LMFDB - Embedded newform 850.2.c.b.749.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [850,2,Mod(749,850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(850, 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("850.749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	850.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:	$6.78728417181$
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 34)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	850.749
Dual form			850.2.c.b.749.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} +2.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +8.00000 q^{21} -6.00000i q^{22} +2.00000 q^{24} +2.00000 q^{26} -4.00000i q^{27} -4.00000i q^{28} -4.00000 q^{31} -1.00000i q^{32} -12.0000i q^{33} +1.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} -4.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} -8.00000i q^{42} +8.00000i q^{43} -6.00000 q^{44} -2.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} -4.00000 q^{54} -4.00000 q^{56} -8.00000i q^{57} -4.00000 q^{61} +4.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -12.0000 q^{66} -8.00000i q^{67} -1.00000i q^{68} -1.00000i q^{72} +2.00000i q^{73} +4.00000 q^{74} -4.00000 q^{76} +24.0000i q^{77} -4.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{82} -8.00000 q^{84} +8.00000 q^{86} +6.00000i q^{88} +6.00000 q^{89} -8.00000 q^{91} +8.00000i q^{93} -2.00000 q^{96} -14.0000i q^{97} +9.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 12 q^{11} + 8 q^{14} + 2 q^{16} + 8 q^{19} + 16 q^{21} + 4 q^{24} + 4 q^{26} - 8 q^{31} + 2 q^{34} + 2 q^{36} + 8 q^{39} + 12 q^{41} - 12 q^{44} - 18 q^{49} + 4 q^{51} - 8 q^{54} - 8 q^{56} - 8 q^{61} - 2 q^{64} - 24 q^{66} + 8 q^{74} - 8 q^{76} - 16 q^{79} - 22 q^{81} - 16 q^{84} + 16 q^{86} + 12 q^{89} - 16 q^{91} - 4 q^{96} - 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 12 * q^11 + 8 * q^14 + 2 * q^16 + 8 * q^19 + 16 * q^21 + 4 * q^24 + 4 * q^26 - 8 * q^31 + 2 * q^34 + 2 * q^36 + 8 * q^39 + 12 * q^41 - 12 * q^44 - 18 * q^49 + 4 * q^51 - 8 * q^54 - 8 * q^56 - 8 * q^61 - 2 * q^64 - 24 * q^66 + 8 * q^74 - 8 * q^76 - 16 * q^79 - 22 * q^81 - 16 * q^84 + 16 * q^86 + 12 * q^89 - 16 * q^91 - 4 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/850\mathbb{Z}\right)^\times$.

$n$	$477$	$751$
$\chi(n)$	$-1$	$1$

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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	2.00000i	0.577350i
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	1.00000i	0.235702i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	− 6.00000i	− 1.27920i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	2.00000	0.408248
$25$	0	0
$26$	2.00000	0.392232
$27$	− 4.00000i	− 0.769800i
$28$	− 4.00000i	− 0.755929i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 12.0000i	− 2.08893i
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	− 4.00000i	− 0.648886i
$39$	4.00000	0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	− 8.00000i	− 1.23443i
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 2.00000i	− 0.288675i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	2.00000	0.280056
$52$	− 2.00000i	− 0.277350i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−4.00000	−0.534522
$57$	− 8.00000i	− 1.05963i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	4.00000i	0.508001i
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−12.0000	−1.47710
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	− 1.00000i	− 0.121268i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 1.00000i	− 0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−4.00000	−0.458831
$77$	24.0000i	2.73505i
$78$	− 4.00000i	− 0.452911i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 6.00000i	− 0.662589i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−8.00000	−0.872872
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	6.00000i	0.639602i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	−2.00000	−0.204124
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	9.00000i	0.909137i
$99$	−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	$0.146794\pi$
$102$	− 2.00000i	− 0.198030i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	4.00000i	0.384900i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	4.00000i	0.377964i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	25.0000	2.27273
$122$	4.00000i	0.362143i
$123$	− 12.0000i	− 1.08200i
$124$	4.00000	0.359211
$125$	0	0
$126$	−4.00000	−0.356348
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	1.00000i	0.0883883i
$129$	16.0000	1.40872
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	12.0000i	1.04447i
$133$	16.0000i	1.38738i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.0000i	1.00349i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	18.0000i	1.48461i
$148$	− 4.00000i	− 0.328798i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000i	0.324443i
$153$	− 1.00000i	− 0.0808452i
$154$	24.0000	1.93398
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	8.00000i	0.636446i
$159$	−12.0000	−0.951662
$160$	0	0
$161$	0	0
$162$	11.0000i	0.864242i
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	0	0
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	8.00000i	0.617213i
$169$	9.00000	0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	− 8.00000i	− 0.609994i
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	8.00000i	0.592999i
$183$	8.00000i	0.591377i
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	6.00000i	0.438763i
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	2.00000i	0.144338i
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	9.00000	0.642857
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	6.00000i	0.426401i
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	− 18.0000i	− 1.26648i
$203$	0	0
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−16.0000	−1.11477
$207$	0	0
$208$	2.00000i	0.138675i
$209$	24.0000	1.66011
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	4.00000	0.272166
$217$	− 16.0000i	− 1.08615i
$218$	− 16.0000i	− 1.08366i
$219$	4.00000	0.270295
$220$	0	0
$221$	−2.00000	−0.134535
$222$	− 8.00000i	− 0.536925i
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−6.00000	−0.399114
$227$	6.00000i	0.398234i	0.979976	+	0.199117i	$0.0638074\pi$
$227$	6.00000i	0.398234i	−0.979976	+	0.199117i	$0.936193\pi$
$228$	8.00000i	0.529813i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	48.0000	3.15817
$232$	0	0
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	0	0
$237$	16.0000i	1.03931i
$238$	4.00000i	0.259281i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	− 25.0000i	− 1.60706i
$243$	10.0000i	0.641500i
$244$	4.00000	0.256074
$245$	0	0
$246$	−12.0000	−0.765092
$247$	8.00000i	0.509028i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	4.00000i	0.251976i
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	− 16.0000i	− 0.996116i
$259$	−16.0000	−0.994192
$260$	0	0
$261$	0	0
$262$	6.00000i	0.370681i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	12.0000	0.738549
$265$	0	0
$266$	16.0000	0.981023
$267$	− 12.0000i	− 0.734388i
$268$	8.00000i	0.488678i
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	1.00000i	0.0606339i
$273$	16.0000i	0.968364i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	2.00000i	0.119952i
$279$	4.00000	0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	0	0
$285$	0	0
$286$	12.0000	0.709575
$287$	24.0000i	1.41668i
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−28.0000	−1.64139
$292$	− 2.00000i	− 0.117041i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	18.0000	1.04978
$295$	0	0
$296$	−4.00000	−0.232495
$297$	− 24.0000i	− 1.39262i
$298$	6.00000i	0.347571i
$299$	0	0
$300$	0	0
$301$	−32.0000	−1.84445
$302$	16.0000i	0.920697i
$303$	− 36.0000i	− 2.06815i
$304$	4.00000	0.229416
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	− 24.0000i	− 1.36753i
$309$	−32.0000	−1.82042
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	4.00000i	0.226455i
$313$	− 34.0000i	− 1.92179i	−0.276907	−	0.960897i	$-0.589309\pi$
$313$	− 34.0000i	− 1.92179i	0.276907	−	0.960897i	$-0.410691\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	12.0000i	0.672927i
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	4.00000i	0.222566i
$324$	11.0000	0.611111
$325$	0	0
$326$	2.00000	0.110770
$327$	− 32.0000i	− 1.76960i
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	0	0
$333$	− 4.00000i	− 0.219199i
$334$	−12.0000	−0.656611
$335$	0	0
$336$	8.00000	0.436436
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	− 9.00000i	− 0.489535i
$339$	−12.0000	−0.651751
$340$	0	0
$341$	−24.0000	−1.29967
$342$	4.00000i	0.216295i
$343$	− 8.00000i	− 0.431959i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	24.0000	1.29025
$347$	− 18.0000i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$	− 18.0000i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	− 6.00000i	− 0.319801i
$353$	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
$356$	−6.00000	−0.317999
$357$	8.00000i	0.423405i
$358$	12.0000i	0.634220i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	4.00000i	0.210235i
$363$	− 50.0000i	− 2.62432i
$364$	8.00000	0.419314
$365$	0	0
$366$	8.00000	0.418167
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	24.0000	1.24602
$372$	− 8.00000i	− 0.414781i
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	0	0
$377$	0	0
$378$	− 16.0000i	− 0.822951i
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	0	0
$381$	32.0000	1.63941
$382$	24.0000i	1.22795i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	−10.0000	−0.508987
$387$	− 8.00000i	− 0.406663i
$388$	14.0000i	0.710742i
$389$	−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
$391$	0	0
$392$	− 9.00000i	− 0.454569i
$393$	12.0000i	0.605320i
$394$	12.0000	0.604551
$395$	0	0
$396$	6.00000	0.301511
$397$	− 20.0000i	− 1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$	− 20.0000i	− 1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	− 16.0000i	− 0.802008i
$399$	32.0000	1.60200
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	16.0000i	0.798007i
$403$	− 8.00000i	− 0.398508i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	0	0
$407$	24.0000i	1.18964i
$408$	2.00000i	0.0990148i
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	16.0000i	0.788263i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	4.00000i	0.195881i
$418$	− 24.0000i	− 1.17388i
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	10.0000i	0.486792i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	− 16.0000i	− 0.774294i
$428$	− 6.00000i	− 0.290021i
$429$	24.0000	1.15873
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	− 4.00000i	− 0.192450i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	−16.0000	−0.766261
$437$	0	0
$438$	− 4.00000i	− 0.191127i
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	2.00000i	0.0951303i
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	8.00000	0.378811
$447$	12.0000i	0.567581i
$448$	− 4.00000i	− 0.188982i
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	6.00000i	0.282216i
$453$	32.0000i	1.50349i
$454$	6.00000	0.281594
$455$	0	0
$456$	8.00000	0.374634
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	14.0000i	0.654177i
$459$	4.00000	0.186704
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	− 48.0000i	− 2.23316i
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	0	0
$465$	0	0
$466$	18.0000	0.833834
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	2.00000i	0.0924500i
$469$	32.0000	1.47762
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	48.0000i	2.20704i
$474$	16.0000	0.734904
$475$	0	0
$476$	4.00000	0.183340
$477$	6.00000i	0.274721i
$478$	24.0000i	1.09773i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	10.0000i	0.455488i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	10.0000	0.453609
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 4.00000i	− 0.181071i
$489$	4.00000	0.180886
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	12.0000i	0.541002i
$493$	0	0
$494$	8.00000	0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	24.0000i	1.07117i
$503$	− 24.0000i	− 1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$	− 24.0000i	− 1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	0	0
$507$	− 18.0000i	− 0.799408i
$508$	− 16.0000i	− 0.709885i
$509$	−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
$511$	−8.00000	−0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	− 16.0000i	− 0.706417i
$514$	−6.00000	−0.264649
$515$	0	0
$516$	−16.0000	−0.704361
$517$	0	0
$518$	16.0000i	0.703000i
$519$	48.0000	2.10697
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	24.0000	1.04645
$527$	− 4.00000i	− 0.174243i
$528$	− 12.0000i	− 0.522233i
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	− 16.0000i	− 0.693688i
$533$	12.0000i	0.519778i
$534$	−12.0000	−0.519291
$535$	0	0
$536$	8.00000	0.345547
$537$	24.0000i	1.03568i
$538$	− 24.0000i	− 1.03471i
$539$	−54.0000	−2.32594
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	− 8.00000i	− 0.343629i
$543$	8.00000i	0.343313i
$544$	1.00000	0.0428746
$545$	0	0
$546$	16.0000	0.684737
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	6.00000i	0.256307i
$549$	4.00000	0.170716
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	−8.00000	−0.339887
$555$	0	0
$556$	2.00000	0.0848189
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	− 4.00000i	− 0.169334i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	12.0000	0.506640
$562$	− 6.00000i	− 0.253095i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	0	0
$566$	14.0000	0.588464
$567$	− 44.0000i	− 1.84783i
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	− 12.0000i	− 0.501745i
$573$	48.0000i	2.00523i
$574$	24.0000	1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	1.00000i	0.0415945i
$579$	−20.0000	−0.831172
$580$	0	0
$581$	0	0
$582$	28.0000i	1.16064i
$583$	− 36.0000i	− 1.49097i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	− 18.0000i	− 0.742307i
$589$	−16.0000	−0.659269
$590$	0	0
$591$	24.0000	0.987228
$592$	4.00000i	0.164399i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	−24.0000	−0.984732
$595$	0	0
$596$	6.00000	0.245770
$597$	− 32.0000i	− 1.30967i
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	32.0000i	1.30422i
$603$	8.00000i	0.325785i
$604$	16.0000	0.651031
$605$	0	0
$606$	−36.0000	−1.46240
$607$	− 20.0000i	− 0.811775i	−0.913923	−	0.405887i	$-0.866962\pi$
$607$	− 20.0000i	− 0.811775i	0.913923	−	0.405887i	$-0.133038\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	1.00000i	0.0404226i
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	−24.0000	−0.966988
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	32.0000i	1.28723i
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	0	0
$622$	− 12.0000i	− 0.481156i
$623$	24.0000i	0.961540i
$624$	4.00000	0.160128
$625$	0	0
$626$	−34.0000	−1.35891
$627$	− 48.0000i	− 1.91694i
$628$	14.0000i	0.558661i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	− 8.00000i	− 0.318223i
$633$	20.0000i	0.794929i
$634$	12.0000	0.476581
$635$	0	0
$636$	12.0000	0.475831
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	− 12.0000i	− 0.473602i
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	48.0000i	1.88707i	0.331266	+	0.943537i	$0.392524\pi$
$647$	48.0000i	1.88707i	−0.331266	+	0.943537i	$0.607476\pi$
$648$	− 11.0000i	− 0.432121i
$649$	0	0
$650$	0	0
$651$	−32.0000	−1.25418
$652$	− 2.00000i	− 0.0783260i
$653$	24.0000i	0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$	24.0000i	0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	−32.0000	−1.25130
$655$	0	0
$656$	6.00000	0.234261
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	16.0000i	0.621858i
$663$	4.00000i	0.155347i
$664$	0	0
$665$	0	0
$666$	−4.00000	−0.154997
$667$	0	0
$668$	12.0000i	0.464294i
$669$	16.0000	0.618596
$670$	0	0
$671$	−24.0000	−0.926510
$672$	− 8.00000i	− 0.308607i
$673$	38.0000i	1.46479i	0.680879	+	0.732396i	$0.261598\pi$
$673$	38.0000i	1.46479i	−0.680879	+	0.732396i	$0.738402\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	12.0000i	0.460857i
$679$	56.0000	2.14908
$680$	0	0
$681$	12.0000	0.459841
$682$	24.0000i	0.919007i
$683$	30.0000i	1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$	30.0000i	1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	−8.00000	−0.305441
$687$	28.0000i	1.06827i
$688$	8.00000i	0.304997i
$689$	12.0000	0.457164
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	− 24.0000i	− 0.912343i
$693$	− 24.0000i	− 0.911685i
$694$	−18.0000	−0.683271
$695$	0	0
$696$	0	0
$697$	6.00000i	0.227266i
$698$	26.0000i	0.984115i
$699$	36.0000	1.36165
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	− 8.00000i	− 0.301941i
$703$	16.0000i	0.603451i
$704$	−6.00000	−0.226134
$705$	0	0
$706$	6.00000	0.225813
$707$	72.0000i	2.70784i
$708$	0	0
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	6.00000i	0.224860i
$713$	0	0
$714$	8.00000	0.299392
$715$	0	0
$716$	12.0000	0.448461
$717$	48.0000i	1.79259i
$718$	24.0000i	0.895672i
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	3.00000i	0.111648i
$723$	20.0000i	0.743808i
$724$	4.00000	0.148659
$725$	0	0
$726$	−50.0000	−1.85567
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−8.00000	−0.295891
$732$	− 8.00000i	− 0.295689i
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	0	0
$737$	− 48.0000i	− 1.76810i
$738$	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
$741$	16.0000	0.587775
$742$	− 24.0000i	− 0.881068i
$743$	− 36.0000i	− 1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$	− 36.0000i	− 1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	−22.0000	−0.805477
$747$	0	0
$748$	− 6.00000i	− 0.219382i
$749$	−24.0000	−0.876941
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	48.0000i	1.74922i
$754$	0	0
$755$	0	0
$756$	−16.0000	−0.581914
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	14.0000i	0.508503i
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	− 32.0000i	− 1.15924i
$763$	64.0000i	2.31696i
$764$	24.0000	0.868290
$765$	0	0
$766$	−24.0000	−0.867155
$767$	0	0
$768$	− 2.00000i	− 0.0721688i
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	10.0000i	0.359908i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	14.0000	0.502571
$777$	32.0000i	1.14799i
$778$	30.0000i	1.07555i
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	12.0000	0.428026
$787$	46.0000i	1.63972i	0.572562	+	0.819861i	$0.305950\pi$
$787$	46.0000i	1.63972i	−0.572562	+	0.819861i	$0.694050\pi$
$788$	− 12.0000i	− 0.427482i
$789$	48.0000	1.70885
$790$	0	0
$791$	24.0000	0.853342
$792$	− 6.00000i	− 0.213201i
$793$	− 8.00000i	− 0.284088i
$794$	−20.0000	−0.709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	− 32.0000i	− 1.13279i
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 30.0000i	− 1.05934i
$803$	12.0000i	0.423471i
$804$	16.0000	0.564276
$805$	0	0
$806$	−8.00000	−0.281788
$807$	− 48.0000i	− 1.68968i
$808$	18.0000i	0.633238i
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	24.0000	0.841200
$815$	0	0
$816$	2.00000	0.0700140
$817$	32.0000i	1.11954i
$818$	− 10.0000i	− 0.349642i
$819$	8.00000	0.279543
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	12.0000i	0.418548i
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	− 6.00000i	− 0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$	− 6.00000i	− 0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	− 2.00000i	− 0.0693375i
$833$	− 9.00000i	− 0.311832i
$834$	4.00000	0.138509
$835$	0	0
$836$	−24.0000	−0.830057
$837$	16.0000i	0.553041i
$838$	− 30.0000i	− 1.03633i
$839$	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
$841$	−29.0000	−1.00000
$842$	− 2.00000i	− 0.0689246i
$843$	− 12.0000i	− 0.413302i
$844$	10.0000	0.344214
$845$	0	0
$846$	0	0
$847$	100.000i	3.43604i
$848$	− 6.00000i	− 0.206041i
$849$	28.0000	0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 40.0000i	− 1.36957i	−0.728743	−	0.684787i	$-0.759895\pi$
$853$	− 40.0000i	− 1.36957i	0.728743	−	0.684787i	$-0.240105\pi$
$854$	−16.0000	−0.547509
$855$	0	0
$856$	−6.00000	−0.205076
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	− 24.0000i	− 0.819346i
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	0	0
$861$	48.0000	1.63584
$862$	− 24.0000i	− 0.817443i
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	2.00000	0.0679628
$867$	2.00000i	0.0679236i
$868$	16.0000i	0.543075i
$869$	−48.0000	−1.62829
$870$	0	0
$871$	16.0000	0.542139
$872$	16.0000i	0.541828i
$873$	14.0000i	0.473828i
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	8.00000i	0.269987i
$879$	12.0000	0.404750
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	− 9.00000i	− 0.303046i
$883$	− 16.0000i	− 0.538443i	−0.963078	−	0.269221i	$-0.913234\pi$
$883$	− 16.0000i	− 0.538443i	0.963078	−	0.269221i	$-0.0867663\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−24.0000	−0.806296
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	8.00000i	0.268462i
$889$	−64.0000	−2.14649
$890$	0	0
$891$	−66.0000	−2.21108
$892$	− 8.00000i	− 0.267860i
$893$	0	0
$894$	12.0000	0.401340
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	− 18.0000i	− 0.600668i
$899$	0	0
$900$	0	0
$901$	6.00000	0.199889
$902$	− 36.0000i	− 1.19867i
$903$	64.0000i	2.12979i
$904$	6.00000	0.199557
$905$	0	0
$906$	32.0000	1.06313
$907$	58.0000i	1.92586i	0.269754	+	0.962929i	$0.413058\pi$
$907$	58.0000i	1.92586i	−0.269754	+	0.962929i	$0.586942\pi$
$908$	− 6.00000i	− 0.199117i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	− 8.00000i	− 0.264906i
$913$	0	0
$914$	−26.0000	−0.860004
$915$	0	0
$916$	14.0000	0.462573
$917$	− 24.0000i	− 0.792550i
$918$	− 4.00000i	− 0.132020i
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	−40.0000	−1.31804
$922$	6.00000i	0.197599i
$923$	0	0
$924$	−48.0000	−1.57908
$925$	0	0
$926$	−16.0000	−0.525793
$927$	16.0000i	0.525509i
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	− 18.0000i	− 0.589610i
$933$	− 24.0000i	− 0.785725i
$934$	−12.0000	−0.392652
$935$	0	0
$936$	2.00000	0.0653720
$937$	58.0000i	1.89478i	0.320085	+	0.947389i	$0.396288\pi$
$937$	58.0000i	1.89478i	−0.320085	+	0.947389i	$0.603712\pi$
$938$	− 32.0000i	− 1.04484i
$939$	−68.0000	−2.21910
$940$	0	0
$941$	−36.0000	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$942$	28.0000i	0.912289i
$943$	0	0
$944$	0	0
$945$	0	0
$946$	48.0000	1.56061
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	− 16.0000i	− 0.519656i
$949$	−4.00000	−0.129845
$950$	0	0
$951$	24.0000	0.778253
$952$	− 4.00000i	− 0.129641i
$953$	− 30.0000i	− 0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$	− 30.0000i	− 0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	24.0000i	0.775405i
$959$	24.0000	0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	8.00000i	0.257930i
$963$	− 6.00000i	− 0.193347i
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	25.0000i	0.803530i
$969$	8.00000	0.256997
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	− 10.0000i	− 0.320750i
$973$	− 8.00000i	− 0.256468i
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−4.00000	−0.128037
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	− 4.00000i	− 0.127906i
$979$	36.0000	1.15056
$980$	0	0
$981$	−16.0000	−0.510841
$982$	12.0000i	0.382935i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	12.0000	0.382546
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	4.00000i	0.127000i
$993$	32.0000i	1.01549i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	14.0000i	0.443162i
$999$	16.0000	0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.c.b.749.1		2
5.2	odd	4		34.2.a.a.1.1	✓	1
5.3	odd	4		850.2.a.e.1.1		1
5.4	even	2	inner	850.2.c.b.749.2		2
15.2	even	4		306.2.a.a.1.1		1
15.8	even	4		7650.2.a.ci.1.1		1
20.3	even	4		6800.2.a.b.1.1		1
20.7	even	4		272.2.a.d.1.1		1
35.27	even	4		1666.2.a.m.1.1		1
40.27	even	4		1088.2.a.d.1.1		1
40.37	odd	4		1088.2.a.l.1.1		1
55.32	even	4		4114.2.a.a.1.1		1
60.47	odd	4		2448.2.a.k.1.1		1
65.12	odd	4		5746.2.a.b.1.1		1
85.2	odd	8		578.2.c.e.327.1		4
85.7	even	16		578.2.d.e.423.2		8
85.12	even	16		578.2.d.e.399.1		8
85.22	even	16		578.2.d.e.399.2		8
85.27	even	16		578.2.d.e.423.1		8
85.32	odd	8		578.2.c.e.327.2		4
85.37	even	16		578.2.d.e.179.1		8
85.42	odd	8		578.2.c.e.251.2		4
85.47	odd	4		578.2.b.a.577.1		2
85.57	even	16		578.2.d.e.155.1		8
85.62	even	16		578.2.d.e.155.2		8
85.67	odd	4		578.2.a.a.1.1		1
85.72	odd	4		578.2.b.a.577.2		2
85.77	odd	8		578.2.c.e.251.1		4
85.82	even	16		578.2.d.e.179.2		8
120.77	even	4		9792.2.a.y.1.1		1
120.107	odd	4		9792.2.a.bj.1.1		1
255.152	even	4		5202.2.a.d.1.1		1
340.67	even	4		4624.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
34.2.a.a.1.1	✓	1	5.2	odd	4
272.2.a.d.1.1		1	20.7	even	4
306.2.a.a.1.1		1	15.2	even	4
578.2.a.a.1.1		1	85.67	odd	4
578.2.b.a.577.1		2	85.47	odd	4
578.2.b.a.577.2		2	85.72	odd	4
578.2.c.e.251.1		4	85.77	odd	8
578.2.c.e.251.2		4	85.42	odd	8
578.2.c.e.327.1		4	85.2	odd	8
578.2.c.e.327.2		4	85.32	odd	8
578.2.d.e.155.1		8	85.57	even	16
578.2.d.e.155.2		8	85.62	even	16
578.2.d.e.179.1		8	85.37	even	16
578.2.d.e.179.2		8	85.82	even	16
578.2.d.e.399.1		8	85.12	even	16
578.2.d.e.399.2		8	85.22	even	16
578.2.d.e.423.1		8	85.27	even	16
578.2.d.e.423.2		8	85.7	even	16
850.2.a.e.1.1		1	5.3	odd	4
850.2.c.b.749.1		2	1.1	even	1	trivial
850.2.c.b.749.2		2	5.4	even	2	inner
1088.2.a.d.1.1		1	40.27	even	4
1088.2.a.l.1.1		1	40.37	odd	4
1666.2.a.m.1.1		1	35.27	even	4
2448.2.a.k.1.1		1	60.47	odd	4
4114.2.a.a.1.1		1	55.32	even	4
4624.2.a.a.1.1		1	340.67	even	4
5202.2.a.d.1.1		1	255.152	even	4
5746.2.a.b.1.1		1	65.12	odd	4
6800.2.a.b.1.1		1	20.3	even	4
7650.2.a.ci.1.1		1	15.8	even	4
9792.2.a.y.1.1		1	120.77	even	4
9792.2.a.bj.1.1		1	120.107	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:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} +2.00000i q^{12} +2.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +8.00000 q^{21} -6.00000i q^{22} +2.00000 q^{24} +2.00000 q^{26} -4.00000i q^{27} -4.00000i q^{28} -4.00000 q^{31} -1.00000i q^{32} -12.0000i q^{33} +1.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} -4.00000i q^{38} +4.00000 q^{39} +6.00000 q^{41} -8.00000i q^{42} +8.00000i q^{43} -6.00000 q^{44} -2.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{52} -6.00000i q^{53} -4.00000 q^{54} -4.00000 q^{56} -8.00000i q^{57} -4.00000 q^{61} +4.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -12.0000 q^{66} -8.00000i q^{67} -1.00000i q^{68} -1.00000i q^{72} +2.00000i q^{73} +4.00000 q^{74} -4.00000 q^{76} +24.0000i q^{77} -4.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{82} -8.00000 q^{84} +8.00000 q^{86} +6.00000i q^{88} +6.00000 q^{89} -8.00000 q^{91} +8.00000i q^{93} -2.00000 q^{96} -14.0000i q^{97} +9.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 12 q^{11} + 8 q^{14} + 2 q^{16} + 8 q^{19} + 16 q^{21} + 4 q^{24} + 4 q^{26} - 8 q^{31} + 2 q^{34} + 2 q^{36} + 8 q^{39} + 12 q^{41} - 12 q^{44} - 18 q^{49} + 4 q^{51} - 8 q^{54} - 8 q^{56} - 8 q^{61} - 2 q^{64} - 24 q^{66} + 8 q^{74} - 8 q^{76} - 16 q^{79} - 22 q^{81} - 16 q^{84} + 16 q^{86} + 12 q^{89} - 16 q^{91} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 12 * q^11 + 8 * q^14 + 2 * q^16 + 8 * q^19 + 16 * q^21 + 4 * q^24 + 4 * q^26 - 8 * q^31 + 2 * q^34 + 2 * q^36 + 8 * q^39 + 12 * q^41 - 12 * q^44 - 18 * q^49 + 4 * q^51 - 8 * q^54 - 8 * q^56 - 8 * q^61 - 2 * q^64 - 24 * q^66 + 8 * q^74 - 8 * q^76 - 16 * q^79 - 22 * q^81 - 16 * q^84 + 16 * q^86 + 12 * q^89 - 16 * q^91 - 4 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more