LMFDB - Embedded newform 448.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("448.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.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:	$71.8519512762$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{3} +96.0000 q^{5} +49.0000 q^{7} -239.000 q^{9} +O(q^{10})$

$q+2.00000 q^{3} +96.0000 q^{5} +49.0000 q^{7} -239.000 q^{9} +720.000 q^{11} -572.000 q^{13} +192.000 q^{15} +1254.00 q^{17} +94.0000 q^{19} +98.0000 q^{21} +96.0000 q^{23} +6091.00 q^{25} -964.000 q^{27} +4374.00 q^{29} -6244.00 q^{31} +1440.00 q^{33} +4704.00 q^{35} +10798.0 q^{37} -1144.00 q^{39} +12006.0 q^{41} +9160.00 q^{43} -22944.0 q^{45} -25836.0 q^{47} +2401.00 q^{49} +2508.00 q^{51} -1014.00 q^{53} +69120.0 q^{55} +188.000 q^{57} -1242.00 q^{59} -7592.00 q^{61} -11711.0 q^{63} -54912.0 q^{65} -41132.0 q^{67} +192.000 q^{69} -37632.0 q^{71} -13438.0 q^{73} +12182.0 q^{75} +35280.0 q^{77} +6248.00 q^{79} +56149.0 q^{81} +25254.0 q^{83} +120384. q^{85} +8748.00 q^{87} -45126.0 q^{89} -28028.0 q^{91} -12488.0 q^{93} +9024.00 q^{95} +107222. q^{97} -172080. q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	2.00000	0.128300	0.0641500	−	0.997940i	$-0.479566\pi$
$3$	2.00000	0.128300	0.0641500	+	0.997940i	$0.479566\pi$
$4$	0	0
$5$	96.0000	1.71730	0.858650	−	0.512562i	$-0.171304\pi$
$5$	96.0000	1.71730	0.858650	+	0.512562i	$0.171304\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−239.000	−0.983539
$10$	0	0
$11$	720.000	1.79412	0.897059	−	0.441912i	$-0.145700\pi$
$11$	720.000	1.79412	0.897059	+	0.441912i	$0.145700\pi$
$12$	0	0
$13$	−572.000	−0.938723	−0.469362	−	0.883006i	$-0.655516\pi$
$13$	−572.000	−0.938723	−0.469362	+	0.883006i	$0.655516\pi$
$14$	0	0
$15$	192.000	0.220330
$16$	0	0
$17$	1254.00	1.05239	0.526193	−	0.850365i	$-0.323619\pi$
$17$	1254.00	1.05239	0.526193	+	0.850365i	$0.323619\pi$
$18$	0	0
$19$	94.0000	0.0597371	0.0298685	−	0.999554i	$-0.490491\pi$
$19$	94.0000	0.0597371	0.0298685	+	0.999554i	$0.490491\pi$
$20$	0	0
$21$	98.0000	0.0484929
$22$	0	0
$23$	96.0000	0.0378400	0.0189200	−	0.999821i	$-0.493977\pi$
$23$	96.0000	0.0378400	0.0189200	+	0.999821i	$0.493977\pi$
$24$	0	0
$25$	6091.00	1.94912
$26$	0	0
$27$	−964.000	−0.254488
$28$	0	0
$29$	4374.00	0.965792	0.482896	−	0.875678i	$-0.339585\pi$
$29$	4374.00	0.965792	0.482896	+	0.875678i	$0.339585\pi$
$30$	0	0
$31$	−6244.00	−1.16697	−0.583484	−	0.812125i	$-0.698311\pi$
$31$	−6244.00	−1.16697	−0.583484	+	0.812125i	$0.698311\pi$
$32$	0	0
$33$	1440.00	0.230185
$34$	0	0
$35$	4704.00	0.649078
$36$	0	0
$37$	10798.0	1.29670	0.648349	−	0.761343i	$-0.275460\pi$
$37$	10798.0	1.29670	0.648349	+	0.761343i	$0.275460\pi$
$38$	0	0
$39$	−1144.00	−0.120438
$40$	0	0
$41$	12006.0	1.11542	0.557710	−	0.830036i	$-0.311680\pi$
$41$	12006.0	1.11542	0.557710	+	0.830036i	$0.311680\pi$
$42$	0	0
$43$	9160.00	0.755482	0.377741	−	0.925911i	$-0.376701\pi$
$43$	9160.00	0.755482	0.377741	+	0.925911i	$0.376701\pi$
$44$	0	0
$45$	−22944.0	−1.68903
$46$	0	0
$47$	−25836.0	−1.70601	−0.853003	−	0.521906i	$-0.825221\pi$
$47$	−25836.0	−1.70601	−0.853003	+	0.521906i	$0.825221\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	2508.00	0.135021
$52$	0	0
$53$	−1014.00	−0.0495848	−0.0247924	−	0.999693i	$-0.507892\pi$
$53$	−1014.00	−0.0495848	−0.0247924	+	0.999693i	$0.507892\pi$
$54$	0	0
$55$	69120.0	3.08104
$56$	0	0
$57$	188.000	0.00766427
$58$	0	0
$59$	−1242.00	−0.0464506	−0.0232253	−	0.999730i	$-0.507394\pi$
$59$	−1242.00	−0.0464506	−0.0232253	+	0.999730i	$0.507394\pi$
$60$	0	0
$61$	−7592.00	−0.261235	−0.130618	−	0.991433i	$-0.541696\pi$
$61$	−7592.00	−0.261235	−0.130618	+	0.991433i	$0.541696\pi$
$62$	0	0
$63$	−11711.0	−0.371743
$64$	0	0
$65$	−54912.0	−1.61207
$66$	0	0
$67$	−41132.0	−1.11942	−0.559710	−	0.828689i	$-0.689087\pi$
$67$	−41132.0	−1.11942	−0.559710	+	0.828689i	$0.689087\pi$
$68$	0	0
$69$	192.000	0.00485488
$70$	0	0
$71$	−37632.0	−0.885955	−0.442977	−	0.896533i	$-0.646078\pi$
$71$	−37632.0	−0.885955	−0.442977	+	0.896533i	$0.646078\pi$
$72$	0	0
$73$	−13438.0	−0.295140	−0.147570	−	0.989052i	$-0.547145\pi$
$73$	−13438.0	−0.295140	−0.147570	+	0.989052i	$0.547145\pi$
$74$	0	0
$75$	12182.0	0.250072
$76$	0	0
$77$	35280.0	0.678113
$78$	0	0
$79$	6248.00	0.112635	0.0563175	−	0.998413i	$-0.482064\pi$
$79$	6248.00	0.112635	0.0563175	+	0.998413i	$0.482064\pi$
$80$	0	0
$81$	56149.0	0.950888
$82$	0	0
$83$	25254.0	0.402379	0.201189	−	0.979552i	$-0.435519\pi$
$83$	25254.0	0.402379	0.201189	+	0.979552i	$0.435519\pi$
$84$	0	0
$85$	120384.	1.80726
$86$	0	0
$87$	8748.00	0.123911
$88$	0	0
$89$	−45126.0	−0.603882	−0.301941	−	0.953327i	$-0.597635\pi$
$89$	−45126.0	−0.603882	−0.301941	+	0.953327i	$0.597635\pi$
$90$	0	0
$91$	−28028.0	−0.354804
$92$	0	0
$93$	−12488.0	−0.149722
$94$	0	0
$95$	9024.00	0.102586
$96$	0	0
$97$	107222.	1.15706	0.578528	−	0.815662i	$-0.303627\pi$
$97$	107222.	1.15706	0.578528	+	0.815662i	$0.303627\pi$
$98$	0	0
$99$	−172080.	−1.76458
$100$	0	0
$101$	−47136.0	−0.459779	−0.229890	−	0.973217i	$-0.573837\pi$
$101$	−47136.0	−0.459779	−0.229890	+	0.973217i	$0.573837\pi$
$102$	0	0
$103$	122204.	1.13499	0.567495	−	0.823377i	$-0.307912\pi$
$103$	122204.	1.13499	0.567495	+	0.823377i	$0.307912\pi$
$104$	0	0
$105$	9408.00	0.0832768
$106$	0	0
$107$	129636.	1.09463	0.547314	−	0.836928i	$-0.315651\pi$
$107$	129636.	1.09463	0.547314	+	0.836928i	$0.315651\pi$
$108$	0	0
$109$	220558.	1.77810	0.889051	−	0.457809i	$-0.151365\pi$
$109$	220558.	1.77810	0.889051	+	0.457809i	$0.151365\pi$
$110$	0	0
$111$	21596.0	0.166366
$112$	0	0
$113$	170694.	1.25754	0.628770	−	0.777591i	$-0.283559\pi$
$113$	170694.	1.25754	0.628770	+	0.777591i	$0.283559\pi$
$114$	0	0
$115$	9216.00	0.0649827
$116$	0	0
$117$	136708.	0.923271
$118$	0	0
$119$	61446.0	0.397765
$120$	0	0
$121$	357349.	2.21886
$122$	0	0
$123$	24012.0	0.143109
$124$	0	0
$125$	284736.	1.62992
$126$	0	0
$127$	−249808.	−1.37435	−0.687175	−	0.726492i	$-0.741149\pi$
$127$	−249808.	−1.37435	−0.687175	+	0.726492i	$0.741149\pi$
$128$	0	0
$129$	18320.0	0.0969284
$130$	0	0
$131$	−12210.0	−0.0621638	−0.0310819	−	0.999517i	$-0.509895\pi$
$131$	−12210.0	−0.0621638	−0.0310819	+	0.999517i	$0.509895\pi$
$132$	0	0
$133$	4606.00	0.0225785
$134$	0	0
$135$	−92544.0	−0.437033
$136$	0	0
$137$	−13902.0	−0.0632814	−0.0316407	−	0.999499i	$-0.510073\pi$
$137$	−13902.0	−0.0632814	−0.0316407	+	0.999499i	$0.510073\pi$
$138$	0	0
$139$	431794.	1.89557	0.947785	−	0.318911i	$-0.103317\pi$
$139$	431794.	1.89557	0.947785	+	0.318911i	$0.103317\pi$
$140$	0	0
$141$	−51672.0	−0.218881
$142$	0	0
$143$	−411840.	−1.68418
$144$	0	0
$145$	419904.	1.65856
$146$	0	0
$147$	4802.00	0.0183286
$148$	0	0
$149$	−326814.	−1.20597	−0.602983	−	0.797754i	$-0.706021\pi$
$149$	−326814.	−1.20597	−0.602983	+	0.797754i	$0.706021\pi$
$150$	0	0
$151$	173480.	0.619166	0.309583	−	0.950872i	$-0.399811\pi$
$151$	173480.	0.619166	0.309583	+	0.950872i	$0.399811\pi$
$152$	0	0
$153$	−299706.	−1.03506
$154$	0	0
$155$	−599424.	−2.00403
$156$	0	0
$157$	54532.0	0.176564	0.0882820	−	0.996096i	$-0.471862\pi$
$157$	54532.0	0.176564	0.0882820	+	0.996096i	$0.471862\pi$
$158$	0	0
$159$	−2028.00	−0.00636173
$160$	0	0
$161$	4704.00	0.0143022
$162$	0	0
$163$	−104960.	−0.309425	−0.154712	−	0.987960i	$-0.549445\pi$
$163$	−104960.	−0.309425	−0.154712	+	0.987960i	$0.549445\pi$
$164$	0	0
$165$	138240.	0.395297
$166$	0	0
$167$	160788.	0.446131	0.223066	−	0.974803i	$-0.428394\pi$
$167$	160788.	0.446131	0.223066	+	0.974803i	$0.428394\pi$
$168$	0	0
$169$	−44109.0	−0.118798
$170$	0	0
$171$	−22466.0	−0.0587537
$172$	0	0
$173$	−360564.	−0.915940	−0.457970	−	0.888968i	$-0.651423\pi$
$173$	−360564.	−0.915940	−0.457970	+	0.888968i	$0.651423\pi$
$174$	0	0
$175$	298459.	0.736698
$176$	0	0
$177$	−2484.00	−0.00595962
$178$	0	0
$179$	312732.	0.729524	0.364762	−	0.931101i	$-0.381150\pi$
$179$	312732.	0.729524	0.364762	+	0.931101i	$0.381150\pi$
$180$	0	0
$181$	123820.	0.280928	0.140464	−	0.990086i	$-0.455141\pi$
$181$	123820.	0.280928	0.140464	+	0.990086i	$0.455141\pi$
$182$	0	0
$183$	−15184.0	−0.0335165
$184$	0	0
$185$	1.03661e6	2.22682
$186$	0	0
$187$	902880.	1.88810
$188$	0	0
$189$	−47236.0	−0.0961875
$190$	0	0
$191$	323448.	0.641536	0.320768	−	0.947158i	$-0.396059\pi$
$191$	323448.	0.641536	0.320768	+	0.947158i	$0.396059\pi$
$192$	0	0
$193$	−619954.	−1.19803	−0.599013	−	0.800739i	$-0.704440\pi$
$193$	−619954.	−1.19803	−0.599013	+	0.800739i	$0.704440\pi$
$194$	0	0
$195$	−109824.	−0.206829
$196$	0	0
$197$	499362.	0.916748	0.458374	−	0.888759i	$-0.348432\pi$
$197$	499362.	0.916748	0.458374	+	0.888759i	$0.348432\pi$
$198$	0	0
$199$	−785932.	−1.40686	−0.703432	−	0.710762i	$-0.748350\pi$
$199$	−785932.	−1.40686	−0.703432	+	0.710762i	$0.748350\pi$
$200$	0	0
$201$	−82264.0	−0.143622
$202$	0	0
$203$	214326.	0.365035
$204$	0	0
$205$	1.15258e6	1.91551
$206$	0	0
$207$	−22944.0	−0.0372172
$208$	0	0
$209$	67680.0	0.107175
$210$	0	0
$211$	−1.06276e6	−1.64335	−0.821676	−	0.569955i	$-0.806961\pi$
$211$	−1.06276e6	−1.64335	−0.821676	+	0.569955i	$0.806961\pi$
$212$	0	0
$213$	−75264.0	−0.113668
$214$	0	0
$215$	879360.	1.29739
$216$	0	0
$217$	−305956.	−0.441072
$218$	0	0
$219$	−26876.0	−0.0378664
$220$	0	0
$221$	−717288.	−0.987900
$222$	0	0
$223$	707720.	0.953014	0.476507	−	0.879171i	$-0.341903\pi$
$223$	707720.	0.953014	0.476507	+	0.879171i	$0.341903\pi$
$224$	0	0
$225$	−1.45575e6	−1.91704
$226$	0	0
$227$	1.04437e6	1.34520	0.672602	−	0.740005i	$-0.265177\pi$
$227$	1.04437e6	1.34520	0.672602	+	0.740005i	$0.265177\pi$
$228$	0	0
$229$	539716.	0.680106	0.340053	−	0.940406i	$-0.389555\pi$
$229$	539716.	0.680106	0.340053	+	0.940406i	$0.389555\pi$
$230$	0	0
$231$	70560.0	0.0870019
$232$	0	0
$233$	177114.	0.213729	0.106864	−	0.994274i	$-0.465919\pi$
$233$	177114.	0.213729	0.106864	+	0.994274i	$0.465919\pi$
$234$	0	0
$235$	−2.48026e6	−2.92972
$236$	0	0
$237$	12496.0	0.0144511
$238$	0	0
$239$	−655464.	−0.742257	−0.371128	−	0.928582i	$-0.621029\pi$
$239$	−655464.	−0.742257	−0.371128	+	0.928582i	$0.621029\pi$
$240$	0	0
$241$	1.38709e6	1.53838	0.769189	−	0.639021i	$-0.220660\pi$
$241$	1.38709e6	1.53838	0.769189	+	0.639021i	$0.220660\pi$
$242$	0	0
$243$	346550.	0.376487
$244$	0	0
$245$	230496.	0.245329
$246$	0	0
$247$	−53768.0	−0.0560766
$248$	0	0
$249$	50508.0	0.0516252
$250$	0	0
$251$	−1.88811e6	−1.89166	−0.945830	−	0.324663i	$-0.894749\pi$
$251$	−1.88811e6	−1.89166	−0.945830	+	0.324663i	$0.894749\pi$
$252$	0	0
$253$	69120.0	0.0678895
$254$	0	0
$255$	240768.	0.231872
$256$	0	0
$257$	346194.	0.326954	0.163477	−	0.986547i	$-0.447729\pi$
$257$	346194.	0.326954	0.163477	+	0.986547i	$0.447729\pi$
$258$	0	0
$259$	529102.	0.490106
$260$	0	0
$261$	−1.04539e6	−0.949895
$262$	0	0
$263$	−929088.	−0.828262	−0.414131	−	0.910217i	$-0.635914\pi$
$263$	−929088.	−0.828262	−0.414131	+	0.910217i	$0.635914\pi$
$264$	0	0
$265$	−97344.0	−0.0851519
$266$	0	0
$267$	−90252.0	−0.0774780
$268$	0	0
$269$	−382068.	−0.321929	−0.160964	−	0.986960i	$-0.551460\pi$
$269$	−382068.	−0.321929	−0.160964	+	0.986960i	$0.551460\pi$
$270$	0	0
$271$	−1.58056e6	−1.30734	−0.653669	−	0.756781i	$-0.726771\pi$
$271$	−1.58056e6	−1.30734	−0.653669	+	0.756781i	$0.726771\pi$
$272$	0	0
$273$	−56056.0	−0.0455214
$274$	0	0
$275$	4.38552e6	3.49695
$276$	0	0
$277$	1.36911e6	1.07211	0.536056	−	0.844182i	$-0.319914\pi$
$277$	1.36911e6	1.07211	0.536056	+	0.844182i	$0.319914\pi$
$278$	0	0
$279$	1.49232e6	1.14776
$280$	0	0
$281$	−394854.	−0.298312	−0.149156	−	0.988814i	$-0.547656\pi$
$281$	−394854.	−0.298312	−0.149156	+	0.988814i	$0.547656\pi$
$282$	0	0
$283$	−673034.	−0.499541	−0.249770	−	0.968305i	$-0.580355\pi$
$283$	−673034.	−0.499541	−0.249770	+	0.968305i	$0.580355\pi$
$284$	0	0
$285$	18048.0	0.0131618
$286$	0	0
$287$	588294.	0.421589
$288$	0	0
$289$	152659.	0.107517
$290$	0	0
$291$	214444.	0.148450
$292$	0	0
$293$	−1.83468e6	−1.24851	−0.624254	−	0.781222i	$-0.714597\pi$
$293$	−1.83468e6	−1.24851	−0.624254	+	0.781222i	$0.714597\pi$
$294$	0	0
$295$	−119232.	−0.0797697
$296$	0	0
$297$	−694080.	−0.456582
$298$	0	0
$299$	−54912.0	−0.0355213
$300$	0	0
$301$	448840.	0.285545
$302$	0	0
$303$	−94272.0	−0.0589897
$304$	0	0
$305$	−728832.	−0.448619
$306$	0	0
$307$	1.51056e6	0.914727	0.457363	−	0.889280i	$-0.348794\pi$
$307$	1.51056e6	0.914727	0.457363	+	0.889280i	$0.348794\pi$
$308$	0	0
$309$	244408.	0.145619
$310$	0	0
$311$	−1.87529e6	−1.09943	−0.549714	−	0.835353i	$-0.685263\pi$
$311$	−1.87529e6	−1.09943	−0.549714	+	0.835353i	$0.685263\pi$
$312$	0	0
$313$	−1.51076e6	−0.871636	−0.435818	−	0.900035i	$-0.643541\pi$
$313$	−1.51076e6	−0.871636	−0.435818	+	0.900035i	$0.643541\pi$
$314$	0	0
$315$	−1.12426e6	−0.638394
$316$	0	0
$317$	−2.02709e6	−1.13299	−0.566495	−	0.824065i	$-0.691701\pi$
$317$	−2.02709e6	−1.13299	−0.566495	+	0.824065i	$0.691701\pi$
$318$	0	0
$319$	3.14928e6	1.73274
$320$	0	0
$321$	259272.	0.140441
$322$	0	0
$323$	117876.	0.0628665
$324$	0	0
$325$	−3.48405e6	−1.82968
$326$	0	0
$327$	441116.	0.228131
$328$	0	0
$329$	−1.26596e6	−0.644810
$330$	0	0
$331$	−1.54009e6	−0.772637	−0.386319	−	0.922365i	$-0.626253\pi$
$331$	−1.54009e6	−0.772637	−0.386319	+	0.922365i	$0.626253\pi$
$332$	0	0
$333$	−2.58072e6	−1.27535
$334$	0	0
$335$	−3.94867e6	−1.92238
$336$	0	0
$337$	1.01166e6	0.485245	0.242622	−	0.970121i	$-0.421992\pi$
$337$	1.01166e6	0.485245	0.242622	+	0.970121i	$0.421992\pi$
$338$	0	0
$339$	341388.	0.161343
$340$	0	0
$341$	−4.49568e6	−2.09368
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	18432.0	0.00833729
$346$	0	0
$347$	2.15748e6	0.961885	0.480942	−	0.876752i	$-0.340295\pi$
$347$	2.15748e6	0.961885	0.480942	+	0.876752i	$0.340295\pi$
$348$	0	0
$349$	1.15798e6	0.508906	0.254453	−	0.967085i	$-0.418105\pi$
$349$	1.15798e6	0.508906	0.254453	+	0.967085i	$0.418105\pi$
$350$	0	0
$351$	551408.	0.238894
$352$	0	0
$353$	−3.17566e6	−1.35643	−0.678215	−	0.734863i	$-0.737246\pi$
$353$	−3.17566e6	−1.35643	−0.678215	+	0.734863i	$0.737246\pi$
$354$	0	0
$355$	−3.61267e6	−1.52145
$356$	0	0
$357$	122892.	0.0510332
$358$	0	0
$359$	−74616.0	−0.0305560	−0.0152780	−	0.999883i	$-0.504863\pi$
$359$	−74616.0	−0.0305560	−0.0152780	+	0.999883i	$0.504863\pi$
$360$	0	0
$361$	−2.46726e6	−0.996431
$362$	0	0
$363$	714698.	0.284679
$364$	0	0
$365$	−1.29005e6	−0.506843
$366$	0	0
$367$	−1.79807e6	−0.696854	−0.348427	−	0.937336i	$-0.613284\pi$
$367$	−1.79807e6	−0.696854	−0.348427	+	0.937336i	$0.613284\pi$
$368$	0	0
$369$	−2.86943e6	−1.09706
$370$	0	0
$371$	−49686.0	−0.0187413
$372$	0	0
$373$	−2.20461e6	−0.820463	−0.410231	−	0.911981i	$-0.634552\pi$
$373$	−2.20461e6	−0.820463	−0.410231	+	0.911981i	$0.634552\pi$
$374$	0	0
$375$	569472.	0.209119
$376$	0	0
$377$	−2.50193e6	−0.906612
$378$	0	0
$379$	177568.	0.0634990	0.0317495	−	0.999496i	$-0.489892\pi$
$379$	177568.	0.0634990	0.0317495	+	0.999496i	$0.489892\pi$
$380$	0	0
$381$	−499616.	−0.176329
$382$	0	0
$383$	−2.87468e6	−1.00137	−0.500683	−	0.865630i	$-0.666918\pi$
$383$	−2.87468e6	−1.00137	−0.500683	+	0.865630i	$0.666918\pi$
$384$	0	0
$385$	3.38688e6	1.16452
$386$	0	0
$387$	−2.18924e6	−0.743046
$388$	0	0
$389$	4.79965e6	1.60818	0.804091	−	0.594506i	$-0.202652\pi$
$389$	4.79965e6	1.60818	0.804091	+	0.594506i	$0.202652\pi$
$390$	0	0
$391$	120384.	0.0398223
$392$	0	0
$393$	−24420.0	−0.00797562
$394$	0	0
$395$	599808.	0.193428
$396$	0	0
$397$	2.81643e6	0.896855	0.448428	−	0.893819i	$-0.351984\pi$
$397$	2.81643e6	0.896855	0.448428	+	0.893819i	$0.351984\pi$
$398$	0	0
$399$	9212.00	0.00289682
$400$	0	0
$401$	−2.83797e6	−0.881347	−0.440673	−	0.897667i	$-0.645260\pi$
$401$	−2.83797e6	−0.881347	−0.440673	+	0.897667i	$0.645260\pi$
$402$	0	0
$403$	3.57157e6	1.09546
$404$	0	0
$405$	5.39030e6	1.63296
$406$	0	0
$407$	7.77456e6	2.32643
$408$	0	0
$409$	154286.	0.0456056	0.0228028	−	0.999740i	$-0.492741\pi$
$409$	154286.	0.0456056	0.0228028	+	0.999740i	$0.492741\pi$
$410$	0	0
$411$	−27804.0	−0.00811900
$412$	0	0
$413$	−60858.0	−0.0175567
$414$	0	0
$415$	2.42438e6	0.691005
$416$	0	0
$417$	863588.	0.243202
$418$	0	0
$419$	−3.72865e6	−1.03757	−0.518783	−	0.854906i	$-0.673615\pi$
$419$	−3.72865e6	−1.03757	−0.518783	+	0.854906i	$0.673615\pi$
$420$	0	0
$421$	2.32623e6	0.639658	0.319829	−	0.947475i	$-0.396374\pi$
$421$	2.32623e6	0.639658	0.319829	+	0.947475i	$0.396374\pi$
$422$	0	0
$423$	6.17480e6	1.67792
$424$	0	0
$425$	7.63811e6	2.05123
$426$	0	0
$427$	−372008.	−0.0987376
$428$	0	0
$429$	−823680.	−0.216080
$430$	0	0
$431$	−2.61482e6	−0.678031	−0.339015	−	0.940781i	$-0.610094\pi$
$431$	−2.61482e6	−0.678031	−0.339015	+	0.940781i	$0.610094\pi$
$432$	0	0
$433$	−1.19226e6	−0.305598	−0.152799	−	0.988257i	$-0.548829\pi$
$433$	−1.19226e6	−0.305598	−0.152799	+	0.988257i	$0.548829\pi$
$434$	0	0
$435$	839808.	0.212793
$436$	0	0
$437$	9024.00	0.00226045
$438$	0	0
$439$	1.05793e6	0.261996	0.130998	−	0.991383i	$-0.458182\pi$
$439$	1.05793e6	0.261996	0.130998	+	0.991383i	$0.458182\pi$
$440$	0	0
$441$	−573839.	−0.140506
$442$	0	0
$443$	−4.12756e6	−0.999272	−0.499636	−	0.866235i	$-0.666533\pi$
$443$	−4.12756e6	−0.999272	−0.499636	+	0.866235i	$0.666533\pi$
$444$	0	0
$445$	−4.33210e6	−1.03705
$446$	0	0
$447$	−653628.	−0.154725
$448$	0	0
$449$	3.75823e6	0.879766	0.439883	−	0.898055i	$-0.355020\pi$
$449$	3.75823e6	0.879766	0.439883	+	0.898055i	$0.355020\pi$
$450$	0	0
$451$	8.64432e6	2.00120
$452$	0	0
$453$	346960.	0.0794390
$454$	0	0
$455$	−2.69069e6	−0.609305
$456$	0	0
$457$	−451114.	−0.101041	−0.0505203	−	0.998723i	$-0.516088\pi$
$457$	−451114.	−0.101041	−0.0505203	+	0.998723i	$0.516088\pi$
$458$	0	0
$459$	−1.20886e6	−0.267820
$460$	0	0
$461$	1.95186e6	0.427756	0.213878	−	0.976860i	$-0.431390\pi$
$461$	1.95186e6	0.427756	0.213878	+	0.976860i	$0.431390\pi$
$462$	0	0
$463$	−7.20218e6	−1.56139	−0.780695	−	0.624913i	$-0.785135\pi$
$463$	−7.20218e6	−1.56139	−0.780695	+	0.624913i	$0.785135\pi$
$464$	0	0
$465$	−1.19885e6	−0.257118
$466$	0	0
$467$	−7.17801e6	−1.52304	−0.761521	−	0.648140i	$-0.775547\pi$
$467$	−7.17801e6	−1.52304	−0.761521	+	0.648140i	$0.775547\pi$
$468$	0	0
$469$	−2.01547e6	−0.423101
$470$	0	0
$471$	109064.	0.0226532
$472$	0	0
$473$	6.59520e6	1.35542
$474$	0	0
$475$	572554.	0.116435
$476$	0	0
$477$	242346.	0.0487686
$478$	0	0
$479$	6.17632e6	1.22996	0.614980	−	0.788543i	$-0.289164\pi$
$479$	6.17632e6	1.22996	0.614980	+	0.788543i	$0.289164\pi$
$480$	0	0
$481$	−6.17646e6	−1.21724
$482$	0	0
$483$	9408.00	0.00183497
$484$	0	0
$485$	1.02933e7	1.98701
$486$	0	0
$487$	7.59330e6	1.45080	0.725401	−	0.688327i	$-0.241655\pi$
$487$	7.59330e6	1.45080	0.725401	+	0.688327i	$0.241655\pi$
$488$	0	0
$489$	−209920.	−0.0396992
$490$	0	0
$491$	1.51878e6	0.284309	0.142155	−	0.989844i	$-0.454597\pi$
$491$	1.51878e6	0.284309	0.142155	+	0.989844i	$0.454597\pi$
$492$	0	0
$493$	5.48500e6	1.01639
$494$	0	0
$495$	−1.65197e7	−3.03032
$496$	0	0
$497$	−1.84397e6	−0.334859
$498$	0	0
$499$	−1.47576e6	−0.265316	−0.132658	−	0.991162i	$-0.542351\pi$
$499$	−1.47576e6	−0.265316	−0.132658	+	0.991162i	$0.542351\pi$
$500$	0	0
$501$	321576.	0.0572386
$502$	0	0
$503$	−1.31309e6	−0.231406	−0.115703	−	0.993284i	$-0.536912\pi$
$503$	−1.31309e6	−0.231406	−0.115703	+	0.993284i	$0.536912\pi$
$504$	0	0
$505$	−4.52506e6	−0.789579
$506$	0	0
$507$	−88218.0	−0.0152418
$508$	0	0
$509$	−4.40932e6	−0.754357	−0.377178	−	0.926141i	$-0.623106\pi$
$509$	−4.40932e6	−0.754357	−0.377178	+	0.926141i	$0.623106\pi$
$510$	0	0
$511$	−658462.	−0.111552
$512$	0	0
$513$	−90616.0	−0.0152024
$514$	0	0
$515$	1.17316e7	1.94912
$516$	0	0
$517$	−1.86019e7	−3.06078
$518$	0	0
$519$	−721128.	−0.117515
$520$	0	0
$521$	2.97629e6	0.480376	0.240188	−	0.970726i	$-0.422791\pi$
$521$	2.97629e6	0.480376	0.240188	+	0.970726i	$0.422791\pi$
$522$	0	0
$523$	−6.34627e6	−1.01453	−0.507265	−	0.861790i	$-0.669343\pi$
$523$	−6.34627e6	−1.01453	−0.507265	+	0.861790i	$0.669343\pi$
$524$	0	0
$525$	596918.	0.0945184
$526$	0	0
$527$	−7.82998e6	−1.22810
$528$	0	0
$529$	−6.42713e6	−0.998568
$530$	0	0
$531$	296838.	0.0456860
$532$	0	0
$533$	−6.86743e6	−1.04707
$534$	0	0
$535$	1.24451e7	1.87980
$536$	0	0
$537$	625464.	0.0935980
$538$	0	0
$539$	1.72872e6	0.256302
$540$	0	0
$541$	1.36667e6	0.200756	0.100378	−	0.994949i	$-0.467995\pi$
$541$	1.36667e6	0.200756	0.100378	+	0.994949i	$0.467995\pi$
$542$	0	0
$543$	247640.	0.0360430
$544$	0	0
$545$	2.11736e7	3.05353
$546$	0	0
$547$	9.55818e6	1.36586	0.682931	−	0.730483i	$-0.260705\pi$
$547$	9.55818e6	1.36586	0.682931	+	0.730483i	$0.260705\pi$
$548$	0	0
$549$	1.81449e6	0.256935
$550$	0	0
$551$	411156.	0.0576936
$552$	0	0
$553$	306152.	0.0425720
$554$	0	0
$555$	2.07322e6	0.285701
$556$	0	0
$557$	−6.94287e6	−0.948202	−0.474101	−	0.880470i	$-0.657227\pi$
$557$	−6.94287e6	−0.948202	−0.474101	+	0.880470i	$0.657227\pi$
$558$	0	0
$559$	−5.23952e6	−0.709189
$560$	0	0
$561$	1.80576e6	0.242244
$562$	0	0
$563$	−5.24662e6	−0.697604	−0.348802	−	0.937196i	$-0.613411\pi$
$563$	−5.24662e6	−0.697604	−0.348802	+	0.937196i	$0.613411\pi$
$564$	0	0
$565$	1.63866e7	2.15958
$566$	0	0
$567$	2.75130e6	0.359402
$568$	0	0
$569$	3.46551e6	0.448731	0.224366	−	0.974505i	$-0.427969\pi$
$569$	3.46551e6	0.448731	0.224366	+	0.974505i	$0.427969\pi$
$570$	0	0
$571$	−4.90069e6	−0.629023	−0.314512	−	0.949254i	$-0.601841\pi$
$571$	−4.90069e6	−0.629023	−0.314512	+	0.949254i	$0.601841\pi$
$572$	0	0
$573$	646896.	0.0823091
$574$	0	0
$575$	584736.	0.0737548
$576$	0	0
$577$	2.28346e6	0.285531	0.142766	−	0.989757i	$-0.454401\pi$
$577$	2.28346e6	0.285531	0.142766	+	0.989757i	$0.454401\pi$
$578$	0	0
$579$	−1.23991e6	−0.153707
$580$	0	0
$581$	1.23745e6	0.152085
$582$	0	0
$583$	−730080.	−0.0889609
$584$	0	0
$585$	1.31240e7	1.58553
$586$	0	0
$587$	−1.03157e7	−1.23568	−0.617838	−	0.786305i	$-0.711991\pi$
$587$	−1.03157e7	−1.23568	−0.617838	+	0.786305i	$0.711991\pi$
$588$	0	0
$589$	−586936.	−0.0697112
$590$	0	0
$591$	998724.	0.117619
$592$	0	0
$593$	3.52838e6	0.412039	0.206020	−	0.978548i	$-0.433949\pi$
$593$	3.52838e6	0.412039	0.206020	+	0.978548i	$0.433949\pi$
$594$	0	0
$595$	5.89882e6	0.683081
$596$	0	0
$597$	−1.57186e6	−0.180501
$598$	0	0
$599$	2.92260e6	0.332815	0.166407	−	0.986057i	$-0.446783\pi$
$599$	2.92260e6	0.332815	0.166407	+	0.986057i	$0.446783\pi$
$600$	0	0
$601$	−1.17567e7	−1.32770	−0.663849	−	0.747866i	$-0.731078\pi$
$601$	−1.17567e7	−1.32770	−0.663849	+	0.747866i	$0.731078\pi$
$602$	0	0
$603$	9.83055e6	1.10099
$604$	0	0
$605$	3.43055e7	3.81044
$606$	0	0
$607$	−4.71491e6	−0.519400	−0.259700	−	0.965689i	$-0.583624\pi$
$607$	−4.71491e6	−0.519400	−0.259700	+	0.965689i	$0.583624\pi$
$608$	0	0
$609$	428652.	0.0468340
$610$	0	0
$611$	1.47782e7	1.60147
$612$	0	0
$613$	−213842.	−0.0229849	−0.0114924	−	0.999934i	$-0.503658\pi$
$613$	−213842.	−0.0229849	−0.0114924	+	0.999934i	$0.503658\pi$
$614$	0	0
$615$	2.30515e6	0.245760
$616$	0	0
$617$	−336666.	−0.0356030	−0.0178015	−	0.999842i	$-0.505667\pi$
$617$	−336666.	−0.0356030	−0.0178015	+	0.999842i	$0.505667\pi$
$618$	0	0
$619$	−1.42655e7	−1.49645	−0.748223	−	0.663447i	$-0.769093\pi$
$619$	−1.42655e7	−1.49645	−0.748223	+	0.663447i	$0.769093\pi$
$620$	0	0
$621$	−92544.0	−0.00962984
$622$	0	0
$623$	−2.21117e6	−0.228246
$624$	0	0
$625$	8.30028e6	0.849949
$626$	0	0
$627$	135360.	0.0137506
$628$	0	0
$629$	1.35407e7	1.36463
$630$	0	0
$631$	−6.59637e6	−0.659525	−0.329763	−	0.944064i	$-0.606969\pi$
$631$	−6.59637e6	−0.659525	−0.329763	+	0.944064i	$0.606969\pi$
$632$	0	0
$633$	−2.12553e6	−0.210842
$634$	0	0
$635$	−2.39816e7	−2.36017
$636$	0	0
$637$	−1.37337e6	−0.134103
$638$	0	0
$639$	8.99405e6	0.871371
$640$	0	0
$641$	−1.02490e7	−0.985225	−0.492613	−	0.870249i	$-0.663958\pi$
$641$	−1.02490e7	−0.985225	−0.492613	+	0.870249i	$0.663958\pi$
$642$	0	0
$643$	4.16543e6	0.397312	0.198656	−	0.980069i	$-0.436342\pi$
$643$	4.16543e6	0.397312	0.198656	+	0.980069i	$0.436342\pi$
$644$	0	0
$645$	1.75872e6	0.166455
$646$	0	0
$647$	−3.35051e6	−0.314666	−0.157333	−	0.987546i	$-0.550290\pi$
$647$	−3.35051e6	−0.314666	−0.157333	+	0.987546i	$0.550290\pi$
$648$	0	0
$649$	−894240.	−0.0833379
$650$	0	0
$651$	−611912.	−0.0565896
$652$	0	0
$653$	9.05408e6	0.830924	0.415462	−	0.909611i	$-0.363620\pi$
$653$	9.05408e6	0.830924	0.415462	+	0.909611i	$0.363620\pi$
$654$	0	0
$655$	−1.17216e6	−0.106754
$656$	0	0
$657$	3.21168e6	0.290281
$658$	0	0
$659$	−6.45382e6	−0.578899	−0.289450	−	0.957193i	$-0.593472\pi$
$659$	−6.45382e6	−0.578899	−0.289450	+	0.957193i	$0.593472\pi$
$660$	0	0
$661$	−1.43167e7	−1.27450	−0.637250	−	0.770657i	$-0.719928\pi$
$661$	−1.43167e7	−1.27450	−0.637250	+	0.770657i	$0.719928\pi$
$662$	0	0
$663$	−1.43458e6	−0.126748
$664$	0	0
$665$	442176.	0.0387740
$666$	0	0
$667$	419904.	0.0365456
$668$	0	0
$669$	1.41544e6	0.122272
$670$	0	0
$671$	−5.46624e6	−0.468686
$672$	0	0
$673$	2.27250e7	1.93404	0.967020	−	0.254701i	$-0.0819771\pi$
$673$	2.27250e7	1.93404	0.967020	+	0.254701i	$0.0819771\pi$
$674$	0	0
$675$	−5.87172e6	−0.496028
$676$	0	0
$677$	8.53249e6	0.715491	0.357746	−	0.933819i	$-0.383546\pi$
$677$	8.53249e6	0.715491	0.357746	+	0.933819i	$0.383546\pi$
$678$	0	0
$679$	5.25388e6	0.437326
$680$	0	0
$681$	2.08873e6	0.172590
$682$	0	0
$683$	2.24921e7	1.84492	0.922461	−	0.386090i	$-0.126175\pi$
$683$	2.24921e7	1.84492	0.922461	+	0.386090i	$0.126175\pi$
$684$	0	0
$685$	−1.33459e6	−0.108673
$686$	0	0
$687$	1.07943e6	0.0872576
$688$	0	0
$689$	580008.	0.0465464
$690$	0	0
$691$	1.26894e7	1.01099	0.505495	−	0.862830i	$-0.331310\pi$
$691$	1.26894e7	1.01099	0.505495	+	0.862830i	$0.331310\pi$
$692$	0	0
$693$	−8.43192e6	−0.666950
$694$	0	0
$695$	4.14522e7	3.25526
$696$	0	0
$697$	1.50555e7	1.17385
$698$	0	0
$699$	354228.	0.0274214
$700$	0	0
$701$	5.13939e6	0.395018	0.197509	−	0.980301i	$-0.436715\pi$
$701$	5.13939e6	0.395018	0.197509	+	0.980301i	$0.436715\pi$
$702$	0	0
$703$	1.01501e6	0.0774610
$704$	0	0
$705$	−4.96051e6	−0.375884
$706$	0	0
$707$	−2.30966e6	−0.173780
$708$	0	0
$709$	1.16065e7	0.867132	0.433566	−	0.901122i	$-0.357255\pi$
$709$	1.16065e7	0.867132	0.433566	+	0.901122i	$0.357255\pi$
$710$	0	0
$711$	−1.49327e6	−0.110781
$712$	0	0
$713$	−599424.	−0.0441581
$714$	0	0
$715$	−3.95366e7	−2.89224
$716$	0	0
$717$	−1.31093e6	−0.0952316
$718$	0	0
$719$	1.50998e7	1.08930	0.544650	−	0.838663i	$-0.316662\pi$
$719$	1.50998e7	1.08930	0.544650	+	0.838663i	$0.316662\pi$
$720$	0	0
$721$	5.98800e6	0.428986
$722$	0	0
$723$	2.77419e6	0.197374
$724$	0	0
$725$	2.66420e7	1.88245
$726$	0	0
$727$	−2.32536e7	−1.63175	−0.815874	−	0.578229i	$-0.803744\pi$
$727$	−2.32536e7	−1.63175	−0.815874	+	0.578229i	$0.803744\pi$
$728$	0	0
$729$	−1.29511e7	−0.902585
$730$	0	0
$731$	1.14866e7	0.795059
$732$	0	0
$733$	−2.37814e7	−1.63485	−0.817423	−	0.576038i	$-0.804598\pi$
$733$	−2.37814e7	−1.63485	−0.817423	+	0.576038i	$0.804598\pi$
$734$	0	0
$735$	460992.	0.0314757
$736$	0	0
$737$	−2.96150e7	−2.00837
$738$	0	0
$739$	2.51392e6	0.169333	0.0846663	−	0.996409i	$-0.473018\pi$
$739$	2.51392e6	0.169333	0.0846663	+	0.996409i	$0.473018\pi$
$740$	0	0
$741$	−107536.	−0.00719463
$742$	0	0
$743$	−2.22646e7	−1.47959	−0.739797	−	0.672830i	$-0.765078\pi$
$743$	−2.22646e7	−1.47959	−0.739797	+	0.672830i	$0.765078\pi$
$744$	0	0
$745$	−3.13741e7	−2.07101
$746$	0	0
$747$	−6.03571e6	−0.395755
$748$	0	0
$749$	6.35216e6	0.413730
$750$	0	0
$751$	2.30108e7	1.48878	0.744392	−	0.667743i	$-0.232740\pi$
$751$	2.30108e7	1.48878	0.744392	+	0.667743i	$0.232740\pi$
$752$	0	0
$753$	−3.77622e6	−0.242700
$754$	0	0
$755$	1.66541e7	1.06329
$756$	0	0
$757$	−1.59335e7	−1.01058	−0.505290	−	0.862950i	$-0.668614\pi$
$757$	−1.59335e7	−1.01058	−0.505290	+	0.862950i	$0.668614\pi$
$758$	0	0
$759$	138240.	0.00871022
$760$	0	0
$761$	1.68629e7	1.05553	0.527764	−	0.849391i	$-0.323031\pi$
$761$	1.68629e7	1.05553	0.527764	+	0.849391i	$0.323031\pi$
$762$	0	0
$763$	1.08073e7	0.672059
$764$	0	0
$765$	−2.87718e7	−1.77751
$766$	0	0
$767$	710424.	0.0436043
$768$	0	0
$769$	−2.75402e7	−1.67939	−0.839694	−	0.543060i	$-0.817265\pi$
$769$	−2.75402e7	−1.67939	−0.839694	+	0.543060i	$0.817265\pi$
$770$	0	0
$771$	692388.	0.0419482
$772$	0	0
$773$	−2.26820e7	−1.36532	−0.682658	−	0.730738i	$-0.739176\pi$
$773$	−2.26820e7	−1.36532	−0.682658	+	0.730738i	$0.739176\pi$
$774$	0	0
$775$	−3.80322e7	−2.27456
$776$	0	0
$777$	1.05820e6	0.0628806
$778$	0	0
$779$	1.12856e6	0.0666320
$780$	0	0
$781$	−2.70950e7	−1.58951
$782$	0	0
$783$	−4.21654e6	−0.245783
$784$	0	0
$785$	5.23507e6	0.303213
$786$	0	0
$787$	2.34266e7	1.34826	0.674129	−	0.738614i	$-0.264519\pi$
$787$	2.34266e7	1.34826	0.674129	+	0.738614i	$0.264519\pi$
$788$	0	0
$789$	−1.85818e6	−0.106266
$790$	0	0
$791$	8.36401e6	0.475306
$792$	0	0
$793$	4.34262e6	0.245228
$794$	0	0
$795$	−194688.	−0.0109250
$796$	0	0
$797$	1.82051e7	1.01519	0.507594	−	0.861596i	$-0.330535\pi$
$797$	1.82051e7	1.01519	0.507594	+	0.861596i	$0.330535\pi$
$798$	0	0
$799$	−3.23983e7	−1.79538
$800$	0	0
$801$	1.07851e7	0.593941
$802$	0	0
$803$	−9.67536e6	−0.529515
$804$	0	0
$805$	451584.	0.0245612
$806$	0	0
$807$	−764136.	−0.0413035
$808$	0	0
$809$	1.47411e7	0.791878	0.395939	−	0.918277i	$-0.370419\pi$
$809$	1.47411e7	0.791878	0.395939	+	0.918277i	$0.370419\pi$
$810$	0	0
$811$	−1.69629e7	−0.905625	−0.452812	−	0.891606i	$-0.649579\pi$
$811$	−1.69629e7	−0.905625	−0.452812	+	0.891606i	$0.649579\pi$
$812$	0	0
$813$	−3.16112e6	−0.167731
$814$	0	0
$815$	−1.00762e7	−0.531375
$816$	0	0
$817$	861040.	0.0451303
$818$	0	0
$819$	6.69869e6	0.348964
$820$	0	0
$821$	−8.03929e6	−0.416255	−0.208128	−	0.978102i	$-0.566737\pi$
$821$	−8.03929e6	−0.416255	−0.208128	+	0.978102i	$0.566737\pi$
$822$	0	0
$823$	386648.	0.0198983	0.00994915	−	0.999951i	$-0.496833\pi$
$823$	386648.	0.0198983	0.00994915	+	0.999951i	$0.496833\pi$
$824$	0	0
$825$	8.77104e6	0.448659
$826$	0	0
$827$	−3.55021e7	−1.80505	−0.902526	−	0.430635i	$-0.858290\pi$
$827$	−3.55021e7	−1.80505	−0.902526	+	0.430635i	$0.858290\pi$
$828$	0	0
$829$	−2.48814e7	−1.25745	−0.628723	−	0.777630i	$-0.716422\pi$
$829$	−2.48814e7	−1.25745	−0.628723	+	0.777630i	$0.716422\pi$
$830$	0	0
$831$	2.73823e6	0.137552
$832$	0	0
$833$	3.01085e6	0.150341
$834$	0	0
$835$	1.54356e7	0.766141
$836$	0	0
$837$	6.01922e6	0.296979
$838$	0	0
$839$	3.41458e7	1.67468	0.837340	−	0.546682i	$-0.184109\pi$
$839$	3.41458e7	1.67468	0.837340	+	0.546682i	$0.184109\pi$
$840$	0	0
$841$	−1.37927e6	−0.0672450
$842$	0	0
$843$	−789708.	−0.0382734
$844$	0	0
$845$	−4.23446e6	−0.204012
$846$	0	0
$847$	1.75101e7	0.838649
$848$	0	0
$849$	−1.34607e6	−0.0640911
$850$	0	0
$851$	1.03661e6	0.0490671
$852$	0	0
$853$	−2.50701e7	−1.17973	−0.589865	−	0.807502i	$-0.700819\pi$
$853$	−2.50701e7	−1.17973	−0.589865	+	0.807502i	$0.700819\pi$
$854$	0	0
$855$	−2.15674e6	−0.100898
$856$	0	0
$857$	−1.81938e7	−0.846199	−0.423099	−	0.906083i	$-0.639058\pi$
$857$	−1.81938e7	−0.846199	−0.423099	+	0.906083i	$0.639058\pi$
$858$	0	0
$859$	−6.91797e6	−0.319886	−0.159943	−	0.987126i	$-0.551131\pi$
$859$	−6.91797e6	−0.319886	−0.159943	+	0.987126i	$0.551131\pi$
$860$	0	0
$861$	1.17659e6	0.0540899
$862$	0	0
$863$	−2.78069e7	−1.27094	−0.635471	−	0.772125i	$-0.719194\pi$
$863$	−2.78069e7	−1.27094	−0.635471	+	0.772125i	$0.719194\pi$
$864$	0	0
$865$	−3.46141e7	−1.57294
$866$	0	0
$867$	305318.	0.0137945
$868$	0	0
$869$	4.49856e6	0.202080
$870$	0	0
$871$	2.35275e7	1.05083
$872$	0	0
$873$	−2.56261e7	−1.13801
$874$	0	0
$875$	1.39521e7	0.616053
$876$	0	0
$877$	−3.79587e6	−0.166653	−0.0833263	−	0.996522i	$-0.526554\pi$
$877$	−3.79587e6	−0.166653	−0.0833263	+	0.996522i	$0.526554\pi$
$878$	0	0
$879$	−3.66936e6	−0.160184
$880$	0	0
$881$	2.48904e7	1.08042	0.540210	−	0.841530i	$-0.318345\pi$
$881$	2.48904e7	1.08042	0.540210	+	0.841530i	$0.318345\pi$
$882$	0	0
$883$	3.13568e6	0.135341	0.0676705	−	0.997708i	$-0.478443\pi$
$883$	3.13568e6	0.135341	0.0676705	+	0.997708i	$0.478443\pi$
$884$	0	0
$885$	−238464.	−0.0102345
$886$	0	0
$887$	−2.02437e7	−0.863933	−0.431966	−	0.901890i	$-0.642180\pi$
$887$	−2.02437e7	−0.863933	−0.431966	+	0.901890i	$0.642180\pi$
$888$	0	0
$889$	−1.22406e7	−0.519455
$890$	0	0
$891$	4.04273e7	1.70600
$892$	0	0
$893$	−2.42858e6	−0.101912
$894$	0	0
$895$	3.00223e7	1.25281
$896$	0	0
$897$	−109824.	−0.00455739
$898$	0	0
$899$	−2.73113e7	−1.12705
$900$	0	0
$901$	−1.27156e6	−0.0521823
$902$	0	0
$903$	897680.	0.0366355
$904$	0	0
$905$	1.18867e7	0.482437
$906$	0	0
$907$	6.86324e6	0.277020	0.138510	−	0.990361i	$-0.455769\pi$
$907$	6.86324e6	0.277020	0.138510	+	0.990361i	$0.455769\pi$
$908$	0	0
$909$	1.12655e7	0.452211
$910$	0	0
$911$	−9.40661e6	−0.375523	−0.187762	−	0.982215i	$-0.560123\pi$
$911$	−9.40661e6	−0.375523	−0.187762	+	0.982215i	$0.560123\pi$
$912$	0	0
$913$	1.81829e7	0.721914
$914$	0	0
$915$	−1.45766e6	−0.0575579
$916$	0	0
$917$	−598290.	−0.0234957
$918$	0	0
$919$	−2.10280e7	−0.821313	−0.410656	−	0.911790i	$-0.634700\pi$
$919$	−2.10280e7	−0.821313	−0.410656	+	0.911790i	$0.634700\pi$
$920$	0	0
$921$	3.02112e6	0.117360
$922$	0	0
$923$	2.15255e7	0.831666
$924$	0	0
$925$	6.57706e7	2.52742
$926$	0	0
$927$	−2.92068e7	−1.11631
$928$	0	0
$929$	−4.30928e7	−1.63819	−0.819096	−	0.573656i	$-0.805525\pi$
$929$	−4.30928e7	−1.63819	−0.819096	+	0.573656i	$0.805525\pi$
$930$	0	0
$931$	225694.	0.00853387
$932$	0	0
$933$	−3.75058e6	−0.141057
$934$	0	0
$935$	8.66765e7	3.24244
$936$	0	0
$937$	−3.85862e6	−0.143576	−0.0717882	−	0.997420i	$-0.522871\pi$
$937$	−3.85862e6	−0.143576	−0.0717882	+	0.997420i	$0.522871\pi$
$938$	0	0
$939$	−3.02152e6	−0.111831
$940$	0	0
$941$	−1.59601e7	−0.587572	−0.293786	−	0.955871i	$-0.594915\pi$
$941$	−1.59601e7	−0.587572	−0.293786	+	0.955871i	$0.594915\pi$
$942$	0	0
$943$	1.15258e6	0.0422076
$944$	0	0
$945$	−4.53466e6	−0.165183
$946$	0	0
$947$	4.66831e6	0.169155	0.0845775	−	0.996417i	$-0.473046\pi$
$947$	4.66831e6	0.169155	0.0845775	+	0.996417i	$0.473046\pi$
$948$	0	0
$949$	7.68654e6	0.277054
$950$	0	0
$951$	−4.05419e6	−0.145363
$952$	0	0
$953$	−3.43457e7	−1.22501	−0.612505	−	0.790466i	$-0.709838\pi$
$953$	−3.43457e7	−1.22501	−0.612505	+	0.790466i	$0.709838\pi$
$954$	0	0
$955$	3.10510e7	1.10171
$956$	0	0
$957$	6.29856e6	0.222311
$958$	0	0
$959$	−681198.	−0.0239181
$960$	0	0
$961$	1.03584e7	0.361813
$962$	0	0
$963$	−3.09830e7	−1.07661
$964$	0	0
$965$	−5.95156e7	−2.05737
$966$	0	0
$967$	−2.28181e7	−0.784718	−0.392359	−	0.919812i	$-0.628341\pi$
$967$	−2.28181e7	−0.784718	−0.392359	+	0.919812i	$0.628341\pi$
$968$	0	0
$969$	235752.	0.00806577
$970$	0	0
$971$	−4.94042e7	−1.68157	−0.840786	−	0.541367i	$-0.817907\pi$
$971$	−4.94042e7	−1.68157	−0.840786	+	0.541367i	$0.817907\pi$
$972$	0	0
$973$	2.11579e7	0.716458
$974$	0	0
$975$	−6.96810e6	−0.234749
$976$	0	0
$977$	7.17542e6	0.240498	0.120249	−	0.992744i	$-0.461631\pi$
$977$	7.17542e6	0.240498	0.120249	+	0.992744i	$0.461631\pi$
$978$	0	0
$979$	−3.24907e7	−1.08343
$980$	0	0
$981$	−5.27134e7	−1.74883
$982$	0	0
$983$	−4.22279e6	−0.139385	−0.0696924	−	0.997569i	$-0.522202\pi$
$983$	−4.22279e6	−0.139385	−0.0696924	+	0.997569i	$0.522202\pi$
$984$	0	0
$985$	4.79388e7	1.57433
$986$	0	0
$987$	−2.53193e6	−0.0827291
$988$	0	0
$989$	879360.	0.0285875
$990$	0	0
$991$	1.65645e7	0.535789	0.267895	−	0.963448i	$-0.413672\pi$
$991$	1.65645e7	0.535789	0.267895	+	0.963448i	$0.413672\pi$
$992$	0	0
$993$	−3.08018e6	−0.0991294
$994$	0	0
$995$	−7.54495e7	−2.41601
$996$	0	0
$997$	4.40973e7	1.40499	0.702496	−	0.711687i	$-0.252069\pi$
$997$	4.40973e7	1.40499	0.702496	+	0.711687i	$0.252069\pi$
$998$	0	0
$999$	−1.04093e7	−0.329994

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.i.1.1		1
4.3	odd	2		448.6.a.h.1.1		1
8.3	odd	2		112.6.a.e.1.1		1
8.5	even	2		28.6.a.a.1.1	✓	1
24.5	odd	2		252.6.a.d.1.1		1
24.11	even	2		1008.6.a.bb.1.1		1
40.13	odd	4		700.6.e.d.449.1		2
40.29	even	2		700.6.a.d.1.1		1
40.37	odd	4		700.6.e.d.449.2		2
56.5	odd	6		196.6.e.e.165.1		2
56.13	odd	2		196.6.a.d.1.1		1
56.27	even	2		784.6.a.f.1.1		1
56.37	even	6		196.6.e.f.165.1		2
56.45	odd	6		196.6.e.e.177.1		2
56.53	even	6		196.6.e.f.177.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.a.a.1.1	✓	1	8.5	even	2
112.6.a.e.1.1		1	8.3	odd	2
196.6.a.d.1.1		1	56.13	odd	2
196.6.e.e.165.1		2	56.5	odd	6
196.6.e.e.177.1		2	56.45	odd	6
196.6.e.f.165.1		2	56.37	even	6
196.6.e.f.177.1		2	56.53	even	6
252.6.a.d.1.1		1	24.5	odd	2
448.6.a.h.1.1		1	4.3	odd	2
448.6.a.i.1.1		1	1.1	even	1	trivial
700.6.a.d.1.1		1	40.29	even	2
700.6.e.d.449.1		2	40.13	odd	4
700.6.e.d.449.2		2	40.37	odd	4
784.6.a.f.1.1		1	56.27	even	2
1008.6.a.bb.1.1		1	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more