LMFDB - Embedded newform 768.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1436.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 12 $ x^3 - 11*x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.28282$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -9.15486 q^{5} -27.4175 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -9.15486 q^{5} -27.4175 q^{7} +9.00000 q^{9} -20.5252 q^{11} +32.0471 q^{13} +27.4646 q^{15} -111.764 q^{17} -129.764 q^{19} +82.2524 q^{21} -9.16510 q^{23} -41.1885 q^{25} -27.0000 q^{27} +41.0606 q^{29} -187.606 q^{31} +61.5755 q^{33} +251.003 q^{35} -114.127 q^{37} -96.1414 q^{39} -282.915 q^{41} +89.3870 q^{43} -82.3937 q^{45} -54.6464 q^{47} +408.717 q^{49} +335.292 q^{51} +726.878 q^{53} +187.905 q^{55} +389.292 q^{57} +216.579 q^{59} +754.222 q^{61} -246.757 q^{63} -293.387 q^{65} -379.433 q^{67} +27.4953 q^{69} -302.080 q^{71} +504.396 q^{73} +123.566 q^{75} +562.748 q^{77} +301.780 q^{79} +81.0000 q^{81} -599.003 q^{83} +1023.18 q^{85} -123.182 q^{87} +277.528 q^{89} -878.651 q^{91} +562.818 q^{93} +1187.97 q^{95} -765.905 q^{97} -184.727 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 10 q^{5} - 14 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 + 10 * q^5 - 14 * q^7 + 27 * q^9	$ 3 q - 9 q^{3} + 10 q^{5} - 14 q^{7} + 27 q^{9} + 52 q^{13} - 30 q^{15} + 26 q^{17} - 28 q^{19} + 42 q^{21} - 164 q^{23} + 53 q^{25} - 81 q^{27} + 174 q^{29} - 318 q^{31} + 92 q^{35} + 296 q^{37} - 156 q^{39} - 118 q^{41} + 260 q^{43} + 90 q^{45} - 204 q^{47} + 327 q^{49} - 78 q^{51} + 1086 q^{53} - 512 q^{55} + 84 q^{57} - 196 q^{59} + 1536 q^{61} - 126 q^{63} - 872 q^{65} + 660 q^{67} + 492 q^{69} + 852 q^{71} - 478 q^{73} - 159 q^{75} + 2304 q^{77} - 22 q^{79} + 243 q^{81} - 1136 q^{83} + 2732 q^{85} - 522 q^{87} + 110 q^{89} + 632 q^{91} + 954 q^{93} + 2552 q^{95} - 1222 q^{97}+O(q^{100}) $ 3 * q - 9 * q^3 + 10 * q^5 - 14 * q^7 + 27 * q^9 + 52 * q^13 - 30 * q^15 + 26 * q^17 - 28 * q^19 + 42 * q^21 - 164 * q^23 + 53 * q^25 - 81 * q^27 + 174 * q^29 - 318 * q^31 + 92 * q^35 + 296 * q^37 - 156 * q^39 - 118 * q^41 + 260 * q^43 + 90 * q^45 - 204 * q^47 + 327 * q^49 - 78 * q^51 + 1086 * q^53 - 512 * q^55 + 84 * q^57 - 196 * q^59 + 1536 * q^61 - 126 * q^63 - 872 * q^65 + 660 * q^67 + 492 * q^69 + 852 * q^71 - 478 * q^73 - 159 * q^75 + 2304 * q^77 - 22 * q^79 + 243 * q^81 - 1136 * q^83 + 2732 * q^85 - 522 * q^87 + 110 * q^89 + 632 * q^91 + 954 * q^93 + 2552 * q^95 - 1222 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−9.15486	−0.818836	−0.409418	−	0.912347i	$-0.634268\pi$
$5$	−9.15486	−0.818836	−0.409418	+	0.912347i	$0.634268\pi$
$6$	0	0
$7$	−27.4175	−1.48040	−0.740202	−	0.672385i	$-0.765270\pi$
$7$	−27.4175	−1.48040	−0.740202	+	0.672385i	$0.765270\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−20.5252	−0.562598	−0.281299	−	0.959620i	$-0.590765\pi$
$11$	−20.5252	−0.562598	−0.281299	+	0.959620i	$0.590765\pi$
$12$	0	0
$13$	32.0471	0.683713	0.341857	−	0.939752i	$-0.388944\pi$
$13$	32.0471	0.683713	0.341857	+	0.939752i	$0.388944\pi$
$14$	0	0
$15$	27.4646	0.472755
$16$	0	0
$17$	−111.764	−1.59452	−0.797258	−	0.603639i	$-0.793717\pi$
$17$	−111.764	−1.59452	−0.797258	+	0.603639i	$0.793717\pi$
$18$	0	0
$19$	−129.764	−1.56684	−0.783419	−	0.621494i	$-0.786526\pi$
$19$	−129.764	−1.56684	−0.783419	+	0.621494i	$0.786526\pi$
$20$	0	0
$21$	82.2524	0.854711
$22$	0	0
$23$	−9.16510	−0.0830893	−0.0415447	−	0.999137i	$-0.513228\pi$
$23$	−9.16510	−0.0830893	−0.0415447	+	0.999137i	$0.513228\pi$
$24$	0	0
$25$	−41.1885	−0.329508
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	41.0606	0.262923	0.131461	−	0.991321i	$-0.458033\pi$
$29$	41.0606	0.262923	0.131461	+	0.991321i	$0.458033\pi$
$30$	0	0
$31$	−187.606	−1.08694	−0.543468	−	0.839430i	$-0.682889\pi$
$31$	−187.606	−1.08694	−0.543468	+	0.839430i	$0.682889\pi$
$32$	0	0
$33$	61.5755	0.324816
$34$	0	0
$35$	251.003	1.21221
$36$	0	0
$37$	−114.127	−0.507093	−0.253546	−	0.967323i	$-0.581597\pi$
$37$	−114.127	−0.507093	−0.253546	+	0.967323i	$0.581597\pi$
$38$	0	0
$39$	−96.1414	−0.394742
$40$	0	0
$41$	−282.915	−1.07766	−0.538828	−	0.842416i	$-0.681133\pi$
$41$	−282.915	−1.07766	−0.538828	+	0.842416i	$0.681133\pi$
$42$	0	0
$43$	89.3870	0.317009	0.158505	−	0.987358i	$-0.449333\pi$
$43$	89.3870	0.317009	0.158505	+	0.987358i	$0.449333\pi$
$44$	0	0
$45$	−82.3937	−0.272945
$46$	0	0
$47$	−54.6464	−0.169596	−0.0847978	−	0.996398i	$-0.527024\pi$
$47$	−54.6464	−0.169596	−0.0847978	+	0.996398i	$0.527024\pi$
$48$	0	0
$49$	408.717	1.19159
$50$	0	0
$51$	335.292	0.920594
$52$	0	0
$53$	726.878	1.88386	0.941928	−	0.335815i	$-0.109012\pi$
$53$	726.878	1.88386	0.941928	+	0.335815i	$0.109012\pi$
$54$	0	0
$55$	187.905	0.460675
$56$	0	0
$57$	389.292	0.904614
$58$	0	0
$59$	216.579	0.477900	0.238950	−	0.971032i	$-0.423197\pi$
$59$	216.579	0.477900	0.238950	+	0.971032i	$0.423197\pi$
$60$	0	0
$61$	754.222	1.58309	0.791543	−	0.611114i	$-0.209278\pi$
$61$	754.222	1.58309	0.791543	+	0.611114i	$0.209278\pi$
$62$	0	0
$63$	−246.757	−0.493468
$64$	0	0
$65$	−293.387	−0.559849
$66$	0	0
$67$	−379.433	−0.691868	−0.345934	−	0.938259i	$-0.612438\pi$
$67$	−379.433	−0.691868	−0.345934	+	0.938259i	$0.612438\pi$
$68$	0	0
$69$	27.4953	0.0479717
$70$	0	0
$71$	−302.080	−0.504933	−0.252467	−	0.967606i	$-0.581242\pi$
$71$	−302.080	−0.504933	−0.252467	+	0.967606i	$0.581242\pi$
$72$	0	0
$73$	504.396	0.808700	0.404350	−	0.914604i	$-0.367498\pi$
$73$	504.396	0.808700	0.404350	+	0.914604i	$0.367498\pi$
$74$	0	0
$75$	123.566	0.190242
$76$	0	0
$77$	562.748	0.832872
$78$	0	0
$79$	301.780	0.429784	0.214892	−	0.976638i	$-0.431060\pi$
$79$	301.780	0.429784	0.214892	+	0.976638i	$0.431060\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−599.003	−0.792158	−0.396079	−	0.918216i	$-0.629629\pi$
$83$	−599.003	−0.792158	−0.396079	+	0.918216i	$0.629629\pi$
$84$	0	0
$85$	1023.18	1.30565
$86$	0	0
$87$	−123.182	−0.151799
$88$	0	0
$89$	277.528	0.330538	0.165269	−	0.986248i	$-0.447151\pi$
$89$	277.528	0.330538	0.165269	+	0.986248i	$0.447151\pi$
$90$	0	0
$91$	−878.651	−1.01217
$92$	0	0
$93$	562.818	0.627543
$94$	0	0
$95$	1187.97	1.28298
$96$	0	0
$97$	−765.905	−0.801710	−0.400855	−	0.916141i	$-0.631287\pi$
$97$	−765.905	−0.801710	−0.400855	+	0.916141i	$0.631287\pi$
$98$	0	0
$99$	−184.727	−0.187533
$100$	0	0
$101$	−201.253	−0.198272	−0.0991360	−	0.995074i	$-0.531608\pi$
$101$	−201.253	−0.198272	−0.0991360	+	0.995074i	$0.531608\pi$
$102$	0	0
$103$	−682.440	−0.652843	−0.326421	−	0.945224i	$-0.605843\pi$
$103$	−682.440	−0.652843	−0.326421	+	0.945224i	$0.605843\pi$
$104$	0	0
$105$	−753.009	−0.699868
$106$	0	0
$107$	−457.252	−0.413123	−0.206562	−	0.978434i	$-0.566227\pi$
$107$	−457.252	−0.413123	−0.206562	+	0.978434i	$0.566227\pi$
$108$	0	0
$109$	625.812	0.549926	0.274963	−	0.961455i	$-0.411334\pi$
$109$	625.812	0.549926	0.274963	+	0.961455i	$0.411334\pi$
$110$	0	0
$111$	342.382	0.292770
$112$	0	0
$113$	981.151	0.816805	0.408402	−	0.912802i	$-0.366086\pi$
$113$	981.151	0.816805	0.408402	+	0.912802i	$0.366086\pi$
$114$	0	0
$115$	83.9052	0.0680365
$116$	0	0
$117$	288.424	0.227904
$118$	0	0
$119$	3064.29	2.36053
$120$	0	0
$121$	−909.717	−0.683484
$122$	0	0
$123$	848.745	0.622185
$124$	0	0
$125$	1521.43	1.08865
$126$	0	0
$127$	−808.055	−0.564593	−0.282296	−	0.959327i	$-0.591096\pi$
$127$	−808.055	−0.564593	−0.282296	+	0.959327i	$0.591096\pi$
$128$	0	0
$129$	−268.161	−0.183025
$130$	0	0
$131$	1110.85	0.740884	0.370442	−	0.928856i	$-0.379206\pi$
$131$	1110.85	0.740884	0.370442	+	0.928856i	$0.379206\pi$
$132$	0	0
$133$	3557.80	2.31955
$134$	0	0
$135$	247.181	0.157585
$136$	0	0
$137$	466.765	0.291084	0.145542	−	0.989352i	$-0.453507\pi$
$137$	466.765	0.291084	0.145542	+	0.989352i	$0.453507\pi$
$138$	0	0
$139$	−351.773	−0.214654	−0.107327	−	0.994224i	$-0.534229\pi$
$139$	−351.773	−0.214654	−0.107327	+	0.994224i	$0.534229\pi$
$140$	0	0
$141$	163.939	0.0979161
$142$	0	0
$143$	−657.773	−0.384656
$144$	0	0
$145$	−375.904	−0.215291
$146$	0	0
$147$	−1226.15	−0.687967
$148$	0	0
$149$	1290.49	0.709540	0.354770	−	0.934954i	$-0.384559\pi$
$149$	1290.49	0.709540	0.354770	+	0.934954i	$0.384559\pi$
$150$	0	0
$151$	−1175.51	−0.633521	−0.316761	−	0.948505i	$-0.602595\pi$
$151$	−1175.51	−0.633521	−0.316761	+	0.948505i	$0.602595\pi$
$152$	0	0
$153$	−1005.88	−0.531505
$154$	0	0
$155$	1717.51	0.890022
$156$	0	0
$157$	−1092.09	−0.555148	−0.277574	−	0.960704i	$-0.589530\pi$
$157$	−1092.09	−0.555148	−0.277574	+	0.960704i	$0.589530\pi$
$158$	0	0
$159$	−2180.63	−1.08764
$160$	0	0
$161$	251.284	0.123006
$162$	0	0
$163$	3626.97	1.74286	0.871430	−	0.490519i	$-0.163193\pi$
$163$	3626.97	1.74286	0.871430	+	0.490519i	$0.163193\pi$
$164$	0	0
$165$	−563.716	−0.265971
$166$	0	0
$167$	−45.8012	−0.0212228	−0.0106114	−	0.999944i	$-0.503378\pi$
$167$	−45.8012	−0.0212228	−0.0106114	+	0.999944i	$0.503378\pi$
$168$	0	0
$169$	−1169.98	−0.532536
$170$	0	0
$171$	−1167.88	−0.522279
$172$	0	0
$173$	2455.02	1.07891	0.539455	−	0.842014i	$-0.318630\pi$
$173$	2455.02	1.07891	0.539455	+	0.842014i	$0.318630\pi$
$174$	0	0
$175$	1129.28	0.487805
$176$	0	0
$177$	−649.736	−0.275916
$178$	0	0
$179$	−1026.28	−0.428533	−0.214267	−	0.976775i	$-0.568736\pi$
$179$	−1026.28	−0.428533	−0.214267	+	0.976775i	$0.568736\pi$
$180$	0	0
$181$	−3699.05	−1.51905	−0.759526	−	0.650477i	$-0.774569\pi$
$181$	−3699.05	−1.51905	−0.759526	+	0.650477i	$0.774569\pi$
$182$	0	0
$183$	−2262.66	−0.913995
$184$	0	0
$185$	1044.82	0.415226
$186$	0	0
$187$	2293.98	0.897071
$188$	0	0
$189$	740.271	0.284904
$190$	0	0
$191$	5108.93	1.93544	0.967721	−	0.252023i	$-0.0810960\pi$
$191$	5108.93	1.93544	0.967721	+	0.252023i	$0.0810960\pi$
$192$	0	0
$193$	−1414.13	−0.527417	−0.263709	−	0.964602i	$-0.584946\pi$
$193$	−1414.13	−0.527417	−0.263709	+	0.964602i	$0.584946\pi$
$194$	0	0
$195$	880.161	0.323229
$196$	0	0
$197$	2816.66	1.01867	0.509337	−	0.860567i	$-0.329891\pi$
$197$	2816.66	1.01867	0.509337	+	0.860567i	$0.329891\pi$
$198$	0	0
$199$	948.556	0.337896	0.168948	−	0.985625i	$-0.445963\pi$
$199$	948.556	0.337896	0.168948	+	0.985625i	$0.445963\pi$
$200$	0	0
$201$	1138.30	0.399450
$202$	0	0
$203$	−1125.78	−0.389232
$204$	0	0
$205$	2590.05	0.882424
$206$	0	0
$207$	−82.4859	−0.0276964
$208$	0	0
$209$	2663.43	0.881499
$210$	0	0
$211$	−4487.28	−1.46406	−0.732032	−	0.681271i	$-0.761428\pi$
$211$	−4487.28	−1.46406	−0.732032	+	0.681271i	$0.761428\pi$
$212$	0	0
$213$	906.239	0.291523
$214$	0	0
$215$	−818.326	−0.259578
$216$	0	0
$217$	5143.68	1.60910
$218$	0	0
$219$	−1513.19	−0.466903
$220$	0	0
$221$	−3581.72	−1.09019
$222$	0	0
$223$	−4590.98	−1.37863	−0.689315	−	0.724462i	$-0.742088\pi$
$223$	−4590.98	−1.37863	−0.689315	+	0.724462i	$0.742088\pi$
$224$	0	0
$225$	−370.697	−0.109836
$226$	0	0
$227$	−2897.47	−0.847189	−0.423594	−	0.905852i	$-0.639232\pi$
$227$	−2897.47	−0.847189	−0.423594	+	0.905852i	$0.639232\pi$
$228$	0	0
$229$	34.6293	0.00999288	0.00499644	−	0.999988i	$-0.498410\pi$
$229$	34.6293	0.00999288	0.00499644	+	0.999988i	$0.498410\pi$
$230$	0	0
$231$	−1688.24	−0.480859
$232$	0	0
$233$	1054.02	0.296355	0.148178	−	0.988961i	$-0.452659\pi$
$233$	1054.02	0.296355	0.148178	+	0.988961i	$0.452659\pi$
$234$	0	0
$235$	500.280	0.138871
$236$	0	0
$237$	−905.341	−0.248136
$238$	0	0
$239$	−654.700	−0.177192	−0.0885962	−	0.996068i	$-0.528238\pi$
$239$	−654.700	−0.177192	−0.0885962	+	0.996068i	$0.528238\pi$
$240$	0	0
$241$	3194.00	0.853707	0.426854	−	0.904321i	$-0.359622\pi$
$241$	3194.00	0.853707	0.426854	+	0.904321i	$0.359622\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−3741.74	−0.975719
$246$	0	0
$247$	−4158.57	−1.07127
$248$	0	0
$249$	1797.01	0.457353
$250$	0	0
$251$	−5042.90	−1.26815	−0.634074	−	0.773273i	$-0.718618\pi$
$251$	−5042.90	−1.26815	−0.634074	+	0.773273i	$0.718618\pi$
$252$	0	0
$253$	188.115	0.0467459
$254$	0	0
$255$	−3069.55	−0.753815
$256$	0	0
$257$	−5166.64	−1.25403	−0.627016	−	0.779007i	$-0.715724\pi$
$257$	−5166.64	−1.25403	−0.627016	+	0.779007i	$0.715724\pi$
$258$	0	0
$259$	3129.08	0.750702
$260$	0	0
$261$	369.545	0.0876409
$262$	0	0
$263$	7366.11	1.72705	0.863524	−	0.504308i	$-0.168252\pi$
$263$	7366.11	1.72705	0.863524	+	0.504308i	$0.168252\pi$
$264$	0	0
$265$	−6654.47	−1.54257
$266$	0	0
$267$	−832.584	−0.190836
$268$	0	0
$269$	−7877.80	−1.78557	−0.892784	−	0.450484i	$-0.851251\pi$
$269$	−7877.80	−1.78557	−0.892784	+	0.450484i	$0.851251\pi$
$270$	0	0
$271$	−5399.92	−1.21041	−0.605206	−	0.796069i	$-0.706909\pi$
$271$	−5399.92	−1.21041	−0.605206	+	0.796069i	$0.706909\pi$
$272$	0	0
$273$	2635.95	0.584378
$274$	0	0
$275$	845.402	0.185381
$276$	0	0
$277$	−4416.07	−0.957892	−0.478946	−	0.877844i	$-0.658981\pi$
$277$	−4416.07	−0.957892	−0.478946	+	0.877844i	$0.658981\pi$
$278$	0	0
$279$	−1688.45	−0.362312
$280$	0	0
$281$	−8068.94	−1.71300	−0.856499	−	0.516148i	$-0.827365\pi$
$281$	−8068.94	−1.71300	−0.856499	+	0.516148i	$0.827365\pi$
$282$	0	0
$283$	5241.13	1.10089	0.550447	−	0.834870i	$-0.314457\pi$
$283$	5241.13	1.10089	0.550447	+	0.834870i	$0.314457\pi$
$284$	0	0
$285$	−3563.92	−0.740730
$286$	0	0
$287$	7756.81	1.59537
$288$	0	0
$289$	7578.21	1.54248
$290$	0	0
$291$	2297.72	0.462868
$292$	0	0
$293$	−6372.75	−1.27065	−0.635324	−	0.772246i	$-0.719133\pi$
$293$	−6372.75	−1.27065	−0.635324	+	0.772246i	$0.719133\pi$
$294$	0	0
$295$	−1982.75	−0.391322
$296$	0	0
$297$	554.180	0.108272
$298$	0	0
$299$	−293.715	−0.0568093
$300$	0	0
$301$	−2450.76	−0.469301
$302$	0	0
$303$	603.760	0.114472
$304$	0	0
$305$	−6904.79	−1.29629
$306$	0	0
$307$	−3810.22	−0.708342	−0.354171	−	0.935181i	$-0.615237\pi$
$307$	−3810.22	−0.708342	−0.354171	+	0.935181i	$0.615237\pi$
$308$	0	0
$309$	2047.32	0.376919
$310$	0	0
$311$	8106.73	1.47810	0.739052	−	0.673648i	$-0.235274\pi$
$311$	8106.73	1.47810	0.739052	+	0.673648i	$0.235274\pi$
$312$	0	0
$313$	559.983	0.101125	0.0505625	−	0.998721i	$-0.483899\pi$
$313$	559.983	0.101125	0.0505625	+	0.998721i	$0.483899\pi$
$314$	0	0
$315$	2259.03	0.404069
$316$	0	0
$317$	5828.98	1.03277	0.516385	−	0.856357i	$-0.327277\pi$
$317$	5828.98	1.03277	0.516385	+	0.856357i	$0.327277\pi$
$318$	0	0
$319$	−842.776	−0.147920
$320$	0	0
$321$	1371.76	0.238517
$322$	0	0
$323$	14503.0	2.49835
$324$	0	0
$325$	−1319.97	−0.225289
$326$	0	0
$327$	−1877.44	−0.317500
$328$	0	0
$329$	1498.26	0.251070
$330$	0	0
$331$	−2847.98	−0.472928	−0.236464	−	0.971640i	$-0.575989\pi$
$331$	−2847.98	−0.472928	−0.236464	+	0.971640i	$0.575989\pi$
$332$	0	0
$333$	−1027.15	−0.169031
$334$	0	0
$335$	3473.66	0.566526
$336$	0	0
$337$	−10127.8	−1.63707	−0.818537	−	0.574454i	$-0.805214\pi$
$337$	−10127.8	−1.63707	−0.818537	+	0.574454i	$0.805214\pi$
$338$	0	0
$339$	−2943.45	−0.471582
$340$	0	0
$341$	3850.65	0.611508
$342$	0	0
$343$	−1801.78	−0.283636
$344$	0	0
$345$	−251.716	−0.0392809
$346$	0	0
$347$	10148.2	1.56999	0.784993	−	0.619505i	$-0.212667\pi$
$347$	10148.2	1.56999	0.784993	+	0.619505i	$0.212667\pi$
$348$	0	0
$349$	−9515.96	−1.45954	−0.729768	−	0.683695i	$-0.760372\pi$
$349$	−9515.96	−1.45954	−0.729768	+	0.683695i	$0.760372\pi$
$350$	0	0
$351$	−865.273	−0.131581
$352$	0	0
$353$	2813.56	0.424223	0.212111	−	0.977245i	$-0.431966\pi$
$353$	2813.56	0.424223	0.212111	+	0.977245i	$0.431966\pi$
$354$	0	0
$355$	2765.50	0.413457
$356$	0	0
$357$	−9192.86	−1.36285
$358$	0	0
$359$	2427.25	0.356839	0.178419	−	0.983955i	$-0.442902\pi$
$359$	2427.25	0.356839	0.178419	+	0.983955i	$0.442902\pi$
$360$	0	0
$361$	9979.71	1.45498
$362$	0	0
$363$	2729.15	0.394610
$364$	0	0
$365$	−4617.67	−0.662192
$366$	0	0
$367$	5021.46	0.714219	0.357109	−	0.934063i	$-0.383762\pi$
$367$	5021.46	0.714219	0.357109	+	0.934063i	$0.383762\pi$
$368$	0	0
$369$	−2546.24	−0.359219
$370$	0	0
$371$	−19929.1	−2.78887
$372$	0	0
$373$	−3182.40	−0.441765	−0.220882	−	0.975300i	$-0.570894\pi$
$373$	−3182.40	−0.441765	−0.220882	+	0.975300i	$0.570894\pi$
$374$	0	0
$375$	−4564.30	−0.628532
$376$	0	0
$377$	1315.87	0.179764
$378$	0	0
$379$	−5868.93	−0.795426	−0.397713	−	0.917510i	$-0.630196\pi$
$379$	−5868.93	−0.795426	−0.397713	+	0.917510i	$0.630196\pi$
$380$	0	0
$381$	2424.16	0.325968
$382$	0	0
$383$	7350.18	0.980618	0.490309	−	0.871549i	$-0.336884\pi$
$383$	7350.18	0.980618	0.490309	+	0.871549i	$0.336884\pi$
$384$	0	0
$385$	−5151.88	−0.681985
$386$	0	0
$387$	804.483	0.105670
$388$	0	0
$389$	−13009.1	−1.69560	−0.847800	−	0.530317i	$-0.822073\pi$
$389$	−13009.1	−1.69560	−0.847800	+	0.530317i	$0.822073\pi$
$390$	0	0
$391$	1024.33	0.132487
$392$	0	0
$393$	−3332.56	−0.427750
$394$	0	0
$395$	−2762.76	−0.351923
$396$	0	0
$397$	−4877.88	−0.616659	−0.308330	−	0.951280i	$-0.599770\pi$
$397$	−4877.88	−0.616659	−0.308330	+	0.951280i	$0.599770\pi$
$398$	0	0
$399$	−10673.4	−1.33919
$400$	0	0
$401$	5552.33	0.691446	0.345723	−	0.938337i	$-0.387634\pi$
$401$	5552.33	0.691446	0.345723	+	0.938337i	$0.387634\pi$
$402$	0	0
$403$	−6012.23	−0.743153
$404$	0	0
$405$	−741.544	−0.0909817
$406$	0	0
$407$	2342.49	0.285289
$408$	0	0
$409$	−6989.27	−0.844981	−0.422491	−	0.906367i	$-0.638844\pi$
$409$	−6989.27	−0.844981	−0.422491	+	0.906367i	$0.638844\pi$
$410$	0	0
$411$	−1400.30	−0.168057
$412$	0	0
$413$	−5938.03	−0.707485
$414$	0	0
$415$	5483.79	0.648647
$416$	0	0
$417$	1055.32	0.123931
$418$	0	0
$419$	10461.0	1.21970	0.609849	−	0.792518i	$-0.291230\pi$
$419$	10461.0	1.21970	0.609849	+	0.792518i	$0.291230\pi$
$420$	0	0
$421$	−4648.55	−0.538139	−0.269070	−	0.963121i	$-0.586716\pi$
$421$	−4648.55	−0.538139	−0.269070	+	0.963121i	$0.586716\pi$
$422$	0	0
$423$	−491.817	−0.0565319
$424$	0	0
$425$	4603.40	0.525406
$426$	0	0
$427$	−20678.8	−2.34360
$428$	0	0
$429$	1973.32	0.222081
$430$	0	0
$431$	12490.7	1.39595	0.697975	−	0.716122i	$-0.254085\pi$
$431$	12490.7	1.39595	0.697975	+	0.716122i	$0.254085\pi$
$432$	0	0
$433$	9446.37	1.04842	0.524208	−	0.851590i	$-0.324362\pi$
$433$	9446.37	1.04842	0.524208	+	0.851590i	$0.324362\pi$
$434$	0	0
$435$	1127.71	0.124298
$436$	0	0
$437$	1189.30	0.130188
$438$	0	0
$439$	−2793.60	−0.303716	−0.151858	−	0.988402i	$-0.548526\pi$
$439$	−2793.60	−0.303716	−0.151858	+	0.988402i	$0.548526\pi$
$440$	0	0
$441$	3678.45	0.397198
$442$	0	0
$443$	7601.37	0.815241	0.407621	−	0.913151i	$-0.366359\pi$
$443$	7601.37	0.815241	0.407621	+	0.913151i	$0.366359\pi$
$444$	0	0
$445$	−2540.73	−0.270657
$446$	0	0
$447$	−3871.48	−0.409653
$448$	0	0
$449$	10708.8	1.12557	0.562785	−	0.826603i	$-0.309730\pi$
$449$	10708.8	1.12557	0.562785	+	0.826603i	$0.309730\pi$
$450$	0	0
$451$	5806.89	0.606287
$452$	0	0
$453$	3526.53	0.365764
$454$	0	0
$455$	8043.92	0.828802
$456$	0	0
$457$	−233.840	−0.0239356	−0.0119678	−	0.999928i	$-0.503810\pi$
$457$	−233.840	−0.0239356	−0.0119678	+	0.999928i	$0.503810\pi$
$458$	0	0
$459$	3017.63	0.306865
$460$	0	0
$461$	−981.307	−0.0991410	−0.0495705	−	0.998771i	$-0.515785\pi$
$461$	−981.307	−0.0991410	−0.0495705	+	0.998771i	$0.515785\pi$
$462$	0	0
$463$	−14082.7	−1.41356	−0.706782	−	0.707431i	$-0.749854\pi$
$463$	−14082.7	−1.41356	−0.706782	+	0.707431i	$0.749854\pi$
$464$	0	0
$465$	−5152.52	−0.513855
$466$	0	0
$467$	−9286.49	−0.920188	−0.460094	−	0.887870i	$-0.652184\pi$
$467$	−9286.49	−0.920188	−0.460094	+	0.887870i	$0.652184\pi$
$468$	0	0
$469$	10403.1	1.02424
$470$	0	0
$471$	3276.27	0.320515
$472$	0	0
$473$	−1834.68	−0.178349
$474$	0	0
$475$	5344.79	0.516286
$476$	0	0
$477$	6541.90	0.627952
$478$	0	0
$479$	−19409.3	−1.85143	−0.925715	−	0.378222i	$-0.876536\pi$
$479$	−19409.3	−1.85143	−0.925715	+	0.378222i	$0.876536\pi$
$480$	0	0
$481$	−3657.46	−0.346706
$482$	0	0
$483$	−753.851	−0.0710174
$484$	0	0
$485$	7011.76	0.656469
$486$	0	0
$487$	−12124.8	−1.12818	−0.564091	−	0.825712i	$-0.690773\pi$
$487$	−12124.8	−1.12818	−0.564091	+	0.825712i	$0.690773\pi$
$488$	0	0
$489$	−10880.9	−1.00624
$490$	0	0
$491$	−5100.69	−0.468820	−0.234410	−	0.972138i	$-0.575316\pi$
$491$	−5100.69	−0.468820	−0.234410	+	0.972138i	$0.575316\pi$
$492$	0	0
$493$	−4589.10	−0.419235
$494$	0	0
$495$	1691.15	0.153558
$496$	0	0
$497$	8282.26	0.747505
$498$	0	0
$499$	85.2797	0.00765058	0.00382529	−	0.999993i	$-0.498782\pi$
$499$	85.2797	0.00765058	0.00382529	+	0.999993i	$0.498782\pi$
$500$	0	0
$501$	137.404	0.0122530
$502$	0	0
$503$	12287.2	1.08918	0.544592	−	0.838701i	$-0.316684\pi$
$503$	12287.2	1.08918	0.544592	+	0.838701i	$0.316684\pi$
$504$	0	0
$505$	1842.45	0.162352
$506$	0	0
$507$	3509.94	0.307460
$508$	0	0
$509$	450.441	0.0392248	0.0196124	−	0.999808i	$-0.493757\pi$
$509$	450.441	0.0392248	0.0196124	+	0.999808i	$0.493757\pi$
$510$	0	0
$511$	−13829.2	−1.19720
$512$	0	0
$513$	3503.63	0.301538
$514$	0	0
$515$	6247.64	0.534571
$516$	0	0
$517$	1121.63	0.0954141
$518$	0	0
$519$	−7365.05	−0.622909
$520$	0	0
$521$	15088.1	1.26876	0.634378	−	0.773023i	$-0.281256\pi$
$521$	15088.1	1.26876	0.634378	+	0.773023i	$0.281256\pi$
$522$	0	0
$523$	17719.4	1.48149	0.740743	−	0.671789i	$-0.234474\pi$
$523$	17719.4	1.48149	0.740743	+	0.671789i	$0.234474\pi$
$524$	0	0
$525$	−3387.85	−0.281634
$526$	0	0
$527$	20967.6	1.73314
$528$	0	0
$529$	−12083.0	−0.993096
$530$	0	0
$531$	1949.21	0.159300
$532$	0	0
$533$	−9066.62	−0.736808
$534$	0	0
$535$	4186.08	0.338280
$536$	0	0
$537$	3078.83	0.247414
$538$	0	0
$539$	−8388.98	−0.670388
$540$	0	0
$541$	12244.5	0.973074	0.486537	−	0.873660i	$-0.338260\pi$
$541$	12244.5	0.973074	0.486537	+	0.873660i	$0.338260\pi$
$542$	0	0
$543$	11097.2	0.877025
$544$	0	0
$545$	−5729.22	−0.450299
$546$	0	0
$547$	7822.46	0.611452	0.305726	−	0.952119i	$-0.401101\pi$
$547$	7822.46	0.611452	0.305726	+	0.952119i	$0.401101\pi$
$548$	0	0
$549$	6787.99	0.527695
$550$	0	0
$551$	−5328.19	−0.411957
$552$	0	0
$553$	−8274.05	−0.636254
$554$	0	0
$555$	−3134.46	−0.239731
$556$	0	0
$557$	16555.5	1.25938	0.629692	−	0.776845i	$-0.283181\pi$
$557$	16555.5	1.25938	0.629692	+	0.776845i	$0.283181\pi$
$558$	0	0
$559$	2864.60	0.216743
$560$	0	0
$561$	−6881.93	−0.517924
$562$	0	0
$563$	12580.7	0.941766	0.470883	−	0.882196i	$-0.343935\pi$
$563$	12580.7	0.941766	0.470883	+	0.882196i	$0.343935\pi$
$564$	0	0
$565$	−8982.30	−0.668829
$566$	0	0
$567$	−2220.81	−0.164489
$568$	0	0
$569$	−2657.93	−0.195828	−0.0979141	−	0.995195i	$-0.531217\pi$
$569$	−2657.93	−0.195828	−0.0979141	+	0.995195i	$0.531217\pi$
$570$	0	0
$571$	17669.0	1.29496	0.647481	−	0.762081i	$-0.275822\pi$
$571$	17669.0	1.29496	0.647481	+	0.762081i	$0.275822\pi$
$572$	0	0
$573$	−15326.8	−1.11743
$574$	0	0
$575$	377.497	0.0273786
$576$	0	0
$577$	14617.7	1.05467	0.527334	−	0.849658i	$-0.323192\pi$
$577$	14617.7	1.05467	0.527334	+	0.849658i	$0.323192\pi$
$578$	0	0
$579$	4242.40	0.304505
$580$	0	0
$581$	16423.1	1.17271
$582$	0	0
$583$	−14919.3	−1.05985
$584$	0	0
$585$	−2640.48	−0.186616
$586$	0	0
$587$	−4096.53	−0.288044	−0.144022	−	0.989574i	$-0.546004\pi$
$587$	−4096.53	−0.288044	−0.144022	+	0.989574i	$0.546004\pi$
$588$	0	0
$589$	24344.5	1.70305
$590$	0	0
$591$	−8449.99	−0.588132
$592$	0	0
$593$	−21988.3	−1.52269	−0.761343	−	0.648349i	$-0.775460\pi$
$593$	−21988.3	−1.52269	−0.761343	+	0.648349i	$0.775460\pi$
$594$	0	0
$595$	−28053.1	−1.93288
$596$	0	0
$597$	−2845.67	−0.195084
$598$	0	0
$599$	−20767.7	−1.41660	−0.708302	−	0.705909i	$-0.750539\pi$
$599$	−20767.7	−1.41660	−0.708302	+	0.705909i	$0.750539\pi$
$600$	0	0
$601$	−5382.61	−0.365326	−0.182663	−	0.983176i	$-0.558472\pi$
$601$	−5382.61	−0.365326	−0.182663	+	0.983176i	$0.558472\pi$
$602$	0	0
$603$	−3414.90	−0.230623
$604$	0	0
$605$	8328.33	0.559661
$606$	0	0
$607$	11165.4	0.746607	0.373304	−	0.927709i	$-0.378225\pi$
$607$	11165.4	0.746607	0.373304	+	0.927709i	$0.378225\pi$
$608$	0	0
$609$	3377.33	0.224723
$610$	0	0
$611$	−1751.26	−0.115955
$612$	0	0
$613$	16413.5	1.08146	0.540731	−	0.841195i	$-0.318148\pi$
$613$	16413.5	1.08146	0.540731	+	0.841195i	$0.318148\pi$
$614$	0	0
$615$	−7770.15	−0.509468
$616$	0	0
$617$	51.5882	0.00336607	0.00168303	−	0.999999i	$-0.499464\pi$
$617$	51.5882	0.00336607	0.00168303	+	0.999999i	$0.499464\pi$
$618$	0	0
$619$	−6349.55	−0.412294	−0.206147	−	0.978521i	$-0.566093\pi$
$619$	−6349.55	−0.412294	−0.206147	+	0.978521i	$0.566093\pi$
$620$	0	0
$621$	247.458	0.0159906
$622$	0	0
$623$	−7609.11	−0.489330
$624$	0	0
$625$	−8779.94	−0.561916
$626$	0	0
$627$	−7990.29	−0.508934
$628$	0	0
$629$	12755.3	0.808567
$630$	0	0
$631$	13379.1	0.844078	0.422039	−	0.906578i	$-0.361315\pi$
$631$	13379.1	0.844078	0.422039	+	0.906578i	$0.361315\pi$
$632$	0	0
$633$	13461.9	0.845277
$634$	0	0
$635$	7397.63	0.462309
$636$	0	0
$637$	13098.2	0.814709
$638$	0	0
$639$	−2718.72	−0.168311
$640$	0	0
$641$	−20406.3	−1.25741	−0.628705	−	0.777644i	$-0.716415\pi$
$641$	−20406.3	−1.25741	−0.628705	+	0.777644i	$0.716415\pi$
$642$	0	0
$643$	19415.1	1.19076	0.595378	−	0.803446i	$-0.297002\pi$
$643$	19415.1	1.19076	0.595378	+	0.803446i	$0.297002\pi$
$644$	0	0
$645$	2454.98	0.149868
$646$	0	0
$647$	8167.12	0.496264	0.248132	−	0.968726i	$-0.420183\pi$
$647$	8167.12	0.496264	0.248132	+	0.968726i	$0.420183\pi$
$648$	0	0
$649$	−4445.31	−0.268866
$650$	0	0
$651$	−15431.0	−0.929017
$652$	0	0
$653$	7444.93	0.446160	0.223080	−	0.974800i	$-0.428389\pi$
$653$	7444.93	0.446160	0.223080	+	0.974800i	$0.428389\pi$
$654$	0	0
$655$	−10169.7	−0.606662
$656$	0	0
$657$	4539.56	0.269567
$658$	0	0
$659$	−23780.4	−1.40569	−0.702846	−	0.711342i	$-0.748088\pi$
$659$	−23780.4	−1.40569	−0.702846	+	0.711342i	$0.748088\pi$
$660$	0	0
$661$	−2528.90	−0.148809	−0.0744046	−	0.997228i	$-0.523706\pi$
$661$	−2528.90	−0.148809	−0.0744046	+	0.997228i	$0.523706\pi$
$662$	0	0
$663$	10745.2	0.629423
$664$	0	0
$665$	−32571.2	−1.89933
$666$	0	0
$667$	−376.324	−0.0218461
$668$	0	0
$669$	13772.9	0.795953
$670$	0	0
$671$	−15480.5	−0.890640
$672$	0	0
$673$	16733.7	0.958447	0.479224	−	0.877693i	$-0.340918\pi$
$673$	16733.7	0.958447	0.479224	+	0.877693i	$0.340918\pi$
$674$	0	0
$675$	1112.09	0.0634139
$676$	0	0
$677$	24191.5	1.37335	0.686673	−	0.726966i	$-0.259070\pi$
$677$	24191.5	1.37335	0.686673	+	0.726966i	$0.259070\pi$
$678$	0	0
$679$	20999.2	1.18685
$680$	0	0
$681$	8692.41	0.489125
$682$	0	0
$683$	−13965.2	−0.782376	−0.391188	−	0.920311i	$-0.627936\pi$
$683$	−13965.2	−0.782376	−0.391188	+	0.920311i	$0.627936\pi$
$684$	0	0
$685$	−4273.17	−0.238350
$686$	0	0
$687$	−103.888	−0.00576939
$688$	0	0
$689$	23294.4	1.28802
$690$	0	0
$691$	−8685.63	−0.478172	−0.239086	−	0.970998i	$-0.576848\pi$
$691$	−8685.63	−0.478172	−0.239086	+	0.970998i	$0.576848\pi$
$692$	0	0
$693$	5064.73	0.277624
$694$	0	0
$695$	3220.43	0.175767
$696$	0	0
$697$	31619.7	1.71834
$698$	0	0
$699$	−3162.05	−0.171101
$700$	0	0
$701$	−25942.2	−1.39775	−0.698876	−	0.715243i	$-0.746316\pi$
$701$	−25942.2	−1.39775	−0.698876	+	0.715243i	$0.746316\pi$
$702$	0	0
$703$	14809.6	0.794532
$704$	0	0
$705$	−1500.84	−0.0801772
$706$	0	0
$707$	5517.86	0.293522
$708$	0	0
$709$	5487.75	0.290687	0.145343	−	0.989381i	$-0.453571\pi$
$709$	5487.75	0.290687	0.145343	+	0.989381i	$0.453571\pi$
$710$	0	0
$711$	2716.02	0.143261
$712$	0	0
$713$	1719.43	0.0903128
$714$	0	0
$715$	6021.82	0.314970
$716$	0	0
$717$	1964.10	0.102302
$718$	0	0
$719$	17141.2	0.889094	0.444547	−	0.895756i	$-0.353365\pi$
$719$	17141.2	0.889094	0.444547	+	0.895756i	$0.353365\pi$
$720$	0	0
$721$	18710.8	0.966470
$722$	0	0
$723$	−9581.99	−0.492888
$724$	0	0
$725$	−1691.23	−0.0866352
$726$	0	0
$727$	15946.4	0.813508	0.406754	−	0.913538i	$-0.366661\pi$
$727$	15946.4	0.813508	0.406754	+	0.913538i	$0.366661\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−9990.26	−0.505476
$732$	0	0
$733$	15914.2	0.801917	0.400958	−	0.916096i	$-0.368677\pi$
$733$	15914.2	0.801917	0.400958	+	0.916096i	$0.368677\pi$
$734$	0	0
$735$	11225.2	0.563332
$736$	0	0
$737$	7787.94	0.389243
$738$	0	0
$739$	13555.4	0.674755	0.337377	−	0.941369i	$-0.390460\pi$
$739$	13555.4	0.674755	0.337377	+	0.941369i	$0.390460\pi$
$740$	0	0
$741$	12475.7	0.618497
$742$	0	0
$743$	1772.73	0.0875303	0.0437652	−	0.999042i	$-0.486065\pi$
$743$	1772.73	0.0875303	0.0437652	+	0.999042i	$0.486065\pi$
$744$	0	0
$745$	−11814.3	−0.580997
$746$	0	0
$747$	−5391.03	−0.264053
$748$	0	0
$749$	12536.7	0.611589
$750$	0	0
$751$	−1006.65	−0.0489124	−0.0244562	−	0.999701i	$-0.507785\pi$
$751$	−1006.65	−0.0489124	−0.0244562	+	0.999701i	$0.507785\pi$
$752$	0	0
$753$	15128.7	0.732165
$754$	0	0
$755$	10761.6	0.518750
$756$	0	0
$757$	28774.0	1.38152	0.690758	−	0.723086i	$-0.257277\pi$
$757$	28774.0	1.38152	0.690758	+	0.723086i	$0.257277\pi$
$758$	0	0
$759$	−564.346	−0.0269887
$760$	0	0
$761$	−18393.5	−0.876166	−0.438083	−	0.898934i	$-0.644342\pi$
$761$	−18393.5	−0.876166	−0.438083	+	0.898934i	$0.644342\pi$
$762$	0	0
$763$	−17158.2	−0.814112
$764$	0	0
$765$	9208.66	0.435215
$766$	0	0
$767$	6940.72	0.326747
$768$	0	0
$769$	−14672.2	−0.688027	−0.344014	−	0.938965i	$-0.611787\pi$
$769$	−14672.2	−0.688027	−0.344014	+	0.938965i	$0.611787\pi$
$770$	0	0
$771$	15499.9	0.724016
$772$	0	0
$773$	−16256.4	−0.756405	−0.378202	−	0.925723i	$-0.623458\pi$
$773$	−16256.4	−0.756405	−0.378202	+	0.925723i	$0.623458\pi$
$774$	0	0
$775$	7727.21	0.358154
$776$	0	0
$777$	−9387.25	−0.433418
$778$	0	0
$779$	36712.2	1.68851
$780$	0	0
$781$	6200.24	0.284074
$782$	0	0
$783$	−1108.64	−0.0505995
$784$	0	0
$785$	9997.92	0.454575
$786$	0	0
$787$	23988.3	1.08652	0.543259	−	0.839565i	$-0.317190\pi$
$787$	23988.3	1.08652	0.543259	+	0.839565i	$0.317190\pi$
$788$	0	0
$789$	−22098.3	−0.997112
$790$	0	0
$791$	−26900.7	−1.20920
$792$	0	0
$793$	24170.6	1.08238
$794$	0	0
$795$	19963.4	0.890602
$796$	0	0
$797$	32966.9	1.46518	0.732589	−	0.680672i	$-0.238312\pi$
$797$	32966.9	1.46518	0.732589	+	0.680672i	$0.238312\pi$
$798$	0	0
$799$	6107.50	0.270423
$800$	0	0
$801$	2497.75	0.110179
$802$	0	0
$803$	−10352.8	−0.454973
$804$	0	0
$805$	−2300.47	−0.100721
$806$	0	0
$807$	23633.4	1.03090
$808$	0	0
$809$	41700.3	1.81224	0.906122	−	0.423017i	$-0.139029\pi$
$809$	41700.3	1.81224	0.906122	+	0.423017i	$0.139029\pi$
$810$	0	0
$811$	5981.80	0.259000	0.129500	−	0.991579i	$-0.458663\pi$
$811$	5981.80	0.259000	0.129500	+	0.991579i	$0.458663\pi$
$812$	0	0
$813$	16199.7	0.698832
$814$	0	0
$815$	−33204.4	−1.42712
$816$	0	0
$817$	−11599.2	−0.496702
$818$	0	0
$819$	−7907.86	−0.337391
$820$	0	0
$821$	9846.06	0.418550	0.209275	−	0.977857i	$-0.432890\pi$
$821$	9846.06	0.418550	0.209275	+	0.977857i	$0.432890\pi$
$822$	0	0
$823$	−47001.9	−1.99074	−0.995372	−	0.0960935i	$-0.969365\pi$
$823$	−47001.9	−1.99074	−0.995372	+	0.0960935i	$0.969365\pi$
$824$	0	0
$825$	−2536.21	−0.107030
$826$	0	0
$827$	21727.4	0.913587	0.456794	−	0.889573i	$-0.348998\pi$
$827$	21727.4	0.913587	0.456794	+	0.889573i	$0.348998\pi$
$828$	0	0
$829$	−22772.3	−0.954058	−0.477029	−	0.878888i	$-0.658286\pi$
$829$	−22772.3	−0.954058	−0.477029	+	0.878888i	$0.658286\pi$
$830$	0	0
$831$	13248.2	0.553039
$832$	0	0
$833$	−45679.8	−1.90002
$834$	0	0
$835$	419.304	0.0173780
$836$	0	0
$837$	5065.36	0.209181
$838$	0	0
$839$	−11010.4	−0.453064	−0.226532	−	0.974004i	$-0.572739\pi$
$839$	−11010.4	−0.453064	−0.226532	+	0.974004i	$0.572739\pi$
$840$	0	0
$841$	−22703.0	−0.930872
$842$	0	0
$843$	24206.8	0.989000
$844$	0	0
$845$	10711.0	0.436059
$846$	0	0
$847$	24942.1	1.01183
$848$	0	0
$849$	−15723.4	−0.635602
$850$	0	0
$851$	1045.99	0.0421340
$852$	0	0
$853$	38177.4	1.53244	0.766219	−	0.642579i	$-0.222136\pi$
$853$	38177.4	1.53244	0.766219	+	0.642579i	$0.222136\pi$
$854$	0	0
$855$	10691.7	0.427661
$856$	0	0
$857$	−8848.01	−0.352675	−0.176337	−	0.984330i	$-0.556425\pi$
$857$	−8848.01	−0.352675	−0.176337	+	0.984330i	$0.556425\pi$
$858$	0	0
$859$	4347.66	0.172690	0.0863448	−	0.996265i	$-0.472481\pi$
$859$	4347.66	0.172690	0.0863448	+	0.996265i	$0.472481\pi$
$860$	0	0
$861$	−23270.4	−0.921085
$862$	0	0
$863$	−33669.9	−1.32808	−0.664042	−	0.747695i	$-0.731161\pi$
$863$	−33669.9	−1.32808	−0.664042	+	0.747695i	$0.731161\pi$
$864$	0	0
$865$	−22475.3	−0.883450
$866$	0	0
$867$	−22734.6	−0.890552
$868$	0	0
$869$	−6194.10	−0.241796
$870$	0	0
$871$	−12159.7	−0.473039
$872$	0	0
$873$	−6893.15	−0.267237
$874$	0	0
$875$	−41713.8	−1.61164
$876$	0	0
$877$	−50102.0	−1.92910	−0.964552	−	0.263892i	$-0.914994\pi$
$877$	−50102.0	−1.92910	−0.964552	+	0.263892i	$0.914994\pi$
$878$	0	0
$879$	19118.2	0.733609
$880$	0	0
$881$	18716.9	0.715766	0.357883	−	0.933766i	$-0.383499\pi$
$881$	18716.9	0.715766	0.357883	+	0.933766i	$0.383499\pi$
$882$	0	0
$883$	7514.19	0.286379	0.143189	−	0.989695i	$-0.454264\pi$
$883$	7514.19	0.286379	0.143189	+	0.989695i	$0.454264\pi$
$884$	0	0
$885$	5948.24	0.225930
$886$	0	0
$887$	15544.6	0.588429	0.294215	−	0.955739i	$-0.404942\pi$
$887$	15544.6	0.588429	0.294215	+	0.955739i	$0.404942\pi$
$888$	0	0
$889$	22154.8	0.835825
$890$	0	0
$891$	−1662.54	−0.0625109
$892$	0	0
$893$	7091.14	0.265729
$894$	0	0
$895$	9395.42	0.350898
$896$	0	0
$897$	881.145	0.0327989
$898$	0	0
$899$	−7703.21	−0.285780
$900$	0	0
$901$	−81238.8	−3.00384
$902$	0	0
$903$	7352.29	0.270951
$904$	0	0
$905$	33864.3	1.24385
$906$	0	0
$907$	−8713.10	−0.318979	−0.159489	−	0.987200i	$-0.550985\pi$
$907$	−8713.10	−0.318979	−0.159489	+	0.987200i	$0.550985\pi$
$908$	0	0
$909$	−1811.28	−0.0660906
$910$	0	0
$911$	1975.97	0.0718627	0.0359313	−	0.999354i	$-0.488560\pi$
$911$	1975.97	0.0718627	0.0359313	+	0.999354i	$0.488560\pi$
$912$	0	0
$913$	12294.6	0.445666
$914$	0	0
$915$	20714.4	0.748411
$916$	0	0
$917$	−30456.8	−1.09681
$918$	0	0
$919$	18430.5	0.661552	0.330776	−	0.943709i	$-0.392689\pi$
$919$	18430.5	0.661552	0.330776	+	0.943709i	$0.392689\pi$
$920$	0	0
$921$	11430.7	0.408961
$922$	0	0
$923$	−9680.79	−0.345230
$924$	0	0
$925$	4700.74	0.167091
$926$	0	0
$927$	−6141.96	−0.217614
$928$	0	0
$929$	12506.8	0.441697	0.220848	−	0.975308i	$-0.429117\pi$
$929$	12506.8	0.441697	0.220848	+	0.975308i	$0.429117\pi$
$930$	0	0
$931$	−53036.7	−1.86703
$932$	0	0
$933$	−24320.2	−0.853384
$934$	0	0
$935$	−21001.0	−0.734554
$936$	0	0
$937$	−39267.5	−1.36906	−0.684532	−	0.728982i	$-0.739994\pi$
$937$	−39267.5	−1.36906	−0.684532	+	0.728982i	$0.739994\pi$
$938$	0	0
$939$	−1679.95	−0.0583846
$940$	0	0
$941$	−23727.2	−0.821981	−0.410991	−	0.911640i	$-0.634817\pi$
$941$	−23727.2	−0.821981	−0.410991	+	0.911640i	$0.634817\pi$
$942$	0	0
$943$	2592.94	0.0895418
$944$	0	0
$945$	−6777.08	−0.233289
$946$	0	0
$947$	23399.8	0.802948	0.401474	−	0.915870i	$-0.368498\pi$
$947$	23399.8	0.802948	0.401474	+	0.915870i	$0.368498\pi$
$948$	0	0
$949$	16164.4	0.552919
$950$	0	0
$951$	−17486.9	−0.596270
$952$	0	0
$953$	−41497.1	−1.41052	−0.705258	−	0.708950i	$-0.749169\pi$
$953$	−41497.1	−1.41052	−0.705258	+	0.708950i	$0.749169\pi$
$954$	0	0
$955$	−46771.6	−1.58481
$956$	0	0
$957$	2528.33	0.0854015
$958$	0	0
$959$	−12797.5	−0.430921
$960$	0	0
$961$	5405.00	0.181431
$962$	0	0
$963$	−4115.27	−0.137708
$964$	0	0
$965$	12946.2	0.431868
$966$	0	0
$967$	49123.6	1.63362	0.816808	−	0.576909i	$-0.195741\pi$
$967$	49123.6	1.63362	0.816808	+	0.576909i	$0.195741\pi$
$968$	0	0
$969$	−43508.9	−1.44242
$970$	0	0
$971$	20346.8	0.672462	0.336231	−	0.941780i	$-0.390848\pi$
$971$	20346.8	0.672462	0.336231	+	0.941780i	$0.390848\pi$
$972$	0	0
$973$	9644.71	0.317775
$974$	0	0
$975$	3959.92	0.130071
$976$	0	0
$977$	40602.1	1.32955	0.664777	−	0.747042i	$-0.268526\pi$
$977$	40602.1	1.32955	0.664777	+	0.747042i	$0.268526\pi$
$978$	0	0
$979$	−5696.32	−0.185960
$980$	0	0
$981$	5632.31	0.183309
$982$	0	0
$983$	−50425.9	−1.63615	−0.818075	−	0.575112i	$-0.804959\pi$
$983$	−50425.9	−1.63615	−0.818075	+	0.575112i	$0.804959\pi$
$984$	0	0
$985$	−25786.2	−0.834127
$986$	0	0
$987$	−4494.79	−0.144955
$988$	0	0
$989$	−819.241	−0.0263401
$990$	0	0
$991$	−8511.62	−0.272836	−0.136418	−	0.990651i	$-0.543559\pi$
$991$	−8511.62	−0.272836	−0.136418	+	0.990651i	$0.543559\pi$
$992$	0	0
$993$	8543.94	0.273045
$994$	0	0
$995$	−8683.90	−0.276681
$996$	0	0
$997$	25302.1	0.803738	0.401869	−	0.915697i	$-0.368361\pi$
$997$	25302.1	0.803738	0.401869	+	0.915697i	$0.368361\pi$
$998$	0	0
$999$	3081.44	0.0975900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.r.1.1		3
3.2	odd	2		2304.4.a.bt.1.3		3
4.3	odd	2		768.4.a.t.1.1		3
8.3	odd	2		768.4.a.q.1.3		3
8.5	even	2		768.4.a.s.1.3		3
12.11	even	2		2304.4.a.bu.1.3		3
16.3	odd	4		96.4.d.a.49.6		6
16.5	even	4		24.4.d.a.13.4	yes	6
16.11	odd	4		96.4.d.a.49.1		6
16.13	even	4		24.4.d.a.13.3	✓	6
24.5	odd	2		2304.4.a.bv.1.1		3
24.11	even	2		2304.4.a.bw.1.1		3
48.5	odd	4		72.4.d.d.37.3		6
48.11	even	4		288.4.d.d.145.5		6
48.29	odd	4		72.4.d.d.37.4		6
48.35	even	4		288.4.d.d.145.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.4.d.a.13.3	✓	6	16.13	even	4
24.4.d.a.13.4	yes	6	16.5	even	4
72.4.d.d.37.3		6	48.5	odd	4
72.4.d.d.37.4		6	48.29	odd	4
96.4.d.a.49.1		6	16.11	odd	4
96.4.d.a.49.6		6	16.3	odd	4
288.4.d.d.145.2		6	48.35	even	4
288.4.d.d.145.5		6	48.11	even	4
768.4.a.q.1.3		3	8.3	odd	2
768.4.a.r.1.1		3	1.1	even	1	trivial
768.4.a.s.1.3		3	8.5	even	2
768.4.a.t.1.1		3	4.3	odd	2
2304.4.a.bt.1.3		3	3.2	odd	2
2304.4.a.bu.1.3		3	12.11	even	2
2304.4.a.bv.1.1		3	24.5	odd	2
2304.4.a.bw.1.1		3	24.11	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -9.15486 q^{5} -27.4175 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -9.15486 q^{5} -27.4175 q^{7} +9.00000 q^{9} -20.5252 q^{11} +32.0471 q^{13} +27.4646 q^{15} -111.764 q^{17} -129.764 q^{19} +82.2524 q^{21} -9.16510 q^{23} -41.1885 q^{25} -27.0000 q^{27} +41.0606 q^{29} -187.606 q^{31} +61.5755 q^{33} +251.003 q^{35} -114.127 q^{37} -96.1414 q^{39} -282.915 q^{41} +89.3870 q^{43} -82.3937 q^{45} -54.6464 q^{47} +408.717 q^{49} +335.292 q^{51} +726.878 q^{53} +187.905 q^{55} +389.292 q^{57} +216.579 q^{59} +754.222 q^{61} -246.757 q^{63} -293.387 q^{65} -379.433 q^{67} +27.4953 q^{69} -302.080 q^{71} +504.396 q^{73} +123.566 q^{75} +562.748 q^{77} +301.780 q^{79} +81.0000 q^{81} -599.003 q^{83} +1023.18 q^{85} -123.182 q^{87} +277.528 q^{89} -878.651 q^{91} +562.818 q^{93} +1187.97 q^{95} -765.905 q^{97} -184.727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 10 q^{5} - 14 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 + 10 * q^5 - 14 * q^7 + 27 * q^9	\( 3 q - 9 q^{3} + 10 q^{5} - 14 q^{7} + 27 q^{9} + 52 q^{13} - 30 q^{15} + 26 q^{17} - 28 q^{19} + 42 q^{21} - 164 q^{23} + 53 q^{25} - 81 q^{27} + 174 q^{29} - 318 q^{31} + 92 q^{35} + 296 q^{37} - 156 q^{39} - 118 q^{41} + 260 q^{43} + 90 q^{45} - 204 q^{47} + 327 q^{49} - 78 q^{51} + 1086 q^{53} - 512 q^{55} + 84 q^{57} - 196 q^{59} + 1536 q^{61} - 126 q^{63} - 872 q^{65} + 660 q^{67} + 492 q^{69} + 852 q^{71} - 478 q^{73} - 159 q^{75} + 2304 q^{77} - 22 q^{79} + 243 q^{81} - 1136 q^{83} + 2732 q^{85} - 522 q^{87} + 110 q^{89} + 632 q^{91} + 954 q^{93} + 2552 q^{95} - 1222 q^{97}+O(q^{100}) \) 3 * q - 9 * q^3 + 10 * q^5 - 14 * q^7 + 27 * q^9 + 52 * q^13 - 30 * q^15 + 26 * q^17 - 28 * q^19 + 42 * q^21 - 164 * q^23 + 53 * q^25 - 81 * q^27 + 174 * q^29 - 318 * q^31 + 92 * q^35 + 296 * q^37 - 156 * q^39 - 118 * q^41 + 260 * q^43 + 90 * q^45 - 204 * q^47 + 327 * q^49 - 78 * q^51 + 1086 * q^53 - 512 * q^55 + 84 * q^57 - 196 * q^59 + 1536 * q^61 - 126 * q^63 - 872 * q^65 + 660 * q^67 + 492 * q^69 + 852 * q^71 - 478 * q^73 - 159 * q^75 + 2304 * q^77 - 22 * q^79 + 243 * q^81 - 1136 * q^83 + 2732 * q^85 - 522 * q^87 + 110 * q^89 + 632 * q^91 + 954 * q^93 + 2552 * q^95 - 1222 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more