LMFDB - Embedded newform 950.2.b.d.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,2,Mod(799,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, 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("950.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	950.b (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:	$7.58578819202$
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 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	950.799
Dual form			950.2.b.d.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +1.00000i q^{12} -3.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.00000i q^{17} +2.00000i q^{18} +1.00000 q^{19} +1.00000 q^{21} -5.00000i q^{23} -1.00000 q^{24} +3.00000 q^{26} -5.00000i q^{27} -1.00000i q^{28} +5.00000 q^{29} +10.0000 q^{31} +1.00000i q^{32} -7.00000 q^{34} -2.00000 q^{36} -2.00000i q^{37} +1.00000i q^{38} -3.00000 q^{39} +2.00000 q^{41} +1.00000i q^{42} +6.00000i q^{43} +5.00000 q^{46} -1.00000i q^{48} +6.00000 q^{49} +7.00000 q^{51} +3.00000i q^{52} +9.00000i q^{53} +5.00000 q^{54} +1.00000 q^{56} -1.00000i q^{57} +5.00000i q^{58} +7.00000 q^{59} -4.00000 q^{61} +10.0000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -7.00000i q^{67} -7.00000i q^{68} -5.00000 q^{69} -2.00000i q^{72} -9.00000i q^{73} +2.00000 q^{74} -1.00000 q^{76} -3.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -2.00000i q^{83} -1.00000 q^{84} -6.00000 q^{86} -5.00000i q^{87} +10.0000 q^{89} +3.00000 q^{91} +5.00000i q^{92} -10.0000i q^{93} +1.00000 q^{96} +18.0000i q^{97} +6.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{14} + 2 q^{16} + 2 q^{19} + 2 q^{21} - 2 q^{24} + 6 q^{26} + 10 q^{29} + 20 q^{31} - 14 q^{34} - 4 q^{36} - 6 q^{39} + 4 q^{41} + 10 q^{46} + 12 q^{49} + 14 q^{51} + 10 q^{54} + 2 q^{56} + 14 q^{59} - 8 q^{61} - 2 q^{64} - 10 q^{69} + 4 q^{74} - 2 q^{76} + 20 q^{79} + 2 q^{81} - 2 q^{84} - 12 q^{86} + 20 q^{89} + 6 q^{91} + 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 - 2 * q^14 + 2 * q^16 + 2 * q^19 + 2 * q^21 - 2 * q^24 + 6 * q^26 + 10 * q^29 + 20 * q^31 - 14 * q^34 - 4 * q^36 - 6 * q^39 + 4 * q^41 + 10 * q^46 + 12 * q^49 + 14 * q^51 + 10 * q^54 + 2 * q^56 + 14 * q^59 - 8 * q^61 - 2 * q^64 - 10 * q^69 + 4 * q^74 - 2 * q^76 + 20 * q^79 + 2 * q^81 - 2 * q^84 - 12 * q^86 + 20 * q^89 + 6 * q^91 + 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	− 1.00000i	− 0.353553i
$9$	2.00000	0.666667
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000i	0.288675i
$13$	− 3.00000i	− 0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	− 3.00000i	− 0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	7.00000i	1.69775i	0.528594	+	0.848875i	$0.322719\pi$
$17$	7.00000i	1.69775i	−0.528594	+	0.848875i	$0.677281\pi$
$18$	2.00000i	0.471405i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	− 5.00000i	− 1.04257i	−0.853382	−	0.521286i	$-0.825452\pi$
$23$	− 5.00000i	− 1.04257i	0.853382	−	0.521286i	$-0.174548\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	3.00000	0.588348
$27$	− 5.00000i	− 0.962250i
$28$	− 1.00000i	− 0.188982i
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−7.00000	−1.20049
$35$	0	0
$36$	−2.00000	−0.333333
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	1.00000i	0.162221i
$39$	−3.00000	−0.480384
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	1.00000i	0.154303i
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	0	0
$46$	5.00000	0.737210
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 1.00000i	− 0.144338i
$49$	6.00000	0.857143
$50$	0	0
$51$	7.00000	0.980196
$52$	3.00000i	0.416025i
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	1.00000	0.133631
$57$	− 1.00000i	− 0.132453i
$58$	5.00000i	0.656532i
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	10.0000i	1.27000i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	− 7.00000i	− 0.848875i
$69$	−5.00000	−0.601929
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 2.00000i	− 0.235702i
$73$	− 9.00000i	− 1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$	− 9.00000i	− 1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	− 3.00000i	− 0.339683i
$79$	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
$82$	2.00000i	0.220863i
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−6.00000	−0.646997
$87$	− 5.00000i	− 0.536056i
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	5.00000i	0.521286i
$93$	− 10.0000i	− 1.03695i
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	18.0000i	1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$	18.0000i	1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	6.00000i	0.606092i
$99$	0	0
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	7.00000i	0.693103i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−9.00000	−0.874157
$107$	− 9.00000i	− 0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$	− 9.00000i	− 0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$	5.00000i	0.481125i
$109$	−13.0000	−1.24517	−0.622587	−	0.782551i	$-0.713918\pi$
$109$	−13.0000	−1.24517	−0.622587	+	0.782551i	$0.713918\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	1.00000i	0.0944911i
$113$	8.00000i	0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$	8.00000i	0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	−5.00000	−0.464238
$117$	− 6.00000i	− 0.554700i
$118$	7.00000i	0.644402i
$119$	−7.00000	−0.641689
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 4.00000i	− 0.362143i
$123$	− 2.00000i	− 0.180334i
$124$	−10.0000	−0.898027
$125$	0	0
$126$	−2.00000	−0.178174
$127$	6.00000i	0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$	6.00000i	0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	6.00000	0.528271
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	1.00000i	0.0867110i
$134$	7.00000	0.604708
$135$	0	0
$136$	7.00000	0.600245
$137$	3.00000i	0.256307i	0.991754	+	0.128154i	$0.0409051\pi$
$137$	3.00000i	0.256307i	−0.991754	+	0.128154i	$0.959095\pi$
$138$	− 5.00000i	− 0.425628i
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	2.00000	0.166667
$145$	0	0
$146$	9.00000	0.744845
$147$	− 6.00000i	− 0.494872i
$148$	2.00000i	0.164399i
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	14.0000i	1.13183i
$154$	0	0
$155$	0	0
$156$	3.00000	0.240192
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	10.0000i	0.795557i
$159$	9.00000	0.713746
$160$	0	0
$161$	5.00000	0.394055
$162$	1.00000i	0.0785674i
$163$	− 18.0000i	− 1.40987i	−0.709273	−	0.704934i	$-0.750976\pi$
$163$	− 18.0000i	− 1.40987i	0.709273	−	0.704934i	$-0.249024\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	2.00000	0.155230
$167$	− 14.0000i	− 1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$	− 14.0000i	− 1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$	− 1.00000i	− 0.0771517i
$169$	4.00000	0.307692
$170$	0	0
$171$	2.00000	0.152944
$172$	− 6.00000i	− 0.457496i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	5.00000	0.379049
$175$	0	0
$176$	0	0
$177$	− 7.00000i	− 0.526152i
$178$	10.0000i	0.749532i
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	3.00000i	0.222375i
$183$	4.00000i	0.295689i
$184$	−5.00000	−0.368605
$185$	0	0
$186$	10.0000	0.733236
$187$	0	0
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	−7.00000	−0.506502	−0.253251	−	0.967401i	$-0.581500\pi$
$191$	−7.00000	−0.506502	−0.253251	+	0.967401i	$0.581500\pi$
$192$	1.00000i	0.0721688i
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	−6.00000	−0.428571
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	0	0
$199$	−17.0000	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	− 4.00000i	− 0.281439i
$203$	5.00000i	0.350931i
$204$	−7.00000	−0.490098
$205$	0	0
$206$	0	0
$207$	− 10.0000i	− 0.695048i
$208$	− 3.00000i	− 0.208013i
$209$	0	0
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	− 9.00000i	− 0.618123i
$213$	0	0
$214$	9.00000	0.615227
$215$	0	0
$216$	−5.00000	−0.340207
$217$	10.0000i	0.678844i
$218$	− 13.0000i	− 0.880471i
$219$	−9.00000	−0.608164
$220$	0	0
$221$	21.0000	1.41261
$222$	− 2.00000i	− 0.134231i
$223$	− 22.0000i	− 1.47323i	−0.676313	−	0.736614i	$-0.736423\pi$
$223$	− 22.0000i	− 1.47323i	0.676313	−	0.736614i	$-0.263577\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−8.00000	−0.532152
$227$	25.0000i	1.65931i	0.558278	+	0.829654i	$0.311462\pi$
$227$	25.0000i	1.65931i	−0.558278	+	0.829654i	$0.688538\pi$
$228$	1.00000i	0.0662266i
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	0	0
$231$	0	0
$232$	− 5.00000i	− 0.328266i
$233$	26.0000i	1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$	26.0000i	1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−7.00000	−0.455661
$237$	− 10.0000i	− 0.649570i
$238$	− 7.00000i	− 0.453743i
$239$	27.0000	1.74648	0.873242	−	0.487286i	$-0.162013\pi$
$239$	27.0000	1.74648	0.873242	+	0.487286i	$0.162013\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	− 11.0000i	− 0.707107i
$243$	− 16.0000i	− 1.02640i
$244$	4.00000	0.256074
$245$	0	0
$246$	2.00000	0.127515
$247$	− 3.00000i	− 0.190885i
$248$	− 10.0000i	− 0.635001i
$249$	−2.00000	−0.126745
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	−6.00000	−0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	6.00000i	0.373544i
$259$	2.00000	0.124274
$260$	0	0
$261$	10.0000	0.618984
$262$	− 20.0000i	− 1.23560i
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	0	0
$265$	0	0
$266$	−1.00000	−0.0613139
$267$	− 10.0000i	− 0.611990i
$268$	7.00000i	0.427593i
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	7.00000i	0.424437i
$273$	− 3.00000i	− 0.181568i
$274$	−3.00000	−0.181237
$275$	0	0
$276$	5.00000	0.300965
$277$	− 28.0000i	− 1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	− 28.0000i	− 1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$	− 12.0000i	− 0.719712i
$279$	20.0000	1.19737
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	− 26.0000i	− 1.54554i	−0.634686	−	0.772770i	$-0.718871\pi$
$283$	− 26.0000i	− 1.54554i	0.634686	−	0.772770i	$-0.281129\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00000i	0.118056i
$288$	2.00000i	0.117851i
$289$	−32.0000	−1.88235
$290$	0	0
$291$	18.0000	1.05518
$292$	9.00000i	0.526685i
$293$	31.0000i	1.81104i	0.424304	+	0.905520i	$0.360519\pi$
$293$	31.0000i	1.81104i	−0.424304	+	0.905520i	$0.639481\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	− 4.00000i	− 0.231714i
$299$	−15.0000	−0.867472
$300$	0	0
$301$	−6.00000	−0.345834
$302$	− 6.00000i	− 0.345261i
$303$	4.00000i	0.229794i
$304$	1.00000	0.0573539
$305$	0	0
$306$	−14.0000	−0.800327
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	33.0000	1.87126	0.935629	−	0.352985i	$-0.114833\pi$
$311$	33.0000	1.87126	0.935629	+	0.352985i	$0.114833\pi$
$312$	3.00000i	0.169842i
$313$	− 29.0000i	− 1.63918i	−0.572953	−	0.819588i	$-0.694202\pi$
$313$	− 29.0000i	− 1.63918i	0.572953	−	0.819588i	$-0.305798\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−10.0000	−0.562544
$317$	− 3.00000i	− 0.168497i	−0.996445	−	0.0842484i	$-0.973151\pi$
$317$	− 3.00000i	− 0.168497i	0.996445	−	0.0842484i	$-0.0268489\pi$
$318$	9.00000i	0.504695i
$319$	0	0
$320$	0	0
$321$	−9.00000	−0.502331
$322$	5.00000i	0.278639i
$323$	7.00000i	0.389490i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	18.0000	0.996928
$327$	13.0000i	0.718902i
$328$	− 2.00000i	− 0.110432i
$329$	0	0
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	2.00000i	0.109764i
$333$	− 4.00000i	− 0.219199i
$334$	14.0000	0.766046
$335$	0	0
$336$	1.00000	0.0545545
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	4.00000i	0.217571i
$339$	8.00000	0.434500
$340$	0	0
$341$	0	0
$342$	2.00000i	0.108148i
$343$	13.0000i	0.701934i
$344$	6.00000	0.323498
$345$	0	0
$346$	−6.00000	−0.322562
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	5.00000i	0.268028i
$349$	−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
$351$	−15.0000	−0.800641
$352$	0	0
$353$	27.0000i	1.43706i	0.695493	+	0.718532i	$0.255186\pi$
$353$	27.0000i	1.43706i	−0.695493	+	0.718532i	$0.744814\pi$
$354$	7.00000	0.372046
$355$	0	0
$356$	−10.0000	−0.529999
$357$	7.00000i	0.370479i
$358$	− 24.0000i	− 1.26844i
$359$	−27.0000	−1.42501	−0.712503	−	0.701669i	$-0.752438\pi$
$359$	−27.0000	−1.42501	−0.712503	+	0.701669i	$0.752438\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 10.0000i	− 0.525588i
$363$	11.0000i	0.577350i
$364$	−3.00000	−0.157243
$365$	0	0
$366$	−4.00000	−0.209083
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	− 5.00000i	− 0.260643i
$369$	4.00000	0.208232
$370$	0	0
$371$	−9.00000	−0.467257
$372$	10.0000i	0.518476i
$373$	3.00000i	0.155334i	0.996979	+	0.0776671i	$0.0247471\pi$
$373$	3.00000i	0.155334i	−0.996979	+	0.0776671i	$0.975253\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15.0000i	− 0.772539i
$378$	5.00000i	0.257172i
$379$	7.00000	0.359566	0.179783	−	0.983706i	$-0.442460\pi$
$379$	7.00000	0.359566	0.179783	+	0.983706i	$0.442460\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	− 7.00000i	− 0.358151i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	12.0000i	0.609994i
$388$	− 18.0000i	− 0.913812i
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	35.0000	1.77003
$392$	− 6.00000i	− 0.303046i
$393$	20.0000i	1.00887i
$394$	−10.0000	−0.503793
$395$	0	0
$396$	0	0
$397$	20.0000i	1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$397$	20.0000i	1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$398$	− 17.0000i	− 0.852133i
$399$	1.00000	0.0500626
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	− 7.00000i	− 0.349128i
$403$	− 30.0000i	− 1.49441i
$404$	4.00000	0.199007
$405$	0	0
$406$	−5.00000	−0.248146
$407$	0	0
$408$	− 7.00000i	− 0.346552i
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	3.00000	0.147979
$412$	0	0
$413$	7.00000i	0.344447i
$414$	10.0000	0.491473
$415$	0	0
$416$	3.00000	0.147087
$417$	12.0000i	0.587643i
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	31.0000	1.51085	0.755424	−	0.655237i	$-0.227431\pi$
$421$	31.0000	1.51085	0.755424	+	0.655237i	$0.227431\pi$
$422$	5.00000i	0.243396i
$423$	0	0
$424$	9.00000	0.437079
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	9.00000i	0.435031i
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	− 5.00000i	− 0.240563i
$433$	12.0000i	0.576683i	0.957528	+	0.288342i	$0.0931039\pi$
$433$	12.0000i	0.576683i	−0.957528	+	0.288342i	$0.906896\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	13.0000	0.622587
$437$	− 5.00000i	− 0.239182i
$438$	− 9.00000i	− 0.430037i
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	21.0000i	0.998868i
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	2.00000	0.0949158
$445$	0	0
$446$	22.0000	1.04173
$447$	4.00000i	0.189194i
$448$	− 1.00000i	− 0.0472456i
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	0	0
$452$	− 8.00000i	− 0.376288i
$453$	6.00000i	0.281905i
$454$	−25.0000	−1.17331
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	41.0000i	1.91790i	0.283577	+	0.958950i	$0.408479\pi$
$457$	41.0000i	1.91790i	−0.283577	+	0.958950i	$0.591521\pi$
$458$	18.0000i	0.841085i
$459$	35.0000	1.63366
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	−26.0000	−1.20443
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	6.00000i	0.277350i
$469$	7.00000	0.323230
$470$	0	0
$471$	−18.0000	−0.829396
$472$	− 7.00000i	− 0.322201i
$473$	0	0
$474$	10.0000	0.459315
$475$	0	0
$476$	7.00000	0.320844
$477$	18.0000i	0.824163i
$478$	27.0000i	1.23495i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	− 28.0000i	− 1.27537i
$483$	− 5.00000i	− 0.227508i
$484$	11.0000	0.500000
$485$	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	$0.985571\pi$
$488$	4.00000i	0.181071i
$489$	−18.0000	−0.813988
$490$	0	0
$491$	14.0000	0.631811	0.315906	−	0.948791i	$-0.397692\pi$
$491$	14.0000	0.631811	0.315906	+	0.948791i	$0.397692\pi$
$492$	2.00000i	0.0901670i
$493$	35.0000i	1.57632i
$494$	3.00000	0.134976
$495$	0	0
$496$	10.0000	0.449013
$497$	0	0
$498$	− 2.00000i	− 0.0896221i
$499$	−42.0000	−1.88018	−0.940089	−	0.340929i	$-0.889258\pi$
$499$	−42.0000	−1.88018	−0.940089	+	0.340929i	$0.889258\pi$
$500$	0	0
$501$	−14.0000	−0.625474
$502$	4.00000i	0.178529i
$503$	− 17.0000i	− 0.757993i	−0.925398	−	0.378996i	$-0.876269\pi$
$503$	− 17.0000i	− 0.757993i	0.925398	−	0.378996i	$-0.123731\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	0	0
$507$	− 4.00000i	− 0.177646i
$508$	− 6.00000i	− 0.266207i
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	9.00000	0.398137
$512$	1.00000i	0.0441942i
$513$	− 5.00000i	− 0.220755i
$514$	−6.00000	−0.264649
$515$	0	0
$516$	−6.00000	−0.264135
$517$	0	0
$518$	2.00000i	0.0878750i
$519$	6.00000	0.263371
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	10.0000i	0.437688i
$523$	− 19.0000i	− 0.830812i	−0.909636	−	0.415406i	$-0.863640\pi$
$523$	− 19.0000i	− 0.830812i	0.909636	−	0.415406i	$-0.136360\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	8.00000	0.348817
$527$	70.0000i	3.04925i
$528$	0	0
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	14.0000	0.607548
$532$	− 1.00000i	− 0.0433555i
$533$	− 6.00000i	− 0.259889i
$534$	10.0000	0.432742
$535$	0	0
$536$	−7.00000	−0.302354
$537$	24.0000i	1.03568i
$538$	10.0000i	0.431131i
$539$	0	0
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	5.00000i	0.214768i
$543$	10.0000i	0.429141i
$544$	−7.00000	−0.300123
$545$	0	0
$546$	3.00000	0.128388
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	− 3.00000i	− 0.128154i
$549$	−8.00000	−0.341432
$550$	0	0
$551$	5.00000	0.213007
$552$	5.00000i	0.212814i
$553$	10.0000i	0.425243i
$554$	28.0000	1.18961
$555$	0	0
$556$	12.0000	0.508913
$557$	− 8.00000i	− 0.338971i	−0.985533	−	0.169485i	$-0.945789\pi$
$557$	− 8.00000i	− 0.338971i	0.985533	−	0.169485i	$-0.0542106\pi$
$558$	20.0000i	0.846668i
$559$	18.0000	0.761319
$560$	0	0
$561$	0	0
$562$	− 26.0000i	− 1.09674i
$563$	− 44.0000i	− 1.85438i	−0.374593	−	0.927189i	$-0.622217\pi$
$563$	− 44.0000i	− 1.85438i	0.374593	−	0.927189i	$-0.377783\pi$
$564$	0	0
$565$	0	0
$566$	26.0000	1.09286
$567$	1.00000i	0.0419961i
$568$	0	0
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	0	0
$573$	7.00000i	0.292429i
$574$	−2.00000	−0.0834784
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	11.0000i	0.457936i	0.973434	+	0.228968i	$0.0735351\pi$
$577$	11.0000i	0.457936i	−0.973434	+	0.228968i	$0.926465\pi$
$578$	− 32.0000i	− 1.33102i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	2.00000	0.0829740
$582$	18.0000i	0.746124i
$583$	0	0
$584$	−9.00000	−0.372423
$585$	0	0
$586$	−31.0000	−1.28060
$587$	− 30.0000i	− 1.23823i	−0.785299	−	0.619116i	$-0.787491\pi$
$587$	− 30.0000i	− 1.23823i	0.785299	−	0.619116i	$-0.212509\pi$
$588$	6.00000i	0.247436i
$589$	10.0000	0.412043
$590$	0	0
$591$	10.0000	0.411345
$592$	− 2.00000i	− 0.0821995i
$593$	− 34.0000i	− 1.39621i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$	− 34.0000i	− 1.39621i	0.715994	−	0.698106i	$-0.245974\pi$
$594$	0	0
$595$	0	0
$596$	4.00000	0.163846
$597$	17.0000i	0.695764i
$598$	− 15.0000i	− 0.613396i
$599$	−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
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	− 6.00000i	− 0.244542i
$603$	− 14.0000i	− 0.570124i
$604$	6.00000	0.244137
$605$	0	0
$606$	−4.00000	−0.162489
$607$	− 22.0000i	− 0.892952i	−0.894795	−	0.446476i	$-0.852679\pi$
$607$	− 22.0000i	− 0.892952i	0.894795	−	0.446476i	$-0.147321\pi$
$608$	1.00000i	0.0405554i
$609$	5.00000	0.202610
$610$	0	0
$611$	0	0
$612$	− 14.0000i	− 0.565916i
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	−25.0000	−1.00322
$622$	33.0000i	1.32318i
$623$	10.0000i	0.400642i
$624$	−3.00000	−0.120096
$625$	0	0
$626$	29.0000	1.15907
$627$	0	0
$628$	18.0000i	0.718278i
$629$	14.0000	0.558217
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	− 10.0000i	− 0.397779i
$633$	− 5.00000i	− 0.198732i
$634$	3.00000	0.119145
$635$	0	0
$636$	−9.00000	−0.356873
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	− 9.00000i	− 0.355202i
$643$	− 2.00000i	− 0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$	− 2.00000i	− 0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$	−5.00000	−0.197028
$645$	0	0
$646$	−7.00000	−0.275411
$647$	9.00000i	0.353827i	0.984226	+	0.176913i	$0.0566112\pi$
$647$	9.00000i	0.353827i	−0.984226	+	0.176913i	$0.943389\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	10.0000	0.391931
$652$	18.0000i	0.704934i
$653$	− 12.0000i	− 0.469596i	−0.972044	−	0.234798i	$-0.924557\pi$
$653$	− 12.0000i	− 0.469596i	0.972044	−	0.234798i	$-0.0754429\pi$
$654$	−13.0000	−0.508340
$655$	0	0
$656$	2.00000	0.0780869
$657$	− 18.0000i	− 0.702247i
$658$	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
$661$	17.0000	0.661223	0.330612	−	0.943767i	$-0.392745\pi$
$661$	17.0000	0.661223	0.330612	+	0.943767i	$0.392745\pi$
$662$	− 17.0000i	− 0.660724i
$663$	− 21.0000i	− 0.815572i
$664$	−2.00000	−0.0776151
$665$	0	0
$666$	4.00000	0.154997
$667$	− 25.0000i	− 0.968004i
$668$	14.0000i	0.541676i
$669$	−22.0000	−0.850569
$670$	0	0
$671$	0	0
$672$	1.00000i	0.0385758i
$673$	− 28.0000i	− 1.07932i	−0.841883	−	0.539660i	$-0.818553\pi$
$673$	− 28.0000i	− 1.07932i	0.841883	−	0.539660i	$-0.181447\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	−4.00000	−0.153846
$677$	− 19.0000i	− 0.730229i	−0.930963	−	0.365115i	$-0.881030\pi$
$677$	− 19.0000i	− 0.730229i	0.930963	−	0.365115i	$-0.118970\pi$
$678$	8.00000i	0.307238i
$679$	−18.0000	−0.690777
$680$	0	0
$681$	25.0000	0.958002
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	−13.0000	−0.496342
$687$	− 18.0000i	− 0.686743i
$688$	6.00000i	0.228748i
$689$	27.0000	1.02862
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	−18.0000	−0.683271
$695$	0	0
$696$	−5.00000	−0.189525
$697$	14.0000i	0.530288i
$698$	− 10.0000i	− 0.378506i
$699$	26.0000	0.983410
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	− 15.0000i	− 0.566139i
$703$	− 2.00000i	− 0.0754314i
$704$	0	0
$705$	0	0
$706$	−27.0000	−1.01616
$707$	− 4.00000i	− 0.150435i
$708$	7.00000i	0.263076i
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	− 10.0000i	− 0.374766i
$713$	− 50.0000i	− 1.87251i
$714$	−7.00000	−0.261968
$715$	0	0
$716$	24.0000	0.896922
$717$	− 27.0000i	− 1.00833i
$718$	− 27.0000i	− 1.00763i
$719$	−29.0000	−1.08152	−0.540759	−	0.841178i	$-0.681863\pi$
$719$	−29.0000	−1.08152	−0.540759	+	0.841178i	$0.681863\pi$
$720$	0	0
$721$	0	0
$722$	1.00000i	0.0372161i
$723$	28.0000i	1.04133i
$724$	10.0000	0.371647
$725$	0	0
$726$	−11.0000	−0.408248
$727$	− 11.0000i	− 0.407967i	−0.978974	−	0.203984i	$-0.934611\pi$
$727$	− 11.0000i	− 0.407967i	0.978974	−	0.203984i	$-0.0653890\pi$
$728$	− 3.00000i	− 0.111187i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−42.0000	−1.55343
$732$	− 4.00000i	− 0.147844i
$733$	24.0000i	0.886460i	0.896408	+	0.443230i	$0.146168\pi$
$733$	24.0000i	0.886460i	−0.896408	+	0.443230i	$0.853832\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	5.00000	0.184302
$737$	0	0
$738$	4.00000i	0.147242i
$739$	−50.0000	−1.83928	−0.919640	−	0.392763i	$-0.871519\pi$
$739$	−50.0000	−1.83928	−0.919640	+	0.392763i	$0.871519\pi$
$740$	0	0
$741$	−3.00000	−0.110208
$742$	− 9.00000i	− 0.330400i
$743$	34.0000i	1.24734i	0.781688	+	0.623670i	$0.214359\pi$
$743$	34.0000i	1.24734i	−0.781688	+	0.623670i	$0.785641\pi$
$744$	−10.0000	−0.366618
$745$	0	0
$746$	−3.00000	−0.109838
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	9.00000	0.328853
$750$	0	0
$751$	46.0000	1.67856	0.839282	−	0.543696i	$-0.182976\pi$
$751$	46.0000	1.67856	0.839282	+	0.543696i	$0.182976\pi$
$752$	0	0
$753$	− 4.00000i	− 0.145768i
$754$	15.0000	0.546268
$755$	0	0
$756$	−5.00000	−0.181848
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	7.00000i	0.254251i
$759$	0	0
$760$	0	0
$761$	−45.0000	−1.63125	−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000	−1.63125	−0.815624	+	0.578582i	$0.803606\pi$
$762$	6.00000i	0.217357i
$763$	− 13.0000i	− 0.470632i
$764$	7.00000	0.253251
$765$	0	0
$766$	0	0
$767$	− 21.0000i	− 0.758266i
$768$	− 1.00000i	− 0.0360844i
$769$	−49.0000	−1.76699	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−49.0000	−1.76699	−0.883493	+	0.468445i	$0.844814\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	2.00000i	0.0719816i
$773$	15.0000i	0.539513i	0.962929	+	0.269756i	$0.0869431\pi$
$773$	15.0000i	0.539513i	−0.962929	+	0.269756i	$0.913057\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	18.0000	0.646162
$777$	− 2.00000i	− 0.0717496i
$778$	− 24.0000i	− 0.860442i
$779$	2.00000	0.0716574
$780$	0	0
$781$	0	0
$782$	35.0000i	1.25160i
$783$	− 25.0000i	− 0.893427i
$784$	6.00000	0.214286
$785$	0	0
$786$	−20.0000	−0.713376
$787$	− 11.0000i	− 0.392108i	−0.980593	−	0.196054i	$-0.937187\pi$
$787$	− 11.0000i	− 0.392108i	0.980593	−	0.196054i	$-0.0628127\pi$
$788$	− 10.0000i	− 0.356235i
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−8.00000	−0.284447
$792$	0	0
$793$	12.0000i	0.426132i
$794$	−20.0000	−0.709773
$795$	0	0
$796$	17.0000	0.602549
$797$	− 5.00000i	− 0.177109i	−0.996071	−	0.0885545i	$-0.971775\pi$
$797$	− 5.00000i	− 0.177109i	0.996071	−	0.0885545i	$-0.0282248\pi$
$798$	1.00000i	0.0353996i
$799$	0	0
$800$	0	0
$801$	20.0000	0.706665
$802$	18.0000i	0.635602i
$803$	0	0
$804$	7.00000	0.246871
$805$	0	0
$806$	30.0000	1.05670
$807$	− 10.0000i	− 0.352017i
$808$	4.00000i	0.140720i
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	11.0000	0.386262	0.193131	−	0.981173i	$-0.438136\pi$
$811$	11.0000	0.386262	0.193131	+	0.981173i	$0.438136\pi$
$812$	− 5.00000i	− 0.175466i
$813$	− 5.00000i	− 0.175358i
$814$	0	0
$815$	0	0
$816$	7.00000	0.245049
$817$	6.00000i	0.209913i
$818$	− 26.0000i	− 0.909069i
$819$	6.00000	0.209657
$820$	0	0
$821$	−56.0000	−1.95441	−0.977207	−	0.212290i	$-0.931908\pi$
$821$	−56.0000	−1.95441	−0.977207	+	0.212290i	$0.931908\pi$
$822$	3.00000i	0.104637i
$823$	− 31.0000i	− 1.08059i	−0.841475	−	0.540296i	$-0.818312\pi$
$823$	− 31.0000i	− 1.08059i	0.841475	−	0.540296i	$-0.181688\pi$
$824$	0	0
$825$	0	0
$826$	−7.00000	−0.243561
$827$	9.00000i	0.312961i	0.987681	+	0.156480i	$0.0500148\pi$
$827$	9.00000i	0.312961i	−0.987681	+	0.156480i	$0.949985\pi$
$828$	10.0000i	0.347524i
$829$	51.0000	1.77130	0.885652	−	0.464350i	$-0.153712\pi$
$829$	51.0000	1.77130	0.885652	+	0.464350i	$0.153712\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	3.00000i	0.104006i
$833$	42.0000i	1.45521i
$834$	−12.0000	−0.415526
$835$	0	0
$836$	0	0
$837$	− 50.0000i	− 1.72825i
$838$	26.0000i	0.898155i
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	31.0000i	1.06833i
$843$	26.0000i	0.895488i
$844$	−5.00000	−0.172107
$845$	0	0
$846$	0	0
$847$	− 11.0000i	− 0.377964i
$848$	9.00000i	0.309061i
$849$	−26.0000	−0.892318
$850$	0	0
$851$	−10.0000	−0.342796
$852$	0	0
$853$	− 46.0000i	− 1.57501i	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	− 46.0000i	− 1.57501i	0.616308	−	0.787505i	$-0.288628\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	−9.00000	−0.307614
$857$	− 24.0000i	− 0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$	− 24.0000i	− 0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$	0	0
$859$	−56.0000	−1.91070	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−56.0000	−1.91070	−0.955348	+	0.295484i	$0.904519\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	12.0000i	0.408722i
$863$	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
$866$	−12.0000	−0.407777
$867$	32.0000i	1.08678i
$868$	− 10.0000i	− 0.339422i
$869$	0	0
$870$	0	0
$871$	−21.0000	−0.711558
$872$	13.0000i	0.440236i
$873$	36.0000i	1.21842i
$874$	5.00000	0.169128
$875$	0	0
$876$	9.00000	0.304082
$877$	5.00000i	0.168838i	0.996430	+	0.0844190i	$0.0269034\pi$
$877$	5.00000i	0.168838i	−0.996430	+	0.0844190i	$0.973097\pi$
$878$	− 10.0000i	− 0.337484i
$879$	31.0000	1.04560
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	12.0000i	0.404061i
$883$	26.0000i	0.874970i	0.899226	+	0.437485i	$0.144131\pi$
$883$	26.0000i	0.874970i	−0.899226	+	0.437485i	$0.855869\pi$
$884$	−21.0000	−0.706306
$885$	0	0
$886$	−24.0000	−0.806296
$887$	− 52.0000i	− 1.74599i	−0.487730	−	0.872995i	$-0.662175\pi$
$887$	− 52.0000i	− 1.74599i	0.487730	−	0.872995i	$-0.337825\pi$
$888$	2.00000i	0.0671156i
$889$	−6.00000	−0.201234
$890$	0	0
$891$	0	0
$892$	22.0000i	0.736614i
$893$	0	0
$894$	−4.00000	−0.133780
$895$	0	0
$896$	1.00000	0.0334077
$897$	15.0000i	0.500835i
$898$	− 14.0000i	− 0.467186i
$899$	50.0000	1.66759
$900$	0	0
$901$	−63.0000	−2.09883
$902$	0	0
$903$	6.00000i	0.199667i
$904$	8.00000	0.266076
$905$	0	0
$906$	−6.00000	−0.199337
$907$	− 5.00000i	− 0.166022i	−0.996549	−	0.0830111i	$-0.973546\pi$
$907$	− 5.00000i	− 0.166022i	0.996549	−	0.0830111i	$-0.0264537\pi$
$908$	− 25.0000i	− 0.829654i
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	− 1.00000i	− 0.0331133i
$913$	0	0
$914$	−41.0000	−1.35616
$915$	0	0
$916$	−18.0000	−0.594737
$917$	− 20.0000i	− 0.660458i
$918$	35.0000i	1.15517i
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	− 10.0000i	− 0.329332i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−16.0000	−0.525793
$927$	0	0
$928$	5.00000i	0.164133i
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	− 26.0000i	− 0.851658i
$933$	− 33.0000i	− 1.08037i
$934$	−8.00000	−0.261768
$935$	0	0
$936$	−6.00000	−0.196116
$937$	13.0000i	0.424691i	0.977195	+	0.212346i	$0.0681103\pi$
$937$	13.0000i	0.424691i	−0.977195	+	0.212346i	$0.931890\pi$
$938$	7.00000i	0.228558i
$939$	−29.0000	−0.946379
$940$	0	0
$941$	−37.0000	−1.20617	−0.603083	−	0.797679i	$-0.706061\pi$
$941$	−37.0000	−1.20617	−0.603083	+	0.797679i	$0.706061\pi$
$942$	− 18.0000i	− 0.586472i
$943$	− 10.0000i	− 0.325645i
$944$	7.00000	0.227831
$945$	0	0
$946$	0	0
$947$	28.0000i	0.909878i	0.890523	+	0.454939i	$0.150339\pi$
$947$	28.0000i	0.909878i	−0.890523	+	0.454939i	$0.849661\pi$
$948$	10.0000i	0.324785i
$949$	−27.0000	−0.876457
$950$	0	0
$951$	−3.00000	−0.0972817
$952$	7.00000i	0.226871i
$953$	44.0000i	1.42530i	0.701520	+	0.712650i	$0.252505\pi$
$953$	44.0000i	1.42530i	−0.701520	+	0.712650i	$0.747495\pi$
$954$	−18.0000	−0.582772
$955$	0	0
$956$	−27.0000	−0.873242
$957$	0	0
$958$	24.0000i	0.775405i
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	69.0000	2.22581
$962$	− 6.00000i	− 0.193448i
$963$	− 18.0000i	− 0.580042i
$964$	28.0000	0.901819
$965$	0	0
$966$	5.00000	0.160872
$967$	− 12.0000i	− 0.385894i	−0.981209	−	0.192947i	$-0.938195\pi$
$967$	− 12.0000i	− 0.385894i	0.981209	−	0.192947i	$-0.0618045\pi$
$968$	11.0000i	0.353553i
$969$	7.00000	0.224872
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	16.0000i	0.513200i
$973$	− 12.0000i	− 0.384702i
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−4.00000	−0.128037
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	− 18.0000i	− 0.575577i
$979$	0	0
$980$	0	0
$981$	−26.0000	−0.830116
$982$	14.0000i	0.446758i
$983$	− 18.0000i	− 0.574111i	−0.957914	−	0.287055i	$-0.907324\pi$
$983$	− 18.0000i	− 0.574111i	0.957914	−	0.287055i	$-0.0926764\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	−35.0000	−1.11463
$987$	0	0
$988$	3.00000i	0.0954427i
$989$	30.0000	0.953945
$990$	0	0
$991$	−46.0000	−1.46124	−0.730619	−	0.682785i	$-0.760768\pi$
$991$	−46.0000	−1.46124	−0.730619	+	0.682785i	$0.760768\pi$
$992$	10.0000i	0.317500i
$993$	17.0000i	0.539479i
$994$	0	0
$995$	0	0
$996$	2.00000	0.0633724
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	− 42.0000i	− 1.32949i
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.2.b.d.799.2		2
5.2	odd	4		190.2.a.a.1.1	✓	1
5.3	odd	4		950.2.a.e.1.1		1
5.4	even	2	inner	950.2.b.d.799.1		2
15.2	even	4		1710.2.a.r.1.1		1
15.8	even	4		8550.2.a.l.1.1		1
20.3	even	4		7600.2.a.g.1.1		1
20.7	even	4		1520.2.a.g.1.1		1
35.27	even	4		9310.2.a.i.1.1		1
40.27	even	4		6080.2.a.i.1.1		1
40.37	odd	4		6080.2.a.r.1.1		1
95.37	even	4		3610.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.a.a.1.1	✓	1	5.2	odd	4
950.2.a.e.1.1		1	5.3	odd	4
950.2.b.d.799.1		2	5.4	even	2	inner
950.2.b.d.799.2		2	1.1	even	1	trivial
1520.2.a.g.1.1		1	20.7	even	4
1710.2.a.r.1.1		1	15.2	even	4
3610.2.a.h.1.1		1	95.37	even	4
6080.2.a.i.1.1		1	40.27	even	4
6080.2.a.r.1.1		1	40.37	odd	4
7600.2.a.g.1.1		1	20.3	even	4
8550.2.a.l.1.1		1	15.8	even	4
9310.2.a.i.1.1		1	35.27	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:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	950.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +1.00000i q^{12} -3.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.00000i q^{17} +2.00000i q^{18} +1.00000 q^{19} +1.00000 q^{21} -5.00000i q^{23} -1.00000 q^{24} +3.00000 q^{26} -5.00000i q^{27} -1.00000i q^{28} +5.00000 q^{29} +10.0000 q^{31} +1.00000i q^{32} -7.00000 q^{34} -2.00000 q^{36} -2.00000i q^{37} +1.00000i q^{38} -3.00000 q^{39} +2.00000 q^{41} +1.00000i q^{42} +6.00000i q^{43} +5.00000 q^{46} -1.00000i q^{48} +6.00000 q^{49} +7.00000 q^{51} +3.00000i q^{52} +9.00000i q^{53} +5.00000 q^{54} +1.00000 q^{56} -1.00000i q^{57} +5.00000i q^{58} +7.00000 q^{59} -4.00000 q^{61} +10.0000i q^{62} +2.00000i q^{63} -1.00000 q^{64} -7.00000i q^{67} -7.00000i q^{68} -5.00000 q^{69} -2.00000i q^{72} -9.00000i q^{73} +2.00000 q^{74} -1.00000 q^{76} -3.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -2.00000i q^{83} -1.00000 q^{84} -6.00000 q^{86} -5.00000i q^{87} +10.0000 q^{89} +3.00000 q^{91} +5.00000i q^{92} -10.0000i q^{93} +1.00000 q^{96} +18.0000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} - 2 q^{14} + 2 q^{16} + 2 q^{19} + 2 q^{21} - 2 q^{24} + 6 q^{26} + 10 q^{29} + 20 q^{31} - 14 q^{34} - 4 q^{36} - 6 q^{39} + 4 q^{41} + 10 q^{46} + 12 q^{49} + 14 q^{51} + 10 q^{54} + 2 q^{56} + 14 q^{59} - 8 q^{61} - 2 q^{64} - 10 q^{69} + 4 q^{74} - 2 q^{76} + 20 q^{79} + 2 q^{81} - 2 q^{84} - 12 q^{86} + 20 q^{89} + 6 q^{91} + 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 - 2 * q^14 + 2 * q^16 + 2 * q^19 + 2 * q^21 - 2 * q^24 + 6 * q^26 + 10 * q^29 + 20 * q^31 - 14 * q^34 - 4 * q^36 - 6 * q^39 + 4 * q^41 + 10 * q^46 + 12 * q^49 + 14 * q^51 + 10 * q^54 + 2 * q^56 + 14 * q^59 - 8 * q^61 - 2 * q^64 - 10 * q^69 + 4 * q^74 - 2 * q^76 + 20 * q^79 + 2 * q^81 - 2 * q^84 - 12 * q^86 + 20 * q^89 + 6 * q^91 + 2 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more