LMFDB - Embedded newform 315.6.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,6,Mod(1,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-1,0,-31] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$50.5209032411$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.53113$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.53113 q^{2} -19.5311 q^{4} +25.0000 q^{5} -49.0000 q^{7} -181.963 q^{8} +88.2782 q^{10} +691.520 q^{11} -502.150 q^{13} -173.025 q^{14} -17.5389 q^{16} +991.313 q^{17} +661.677 q^{19} -488.278 q^{20} +2441.84 q^{22} -3415.08 q^{23} +625.000 q^{25} -1773.16 q^{26} +957.025 q^{28} -6751.92 q^{29} -3922.76 q^{31} +5760.89 q^{32} +3500.46 q^{34} -1225.00 q^{35} +627.222 q^{37} +2336.47 q^{38} -4549.08 q^{40} -16277.9 q^{41} -17277.7 q^{43} -13506.2 q^{44} -12059.1 q^{46} +4295.47 q^{47} +2401.00 q^{49} +2206.96 q^{50} +9807.55 q^{52} +25960.9 q^{53} +17288.0 q^{55} +8916.19 q^{56} -23841.9 q^{58} -8902.63 q^{59} -48924.6 q^{61} -13851.8 q^{62} +20903.7 q^{64} -12553.7 q^{65} -4257.80 q^{67} -19361.5 q^{68} -4325.63 q^{70} -18990.9 q^{71} +10132.5 q^{73} +2214.80 q^{74} -12923.3 q^{76} -33884.5 q^{77} -96986.5 q^{79} -438.472 q^{80} -57479.2 q^{82} -70732.1 q^{83} +24782.8 q^{85} -61009.8 q^{86} -125831. q^{88} -4241.12 q^{89} +24605.3 q^{91} +66700.3 q^{92} +15167.9 q^{94} +16541.9 q^{95} -104376. q^{97} +8478.24 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 31 q^{4} + 50 q^{5} - 98 q^{7} + 15 q^{8} - 25 q^{10} + 601 q^{11} - 577 q^{13} + 49 q^{14} - 543 q^{16} - 41 q^{17} + 630 q^{19} - 775 q^{20} + 2852 q^{22} + 442 q^{23} + 1250 q^{25} - 1434 q^{26}+ \cdots - 2401 q^{98}+O(q^{100}) $ 2 * q - q^2 - 31 * q^4 + 50 * q^5 - 98 * q^7 + 15 * q^8 - 25 * q^10 + 601 * q^11 - 577 * q^13 + 49 * q^14 - 543 * q^16 - 41 * q^17 + 630 * q^19 - 775 * q^20 + 2852 * q^22 + 442 * q^23 + 1250 * q^25 - 1434 * q^26 + 1519 * q^28 - 5885 * q^29 - 396 * q^31 + 1839 * q^32 + 8178 * q^34 - 2450 * q^35 - 8904 * q^37 + 2480 * q^38 + 375 * q^40 - 1774 * q^41 - 27122 * q^43 - 12468 * q^44 - 29536 * q^46 + 21289 * q^47 + 4802 * q^49 - 625 * q^50 + 10666 * q^52 + 55582 * q^53 + 15025 * q^55 - 735 * q^56 - 27770 * q^58 - 59600 * q^59 - 51846 * q^61 - 29832 * q^62 + 55489 * q^64 - 14425 * q^65 - 45344 * q^67 - 7522 * q^68 + 1225 * q^70 - 80744 * q^71 - 13532 * q^73 + 45402 * q^74 - 12560 * q^76 - 29449 * q^77 - 51795 * q^79 - 13575 * q^80 - 123198 * q^82 - 109828 * q^83 - 1025 * q^85 - 16404 * q^86 - 143660 * q^88 + 37650 * q^89 + 28273 * q^91 + 22464 * q^92 - 61832 * q^94 + 15750 * q^95 - 96339 * q^97 - 2401 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.53113	0.624221	0.312111	−	0.950046i	$-0.398964\pi$
$2$	3.53113	0.624221	0.312111	+	0.950046i	$0.398964\pi$
$3$	0	0
$3$	0	0
$4$	−19.5311	−0.610348
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−181.963	−1.00521
$9$	0	0
$10$	88.2782	0.279160
$11$	691.520	1.72315	0.861574	−	0.507632i	$-0.169479\pi$
$11$	691.520	1.72315	0.861574	+	0.507632i	$0.169479\pi$
$12$	0	0
$13$	−502.150	−0.824091	−0.412045	−	0.911163i	$-0.635185\pi$
$13$	−502.150	−0.824091	−0.412045	+	0.911163i	$0.635185\pi$
$14$	−173.025	−0.235933
$15$	0	0
$16$	−17.5389	−0.0171278
$17$	991.313	0.831934	0.415967	−	0.909380i	$-0.363443\pi$
$17$	991.313	0.831934	0.415967	+	0.909380i	$0.363443\pi$
$18$	0	0
$19$	661.677	0.420496	0.210248	−	0.977648i	$-0.432573\pi$
$19$	661.677	0.420496	0.210248	+	0.977648i	$0.432573\pi$
$20$	−488.278	−0.272956
$21$	0	0
$22$	2441.84	1.07563
$23$	−3415.08	−1.34611	−0.673056	−	0.739592i	$-0.735019\pi$
$23$	−3415.08	−1.34611	−0.673056	+	0.739592i	$0.735019\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−1773.16	−0.514415
$27$	0	0
$28$	957.025	0.230690
$29$	−6751.92	−1.49084	−0.745422	−	0.666593i	$-0.767752\pi$
$29$	−6751.92	−1.49084	−0.745422	+	0.666593i	$0.767752\pi$
$30$	0	0
$31$	−3922.76	−0.733142	−0.366571	−	0.930390i	$-0.619468\pi$
$31$	−3922.76	−0.733142	−0.366571	+	0.930390i	$0.619468\pi$
$32$	5760.89	0.994522
$33$	0	0
$34$	3500.46	0.519311
$35$	−1225.00	−0.169031
$36$	0	0
$37$	627.222	0.0753212	0.0376606	−	0.999291i	$-0.488009\pi$
$37$	627.222	0.0753212	0.0376606	+	0.999291i	$0.488009\pi$
$38$	2336.47	0.262483
$39$	0	0
$40$	−4549.08	−0.449545
$41$	−16277.9	−1.51230	−0.756149	−	0.654399i	$-0.772922\pi$
$41$	−16277.9	−1.51230	−0.756149	+	0.654399i	$0.772922\pi$
$42$	0	0
$43$	−17277.7	−1.42500	−0.712500	−	0.701672i	$-0.752437\pi$
$43$	−17277.7	−1.42500	−0.712500	+	0.701672i	$0.752437\pi$
$44$	−13506.2	−1.05172
$45$	0	0
$46$	−12059.1	−0.840272
$47$	4295.47	0.283639	0.141820	−	0.989893i	$-0.454705\pi$
$47$	4295.47	0.283639	0.141820	+	0.989893i	$0.454705\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	2206.96	0.124844
$51$	0	0
$52$	9807.55	0.502982
$53$	25960.9	1.26949	0.634745	−	0.772721i	$-0.281105\pi$
$53$	25960.9	1.26949	0.634745	+	0.772721i	$0.281105\pi$
$54$	0	0
$55$	17288.0	0.770615
$56$	8916.19	0.379935
$57$	0	0
$58$	−23841.9	−0.930616
$59$	−8902.63	−0.332957	−0.166479	−	0.986045i	$-0.553240\pi$
$59$	−8902.63	−0.332957	−0.166479	+	0.986045i	$0.553240\pi$
$60$	0	0
$61$	−48924.6	−1.68346	−0.841730	−	0.539898i	$-0.818463\pi$
$61$	−48924.6	−1.68346	−0.841730	+	0.539898i	$0.818463\pi$
$62$	−13851.8	−0.457643
$63$	0	0
$64$	20903.7	0.637930
$65$	−12553.7	−0.368545
$66$	0	0
$67$	−4257.80	−0.115877	−0.0579387	−	0.998320i	$-0.518453\pi$
$67$	−4257.80	−0.115877	−0.0579387	+	0.998320i	$0.518453\pi$
$68$	−19361.5	−0.507769
$69$	0	0
$70$	−4325.63	−0.105513
$71$	−18990.9	−0.447095	−0.223547	−	0.974693i	$-0.571764\pi$
$71$	−18990.9	−0.447095	−0.223547	+	0.974693i	$0.571764\pi$
$72$	0	0
$73$	10132.5	0.222541	0.111270	−	0.993790i	$-0.464508\pi$
$73$	10132.5	0.222541	0.111270	+	0.993790i	$0.464508\pi$
$74$	2214.80	0.0470171
$75$	0	0
$76$	−12923.3	−0.256649
$77$	−33884.5	−0.651289
$78$	0	0
$79$	−96986.5	−1.74841	−0.874205	−	0.485557i	$-0.838617\pi$
$79$	−96986.5	−1.74841	−0.874205	+	0.485557i	$0.838617\pi$
$80$	−438.472	−0.00765979
$81$	0	0
$82$	−57479.2	−0.944009
$83$	−70732.1	−1.12699	−0.563497	−	0.826118i	$-0.690544\pi$
$83$	−70732.1	−1.12699	−0.563497	+	0.826118i	$0.690544\pi$
$84$	0	0
$85$	24782.8	0.372052
$86$	−61009.8	−0.889515
$87$	0	0
$88$	−125831.	−1.73213
$89$	−4241.12	−0.0567552	−0.0283776	−	0.999597i	$-0.509034\pi$
$89$	−4241.12	−0.0567552	−0.0283776	+	0.999597i	$0.509034\pi$
$90$	0	0
$91$	24605.3	0.311477
$92$	66700.3	0.821596
$93$	0	0
$94$	15167.9	0.177054
$95$	16541.9	0.188052
$96$	0	0
$97$	−104376.	−1.12634	−0.563170	−	0.826341i	$-0.690418\pi$
$97$	−104376.	−1.12634	−0.563170	+	0.826341i	$0.690418\pi$
$98$	8478.24	0.0891745
$99$	0	0
$100$	−12207.0	−0.122070
$101$	45715.1	0.445919	0.222959	−	0.974828i	$-0.428428\pi$
$101$	45715.1	0.445919	0.222959	+	0.974828i	$0.428428\pi$
$102$	0	0
$103$	89278.1	0.829186	0.414593	−	0.910007i	$-0.363924\pi$
$103$	89278.1	0.829186	0.414593	+	0.910007i	$0.363924\pi$
$104$	91372.7	0.828387
$105$	0	0
$106$	91671.2	0.792443
$107$	106330.	0.897834	0.448917	−	0.893573i	$-0.351810\pi$
$107$	106330.	0.897834	0.448917	+	0.893573i	$0.351810\pi$
$108$	0	0
$109$	−49816.5	−0.401613	−0.200806	−	0.979631i	$-0.564356\pi$
$109$	−49816.5	−0.401613	−0.200806	+	0.979631i	$0.564356\pi$
$110$	61046.1	0.481035
$111$	0	0
$112$	859.405	0.00647370
$113$	37160.7	0.273771	0.136886	−	0.990587i	$-0.456291\pi$
$113$	37160.7	0.273771	0.136886	+	0.990587i	$0.456291\pi$
$114$	0	0
$115$	−85377.0	−0.601999
$116$	131873.	0.909933
$117$	0	0
$118$	−31436.3	−0.207839
$119$	−48574.4	−0.314441
$120$	0	0
$121$	317148.	1.96924
$122$	−172759.	−1.05085
$123$	0	0
$124$	76616.0	0.447471
$125$	15625.0	0.0894427
$126$	0	0
$127$	−46510.2	−0.255882	−0.127941	−	0.991782i	$-0.540837\pi$
$127$	−46510.2	−0.255882	−0.127941	+	0.991782i	$0.540837\pi$
$128$	−110535.	−0.596313
$129$	0	0
$130$	−44328.9	−0.230053
$131$	−381771.	−1.94368	−0.971839	−	0.235646i	$-0.924279\pi$
$131$	−381771.	−1.94368	−0.971839	+	0.235646i	$0.924279\pi$
$132$	0	0
$133$	−32422.2	−0.158933
$134$	−15034.9	−0.0723331
$135$	0	0
$136$	−180382.	−0.836271
$137$	1894.54	0.00862389	0.00431194	−	0.999991i	$-0.498627\pi$
$137$	1894.54	0.00862389	0.00431194	+	0.999991i	$0.498627\pi$
$138$	0	0
$139$	201798.	0.885889	0.442944	−	0.896549i	$-0.353934\pi$
$139$	201798.	0.885889	0.442944	+	0.896549i	$0.353934\pi$
$140$	23925.6	0.103168
$141$	0	0
$142$	−67059.3	−0.279086
$143$	−347246.	−1.42003
$144$	0	0
$145$	−168798.	−0.666726
$146$	35779.1	0.138915
$147$	0	0
$148$	−12250.4	−0.0459721
$149$	466237.	1.72045	0.860224	−	0.509917i	$-0.170324\pi$
$149$	466237.	1.72045	0.860224	+	0.509917i	$0.170324\pi$
$150$	0	0
$151$	−122212.	−0.436185	−0.218093	−	0.975928i	$-0.569983\pi$
$151$	−122212.	−0.436185	−0.218093	+	0.975928i	$0.569983\pi$
$152$	−120401.	−0.422688
$153$	0	0
$154$	−119650.	−0.406548
$155$	−98069.1	−0.327871
$156$	0	0
$157$	−410638.	−1.32957	−0.664784	−	0.747036i	$-0.731476\pi$
$157$	−410638.	−1.32957	−0.664784	+	0.747036i	$0.731476\pi$
$158$	−342472.	−1.09139
$159$	0	0
$160$	144022.	0.444764
$161$	167339.	0.508782
$162$	0	0
$163$	78525.4	0.231495	0.115747	−	0.993279i	$-0.463074\pi$
$163$	78525.4	0.231495	0.115747	+	0.993279i	$0.463074\pi$
$164$	317925.	0.923028
$165$	0	0
$166$	−249764.	−0.703494
$167$	597714.	1.65845	0.829224	−	0.558916i	$-0.188783\pi$
$167$	597714.	1.65845	0.829224	+	0.558916i	$0.188783\pi$
$168$	0	0
$169$	−119139.	−0.320875
$170$	87511.4	0.232243
$171$	0	0
$172$	337453.	0.869745
$173$	59874.0	0.152098	0.0760490	−	0.997104i	$-0.475769\pi$
$173$	59874.0	0.152098	0.0760490	+	0.997104i	$0.475769\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	−12128.5	−0.0295138
$177$	0	0
$178$	−14975.9	−0.0354278
$179$	−616812.	−1.43887	−0.719433	−	0.694562i	$-0.755598\pi$
$179$	−616812.	−1.43887	−0.719433	+	0.694562i	$0.755598\pi$
$180$	0	0
$181$	−37287.0	−0.0845981	−0.0422990	−	0.999105i	$-0.513468\pi$
$181$	−37287.0	−0.0845981	−0.0422990	+	0.999105i	$0.513468\pi$
$182$	86884.6	0.194431
$183$	0	0
$184$	621418.	1.35313
$185$	15680.6	0.0336847
$186$	0	0
$187$	685513.	1.43355
$188$	−83895.4	−0.173119
$189$	0	0
$190$	58411.7	0.117386
$191$	−326760.	−0.648106	−0.324053	−	0.946039i	$-0.605046\pi$
$191$	−326760.	−0.648106	−0.324053	+	0.946039i	$0.605046\pi$
$192$	0	0
$193$	−265735.	−0.513518	−0.256759	−	0.966475i	$-0.582655\pi$
$193$	−265735.	−0.513518	−0.256759	+	0.966475i	$0.582655\pi$
$194$	−368563.	−0.703085
$195$	0	0
$196$	−46894.2	−0.0871925
$197$	517865.	0.950716	0.475358	−	0.879792i	$-0.342318\pi$
$197$	517865.	0.950716	0.475358	+	0.879792i	$0.342318\pi$
$198$	0	0
$199$	−148687.	−0.266158	−0.133079	−	0.991105i	$-0.542486\pi$
$199$	−148687.	−0.266158	−0.133079	+	0.991105i	$0.542486\pi$
$200$	−113727.	−0.201043
$201$	0	0
$202$	161426.	0.278352
$203$	330844.	0.563486
$204$	0	0
$205$	−406946.	−0.676320
$206$	315252.	0.517595
$207$	0	0
$208$	8807.15	0.0141149
$209$	457563.	0.724577
$210$	0	0
$211$	7443.09	0.0115093	0.00575463	−	0.999983i	$-0.498168\pi$
$211$	7443.09	0.0115093	0.00575463	+	0.999983i	$0.498168\pi$
$212$	−507045.	−0.774831
$213$	0	0
$214$	375465.	0.560447
$215$	−431943.	−0.637279
$216$	0	0
$217$	192215.	0.277101
$218$	−175909.	−0.250695
$219$	0	0
$220$	−337654.	−0.470343
$221$	−497788.	−0.685589
$222$	0	0
$223$	119157.	0.160456	0.0802280	−	0.996777i	$-0.474435\pi$
$223$	119157.	0.160456	0.0802280	+	0.996777i	$0.474435\pi$
$224$	−282283.	−0.375894
$225$	0	0
$226$	131219.	0.170894
$227$	388843.	0.500852	0.250426	−	0.968136i	$-0.419429\pi$
$227$	388843.	0.500852	0.250426	+	0.968136i	$0.419429\pi$
$228$	0	0
$229$	732622.	0.923191	0.461595	−	0.887091i	$-0.347277\pi$
$229$	732622.	0.923191	0.461595	+	0.887091i	$0.347277\pi$
$230$	−301477.	−0.375781
$231$	0	0
$232$	1.22860e6	1.49862
$233$	1.12639e6	1.35925	0.679626	−	0.733559i	$-0.262142\pi$
$233$	1.12639e6	1.35925	0.679626	+	0.733559i	$0.262142\pi$
$234$	0	0
$235$	107387.	0.126847
$236$	173878.	0.203220
$237$	0	0
$238$	−171522.	−0.196281
$239$	−772317.	−0.874583	−0.437292	−	0.899320i	$-0.644062\pi$
$239$	−772317.	−0.874583	−0.437292	+	0.899320i	$0.644062\pi$
$240$	0	0
$241$	1.40297e6	1.55598	0.777991	−	0.628275i	$-0.216239\pi$
$241$	1.40297e6	1.55598	0.777991	+	0.628275i	$0.216239\pi$
$242$	1.11989e6	1.22924
$243$	0	0
$244$	955553.	1.02750
$245$	60025.0	0.0638877
$246$	0	0
$247$	−332261.	−0.346527
$248$	713798.	0.736964
$249$	0	0
$250$	55173.9	0.0558320
$251$	1.63922e6	1.64230	0.821151	−	0.570712i	$-0.193333\pi$
$251$	1.63922e6	1.64230	0.821151	+	0.570712i	$0.193333\pi$
$252$	0	0
$253$	−2.36159e6	−2.31955
$254$	−164234.	−0.159727
$255$	0	0
$256$	−1.05923e6	−1.01016
$257$	223664.	0.211234	0.105617	−	0.994407i	$-0.466318\pi$
$257$	223664.	0.211234	0.105617	+	0.994407i	$0.466318\pi$
$258$	0	0
$259$	−30733.9	−0.0284687
$260$	245189.	0.224940
$261$	0	0
$262$	−1.34808e6	−1.21329
$263$	−299519.	−0.267014	−0.133507	−	0.991048i	$-0.542624\pi$
$263$	−299519.	−0.267014	−0.133507	+	0.991048i	$0.542624\pi$
$264$	0	0
$265$	649022.	0.567734
$266$	−114487.	−0.0992091
$267$	0	0
$268$	83159.7	0.0707255
$269$	−134341.	−0.113195	−0.0565974	−	0.998397i	$-0.518025\pi$
$269$	−134341.	−0.113195	−0.0565974	+	0.998397i	$0.518025\pi$
$270$	0	0
$271$	1.93414e6	1.59980	0.799898	−	0.600136i	$-0.204887\pi$
$271$	1.93414e6	1.59980	0.799898	+	0.600136i	$0.204887\pi$
$272$	−17386.5	−0.0142492
$273$	0	0
$274$	6689.88	0.00538321
$275$	432200.	0.344630
$276$	0	0
$277$	177599.	0.139072	0.0695362	−	0.997579i	$-0.477848\pi$
$277$	177599.	0.139072	0.0695362	+	0.997579i	$0.477848\pi$
$278$	712574.	0.552991
$279$	0	0
$280$	222905.	0.169912
$281$	−1.85131e6	−1.39867	−0.699333	−	0.714796i	$-0.746519\pi$
$281$	−1.85131e6	−1.39867	−0.699333	+	0.714796i	$0.746519\pi$
$282$	0	0
$283$	2.39851e6	1.78023	0.890114	−	0.455737i	$-0.150624\pi$
$283$	2.39851e6	1.78023	0.890114	+	0.455737i	$0.150624\pi$
$284$	370914.	0.272883
$285$	0	0
$286$	−1.22617e6	−0.886413
$287$	797615.	0.571595
$288$	0	0
$289$	−437155.	−0.307887
$290$	−596047.	−0.416184
$291$	0	0
$292$	−197899.	−0.135827
$293$	−2.49922e6	−1.70073	−0.850364	−	0.526196i	$-0.823618\pi$
$293$	−2.49922e6	−1.70073	−0.850364	+	0.526196i	$0.823618\pi$
$294$	0	0
$295$	−222566.	−0.148903
$296$	−114131.	−0.0757139
$297$	0	0
$298$	1.64634e6	1.07394
$299$	1.71488e6	1.10932
$300$	0	0
$301$	846607.	0.538599
$302$	−431546.	−0.272276
$303$	0	0
$304$	−11605.1	−0.00720218
$305$	−1.22312e6	−0.752866
$306$	0	0
$307$	3.07195e6	1.86024	0.930119	−	0.367258i	$-0.119703\pi$
$307$	3.07195e6	1.86024	0.930119	+	0.367258i	$0.119703\pi$
$308$	661802.	0.397513
$309$	0	0
$310$	−346295.	−0.204664
$311$	−661233.	−0.387662	−0.193831	−	0.981035i	$-0.562091\pi$
$311$	−661233.	−0.387662	−0.193831	+	0.981035i	$0.562091\pi$
$312$	0	0
$313$	−3.29393e6	−1.90043	−0.950217	−	0.311588i	$-0.899139\pi$
$313$	−3.29393e6	−1.90043	−0.950217	+	0.311588i	$0.899139\pi$
$314$	−1.45002e6	−0.829944
$315$	0	0
$316$	1.89426e6	1.06714
$317$	−639724.	−0.357556	−0.178778	−	0.983889i	$-0.557214\pi$
$317$	−639724.	−0.357556	−0.178778	+	0.983889i	$0.557214\pi$
$318$	0	0
$319$	−4.66908e6	−2.56894
$320$	522592.	0.285291
$321$	0	0
$322$	590895.	0.317593
$323$	655929.	0.349825
$324$	0	0
$325$	−313844.	−0.164818
$326$	277283.	0.144504
$327$	0	0
$328$	2.96197e6	1.52018
$329$	−210478.	−0.107206
$330$	0	0
$331$	−1.13876e6	−0.571298	−0.285649	−	0.958334i	$-0.592209\pi$
$331$	−1.13876e6	−0.571298	−0.285649	+	0.958334i	$0.592209\pi$
$332$	1.38148e6	0.687858
$333$	0	0
$334$	2.11060e6	1.03524
$335$	−106445.	−0.0518219
$336$	0	0
$337$	685493.	0.328797	0.164399	−	0.986394i	$-0.447432\pi$
$337$	685493.	0.328797	0.164399	+	0.986394i	$0.447432\pi$
$338$	−420694.	−0.200297
$339$	0	0
$340$	−484037.	−0.227081
$341$	−2.71267e6	−1.26331
$342$	0	0
$343$	−117649.	−0.0539949
$344$	3.14390e6	1.43243
$345$	0	0
$346$	211423.	0.0949428
$347$	1.25151e6	0.557970	0.278985	−	0.960295i	$-0.410002\pi$
$347$	1.25151e6	0.557970	0.278985	+	0.960295i	$0.410002\pi$
$348$	0	0
$349$	−3.16606e6	−1.39141	−0.695706	−	0.718327i	$-0.744908\pi$
$349$	−3.16606e6	−1.39141	−0.695706	+	0.718327i	$0.744908\pi$
$350$	−108141.	−0.0471867
$351$	0	0
$352$	3.98376e6	1.71371
$353$	2.43368e6	1.03951	0.519754	−	0.854316i	$-0.326024\pi$
$353$	2.43368e6	1.03951	0.519754	+	0.854316i	$0.326024\pi$
$354$	0	0
$355$	−474772.	−0.199947
$356$	82833.8	0.0346404
$357$	0	0
$358$	−2.17804e6	−0.898171
$359$	2.13021e6	0.872341	0.436170	−	0.899864i	$-0.356335\pi$
$359$	2.13021e6	0.872341	0.436170	+	0.899864i	$0.356335\pi$
$360$	0	0
$361$	−2.03828e6	−0.823183
$362$	−131665.	−0.0528079
$363$	0	0
$364$	−480570.	−0.190109
$365$	253312.	0.0995232
$366$	0	0
$367$	−3.10976e6	−1.20521	−0.602604	−	0.798041i	$-0.705870\pi$
$367$	−3.10976e6	−1.20521	−0.602604	+	0.798041i	$0.705870\pi$
$368$	59896.7	0.0230559
$369$	0	0
$370$	55370.1	0.0210267
$371$	−1.27208e6	−0.479822
$372$	0	0
$373$	−3.15189e6	−1.17300	−0.586502	−	0.809948i	$-0.699495\pi$
$373$	−3.15189e6	−1.17300	−0.586502	+	0.809948i	$0.699495\pi$
$374$	2.42063e6	0.894849
$375$	0	0
$376$	−781617.	−0.285118
$377$	3.39047e6	1.22859
$378$	0	0
$379$	342350.	0.122426	0.0612129	−	0.998125i	$-0.480503\pi$
$379$	342350.	0.122426	0.0612129	+	0.998125i	$0.480503\pi$
$380$	−323083.	−0.114777
$381$	0	0
$382$	−1.15383e6	−0.404562
$383$	−3.69387e6	−1.28672	−0.643361	−	0.765563i	$-0.722461\pi$
$383$	−3.69387e6	−1.28672	−0.643361	+	0.765563i	$0.722461\pi$
$384$	0	0
$385$	−847111.	−0.291265
$386$	−938345.	−0.320549
$387$	0	0
$388$	2.03857e6	0.687459
$389$	−2.05313e6	−0.687928	−0.343964	−	0.938983i	$-0.611770\pi$
$389$	−2.05313e6	−0.687928	−0.343964	+	0.938983i	$0.611770\pi$
$390$	0	0
$391$	−3.38541e6	−1.11988
$392$	−436893.	−0.143602
$393$	0	0
$394$	1.82865e6	0.593457
$395$	−2.42466e6	−0.781913
$396$	0	0
$397$	−2.28107e6	−0.726377	−0.363189	−	0.931716i	$-0.618312\pi$
$397$	−2.28107e6	−0.726377	−0.363189	+	0.931716i	$0.618312\pi$
$398$	−525031.	−0.166141
$399$	0	0
$400$	−10961.8	−0.00342556
$401$	−3.32082e6	−1.03130	−0.515649	−	0.856800i	$-0.672449\pi$
$401$	−3.32082e6	−1.03130	−0.515649	+	0.856800i	$0.672449\pi$
$402$	0	0
$403$	1.96981e6	0.604175
$404$	−892867.	−0.272166
$405$	0	0
$406$	1.16825e6	0.351740
$407$	433737.	0.129790
$408$	0	0
$409$	5.15938e6	1.52507	0.762534	−	0.646948i	$-0.223955\pi$
$409$	5.15938e6	1.52507	0.762534	+	0.646948i	$0.223955\pi$
$410$	−1.43698e6	−0.422174
$411$	0	0
$412$	−1.74370e6	−0.506092
$413$	436229.	0.125846
$414$	0	0
$415$	−1.76830e6	−0.504007
$416$	−2.89283e6	−0.819576
$417$	0	0
$418$	1.61571e6	0.452297
$419$	4.85187e6	1.35012	0.675062	−	0.737761i	$-0.264117\pi$
$419$	4.85187e6	1.35012	0.675062	+	0.737761i	$0.264117\pi$
$420$	0	0
$421$	−6.14767e6	−1.69046	−0.845231	−	0.534401i	$-0.820537\pi$
$421$	−6.14767e6	−1.69046	−0.845231	+	0.534401i	$0.820537\pi$
$422$	26282.5	0.00718432
$423$	0	0
$424$	−4.72392e6	−1.27611
$425$	619571.	0.166387
$426$	0	0
$427$	2.39731e6	0.636288
$428$	−2.07674e6	−0.547991
$429$	0	0
$430$	−1.52524e6	−0.397803
$431$	−3.55411e6	−0.921590	−0.460795	−	0.887507i	$-0.652436\pi$
$431$	−3.55411e6	−0.921590	−0.460795	+	0.887507i	$0.652436\pi$
$432$	0	0
$433$	2.82650e6	0.724485	0.362243	−	0.932084i	$-0.382011\pi$
$433$	2.82650e6	0.724485	0.362243	+	0.932084i	$0.382011\pi$
$434$	678737.	0.172973
$435$	0	0
$436$	972973.	0.245123
$437$	−2.25968e6	−0.566035
$438$	0	0
$439$	4.64410e6	1.15011	0.575056	−	0.818114i	$-0.304980\pi$
$439$	4.64410e6	1.15011	0.575056	+	0.818114i	$0.304980\pi$
$440$	−3.14578e6	−0.774633
$441$	0	0
$442$	−1.75775e6	−0.427959
$443$	6.15534e6	1.49019	0.745097	−	0.666957i	$-0.232403\pi$
$443$	6.15534e6	1.49019	0.745097	+	0.666957i	$0.232403\pi$
$444$	0	0
$445$	−106028.	−0.0253817
$446$	420757.	0.100160
$447$	0	0
$448$	−1.02428e6	−0.241115
$449$	−3.67035e6	−0.859196	−0.429598	−	0.903020i	$-0.641345\pi$
$449$	−3.67035e6	−0.859196	−0.429598	+	0.903020i	$0.641345\pi$
$450$	0	0
$451$	−1.12565e7	−2.60591
$452$	−725791.	−0.167096
$453$	0	0
$454$	1.37305e6	0.312642
$455$	615134.	0.139297
$456$	0	0
$457$	−866327.	−0.194040	−0.0970201	−	0.995282i	$-0.530931\pi$
$457$	−866327.	−0.194040	−0.0970201	+	0.995282i	$0.530931\pi$
$458$	2.58698e6	0.576275
$459$	0	0
$460$	1.66751e6	0.367429
$461$	1.88572e6	0.413261	0.206631	−	0.978419i	$-0.433750\pi$
$461$	1.88572e6	0.413261	0.206631	+	0.978419i	$0.433750\pi$
$462$	0	0
$463$	−8.49017e6	−1.84062	−0.920309	−	0.391192i	$-0.872063\pi$
$463$	−8.49017e6	−1.84062	−0.920309	+	0.391192i	$0.872063\pi$
$464$	118421.	0.0255349
$465$	0	0
$466$	3.97744e6	0.848474
$467$	−1.82738e6	−0.387737	−0.193868	−	0.981028i	$-0.562104\pi$
$467$	−1.82738e6	−0.387737	−0.193868	+	0.981028i	$0.562104\pi$
$468$	0	0
$469$	208632.	0.0437975
$470$	379197.	0.0791808
$471$	0	0
$472$	1.61995e6	0.334693
$473$	−1.19479e7	−2.45549
$474$	0	0
$475$	413548.	0.0840992
$476$	948712.	0.191919
$477$	0	0
$478$	−2.72715e6	−0.545933
$479$	−4.68744e6	−0.933463	−0.466731	−	0.884399i	$-0.654568\pi$
$479$	−4.68744e6	−0.933463	−0.466731	+	0.884399i	$0.654568\pi$
$480$	0	0
$481$	−314960.	−0.0620715
$482$	4.95405e6	0.971277
$483$	0	0
$484$	−6.19426e6	−1.20192
$485$	−2.60939e6	−0.503714
$486$	0	0
$487$	4.59651e6	0.878225	0.439112	−	0.898432i	$-0.355293\pi$
$487$	4.59651e6	0.878225	0.439112	+	0.898432i	$0.355293\pi$
$488$	8.90247e6	1.69224
$489$	0	0
$490$	211956.	0.0398800
$491$	−6.62099e6	−1.23942	−0.619711	−	0.784830i	$-0.712750\pi$
$491$	−6.62099e6	−1.23942	−0.619711	+	0.784830i	$0.712750\pi$
$492$	0	0
$493$	−6.69327e6	−1.24028
$494$	−1.17326e6	−0.216310
$495$	0	0
$496$	68800.9	0.0125571
$497$	930554.	0.168986
$498$	0	0
$499$	−4.52632e6	−0.813756	−0.406878	−	0.913482i	$-0.633383\pi$
$499$	−4.52632e6	−0.813756	−0.406878	+	0.913482i	$0.633383\pi$
$500$	−305174.	−0.0545912
$501$	0	0
$502$	5.78829e6	1.02516
$503$	−3.83316e6	−0.675518	−0.337759	−	0.941233i	$-0.609669\pi$
$503$	−3.83316e6	−0.675518	−0.337759	+	0.941233i	$0.609669\pi$
$504$	0	0
$505$	1.14288e6	0.199421
$506$	−8.33909e6	−1.44791
$507$	0	0
$508$	908397.	0.156177
$509$	3.25460e6	0.556806	0.278403	−	0.960464i	$-0.410195\pi$
$509$	3.25460e6	0.556806	0.278403	+	0.960464i	$0.410195\pi$
$510$	0	0
$511$	−496492.	−0.0841124
$512$	−203165.	−0.0342511
$513$	0	0
$514$	789788.	0.131857
$515$	2.23195e6	0.370823
$516$	0	0
$517$	2.97040e6	0.488752
$518$	−108525.	−0.0177708
$519$	0	0
$520$	2.28432e6	0.370466
$521$	1.07842e6	0.174057	0.0870287	−	0.996206i	$-0.472263\pi$
$521$	1.07842e6	0.174057	0.0870287	+	0.996206i	$0.472263\pi$
$522$	0	0
$523$	408626.	0.0653238	0.0326619	−	0.999466i	$-0.489602\pi$
$523$	408626.	0.0653238	0.0326619	+	0.999466i	$0.489602\pi$
$524$	7.45641e6	1.18632
$525$	0	0
$526$	−1.05764e6	−0.166676
$527$	−3.88869e6	−0.609925
$528$	0	0
$529$	5.22642e6	0.812016
$530$	2.29178e6	0.354391
$531$	0	0
$532$	633242.	0.0970042
$533$	8.17392e6	1.24627
$534$	0	0
$535$	2.65825e6	0.401524
$536$	774763.	0.116481
$537$	0	0
$538$	−474374.	−0.0706586
$539$	1.66034e6	0.246164
$540$	0	0
$541$	1.13918e7	1.67340	0.836701	−	0.547659i	$-0.184481\pi$
$541$	1.13918e7	1.67340	0.836701	+	0.547659i	$0.184481\pi$
$542$	6.82970e6	0.998627
$543$	0	0
$544$	5.71084e6	0.827376
$545$	−1.24541e6	−0.179607
$546$	0	0
$547$	−3.44866e6	−0.492813	−0.246406	−	0.969167i	$-0.579250\pi$
$547$	−3.44866e6	−0.492813	−0.246406	+	0.969167i	$0.579250\pi$
$548$	−37002.6	−0.00526357
$549$	0	0
$550$	1.52615e6	0.215125
$551$	−4.46759e6	−0.626894
$552$	0	0
$553$	4.75234e6	0.660837
$554$	627124.	0.0868119
$555$	0	0
$556$	−3.94134e6	−0.540700
$557$	4.69772e6	0.641577	0.320789	−	0.947151i	$-0.396052\pi$
$557$	4.69772e6	0.641577	0.320789	+	0.947151i	$0.396052\pi$
$558$	0	0
$559$	8.67599e6	1.17433
$560$	21485.1	0.00289513
$561$	0	0
$562$	−6.53722e6	−0.873077
$563$	561642.	0.0746772	0.0373386	−	0.999303i	$-0.488112\pi$
$563$	561642.	0.0746772	0.0373386	+	0.999303i	$0.488112\pi$
$564$	0	0
$565$	929018.	0.122434
$566$	8.46945e6	1.11126
$567$	0	0
$568$	3.45564e6	0.449426
$569$	−5.19019e6	−0.672051	−0.336026	−	0.941853i	$-0.609083\pi$
$569$	−5.19019e6	−0.672051	−0.336026	+	0.941853i	$0.609083\pi$
$570$	0	0
$571$	−5.20274e6	−0.667793	−0.333897	−	0.942610i	$-0.608364\pi$
$571$	−5.20274e6	−0.667793	−0.333897	+	0.942610i	$0.608364\pi$
$572$	6.78211e6	0.866712
$573$	0	0
$574$	2.81648e6	0.356802
$575$	−2.13442e6	−0.269222
$576$	0	0
$577$	8.70662e6	1.08870	0.544352	−	0.838857i	$-0.316775\pi$
$577$	8.70662e6	1.08870	0.544352	+	0.838857i	$0.316775\pi$
$578$	−1.54365e6	−0.192189
$579$	0	0
$580$	3.29681e6	0.406934
$581$	3.46588e6	0.425964
$582$	0	0
$583$	1.79524e7	2.18752
$584$	−1.84374e6	−0.223701
$585$	0	0
$586$	−8.82505e6	−1.06163
$587$	4.35717e6	0.521926	0.260963	−	0.965349i	$-0.415960\pi$
$587$	4.35717e6	0.521926	0.260963	+	0.965349i	$0.415960\pi$
$588$	0	0
$589$	−2.59560e6	−0.308283
$590$	−785908.	−0.0929484
$591$	0	0
$592$	−11000.8	−0.00129009
$593$	3.22991e6	0.377184	0.188592	−	0.982055i	$-0.439608\pi$
$593$	3.22991e6	0.377184	0.188592	+	0.982055i	$0.439608\pi$
$594$	0	0
$595$	−1.21436e6	−0.140622
$596$	−9.10614e6	−1.05007
$597$	0	0
$598$	6.05547e6	0.692460
$599$	7.23988e6	0.824450	0.412225	−	0.911082i	$-0.364752\pi$
$599$	7.23988e6	0.824450	0.412225	+	0.911082i	$0.364752\pi$
$600$	0	0
$601$	−1.06837e7	−1.20652	−0.603262	−	0.797543i	$-0.706133\pi$
$601$	−1.06837e7	−1.20652	−0.603262	+	0.797543i	$0.706133\pi$
$602$	2.98948e6	0.336205
$603$	0	0
$604$	2.38693e6	0.266225
$605$	7.92871e6	0.880671
$606$	0	0
$607$	2.51528e6	0.277086	0.138543	−	0.990356i	$-0.455758\pi$
$607$	2.51528e6	0.277086	0.138543	+	0.990356i	$0.455758\pi$
$608$	3.81185e6	0.418193
$609$	0	0
$610$	−4.31898e6	−0.469955
$611$	−2.15697e6	−0.233744
$612$	0	0
$613$	213999.	0.0230017	0.0115009	−	0.999934i	$-0.496339\pi$
$613$	213999.	0.0230017	0.0115009	+	0.999934i	$0.496339\pi$
$614$	1.08475e7	1.16120
$615$	0	0
$616$	6.16572e6	0.654684
$617$	127951.	0.0135310	0.00676552	−	0.999977i	$-0.497846\pi$
$617$	127951.	0.0135310	0.00676552	+	0.999977i	$0.497846\pi$
$618$	0	0
$619$	−1.23980e7	−1.30054	−0.650272	−	0.759701i	$-0.725345\pi$
$619$	−1.23980e7	−1.30054	−0.650272	+	0.759701i	$0.725345\pi$
$620$	1.91540e6	0.200115
$621$	0	0
$622$	−2.33490e6	−0.241987
$623$	207815.	0.0214514
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.16313e7	−1.18629
$627$	0	0
$628$	8.02023e6	0.811499
$629$	621774.	0.0626622
$630$	0	0
$631$	−5.87683e6	−0.587584	−0.293792	−	0.955869i	$-0.594917\pi$
$631$	−5.87683e6	−0.587584	−0.293792	+	0.955869i	$0.594917\pi$
$632$	1.76480e7	1.75753
$633$	0	0
$634$	−2.25895e6	−0.223194
$635$	−1.16275e6	−0.114434
$636$	0	0
$637$	−1.20566e6	−0.117727
$638$	−1.64871e7	−1.60359
$639$	0	0
$640$	−2.76337e6	−0.266679
$641$	2.60122e6	0.250053	0.125026	−	0.992153i	$-0.460098\pi$
$641$	2.60122e6	0.250053	0.125026	+	0.992153i	$0.460098\pi$
$642$	0	0
$643$	1.31345e7	1.25282	0.626408	−	0.779495i	$-0.284524\pi$
$643$	1.31345e7	1.25282	0.626408	+	0.779495i	$0.284524\pi$
$644$	−3.26832e6	−0.310534
$645$	0	0
$646$	2.31617e6	0.218368
$647$	−5.54662e6	−0.520916	−0.260458	−	0.965485i	$-0.583874\pi$
$647$	−5.54662e6	−0.520916	−0.260458	+	0.965485i	$0.583874\pi$
$648$	0	0
$649$	−6.15634e6	−0.573734
$650$	−1.10822e6	−0.102883
$651$	0	0
$652$	−1.53369e6	−0.141292
$653$	1.54993e7	1.42242	0.711212	−	0.702978i	$-0.248147\pi$
$653$	1.54993e7	1.42242	0.711212	+	0.702978i	$0.248147\pi$
$654$	0	0
$655$	−9.54427e6	−0.869239
$656$	285495.	0.0259024
$657$	0	0
$658$	−743225.	−0.0669200
$659$	4.30145e6	0.385835	0.192917	−	0.981215i	$-0.438205\pi$
$659$	4.30145e6	0.385835	0.192917	+	0.981215i	$0.438205\pi$
$660$	0	0
$661$	861980.	0.0767350	0.0383675	−	0.999264i	$-0.487784\pi$
$661$	861980.	0.0767350	0.0383675	+	0.999264i	$0.487784\pi$
$662$	−4.02111e6	−0.356616
$663$	0	0
$664$	1.28706e7	1.13287
$665$	−810554.	−0.0710768
$666$	0	0
$667$	2.30583e7	2.00684
$668$	−1.16740e7	−1.01223
$669$	0	0
$670$	−375871.	−0.0323484
$671$	−3.38323e7	−2.90085
$672$	0	0
$673$	−953818.	−0.0811760	−0.0405880	−	0.999176i	$-0.512923\pi$
$673$	−953818.	−0.0811760	−0.0405880	+	0.999176i	$0.512923\pi$
$674$	2.42056e6	0.205242
$675$	0	0
$676$	2.32691e6	0.195845
$677$	1.48606e7	1.24613	0.623065	−	0.782170i	$-0.285887\pi$
$677$	1.48606e7	1.24613	0.623065	+	0.782170i	$0.285887\pi$
$678$	0	0
$679$	5.11440e6	0.425716
$680$	−4.50956e6	−0.373992
$681$	0	0
$682$	−9.57878e6	−0.788586
$683$	1.57605e7	1.29276	0.646381	−	0.763015i	$-0.276282\pi$
$683$	1.57605e7	1.29276	0.646381	+	0.763015i	$0.276282\pi$
$684$	0	0
$685$	47363.6	0.00385672
$686$	−415434.	−0.0337048
$687$	0	0
$688$	303032.	0.0244071
$689$	−1.30362e7	−1.04618
$690$	0	0
$691$	1.59740e7	1.27267	0.636337	−	0.771411i	$-0.280449\pi$
$691$	1.59740e7	1.27267	0.636337	+	0.771411i	$0.280449\pi$
$692$	−1.16941e6	−0.0928326
$693$	0	0
$694$	4.41925e6	0.348297
$695$	5.04494e6	0.396182
$696$	0	0
$697$	−1.61365e7	−1.25813
$698$	−1.11798e7	−0.868549
$699$	0	0
$700$	598141.	0.0461380
$701$	1.85736e7	1.42758	0.713790	−	0.700360i	$-0.246977\pi$
$701$	1.85736e7	1.42758	0.713790	+	0.700360i	$0.246977\pi$
$702$	0	0
$703$	415019.	0.0316723
$704$	1.44553e7	1.09925
$705$	0	0
$706$	8.59365e6	0.648883
$707$	−2.24004e6	−0.168541
$708$	0	0
$709$	−1.71029e7	−1.27778	−0.638888	−	0.769300i	$-0.720605\pi$
$709$	−1.71029e7	−1.27778	−0.638888	+	0.769300i	$0.720605\pi$
$710$	−1.67648e6	−0.124811
$711$	0	0
$712$	771727.	0.0570511
$713$	1.33965e7	0.986890
$714$	0	0
$715$	−8.68116e6	−0.635057
$716$	1.20470e7	0.878208
$717$	0	0
$718$	7.52204e6	0.544534
$719$	1.40945e7	1.01678	0.508390	−	0.861127i	$-0.330241\pi$
$719$	1.40945e7	1.01678	0.508390	+	0.861127i	$0.330241\pi$
$720$	0	0
$721$	−4.37463e6	−0.313403
$722$	−7.19744e6	−0.513848
$723$	0	0
$724$	728256.	0.0516343
$725$	−4.21995e6	−0.298169
$726$	0	0
$727$	2.24196e7	1.57323	0.786616	−	0.617443i	$-0.211831\pi$
$727$	2.24196e7	1.57323	0.786616	+	0.617443i	$0.211831\pi$
$728$	−4.47726e6	−0.313101
$729$	0	0
$730$	894478.	0.0621245
$731$	−1.71276e7	−1.18551
$732$	0	0
$733$	−8.02464e6	−0.551653	−0.275826	−	0.961208i	$-0.588951\pi$
$733$	−8.02464e6	−0.551653	−0.275826	+	0.961208i	$0.588951\pi$
$734$	−1.09810e7	−0.752316
$735$	0	0
$736$	−1.96739e7	−1.33874
$737$	−2.94435e6	−0.199674
$738$	0	0
$739$	1.15678e7	0.779181	0.389591	−	0.920988i	$-0.372616\pi$
$739$	1.15678e7	0.779181	0.389591	+	0.920988i	$0.372616\pi$
$740$	−306259.	−0.0205594
$741$	0	0
$742$	−4.49189e6	−0.299515
$743$	−3.79387e6	−0.252122	−0.126061	−	0.992023i	$-0.540233\pi$
$743$	−3.79387e6	−0.252122	−0.126061	+	0.992023i	$0.540233\pi$
$744$	0	0
$745$	1.16559e7	0.769407
$746$	−1.11297e7	−0.732214
$747$	0	0
$748$	−1.33888e7	−0.874961
$749$	−5.21017e6	−0.339349
$750$	0	0
$751$	−9.07884e6	−0.587395	−0.293698	−	0.955898i	$-0.594886\pi$
$751$	−9.07884e6	−0.587395	−0.293698	+	0.955898i	$0.594886\pi$
$752$	−75337.8	−0.00485812
$753$	0	0
$754$	1.19722e7	0.766912
$755$	−3.05530e6	−0.195068
$756$	0	0
$757$	−2.33210e7	−1.47913	−0.739565	−	0.673085i	$-0.764969\pi$
$757$	−2.33210e7	−1.47913	−0.739565	+	0.673085i	$0.764969\pi$
$758$	1.20888e6	0.0764208
$759$	0	0
$760$	−3.01002e6	−0.189032
$761$	−1.44859e7	−0.906745	−0.453373	−	0.891321i	$-0.649779\pi$
$761$	−1.44859e7	−0.906745	−0.453373	+	0.891321i	$0.649779\pi$
$762$	0	0
$763$	2.44101e6	0.151795
$764$	6.38200e6	0.395570
$765$	0	0
$766$	−1.30435e7	−0.803200
$767$	4.47045e6	0.274387
$768$	0	0
$769$	1.59424e7	0.972162	0.486081	−	0.873914i	$-0.338426\pi$
$769$	1.59424e7	0.972162	0.486081	+	0.873914i	$0.338426\pi$
$770$	−2.99126e6	−0.181814
$771$	0	0
$772$	5.19011e6	0.313425
$773$	3.36066e6	0.202290	0.101145	−	0.994872i	$-0.467749\pi$
$773$	3.36066e6	0.202290	0.101145	+	0.994872i	$0.467749\pi$
$774$	0	0
$775$	−2.45173e6	−0.146628
$776$	1.89925e7	1.13221
$777$	0	0
$778$	−7.24988e6	−0.429419
$779$	−1.07707e7	−0.635916
$780$	0	0
$781$	−1.31326e7	−0.770411
$782$	−1.19543e7	−0.699050
$783$	0	0
$784$	−42110.9	−0.00244683
$785$	−1.02660e7	−0.594601
$786$	0	0
$787$	−3.28918e6	−0.189300	−0.0946500	−	0.995511i	$-0.530173\pi$
$787$	−3.28918e6	−0.189300	−0.0946500	+	0.995511i	$0.530173\pi$
$788$	−1.01145e7	−0.580268
$789$	0	0
$790$	−8.56179e6	−0.488087
$791$	−1.82088e6	−0.103476
$792$	0	0
$793$	2.45675e7	1.38732
$794$	−8.05475e6	−0.453420
$795$	0	0
$796$	2.90402e6	0.162449
$797$	2.51939e6	0.140492	0.0702458	−	0.997530i	$-0.477622\pi$
$797$	2.51939e6	0.140492	0.0702458	+	0.997530i	$0.477622\pi$
$798$	0	0
$799$	4.25816e6	0.235969
$800$	3.60055e6	0.198904
$801$	0	0
$802$	−1.17262e7	−0.643758
$803$	7.00682e6	0.383470
$804$	0	0
$805$	4.18347e6	0.227534
$806$	6.95567e6	0.377139
$807$	0	0
$808$	−8.31845e6	−0.448244
$809$	−8.93808e6	−0.480146	−0.240073	−	0.970755i	$-0.577171\pi$
$809$	−8.93808e6	−0.480146	−0.240073	+	0.970755i	$0.577171\pi$
$810$	0	0
$811$	3.01341e7	1.60881	0.804406	−	0.594080i	$-0.202484\pi$
$811$	3.01341e7	1.60881	0.804406	+	0.594080i	$0.202484\pi$
$812$	−6.46176e6	−0.343922
$813$	0	0
$814$	1.53158e6	0.0810174
$815$	1.96313e6	0.103528
$816$	0	0
$817$	−1.14323e7	−0.599207
$818$	1.82184e7	0.951980
$819$	0	0
$820$	7.94812e6	0.412791
$821$	3.25440e7	1.68505	0.842524	−	0.538658i	$-0.181069\pi$
$821$	3.25440e7	1.68505	0.842524	+	0.538658i	$0.181069\pi$
$822$	0	0
$823$	−499640.	−0.0257133	−0.0128566	−	0.999917i	$-0.504093\pi$
$823$	−499640.	−0.0257133	−0.0128566	+	0.999917i	$0.504093\pi$
$824$	−1.62453e7	−0.833509
$825$	0	0
$826$	1.54038e6	0.0785557
$827$	4.64084e6	0.235957	0.117978	−	0.993016i	$-0.462359\pi$
$827$	4.64084e6	0.235957	0.117978	+	0.993016i	$0.462359\pi$
$828$	0	0
$829$	−3.08794e7	−1.56057	−0.780283	−	0.625427i	$-0.784925\pi$
$829$	−3.08794e7	−1.56057	−0.780283	+	0.625427i	$0.784925\pi$
$830$	−6.24411e6	−0.314612
$831$	0	0
$832$	−1.04968e7	−0.525712
$833$	2.38014e6	0.118848
$834$	0	0
$835$	1.49428e7	0.741681
$836$	−8.93671e6	−0.442244
$837$	0	0
$838$	1.71326e7	0.842777
$839$	−1.93485e7	−0.948948	−0.474474	−	0.880269i	$-0.657362\pi$
$839$	−1.93485e7	−0.948948	−0.474474	+	0.880269i	$0.657362\pi$
$840$	0	0
$841$	2.50772e7	1.22261
$842$	−2.17082e7	−1.05522
$843$	0	0
$844$	−145372.	−0.00702465
$845$	−2.97846e6	−0.143500
$846$	0	0
$847$	−1.55403e7	−0.744303
$848$	−455325.	−0.0217436
$849$	0	0
$850$	2.18778e6	0.103862
$851$	−2.14201e6	−0.101391
$852$	0	0
$853$	−3.05998e7	−1.43994	−0.719972	−	0.694003i	$-0.755845\pi$
$853$	−3.05998e7	−1.43994	−0.719972	+	0.694003i	$0.755845\pi$
$854$	8.46520e6	0.397185
$855$	0	0
$856$	−1.93481e7	−0.902515
$857$	1.33039e7	0.618766	0.309383	−	0.950937i	$-0.399877\pi$
$857$	1.33039e7	0.618766	0.309383	+	0.950937i	$0.399877\pi$
$858$	0	0
$859$	−1.27708e7	−0.590520	−0.295260	−	0.955417i	$-0.595406\pi$
$859$	−1.27708e7	−0.590520	−0.295260	+	0.955417i	$0.595406\pi$
$860$	8.43633e6	0.388962
$861$	0	0
$862$	−1.25500e7	−0.575276
$863$	−1.37987e7	−0.630684	−0.315342	−	0.948978i	$-0.602119\pi$
$863$	−1.37987e7	−0.630684	−0.315342	+	0.948978i	$0.602119\pi$
$864$	0	0
$865$	1.49685e6	0.0680203
$866$	9.98074e6	0.452239
$867$	0	0
$868$	−3.75418e6	−0.169128
$869$	−6.70680e7	−3.01277
$870$	0	0
$871$	2.13806e6	0.0954934
$872$	9.06477e6	0.403706
$873$	0	0
$874$	−7.97922e6	−0.353331
$875$	−765625.	−0.0338062
$876$	0	0
$877$	6.68992e6	0.293712	0.146856	−	0.989158i	$-0.453085\pi$
$877$	6.68992e6	0.293712	0.146856	+	0.989158i	$0.453085\pi$
$878$	1.63989e7	0.717924
$879$	0	0
$880$	−303212.	−0.0131990
$881$	7.25051e6	0.314723	0.157362	−	0.987541i	$-0.449701\pi$
$881$	7.25051e6	0.314723	0.157362	+	0.987541i	$0.449701\pi$
$882$	0	0
$883$	−2.55944e7	−1.10470	−0.552348	−	0.833613i	$-0.686268\pi$
$883$	−2.55944e7	−1.10470	−0.552348	+	0.833613i	$0.686268\pi$
$884$	9.72236e6	0.418447
$885$	0	0
$886$	2.17353e7	0.930210
$887$	2.33204e7	0.995240	0.497620	−	0.867395i	$-0.334207\pi$
$887$	2.33204e7	0.995240	0.497620	+	0.867395i	$0.334207\pi$
$888$	0	0
$889$	2.27900e6	0.0967141
$890$	−374398.	−0.0158438
$891$	0	0
$892$	−2.32726e6	−0.0979340
$893$	2.84222e6	0.119269
$894$	0	0
$895$	−1.54203e7	−0.643480
$896$	5.41620e6	0.225385
$897$	0	0
$898$	−1.29605e7	−0.536328
$899$	2.64862e7	1.09300
$900$	0	0
$901$	2.57354e7	1.05613
$902$	−3.97480e7	−1.62667
$903$	0	0
$904$	−6.76188e6	−0.275199
$905$	−932174.	−0.0378334
$906$	0	0
$907$	3.71721e7	1.50037	0.750187	−	0.661226i	$-0.229964\pi$
$907$	3.71721e7	1.50037	0.750187	+	0.661226i	$0.229964\pi$
$908$	−7.59453e6	−0.305694
$909$	0	0
$910$	2.17212e6	0.0869520
$911$	2.89923e7	1.15741	0.578704	−	0.815537i	$-0.303559\pi$
$911$	2.89923e7	1.15741	0.578704	+	0.815537i	$0.303559\pi$
$912$	0	0
$913$	−4.89127e7	−1.94198
$914$	−3.05911e6	−0.121124
$915$	0	0
$916$	−1.43089e7	−0.563468
$917$	1.87068e7	0.734641
$918$	0	0
$919$	−1.25020e6	−0.0488306	−0.0244153	−	0.999702i	$-0.507772\pi$
$919$	−1.25020e6	−0.0488306	−0.0244153	+	0.999702i	$0.507772\pi$
$920$	1.55355e7	0.605138
$921$	0	0
$922$	6.65872e6	0.257966
$923$	9.53627e6	0.368447
$924$	0	0
$925$	392014.	0.0150642
$926$	−2.99799e7	−1.14895
$927$	0	0
$928$	−3.88970e7	−1.48268
$929$	−6.87875e6	−0.261499	−0.130749	−	0.991415i	$-0.541738\pi$
$929$	−6.87875e6	−0.261499	−0.130749	+	0.991415i	$0.541738\pi$
$930$	0	0
$931$	1.58869e6	0.0600709
$932$	−2.19997e7	−0.829616
$933$	0	0
$934$	−6.45272e6	−0.242034
$935$	1.71378e7	0.641101
$936$	0	0
$937$	2.18600e7	0.813396	0.406698	−	0.913563i	$-0.366680\pi$
$937$	2.18600e7	0.813396	0.406698	+	0.913563i	$0.366680\pi$
$938$	736708.	0.0273393
$939$	0	0
$940$	−2.09739e6	−0.0774210
$941$	−2.26450e6	−0.0833679	−0.0416840	−	0.999131i	$-0.513272\pi$
$941$	−2.26450e6	−0.0833679	−0.0416840	+	0.999131i	$0.513272\pi$
$942$	0	0
$943$	5.55901e7	2.03572
$944$	156142.	0.00570283
$945$	0	0
$946$	−4.21895e7	−1.53277
$947$	2.45846e7	0.890815	0.445408	−	0.895328i	$-0.353059\pi$
$947$	2.45846e7	0.890815	0.445408	+	0.895328i	$0.353059\pi$
$948$	0	0
$949$	−5.08803e6	−0.183394
$950$	1.46029e6	0.0524965
$951$	0	0
$952$	8.83874e6	0.316081
$953$	−4.59681e7	−1.63955	−0.819774	−	0.572687i	$-0.805901\pi$
$953$	−4.59681e7	−1.63955	−0.819774	+	0.572687i	$0.805901\pi$
$954$	0	0
$955$	−8.16901e6	−0.289842
$956$	1.50842e7	0.533800
$957$	0	0
$958$	−1.65520e7	−0.582687
$959$	−92832.6	−0.00325952
$960$	0	0
$961$	−1.32411e7	−0.462503
$962$	−1.11216e6	−0.0387463
$963$	0	0
$964$	−2.74015e7	−0.949690
$965$	−6.64338e6	−0.229652
$966$	0	0
$967$	−5.30202e7	−1.82337	−0.911686	−	0.410888i	$-0.865219\pi$
$967$	−5.30202e7	−1.82337	−0.911686	+	0.410888i	$0.865219\pi$
$968$	−5.77093e7	−1.97951
$969$	0	0
$970$	−9.21409e6	−0.314429
$971$	5.43523e7	1.84999	0.924996	−	0.379976i	$-0.124068\pi$
$971$	5.43523e7	1.84999	0.924996	+	0.379976i	$0.124068\pi$
$972$	0	0
$973$	−9.88809e6	−0.334835
$974$	1.62309e7	0.548207
$975$	0	0
$976$	858083.	0.0288340
$977$	−1.08929e7	−0.365097	−0.182549	−	0.983197i	$-0.558435\pi$
$977$	−1.08929e7	−0.365097	−0.182549	+	0.983197i	$0.558435\pi$
$978$	0	0
$979$	−2.93282e6	−0.0977976
$980$	−1.17236e6	−0.0389937
$981$	0	0
$982$	−2.33796e7	−0.773674
$983$	3.42136e7	1.12932	0.564658	−	0.825325i	$-0.309008\pi$
$983$	3.42136e7	1.12932	0.564658	+	0.825325i	$0.309008\pi$
$984$	0	0
$985$	1.29466e7	0.425173
$986$	−2.36348e7	−0.774211
$987$	0	0
$988$	6.48943e6	0.211502
$989$	5.90047e7	1.91821
$990$	0	0
$991$	4.22117e6	0.136537	0.0682683	−	0.997667i	$-0.478253\pi$
$991$	4.22117e6	0.136537	0.0682683	+	0.997667i	$0.478253\pi$
$992$	−2.25986e7	−0.729125
$993$	0	0
$994$	3.28591e6	0.105485
$995$	−3.71716e6	−0.119029
$996$	0	0
$997$	4.30323e7	1.37106	0.685530	−	0.728044i	$-0.259571\pi$
$997$	4.30323e7	1.37106	0.685530	+	0.728044i	$0.259571\pi$
$998$	−1.59830e7	−0.507964
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.6.a.c.1.2		2
3.2	odd	2		35.6.a.b.1.1	✓	2
12.11	even	2		560.6.a.l.1.1		2
15.2	even	4		175.6.b.d.99.2		4
15.8	even	4		175.6.b.d.99.3		4
15.14	odd	2		175.6.a.d.1.2		2
21.20	even	2		245.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.a.b.1.1	✓	2	3.2	odd	2
175.6.a.d.1.2		2	15.14	odd	2
175.6.b.d.99.2		4	15.2	even	4
175.6.b.d.99.3		4	15.8	even	4
245.6.a.c.1.1		2	21.20	even	2
315.6.a.c.1.2		2	1.1	even	1	trivial
560.6.a.l.1.1		2	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+3.53113 q^{2} -19.5311 q^{4} +25.0000 q^{5} -49.0000 q^{7} -181.963 q^{8} +88.2782 q^{10} +691.520 q^{11} -502.150 q^{13} -173.025 q^{14} -17.5389 q^{16} +991.313 q^{17} +661.677 q^{19} -488.278 q^{20} +2441.84 q^{22} -3415.08 q^{23} +625.000 q^{25} -1773.16 q^{26} +957.025 q^{28} -6751.92 q^{29} -3922.76 q^{31} +5760.89 q^{32} +3500.46 q^{34} -1225.00 q^{35} +627.222 q^{37} +2336.47 q^{38} -4549.08 q^{40} -16277.9 q^{41} -17277.7 q^{43} -13506.2 q^{44} -12059.1 q^{46} +4295.47 q^{47} +2401.00 q^{49} +2206.96 q^{50} +9807.55 q^{52} +25960.9 q^{53} +17288.0 q^{55} +8916.19 q^{56} -23841.9 q^{58} -8902.63 q^{59} -48924.6 q^{61} -13851.8 q^{62} +20903.7 q^{64} -12553.7 q^{65} -4257.80 q^{67} -19361.5 q^{68} -4325.63 q^{70} -18990.9 q^{71} +10132.5 q^{73} +2214.80 q^{74} -12923.3 q^{76} -33884.5 q^{77} -96986.5 q^{79} -438.472 q^{80} -57479.2 q^{82} -70732.1 q^{83} +24782.8 q^{85} -61009.8 q^{86} -125831. q^{88} -4241.12 q^{89} +24605.3 q^{91} +66700.3 q^{92} +15167.9 q^{94} +16541.9 q^{95} -104376. q^{97} +8478.24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 31 q^{4} + 50 q^{5} - 98 q^{7} + 15 q^{8} - 25 q^{10} + 601 q^{11} - 577 q^{13} + 49 q^{14} - 543 q^{16} - 41 q^{17} + 630 q^{19} - 775 q^{20} + 2852 q^{22} + 442 q^{23} + 1250 q^{25} - 1434 q^{26}+ \cdots - 2401 q^{98}+O(q^{100}) \) 2 * q - q^2 - 31 * q^4 + 50 * q^5 - 98 * q^7 + 15 * q^8 - 25 * q^10 + 601 * q^11 - 577 * q^13 + 49 * q^14 - 543 * q^16 - 41 * q^17 + 630 * q^19 - 775 * q^20 + 2852 * q^22 + 442 * q^23 + 1250 * q^25 - 1434 * q^26 + 1519 * q^28 - 5885 * q^29 - 396 * q^31 + 1839 * q^32 + 8178 * q^34 - 2450 * q^35 - 8904 * q^37 + 2480 * q^38 + 375 * q^40 - 1774 * q^41 - 27122 * q^43 - 12468 * q^44 - 29536 * q^46 + 21289 * q^47 + 4802 * q^49 - 625 * q^50 + 10666 * q^52 + 55582 * q^53 + 15025 * q^55 - 735 * q^56 - 27770 * q^58 - 59600 * q^59 - 51846 * q^61 - 29832 * q^62 + 55489 * q^64 - 14425 * q^65 - 45344 * q^67 - 7522 * q^68 + 1225 * q^70 - 80744 * q^71 - 13532 * q^73 + 45402 * q^74 - 12560 * q^76 - 29449 * q^77 - 51795 * q^79 - 13575 * q^80 - 123198 * q^82 - 109828 * q^83 - 1025 * q^85 - 16404 * q^86 - 143660 * q^88 + 37650 * q^89 + 28273 * q^91 + 22464 * q^92 - 61832 * q^94 + 15750 * q^95 - 96339 * q^97 - 2401 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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