LMFDB - Embedded newform 7406.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7406,2,Mod(1,7406)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7406, 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("7406.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7406 = 2 \cdot 7 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7406.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:	$59.1372077370$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7406.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -6.00000 q^{11} +2.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} -2.00000 q^{21} -6.00000 q^{22} +2.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} -10.0000 q^{29} +4.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} -12.0000 q^{33} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} -4.00000 q^{38} -8.00000 q^{39} +2.00000 q^{40} -2.00000 q^{41} -2.00000 q^{42} -6.00000 q^{43} -6.00000 q^{44} +2.00000 q^{45} +12.0000 q^{47} +2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.00000 q^{51} -4.00000 q^{52} -12.0000 q^{53} -4.00000 q^{54} -12.0000 q^{55} -1.00000 q^{56} -8.00000 q^{57} -10.0000 q^{58} -6.00000 q^{59} +4.00000 q^{60} +6.00000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} -12.0000 q^{66} +2.00000 q^{67} +2.00000 q^{68} -2.00000 q^{70} +16.0000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +8.00000 q^{74} -2.00000 q^{75} -4.00000 q^{76} +6.00000 q^{77} -8.00000 q^{78} +2.00000 q^{80} -11.0000 q^{81} -2.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} +4.00000 q^{85} -6.00000 q^{86} -20.0000 q^{87} -6.00000 q^{88} +6.00000 q^{89} +2.00000 q^{90} +4.00000 q^{91} -16.0000 q^{93} +12.0000 q^{94} -8.00000 q^{95} +2.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	2.00000	0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−1.00000	−0.267261
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	−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
$21$	−2.00000	−0.436436
$22$	−6.00000	−1.27920
$23$	0	0
$23$	0	0
$24$	2.00000	0.408248
$25$	−1.00000	−0.200000
$26$	−4.00000	−0.784465
$27$	−4.00000	−0.769800
$28$	−1.00000	−0.188982
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	4.00000	0.730297
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	−12.0000	−2.08893
$34$	2.00000	0.342997
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−4.00000	−0.648886
$39$	−8.00000	−1.28103
$40$	2.00000	0.316228
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−2.00000	−0.308607
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−6.00000	−0.904534
$45$	2.00000	0.298142
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	2.00000	0.288675
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	4.00000	0.560112
$52$	−4.00000	−0.554700
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	−4.00000	−0.544331
$55$	−12.0000	−1.61808
$56$	−1.00000	−0.133631
$57$	−8.00000	−1.05963
$58$	−10.0000	−1.31306
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	4.00000	0.516398
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−8.00000	−1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	−12.0000	−1.47710
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	−2.00000	−0.239046
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	1.00000	0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	8.00000	0.929981
$75$	−2.00000	−0.230940
$76$	−4.00000	−0.458831
$77$	6.00000	0.683763
$78$	−8.00000	−0.905822
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	2.00000	0.223607
$81$	−11.0000	−1.22222
$82$	−2.00000	−0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−2.00000	−0.218218
$85$	4.00000	0.433861
$86$	−6.00000	−0.646997
$87$	−20.0000	−2.14423
$88$	−6.00000	−0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	2.00000	0.210819
$91$	4.00000	0.419314
$92$	0	0
$93$	−16.0000	−1.65912
$94$	12.0000	1.23771
$95$	−8.00000	−0.820783
$96$	2.00000	0.204124
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	1.00000	0.101015
$99$	−6.00000	−0.603023
$100$	−1.00000	−0.100000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	4.00000	0.396059
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−4.00000	−0.392232
$105$	−4.00000	−0.390360
$106$	−12.0000	−1.16554
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−4.00000	−0.384900
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	−12.0000	−1.14416
$111$	16.0000	1.51865
$112$	−1.00000	−0.0944911
$113$	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
$115$	0	0
$116$	−10.0000	−0.928477
$117$	−4.00000	−0.369800
$118$	−6.00000	−0.552345
$119$	−2.00000	−0.183340
$120$	4.00000	0.365148
$121$	25.0000	2.27273
$122$	6.00000	0.543214
$123$	−4.00000	−0.360668
$124$	−8.00000	−0.718421
$125$	−12.0000	−1.07331
$126$	−1.00000	−0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−12.0000	−1.05654
$130$	−8.00000	−0.701646
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	−12.0000	−1.04447
$133$	4.00000	0.346844
$134$	2.00000	0.172774
$135$	−8.00000	−0.688530
$136$	2.00000	0.171499
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	−2.00000	−0.169031
$141$	24.0000	2.02116
$142$	16.0000	1.34269
$143$	24.0000	2.00698
$144$	1.00000	0.0833333
$145$	−20.0000	−1.66091
$146$	2.00000	0.165521
$147$	2.00000	0.164957
$148$	8.00000	0.657596
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	−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	$0.605538\pi$
$152$	−4.00000	−0.324443
$153$	2.00000	0.161690
$154$	6.00000	0.483494
$155$	−16.0000	−1.28515
$156$	−8.00000	−0.640513
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	0	0
$159$	−24.0000	−1.90332
$160$	2.00000	0.158114
$161$	0	0
$162$	−11.0000	−0.864242
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−2.00000	−0.156174
$165$	−24.0000	−1.86840
$166$	−4.00000	−0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−2.00000	−0.154303
$169$	3.00000	0.230769
$170$	4.00000	0.306786
$171$	−4.00000	−0.305888
$172$	−6.00000	−0.457496
$173$	24.0000	1.82469	0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000	1.82469	0.912343	+	0.409426i	$0.134271\pi$
$174$	−20.0000	−1.51620
$175$	1.00000	0.0755929
$176$	−6.00000	−0.452267
$177$	−12.0000	−0.901975
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	2.00000	0.149071
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	4.00000	0.296500
$183$	12.0000	0.887066
$184$	0	0
$185$	16.0000	1.17634
$186$	−16.0000	−1.17318
$187$	−12.0000	−0.877527
$188$	12.0000	0.875190
$189$	4.00000	0.290957
$190$	−8.00000	−0.580381
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	2.00000	0.144338
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−2.00000	−0.143592
$195$	−16.0000	−1.14578
$196$	1.00000	0.0714286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−6.00000	−0.426401
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−1.00000	−0.0707107
$201$	4.00000	0.282138
$202$	8.00000	0.562878
$203$	10.0000	0.701862
$204$	4.00000	0.280056
$205$	−4.00000	−0.279372
$206$	8.00000	0.557386
$207$	0	0
$208$	−4.00000	−0.277350
$209$	24.0000	1.66011
$210$	−4.00000	−0.276026
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−12.0000	−0.824163
$213$	32.0000	2.19260
$214$	−6.00000	−0.410152
$215$	−12.0000	−0.818393
$216$	−4.00000	−0.272166
$217$	8.00000	0.543075
$218$	−4.00000	−0.270914
$219$	4.00000	0.270295
$220$	−12.0000	−0.809040
$221$	−8.00000	−0.538138
$222$	16.0000	1.07385
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−1.00000	−0.0668153
$225$	−1.00000	−0.0666667
$226$	14.0000	0.931266
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−8.00000	−0.529813
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	−10.0000	−0.656532
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−4.00000	−0.261488
$235$	24.0000	1.56559
$236$	−6.00000	−0.390567
$237$	0	0
$238$	−2.00000	−0.129641
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	4.00000	0.258199
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	25.0000	1.60706
$243$	−10.0000	−0.641500
$244$	6.00000	0.384111
$245$	2.00000	0.127775
$246$	−4.00000	−0.255031
$247$	16.0000	1.01806
$248$	−8.00000	−0.508001
$249$	−8.00000	−0.506979
$250$	−12.0000	−0.758947
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	8.00000	0.501965
$255$	8.00000	0.500979
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−12.0000	−0.747087
$259$	−8.00000	−0.497096
$260$	−8.00000	−0.496139
$261$	−10.0000	−0.618984
$262$	10.0000	0.617802
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	−12.0000	−0.738549
$265$	−24.0000	−1.47431
$266$	4.00000	0.245256
$267$	12.0000	0.734388
$268$	2.00000	0.122169
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	−8.00000	−0.486864
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	2.00000	0.121268
$273$	8.00000	0.484182
$274$	−6.00000	−0.362473
$275$	6.00000	0.361814
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	−10.0000	−0.599760
$279$	−8.00000	−0.478947
$280$	−2.00000	−0.119523
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	24.0000	1.42918
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	16.0000	0.949425
$285$	−16.0000	−0.947758
$286$	24.0000	1.41915
$287$	2.00000	0.118056
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−20.0000	−1.17444
$291$	−4.00000	−0.234484
$292$	2.00000	0.117041
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	2.00000	0.116642
$295$	−12.0000	−0.698667
$296$	8.00000	0.464991
$297$	24.0000	1.39262
$298$	4.00000	0.231714
$299$	0	0
$300$	−2.00000	−0.115470
$301$	6.00000	0.345834
$302$	−8.00000	−0.460348
$303$	16.0000	0.919176
$304$	−4.00000	−0.229416
$305$	12.0000	0.687118
$306$	2.00000	0.114332
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	6.00000	0.341882
$309$	16.0000	0.910208
$310$	−16.0000	−0.908739
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	−8.00000	−0.452911
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	6.00000	0.338600
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	−24.0000	−1.34585
$319$	60.0000	3.35936
$320$	2.00000	0.111803
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−8.00000	−0.445132
$324$	−11.0000	−0.611111
$325$	4.00000	0.221880
$326$	20.0000	1.10770
$327$	−8.00000	−0.442401
$328$	−2.00000	−0.110432
$329$	−12.0000	−0.661581
$330$	−24.0000	−1.32116
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−4.00000	−0.219529
$333$	8.00000	0.438397
$334$	−12.0000	−0.656611
$335$	4.00000	0.218543
$336$	−2.00000	−0.109109
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	3.00000	0.163178
$339$	28.0000	1.52075
$340$	4.00000	0.216930
$341$	48.0000	2.59935
$342$	−4.00000	−0.216295
$343$	−1.00000	−0.0539949
$344$	−6.00000	−0.323498
$345$	0	0
$346$	24.0000	1.29025
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−20.0000	−1.07211
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	1.00000	0.0534522
$351$	16.0000	0.854017
$352$	−6.00000	−0.319801
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−12.0000	−0.637793
$355$	32.0000	1.69838
$356$	6.00000	0.317999
$357$	−4.00000	−0.211702
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	2.00000	0.105409
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	50.0000	2.62432
$364$	4.00000	0.209657
$365$	4.00000	0.209370
$366$	12.0000	0.627250
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	16.0000	0.831800
$371$	12.0000	0.623009
$372$	−16.0000	−0.829561
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	−12.0000	−0.620505
$375$	−24.0000	−1.23935
$376$	12.0000	0.618853
$377$	40.0000	2.06010
$378$	4.00000	0.205738
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	−8.00000	−0.410391
$381$	16.0000	0.819705
$382$	−16.0000	−0.818631
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	2.00000	0.102062
$385$	12.0000	0.611577
$386$	−2.00000	−0.101797
$387$	−6.00000	−0.304997
$388$	−2.00000	−0.101535
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	−16.0000	−0.810191
$391$	0	0
$392$	1.00000	0.0505076
$393$	20.0000	1.00887
$394$	2.00000	0.100759
$395$	0	0
$396$	−6.00000	−0.301511
$397$	28.0000	1.40528	0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000	1.40528	0.702640	+	0.711546i	$0.252005\pi$
$398$	16.0000	0.802008
$399$	8.00000	0.400501
$400$	−1.00000	−0.0500000
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	4.00000	0.199502
$403$	32.0000	1.59403
$404$	8.00000	0.398015
$405$	−22.0000	−1.09319
$406$	10.0000	0.496292
$407$	−48.0000	−2.37927
$408$	4.00000	0.198030
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	−4.00000	−0.197546
$411$	−12.0000	−0.591916
$412$	8.00000	0.394132
$413$	6.00000	0.295241
$414$	0	0
$415$	−8.00000	−0.392705
$416$	−4.00000	−0.196116
$417$	−20.0000	−0.979404
$418$	24.0000	1.17388
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	−4.00000	−0.195180
$421$	12.0000	0.584844	0.292422	−	0.956289i	$-0.405539\pi$
$421$	12.0000	0.584844	0.292422	+	0.956289i	$0.405539\pi$
$422$	−12.0000	−0.584151
$423$	12.0000	0.583460
$424$	−12.0000	−0.582772
$425$	−2.00000	−0.0970143
$426$	32.0000	1.55041
$427$	−6.00000	−0.290360
$428$	−6.00000	−0.290021
$429$	48.0000	2.31746
$430$	−12.0000	−0.578691
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−4.00000	−0.192450
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	8.00000	0.384012
$435$	−40.0000	−1.91785
$436$	−4.00000	−0.191565
$437$	0	0
$438$	4.00000	0.191127
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−12.0000	−0.572078
$441$	1.00000	0.0476190
$442$	−8.00000	−0.380521
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	16.0000	0.759326
$445$	12.0000	0.568855
$446$	4.00000	0.189405
$447$	8.00000	0.378387
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−1.00000	−0.0471405
$451$	12.0000	0.565058
$452$	14.0000	0.658505
$453$	−16.0000	−0.751746
$454$	−24.0000	−1.12638
$455$	8.00000	0.375046
$456$	−8.00000	−0.374634
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	−22.0000	−1.02799
$459$	−8.00000	−0.373408
$460$	0	0
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	12.0000	0.558291
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−10.0000	−0.464238
$465$	−32.0000	−1.48396
$466$	−6.00000	−0.277945
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	−4.00000	−0.184900
$469$	−2.00000	−0.0923514
$470$	24.0000	1.10704
$471$	12.0000	0.552931
$472$	−6.00000	−0.276172
$473$	36.0000	1.65528
$474$	0	0
$475$	4.00000	0.183533
$476$	−2.00000	−0.0916698
$477$	−12.0000	−0.549442
$478$	−24.0000	−1.09773
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	4.00000	0.182574
$481$	−32.0000	−1.45907
$482$	6.00000	0.273293
$483$	0	0
$484$	25.0000	1.13636
$485$	−4.00000	−0.181631
$486$	−10.0000	−0.453609
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	6.00000	0.271607
$489$	40.0000	1.80886
$490$	2.00000	0.0903508
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	−4.00000	−0.180334
$493$	−20.0000	−0.900755
$494$	16.0000	0.719874
$495$	−12.0000	−0.539360
$496$	−8.00000	−0.359211
$497$	−16.0000	−0.717698
$498$	−8.00000	−0.358489
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	−12.0000	−0.536656
$501$	−24.0000	−1.07224
$502$	0	0
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	−1.00000	−0.0445435
$505$	16.0000	0.711991
$506$	0	0
$507$	6.00000	0.266469
$508$	8.00000	0.354943
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	8.00000	0.354246
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	16.0000	0.706417
$514$	2.00000	0.0882162
$515$	16.0000	0.705044
$516$	−12.0000	−0.528271
$517$	−72.0000	−3.16656
$518$	−8.00000	−0.351500
$519$	48.0000	2.10697
$520$	−8.00000	−0.350823
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−10.0000	−0.437688
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	10.0000	0.436852
$525$	2.00000	0.0872872
$526$	−28.0000	−1.22086
$527$	−16.0000	−0.696971
$528$	−12.0000	−0.522233
$529$	0	0
$530$	−24.0000	−1.04249
$531$	−6.00000	−0.260378
$532$	4.00000	0.173422
$533$	8.00000	0.346518
$534$	12.0000	0.519291
$535$	−12.0000	−0.518805
$536$	2.00000	0.0863868
$537$	0	0
$538$	−24.0000	−1.03471
$539$	−6.00000	−0.258438
$540$	−8.00000	−0.344265
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	4.00000	0.171656
$544$	2.00000	0.0857493
$545$	−8.00000	−0.342682
$546$	8.00000	0.342368
$547$	−32.0000	−1.36822	−0.684111	−	0.729378i	$-0.739809\pi$
$547$	−32.0000	−1.36822	−0.684111	+	0.729378i	$0.739809\pi$
$548$	−6.00000	−0.256307
$549$	6.00000	0.256074
$550$	6.00000	0.255841
$551$	40.0000	1.70406
$552$	0	0
$553$	0	0
$554$	−14.0000	−0.594803
$555$	32.0000	1.35832
$556$	−10.0000	−0.424094
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	−8.00000	−0.338667
$559$	24.0000	1.01509
$560$	−2.00000	−0.0845154
$561$	−24.0000	−1.01328
$562$	−2.00000	−0.0843649
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	24.0000	1.01058
$565$	28.0000	1.17797
$566$	−24.0000	−1.00880
$567$	11.0000	0.461957
$568$	16.0000	0.671345
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−16.0000	−0.670166
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	24.0000	1.00349
$573$	−32.0000	−1.33682
$574$	2.00000	0.0834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	−13.0000	−0.540729
$579$	−4.00000	−0.166234
$580$	−20.0000	−0.830455
$581$	4.00000	0.165948
$582$	−4.00000	−0.165805
$583$	72.0000	2.98194
$584$	2.00000	0.0827606
$585$	−8.00000	−0.330759
$586$	26.0000	1.07405
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	2.00000	0.0824786
$589$	32.0000	1.31854
$590$	−12.0000	−0.494032
$591$	4.00000	0.164538
$592$	8.00000	0.328798
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	24.0000	0.984732
$595$	−4.00000	−0.163984
$596$	4.00000	0.163846
$597$	32.0000	1.30967
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	−2.00000	−0.0816497
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	6.00000	0.244542
$603$	2.00000	0.0814463
$604$	−8.00000	−0.325515
$605$	50.0000	2.03279
$606$	16.0000	0.649956
$607$	36.0000	1.46119	0.730597	−	0.682808i	$-0.239242\pi$
$607$	36.0000	1.46119	0.730597	+	0.682808i	$0.239242\pi$
$608$	−4.00000	−0.162221
$609$	20.0000	0.810441
$610$	12.0000	0.485866
$611$	−48.0000	−1.94187
$612$	2.00000	0.0808452
$613$	−36.0000	−1.45403	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	−36.0000	−1.45403	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−2.00000	−0.0807134
$615$	−8.00000	−0.322591
$616$	6.00000	0.241747
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	16.0000	0.643614
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	−16.0000	−0.642575
$621$	0	0
$622$	28.0000	1.12270
$623$	−6.00000	−0.240385
$624$	−8.00000	−0.320256
$625$	−19.0000	−0.760000
$626$	−2.00000	−0.0799361
$627$	48.0000	1.91694
$628$	6.00000	0.239426
$629$	16.0000	0.637962
$630$	−2.00000	−0.0796819
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	−22.0000	−0.873732
$635$	16.0000	0.634941
$636$	−24.0000	−0.951662
$637$	−4.00000	−0.158486
$638$	60.0000	2.37542
$639$	16.0000	0.632950
$640$	2.00000	0.0790569
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−12.0000	−0.473602
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	−8.00000	−0.314756
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	−11.0000	−0.432121
$649$	36.0000	1.41312
$650$	4.00000	0.156893
$651$	16.0000	0.627089
$652$	20.0000	0.783260
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	−8.00000	−0.312825
$655$	20.0000	0.781465
$656$	−2.00000	−0.0780869
$657$	2.00000	0.0780274
$658$	−12.0000	−0.467809
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	−24.0000	−0.934199
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	−8.00000	−0.310929
$663$	−16.0000	−0.621389
$664$	−4.00000	−0.155230
$665$	8.00000	0.310227
$666$	8.00000	0.309994
$667$	0	0
$668$	−12.0000	−0.464294
$669$	8.00000	0.309298
$670$	4.00000	0.154533
$671$	−36.0000	−1.38976
$672$	−2.00000	−0.0771517
$673$	−50.0000	−1.92736	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000	−1.92736	−0.963679	+	0.267063i	$0.913947\pi$
$674$	6.00000	0.231111
$675$	4.00000	0.153960
$676$	3.00000	0.115385
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	28.0000	1.07533
$679$	2.00000	0.0767530
$680$	4.00000	0.153393
$681$	−48.0000	−1.83936
$682$	48.0000	1.83801
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−4.00000	−0.152944
$685$	−12.0000	−0.458496
$686$	−1.00000	−0.0381802
$687$	−44.0000	−1.67870
$688$	−6.00000	−0.228748
$689$	48.0000	1.82865
$690$	0	0
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	24.0000	0.912343
$693$	6.00000	0.227921
$694$	−12.0000	−0.455514
$695$	−20.0000	−0.758643
$696$	−20.0000	−0.758098
$697$	−4.00000	−0.151511
$698$	−8.00000	−0.302804
$699$	−12.0000	−0.453882
$700$	1.00000	0.0377964
$701$	4.00000	0.151078	0.0755390	−	0.997143i	$-0.475932\pi$
$701$	4.00000	0.151078	0.0755390	+	0.997143i	$0.475932\pi$
$702$	16.0000	0.603881
$703$	−32.0000	−1.20690
$704$	−6.00000	−0.226134
$705$	48.0000	1.80778
$706$	−14.0000	−0.526897
$707$	−8.00000	−0.300871
$708$	−12.0000	−0.450988
$709$	−32.0000	−1.20179	−0.600893	−	0.799330i	$-0.705188\pi$
$709$	−32.0000	−1.20179	−0.600893	+	0.799330i	$0.705188\pi$
$710$	32.0000	1.20094
$711$	0	0
$712$	6.00000	0.224860
$713$	0	0
$714$	−4.00000	−0.149696
$715$	48.0000	1.79510
$716$	0	0
$717$	−48.0000	−1.79259
$718$	12.0000	0.447836
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	2.00000	0.0745356
$721$	−8.00000	−0.297936
$722$	−3.00000	−0.111648
$723$	12.0000	0.446285
$724$	2.00000	0.0743294
$725$	10.0000	0.371391
$726$	50.0000	1.85567
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	4.00000	0.148250
$729$	13.0000	0.481481
$730$	4.00000	0.148047
$731$	−12.0000	−0.443836
$732$	12.0000	0.443533
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	8.00000	0.295285
$735$	4.00000	0.147542
$736$	0	0
$737$	−12.0000	−0.442026
$738$	−2.00000	−0.0736210
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	16.0000	0.588172
$741$	32.0000	1.17555
$742$	12.0000	0.440534
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−16.0000	−0.586588
$745$	8.00000	0.293097
$746$	4.00000	0.146450
$747$	−4.00000	−0.146352
$748$	−12.0000	−0.438763
$749$	6.00000	0.219235
$750$	−24.0000	−0.876356
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	40.0000	1.45671
$755$	−16.0000	−0.582300
$756$	4.00000	0.145479
$757$	28.0000	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$758$	−26.0000	−0.944363
$759$	0	0
$760$	−8.00000	−0.290191
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	16.0000	0.579619
$763$	4.00000	0.144810
$764$	−16.0000	−0.578860
$765$	4.00000	0.144620
$766$	−8.00000	−0.289052
$767$	24.0000	0.866590
$768$	2.00000	0.0721688
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	12.0000	0.432450
$771$	4.00000	0.144056
$772$	−2.00000	−0.0719816
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	−6.00000	−0.215666
$775$	8.00000	0.287368
$776$	−2.00000	−0.0717958
$777$	−16.0000	−0.573997
$778$	4.00000	0.143407
$779$	8.00000	0.286630
$780$	−16.0000	−0.572892
$781$	−96.0000	−3.43515
$782$	0	0
$783$	40.0000	1.42948
$784$	1.00000	0.0357143
$785$	12.0000	0.428298
$786$	20.0000	0.713376
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	2.00000	0.0712470
$789$	−56.0000	−1.99365
$790$	0	0
$791$	−14.0000	−0.497783
$792$	−6.00000	−0.213201
$793$	−24.0000	−0.852265
$794$	28.0000	0.993683
$795$	−48.0000	−1.70238
$796$	16.0000	0.567105
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	8.00000	0.283197
$799$	24.0000	0.849059
$800$	−1.00000	−0.0353553
$801$	6.00000	0.212000
$802$	26.0000	0.918092
$803$	−12.0000	−0.423471
$804$	4.00000	0.141069
$805$	0	0
$806$	32.0000	1.12715
$807$	−48.0000	−1.68968
$808$	8.00000	0.281439
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	−22.0000	−0.773001
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	10.0000	0.350931
$813$	0	0
$814$	−48.0000	−1.68240
$815$	40.0000	1.40114
$816$	4.00000	0.140028
$817$	24.0000	0.839654
$818$	22.0000	0.769212
$819$	4.00000	0.139771
$820$	−4.00000	−0.139686
$821$	14.0000	0.488603	0.244302	−	0.969699i	$-0.421441\pi$
$821$	14.0000	0.488603	0.244302	+	0.969699i	$0.421441\pi$
$822$	−12.0000	−0.418548
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	8.00000	0.278693
$825$	12.0000	0.417786
$826$	6.00000	0.208767
$827$	10.0000	0.347734	0.173867	−	0.984769i	$-0.444374\pi$
$827$	10.0000	0.347734	0.173867	+	0.984769i	$0.444374\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	−8.00000	−0.277684
$831$	−28.0000	−0.971309
$832$	−4.00000	−0.138675
$833$	2.00000	0.0692959
$834$	−20.0000	−0.692543
$835$	−24.0000	−0.830554
$836$	24.0000	0.830057
$837$	32.0000	1.10608
$838$	0	0
$839$	32.0000	1.10476	0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000	1.10476	0.552381	+	0.833592i	$0.313719\pi$
$840$	−4.00000	−0.138013
$841$	71.0000	2.44828
$842$	12.0000	0.413547
$843$	−4.00000	−0.137767
$844$	−12.0000	−0.413057
$845$	6.00000	0.206406
$846$	12.0000	0.412568
$847$	−25.0000	−0.859010
$848$	−12.0000	−0.412082
$849$	−48.0000	−1.64736
$850$	−2.00000	−0.0685994
$851$	0	0
$852$	32.0000	1.09630
$853$	−12.0000	−0.410872	−0.205436	−	0.978671i	$-0.565861\pi$
$853$	−12.0000	−0.410872	−0.205436	+	0.978671i	$0.565861\pi$
$854$	−6.00000	−0.205316
$855$	−8.00000	−0.273594
$856$	−6.00000	−0.205076
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	48.0000	1.63869
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−12.0000	−0.409197
$861$	4.00000	0.136320
$862$	−12.0000	−0.408722
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−4.00000	−0.136083
$865$	48.0000	1.63205
$866$	34.0000	1.15537
$867$	−26.0000	−0.883006
$868$	8.00000	0.271538
$869$	0	0
$870$	−40.0000	−1.35613
$871$	−8.00000	−0.271070
$872$	−4.00000	−0.135457
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	12.0000	0.405674
$876$	4.00000	0.135147
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	8.00000	0.269987
$879$	52.0000	1.75392
$880$	−12.0000	−0.404520
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	1.00000	0.0336718
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	−8.00000	−0.269069
$885$	−24.0000	−0.806751
$886$	16.0000	0.537531
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	16.0000	0.536925
$889$	−8.00000	−0.268311
$890$	12.0000	0.402241
$891$	66.0000	2.21108
$892$	4.00000	0.133930
$893$	−48.0000	−1.60626
$894$	8.00000	0.267560
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−18.0000	−0.600668
$899$	80.0000	2.66815
$900$	−1.00000	−0.0333333
$901$	−24.0000	−0.799556
$902$	12.0000	0.399556
$903$	12.0000	0.399335
$904$	14.0000	0.465633
$905$	4.00000	0.132964
$906$	−16.0000	−0.531564
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−24.0000	−0.796468
$909$	8.00000	0.265343
$910$	8.00000	0.265197
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	−8.00000	−0.264906
$913$	24.0000	0.794284
$914$	26.0000	0.860004
$915$	24.0000	0.793416
$916$	−22.0000	−0.726900
$917$	−10.0000	−0.330229
$918$	−8.00000	−0.264039
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	8.00000	0.263466
$923$	−64.0000	−2.10659
$924$	12.0000	0.394771
$925$	−8.00000	−0.263038
$926$	−32.0000	−1.05159
$927$	8.00000	0.262754
$928$	−10.0000	−0.328266
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	−32.0000	−1.04932
$931$	−4.00000	−0.131095
$932$	−6.00000	−0.196537
$933$	56.0000	1.83336
$934$	−28.0000	−0.916188
$935$	−24.0000	−0.784884
$936$	−4.00000	−0.130744
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	−2.00000	−0.0653023
$939$	−4.00000	−0.130535
$940$	24.0000	0.782794
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	12.0000	0.390981
$943$	0	0
$944$	−6.00000	−0.195283
$945$	8.00000	0.260240
$946$	36.0000	1.17046
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	4.00000	0.129777
$951$	−44.0000	−1.42680
$952$	−2.00000	−0.0648204
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	−12.0000	−0.388514
$955$	−32.0000	−1.03550
$956$	−24.0000	−0.776215
$957$	120.000	3.87905
$958$	32.0000	1.03387
$959$	6.00000	0.193750
$960$	4.00000	0.129099
$961$	33.0000	1.06452
$962$	−32.0000	−1.03172
$963$	−6.00000	−0.193347
$964$	6.00000	0.193247
$965$	−4.00000	−0.128765
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	25.0000	0.803530
$969$	−16.0000	−0.513994
$970$	−4.00000	−0.128432
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	−10.0000	−0.320750
$973$	10.0000	0.320585
$974$	−24.0000	−0.769010
$975$	8.00000	0.256205
$976$	6.00000	0.192055
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	40.0000	1.27906
$979$	−36.0000	−1.15056
$980$	2.00000	0.0638877
$981$	−4.00000	−0.127710
$982$	16.0000	0.510581
$983$	−40.0000	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$983$	−40.0000	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$984$	−4.00000	−0.127515
$985$	4.00000	0.127451
$986$	−20.0000	−0.636930
$987$	−24.0000	−0.763928
$988$	16.0000	0.509028
$989$	0	0
$990$	−12.0000	−0.381385
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	−8.00000	−0.254000
$993$	−16.0000	−0.507745
$994$	−16.0000	−0.507489
$995$	32.0000	1.01447
$996$	−8.00000	−0.253490
$997$	12.0000	0.380044	0.190022	−	0.981780i	$-0.439144\pi$
$997$	12.0000	0.380044	0.190022	+	0.981780i	$0.439144\pi$
$998$	24.0000	0.759707
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7406.2.a.i.1.1		1
23.22	odd	2		322.2.a.d.1.1	✓	1
69.68	even	2		2898.2.a.h.1.1		1
92.91	even	2		2576.2.a.b.1.1		1
115.114	odd	2		8050.2.a.a.1.1		1
161.160	even	2		2254.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.d.1.1	✓	1	23.22	odd	2
2254.2.a.f.1.1		1	161.160	even	2
2576.2.a.b.1.1		1	92.91	even	2
2898.2.a.h.1.1		1	69.68	even	2
7406.2.a.i.1.1		1	1.1	even	1	trivial
8050.2.a.a.1.1		1	115.114	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:	\( N \)	\(=\)	\( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7406.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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