LMFDB - Embedded newform 325.6.a.i.1.7

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 181 x^{7} + 688 x^{6} + 10455 x^{5} - 37904 x^{4} - 197375 x^{3} + 702868 x^{2} + \cdots - 366960 $ x^9 - 4x^8 - 181x^7 + 688x^6 + 10455x^5 - 37904x^4 - 197375x^3 + 702868x^2 + 262812x - 366960
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-6.50844$ 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)

$f(q)$	$=$	\(q+7.50844 q^{2} +17.3551 q^{3} +24.3767 q^{4} +130.310 q^{6} -149.455 q^{7} -57.2388 q^{8} +58.2000 q^{9} -492.989 q^{11} +423.061 q^{12} -169.000 q^{13} -1122.18 q^{14} -1209.83 q^{16} +1824.24 q^{17} +436.991 q^{18} +424.035 q^{19} -2593.81 q^{21} -3701.58 q^{22} -3660.46 q^{23} -993.387 q^{24} -1268.93 q^{26} -3207.23 q^{27} -3643.23 q^{28} -8193.74 q^{29} +8113.24 q^{31} -7252.30 q^{32} -8555.87 q^{33} +13697.2 q^{34} +1418.73 q^{36} -6297.67 q^{37} +3183.84 q^{38} -2933.01 q^{39} +4298.36 q^{41} -19475.5 q^{42} +5822.12 q^{43} -12017.5 q^{44} -27484.4 q^{46} -8502.70 q^{47} -20996.7 q^{48} +5529.90 q^{49} +31660.0 q^{51} -4119.67 q^{52} +7049.74 q^{53} -24081.3 q^{54} +8554.65 q^{56} +7359.17 q^{57} -61522.2 q^{58} -21509.2 q^{59} +35743.1 q^{61} +60917.8 q^{62} -8698.30 q^{63} -15738.9 q^{64} -64241.3 q^{66} +15239.3 q^{67} +44469.1 q^{68} -63527.7 q^{69} +69104.1 q^{71} -3331.30 q^{72} -4134.60 q^{73} -47285.7 q^{74} +10336.6 q^{76} +73679.8 q^{77} -22022.4 q^{78} -16065.3 q^{79} -69804.4 q^{81} +32274.0 q^{82} -115739. q^{83} -63228.7 q^{84} +43715.0 q^{86} -142203. q^{87} +28218.1 q^{88} -35250.6 q^{89} +25258.0 q^{91} -89230.1 q^{92} +140806. q^{93} -63842.1 q^{94} -125864. q^{96} +32169.5 q^{97} +41520.9 q^{98} -28691.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 11 q^{3} + 91 q^{4} - 83 q^{6} - 12 q^{7} + 639 q^{8} + 562 q^{9} - 1422 q^{11} - 1567 q^{12} - 1521 q^{13} - 342 q^{14} - 1061 q^{16} - 648 q^{17} - 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) $ 9 * q + 5 * q^2 + 11 * q^3 + 91 * q^4 - 83 * q^6 - 12 * q^7 + 639 * q^8 + 562 * q^9 - 1422 * q^11 - 1567 * q^12 - 1521 * q^13 - 342 * q^14 - 1061 * q^16 - 648 * q^17 - 418 * q^18 - 408 * q^19 - 3912 * q^21 - 4345 * q^22 - 1839 * q^23 - 8469 * q^24 - 845 * q^26 + 7649 * q^27 + 2836 * q^28 - 8737 * q^29 + 748 * q^31 + 423 * q^32 + 356 * q^33 - 17789 * q^34 + 512 * q^36 + 15486 * q^37 + 3425 * q^38 - 1859 * q^39 - 28676 * q^41 - 6876 * q^42 - 28665 * q^43 - 30599 * q^44 - 12056 * q^46 + 29452 * q^47 - 64759 * q^48 - 40907 * q^49 - 31006 * q^51 - 15379 * q^52 - 75977 * q^53 - 102761 * q^54 - 23002 * q^56 + 38038 * q^57 - 142384 * q^58 - 88142 * q^59 + 28165 * q^61 + 137308 * q^62 - 41492 * q^63 - 100845 * q^64 + 42577 * q^66 + 94754 * q^67 - 89267 * q^68 - 181747 * q^69 - 70562 * q^71 + 263778 * q^72 - 60602 * q^73 - 135676 * q^74 + 46373 * q^76 + 140292 * q^77 + 14027 * q^78 - 164073 * q^79 - 69935 * q^81 + 72887 * q^82 + 22458 * q^83 - 345656 * q^84 - 294920 * q^86 + 87031 * q^87 - 430607 * q^88 - 252698 * q^89 + 2028 * q^91 + 237824 * q^92 - 56556 * q^93 - 501606 * q^94 - 319181 * q^96 - 137986 * q^97 - 378699 * q^98 - 757776 * 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 $

$2$	7.50844	1.32732	0.663659	−	0.748035i	$-0.269003\pi$
$2$	7.50844	1.32732	0.663659	+	0.748035i	$0.269003\pi$
$3$	17.3551	1.11333	0.556666	−	0.830737i	$-0.312081\pi$
$3$	17.3551	1.11333	0.556666	+	0.830737i	$0.312081\pi$
$4$	24.3767	0.761773
$5$	0	0
$5$	0	0
$6$	130.310	1.47774
$7$	−149.455	−1.15283	−0.576416	−	0.817156i	$-0.695549\pi$
$7$	−149.455	−1.15283	−0.576416	+	0.817156i	$0.695549\pi$
$8$	−57.2388	−0.316203
$9$	58.2000	0.239506
$10$	0	0
$11$	−492.989	−1.22844	−0.614222	−	0.789133i	$-0.710530\pi$
$11$	−492.989	−1.22844	−0.614222	+	0.789133i	$0.710530\pi$
$12$	423.061	0.848106
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−1122.18	−1.53018
$15$	0	0
$16$	−1209.83	−1.18147
$17$	1824.24	1.53095	0.765475	−	0.643466i	$-0.222504\pi$
$17$	1824.24	1.53095	0.765475	+	0.643466i	$0.222504\pi$
$18$	436.991	0.317901
$19$	424.035	0.269474	0.134737	−	0.990881i	$-0.456981\pi$
$19$	424.035	0.269474	0.134737	+	0.990881i	$0.456981\pi$
$20$	0	0
$21$	−2593.81	−1.28348
$22$	−3701.58	−1.63054
$23$	−3660.46	−1.44283	−0.721417	−	0.692501i	$-0.756509\pi$
$23$	−3660.46	−1.44283	−0.721417	+	0.692501i	$0.756509\pi$
$24$	−993.387	−0.352039
$25$	0	0
$26$	−1268.93	−0.368132
$27$	−3207.23	−0.846682
$28$	−3643.23	−0.878197
$29$	−8193.74	−1.80920	−0.904601	−	0.426259i	$-0.859831\pi$
$29$	−8193.74	−1.80920	−0.904601	+	0.426259i	$0.859831\pi$
$30$	0	0
$31$	8113.24	1.51632	0.758158	−	0.652070i	$-0.226099\pi$
$31$	8113.24	1.51632	0.758158	+	0.652070i	$0.226099\pi$
$32$	−7252.30	−1.25199
$33$	−8555.87	−1.36766
$34$	13697.2	2.03206
$35$	0	0
$36$	1418.73	0.182449
$37$	−6297.67	−0.756268	−0.378134	−	0.925751i	$-0.623434\pi$
$37$	−6297.67	−0.756268	−0.378134	+	0.925751i	$0.623434\pi$
$38$	3183.84	0.357678
$39$	−2933.01	−0.308782
$40$	0	0
$41$	4298.36	0.399341	0.199670	−	0.979863i	$-0.436013\pi$
$41$	4298.36	0.399341	0.199670	+	0.979863i	$0.436013\pi$
$42$	−19475.5	−1.70359
$43$	5822.12	0.480186	0.240093	−	0.970750i	$-0.422822\pi$
$43$	5822.12	0.480186	0.240093	+	0.970750i	$0.422822\pi$
$44$	−12017.5	−0.935795
$45$	0	0
$46$	−27484.4	−1.91510
$47$	−8502.70	−0.561452	−0.280726	−	0.959788i	$-0.590575\pi$
$47$	−8502.70	−0.561452	−0.280726	+	0.959788i	$0.590575\pi$
$48$	−20996.7	−1.31537
$49$	5529.90	0.329023
$50$	0	0
$51$	31660.0	1.70445
$52$	−4119.67	−0.211278
$53$	7049.74	0.344733	0.172367	−	0.985033i	$-0.444859\pi$
$53$	7049.74	0.344733	0.172367	+	0.985033i	$0.444859\pi$
$54$	−24081.3	−1.12382
$55$	0	0
$56$	8554.65	0.364529
$57$	7359.17	0.300014
$58$	−61522.2	−2.40139
$59$	−21509.2	−0.804440	−0.402220	−	0.915543i	$-0.631761\pi$
$59$	−21509.2	−0.804440	−0.402220	+	0.915543i	$0.631761\pi$
$60$	0	0
$61$	35743.1	1.22989	0.614947	−	0.788568i	$-0.289177\pi$
$61$	35743.1	1.22989	0.614947	+	0.788568i	$0.289177\pi$
$62$	60917.8	2.01263
$63$	−8698.30	−0.276110
$64$	−15738.9	−0.480314
$65$	0	0
$66$	−64241.3	−1.81533
$67$	15239.3	0.414742	0.207371	−	0.978262i	$-0.433509\pi$
$67$	15239.3	0.414742	0.207371	+	0.978262i	$0.433509\pi$
$68$	44469.1	1.16624
$69$	−63527.7	−1.60635
$70$	0	0
$71$	69104.1	1.62689	0.813444	−	0.581643i	$-0.197590\pi$
$71$	69104.1	1.62689	0.813444	+	0.581643i	$0.197590\pi$
$72$	−3331.30	−0.0757325
$73$	−4134.60	−0.0908085	−0.0454043	−	0.998969i	$-0.514458\pi$
$73$	−4134.60	−0.0908085	−0.0454043	+	0.998969i	$0.514458\pi$
$74$	−47285.7	−1.00381
$75$	0	0
$76$	10336.6	0.205278
$77$	73679.8	1.41619
$78$	−22022.4	−0.409853
$79$	−16065.3	−0.289615	−0.144808	−	0.989460i	$-0.546256\pi$
$79$	−16065.3	−0.289615	−0.144808	+	0.989460i	$0.546256\pi$
$80$	0	0
$81$	−69804.4	−1.18214
$82$	32274.0	0.530052
$83$	−115739.	−1.84409	−0.922047	−	0.387079i	$-0.873484\pi$
$83$	−115739.	−1.84409	−0.922047	+	0.387079i	$0.873484\pi$
$84$	−63228.7	−0.977724
$85$	0	0
$86$	43715.0	0.637360
$87$	−142203.	−2.01424
$88$	28218.1	0.388438
$89$	−35250.6	−0.471728	−0.235864	−	0.971786i	$-0.575792\pi$
$89$	−35250.6	−0.471728	−0.235864	+	0.971786i	$0.575792\pi$
$90$	0	0
$91$	25258.0	0.319738
$92$	−89230.1	−1.09911
$93$	140806.	1.68816
$94$	−63842.1	−0.745225
$95$	0	0
$96$	−125864.	−1.39388
$97$	32169.5	0.347148	0.173574	−	0.984821i	$-0.444468\pi$
$97$	32169.5	0.347148	0.173574	+	0.984821i	$0.444468\pi$
$98$	41520.9	0.436719
$99$	−28691.9	−0.294220
$100$	0	0
$101$	−38266.9	−0.373267	−0.186634	−	0.982430i	$-0.559758\pi$
$101$	−38266.9	−0.373267	−0.186634	+	0.982430i	$0.559758\pi$
$102$	237717.	2.26235
$103$	52058.1	0.483499	0.241750	−	0.970339i	$-0.422279\pi$
$103$	52058.1	0.483499	0.241750	+	0.970339i	$0.422279\pi$
$104$	9673.37	0.0876989
$105$	0	0
$106$	52932.6	0.457571
$107$	121233.	1.02367	0.511836	−	0.859083i	$-0.328966\pi$
$107$	121233.	1.02367	0.511836	+	0.859083i	$0.328966\pi$
$108$	−78181.7	−0.644979
$109$	140412.	1.13198	0.565989	−	0.824413i	$-0.308494\pi$
$109$	140412.	1.13198	0.565989	+	0.824413i	$0.308494\pi$
$110$	0	0
$111$	−109297.	−0.841977
$112$	180816.	1.36204
$113$	8725.69	0.0642841	0.0321421	−	0.999483i	$-0.489767\pi$
$113$	8725.69	0.0642841	0.0321421	+	0.999483i	$0.489767\pi$
$114$	55255.9	0.398214
$115$	0	0
$116$	−199737.	−1.37820
$117$	−9835.80	−0.0664270
$118$	−161500.	−1.06775
$119$	−272643.	−1.76493
$120$	0	0
$121$	81986.8	0.509074
$122$	268375.	1.63246
$123$	74598.6	0.444598
$124$	197774.	1.15509
$125$	0	0
$126$	−65310.7	−0.366486
$127$	−125413.	−0.689974	−0.344987	−	0.938607i	$-0.612117\pi$
$127$	−125413.	−0.689974	−0.344987	+	0.938607i	$0.612117\pi$
$128$	113899.	0.614461
$129$	101044.	0.534606
$130$	0	0
$131$	−392187.	−1.99671	−0.998354	−	0.0573546i	$-0.981733\pi$
$131$	−392187.	−1.99671	−0.998354	+	0.0573546i	$0.981733\pi$
$132$	−208564.	−1.04185
$133$	−63374.3	−0.310659
$134$	114423.	0.550494
$135$	0	0
$136$	−104418.	−0.484091
$137$	10040.7	0.0457051	0.0228525	−	0.999739i	$-0.492725\pi$
$137$	10040.7	0.0457051	0.0228525	+	0.999739i	$0.492725\pi$
$138$	−476994.	−2.13214
$139$	−413226.	−1.81406	−0.907028	−	0.421071i	$-0.861654\pi$
$139$	−413226.	−1.81406	−0.907028	+	0.421071i	$0.861654\pi$
$140$	0	0
$141$	−147565.	−0.625081
$142$	518864.	2.15940
$143$	83315.1	0.340709
$144$	−70412.1	−0.282970
$145$	0	0
$146$	−31044.4	−0.120532
$147$	95972.0	0.366312
$148$	−153517.	−0.576105
$149$	111303.	0.410716	0.205358	−	0.978687i	$-0.434164\pi$
$149$	111303.	0.410716	0.205358	+	0.978687i	$0.434164\pi$
$150$	0	0
$151$	329006.	1.17425	0.587127	−	0.809495i	$-0.300259\pi$
$151$	329006.	1.17425	0.587127	+	0.809495i	$0.300259\pi$
$152$	−24271.3	−0.0852086
$153$	106171.	0.366672
$154$	553221.	1.87973
$155$	0	0
$156$	−71497.3	−0.235222
$157$	324164.	1.04958	0.524790	−	0.851232i	$-0.324144\pi$
$157$	324164.	1.04958	0.524790	+	0.851232i	$0.324144\pi$
$158$	−120626.	−0.384412
$159$	122349.	0.383803
$160$	0	0
$161$	547076.	1.66335
$162$	−524122.	−1.56908
$163$	−471803.	−1.39089	−0.695443	−	0.718581i	$-0.744792\pi$
$163$	−471803.	−1.39089	−0.695443	+	0.718581i	$0.744792\pi$
$164$	104780.	0.304207
$165$	0	0
$166$	−869017.	−2.44770
$167$	318786.	0.884522	0.442261	−	0.896886i	$-0.354177\pi$
$167$	318786.	0.884522	0.442261	+	0.896886i	$0.354177\pi$
$168$	148467.	0.405842
$169$	28561.0	0.0769231
$170$	0	0
$171$	24678.8	0.0645407
$172$	141924.	0.365793
$173$	−340132.	−0.864036	−0.432018	−	0.901865i	$-0.642198\pi$
$173$	−340132.	−0.864036	−0.432018	+	0.901865i	$0.642198\pi$
$174$	−1.06773e6	−2.67354
$175$	0	0
$176$	596433.	1.45138
$177$	−373294.	−0.895608
$178$	−264677.	−0.626134
$179$	−749176.	−1.74764	−0.873819	−	0.486251i	$-0.838364\pi$
$179$	−749176.	−1.74764	−0.873819	+	0.486251i	$0.838364\pi$
$180$	0	0
$181$	−496170.	−1.12573	−0.562865	−	0.826549i	$-0.690301\pi$
$181$	−496170.	−1.12573	−0.562865	+	0.826549i	$0.690301\pi$
$182$	189648.	0.424394
$183$	620326.	1.36928
$184$	209521.	0.456228
$185$	0	0
$186$	1.05724e6	2.24073
$187$	−899332.	−1.88068
$188$	−207268.	−0.427699
$189$	479337.	0.976082
$190$	0	0
$191$	784516.	1.55603	0.778015	−	0.628245i	$-0.216227\pi$
$191$	784516.	1.55603	0.778015	+	0.628245i	$0.216227\pi$
$192$	−273151.	−0.534748
$193$	394155.	0.761683	0.380842	−	0.924640i	$-0.375634\pi$
$193$	394155.	0.761683	0.380842	+	0.924640i	$0.375634\pi$
$194$	241543.	0.460776
$195$	0	0
$196$	134801.	0.250641
$197$	−913374.	−1.67681	−0.838403	−	0.545051i	$-0.816510\pi$
$197$	−913374.	−1.67681	−0.838403	+	0.545051i	$0.816510\pi$
$198$	−215432.	−0.390523
$199$	−340138.	−0.608866	−0.304433	−	0.952534i	$-0.598467\pi$
$199$	−340138.	−0.608866	−0.304433	+	0.952534i	$0.598467\pi$
$200$	0	0
$201$	264480.	0.461745
$202$	−287325.	−0.495444
$203$	1.22460e6	2.08571
$204$	771767.	1.29841
$205$	0	0
$206$	390876.	0.641757
$207$	−213039.	−0.345567
$208$	204461.	0.327682
$209$	−209044.	−0.331034
$210$	0	0
$211$	−582499.	−0.900719	−0.450360	−	0.892847i	$-0.648704\pi$
$211$	−582499.	−0.900719	−0.450360	+	0.892847i	$0.648704\pi$
$212$	171850.	0.262609
$213$	1.19931e6	1.81127
$214$	910269.	1.35874
$215$	0	0
$216$	183578.	0.267723
$217$	−1.21257e6	−1.74806
$218$	1.05428e6	1.50249
$219$	−71756.5	−0.101100
$220$	0	0
$221$	−308297.	−0.424609
$222$	−820649.	−1.11757
$223$	580068.	0.781119	0.390559	−	0.920578i	$-0.372282\pi$
$223$	580068.	0.781119	0.390559	+	0.920578i	$0.372282\pi$
$224$	1.08389e6	1.44333
$225$	0	0
$226$	65516.4	0.0853255
$227$	833624.	1.07376	0.536878	−	0.843660i	$-0.319604\pi$
$227$	833624.	1.07376	0.536878	+	0.843660i	$0.319604\pi$
$228$	179393.	0.228543
$229$	−10222.2	−0.0128811	−0.00644057	−	0.999979i	$-0.502050\pi$
$229$	−10222.2	−0.0128811	−0.00644057	+	0.999979i	$0.502050\pi$
$230$	0	0
$231$	1.27872e6	1.57669
$232$	469000.	0.572075
$233$	−932235.	−1.12496	−0.562478	−	0.826812i	$-0.690152\pi$
$233$	−932235.	−1.12496	−0.562478	+	0.826812i	$0.690152\pi$
$234$	−73851.5	−0.0881698
$235$	0	0
$236$	−524324.	−0.612801
$237$	−278815.	−0.322438
$238$	−2.04713e6	−2.34262
$239$	−1.35733e6	−1.53706	−0.768530	−	0.639814i	$-0.779012\pi$
$239$	−1.35733e6	−1.53706	−0.768530	+	0.639814i	$0.779012\pi$
$240$	0	0
$241$	59546.1	0.0660406	0.0330203	−	0.999455i	$-0.489487\pi$
$241$	59546.1	0.0660406	0.0330203	+	0.999455i	$0.489487\pi$
$242$	615594.	0.675703
$243$	−432107.	−0.469435
$244$	871301.	0.936901
$245$	0	0
$246$	560119.	0.590123
$247$	−71661.9	−0.0747387
$248$	−464392.	−0.479464
$249$	−2.00866e6	−2.05309
$250$	0	0
$251$	−1.20903e6	−1.21130	−0.605651	−	0.795731i	$-0.707087\pi$
$251$	−1.20903e6	−1.21130	−0.605651	+	0.795731i	$0.707087\pi$
$252$	−212036.	−0.210333
$253$	1.80457e6	1.77244
$254$	−941656.	−0.915815
$255$	0	0
$256$	1.35885e6	1.29590
$257$	1.91933e6	1.81266	0.906329	−	0.422573i	$-0.138873\pi$
$257$	1.91933e6	1.81266	0.906329	+	0.422573i	$0.138873\pi$
$258$	758679.	0.709593
$259$	941221.	0.871851
$260$	0	0
$261$	−476875.	−0.433315
$262$	−2.94471e6	−2.65027
$263$	637187.	0.568038	0.284019	−	0.958819i	$-0.408332\pi$
$263$	637187.	0.568038	0.284019	+	0.958819i	$0.408332\pi$
$264$	489728.	0.432460
$265$	0	0
$266$	−475842.	−0.412343
$267$	−611779.	−0.525190
$268$	371484.	0.315939
$269$	250073.	0.210711	0.105355	−	0.994435i	$-0.466402\pi$
$269$	250073.	0.210711	0.105355	+	0.994435i	$0.466402\pi$
$270$	0	0
$271$	994132.	0.822282	0.411141	−	0.911572i	$-0.365130\pi$
$271$	994132.	0.822282	0.411141	+	0.911572i	$0.365130\pi$
$272$	−2.20703e6	−1.80878
$273$	438355.	0.355975
$274$	75390.4	0.0606652
$275$	0	0
$276$	−1.54860e6	−1.22368
$277$	1.80516e6	1.41356	0.706782	−	0.707432i	$-0.250146\pi$
$277$	1.80516e6	1.41356	0.706782	+	0.707432i	$0.250146\pi$
$278$	−3.10268e6	−2.40783
$279$	472190.	0.363167
$280$	0	0
$281$	−349545.	−0.264081	−0.132040	−	0.991244i	$-0.542153\pi$
$281$	−349545.	−0.264081	−0.132040	+	0.991244i	$0.542153\pi$
$282$	−1.10799e6	−0.829682
$283$	−204139.	−0.151516	−0.0757582	−	0.997126i	$-0.524138\pi$
$283$	−204139.	−0.151516	−0.0757582	+	0.997126i	$0.524138\pi$
$284$	1.68453e6	1.23932
$285$	0	0
$286$	625567.	0.452229
$287$	−642413.	−0.460373
$288$	−422084.	−0.299859
$289$	1.90801e6	1.34381
$290$	0	0
$291$	558305.	0.386491
$292$	−100788.	−0.0691755
$293$	78795.6	0.0536207	0.0268104	−	0.999641i	$-0.491465\pi$
$293$	78795.6	0.0536207	0.0268104	+	0.999641i	$0.491465\pi$
$294$	720600.	0.486212
$295$	0	0
$296$	360471.	0.239134
$297$	1.58113e6	1.04010
$298$	835713.	0.545150
$299$	618618.	0.400170
$300$	0	0
$301$	−870146.	−0.553575
$302$	2.47033e6	1.55861
$303$	−664126.	−0.415570
$304$	−513010.	−0.318377
$305$	0	0
$306$	797179.	0.486690
$307$	−2.05653e6	−1.24534	−0.622672	−	0.782483i	$-0.713953\pi$
$307$	−2.05653e6	−1.24534	−0.622672	+	0.782483i	$0.713953\pi$
$308$	1.79607e6	1.07882
$309$	903475.	0.538295
$310$	0	0
$311$	−3119.84	−0.00182908	−0.000914538	−	1.00000i	$-0.500291\pi$
$311$	−3119.84	−0.00182908	−0.000914538		1.00000i	$0.500291\pi$
$312$	167882.	0.0976379
$313$	−984315.	−0.567902	−0.283951	−	0.958839i	$-0.591645\pi$
$313$	−984315.	−0.567902	−0.283951	+	0.958839i	$0.591645\pi$
$314$	2.43397e6	1.39313
$315$	0	0
$316$	−391620.	−0.220621
$317$	1.90110e6	1.06257	0.531283	−	0.847194i	$-0.321710\pi$
$317$	1.90110e6	1.06257	0.531283	+	0.847194i	$0.321710\pi$
$318$	918651.	0.509428
$319$	4.03942e6	2.22250
$320$	0	0
$321$	2.10401e6	1.13969
$322$	4.10769e6	2.20779
$323$	773543.	0.412552
$324$	−1.70160e6	−0.900525
$325$	0	0
$326$	−3.54251e6	−1.84615
$327$	2.43687e6	1.26027
$328$	−246033.	−0.126273
$329$	1.27077e6	0.647260
$330$	0	0
$331$	−1.74888e6	−0.877386	−0.438693	−	0.898637i	$-0.644558\pi$
$331$	−1.74888e6	−0.877386	−0.438693	+	0.898637i	$0.644558\pi$
$332$	−2.82133e6	−1.40478
$333$	−366524.	−0.181131
$334$	2.39359e6	1.17404
$335$	0	0
$336$	3.13808e6	1.51640
$337$	1.09525e6	0.525338	0.262669	−	0.964886i	$-0.415397\pi$
$337$	1.09525e6	0.525338	0.262669	+	0.964886i	$0.415397\pi$
$338$	214449.	0.102101
$339$	151435.	0.0715695
$340$	0	0
$341$	−3.99973e6	−1.86271
$342$	185300.	0.0856661
$343$	1.68542e6	0.773524
$344$	−333251.	−0.151836
$345$	0	0
$346$	−2.55386e6	−1.14685
$347$	1.99055e6	0.887461	0.443730	−	0.896160i	$-0.353655\pi$
$347$	1.99055e6	0.887461	0.443730	+	0.896160i	$0.353655\pi$
$348$	−3.46645e6	−1.53439
$349$	−846086.	−0.371835	−0.185918	−	0.982565i	$-0.559526\pi$
$349$	−846086.	−0.371835	−0.185918	+	0.982565i	$0.559526\pi$
$350$	0	0
$351$	542021.	0.234827
$352$	3.57530e6	1.53800
$353$	2.61806e6	1.11826	0.559131	−	0.829079i	$-0.311135\pi$
$353$	2.61806e6	1.11826	0.559131	+	0.829079i	$0.311135\pi$
$354$	−2.80286e6	−1.18876
$355$	0	0
$356$	−859296.	−0.359350
$357$	−4.73175e6	−1.96495
$358$	−5.62515e6	−2.31967
$359$	−1.63224e6	−0.668418	−0.334209	−	0.942499i	$-0.608469\pi$
$359$	−1.63224e6	−0.668418	−0.334209	+	0.942499i	$0.608469\pi$
$360$	0	0
$361$	−2.29629e6	−0.927384
$362$	−3.72546e6	−1.49420
$363$	1.42289e6	0.566768
$364$	615706.	0.243568
$365$	0	0
$366$	4.65768e6	1.81747
$367$	−3.09644e6	−1.20005	−0.600023	−	0.799983i	$-0.704842\pi$
$367$	−3.09644e6	−1.20005	−0.600023	+	0.799983i	$0.704842\pi$
$368$	4.42854e6	1.70467
$369$	250165.	0.0956445
$370$	0	0
$371$	−1.05362e6	−0.397420
$372$	3.43239e6	1.28600
$373$	−2.44663e6	−0.910533	−0.455266	−	0.890355i	$-0.650456\pi$
$373$	−2.44663e6	−0.910533	−0.455266	+	0.890355i	$0.650456\pi$
$374$	−6.75258e6	−2.49627
$375$	0	0
$376$	486685.	0.177533
$377$	1.38474e6	0.501782
$378$	3.59907e6	1.29557
$379$	−4.31307e6	−1.54237	−0.771184	−	0.636612i	$-0.780335\pi$
$379$	−4.31307e6	−1.54237	−0.771184	+	0.636612i	$0.780335\pi$
$380$	0	0
$381$	−2.17656e6	−0.768170
$382$	5.89049e6	2.06535
$383$	4.18340e6	1.45724	0.728622	−	0.684916i	$-0.240161\pi$
$383$	4.18340e6	1.45724	0.728622	+	0.684916i	$0.240161\pi$
$384$	1.97673e6	0.684098
$385$	0	0
$386$	2.95949e6	1.01100
$387$	338847.	0.115008
$388$	784187.	0.264448
$389$	−3.01512e6	−1.01025	−0.505126	−	0.863045i	$-0.668554\pi$
$389$	−3.01512e6	−1.01025	−0.505126	+	0.863045i	$0.668554\pi$
$390$	0	0
$391$	−6.67758e6	−2.20891
$392$	−316525.	−0.104038
$393$	−6.80644e6	−2.22300
$394$	−6.85801e6	−2.22566
$395$	0	0
$396$	−699416.	−0.224129
$397$	−183544.	−0.0584471	−0.0292236	−	0.999573i	$-0.509303\pi$
$397$	−183544.	−0.0584471	−0.0292236	+	0.999573i	$0.509303\pi$
$398$	−2.55390e6	−0.808159
$399$	−1.09987e6	−0.345866
$400$	0	0
$401$	−3.10417e6	−0.964015	−0.482008	−	0.876167i	$-0.660092\pi$
$401$	−3.10417e6	−0.964015	−0.482008	+	0.876167i	$0.660092\pi$
$402$	1.98583e6	0.612882
$403$	−1.37114e6	−0.420551
$404$	−932822.	−0.284345
$405$	0	0
$406$	9.19483e6	2.76840
$407$	3.10468e6	0.929033
$408$	−1.81218e6	−0.538953
$409$	2.45114e6	0.724535	0.362268	−	0.932074i	$-0.382003\pi$
$409$	2.45114e6	0.724535	0.362268	+	0.932074i	$0.382003\pi$
$410$	0	0
$411$	174258.	0.0508849
$412$	1.26901e6	0.368317
$413$	3.21466e6	0.927385
$414$	−1.59959e6	−0.458678
$415$	0	0
$416$	1.22564e6	0.347240
$417$	−7.17158e6	−2.01964
$418$	−1.56960e6	−0.439387
$419$	−1.38177e6	−0.384503	−0.192251	−	0.981346i	$-0.561579\pi$
$419$	−1.38177e6	−0.384503	−0.192251	+	0.981346i	$0.561579\pi$
$420$	0	0
$421$	−4.38636e6	−1.20614	−0.603071	−	0.797687i	$-0.706057\pi$
$421$	−4.38636e6	−1.20614	−0.603071	+	0.797687i	$0.706057\pi$
$422$	−4.37366e6	−1.19554
$423$	−494857.	−0.134471
$424$	−403519.	−0.109006
$425$	0	0
$426$	9.00495e6	2.40413
$427$	−5.34200e6	−1.41786
$428$	2.95526e6	0.779805
$429$	1.44594e6	0.379322
$430$	0	0
$431$	1.36275e6	0.353365	0.176682	−	0.984268i	$-0.443463\pi$
$431$	1.36275e6	0.353365	0.176682	+	0.984268i	$0.443463\pi$
$432$	3.88020e6	1.00033
$433$	−5.59344e6	−1.43370	−0.716851	−	0.697226i	$-0.754417\pi$
$433$	−5.59344e6	−1.43370	−0.716851	+	0.697226i	$0.754417\pi$
$434$	−9.10449e6	−2.32023
$435$	0	0
$436$	3.42279e6	0.862310
$437$	−1.55216e6	−0.388807
$438$	−538780.	−0.134192
$439$	−4.86469e6	−1.20474	−0.602371	−	0.798216i	$-0.705777\pi$
$439$	−4.86469e6	−1.20474	−0.602371	+	0.798216i	$0.705777\pi$
$440$	0	0
$441$	321840.	0.0788031
$442$	−2.31483e6	−0.563591
$443$	−4.59200e6	−1.11171	−0.555857	−	0.831278i	$-0.687610\pi$
$443$	−4.59200e6	−1.11171	−0.555857	+	0.831278i	$0.687610\pi$
$444$	−2.66430e6	−0.641395
$445$	0	0
$446$	4.35541e6	1.03679
$447$	1.93168e6	0.457263
$448$	2.35227e6	0.553722
$449$	−351998.	−0.0823994	−0.0411997	−	0.999151i	$-0.513118\pi$
$449$	−351998.	−0.0823994	−0.0411997	+	0.999151i	$0.513118\pi$
$450$	0	0
$451$	−2.11904e6	−0.490567
$452$	212704.	0.0489699
$453$	5.70994e6	1.30733
$454$	6.25922e6	1.42521
$455$	0	0
$456$	−421231.	−0.0948654
$457$	546819.	0.122477	0.0612383	−	0.998123i	$-0.480495\pi$
$457$	546819.	0.122477	0.0612383	+	0.998123i	$0.480495\pi$
$458$	−76752.6	−0.0170974
$459$	−5.85076e6	−1.29623
$460$	0	0
$461$	−6.35474e6	−1.39266	−0.696331	−	0.717721i	$-0.745185\pi$
$461$	−6.35474e6	−1.39266	−0.696331	+	0.717721i	$0.745185\pi$
$462$	9.60121e6	2.09277
$463$	2.90818e6	0.630476	0.315238	−	0.949013i	$-0.397916\pi$
$463$	2.90818e6	0.630476	0.315238	+	0.949013i	$0.397916\pi$
$464$	9.91303e6	2.13753
$465$	0	0
$466$	−6.99963e6	−1.49317
$467$	−3.33677e6	−0.708002	−0.354001	−	0.935245i	$-0.615179\pi$
$467$	−3.33677e6	−0.708002	−0.354001	+	0.935245i	$0.615179\pi$
$468$	−239765.	−0.0506023
$469$	−2.27759e6	−0.478128
$470$	0	0
$471$	5.62590e6	1.16853
$472$	1.23116e6	0.254366
$473$	−2.87024e6	−0.589882
$474$	−2.09347e6	−0.427978
$475$	0	0
$476$	−6.64615e6	−1.34447
$477$	410295.	0.0825658
$478$	−1.01914e7	−2.04017
$479$	1.93937e6	0.386208	0.193104	−	0.981178i	$-0.438144\pi$
$479$	1.93937e6	0.386208	0.193104	+	0.981178i	$0.438144\pi$
$480$	0	0
$481$	1.06431e6	0.209751
$482$	447099.	0.0876568
$483$	9.49456e6	1.85185
$484$	1.99857e6	0.387799
$485$	0	0
$486$	−3.24445e6	−0.623089
$487$	−5.88271e6	−1.12397	−0.561985	−	0.827147i	$-0.689962\pi$
$487$	−5.88271e6	−1.12397	−0.561985	+	0.827147i	$0.689962\pi$
$488$	−2.04590e6	−0.388896
$489$	−8.18820e6	−1.54852
$490$	0	0
$491$	8.39552e6	1.57161	0.785804	−	0.618476i	$-0.212250\pi$
$491$	8.39552e6	1.57161	0.785804	+	0.618476i	$0.212250\pi$
$492$	1.81847e6	0.338683
$493$	−1.49474e7	−2.76980
$494$	−538069.	−0.0992021
$495$	0	0
$496$	−9.81564e6	−1.79149
$497$	−1.03280e7	−1.87553
$498$	−1.50819e7	−2.72510
$499$	8.24513e6	1.48234	0.741168	−	0.671320i	$-0.234272\pi$
$499$	8.24513e6	1.48234	0.741168	+	0.671320i	$0.234272\pi$
$500$	0	0
$501$	5.53257e6	0.984766
$502$	−9.07792e6	−1.60778
$503$	8.99706e6	1.58555	0.792776	−	0.609513i	$-0.208635\pi$
$503$	8.99706e6	1.58555	0.792776	+	0.609513i	$0.208635\pi$
$504$	497880.	0.0873069
$505$	0	0
$506$	1.35495e7	2.35259
$507$	495679.	0.0856409
$508$	−3.05716e6	−0.525604
$509$	6.27444e6	1.07345	0.536723	−	0.843758i	$-0.319662\pi$
$509$	6.27444e6	1.07345	0.536723	+	0.843758i	$0.319662\pi$
$510$	0	0
$511$	617938.	0.104687
$512$	6.55808e6	1.10561
$513$	−1.35998e6	−0.228159
$514$	1.44111e7	2.40597
$515$	0	0
$516$	2.46311e6	0.407249
$517$	4.19174e6	0.689712
$518$	7.06710e6	1.15722
$519$	−5.90302e6	−0.961958
$520$	0	0
$521$	−8.31964e6	−1.34280	−0.671398	−	0.741097i	$-0.734306\pi$
$521$	−8.31964e6	−1.34280	−0.671398	+	0.741097i	$0.734306\pi$
$522$	−3.58059e6	−0.575147
$523$	−8.50917e6	−1.36029	−0.680147	−	0.733076i	$-0.738084\pi$
$523$	−8.50917e6	−1.36029	−0.680147	+	0.733076i	$0.738084\pi$
$524$	−9.56023e6	−1.52104
$525$	0	0
$526$	4.78428e6	0.753967
$527$	1.48005e7	2.32140
$528$	1.03512e7	1.61586
$529$	6.96264e6	1.08177
$530$	0	0
$531$	−1.25183e6	−0.192668
$532$	−1.54486e6	−0.236652
$533$	−726423.	−0.110757
$534$	−4.59351e6	−0.697094
$535$	0	0
$536$	−872280.	−0.131143
$537$	−1.30020e7	−1.94570
$538$	1.87766e6	0.279680
$539$	−2.72618e6	−0.404187
$540$	0	0
$541$	−1.15207e6	−0.169233	−0.0846164	−	0.996414i	$-0.526966\pi$
$541$	−1.15207e6	−0.169233	−0.0846164	+	0.996414i	$0.526966\pi$
$542$	7.46438e6	1.09143
$543$	−8.61109e6	−1.25331
$544$	−1.32300e7	−1.91673
$545$	0	0
$546$	3.29136e6	0.472491
$547$	−943671.	−0.134850	−0.0674252	−	0.997724i	$-0.521478\pi$
$547$	−943671.	−0.134850	−0.0674252	+	0.997724i	$0.521478\pi$
$548$	244761.	0.0348169
$549$	2.08025e6	0.294567
$550$	0	0
$551$	−3.47443e6	−0.487534
$552$	3.63625e6	0.507933
$553$	2.40105e6	0.333878
$554$	1.35539e7	1.87625
$555$	0	0
$556$	−1.00731e7	−1.38190
$557$	−1.19731e7	−1.63519	−0.817597	−	0.575791i	$-0.804694\pi$
$557$	−1.19731e7	−1.63519	−0.817597	+	0.575791i	$0.804694\pi$
$558$	3.54541e6	0.482038
$559$	−983938.	−0.133180
$560$	0	0
$561$	−1.56080e7	−2.09382
$562$	−2.62454e6	−0.350519
$563$	−3.19566e6	−0.424903	−0.212452	−	0.977172i	$-0.568145\pi$
$563$	−3.19566e6	−0.424903	−0.212452	+	0.977172i	$0.568145\pi$
$564$	−3.59716e6	−0.476170
$565$	0	0
$566$	−1.53276e6	−0.201110
$567$	1.04326e7	1.36281
$568$	−3.95544e6	−0.514427
$569$	3.18417e6	0.412303	0.206151	−	0.978520i	$-0.433906\pi$
$569$	3.18417e6	0.412303	0.206151	+	0.978520i	$0.433906\pi$
$570$	0	0
$571$	−6.71439e6	−0.861820	−0.430910	−	0.902395i	$-0.641807\pi$
$571$	−6.71439e6	−0.861820	−0.430910	+	0.902395i	$0.641807\pi$
$572$	2.03095e6	0.259543
$573$	1.36154e7	1.73238
$574$	−4.82352e6	−0.611061
$575$	0	0
$576$	−916005.	−0.115038
$577$	4.75161e6	0.594157	0.297078	−	0.954853i	$-0.403988\pi$
$577$	4.75161e6	0.594157	0.297078	+	0.954853i	$0.403988\pi$
$578$	1.43262e7	1.78366
$579$	6.84061e6	0.848005
$580$	0	0
$581$	1.72977e7	2.12593
$582$	4.19200e6	0.512996
$583$	−3.47544e6	−0.423486
$584$	236660.	0.0287139
$585$	0	0
$586$	591632.	0.0711718
$587$	−639190.	−0.0765658	−0.0382829	−	0.999267i	$-0.512189\pi$
$587$	−639190.	−0.0765658	−0.0382829	+	0.999267i	$0.512189\pi$
$588$	2.33948e6	0.279047
$589$	3.44030e6	0.408609
$590$	0	0
$591$	−1.58517e7	−1.86684
$592$	7.61911e6	0.893512
$593$	1.37403e6	0.160458	0.0802288	−	0.996776i	$-0.474435\pi$
$593$	1.37403e6	0.160458	0.0802288	+	0.996776i	$0.474435\pi$
$594$	1.18718e7	1.38054
$595$	0	0
$596$	2.71320e6	0.312872
$597$	−5.90313e6	−0.677870
$598$	4.64486e6	0.531153
$599$	−1.28982e7	−1.46880	−0.734401	−	0.678716i	$-0.762537\pi$
$599$	−1.28982e7	−1.46880	−0.734401	+	0.678716i	$0.762537\pi$
$600$	0	0
$601$	6.50916e6	0.735087	0.367543	−	0.930006i	$-0.380199\pi$
$601$	6.50916e6	0.735087	0.367543	+	0.930006i	$0.380199\pi$
$602$	−6.53345e6	−0.734769
$603$	886926.	0.0993332
$604$	8.02010e6	0.894515
$605$	0	0
$606$	−4.98655e6	−0.551593
$607$	6.58784e6	0.725724	0.362862	−	0.931843i	$-0.381800\pi$
$607$	6.58784e6	0.725724	0.362862	+	0.931843i	$0.381800\pi$
$608$	−3.07523e6	−0.337379
$609$	2.12530e7	2.32208
$610$	0	0
$611$	1.43696e6	0.155719
$612$	2.58810e6	0.279321
$613$	−6.83056e6	−0.734184	−0.367092	−	0.930185i	$-0.619647\pi$
$613$	−6.83056e6	−0.734184	−0.367092	+	0.930185i	$0.619647\pi$
$614$	−1.54413e7	−1.65297
$615$	0	0
$616$	−4.21735e6	−0.447803
$617$	−5.07265e6	−0.536441	−0.268221	−	0.963357i	$-0.586436\pi$
$617$	−5.07265e6	−0.536441	−0.268221	+	0.963357i	$0.586436\pi$
$618$	6.78369e6	0.714488
$619$	2.32710e6	0.244112	0.122056	−	0.992523i	$-0.461051\pi$
$619$	2.32710e6	0.244112	0.122056	+	0.992523i	$0.461051\pi$
$620$	0	0
$621$	1.17399e7	1.22162
$622$	−23425.2	−0.00242776
$623$	5.26840e6	0.543824
$624$	3.54845e6	0.364819
$625$	0	0
$626$	−7.39067e6	−0.753786
$627$	−3.62799e6	−0.368551
$628$	7.90206e6	0.799542
$629$	−1.14885e7	−1.15781
$630$	0	0
$631$	1.62914e7	1.62886	0.814432	−	0.580259i	$-0.197049\pi$
$631$	1.62914e7	1.62886	0.814432	+	0.580259i	$0.197049\pi$
$632$	919560.	0.0915772
$633$	−1.01093e7	−1.00280
$634$	1.42743e7	1.41036
$635$	0	0
$636$	2.98247e6	0.292370
$637$	−934552.	−0.0912547
$638$	3.03298e7	2.94997
$639$	4.02186e6	0.389650
$640$	0	0
$641$	9.20916e6	0.885268	0.442634	−	0.896702i	$-0.354044\pi$
$641$	9.20916e6	0.885268	0.442634	+	0.896702i	$0.354044\pi$
$642$	1.57978e7	1.51272
$643$	1.30502e7	1.24477	0.622387	−	0.782710i	$-0.286163\pi$
$643$	1.30502e7	1.24477	0.622387	+	0.782710i	$0.286163\pi$
$644$	1.33359e7	1.26709
$645$	0	0
$646$	5.80811e6	0.547587
$647$	−4.14869e6	−0.389628	−0.194814	−	0.980840i	$-0.562410\pi$
$647$	−4.14869e6	−0.389628	−0.194814	+	0.980840i	$0.562410\pi$
$648$	3.99552e6	0.373797
$649$	1.06038e7	0.988210
$650$	0	0
$651$	−2.10442e7	−1.94617
$652$	−1.15010e7	−1.05954
$653$	1.41931e7	1.30255	0.651273	−	0.758844i	$-0.274235\pi$
$653$	1.41931e7	1.30255	0.651273	+	0.758844i	$0.274235\pi$
$654$	1.82971e7	1.67277
$655$	0	0
$656$	−5.20029e6	−0.471811
$657$	−240634.	−0.0217492
$658$	9.54154e6	0.859120
$659$	7.37184e6	0.661245	0.330622	−	0.943763i	$-0.392741\pi$
$659$	7.37184e6	0.661245	0.330622	+	0.943763i	$0.392741\pi$
$660$	0	0
$661$	1.77538e7	1.58047	0.790235	−	0.612803i	$-0.209958\pi$
$661$	1.77538e7	1.58047	0.790235	+	0.612803i	$0.209958\pi$
$662$	−1.31314e7	−1.16457
$663$	−5.35054e6	−0.472730
$664$	6.62474e6	0.583108
$665$	0	0
$666$	−2.75203e6	−0.240418
$667$	2.99929e7	2.61038
$668$	7.77097e6	0.673805
$669$	1.00672e7	0.869644
$670$	0	0
$671$	−1.76210e7	−1.51086
$672$	1.88111e7	1.60691
$673$	−274222.	−0.0233381	−0.0116690	−	0.999932i	$-0.503714\pi$
$673$	−274222.	−0.0233381	−0.0116690	+	0.999932i	$0.503714\pi$
$674$	8.22363e6	0.697290
$675$	0	0
$676$	696224.	0.0585979
$677$	−3.03927e6	−0.254857	−0.127429	−	0.991848i	$-0.540672\pi$
$677$	−3.03927e6	−0.254857	−0.127429	+	0.991848i	$0.540672\pi$
$678$	1.13704e6	0.0949955
$679$	−4.80790e6	−0.400204
$680$	0	0
$681$	1.44676e7	1.19545
$682$	−3.00318e7	−2.47241
$683$	1.26346e7	1.03636	0.518178	−	0.855273i	$-0.326610\pi$
$683$	1.26346e7	1.03636	0.518178	+	0.855273i	$0.326610\pi$
$684$	601589.	0.0491654
$685$	0	0
$686$	1.26549e7	1.02671
$687$	−177407.	−0.0143410
$688$	−7.04377e6	−0.567328
$689$	−1.19141e6	−0.0956119
$690$	0	0
$691$	1.53558e7	1.22343	0.611713	−	0.791080i	$-0.290481\pi$
$691$	1.53558e7	1.22343	0.611713	+	0.791080i	$0.290481\pi$
$692$	−8.29130e6	−0.658199
$693$	4.28816e6	0.339186
$694$	1.49459e7	1.17794
$695$	0	0
$696$	8.13955e6	0.636909
$697$	7.84127e6	0.611370
$698$	−6.35279e6	−0.493544
$699$	−1.61790e7	−1.25245
$700$	0	0
$701$	−2.13816e7	−1.64341	−0.821703	−	0.569916i	$-0.806976\pi$
$701$	−2.13816e7	−1.64341	−0.821703	+	0.569916i	$0.806976\pi$
$702$	4.06974e6	0.311690
$703$	−2.67043e6	−0.203795
$704$	7.75911e6	0.590038
$705$	0	0
$706$	1.96576e7	1.48429
$707$	5.71919e6	0.430315
$708$	−9.09970e6	−0.682250
$709$	−2.39380e7	−1.78843	−0.894214	−	0.447639i	$-0.852265\pi$
$709$	−2.39380e7	−1.78843	−0.894214	+	0.447639i	$0.852265\pi$
$710$	0	0
$711$	−935001.	−0.0693646
$712$	2.01771e6	0.149162
$713$	−2.96982e7	−2.18779
$714$	−3.55281e7	−2.60811
$715$	0	0
$716$	−1.82625e7	−1.33130
$717$	−2.35566e7	−1.71126
$718$	−1.22556e7	−0.887203
$719$	−2.50209e6	−0.180501	−0.0902507	−	0.995919i	$-0.528767\pi$
$719$	−2.50209e6	−0.180501	−0.0902507	+	0.995919i	$0.528767\pi$
$720$	0	0
$721$	−7.78037e6	−0.557394
$722$	−1.72416e7	−1.23093
$723$	1.03343e6	0.0735250
$724$	−1.20950e7	−0.857550
$725$	0	0
$726$	1.06837e7	0.752281
$727$	−2.78011e7	−1.95086	−0.975429	−	0.220315i	$-0.929291\pi$
$727$	−2.78011e7	−1.95086	−0.975429	+	0.220315i	$0.929291\pi$
$728$	−1.44574e6	−0.101102
$729$	9.46320e6	0.659506
$730$	0	0
$731$	1.06210e7	0.735141
$732$	1.51215e7	1.04308
$733$	−4.46741e6	−0.307111	−0.153556	−	0.988140i	$-0.549072\pi$
$733$	−4.46741e6	−0.307111	−0.153556	+	0.988140i	$0.549072\pi$
$734$	−2.32495e7	−1.59284
$735$	0	0
$736$	2.65468e7	1.80641
$737$	−7.51280e6	−0.509487
$738$	1.87835e6	0.126951
$739$	−1.23320e7	−0.830656	−0.415328	−	0.909672i	$-0.636333\pi$
$739$	−1.23320e7	−0.830656	−0.415328	+	0.909672i	$0.636333\pi$
$740$	0	0
$741$	−1.24370e6	−0.0832090
$742$	−7.91106e6	−0.527503
$743$	9.81721e6	0.652403	0.326202	−	0.945300i	$-0.394231\pi$
$743$	9.81721e6	0.652403	0.326202	+	0.945300i	$0.394231\pi$
$744$	−8.05958e6	−0.533802
$745$	0	0
$746$	−1.83704e7	−1.20857
$747$	−6.73598e6	−0.441672
$748$	−2.19228e7	−1.43265
$749$	−1.81189e7	−1.18012
$750$	0	0
$751$	4.20655e6	0.272161	0.136081	−	0.990698i	$-0.456549\pi$
$751$	4.20655e6	0.272161	0.136081	+	0.990698i	$0.456549\pi$
$752$	1.02868e7	0.663341
$753$	−2.09828e7	−1.34858
$754$	1.03973e7	0.666025
$755$	0	0
$756$	1.16847e7	0.743553
$757$	2.89772e7	1.83788	0.918939	−	0.394400i	$-0.129048\pi$
$757$	2.89772e7	1.83788	0.918939	+	0.394400i	$0.129048\pi$
$758$	−3.23844e7	−2.04721
$759$	3.13185e7	1.97331
$760$	0	0
$761$	−1.57354e6	−0.0984957	−0.0492478	−	0.998787i	$-0.515682\pi$
$761$	−1.57354e6	−0.0984957	−0.0492478	+	0.998787i	$0.515682\pi$
$762$	−1.63425e7	−1.01961
$763$	−2.09853e7	−1.30498
$764$	1.91239e7	1.18534
$765$	0	0
$766$	3.14108e7	1.93423
$767$	3.63505e6	0.223112
$768$	2.35830e7	1.44276
$769$	−1.98322e7	−1.20936	−0.604680	−	0.796468i	$-0.706699\pi$
$769$	−1.98322e7	−1.20936	−0.604680	+	0.796468i	$0.706699\pi$
$770$	0	0
$771$	3.33101e7	2.01809
$772$	9.60822e6	0.580230
$773$	2.42031e7	1.45687	0.728437	−	0.685112i	$-0.240247\pi$
$773$	2.42031e7	1.45687	0.728437	+	0.685112i	$0.240247\pi$
$774$	2.54421e6	0.152652
$775$	0	0
$776$	−1.84134e6	−0.109769
$777$	1.63350e7	0.970658
$778$	−2.26388e7	−1.34093
$779$	1.82266e6	0.107612
$780$	0	0
$781$	−3.40675e7	−1.99854
$782$	−5.01382e7	−2.93192
$783$	2.62792e7	1.53182
$784$	−6.69024e6	−0.388733
$785$	0	0
$786$	−5.11058e7	−2.95062
$787$	1.03509e6	0.0595721	0.0297860	−	0.999556i	$-0.490517\pi$
$787$	1.03509e6	0.0595721	0.0297860	+	0.999556i	$0.490517\pi$
$788$	−2.22651e7	−1.27735
$789$	1.10584e7	0.632414
$790$	0	0
$791$	−1.30410e6	−0.0741089
$792$	1.64229e6	0.0930331
$793$	−6.04059e6	−0.341111
$794$	−1.37813e6	−0.0775779
$795$	0	0
$796$	−8.29144e6	−0.463818
$797$	−2.57309e7	−1.43486	−0.717429	−	0.696632i	$-0.754681\pi$
$797$	−2.57309e7	−1.43486	−0.717429	+	0.696632i	$0.754681\pi$
$798$	−8.25829e6	−0.459074
$799$	−1.55110e7	−0.859554
$800$	0	0
$801$	−2.05159e6	−0.112982
$802$	−2.33075e7	−1.27955
$803$	2.03831e6	0.111553
$804$	6.44715e6	0.351745
$805$	0	0
$806$	−1.02951e7	−0.558204
$807$	4.34005e6	0.234591
$808$	2.19035e6	0.118028
$809$	3.62907e7	1.94950	0.974751	−	0.223297i	$-0.0716819\pi$
$809$	3.62907e7	1.94950	0.974751	+	0.223297i	$0.0716819\pi$
$810$	0	0
$811$	−2.80415e7	−1.49709	−0.748546	−	0.663083i	$-0.769248\pi$
$811$	−2.80415e7	−1.49709	−0.748546	+	0.663083i	$0.769248\pi$
$812$	2.98517e7	1.58884
$813$	1.72533e7	0.915472
$814$	2.33113e7	1.23312
$815$	0	0
$816$	−3.83032e7	−2.01377
$817$	2.46878e6	0.129398
$818$	1.84042e7	0.961688
$819$	1.47001e6	0.0765793
$820$	0	0
$821$	−1.73781e7	−0.899795	−0.449897	−	0.893080i	$-0.648539\pi$
$821$	−1.73781e7	−0.899795	−0.449897	+	0.893080i	$0.648539\pi$
$822$	1.30841e6	0.0675404
$823$	8.26775e6	0.425488	0.212744	−	0.977108i	$-0.431760\pi$
$823$	8.26775e6	0.425488	0.212744	+	0.977108i	$0.431760\pi$
$824$	−2.97975e6	−0.152884
$825$	0	0
$826$	2.41371e7	1.23094
$827$	−9.80834e6	−0.498691	−0.249346	−	0.968415i	$-0.580216\pi$
$827$	−9.80834e6	−0.498691	−0.249346	+	0.968415i	$0.580216\pi$
$828$	−5.19319e6	−0.263244
$829$	2.34318e7	1.18419	0.592093	−	0.805870i	$-0.298302\pi$
$829$	2.34318e7	1.18419	0.592093	+	0.805870i	$0.298302\pi$
$830$	0	0
$831$	3.13287e7	1.57376
$832$	2.65988e6	0.133215
$833$	1.00879e7	0.503718
$834$	−5.38474e7	−2.68071
$835$	0	0
$836$	−5.09582e6	−0.252173
$837$	−2.60210e7	−1.28384
$838$	−1.03749e7	−0.510357
$839$	−1.14704e7	−0.562568	−0.281284	−	0.959625i	$-0.590760\pi$
$839$	−1.14704e7	−0.562568	−0.281284	+	0.959625i	$0.590760\pi$
$840$	0	0
$841$	4.66262e7	2.27321
$842$	−3.29347e7	−1.60093
$843$	−6.06639e6	−0.294010
$844$	−1.41994e7	−0.686144
$845$	0	0
$846$	−3.71561e6	−0.178486
$847$	−1.22534e7	−0.586877
$848$	−8.52899e6	−0.407294
$849$	−3.54285e6	−0.168688
$850$	0	0
$851$	2.30524e7	1.09117
$852$	2.92352e7	1.37977
$853$	2.31885e7	1.09119	0.545594	−	0.838049i	$-0.316304\pi$
$853$	2.31885e7	1.09119	0.545594	+	0.838049i	$0.316304\pi$
$854$	−4.01101e7	−1.88196
$855$	0	0
$856$	−6.93922e6	−0.323688
$857$	3.35616e7	1.56096	0.780478	−	0.625184i	$-0.214976\pi$
$857$	3.35616e7	1.56096	0.780478	+	0.625184i	$0.214976\pi$
$858$	1.08568e7	0.503481
$859$	−2.84860e7	−1.31719	−0.658594	−	0.752498i	$-0.728849\pi$
$859$	−2.84860e7	−1.31719	−0.658594	+	0.752498i	$0.728849\pi$
$860$	0	0
$861$	−1.11492e7	−0.512547
$862$	1.02321e7	0.469028
$863$	−3.40787e7	−1.55760	−0.778799	−	0.627273i	$-0.784171\pi$
$863$	−3.40787e7	−1.55760	−0.778799	+	0.627273i	$0.784171\pi$
$864$	2.32598e7	1.06004
$865$	0	0
$866$	−4.19980e7	−1.90298
$867$	3.31138e7	1.49610
$868$	−2.95584e7	−1.33162
$869$	7.92002e6	0.355776
$870$	0	0
$871$	−2.57544e6	−0.115029
$872$	−8.03702e6	−0.357935
$873$	1.87226e6	0.0831441
$874$	−1.16543e7	−0.516070
$875$	0	0
$876$	−1.74919e6	−0.0770152
$877$	1.29489e7	0.568505	0.284252	−	0.958750i	$-0.408255\pi$
$877$	1.29489e7	0.568505	0.284252	+	0.958750i	$0.408255\pi$
$878$	−3.65263e7	−1.59908
$879$	1.36751e6	0.0596976
$880$	0	0
$881$	8.39894e6	0.364573	0.182287	−	0.983245i	$-0.441650\pi$
$881$	8.39894e6	0.364573	0.182287	+	0.983245i	$0.441650\pi$
$882$	2.41652e6	0.104597
$883$	2.38721e7	1.03036	0.515179	−	0.857083i	$-0.327725\pi$
$883$	2.38721e7	1.03036	0.515179	+	0.857083i	$0.327725\pi$
$884$	−7.51528e6	−0.323456
$885$	0	0
$886$	−3.44788e7	−1.47560
$887$	−8.56803e6	−0.365655	−0.182828	−	0.983145i	$-0.558525\pi$
$887$	−8.56803e6	−0.365655	−0.182828	+	0.983145i	$0.558525\pi$
$888$	6.25602e6	0.266236
$889$	1.87436e7	0.795425
$890$	0	0
$891$	3.44128e7	1.45220
$892$	1.41402e7	0.595035
$893$	−3.60544e6	−0.151297
$894$	1.45039e7	0.606933
$895$	0	0
$896$	−1.70228e7	−0.708370
$897$	1.07362e7	0.445522
$898$	−2.64295e6	−0.109370
$899$	−6.64777e7	−2.74332
$900$	0	0
$901$	1.28605e7	0.527769
$902$	−1.59107e7	−0.651139
$903$	−1.51015e7	−0.616312
$904$	−499449.	−0.0203268
$905$	0	0
$906$	4.28728e7	1.73525
$907$	2.84173e7	1.14700	0.573502	−	0.819204i	$-0.305584\pi$
$907$	2.84173e7	1.14700	0.573502	+	0.819204i	$0.305584\pi$
$908$	2.03210e7	0.817958
$909$	−2.22713e6	−0.0893997
$910$	0	0
$911$	−1.90880e7	−0.762015	−0.381008	−	0.924572i	$-0.624423\pi$
$911$	−1.90880e7	−0.762015	−0.381008	+	0.924572i	$0.624423\pi$
$912$	−8.90335e6	−0.354459
$913$	5.70578e7	2.26536
$914$	4.10576e6	0.162565
$915$	0	0
$916$	−249183.	−0.00981251
$917$	5.86144e7	2.30187
$918$	−4.39301e7	−1.72050
$919$	−1.95252e7	−0.762619	−0.381310	−	0.924447i	$-0.624527\pi$
$919$	−1.95252e7	−0.762619	−0.381310	+	0.924447i	$0.624527\pi$
$920$	0	0
$921$	−3.56913e7	−1.38648
$922$	−4.77142e7	−1.84850
$923$	−1.16786e7	−0.451218
$924$	3.11710e7	1.20108
$925$	0	0
$926$	2.18359e7	0.836843
$927$	3.02978e6	0.115801
$928$	5.94235e7	2.26510
$929$	−1.62698e7	−0.618504	−0.309252	−	0.950980i	$-0.600079\pi$
$929$	−1.62698e7	−0.618504	−0.309252	+	0.950980i	$0.600079\pi$
$930$	0	0
$931$	2.34487e6	0.0886634
$932$	−2.27248e7	−0.856961
$933$	−54145.2	−0.00203637
$934$	−2.50540e7	−0.939744
$935$	0	0
$936$	562990.	0.0210044
$937$	3.49494e7	1.30044	0.650221	−	0.759745i	$-0.274676\pi$
$937$	3.49494e7	1.30044	0.650221	+	0.759745i	$0.274676\pi$
$938$	−1.71012e7	−0.634628
$939$	−1.70829e7	−0.632263
$940$	0	0
$941$	2.04858e7	0.754187	0.377094	−	0.926175i	$-0.376923\pi$
$941$	2.04858e7	0.754187	0.377094	+	0.926175i	$0.376923\pi$
$942$	4.22418e7	1.55101
$943$	−1.57340e7	−0.576182
$944$	2.60225e7	0.950426
$945$	0	0
$946$	−2.15510e7	−0.782961
$947$	3.59599e7	1.30300	0.651498	−	0.758650i	$-0.274141\pi$
$947$	3.59599e7	1.30300	0.651498	+	0.758650i	$0.274141\pi$
$948$	−6.79661e6	−0.245624
$949$	698748.	0.0251858
$950$	0	0
$951$	3.29938e7	1.18299
$952$	1.56058e7	0.558076
$953$	−2.09887e7	−0.748606	−0.374303	−	0.927306i	$-0.622118\pi$
$953$	−2.09887e7	−0.748606	−0.374303	+	0.927306i	$0.622118\pi$
$954$	3.08068e6	0.109591
$955$	0	0
$956$	−3.30873e7	−1.17089
$957$	7.01046e7	2.47438
$958$	1.45616e7	0.512621
$959$	−1.50064e6	−0.0526903
$960$	0	0
$961$	3.71955e7	1.29922
$962$	7.99129e6	0.278406
$963$	7.05574e6	0.245176
$964$	1.45154e6	0.0503079
$965$	0	0
$966$	7.12894e7	2.45800
$967$	5.30179e6	0.182329	0.0911646	−	0.995836i	$-0.470941\pi$
$967$	5.30179e6	0.182329	0.0911646	+	0.995836i	$0.470941\pi$
$968$	−4.69283e6	−0.160971
$969$	1.34249e7	0.459306
$970$	0	0
$971$	−5.19084e7	−1.76681	−0.883405	−	0.468611i	$-0.844755\pi$
$971$	−5.19084e7	−1.76681	−0.883405	+	0.468611i	$0.844755\pi$
$972$	−1.05334e7	−0.357603
$973$	6.17588e7	2.09130
$974$	−4.41700e7	−1.49187
$975$	0	0
$976$	−4.32431e7	−1.45309
$977$	−5.09506e7	−1.70771	−0.853853	−	0.520515i	$-0.825740\pi$
$977$	−5.09506e7	−1.70771	−0.853853	+	0.520515i	$0.825740\pi$
$978$	−6.14806e7	−2.05537
$979$	1.73782e7	0.579492
$980$	0	0
$981$	8.17198e6	0.271116
$982$	6.30373e7	2.08602
$983$	4.03520e7	1.33193	0.665964	−	0.745984i	$-0.268020\pi$
$983$	4.03520e7	1.33193	0.665964	+	0.745984i	$0.268020\pi$
$984$	−4.26994e6	−0.140583
$985$	0	0
$986$	−1.12232e8	−3.67640
$987$	2.20544e7	0.720614
$988$	−1.74688e6	−0.0569340
$989$	−2.13116e7	−0.692829
$990$	0	0
$991$	−2.26833e7	−0.733705	−0.366852	−	0.930279i	$-0.619565\pi$
$991$	−2.26833e7	−0.733705	−0.366852	+	0.930279i	$0.619565\pi$
$992$	−5.88396e7	−1.89841
$993$	−3.03521e7	−0.976821
$994$	−7.75470e7	−2.48943
$995$	0	0
$996$	−4.89645e7	−1.56399
$997$	840910.	0.0267924	0.0133962	−	0.999910i	$-0.495736\pi$
$997$	840910.	0.0267924	0.0133962	+	0.999910i	$0.495736\pi$
$998$	6.19081e7	1.96753
$999$	2.01981e7	0.640318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.i.1.7	yes	9
5.2	odd	4		325.6.b.h.274.15		18
5.3	odd	4		325.6.b.h.274.4		18
5.4	even	2		325.6.a.h.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.3	✓	9	5.4	even	2
325.6.a.i.1.7	yes	9	1.1	even	1	trivial
325.6.b.h.274.4		18	5.3	odd	4
325.6.b.h.274.15		18	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)

\(f(q)\)	\(=\)	\(q+7.50844 q^{2} +17.3551 q^{3} +24.3767 q^{4} +130.310 q^{6} -149.455 q^{7} -57.2388 q^{8} +58.2000 q^{9} -492.989 q^{11} +423.061 q^{12} -169.000 q^{13} -1122.18 q^{14} -1209.83 q^{16} +1824.24 q^{17} +436.991 q^{18} +424.035 q^{19} -2593.81 q^{21} -3701.58 q^{22} -3660.46 q^{23} -993.387 q^{24} -1268.93 q^{26} -3207.23 q^{27} -3643.23 q^{28} -8193.74 q^{29} +8113.24 q^{31} -7252.30 q^{32} -8555.87 q^{33} +13697.2 q^{34} +1418.73 q^{36} -6297.67 q^{37} +3183.84 q^{38} -2933.01 q^{39} +4298.36 q^{41} -19475.5 q^{42} +5822.12 q^{43} -12017.5 q^{44} -27484.4 q^{46} -8502.70 q^{47} -20996.7 q^{48} +5529.90 q^{49} +31660.0 q^{51} -4119.67 q^{52} +7049.74 q^{53} -24081.3 q^{54} +8554.65 q^{56} +7359.17 q^{57} -61522.2 q^{58} -21509.2 q^{59} +35743.1 q^{61} +60917.8 q^{62} -8698.30 q^{63} -15738.9 q^{64} -64241.3 q^{66} +15239.3 q^{67} +44469.1 q^{68} -63527.7 q^{69} +69104.1 q^{71} -3331.30 q^{72} -4134.60 q^{73} -47285.7 q^{74} +10336.6 q^{76} +73679.8 q^{77} -22022.4 q^{78} -16065.3 q^{79} -69804.4 q^{81} +32274.0 q^{82} -115739. q^{83} -63228.7 q^{84} +43715.0 q^{86} -142203. q^{87} +28218.1 q^{88} -35250.6 q^{89} +25258.0 q^{91} -89230.1 q^{92} +140806. q^{93} -63842.1 q^{94} -125864. q^{96} +32169.5 q^{97} +41520.9 q^{98} -28691.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{3} + 91 q^{4} - 83 q^{6} - 12 q^{7} + 639 q^{8} + 562 q^{9} - 1422 q^{11} - 1567 q^{12} - 1521 q^{13} - 342 q^{14} - 1061 q^{16} - 648 q^{17} - 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) \) 9 * q + 5 * q^2 + 11 * q^3 + 91 * q^4 - 83 * q^6 - 12 * q^7 + 639 * q^8 + 562 * q^9 - 1422 * q^11 - 1567 * q^12 - 1521 * q^13 - 342 * q^14 - 1061 * q^16 - 648 * q^17 - 418 * q^18 - 408 * q^19 - 3912 * q^21 - 4345 * q^22 - 1839 * q^23 - 8469 * q^24 - 845 * q^26 + 7649 * q^27 + 2836 * q^28 - 8737 * q^29 + 748 * q^31 + 423 * q^32 + 356 * q^33 - 17789 * q^34 + 512 * q^36 + 15486 * q^37 + 3425 * q^38 - 1859 * q^39 - 28676 * q^41 - 6876 * q^42 - 28665 * q^43 - 30599 * q^44 - 12056 * q^46 + 29452 * q^47 - 64759 * q^48 - 40907 * q^49 - 31006 * q^51 - 15379 * q^52 - 75977 * q^53 - 102761 * q^54 - 23002 * q^56 + 38038 * q^57 - 142384 * q^58 - 88142 * q^59 + 28165 * q^61 + 137308 * q^62 - 41492 * q^63 - 100845 * q^64 + 42577 * q^66 + 94754 * q^67 - 89267 * q^68 - 181747 * q^69 - 70562 * q^71 + 263778 * q^72 - 60602 * q^73 - 135676 * q^74 + 46373 * q^76 + 140292 * q^77 + 14027 * q^78 - 164073 * q^79 - 69935 * q^81 + 72887 * q^82 + 22458 * q^83 - 345656 * q^84 - 294920 * q^86 + 87031 * q^87 - 430607 * q^88 - 252698 * q^89 + 2028 * q^91 + 237824 * q^92 - 56556 * q^93 - 501606 * q^94 - 319181 * q^96 - 137986 * q^97 - 378699 * q^98 - 757776 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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