LMFDB - Embedded newform 975.6.a.u.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,6,Mod(1,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, 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("975.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	975.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:	$156.374224318$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 286 x^{9} + 442 x^{8} + 28715 x^{7} - 29138 x^{6} - 1208172 x^{5} + 509768 x^{4} + \cdots + 55036800 $ x^11 - 2x^10 - 286x^9 + 442x^8 + 28715x^7 - 29138x^6 - 1208172x^5 + 509768x^4 + 20144528x^3 - 3093984x^2 - 98068800x + 55036800
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3\cdot 5^{3}\cdot 11 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.29989$ of defining polynomial
Character	$\chi$	$=$	975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.29989 q^{2} -9.00000 q^{3} -26.7105 q^{4} -20.6990 q^{6} -228.917 q^{7} -135.028 q^{8} +81.0000 q^{9} +59.1539 q^{11} +240.395 q^{12} +169.000 q^{13} -526.484 q^{14} +544.187 q^{16} -2342.37 q^{17} +186.291 q^{18} +2145.80 q^{19} +2060.25 q^{21} +136.047 q^{22} -1152.00 q^{23} +1215.25 q^{24} +388.681 q^{26} -729.000 q^{27} +6114.49 q^{28} +5872.31 q^{29} +4568.68 q^{31} +5572.46 q^{32} -532.385 q^{33} -5387.20 q^{34} -2163.55 q^{36} +9110.68 q^{37} +4935.11 q^{38} -1521.00 q^{39} -8258.68 q^{41} +4738.36 q^{42} +498.716 q^{43} -1580.03 q^{44} -2649.47 q^{46} -18526.7 q^{47} -4897.69 q^{48} +35596.0 q^{49} +21081.4 q^{51} -4514.08 q^{52} -10763.3 q^{53} -1676.62 q^{54} +30910.2 q^{56} -19312.2 q^{57} +13505.7 q^{58} +3060.98 q^{59} +40411.1 q^{61} +10507.5 q^{62} -18542.3 q^{63} -4597.96 q^{64} -1224.43 q^{66} +69675.2 q^{67} +62566.0 q^{68} +10368.0 q^{69} -72896.6 q^{71} -10937.2 q^{72} +5906.20 q^{73} +20953.6 q^{74} -57315.5 q^{76} -13541.3 q^{77} -3498.13 q^{78} +93605.5 q^{79} +6561.00 q^{81} -18994.1 q^{82} +28221.8 q^{83} -55030.4 q^{84} +1146.99 q^{86} -52850.8 q^{87} -7987.41 q^{88} +34970.9 q^{89} -38687.0 q^{91} +30770.5 q^{92} -41118.1 q^{93} -42609.3 q^{94} -50152.1 q^{96} +67239.7 q^{97} +81867.0 q^{98} +4791.46 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} - 99 q^{3} + 224 q^{4} - 18 q^{6} - 55 q^{7} + 270 q^{8} + 891 q^{9} - 125 q^{11} - 2016 q^{12} + 1859 q^{13} - 1311 q^{14} + 5756 q^{16} - 4507 q^{17} + 162 q^{18} + 142 q^{19} + 495 q^{21}+ \cdots - 10125 q^{99}+O(q^{100}) $ 11 * q + 2 * q^2 - 99 * q^3 + 224 * q^4 - 18 * q^6 - 55 * q^7 + 270 * q^8 + 891 * q^9 - 125 * q^11 - 2016 * q^12 + 1859 * q^13 - 1311 * q^14 + 5756 * q^16 - 4507 * q^17 + 162 * q^18 + 142 * q^19 + 495 * q^21 - 5975 * q^22 - 4672 * q^23 - 2430 * q^24 + 338 * q^26 - 8019 * q^27 - 4149 * q^28 + 5197 * q^29 - 277 * q^31 + 2272 * q^32 + 1125 * q^33 - 21225 * q^34 + 18144 * q^36 - 1696 * q^37 - 27485 * q^38 - 16731 * q^39 + 2040 * q^41 + 11799 * q^42 - 31436 * q^43 - 39883 * q^44 + 67530 * q^46 + 15499 * q^47 - 51804 * q^48 + 29212 * q^49 + 40563 * q^51 + 37856 * q^52 - 119577 * q^53 - 1458 * q^54 + 1677 * q^56 - 1278 * q^57 - 91154 * q^58 + 17115 * q^59 + 26273 * q^61 - 45211 * q^62 - 4455 * q^63 + 137480 * q^64 + 53775 * q^66 + 26313 * q^67 - 191297 * q^68 + 42048 * q^69 - 105756 * q^71 + 21870 * q^72 - 103230 * q^73 - 6501 * q^74 + 124495 * q^76 - 104415 * q^77 - 3042 * q^78 + 210732 * q^79 + 72171 * q^81 - 60283 * q^82 - 144247 * q^83 + 37341 * q^84 - 42260 * q^86 - 46773 * q^87 - 415403 * q^88 + 117088 * q^89 - 9295 * q^91 - 70700 * q^92 + 2493 * q^93 + 383414 * q^94 - 20448 * q^96 - 112934 * q^97 - 159305 * q^98 - 10125 * q^99

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 $

$2$	2.29989	0.406567	0.203283	−	0.979120i	$-0.434839\pi$
$2$	2.29989	0.406567	0.203283	+	0.979120i	$0.434839\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−26.7105	−0.834703
$5$	0	0
$5$	0	0
$6$	−20.6990	−0.234732
$7$	−228.917	−1.76577	−0.882883	−	0.469593i	$-0.844401\pi$
$7$	−228.917	−1.76577	−0.882883	+	0.469593i	$0.844401\pi$
$8$	−135.028	−0.745930
$9$	81.0000	0.333333
$10$	0	0
$11$	59.1539	0.147401	0.0737007	−	0.997280i	$-0.476519\pi$
$11$	59.1539	0.147401	0.0737007	+	0.997280i	$0.476519\pi$
$12$	240.395	0.481916
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−526.484	−0.717902
$15$	0	0
$16$	544.187	0.531433
$17$	−2342.37	−1.96577	−0.982887	−	0.184208i	$-0.941028\pi$
$17$	−2342.37	−1.96577	−0.982887	+	0.184208i	$0.941028\pi$
$18$	186.291	0.135522
$19$	2145.80	1.36366	0.681830	−	0.731511i	$-0.261184\pi$
$19$	2145.80	1.36366	0.681830	+	0.731511i	$0.261184\pi$
$20$	0	0
$21$	2060.25	1.01947
$22$	136.047	0.0599285
$23$	−1152.00	−0.454080	−0.227040	−	0.973885i	$-0.572905\pi$
$23$	−1152.00	−0.454080	−0.227040	+	0.973885i	$0.572905\pi$
$24$	1215.25	0.430663
$25$	0	0
$26$	388.681	0.112761
$27$	−729.000	−0.192450
$28$	6114.49	1.47389
$29$	5872.31	1.29662	0.648312	−	0.761374i	$-0.275475\pi$
$29$	5872.31	1.29662	0.648312	+	0.761374i	$0.275475\pi$
$30$	0	0
$31$	4568.68	0.853859	0.426930	−	0.904285i	$-0.359595\pi$
$31$	4568.68	0.853859	0.426930	+	0.904285i	$0.359595\pi$
$32$	5572.46	0.961993
$33$	−532.385	−0.0851022
$34$	−5387.20	−0.799219
$35$	0	0
$36$	−2163.55	−0.278234
$37$	9110.68	1.09407	0.547037	−	0.837109i	$-0.315756\pi$
$37$	9110.68	1.09407	0.547037	+	0.837109i	$0.315756\pi$
$38$	4935.11	0.554419
$39$	−1521.00	−0.160128
$40$	0	0
$41$	−8258.68	−0.767275	−0.383638	−	0.923484i	$-0.625329\pi$
$41$	−8258.68	−0.767275	−0.383638	+	0.923484i	$0.625329\pi$
$42$	4738.36	0.414481
$43$	498.716	0.0411322	0.0205661	−	0.999788i	$-0.493453\pi$
$43$	498.716	0.0411322	0.0205661	+	0.999788i	$0.493453\pi$
$44$	−1580.03	−0.123036
$45$	0	0
$46$	−2649.47	−0.184614
$47$	−18526.7	−1.22336	−0.611678	−	0.791107i	$-0.709505\pi$
$47$	−18526.7	−1.22336	−0.611678	+	0.791107i	$0.709505\pi$
$48$	−4897.69	−0.306823
$49$	35596.0	2.11793
$50$	0	0
$51$	21081.4	1.13494
$52$	−4514.08	−0.231505
$53$	−10763.3	−0.526327	−0.263164	−	0.964751i	$-0.584766\pi$
$53$	−10763.3	−0.526327	−0.263164	+	0.964751i	$0.584766\pi$
$54$	−1676.62	−0.0782438
$55$	0	0
$56$	30910.2	1.31714
$57$	−19312.2	−0.787309
$58$	13505.7	0.527165
$59$	3060.98	0.114480	0.0572401	−	0.998360i	$-0.481770\pi$
$59$	3060.98	0.114480	0.0572401	+	0.998360i	$0.481770\pi$
$60$	0	0
$61$	40411.1	1.39052	0.695258	−	0.718760i	$-0.255290\pi$
$61$	40411.1	1.39052	0.695258	+	0.718760i	$0.255290\pi$
$62$	10507.5	0.347151
$63$	−18542.3	−0.588589
$64$	−4597.96	−0.140318
$65$	0	0
$66$	−1224.43	−0.0345997
$67$	69675.2	1.89623	0.948115	−	0.317926i	$-0.102986\pi$
$67$	69675.2	1.89623	0.948115	+	0.317926i	$0.102986\pi$
$68$	62566.0	1.64084
$69$	10368.0	0.262163
$70$	0	0
$71$	−72896.6	−1.71618	−0.858088	−	0.513503i	$-0.828347\pi$
$71$	−72896.6	−1.71618	−0.858088	+	0.513503i	$0.828347\pi$
$72$	−10937.2	−0.248643
$73$	5906.20	0.129718	0.0648591	−	0.997894i	$-0.479340\pi$
$73$	5906.20	0.129718	0.0648591	+	0.997894i	$0.479340\pi$
$74$	20953.6	0.444814
$75$	0	0
$76$	−57315.5	−1.13825
$77$	−13541.3	−0.260276
$78$	−3498.13	−0.0651028
$79$	93605.5	1.68746	0.843731	−	0.536767i	$-0.180355\pi$
$79$	93605.5	1.68746	0.843731	+	0.536767i	$0.180355\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−18994.1	−0.311949
$83$	28221.8	0.449665	0.224833	−	0.974397i	$-0.427816\pi$
$83$	28221.8	0.449665	0.224833	+	0.974397i	$0.427816\pi$
$84$	−55030.4	−0.850951
$85$	0	0
$86$	1146.99	0.0167230
$87$	−52850.8	−0.748607
$88$	−7987.41	−0.109951
$89$	34970.9	0.467985	0.233992	−	0.972238i	$-0.424821\pi$
$89$	34970.9	0.467985	0.233992	+	0.972238i	$0.424821\pi$
$90$	0	0
$91$	−38687.0	−0.489735
$92$	30770.5	0.379022
$93$	−41118.1	−0.492976
$94$	−42609.3	−0.497376
$95$	0	0
$96$	−50152.1	−0.555407
$97$	67239.7	0.725599	0.362799	−	0.931867i	$-0.381821\pi$
$97$	67239.7	0.725599	0.362799	+	0.931867i	$0.381821\pi$
$98$	81867.0	0.861080
$99$	4791.46	0.0491338
$100$	0	0
$101$	−162082.	−1.58100	−0.790498	−	0.612465i	$-0.790178\pi$
$101$	−162082.	−1.58100	−0.790498	+	0.612465i	$0.790178\pi$
$102$	48484.8	0.461429
$103$	−105332.	−0.978289	−0.489145	−	0.872203i	$-0.662691\pi$
$103$	−105332.	−0.978289	−0.489145	+	0.872203i	$0.662691\pi$
$104$	−22819.7	−0.206884
$105$	0	0
$106$	−24754.4	−0.213987
$107$	234277.	1.97820	0.989100	−	0.147244i	$-0.0470403\pi$
$107$	234277.	1.97820	0.989100	+	0.147244i	$0.0470403\pi$
$108$	19472.0	0.160639
$109$	−205781.	−1.65897	−0.829487	−	0.558525i	$-0.811367\pi$
$109$	−205781.	−1.65897	−0.829487	+	0.558525i	$0.811367\pi$
$110$	0	0
$111$	−81996.1	−0.631663
$112$	−124574.	−0.938386
$113$	−161817.	−1.19214	−0.596070	−	0.802933i	$-0.703272\pi$
$113$	−161817.	−1.19214	−0.596070	+	0.802933i	$0.703272\pi$
$114$	−44416.0	−0.320094
$115$	0	0
$116$	−156853.	−1.08230
$117$	13689.0	0.0924500
$118$	7039.91	0.0465438
$119$	536209.	3.47110
$120$	0	0
$121$	−157552.	−0.978273
$122$	92941.1	0.565338
$123$	74328.1	0.442986
$124$	−122032.	−0.712719
$125$	0	0
$126$	−42645.2	−0.239301
$127$	−2226.71	−0.0122505	−0.00612527	−	0.999981i	$-0.501950\pi$
$127$	−2226.71	−0.0122505	−0.00612527	+	0.999981i	$0.501950\pi$
$128$	−188893.	−1.01904
$129$	−4488.44	−0.0237477
$130$	0	0
$131$	−180748.	−0.920228	−0.460114	−	0.887860i	$-0.652191\pi$
$131$	−180748.	−0.920228	−0.460114	+	0.887860i	$0.652191\pi$
$132$	14220.3	0.0710351
$133$	−491211.	−2.40790
$134$	160245.	0.770945
$135$	0	0
$136$	316285.	1.46633
$137$	105032.	0.478100	0.239050	−	0.971007i	$-0.423164\pi$
$137$	105032.	0.478100	0.239050	+	0.971007i	$0.423164\pi$
$138$	23845.2	0.106587
$139$	−356298.	−1.56414	−0.782072	−	0.623188i	$-0.785837\pi$
$139$	−356298.	−1.56414	−0.782072	+	0.623188i	$0.785837\pi$
$140$	0	0
$141$	166740.	0.706305
$142$	−167654.	−0.697740
$143$	9997.00	0.0408818
$144$	44079.2	0.177144
$145$	0	0
$146$	13583.6	0.0527391
$147$	−320364.	−1.22279
$148$	−243351.	−0.913226
$149$	295683.	1.09109	0.545545	−	0.838082i	$-0.316323\pi$
$149$	295683.	1.09109	0.545545	+	0.838082i	$0.316323\pi$
$150$	0	0
$151$	122209.	0.436174	0.218087	−	0.975929i	$-0.430018\pi$
$151$	122209.	0.436174	0.218087	+	0.975929i	$0.430018\pi$
$152$	−289743.	−1.01719
$153$	−189732.	−0.655258
$154$	−31143.6	−0.105820
$155$	0	0
$156$	40626.7	0.133660
$157$	−67878.4	−0.219777	−0.109889	−	0.993944i	$-0.535049\pi$
$157$	−67878.4	−0.219777	−0.109889	+	0.993944i	$0.535049\pi$
$158$	215282.	0.686066
$159$	96869.8	0.303875
$160$	0	0
$161$	263712.	0.801799
$162$	15089.6	0.0451741
$163$	157781.	0.465142	0.232571	−	0.972579i	$-0.425286\pi$
$163$	157781.	0.465142	0.232571	+	0.972579i	$0.425286\pi$
$164$	220594.	0.640447
$165$	0	0
$166$	64907.0	0.182819
$167$	−548705.	−1.52247	−0.761233	−	0.648478i	$-0.775406\pi$
$167$	−548705.	−1.52247	−0.761233	+	0.648478i	$0.775406\pi$
$168$	−278191.	−0.760450
$169$	28561.0	0.0769231
$170$	0	0
$171$	173810.	0.454553
$172$	−13321.0	−0.0343332
$173$	471036.	1.19657	0.598285	−	0.801283i	$-0.295849\pi$
$173$	471036.	1.19657	0.598285	+	0.801283i	$0.295849\pi$
$174$	−121551.	−0.304359
$175$	0	0
$176$	32190.8	0.0783339
$177$	−27548.8	−0.0660951
$178$	80429.2	0.190267
$179$	−371293.	−0.866133	−0.433066	−	0.901362i	$-0.642568\pi$
$179$	−371293.	−0.866133	−0.433066	+	0.901362i	$0.642568\pi$
$180$	0	0
$181$	510409.	1.15804	0.579018	−	0.815315i	$-0.303436\pi$
$181$	510409.	1.15804	0.579018	+	0.815315i	$0.303436\pi$
$182$	−88975.8	−0.199110
$183$	−363700.	−0.802815
$184$	155552.	0.338712
$185$	0	0
$186$	−94567.1	−0.200428
$187$	−138560.	−0.289758
$188$	494857.	1.02114
$189$	166881.	0.339822
$190$	0	0
$191$	60794.2	0.120581	0.0602905	−	0.998181i	$-0.480797\pi$
$191$	60794.2	0.120581	0.0602905	+	0.998181i	$0.480797\pi$
$192$	41381.6	0.0810129
$193$	473665.	0.915331	0.457666	−	0.889124i	$-0.348686\pi$
$193$	473665.	0.915331	0.457666	+	0.889124i	$0.348686\pi$
$194$	154644.	0.295004
$195$	0	0
$196$	−950788.	−1.76784
$197$	471821.	0.866186	0.433093	−	0.901349i	$-0.357422\pi$
$197$	471821.	0.866186	0.433093	+	0.901349i	$0.357422\pi$
$198$	11019.8	0.0199762
$199$	874843.	1.56602	0.783010	−	0.622009i	$-0.213683\pi$
$199$	874843.	1.56602	0.783010	+	0.622009i	$0.213683\pi$
$200$	0	0
$201$	−627077.	−1.09479
$202$	−372770.	−0.642780
$203$	−1.34427e6	−2.28954
$204$	−563094.	−0.947339
$205$	0	0
$206$	−242252.	−0.397740
$207$	−93311.9	−0.151360
$208$	91967.7	0.147393
$209$	126933.	0.201005
$210$	0	0
$211$	−37674.0	−0.0582554	−0.0291277	−	0.999576i	$-0.509273\pi$
$211$	−37674.0	−0.0582554	−0.0291277	+	0.999576i	$0.509273\pi$
$212$	287493.	0.439327
$213$	656070.	0.990834
$214$	538811.	0.804271
$215$	0	0
$216$	98435.2	0.143554
$217$	−1.04585e6	−1.50772
$218$	−473274.	−0.674484
$219$	−53155.8	−0.0748928
$220$	0	0
$221$	−395861.	−0.545208
$222$	−188582.	−0.256813
$223$	−1.13356e6	−1.52645	−0.763225	−	0.646133i	$-0.776385\pi$
$223$	−1.13356e6	−1.52645	−0.763225	+	0.646133i	$0.776385\pi$
$224$	−1.27563e6	−1.69865
$225$	0	0
$226$	−372160.	−0.484685
$227$	329692.	0.424663	0.212331	−	0.977198i	$-0.431894\pi$
$227$	329692.	0.424663	0.212331	+	0.977198i	$0.431894\pi$
$228$	515839.	0.657170
$229$	−66202.8	−0.0834233	−0.0417117	−	0.999130i	$-0.513281\pi$
$229$	−66202.8	−0.0834233	−0.0417117	+	0.999130i	$0.513281\pi$
$230$	0	0
$231$	121872.	0.150271
$232$	−792925.	−0.967191
$233$	−847118.	−1.02224	−0.511121	−	0.859509i	$-0.670770\pi$
$233$	−847118.	−1.02224	−0.511121	+	0.859509i	$0.670770\pi$
$234$	31483.2	0.0375871
$235$	0	0
$236$	−81760.3	−0.0955569
$237$	−842450.	−0.974256
$238$	1.23322e6	1.41123
$239$	646882.	0.732538	0.366269	−	0.930509i	$-0.380635\pi$
$239$	646882.	0.732538	0.366269	+	0.930509i	$0.380635\pi$
$240$	0	0
$241$	−1.51311e6	−1.67814	−0.839068	−	0.544027i	$-0.816899\pi$
$241$	−1.51311e6	−1.67814	−0.839068	+	0.544027i	$0.816899\pi$
$242$	−362352.	−0.397733
$243$	−59049.0	−0.0641500
$244$	−1.07940e6	−1.16067
$245$	0	0
$246$	170947.	0.180104
$247$	362641.	0.378211
$248$	−616898.	−0.636919
$249$	−253996.	−0.259614
$250$	0	0
$251$	−404872.	−0.405633	−0.202816	−	0.979217i	$-0.565009\pi$
$251$	−404872.	−0.405633	−0.202816	+	0.979217i	$0.565009\pi$
$252$	495274.	0.491297
$253$	−68145.1	−0.0669320
$254$	−5121.19	−0.00498066
$255$	0	0
$256$	−287300.	−0.273990
$257$	−1.76267e6	−1.66471	−0.832353	−	0.554245i	$-0.813007\pi$
$257$	−1.76267e6	−1.66471	−0.832353	+	0.554245i	$0.813007\pi$
$258$	−10322.9	−0.00965503
$259$	−2.08559e6	−1.93188
$260$	0	0
$261$	475658.	0.432208
$262$	−415701.	−0.374134
$263$	567933.	0.506300	0.253150	−	0.967427i	$-0.418533\pi$
$263$	567933.	0.506300	0.253150	+	0.967427i	$0.418533\pi$
$264$	71886.7	0.0634802
$265$	0	0
$266$	−1.12973e6	−0.978974
$267$	−314738.	−0.270191
$268$	−1.86106e6	−1.58279
$269$	−242686.	−0.204486	−0.102243	−	0.994759i	$-0.532602\pi$
$269$	−242686.	−0.204486	−0.102243	+	0.994759i	$0.532602\pi$
$270$	0	0
$271$	389530.	0.322194	0.161097	−	0.986939i	$-0.448497\pi$
$271$	389530.	0.322194	0.161097	+	0.986939i	$0.448497\pi$
$272$	−1.27469e6	−1.04468
$273$	348183.	0.282749
$274$	241561.	0.194380
$275$	0	0
$276$	−276934.	−0.218828
$277$	−1.13298e6	−0.887206	−0.443603	−	0.896223i	$-0.646300\pi$
$277$	−1.13298e6	−0.887206	−0.443603	+	0.896223i	$0.646300\pi$
$278$	−819447.	−0.635929
$279$	370063.	0.284620
$280$	0	0
$281$	886440.	0.669705	0.334852	−	0.942271i	$-0.391314\pi$
$281$	886440.	0.669705	0.334852	+	0.942271i	$0.391314\pi$
$282$	383484.	0.287160
$283$	1.37650e6	1.02167	0.510836	−	0.859678i	$-0.329336\pi$
$283$	1.37650e6	1.02167	0.510836	+	0.859678i	$0.329336\pi$
$284$	1.94711e6	1.43250
$285$	0	0
$286$	22992.0	0.0166212
$287$	1.89055e6	1.35483
$288$	451369.	0.320664
$289$	4.06685e6	2.86427
$290$	0	0
$291$	−605157.	−0.418925
$292$	−157757.	−0.108276
$293$	−239467.	−0.162958	−0.0814790	−	0.996675i	$-0.525964\pi$
$293$	−239467.	−0.162958	−0.0814790	+	0.996675i	$0.525964\pi$
$294$	−736803.	−0.497145
$295$	0	0
$296$	−1.23019e6	−0.816102
$297$	−43123.2	−0.0283674
$298$	680038.	0.443601
$299$	−194688.	−0.125939
$300$	0	0
$301$	−114165.	−0.0726299
$302$	281067.	0.177334
$303$	1.45874e6	0.912788
$304$	1.16772e6	0.724694
$305$	0	0
$306$	−436363.	−0.266406
$307$	−1.67234e6	−1.01269	−0.506347	−	0.862330i	$-0.669004\pi$
$307$	−1.67234e6	−1.01269	−0.506347	+	0.862330i	$0.669004\pi$
$308$	361696.	0.217253
$309$	947988.	0.564816
$310$	0	0
$311$	−1.00067e6	−0.586664	−0.293332	−	0.956011i	$-0.594764\pi$
$311$	−1.00067e6	−0.586664	−0.293332	+	0.956011i	$0.594764\pi$
$312$	205377.	0.119444
$313$	612773.	0.353540	0.176770	−	0.984252i	$-0.443435\pi$
$313$	612773.	0.353540	0.176770	+	0.984252i	$0.443435\pi$
$314$	−156113.	−0.0893541
$315$	0	0
$316$	−2.50025e6	−1.40853
$317$	−1.52019e6	−0.849670	−0.424835	−	0.905271i	$-0.639668\pi$
$317$	−1.52019e6	−0.849670	−0.424835	+	0.905271i	$0.639668\pi$
$318$	222790.	0.123546
$319$	347370.	0.191124
$320$	0	0
$321$	−2.10849e6	−1.14211
$322$	606509.	0.325985
$323$	−5.02627e6	−2.68065
$324$	−175248.	−0.0927448
$325$	0	0
$326$	362879.	0.189111
$327$	1.85203e6	0.957810
$328$	1.11515e6	0.572333
$329$	4.24108e6	2.16016
$330$	0	0
$331$	−1.70279e6	−0.854260	−0.427130	−	0.904190i	$-0.640475\pi$
$331$	−1.70279e6	−0.854260	−0.427130	+	0.904190i	$0.640475\pi$
$332$	−753819.	−0.375337
$333$	737965.	0.364691
$334$	−1.26196e6	−0.618985
$335$	0	0
$336$	1.12116e6	0.541778
$337$	−2.16136e6	−1.03670	−0.518350	−	0.855168i	$-0.673454\pi$
$337$	−2.16136e6	−1.03670	−0.518350	+	0.855168i	$0.673454\pi$
$338$	65687.2	0.0312744
$339$	1.45635e6	0.688282
$340$	0	0
$341$	270255.	0.125860
$342$	399744.	0.184806
$343$	−4.30113e6	−1.97400
$344$	−67340.5	−0.0306817
$345$	0	0
$346$	1.08333e6	0.486486
$347$	−1.30185e6	−0.580413	−0.290207	−	0.956964i	$-0.593724\pi$
$347$	−1.30185e6	−0.580413	−0.290207	+	0.956964i	$0.593724\pi$
$348$	1.41167e6	0.624865
$349$	−806129.	−0.354275	−0.177138	−	0.984186i	$-0.556684\pi$
$349$	−806129.	−0.354275	−0.177138	+	0.984186i	$0.556684\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	329632.	0.141799
$353$	1.21170e6	0.517555	0.258778	−	0.965937i	$-0.416680\pi$
$353$	1.21170e6	0.517555	0.258778	+	0.965937i	$0.416680\pi$
$354$	−63359.2	−0.0268721
$355$	0	0
$356$	−934091.	−0.390629
$357$	−4.82588e6	−2.00404
$358$	−853933.	−0.352141
$359$	40339.1	0.0165193	0.00825963	−	0.999966i	$-0.497371\pi$
$359$	40339.1	0.0165193	0.00825963	+	0.999966i	$0.497371\pi$
$360$	0	0
$361$	2.12837e6	0.859567
$362$	1.17388e6	0.470819
$363$	1.41797e6	0.564806
$364$	1.03335e6	0.408784
$365$	0	0
$366$	−836470.	−0.326398
$367$	−3.28931e6	−1.27479	−0.637396	−	0.770537i	$-0.719988\pi$
$367$	−3.28931e6	−1.27479	−0.637396	+	0.770537i	$0.719988\pi$
$368$	−626903.	−0.241313
$369$	−668953.	−0.255758
$370$	0	0
$371$	2.46390e6	0.929371
$372$	1.09829e6	0.411489
$373$	1.73305e6	0.644970	0.322485	−	0.946575i	$-0.395482\pi$
$373$	1.73305e6	0.644970	0.322485	+	0.946575i	$0.395482\pi$
$374$	−318674.	−0.117806
$375$	0	0
$376$	2.50162e6	0.912538
$377$	992421.	0.359619
$378$	383807.	0.138160
$379$	425983.	0.152333	0.0761665	−	0.997095i	$-0.475732\pi$
$379$	425983.	0.152333	0.0761665	+	0.997095i	$0.475732\pi$
$380$	0	0
$381$	20040.4	0.00707285
$382$	139820.	0.0490242
$383$	4.18083e6	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$383$	4.18083e6	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$384$	1.70004e6	0.588344
$385$	0	0
$386$	1.08938e6	0.372143
$387$	40396.0	0.0137107
$388$	−1.79601e6	−0.605660
$389$	−1.92802e6	−0.646008	−0.323004	−	0.946398i	$-0.604693\pi$
$389$	−1.92802e6	−0.646008	−0.323004	+	0.946398i	$0.604693\pi$
$390$	0	0
$391$	2.69841e6	0.892619
$392$	−4.80645e6	−1.57983
$393$	1.62673e6	0.531294
$394$	1.08514e6	0.352163
$395$	0	0
$396$	−127982.	−0.0410121
$397$	1.51118e6	0.481216	0.240608	−	0.970622i	$-0.422653\pi$
$397$	1.51118e6	0.481216	0.240608	+	0.970622i	$0.422653\pi$
$398$	2.01204e6	0.636692
$399$	4.42090e6	1.39020
$400$	0	0
$401$	−201369.	−0.0625362	−0.0312681	−	0.999511i	$-0.509955\pi$
$401$	−201369.	−0.0625362	−0.0312681	+	0.999511i	$0.509955\pi$
$402$	−1.44221e6	−0.445105
$403$	772107.	0.236818
$404$	4.32928e6	1.31966
$405$	0	0
$406$	−3.09168e6	−0.930850
$407$	538932.	0.161268
$408$	−2.84657e6	−0.846586
$409$	3.36373e6	0.994291	0.497145	−	0.867667i	$-0.334382\pi$
$409$	3.36373e6	0.994291	0.497145	+	0.867667i	$0.334382\pi$
$410$	0	0
$411$	−945286.	−0.276031
$412$	2.81347e6	0.816581
$413$	−700710.	−0.202145
$414$	−214607.	−0.0615379
$415$	0	0
$416$	941745.	0.266809
$417$	3.20668e6	0.903059
$418$	291931.	0.0817221
$419$	−1.76055e6	−0.489906	−0.244953	−	0.969535i	$-0.578772\pi$
$419$	−1.76055e6	−0.489906	−0.244953	+	0.969535i	$0.578772\pi$
$420$	0	0
$421$	2.40677e6	0.661803	0.330901	−	0.943665i	$-0.392647\pi$
$421$	2.40677e6	0.661803	0.330901	+	0.943665i	$0.392647\pi$
$422$	−86646.1	−0.0236847
$423$	−1.50066e6	−0.407786
$424$	1.45334e6	0.392603
$425$	0	0
$426$	1.50889e6	0.402840
$427$	−9.25079e6	−2.45533
$428$	−6.25766e6	−1.65121
$429$	−89973.0	−0.0236031
$430$	0	0
$431$	3.02707e6	0.784927	0.392463	−	0.919768i	$-0.371623\pi$
$431$	3.02707e6	0.784927	0.392463	+	0.919768i	$0.371623\pi$
$432$	−396713.	−0.102274
$433$	−998142.	−0.255842	−0.127921	−	0.991784i	$-0.540830\pi$
$433$	−998142.	−0.255842	−0.127921	+	0.991784i	$0.540830\pi$
$434$	−2.40534e6	−0.612987
$435$	0	0
$436$	5.49652e6	1.38475
$437$	−2.47196e6	−0.619210
$438$	−122252.	−0.0304489
$439$	2.45313e6	0.607519	0.303759	−	0.952749i	$-0.401758\pi$
$439$	2.45313e6	0.607519	0.303759	+	0.952749i	$0.401758\pi$
$440$	0	0
$441$	2.88328e6	0.705977
$442$	−910437.	−0.221663
$443$	2.53464e6	0.613631	0.306815	−	0.951769i	$-0.400737\pi$
$443$	2.53464e6	0.613631	0.306815	+	0.951769i	$0.400737\pi$
$444$	2.19016e6	0.527252
$445$	0	0
$446$	−2.60706e6	−0.620604
$447$	−2.66114e6	−0.629941
$448$	1.05255e6	0.247770
$449$	−1.70620e6	−0.399405	−0.199703	−	0.979857i	$-0.563998\pi$
$449$	−1.70620e6	−0.399405	−0.199703	+	0.979857i	$0.563998\pi$
$450$	0	0
$451$	−488533.	−0.113097
$452$	4.32220e6	0.995083
$453$	−1.09988e6	−0.251825
$454$	758256.	0.172654
$455$	0	0
$456$	2.60769e6	0.587277
$457$	−382585.	−0.0856915	−0.0428458	−	0.999082i	$-0.513642\pi$
$457$	−382585.	−0.0856915	−0.0428458	+	0.999082i	$0.513642\pi$
$458$	−152259.	−0.0339172
$459$	1.70759e6	0.378313
$460$	0	0
$461$	−7.16362e6	−1.56993	−0.784964	−	0.619541i	$-0.787319\pi$
$461$	−7.16362e6	−1.56993	−0.784964	+	0.619541i	$0.787319\pi$
$462$	280292.	0.0610950
$463$	3.22317e6	0.698764	0.349382	−	0.936980i	$-0.386392\pi$
$463$	3.22317e6	0.698764	0.349382	+	0.936980i	$0.386392\pi$
$464$	3.19564e6	0.689069
$465$	0	0
$466$	−1.94828e6	−0.415610
$467$	−5.46265e6	−1.15907	−0.579537	−	0.814946i	$-0.696767\pi$
$467$	−5.46265e6	−1.15907	−0.579537	+	0.814946i	$0.696767\pi$
$468$	−365640.	−0.0771683
$469$	−1.59498e7	−3.34830
$470$	0	0
$471$	610906.	0.126888
$472$	−413317.	−0.0853941
$473$	29501.0	0.00606294
$474$	−1.93754e6	−0.396100
$475$	0	0
$476$	−1.43224e7	−2.89734
$477$	−871828.	−0.175442
$478$	1.48776e6	0.297826
$479$	−5.39101e6	−1.07357	−0.536786	−	0.843718i	$-0.680362\pi$
$479$	−5.39101e6	−1.07357	−0.536786	+	0.843718i	$0.680362\pi$
$480$	0	0
$481$	1.53970e6	0.303441
$482$	−3.47998e6	−0.682275
$483$	−2.37341e6	−0.462919
$484$	4.20829e6	0.816568
$485$	0	0
$486$	−135806.	−0.0260813
$487$	7.84939e6	1.49973	0.749866	−	0.661590i	$-0.230118\pi$
$487$	7.84939e6	1.49973	0.749866	+	0.661590i	$0.230118\pi$
$488$	−5.45662e6	−1.03723
$489$	−1.42003e6	−0.268550
$490$	0	0
$491$	1.12840e6	0.211231	0.105616	−	0.994407i	$-0.466319\pi$
$491$	1.12840e6	0.211231	0.105616	+	0.994407i	$0.466319\pi$
$492$	−1.98534e6	−0.369762
$493$	−1.37552e7	−2.54887
$494$	834034.	0.153768
$495$	0	0
$496$	2.48622e6	0.453769
$497$	1.66873e7	3.03036
$498$	−584163.	−0.105551
$499$	9.57003e6	1.72053	0.860265	−	0.509848i	$-0.170298\pi$
$499$	9.57003e6	1.72053	0.860265	+	0.509848i	$0.170298\pi$
$500$	0	0
$501$	4.93835e6	0.878997
$502$	−931160.	−0.164917
$503$	8.58409e6	1.51278	0.756388	−	0.654123i	$-0.226962\pi$
$503$	8.58409e6	1.51278	0.756388	+	0.654123i	$0.226962\pi$
$504$	2.50372e6	0.439046
$505$	0	0
$506$	−156726.	−0.0272123
$507$	−257049.	−0.0444116
$508$	59476.6	0.0102256
$509$	−7.23162e6	−1.23720	−0.618602	−	0.785705i	$-0.712300\pi$
$509$	−7.23162e6	−1.23720	−0.618602	+	0.785705i	$0.712300\pi$
$510$	0	0
$511$	−1.35203e6	−0.229052
$512$	5.38383e6	0.907646
$513$	−1.56429e6	−0.262436
$514$	−4.05394e6	−0.676815
$515$	0	0
$516$	119889.	0.0198223
$517$	−1.09592e6	−0.180324
$518$	−4.79663e6	−0.785437
$519$	−4.23932e6	−0.690841
$520$	0	0
$521$	−1.15289e7	−1.86078	−0.930389	−	0.366573i	$-0.880531\pi$
$521$	−1.15289e7	−1.86078	−0.930389	+	0.366573i	$0.880531\pi$
$522$	1.09396e6	0.175722
$523$	9.32605e6	1.49088	0.745442	−	0.666571i	$-0.232239\pi$
$523$	9.32605e6	1.49088	0.745442	+	0.666571i	$0.232239\pi$
$524$	4.82787e6	0.768117
$525$	0	0
$526$	1.30618e6	0.205845
$527$	−1.07015e7	−1.67850
$528$	−289717.	−0.0452261
$529$	−5.10924e6	−0.793812
$530$	0	0
$531$	247939.	0.0381600
$532$	1.31205e7	2.00989
$533$	−1.39572e6	−0.212804
$534$	−723863.	−0.109851
$535$	0	0
$536$	−9.40808e6	−1.41445
$537$	3.34164e6	0.500062
$538$	−558151.	−0.0831373
$539$	2.10564e6	0.312186
$540$	0	0
$541$	−3.26784e6	−0.480029	−0.240014	−	0.970769i	$-0.577152\pi$
$541$	−3.26784e6	−0.480029	−0.240014	+	0.970769i	$0.577152\pi$
$542$	895876.	0.130994
$543$	−4.59368e6	−0.668592
$544$	−1.30528e7	−1.89106
$545$	0	0
$546$	800782.	0.114956
$547$	−5.16606e6	−0.738229	−0.369114	−	0.929384i	$-0.620339\pi$
$547$	−5.16606e6	−0.738229	−0.369114	+	0.929384i	$0.620339\pi$
$548$	−2.80545e6	−0.399072
$549$	3.27330e6	0.463505
$550$	0	0
$551$	1.26008e7	1.76815
$552$	−1.39997e6	−0.195555
$553$	−2.14279e7	−2.97966
$554$	−2.60574e6	−0.360709
$555$	0	0
$556$	9.51691e6	1.30560
$557$	7.71027e6	1.05301	0.526504	−	0.850173i	$-0.323503\pi$
$557$	7.71027e6	1.05301	0.526504	+	0.850173i	$0.323503\pi$
$558$	851104.	0.115717
$559$	84283.0	0.0114080
$560$	0	0
$561$	1.24704e6	0.167292
$562$	2.03871e6	0.272280
$563$	−8.33451e6	−1.10818	−0.554088	−	0.832458i	$-0.686933\pi$
$563$	−8.33451e6	−1.10818	−0.554088	+	0.832458i	$0.686933\pi$
$564$	−4.45371e6	−0.589555
$565$	0	0
$566$	3.16581e6	0.415378
$567$	−1.50193e6	−0.196196
$568$	9.84306e6	1.28015
$569$	−1.88775e6	−0.244435	−0.122218	−	0.992503i	$-0.539001\pi$
$569$	−1.88775e6	−0.244435	−0.122218	+	0.992503i	$0.539001\pi$
$570$	0	0
$571$	7.84065e6	1.00638	0.503190	−	0.864176i	$-0.332160\pi$
$571$	7.84065e6	1.00638	0.503190	+	0.864176i	$0.332160\pi$
$572$	−267025.	−0.0341241
$573$	−547148.	−0.0696174
$574$	4.34806e6	0.550828
$575$	0	0
$576$	−372434.	−0.0467728
$577$	−4.39327e6	−0.549349	−0.274674	−	0.961537i	$-0.588570\pi$
$577$	−4.39327e6	−0.549349	−0.274674	+	0.961537i	$0.588570\pi$
$578$	9.35331e6	1.16452
$579$	−4.26299e6	−0.528467
$580$	0	0
$581$	−6.46045e6	−0.794004
$582$	−1.39180e6	−0.170321
$583$	−636691.	−0.0775814
$584$	−797500.	−0.0967606
$585$	0	0
$586$	−550747.	−0.0662534
$587$	1.21559e7	1.45610	0.728049	−	0.685525i	$-0.240427\pi$
$587$	1.21559e7	1.45610	0.728049	+	0.685525i	$0.240427\pi$
$588$	8.55709e6	1.02066
$589$	9.80349e6	1.16437
$590$	0	0
$591$	−4.24639e6	−0.500093
$592$	4.95792e6	0.581427
$593$	1.27303e7	1.48663	0.743315	−	0.668941i	$-0.233252\pi$
$593$	1.27303e7	1.48663	0.743315	+	0.668941i	$0.233252\pi$
$594$	−99178.5	−0.0115332
$595$	0	0
$596$	−7.89784e6	−0.910736
$597$	−7.87359e6	−0.904142
$598$	−447760.	−0.0512027
$599$	4.00848e6	0.456470	0.228235	−	0.973606i	$-0.426705\pi$
$599$	4.00848e6	0.456470	0.228235	+	0.973606i	$0.426705\pi$
$600$	0	0
$601$	2.62995e6	0.297003	0.148502	−	0.988912i	$-0.452555\pi$
$601$	2.62995e6	0.297003	0.148502	+	0.988912i	$0.452555\pi$
$602$	−262566.	−0.0295289
$603$	5.64369e6	0.632077
$604$	−3.26426e6	−0.364076
$605$	0	0
$606$	3.35493e6	0.371109
$607$	−6.42797e6	−0.708112	−0.354056	−	0.935224i	$-0.615198\pi$
$607$	−6.42797e6	−0.708112	−0.354056	+	0.935224i	$0.615198\pi$
$608$	1.19574e7	1.31183
$609$	1.20985e7	1.32186
$610$	0	0
$611$	−3.13101e6	−0.339298
$612$	5.06784e6	0.546946
$613$	−1.20038e7	−1.29023	−0.645116	−	0.764084i	$-0.723191\pi$
$613$	−1.20038e7	−1.29023	−0.645116	+	0.764084i	$0.723191\pi$
$614$	−3.84619e6	−0.411728
$615$	0	0
$616$	1.82845e6	0.194148
$617$	8.45943e6	0.894599	0.447300	−	0.894384i	$-0.352386\pi$
$617$	8.45943e6	0.894599	0.447300	+	0.894384i	$0.352386\pi$
$618$	2.18027e6	0.229635
$619$	1.08577e7	1.13897	0.569483	−	0.822003i	$-0.307144\pi$
$619$	1.08577e7	1.13897	0.569483	+	0.822003i	$0.307144\pi$
$620$	0	0
$621$	839807.	0.0873877
$622$	−2.30143e6	−0.238518
$623$	−8.00544e6	−0.826352
$624$	−827709.	−0.0850974
$625$	0	0
$626$	1.40931e6	0.143738
$627$	−1.14239e6	−0.116050
$628$	1.81307e6	0.183449
$629$	−2.13406e7	−2.15070
$630$	0	0
$631$	1.62647e7	1.62620	0.813099	−	0.582126i	$-0.197779\pi$
$631$	1.62647e7	1.62620	0.813099	+	0.582126i	$0.197779\pi$
$632$	−1.26393e7	−1.25873
$633$	339066.	0.0336337
$634$	−3.49627e6	−0.345448
$635$	0	0
$636$	−2.58744e6	−0.253646
$637$	6.01573e6	0.587408
$638$	798913.	0.0777048
$639$	−5.90463e6	−0.572058
$640$	0	0
$641$	−1.24335e7	−1.19523	−0.597613	−	0.801785i	$-0.703884\pi$
$641$	−1.24335e7	−1.19523	−0.597613	+	0.801785i	$0.703884\pi$
$642$	−4.84930e6	−0.464346
$643$	5.47279e6	0.522013	0.261006	−	0.965337i	$-0.415946\pi$
$643$	5.47279e6	0.522013	0.261006	+	0.965337i	$0.415946\pi$
$644$	−7.04388e6	−0.669264
$645$	0	0
$646$	−1.15599e7	−1.08986
$647$	1.77694e7	1.66883	0.834414	−	0.551138i	$-0.185806\pi$
$647$	1.77694e7	1.66883	0.834414	+	0.551138i	$0.185806\pi$
$648$	−885917.	−0.0828811
$649$	181069.	0.0168745
$650$	0	0
$651$	9.41264e6	0.870480
$652$	−4.21441e6	−0.388256
$653$	2.12465e7	1.94987	0.974933	−	0.222498i	$-0.0714210\pi$
$653$	2.12465e7	1.94987	0.974933	+	0.222498i	$0.0714210\pi$
$654$	4.25947e6	0.389414
$655$	0	0
$656$	−4.49427e6	−0.407755
$657$	478402.	0.0432394
$658$	9.75401e6	0.878250
$659$	2.08310e7	1.86852	0.934258	−	0.356597i	$-0.116063\pi$
$659$	2.08310e7	1.86852	0.934258	+	0.356597i	$0.116063\pi$
$660$	0	0
$661$	1.36207e7	1.21254	0.606272	−	0.795258i	$-0.292664\pi$
$661$	1.36207e7	1.21254	0.606272	+	0.795258i	$0.292664\pi$
$662$	−3.91622e6	−0.347314
$663$	3.56275e6	0.314776
$664$	−3.81073e6	−0.335419
$665$	0	0
$666$	1.69724e6	0.148271
$667$	−6.76490e6	−0.588771
$668$	1.46562e7	1.27081
$669$	1.02020e7	0.881296
$670$	0	0
$671$	2.39047e6	0.204964
$672$	1.14807e7	0.980718
$673$	−6.82294e6	−0.580676	−0.290338	−	0.956924i	$-0.593768\pi$
$673$	−6.82294e6	−0.580676	−0.290338	+	0.956924i	$0.593768\pi$
$674$	−4.97090e6	−0.421488
$675$	0	0
$676$	−762879.	−0.0642079
$677$	7.93083e6	0.665038	0.332519	−	0.943096i	$-0.392101\pi$
$677$	7.93083e6	0.665038	0.332519	+	0.943096i	$0.392101\pi$
$678$	3.34944e6	0.279833
$679$	−1.53923e7	−1.28124
$680$	0	0
$681$	−2.96723e6	−0.245179
$682$	621557.	0.0511705
$683$	−1.34275e7	−1.10140	−0.550698	−	0.834705i	$-0.685638\pi$
$683$	−1.34275e7	−1.10140	−0.550698	+	0.834705i	$0.685638\pi$
$684$	−4.64256e6	−0.379417
$685$	0	0
$686$	−9.89213e6	−0.802564
$687$	595825.	0.0481645
$688$	271395.	0.0218590
$689$	−1.81900e6	−0.145977
$690$	0	0
$691$	−6.45401e6	−0.514203	−0.257101	−	0.966384i	$-0.582767\pi$
$691$	−6.45401e6	−0.514203	−0.257101	+	0.966384i	$0.582767\pi$
$692$	−1.25816e7	−0.998782
$693$	−1.09685e6	−0.0867587
$694$	−2.99411e6	−0.235977
$695$	0	0
$696$	7.13633e6	0.558408
$697$	1.93449e7	1.50829
$698$	−1.85401e6	−0.144037
$699$	7.62406e6	0.590192
$700$	0	0
$701$	−1.35502e7	−1.04148	−0.520740	−	0.853715i	$-0.674344\pi$
$701$	−1.35502e7	−1.04148	−0.520740	+	0.853715i	$0.674344\pi$
$702$	−283349.	−0.0217009
$703$	1.95497e7	1.49194
$704$	−271987.	−0.0206831
$705$	0	0
$706$	2.78677e6	0.210421
$707$	3.71033e7	2.79167
$708$	735842.	0.0551698
$709$	1.41562e6	0.105762	0.0528812	−	0.998601i	$-0.483160\pi$
$709$	1.41562e6	0.105762	0.0528812	+	0.998601i	$0.483160\pi$
$710$	0	0
$711$	7.58205e6	0.562487
$712$	−4.72204e6	−0.349084
$713$	−5.26311e6	−0.387720
$714$	−1.10990e7	−0.814776
$715$	0	0
$716$	9.91743e6	0.722964
$717$	−5.82194e6	−0.422931
$718$	92775.6	0.00671619
$719$	6.11475e6	0.441120	0.220560	−	0.975373i	$-0.429212\pi$
$719$	6.11475e6	0.441120	0.220560	+	0.975373i	$0.429212\pi$
$720$	0	0
$721$	2.41123e7	1.72743
$722$	4.89502e6	0.349472
$723$	1.36180e7	0.968872
$724$	−1.36333e7	−0.966616
$725$	0	0
$726$	3.26117e6	0.229631
$727$	−1.44079e7	−1.01103	−0.505515	−	0.862818i	$-0.668697\pi$
$727$	−1.44079e7	−1.01103	−0.505515	+	0.862818i	$0.668697\pi$
$728$	5.22382e6	0.365308
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.16818e6	−0.0808567
$732$	9.71461e6	0.670112
$733$	−9.95927e6	−0.684649	−0.342324	−	0.939582i	$-0.611214\pi$
$733$	−9.95927e6	−0.684649	−0.342324	+	0.939582i	$0.611214\pi$
$734$	−7.56504e6	−0.518288
$735$	0	0
$736$	−6.41946e6	−0.436822
$737$	4.12156e6	0.279507
$738$	−1.53852e6	−0.103983
$739$	−5.09810e6	−0.343398	−0.171699	−	0.985149i	$-0.554926\pi$
$739$	−5.09810e6	−0.343398	−0.171699	+	0.985149i	$0.554926\pi$
$740$	0	0
$741$	−3.26377e6	−0.218360
$742$	5.66671e6	0.377852
$743$	−1.43927e7	−0.956465	−0.478232	−	0.878233i	$-0.658722\pi$
$743$	−1.43927e7	−0.956465	−0.478232	+	0.878233i	$0.658722\pi$
$744$	5.55208e6	0.367725
$745$	0	0
$746$	3.98583e6	0.262224
$747$	2.28597e6	0.149888
$748$	3.70102e6	0.241862
$749$	−5.36300e7	−3.49304
$750$	0	0
$751$	−5.64574e6	−0.365276	−0.182638	−	0.983180i	$-0.558464\pi$
$751$	−5.64574e6	−0.365276	−0.182638	+	0.983180i	$0.558464\pi$
$752$	−1.00820e7	−0.650132
$753$	3.64384e6	0.234192
$754$	2.28246e6	0.146209
$755$	0	0
$756$	−4.45746e6	−0.283650
$757$	8.05893e6	0.511137	0.255569	−	0.966791i	$-0.417737\pi$
$757$	8.05893e6	0.511137	0.255569	+	0.966791i	$0.417737\pi$
$758$	979714.	0.0619336
$759$	613306.	0.0386432
$760$	0	0
$761$	1.08877e7	0.681513	0.340757	−	0.940152i	$-0.389317\pi$
$761$	1.08877e7	0.681513	0.340757	+	0.940152i	$0.389317\pi$
$762$	46090.8	0.00287559
$763$	4.71069e7	2.92936
$764$	−1.62384e6	−0.100649
$765$	0	0
$766$	9.61546e6	0.592104
$767$	517305.	0.0317511
$768$	2.58570e6	0.158188
$769$	−1.39723e7	−0.852024	−0.426012	−	0.904718i	$-0.640082\pi$
$769$	−1.39723e7	−0.852024	−0.426012	+	0.904718i	$0.640082\pi$
$770$	0	0
$771$	1.58640e7	0.961119
$772$	−1.26518e7	−0.764030
$773$	−1.39598e7	−0.840291	−0.420145	−	0.907457i	$-0.638021\pi$
$773$	−1.39598e7	−0.840291	−0.420145	+	0.907457i	$0.638021\pi$
$774$	92906.3	0.00557433
$775$	0	0
$776$	−9.07922e6	−0.541246
$777$	1.87703e7	1.11537
$778$	−4.43424e6	−0.262645
$779$	−1.77215e7	−1.04630
$780$	0	0
$781$	−4.31212e6	−0.252966
$782$	6.20604e6	0.362909
$783$	−4.28092e6	−0.249536
$784$	1.93709e7	1.12554
$785$	0	0
$786$	3.74131e6	0.216007
$787$	1.67531e7	0.964182	0.482091	−	0.876121i	$-0.339877\pi$
$787$	1.67531e7	0.964182	0.482091	+	0.876121i	$0.339877\pi$
$788$	−1.26026e7	−0.723009
$789$	−5.11140e6	−0.292313
$790$	0	0
$791$	3.70426e7	2.10504
$792$	−646980.	−0.0366503
$793$	6.82948e6	0.385660
$794$	3.47555e6	0.195646
$795$	0	0
$796$	−2.33675e7	−1.30716
$797$	−3.77172e6	−0.210326	−0.105163	−	0.994455i	$-0.533536\pi$
$797$	−3.77172e6	−0.210326	−0.105163	+	0.994455i	$0.533536\pi$
$798$	1.01676e7	0.565211
$799$	4.33964e7	2.40484
$800$	0	0
$801$	2.83264e6	0.155995
$802$	−463126.	−0.0254251
$803$	349374.	0.0191206
$804$	1.67495e7	0.913824
$805$	0	0
$806$	1.77576e6	0.0962824
$807$	2.18417e6	0.118060
$808$	2.18855e7	1.17931
$809$	−1.66028e7	−0.891889	−0.445944	−	0.895061i	$-0.647132\pi$
$809$	−1.66028e7	−0.891889	−0.445944	+	0.895061i	$0.647132\pi$
$810$	0	0
$811$	−1.33967e7	−0.715229	−0.357614	−	0.933869i	$-0.616410\pi$
$811$	−1.33967e7	−0.715229	−0.357614	+	0.933869i	$0.616410\pi$
$812$	3.59062e7	1.91108
$813$	−3.50577e6	−0.186019
$814$	1.23948e6	0.0655662
$815$	0	0
$816$	1.14722e7	0.603145
$817$	1.07015e6	0.0560903
$818$	7.73622e6	0.404246
$819$	−3.13365e6	−0.163245
$820$	0	0
$821$	−3.43509e7	−1.77861	−0.889303	−	0.457319i	$-0.848810\pi$
$821$	−3.43509e7	−1.77861	−0.889303	+	0.457319i	$0.848810\pi$
$822$	−2.17405e6	−0.112225
$823$	2.61045e7	1.34343	0.671716	−	0.740809i	$-0.265558\pi$
$823$	2.61045e7	1.34343	0.671716	+	0.740809i	$0.265558\pi$
$824$	1.42227e7	0.729735
$825$	0	0
$826$	−1.61156e6	−0.0821855
$827$	−1.80085e7	−0.915619	−0.457809	−	0.889050i	$-0.651366\pi$
$827$	−1.80085e7	−0.915619	−0.457809	+	0.889050i	$0.651366\pi$
$828$	2.49241e6	0.126341
$829$	603123.	0.0304803	0.0152402	−	0.999884i	$-0.495149\pi$
$829$	603123.	0.0304803	0.0152402	+	0.999884i	$0.495149\pi$
$830$	0	0
$831$	1.01969e7	0.512229
$832$	−777054.	−0.0389173
$833$	−8.33792e7	−4.16337
$834$	7.37502e6	0.367154
$835$	0	0
$836$	−3.39043e6	−0.167780
$837$	−3.33057e6	−0.164325
$838$	−4.04906e6	−0.199179
$839$	−1.88617e7	−0.925075	−0.462537	−	0.886600i	$-0.653061\pi$
$839$	−1.88617e7	−0.925075	−0.462537	+	0.886600i	$0.653061\pi$
$840$	0	0
$841$	1.39729e7	0.681236
$842$	5.53530e6	0.269067
$843$	−7.97796e6	−0.386654
$844$	1.00629e6	0.0486259
$845$	0	0
$846$	−3.45136e6	−0.165792
$847$	3.60663e7	1.72740
$848$	−5.85725e6	−0.279708
$849$	−1.23885e7	−0.589862
$850$	0	0
$851$	−1.04955e7	−0.496797
$852$	−1.75240e7	−0.827053
$853$	−3.84677e7	−1.81019	−0.905095	−	0.425209i	$-0.860200\pi$
$853$	−3.84677e7	−1.81019	−0.905095	+	0.425209i	$0.860200\pi$
$854$	−2.12758e7	−0.998255
$855$	0	0
$856$	−3.16339e7	−1.47560
$857$	1.67613e7	0.779571	0.389785	−	0.920906i	$-0.372549\pi$
$857$	1.67613e7	0.779571	0.389785	+	0.920906i	$0.372549\pi$
$858$	−206928.	−0.00959624
$859$	−4.29704e6	−0.198695	−0.0993474	−	0.995053i	$-0.531676\pi$
$859$	−4.29704e6	−0.198695	−0.0993474	+	0.995053i	$0.531676\pi$
$860$	0	0
$861$	−1.70150e7	−0.782210
$862$	6.96193e6	0.319125
$863$	−3.78429e7	−1.72965	−0.864824	−	0.502076i	$-0.832570\pi$
$863$	−3.78429e7	−1.72965	−0.864824	+	0.502076i	$0.832570\pi$
$864$	−4.06232e6	−0.185136
$865$	0	0
$866$	−2.29562e6	−0.104017
$867$	−3.66017e7	−1.65369
$868$	2.79351e7	1.25850
$869$	5.53713e6	0.248734
$870$	0	0
$871$	1.17751e7	0.525920
$872$	2.77862e7	1.23748
$873$	5.44642e6	0.241866
$874$	−5.68524e6	−0.251750
$875$	0	0
$876$	1.41982e6	0.0625133
$877$	636892.	0.0279619	0.0139809	−	0.999902i	$-0.495550\pi$
$877$	636892.	0.0279619	0.0139809	+	0.999902i	$0.495550\pi$
$878$	5.64193e6	0.246997
$879$	2.15520e6	0.0940839
$880$	0	0
$881$	−1.61380e7	−0.700501	−0.350251	−	0.936656i	$-0.613904\pi$
$881$	−1.61380e7	−0.700501	−0.350251	+	0.936656i	$0.613904\pi$
$882$	6.63123e6	0.287027
$883$	−4.30661e7	−1.85880	−0.929401	−	0.369070i	$-0.879676\pi$
$883$	−4.30661e7	−1.85880	−0.929401	+	0.369070i	$0.879676\pi$
$884$	1.05736e7	0.455087
$885$	0	0
$886$	5.82939e6	0.249482
$887$	1.14749e7	0.489710	0.244855	−	0.969560i	$-0.421260\pi$
$887$	1.14749e7	0.489710	0.244855	+	0.969560i	$0.421260\pi$
$888$	1.10717e7	0.471177
$889$	509733.	0.0216316
$890$	0	0
$891$	388108.	0.0163779
$892$	3.02780e7	1.27413
$893$	−3.97546e7	−1.66824
$894$	−6.12034e6	−0.256113
$895$	0	0
$896$	4.32409e7	1.79939
$897$	1.75219e6	0.0727110
$898$	−3.92407e6	−0.162385
$899$	2.68287e7	1.10714
$900$	0	0
$901$	2.52117e7	1.03464
$902$	−1.12357e6	−0.0459816
$903$	1.02748e6	0.0419329
$904$	2.18497e7	0.889252
$905$	0	0
$906$	−2.52960e6	−0.102384
$907$	−2.78914e7	−1.12578	−0.562888	−	0.826533i	$-0.690310\pi$
$907$	−2.78914e7	−1.12578	−0.562888	+	0.826533i	$0.690310\pi$
$908$	−8.80624e6	−0.354467
$909$	−1.31286e7	−0.526998
$910$	0	0
$911$	3.28566e7	1.31168	0.655838	−	0.754901i	$-0.272315\pi$
$911$	3.28566e7	1.31168	0.655838	+	0.754901i	$0.272315\pi$
$912$	−1.05095e7	−0.418402
$913$	1.66943e6	0.0662813
$914$	−879904.	−0.0348393
$915$	0	0
$916$	1.76831e6	0.0696337
$917$	4.13763e7	1.62491
$918$	3.92727e6	0.153810
$919$	−4.82181e7	−1.88331	−0.941655	−	0.336580i	$-0.890730\pi$
$919$	−4.82181e7	−1.88331	−0.941655	+	0.336580i	$0.890730\pi$
$920$	0	0
$921$	1.50510e7	0.584679
$922$	−1.64755e7	−0.638281
$923$	−1.23195e7	−0.475981
$924$	−3.25526e6	−0.125431
$925$	0	0
$926$	7.41293e6	0.284094
$927$	−8.53189e6	−0.326096
$928$	3.27232e7	1.24734
$929$	3.03481e7	1.15370	0.576850	−	0.816850i	$-0.304282\pi$
$929$	3.03481e7	1.15370	0.576850	+	0.816850i	$0.304282\pi$
$930$	0	0
$931$	7.63821e7	2.88813
$932$	2.26269e7	0.853269
$933$	9.00601e6	0.338710
$934$	−1.25635e7	−0.471241
$935$	0	0
$936$	−1.84839e6	−0.0689612
$937$	1.60933e7	0.598820	0.299410	−	0.954125i	$-0.403210\pi$
$937$	1.60933e7	0.598820	0.299410	+	0.954125i	$0.403210\pi$
$938$	−3.66829e7	−1.36131
$939$	−5.51496e6	−0.204116
$940$	0	0
$941$	−1.68509e6	−0.0620366	−0.0310183	−	0.999519i	$-0.509875\pi$
$941$	−1.68509e6	−0.0620366	−0.0310183	+	0.999519i	$0.509875\pi$
$942$	1.40502e6	0.0515886
$943$	9.51399e6	0.348404
$944$	1.66575e6	0.0608385
$945$	0	0
$946$	67849.0	0.00246499
$947$	8.47809e6	0.307201	0.153601	−	0.988133i	$-0.450913\pi$
$947$	8.47809e6	0.307201	0.153601	+	0.988133i	$0.450913\pi$
$948$	2.25023e7	0.813215
$949$	998147.	0.0359773
$950$	0	0
$951$	1.36817e7	0.490557
$952$	−7.24031e7	−2.58920
$953$	2.42401e7	0.864573	0.432286	−	0.901736i	$-0.357707\pi$
$953$	2.42401e7	0.864573	0.432286	+	0.901736i	$0.357707\pi$
$954$	−2.00511e6	−0.0713291
$955$	0	0
$956$	−1.72785e7	−0.611452
$957$	−3.12633e6	−0.110346
$958$	−1.23987e7	−0.436479
$959$	−2.40436e7	−0.844213
$960$	0	0
$961$	−7.75633e6	−0.270924
$962$	3.54115e6	0.123369
$963$	1.89764e7	0.659400
$964$	4.04159e7	1.40075
$965$	0	0
$966$	−5.45858e6	−0.188207
$967$	2.22310e7	0.764528	0.382264	−	0.924053i	$-0.375144\pi$
$967$	2.22310e7	0.764528	0.382264	+	0.924053i	$0.375144\pi$
$968$	2.12739e7	0.729723
$969$	4.52364e7	1.54767
$970$	0	0
$971$	−2.48740e7	−0.846636	−0.423318	−	0.905981i	$-0.639135\pi$
$971$	−2.48740e7	−0.846636	−0.423318	+	0.905981i	$0.639135\pi$
$972$	1.57723e6	0.0535462
$973$	8.15628e7	2.76191
$974$	1.80527e7	0.609741
$975$	0	0
$976$	2.19912e7	0.738966
$977$	−3.30020e7	−1.10613	−0.553063	−	0.833140i	$-0.686541\pi$
$977$	−3.30020e7	−1.10613	−0.553063	+	0.833140i	$0.686541\pi$
$978$	−3.26591e6	−0.109184
$979$	2.06866e6	0.0689816
$980$	0	0
$981$	−1.66683e7	−0.552992
$982$	2.59519e6	0.0858797
$983$	6.32809e6	0.208876	0.104438	−	0.994531i	$-0.466696\pi$
$983$	6.32809e6	0.208876	0.104438	+	0.994531i	$0.466696\pi$
$984$	−1.00364e7	−0.330437
$985$	0	0
$986$	−3.16353e7	−1.03629
$987$	−3.81697e7	−1.24717
$988$	−9.68632e6	−0.315694
$989$	−574520.	−0.0186773
$990$	0	0
$991$	1.37930e7	0.446145	0.223072	−	0.974802i	$-0.428391\pi$
$991$	1.37930e7	0.446145	0.223072	+	0.974802i	$0.428391\pi$
$992$	2.54588e7	0.821407
$993$	1.53251e7	0.493207
$994$	3.83789e7	1.23205
$995$	0	0
$996$	6.78437e6	0.216701
$997$	−4.62221e7	−1.47269	−0.736346	−	0.676605i	$-0.763450\pi$
$997$	−4.62221e7	−1.47269	−0.736346	+	0.676605i	$0.763450\pi$
$998$	2.20100e7	0.699510
$999$	−6.64168e6	−0.210554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.6.a.u.1.7	yes	11
5.4	even	2		975.6.a.r.1.5	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.6.a.r.1.5	✓	11	5.4	even	2
975.6.a.u.1.7	yes	11	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

\(f(q)\)	\(=\)	\(q+2.29989 q^{2} -9.00000 q^{3} -26.7105 q^{4} -20.6990 q^{6} -228.917 q^{7} -135.028 q^{8} +81.0000 q^{9} +59.1539 q^{11} +240.395 q^{12} +169.000 q^{13} -526.484 q^{14} +544.187 q^{16} -2342.37 q^{17} +186.291 q^{18} +2145.80 q^{19} +2060.25 q^{21} +136.047 q^{22} -1152.00 q^{23} +1215.25 q^{24} +388.681 q^{26} -729.000 q^{27} +6114.49 q^{28} +5872.31 q^{29} +4568.68 q^{31} +5572.46 q^{32} -532.385 q^{33} -5387.20 q^{34} -2163.55 q^{36} +9110.68 q^{37} +4935.11 q^{38} -1521.00 q^{39} -8258.68 q^{41} +4738.36 q^{42} +498.716 q^{43} -1580.03 q^{44} -2649.47 q^{46} -18526.7 q^{47} -4897.69 q^{48} +35596.0 q^{49} +21081.4 q^{51} -4514.08 q^{52} -10763.3 q^{53} -1676.62 q^{54} +30910.2 q^{56} -19312.2 q^{57} +13505.7 q^{58} +3060.98 q^{59} +40411.1 q^{61} +10507.5 q^{62} -18542.3 q^{63} -4597.96 q^{64} -1224.43 q^{66} +69675.2 q^{67} +62566.0 q^{68} +10368.0 q^{69} -72896.6 q^{71} -10937.2 q^{72} +5906.20 q^{73} +20953.6 q^{74} -57315.5 q^{76} -13541.3 q^{77} -3498.13 q^{78} +93605.5 q^{79} +6561.00 q^{81} -18994.1 q^{82} +28221.8 q^{83} -55030.4 q^{84} +1146.99 q^{86} -52850.8 q^{87} -7987.41 q^{88} +34970.9 q^{89} -38687.0 q^{91} +30770.5 q^{92} -41118.1 q^{93} -42609.3 q^{94} -50152.1 q^{96} +67239.7 q^{97} +81867.0 q^{98} +4791.46 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} - 99 q^{3} + 224 q^{4} - 18 q^{6} - 55 q^{7} + 270 q^{8} + 891 q^{9} - 125 q^{11} - 2016 q^{12} + 1859 q^{13} - 1311 q^{14} + 5756 q^{16} - 4507 q^{17} + 162 q^{18} + 142 q^{19} + 495 q^{21}+ \cdots - 10125 q^{99}+O(q^{100}) \) 11 * q + 2 * q^2 - 99 * q^3 + 224 * q^4 - 18 * q^6 - 55 * q^7 + 270 * q^8 + 891 * q^9 - 125 * q^11 - 2016 * q^12 + 1859 * q^13 - 1311 * q^14 + 5756 * q^16 - 4507 * q^17 + 162 * q^18 + 142 * q^19 + 495 * q^21 - 5975 * q^22 - 4672 * q^23 - 2430 * q^24 + 338 * q^26 - 8019 * q^27 - 4149 * q^28 + 5197 * q^29 - 277 * q^31 + 2272 * q^32 + 1125 * q^33 - 21225 * q^34 + 18144 * q^36 - 1696 * q^37 - 27485 * q^38 - 16731 * q^39 + 2040 * q^41 + 11799 * q^42 - 31436 * q^43 - 39883 * q^44 + 67530 * q^46 + 15499 * q^47 - 51804 * q^48 + 29212 * q^49 + 40563 * q^51 + 37856 * q^52 - 119577 * q^53 - 1458 * q^54 + 1677 * q^56 - 1278 * q^57 - 91154 * q^58 + 17115 * q^59 + 26273 * q^61 - 45211 * q^62 - 4455 * q^63 + 137480 * q^64 + 53775 * q^66 + 26313 * q^67 - 191297 * q^68 + 42048 * q^69 - 105756 * q^71 + 21870 * q^72 - 103230 * q^73 - 6501 * q^74 + 124495 * q^76 - 104415 * q^77 - 3042 * q^78 + 210732 * q^79 + 72171 * q^81 - 60283 * q^82 - 144247 * q^83 + 37341 * q^84 - 42260 * q^86 - 46773 * q^87 - 415403 * q^88 + 117088 * q^89 - 9295 * q^91 - 70700 * q^92 + 2493 * q^93 + 383414 * q^94 - 20448 * q^96 - 112934 * q^97 - 159305 * q^98 - 10125 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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