LMFDB - Embedded newform 495.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 129x^{3} + 45x^{2} + 2924x - 5216 $ x^5 - x^4 - 129x^3 + 45x^2 + 2924*x - 5216
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.57133$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.57133 q^{2} +41.4676 q^{4} -25.0000 q^{5} +3.71474 q^{7} -81.1502 q^{8} +O(q^{10})$	\(q-8.57133 q^{2} +41.4676 q^{4} -25.0000 q^{5} +3.71474 q^{7} -81.1502 q^{8} +214.283 q^{10} +121.000 q^{11} +94.0133 q^{13} -31.8402 q^{14} -631.400 q^{16} -196.903 q^{17} -425.349 q^{19} -1036.69 q^{20} -1037.13 q^{22} -518.668 q^{23} +625.000 q^{25} -805.819 q^{26} +154.041 q^{28} +378.739 q^{29} +5098.68 q^{31} +8008.74 q^{32} +1687.72 q^{34} -92.8684 q^{35} -4418.94 q^{37} +3645.81 q^{38} +2028.76 q^{40} -12225.3 q^{41} -12069.9 q^{43} +5017.58 q^{44} +4445.67 q^{46} -1450.03 q^{47} -16793.2 q^{49} -5357.08 q^{50} +3898.51 q^{52} +24797.4 q^{53} -3025.00 q^{55} -301.452 q^{56} -3246.30 q^{58} +22302.7 q^{59} +52941.4 q^{61} -43702.5 q^{62} -48440.7 q^{64} -2350.33 q^{65} +65036.7 q^{67} -8165.11 q^{68} +796.005 q^{70} -60745.5 q^{71} +54856.0 q^{73} +37876.1 q^{74} -17638.2 q^{76} +449.483 q^{77} +40298.7 q^{79} +15785.0 q^{80} +104787. q^{82} -74192.7 q^{83} +4922.58 q^{85} +103455. q^{86} -9819.17 q^{88} -551.359 q^{89} +349.235 q^{91} -21507.9 q^{92} +12428.6 q^{94} +10633.7 q^{95} +82992.9 q^{97} +143940. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 9 q^{2} + 115 q^{4} - 125 q^{5} + 70 q^{7} - 753 q^{8}+O(q^{10}) $ 5 * q - 9 * q^2 + 115 * q^4 - 125 * q^5 + 70 * q^7 - 753 * q^8	$ 5 q - 9 q^{2} + 115 q^{4} - 125 q^{5} + 70 q^{7} - 753 q^{8} + 225 q^{10} + 605 q^{11} + 1498 q^{13} - 2113 q^{14} + 4883 q^{16} - 3874 q^{17} + 882 q^{19} - 2875 q^{20} - 1089 q^{22} + 5344 q^{23} + 3125 q^{25} + 4478 q^{26} + 12565 q^{28} - 5318 q^{29} - 7916 q^{31} - 21385 q^{32} - 18605 q^{34} - 1750 q^{35} - 1788 q^{37} + 34421 q^{38} + 18825 q^{40} - 5854 q^{41} - 4364 q^{43} + 13915 q^{44} - 33834 q^{46} - 46452 q^{47} - 34217 q^{49} - 5625 q^{50} + 3222 q^{52} - 4412 q^{53} - 15125 q^{55} - 115575 q^{56} - 58221 q^{58} - 17896 q^{59} - 35930 q^{61} + 19627 q^{62} + 14779 q^{64} - 37450 q^{65} + 73136 q^{67} + 83409 q^{68} + 52825 q^{70} - 43612 q^{71} + 142306 q^{73} + 95609 q^{74} - 6617 q^{76} + 8470 q^{77} - 46504 q^{79} - 122075 q^{80} + 175798 q^{82} - 81604 q^{83} + 96850 q^{85} + 101788 q^{86} - 91113 q^{88} - 8664 q^{89} - 203380 q^{91} - 251174 q^{92} - 71458 q^{94} - 22050 q^{95} - 22230 q^{97} - 59962 q^{98}+O(q^{100}) $ 5 * q - 9 * q^2 + 115 * q^4 - 125 * q^5 + 70 * q^7 - 753 * q^8 + 225 * q^10 + 605 * q^11 + 1498 * q^13 - 2113 * q^14 + 4883 * q^16 - 3874 * q^17 + 882 * q^19 - 2875 * q^20 - 1089 * q^22 + 5344 * q^23 + 3125 * q^25 + 4478 * q^26 + 12565 * q^28 - 5318 * q^29 - 7916 * q^31 - 21385 * q^32 - 18605 * q^34 - 1750 * q^35 - 1788 * q^37 + 34421 * q^38 + 18825 * q^40 - 5854 * q^41 - 4364 * q^43 + 13915 * q^44 - 33834 * q^46 - 46452 * q^47 - 34217 * q^49 - 5625 * q^50 + 3222 * q^52 - 4412 * q^53 - 15125 * q^55 - 115575 * q^56 - 58221 * q^58 - 17896 * q^59 - 35930 * q^61 + 19627 * q^62 + 14779 * q^64 - 37450 * q^65 + 73136 * q^67 + 83409 * q^68 + 52825 * q^70 - 43612 * q^71 + 142306 * q^73 + 95609 * q^74 - 6617 * q^76 + 8470 * q^77 - 46504 * q^79 - 122075 * q^80 + 175798 * q^82 - 81604 * q^83 + 96850 * q^85 + 101788 * q^86 - 91113 * q^88 - 8664 * q^89 - 203380 * q^91 - 251174 * q^92 - 71458 * q^94 - 22050 * q^95 - 22230 * q^97 - 59962 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−8.57133	−1.51521	−0.757605	−	0.652713i	$-0.773631\pi$
$2$	−8.57133	−1.51521	−0.757605	+	0.652713i	$0.773631\pi$
$3$	0	0
$3$	0	0
$4$	41.4676	1.29586
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	3.71474	0.0286538	0.0143269	−	0.999897i	$-0.495439\pi$
$7$	3.71474	0.0286538	0.0143269	+	0.999897i	$0.495439\pi$
$8$	−81.1502	−0.448296
$9$	0	0
$10$	214.283	0.677623
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	94.0133	0.154288	0.0771438	−	0.997020i	$-0.475420\pi$
$13$	94.0133	0.154288	0.0771438	+	0.997020i	$0.475420\pi$
$14$	−31.8402	−0.0434166
$15$	0	0
$16$	−631.400	−0.616601
$17$	−196.903	−0.165246	−0.0826229	−	0.996581i	$-0.526330\pi$
$17$	−196.903	−0.165246	−0.0826229	+	0.996581i	$0.526330\pi$
$18$	0	0
$19$	−425.349	−0.270310	−0.135155	−	0.990824i	$-0.543153\pi$
$19$	−425.349	−0.270310	−0.135155	+	0.990824i	$0.543153\pi$
$20$	−1036.69	−0.579528
$21$	0	0
$22$	−1037.13	−0.456853
$23$	−518.668	−0.204442	−0.102221	−	0.994762i	$-0.532595\pi$
$23$	−518.668	−0.204442	−0.102221	+	0.994762i	$0.532595\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−805.819	−0.233778
$27$	0	0
$28$	154.041	0.0371315
$29$	378.739	0.0836268	0.0418134	−	0.999125i	$-0.486687\pi$
$29$	378.739	0.0836268	0.0418134	+	0.999125i	$0.486687\pi$
$30$	0	0
$31$	5098.68	0.952914	0.476457	−	0.879198i	$-0.341921\pi$
$31$	5098.68	0.952914	0.476457	+	0.879198i	$0.341921\pi$
$32$	8008.74	1.38258
$33$	0	0
$34$	1687.72	0.250382
$35$	−92.8684	−0.0128144
$36$	0	0
$37$	−4418.94	−0.530656	−0.265328	−	0.964158i	$-0.585480\pi$
$37$	−4418.94	−0.530656	−0.265328	+	0.964158i	$0.585480\pi$
$38$	3645.81	0.409576
$39$	0	0
$40$	2028.76	0.200484
$41$	−12225.3	−1.13579	−0.567896	−	0.823101i	$-0.692242\pi$
$41$	−12225.3	−1.13579	−0.567896	+	0.823101i	$0.692242\pi$
$42$	0	0
$43$	−12069.9	−0.995477	−0.497738	−	0.867327i	$-0.665836\pi$
$43$	−12069.9	−0.995477	−0.497738	+	0.867327i	$0.665836\pi$
$44$	5017.58	0.390718
$45$	0	0
$46$	4445.67	0.309772
$47$	−1450.03	−0.0957483	−0.0478741	−	0.998853i	$-0.515245\pi$
$47$	−1450.03	−0.0957483	−0.0478741	+	0.998853i	$0.515245\pi$
$48$	0	0
$49$	−16793.2	−0.999179
$50$	−5357.08	−0.303042
$51$	0	0
$52$	3898.51	0.199936
$53$	24797.4	1.21259	0.606297	−	0.795238i	$-0.292654\pi$
$53$	24797.4	1.21259	0.606297	+	0.795238i	$0.292654\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	−301.452	−0.0128454
$57$	0	0
$58$	−3246.30	−0.126712
$59$	22302.7	0.834119	0.417060	−	0.908879i	$-0.363061\pi$
$59$	22302.7	0.834119	0.417060	+	0.908879i	$0.363061\pi$
$60$	0	0
$61$	52941.4	1.82167	0.910837	−	0.412766i	$-0.135437\pi$
$61$	52941.4	1.82167	0.910837	+	0.412766i	$0.135437\pi$
$62$	−43702.5	−1.44387
$63$	0	0
$64$	−48440.7	−1.47829
$65$	−2350.33	−0.0689995
$66$	0	0
$67$	65036.7	1.76999	0.884997	−	0.465597i	$-0.154161\pi$
$67$	65036.7	1.76999	0.884997	+	0.465597i	$0.154161\pi$
$68$	−8165.11	−0.214136
$69$	0	0
$70$	796.005	0.0194165
$71$	−60745.5	−1.43011	−0.715053	−	0.699071i	$-0.753597\pi$
$71$	−60745.5	−1.43011	−0.715053	+	0.699071i	$0.753597\pi$
$72$	0	0
$73$	54856.0	1.20481	0.602403	−	0.798192i	$-0.294210\pi$
$73$	54856.0	1.20481	0.602403	+	0.798192i	$0.294210\pi$
$74$	37876.1	0.804056
$75$	0	0
$76$	−17638.2	−0.350284
$77$	449.483	0.00863946
$78$	0	0
$79$	40298.7	0.726479	0.363240	−	0.931696i	$-0.381671\pi$
$79$	40298.7	0.726479	0.363240	+	0.931696i	$0.381671\pi$
$80$	15785.0	0.275752
$81$	0	0
$82$	104787.	1.72096
$83$	−74192.7	−1.18213	−0.591066	−	0.806623i	$-0.701293\pi$
$83$	−74192.7	−1.18213	−0.591066	+	0.806623i	$0.701293\pi$
$84$	0	0
$85$	4922.58	0.0739002
$86$	103455.	1.50836
$87$	0	0
$88$	−9819.17	−0.135166
$89$	−551.359	−0.00737836	−0.00368918	−	0.999993i	$-0.501174\pi$
$89$	−551.359	−0.00737836	−0.00368918	+	0.999993i	$0.501174\pi$
$90$	0	0
$91$	349.235	0.00442093
$92$	−21507.9	−0.264929
$93$	0	0
$94$	12428.6	0.145079
$95$	10633.7	0.120886
$96$	0	0
$97$	82992.9	0.895594	0.447797	−	0.894135i	$-0.352209\pi$
$97$	82992.9	0.895594	0.447797	+	0.894135i	$0.352209\pi$
$98$	143940.	1.51397
$99$	0	0
$100$	25917.3	0.259173
$101$	16805.0	0.163921	0.0819605	−	0.996636i	$-0.473882\pi$
$101$	16805.0	0.163921	0.0819605	+	0.996636i	$0.473882\pi$
$102$	0	0
$103$	−16517.4	−0.153409	−0.0767043	−	0.997054i	$-0.524440\pi$
$103$	−16517.4	−0.153409	−0.0767043	+	0.997054i	$0.524440\pi$
$104$	−7629.20	−0.0691665
$105$	0	0
$106$	−212546.	−1.83734
$107$	127625.	1.07765	0.538825	−	0.842418i	$-0.318868\pi$
$107$	127625.	1.07765	0.538825	+	0.842418i	$0.318868\pi$
$108$	0	0
$109$	−155192.	−1.25113	−0.625567	−	0.780171i	$-0.715132\pi$
$109$	−155192.	−1.25113	−0.625567	+	0.780171i	$0.715132\pi$
$110$	25928.3	0.204311
$111$	0	0
$112$	−2345.48	−0.0176680
$113$	64553.7	0.475582	0.237791	−	0.971316i	$-0.423577\pi$
$113$	64553.7	0.475582	0.237791	+	0.971316i	$0.423577\pi$
$114$	0	0
$115$	12966.7	0.0914291
$116$	15705.4	0.108369
$117$	0	0
$118$	−191164.	−1.26387
$119$	−731.444	−0.00473493
$120$	0	0
$121$	14641.0	0.0909091
$122$	−453778.	−2.76022
$123$	0	0
$124$	211430.	1.23485
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	153990.	0.847193	0.423596	−	0.905851i	$-0.360768\pi$
$127$	153990.	0.847193	0.423596	+	0.905851i	$0.360768\pi$
$128$	158922.	0.857350
$129$	0	0
$130$	20145.5	0.104549
$131$	−134029.	−0.682372	−0.341186	−	0.939996i	$-0.610829\pi$
$131$	−134029.	−0.682372	−0.341186	+	0.939996i	$0.610829\pi$
$132$	0	0
$133$	−1580.06	−0.00774541
$134$	−557451.	−2.68191
$135$	0	0
$136$	15978.7	0.0740790
$137$	−100996.	−0.459730	−0.229865	−	0.973223i	$-0.573828\pi$
$137$	−100996.	−0.459730	−0.229865	+	0.973223i	$0.573828\pi$
$138$	0	0
$139$	−298270.	−1.30940	−0.654700	−	0.755889i	$-0.727205\pi$
$139$	−298270.	−1.30940	−0.654700	+	0.755889i	$0.727205\pi$
$140$	−3851.03	−0.0166057
$141$	0	0
$142$	520669.	2.16691
$143$	11375.6	0.0465195
$144$	0	0
$145$	−9468.48	−0.0373990
$146$	−470189.	−1.82554
$147$	0	0
$148$	−183243.	−0.687658
$149$	−169128.	−0.624093	−0.312047	−	0.950067i	$-0.601014\pi$
$149$	−169128.	−0.624093	−0.312047	+	0.950067i	$0.601014\pi$
$150$	0	0
$151$	−161845.	−0.577641	−0.288821	−	0.957383i	$-0.593263\pi$
$151$	−161845.	−0.577641	−0.288821	+	0.957383i	$0.593263\pi$
$152$	34517.2	0.121179
$153$	0	0
$154$	−3852.67	−0.0130906
$155$	−127467.	−0.426156
$156$	0	0
$157$	−421890.	−1.36600	−0.683000	−	0.730419i	$-0.739325\pi$
$157$	−421890.	−1.36600	−0.683000	+	0.730419i	$0.739325\pi$
$158$	−345413.	−1.10077
$159$	0	0
$160$	−200218.	−0.618307
$161$	−1926.71	−0.00585804
$162$	0	0
$163$	−380794.	−1.12259	−0.561295	−	0.827616i	$-0.689697\pi$
$163$	−380794.	−1.12259	−0.561295	+	0.827616i	$0.689697\pi$
$164$	−506953.	−1.47183
$165$	0	0
$166$	635930.	1.79118
$167$	652565.	1.81064	0.905321	−	0.424727i	$-0.139630\pi$
$167$	652565.	1.81064	0.905321	+	0.424727i	$0.139630\pi$
$168$	0	0
$169$	−362454.	−0.976195
$170$	−42193.1	−0.111974
$171$	0	0
$172$	−500509.	−1.29000
$173$	−541735.	−1.37617	−0.688084	−	0.725631i	$-0.741548\pi$
$173$	−541735.	−1.37617	−0.688084	+	0.725631i	$0.741548\pi$
$174$	0	0
$175$	2321.71	0.00573077
$176$	−76399.3	−0.185912
$177$	0	0
$178$	4725.88	0.0111798
$179$	−185595.	−0.432945	−0.216472	−	0.976289i	$-0.569455\pi$
$179$	−185595.	−0.432945	−0.216472	+	0.976289i	$0.569455\pi$
$180$	0	0
$181$	−194784.	−0.441933	−0.220967	−	0.975281i	$-0.570921\pi$
$181$	−194784.	−0.441933	−0.220967	+	0.975281i	$0.570921\pi$
$182$	−2993.40	−0.00669865
$183$	0	0
$184$	42090.0	0.0916504
$185$	110473.	0.237317
$186$	0	0
$187$	−23825.3	−0.0498235
$188$	−60129.1	−0.124077
$189$	0	0
$190$	−91145.1	−0.183168
$191$	−71148.2	−0.141117	−0.0705586	−	0.997508i	$-0.522478\pi$
$191$	−71148.2	−0.141117	−0.0705586	+	0.997508i	$0.522478\pi$
$192$	0	0
$193$	627831.	1.21325	0.606624	−	0.794989i	$-0.292524\pi$
$193$	627831.	1.21325	0.606624	+	0.794989i	$0.292524\pi$
$194$	−711359.	−1.35701
$195$	0	0
$196$	−696374.	−1.29480
$197$	−802033.	−1.47240	−0.736201	−	0.676763i	$-0.763382\pi$
$197$	−802033.	−1.47240	−0.736201	+	0.676763i	$0.763382\pi$
$198$	0	0
$199$	−1.02974e6	−1.84329	−0.921647	−	0.388030i	$-0.873156\pi$
$199$	−1.02974e6	−1.84329	−0.921647	+	0.388030i	$0.873156\pi$
$200$	−50718.9	−0.0896592
$201$	0	0
$202$	−144041.	−0.248375
$203$	1406.92	0.00239623
$204$	0	0
$205$	305631.	0.507941
$206$	141576.	0.232446
$207$	0	0
$208$	−59360.0	−0.0951339
$209$	−51467.2	−0.0815014
$210$	0	0
$211$	−642370.	−0.993297	−0.496648	−	0.867952i	$-0.665436\pi$
$211$	−642370.	−0.993297	−0.496648	+	0.867952i	$0.665436\pi$
$212$	1.02829e6	1.57136
$213$	0	0
$214$	−1.09392e6	−1.63287
$215$	301747.	0.445191
$216$	0	0
$217$	18940.3	0.0273047
$218$	1.33020e6	1.89573
$219$	0	0
$220$	−125440.	−0.174734
$221$	−18511.5	−0.0254954
$222$	0	0
$223$	−1.29531e6	−1.74427	−0.872133	−	0.489269i	$-0.837264\pi$
$223$	−1.29531e6	−1.74427	−0.872133	+	0.489269i	$0.837264\pi$
$224$	29750.3	0.0396161
$225$	0	0
$226$	−553311.	−0.720607
$227$	−51821.2	−0.0667487	−0.0333743	−	0.999443i	$-0.510625\pi$
$227$	−51821.2	−0.0667487	−0.0333743	+	0.999443i	$0.510625\pi$
$228$	0	0
$229$	−543679.	−0.685100	−0.342550	−	0.939500i	$-0.611291\pi$
$229$	−543679.	−0.685100	−0.342550	+	0.939500i	$0.611291\pi$
$230$	−111142.	−0.138534
$231$	0	0
$232$	−30734.8	−0.0374895
$233$	−567463.	−0.684775	−0.342387	−	0.939559i	$-0.611236\pi$
$233$	−567463.	−0.684775	−0.342387	+	0.939559i	$0.611236\pi$
$234$	0	0
$235$	36250.6	0.0428199
$236$	924842.	1.08090
$237$	0	0
$238$	6269.44	0.00717442
$239$	−244565.	−0.276948	−0.138474	−	0.990366i	$-0.544220\pi$
$239$	−244565.	−0.276948	−0.138474	+	0.990366i	$0.544220\pi$
$240$	0	0
$241$	1.25296e6	1.38961	0.694806	−	0.719197i	$-0.255490\pi$
$241$	1.25296e6	1.38961	0.694806	+	0.719197i	$0.255490\pi$
$242$	−125493.	−0.137746
$243$	0	0
$244$	2.19535e6	2.36064
$245$	419830.	0.446846
$246$	0	0
$247$	−39988.5	−0.0417054
$248$	−413759.	−0.427188
$249$	0	0
$250$	133927.	0.135525
$251$	1.28427e6	1.28668	0.643340	−	0.765581i	$-0.277548\pi$
$251$	1.28427e6	1.28668	0.643340	+	0.765581i	$0.277548\pi$
$252$	0	0
$253$	−62758.8	−0.0616415
$254$	−1.31990e6	−1.28368
$255$	0	0
$256$	187934.	0.179228
$257$	−190501.	−0.179914	−0.0899569	−	0.995946i	$-0.528673\pi$
$257$	−190501.	−0.179914	−0.0899569	+	0.995946i	$0.528673\pi$
$258$	0	0
$259$	−16415.2	−0.0152053
$260$	−97462.8	−0.0894140
$261$	0	0
$262$	1.14881e6	1.03394
$263$	631347.	0.562832	0.281416	−	0.959586i	$-0.409196\pi$
$263$	631347.	0.562832	0.281416	+	0.959586i	$0.409196\pi$
$264$	0	0
$265$	−619934.	−0.542289
$266$	13543.2	0.0117359
$267$	0	0
$268$	2.69692e6	2.29367
$269$	−1.32229e6	−1.11415	−0.557077	−	0.830461i	$-0.688077\pi$
$269$	−1.32229e6	−1.11415	−0.557077	+	0.830461i	$0.688077\pi$
$270$	0	0
$271$	−2.26680e6	−1.87495	−0.937475	−	0.348053i	$-0.886843\pi$
$271$	−2.26680e6	−1.87495	−0.937475	+	0.348053i	$0.886843\pi$
$272$	124325.	0.101891
$273$	0	0
$274$	865669.	0.696588
$275$	75625.0	0.0603023
$276$	0	0
$277$	1.21741e6	0.953315	0.476658	−	0.879089i	$-0.341848\pi$
$277$	1.21741e6	0.953315	0.476658	+	0.879089i	$0.341848\pi$
$278$	2.55657e6	1.98402
$279$	0	0
$280$	7536.29	0.00574464
$281$	1.94463e6	1.46917	0.734584	−	0.678518i	$-0.237377\pi$
$281$	1.94463e6	1.46917	0.734584	+	0.678518i	$0.237377\pi$
$282$	0	0
$283$	1.61891e6	1.20159	0.600796	−	0.799402i	$-0.294850\pi$
$283$	1.61891e6	1.20159	0.600796	+	0.799402i	$0.294850\pi$
$284$	−2.51897e6	−1.85322
$285$	0	0
$286$	−97504.1	−0.0704868
$287$	−45413.6	−0.0325448
$288$	0	0
$289$	−1.38109e6	−0.972694
$290$	81157.4	0.0566674
$291$	0	0
$292$	2.27475e6	1.56126
$293$	−232000.	−0.157877	−0.0789384	−	0.996879i	$-0.525153\pi$
$293$	−232000.	−0.157877	−0.0789384	+	0.996879i	$0.525153\pi$
$294$	0	0
$295$	−557568.	−0.373029
$296$	358598.	0.237891
$297$	0	0
$298$	1.44965e6	0.945633
$299$	−48761.7	−0.0315428
$300$	0	0
$301$	−44836.4	−0.0285242
$302$	1.38723e6	0.875248
$303$	0	0
$304$	268565.	0.166673
$305$	−1.32353e6	−0.814678
$306$	0	0
$307$	−83735.6	−0.0507066	−0.0253533	−	0.999679i	$-0.508071\pi$
$307$	−83735.6	−0.0507066	−0.0253533	+	0.999679i	$0.508071\pi$
$308$	18639.0	0.0111956
$309$	0	0
$310$	1.09256e6	0.645717
$311$	−467964.	−0.274354	−0.137177	−	0.990547i	$-0.543803\pi$
$311$	−467964.	−0.274354	−0.137177	+	0.990547i	$0.543803\pi$
$312$	0	0
$313$	2.81546e6	1.62439	0.812193	−	0.583389i	$-0.198274\pi$
$313$	2.81546e6	1.62439	0.812193	+	0.583389i	$0.198274\pi$
$314$	3.61616e6	2.06978
$315$	0	0
$316$	1.67109e6	0.941418
$317$	26046.1	0.0145578	0.00727889	−	0.999974i	$-0.497683\pi$
$317$	26046.1	0.0145578	0.00727889	+	0.999974i	$0.497683\pi$
$318$	0	0
$319$	45827.5	0.0252144
$320$	1.21102e6	0.661113
$321$	0	0
$322$	16514.5	0.00887617
$323$	83752.6	0.0446675
$324$	0	0
$325$	58758.3	0.0308575
$326$	3.26391e6	1.70096
$327$	0	0
$328$	992082.	0.509170
$329$	−5386.46	−0.00274356
$330$	0	0
$331$	−3.14034e6	−1.57546	−0.787729	−	0.616022i	$-0.788743\pi$
$331$	−3.14034e6	−1.57546	−0.787729	+	0.616022i	$0.788743\pi$
$332$	−3.07660e6	−1.53188
$333$	0	0
$334$	−5.59335e6	−2.74351
$335$	−1.62592e6	−0.791565
$336$	0	0
$337$	1.01090e6	0.484880	0.242440	−	0.970166i	$-0.422052\pi$
$337$	1.01090e6	0.484880	0.242440	+	0.970166i	$0.422052\pi$
$338$	3.10672e6	1.47914
$339$	0	0
$340$	204128.	0.0957646
$341$	616941.	0.287315
$342$	0	0
$343$	−124816.	−0.0572842
$344$	979472.	0.446268
$345$	0	0
$346$	4.64339e6	2.08518
$347$	−1.73452e6	−0.773315	−0.386657	−	0.922223i	$-0.626370\pi$
$347$	−1.73452e6	−0.773315	−0.386657	+	0.922223i	$0.626370\pi$
$348$	0	0
$349$	−839301.	−0.368854	−0.184427	−	0.982846i	$-0.559043\pi$
$349$	−839301.	−0.368854	−0.184427	+	0.982846i	$0.559043\pi$
$350$	−19900.1	−0.00868332
$351$	0	0
$352$	969057.	0.416862
$353$	−1.53771e6	−0.656808	−0.328404	−	0.944537i	$-0.606511\pi$
$353$	−1.53771e6	−0.656808	−0.328404	+	0.944537i	$0.606511\pi$
$354$	0	0
$355$	1.51864e6	0.639563
$356$	−22863.6	−0.00956135
$357$	0	0
$358$	1.59079e6	0.656003
$359$	−3.58199e6	−1.46686	−0.733430	−	0.679765i	$-0.762082\pi$
$359$	−3.58199e6	−1.46686	−0.733430	+	0.679765i	$0.762082\pi$
$360$	0	0
$361$	−2.29518e6	−0.926933
$362$	1.66956e6	0.669622
$363$	0	0
$364$	14481.9	0.00572893
$365$	−1.37140e6	−0.538806
$366$	0	0
$367$	−2.65471e6	−1.02885	−0.514424	−	0.857536i	$-0.671994\pi$
$367$	−2.65471e6	−1.02885	−0.514424	+	0.857536i	$0.671994\pi$
$368$	327487.	0.126059
$369$	0	0
$370$	−946904.	−0.359585
$371$	92115.6	0.0347455
$372$	0	0
$373$	1.09038e6	0.405793	0.202897	−	0.979200i	$-0.434964\pi$
$373$	1.09038e6	0.405793	0.202897	+	0.979200i	$0.434964\pi$
$374$	204214.	0.0754931
$375$	0	0
$376$	117670.	0.0429236
$377$	35606.5	0.0129026
$378$	0	0
$379$	2.62200e6	0.937637	0.468819	−	0.883295i	$-0.344680\pi$
$379$	2.62200e6	0.937637	0.468819	+	0.883295i	$0.344680\pi$
$380$	440955.	0.156652
$381$	0	0
$382$	609834.	0.213822
$383$	2.67839e6	0.932990	0.466495	−	0.884524i	$-0.345516\pi$
$383$	2.67839e6	0.932990	0.466495	+	0.884524i	$0.345516\pi$
$384$	0	0
$385$	−11237.1	−0.00386368
$386$	−5.38134e6	−1.83833
$387$	0	0
$388$	3.44152e6	1.16057
$389$	3.20686e6	1.07450	0.537250	−	0.843423i	$-0.319463\pi$
$389$	3.20686e6	1.07450	0.537250	+	0.843423i	$0.319463\pi$
$390$	0	0
$391$	102127.	0.0337832
$392$	1.36277e6	0.447928
$393$	0	0
$394$	6.87448e6	2.23100
$395$	−1.00747e6	−0.324891
$396$	0	0
$397$	−4.87414e6	−1.55211	−0.776053	−	0.630668i	$-0.782781\pi$
$397$	−4.87414e6	−1.55211	−0.776053	+	0.630668i	$0.782781\pi$
$398$	8.82623e6	2.79298
$399$	0	0
$400$	−394625.	−0.123320
$401$	−3.60536e6	−1.11966	−0.559831	−	0.828606i	$-0.689134\pi$
$401$	−3.60536e6	−1.11966	−0.559831	+	0.828606i	$0.689134\pi$
$402$	0	0
$403$	479344.	0.147023
$404$	696863.	0.212419
$405$	0	0
$406$	−12059.1	−0.00363079
$407$	−534691.	−0.159999
$408$	0	0
$409$	−1.61929e6	−0.478649	−0.239325	−	0.970940i	$-0.576926\pi$
$409$	−1.61929e6	−0.478649	−0.239325	+	0.970940i	$0.576926\pi$
$410$	−2.61967e6	−0.769638
$411$	0	0
$412$	−684939.	−0.198797
$413$	82848.8	0.0239007
$414$	0	0
$415$	1.85482e6	0.528666
$416$	752928.	0.213314
$417$	0	0
$418$	441142.	0.123492
$419$	−3.67267e6	−1.02199	−0.510995	−	0.859583i	$-0.670723\pi$
$419$	−3.67267e6	−1.02199	−0.510995	+	0.859583i	$0.670723\pi$
$420$	0	0
$421$	3.38719e6	0.931395	0.465697	−	0.884944i	$-0.345804\pi$
$421$	3.38719e6	0.931395	0.465697	+	0.884944i	$0.345804\pi$
$422$	5.50596e6	1.50505
$423$	0	0
$424$	−2.01231e6	−0.543601
$425$	−123065.	−0.0330492
$426$	0	0
$427$	196663.	0.0521980
$428$	5.29233e6	1.39649
$429$	0	0
$430$	−2.58637e6	−0.674558
$431$	4.13106e6	1.07120	0.535598	−	0.844473i	$-0.320086\pi$
$431$	4.13106e6	1.07120	0.535598	+	0.844473i	$0.320086\pi$
$432$	0	0
$433$	−5.97405e6	−1.53126	−0.765631	−	0.643280i	$-0.777573\pi$
$433$	−5.97405e6	−1.53126	−0.765631	+	0.643280i	$0.777573\pi$
$434$	−162343.	−0.0413723
$435$	0	0
$436$	−6.43545e6	−1.62130
$437$	220615.	0.0552626
$438$	0	0
$439$	−3.38748e6	−0.838911	−0.419456	−	0.907776i	$-0.637779\pi$
$439$	−3.38748e6	−0.838911	−0.419456	+	0.907776i	$0.637779\pi$
$440$	245479.	0.0604482
$441$	0	0
$442$	158668.	0.0386309
$443$	3.50641e6	0.848893	0.424447	−	0.905453i	$-0.360469\pi$
$443$	3.50641e6	0.848893	0.424447	+	0.905453i	$0.360469\pi$
$444$	0	0
$445$	13784.0	0.00329970
$446$	1.11026e7	2.64293
$447$	0	0
$448$	−179945.	−0.0423588
$449$	−7.19124e6	−1.68340	−0.841701	−	0.539943i	$-0.818446\pi$
$449$	−7.19124e6	−1.68340	−0.841701	+	0.539943i	$0.818446\pi$
$450$	0	0
$451$	−1.47926e6	−0.342454
$452$	2.67689e6	0.616289
$453$	0	0
$454$	444176.	0.101138
$455$	−8730.87	−0.00197710
$456$	0	0
$457$	6.32089e6	1.41575	0.707877	−	0.706336i	$-0.249653\pi$
$457$	6.32089e6	1.41575	0.707877	+	0.706336i	$0.249653\pi$
$458$	4.66005e6	1.03807
$459$	0	0
$460$	537698.	0.118480
$461$	5.31660e6	1.16515	0.582574	−	0.812777i	$-0.302045\pi$
$461$	5.31660e6	1.16515	0.582574	+	0.812777i	$0.302045\pi$
$462$	0	0
$463$	2.17724e6	0.472013	0.236007	−	0.971751i	$-0.424161\pi$
$463$	2.17724e6	0.472013	0.236007	+	0.971751i	$0.424161\pi$
$464$	−239136.	−0.0515644
$465$	0	0
$466$	4.86391e6	1.03758
$467$	−67604.5	−0.0143444	−0.00717222	−	0.999974i	$-0.502283\pi$
$467$	−67604.5	−0.0143444	−0.00717222	+	0.999974i	$0.502283\pi$
$468$	0	0
$469$	241594.	0.0507171
$470$	−310716.	−0.0648812
$471$	0	0
$472$	−1.80987e6	−0.373932
$473$	−1.46045e6	−0.300148
$474$	0	0
$475$	−265843.	−0.0540619
$476$	−30331.2	−0.00613582
$477$	0	0
$478$	2.09624e6	0.419635
$479$	−2.70352e6	−0.538382	−0.269191	−	0.963087i	$-0.586756\pi$
$479$	−2.70352e6	−0.538382	−0.269191	+	0.963087i	$0.586756\pi$
$480$	0	0
$481$	−415439.	−0.0818737
$482$	−1.07395e7	−2.10556
$483$	0	0
$484$	607128.	0.117806
$485$	−2.07482e6	−0.400522
$486$	0	0
$487$	−9.64211e6	−1.84225	−0.921127	−	0.389262i	$-0.872730\pi$
$487$	−9.64211e6	−1.84225	−0.921127	+	0.389262i	$0.872730\pi$
$488$	−4.29620e6	−0.816649
$489$	0	0
$490$	−3.59850e6	−0.677066
$491$	−6.66155e6	−1.24701	−0.623507	−	0.781818i	$-0.714293\pi$
$491$	−6.66155e6	−1.24701	−0.623507	+	0.781818i	$0.714293\pi$
$492$	0	0
$493$	−74575.0	−0.0138190
$494$	342754.	0.0631925
$495$	0	0
$496$	−3.21931e6	−0.587568
$497$	−225653.	−0.0409780
$498$	0	0
$499$	4.19798e6	0.754725	0.377362	−	0.926066i	$-0.376831\pi$
$499$	4.19798e6	0.754725	0.377362	+	0.926066i	$0.376831\pi$
$500$	−647932.	−0.115906
$501$	0	0
$502$	−1.10079e7	−1.94959
$503$	4.23018e6	0.745486	0.372743	−	0.927935i	$-0.378417\pi$
$503$	4.23018e6	0.745486	0.372743	+	0.927935i	$0.378417\pi$
$504$	0	0
$505$	−420124.	−0.0733077
$506$	537926.	0.0933999
$507$	0	0
$508$	6.38559e6	1.09785
$509$	5.17751e6	0.885782	0.442891	−	0.896575i	$-0.353953\pi$
$509$	5.17751e6	0.885782	0.442891	+	0.896575i	$0.353953\pi$
$510$	0	0
$511$	203776.	0.0345223
$512$	−6.69634e6	−1.12892
$513$	0	0
$514$	1.63285e6	0.272607
$515$	412936.	0.0686064
$516$	0	0
$517$	−175453.	−0.0288692
$518$	140700.	0.0230393
$519$	0	0
$520$	190730.	0.0309322
$521$	−7.76284e6	−1.25293	−0.626465	−	0.779450i	$-0.715499\pi$
$521$	−7.76284e6	−1.25293	−0.626465	+	0.779450i	$0.715499\pi$
$522$	0	0
$523$	3.35346e6	0.536092	0.268046	−	0.963406i	$-0.413622\pi$
$523$	3.35346e6	0.536092	0.268046	+	0.963406i	$0.413622\pi$
$524$	−5.55788e6	−0.884262
$525$	0	0
$526$	−5.41148e6	−0.852808
$527$	−1.00395e6	−0.157465
$528$	0	0
$529$	−6.16733e6	−0.958204
$530$	5.31366e6	0.821682
$531$	0	0
$532$	−65521.3	−0.0100370
$533$	−1.14934e6	−0.175238
$534$	0	0
$535$	−3.19064e6	−0.481940
$536$	−5.27774e6	−0.793480
$537$	0	0
$538$	1.13338e7	1.68818
$539$	−2.03198e6	−0.301264
$540$	0	0
$541$	2.87042e6	0.421650	0.210825	−	0.977524i	$-0.432385\pi$
$541$	2.87042e6	0.421650	0.210825	+	0.977524i	$0.432385\pi$
$542$	1.94295e7	2.84094
$543$	0	0
$544$	−1.57695e6	−0.228465
$545$	3.87981e6	0.559524
$546$	0	0
$547$	−4.40734e6	−0.629808	−0.314904	−	0.949124i	$-0.601972\pi$
$547$	−4.40734e6	−0.629808	−0.314904	+	0.949124i	$0.601972\pi$
$548$	−4.18806e6	−0.595747
$549$	0	0
$550$	−648207.	−0.0913706
$551$	−161096.	−0.0226051
$552$	0	0
$553$	149699.	0.0208164
$554$	−1.04348e7	−1.44447
$555$	0	0
$556$	−1.23685e7	−1.69680
$557$	−1.15892e7	−1.58276	−0.791379	−	0.611325i	$-0.790637\pi$
$557$	−1.15892e7	−1.58276	−0.791379	+	0.611325i	$0.790637\pi$
$558$	0	0
$559$	−1.13473e6	−0.153590
$560$	58637.1	0.00790137
$561$	0	0
$562$	−1.66681e7	−2.22610
$563$	2.32310e6	0.308885	0.154443	−	0.988002i	$-0.450642\pi$
$563$	2.32310e6	0.308885	0.154443	+	0.988002i	$0.450642\pi$
$564$	0	0
$565$	−1.61384e6	−0.212687
$566$	−1.38762e7	−1.82067
$567$	0	0
$568$	4.92951e6	0.641110
$569$	1.40597e6	0.182052	0.0910260	−	0.995849i	$-0.470985\pi$
$569$	1.40597e6	0.182052	0.0910260	+	0.995849i	$0.470985\pi$
$570$	0	0
$571$	−5.52531e6	−0.709196	−0.354598	−	0.935019i	$-0.615382\pi$
$571$	−5.52531e6	−0.709196	−0.354598	+	0.935019i	$0.615382\pi$
$572$	471720.	0.0602829
$573$	0	0
$574$	389255.	0.0493122
$575$	−324167.	−0.0408884
$576$	0	0
$577$	3.12727e6	0.391044	0.195522	−	0.980699i	$-0.437360\pi$
$577$	3.12727e6	0.391044	0.195522	+	0.980699i	$0.437360\pi$
$578$	1.18377e7	1.47384
$579$	0	0
$580$	−392636.	−0.0484640
$581$	−275606.	−0.0338726
$582$	0	0
$583$	3.00048e6	0.365611
$584$	−4.45158e6	−0.540110
$585$	0	0
$586$	1.98854e6	0.239216
$587$	−7.45555e6	−0.893068	−0.446534	−	0.894767i	$-0.647342\pi$
$587$	−7.45555e6	−0.893068	−0.446534	+	0.894767i	$0.647342\pi$
$588$	0	0
$589$	−2.16872e6	−0.257582
$590$	4.77910e6	0.565218
$591$	0	0
$592$	2.79011e6	0.327203
$593$	−7.46375e6	−0.871607	−0.435803	−	0.900042i	$-0.643536\pi$
$593$	−7.46375e6	−0.871607	−0.435803	+	0.900042i	$0.643536\pi$
$594$	0	0
$595$	18286.1	0.00211752
$596$	−7.01333e6	−0.808740
$597$	0	0
$598$	417952.	0.0477940
$599$	−314190.	−0.0357787	−0.0178894	−	0.999840i	$-0.505695\pi$
$599$	−314190.	−0.0357787	−0.0178894	+	0.999840i	$0.505695\pi$
$600$	0	0
$601$	1.42821e7	1.61290	0.806450	−	0.591303i	$-0.201386\pi$
$601$	1.42821e7	1.61290	0.806450	+	0.591303i	$0.201386\pi$
$602$	384307.	0.0432202
$603$	0	0
$604$	−6.71135e6	−0.748544
$605$	−366025.	−0.0406558
$606$	0	0
$607$	−188944.	−0.0208142	−0.0104071	−	0.999946i	$-0.503313\pi$
$607$	−188944.	−0.0208142	−0.0104071	+	0.999946i	$0.503313\pi$
$608$	−3.40651e6	−0.373724
$609$	0	0
$610$	1.13444e7	1.23441
$611$	−136322.	−0.0147728
$612$	0	0
$613$	3.63784e6	0.391014	0.195507	−	0.980702i	$-0.437365\pi$
$613$	3.63784e6	0.391014	0.195507	+	0.980702i	$0.437365\pi$
$614$	717725.	0.0768312
$615$	0	0
$616$	−36475.6	−0.00387303
$617$	−6.34013e6	−0.670479	−0.335240	−	0.942133i	$-0.608817\pi$
$617$	−6.34013e6	−0.670479	−0.335240	+	0.942133i	$0.608817\pi$
$618$	0	0
$619$	−1.42152e6	−0.149117	−0.0745583	−	0.997217i	$-0.523755\pi$
$619$	−1.42152e6	−0.149117	−0.0745583	+	0.997217i	$0.523755\pi$
$620$	−5.28576e6	−0.552240
$621$	0	0
$622$	4.01107e6	0.415704
$623$	−2048.15	−0.000211418 0
$624$	0	0
$625$	390625.	0.0400000
$626$	−2.41323e7	−2.46129
$627$	0	0
$628$	−1.74948e7	−1.77015
$629$	870103.	0.0876888
$630$	0	0
$631$	1.68033e7	1.68005	0.840025	−	0.542547i	$-0.182540\pi$
$631$	1.68033e7	1.68005	0.840025	+	0.542547i	$0.182540\pi$
$632$	−3.27025e6	−0.325678
$633$	0	0
$634$	−223250.	−0.0220581
$635$	−3.84974e6	−0.378876
$636$	0	0
$637$	−1.57878e6	−0.154161
$638$	−392802.	−0.0382052
$639$	0	0
$640$	−3.97304e6	−0.383418
$641$	3.74097e6	0.359616	0.179808	−	0.983702i	$-0.442452\pi$
$641$	3.74097e6	0.359616	0.179808	+	0.983702i	$0.442452\pi$
$642$	0	0
$643$	−5.88753e6	−0.561573	−0.280786	−	0.959770i	$-0.590595\pi$
$643$	−5.88753e6	−0.561573	−0.280786	+	0.959770i	$0.590595\pi$
$644$	−79896.3	−0.00759122
$645$	0	0
$646$	−717871.	−0.0676807
$647$	−5.19398e6	−0.487797	−0.243899	−	0.969801i	$-0.578426\pi$
$647$	−5.19398e6	−0.487797	−0.243899	+	0.969801i	$0.578426\pi$
$648$	0	0
$649$	2.69863e6	0.251496
$650$	−503637.	−0.0467556
$651$	0	0
$652$	−1.57906e7	−1.45472
$653$	5.73999e6	0.526779	0.263390	−	0.964690i	$-0.415160\pi$
$653$	5.73999e6	0.526779	0.263390	+	0.964690i	$0.415160\pi$
$654$	0	0
$655$	3.35073e6	0.305166
$656$	7.71902e6	0.700330
$657$	0	0
$658$	46169.1	0.00415707
$659$	1.18769e7	1.06534	0.532671	−	0.846323i	$-0.321188\pi$
$659$	1.18769e7	1.06534	0.532671	+	0.846323i	$0.321188\pi$
$660$	0	0
$661$	−1.10514e7	−0.983813	−0.491907	−	0.870648i	$-0.663700\pi$
$661$	−1.10514e7	−0.983813	−0.491907	+	0.870648i	$0.663700\pi$
$662$	2.69169e7	2.38715
$663$	0	0
$664$	6.02075e6	0.529945
$665$	39501.5	0.00346385
$666$	0	0
$667$	−196440.	−0.0170968
$668$	2.70603e7	2.34635
$669$	0	0
$670$	1.39363e7	1.19939
$671$	6.40591e6	0.549255
$672$	0	0
$673$	1.81997e7	1.54891	0.774454	−	0.632631i	$-0.218025\pi$
$673$	1.81997e7	1.54891	0.774454	+	0.632631i	$0.218025\pi$
$674$	−8.66477e6	−0.734695
$675$	0	0
$676$	−1.50301e7	−1.26502
$677$	1.94815e6	0.163362	0.0816810	−	0.996659i	$-0.473971\pi$
$677$	1.94815e6	0.163362	0.0816810	+	0.996659i	$0.473971\pi$
$678$	0	0
$679$	308297.	0.0256622
$680$	−399468.	−0.0331291
$681$	0	0
$682$	−5.28800e6	−0.435342
$683$	−2.17374e7	−1.78302	−0.891510	−	0.453001i	$-0.850354\pi$
$683$	−2.17374e7	−1.78302	−0.891510	+	0.453001i	$0.850354\pi$
$684$	0	0
$685$	2.52490e6	0.205597
$686$	1.06984e6	0.0867976
$687$	0	0
$688$	7.62090e6	0.613812
$689$	2.33128e6	0.187088
$690$	0	0
$691$	−1.99747e6	−0.159142	−0.0795711	−	0.996829i	$-0.525355\pi$
$691$	−1.99747e6	−0.159142	−0.0795711	+	0.996829i	$0.525355\pi$
$692$	−2.24645e7	−1.78333
$693$	0	0
$694$	1.48672e7	1.17174
$695$	7.45675e6	0.585582
$696$	0	0
$697$	2.40719e6	0.187685
$698$	7.19393e6	0.558891
$699$	0	0
$700$	96275.8	0.00742629
$701$	−6.76258e6	−0.519778	−0.259889	−	0.965639i	$-0.583686\pi$
$701$	−6.76258e6	−0.519778	−0.259889	+	0.965639i	$0.583686\pi$
$702$	0	0
$703$	1.87959e6	0.143442
$704$	−5.86133e6	−0.445722
$705$	0	0
$706$	1.31802e7	0.995203
$707$	62426.1	0.00469697
$708$	0	0
$709$	−1.31058e7	−0.979148	−0.489574	−	0.871962i	$-0.662848\pi$
$709$	−1.31058e7	−0.979148	−0.489574	+	0.871962i	$0.662848\pi$
$710$	−1.30167e7	−0.969072
$711$	0	0
$712$	44742.9	0.00330769
$713$	−2.64452e6	−0.194816
$714$	0	0
$715$	−284390.	−0.0208041
$716$	−7.69617e6	−0.561037
$717$	0	0
$718$	3.07024e7	2.22260
$719$	−1.24991e7	−0.901691	−0.450845	−	0.892602i	$-0.648877\pi$
$719$	−1.24991e7	−0.901691	−0.450845	+	0.892602i	$0.648877\pi$
$720$	0	0
$721$	−61357.9	−0.00439574
$722$	1.96727e7	1.40450
$723$	0	0
$724$	−8.07723e6	−0.572685
$725$	236712.	0.0167254
$726$	0	0
$727$	1.55739e7	1.09285	0.546424	−	0.837508i	$-0.315989\pi$
$727$	1.55739e7	1.09285	0.546424	+	0.837508i	$0.315989\pi$
$728$	−28340.5	−0.00198189
$729$	0	0
$730$	1.17547e7	0.816404
$731$	2.37660e6	0.164498
$732$	0	0
$733$	926307.	0.0636789	0.0318394	−	0.999493i	$-0.489863\pi$
$733$	926307.	0.0636789	0.0318394	+	0.999493i	$0.489863\pi$
$734$	2.27544e7	1.55892
$735$	0	0
$736$	−4.15387e6	−0.282656
$737$	7.86944e6	0.533673
$738$	0	0
$739$	3.44944e6	0.232347	0.116174	−	0.993229i	$-0.462937\pi$
$739$	3.44944e6	0.232347	0.116174	+	0.993229i	$0.462937\pi$
$740$	4.58107e6	0.307530
$741$	0	0
$742$	−789553.	−0.0526468
$743$	4.52822e6	0.300923	0.150462	−	0.988616i	$-0.451924\pi$
$743$	4.52822e6	0.300923	0.150462	+	0.988616i	$0.451924\pi$
$744$	0	0
$745$	4.22820e6	0.279103
$746$	−9.34598e6	−0.614862
$747$	0	0
$748$	−987979.	−0.0645645
$749$	474095.	0.0308788
$750$	0	0
$751$	1.53285e7	0.991745	0.495873	−	0.868395i	$-0.334848\pi$
$751$	1.53285e7	0.991745	0.495873	+	0.868395i	$0.334848\pi$
$752$	915545.	0.0590385
$753$	0	0
$754$	−305195.	−0.0195501
$755$	4.04614e6	0.258329
$756$	0	0
$757$	1.46683e7	0.930339	0.465169	−	0.885222i	$-0.345993\pi$
$757$	1.46683e7	0.930339	0.465169	+	0.885222i	$0.345993\pi$
$758$	−2.24740e7	−1.42072
$759$	0	0
$760$	−862929.	−0.0541927
$761$	6.76195e6	0.423263	0.211632	−	0.977350i	$-0.432122\pi$
$761$	6.76195e6	0.423263	0.211632	+	0.977350i	$0.432122\pi$
$762$	0	0
$763$	−576498.	−0.0358498
$764$	−2.95035e6	−0.182869
$765$	0	0
$766$	−2.29574e7	−1.41368
$767$	2.09675e6	0.128694
$768$	0	0
$769$	−2.65717e7	−1.62033	−0.810166	−	0.586201i	$-0.800623\pi$
$769$	−2.65717e7	−1.62033	−0.810166	+	0.586201i	$0.800623\pi$
$770$	96316.7	0.00585429
$771$	0	0
$772$	2.60347e7	1.57220
$773$	1.30745e7	0.787000	0.393500	−	0.919325i	$-0.371264\pi$
$773$	1.30745e7	0.787000	0.393500	+	0.919325i	$0.371264\pi$
$774$	0	0
$775$	3.18668e6	0.190583
$776$	−6.73489e6	−0.401491
$777$	0	0
$778$	−2.74871e7	−1.62809
$779$	5.20000e6	0.307015
$780$	0	0
$781$	−7.35020e6	−0.431193
$782$	−875367.	−0.0511886
$783$	0	0
$784$	1.06032e7	0.616095
$785$	1.05473e7	0.610893
$786$	0	0
$787$	3.73173e6	0.214770	0.107385	−	0.994218i	$-0.465752\pi$
$787$	3.73173e6	0.214770	0.107385	+	0.994218i	$0.465752\pi$
$788$	−3.32584e7	−1.90803
$789$	0	0
$790$	8.63533e6	0.492279
$791$	239800.	0.0136272
$792$	0	0
$793$	4.97720e6	0.281062
$794$	4.17778e7	2.35177
$795$	0	0
$796$	−4.27008e7	−2.38866
$797$	−2.53825e7	−1.41543	−0.707714	−	0.706499i	$-0.750273\pi$
$797$	−2.53825e7	−1.41543	−0.707714	+	0.706499i	$0.750273\pi$
$798$	0	0
$799$	285515.	0.0158220
$800$	5.00546e6	0.276515
$801$	0	0
$802$	3.09027e7	1.69653
$803$	6.63758e6	0.363263
$804$	0	0
$805$	48167.8	0.00261980
$806$	−4.10862e6	−0.222771
$807$	0	0
$808$	−1.36373e6	−0.0734851
$809$	2.24328e7	1.20507	0.602534	−	0.798093i	$-0.294158\pi$
$809$	2.24328e7	1.20507	0.602534	+	0.798093i	$0.294158\pi$
$810$	0	0
$811$	−2.48963e7	−1.32918	−0.664588	−	0.747210i	$-0.731393\pi$
$811$	−2.48963e7	−1.32918	−0.664588	+	0.747210i	$0.731393\pi$
$812$	58341.5	0.00310519
$813$	0	0
$814$	4.58301e6	0.242432
$815$	9.51986e6	0.502038
$816$	0	0
$817$	5.13390e6	0.269087
$818$	1.38795e7	0.725255
$819$	0	0
$820$	1.26738e7	0.658223
$821$	4.11968e6	0.213307	0.106654	−	0.994296i	$-0.465986\pi$
$821$	4.11968e6	0.213307	0.106654	+	0.994296i	$0.465986\pi$
$822$	0	0
$823$	−1.14381e7	−0.588648	−0.294324	−	0.955706i	$-0.595095\pi$
$823$	−1.14381e7	−0.588648	−0.294324	+	0.955706i	$0.595095\pi$
$824$	1.34039e6	0.0687724
$825$	0	0
$826$	−710124.	−0.0362146
$827$	−3.28841e6	−0.167195	−0.0835974	−	0.996500i	$-0.526641\pi$
$827$	−3.28841e6	−0.167195	−0.0835974	+	0.996500i	$0.526641\pi$
$828$	0	0
$829$	−8.29845e6	−0.419383	−0.209691	−	0.977768i	$-0.567246\pi$
$829$	−8.29845e6	−0.419383	−0.209691	+	0.977768i	$0.567246\pi$
$830$	−1.58982e7	−0.801040
$831$	0	0
$832$	−4.55407e6	−0.228082
$833$	3.30664e6	0.165110
$834$	0	0
$835$	−1.63141e7	−0.809744
$836$	−2.13422e6	−0.105615
$837$	0	0
$838$	3.14797e7	1.54853
$839$	1.43568e7	0.704130	0.352065	−	0.935976i	$-0.385480\pi$
$839$	1.43568e7	0.704130	0.352065	+	0.935976i	$0.385480\pi$
$840$	0	0
$841$	−2.03677e7	−0.993007
$842$	−2.90327e7	−1.41126
$843$	0	0
$844$	−2.66376e7	−1.28718
$845$	9.06136e6	0.436568
$846$	0	0
$847$	54387.5	0.00260489
$848$	−1.56570e7	−0.747687
$849$	0	0
$850$	1.05483e6	0.0500765
$851$	2.29196e6	0.108488
$852$	0	0
$853$	−2.78233e7	−1.30929	−0.654646	−	0.755935i	$-0.727182\pi$
$853$	−2.78233e7	−1.30929	−0.654646	+	0.755935i	$0.727182\pi$
$854$	−1.68567e6	−0.0790909
$855$	0	0
$856$	−1.03568e7	−0.483106
$857$	−1.42355e7	−0.662096	−0.331048	−	0.943614i	$-0.607402\pi$
$857$	−1.42355e7	−0.662096	−0.331048	+	0.943614i	$0.607402\pi$
$858$	0	0
$859$	4.22812e6	0.195508	0.0977540	−	0.995211i	$-0.468834\pi$
$859$	4.22812e6	0.195508	0.0977540	+	0.995211i	$0.468834\pi$
$860$	1.25127e7	0.576907
$861$	0	0
$862$	−3.54087e7	−1.62309
$863$	−4.11087e7	−1.87891	−0.939456	−	0.342670i	$-0.888669\pi$
$863$	−4.11087e7	−1.87891	−0.939456	+	0.342670i	$0.888669\pi$
$864$	0	0
$865$	1.35434e7	0.615441
$866$	5.12056e7	2.32018
$867$	0	0
$868$	785408.	0.0353831
$869$	4.87614e6	0.219042
$870$	0	0
$871$	6.11432e6	0.273088
$872$	1.25939e7	0.560878
$873$	0	0
$874$	−1.89096e6	−0.0837344
$875$	−58042.8	−0.00256288
$876$	0	0
$877$	−1.50841e6	−0.0662247	−0.0331124	−	0.999452i	$-0.510542\pi$
$877$	−1.50841e6	−0.0662247	−0.0331124	+	0.999452i	$0.510542\pi$
$878$	2.90352e7	1.27113
$879$	0	0
$880$	1.90998e6	0.0831425
$881$	−3.81167e7	−1.65454	−0.827268	−	0.561808i	$-0.810106\pi$
$881$	−3.81167e7	−1.65454	−0.827268	+	0.561808i	$0.810106\pi$
$882$	0	0
$883$	−2.07323e7	−0.894842	−0.447421	−	0.894323i	$-0.647657\pi$
$883$	−2.07323e7	−0.894842	−0.447421	+	0.894323i	$0.647657\pi$
$884$	−767629.	−0.0330385
$885$	0	0
$886$	−3.00546e7	−1.28625
$887$	2.93790e7	1.25380	0.626900	−	0.779100i	$-0.284323\pi$
$887$	2.93790e7	1.25380	0.626900	+	0.779100i	$0.284323\pi$
$888$	0	0
$889$	572031.	0.0242753
$890$	−118147.	−0.00499974
$891$	0	0
$892$	−5.37136e7	−2.26033
$893$	616767.	0.0258817
$894$	0	0
$895$	4.63986e6	0.193619
$896$	590352.	0.0245664
$897$	0	0
$898$	6.16385e7	2.55071
$899$	1.93107e6	0.0796892
$900$	0	0
$901$	−4.88268e6	−0.200376
$902$	1.26792e7	0.518890
$903$	0	0
$904$	−5.23855e6	−0.213201
$905$	4.86960e6	0.197639
$906$	0	0
$907$	2.30519e7	0.930441	0.465220	−	0.885195i	$-0.345975\pi$
$907$	2.30519e7	0.930441	0.465220	+	0.885195i	$0.345975\pi$
$908$	−2.14890e6	−0.0864972
$909$	0	0
$910$	74835.1	0.00299573
$911$	−2.54013e7	−1.01405	−0.507025	−	0.861931i	$-0.669255\pi$
$911$	−2.54013e7	−1.01405	−0.507025	+	0.861931i	$0.669255\pi$
$912$	0	0
$913$	−8.97732e6	−0.356426
$914$	−5.41784e7	−2.14516
$915$	0	0
$916$	−2.25451e7	−0.887797
$917$	−497883.	−0.0195526
$918$	0	0
$919$	250887.	0.00979919	0.00489959	−	0.999988i	$-0.498440\pi$
$919$	250887.	0.00979919	0.00489959	+	0.999988i	$0.498440\pi$
$920$	−1.05225e6	−0.0409873
$921$	0	0
$922$	−4.55703e7	−1.76545
$923$	−5.71088e6	−0.220648
$924$	0	0
$925$	−2.76183e6	−0.106131
$926$	−1.86619e7	−0.715200
$927$	0	0
$928$	3.03322e6	0.115620
$929$	−2.87097e7	−1.09141	−0.545707	−	0.837976i	$-0.683739\pi$
$929$	−2.87097e7	−1.09141	−0.545707	+	0.837976i	$0.683739\pi$
$930$	0	0
$931$	7.14297e6	0.270088
$932$	−2.35313e7	−0.887375
$933$	0	0
$934$	579460.	0.0217348
$935$	595632.	0.0222817
$936$	0	0
$937$	4.58299e7	1.70530	0.852649	−	0.522484i	$-0.174995\pi$
$937$	4.58299e7	1.70530	0.852649	+	0.522484i	$0.174995\pi$
$938$	−2.07078e6	−0.0768471
$939$	0	0
$940$	1.50323e6	0.0554888
$941$	−1.33338e7	−0.490887	−0.245443	−	0.969411i	$-0.578934\pi$
$941$	−1.33338e7	−0.490887	−0.245443	+	0.969411i	$0.578934\pi$
$942$	0	0
$943$	6.34085e6	0.232203
$944$	−1.40819e7	−0.514319
$945$	0	0
$946$	1.25180e7	0.454787
$947$	3.58301e7	1.29829	0.649147	−	0.760663i	$-0.275126\pi$
$947$	3.58301e7	1.29829	0.649147	+	0.760663i	$0.275126\pi$
$948$	0	0
$949$	5.15720e6	0.185887
$950$	2.27863e6	0.0819152
$951$	0	0
$952$	59356.8	0.00212265
$953$	−1.71351e7	−0.611158	−0.305579	−	0.952167i	$-0.598850\pi$
$953$	−1.71351e7	−0.611158	−0.305579	+	0.952167i	$0.598850\pi$
$954$	0	0
$955$	1.77870e6	0.0631096
$956$	−1.01415e7	−0.358887
$957$	0	0
$958$	2.31727e7	0.815761
$959$	−375173.	−0.0131730
$960$	0	0
$961$	−2.63256e6	−0.0919540
$962$	3.56086e6	0.124056
$963$	0	0
$964$	5.19572e7	1.80075
$965$	−1.56958e7	−0.542581
$966$	0	0
$967$	−5.18927e7	−1.78460	−0.892298	−	0.451448i	$-0.850908\pi$
$967$	−5.18927e7	−1.78460	−0.892298	+	0.451448i	$0.850908\pi$
$968$	−1.18812e6	−0.0407542
$969$	0	0
$970$	1.77840e7	0.606875
$971$	8.70846e6	0.296410	0.148205	−	0.988957i	$-0.452650\pi$
$971$	8.70846e6	0.296410	0.148205	+	0.988957i	$0.452650\pi$
$972$	0	0
$973$	−1.10799e6	−0.0375193
$974$	8.26456e7	2.79140
$975$	0	0
$976$	−3.34272e7	−1.12325
$977$	−2.09303e7	−0.701519	−0.350760	−	0.936466i	$-0.614077\pi$
$977$	−2.09303e7	−0.701519	−0.350760	+	0.936466i	$0.614077\pi$
$978$	0	0
$979$	−66714.5	−0.00222466
$980$	1.74094e7	0.579052
$981$	0	0
$982$	5.70983e7	1.88949
$983$	−1.93101e7	−0.637382	−0.318691	−	0.947859i	$-0.603243\pi$
$983$	−1.93101e7	−0.637382	−0.318691	+	0.947859i	$0.603243\pi$
$984$	0	0
$985$	2.00508e7	0.658478
$986$	639207.	0.0209387
$987$	0	0
$988$	−1.65823e6	−0.0540445
$989$	6.26025e6	0.203517
$990$	0	0
$991$	1.51318e7	0.489449	0.244725	−	0.969593i	$-0.421302\pi$
$991$	1.51318e7	0.489449	0.244725	+	0.969593i	$0.421302\pi$
$992$	4.08340e7	1.31748
$993$	0	0
$994$	1.93415e6	0.0620903
$995$	2.57435e7	0.824346
$996$	0	0
$997$	−4.47054e7	−1.42437	−0.712184	−	0.701993i	$-0.752294\pi$
$997$	−4.47054e7	−1.42437	−0.712184	+	0.701993i	$0.752294\pi$
$998$	−3.59822e7	−1.14357
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.h.1.2		5
3.2	odd	2		55.6.a.c.1.4	✓	5
12.11	even	2		880.6.a.r.1.2		5
15.2	even	4		275.6.b.e.199.8		10
15.8	even	4		275.6.b.e.199.3		10
15.14	odd	2		275.6.a.e.1.2		5
33.32	even	2		605.6.a.d.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.a.c.1.4	✓	5	3.2	odd	2
275.6.a.e.1.2		5	15.14	odd	2
275.6.b.e.199.3		10	15.8	even	4
275.6.b.e.199.8		10	15.2	even	4
495.6.a.h.1.2		5	1.1	even	1	trivial
605.6.a.d.1.2		5	33.32	even	2
880.6.a.r.1.2		5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-8.57133 q^{2} +41.4676 q^{4} -25.0000 q^{5} +3.71474 q^{7} -81.1502 q^{8} +O(q^{10})\)	\(q-8.57133 q^{2} +41.4676 q^{4} -25.0000 q^{5} +3.71474 q^{7} -81.1502 q^{8} +214.283 q^{10} +121.000 q^{11} +94.0133 q^{13} -31.8402 q^{14} -631.400 q^{16} -196.903 q^{17} -425.349 q^{19} -1036.69 q^{20} -1037.13 q^{22} -518.668 q^{23} +625.000 q^{25} -805.819 q^{26} +154.041 q^{28} +378.739 q^{29} +5098.68 q^{31} +8008.74 q^{32} +1687.72 q^{34} -92.8684 q^{35} -4418.94 q^{37} +3645.81 q^{38} +2028.76 q^{40} -12225.3 q^{41} -12069.9 q^{43} +5017.58 q^{44} +4445.67 q^{46} -1450.03 q^{47} -16793.2 q^{49} -5357.08 q^{50} +3898.51 q^{52} +24797.4 q^{53} -3025.00 q^{55} -301.452 q^{56} -3246.30 q^{58} +22302.7 q^{59} +52941.4 q^{61} -43702.5 q^{62} -48440.7 q^{64} -2350.33 q^{65} +65036.7 q^{67} -8165.11 q^{68} +796.005 q^{70} -60745.5 q^{71} +54856.0 q^{73} +37876.1 q^{74} -17638.2 q^{76} +449.483 q^{77} +40298.7 q^{79} +15785.0 q^{80} +104787. q^{82} -74192.7 q^{83} +4922.58 q^{85} +103455. q^{86} -9819.17 q^{88} -551.359 q^{89} +349.235 q^{91} -21507.9 q^{92} +12428.6 q^{94} +10633.7 q^{95} +82992.9 q^{97} +143940. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 9 q^{2} + 115 q^{4} - 125 q^{5} + 70 q^{7} - 753 q^{8}+O(q^{10}) \) 5 * q - 9 * q^2 + 115 * q^4 - 125 * q^5 + 70 * q^7 - 753 * q^8	\( 5 q - 9 q^{2} + 115 q^{4} - 125 q^{5} + 70 q^{7} - 753 q^{8} + 225 q^{10} + 605 q^{11} + 1498 q^{13} - 2113 q^{14} + 4883 q^{16} - 3874 q^{17} + 882 q^{19} - 2875 q^{20} - 1089 q^{22} + 5344 q^{23} + 3125 q^{25} + 4478 q^{26} + 12565 q^{28} - 5318 q^{29} - 7916 q^{31} - 21385 q^{32} - 18605 q^{34} - 1750 q^{35} - 1788 q^{37} + 34421 q^{38} + 18825 q^{40} - 5854 q^{41} - 4364 q^{43} + 13915 q^{44} - 33834 q^{46} - 46452 q^{47} - 34217 q^{49} - 5625 q^{50} + 3222 q^{52} - 4412 q^{53} - 15125 q^{55} - 115575 q^{56} - 58221 q^{58} - 17896 q^{59} - 35930 q^{61} + 19627 q^{62} + 14779 q^{64} - 37450 q^{65} + 73136 q^{67} + 83409 q^{68} + 52825 q^{70} - 43612 q^{71} + 142306 q^{73} + 95609 q^{74} - 6617 q^{76} + 8470 q^{77} - 46504 q^{79} - 122075 q^{80} + 175798 q^{82} - 81604 q^{83} + 96850 q^{85} + 101788 q^{86} - 91113 q^{88} - 8664 q^{89} - 203380 q^{91} - 251174 q^{92} - 71458 q^{94} - 22050 q^{95} - 22230 q^{97} - 59962 q^{98}+O(q^{100}) \) 5 * q - 9 * q^2 + 115 * q^4 - 125 * q^5 + 70 * q^7 - 753 * q^8 + 225 * q^10 + 605 * q^11 + 1498 * q^13 - 2113 * q^14 + 4883 * q^16 - 3874 * q^17 + 882 * q^19 - 2875 * q^20 - 1089 * q^22 + 5344 * q^23 + 3125 * q^25 + 4478 * q^26 + 12565 * q^28 - 5318 * q^29 - 7916 * q^31 - 21385 * q^32 - 18605 * q^34 - 1750 * q^35 - 1788 * q^37 + 34421 * q^38 + 18825 * q^40 - 5854 * q^41 - 4364 * q^43 + 13915 * q^44 - 33834 * q^46 - 46452 * q^47 - 34217 * q^49 - 5625 * q^50 + 3222 * q^52 - 4412 * q^53 - 15125 * q^55 - 115575 * q^56 - 58221 * q^58 - 17896 * q^59 - 35930 * q^61 + 19627 * q^62 + 14779 * q^64 - 37450 * q^65 + 73136 * q^67 + 83409 * q^68 + 52825 * q^70 - 43612 * q^71 + 142306 * q^73 + 95609 * q^74 - 6617 * q^76 + 8470 * q^77 - 46504 * q^79 - 122075 * q^80 + 175798 * q^82 - 81604 * q^83 + 96850 * q^85 + 101788 * q^86 - 91113 * q^88 - 8664 * q^89 - 203380 * q^91 - 251174 * q^92 - 71458 * q^94 - 22050 * q^95 - 22230 * q^97 - 59962 * q^98

Introduction

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