LMFDB - Embedded newform 275.6.a.l.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,6,Mod(1,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("275.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	275.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:	$44.1055504486$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 352 x^{12} + 47255 x^{10} - 3018020 x^{8} + 93670492 x^{6} - 1312532272 x^{4} + \cdots - 10706454784 $ x^14 - 352x^12 + 47255x^10 - 3018020x^8 + 93670492x^6 - 1312532272x^4 + 7314926400x^2 - 10706454784
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.98693$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.98693 q^{2} +2.19418 q^{3} -16.1044 q^{4} -8.74803 q^{6} +90.9095 q^{7} +191.789 q^{8} -238.186 q^{9} +O(q^{10})$	\(q-3.98693 q^{2} +2.19418 q^{3} -16.1044 q^{4} -8.74803 q^{6} +90.9095 q^{7} +191.789 q^{8} -238.186 q^{9} -121.000 q^{11} -35.3359 q^{12} +533.632 q^{13} -362.450 q^{14} -249.308 q^{16} -1599.02 q^{17} +949.629 q^{18} +2530.14 q^{19} +199.472 q^{21} +482.418 q^{22} -3015.96 q^{23} +420.819 q^{24} -2127.55 q^{26} -1055.81 q^{27} -1464.04 q^{28} -8585.18 q^{29} +5151.17 q^{31} -5143.27 q^{32} -265.496 q^{33} +6375.17 q^{34} +3835.84 q^{36} +10772.3 q^{37} -10087.5 q^{38} +1170.88 q^{39} +3608.14 q^{41} -795.279 q^{42} -9224.88 q^{43} +1948.63 q^{44} +12024.4 q^{46} +14482.8 q^{47} -547.025 q^{48} -8542.47 q^{49} -3508.53 q^{51} -8593.82 q^{52} -14087.6 q^{53} +4209.43 q^{54} +17435.4 q^{56} +5551.58 q^{57} +34228.5 q^{58} +47154.1 q^{59} +11407.2 q^{61} -20537.4 q^{62} -21653.3 q^{63} +28483.7 q^{64} +1058.51 q^{66} -16964.5 q^{67} +25751.2 q^{68} -6617.55 q^{69} +70000.1 q^{71} -45681.3 q^{72} -14601.9 q^{73} -42948.5 q^{74} -40746.4 q^{76} -11000.0 q^{77} -4668.23 q^{78} +36856.2 q^{79} +55562.5 q^{81} -14385.4 q^{82} +48596.1 q^{83} -3212.37 q^{84} +36778.9 q^{86} -18837.4 q^{87} -23206.4 q^{88} -48077.2 q^{89} +48512.2 q^{91} +48570.2 q^{92} +11302.6 q^{93} -57741.9 q^{94} -11285.2 q^{96} +153399. q^{97} +34058.2 q^{98} +28820.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 256 q^{4} + 14 q^{6} + 1550 q^{9}+O(q^{10}) $ 14 * q + 256 * q^4 + 14 * q^6 + 1550 * q^9	$ 14 q + 256 q^{4} + 14 q^{6} + 1550 q^{9} - 1694 q^{11} + 3094 q^{14} + 5540 q^{16} + 3940 q^{19} + 12776 q^{21} + 14438 q^{24} - 4696 q^{26} - 1132 q^{29} + 3740 q^{31} + 23834 q^{34} + 59986 q^{36} + 10800 q^{39} + 30740 q^{41} - 30976 q^{44} + 82544 q^{46} + 21746 q^{49} + 104916 q^{51} - 17210 q^{54} + 194170 q^{56} - 39076 q^{59} + 246740 q^{61} - 5480 q^{64} - 1694 q^{66} + 177988 q^{69} + 57468 q^{71} + 247910 q^{74} + 581730 q^{76} + 252200 q^{79} + 605134 q^{81} + 752886 q^{84} + 565312 q^{86} + 359692 q^{89} + 305800 q^{91} + 710788 q^{94} + 1094678 q^{96} - 187550 q^{99}+O(q^{100}) $ 14 * q + 256 * q^4 + 14 * q^6 + 1550 * q^9 - 1694 * q^11 + 3094 * q^14 + 5540 * q^16 + 3940 * q^19 + 12776 * q^21 + 14438 * q^24 - 4696 * q^26 - 1132 * q^29 + 3740 * q^31 + 23834 * q^34 + 59986 * q^36 + 10800 * q^39 + 30740 * q^41 - 30976 * q^44 + 82544 * q^46 + 21746 * q^49 + 104916 * q^51 - 17210 * q^54 + 194170 * q^56 - 39076 * q^59 + 246740 * q^61 - 5480 * q^64 - 1694 * q^66 + 177988 * q^69 + 57468 * q^71 + 247910 * q^74 + 581730 * q^76 + 252200 * q^79 + 605134 * q^81 + 752886 * q^84 + 565312 * q^86 + 359692 * q^89 + 305800 * q^91 + 710788 * q^94 + 1094678 * q^96 - 187550 * q^99

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 $

$2$	−3.98693	−0.704796	−0.352398	−	0.935850i	$-0.614634\pi$
$2$	−3.98693	−0.704796	−0.352398	+	0.935850i	$0.614634\pi$
$3$	2.19418	0.140757	0.0703783	−	0.997520i	$-0.477579\pi$
$3$	2.19418	0.140757	0.0703783	+	0.997520i	$0.477579\pi$
$4$	−16.1044	−0.503262
$5$	0	0
$5$	0	0
$6$	−8.74803	−0.0992047
$7$	90.9095	0.701236	0.350618	−	0.936519i	$-0.385972\pi$
$7$	90.9095	0.701236	0.350618	+	0.936519i	$0.385972\pi$
$8$	191.789	1.05949
$9$	−238.186	−0.980188
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−35.3359	−0.0708375
$13$	533.632	0.875757	0.437878	−	0.899034i	$-0.355730\pi$
$13$	533.632	0.875757	0.437878	+	0.899034i	$0.355730\pi$
$14$	−362.450	−0.494228
$15$	0	0
$16$	−249.308	−0.243465
$17$	−1599.02	−1.34193	−0.670966	−	0.741488i	$-0.734121\pi$
$17$	−1599.02	−1.34193	−0.670966	+	0.741488i	$0.734121\pi$
$18$	949.629	0.690832
$19$	2530.14	1.60791	0.803953	−	0.594692i	$-0.202726\pi$
$19$	2530.14	1.60791	0.803953	+	0.594692i	$0.202726\pi$
$20$	0	0
$21$	199.472	0.0987035
$22$	482.418	0.212504
$23$	−3015.96	−1.18879	−0.594396	−	0.804173i	$-0.702609\pi$
$23$	−3015.96	−1.18879	−0.594396	+	0.804173i	$0.702609\pi$
$24$	420.819	0.149131
$25$	0	0
$26$	−2127.55	−0.617230
$27$	−1055.81	−0.278724
$28$	−1464.04	−0.352906
$29$	−8585.18	−1.89563	−0.947817	−	0.318816i	$-0.896715\pi$
$29$	−8585.18	−1.89563	−0.947817	+	0.318816i	$0.896715\pi$
$30$	0	0
$31$	5151.17	0.962724	0.481362	−	0.876522i	$-0.340142\pi$
$31$	5151.17	0.962724	0.481362	+	0.876522i	$0.340142\pi$
$32$	−5143.27	−0.887901
$33$	−265.496	−0.0424397
$34$	6375.17	0.945789
$35$	0	0
$36$	3835.84	0.493292
$37$	10772.3	1.29362	0.646808	−	0.762653i	$-0.276104\pi$
$37$	10772.3	1.29362	0.646808	+	0.762653i	$0.276104\pi$
$38$	−10087.5	−1.13325
$39$	1170.88	0.123269
$40$	0	0
$41$	3608.14	0.335215	0.167608	−	0.985854i	$-0.446396\pi$
$41$	3608.14	0.335215	0.167608	+	0.985854i	$0.446396\pi$
$42$	−795.279	−0.0695659
$43$	−9224.88	−0.760834	−0.380417	−	0.924815i	$-0.624219\pi$
$43$	−9224.88	−0.760834	−0.380417	+	0.924815i	$0.624219\pi$
$44$	1948.63	0.151739
$45$	0	0
$46$	12024.4	0.837856
$47$	14482.8	0.956330	0.478165	−	0.878270i	$-0.341302\pi$
$47$	14482.8	0.956330	0.478165	+	0.878270i	$0.341302\pi$
$48$	−547.025	−0.0342692
$49$	−8542.47	−0.508269
$50$	0	0
$51$	−3508.53	−0.188886
$52$	−8593.82	−0.440736
$53$	−14087.6	−0.688884	−0.344442	−	0.938808i	$-0.611932\pi$
$53$	−14087.6	−0.688884	−0.344442	+	0.938808i	$0.611932\pi$
$54$	4209.43	0.196444
$55$	0	0
$56$	17435.4	0.742955
$57$	5551.58	0.226323
$58$	34228.5	1.33603
$59$	47154.1	1.76356	0.881778	−	0.471665i	$-0.156347\pi$
$59$	47154.1	1.76356	0.881778	+	0.471665i	$0.156347\pi$
$60$	0	0
$61$	11407.2	0.392512	0.196256	−	0.980553i	$-0.437122\pi$
$61$	11407.2	0.392512	0.196256	+	0.980553i	$0.437122\pi$
$62$	−20537.4	−0.678524
$63$	−21653.3	−0.687342
$64$	28483.7	0.869254
$65$	0	0
$66$	1058.51	0.0299113
$67$	−16964.5	−0.461693	−0.230846	−	0.972990i	$-0.574149\pi$
$67$	−16964.5	−0.461693	−0.230846	+	0.972990i	$0.574149\pi$
$68$	25751.2	0.675344
$69$	−6617.55	−0.167330
$70$	0	0
$71$	70000.1	1.64798	0.823992	−	0.566602i	$-0.191742\pi$
$71$	70000.1	1.64798	0.823992	+	0.566602i	$0.191742\pi$
$72$	−45681.3	−1.03850
$73$	−14601.9	−0.320702	−0.160351	−	0.987060i	$-0.551263\pi$
$73$	−14601.9	−0.320702	−0.160351	+	0.987060i	$0.551263\pi$
$74$	−42948.5	−0.911735
$75$	0	0
$76$	−40746.4	−0.809199
$77$	−11000.0	−0.211431
$78$	−4668.23	−0.0868792
$79$	36856.2	0.664421	0.332210	−	0.943205i	$-0.392206\pi$
$79$	36856.2	0.664421	0.332210	+	0.943205i	$0.392206\pi$
$80$	0	0
$81$	55562.5	0.940955
$82$	−14385.4	−0.236258
$83$	48596.1	0.774294	0.387147	−	0.922018i	$-0.373461\pi$
$83$	48596.1	0.774294	0.387147	+	0.922018i	$0.373461\pi$
$84$	−3212.37	−0.0496738
$85$	0	0
$86$	36778.9	0.536233
$87$	−18837.4	−0.266823
$88$	−23206.4	−0.319449
$89$	−48077.2	−0.643375	−0.321687	−	0.946846i	$-0.604250\pi$
$89$	−48077.2	−0.643375	−0.321687	+	0.946846i	$0.604250\pi$
$90$	0	0
$91$	48512.2	0.614112
$92$	48570.2	0.598274
$93$	11302.6	0.135510
$94$	−57741.9	−0.674018
$95$	0	0
$96$	−11285.2	−0.124978
$97$	153399.	1.65536	0.827682	−	0.561198i	$-0.189659\pi$
$97$	153399.	1.65536	0.827682	+	0.561198i	$0.189659\pi$
$98$	34058.2	0.358226
$99$	28820.5	0.295538
$100$	0	0
$101$	25004.5	0.243902	0.121951	−	0.992536i	$-0.461085\pi$
$101$	25004.5	0.243902	0.121951	+	0.992536i	$0.461085\pi$
$102$	13988.3	0.133126
$103$	137496.	1.27701	0.638507	−	0.769616i	$-0.279552\pi$
$103$	137496.	1.27701	0.638507	+	0.769616i	$0.279552\pi$
$104$	102345.	0.927859
$105$	0	0
$106$	56166.1	0.485523
$107$	220391.	1.86095	0.930474	−	0.366358i	$-0.119396\pi$
$107$	220391.	1.86095	0.930474	+	0.366358i	$0.119396\pi$
$108$	17003.1	0.140272
$109$	225704.	1.81959	0.909796	−	0.415056i	$-0.136238\pi$
$109$	225704.	1.81959	0.909796	+	0.415056i	$0.136238\pi$
$110$	0	0
$111$	23636.4	0.182085
$112$	−22664.4	−0.170726
$113$	−98344.0	−0.724522	−0.362261	−	0.932077i	$-0.617995\pi$
$113$	−98344.0	−0.724522	−0.362261	+	0.932077i	$0.617995\pi$
$114$	−22133.8	−0.159512
$115$	0	0
$116$	138259.	0.954001
$117$	−127103.	−0.858406
$118$	−188000.	−1.24295
$119$	−145366.	−0.941011
$120$	0	0
$121$	14641.0	0.0909091
$122$	−45479.6	−0.276641
$123$	7916.90	0.0471838
$124$	−82956.5	−0.484503
$125$	0	0
$126$	86330.3	0.484436
$127$	−22651.8	−0.124622	−0.0623108	−	0.998057i	$-0.519847\pi$
$127$	−22651.8	−0.124622	−0.0623108	+	0.998057i	$0.519847\pi$
$128$	51022.2	0.275254
$129$	−20241.0	−0.107092
$130$	0	0
$131$	45526.9	0.231787	0.115894	−	0.993262i	$-0.463027\pi$
$131$	45526.9	0.231787	0.115894	+	0.993262i	$0.463027\pi$
$132$	4275.65	0.0213583
$133$	230014.	1.12752
$134$	67636.1	0.325399
$135$	0	0
$136$	−306674.	−1.42177
$137$	−97204.5	−0.442471	−0.221235	−	0.975220i	$-0.571009\pi$
$137$	−97204.5	−0.442471	−0.221235	+	0.975220i	$0.571009\pi$
$138$	26383.7	0.117934
$139$	−4897.20	−0.0214986	−0.0107493	−	0.999942i	$-0.503422\pi$
$139$	−4897.20	−0.0214986	−0.0107493	+	0.999942i	$0.503422\pi$
$140$	0	0
$141$	31777.8	0.134610
$142$	−279086.	−1.16149
$143$	−64569.5	−0.264051
$144$	59381.5	0.238641
$145$	0	0
$146$	58216.6	0.226029
$147$	−18743.7	−0.0715421
$148$	−173482.	−0.651028
$149$	72021.5	0.265764	0.132882	−	0.991132i	$-0.457577\pi$
$149$	72021.5	0.265764	0.132882	+	0.991132i	$0.457577\pi$
$150$	0	0
$151$	120149.	0.428822	0.214411	−	0.976743i	$-0.431217\pi$
$151$	120149.	0.428822	0.214411	+	0.976743i	$0.431217\pi$
$152$	485253.	1.70357
$153$	380863.	1.31535
$154$	43856.4	0.149015
$155$	0	0
$156$	−18856.4	−0.0620364
$157$	−147799.	−0.478545	−0.239273	−	0.970952i	$-0.576909\pi$
$157$	−147799.	−0.478545	−0.239273	+	0.970952i	$0.576909\pi$
$158$	−146943.	−0.468281
$159$	−30910.6	−0.0969650
$160$	0	0
$161$	−274179.	−0.833623
$162$	−221524.	−0.663182
$163$	−47145.7	−0.138987	−0.0694933	−	0.997582i	$-0.522138\pi$
$163$	−47145.7	−0.138987	−0.0694933	+	0.997582i	$0.522138\pi$
$164$	−58106.9	−0.168701
$165$	0	0
$166$	−193749.	−0.545720
$167$	−181485.	−0.503559	−0.251779	−	0.967785i	$-0.581016\pi$
$167$	−181485.	−0.503559	−0.251779	+	0.967785i	$0.581016\pi$
$168$	38256.4	0.104576
$169$	−86529.8	−0.233050
$170$	0	0
$171$	−602643.	−1.57605
$172$	148561.	0.382899
$173$	−680704.	−1.72919	−0.864596	−	0.502468i	$-0.832425\pi$
$173$	−680704.	−1.72919	−0.864596	+	0.502468i	$0.832425\pi$
$174$	75103.4	0.188056
$175$	0	0
$176$	30166.2	0.0734073
$177$	103464.	0.248232
$178$	191680.	0.453448
$179$	372370.	0.868646	0.434323	−	0.900757i	$-0.356988\pi$
$179$	372370.	0.868646	0.434323	+	0.900757i	$0.356988\pi$
$180$	0	0
$181$	−187574.	−0.425575	−0.212788	−	0.977098i	$-0.568254\pi$
$181$	−187574.	−0.425575	−0.212788	+	0.977098i	$0.568254\pi$
$182$	−193415.	−0.432824
$183$	25029.3	0.0552487
$184$	−578427.	−1.25952
$185$	0	0
$186$	−45062.6	−0.0955067
$187$	193481.	0.404608
$188$	−233237.	−0.481285
$189$	−95982.8	−0.195452
$190$	0	0
$191$	327121.	0.648821	0.324411	−	0.945916i	$-0.394834\pi$
$191$	327121.	0.648821	0.324411	+	0.945916i	$0.394834\pi$
$192$	62498.3	0.122353
$193$	593369.	1.14665	0.573326	−	0.819327i	$-0.305653\pi$
$193$	593369.	1.14665	0.573326	+	0.819327i	$0.305653\pi$
$194$	−611591.	−1.16669
$195$	0	0
$196$	137571.	0.255792
$197$	−699226.	−1.28367	−0.641833	−	0.766844i	$-0.721826\pi$
$197$	−699226.	−1.28367	−0.641833	+	0.766844i	$0.721826\pi$
$198$	−114905.	−0.208294
$199$	203765.	0.364751	0.182376	−	0.983229i	$-0.441621\pi$
$199$	203765.	0.364751	0.182376	+	0.983229i	$0.441621\pi$
$200$	0	0
$201$	−37223.0	−0.0649863
$202$	−99691.2	−0.171901
$203$	−780474.	−1.32929
$204$	56502.7	0.0950592
$205$	0	0
$206$	−548185.	−0.900035
$207$	718358.	1.16524
$208$	−133039.	−0.213216
$209$	−306147.	−0.484802
$210$	0	0
$211$	−395743.	−0.611937	−0.305969	−	0.952042i	$-0.598980\pi$
$211$	−395743.	−0.611937	−0.305969	+	0.952042i	$0.598980\pi$
$212$	226872.	0.346690
$213$	153593.	0.231965
$214$	−878683.	−1.31159
$215$	0	0
$216$	−202492.	−0.295307
$217$	468290.	0.675096
$218$	−899868.	−1.28244
$219$	−32039.1	−0.0451409
$220$	0	0
$221$	−853287.	−1.17521
$222$	−94236.6	−0.128333
$223$	−477978.	−0.643644	−0.321822	−	0.946800i	$-0.604295\pi$
$223$	−477978.	−0.643644	−0.321822	+	0.946800i	$0.604295\pi$
$224$	−467572.	−0.622628
$225$	0	0
$226$	392091.	0.510641
$227$	588034.	0.757422	0.378711	−	0.925515i	$-0.376367\pi$
$227$	588034.	0.757422	0.378711	+	0.925515i	$0.376367\pi$
$228$	−89404.9	−0.113900
$229$	388983.	0.490165	0.245082	−	0.969502i	$-0.421185\pi$
$229$	388983.	0.490165	0.245082	+	0.969502i	$0.421185\pi$
$230$	0	0
$231$	−24136.1	−0.0297602
$232$	−1.64654e6	−2.00841
$233$	501987.	0.605763	0.302882	−	0.953028i	$-0.402051\pi$
$233$	501987.	0.605763	0.302882	+	0.953028i	$0.402051\pi$
$234$	506753.	0.605001
$235$	0	0
$236$	−759388.	−0.887531
$237$	80869.1	0.0935216
$238$	579563.	0.663221
$239$	481013.	0.544705	0.272353	−	0.962197i	$-0.412198\pi$
$239$	481013.	0.544705	0.272353	+	0.962197i	$0.412198\pi$
$240$	0	0
$241$	−463041.	−0.513544	−0.256772	−	0.966472i	$-0.582659\pi$
$241$	−463041.	−0.513544	−0.256772	+	0.966472i	$0.582659\pi$
$242$	−58372.6	−0.0640724
$243$	378475.	0.411170
$244$	−183706.	−0.197537
$245$	0	0
$246$	−31564.1	−0.0332549
$247$	1.35016e6	1.40814
$248$	987937.	1.02000
$249$	106628.	0.108987
$250$	0	0
$251$	401567.	0.402321	0.201161	−	0.979558i	$-0.435529\pi$
$251$	401567.	0.402321	0.201161	+	0.979558i	$0.435529\pi$
$252$	348714.	0.345914
$253$	364931.	0.358434
$254$	90311.0	0.0878328
$255$	0	0
$256$	−1.11490e6	−1.06325
$257$	−1.25197e6	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$257$	−1.25197e6	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$258$	80699.5	0.0754783
$259$	979306.	0.907129
$260$	0	0
$261$	2.04487e6	1.85808
$262$	−181513.	−0.163363
$263$	1.27396e6	1.13571	0.567853	−	0.823130i	$-0.307774\pi$
$263$	1.27396e6	1.13571	0.567853	+	0.823130i	$0.307774\pi$
$264$	−50919.1	−0.0449646
$265$	0	0
$266$	−917049.	−0.794673
$267$	−105490.	−0.0905593
$268$	273202.	0.232353
$269$	863140.	0.727278	0.363639	−	0.931540i	$-0.381534\pi$
$269$	863140.	0.727278	0.363639	+	0.931540i	$0.381534\pi$
$270$	0	0
$271$	−265882.	−0.219920	−0.109960	−	0.993936i	$-0.535072\pi$
$271$	−265882.	−0.219920	−0.109960	+	0.993936i	$0.535072\pi$
$272$	398647.	0.326713
$273$	106444.	0.0864403
$274$	387547.	0.311852
$275$	0	0
$276$	106572.	0.0842110
$277$	−269789.	−0.211264	−0.105632	−	0.994405i	$-0.533687\pi$
$277$	−269789.	−0.211264	−0.105632	+	0.994405i	$0.533687\pi$
$278$	19524.8	0.0151521
$279$	−1.22693e6	−0.943650
$280$	0	0
$281$	453679.	0.342754	0.171377	−	0.985206i	$-0.445178\pi$
$281$	453679.	0.342754	0.171377	+	0.985206i	$0.445178\pi$
$282$	−126696.	−0.0948724
$283$	10472.1	0.00777264	0.00388632	−	0.999992i	$-0.498763\pi$
$283$	10472.1	0.00777264	0.00388632	+	0.999992i	$0.498763\pi$
$284$	−1.12731e6	−0.829368
$285$	0	0
$286$	257434.	0.186102
$287$	328014.	0.235065
$288$	1.22505e6	0.870309
$289$	1.13700e6	0.800784
$290$	0	0
$291$	336585.	0.233003
$292$	235154.	0.161397
$293$	−339136.	−0.230783	−0.115392	−	0.993320i	$-0.536812\pi$
$293$	−339136.	−0.230783	−0.115392	+	0.993320i	$0.536812\pi$
$294$	74729.8	0.0504226
$295$	0	0
$296$	2.06601e6	1.37058
$297$	127753.	0.0840386
$298$	−287145.	−0.187310
$299$	−1.60941e6	−1.04109
$300$	0	0
$301$	−838629.	−0.533524
$302$	−479025.	−0.302232
$303$	54864.3	0.0343308
$304$	−630784.	−0.391468
$305$	0	0
$306$	−1.51847e6	−0.927051
$307$	821082.	0.497211	0.248605	−	0.968605i	$-0.420028\pi$
$307$	821082.	0.497211	0.248605	+	0.968605i	$0.420028\pi$
$308$	177149.	0.106405
$309$	301690.	0.179748
$310$	0	0
$311$	2.41897e6	1.41817	0.709087	−	0.705121i	$-0.249107\pi$
$311$	2.41897e6	1.41817	0.709087	+	0.705121i	$0.249107\pi$
$312$	224562.	0.130602
$313$	433767.	0.250263	0.125131	−	0.992140i	$-0.460065\pi$
$313$	433767.	0.250263	0.125131	+	0.992140i	$0.460065\pi$
$314$	589265.	0.337277
$315$	0	0
$316$	−593547.	−0.334378
$317$	−2.12363e6	−1.18695	−0.593473	−	0.804854i	$-0.702243\pi$
$317$	−2.12363e6	−1.18695	−0.593473	+	0.804854i	$0.702243\pi$
$318$	123238.	0.0683406
$319$	1.03881e6	0.571555
$320$	0	0
$321$	483577.	0.261941
$322$	1.09313e6	0.587534
$323$	−4.04574e6	−2.15770
$324$	−894800.	−0.473547
$325$	0	0
$326$	187966.	0.0979572
$327$	495236.	0.256119
$328$	692001.	0.355158
$329$	1.31662e6	0.670613
$330$	0	0
$331$	763933.	0.383253	0.191626	−	0.981468i	$-0.438624\pi$
$331$	763933.	0.383253	0.191626	+	0.981468i	$0.438624\pi$
$332$	−782611.	−0.389673
$333$	−2.56581e6	−1.26799
$334$	723569.	0.354906
$335$	0	0
$336$	−49729.8	−0.0240308
$337$	1.99588e6	0.957324	0.478662	−	0.877999i	$-0.341122\pi$
$337$	1.99588e6	0.957324	0.478662	+	0.877999i	$0.341122\pi$
$338$	344988.	0.164253
$339$	−215784.	−0.101981
$340$	0	0
$341$	−623292.	−0.290272
$342$	2.40270e6	1.11079
$343$	−2.30451e6	−1.05765
$344$	−1.76923e6	−0.806098
$345$	0	0
$346$	2.71392e6	1.21873
$347$	−651996.	−0.290684	−0.145342	−	0.989381i	$-0.546428\pi$
$347$	−651996.	−0.290684	−0.145342	+	0.989381i	$0.546428\pi$
$348$	303365.	0.134282
$349$	274136.	0.120477	0.0602383	−	0.998184i	$-0.480814\pi$
$349$	274136.	0.120477	0.0602383	+	0.998184i	$0.480814\pi$
$350$	0	0
$351$	−563412.	−0.244095
$352$	622336.	0.267712
$353$	2.59169e6	1.10700	0.553498	−	0.832851i	$-0.313293\pi$
$353$	2.59169e6	1.10700	0.553498	+	0.832851i	$0.313293\pi$
$354$	−412505.	−0.174953
$355$	0	0
$356$	774254.	0.323786
$357$	−318958.	−0.132454
$358$	−1.48461e6	−0.612218
$359$	1.82934e6	0.749132	0.374566	−	0.927200i	$-0.377792\pi$
$359$	1.82934e6	0.749132	0.374566	+	0.927200i	$0.377792\pi$
$360$	0	0
$361$	3.92552e6	1.58536
$362$	747845.	0.299944
$363$	32125.0	0.0127961
$364$	−781260.	−0.309059
$365$	0	0
$366$	−99790.2	−0.0389390
$367$	−1.90734e6	−0.739203	−0.369601	−	0.929190i	$-0.620506\pi$
$367$	−1.90734e6	−0.739203	−0.369601	+	0.929190i	$0.620506\pi$
$368$	751902.	0.289429
$369$	−859407.	−0.328574
$370$	0	0
$371$	−1.28069e6	−0.483070
$372$	−182021.	−0.0681969
$373$	−3.03699e6	−1.13024	−0.565122	−	0.825008i	$-0.691171\pi$
$373$	−3.03699e6	−1.13024	−0.565122	+	0.825008i	$0.691171\pi$
$374$	−771395.	−0.285166
$375$	0	0
$376$	2.77764e6	1.01323
$377$	−4.58133e6	−1.66011
$378$	382677.	0.137753
$379$	3.09949e6	1.10839	0.554195	−	0.832387i	$-0.313026\pi$
$379$	3.09949e6	1.10839	0.554195	+	0.832387i	$0.313026\pi$
$380$	0	0
$381$	−49702.0	−0.0175413
$382$	−1.30421e6	−0.457287
$383$	5.38627e6	1.87625	0.938127	−	0.346292i	$-0.112559\pi$
$383$	5.38627e6	1.87625	0.938127	+	0.346292i	$0.112559\pi$
$384$	111952.	0.0387438
$385$	0	0
$386$	−2.36572e6	−0.808156
$387$	2.19723e6	0.745760
$388$	−2.47040e6	−0.833082
$389$	3.81576e6	1.27852	0.639260	−	0.768991i	$-0.279241\pi$
$389$	3.81576e6	1.27852	0.639260	+	0.768991i	$0.279241\pi$
$390$	0	0
$391$	4.82257e6	1.59528
$392$	−1.63835e6	−0.538507
$393$	99894.1	0.0326256
$394$	2.78776e6	0.904723
$395$	0	0
$396$	−464136.	−0.148733
$397$	−187812.	−0.0598063	−0.0299032	−	0.999553i	$-0.509520\pi$
$397$	−187812.	−0.0598063	−0.0299032	+	0.999553i	$0.509520\pi$
$398$	−812397.	−0.257075
$399$	504691.	0.158706
$400$	0	0
$401$	2.01719e6	0.626448	0.313224	−	0.949679i	$-0.398591\pi$
$401$	2.01719e6	0.626448	0.313224	+	0.949679i	$0.398591\pi$
$402$	148406.	0.0458021
$403$	2.74883e6	0.843112
$404$	−402683.	−0.122747
$405$	0	0
$406$	3.11169e6	0.936875
$407$	−1.30345e6	−0.390040
$408$	−672896.	−0.200123
$409$	4.21012e6	1.24447	0.622237	−	0.782829i	$-0.286224\pi$
$409$	4.21012e6	1.24447	0.622237	+	0.782829i	$0.286224\pi$
$410$	0	0
$411$	−213284.	−0.0622807
$412$	−2.21428e6	−0.642673
$413$	4.28675e6	1.23667
$414$	−2.86404e6	−0.821256
$415$	0	0
$416$	−2.74461e6	−0.777585
$417$	−10745.3	−0.00302607
$418$	1.22059e6	0.341687
$419$	−4.57372e6	−1.27273	−0.636363	−	0.771390i	$-0.719562\pi$
$419$	−4.57372e6	−1.27273	−0.636363	+	0.771390i	$0.719562\pi$
$420$	0	0
$421$	−4.03432e6	−1.10934	−0.554670	−	0.832070i	$-0.687156\pi$
$421$	−4.03432e6	−1.10934	−0.554670	+	0.832070i	$0.687156\pi$
$422$	1.57780e6	0.431291
$423$	−3.44959e6	−0.937383
$424$	−2.70184e6	−0.729868
$425$	0	0
$426$	−612363.	−0.163488
$427$	1.03702e6	0.275244
$428$	−3.54926e6	−0.936545
$429$	−141677.	−0.0371669
$430$	0	0
$431$	306364.	0.0794411	0.0397205	−	0.999211i	$-0.487353\pi$
$431$	306364.	0.0794411	0.0397205	+	0.999211i	$0.487353\pi$
$432$	263221.	0.0678595
$433$	−2.76255e6	−0.708093	−0.354046	−	0.935228i	$-0.615194\pi$
$433$	−2.76255e6	−0.708093	−0.354046	+	0.935228i	$0.615194\pi$
$434$	−1.86704e6	−0.475805
$435$	0	0
$436$	−3.63483e6	−0.915732
$437$	−7.63080e6	−1.91147
$438$	127738.	0.0318151
$439$	1.84274e6	0.456356	0.228178	−	0.973619i	$-0.426723\pi$
$439$	1.84274e6	0.456356	0.228178	+	0.973619i	$0.426723\pi$
$440$	0	0
$441$	2.03469e6	0.498199
$442$	3.40199e6	0.828281
$443$	574713.	0.139137	0.0695684	−	0.997577i	$-0.477838\pi$
$443$	574713.	0.139137	0.0695684	+	0.997577i	$0.477838\pi$
$444$	−380650.	−0.0916365
$445$	0	0
$446$	1.90566e6	0.453638
$447$	158028.	0.0374080
$448$	2.58944e6	0.609552
$449$	5.24929e6	1.22881	0.614404	−	0.788991i	$-0.289396\pi$
$449$	5.24929e6	1.22881	0.614404	+	0.788991i	$0.289396\pi$
$450$	0	0
$451$	−436585.	−0.101071
$452$	1.58377e6	0.364625
$453$	263628.	0.0603596
$454$	−2.34445e6	−0.533828
$455$	0	0
$456$	1.06473e6	0.239788
$457$	7.46219e6	1.67138	0.835691	−	0.549200i	$-0.185068\pi$
$457$	7.46219e6	1.67138	0.835691	+	0.549200i	$0.185068\pi$
$458$	−1.55085e6	−0.345466
$459$	1.68825e6	0.374029
$460$	0	0
$461$	−2.68536e6	−0.588506	−0.294253	−	0.955728i	$-0.595071\pi$
$461$	−2.68536e6	−0.588506	−0.294253	+	0.955728i	$0.595071\pi$
$462$	96228.7	0.0209749
$463$	−6.34877e6	−1.37638	−0.688188	−	0.725533i	$-0.741593\pi$
$463$	−6.34877e6	−1.37638	−0.688188	+	0.725533i	$0.741593\pi$
$464$	2.14035e6	0.461519
$465$	0	0
$466$	−2.00139e6	−0.426940
$467$	−4.68768e6	−0.994641	−0.497320	−	0.867567i	$-0.665683\pi$
$467$	−4.68768e6	−0.994641	−0.497320	+	0.867567i	$0.665683\pi$
$468$	2.04692e6	0.432003
$469$	−1.54223e6	−0.323755
$470$	0	0
$471$	−324298.	−0.0673584
$472$	9.04362e6	1.86848
$473$	1.11621e6	0.229400
$474$	−322419.	−0.0659137
$475$	0	0
$476$	2.34103e6	0.473576
$477$	3.35546e6	0.675236
$478$	−1.91776e6	−0.383906
$479$	−795999.	−0.158516	−0.0792581	−	0.996854i	$-0.525255\pi$
$479$	−795999.	−0.158516	−0.0792581	+	0.996854i	$0.525255\pi$
$480$	0	0
$481$	5.74846e6	1.13289
$482$	1.84611e6	0.361944
$483$	−601598.	−0.117338
$484$	−235784.	−0.0457511
$485$	0	0
$486$	−1.50895e6	−0.289791
$487$	−1.83464e6	−0.350532	−0.175266	−	0.984521i	$-0.556079\pi$
$487$	−1.83464e6	−0.350532	−0.175266	+	0.984521i	$0.556079\pi$
$488$	2.18777e6	0.415864
$489$	−103446.	−0.0195633
$490$	0	0
$491$	−5.40439e6	−1.01168	−0.505840	−	0.862627i	$-0.668817\pi$
$491$	−5.40439e6	−1.01168	−0.505840	+	0.862627i	$0.668817\pi$
$492$	−127497.	−0.0237458
$493$	1.37278e7	2.54381
$494$	−5.38301e6	−0.992448
$495$	0	0
$496$	−1.28423e6	−0.234389
$497$	6.36367e6	1.15562
$498$	−425120.	−0.0768136
$499$	−3.85216e6	−0.692553	−0.346277	−	0.938132i	$-0.612554\pi$
$499$	−3.85216e6	−0.692553	−0.346277	+	0.938132i	$0.612554\pi$
$500$	0	0
$501$	−398211.	−0.0708792
$502$	−1.60102e6	−0.283555
$503$	−5.01187e6	−0.883242	−0.441621	−	0.897202i	$-0.645596\pi$
$503$	−5.01187e6	−0.883242	−0.441621	+	0.897202i	$0.645596\pi$
$504$	−4.15287e6	−0.728235
$505$	0	0
$506$	−1.45495e6	−0.252623
$507$	−189862.	−0.0328033
$508$	364793.	0.0627173
$509$	−8.08491e6	−1.38319	−0.691594	−	0.722287i	$-0.743091\pi$
$509$	−8.08491e6	−1.38319	−0.691594	+	0.722287i	$0.743091\pi$
$510$	0	0
$511$	−1.32745e6	−0.224887
$512$	2.81232e6	0.474121
$513$	−2.67134e6	−0.448163
$514$	4.99153e6	0.833347
$515$	0	0
$516$	325970.	0.0538955
$517$	−1.75242e6	−0.288344
$518$	−3.90442e6	−0.639341
$519$	−1.49359e6	−0.243395
$520$	0	0
$521$	4.18229e6	0.675025	0.337513	−	0.941321i	$-0.390414\pi$
$521$	4.18229e6	0.675025	0.337513	+	0.941321i	$0.390414\pi$
$522$	−8.15273e6	−1.30956
$523$	6.29712e6	1.00667	0.503336	−	0.864091i	$-0.332106\pi$
$523$	6.29712e6	1.00667	0.503336	+	0.864091i	$0.332106\pi$
$524$	−733183.	−0.116650
$525$	0	0
$526$	−5.07919e6	−0.800442
$527$	−8.23681e6	−1.29191
$528$	66190.1	0.0103326
$529$	2.65966e6	0.413225
$530$	0	0
$531$	−1.12314e7	−1.72861
$532$	−3.70423e6	−0.567439
$533$	1.92542e6	0.293567
$534$	420581.	0.0638258
$535$	0	0
$536$	−3.25359e6	−0.489160
$537$	817047.	0.122268
$538$	−3.44128e6	−0.512583
$539$	1.03364e6	0.153249
$540$	0	0
$541$	−863454.	−0.126837	−0.0634185	−	0.997987i	$-0.520200\pi$
$541$	−863454.	−0.126837	−0.0634185	+	0.997987i	$0.520200\pi$
$542$	1.06005e6	0.154999
$543$	−411571.	−0.0599025
$544$	8.22418e6	1.19150
$545$	0	0
$546$	−424386.	−0.0609228
$547$	−2.32481e6	−0.332215	−0.166108	−	0.986108i	$-0.553120\pi$
$547$	−2.32481e6	−0.332215	−0.166108	+	0.986108i	$0.553120\pi$
$548$	1.56542e6	0.222679
$549$	−2.71702e6	−0.384736
$550$	0	0
$551$	−2.17217e7	−3.04800
$552$	−1.26917e6	−0.177285
$553$	3.35058e6	0.465916
$554$	1.07563e6	0.148898
$555$	0	0
$556$	78866.4	0.0108194
$557$	−2.37211e6	−0.323964	−0.161982	−	0.986794i	$-0.551789\pi$
$557$	−2.37211e6	−0.323964	−0.161982	+	0.986794i	$0.551789\pi$
$558$	4.89170e6	0.665081
$559$	−4.92269e6	−0.666305
$560$	0	0
$561$	424532.	0.0569512
$562$	−1.80878e6	−0.241572
$563$	3.70497e6	0.492621	0.246311	−	0.969191i	$-0.420782\pi$
$563$	3.70497e6	0.492621	0.246311	+	0.969191i	$0.420782\pi$
$564$	−511763.	−0.0677440
$565$	0	0
$566$	−41751.6	−0.00547813
$567$	5.05115e6	0.659831
$568$	1.34252e7	1.74603
$569$	−4.79504e6	−0.620886	−0.310443	−	0.950592i	$-0.600477\pi$
$569$	−4.79504e6	−0.620886	−0.310443	+	0.950592i	$0.600477\pi$
$570$	0	0
$571$	1.37138e7	1.76023	0.880114	−	0.474763i	$-0.157466\pi$
$571$	1.37138e7	1.76023	0.880114	+	0.474763i	$0.157466\pi$
$572$	1.03985e6	0.132887
$573$	717762.	0.0913259
$574$	−1.30777e6	−0.165673
$575$	0	0
$576$	−6.78441e6	−0.852031
$577$	5.34197e6	0.667977	0.333989	−	0.942577i	$-0.391605\pi$
$577$	5.34197e6	0.667977	0.333989	+	0.942577i	$0.391605\pi$
$578$	−4.53313e6	−0.564390
$579$	1.30196e6	0.161399
$580$	0	0
$581$	4.41784e6	0.542963
$582$	−1.34194e6	−0.164220
$583$	1.70460e6	0.207706
$584$	−2.80047e6	−0.339781
$585$	0	0
$586$	1.35211e6	0.162655
$587$	5.07416e6	0.607811	0.303906	−	0.952702i	$-0.401709\pi$
$587$	5.07416e6	0.607811	0.303906	+	0.952702i	$0.401709\pi$
$588$	301856.	0.0360045
$589$	1.30332e7	1.54797
$590$	0	0
$591$	−1.53423e6	−0.180684
$592$	−2.68562e6	−0.314949
$593$	−7.73891e6	−0.903739	−0.451870	−	0.892084i	$-0.649243\pi$
$593$	−7.73891e6	−0.903739	−0.451870	+	0.892084i	$0.649243\pi$
$594$	−509341.	−0.0592301
$595$	0	0
$596$	−1.15986e6	−0.133749
$597$	447097.	0.0513412
$598$	6.41661e6	0.733758
$599$	5.04688e6	0.574720	0.287360	−	0.957823i	$-0.407222\pi$
$599$	5.04688e6	0.574720	0.287360	+	0.957823i	$0.407222\pi$
$600$	0	0
$601$	−4.89498e6	−0.552796	−0.276398	−	0.961043i	$-0.589141\pi$
$601$	−4.89498e6	−0.552796	−0.276398	+	0.961043i	$0.589141\pi$
$602$	3.34355e6	0.376025
$603$	4.04069e6	0.452545
$604$	−1.93493e6	−0.215810
$605$	0	0
$606$	−218740.	−0.0241962
$607$	−9.09514e6	−1.00193	−0.500965	−	0.865467i	$-0.667022\pi$
$607$	−9.09514e6	−1.00193	−0.500965	+	0.865467i	$0.667022\pi$
$608$	−1.30132e7	−1.42766
$609$	−1.71250e6	−0.187106
$610$	0	0
$611$	7.72849e6	0.837513
$612$	−6.13357e6	−0.661964
$613$	−1.68653e7	−1.81277	−0.906385	−	0.422452i	$-0.861170\pi$
$613$	−1.68653e7	−1.81277	−0.906385	+	0.422452i	$0.861170\pi$
$614$	−3.27360e6	−0.350432
$615$	0	0
$616$	−2.10969e6	−0.224009
$617$	1.33815e7	1.41512	0.707560	−	0.706654i	$-0.249796\pi$
$617$	1.33815e7	1.41512	0.707560	+	0.706654i	$0.249796\pi$
$618$	−1.20282e6	−0.126686
$619$	1.36962e7	1.43672	0.718360	−	0.695671i	$-0.244893\pi$
$619$	1.36962e7	1.43672	0.718360	+	0.695671i	$0.244893\pi$
$620$	0	0
$621$	3.18427e6	0.331345
$622$	−9.64426e6	−0.999523
$623$	−4.37067e6	−0.451157
$624$	−291910.	−0.0300115
$625$	0	0
$626$	−1.72940e6	−0.176384
$627$	−671741.	−0.0682391
$628$	2.38022e6	0.240834
$629$	−1.72251e7	−1.73594
$630$	0	0
$631$	−1.04869e6	−0.104851	−0.0524254	−	0.998625i	$-0.516695\pi$
$631$	−1.04869e6	−0.104851	−0.0524254	+	0.998625i	$0.516695\pi$
$632$	7.06861e6	0.703949
$633$	−868330.	−0.0861342
$634$	8.46676e6	0.836554
$635$	0	0
$636$	497797.	0.0487988
$637$	−4.55854e6	−0.445120
$638$	−4.14165e6	−0.402830
$639$	−1.66730e7	−1.61533
$640$	0	0
$641$	−9.66293e6	−0.928889	−0.464444	−	0.885602i	$-0.653746\pi$
$641$	−9.66293e6	−0.928889	−0.464444	+	0.885602i	$0.653746\pi$
$642$	−1.92799e6	−0.184615
$643$	−8.87055e6	−0.846102	−0.423051	−	0.906106i	$-0.639041\pi$
$643$	−8.87055e6	−0.846102	−0.423051	+	0.906106i	$0.639041\pi$
$644$	4.41549e6	0.419531
$645$	0	0
$646$	1.61301e7	1.52074
$647$	−1.95886e6	−0.183969	−0.0919843	−	0.995760i	$-0.529321\pi$
$647$	−1.95886e6	−0.183969	−0.0919843	+	0.995760i	$0.529321\pi$
$648$	1.06563e7	0.996936
$649$	−5.70564e6	−0.531732
$650$	0	0
$651$	1.02751e6	0.0950242
$652$	759252.	0.0699467
$653$	−1.06165e7	−0.974311	−0.487155	−	0.873315i	$-0.661965\pi$
$653$	−1.06165e7	−0.974311	−0.487155	+	0.873315i	$0.661965\pi$
$654$	−1.97447e6	−0.180512
$655$	0	0
$656$	−899537.	−0.0816130
$657$	3.47795e6	0.314348
$658$	−5.24928e6	−0.472645
$659$	549361.	0.0492770	0.0246385	−	0.999696i	$-0.492157\pi$
$659$	549361.	0.0492770	0.0246385	+	0.999696i	$0.492157\pi$
$660$	0	0
$661$	−1.35330e7	−1.20473	−0.602365	−	0.798221i	$-0.705775\pi$
$661$	−1.35330e7	−1.20473	−0.602365	+	0.798221i	$0.705775\pi$
$662$	−3.04575e6	−0.270115
$663$	−1.87226e6	−0.165418
$664$	9.32018e6	0.820360
$665$	0	0
$666$	1.02297e7	0.893671
$667$	2.58925e7	2.25351
$668$	2.92271e6	0.253422
$669$	−1.04877e6	−0.0905972
$670$	0	0
$671$	−1.38027e6	−0.118347
$672$	−1.02594e6	−0.0876389
$673$	−1.81588e7	−1.54543	−0.772714	−	0.634755i	$-0.781101\pi$
$673$	−1.81588e7	−1.54543	−0.772714	+	0.634755i	$0.781101\pi$
$674$	−7.95742e6	−0.674718
$675$	0	0
$676$	1.39351e6	0.117285
$677$	−8.73920e6	−0.732824	−0.366412	−	0.930453i	$-0.619414\pi$
$677$	−8.73920e6	−0.732824	−0.366412	+	0.930453i	$0.619414\pi$
$678$	860316.	0.0718760
$679$	1.39454e7	1.16080
$680$	0	0
$681$	1.29025e6	0.106612
$682$	2.48502e6	0.204583
$683$	7.60223e6	0.623575	0.311788	−	0.950152i	$-0.399072\pi$
$683$	7.60223e6	0.623575	0.311788	+	0.950152i	$0.399072\pi$
$684$	9.70521e6	0.793167
$685$	0	0
$686$	9.18790e6	0.745429
$687$	853498.	0.0689939
$688$	2.29983e6	0.185236
$689$	−7.51758e6	−0.603295
$690$	0	0
$691$	−5.58643e6	−0.445081	−0.222541	−	0.974923i	$-0.571435\pi$
$691$	−5.58643e6	−0.445081	−0.222541	+	0.974923i	$0.571435\pi$
$692$	1.09623e7	0.870237
$693$	2.62005e6	0.207242
$694$	2.59946e6	0.204873
$695$	0	0
$696$	−3.61280e6	−0.282697
$697$	−5.76948e6	−0.449836
$698$	−1.09296e6	−0.0849114
$699$	1.10145e6	0.0852652
$700$	0	0
$701$	1.78672e7	1.37329	0.686645	−	0.726993i	$-0.259083\pi$
$701$	1.78672e7	1.37329	0.686645	+	0.726993i	$0.259083\pi$
$702$	2.24629e6	0.172037
$703$	2.72555e7	2.08001
$704$	−3.44653e6	−0.262090
$705$	0	0
$706$	−1.03329e7	−0.780206
$707$	2.27315e6	0.171033
$708$	−1.66623e6	−0.124926
$709$	−1.16435e7	−0.869894	−0.434947	−	0.900456i	$-0.643233\pi$
$709$	−1.16435e7	−0.869894	−0.434947	+	0.900456i	$0.643233\pi$
$710$	0	0
$711$	−8.77862e6	−0.651257
$712$	−9.22067e6	−0.681652
$713$	−1.55357e7	−1.14448
$714$	1.27166e6	0.0933527
$715$	0	0
$716$	−5.99680e6	−0.437157
$717$	1.05543e6	0.0766709
$718$	−7.29345e6	−0.527985
$719$	−1.67754e7	−1.21018	−0.605090	−	0.796157i	$-0.706863\pi$
$719$	−1.67754e7	−1.21018	−0.605090	+	0.796157i	$0.706863\pi$
$720$	0	0
$721$	1.24997e7	0.895488
$722$	−1.56508e7	−1.11736
$723$	−1.01600e6	−0.0722846
$724$	3.02077e6	0.214176
$725$	0	0
$726$	−128080.	−0.00901861
$727$	−2.15242e7	−1.51039	−0.755197	−	0.655498i	$-0.772459\pi$
$727$	−2.15242e7	−1.51039	−0.755197	+	0.655498i	$0.772459\pi$
$728$	9.30410e6	0.650648
$729$	−1.26712e7	−0.883080
$730$	0	0
$731$	1.47507e7	1.02099
$732$	−403083.	−0.0278046
$733$	−5.66516e6	−0.389450	−0.194725	−	0.980858i	$-0.562381\pi$
$733$	−5.66516e6	−0.389450	−0.194725	+	0.980858i	$0.562381\pi$
$734$	7.60444e6	0.520987
$735$	0	0
$736$	1.55119e7	1.05553
$737$	2.05270e6	0.139206
$738$	3.42640e6	0.231578
$739$	2.04439e7	1.37706	0.688530	−	0.725208i	$-0.258256\pi$
$739$	2.04439e7	1.37706	0.688530	+	0.725208i	$0.258256\pi$
$740$	0	0
$741$	2.96250e6	0.198204
$742$	5.10603e6	0.340466
$743$	2.00010e7	1.32917	0.664583	−	0.747214i	$-0.268609\pi$
$743$	2.00010e7	1.32917	0.664583	+	0.747214i	$0.268609\pi$
$744$	2.16771e6	0.143572
$745$	0	0
$746$	1.21083e7	0.796591
$747$	−1.15749e7	−0.758954
$748$	−3.11590e6	−0.203624
$749$	2.00356e7	1.30496
$750$	0	0
$751$	1.47046e7	0.951381	0.475690	−	0.879613i	$-0.342198\pi$
$751$	1.47046e7	0.951381	0.475690	+	0.879613i	$0.342198\pi$
$752$	−3.61067e6	−0.232832
$753$	881108.	0.0566294
$754$	1.82654e7	1.17004
$755$	0	0
$756$	1.54575e6	0.0983634
$757$	6.06609e6	0.384742	0.192371	−	0.981322i	$-0.438382\pi$
$757$	6.06609e6	0.384742	0.192371	+	0.981322i	$0.438382\pi$
$758$	−1.23574e7	−0.781188
$759$	800723.	0.0504520
$760$	0	0
$761$	1.60732e7	1.00610	0.503048	−	0.864258i	$-0.332212\pi$
$761$	1.60732e7	1.00610	0.503048	+	0.864258i	$0.332212\pi$
$762$	198158.	0.0123630
$763$	2.05187e7	1.27596
$764$	−5.26809e6	−0.326527
$765$	0	0
$766$	−2.14747e7	−1.32238
$767$	2.51629e7	1.54445
$768$	−2.44629e6	−0.149660
$769$	1.37349e7	0.837547	0.418773	−	0.908091i	$-0.362460\pi$
$769$	1.37349e7	0.837547	0.418773	+	0.908091i	$0.362460\pi$
$770$	0	0
$771$	−2.74705e6	−0.166430
$772$	−9.55585e6	−0.577067
$773$	2.30727e7	1.38883	0.694416	−	0.719573i	$-0.255663\pi$
$773$	2.30727e7	1.38883	0.694416	+	0.719573i	$0.255663\pi$
$774$	−8.76021e6	−0.525608
$775$	0	0
$776$	2.94202e7	1.75385
$777$	2.14877e6	0.127684
$778$	−1.52132e7	−0.901096
$779$	9.12911e6	0.538995
$780$	0	0
$781$	−8.47001e6	−0.496886
$782$	−1.92272e7	−1.12435
$783$	9.06429e6	0.528359
$784$	2.12970e6	0.123745
$785$	0	0
$786$	−398271.	−0.0229944
$787$	6.11082e6	0.351692	0.175846	−	0.984418i	$-0.443734\pi$
$787$	6.11082e6	0.351692	0.175846	+	0.984418i	$0.443734\pi$
$788$	1.12606e7	0.646021
$789$	2.79529e6	0.159858
$790$	0	0
$791$	−8.94040e6	−0.508061
$792$	5.52744e6	0.313120
$793$	6.08723e6	0.343745
$794$	748793.	0.0421513
$795$	0	0
$796$	−3.28151e6	−0.183566
$797$	2.96316e7	1.65238	0.826188	−	0.563395i	$-0.190505\pi$
$797$	2.96316e7	1.65238	0.826188	+	0.563395i	$0.190505\pi$
$798$	−2.01217e6	−0.111855
$799$	−2.31582e7	−1.28333
$800$	0	0
$801$	1.14513e7	0.630628
$802$	−8.04238e6	−0.441518
$803$	1.76683e6	0.0966952
$804$	599455.	0.0327052
$805$	0	0
$806$	−1.09594e7	−0.594222
$807$	1.89388e6	0.102369
$808$	4.79559e6	0.258412
$809$	−2.07185e7	−1.11298	−0.556489	−	0.830855i	$-0.687852\pi$
$809$	−2.07185e7	−1.11298	−0.556489	+	0.830855i	$0.687852\pi$
$810$	0	0
$811$	4.27788e6	0.228390	0.114195	−	0.993458i	$-0.463571\pi$
$811$	4.27788e6	0.228390	0.114195	+	0.993458i	$0.463571\pi$
$812$	1.25691e7	0.668979
$813$	−583392.	−0.0309552
$814$	5.19677e6	0.274898
$815$	0	0
$816$	874703.	0.0459870
$817$	−2.33403e7	−1.22335
$818$	−1.67854e7	−0.877100
$819$	−1.15549e7	−0.601945
$820$	0	0
$821$	−1.14051e7	−0.590531	−0.295265	−	0.955415i	$-0.595408\pi$
$821$	−1.14051e7	−0.590531	−0.295265	+	0.955415i	$0.595408\pi$
$822$	850348.	0.0438952
$823$	−3.13322e7	−1.61247	−0.806233	−	0.591598i	$-0.798497\pi$
$823$	−3.13322e7	−1.61247	−0.806233	+	0.591598i	$0.798497\pi$
$824$	2.63701e7	1.35299
$825$	0	0
$826$	−1.70910e7	−0.871599
$827$	−1.84821e7	−0.939696	−0.469848	−	0.882747i	$-0.655691\pi$
$827$	−1.84821e7	−0.939696	−0.469848	+	0.882747i	$0.655691\pi$
$828$	−1.15687e7	−0.586421
$829$	2.73374e7	1.38156	0.690782	−	0.723063i	$-0.257266\pi$
$829$	2.73374e7	1.38156	0.690782	+	0.723063i	$0.257266\pi$
$830$	0	0
$831$	−591965.	−0.0297368
$832$	1.51998e7	0.761255
$833$	1.36596e7	0.682062
$834$	42840.8	0.00213276
$835$	0	0
$836$	4.93032e6	0.243983
$837$	−5.43864e6	−0.268335
$838$	1.82351e7	0.897012
$839$	3.10089e6	0.152083	0.0760417	−	0.997105i	$-0.475772\pi$
$839$	3.10089e6	0.152083	0.0760417	+	0.997105i	$0.475772\pi$
$840$	0	0
$841$	5.31941e7	2.59342
$842$	1.60845e7	0.781859
$843$	995452.	0.0482449
$844$	6.37320e6	0.307965
$845$	0	0
$846$	1.37533e7	0.660664
$847$	1.33101e6	0.0637487
$848$	3.51214e6	0.167719
$849$	22977.7	0.00109405
$850$	0	0
$851$	−3.24889e7	−1.53784
$852$	−2.47352e6	−0.116739
$853$	1.43661e7	0.676032	0.338016	−	0.941140i	$-0.390244\pi$
$853$	1.43661e7	0.676032	0.338016	+	0.941140i	$0.390244\pi$
$854$	−4.13452e6	−0.193991
$855$	0	0
$856$	4.22685e7	1.97166
$857$	−1.49076e7	−0.693353	−0.346677	−	0.937985i	$-0.612690\pi$
$857$	−1.49076e7	−0.693353	−0.346677	+	0.937985i	$0.612690\pi$
$858$	564856.	0.0261951
$859$	−4.03264e7	−1.86469	−0.932344	−	0.361571i	$-0.882240\pi$
$859$	−4.03264e7	−1.86469	−0.932344	+	0.361571i	$0.882240\pi$
$860$	0	0
$861$	719721.	0.0330869
$862$	−1.22145e6	−0.0559898
$863$	327785.	0.0149817	0.00749086	−	0.999972i	$-0.497616\pi$
$863$	327785.	0.0149817	0.00749086	+	0.999972i	$0.497616\pi$
$864$	5.43030e6	0.247480
$865$	0	0
$866$	1.10141e7	0.499061
$867$	2.49478e6	0.112716
$868$	−7.54153e6	−0.339751
$869$	−4.45960e6	−0.200330
$870$	0	0
$871$	−9.05278e6	−0.404331
$872$	4.32876e7	1.92785
$873$	−3.65374e7	−1.62257
$874$	3.04235e7	1.34719
$875$	0	0
$876$	515970.	0.0227177
$877$	1.74545e7	0.766317	0.383159	−	0.923683i	$-0.374836\pi$
$877$	1.74545e7	0.766317	0.383159	+	0.923683i	$0.374836\pi$
$878$	−7.34688e6	−0.321638
$879$	−744124.	−0.0324843
$880$	0	0
$881$	−3.31899e7	−1.44068	−0.720339	−	0.693622i	$-0.756014\pi$
$881$	−3.31899e7	−1.44068	−0.720339	+	0.693622i	$0.756014\pi$
$882$	−8.11218e6	−0.351128
$883$	97139.3	0.00419269	0.00209635	−	0.999998i	$-0.499333\pi$
$883$	97139.3	0.00419269	0.00209635	+	0.999998i	$0.499333\pi$
$884$	1.37417e7	0.591438
$885$	0	0
$886$	−2.29134e6	−0.0980631
$887$	1.86238e6	0.0794803	0.0397401	−	0.999210i	$-0.487347\pi$
$887$	1.86238e6	0.0794803	0.0397401	+	0.999210i	$0.487347\pi$
$888$	4.53320e6	0.192918
$889$	−2.05926e6	−0.0873890
$890$	0	0
$891$	−6.72306e6	−0.283709
$892$	7.69755e6	0.323922
$893$	3.66435e7	1.53769
$894$	−630046.	−0.0263650
$895$	0	0
$896$	4.63840e6	0.193018
$897$	−3.53134e6	−0.146541
$898$	−2.09285e7	−0.866060
$899$	−4.42237e7	−1.82497
$900$	0	0
$901$	2.25263e7	0.924437
$902$	1.74063e6	0.0712346
$903$	−1.84010e6	−0.0750970
$904$	−1.88613e7	−0.767627
$905$	0	0
$906$	−1.05107e6	−0.0425412
$907$	−3.38081e7	−1.36459	−0.682295	−	0.731077i	$-0.739018\pi$
$907$	−3.38081e7	−1.36459	−0.682295	+	0.731077i	$0.739018\pi$
$908$	−9.46994e6	−0.381182
$909$	−5.95571e6	−0.239069
$910$	0	0
$911$	−4.30899e7	−1.72020	−0.860102	−	0.510123i	$-0.829600\pi$
$911$	−4.30899e7	−1.72020	−0.860102	+	0.510123i	$0.829600\pi$
$912$	−1.38405e6	−0.0551017
$913$	−5.88013e6	−0.233458
$914$	−2.97512e7	−1.17798
$915$	0	0
$916$	−6.26434e6	−0.246681
$917$	4.13883e6	0.162538
$918$	−6.73095e6	−0.263615
$919$	1.53572e7	0.599824	0.299912	−	0.953967i	$-0.403043\pi$
$919$	1.53572e7	0.599824	0.299912	+	0.953967i	$0.403043\pi$
$920$	0	0
$921$	1.80160e6	0.0699857
$922$	1.07064e7	0.414777
$923$	3.73543e7	1.44323
$924$	388697.	0.0149772
$925$	0	0
$926$	2.53121e7	0.970064
$927$	−3.27495e7	−1.25171
$928$	4.41559e7	1.68313
$929$	3.93862e7	1.49729	0.748644	−	0.662972i	$-0.230705\pi$
$929$	3.93862e7	1.49729	0.748644	+	0.662972i	$0.230705\pi$
$930$	0	0
$931$	−2.16137e7	−0.817248
$932$	−8.08420e6	−0.304858
$933$	5.30765e6	0.199617
$934$	1.86895e7	0.701019
$935$	0	0
$936$	−2.43770e7	−0.909476
$937$	3.81263e7	1.41865	0.709325	−	0.704882i	$-0.249000\pi$
$937$	3.81263e7	1.41865	0.709325	+	0.704882i	$0.249000\pi$
$938$	6.14876e6	0.228182
$939$	951762.	0.0352261
$940$	0	0
$941$	4.40994e7	1.62352	0.811761	−	0.583989i	$-0.198509\pi$
$941$	4.40994e7	1.62352	0.811761	+	0.583989i	$0.198509\pi$
$942$	1.29295e6	0.0474739
$943$	−1.08820e7	−0.398501
$944$	−1.17559e7	−0.429363
$945$	0	0
$946$	−4.45025e6	−0.161680
$947$	3.53205e7	1.27983	0.639915	−	0.768446i	$-0.278969\pi$
$947$	3.53205e7	1.27983	0.639915	+	0.768446i	$0.278969\pi$
$948$	−1.30235e6	−0.0470659
$949$	−7.79202e6	−0.280857
$950$	0	0
$951$	−4.65962e6	−0.167070
$952$	−2.78795e7	−0.996995
$953$	−1.95033e7	−0.695626	−0.347813	−	0.937564i	$-0.613076\pi$
$953$	−1.95033e7	−0.695626	−0.347813	+	0.937564i	$0.613076\pi$
$954$	−1.33780e7	−0.475904
$955$	0	0
$956$	−7.74642e6	−0.274130
$957$	2.27933e6	0.0804501
$958$	3.17359e6	0.111722
$959$	−8.83680e6	−0.310276
$960$	0	0
$961$	−2.09459e6	−0.0731630
$962$	−2.29187e7	−0.798458
$963$	−5.24939e7	−1.82408
$964$	7.45700e6	0.258447
$965$	0	0
$966$	2.39853e6	0.0826993
$967$	9.54346e6	0.328201	0.164100	−	0.986444i	$-0.447528\pi$
$967$	9.54346e6	0.328201	0.164100	+	0.986444i	$0.447528\pi$
$968$	2.80798e6	0.0963176
$969$	−8.87707e6	−0.303711
$970$	0	0
$971$	−4.16714e7	−1.41837	−0.709186	−	0.705021i	$-0.750938\pi$
$971$	−4.16714e7	−1.41837	−0.709186	+	0.705021i	$0.750938\pi$
$972$	−6.09511e6	−0.206926
$973$	−445201.	−0.0150756
$974$	7.31457e6	0.247054
$975$	0	0
$976$	−2.84389e6	−0.0955628
$977$	−2.59092e7	−0.868395	−0.434197	−	0.900818i	$-0.642968\pi$
$977$	−2.59092e7	−0.868395	−0.434197	+	0.900818i	$0.642968\pi$
$978$	412432.	0.0137881
$979$	5.81734e6	0.193985
$980$	0	0
$981$	−5.37595e7	−1.78354
$982$	2.15469e7	0.713028
$983$	−1.17591e6	−0.0388143	−0.0194071	−	0.999812i	$-0.506178\pi$
$983$	−1.17591e6	−0.0388143	−0.0194071	+	0.999812i	$0.506178\pi$
$984$	1.51837e6	0.0499909
$985$	0	0
$986$	−5.47319e7	−1.79287
$987$	2.88891e6	0.0943931
$988$	−2.17436e7	−0.708662
$989$	2.78219e7	0.904472
$990$	0	0
$991$	−3.35602e7	−1.08553	−0.542763	−	0.839886i	$-0.682622\pi$
$991$	−3.35602e7	−1.08553	−0.542763	+	0.839886i	$0.682622\pi$
$992$	−2.64939e7	−0.854803
$993$	1.67621e6	0.0539454
$994$	−2.53715e7	−0.814480
$995$	0	0
$996$	−1.71719e6	−0.0548491
$997$	−4.73106e7	−1.50737	−0.753686	−	0.657235i	$-0.771726\pi$
$997$	−4.73106e7	−1.50737	−0.753686	+	0.657235i	$0.771726\pi$
$998$	1.53583e7	0.488109
$999$	−1.13735e7	−0.360562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.6.a.l.1.5		14
5.2	odd	4		55.6.b.b.34.5	✓	14
5.3	odd	4		55.6.b.b.34.10	yes	14
5.4	even	2	inner	275.6.a.l.1.10		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.b.b.34.5	✓	14	5.2	odd	4
55.6.b.b.34.10	yes	14	5.3	odd	4
275.6.a.l.1.5		14	1.1	even	1	trivial
275.6.a.l.1.10		14	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q-3.98693 q^{2} +2.19418 q^{3} -16.1044 q^{4} -8.74803 q^{6} +90.9095 q^{7} +191.789 q^{8} -238.186 q^{9} +O(q^{10})\)	\(q-3.98693 q^{2} +2.19418 q^{3} -16.1044 q^{4} -8.74803 q^{6} +90.9095 q^{7} +191.789 q^{8} -238.186 q^{9} -121.000 q^{11} -35.3359 q^{12} +533.632 q^{13} -362.450 q^{14} -249.308 q^{16} -1599.02 q^{17} +949.629 q^{18} +2530.14 q^{19} +199.472 q^{21} +482.418 q^{22} -3015.96 q^{23} +420.819 q^{24} -2127.55 q^{26} -1055.81 q^{27} -1464.04 q^{28} -8585.18 q^{29} +5151.17 q^{31} -5143.27 q^{32} -265.496 q^{33} +6375.17 q^{34} +3835.84 q^{36} +10772.3 q^{37} -10087.5 q^{38} +1170.88 q^{39} +3608.14 q^{41} -795.279 q^{42} -9224.88 q^{43} +1948.63 q^{44} +12024.4 q^{46} +14482.8 q^{47} -547.025 q^{48} -8542.47 q^{49} -3508.53 q^{51} -8593.82 q^{52} -14087.6 q^{53} +4209.43 q^{54} +17435.4 q^{56} +5551.58 q^{57} +34228.5 q^{58} +47154.1 q^{59} +11407.2 q^{61} -20537.4 q^{62} -21653.3 q^{63} +28483.7 q^{64} +1058.51 q^{66} -16964.5 q^{67} +25751.2 q^{68} -6617.55 q^{69} +70000.1 q^{71} -45681.3 q^{72} -14601.9 q^{73} -42948.5 q^{74} -40746.4 q^{76} -11000.0 q^{77} -4668.23 q^{78} +36856.2 q^{79} +55562.5 q^{81} -14385.4 q^{82} +48596.1 q^{83} -3212.37 q^{84} +36778.9 q^{86} -18837.4 q^{87} -23206.4 q^{88} -48077.2 q^{89} +48512.2 q^{91} +48570.2 q^{92} +11302.6 q^{93} -57741.9 q^{94} -11285.2 q^{96} +153399. q^{97} +34058.2 q^{98} +28820.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 256 q^{4} + 14 q^{6} + 1550 q^{9}+O(q^{10}) \) 14 * q + 256 * q^4 + 14 * q^6 + 1550 * q^9	\( 14 q + 256 q^{4} + 14 q^{6} + 1550 q^{9} - 1694 q^{11} + 3094 q^{14} + 5540 q^{16} + 3940 q^{19} + 12776 q^{21} + 14438 q^{24} - 4696 q^{26} - 1132 q^{29} + 3740 q^{31} + 23834 q^{34} + 59986 q^{36} + 10800 q^{39} + 30740 q^{41} - 30976 q^{44} + 82544 q^{46} + 21746 q^{49} + 104916 q^{51} - 17210 q^{54} + 194170 q^{56} - 39076 q^{59} + 246740 q^{61} - 5480 q^{64} - 1694 q^{66} + 177988 q^{69} + 57468 q^{71} + 247910 q^{74} + 581730 q^{76} + 252200 q^{79} + 605134 q^{81} + 752886 q^{84} + 565312 q^{86} + 359692 q^{89} + 305800 q^{91} + 710788 q^{94} + 1094678 q^{96} - 187550 q^{99}+O(q^{100}) \) 14 * q + 256 * q^4 + 14 * q^6 + 1550 * q^9 - 1694 * q^11 + 3094 * q^14 + 5540 * q^16 + 3940 * q^19 + 12776 * q^21 + 14438 * q^24 - 4696 * q^26 - 1132 * q^29 + 3740 * q^31 + 23834 * q^34 + 59986 * q^36 + 10800 * q^39 + 30740 * q^41 - 30976 * q^44 + 82544 * q^46 + 21746 * q^49 + 104916 * q^51 - 17210 * q^54 + 194170 * q^56 - 39076 * q^59 + 246740 * q^61 - 5480 * q^64 - 1694 * q^66 + 177988 * q^69 + 57468 * q^71 + 247910 * q^74 + 581730 * q^76 + 252200 * q^79 + 605134 * q^81 + 752886 * q^84 + 565312 * q^86 + 359692 * q^89 + 305800 * q^91 + 710788 * q^94 + 1094678 * q^96 - 187550 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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