0.196537i
$933$	18.0000i	0.589294i
$934$	−12.0000	−0.392652
$935$	0	0
$936$	8.00000	0.261488
$937$	13.0000i	0.424691i	0.977195	+	0.212346i	$0.0681103\pi$
$937$	13.0000i	0.424691i	−0.977195	+	0.212346i	$0.931890\pi$
$938$	0	0
$939$	−26.0000	−0.848478
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	− 18.0000i	− 0.586161i
$944$	0	0
$945$	0	0
$946$	−12.0000	−0.390154
$947$	− 48.0000i	− 1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$	− 48.0000i	− 1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$	− 10.0000i	− 0.324785i
$949$	44.0000	1.42830
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	39.0000i	1.26333i	0.775240	+	0.631667i	$0.217629\pi$
$953$	39.0000i	1.26333i	−0.775240	+	0.631667i	$0.782371\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	0	0
$957$	0	0
$958$	− 30.0000i	− 0.969256i
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 8.00000i	− 0.257930i
$963$	− 6.00000i	− 0.193347i
$964$	17.0000	0.547533
$965$	0	0
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	−15.0000	−0.481869
$970$	0	0
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	− 16.0000i	− 0.513200i
$973$	0	0
$974$	2.00000	0.0640841
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	57.0000i	1.82359i	0.410644	+	0.911796i	$0.365304\pi$
$977$	57.0000i	1.82359i	−0.410644	+	0.911796i	$0.634696\pi$
$978$	− 11.0000i	− 0.351741i
$979$	−45.0000	−1.43821
$980$	0	0
$981$	20.0000	0.638551
$982$	− 12.0000i	− 0.382935i
$983$	6.00000i	0.191370i	0.995412	+	0.0956851i	$0.0305042\pi$
$983$	6.00000i	0.191370i	−0.995412	+	0.0956851i	$0.969496\pi$
$984$	−3.00000	−0.0956365
$985$	0	0
$986$	0	0
$987$	0	0
$988$	20.0000i	0.636285i
$989$	24.0000	0.763156
$990$	0	0
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	2.00000i	0.0635001i
$993$	17.0000i	0.539479i
$994$	0	0
$995$	0	0
$996$	−9.00000	−0.285176
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	20.0000i	0.633089i
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.c.m.99.1		2
5.2	odd	4		2450.2.a.bd.1.1		1
5.3	odd	4		2450.2.a.g.1.1		1
5.4	even	2	inner	2450.2.c.m.99.2		2
7.6	odd	2		50.2.b.a.49.1		2
21.20	even	2		450.2.c.c.199.2		2
28.27	even	2		400.2.c.c.49.2		2
35.13	even	4		50.2.a.a.1.1	✓	1
35.27	even	4		50.2.a.b.1.1	yes	1
35.34	odd	2		50.2.b.a.49.2		2
56.13	odd	2		1600.2.c.i.449.2		2
56.27	even	2		1600.2.c.h.449.1		2
84.83	odd	2		3600.2.f.f.2449.1		2
105.62	odd	4		450.2.a.c.1.1		1
105.83	odd	4		450.2.a.g.1.1		1
105.104	even	2		450.2.c.c.199.1		2
140.27	odd	4		400.2.a.f.1.1		1
140.83	odd	4		400.2.a.d.1.1		1
140.139	even	2		400.2.c.c.49.1		2
280.13	even	4		1600.2.a.j.1.1		1
280.27	odd	4		1600.2.a.i.1.1		1
280.69	odd	2		1600.2.c.i.449.1		2
280.83	odd	4		1600.2.a.p.1.1		1
280.139	even	2		1600.2.c.h.449.2		2
280.237	even	4		1600.2.a.q.1.1		1
385.153	odd	4		6050.2.a.bi.1.1		1
385.307	odd	4		6050.2.a.h.1.1		1
420.83	even	4		3600.2.a.l.1.1		1
420.167	even	4		3600.2.a.bc.1.1		1
420.419	odd	2		3600.2.f.f.2449.2		2
455.272	even	4		8450.2.a.d.1.1		1
455.363	even	4		8450.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.a.a.1.1	✓	1	35.13	even	4
50.2.a.b.1.1	yes	1	35.27	even	4
50.2.b.a.49.1		2	7.6	odd	2
50.2.b.a.49.2		2	35.34	odd	2
400.2.a.d.1.1		1	140.83	odd	4
400.2.a.f.1.1		1	140.27	odd	4
400.2.c.c.49.1		2	140.139	even	2
400.2.c.c.49.2		2	28.27	even	2
450.2.a.c.1.1		1	105.62	odd	4
450.2.a.g.1.1		1	105.83	odd	4
450.2.c.c.199.1		2	105.104	even	2
450.2.c.c.199.2		2	21.20	even	2
1600.2.a.i.1.1		1	280.27	odd	4
1600.2.a.j.1.1		1	280.13	even	4
1600.2.a.p.1.1		1	280.83	odd	4
1600.2.a.q.1.1		1	280.237	even	4
1600.2.c.h.449.1		2	56.27	even	2
1600.2.c.h.449.2		2	280.139	even	2
1600.2.c.i.449.1		2	280.69	odd	2
1600.2.c.i.449.2		2	56.13	odd	2
2450.2.a.g.1.1		1	5.3	odd	4
2450.2.a.bd.1.1		1	5.2	odd	4
2450.2.c.m.99.1		2	1.1	even	1	trivial
2450.2.c.m.99.2		2	5.4	even	2	inner
3600.2.a.l.1.1		1	420.83	even	4
3600.2.a.bc.1.1		1	420.167	even	4
3600.2.f.f.2449.1		2	84.83	odd	2
3600.2.f.f.2449.2		2	420.419	odd	2
6050.2.a.h.1.1		1	385.307	odd	4
6050.2.a.bi.1.1		1	385.153	odd	4
8450.2.a.d.1.1		1	455.272	even	4
8450.2.a.v.1.1		1	455.363	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{8} +2.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{18} +5.00000 q^{19} +3.00000i q^{22} -6.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} +5.00000i q^{27} -2.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} +3.00000 q^{34} -2.00000 q^{36} +2.00000i q^{37} -5.00000i q^{38} +4.00000 q^{39} +3.00000 q^{41} +4.00000i q^{43} +3.00000 q^{44} -6.00000 q^{46} -12.0000i q^{47} +1.00000i q^{48} -3.00000 q^{51} +4.00000i q^{52} -6.00000i q^{53} +5.00000 q^{54} +5.00000i q^{57} -2.00000 q^{61} +2.00000i q^{62} -1.00000 q^{64} -3.00000 q^{66} -13.0000i q^{67} -3.00000i q^{68} +6.00000 q^{69} +12.0000 q^{71} +2.00000i q^{72} +11.0000i q^{73} +2.00000 q^{74} -5.00000 q^{76} -4.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} -3.00000i q^{82} -9.00000i q^{83} +4.00000 q^{86} -3.00000i q^{88} +15.0000 q^{89} +6.00000i q^{92} -2.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} - 6 q^{11} + 2 q^{16} + 10 q^{19} - 2 q^{24} - 8 q^{26} - 4 q^{31} + 6 q^{34} - 4 q^{36} + 8 q^{39} + 6 q^{41} + 6 q^{44} - 12 q^{46} - 6 q^{51} + 10 q^{54} - 4 q^{61} - 2 q^{64} - 6 q^{66} + 12 q^{69} + 24 q^{71} + 4 q^{74} - 10 q^{76} + 20 q^{79} + 2 q^{81} + 8 q^{86} + 30 q^{89} - 24 q^{94} + 2 q^{96} - 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 - 6 * q^11 + 2 * q^16 + 10 * q^19 - 2 * q^24 - 8 * q^26 - 4 * q^31 + 6 * q^34 - 4 * q^36 + 8 * q^39 + 6 * q^41 + 6 * q^44 - 12 * q^46 - 6 * q^51 + 10 * q^54 - 4 * q^61 - 2 * q^64 - 6 * q^66 + 12 * q^69 + 24 * q^71 + 4 * q^74 - 10 * q^76 + 20 * q^79 + 2 * q^81 + 8 * q^86 + 30 * q^89 - 24 * q^94 + 2 * q^96 - 12 * q^99

Introduction

L-functions

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Database

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