LMFDB - Embedded newform 325.6.a.k.1.4

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:	$11$
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} - 5 x^{10} - 257 x^{9} + 1165 x^{8} + 22234 x^{7} - 90282 x^{6} - 751180 x^{5} + 2564400 x^{4} + \cdots + 44115200 $ x^11 - 5x^10 - 257x^9 + 1165x^8 + 22234x^7 - 90282x^6 - 751180x^5 + 2564400x^4 + 9571584x^3 - 23191072x^2 - 33154496x + 44115200
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.78432$ 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)

$f(q)$	$=$	\(q-3.78432 q^{2} +8.20788 q^{3} -17.6789 q^{4} -31.0612 q^{6} +88.9969 q^{7} +188.001 q^{8} -175.631 q^{9} -156.575 q^{11} -145.107 q^{12} +169.000 q^{13} -336.793 q^{14} -145.729 q^{16} +447.890 q^{17} +664.642 q^{18} -269.669 q^{19} +730.476 q^{21} +592.531 q^{22} +1371.05 q^{23} +1543.09 q^{24} -639.550 q^{26} -3436.07 q^{27} -1573.37 q^{28} -3698.11 q^{29} +797.974 q^{31} -5464.54 q^{32} -1285.15 q^{33} -1694.96 q^{34} +3104.97 q^{36} -4394.83 q^{37} +1020.51 q^{38} +1387.13 q^{39} +14320.9 q^{41} -2764.35 q^{42} +11337.1 q^{43} +2768.09 q^{44} -5188.50 q^{46} -9974.03 q^{47} -1196.13 q^{48} -8886.54 q^{49} +3676.23 q^{51} -2987.74 q^{52} +11587.1 q^{53} +13003.2 q^{54} +16731.5 q^{56} -2213.41 q^{57} +13994.8 q^{58} +7133.45 q^{59} +16629.8 q^{61} -3019.79 q^{62} -15630.6 q^{63} +25342.9 q^{64} +4863.42 q^{66} +4211.88 q^{67} -7918.22 q^{68} +11253.4 q^{69} +12871.4 q^{71} -33018.7 q^{72} -13097.1 q^{73} +16631.4 q^{74} +4767.46 q^{76} -13934.7 q^{77} -5249.35 q^{78} +74729.2 q^{79} +14475.4 q^{81} -54194.7 q^{82} +8547.83 q^{83} -12914.0 q^{84} -42903.0 q^{86} -30353.6 q^{87} -29436.3 q^{88} +59586.0 q^{89} +15040.5 q^{91} -24238.8 q^{92} +6549.68 q^{93} +37744.9 q^{94} -44852.3 q^{96} -17376.2 q^{97} +33629.5 q^{98} +27499.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 5 q^{2} + 11 q^{3} + 187 q^{4} + 351 q^{6} + 208 q^{7} + 165 q^{8} + 1372 q^{9} + 1276 q^{11} + 1533 q^{12} + 1859 q^{13} + 578 q^{14} + 5707 q^{16} + 2218 q^{17} - 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) $ 11 * q + 5 * q^2 + 11 * q^3 + 187 * q^4 + 351 * q^6 + 208 * q^7 + 165 * q^8 + 1372 * q^9 + 1276 * q^11 + 1533 * q^12 + 1859 * q^13 + 578 * q^14 + 5707 * q^16 + 2218 * q^17 - 6776 * q^18 + 3520 * q^19 + 1706 * q^21 - 6785 * q^22 - 1215 * q^23 + 7723 * q^24 + 845 * q^26 + 161 * q^27 + 12746 * q^28 + 18425 * q^29 + 9068 * q^31 + 21165 * q^32 + 19748 * q^33 - 14455 * q^34 + 37684 * q^36 - 2280 * q^37 - 23783 * q^38 + 1859 * q^39 - 10010 * q^41 - 44184 * q^42 + 6763 * q^43 + 60651 * q^44 - 9858 * q^46 + 4110 * q^47 + 63289 * q^48 + 73313 * q^49 + 36606 * q^51 + 31603 * q^52 + 15633 * q^53 + 48013 * q^54 - 1262 * q^56 - 205974 * q^57 + 77652 * q^58 + 131642 * q^59 - 43365 * q^61 - 115940 * q^62 + 218466 * q^63 + 180071 * q^64 - 58607 * q^66 - 100904 * q^67 + 371277 * q^68 + 59345 * q^69 + 68420 * q^71 - 210708 * q^72 + 124436 * q^73 + 371396 * q^74 + 22517 * q^76 - 150978 * q^77 + 59319 * q^78 + 52239 * q^79 + 107507 * q^81 - 351137 * q^82 + 211538 * q^83 + 852716 * q^84 + 202246 * q^86 - 28273 * q^87 - 28107 * q^88 + 327712 * q^89 + 35152 * q^91 - 413252 * q^92 + 243338 * q^93 + 649974 * q^94 + 899663 * q^96 - 110796 * q^97 + 27143 * q^98 + 426698 * 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 $

$2$	−3.78432	−0.668979	−0.334490	−	0.942399i	$-0.608564\pi$
$2$	−3.78432	−0.668979	−0.334490	+	0.942399i	$0.608564\pi$
$3$	8.20788	0.526536	0.263268	−	0.964723i	$-0.415200\pi$
$3$	8.20788	0.526536	0.263268	+	0.964723i	$0.415200\pi$
$4$	−17.6789	−0.552467
$5$	0	0
$5$	0	0
$6$	−31.0612	−0.352241
$7$	88.9969	0.686483	0.343242	−	0.939247i	$-0.388475\pi$
$7$	88.9969	0.686483	0.343242	+	0.939247i	$0.388475\pi$
$8$	188.001	1.03857
$9$	−175.631	−0.722760
$10$	0	0
$11$	−156.575	−0.390159	−0.195080	−	0.980787i	$-0.562497\pi$
$11$	−156.575	−0.390159	−0.195080	+	0.980787i	$0.562497\pi$
$12$	−145.107	−0.290893
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−336.793	−0.459243
$15$	0	0
$16$	−145.729	−0.142314
$17$	447.890	0.375880	0.187940	−	0.982181i	$-0.439819\pi$
$17$	447.890	0.375880	0.187940	+	0.982181i	$0.439819\pi$
$18$	664.642	0.483512
$19$	−269.669	−0.171375	−0.0856875	−	0.996322i	$-0.527309\pi$
$19$	−269.669	−0.171375	−0.0856875	+	0.996322i	$0.527309\pi$
$20$	0	0
$21$	730.476	0.361458
$22$	592.531	0.261008
$23$	1371.05	0.540424	0.270212	−	0.962801i	$-0.412906\pi$
$23$	1371.05	0.540424	0.270212	+	0.962801i	$0.412906\pi$
$24$	1543.09	0.546843
$25$	0	0
$26$	−639.550	−0.185541
$27$	−3436.07	−0.907095
$28$	−1573.37	−0.379259
$29$	−3698.11	−0.816554	−0.408277	−	0.912858i	$-0.633870\pi$
$29$	−3698.11	−0.816554	−0.408277	+	0.912858i	$0.633870\pi$
$30$	0	0
$31$	797.974	0.149137	0.0745684	−	0.997216i	$-0.476242\pi$
$31$	797.974	0.149137	0.0745684	+	0.997216i	$0.476242\pi$
$32$	−5464.54	−0.943363
$33$	−1285.15	−0.205433
$34$	−1694.96	−0.251456
$35$	0	0
$36$	3104.97	0.399301
$37$	−4394.83	−0.527761	−0.263881	−	0.964555i	$-0.585003\pi$
$37$	−4394.83	−0.527761	−0.263881	+	0.964555i	$0.585003\pi$
$38$	1020.51	0.114646
$39$	1387.13	0.146035
$40$	0	0
$41$	14320.9	1.33048	0.665242	−	0.746628i	$-0.268328\pi$
$41$	14320.9	1.33048	0.665242	+	0.746628i	$0.268328\pi$
$42$	−2764.35	−0.241808
$43$	11337.1	0.935037	0.467519	−	0.883983i	$-0.345148\pi$
$43$	11337.1	0.935037	0.467519	+	0.883983i	$0.345148\pi$
$44$	2768.09	0.215550
$45$	0	0
$46$	−5188.50	−0.361532
$47$	−9974.03	−0.658607	−0.329303	−	0.944224i	$-0.606814\pi$
$47$	−9974.03	−0.658607	−0.329303	+	0.944224i	$0.606814\pi$
$48$	−1196.13	−0.0749331
$49$	−8886.54	−0.528741
$50$	0	0
$51$	3676.23	0.197914
$52$	−2987.74	−0.153227
$53$	11587.1	0.566611	0.283305	−	0.959030i	$-0.408569\pi$
$53$	11587.1	0.566611	0.283305	+	0.959030i	$0.408569\pi$
$54$	13003.2	0.606827
$55$	0	0
$56$	16731.5	0.712960
$57$	−2213.41	−0.0902350
$58$	13994.8	0.546258
$59$	7133.45	0.266790	0.133395	−	0.991063i	$-0.457412\pi$
$59$	7133.45	0.266790	0.133395	+	0.991063i	$0.457412\pi$
$60$	0	0
$61$	16629.8	0.572220	0.286110	−	0.958197i	$-0.407638\pi$
$61$	16629.8	0.572220	0.286110	+	0.958197i	$0.407638\pi$
$62$	−3019.79	−0.0997694
$63$	−15630.6	−0.496163
$64$	25342.9	0.773404
$65$	0	0
$66$	4863.42	0.137430
$67$	4211.88	0.114628	0.0573138	−	0.998356i	$-0.481746\pi$
$67$	4211.88	0.114628	0.0573138	+	0.998356i	$0.481746\pi$
$68$	−7918.22	−0.207661
$69$	11253.4	0.284552
$70$	0	0
$71$	12871.4	0.303026	0.151513	−	0.988455i	$-0.451585\pi$
$71$	12871.4	0.303026	0.151513	+	0.988455i	$0.451585\pi$
$72$	−33018.7	−0.750636
$73$	−13097.1	−0.287653	−0.143826	−	0.989603i	$-0.545941\pi$
$73$	−13097.1	−0.287653	−0.143826	+	0.989603i	$0.545941\pi$
$74$	16631.4	0.353061
$75$	0	0
$76$	4767.46	0.0946790
$77$	−13934.7	−0.267838
$78$	−5249.35	−0.0976942
$79$	74729.2	1.34717	0.673586	−	0.739109i	$-0.264753\pi$
$79$	74729.2	1.34717	0.673586	+	0.739109i	$0.264753\pi$
$80$	0	0
$81$	14475.4	0.245143
$82$	−54194.7	−0.890066
$83$	8547.83	0.136195	0.0680974	−	0.997679i	$-0.478307\pi$
$83$	8547.83	0.136195	0.0680974	+	0.997679i	$0.478307\pi$
$84$	−12914.0	−0.199694
$85$	0	0
$86$	−42903.0	−0.625521
$87$	−30353.6	−0.429945
$88$	−29436.3	−0.405207
$89$	59586.0	0.797387	0.398694	−	0.917084i	$-0.369464\pi$
$89$	59586.0	0.797387	0.398694	+	0.917084i	$0.369464\pi$
$90$	0	0
$91$	15040.5	0.190396
$92$	−24238.8	−0.298566
$93$	6549.68	0.0785258
$94$	37744.9	0.440594
$95$	0	0
$96$	−44852.3	−0.496714
$97$	−17376.2	−0.187510	−0.0937552	−	0.995595i	$-0.529887\pi$
$97$	−17376.2	−0.187510	−0.0937552	+	0.995595i	$0.529887\pi$
$98$	33629.5	0.353716
$99$	27499.5	0.281992
$100$	0	0
$101$	68669.7	0.669825	0.334913	−	0.942249i	$-0.391293\pi$
$101$	68669.7	0.669825	0.334913	+	0.942249i	$0.391293\pi$
$102$	−13912.0	−0.132400
$103$	191203.	1.77584	0.887918	−	0.460002i	$-0.152151\pi$
$103$	191203.	1.77584	0.887918	+	0.460002i	$0.152151\pi$
$104$	31772.2	0.288047
$105$	0	0
$106$	−43849.2	−0.379051
$107$	−46117.6	−0.389410	−0.194705	−	0.980862i	$-0.562375\pi$
$107$	−46117.6	−0.389410	−0.194705	+	0.980862i	$0.562375\pi$
$108$	60746.1	0.501140
$109$	187271.	1.50975	0.754873	−	0.655871i	$-0.227698\pi$
$109$	187271.	1.50975	0.754873	+	0.655871i	$0.227698\pi$
$110$	0	0
$111$	−36072.2	−0.277885
$112$	−12969.4	−0.0976959
$113$	189262.	1.39434	0.697169	−	0.716907i	$-0.254443\pi$
$113$	189262.	1.39434	0.697169	+	0.716907i	$0.254443\pi$
$114$	8376.25	0.0603653
$115$	0	0
$116$	65378.7	0.451119
$117$	−29681.6	−0.200458
$118$	−26995.2	−0.178477
$119$	39860.9	0.258035
$120$	0	0
$121$	−136535.	−0.847776
$122$	−62932.5	−0.382803
$123$	117544.	0.700547
$124$	−14107.3	−0.0823931
$125$	0	0
$126$	59151.2	0.331923
$127$	183029.	1.00695	0.503477	−	0.864009i	$-0.332054\pi$
$127$	183029.	1.00695	0.503477	+	0.864009i	$0.332054\pi$
$128$	78959.8	0.425972
$129$	93053.1	0.492331
$130$	0	0
$131$	221035.	1.12534	0.562668	−	0.826683i	$-0.309775\pi$
$131$	221035.	1.12534	0.562668	+	0.826683i	$0.309775\pi$
$132$	22720.1	0.113495
$133$	−23999.7	−0.117646
$134$	−15939.1	−0.0766835
$135$	0	0
$136$	84203.7	0.390377
$137$	353821.	1.61058	0.805290	−	0.592881i	$-0.202009\pi$
$137$	353821.	1.61058	0.805290	+	0.592881i	$0.202009\pi$
$138$	−42586.6	−0.190360
$139$	282625.	1.24072	0.620359	−	0.784318i	$-0.286987\pi$
$139$	282625.	1.24072	0.620359	+	0.784318i	$0.286987\pi$
$140$	0	0
$141$	−81865.7	−0.346780
$142$	−48709.4	−0.202718
$143$	−26461.2	−0.108211
$144$	25594.5	0.102859
$145$	0	0
$146$	49563.7	0.192434
$147$	−72939.7	−0.278401
$148$	77695.9	0.291571
$149$	−77579.6	−0.286274	−0.143137	−	0.989703i	$-0.545719\pi$
$149$	−77579.6	−0.286274	−0.143137	+	0.989703i	$0.545719\pi$
$150$	0	0
$151$	−209693.	−0.748412	−0.374206	−	0.927346i	$-0.622085\pi$
$151$	−209693.	−0.748412	−0.374206	+	0.927346i	$0.622085\pi$
$152$	−50698.0	−0.177985
$153$	−78663.3	−0.271671
$154$	52733.5	0.179178
$155$	0	0
$156$	−24523.0	−0.0806793
$157$	127745.	0.413612	0.206806	−	0.978382i	$-0.433693\pi$
$157$	127745.	0.413612	0.206806	+	0.978382i	$0.433693\pi$
$158$	−282799.	−0.901230
$159$	95105.4	0.298341
$160$	0	0
$161$	122019.	0.370992
$162$	−54779.6	−0.163995
$163$	33444.3	0.0985947	0.0492973	−	0.998784i	$-0.484302\pi$
$163$	33444.3	0.0985947	0.0492973	+	0.998784i	$0.484302\pi$
$164$	−253178.	−0.735048
$165$	0	0
$166$	−32347.7	−0.0911115
$167$	−317479.	−0.880893	−0.440447	−	0.897779i	$-0.645180\pi$
$167$	−317479.	−0.880893	−0.440447	+	0.897779i	$0.645180\pi$
$168$	137330.	0.375399
$169$	28561.0	0.0769231
$170$	0	0
$171$	47362.2	0.123863
$172$	−200427.	−0.516577
$173$	84505.1	0.214668	0.107334	−	0.994223i	$-0.465769\pi$
$173$	84505.1	0.214668	0.107334	+	0.994223i	$0.465769\pi$
$174$	114868.	0.287624
$175$	0	0
$176$	22817.6	0.0555249
$177$	58550.5	0.140474
$178$	−225492.	−0.533435
$179$	−734183.	−1.71266	−0.856331	−	0.516428i	$-0.827261\pi$
$179$	−734183.	−1.71266	−0.856331	+	0.516428i	$0.827261\pi$
$180$	0	0
$181$	363356.	0.824396	0.412198	−	0.911094i	$-0.364761\pi$
$181$	363356.	0.824396	0.412198	+	0.911094i	$0.364761\pi$
$182$	−56918.0	−0.127371
$183$	136496.	0.301294
$184$	257759.	0.561267
$185$	0	0
$186$	−24786.1	−0.0525321
$187$	−70128.6	−0.146653
$188$	176330.	0.363858
$189$	−305800.	−0.622705
$190$	0	0
$191$	54142.3	0.107387	0.0536937	−	0.998557i	$-0.482901\pi$
$191$	54142.3	0.107387	0.0536937	+	0.998557i	$0.482901\pi$
$192$	208011.	0.407225
$193$	−208911.	−0.403709	−0.201855	−	0.979416i	$-0.564697\pi$
$193$	−208911.	−0.403709	−0.201855	+	0.979416i	$0.564697\pi$
$194$	65757.1	0.125441
$195$	0	0
$196$	157105.	0.292112
$197$	581675.	1.06786	0.533931	−	0.845528i	$-0.320714\pi$
$197$	581675.	1.06786	0.533931	+	0.845528i	$0.320714\pi$
$198$	−104067.	−0.188646
$199$	−624788.	−1.11841	−0.559204	−	0.829030i	$-0.688893\pi$
$199$	−624788.	−1.11841	−0.559204	+	0.829030i	$0.688893\pi$
$200$	0	0
$201$	34570.6	0.0603555
$202$	−259868.	−0.448099
$203$	−329121.	−0.560551
$204$	−64991.8	−0.109341
$205$	0	0
$206$	−723575.	−1.18800
$207$	−240799.	−0.390597
$208$	−24628.2	−0.0394707
$209$	42223.6	0.0668635
$210$	0	0
$211$	628250.	0.971464	0.485732	−	0.874108i	$-0.338553\pi$
$211$	628250.	0.971464	0.485732	+	0.874108i	$0.338553\pi$
$212$	−204847.	−0.313034
$213$	105647.	0.159554
$214$	174524.	0.260507
$215$	0	0
$216$	−645984.	−0.942079
$217$	71017.3	0.102380
$218$	−708693.	−1.00999
$219$	−107500.	−0.151459
$220$	0	0
$221$	75693.4	0.104250
$222$	136509.	0.185899
$223$	−1.09757e6	−1.47798	−0.738992	−	0.673715i	$-0.764698\pi$
$223$	−1.09757e6	−1.47798	−0.738992	+	0.673715i	$0.764698\pi$
$224$	−486328.	−0.647603
$225$	0	0
$226$	−716229.	−0.932783
$227$	161937.	0.208584	0.104292	−	0.994547i	$-0.466742\pi$
$227$	161937.	0.208584	0.104292	+	0.994547i	$0.466742\pi$
$228$	39130.8	0.0498518
$229$	−645210.	−0.813040	−0.406520	−	0.913642i	$-0.633258\pi$
$229$	−645210.	−0.813040	−0.406520	+	0.913642i	$0.633258\pi$
$230$	0	0
$231$	−114375.	−0.141026
$232$	−695248.	−0.848047
$233$	−839907.	−1.01354	−0.506770	−	0.862081i	$-0.669161\pi$
$233$	−839907.	−1.01354	−0.506770	+	0.862081i	$0.669161\pi$
$234$	112325.	0.134102
$235$	0	0
$236$	−126112.	−0.147393
$237$	613369.	0.709334
$238$	−150846.	−0.172620
$239$	−532168.	−0.602635	−0.301317	−	0.953524i	$-0.597426\pi$
$239$	−532168.	−0.602635	−0.301317	+	0.953524i	$0.597426\pi$
$240$	0	0
$241$	−11282.2	−0.0125127	−0.00625633	−	0.999980i	$-0.501991\pi$
$241$	−11282.2	−0.0125127	−0.00625633	+	0.999980i	$0.501991\pi$
$242$	516692.	0.567144
$243$	953778.	1.03617
$244$	−293998.	−0.316133
$245$	0	0
$246$	−444824.	−0.468652
$247$	−45574.1	−0.0475308
$248$	150020.	0.154889
$249$	70159.5	0.0717114
$250$	0	0
$251$	−592652.	−0.593766	−0.296883	−	0.954914i	$-0.595947\pi$
$251$	−592652.	−0.593766	−0.296883	+	0.954914i	$0.595947\pi$
$252$	276332.	0.274114
$253$	−214673.	−0.210851
$254$	−692639.	−0.673632
$255$	0	0
$256$	−1.10978e6	−1.05837
$257$	−161530.	−0.152552	−0.0762762	−	0.997087i	$-0.524303\pi$
$257$	−161530.	−0.152552	−0.0762762	+	0.997087i	$0.524303\pi$
$258$	−352143.	−0.329359
$259$	−391126.	−0.362299
$260$	0	0
$261$	649502.	0.590173
$262$	−836466.	−0.752826
$263$	−377791.	−0.336793	−0.168396	−	0.985719i	$-0.553859\pi$
$263$	−377791.	−0.336793	−0.168396	+	0.985719i	$0.553859\pi$
$264$	−241610.	−0.213356
$265$	0	0
$266$	90822.6	0.0787027
$267$	489075.	0.419853
$268$	−74461.6	−0.0633280
$269$	−689.090	−0.000580624 0	−0.000290312	−	1.00000i	$-0.500092\pi$
$269$	−689.090	−0.000580624 0	−0.000290312		1.00000i	$0.500092\pi$
$270$	0	0
$271$	625466.	0.517345	0.258673	−	0.965965i	$-0.416715\pi$
$271$	625466.	0.517345	0.258673	+	0.965965i	$0.416715\pi$
$272$	−65270.6	−0.0534928
$273$	123450.	0.100250
$274$	−1.33897e6	−1.07744
$275$	0	0
$276$	−198949.	−0.157206
$277$	744000.	0.582604	0.291302	−	0.956631i	$-0.405912\pi$
$277$	744000.	0.582604	0.291302	+	0.956631i	$0.405912\pi$
$278$	−1.06954e6	−0.830015
$279$	−140149.	−0.107790
$280$	0	0
$281$	1.43786e6	1.08631	0.543153	−	0.839634i	$-0.317230\pi$
$281$	1.43786e6	1.08631	0.543153	+	0.839634i	$0.317230\pi$
$282$	309806.	0.231989
$283$	−1.24662e6	−0.925267	−0.462634	−	0.886550i	$-0.653095\pi$
$283$	−1.24662e6	−0.925267	−0.462634	+	0.886550i	$0.653095\pi$
$284$	−227553.	−0.167412
$285$	0	0
$286$	100138.	0.0723907
$287$	1.27451e6	0.913355
$288$	959742.	0.681825
$289$	−1.21925e6	−0.858714
$290$	0	0
$291$	−142622.	−0.0987309
$292$	231543.	0.158919
$293$	1.74658e6	1.18855	0.594276	−	0.804261i	$-0.297439\pi$
$293$	1.74658e6	1.18855	0.594276	+	0.804261i	$0.297439\pi$
$294$	276027.	0.186244
$295$	0	0
$296$	−826231.	−0.548116
$297$	538004.	0.353911
$298$	293586.	0.191511
$299$	231708.	0.149887
$300$	0	0
$301$	1.00896e6	0.641888
$302$	793543.	0.500672
$303$	563632.	0.352687
$304$	39298.6	0.0243890
$305$	0	0
$306$	297687.	0.181742
$307$	−239330.	−0.144928	−0.0724638	−	0.997371i	$-0.523086\pi$
$307$	−239330.	−0.144928	−0.0724638	+	0.997371i	$0.523086\pi$
$308$	246351.	0.147972
$309$	1.56937e6	0.935041
$310$	0	0
$311$	1.10338e6	0.646883	0.323441	−	0.946248i	$-0.395160\pi$
$311$	1.10338e6	0.646883	0.323441	+	0.946248i	$0.395160\pi$
$312$	260782.	0.151667
$313$	−934831.	−0.539352	−0.269676	−	0.962951i	$-0.586917\pi$
$313$	−934831.	−0.539352	−0.269676	+	0.962951i	$0.586917\pi$
$314$	−483426.	−0.276698
$315$	0	0
$316$	−1.32113e6	−0.744268
$317$	2.65437e6	1.48359	0.741794	−	0.670628i	$-0.233975\pi$
$317$	2.65437e6	1.48359	0.741794	+	0.670628i	$0.233975\pi$
$318$	−359909.	−0.199584
$319$	579033.	0.318586
$320$	0	0
$321$	−378528.	−0.205038
$322$	−461761.	−0.248186
$323$	−120782.	−0.0644164
$324$	−255910.	−0.135433
$325$	0	0
$326$	−126564.	−0.0659578
$327$	1.53710e6	0.794935
$328$	2.69234e6	1.38180
$329$	−887659.	−0.452123
$330$	0	0
$331$	648335.	0.325259	0.162630	−	0.986687i	$-0.448002\pi$
$331$	648335.	0.325259	0.162630	+	0.986687i	$0.448002\pi$
$332$	−151117.	−0.0752431
$333$	771867.	0.381445
$334$	1.20144e6	0.589299
$335$	0	0
$336$	−106452.	−0.0514404
$337$	−761016.	−0.365022	−0.182511	−	0.983204i	$-0.558423\pi$
$337$	−761016.	−0.365022	−0.182511	+	0.983204i	$0.558423\pi$
$338$	−108084.	−0.0514599
$339$	1.55344e6	0.734169
$340$	0	0
$341$	−124943.	−0.0581871
$342$	−179234.	−0.0828617
$343$	−2.28665e6	−1.04945
$344$	2.13138e6	0.971100
$345$	0	0
$346$	−319794.	−0.143609
$347$	−1.06891e6	−0.476561	−0.238281	−	0.971196i	$-0.576584\pi$
$347$	−1.06891e6	−0.476561	−0.238281	+	0.971196i	$0.576584\pi$
$348$	536620.	0.237530
$349$	1.57765e6	0.693342	0.346671	−	0.937987i	$-0.387312\pi$
$349$	1.57765e6	0.693342	0.346671	+	0.937987i	$0.387312\pi$
$350$	0	0
$351$	−580696.	−0.251583
$352$	855613.	0.368062
$353$	109374.	0.0467173	0.0233587	−	0.999727i	$-0.492564\pi$
$353$	109374.	0.0467173	0.0233587	+	0.999727i	$0.492564\pi$
$354$	−221574.	−0.0939745
$355$	0	0
$356$	−1.05342e6	−0.440530
$357$	327173.	0.135865
$358$	2.77838e6	1.14573
$359$	3.75166e6	1.53634	0.768169	−	0.640247i	$-0.221168\pi$
$359$	3.75166e6	1.53634	0.768169	+	0.640247i	$0.221168\pi$
$360$	0	0
$361$	−2.40338e6	−0.970631
$362$	−1.37506e6	−0.551504
$363$	−1.12066e6	−0.446384
$364$	−265900.	−0.105188
$365$	0	0
$366$	−516543.	−0.201560
$367$	−2.12587e6	−0.823895	−0.411947	−	0.911208i	$-0.635151\pi$
$367$	−2.12587e6	−0.823895	−0.411947	+	0.911208i	$0.635151\pi$
$368$	−199802.	−0.0769096
$369$	−2.51518e6	−0.961621
$370$	0	0
$371$	1.03122e6	0.388969
$372$	−115791.	−0.0433829
$373$	−725713.	−0.270080	−0.135040	−	0.990840i	$-0.543116\pi$
$373$	−725713.	−0.270080	−0.135040	+	0.990840i	$0.543116\pi$
$374$	265389.	0.0981078
$375$	0	0
$376$	−1.87513e6	−0.684008
$377$	−624981.	−0.226471
$378$	1.15724e6	0.416577
$379$	1.40410e6	0.502111	0.251056	−	0.967973i	$-0.419222\pi$
$379$	1.40410e6	0.502111	0.251056	+	0.967973i	$0.419222\pi$
$380$	0	0
$381$	1.50228e6	0.530197
$382$	−204892.	−0.0718400
$383$	−2.82085e6	−0.982615	−0.491307	−	0.870986i	$-0.663481\pi$
$383$	−2.82085e6	−0.982615	−0.491307	+	0.870986i	$0.663481\pi$
$384$	648093.	0.224290
$385$	0	0
$386$	790587.	0.270073
$387$	−1.99113e6	−0.675808
$388$	307193.	0.103593
$389$	−1.71293e6	−0.573940	−0.286970	−	0.957940i	$-0.592648\pi$
$389$	−1.71293e6	−0.573940	−0.286970	+	0.957940i	$0.592648\pi$
$390$	0	0
$391$	614081.	0.203135
$392$	−1.67068e6	−0.549133
$393$	1.81423e6	0.592530
$394$	−2.20124e6	−0.714377
$395$	0	0
$396$	−486161.	−0.155791
$397$	−3.80080e6	−1.21032	−0.605158	−	0.796106i	$-0.706890\pi$
$397$	−3.80080e6	−1.21032	−0.605158	+	0.796106i	$0.706890\pi$
$398$	2.36440e6	0.748191
$399$	−196987.	−0.0619448
$400$	0	0
$401$	1.09440e6	0.339871	0.169936	−	0.985455i	$-0.445644\pi$
$401$	1.09440e6	0.339871	0.169936	+	0.985455i	$0.445644\pi$
$402$	−130826.	−0.0403766
$403$	134858.	0.0413631
$404$	−1.21401e6	−0.370056
$405$	0	0
$406$	1.24550e6	0.374997
$407$	688122.	0.205911
$408$	691134.	0.205547
$409$	6.63162e6	1.96025	0.980125	−	0.198380i	$-0.0635679\pi$
$409$	6.63162e6	1.96025	0.980125	+	0.198380i	$0.0635679\pi$
$410$	0	0
$411$	2.90412e6	0.848028
$412$	−3.38027e6	−0.981090
$413$	634855.	0.183147
$414$	911260.	0.261301
$415$	0	0
$416$	−923508.	−0.261642
$417$	2.31975e6	0.653283
$418$	−159787.	−0.0447303
$419$	−1.50034e6	−0.417499	−0.208749	−	0.977969i	$-0.566939\pi$
$419$	−1.50034e6	−0.417499	−0.208749	+	0.977969i	$0.566939\pi$
$420$	0	0
$421$	−1.91392e6	−0.526282	−0.263141	−	0.964757i	$-0.584758\pi$
$421$	−1.91392e6	−0.526282	−0.263141	+	0.964757i	$0.584758\pi$
$422$	−2.37750e6	−0.649889
$423$	1.75175e6	0.476015
$424$	2.17838e6	0.588464
$425$	0	0
$426$	−399801.	−0.106738
$427$	1.48000e6	0.392820
$428$	815311.	0.215136
$429$	−217191.	−0.0569768
$430$	0	0
$431$	6.04255e6	1.56685	0.783424	−	0.621488i	$-0.213471\pi$
$431$	6.04255e6	1.56685	0.783424	+	0.621488i	$0.213471\pi$
$432$	500735.	0.129092
$433$	−1.81284e6	−0.464666	−0.232333	−	0.972636i	$-0.574636\pi$
$433$	−1.81284e6	−0.464666	−0.232333	+	0.972636i	$0.574636\pi$
$434$	−268752.	−0.0684900
$435$	0	0
$436$	−3.31075e6	−0.834085
$437$	−369731.	−0.0926151
$438$	406813.	0.101323
$439$	3.45414e6	0.855418	0.427709	−	0.903916i	$-0.359321\pi$
$439$	3.45414e6	0.855418	0.427709	+	0.903916i	$0.359321\pi$
$440$	0	0
$441$	1.56075e6	0.382153
$442$	−286448.	−0.0697413
$443$	−1.64299e6	−0.397764	−0.198882	−	0.980023i	$-0.563731\pi$
$443$	−1.64299e6	−0.397764	−0.198882	+	0.980023i	$0.563731\pi$
$444$	637718.	0.153522
$445$	0	0
$446$	4.15355e6	0.988740
$447$	−636764.	−0.150733
$448$	2.25544e6	0.530929
$449$	3.90173e6	0.913359	0.456680	−	0.889631i	$-0.349039\pi$
$449$	3.90173e6	0.913359	0.456680	+	0.889631i	$0.349039\pi$
$450$	0	0
$451$	−2.24230e6	−0.519101
$452$	−3.34596e6	−0.770325
$453$	−1.72113e6	−0.394066
$454$	−612821.	−0.139539
$455$	0	0
$456$	−416123.	−0.0937152
$457$	−6.09815e6	−1.36586	−0.682932	−	0.730482i	$-0.739296\pi$
$457$	−6.09815e6	−1.36586	−0.682932	+	0.730482i	$0.739296\pi$
$458$	2.44168e6	0.543907
$459$	−1.53898e6	−0.340959
$460$	0	0
$461$	8.52589e6	1.86848	0.934238	−	0.356651i	$-0.116081\pi$
$461$	8.52589e6	1.86848	0.934238	+	0.356651i	$0.116081\pi$
$462$	432830.	0.0943436
$463$	−2.70655e6	−0.586765	−0.293382	−	0.955995i	$-0.594781\pi$
$463$	−2.70655e6	−0.586765	−0.293382	+	0.955995i	$0.594781\pi$
$464$	538922.	0.116207
$465$	0	0
$466$	3.17847e6	0.678038
$467$	−350172.	−0.0743001	−0.0371501	−	0.999310i	$-0.511828\pi$
$467$	−350172.	−0.0743001	−0.0371501	+	0.999310i	$0.511828\pi$
$468$	524739.	0.110746
$469$	374845.	0.0786900
$470$	0	0
$471$	1.04851e6	0.217782
$472$	1.34110e6	0.277080
$473$	−1.77510e6	−0.364813
$474$	−2.32118e6	−0.474529
$475$	0	0
$476$	−704698.	−0.142556
$477$	−2.03505e6	−0.409524
$478$	2.01389e6	0.403150
$479$	8.49125e6	1.69096	0.845479	−	0.534009i	$-0.179315\pi$
$479$	8.49125e6	1.69096	0.845479	+	0.534009i	$0.179315\pi$
$480$	0	0
$481$	−742726.	−0.146375
$482$	42695.3	0.00837071
$483$	1.00152e6	0.195341
$484$	2.41380e6	0.468368
$485$	0	0
$486$	−3.60940e6	−0.693177
$487$	3034.65	0.000579811 0	0.000289905	−	1.00000i	$-0.499908\pi$
$487$	3034.65	0.000579811 0	0.000289905		1.00000i	$0.499908\pi$
$488$	3.12642e6	0.594289
$489$	274507.	0.0519136
$490$	0	0
$491$	−3.43114e6	−0.642295	−0.321147	−	0.947029i	$-0.604068\pi$
$491$	−3.43114e6	−0.642295	−0.321147	+	0.947029i	$0.604068\pi$
$492$	−2.07805e6	−0.387029
$493$	−1.65635e6	−0.306926
$494$	172467.	0.0317971
$495$	0	0
$496$	−116288.	−0.0212242
$497$	1.14551e6	0.208022
$498$	−265506.	−0.0479735
$499$	−1.11566e6	−0.200577	−0.100289	−	0.994958i	$-0.531977\pi$
$499$	−1.11566e6	−0.200577	−0.100289	+	0.994958i	$0.531977\pi$
$500$	0	0
$501$	−2.60583e6	−0.463822
$502$	2.24278e6	0.397217
$503$	4.99165e6	0.879679	0.439840	−	0.898076i	$-0.355035\pi$
$503$	4.99165e6	0.879679	0.439840	+	0.898076i	$0.355035\pi$
$504$	−2.93857e6	−0.515299
$505$	0	0
$506$	812391.	0.141055
$507$	234425.	0.0405027
$508$	−3.23575e6	−0.556309
$509$	−398811.	−0.0682296	−0.0341148	−	0.999418i	$-0.510861\pi$
$509$	−398811.	−0.0682296	−0.0341148	+	0.999418i	$0.510861\pi$
$510$	0	0
$511$	−1.16560e6	−0.197469
$512$	1.67305e6	0.282056
$513$	926602.	0.155453
$514$	611279.	0.102054
$515$	0	0
$516$	−1.64508e6	−0.271996
$517$	1.56169e6	0.256962
$518$	1.48015e6	0.242371
$519$	693607.	0.113030
$520$	0	0
$521$	9.33110e6	1.50605	0.753023	−	0.657994i	$-0.228595\pi$
$521$	9.33110e6	1.50605	0.753023	+	0.657994i	$0.228595\pi$
$522$	−2.45792e6	−0.394813
$523$	−4.93297e6	−0.788596	−0.394298	−	0.918983i	$-0.629012\pi$
$523$	−4.93297e6	−0.788596	−0.394298	+	0.918983i	$0.629012\pi$
$524$	−3.90766e6	−0.621711
$525$	0	0
$526$	1.42968e6	0.225307
$527$	357405.	0.0560575
$528$	187284.	0.0292359
$529$	−4.55656e6	−0.707942
$530$	0	0
$531$	−1.25285e6	−0.192825
$532$	424290.	0.0649955
$533$	2.42023e6	0.369010
$534$	−1.85081e6	−0.280873
$535$	0	0
$536$	791838.	0.119049
$537$	−6.02608e6	−0.901777
$538$	2607.73	0.000388425 0
$539$	1.39141e6	0.206293
$540$	0	0
$541$	2.72340e6	0.400054	0.200027	−	0.979790i	$-0.435897\pi$
$541$	2.72340e6	0.400054	0.200027	+	0.979790i	$0.435897\pi$
$542$	−2.36696e6	−0.346093
$543$	2.98238e6	0.434074
$544$	−2.44751e6	−0.354591
$545$	0	0
$546$	−467176.	−0.0670654
$547$	1.28669e7	1.83867	0.919337	−	0.393472i	$-0.128726\pi$
$547$	1.28669e7	1.83867	0.919337	+	0.393472i	$0.128726\pi$
$548$	−6.25518e6	−0.889792
$549$	−2.92071e6	−0.413578
$550$	0	0
$551$	997266.	0.139937
$552$	2.11566e6	0.295527
$553$	6.65067e6	0.924811
$554$	−2.81553e6	−0.389750
$555$	0	0
$556$	−4.99651e6	−0.685456
$557$	1.03791e7	1.41750	0.708751	−	0.705459i	$-0.249259\pi$
$557$	1.03791e7	1.41750	0.708751	+	0.705459i	$0.249259\pi$
$558$	530368.	0.0721094
$559$	1.91596e6	0.259333
$560$	0	0
$561$	−575607.	−0.0772180
$562$	−5.44133e6	−0.726716
$563$	−3.37237e6	−0.448399	−0.224199	−	0.974543i	$-0.571977\pi$
$563$	−3.37237e6	−0.448399	−0.224199	+	0.974543i	$0.571977\pi$
$564$	1.44730e6	0.191584
$565$	0	0
$566$	4.71760e6	0.618984
$567$	1.28827e6	0.168286
$568$	2.41983e6	0.314713
$569$	−9.65718e6	−1.25046	−0.625230	−	0.780441i	$-0.714995\pi$
$569$	−9.65718e6	−1.25046	−0.625230	+	0.780441i	$0.714995\pi$
$570$	0	0
$571$	−3.19460e6	−0.410040	−0.205020	−	0.978758i	$-0.565726\pi$
$571$	−3.19460e6	−0.410040	−0.205020	+	0.978758i	$0.565726\pi$
$572$	467807.	0.0597828
$573$	444394.	0.0565433
$574$	−4.82316e6	−0.611016
$575$	0	0
$576$	−4.45099e6	−0.558986
$577$	1.01384e7	1.26774	0.633872	−	0.773438i	$-0.281465\pi$
$577$	1.01384e7	1.26774	0.633872	+	0.773438i	$0.281465\pi$
$578$	4.61403e6	0.574462
$579$	−1.71472e6	−0.212567
$580$	0	0
$581$	760731.	0.0934955
$582$	539726.	0.0660489
$583$	−1.81425e6	−0.221068
$584$	−2.46227e6	−0.298747
$585$	0	0
$586$	−6.60960e6	−0.795117
$587$	777799.	0.0931691	0.0465846	−	0.998914i	$-0.485166\pi$
$587$	777799.	0.0931691	0.0465846	+	0.998914i	$0.485166\pi$
$588$	1.28950e6	0.153807
$589$	−215189.	−0.0255583
$590$	0	0
$591$	4.77432e6	0.562267
$592$	640454.	0.0751076
$593$	−8.28738e6	−0.967789	−0.483894	−	0.875126i	$-0.660778\pi$
$593$	−8.28738e6	−0.967789	−0.483894	+	0.875126i	$0.660778\pi$
$594$	−2.03598e6	−0.236759
$595$	0	0
$596$	1.37153e6	0.158157
$597$	−5.12818e6	−0.588881
$598$	−876856.	−0.100271
$599$	1.39700e7	1.59085	0.795423	−	0.606054i	$-0.207249\pi$
$599$	1.39700e7	1.59085	0.795423	+	0.606054i	$0.207249\pi$
$600$	0	0
$601$	−1.27397e7	−1.43871	−0.719355	−	0.694643i	$-0.755562\pi$
$601$	−1.27397e7	−1.43871	−0.719355	+	0.694643i	$0.755562\pi$
$602$	−3.81824e6	−0.429409
$603$	−739736.	−0.0828483
$604$	3.70714e6	0.413473
$605$	0	0
$606$	−2.13296e6	−0.235940
$607$	−6.25976e6	−0.689583	−0.344791	−	0.938679i	$-0.612050\pi$
$607$	−6.25976e6	−0.689583	−0.344791	+	0.938679i	$0.612050\pi$
$608$	1.47362e6	0.161669
$609$	−2.70138e6	−0.295150
$610$	0	0
$611$	−1.68561e6	−0.182665
$612$	1.39068e6	0.150089
$613$	−1.17405e7	−1.26193	−0.630964	−	0.775812i	$-0.717340\pi$
$613$	−1.17405e7	−1.26193	−0.630964	+	0.775812i	$0.717340\pi$
$614$	905700.	0.0969535
$615$	0	0
$616$	−2.61974e6	−0.278168
$617$	1.57982e7	1.67069	0.835343	−	0.549729i	$-0.185269\pi$
$617$	1.57982e7	1.67069	0.835343	+	0.549729i	$0.185269\pi$
$618$	−5.93901e6	−0.625523
$619$	3.16596e6	0.332108	0.166054	−	0.986117i	$-0.446897\pi$
$619$	3.16596e6	0.332108	0.166054	+	0.986117i	$0.446897\pi$
$620$	0	0
$621$	−4.71103e6	−0.490216
$622$	−4.17556e6	−0.432751
$623$	5.30297e6	0.547393
$624$	−202145.	−0.0207827
$625$	0	0
$626$	3.53770e6	0.360815
$627$	346566.	0.0352060
$628$	−2.25839e6	−0.228507
$629$	−1.96840e6	−0.198375
$630$	0	0
$631$	−1.95795e7	−1.95762	−0.978811	−	0.204764i	$-0.934357\pi$
$631$	−1.95795e7	−1.95762	−0.978811	+	0.204764i	$0.934357\pi$
$632$	1.40492e7	1.39913
$633$	5.15660e6	0.511510
$634$	−1.00450e7	−0.992489
$635$	0	0
$636$	−1.68136e6	−0.164823
$637$	−1.50183e6	−0.146646
$638$	−2.19125e6	−0.213127
$639$	−2.26061e6	−0.219015
$640$	0	0
$641$	−7.23060e6	−0.695072	−0.347536	−	0.937667i	$-0.612981\pi$
$641$	−7.23060e6	−0.695072	−0.347536	+	0.937667i	$0.612981\pi$
$642$	1.43247e6	0.137166
$643$	−1.17438e7	−1.12016	−0.560081	−	0.828438i	$-0.689230\pi$
$643$	−1.17438e7	−1.12016	−0.560081	+	0.828438i	$0.689230\pi$
$644$	−2.15718e6	−0.204961
$645$	0	0
$646$	457078.	0.0430932
$647$	9.24080e6	0.867858	0.433929	−	0.900947i	$-0.357127\pi$
$647$	9.24080e6	0.867858	0.433929	+	0.900947i	$0.357127\pi$
$648$	2.72139e6	0.254597
$649$	−1.11692e6	−0.104091
$650$	0	0
$651$	582901.	0.0539067
$652$	−591260.	−0.0544703
$653$	−1.11349e7	−1.02188	−0.510942	−	0.859615i	$-0.670703\pi$
$653$	−1.11349e7	−1.02188	−0.510942	+	0.859615i	$0.670703\pi$
$654$	−5.81686e6	−0.531795
$655$	0	0
$656$	−2.08697e6	−0.189346
$657$	2.30026e6	0.207904
$658$	3.35918e6	0.302461
$659$	8.65776e6	0.776590	0.388295	−	0.921535i	$-0.373064\pi$
$659$	8.65776e6	0.776590	0.388295	+	0.921535i	$0.373064\pi$
$660$	0	0
$661$	−1.23594e7	−1.10025	−0.550127	−	0.835081i	$-0.685421\pi$
$661$	−1.23594e7	−1.10025	−0.550127	+	0.835081i	$0.685421\pi$
$662$	−2.45351e6	−0.217592
$663$	621282.	0.0548915
$664$	1.60700e6	0.141448
$665$	0	0
$666$	−2.92099e6	−0.255179
$667$	−5.07030e6	−0.441285
$668$	5.61268e6	0.486664
$669$	−9.00871e6	−0.778211
$670$	0	0
$671$	−2.60382e6	−0.223257
$672$	−3.99172e6	−0.340986
$673$	−6.11057e6	−0.520048	−0.260024	−	0.965602i	$-0.583731\pi$
$673$	−6.11057e6	−0.520048	−0.260024	+	0.965602i	$0.583731\pi$
$674$	2.87993e6	0.244192
$675$	0	0
$676$	−504928.	−0.0424974
$677$	1.48518e7	1.24540	0.622698	−	0.782462i	$-0.286036\pi$
$677$	1.48518e7	1.24540	0.622698	+	0.782462i	$0.286036\pi$
$678$	−5.87872e6	−0.491144
$679$	−1.54643e6	−0.128723
$680$	0	0
$681$	1.32916e6	0.109827
$682$	472825.	0.0389260
$683$	−6.13145e6	−0.502934	−0.251467	−	0.967866i	$-0.580913\pi$
$683$	−6.13145e6	−0.502934	−0.251467	+	0.967866i	$0.580913\pi$
$684$	−837313.	−0.0684302
$685$	0	0
$686$	8.65340e6	0.702064
$687$	−5.29580e6	−0.428095
$688$	−1.65214e6	−0.133068
$689$	1.95822e6	0.157150
$690$	0	0
$691$	2.33358e7	1.85920	0.929602	−	0.368566i	$-0.120151\pi$
$691$	2.33358e7	1.85920	0.929602	+	0.368566i	$0.120151\pi$
$692$	−1.49396e6	−0.118597
$693$	2.44737e6	0.193582
$694$	4.04511e6	0.318809
$695$	0	0
$696$	−5.70651e6	−0.446527
$697$	6.41417e6	0.500102
$698$	−5.97033e6	−0.463831
$699$	−6.89385e6	−0.533665
$700$	0	0
$701$	2.53633e7	1.94944	0.974721	−	0.223426i	$-0.0717242\pi$
$701$	2.53633e7	1.94944	0.974721	+	0.223426i	$0.0717242\pi$
$702$	2.19754e6	0.168304
$703$	1.18515e6	0.0904450
$704$	−3.96807e6	−0.301751
$705$	0	0
$706$	−413907.	−0.0312529
$707$	6.11139e6	0.459824
$708$	−1.03511e6	−0.0776075
$709$	−1.64964e7	−1.23246	−0.616232	−	0.787565i	$-0.711342\pi$
$709$	−1.64964e7	−1.23246	−0.616232	+	0.787565i	$0.711342\pi$
$710$	0	0
$711$	−1.31248e7	−0.973682
$712$	1.12022e7	0.828141
$713$	1.09407e6	0.0805971
$714$	−1.23813e6	−0.0908907
$715$	0	0
$716$	1.29796e7	0.946189
$717$	−4.36797e6	−0.317309
$718$	−1.41975e7	−1.02778
$719$	−9.96268e6	−0.718710	−0.359355	−	0.933201i	$-0.617003\pi$
$719$	−9.96268e6	−0.718710	−0.359355	+	0.933201i	$0.617003\pi$
$720$	0	0
$721$	1.70165e7	1.21908
$722$	9.09514e6	0.649332
$723$	−92602.6	−0.00658836
$724$	−6.42375e6	−0.455452
$725$	0	0
$726$	4.24095e6	0.298622
$727$	−2.53565e7	−1.77932	−0.889658	−	0.456627i	$-0.849057\pi$
$727$	−2.53565e7	−1.77932	−0.889658	+	0.456627i	$0.849057\pi$
$728$	2.82762e6	0.197739
$729$	4.31096e6	0.300438
$730$	0	0
$731$	5.07775e6	0.351462
$732$	−2.41310e6	−0.166455
$733$	1.70653e7	1.17315	0.586576	−	0.809894i	$-0.300476\pi$
$733$	1.70653e7	1.17315	0.586576	+	0.809894i	$0.300476\pi$
$734$	8.04497e6	0.551168
$735$	0	0
$736$	−7.49218e6	−0.509816
$737$	−659478.	−0.0447230
$738$	9.51826e6	0.643304
$739$	−1.40560e7	−0.946781	−0.473391	−	0.880853i	$-0.656970\pi$
$739$	−1.40560e7	−0.946781	−0.473391	+	0.880853i	$0.656970\pi$
$740$	0	0
$741$	−374067.	−0.0250267
$742$	−3.90245e6	−0.260212
$743$	−1.55718e7	−1.03483	−0.517413	−	0.855736i	$-0.673105\pi$
$743$	−1.55718e7	−1.03483	−0.517413	+	0.855736i	$0.673105\pi$
$744$	1.23135e6	0.0815544
$745$	0	0
$746$	2.74633e6	0.180678
$747$	−1.50126e6	−0.0984362
$748$	1.23980e6	0.0810209
$749$	−4.10433e6	−0.267324
$750$	0	0
$751$	−5.89761e6	−0.381572	−0.190786	−	0.981632i	$-0.561104\pi$
$751$	−5.89761e6	−0.381572	−0.190786	+	0.981632i	$0.561104\pi$
$752$	1.45351e6	0.0937287
$753$	−4.86441e6	−0.312639
$754$	2.36513e6	0.151505
$755$	0	0
$756$	5.40622e6	0.344024
$757$	−2.57205e6	−0.163132	−0.0815660	−	0.996668i	$-0.525992\pi$
$757$	−2.57205e6	−0.163132	−0.0815660	+	0.996668i	$0.525992\pi$
$758$	−5.31356e6	−0.335902
$759$	−1.76201e6	−0.111021
$760$	0	0
$761$	−2.56012e7	−1.60250	−0.801252	−	0.598328i	$-0.795832\pi$
$761$	−2.56012e7	−1.60250	−0.801252	+	0.598328i	$0.795832\pi$
$762$	−5.68509e6	−0.354691
$763$	1.66665e7	1.03642
$764$	−957179.	−0.0593280
$765$	0	0
$766$	1.06750e7	0.657349
$767$	1.20555e6	0.0739942
$768$	−9.10895e6	−0.557270
$769$	−1.15095e7	−0.701842	−0.350921	−	0.936405i	$-0.614131\pi$
$769$	−1.15095e7	−0.701842	−0.350921	+	0.936405i	$0.614131\pi$
$770$	0	0
$771$	−1.32582e6	−0.0803243
$772$	3.69333e6	0.223036
$773$	−5.18479e6	−0.312092	−0.156046	−	0.987750i	$-0.549875\pi$
$773$	−5.18479e6	−0.312092	−0.156046	+	0.987750i	$0.549875\pi$
$774$	7.53509e6	0.452101
$775$	0	0
$776$	−3.26674e6	−0.194742
$777$	−3.21032e6	−0.190764
$778$	6.48229e6	0.383954
$779$	−3.86190e6	−0.228012
$780$	0	0
$781$	−2.01534e6	−0.118228
$782$	−2.32388e6	−0.135893
$783$	1.27070e7	0.740692
$784$	1.29503e6	0.0752469
$785$	0	0
$786$	−6.86561e6	−0.396390
$787$	−1.28463e7	−0.739333	−0.369667	−	0.929165i	$-0.620528\pi$
$787$	−1.28463e7	−0.739333	−0.369667	+	0.929165i	$0.620528\pi$
$788$	−1.02834e7	−0.589958
$789$	−3.10087e6	−0.177333
$790$	0	0
$791$	1.68438e7	0.957190
$792$	5.16992e6	0.292867
$793$	2.81044e6	0.158705
$794$	1.43834e7	0.809676
$795$	0	0
$796$	1.10456e7	0.617883
$797$	−2.11768e7	−1.18090	−0.590452	−	0.807073i	$-0.701050\pi$
$797$	−2.11768e7	−1.18090	−0.590452	+	0.807073i	$0.701050\pi$
$798$	745461.	0.0414398
$799$	−4.46727e6	−0.247557
$800$	0	0
$801$	−1.04651e7	−0.576320
$802$	−4.14155e6	−0.227367
$803$	2.05069e6	0.112230
$804$	−611172.	−0.0333444
$805$	0	0
$806$	−510344.	−0.0276711
$807$	−5655.97	−0.000305719 0
$808$	1.29100e7	0.695659
$809$	1.58537e6	0.0851648	0.0425824	−	0.999093i	$-0.486442\pi$
$809$	1.58537e6	0.0851648	0.0425824	+	0.999093i	$0.486442\pi$
$810$	0	0
$811$	−2.18155e7	−1.16470	−0.582350	−	0.812938i	$-0.697867\pi$
$811$	−2.18155e7	−1.16470	−0.582350	+	0.812938i	$0.697867\pi$
$812$	5.81850e6	0.309686
$813$	5.13375e6	0.272401
$814$	−2.60407e6	−0.137750
$815$	0	0
$816$	−535733.	−0.0281659
$817$	−3.05725e6	−0.160242
$818$	−2.50962e7	−1.31137
$819$	−2.64157e6	−0.137611
$820$	0	0
$821$	3.09541e7	1.60273	0.801365	−	0.598176i	$-0.204108\pi$
$821$	3.09541e7	1.60273	0.801365	+	0.598176i	$0.204108\pi$
$822$	−1.09901e7	−0.567313
$823$	2.68746e7	1.38307	0.691533	−	0.722345i	$-0.256936\pi$
$823$	2.68746e7	1.38307	0.691533	+	0.722345i	$0.256936\pi$
$824$	3.59464e7	1.84433
$825$	0	0
$826$	−2.40249e6	−0.122521
$827$	9.94152e6	0.505462	0.252731	−	0.967537i	$-0.418671\pi$
$827$	9.94152e6	0.505462	0.252731	+	0.967537i	$0.418671\pi$
$828$	4.25707e6	0.215792
$829$	7.20887e6	0.364318	0.182159	−	0.983269i	$-0.441691\pi$
$829$	7.20887e6	0.364318	0.182159	+	0.983269i	$0.441691\pi$
$830$	0	0
$831$	6.10666e6	0.306762
$832$	4.28295e6	0.214504
$833$	−3.98020e6	−0.198743
$834$	−8.77867e6	−0.437032
$835$	0	0
$836$	−746468.	−0.0369399
$837$	−2.74190e6	−0.135281
$838$	5.67777e6	0.279298
$839$	1.96670e7	0.964569	0.482285	−	0.876015i	$-0.339807\pi$
$839$	1.96670e7	0.964569	0.482285	+	0.876015i	$0.339807\pi$
$840$	0	0
$841$	−6.83513e6	−0.333240
$842$	7.24288e6	0.352072
$843$	1.18018e7	0.571979
$844$	−1.11068e7	−0.536702
$845$	0	0
$846$	−6.62917e6	−0.318444
$847$	−1.21512e7	−0.581984
$848$	−1.68858e6	−0.0806364
$849$	−1.02321e7	−0.487186
$850$	0	0
$851$	−6.02554e6	−0.285215
$852$	−1.86772e6	−0.0881482
$853$	2.60044e7	1.22370	0.611848	−	0.790975i	$-0.290426\pi$
$853$	2.60044e7	1.22370	0.611848	+	0.790975i	$0.290426\pi$
$854$	−5.60080e6	−0.262788
$855$	0	0
$856$	−8.67015e6	−0.404429
$857$	−2.74925e7	−1.27868	−0.639341	−	0.768923i	$-0.720793\pi$
$857$	−2.74925e7	−1.27868	−0.639341	+	0.768923i	$0.720793\pi$
$858$	821919.	0.0381163
$859$	3.86634e7	1.78779	0.893896	−	0.448274i	$-0.147961\pi$
$859$	3.86634e7	1.78779	0.893896	+	0.448274i	$0.147961\pi$
$860$	0	0
$861$	1.04611e7	0.480914
$862$	−2.28669e7	−1.04819
$863$	8.41848e6	0.384775	0.192387	−	0.981319i	$-0.438377\pi$
$863$	8.41848e6	0.384775	0.192387	+	0.981319i	$0.438377\pi$
$864$	1.87766e7	0.855720
$865$	0	0
$866$	6.86038e6	0.310852
$867$	−1.00075e7	−0.452144
$868$	−1.25551e6	−0.0565615
$869$	−1.17008e7	−0.525611
$870$	0	0
$871$	711808.	0.0317920
$872$	3.52071e7	1.56797
$873$	3.05179e6	0.135525
$874$	1.39918e6	0.0619576
$875$	0	0
$876$	1.90048e6	0.0836763
$877$	1.24226e7	0.545399	0.272699	−	0.962099i	$-0.412084\pi$
$877$	1.24226e7	0.545399	0.272699	+	0.962099i	$0.412084\pi$
$878$	−1.30716e7	−0.572257
$879$	1.43357e7	0.625815
$880$	0	0
$881$	7.24684e6	0.314564	0.157282	−	0.987554i	$-0.449727\pi$
$881$	7.24684e6	0.314564	0.157282	+	0.987554i	$0.449727\pi$
$882$	−5.90637e6	−0.255652
$883$	1.58232e7	0.682958	0.341479	−	0.939889i	$-0.389072\pi$
$883$	1.58232e7	0.682958	0.341479	+	0.939889i	$0.389072\pi$
$884$	−1.33818e6	−0.0575949
$885$	0	0
$886$	6.21760e6	0.266096
$887$	−1.85905e7	−0.793382	−0.396691	−	0.917952i	$-0.629842\pi$
$887$	−1.85905e7	−0.793382	−0.396691	+	0.917952i	$0.629842\pi$
$888$	−6.78161e6	−0.288603
$889$	1.62890e7	0.691257
$890$	0	0
$891$	−2.26650e6	−0.0956446
$892$	1.94038e7	0.816537
$893$	2.68969e6	0.112869
$894$	2.40972e6	0.100838
$895$	0	0
$896$	7.02718e6	0.292423
$897$	1.90183e6	0.0789206
$898$	−1.47654e7	−0.611018
$899$	−2.95100e6	−0.121778
$900$	0	0
$901$	5.18974e6	0.212978
$902$	8.48556e6	0.347267
$903$	8.28144e6	0.337977
$904$	3.55815e7	1.44811
$905$	0	0
$906$	6.51331e6	0.263622
$907$	−4.30877e7	−1.73914	−0.869571	−	0.493808i	$-0.835605\pi$
$907$	−4.30877e7	−1.73914	−0.869571	+	0.493808i	$0.835605\pi$
$908$	−2.86288e6	−0.115236
$909$	−1.20605e7	−0.484123
$910$	0	0
$911$	870987.	0.0347709	0.0173854	−	0.999849i	$-0.494466\pi$
$911$	870987.	0.0347709	0.0173854	+	0.999849i	$0.494466\pi$
$912$	322558.	0.0128417
$913$	−1.33838e6	−0.0531377
$914$	2.30773e7	0.913735
$915$	0	0
$916$	1.14066e7	0.449178
$917$	1.96714e7	0.772524
$918$	5.82400e6	0.228094
$919$	−7.81477e6	−0.305230	−0.152615	−	0.988286i	$-0.548769\pi$
$919$	−7.81477e6	−0.305230	−0.152615	+	0.988286i	$0.548769\pi$
$920$	0	0
$921$	−1.96439e6	−0.0763095
$922$	−3.22647e7	−1.24997
$923$	2.17526e6	0.0840442
$924$	2.02202e6	0.0779123
$925$	0	0
$926$	1.02425e7	0.392533
$927$	−3.35812e7	−1.28350
$928$	2.02085e7	0.770307
$929$	6.93323e6	0.263570	0.131785	−	0.991278i	$-0.457929\pi$
$929$	6.93323e6	0.263570	0.131785	+	0.991278i	$0.457929\pi$
$930$	0	0
$931$	2.39643e6	0.0906129
$932$	1.48487e7	0.559948
$933$	9.05644e6	0.340607
$934$	1.32516e6	0.0497052
$935$	0	0
$936$	−5.58017e6	−0.208189
$937$	3.31684e7	1.23417	0.617085	−	0.786896i	$-0.288313\pi$
$937$	3.31684e7	1.23417	0.617085	+	0.786896i	$0.288313\pi$
$938$	−1.41853e6	−0.0526420
$939$	−7.67298e6	−0.283988
$940$	0	0
$941$	−5.00778e6	−0.184362	−0.0921809	−	0.995742i	$-0.529384\pi$
$941$	−5.00778e6	−0.184362	−0.0921809	+	0.995742i	$0.529384\pi$
$942$	−3.96790e6	−0.145691
$943$	1.96347e7	0.719025
$944$	−1.03955e6	−0.0379678
$945$	0	0
$946$	6.71756e6	0.244053
$947$	7.37028e6	0.267060	0.133530	−	0.991045i	$-0.457369\pi$
$947$	7.37028e6	0.267060	0.133530	+	0.991045i	$0.457369\pi$
$948$	−1.08437e7	−0.391883
$949$	−2.21341e6	−0.0797805
$950$	0	0
$951$	2.17867e7	0.781162
$952$	7.49388e6	0.267987
$953$	−7.45227e6	−0.265801	−0.132900	−	0.991129i	$-0.542429\pi$
$953$	−7.45227e6	−0.265801	−0.132900	+	0.991129i	$0.542429\pi$
$954$	7.70127e6	0.273963
$955$	0	0
$956$	9.40817e6	0.332936
$957$	4.75263e6	0.167747
$958$	−3.21336e7	−1.13122
$959$	3.14890e7	1.10564
$960$	0	0
$961$	−2.79924e7	−0.977758
$962$	2.81071e6	0.0979216
$963$	8.09967e6	0.281450
$964$	199457.	0.00691283
$965$	0	0
$966$	−3.79007e6	−0.130679
$967$	4.78544e7	1.64572	0.822860	−	0.568244i	$-0.192377\pi$
$967$	4.78544e7	1.64572	0.822860	+	0.568244i	$0.192377\pi$
$968$	−2.56687e7	−0.880473
$969$	−991365.	−0.0339175
$970$	0	0
$971$	−2.54187e7	−0.865177	−0.432588	−	0.901592i	$-0.642400\pi$
$971$	−2.54187e7	−0.865177	−0.432588	+	0.901592i	$0.642400\pi$
$972$	−1.68618e7	−0.572450
$973$	2.51527e7	0.851733
$974$	−11484.1	−0.000387881 0
$975$	0	0
$976$	−2.42345e6	−0.0814347
$977$	−3.10699e7	−1.04137	−0.520683	−	0.853750i	$-0.674323\pi$
$977$	−3.10699e7	−1.04137	−0.520683	+	0.853750i	$0.674323\pi$
$978$	−1.03882e6	−0.0347291
$979$	−9.32970e6	−0.311108
$980$	0	0
$981$	−3.28905e7	−1.09118
$982$	1.29845e7	0.429682
$983$	−1.28575e6	−0.0424398	−0.0212199	−	0.999775i	$-0.506755\pi$
$983$	−1.28575e6	−0.0424398	−0.0212199	+	0.999775i	$0.506755\pi$
$984$	2.20984e7	0.727566
$985$	0	0
$986$	6.26814e6	0.205327
$987$	−7.28579e6	−0.238059
$988$	805701.	0.0262592
$989$	1.55437e7	0.505317
$990$	0	0
$991$	−4.43766e6	−0.143539	−0.0717695	−	0.997421i	$-0.522865\pi$
$991$	−4.43766e6	−0.143539	−0.0717695	+	0.997421i	$0.522865\pi$
$992$	−4.36057e6	−0.140690
$993$	5.32146e6	0.171261
$994$	−4.33499e6	−0.139162
$995$	0	0
$996$	−1.24035e6	−0.0396182
$997$	2.72549e7	0.868375	0.434187	−	0.900823i	$-0.357036\pi$
$997$	2.72549e7	0.868375	0.434187	+	0.900823i	$0.357036\pi$
$998$	4.22202e6	0.134182
$999$	1.51009e7	0.478729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.k.1.4	yes	11
5.2	odd	4		325.6.b.i.274.8		22
5.3	odd	4		325.6.b.i.274.15		22
5.4	even	2		325.6.a.j.1.8	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.8	✓	11	5.4	even	2
325.6.a.k.1.4	yes	11	1.1	even	1	trivial
325.6.b.i.274.8		22	5.2	odd	4
325.6.b.i.274.15		22	5.3	odd	4

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)

\(f(q)\)	\(=\)	\(q-3.78432 q^{2} +8.20788 q^{3} -17.6789 q^{4} -31.0612 q^{6} +88.9969 q^{7} +188.001 q^{8} -175.631 q^{9} -156.575 q^{11} -145.107 q^{12} +169.000 q^{13} -336.793 q^{14} -145.729 q^{16} +447.890 q^{17} +664.642 q^{18} -269.669 q^{19} +730.476 q^{21} +592.531 q^{22} +1371.05 q^{23} +1543.09 q^{24} -639.550 q^{26} -3436.07 q^{27} -1573.37 q^{28} -3698.11 q^{29} +797.974 q^{31} -5464.54 q^{32} -1285.15 q^{33} -1694.96 q^{34} +3104.97 q^{36} -4394.83 q^{37} +1020.51 q^{38} +1387.13 q^{39} +14320.9 q^{41} -2764.35 q^{42} +11337.1 q^{43} +2768.09 q^{44} -5188.50 q^{46} -9974.03 q^{47} -1196.13 q^{48} -8886.54 q^{49} +3676.23 q^{51} -2987.74 q^{52} +11587.1 q^{53} +13003.2 q^{54} +16731.5 q^{56} -2213.41 q^{57} +13994.8 q^{58} +7133.45 q^{59} +16629.8 q^{61} -3019.79 q^{62} -15630.6 q^{63} +25342.9 q^{64} +4863.42 q^{66} +4211.88 q^{67} -7918.22 q^{68} +11253.4 q^{69} +12871.4 q^{71} -33018.7 q^{72} -13097.1 q^{73} +16631.4 q^{74} +4767.46 q^{76} -13934.7 q^{77} -5249.35 q^{78} +74729.2 q^{79} +14475.4 q^{81} -54194.7 q^{82} +8547.83 q^{83} -12914.0 q^{84} -42903.0 q^{86} -30353.6 q^{87} -29436.3 q^{88} +59586.0 q^{89} +15040.5 q^{91} -24238.8 q^{92} +6549.68 q^{93} +37744.9 q^{94} -44852.3 q^{96} -17376.2 q^{97} +33629.5 q^{98} +27499.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 5 q^{2} + 11 q^{3} + 187 q^{4} + 351 q^{6} + 208 q^{7} + 165 q^{8} + 1372 q^{9} + 1276 q^{11} + 1533 q^{12} + 1859 q^{13} + 578 q^{14} + 5707 q^{16} + 2218 q^{17} - 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 11 * q + 5 * q^2 + 11 * q^3 + 187 * q^4 + 351 * q^6 + 208 * q^7 + 165 * q^8 + 1372 * q^9 + 1276 * q^11 + 1533 * q^12 + 1859 * q^13 + 578 * q^14 + 5707 * q^16 + 2218 * q^17 - 6776 * q^18 + 3520 * q^19 + 1706 * q^21 - 6785 * q^22 - 1215 * q^23 + 7723 * q^24 + 845 * q^26 + 161 * q^27 + 12746 * q^28 + 18425 * q^29 + 9068 * q^31 + 21165 * q^32 + 19748 * q^33 - 14455 * q^34 + 37684 * q^36 - 2280 * q^37 - 23783 * q^38 + 1859 * q^39 - 10010 * q^41 - 44184 * q^42 + 6763 * q^43 + 60651 * q^44 - 9858 * q^46 + 4110 * q^47 + 63289 * q^48 + 73313 * q^49 + 36606 * q^51 + 31603 * q^52 + 15633 * q^53 + 48013 * q^54 - 1262 * q^56 - 205974 * q^57 + 77652 * q^58 + 131642 * q^59 - 43365 * q^61 - 115940 * q^62 + 218466 * q^63 + 180071 * q^64 - 58607 * q^66 - 100904 * q^67 + 371277 * q^68 + 59345 * q^69 + 68420 * q^71 - 210708 * q^72 + 124436 * q^73 + 371396 * q^74 + 22517 * q^76 - 150978 * q^77 + 59319 * q^78 + 52239 * q^79 + 107507 * q^81 - 351137 * q^82 + 211538 * q^83 + 852716 * q^84 + 202246 * q^86 - 28273 * q^87 - 28107 * q^88 + 327712 * q^89 + 35152 * q^91 - 413252 * q^92 + 243338 * q^93 + 649974 * q^94 + 899663 * q^96 - 110796 * q^97 + 27143 * q^98 + 426698 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more