LMFDB - Embedded newform 75.8.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,8,Mod(1,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$23.4288769113$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	75.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-6.00000 q^{2} +27.0000 q^{3} -92.0000 q^{4} -162.000 q^{6} +64.0000 q^{7} +1320.00 q^{8} +729.000 q^{9} -948.000 q^{11} -2484.00 q^{12} +5098.00 q^{13} -384.000 q^{14} +3856.00 q^{16} -28386.0 q^{17} -4374.00 q^{18} -8620.00 q^{19} +1728.00 q^{21} +5688.00 q^{22} +15288.0 q^{23} +35640.0 q^{24} -30588.0 q^{26} +19683.0 q^{27} -5888.00 q^{28} +36510.0 q^{29} -276808. q^{31} -192096. q^{32} -25596.0 q^{33} +170316. q^{34} -67068.0 q^{36} -268526. q^{37} +51720.0 q^{38} +137646. q^{39} -629718. q^{41} -10368.0 q^{42} -685772. q^{43} +87216.0 q^{44} -91728.0 q^{46} -583296. q^{47} +104112. q^{48} -819447. q^{49} -766422. q^{51} -469016. q^{52} +428058. q^{53} -118098. q^{54} +84480.0 q^{56} -232740. q^{57} -219060. q^{58} +1.30638e6 q^{59} +300662. q^{61} +1.66085e6 q^{62} +46656.0 q^{63} +659008. q^{64} +153576. q^{66} +507244. q^{67} +2.61151e6 q^{68} +412776. q^{69} +5.56063e6 q^{71} +962280. q^{72} -1.36908e6 q^{73} +1.61116e6 q^{74} +793040. q^{76} -60672.0 q^{77} -825876. q^{78} -6.91372e6 q^{79} +531441. q^{81} +3.77831e6 q^{82} +4.37675e6 q^{83} -158976. q^{84} +4.11463e6 q^{86} +985770. q^{87} -1.25136e6 q^{88} -8.52831e6 q^{89} +326272. q^{91} -1.40650e6 q^{92} -7.47382e6 q^{93} +3.49978e6 q^{94} -5.18659e6 q^{96} +8.82681e6 q^{97} +4.91668e6 q^{98} -691092. q^{99} +O(q^{100})\)

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 $

$2$	−6.00000	−0.530330	−0.265165	−	0.964203i	$-0.585426\pi$
$2$	−6.00000	−0.530330	−0.265165	+	0.964203i	$0.585426\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−92.0000	−0.718750
$5$	0	0
$5$	0	0
$6$	−162.000	−0.306186
$7$	64.0000	0.0705240	0.0352620	−	0.999378i	$-0.488773\pi$
$7$	64.0000	0.0705240	0.0352620	+	0.999378i	$0.488773\pi$
$8$	1320.00	0.911505
$9$	729.000	0.333333
$10$	0	0
$11$	−948.000	−0.214750	−0.107375	−	0.994219i	$-0.534245\pi$
$11$	−948.000	−0.214750	−0.107375	+	0.994219i	$0.534245\pi$
$12$	−2484.00	−0.414971
$13$	5098.00	0.643573	0.321787	−	0.946812i	$-0.395717\pi$
$13$	5098.00	0.643573	0.321787	+	0.946812i	$0.395717\pi$
$14$	−384.000	−0.0374010
$15$	0	0
$16$	3856.00	0.235352
$17$	−28386.0	−1.40131	−0.700653	−	0.713502i	$-0.747108\pi$
$17$	−28386.0	−1.40131	−0.700653	+	0.713502i	$0.747108\pi$
$18$	−4374.00	−0.176777
$19$	−8620.00	−0.288317	−0.144158	−	0.989555i	$-0.546047\pi$
$19$	−8620.00	−0.288317	−0.144158	+	0.989555i	$0.546047\pi$
$20$	0	0
$21$	1728.00	0.0407170
$22$	5688.00	0.113889
$23$	15288.0	0.262001	0.131001	−	0.991382i	$-0.458181\pi$
$23$	15288.0	0.262001	0.131001	+	0.991382i	$0.458181\pi$
$24$	35640.0	0.526258
$25$	0	0
$26$	−30588.0	−0.341306
$27$	19683.0	0.192450
$28$	−5888.00	−0.0506891
$29$	36510.0	0.277983	0.138992	−	0.990294i	$-0.455614\pi$
$29$	36510.0	0.277983	0.138992	+	0.990294i	$0.455614\pi$
$30$	0	0
$31$	−276808.	−1.66883	−0.834416	−	0.551135i	$-0.814195\pi$
$31$	−276808.	−1.66883	−0.834416	+	0.551135i	$0.814195\pi$
$32$	−192096.	−1.03632
$33$	−25596.0	−0.123986
$34$	170316.	0.743155
$35$	0	0
$36$	−67068.0	−0.239583
$37$	−268526.	−0.871526	−0.435763	−	0.900061i	$-0.643521\pi$
$37$	−268526.	−0.871526	−0.435763	+	0.900061i	$0.643521\pi$
$38$	51720.0	0.152903
$39$	137646.	0.371567
$40$	0	0
$41$	−629718.	−1.42693	−0.713465	−	0.700691i	$-0.752875\pi$
$41$	−629718.	−1.42693	−0.713465	+	0.700691i	$0.752875\pi$
$42$	−10368.0	−0.0215935
$43$	−685772.	−1.31535	−0.657673	−	0.753303i	$-0.728459\pi$
$43$	−685772.	−1.31535	−0.657673	+	0.753303i	$0.728459\pi$
$44$	87216.0	0.154352
$45$	0	0
$46$	−91728.0	−0.138947
$47$	−583296.	−0.819495	−0.409748	−	0.912199i	$-0.634383\pi$
$47$	−583296.	−0.819495	−0.409748	+	0.912199i	$0.634383\pi$
$48$	104112.	0.135880
$49$	−819447.	−0.995026
$50$	0	0
$51$	−766422.	−0.809044
$52$	−469016.	−0.462568
$53$	428058.	0.394945	0.197473	−	0.980308i	$-0.436727\pi$
$53$	428058.	0.394945	0.197473	+	0.980308i	$0.436727\pi$
$54$	−118098.	−0.102062
$55$	0	0
$56$	84480.0	0.0642830
$57$	−232740.	−0.166460
$58$	−219060.	−0.147423
$59$	1.30638e6	0.828109	0.414054	−	0.910252i	$-0.364112\pi$
$59$	1.30638e6	0.828109	0.414054	+	0.910252i	$0.364112\pi$
$60$	0	0
$61$	300662.	0.169599	0.0847997	−	0.996398i	$-0.472975\pi$
$61$	300662.	0.169599	0.0847997	+	0.996398i	$0.472975\pi$
$62$	1.66085e6	0.885032
$63$	46656.0	0.0235080
$64$	659008.	0.314240
$65$	0	0
$66$	153576.	0.0657536
$67$	507244.	0.206042	0.103021	−	0.994679i	$-0.467149\pi$
$67$	507244.	0.206042	0.103021	+	0.994679i	$0.467149\pi$
$68$	2.61151e6	1.00719
$69$	412776.	0.151266
$70$	0	0
$71$	5.56063e6	1.84383	0.921913	−	0.387397i	$-0.126626\pi$
$71$	5.56063e6	1.84383	0.921913	+	0.387397i	$0.126626\pi$
$72$	962280.	0.303835
$73$	−1.36908e6	−0.411907	−0.205954	−	0.978562i	$-0.566030\pi$
$73$	−1.36908e6	−0.411907	−0.205954	+	0.978562i	$0.566030\pi$
$74$	1.61116e6	0.462196
$75$	0	0
$76$	793040.	0.207228
$77$	−60672.0	−0.0151451
$78$	−825876.	−0.197053
$79$	−6.91372e6	−1.57767	−0.788836	−	0.614603i	$-0.789316\pi$
$79$	−6.91372e6	−1.57767	−0.788836	+	0.614603i	$0.789316\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	3.77831e6	0.756744
$83$	4.37675e6	0.840191	0.420096	−	0.907480i	$-0.361997\pi$
$83$	4.37675e6	0.840191	0.420096	+	0.907480i	$0.361997\pi$
$84$	−158976.	−0.0292654
$85$	0	0
$86$	4.11463e6	0.697568
$87$	985770.	0.160494
$88$	−1.25136e6	−0.195746
$89$	−8.52831e6	−1.28232	−0.641162	−	0.767405i	$-0.721547\pi$
$89$	−8.52831e6	−1.28232	−0.641162	+	0.767405i	$0.721547\pi$
$90$	0	0
$91$	326272.	0.0453874
$92$	−1.40650e6	−0.188313
$93$	−7.47382e6	−0.963501
$94$	3.49978e6	0.434603
$95$	0	0
$96$	−5.18659e6	−0.598319
$97$	8.82681e6	0.981981	0.490990	−	0.871165i	$-0.336635\pi$
$97$	8.82681e6	0.981981	0.490990	+	0.871165i	$0.336635\pi$
$98$	4.91668e6	0.527692
$99$	−691092.	−0.0715835
$100$	0	0
$101$	1.19864e7	1.15762	0.578808	−	0.815464i	$-0.303518\pi$
$101$	1.19864e7	1.15762	0.578808	+	0.815464i	$0.303518\pi$
$102$	4.59853e6	0.429061
$103$	−7.20939e6	−0.650082	−0.325041	−	0.945700i	$-0.605378\pi$
$103$	−7.20939e6	−0.650082	−0.325041	+	0.945700i	$0.605378\pi$
$104$	6.72936e6	0.586620
$105$	0	0
$106$	−2.56835e6	−0.209451
$107$	−1.14261e7	−0.901683	−0.450842	−	0.892604i	$-0.648876\pi$
$107$	−1.14261e7	−0.901683	−0.450842	+	0.892604i	$0.648876\pi$
$108$	−1.81084e6	−0.138324
$109$	4.02095e6	0.297397	0.148698	−	0.988883i	$-0.452492\pi$
$109$	4.02095e6	0.297397	0.148698	+	0.988883i	$0.452492\pi$
$110$	0	0
$111$	−7.25020e6	−0.503176
$112$	246784.	0.0165979
$113$	1.77063e7	1.15439	0.577197	−	0.816605i	$-0.304147\pi$
$113$	1.77063e7	1.15439	0.577197	+	0.816605i	$0.304147\pi$
$114$	1.39644e6	0.0882786
$115$	0	0
$116$	−3.35892e6	−0.199801
$117$	3.71644e6	0.214524
$118$	−7.83828e6	−0.439171
$119$	−1.81670e6	−0.0988257
$120$	0	0
$121$	−1.85885e7	−0.953882
$122$	−1.80397e6	−0.0899436
$123$	−1.70024e7	−0.823838
$124$	2.54663e7	1.19947
$125$	0	0
$126$	−279936.	−0.0124670
$127$	−1.67883e7	−0.727267	−0.363633	−	0.931542i	$-0.618464\pi$
$127$	−1.67883e7	−0.727267	−0.363633	+	0.931542i	$0.618464\pi$
$128$	2.06342e7	0.869668
$129$	−1.85158e7	−0.759416
$130$	0	0
$131$	1.68268e7	0.653960	0.326980	−	0.945031i	$-0.393969\pi$
$131$	1.68268e7	0.653960	0.326980	+	0.945031i	$0.393969\pi$
$132$	2.35483e6	0.0891151
$133$	−551680.	−0.0203332
$134$	−3.04346e6	−0.109270
$135$	0	0
$136$	−3.74695e7	−1.27730
$137$	−2.80449e7	−0.931820	−0.465910	−	0.884832i	$-0.654273\pi$
$137$	−2.80449e7	−0.931820	−0.465910	+	0.884832i	$0.654273\pi$
$138$	−2.47666e6	−0.0802212
$139$	−1.18273e7	−0.373537	−0.186769	−	0.982404i	$-0.559801\pi$
$139$	−1.18273e7	−0.373537	−0.186769	+	0.982404i	$0.559801\pi$
$140$	0	0
$141$	−1.57490e7	−0.473136
$142$	−3.33638e7	−0.977836
$143$	−4.83290e6	−0.138208
$144$	2.81102e6	0.0784505
$145$	0	0
$146$	8.21449e6	0.218447
$147$	−2.21251e7	−0.574479
$148$	2.47044e7	0.626409
$149$	2.07846e7	0.514743	0.257371	−	0.966313i	$-0.417144\pi$
$149$	2.07846e7	0.514743	0.257371	+	0.966313i	$0.417144\pi$
$150$	0	0
$151$	76112.0	0.00179901	0.000899505	−	1.00000i	$-0.499714\pi$
$151$	76112.0	0.00179901	0.000899505		1.00000i	$0.499714\pi$
$152$	−1.13784e7	−0.262802
$153$	−2.06934e7	−0.467102
$154$	364032.	0.00803188
$155$	0	0
$156$	−1.26634e7	−0.267064
$157$	3.21825e7	0.663698	0.331849	−	0.943332i	$-0.392328\pi$
$157$	3.21825e7	0.663698	0.331849	+	0.943332i	$0.392328\pi$
$158$	4.14823e7	0.836687
$159$	1.15576e7	0.228022
$160$	0	0
$161$	978432.	0.0184774
$162$	−3.18865e6	−0.0589256
$163$	−5.83435e7	−1.05520	−0.527601	−	0.849492i	$-0.676908\pi$
$163$	−5.83435e7	−1.05520	−0.527601	+	0.849492i	$0.676908\pi$
$164$	5.79341e7	1.02561
$165$	0	0
$166$	−2.62605e7	−0.445579
$167$	2.58365e7	0.429266	0.214633	−	0.976695i	$-0.431145\pi$
$167$	2.58365e7	0.429266	0.214633	+	0.976695i	$0.431145\pi$
$168$	2.28096e6	0.0371138
$169$	−3.67589e7	−0.585813
$170$	0	0
$171$	−6.28398e6	−0.0961055
$172$	6.30910e7	0.945405
$173$	−6.35201e7	−0.932716	−0.466358	−	0.884596i	$-0.654434\pi$
$173$	−6.35201e7	−0.932716	−0.466358	+	0.884596i	$0.654434\pi$
$174$	−5.91462e6	−0.0851147
$175$	0	0
$176$	−3.65549e6	−0.0505418
$177$	3.52723e7	0.478109
$178$	5.11699e7	0.680055
$179$	−8.09559e7	−1.05503	−0.527513	−	0.849547i	$-0.676875\pi$
$179$	−8.09559e7	−1.05503	−0.527513	+	0.849547i	$0.676875\pi$
$180$	0	0
$181$	6.45032e7	0.808549	0.404274	−	0.914638i	$-0.367524\pi$
$181$	6.45032e7	0.808549	0.404274	+	0.914638i	$0.367524\pi$
$182$	−1.95763e6	−0.0240703
$183$	8.11787e6	0.0979182
$184$	2.01802e7	0.238815
$185$	0	0
$186$	4.48429e7	0.510973
$187$	2.69099e7	0.300931
$188$	5.36632e7	0.589012
$189$	1.25971e6	0.0135723
$190$	0	0
$191$	5.68274e7	0.590121	0.295060	−	0.955479i	$-0.404660\pi$
$191$	5.68274e7	0.590121	0.295060	+	0.955479i	$0.404660\pi$
$192$	1.77932e7	0.181426
$193$	−1.16377e8	−1.16524	−0.582621	−	0.812744i	$-0.697973\pi$
$193$	−1.16377e8	−1.16524	−0.582621	+	0.812744i	$0.697973\pi$
$194$	−5.29609e7	−0.520774
$195$	0	0
$196$	7.53891e7	0.715175
$197$	1.18816e8	1.10724	0.553622	−	0.832768i	$-0.313245\pi$
$197$	1.18816e8	1.10724	0.553622	+	0.832768i	$0.313245\pi$
$198$	4.14655e6	0.0379629
$199$	−9.50106e7	−0.854646	−0.427323	−	0.904099i	$-0.640543\pi$
$199$	−9.50106e7	−0.854646	−0.427323	+	0.904099i	$0.640543\pi$
$200$	0	0
$201$	1.36956e7	0.118958
$202$	−7.19185e7	−0.613919
$203$	2.33664e6	0.0196045
$204$	7.05108e7	0.581501
$205$	0	0
$206$	4.32564e7	0.344758
$207$	1.11450e7	0.0873337
$208$	1.96579e7	0.151466
$209$	8.17176e6	0.0619161
$210$	0	0
$211$	1.79246e8	1.31360	0.656798	−	0.754067i	$-0.271910\pi$
$211$	1.79246e8	1.31360	0.656798	+	0.754067i	$0.271910\pi$
$212$	−3.93813e7	−0.283867
$213$	1.50137e8	1.06453
$214$	6.85565e7	0.478190
$215$	0	0
$216$	2.59816e7	0.175419
$217$	−1.77157e7	−0.117693
$218$	−2.41257e7	−0.157718
$219$	−3.69652e7	−0.237815
$220$	0	0
$221$	−1.44712e8	−0.901843
$222$	4.35012e7	0.266849
$223$	2.06537e8	1.24718	0.623592	−	0.781750i	$-0.285673\pi$
$223$	2.06537e8	1.24718	0.623592	+	0.781750i	$0.285673\pi$
$224$	−1.22941e7	−0.0730853
$225$	0	0
$226$	−1.06238e8	−0.612209
$227$	−4.33954e7	−0.246237	−0.123118	−	0.992392i	$-0.539290\pi$
$227$	−4.33954e7	−0.246237	−0.123118	+	0.992392i	$0.539290\pi$
$228$	2.14121e7	0.119643
$229$	−3.61931e7	−0.199160	−0.0995799	−	0.995030i	$-0.531750\pi$
$229$	−3.61931e7	−0.199160	−0.0995799	+	0.995030i	$0.531750\pi$
$230$	0	0
$231$	−1.63814e6	−0.00874400
$232$	4.81932e7	0.253383
$233$	−9.22347e7	−0.477693	−0.238846	−	0.971057i	$-0.576769\pi$
$233$	−9.22347e7	−0.477693	−0.238846	+	0.971057i	$0.576769\pi$
$234$	−2.22987e7	−0.113769
$235$	0	0
$236$	−1.20187e8	−0.595203
$237$	−1.86670e8	−0.910870
$238$	1.09002e7	0.0524102
$239$	4.98468e7	0.236181	0.118090	−	0.993003i	$-0.462323\pi$
$239$	4.98468e7	0.236181	0.118090	+	0.993003i	$0.462323\pi$
$240$	0	0
$241$	1.99374e8	0.917506	0.458753	−	0.888564i	$-0.348296\pi$
$241$	1.99374e8	0.917506	0.458753	+	0.888564i	$0.348296\pi$
$242$	1.11531e8	0.505872
$243$	1.43489e7	0.0641500
$244$	−2.76609e7	−0.121900
$245$	0	0
$246$	1.02014e8	0.436906
$247$	−4.39448e7	−0.185553
$248$	−3.65387e8	−1.52115
$249$	1.18172e8	0.485085
$250$	0	0
$251$	−3.94678e8	−1.57538	−0.787689	−	0.616073i	$-0.788723\pi$
$251$	−3.94678e8	−1.57538	−0.787689	+	0.616073i	$0.788723\pi$
$252$	−4.29235e6	−0.0168964
$253$	−1.44930e7	−0.0562649
$254$	1.00730e8	0.385691
$255$	0	0
$256$	−2.08158e8	−0.775451
$257$	1.42885e8	0.525076	0.262538	−	0.964922i	$-0.415441\pi$
$257$	1.42885e8	0.525076	0.262538	+	0.964922i	$0.415441\pi$
$258$	1.11095e8	0.402741
$259$	−1.71857e7	−0.0614635
$260$	0	0
$261$	2.66158e7	0.0926611
$262$	−1.00961e8	−0.346815
$263$	−4.40241e8	−1.49226	−0.746131	−	0.665799i	$-0.768091\pi$
$263$	−4.40241e8	−1.49226	−0.746131	+	0.665799i	$0.768091\pi$
$264$	−3.37867e7	−0.113014
$265$	0	0
$266$	3.31008e6	0.0107833
$267$	−2.30264e8	−0.740350
$268$	−4.66664e7	−0.148092
$269$	2.75405e8	0.862657	0.431329	−	0.902195i	$-0.358045\pi$
$269$	2.75405e8	0.862657	0.431329	+	0.902195i	$0.358045\pi$
$270$	0	0
$271$	−4.24670e8	−1.29616	−0.648080	−	0.761572i	$-0.724428\pi$
$271$	−4.24670e8	−1.29616	−0.648080	+	0.761572i	$0.724428\pi$
$272$	−1.09456e8	−0.329800
$273$	8.80934e6	0.0262044
$274$	1.68269e8	0.494172
$275$	0	0
$276$	−3.79754e7	−0.108723
$277$	−5.16158e8	−1.45916	−0.729581	−	0.683894i	$-0.760285\pi$
$277$	−5.16158e8	−1.45916	−0.729581	+	0.683894i	$0.760285\pi$
$278$	7.09638e7	0.198098
$279$	−2.01793e8	−0.556277
$280$	0	0
$281$	−3.11043e8	−0.836273	−0.418137	−	0.908384i	$-0.637317\pi$
$281$	−3.11043e8	−0.836273	−0.418137	+	0.908384i	$0.637317\pi$
$282$	9.44940e7	0.250918
$283$	5.94308e8	1.55869	0.779344	−	0.626596i	$-0.215552\pi$
$283$	5.94308e8	1.55869	0.779344	+	0.626596i	$0.215552\pi$
$284$	−5.11578e8	−1.32525
$285$	0	0
$286$	2.89974e7	0.0732957
$287$	−4.03020e7	−0.100633
$288$	−1.40038e8	−0.345440
$289$	3.95426e8	0.963658
$290$	0	0
$291$	2.38324e8	0.566947
$292$	1.25956e8	0.296058
$293$	−1.15515e8	−0.268288	−0.134144	−	0.990962i	$-0.542828\pi$
$293$	−1.15515e8	−0.268288	−0.134144	+	0.990962i	$0.542828\pi$
$294$	1.32750e8	0.304663
$295$	0	0
$296$	−3.54454e8	−0.794400
$297$	−1.86595e7	−0.0413287
$298$	−1.24708e8	−0.272984
$299$	7.79382e7	0.168617
$300$	0	0
$301$	−4.38894e7	−0.0927635
$302$	−456672.	−0.000954070 0
$303$	3.23633e8	0.668350
$304$	−3.32387e7	−0.0678558
$305$	0	0
$306$	1.24160e8	0.247718
$307$	2.60600e8	0.514032	0.257016	−	0.966407i	$-0.417261\pi$
$307$	2.60600e8	0.514032	0.257016	+	0.966407i	$0.417261\pi$
$308$	5.58182e6	0.0108855
$309$	−1.94654e8	−0.375325
$310$	0	0
$311$	5.76795e8	1.08733	0.543663	−	0.839303i	$-0.317037\pi$
$311$	5.76795e8	1.08733	0.543663	+	0.839303i	$0.317037\pi$
$312$	1.81693e8	0.338685
$313$	4.60074e8	0.848053	0.424026	−	0.905650i	$-0.360616\pi$
$313$	4.60074e8	0.848053	0.424026	+	0.905650i	$0.360616\pi$
$314$	−1.93095e8	−0.351979
$315$	0	0
$316$	6.36062e8	1.13395
$317$	−6.25561e7	−0.110297	−0.0551483	−	0.998478i	$-0.517563\pi$
$317$	−6.25561e7	−0.110297	−0.0551483	+	0.998478i	$0.517563\pi$
$318$	−6.93454e7	−0.120927
$319$	−3.46115e7	−0.0596970
$320$	0	0
$321$	−3.08504e8	−0.520587
$322$	−5.87059e6	−0.00979910
$323$	2.44687e8	0.404020
$324$	−4.88926e7	−0.0798611
$325$	0	0
$326$	3.50061e8	0.559606
$327$	1.08566e8	0.171702
$328$	−8.31228e8	−1.30065
$329$	−3.73309e7	−0.0577941
$330$	0	0
$331$	6.84236e8	1.03707	0.518535	−	0.855057i	$-0.326478\pi$
$331$	6.84236e8	1.03707	0.518535	+	0.855057i	$0.326478\pi$
$332$	−4.02661e8	−0.603888
$333$	−1.95755e8	−0.290509
$334$	−1.55019e8	−0.227652
$335$	0	0
$336$	6.66317e6	0.00958282
$337$	6.26313e8	0.891429	0.445714	−	0.895175i	$-0.352950\pi$
$337$	6.26313e8	0.891429	0.445714	+	0.895175i	$0.352950\pi$
$338$	2.20553e8	0.310674
$339$	4.78071e8	0.666489
$340$	0	0
$341$	2.62414e8	0.358382
$342$	3.77039e7	0.0509677
$343$	−1.05151e8	−0.140697
$344$	−9.05219e8	−1.19894
$345$	0	0
$346$	3.81120e8	0.494647
$347$	1.25340e9	1.61041	0.805203	−	0.593000i	$-0.202057\pi$
$347$	1.25340e9	1.61041	0.805203	+	0.593000i	$0.202057\pi$
$348$	−9.06908e7	−0.115355
$349$	2.65350e8	0.334142	0.167071	−	0.985945i	$-0.446569\pi$
$349$	2.65350e8	0.334142	0.167071	+	0.985945i	$0.446569\pi$
$350$	0	0
$351$	1.00344e8	0.123856
$352$	1.82107e8	0.222550
$353$	5.69636e8	0.689264	0.344632	−	0.938738i	$-0.388004\pi$
$353$	5.69636e8	0.689264	0.344632	+	0.938738i	$0.388004\pi$
$354$	−2.11634e8	−0.253556
$355$	0	0
$356$	7.84605e8	0.921671
$357$	−4.90510e7	−0.0570570
$358$	4.85735e8	0.559512
$359$	9.32541e8	1.06374	0.531872	−	0.846825i	$-0.321489\pi$
$359$	9.32541e8	1.06374	0.531872	+	0.846825i	$0.321489\pi$
$360$	0	0
$361$	−8.19567e8	−0.916874
$362$	−3.87019e8	−0.428798
$363$	−5.01889e8	−0.550724
$364$	−3.00170e7	−0.0326222
$365$	0	0
$366$	−4.87072e7	−0.0519290
$367$	8.52565e8	0.900318	0.450159	−	0.892948i	$-0.351367\pi$
$367$	8.52565e8	0.900318	0.450159	+	0.892948i	$0.351367\pi$
$368$	5.89505e7	0.0616624
$369$	−4.59064e8	−0.475643
$370$	0	0
$371$	2.73957e7	0.0278531
$372$	6.87591e8	0.692516
$373$	−3.81183e8	−0.380323	−0.190162	−	0.981753i	$-0.560901\pi$
$373$	−3.81183e8	−0.380323	−0.190162	+	0.981753i	$0.560901\pi$
$374$	−1.61460e8	−0.159593
$375$	0	0
$376$	−7.69951e8	−0.746974
$377$	1.86128e8	0.178903
$378$	−7.55827e6	−0.00719782
$379$	−1.48353e9	−1.39978	−0.699889	−	0.714251i	$-0.746767\pi$
$379$	−1.48353e9	−1.39978	−0.699889	+	0.714251i	$0.746767\pi$
$380$	0	0
$381$	−4.53284e8	−0.419888
$382$	−3.40964e8	−0.312959
$383$	7.61930e8	0.692978	0.346489	−	0.938054i	$-0.387374\pi$
$383$	7.61930e8	0.692978	0.346489	+	0.938054i	$0.387374\pi$
$384$	5.57124e8	0.502103
$385$	0	0
$386$	6.98262e8	0.617963
$387$	−4.99928e8	−0.438449
$388$	−8.12067e8	−0.705799
$389$	1.60902e9	1.38592	0.692959	−	0.720977i	$-0.256307\pi$
$389$	1.60902e9	1.38592	0.692959	+	0.720977i	$0.256307\pi$
$390$	0	0
$391$	−4.33965e8	−0.367144
$392$	−1.08167e9	−0.906971
$393$	4.54323e8	0.377564
$394$	−7.12896e8	−0.587205
$395$	0	0
$396$	6.35805e7	0.0514506
$397$	−1.88016e9	−1.50809	−0.754046	−	0.656822i	$-0.771900\pi$
$397$	−1.88016e9	−1.50809	−0.754046	+	0.656822i	$0.771900\pi$
$398$	5.70064e8	0.453245
$399$	−1.48954e7	−0.0117394
$400$	0	0
$401$	2.68592e8	0.208012	0.104006	−	0.994577i	$-0.466834\pi$
$401$	2.68592e8	0.208012	0.104006	+	0.994577i	$0.466834\pi$
$402$	−8.21735e7	−0.0630871
$403$	−1.41117e9	−1.07402
$404$	−1.10275e9	−0.832037
$405$	0	0
$406$	−1.40198e7	−0.0103969
$407$	2.54563e8	0.187161
$408$	−1.01168e9	−0.737448
$409$	8.99478e7	0.0650069	0.0325034	−	0.999472i	$-0.489652\pi$
$409$	8.99478e7	0.0650069	0.0325034	+	0.999472i	$0.489652\pi$
$410$	0	0
$411$	−7.57212e8	−0.537986
$412$	6.63264e8	0.467247
$413$	8.36083e7	0.0584015
$414$	−6.68697e7	−0.0463157
$415$	0	0
$416$	−9.79305e8	−0.666947
$417$	−3.19337e8	−0.215662
$418$	−4.90306e7	−0.0328360
$419$	1.69054e9	1.12273	0.561367	−	0.827567i	$-0.310276\pi$
$419$	1.69054e9	1.12273	0.561367	+	0.827567i	$0.310276\pi$
$420$	0	0
$421$	−1.13333e9	−0.740232	−0.370116	−	0.928985i	$-0.620682\pi$
$421$	−1.13333e9	−0.740232	−0.370116	+	0.928985i	$0.620682\pi$
$422$	−1.07548e9	−0.696639
$423$	−4.25223e8	−0.273165
$424$	5.65037e8	0.359995
$425$	0	0
$426$	−9.00822e8	−0.564554
$427$	1.92424e7	0.0119608
$428$	1.05120e9	0.648085
$429$	−1.30488e8	−0.0797942
$430$	0	0
$431$	2.19943e9	1.32324	0.661621	−	0.749839i	$-0.269869\pi$
$431$	2.19943e9	1.32324	0.661621	+	0.749839i	$0.269869\pi$
$432$	7.58976e7	0.0452934
$433$	1.51738e8	0.0898227	0.0449114	−	0.998991i	$-0.485699\pi$
$433$	1.51738e8	0.0898227	0.0449114	+	0.998991i	$0.485699\pi$
$434$	1.06294e8	0.0624160
$435$	0	0
$436$	−3.69927e8	−0.213754
$437$	−1.31783e8	−0.0755393
$438$	2.21791e8	0.126120
$439$	9.90763e8	0.558912	0.279456	−	0.960158i	$-0.409846\pi$
$439$	9.90763e8	0.558912	0.279456	+	0.960158i	$0.409846\pi$
$440$	0	0
$441$	−5.97377e8	−0.331675
$442$	8.68271e8	0.478275
$443$	1.77376e9	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$443$	1.77376e9	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$444$	6.67019e8	0.361658
$445$	0	0
$446$	−1.23922e9	−0.661419
$447$	5.61185e8	0.297187
$448$	4.21765e7	0.0221614
$449$	−2.77010e8	−0.144422	−0.0722110	−	0.997389i	$-0.523006\pi$
$449$	−2.77010e8	−0.144422	−0.0722110	+	0.997389i	$0.523006\pi$
$450$	0	0
$451$	5.96973e8	0.306434
$452$	−1.62898e9	−0.829720
$453$	2.05502e6	0.00103866
$454$	2.60372e8	0.130587
$455$	0	0
$456$	−3.07217e8	−0.151729
$457$	−2.94758e9	−1.44464	−0.722320	−	0.691559i	$-0.756924\pi$
$457$	−2.94758e9	−1.44464	−0.722320	+	0.691559i	$0.756924\pi$
$458$	2.17159e8	0.105620
$459$	−5.58722e8	−0.269681
$460$	0	0
$461$	−2.76687e9	−1.31533	−0.657667	−	0.753309i	$-0.728457\pi$
$461$	−2.76687e9	−1.31533	−0.657667	+	0.753309i	$0.728457\pi$
$462$	9.82886e6	0.00463721
$463$	−4.63553e8	−0.217053	−0.108527	−	0.994094i	$-0.534613\pi$
$463$	−4.63553e8	−0.217053	−0.108527	+	0.994094i	$0.534613\pi$
$464$	1.40783e8	0.0654238
$465$	0	0
$466$	5.53408e8	0.253335
$467$	4.17922e8	0.189883	0.0949415	−	0.995483i	$-0.469734\pi$
$467$	4.17922e8	0.189883	0.0949415	+	0.995483i	$0.469734\pi$
$468$	−3.41913e8	−0.154189
$469$	3.24636e7	0.0145309
$470$	0	0
$471$	8.68927e8	0.383186
$472$	1.72442e9	0.754825
$473$	6.50112e8	0.282471
$474$	1.12002e9	0.483062
$475$	0	0
$476$	1.67137e8	0.0710310
$477$	3.12054e8	0.131648
$478$	−2.99081e8	−0.125254
$479$	−1.50973e9	−0.627660	−0.313830	−	0.949479i	$-0.601612\pi$
$479$	−1.50973e9	−0.627660	−0.313830	+	0.949479i	$0.601612\pi$
$480$	0	0
$481$	−1.36895e9	−0.560891
$482$	−1.19624e9	−0.486581
$483$	2.64177e7	0.0106679
$484$	1.71014e9	0.685603
$485$	0	0
$486$	−8.60934e7	−0.0340207
$487$	−9.29460e8	−0.364653	−0.182326	−	0.983238i	$-0.558363\pi$
$487$	−9.29460e8	−0.364653	−0.182326	+	0.983238i	$0.558363\pi$
$488$	3.96874e8	0.154591
$489$	−1.57527e9	−0.609221
$490$	0	0
$491$	5.12803e9	1.95508	0.977541	−	0.210743i	$-0.0675885\pi$
$491$	5.12803e9	1.95508	0.977541	+	0.210743i	$0.0675885\pi$
$492$	1.56422e9	0.592134
$493$	−1.03637e9	−0.389540
$494$	2.63669e8	0.0984043
$495$	0	0
$496$	−1.06737e9	−0.392762
$497$	3.55880e8	0.130034
$498$	−7.09033e8	−0.257255
$499$	−4.10649e8	−0.147951	−0.0739757	−	0.997260i	$-0.523569\pi$
$499$	−4.10649e8	−0.147951	−0.0739757	+	0.997260i	$0.523569\pi$
$500$	0	0
$501$	6.97586e8	0.247837
$502$	2.36807e9	0.835470
$503$	−5.02041e9	−1.75894	−0.879470	−	0.475954i	$-0.842103\pi$
$503$	−5.02041e9	−1.75894	−0.879470	+	0.475954i	$0.842103\pi$
$504$	6.15859e7	0.0214277
$505$	0	0
$506$	8.69581e7	0.0298389
$507$	−9.92491e8	−0.338219
$508$	1.54452e9	0.522723
$509$	−3.24926e9	−1.09212	−0.546062	−	0.837745i	$-0.683874\pi$
$509$	−3.24926e9	−1.09212	−0.546062	+	0.837745i	$0.683874\pi$
$510$	0	0
$511$	−8.76212e7	−0.0290493
$512$	−1.39223e9	−0.458423
$513$	−1.69667e8	−0.0554866
$514$	−8.57312e8	−0.278463
$515$	0	0
$516$	1.70346e9	0.545830
$517$	5.52965e8	0.175987
$518$	1.03114e8	0.0325959
$519$	−1.71504e9	−0.538504
$520$	0	0
$521$	−2.10950e9	−0.653503	−0.326752	−	0.945110i	$-0.605954\pi$
$521$	−2.10950e9	−0.653503	−0.326752	+	0.945110i	$0.605954\pi$
$522$	−1.59695e8	−0.0491410
$523$	5.28911e9	1.61669	0.808345	−	0.588709i	$-0.200364\pi$
$523$	5.28911e9	1.61669	0.808345	+	0.588709i	$0.200364\pi$
$524$	−1.54806e9	−0.470034
$525$	0	0
$526$	2.64144e9	0.791391
$527$	7.85747e9	2.33854
$528$	−9.86982e7	−0.0291803
$529$	−3.17110e9	−0.931355
$530$	0	0
$531$	9.52351e8	0.276036
$532$	5.07546e7	0.0146145
$533$	−3.21030e9	−0.918334
$534$	1.38159e9	0.392630
$535$	0	0
$536$	6.69562e8	0.187808
$537$	−2.18581e9	−0.609119
$538$	−1.65243e9	−0.457493
$539$	7.76836e8	0.213682
$540$	0	0
$541$	3.04614e9	0.827101	0.413551	−	0.910481i	$-0.364288\pi$
$541$	3.04614e9	0.827101	0.413551	+	0.910481i	$0.364288\pi$
$542$	2.54802e9	0.687393
$543$	1.74159e9	0.466816
$544$	5.45284e9	1.45220
$545$	0	0
$546$	−5.28561e7	−0.0138970
$547$	4.85537e9	1.26843	0.634215	−	0.773157i	$-0.281323\pi$
$547$	4.85537e9	1.26843	0.634215	+	0.773157i	$0.281323\pi$
$548$	2.58013e9	0.669746
$549$	2.19183e8	0.0565331
$550$	0	0
$551$	−3.14716e8	−0.0801472
$552$	5.44864e8	0.137880
$553$	−4.42478e8	−0.111264
$554$	3.09695e9	0.773838
$555$	0	0
$556$	1.08811e9	0.268480
$557$	−1.27762e9	−0.313263	−0.156631	−	0.987657i	$-0.550063\pi$
$557$	−1.27762e9	−0.313263	−0.156631	+	0.987657i	$0.550063\pi$
$558$	1.21076e9	0.295011
$559$	−3.49607e9	−0.846522
$560$	0	0
$561$	7.26568e8	0.173743
$562$	1.86626e9	0.443501
$563$	−4.71265e9	−1.11297	−0.556487	−	0.830856i	$-0.687851\pi$
$563$	−4.71265e9	−1.11297	−0.556487	+	0.830856i	$0.687851\pi$
$564$	1.44891e9	0.340066
$565$	0	0
$566$	−3.56585e9	−0.826619
$567$	3.40122e7	0.00783600
$568$	7.34003e9	1.68066
$569$	4.57800e9	1.04180	0.520898	−	0.853619i	$-0.325597\pi$
$569$	4.57800e9	1.04180	0.520898	+	0.853619i	$0.325597\pi$
$570$	0	0
$571$	4.95119e9	1.11297	0.556485	−	0.830858i	$-0.312150\pi$
$571$	4.95119e9	1.11297	0.556485	+	0.830858i	$0.312150\pi$
$572$	4.44627e8	0.0993367
$573$	1.53434e9	0.340706
$574$	2.41812e8	0.0533686
$575$	0	0
$576$	4.80417e8	0.104747
$577$	−8.51847e9	−1.84606	−0.923031	−	0.384725i	$-0.874296\pi$
$577$	−8.51847e9	−1.84606	−0.923031	+	0.384725i	$0.874296\pi$
$578$	−2.37256e9	−0.511057
$579$	−3.14218e9	−0.672753
$580$	0	0
$581$	2.80112e8	0.0592536
$582$	−1.42994e9	−0.300669
$583$	−4.05799e8	−0.0848147
$584$	−1.80719e9	−0.375455
$585$	0	0
$586$	6.93088e8	0.142281
$587$	5.62247e8	0.114735	0.0573673	−	0.998353i	$-0.481729\pi$
$587$	5.62247e8	0.114735	0.0573673	+	0.998353i	$0.481729\pi$
$588$	2.03551e9	0.412907
$589$	2.38608e9	0.481152
$590$	0	0
$591$	3.20803e9	0.639268
$592$	−1.03544e9	−0.205115
$593$	−3.62110e9	−0.713099	−0.356549	−	0.934277i	$-0.616047\pi$
$593$	−3.62110e9	−0.713099	−0.356549	+	0.934277i	$0.616047\pi$
$594$	1.11957e8	0.0219179
$595$	0	0
$596$	−1.91219e9	−0.369971
$597$	−2.56529e9	−0.493430
$598$	−4.67629e8	−0.0894227
$599$	−7.48104e9	−1.42222	−0.711112	−	0.703079i	$-0.751808\pi$
$599$	−7.48104e9	−1.42222	−0.711112	+	0.703079i	$0.751808\pi$
$600$	0	0
$601$	−5.81270e9	−1.09224	−0.546119	−	0.837707i	$-0.683895\pi$
$601$	−5.81270e9	−1.09224	−0.546119	+	0.837707i	$0.683895\pi$
$602$	2.63336e8	0.0491953
$603$	3.69781e8	0.0686806
$604$	−7.00230e6	−0.00129304
$605$	0	0
$606$	−1.94180e9	−0.354446
$607$	−3.84051e9	−0.696993	−0.348497	−	0.937310i	$-0.613308\pi$
$607$	−3.84051e9	−0.696993	−0.348497	+	0.937310i	$0.613308\pi$
$608$	1.65587e9	0.298788
$609$	6.30893e7	0.0113187
$610$	0	0
$611$	−2.97364e9	−0.527405
$612$	1.90379e9	0.335730
$613$	−1.70484e9	−0.298932	−0.149466	−	0.988767i	$-0.547755\pi$
$613$	−1.70484e9	−0.298932	−0.149466	+	0.988767i	$0.547755\pi$
$614$	−1.56360e9	−0.272606
$615$	0	0
$616$	−8.00870e7	−0.0138048
$617$	2.80809e9	0.481297	0.240649	−	0.970612i	$-0.422640\pi$
$617$	2.80809e9	0.481297	0.240649	+	0.970612i	$0.422640\pi$
$618$	1.16792e9	0.199046
$619$	−2.54365e9	−0.431063	−0.215532	−	0.976497i	$-0.569148\pi$
$619$	−2.54365e9	−0.431063	−0.215532	+	0.976497i	$0.569148\pi$
$620$	0	0
$621$	3.00914e8	0.0504222
$622$	−3.46077e9	−0.576642
$623$	−5.45812e8	−0.0904346
$624$	5.30763e8	0.0874489
$625$	0	0
$626$	−2.76045e9	−0.449748
$627$	2.20638e8	0.0357473
$628$	−2.96079e9	−0.477033
$629$	7.62238e9	1.22127
$630$	0	0
$631$	−1.51146e8	−0.0239494	−0.0119747	−	0.999928i	$-0.503812\pi$
$631$	−1.51146e8	−0.0239494	−0.0119747	+	0.999928i	$0.503812\pi$
$632$	−9.12611e9	−1.43806
$633$	4.83965e9	0.758405
$634$	3.75337e8	0.0584936
$635$	0	0
$636$	−1.06330e9	−0.163891
$637$	−4.17754e9	−0.640373
$638$	2.07669e8	0.0316591
$639$	4.05370e9	0.614609
$640$	0	0
$641$	−1.23625e10	−1.85397	−0.926987	−	0.375094i	$-0.877610\pi$
$641$	−1.23625e10	−1.85397	−0.926987	+	0.375094i	$0.877610\pi$
$642$	1.85102e9	0.276083
$643$	−2.86744e9	−0.425359	−0.212680	−	0.977122i	$-0.568219\pi$
$643$	−2.86744e9	−0.425359	−0.212680	+	0.977122i	$0.568219\pi$
$644$	−9.00157e7	−0.0132806
$645$	0	0
$646$	−1.46812e9	−0.214264
$647$	4.10640e9	0.596068	0.298034	−	0.954555i	$-0.403669\pi$
$647$	4.10640e9	0.596068	0.298034	+	0.954555i	$0.403669\pi$
$648$	7.01502e8	0.101278
$649$	−1.23845e9	−0.177837
$650$	0	0
$651$	−4.78324e8	−0.0679499
$652$	5.36760e9	0.758427
$653$	−6.91100e9	−0.971280	−0.485640	−	0.874159i	$-0.661413\pi$
$653$	−6.91100e9	−0.971280	−0.485640	+	0.874159i	$0.661413\pi$
$654$	−6.51394e8	−0.0910587
$655$	0	0
$656$	−2.42819e9	−0.335830
$657$	−9.98061e8	−0.137302
$658$	2.23986e8	0.0306499
$659$	3.42444e9	0.466112	0.233056	−	0.972463i	$-0.425127\pi$
$659$	3.42444e9	0.466112	0.233056	+	0.972463i	$0.425127\pi$
$660$	0	0
$661$	−6.76437e9	−0.911008	−0.455504	−	0.890234i	$-0.650541\pi$
$661$	−6.76437e9	−0.911008	−0.455504	+	0.890234i	$0.650541\pi$
$662$	−4.10541e9	−0.549989
$663$	−3.90722e9	−0.520679
$664$	5.77731e9	0.765839
$665$	0	0
$666$	1.17453e9	0.154065
$667$	5.58165e8	0.0728320
$668$	−2.37696e9	−0.308535
$669$	5.57650e9	0.720062
$670$	0	0
$671$	−2.85028e8	−0.0364215
$672$	−3.31942e8	−0.0421958
$673$	1.74959e9	0.221250	0.110625	−	0.993862i	$-0.464715\pi$
$673$	1.74959e9	0.221250	0.110625	+	0.993862i	$0.464715\pi$
$674$	−3.75788e9	−0.472752
$675$	0	0
$676$	3.38182e9	0.421053
$677$	−8.30011e9	−1.02807	−0.514036	−	0.857769i	$-0.671850\pi$
$677$	−8.30011e9	−1.02807	−0.514036	+	0.857769i	$0.671850\pi$
$678$	−2.86842e9	−0.353459
$679$	5.64916e8	0.0692532
$680$	0	0
$681$	−1.17168e9	−0.142165
$682$	−1.57448e9	−0.190061
$683$	1.21232e10	1.45594	0.727969	−	0.685610i	$-0.240464\pi$
$683$	1.21232e10	1.45594	0.727969	+	0.685610i	$0.240464\pi$
$684$	5.78126e8	0.0690759
$685$	0	0
$686$	6.30908e8	0.0746160
$687$	−9.77213e8	−0.114985
$688$	−2.64434e9	−0.309569
$689$	2.18224e9	0.254176
$690$	0	0
$691$	8.21846e9	0.947583	0.473791	−	0.880637i	$-0.342885\pi$
$691$	8.21846e9	0.947583	0.473791	+	0.880637i	$0.342885\pi$
$692$	5.84385e9	0.670390
$693$	−4.42299e7	−0.00504835
$694$	−7.52038e9	−0.854046
$695$	0	0
$696$	1.30122e9	0.146291
$697$	1.78752e10	1.99957
$698$	−1.59210e9	−0.177205
$699$	−2.49034e9	−0.275796
$700$	0	0
$701$	4.72231e9	0.517775	0.258888	−	0.965907i	$-0.416644\pi$
$701$	4.72231e9	0.517775	0.258888	+	0.965907i	$0.416644\pi$
$702$	−6.02064e8	−0.0656844
$703$	2.31469e9	0.251275
$704$	−6.24740e8	−0.0674831
$705$	0	0
$706$	−3.41781e9	−0.365537
$707$	7.67131e8	0.0816397
$708$	−3.24505e9	−0.343641
$709$	2.78975e9	0.293970	0.146985	−	0.989139i	$-0.453043\pi$
$709$	2.78975e9	0.293970	0.146985	+	0.989139i	$0.453043\pi$
$710$	0	0
$711$	−5.04010e9	−0.525891
$712$	−1.12574e10	−1.16885
$713$	−4.23184e9	−0.437236
$714$	2.94306e8	0.0302591
$715$	0	0
$716$	7.44794e9	0.758299
$717$	1.34586e9	0.136359
$718$	−5.59524e9	−0.564136
$719$	1.51985e9	0.152493	0.0762463	−	0.997089i	$-0.475706\pi$
$719$	1.51985e9	0.152493	0.0762463	+	0.997089i	$0.475706\pi$
$720$	0	0
$721$	−4.61401e8	−0.0458464
$722$	4.91740e9	0.486246
$723$	5.38310e9	0.529722
$724$	−5.93429e9	−0.581144
$725$	0	0
$726$	3.01133e9	0.292066
$727$	8.11761e9	0.783534	0.391767	−	0.920065i	$-0.371864\pi$
$727$	8.11761e9	0.783534	0.391767	+	0.920065i	$0.371864\pi$
$728$	4.30679e8	0.0413708
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.94663e10	1.84320
$732$	−7.46844e8	−0.0703787
$733$	1.03241e10	0.968249	0.484124	−	0.874999i	$-0.339138\pi$
$733$	1.03241e10	0.968249	0.484124	+	0.874999i	$0.339138\pi$
$734$	−5.11539e9	−0.477466
$735$	0	0
$736$	−2.93676e9	−0.271517
$737$	−4.80867e8	−0.0442475
$738$	2.75439e9	0.252248
$739$	−1.35365e10	−1.23382	−0.616908	−	0.787035i	$-0.711615\pi$
$739$	−1.35365e10	−1.23382	−0.616908	+	0.787035i	$0.711615\pi$
$740$	0	0
$741$	−1.18651e9	−0.107129
$742$	−1.64374e8	−0.0147713
$743$	1.71936e10	1.53782	0.768910	−	0.639356i	$-0.220799\pi$
$743$	1.71936e10	1.53782	0.768910	+	0.639356i	$0.220799\pi$
$744$	−9.86544e9	−0.878236
$745$	0	0
$746$	2.28710e9	0.201697
$747$	3.19065e9	0.280064
$748$	−2.47571e9	−0.216294
$749$	−7.31269e8	−0.0635903
$750$	0	0
$751$	1.12478e10	0.969013	0.484506	−	0.874788i	$-0.338999\pi$
$751$	1.12478e10	0.969013	0.484506	+	0.874788i	$0.338999\pi$
$752$	−2.24919e9	−0.192870
$753$	−1.06563e10	−0.909545
$754$	−1.11677e9	−0.0948775
$755$	0	0
$756$	−1.15894e8	−0.00975512
$757$	−1.63068e10	−1.36626	−0.683131	−	0.730296i	$-0.739382\pi$
$757$	−1.63068e10	−1.36626	−0.683131	+	0.730296i	$0.739382\pi$
$758$	8.90118e9	0.742345
$759$	−3.91312e8	−0.0324845
$760$	0	0
$761$	6.14069e9	0.505093	0.252546	−	0.967585i	$-0.418732\pi$
$761$	6.14069e9	0.505093	0.252546	+	0.967585i	$0.418732\pi$
$762$	2.71970e9	0.222679
$763$	2.57341e8	0.0209736
$764$	−5.22812e9	−0.424149
$765$	0	0
$766$	−4.57158e9	−0.367507
$767$	6.65993e9	0.532949
$768$	−5.62028e9	−0.447707
$769$	2.45069e10	1.94333	0.971664	−	0.236368i	$-0.0759569\pi$
$769$	2.45069e10	1.94333	0.971664	+	0.236368i	$0.0759569\pi$
$770$	0	0
$771$	3.85791e9	0.303153
$772$	1.07067e10	0.837518
$773$	1.01722e10	0.792110	0.396055	−	0.918227i	$-0.370379\pi$
$773$	1.01722e10	0.792110	0.396055	+	0.918227i	$0.370379\pi$
$774$	2.99957e9	0.232523
$775$	0	0
$776$	1.16514e10	0.895080
$777$	−4.64013e8	−0.0354860
$778$	−9.65411e9	−0.734994
$779$	5.42817e9	0.411408
$780$	0	0
$781$	−5.27148e9	−0.395962
$782$	2.60379e9	0.194707
$783$	7.18626e8	0.0534979
$784$	−3.15979e9	−0.234181
$785$	0	0
$786$	−2.72594e9	−0.200234
$787$	9.79135e9	0.716030	0.358015	−	0.933716i	$-0.383454\pi$
$787$	9.79135e9	0.716030	0.358015	+	0.933716i	$0.383454\pi$
$788$	−1.09311e10	−0.795832
$789$	−1.18865e10	−0.861558
$790$	0	0
$791$	1.13320e9	0.0814124
$792$	−9.12241e8	−0.0652487
$793$	1.53277e9	0.109150
$794$	1.12809e10	0.799786
$795$	0	0
$796$	8.74098e9	0.614277
$797$	9.75782e9	0.682729	0.341365	−	0.939931i	$-0.389111\pi$
$797$	9.75782e9	0.682729	0.341365	+	0.939931i	$0.389111\pi$
$798$	8.93722e7	0.00622576
$799$	1.65574e10	1.14836
$800$	0	0
$801$	−6.21714e9	−0.427442
$802$	−1.61155e9	−0.110315
$803$	1.29789e9	0.0884572
$804$	−1.25999e9	−0.0855012
$805$	0	0
$806$	8.46700e9	0.569583
$807$	7.43592e9	0.498055
$808$	1.58221e10	1.05517
$809$	−2.78706e9	−0.185066	−0.0925330	−	0.995710i	$-0.529496\pi$
$809$	−2.78706e9	−0.185066	−0.0925330	+	0.995710i	$0.529496\pi$
$810$	0	0
$811$	−7.99983e9	−0.526633	−0.263316	−	0.964710i	$-0.584816\pi$
$811$	−7.99983e9	−0.526633	−0.263316	+	0.964710i	$0.584816\pi$
$812$	−2.14971e8	−0.0140907
$813$	−1.14661e10	−0.748339
$814$	−1.52738e9	−0.0992569
$815$	0	0
$816$	−2.95532e9	−0.190410
$817$	5.91135e9	0.379236
$818$	−5.39687e8	−0.0344751
$819$	2.37852e8	0.0151291
$820$	0	0
$821$	−1.02402e10	−0.645813	−0.322906	−	0.946431i	$-0.604660\pi$
$821$	−1.02402e10	−0.645813	−0.322906	+	0.946431i	$0.604660\pi$
$822$	4.54327e9	0.285310
$823$	−2.78682e10	−1.74265	−0.871324	−	0.490707i	$-0.836738\pi$
$823$	−2.78682e10	−1.74265	−0.871324	+	0.490707i	$0.836738\pi$
$824$	−9.51640e9	−0.592553
$825$	0	0
$826$	−5.01650e8	−0.0309721
$827$	−2.35125e10	−1.44554	−0.722769	−	0.691090i	$-0.757131\pi$
$827$	−2.35125e10	−1.44554	−0.722769	+	0.691090i	$0.757131\pi$
$828$	−1.02534e9	−0.0627711
$829$	−1.28598e10	−0.783960	−0.391980	−	0.919974i	$-0.628210\pi$
$829$	−1.28598e10	−0.783960	−0.391980	+	0.919974i	$0.628210\pi$
$830$	0	0
$831$	−1.39363e10	−0.842448
$832$	3.35962e9	0.202236
$833$	2.32608e10	1.39434
$834$	1.91602e9	0.114372
$835$	0	0
$836$	−7.51802e8	−0.0445022
$837$	−5.44841e9	−0.321167
$838$	−1.01433e10	−0.595420
$839$	−7.99832e9	−0.467554	−0.233777	−	0.972290i	$-0.575109\pi$
$839$	−7.99832e9	−0.467554	−0.233777	+	0.972290i	$0.575109\pi$
$840$	0	0
$841$	−1.59169e10	−0.922725
$842$	6.79996e9	0.392567
$843$	−8.39816e9	−0.482822
$844$	−1.64907e10	−0.944147
$845$	0	0
$846$	2.55134e9	0.144868
$847$	−1.18966e9	−0.0672716
$848$	1.65059e9	0.0929510
$849$	1.60463e10	0.899909
$850$	0	0
$851$	−4.10523e9	−0.228341
$852$	−1.38126e10	−0.765133
$853$	−4.20827e9	−0.232157	−0.116079	−	0.993240i	$-0.537032\pi$
$853$	−4.20827e9	−0.232157	−0.116079	+	0.993240i	$0.537032\pi$
$854$	−1.15454e8	−0.00634318
$855$	0	0
$856$	−1.50824e10	−0.821888
$857$	−3.19307e10	−1.73291	−0.866453	−	0.499259i	$-0.833606\pi$
$857$	−3.19307e10	−1.73291	−0.866453	+	0.499259i	$0.833606\pi$
$858$	7.82930e8	0.0423173
$859$	2.18002e10	1.17350	0.586752	−	0.809767i	$-0.300406\pi$
$859$	2.18002e10	1.17350	0.586752	+	0.809767i	$0.300406\pi$
$860$	0	0
$861$	−1.08815e9	−0.0581004
$862$	−1.31966e10	−0.701755
$863$	−1.04728e10	−0.554657	−0.277329	−	0.960775i	$-0.589449\pi$
$863$	−1.04728e10	−0.554657	−0.277329	+	0.960775i	$0.589449\pi$
$864$	−3.78103e9	−0.199440
$865$	0	0
$866$	−9.10427e8	−0.0476357
$867$	1.06765e10	0.556368
$868$	1.62985e9	0.0845916
$869$	6.55421e9	0.338806
$870$	0	0
$871$	2.58593e9	0.132603
$872$	5.30765e9	0.271078
$873$	6.43475e9	0.327327
$874$	7.90695e8	0.0400608
$875$	0	0
$876$	3.40080e9	0.170929
$877$	1.77787e10	0.890024	0.445012	−	0.895525i	$-0.353199\pi$
$877$	1.77787e10	0.890024	0.445012	+	0.895525i	$0.353199\pi$
$878$	−5.94458e9	−0.296408
$879$	−3.11890e9	−0.154896
$880$	0	0
$881$	−7.64253e9	−0.376549	−0.188274	−	0.982116i	$-0.560289\pi$
$881$	−7.64253e9	−0.376549	−0.188274	+	0.982116i	$0.560289\pi$
$882$	3.58426e9	0.175897
$883$	2.76375e10	1.35094	0.675472	−	0.737386i	$-0.263940\pi$
$883$	2.76375e10	1.35094	0.675472	+	0.737386i	$0.263940\pi$
$884$	1.33135e10	0.648200
$885$	0	0
$886$	−1.06425e10	−0.514076
$887$	−3.23087e10	−1.55449	−0.777243	−	0.629200i	$-0.783383\pi$
$887$	−3.23087e10	−1.55449	−0.777243	+	0.629200i	$0.783383\pi$
$888$	−9.57027e9	−0.458647
$889$	−1.07445e9	−0.0512897
$890$	0	0
$891$	−5.03806e8	−0.0238612
$892$	−1.90014e10	−0.896414
$893$	5.02801e9	0.236274
$894$	−3.36711e9	−0.157607
$895$	0	0
$896$	1.32059e9	0.0613325
$897$	2.10433e9	0.0973511
$898$	1.66206e9	0.0765914
$899$	−1.01063e10	−0.463908
$900$	0	0
$901$	−1.21509e10	−0.553439
$902$	−3.58184e9	−0.162511
$903$	−1.18501e9	−0.0535570
$904$	2.33723e10	1.05223
$905$	0	0
$906$	−1.23301e7	−0.000550832 0
$907$	−2.27142e10	−1.01082	−0.505409	−	0.862880i	$-0.668658\pi$
$907$	−2.27142e10	−1.01082	−0.505409	+	0.862880i	$0.668658\pi$
$908$	3.99238e9	0.176983
$909$	8.73810e9	0.385872
$910$	0	0
$911$	7.50925e9	0.329065	0.164533	−	0.986372i	$-0.447388\pi$
$911$	7.50925e9	0.329065	0.164533	+	0.986372i	$0.447388\pi$
$912$	−8.97445e8	−0.0391765
$913$	−4.14916e9	−0.180431
$914$	1.76855e10	0.766136
$915$	0	0
$916$	3.32976e9	0.143146
$917$	1.07691e9	0.0461199
$918$	3.35233e9	0.143020
$919$	−2.49374e10	−1.05986	−0.529928	−	0.848043i	$-0.677781\pi$
$919$	−2.49374e10	−1.05986	−0.529928	+	0.848043i	$0.677781\pi$
$920$	0	0
$921$	7.03619e9	0.296776
$922$	1.66012e10	0.697561
$923$	2.83481e10	1.18664
$924$	1.50709e8	0.00628475
$925$	0	0
$926$	2.78132e9	0.115110
$927$	−5.25565e9	−0.216694
$928$	−7.01342e9	−0.288079
$929$	−8.66205e9	−0.354459	−0.177229	−	0.984170i	$-0.556713\pi$
$929$	−8.66205e9	−0.354459	−0.177229	+	0.984170i	$0.556713\pi$
$930$	0	0
$931$	7.06363e9	0.286883
$932$	8.48559e9	0.343342
$933$	1.55735e10	0.627768
$934$	−2.50753e9	−0.100701
$935$	0	0
$936$	4.90570e9	0.195540
$937$	−2.82655e10	−1.12245	−0.561226	−	0.827663i	$-0.689670\pi$
$937$	−2.82655e10	−1.12245	−0.561226	+	0.827663i	$0.689670\pi$
$938$	−1.94782e8	−0.00770616
$939$	1.24220e10	0.489623
$940$	0	0
$941$	−4.67082e10	−1.82738	−0.913691	−	0.406410i	$-0.866780\pi$
$941$	−4.67082e10	−1.82738	−0.913691	+	0.406410i	$0.866780\pi$
$942$	−5.21356e9	−0.203215
$943$	−9.62713e9	−0.373857
$944$	5.03740e9	0.194897
$945$	0	0
$946$	−3.90067e9	−0.149803
$947$	4.67392e10	1.78837	0.894184	−	0.447701i	$-0.147757\pi$
$947$	4.67392e10	1.78837	0.894184	+	0.447701i	$0.147757\pi$
$948$	1.71737e10	0.654688
$949$	−6.97958e9	−0.265093
$950$	0	0
$951$	−1.68902e9	−0.0636798
$952$	−2.39805e9	−0.0900801
$953$	−3.82420e10	−1.43125	−0.715625	−	0.698484i	$-0.753858\pi$
$953$	−3.82420e10	−1.43125	−0.715625	+	0.698484i	$0.753858\pi$
$954$	−1.87233e9	−0.0698171
$955$	0	0
$956$	−4.58591e9	−0.169755
$957$	−9.34510e8	−0.0344661
$958$	9.05837e9	0.332867
$959$	−1.79487e9	−0.0657157
$960$	0	0
$961$	4.91101e10	1.78500
$962$	8.21367e9	0.297457
$963$	−8.32961e9	−0.300561
$964$	−1.83424e10	−0.659457
$965$	0	0
$966$	−1.58506e8	−0.00565752
$967$	4.90012e10	1.74267	0.871333	−	0.490692i	$-0.163256\pi$
$967$	4.90012e10	1.74267	0.871333	+	0.490692i	$0.163256\pi$
$968$	−2.45368e10	−0.869468
$969$	6.60656e9	0.233261
$970$	0	0
$971$	2.72929e10	0.956713	0.478357	−	0.878166i	$-0.341233\pi$
$971$	2.72929e10	0.956713	0.478357	+	0.878166i	$0.341233\pi$
$972$	−1.32010e9	−0.0461078
$973$	−7.56947e8	−0.0263433
$974$	5.57676e9	0.193386
$975$	0	0
$976$	1.15935e9	0.0399155
$977$	−3.94482e9	−0.135331	−0.0676653	−	0.997708i	$-0.521555\pi$
$977$	−3.94482e9	−0.135331	−0.0676653	+	0.997708i	$0.521555\pi$
$978$	9.45165e9	0.323088
$979$	8.08484e9	0.275380
$980$	0	0
$981$	2.93127e9	0.0991322
$982$	−3.07682e10	−1.03684
$983$	−4.74320e8	−0.0159270	−0.00796351	−	0.999968i	$-0.502535\pi$
$983$	−4.74320e8	−0.0159270	−0.00796351	+	0.999968i	$0.502535\pi$
$984$	−2.24431e10	−0.750933
$985$	0	0
$986$	6.21824e9	0.206585
$987$	−1.00794e9	−0.0333674
$988$	4.04292e9	0.133366
$989$	−1.04841e10	−0.344622
$990$	0	0
$991$	1.22197e10	0.398843	0.199421	−	0.979914i	$-0.436094\pi$
$991$	1.22197e10	0.398843	0.199421	+	0.979914i	$0.436094\pi$
$992$	5.31737e10	1.72944
$993$	1.84744e10	0.598752
$994$	−2.13528e9	−0.0689609
$995$	0	0
$996$	−1.08718e10	−0.348655
$997$	3.60690e10	1.15266	0.576330	−	0.817217i	$-0.304484\pi$
$997$	3.60690e10	1.15266	0.576330	+	0.817217i	$0.304484\pi$
$998$	2.46390e9	0.0784631
$999$	−5.28540e9	−0.167725

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.8.a.a.1.1		1
3.2	odd	2		225.8.a.i.1.1		1
5.2	odd	4		75.8.b.c.49.1		2
5.3	odd	4		75.8.b.c.49.2		2
5.4	even	2		3.8.a.a.1.1	✓	1
15.2	even	4		225.8.b.f.199.2		2
15.8	even	4		225.8.b.f.199.1		2
15.14	odd	2		9.8.a.a.1.1		1
20.19	odd	2		48.8.a.g.1.1		1
35.4	even	6		147.8.e.b.79.1		2
35.9	even	6		147.8.e.b.67.1		2
35.19	odd	6		147.8.e.a.67.1		2
35.24	odd	6		147.8.e.a.79.1		2
35.34	odd	2		147.8.a.b.1.1		1
40.19	odd	2		192.8.a.a.1.1		1
40.29	even	2		192.8.a.i.1.1		1
45.4	even	6		81.8.c.a.55.1		2
45.14	odd	6		81.8.c.c.55.1		2
45.29	odd	6		81.8.c.c.28.1		2
45.34	even	6		81.8.c.a.28.1		2
55.54	odd	2		363.8.a.b.1.1		1
60.59	even	2		144.8.a.b.1.1		1
65.64	even	2		507.8.a.a.1.1		1
105.104	even	2		441.8.a.a.1.1		1
120.29	odd	2		576.8.a.w.1.1		1
120.59	even	2		576.8.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.8.a.a.1.1	✓	1	5.4	even	2
9.8.a.a.1.1		1	15.14	odd	2
48.8.a.g.1.1		1	20.19	odd	2
75.8.a.a.1.1		1	1.1	even	1	trivial
75.8.b.c.49.1		2	5.2	odd	4
75.8.b.c.49.2		2	5.3	odd	4
81.8.c.a.28.1		2	45.34	even	6
81.8.c.a.55.1		2	45.4	even	6
81.8.c.c.28.1		2	45.29	odd	6
81.8.c.c.55.1		2	45.14	odd	6
144.8.a.b.1.1		1	60.59	even	2
147.8.a.b.1.1		1	35.34	odd	2
147.8.e.a.67.1		2	35.19	odd	6
147.8.e.a.79.1		2	35.24	odd	6
147.8.e.b.67.1		2	35.9	even	6
147.8.e.b.79.1		2	35.4	even	6
192.8.a.a.1.1		1	40.19	odd	2
192.8.a.i.1.1		1	40.29	even	2
225.8.a.i.1.1		1	3.2	odd	2
225.8.b.f.199.1		2	15.8	even	4
225.8.b.f.199.2		2	15.2	even	4
363.8.a.b.1.1		1	55.54	odd	2
441.8.a.a.1.1		1	105.104	even	2
507.8.a.a.1.1		1	65.64	even	2
576.8.a.w.1.1		1	120.29	odd	2
576.8.a.x.1.1		1	120.59	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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