LMFDB - Embedded newform 325.6.a.m.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 356 x^{13} + 802 x^{12} + 49252 x^{11} - 78702 x^{10} - 3324132 x^{9} + \cdots + 3541906480 $ x^15 - 3x^14 - 356x^13 + 802x^12 + 49252x^11 - 78702x^10 - 3324132x^9 + 3620256x^8 + 113209941x^7 - 86135763x^6 - 1838131320x^5 + 957574230x^4 + 11487842206x^3 - 2740914436x^2 - 8383778456x + 3541906480
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$4.30124$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.30124 q^{2} -24.0437 q^{3} -21.1018 q^{4} +79.3742 q^{6} +91.7264 q^{7} +175.302 q^{8} +335.101 q^{9} -45.5406 q^{11} +507.366 q^{12} -169.000 q^{13} -302.811 q^{14} +96.5440 q^{16} +745.700 q^{17} -1106.25 q^{18} +306.010 q^{19} -2205.45 q^{21} +150.340 q^{22} +704.422 q^{23} -4214.91 q^{24} +557.910 q^{26} -2214.45 q^{27} -1935.59 q^{28} -6802.33 q^{29} -7605.25 q^{31} -5928.38 q^{32} +1094.97 q^{33} -2461.74 q^{34} -7071.24 q^{36} -109.012 q^{37} -1010.21 q^{38} +4063.39 q^{39} -13355.2 q^{41} +7280.71 q^{42} +4474.54 q^{43} +960.988 q^{44} -2325.47 q^{46} +4860.59 q^{47} -2321.28 q^{48} -8393.26 q^{49} -17929.4 q^{51} +3566.21 q^{52} +40100.0 q^{53} +7310.45 q^{54} +16079.8 q^{56} -7357.62 q^{57} +22456.1 q^{58} -29655.8 q^{59} +51545.6 q^{61} +25106.8 q^{62} +30737.6 q^{63} +16481.6 q^{64} -3614.75 q^{66} -51877.6 q^{67} -15735.6 q^{68} -16936.9 q^{69} +42271.1 q^{71} +58743.9 q^{72} +45219.7 q^{73} +359.876 q^{74} -6457.36 q^{76} -4177.27 q^{77} -13414.2 q^{78} +62532.7 q^{79} -28185.8 q^{81} +44088.9 q^{82} -46913.1 q^{83} +46538.9 q^{84} -14771.5 q^{86} +163553. q^{87} -7983.35 q^{88} -63387.4 q^{89} -15501.8 q^{91} -14864.6 q^{92} +182859. q^{93} -16046.0 q^{94} +142540. q^{96} -132037. q^{97} +27708.2 q^{98} -15260.7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 12 q^{2} + 36 q^{3} + 250 q^{4} - 202 q^{6} + 306 q^{7} + 576 q^{8} + 963 q^{9} - 326 q^{11} + 2702 q^{12} - 2535 q^{13} + 1334 q^{14} + 6046 q^{16} + 2722 q^{17} + 2036 q^{18} - 2624 q^{19} + 3530 q^{21}+ \cdots - 260488 q^{99}+O(q^{100}) $ 15 * q + 12 * q^2 + 36 * q^3 + 250 * q^4 - 202 * q^6 + 306 * q^7 + 576 * q^8 + 963 * q^9 - 326 * q^11 + 2702 * q^12 - 2535 * q^13 + 1334 * q^14 + 6046 * q^16 + 2722 * q^17 + 2036 * q^18 - 2624 * q^19 + 3530 * q^21 + 7004 * q^22 + 9174 * q^23 + 2740 * q^24 - 2028 * q^26 + 12594 * q^27 + 14880 * q^28 - 3042 * q^29 - 11146 * q^31 + 22054 * q^32 + 18582 * q^33 + 23030 * q^34 + 25172 * q^36 + 10192 * q^37 + 23626 * q^38 - 6084 * q^39 - 7928 * q^41 + 13498 * q^42 + 45934 * q^43 + 22758 * q^44 + 22698 * q^46 + 38730 * q^47 + 103564 * q^48 + 70543 * q^49 + 48084 * q^51 - 42250 * q^52 + 85340 * q^53 + 2482 * q^54 + 33578 * q^56 - 11962 * q^57 + 102450 * q^58 - 62312 * q^59 - 29530 * q^61 + 8266 * q^62 + 209670 * q^63 + 15066 * q^64 - 1240 * q^66 + 43018 * q^67 + 281646 * q^68 + 199340 * q^69 - 22868 * q^71 - 54972 * q^72 + 165124 * q^73 - 258566 * q^74 - 139240 * q^76 + 86270 * q^77 + 34138 * q^78 + 129584 * q^79 + 57731 * q^81 + 95426 * q^82 + 375574 * q^83 - 15182 * q^84 - 241780 * q^86 + 373706 * q^87 + 881886 * q^88 - 167638 * q^89 - 51714 * q^91 + 672054 * q^92 + 538530 * q^93 + 104250 * q^94 - 46370 * q^96 + 265450 * q^97 + 1324762 * q^98 - 260488 * 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 $

$2$	−3.30124	−0.583583	−0.291791	−	0.956482i	$-0.594251\pi$
$2$	−3.30124	−0.583583	−0.291791	+	0.956482i	$0.594251\pi$
$3$	−24.0437	−1.54241	−0.771203	−	0.636589i	$-0.780345\pi$
$3$	−24.0437	−1.54241	−0.771203	+	0.636589i	$0.780345\pi$
$4$	−21.1018	−0.659431
$5$	0	0
$5$	0	0
$6$	79.3742	0.900121
$7$	91.7264	0.707537	0.353769	−	0.935333i	$-0.384900\pi$
$7$	91.7264	0.707537	0.353769	+	0.935333i	$0.384900\pi$
$8$	175.302	0.968415
$9$	335.101	1.37902
$10$	0	0
$11$	−45.5406	−0.113479	−0.0567397	−	0.998389i	$-0.518071\pi$
$11$	−45.5406	−0.113479	−0.0567397	+	0.998389i	$0.518071\pi$
$12$	507.366	1.01711
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−302.811	−0.412906
$15$	0	0
$16$	96.5440	0.0942812
$17$	745.700	0.625809	0.312905	−	0.949785i	$-0.398698\pi$
$17$	745.700	0.625809	0.312905	+	0.949785i	$0.398698\pi$
$18$	−1106.25	−0.804770
$19$	306.010	0.194469	0.0972347	−	0.995261i	$-0.469000\pi$
$19$	306.010	0.194469	0.0972347	+	0.995261i	$0.469000\pi$
$20$	0	0
$21$	−2205.45	−1.09131
$22$	150.340	0.0662246
$23$	704.422	0.277660	0.138830	−	0.990316i	$-0.455666\pi$
$23$	704.422	0.277660	0.138830	+	0.990316i	$0.455666\pi$
$24$	−4214.91	−1.49369
$25$	0	0
$26$	557.910	0.161857
$27$	−2214.45	−0.584598
$28$	−1935.59	−0.466572
$29$	−6802.33	−1.50198	−0.750988	−	0.660316i	$-0.770422\pi$
$29$	−6802.33	−1.50198	−0.750988	+	0.660316i	$0.770422\pi$
$30$	0	0
$31$	−7605.25	−1.42138	−0.710689	−	0.703507i	$-0.751616\pi$
$31$	−7605.25	−1.42138	−0.710689	+	0.703507i	$0.751616\pi$
$32$	−5928.38	−1.02344
$33$	1094.97	0.175031
$34$	−2461.74	−0.365211
$35$	0	0
$36$	−7071.24	−0.909367
$37$	−109.012	−0.0130909	−0.00654547	−	0.999979i	$-0.502084\pi$
$37$	−109.012	−0.0130909	−0.00654547	+	0.999979i	$0.502084\pi$
$38$	−1010.21	−0.113489
$39$	4063.39	0.427787
$40$	0	0
$41$	−13355.2	−1.24077	−0.620386	−	0.784296i	$-0.713024\pi$
$41$	−13355.2	−1.24077	−0.620386	+	0.784296i	$0.713024\pi$
$42$	7280.71	0.636869
$43$	4474.54	0.369044	0.184522	−	0.982828i	$-0.440926\pi$
$43$	4474.54	0.369044	0.184522	+	0.982828i	$0.440926\pi$
$44$	960.988	0.0748319
$45$	0	0
$46$	−2325.47	−0.162038
$47$	4860.59	0.320955	0.160478	−	0.987039i	$-0.448697\pi$
$47$	4860.59	0.320955	0.160478	+	0.987039i	$0.448697\pi$
$48$	−2321.28	−0.145420
$49$	−8393.26	−0.499391
$50$	0	0
$51$	−17929.4	−0.965252
$52$	3566.21	0.182893
$53$	40100.0	1.96090	0.980448	−	0.196777i	$-0.0630473\pi$
$53$	40100.0	1.96090	0.980448	+	0.196777i	$0.0630473\pi$
$54$	7310.45	0.341161
$55$	0	0
$56$	16079.8	0.685190
$57$	−7357.62	−0.299951
$58$	22456.1	0.876526
$59$	−29655.8	−1.10912	−0.554561	−	0.832143i	$-0.687114\pi$
$59$	−29655.8	−1.10912	−0.554561	+	0.832143i	$0.687114\pi$
$60$	0	0
$61$	51545.6	1.77365	0.886823	−	0.462109i	$-0.152907\pi$
$61$	51545.6	1.77365	0.886823	+	0.462109i	$0.152907\pi$
$62$	25106.8	0.829491
$63$	30737.6	0.975706
$64$	16481.6	0.502978
$65$	0	0
$66$	−3614.75	−0.102145
$67$	−51877.6	−1.41186	−0.705932	−	0.708280i	$-0.749472\pi$
$67$	−51877.6	−1.41186	−0.705932	+	0.708280i	$0.749472\pi$
$68$	−15735.6	−0.412678
$69$	−16936.9	−0.428265
$70$	0	0
$71$	42271.1	0.995171	0.497585	−	0.867415i	$-0.334220\pi$
$71$	42271.1	0.995171	0.497585	+	0.867415i	$0.334220\pi$
$72$	58743.9	1.33546
$73$	45219.7	0.993162	0.496581	−	0.867990i	$-0.334588\pi$
$73$	45219.7	0.993162	0.496581	+	0.867990i	$0.334588\pi$
$74$	359.876	0.00763964
$75$	0	0
$76$	−6457.36	−0.128239
$77$	−4177.27	−0.0802909
$78$	−13414.2	−0.249649
$79$	62532.7	1.12730	0.563650	−	0.826014i	$-0.309397\pi$
$79$	62532.7	1.12730	0.563650	+	0.826014i	$0.309397\pi$
$80$	0	0
$81$	−28185.8	−0.477329
$82$	44088.9	0.724093
$83$	−46913.1	−0.747479	−0.373740	−	0.927534i	$-0.621925\pi$
$83$	−46913.1	−0.747479	−0.373740	+	0.927534i	$0.621925\pi$
$84$	46538.9	0.719644
$85$	0	0
$86$	−14771.5	−0.215367
$87$	163553.	2.31666
$88$	−7983.35	−0.109895
$89$	−63387.4	−0.848258	−0.424129	−	0.905602i	$-0.639420\pi$
$89$	−63387.4	−0.848258	−0.424129	+	0.905602i	$0.639420\pi$
$90$	0	0
$91$	−15501.8	−0.196236
$92$	−14864.6	−0.183098
$93$	182859.	2.19234
$94$	−16046.0	−0.187304
$95$	0	0
$96$	142540.	1.57855
$97$	−132037.	−1.42484	−0.712421	−	0.701752i	$-0.752401\pi$
$97$	−132037.	−1.42484	−0.712421	+	0.701752i	$0.752401\pi$
$98$	27708.2	0.291436
$99$	−15260.7	−0.156490
$100$	0	0
$101$	174161.	1.69882	0.849408	−	0.527736i	$-0.176959\pi$
$101$	174161.	1.69882	0.849408	+	0.527736i	$0.176959\pi$
$102$	59189.3	0.563304
$103$	−41504.6	−0.385481	−0.192741	−	0.981250i	$-0.561738\pi$
$103$	−41504.6	−0.385481	−0.192741	+	0.981250i	$0.561738\pi$
$104$	−29626.0	−0.268590
$105$	0	0
$106$	−132380.	−1.14435
$107$	7340.14	0.0619791	0.0309895	−	0.999520i	$-0.490134\pi$
$107$	7340.14	0.0619791	0.0309895	+	0.999520i	$0.490134\pi$
$108$	46729.0	0.385502
$109$	51340.4	0.413898	0.206949	−	0.978352i	$-0.433647\pi$
$109$	51340.4	0.413898	0.206949	+	0.978352i	$0.433647\pi$
$110$	0	0
$111$	2621.06	0.0201915
$112$	8855.63	0.0667075
$113$	−145027.	−1.06845	−0.534223	−	0.845344i	$-0.679396\pi$
$113$	−145027.	−1.06845	−0.534223	+	0.845344i	$0.679396\pi$
$114$	24289.3	0.175046
$115$	0	0
$116$	143541.	0.990450
$117$	−56632.1	−0.382470
$118$	97900.9	0.647264
$119$	68400.4	0.442783
$120$	0	0
$121$	−158977.	−0.987122
$122$	−170164.	−1.03507
$123$	321110.	1.91378
$124$	160485.	0.937301
$125$	0	0
$126$	−101472.	−0.569405
$127$	69448.0	0.382077	0.191038	−	0.981583i	$-0.438814\pi$
$127$	69448.0	0.382077	0.191038	+	0.981583i	$0.438814\pi$
$128$	135298.	0.729907
$129$	−107585.	−0.569215
$130$	0	0
$131$	−264141.	−1.34480	−0.672400	−	0.740188i	$-0.734736\pi$
$131$	−264141.	−1.34480	−0.672400	+	0.740188i	$0.734736\pi$
$132$	−23105.8	−0.115421
$133$	28069.2	0.137594
$134$	171260.	0.823939
$135$	0	0
$136$	130723.	0.606043
$137$	−48840.0	−0.222318	−0.111159	−	0.993803i	$-0.535456\pi$
$137$	−48840.0	−0.222318	−0.111159	+	0.993803i	$0.535456\pi$
$138$	55912.9	0.249928
$139$	193453.	0.849255	0.424627	−	0.905368i	$-0.360405\pi$
$139$	193453.	0.849255	0.424627	+	0.905368i	$0.360405\pi$
$140$	0	0
$141$	−116867.	−0.495043
$142$	−139547.	−0.580764
$143$	7696.36	0.0314735
$144$	32352.0	0.130015
$145$	0	0
$146$	−149281.	−0.579592
$147$	201805.	0.770264
$148$	2300.35	0.00863258
$149$	−133014.	−0.490830	−0.245415	−	0.969418i	$-0.578924\pi$
$149$	−133014.	−0.490830	−0.245415	+	0.969418i	$0.578924\pi$
$150$	0	0
$151$	−310303.	−1.10750	−0.553749	−	0.832684i	$-0.686803\pi$
$151$	−310303.	−1.10750	−0.553749	+	0.832684i	$0.686803\pi$
$152$	53644.1	0.188327
$153$	249885.	0.863002
$154$	13790.2	0.0468564
$155$	0	0
$156$	−85744.9	−0.282096
$157$	−49697.8	−0.160912	−0.0804559	−	0.996758i	$-0.525638\pi$
$157$	−49697.8	−0.160912	−0.0804559	+	0.996758i	$0.525638\pi$
$158$	−206436.	−0.657873
$159$	−964154.	−3.02450
$160$	0	0
$161$	64614.1	0.196455
$162$	93048.2	0.278561
$163$	658943.	1.94258	0.971290	−	0.237899i	$-0.0764588\pi$
$163$	658943.	1.94258	0.971290	+	0.237899i	$0.0764588\pi$
$164$	281820.	0.818205
$165$	0	0
$166$	154872.	0.436216
$167$	655469.	1.81870	0.909349	−	0.416034i	$-0.136580\pi$
$167$	655469.	1.81870	0.909349	+	0.416034i	$0.136580\pi$
$168$	−386619.	−1.05684
$169$	28561.0	0.0769231
$170$	0	0
$171$	102544.	0.268177
$172$	−94420.9	−0.243359
$173$	−280036.	−0.711376	−0.355688	−	0.934605i	$-0.615753\pi$
$173$	−280036.	−0.711376	−0.355688	+	0.934605i	$0.615753\pi$
$174$	−539929.	−1.35196
$175$	0	0
$176$	−4396.67	−0.0106990
$177$	713036.	1.71072
$178$	209257.	0.495028
$179$	261227.	0.609377	0.304689	−	0.952452i	$-0.401448\pi$
$179$	261227.	0.609377	0.304689	+	0.952452i	$0.401448\pi$
$180$	0	0
$181$	−112896.	−0.256143	−0.128071	−	0.991765i	$-0.540879\pi$
$181$	−112896.	−0.256143	−0.128071	+	0.991765i	$0.540879\pi$
$182$	51175.1	0.114520
$183$	−1.23935e6	−2.73568
$184$	123486.	0.268890
$185$	0	0
$186$	−603660.	−1.27941
$187$	−33959.6	−0.0710164
$188$	−102567.	−0.211648
$189$	−203124.	−0.413625
$190$	0	0
$191$	790161.	1.56723	0.783614	−	0.621247i	$-0.213374\pi$
$191$	790161.	1.56723	0.783614	+	0.621247i	$0.213374\pi$
$192$	−396279.	−0.775797
$193$	272267.	0.526140	0.263070	−	0.964777i	$-0.415265\pi$
$193$	272267.	0.526140	0.263070	+	0.964777i	$0.415265\pi$
$194$	435887.	0.831513
$195$	0	0
$196$	177113.	0.329314
$197$	−304065.	−0.558215	−0.279107	−	0.960260i	$-0.590039\pi$
$197$	−304065.	−0.558215	−0.279107	+	0.960260i	$0.590039\pi$
$198$	50379.3	0.0913248
$199$	291885.	0.522492	0.261246	−	0.965272i	$-0.415867\pi$
$199$	291885.	0.522492	0.261246	+	0.965272i	$0.415867\pi$
$200$	0	0
$201$	1.24733e6	2.17767
$202$	−574946.	−0.991400
$203$	−623953.	−1.06270
$204$	378343.	0.636518
$205$	0	0
$206$	137017.	0.224960
$207$	236053.	0.382898
$208$	−16315.9	−0.0261489
$209$	−13935.9	−0.0220683
$210$	0	0
$211$	447521.	0.692003	0.346001	−	0.938234i	$-0.387539\pi$
$211$	447521.	0.692003	0.346001	+	0.938234i	$0.387539\pi$
$212$	−846182.	−1.29308
$213$	−1.01635e6	−1.53496
$214$	−24231.6	−0.0361699
$215$	0	0
$216$	−388198.	−0.566134
$217$	−697603.	−1.00568
$218$	−169487.	−0.241543
$219$	−1.08725e6	−1.53186
$220$	0	0
$221$	−126023.	−0.173568
$222$	−8652.75	−0.0117834
$223$	158.327	0.000213203 0	0.000106602	−	1.00000i	$-0.499966\pi$
$223$	158.327	0.000213203 0	0.000106602		1.00000i	$0.499966\pi$
$224$	−543789.	−0.724119
$225$	0	0
$226$	478769.	0.623526
$227$	1.13950e6	1.46774	0.733870	−	0.679290i	$-0.237712\pi$
$227$	1.13950e6	1.46774	0.733870	+	0.679290i	$0.237712\pi$
$228$	155259.	0.197797
$229$	693287.	0.873624	0.436812	−	0.899553i	$-0.356108\pi$
$229$	693287.	0.873624	0.436812	+	0.899553i	$0.356108\pi$
$230$	0	0
$231$	100437.	0.123841
$232$	−1.19246e6	−1.45454
$233$	−399786.	−0.482434	−0.241217	−	0.970471i	$-0.577547\pi$
$233$	−399786.	−0.482434	−0.241217	+	0.970471i	$0.577547\pi$
$234$	186956.	0.223203
$235$	0	0
$236$	625791.	0.731390
$237$	−1.50352e6	−1.73876
$238$	−225806.	−0.258401
$239$	445444.	0.504427	0.252213	−	0.967672i	$-0.418842\pi$
$239$	445444.	0.504427	0.252213	+	0.967672i	$0.418842\pi$
$240$	0	0
$241$	−1.23166e6	−1.36600	−0.682999	−	0.730420i	$-0.739324\pi$
$241$	−1.23166e6	−1.36600	−0.682999	+	0.730420i	$0.739324\pi$
$242$	524822.	0.576067
$243$	1.21580e6	1.32083
$244$	−1.08771e6	−1.16960
$245$	0	0
$246$	−1.06006e6	−1.11685
$247$	−51715.7	−0.0539361
$248$	−1.33321e6	−1.37648
$249$	1.12797e6	1.15292
$250$	0	0
$251$	615483.	0.616640	0.308320	−	0.951283i	$-0.400233\pi$
$251$	615483.	0.616640	0.308320	+	0.951283i	$0.400233\pi$
$252$	−648619.	−0.643411
$253$	−32079.8	−0.0315087
$254$	−229265.	−0.222973
$255$	0	0
$256$	−974063.	−0.928939
$257$	−844218.	−0.797300	−0.398650	−	0.917103i	$-0.630521\pi$
$257$	−844218.	−0.797300	−0.398650	+	0.917103i	$0.630521\pi$
$258$	355163.	0.332184
$259$	−9999.30	−0.00926233
$260$	0	0
$261$	−2.27947e6	−2.07125
$262$	871993.	0.784802
$263$	586640.	0.522976	0.261488	−	0.965207i	$-0.415787\pi$
$263$	586640.	0.522976	0.261488	+	0.965207i	$0.415787\pi$
$264$	191950.	0.169503
$265$	0	0
$266$	−92663.2	−0.0802977
$267$	1.52407e6	1.30836
$268$	1.09471e6	0.931027
$269$	2.07403e6	1.74757	0.873786	−	0.486310i	$-0.161657\pi$
$269$	2.07403e6	1.74757	0.873786	+	0.486310i	$0.161657\pi$
$270$	0	0
$271$	1.92221e6	1.58993	0.794963	−	0.606658i	$-0.207490\pi$
$271$	1.92221e6	1.58993	0.794963	+	0.606658i	$0.207490\pi$
$272$	71992.9	0.0590021
$273$	372720.	0.302675
$274$	161233.	0.129741
$275$	0	0
$276$	357400.	0.282411
$277$	1.70508e6	1.33520	0.667599	−	0.744521i	$-0.267322\pi$
$277$	1.70508e6	1.33520	0.667599	+	0.744521i	$0.267322\pi$
$278$	−638634.	−0.495610
$279$	−2.54853e6	−1.96010
$280$	0	0
$281$	−961277.	−0.726244	−0.363122	−	0.931742i	$-0.618289\pi$
$281$	−961277.	−0.726244	−0.363122	+	0.931742i	$0.618289\pi$
$282$	385805.	0.288899
$283$	−1.84545e6	−1.36973	−0.684867	−	0.728668i	$-0.740140\pi$
$283$	−1.84545e6	−1.36973	−0.684867	+	0.728668i	$0.740140\pi$
$284$	−891996.	−0.656247
$285$	0	0
$286$	−25407.5	−0.0183674
$287$	−1.22503e6	−0.877893
$288$	−1.98661e6	−1.41134
$289$	−863788.	−0.608363
$290$	0	0
$291$	3.17467e6	2.19769
$292$	−954217.	−0.654923
$293$	425094.	0.289278	0.144639	−	0.989484i	$-0.453798\pi$
$293$	425094.	0.289278	0.144639	+	0.989484i	$0.453798\pi$
$294$	−666208.	−0.449512
$295$	0	0
$296$	−19110.0	−0.0126775
$297$	100848.	0.0663398
$298$	439111.	0.286440
$299$	−119047.	−0.0770090
$300$	0	0
$301$	410434.	0.261112
$302$	1.02438e6	0.646317
$303$	−4.18747e6	−2.62027
$304$	29543.4	0.0183348
$305$	0	0
$306$	−824931.	−0.503633
$307$	−669386.	−0.405351	−0.202675	−	0.979246i	$-0.564964\pi$
$307$	−669386.	−0.405351	−0.202675	+	0.979246i	$0.564964\pi$
$308$	88148.0	0.0529463
$309$	997926.	0.594569
$310$	0	0
$311$	−1.10103e6	−0.645506	−0.322753	−	0.946483i	$-0.604608\pi$
$311$	−1.10103e6	−0.645506	−0.322753	+	0.946483i	$0.604608\pi$
$312$	712320.	0.414275
$313$	698228.	0.402844	0.201422	−	0.979505i	$-0.435444\pi$
$313$	698228.	0.402844	0.201422	+	0.979505i	$0.435444\pi$
$314$	164064.	0.0939053
$315$	0	0
$316$	−1.31955e6	−0.743377
$317$	−1.25102e6	−0.699226	−0.349613	−	0.936894i	$-0.613687\pi$
$317$	−1.25102e6	−0.699226	−0.349613	+	0.936894i	$0.613687\pi$
$318$	3.18290e6	1.76504
$319$	309782.	0.170443
$320$	0	0
$321$	−176484.	−0.0955969
$322$	−213307.	−0.114648
$323$	228192.	0.121701
$324$	594771.	0.314766
$325$	0	0
$326$	−2.17533e6	−1.13366
$327$	−1.23441e6	−0.638398
$328$	−2.34120e6	−1.20158
$329$	445845.	0.227088
$330$	0	0
$331$	−2.38101e6	−1.19452	−0.597258	−	0.802049i	$-0.703743\pi$
$331$	−2.38101e6	−1.19452	−0.597258	+	0.802049i	$0.703743\pi$
$332$	989952.	0.492911
$333$	−36530.1	−0.0180526
$334$	−2.16386e6	−1.06136
$335$	0	0
$336$	−212923.	−0.102890
$337$	2.38010e6	1.14161	0.570807	−	0.821084i	$-0.306630\pi$
$337$	2.38010e6	1.14161	0.570807	+	0.821084i	$0.306630\pi$
$338$	−94286.8	−0.0448910
$339$	3.48699e6	1.64798
$340$	0	0
$341$	346348.	0.161297
$342$	−338523.	−0.156503
$343$	−2.31153e6	−1.06088
$344$	784396.	0.357387
$345$	0	0
$346$	924468.	0.415147
$347$	3.45043e6	1.53833	0.769166	−	0.639049i	$-0.220672\pi$
$347$	3.45043e6	1.53833	0.769166	+	0.639049i	$0.220672\pi$
$348$	−3.45127e6	−1.52768
$349$	3.27121e6	1.43762	0.718811	−	0.695206i	$-0.244687\pi$
$349$	3.27121e6	1.43762	0.718811	+	0.695206i	$0.244687\pi$
$350$	0	0
$351$	374243.	0.162138
$352$	269982.	0.116139
$353$	4.08960e6	1.74680	0.873401	−	0.487001i	$-0.161909\pi$
$353$	4.08960e6	1.74680	0.873401	+	0.487001i	$0.161909\pi$
$354$	−2.35390e6	−0.998345
$355$	0	0
$356$	1.33759e6	0.559368
$357$	−1.64460e6	−0.682952
$358$	−862375.	−0.355622
$359$	2.45540e6	1.00551	0.502754	−	0.864430i	$-0.332320\pi$
$359$	2.45540e6	1.00551	0.502754	+	0.864430i	$0.332320\pi$
$360$	0	0
$361$	−2.38246e6	−0.962182
$362$	372697.	0.149480
$363$	3.82240e6	1.52254
$364$	327115.	0.129404
$365$	0	0
$366$	4.09139e6	1.59650
$367$	−328555.	−0.127333	−0.0636667	−	0.997971i	$-0.520279\pi$
$367$	−328555.	−0.127333	−0.0636667	+	0.997971i	$0.520279\pi$
$368$	68007.7	0.0261781
$369$	−4.47536e6	−1.71105
$370$	0	0
$371$	3.67823e6	1.38741
$372$	−3.85865e6	−1.44570
$373$	5.17656e6	1.92650	0.963250	−	0.268606i	$-0.0865630\pi$
$373$	5.17656e6	1.92650	0.963250	+	0.268606i	$0.0865630\pi$
$374$	112109.	0.0414439
$375$	0	0
$376$	852071.	0.310818
$377$	1.14959e6	0.416573
$378$	670561.	0.241384
$379$	3.24314e6	1.15976	0.579880	−	0.814702i	$-0.303100\pi$
$379$	3.24314e6	1.15976	0.579880	+	0.814702i	$0.303100\pi$
$380$	0	0
$381$	−1.66979e6	−0.589318
$382$	−2.60851e6	−0.914607
$383$	4.53434e6	1.57949	0.789745	−	0.613435i	$-0.210213\pi$
$383$	4.53434e6	1.57949	0.789745	+	0.613435i	$0.210213\pi$
$384$	−3.25308e6	−1.12581
$385$	0	0
$386$	−898819.	−0.307046
$387$	1.49942e6	0.508917
$388$	2.78622e6	0.939586
$389$	1.77865e6	0.595958	0.297979	−	0.954572i	$-0.403687\pi$
$389$	1.77865e6	0.595958	0.297979	+	0.954572i	$0.403687\pi$
$390$	0	0
$391$	525288.	0.173762
$392$	−1.47135e6	−0.483618
$393$	6.35094e6	2.07423
$394$	1.00379e6	0.325764
$395$	0	0
$396$	322028.	0.103194
$397$	5.19763e6	1.65512	0.827560	−	0.561378i	$-0.189729\pi$
$397$	5.19763e6	1.65512	0.827560	+	0.561378i	$0.189729\pi$
$398$	−963583.	−0.304917
$399$	−674888.	−0.212226
$400$	0	0
$401$	4.03901e6	1.25434	0.627168	−	0.778884i	$-0.284214\pi$
$401$	4.03901e6	1.25434	0.627168	+	0.778884i	$0.284214\pi$
$402$	−4.11774e6	−1.27085
$403$	1.28529e6	0.394219
$404$	−3.67510e6	−1.12025
$405$	0	0
$406$	2.05982e6	0.620175
$407$	4964.48	0.00148555
$408$	−3.14306e6	−0.934765
$409$	−2.23547e6	−0.660787	−0.330393	−	0.943843i	$-0.607181\pi$
$409$	−2.23547e6	−0.660787	−0.330393	+	0.943843i	$0.607181\pi$
$410$	0	0
$411$	1.17430e6	0.342904
$412$	875823.	0.254199
$413$	−2.72022e6	−0.784746
$414$	−779267.	−0.223452
$415$	0	0
$416$	1.00190e6	0.283850
$417$	−4.65133e6	−1.30990
$418$	46005.7	0.0128787
$419$	1.52835e6	0.425291	0.212646	−	0.977129i	$-0.431792\pi$
$419$	1.52835e6	0.425291	0.212646	+	0.977129i	$0.431792\pi$
$420$	0	0
$421$	−2.80885e6	−0.772367	−0.386183	−	0.922422i	$-0.626207\pi$
$421$	−2.80885e6	−0.772367	−0.386183	+	0.922422i	$0.626207\pi$
$422$	−1.47738e6	−0.403841
$423$	1.62879e6	0.442603
$424$	7.02961e6	1.89896
$425$	0	0
$426$	3.35523e6	0.895774
$427$	4.72809e6	1.25492
$428$	−154890.	−0.0408710
$429$	−185049.	−0.0485449
$430$	0	0
$431$	−2.64484e6	−0.685813	−0.342906	−	0.939370i	$-0.611411\pi$
$431$	−2.64484e6	−0.685813	−0.342906	+	0.939370i	$0.611411\pi$
$432$	−213792.	−0.0551166
$433$	−6.38548e6	−1.63672	−0.818359	−	0.574708i	$-0.805116\pi$
$433$	−6.38548e6	−1.63672	−0.818359	+	0.574708i	$0.805116\pi$
$434$	2.30295e6	0.586896
$435$	0	0
$436$	−1.08337e6	−0.272937
$437$	215560.	0.0539964
$438$	3.58927e6	0.893967
$439$	−1.84296e6	−0.456409	−0.228205	−	0.973613i	$-0.573286\pi$
$439$	−1.84296e6	−0.456409	−0.228205	+	0.973613i	$0.573286\pi$
$440$	0	0
$441$	−2.81259e6	−0.688669
$442$	416033.	0.101291
$443$	610862.	0.147888	0.0739442	−	0.997262i	$-0.476441\pi$
$443$	610862.	0.147888	0.0739442	+	0.997262i	$0.476441\pi$
$444$	−55309.1	−0.0133149
$445$	0	0
$446$	−522.676	−0.000124422 0
$447$	3.19815e6	0.757059
$448$	1.51180e6	0.355876
$449$	−1.32831e6	−0.310945	−0.155473	−	0.987840i	$-0.549690\pi$
$449$	−1.32831e6	−0.310945	−0.155473	+	0.987840i	$0.549690\pi$
$450$	0	0
$451$	608206.	0.140802
$452$	3.06033e6	0.704567
$453$	7.46083e6	1.70821
$454$	−3.76176e6	−0.856547
$455$	0	0
$456$	−1.28980e6	−0.290477
$457$	−1.05715e6	−0.236780	−0.118390	−	0.992967i	$-0.537773\pi$
$457$	−1.05715e6	−0.236780	−0.118390	+	0.992967i	$0.537773\pi$
$458$	−2.28871e6	−0.509832
$459$	−1.65132e6	−0.365847
$460$	0	0
$461$	−88191.8	−0.0193275	−0.00966376	−	0.999953i	$-0.503076\pi$
$461$	−88191.8	−0.0193275	−0.00966376	+	0.999953i	$0.503076\pi$
$462$	−331568.	−0.0722715
$463$	−5.10298e6	−1.10630	−0.553148	−	0.833083i	$-0.686573\pi$
$463$	−5.10298e6	−1.10630	−0.553148	+	0.833083i	$0.686573\pi$
$464$	−656724.	−0.141608
$465$	0	0
$466$	1.31979e6	0.281540
$467$	4.32926e6	0.918590	0.459295	−	0.888284i	$-0.348102\pi$
$467$	4.32926e6	0.918590	0.459295	+	0.888284i	$0.348102\pi$
$468$	1.19504e6	0.252213
$469$	−4.75855e6	−0.998946
$470$	0	0
$471$	1.19492e6	0.248191
$472$	−5.19872e6	−1.07409
$473$	−203773.	−0.0418788
$474$	4.96348e6	1.01471
$475$	0	0
$476$	−1.44337e6	−0.291985
$477$	1.34376e7	2.70411
$478$	−1.47052e6	−0.294375
$479$	2.88997e6	0.575513	0.287756	−	0.957704i	$-0.407091\pi$
$479$	2.88997e6	0.575513	0.287756	+	0.957704i	$0.407091\pi$
$480$	0	0
$481$	18423.1	0.00363077
$482$	4.06602e6	0.797172
$483$	−1.55356e6	−0.303013
$484$	3.35470e6	0.650940
$485$	0	0
$486$	−4.01366e6	−0.770815
$487$	3.37347e6	0.644546	0.322273	−	0.946647i	$-0.395553\pi$
$487$	3.37347e6	0.644546	0.322273	+	0.946647i	$0.395553\pi$
$488$	9.03604e6	1.71763
$489$	−1.58434e7	−2.99625
$490$	0	0
$491$	5.24844e6	0.982486	0.491243	−	0.871023i	$-0.336543\pi$
$491$	5.24844e6	0.982486	0.491243	+	0.871023i	$0.336543\pi$
$492$	−6.77600e6	−1.26200
$493$	−5.07250e6	−0.939950
$494$	170726.	0.0314762
$495$	0	0
$496$	−734241.	−0.134009
$497$	3.87738e6	0.704120
$498$	−3.72369e6	−0.672822
$499$	−3.51772e6	−0.632427	−0.316213	−	0.948688i	$-0.602412\pi$
$499$	−3.51772e6	−0.632427	−0.316213	+	0.948688i	$0.602412\pi$
$500$	0	0
$501$	−1.57599e7	−2.80517
$502$	−2.03186e6	−0.359860
$503$	8.97569e6	1.58179	0.790893	−	0.611954i	$-0.209616\pi$
$503$	8.97569e6	1.58179	0.790893	+	0.611954i	$0.209616\pi$
$504$	5.38836e6	0.944889
$505$	0	0
$506$	105903.	0.0183879
$507$	−686713.	−0.118647
$508$	−1.46548e6	−0.251953
$509$	−9.09941e6	−1.55675	−0.778375	−	0.627800i	$-0.783955\pi$
$509$	−9.09941e6	−1.55675	−0.778375	+	0.627800i	$0.783955\pi$
$510$	0	0
$511$	4.14784e6	0.702700
$512$	−1.11393e6	−0.187794
$513$	−677645.	−0.113686
$514$	2.78697e6	0.465290
$515$	0	0
$516$	2.27023e6	0.375358
$517$	−221354.	−0.0364218
$518$	33010.1	0.00540533
$519$	6.73312e6	1.09723
$520$	0	0
$521$	5.26898e6	0.850418	0.425209	−	0.905095i	$-0.360201\pi$
$521$	5.26898e6	0.850418	0.425209	+	0.905095i	$0.360201\pi$
$522$	7.52508e6	1.20874
$523$	−4.16491e6	−0.665811	−0.332906	−	0.942960i	$-0.608029\pi$
$523$	−4.16491e6	−0.665811	−0.332906	+	0.942960i	$0.608029\pi$
$524$	5.57385e6	0.886803
$525$	0	0
$526$	−1.93664e6	−0.305200
$527$	−5.67124e6	−0.889511
$528$	105712.	0.0165022
$529$	−5.94013e6	−0.922905
$530$	0	0
$531$	−9.93769e6	−1.52950
$532$	−592311.	−0.0907341
$533$	2.25704e6	0.344129
$534$	−5.03132e6	−0.763535
$535$	0	0
$536$	−9.09424e6	−1.36727
$537$	−6.28088e6	−0.939907
$538$	−6.84689e6	−1.01985
$539$	382234.	0.0566706
$540$	0	0
$541$	−3.03221e6	−0.445416	−0.222708	−	0.974885i	$-0.571490\pi$
$541$	−3.03221e6	−0.445416	−0.222708	+	0.974885i	$0.571490\pi$
$542$	−6.34567e6	−0.927853
$543$	2.71444e6	0.395076
$544$	−4.42079e6	−0.640476
$545$	0	0
$546$	−1.23044e6	−0.176636
$547$	−3.32454e6	−0.475076	−0.237538	−	0.971378i	$-0.576340\pi$
$547$	−3.32454e6	−0.475076	−0.237538	+	0.971378i	$0.576340\pi$
$548$	1.03061e6	0.146603
$549$	1.72730e7	2.44589
$550$	0	0
$551$	−2.08158e6	−0.292088
$552$	−2.96908e6	−0.414738
$553$	5.73590e6	0.797607
$554$	−5.62888e6	−0.779198
$555$	0	0
$556$	−4.08220e6	−0.560025
$557$	7.78685e6	1.06347	0.531733	−	0.846912i	$-0.321541\pi$
$557$	7.78685e6	1.06347	0.531733	+	0.846912i	$0.321541\pi$
$558$	8.41331e6	1.14388
$559$	−756198.	−0.102354
$560$	0	0
$561$	816516.	0.109536
$562$	3.17341e6	0.423824
$563$	4.68638e6	0.623112	0.311556	−	0.950228i	$-0.399150\pi$
$563$	4.68638e6	0.623112	0.311556	+	0.950228i	$0.399150\pi$
$564$	2.46610e6	0.326447
$565$	0	0
$566$	6.09227e6	0.799353
$567$	−2.58538e6	−0.337728
$568$	7.41020e6	0.963738
$569$	−2.10415e6	−0.272456	−0.136228	−	0.990678i	$-0.543498\pi$
$569$	−2.10415e6	−0.272456	−0.136228	+	0.990678i	$0.543498\pi$
$570$	0	0
$571$	1.20728e7	1.54959	0.774796	−	0.632211i	$-0.217852\pi$
$571$	1.20728e7	1.54959	0.774796	+	0.632211i	$0.217852\pi$
$572$	−162407.	−0.0207546
$573$	−1.89984e7	−2.41730
$574$	4.04412e6	0.512323
$575$	0	0
$576$	5.52300e6	0.693615
$577$	−1.02267e7	−1.27878	−0.639390	−	0.768883i	$-0.720813\pi$
$577$	−1.02267e7	−1.27878	−0.639390	+	0.768883i	$0.720813\pi$
$578$	2.85157e6	0.355030
$579$	−6.54631e6	−0.811522
$580$	0	0
$581$	−4.30317e6	−0.528869
$582$	−1.04803e7	−1.28253
$583$	−1.82618e6	−0.222521
$584$	7.92709e6	0.961794
$585$	0	0
$586$	−1.40334e6	−0.168818
$587$	−2.24352e6	−0.268741	−0.134371	−	0.990931i	$-0.542901\pi$
$587$	−2.24352e6	−0.268741	−0.134371	+	0.990931i	$0.542901\pi$
$588$	−4.25846e6	−0.507936
$589$	−2.32728e6	−0.276414
$590$	0	0
$591$	7.31086e6	0.860994
$592$	−10524.5	−0.00123423
$593$	590374.	0.0689430	0.0344715	−	0.999406i	$-0.489025\pi$
$593$	590374.	0.0689430	0.0344715	+	0.999406i	$0.489025\pi$
$594$	−332922.	−0.0387148
$595$	0	0
$596$	2.80683e6	0.323669
$597$	−7.01801e6	−0.805894
$598$	393004.	0.0449411
$599$	2.71185e6	0.308815	0.154407	−	0.988007i	$-0.450653\pi$
$599$	2.71185e6	0.308815	0.154407	+	0.988007i	$0.450653\pi$
$600$	0	0
$601$	−9.99011e6	−1.12820	−0.564098	−	0.825708i	$-0.690776\pi$
$601$	−9.99011e6	−1.12820	−0.564098	+	0.825708i	$0.690776\pi$
$602$	−1.35494e6	−0.152380
$603$	−1.73842e7	−1.94698
$604$	6.54795e6	0.730319
$605$	0	0
$606$	1.38239e7	1.52914
$607$	−4.27804e6	−0.471273	−0.235637	−	0.971841i	$-0.575718\pi$
$607$	−4.27804e6	−0.471273	−0.235637	+	0.971841i	$0.575718\pi$
$608$	−1.81414e6	−0.199027
$609$	1.50022e7	1.63912
$610$	0	0
$611$	−821440.	−0.0890170
$612$	−5.27302e6	−0.569090
$613$	−645645.	−0.0693973	−0.0346986	−	0.999398i	$-0.511047\pi$
$613$	−645645.	−0.0693973	−0.0346986	+	0.999398i	$0.511047\pi$
$614$	2.20981e6	0.236556
$615$	0	0
$616$	−732284.	−0.0777549
$617$	2.65549e6	0.280822	0.140411	−	0.990093i	$-0.455158\pi$
$617$	2.65549e6	0.280822	0.140411	+	0.990093i	$0.455158\pi$
$618$	−3.29440e6	−0.346980
$619$	−1.11716e7	−1.17190	−0.585948	−	0.810349i	$-0.699278\pi$
$619$	−1.11716e7	−1.17190	−0.585948	+	0.810349i	$0.699278\pi$
$620$	0	0
$621$	−1.55991e6	−0.162320
$622$	3.63478e6	0.376706
$623$	−5.81430e6	−0.600174
$624$	392296.	0.0403322
$625$	0	0
$626$	−2.30502e6	−0.235092
$627$	335070.	0.0340382
$628$	1.04871e6	0.106110
$629$	−81290.4	−0.00819243
$630$	0	0
$631$	−1.49550e6	−0.149525	−0.0747624	−	0.997201i	$-0.523820\pi$
$631$	−1.49550e6	−0.149525	−0.0747624	+	0.997201i	$0.523820\pi$
$632$	1.09621e7	1.09169
$633$	−1.07601e7	−1.06735
$634$	4.12993e6	0.408056
$635$	0	0
$636$	2.03454e7	1.99445
$637$	1.41846e6	0.138506
$638$	−1.02267e6	−0.0994676
$639$	1.41651e7	1.37236
$640$	0	0
$641$	9.23788e6	0.888030	0.444015	−	0.896019i	$-0.353554\pi$
$641$	9.23788e6	0.888030	0.444015	+	0.896019i	$0.353554\pi$
$642$	582618.	0.0557887
$643$	−3.69634e6	−0.352569	−0.176285	−	0.984339i	$-0.556408\pi$
$643$	−3.69634e6	−0.352569	−0.176285	+	0.984339i	$0.556408\pi$
$644$	−1.36347e6	−0.129548
$645$	0	0
$646$	−753316.	−0.0710225
$647$	1.94037e7	1.82232	0.911159	−	0.412055i	$-0.135189\pi$
$647$	1.94037e7	1.82232	0.911159	+	0.412055i	$0.135189\pi$
$648$	−4.94103e6	−0.462253
$649$	1.35054e6	0.125862
$650$	0	0
$651$	1.67730e7	1.55116
$652$	−1.39049e7	−1.28100
$653$	−6.16635e6	−0.565907	−0.282953	−	0.959134i	$-0.591314\pi$
$653$	−6.16635e6	−0.565907	−0.282953	+	0.959134i	$0.591314\pi$
$654$	4.07510e6	0.372558
$655$	0	0
$656$	−1.28937e6	−0.116982
$657$	1.51532e7	1.36959
$658$	−1.47184e6	−0.132525
$659$	6.87190e6	0.616401	0.308200	−	0.951321i	$-0.400273\pi$
$659$	6.87190e6	0.616401	0.308200	+	0.951321i	$0.400273\pi$
$660$	0	0
$661$	−3.87812e6	−0.345237	−0.172618	−	0.984989i	$-0.555223\pi$
$661$	−3.87812e6	−0.345237	−0.172618	+	0.984989i	$0.555223\pi$
$662$	7.86030e6	0.697099
$663$	3.03007e6	0.267713
$664$	−8.22396e6	−0.723870
$665$	0	0
$666$	120595.	0.0105352
$667$	−4.79171e6	−0.417038
$668$	−1.38316e7	−1.19931
$669$	−3806.78	−0.000328846 0
$670$	0	0
$671$	−2.34742e6	−0.201272
$672$	1.30747e7	1.11689
$673$	−6.95400e6	−0.591830	−0.295915	−	0.955214i	$-0.595625\pi$
$673$	−6.95400e6	−0.591830	−0.295915	+	0.955214i	$0.595625\pi$
$674$	−7.85727e6	−0.666227
$675$	0	0
$676$	−602689.	−0.0507255
$677$	−1.45095e7	−1.21669	−0.608345	−	0.793672i	$-0.708166\pi$
$677$	−1.45095e7	−1.21669	−0.608345	+	0.793672i	$0.708166\pi$
$678$	−1.15114e7	−0.961731
$679$	−1.21113e7	−1.00813
$680$	0	0
$681$	−2.73978e7	−2.26385
$682$	−1.14338e6	−0.0941301
$683$	1.77058e7	1.45232	0.726162	−	0.687524i	$-0.241302\pi$
$683$	1.77058e7	1.45232	0.726162	+	0.687524i	$0.241302\pi$
$684$	−2.16387e6	−0.176844
$685$	0	0
$686$	7.63092e6	0.619108
$687$	−1.66692e7	−1.34748
$688$	431990.	0.0347939
$689$	−6.77690e6	−0.543855
$690$	0	0
$691$	−8.82763e6	−0.703313	−0.351657	−	0.936129i	$-0.614382\pi$
$691$	−8.82763e6	−0.703313	−0.351657	+	0.936129i	$0.614382\pi$
$692$	5.90927e6	0.469104
$693$	−1.39981e6	−0.110722
$694$	−1.13907e7	−0.897744
$695$	0	0
$696$	2.86712e7	2.24348
$697$	−9.95901e6	−0.776487
$698$	−1.07990e7	−0.838971
$699$	9.61236e6	0.744110
$700$	0	0
$701$	1.38056e7	1.06111	0.530553	−	0.847652i	$-0.321984\pi$
$701$	1.38056e7	1.06111	0.530553	+	0.847652i	$0.321984\pi$
$702$	−1.23547e6	−0.0946211
$703$	−33358.8	−0.00254579
$704$	−750581.	−0.0570776
$705$	0	0
$706$	−1.35007e7	−1.01940
$707$	1.59751e7	1.20198
$708$	−1.50463e7	−1.12810
$709$	2.05869e6	0.153806	0.0769032	−	0.997039i	$-0.475497\pi$
$709$	2.05869e6	0.153806	0.0769032	+	0.997039i	$0.475497\pi$
$710$	0	0
$711$	2.09548e7	1.55457
$712$	−1.11119e7	−0.821466
$713$	−5.35731e6	−0.394660
$714$	5.42923e6	0.398559
$715$	0	0
$716$	−5.51237e6	−0.401842
$717$	−1.07101e7	−0.778031
$718$	−8.10585e6	−0.586797
$719$	−7.96314e6	−0.574463	−0.287232	−	0.957861i	$-0.592735\pi$
$719$	−7.96314e6	−0.574463	−0.287232	+	0.957861i	$0.592735\pi$
$720$	0	0
$721$	−3.80707e6	−0.272743
$722$	7.86507e6	0.561512
$723$	2.96138e7	2.10692
$724$	2.38231e6	0.168909
$725$	0	0
$726$	−1.26187e7	−0.888530
$727$	−3.82952e6	−0.268725	−0.134363	−	0.990932i	$-0.542899\pi$
$727$	−3.82952e6	−0.268725	−0.134363	+	0.990932i	$0.542899\pi$
$728$	−2.71749e6	−0.190037
$729$	−2.23833e7	−1.55993
$730$	0	0
$731$	3.33667e6	0.230951
$732$	2.61525e7	1.80400
$733$	1.45934e7	1.00322	0.501611	−	0.865093i	$-0.332741\pi$
$733$	1.45934e7	1.00322	0.501611	+	0.865093i	$0.332741\pi$
$734$	1.08464e6	0.0743096
$735$	0	0
$736$	−4.17608e6	−0.284167
$737$	2.36254e6	0.160217
$738$	1.47742e7	0.998537
$739$	1.03685e7	0.698403	0.349202	−	0.937048i	$-0.386453\pi$
$739$	1.03685e7	0.698403	0.349202	+	0.937048i	$0.386453\pi$
$740$	0	0
$741$	1.24344e6	0.0831914
$742$	−1.21427e7	−0.809667
$743$	5.40170e6	0.358970	0.179485	−	0.983761i	$-0.442557\pi$
$743$	5.40170e6	0.358970	0.179485	+	0.983761i	$0.442557\pi$
$744$	3.20555e7	2.12310
$745$	0	0
$746$	−1.70891e7	−1.12427
$747$	−1.57206e7	−1.03079
$748$	716609.	0.0468305
$749$	673285.	0.0438525
$750$	0	0
$751$	−5.29620e6	−0.342661	−0.171330	−	0.985214i	$-0.554807\pi$
$751$	−5.29620e6	−0.342661	−0.171330	+	0.985214i	$0.554807\pi$
$752$	469261.	0.0302601
$753$	−1.47985e7	−0.951109
$754$	−3.79509e6	−0.243105
$755$	0	0
$756$	4.28628e6	0.272757
$757$	−8.87136e6	−0.562665	−0.281333	−	0.959610i	$-0.590776\pi$
$757$	−8.87136e6	−0.562665	−0.281333	+	0.959610i	$0.590776\pi$
$758$	−1.07064e7	−0.676815
$759$	771318.	0.0485992
$760$	0	0
$761$	−2.06961e7	−1.29547	−0.647734	−	0.761867i	$-0.724283\pi$
$761$	−2.06961e7	−1.29547	−0.647734	+	0.761867i	$0.724283\pi$
$762$	5.51238e6	0.343916
$763$	4.70927e6	0.292848
$764$	−1.66738e7	−1.03348
$765$	0	0
$766$	−1.49689e7	−0.921763
$767$	5.01183e6	0.307615
$768$	2.34201e7	1.43280
$769$	5.78756e6	0.352923	0.176461	−	0.984308i	$-0.443535\pi$
$769$	5.78756e6	0.352923	0.176461	+	0.984308i	$0.443535\pi$
$770$	0	0
$771$	2.02981e7	1.22976
$772$	−5.74532e6	−0.346953
$773$	1.74975e7	1.05324	0.526619	−	0.850101i	$-0.323459\pi$
$773$	1.74975e7	1.05324	0.526619	+	0.850101i	$0.323459\pi$
$774$	−4.94996e6	−0.296995
$775$	0	0
$776$	−2.31464e7	−1.37984
$777$	240420.	0.0142863
$778$	−5.87175e6	−0.347791
$779$	−4.08684e6	−0.241292
$780$	0	0
$781$	−1.92505e6	−0.112931
$782$	−1.73410e6	−0.101405
$783$	1.50635e7	0.878052
$784$	−810319.	−0.0470832
$785$	0	0
$786$	−2.09660e7	−1.21048
$787$	1.50075e7	0.863719	0.431860	−	0.901941i	$-0.357858\pi$
$787$	1.50075e7	0.863719	0.431860	+	0.901941i	$0.357858\pi$
$788$	6.41633e6	0.368104
$789$	−1.41050e7	−0.806642
$790$	0	0
$791$	−1.33028e7	−0.755965
$792$	−2.67523e6	−0.151547
$793$	−8.71121e6	−0.491921
$794$	−1.71586e7	−0.965899
$795$	0	0
$796$	−6.15930e6	−0.344547
$797$	2.52578e7	1.40848	0.704240	−	0.709962i	$-0.251288\pi$
$797$	2.52578e7	1.40848	0.704240	+	0.709962i	$0.251288\pi$
$798$	2.22797e6	0.123852
$799$	3.62455e6	0.200857
$800$	0	0
$801$	−2.12412e7	−1.16976
$802$	−1.33337e7	−0.732008
$803$	−2.05933e6	−0.112703
$804$	−2.63209e7	−1.43602
$805$	0	0
$806$	−4.24304e6	−0.230059
$807$	−4.98675e7	−2.69547
$808$	3.05307e7	1.64516
$809$	4.92229e6	0.264421	0.132210	−	0.991222i	$-0.457793\pi$
$809$	4.92229e6	0.264421	0.132210	+	0.991222i	$0.457793\pi$
$810$	0	0
$811$	9.00480e6	0.480753	0.240376	−	0.970680i	$-0.422729\pi$
$811$	9.00480e6	0.480753	0.240376	+	0.970680i	$0.422729\pi$
$812$	1.31665e7	0.700780
$813$	−4.62170e7	−2.45231
$814$	−16388.9	−0.000866942 0
$815$	0	0
$816$	−1.73098e6	−0.0910052
$817$	1.36925e6	0.0717677
$818$	7.37984e6	0.385624
$819$	−5.19466e6	−0.270612
$820$	0	0
$821$	2.32720e7	1.20497	0.602486	−	0.798130i	$-0.294177\pi$
$821$	2.32720e7	1.20497	0.602486	+	0.798130i	$0.294177\pi$
$822$	−3.87663e6	−0.200113
$823$	2.00591e7	1.03232	0.516158	−	0.856493i	$-0.327362\pi$
$823$	2.00591e7	1.03232	0.516158	+	0.856493i	$0.327362\pi$
$824$	−7.27584e6	−0.373306
$825$	0	0
$826$	8.98010e6	0.457964
$827$	−9.70394e6	−0.493383	−0.246692	−	0.969094i	$-0.579343\pi$
$827$	−9.70394e6	−0.493383	−0.246692	+	0.969094i	$0.579343\pi$
$828$	−4.98114e6	−0.252495
$829$	−2.31044e7	−1.16764	−0.583819	−	0.811884i	$-0.698442\pi$
$829$	−2.31044e7	−1.16764	−0.583819	+	0.811884i	$0.698442\pi$
$830$	0	0
$831$	−4.09965e7	−2.05942
$832$	−2.78539e6	−0.139501
$833$	−6.25886e6	−0.312523
$834$	1.53552e7	0.764432
$835$	0	0
$836$	294072.	0.0145525
$837$	1.68415e7	0.830934
$838$	−5.04544e6	−0.248193
$839$	2.47020e6	0.121151	0.0605755	−	0.998164i	$-0.480706\pi$
$839$	2.47020e6	0.121151	0.0605755	+	0.998164i	$0.480706\pi$
$840$	0	0
$841$	2.57605e7	1.25593
$842$	9.27270e6	0.450740
$843$	2.31127e7	1.12016
$844$	−9.44351e6	−0.456328
$845$	0	0
$846$	−5.37703e6	−0.258295
$847$	−1.45824e7	−0.698426
$848$	3.87141e6	0.184876
$849$	4.43715e7	2.11269
$850$	0	0
$851$	−76790.6	−0.00363483
$852$	2.14469e7	1.01220
$853$	−2.26066e7	−1.06381	−0.531903	−	0.846805i	$-0.678523\pi$
$853$	−2.26066e7	−1.06381	−0.531903	+	0.846805i	$0.678523\pi$
$854$	−1.56086e7	−0.732350
$855$	0	0
$856$	1.28674e6	0.0600215
$857$	−352440.	−0.0163920	−0.00819602	−	0.999966i	$-0.502609\pi$
$857$	−352440.	−0.0163920	−0.00819602	+	0.999966i	$0.502609\pi$
$858$	610892.	0.0283300
$859$	8.16362e6	0.377485	0.188743	−	0.982027i	$-0.439559\pi$
$859$	8.16362e6	0.377485	0.188743	+	0.982027i	$0.439559\pi$
$860$	0	0
$861$	2.94543e7	1.35407
$862$	8.73124e6	0.400228
$863$	2.20483e7	1.00774	0.503869	−	0.863780i	$-0.331910\pi$
$863$	2.20483e7	1.00774	0.503869	+	0.863780i	$0.331910\pi$
$864$	1.31281e7	0.598299
$865$	0	0
$866$	2.10800e7	0.955160
$867$	2.07687e7	0.938343
$868$	1.47207e7	0.663175
$869$	−2.84778e6	−0.127925
$870$	0	0
$871$	8.76731e6	0.391581
$872$	9.00007e6	0.400825
$873$	−4.42458e7	−1.96488
$874$	−711616.	−0.0315113
$875$	0	0
$876$	2.29429e7	1.01016
$877$	−3.29930e7	−1.44852	−0.724258	−	0.689529i	$-0.757818\pi$
$877$	−3.29930e7	−1.44852	−0.724258	+	0.689529i	$0.757818\pi$
$878$	6.08405e6	0.266352
$879$	−1.02208e7	−0.446185
$880$	0	0
$881$	−2.33438e7	−1.01328	−0.506642	−	0.862156i	$-0.669114\pi$
$881$	−2.33438e7	−1.01328	−0.506642	+	0.862156i	$0.669114\pi$
$882$	9.28504e6	0.401895
$883$	3.71832e6	0.160489	0.0802444	−	0.996775i	$-0.474430\pi$
$883$	3.71832e6	0.160489	0.0802444	+	0.996775i	$0.474430\pi$
$884$	2.65932e6	0.114456
$885$	0	0
$886$	−2.01660e6	−0.0863051
$887$	5.38193e6	0.229683	0.114841	−	0.993384i	$-0.463364\pi$
$887$	5.38193e6	0.229683	0.114841	+	0.993384i	$0.463364\pi$
$888$	459477.	0.0195538
$889$	6.37022e6	0.270334
$890$	0	0
$891$	1.28360e6	0.0541670
$892$	−3340.99	−0.000140593 0
$893$	1.48739e6	0.0624160
$894$	−1.05579e7	−0.441807
$895$	0	0
$896$	1.24104e7	0.516436
$897$	2.86234e6	0.118779
$898$	4.38507e6	0.181462
$899$	5.17334e7	2.13487
$900$	0	0
$901$	2.99026e7	1.22715
$902$	−2.00783e6	−0.0821697
$903$	−9.86836e6	−0.402741
$904$	−2.54235e7	−1.03470
$905$	0	0
$906$	−2.46300e7	−0.996883
$907$	2.68436e7	1.08348	0.541742	−	0.840545i	$-0.317765\pi$
$907$	2.68436e7	1.08348	0.541742	+	0.840545i	$0.317765\pi$
$908$	−2.40455e7	−0.967873
$909$	5.83614e7	2.34270
$910$	0	0
$911$	1.10032e7	0.439260	0.219630	−	0.975583i	$-0.429515\pi$
$911$	1.10032e7	0.439260	0.219630	+	0.975583i	$0.429515\pi$
$912$	−710334.	−0.0282797
$913$	2.13645e6	0.0848235
$914$	3.48990e6	0.138181
$915$	0	0
$916$	−1.46296e7	−0.576095
$917$	−2.42287e7	−0.951496
$918$	5.45140e6	0.213502
$919$	4.98823e7	1.94831	0.974153	−	0.225887i	$-0.0725282\pi$
$919$	4.98823e7	1.94831	0.974153	+	0.225887i	$0.0725282\pi$
$920$	0	0
$921$	1.60945e7	0.625215
$922$	291143.	0.0112792
$923$	−7.14381e6	−0.276011
$924$	−2.11941e6	−0.0816647
$925$	0	0
$926$	1.68462e7	0.645615
$927$	−1.39082e7	−0.531585
$928$	4.03268e7	1.53718
$929$	−4.09032e7	−1.55495	−0.777477	−	0.628911i	$-0.783501\pi$
$929$	−4.09032e7	−1.55495	−0.777477	+	0.628911i	$0.783501\pi$
$930$	0	0
$931$	−2.56842e6	−0.0971163
$932$	8.43622e6	0.318132
$933$	2.64730e7	0.995632
$934$	−1.42919e7	−0.536073
$935$	0	0
$936$	−9.92771e6	−0.370390
$937$	4.44775e7	1.65498	0.827488	−	0.561483i	$-0.189769\pi$
$937$	4.44775e7	1.65498	0.827488	+	0.561483i	$0.189769\pi$
$938$	1.57091e7	0.582968
$939$	−1.67880e7	−0.621348
$940$	0	0
$941$	8.18280e6	0.301251	0.150625	−	0.988591i	$-0.451871\pi$
$941$	8.18280e6	0.301251	0.150625	+	0.988591i	$0.451871\pi$
$942$	−3.94472e6	−0.144840
$943$	−9.40773e6	−0.344513
$944$	−2.86309e6	−0.104569
$945$	0	0
$946$	672705.	0.0244397
$947$	3.72825e7	1.35092	0.675461	−	0.737396i	$-0.263945\pi$
$947$	3.72825e7	1.35092	0.675461	+	0.737396i	$0.263945\pi$
$948$	3.17270e7	1.14659
$949$	−7.64213e6	−0.275454
$950$	0	0
$951$	3.00793e7	1.07849
$952$	1.19907e7	0.428798
$953$	4.11870e7	1.46902	0.734511	−	0.678597i	$-0.237412\pi$
$953$	4.11870e7	1.46902	0.734511	+	0.678597i	$0.237412\pi$
$954$	−4.43606e7	−1.57807
$955$	0	0
$956$	−9.39967e6	−0.332635
$957$	−7.44832e6	−0.262893
$958$	−9.54050e6	−0.335859
$959$	−4.47992e6	−0.157298
$960$	0	0
$961$	2.92107e7	1.02031
$962$	−60819.0	−0.00211886
$963$	2.45969e6	0.0854702
$964$	2.59903e7	0.900781
$965$	0	0
$966$	5.12869e6	0.176833
$967$	2.42760e7	0.834856	0.417428	−	0.908710i	$-0.362932\pi$
$967$	2.42760e7	0.834856	0.417428	+	0.908710i	$0.362932\pi$
$968$	−2.78690e7	−0.955944
$969$	−5.48658e6	−0.187712
$970$	0	0
$971$	−2.85115e6	−0.0970446	−0.0485223	−	0.998822i	$-0.515451\pi$
$971$	−2.85115e6	−0.0970446	−0.0485223	+	0.998822i	$0.515451\pi$
$972$	−2.56557e7	−0.870999
$973$	1.77447e7	0.600879
$974$	−1.11366e7	−0.376146
$975$	0	0
$976$	4.97642e6	0.167222
$977$	3.23237e7	1.08339	0.541695	−	0.840575i	$-0.317783\pi$
$977$	3.23237e7	1.08339	0.541695	+	0.840575i	$0.317783\pi$
$978$	5.23031e7	1.74856
$979$	2.88670e6	0.0962597
$980$	0	0
$981$	1.72042e7	0.570772
$982$	−1.73264e7	−0.573362
$983$	1.56769e7	0.517461	0.258731	−	0.965950i	$-0.416696\pi$
$983$	1.56769e7	0.517461	0.258731	+	0.965950i	$0.416696\pi$
$984$	5.62912e7	1.85333
$985$	0	0
$986$	1.67455e7	0.548538
$987$	−1.07198e7	−0.350262
$988$	1.09129e6	0.0355672
$989$	3.15197e6	0.102469
$990$	0	0
$991$	5.10915e7	1.65259	0.826294	−	0.563239i	$-0.190445\pi$
$991$	5.10915e7	1.65259	0.826294	+	0.563239i	$0.190445\pi$
$992$	4.50868e7	1.45469
$993$	5.72485e7	1.84243
$994$	−1.28002e7	−0.410912
$995$	0	0
$996$	−2.38021e7	−0.760269
$997$	−5.33407e7	−1.69950	−0.849749	−	0.527188i	$-0.823247\pi$
$997$	−5.33407e7	−1.69950	−0.849749	+	0.527188i	$0.823247\pi$
$998$	1.16129e7	0.369073
$999$	241403.	0.00765294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.m.1.6		15
5.2	odd	4		65.6.b.a.14.12	✓	30
5.3	odd	4		65.6.b.a.14.19	yes	30
5.4	even	2		325.6.a.l.1.10		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.b.a.14.12	✓	30	5.2	odd	4
65.6.b.a.14.19	yes	30	5.3	odd	4
325.6.a.l.1.10		15	5.4	even	2
325.6.a.m.1.6		15	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q-3.30124 q^{2} -24.0437 q^{3} -21.1018 q^{4} +79.3742 q^{6} +91.7264 q^{7} +175.302 q^{8} +335.101 q^{9} -45.5406 q^{11} +507.366 q^{12} -169.000 q^{13} -302.811 q^{14} +96.5440 q^{16} +745.700 q^{17} -1106.25 q^{18} +306.010 q^{19} -2205.45 q^{21} +150.340 q^{22} +704.422 q^{23} -4214.91 q^{24} +557.910 q^{26} -2214.45 q^{27} -1935.59 q^{28} -6802.33 q^{29} -7605.25 q^{31} -5928.38 q^{32} +1094.97 q^{33} -2461.74 q^{34} -7071.24 q^{36} -109.012 q^{37} -1010.21 q^{38} +4063.39 q^{39} -13355.2 q^{41} +7280.71 q^{42} +4474.54 q^{43} +960.988 q^{44} -2325.47 q^{46} +4860.59 q^{47} -2321.28 q^{48} -8393.26 q^{49} -17929.4 q^{51} +3566.21 q^{52} +40100.0 q^{53} +7310.45 q^{54} +16079.8 q^{56} -7357.62 q^{57} +22456.1 q^{58} -29655.8 q^{59} +51545.6 q^{61} +25106.8 q^{62} +30737.6 q^{63} +16481.6 q^{64} -3614.75 q^{66} -51877.6 q^{67} -15735.6 q^{68} -16936.9 q^{69} +42271.1 q^{71} +58743.9 q^{72} +45219.7 q^{73} +359.876 q^{74} -6457.36 q^{76} -4177.27 q^{77} -13414.2 q^{78} +62532.7 q^{79} -28185.8 q^{81} +44088.9 q^{82} -46913.1 q^{83} +46538.9 q^{84} -14771.5 q^{86} +163553. q^{87} -7983.35 q^{88} -63387.4 q^{89} -15501.8 q^{91} -14864.6 q^{92} +182859. q^{93} -16046.0 q^{94} +142540. q^{96} -132037. q^{97} +27708.2 q^{98} -15260.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 12 q^{2} + 36 q^{3} + 250 q^{4} - 202 q^{6} + 306 q^{7} + 576 q^{8} + 963 q^{9} - 326 q^{11} + 2702 q^{12} - 2535 q^{13} + 1334 q^{14} + 6046 q^{16} + 2722 q^{17} + 2036 q^{18} - 2624 q^{19} + 3530 q^{21}+ \cdots - 260488 q^{99}+O(q^{100}) \) 15 * q + 12 * q^2 + 36 * q^3 + 250 * q^4 - 202 * q^6 + 306 * q^7 + 576 * q^8 + 963 * q^9 - 326 * q^11 + 2702 * q^12 - 2535 * q^13 + 1334 * q^14 + 6046 * q^16 + 2722 * q^17 + 2036 * q^18 - 2624 * q^19 + 3530 * q^21 + 7004 * q^22 + 9174 * q^23 + 2740 * q^24 - 2028 * q^26 + 12594 * q^27 + 14880 * q^28 - 3042 * q^29 - 11146 * q^31 + 22054 * q^32 + 18582 * q^33 + 23030 * q^34 + 25172 * q^36 + 10192 * q^37 + 23626 * q^38 - 6084 * q^39 - 7928 * q^41 + 13498 * q^42 + 45934 * q^43 + 22758 * q^44 + 22698 * q^46 + 38730 * q^47 + 103564 * q^48 + 70543 * q^49 + 48084 * q^51 - 42250 * q^52 + 85340 * q^53 + 2482 * q^54 + 33578 * q^56 - 11962 * q^57 + 102450 * q^58 - 62312 * q^59 - 29530 * q^61 + 8266 * q^62 + 209670 * q^63 + 15066 * q^64 - 1240 * q^66 + 43018 * q^67 + 281646 * q^68 + 199340 * q^69 - 22868 * q^71 - 54972 * q^72 + 165124 * q^73 - 258566 * q^74 - 139240 * q^76 + 86270 * q^77 + 34138 * q^78 + 129584 * q^79 + 57731 * q^81 + 95426 * q^82 + 375574 * q^83 - 15182 * q^84 - 241780 * q^86 + 373706 * q^87 + 881886 * q^88 - 167638 * q^89 - 51714 * q^91 + 672054 * q^92 + 538530 * q^93 + 104250 * q^94 - 46370 * q^96 + 265450 * q^97 + 1324762 * q^98 - 260488 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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