LMFDB - Embedded newform 7098.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7098.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:	$56.6778153547$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} -3.00000 q^{19} -4.00000 q^{20} +1.00000 q^{21} +3.00000 q^{22} +6.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -1.00000 q^{27} -1.00000 q^{28} -9.00000 q^{29} +4.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -5.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} -3.00000 q^{38} -4.00000 q^{40} -5.00000 q^{41} +1.00000 q^{42} +3.00000 q^{44} -4.00000 q^{45} +6.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +11.0000 q^{50} +5.00000 q^{51} -11.0000 q^{53} -1.00000 q^{54} -12.0000 q^{55} -1.00000 q^{56} +3.00000 q^{57} -9.00000 q^{58} -2.00000 q^{59} +4.00000 q^{60} -1.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} -2.00000 q^{67} -5.00000 q^{68} -6.00000 q^{69} +4.00000 q^{70} -6.00000 q^{71} +1.00000 q^{72} -12.0000 q^{73} -4.00000 q^{74} -11.0000 q^{75} -3.00000 q^{76} -3.00000 q^{77} +11.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -5.00000 q^{82} -6.00000 q^{83} +1.00000 q^{84} +20.0000 q^{85} +9.00000 q^{87} +3.00000 q^{88} -7.00000 q^{89} -4.00000 q^{90} +6.00000 q^{92} -4.00000 q^{93} +3.00000 q^{94} +12.0000 q^{95} -1.00000 q^{96} +12.0000 q^{97} +1.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	1.00000	0.235702
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−4.00000	−0.894427
$21$	1.00000	0.218218
$22$	3.00000	0.639602
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−1.00000	−0.204124
$25$	11.0000	2.20000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	4.00000	0.730297
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	−3.00000	−0.522233
$34$	−5.00000	−0.857493
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−3.00000	−0.486664
$39$	0	0
$40$	−4.00000	−0.632456
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	1.00000	0.154303
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	3.00000	0.452267
$45$	−4.00000	−0.596285
$46$	6.00000	0.884652
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	11.0000	1.55563
$51$	5.00000	0.700140
$52$	0	0
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	−1.00000	−0.136083
$55$	−12.0000	−1.61808
$56$	−1.00000	−0.133631
$57$	3.00000	0.397360
$58$	−9.00000	−1.18176
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	4.00000	0.516398
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	4.00000	0.508001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−5.00000	−0.606339
$69$	−6.00000	−0.722315
$70$	4.00000	0.478091
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	1.00000	0.117851
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	−4.00000	−0.464991
$75$	−11.0000	−1.27017
$76$	−3.00000	−0.344124
$77$	−3.00000	−0.341882
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−5.00000	−0.552158
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	1.00000	0.109109
$85$	20.0000	2.16930
$86$	0	0
$87$	9.00000	0.964901
$88$	3.00000	0.319801
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	−4.00000	−0.421637
$91$	0	0
$92$	6.00000	0.625543
$93$	−4.00000	−0.414781
$94$	3.00000	0.309426
$95$	12.0000	1.23117
$96$	−1.00000	−0.102062
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	1.00000	0.101015
$99$	3.00000	0.301511
$100$	11.0000	1.10000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	5.00000	0.495074
$103$	20.0000	1.97066	0.985329	−	0.170664i	$-0.0545913\pi$
$103$	20.0000	1.97066	0.985329	+	0.170664i	$0.0545913\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	−11.0000	−1.06841
$107$	17.0000	1.64345	0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000	1.64345	0.821726	+	0.569883i	$0.193011\pi$
$108$	−1.00000	−0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−12.0000	−1.14416
$111$	4.00000	0.379663
$112$	−1.00000	−0.0944911
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	3.00000	0.280976
$115$	−24.0000	−2.23801
$116$	−9.00000	−0.835629
$117$	0	0
$118$	−2.00000	−0.184115
$119$	5.00000	0.458349
$120$	4.00000	0.365148
$121$	−2.00000	−0.181818
$122$	−1.00000	−0.0905357
$123$	5.00000	0.450835
$124$	4.00000	0.359211
$125$	−24.0000	−2.14663
$126$	−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
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−3.00000	−0.261116
$133$	3.00000	0.260133
$134$	−2.00000	−0.172774
$135$	4.00000	0.344265
$136$	−5.00000	−0.428746
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−6.00000	−0.510754
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	4.00000	0.338062
$141$	−3.00000	−0.252646
$142$	−6.00000	−0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	36.0000	2.98964
$146$	−12.0000	−0.993127
$147$	−1.00000	−0.0824786
$148$	−4.00000	−0.328798
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−11.0000	−0.898146
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	−3.00000	−0.243332
$153$	−5.00000	−0.404226
$154$	−3.00000	−0.241747
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	11.0000	0.875113
$159$	11.0000	0.872357
$160$	−4.00000	−0.316228
$161$	−6.00000	−0.472866
$162$	1.00000	0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	−5.00000	−0.390434
$165$	12.0000	0.934199
$166$	−6.00000	−0.465690
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	20.0000	1.53393
$171$	−3.00000	−0.229416
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	9.00000	0.682288
$175$	−11.0000	−0.831522
$176$	3.00000	0.226134
$177$	2.00000	0.150329
$178$	−7.00000	−0.524672
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	−4.00000	−0.298142
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	6.00000	0.442326
$185$	16.0000	1.17634
$186$	−4.00000	−0.293294
$187$	−15.0000	−1.09691
$188$	3.00000	0.218797
$189$	1.00000	0.0727393
$190$	12.0000	0.870572
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−1.00000	−0.0721688
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	1.00000	0.0714286
$197$	19.0000	1.35369	0.676847	−	0.736124i	$-0.263346\pi$
$197$	19.0000	1.35369	0.676847	+	0.736124i	$0.263346\pi$
$198$	3.00000	0.213201
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	11.0000	0.777817
$201$	2.00000	0.141069
$202$	10.0000	0.703598
$203$	9.00000	0.631676
$204$	5.00000	0.350070
$205$	20.0000	1.39686
$206$	20.0000	1.39347
$207$	6.00000	0.417029
$208$	0	0
$209$	−9.00000	−0.622543
$210$	−4.00000	−0.276026
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	−11.0000	−0.755483
$213$	6.00000	0.411113
$214$	17.0000	1.16210
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−4.00000	−0.271538
$218$	2.00000	0.135457
$219$	12.0000	0.810885
$220$	−12.0000	−0.809040
$221$	0	0
$222$	4.00000	0.268462
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−1.00000	−0.0668153
$225$	11.0000	0.733333
$226$	12.0000	0.798228
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	3.00000	0.198680
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	−24.0000	−1.58251
$231$	3.00000	0.197386
$232$	−9.00000	−0.590879
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	−2.00000	−0.130189
$237$	−11.0000	−0.714527
$238$	5.00000	0.324102
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	4.00000	0.258199
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	−2.00000	−0.128565
$243$	−1.00000	−0.0641500
$244$	−1.00000	−0.0640184
$245$	−4.00000	−0.255551
$246$	5.00000	0.318788
$247$	0	0
$248$	4.00000	0.254000
$249$	6.00000	0.380235
$250$	−24.0000	−1.51789
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	−1.00000	−0.0629941
$253$	18.0000	1.13165
$254$	8.00000	0.501965
$255$	−20.0000	−1.25245
$256$	1.00000	0.0625000
$257$	7.00000	0.436648	0.218324	−	0.975876i	$-0.429941\pi$
$257$	7.00000	0.436648	0.218324	+	0.975876i	$0.429941\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−9.00000	−0.557086
$262$	12.0000	0.741362
$263$	−26.0000	−1.60323	−0.801614	−	0.597841i	$-0.796025\pi$
$263$	−26.0000	−1.60323	−0.801614	+	0.597841i	$0.796025\pi$
$264$	−3.00000	−0.184637
$265$	44.0000	2.70290
$266$	3.00000	0.183942
$267$	7.00000	0.428393
$268$	−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	$0.761251\pi$
$270$	4.00000	0.243432
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	18.0000	1.08742
$275$	33.0000	1.98997
$276$	−6.00000	−0.361158
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	13.0000	0.779688
$279$	4.00000	0.239474
$280$	4.00000	0.239046
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	−3.00000	−0.178647
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−6.00000	−0.356034
$285$	−12.0000	−0.710819
$286$	0	0
$287$	5.00000	0.295141
$288$	1.00000	0.0589256
$289$	8.00000	0.470588
$290$	36.0000	2.11399
$291$	−12.0000	−0.703452
$292$	−12.0000	−0.702247
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	−1.00000	−0.0583212
$295$	8.00000	0.465778
$296$	−4.00000	−0.232495
$297$	−3.00000	−0.174078
$298$	−18.0000	−1.04271
$299$	0	0
$300$	−11.0000	−0.635085
$301$	0	0
$302$	11.0000	0.632979
$303$	−10.0000	−0.574485
$304$	−3.00000	−0.172062
$305$	4.00000	0.229039
$306$	−5.00000	−0.285831
$307$	−21.0000	−1.19853	−0.599267	−	0.800549i	$-0.704541\pi$
$307$	−21.0000	−1.19853	−0.599267	+	0.800549i	$0.704541\pi$
$308$	−3.00000	−0.170941
$309$	−20.0000	−1.13776
$310$	−16.0000	−0.908739
$311$	−19.0000	−1.07739	−0.538696	−	0.842500i	$-0.681083\pi$
$311$	−19.0000	−1.07739	−0.538696	+	0.842500i	$0.681083\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−2.00000	−0.112867
$315$	4.00000	0.225374
$316$	11.0000	0.618798
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	11.0000	0.616849
$319$	−27.0000	−1.51171
$320$	−4.00000	−0.223607
$321$	−17.0000	−0.948847
$322$	−6.00000	−0.334367
$323$	15.0000	0.834622
$324$	1.00000	0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	−2.00000	−0.110600
$328$	−5.00000	−0.276079
$329$	−3.00000	−0.165395
$330$	12.0000	0.660578
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	−6.00000	−0.329293
$333$	−4.00000	−0.219199
$334$	16.0000	0.875481
$335$	8.00000	0.437087
$336$	1.00000	0.0545545
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	20.0000	1.08465
$341$	12.0000	0.649836
$342$	−3.00000	−0.162221
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	24.0000	1.29212
$346$	−2.00000	−0.107521
$347$	11.0000	0.590511	0.295255	−	0.955418i	$-0.404595\pi$
$347$	11.0000	0.590511	0.295255	+	0.955418i	$0.404595\pi$
$348$	9.00000	0.482451
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	−11.0000	−0.587975
$351$	0	0
$352$	3.00000	0.159901
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	2.00000	0.106299
$355$	24.0000	1.27379
$356$	−7.00000	−0.370999
$357$	−5.00000	−0.264628
$358$	24.0000	1.26844
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	−4.00000	−0.210819
$361$	−10.0000	−0.526316
$362$	−5.00000	−0.262794
$363$	2.00000	0.104973
$364$	0	0
$365$	48.0000	2.51243
$366$	1.00000	0.0522708
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	6.00000	0.312772
$369$	−5.00000	−0.260290
$370$	16.0000	0.831800
$371$	11.0000	0.571092
$372$	−4.00000	−0.207390
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	−15.0000	−0.775632
$375$	24.0000	1.23935
$376$	3.00000	0.154713
$377$	0	0
$378$	1.00000	0.0514344
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	12.0000	0.615587
$381$	−8.00000	−0.409852
$382$	−6.00000	−0.306987
$383$	11.0000	0.562074	0.281037	−	0.959697i	$-0.409322\pi$
$383$	11.0000	0.562074	0.281037	+	0.959697i	$0.409322\pi$
$384$	−1.00000	−0.0510310
$385$	12.0000	0.611577
$386$	−13.0000	−0.661683
$387$	0	0
$388$	12.0000	0.609208
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−30.0000	−1.51717
$392$	1.00000	0.0505076
$393$	−12.0000	−0.605320
$394$	19.0000	0.957206
$395$	−44.0000	−2.21388
$396$	3.00000	0.150756
$397$	23.0000	1.15434	0.577168	−	0.816625i	$-0.304158\pi$
$397$	23.0000	1.15434	0.577168	+	0.816625i	$0.304158\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	11.0000	0.550000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	2.00000	0.0997509
$403$	0	0
$404$	10.0000	0.497519
$405$	−4.00000	−0.198762
$406$	9.00000	0.446663
$407$	−12.0000	−0.594818
$408$	5.00000	0.247537
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	20.0000	0.987730
$411$	−18.0000	−0.887875
$412$	20.0000	0.985329
$413$	2.00000	0.0984136
$414$	6.00000	0.294884
$415$	24.0000	1.17811
$416$	0	0
$417$	−13.0000	−0.636613
$418$	−9.00000	−0.440204
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	−4.00000	−0.195180
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	14.0000	0.681509
$423$	3.00000	0.145865
$424$	−11.0000	−0.534207
$425$	−55.0000	−2.66789
$426$	6.00000	0.290701
$427$	1.00000	0.0483934
$428$	17.0000	0.821726
$429$	0	0
$430$	0	0
$431$	−38.0000	−1.83040	−0.915198	−	0.403005i	$-0.867966\pi$
$431$	−38.0000	−1.83040	−0.915198	+	0.403005i	$0.867966\pi$
$432$	−1.00000	−0.0481125
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	−4.00000	−0.192006
$435$	−36.0000	−1.72607
$436$	2.00000	0.0957826
$437$	−18.0000	−0.861057
$438$	12.0000	0.573382
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	−12.0000	−0.572078
$441$	1.00000	0.0476190
$442$	0	0
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	4.00000	0.189832
$445$	28.0000	1.32733
$446$	16.0000	0.757622
$447$	18.0000	0.851371
$448$	−1.00000	−0.0472456
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	11.0000	0.518545
$451$	−15.0000	−0.706322
$452$	12.0000	0.564433
$453$	−11.0000	−0.516825
$454$	−14.0000	−0.657053
$455$	0	0
$456$	3.00000	0.140488
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	5.00000	0.233635
$459$	5.00000	0.233380
$460$	−24.0000	−1.11901
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	3.00000	0.139573
$463$	−1.00000	−0.0464739	−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000	−0.0464739	−0.0232370	+	0.999730i	$0.507397\pi$
$464$	−9.00000	−0.417815
$465$	16.0000	0.741982
$466$	4.00000	0.185296
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	−12.0000	−0.553519
$471$	2.00000	0.0921551
$472$	−2.00000	−0.0920575
$473$	0	0
$474$	−11.0000	−0.505247
$475$	−33.0000	−1.51414
$476$	5.00000	0.229175
$477$	−11.0000	−0.503655
$478$	26.0000	1.18921
$479$	33.0000	1.50781	0.753904	−	0.656984i	$-0.228168\pi$
$479$	33.0000	1.50781	0.753904	+	0.656984i	$0.228168\pi$
$480$	4.00000	0.182574
$481$	0	0
$482$	−12.0000	−0.546585
$483$	6.00000	0.273009
$484$	−2.00000	−0.0909091
$485$	−48.0000	−2.17957
$486$	−1.00000	−0.0453609
$487$	21.0000	0.951601	0.475800	−	0.879553i	$-0.342158\pi$
$487$	21.0000	0.951601	0.475800	+	0.879553i	$0.342158\pi$
$488$	−1.00000	−0.0452679
$489$	20.0000	0.904431
$490$	−4.00000	−0.180702
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	5.00000	0.225417
$493$	45.0000	2.02670
$494$	0	0
$495$	−12.0000	−0.539360
$496$	4.00000	0.179605
$497$	6.00000	0.269137
$498$	6.00000	0.268866
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	−24.0000	−1.07331
$501$	−16.0000	−0.714827
$502$	14.0000	0.624851
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	−1.00000	−0.0445435
$505$	−40.0000	−1.77998
$506$	18.0000	0.800198
$507$	0	0
$508$	8.00000	0.354943
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	−20.0000	−0.885615
$511$	12.0000	0.530849
$512$	1.00000	0.0441942
$513$	3.00000	0.132453
$514$	7.00000	0.308757
$515$	−80.0000	−3.52522
$516$	0	0
$517$	9.00000	0.395820
$518$	4.00000	0.175750
$519$	2.00000	0.0877903
$520$	0	0
$521$	−5.00000	−0.219054	−0.109527	−	0.993984i	$-0.534934\pi$
$521$	−5.00000	−0.219054	−0.109527	+	0.993984i	$0.534934\pi$
$522$	−9.00000	−0.393919
$523$	27.0000	1.18063	0.590314	−	0.807174i	$-0.299004\pi$
$523$	27.0000	1.18063	0.590314	+	0.807174i	$0.299004\pi$
$524$	12.0000	0.524222
$525$	11.0000	0.480079
$526$	−26.0000	−1.13365
$527$	−20.0000	−0.871214
$528$	−3.00000	−0.130558
$529$	13.0000	0.565217
$530$	44.0000	1.91124
$531$	−2.00000	−0.0867926
$532$	3.00000	0.130066
$533$	0	0
$534$	7.00000	0.302920
$535$	−68.0000	−2.93990
$536$	−2.00000	−0.0863868
$537$	−24.0000	−1.03568
$538$	−24.0000	−1.03471
$539$	3.00000	0.129219
$540$	4.00000	0.172133
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	12.0000	0.515444
$543$	5.00000	0.214571
$544$	−5.00000	−0.214373
$545$	−8.00000	−0.342682
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	18.0000	0.768922
$549$	−1.00000	−0.0426790
$550$	33.0000	1.40712
$551$	27.0000	1.15024
$552$	−6.00000	−0.255377
$553$	−11.0000	−0.467768
$554$	14.0000	0.594803
$555$	−16.0000	−0.679162
$556$	13.0000	0.551323
$557$	−11.0000	−0.466085	−0.233042	−	0.972467i	$-0.574868\pi$
$557$	−11.0000	−0.466085	−0.233042	+	0.972467i	$0.574868\pi$
$558$	4.00000	0.169334
$559$	0	0
$560$	4.00000	0.169031
$561$	15.0000	0.633300
$562$	−30.0000	−1.26547
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−3.00000	−0.126323
$565$	−48.0000	−2.01938
$566$	−4.00000	−0.168133
$567$	−1.00000	−0.0419961
$568$	−6.00000	−0.251754
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	−12.0000	−0.502625
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	5.00000	0.208696
$575$	66.0000	2.75239
$576$	1.00000	0.0416667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	8.00000	0.332756
$579$	13.0000	0.540262
$580$	36.0000	1.49482
$581$	6.00000	0.248922
$582$	−12.0000	−0.497416
$583$	−33.0000	−1.36672
$584$	−12.0000	−0.496564
$585$	0	0
$586$	−24.0000	−0.991431
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−1.00000	−0.0412393
$589$	−12.0000	−0.494451
$590$	8.00000	0.329355
$591$	−19.0000	−0.781556
$592$	−4.00000	−0.164399
$593$	−41.0000	−1.68367	−0.841834	−	0.539736i	$-0.818524\pi$
$593$	−41.0000	−1.68367	−0.841834	+	0.539736i	$0.818524\pi$
$594$	−3.00000	−0.123091
$595$	−20.0000	−0.819920
$596$	−18.0000	−0.737309
$597$	0	0
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	−11.0000	−0.449073
$601$	−44.0000	−1.79480	−0.897399	−	0.441221i	$-0.854546\pi$
$601$	−44.0000	−1.79480	−0.897399	+	0.441221i	$0.854546\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	11.0000	0.447584
$605$	8.00000	0.325246
$606$	−10.0000	−0.406222
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−3.00000	−0.121666
$609$	−9.00000	−0.364698
$610$	4.00000	0.161955
$611$	0	0
$612$	−5.00000	−0.202113
$613$	−8.00000	−0.323117	−0.161558	−	0.986863i	$-0.551652\pi$
$613$	−8.00000	−0.323117	−0.161558	+	0.986863i	$0.551652\pi$
$614$	−21.0000	−0.847491
$615$	−20.0000	−0.806478
$616$	−3.00000	−0.120873
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	−20.0000	−0.804518
$619$	−17.0000	−0.683288	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−17.0000	−0.683288	−0.341644	+	0.939829i	$0.610984\pi$
$620$	−16.0000	−0.642575
$621$	−6.00000	−0.240772
$622$	−19.0000	−0.761831
$623$	7.00000	0.280449
$624$	0	0
$625$	41.0000	1.64000
$626$	−14.0000	−0.559553
$627$	9.00000	0.359425
$628$	−2.00000	−0.0798087
$629$	20.0000	0.797452
$630$	4.00000	0.159364
$631$	13.0000	0.517522	0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000	0.517522	0.258761	+	0.965941i	$0.416686\pi$
$632$	11.0000	0.437557
$633$	−14.0000	−0.556450
$634$	18.0000	0.714871
$635$	−32.0000	−1.26988
$636$	11.0000	0.436178
$637$	0	0
$638$	−27.0000	−1.06894
$639$	−6.00000	−0.237356
$640$	−4.00000	−0.158114
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−17.0000	−0.670936
$643$	17.0000	0.670415	0.335207	−	0.942144i	$-0.391194\pi$
$643$	17.0000	0.670415	0.335207	+	0.942144i	$0.391194\pi$
$644$	−6.00000	−0.236433
$645$	0	0
$646$	15.0000	0.590167
$647$	15.0000	0.589711	0.294855	−	0.955542i	$-0.404729\pi$
$647$	15.0000	0.589711	0.294855	+	0.955542i	$0.404729\pi$
$648$	1.00000	0.0392837
$649$	−6.00000	−0.235521
$650$	0	0
$651$	4.00000	0.156772
$652$	−20.0000	−0.783260
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	−2.00000	−0.0782062
$655$	−48.0000	−1.87552
$656$	−5.00000	−0.195217
$657$	−12.0000	−0.468165
$658$	−3.00000	−0.116952
$659$	39.0000	1.51922	0.759612	−	0.650376i	$-0.225389\pi$
$659$	39.0000	1.51922	0.759612	+	0.650376i	$0.225389\pi$
$660$	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	$0.437699\pi$
$662$	−2.00000	−0.0777322
$663$	0	0
$664$	−6.00000	−0.232845
$665$	−12.0000	−0.465340
$666$	−4.00000	−0.154997
$667$	−54.0000	−2.09089
$668$	16.0000	0.619059
$669$	−16.0000	−0.618596
$670$	8.00000	0.309067
$671$	−3.00000	−0.115814
$672$	1.00000	0.0385758
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	31.0000	1.19408
$675$	−11.0000	−0.423390
$676$	0	0
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	−12.0000	−0.460857
$679$	−12.0000	−0.460518
$680$	20.0000	0.766965
$681$	14.0000	0.536481
$682$	12.0000	0.459504
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	−3.00000	−0.114708
$685$	−72.0000	−2.75098
$686$	−1.00000	−0.0381802
$687$	−5.00000	−0.190762
$688$	0	0
$689$	0	0
$690$	24.0000	0.913664
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−2.00000	−0.0760286
$693$	−3.00000	−0.113961
$694$	11.0000	0.417554
$695$	−52.0000	−1.97247
$696$	9.00000	0.341144
$697$	25.0000	0.946943
$698$	34.0000	1.28692
$699$	−4.00000	−0.151294
$700$	−11.0000	−0.415761
$701$	−21.0000	−0.793159	−0.396580	−	0.918000i	$-0.629803\pi$
$701$	−21.0000	−0.793159	−0.396580	+	0.918000i	$0.629803\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	3.00000	0.113067
$705$	12.0000	0.451946
$706$	18.0000	0.677439
$707$	−10.0000	−0.376089
$708$	2.00000	0.0751646
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	24.0000	0.900704
$711$	11.0000	0.412532
$712$	−7.00000	−0.262336
$713$	24.0000	0.898807
$714$	−5.00000	−0.187120
$715$	0	0
$716$	24.0000	0.896922
$717$	−26.0000	−0.970988
$718$	20.0000	0.746393
$719$	−1.00000	−0.0372937	−0.0186469	−	0.999826i	$-0.505936\pi$
$719$	−1.00000	−0.0372937	−0.0186469	+	0.999826i	$0.505936\pi$
$720$	−4.00000	−0.149071
$721$	−20.0000	−0.744839
$722$	−10.0000	−0.372161
$723$	12.0000	0.446285
$724$	−5.00000	−0.185824
$725$	−99.0000	−3.67677
$726$	2.00000	0.0742270
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	48.0000	1.77656
$731$	0	0
$732$	1.00000	0.0369611
$733$	−13.0000	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$734$	18.0000	0.664392
$735$	4.00000	0.147542
$736$	6.00000	0.221163
$737$	−6.00000	−0.221013
$738$	−5.00000	−0.184053
$739$	54.0000	1.98642	0.993211	−	0.116326i	$-0.0371118\pi$
$739$	54.0000	1.98642	0.993211	+	0.116326i	$0.0371118\pi$
$740$	16.0000	0.588172
$741$	0	0
$742$	11.0000	0.403823
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	−4.00000	−0.146647
$745$	72.0000	2.63788
$746$	34.0000	1.24483
$747$	−6.00000	−0.219529
$748$	−15.0000	−0.548454
$749$	−17.0000	−0.621166
$750$	24.0000	0.876356
$751$	31.0000	1.13121	0.565603	−	0.824678i	$-0.308643\pi$
$751$	31.0000	1.13121	0.565603	+	0.824678i	$0.308643\pi$
$752$	3.00000	0.109399
$753$	−14.0000	−0.510188
$754$	0	0
$755$	−44.0000	−1.60132
$756$	1.00000	0.0363696
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	18.0000	0.653789
$759$	−18.0000	−0.653359
$760$	12.0000	0.435286
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	−8.00000	−0.289809
$763$	−2.00000	−0.0724049
$764$	−6.00000	−0.217072
$765$	20.0000	0.723102
$766$	11.0000	0.397446
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	12.0000	0.432450
$771$	−7.00000	−0.252099
$772$	−13.0000	−0.467880
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	44.0000	1.58053
$776$	12.0000	0.430775
$777$	−4.00000	−0.143499
$778$	−26.0000	−0.932145
$779$	15.0000	0.537431
$780$	0	0
$781$	−18.0000	−0.644091
$782$	−30.0000	−1.07280
$783$	9.00000	0.321634
$784$	1.00000	0.0357143
$785$	8.00000	0.285532
$786$	−12.0000	−0.428026
$787$	11.0000	0.392108	0.196054	−	0.980593i	$-0.437187\pi$
$787$	11.0000	0.392108	0.196054	+	0.980593i	$0.437187\pi$
$788$	19.0000	0.676847
$789$	26.0000	0.925625
$790$	−44.0000	−1.56545
$791$	−12.0000	−0.426671
$792$	3.00000	0.106600
$793$	0	0
$794$	23.0000	0.816239
$795$	−44.0000	−1.56052
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	−3.00000	−0.106199
$799$	−15.0000	−0.530662
$800$	11.0000	0.388909
$801$	−7.00000	−0.247333
$802$	−18.0000	−0.635602
$803$	−36.0000	−1.27041
$804$	2.00000	0.0705346
$805$	24.0000	0.845889
$806$	0	0
$807$	24.0000	0.844840
$808$	10.0000	0.351799
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	−4.00000	−0.140546
$811$	−24.0000	−0.842754	−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000	−0.842754	−0.421377	+	0.906886i	$0.638453\pi$
$812$	9.00000	0.315838
$813$	−12.0000	−0.420858
$814$	−12.0000	−0.420600
$815$	80.0000	2.80228
$816$	5.00000	0.175035
$817$	0	0
$818$	2.00000	0.0699284
$819$	0	0
$820$	20.0000	0.698430
$821$	15.0000	0.523504	0.261752	−	0.965135i	$-0.415700\pi$
$821$	15.0000	0.523504	0.261752	+	0.965135i	$0.415700\pi$
$822$	−18.0000	−0.627822
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	20.0000	0.696733
$825$	−33.0000	−1.14891
$826$	2.00000	0.0695889
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	6.00000	0.208514
$829$	5.00000	0.173657	0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000	0.173657	0.0868286	+	0.996223i	$0.472327\pi$
$830$	24.0000	0.833052
$831$	−14.0000	−0.485655
$832$	0	0
$833$	−5.00000	−0.173240
$834$	−13.0000	−0.450153
$835$	−64.0000	−2.21481
$836$	−9.00000	−0.311272
$837$	−4.00000	−0.138260
$838$	18.0000	0.621800
$839$	−44.0000	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$840$	−4.00000	−0.138013
$841$	52.0000	1.79310
$842$	18.0000	0.620321
$843$	30.0000	1.03325
$844$	14.0000	0.481900
$845$	0	0
$846$	3.00000	0.103142
$847$	2.00000	0.0687208
$848$	−11.0000	−0.377742
$849$	4.00000	0.137280
$850$	−55.0000	−1.88648
$851$	−24.0000	−0.822709
$852$	6.00000	0.205557
$853$	−49.0000	−1.67773	−0.838864	−	0.544341i	$-0.816780\pi$
$853$	−49.0000	−1.67773	−0.838864	+	0.544341i	$0.816780\pi$
$854$	1.00000	0.0342193
$855$	12.0000	0.410391
$856$	17.0000	0.581048
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−5.00000	−0.170598	−0.0852989	−	0.996355i	$-0.527185\pi$
$859$	−5.00000	−0.170598	−0.0852989	+	0.996355i	$0.527185\pi$
$860$	0	0
$861$	−5.00000	−0.170400
$862$	−38.0000	−1.29429
$863$	14.0000	0.476566	0.238283	−	0.971196i	$-0.423415\pi$
$863$	14.0000	0.476566	0.238283	+	0.971196i	$0.423415\pi$
$864$	−1.00000	−0.0340207
$865$	8.00000	0.272008
$866$	34.0000	1.15537
$867$	−8.00000	−0.271694
$868$	−4.00000	−0.135769
$869$	33.0000	1.11945
$870$	−36.0000	−1.22051
$871$	0	0
$872$	2.00000	0.0677285
$873$	12.0000	0.406138
$874$	−18.0000	−0.608859
$875$	24.0000	0.811348
$876$	12.0000	0.405442
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	40.0000	1.34993
$879$	24.0000	0.809500
$880$	−12.0000	−0.404520
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	1.00000	0.0336718
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	−15.0000	−0.503935
$887$	39.0000	1.30949	0.654746	−	0.755849i	$-0.272776\pi$
$887$	39.0000	1.30949	0.654746	+	0.755849i	$0.272776\pi$
$888$	4.00000	0.134231
$889$	−8.00000	−0.268311
$890$	28.0000	0.938562
$891$	3.00000	0.100504
$892$	16.0000	0.535720
$893$	−9.00000	−0.301174
$894$	18.0000	0.602010
$895$	−96.0000	−3.20893
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−20.0000	−0.667409
$899$	−36.0000	−1.20067
$900$	11.0000	0.366667
$901$	55.0000	1.83232
$902$	−15.0000	−0.499445
$903$	0	0
$904$	12.0000	0.399114
$905$	20.0000	0.664822
$906$	−11.0000	−0.365451
$907$	−14.0000	−0.464862	−0.232431	−	0.972613i	$-0.574668\pi$
$907$	−14.0000	−0.464862	−0.232431	+	0.972613i	$0.574668\pi$
$908$	−14.0000	−0.464606
$909$	10.0000	0.331679
$910$	0	0
$911$	−2.00000	−0.0662630	−0.0331315	−	0.999451i	$-0.510548\pi$
$911$	−2.00000	−0.0662630	−0.0331315	+	0.999451i	$0.510548\pi$
$912$	3.00000	0.0993399
$913$	−18.0000	−0.595713
$914$	2.00000	0.0661541
$915$	−4.00000	−0.132236
$916$	5.00000	0.165205
$917$	−12.0000	−0.396275
$918$	5.00000	0.165025
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	−24.0000	−0.791257
$921$	21.0000	0.691974
$922$	32.0000	1.05386
$923$	0	0
$924$	3.00000	0.0986928
$925$	−44.0000	−1.44671
$926$	−1.00000	−0.0328620
$927$	20.0000	0.656886
$928$	−9.00000	−0.295439
$929$	23.0000	0.754606	0.377303	−	0.926090i	$-0.376852\pi$
$929$	23.0000	0.754606	0.377303	+	0.926090i	$0.376852\pi$
$930$	16.0000	0.524661
$931$	−3.00000	−0.0983210
$932$	4.00000	0.131024
$933$	19.0000	0.622032
$934$	−12.0000	−0.392652
$935$	60.0000	1.96221
$936$	0	0
$937$	44.0000	1.43742	0.718709	−	0.695311i	$-0.244734\pi$
$937$	44.0000	1.43742	0.718709	+	0.695311i	$0.244734\pi$
$938$	2.00000	0.0653023
$939$	14.0000	0.456873
$940$	−12.0000	−0.391397
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	2.00000	0.0651635
$943$	−30.0000	−0.976934
$944$	−2.00000	−0.0650945
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	−11.0000	−0.357263
$949$	0	0
$950$	−33.0000	−1.07066
$951$	−18.0000	−0.583690
$952$	5.00000	0.162051
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	−11.0000	−0.356138
$955$	24.0000	0.776622
$956$	26.0000	0.840900
$957$	27.0000	0.872786
$958$	33.0000	1.06618
$959$	−18.0000	−0.581250
$960$	4.00000	0.129099
$961$	−15.0000	−0.483871
$962$	0	0
$963$	17.0000	0.547817
$964$	−12.0000	−0.386494
$965$	52.0000	1.67394
$966$	6.00000	0.193047
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−2.00000	−0.0642824
$969$	−15.0000	−0.481869
$970$	−48.0000	−1.54119
$971$	−50.0000	−1.60458	−0.802288	−	0.596937i	$-0.796384\pi$
$971$	−50.0000	−1.60458	−0.802288	+	0.596937i	$0.796384\pi$
$972$	−1.00000	−0.0320750
$973$	−13.0000	−0.416761
$974$	21.0000	0.672883
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	4.00000	0.127971	0.0639857	−	0.997951i	$-0.479619\pi$
$977$	4.00000	0.127971	0.0639857	+	0.997951i	$0.479619\pi$
$978$	20.0000	0.639529
$979$	−21.0000	−0.671163
$980$	−4.00000	−0.127775
$981$	2.00000	0.0638551
$982$	−20.0000	−0.638226
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	5.00000	0.159394
$985$	−76.0000	−2.42156
$986$	45.0000	1.43309
$987$	3.00000	0.0954911
$988$	0	0
$989$	0	0
$990$	−12.0000	−0.381385
$991$	33.0000	1.04828	0.524140	−	0.851632i	$-0.324387\pi$
$991$	33.0000	1.04828	0.524140	+	0.851632i	$0.324387\pi$
$992$	4.00000	0.127000
$993$	2.00000	0.0634681
$994$	6.00000	0.190308
$995$	0	0
$996$	6.00000	0.190117
$997$	−13.0000	−0.411714	−0.205857	−	0.978582i	$-0.565998\pi$
$997$	−13.0000	−0.411714	−0.205857	+	0.978582i	$0.565998\pi$
$998$	−40.0000	−1.26618
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.q.1.1		1
13.4	even	6		546.2.l.h.211.1	✓	2
13.10	even	6		546.2.l.h.295.1	yes	2
13.12	even	2		7098.2.a.g.1.1		1
39.17	odd	6		1638.2.r.a.757.1		2
39.23	odd	6		1638.2.r.a.1387.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.l.h.211.1	✓	2	13.4	even	6
546.2.l.h.295.1	yes	2	13.10	even	6
1638.2.r.a.757.1		2	39.17	odd	6
1638.2.r.a.1387.1		2	39.23	odd	6
7098.2.a.g.1.1		1	13.12	even	2
7098.2.a.q.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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