LMFDB - Embedded newform 850.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [850,2,Mod(1,850)]

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([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	850.a (trivial)

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:	yes
Analytic conductor:	$6.78728417181$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -4.00000 q^{11} -3.00000 q^{12} +3.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -6.00000 q^{18} +6.00000 q^{19} +3.00000 q^{21} +4.00000 q^{22} +3.00000 q^{24} -3.00000 q^{26} -9.00000 q^{27} -1.00000 q^{28} -9.00000 q^{31} -1.00000 q^{32} +12.0000 q^{33} -1.00000 q^{34} +6.00000 q^{36} +4.00000 q^{37} -6.00000 q^{38} -9.00000 q^{39} +6.00000 q^{41} -3.00000 q^{42} +12.0000 q^{43} -4.00000 q^{44} -10.0000 q^{47} -3.00000 q^{48} -6.00000 q^{49} -3.00000 q^{51} +3.00000 q^{52} -9.00000 q^{53} +9.00000 q^{54} +1.00000 q^{56} -18.0000 q^{57} -14.0000 q^{61} +9.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} -12.0000 q^{66} -8.00000 q^{67} +1.00000 q^{68} -15.0000 q^{71} -6.00000 q^{72} -12.0000 q^{73} -4.00000 q^{74} +6.00000 q^{76} +4.00000 q^{77} +9.00000 q^{78} +3.00000 q^{79} +9.00000 q^{81} -6.00000 q^{82} +3.00000 q^{84} -12.0000 q^{86} +4.00000 q^{88} -6.00000 q^{89} -3.00000 q^{91} +27.0000 q^{93} +10.0000 q^{94} +3.00000 q^{96} +16.0000 q^{97} +6.00000 q^{98} -24.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	3.00000	1.22474
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	6.00000	2.00000
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−3.00000	−0.866025
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−6.00000	−1.41421
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	−3.00000	−0.588348
$27$	−9.00000	−1.73205
$28$	−1.00000	−0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	−1.00000	−0.176777
$33$	12.0000	2.08893
$34$	−1.00000	−0.171499
$35$	0	0
$36$	6.00000	1.00000
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−6.00000	−0.973329
$39$	−9.00000	−1.44115
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−3.00000	−0.462910
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	−3.00000	−0.433013
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	3.00000	0.416025
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	9.00000	1.22474
$55$	0	0
$56$	1.00000	0.133631
$57$	−18.0000	−2.38416
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	9.00000	1.14300
$63$	−6.00000	−0.755929
$64$	1.00000	0.125000
$65$	0	0
$66$	−12.0000	−1.47710
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	−6.00000	−0.707107
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	6.00000	0.688247
$77$	4.00000	0.455842
$78$	9.00000	1.01905
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	−6.00000	−0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	−12.0000	−1.29399
$87$	0	0
$88$	4.00000	0.426401
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	27.0000	2.79977
$94$	10.0000	1.03142
$95$	0	0
$96$	3.00000	0.306186
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	6.00000	0.606092
$99$	−24.0000	−2.41209
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	3.00000	0.297044
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	9.00000	0.874157
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	−9.00000	−0.866025
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	−1.00000	−0.0944911
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	18.0000	1.68585
$115$	0	0
$116$	0	0
$117$	18.0000	1.66410
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	5.00000	0.454545
$122$	14.0000	1.26750
$123$	−18.0000	−1.62301
$124$	−9.00000	−0.808224
$125$	0	0
$126$	6.00000	0.534522
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	−36.0000	−3.16962
$130$	0	0
$131$	19.0000	1.66004	0.830019	−	0.557735i	$-0.188330\pi$
$131$	19.0000	1.66004	0.830019	+	0.557735i	$0.188330\pi$
$132$	12.0000	1.04447
$133$	−6.00000	−0.520266
$134$	8.00000	0.691095
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−11.0000	−0.939793	−0.469897	−	0.882721i	$-0.655709\pi$
$137$	−11.0000	−0.939793	−0.469897	+	0.882721i	$0.655709\pi$
$138$	0	0
$139$	17.0000	1.44192	0.720961	−	0.692976i	$-0.243701\pi$
$139$	17.0000	1.44192	0.720961	+	0.692976i	$0.243701\pi$
$140$	0	0
$141$	30.0000	2.52646
$142$	15.0000	1.25877
$143$	−12.0000	−1.00349
$144$	6.00000	0.500000
$145$	0	0
$146$	12.0000	0.993127
$147$	18.0000	1.48461
$148$	4.00000	0.328798
$149$	1.00000	0.0819232	0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000	0.0819232	0.0409616	+	0.999161i	$0.486958\pi$
$150$	0	0
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	−6.00000	−0.486664
$153$	6.00000	0.485071
$154$	−4.00000	−0.322329
$155$	0	0
$156$	−9.00000	−0.720577
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	−3.00000	−0.238667
$159$	27.0000	2.14124
$160$	0	0
$161$	0	0
$162$	−9.00000	−0.707107
$163$	−17.0000	−1.33154	−0.665771	−	0.746156i	$-0.731897\pi$
$163$	−17.0000	−1.33154	−0.665771	+	0.746156i	$0.731897\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−3.00000	−0.231455
$169$	−4.00000	−0.307692
$170$	0	0
$171$	36.0000	2.75299
$172$	12.0000	0.914991
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	6.00000	0.449719
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	3.00000	0.222375
$183$	42.0000	3.10473
$184$	0	0
$185$	0	0
$186$	−27.0000	−1.97974
$187$	−4.00000	−0.292509
$188$	−10.0000	−0.729325
$189$	9.00000	0.654654
$190$	0	0
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	−3.00000	−0.216506
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	24.0000	1.70561
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	24.0000	1.69283
$202$	−3.00000	−0.211079
$203$	0	0
$204$	−3.00000	−0.210042
$205$	0	0
$206$	14.0000	0.975426
$207$	0	0
$208$	3.00000	0.208013
$209$	−24.0000	−1.66011
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−9.00000	−0.618123
$213$	45.0000	3.08335
$214$	9.00000	0.615227
$215$	0	0
$216$	9.00000	0.612372
$217$	9.00000	0.610960
$218$	−4.00000	−0.270914
$219$	36.0000	2.43265
$220$	0	0
$221$	3.00000	0.201802
$222$	12.0000	0.805387
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	4.00000	0.266076
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	−18.0000	−1.19208
$229$	29.0000	1.91637	0.958187	−	0.286143i	$-0.0923732\pi$
$229$	29.0000	1.91637	0.958187	+	0.286143i	$0.0923732\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	−18.0000	−1.17670
$235$	0	0
$236$	0	0
$237$	−9.00000	−0.584613
$238$	1.00000	0.0648204
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−14.0000	−0.896258
$245$	0	0
$246$	18.0000	1.14764
$247$	18.0000	1.14531
$248$	9.00000	0.571501
$249$	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	$0.773547\pi$
$252$	−6.00000	−0.377964
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	36.0000	2.24126
$259$	−4.00000	−0.248548
$260$	0	0
$261$	0	0
$262$	−19.0000	−1.17382
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	6.00000	0.367884
$267$	18.0000	1.10158
$268$	−8.00000	−0.488678
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	1.00000	0.0606339
$273$	9.00000	0.544705
$274$	11.0000	0.664534
$275$	0	0
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−17.0000	−1.01959
$279$	−54.0000	−3.23290
$280$	0	0
$281$	1.00000	0.0596550	0.0298275	−	0.999555i	$-0.490504\pi$
$281$	1.00000	0.0596550	0.0298275	+	0.999555i	$0.490504\pi$
$282$	−30.0000	−1.78647
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−15.0000	−0.890086
$285$	0	0
$286$	12.0000	0.709575
$287$	−6.00000	−0.354169
$288$	−6.00000	−0.353553
$289$	1.00000	0.0588235
$290$	0	0
$291$	−48.0000	−2.81381
$292$	−12.0000	−0.702247
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	−4.00000	−0.232495
$297$	36.0000	2.08893
$298$	−1.00000	−0.0579284
$299$	0	0
$300$	0	0
$301$	−12.0000	−0.691669
$302$	6.00000	0.345261
$303$	−9.00000	−0.517036
$304$	6.00000	0.344124
$305$	0	0
$306$	−6.00000	−0.342997
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	4.00000	0.227921
$309$	42.0000	2.38930
$310$	0	0
$311$	27.0000	1.53103	0.765515	−	0.643418i	$-0.222484\pi$
$311$	27.0000	1.53103	0.765515	+	0.643418i	$0.222484\pi$
$312$	9.00000	0.509525
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−1.00000	−0.0564333
$315$	0	0
$316$	3.00000	0.168763
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−27.0000	−1.51408
$319$	0	0
$320$	0	0
$321$	27.0000	1.50699
$322$	0	0
$323$	6.00000	0.333849
$324$	9.00000	0.500000
$325$	0	0
$326$	17.0000	0.941543
$327$	−12.0000	−0.663602
$328$	−6.00000	−0.331295
$329$	10.0000	0.551318
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	0	0
$333$	24.0000	1.31519
$334$	−8.00000	−0.437741
$335$	0	0
$336$	3.00000	0.163663
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	4.00000	0.217571
$339$	12.0000	0.651751
$340$	0	0
$341$	36.0000	1.94951
$342$	−36.0000	−1.94666
$343$	13.0000	0.701934
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	−9.00000	−0.481759	−0.240879	−	0.970555i	$-0.577436\pi$
$349$	−9.00000	−0.481759	−0.240879	+	0.970555i	$0.577436\pi$
$350$	0	0
$351$	−27.0000	−1.44115
$352$	4.00000	0.213201
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	3.00000	0.158777
$358$	−2.00000	−0.105703
$359$	−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
$361$	17.0000	0.894737
$362$	14.0000	0.735824
$363$	−15.0000	−0.787296
$364$	−3.00000	−0.157243
$365$	0	0
$366$	−42.0000	−2.19538
$367$	1.00000	0.0521996	0.0260998	−	0.999659i	$-0.491691\pi$
$367$	1.00000	0.0521996	0.0260998	+	0.999659i	$0.491691\pi$
$368$	0	0
$369$	36.0000	1.87409
$370$	0	0
$371$	9.00000	0.467257
$372$	27.0000	1.39988
$373$	7.00000	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$373$	7.00000	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	10.0000	0.515711
$377$	0	0
$378$	−9.00000	−0.462910
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	0	0
$381$	−48.0000	−2.45911
$382$	14.0000	0.716302
$383$	34.0000	1.73732	0.868659	−	0.495410i	$-0.164982\pi$
$383$	34.0000	1.73732	0.868659	+	0.495410i	$0.164982\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	0	0
$387$	72.0000	3.65997
$388$	16.0000	0.812277
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	0	0
$392$	6.00000	0.303046
$393$	−57.0000	−2.87527
$394$	18.0000	0.906827
$395$	0	0
$396$	−24.0000	−1.20605
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	16.0000	0.802008
$399$	18.0000	0.901127
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	−24.0000	−1.19701
$403$	−27.0000	−1.34497
$404$	3.00000	0.149256
$405$	0	0
$406$	0	0
$407$	−16.0000	−0.793091
$408$	3.00000	0.148522
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	0	0
$411$	33.0000	1.62777
$412$	−14.0000	−0.689730
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−3.00000	−0.147087
$417$	−51.0000	−2.49748
$418$	24.0000	1.17388
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	−5.00000	−0.243396
$423$	−60.0000	−2.91730
$424$	9.00000	0.437079
$425$	0	0
$426$	−45.0000	−2.18026
$427$	14.0000	0.677507
$428$	−9.00000	−0.435031
$429$	36.0000	1.73810
$430$	0	0
$431$	9.00000	0.433515	0.216757	−	0.976226i	$-0.430452\pi$
$431$	9.00000	0.433515	0.216757	+	0.976226i	$0.430452\pi$
$432$	−9.00000	−0.433013
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−9.00000	−0.432014
$435$	0	0
$436$	4.00000	0.191565
$437$	0	0
$438$	−36.0000	−1.72015
$439$	−7.00000	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$439$	−7.00000	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	−3.00000	−0.142695
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	−12.0000	−0.569495
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	−3.00000	−0.141895
$448$	−1.00000	−0.0472456
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	−4.00000	−0.188144
$453$	18.0000	0.845714
$454$	−11.0000	−0.516256
$455$	0	0
$456$	18.0000	0.842927
$457$	−21.0000	−0.982339	−0.491169	−	0.871064i	$-0.663430\pi$
$457$	−21.0000	−0.982339	−0.491169	+	0.871064i	$0.663430\pi$
$458$	−29.0000	−1.35508
$459$	−9.00000	−0.420084
$460$	0	0
$461$	34.0000	1.58354	0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000	1.58354	0.791769	+	0.610821i	$0.209160\pi$
$462$	12.0000	0.558291
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	0	0
$466$	8.00000	0.370593
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	18.0000	0.832050
$469$	8.00000	0.369406
$470$	0	0
$471$	−3.00000	−0.138233
$472$	0	0
$473$	−48.0000	−2.20704
$474$	9.00000	0.413384
$475$	0	0
$476$	−1.00000	−0.0458349
$477$	−54.0000	−2.47249
$478$	16.0000	0.731823
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	−10.0000	−0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	14.0000	0.633750
$489$	51.0000	2.30630
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	−18.0000	−0.811503
$493$	0	0
$494$	−18.0000	−0.809858
$495$	0	0
$496$	−9.00000	−0.404112
$497$	15.0000	0.672842
$498$	0	0
$499$	−1.00000	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$499$	−1.00000	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	24.0000	1.07117
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	6.00000	0.267261
$505$	0	0
$506$	0	0
$507$	12.0000	0.532939
$508$	16.0000	0.709885
$509$	25.0000	1.10811	0.554053	−	0.832482i	$-0.313081\pi$
$509$	25.0000	1.10811	0.554053	+	0.832482i	$0.313081\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	−1.00000	−0.0441942
$513$	−54.0000	−2.38416
$514$	21.0000	0.926270
$515$	0	0
$516$	−36.0000	−1.58481
$517$	40.0000	1.75920
$518$	4.00000	0.175750
$519$	−18.0000	−0.790112
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	19.0000	0.830019
$525$	0	0
$526$	24.0000	1.04645
$527$	−9.00000	−0.392046
$528$	12.0000	0.522233
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	−6.00000	−0.260133
$533$	18.0000	0.779667
$534$	−18.0000	−0.778936
$535$	0	0
$536$	8.00000	0.345547
$537$	−6.00000	−0.258919
$538$	14.0000	0.603583
$539$	24.0000	1.03375
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	2.00000	0.0859074
$543$	42.0000	1.80239
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−9.00000	−0.385164
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	−11.0000	−0.469897
$549$	−84.0000	−3.58503
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−3.00000	−0.127573
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	17.0000	0.720961
$557$	−25.0000	−1.05928	−0.529642	−	0.848221i	$-0.677674\pi$
$557$	−25.0000	−1.05928	−0.529642	+	0.848221i	$0.677674\pi$
$558$	54.0000	2.28600
$559$	36.0000	1.52264
$560$	0	0
$561$	12.0000	0.506640
$562$	−1.00000	−0.0421825
$563$	34.0000	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$563$	34.0000	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$564$	30.0000	1.26323
$565$	0	0
$566$	−16.0000	−0.672530
$567$	−9.00000	−0.377964
$568$	15.0000	0.629386
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	−9.00000	−0.376638	−0.188319	−	0.982108i	$-0.560304\pi$
$571$	−9.00000	−0.376638	−0.188319	+	0.982108i	$0.560304\pi$
$572$	−12.0000	−0.501745
$573$	42.0000	1.75458
$574$	6.00000	0.250435
$575$	0	0
$576$	6.00000	0.250000
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	0	0
$581$	0	0
$582$	48.0000	1.98966
$583$	36.0000	1.49097
$584$	12.0000	0.496564
$585$	0	0
$586$	26.0000	1.07405
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	18.0000	0.742307
$589$	−54.0000	−2.22503
$590$	0	0
$591$	54.0000	2.22126
$592$	4.00000	0.164399
$593$	25.0000	1.02663	0.513313	−	0.858201i	$-0.328418\pi$
$593$	25.0000	1.02663	0.513313	+	0.858201i	$0.328418\pi$
$594$	−36.0000	−1.47710
$595$	0	0
$596$	1.00000	0.0409616
$597$	48.0000	1.96451
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	24.0000	0.978980	0.489490	−	0.872009i	$-0.337183\pi$
$601$	24.0000	0.978980	0.489490	+	0.872009i	$0.337183\pi$
$602$	12.0000	0.489083
$603$	−48.0000	−1.95471
$604$	−6.00000	−0.244137
$605$	0	0
$606$	9.00000	0.365600
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	0	0
$611$	−30.0000	−1.21367
$612$	6.00000	0.242536
$613$	−19.0000	−0.767403	−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000	−0.767403	−0.383701	+	0.923457i	$0.625351\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−20.0000	−0.805170	−0.402585	−	0.915383i	$-0.631888\pi$
$617$	−20.0000	−0.805170	−0.402585	+	0.915383i	$0.631888\pi$
$618$	−42.0000	−1.68949
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	−27.0000	−1.08260
$623$	6.00000	0.240385
$624$	−9.00000	−0.360288
$625$	0	0
$626$	−14.0000	−0.559553
$627$	72.0000	2.87540
$628$	1.00000	0.0399043
$629$	4.00000	0.159490
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−3.00000	−0.119334
$633$	−15.0000	−0.596196
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	27.0000	1.07062
$637$	−18.0000	−0.713186
$638$	0	0
$639$	−90.0000	−3.56034
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	−27.0000	−1.06561
$643$	11.0000	0.433798	0.216899	−	0.976194i	$-0.430406\pi$
$643$	11.0000	0.433798	0.216899	+	0.976194i	$0.430406\pi$
$644$	0	0
$645$	0	0
$646$	−6.00000	−0.236067
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	−9.00000	−0.353553
$649$	0	0
$650$	0	0
$651$	−27.0000	−1.05821
$652$	−17.0000	−0.665771
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	12.0000	0.469237
$655$	0	0
$656$	6.00000	0.234261
$657$	−72.0000	−2.80899
$658$	−10.0000	−0.389841
$659$	18.0000	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$660$	0	0
$661$	−1.00000	−0.0388955	−0.0194477	−	0.999811i	$-0.506191\pi$
$661$	−1.00000	−0.0388955	−0.0194477	+	0.999811i	$0.506191\pi$
$662$	26.0000	1.01052
$663$	−9.00000	−0.349531
$664$	0	0
$665$	0	0
$666$	−24.0000	−0.929981
$667$	0	0
$668$	8.00000	0.309529
$669$	−6.00000	−0.231973
$670$	0	0
$671$	56.0000	2.16186
$672$	−3.00000	−0.115728
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	−4.00000	−0.153846
$677$	−28.0000	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$677$	−28.0000	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$678$	−12.0000	−0.460857
$679$	−16.0000	−0.614024
$680$	0	0
$681$	−33.0000	−1.26456
$682$	−36.0000	−1.37851
$683$	−45.0000	−1.72188	−0.860939	−	0.508709i	$-0.830123\pi$
$683$	−45.0000	−1.72188	−0.860939	+	0.508709i	$0.830123\pi$
$684$	36.0000	1.37649
$685$	0	0
$686$	−13.0000	−0.496342
$687$	−87.0000	−3.31926
$688$	12.0000	0.457496
$689$	−27.0000	−1.02862
$690$	0	0
$691$	25.0000	0.951045	0.475522	−	0.879704i	$-0.342259\pi$
$691$	25.0000	0.951045	0.475522	+	0.879704i	$0.342259\pi$
$692$	6.00000	0.228086
$693$	24.0000	0.911685
$694$	3.00000	0.113878
$695$	0	0
$696$	0	0
$697$	6.00000	0.227266
$698$	9.00000	0.340655
$699$	24.0000	0.907763
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	27.0000	1.01905
$703$	24.0000	0.905177
$704$	−4.00000	−0.150756
$705$	0	0
$706$	21.0000	0.790345
$707$	−3.00000	−0.112827
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	18.0000	0.675053
$712$	6.00000	0.224860
$713$	0	0
$714$	−3.00000	−0.112272
$715$	0	0
$716$	2.00000	0.0747435
$717$	48.0000	1.79259
$718$	6.00000	0.223918
$719$	23.0000	0.857755	0.428878	−	0.903363i	$-0.358909\pi$
$719$	23.0000	0.857755	0.428878	+	0.903363i	$0.358909\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	−17.0000	−0.632674
$723$	−30.0000	−1.11571
$724$	−14.0000	−0.520306
$725$	0	0
$726$	15.0000	0.556702
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	3.00000	0.111187
$729$	−27.0000	−1.00000
$730$	0	0
$731$	12.0000	0.443836
$732$	42.0000	1.55236
$733$	7.00000	0.258551	0.129275	−	0.991609i	$-0.458735\pi$
$733$	7.00000	0.258551	0.129275	+	0.991609i	$0.458735\pi$
$734$	−1.00000	−0.0369107
$735$	0	0
$736$	0	0
$737$	32.0000	1.17874
$738$	−36.0000	−1.32518
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	−54.0000	−1.98374
$742$	−9.00000	−0.330400
$743$	11.0000	0.403551	0.201775	−	0.979432i	$-0.435329\pi$
$743$	11.0000	0.403551	0.201775	+	0.979432i	$0.435329\pi$
$744$	−27.0000	−0.989868
$745$	0	0
$746$	−7.00000	−0.256288
$747$	0	0
$748$	−4.00000	−0.146254
$749$	9.00000	0.328853
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−10.0000	−0.364662
$753$	72.0000	2.62383
$754$	0	0
$755$	0	0
$756$	9.00000	0.327327
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−19.0000	−0.690111
$759$	0	0
$760$	0	0
$761$	41.0000	1.48625	0.743124	−	0.669153i	$-0.233343\pi$
$761$	41.0000	1.48625	0.743124	+	0.669153i	$0.233343\pi$
$762$	48.0000	1.73886
$763$	−4.00000	−0.144810
$764$	−14.0000	−0.506502
$765$	0	0
$766$	−34.0000	−1.22847
$767$	0	0
$768$	−3.00000	−0.108253
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	63.0000	2.26889
$772$	0	0
$773$	17.0000	0.611448	0.305724	−	0.952120i	$-0.401102\pi$
$773$	17.0000	0.611448	0.305724	+	0.952120i	$0.401102\pi$
$774$	−72.0000	−2.58799
$775$	0	0
$776$	−16.0000	−0.574367
$777$	12.0000	0.430498
$778$	−30.0000	−1.07555
$779$	36.0000	1.28983
$780$	0	0
$781$	60.0000	2.14697
$782$	0	0
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	57.0000	2.03312
$787$	21.0000	0.748569	0.374285	−	0.927314i	$-0.377888\pi$
$787$	21.0000	0.748569	0.374285	+	0.927314i	$0.377888\pi$
$788$	−18.0000	−0.641223
$789$	72.0000	2.56327
$790$	0	0
$791$	4.00000	0.142224
$792$	24.0000	0.852803
$793$	−42.0000	−1.49146
$794$	20.0000	0.709773
$795$	0	0
$796$	−16.0000	−0.567105
$797$	3.00000	0.106265	0.0531327	−	0.998587i	$-0.483079\pi$
$797$	3.00000	0.106265	0.0531327	+	0.998587i	$0.483079\pi$
$798$	−18.0000	−0.637193
$799$	−10.0000	−0.353775
$800$	0	0
$801$	−36.0000	−1.27200
$802$	10.0000	0.353112
$803$	48.0000	1.69388
$804$	24.0000	0.846415
$805$	0	0
$806$	27.0000	0.951034
$807$	42.0000	1.47847
$808$	−3.00000	−0.105540
$809$	−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
$811$	−27.0000	−0.948098	−0.474049	−	0.880498i	$-0.657208\pi$
$811$	−27.0000	−0.948098	−0.474049	+	0.880498i	$0.657208\pi$
$812$	0	0
$813$	6.00000	0.210429
$814$	16.0000	0.560800
$815$	0	0
$816$	−3.00000	−0.105021
$817$	72.0000	2.51896
$818$	−35.0000	−1.22375
$819$	−18.0000	−0.628971
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−33.0000	−1.15101
$823$	51.0000	1.77775	0.888874	−	0.458151i	$-0.151488\pi$
$823$	51.0000	1.77775	0.888874	+	0.458151i	$0.151488\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	0	0
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	0	0
$829$	5.00000	0.173657	0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000	0.173657	0.0868286	+	0.996223i	$0.472327\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	3.00000	0.104006
$833$	−6.00000	−0.207888
$834$	51.0000	1.76599
$835$	0	0
$836$	−24.0000	−0.830057
$837$	81.0000	2.79977
$838$	20.0000	0.690889
$839$	9.00000	0.310715	0.155357	−	0.987858i	$-0.450347\pi$
$839$	9.00000	0.310715	0.155357	+	0.987858i	$0.450347\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	13.0000	0.448010
$843$	−3.00000	−0.103325
$844$	5.00000	0.172107
$845$	0	0
$846$	60.0000	2.06284
$847$	−5.00000	−0.171802
$848$	−9.00000	−0.309061
$849$	−48.0000	−1.64736
$850$	0	0
$851$	0	0
$852$	45.0000	1.54167
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	−14.0000	−0.479070
$855$	0	0
$856$	9.00000	0.307614
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	−36.0000	−1.22902
$859$	−26.0000	−0.887109	−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000	−0.887109	−0.443554	+	0.896248i	$0.646283\pi$
$860$	0	0
$861$	18.0000	0.613438
$862$	−9.00000	−0.306541
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	9.00000	0.306186
$865$	0	0
$866$	2.00000	0.0679628
$867$	−3.00000	−0.101885
$868$	9.00000	0.305480
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−24.0000	−0.813209
$872$	−4.00000	−0.135457
$873$	96.0000	3.24911
$874$	0	0
$875$	0	0
$876$	36.0000	1.21633
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	7.00000	0.236239
$879$	78.0000	2.63087
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	36.0000	1.21218
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	6.00000	0.201574
$887$	51.0000	1.71241	0.856206	−	0.516634i	$-0.172815\pi$
$887$	51.0000	1.71241	0.856206	+	0.516634i	$0.172815\pi$
$888$	12.0000	0.402694
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−36.0000	−1.20605
$892$	2.00000	0.0669650
$893$	−60.0000	−2.00782
$894$	3.00000	0.100335
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	−12.0000	−0.400445
$899$	0	0
$900$	0	0
$901$	−9.00000	−0.299833
$902$	24.0000	0.799113
$903$	36.0000	1.19800
$904$	4.00000	0.133038
$905$	0	0
$906$	−18.0000	−0.598010
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	11.0000	0.365048
$909$	18.0000	0.597022
$910$	0	0
$911$	−33.0000	−1.09334	−0.546669	−	0.837349i	$-0.684105\pi$
$911$	−33.0000	−1.09334	−0.546669	+	0.837349i	$0.684105\pi$
$912$	−18.0000	−0.596040
$913$	0	0
$914$	21.0000	0.694618
$915$	0	0
$916$	29.0000	0.958187
$917$	−19.0000	−0.627435
$918$	9.00000	0.297044
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	60.0000	1.97707
$922$	−34.0000	−1.11973
$923$	−45.0000	−1.48119
$924$	−12.0000	−0.394771
$925$	0	0
$926$	4.00000	0.131448
$927$	−84.0000	−2.75892
$928$	0	0
$929$	−48.0000	−1.57483	−0.787414	−	0.616424i	$-0.788581\pi$
$929$	−48.0000	−1.57483	−0.787414	+	0.616424i	$0.788581\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	−8.00000	−0.262049
$933$	−81.0000	−2.65182
$934$	−18.0000	−0.588978
$935$	0	0
$936$	−18.0000	−0.588348
$937$	33.0000	1.07806	0.539032	−	0.842286i	$-0.318790\pi$
$937$	33.0000	1.07806	0.539032	+	0.842286i	$0.318790\pi$
$938$	−8.00000	−0.261209
$939$	−42.0000	−1.37062
$940$	0	0
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	3.00000	0.0977453
$943$	0	0
$944$	0	0
$945$	0	0
$946$	48.0000	1.56061
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−9.00000	−0.292306
$949$	−36.0000	−1.16861
$950$	0	0
$951$	−6.00000	−0.194563
$952$	1.00000	0.0324102
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	54.0000	1.74831
$955$	0	0
$956$	−16.0000	−0.517477
$957$	0	0
$958$	36.0000	1.16311
$959$	11.0000	0.355209
$960$	0	0
$961$	50.0000	1.61290
$962$	−12.0000	−0.386896
$963$	−54.0000	−1.74013
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	−60.0000	−1.92947	−0.964735	−	0.263223i	$-0.915214\pi$
$967$	−60.0000	−1.92947	−0.964735	+	0.263223i	$0.915214\pi$
$968$	−5.00000	−0.160706
$969$	−18.0000	−0.578243
$970$	0	0
$971$	16.0000	0.513464	0.256732	−	0.966483i	$-0.417354\pi$
$971$	16.0000	0.513464	0.256732	+	0.966483i	$0.417354\pi$
$972$	0	0
$973$	−17.0000	−0.544995
$974$	−12.0000	−0.384505
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	−51.0000	−1.63080
$979$	24.0000	0.767043
$980$	0	0
$981$	24.0000	0.766261
$982$	−28.0000	−0.893516
$983$	45.0000	1.43528	0.717639	−	0.696416i	$-0.245223\pi$
$983$	45.0000	1.43528	0.717639	+	0.696416i	$0.245223\pi$
$984$	18.0000	0.573819
$985$	0	0
$986$	0	0
$987$	−30.0000	−0.954911
$988$	18.0000	0.572656
$989$	0	0
$990$	0	0
$991$	−61.0000	−1.93773	−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−61.0000	−1.93773	−0.968864	+	0.247592i	$0.920361\pi$
$992$	9.00000	0.285750
$993$	78.0000	2.47526
$994$	−15.0000	−0.475771
$995$	0	0
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	1.00000	0.0316544
$999$	−36.0000	−1.13899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.a.a.1.1	✓	1
3.2	odd	2		7650.2.a.bu.1.1		1
4.3	odd	2		6800.2.a.y.1.1		1
5.2	odd	4		850.2.c.h.749.1		2
5.3	odd	4		850.2.c.h.749.2		2
5.4	even	2		850.2.a.l.1.1	yes	1
15.14	odd	2		7650.2.a.w.1.1		1
20.19	odd	2		6800.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
850.2.a.a.1.1	✓	1	1.1	even	1	trivial
850.2.a.l.1.1	yes	1	5.4	even	2
850.2.c.h.749.1		2	5.2	odd	4
850.2.c.h.749.2		2	5.3	odd	4
6800.2.a.a.1.1		1	20.19	odd	2
6800.2.a.y.1.1		1	4.3	odd	2
7650.2.a.w.1.1		1	15.14	odd	2
7650.2.a.bu.1.1		1	3.2	odd	2

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.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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