LMFDB - Embedded newform 1815.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.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:	$107.088466660$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.23612.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 26 $ x^3 - x^2 - 20*x + 26
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.59056$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.59056 q^{2} -3.00000 q^{3} +4.89212 q^{4} -5.00000 q^{5} +10.7717 q^{6} +16.1465 q^{7} +11.1590 q^{8} +9.00000 q^{9} +17.9528 q^{10} -14.6764 q^{12} +54.1214 q^{13} -57.9749 q^{14} +15.0000 q^{15} -79.2041 q^{16} +107.010 q^{17} -32.3150 q^{18} -48.7496 q^{19} -24.4606 q^{20} -48.4394 q^{21} +11.9498 q^{23} -33.4771 q^{24} +25.0000 q^{25} -194.326 q^{26} -27.0000 q^{27} +78.9905 q^{28} -239.733 q^{29} -53.8584 q^{30} -82.0851 q^{31} +195.115 q^{32} -384.224 q^{34} -80.7324 q^{35} +44.0291 q^{36} -21.7573 q^{37} +175.038 q^{38} -162.364 q^{39} -55.7952 q^{40} +124.835 q^{41} +173.925 q^{42} -224.459 q^{43} -45.0000 q^{45} -42.9064 q^{46} -186.832 q^{47} +237.612 q^{48} -82.2913 q^{49} -89.7640 q^{50} -321.029 q^{51} +264.768 q^{52} +233.997 q^{53} +96.9451 q^{54} +180.179 q^{56} +146.249 q^{57} +860.774 q^{58} +232.936 q^{59} +73.3818 q^{60} -163.849 q^{61} +294.731 q^{62} +145.318 q^{63} -66.9386 q^{64} -270.607 q^{65} -876.918 q^{67} +523.503 q^{68} -35.8493 q^{69} +289.874 q^{70} -733.141 q^{71} +100.431 q^{72} -1161.97 q^{73} +78.1208 q^{74} -75.0000 q^{75} -238.489 q^{76} +582.978 q^{78} +588.831 q^{79} +396.021 q^{80} +81.0000 q^{81} -448.226 q^{82} +1161.06 q^{83} -236.971 q^{84} -535.048 q^{85} +805.933 q^{86} +719.198 q^{87} -1042.16 q^{89} +161.575 q^{90} +873.869 q^{91} +58.4597 q^{92} +246.255 q^{93} +670.831 q^{94} +243.748 q^{95} -585.345 q^{96} +1546.63 q^{97} +295.472 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} - 12 q^{6} + 4 q^{7} + 48 q^{8} + 27 q^{9} - 20 q^{10} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} + 218 q^{17} + 36 q^{18} - 146 q^{19} - 110 q^{20} - 12 q^{21}+ \cdots - 572 q^{98}+O(q^{100}) $ 3 * q + 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 - 12 * q^6 + 4 * q^7 + 48 * q^8 + 27 * q^9 - 20 * q^10 - 66 * q^12 - 56 * q^14 + 45 * q^15 + 50 * q^16 + 218 * q^17 + 36 * q^18 - 146 * q^19 - 110 * q^20 - 12 * q^21 - 200 * q^23 - 144 * q^24 + 75 * q^25 - 508 * q^26 - 81 * q^27 + 340 * q^28 - 68 * q^29 + 60 * q^30 - 68 * q^31 + 688 * q^32 - 176 * q^34 - 20 * q^35 + 198 * q^36 - 390 * q^37 - 316 * q^38 - 240 * q^40 + 196 * q^41 + 168 * q^42 + 524 * q^43 - 135 * q^45 - 1160 * q^46 - 60 * q^47 - 150 * q^48 - 157 * q^49 + 100 * q^50 - 654 * q^51 - 1020 * q^52 - 158 * q^53 - 108 * q^54 + 1368 * q^56 + 438 * q^57 + 1092 * q^58 - 1044 * q^59 + 330 * q^60 - 642 * q^61 - 88 * q^62 + 36 * q^63 + 1166 * q^64 - 236 * q^67 - 144 * q^68 + 600 * q^69 + 280 * q^70 - 544 * q^71 + 432 * q^72 - 900 * q^73 - 1536 * q^74 - 225 * q^75 - 1996 * q^76 + 1524 * q^78 + 1586 * q^79 - 250 * q^80 + 243 * q^81 + 380 * q^82 + 1582 * q^83 - 1020 * q^84 - 1090 * q^85 + 3568 * q^86 + 204 * q^87 - 2122 * q^89 - 180 * q^90 - 8 * q^91 - 4128 * q^92 + 204 * q^93 + 2152 * q^94 + 730 * q^95 - 2064 * q^96 + 618 * q^97 - 572 * 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 $

$2$	−3.59056	−1.26945	−0.634727	−	0.772736i	$-0.718888\pi$
$2$	−3.59056	−1.26945	−0.634727	+	0.772736i	$0.718888\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.89212	0.611515
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	10.7717	0.732920
$7$	16.1465	0.871828	0.435914	−	0.899988i	$-0.356425\pi$
$7$	16.1465	0.871828	0.435914	+	0.899988i	$0.356425\pi$
$8$	11.1590	0.493164
$9$	9.00000	0.333333
$10$	17.9528	0.567717
$11$	0	0
$11$	0	0
$12$	−14.6764	−0.353058
$13$	54.1214	1.15466	0.577329	−	0.816511i	$-0.304095\pi$
$13$	54.1214	1.15466	0.577329	+	0.816511i	$0.304095\pi$
$14$	−57.9749	−1.10675
$15$	15.0000	0.258199
$16$	−79.2041	−1.23756
$17$	107.010	1.52668	0.763342	−	0.645995i	$-0.223557\pi$
$17$	107.010	1.52668	0.763342	+	0.645995i	$0.223557\pi$
$18$	−32.3150	−0.423152
$19$	−48.7496	−0.588628	−0.294314	−	0.955709i	$-0.595091\pi$
$19$	−48.7496	−0.588628	−0.294314	+	0.955709i	$0.595091\pi$
$20$	−24.4606	−0.273478
$21$	−48.4394	−0.503350
$22$	0	0
$23$	11.9498	0.108335	0.0541674	−	0.998532i	$-0.482750\pi$
$23$	11.9498	0.108335	0.0541674	+	0.998532i	$0.482750\pi$
$24$	−33.4771	−0.284729
$25$	25.0000	0.200000
$26$	−194.326	−1.46579
$27$	−27.0000	−0.192450
$28$	78.9905	0.533136
$29$	−239.733	−1.53508	−0.767538	−	0.641003i	$-0.778519\pi$
$29$	−239.733	−1.53508	−0.767538	+	0.641003i	$0.778519\pi$
$30$	−53.8584	−0.327772
$31$	−82.0851	−0.475578	−0.237789	−	0.971317i	$-0.576423\pi$
$31$	−82.0851	−0.475578	−0.237789	+	0.971317i	$0.576423\pi$
$32$	195.115	1.07787
$33$	0	0
$34$	−384.224	−1.93806
$35$	−80.7324	−0.389893
$36$	44.0291	0.203838
$37$	−21.7573	−0.0966723	−0.0483361	−	0.998831i	$-0.515392\pi$
$37$	−21.7573	−0.0966723	−0.0483361	+	0.998831i	$0.515392\pi$
$38$	175.038	0.747236
$39$	−162.364	−0.666643
$40$	−55.7952	−0.220550
$41$	124.835	0.475510	0.237755	−	0.971325i	$-0.423588\pi$
$41$	124.835	0.475510	0.237755	+	0.971325i	$0.423588\pi$
$42$	173.925	0.638980
$43$	−224.459	−0.796039	−0.398019	−	0.917377i	$-0.630302\pi$
$43$	−224.459	−0.796039	−0.398019	+	0.917377i	$0.630302\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	−42.9064	−0.137526
$47$	−186.832	−0.579835	−0.289917	−	0.957052i	$-0.593628\pi$
$47$	−186.832	−0.579835	−0.289917	+	0.957052i	$0.593628\pi$
$48$	237.612	0.714508
$49$	−82.2913	−0.239916
$50$	−89.7640	−0.253891
$51$	−321.029	−0.881431
$52$	264.768	0.706091
$53$	233.997	0.606453	0.303226	−	0.952919i	$-0.401936\pi$
$53$	233.997	0.606453	0.303226	+	0.952919i	$0.401936\pi$
$54$	96.9451	0.244307
$55$	0	0
$56$	180.179	0.429954
$57$	146.249	0.339844
$58$	860.774	1.94871
$59$	232.936	0.513996	0.256998	−	0.966412i	$-0.417267\pi$
$59$	232.936	0.513996	0.256998	+	0.966412i	$0.417267\pi$
$60$	73.3818	0.157892
$61$	−163.849	−0.343913	−0.171957	−	0.985105i	$-0.555009\pi$
$61$	−163.849	−0.343913	−0.171957	+	0.985105i	$0.555009\pi$
$62$	294.731	0.603725
$63$	145.318	0.290609
$64$	−66.9386	−0.130739
$65$	−270.607	−0.516379
$66$	0	0
$67$	−876.918	−1.59899	−0.799497	−	0.600670i	$-0.794900\pi$
$67$	−876.918	−1.59899	−0.799497	+	0.600670i	$0.794900\pi$
$68$	523.503	0.933590
$69$	−35.8493	−0.0625471
$70$	289.874	0.494952
$71$	−733.141	−1.22546	−0.612731	−	0.790291i	$-0.709929\pi$
$71$	−733.141	−1.22546	−0.612731	+	0.790291i	$0.709929\pi$
$72$	100.431	0.164388
$73$	−1161.97	−1.86299	−0.931496	−	0.363750i	$-0.881496\pi$
$73$	−1161.97	−1.86299	−0.931496	+	0.363750i	$0.881496\pi$
$74$	78.1208	0.122721
$75$	−75.0000	−0.115470
$76$	−238.489	−0.359954
$77$	0	0
$78$	582.978	0.846272
$79$	588.831	0.838591	0.419296	−	0.907850i	$-0.362277\pi$
$79$	588.831	0.838591	0.419296	+	0.907850i	$0.362277\pi$
$80$	396.021	0.553456
$81$	81.0000	0.111111
$82$	−448.226	−0.603638
$83$	1161.06	1.53546	0.767731	−	0.640772i	$-0.221386\pi$
$83$	1161.06	1.53546	0.767731	+	0.640772i	$0.221386\pi$
$84$	−236.971	−0.307806
$85$	−535.048	−0.682754
$86$	805.933	1.01053
$87$	719.198	0.886277
$88$	0	0
$89$	−1042.16	−1.24122	−0.620610	−	0.784120i	$-0.713115\pi$
$89$	−1042.16	−1.24122	−0.620610	+	0.784120i	$0.713115\pi$
$90$	161.575	0.189239
$91$	873.869	1.00666
$92$	58.4597	0.0662483
$93$	246.255	0.274575
$94$	670.831	0.736074
$95$	243.748	0.263242
$96$	−585.345	−0.622307
$97$	1546.63	1.61893	0.809464	−	0.587169i	$-0.199758\pi$
$97$	1546.63	1.61893	0.809464	+	0.587169i	$0.199758\pi$
$98$	295.472	0.304563
$99$	0	0
$100$	122.303	0.122303
$101$	662.282	0.652470	0.326235	−	0.945289i	$-0.394220\pi$
$101$	662.282	0.652470	0.326235	+	0.945289i	$0.394220\pi$
$102$	1152.67	1.11894
$103$	399.592	0.382262	0.191131	−	0.981565i	$-0.438784\pi$
$103$	399.592	0.382262	0.191131	+	0.981565i	$0.438784\pi$
$104$	603.942	0.569436
$105$	242.197	0.225105
$106$	−840.181	−0.769864
$107$	1591.22	1.43765	0.718827	−	0.695189i	$-0.244679\pi$
$107$	1591.22	1.43765	0.718827	+	0.695189i	$0.244679\pi$
$108$	−132.087	−0.117686
$109$	−755.128	−0.663561	−0.331780	−	0.943357i	$-0.607649\pi$
$109$	−755.128	−0.663561	−0.331780	+	0.943357i	$0.607649\pi$
$110$	0	0
$111$	65.2718	0.0558138
$112$	−1278.87	−1.07894
$113$	−1145.65	−0.953753	−0.476876	−	0.878970i	$-0.658231\pi$
$113$	−1145.65	−0.953753	−0.476876	+	0.878970i	$0.658231\pi$
$114$	−525.115	−0.431417
$115$	−59.7488	−0.0484488
$116$	−1172.80	−0.938722
$117$	487.092	0.384886
$118$	−836.372	−0.652494
$119$	1727.83	1.33101
$120$	167.386	0.127334
$121$	0	0
$122$	588.309	0.436582
$123$	−374.504	−0.274536
$124$	−401.570	−0.290823
$125$	−125.000	−0.0894427
$126$	−521.774	−0.368915
$127$	−1461.29	−1.02101	−0.510505	−	0.859875i	$-0.670542\pi$
$127$	−1461.29	−1.02101	−0.510505	+	0.859875i	$0.670542\pi$
$128$	−1320.57	−0.911900
$129$	673.377	0.459593
$130$	971.630	0.655520
$131$	1524.94	1.01706	0.508528	−	0.861045i	$-0.330190\pi$
$131$	1524.94	1.01706	0.508528	+	0.861045i	$0.330190\pi$
$132$	0	0
$133$	−787.134	−0.513182
$134$	3148.63	2.02985
$135$	135.000	0.0860663
$136$	1194.12	0.752906
$137$	−2125.68	−1.32561	−0.662805	−	0.748792i	$-0.730634\pi$
$137$	−2125.68	−1.32561	−0.662805	+	0.748792i	$0.730634\pi$
$138$	128.719	0.0794007
$139$	1774.28	1.08268	0.541339	−	0.840805i	$-0.317918\pi$
$139$	1774.28	1.08268	0.541339	+	0.840805i	$0.317918\pi$
$140$	−394.952	−0.238425
$141$	560.496	0.334768
$142$	2632.39	1.55567
$143$	0	0
$144$	−712.837	−0.412521
$145$	1198.66	0.686507
$146$	4172.13	2.36498
$147$	246.874	0.138516
$148$	−106.439	−0.0591165
$149$	−1575.78	−0.866393	−0.433197	−	0.901299i	$-0.642614\pi$
$149$	−1575.78	−0.866393	−0.433197	+	0.901299i	$0.642614\pi$
$150$	269.292	0.146584
$151$	−420.978	−0.226879	−0.113439	−	0.993545i	$-0.536187\pi$
$151$	−420.978	−0.226879	−0.113439	+	0.993545i	$0.536187\pi$
$152$	−543.998	−0.290290
$153$	963.086	0.508895
$154$	0	0
$155$	410.425	0.212685
$156$	−794.304	−0.407662
$157$	−2224.30	−1.13069	−0.565345	−	0.824854i	$-0.691257\pi$
$157$	−2224.30	−1.13069	−0.565345	+	0.824854i	$0.691257\pi$
$158$	−2114.23	−1.06455
$159$	−701.992	−0.350136
$160$	−975.574	−0.482037
$161$	192.947	0.0944492
$162$	−290.835	−0.141051
$163$	−3093.37	−1.48645	−0.743226	−	0.669040i	$-0.766705\pi$
$163$	−3093.37	−1.48645	−0.743226	+	0.669040i	$0.766705\pi$
$164$	610.706	0.290781
$165$	0	0
$166$	−4168.87	−1.94920
$167$	2416.43	1.11970	0.559848	−	0.828595i	$-0.310860\pi$
$167$	2416.43	1.11970	0.559848	+	0.828595i	$0.310860\pi$
$168$	−540.537	−0.248234
$169$	732.122	0.333237
$170$	1921.12	0.866725
$171$	−438.746	−0.196209
$172$	−1098.08	−0.486789
$173$	3758.02	1.65154	0.825771	−	0.564005i	$-0.190740\pi$
$173$	3758.02	1.65154	0.825771	+	0.564005i	$0.190740\pi$
$174$	−2582.32	−1.12509
$175$	403.662	0.174366
$176$	0	0
$177$	−698.809	−0.296756
$178$	3741.93	1.57567
$179$	2533.99	1.05810	0.529049	−	0.848591i	$-0.322549\pi$
$179$	2533.99	1.05810	0.529049	+	0.848591i	$0.322549\pi$
$180$	−220.145	−0.0911592
$181$	−13.8995	−0.00570798	−0.00285399	−	0.999996i	$-0.500908\pi$
$181$	−13.8995	−0.00570798	−0.00285399	+	0.999996i	$0.500908\pi$
$182$	−3137.68	−1.27791
$183$	491.547	0.198558
$184$	133.348	0.0534268
$185$	108.786	0.0432332
$186$	−884.194	−0.348561
$187$	0	0
$188$	−914.004	−0.354577
$189$	−435.955	−0.167783
$190$	−875.192	−0.334174
$191$	3495.39	1.32417	0.662087	−	0.749427i	$-0.269671\pi$
$191$	3495.39	1.32417	0.662087	+	0.749427i	$0.269671\pi$
$192$	200.816	0.0754824
$193$	−3469.33	−1.29393	−0.646963	−	0.762522i	$-0.723961\pi$
$193$	−3469.33	−1.29393	−0.646963	+	0.762522i	$0.723961\pi$
$194$	−5553.25	−2.05516
$195$	811.820	0.298132
$196$	−402.579	−0.146712
$197$	−3638.39	−1.31586	−0.657930	−	0.753079i	$-0.728568\pi$
$197$	−3638.39	−1.31586	−0.657930	+	0.753079i	$0.728568\pi$
$198$	0	0
$199$	51.6049	0.0183828	0.00919140	−	0.999958i	$-0.497074\pi$
$199$	51.6049	0.0183828	0.00919140	+	0.999958i	$0.497074\pi$
$200$	278.976	0.0986329
$201$	2630.75	0.923179
$202$	−2377.96	−0.828281
$203$	−3870.84	−1.33832
$204$	−1570.51	−0.539008
$205$	−624.173	−0.212654
$206$	−1434.76	−0.485265
$207$	107.548	0.0361116
$208$	−4286.63	−1.42896
$209$	0	0
$210$	−869.623	−0.285761
$211$	−2084.57	−0.680131	−0.340065	−	0.940402i	$-0.610449\pi$
$211$	−2084.57	−0.680131	−0.340065	+	0.940402i	$0.610449\pi$
$212$	1144.74	0.370855
$213$	2199.42	0.707521
$214$	−5713.37	−1.82504
$215$	1122.29	0.355999
$216$	−301.294	−0.0949095
$217$	−1325.38	−0.414622
$218$	2711.33	0.842361
$219$	3485.91	1.07560
$220$	0	0
$221$	5791.50	1.76280
$222$	−234.362	−0.0708531
$223$	1887.36	0.566757	0.283378	−	0.959008i	$-0.408545\pi$
$223$	1887.36	0.566757	0.283378	+	0.959008i	$0.408545\pi$
$224$	3150.42	0.939715
$225$	225.000	0.0666667
$226$	4113.54	1.21075
$227$	1150.73	0.336462	0.168231	−	0.985748i	$-0.446194\pi$
$227$	1150.73	0.336462	0.168231	+	0.985748i	$0.446194\pi$
$228$	715.466	0.207820
$229$	4106.79	1.18508	0.592542	−	0.805540i	$-0.298125\pi$
$229$	4106.79	1.18508	0.592542	+	0.805540i	$0.298125\pi$
$230$	214.532	0.0615035
$231$	0	0
$232$	−2675.18	−0.757045
$233$	5733.58	1.61210	0.806050	−	0.591848i	$-0.201601\pi$
$233$	5733.58	1.61210	0.806050	+	0.591848i	$0.201601\pi$
$234$	−1748.93	−0.488596
$235$	934.159	0.259310
$236$	1139.55	0.314316
$237$	−1766.49	−0.484161
$238$	−6203.86	−1.68965
$239$	−6036.18	−1.63367	−0.816837	−	0.576868i	$-0.804275\pi$
$239$	−6036.18	−1.63367	−0.816837	+	0.576868i	$0.804275\pi$
$240$	−1188.06	−0.319538
$241$	−3720.90	−0.994540	−0.497270	−	0.867596i	$-0.665664\pi$
$241$	−3720.90	−0.994540	−0.497270	+	0.867596i	$0.665664\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	−801.568	−0.210308
$245$	411.457	0.107294
$246$	1344.68	0.348511
$247$	−2638.39	−0.679664
$248$	−915.990	−0.234538
$249$	−3483.19	−0.886500
$250$	448.820	0.113543
$251$	−3809.88	−0.958077	−0.479038	−	0.877794i	$-0.659015\pi$
$251$	−3809.88	−0.958077	−0.479038	+	0.877794i	$0.659015\pi$
$252$	710.914	0.177712
$253$	0	0
$254$	5246.84	1.29613
$255$	1605.14	0.394188
$256$	5277.10	1.28835
$257$	1225.95	0.297559	0.148780	−	0.988870i	$-0.452465\pi$
$257$	1225.95	0.297559	0.148780	+	0.988870i	$0.452465\pi$
$258$	−2417.80	−0.583433
$259$	−351.303	−0.0842816
$260$	−1323.84	−0.315773
$261$	−2157.59	−0.511692
$262$	−5475.38	−1.29111
$263$	−7397.68	−1.73445	−0.867225	−	0.497916i	$-0.834099\pi$
$263$	−7397.68	−1.73445	−0.867225	+	0.497916i	$0.834099\pi$
$264$	0	0
$265$	−1169.99	−0.271214
$266$	2826.25	0.651461
$267$	3126.47	0.716618
$268$	−4289.99	−0.977808
$269$	−3214.48	−0.728589	−0.364295	−	0.931284i	$-0.618690\pi$
$269$	−3214.48	−0.728589	−0.364295	+	0.931284i	$0.618690\pi$
$270$	−484.726	−0.109257
$271$	7377.37	1.65367	0.826833	−	0.562448i	$-0.190140\pi$
$271$	7377.37	1.65367	0.826833	+	0.562448i	$0.190140\pi$
$272$	−8475.60	−1.88937
$273$	−2621.61	−0.581197
$274$	7632.36	1.68280
$275$	0	0
$276$	−175.379	−0.0382485
$277$	810.606	0.175829	0.0879144	−	0.996128i	$-0.471980\pi$
$277$	810.606	0.175829	0.0879144	+	0.996128i	$0.471980\pi$
$278$	−6370.65	−1.37441
$279$	−738.766	−0.158526
$280$	−900.895	−0.192281
$281$	−1114.72	−0.236651	−0.118325	−	0.992975i	$-0.537753\pi$
$281$	−1114.72	−0.236651	−0.118325	+	0.992975i	$0.537753\pi$
$282$	−2012.49	−0.424972
$283$	−2265.40	−0.475844	−0.237922	−	0.971284i	$-0.576466\pi$
$283$	−2265.40	−0.475844	−0.237922	+	0.971284i	$0.576466\pi$
$284$	−3586.61	−0.749388
$285$	−731.244	−0.151983
$286$	0	0
$287$	2015.64	0.414563
$288$	1756.03	0.359289
$289$	6538.04	1.33076
$290$	−4303.87	−0.871490
$291$	−4639.88	−0.934689
$292$	−5684.50	−1.13925
$293$	−3802.06	−0.758084	−0.379042	−	0.925380i	$-0.623746\pi$
$293$	−3802.06	−0.758084	−0.379042	+	0.925380i	$0.623746\pi$
$294$	−886.416	−0.175840
$295$	−1164.68	−0.229866
$296$	−242.790	−0.0476753
$297$	0	0
$298$	5657.92	1.09985
$299$	646.738	0.125090
$300$	−366.909	−0.0706116
$301$	−3624.22	−0.694009
$302$	1511.55	0.288013
$303$	−1986.85	−0.376704
$304$	3861.17	0.728465
$305$	819.244	0.153803
$306$	−3458.02	−0.646019
$307$	1356.35	0.252153	0.126077	−	0.992021i	$-0.459761\pi$
$307$	1356.35	0.252153	0.126077	+	0.992021i	$0.459761\pi$
$308$	0	0
$309$	−1198.78	−0.220699
$310$	−1473.66	−0.269994
$311$	−8078.07	−1.47288	−0.736440	−	0.676503i	$-0.763495\pi$
$311$	−8078.07	−1.47288	−0.736440	+	0.676503i	$0.763495\pi$
$312$	−1811.83	−0.328764
$313$	5761.54	1.04045	0.520226	−	0.854029i	$-0.325848\pi$
$313$	5761.54	1.04045	0.520226	+	0.854029i	$0.325848\pi$
$314$	7986.48	1.43536
$315$	−726.591	−0.129964
$316$	2880.63	0.512811
$317$	5107.00	0.904851	0.452426	−	0.891802i	$-0.350559\pi$
$317$	5107.00	0.904851	0.452426	+	0.891802i	$0.350559\pi$
$318$	2520.54	0.444481
$319$	0	0
$320$	334.693	0.0584684
$321$	−4773.66	−0.830030
$322$	−692.786	−0.119899
$323$	−5216.67	−0.898648
$324$	396.262	0.0679461
$325$	1353.03	0.230932
$326$	11106.9	1.88698
$327$	2265.38	0.383107
$328$	1393.03	0.234504
$329$	−3016.68	−0.505516
$330$	0	0
$331$	−2780.94	−0.461796	−0.230898	−	0.972978i	$-0.574166\pi$
$331$	−2780.94	−0.461796	−0.230898	+	0.972978i	$0.574166\pi$
$332$	5680.06	0.938958
$333$	−195.816	−0.0322241
$334$	−8676.35	−1.42140
$335$	4384.59	0.715092
$336$	3836.60	0.622928
$337$	−4939.28	−0.798397	−0.399198	−	0.916865i	$-0.630712\pi$
$337$	−4939.28	−0.798397	−0.399198	+	0.916865i	$0.630712\pi$
$338$	−2628.73	−0.423029
$339$	3436.96	0.550649
$340$	−2617.52	−0.417514
$341$	0	0
$342$	1575.34	0.249079
$343$	−6866.96	−1.08099
$344$	−2504.74	−0.392578
$345$	179.247	0.0279719
$346$	−13493.4	−2.09656
$347$	2711.58	0.419496	0.209748	−	0.977755i	$-0.432736\pi$
$347$	2711.58	0.419496	0.209748	+	0.977755i	$0.432736\pi$
$348$	3518.40	0.541971
$349$	−5496.03	−0.842967	−0.421484	−	0.906836i	$-0.638490\pi$
$349$	−5496.03	−0.842967	−0.421484	+	0.906836i	$0.638490\pi$
$350$	−1449.37	−0.221349
$351$	−1461.28	−0.222214
$352$	0	0
$353$	6372.23	0.960792	0.480396	−	0.877052i	$-0.340493\pi$
$353$	6372.23	0.960792	0.480396	+	0.877052i	$0.340493\pi$
$354$	2509.12	0.376718
$355$	3665.71	0.548043
$356$	−5098.36	−0.759024
$357$	−5183.48	−0.768456
$358$	−9098.46	−1.34321
$359$	−630.622	−0.0927101	−0.0463551	−	0.998925i	$-0.514761\pi$
$359$	−630.622	−0.0927101	−0.0463551	+	0.998925i	$0.514761\pi$
$360$	−502.157	−0.0735166
$361$	−4482.48	−0.653518
$362$	49.9071	0.00724602
$363$	0	0
$364$	4275.07	0.615590
$365$	5809.86	0.833156
$366$	−1764.93	−0.252061
$367$	−5374.49	−0.764431	−0.382216	−	0.924073i	$-0.624839\pi$
$367$	−5374.49	−0.764431	−0.382216	+	0.924073i	$0.624839\pi$
$368$	−946.471	−0.134071
$369$	1123.51	0.158503
$370$	−390.604	−0.0548825
$371$	3778.23	0.528722
$372$	1204.71	0.167907
$373$	−7520.12	−1.04391	−0.521953	−	0.852974i	$-0.674796\pi$
$373$	−7520.12	−1.04391	−0.521953	+	0.852974i	$0.674796\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	−2084.86	−0.285954
$377$	−12974.7	−1.77249
$378$	1565.32	0.212993
$379$	−12509.1	−1.69538	−0.847690	−	0.530492i	$-0.822007\pi$
$379$	−12509.1	−1.69538	−0.847690	+	0.530492i	$0.822007\pi$
$380$	1192.44	0.160977
$381$	4383.86	0.589481
$382$	−12550.4	−1.68098
$383$	−11149.9	−1.48755	−0.743775	−	0.668430i	$-0.766967\pi$
$383$	−11149.9	−1.48755	−0.743775	+	0.668430i	$0.766967\pi$
$384$	3961.72	0.526486
$385$	0	0
$386$	12456.8	1.64258
$387$	−2020.13	−0.265346
$388$	7566.28	0.989999
$389$	3194.22	0.416333	0.208166	−	0.978093i	$-0.433250\pi$
$389$	3194.22	0.416333	0.208166	+	0.978093i	$0.433250\pi$
$390$	−2914.89	−0.378465
$391$	1278.74	0.165393
$392$	−918.292	−0.118318
$393$	−4574.81	−0.587198
$394$	13063.8	1.67042
$395$	−2944.16	−0.375029
$396$	0	0
$397$	−584.410	−0.0738809	−0.0369404	−	0.999317i	$-0.511761\pi$
$397$	−584.410	−0.0738809	−0.0369404	+	0.999317i	$0.511761\pi$
$398$	−185.291	−0.0233361
$399$	2361.40	0.296286
$400$	−1980.10	−0.247513
$401$	−6951.24	−0.865657	−0.432829	−	0.901476i	$-0.642484\pi$
$401$	−6951.24	−0.865657	−0.432829	+	0.901476i	$0.642484\pi$
$402$	−9445.88	−1.17193
$403$	−4442.56	−0.549130
$404$	3239.96	0.398995
$405$	−405.000	−0.0496904
$406$	13898.5	1.69894
$407$	0	0
$408$	−3582.37	−0.434690
$409$	−11754.1	−1.42104	−0.710519	−	0.703678i	$-0.751540\pi$
$409$	−11754.1	−1.42104	−0.710519	+	0.703678i	$0.751540\pi$
$410$	2241.13	0.269955
$411$	6377.03	0.765342
$412$	1954.85	0.233759
$413$	3761.10	0.448116
$414$	−386.157	−0.0458420
$415$	−5805.32	−0.686680
$416$	10559.9	1.24457
$417$	−5322.83	−0.625084
$418$	0	0
$419$	11829.8	1.37929	0.689646	−	0.724146i	$-0.257766\pi$
$419$	11829.8	1.37929	0.689646	+	0.724146i	$0.257766\pi$
$420$	1184.86	0.137655
$421$	−2000.07	−0.231538	−0.115769	−	0.993276i	$-0.536933\pi$
$421$	−2000.07	−0.231538	−0.115769	+	0.993276i	$0.536933\pi$
$422$	7484.76	0.863395
$423$	−1681.49	−0.193278
$424$	2611.18	0.299081
$425$	2675.24	0.305337
$426$	−7897.16	−0.898166
$427$	−2645.58	−0.299833
$428$	7784.43	0.879147
$429$	0	0
$430$	−4029.67	−0.451925
$431$	8064.11	0.901240	0.450620	−	0.892716i	$-0.351203\pi$
$431$	8064.11	0.901240	0.450620	+	0.892716i	$0.351203\pi$
$432$	2138.51	0.238169
$433$	−10710.3	−1.18869	−0.594345	−	0.804210i	$-0.702589\pi$
$433$	−10710.3	−1.18869	−0.594345	+	0.804210i	$0.702589\pi$
$434$	4758.87	0.526344
$435$	−3595.99	−0.396355
$436$	−3694.18	−0.405777
$437$	−582.546	−0.0637688
$438$	−12516.4	−1.36542
$439$	−4658.08	−0.506419	−0.253210	−	0.967411i	$-0.581486\pi$
$439$	−4658.08	−0.506419	−0.253210	+	0.967411i	$0.581486\pi$
$440$	0	0
$441$	−740.622	−0.0799721
$442$	−20794.7	−2.23779
$443$	2094.73	0.224658	0.112329	−	0.993671i	$-0.464169\pi$
$443$	2094.73	0.224658	0.112329	+	0.993671i	$0.464169\pi$
$444$	319.318	0.0341310
$445$	5210.79	0.555090
$446$	−6776.67	−0.719472
$447$	4727.33	0.500212
$448$	−1080.82	−0.113982
$449$	−1402.21	−0.147382	−0.0736909	−	0.997281i	$-0.523478\pi$
$449$	−1402.21	−0.147382	−0.0736909	+	0.997281i	$0.523478\pi$
$450$	−807.876	−0.0846303
$451$	0	0
$452$	−5604.67	−0.583234
$453$	1262.93	0.130989
$454$	−4131.78	−0.427124
$455$	−4369.35	−0.450194
$456$	1632.00	0.167599
$457$	4914.19	0.503011	0.251506	−	0.967856i	$-0.419074\pi$
$457$	4914.19	0.503011	0.251506	+	0.967856i	$0.419074\pi$
$458$	−14745.7	−1.50441
$459$	−2889.26	−0.293810
$460$	−292.298	−0.0296271
$461$	2214.08	0.223688	0.111844	−	0.993726i	$-0.464324\pi$
$461$	2214.08	0.223688	0.111844	+	0.993726i	$0.464324\pi$
$462$	0	0
$463$	−5567.02	−0.558793	−0.279396	−	0.960176i	$-0.590134\pi$
$463$	−5567.02	−0.558793	−0.279396	+	0.960176i	$0.590134\pi$
$464$	18987.8	1.89976
$465$	−1231.28	−0.122794
$466$	−20586.8	−2.04649
$467$	497.054	0.0492525	0.0246263	−	0.999697i	$-0.492160\pi$
$467$	497.054	0.0492525	0.0246263	+	0.999697i	$0.492160\pi$
$468$	2382.91	0.235364
$469$	−14159.1	−1.39405
$470$	−3354.15	−0.329182
$471$	6672.90	0.652804
$472$	2599.35	0.253484
$473$	0	0
$474$	6342.70	0.614620
$475$	−1218.74	−0.117726
$476$	8452.73	0.813929
$477$	2105.97	0.202151
$478$	21673.3	2.07388
$479$	9349.28	0.891815	0.445908	−	0.895079i	$-0.352881\pi$
$479$	9349.28	0.891815	0.445908	+	0.895079i	$0.352881\pi$
$480$	2926.72	0.278304
$481$	−1177.53	−0.111624
$482$	13360.1	1.26252
$483$	−578.840	−0.0545303
$484$	0	0
$485$	−7733.13	−0.724007
$486$	872.506	0.0814355
$487$	197.750	0.0184003	0.00920013	−	0.999958i	$-0.497071\pi$
$487$	197.750	0.0184003	0.00920013	+	0.999958i	$0.497071\pi$
$488$	−1828.40	−0.169606
$489$	9280.12	0.858204
$490$	−1477.36	−0.136205
$491$	9997.05	0.918861	0.459430	−	0.888214i	$-0.348054\pi$
$491$	9997.05	0.918861	0.459430	+	0.888214i	$0.348054\pi$
$492$	−1832.12	−0.167883
$493$	−25653.7	−2.34358
$494$	9473.31	0.862803
$495$	0	0
$496$	6501.48	0.588558
$497$	−11837.6	−1.06839
$498$	12506.6	1.12537
$499$	−8714.73	−0.781813	−0.390907	−	0.920430i	$-0.627838\pi$
$499$	−8714.73	−0.781813	−0.390907	+	0.920430i	$0.627838\pi$
$500$	−611.515	−0.0546955
$501$	−7249.30	−0.646457
$502$	13679.6	1.21623
$503$	−5978.53	−0.529959	−0.264979	−	0.964254i	$-0.585365\pi$
$503$	−5978.53	−0.529959	−0.264979	+	0.964254i	$0.585365\pi$
$504$	1621.61	0.143318
$505$	−3311.41	−0.291794
$506$	0	0
$507$	−2196.36	−0.192394
$508$	−7148.79	−0.624363
$509$	−8205.79	−0.714569	−0.357284	−	0.933996i	$-0.616297\pi$
$509$	−8205.79	−0.714569	−0.357284	+	0.933996i	$0.616297\pi$
$510$	−5763.36	−0.500404
$511$	−18761.7	−1.62421
$512$	−8383.17	−0.723608
$513$	1316.24	0.113281
$514$	−4401.85	−0.377738
$515$	−1997.96	−0.170953
$516$	3294.24	0.281048
$517$	0	0
$518$	1261.38	0.106992
$519$	−11274.1	−0.953519
$520$	−3019.71	−0.254660
$521$	−5266.06	−0.442822	−0.221411	−	0.975181i	$-0.571066\pi$
$521$	−5266.06	−0.442822	−0.221411	+	0.975181i	$0.571066\pi$
$522$	7746.97	0.649570
$523$	22398.6	1.87270	0.936350	−	0.351068i	$-0.114181\pi$
$523$	22398.6	1.87270	0.936350	+	0.351068i	$0.114181\pi$
$524$	7460.18	0.621945
$525$	−1210.99	−0.100670
$526$	26561.8	2.20181
$527$	−8783.89	−0.726057
$528$	0	0
$529$	−12024.2	−0.988264
$530$	4200.90	0.344294
$531$	2096.43	0.171332
$532$	−3850.75	−0.313818
$533$	6756.22	0.549052
$534$	−11225.8	−0.909714
$535$	−7956.10	−0.642938
$536$	−9785.56	−0.788566
$537$	−7601.98	−0.610893
$538$	11541.8	0.924911
$539$	0	0
$540$	660.436	0.0526308
$541$	13030.2	1.03551	0.517757	−	0.855528i	$-0.326767\pi$
$541$	13030.2	1.03551	0.517757	+	0.855528i	$0.326767\pi$
$542$	−26488.9	−2.09925
$543$	41.6986	0.00329550
$544$	20879.1	1.64556
$545$	3775.64	0.296753
$546$	9413.04	0.737804
$547$	−10448.9	−0.816747	−0.408374	−	0.912815i	$-0.633904\pi$
$547$	−10448.9	−0.816747	−0.408374	+	0.912815i	$0.633904\pi$
$548$	−10399.1	−0.810631
$549$	−1474.64	−0.114638
$550$	0	0
$551$	11686.9	0.903588
$552$	−400.044	−0.0308460
$553$	9507.55	0.731107
$554$	−2910.53	−0.223207
$555$	−326.359	−0.0249607
$556$	8679.97	0.662073
$557$	11448.7	0.870911	0.435456	−	0.900210i	$-0.356587\pi$
$557$	11448.7	0.870911	0.435456	+	0.900210i	$0.356587\pi$
$558$	2652.58	0.201242
$559$	−12148.0	−0.919153
$560$	6394.34	0.482518
$561$	0	0
$562$	4002.49	0.300418
$563$	26035.8	1.94898	0.974492	−	0.224422i	$-0.0720494\pi$
$563$	26035.8	1.94898	0.974492	+	0.224422i	$0.0720494\pi$
$564$	2742.01	0.204715
$565$	5728.27	0.426531
$566$	8134.05	0.604063
$567$	1307.86	0.0968697
$568$	−8181.15	−0.604354
$569$	−25075.0	−1.84745	−0.923725	−	0.383057i	$-0.874871\pi$
$569$	−25075.0	−1.84745	−0.923725	+	0.383057i	$0.874871\pi$
$570$	2625.57	0.192935
$571$	−19056.6	−1.39666	−0.698330	−	0.715776i	$-0.746073\pi$
$571$	−19056.6	−1.39666	−0.698330	+	0.715776i	$0.746073\pi$
$572$	0	0
$573$	−10486.2	−0.764512
$574$	−7237.28	−0.526268
$575$	298.744	0.0216669
$576$	−602.447	−0.0435798
$577$	−6482.55	−0.467716	−0.233858	−	0.972271i	$-0.575135\pi$
$577$	−6482.55	−0.467716	−0.233858	+	0.972271i	$0.575135\pi$
$578$	−23475.2	−1.68934
$579$	10408.0	0.747048
$580$	5864.00	0.419809
$581$	18747.1	1.33866
$582$	16659.8	1.18654
$583$	0	0
$584$	−12966.5	−0.918762
$585$	−2435.46	−0.172126
$586$	13651.5	0.962353
$587$	24650.3	1.73327	0.866633	−	0.498946i	$-0.166280\pi$
$587$	24650.3	1.73327	0.866633	+	0.498946i	$0.166280\pi$
$588$	1207.74	0.0847045
$589$	4001.61	0.279938
$590$	4181.86	0.291804
$591$	10915.2	0.759712
$592$	1723.27	0.119638
$593$	−23163.2	−1.60405	−0.802024	−	0.597292i	$-0.796243\pi$
$593$	−23163.2	−1.60405	−0.802024	+	0.597292i	$0.796243\pi$
$594$	0	0
$595$	−8639.13	−0.595244
$596$	−7708.88	−0.529812
$597$	−154.815	−0.0106133
$598$	−2322.15	−0.158796
$599$	5355.44	0.365304	0.182652	−	0.983178i	$-0.441532\pi$
$599$	5355.44	0.365304	0.182652	+	0.983178i	$0.441532\pi$
$600$	−836.928	−0.0569457
$601$	20417.6	1.38578	0.692889	−	0.721045i	$-0.256338\pi$
$601$	20417.6	1.38578	0.692889	+	0.721045i	$0.256338\pi$
$602$	13013.0	0.881012
$603$	−7892.26	−0.532998
$604$	−2059.48	−0.138740
$605$	0	0
$606$	7133.89	0.478208
$607$	6749.81	0.451345	0.225673	−	0.974203i	$-0.427542\pi$
$607$	6749.81	0.451345	0.225673	+	0.974203i	$0.427542\pi$
$608$	−9511.77	−0.634463
$609$	11612.5	0.772681
$610$	−2941.55	−0.195245
$611$	−10111.6	−0.669511
$612$	4711.53	0.311197
$613$	30321.0	1.99780	0.998901	−	0.0468613i	$-0.0149219\pi$
$613$	30321.0	1.99780	0.998901	+	0.0468613i	$0.0149219\pi$
$614$	−4870.06	−0.320097
$615$	1872.52	0.122776
$616$	0	0
$617$	15236.8	0.994182	0.497091	−	0.867699i	$-0.334402\pi$
$617$	15236.8	0.994182	0.497091	+	0.867699i	$0.334402\pi$
$618$	4304.28	0.280168
$619$	−20875.4	−1.35550	−0.677749	−	0.735293i	$-0.737044\pi$
$619$	−20875.4	−1.35550	−0.677749	+	0.735293i	$0.737044\pi$
$620$	2007.85	0.130060
$621$	−322.644	−0.0208490
$622$	29004.8	1.86975
$623$	−16827.2	−1.08213
$624$	12859.9	0.825013
$625$	625.000	0.0400000
$626$	−20687.2	−1.32081
$627$	0	0
$628$	−10881.5	−0.691434
$629$	−2328.24	−0.147588
$630$	2608.87	0.164984
$631$	27966.1	1.76437	0.882183	−	0.470907i	$-0.156073\pi$
$631$	27966.1	1.76437	0.882183	+	0.470907i	$0.156073\pi$
$632$	6570.79	0.413563
$633$	6253.70	0.392674
$634$	−18337.0	−1.14867
$635$	7306.44	0.456610
$636$	−3434.23	−0.214113
$637$	−4453.72	−0.277022
$638$	0	0
$639$	−6598.27	−0.408487
$640$	6602.86	0.407814
$641$	−17992.0	−1.10865	−0.554323	−	0.832301i	$-0.687023\pi$
$641$	−17992.0	−1.10865	−0.554323	+	0.832301i	$0.687023\pi$
$642$	17140.1	1.05369
$643$	−9448.64	−0.579499	−0.289750	−	0.957102i	$-0.593572\pi$
$643$	−9448.64	−0.579499	−0.289750	+	0.957102i	$0.593572\pi$
$644$	943.918	0.0577571
$645$	−3366.88	−0.205536
$646$	18730.8	1.14079
$647$	−7429.22	−0.451426	−0.225713	−	0.974194i	$-0.572471\pi$
$647$	−7429.22	−0.451426	−0.225713	+	0.974194i	$0.572471\pi$
$648$	903.882	0.0547960
$649$	0	0
$650$	−4858.15	−0.293157
$651$	3976.15	0.239382
$652$	−15133.2	−0.908988
$653$	−4488.63	−0.268995	−0.134497	−	0.990914i	$-0.542942\pi$
$653$	−4488.63	−0.268995	−0.134497	+	0.990914i	$0.542942\pi$
$654$	−8134.00	−0.486337
$655$	−7624.69	−0.454842
$656$	−9887.42	−0.588474
$657$	−10457.7	−0.620998
$658$	10831.6	0.641730
$659$	25326.7	1.49710	0.748550	−	0.663078i	$-0.230750\pi$
$659$	25326.7	1.49710	0.748550	+	0.663078i	$0.230750\pi$
$660$	0	0
$661$	15192.3	0.893969	0.446984	−	0.894542i	$-0.352498\pi$
$661$	15192.3	0.893969	0.446984	+	0.894542i	$0.352498\pi$
$662$	9985.15	0.586229
$663$	−17374.5	−1.01775
$664$	12956.4	0.757235
$665$	3935.67	0.229502
$666$	703.087	0.0409070
$667$	−2864.75	−0.166302
$668$	11821.5	0.684711
$669$	−5662.07	−0.327217
$670$	−15743.1	−0.907776
$671$	0	0
$672$	−9451.25	−0.542545
$673$	−11718.2	−0.671180	−0.335590	−	0.942008i	$-0.608936\pi$
$673$	−11718.2	−0.671180	−0.335590	+	0.942008i	$0.608936\pi$
$674$	17734.8	1.01353
$675$	−675.000	−0.0384900
$676$	3581.63	0.203779
$677$	15649.1	0.888397	0.444198	−	0.895928i	$-0.353489\pi$
$677$	15649.1	0.888397	0.444198	+	0.895928i	$0.353489\pi$
$678$	−12340.6	−0.699024
$679$	24972.6	1.41143
$680$	−5970.61	−0.336710
$681$	−3452.20	−0.194257
$682$	0	0
$683$	−18162.8	−1.01754	−0.508770	−	0.860903i	$-0.669899\pi$
$683$	−18162.8	−1.01754	−0.508770	+	0.860903i	$0.669899\pi$
$684$	−2146.40	−0.119985
$685$	10628.4	0.592831
$686$	24656.2	1.37227
$687$	−12320.4	−0.684208
$688$	17778.1	0.985149
$689$	12664.2	0.700246
$690$	−643.595	−0.0355091
$691$	29606.7	1.62995	0.814973	−	0.579500i	$-0.196752\pi$
$691$	29606.7	1.62995	0.814973	+	0.579500i	$0.196752\pi$
$692$	18384.7	1.00994
$693$	0	0
$694$	−9736.08	−0.532531
$695$	−8871.38	−0.484188
$696$	8025.55	0.437080
$697$	13358.5	0.725953
$698$	19733.8	1.07011
$699$	−17200.7	−0.930746
$700$	1974.76	0.106627
$701$	26164.7	1.40974	0.704870	−	0.709336i	$-0.251005\pi$
$701$	26164.7	1.40974	0.704870	+	0.709336i	$0.251005\pi$
$702$	5246.80	0.282091
$703$	1060.66	0.0569040
$704$	0	0
$705$	−2802.48	−0.149713
$706$	−22879.9	−1.21968
$707$	10693.5	0.568842
$708$	−3418.66	−0.181470
$709$	−14508.9	−0.768535	−0.384268	−	0.923222i	$-0.625546\pi$
$709$	−14508.9	−0.768535	−0.384268	+	0.923222i	$0.625546\pi$
$710$	−13161.9	−0.695716
$711$	5299.48	0.279530
$712$	−11629.5	−0.612125
$713$	−980.898	−0.0515216
$714$	18611.6	0.975520
$715$	0	0
$716$	12396.6	0.647043
$717$	18108.5	0.943202
$718$	2264.28	0.117691
$719$	−2545.80	−0.132048	−0.0660239	−	0.997818i	$-0.521031\pi$
$719$	−2545.80	−0.132048	−0.0660239	+	0.997818i	$0.521031\pi$
$720$	3564.19	0.184485
$721$	6452.01	0.333267
$722$	16094.6	0.829611
$723$	11162.7	0.574198
$724$	−67.9982	−0.00349051
$725$	−5993.31	−0.307015
$726$	0	0
$727$	−18984.8	−0.968511	−0.484255	−	0.874927i	$-0.660909\pi$
$727$	−18984.8	−0.968511	−0.484255	+	0.874927i	$0.660909\pi$
$728$	9751.54	0.496451
$729$	729.000	0.0370370
$730$	−20860.6	−1.05765
$731$	−24019.2	−1.21530
$732$	2404.70	0.121421
$733$	−36740.4	−1.85134	−0.925672	−	0.378326i	$-0.876500\pi$
$733$	−36740.4	−1.85134	−0.925672	+	0.378326i	$0.876500\pi$
$734$	19297.4	0.970411
$735$	−1234.37	−0.0619461
$736$	2331.58	0.116770
$737$	0	0
$738$	−4034.04	−0.201213
$739$	−1755.17	−0.0873682	−0.0436841	−	0.999045i	$-0.513910\pi$
$739$	−1755.17	−0.0873682	−0.0436841	+	0.999045i	$0.513910\pi$
$740$	532.196	0.0264377
$741$	7915.18	0.392404
$742$	−13566.0	−0.671189
$743$	−17972.6	−0.887419	−0.443709	−	0.896171i	$-0.646338\pi$
$743$	−17972.6	−0.887419	−0.443709	+	0.896171i	$0.646338\pi$
$744$	2747.97	0.135411
$745$	7878.88	0.387463
$746$	27001.4	1.32519
$747$	10449.6	0.511821
$748$	0	0
$749$	25692.6	1.25339
$750$	−1346.46	−0.0655544
$751$	22704.9	1.10321	0.551606	−	0.834105i	$-0.314015\pi$
$751$	22704.9	1.10321	0.551606	+	0.834105i	$0.314015\pi$
$752$	14797.9	0.717583
$753$	11429.6	0.553146
$754$	46586.3	2.25010
$755$	2104.89	0.101463
$756$	−2132.74	−0.102602
$757$	39860.3	1.91380	0.956901	−	0.290414i	$-0.0937929\pi$
$757$	39860.3	1.91380	0.956901	+	0.290414i	$0.0937929\pi$
$758$	44914.6	2.15221
$759$	0	0
$760$	2719.99	0.129822
$761$	−19631.4	−0.935135	−0.467568	−	0.883957i	$-0.654870\pi$
$761$	−19631.4	−0.935135	−0.467568	+	0.883957i	$0.654870\pi$
$762$	−15740.5	−0.748319
$763$	−12192.7	−0.578511
$764$	17099.8	0.809752
$765$	−4815.43	−0.227585
$766$	40034.3	1.88838
$767$	12606.8	0.593490
$768$	−15831.3	−0.743832
$769$	−35029.9	−1.64267	−0.821333	−	0.570449i	$-0.806769\pi$
$769$	−35029.9	−1.64267	−0.821333	+	0.570449i	$0.806769\pi$
$770$	0	0
$771$	−3677.86	−0.171796
$772$	−16972.4	−0.791255
$773$	−26736.9	−1.24406	−0.622032	−	0.782992i	$-0.713693\pi$
$773$	−26736.9	−1.24406	−0.622032	+	0.782992i	$0.713693\pi$
$774$	7253.40	0.336845
$775$	−2052.13	−0.0951156
$776$	17258.8	0.798398
$777$	1053.91	0.0486600
$778$	−11469.0	−0.528515
$779$	−6085.64	−0.279898
$780$	3971.52	0.182312
$781$	0	0
$782$	−4591.39	−0.209959
$783$	6472.78	0.295426
$784$	6517.81	0.296912
$785$	11121.5	0.505660
$786$	16426.1	0.745421
$787$	−4799.45	−0.217385	−0.108693	−	0.994075i	$-0.534666\pi$
$787$	−4799.45	−0.217385	−0.108693	+	0.994075i	$0.534666\pi$
$788$	−17799.4	−0.804668
$789$	22193.0	1.00139
$790$	10571.2	0.476083
$791$	−18498.3	−0.831508
$792$	0	0
$793$	−8867.72	−0.397102
$794$	2098.36	0.0937884
$795$	3509.96	0.156585
$796$	252.457	0.0112413
$797$	38438.9	1.70838	0.854189	−	0.519963i	$-0.174054\pi$
$797$	38438.9	1.70838	0.854189	+	0.519963i	$0.174054\pi$
$798$	−8478.76	−0.376121
$799$	−19992.8	−0.885224
$800$	4877.87	0.215574
$801$	−9379.42	−0.413740
$802$	24958.9	1.09891
$803$	0	0
$804$	12870.0	0.564538
$805$	−964.733	−0.0422390
$806$	15951.3	0.697096
$807$	9643.45	0.420651
$808$	7390.42	0.321775
$809$	−9960.91	−0.432889	−0.216444	−	0.976295i	$-0.569446\pi$
$809$	−9960.91	−0.432889	−0.216444	+	0.976295i	$0.569446\pi$
$810$	1454.18	0.0630797
$811$	34199.9	1.48079	0.740396	−	0.672171i	$-0.234638\pi$
$811$	34199.9	1.48079	0.740396	+	0.672171i	$0.234638\pi$
$812$	−18936.6	−0.818404
$813$	−22132.1	−0.954744
$814$	0	0
$815$	15466.9	0.664762
$816$	25426.8	1.09083
$817$	10942.3	0.468570
$818$	42203.9	1.80394
$819$	7864.82	0.335555
$820$	−3053.53	−0.130041
$821$	31796.4	1.35165	0.675823	−	0.737064i	$-0.263788\pi$
$821$	31796.4	1.35165	0.675823	+	0.737064i	$0.263788\pi$
$822$	−22897.1	−0.971567
$823$	−10783.6	−0.456733	−0.228366	−	0.973575i	$-0.573338\pi$
$823$	−10783.6	−0.456733	−0.228366	+	0.973575i	$0.573338\pi$
$824$	4459.07	0.188518
$825$	0	0
$826$	−13504.5	−0.568862
$827$	−44197.5	−1.85840	−0.929201	−	0.369576i	$-0.879503\pi$
$827$	−44197.5	−1.85840	−0.929201	+	0.369576i	$0.879503\pi$
$828$	526.137	0.0220828
$829$	22487.7	0.942137	0.471069	−	0.882097i	$-0.343868\pi$
$829$	22487.7	0.942137	0.471069	+	0.882097i	$0.343868\pi$
$830$	20844.4	0.871709
$831$	−2431.82	−0.101515
$832$	−3622.81	−0.150959
$833$	−8805.96	−0.366276
$834$	19111.9	0.793516
$835$	−12082.2	−0.500743
$836$	0	0
$837$	2216.30	0.0915250
$838$	−42475.6	−1.75095
$839$	22491.0	0.925477	0.462738	−	0.886495i	$-0.346867\pi$
$839$	22491.0	0.925477	0.462738	+	0.886495i	$0.346867\pi$
$840$	2702.69	0.111014
$841$	33082.7	1.35646
$842$	7181.37	0.293927
$843$	3344.17	0.136630
$844$	−10198.0	−0.415910
$845$	−3660.61	−0.149028
$846$	6037.48	0.245358
$847$	0	0
$848$	−18533.5	−0.750524
$849$	6796.19	0.274729
$850$	−9605.60	−0.387611
$851$	−259.994	−0.0104730
$852$	10759.8	0.432660
$853$	−22327.9	−0.896241	−0.448120	−	0.893973i	$-0.647906\pi$
$853$	−22327.9	−0.896241	−0.448120	+	0.893973i	$0.647906\pi$
$854$	9499.12	0.380624
$855$	2193.73	0.0877474
$856$	17756.5	0.708999
$857$	14505.9	0.578193	0.289096	−	0.957300i	$-0.406645\pi$
$857$	14505.9	0.578193	0.289096	+	0.957300i	$0.406645\pi$
$858$	0	0
$859$	−8411.45	−0.334104	−0.167052	−	0.985948i	$-0.553425\pi$
$859$	−8411.45	−0.334104	−0.167052	+	0.985948i	$0.553425\pi$
$860$	5490.40	0.217699
$861$	−6046.92	−0.239348
$862$	−28954.7	−1.14408
$863$	32499.6	1.28192	0.640961	−	0.767573i	$-0.278536\pi$
$863$	32499.6	1.28192	0.640961	+	0.767573i	$0.278536\pi$
$864$	−5268.10	−0.207436
$865$	−18790.1	−0.738592
$866$	38455.9	1.50899
$867$	−19614.1	−0.768316
$868$	−6483.94	−0.253548
$869$	0	0
$870$	12911.6	0.503155
$871$	−47460.0	−1.84629
$872$	−8426.50	−0.327245
$873$	13919.6	0.539643
$874$	2091.67	0.0809516
$875$	−2018.31	−0.0779786
$876$	17053.5	0.657745
$877$	32183.1	1.23916	0.619581	−	0.784933i	$-0.287303\pi$
$877$	32183.1	1.23916	0.619581	+	0.784933i	$0.287303\pi$
$878$	16725.1	0.642876
$879$	11406.2	0.437680
$880$	0	0
$881$	−6246.34	−0.238870	−0.119435	−	0.992842i	$-0.538108\pi$
$881$	−6246.34	−0.238870	−0.119435	+	0.992842i	$0.538108\pi$
$882$	2659.25	0.101521
$883$	−11801.1	−0.449762	−0.224881	−	0.974386i	$-0.572199\pi$
$883$	−11801.1	−0.449762	−0.224881	+	0.974386i	$0.572199\pi$
$884$	28332.7	1.07798
$885$	3494.05	0.132713
$886$	−7521.24	−0.285193
$887$	−32375.1	−1.22553	−0.612767	−	0.790264i	$-0.709944\pi$
$887$	−32375.1	−1.22553	−0.612767	+	0.790264i	$0.709944\pi$
$888$	728.371	0.0275254
$889$	−23594.6	−0.890145
$890$	−18709.7	−0.704662
$891$	0	0
$892$	9233.17	0.346580
$893$	9107.98	0.341307
$894$	−16973.8	−0.634997
$895$	−12670.0	−0.473196
$896$	−21322.6	−0.795020
$897$	−1940.21	−0.0722205
$898$	5034.72	0.187095
$899$	19678.5	0.730049
$900$	1100.73	0.0407677
$901$	25039.9	0.925861
$902$	0	0
$903$	10872.7	0.400686
$904$	−12784.4	−0.470357
$905$	69.4977	0.00255269
$906$	−4534.64	−0.166284
$907$	−19592.8	−0.717277	−0.358638	−	0.933477i	$-0.616759\pi$
$907$	−19592.8	−0.717277	−0.358638	+	0.933477i	$0.616759\pi$
$908$	5629.53	0.205752
$909$	5960.54	0.217490
$910$	15688.4	0.571500
$911$	18673.8	0.679133	0.339567	−	0.940582i	$-0.389720\pi$
$911$	18673.8	0.679133	0.339567	+	0.940582i	$0.389720\pi$
$912$	−11583.5	−0.420579
$913$	0	0
$914$	−17644.7	−0.638550
$915$	−2457.73	−0.0887980
$916$	20090.9	0.724696
$917$	24622.4	0.886698
$918$	10374.1	0.372979
$919$	4572.90	0.164142	0.0820708	−	0.996627i	$-0.473847\pi$
$919$	4572.90	0.164142	0.0820708	+	0.996627i	$0.473847\pi$
$920$	−666.739	−0.0238932
$921$	−4069.05	−0.145581
$922$	−7949.78	−0.283961
$923$	−39678.6	−1.41499
$924$	0	0
$925$	−543.932	−0.0193345
$926$	19988.7	0.709362
$927$	3596.33	0.127421
$928$	−46775.4	−1.65461
$929$	−44222.1	−1.56176	−0.780882	−	0.624679i	$-0.785230\pi$
$929$	−44222.1	−1.56176	−0.780882	+	0.624679i	$0.785230\pi$
$930$	4420.97	0.155881
$931$	4011.67	0.141221
$932$	28049.3	0.985823
$933$	24234.2	0.850367
$934$	−1784.70	−0.0625238
$935$	0	0
$936$	5435.48	0.189812
$937$	−9218.62	−0.321408	−0.160704	−	0.987003i	$-0.551376\pi$
$937$	−9218.62	−0.321408	−0.160704	+	0.987003i	$0.551376\pi$
$938$	50839.2	1.76968
$939$	−17284.6	−0.600705
$940$	4570.02	0.158572
$941$	26484.9	0.917516	0.458758	−	0.888561i	$-0.348294\pi$
$941$	26484.9	0.917516	0.458758	+	0.888561i	$0.348294\pi$
$942$	−23959.4	−0.828705
$943$	1491.75	0.0515142
$944$	−18449.5	−0.636103
$945$	2179.77	0.0750350
$946$	0	0
$947$	−44972.4	−1.54320	−0.771599	−	0.636110i	$-0.780543\pi$
$947$	−44972.4	−1.54320	−0.771599	+	0.636110i	$0.780543\pi$
$948$	−8641.90	−0.296072
$949$	−62887.5	−2.15112
$950$	4375.96	0.149447
$951$	−15321.0	−0.522416
$952$	19280.9	0.656404
$953$	2052.50	0.0697659	0.0348829	−	0.999391i	$-0.488894\pi$
$953$	2052.50	0.0697659	0.0348829	+	0.999391i	$0.488894\pi$
$954$	−7561.63	−0.256621
$955$	−17476.9	−0.592189
$956$	−29529.7	−0.999016
$957$	0	0
$958$	−33569.2	−1.13212
$959$	−34322.2	−1.15570
$960$	−1004.08	−0.0337568
$961$	−23053.0	−0.773826
$962$	4228.00	0.141701
$963$	14321.0	0.479218
$964$	−18203.1	−0.608176
$965$	17346.6	0.578661
$966$	2078.36	0.0692237
$967$	−14950.9	−0.497195	−0.248598	−	0.968607i	$-0.579970\pi$
$967$	−14950.9	−0.497195	−0.248598	+	0.968607i	$0.579970\pi$
$968$	0	0
$969$	15650.0	0.518835
$970$	27766.3	0.919094
$971$	26751.9	0.884151	0.442076	−	0.896978i	$-0.354242\pi$
$971$	26751.9	0.884151	0.442076	+	0.896978i	$0.354242\pi$
$972$	−1188.78	−0.0392287
$973$	28648.3	0.943908
$974$	−710.034	−0.0233583
$975$	−4059.10	−0.133329
$976$	12977.5	0.425615
$977$	−56893.7	−1.86304	−0.931520	−	0.363690i	$-0.881517\pi$
$977$	−56893.7	−1.86304	−0.931520	+	0.363690i	$0.881517\pi$
$978$	−33320.8	−1.08945
$979$	0	0
$980$	2012.89	0.0656118
$981$	−6796.15	−0.221187
$982$	−35895.0	−1.16645
$983$	−34633.1	−1.12373	−0.561863	−	0.827230i	$-0.689915\pi$
$983$	−34633.1	−1.12373	−0.561863	+	0.827230i	$0.689915\pi$
$984$	−4179.10	−0.135391
$985$	18191.9	0.588470
$986$	92111.0	2.97506
$987$	9050.03	0.291860
$988$	−12907.3	−0.415625
$989$	−2682.23	−0.0862386
$990$	0	0
$991$	1961.79	0.0628843	0.0314422	−	0.999506i	$-0.489990\pi$
$991$	1961.79	0.0628843	0.0314422	+	0.999506i	$0.489990\pi$
$992$	−16016.0	−0.512610
$993$	8342.83	0.266618
$994$	42503.8	1.35628
$995$	−258.025	−0.00822103
$996$	−17040.2	−0.542108
$997$	−13096.5	−0.416017	−0.208009	−	0.978127i	$-0.566698\pi$
$997$	−13096.5	−0.416017	−0.208009	+	0.978127i	$0.566698\pi$
$998$	31290.8	0.992476
$999$	587.447	0.0186046

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.s.1.1		3
11.10	odd	2		165.4.a.d.1.3	✓	3
33.32	even	2		495.4.a.l.1.1		3
55.32	even	4		825.4.c.l.199.5		6
55.43	even	4		825.4.c.l.199.2		6
55.54	odd	2		825.4.a.s.1.1		3
165.164	even	2		2475.4.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.d.1.3	✓	3	11.10	odd	2
495.4.a.l.1.1		3	33.32	even	2
825.4.a.s.1.1		3	55.54	odd	2
825.4.c.l.199.2		6	55.43	even	4
825.4.c.l.199.5		6	55.32	even	4
1815.4.a.s.1.1		3	1.1	even	1	trivial
2475.4.a.s.1.3		3	165.164	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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q-3.59056 q^{2} -3.00000 q^{3} +4.89212 q^{4} -5.00000 q^{5} +10.7717 q^{6} +16.1465 q^{7} +11.1590 q^{8} +9.00000 q^{9} +17.9528 q^{10} -14.6764 q^{12} +54.1214 q^{13} -57.9749 q^{14} +15.0000 q^{15} -79.2041 q^{16} +107.010 q^{17} -32.3150 q^{18} -48.7496 q^{19} -24.4606 q^{20} -48.4394 q^{21} +11.9498 q^{23} -33.4771 q^{24} +25.0000 q^{25} -194.326 q^{26} -27.0000 q^{27} +78.9905 q^{28} -239.733 q^{29} -53.8584 q^{30} -82.0851 q^{31} +195.115 q^{32} -384.224 q^{34} -80.7324 q^{35} +44.0291 q^{36} -21.7573 q^{37} +175.038 q^{38} -162.364 q^{39} -55.7952 q^{40} +124.835 q^{41} +173.925 q^{42} -224.459 q^{43} -45.0000 q^{45} -42.9064 q^{46} -186.832 q^{47} +237.612 q^{48} -82.2913 q^{49} -89.7640 q^{50} -321.029 q^{51} +264.768 q^{52} +233.997 q^{53} +96.9451 q^{54} +180.179 q^{56} +146.249 q^{57} +860.774 q^{58} +232.936 q^{59} +73.3818 q^{60} -163.849 q^{61} +294.731 q^{62} +145.318 q^{63} -66.9386 q^{64} -270.607 q^{65} -876.918 q^{67} +523.503 q^{68} -35.8493 q^{69} +289.874 q^{70} -733.141 q^{71} +100.431 q^{72} -1161.97 q^{73} +78.1208 q^{74} -75.0000 q^{75} -238.489 q^{76} +582.978 q^{78} +588.831 q^{79} +396.021 q^{80} +81.0000 q^{81} -448.226 q^{82} +1161.06 q^{83} -236.971 q^{84} -535.048 q^{85} +805.933 q^{86} +719.198 q^{87} -1042.16 q^{89} +161.575 q^{90} +873.869 q^{91} +58.4597 q^{92} +246.255 q^{93} +670.831 q^{94} +243.748 q^{95} -585.345 q^{96} +1546.63 q^{97} +295.472 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} - 12 q^{6} + 4 q^{7} + 48 q^{8} + 27 q^{9} - 20 q^{10} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} + 218 q^{17} + 36 q^{18} - 146 q^{19} - 110 q^{20} - 12 q^{21}+ \cdots - 572 q^{98}+O(q^{100}) \) 3 * q + 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 - 12 * q^6 + 4 * q^7 + 48 * q^8 + 27 * q^9 - 20 * q^10 - 66 * q^12 - 56 * q^14 + 45 * q^15 + 50 * q^16 + 218 * q^17 + 36 * q^18 - 146 * q^19 - 110 * q^20 - 12 * q^21 - 200 * q^23 - 144 * q^24 + 75 * q^25 - 508 * q^26 - 81 * q^27 + 340 * q^28 - 68 * q^29 + 60 * q^30 - 68 * q^31 + 688 * q^32 - 176 * q^34 - 20 * q^35 + 198 * q^36 - 390 * q^37 - 316 * q^38 - 240 * q^40 + 196 * q^41 + 168 * q^42 + 524 * q^43 - 135 * q^45 - 1160 * q^46 - 60 * q^47 - 150 * q^48 - 157 * q^49 + 100 * q^50 - 654 * q^51 - 1020 * q^52 - 158 * q^53 - 108 * q^54 + 1368 * q^56 + 438 * q^57 + 1092 * q^58 - 1044 * q^59 + 330 * q^60 - 642 * q^61 - 88 * q^62 + 36 * q^63 + 1166 * q^64 - 236 * q^67 - 144 * q^68 + 600 * q^69 + 280 * q^70 - 544 * q^71 + 432 * q^72 - 900 * q^73 - 1536 * q^74 - 225 * q^75 - 1996 * q^76 + 1524 * q^78 + 1586 * q^79 - 250 * q^80 + 243 * q^81 + 380 * q^82 + 1582 * q^83 - 1020 * q^84 - 1090 * q^85 + 3568 * q^86 + 204 * q^87 - 2122 * q^89 - 180 * q^90 - 8 * q^91 - 4128 * q^92 + 204 * q^93 + 2152 * q^94 + 730 * q^95 - 2064 * q^96 + 618 * q^97 - 572 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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