LMFDB - Embedded newform 1078.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, 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("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.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:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1078.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{6} -1.00000 q^{8} +6.00000 q^{9} -4.00000 q^{10} -1.00000 q^{11} +3.00000 q^{12} +1.00000 q^{13} +12.0000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -6.00000 q^{18} -6.00000 q^{19} +4.00000 q^{20} +1.00000 q^{22} -2.00000 q^{23} -3.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} +9.00000 q^{27} +1.00000 q^{29} -12.0000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +2.00000 q^{34} +6.00000 q^{36} -2.00000 q^{37} +6.00000 q^{38} +3.00000 q^{39} -4.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} +24.0000 q^{45} +2.00000 q^{46} -2.00000 q^{47} +3.00000 q^{48} -11.0000 q^{50} -6.00000 q^{51} +1.00000 q^{52} -12.0000 q^{53} -9.00000 q^{54} -4.00000 q^{55} -18.0000 q^{57} -1.00000 q^{58} -9.00000 q^{59} +12.0000 q^{60} +5.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +3.00000 q^{66} -9.00000 q^{67} -2.00000 q^{68} -6.00000 q^{69} +4.00000 q^{71} -6.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +33.0000 q^{75} -6.00000 q^{76} -3.00000 q^{78} -15.0000 q^{79} +4.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +6.00000 q^{83} -8.00000 q^{85} -4.00000 q^{86} +3.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} -24.0000 q^{90} -2.00000 q^{92} -12.0000 q^{93} +2.00000 q^{94} -24.0000 q^{95} -3.00000 q^{96} +5.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	−3.00000	−1.22474
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	6.00000	2.00000
$10$	−4.00000	−1.26491
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	3.00000	0.866025
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	12.0000	3.09839
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−6.00000	−1.41421
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	1.00000	0.213201
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	−3.00000	−0.612372
$25$	11.0000	2.20000
$26$	−1.00000	−0.196116
$27$	9.00000	1.73205
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	−12.0000	−2.19089
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	2.00000	0.342997
$35$	0	0
$36$	6.00000	1.00000
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	6.00000	0.973329
$39$	3.00000	0.480384
$40$	−4.00000	−0.632456
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.00000	−0.150756
$45$	24.0000	3.57771
$46$	2.00000	0.294884
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	3.00000	0.433013
$49$	0	0
$50$	−11.0000	−1.55563
$51$	−6.00000	−0.840168
$52$	1.00000	0.138675
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	−9.00000	−1.22474
$55$	−4.00000	−0.539360
$56$	0	0
$57$	−18.0000	−2.38416
$58$	−1.00000	−0.131306
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	12.0000	1.54919
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	3.00000	0.369274
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	−2.00000	−0.242536
$69$	−6.00000	−0.722315
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	−6.00000	−0.707107
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	2.00000	0.232495
$75$	33.0000	3.81051
$76$	−6.00000	−0.688247
$77$	0	0
$78$	−3.00000	−0.339683
$79$	−15.0000	−1.68763	−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000	−1.68763	−0.843816	+	0.536633i	$0.819696\pi$
$80$	4.00000	0.447214
$81$	9.00000	1.00000
$82$	−2.00000	−0.220863
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	−4.00000	−0.431331
$87$	3.00000	0.321634
$88$	1.00000	0.106600
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−24.0000	−2.52982
$91$	0	0
$92$	−2.00000	−0.208514
$93$	−12.0000	−1.24434
$94$	2.00000	0.206284
$95$	−24.0000	−2.46235
$96$	−3.00000	−0.306186
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	11.0000	1.10000
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	6.00000	0.594089
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	12.0000	1.16554
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	9.00000	0.866025
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	4.00000	0.381385
$111$	−6.00000	−0.569495
$112$	0	0
$113$	17.0000	1.59923	0.799613	−	0.600516i	$-0.205038\pi$
$113$	17.0000	1.59923	0.799613	+	0.600516i	$0.205038\pi$
$114$	18.0000	1.68585
$115$	−8.00000	−0.746004
$116$	1.00000	0.0928477
$117$	6.00000	0.554700
$118$	9.00000	0.828517
$119$	0	0
$120$	−12.0000	−1.09545
$121$	1.00000	0.0909091
$122$	−5.00000	−0.452679
$123$	6.00000	0.541002
$124$	−4.00000	−0.359211
$125$	24.0000	2.14663
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	−1.00000	−0.0883883
$129$	12.0000	1.05654
$130$	−4.00000	−0.350823
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	−3.00000	−0.261116
$133$	0	0
$134$	9.00000	0.777482
$135$	36.0000	3.09839
$136$	2.00000	0.171499
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	6.00000	0.510754
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−4.00000	−0.335673
$143$	−1.00000	−0.0836242
$144$	6.00000	0.500000
$145$	4.00000	0.332182
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−33.0000	−2.69444
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	6.00000	0.486664
$153$	−12.0000	−0.970143
$154$	0	0
$155$	−16.0000	−1.28515
$156$	3.00000	0.240192
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	15.0000	1.19334
$159$	−36.0000	−2.85499
$160$	−4.00000	−0.316228
$161$	0	0
$162$	−9.00000	−0.707107
$163$	13.0000	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$164$	2.00000	0.156174
$165$	−12.0000	−0.934199
$166$	−6.00000	−0.465690
$167$	17.0000	1.31550	0.657750	−	0.753237i	$-0.271508\pi$
$167$	17.0000	1.31550	0.657750	+	0.753237i	$0.271508\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	8.00000	0.613572
$171$	−36.0000	−2.75299
$172$	4.00000	0.304997
$173$	5.00000	0.380143	0.190071	−	0.981770i	$-0.439128\pi$
$173$	5.00000	0.380143	0.190071	+	0.981770i	$0.439128\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−27.0000	−2.02944
$178$	6.00000	0.449719
$179$	13.0000	0.971666	0.485833	−	0.874052i	$-0.338516\pi$
$179$	13.0000	0.971666	0.485833	+	0.874052i	$0.338516\pi$
$180$	24.0000	1.78885
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	15.0000	1.10883
$184$	2.00000	0.147442
$185$	−8.00000	−0.588172
$186$	12.0000	0.879883
$187$	2.00000	0.146254
$188$	−2.00000	−0.145865
$189$	0	0
$190$	24.0000	1.74114
$191$	14.0000	1.01300	0.506502	−	0.862239i	$-0.330938\pi$
$191$	14.0000	1.01300	0.506502	+	0.862239i	$0.330938\pi$
$192$	3.00000	0.216506
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	−5.00000	−0.358979
$195$	12.0000	0.859338
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	6.00000	0.426401
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	−11.0000	−0.777817
$201$	−27.0000	−1.90443
$202$	−15.0000	−1.05540
$203$	0	0
$204$	−6.00000	−0.420084
$205$	8.00000	0.558744
$206$	12.0000	0.836080
$207$	−12.0000	−0.834058
$208$	1.00000	0.0693375
$209$	6.00000	0.415029
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	−12.0000	−0.824163
$213$	12.0000	0.822226
$214$	−8.00000	−0.546869
$215$	16.0000	1.09119
$216$	−9.00000	−0.612372
$217$	0	0
$218$	−10.0000	−0.677285
$219$	6.00000	0.405442
$220$	−4.00000	−0.269680
$221$	−2.00000	−0.134535
$222$	6.00000	0.402694
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	0	0
$225$	66.0000	4.40000
$226$	−17.0000	−1.13082
$227$	−10.0000	−0.663723	−0.331862	−	0.943328i	$-0.607677\pi$
$227$	−10.0000	−0.663723	−0.331862	+	0.943328i	$0.607677\pi$
$228$	−18.0000	−1.19208
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−6.00000	−0.392232
$235$	−8.00000	−0.521862
$236$	−9.00000	−0.585850
$237$	−45.0000	−2.92306
$238$	0	0
$239$	19.0000	1.22901	0.614504	−	0.788914i	$-0.289356\pi$
$239$	19.0000	1.22901	0.614504	+	0.788914i	$0.289356\pi$
$240$	12.0000	0.774597
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	5.00000	0.320092
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−6.00000	−0.381771
$248$	4.00000	0.254000
$249$	18.0000	1.14070
$250$	−24.0000	−1.51789
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	−5.00000	−0.313728
$255$	−24.0000	−1.50294
$256$	1.00000	0.0625000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	4.00000	0.248069
$261$	6.00000	0.371391
$262$	−18.0000	−1.11204
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	3.00000	0.184637
$265$	−48.0000	−2.94862
$266$	0	0
$267$	−18.0000	−1.10158
$268$	−9.00000	−0.549762
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	−36.0000	−2.19089
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	9.00000	0.543710
$275$	−11.0000	−0.663325
$276$	−6.00000	−0.361158
$277$	−3.00000	−0.180253	−0.0901263	−	0.995930i	$-0.528727\pi$
$277$	−3.00000	−0.180253	−0.0901263	+	0.995930i	$0.528727\pi$
$278$	8.00000	0.479808
$279$	−24.0000	−1.43684
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	6.00000	0.357295
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	4.00000	0.237356
$285$	−72.0000	−4.26491
$286$	1.00000	0.0591312
$287$	0	0
$288$	−6.00000	−0.353553
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	15.0000	0.879316
$292$	2.00000	0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−36.0000	−2.09600
$296$	2.00000	0.116248
$297$	−9.00000	−0.522233
$298$	10.0000	0.579284
$299$	−2.00000	−0.115663
$300$	33.0000	1.90526
$301$	0	0
$302$	−9.00000	−0.517892
$303$	45.0000	2.58518
$304$	−6.00000	−0.344124
$305$	20.0000	1.14520
$306$	12.0000	0.685994
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	0	0
$309$	−36.0000	−2.04797
$310$	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	$0.208065\pi$
$312$	−3.00000	−0.169842
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−15.0000	−0.843816
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	36.0000	2.01878
$319$	−1.00000	−0.0559893
$320$	4.00000	0.223607
$321$	24.0000	1.33955
$322$	0	0
$323$	12.0000	0.667698
$324$	9.00000	0.500000
$325$	11.0000	0.610170
$326$	−13.0000	−0.720003
$327$	30.0000	1.65900
$328$	−2.00000	−0.110432
$329$	0	0
$330$	12.0000	0.660578
$331$	7.00000	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$332$	6.00000	0.329293
$333$	−12.0000	−0.657596
$334$	−17.0000	−0.930199
$335$	−36.0000	−1.96689
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	12.0000	0.652714
$339$	51.0000	2.76994
$340$	−8.00000	−0.433861
$341$	4.00000	0.216612
$342$	36.0000	1.94666
$343$	0	0
$344$	−4.00000	−0.215666
$345$	−24.0000	−1.29212
$346$	−5.00000	−0.268802
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	3.00000	0.160817
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	9.00000	0.480384
$352$	1.00000	0.0533002
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	27.0000	1.43503
$355$	16.0000	0.849192
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−13.0000	−0.687071
$359$	−31.0000	−1.63612	−0.818059	−	0.575135i	$-0.804950\pi$
$359$	−31.0000	−1.63612	−0.818059	+	0.575135i	$0.804950\pi$
$360$	−24.0000	−1.26491
$361$	17.0000	0.894737
$362$	−22.0000	−1.15629
$363$	3.00000	0.157459
$364$	0	0
$365$	8.00000	0.418739
$366$	−15.0000	−0.784063
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	−2.00000	−0.104257
$369$	12.0000	0.624695
$370$	8.00000	0.415900
$371$	0	0
$372$	−12.0000	−0.622171
$373$	−7.00000	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$373$	−7.00000	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$374$	−2.00000	−0.103418
$375$	72.0000	3.71806
$376$	2.00000	0.103142
$377$	1.00000	0.0515026
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	−24.0000	−1.23117
$381$	15.0000	0.768473
$382$	−14.0000	−0.716302
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	22.0000	1.11977
$387$	24.0000	1.21999
$388$	5.00000	0.253837
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	−12.0000	−0.607644
$391$	4.00000	0.202289
$392$	0	0
$393$	54.0000	2.72394
$394$	3.00000	0.151138
$395$	−60.0000	−3.01893
$396$	−6.00000	−0.301511
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	11.0000	0.550000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	27.0000	1.34664
$403$	−4.00000	−0.199254
$404$	15.0000	0.746278
$405$	36.0000	1.78885
$406$	0	0
$407$	2.00000	0.0991363
$408$	6.00000	0.297044
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−8.00000	−0.395092
$411$	−27.0000	−1.33181
$412$	−12.0000	−0.591198
$413$	0	0
$414$	12.0000	0.589768
$415$	24.0000	1.17811
$416$	−1.00000	−0.0490290
$417$	−24.0000	−1.17529
$418$	−6.00000	−0.293470
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	14.0000	0.681509
$423$	−12.0000	−0.583460
$424$	12.0000	0.582772
$425$	−22.0000	−1.06716
$426$	−12.0000	−0.581402
$427$	0	0
$428$	8.00000	0.386695
$429$	−3.00000	−0.144841
$430$	−16.0000	−0.771589
$431$	1.00000	0.0481683	0.0240842	−	0.999710i	$-0.492333\pi$
$431$	1.00000	0.0481683	0.0240842	+	0.999710i	$0.492333\pi$
$432$	9.00000	0.433013
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	10.0000	0.478913
$437$	12.0000	0.574038
$438$	−6.00000	−0.286691
$439$	5.00000	0.238637	0.119318	−	0.992856i	$-0.461929\pi$
$439$	5.00000	0.238637	0.119318	+	0.992856i	$0.461929\pi$
$440$	4.00000	0.190693
$441$	0	0
$442$	2.00000	0.0951303
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−6.00000	−0.284747
$445$	−24.0000	−1.13771
$446$	−26.0000	−1.23114
$447$	−30.0000	−1.41895
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	−66.0000	−3.11127
$451$	−2.00000	−0.0941763
$452$	17.0000	0.799613
$453$	27.0000	1.26857
$454$	10.0000	0.469323
$455$	0	0
$456$	18.0000	0.842927
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	−16.0000	−0.747631
$459$	−18.0000	−0.840168
$460$	−8.00000	−0.373002
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	1.00000	0.0464238
$465$	−48.0000	−2.22595
$466$	−6.00000	−0.277945
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	6.00000	0.277350
$469$	0	0
$470$	8.00000	0.369012
$471$	−12.0000	−0.552931
$472$	9.00000	0.414259
$473$	−4.00000	−0.183920
$474$	45.0000	2.06692
$475$	−66.0000	−3.02829
$476$	0	0
$477$	−72.0000	−3.29665
$478$	−19.0000	−0.869040
$479$	37.0000	1.69057	0.845287	−	0.534313i	$-0.179430\pi$
$479$	37.0000	1.69057	0.845287	+	0.534313i	$0.179430\pi$
$480$	−12.0000	−0.547723
$481$	−2.00000	−0.0911922
$482$	30.0000	1.36646
$483$	0	0
$484$	1.00000	0.0454545
$485$	20.0000	0.908153
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−5.00000	−0.226339
$489$	39.0000	1.76364
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	6.00000	0.270501
$493$	−2.00000	−0.0900755
$494$	6.00000	0.269953
$495$	−24.0000	−1.07872
$496$	−4.00000	−0.179605
$497$	0	0
$498$	−18.0000	−0.806599
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	24.0000	1.07331
$501$	51.0000	2.27851
$502$	12.0000	0.535586
$503$	21.0000	0.936344	0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000	0.936344	0.468172	+	0.883637i	$0.344913\pi$
$504$	0	0
$505$	60.0000	2.66996
$506$	−2.00000	−0.0889108
$507$	−36.0000	−1.59882
$508$	5.00000	0.221839
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	24.0000	1.06274
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−54.0000	−2.38416
$514$	−21.0000	−0.926270
$515$	−48.0000	−2.11513
$516$	12.0000	0.528271
$517$	2.00000	0.0879599
$518$	0	0
$519$	15.0000	0.658427
$520$	−4.00000	−0.175412
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	−6.00000	−0.262613
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	−3.00000	−0.130806
$527$	8.00000	0.348485
$528$	−3.00000	−0.130558
$529$	−19.0000	−0.826087
$530$	48.0000	2.08499
$531$	−54.0000	−2.34340
$532$	0	0
$533$	2.00000	0.0866296
$534$	18.0000	0.778936
$535$	32.0000	1.38348
$536$	9.00000	0.388741
$537$	39.0000	1.68297
$538$	12.0000	0.517357
$539$	0	0
$540$	36.0000	1.54919
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	25.0000	1.07384
$543$	66.0000	2.83233
$544$	2.00000	0.0857493
$545$	40.0000	1.71341
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	−9.00000	−0.384461
$549$	30.0000	1.28037
$550$	11.0000	0.469042
$551$	−6.00000	−0.255609
$552$	6.00000	0.255377
$553$	0	0
$554$	3.00000	0.127458
$555$	−24.0000	−1.01874
$556$	−8.00000	−0.339276
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	24.0000	1.01600
$559$	4.00000	0.169182
$560$	0	0
$561$	6.00000	0.253320
$562$	10.0000	0.421825
$563$	−26.0000	−1.09577	−0.547885	−	0.836554i	$-0.684567\pi$
$563$	−26.0000	−1.09577	−0.547885	+	0.836554i	$0.684567\pi$
$564$	−6.00000	−0.252646
$565$	68.0000	2.86078
$566$	−6.00000	−0.252199
$567$	0	0
$568$	−4.00000	−0.167836
$569$	36.0000	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$570$	72.0000	3.01575
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−1.00000	−0.0418121
$573$	42.0000	1.75458
$574$	0	0
$575$	−22.0000	−0.917463
$576$	6.00000	0.250000
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	13.0000	0.540729
$579$	−66.0000	−2.74287
$580$	4.00000	0.166091
$581$	0	0
$582$	−15.0000	−0.621770
$583$	12.0000	0.496989
$584$	−2.00000	−0.0827606
$585$	24.0000	0.992278
$586$	−6.00000	−0.247858
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	36.0000	1.48210
$591$	−9.00000	−0.370211
$592$	−2.00000	−0.0821995
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	9.00000	0.369274
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−30.0000	−1.22782
$598$	2.00000	0.0817861
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	−33.0000	−1.34722
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−54.0000	−2.19905
$604$	9.00000	0.366205
$605$	4.00000	0.162623
$606$	−45.0000	−1.82800
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	−20.0000	−0.809776
$611$	−2.00000	−0.0809113
$612$	−12.0000	−0.485071
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	32.0000	1.29141
$615$	24.0000	0.967773
$616$	0	0
$617$	−27.0000	−1.08698	−0.543490	−	0.839416i	$-0.682897\pi$
$617$	−27.0000	−1.08698	−0.543490	+	0.839416i	$0.682897\pi$
$618$	36.0000	1.44813
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	−16.0000	−0.642575
$621$	−18.0000	−0.722315
$622$	−28.0000	−1.12270
$623$	0	0
$624$	3.00000	0.120096
$625$	41.0000	1.64000
$626$	1.00000	0.0399680
$627$	18.0000	0.718851
$628$	−4.00000	−0.159617
$629$	4.00000	0.159490
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	15.0000	0.596668
$633$	−42.0000	−1.66935
$634$	12.0000	0.476581
$635$	20.0000	0.793676
$636$	−36.0000	−1.42749
$637$	0	0
$638$	1.00000	0.0395904
$639$	24.0000	0.949425
$640$	−4.00000	−0.158114
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	−24.0000	−0.947204
$643$	−1.00000	−0.0394362	−0.0197181	−	0.999806i	$-0.506277\pi$
$643$	−1.00000	−0.0394362	−0.0197181	+	0.999806i	$0.506277\pi$
$644$	0	0
$645$	48.0000	1.89000
$646$	−12.0000	−0.472134
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−9.00000	−0.353553
$649$	9.00000	0.353281
$650$	−11.0000	−0.431455
$651$	0	0
$652$	13.0000	0.509119
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	−30.0000	−1.17309
$655$	72.0000	2.81327
$656$	2.00000	0.0780869
$657$	12.0000	0.468165
$658$	0	0
$659$	−32.0000	−1.24654	−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000	−1.24654	−0.623272	+	0.782006i	$0.714197\pi$
$660$	−12.0000	−0.467099
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−7.00000	−0.272063
$663$	−6.00000	−0.233021
$664$	−6.00000	−0.232845
$665$	0	0
$666$	12.0000	0.464991
$667$	−2.00000	−0.0774403
$668$	17.0000	0.657750
$669$	78.0000	3.01565
$670$	36.0000	1.39080
$671$	−5.00000	−0.193023
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	30.0000	1.15556
$675$	99.0000	3.81051
$676$	−12.0000	−0.461538
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	−51.0000	−1.95864
$679$	0	0
$680$	8.00000	0.306786
$681$	−30.0000	−1.14960
$682$	−4.00000	−0.153168
$683$	−33.0000	−1.26271	−0.631355	−	0.775494i	$-0.717501\pi$
$683$	−33.0000	−1.26271	−0.631355	+	0.775494i	$0.717501\pi$
$684$	−36.0000	−1.37649
$685$	−36.0000	−1.37549
$686$	0	0
$687$	48.0000	1.83131
$688$	4.00000	0.152499
$689$	−12.0000	−0.457164
$690$	24.0000	0.913664
$691$	−15.0000	−0.570627	−0.285313	−	0.958434i	$-0.592098\pi$
$691$	−15.0000	−0.570627	−0.285313	+	0.958434i	$0.592098\pi$
$692$	5.00000	0.190071
$693$	0	0
$694$	28.0000	1.06287
$695$	−32.0000	−1.21383
$696$	−3.00000	−0.113715
$697$	−4.00000	−0.151511
$698$	2.00000	0.0757011
$699$	18.0000	0.680823
$700$	0	0
$701$	39.0000	1.47301	0.736505	−	0.676432i	$-0.236475\pi$
$701$	39.0000	1.47301	0.736505	+	0.676432i	$0.236475\pi$
$702$	−9.00000	−0.339683
$703$	12.0000	0.452589
$704$	−1.00000	−0.0376889
$705$	−24.0000	−0.903892
$706$	−18.0000	−0.677439
$707$	0	0
$708$	−27.0000	−1.01472
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	−16.0000	−0.600469
$711$	−90.0000	−3.37526
$712$	6.00000	0.224860
$713$	8.00000	0.299602
$714$	0	0
$715$	−4.00000	−0.149592
$716$	13.0000	0.485833
$717$	57.0000	2.12870
$718$	31.0000	1.15691
$719$	26.0000	0.969636	0.484818	−	0.874615i	$-0.338886\pi$
$719$	26.0000	0.969636	0.484818	+	0.874615i	$0.338886\pi$
$720$	24.0000	0.894427
$721$	0	0
$722$	−17.0000	−0.632674
$723$	−90.0000	−3.34714
$724$	22.0000	0.817624
$725$	11.0000	0.408530
$726$	−3.00000	−0.111340
$727$	34.0000	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$727$	34.0000	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−8.00000	−0.296093
$731$	−8.00000	−0.295891
$732$	15.0000	0.554416
$733$	1.00000	0.0369358	0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000	0.0369358	0.0184679	+	0.999829i	$0.494121\pi$
$734$	−14.0000	−0.516749
$735$	0	0
$736$	2.00000	0.0737210
$737$	9.00000	0.331519
$738$	−12.0000	−0.441726
$739$	18.0000	0.662141	0.331070	−	0.943606i	$-0.392590\pi$
$739$	18.0000	0.662141	0.331070	+	0.943606i	$0.392590\pi$
$740$	−8.00000	−0.294086
$741$	−18.0000	−0.661247
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	12.0000	0.439941
$745$	−40.0000	−1.46549
$746$	7.00000	0.256288
$747$	36.0000	1.31717
$748$	2.00000	0.0731272
$749$	0	0
$750$	−72.0000	−2.62907
$751$	−44.0000	−1.60558	−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000	−1.60558	−0.802791	+	0.596260i	$0.796653\pi$
$752$	−2.00000	−0.0729325
$753$	−36.0000	−1.31191
$754$	−1.00000	−0.0364179
$755$	36.0000	1.31017
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	29.0000	1.05333
$759$	6.00000	0.217786
$760$	24.0000	0.870572
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	−15.0000	−0.543393
$763$	0	0
$764$	14.0000	0.506502
$765$	−48.0000	−1.73544
$766$	8.00000	0.289052
$767$	−9.00000	−0.324971
$768$	3.00000	0.108253
$769$	38.0000	1.37032	0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	0	0
$771$	63.0000	2.26889
$772$	−22.0000	−0.791797
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	−24.0000	−0.862662
$775$	−44.0000	−1.58053
$776$	−5.00000	−0.179490
$777$	0	0
$778$	0	0
$779$	−12.0000	−0.429945
$780$	12.0000	0.429669
$781$	−4.00000	−0.143131
$782$	−4.00000	−0.143040
$783$	9.00000	0.321634
$784$	0	0
$785$	−16.0000	−0.571064
$786$	−54.0000	−1.92612
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	−3.00000	−0.106871
$789$	9.00000	0.320408
$790$	60.0000	2.13470
$791$	0	0
$792$	6.00000	0.213201
$793$	5.00000	0.177555
$794$	−30.0000	−1.06466
$795$	−144.000	−5.10715
$796$	−10.0000	−0.354441
$797$	−38.0000	−1.34603	−0.673015	−	0.739629i	$-0.735001\pi$
$797$	−38.0000	−1.34603	−0.673015	+	0.739629i	$0.735001\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	−11.0000	−0.388909
$801$	−36.0000	−1.27200
$802$	−15.0000	−0.529668
$803$	−2.00000	−0.0705785
$804$	−27.0000	−0.952217
$805$	0	0
$806$	4.00000	0.140894
$807$	−36.0000	−1.26726
$808$	−15.0000	−0.527698
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	−36.0000	−1.26491
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	−75.0000	−2.63036
$814$	−2.00000	−0.0701000
$815$	52.0000	1.82148
$816$	−6.00000	−0.210042
$817$	−24.0000	−0.839654
$818$	−32.0000	−1.11885
$819$	0	0
$820$	8.00000	0.279372
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	27.0000	0.941733
$823$	38.0000	1.32460	0.662298	−	0.749240i	$-0.269581\pi$
$823$	38.0000	1.32460	0.662298	+	0.749240i	$0.269581\pi$
$824$	12.0000	0.418040
$825$	−33.0000	−1.14891
$826$	0	0
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	−12.0000	−0.417029
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	−24.0000	−0.833052
$831$	−9.00000	−0.312207
$832$	1.00000	0.0346688
$833$	0	0
$834$	24.0000	0.831052
$835$	68.0000	2.35324
$836$	6.00000	0.207514
$837$	−36.0000	−1.24434
$838$	−20.0000	−0.690889
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	20.0000	0.689246
$843$	−30.0000	−1.03325
$844$	−14.0000	−0.481900
$845$	−48.0000	−1.65125
$846$	12.0000	0.412568
$847$	0	0
$848$	−12.0000	−0.412082
$849$	18.0000	0.617758
$850$	22.0000	0.754594
$851$	4.00000	0.137118
$852$	12.0000	0.411113
$853$	−50.0000	−1.71197	−0.855984	−	0.517003i	$-0.827048\pi$
$853$	−50.0000	−1.71197	−0.855984	+	0.517003i	$0.827048\pi$
$854$	0	0
$855$	−144.000	−4.92470
$856$	−8.00000	−0.273434
$857$	4.00000	0.136637	0.0683187	−	0.997664i	$-0.478237\pi$
$857$	4.00000	0.136637	0.0683187	+	0.997664i	$0.478237\pi$
$858$	3.00000	0.102418
$859$	17.0000	0.580033	0.290016	−	0.957022i	$-0.406339\pi$
$859$	17.0000	0.580033	0.290016	+	0.957022i	$0.406339\pi$
$860$	16.0000	0.545595
$861$	0	0
$862$	−1.00000	−0.0340601
$863$	10.0000	0.340404	0.170202	−	0.985409i	$-0.445558\pi$
$863$	10.0000	0.340404	0.170202	+	0.985409i	$0.445558\pi$
$864$	−9.00000	−0.306186
$865$	20.0000	0.680020
$866$	−10.0000	−0.339814
$867$	−39.0000	−1.32451
$868$	0	0
$869$	15.0000	0.508840
$870$	−12.0000	−0.406838
$871$	−9.00000	−0.304953
$872$	−10.0000	−0.338643
$873$	30.0000	1.01535
$874$	−12.0000	−0.405906
$875$	0	0
$876$	6.00000	0.202721
$877$	25.0000	0.844190	0.422095	−	0.906552i	$-0.361295\pi$
$877$	25.0000	0.844190	0.422095	+	0.906552i	$0.361295\pi$
$878$	−5.00000	−0.168742
$879$	18.0000	0.607125
$880$	−4.00000	−0.134840
$881$	−9.00000	−0.303218	−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000	−0.303218	−0.151609	+	0.988441i	$0.548445\pi$
$882$	0	0
$883$	1.00000	0.0336527	0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000	0.0336527	0.0168263	+	0.999858i	$0.494644\pi$
$884$	−2.00000	−0.0672673
$885$	−108.000	−3.63038
$886$	12.0000	0.403148
$887$	17.0000	0.570804	0.285402	−	0.958408i	$-0.407873\pi$
$887$	17.0000	0.570804	0.285402	+	0.958408i	$0.407873\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	24.0000	0.804482
$891$	−9.00000	−0.301511
$892$	26.0000	0.870544
$893$	12.0000	0.401565
$894$	30.0000	1.00335
$895$	52.0000	1.73817
$896$	0	0
$897$	−6.00000	−0.200334
$898$	−30.0000	−1.00111
$899$	−4.00000	−0.133407
$900$	66.0000	2.20000
$901$	24.0000	0.799556
$902$	2.00000	0.0665927
$903$	0	0
$904$	−17.0000	−0.565412
$905$	88.0000	2.92522
$906$	−27.0000	−0.897015
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−10.0000	−0.331862
$909$	90.0000	2.98511
$910$	0	0
$911$	−54.0000	−1.78910	−0.894550	−	0.446968i	$-0.852504\pi$
$911$	−54.0000	−1.78910	−0.894550	+	0.446968i	$0.852504\pi$
$912$	−18.0000	−0.596040
$913$	−6.00000	−0.198571
$914$	−2.00000	−0.0661541
$915$	60.0000	1.98354
$916$	16.0000	0.528655
$917$	0	0
$918$	18.0000	0.594089
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	8.00000	0.263752
$921$	−96.0000	−3.16331
$922$	−3.00000	−0.0987997
$923$	4.00000	0.131662
$924$	0	0
$925$	−22.0000	−0.723356
$926$	14.0000	0.460069
$927$	−72.0000	−2.36479
$928$	−1.00000	−0.0328266
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	48.0000	1.57398
$931$	0	0
$932$	6.00000	0.196537
$933$	84.0000	2.75004
$934$	−12.0000	−0.392652
$935$	8.00000	0.261628
$936$	−6.00000	−0.196116
$937$	12.0000	0.392023	0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000	0.392023	0.196011	+	0.980602i	$0.437201\pi$
$938$	0	0
$939$	−3.00000	−0.0979013
$940$	−8.00000	−0.260931
$941$	−55.0000	−1.79295	−0.896474	−	0.443096i	$-0.853880\pi$
$941$	−55.0000	−1.79295	−0.896474	+	0.443096i	$0.853880\pi$
$942$	12.0000	0.390981
$943$	−4.00000	−0.130258
$944$	−9.00000	−0.292925
$945$	0	0
$946$	4.00000	0.130051
$947$	−16.0000	−0.519930	−0.259965	−	0.965618i	$-0.583711\pi$
$947$	−16.0000	−0.519930	−0.259965	+	0.965618i	$0.583711\pi$
$948$	−45.0000	−1.46153
$949$	2.00000	0.0649227
$950$	66.0000	2.14132
$951$	−36.0000	−1.16738
$952$	0	0
$953$	56.0000	1.81402	0.907009	−	0.421111i	$-0.138360\pi$
$953$	56.0000	1.81402	0.907009	+	0.421111i	$0.138360\pi$
$954$	72.0000	2.33109
$955$	56.0000	1.81212
$956$	19.0000	0.614504
$957$	−3.00000	−0.0969762
$958$	−37.0000	−1.19542
$959$	0	0
$960$	12.0000	0.387298
$961$	−15.0000	−0.483871
$962$	2.00000	0.0644826
$963$	48.0000	1.54678
$964$	−30.0000	−0.966235
$965$	−88.0000	−2.83282
$966$	0	0
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	−1.00000	−0.0321412
$969$	36.0000	1.15649
$970$	−20.0000	−0.642161
$971$	13.0000	0.417190	0.208595	−	0.978002i	$-0.433111\pi$
$971$	13.0000	0.417190	0.208595	+	0.978002i	$0.433111\pi$
$972$	0	0
$973$	0	0
$974$	4.00000	0.128168
$975$	33.0000	1.05685
$976$	5.00000	0.160046
$977$	14.0000	0.447900	0.223950	−	0.974601i	$-0.428105\pi$
$977$	14.0000	0.447900	0.223950	+	0.974601i	$0.428105\pi$
$978$	−39.0000	−1.24708
$979$	6.00000	0.191761
$980$	0	0
$981$	60.0000	1.91565
$982$	−18.0000	−0.574403
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	−6.00000	−0.191273
$985$	−12.0000	−0.382352
$986$	2.00000	0.0636930
$987$	0	0
$988$	−6.00000	−0.190885
$989$	−8.00000	−0.254385
$990$	24.0000	0.762770
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	4.00000	0.127000
$993$	21.0000	0.666415
$994$	0	0
$995$	−40.0000	−1.26809
$996$	18.0000	0.570352
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	16.0000	0.506471
$999$	−18.0000	−0.569495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.a.f.1.1		1
3.2	odd	2		9702.2.a.bb.1.1		1
4.3	odd	2		8624.2.a.d.1.1		1
7.2	even	3		1078.2.e.g.67.1		2
7.3	odd	6		154.2.e.d.23.1	✓	2
7.4	even	3		1078.2.e.g.177.1		2
7.5	odd	6		154.2.e.d.67.1	yes	2
7.6	odd	2		1078.2.a.a.1.1		1
21.5	even	6		1386.2.k.a.991.1		2
21.17	even	6		1386.2.k.a.793.1		2
21.20	even	2		9702.2.a.cg.1.1		1
28.3	even	6		1232.2.q.a.177.1		2
28.19	even	6		1232.2.q.a.529.1		2
28.27	even	2		8624.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.d.23.1	✓	2	7.3	odd	6
154.2.e.d.67.1	yes	2	7.5	odd	6
1078.2.a.a.1.1		1	7.6	odd	2
1078.2.a.f.1.1		1	1.1	even	1	trivial
1078.2.e.g.67.1		2	7.2	even	3
1078.2.e.g.177.1		2	7.4	even	3
1232.2.q.a.177.1		2	28.3	even	6
1232.2.q.a.529.1		2	28.19	even	6
1386.2.k.a.793.1		2	21.17	even	6
1386.2.k.a.991.1		2	21.5	even	6
8624.2.a.d.1.1		1	4.3	odd	2
8624.2.a.bd.1.1		1	28.27	even	2
9702.2.a.bb.1.1		1	3.2	odd	2
9702.2.a.cg.1.1		1	21.20	even	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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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