LMFDB - Embedded newform 185.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [185,2,Mod(1,185)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(185, 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("185.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 185 = 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	185.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:	$1.47723243739$
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	$\chi$	$=$	185.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -5.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -5.00000 q^{7} -2.00000 q^{9} +2.00000 q^{10} +3.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} +10.0000 q^{14} -1.00000 q^{15} -4.00000 q^{16} -4.00000 q^{17} +4.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -5.00000 q^{21} -6.00000 q^{22} -2.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} -5.00000 q^{27} -10.0000 q^{28} +2.00000 q^{29} +2.00000 q^{30} +8.00000 q^{32} +3.00000 q^{33} +8.00000 q^{34} +5.00000 q^{35} -4.00000 q^{36} -1.00000 q^{37} +8.00000 q^{38} -2.00000 q^{39} +7.00000 q^{41} +10.0000 q^{42} -10.0000 q^{43} +6.00000 q^{44} +2.00000 q^{45} +4.00000 q^{46} +11.0000 q^{47} -4.00000 q^{48} +18.0000 q^{49} -2.00000 q^{50} -4.00000 q^{51} -4.00000 q^{52} -3.00000 q^{53} +10.0000 q^{54} -3.00000 q^{55} -4.00000 q^{57} -4.00000 q^{58} -2.00000 q^{60} -4.00000 q^{61} +10.0000 q^{63} -8.00000 q^{64} +2.00000 q^{65} -6.00000 q^{66} +16.0000 q^{67} -8.00000 q^{68} -2.00000 q^{69} -10.0000 q^{70} -15.0000 q^{71} +11.0000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} -15.0000 q^{77} +4.00000 q^{78} -12.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -14.0000 q^{82} -3.00000 q^{83} -10.0000 q^{84} +4.00000 q^{85} +20.0000 q^{86} +2.00000 q^{87} -4.00000 q^{89} -4.00000 q^{90} +10.0000 q^{91} -4.00000 q^{92} -22.0000 q^{94} +4.00000 q^{95} +8.00000 q^{96} +8.00000 q^{97} -36.0000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	2.00000	1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.00000	−0.816497
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	2.00000	0.632456
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	2.00000	0.577350
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	10.0000	2.67261
$15$	−1.00000	−0.258199
$16$	−4.00000	−1.00000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	4.00000	0.942809
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−2.00000	−0.447214
$21$	−5.00000	−1.09109
$22$	−6.00000	−1.27920
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	−5.00000	−0.962250
$28$	−10.0000	−1.88982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	8.00000	1.41421
$33$	3.00000	0.522233
$34$	8.00000	1.37199
$35$	5.00000	0.845154
$36$	−4.00000	−0.666667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	8.00000	1.29777
$39$	−2.00000	−0.320256
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	10.0000	1.54303
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	6.00000	0.904534
$45$	2.00000	0.298142
$46$	4.00000	0.589768
$47$	11.0000	1.60451	0.802257	−	0.596978i	$-0.203632\pi$
$47$	11.0000	1.60451	0.802257	+	0.596978i	$0.203632\pi$
$48$	−4.00000	−0.577350
$49$	18.0000	2.57143
$50$	−2.00000	−0.282843
$51$	−4.00000	−0.560112
$52$	−4.00000	−0.554700
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	10.0000	1.36083
$55$	−3.00000	−0.404520
$56$	0	0
$57$	−4.00000	−0.529813
$58$	−4.00000	−0.525226
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−2.00000	−0.258199
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	10.0000	1.25988
$64$	−8.00000	−1.00000
$65$	2.00000	0.248069
$66$	−6.00000	−0.738549
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	−8.00000	−0.970143
$69$	−2.00000	−0.240772
$70$	−10.0000	−1.19523
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	−8.00000	−0.917663
$77$	−15.0000	−1.70941
$78$	4.00000	0.452911
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−14.0000	−1.54604
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	−10.0000	−1.09109
$85$	4.00000	0.433861
$86$	20.0000	2.15666
$87$	2.00000	0.214423
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	−4.00000	−0.421637
$91$	10.0000	1.04828
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−22.0000	−2.26913
$95$	4.00000	0.410391
$96$	8.00000	0.816497
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−36.0000	−3.63655
$99$	−6.00000	−0.603023
$100$	2.00000	0.200000
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	8.00000	0.792118
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	5.00000	0.487950
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−10.0000	−0.962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	6.00000	0.572078
$111$	−1.00000	−0.0949158
$112$	20.0000	1.88982
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	8.00000	0.749269
$115$	2.00000	0.186501
$116$	4.00000	0.371391
$117$	4.00000	0.369800
$118$	0	0
$119$	20.0000	1.83340
$120$	0	0
$121$	−2.00000	−0.181818
$122$	8.00000	0.724286
$123$	7.00000	0.631169
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−20.0000	−1.78174
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	−4.00000	−0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	6.00000	0.522233
$133$	20.0000	1.73422
$134$	−32.0000	−2.76438
$135$	5.00000	0.430331
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	4.00000	0.340503
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	10.0000	0.845154
$141$	11.0000	0.926367
$142$	30.0000	2.51754
$143$	−6.00000	−0.501745
$144$	8.00000	0.666667
$145$	−2.00000	−0.166091
$146$	−22.0000	−1.82073
$147$	18.0000	1.48461
$148$	−2.00000	−0.164399
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	−2.00000	−0.163299
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	8.00000	0.646762
$154$	30.0000	2.41747
$155$	0	0
$156$	−4.00000	−0.320256
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	24.0000	1.90934
$159$	−3.00000	−0.237915
$160$	−8.00000	−0.632456
$161$	10.0000	0.788110
$162$	−2.00000	−0.157135
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	14.0000	1.09322
$165$	−3.00000	−0.233550
$166$	6.00000	0.465690
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−8.00000	−0.613572
$171$	8.00000	0.611775
$172$	−20.0000	−1.52499
$173$	5.00000	0.380143	0.190071	−	0.981770i	$-0.439128\pi$
$173$	5.00000	0.380143	0.190071	+	0.981770i	$0.439128\pi$
$174$	−4.00000	−0.303239
$175$	−5.00000	−0.377964
$176$	−12.0000	−0.904534
$177$	0	0
$178$	8.00000	0.599625
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	4.00000	0.298142
$181$	13.0000	0.966282	0.483141	−	0.875542i	$-0.339496\pi$
$181$	13.0000	0.966282	0.483141	+	0.875542i	$0.339496\pi$
$182$	−20.0000	−1.48250
$183$	−4.00000	−0.295689
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	−12.0000	−0.877527
$188$	22.0000	1.60451
$189$	25.0000	1.81848
$190$	−8.00000	−0.580381
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	−8.00000	−0.577350
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−16.0000	−1.14873
$195$	2.00000	0.143223
$196$	36.0000	2.57143
$197$	−17.0000	−1.21120	−0.605600	−	0.795769i	$-0.707067\pi$
$197$	−17.0000	−1.21120	−0.605600	+	0.795769i	$0.707067\pi$
$198$	12.0000	0.852803
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	10.0000	0.703598
$203$	−10.0000	−0.701862
$204$	−8.00000	−0.560112
$205$	−7.00000	−0.488901
$206$	20.0000	1.39347
$207$	4.00000	0.278019
$208$	8.00000	0.554700
$209$	−12.0000	−0.830057
$210$	−10.0000	−0.690066
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	−6.00000	−0.412082
$213$	−15.0000	−1.02778
$214$	24.0000	1.64061
$215$	10.0000	0.681994
$216$	0	0
$217$	0	0
$218$	32.0000	2.16731
$219$	11.0000	0.743311
$220$	−6.00000	−0.404520
$221$	8.00000	0.538138
$222$	2.00000	0.134231
$223$	−29.0000	−1.94198	−0.970992	−	0.239113i	$-0.923143\pi$
$223$	−29.0000	−1.94198	−0.970992	+	0.239113i	$0.923143\pi$
$224$	−40.0000	−2.67261
$225$	−2.00000	−0.133333
$226$	−28.0000	−1.86253
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	−8.00000	−0.529813
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	−4.00000	−0.263752
$231$	−15.0000	−0.986928
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	−8.00000	−0.522976
$235$	−11.0000	−0.717561
$236$	0	0
$237$	−12.0000	−0.779484
$238$	−40.0000	−2.59281
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	4.00000	0.258199
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	4.00000	0.257130
$243$	16.0000	1.02640
$244$	−8.00000	−0.512148
$245$	−18.0000	−1.14998
$246$	−14.0000	−0.892607
$247$	8.00000	0.509028
$248$	0	0
$249$	−3.00000	−0.190117
$250$	2.00000	0.126491
$251$	22.0000	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$252$	20.0000	1.25988
$253$	−6.00000	−0.377217
$254$	22.0000	1.38040
$255$	4.00000	0.250490
$256$	16.0000	1.00000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	20.0000	1.24515
$259$	5.00000	0.310685
$260$	4.00000	0.248069
$261$	−4.00000	−0.247594
$262$	−24.0000	−1.48272
$263$	7.00000	0.431638	0.215819	−	0.976433i	$-0.430758\pi$
$263$	7.00000	0.431638	0.215819	+	0.976433i	$0.430758\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	−40.0000	−2.45256
$267$	−4.00000	−0.244796
$268$	32.0000	1.95471
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−10.0000	−0.608581
$271$	9.00000	0.546711	0.273356	−	0.961913i	$-0.411866\pi$
$271$	9.00000	0.546711	0.273356	+	0.961913i	$0.411866\pi$
$272$	16.0000	0.970143
$273$	10.0000	0.605228
$274$	12.0000	0.724947
$275$	3.00000	0.180907
$276$	−4.00000	−0.240772
$277$	32.0000	1.92269	0.961347	−	0.275340i	$-0.0887905\pi$
$277$	32.0000	1.92269	0.961347	+	0.275340i	$0.0887905\pi$
$278$	8.00000	0.479808
$279$	0	0
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	−22.0000	−1.31008
$283$	−28.0000	−1.66443	−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000	−1.66443	−0.832214	+	0.554455i	$0.812927\pi$
$284$	−30.0000	−1.78017
$285$	4.00000	0.236940
$286$	12.0000	0.709575
$287$	−35.0000	−2.06598
$288$	−16.0000	−0.942809
$289$	−1.00000	−0.0588235
$290$	4.00000	0.234888
$291$	8.00000	0.468968
$292$	22.0000	1.28745
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	−36.0000	−2.09956
$295$	0	0
$296$	0	0
$297$	−15.0000	−0.870388
$298$	−6.00000	−0.347571
$299$	4.00000	0.231326
$300$	2.00000	0.115470
$301$	50.0000	2.88195
$302$	16.0000	0.920697
$303$	−5.00000	−0.287242
$304$	16.0000	0.917663
$305$	4.00000	0.229039
$306$	−16.0000	−0.914659
$307$	19.0000	1.08439	0.542194	−	0.840254i	$-0.317594\pi$
$307$	19.0000	1.08439	0.542194	+	0.840254i	$0.317594\pi$
$308$	−30.0000	−1.70941
$309$	−10.0000	−0.568880
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−6.00000	−0.338600
$315$	−10.0000	−0.563436
$316$	−24.0000	−1.35011
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	6.00000	0.336463
$319$	6.00000	0.335936
$320$	8.00000	0.447214
$321$	−12.0000	−0.669775
$322$	−20.0000	−1.11456
$323$	16.0000	0.890264
$324$	2.00000	0.111111
$325$	−2.00000	−0.110940
$326$	4.00000	0.221540
$327$	−16.0000	−0.884802
$328$	0	0
$329$	−55.0000	−3.03225
$330$	6.00000	0.330289
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−6.00000	−0.329293
$333$	2.00000	0.109599
$334$	24.0000	1.31322
$335$	−16.0000	−0.874173
$336$	20.0000	1.09109
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	18.0000	0.979071
$339$	14.0000	0.760376
$340$	8.00000	0.433861
$341$	0	0
$342$	−16.0000	−0.865181
$343$	−55.0000	−2.96972
$344$	0	0
$345$	2.00000	0.107676
$346$	−10.0000	−0.537603
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	4.00000	0.214423
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	10.0000	0.534522
$351$	10.0000	0.533761
$352$	24.0000	1.27920
$353$	−32.0000	−1.70319	−0.851594	−	0.524202i	$-0.824364\pi$
$353$	−32.0000	−1.70319	−0.851594	+	0.524202i	$0.824364\pi$
$354$	0	0
$355$	15.0000	0.796117
$356$	−8.00000	−0.423999
$357$	20.0000	1.05851
$358$	20.0000	1.05703
$359$	9.00000	0.475002	0.237501	−	0.971387i	$-0.423672\pi$
$359$	9.00000	0.475002	0.237501	+	0.971387i	$0.423672\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−26.0000	−1.36653
$363$	−2.00000	−0.104973
$364$	20.0000	1.04828
$365$	−11.0000	−0.575766
$366$	8.00000	0.418167
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	8.00000	0.417029
$369$	−14.0000	−0.728811
$370$	−2.00000	−0.103975
$371$	15.0000	0.778761
$372$	0	0
$373$	−15.0000	−0.776671	−0.388335	−	0.921518i	$-0.626950\pi$
$373$	−15.0000	−0.776671	−0.388335	+	0.921518i	$0.626950\pi$
$374$	24.0000	1.24101
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−4.00000	−0.206010
$378$	−50.0000	−2.57172
$379$	7.00000	0.359566	0.179783	−	0.983706i	$-0.442460\pi$
$379$	7.00000	0.359566	0.179783	+	0.983706i	$0.442460\pi$
$380$	8.00000	0.410391
$381$	−11.0000	−0.563547
$382$	40.0000	2.04658
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	15.0000	0.764471
$386$	−28.0000	−1.42516
$387$	20.0000	1.01666
$388$	16.0000	0.812277
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	−4.00000	−0.202548
$391$	8.00000	0.404577
$392$	0	0
$393$	12.0000	0.605320
$394$	34.0000	1.71290
$395$	12.0000	0.603786
$396$	−12.0000	−0.603023
$397$	15.0000	0.752828	0.376414	−	0.926451i	$-0.377157\pi$
$397$	15.0000	0.752828	0.376414	+	0.926451i	$0.377157\pi$
$398$	28.0000	1.40351
$399$	20.0000	1.00125
$400$	−4.00000	−0.200000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−32.0000	−1.59601
$403$	0	0
$404$	−10.0000	−0.497519
$405$	−1.00000	−0.0496904
$406$	20.0000	0.992583
$407$	−3.00000	−0.148704
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	14.0000	0.691411
$411$	−6.00000	−0.295958
$412$	−20.0000	−0.985329
$413$	0	0
$414$	−8.00000	−0.393179
$415$	3.00000	0.147264
$416$	−16.0000	−0.784465
$417$	−4.00000	−0.195881
$418$	24.0000	1.17388
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	10.0000	0.487950
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	26.0000	1.26566
$423$	−22.0000	−1.06968
$424$	0	0
$425$	−4.00000	−0.194029
$426$	30.0000	1.45350
$427$	20.0000	0.967868
$428$	−24.0000	−1.16008
$429$	−6.00000	−0.289683
$430$	−20.0000	−0.964486
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	20.0000	0.962250
$433$	13.0000	0.624740	0.312370	−	0.949960i	$-0.398877\pi$
$433$	13.0000	0.624740	0.312370	+	0.949960i	$0.398877\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	−32.0000	−1.53252
$437$	8.00000	0.382692
$438$	−22.0000	−1.05120
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	−16.0000	−0.761042
$443$	21.0000	0.997740	0.498870	−	0.866677i	$-0.333748\pi$
$443$	21.0000	0.997740	0.498870	+	0.866677i	$0.333748\pi$
$444$	−2.00000	−0.0949158
$445$	4.00000	0.189618
$446$	58.0000	2.74638
$447$	3.00000	0.141895
$448$	40.0000	1.88982
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	4.00000	0.188562
$451$	21.0000	0.988851
$452$	28.0000	1.31701
$453$	−8.00000	−0.375873
$454$	16.0000	0.750917
$455$	−10.0000	−0.468807
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	−30.0000	−1.40181
$459$	20.0000	0.933520
$460$	4.00000	0.186501
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	30.0000	1.39573
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−44.0000	−2.03826
$467$	−34.0000	−1.57333	−0.786666	−	0.617379i	$-0.788195\pi$
$467$	−34.0000	−1.57333	−0.786666	+	0.617379i	$0.788195\pi$
$468$	8.00000	0.369800
$469$	−80.0000	−3.69406
$470$	22.0000	1.01478
$471$	3.00000	0.138233
$472$	0	0
$473$	−30.0000	−1.37940
$474$	24.0000	1.10236
$475$	−4.00000	−0.183533
$476$	40.0000	1.83340
$477$	6.00000	0.274721
$478$	−36.0000	−1.64660
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	−8.00000	−0.365148
$481$	2.00000	0.0911922
$482$	28.0000	1.27537
$483$	10.0000	0.455016
$484$	−4.00000	−0.181818
$485$	−8.00000	−0.363261
$486$	−32.0000	−1.45155
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	36.0000	1.62631
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	14.0000	0.631169
$493$	−8.00000	−0.360302
$494$	−16.0000	−0.719874
$495$	6.00000	0.269680
$496$	0	0
$497$	75.0000	3.36421
$498$	6.00000	0.268866
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	−2.00000	−0.0894427
$501$	−12.0000	−0.536120
$502$	−44.0000	−1.96382
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	5.00000	0.222497
$506$	12.0000	0.533465
$507$	−9.00000	−0.399704
$508$	−22.0000	−0.976092
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	−8.00000	−0.354246
$511$	−55.0000	−2.43306
$512$	−32.0000	−1.41421
$513$	20.0000	0.883022
$514$	−24.0000	−1.05859
$515$	10.0000	0.440653
$516$	−20.0000	−0.880451
$517$	33.0000	1.45134
$518$	−10.0000	−0.439375
$519$	5.00000	0.219476
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	8.00000	0.350150
$523$	−2.00000	−0.0874539	−0.0437269	−	0.999044i	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	24.0000	1.04844
$525$	−5.00000	−0.218218
$526$	−14.0000	−0.610429
$527$	0	0
$528$	−12.0000	−0.522233
$529$	−19.0000	−0.826087
$530$	−6.00000	−0.260623
$531$	0	0
$532$	40.0000	1.73422
$533$	−14.0000	−0.606407
$534$	8.00000	0.346194
$535$	12.0000	0.518805
$536$	0	0
$537$	−10.0000	−0.431532
$538$	−36.0000	−1.55207
$539$	54.0000	2.32594
$540$	10.0000	0.430331
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	−18.0000	−0.773166
$543$	13.0000	0.557883
$544$	−32.0000	−1.37199
$545$	16.0000	0.685365
$546$	−20.0000	−0.855921
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	−12.0000	−0.512615
$549$	8.00000	0.341432
$550$	−6.00000	−0.255841
$551$	−8.00000	−0.340811
$552$	0	0
$553$	60.0000	2.55146
$554$	−64.0000	−2.71910
$555$	1.00000	0.0424476
$556$	−8.00000	−0.339276
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	−20.0000	−0.845154
$561$	−12.0000	−0.506640
$562$	−16.0000	−0.674919
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	22.0000	0.926367
$565$	−14.0000	−0.588984
$566$	56.0000	2.35386
$567$	−5.00000	−0.209980
$568$	0	0
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	−8.00000	−0.335083
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	−12.0000	−0.501745
$573$	−20.0000	−0.835512
$574$	70.0000	2.92174
$575$	−2.00000	−0.0834058
$576$	16.0000	0.666667
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	2.00000	0.0831890
$579$	14.0000	0.581820
$580$	−4.00000	−0.166091
$581$	15.0000	0.622305
$582$	−16.0000	−0.663221
$583$	−9.00000	−0.372742
$584$	0	0
$585$	−4.00000	−0.165380
$586$	4.00000	0.165238
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	36.0000	1.48461
$589$	0	0
$590$	0	0
$591$	−17.0000	−0.699287
$592$	4.00000	0.164399
$593$	−17.0000	−0.698106	−0.349053	−	0.937103i	$-0.613497\pi$
$593$	−17.0000	−0.698106	−0.349053	+	0.937103i	$0.613497\pi$
$594$	30.0000	1.23091
$595$	−20.0000	−0.819920
$596$	6.00000	0.245770
$597$	−14.0000	−0.572982
$598$	−8.00000	−0.327144
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−100.000	−4.07570
$603$	−32.0000	−1.30314
$604$	−16.0000	−0.651031
$605$	2.00000	0.0813116
$606$	10.0000	0.406222
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	−32.0000	−1.29777
$609$	−10.0000	−0.405220
$610$	−8.00000	−0.323911
$611$	−22.0000	−0.890025
$612$	16.0000	0.646762
$613$	−37.0000	−1.49442	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−37.0000	−1.49442	−0.747208	+	0.664590i	$0.768606\pi$
$614$	−38.0000	−1.53356
$615$	−7.00000	−0.282267
$616$	0	0
$617$	5.00000	0.201292	0.100646	−	0.994922i	$-0.467909\pi$
$617$	5.00000	0.201292	0.100646	+	0.994922i	$0.467909\pi$
$618$	20.0000	0.804518
$619$	39.0000	1.56754	0.783771	−	0.621050i	$-0.213294\pi$
$619$	39.0000	1.56754	0.783771	+	0.621050i	$0.213294\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	−24.0000	−0.962312
$623$	20.0000	0.801283
$624$	8.00000	0.320256
$625$	1.00000	0.0400000
$626$	28.0000	1.11911
$627$	−12.0000	−0.479234
$628$	6.00000	0.239426
$629$	4.00000	0.159490
$630$	20.0000	0.796819
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	4.00000	0.158860
$635$	11.0000	0.436522
$636$	−6.00000	−0.237915
$637$	−36.0000	−1.42637
$638$	−12.0000	−0.475085
$639$	30.0000	1.18678
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	24.0000	0.947204
$643$	−22.0000	−0.867595	−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000	−0.867595	−0.433798	+	0.901010i	$0.642827\pi$
$644$	20.0000	0.788110
$645$	10.0000	0.393750
$646$	−32.0000	−1.25902
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	0	0
$650$	4.00000	0.156893
$651$	0	0
$652$	−4.00000	−0.156652
$653$	20.0000	0.782660	0.391330	−	0.920250i	$-0.372015\pi$
$653$	20.0000	0.782660	0.391330	+	0.920250i	$0.372015\pi$
$654$	32.0000	1.25130
$655$	−12.0000	−0.468879
$656$	−28.0000	−1.09322
$657$	−22.0000	−0.858302
$658$	110.000	4.28825
$659$	17.0000	0.662226	0.331113	−	0.943591i	$-0.392576\pi$
$659$	17.0000	0.662226	0.331113	+	0.943591i	$0.392576\pi$
$660$	−6.00000	−0.233550
$661$	36.0000	1.40024	0.700119	−	0.714026i	$-0.253130\pi$
$661$	36.0000	1.40024	0.700119	+	0.714026i	$0.253130\pi$
$662$	−20.0000	−0.777322
$663$	8.00000	0.310694
$664$	0	0
$665$	−20.0000	−0.775567
$666$	−4.00000	−0.154997
$667$	−4.00000	−0.154881
$668$	−24.0000	−0.928588
$669$	−29.0000	−1.12120
$670$	32.0000	1.23627
$671$	−12.0000	−0.463255
$672$	−40.0000	−1.54303
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	26.0000	1.00148
$675$	−5.00000	−0.192450
$676$	−18.0000	−0.692308
$677$	−31.0000	−1.19143	−0.595713	−	0.803197i	$-0.703131\pi$
$677$	−31.0000	−1.19143	−0.595713	+	0.803197i	$0.703131\pi$
$678$	−28.0000	−1.07533
$679$	−40.0000	−1.53506
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	−14.0000	−0.535695	−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000	−0.535695	−0.267848	+	0.963461i	$0.586312\pi$
$684$	16.0000	0.611775
$685$	6.00000	0.229248
$686$	110.000	4.19982
$687$	15.0000	0.572286
$688$	40.0000	1.52499
$689$	6.00000	0.228582
$690$	−4.00000	−0.152277
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	10.0000	0.380143
$693$	30.0000	1.13961
$694$	44.0000	1.67022
$695$	4.00000	0.151729
$696$	0	0
$697$	−28.0000	−1.06058
$698$	20.0000	0.757011
$699$	22.0000	0.832116
$700$	−10.0000	−0.377964
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	−20.0000	−0.754851
$703$	4.00000	0.150863
$704$	−24.0000	−0.904534
$705$	−11.0000	−0.414284
$706$	64.0000	2.40867
$707$	25.0000	0.940222
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	−30.0000	−1.12588
$711$	24.0000	0.900070
$712$	0	0
$713$	0	0
$714$	−40.0000	−1.49696
$715$	6.00000	0.224387
$716$	−20.0000	−0.747435
$717$	18.0000	0.672222
$718$	−18.0000	−0.671754
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	−8.00000	−0.298142
$721$	50.0000	1.86210
$722$	6.00000	0.223297
$723$	−14.0000	−0.520666
$724$	26.0000	0.966282
$725$	2.00000	0.0742781
$726$	4.00000	0.148454
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	22.0000	0.814257
$731$	40.0000	1.47945
$732$	−8.00000	−0.295689
$733$	11.0000	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$733$	11.0000	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$734$	−32.0000	−1.18114
$735$	−18.0000	−0.663940
$736$	−16.0000	−0.589768
$737$	48.0000	1.76810
$738$	28.0000	1.03069
$739$	−9.00000	−0.331070	−0.165535	−	0.986204i	$-0.552935\pi$
$739$	−9.00000	−0.331070	−0.165535	+	0.986204i	$0.552935\pi$
$740$	2.00000	0.0735215
$741$	8.00000	0.293887
$742$	−30.0000	−1.10133
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	30.0000	1.09838
$747$	6.00000	0.219529
$748$	−24.0000	−0.877527
$749$	60.0000	2.19235
$750$	2.00000	0.0730297
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	−44.0000	−1.60451
$753$	22.0000	0.801725
$754$	8.00000	0.291343
$755$	8.00000	0.291150
$756$	50.0000	1.81848
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	−14.0000	−0.508503
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−43.0000	−1.55875	−0.779374	−	0.626559i	$-0.784463\pi$
$761$	−43.0000	−1.55875	−0.779374	+	0.626559i	$0.784463\pi$
$762$	22.0000	0.796976
$763$	80.0000	2.89619
$764$	−40.0000	−1.44715
$765$	−8.00000	−0.289241
$766$	−48.0000	−1.73431
$767$	0	0
$768$	16.0000	0.577350
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	−30.0000	−1.08112
$771$	12.0000	0.432169
$772$	28.0000	1.00774
$773$	51.0000	1.83434	0.917171	−	0.398493i	$-0.130467\pi$
$773$	51.0000	1.83434	0.917171	+	0.398493i	$0.130467\pi$
$774$	−40.0000	−1.43777
$775$	0	0
$776$	0	0
$777$	5.00000	0.179374
$778$	−24.0000	−0.860442
$779$	−28.0000	−1.00320
$780$	4.00000	0.143223
$781$	−45.0000	−1.61023
$782$	−16.0000	−0.572159
$783$	−10.0000	−0.357371
$784$	−72.0000	−2.57143
$785$	−3.00000	−0.107075
$786$	−24.0000	−0.856052
$787$	47.0000	1.67537	0.837685	−	0.546154i	$-0.183909\pi$
$787$	47.0000	1.67537	0.837685	+	0.546154i	$0.183909\pi$
$788$	−34.0000	−1.21120
$789$	7.00000	0.249207
$790$	−24.0000	−0.853882
$791$	−70.0000	−2.48891
$792$	0	0
$793$	8.00000	0.284088
$794$	−30.0000	−1.06466
$795$	3.00000	0.106399
$796$	−28.0000	−0.992434
$797$	−28.0000	−0.991811	−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000	−0.991811	−0.495905	+	0.868377i	$0.665164\pi$
$798$	−40.0000	−1.41598
$799$	−44.0000	−1.55661
$800$	8.00000	0.282843
$801$	8.00000	0.282666
$802$	36.0000	1.27120
$803$	33.0000	1.16454
$804$	32.0000	1.12855
$805$	−10.0000	−0.352454
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	2.00000	0.0702728
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	−20.0000	−0.701862
$813$	9.00000	0.315644
$814$	6.00000	0.210300
$815$	2.00000	0.0700569
$816$	16.0000	0.560112
$817$	40.0000	1.39942
$818$	8.00000	0.279713
$819$	−20.0000	−0.698857
$820$	−14.0000	−0.488901
$821$	−39.0000	−1.36111	−0.680555	−	0.732697i	$-0.738261\pi$
$821$	−39.0000	−1.36111	−0.680555	+	0.732697i	$0.738261\pi$
$822$	12.0000	0.418548
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	3.00000	0.104447
$826$	0	0
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	8.00000	0.278019
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	−6.00000	−0.208263
$831$	32.0000	1.11007
$832$	16.0000	0.554700
$833$	−72.0000	−2.49465
$834$	8.00000	0.277017
$835$	12.0000	0.415277
$836$	−24.0000	−0.830057
$837$	0	0
$838$	18.0000	0.621800
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	56.0000	1.92989
$843$	8.00000	0.275535
$844$	−26.0000	−0.894957
$845$	9.00000	0.309609
$846$	44.0000	1.51275
$847$	10.0000	0.343604
$848$	12.0000	0.412082
$849$	−28.0000	−0.960958
$850$	8.00000	0.274398
$851$	2.00000	0.0685591
$852$	−30.0000	−1.02778
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	−40.0000	−1.36877
$855$	−8.00000	−0.273594
$856$	0	0
$857$	8.00000	0.273275	0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000	0.273275	0.136637	+	0.990621i	$0.456370\pi$
$858$	12.0000	0.409673
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	20.0000	0.681994
$861$	−35.0000	−1.19280
$862$	−36.0000	−1.22616
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−40.0000	−1.36083
$865$	−5.00000	−0.170005
$866$	−26.0000	−0.883516
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−36.0000	−1.22122
$870$	4.00000	0.135613
$871$	−32.0000	−1.08428
$872$	0	0
$873$	−16.0000	−0.541518
$874$	−16.0000	−0.541208
$875$	5.00000	0.169031
$876$	22.0000	0.743311
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	−8.00000	−0.269987
$879$	−2.00000	−0.0674583
$880$	12.0000	0.404520
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	72.0000	2.42437
$883$	−32.0000	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$884$	16.0000	0.538138
$885$	0	0
$886$	−42.0000	−1.41102
$887$	37.0000	1.24234	0.621169	−	0.783676i	$-0.286658\pi$
$887$	37.0000	1.24234	0.621169	+	0.783676i	$0.286658\pi$
$888$	0	0
$889$	55.0000	1.84464
$890$	−8.00000	−0.268161
$891$	3.00000	0.100504
$892$	−58.0000	−1.94198
$893$	−44.0000	−1.47240
$894$	−6.00000	−0.200670
$895$	10.0000	0.334263
$896$	0	0
$897$	4.00000	0.133556
$898$	−40.0000	−1.33482
$899$	0	0
$900$	−4.00000	−0.133333
$901$	12.0000	0.399778
$902$	−42.0000	−1.39845
$903$	50.0000	1.66390
$904$	0	0
$905$	−13.0000	−0.432135
$906$	16.0000	0.531564
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	−16.0000	−0.530979
$909$	10.0000	0.331679
$910$	20.0000	0.662994
$911$	38.0000	1.25900	0.629498	−	0.777002i	$-0.283261\pi$
$911$	38.0000	1.25900	0.629498	+	0.777002i	$0.283261\pi$
$912$	16.0000	0.529813
$913$	−9.00000	−0.297857
$914$	36.0000	1.19077
$915$	4.00000	0.132236
$916$	30.0000	0.991228
$917$	−60.0000	−1.98137
$918$	−40.0000	−1.32020
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	19.0000	0.626071
$922$	−36.0000	−1.18560
$923$	30.0000	0.987462
$924$	−30.0000	−0.986928
$925$	−1.00000	−0.0328798
$926$	−68.0000	−2.23462
$927$	20.0000	0.656886
$928$	16.0000	0.525226
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−72.0000	−2.35970
$932$	44.0000	1.44127
$933$	12.0000	0.392862
$934$	68.0000	2.22503
$935$	12.0000	0.392442
$936$	0	0
$937$	9.00000	0.294017	0.147009	−	0.989135i	$-0.453036\pi$
$937$	9.00000	0.294017	0.147009	+	0.989135i	$0.453036\pi$
$938$	160.000	5.22419
$939$	−14.0000	−0.456873
$940$	−22.0000	−0.717561
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	−6.00000	−0.195491
$943$	−14.0000	−0.455903
$944$	0	0
$945$	−25.0000	−0.813250
$946$	60.0000	1.95077
$947$	−40.0000	−1.29983	−0.649913	−	0.760009i	$-0.725195\pi$
$947$	−40.0000	−1.29983	−0.649913	+	0.760009i	$0.725195\pi$
$948$	−24.0000	−0.779484
$949$	−22.0000	−0.714150
$950$	8.00000	0.259554
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	−12.0000	−0.388514
$955$	20.0000	0.647185
$956$	36.0000	1.16432
$957$	6.00000	0.193952
$958$	−36.0000	−1.16311
$959$	30.0000	0.968751
$960$	8.00000	0.258199
$961$	−31.0000	−1.00000
$962$	−4.00000	−0.128965
$963$	24.0000	0.773389
$964$	−28.0000	−0.901819
$965$	−14.0000	−0.450676
$966$	−20.0000	−0.643489
$967$	10.0000	0.321578	0.160789	−	0.986989i	$-0.448596\pi$
$967$	10.0000	0.321578	0.160789	+	0.986989i	$0.448596\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	16.0000	0.513729
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	32.0000	1.02640
$973$	20.0000	0.641171
$974$	24.0000	0.769010
$975$	−2.00000	−0.0640513
$976$	16.0000	0.512148
$977$	−20.0000	−0.639857	−0.319928	−	0.947442i	$-0.603659\pi$
$977$	−20.0000	−0.639857	−0.319928	+	0.947442i	$0.603659\pi$
$978$	4.00000	0.127906
$979$	−12.0000	−0.383522
$980$	−36.0000	−1.14998
$981$	32.0000	1.02168
$982$	8.00000	0.255290
$983$	−19.0000	−0.606006	−0.303003	−	0.952990i	$-0.597989\pi$
$983$	−19.0000	−0.606006	−0.303003	+	0.952990i	$0.597989\pi$
$984$	0	0
$985$	17.0000	0.541665
$986$	16.0000	0.509544
$987$	−55.0000	−1.75067
$988$	16.0000	0.509028
$989$	20.0000	0.635963
$990$	−12.0000	−0.381385
$991$	−22.0000	−0.698853	−0.349427	−	0.936964i	$-0.613624\pi$
$991$	−22.0000	−0.698853	−0.349427	+	0.936964i	$0.613624\pi$
$992$	0	0
$993$	10.0000	0.317340
$994$	−150.000	−4.75771
$995$	14.0000	0.443830
$996$	−6.00000	−0.190117
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	32.0000	1.01294
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	185.2.a.a.1.1	✓	1
3.2	odd	2		1665.2.a.f.1.1		1
4.3	odd	2		2960.2.a.e.1.1		1
5.2	odd	4		925.2.b.a.149.1		2
5.3	odd	4		925.2.b.a.149.2		2
5.4	even	2		925.2.a.d.1.1		1
7.6	odd	2		9065.2.a.b.1.1		1
15.14	odd	2		8325.2.a.g.1.1		1
37.36	even	2		6845.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.a.1.1	✓	1	1.1	even	1	trivial
925.2.a.d.1.1		1	5.4	even	2
925.2.b.a.149.1		2	5.2	odd	4
925.2.b.a.149.2		2	5.3	odd	4
1665.2.a.f.1.1		1	3.2	odd	2
2960.2.a.e.1.1		1	4.3	odd	2
6845.2.a.e.1.1		1	37.36	even	2
8325.2.a.g.1.1		1	15.14	odd	2
9065.2.a.b.1.1		1	7.6	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:	\( N \)	\(=\)	\( 185 = 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	185.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more