LMFDB - Embedded newform 207.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.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:	$33.1994507013$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 39x^{2} - 30x + 20 $ x^4 - 39x^2 - 30x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.56547$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.56547 q^{2} +25.2363 q^{4} -40.1532 q^{5} +194.921 q^{7} +51.1704 q^{8} +O(q^{10})$	\(q-7.56547 q^{2} +25.2363 q^{4} -40.1532 q^{5} +194.921 q^{7} +51.1704 q^{8} +303.778 q^{10} +39.1915 q^{11} +705.021 q^{13} -1474.67 q^{14} -1194.69 q^{16} -1222.51 q^{17} +1891.46 q^{19} -1013.32 q^{20} -296.502 q^{22} -529.000 q^{23} -1512.72 q^{25} -5333.81 q^{26} +4919.10 q^{28} +8579.39 q^{29} -9582.72 q^{31} +7400.94 q^{32} +9248.86 q^{34} -7826.73 q^{35} -8144.49 q^{37} -14309.8 q^{38} -2054.66 q^{40} -4834.24 q^{41} +10631.5 q^{43} +989.049 q^{44} +4002.13 q^{46} -222.835 q^{47} +21187.4 q^{49} +11444.4 q^{50} +17792.1 q^{52} -6577.44 q^{53} -1573.66 q^{55} +9974.21 q^{56} -64907.1 q^{58} +30780.8 q^{59} -8695.20 q^{61} +72497.8 q^{62} -17761.5 q^{64} -28308.9 q^{65} +48398.0 q^{67} -30851.6 q^{68} +59212.8 q^{70} +60285.7 q^{71} +30638.2 q^{73} +61616.8 q^{74} +47733.4 q^{76} +7639.26 q^{77} +50285.3 q^{79} +47970.7 q^{80} +36573.3 q^{82} +35671.7 q^{83} +49087.7 q^{85} -80432.0 q^{86} +2005.44 q^{88} +77619.2 q^{89} +137424. q^{91} -13350.0 q^{92} +1685.85 q^{94} -75948.1 q^{95} -83069.4 q^{97} -160292. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8	$ 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8} + 642 q^{10} - 32 q^{11} + 1364 q^{13} - 2754 q^{14} + 18 q^{16} - 278 q^{17} + 2862 q^{19} - 3830 q^{20} + 3176 q^{22} - 2116 q^{23} + 5944 q^{25} - 6996 q^{26} + 4738 q^{28} + 5180 q^{29} - 1788 q^{31} + 7352 q^{32} + 15818 q^{34} - 11768 q^{35} + 3348 q^{37} - 1050 q^{38} - 13462 q^{40} + 17664 q^{41} + 25398 q^{43} + 16848 q^{44} + 2116 q^{46} + 26040 q^{47} + 55720 q^{49} + 35256 q^{50} - 2752 q^{52} + 32006 q^{53} + 34904 q^{55} + 68542 q^{56} - 40804 q^{58} + 61136 q^{59} + 35844 q^{61} + 47524 q^{62} - 35142 q^{64} + 48036 q^{65} + 73458 q^{67} - 17910 q^{68} - 59104 q^{70} - 24432 q^{71} + 122512 q^{73} + 20828 q^{74} + 56834 q^{76} - 159496 q^{77} + 90170 q^{79} + 36546 q^{80} + 84144 q^{82} - 28592 q^{83} + 355124 q^{85} - 103778 q^{86} - 150776 q^{88} + 27926 q^{89} + 334180 q^{91} + 24334 q^{92} + 113632 q^{94} - 113392 q^{95} + 16580 q^{97} - 49688 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8 + 642 * q^10 - 32 * q^11 + 1364 * q^13 - 2754 * q^14 + 18 * q^16 - 278 * q^17 + 2862 * q^19 - 3830 * q^20 + 3176 * q^22 - 2116 * q^23 + 5944 * q^25 - 6996 * q^26 + 4738 * q^28 + 5180 * q^29 - 1788 * q^31 + 7352 * q^32 + 15818 * q^34 - 11768 * q^35 + 3348 * q^37 - 1050 * q^38 - 13462 * q^40 + 17664 * q^41 + 25398 * q^43 + 16848 * q^44 + 2116 * q^46 + 26040 * q^47 + 55720 * q^49 + 35256 * q^50 - 2752 * q^52 + 32006 * q^53 + 34904 * q^55 + 68542 * q^56 - 40804 * q^58 + 61136 * q^59 + 35844 * q^61 + 47524 * q^62 - 35142 * q^64 + 48036 * q^65 + 73458 * q^67 - 17910 * q^68 - 59104 * q^70 - 24432 * q^71 + 122512 * q^73 + 20828 * q^74 + 56834 * q^76 - 159496 * q^77 + 90170 * q^79 + 36546 * q^80 + 84144 * q^82 - 28592 * q^83 + 355124 * q^85 - 103778 * q^86 - 150776 * q^88 + 27926 * q^89 + 334180 * q^91 + 24334 * q^92 + 113632 * q^94 - 113392 * q^95 + 16580 * q^97 - 49688 * q^98

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 $

$2$	−7.56547	−1.33740	−0.668699	−	0.743533i	$-0.733149\pi$
$2$	−7.56547	−1.33740	−0.668699	+	0.743533i	$0.733149\pi$
$3$	0	0
$3$	0	0
$4$	25.2363	0.788635
$5$	−40.1532	−0.718283	−0.359141	−	0.933283i	$-0.616930\pi$
$5$	−40.1532	−0.718283	−0.359141	+	0.933283i	$0.616930\pi$
$6$	0	0
$7$	194.921	1.50354	0.751769	−	0.659426i	$-0.229201\pi$
$7$	194.921	1.50354	0.751769	+	0.659426i	$0.229201\pi$
$8$	51.1704	0.282679
$9$	0	0
$10$	303.778	0.960630
$11$	39.1915	0.0976585	0.0488292	−	0.998807i	$-0.484451\pi$
$11$	39.1915	0.0976585	0.0488292	+	0.998807i	$0.484451\pi$
$12$	0	0
$13$	705.021	1.15703	0.578514	−	0.815673i	$-0.303633\pi$
$13$	705.021	1.15703	0.578514	+	0.815673i	$0.303633\pi$
$14$	−1474.67	−2.01083
$15$	0	0
$16$	−1194.69	−1.16669
$17$	−1222.51	−1.02596	−0.512979	−	0.858401i	$-0.671458\pi$
$17$	−1222.51	−1.02596	−0.512979	+	0.858401i	$0.671458\pi$
$18$	0	0
$19$	1891.46	1.20202	0.601011	−	0.799241i	$-0.294765\pi$
$19$	1891.46	1.20202	0.601011	+	0.799241i	$0.294765\pi$
$20$	−1013.32	−0.566463
$21$	0	0
$22$	−296.502	−0.130608
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−1512.72	−0.484070
$26$	−5333.81	−1.54741
$27$	0	0
$28$	4919.10	1.18574
$29$	8579.39	1.89436	0.947178	−	0.320709i	$-0.103921\pi$
$29$	8579.39	1.89436	0.947178	+	0.320709i	$0.103921\pi$
$30$	0	0
$31$	−9582.72	−1.79095	−0.895477	−	0.445107i	$-0.853165\pi$
$31$	−9582.72	−1.79095	−0.895477	+	0.445107i	$0.853165\pi$
$32$	7400.94	1.27765
$33$	0	0
$34$	9248.86	1.37212
$35$	−7826.73	−1.07997
$36$	0	0
$37$	−8144.49	−0.978046	−0.489023	−	0.872271i	$-0.662647\pi$
$37$	−8144.49	−0.978046	−0.489023	+	0.872271i	$0.662647\pi$
$38$	−14309.8	−1.60758
$39$	0	0
$40$	−2054.66	−0.203044
$41$	−4834.24	−0.449127	−0.224563	−	0.974459i	$-0.572096\pi$
$41$	−4834.24	−0.449127	−0.224563	+	0.974459i	$0.572096\pi$
$42$	0	0
$43$	10631.5	0.876843	0.438422	−	0.898769i	$-0.355538\pi$
$43$	10631.5	0.876843	0.438422	+	0.898769i	$0.355538\pi$
$44$	989.049	0.0770169
$45$	0	0
$46$	4002.13	0.278867
$47$	−222.835	−0.0147142	−0.00735712	−	0.999973i	$-0.502342\pi$
$47$	−222.835	−0.0147142	−0.00735712	+	0.999973i	$0.502342\pi$
$48$	0	0
$49$	21187.4	1.26063
$50$	11444.4	0.647394
$51$	0	0
$52$	17792.1	0.912472
$53$	−6577.44	−0.321638	−0.160819	−	0.986984i	$-0.551414\pi$
$53$	−6577.44	−0.321638	−0.160819	+	0.986984i	$0.551414\pi$
$54$	0	0
$55$	−1573.66	−0.0701464
$56$	9974.21	0.425019
$57$	0	0
$58$	−64907.1	−2.53351
$59$	30780.8	1.15120	0.575600	−	0.817732i	$-0.304769\pi$
$59$	30780.8	1.15120	0.575600	+	0.817732i	$0.304769\pi$
$60$	0	0
$61$	−8695.20	−0.299196	−0.149598	−	0.988747i	$-0.547798\pi$
$61$	−8695.20	−0.299196	−0.149598	+	0.988747i	$0.547798\pi$
$62$	72497.8	2.39522
$63$	0	0
$64$	−17761.5	−0.542038
$65$	−28308.9	−0.831073
$66$	0	0
$67$	48398.0	1.31716	0.658582	−	0.752509i	$-0.271156\pi$
$67$	48398.0	1.31716	0.658582	+	0.752509i	$0.271156\pi$
$68$	−30851.6	−0.809107
$69$	0	0
$70$	59212.8	1.44434
$71$	60285.7	1.41928	0.709640	−	0.704564i	$-0.248858\pi$
$71$	60285.7	1.41928	0.709640	+	0.704564i	$0.248858\pi$
$72$	0	0
$73$	30638.2	0.672908	0.336454	−	0.941700i	$-0.390772\pi$
$73$	30638.2	0.672908	0.336454	+	0.941700i	$0.390772\pi$
$74$	61616.8	1.30804
$75$	0	0
$76$	47733.4	0.947957
$77$	7639.26	0.146833
$78$	0	0
$79$	50285.3	0.906512	0.453256	−	0.891380i	$-0.350262\pi$
$79$	50285.3	0.906512	0.453256	+	0.891380i	$0.350262\pi$
$80$	47970.7	0.838013
$81$	0	0
$82$	36573.3	0.600661
$83$	35671.7	0.568366	0.284183	−	0.958770i	$-0.408278\pi$
$83$	35671.7	0.568366	0.284183	+	0.958770i	$0.408278\pi$
$84$	0	0
$85$	49087.7	0.736929
$86$	−80432.0	−1.17269
$87$	0	0
$88$	2005.44	0.0276060
$89$	77619.2	1.03871	0.519355	−	0.854559i	$-0.326172\pi$
$89$	77619.2	1.03871	0.519355	+	0.854559i	$0.326172\pi$
$90$	0	0
$91$	137424.	1.73963
$92$	−13350.0	−0.164442
$93$	0	0
$94$	1685.85	0.0196788
$95$	−75948.1	−0.863392
$96$	0	0
$97$	−83069.4	−0.896421	−0.448210	−	0.893928i	$-0.647938\pi$
$97$	−83069.4	−0.896421	−0.448210	+	0.893928i	$0.647938\pi$
$98$	−160292.	−1.68596
$99$	0	0
$100$	−38175.4	−0.381754
$101$	−30834.9	−0.300773	−0.150387	−	0.988627i	$-0.548052\pi$
$101$	−30834.9	−0.300773	−0.150387	+	0.988627i	$0.548052\pi$
$102$	0	0
$103$	141687.	1.31594	0.657970	−	0.753044i	$-0.271415\pi$
$103$	141687.	1.31594	0.657970	+	0.753044i	$0.271415\pi$
$104$	36076.2	0.327068
$105$	0	0
$106$	49761.4	0.430158
$107$	−132757.	−1.12098	−0.560488	−	0.828162i	$-0.689386\pi$
$107$	−132757.	−1.12098	−0.560488	+	0.828162i	$0.689386\pi$
$108$	0	0
$109$	−113030.	−0.911232	−0.455616	−	0.890176i	$-0.650581\pi$
$109$	−113030.	−0.911232	−0.455616	+	0.890176i	$0.650581\pi$
$110$	11905.5	0.0938137
$111$	0	0
$112$	−232871.	−1.75416
$113$	−46155.5	−0.340038	−0.170019	−	0.985441i	$-0.554383\pi$
$113$	−46155.5	−0.340038	−0.170019	+	0.985441i	$0.554383\pi$
$114$	0	0
$115$	21241.1	0.149772
$116$	216512.	1.49396
$117$	0	0
$118$	−232872.	−1.53961
$119$	−238293.	−1.54257
$120$	0	0
$121$	−159515.	−0.990463
$122$	65783.3	0.400144
$123$	0	0
$124$	−241833.	−1.41241
$125$	186219.	1.06598
$126$	0	0
$127$	−223792.	−1.23122	−0.615609	−	0.788051i	$-0.711090\pi$
$127$	−223792.	−1.23122	−0.615609	+	0.788051i	$0.711090\pi$
$128$	−102456.	−0.552730
$129$	0	0
$130$	214170.	1.11148
$131$	249499.	1.27025	0.635127	−	0.772407i	$-0.280948\pi$
$131$	249499.	1.27025	0.635127	+	0.772407i	$0.280948\pi$
$132$	0	0
$133$	368685.	1.80729
$134$	−366153.	−1.76157
$135$	0	0
$136$	−62556.3	−0.290017
$137$	25829.5	0.117575	0.0587875	−	0.998271i	$-0.481277\pi$
$137$	25829.5	0.117575	0.0587875	+	0.998271i	$0.481277\pi$
$138$	0	0
$139$	79301.9	0.348134	0.174067	−	0.984734i	$-0.444309\pi$
$139$	79301.9	0.348134	0.174067	+	0.984734i	$0.444309\pi$
$140$	−197518.	−0.851699
$141$	0	0
$142$	−456089.	−1.89814
$143$	27630.8	0.112994
$144$	0	0
$145$	−344490.	−1.36068
$146$	−231792.	−0.899947
$147$	0	0
$148$	−205537.	−0.771321
$149$	338110.	1.24765	0.623825	−	0.781564i	$-0.285578\pi$
$149$	338110.	1.24765	0.623825	+	0.781564i	$0.285578\pi$
$150$	0	0
$151$	494788.	1.76594	0.882971	−	0.469427i	$-0.155539\pi$
$151$	494788.	1.76594	0.882971	+	0.469427i	$0.155539\pi$
$152$	96786.6	0.339787
$153$	0	0
$154$	−57794.6	−0.196375
$155$	384777.	1.28641
$156$	0	0
$157$	195963.	0.634490	0.317245	−	0.948344i	$-0.397242\pi$
$157$	195963.	0.634490	0.317245	+	0.948344i	$0.397242\pi$
$158$	−380432.	−1.21237
$159$	0	0
$160$	−297172.	−0.917714
$161$	−103113.	−0.313509
$162$	0	0
$163$	223223.	0.658067	0.329034	−	0.944318i	$-0.393277\pi$
$163$	223223.	0.658067	0.329034	+	0.944318i	$0.393277\pi$
$164$	−121999.	−0.354197
$165$	0	0
$166$	−269873.	−0.760132
$167$	232525.	0.645176	0.322588	−	0.946539i	$-0.395447\pi$
$167$	232525.	0.645176	0.322588	+	0.946539i	$0.395447\pi$
$168$	0	0
$169$	125762.	0.338713
$170$	−371372.	−0.985567
$171$	0	0
$172$	268299.	0.691509
$173$	−129688.	−0.329445	−0.164723	−	0.986340i	$-0.552673\pi$
$173$	−129688.	−0.329445	−0.164723	+	0.986340i	$0.552673\pi$
$174$	0	0
$175$	−294861.	−0.727817
$176$	−46821.7	−0.113937
$177$	0	0
$178$	−587225.	−1.38917
$179$	−722275.	−1.68489	−0.842443	−	0.538786i	$-0.818883\pi$
$179$	−722275.	−1.68489	−0.842443	+	0.538786i	$0.818883\pi$
$180$	0	0
$181$	−126793.	−0.287672	−0.143836	−	0.989602i	$-0.545944\pi$
$181$	−126793.	−0.287672	−0.143836	+	0.989602i	$0.545944\pi$
$182$	−1.03967e6	−2.32659
$183$	0	0
$184$	−27069.1	−0.0589427
$185$	327027.	0.702514
$186$	0	0
$187$	−47912.0	−0.100194
$188$	−5623.53	−0.0116042
$189$	0	0
$190$	574583.	1.15470
$191$	582441.	1.15523	0.577615	−	0.816310i	$-0.303984\pi$
$191$	582441.	1.15523	0.577615	+	0.816310i	$0.303984\pi$
$192$	0	0
$193$	−441777.	−0.853709	−0.426855	−	0.904320i	$-0.640378\pi$
$193$	−441777.	−0.853709	−0.426855	+	0.904320i	$0.640378\pi$
$194$	628459.	1.19887
$195$	0	0
$196$	534691.	0.994175
$197$	149838.	0.275078	0.137539	−	0.990496i	$-0.456081\pi$
$197$	149838.	0.275078	0.137539	+	0.990496i	$0.456081\pi$
$198$	0	0
$199$	−48644.4	−0.0870764	−0.0435382	−	0.999052i	$-0.513863\pi$
$199$	−48644.4	−0.0870764	−0.0435382	+	0.999052i	$0.513863\pi$
$200$	−77406.4	−0.136836
$201$	0	0
$202$	233280.	0.402254
$203$	1.67231e6	2.84824
$204$	0	0
$205$	194111.	0.322600
$206$	−1.07193e6	−1.75994
$207$	0	0
$208$	−842282.	−1.34989
$209$	74129.0	0.117388
$210$	0	0
$211$	921315.	1.42463	0.712315	−	0.701860i	$-0.247647\pi$
$211$	921315.	1.42463	0.712315	+	0.701860i	$0.247647\pi$
$212$	−165990.	−0.253655
$213$	0	0
$214$	1.00437e6	1.49919
$215$	−426888.	−0.629822
$216$	0	0
$217$	−1.86788e6	−2.69277
$218$	855128.	1.21868
$219$	0	0
$220$	−39713.5	−0.0553199
$221$	−861895.	−1.18706
$222$	0	0
$223$	998446.	1.34450	0.672252	−	0.740322i	$-0.265327\pi$
$223$	998446.	1.34450	0.672252	+	0.740322i	$0.265327\pi$
$224$	1.44260e6	1.92100
$225$	0	0
$226$	349188.	0.454766
$227$	376703.	0.485215	0.242608	−	0.970125i	$-0.421997\pi$
$227$	376703.	0.485215	0.242608	+	0.970125i	$0.421997\pi$
$228$	0	0
$229$	814218.	1.02601	0.513005	−	0.858385i	$-0.328532\pi$
$229$	814218.	1.02601	0.513005	+	0.858385i	$0.328532\pi$
$230$	−160699.	−0.200305
$231$	0	0
$232$	439011.	0.535495
$233$	−255689.	−0.308547	−0.154274	−	0.988028i	$-0.549304\pi$
$233$	−255689.	−0.308547	−0.154274	+	0.988028i	$0.549304\pi$
$234$	0	0
$235$	8947.53	0.0105690
$236$	776795.	0.907876
$237$	0	0
$238$	1.80280e6	2.06303
$239$	1.27483e6	1.44364	0.721818	−	0.692083i	$-0.243307\pi$
$239$	1.27483e6	1.44364	0.721818	+	0.692083i	$0.243307\pi$
$240$	0	0
$241$	−1.47036e6	−1.63073	−0.815365	−	0.578948i	$-0.803464\pi$
$241$	−1.47036e6	−1.63073	−0.815365	+	0.578948i	$0.803464\pi$
$242$	1.20681e6	1.32464
$243$	0	0
$244$	−219435.	−0.235956
$245$	−850741.	−0.905487
$246$	0	0
$247$	1.33352e6	1.39077
$248$	−490352.	−0.506266
$249$	0	0
$250$	−1.40884e6	−1.42564
$251$	−1.25961e6	−1.26197	−0.630987	−	0.775793i	$-0.717350\pi$
$251$	−1.25961e6	−1.26197	−0.630987	+	0.775793i	$0.717350\pi$
$252$	0	0
$253$	−20732.3	−0.0203632
$254$	1.69309e6	1.64663
$255$	0	0
$256$	1.34350e6	1.28126
$257$	−400710.	−0.378440	−0.189220	−	0.981935i	$-0.560596\pi$
$257$	−400710.	−0.378440	−0.189220	+	0.981935i	$0.560596\pi$
$258$	0	0
$259$	−1.58753e6	−1.47053
$260$	−714412.	−0.655413
$261$	0	0
$262$	−1.88758e6	−1.69884
$263$	−1.10828e6	−0.988009	−0.494004	−	0.869459i	$-0.664467\pi$
$263$	−1.10828e6	−0.988009	−0.494004	+	0.869459i	$0.664467\pi$
$264$	0	0
$265$	264105.	0.231027
$266$	−2.78928e6	−2.41706
$267$	0	0
$268$	1.22139e6	1.03876
$269$	1.22073e6	1.02858	0.514289	−	0.857617i	$-0.328056\pi$
$269$	1.22073e6	1.02858	0.514289	+	0.857617i	$0.328056\pi$
$270$	0	0
$271$	−606547.	−0.501697	−0.250848	−	0.968026i	$-0.580710\pi$
$271$	−606547.	−0.501697	−0.250848	+	0.968026i	$0.580710\pi$
$272$	1.46052e6	1.19698
$273$	0	0
$274$	−195413.	−0.157245
$275$	−59285.7	−0.0472735
$276$	0	0
$277$	1.73194e6	1.35623	0.678116	−	0.734955i	$-0.262797\pi$
$277$	1.73194e6	1.35623	0.678116	+	0.734955i	$0.262797\pi$
$278$	−599956.	−0.465594
$279$	0	0
$280$	−400497.	−0.305284
$281$	1.03139e6	0.779212	0.389606	−	0.920982i	$-0.372611\pi$
$281$	1.03139e6	0.779212	0.389606	+	0.920982i	$0.372611\pi$
$282$	0	0
$283$	1.83231e6	1.35998	0.679990	−	0.733221i	$-0.261984\pi$
$283$	1.83231e6	1.35998	0.679990	+	0.733221i	$0.261984\pi$
$284$	1.52139e6	1.11929
$285$	0	0
$286$	−209040.	−0.151117
$287$	−942298.	−0.675279
$288$	0	0
$289$	74672.5	0.0525915
$290$	2.60623e6	1.81978
$291$	0	0
$292$	773195.	0.530679
$293$	277123.	0.188584	0.0942919	−	0.995545i	$-0.469941\pi$
$293$	277123.	0.188584	0.0942919	+	0.995545i	$0.469941\pi$
$294$	0	0
$295$	−1.23595e6	−0.826887
$296$	−416757.	−0.276473
$297$	0	0
$298$	−2.55796e6	−1.66860
$299$	−372956.	−0.241257
$300$	0	0
$301$	2.07230e6	1.31837
$302$	−3.74330e6	−2.36177
$303$	0	0
$304$	−2.25971e6	−1.40239
$305$	349141.	0.214907
$306$	0	0
$307$	1.98065e6	1.19939	0.599696	−	0.800228i	$-0.295288\pi$
$307$	1.98065e6	1.19939	0.599696	+	0.800228i	$0.295288\pi$
$308$	192787.	0.115798
$309$	0	0
$310$	−2.91102e6	−1.72045
$311$	−2.56169e6	−1.50185	−0.750925	−	0.660388i	$-0.770392\pi$
$311$	−2.56169e6	−1.50185	−0.750925	+	0.660388i	$0.770392\pi$
$312$	0	0
$313$	2.46321e6	1.42115	0.710576	−	0.703621i	$-0.248435\pi$
$313$	2.46321e6	1.42115	0.710576	+	0.703621i	$0.248435\pi$
$314$	−1.48255e6	−0.848566
$315$	0	0
$316$	1.26902e6	0.714907
$317$	3.04669e6	1.70287	0.851433	−	0.524464i	$-0.175734\pi$
$317$	3.04669e6	1.70287	0.851433	+	0.524464i	$0.175734\pi$
$318$	0	0
$319$	336239.	0.185000
$320$	713181.	0.389336
$321$	0	0
$322$	780101.	0.419287
$323$	−2.31232e6	−1.23323
$324$	0	0
$325$	−1.06650e6	−0.560082
$326$	−1.68879e6	−0.880098
$327$	0	0
$328$	−247370.	−0.126959
$329$	−43435.2	−0.0221234
$330$	0	0
$331$	−2.59437e6	−1.30155	−0.650776	−	0.759270i	$-0.725556\pi$
$331$	−2.59437e6	−1.30155	−0.650776	+	0.759270i	$0.725556\pi$
$332$	900222.	0.448234
$333$	0	0
$334$	−1.75916e6	−0.862857
$335$	−1.94333e6	−0.946097
$336$	0	0
$337$	2.23509e6	1.07206	0.536032	−	0.844198i	$-0.319923\pi$
$337$	2.23509e6	1.07206	0.536032	+	0.844198i	$0.319923\pi$
$338$	−951445.	−0.452994
$339$	0	0
$340$	1.23879e6	0.581168
$341$	−375561.	−0.174902
$342$	0	0
$343$	853826.	0.391863
$344$	544017.	0.247865
$345$	0	0
$346$	981148.	0.440600
$347$	−1.61546e6	−0.720230	−0.360115	−	0.932908i	$-0.617263\pi$
$347$	−1.61546e6	−0.720230	−0.360115	+	0.932908i	$0.617263\pi$
$348$	0	0
$349$	−2.56381e6	−1.12674	−0.563369	−	0.826205i	$-0.690495\pi$
$349$	−2.56381e6	−1.12674	−0.563369	+	0.826205i	$0.690495\pi$
$350$	2.23076e6	0.973382
$351$	0	0
$352$	290054.	0.124773
$353$	1.64946e6	0.704541	0.352271	−	0.935898i	$-0.385410\pi$
$353$	1.64946e6	0.704541	0.352271	+	0.935898i	$0.385410\pi$
$354$	0	0
$355$	−2.42066e6	−1.01944
$356$	1.95882e6	0.819162
$357$	0	0
$358$	5.46435e6	2.25336
$359$	−1.30522e6	−0.534500	−0.267250	−	0.963627i	$-0.586115\pi$
$359$	−1.30522e6	−0.534500	−0.267250	+	0.963627i	$0.586115\pi$
$360$	0	0
$361$	1.10151e6	0.444857
$362$	959247.	0.384732
$363$	0	0
$364$	3.46807e6	1.37194
$365$	−1.23022e6	−0.483338
$366$	0	0
$367$	5.04859e6	1.95661	0.978306	−	0.207164i	$-0.0664233\pi$
$367$	5.04859e6	1.95661	0.978306	+	0.207164i	$0.0664233\pi$
$368$	631991.	0.243272
$369$	0	0
$370$	−2.47412e6	−0.939541
$371$	−1.28208e6	−0.483595
$372$	0	0
$373$	512860.	0.190865	0.0954326	−	0.995436i	$-0.469577\pi$
$373$	512860.	0.190865	0.0954326	+	0.995436i	$0.469577\pi$
$374$	362476.	0.133999
$375$	0	0
$376$	−11402.5	−0.00415941
$377$	6.04865e6	2.19182
$378$	0	0
$379$	1.34067e6	0.479429	0.239714	−	0.970843i	$-0.422946\pi$
$379$	1.34067e6	0.479429	0.239714	+	0.970843i	$0.422946\pi$
$380$	−1.91665e6	−0.680901
$381$	0	0
$382$	−4.40644e6	−1.54500
$383$	5.32498e6	1.85490	0.927451	−	0.373944i	$-0.121995\pi$
$383$	5.32498e6	1.85490	0.927451	+	0.373944i	$0.121995\pi$
$384$	0	0
$385$	−306741.	−0.105468
$386$	3.34225e6	1.14175
$387$	0	0
$388$	−2.09637e6	−0.706949
$389$	−4.22472e6	−1.41554	−0.707772	−	0.706441i	$-0.750300\pi$
$389$	−4.22472e6	−1.41554	−0.707772	+	0.706441i	$0.750300\pi$
$390$	0	0
$391$	646708.	0.213927
$392$	1.08417e6	0.356353
$393$	0	0
$394$	−1.13359e6	−0.367889
$395$	−2.01912e6	−0.651132
$396$	0	0
$397$	−451747.	−0.143853	−0.0719266	−	0.997410i	$-0.522915\pi$
$397$	−451747.	−0.143853	−0.0719266	+	0.997410i	$0.522915\pi$
$398$	368018.	0.116456
$399$	0	0
$400$	1.80723e6	0.564759
$401$	−1.72721e6	−0.536395	−0.268197	−	0.963364i	$-0.586428\pi$
$401$	−1.72721e6	−0.536395	−0.268197	+	0.963364i	$0.586428\pi$
$402$	0	0
$403$	−6.75602e6	−2.07218
$404$	−778159.	−0.237200
$405$	0	0
$406$	−1.26518e7	−3.80923
$407$	−319194.	−0.0955145
$408$	0	0
$409$	−1.57103e6	−0.464384	−0.232192	−	0.972670i	$-0.574590\pi$
$409$	−1.57103e6	−0.464384	−0.232192	+	0.972670i	$0.574590\pi$
$410$	−1.46854e6	−0.431445
$411$	0	0
$412$	3.57565e6	1.03780
$413$	5.99985e6	1.73087
$414$	0	0
$415$	−1.43233e6	−0.408248
$416$	5.21782e6	1.47828
$417$	0	0
$418$	−560821.	−0.156994
$419$	1.93316e6	0.537938	0.268969	−	0.963149i	$-0.413317\pi$
$419$	1.93316e6	0.537938	0.268969	+	0.963149i	$0.413317\pi$
$420$	0	0
$421$	−188965.	−0.0519610	−0.0259805	−	0.999662i	$-0.508271\pi$
$421$	−188965.	−0.0519610	−0.0259805	+	0.999662i	$0.508271\pi$
$422$	−6.97018e6	−1.90530
$423$	0	0
$424$	−336570.	−0.0909204
$425$	1.84931e6	0.496636
$426$	0	0
$427$	−1.69488e6	−0.449852
$428$	−3.35029e6	−0.884041
$429$	0	0
$430$	3.22961e6	0.842323
$431$	3.11533e6	0.807812	0.403906	−	0.914800i	$-0.367652\pi$
$431$	3.11533e6	0.807812	0.403906	+	0.914800i	$0.367652\pi$
$432$	0	0
$433$	−2.18147e6	−0.559151	−0.279576	−	0.960124i	$-0.590194\pi$
$433$	−2.18147e6	−0.559151	−0.279576	+	0.960124i	$0.590194\pi$
$434$	1.41314e7	3.60130
$435$	0	0
$436$	−2.85247e6	−0.718629
$437$	−1.00058e6	−0.250639
$438$	0	0
$439$	−3.03543e6	−0.751725	−0.375863	−	0.926675i	$-0.622654\pi$
$439$	−3.03543e6	−0.751725	−0.375863	+	0.926675i	$0.622654\pi$
$440$	−80525.1	−0.0198289
$441$	0	0
$442$	6.52064e6	1.58758
$443$	3.62547e6	0.877719	0.438859	−	0.898556i	$-0.355383\pi$
$443$	3.62547e6	0.877719	0.438859	+	0.898556i	$0.355383\pi$
$444$	0	0
$445$	−3.11666e6	−0.746087
$446$	−7.55371e6	−1.79814
$447$	0	0
$448$	−3.46209e6	−0.814974
$449$	−6.84914e6	−1.60332	−0.801660	−	0.597781i	$-0.796049\pi$
$449$	−6.84914e6	−1.60332	−0.801660	+	0.597781i	$0.796049\pi$
$450$	0	0
$451$	−189461.	−0.0438610
$452$	−1.16480e6	−0.268166
$453$	0	0
$454$	−2.84993e6	−0.648926
$455$	−5.51801e6	−1.24955
$456$	0	0
$457$	4.07502e6	0.912725	0.456362	−	0.889794i	$-0.349152\pi$
$457$	4.07502e6	0.912725	0.456362	+	0.889794i	$0.349152\pi$
$458$	−6.15994e6	−1.37219
$459$	0	0
$460$	536046.	0.118116
$461$	−1.10815e6	−0.242854	−0.121427	−	0.992600i	$-0.538747\pi$
$461$	−1.10815e6	−0.242854	−0.121427	+	0.992600i	$0.538747\pi$
$462$	0	0
$463$	3.77893e6	0.819251	0.409625	−	0.912254i	$-0.365659\pi$
$463$	3.77893e6	0.819251	0.409625	+	0.912254i	$0.365659\pi$
$464$	−1.02497e7	−2.21013
$465$	0	0
$466$	1.93440e6	0.412650
$467$	−8.40396e6	−1.78317	−0.891583	−	0.452857i	$-0.850405\pi$
$467$	−8.40396e6	−1.78317	−0.891583	+	0.452857i	$0.850405\pi$
$468$	0	0
$469$	9.43380e6	1.98041
$470$	−67692.3	−0.0141350
$471$	0	0
$472$	1.57507e6	0.325420
$473$	416663.	0.0856312
$474$	0	0
$475$	−2.86124e6	−0.581863
$476$	−6.01365e6	−1.21652
$477$	0	0
$478$	−9.64470e6	−1.93072
$479$	−6.44233e6	−1.28293	−0.641467	−	0.767151i	$-0.721674\pi$
$479$	−6.44233e6	−1.28293	−0.641467	+	0.767151i	$0.721674\pi$
$480$	0	0
$481$	−5.74203e6	−1.13163
$482$	1.11240e7	2.18094
$483$	0	0
$484$	−4.02557e6	−0.781114
$485$	3.33551e6	0.643884
$486$	0	0
$487$	−2.57622e6	−0.492221	−0.246111	−	0.969242i	$-0.579153\pi$
$487$	−2.57622e6	−0.492221	−0.246111	+	0.969242i	$0.579153\pi$
$488$	−444937.	−0.0845764
$489$	0	0
$490$	6.43625e6	1.21100
$491$	443058.	0.0829386	0.0414693	−	0.999140i	$-0.486796\pi$
$491$	443058.	0.0829386	0.0414693	+	0.999140i	$0.486796\pi$
$492$	0	0
$493$	−1.04884e7	−1.94353
$494$	−1.00887e7	−1.86002
$495$	0	0
$496$	1.14484e7	2.08949
$497$	1.17510e7	2.13394
$498$	0	0
$499$	6.36169e6	1.14372	0.571862	−	0.820350i	$-0.306221\pi$
$499$	6.36169e6	1.14372	0.571862	+	0.820350i	$0.306221\pi$
$500$	4.69949e6	0.840671
$501$	0	0
$502$	9.52952e6	1.68776
$503$	−4.69139e6	−0.826763	−0.413382	−	0.910558i	$-0.635652\pi$
$503$	−4.69139e6	−0.826763	−0.413382	+	0.910558i	$0.635652\pi$
$504$	0	0
$505$	1.23812e6	0.216040
$506$	156850.	0.0272337
$507$	0	0
$508$	−5.64769e6	−0.970982
$509$	−3.35385e6	−0.573785	−0.286893	−	0.957963i	$-0.592622\pi$
$509$	−3.35385e6	−0.573785	−0.286893	+	0.957963i	$0.592622\pi$
$510$	0	0
$511$	5.97204e6	1.01174
$512$	−6.88558e6	−1.16082
$513$	0	0
$514$	3.03156e6	0.506125
$515$	−5.68918e6	−0.945217
$516$	0	0
$517$	−8733.22	−0.00143697
$518$	1.20104e7	1.96668
$519$	0	0
$520$	−1.44858e6	−0.234927
$521$	−3.64554e6	−0.588393	−0.294196	−	0.955745i	$-0.595052\pi$
$521$	−3.64554e6	−0.588393	−0.294196	+	0.955745i	$0.595052\pi$
$522$	0	0
$523$	−9.63934e6	−1.54097	−0.770483	−	0.637461i	$-0.779985\pi$
$523$	−9.63934e6	−1.54097	−0.770483	+	0.637461i	$0.779985\pi$
$524$	6.29644e6	1.00177
$525$	0	0
$526$	8.38467e6	1.32136
$527$	1.17150e7	1.83745
$528$	0	0
$529$	279841.	0.0434783
$530$	−1.99808e6	−0.308975
$531$	0	0
$532$	9.30426e6	1.42529
$533$	−3.40824e6	−0.519652
$534$	0	0
$535$	5.33060e6	0.805178
$536$	2.47654e6	0.372335
$537$	0	0
$538$	−9.23537e6	−1.37562
$539$	830364.	0.123111
$540$	0	0
$541$	1.04511e7	1.53521	0.767604	−	0.640925i	$-0.221449\pi$
$541$	1.04511e7	1.53521	0.767604	+	0.640925i	$0.221449\pi$
$542$	4.58881e6	0.670969
$543$	0	0
$544$	−9.04772e6	−1.31082
$545$	4.53854e6	0.654522
$546$	0	0
$547$	−6.35425e6	−0.908021	−0.454011	−	0.890996i	$-0.650007\pi$
$547$	−6.35425e6	−0.908021	−0.454011	+	0.890996i	$0.650007\pi$
$548$	651843.	0.0927238
$549$	0	0
$550$	448524.	0.0632235
$551$	1.62276e7	2.27706
$552$	0	0
$553$	9.80169e6	1.36298
$554$	−1.31030e7	−1.81382
$555$	0	0
$556$	2.00129e6	0.274551
$557$	283483.	0.0387159	0.0193580	−	0.999813i	$-0.493838\pi$
$557$	283483.	0.0387159	0.0193580	+	0.999813i	$0.493838\pi$
$558$	0	0
$559$	7.49541e6	1.01453
$560$	9.35051e6	1.25999
$561$	0	0
$562$	−7.80292e6	−1.04212
$563$	−1.05154e7	−1.39815	−0.699076	−	0.715047i	$-0.746405\pi$
$563$	−1.05154e7	−1.39815	−0.699076	+	0.715047i	$0.746405\pi$
$564$	0	0
$565$	1.85329e6	0.244243
$566$	−1.38623e7	−1.81884
$567$	0	0
$568$	3.08484e6	0.401201
$569$	−6.83667e6	−0.885246	−0.442623	−	0.896708i	$-0.645952\pi$
$569$	−6.83667e6	−0.885246	−0.442623	+	0.896708i	$0.645952\pi$
$570$	0	0
$571$	3.11227e6	0.399473	0.199736	−	0.979850i	$-0.435991\pi$
$571$	3.11227e6	0.399473	0.199736	+	0.979850i	$0.435991\pi$
$572$	697300.	0.0891107
$573$	0	0
$574$	7.12892e6	0.903117
$575$	800228.	0.100936
$576$	0	0
$577$	−1.10234e7	−1.37841	−0.689203	−	0.724569i	$-0.742039\pi$
$577$	−1.10234e7	−1.37841	−0.689203	+	0.724569i	$0.742039\pi$
$578$	−564932.	−0.0703358
$579$	0	0
$580$	−8.69367e6	−1.07308
$581$	6.95317e6	0.854560
$582$	0	0
$583$	−257780.	−0.0314107
$584$	1.56777e6	0.190217
$585$	0	0
$586$	−2.09657e6	−0.252212
$587$	2.79249e6	0.334500	0.167250	−	0.985915i	$-0.446511\pi$
$587$	2.79249e6	0.334500	0.167250	+	0.985915i	$0.446511\pi$
$588$	0	0
$589$	−1.81253e7	−2.15277
$590$	9.35054e6	1.10588
$591$	0	0
$592$	9.73014e6	1.14108
$593$	1.65811e7	1.93632	0.968161	−	0.250328i	$-0.0805384\pi$
$593$	1.65811e7	1.93632	0.968161	+	0.250328i	$0.0805384\pi$
$594$	0	0
$595$	9.56825e6	1.10800
$596$	8.53266e6	0.983940
$597$	0	0
$598$	2.82159e6	0.322657
$599$	−6.89907e6	−0.785640	−0.392820	−	0.919615i	$-0.628500\pi$
$599$	−6.89907e6	−0.785640	−0.392820	+	0.919615i	$0.628500\pi$
$600$	0	0
$601$	−807441.	−0.0911853	−0.0455927	−	0.998960i	$-0.514518\pi$
$601$	−807441.	−0.0911853	−0.0455927	+	0.998960i	$0.514518\pi$
$602$	−1.56779e7	−1.76318
$603$	0	0
$604$	1.24866e7	1.39268
$605$	6.40504e6	0.711432
$606$	0	0
$607$	−1.46288e6	−0.161153	−0.0805765	−	0.996748i	$-0.525676\pi$
$607$	−1.46288e6	−0.161153	−0.0805765	+	0.996748i	$0.525676\pi$
$608$	1.39986e7	1.53576
$609$	0	0
$610$	−2.64141e6	−0.287416
$611$	−157103.	−0.0170248
$612$	0	0
$613$	−1.41124e7	−1.51687	−0.758435	−	0.651749i	$-0.774036\pi$
$613$	−1.41124e7	−1.51687	−0.758435	+	0.651749i	$0.774036\pi$
$614$	−1.49845e7	−1.60406
$615$	0	0
$616$	390904.	0.0415067
$617$	1.32361e7	1.39974	0.699871	−	0.714269i	$-0.253241\pi$
$617$	1.32361e7	1.39974	0.699871	+	0.714269i	$0.253241\pi$
$618$	0	0
$619$	3.89949e6	0.409055	0.204527	−	0.978861i	$-0.434434\pi$
$619$	3.89949e6	0.409055	0.204527	+	0.978861i	$0.434434\pi$
$620$	9.71036e6	1.01451
$621$	0	0
$622$	1.93804e7	2.00857
$623$	1.51296e7	1.56174
$624$	0	0
$625$	−2.75007e6	−0.281607
$626$	−1.86353e7	−1.90065
$627$	0	0
$628$	4.94538e6	0.500381
$629$	9.95671e6	1.00344
$630$	0	0
$631$	−7.32659e6	−0.732535	−0.366268	−	0.930510i	$-0.619365\pi$
$631$	−7.32659e6	−0.732535	−0.366268	+	0.930510i	$0.619365\pi$
$632$	2.57312e6	0.256252
$633$	0	0
$634$	−2.30497e7	−2.27741
$635$	8.98597e6	0.884363
$636$	0	0
$637$	1.49375e7	1.45858
$638$	−2.54381e6	−0.247419
$639$	0	0
$640$	4.11394e6	0.397016
$641$	−4.68243e6	−0.450118	−0.225059	−	0.974345i	$-0.572258\pi$
$641$	−4.68243e6	−0.450118	−0.225059	+	0.974345i	$0.572258\pi$
$642$	0	0
$643$	5.15232e6	0.491445	0.245723	−	0.969340i	$-0.420975\pi$
$643$	5.15232e6	0.491445	0.245723	+	0.969340i	$0.420975\pi$
$644$	−2.60220e6	−0.247244
$645$	0	0
$646$	1.74938e7	1.64931
$647$	5.60768e6	0.526650	0.263325	−	0.964707i	$-0.415181\pi$
$647$	5.60768e6	0.526650	0.263325	+	0.964707i	$0.415181\pi$
$648$	0	0
$649$	1.20635e6	0.112424
$650$	8.06856e6	0.749053
$651$	0	0
$652$	5.63333e6	0.518975
$653$	−7.57877e6	−0.695530	−0.347765	−	0.937582i	$-0.613059\pi$
$653$	−7.57877e6	−0.695530	−0.347765	+	0.937582i	$0.613059\pi$
$654$	0	0
$655$	−1.00182e7	−0.912402
$656$	5.77542e6	0.523992
$657$	0	0
$658$	328608.	0.0295878
$659$	−1.64530e7	−1.47581	−0.737905	−	0.674904i	$-0.764185\pi$
$659$	−1.64530e7	−1.47581	−0.737905	+	0.674904i	$0.764185\pi$
$660$	0	0
$661$	1.72074e6	0.153184	0.0765918	−	0.997063i	$-0.475596\pi$
$661$	1.72074e6	0.153184	0.0765918	+	0.997063i	$0.475596\pi$
$662$	1.96276e7	1.74069
$663$	0	0
$664$	1.82533e6	0.160665
$665$	−1.48039e7	−1.29814
$666$	0	0
$667$	−4.53850e6	−0.395000
$668$	5.86807e6	0.508808
$669$	0	0
$670$	1.47022e7	1.26531
$671$	−340778.	−0.0292190
$672$	0	0
$673$	8.54705e6	0.727408	0.363704	−	0.931514i	$-0.381512\pi$
$673$	8.54705e6	0.727408	0.363704	+	0.931514i	$0.381512\pi$
$674$	−1.69095e7	−1.43378
$675$	0	0
$676$	3.17376e6	0.267121
$677$	8.31269e6	0.697060	0.348530	−	0.937298i	$-0.386681\pi$
$677$	8.31269e6	0.697060	0.348530	+	0.937298i	$0.386681\pi$
$678$	0	0
$679$	−1.61920e7	−1.34780
$680$	2.51184e6	0.208314
$681$	0	0
$682$	2.84129e6	0.233914
$683$	1.80054e6	0.147690	0.0738451	−	0.997270i	$-0.476473\pi$
$683$	1.80054e6	0.147690	0.0738451	+	0.997270i	$0.476473\pi$
$684$	0	0
$685$	−1.03714e6	−0.0844522
$686$	−6.45960e6	−0.524077
$687$	0	0
$688$	−1.27013e7	−1.02300
$689$	−4.63723e6	−0.372144
$690$	0	0
$691$	−1.38259e7	−1.10153	−0.550766	−	0.834660i	$-0.685664\pi$
$691$	−1.38259e7	−1.10153	−0.550766	+	0.834660i	$0.685664\pi$
$692$	−3.27284e6	−0.259812
$693$	0	0
$694$	1.22217e7	0.963235
$695$	−3.18423e6	−0.250059
$696$	0	0
$697$	5.90991e6	0.460786
$698$	1.93965e7	1.50690
$699$	0	0
$700$	−7.44121e6	−0.573982
$701$	−2.42009e7	−1.86010	−0.930051	−	0.367430i	$-0.880238\pi$
$701$	−2.42009e7	−1.86010	−0.930051	+	0.367430i	$0.880238\pi$
$702$	0	0
$703$	−1.54049e7	−1.17563
$704$	−696099.	−0.0529346
$705$	0	0
$706$	−1.24790e7	−0.942252
$707$	−6.01038e6	−0.452224
$708$	0	0
$709$	−1.13875e7	−0.850768	−0.425384	−	0.905013i	$-0.639861\pi$
$709$	−1.13875e7	−0.850768	−0.425384	+	0.905013i	$0.639861\pi$
$710$	1.83135e7	1.36340
$711$	0	0
$712$	3.97181e6	0.293622
$713$	5.06926e6	0.373440
$714$	0	0
$715$	−1.10947e6	−0.0811613
$716$	−1.82276e7	−1.32876
$717$	0	0
$718$	9.87459e6	0.714839
$719$	1.24869e7	0.900807	0.450403	−	0.892825i	$-0.351280\pi$
$719$	1.24869e7	0.900807	0.450403	+	0.892825i	$0.351280\pi$
$720$	0	0
$721$	2.76178e7	1.97857
$722$	−8.33344e6	−0.594951
$723$	0	0
$724$	−3.19978e6	−0.226868
$725$	−1.29782e7	−0.917000
$726$	0	0
$727$	−1.69620e7	−1.19026	−0.595128	−	0.803631i	$-0.702899\pi$
$727$	−1.69620e7	−1.19026	−0.595128	+	0.803631i	$0.702899\pi$
$728$	7.03203e6	0.491759
$729$	0	0
$730$	9.30721e6	0.646416
$731$	−1.29971e7	−0.899605
$732$	0	0
$733$	1.73348e7	1.19168	0.595839	−	0.803104i	$-0.296820\pi$
$733$	1.73348e7	1.19168	0.595839	+	0.803104i	$0.296820\pi$
$734$	−3.81949e7	−2.61677
$735$	0	0
$736$	−3.91510e6	−0.266408
$737$	1.89679e6	0.128632
$738$	0	0
$739$	2.67348e6	0.180080	0.0900400	−	0.995938i	$-0.471301\pi$
$739$	2.67348e6	0.180080	0.0900400	+	0.995938i	$0.471301\pi$
$740$	8.25297e6	0.554027
$741$	0	0
$742$	9.69957e6	0.646759
$743$	−1.67454e7	−1.11281	−0.556407	−	0.830910i	$-0.687820\pi$
$743$	−1.67454e7	−1.11281	−0.556407	+	0.830910i	$0.687820\pi$
$744$	0	0
$745$	−1.35762e7	−0.896165
$746$	−3.88002e6	−0.255263
$747$	0	0
$748$	−1.20912e6	−0.0790162
$749$	−2.58771e7	−1.68543
$750$	0	0
$751$	−2.86431e7	−1.85319	−0.926595	−	0.376062i	$-0.877278\pi$
$751$	−2.86431e7	−1.85319	−0.926595	+	0.376062i	$0.877278\pi$
$752$	266218.	0.0171670
$753$	0	0
$754$	−4.57609e7	−2.93134
$755$	−1.98673e7	−1.26845
$756$	0	0
$757$	2.69646e7	1.71023	0.855113	−	0.518441i	$-0.173487\pi$
$757$	2.69646e7	1.71023	0.855113	+	0.518441i	$0.173487\pi$
$758$	−1.01428e7	−0.641188
$759$	0	0
$760$	−3.88630e6	−0.244063
$761$	−4.94085e6	−0.309272	−0.154636	−	0.987972i	$-0.549420\pi$
$761$	−4.94085e6	−0.309272	−0.154636	+	0.987972i	$0.549420\pi$
$762$	0	0
$763$	−2.20320e7	−1.37007
$764$	1.46987e7	0.911054
$765$	0	0
$766$	−4.02860e7	−2.48074
$767$	2.17011e7	1.33197
$768$	0	0
$769$	3.74373e6	0.228291	0.114145	−	0.993464i	$-0.463587\pi$
$769$	3.74373e6	0.228291	0.114145	+	0.993464i	$0.463587\pi$
$770$	2.32064e6	0.141053
$771$	0	0
$772$	−1.11488e7	−0.673265
$773$	−2.17718e7	−1.31053	−0.655264	−	0.755400i	$-0.727443\pi$
$773$	−2.17718e7	−1.31053	−0.655264	+	0.755400i	$0.727443\pi$
$774$	0	0
$775$	1.44960e7	0.866947
$776$	−4.25070e6	−0.253400
$777$	0	0
$778$	3.19620e7	1.89315
$779$	−9.14376e6	−0.539860
$780$	0	0
$781$	2.36268e6	0.138605
$782$	−4.89265e6	−0.286106
$783$	0	0
$784$	−2.53123e7	−1.47076
$785$	−7.86855e6	−0.455743
$786$	0	0
$787$	−2.13696e7	−1.22987	−0.614936	−	0.788577i	$-0.710818\pi$
$787$	−2.13696e7	−1.22987	−0.614936	+	0.788577i	$0.710818\pi$
$788$	3.78135e6	0.216936
$789$	0	0
$790$	1.52756e7	0.870823
$791$	−8.99670e6	−0.511260
$792$	0	0
$793$	−6.13030e6	−0.346178
$794$	3.41768e6	0.192389
$795$	0	0
$796$	−1.22761e6	−0.0686715
$797$	−2.80148e7	−1.56222	−0.781108	−	0.624396i	$-0.785345\pi$
$797$	−2.80148e7	−1.56222	−0.781108	+	0.624396i	$0.785345\pi$
$798$	0	0
$799$	272417.	0.0150962
$800$	−1.11955e7	−0.618472
$801$	0	0
$802$	1.30672e7	0.717374
$803$	1.20076e6	0.0657152
$804$	0	0
$805$	4.14034e6	0.225188
$806$	5.11124e7	2.77134
$807$	0	0
$808$	−1.57783e6	−0.0850223
$809$	3.12241e7	1.67733	0.838665	−	0.544648i	$-0.183337\pi$
$809$	3.12241e7	1.67733	0.838665	+	0.544648i	$0.183337\pi$
$810$	0	0
$811$	1.30079e7	0.694471	0.347235	−	0.937778i	$-0.387120\pi$
$811$	1.30079e7	0.694471	0.347235	+	0.937778i	$0.387120\pi$
$812$	4.22029e7	2.24622
$813$	0	0
$814$	2.41486e6	0.127741
$815$	−8.96313e6	−0.472678
$816$	0	0
$817$	2.01090e7	1.05399
$818$	1.18856e7	0.621066
$819$	0	0
$820$	4.89863e6	0.254414
$821$	−9.48385e6	−0.491051	−0.245526	−	0.969390i	$-0.578961\pi$
$821$	−9.48385e6	−0.491051	−0.245526	+	0.969390i	$0.578961\pi$
$822$	0	0
$823$	−1.57317e7	−0.809610	−0.404805	−	0.914403i	$-0.632661\pi$
$823$	−1.57317e7	−0.809610	−0.404805	+	0.914403i	$0.632661\pi$
$824$	7.25017e6	0.371989
$825$	0	0
$826$	−4.53916e7	−2.31487
$827$	1.68809e7	0.858284	0.429142	−	0.903237i	$-0.358816\pi$
$827$	1.68809e7	0.858284	0.429142	+	0.903237i	$0.358816\pi$
$828$	0	0
$829$	2.04228e7	1.03212	0.516059	−	0.856553i	$-0.327399\pi$
$829$	2.04228e7	1.03212	0.516059	+	0.856553i	$0.327399\pi$
$830$	1.08363e7	0.545990
$831$	0	0
$832$	−1.25222e7	−0.627152
$833$	−2.59018e7	−1.29335
$834$	0	0
$835$	−9.33662e6	−0.463419
$836$	1.87074e6	0.0925760
$837$	0	0
$838$	−1.46252e7	−0.719437
$839$	3.17685e7	1.55809	0.779044	−	0.626970i	$-0.215705\pi$
$839$	3.17685e7	1.55809	0.779044	+	0.626970i	$0.215705\pi$
$840$	0	0
$841$	5.30948e7	2.58858
$842$	1.42961e6	0.0694925
$843$	0	0
$844$	2.32506e7	1.12351
$845$	−5.04973e6	−0.243291
$846$	0	0
$847$	−3.10929e7	−1.48920
$848$	7.85800e6	0.375252
$849$	0	0
$850$	−1.39909e7	−0.664200
$851$	4.30843e6	0.203937
$852$	0	0
$853$	−1.26754e7	−0.596471	−0.298236	−	0.954492i	$-0.596398\pi$
$853$	−1.26754e7	−0.596471	−0.298236	+	0.954492i	$0.596398\pi$
$854$	1.28226e7	0.601631
$855$	0	0
$856$	−6.79321e6	−0.316877
$857$	1.77408e7	0.825126	0.412563	−	0.910929i	$-0.364634\pi$
$857$	1.77408e7	0.825126	0.412563	+	0.910929i	$0.364634\pi$
$858$	0	0
$859$	3.98531e6	0.184280	0.0921401	−	0.995746i	$-0.470629\pi$
$859$	3.98531e6	0.184280	0.0921401	+	0.995746i	$0.470629\pi$
$860$	−1.07731e7	−0.496699
$861$	0	0
$862$	−2.35689e7	−1.08037
$863$	−3.42751e7	−1.56658	−0.783289	−	0.621658i	$-0.786459\pi$
$863$	−3.42751e7	−1.56658	−0.783289	+	0.621658i	$0.786459\pi$
$864$	0	0
$865$	5.20738e6	0.236635
$866$	1.65038e7	0.747808
$867$	0	0
$868$	−4.71383e7	−2.12361
$869$	1.97076e6	0.0885286
$870$	0	0
$871$	3.41216e7	1.52400
$872$	−5.78381e6	−0.257586
$873$	0	0
$874$	7.56986e6	0.335204
$875$	3.62981e7	1.60274
$876$	0	0
$877$	−3.57889e7	−1.57126	−0.785632	−	0.618694i	$-0.787662\pi$
$877$	−3.57889e7	−1.57126	−0.785632	+	0.618694i	$0.787662\pi$
$878$	2.29645e7	1.00536
$879$	0	0
$880$	1.88004e6	0.0818391
$881$	−5.76071e6	−0.250055	−0.125028	−	0.992153i	$-0.539902\pi$
$881$	−5.76071e6	−0.250055	−0.125028	+	0.992153i	$0.539902\pi$
$882$	0	0
$883$	−4.41397e7	−1.90514	−0.952572	−	0.304312i	$-0.901573\pi$
$883$	−4.41397e7	−1.90514	−0.952572	+	0.304312i	$0.901573\pi$
$884$	−2.17511e7	−0.936159
$885$	0	0
$886$	−2.74284e7	−1.17386
$887$	9.63127e6	0.411031	0.205515	−	0.978654i	$-0.434113\pi$
$887$	9.63127e6	0.411031	0.205515	+	0.978654i	$0.434113\pi$
$888$	0	0
$889$	−4.36218e7	−1.85118
$890$	2.35790e7	0.997816
$891$	0	0
$892$	2.51971e7	1.06032
$893$	−421482.	−0.0176869
$894$	0	0
$895$	2.90017e7	1.21022
$896$	−1.99709e7	−0.831050
$897$	0	0
$898$	5.18169e7	2.14428
$899$	−8.22139e7	−3.39270
$900$	0	0
$901$	8.04098e6	0.329987
$902$	1.43336e6	0.0586597
$903$	0	0
$904$	−2.36180e6	−0.0961217
$905$	5.09114e6	0.206630
$906$	0	0
$907$	1.72941e7	0.698037	0.349019	−	0.937116i	$-0.386515\pi$
$907$	1.72941e7	0.698037	0.349019	+	0.937116i	$0.386515\pi$
$908$	9.50659e6	0.382658
$909$	0	0
$910$	4.17463e7	1.67115
$911$	3.64206e7	1.45396	0.726978	−	0.686660i	$-0.240924\pi$
$911$	3.64206e7	1.45396	0.726978	+	0.686660i	$0.240924\pi$
$912$	0	0
$913$	1.39803e6	0.0555058
$914$	−3.08295e7	−1.22068
$915$	0	0
$916$	2.05479e7	0.809148
$917$	4.86327e7	1.90988
$918$	0	0
$919$	3.59707e7	1.40495	0.702474	−	0.711710i	$-0.252079\pi$
$919$	3.59707e7	1.40495	0.702474	+	0.711710i	$0.252079\pi$
$920$	1.08691e6	0.0423375
$921$	0	0
$922$	8.38366e6	0.324793
$923$	4.25027e7	1.64215
$924$	0	0
$925$	1.23203e7	0.473443
$926$	−2.85894e7	−1.09566
$927$	0	0
$928$	6.34956e7	2.42032
$929$	1.66420e7	0.632654	0.316327	−	0.948650i	$-0.397550\pi$
$929$	1.66420e7	0.632654	0.316327	+	0.948650i	$0.397550\pi$
$930$	0	0
$931$	4.00750e7	1.51530
$932$	−6.45264e6	−0.243331
$933$	0	0
$934$	6.35799e7	2.38480
$935$	1.92382e6	0.0719673
$936$	0	0
$937$	1.14846e7	0.427332	0.213666	−	0.976907i	$-0.431460\pi$
$937$	1.14846e7	0.427332	0.213666	+	0.976907i	$0.431460\pi$
$938$	−7.13711e7	−2.64859
$939$	0	0
$940$	225803.	0.00833508
$941$	−4.50411e7	−1.65819	−0.829095	−	0.559107i	$-0.811144\pi$
$941$	−4.50411e7	−1.65819	−0.829095	+	0.559107i	$0.811144\pi$
$942$	0	0
$943$	2.55731e6	0.0936494
$944$	−3.67736e7	−1.34309
$945$	0	0
$946$	−3.15225e6	−0.114523
$947$	−3.18096e7	−1.15261	−0.576307	−	0.817234i	$-0.695507\pi$
$947$	−3.18096e7	−1.15261	−0.576307	+	0.817234i	$0.695507\pi$
$948$	0	0
$949$	2.16006e7	0.778573
$950$	2.16466e7	0.778182
$951$	0	0
$952$	−1.21936e7	−0.436052
$953$	3.14562e7	1.12195	0.560975	−	0.827833i	$-0.310426\pi$
$953$	3.14562e7	1.12195	0.560975	+	0.827833i	$0.310426\pi$
$954$	0	0
$955$	−2.33869e7	−0.829781
$956$	3.21720e7	1.13850
$957$	0	0
$958$	4.87393e7	1.71579
$959$	5.03473e6	0.176779
$960$	0	0
$961$	6.31993e7	2.20752
$962$	4.34412e7	1.51344
$963$	0	0
$964$	−3.71065e7	−1.28605
$965$	1.77388e7	0.613205
$966$	0	0
$967$	−1.39347e7	−0.479216	−0.239608	−	0.970870i	$-0.577019\pi$
$967$	−1.39347e7	−0.479216	−0.239608	+	0.970870i	$0.577019\pi$
$968$	−8.16245e6	−0.279983
$969$	0	0
$970$	−2.52347e7	−0.861129
$971$	6.17798e6	0.210280	0.105140	−	0.994457i	$-0.466471\pi$
$971$	6.17798e6	0.210280	0.105140	+	0.994457i	$0.466471\pi$
$972$	0	0
$973$	1.54576e7	0.523433
$974$	1.94903e7	0.658296
$975$	0	0
$976$	1.03881e7	0.349068
$977$	2.28927e7	0.767293	0.383646	−	0.923480i	$-0.374668\pi$
$977$	2.28927e7	0.767293	0.383646	+	0.923480i	$0.374668\pi$
$978$	0	0
$979$	3.04201e6	0.101439
$980$	−2.14696e7	−0.714099
$981$	0	0
$982$	−3.35194e6	−0.110922
$983$	3.76941e7	1.24420	0.622100	−	0.782938i	$-0.286280\pi$
$983$	3.76941e7	1.24420	0.622100	+	0.782938i	$0.286280\pi$
$984$	0	0
$985$	−6.01647e6	−0.197584
$986$	7.93496e7	2.59928
$987$	0	0
$988$	3.36531e7	1.09681
$989$	−5.62405e6	−0.182835
$990$	0	0
$991$	−2.55042e7	−0.824951	−0.412475	−	0.910969i	$-0.635336\pi$
$991$	−2.55042e7	−0.824951	−0.412475	+	0.910969i	$0.635336\pi$
$992$	−7.09211e7	−2.28821
$993$	0	0
$994$	−8.89016e7	−2.85393
$995$	1.95323e6	0.0625455
$996$	0	0
$997$	−2.40517e7	−0.766315	−0.383157	−	0.923683i	$-0.625163\pi$
$997$	−2.40517e7	−0.766315	−0.383157	+	0.923683i	$0.625163\pi$
$998$	−4.81292e7	−1.52961
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.d.1.1		4
3.2	odd	2		69.6.a.c.1.4	✓	4
12.11	even	2		1104.6.a.n.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.c.1.4	✓	4	3.2	odd	2
207.6.a.d.1.1		4	1.1	even	1	trivial
1104.6.a.n.1.4		4	12.11	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-7.56547 q^{2} +25.2363 q^{4} -40.1532 q^{5} +194.921 q^{7} +51.1704 q^{8} +O(q^{10})\)	\(q-7.56547 q^{2} +25.2363 q^{4} -40.1532 q^{5} +194.921 q^{7} +51.1704 q^{8} +303.778 q^{10} +39.1915 q^{11} +705.021 q^{13} -1474.67 q^{14} -1194.69 q^{16} -1222.51 q^{17} +1891.46 q^{19} -1013.32 q^{20} -296.502 q^{22} -529.000 q^{23} -1512.72 q^{25} -5333.81 q^{26} +4919.10 q^{28} +8579.39 q^{29} -9582.72 q^{31} +7400.94 q^{32} +9248.86 q^{34} -7826.73 q^{35} -8144.49 q^{37} -14309.8 q^{38} -2054.66 q^{40} -4834.24 q^{41} +10631.5 q^{43} +989.049 q^{44} +4002.13 q^{46} -222.835 q^{47} +21187.4 q^{49} +11444.4 q^{50} +17792.1 q^{52} -6577.44 q^{53} -1573.66 q^{55} +9974.21 q^{56} -64907.1 q^{58} +30780.8 q^{59} -8695.20 q^{61} +72497.8 q^{62} -17761.5 q^{64} -28308.9 q^{65} +48398.0 q^{67} -30851.6 q^{68} +59212.8 q^{70} +60285.7 q^{71} +30638.2 q^{73} +61616.8 q^{74} +47733.4 q^{76} +7639.26 q^{77} +50285.3 q^{79} +47970.7 q^{80} +36573.3 q^{82} +35671.7 q^{83} +49087.7 q^{85} -80432.0 q^{86} +2005.44 q^{88} +77619.2 q^{89} +137424. q^{91} -13350.0 q^{92} +1685.85 q^{94} -75948.1 q^{95} -83069.4 q^{97} -160292. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8	\( 4 q - 4 q^{2} - 46 q^{4} + 122 q^{5} + 62 q^{7} - 72 q^{8} + 642 q^{10} - 32 q^{11} + 1364 q^{13} - 2754 q^{14} + 18 q^{16} - 278 q^{17} + 2862 q^{19} - 3830 q^{20} + 3176 q^{22} - 2116 q^{23} + 5944 q^{25} - 6996 q^{26} + 4738 q^{28} + 5180 q^{29} - 1788 q^{31} + 7352 q^{32} + 15818 q^{34} - 11768 q^{35} + 3348 q^{37} - 1050 q^{38} - 13462 q^{40} + 17664 q^{41} + 25398 q^{43} + 16848 q^{44} + 2116 q^{46} + 26040 q^{47} + 55720 q^{49} + 35256 q^{50} - 2752 q^{52} + 32006 q^{53} + 34904 q^{55} + 68542 q^{56} - 40804 q^{58} + 61136 q^{59} + 35844 q^{61} + 47524 q^{62} - 35142 q^{64} + 48036 q^{65} + 73458 q^{67} - 17910 q^{68} - 59104 q^{70} - 24432 q^{71} + 122512 q^{73} + 20828 q^{74} + 56834 q^{76} - 159496 q^{77} + 90170 q^{79} + 36546 q^{80} + 84144 q^{82} - 28592 q^{83} + 355124 q^{85} - 103778 q^{86} - 150776 q^{88} + 27926 q^{89} + 334180 q^{91} + 24334 q^{92} + 113632 q^{94} - 113392 q^{95} + 16580 q^{97} - 49688 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 46 * q^4 + 122 * q^5 + 62 * q^7 - 72 * q^8 + 642 * q^10 - 32 * q^11 + 1364 * q^13 - 2754 * q^14 + 18 * q^16 - 278 * q^17 + 2862 * q^19 - 3830 * q^20 + 3176 * q^22 - 2116 * q^23 + 5944 * q^25 - 6996 * q^26 + 4738 * q^28 + 5180 * q^29 - 1788 * q^31 + 7352 * q^32 + 15818 * q^34 - 11768 * q^35 + 3348 * q^37 - 1050 * q^38 - 13462 * q^40 + 17664 * q^41 + 25398 * q^43 + 16848 * q^44 + 2116 * q^46 + 26040 * q^47 + 55720 * q^49 + 35256 * q^50 - 2752 * q^52 + 32006 * q^53 + 34904 * q^55 + 68542 * q^56 - 40804 * q^58 + 61136 * q^59 + 35844 * q^61 + 47524 * q^62 - 35142 * q^64 + 48036 * q^65 + 73458 * q^67 - 17910 * q^68 - 59104 * q^70 - 24432 * q^71 + 122512 * q^73 + 20828 * q^74 + 56834 * q^76 - 159496 * q^77 + 90170 * q^79 + 36546 * q^80 + 84144 * q^82 - 28592 * q^83 + 355124 * q^85 - 103778 * q^86 - 150776 * q^88 + 27926 * q^89 + 334180 * q^91 + 24334 * q^92 + 113632 * q^94 - 113392 * q^95 + 16580 * q^97 - 49688 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

Properties

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