LMFDB - Embedded newform 720.6.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(1,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("720.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.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:	$115.476350265$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{409}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 102 $ x^2 - x - 102
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$10.6119$ of defining polynomial
Character	$\chi$	$=$	720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-25.0000 q^{5} -105.790 q^{7} +O(q^{10})$	$q-25.0000 q^{5} -105.790 q^{7} +447.580 q^{11} +276.210 q^{13} -1826.95 q^{17} +1371.27 q^{19} -1122.63 q^{23} +625.000 q^{25} -1621.16 q^{29} +443.690 q^{31} +2644.75 q^{35} +12585.3 q^{37} -1686.86 q^{41} +8867.16 q^{43} +2777.83 q^{47} -5615.48 q^{49} +30152.2 q^{53} -11189.5 q^{55} -33133.6 q^{59} +25965.4 q^{61} -6905.25 q^{65} +19395.2 q^{67} -52846.0 q^{71} +35710.0 q^{73} -47349.5 q^{77} -91820.6 q^{79} -20272.9 q^{83} +45673.7 q^{85} -126629. q^{89} -29220.3 q^{91} -34281.7 q^{95} -138578. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 50 q^{5} + 112 q^{7}+O(q^{10}) $ 2 * q - 50 * q^5 + 112 * q^7	$ 2 q - 50 q^{5} + 112 q^{7} + 248 q^{11} + 876 q^{13} - 2036 q^{17} - 1464 q^{19} - 3216 q^{23} + 1250 q^{25} - 1948 q^{29} - 2672 q^{31} - 2800 q^{35} + 8668 q^{37} + 7628 q^{41} + 16440 q^{43} - 19360 q^{47} + 25010 q^{49} + 14356 q^{53} - 6200 q^{55} - 904 q^{59} + 20220 q^{61} - 21900 q^{65} + 12904 q^{67} - 40976 q^{71} + 59124 q^{73} - 90816 q^{77} - 107600 q^{79} - 122088 q^{83} + 50900 q^{85} - 103764 q^{89} + 101408 q^{91} + 36600 q^{95} - 24764 q^{97}+O(q^{100}) $ 2 * q - 50 * q^5 + 112 * q^7 + 248 * q^11 + 876 * q^13 - 2036 * q^17 - 1464 * q^19 - 3216 * q^23 + 1250 * q^25 - 1948 * q^29 - 2672 * q^31 - 2800 * q^35 + 8668 * q^37 + 7628 * q^41 + 16440 * q^43 - 19360 * q^47 + 25010 * q^49 + 14356 * q^53 - 6200 * q^55 - 904 * q^59 + 20220 * q^61 - 21900 * q^65 + 12904 * q^67 - 40976 * q^71 + 59124 * q^73 - 90816 * q^77 - 107600 * q^79 - 122088 * q^83 + 50900 * q^85 - 103764 * q^89 + 101408 * q^91 + 36600 * q^95 - 24764 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−105.790	−0.816017	−0.408009	−	0.912978i	$-0.633777\pi$
$7$	−105.790	−0.816017	−0.408009	+	0.912978i	$0.633777\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	447.580	1.11529	0.557646	−	0.830079i	$-0.311705\pi$
$11$	447.580	1.11529	0.557646	+	0.830079i	$0.311705\pi$
$12$	0	0
$13$	276.210	0.453295	0.226648	−	0.973977i	$-0.427223\pi$
$13$	276.210	0.453295	0.226648	+	0.973977i	$0.427223\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1826.95	−1.53322	−0.766610	−	0.642113i	$-0.778058\pi$
$17$	−1826.95	−1.53322	−0.766610	+	0.642113i	$0.778058\pi$
$18$	0	0
$19$	1371.27	0.871443	0.435721	−	0.900082i	$-0.356493\pi$
$19$	1371.27	0.871443	0.435721	+	0.900082i	$0.356493\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1122.63	−0.442504	−0.221252	−	0.975217i	$-0.571014\pi$
$23$	−1122.63	−0.442504	−0.221252	+	0.975217i	$0.571014\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1621.16	−0.357957	−0.178979	−	0.983853i	$-0.557279\pi$
$29$	−1621.16	−0.357957	−0.178979	+	0.983853i	$0.557279\pi$
$30$	0	0
$31$	443.690	0.0829231	0.0414615	−	0.999140i	$-0.486799\pi$
$31$	443.690	0.0829231	0.0414615	+	0.999140i	$0.486799\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2644.75	0.364934
$36$	0	0
$37$	12585.3	1.51133	0.755664	−	0.654959i	$-0.227314\pi$
$37$	12585.3	1.51133	0.755664	+	0.654959i	$0.227314\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1686.86	−0.156718	−0.0783591	−	0.996925i	$-0.524968\pi$
$41$	−1686.86	−0.156718	−0.0783591	+	0.996925i	$0.524968\pi$
$42$	0	0
$43$	8867.16	0.731330	0.365665	−	0.930747i	$-0.380842\pi$
$43$	8867.16	0.731330	0.365665	+	0.930747i	$0.380842\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2777.83	0.183426	0.0917130	−	0.995785i	$-0.470766\pi$
$47$	2777.83	0.183426	0.0917130	+	0.995785i	$0.470766\pi$
$48$	0	0
$49$	−5615.48	−0.334115
$50$	0	0
$51$	0	0
$52$	0	0
$53$	30152.2	1.47445	0.737223	−	0.675649i	$-0.236137\pi$
$53$	30152.2	1.47445	0.737223	+	0.675649i	$0.236137\pi$
$54$	0	0
$55$	−11189.5	−0.498774
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−33133.6	−1.23919	−0.619596	−	0.784921i	$-0.712703\pi$
$59$	−33133.6	−1.23919	−0.619596	+	0.784921i	$0.712703\pi$
$60$	0	0
$61$	25965.4	0.893451	0.446726	−	0.894671i	$-0.352590\pi$
$61$	25965.4	0.893451	0.446726	+	0.894671i	$0.352590\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6905.25	−0.202720
$66$	0	0
$67$	19395.2	0.527846	0.263923	−	0.964544i	$-0.414984\pi$
$67$	19395.2	0.527846	0.263923	+	0.964544i	$0.414984\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−52846.0	−1.24413	−0.622066	−	0.782965i	$-0.713706\pi$
$71$	−52846.0	−1.24413	−0.622066	+	0.782965i	$0.713706\pi$
$72$	0	0
$73$	35710.0	0.784301	0.392151	−	0.919901i	$-0.371731\pi$
$73$	35710.0	0.784301	0.392151	+	0.919901i	$0.371731\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−47349.5	−0.910099
$78$	0	0
$79$	−91820.6	−1.65528	−0.827642	−	0.561256i	$-0.810318\pi$
$79$	−91820.6	−1.65528	−0.827642	+	0.561256i	$0.810318\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−20272.9	−0.323014	−0.161507	−	0.986872i	$-0.551635\pi$
$83$	−20272.9	−0.323014	−0.161507	+	0.986872i	$0.551635\pi$
$84$	0	0
$85$	45673.7	0.685677
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−126629.	−1.69456	−0.847282	−	0.531143i	$-0.821763\pi$
$89$	−126629.	−1.69456	−0.847282	+	0.531143i	$0.821763\pi$
$90$	0	0
$91$	−29220.3	−0.369897
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−34281.7	−0.389721
$96$	0	0
$97$	−138578.	−1.49543	−0.747714	−	0.664021i	$-0.768849\pi$
$97$	−138578.	−1.49543	−0.747714	+	0.664021i	$0.768849\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	24568.9	0.239653	0.119826	−	0.992795i	$-0.461766\pi$
$101$	24568.9	0.239653	0.119826	+	0.992795i	$0.461766\pi$
$102$	0	0
$103$	134516.	1.24934	0.624672	−	0.780887i	$-0.285233\pi$
$103$	134516.	1.24934	0.624672	+	0.780887i	$0.285233\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−69262.6	−0.584843	−0.292422	−	0.956289i	$-0.594461\pi$
$107$	−69262.6	−0.584843	−0.292422	+	0.956289i	$0.594461\pi$
$108$	0	0
$109$	−68878.7	−0.555289	−0.277644	−	0.960684i	$-0.589554\pi$
$109$	−68878.7	−0.555289	−0.277644	+	0.960684i	$0.589554\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14831.2	−0.109265	−0.0546325	−	0.998507i	$-0.517399\pi$
$113$	−14831.2	−0.109265	−0.0546325	+	0.998507i	$0.517399\pi$
$114$	0	0
$115$	28065.8	0.197894
$116$	0	0
$117$	0	0
$118$	0	0
$119$	193273.	1.25113
$120$	0	0
$121$	39276.8	0.243878
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	247133.	1.35963	0.679817	−	0.733382i	$-0.262059\pi$
$127$	247133.	1.35963	0.679817	+	0.733382i	$0.262059\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−48024.5	−0.244503	−0.122252	−	0.992499i	$-0.539011\pi$
$131$	−48024.5	−0.244503	−0.122252	+	0.992499i	$0.539011\pi$
$132$	0	0
$133$	−145067.	−0.711113
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−250261.	−1.13918	−0.569590	−	0.821929i	$-0.692898\pi$
$137$	−250261.	−1.13918	−0.569590	+	0.821929i	$0.692898\pi$
$138$	0	0
$139$	−19048.7	−0.0836234	−0.0418117	−	0.999126i	$-0.513313\pi$
$139$	−19048.7	−0.0836234	−0.0418117	+	0.999126i	$0.513313\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	123626.	0.505557
$144$	0	0
$145$	40529.0	0.160083
$146$	0	0
$147$	0	0
$148$	0	0
$149$	180224.	0.665039	0.332519	−	0.943096i	$-0.392101\pi$
$149$	180224.	0.665039	0.332519	+	0.943096i	$0.392101\pi$
$150$	0	0
$151$	−553375.	−1.97504	−0.987522	−	0.157479i	$-0.949663\pi$
$151$	−553375.	−1.97504	−0.987522	+	0.157479i	$0.949663\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−11092.2	−0.0370843
$156$	0	0
$157$	−544773.	−1.76387	−0.881935	−	0.471372i	$-0.843759\pi$
$157$	−544773.	−1.76387	−0.881935	+	0.471372i	$0.843759\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	118763.	0.361091
$162$	0	0
$163$	−401608.	−1.18395	−0.591975	−	0.805957i	$-0.701651\pi$
$163$	−401608.	−1.18395	−0.591975	+	0.805957i	$0.701651\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	411892.	1.14286	0.571428	−	0.820652i	$-0.306389\pi$
$167$	411892.	1.14286	0.571428	+	0.820652i	$0.306389\pi$
$168$	0	0
$169$	−295001.	−0.794524
$170$	0	0
$171$	0	0
$172$	0	0
$173$	391028.	0.993329	0.496665	−	0.867943i	$-0.334558\pi$
$173$	391028.	0.993329	0.496665	+	0.867943i	$0.334558\pi$
$174$	0	0
$175$	−66118.7	−0.163203
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−562896.	−1.31309	−0.656546	−	0.754286i	$-0.727983\pi$
$179$	−562896.	−1.31309	−0.656546	+	0.754286i	$0.727983\pi$
$180$	0	0
$181$	−20889.1	−0.0473939	−0.0236970	−	0.999719i	$-0.507544\pi$
$181$	−20889.1	−0.0473939	−0.0236970	+	0.999719i	$0.507544\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−314632.	−0.675887
$186$	0	0
$187$	−817706.	−1.70999
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−464789.	−0.921875	−0.460938	−	0.887433i	$-0.652487\pi$
$191$	−464789.	−0.921875	−0.460938	+	0.887433i	$0.652487\pi$
$192$	0	0
$193$	−306696.	−0.592673	−0.296336	−	0.955084i	$-0.595765\pi$
$193$	−306696.	−0.592673	−0.296336	+	0.955084i	$0.595765\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	119348.	0.219104	0.109552	−	0.993981i	$-0.465058\pi$
$197$	119348.	0.219104	0.109552	+	0.993981i	$0.465058\pi$
$198$	0	0
$199$	−224688.	−0.402204	−0.201102	−	0.979570i	$-0.564452\pi$
$199$	−224688.	−0.402204	−0.201102	+	0.979570i	$0.564452\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	171502.	0.292099
$204$	0	0
$205$	42171.5	0.0700865
$206$	0	0
$207$	0	0
$208$	0	0
$209$	613753.	0.971914
$210$	0	0
$211$	362491.	0.560520	0.280260	−	0.959924i	$-0.409579\pi$
$211$	362491.	0.560520	0.280260	+	0.959924i	$0.409579\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−221679.	−0.327061
$216$	0	0
$217$	−46937.9	−0.0676667
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−504622.	−0.695001
$222$	0	0
$223$	−748336.	−1.00771	−0.503854	−	0.863789i	$-0.668085\pi$
$223$	−748336.	−1.00771	−0.503854	+	0.863789i	$0.668085\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.22456e6	1.57731	0.788654	−	0.614837i	$-0.210778\pi$
$227$	1.22456e6	1.57731	0.788654	+	0.614837i	$0.210778\pi$
$228$	0	0
$229$	−1.22358e6	−1.54186	−0.770929	−	0.636921i	$-0.780208\pi$
$229$	−1.22358e6	−1.54186	−0.770929	+	0.636921i	$0.780208\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−548513.	−0.661907	−0.330954	−	0.943647i	$-0.607370\pi$
$233$	−548513.	−0.661907	−0.330954	+	0.943647i	$0.607370\pi$
$234$	0	0
$235$	−69445.7	−0.0820306
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−858038.	−0.971654	−0.485827	−	0.874055i	$-0.661482\pi$
$239$	−858038.	−0.971654	−0.485827	+	0.874055i	$0.661482\pi$
$240$	0	0
$241$	1.31732e6	1.46100	0.730499	−	0.682914i	$-0.239288\pi$
$241$	1.31732e6	1.46100	0.730499	+	0.682914i	$0.239288\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	140387.	0.149421
$246$	0	0
$247$	378758.	0.395021
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−905122.	−0.906824	−0.453412	−	0.891301i	$-0.649793\pi$
$251$	−905122.	−0.906824	−0.453412	+	0.891301i	$0.649793\pi$
$252$	0	0
$253$	−502467.	−0.493521
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−296690.	−0.280202	−0.140101	−	0.990137i	$-0.544743\pi$
$257$	−296690.	−0.280202	−0.140101	+	0.990137i	$0.544743\pi$
$258$	0	0
$259$	−1.33140e6	−1.23327
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−295333.	−0.263283	−0.131641	−	0.991297i	$-0.542025\pi$
$263$	−295333.	−0.263283	−0.131641	+	0.991297i	$0.542025\pi$
$264$	0	0
$265$	−753804.	−0.659393
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−262660.	−0.221316	−0.110658	−	0.993859i	$-0.535296\pi$
$269$	−262660.	−0.221316	−0.110658	+	0.993859i	$0.535296\pi$
$270$	0	0
$271$	689179.	0.570044	0.285022	−	0.958521i	$-0.407999\pi$
$271$	689179.	0.570044	0.285022	+	0.958521i	$0.407999\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	279737.	0.223059
$276$	0	0
$277$	−1.14961e6	−0.900225	−0.450112	−	0.892972i	$-0.648616\pi$
$277$	−1.14961e6	−0.900225	−0.450112	+	0.892972i	$0.648616\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.41477e6	−1.82436	−0.912179	−	0.409792i	$-0.865601\pi$
$281$	−2.41477e6	−1.82436	−0.912179	+	0.409792i	$0.865601\pi$
$282$	0	0
$283$	−922080.	−0.684388	−0.342194	−	0.939629i	$-0.611170\pi$
$283$	−922080.	−0.684388	−0.342194	+	0.939629i	$0.611170\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	178453.	0.127885
$288$	0	0
$289$	1.91789e6	1.35076
$290$	0	0
$291$	0	0
$292$	0	0
$293$	347703.	0.236613	0.118307	−	0.992977i	$-0.462253\pi$
$293$	347703.	0.236613	0.118307	+	0.992977i	$0.462253\pi$
$294$	0	0
$295$	828339.	0.554183
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−310082.	−0.200585
$300$	0	0
$301$	−938057.	−0.596778
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−649135.	−0.399563
$306$	0	0
$307$	−1.66131e6	−1.00602	−0.503008	−	0.864282i	$-0.667774\pi$
$307$	−1.66131e6	−1.00602	−0.503008	+	0.864282i	$0.667774\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	335376.	0.196622	0.0983108	−	0.995156i	$-0.468656\pi$
$311$	335376.	0.196622	0.0983108	+	0.995156i	$0.468656\pi$
$312$	0	0
$313$	2.99160e6	1.72601	0.863004	−	0.505196i	$-0.168580\pi$
$313$	2.99160e6	1.72601	0.863004	+	0.505196i	$0.168580\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.35871e6	0.759412	0.379706	−	0.925107i	$-0.376025\pi$
$317$	1.35871e6	0.759412	0.379706	+	0.925107i	$0.376025\pi$
$318$	0	0
$319$	−725599.	−0.399227
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.50524e6	−1.33611
$324$	0	0
$325$	172631.	0.0906590
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−293866.	−0.149679
$330$	0	0
$331$	3.46060e6	1.73613	0.868063	−	0.496453i	$-0.165365\pi$
$331$	3.46060e6	1.73613	0.868063	+	0.496453i	$0.165365\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−484880.	−0.236060
$336$	0	0
$337$	−2.15998e6	−1.03603	−0.518017	−	0.855370i	$-0.673330\pi$
$337$	−2.15998e6	−1.03603	−0.518017	+	0.855370i	$0.673330\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	198587.	0.0924835
$342$	0	0
$343$	2.37207e6	1.08866
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.17748e6	−0.524966	−0.262483	−	0.964937i	$-0.584541\pi$
$347$	−1.17748e6	−0.524966	−0.262483	+	0.964937i	$0.584541\pi$
$348$	0	0
$349$	−437128.	−0.192108	−0.0960539	−	0.995376i	$-0.530622\pi$
$349$	−437128.	−0.192108	−0.0960539	+	0.995376i	$0.530622\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−511587.	−0.218516	−0.109258	−	0.994013i	$-0.534847\pi$
$353$	−511587.	−0.218516	−0.109258	+	0.994013i	$0.534847\pi$
$354$	0	0
$355$	1.32115e6	0.556392
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−396627.	−0.162422	−0.0812112	−	0.996697i	$-0.525879\pi$
$359$	−396627.	−0.162422	−0.0812112	+	0.996697i	$0.525879\pi$
$360$	0	0
$361$	−595718.	−0.240587
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−892750.	−0.350750
$366$	0	0
$367$	−3.07015e6	−1.18986	−0.594928	−	0.803779i	$-0.702819\pi$
$367$	−3.07015e6	−1.18986	−0.594928	+	0.803779i	$0.702819\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.18980e6	−1.20317
$372$	0	0
$373$	−2.96347e6	−1.10288	−0.551440	−	0.834215i	$-0.685921\pi$
$373$	−2.96347e6	−1.10288	−0.551440	+	0.834215i	$0.685921\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−447781.	−0.162260
$378$	0	0
$379$	−48151.7	−0.0172192	−0.00860961	−	0.999963i	$-0.502741\pi$
$379$	−48151.7	−0.0172192	−0.00860961	+	0.999963i	$0.502741\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	750637.	0.261477	0.130738	−	0.991417i	$-0.458265\pi$
$383$	750637.	0.261477	0.130738	+	0.991417i	$0.458265\pi$
$384$	0	0
$385$	1.18374e6	0.407008
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.42194e6	1.14656	0.573281	−	0.819359i	$-0.305670\pi$
$389$	3.42194e6	1.14656	0.573281	+	0.819359i	$0.305670\pi$
$390$	0	0
$391$	2.05099e6	0.678456
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.29552e6	0.740266
$396$	0	0
$397$	972175.	0.309577	0.154788	−	0.987948i	$-0.450530\pi$
$397$	972175.	0.309577	0.154788	+	0.987948i	$0.450530\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.73421e6	−1.78079	−0.890394	−	0.455190i	$-0.849571\pi$
$401$	−5.73421e6	−1.78079	−0.890394	+	0.455190i	$0.849571\pi$
$402$	0	0
$403$	122552.	0.0375886
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.63292e6	1.68557
$408$	0	0
$409$	3.62492e6	1.07149	0.535747	−	0.844379i	$-0.320030\pi$
$409$	3.62492e6	1.07149	0.535747	+	0.844379i	$0.320030\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.50520e6	1.01120
$414$	0	0
$415$	506823.	0.144456
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.93470e6	1.37317	0.686587	−	0.727048i	$-0.259108\pi$
$419$	4.93470e6	1.37317	0.686587	+	0.727048i	$0.259108\pi$
$420$	0	0
$421$	−1.23702e6	−0.340150	−0.170075	−	0.985431i	$-0.554401\pi$
$421$	−1.23702e6	−0.340150	−0.170075	+	0.985431i	$0.554401\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.14184e6	−0.306644
$426$	0	0
$427$	−2.74688e6	−0.729072
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.06478e6	−0.535402	−0.267701	−	0.963502i	$-0.586264\pi$
$431$	−2.06478e6	−0.535402	−0.267701	+	0.963502i	$0.586264\pi$
$432$	0	0
$433$	−2.00096e6	−0.512882	−0.256441	−	0.966560i	$-0.582550\pi$
$433$	−2.00096e6	−0.512882	−0.256441	+	0.966560i	$0.582550\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.53943e6	−0.385617
$438$	0	0
$439$	1.52639e6	0.378010	0.189005	−	0.981976i	$-0.439474\pi$
$439$	1.52639e6	0.378010	0.189005	+	0.981976i	$0.439474\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.34148e6	1.29316	0.646580	−	0.762846i	$-0.276199\pi$
$443$	5.34148e6	1.29316	0.646580	+	0.762846i	$0.276199\pi$
$444$	0	0
$445$	3.16572e6	0.757832
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.63920e6	1.32008	0.660042	−	0.751229i	$-0.270539\pi$
$449$	5.63920e6	1.32008	0.660042	+	0.751229i	$0.270539\pi$
$450$	0	0
$451$	−755005.	−0.174787
$452$	0	0
$453$	0	0
$454$	0	0
$455$	730506.	0.165423
$456$	0	0
$457$	−1.58458e6	−0.354915	−0.177457	−	0.984128i	$-0.556787\pi$
$457$	−1.58458e6	−0.354915	−0.177457	+	0.984128i	$0.556787\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−280734.	−0.0615238	−0.0307619	−	0.999527i	$-0.509793\pi$
$461$	−280734.	−0.0615238	−0.0307619	+	0.999527i	$0.509793\pi$
$462$	0	0
$463$	1.58025e6	0.342590	0.171295	−	0.985220i	$-0.445205\pi$
$463$	1.58025e6	0.342590	0.171295	+	0.985220i	$0.445205\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−432682.	−0.0918071	−0.0459036	−	0.998946i	$-0.514617\pi$
$467$	−432682.	−0.0918071	−0.0459036	+	0.998946i	$0.514617\pi$
$468$	0	0
$469$	−2.05182e6	−0.430732
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.96876e6	0.815647
$474$	0	0
$475$	857044.	0.174289
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.30302e6	0.856908	0.428454	−	0.903564i	$-0.359058\pi$
$479$	4.30302e6	0.856908	0.428454	+	0.903564i	$0.359058\pi$
$480$	0	0
$481$	3.47618e6	0.685078
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.46445e6	0.668776
$486$	0	0
$487$	2.16178e6	0.413036	0.206518	−	0.978443i	$-0.433787\pi$
$487$	2.16178e6	0.413036	0.206518	+	0.978443i	$0.433787\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.58618e6	−0.858514	−0.429257	−	0.903182i	$-0.641225\pi$
$491$	−4.58618e6	−0.858514	−0.429257	+	0.903182i	$0.641225\pi$
$492$	0	0
$493$	2.96178e6	0.548827
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.59058e6	1.01523
$498$	0	0
$499$	−3.89509e6	−0.700271	−0.350136	−	0.936699i	$-0.613865\pi$
$499$	−3.89509e6	−0.700271	−0.350136	+	0.936699i	$0.613865\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.16926e6	−0.382290	−0.191145	−	0.981562i	$-0.561220\pi$
$503$	−2.16926e6	−0.382290	−0.191145	+	0.981562i	$0.561220\pi$
$504$	0	0
$505$	−614223.	−0.107176
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.42112e6	−0.756376	−0.378188	−	0.925729i	$-0.623453\pi$
$509$	−4.42112e6	−0.756376	−0.378188	+	0.925729i	$0.623453\pi$
$510$	0	0
$511$	−3.77776e6	−0.640004
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.36291e6	−0.558724
$516$	0	0
$517$	1.24330e6	0.204574
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.88426e6	0.465522	0.232761	−	0.972534i	$-0.425224\pi$
$521$	2.88426e6	0.465522	0.232761	+	0.972534i	$0.425224\pi$
$522$	0	0
$523$	−5.08193e6	−0.812408	−0.406204	−	0.913782i	$-0.633148\pi$
$523$	−5.08193e6	−0.812408	−0.406204	+	0.913782i	$0.633148\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−810599.	−0.127139
$528$	0	0
$529$	−5.17604e6	−0.804190
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−465928.	−0.0710396
$534$	0	0
$535$	1.73157e6	0.261550
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.51338e6	−0.372637
$540$	0	0
$541$	1.19634e7	1.75737	0.878685	−	0.477402i	$-0.158421\pi$
$541$	1.19634e7	1.75737	0.878685	+	0.477402i	$0.158421\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.72197e6	0.248333
$546$	0	0
$547$	9.41845e6	1.34589	0.672947	−	0.739690i	$-0.265028\pi$
$547$	9.41845e6	1.34589	0.672947	+	0.739690i	$0.265028\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.22305e6	−0.311939
$552$	0	0
$553$	9.71371e6	1.35074
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.13295e6	0.837589	0.418795	−	0.908081i	$-0.362453\pi$
$557$	6.13295e6	0.837589	0.418795	+	0.908081i	$0.362453\pi$
$558$	0	0
$559$	2.44920e6	0.331508
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.25000e7	1.66204	0.831019	−	0.556245i	$-0.187758\pi$
$563$	1.25000e7	1.66204	0.831019	+	0.556245i	$0.187758\pi$
$564$	0	0
$565$	370781.	0.0488648
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.54884e6	0.459521	0.229761	−	0.973247i	$-0.426206\pi$
$569$	3.54884e6	0.459521	0.229761	+	0.973247i	$0.426206\pi$
$570$	0	0
$571$	−6.09787e6	−0.782687	−0.391344	−	0.920245i	$-0.627990\pi$
$571$	−6.09787e6	−0.782687	−0.391344	+	0.920245i	$0.627990\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−701644.	−0.0885008
$576$	0	0
$577$	−408501.	−0.0510803	−0.0255401	−	0.999674i	$-0.508131\pi$
$577$	−408501.	−0.0510803	−0.0255401	+	0.999674i	$0.508131\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.14467e6	0.263585
$582$	0	0
$583$	1.34955e7	1.64444
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.07435e6	−0.607835	−0.303917	−	0.952698i	$-0.598295\pi$
$587$	−5.07435e6	−0.607835	−0.303917	+	0.952698i	$0.598295\pi$
$588$	0	0
$589$	608419.	0.0722627
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.10529e6	0.712968	0.356484	−	0.934301i	$-0.383975\pi$
$593$	6.10529e6	0.712968	0.356484	+	0.934301i	$0.383975\pi$
$594$	0	0
$595$	−4.83183e6	−0.559524
$596$	0	0
$597$	0	0
$598$	0	0
$599$	993739.	0.113163	0.0565816	−	0.998398i	$-0.481980\pi$
$599$	993739.	0.113163	0.0565816	+	0.998398i	$0.481980\pi$
$600$	0	0
$601$	−1.45037e6	−0.163792	−0.0818959	−	0.996641i	$-0.526098\pi$
$601$	−1.45037e6	−0.163792	−0.0818959	+	0.996641i	$0.526098\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−981921.	−0.109066
$606$	0	0
$607$	1.27149e6	0.140068	0.0700341	−	0.997545i	$-0.477689\pi$
$607$	1.27149e6	0.140068	0.0700341	+	0.997545i	$0.477689\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	767264.	0.0831461
$612$	0	0
$613$	3.99513e6	0.429417	0.214709	−	0.976678i	$-0.431120\pi$
$613$	3.99513e6	0.429417	0.214709	+	0.976678i	$0.431120\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.17424e6	−0.547184	−0.273592	−	0.961846i	$-0.588212\pi$
$617$	−5.17424e6	−0.547184	−0.273592	+	0.961846i	$0.588212\pi$
$618$	0	0
$619$	4.91770e6	0.515864	0.257932	−	0.966163i	$-0.416959\pi$
$619$	4.91770e6	0.515864	0.257932	+	0.966163i	$0.416959\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.33961e7	1.38279
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.29927e7	−2.31720
$630$	0	0
$631$	1.71486e7	1.71457	0.857286	−	0.514841i	$-0.172149\pi$
$631$	1.71486e7	1.71457	0.857286	+	0.514841i	$0.172149\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.17833e6	−0.608047
$636$	0	0
$637$	−1.55105e6	−0.151453
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.87181e7	−1.79936	−0.899679	−	0.436552i	$-0.856199\pi$
$641$	−1.87181e7	−1.79936	−0.899679	+	0.436552i	$0.856199\pi$
$642$	0	0
$643$	−4.37631e6	−0.417427	−0.208714	−	0.977977i	$-0.566928\pi$
$643$	−4.37631e6	−0.417427	−0.208714	+	0.977977i	$0.566928\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−787451.	−0.0739542	−0.0369771	−	0.999316i	$-0.511773\pi$
$647$	−787451.	−0.0739542	−0.0369771	+	0.999316i	$0.511773\pi$
$648$	0	0
$649$	−1.48299e7	−1.38206
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.20974e6	−0.478116	−0.239058	−	0.971005i	$-0.576839\pi$
$653$	−5.20974e6	−0.478116	−0.239058	+	0.971005i	$0.576839\pi$
$654$	0	0
$655$	1.20061e6	0.109345
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5.95464e6	−0.534124	−0.267062	−	0.963679i	$-0.586053\pi$
$659$	−5.95464e6	−0.534124	−0.267062	+	0.963679i	$0.586053\pi$
$660$	0	0
$661$	1.16361e7	1.03587	0.517934	−	0.855421i	$-0.326701\pi$
$661$	1.16361e7	1.03587	0.517934	+	0.855421i	$0.326701\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.62667e6	0.318019
$666$	0	0
$667$	1.81996e6	0.158397
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.16216e7	0.996460
$672$	0	0
$673$	−888669.	−0.0756315	−0.0378157	−	0.999285i	$-0.512040\pi$
$673$	−888669.	−0.0756315	−0.0378157	+	0.999285i	$0.512040\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.10703e7	−0.928299	−0.464149	−	0.885757i	$-0.653640\pi$
$677$	−1.10703e7	−0.928299	−0.464149	+	0.885757i	$0.653640\pi$
$678$	0	0
$679$	1.46602e7	1.22030
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.70735e6	−0.550173	−0.275087	−	0.961419i	$-0.588707\pi$
$683$	−6.70735e6	−0.550173	−0.275087	+	0.961419i	$0.588707\pi$
$684$	0	0
$685$	6.25654e6	0.509457
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.32833e6	0.668359
$690$	0	0
$691$	−1.18223e7	−0.941901	−0.470950	−	0.882160i	$-0.656089\pi$
$691$	−1.18223e7	−0.941901	−0.470950	+	0.882160i	$0.656089\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	476217.	0.0373975
$696$	0	0
$697$	3.08181e6	0.240283
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.39654e7	1.07339	0.536695	−	0.843777i	$-0.319673\pi$
$701$	1.39654e7	1.07339	0.536695	+	0.843777i	$0.319673\pi$
$702$	0	0
$703$	1.72578e7	1.31704
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.59915e6	−0.195561
$708$	0	0
$709$	9.11846e6	0.681249	0.340625	−	0.940199i	$-0.389362\pi$
$709$	9.11846e6	0.681249	0.340625	+	0.940199i	$0.389362\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−498100.	−0.0366938
$714$	0	0
$715$	−3.09065e6	−0.226092
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.48606e6	0.323625	0.161813	−	0.986822i	$-0.448266\pi$
$719$	4.48606e6	0.323625	0.161813	+	0.986822i	$0.448266\pi$
$720$	0	0
$721$	−1.42305e7	−1.01949
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.01322e6	−0.0715914
$726$	0	0
$727$	2.53427e6	0.177835	0.0889175	−	0.996039i	$-0.471659\pi$
$727$	2.53427e6	0.177835	0.0889175	+	0.996039i	$0.471659\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.61999e7	−1.12129
$732$	0	0
$733$	−6.77813e6	−0.465961	−0.232981	−	0.972481i	$-0.574848\pi$
$733$	−6.77813e6	−0.465961	−0.232981	+	0.972481i	$0.574848\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.68090e6	0.588703
$738$	0	0
$739$	7.73044e6	0.520707	0.260353	−	0.965513i	$-0.416161\pi$
$739$	7.73044e6	0.520707	0.260353	+	0.965513i	$0.416161\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.83687e6	−0.188525	−0.0942623	−	0.995547i	$-0.530049\pi$
$743$	−2.83687e6	−0.188525	−0.0942623	+	0.995547i	$0.530049\pi$
$744$	0	0
$745$	−4.50560e6	−0.297414
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.32729e6	0.477242
$750$	0	0
$751$	−3.97981e6	−0.257491	−0.128746	−	0.991678i	$-0.541095\pi$
$751$	−3.97981e6	−0.257491	−0.128746	+	0.991678i	$0.541095\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.38344e7	0.883267
$756$	0	0
$757$	1.72106e7	1.09158	0.545789	−	0.837922i	$-0.316230\pi$
$757$	1.72106e7	1.09158	0.545789	+	0.837922i	$0.316230\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.81793e6	0.614552	0.307276	−	0.951621i	$-0.400583\pi$
$761$	9.81793e6	0.614552	0.307276	+	0.951621i	$0.400583\pi$
$762$	0	0
$763$	7.28668e6	0.453125
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.15183e6	−0.561719
$768$	0	0
$769$	8.65401e6	0.527718	0.263859	−	0.964561i	$-0.415005\pi$
$769$	8.65401e6	0.527718	0.263859	+	0.964561i	$0.415005\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.44011e7	1.46879	0.734396	−	0.678722i	$-0.237466\pi$
$773$	2.44011e7	1.46879	0.734396	+	0.678722i	$0.237466\pi$
$774$	0	0
$775$	277306.	0.0165846
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.31314e6	−0.136571
$780$	0	0
$781$	−2.36528e7	−1.38757
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.36193e7	0.788826
$786$	0	0
$787$	−2.10471e7	−1.21131	−0.605655	−	0.795728i	$-0.707089\pi$
$787$	−2.10471e7	−1.21131	−0.605655	+	0.795728i	$0.707089\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.56900e6	0.0891621
$792$	0	0
$793$	7.17191e6	0.404997
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.60267e7	−1.45135	−0.725676	−	0.688036i	$-0.758473\pi$
$797$	−2.60267e7	−1.45135	−0.725676	+	0.688036i	$0.758473\pi$
$798$	0	0
$799$	−5.07495e6	−0.281232
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.59831e7	0.874726
$804$	0	0
$805$	−2.96908e6	−0.161485
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.30770e7	−1.77687	−0.888434	−	0.459004i	$-0.848206\pi$
$809$	−3.30770e7	−1.77687	−0.888434	+	0.459004i	$0.848206\pi$
$810$	0	0
$811$	−2.88531e7	−1.54042	−0.770212	−	0.637788i	$-0.779849\pi$
$811$	−2.88531e7	−1.54042	−0.770212	+	0.637788i	$0.779849\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.00402e7	0.529478
$816$	0	0
$817$	1.21593e7	0.637312
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.82331e7	−0.944069	−0.472034	−	0.881580i	$-0.656480\pi$
$821$	−1.82331e7	−0.944069	−0.472034	+	0.881580i	$0.656480\pi$
$822$	0	0
$823$	−1.08726e7	−0.559541	−0.279771	−	0.960067i	$-0.590258\pi$
$823$	−1.08726e7	−0.559541	−0.279771	+	0.960067i	$0.590258\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.22275e7	−1.63856	−0.819282	−	0.573391i	$-0.805628\pi$
$827$	−3.22275e7	−1.63856	−0.819282	+	0.573391i	$0.805628\pi$
$828$	0	0
$829$	2.82837e7	1.42939	0.714693	−	0.699439i	$-0.246567\pi$
$829$	2.82837e7	1.42939	0.714693	+	0.699439i	$0.246567\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.02592e7	0.512272
$834$	0	0
$835$	−1.02973e7	−0.511101
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.23138e7	−0.603933	−0.301967	−	0.953319i	$-0.597643\pi$
$839$	−1.23138e7	−0.603933	−0.301967	+	0.953319i	$0.597643\pi$
$840$	0	0
$841$	−1.78830e7	−0.871867
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.37503e6	0.355322
$846$	0	0
$847$	−4.15510e6	−0.199009
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.41286e7	−0.668769
$852$	0	0
$853$	3.80421e6	0.179016	0.0895080	−	0.995986i	$-0.471471\pi$
$853$	3.80421e6	0.179016	0.0895080	+	0.995986i	$0.471471\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.86121e7	1.33075	0.665376	−	0.746508i	$-0.268271\pi$
$857$	2.86121e7	1.33075	0.665376	+	0.746508i	$0.268271\pi$
$858$	0	0
$859$	−2.86987e7	−1.32702	−0.663512	−	0.748165i	$-0.730935\pi$
$859$	−2.86987e7	−1.32702	−0.663512	+	0.748165i	$0.730935\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.51054e7	1.14747	0.573734	−	0.819042i	$-0.305494\pi$
$863$	2.51054e7	1.14747	0.573734	+	0.819042i	$0.305494\pi$
$864$	0	0
$865$	−9.77571e6	−0.444230
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.10971e7	−1.84613
$870$	0	0
$871$	5.35715e6	0.239270
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.65297e6	0.0729868
$876$	0	0
$877$	−3.22931e7	−1.41779	−0.708893	−	0.705316i	$-0.750805\pi$
$877$	−3.22931e7	−1.41779	−0.708893	+	0.705316i	$0.750805\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.21912e6	−0.313361	−0.156680	−	0.987649i	$-0.550079\pi$
$881$	−7.21912e6	−0.313361	−0.156680	+	0.987649i	$0.550079\pi$
$882$	0	0
$883$	−1.54973e7	−0.668890	−0.334445	−	0.942415i	$-0.608549\pi$
$883$	−1.54973e7	−0.668890	−0.334445	+	0.942415i	$0.608549\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.01737e7	−1.71448	−0.857242	−	0.514914i	$-0.827824\pi$
$887$	−4.01737e7	−1.71448	−0.857242	+	0.514914i	$0.827824\pi$
$888$	0	0
$889$	−2.61442e7	−1.10948
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.80915e6	0.159845
$894$	0	0
$895$	1.40724e7	0.587233
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−719292.	−0.0296829
$900$	0	0
$901$	−5.50865e7	−2.26065
$902$	0	0
$903$	0	0
$904$	0	0
$905$	522227.	0.0211952
$906$	0	0
$907$	−3.33663e7	−1.34676	−0.673380	−	0.739297i	$-0.735158\pi$
$907$	−3.33663e7	−1.34676	−0.673380	+	0.739297i	$0.735158\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.88206e7	1.15055	0.575277	−	0.817958i	$-0.304894\pi$
$911$	2.88206e7	1.15055	0.575277	+	0.817958i	$0.304894\pi$
$912$	0	0
$913$	−9.07375e6	−0.360255
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.08051e6	0.199519
$918$	0	0
$919$	−9.79583e6	−0.382607	−0.191303	−	0.981531i	$-0.561271\pi$
$919$	−9.79583e6	−0.382607	−0.191303	+	0.981531i	$0.561271\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.45966e7	−0.563959
$924$	0	0
$925$	7.86581e6	0.302266
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.71065e7	1.79078	0.895389	−	0.445284i	$-0.146897\pi$
$929$	4.71065e7	1.79078	0.895389	+	0.445284i	$0.146897\pi$
$930$	0	0
$931$	−7.70034e6	−0.291163
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.04427e7	0.764730
$936$	0	0
$937$	548115.	0.0203950	0.0101975	−	0.999948i	$-0.496754\pi$
$937$	548115.	0.0203950	0.0101975	+	0.999948i	$0.496754\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.27644e7	−0.469922	−0.234961	−	0.972005i	$-0.575496\pi$
$941$	−1.27644e7	−0.469922	−0.234961	+	0.972005i	$0.575496\pi$
$942$	0	0
$943$	1.89372e6	0.0693484
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.06700e7	0.386624	0.193312	−	0.981137i	$-0.438077\pi$
$947$	1.06700e7	0.386624	0.193312	+	0.981137i	$0.438077\pi$
$948$	0	0
$949$	9.86346e6	0.355520
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6.61145e6	−0.235811	−0.117906	−	0.993025i	$-0.537618\pi$
$953$	−6.61145e6	−0.235811	−0.117906	+	0.993025i	$0.537618\pi$
$954$	0	0
$955$	1.16197e7	0.412275
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.64752e7	0.929591
$960$	0	0
$961$	−2.84323e7	−0.993124
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.66740e6	0.265051
$966$	0	0
$967$	3.19820e7	1.09987	0.549933	−	0.835209i	$-0.314653\pi$
$967$	3.19820e7	1.09987	0.549933	+	0.835209i	$0.314653\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.08607e6	−0.0369666	−0.0184833	−	0.999829i	$-0.505884\pi$
$971$	−1.08607e6	−0.0369666	−0.0184833	+	0.999829i	$0.505884\pi$
$972$	0	0
$973$	2.01516e6	0.0682382
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.40058e7	−1.13977	−0.569884	−	0.821725i	$-0.693012\pi$
$977$	−3.40058e7	−1.13977	−0.569884	+	0.821725i	$0.693012\pi$
$978$	0	0
$979$	−5.66766e7	−1.88994
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.75262e7	0.908579	0.454290	−	0.890854i	$-0.349893\pi$
$983$	2.75262e7	0.908579	0.454290	+	0.890854i	$0.349893\pi$
$984$	0	0
$985$	−2.98370e6	−0.0979861
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.95454e6	−0.323616
$990$	0	0
$991$	−6.12961e7	−1.98266	−0.991331	−	0.131389i	$-0.958056\pi$
$991$	−6.12961e7	−1.98266	−0.991331	+	0.131389i	$0.958056\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.61719e6	0.179871
$996$	0	0
$997$	−1.40079e7	−0.446307	−0.223153	−	0.974783i	$-0.571635\pi$
$997$	−1.40079e7	−0.446307	−0.223153	+	0.974783i	$0.571635\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.a.bd.1.1		2
3.2	odd	2		240.6.a.q.1.1		2
4.3	odd	2		45.6.a.e.1.2		2
12.11	even	2		15.6.a.c.1.1	✓	2
20.3	even	4		225.6.b.g.199.1		4
20.7	even	4		225.6.b.g.199.4		4
20.19	odd	2		225.6.a.m.1.1		2
24.5	odd	2		960.6.a.bf.1.1		2
24.11	even	2		960.6.a.bj.1.2		2
60.23	odd	4		75.6.b.e.49.4		4
60.47	odd	4		75.6.b.e.49.1		4
60.59	even	2		75.6.a.h.1.2		2
84.83	odd	2		735.6.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.6.a.c.1.1	✓	2	12.11	even	2
45.6.a.e.1.2		2	4.3	odd	2
75.6.a.h.1.2		2	60.59	even	2
75.6.b.e.49.1		4	60.47	odd	4
75.6.b.e.49.4		4	60.23	odd	4
225.6.a.m.1.1		2	20.19	odd	2
225.6.b.g.199.1		4	20.3	even	4
225.6.b.g.199.4		4	20.7	even	4
240.6.a.q.1.1		2	3.2	odd	2
720.6.a.bd.1.1		2	1.1	even	1	trivial
735.6.a.g.1.1		2	84.83	odd	2
960.6.a.bf.1.1		2	24.5	odd	2
960.6.a.bj.1.2		2	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)

\(f(q)\)	\(=\)	\(q-25.0000 q^{5} -105.790 q^{7} +O(q^{10})\)	\(q-25.0000 q^{5} -105.790 q^{7} +447.580 q^{11} +276.210 q^{13} -1826.95 q^{17} +1371.27 q^{19} -1122.63 q^{23} +625.000 q^{25} -1621.16 q^{29} +443.690 q^{31} +2644.75 q^{35} +12585.3 q^{37} -1686.86 q^{41} +8867.16 q^{43} +2777.83 q^{47} -5615.48 q^{49} +30152.2 q^{53} -11189.5 q^{55} -33133.6 q^{59} +25965.4 q^{61} -6905.25 q^{65} +19395.2 q^{67} -52846.0 q^{71} +35710.0 q^{73} -47349.5 q^{77} -91820.6 q^{79} -20272.9 q^{83} +45673.7 q^{85} -126629. q^{89} -29220.3 q^{91} -34281.7 q^{95} -138578. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 50 q^{5} + 112 q^{7}+O(q^{10}) \) 2 * q - 50 * q^5 + 112 * q^7	\( 2 q - 50 q^{5} + 112 q^{7} + 248 q^{11} + 876 q^{13} - 2036 q^{17} - 1464 q^{19} - 3216 q^{23} + 1250 q^{25} - 1948 q^{29} - 2672 q^{31} - 2800 q^{35} + 8668 q^{37} + 7628 q^{41} + 16440 q^{43} - 19360 q^{47} + 25010 q^{49} + 14356 q^{53} - 6200 q^{55} - 904 q^{59} + 20220 q^{61} - 21900 q^{65} + 12904 q^{67} - 40976 q^{71} + 59124 q^{73} - 90816 q^{77} - 107600 q^{79} - 122088 q^{83} + 50900 q^{85} - 103764 q^{89} + 101408 q^{91} + 36600 q^{95} - 24764 q^{97}+O(q^{100}) \) 2 * q - 50 * q^5 + 112 * q^7 + 248 * q^11 + 876 * q^13 - 2036 * q^17 - 1464 * q^19 - 3216 * q^23 + 1250 * q^25 - 1948 * q^29 - 2672 * q^31 - 2800 * q^35 + 8668 * q^37 + 7628 * q^41 + 16440 * q^43 - 19360 * q^47 + 25010 * q^49 + 14356 * q^53 - 6200 * q^55 - 904 * q^59 + 20220 * q^61 - 21900 * q^65 + 12904 * q^67 - 40976 * q^71 + 59124 * q^73 - 90816 * q^77 - 107600 * q^79 - 122088 * q^83 + 50900 * q^85 - 103764 * q^89 + 101408 * q^91 + 36600 * q^95 - 24764 * q^97

Introduction

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