LMFDB - Embedded newform 75.10.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,10,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	75.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:	$38.6276877123$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
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)

$f(q)$

$=$

$q+4.00000 q^{2} -81.0000 q^{3} -496.000 q^{4} -324.000 q^{6} +7680.00 q^{7} -4032.00 q^{8} +6561.00 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -81.0000 q^{3} -496.000 q^{4} -324.000 q^{6} +7680.00 q^{7} -4032.00 q^{8} +6561.00 q^{9} -86404.0 q^{11} +40176.0 q^{12} +149978. q^{13} +30720.0 q^{14} +237824. q^{16} +207622. q^{17} +26244.0 q^{18} +716284. q^{19} -622080. q^{21} -345616. q^{22} -1.36992e6 q^{23} +326592. q^{24} +599912. q^{26} -531441. q^{27} -3.80928e6 q^{28} -3.19440e6 q^{29} -2.34900e6 q^{31} +3.01568e6 q^{32} +6.99872e6 q^{33} +830488. q^{34} -3.25426e6 q^{36} -1.87357e7 q^{37} +2.86514e6 q^{38} -1.21482e7 q^{39} -2.92826e7 q^{41} -2.48832e6 q^{42} +1.51672e6 q^{43} +4.28564e7 q^{44} -5.47968e6 q^{46} -615752. q^{47} -1.92637e7 q^{48} +1.86288e7 q^{49} -1.68174e7 q^{51} -7.43891e7 q^{52} -4.74743e6 q^{53} -2.12576e6 q^{54} -3.09658e7 q^{56} -5.80190e7 q^{57} -1.27776e7 q^{58} +6.06161e7 q^{59} -1.26746e8 q^{61} -9.39600e6 q^{62} +5.03885e7 q^{63} -1.09703e8 q^{64} +2.79949e7 q^{66} +1.11183e8 q^{67} -1.02981e8 q^{68} +1.10964e8 q^{69} -1.75552e8 q^{71} -2.64540e7 q^{72} +6.12334e7 q^{73} -7.49428e7 q^{74} -3.55277e8 q^{76} -6.63583e8 q^{77} -4.85929e7 q^{78} +2.34431e8 q^{79} +4.30467e7 q^{81} -1.17131e8 q^{82} -1.18910e8 q^{83} +3.08552e8 q^{84} +6.06690e6 q^{86} +2.58747e8 q^{87} +3.48381e8 q^{88} -3.16534e8 q^{89} +1.15183e9 q^{91} +6.79480e8 q^{92} +1.90269e8 q^{93} -2.46301e6 q^{94} -2.44270e8 q^{96} -2.42912e8 q^{97} +7.45152e7 q^{98} -5.66897e8 q^{99} +O(q^{100})\)

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 $

$2$	4.00000	0.176777	0.0883883	−	0.996086i	$-0.471828\pi$
$2$	4.00000	0.176777	0.0883883	+	0.996086i	$0.471828\pi$
$3$	−81.0000	−0.577350
$3$	−81.0000	−0.577350
$4$	−496.000	−0.968750
$5$	0	0
$5$	0	0
$6$	−324.000	−0.102062
$7$	7680.00	1.20898	0.604491	−	0.796612i	$-0.293376\pi$
$7$	7680.00	1.20898	0.604491	+	0.796612i	$0.293376\pi$
$8$	−4032.00	−0.348029
$9$	6561.00	0.333333
$10$	0	0
$11$	−86404.0	−1.77937	−0.889686	−	0.456573i	$-0.849077\pi$
$11$	−86404.0	−1.77937	−0.889686	+	0.456573i	$0.849077\pi$
$12$	40176.0	0.559308
$13$	149978.	1.45641	0.728203	−	0.685361i	$-0.240356\pi$
$13$	149978.	1.45641	0.728203	+	0.685361i	$0.240356\pi$
$14$	30720.0	0.213720
$15$	0	0
$16$	237824.	0.907227
$17$	207622.	0.602911	0.301456	−	0.953480i	$-0.402528\pi$
$17$	207622.	0.602911	0.301456	+	0.953480i	$0.402528\pi$
$18$	26244.0	0.0589256
$19$	716284.	1.26094	0.630469	−	0.776214i	$-0.282862\pi$
$19$	716284.	1.26094	0.630469	+	0.776214i	$0.282862\pi$
$20$	0	0
$21$	−622080.	−0.698006
$22$	−345616.	−0.314552
$23$	−1.36992e6	−1.02075	−0.510376	−	0.859952i	$-0.670494\pi$
$23$	−1.36992e6	−1.02075	−0.510376	+	0.859952i	$0.670494\pi$
$24$	326592.	0.200935
$25$	0	0
$26$	599912.	0.257459
$27$	−531441.	−0.192450
$28$	−3.80928e6	−1.17120
$29$	−3.19440e6	−0.838684	−0.419342	−	0.907828i	$-0.637739\pi$
$29$	−3.19440e6	−0.838684	−0.419342	+	0.907828i	$0.637739\pi$
$30$	0	0
$31$	−2.34900e6	−0.456831	−0.228415	−	0.973564i	$-0.573354\pi$
$31$	−2.34900e6	−0.456831	−0.228415	+	0.973564i	$0.573354\pi$
$32$	3.01568e6	0.508406
$33$	6.99872e6	1.02732
$34$	830488.	0.106581
$35$	0	0
$36$	−3.25426e6	−0.322917
$37$	−1.87357e7	−1.64347	−0.821736	−	0.569868i	$-0.806994\pi$
$37$	−1.87357e7	−1.64347	−0.821736	+	0.569868i	$0.806994\pi$
$38$	2.86514e6	0.222905
$39$	−1.21482e7	−0.840856
$40$	0	0
$41$	−2.92826e7	−1.61839	−0.809194	−	0.587541i	$-0.800096\pi$
$41$	−2.92826e7	−1.61839	−0.809194	+	0.587541i	$0.800096\pi$
$42$	−2.48832e6	−0.123391
$43$	1.51672e6	0.0676548	0.0338274	−	0.999428i	$-0.489230\pi$
$43$	1.51672e6	0.0676548	0.0338274	+	0.999428i	$0.489230\pi$
$44$	4.28564e7	1.72377
$45$	0	0
$46$	−5.47968e6	−0.180445
$47$	−615752.	−0.0184063	−0.00920313	−	0.999958i	$-0.502929\pi$
$47$	−615752.	−0.0184063	−0.00920313	+	0.999958i	$0.502929\pi$
$48$	−1.92637e7	−0.523788
$49$	1.86288e7	0.461639
$50$	0	0
$51$	−1.68174e7	−0.348091
$52$	−7.43891e7	−1.41089
$53$	−4.74743e6	−0.0826451	−0.0413226	−	0.999146i	$-0.513157\pi$
$53$	−4.74743e6	−0.0826451	−0.0413226	+	0.999146i	$0.513157\pi$
$54$	−2.12576e6	−0.0340207
$55$	0	0
$56$	−3.09658e7	−0.420761
$57$	−5.80190e7	−0.728003
$58$	−1.27776e7	−0.148260
$59$	6.06161e7	0.651259	0.325630	−	0.945497i	$-0.394424\pi$
$59$	6.06161e7	0.651259	0.325630	+	0.945497i	$0.394424\pi$
$60$	0	0
$61$	−1.26746e8	−1.17206	−0.586029	−	0.810290i	$-0.699309\pi$
$61$	−1.26746e8	−1.17206	−0.586029	+	0.810290i	$0.699309\pi$
$62$	−9.39600e6	−0.0807570
$63$	5.03885e7	0.402994
$64$	−1.09703e8	−0.817352
$65$	0	0
$66$	2.79949e7	0.181606
$67$	1.11183e8	0.674063	0.337031	−	0.941493i	$-0.390577\pi$
$67$	1.11183e8	0.674063	0.337031	+	0.941493i	$0.390577\pi$
$68$	−1.02981e8	−0.584070
$69$	1.10964e8	0.589331
$70$	0	0
$71$	−1.75552e8	−0.819865	−0.409932	−	0.912116i	$-0.634448\pi$
$71$	−1.75552e8	−0.819865	−0.409932	+	0.912116i	$0.634448\pi$
$72$	−2.64540e7	−0.116010
$73$	6.12334e7	0.252369	0.126184	−	0.992007i	$-0.459727\pi$
$73$	6.12334e7	0.252369	0.126184	+	0.992007i	$0.459727\pi$
$74$	−7.49428e7	−0.290528
$75$	0	0
$76$	−3.55277e8	−1.22153
$77$	−6.63583e8	−2.15123
$78$	−4.85929e7	−0.148644
$79$	2.34431e8	0.677163	0.338582	−	0.940937i	$-0.390053\pi$
$79$	2.34431e8	0.677163	0.338582	+	0.940937i	$0.390053\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	−1.17131e8	−0.286093
$83$	−1.18910e8	−0.275023	−0.137511	−	0.990500i	$-0.543910\pi$
$83$	−1.18910e8	−0.275023	−0.137511	+	0.990500i	$0.543910\pi$
$84$	3.08552e8	0.676194
$85$	0	0
$86$	6.06690e6	0.0119598
$87$	2.58747e8	0.484215
$88$	3.48381e8	0.619273
$89$	−3.16534e8	−0.534768	−0.267384	−	0.963590i	$-0.586159\pi$
$89$	−3.16534e8	−0.534768	−0.267384	+	0.963590i	$0.586159\pi$
$90$	0	0
$91$	1.15183e9	1.76077
$92$	6.79480e8	0.988853
$93$	1.90269e8	0.263751
$94$	−2.46301e6	−0.00325380
$95$	0	0
$96$	−2.44270e8	−0.293528
$97$	−2.42912e8	−0.278597	−0.139299	−	0.990250i	$-0.544485\pi$
$97$	−2.42912e8	−0.278597	−0.139299	+	0.990250i	$0.544485\pi$
$98$	7.45152e7	0.0816070
$99$	−5.66897e8	−0.593124
$100$	0	0
$101$	−6.53803e8	−0.625173	−0.312587	−	0.949889i	$-0.601195\pi$
$101$	−6.53803e8	−0.625173	−0.312587	+	0.949889i	$0.601195\pi$
$102$	−6.72695e7	−0.0615343
$103$	−1.40420e9	−1.22931	−0.614656	−	0.788795i	$-0.710705\pi$
$103$	−1.40420e9	−1.22931	−0.614656	+	0.788795i	$0.710705\pi$
$104$	−6.04711e8	−0.506872
$105$	0	0
$106$	−1.89897e7	−0.0146097
$107$	1.83854e9	1.35595	0.677977	−	0.735083i	$-0.262857\pi$
$107$	1.83854e9	1.35595	0.677977	+	0.735083i	$0.262857\pi$
$108$	2.63595e8	0.186436
$109$	−9.33452e8	−0.633392	−0.316696	−	0.948527i	$-0.602574\pi$
$109$	−9.33452e8	−0.633392	−0.316696	+	0.948527i	$0.602574\pi$
$110$	0	0
$111$	1.51759e9	0.948859
$112$	1.82649e9	1.09682
$113$	9.28534e7	0.0535728	0.0267864	−	0.999641i	$-0.491473\pi$
$113$	9.28534e7	0.0535728	0.0267864	+	0.999641i	$0.491473\pi$
$114$	−2.32076e8	−0.128694
$115$	0	0
$116$	1.58442e9	0.812476
$117$	9.84006e8	0.485469
$118$	2.42464e8	0.115127
$119$	1.59454e9	0.728909
$120$	0	0
$121$	5.10770e9	2.16616
$122$	−5.06983e8	−0.207192
$123$	2.37189e9	0.934377
$124$	1.16510e9	0.442555
$125$	0	0
$126$	2.01554e8	0.0712400
$127$	−1.73819e9	−0.592900	−0.296450	−	0.955048i	$-0.595803\pi$
$127$	−1.73819e9	−0.592900	−0.296450	+	0.955048i	$0.595803\pi$
$128$	−1.98284e9	−0.652894
$129$	−1.22855e8	−0.0390605
$130$	0	0
$131$	−2.49730e9	−0.740882	−0.370441	−	0.928856i	$-0.620793\pi$
$131$	−2.49730e9	−0.740882	−0.370441	+	0.928856i	$0.620793\pi$
$132$	−3.47137e9	−0.995217
$133$	5.50106e9	1.52445
$134$	4.44731e8	0.119159
$135$	0	0
$136$	−8.37132e8	−0.209831
$137$	7.96226e9	1.93105	0.965526	−	0.260306i	$-0.0838234\pi$
$137$	7.96226e9	1.93105	0.965526	+	0.260306i	$0.0838234\pi$
$138$	4.43854e8	0.104180
$139$	−2.85565e9	−0.648842	−0.324421	−	0.945913i	$-0.605169\pi$
$139$	−2.85565e9	−0.648842	−0.324421	+	0.945913i	$0.605169\pi$
$140$	0	0
$141$	4.98759e7	0.0106269
$142$	−7.02206e8	−0.144933
$143$	−1.29587e10	−2.59149
$144$	1.56036e9	0.302409
$145$	0	0
$146$	2.44933e8	0.0446129
$147$	−1.50893e9	−0.266527
$148$	9.29291e9	1.59211
$149$	−9.63383e9	−1.60126	−0.800628	−	0.599161i	$-0.795501\pi$
$149$	−9.63383e9	−1.60126	−0.800628	+	0.599161i	$0.795501\pi$
$150$	0	0
$151$	−5.38292e9	−0.842601	−0.421300	−	0.906921i	$-0.638426\pi$
$151$	−5.38292e9	−0.842601	−0.421300	+	0.906921i	$0.638426\pi$
$152$	−2.88806e9	−0.438843
$153$	1.36221e9	0.200970
$154$	−2.65433e9	−0.380287
$155$	0	0
$156$	6.02552e9	0.814580
$157$	−5.19434e8	−0.0682310	−0.0341155	−	0.999418i	$-0.510861\pi$
$157$	−5.19434e8	−0.0682310	−0.0341155	+	0.999418i	$0.510861\pi$
$158$	9.37725e8	0.119707
$159$	3.84542e8	0.0477152
$160$	0	0
$161$	−1.05210e10	−1.23407
$162$	1.72187e8	0.0196419
$163$	−9.41239e9	−1.04437	−0.522187	−	0.852831i	$-0.674884\pi$
$163$	−9.41239e9	−1.04437	−0.522187	+	0.852831i	$0.674884\pi$
$164$	1.45242e10	1.56781
$165$	0	0
$166$	−4.75642e8	−0.0486176
$167$	−9.37241e9	−0.932453	−0.466227	−	0.884665i	$-0.654387\pi$
$167$	−9.37241e9	−0.932453	−0.466227	+	0.884665i	$0.654387\pi$
$168$	2.50823e9	0.242927
$169$	1.18889e10	1.12112
$170$	0	0
$171$	4.69954e9	0.420313
$172$	−7.52295e8	−0.0655406
$173$	−1.23573e10	−1.04886	−0.524428	−	0.851455i	$-0.675721\pi$
$173$	−1.23573e10	−1.04886	−0.524428	+	0.851455i	$0.675721\pi$
$174$	1.03499e9	0.0855979
$175$	0	0
$176$	−2.05489e10	−1.61429
$177$	−4.90990e9	−0.376005
$178$	−1.26614e9	−0.0945346
$179$	−6.66040e8	−0.0484910	−0.0242455	−	0.999706i	$-0.507718\pi$
$179$	−6.66040e8	−0.0484910	−0.0242455	+	0.999706i	$0.507718\pi$
$180$	0	0
$181$	5.27207e9	0.365113	0.182557	−	0.983195i	$-0.441563\pi$
$181$	5.27207e9	0.365113	0.182557	+	0.983195i	$0.441563\pi$
$182$	4.60732e9	0.311263
$183$	1.02664e10	0.676688
$184$	5.52352e9	0.355251
$185$	0	0
$186$	7.61076e8	0.0466251
$187$	−1.79394e10	−1.07280
$188$	3.05413e8	0.0178311
$189$	−4.08147e9	−0.232669
$190$	0	0
$191$	−2.93896e10	−1.59788	−0.798939	−	0.601412i	$-0.794605\pi$
$191$	−2.93896e10	−1.59788	−0.798939	+	0.601412i	$0.794605\pi$
$192$	8.88596e9	0.471899
$193$	1.48746e10	0.771681	0.385841	−	0.922565i	$-0.373911\pi$
$193$	1.48746e10	0.771681	0.385841	+	0.922565i	$0.373911\pi$
$194$	−9.71649e8	−0.0492495
$195$	0	0
$196$	−9.23988e9	−0.447213
$197$	−4.98675e9	−0.235895	−0.117948	−	0.993020i	$-0.537632\pi$
$197$	−4.98675e9	−0.235895	−0.117948	+	0.993020i	$0.537632\pi$
$198$	−2.26759e9	−0.104851
$199$	1.45527e10	0.657816	0.328908	−	0.944362i	$-0.393319\pi$
$199$	1.45527e10	0.657816	0.328908	+	0.944362i	$0.393319\pi$
$200$	0	0
$201$	−9.00579e9	−0.389170
$202$	−2.61521e9	−0.110516
$203$	−2.45330e10	−1.01395
$204$	8.34142e9	0.337213
$205$	0	0
$206$	−5.61681e9	−0.217314
$207$	−8.98805e9	−0.340250
$208$	3.56684e10	1.32129
$209$	−6.18898e10	−2.24368
$210$	0	0
$211$	5.15407e10	1.79011	0.895054	−	0.445959i	$-0.147137\pi$
$211$	5.15407e10	1.79011	0.895054	+	0.445959i	$0.147137\pi$
$212$	2.35473e9	0.0800624
$213$	1.42197e10	0.473349
$214$	7.35414e9	0.239701
$215$	0	0
$216$	2.14277e9	0.0669782
$217$	−1.80403e10	−0.552300
$218$	−3.73381e9	−0.111969
$219$	−4.95990e9	−0.145705
$220$	0	0
$221$	3.11387e10	0.878083
$222$	6.07037e9	0.167736
$223$	−4.61272e10	−1.24907	−0.624533	−	0.780998i	$-0.714711\pi$
$223$	−4.61272e10	−1.24907	−0.624533	+	0.780998i	$0.714711\pi$
$224$	2.31604e10	0.614654
$225$	0	0
$226$	3.71414e8	0.00947043
$227$	3.75833e10	0.939460	0.469730	−	0.882810i	$-0.344351\pi$
$227$	3.75833e10	0.939460	0.469730	+	0.882810i	$0.344351\pi$
$228$	2.87774e10	0.705253
$229$	−6.41082e10	−1.54047	−0.770236	−	0.637759i	$-0.779862\pi$
$229$	−6.41082e10	−1.54047	−0.770236	+	0.637759i	$0.779862\pi$
$230$	0	0
$231$	5.37502e10	1.24201
$232$	1.28798e10	0.291887
$233$	−6.96578e10	−1.54835	−0.774174	−	0.632973i	$-0.781834\pi$
$233$	−6.96578e10	−1.54835	−0.774174	+	0.632973i	$0.781834\pi$
$234$	3.93602e9	0.0858195
$235$	0	0
$236$	−3.00656e10	−0.630907
$237$	−1.89889e10	−0.390960
$238$	6.37815e9	0.128854
$239$	6.65825e10	1.31999	0.659993	−	0.751272i	$-0.270559\pi$
$239$	6.65825e10	1.31999	0.659993	+	0.751272i	$0.270559\pi$
$240$	0	0
$241$	−4.41659e10	−0.843354	−0.421677	−	0.906746i	$-0.638558\pi$
$241$	−4.41659e10	−0.843354	−0.421677	+	0.906746i	$0.638558\pi$
$242$	2.04308e10	0.382927
$243$	−3.48678e9	−0.0641500
$244$	6.28659e10	1.13543
$245$	0	0
$246$	9.48757e9	0.165176
$247$	1.07427e11	1.83644
$248$	9.47117e9	0.158990
$249$	9.63174e9	0.158784
$250$	0	0
$251$	8.36236e10	1.32983	0.664916	−	0.746918i	$-0.268467\pi$
$251$	8.36236e10	1.32983	0.664916	+	0.746918i	$0.268467\pi$
$252$	−2.49927e10	−0.390401
$253$	1.18367e11	1.81630
$254$	−6.95277e9	−0.104811
$255$	0	0
$256$	4.82367e10	0.701936
$257$	8.65274e10	1.23724	0.618621	−	0.785690i	$-0.287692\pi$
$257$	8.65274e10	1.23724	0.618621	+	0.785690i	$0.287692\pi$
$258$	−4.91419e8	−0.00690499
$259$	−1.43890e11	−1.98693
$260$	0	0
$261$	−2.09585e10	−0.279561
$262$	−9.98919e9	−0.130971
$263$	9.61535e10	1.23927	0.619633	−	0.784892i	$-0.287282\pi$
$263$	9.61535e10	1.23927	0.619633	+	0.784892i	$0.287282\pi$
$264$	−2.82189e10	−0.357538
$265$	0	0
$266$	2.20042e10	0.269488
$267$	2.56393e10	0.308749
$268$	−5.51466e10	−0.652998
$269$	−1.09505e10	−0.127511	−0.0637557	−	0.997966i	$-0.520308\pi$
$269$	−1.09505e10	−0.127511	−0.0637557	+	0.997966i	$0.520308\pi$
$270$	0	0
$271$	7.80287e10	0.878805	0.439403	−	0.898290i	$-0.355190\pi$
$271$	7.80287e10	0.878805	0.439403	+	0.898290i	$0.355190\pi$
$272$	4.93775e10	0.546977
$273$	−9.32983e10	−1.01658
$274$	3.18491e10	0.341365
$275$	0	0
$276$	−5.50379e10	−0.570914
$277$	−6.56840e10	−0.670349	−0.335174	−	0.942156i	$-0.608795\pi$
$277$	−6.56840e10	−0.670349	−0.335174	+	0.942156i	$0.608795\pi$
$278$	−1.14226e10	−0.114700
$279$	−1.54118e10	−0.152277
$280$	0	0
$281$	−6.44906e10	−0.617046	−0.308523	−	0.951217i	$-0.599835\pi$
$281$	−6.44906e10	−0.617046	−0.308523	+	0.951217i	$0.599835\pi$
$282$	1.99504e8	0.00187858
$283$	−9.63133e10	−0.892580	−0.446290	−	0.894888i	$-0.647255\pi$
$283$	−9.63133e10	−0.892580	−0.446290	+	0.894888i	$0.647255\pi$
$284$	8.70736e10	0.794244
$285$	0	0
$286$	−5.18348e10	−0.458115
$287$	−2.24891e11	−1.95660
$288$	1.97859e10	0.169469
$289$	−7.54810e10	−0.636498
$290$	0	0
$291$	1.96759e10	0.160848
$292$	−3.03717e10	−0.244482
$293$	−8.16308e10	−0.647068	−0.323534	−	0.946217i	$-0.604871\pi$
$293$	−8.16308e10	−0.647068	−0.323534	+	0.946217i	$0.604871\pi$
$294$	−6.03573e9	−0.0471158
$295$	0	0
$296$	7.55424e10	0.571976
$297$	4.59186e10	0.342440
$298$	−3.85353e10	−0.283065
$299$	−2.05458e11	−1.48663
$300$	0	0
$301$	1.16484e10	0.0817935
$302$	−2.15317e10	−0.148952
$303$	5.29580e10	0.360944
$304$	1.70350e11	1.14396
$305$	0	0
$306$	5.44883e9	0.0355269
$307$	2.95582e10	0.189914	0.0949568	−	0.995481i	$-0.469729\pi$
$307$	2.95582e10	0.189914	0.0949568	+	0.995481i	$0.469729\pi$
$308$	3.29137e11	2.08400
$309$	1.13740e11	0.709744
$310$	0	0
$311$	−3.99071e10	−0.241896	−0.120948	−	0.992659i	$-0.538593\pi$
$311$	−3.99071e10	−0.241896	−0.120948	+	0.992659i	$0.538593\pi$
$312$	4.89816e10	0.292643
$313$	−1.85371e11	−1.09167	−0.545836	−	0.837892i	$-0.683788\pi$
$313$	−1.85371e11	−1.09167	−0.545836	+	0.837892i	$0.683788\pi$
$314$	−2.07774e9	−0.0120617
$315$	0	0
$316$	−1.16278e11	−0.656002
$317$	−2.68895e11	−1.49560	−0.747800	−	0.663924i	$-0.768890\pi$
$317$	−2.68895e11	−1.49560	−0.747800	+	0.663924i	$0.768890\pi$
$318$	1.53817e9	0.00843493
$319$	2.76009e11	1.49233
$320$	0	0
$321$	−1.48921e11	−0.782860
$322$	−4.20839e10	−0.218155
$323$	1.48716e11	0.760234
$324$	−2.13512e10	−0.107639
$325$	0	0
$326$	−3.76496e10	−0.184621
$327$	7.56096e10	0.365689
$328$	1.18068e11	0.563246
$329$	−4.72898e9	−0.0222528
$330$	0	0
$331$	−4.29099e11	−1.96486	−0.982430	−	0.186629i	$-0.940244\pi$
$331$	−4.29099e11	−1.96486	−0.982430	+	0.186629i	$0.940244\pi$
$332$	5.89796e10	0.266428
$333$	−1.22925e11	−0.547824
$334$	−3.74896e10	−0.164836
$335$	0	0
$336$	−1.47946e11	−0.633250
$337$	2.02598e10	0.0855657	0.0427828	−	0.999084i	$-0.486378\pi$
$337$	2.02598e10	0.0855657	0.0427828	+	0.999084i	$0.486378\pi$
$338$	4.75556e10	0.198188
$339$	−7.52112e9	−0.0309303
$340$	0	0
$341$	2.02963e11	0.812872
$342$	1.87982e10	0.0743015
$343$	−1.66847e11	−0.650869
$344$	−6.11543e9	−0.0235458
$345$	0	0
$346$	−4.94292e10	−0.185413
$347$	−2.92783e10	−0.108409	−0.0542043	−	0.998530i	$-0.517262\pi$
$347$	−2.92783e10	−0.108409	−0.0542043	+	0.998530i	$0.517262\pi$
$348$	−1.28338e11	−0.469083
$349$	7.05132e10	0.254423	0.127211	−	0.991876i	$-0.459397\pi$
$349$	7.05132e10	0.254423	0.127211	+	0.991876i	$0.459397\pi$
$350$	0	0
$351$	−7.97045e10	−0.280285
$352$	−2.60567e11	−0.904643
$353$	6.57350e10	0.225325	0.112663	−	0.993633i	$-0.464062\pi$
$353$	6.57350e10	0.225325	0.112663	+	0.993633i	$0.464062\pi$
$354$	−1.96396e10	−0.0664688
$355$	0	0
$356$	1.57001e11	0.518057
$357$	−1.29157e11	−0.420836
$358$	−2.66416e9	−0.00857209
$359$	5.81702e11	1.84831	0.924157	−	0.382013i	$-0.124769\pi$
$359$	5.81702e11	1.84831	0.924157	+	0.382013i	$0.124769\pi$
$360$	0	0
$361$	1.90375e11	0.589967
$362$	2.10883e10	0.0645435
$363$	−4.13724e11	−1.25064
$364$	−5.71308e11	−1.70575
$365$	0	0
$366$	4.10656e10	0.119623
$367$	4.17070e11	1.20008	0.600042	−	0.799969i	$-0.295151\pi$
$367$	4.17070e11	1.20008	0.600042	+	0.799969i	$0.295151\pi$
$368$	−3.25800e11	−0.926053
$369$	−1.92123e11	−0.539463
$370$	0	0
$371$	−3.64603e10	−0.0999165
$372$	−9.43734e10	−0.255509
$373$	7.60417e10	0.203405	0.101703	−	0.994815i	$-0.467571\pi$
$373$	7.60417e10	0.203405	0.101703	+	0.994815i	$0.467571\pi$
$374$	−7.17575e10	−0.189647
$375$	0	0
$376$	2.48271e9	0.00640591
$377$	−4.79090e11	−1.22147
$378$	−1.63259e10	−0.0411304
$379$	−1.79180e11	−0.446080	−0.223040	−	0.974809i	$-0.571598\pi$
$379$	−1.79180e11	−0.446080	−0.223040	+	0.974809i	$0.571598\pi$
$380$	0	0
$381$	1.40794e11	0.342311
$382$	−1.17558e11	−0.282467
$383$	7.95018e11	1.88792	0.943958	−	0.330066i	$-0.107071\pi$
$383$	7.95018e11	1.88792	0.943958	+	0.330066i	$0.107071\pi$
$384$	1.60610e11	0.376949
$385$	0	0
$386$	5.94985e10	0.136415
$387$	9.95123e9	0.0225516
$388$	1.20484e11	0.269891
$389$	−1.79533e11	−0.397532	−0.198766	−	0.980047i	$-0.563693\pi$
$389$	−1.79533e11	−0.397532	−0.198766	+	0.980047i	$0.563693\pi$
$390$	0	0
$391$	−2.84426e11	−0.615422
$392$	−7.51113e10	−0.160664
$393$	2.02281e11	0.427749
$394$	−1.99470e10	−0.0417008
$395$	0	0
$396$	2.81181e11	0.574589
$397$	3.43730e11	0.694480	0.347240	−	0.937776i	$-0.387119\pi$
$397$	3.43730e11	0.694480	0.347240	+	0.937776i	$0.387119\pi$
$398$	5.82108e10	0.116287
$399$	−4.45586e11	−0.880143
$400$	0	0
$401$	7.72080e11	1.49112	0.745560	−	0.666438i	$-0.232182\pi$
$401$	7.72080e11	1.49112	0.745560	+	0.666438i	$0.232182\pi$
$402$	−3.60232e10	−0.0687963
$403$	−3.52298e11	−0.665331
$404$	3.24286e11	0.605637
$405$	0	0
$406$	−9.81320e10	−0.179244
$407$	1.61884e12	2.92435
$408$	6.78077e10	0.121146
$409$	2.60632e11	0.460546	0.230273	−	0.973126i	$-0.426038\pi$
$409$	2.60632e11	0.460546	0.230273	+	0.973126i	$0.426038\pi$
$410$	0	0
$411$	−6.44943e11	−1.11489
$412$	6.96484e11	1.19090
$413$	4.65531e11	0.787361
$414$	−3.59522e10	−0.0601483
$415$	0	0
$416$	4.52286e11	0.740445
$417$	2.31308e11	0.374609
$418$	−2.47559e11	−0.396630
$419$	−5.60166e11	−0.887879	−0.443939	−	0.896057i	$-0.646419\pi$
$419$	−5.60166e11	−0.887879	−0.443939	+	0.896057i	$0.646419\pi$
$420$	0	0
$421$	1.68321e11	0.261137	0.130569	−	0.991439i	$-0.458320\pi$
$421$	1.68321e11	0.261137	0.130569	+	0.991439i	$0.458320\pi$
$422$	2.06163e11	0.316449
$423$	−4.03995e9	−0.00613542
$424$	1.91416e10	0.0287629
$425$	0	0
$426$	5.68787e10	0.0836771
$427$	−9.73407e11	−1.41700
$428$	−9.11913e11	−1.31358
$429$	1.04965e12	1.49620
$430$	0	0
$431$	4.48383e11	0.625895	0.312948	−	0.949770i	$-0.398684\pi$
$431$	4.48383e11	0.625895	0.312948	+	0.949770i	$0.398684\pi$
$432$	−1.26389e11	−0.174596
$433$	1.08485e12	1.48311	0.741556	−	0.670891i	$-0.234088\pi$
$433$	1.08485e12	1.48311	0.741556	+	0.670891i	$0.234088\pi$
$434$	−7.21613e10	−0.0976339
$435$	0	0
$436$	4.62992e11	0.613599
$437$	−9.81252e11	−1.28711
$438$	−1.98396e10	−0.0257573
$439$	4.60548e11	0.591814	0.295907	−	0.955217i	$-0.404378\pi$
$439$	4.60548e11	0.591814	0.295907	+	0.955217i	$0.404378\pi$
$440$	0	0
$441$	1.22224e11	0.153880
$442$	1.24555e11	0.155225
$443$	1.32095e10	0.0162956	0.00814779	−	0.999967i	$-0.497406\pi$
$443$	1.32095e10	0.0162956	0.00814779	+	0.999967i	$0.497406\pi$
$444$	−7.52726e11	−0.919207
$445$	0	0
$446$	−1.84509e11	−0.220806
$447$	7.80341e11	0.924486
$448$	−8.42520e11	−0.988165
$449$	−6.91889e11	−0.803393	−0.401696	−	0.915773i	$-0.631579\pi$
$449$	−6.91889e11	−0.803393	−0.401696	+	0.915773i	$0.631579\pi$
$450$	0	0
$451$	2.53014e12	2.87971
$452$	−4.60553e10	−0.0518987
$453$	4.36017e11	0.486476
$454$	1.50333e11	0.166075
$455$	0	0
$456$	2.33933e11	0.253366
$457$	3.73135e11	0.400168	0.200084	−	0.979779i	$-0.435878\pi$
$457$	3.73135e11	0.400168	0.200084	+	0.979779i	$0.435878\pi$
$458$	−2.56433e11	−0.272320
$459$	−1.10339e11	−0.116030
$460$	0	0
$461$	−1.45940e12	−1.50494	−0.752470	−	0.658627i	$-0.771138\pi$
$461$	−1.45940e12	−1.50494	−0.752470	+	0.658627i	$0.771138\pi$
$462$	2.15001e11	0.219559
$463$	−1.34213e11	−0.135732	−0.0678658	−	0.997694i	$-0.521619\pi$
$463$	−1.34213e11	−0.135732	−0.0678658	+	0.997694i	$0.521619\pi$
$464$	−7.59705e11	−0.760877
$465$	0	0
$466$	−2.78631e11	−0.273712
$467$	3.64531e10	0.0354657	0.0177329	−	0.999843i	$-0.494355\pi$
$467$	3.64531e10	0.0354657	0.0177329	+	0.999843i	$0.494355\pi$
$468$	−4.88067e11	−0.470298
$469$	8.53883e11	0.814930
$470$	0	0
$471$	4.20741e10	0.0393932
$472$	−2.44404e11	−0.226657
$473$	−1.31051e11	−0.120383
$474$	−7.59557e10	−0.0691127
$475$	0	0
$476$	−7.90890e11	−0.706131
$477$	−3.11479e10	−0.0275484
$478$	2.66330e11	0.233343
$479$	8.82280e11	0.765767	0.382883	−	0.923797i	$-0.374931\pi$
$479$	8.82280e11	0.765767	0.382883	+	0.923797i	$0.374931\pi$
$480$	0	0
$481$	−2.80994e12	−2.39356
$482$	−1.76663e11	−0.149085
$483$	8.52200e11	0.712491
$484$	−2.53342e12	−2.09847
$485$	0	0
$486$	−1.39471e10	−0.0113402
$487$	−5.09840e11	−0.410727	−0.205364	−	0.978686i	$-0.565838\pi$
$487$	−5.09840e11	−0.410727	−0.205364	+	0.978686i	$0.565838\pi$
$488$	5.11039e11	0.407910
$489$	7.62404e11	0.602969
$490$	0	0
$491$	−1.52131e12	−1.18128	−0.590638	−	0.806937i	$-0.701124\pi$
$491$	−1.52131e12	−1.18128	−0.590638	+	0.806937i	$0.701124\pi$
$492$	−1.17646e12	−0.905178
$493$	−6.63228e11	−0.505652
$494$	4.29707e11	0.324640
$495$	0	0
$496$	−5.58649e11	−0.414449
$497$	−1.34824e12	−0.991202
$498$	3.85270e10	0.0280694
$499$	7.03413e11	0.507876	0.253938	−	0.967220i	$-0.418274\pi$
$499$	7.03413e11	0.507876	0.253938	+	0.967220i	$0.418274\pi$
$500$	0	0
$501$	7.59165e11	0.538352
$502$	3.34494e11	0.235083
$503$	−3.78018e8	−0.000263304 0	−0.000131652	−	1.00000i	$-0.500042\pi$
$503$	−3.78018e8	−0.000263304 0	−0.000131652		1.00000i	$0.500042\pi$
$504$	−2.03166e11	−0.140254
$505$	0	0
$506$	4.73466e11	0.321079
$507$	−9.63001e11	−0.647278
$508$	8.62144e11	0.574372
$509$	1.32057e12	0.872027	0.436013	−	0.899940i	$-0.356390\pi$
$509$	1.32057e12	0.872027	0.436013	+	0.899940i	$0.356390\pi$
$510$	0	0
$511$	4.70272e11	0.305109
$512$	1.20816e12	0.776980
$513$	−3.80663e11	−0.242668
$514$	3.46110e11	0.218715
$515$	0	0
$516$	6.09359e10	0.0378399
$517$	5.32034e10	0.0327516
$518$	−5.75561e11	−0.351243
$519$	1.00094e12	0.605558
$520$	0	0
$521$	−1.31853e12	−0.784009	−0.392005	−	0.919963i	$-0.628218\pi$
$521$	−1.31853e12	−0.784009	−0.392005	+	0.919963i	$0.628218\pi$
$522$	−8.38339e10	−0.0494200
$523$	1.69211e12	0.988945	0.494472	−	0.869193i	$-0.335361\pi$
$523$	1.69211e12	0.988945	0.494472	+	0.869193i	$0.335361\pi$
$524$	1.23866e12	0.717730
$525$	0	0
$526$	3.84614e11	0.219073
$527$	−4.87704e11	−0.275428
$528$	1.66446e12	0.932013
$529$	7.55281e10	0.0419332
$530$	0	0
$531$	3.97702e11	0.217086
$532$	−2.72853e12	−1.47681
$533$	−4.39175e12	−2.35703
$534$	1.02557e11	0.0545796
$535$	0	0
$536$	−4.48288e11	−0.234594
$537$	5.39492e10	0.0279963
$538$	−4.38021e10	−0.0225411
$539$	−1.60960e12	−0.821427
$540$	0	0
$541$	−1.86369e12	−0.935373	−0.467687	−	0.883894i	$-0.654912\pi$
$541$	−1.86369e12	−0.935373	−0.467687	+	0.883894i	$0.654912\pi$
$542$	3.12115e11	0.155352
$543$	−4.27037e11	−0.210798
$544$	6.26122e11	0.306523
$545$	0	0
$546$	−3.73193e11	−0.179708
$547$	−4.37242e11	−0.208823	−0.104412	−	0.994534i	$-0.533296\pi$
$547$	−4.37242e11	−0.208823	−0.104412	+	0.994534i	$0.533296\pi$
$548$	−3.94928e12	−1.87071
$549$	−8.31578e11	−0.390686
$550$	0	0
$551$	−2.28810e12	−1.05753
$552$	−4.47405e11	−0.205104
$553$	1.80043e12	0.818679
$554$	−2.62736e11	−0.118502
$555$	0	0
$556$	1.41640e12	0.628565
$557$	7.09146e11	0.312167	0.156084	−	0.987744i	$-0.450113\pi$
$557$	7.09146e11	0.312167	0.156084	+	0.987744i	$0.450113\pi$
$558$	−6.16472e10	−0.0269190
$559$	2.27475e11	0.0985328
$560$	0	0
$561$	1.45309e12	0.619383
$562$	−2.57962e11	−0.109079
$563$	−3.77472e10	−0.0158342	−0.00791711	−	0.999969i	$-0.502520\pi$
$563$	−3.77472e10	−0.0158342	−0.00791711	+	0.999969i	$0.502520\pi$
$564$	−2.47385e10	−0.0102948
$565$	0	0
$566$	−3.85253e11	−0.157787
$567$	3.30599e11	0.134331
$568$	7.07824e11	0.285337
$569$	−3.56270e11	−0.142487	−0.0712433	−	0.997459i	$-0.522697\pi$
$569$	−3.56270e11	−0.142487	−0.0712433	+	0.997459i	$0.522697\pi$
$570$	0	0
$571$	3.87932e12	1.52719	0.763596	−	0.645694i	$-0.223432\pi$
$571$	3.87932e12	1.52719	0.763596	+	0.645694i	$0.223432\pi$
$572$	6.42751e12	2.51050
$573$	2.38056e12	0.922535
$574$	−8.99562e11	−0.345882
$575$	0	0
$576$	−7.19762e11	−0.272451
$577$	4.28876e12	1.61080	0.805399	−	0.592734i	$-0.201951\pi$
$577$	4.28876e12	1.61080	0.805399	+	0.592734i	$0.201951\pi$
$578$	−3.01924e11	−0.112518
$579$	−1.20484e12	−0.445530
$580$	0	0
$581$	−9.13232e11	−0.332498
$582$	7.87036e10	0.0284342
$583$	4.10197e11	0.147056
$584$	−2.46893e11	−0.0878316
$585$	0	0
$586$	−3.26523e11	−0.114387
$587$	−4.43245e12	−1.54089	−0.770447	−	0.637504i	$-0.779967\pi$
$587$	−4.43245e12	−1.54089	−0.770447	+	0.637504i	$0.779967\pi$
$588$	7.48430e11	0.258198
$589$	−1.68255e12	−0.576036
$590$	0	0
$591$	4.03927e11	0.136194
$592$	−4.45580e12	−1.49100
$593$	5.10104e12	1.69400	0.846998	−	0.531596i	$-0.178407\pi$
$593$	5.10104e12	1.69400	0.846998	+	0.531596i	$0.178407\pi$
$594$	1.83675e11	0.0605355
$595$	0	0
$596$	4.77838e12	1.55122
$597$	−1.17877e12	−0.379790
$598$	−8.21831e11	−0.262801
$599$	−7.04599e11	−0.223626	−0.111813	−	0.993729i	$-0.535666\pi$
$599$	−7.04599e11	−0.223626	−0.111813	+	0.993729i	$0.535666\pi$
$600$	0	0
$601$	−1.73879e12	−0.543641	−0.271821	−	0.962348i	$-0.587626\pi$
$601$	−1.73879e12	−0.543641	−0.271821	+	0.962348i	$0.587626\pi$
$602$	4.65938e10	0.0144592
$603$	7.29469e11	0.224688
$604$	2.66993e12	0.816269
$605$	0	0
$606$	2.11832e11	0.0638065
$607$	5.78292e11	0.172901	0.0864507	−	0.996256i	$-0.472447\pi$
$607$	5.78292e11	0.172901	0.0864507	+	0.996256i	$0.472447\pi$
$608$	2.16008e12	0.641068
$609$	1.98717e12	0.585407
$610$	0	0
$611$	−9.23493e10	−0.0268070
$612$	−6.75655e11	−0.194690
$613$	−3.74595e12	−1.07150	−0.535748	−	0.844378i	$-0.679970\pi$
$613$	−3.74595e12	−1.07150	−0.535748	+	0.844378i	$0.679970\pi$
$614$	1.18233e11	0.0335723
$615$	0	0
$616$	2.67557e12	0.748691
$617$	3.94875e12	1.09692	0.548461	−	0.836176i	$-0.315214\pi$
$617$	3.94875e12	1.09692	0.548461	+	0.836176i	$0.315214\pi$
$618$	4.54962e11	0.125466
$619$	3.42253e12	0.937000	0.468500	−	0.883463i	$-0.344795\pi$
$619$	3.42253e12	0.937000	0.468500	+	0.883463i	$0.344795\pi$
$620$	0	0
$621$	7.28032e11	0.196444
$622$	−1.59628e11	−0.0427615
$623$	−2.43098e12	−0.646526
$624$	−2.88914e12	−0.762847
$625$	0	0
$626$	−7.41484e11	−0.192982
$627$	5.01307e12	1.29539
$628$	2.57639e11	0.0660988
$629$	−3.88995e12	−0.990868
$630$	0	0
$631$	5.84755e12	1.46839	0.734196	−	0.678938i	$-0.237560\pi$
$631$	5.84755e12	1.46839	0.734196	+	0.678938i	$0.237560\pi$
$632$	−9.45226e11	−0.235673
$633$	−4.17479e12	−1.03352
$634$	−1.07558e12	−0.264387
$635$	0	0
$636$	−1.90733e11	−0.0462241
$637$	2.79391e12	0.672334
$638$	1.10404e12	0.263809
$639$	−1.15179e12	−0.273288
$640$	0	0
$641$	2.66671e12	0.623899	0.311950	−	0.950099i	$-0.399018\pi$
$641$	2.66671e12	0.623899	0.311950	+	0.950099i	$0.399018\pi$
$642$	−5.95685e11	−0.138391
$643$	−9.68716e10	−0.0223484	−0.0111742	−	0.999938i	$-0.503557\pi$
$643$	−9.68716e10	−0.0223484	−0.0111742	+	0.999938i	$0.503557\pi$
$644$	5.21841e12	1.19551
$645$	0	0
$646$	5.94865e11	0.134392
$647$	−4.47368e10	−0.0100368	−0.00501840	−	0.999987i	$-0.501597\pi$
$647$	−4.47368e10	−0.0100368	−0.00501840	+	0.999987i	$0.501597\pi$
$648$	−1.73564e11	−0.0386699
$649$	−5.23747e12	−1.15883
$650$	0	0
$651$	1.46127e12	0.318871
$652$	4.66855e12	1.01174
$653$	−4.95385e12	−1.06619	−0.533094	−	0.846056i	$-0.678971\pi$
$653$	−4.95385e12	−1.06619	−0.533094	+	0.846056i	$0.678971\pi$
$654$	3.02438e11	0.0646453
$655$	0	0
$656$	−6.96411e12	−1.46824
$657$	4.01752e11	0.0841228
$658$	−1.89159e10	−0.00393378
$659$	−5.85077e12	−1.20845	−0.604225	−	0.796814i	$-0.706517\pi$
$659$	−5.85077e12	−1.20845	−0.604225	+	0.796814i	$0.706517\pi$
$660$	0	0
$661$	−8.81007e11	−0.179503	−0.0897517	−	0.995964i	$-0.528607\pi$
$661$	−8.81007e11	−0.179503	−0.0897517	+	0.995964i	$0.528607\pi$
$662$	−1.71640e12	−0.347342
$663$	−2.52224e12	−0.506962
$664$	4.79447e11	0.0957159
$665$	0	0
$666$	−4.91700e11	−0.0968425
$667$	4.37608e12	0.856088
$668$	4.64872e12	0.903314
$669$	3.73630e12	0.721149
$670$	0	0
$671$	1.09513e13	2.08553
$672$	−1.87599e12	−0.354870
$673$	8.66521e12	1.62821	0.814107	−	0.580715i	$-0.197227\pi$
$673$	8.66521e12	1.62821	0.814107	+	0.580715i	$0.197227\pi$
$674$	8.10390e10	0.0151260
$675$	0	0
$676$	−5.89689e12	−1.08608
$677$	−8.98549e12	−1.64397	−0.821983	−	0.569512i	$-0.807132\pi$
$677$	−8.98549e12	−1.64397	−0.821983	+	0.569512i	$0.807132\pi$
$678$	−3.00845e10	−0.00546776
$679$	−1.86557e12	−0.336819
$680$	0	0
$681$	−3.04425e12	−0.542397
$682$	8.11852e11	0.143697
$683$	3.86477e12	0.679564	0.339782	−	0.940504i	$-0.389647\pi$
$683$	3.86477e12	0.679564	0.339782	+	0.940504i	$0.389647\pi$
$684$	−2.33097e12	−0.407178
$685$	0	0
$686$	−6.67386e11	−0.115059
$687$	5.19276e12	0.889392
$688$	3.60713e11	0.0613782
$689$	−7.12010e11	−0.120365
$690$	0	0
$691$	−7.08564e12	−1.18230	−0.591150	−	0.806561i	$-0.701326\pi$
$691$	−7.08564e12	−1.18230	−0.591150	+	0.806561i	$0.701326\pi$
$692$	6.12922e12	1.01608
$693$	−4.35377e12	−0.717077
$694$	−1.17113e11	−0.0191641
$695$	0	0
$696$	−1.04327e12	−0.168521
$697$	−6.07972e12	−0.975744
$698$	2.82053e11	0.0449760
$699$	5.64229e12	0.893939
$700$	0	0
$701$	4.69380e12	0.734165	0.367083	−	0.930188i	$-0.380357\pi$
$701$	4.69380e12	0.734165	0.367083	+	0.930188i	$0.380357\pi$
$702$	−3.18818e11	−0.0495479
$703$	−1.34201e13	−2.07232
$704$	9.47879e12	1.45437
$705$	0	0
$706$	2.62940e11	0.0398323
$707$	−5.02120e12	−0.755824
$708$	2.43531e12	0.364254
$709$	1.06645e13	1.58501	0.792503	−	0.609868i	$-0.208777\pi$
$709$	1.06645e13	1.58501	0.792503	+	0.609868i	$0.208777\pi$
$710$	0	0
$711$	1.53810e12	0.225721
$712$	1.27627e12	0.186115
$713$	3.21794e12	0.466311
$714$	−5.16630e11	−0.0743940
$715$	0	0
$716$	3.30356e11	0.0469757
$717$	−5.39318e12	−0.762095
$718$	2.32681e12	0.326739
$719$	−8.15663e12	−1.13823	−0.569116	−	0.822257i	$-0.692715\pi$
$719$	−8.15663e12	−1.13823	−0.569116	+	0.822257i	$0.692715\pi$
$720$	0	0
$721$	−1.07843e13	−1.48622
$722$	7.61500e11	0.104292
$723$	3.57743e12	0.486911
$724$	−2.61495e12	−0.353703
$725$	0	0
$726$	−1.65490e12	−0.221083
$727$	−6.64771e12	−0.882606	−0.441303	−	0.897358i	$-0.645484\pi$
$727$	−6.64771e12	−0.882606	−0.441303	+	0.897358i	$0.645484\pi$
$728$	−4.64418e12	−0.612799
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	3.14905e11	0.0407898
$732$	−5.09213e12	−0.655541
$733$	−7.07821e12	−0.905640	−0.452820	−	0.891602i	$-0.649582\pi$
$733$	−7.07821e12	−0.905640	−0.452820	+	0.891602i	$0.649582\pi$
$734$	1.66828e12	0.212147
$735$	0	0
$736$	−4.13124e12	−0.518956
$737$	−9.60663e12	−1.19941
$738$	−7.68493e11	−0.0953644
$739$	−2.61052e12	−0.321979	−0.160989	−	0.986956i	$-0.551468\pi$
$739$	−2.61052e12	−0.321979	−0.160989	+	0.986956i	$0.551468\pi$
$740$	0	0
$741$	−8.70157e12	−1.06027
$742$	−1.45841e11	−0.0176629
$743$	1.41841e13	1.70747	0.853734	−	0.520709i	$-0.174333\pi$
$743$	1.41841e13	1.70747	0.853734	+	0.520709i	$0.174333\pi$
$744$	−7.67165e11	−0.0917932
$745$	0	0
$746$	3.04167e11	0.0359573
$747$	−7.80171e11	−0.0916742
$748$	8.89793e12	1.03928
$749$	1.41199e13	1.63932
$750$	0	0
$751$	8.44355e11	0.0968603	0.0484301	−	0.998827i	$-0.484578\pi$
$751$	8.44355e11	0.0968603	0.0484301	+	0.998827i	$0.484578\pi$
$752$	−1.46441e11	−0.0166986
$753$	−6.77351e12	−0.767779
$754$	−1.91636e12	−0.215927
$755$	0	0
$756$	2.02441e12	0.225398
$757$	−9.05305e12	−1.00199	−0.500995	−	0.865450i	$-0.667032\pi$
$757$	−9.05305e12	−1.00199	−0.500995	+	0.865450i	$0.667032\pi$
$758$	−7.16719e11	−0.0788565
$759$	−9.58769e12	−1.04864
$760$	0	0
$761$	−6.97701e12	−0.754116	−0.377058	−	0.926190i	$-0.623064\pi$
$761$	−6.97701e12	−0.754116	−0.377058	+	0.926190i	$0.623064\pi$
$762$	5.63174e11	0.0605126
$763$	−7.16891e12	−0.765760
$764$	1.45772e13	1.54794
$765$	0	0
$766$	3.18007e12	0.333739
$767$	9.09108e12	0.948498
$768$	−3.90717e12	−0.405263
$769$	−1.07233e13	−1.10576	−0.552879	−	0.833261i	$-0.686471\pi$
$769$	−1.07233e13	−1.10576	−0.552879	+	0.833261i	$0.686471\pi$
$770$	0	0
$771$	−7.00872e12	−0.714322
$772$	−7.37781e12	−0.747566
$773$	1.37568e13	1.38583	0.692916	−	0.721019i	$-0.256326\pi$
$773$	1.37568e13	1.38583	0.692916	+	0.721019i	$0.256326\pi$
$774$	3.98049e10	0.00398660
$775$	0	0
$776$	9.79422e11	0.0969599
$777$	1.16551e13	1.14715
$778$	−7.18134e11	−0.0702744
$779$	−2.09747e13	−2.04069
$780$	0	0
$781$	1.51684e13	1.45884
$782$	−1.13770e12	−0.108792
$783$	1.69764e12	0.161405
$784$	4.43037e12	0.418811
$785$	0	0
$786$	8.09124e11	0.0756160
$787$	1.27539e13	1.18510	0.592551	−	0.805533i	$-0.298121\pi$
$787$	1.27539e13	1.18510	0.592551	+	0.805533i	$0.298121\pi$
$788$	2.47343e12	0.228524
$789$	−7.78843e12	−0.715490
$790$	0	0
$791$	7.13114e11	0.0647686
$792$	2.28573e12	0.206424
$793$	−1.90091e13	−1.70699
$794$	1.37492e12	0.122768
$795$	0	0
$796$	−7.21814e12	−0.637260
$797$	−1.27845e13	−1.12233	−0.561164	−	0.827704i	$-0.689646\pi$
$797$	−1.27845e13	−1.12233	−0.561164	+	0.827704i	$0.689646\pi$
$798$	−1.78234e12	−0.155589
$799$	−1.27844e11	−0.0110973
$800$	0	0
$801$	−2.07678e12	−0.178256
$802$	3.08832e12	0.263595
$803$	−5.29081e12	−0.449057
$804$	4.46687e12	0.377009
$805$	0	0
$806$	−1.40919e12	−0.117615
$807$	8.86992e11	0.0736188
$808$	2.63613e12	0.217579
$809$	6.03746e12	0.495548	0.247774	−	0.968818i	$-0.420301\pi$
$809$	6.03746e12	0.495548	0.247774	+	0.968818i	$0.420301\pi$
$810$	0	0
$811$	−2.50043e12	−0.202965	−0.101482	−	0.994837i	$-0.532359\pi$
$811$	−2.50043e12	−0.202965	−0.101482	+	0.994837i	$0.532359\pi$
$812$	1.21684e13	0.982269
$813$	−6.32033e12	−0.507379
$814$	6.47536e12	0.516957
$815$	0	0
$816$	−3.99958e12	−0.315797
$817$	1.08641e12	0.0853085
$818$	1.04253e12	0.0814138
$819$	7.55716e12	0.586923
$820$	0	0
$821$	−4.21082e12	−0.323461	−0.161731	−	0.986835i	$-0.551708\pi$
$821$	−4.21082e12	−0.323461	−0.161731	+	0.986835i	$0.551708\pi$
$822$	−2.57977e12	−0.197087
$823$	−2.08206e11	−0.0158196	−0.00790978	−	0.999969i	$-0.502518\pi$
$823$	−2.08206e11	−0.0158196	−0.00790978	+	0.999969i	$0.502518\pi$
$824$	5.66174e12	0.427836
$825$	0	0
$826$	1.86213e12	0.139187
$827$	9.26106e12	0.688472	0.344236	−	0.938883i	$-0.388138\pi$
$827$	9.26106e12	0.688472	0.344236	+	0.938883i	$0.388138\pi$
$828$	4.45807e12	0.329618
$829$	2.42762e13	1.78519	0.892597	−	0.450856i	$-0.148881\pi$
$829$	2.42762e13	1.78519	0.892597	+	0.450856i	$0.148881\pi$
$830$	0	0
$831$	5.32041e12	0.387026
$832$	−1.64531e13	−1.19040
$833$	3.86775e12	0.278327
$834$	9.25231e11	0.0662221
$835$	0	0
$836$	3.06973e13	2.17356
$837$	1.24835e12	0.0879171
$838$	−2.24066e12	−0.156956
$839$	−1.25546e13	−0.874728	−0.437364	−	0.899285i	$-0.644088\pi$
$839$	−1.25546e13	−0.874728	−0.437364	+	0.899285i	$0.644088\pi$
$840$	0	0
$841$	−4.30294e12	−0.296608
$842$	6.73284e11	0.0461630
$843$	5.22374e12	0.356252
$844$	−2.55642e13	−1.73417
$845$	0	0
$846$	−1.61598e10	−0.00108460
$847$	3.92272e13	2.61886
$848$	−1.12905e12	−0.0749778
$849$	7.80137e12	0.515331
$850$	0	0
$851$	2.56664e13	1.67758
$852$	−7.05296e12	−0.458557
$853$	1.36320e13	0.881632	0.440816	−	0.897597i	$-0.354689\pi$
$853$	1.36320e13	0.881632	0.440816	+	0.897597i	$0.354689\pi$
$854$	−3.89363e12	−0.250492
$855$	0	0
$856$	−7.41297e12	−0.471911
$857$	−1.73987e13	−1.10180	−0.550901	−	0.834570i	$-0.685716\pi$
$857$	−1.73987e13	−1.10180	−0.550901	+	0.834570i	$0.685716\pi$
$858$	4.19862e12	0.264493
$859$	−6.43355e10	−0.00403163	−0.00201582	−	0.999998i	$-0.500642\pi$
$859$	−6.43355e10	−0.00403163	−0.00201582	+	0.999998i	$0.500642\pi$
$860$	0	0
$861$	1.82161e13	1.12965
$862$	1.79353e12	0.110644
$863$	−3.32120e12	−0.203820	−0.101910	−	0.994794i	$-0.532495\pi$
$863$	−3.32120e12	−0.203820	−0.101910	+	0.994794i	$0.532495\pi$
$864$	−1.60266e12	−0.0978427
$865$	0	0
$866$	4.33940e12	0.262180
$867$	6.11396e12	0.367482
$868$	8.94800e12	0.535041
$869$	−2.02558e13	−1.20493
$870$	0	0
$871$	1.66750e13	0.981709
$872$	3.76368e12	0.220439
$873$	−1.59375e12	−0.0928657
$874$	−3.92501e12	−0.227530
$875$	0	0
$876$	2.46011e12	0.141152
$877$	−1.95832e12	−0.111786	−0.0558928	−	0.998437i	$-0.517800\pi$
$877$	−1.95832e12	−0.111786	−0.0558928	+	0.998437i	$0.517800\pi$
$878$	1.84219e12	0.104619
$879$	6.61210e12	0.373585
$880$	0	0
$881$	−3.02253e13	−1.69036	−0.845179	−	0.534483i	$-0.820506\pi$
$881$	−3.02253e13	−1.69036	−0.845179	+	0.534483i	$0.820506\pi$
$882$	4.88894e11	0.0272023
$883$	9.30097e12	0.514879	0.257439	−	0.966294i	$-0.417121\pi$
$883$	9.30097e12	0.514879	0.257439	+	0.966294i	$0.417121\pi$
$884$	−1.54448e13	−0.850643
$885$	0	0
$886$	5.28380e10	0.00288068
$887$	−9.27171e12	−0.502925	−0.251463	−	0.967867i	$-0.580912\pi$
$887$	−9.27171e12	−0.502925	−0.251463	+	0.967867i	$0.580912\pi$
$888$	−6.11893e12	−0.330231
$889$	−1.33493e13	−0.716805
$890$	0	0
$891$	−3.71941e12	−0.197708
$892$	2.28791e13	1.21003
$893$	−4.41053e11	−0.0232092
$894$	3.12136e12	0.163428
$895$	0	0
$896$	−1.52282e13	−0.789338
$897$	1.66421e13	0.858305
$898$	−2.76756e12	−0.142021
$899$	7.50365e12	0.383137
$900$	0	0
$901$	−9.85671e11	−0.0498276
$902$	1.01205e13	0.509066
$903$	−9.43524e11	−0.0472235
$904$	−3.74385e11	−0.0186449
$905$	0	0
$906$	1.74407e12	0.0859976
$907$	−1.32868e12	−0.0651908	−0.0325954	−	0.999469i	$-0.510377\pi$
$907$	−1.32868e12	−0.0651908	−0.0325954	+	0.999469i	$0.510377\pi$
$908$	−1.86413e13	−0.910102
$909$	−4.28960e12	−0.208391
$910$	0	0
$911$	2.71297e12	0.130501	0.0652503	−	0.997869i	$-0.479215\pi$
$911$	2.71297e12	0.130501	0.0652503	+	0.997869i	$0.479215\pi$
$912$	−1.37983e13	−0.660464
$913$	1.02743e13	0.489368
$914$	1.49254e12	0.0707404
$915$	0	0
$916$	3.17977e13	1.49233
$917$	−1.91792e13	−0.895714
$918$	−4.41355e11	−0.0205114
$919$	1.32139e12	0.0611100	0.0305550	−	0.999533i	$-0.490273\pi$
$919$	1.32139e12	0.0611100	0.0305550	+	0.999533i	$0.490273\pi$
$920$	0	0
$921$	−2.39422e12	−0.109647
$922$	−5.83758e12	−0.266038
$923$	−2.63289e13	−1.19406
$924$	−2.66601e13	−1.20320
$925$	0	0
$926$	−5.36853e11	−0.0239942
$927$	−9.21297e12	−0.409771
$928$	−9.63329e12	−0.426392
$929$	1.16124e12	0.0511507	0.0255754	−	0.999673i	$-0.491858\pi$
$929$	1.16124e12	0.0511507	0.0255754	+	0.999673i	$0.491858\pi$
$930$	0	0
$931$	1.33435e13	0.582098
$932$	3.45503e13	1.49996
$933$	3.23247e12	0.139659
$934$	1.45813e11	0.00626952
$935$	0	0
$936$	−3.96751e12	−0.168957
$937$	−3.40914e13	−1.44483	−0.722415	−	0.691460i	$-0.756968\pi$
$937$	−3.40914e13	−1.44483	−0.722415	+	0.691460i	$0.756968\pi$
$938$	3.41553e12	0.144061
$939$	1.50151e13	0.630277
$940$	0	0
$941$	−1.60215e12	−0.0666114	−0.0333057	−	0.999445i	$-0.510603\pi$
$941$	−1.60215e12	−0.0666114	−0.0333057	+	0.999445i	$0.510603\pi$
$942$	1.68297e11	0.00696380
$943$	4.01149e13	1.65197
$944$	1.44160e13	0.590840
$945$	0	0
$946$	−5.24204e11	−0.0212809
$947$	3.38850e13	1.36909	0.684546	−	0.728969i	$-0.260000\pi$
$947$	3.38850e13	1.36909	0.684546	+	0.728969i	$0.260000\pi$
$948$	9.41851e12	0.378743
$949$	9.18366e12	0.367551
$950$	0	0
$951$	2.17805e13	0.863485
$952$	−6.42917e12	−0.253682
$953$	−2.15757e13	−0.847320	−0.423660	−	0.905821i	$-0.639255\pi$
$953$	−2.15757e13	−0.847320	−0.423660	+	0.905821i	$0.639255\pi$
$954$	−1.24592e11	−0.00486991
$955$	0	0
$956$	−3.30249e13	−1.27874
$957$	−2.23567e13	−0.861598
$958$	3.52912e12	0.135370
$959$	6.11502e13	2.33461
$960$	0	0
$961$	−2.09218e13	−0.791306
$962$	−1.12398e13	−0.423126
$963$	1.20626e13	0.451985
$964$	2.19063e13	0.816999
$965$	0	0
$966$	3.40880e12	0.125952
$967$	−3.06249e13	−1.12630	−0.563152	−	0.826353i	$-0.690411\pi$
$967$	−3.06249e13	−1.12630	−0.563152	+	0.826353i	$0.690411\pi$
$968$	−2.05943e13	−0.753888
$969$	−1.20460e13	−0.438921
$970$	0	0
$971$	1.92365e12	0.0694447	0.0347224	−	0.999397i	$-0.488945\pi$
$971$	1.92365e12	0.0694447	0.0347224	+	0.999397i	$0.488945\pi$
$972$	1.72945e12	0.0621453
$973$	−2.19314e13	−0.784438
$974$	−2.03936e12	−0.0726070
$975$	0	0
$976$	−3.01432e13	−1.06332
$977$	1.83893e13	0.645714	0.322857	−	0.946448i	$-0.395357\pi$
$977$	1.83893e13	0.645714	0.322857	+	0.946448i	$0.395357\pi$
$978$	3.04962e12	0.106591
$979$	2.73498e13	0.951552
$980$	0	0
$981$	−6.12438e12	−0.211131
$982$	−6.08524e12	−0.208822
$983$	4.63273e13	1.58251	0.791254	−	0.611488i	$-0.209429\pi$
$983$	4.63273e13	1.58251	0.791254	+	0.611488i	$0.209429\pi$
$984$	−9.56347e12	−0.325190
$985$	0	0
$986$	−2.65291e12	−0.0893875
$987$	3.83047e11	0.0128477
$988$	−5.32837e13	−1.77905
$989$	−2.07779e12	−0.0690587
$990$	0	0
$991$	2.76252e12	0.0909857	0.0454929	−	0.998965i	$-0.485514\pi$
$991$	2.76252e12	0.0909857	0.0454929	+	0.998965i	$0.485514\pi$
$992$	−7.08383e12	−0.232255
$993$	3.47570e13	1.13441
$994$	−5.39295e12	−0.175221
$995$	0	0
$996$	−4.77734e12	−0.153822
$997$	1.74502e13	0.559337	0.279668	−	0.960097i	$-0.409776\pi$
$997$	1.74502e13	0.559337	0.279668	+	0.960097i	$0.409776\pi$
$998$	2.81365e12	0.0897807
$999$	9.95692e12	0.316286

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.10.a.c.1.1		1
3.2	odd	2		225.10.a.c.1.1		1
5.2	odd	4		75.10.b.d.49.2		2
5.3	odd	4		75.10.b.d.49.1		2
5.4	even	2		15.10.a.a.1.1	✓	1
15.2	even	4		225.10.b.e.199.1		2
15.8	even	4		225.10.b.e.199.2		2
15.14	odd	2		45.10.a.b.1.1		1
20.19	odd	2		240.10.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.a.1.1	✓	1	5.4	even	2
45.10.a.b.1.1		1	15.14	odd	2
75.10.a.c.1.1		1	1.1	even	1	trivial
75.10.b.d.49.1		2	5.3	odd	4
75.10.b.d.49.2		2	5.2	odd	4
225.10.a.c.1.1		1	3.2	odd	2
225.10.b.e.199.1		2	15.2	even	4
225.10.b.e.199.2		2	15.8	even	4
240.10.a.c.1.1		1	20.19	odd	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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