LMFDB - Embedded newform 7098.2.a.c.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{12} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} +3.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} -1.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} +2.00000 q^{40} -10.0000 q^{41} +1.00000 q^{42} +3.00000 q^{43} -2.00000 q^{45} -3.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} +7.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -6.00000 q^{57} -6.00000 q^{58} -7.00000 q^{59} +2.00000 q^{60} +11.0000 q^{61} -5.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -13.0000 q^{67} +1.00000 q^{68} -3.00000 q^{69} +2.00000 q^{70} -3.00000 q^{71} -1.00000 q^{72} +12.0000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +6.00000 q^{76} -4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -15.0000 q^{83} -1.00000 q^{84} -2.00000 q^{85} -3.00000 q^{86} -6.00000 q^{87} -11.0000 q^{89} +2.00000 q^{90} +3.00000 q^{92} -5.00000 q^{93} -6.00000 q^{94} -12.0000 q^{95} +1.00000 q^{96} +12.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−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
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−2.00000	−0.447214
$21$	−1.00000	−0.218218
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−2.00000	−0.365148
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	−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	$0.552568\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	2.00000	0.316228
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	1.00000	0.154303
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−3.00000	−0.442326
$47$	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
$51$	−1.00000	−0.140028
$52$	0	0
$53$	7.00000	0.961524	0.480762	−	0.876851i	$-0.340360\pi$
$53$	7.00000	0.961524	0.480762	+	0.876851i	$0.340360\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−6.00000	−0.794719
$58$	−6.00000	−0.787839
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	2.00000	0.258199
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	−5.00000	−0.635001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	1.00000	0.121268
$69$	−3.00000	−0.361158
$70$	2.00000	0.239046
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	−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
$76$	6.00000	0.688247
$77$	0	0
$78$	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
$82$	10.0000	1.10432
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	−1.00000	−0.109109
$85$	−2.00000	−0.216930
$86$	−3.00000	−0.323498
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	3.00000	0.312772
$93$	−5.00000	−0.518476
$94$	−6.00000	−0.618853
$95$	−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
$99$	0	0
$100$	−1.00000	−0.100000
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	1.00000	0.0990148
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	−7.00000	−0.679900
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	−1.00000	−0.0962250
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	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	$0.309095\pi$
$114$	6.00000	0.561951
$115$	−6.00000	−0.559503
$116$	6.00000	0.557086
$117$	0	0
$118$	7.00000	0.644402
$119$	1.00000	0.0916698
$120$	−2.00000	−0.182574
$121$	−11.0000	−1.00000
$122$	−11.0000	−0.995893
$123$	10.0000	0.901670
$124$	5.00000	0.449013
$125$	12.0000	1.07331
$126$	−1.00000	−0.0890871
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−1.00000	−0.0883883
$129$	−3.00000	−0.264135
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	13.0000	1.12303
$135$	2.00000	0.172133
$136$	−1.00000	−0.0857493
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	3.00000	0.255377
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−2.00000	−0.169031
$141$	−6.00000	−0.505291
$142$	3.00000	0.251754
$143$	0	0
$144$	1.00000	0.0833333
$145$	−12.0000	−0.996546
$146$	−12.0000	−0.993127
$147$	−1.00000	−0.0824786
$148$	−2.00000	−0.164399
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	−1.00000	−0.0816497
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−6.00000	−0.486664
$153$	1.00000	0.0808452
$154$	0	0
$155$	−10.0000	−0.803219
$156$	0	0
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	4.00000	0.318223
$159$	−7.00000	−0.555136
$160$	2.00000	0.158114
$161$	3.00000	0.236433
$162$	−1.00000	−0.0785674
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	15.0000	1.16423
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	2.00000	0.153393
$171$	6.00000	0.458831
$172$	3.00000	0.228748
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	6.00000	0.454859
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	7.00000	0.526152
$178$	11.0000	0.824485
$179$	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
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	−11.0000	−0.813143
$184$	−3.00000	−0.221163
$185$	4.00000	0.294086
$186$	5.00000	0.366618
$187$	0	0
$188$	6.00000	0.437595
$189$	−1.00000	−0.0727393
$190$	12.0000	0.870572
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	−1.00000	−0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	1.00000	0.0714286
$197$	−13.0000	−0.926212	−0.463106	−	0.886303i	$-0.653265\pi$
$197$	−13.0000	−0.926212	−0.463106	+	0.886303i	$0.653265\pi$
$198$	0	0
$199$	9.00000	0.637993	0.318997	−	0.947756i	$-0.396654\pi$
$199$	9.00000	0.637993	0.318997	+	0.947756i	$0.396654\pi$
$200$	1.00000	0.0707107
$201$	13.0000	0.916949
$202$	−4.00000	−0.281439
$203$	6.00000	0.421117
$204$	−1.00000	−0.0700140
$205$	20.0000	1.39686
$206$	1.00000	0.0696733
$207$	3.00000	0.208514
$208$	0	0
$209$	0	0
$210$	−2.00000	−0.138013
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	7.00000	0.480762
$213$	3.00000	0.205557
$214$	−2.00000	−0.136717
$215$	−6.00000	−0.409197
$216$	1.00000	0.0680414
$217$	5.00000	0.339422
$218$	−4.00000	−0.270914
$219$	−12.0000	−0.810885
$220$	0	0
$221$	0	0
$222$	−2.00000	−0.134231
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	−1.00000	−0.0668153
$225$	−1.00000	−0.0666667
$226$	−12.0000	−0.798228
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	−6.00000	−0.397360
$229$	−17.0000	−1.12339	−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000	−1.12339	−0.561696	+	0.827344i	$0.689851\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	−7.00000	−0.455661
$237$	4.00000	0.259828
$238$	−1.00000	−0.0648204
$239$	−29.0000	−1.87585	−0.937927	−	0.346833i	$-0.887257\pi$
$239$	−29.0000	−1.87585	−0.937927	+	0.346833i	$0.887257\pi$
$240$	2.00000	0.129099
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	11.0000	0.707107
$243$	−1.00000	−0.0641500
$244$	11.0000	0.704203
$245$	−2.00000	−0.127775
$246$	−10.0000	−0.637577
$247$	0	0
$248$	−5.00000	−0.317500
$249$	15.0000	0.950586
$250$	−12.0000	−0.758947
$251$	−7.00000	−0.441836	−0.220918	−	0.975292i	$-0.570905\pi$
$251$	−7.00000	−0.441836	−0.220918	+	0.975292i	$0.570905\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	4.00000	0.250982
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−11.0000	−0.686161	−0.343081	−	0.939306i	$-0.611470\pi$
$257$	−11.0000	−0.686161	−0.343081	+	0.939306i	$0.611470\pi$
$258$	3.00000	0.186772
$259$	−2.00000	−0.124274
$260$	0	0
$261$	6.00000	0.371391
$262$	15.0000	0.926703
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	0	0
$265$	−14.0000	−0.860013
$266$	−6.00000	−0.367884
$267$	11.0000	0.673189
$268$	−13.0000	−0.794101
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−2.00000	−0.121716
$271$	27.0000	1.64013	0.820067	−	0.572268i	$-0.193936\pi$
$271$	27.0000	1.64013	0.820067	+	0.572268i	$0.193936\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	−3.00000	−0.180579
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	−4.00000	−0.239904
$279$	5.00000	0.299342
$280$	2.00000	0.119523
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	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	$0.363394\pi$
$284$	−3.00000	−0.178017
$285$	12.0000	0.710819
$286$	0	0
$287$	−10.0000	−0.590281
$288$	−1.00000	−0.0589256
$289$	−16.0000	−0.941176
$290$	12.0000	0.704664
$291$	−12.0000	−0.703452
$292$	12.0000	0.702247
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	1.00000	0.0583212
$295$	14.0000	0.815112
$296$	2.00000	0.116248
$297$	0	0
$298$	15.0000	0.868927
$299$	0	0
$300$	1.00000	0.0577350
$301$	3.00000	0.172917
$302$	−4.00000	−0.230174
$303$	−4.00000	−0.229794
$304$	6.00000	0.344124
$305$	−22.0000	−1.25972
$306$	−1.00000	−0.0571662
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	0	0
$309$	1.00000	0.0568880
$310$	10.0000	0.567962
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	−32.0000	−1.80875	−0.904373	−	0.426742i	$-0.859661\pi$
$313$	−32.0000	−1.80875	−0.904373	+	0.426742i	$0.859661\pi$
$314$	−22.0000	−1.24153
$315$	−2.00000	−0.112687
$316$	−4.00000	−0.225018
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	7.00000	0.392541
$319$	0	0
$320$	−2.00000	−0.111803
$321$	−2.00000	−0.111629
$322$	−3.00000	−0.167183
$323$	6.00000	0.333849
$324$	1.00000	0.0555556
$325$	0	0
$326$	−5.00000	−0.276924
$327$	−4.00000	−0.221201
$328$	10.0000	0.552158
$329$	6.00000	0.330791
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−15.0000	−0.823232
$333$	−2.00000	−0.109599
$334$	−8.00000	−0.437741
$335$	26.0000	1.42053
$336$	−1.00000	−0.0545545
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	−2.00000	−0.108465
$341$	0	0
$342$	−6.00000	−0.324443
$343$	1.00000	0.0539949
$344$	−3.00000	−0.161749
$345$	6.00000	0.323029
$346$	−4.00000	−0.215041
$347$	26.0000	1.39575	0.697877	−	0.716218i	$-0.254128\pi$
$347$	26.0000	1.39575	0.697877	+	0.716218i	$0.254128\pi$
$348$	−6.00000	−0.321634
$349$	29.0000	1.55233	0.776167	−	0.630527i	$-0.217161\pi$
$349$	29.0000	1.55233	0.776167	+	0.630527i	$0.217161\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	0	0
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	−7.00000	−0.372046
$355$	6.00000	0.318447
$356$	−11.0000	−0.582999
$357$	−1.00000	−0.0529256
$358$	−6.00000	−0.317110
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	2.00000	0.105409
$361$	17.0000	0.894737
$362$	−10.0000	−0.525588
$363$	11.0000	0.577350
$364$	0	0
$365$	−24.0000	−1.25622
$366$	11.0000	0.574979
$367$	15.0000	0.782994	0.391497	−	0.920179i	$-0.371957\pi$
$367$	15.0000	0.782994	0.391497	+	0.920179i	$0.371957\pi$
$368$	3.00000	0.156386
$369$	−10.0000	−0.520579
$370$	−4.00000	−0.207950
$371$	7.00000	0.363422
$372$	−5.00000	−0.259238
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	−6.00000	−0.309426
$377$	0	0
$378$	1.00000	0.0514344
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	−12.0000	−0.615587
$381$	4.00000	0.204926
$382$	9.00000	0.460480
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	3.00000	0.152499
$388$	12.0000	0.609208
$389$	−5.00000	−0.253510	−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000	−0.253510	−0.126755	+	0.991934i	$0.540456\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	−1.00000	−0.0505076
$393$	15.0000	0.756650
$394$	13.0000	0.654931
$395$	8.00000	0.402524
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	−9.00000	−0.451129
$399$	−6.00000	−0.300376
$400$	−1.00000	−0.0500000
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	−13.0000	−0.648381
$403$	0	0
$404$	4.00000	0.199007
$405$	−2.00000	−0.0993808
$406$	−6.00000	−0.297775
$407$	0	0
$408$	1.00000	0.0495074
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	−20.0000	−0.987730
$411$	12.0000	0.591916
$412$	−1.00000	−0.0492665
$413$	−7.00000	−0.344447
$414$	−3.00000	−0.147442
$415$	30.0000	1.47264
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	2.00000	0.0975900
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	4.00000	0.194717
$423$	6.00000	0.291730
$424$	−7.00000	−0.339950
$425$	−1.00000	−0.0485071
$426$	−3.00000	−0.145350
$427$	11.0000	0.532327
$428$	2.00000	0.0966736
$429$	0	0
$430$	6.00000	0.289346
$431$	−13.0000	−0.626188	−0.313094	−	0.949722i	$-0.601365\pi$
$431$	−13.0000	−0.626188	−0.313094	+	0.949722i	$0.601365\pi$
$432$	−1.00000	−0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	−5.00000	−0.240008
$435$	12.0000	0.575356
$436$	4.00000	0.191565
$437$	18.0000	0.861057
$438$	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	$0.469569\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	2.00000	0.0949158
$445$	22.0000	1.04290
$446$	7.00000	0.331460
$447$	15.0000	0.709476
$448$	1.00000	0.0472456
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	12.0000	0.564433
$453$	−4.00000	−0.187936
$454$	−20.0000	−0.938647
$455$	0	0
$456$	6.00000	0.280976
$457$	−41.0000	−1.91790	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−41.0000	−1.91790	−0.958950	+	0.283577i	$0.908479\pi$
$458$	17.0000	0.794358
$459$	−1.00000	−0.0466760
$460$	−6.00000	−0.279751
$461$	16.0000	0.745194	0.372597	−	0.927993i	$-0.378467\pi$
$461$	16.0000	0.745194	0.372597	+	0.927993i	$0.378467\pi$
$462$	0	0
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	6.00000	0.278543
$465$	10.0000	0.463739
$466$	8.00000	0.370593
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	0	0
$469$	−13.0000	−0.600284
$470$	12.0000	0.553519
$471$	−22.0000	−1.01371
$472$	7.00000	0.322201
$473$	0	0
$474$	−4.00000	−0.183726
$475$	−6.00000	−0.275299
$476$	1.00000	0.0458349
$477$	7.00000	0.320508
$478$	29.0000	1.32643
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−2.00000	−0.0912871
$481$	0	0
$482$	18.0000	0.819878
$483$	−3.00000	−0.136505
$484$	−11.0000	−0.500000
$485$	−24.0000	−1.08978
$486$	1.00000	0.0453609
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	−11.0000	−0.497947
$489$	−5.00000	−0.226108
$490$	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	$0.557777\pi$
$492$	10.0000	0.450835
$493$	6.00000	0.270226
$494$	0	0
$495$	0	0
$496$	5.00000	0.224507
$497$	−3.00000	−0.134568
$498$	−15.0000	−0.672166
$499$	−17.0000	−0.761025	−0.380512	−	0.924776i	$-0.624252\pi$
$499$	−17.0000	−0.761025	−0.380512	+	0.924776i	$0.624252\pi$
$500$	12.0000	0.536656
$501$	−8.00000	−0.357414
$502$	7.00000	0.312425
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	−1.00000	−0.0445435
$505$	−8.00000	−0.355995
$506$	0	0
$507$	0	0
$508$	−4.00000	−0.177471
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	−2.00000	−0.0885615
$511$	12.0000	0.530849
$512$	−1.00000	−0.0441942
$513$	−6.00000	−0.264906
$514$	11.0000	0.485189
$515$	2.00000	0.0881305
$516$	−3.00000	−0.132068
$517$	0	0
$518$	2.00000	0.0878750
$519$	−4.00000	−0.175581
$520$	0	0
$521$	34.0000	1.48957	0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000	1.48957	0.744784	+	0.667306i	$0.232553\pi$
$522$	−6.00000	−0.262613
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	−15.0000	−0.655278
$525$	1.00000	0.0436436
$526$	−28.0000	−1.22086
$527$	5.00000	0.217803
$528$	0	0
$529$	−14.0000	−0.608696
$530$	14.0000	0.608121
$531$	−7.00000	−0.303774
$532$	6.00000	0.260133
$533$	0	0
$534$	−11.0000	−0.476017
$535$	−4.00000	−0.172935
$536$	13.0000	0.561514
$537$	−6.00000	−0.258919
$538$	6.00000	0.258678
$539$	0	0
$540$	2.00000	0.0860663
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−27.0000	−1.15975
$543$	−10.0000	−0.429141
$544$	−1.00000	−0.0428746
$545$	−8.00000	−0.342682
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−12.0000	−0.512615
$549$	11.0000	0.469469
$550$	0	0
$551$	36.0000	1.53365
$552$	3.00000	0.127688
$553$	−4.00000	−0.170097
$554$	4.00000	0.169944
$555$	−4.00000	−0.169791
$556$	4.00000	0.169638
$557$	−31.0000	−1.31351	−0.656756	−	0.754103i	$-0.728072\pi$
$557$	−31.0000	−1.31351	−0.656756	+	0.754103i	$0.728072\pi$
$558$	−5.00000	−0.211667
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−18.0000	−0.759284
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	−6.00000	−0.252646
$565$	−24.0000	−1.00969
$566$	−14.0000	−0.588464
$567$	1.00000	0.0419961
$568$	3.00000	0.125877
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	−12.0000	−0.502625
$571$	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
$573$	9.00000	0.375980
$574$	10.0000	0.417392
$575$	−3.00000	−0.125109
$576$	1.00000	0.0416667
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	16.0000	0.665512
$579$	−10.0000	−0.415586
$580$	−12.0000	−0.498273
$581$	−15.0000	−0.622305
$582$	12.0000	0.497416
$583$	0	0
$584$	−12.0000	−0.496564
$585$	0	0
$586$	−24.0000	−0.991431
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	−1.00000	−0.0412393
$589$	30.0000	1.23613
$590$	−14.0000	−0.576371
$591$	13.0000	0.534749
$592$	−2.00000	−0.0821995
$593$	35.0000	1.43728	0.718639	−	0.695383i	$-0.244765\pi$
$593$	35.0000	1.43728	0.718639	+	0.695383i	$0.244765\pi$
$594$	0	0
$595$	−2.00000	−0.0819920
$596$	−15.0000	−0.614424
$597$	−9.00000	−0.368345
$598$	0	0
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	−1.00000	−0.0408248
$601$	46.0000	1.87638	0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000	1.87638	0.938190	+	0.346122i	$0.112502\pi$
$602$	−3.00000	−0.122271
$603$	−13.0000	−0.529401
$604$	4.00000	0.162758
$605$	22.0000	0.894427
$606$	4.00000	0.162489
$607$	−35.0000	−1.42061	−0.710303	−	0.703896i	$-0.751442\pi$
$607$	−35.0000	−1.42061	−0.710303	+	0.703896i	$0.751442\pi$
$608$	−6.00000	−0.243332
$609$	−6.00000	−0.243132
$610$	22.0000	0.890754
$611$	0	0
$612$	1.00000	0.0404226
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	18.0000	0.726421
$615$	−20.0000	−0.806478
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	−1.00000	−0.0402259
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	−10.0000	−0.401610
$621$	−3.00000	−0.120386
$622$	−14.0000	−0.561349
$623$	−11.0000	−0.440706
$624$	0	0
$625$	−19.0000	−0.760000
$626$	32.0000	1.27898
$627$	0	0
$628$	22.0000	0.877896
$629$	−2.00000	−0.0797452
$630$	2.00000	0.0796819
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	4.00000	0.159111
$633$	4.00000	0.158986
$634$	−3.00000	−0.119145
$635$	8.00000	0.317470
$636$	−7.00000	−0.277568
$637$	0	0
$638$	0	0
$639$	−3.00000	−0.118678
$640$	2.00000	0.0790569
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	2.00000	0.0789337
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	3.00000	0.118217
$645$	6.00000	0.236250
$646$	−6.00000	−0.236067
$647$	−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
$649$	0	0
$650$	0	0
$651$	−5.00000	−0.195965
$652$	5.00000	0.195815
$653$	15.0000	0.586995	0.293498	−	0.955960i	$-0.405181\pi$
$653$	15.0000	0.586995	0.293498	+	0.955960i	$0.405181\pi$
$654$	4.00000	0.156412
$655$	30.0000	1.17220
$656$	−10.0000	−0.390434
$657$	12.0000	0.468165
$658$	−6.00000	−0.233904
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	23.0000	0.894596	0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000	0.894596	0.447298	+	0.894385i	$0.352386\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	15.0000	0.582113
$665$	−12.0000	−0.465340
$666$	2.00000	0.0774984
$667$	18.0000	0.696963
$668$	8.00000	0.309529
$669$	7.00000	0.270636
$670$	−26.0000	−1.00447
$671$	0	0
$672$	1.00000	0.0385758
$673$	−25.0000	−0.963679	−0.481840	−	0.876259i	$-0.660031\pi$
$673$	−25.0000	−0.963679	−0.481840	+	0.876259i	$0.660031\pi$
$674$	−34.0000	−1.30963
$675$	1.00000	0.0384900
$676$	0	0
$677$	4.00000	0.153732	0.0768662	−	0.997041i	$-0.475509\pi$
$677$	4.00000	0.153732	0.0768662	+	0.997041i	$0.475509\pi$
$678$	12.0000	0.460857
$679$	12.0000	0.460518
$680$	2.00000	0.0766965
$681$	−20.0000	−0.766402
$682$	0	0
$683$	46.0000	1.76014	0.880071	−	0.474843i	$-0.157495\pi$
$683$	46.0000	1.76014	0.880071	+	0.474843i	$0.157495\pi$
$684$	6.00000	0.229416
$685$	24.0000	0.916993
$686$	−1.00000	−0.0381802
$687$	17.0000	0.648590
$688$	3.00000	0.114374
$689$	0	0
$690$	−6.00000	−0.228416
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	4.00000	0.152057
$693$	0	0
$694$	−26.0000	−0.986947
$695$	−8.00000	−0.303457
$696$	6.00000	0.227429
$697$	−10.0000	−0.378777
$698$	−29.0000	−1.09767
$699$	8.00000	0.302588
$700$	−1.00000	−0.0377964
$701$	−27.0000	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$701$	−27.0000	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	12.0000	0.451946
$706$	21.0000	0.790345
$707$	4.00000	0.150435
$708$	7.00000	0.263076
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	−6.00000	−0.225176
$711$	−4.00000	−0.150012
$712$	11.0000	0.412242
$713$	15.0000	0.561754
$714$	1.00000	0.0374241
$715$	0	0
$716$	6.00000	0.224231
$717$	29.0000	1.08302
$718$	−16.0000	−0.597115
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	−2.00000	−0.0745356
$721$	−1.00000	−0.0372419
$722$	−17.0000	−0.632674
$723$	18.0000	0.669427
$724$	10.0000	0.371647
$725$	−6.00000	−0.222834
$726$	−11.0000	−0.408248
$727$	33.0000	1.22390	0.611951	−	0.790896i	$-0.290385\pi$
$727$	33.0000	1.22390	0.611951	+	0.790896i	$0.290385\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	24.0000	0.888280
$731$	3.00000	0.110959
$732$	−11.0000	−0.406572
$733$	25.0000	0.923396	0.461698	−	0.887037i	$-0.347240\pi$
$733$	25.0000	0.923396	0.461698	+	0.887037i	$0.347240\pi$
$734$	−15.0000	−0.553660
$735$	2.00000	0.0737711
$736$	−3.00000	−0.110581
$737$	0	0
$738$	10.0000	0.368105
$739$	9.00000	0.331070	0.165535	−	0.986204i	$-0.447065\pi$
$739$	9.00000	0.331070	0.165535	+	0.986204i	$0.447065\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	−7.00000	−0.256978
$743$	43.0000	1.57752	0.788759	−	0.614703i	$-0.210724\pi$
$743$	43.0000	1.57752	0.788759	+	0.614703i	$0.210724\pi$
$744$	5.00000	0.183309
$745$	30.0000	1.09911
$746$	8.00000	0.292901
$747$	−15.0000	−0.548821
$748$	0	0
$749$	2.00000	0.0730784
$750$	12.0000	0.438178
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	6.00000	0.218797
$753$	7.00000	0.255094
$754$	0	0
$755$	−8.00000	−0.291150
$756$	−1.00000	−0.0363696
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−12.0000	−0.435860
$759$	0	0
$760$	12.0000	0.435286
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	−4.00000	−0.144905
$763$	4.00000	0.144810
$764$	−9.00000	−0.325609
$765$	−2.00000	−0.0723102
$766$	−4.00000	−0.144526
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	11.0000	0.396155
$772$	10.0000	0.359908
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	−3.00000	−0.107833
$775$	−5.00000	−0.179605
$776$	−12.0000	−0.430775
$777$	2.00000	0.0717496
$778$	5.00000	0.179259
$779$	−60.0000	−2.14972
$780$	0	0
$781$	0	0
$782$	−3.00000	−0.107280
$783$	−6.00000	−0.214423
$784$	1.00000	0.0357143
$785$	−44.0000	−1.57043
$786$	−15.0000	−0.535032
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−13.0000	−0.463106
$789$	−28.0000	−0.996826
$790$	−8.00000	−0.284627
$791$	12.0000	0.426671
$792$	0	0
$793$	0	0
$794$	−13.0000	−0.461353
$795$	14.0000	0.496529
$796$	9.00000	0.318997
$797$	−20.0000	−0.708436	−0.354218	−	0.935163i	$-0.615253\pi$
$797$	−20.0000	−0.708436	−0.354218	+	0.935163i	$0.615253\pi$
$798$	6.00000	0.212398
$799$	6.00000	0.212265
$800$	1.00000	0.0353553
$801$	−11.0000	−0.388666
$802$	24.0000	0.847469
$803$	0	0
$804$	13.0000	0.458475
$805$	−6.00000	−0.211472
$806$	0	0
$807$	6.00000	0.211210
$808$	−4.00000	−0.140720
$809$	28.0000	0.984428	0.492214	−	0.870474i	$-0.336188\pi$
$809$	28.0000	0.984428	0.492214	+	0.870474i	$0.336188\pi$
$810$	2.00000	0.0702728
$811$	−24.0000	−0.842754	−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000	−0.842754	−0.421377	+	0.906886i	$0.638453\pi$
$812$	6.00000	0.210559
$813$	−27.0000	−0.946931
$814$	0	0
$815$	−10.0000	−0.350285
$816$	−1.00000	−0.0350070
$817$	18.0000	0.629740
$818$	2.00000	0.0699284
$819$	0	0
$820$	20.0000	0.698430
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	−12.0000	−0.418548
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	1.00000	0.0348367
$825$	0	0
$826$	7.00000	0.243561
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	3.00000	0.104257
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	−30.0000	−1.04132
$831$	4.00000	0.138758
$832$	0	0
$833$	1.00000	0.0346479
$834$	4.00000	0.138509
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−5.00000	−0.172825
$838$	−15.0000	−0.518166
$839$	−34.0000	−1.17381	−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000	−1.17381	−0.586905	+	0.809656i	$0.699654\pi$
$840$	−2.00000	−0.0690066
$841$	7.00000	0.241379
$842$	−6.00000	−0.206774
$843$	−18.0000	−0.619953
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−6.00000	−0.206284
$847$	−11.0000	−0.377964
$848$	7.00000	0.240381
$849$	−14.0000	−0.480479
$850$	1.00000	0.0342997
$851$	−6.00000	−0.205677
$852$	3.00000	0.102778
$853$	−23.0000	−0.787505	−0.393753	−	0.919216i	$-0.628823\pi$
$853$	−23.0000	−0.787505	−0.393753	+	0.919216i	$0.628823\pi$
$854$	−11.0000	−0.376412
$855$	−12.0000	−0.410391
$856$	−2.00000	−0.0683586
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−6.00000	−0.204598
$861$	10.0000	0.340799
$862$	13.0000	0.442782
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	1.00000	0.0340207
$865$	−8.00000	−0.272008
$866$	−16.0000	−0.543702
$867$	16.0000	0.543388
$868$	5.00000	0.169711
$869$	0	0
$870$	−12.0000	−0.406838
$871$	0	0
$872$	−4.00000	−0.135457
$873$	12.0000	0.406138
$874$	−18.0000	−0.608859
$875$	12.0000	0.405674
$876$	−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	$0.750907\pi$
$878$	−4.00000	−0.134993
$879$	−24.0000	−0.809500
$880$	0	0
$881$	−1.00000	−0.0336909	−0.0168454	−	0.999858i	$-0.505362\pi$
$881$	−1.00000	−0.0336909	−0.0168454	+	0.999858i	$0.505362\pi$
$882$	−1.00000	−0.0336718
$883$	25.0000	0.841317	0.420658	−	0.907219i	$-0.361799\pi$
$883$	25.0000	0.841317	0.420658	+	0.907219i	$0.361799\pi$
$884$	0	0
$885$	−14.0000	−0.470605
$886$	−18.0000	−0.604722
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	−2.00000	−0.0671156
$889$	−4.00000	−0.134156
$890$	−22.0000	−0.737442
$891$	0	0
$892$	−7.00000	−0.234377
$893$	36.0000	1.20469
$894$	−15.0000	−0.501675
$895$	−12.0000	−0.401116
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−14.0000	−0.467186
$899$	30.0000	1.00056
$900$	−1.00000	−0.0333333
$901$	7.00000	0.233204
$902$	0	0
$903$	−3.00000	−0.0998337
$904$	−12.0000	−0.399114
$905$	−20.0000	−0.664822
$906$	4.00000	0.132891
$907$	−41.0000	−1.36138	−0.680691	−	0.732570i	$-0.738320\pi$
$907$	−41.0000	−1.36138	−0.680691	+	0.732570i	$0.738320\pi$
$908$	20.0000	0.663723
$909$	4.00000	0.132672
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	−6.00000	−0.198680
$913$	0	0
$914$	41.0000	1.35616
$915$	22.0000	0.727298
$916$	−17.0000	−0.561696
$917$	−15.0000	−0.495344
$918$	1.00000	0.0330049
$919$	58.0000	1.91324	0.956622	−	0.291333i	$-0.0940987\pi$
$919$	58.0000	1.91324	0.956622	+	0.291333i	$0.0940987\pi$
$920$	6.00000	0.197814
$921$	18.0000	0.593120
$922$	−16.0000	−0.526932
$923$	0	0
$924$	0	0
$925$	2.00000	0.0657596
$926$	−34.0000	−1.11731
$927$	−1.00000	−0.0328443
$928$	−6.00000	−0.196960
$929$	43.0000	1.41078	0.705392	−	0.708817i	$-0.250771\pi$
$929$	43.0000	1.41078	0.705392	+	0.708817i	$0.250771\pi$
$930$	−10.0000	−0.327913
$931$	6.00000	0.196642
$932$	−8.00000	−0.262049
$933$	−14.0000	−0.458339
$934$	−27.0000	−0.883467
$935$	0	0
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	13.0000	0.424465
$939$	32.0000	1.04428
$940$	−12.0000	−0.391397
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	22.0000	0.716799
$943$	−30.0000	−0.976934
$944$	−7.00000	−0.227831
$945$	2.00000	0.0650600
$946$	0	0
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	4.00000	0.129914
$949$	0	0
$950$	6.00000	0.194666
$951$	−3.00000	−0.0972817
$952$	−1.00000	−0.0324102
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	−7.00000	−0.226633
$955$	18.0000	0.582466
$956$	−29.0000	−0.937927
$957$	0	0
$958$	0	0
$959$	−12.0000	−0.387500
$960$	2.00000	0.0645497
$961$	−6.00000	−0.193548
$962$	0	0
$963$	2.00000	0.0644491
$964$	−18.0000	−0.579741
$965$	−20.0000	−0.643823
$966$	3.00000	0.0965234
$967$	54.0000	1.73652	0.868261	−	0.496107i	$-0.165238\pi$
$967$	54.0000	1.73652	0.868261	+	0.496107i	$0.165238\pi$
$968$	11.0000	0.353553
$969$	−6.00000	−0.192748
$970$	24.0000	0.770594
$971$	−47.0000	−1.50830	−0.754151	−	0.656701i	$-0.771951\pi$
$971$	−47.0000	−1.50830	−0.754151	+	0.656701i	$0.771951\pi$
$972$	−1.00000	−0.0320750
$973$	4.00000	0.128234
$974$	−12.0000	−0.384505
$975$	0	0
$976$	11.0000	0.352101
$977$	44.0000	1.40768	0.703842	−	0.710356i	$-0.251466\pi$
$977$	44.0000	1.40768	0.703842	+	0.710356i	$0.251466\pi$
$978$	5.00000	0.159882
$979$	0	0
$980$	−2.00000	−0.0638877
$981$	4.00000	0.127710
$982$	8.00000	0.255290
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	−10.0000	−0.318788
$985$	26.0000	0.828429
$986$	−6.00000	−0.191079
$987$	−6.00000	−0.190982
$988$	0	0
$989$	9.00000	0.286183
$990$	0	0
$991$	−18.0000	−0.571789	−0.285894	−	0.958261i	$-0.592291\pi$
$991$	−18.0000	−0.571789	−0.285894	+	0.958261i	$0.592291\pi$
$992$	−5.00000	−0.158750
$993$	−20.0000	−0.634681
$994$	3.00000	0.0951542
$995$	−18.0000	−0.570638
$996$	15.0000	0.475293
$997$	−25.0000	−0.791758	−0.395879	−	0.918303i	$-0.629560\pi$
$997$	−25.0000	−0.791758	−0.395879	+	0.918303i	$0.629560\pi$
$998$	17.0000	0.538126
$999$	2.00000	0.0632772

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.l.f.211.1	✓	2	13.9	even	3
546.2.l.f.295.1	yes	2	13.3	even	3
1638.2.r.j.757.1		2	39.35	odd	6
1638.2.r.j.1387.1		2	39.29	odd	6
7098.2.a.c.1.1		1	1.1	even	1	trivial
7098.2.a.u.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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