LMFDB - Embedded newform 207.6.a.h.1.7

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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 251 x^{8} + 296 x^{7} + 21169 x^{6} - 9042 x^{5} - 685949 x^{4} - 270764 x^{3} + \cdots + 5446216 $ x^10 - 2x^9 - 251x^8 + 296x^7 + 21169x^6 - 9042x^5 - 685949x^4 - 270764x^3 + 6230110x^2 + 11573432*x + 5446216
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$4.70922$ 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)

$f(q)$	$=$	$q+3.70922 q^{2} -18.2417 q^{4} +28.0072 q^{5} -2.54570 q^{7} -186.357 q^{8} +O(q^{10})$	\(q+3.70922 q^{2} -18.2417 q^{4} +28.0072 q^{5} -2.54570 q^{7} -186.357 q^{8} +103.885 q^{10} +419.791 q^{11} -176.339 q^{13} -9.44254 q^{14} -107.505 q^{16} +688.748 q^{17} -3059.20 q^{19} -510.899 q^{20} +1557.10 q^{22} -529.000 q^{23} -2340.60 q^{25} -654.078 q^{26} +46.4378 q^{28} -7240.01 q^{29} -2824.49 q^{31} +5564.68 q^{32} +2554.72 q^{34} -71.2978 q^{35} +12253.4 q^{37} -11347.2 q^{38} -5219.35 q^{40} -16527.6 q^{41} -13253.1 q^{43} -7657.70 q^{44} -1962.18 q^{46} +9209.62 q^{47} -16800.5 q^{49} -8681.78 q^{50} +3216.72 q^{52} -10343.8 q^{53} +11757.2 q^{55} +474.409 q^{56} -26854.8 q^{58} -35190.0 q^{59} -28033.8 q^{61} -10476.6 q^{62} +24080.8 q^{64} -4938.75 q^{65} +64067.7 q^{67} -12563.9 q^{68} -264.459 q^{70} -57099.1 q^{71} +74280.3 q^{73} +45450.4 q^{74} +55805.0 q^{76} -1068.66 q^{77} +3833.20 q^{79} -3010.92 q^{80} -61304.5 q^{82} +15687.9 q^{83} +19289.9 q^{85} -49158.8 q^{86} -78231.1 q^{88} -31766.3 q^{89} +448.904 q^{91} +9649.86 q^{92} +34160.5 q^{94} -85679.6 q^{95} -741.484 q^{97} -62316.8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8}+O(q^{10}) $ 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8	$ 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8} - 250 q^{10} - 460 q^{11} + 464 q^{13} - 3676 q^{14} + 4612 q^{16} - 4756 q^{17} - 1780 q^{19} - 10314 q^{20} - 4214 q^{22} - 5290 q^{23} + 1330 q^{25} + 5152 q^{26} + 7072 q^{28} - 4048 q^{29} + 2816 q^{31} - 27436 q^{32} + 420 q^{34} - 9452 q^{35} + 2872 q^{37} - 31038 q^{38} + 2618 q^{40} - 34056 q^{41} + 7316 q^{43} - 33562 q^{44} + 4232 q^{46} - 49300 q^{47} + 45118 q^{49} - 44764 q^{50} - 25120 q^{52} - 86676 q^{53} - 2120 q^{55} - 290684 q^{56} - 87408 q^{58} - 67100 q^{59} - 40432 q^{61} - 230992 q^{62} + 136776 q^{64} - 184000 q^{65} - 50108 q^{67} - 270592 q^{68} + 117456 q^{70} - 238584 q^{71} - 13804 q^{73} - 150074 q^{74} - 197622 q^{76} - 116248 q^{77} - 9228 q^{79} - 313010 q^{80} - 68604 q^{82} - 155300 q^{83} + 80444 q^{85} + 80914 q^{86} - 237738 q^{88} - 213732 q^{89} - 264352 q^{91} - 101568 q^{92} + 140280 q^{94} + 123612 q^{95} + 42516 q^{97} + 151388 q^{98}+O(q^{100}) $ 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8 - 250 * q^10 - 460 * q^11 + 464 * q^13 - 3676 * q^14 + 4612 * q^16 - 4756 * q^17 - 1780 * q^19 - 10314 * q^20 - 4214 * q^22 - 5290 * q^23 + 1330 * q^25 + 5152 * q^26 + 7072 * q^28 - 4048 * q^29 + 2816 * q^31 - 27436 * q^32 + 420 * q^34 - 9452 * q^35 + 2872 * q^37 - 31038 * q^38 + 2618 * q^40 - 34056 * q^41 + 7316 * q^43 - 33562 * q^44 + 4232 * q^46 - 49300 * q^47 + 45118 * q^49 - 44764 * q^50 - 25120 * q^52 - 86676 * q^53 - 2120 * q^55 - 290684 * q^56 - 87408 * q^58 - 67100 * q^59 - 40432 * q^61 - 230992 * q^62 + 136776 * q^64 - 184000 * q^65 - 50108 * q^67 - 270592 * q^68 + 117456 * q^70 - 238584 * q^71 - 13804 * q^73 - 150074 * q^74 - 197622 * q^76 - 116248 * q^77 - 9228 * q^79 - 313010 * q^80 - 68604 * q^82 - 155300 * q^83 + 80444 * q^85 + 80914 * q^86 - 237738 * q^88 - 213732 * q^89 - 264352 * q^91 - 101568 * q^92 + 140280 * q^94 + 123612 * q^95 + 42516 * q^97 + 151388 * 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 $

$2$	3.70922	0.655703	0.327852	−	0.944729i	$-0.393675\pi$
$2$	3.70922	0.655703	0.327852	+	0.944729i	$0.393675\pi$
$3$	0	0
$3$	0	0
$4$	−18.2417	−0.570053
$5$	28.0072	0.501008	0.250504	−	0.968116i	$-0.419404\pi$
$5$	28.0072	0.501008	0.250504	+	0.968116i	$0.419404\pi$
$6$	0	0
$7$	−2.54570	−0.0196364	−0.00981819	−	0.999952i	$-0.503125\pi$
$7$	−2.54570	−0.0196364	−0.00981819	+	0.999952i	$0.503125\pi$
$8$	−186.357	−1.02949
$9$	0	0
$10$	103.885	0.328512
$11$	419.791	1.04605	0.523024	−	0.852318i	$-0.324804\pi$
$11$	419.791	1.04605	0.523024	+	0.852318i	$0.324804\pi$
$12$	0	0
$13$	−176.339	−0.289394	−0.144697	−	0.989476i	$-0.546221\pi$
$13$	−176.339	−0.289394	−0.144697	+	0.989476i	$0.546221\pi$
$14$	−9.44254	−0.0128756
$15$	0	0
$16$	−107.505	−0.104986
$17$	688.748	0.578014	0.289007	−	0.957327i	$-0.406675\pi$
$17$	688.748	0.578014	0.289007	+	0.957327i	$0.406675\pi$
$18$	0	0
$19$	−3059.20	−1.94412	−0.972061	−	0.234727i	$-0.924580\pi$
$19$	−3059.20	−1.94412	−0.972061	+	0.234727i	$0.924580\pi$
$20$	−510.899	−0.285601
$21$	0	0
$22$	1557.10	0.685896
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−2340.60	−0.748991
$26$	−654.078	−0.189756
$27$	0	0
$28$	46.4378	0.0111938
$29$	−7240.01	−1.59862	−0.799308	−	0.600921i	$-0.794801\pi$
$29$	−7240.01	−1.59862	−0.799308	+	0.600921i	$0.794801\pi$
$30$	0	0
$31$	−2824.49	−0.527880	−0.263940	−	0.964539i	$-0.585022\pi$
$31$	−2824.49	−0.527880	−0.263940	+	0.964539i	$0.585022\pi$
$32$	5564.68	0.960649
$33$	0	0
$34$	2554.72	0.379006
$35$	−71.2978	−0.00983798
$36$	0	0
$37$	12253.4	1.47147	0.735734	−	0.677270i	$-0.236837\pi$
$37$	12253.4	1.47147	0.735734	+	0.677270i	$0.236837\pi$
$38$	−11347.2	−1.27477
$39$	0	0
$40$	−5219.35	−0.515782
$41$	−16527.6	−1.53550	−0.767751	−	0.640748i	$-0.778624\pi$
$41$	−16527.6	−1.53550	−0.767751	+	0.640748i	$0.778624\pi$
$42$	0	0
$43$	−13253.1	−1.09307	−0.546534	−	0.837437i	$-0.684053\pi$
$43$	−13253.1	−1.09307	−0.546534	+	0.837437i	$0.684053\pi$
$44$	−7657.70	−0.596303
$45$	0	0
$46$	−1962.18	−0.136724
$47$	9209.62	0.608131	0.304065	−	0.952651i	$-0.401656\pi$
$47$	9209.62	0.608131	0.304065	+	0.952651i	$0.401656\pi$
$48$	0	0
$49$	−16800.5	−0.999614
$50$	−8681.78	−0.491116
$51$	0	0
$52$	3216.72	0.164970
$53$	−10343.8	−0.505811	−0.252906	−	0.967491i	$-0.581386\pi$
$53$	−10343.8	−0.505811	−0.252906	+	0.967491i	$0.581386\pi$
$54$	0	0
$55$	11757.2	0.524078
$56$	474.409	0.0202154
$57$	0	0
$58$	−26854.8	−1.04822
$59$	−35190.0	−1.31610	−0.658050	−	0.752974i	$-0.728619\pi$
$59$	−35190.0	−1.31610	−0.658050	+	0.752974i	$0.728619\pi$
$60$	0	0
$61$	−28033.8	−0.964622	−0.482311	−	0.876000i	$-0.660202\pi$
$61$	−28033.8	−0.964622	−0.482311	+	0.876000i	$0.660202\pi$
$62$	−10476.6	−0.346133
$63$	0	0
$64$	24080.8	0.734887
$65$	−4938.75	−0.144988
$66$	0	0
$67$	64067.7	1.74362	0.871811	−	0.489843i	$-0.162946\pi$
$67$	64067.7	1.74362	0.871811	+	0.489843i	$0.162946\pi$
$68$	−12563.9	−0.329499
$69$	0	0
$70$	−264.459	−0.00645080
$71$	−57099.1	−1.34426	−0.672131	−	0.740432i	$-0.734621\pi$
$71$	−57099.1	−1.34426	−0.672131	+	0.740432i	$0.734621\pi$
$72$	0	0
$73$	74280.3	1.63142	0.815711	−	0.578459i	$-0.196346\pi$
$73$	74280.3	1.63142	0.815711	+	0.578459i	$0.196346\pi$
$74$	45450.4	0.964846
$75$	0	0
$76$	55805.0	1.10825
$77$	−1068.66	−0.0205406
$78$	0	0
$79$	3833.20	0.0691025	0.0345513	−	0.999403i	$-0.489000\pi$
$79$	3833.20	0.0691025	0.0345513	+	0.999403i	$0.489000\pi$
$80$	−3010.92	−0.0525987
$81$	0	0
$82$	−61304.5	−1.00683
$83$	15687.9	0.249960	0.124980	−	0.992159i	$-0.460113\pi$
$83$	15687.9	0.249960	0.124980	+	0.992159i	$0.460113\pi$
$84$	0	0
$85$	19289.9	0.289590
$86$	−49158.8	−0.716729
$87$	0	0
$88$	−78231.1	−1.07689
$89$	−31766.3	−0.425100	−0.212550	−	0.977150i	$-0.568177\pi$
$89$	−31766.3	−0.425100	−0.212550	+	0.977150i	$0.568177\pi$
$90$	0	0
$91$	448.904	0.00568264
$92$	9649.86	0.118864
$93$	0	0
$94$	34160.5	0.398753
$95$	−85679.6	−0.974021
$96$	0	0
$97$	−741.484	−0.00800152	−0.00400076	−	0.999992i	$-0.501273\pi$
$97$	−741.484	−0.00800152	−0.00400076	+	0.999992i	$0.501273\pi$
$98$	−62316.8	−0.655450
$99$	0	0
$100$	42696.5	0.426965
$101$	−169038.	−1.64885	−0.824423	−	0.565975i	$-0.808500\pi$
$101$	−169038.	−1.64885	−0.824423	+	0.565975i	$0.808500\pi$
$102$	0	0
$103$	−89406.2	−0.830376	−0.415188	−	0.909736i	$-0.636284\pi$
$103$	−89406.2	−0.830376	−0.415188	+	0.909736i	$0.636284\pi$
$104$	32862.0	0.297927
$105$	0	0
$106$	−38367.2	−0.331662
$107$	116497.	0.983687	0.491843	−	0.870684i	$-0.336323\pi$
$107$	116497.	0.983687	0.491843	+	0.870684i	$0.336323\pi$
$108$	0	0
$109$	111980.	0.902761	0.451380	−	0.892332i	$-0.350932\pi$
$109$	111980.	0.902761	0.451380	+	0.892332i	$0.350932\pi$
$110$	43609.9	0.343639
$111$	0	0
$112$	273.676	0.00206154
$113$	205819.	1.51632	0.758159	−	0.652070i	$-0.226099\pi$
$113$	205819.	1.51632	0.758159	+	0.652070i	$0.226099\pi$
$114$	0	0
$115$	−14815.8	−0.104467
$116$	132070.	0.911297
$117$	0	0
$118$	−130527.	−0.862971
$119$	−1753.34	−0.0113501
$120$	0	0
$121$	15173.3	0.0942145
$122$	−103983.	−0.632506
$123$	0	0
$124$	51523.4	0.300920
$125$	−153076.	−0.876258
$126$	0	0
$127$	−49424.8	−0.271916	−0.135958	−	0.990715i	$-0.543411\pi$
$127$	−49424.8	−0.271916	−0.135958	+	0.990715i	$0.543411\pi$
$128$	−88748.9	−0.478782
$129$	0	0
$130$	−18318.9	−0.0950694
$131$	130940.	0.666646	0.333323	−	0.942813i	$-0.391830\pi$
$131$	130940.	0.666646	0.333323	+	0.942813i	$0.391830\pi$
$132$	0	0
$133$	7787.79	0.0381755
$134$	237641.	1.14330
$135$	0	0
$136$	−128353.	−0.595059
$137$	285107.	1.29779	0.648897	−	0.760876i	$-0.275231\pi$
$137$	285107.	1.29779	0.648897	+	0.760876i	$0.275231\pi$
$138$	0	0
$139$	16728.1	0.0734362	0.0367181	−	0.999326i	$-0.488310\pi$
$139$	16728.1	0.0734362	0.0367181	+	0.999326i	$0.488310\pi$
$140$	1300.59	0.00560817
$141$	0	0
$142$	−211793.	−0.881436
$143$	−74025.3	−0.302719
$144$	0	0
$145$	−202772.	−0.800919
$146$	275522.	1.06973
$147$	0	0
$148$	−223522.	−0.838815
$149$	126605.	0.467182	0.233591	−	0.972335i	$-0.424952\pi$
$149$	126605.	0.467182	0.233591	+	0.972335i	$0.424952\pi$
$150$	0	0
$151$	376891.	1.34516	0.672579	−	0.740026i	$-0.265187\pi$
$151$	376891.	1.34516	0.672579	+	0.740026i	$0.265187\pi$
$152$	570104.	2.00145
$153$	0	0
$154$	−3963.89	−0.0134685
$155$	−79105.9	−0.264472
$156$	0	0
$157$	318586.	1.03152	0.515760	−	0.856733i	$-0.327510\pi$
$157$	318586.	1.03152	0.515760	+	0.856733i	$0.327510\pi$
$158$	14218.2	0.0453107
$159$	0	0
$160$	155851.	0.481293
$161$	1346.67	0.00409447
$162$	0	0
$163$	−301215.	−0.887988	−0.443994	−	0.896030i	$-0.646439\pi$
$163$	−301215.	−0.887988	−0.443994	+	0.896030i	$0.646439\pi$
$164$	301492.	0.875318
$165$	0	0
$166$	58189.9	0.163899
$167$	−644636.	−1.78864	−0.894321	−	0.447425i	$-0.852341\pi$
$167$	−644636.	−1.78864	−0.894321	+	0.447425i	$0.852341\pi$
$168$	0	0
$169$	−340198.	−0.916251
$170$	71550.5	0.189885
$171$	0	0
$172$	241760.	0.623108
$173$	363211.	0.922664	0.461332	−	0.887228i	$-0.347372\pi$
$173$	363211.	0.922664	0.461332	+	0.887228i	$0.347372\pi$
$174$	0	0
$175$	5958.45	0.0147075
$176$	−45129.8	−0.109820
$177$	0	0
$178$	−117828.	−0.278739
$179$	127136.	0.296576	0.148288	−	0.988944i	$-0.452624\pi$
$179$	127136.	0.296576	0.148288	+	0.988944i	$0.452624\pi$
$180$	0	0
$181$	445356.	1.01044	0.505220	−	0.862990i	$-0.331411\pi$
$181$	445356.	1.01044	0.505220	+	0.862990i	$0.331411\pi$
$182$	1665.08	0.00372613
$183$	0	0
$184$	98583.1	0.214663
$185$	343182.	0.737217
$186$	0	0
$187$	289130.	0.604630
$188$	−167999.	−0.346667
$189$	0	0
$190$	−317804.	−0.638668
$191$	658753.	1.30659	0.653295	−	0.757104i	$-0.273386\pi$
$191$	658753.	1.30659	0.653295	+	0.757104i	$0.273386\pi$
$192$	0	0
$193$	−120143.	−0.232170	−0.116085	−	0.993239i	$-0.537035\pi$
$193$	−120143.	−0.232170	−0.116085	+	0.993239i	$0.537035\pi$
$194$	−2750.33	−0.00524662
$195$	0	0
$196$	306470.	0.569834
$197$	−728986.	−1.33830	−0.669150	−	0.743127i	$-0.733342\pi$
$197$	−728986.	−1.33830	−0.669150	+	0.743127i	$0.733342\pi$
$198$	0	0
$199$	1.02356e6	1.83224	0.916120	−	0.400905i	$-0.131304\pi$
$199$	1.02356e6	1.83224	0.916120	+	0.400905i	$0.131304\pi$
$200$	436188.	0.771078
$201$	0	0
$202$	−626997.	−1.08115
$203$	18430.9	0.0313910
$204$	0	0
$205$	−462892.	−0.769299
$206$	−331627.	−0.544480
$207$	0	0
$208$	18957.3	0.0303822
$209$	−1.28422e6	−2.03364
$210$	0	0
$211$	−474714.	−0.734051	−0.367026	−	0.930211i	$-0.619624\pi$
$211$	−474714.	−0.734051	−0.367026	+	0.930211i	$0.619624\pi$
$212$	188688.	0.288339
$213$	0	0
$214$	432114.	0.645006
$215$	−371183.	−0.547636
$216$	0	0
$217$	7190.28	0.0103657
$218$	415357.	0.591943
$219$	0	0
$220$	−214471.	−0.298752
$221$	−121453.	−0.167274
$222$	0	0
$223$	88378.4	0.119010	0.0595051	−	0.998228i	$-0.481048\pi$
$223$	88378.4	0.119010	0.0595051	+	0.998228i	$0.481048\pi$
$224$	−14166.0	−0.0188637
$225$	0	0
$226$	763429.	0.994254
$227$	608695.	0.784035	0.392017	−	0.919958i	$-0.371777\pi$
$227$	608695.	0.784035	0.392017	+	0.919958i	$0.371777\pi$
$228$	0	0
$229$	−735727.	−0.927103	−0.463552	−	0.886070i	$-0.653425\pi$
$229$	−735727.	−0.927103	−0.463552	+	0.886070i	$0.653425\pi$
$230$	−54955.0	−0.0684996
$231$	0	0
$232$	1.34923e6	1.64576
$233$	−671354.	−0.810143	−0.405071	−	0.914285i	$-0.632753\pi$
$233$	−671354.	−0.810143	−0.405071	+	0.914285i	$0.632753\pi$
$234$	0	0
$235$	257936.	0.304678
$236$	641925.	0.750248
$237$	0	0
$238$	−6503.53	−0.00744230
$239$	−293973.	−0.332899	−0.166449	−	0.986050i	$-0.553230\pi$
$239$	−293973.	−0.332899	−0.166449	+	0.986050i	$0.553230\pi$
$240$	0	0
$241$	−829861.	−0.920370	−0.460185	−	0.887823i	$-0.652217\pi$
$241$	−829861.	−0.920370	−0.460185	+	0.887823i	$0.652217\pi$
$242$	56281.2	0.0617768
$243$	0	0
$244$	511384.	0.549886
$245$	−470535.	−0.500815
$246$	0	0
$247$	539455.	0.562617
$248$	526364.	0.543447
$249$	0	0
$250$	−567792.	−0.574565
$251$	−55249.0	−0.0553529	−0.0276764	−	0.999617i	$-0.508811\pi$
$251$	−55249.0	−0.0553529	−0.0276764	+	0.999617i	$0.508811\pi$
$252$	0	0
$253$	−222069.	−0.218116
$254$	−183327.	−0.178296
$255$	0	0
$256$	−1.09977e6	−1.04883
$257$	−2.00872e6	−1.89709	−0.948544	−	0.316645i	$-0.897444\pi$
$257$	−2.00872e6	−1.89709	−0.948544	+	0.316645i	$0.897444\pi$
$258$	0	0
$259$	−31193.3	−0.0288943
$260$	90091.2	0.0826512
$261$	0	0
$262$	485686.	0.437122
$263$	−1.10159e6	−0.982047	−0.491023	−	0.871146i	$-0.663377\pi$
$263$	−1.10159e6	−0.982047	−0.491023	+	0.871146i	$0.663377\pi$
$264$	0	0
$265$	−289699.	−0.253415
$266$	28886.6	0.0250318
$267$	0	0
$268$	−1.16870e6	−0.993957
$269$	216461.	0.182389	0.0911947	−	0.995833i	$-0.470931\pi$
$269$	216461.	0.182389	0.0911947	+	0.995833i	$0.470931\pi$
$270$	0	0
$271$	1.16127e6	0.960531	0.480265	−	0.877123i	$-0.340540\pi$
$271$	1.16127e6	0.960531	0.480265	+	0.877123i	$0.340540\pi$
$272$	−74044.2	−0.0606832
$273$	0	0
$274$	1.05752e6	0.850968
$275$	−982561.	−0.783480
$276$	0	0
$277$	−128134.	−0.100338	−0.0501690	−	0.998741i	$-0.515976\pi$
$277$	−128134.	−0.100338	−0.0501690	+	0.998741i	$0.515976\pi$
$278$	62048.2	0.0481523
$279$	0	0
$280$	13286.9	0.0101281
$281$	−1.55063e6	−1.17150	−0.585749	−	0.810492i	$-0.699200\pi$
$281$	−1.55063e6	−1.17150	−0.585749	+	0.810492i	$0.699200\pi$
$282$	0	0
$283$	−226590.	−0.168180	−0.0840900	−	0.996458i	$-0.526798\pi$
$283$	−226590.	−0.168180	−0.0840900	+	0.996458i	$0.526798\pi$
$284$	1.04159e6	0.766301
$285$	0	0
$286$	−274576.	−0.198494
$287$	42074.3	0.0301517
$288$	0	0
$289$	−945483.	−0.665900
$290$	−752127.	−0.525165
$291$	0	0
$292$	−1.35500e6	−0.929998
$293$	322570.	0.219510	0.109755	−	0.993959i	$-0.464993\pi$
$293$	322570.	0.219510	0.109755	+	0.993959i	$0.464993\pi$
$294$	0	0
$295$	−985573.	−0.659377
$296$	−2.28350e6	−1.51486
$297$	0	0
$298$	469606.	0.306333
$299$	93283.1	0.0603427
$300$	0	0
$301$	33738.5	0.0214639
$302$	1.39797e6	0.882024
$303$	0	0
$304$	328880.	0.204105
$305$	−785147.	−0.483283
$306$	0	0
$307$	−930722.	−0.563604	−0.281802	−	0.959473i	$-0.590932\pi$
$307$	−930722.	−0.563604	−0.281802	+	0.959473i	$0.590932\pi$
$308$	19494.2	0.0117092
$309$	0	0
$310$	−293421.	−0.173415
$311$	−2.93697e6	−1.72186	−0.860932	−	0.508720i	$-0.830119\pi$
$311$	−2.93697e6	−1.72186	−0.860932	+	0.508720i	$0.830119\pi$
$312$	0	0
$313$	35202.7	0.0203103	0.0101551	−	0.999948i	$-0.496767\pi$
$313$	35202.7	0.0203103	0.0101551	+	0.999948i	$0.496767\pi$
$314$	1.18170e6	0.676371
$315$	0	0
$316$	−69924.1	−0.0393921
$317$	647766.	0.362051	0.181025	−	0.983478i	$-0.442058\pi$
$317$	647766.	0.362051	0.181025	+	0.983478i	$0.442058\pi$
$318$	0	0
$319$	−3.03929e6	−1.67223
$320$	674435.	0.368184
$321$	0	0
$322$	4995.10	0.00268476
$323$	−2.10702e6	−1.12373
$324$	0	0
$325$	412737.	0.216753
$326$	−1.11727e6	−0.582257
$327$	0	0
$328$	3.08004e6	1.58078
$329$	−23444.9	−0.0119415
$330$	0	0
$331$	808484.	0.405603	0.202802	−	0.979220i	$-0.434995\pi$
$331$	808484.	0.405603	0.202802	+	0.979220i	$0.434995\pi$
$332$	−286174.	−0.142490
$333$	0	0
$334$	−2.39110e6	−1.17282
$335$	1.79436e6	0.873568
$336$	0	0
$337$	−1.20728e6	−0.579072	−0.289536	−	0.957167i	$-0.593501\pi$
$337$	−1.20728e6	−0.579072	−0.289536	+	0.957167i	$0.593501\pi$
$338$	−1.26187e6	−0.600789
$339$	0	0
$340$	−351881.	−0.165081
$341$	−1.18569e6	−0.552187
$342$	0	0
$343$	85554.5	0.0392652
$344$	2.46982e6	1.12530
$345$	0	0
$346$	1.34723e6	0.604994
$347$	2.72187e6	1.21351	0.606756	−	0.794888i	$-0.292470\pi$
$347$	2.72187e6	1.21351	0.606756	+	0.794888i	$0.292470\pi$
$348$	0	0
$349$	560130.	0.246164	0.123082	−	0.992396i	$-0.460722\pi$
$349$	560130.	0.246164	0.123082	+	0.992396i	$0.460722\pi$
$350$	22101.2	0.00964374
$351$	0	0
$352$	2.33600e6	1.00488
$353$	4.01855e6	1.71646	0.858229	−	0.513267i	$-0.171565\pi$
$353$	4.01855e6	1.71646	0.858229	+	0.513267i	$0.171565\pi$
$354$	0	0
$355$	−1.59919e6	−0.673486
$356$	579471.	0.242330
$357$	0	0
$358$	471575.	0.194466
$359$	−3.13021e6	−1.28185	−0.640925	−	0.767603i	$-0.721449\pi$
$359$	−3.13021e6	−1.28185	−0.640925	+	0.767603i	$0.721449\pi$
$360$	0	0
$361$	6.88260e6	2.77961
$362$	1.65192e6	0.662549
$363$	0	0
$364$	−8188.78	−0.00323941
$365$	2.08038e6	0.817355
$366$	0	0
$367$	2.91054e6	1.12800	0.564000	−	0.825775i	$-0.309262\pi$
$367$	2.91054e6	1.12800	0.564000	+	0.825775i	$0.309262\pi$
$368$	56870.4	0.0218910
$369$	0	0
$370$	1.27294e6	0.483396
$371$	26332.1	0.00993230
$372$	0	0
$373$	1.13055e6	0.420743	0.210372	−	0.977621i	$-0.432533\pi$
$373$	1.13055e6	0.420743	0.210372	+	0.977621i	$0.432533\pi$
$374$	1.07245e6	0.396458
$375$	0	0
$376$	−1.71628e6	−0.626064
$377$	1.27669e6	0.462629
$378$	0	0
$379$	−2.43769e6	−0.871727	−0.435863	−	0.900013i	$-0.643557\pi$
$379$	−2.43769e6	−0.871727	−0.435863	+	0.900013i	$0.643557\pi$
$380$	1.56294e6	0.555244
$381$	0	0
$382$	2.44346e6	0.856735
$383$	−2.60318e6	−0.906792	−0.453396	−	0.891309i	$-0.649788\pi$
$383$	−2.60318e6	−0.906792	−0.453396	+	0.891309i	$0.649788\pi$
$384$	0	0
$385$	−29930.2	−0.0102910
$386$	−445638.	−0.152235
$387$	0	0
$388$	13525.9	0.00456129
$389$	−2.98208e6	−0.999184	−0.499592	−	0.866261i	$-0.666517\pi$
$389$	−2.98208e6	−0.999184	−0.499592	+	0.866261i	$0.666517\pi$
$390$	0	0
$391$	−364348.	−0.120524
$392$	3.13090e6	1.02909
$393$	0	0
$394$	−2.70397e6	−0.877528
$395$	107357.	0.0346209
$396$	0	0
$397$	−3.47355e6	−1.10611	−0.553054	−	0.833145i	$-0.686538\pi$
$397$	−3.47355e6	−1.10611	−0.553054	+	0.833145i	$0.686538\pi$
$398$	3.79662e6	1.20140
$399$	0	0
$400$	251627.	0.0786334
$401$	1.75661e6	0.545525	0.272763	−	0.962081i	$-0.412063\pi$
$401$	1.75661e6	0.545525	0.272763	+	0.962081i	$0.412063\pi$
$402$	0	0
$403$	498066.	0.152765
$404$	3.08353e6	0.939930
$405$	0	0
$406$	68364.1	0.0205832
$407$	5.14385e6	1.53923
$408$	0	0
$409$	377309.	0.111529	0.0557646	−	0.998444i	$-0.482240\pi$
$409$	377309.	0.111529	0.0557646	+	0.998444i	$0.482240\pi$
$410$	−1.71697e6	−0.504432
$411$	0	0
$412$	1.63092e6	0.473359
$413$	89583.0	0.0258435
$414$	0	0
$415$	439374.	0.125232
$416$	−981267.	−0.278006
$417$	0	0
$418$	−4.76346e6	−1.33347
$419$	3.37374e6	0.938808	0.469404	−	0.882984i	$-0.344469\pi$
$419$	3.37374e6	0.938808	0.469404	+	0.882984i	$0.344469\pi$
$420$	0	0
$421$	−3.05947e6	−0.841280	−0.420640	−	0.907228i	$-0.638194\pi$
$421$	−3.05947e6	−0.841280	−0.420640	+	0.907228i	$0.638194\pi$
$422$	−1.76082e6	−0.481320
$423$	0	0
$424$	1.92764e6	0.520727
$425$	−1.61208e6	−0.432927
$426$	0	0
$427$	71365.5	0.0189417
$428$	−2.12511e6	−0.560754
$429$	0	0
$430$	−1.37680e6	−0.359087
$431$	5.13524e6	1.33158	0.665791	−	0.746139i	$-0.268094\pi$
$431$	5.13524e6	1.33158	0.665791	+	0.746139i	$0.268094\pi$
$432$	0	0
$433$	6.02130e6	1.54337	0.771686	−	0.636004i	$-0.219414\pi$
$433$	6.02130e6	1.54337	0.771686	+	0.636004i	$0.219414\pi$
$434$	26670.3	0.00679679
$435$	0	0
$436$	−2.04270e6	−0.514622
$437$	1.61832e6	0.405378
$438$	0	0
$439$	1.70668e6	0.422661	0.211330	−	0.977415i	$-0.432220\pi$
$439$	1.70668e6	0.422661	0.211330	+	0.977415i	$0.432220\pi$
$440$	−2.19103e6	−0.539532
$441$	0	0
$442$	−450495.	−0.109682
$443$	6.41456e6	1.55295	0.776476	−	0.630147i	$-0.217006\pi$
$443$	6.41456e6	1.55295	0.776476	+	0.630147i	$0.217006\pi$
$444$	0	0
$445$	−889684.	−0.212979
$446$	327815.	0.0780353
$447$	0	0
$448$	−61302.3	−0.0144305
$449$	−249022.	−0.0582936	−0.0291468	−	0.999575i	$-0.509279\pi$
$449$	−249022.	−0.0582936	−0.0291468	+	0.999575i	$0.509279\pi$
$450$	0	0
$451$	−6.93814e6	−1.60621
$452$	−3.75450e6	−0.864382
$453$	0	0
$454$	2.25778e6	0.514094
$455$	12572.5	0.00284705
$456$	0	0
$457$	3.25907e6	0.729966	0.364983	−	0.931014i	$-0.381075\pi$
$457$	3.25907e6	0.729966	0.364983	+	0.931014i	$0.381075\pi$
$458$	−2.72897e6	−0.607905
$459$	0	0
$460$	270266.	0.0595520
$461$	−3.44746e6	−0.755521	−0.377761	−	0.925903i	$-0.623306\pi$
$461$	−3.44746e6	−0.755521	−0.377761	+	0.925903i	$0.623306\pi$
$462$	0	0
$463$	2.31741e6	0.502400	0.251200	−	0.967935i	$-0.419175\pi$
$463$	2.31741e6	0.502400	0.251200	+	0.967935i	$0.419175\pi$
$464$	778340.	0.167832
$465$	0	0
$466$	−2.49020e6	−0.531213
$467$	−4.01087e6	−0.851034	−0.425517	−	0.904950i	$-0.639908\pi$
$467$	−4.01087e6	−0.851034	−0.425517	+	0.904950i	$0.639908\pi$
$468$	0	0
$469$	−163097.	−0.0342384
$470$	956739.	0.199779
$471$	0	0
$472$	6.55791e6	1.35491
$473$	−5.56355e6	−1.14340
$474$	0	0
$475$	7.16035e6	1.45613
$476$	31984.0	0.00647016
$477$	0	0
$478$	−1.09041e6	−0.218283
$479$	8.46170e6	1.68507	0.842536	−	0.538640i	$-0.181062\pi$
$479$	8.46170e6	1.68507	0.842536	+	0.538640i	$0.181062\pi$
$480$	0	0
$481$	−2.16074e6	−0.425833
$482$	−3.07813e6	−0.603490
$483$	0	0
$484$	−276788.	−0.0537073
$485$	−20766.9	−0.00400883
$486$	0	0
$487$	−3.49903e6	−0.668537	−0.334268	−	0.942478i	$-0.608489\pi$
$487$	−3.49903e6	−0.668537	−0.334268	+	0.942478i	$0.608489\pi$
$488$	5.22430e6	0.993068
$489$	0	0
$490$	−1.74532e6	−0.328386
$491$	297532.	0.0556967	0.0278484	−	0.999612i	$-0.491134\pi$
$491$	297532.	0.0556967	0.0278484	+	0.999612i	$0.491134\pi$
$492$	0	0
$493$	−4.98655e6	−0.924023
$494$	2.00095e6	0.368909
$495$	0	0
$496$	303647.	0.0554199
$497$	145357.	0.0263964
$498$	0	0
$499$	−8.52738e6	−1.53308	−0.766539	−	0.642198i	$-0.778023\pi$
$499$	−8.52738e6	−1.53308	−0.766539	+	0.642198i	$0.778023\pi$
$500$	2.79237e6	0.499514
$501$	0	0
$502$	−204931.	−0.0362951
$503$	1.77175e6	0.312235	0.156118	−	0.987738i	$-0.450102\pi$
$503$	1.77175e6	0.312235	0.156118	+	0.987738i	$0.450102\pi$
$504$	0	0
$505$	−4.73427e6	−0.826084
$506$	−823703.	−0.143019
$507$	0	0
$508$	901592.	0.155007
$509$	2.05853e6	0.352178	0.176089	−	0.984374i	$-0.443655\pi$
$509$	2.05853e6	0.352178	0.176089	+	0.984374i	$0.443655\pi$
$510$	0	0
$511$	−189095.	−0.0320352
$512$	−1.23933e6	−0.208936
$513$	0	0
$514$	−7.45080e6	−1.24393
$515$	−2.50402e6	−0.416025
$516$	0	0
$517$	3.86611e6	0.636133
$518$	−115703.	−0.0189461
$519$	0	0
$520$	920372.	0.149264
$521$	−8.50254e6	−1.37232	−0.686159	−	0.727452i	$-0.740704\pi$
$521$	−8.50254e6	−1.37232	−0.686159	+	0.727452i	$0.740704\pi$
$522$	0	0
$523$	3.82452e6	0.611396	0.305698	−	0.952129i	$-0.401110\pi$
$523$	3.82452e6	0.611396	0.305698	+	0.952129i	$0.401110\pi$
$524$	−2.38857e6	−0.380024
$525$	0	0
$526$	−4.08605e6	−0.643931
$527$	−1.94536e6	−0.305122
$528$	0	0
$529$	279841.	0.0434783
$530$	−1.07456e6	−0.166165
$531$	0	0
$532$	−142063.	−0.0217621
$533$	2.91445e6	0.444364
$534$	0	0
$535$	3.26277e6	0.492835
$536$	−1.19395e7	−1.79504
$537$	0	0
$538$	802902.	0.119593
$539$	−7.05270e6	−1.04564
$540$	0	0
$541$	1.76283e6	0.258951	0.129475	−	0.991583i	$-0.458671\pi$
$541$	1.76283e6	0.258951	0.129475	+	0.991583i	$0.458671\pi$
$542$	4.30742e6	0.629823
$543$	0	0
$544$	3.83266e6	0.555269
$545$	3.13623e6	0.452290
$546$	0	0
$547$	−5.91089e6	−0.844665	−0.422332	−	0.906441i	$-0.638788\pi$
$547$	−5.91089e6	−0.844665	−0.422332	+	0.906441i	$0.638788\pi$
$548$	−5.20083e6	−0.739812
$549$	0	0
$550$	−3.64453e6	−0.513730
$551$	2.21486e7	3.10791
$552$	0	0
$553$	−9758.16	−0.00135692
$554$	−475277.	−0.0657920
$555$	0	0
$556$	−305149.	−0.0418625
$557$	−1.30751e7	−1.78569	−0.892845	−	0.450364i	$-0.851294\pi$
$557$	−1.30751e7	−1.78569	−0.892845	+	0.450364i	$0.851294\pi$
$558$	0	0
$559$	2.33704e6	0.316327
$560$	7664.90	0.00103285
$561$	0	0
$562$	−5.75162e6	−0.768155
$563$	−9.57017e6	−1.27247	−0.636237	−	0.771494i	$-0.719510\pi$
$563$	−9.57017e6	−1.27247	−0.636237	+	0.771494i	$0.719510\pi$
$564$	0	0
$565$	5.76442e6	0.759687
$566$	−840471.	−0.110276
$567$	0	0
$568$	1.06408e7	1.38390
$569$	−497195.	−0.0643793	−0.0321897	−	0.999482i	$-0.510248\pi$
$569$	−497195.	−0.0643793	−0.0321897	+	0.999482i	$0.510248\pi$
$570$	0	0
$571$	−2.53185e6	−0.324974	−0.162487	−	0.986711i	$-0.551952\pi$
$571$	−2.53185e6	−0.324974	−0.162487	+	0.986711i	$0.551952\pi$
$572$	1.35035e6	0.172566
$573$	0	0
$574$	156063.	0.0197706
$575$	1.23818e6	0.156175
$576$	0	0
$577$	9.24807e6	1.15641	0.578205	−	0.815892i	$-0.303753\pi$
$577$	9.24807e6	1.15641	0.578205	+	0.815892i	$0.303753\pi$
$578$	−3.50700e6	−0.436633
$579$	0	0
$580$	3.69891e6	0.456567
$581$	−39936.7	−0.00490830
$582$	0	0
$583$	−4.34221e6	−0.529102
$584$	−1.38427e7	−1.67953
$585$	0	0
$586$	1.19648e6	0.143934
$587$	1.46944e6	0.176018	0.0880089	−	0.996120i	$-0.471950\pi$
$587$	1.46944e6	0.176018	0.0880089	+	0.996120i	$0.471950\pi$
$588$	0	0
$589$	8.64066e6	1.02626
$590$	−3.65570e6	−0.432355
$591$	0	0
$592$	−1.31730e6	−0.154483
$593$	−5.86571e6	−0.684989	−0.342495	−	0.939520i	$-0.611272\pi$
$593$	−5.86571e6	−0.684989	−0.342495	+	0.939520i	$0.611272\pi$
$594$	0	0
$595$	−49106.2	−0.00568649
$596$	−2.30950e6	−0.266319
$597$	0	0
$598$	346007.	0.0395669
$599$	5.13745e6	0.585034	0.292517	−	0.956260i	$-0.405507\pi$
$599$	5.13745e6	0.585034	0.292517	+	0.956260i	$0.405507\pi$
$600$	0	0
$601$	−6.94742e6	−0.784580	−0.392290	−	0.919842i	$-0.628317\pi$
$601$	−6.94742e6	−0.784580	−0.392290	+	0.919842i	$0.628317\pi$
$602$	125143.	0.0140740
$603$	0	0
$604$	−6.87513e6	−0.766812
$605$	424963.	0.0472022
$606$	0	0
$607$	1.06646e7	1.17483	0.587413	−	0.809287i	$-0.300146\pi$
$607$	1.06646e7	1.17483	0.587413	+	0.809287i	$0.300146\pi$
$608$	−1.70234e7	−1.86762
$609$	0	0
$610$	−2.91228e6	−0.316890
$611$	−1.62401e6	−0.175989
$612$	0	0
$613$	−1.02692e7	−1.10379	−0.551893	−	0.833915i	$-0.686094\pi$
$613$	−1.02692e7	−1.10379	−0.551893	+	0.833915i	$0.686094\pi$
$614$	−3.45225e6	−0.369557
$615$	0	0
$616$	199153.	0.0211463
$617$	59395.1	0.00628113	0.00314057	−	0.999995i	$-0.499000\pi$
$617$	59395.1	0.00628113	0.00314057	+	0.999995i	$0.499000\pi$
$618$	0	0
$619$	1.89387e7	1.98666	0.993329	−	0.115311i	$-0.0367865\pi$
$619$	1.89387e7	1.98666	0.993329	+	0.115311i	$0.0367865\pi$
$620$	1.44303e6	0.150763
$621$	0	0
$622$	−1.08939e7	−1.12903
$623$	80867.3	0.00834743
$624$	0	0
$625$	3.02714e6	0.309979
$626$	130575.	0.0133175
$627$	0	0
$628$	−5.81155e6	−0.588021
$629$	8.43948e6	0.850529
$630$	0	0
$631$	8.00077e6	0.799942	0.399971	−	0.916528i	$-0.369020\pi$
$631$	8.00077e6	0.799942	0.399971	+	0.916528i	$0.369020\pi$
$632$	−714345.	−0.0711403
$633$	0	0
$634$	2.40270e6	0.237398
$635$	−1.38425e6	−0.136232
$636$	0	0
$637$	2.96258e6	0.289282
$638$	−1.12734e7	−1.09649
$639$	0	0
$640$	−2.48561e6	−0.239874
$641$	7.69346e6	0.739566	0.369783	−	0.929118i	$-0.379432\pi$
$641$	7.69346e6	0.739566	0.369783	+	0.929118i	$0.379432\pi$
$642$	0	0
$643$	2.37208e6	0.226257	0.113128	−	0.993580i	$-0.463913\pi$
$643$	2.37208e6	0.226257	0.113128	+	0.993580i	$0.463913\pi$
$644$	−24565.6	−0.00233407
$645$	0	0
$646$	−7.81539e6	−0.736833
$647$	−3.97133e6	−0.372972	−0.186486	−	0.982458i	$-0.559710\pi$
$647$	−3.97133e6	−0.372972	−0.186486	+	0.982458i	$0.559710\pi$
$648$	0	0
$649$	−1.47724e7	−1.37670
$650$	1.53093e6	0.142126
$651$	0	0
$652$	5.49467e6	0.506201
$653$	−4.09240e6	−0.375573	−0.187787	−	0.982210i	$-0.560131\pi$
$653$	−4.09240e6	−0.375573	−0.187787	+	0.982210i	$0.560131\pi$
$654$	0	0
$655$	3.66727e6	0.333995
$656$	1.77681e6	0.161206
$657$	0	0
$658$	−86962.2	−0.00783007
$659$	4.63360e6	0.415629	0.207814	−	0.978168i	$-0.433365\pi$
$659$	4.63360e6	0.415629	0.207814	+	0.978168i	$0.433365\pi$
$660$	0	0
$661$	5.40785e6	0.481417	0.240708	−	0.970597i	$-0.422620\pi$
$661$	5.40785e6	0.481417	0.240708	+	0.970597i	$0.422620\pi$
$662$	2.99884e6	0.265955
$663$	0	0
$664$	−2.92356e6	−0.257331
$665$	218114.	0.0191262
$666$	0	0
$667$	3.82997e6	0.333335
$668$	1.17593e7	1.01962
$669$	0	0
$670$	6.65566e6	0.572801
$671$	−1.17683e7	−1.00904
$672$	0	0
$673$	1.29325e7	1.10064	0.550321	−	0.834953i	$-0.314505\pi$
$673$	1.29325e7	1.10064	0.550321	+	0.834953i	$0.314505\pi$
$674$	−4.47806e6	−0.379699
$675$	0	0
$676$	6.20579e6	0.522312
$677$	6.95322e6	0.583061	0.291531	−	0.956562i	$-0.405836\pi$
$677$	6.95322e6	0.583061	0.291531	+	0.956562i	$0.405836\pi$
$678$	0	0
$679$	1887.59	0.000157121 0
$680$	−3.59482e6	−0.298129
$681$	0	0
$682$	−4.39799e6	−0.362071
$683$	1.94686e7	1.59692	0.798458	−	0.602050i	$-0.205649\pi$
$683$	1.94686e7	1.59692	0.798458	+	0.602050i	$0.205649\pi$
$684$	0	0
$685$	7.98503e6	0.650205
$686$	317340.	0.0257463
$687$	0	0
$688$	1.42478e6	0.114757
$689$	1.82400e6	0.146379
$690$	0	0
$691$	1.24405e7	0.991154	0.495577	−	0.868564i	$-0.334957\pi$
$691$	1.24405e7	0.991154	0.495577	+	0.868564i	$0.334957\pi$
$692$	−6.62559e6	−0.525968
$693$	0	0
$694$	1.00960e7	0.795704
$695$	468508.	0.0367921
$696$	0	0
$697$	−1.13834e7	−0.887542
$698$	2.07764e6	0.161411
$699$	0	0
$700$	−108692.	−0.00838405
$701$	−1.29918e7	−0.998561	−0.499281	−	0.866440i	$-0.666402\pi$
$701$	−1.29918e7	−0.998561	−0.499281	+	0.866440i	$0.666402\pi$
$702$	0	0
$703$	−3.74855e7	−2.86071
$704$	1.01089e7	0.768726
$705$	0	0
$706$	1.49057e7	1.12549
$707$	430318.	0.0323774
$708$	0	0
$709$	−1.23928e7	−0.925875	−0.462938	−	0.886391i	$-0.653205\pi$
$709$	−1.23928e7	−0.925875	−0.462938	+	0.886391i	$0.653205\pi$
$710$	−5.93173e6	−0.441607
$711$	0	0
$712$	5.91988e6	0.437636
$713$	1.49415e6	0.110071
$714$	0	0
$715$	−2.07324e6	−0.151665
$716$	−2.31918e6	−0.169064
$717$	0	0
$718$	−1.16106e7	−0.840513
$719$	2.12541e7	1.53328	0.766640	−	0.642078i	$-0.221927\pi$
$719$	2.12541e7	1.53328	0.766640	+	0.642078i	$0.221927\pi$
$720$	0	0
$721$	227601.	0.0163056
$722$	2.55290e7	1.82260
$723$	0	0
$724$	−8.12405e6	−0.576005
$725$	1.69460e7	1.19735
$726$	0	0
$727$	−8.68787e6	−0.609645	−0.304823	−	0.952409i	$-0.598597\pi$
$727$	−8.68787e6	−0.609645	−0.304823	+	0.952409i	$0.598597\pi$
$728$	−83656.6	−0.00585022
$729$	0	0
$730$	7.71659e6	0.535943
$731$	−9.12808e6	−0.631809
$732$	0	0
$733$	−2.42075e7	−1.66414	−0.832070	−	0.554671i	$-0.812844\pi$
$733$	−2.42075e7	−1.66414	−0.832070	+	0.554671i	$0.812844\pi$
$734$	1.07958e7	0.739633
$735$	0	0
$736$	−2.94371e6	−0.200309
$737$	2.68950e7	1.82391
$738$	0	0
$739$	−1.67373e7	−1.12739	−0.563697	−	0.825982i	$-0.690621\pi$
$739$	−1.67373e7	−1.12739	−0.563697	+	0.825982i	$0.690621\pi$
$740$	−6.26023e6	−0.420253
$741$	0	0
$742$	97671.3	0.00651264
$743$	9.60546e6	0.638332	0.319166	−	0.947699i	$-0.396597\pi$
$743$	9.60546e6	0.638332	0.319166	+	0.947699i	$0.396597\pi$
$744$	0	0
$745$	3.54586e6	0.234062
$746$	4.19345e6	0.275883
$747$	0	0
$748$	−5.27423e6	−0.344671
$749$	−296567.	−0.0193160
$750$	0	0
$751$	2.58992e6	0.167566	0.0837830	−	0.996484i	$-0.473300\pi$
$751$	2.58992e6	0.167566	0.0837830	+	0.996484i	$0.473300\pi$
$752$	−990084.	−0.0638451
$753$	0	0
$754$	4.73553e6	0.303347
$755$	1.05557e7	0.673934
$756$	0	0
$757$	2.17648e7	1.38043	0.690215	−	0.723605i	$-0.257516\pi$
$757$	2.17648e7	1.38043	0.690215	+	0.723605i	$0.257516\pi$
$758$	−9.04192e6	−0.571594
$759$	0	0
$760$	1.59670e7	1.00274
$761$	−1.72843e7	−1.08190	−0.540952	−	0.841053i	$-0.681936\pi$
$761$	−1.72843e7	−1.08190	−0.540952	+	0.841053i	$0.681936\pi$
$762$	0	0
$763$	−285066.	−0.0177270
$764$	−1.20168e7	−0.744826
$765$	0	0
$766$	−9.65577e6	−0.594586
$767$	6.20535e6	0.380871
$768$	0	0
$769$	−2.45978e7	−1.49996	−0.749980	−	0.661460i	$-0.769937\pi$
$769$	−2.45978e7	−1.49996	−0.749980	+	0.661460i	$0.769937\pi$
$770$	−111017.	−0.00674784
$771$	0	0
$772$	2.19162e6	0.132349
$773$	−1.76885e7	−1.06474	−0.532369	−	0.846513i	$-0.678698\pi$
$773$	−1.76885e7	−1.06474	−0.532369	+	0.846513i	$0.678698\pi$
$774$	0	0
$775$	6.61098e6	0.395377
$776$	138181.	0.00823748
$777$	0	0
$778$	−1.10612e7	−0.655168
$779$	5.05612e7	2.98520
$780$	0	0
$781$	−2.39697e7	−1.40616
$782$	−1.35145e6	−0.0790281
$783$	0	0
$784$	1.80615e6	0.104945
$785$	8.92270e6	0.516800
$786$	0	0
$787$	6.45182e6	0.371318	0.185659	−	0.982614i	$-0.440558\pi$
$787$	6.45182e6	0.371318	0.185659	+	0.982614i	$0.440558\pi$
$788$	1.32980e7	0.762903
$789$	0	0
$790$	398211.	0.0227010
$791$	−523954.	−0.0297750
$792$	0	0
$793$	4.94344e6	0.279155
$794$	−1.28842e7	−0.725278
$795$	0	0
$796$	−1.86715e7	−1.04447
$797$	−2.94207e7	−1.64062	−0.820308	−	0.571922i	$-0.806198\pi$
$797$	−2.94207e7	−1.64062	−0.820308	+	0.571922i	$0.806198\pi$
$798$	0	0
$799$	6.34311e6	0.351508
$800$	−1.30247e7	−0.719518
$801$	0	0
$802$	6.51565e6	0.357702
$803$	3.11822e7	1.70654
$804$	0	0
$805$	37716.5	0.00205136
$806$	1.84743e6	0.100169
$807$	0	0
$808$	3.15014e7	1.69747
$809$	−3.48377e7	−1.87145	−0.935726	−	0.352728i	$-0.885254\pi$
$809$	−3.48377e7	−1.87145	−0.935726	+	0.352728i	$0.885254\pi$
$810$	0	0
$811$	−1.14935e7	−0.613620	−0.306810	−	0.951771i	$-0.599262\pi$
$811$	−1.14935e7	−0.613620	−0.306810	+	0.951771i	$0.599262\pi$
$812$	−336211.	−0.0178946
$813$	0	0
$814$	1.90797e7	1.00927
$815$	−8.43618e6	−0.444889
$816$	0	0
$817$	4.05440e7	2.12506
$818$	1.39952e6	0.0731300
$819$	0	0
$820$	8.44394e6	0.438541
$821$	1.52928e7	0.791825	0.395912	−	0.918288i	$-0.370428\pi$
$821$	1.52928e7	0.791825	0.395912	+	0.918288i	$0.370428\pi$
$822$	0	0
$823$	−1.91693e7	−0.986522	−0.493261	−	0.869881i	$-0.664195\pi$
$823$	−1.91693e7	−0.986522	−0.493261	+	0.869881i	$0.664195\pi$
$824$	1.66615e7	0.854863
$825$	0	0
$826$	332283.	0.0169456
$827$	2.45097e7	1.24616	0.623080	−	0.782158i	$-0.285881\pi$
$827$	2.45097e7	1.24616	0.623080	+	0.782158i	$0.285881\pi$
$828$	0	0
$829$	−3.48721e7	−1.76235	−0.881174	−	0.472793i	$-0.843246\pi$
$829$	−3.48721e7	−1.76235	−0.881174	+	0.472793i	$0.843246\pi$
$830$	1.62974e6	0.0821149
$831$	0	0
$832$	−4.24637e6	−0.212671
$833$	−1.15713e7	−0.577791
$834$	0	0
$835$	−1.80545e7	−0.896124
$836$	2.34264e7	1.15929
$837$	0	0
$838$	1.25139e7	0.615579
$839$	2.79370e7	1.37017	0.685085	−	0.728464i	$-0.259765\pi$
$839$	2.79370e7	1.37017	0.685085	+	0.728464i	$0.259765\pi$
$840$	0	0
$841$	3.19066e7	1.55557
$842$	−1.13482e7	−0.551630
$843$	0	0
$844$	8.65960e6	0.418448
$845$	−9.52798e6	−0.459049
$846$	0	0
$847$	−38626.7	−0.00185003
$848$	1.11201e6	0.0531030
$849$	0	0
$850$	−5.97956e6	−0.283872
$851$	−6.48203e6	−0.306822
$852$	0	0
$853$	7.32343e6	0.344621	0.172311	−	0.985043i	$-0.444877\pi$
$853$	7.32343e6	0.344621	0.172311	+	0.985043i	$0.444877\pi$
$854$	264710.	0.0124201
$855$	0	0
$856$	−2.17102e7	−1.01269
$857$	−7.95182e6	−0.369840	−0.184920	−	0.982754i	$-0.559203\pi$
$857$	−7.95182e6	−0.369840	−0.184920	+	0.982754i	$0.559203\pi$
$858$	0	0
$859$	−1.01045e7	−0.467233	−0.233616	−	0.972329i	$-0.575056\pi$
$859$	−1.01045e7	−0.467233	−0.233616	+	0.972329i	$0.575056\pi$
$860$	6.77101e6	0.312182
$861$	0	0
$862$	1.90477e7	0.873122
$863$	−2.78987e7	−1.27514	−0.637568	−	0.770394i	$-0.720060\pi$
$863$	−2.78987e7	−1.27514	−0.637568	+	0.770394i	$0.720060\pi$
$864$	0	0
$865$	1.01725e7	0.462262
$866$	2.23343e7	1.01199
$867$	0	0
$868$	−131163.	−0.00590898
$869$	1.60914e6	0.0722845
$870$	0	0
$871$	−1.12976e7	−0.504593
$872$	−2.08682e7	−0.929382
$873$	0	0
$874$	6.00269e6	0.265807
$875$	389685.	0.0172065
$876$	0	0
$877$	2.83760e7	1.24581	0.622905	−	0.782298i	$-0.285952\pi$
$877$	2.83760e7	1.24581	0.622905	+	0.782298i	$0.285952\pi$
$878$	6.33046e6	0.277140
$879$	0	0
$880$	−1.26396e6	−0.0550207
$881$	−1.82882e6	−0.0793838	−0.0396919	−	0.999212i	$-0.512638\pi$
$881$	−1.82882e6	−0.0793838	−0.0396919	+	0.999212i	$0.512638\pi$
$882$	0	0
$883$	2.28755e6	0.0987347	0.0493673	−	0.998781i	$-0.484279\pi$
$883$	2.28755e6	0.0987347	0.0493673	+	0.998781i	$0.484279\pi$
$884$	2.21551e6	0.0953548
$885$	0	0
$886$	2.37930e7	1.01827
$887$	−1.61418e7	−0.688881	−0.344440	−	0.938808i	$-0.611931\pi$
$887$	−1.61418e7	−0.688881	−0.344440	+	0.938808i	$0.611931\pi$
$888$	0	0
$889$	125820.	0.00533945
$890$	−3.30003e6	−0.139651
$891$	0	0
$892$	−1.61217e6	−0.0678421
$893$	−2.81741e7	−1.18228
$894$	0	0
$895$	3.56072e6	0.148587
$896$	225928.	0.00940155
$897$	0	0
$898$	−923675.	−0.0382233
$899$	2.04493e7	0.843878
$900$	0	0
$901$	−7.12424e6	−0.292366
$902$	−2.57351e7	−1.05320
$903$	0	0
$904$	−3.83560e7	−1.56103
$905$	1.24732e7	0.506239
$906$	0	0
$907$	2.38437e7	0.962400	0.481200	−	0.876611i	$-0.340201\pi$
$907$	2.38437e7	0.962400	0.481200	+	0.876611i	$0.340201\pi$
$908$	−1.11036e7	−0.446942
$909$	0	0
$910$	46634.3	0.00186682
$911$	−7.23611e6	−0.288875	−0.144437	−	0.989514i	$-0.546137\pi$
$911$	−7.23611e6	−0.288875	−0.144437	+	0.989514i	$0.546137\pi$
$912$	0	0
$913$	6.58564e6	0.261470
$914$	1.20886e7	0.478641
$915$	0	0
$916$	1.34209e7	0.528498
$917$	−333334.	−0.0130905
$918$	0	0
$919$	−2.62591e6	−0.102563	−0.0512815	−	0.998684i	$-0.516331\pi$
$919$	−2.62591e6	−0.102563	−0.0512815	+	0.998684i	$0.516331\pi$
$920$	2.76103e6	0.107548
$921$	0	0
$922$	−1.27874e7	−0.495398
$923$	1.00688e7	0.389021
$924$	0	0
$925$	−2.86802e7	−1.10212
$926$	8.59577e6	0.329426
$927$	0	0
$928$	−4.02883e7	−1.53571
$929$	−3.79641e6	−0.144322	−0.0721612	−	0.997393i	$-0.522990\pi$
$929$	−3.79641e6	−0.144322	−0.0721612	+	0.997393i	$0.522990\pi$
$930$	0	0
$931$	5.13961e7	1.94337
$932$	1.22466e7	0.461825
$933$	0	0
$934$	−1.48772e7	−0.558026
$935$	8.09773e6	0.302924
$936$	0	0
$937$	5.18905e6	0.193081	0.0965404	−	0.995329i	$-0.469222\pi$
$937$	5.18905e6	0.193081	0.0965404	+	0.995329i	$0.469222\pi$
$938$	−604962.	−0.0224502
$939$	0	0
$940$	−4.70519e6	−0.173683
$941$	−1.22418e7	−0.450685	−0.225342	−	0.974280i	$-0.572350\pi$
$941$	−1.22418e7	−0.450685	−0.225342	+	0.974280i	$0.572350\pi$
$942$	0	0
$943$	8.74311e6	0.320174
$944$	3.78311e6	0.138172
$945$	0	0
$946$	−2.06364e7	−0.749732
$947$	−3.29016e7	−1.19218	−0.596090	−	0.802918i	$-0.703280\pi$
$947$	−3.29016e7	−1.19218	−0.596090	+	0.802918i	$0.703280\pi$
$948$	0	0
$949$	−1.30985e7	−0.472123
$950$	2.65593e7	0.954789
$951$	0	0
$952$	326749.	0.0116848
$953$	−4.35744e7	−1.55417	−0.777086	−	0.629394i	$-0.783303\pi$
$953$	−4.35744e7	−1.55417	−0.777086	+	0.629394i	$0.783303\pi$
$954$	0	0
$955$	1.84498e7	0.654612
$956$	5.36257e6	0.189770
$957$	0	0
$958$	3.13863e7	1.10491
$959$	−725795.	−0.0254840
$960$	0	0
$961$	−2.06514e7	−0.721343
$962$	−8.01465e6	−0.279220
$963$	0	0
$964$	1.51381e7	0.524660
$965$	−3.36488e6	−0.116319
$966$	0	0
$967$	4.38755e6	0.150888	0.0754442	−	0.997150i	$-0.475963\pi$
$967$	4.38755e6	0.150888	0.0754442	+	0.997150i	$0.475963\pi$
$968$	−2.82767e6	−0.0969928
$969$	0	0
$970$	−77028.9	−0.00262860
$971$	−2.03209e7	−0.691663	−0.345831	−	0.938297i	$-0.612403\pi$
$971$	−2.03209e7	−0.691663	−0.345831	+	0.938297i	$0.612403\pi$
$972$	0	0
$973$	−42584.7	−0.00144202
$974$	−1.29787e7	−0.438362
$975$	0	0
$976$	3.01378e6	0.101272
$977$	−5.66302e7	−1.89807	−0.949034	−	0.315175i	$-0.897937\pi$
$977$	−5.66302e7	−1.89807	−0.949034	+	0.315175i	$0.897937\pi$
$978$	0	0
$979$	−1.33352e7	−0.444675
$980$	8.58337e6	0.285491
$981$	0	0
$982$	1.10361e6	0.0365205
$983$	−3.61794e7	−1.19420	−0.597100	−	0.802167i	$-0.703681\pi$
$983$	−3.61794e7	−1.19420	−0.597100	+	0.802167i	$0.703681\pi$
$984$	0	0
$985$	−2.04169e7	−0.670499
$986$	−1.84962e7	−0.605885
$987$	0	0
$988$	−9.84057e6	−0.320721
$989$	7.01091e6	0.227921
$990$	0	0
$991$	5.70206e7	1.84437	0.922184	−	0.386750i	$-0.126403\pi$
$991$	5.70206e7	1.84437	0.922184	+	0.386750i	$0.126403\pi$
$992$	−1.57173e7	−0.507108
$993$	0	0
$994$	539161.	0.0173082
$995$	2.86671e7	0.917966
$996$	0	0
$997$	−3.60612e7	−1.14895	−0.574477	−	0.818521i	$-0.694795\pi$
$997$	−3.60612e7	−1.14895	−0.574477	+	0.818521i	$0.694795\pi$
$998$	−3.16299e7	−1.00524
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.h.1.7	✓	10
3.2	odd	2		207.6.a.i.1.4	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.7	✓	10	1.1	even	1	trivial
207.6.a.i.1.4	yes	10	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+3.70922 q^{2} -18.2417 q^{4} +28.0072 q^{5} -2.54570 q^{7} -186.357 q^{8} +O(q^{10})\)	\(q+3.70922 q^{2} -18.2417 q^{4} +28.0072 q^{5} -2.54570 q^{7} -186.357 q^{8} +103.885 q^{10} +419.791 q^{11} -176.339 q^{13} -9.44254 q^{14} -107.505 q^{16} +688.748 q^{17} -3059.20 q^{19} -510.899 q^{20} +1557.10 q^{22} -529.000 q^{23} -2340.60 q^{25} -654.078 q^{26} +46.4378 q^{28} -7240.01 q^{29} -2824.49 q^{31} +5564.68 q^{32} +2554.72 q^{34} -71.2978 q^{35} +12253.4 q^{37} -11347.2 q^{38} -5219.35 q^{40} -16527.6 q^{41} -13253.1 q^{43} -7657.70 q^{44} -1962.18 q^{46} +9209.62 q^{47} -16800.5 q^{49} -8681.78 q^{50} +3216.72 q^{52} -10343.8 q^{53} +11757.2 q^{55} +474.409 q^{56} -26854.8 q^{58} -35190.0 q^{59} -28033.8 q^{61} -10476.6 q^{62} +24080.8 q^{64} -4938.75 q^{65} +64067.7 q^{67} -12563.9 q^{68} -264.459 q^{70} -57099.1 q^{71} +74280.3 q^{73} +45450.4 q^{74} +55805.0 q^{76} -1068.66 q^{77} +3833.20 q^{79} -3010.92 q^{80} -61304.5 q^{82} +15687.9 q^{83} +19289.9 q^{85} -49158.8 q^{86} -78231.1 q^{88} -31766.3 q^{89} +448.904 q^{91} +9649.86 q^{92} +34160.5 q^{94} -85679.6 q^{95} -741.484 q^{97} -62316.8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8}+O(q^{10}) \) 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8	\( 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8} - 250 q^{10} - 460 q^{11} + 464 q^{13} - 3676 q^{14} + 4612 q^{16} - 4756 q^{17} - 1780 q^{19} - 10314 q^{20} - 4214 q^{22} - 5290 q^{23} + 1330 q^{25} + 5152 q^{26} + 7072 q^{28} - 4048 q^{29} + 2816 q^{31} - 27436 q^{32} + 420 q^{34} - 9452 q^{35} + 2872 q^{37} - 31038 q^{38} + 2618 q^{40} - 34056 q^{41} + 7316 q^{43} - 33562 q^{44} + 4232 q^{46} - 49300 q^{47} + 45118 q^{49} - 44764 q^{50} - 25120 q^{52} - 86676 q^{53} - 2120 q^{55} - 290684 q^{56} - 87408 q^{58} - 67100 q^{59} - 40432 q^{61} - 230992 q^{62} + 136776 q^{64} - 184000 q^{65} - 50108 q^{67} - 270592 q^{68} + 117456 q^{70} - 238584 q^{71} - 13804 q^{73} - 150074 q^{74} - 197622 q^{76} - 116248 q^{77} - 9228 q^{79} - 313010 q^{80} - 68604 q^{82} - 155300 q^{83} + 80444 q^{85} + 80914 q^{86} - 237738 q^{88} - 213732 q^{89} - 264352 q^{91} - 101568 q^{92} + 140280 q^{94} + 123612 q^{95} + 42516 q^{97} + 151388 q^{98}+O(q^{100}) \) 10 * q - 8 * q^2 + 192 * q^4 - 100 * q^5 + 20 * q^7 - 384 * q^8 - 250 * q^10 - 460 * q^11 + 464 * q^13 - 3676 * q^14 + 4612 * q^16 - 4756 * q^17 - 1780 * q^19 - 10314 * q^20 - 4214 * q^22 - 5290 * q^23 + 1330 * q^25 + 5152 * q^26 + 7072 * q^28 - 4048 * q^29 + 2816 * q^31 - 27436 * q^32 + 420 * q^34 - 9452 * q^35 + 2872 * q^37 - 31038 * q^38 + 2618 * q^40 - 34056 * q^41 + 7316 * q^43 - 33562 * q^44 + 4232 * q^46 - 49300 * q^47 + 45118 * q^49 - 44764 * q^50 - 25120 * q^52 - 86676 * q^53 - 2120 * q^55 - 290684 * q^56 - 87408 * q^58 - 67100 * q^59 - 40432 * q^61 - 230992 * q^62 + 136776 * q^64 - 184000 * q^65 - 50108 * q^67 - 270592 * q^68 + 117456 * q^70 - 238584 * q^71 - 13804 * q^73 - 150074 * q^74 - 197622 * q^76 - 116248 * q^77 - 9228 * q^79 - 313010 * q^80 - 68604 * q^82 - 155300 * q^83 + 80444 * q^85 + 80914 * q^86 - 237738 * q^88 - 213732 * q^89 - 264352 * q^91 - 101568 * q^92 + 140280 * q^94 + 123612 * q^95 + 42516 * q^97 + 151388 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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