LMFDB - Embedded newform 850.2.c.c.749.2

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 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} -1.00000i q^{12} +5.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +2.00000i q^{18} +1.00000 q^{19} +2.00000 q^{21} +6.00000i q^{23} +1.00000 q^{24} -5.00000 q^{26} +5.00000i q^{27} +2.00000i q^{28} +9.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} -1.00000 q^{34} -2.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} -5.00000 q^{39} -6.00000 q^{41} +2.00000i q^{42} +2.00000i q^{43} -6.00000 q^{46} +9.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -1.00000 q^{51} -5.00000i q^{52} -9.00000i q^{53} -5.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} +9.00000i q^{58} -3.00000 q^{59} -7.00000 q^{61} -1.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -14.0000i q^{67} -1.00000i q^{68} -6.00000 q^{69} +3.00000 q^{71} -2.00000i q^{72} +11.0000i q^{73} -4.00000 q^{74} -1.00000 q^{76} -5.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -2.00000 q^{84} -2.00000 q^{86} +9.00000i q^{87} +9.00000 q^{89} +10.0000 q^{91} -6.00000i q^{92} -1.00000i q^{93} -9.00000 q^{94} -1.00000 q^{96} +7.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{14} + 2 q^{16} + 2 q^{19} + 4 q^{21} + 2 q^{24} - 10 q^{26} + 18 q^{29} - 2 q^{31} - 2 q^{34} - 4 q^{36} - 10 q^{39} - 12 q^{41} - 12 q^{46} + 6 q^{49} - 2 q^{51} - 10 q^{54} - 4 q^{56} - 6 q^{59} - 14 q^{61} - 2 q^{64} - 12 q^{69} + 6 q^{71} - 8 q^{74} - 2 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} - 4 q^{86} + 18 q^{89} + 20 q^{91} - 18 q^{94} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 + 4 * q^14 + 2 * q^16 + 2 * q^19 + 4 * q^21 + 2 * q^24 - 10 * q^26 + 18 * q^29 - 2 * q^31 - 2 * q^34 - 4 * q^36 - 10 * q^39 - 12 * q^41 - 12 * q^46 + 6 * q^49 - 2 * q^51 - 10 * q^54 - 4 * q^56 - 6 * q^59 - 14 * q^61 - 2 * q^64 - 12 * q^69 + 6 * q^71 - 8 * q^74 - 2 * q^76 - 16 * q^79 + 2 * q^81 - 4 * q^84 - 4 * q^86 + 18 * q^89 + 20 * q^91 - 18 * q^94 - 2 * q^96

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
$3$	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
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 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
$13$	5.00000i	1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$	5.00000i	1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	2.00000i	0.471405i
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−5.00000	−0.980581
$27$	5.00000i	0.962250i
$28$	2.00000i	0.377964i
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−2.00000	−0.333333
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	1.00000i	0.162221i
$39$	−5.00000	−0.800641
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	2.00000i	0.308607i
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	−6.00000	−0.884652
$47$	9.00000i	1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$	9.00000i	1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$	1.00000i	0.144338i
$49$	3.00000	0.428571
$50$	0	0
$51$	−1.00000	−0.140028
$52$	− 5.00000i	− 0.693375i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	−2.00000	−0.267261
$57$	1.00000i	0.132453i
$58$	9.00000i	1.18176i
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	− 1.00000i	− 0.127000i
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 14.0000i	− 1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	− 14.0000i	− 1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$	− 1.00000i	− 0.121268i
$69$	−6.00000	−0.722315
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	− 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	$0.777388\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	− 5.00000i	− 0.566139i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−2.00000	−0.215666
$87$	9.00000i	0.964901i
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	− 6.00000i	− 0.625543i
$93$	− 1.00000i	− 0.103695i
$94$	−9.00000	−0.928279
$95$	0	0
$96$	−1.00000	−0.102062
$97$	7.00000i	0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$	7.00000i	0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	− 1.00000i	− 0.0990148i
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	9.00000	0.874157
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	− 5.00000i	− 0.481125i
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	− 2.00000i	− 0.188982i
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	−9.00000	−0.835629
$117$	10.0000i	0.924500i
$118$	− 3.00000i	− 0.276172i
$119$	2.00000	0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 7.00000i	− 0.633750i
$123$	− 6.00000i	− 0.541002i
$124$	1.00000	0.0898027
$125$	0	0
$126$	4.00000	0.356348
$127$	7.00000i	0.621150i	0.950549	+	0.310575i	$0.100522\pi$
$127$	7.00000i	0.621150i	−0.950549	+	0.310575i	$0.899478\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	14.0000	1.20942
$135$	0	0
$136$	1.00000	0.0857493
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	− 6.00000i	− 0.510754i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	3.00000i	0.251754i
$143$	0	0
$144$	2.00000	0.166667
$145$	0	0
$146$	−11.0000	−0.910366
$147$	3.00000i	0.247436i
$148$	− 4.00000i	− 0.328798i
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	5.00000	0.400320
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	− 8.00000i	− 0.636446i
$159$	9.00000	0.713746
$160$	0	0
$161$	12.0000	0.945732
$162$	1.00000i	0.0785674i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	− 6.00000i	− 0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$	− 6.00000i	− 0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$	− 2.00000i	− 0.154303i
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	− 2.00000i	− 0.152499i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	−9.00000	−0.682288
$175$	0	0
$176$	0	0
$177$	− 3.00000i	− 0.225494i
$178$	9.00000i	0.674579i
$179$	−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
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	10.0000i	0.741249i
$183$	− 7.00000i	− 0.517455i
$184$	6.00000	0.442326
$185$	0	0
$186$	1.00000	0.0733236
$187$	0	0
$188$	− 9.00000i	− 0.656392i
$189$	10.0000	0.727393
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	−7.00000	−0.502571
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 24.0000i	− 1.70993i	−0.518686	−	0.854965i	$-0.673579\pi$
$197$	− 24.0000i	− 1.70993i	0.518686	−	0.854965i	$-0.326421\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	18.0000i	1.26648i
$203$	− 18.0000i	− 1.26335i
$204$	1.00000	0.0700140
$205$	0	0
$206$	16.0000	1.11477
$207$	12.0000i	0.834058i
$208$	5.00000i	0.346688i
$209$	0	0
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	9.00000i	0.618123i
$213$	3.00000i	0.205557i
$214$	12.0000	0.820303
$215$	0	0
$216$	5.00000	0.340207
$217$	2.00000i	0.135769i
$218$	7.00000i	0.474100i
$219$	−11.0000	−0.743311
$220$	0	0
$221$	−5.00000	−0.336336
$222$	− 4.00000i	− 0.268462i
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−9.00000	−0.598671
$227$	− 27.0000i	− 1.79205i	−0.444001	−	0.896026i	$-0.646441\pi$
$227$	− 27.0000i	− 1.79205i	0.444001	−	0.896026i	$-0.353559\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	−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
$231$	0	0
$232$	− 9.00000i	− 0.590879i
$233$	9.00000i	0.589610i	0.955557	+	0.294805i	$0.0952546\pi$
$233$	9.00000i	0.589610i	−0.955557	+	0.294805i	$0.904745\pi$
$234$	−10.0000	−0.653720
$235$	0	0
$236$	3.00000	0.195283
$237$	− 8.00000i	− 0.519656i
$238$	2.00000i	0.129641i
$239$	30.0000	1.94054	0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000	1.94054	0.970269	+	0.242028i	$0.0778125\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	− 11.0000i	− 0.707107i
$243$	16.0000i	1.02640i
$244$	7.00000	0.448129
$245$	0	0
$246$	6.00000	0.382546
$247$	5.00000i	0.318142i
$248$	1.00000i	0.0635001i
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	4.00000i	0.251976i
$253$	0	0
$254$	−7.00000	−0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	− 2.00000i	− 0.124515i
$259$	8.00000	0.497096
$260$	0	0
$261$	18.0000	1.11417
$262$	6.00000i	0.370681i
$263$	3.00000i	0.184988i	0.995713	+	0.0924940i	$0.0294839\pi$
$263$	3.00000i	0.184988i	−0.995713	+	0.0924940i	$0.970516\pi$
$264$	0	0
$265$	0	0
$266$	2.00000	0.122628
$267$	9.00000i	0.550791i
$268$	14.0000i	0.855186i
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	1.00000i	0.0606339i
$273$	10.0000i	0.605228i
$274$	−6.00000	−0.362473
$275$	0	0
$276$	6.00000	0.361158
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	− 20.0000i	− 1.19952i
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−21.0000	−1.25275	−0.626377	−	0.779520i	$-0.715463\pi$
$281$	−21.0000	−1.25275	−0.626377	+	0.779520i	$0.715463\pi$
$282$	− 9.00000i	− 0.535942i
$283$	11.0000i	0.653882i	0.945045	+	0.326941i	$0.106018\pi$
$283$	11.0000i	0.653882i	−0.945045	+	0.326941i	$0.893982\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	2.00000i	0.117851i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−7.00000	−0.410347
$292$	− 11.0000i	− 0.643726i
$293$	− 9.00000i	− 0.525786i	−0.964825	−	0.262893i	$-0.915323\pi$
$293$	− 9.00000i	− 0.525786i	0.964825	−	0.262893i	$-0.0846766\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	4.00000	0.232495
$297$	0	0
$298$	12.0000i	0.695141i
$299$	−30.0000	−1.73494
$300$	0	0
$301$	4.00000	0.230556
$302$	14.0000i	0.805609i
$303$	18.0000i	1.03407i
$304$	1.00000	0.0573539
$305$	0	0
$306$	−2.00000	−0.114332
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	5.00000i	0.283069i
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	− 24.0000i	− 1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$	− 24.0000i	− 1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$	9.00000i	0.504695i
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	12.0000i	0.668734i
$323$	1.00000i	0.0556415i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	7.00000i	0.387101i
$328$	6.00000i	0.331295i
$329$	18.0000	0.992372
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	8.00000i	0.438397i
$334$	6.00000	0.328305
$335$	0	0
$336$	2.00000	0.109109
$337$	13.0000i	0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$	13.0000i	0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$	− 12.0000i	− 0.652714i
$339$	−9.00000	−0.488813
$340$	0	0
$341$	0	0
$342$	2.00000i	0.108148i
$343$	− 20.0000i	− 1.07990i
$344$	2.00000	0.107833
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 33.0000i	− 1.77153i	−0.464131	−	0.885766i	$-0.653633\pi$
$347$	− 33.0000i	− 1.77153i	0.464131	−	0.885766i	$-0.346367\pi$
$348$	− 9.00000i	− 0.482451i
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	−25.0000	−1.33440
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	3.00000	0.159448
$355$	0	0
$356$	−9.00000	−0.476999
$357$	2.00000i	0.105851i
$358$	− 12.0000i	− 0.634220i
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	2.00000i	0.105118i
$363$	− 11.0000i	− 0.577350i
$364$	−10.0000	−0.524142
$365$	0	0
$366$	7.00000	0.365896
$367$	− 26.0000i	− 1.35719i	−0.734513	−	0.678594i	$-0.762589\pi$
$367$	− 26.0000i	− 1.35719i	0.734513	−	0.678594i	$-0.237411\pi$
$368$	6.00000i	0.312772i
$369$	−12.0000	−0.624695
$370$	0	0
$371$	−18.0000	−0.934513
$372$	1.00000i	0.0518476i
$373$	14.0000i	0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$	14.0000i	0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	0	0
$376$	9.00000	0.464140
$377$	45.0000i	2.31762i
$378$	10.0000i	0.514344i
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	24.0000i	1.22795i
$383$	− 27.0000i	− 1.37964i	−0.723983	−	0.689818i	$-0.757691\pi$
$383$	− 27.0000i	− 1.37964i	0.723983	−	0.689818i	$-0.242309\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	4.00000i	0.203331i
$388$	− 7.00000i	− 0.355371i
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	− 3.00000i	− 0.151523i
$393$	6.00000i	0.302660i
$394$	24.0000	1.20910
$395$	0	0
$396$	0	0
$397$	16.0000i	0.803017i	0.915855	+	0.401508i	$0.131514\pi$
$397$	16.0000i	0.803017i	−0.915855	+	0.401508i	$0.868486\pi$
$398$	7.00000i	0.350878i
$399$	2.00000	0.100125
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	14.0000i	0.698257i
$403$	− 5.00000i	− 0.249068i
$404$	−18.0000	−0.895533
$405$	0	0
$406$	18.0000	0.893325
$407$	0	0
$408$	1.00000i	0.0495074i
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	16.0000i	0.788263i
$413$	6.00000i	0.295241i
$414$	−12.0000	−0.589768
$415$	0	0
$416$	−5.00000	−0.245145
$417$	− 20.0000i	− 0.979404i
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	− 22.0000i	− 1.07094i
$423$	18.0000i	0.875190i
$424$	−9.00000	−0.437079
$425$	0	0
$426$	−3.00000	−0.145350
$427$	14.0000i	0.677507i
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	5.00000i	0.240563i
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−7.00000	−0.335239
$437$	6.00000i	0.287019i
$438$	− 11.0000i	− 0.525600i
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	− 5.00000i	− 0.237826i
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	19.0000	0.899676
$447$	12.0000i	0.567581i
$448$	2.00000i	0.0944911i
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	− 9.00000i	− 0.423324i
$453$	14.0000i	0.657777i
$454$	27.0000	1.26717
$455$	0	0
$456$	1.00000	0.0468293
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	− 14.0000i	− 0.654177i
$459$	−5.00000	−0.233380
$460$	0	0
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	0	0
$463$	− 25.0000i	− 1.16185i	−0.813958	−	0.580924i	$-0.802691\pi$
$463$	− 25.0000i	− 1.16185i	0.813958	−	0.580924i	$-0.197309\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	−9.00000	−0.416917
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	− 10.0000i	− 0.462250i
$469$	−28.0000	−1.29292
$470$	0	0
$471$	2.00000	0.0921551
$472$	3.00000i	0.138086i
$473$	0	0
$474$	8.00000	0.367452
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	− 18.0000i	− 0.824163i
$478$	30.0000i	1.37217i
$479$	−9.00000	−0.411220	−0.205610	−	0.978634i	$-0.565918\pi$
$479$	−9.00000	−0.411220	−0.205610	+	0.978634i	$0.565918\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	26.0000i	1.18427i
$483$	12.0000i	0.546019i
$484$	11.0000	0.500000
$485$	0	0
$486$	−16.0000	−0.725775
$487$	34.0000i	1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$487$	34.0000i	1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$488$	7.00000i	0.316875i
$489$	16.0000	0.723545
$490$	0	0
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	6.00000i	0.270501i
$493$	9.00000i	0.405340i
$494$	−5.00000	−0.224961
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	− 12.0000i	− 0.535586i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	− 7.00000i	− 0.310575i
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	22.0000	0.973223
$512$	1.00000i	0.0441942i
$513$	5.00000i	0.220755i
$514$	18.0000	0.793946
$515$	0	0
$516$	2.00000	0.0880451
$517$	0	0
$518$	8.00000i	0.351500i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	18.0000i	0.787839i
$523$	38.0000i	1.66162i	0.556553	+	0.830812i	$0.312124\pi$
$523$	38.0000i	1.66162i	−0.556553	+	0.830812i	$0.687876\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−3.00000	−0.130806
$527$	− 1.00000i	− 0.0435607i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−6.00000	−0.260378
$532$	2.00000i	0.0867110i
$533$	− 30.0000i	− 1.29944i
$534$	−9.00000	−0.389468
$535$	0	0
$536$	−14.0000	−0.604708
$537$	− 12.0000i	− 0.517838i
$538$	3.00000i	0.129339i
$539$	0	0
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	− 28.0000i	− 1.20270i
$543$	2.00000i	0.0858282i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−10.0000	−0.427960
$547$	7.00000i	0.299298i	0.988739	+	0.149649i	$0.0478144\pi$
$547$	7.00000i	0.299298i	−0.988739	+	0.149649i	$0.952186\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−14.0000	−0.597505
$550$	0	0
$551$	9.00000	0.383413
$552$	6.00000i	0.255377i
$553$	16.0000i	0.680389i
$554$	14.0000	0.594803
$555$	0	0
$556$	20.0000	0.848189
$557$	− 21.0000i	− 0.889799i	−0.895581	−	0.444899i	$-0.853239\pi$
$557$	− 21.0000i	− 0.889799i	0.895581	−	0.444899i	$-0.146761\pi$
$558$	− 2.00000i	− 0.0846668i
$559$	−10.0000	−0.422955
$560$	0	0
$561$	0	0
$562$	− 21.0000i	− 0.885832i
$563$	18.0000i	0.758610i	0.925272	+	0.379305i	$0.123837\pi$
$563$	18.0000i	0.758610i	−0.925272	+	0.379305i	$0.876163\pi$
$564$	9.00000	0.378968
$565$	0	0
$566$	−11.0000	−0.462364
$567$	− 2.00000i	− 0.0839921i
$568$	− 3.00000i	− 0.125877i
$569$	21.0000	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	0	0
$573$	24.0000i	1.00261i
$574$	−12.0000	−0.500870
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	− 44.0000i	− 1.83174i	−0.401470	−	0.915872i	$-0.631501\pi$
$577$	− 44.0000i	− 1.83174i	0.401470	−	0.915872i	$-0.368499\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	0	0
$582$	− 7.00000i	− 0.290159i
$583$	0	0
$584$	11.0000	0.455183
$585$	0	0
$586$	9.00000	0.371787
$587$	− 30.0000i	− 1.23823i	−0.785299	−	0.619116i	$-0.787491\pi$
$587$	− 30.0000i	− 1.23823i	0.785299	−	0.619116i	$-0.212509\pi$
$588$	− 3.00000i	− 0.123718i
$589$	−1.00000	−0.0412043
$590$	0	0
$591$	24.0000	0.987228
$592$	4.00000i	0.164399i
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	7.00000i	0.286491i
$598$	− 30.0000i	− 1.22679i
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	4.00000i	0.163028i
$603$	− 28.0000i	− 1.14025i
$604$	−14.0000	−0.569652
$605$	0	0
$606$	−18.0000	−0.731200
$607$	− 2.00000i	− 0.0811775i	−0.999176	−	0.0405887i	$-0.987077\pi$
$607$	− 2.00000i	− 0.0811775i	0.999176	−	0.0405887i	$-0.0129233\pi$
$608$	1.00000i	0.0405554i
$609$	18.0000	0.729397
$610$	0	0
$611$	−45.0000	−1.82051
$612$	− 2.00000i	− 0.0808452i
$613$	5.00000i	0.201948i	0.994889	+	0.100974i	$0.0321959\pi$
$613$	5.00000i	0.201948i	−0.994889	+	0.100974i	$0.967804\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	15.0000i	0.603877i	0.953327	+	0.301939i	$0.0976338\pi$
$617$	15.0000i	0.603877i	−0.953327	+	0.301939i	$0.902366\pi$
$618$	16.0000i	0.643614i
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−30.0000	−1.20386
$622$	− 24.0000i	− 0.962312i
$623$	− 18.0000i	− 0.721155i
$624$	−5.00000	−0.200160
$625$	0	0
$626$	10.0000	0.399680
$627$	0	0
$628$	2.00000i	0.0798087i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	8.00000i	0.318223i
$633$	− 22.0000i	− 0.874421i
$634$	24.0000	0.953162
$635$	0	0
$636$	−9.00000	−0.356873
$637$	15.0000i	0.594322i
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	12.0000i	0.473602i
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−1.00000	−0.0393445
$647$	27.0000i	1.06148i	0.847535	+	0.530740i	$0.178086\pi$
$647$	27.0000i	1.06148i	−0.847535	+	0.530740i	$0.821914\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	16.0000i	0.626608i
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	−7.00000	−0.273722
$655$	0	0
$656$	−6.00000	−0.234261
$657$	22.0000i	0.858302i
$658$	18.0000i	0.701713i
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	0	0
$661$	44.0000	1.71140	0.855701	−	0.517471i	$-0.173126\pi$
$661$	44.0000	1.71140	0.855701	+	0.517471i	$0.173126\pi$
$662$	− 7.00000i	− 0.272063i
$663$	− 5.00000i	− 0.194184i
$664$	0	0
$665$	0	0
$666$	−8.00000	−0.309994
$667$	54.0000i	2.09089i
$668$	6.00000i	0.232147i
$669$	19.0000	0.734582
$670$	0	0
$671$	0	0
$672$	2.00000i	0.0771517i
$673$	− 25.0000i	− 0.963679i	−0.876259	−	0.481840i	$-0.839969\pi$
$673$	− 25.0000i	− 0.963679i	0.876259	−	0.481840i	$-0.160031\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	12.0000	0.461538
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	− 9.00000i	− 0.345643i
$679$	14.0000	0.537271
$680$	0	0
$681$	27.0000	1.03464
$682$	0	0
$683$	− 3.00000i	− 0.114792i	−0.998351	−	0.0573959i	$-0.981720\pi$
$683$	− 3.00000i	− 0.114792i	0.998351	−	0.0573959i	$-0.0182797\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	20.0000	0.763604
$687$	− 14.0000i	− 0.534133i
$688$	2.00000i	0.0762493i
$689$	45.0000	1.71436
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	33.0000	1.25266
$695$	0	0
$696$	9.00000	0.341144
$697$	− 6.00000i	− 0.227266i
$698$	− 8.00000i	− 0.302804i
$699$	−9.00000	−0.340411
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	− 25.0000i	− 0.943564i
$703$	4.00000i	0.150863i
$704$	0	0
$705$	0	0
$706$	6.00000	0.225813
$707$	− 36.0000i	− 1.35392i
$708$	3.00000i	0.112747i
$709$	13.0000	0.488225	0.244113	−	0.969747i	$-0.421503\pi$
$709$	13.0000	0.488225	0.244113	+	0.969747i	$0.421503\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	− 9.00000i	− 0.337289i
$713$	− 6.00000i	− 0.224702i
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	12.0000	0.448461
$717$	30.0000i	1.12037i
$718$	− 6.00000i	− 0.223918i
$719$	−27.0000	−1.00693	−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000	−1.00693	−0.503465	+	0.864016i	$0.667942\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	− 18.0000i	− 0.669891i
$723$	26.0000i	0.966950i
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	11.0000	0.408248
$727$	− 23.0000i	− 0.853023i	−0.904482	−	0.426511i	$-0.859742\pi$
$727$	− 23.0000i	− 0.853023i	0.904482	−	0.426511i	$-0.140258\pi$
$728$	− 10.0000i	− 0.370625i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−2.00000	−0.0739727
$732$	7.00000i	0.258727i
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	−6.00000	−0.221163
$737$	0	0
$738$	− 12.0000i	− 0.441726i
$739$	43.0000	1.58178	0.790890	−	0.611958i	$-0.209618\pi$
$739$	43.0000	1.58178	0.790890	+	0.611958i	$0.209618\pi$
$740$	0	0
$741$	−5.00000	−0.183680
$742$	− 18.0000i	− 0.660801i
$743$	− 18.0000i	− 0.660356i	−0.943919	−	0.330178i	$-0.892891\pi$
$743$	− 18.0000i	− 0.660356i	0.943919	−	0.330178i	$-0.107109\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−14.0000	−0.512576
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	53.0000	1.93400	0.966999	−	0.254781i	$-0.0820034\pi$
$751$	53.0000	1.93400	0.966999	+	0.254781i	$0.0820034\pi$
$752$	9.00000i	0.328196i
$753$	− 12.0000i	− 0.437304i
$754$	−45.0000	−1.63880
$755$	0	0
$756$	−10.0000	−0.363696
$757$	− 41.0000i	− 1.49017i	−0.666969	−	0.745085i	$-0.732409\pi$
$757$	− 41.0000i	− 1.49017i	0.666969	−	0.745085i	$-0.267591\pi$
$758$	16.0000i	0.581146i
$759$	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	$0.465315\pi$
$762$	− 7.00000i	− 0.253583i
$763$	− 14.0000i	− 0.506834i
$764$	−24.0000	−0.868290
$765$	0	0
$766$	27.0000	0.975550
$767$	− 15.0000i	− 0.541619i
$768$	1.00000i	0.0360844i
$769$	−41.0000	−1.47850	−0.739249	−	0.673432i	$-0.764819\pi$
$769$	−41.0000	−1.47850	−0.739249	+	0.673432i	$0.764819\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	− 2.00000i	− 0.0719816i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	7.00000	0.251285
$777$	8.00000i	0.286998i
$778$	24.0000i	0.860442i
$779$	−6.00000	−0.214972
$780$	0	0
$781$	0	0
$782$	− 6.00000i	− 0.214560i
$783$	45.0000i	1.60817i
$784$	3.00000	0.107143
$785$	0	0
$786$	−6.00000	−0.214013
$787$	7.00000i	0.249523i	0.992187	+	0.124762i	$0.0398166\pi$
$787$	7.00000i	0.249523i	−0.992187	+	0.124762i	$0.960183\pi$
$788$	24.0000i	0.854965i
$789$	−3.00000	−0.106803
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	− 35.0000i	− 1.24289i
$794$	−16.0000	−0.567819
$795$	0	0
$796$	−7.00000	−0.248108
$797$	54.0000i	1.91278i	0.292096	+	0.956389i	$0.405647\pi$
$797$	54.0000i	1.91278i	−0.292096	+	0.956389i	$0.594353\pi$
$798$	2.00000i	0.0707992i
$799$	−9.00000	−0.318397
$800$	0	0
$801$	18.0000	0.635999
$802$	− 24.0000i	− 0.847469i
$803$	0	0
$804$	−14.0000	−0.493742
$805$	0	0
$806$	5.00000	0.176117
$807$	3.00000i	0.105605i
$808$	− 18.0000i	− 0.633238i
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	14.0000	0.491606	0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000	0.491606	0.245803	+	0.969320i	$0.420948\pi$
$812$	18.0000i	0.631676i
$813$	− 28.0000i	− 0.982003i
$814$	0	0
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	2.00000i	0.0699711i
$818$	− 5.00000i	− 0.174821i
$819$	20.0000	0.698857
$820$	0	0
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	− 6.00000i	− 0.209274i
$823$	14.0000i	0.488009i	0.969774	+	0.244005i	$0.0784612\pi$
$823$	14.0000i	0.488009i	−0.969774	+	0.244005i	$0.921539\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	−6.00000	−0.208767
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	− 12.0000i	− 0.417029i
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	− 5.00000i	− 0.173344i
$833$	3.00000i	0.103944i
$834$	20.0000	0.692543
$835$	0	0
$836$	0	0
$837$	− 5.00000i	− 0.172825i
$838$	− 6.00000i	− 0.207267i
$839$	39.0000	1.34643	0.673215	−	0.739447i	$-0.264913\pi$
$839$	39.0000	1.34643	0.673215	+	0.739447i	$0.264913\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	− 16.0000i	− 0.551396i
$843$	− 21.0000i	− 0.723278i
$844$	22.0000	0.757271
$845$	0	0
$846$	−18.0000	−0.618853
$847$	22.0000i	0.755929i
$848$	− 9.00000i	− 0.309061i
$849$	−11.0000	−0.377519
$850$	0	0
$851$	−24.0000	−0.822709
$852$	− 3.00000i	− 0.102778i
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	−14.0000	−0.479070
$855$	0	0
$856$	−12.0000	−0.410152
$857$	15.0000i	0.512390i	0.966625	+	0.256195i	$0.0824690\pi$
$857$	15.0000i	0.512390i	−0.966625	+	0.256195i	$0.917531\pi$
$858$	0	0
$859$	−47.0000	−1.60362	−0.801810	−	0.597580i	$-0.796129\pi$
$859$	−47.0000	−1.60362	−0.801810	+	0.597580i	$0.796129\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−26.0000	−0.883516
$867$	− 1.00000i	− 0.0339618i
$868$	− 2.00000i	− 0.0678844i
$869$	0	0
$870$	0	0
$871$	70.0000	2.37186
$872$	− 7.00000i	− 0.237050i
$873$	14.0000i	0.473828i
$874$	−6.00000	−0.202953
$875$	0	0
$876$	11.0000	0.371656
$877$	40.0000i	1.35070i	0.737496	+	0.675352i	$0.236008\pi$
$877$	40.0000i	1.35070i	−0.737496	+	0.675352i	$0.763992\pi$
$878$	− 32.0000i	− 1.07995i
$879$	9.00000	0.303562
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	6.00000i	0.202031i
$883$	− 34.0000i	− 1.14419i	−0.820187	−	0.572096i	$-0.806131\pi$
$883$	− 34.0000i	− 1.14419i	0.820187	−	0.572096i	$-0.193869\pi$
$884$	5.00000	0.168168
$885$	0	0
$886$	−24.0000	−0.806296
$887$	42.0000i	1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$	42.0000i	1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$	4.00000i	0.134231i
$889$	14.0000	0.469545
$890$	0	0
$891$	0	0
$892$	19.0000i	0.636167i
$893$	9.00000i	0.301174i
$894$	−12.0000	−0.401340
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	− 30.0000i	− 1.00167i
$898$	0	0
$899$	−9.00000	−0.300167
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	4.00000i	0.133112i
$904$	9.00000	0.299336
$905$	0	0
$906$	−14.0000	−0.465119
$907$	− 23.0000i	− 0.763702i	−0.924224	−	0.381851i	$-0.875287\pi$
$907$	− 23.0000i	− 0.763702i	0.924224	−	0.381851i	$-0.124713\pi$
$908$	27.0000i	0.896026i
$909$	36.0000	1.19404
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	1.00000i	0.0331133i
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	14.0000	0.462573
$917$	− 12.0000i	− 0.396275i
$918$	− 5.00000i	− 0.165025i
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	42.0000i	1.38320i
$923$	15.0000i	0.493731i
$924$	0	0
$925$	0	0
$926$	25.0000	0.821551
$927$	− 32.0000i	− 1.05102i
$928$	9.00000i	0.295439i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	− 9.00000i	− 0.294805i
$933$	− 24.0000i	− 0.785725i
$934$	0	0
$935$	0	0
$936$	10.0000	0.326860
$937$	− 8.00000i	− 0.261349i	−0.991425	−	0.130674i	$-0.958286\pi$
$937$	− 8.00000i	− 0.261349i	0.991425	−	0.130674i	$-0.0417142\pi$
$938$	− 28.0000i	− 0.914232i
$939$	10.0000	0.326338
$940$	0	0
$941$	−45.0000	−1.46696	−0.733479	−	0.679712i	$-0.762105\pi$
$941$	−45.0000	−1.46696	−0.733479	+	0.679712i	$0.762105\pi$
$942$	2.00000i	0.0651635i
$943$	− 36.0000i	− 1.17232i
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	0	0
$947$	− 3.00000i	− 0.0974869i	−0.998811	−	0.0487435i	$-0.984478\pi$
$947$	− 3.00000i	− 0.0974869i	0.998811	−	0.0487435i	$-0.0155217\pi$
$948$	8.00000i	0.259828i
$949$	−55.0000	−1.78538
$950$	0	0
$951$	24.0000	0.778253
$952$	− 2.00000i	− 0.0648204i
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	18.0000	0.582772
$955$	0	0
$956$	−30.0000	−0.970269
$957$	0	0
$958$	− 9.00000i	− 0.290777i
$959$	12.0000	0.387500
$960$	0	0
$961$	−30.0000	−0.967742
$962$	− 20.0000i	− 0.644826i
$963$	− 24.0000i	− 0.773389i
$964$	−26.0000	−0.837404
$965$	0	0
$966$	−12.0000	−0.386094
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	11.0000i	0.353553i
$969$	−1.00000	−0.0321246
$970$	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
$973$	40.0000i	1.28234i
$974$	−34.0000	−1.08943
$975$	0	0
$976$	−7.00000	−0.224065
$977$	12.0000i	0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$	12.0000i	0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$	16.0000i	0.511624i
$979$	0	0
$980$	0	0
$981$	14.0000	0.446986
$982$	33.0000i	1.05307i
$983$	18.0000i	0.574111i	0.957914	+	0.287055i	$0.0926764\pi$
$983$	18.0000i	0.574111i	−0.957914	+	0.287055i	$0.907324\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	−9.00000	−0.286618
$987$	18.0000i	0.572946i
$988$	− 5.00000i	− 0.159071i
$989$	−12.0000	−0.381578
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	− 1.00000i	− 0.0317500i
$993$	− 7.00000i	− 0.222138i
$994$	6.00000	0.190308
$995$	0	0
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	− 38.0000i	− 1.20287i
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.c.c.749.2		2
5.2	odd	4		170.2.a.c.1.1	✓	1
5.3	odd	4		850.2.a.g.1.1		1
5.4	even	2	inner	850.2.c.c.749.1		2
15.2	even	4		1530.2.a.l.1.1		1
15.8	even	4		7650.2.a.i.1.1		1
20.3	even	4		6800.2.a.s.1.1		1
20.7	even	4		1360.2.a.e.1.1		1
35.27	even	4		8330.2.a.d.1.1		1
40.27	even	4		5440.2.a.p.1.1		1
40.37	odd	4		5440.2.a.i.1.1		1
85.47	odd	4		2890.2.b.h.2311.2		2
85.67	odd	4		2890.2.a.e.1.1		1
85.72	odd	4		2890.2.b.h.2311.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.c.1.1	✓	1	5.2	odd	4
850.2.a.g.1.1		1	5.3	odd	4
850.2.c.c.749.1		2	5.4	even	2	inner
850.2.c.c.749.2		2	1.1	even	1	trivial
1360.2.a.e.1.1		1	20.7	even	4
1530.2.a.l.1.1		1	15.2	even	4
2890.2.a.e.1.1		1	85.67	odd	4
2890.2.b.h.2311.1		2	85.72	odd	4
2890.2.b.h.2311.2		2	85.47	odd	4
5440.2.a.i.1.1		1	40.37	odd	4
5440.2.a.p.1.1		1	40.27	even	4
6800.2.a.s.1.1		1	20.3	even	4
7650.2.a.i.1.1		1	15.8	even	4
8330.2.a.d.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:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} -1.00000i q^{12} +5.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +2.00000i q^{18} +1.00000 q^{19} +2.00000 q^{21} +6.00000i q^{23} +1.00000 q^{24} -5.00000 q^{26} +5.00000i q^{27} +2.00000i q^{28} +9.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} -1.00000 q^{34} -2.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} -5.00000 q^{39} -6.00000 q^{41} +2.00000i q^{42} +2.00000i q^{43} -6.00000 q^{46} +9.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} -1.00000 q^{51} -5.00000i q^{52} -9.00000i q^{53} -5.00000 q^{54} -2.00000 q^{56} +1.00000i q^{57} +9.00000i q^{58} -3.00000 q^{59} -7.00000 q^{61} -1.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -14.0000i q^{67} -1.00000i q^{68} -6.00000 q^{69} +3.00000 q^{71} -2.00000i q^{72} +11.0000i q^{73} -4.00000 q^{74} -1.00000 q^{76} -5.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -2.00000 q^{84} -2.00000 q^{86} +9.00000i q^{87} +9.00000 q^{89} +10.0000 q^{91} -6.00000i q^{92} -1.00000i q^{93} -9.00000 q^{94} -1.00000 q^{96} +7.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} + 4 q^{14} + 2 q^{16} + 2 q^{19} + 4 q^{21} + 2 q^{24} - 10 q^{26} + 18 q^{29} - 2 q^{31} - 2 q^{34} - 4 q^{36} - 10 q^{39} - 12 q^{41} - 12 q^{46} + 6 q^{49} - 2 q^{51} - 10 q^{54} - 4 q^{56} - 6 q^{59} - 14 q^{61} - 2 q^{64} - 12 q^{69} + 6 q^{71} - 8 q^{74} - 2 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} - 4 q^{86} + 18 q^{89} + 20 q^{91} - 18 q^{94} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 + 4 * q^9 + 4 * q^14 + 2 * q^16 + 2 * q^19 + 4 * q^21 + 2 * q^24 - 10 * q^26 + 18 * q^29 - 2 * q^31 - 2 * q^34 - 4 * q^36 - 10 * q^39 - 12 * q^41 - 12 * q^46 + 6 * q^49 - 2 * q^51 - 10 * q^54 - 4 * q^56 - 6 * q^59 - 14 * q^61 - 2 * q^64 - 12 * q^69 + 6 * q^71 - 8 * q^74 - 2 * q^76 - 16 * q^79 + 2 * q^81 - 4 * q^84 - 4 * q^86 + 18 * q^89 + 20 * q^91 - 18 * q^94 - 2 * q^96

Introduction

L-functions

Modular forms

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