LMFDB - Embedded newform 7098.2.a.h.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +7.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} -9.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -7.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} +2.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} -5.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +7.00000 q^{51} +3.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +1.00000 q^{56} +2.00000 q^{57} +2.00000 q^{58} +15.0000 q^{59} -2.00000 q^{60} -7.00000 q^{61} +9.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -5.00000 q^{67} +7.00000 q^{68} -1.00000 q^{69} -2.00000 q^{70} +1.00000 q^{71} -1.00000 q^{72} +12.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +2.00000 q^{76} +4.00000 q^{77} -4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -1.00000 q^{83} -1.00000 q^{84} -14.0000 q^{85} +5.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} +3.00000 q^{89} +2.00000 q^{90} -1.00000 q^{92} -9.00000 q^{93} -6.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -16.0000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	−1.00000	−0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−2.00000	−0.447214
$21$	−1.00000	−0.218218
$22$	4.00000	0.852803
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	2.00000	0.365148
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	−7.00000	−1.20049
$35$	2.00000	0.338062
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	2.00000	0.316228
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	1.00000	0.154303
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	−4.00000	−0.603023
$45$	−2.00000	−0.298142
$46$	1.00000	0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	7.00000	0.980196
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	−1.00000	−0.136083
$55$	8.00000	1.07872
$56$	1.00000	0.133631
$57$	2.00000	0.264906
$58$	2.00000	0.262613
$59$	15.0000	1.95283	0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000	1.95283	0.976417	+	0.215894i	$0.0692665\pi$
$60$	−2.00000	−0.258199
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	9.00000	1.14300
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	7.00000	0.848875
$69$	−1.00000	−0.120386
$70$	−2.00000	−0.239046
$71$	1.00000	0.118678	0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000	0.118678	0.0593391	+	0.998238i	$0.481101\pi$
$72$	−1.00000	−0.117851
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	2.00000	0.229416
$77$	4.00000	0.455842
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	−1.00000	−0.109109
$85$	−14.0000	−1.51851
$86$	5.00000	0.539164
$87$	−2.00000	−0.214423
$88$	4.00000	0.426401
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	−1.00000	−0.104257
$93$	−9.00000	−0.933257
$94$	−6.00000	−0.618853
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	−1.00000	−0.101015
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−7.00000	−0.693103
$103$	−11.0000	−1.08386	−0.541931	−	0.840423i	$-0.682307\pi$
$103$	−11.0000	−1.08386	−0.541931	+	0.840423i	$0.682307\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	−3.00000	−0.291386
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	1.00000	0.0962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	−8.00000	−0.762770
$111$	−2.00000	−0.189832
$112$	−1.00000	−0.0944911
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	−2.00000	−0.187317
$115$	2.00000	0.186501
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−15.0000	−1.38086
$119$	−7.00000	−0.641689
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	7.00000	0.633750
$123$	2.00000	0.180334
$124$	−9.00000	−0.808224
$125$	12.0000	1.07331
$126$	1.00000	0.0890871
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	−5.00000	−0.440225
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	−4.00000	−0.348155
$133$	−2.00000	−0.173422
$134$	5.00000	0.431934
$135$	−2.00000	−0.172133
$136$	−7.00000	−0.600245
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	1.00000	0.0851257
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	2.00000	0.169031
$141$	6.00000	0.505291
$142$	−1.00000	−0.0839181
$143$	0	0
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−12.0000	−0.993127
$147$	1.00000	0.0824786
$148$	−2.00000	−0.164399
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	1.00000	0.0816497
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	−2.00000	−0.162221
$153$	7.00000	0.565916
$154$	−4.00000	−0.322329
$155$	18.0000	1.44579
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	4.00000	0.318223
$159$	3.00000	0.237915
$160$	2.00000	0.158114
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	21.0000	1.64485	0.822423	−	0.568876i	$-0.192621\pi$
$163$	21.0000	1.64485	0.822423	+	0.568876i	$0.192621\pi$
$164$	2.00000	0.156174
$165$	8.00000	0.622799
$166$	1.00000	0.0776151
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	14.0000	1.07375
$171$	2.00000	0.152944
$172$	−5.00000	−0.381246
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	2.00000	0.151620
$175$	1.00000	0.0755929
$176$	−4.00000	−0.301511
$177$	15.0000	1.12747
$178$	−3.00000	−0.224860
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	−2.00000	−0.149071
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−7.00000	−0.517455
$184$	1.00000	0.0737210
$185$	4.00000	0.294086
$186$	9.00000	0.659912
$187$	−28.0000	−2.04756
$188$	6.00000	0.437595
$189$	−1.00000	−0.0727393
$190$	4.00000	0.290191
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	1.00000	0.0721688
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	1.00000	0.0714286
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	4.00000	0.284268
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	1.00000	0.0707107
$201$	−5.00000	−0.352673
$202$	0	0
$203$	2.00000	0.140372
$204$	7.00000	0.490098
$205$	−4.00000	−0.279372
$206$	11.0000	0.766406
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−8.00000	−0.553372
$210$	−2.00000	−0.138013
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	3.00000	0.206041
$213$	1.00000	0.0685189
$214$	6.00000	0.410152
$215$	10.0000	0.681994
$216$	−1.00000	−0.0680414
$217$	9.00000	0.610960
$218$	16.0000	1.08366
$219$	12.0000	0.810885
$220$	8.00000	0.539360
$221$	0	0
$222$	2.00000	0.134231
$223$	3.00000	0.200895	0.100447	−	0.994942i	$-0.467973\pi$
$223$	3.00000	0.200895	0.100447	+	0.994942i	$0.467973\pi$
$224$	1.00000	0.0668153
$225$	−1.00000	−0.0666667
$226$	8.00000	0.532152
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	2.00000	0.132453
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	−2.00000	−0.131876
$231$	4.00000	0.263181
$232$	2.00000	0.131306
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	15.0000	0.976417
$237$	−4.00000	−0.259828
$238$	7.00000	0.453743
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	−2.00000	−0.129099
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	−7.00000	−0.448129
$245$	−2.00000	−0.127775
$246$	−2.00000	−0.127515
$247$	0	0
$248$	9.00000	0.571501
$249$	−1.00000	−0.0633724
$250$	−12.0000	−0.758947
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	−16.0000	−1.00393
$255$	−14.0000	−0.876714
$256$	1.00000	0.0625000
$257$	27.0000	1.68421	0.842107	−	0.539311i	$-0.181315\pi$
$257$	27.0000	1.68421	0.842107	+	0.539311i	$0.181315\pi$
$258$	5.00000	0.311286
$259$	2.00000	0.124274
$260$	0	0
$261$	−2.00000	−0.123797
$262$	−15.0000	−0.926703
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	4.00000	0.246183
$265$	−6.00000	−0.368577
$266$	2.00000	0.122628
$267$	3.00000	0.183597
$268$	−5.00000	−0.305424
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	2.00000	0.121716
$271$	9.00000	0.546711	0.273356	−	0.961913i	$-0.411866\pi$
$271$	9.00000	0.546711	0.273356	+	0.961913i	$0.411866\pi$
$272$	7.00000	0.424437
$273$	0	0
$274$	−4.00000	−0.241649
$275$	4.00000	0.241209
$276$	−1.00000	−0.0601929
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−16.0000	−0.959616
$279$	−9.00000	−0.538816
$280$	−2.00000	−0.119523
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	−6.00000	−0.357295
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	1.00000	0.0593391
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−2.00000	−0.118056
$288$	−1.00000	−0.0589256
$289$	32.0000	1.88235
$290$	−4.00000	−0.234888
$291$	−16.0000	−0.937937
$292$	12.0000	0.702247
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	−1.00000	−0.0583212
$295$	−30.0000	−1.74667
$296$	2.00000	0.116248
$297$	−4.00000	−0.232104
$298$	−21.0000	−1.21650
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	5.00000	0.288195
$302$	−20.0000	−1.15087
$303$	0	0
$304$	2.00000	0.114708
$305$	14.0000	0.801638
$306$	−7.00000	−0.400163
$307$	30.0000	1.71219	0.856095	−	0.516818i	$-0.172884\pi$
$307$	30.0000	1.71219	0.856095	+	0.516818i	$0.172884\pi$
$308$	4.00000	0.227921
$309$	−11.0000	−0.625768
$310$	−18.0000	−1.02233
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	−10.0000	−0.564333
$315$	2.00000	0.112687
$316$	−4.00000	−0.225018
$317$	23.0000	1.29181	0.645904	−	0.763418i	$-0.276480\pi$
$317$	23.0000	1.29181	0.645904	+	0.763418i	$0.276480\pi$
$318$	−3.00000	−0.168232
$319$	8.00000	0.447914
$320$	−2.00000	−0.111803
$321$	−6.00000	−0.334887
$322$	−1.00000	−0.0557278
$323$	14.0000	0.778981
$324$	1.00000	0.0555556
$325$	0	0
$326$	−21.0000	−1.16308
$327$	−16.0000	−0.884802
$328$	−2.00000	−0.110432
$329$	−6.00000	−0.330791
$330$	−8.00000	−0.440386
$331$	36.0000	1.97874	0.989369	−	0.145424i	$-0.0464545\pi$
$331$	36.0000	1.97874	0.989369	+	0.145424i	$0.0464545\pi$
$332$	−1.00000	−0.0548821
$333$	−2.00000	−0.109599
$334$	0	0
$335$	10.0000	0.546358
$336$	−1.00000	−0.0545545
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	−8.00000	−0.434500
$340$	−14.0000	−0.759257
$341$	36.0000	1.94951
$342$	−2.00000	−0.108148
$343$	−1.00000	−0.0539949
$344$	5.00000	0.269582
$345$	2.00000	0.107676
$346$	16.0000	0.860165
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	−2.00000	−0.107211
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	4.00000	0.213201
$353$	−35.0000	−1.86286	−0.931431	−	0.363918i	$-0.881439\pi$
$353$	−35.0000	−1.86286	−0.931431	+	0.363918i	$0.881439\pi$
$354$	−15.0000	−0.797241
$355$	−2.00000	−0.106149
$356$	3.00000	0.159000
$357$	−7.00000	−0.370479
$358$	−6.00000	−0.317110
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	2.00000	0.105409
$361$	−15.0000	−0.789474
$362$	2.00000	0.105118
$363$	5.00000	0.262432
$364$	0	0
$365$	−24.0000	−1.25622
$366$	7.00000	0.365896
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	−1.00000	−0.0521286
$369$	2.00000	0.104116
$370$	−4.00000	−0.207950
$371$	−3.00000	−0.155752
$372$	−9.00000	−0.466628
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	28.0000	1.44785
$375$	12.0000	0.619677
$376$	−6.00000	−0.309426
$377$	0	0
$378$	1.00000	0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−4.00000	−0.205196
$381$	16.0000	0.819705
$382$	−3.00000	−0.153493
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−1.00000	−0.0510310
$385$	−8.00000	−0.407718
$386$	22.0000	1.11977
$387$	−5.00000	−0.254164
$388$	−16.0000	−0.812277
$389$	−1.00000	−0.0507020	−0.0253510	−	0.999679i	$-0.508070\pi$
$389$	−1.00000	−0.0507020	−0.0253510	+	0.999679i	$0.508070\pi$
$390$	0	0
$391$	−7.00000	−0.354005
$392$	−1.00000	−0.0505076
$393$	15.0000	0.756650
$394$	9.00000	0.453413
$395$	8.00000	0.402524
$396$	−4.00000	−0.201008
$397$	−1.00000	−0.0501886	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000	−0.0501886	−0.0250943	+	0.999685i	$0.507989\pi$
$398$	5.00000	0.250627
$399$	−2.00000	−0.100125
$400$	−1.00000	−0.0500000
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	5.00000	0.249377
$403$	0	0
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	−2.00000	−0.0992583
$407$	8.00000	0.396545
$408$	−7.00000	−0.346552
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	4.00000	0.197546
$411$	4.00000	0.197305
$412$	−11.0000	−0.541931
$413$	−15.0000	−0.738102
$414$	1.00000	0.0491473
$415$	2.00000	0.0981761
$416$	0	0
$417$	16.0000	0.783523
$418$	8.00000	0.391293
$419$	−31.0000	−1.51445	−0.757225	−	0.653155i	$-0.773445\pi$
$419$	−31.0000	−1.51445	−0.757225	+	0.653155i	$0.773445\pi$
$420$	2.00000	0.0975900
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	−4.00000	−0.194717
$423$	6.00000	0.291730
$424$	−3.00000	−0.145693
$425$	−7.00000	−0.339550
$426$	−1.00000	−0.0484502
$427$	7.00000	0.338754
$428$	−6.00000	−0.290021
$429$	0	0
$430$	−10.0000	−0.482243
$431$	−1.00000	−0.0481683	−0.0240842	−	0.999710i	$-0.507667\pi$
$431$	−1.00000	−0.0481683	−0.0240842	+	0.999710i	$0.507667\pi$
$432$	1.00000	0.0481125
$433$	12.0000	0.576683	0.288342	−	0.957528i	$-0.406896\pi$
$433$	12.0000	0.576683	0.288342	+	0.957528i	$0.406896\pi$
$434$	−9.00000	−0.432014
$435$	4.00000	0.191785
$436$	−16.0000	−0.766261
$437$	−2.00000	−0.0956730
$438$	−12.0000	−0.573382
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	−8.00000	−0.381385
$441$	1.00000	0.0476190
$442$	0	0
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	−2.00000	−0.0949158
$445$	−6.00000	−0.284427
$446$	−3.00000	−0.142054
$447$	21.0000	0.993266
$448$	−1.00000	−0.0472456
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	1.00000	0.0471405
$451$	−8.00000	−0.376705
$452$	−8.00000	−0.376288
$453$	20.0000	0.939682
$454$	−28.0000	−1.31411
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	−9.00000	−0.421002	−0.210501	−	0.977594i	$-0.567510\pi$
$457$	−9.00000	−0.421002	−0.210501	+	0.977594i	$0.567510\pi$
$458$	−13.0000	−0.607450
$459$	7.00000	0.326732
$460$	2.00000	0.0932505
$461$	−20.0000	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$462$	−4.00000	−0.186097
$463$	18.0000	0.836531	0.418265	−	0.908325i	$-0.362638\pi$
$463$	18.0000	0.836531	0.418265	+	0.908325i	$0.362638\pi$
$464$	−2.00000	−0.0928477
$465$	18.0000	0.834730
$466$	16.0000	0.741186
$467$	37.0000	1.71216	0.856078	−	0.516847i	$-0.172894\pi$
$467$	37.0000	1.71216	0.856078	+	0.516847i	$0.172894\pi$
$468$	0	0
$469$	5.00000	0.230879
$470$	12.0000	0.553519
$471$	10.0000	0.460776
$472$	−15.0000	−0.690431
$473$	20.0000	0.919601
$474$	4.00000	0.183726
$475$	−2.00000	−0.0917663
$476$	−7.00000	−0.320844
$477$	3.00000	0.137361
$478$	9.00000	0.411650
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	2.00000	0.0912871
$481$	0	0
$482$	−14.0000	−0.637683
$483$	1.00000	0.0455016
$484$	5.00000	0.227273
$485$	32.0000	1.45305
$486$	−1.00000	−0.0453609
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	7.00000	0.316875
$489$	21.0000	0.949653
$490$	2.00000	0.0903508
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	2.00000	0.0901670
$493$	−14.0000	−0.630528
$494$	0	0
$495$	8.00000	0.359573
$496$	−9.00000	−0.404112
$497$	−1.00000	−0.0448561
$498$	1.00000	0.0448111
$499$	7.00000	0.313363	0.156682	−	0.987649i	$-0.449920\pi$
$499$	7.00000	0.313363	0.156682	+	0.987649i	$0.449920\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−23.0000	−1.02654
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	−4.00000	−0.177822
$507$	0	0
$508$	16.0000	0.709885
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	14.0000	0.619930
$511$	−12.0000	−0.530849
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	−27.0000	−1.19092
$515$	22.0000	0.969436
$516$	−5.00000	−0.220113
$517$	−24.0000	−1.05552
$518$	−2.00000	−0.0878750
$519$	−16.0000	−0.702322
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	2.00000	0.0875376
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	15.0000	0.655278
$525$	1.00000	0.0436436
$526$	−4.00000	−0.174408
$527$	−63.0000	−2.74432
$528$	−4.00000	−0.174078
$529$	−22.0000	−0.956522
$530$	6.00000	0.260623
$531$	15.0000	0.650945
$532$	−2.00000	−0.0867110
$533$	0	0
$534$	−3.00000	−0.129823
$535$	12.0000	0.518805
$536$	5.00000	0.215967
$537$	6.00000	0.258919
$538$	−18.0000	−0.776035
$539$	−4.00000	−0.172292
$540$	−2.00000	−0.0860663
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	−9.00000	−0.386583
$543$	−2.00000	−0.0858282
$544$	−7.00000	−0.300123
$545$	32.0000	1.37073
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	4.00000	0.170872
$549$	−7.00000	−0.298753
$550$	−4.00000	−0.170561
$551$	−4.00000	−0.170406
$552$	1.00000	0.0425628
$553$	4.00000	0.170097
$554$	8.00000	0.339887
$555$	4.00000	0.169791
$556$	16.0000	0.678551
$557$	13.0000	0.550828	0.275414	−	0.961326i	$-0.411185\pi$
$557$	13.0000	0.550828	0.275414	+	0.961326i	$0.411185\pi$
$558$	9.00000	0.381000
$559$	0	0
$560$	2.00000	0.0845154
$561$	−28.0000	−1.18216
$562$	26.0000	1.09674
$563$	−40.0000	−1.68580	−0.842900	−	0.538071i	$-0.819153\pi$
$563$	−40.0000	−1.68580	−0.842900	+	0.538071i	$0.819153\pi$
$564$	6.00000	0.252646
$565$	16.0000	0.673125
$566$	−14.0000	−0.588464
$567$	−1.00000	−0.0419961
$568$	−1.00000	−0.0419591
$569$	26.0000	1.08998	0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000	1.08998	0.544988	+	0.838444i	$0.316534\pi$
$570$	4.00000	0.167542
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	0	0
$573$	3.00000	0.125327
$574$	2.00000	0.0834784
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	−32.0000	−1.33102
$579$	−22.0000	−0.914289
$580$	4.00000	0.166091
$581$	1.00000	0.0414870
$582$	16.0000	0.663221
$583$	−12.0000	−0.496989
$584$	−12.0000	−0.496564
$585$	0	0
$586$	16.0000	0.660954
$587$	45.0000	1.85735	0.928674	−	0.370896i	$-0.120949\pi$
$587$	45.0000	1.85735	0.928674	+	0.370896i	$0.120949\pi$
$588$	1.00000	0.0412393
$589$	−18.0000	−0.741677
$590$	30.0000	1.23508
$591$	−9.00000	−0.370211
$592$	−2.00000	−0.0821995
$593$	29.0000	1.19089	0.595444	−	0.803397i	$-0.296976\pi$
$593$	29.0000	1.19089	0.595444	+	0.803397i	$0.296976\pi$
$594$	4.00000	0.164122
$595$	14.0000	0.573944
$596$	21.0000	0.860194
$597$	−5.00000	−0.204636
$598$	0	0
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	1.00000	0.0408248
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	−5.00000	−0.203785
$603$	−5.00000	−0.203616
$604$	20.0000	0.813788
$605$	−10.0000	−0.406558
$606$	0	0
$607$	47.0000	1.90767	0.953836	−	0.300329i	$-0.0970966\pi$
$607$	47.0000	1.90767	0.953836	+	0.300329i	$0.0970966\pi$
$608$	−2.00000	−0.0811107
$609$	2.00000	0.0810441
$610$	−14.0000	−0.566843
$611$	0	0
$612$	7.00000	0.282958
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−30.0000	−1.21070
$615$	−4.00000	−0.161296
$616$	−4.00000	−0.161165
$617$	−20.0000	−0.805170	−0.402585	−	0.915383i	$-0.631888\pi$
$617$	−20.0000	−0.805170	−0.402585	+	0.915383i	$0.631888\pi$
$618$	11.0000	0.442485
$619$	−22.0000	−0.884255	−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000	−0.884255	−0.442127	+	0.896952i	$0.645776\pi$
$620$	18.0000	0.722897
$621$	−1.00000	−0.0401286
$622$	−30.0000	−1.20289
$623$	−3.00000	−0.120192
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	10.0000	0.399043
$629$	−14.0000	−0.558217
$630$	−2.00000	−0.0796819
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	4.00000	0.159111
$633$	4.00000	0.158986
$634$	−23.0000	−0.913447
$635$	−32.0000	−1.26988
$636$	3.00000	0.118958
$637$	0	0
$638$	−8.00000	−0.316723
$639$	1.00000	0.0395594
$640$	2.00000	0.0790569
$641$	32.0000	1.26392	0.631962	−	0.774999i	$-0.282250\pi$
$641$	32.0000	1.26392	0.631962	+	0.774999i	$0.282250\pi$
$642$	6.00000	0.236801
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	1.00000	0.0394055
$645$	10.0000	0.393750
$646$	−14.0000	−0.550823
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	−1.00000	−0.0392837
$649$	−60.0000	−2.35521
$650$	0	0
$651$	9.00000	0.352738
$652$	21.0000	0.822423
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	16.0000	0.625650
$655$	−30.0000	−1.17220
$656$	2.00000	0.0780869
$657$	12.0000	0.468165
$658$	6.00000	0.233904
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	8.00000	0.311400
$661$	−35.0000	−1.36134	−0.680671	−	0.732589i	$-0.738312\pi$
$661$	−35.0000	−1.36134	−0.680671	+	0.732589i	$0.738312\pi$
$662$	−36.0000	−1.39918
$663$	0	0
$664$	1.00000	0.0388075
$665$	4.00000	0.155113
$666$	2.00000	0.0774984
$667$	2.00000	0.0774403
$668$	0	0
$669$	3.00000	0.115987
$670$	−10.0000	−0.386334
$671$	28.0000	1.08093
$672$	1.00000	0.0385758
$673$	−33.0000	−1.27206	−0.636028	−	0.771666i	$-0.719424\pi$
$673$	−33.0000	−1.27206	−0.636028	+	0.771666i	$0.719424\pi$
$674$	−18.0000	−0.693334
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	28.0000	1.07613	0.538064	−	0.842904i	$-0.319156\pi$
$677$	28.0000	1.07613	0.538064	+	0.842904i	$0.319156\pi$
$678$	8.00000	0.307238
$679$	16.0000	0.614024
$680$	14.0000	0.536875
$681$	28.0000	1.07296
$682$	−36.0000	−1.37851
$683$	22.0000	0.841807	0.420903	−	0.907106i	$-0.361713\pi$
$683$	22.0000	0.841807	0.420903	+	0.907106i	$0.361713\pi$
$684$	2.00000	0.0764719
$685$	−8.00000	−0.305664
$686$	1.00000	0.0381802
$687$	13.0000	0.495981
$688$	−5.00000	−0.190623
$689$	0	0
$690$	−2.00000	−0.0761387
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	−16.0000	−0.608229
$693$	4.00000	0.151947
$694$	−10.0000	−0.379595
$695$	−32.0000	−1.21383
$696$	2.00000	0.0758098
$697$	14.0000	0.530288
$698$	25.0000	0.946264
$699$	−16.0000	−0.605176
$700$	1.00000	0.0377964
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	−4.00000	−0.150756
$705$	−12.0000	−0.451946
$706$	35.0000	1.31724
$707$	0	0
$708$	15.0000	0.563735
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	2.00000	0.0750587
$711$	−4.00000	−0.150012
$712$	−3.00000	−0.112430
$713$	9.00000	0.337053
$714$	7.00000	0.261968
$715$	0	0
$716$	6.00000	0.224231
$717$	−9.00000	−0.336111
$718$	24.0000	0.895672
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	−2.00000	−0.0745356
$721$	11.0000	0.409661
$722$	15.0000	0.558242
$723$	14.0000	0.520666
$724$	−2.00000	−0.0743294
$725$	2.00000	0.0742781
$726$	−5.00000	−0.185567
$727$	−21.0000	−0.778847	−0.389423	−	0.921059i	$-0.627326\pi$
$727$	−21.0000	−0.778847	−0.389423	+	0.921059i	$0.627326\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	24.0000	0.888280
$731$	−35.0000	−1.29452
$732$	−7.00000	−0.258727
$733$	−21.0000	−0.775653	−0.387826	−	0.921732i	$-0.626774\pi$
$733$	−21.0000	−0.775653	−0.387826	+	0.921732i	$0.626774\pi$
$734$	3.00000	0.110732
$735$	−2.00000	−0.0737711
$736$	1.00000	0.0368605
$737$	20.0000	0.736709
$738$	−2.00000	−0.0736210
$739$	49.0000	1.80249	0.901247	−	0.433306i	$-0.142653\pi$
$739$	49.0000	1.80249	0.901247	+	0.433306i	$0.142653\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	3.00000	0.110133
$743$	−41.0000	−1.50414	−0.752072	−	0.659081i	$-0.770945\pi$
$743$	−41.0000	−1.50414	−0.752072	+	0.659081i	$0.770945\pi$
$744$	9.00000	0.329956
$745$	−42.0000	−1.53876
$746$	−4.00000	−0.146450
$747$	−1.00000	−0.0365881
$748$	−28.0000	−1.02378
$749$	6.00000	0.219235
$750$	−12.0000	−0.438178
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	6.00000	0.218797
$753$	23.0000	0.838167
$754$	0	0
$755$	−40.0000	−1.45575
$756$	−1.00000	−0.0363696
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	20.0000	0.726433
$759$	4.00000	0.145191
$760$	4.00000	0.145095
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	−16.0000	−0.579619
$763$	16.0000	0.579239
$764$	3.00000	0.108536
$765$	−14.0000	−0.506171
$766$	−24.0000	−0.867155
$767$	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	$0.477023\pi$
$770$	8.00000	0.288300
$771$	27.0000	0.972381
$772$	−22.0000	−0.791797
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	5.00000	0.179721
$775$	9.00000	0.323290
$776$	16.0000	0.574367
$777$	2.00000	0.0717496
$778$	1.00000	0.0358517
$779$	4.00000	0.143315
$780$	0	0
$781$	−4.00000	−0.143131
$782$	7.00000	0.250319
$783$	−2.00000	−0.0714742
$784$	1.00000	0.0357143
$785$	−20.0000	−0.713831
$786$	−15.0000	−0.535032
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	−9.00000	−0.320612
$789$	4.00000	0.142404
$790$	−8.00000	−0.284627
$791$	8.00000	0.284447
$792$	4.00000	0.142134
$793$	0	0
$794$	1.00000	0.0354887
$795$	−6.00000	−0.212798
$796$	−5.00000	−0.177220
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	2.00000	0.0707992
$799$	42.0000	1.48585
$800$	1.00000	0.0353553
$801$	3.00000	0.106000
$802$	−32.0000	−1.12996
$803$	−48.0000	−1.69388
$804$	−5.00000	−0.176336
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	18.0000	0.633630
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	2.00000	0.0702728
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	2.00000	0.0701862
$813$	9.00000	0.315644
$814$	−8.00000	−0.280400
$815$	−42.0000	−1.47120
$816$	7.00000	0.245049
$817$	−10.0000	−0.349856
$818$	−14.0000	−0.489499
$819$	0	0
$820$	−4.00000	−0.139686
$821$	43.0000	1.50071	0.750355	−	0.661035i	$-0.229882\pi$
$821$	43.0000	1.50071	0.750355	+	0.661035i	$0.229882\pi$
$822$	−4.00000	−0.139516
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	11.0000	0.383203
$825$	4.00000	0.139262
$826$	15.0000	0.521917
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	−1.00000	−0.0347524
$829$	42.0000	1.45872	0.729360	−	0.684130i	$-0.239818\pi$
$829$	42.0000	1.45872	0.729360	+	0.684130i	$0.239818\pi$
$830$	−2.00000	−0.0694210
$831$	−8.00000	−0.277517
$832$	0	0
$833$	7.00000	0.242536
$834$	−16.0000	−0.554035
$835$	0	0
$836$	−8.00000	−0.276686
$837$	−9.00000	−0.311086
$838$	31.0000	1.07088
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	−2.00000	−0.0690066
$841$	−25.0000	−0.862069
$842$	−34.0000	−1.17172
$843$	−26.0000	−0.895488
$844$	4.00000	0.137686
$845$	0	0
$846$	−6.00000	−0.206284
$847$	−5.00000	−0.171802
$848$	3.00000	0.103020
$849$	14.0000	0.480479
$850$	7.00000	0.240098
$851$	2.00000	0.0685591
$852$	1.00000	0.0342594
$853$	−29.0000	−0.992941	−0.496471	−	0.868054i	$-0.665371\pi$
$853$	−29.0000	−0.992941	−0.496471	+	0.868054i	$0.665371\pi$
$854$	−7.00000	−0.239535
$855$	−4.00000	−0.136797
$856$	6.00000	0.205076
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	10.0000	0.340997
$861$	−2.00000	−0.0681598
$862$	1.00000	0.0340601
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	−1.00000	−0.0340207
$865$	32.0000	1.08803
$866$	−12.0000	−0.407777
$867$	32.0000	1.08678
$868$	9.00000	0.305480
$869$	16.0000	0.542763
$870$	−4.00000	−0.135613
$871$	0	0
$872$	16.0000	0.541828
$873$	−16.0000	−0.541518
$874$	2.00000	0.0676510
$875$	−12.0000	−0.405674
$876$	12.0000	0.405442
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	−4.00000	−0.134993
$879$	−16.0000	−0.539667
$880$	8.00000	0.269680
$881$	−47.0000	−1.58347	−0.791735	−	0.610865i	$-0.790822\pi$
$881$	−47.0000	−1.58347	−0.791735	+	0.610865i	$0.790822\pi$
$882$	−1.00000	−0.0336718
$883$	−7.00000	−0.235569	−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−7.00000	−0.235569	−0.117784	+	0.993039i	$0.537579\pi$
$884$	0	0
$885$	−30.0000	−1.00844
$886$	6.00000	0.201574
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	2.00000	0.0671156
$889$	−16.0000	−0.536623
$890$	6.00000	0.201120
$891$	−4.00000	−0.134005
$892$	3.00000	0.100447
$893$	12.0000	0.401565
$894$	−21.0000	−0.702345
$895$	−12.0000	−0.401116
$896$	1.00000	0.0334077
$897$	0	0
$898$	−18.0000	−0.600668
$899$	18.0000	0.600334
$900$	−1.00000	−0.0333333
$901$	21.0000	0.699611
$902$	8.00000	0.266371
$903$	5.00000	0.166390
$904$	8.00000	0.266076
$905$	4.00000	0.132964
$906$	−20.0000	−0.664455
$907$	39.0000	1.29497	0.647487	−	0.762077i	$-0.275820\pi$
$907$	39.0000	1.29497	0.647487	+	0.762077i	$0.275820\pi$
$908$	28.0000	0.929213
$909$	0	0
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	2.00000	0.0662266
$913$	4.00000	0.132381
$914$	9.00000	0.297694
$915$	14.0000	0.462826
$916$	13.0000	0.429532
$917$	−15.0000	−0.495344
$918$	−7.00000	−0.231034
$919$	−46.0000	−1.51740	−0.758700	−	0.651440i	$-0.774165\pi$
$919$	−46.0000	−1.51740	−0.758700	+	0.651440i	$0.774165\pi$
$920$	−2.00000	−0.0659380
$921$	30.0000	0.988534
$922$	20.0000	0.658665
$923$	0	0
$924$	4.00000	0.131590
$925$	2.00000	0.0657596
$926$	−18.0000	−0.591517
$927$	−11.0000	−0.361287
$928$	2.00000	0.0656532
$929$	−27.0000	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$930$	−18.0000	−0.590243
$931$	2.00000	0.0655474
$932$	−16.0000	−0.524097
$933$	30.0000	0.982156
$934$	−37.0000	−1.21068
$935$	56.0000	1.83140
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	−5.00000	−0.163256
$939$	0	0
$940$	−12.0000	−0.391397
$941$	−28.0000	−0.912774	−0.456387	−	0.889781i	$-0.650857\pi$
$941$	−28.0000	−0.912774	−0.456387	+	0.889781i	$0.650857\pi$
$942$	−10.0000	−0.325818
$943$	−2.00000	−0.0651290
$944$	15.0000	0.488208
$945$	2.00000	0.0650600
$946$	−20.0000	−0.650256
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	2.00000	0.0648886
$951$	23.0000	0.745826
$952$	7.00000	0.226871
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	−3.00000	−0.0971286
$955$	−6.00000	−0.194155
$956$	−9.00000	−0.291081
$957$	8.00000	0.258603
$958$	0	0
$959$	−4.00000	−0.129167
$960$	−2.00000	−0.0645497
$961$	50.0000	1.61290
$962$	0	0
$963$	−6.00000	−0.193347
$964$	14.0000	0.450910
$965$	44.0000	1.41641
$966$	−1.00000	−0.0321745
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	−5.00000	−0.160706
$969$	14.0000	0.449745
$970$	−32.0000	−1.02746
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	1.00000	0.0320750
$973$	−16.0000	−0.512936
$974$	28.0000	0.897178
$975$	0	0
$976$	−7.00000	−0.224065
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	−21.0000	−0.671506
$979$	−12.0000	−0.383522
$980$	−2.00000	−0.0638877
$981$	−16.0000	−0.510841
$982$	8.00000	0.255290
$983$	−46.0000	−1.46717	−0.733586	−	0.679597i	$-0.762155\pi$
$983$	−46.0000	−1.46717	−0.733586	+	0.679597i	$0.762155\pi$
$984$	−2.00000	−0.0637577
$985$	18.0000	0.573528
$986$	14.0000	0.445851
$987$	−6.00000	−0.190982
$988$	0	0
$989$	5.00000	0.158991
$990$	−8.00000	−0.254257
$991$	18.0000	0.571789	0.285894	−	0.958261i	$-0.407709\pi$
$991$	18.0000	0.571789	0.285894	+	0.958261i	$0.407709\pi$
$992$	9.00000	0.285750
$993$	36.0000	1.14243
$994$	1.00000	0.0317181
$995$	10.0000	0.317021
$996$	−1.00000	−0.0316862
$997$	−35.0000	−1.10846	−0.554231	−	0.832363i	$-0.686987\pi$
$997$	−35.0000	−1.10846	−0.554231	+	0.832363i	$0.686987\pi$
$998$	−7.00000	−0.221581
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.h.1.1		1
13.3	even	3		546.2.l.d.295.1	yes	2
13.9	even	3		546.2.l.d.211.1	✓	2
13.12	even	2		7098.2.a.bf.1.1		1
39.29	odd	6		1638.2.r.k.1387.1		2
39.35	odd	6		1638.2.r.k.757.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.l.d.211.1	✓	2	13.9	even	3
546.2.l.d.295.1	yes	2	13.3	even	3
1638.2.r.k.757.1		2	39.35	odd	6
1638.2.r.k.1387.1		2	39.29	odd	6
7098.2.a.h.1.1		1	1.1	even	1	trivial
7098.2.a.bf.1.1		1	13.12	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:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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