LMFDB - Embedded newform 325.6.a.j.1.11

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			1.11
Root			$-10.1023$ 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)

$f(q)$	$=$	\(q+10.1023 q^{2} +24.5176 q^{3} +70.0572 q^{4} +247.685 q^{6} +72.1186 q^{7} +384.466 q^{8} +358.110 q^{9} +127.428 q^{11} +1717.63 q^{12} -169.000 q^{13} +728.566 q^{14} +1642.18 q^{16} -2152.48 q^{17} +3617.75 q^{18} +2726.20 q^{19} +1768.17 q^{21} +1287.32 q^{22} -2658.77 q^{23} +9426.17 q^{24} -1707.29 q^{26} +2822.22 q^{27} +5052.43 q^{28} -5885.60 q^{29} +1366.14 q^{31} +4286.91 q^{32} +3124.23 q^{33} -21745.0 q^{34} +25088.2 q^{36} +481.670 q^{37} +27541.0 q^{38} -4143.47 q^{39} +7385.44 q^{41} +17862.7 q^{42} -10167.0 q^{43} +8927.26 q^{44} -26859.8 q^{46} +16361.7 q^{47} +40262.2 q^{48} -11605.9 q^{49} -52773.4 q^{51} -11839.7 q^{52} +9061.97 q^{53} +28511.0 q^{54} +27727.2 q^{56} +66839.8 q^{57} -59458.3 q^{58} +29585.4 q^{59} +16410.0 q^{61} +13801.2 q^{62} +25826.4 q^{63} -9241.89 q^{64} +31562.0 q^{66} +61557.7 q^{67} -150796. q^{68} -65186.5 q^{69} -5229.46 q^{71} +137681. q^{72} -67851.0 q^{73} +4865.99 q^{74} +190990. q^{76} +9189.95 q^{77} -41858.7 q^{78} -89505.0 q^{79} -17826.8 q^{81} +74610.2 q^{82} -78989.9 q^{83} +123873. q^{84} -102711. q^{86} -144300. q^{87} +48991.9 q^{88} +111252. q^{89} -12188.0 q^{91} -186266. q^{92} +33494.5 q^{93} +165292. q^{94} +105105. q^{96} +71553.7 q^{97} -117247. q^{98} +45633.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	10.1023	1.78586	0.892929	−	0.450198i	$-0.148647\pi$
$2$	10.1023	1.78586	0.892929	+	0.450198i	$0.148647\pi$
$3$	24.5176	1.57280	0.786401	−	0.617717i	$-0.211942\pi$
$3$	24.5176	1.57280	0.786401	+	0.617717i	$0.211942\pi$
$4$	70.0572	2.18929
$5$	0	0
$5$	0	0
$6$	247.685	2.80880
$7$	72.1186	0.556291	0.278146	−	0.960539i	$-0.410280\pi$
$7$	72.1186	0.556291	0.278146	+	0.960539i	$0.410280\pi$
$8$	384.466	2.12390
$9$	358.110	1.47370
$10$	0	0
$11$	127.428	0.317529	0.158765	−	0.987316i	$-0.449249\pi$
$11$	127.428	0.317529	0.158765	+	0.987316i	$0.449249\pi$
$12$	1717.63	3.44331
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	728.566	0.993457
$15$	0	0
$16$	1642.18	1.60369
$17$	−2152.48	−1.80641	−0.903204	−	0.429211i	$-0.858792\pi$
$17$	−2152.48	−1.80641	−0.903204	+	0.429211i	$0.858792\pi$
$18$	3617.75	2.63183
$19$	2726.20	1.73250	0.866252	−	0.499608i	$-0.166523\pi$
$19$	2726.20	1.73250	0.866252	+	0.499608i	$0.166523\pi$
$20$	0	0
$21$	1768.17	0.874936
$22$	1287.32	0.567062
$23$	−2658.77	−1.04800	−0.524000	−	0.851719i	$-0.675561\pi$
$23$	−2658.77	−1.04800	−0.524000	+	0.851719i	$0.675561\pi$
$24$	9426.17	3.34047
$25$	0	0
$26$	−1707.29	−0.495308
$27$	2822.22	0.745044
$28$	5052.43	1.21788
$29$	−5885.60	−1.29956	−0.649779	−	0.760123i	$-0.725139\pi$
$29$	−5885.60	−1.29956	−0.649779	+	0.760123i	$0.725139\pi$
$30$	0	0
$31$	1366.14	0.255324	0.127662	−	0.991818i	$-0.459253\pi$
$31$	1366.14	0.255324	0.127662	+	0.991818i	$0.459253\pi$
$32$	4286.91	0.740064
$33$	3124.23	0.499411
$34$	−21745.0	−3.22599
$35$	0	0
$36$	25088.2	3.22636
$37$	481.670	0.0578423	0.0289211	−	0.999582i	$-0.490793\pi$
$37$	481.670	0.0578423	0.0289211	+	0.999582i	$0.490793\pi$
$38$	27541.0	3.09400
$39$	−4143.47	−0.436217
$40$	0	0
$41$	7385.44	0.686146	0.343073	−	0.939309i	$-0.388532\pi$
$41$	7385.44	0.686146	0.343073	+	0.939309i	$0.388532\pi$
$42$	17862.7	1.56251
$43$	−10167.0	−0.838538	−0.419269	−	0.907862i	$-0.637714\pi$
$43$	−10167.0	−0.838538	−0.419269	+	0.907862i	$0.637714\pi$
$44$	8927.26	0.695163
$45$	0	0
$46$	−26859.8	−1.87158
$47$	16361.7	1.08040	0.540200	−	0.841537i	$-0.318349\pi$
$47$	16361.7	1.08040	0.540200	+	0.841537i	$0.318349\pi$
$48$	40262.2	2.52229
$49$	−11605.9	−0.690540
$50$	0	0
$51$	−52773.4	−2.84112
$52$	−11839.7	−0.607199
$53$	9061.97	0.443132	0.221566	−	0.975145i	$-0.428883\pi$
$53$	9061.97	0.443132	0.221566	+	0.975145i	$0.428883\pi$
$54$	28511.0	1.33054
$55$	0	0
$56$	27727.2	1.18151
$57$	66839.8	2.72488
$58$	−59458.3	−2.32082
$59$	29585.4	1.10649	0.553245	−	0.833019i	$-0.313389\pi$
$59$	29585.4	1.10649	0.553245	+	0.833019i	$0.313389\pi$
$60$	0	0
$61$	16410.0	0.564656	0.282328	−	0.959318i	$-0.408893\pi$
$61$	16410.0	0.564656	0.282328	+	0.959318i	$0.408893\pi$
$62$	13801.2	0.455972
$63$	25826.4	0.819809
$64$	−9241.89	−0.282040
$65$	0	0
$66$	31562.0	0.891876
$67$	61557.7	1.67531	0.837656	−	0.546198i	$-0.183926\pi$
$67$	61557.7	1.67531	0.837656	+	0.546198i	$0.183926\pi$
$68$	−150796.	−3.95475
$69$	−65186.5	−1.64829
$70$	0	0
$71$	−5229.46	−0.123115	−0.0615575	−	0.998104i	$-0.519607\pi$
$71$	−5229.46	−0.123115	−0.0615575	+	0.998104i	$0.519607\pi$
$72$	137681.	3.13000
$73$	−67851.0	−1.49021	−0.745107	−	0.666944i	$-0.767602\pi$
$73$	−67851.0	−1.49021	−0.745107	+	0.666944i	$0.767602\pi$
$74$	4865.99	0.103298
$75$	0	0
$76$	190990.	3.79295
$77$	9189.95	0.176639
$78$	−41858.7	−0.779021
$79$	−89505.0	−1.61354	−0.806769	−	0.590867i	$-0.798786\pi$
$79$	−89505.0	−1.61354	−0.806769	+	0.590867i	$0.798786\pi$
$80$	0	0
$81$	−17826.8	−0.301899
$82$	74610.2	1.22536
$83$	−78989.9	−1.25857	−0.629283	−	0.777176i	$-0.716652\pi$
$83$	−78989.9	−1.25857	−0.629283	+	0.777176i	$0.716652\pi$
$84$	123873.	1.91549
$85$	0	0
$86$	−102711.	−1.49751
$87$	−144300.	−2.04395
$88$	48991.9	0.674399
$89$	111252.	1.48879	0.744394	−	0.667741i	$-0.232738\pi$
$89$	111252.	1.48879	0.744394	+	0.667741i	$0.232738\pi$
$90$	0	0
$91$	−12188.0	−0.154287
$92$	−186266.	−2.29437
$93$	33494.5	0.401574
$94$	165292.	1.92944
$95$	0	0
$96$	105105.	1.16397
$97$	71553.7	0.772151	0.386076	−	0.922467i	$-0.373830\pi$
$97$	71553.7	0.772151	0.386076	+	0.922467i	$0.373830\pi$
$98$	−117247.	−1.23321
$99$	45633.3	0.467945
$100$	0	0
$101$	150676.	1.46974	0.734868	−	0.678210i	$-0.237244\pi$
$101$	150676.	1.46974	0.734868	+	0.678210i	$0.237244\pi$
$102$	−533135.	−5.07384
$103$	−160191.	−1.48781	−0.743903	−	0.668288i	$-0.767027\pi$
$103$	−160191.	−1.48781	−0.743903	+	0.668288i	$0.767027\pi$
$104$	−64974.8	−0.589063
$105$	0	0
$106$	91547.1	0.791371
$107$	−78017.7	−0.658769	−0.329385	−	0.944196i	$-0.606841\pi$
$107$	−78017.7	−0.658769	−0.329385	+	0.944196i	$0.606841\pi$
$108$	197717.	1.63111
$109$	−219872.	−1.77257	−0.886284	−	0.463141i	$-0.846722\pi$
$109$	−219872.	−1.77257	−0.886284	+	0.463141i	$0.846722\pi$
$110$	0	0
$111$	11809.4	0.0909745
$112$	118432.	0.892118
$113$	171300.	1.26201	0.631004	−	0.775780i	$-0.282643\pi$
$113$	171300.	1.26201	0.631004	+	0.775780i	$0.282643\pi$
$114$	675238.	4.86625
$115$	0	0
$116$	−412328.	−2.84510
$117$	−60520.6	−0.408732
$118$	298882.	1.97603
$119$	−155234.	−1.00489
$120$	0	0
$121$	−144813.	−0.899175
$122$	165779.	1.00839
$123$	181073.	1.07917
$124$	95708.0	0.558977
$125$	0	0
$126$	260907.	1.46406
$127$	98906.0	0.544143	0.272072	−	0.962277i	$-0.412291\pi$
$127$	98906.0	0.544143	0.272072	+	0.962277i	$0.412291\pi$
$128$	−230546.	−1.24375
$129$	−249271.	−1.31885
$130$	0	0
$131$	266465.	1.35663	0.678317	−	0.734769i	$-0.262710\pi$
$131$	266465.	1.35663	0.678317	+	0.734769i	$0.262710\pi$
$132$	218875.	1.09335
$133$	196610.	0.963777
$134$	621877.	2.99187
$135$	0	0
$136$	−827554.	−3.83662
$137$	−35452.7	−0.161379	−0.0806896	−	0.996739i	$-0.525712\pi$
$137$	−35452.7	−0.161379	−0.0806896	+	0.996739i	$0.525712\pi$
$138$	−658536.	−2.94362
$139$	−63698.3	−0.279635	−0.139817	−	0.990177i	$-0.544652\pi$
$139$	−63698.3	−0.279635	−0.139817	+	0.990177i	$0.544652\pi$
$140$	0	0
$141$	401150.	1.69926
$142$	−52829.7	−0.219866
$143$	−21535.4	−0.0880668
$144$	588081.	2.36336
$145$	0	0
$146$	−685453.	−2.66131
$147$	−284548.	−1.08608
$148$	33744.5	0.126633
$149$	−26631.1	−0.0982707	−0.0491353	−	0.998792i	$-0.515647\pi$
$149$	−26631.1	−0.0982707	−0.0491353	+	0.998792i	$0.515647\pi$
$150$	0	0
$151$	448794.	1.60179	0.800893	−	0.598808i	$-0.204359\pi$
$151$	448794.	1.60179	0.800893	+	0.598808i	$0.204359\pi$
$152$	1.04813e6	3.67966
$153$	−770824.	−2.66211
$154$	92839.9	0.315452
$155$	0	0
$156$	−290280.	−0.955003
$157$	16497.9	0.0534170	0.0267085	−	0.999643i	$-0.491497\pi$
$157$	16497.9	0.0534170	0.0267085	+	0.999643i	$0.491497\pi$
$158$	−904209.	−2.88155
$159$	222177.	0.696959
$160$	0	0
$161$	−191747.	−0.582993
$162$	−180093.	−0.539149
$163$	−83354.3	−0.245731	−0.122865	−	0.992423i	$-0.539208\pi$
$163$	−83354.3	−0.245731	−0.122865	+	0.992423i	$0.539208\pi$
$164$	517403.	1.50217
$165$	0	0
$166$	−797982.	−2.24762
$167$	−215768.	−0.598680	−0.299340	−	0.954146i	$-0.596767\pi$
$167$	−215768.	−0.598680	−0.299340	+	0.954146i	$0.596767\pi$
$168$	679802.	1.85827
$169$	28561.0	0.0769231
$170$	0	0
$171$	976281.	2.55320
$172$	−712273.	−1.83580
$173$	−42375.0	−0.107645	−0.0538225	−	0.998551i	$-0.517141\pi$
$173$	−42375.0	−0.107645	−0.0538225	+	0.998551i	$0.517141\pi$
$174$	−1.45777e6	−3.65020
$175$	0	0
$176$	209260.	0.509218
$177$	725362.	1.74029
$178$	1.12391e6	2.65876
$179$	147708.	0.344566	0.172283	−	0.985047i	$-0.444886\pi$
$179$	147708.	0.344566	0.172283	+	0.985047i	$0.444886\pi$
$180$	0	0
$181$	−742092.	−1.68369	−0.841843	−	0.539722i	$-0.818529\pi$
$181$	−742092.	−1.68369	−0.841843	+	0.539722i	$0.818529\pi$
$182$	−123128.	−0.275535
$183$	402333.	0.888092
$184$	−1.02221e6	−2.22584
$185$	0	0
$186$	338372.	0.717154
$187$	−274286.	−0.573588
$188$	1.14626e6	2.36531
$189$	203535.	0.414461
$190$	0	0
$191$	698156.	1.38474	0.692372	−	0.721541i	$-0.256566\pi$
$191$	698156.	1.38474	0.692372	+	0.721541i	$0.256566\pi$
$192$	−226589.	−0.443593
$193$	−190607.	−0.368338	−0.184169	−	0.982895i	$-0.558959\pi$
$193$	−190607.	−0.368338	−0.184169	+	0.982895i	$0.558959\pi$
$194$	722859.	1.37895
$195$	0	0
$196$	−813077.	−1.51179
$197$	817552.	1.50089	0.750447	−	0.660931i	$-0.229838\pi$
$197$	817552.	1.50089	0.750447	+	0.660931i	$0.229838\pi$
$198$	461003.	0.835682
$199$	−288709.	−0.516806	−0.258403	−	0.966037i	$-0.583196\pi$
$199$	−288709.	−0.516806	−0.258403	+	0.966037i	$0.583196\pi$
$200$	0	0
$201$	1.50924e6	2.63493
$202$	1.52218e6	2.62474
$203$	−424461.	−0.722933
$204$	−3.69716e6	−6.22003
$205$	0	0
$206$	−1.61831e6	−2.65701
$207$	−952132.	−1.54444
$208$	−277528.	−0.444783
$209$	347395.	0.550121
$210$	0	0
$211$	134806.	0.208451	0.104225	−	0.994554i	$-0.466764\pi$
$211$	134806.	0.208451	0.104225	+	0.994554i	$0.466764\pi$
$212$	634856.	0.970143
$213$	−128213.	−0.193635
$214$	−788161.	−1.17647
$215$	0	0
$216$	1.08505e6	1.58240
$217$	98524.3	0.142035
$218$	−2.22122e6	−3.16556
$219$	−1.66354e6	−2.34381
$220$	0	0
$221$	363768.	0.501007
$222$	119302.	0.162467
$223$	320971.	0.432218	0.216109	−	0.976369i	$-0.430663\pi$
$223$	320971.	0.432218	0.216109	+	0.976369i	$0.430663\pi$
$224$	309166.	0.411691
$225$	0	0
$226$	1.73053e6	2.25377
$227$	−28722.2	−0.0369958	−0.0184979	−	0.999829i	$-0.505888\pi$
$227$	−28722.2	−0.0369958	−0.0184979	+	0.999829i	$0.505888\pi$
$228$	4.68261e6	5.96555
$229$	850808.	1.07212	0.536059	−	0.844180i	$-0.319912\pi$
$229$	850808.	1.07212	0.536059	+	0.844180i	$0.319912\pi$
$230$	0	0
$231$	225315.	0.277818
$232$	−2.26281e6	−2.76013
$233$	432423.	0.521818	0.260909	−	0.965363i	$-0.415978\pi$
$233$	432423.	0.521818	0.260909	+	0.965363i	$0.415978\pi$
$234$	−611400.	−0.729937
$235$	0	0
$236$	2.07267e6	2.42242
$237$	−2.19444e6	−2.53778
$238$	−1.56822e6	−1.79459
$239$	530968.	0.601276	0.300638	−	0.953738i	$-0.402800\pi$
$239$	530968.	0.601276	0.300638	+	0.953738i	$0.402800\pi$
$240$	0	0
$241$	−741085.	−0.821912	−0.410956	−	0.911655i	$-0.634805\pi$
$241$	−741085.	−0.821912	−0.410956	+	0.911655i	$0.634805\pi$
$242$	−1.46295e6	−1.60580
$243$	−1.12287e6	−1.21987
$244$	1.14964e6	1.23619
$245$	0	0
$246$	1.82926e6	1.92725
$247$	−460728.	−0.480510
$248$	525236.	0.542282
$249$	−1.93664e6	−1.97948
$250$	0	0
$251$	1.09761e6	1.09968	0.549839	−	0.835271i	$-0.314689\pi$
$251$	1.09761e6	1.09968	0.549839	+	0.835271i	$0.314689\pi$
$252$	1.80933e6	1.79480
$253$	−338802.	−0.332770
$254$	999182.	0.971762
$255$	0	0
$256$	−2.03331e6	−1.93912
$257$	379916.	0.358802	0.179401	−	0.983776i	$-0.442584\pi$
$257$	379916.	0.358802	0.179401	+	0.983776i	$0.442584\pi$
$258$	−2.51822e6	−2.35529
$259$	34737.4	0.0321772
$260$	0	0
$261$	−2.10769e6	−1.91516
$262$	2.69192e6	2.42275
$263$	−261845.	−0.233429	−0.116714	−	0.993166i	$-0.537236\pi$
$263$	−261845.	−0.233429	−0.116714	+	0.993166i	$0.537236\pi$
$264$	1.20116e6	1.06070
$265$	0	0
$266$	1.98622e6	1.72117
$267$	2.72763e6	2.34157
$268$	4.31256e6	3.66774
$269$	−892153.	−0.751724	−0.375862	−	0.926676i	$-0.622653\pi$
$269$	−892153.	−0.751724	−0.375862	+	0.926676i	$0.622653\pi$
$270$	0	0
$271$	−1.54998e6	−1.28204	−0.641021	−	0.767524i	$-0.721489\pi$
$271$	−1.54998e6	−1.28204	−0.641021	+	0.767524i	$0.721489\pi$
$272$	−3.53475e6	−2.89692
$273$	−298821.	−0.242664
$274$	−358155.	−0.288200
$275$	0	0
$276$	−4.56678e6	−3.60859
$277$	−797920.	−0.624828	−0.312414	−	0.949946i	$-0.601138\pi$
$277$	−797920.	−0.624828	−0.312414	+	0.949946i	$0.601138\pi$
$278$	−643502.	−0.499388
$279$	489229.	0.376272
$280$	0	0
$281$	1.16864e6	0.882907	0.441453	−	0.897284i	$-0.354463\pi$
$281$	1.16864e6	0.882907	0.441453	+	0.897284i	$0.354463\pi$
$282$	4.05255e6	3.03463
$283$	−1.09370e6	−0.811771	−0.405886	−	0.913924i	$-0.633037\pi$
$283$	−1.09370e6	−0.811771	−0.405886	+	0.913924i	$0.633037\pi$
$284$	−366361.	−0.269534
$285$	0	0
$286$	−217557.	−0.157275
$287$	532628.	0.381697
$288$	1.53519e6	1.09064
$289$	3.21329e6	2.26311
$290$	0	0
$291$	1.75432e6	1.21444
$292$	−4.75345e6	−3.26251
$293$	−1.02050e6	−0.694451	−0.347226	−	0.937782i	$-0.612876\pi$
$293$	−1.02050e6	−0.694451	−0.347226	+	0.937782i	$0.612876\pi$
$294$	−2.87460e6	−1.93959
$295$	0	0
$296$	185186.	0.122851
$297$	359631.	0.236573
$298$	−269037.	−0.175497
$299$	449332.	0.290663
$300$	0	0
$301$	−733232.	−0.466472
$302$	4.53386e6	2.86056
$303$	3.69420e6	2.31160
$304$	4.47691e6	2.77840
$305$	0	0
$306$	−7.78712e6	−4.75415
$307$	−3.21266e6	−1.94544	−0.972722	−	0.231975i	$-0.925481\pi$
$307$	−3.21266e6	−1.94544	−0.972722	+	0.231975i	$0.925481\pi$
$308$	643822.	0.386713
$309$	−3.92750e6	−2.34002
$310$	0	0
$311$	−2.57875e6	−1.51185	−0.755924	−	0.654659i	$-0.772812\pi$
$311$	−2.57875e6	−1.51185	−0.755924	+	0.654659i	$0.772812\pi$
$312$	−1.59302e6	−0.926479
$313$	2.26878e6	1.30898	0.654488	−	0.756072i	$-0.272884\pi$
$313$	2.26878e6	1.30898	0.654488	+	0.756072i	$0.272884\pi$
$314$	166667.	0.0953952
$315$	0	0
$316$	−6.27046e6	−3.53250
$317$	200774.	0.112217	0.0561086	−	0.998425i	$-0.482131\pi$
$317$	200774.	0.112217	0.0561086	+	0.998425i	$0.482131\pi$
$318$	2.24451e6	1.24467
$319$	−749991.	−0.412648
$320$	0	0
$321$	−1.91280e6	−1.03611
$322$	−1.93709e6	−1.04114
$323$	−5.86808e6	−3.12961
$324$	−1.24890e6	−0.660943
$325$	0	0
$326$	−842073.	−0.438840
$327$	−5.39072e6	−2.78790
$328$	2.83945e6	1.45730
$329$	1.17999e6	0.601017
$330$	0	0
$331$	−1.31065e6	−0.657530	−0.328765	−	0.944412i	$-0.606632\pi$
$331$	−1.31065e6	−0.657530	−0.328765	+	0.944412i	$0.606632\pi$
$332$	−5.53381e6	−2.75536
$333$	172491.	0.0852425
$334$	−2.17976e6	−1.06916
$335$	0	0
$336$	2.90365e6	1.40313
$337$	−1.35360e6	−0.649255	−0.324628	−	0.945842i	$-0.605239\pi$
$337$	−1.35360e6	−0.649255	−0.324628	+	0.945842i	$0.605239\pi$
$338$	288533.	0.137374
$339$	4.19986e6	1.98489
$340$	0	0
$341$	174085.	0.0810729
$342$	9.86272e6	4.55965
$343$	−2.04910e6	−0.940433
$344$	−3.90888e6	−1.78097
$345$	0	0
$346$	−428086.	−0.192239
$347$	231416.	0.103174	0.0515869	−	0.998669i	$-0.483572\pi$
$347$	231416.	0.103174	0.0515869	+	0.998669i	$0.483572\pi$
$348$	−1.01093e7	−4.47478
$349$	112062.	0.0492486	0.0246243	−	0.999697i	$-0.492161\pi$
$349$	112062.	0.0492486	0.0246243	+	0.999697i	$0.492161\pi$
$350$	0	0
$351$	−476955.	−0.206638
$352$	546273.	0.234992
$353$	1.64207e6	0.701383	0.350692	−	0.936491i	$-0.385947\pi$
$353$	1.64207e6	0.701383	0.350692	+	0.936491i	$0.385947\pi$
$354$	7.32785e6	3.10791
$355$	0	0
$356$	7.79400e6	3.25938
$357$	−3.80595e6	−1.58049
$358$	1.49220e6	0.615346
$359$	−1.14212e6	−0.467708	−0.233854	−	0.972272i	$-0.575134\pi$
$359$	−1.14212e6	−0.467708	−0.233854	+	0.972272i	$0.575134\pi$
$360$	0	0
$361$	4.95608e6	2.00157
$362$	−7.49686e6	−3.00682
$363$	−3.55046e6	−1.41422
$364$	−853860.	−0.337779
$365$	0	0
$366$	4.06450e6	1.58600
$367$	3.01075e6	1.16683	0.583417	−	0.812173i	$-0.301715\pi$
$367$	3.01075e6	1.16683	0.583417	+	0.812173i	$0.301715\pi$
$368$	−4.36617e6	−1.68067
$369$	2.64480e6	1.01118
$370$	0	0
$371$	653537.	0.246510
$372$	2.34653e6	0.879161
$373$	2.87230e6	1.06895	0.534476	−	0.845183i	$-0.320509\pi$
$373$	2.87230e6	1.06895	0.534476	+	0.845183i	$0.320509\pi$
$374$	−2.77093e6	−1.02435
$375$	0	0
$376$	6.29054e6	2.29466
$377$	994666.	0.360432
$378$	2.05618e6	0.740169
$379$	382184.	0.136670	0.0683352	−	0.997662i	$-0.478231\pi$
$379$	382184.	0.136670	0.0683352	+	0.997662i	$0.478231\pi$
$380$	0	0
$381$	2.42493e6	0.855830
$382$	7.05301e6	2.47295
$383$	789967.	0.275177	0.137588	−	0.990489i	$-0.456065\pi$
$383$	789967.	0.275177	0.137588	+	0.990489i	$0.456065\pi$
$384$	−5.65242e6	−1.95617
$385$	0	0
$386$	−1.92558e6	−0.657799
$387$	−3.64092e6	−1.23576
$388$	5.01285e6	1.69046
$389$	4.32009e6	1.44750	0.723751	−	0.690061i	$-0.242416\pi$
$389$	4.32009e6	1.44750	0.723751	+	0.690061i	$0.242416\pi$
$390$	0	0
$391$	5.72293e6	1.89311
$392$	−4.46208e6	−1.46664
$393$	6.53308e6	2.13372
$394$	8.25919e6	2.68038
$395$	0	0
$396$	3.19694e6	1.02446
$397$	3.54302e6	1.12823	0.564115	−	0.825696i	$-0.309217\pi$
$397$	3.54302e6	1.12823	0.564115	+	0.825696i	$0.309217\pi$
$398$	−2.91664e6	−0.922943
$399$	4.82039e6	1.51583
$400$	0	0
$401$	3.78071e6	1.17412	0.587060	−	0.809543i	$-0.300285\pi$
$401$	3.78071e6	1.17412	0.587060	+	0.809543i	$0.300285\pi$
$402$	1.52469e7	4.70561
$403$	−230878.	−0.0708141
$404$	1.05559e7	3.21768
$405$	0	0
$406$	−4.28805e6	−1.29105
$407$	61378.4	0.0183666
$408$	−2.02896e7	−6.03425
$409$	5.36133e6	1.58476	0.792381	−	0.610026i	$-0.208841\pi$
$409$	5.36133e6	1.58476	0.792381	+	0.610026i	$0.208841\pi$
$410$	0	0
$411$	−869212.	−0.253817
$412$	−1.12226e7	−3.25723
$413$	2.13366e6	0.615531
$414$	−9.61876e6	−2.75815
$415$	0	0
$416$	−724488.	−0.205257
$417$	−1.56173e6	−0.439810
$418$	3.50950e6	0.982437
$419$	−4.19124e6	−1.16629	−0.583146	−	0.812368i	$-0.698178\pi$
$419$	−4.19124e6	−1.16629	−0.583146	+	0.812368i	$0.698178\pi$
$420$	0	0
$421$	−1.98109e6	−0.544752	−0.272376	−	0.962191i	$-0.587809\pi$
$421$	−1.98109e6	−0.544752	−0.272376	+	0.962191i	$0.587809\pi$
$422$	1.36186e6	0.372264
$423$	5.85930e6	1.59219
$424$	3.48402e6	0.941166
$425$	0	0
$426$	−1.29526e6	−0.345805
$427$	1.18347e6	0.314113
$428$	−5.46570e6	−1.44224
$429$	−527994.	−0.138512
$430$	0	0
$431$	4.49072e6	1.16446	0.582228	−	0.813026i	$-0.302181\pi$
$431$	4.49072e6	1.16446	0.582228	+	0.813026i	$0.302181\pi$
$432$	4.63459e6	1.19482
$433$	1.75380e6	0.449531	0.224765	−	0.974413i	$-0.427838\pi$
$433$	1.75380e6	0.449531	0.224765	+	0.974413i	$0.427838\pi$
$434$	995325.	0.253653
$435$	0	0
$436$	−1.54036e7	−3.88066
$437$	−7.24834e6	−1.81566
$438$	−1.68056e7	−4.18571
$439$	−5.04733e6	−1.24997	−0.624987	−	0.780635i	$-0.714896\pi$
$439$	−5.04733e6	−1.24997	−0.624987	+	0.780635i	$0.714896\pi$
$440$	0	0
$441$	−4.15619e6	−1.01765
$442$	3.67491e6	0.894728
$443$	3.47296e6	0.840795	0.420398	−	0.907340i	$-0.361891\pi$
$443$	3.47296e6	0.840795	0.420398	+	0.907340i	$0.361891\pi$
$444$	827331.	0.199169
$445$	0	0
$446$	3.24255e6	0.771880
$447$	−652930.	−0.154560
$448$	−666513.	−0.156897
$449$	298830.	0.0699533	0.0349767	−	0.999388i	$-0.488864\pi$
$449$	298830.	0.0699533	0.0349767	+	0.999388i	$0.488864\pi$
$450$	0	0
$451$	941114.	0.217872
$452$	1.20008e7	2.76290
$453$	1.10033e7	2.51929
$454$	−290161.	−0.0660693
$455$	0	0
$456$	2.56977e7	5.78737
$457$	4.62298e6	1.03546	0.517728	−	0.855545i	$-0.326778\pi$
$457$	4.62298e6	1.03546	0.517728	+	0.855545i	$0.326778\pi$
$458$	8.59515e6	1.91465
$459$	−6.07476e6	−1.34585
$460$	0	0
$461$	1.73228e6	0.379634	0.189817	−	0.981820i	$-0.439211\pi$
$461$	1.73228e6	0.379634	0.189817	+	0.981820i	$0.439211\pi$
$462$	2.27621e6	0.496143
$463$	−3.98725e6	−0.864412	−0.432206	−	0.901775i	$-0.642265\pi$
$463$	−3.98725e6	−0.864412	−0.432206	+	0.901775i	$0.642265\pi$
$464$	−9.66520e6	−2.08409
$465$	0	0
$466$	4.36848e6	0.931892
$467$	4.73552e6	1.00479	0.502395	−	0.864638i	$-0.332452\pi$
$467$	4.73552e6	1.00479	0.502395	+	0.864638i	$0.332452\pi$
$468$	−4.23990e6	−0.894832
$469$	4.43946e6	0.931961
$470$	0	0
$471$	404488.	0.0840144
$472$	1.13746e7	2.35007
$473$	−1.29557e6	−0.266261
$474$	−2.21690e7	−4.53211
$475$	0	0
$476$	−1.08752e7	−2.19999
$477$	3.24519e6	0.653046
$478$	5.36402e6	1.07379
$479$	974320.	0.194027	0.0970137	−	0.995283i	$-0.469071\pi$
$479$	974320.	0.194027	0.0970137	+	0.995283i	$0.469071\pi$
$480$	0	0
$481$	−81402.3	−0.0160426
$482$	−7.48669e6	−1.46782
$483$	−4.70116e6	−0.916932
$484$	−1.01452e7	−1.96855
$485$	0	0
$486$	−1.13436e7	−2.17852
$487$	−834741.	−0.159489	−0.0797443	−	0.996815i	$-0.525410\pi$
$487$	−834741.	−0.159489	−0.0797443	+	0.996815i	$0.525410\pi$
$488$	6.30909e6	1.19927
$489$	−2.04364e6	−0.386485
$490$	0	0
$491$	829646.	0.155306	0.0776532	−	0.996980i	$-0.475257\pi$
$491$	829646.	0.155306	0.0776532	+	0.996980i	$0.475257\pi$
$492$	1.26855e7	2.36262
$493$	1.26686e7	2.34753
$494$	−4.65443e6	−0.858122
$495$	0	0
$496$	2.24345e6	0.409460
$497$	−377141.	−0.0684878
$498$	−1.95646e7	−3.53506
$499$	8.32784e6	1.49720	0.748602	−	0.663019i	$-0.230725\pi$
$499$	8.32784e6	1.49720	0.748602	+	0.663019i	$0.230725\pi$
$500$	0	0
$501$	−5.29009e6	−0.941605
$502$	1.10885e7	1.96387
$503$	6.37207e6	1.12295	0.561475	−	0.827493i	$-0.310234\pi$
$503$	6.37207e6	1.12295	0.561475	+	0.827493i	$0.310234\pi$
$504$	9.92939e6	1.74119
$505$	0	0
$506$	−3.42269e6	−0.594281
$507$	700246.	0.120985
$508$	6.92907e6	1.19129
$509$	3.38691e6	0.579441	0.289721	−	0.957111i	$-0.406438\pi$
$509$	3.38691e6	0.579441	0.289721	+	0.957111i	$0.406438\pi$
$510$	0	0
$511$	−4.89332e6	−0.828994
$512$	−1.31637e7	−2.21924
$513$	7.69395e6	1.29079
$514$	3.83804e6	0.640769
$515$	0	0
$516$	−1.74632e7	−2.88735
$517$	2.08495e6	0.343059
$518$	350929.	0.0574638
$519$	−1.03893e6	−0.169304
$520$	0	0
$521$	8.84049e6	1.42686	0.713431	−	0.700725i	$-0.247140\pi$
$521$	8.84049e6	1.42686	0.713431	+	0.700725i	$0.247140\pi$
$522$	−2.12926e7	−3.42021
$523$	975493.	0.155944	0.0779722	−	0.996956i	$-0.475155\pi$
$523$	975493.	0.155944	0.0779722	+	0.996956i	$0.475155\pi$
$524$	1.86678e7	2.97006
$525$	0	0
$526$	−2.64524e6	−0.416870
$527$	−2.94059e6	−0.461219
$528$	5.13054e6	0.800900
$529$	632708.	0.0983024
$530$	0	0
$531$	1.05948e7	1.63064
$532$	1.37739e7	2.10998
$533$	−1.24814e6	−0.190303
$534$	2.75554e7	4.18171
$535$	0	0
$536$	2.36669e7	3.55819
$537$	3.62145e6	0.541934
$538$	−9.01283e6	−1.34247
$539$	−1.47892e6	−0.219267
$540$	0	0
$541$	−2.36388e6	−0.347242	−0.173621	−	0.984813i	$-0.555547\pi$
$541$	−2.36388e6	−0.347242	−0.173621	+	0.984813i	$0.555547\pi$
$542$	−1.56584e7	−2.28954
$543$	−1.81943e7	−2.64811
$544$	−9.22747e6	−1.33686
$545$	0	0
$546$	−3.01879e6	−0.433362
$547$	4.75103e6	0.678922	0.339461	−	0.940620i	$-0.389755\pi$
$547$	4.75103e6	0.678922	0.339461	+	0.940620i	$0.389755\pi$
$548$	−2.48371e6	−0.353305
$549$	5.87659e6	0.832136
$550$	0	0
$551$	−1.60453e7	−2.25149
$552$	−2.50620e7	−3.50081
$553$	−6.45497e6	−0.897597
$554$	−8.06086e6	−1.11585
$555$	0	0
$556$	−4.46252e6	−0.612200
$557$	8.56354e6	1.16954	0.584770	−	0.811199i	$-0.301185\pi$
$557$	8.56354e6	1.16954	0.584770	+	0.811199i	$0.301185\pi$
$558$	4.94236e6	0.671968
$559$	1.71823e6	0.232569
$560$	0	0
$561$	−6.72482e6	−0.902139
$562$	1.18060e7	1.57675
$563$	−3.06139e6	−0.407049	−0.203525	−	0.979070i	$-0.565240\pi$
$563$	−3.06139e6	−0.407049	−0.203525	+	0.979070i	$0.565240\pi$
$564$	2.81034e7	3.72016
$565$	0	0
$566$	−1.10490e7	−1.44971
$567$	−1.28565e6	−0.167944
$568$	−2.01055e6	−0.261483
$569$	1.01121e6	0.130937	0.0654685	−	0.997855i	$-0.479146\pi$
$569$	1.01121e6	0.130937	0.0654685	+	0.997855i	$0.479146\pi$
$570$	0	0
$571$	4.63373e6	0.594759	0.297379	−	0.954759i	$-0.403887\pi$
$571$	4.63373e6	0.594759	0.297379	+	0.954759i	$0.403887\pi$
$572$	−1.50871e6	−0.192803
$573$	1.71171e7	2.17793
$574$	5.38078e6	0.681657
$575$	0	0
$576$	−3.30962e6	−0.415644
$577$	328589.	0.0410879	0.0205439	−	0.999789i	$-0.493460\pi$
$577$	328589.	0.0410879	0.0205439	+	0.999789i	$0.493460\pi$
$578$	3.24618e7	4.04159
$579$	−4.67323e6	−0.579323
$580$	0	0
$581$	−5.69664e6	−0.700130
$582$	1.77227e7	2.16882
$583$	1.15475e6	0.140707
$584$	−2.60864e7	−3.16506
$585$	0	0
$586$	−1.03094e7	−1.24019
$587$	−1.02498e7	−1.22778	−0.613888	−	0.789393i	$-0.710395\pi$
$587$	−1.02498e7	−1.22778	−0.613888	+	0.789393i	$0.710395\pi$
$588$	−1.99347e7	−2.37775
$589$	3.72438e6	0.442350
$590$	0	0
$591$	2.00444e7	2.36061
$592$	790988.	0.0927611
$593$	−2.82390e6	−0.329771	−0.164886	−	0.986313i	$-0.552725\pi$
$593$	−2.82390e6	−0.329771	−0.164886	+	0.986313i	$0.552725\pi$
$594$	3.63311e6	0.422486
$595$	0	0
$596$	−1.86570e6	−0.215143
$597$	−7.07844e6	−0.812834
$598$	4.53930e6	0.519082
$599$	−1.52502e7	−1.73664	−0.868318	−	0.496007i	$-0.834799\pi$
$599$	−1.52502e7	−1.73664	−0.868318	+	0.496007i	$0.834799\pi$
$600$	0	0
$601$	−6.06769e6	−0.685231	−0.342616	−	0.939476i	$-0.611313\pi$
$601$	−6.06769e6	−0.685231	−0.342616	+	0.939476i	$0.611313\pi$
$602$	−7.40735e6	−0.833052
$603$	2.20445e7	2.46891
$604$	3.14412e7	3.50677
$605$	0	0
$606$	3.73200e7	4.12820
$607$	2.77430e6	0.305620	0.152810	−	0.988256i	$-0.451168\pi$
$607$	2.77430e6	0.305620	0.152810	+	0.988256i	$0.451168\pi$
$608$	1.16870e7	1.28216
$609$	−1.04067e7	−1.13703
$610$	0	0
$611$	−2.76513e6	−0.299649
$612$	−5.40017e7	−5.82813
$613$	−694666.	−0.0746664	−0.0373332	−	0.999303i	$-0.511886\pi$
$613$	−694666.	−0.0746664	−0.0373332	+	0.999303i	$0.511886\pi$
$614$	−3.24554e7	−3.47428
$615$	0	0
$616$	3.53322e6	0.375163
$617$	6.78058e6	0.717057	0.358529	−	0.933519i	$-0.383279\pi$
$617$	6.78058e6	0.717057	0.358529	+	0.933519i	$0.383279\pi$
$618$	−3.96769e7	−4.17895
$619$	−2.60889e6	−0.273671	−0.136835	−	0.990594i	$-0.543693\pi$
$619$	−2.60889e6	−0.273671	−0.136835	+	0.990594i	$0.543693\pi$
$620$	0	0
$621$	−7.50363e6	−0.780805
$622$	−2.60514e7	−2.69994
$623$	8.02334e6	0.828200
$624$	−6.80431e6	−0.699556
$625$	0	0
$626$	2.29200e7	2.33765
$627$	8.51728e6	0.865231
$628$	1.15580e6	0.116945
$629$	−1.03678e6	−0.104487
$630$	0	0
$631$	1.81425e7	1.81395	0.906973	−	0.421190i	$-0.138387\pi$
$631$	1.81425e7	1.81395	0.906973	+	0.421190i	$0.138387\pi$
$632$	−3.44116e7	−3.42699
$633$	3.30512e6	0.327852
$634$	2.02829e6	0.200404
$635$	0	0
$636$	1.55651e7	1.52584
$637$	1.96140e6	0.191521
$638$	−7.57666e6	−0.736930
$639$	−1.87272e6	−0.181435
$640$	0	0
$641$	−2.04257e7	−1.96350	−0.981750	−	0.190174i	$-0.939095\pi$
$641$	−2.04257e7	−1.96350	−0.981750	+	0.190174i	$0.939095\pi$
$642$	−1.93238e7	−1.85035
$643$	−1.26717e7	−1.20867	−0.604335	−	0.796730i	$-0.706561\pi$
$643$	−1.26717e7	−1.20867	−0.604335	+	0.796730i	$0.706561\pi$
$644$	−1.34332e7	−1.27634
$645$	0	0
$646$	−5.92813e7	−5.58903
$647$	8.01621e6	0.752850	0.376425	−	0.926447i	$-0.377153\pi$
$647$	8.01621e6	0.752850	0.376425	+	0.926447i	$0.377153\pi$
$648$	−6.85382e6	−0.641202
$649$	3.77002e6	0.351343
$650$	0	0
$651$	2.41557e6	0.223392
$652$	−5.83957e6	−0.537975
$653$	−1.98662e7	−1.82319	−0.911597	−	0.411086i	$-0.865150\pi$
$653$	−1.98662e7	−1.82319	−0.911597	+	0.411086i	$0.865150\pi$
$654$	−5.44588e7	−4.97879
$655$	0	0
$656$	1.21282e7	1.10037
$657$	−2.42981e7	−2.19614
$658$	1.19206e7	1.07333
$659$	1.33714e6	0.119940	0.0599699	−	0.998200i	$-0.480900\pi$
$659$	1.33714e6	0.119940	0.0599699	+	0.998200i	$0.480900\pi$
$660$	0	0
$661$	−9.97249e6	−0.887769	−0.443884	−	0.896084i	$-0.646400\pi$
$661$	−9.97249e6	−0.887769	−0.443884	+	0.896084i	$0.646400\pi$
$662$	−1.32406e7	−1.17426
$663$	8.91871e6	0.787985
$664$	−3.03690e7	−2.67307
$665$	0	0
$666$	1.74256e6	0.152231
$667$	1.56484e7	1.36194
$668$	−1.51161e7	−1.31068
$669$	7.86941e6	0.679793
$670$	0	0
$671$	2.09110e6	0.179295
$672$	7.57999e6	0.647509
$673$	−1.31355e7	−1.11792	−0.558958	−	0.829196i	$-0.688799\pi$
$673$	−1.31355e7	−1.11792	−0.558958	+	0.829196i	$0.688799\pi$
$674$	−1.36745e7	−1.15948
$675$	0	0
$676$	2.00090e6	0.168407
$677$	1.58935e7	1.33275	0.666376	−	0.745616i	$-0.267845\pi$
$677$	1.58935e7	1.33275	0.666376	+	0.745616i	$0.267845\pi$
$678$	4.24284e7	3.54473
$679$	5.16035e6	0.429541
$680$	0	0
$681$	−704198.	−0.0581871
$682$	1.75867e6	0.144785
$683$	−1.90939e7	−1.56618	−0.783092	−	0.621905i	$-0.786359\pi$
$683$	−1.90939e7	−1.56618	−0.783092	+	0.621905i	$0.786359\pi$
$684$	6.83955e7	5.58968
$685$	0	0
$686$	−2.07007e7	−1.67948
$687$	2.08597e7	1.68623
$688$	−1.66961e7	−1.34475
$689$	−1.53147e6	−0.122903
$690$	0	0
$691$	2.12588e7	1.69373	0.846863	−	0.531810i	$-0.178488\pi$
$691$	2.12588e7	1.69373	0.846863	+	0.531810i	$0.178488\pi$
$692$	−2.96867e6	−0.235666
$693$	3.29101e6	0.260313
$694$	2.33784e6	0.184254
$695$	0	0
$696$	−5.54787e7	−4.34113
$697$	−1.58970e7	−1.23946
$698$	1.13209e6	0.0879510
$699$	1.06020e7	0.820716
$700$	0	0
$701$	9.34437e6	0.718216	0.359108	−	0.933296i	$-0.383081\pi$
$701$	9.34437e6	0.718216	0.359108	+	0.933296i	$0.383081\pi$
$702$	−4.81836e6	−0.369026
$703$	1.31313e6	0.100212
$704$	−1.17768e6	−0.0895560
$705$	0	0
$706$	1.65888e7	1.25257
$707$	1.08665e7	0.817602
$708$	5.08168e7	3.80999
$709$	2.84238e6	0.212357	0.106178	−	0.994347i	$-0.466139\pi$
$709$	2.84238e6	0.212357	0.106178	+	0.994347i	$0.466139\pi$
$710$	0	0
$711$	−3.20526e7	−2.37788
$712$	4.27727e7	3.16203
$713$	−3.63226e6	−0.267579
$714$	−3.84489e7	−2.82253
$715$	0	0
$716$	1.03480e7	0.754354
$717$	1.30180e7	0.945687
$718$	−1.15381e7	−0.835260
$719$	1.63490e6	0.117942	0.0589712	−	0.998260i	$-0.481218\pi$
$719$	1.63490e6	0.117942	0.0589712	+	0.998260i	$0.481218\pi$
$720$	0	0
$721$	−1.15528e7	−0.827653
$722$	5.00680e7	3.57451
$723$	−1.81696e7	−1.29270
$724$	−5.19889e7	−3.68607
$725$	0	0
$726$	−3.58679e7	−2.52560
$727$	−1.60243e7	−1.12446	−0.562230	−	0.826981i	$-0.690057\pi$
$727$	−1.60243e7	−1.12446	−0.562230	+	0.826981i	$0.690057\pi$
$728$	−4.68589e6	−0.327691
$729$	−2.31981e7	−1.61672
$730$	0	0
$731$	2.18843e7	1.51474
$732$	2.81863e7	1.94429
$733$	1.85018e7	1.27190	0.635951	−	0.771730i	$-0.280608\pi$
$733$	1.85018e7	1.27190	0.635951	+	0.771730i	$0.280608\pi$
$734$	3.04156e7	2.08380
$735$	0	0
$736$	−1.13979e7	−0.775587
$737$	7.84419e6	0.531961
$738$	2.67187e7	1.80582
$739$	8.93389e6	0.601769	0.300884	−	0.953661i	$-0.402718\pi$
$739$	8.93389e6	0.601769	0.300884	+	0.953661i	$0.402718\pi$
$740$	0	0
$741$	−1.12959e7	−0.755747
$742$	6.60225e6	0.440233
$743$	−1.81525e7	−1.20633	−0.603164	−	0.797617i	$-0.706093\pi$
$743$	−1.81525e7	−1.20633	−0.603164	+	0.797617i	$0.706093\pi$
$744$	1.28775e7	0.852901
$745$	0	0
$746$	2.90170e7	1.90900
$747$	−2.82871e7	−1.85476
$748$	−1.92157e7	−1.25575
$749$	−5.62653e6	−0.366468
$750$	0	0
$751$	−4.74200e6	−0.306805	−0.153402	−	0.988164i	$-0.549023\pi$
$751$	−4.74200e6	−0.306805	−0.153402	+	0.988164i	$0.549023\pi$
$752$	2.68689e7	1.73263
$753$	2.69108e7	1.72958
$754$	1.00484e7	0.643681
$755$	0	0
$756$	1.42591e7	0.907375
$757$	−1.70743e6	−0.108294	−0.0541469	−	0.998533i	$-0.517244\pi$
$757$	−1.70743e6	−0.108294	−0.0541469	+	0.998533i	$0.517244\pi$
$758$	3.86095e6	0.244074
$759$	−8.30660e6	−0.523382
$760$	0	0
$761$	1.53942e7	0.963596	0.481798	−	0.876282i	$-0.339984\pi$
$761$	1.53942e7	0.963596	0.481798	+	0.876282i	$0.339984\pi$
$762$	2.44975e7	1.52839
$763$	−1.58568e7	−0.986065
$764$	4.89109e7	3.03160
$765$	0	0
$766$	7.98051e6	0.491427
$767$	−4.99993e6	−0.306885
$768$	−4.98518e7	−3.04984
$769$	2.79442e7	1.70402	0.852012	−	0.523522i	$-0.175382\pi$
$769$	2.79442e7	1.70402	0.852012	+	0.523522i	$0.175382\pi$
$770$	0	0
$771$	9.31461e6	0.564325
$772$	−1.33534e7	−0.806398
$773$	−2.09934e6	−0.126367	−0.0631834	−	0.998002i	$-0.520125\pi$
$773$	−2.09934e6	−0.126367	−0.0631834	+	0.998002i	$0.520125\pi$
$774$	−3.67818e7	−2.20689
$775$	0	0
$776$	2.75100e7	1.63997
$777$	851676.	0.0506083
$778$	4.36430e7	2.58503
$779$	2.01342e7	1.18875
$780$	0	0
$781$	−666380.	−0.0390926
$782$	5.78150e7	3.38083
$783$	−1.66105e7	−0.968227
$784$	−1.90590e7	−1.10741
$785$	0	0
$786$	6.59994e7	3.81051
$787$	−2.97406e7	−1.71164	−0.855820	−	0.517274i	$-0.826947\pi$
$787$	−2.97406e7	−1.71164	−0.855820	+	0.517274i	$0.826947\pi$
$788$	5.72754e7	3.28589
$789$	−6.41979e6	−0.367137
$790$	0	0
$791$	1.23539e7	0.702044
$792$	1.75445e7	0.993866
$793$	−2.77329e6	−0.156607
$794$	3.57928e7	2.01486
$795$	0	0
$796$	−2.02261e7	−1.13144
$797$	−1.67730e7	−0.935329	−0.467664	−	0.883906i	$-0.654904\pi$
$797$	−1.67730e7	−0.935329	−0.467664	+	0.883906i	$0.654904\pi$
$798$	4.86972e7	2.70705
$799$	−3.52182e7	−1.95164
$800$	0	0
$801$	3.98405e7	2.19403
$802$	3.81940e7	2.09681
$803$	−8.64613e6	−0.473187
$804$	1.05733e8	5.76862
$805$	0	0
$806$	−2.33241e6	−0.126464
$807$	−2.18734e7	−1.18231
$808$	5.79297e7	3.12157
$809$	−2.01463e7	−1.08224	−0.541120	−	0.840945i	$-0.682000\pi$
$809$	−2.01463e7	−1.08224	−0.541120	+	0.840945i	$0.682000\pi$
$810$	0	0
$811$	3.28200e6	0.175221	0.0876105	−	0.996155i	$-0.472077\pi$
$811$	3.28200e6	0.175221	0.0876105	+	0.996155i	$0.472077\pi$
$812$	−2.97365e7	−1.58271
$813$	−3.80016e7	−2.01640
$814$	620065.	0.0328002
$815$	0	0
$816$	−8.66634e7	−4.55628
$817$	−2.77174e7	−1.45277
$818$	5.41619e7	2.83016
$819$	−4.36466e6	−0.227374
$820$	0	0
$821$	698069.	0.0361444	0.0180722	−	0.999837i	$-0.494247\pi$
$821$	698069.	0.0361444	0.0180722	+	0.999837i	$0.494247\pi$
$822$	−8.78107e6	−0.453282
$823$	−2.24014e7	−1.15286	−0.576428	−	0.817148i	$-0.695554\pi$
$823$	−2.24014e7	−1.15286	−0.576428	+	0.817148i	$0.695554\pi$
$824$	−6.15882e7	−3.15994
$825$	0	0
$826$	2.15549e7	1.09925
$827$	−1.30401e7	−0.663005	−0.331503	−	0.943454i	$-0.607556\pi$
$827$	−1.30401e7	−0.663005	−0.331503	+	0.943454i	$0.607556\pi$
$828$	−6.67037e7	−3.38123
$829$	−3.65981e7	−1.84958	−0.924788	−	0.380483i	$-0.875758\pi$
$829$	−3.65981e7	−1.84958	−0.924788	+	0.380483i	$0.875758\pi$
$830$	0	0
$831$	−1.95631e7	−0.982730
$832$	1.56188e6	0.0782239
$833$	2.49814e7	1.24740
$834$	−1.57771e7	−0.785438
$835$	0	0
$836$	2.43375e7	1.20437
$837$	3.85556e6	0.190228
$838$	−4.23413e7	−2.08283
$839$	2.56393e7	1.25748	0.628740	−	0.777616i	$-0.283571\pi$
$839$	2.56393e7	1.25748	0.628740	+	0.777616i	$0.283571\pi$
$840$	0	0
$841$	1.41291e7	0.688850
$842$	−2.00136e7	−0.972849
$843$	2.86522e7	1.38864
$844$	9.44414e6	0.456359
$845$	0	0
$846$	5.91927e7	2.84343
$847$	−1.04437e7	−0.500203
$848$	1.48814e7	0.710646
$849$	−2.68149e7	−1.27675
$850$	0	0
$851$	−1.28065e6	−0.0606187
$852$	−8.98227e6	−0.423923
$853$	−1.27542e7	−0.600178	−0.300089	−	0.953911i	$-0.597016\pi$
$853$	−1.27542e7	−0.600178	−0.300089	+	0.953911i	$0.597016\pi$
$854$	1.19558e7	0.560961
$855$	0	0
$856$	−2.99952e7	−1.39916
$857$	3.05245e7	1.41970	0.709850	−	0.704353i	$-0.248763\pi$
$857$	3.05245e7	1.41970	0.709850	+	0.704353i	$0.248763\pi$
$858$	−5.33398e6	−0.247362
$859$	3.68731e7	1.70501	0.852504	−	0.522721i	$-0.175083\pi$
$859$	3.68731e7	1.70501	0.852504	+	0.522721i	$0.175083\pi$
$860$	0	0
$861$	1.30587e7	0.600334
$862$	4.53668e7	2.07955
$863$	2.27261e7	1.03872	0.519360	−	0.854555i	$-0.326170\pi$
$863$	2.27261e7	1.03872	0.519360	+	0.854555i	$0.326170\pi$
$864$	1.20986e7	0.551380
$865$	0	0
$866$	1.77174e7	0.802798
$867$	7.87821e7	3.55942
$868$	6.90233e6	0.310954
$869$	−1.14055e7	−0.512346
$870$	0	0
$871$	−1.04033e7	−0.464648
$872$	−8.45333e7	−3.76475
$873$	2.56241e7	1.13792
$874$	−7.32252e7	−3.24251
$875$	0	0
$876$	−1.16543e8	−5.13128
$877$	471311.	0.0206923	0.0103461	−	0.999946i	$-0.496707\pi$
$877$	471311.	0.0206923	0.0103461	+	0.999946i	$0.496707\pi$
$878$	−5.09899e7	−2.23227
$879$	−2.50200e7	−1.09223
$880$	0	0
$881$	3.32241e7	1.44216	0.721080	−	0.692852i	$-0.243646\pi$
$881$	3.32241e7	1.44216	0.721080	+	0.692852i	$0.243646\pi$
$882$	−4.19873e7	−1.81738
$883$	−1.34686e6	−0.0581328	−0.0290664	−	0.999577i	$-0.509253\pi$
$883$	−1.34686e6	−0.0581328	−0.0290664	+	0.999577i	$0.509253\pi$
$884$	2.54846e7	1.09685
$885$	0	0
$886$	3.50850e7	1.50154
$887$	−4.53882e7	−1.93702	−0.968509	−	0.248980i	$-0.919905\pi$
$887$	−4.53882e7	−1.93702	−0.968509	+	0.248980i	$0.919905\pi$
$888$	4.54031e6	0.193220
$889$	7.13296e6	0.302702
$890$	0	0
$891$	−2.27164e6	−0.0958618
$892$	2.24863e7	0.946249
$893$	4.46054e7	1.87180
$894$	−6.59612e6	−0.276023
$895$	0	0
$896$	−1.66266e7	−0.691886
$897$	1.10165e7	0.457155
$898$	3.01888e6	0.124927
$899$	−8.04056e6	−0.331808
$900$	0	0
$901$	−1.95057e7	−0.800477
$902$	9.50744e6	0.389088
$903$	−1.79771e7	−0.733667
$904$	6.58592e7	2.68037
$905$	0	0
$906$	1.11159e8	4.49909
$907$	−5.01481e6	−0.202412	−0.101206	−	0.994865i	$-0.532270\pi$
$907$	−5.01481e6	−0.202412	−0.101206	+	0.994865i	$0.532270\pi$
$908$	−2.01219e6	−0.0809945
$909$	5.39585e7	2.16596
$910$	0	0
$911$	2.48880e7	0.993560	0.496780	−	0.867877i	$-0.334516\pi$
$911$	2.48880e7	0.993560	0.496780	+	0.867877i	$0.334516\pi$
$912$	1.09763e8	4.36987
$913$	−1.00655e7	−0.399632
$914$	4.67029e7	1.84918
$915$	0	0
$916$	5.96052e7	2.34717
$917$	1.92171e7	0.754684
$918$	−6.13693e7	−2.40350
$919$	−3.35944e7	−1.31213	−0.656067	−	0.754703i	$-0.727781\pi$
$919$	−3.35944e7	−1.31213	−0.656067	+	0.754703i	$0.727781\pi$
$920$	0	0
$921$	−7.87665e7	−3.05980
$922$	1.75000e7	0.677972
$923$	883778.	0.0341459
$924$	1.57849e7	0.608223
$925$	0	0
$926$	−4.02805e7	−1.54372
$927$	−5.73662e7	−2.19259
$928$	−2.52310e7	−0.961756
$929$	2.73986e7	1.04157	0.520785	−	0.853688i	$-0.325639\pi$
$929$	2.73986e7	1.04157	0.520785	+	0.853688i	$0.325639\pi$
$930$	0	0
$931$	−3.16400e7	−1.19636
$932$	3.02943e7	1.14241
$933$	−6.32246e7	−2.37784
$934$	4.78398e7	1.79441
$935$	0	0
$936$	−2.32681e7	−0.868105
$937$	−4.73279e7	−1.76104	−0.880519	−	0.474011i	$-0.842806\pi$
$937$	−4.73279e7	−1.76104	−0.880519	+	0.474011i	$0.842806\pi$
$938$	4.48489e7	1.66435
$939$	5.56250e7	2.05876
$940$	0	0
$941$	−2.81830e7	−1.03756	−0.518780	−	0.854908i	$-0.673614\pi$
$941$	−2.81830e7	−1.03756	−0.518780	+	0.854908i	$0.673614\pi$
$942$	4.08628e6	0.150038
$943$	−1.96362e7	−0.719081
$944$	4.85845e7	1.77447
$945$	0	0
$946$	−1.30882e7	−0.475503
$947$	1.35455e7	0.490817	0.245409	−	0.969420i	$-0.421078\pi$
$947$	1.35455e7	0.490817	0.245409	+	0.969420i	$0.421078\pi$
$948$	−1.53736e8	−5.55592
$949$	1.14668e7	0.413311
$950$	0	0
$951$	4.92249e6	0.176495
$952$	−5.96821e7	−2.13428
$953$	−2.85179e7	−1.01715	−0.508576	−	0.861017i	$-0.669828\pi$
$953$	−2.85179e7	−1.01715	−0.508576	+	0.861017i	$0.669828\pi$
$954$	3.27840e7	1.16625
$955$	0	0
$956$	3.71981e7	1.31637
$957$	−1.83879e7	−0.649013
$958$	9.84291e6	0.346505
$959$	−2.55680e6	−0.0897738
$960$	0	0
$961$	−2.67628e7	−0.934810
$962$	−822353.	−0.0286497
$963$	−2.79389e7	−0.970832
$964$	−5.19183e7	−1.79940
$965$	0	0
$966$	−4.74927e7	−1.63751
$967$	2.08677e7	0.717644	0.358822	−	0.933406i	$-0.383179\pi$
$967$	2.08677e7	0.717644	0.358822	+	0.933406i	$0.383179\pi$
$968$	−5.56757e7	−1.90975
$969$	−1.43871e8	−4.92225
$970$	0	0
$971$	−3.67948e7	−1.25239	−0.626194	−	0.779667i	$-0.715388\pi$
$971$	−3.67948e7	−1.25239	−0.626194	+	0.779667i	$0.715388\pi$
$972$	−7.86651e7	−2.67065
$973$	−4.59383e6	−0.155558
$974$	−8.43284e6	−0.284824
$975$	0	0
$976$	2.69481e7	0.905533
$977$	1.31397e7	0.440403	0.220202	−	0.975454i	$-0.429328\pi$
$977$	1.31397e7	0.440403	0.220202	+	0.975454i	$0.429328\pi$
$978$	−2.06456e7	−0.690208
$979$	1.41766e7	0.472734
$980$	0	0
$981$	−7.87383e7	−2.61224
$982$	8.38137e6	0.277355
$983$	−2.66889e7	−0.880941	−0.440470	−	0.897767i	$-0.645188\pi$
$983$	−2.66889e7	−0.880941	−0.440470	+	0.897767i	$0.645188\pi$
$984$	6.96164e7	2.29205
$985$	0	0
$986$	1.27982e8	4.19236
$987$	2.89304e7	0.945281
$988$	−3.22773e7	−1.05197
$989$	2.70318e7	0.878787
$990$	0	0
$991$	−2.39103e7	−0.773394	−0.386697	−	0.922207i	$-0.626384\pi$
$991$	−2.39103e7	−0.773394	−0.386697	+	0.922207i	$0.626384\pi$
$992$	5.85653e6	0.188956
$993$	−3.21339e7	−1.03416
$994$	−3.81001e6	−0.122309
$995$	0	0
$996$	−1.35675e8	−4.33364
$997$	−3.32793e7	−1.06032	−0.530160	−	0.847898i	$-0.677868\pi$
$997$	−3.32793e7	−1.06032	−0.530160	+	0.847898i	$0.677868\pi$
$998$	8.41307e7	2.67379
$999$	1.35938e6	0.0430950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.j.1.11	✓	11
5.2	odd	4		325.6.b.i.274.21		22
5.3	odd	4		325.6.b.i.274.2		22
5.4	even	2		325.6.a.k.1.1	yes	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.11	✓	11	1.1	even	1	trivial
325.6.a.k.1.1	yes	11	5.4	even	2
325.6.b.i.274.2		22	5.3	odd	4
325.6.b.i.274.21		22	5.2	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)

\(f(q)\)	\(=\)	\(q+10.1023 q^{2} +24.5176 q^{3} +70.0572 q^{4} +247.685 q^{6} +72.1186 q^{7} +384.466 q^{8} +358.110 q^{9} +127.428 q^{11} +1717.63 q^{12} -169.000 q^{13} +728.566 q^{14} +1642.18 q^{16} -2152.48 q^{17} +3617.75 q^{18} +2726.20 q^{19} +1768.17 q^{21} +1287.32 q^{22} -2658.77 q^{23} +9426.17 q^{24} -1707.29 q^{26} +2822.22 q^{27} +5052.43 q^{28} -5885.60 q^{29} +1366.14 q^{31} +4286.91 q^{32} +3124.23 q^{33} -21745.0 q^{34} +25088.2 q^{36} +481.670 q^{37} +27541.0 q^{38} -4143.47 q^{39} +7385.44 q^{41} +17862.7 q^{42} -10167.0 q^{43} +8927.26 q^{44} -26859.8 q^{46} +16361.7 q^{47} +40262.2 q^{48} -11605.9 q^{49} -52773.4 q^{51} -11839.7 q^{52} +9061.97 q^{53} +28511.0 q^{54} +27727.2 q^{56} +66839.8 q^{57} -59458.3 q^{58} +29585.4 q^{59} +16410.0 q^{61} +13801.2 q^{62} +25826.4 q^{63} -9241.89 q^{64} +31562.0 q^{66} +61557.7 q^{67} -150796. q^{68} -65186.5 q^{69} -5229.46 q^{71} +137681. q^{72} -67851.0 q^{73} +4865.99 q^{74} +190990. q^{76} +9189.95 q^{77} -41858.7 q^{78} -89505.0 q^{79} -17826.8 q^{81} +74610.2 q^{82} -78989.9 q^{83} +123873. q^{84} -102711. q^{86} -144300. q^{87} +48991.9 q^{88} +111252. q^{89} -12188.0 q^{91} -186266. q^{92} +33494.5 q^{93} +165292. q^{94} +105105. q^{96} +71553.7 q^{97} -117247. q^{98} +45633.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 5 q^{2} - 11 q^{3} + 187 q^{4} + 351 q^{6} - 208 q^{7} - 165 q^{8} + 1372 q^{9} + 1276 q^{11} - 1533 q^{12} - 1859 q^{13} + 578 q^{14} + 5707 q^{16} - 2218 q^{17} + 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 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

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