LMFDB - Embedded newform 275.6.a.l.1.1

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			1.1
Root			$-10.3818$ 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)

$f(q)$	$=$	$q-10.3818 q^{2} -3.25402 q^{3} +75.7815 q^{4} +33.7825 q^{6} -50.5985 q^{7} -454.530 q^{8} -232.411 q^{9} +O(q^{10})$	\(q-10.3818 q^{2} -3.25402 q^{3} +75.7815 q^{4} +33.7825 q^{6} -50.5985 q^{7} -454.530 q^{8} -232.411 q^{9} -121.000 q^{11} -246.594 q^{12} -990.313 q^{13} +525.303 q^{14} +2293.83 q^{16} -427.441 q^{17} +2412.85 q^{18} +2338.79 q^{19} +164.648 q^{21} +1256.20 q^{22} -4201.10 q^{23} +1479.05 q^{24} +10281.2 q^{26} +1547.00 q^{27} -3834.43 q^{28} +1541.00 q^{29} -6335.60 q^{31} -9269.08 q^{32} +393.736 q^{33} +4437.60 q^{34} -17612.5 q^{36} +1212.35 q^{37} -24280.8 q^{38} +3222.50 q^{39} -14978.1 q^{41} -1709.34 q^{42} +1792.48 q^{43} -9169.56 q^{44} +43614.9 q^{46} +1669.75 q^{47} -7464.16 q^{48} -14246.8 q^{49} +1390.90 q^{51} -75047.4 q^{52} -11936.6 q^{53} -16060.6 q^{54} +22998.5 q^{56} -7610.45 q^{57} -15998.3 q^{58} -16390.0 q^{59} -6937.73 q^{61} +65774.9 q^{62} +11759.7 q^{63} +22827.1 q^{64} -4087.68 q^{66} -4862.12 q^{67} -32392.1 q^{68} +13670.4 q^{69} -18827.8 q^{71} +105638. q^{72} +66194.3 q^{73} -12586.3 q^{74} +177237. q^{76} +6122.41 q^{77} -33455.3 q^{78} +83073.7 q^{79} +51442.0 q^{81} +155499. q^{82} -5982.11 q^{83} +12477.3 q^{84} -18609.1 q^{86} -5014.43 q^{87} +54998.2 q^{88} +48499.0 q^{89} +50108.3 q^{91} -318365. q^{92} +20616.2 q^{93} -17335.0 q^{94} +30161.7 q^{96} -109206. q^{97} +147907. q^{98} +28121.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−10.3818	−1.83526	−0.917629	−	0.397438i	$-0.869899\pi$
$2$	−10.3818	−1.83526	−0.917629	+	0.397438i	$0.869899\pi$
$3$	−3.25402	−0.208745	−0.104373	−	0.994538i	$-0.533283\pi$
$3$	−3.25402	−0.208745	−0.104373	+	0.994538i	$0.533283\pi$
$4$	75.7815	2.36817
$5$	0	0
$5$	0	0
$6$	33.7825	0.383102
$7$	−50.5985	−0.390294	−0.195147	−	0.980774i	$-0.562518\pi$
$7$	−50.5985	−0.390294	−0.195147	+	0.980774i	$0.562518\pi$
$8$	−454.530	−2.51095
$9$	−232.411	−0.956425
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−246.594	−0.494345
$13$	−990.313	−1.62523	−0.812614	−	0.582803i	$-0.801956\pi$
$13$	−990.313	−1.62523	−0.812614	+	0.582803i	$0.801956\pi$
$14$	525.303	0.716291
$15$	0	0
$16$	2293.83	2.24007
$17$	−427.441	−0.358719	−0.179359	−	0.983784i	$-0.557402\pi$
$17$	−427.441	−0.358719	−0.179359	+	0.983784i	$0.557402\pi$
$18$	2412.85	1.75529
$19$	2338.79	1.48630	0.743150	−	0.669124i	$-0.233331\pi$
$19$	2338.79	1.48630	0.743150	+	0.669124i	$0.233331\pi$
$20$	0	0
$21$	164.648	0.0814721
$22$	1256.20	0.553351
$23$	−4201.10	−1.65593	−0.827967	−	0.560777i	$-0.810502\pi$
$23$	−4201.10	−1.65593	−0.827967	+	0.560777i	$0.810502\pi$
$24$	1479.05	0.524149
$25$	0	0
$26$	10281.2	2.98271
$27$	1547.00	0.408395
$28$	−3834.43	−0.924284
$29$	1541.00	0.340257	0.170128	−	0.985422i	$-0.445582\pi$
$29$	1541.00	0.340257	0.170128	+	0.985422i	$0.445582\pi$
$30$	0	0
$31$	−6335.60	−1.18409	−0.592044	−	0.805906i	$-0.701679\pi$
$31$	−6335.60	−1.18409	−0.592044	+	0.805906i	$0.701679\pi$
$32$	−9269.08	−1.60015
$33$	393.736	0.0629391
$34$	4437.60	0.658341
$35$	0	0
$36$	−17612.5	−2.26498
$37$	1212.35	0.145587	0.0727935	−	0.997347i	$-0.476809\pi$
$37$	1212.35	0.145587	0.0727935	+	0.997347i	$0.476809\pi$
$38$	−24280.8	−2.72775
$39$	3222.50	0.339259
$40$	0	0
$41$	−14978.1	−1.39154	−0.695771	−	0.718264i	$-0.744937\pi$
$41$	−14978.1	−1.39154	−0.695771	+	0.718264i	$0.744937\pi$
$42$	−1709.34	−0.149522
$43$	1792.48	0.147837	0.0739185	−	0.997264i	$-0.476450\pi$
$43$	1792.48	0.147837	0.0739185	+	0.997264i	$0.476450\pi$
$44$	−9169.56	−0.714031
$45$	0	0
$46$	43614.9	3.03907
$47$	1669.75	0.110257	0.0551286	−	0.998479i	$-0.482443\pi$
$47$	1669.75	0.110257	0.0551286	+	0.998479i	$0.482443\pi$
$48$	−7464.16	−0.467604
$49$	−14246.8	−0.847670
$50$	0	0
$51$	1390.90	0.0748808
$52$	−75047.4	−3.84882
$53$	−11936.6	−0.583702	−0.291851	−	0.956464i	$-0.594271\pi$
$53$	−11936.6	−0.583702	−0.291851	+	0.956464i	$0.594271\pi$
$54$	−16060.6	−0.749510
$55$	0	0
$56$	22998.5	0.980009
$57$	−7610.45	−0.310258
$58$	−15998.3	−0.624459
$59$	−16390.0	−0.612982	−0.306491	−	0.951874i	$-0.599155\pi$
$59$	−16390.0	−0.612982	−0.306491	+	0.951874i	$0.599155\pi$
$60$	0	0
$61$	−6937.73	−0.238722	−0.119361	−	0.992851i	$-0.538085\pi$
$61$	−6937.73	−0.238722	−0.119361	+	0.992851i	$0.538085\pi$
$62$	65774.9	2.17311
$63$	11759.7	0.373287
$64$	22827.1	0.696627
$65$	0	0
$66$	−4087.68	−0.115509
$67$	−4862.12	−0.132324	−0.0661620	−	0.997809i	$-0.521075\pi$
$67$	−4862.12	−0.132324	−0.0661620	+	0.997809i	$0.521075\pi$
$68$	−32392.1	−0.849507
$69$	13670.4	0.345668
$70$	0	0
$71$	−18827.8	−0.443255	−0.221628	−	0.975131i	$-0.571137\pi$
$71$	−18827.8	−0.443255	−0.221628	+	0.975131i	$0.571137\pi$
$72$	105638.	2.40154
$73$	66194.3	1.45383	0.726914	−	0.686728i	$-0.240954\pi$
$73$	66194.3	1.45383	0.726914	+	0.686728i	$0.240954\pi$
$74$	−12586.3	−0.267190
$75$	0	0
$76$	177237.	3.51982
$77$	6122.41	0.117678
$78$	−33455.3	−0.622627
$79$	83073.7	1.49760	0.748800	−	0.662796i	$-0.230630\pi$
$79$	83073.7	1.49760	0.748800	+	0.662796i	$0.230630\pi$
$80$	0	0
$81$	51442.0	0.871175
$82$	155499.	2.55384
$83$	−5982.11	−0.0953145	−0.0476573	−	0.998864i	$-0.515176\pi$
$83$	−5982.11	−0.0953145	−0.0476573	+	0.998864i	$0.515176\pi$
$84$	12477.3	0.192940
$85$	0	0
$86$	−18609.1	−0.271319
$87$	−5014.43	−0.0710271
$88$	54998.2	0.757080
$89$	48499.0	0.649019	0.324510	−	0.945882i	$-0.394801\pi$
$89$	48499.0	0.649019	0.324510	+	0.945882i	$0.394801\pi$
$90$	0	0
$91$	50108.3	0.634317
$92$	−318365.	−3.92154
$93$	20616.2	0.247173
$94$	−17335.0	−0.202351
$95$	0	0
$96$	30161.7	0.334025
$97$	−109206.	−1.17847	−0.589236	−	0.807961i	$-0.700571\pi$
$97$	−109206.	−1.17847	−0.589236	+	0.807961i	$0.700571\pi$
$98$	147907.	1.55569
$99$	28121.8	0.288373
$100$	0	0
$101$	−82441.8	−0.804163	−0.402081	−	0.915604i	$-0.631713\pi$
$101$	−82441.8	−0.804163	−0.402081	+	0.915604i	$0.631713\pi$
$102$	−14440.0	−0.137426
$103$	166529.	1.54666	0.773332	−	0.634001i	$-0.218589\pi$
$103$	166529.	1.54666	0.773332	+	0.634001i	$0.218589\pi$
$104$	450127.	4.08086
$105$	0	0
$106$	123923.	1.07124
$107$	145595.	1.22938	0.614690	−	0.788769i	$-0.289281\pi$
$107$	145595.	1.22938	0.614690	+	0.788769i	$0.289281\pi$
$108$	117234.	0.967149
$109$	−227065.	−1.83056	−0.915282	−	0.402814i	$-0.868032\pi$
$109$	−227065.	−1.83056	−0.915282	+	0.402814i	$0.868032\pi$
$110$	0	0
$111$	−3945.00	−0.0303906
$112$	−116064.	−0.874286
$113$	−179567.	−1.32291	−0.661454	−	0.749986i	$-0.730060\pi$
$113$	−179567.	−1.32291	−0.661454	+	0.749986i	$0.730060\pi$
$114$	79010.1	0.569404
$115$	0	0
$116$	116779.	0.805787
$117$	230160.	1.55441
$118$	170157.	1.12498
$119$	21627.9	0.140006
$120$	0	0
$121$	14641.0	0.0909091
$122$	72026.0	0.438117
$123$	48738.9	0.290478
$124$	−480122.	−2.80412
$125$	0	0
$126$	−122086.	−0.685079
$127$	217036.	1.19405	0.597026	−	0.802222i	$-0.296349\pi$
$127$	217036.	1.19405	0.597026	+	0.802222i	$0.296349\pi$
$128$	59624.8	0.321664
$129$	−5832.76	−0.0308603
$130$	0	0
$131$	−187469.	−0.954444	−0.477222	−	0.878783i	$-0.658356\pi$
$131$	−187469.	−0.954444	−0.477222	+	0.878783i	$0.658356\pi$
$132$	29837.9	0.149051
$133$	−118339.	−0.580095
$134$	50477.5	0.242849
$135$	0	0
$136$	194285.	0.900724
$137$	175520.	0.798959	0.399479	−	0.916742i	$-0.369191\pi$
$137$	175520.	0.798959	0.399479	+	0.916742i	$0.369191\pi$
$138$	−141924.	−0.634391
$139$	343197.	1.50663	0.753316	−	0.657659i	$-0.228453\pi$
$139$	343197.	1.50663	0.753316	+	0.657659i	$0.228453\pi$
$140$	0	0
$141$	−5433.40	−0.0230157
$142$	195466.	0.813488
$143$	119828.	0.490024
$144$	−533112.	−2.14246
$145$	0	0
$146$	−687215.	−2.66815
$147$	46359.3	0.176947
$148$	91873.5	0.344775
$149$	−307566.	−1.13494	−0.567470	−	0.823394i	$-0.692078\pi$
$149$	−307566.	−1.13494	−0.567470	+	0.823394i	$0.692078\pi$
$150$	0	0
$151$	355275.	1.26801	0.634004	−	0.773330i	$-0.281410\pi$
$151$	355275.	1.26801	0.634004	+	0.773330i	$0.281410\pi$
$152$	−1.06305e6	−3.73203
$153$	99342.1	0.343088
$154$	−63561.6	−0.215970
$155$	0	0
$156$	244206.	0.803423
$157$	165270.	0.535113	0.267557	−	0.963542i	$-0.413784\pi$
$157$	165270.	0.535113	0.267557	+	0.963542i	$0.413784\pi$
$158$	−862454.	−2.74848
$159$	38841.9	0.121845
$160$	0	0
$161$	212569.	0.646302
$162$	−534060.	−1.59883
$163$	−379294.	−1.11817	−0.559085	−	0.829111i	$-0.688847\pi$
$163$	−379294.	−1.11817	−0.559085	+	0.829111i	$0.688847\pi$
$164$	−1.13506e6	−3.29541
$165$	0	0
$166$	62105.0	0.174927
$167$	32375.9	0.0898319	0.0449159	−	0.998991i	$-0.485698\pi$
$167$	32375.9	0.0898319	0.0449159	+	0.998991i	$0.485698\pi$
$168$	−74837.6	−0.204572
$169$	609427.	1.64136
$170$	0	0
$171$	−543561.	−1.42154
$172$	135837.	0.350103
$173$	−50258.6	−0.127672	−0.0638360	−	0.997960i	$-0.520333\pi$
$173$	−50258.6	−0.127672	−0.0638360	+	0.997960i	$0.520333\pi$
$174$	52058.8	0.130353
$175$	0	0
$176$	−277553.	−0.675406
$177$	53333.2	0.127957
$178$	−503506.	−1.19112
$179$	−384940.	−0.897967	−0.448984	−	0.893540i	$-0.648214\pi$
$179$	−384940.	−0.897967	−0.448984	+	0.893540i	$0.648214\pi$
$180$	0	0
$181$	724705.	1.64424	0.822120	−	0.569315i	$-0.192791\pi$
$181$	724705.	1.64424	0.822120	+	0.569315i	$0.192791\pi$
$182$	−520214.	−1.16414
$183$	22575.5	0.0498321
$184$	1.90953e6	4.15797
$185$	0	0
$186$	−214033.	−0.453626
$187$	51720.4	0.108158
$188$	126536.	0.261108
$189$	−78275.7	−0.159394
$190$	0	0
$191$	−15927.2	−0.0315905	−0.0157952	−	0.999875i	$-0.505028\pi$
$191$	−15927.2	−0.0315905	−0.0157952	+	0.999875i	$0.505028\pi$
$192$	−74279.7	−0.145418
$193$	492233.	0.951212	0.475606	−	0.879658i	$-0.342229\pi$
$193$	492233.	0.951212	0.475606	+	0.879658i	$0.342229\pi$
$194$	1.13376e6	2.16280
$195$	0	0
$196$	−1.07964e6	−2.00743
$197$	435505.	0.799517	0.399758	−	0.916621i	$-0.369094\pi$
$197$	435505.	0.799517	0.399758	+	0.916621i	$0.369094\pi$
$198$	−291954.	−0.529239
$199$	590146.	1.05640	0.528198	−	0.849121i	$-0.322868\pi$
$199$	590146.	1.05640	0.528198	+	0.849121i	$0.322868\pi$
$200$	0	0
$201$	15821.4	0.0276220
$202$	855893.	1.47585
$203$	−77972.1	−0.132800
$204$	105405.	0.177331
$205$	0	0
$206$	−1.72887e6	−2.83853
$207$	976382.	1.58378
$208$	−2.27161e6	−3.64062
$209$	−282993.	−0.448137
$210$	0	0
$211$	218014.	0.337116	0.168558	−	0.985692i	$-0.446089\pi$
$211$	218014.	0.337116	0.168558	+	0.985692i	$0.446089\pi$
$212$	−904575.	−1.38231
$213$	61266.0	0.0925275
$214$	−1.51153e6	−2.25623
$215$	0	0
$216$	−703157.	−1.02546
$217$	320572.	0.462142
$218$	2.35735e6	3.35956
$219$	−215397.	−0.303480
$220$	0	0
$221$	423300.	0.582999
$222$	40956.1	0.0557746
$223$	614763.	0.827838	0.413919	−	0.910314i	$-0.364160\pi$
$223$	614763.	0.827838	0.413919	+	0.910314i	$0.364160\pi$
$224$	469001.	0.624531
$225$	0	0
$226$	1.86422e6	2.42788
$227$	−1.07088e6	−1.37935	−0.689675	−	0.724119i	$-0.742247\pi$
$227$	−1.07088e6	−1.37935	−0.689675	+	0.724119i	$0.742247\pi$
$228$	−576732.	−0.734745
$229$	−134715.	−0.169757	−0.0848784	−	0.996391i	$-0.527050\pi$
$229$	−134715.	−0.169757	−0.0848784	+	0.996391i	$0.527050\pi$
$230$	0	0
$231$	−19922.4	−0.0245648
$232$	−700430.	−0.854368
$233$	17916.1	0.0216199	0.0108099	−	0.999942i	$-0.496559\pi$
$233$	17916.1	0.0216199	0.0108099	+	0.999942i	$0.496559\pi$
$234$	−2.38947e6	−2.85274
$235$	0	0
$236$	−1.24206e6	−1.45165
$237$	−270323.	−0.312617
$238$	−224536.	−0.256947
$239$	878276.	0.994572	0.497286	−	0.867587i	$-0.334330\pi$
$239$	878276.	0.994572	0.497286	+	0.867587i	$0.334330\pi$
$240$	0	0
$241$	480494.	0.532900	0.266450	−	0.963849i	$-0.414149\pi$
$241$	480494.	0.532900	0.266450	+	0.963849i	$0.414149\pi$
$242$	−152000.	−0.166842
$243$	−543313.	−0.590248
$244$	−525751.	−0.565335
$245$	0	0
$246$	−505997.	−0.533102
$247$	−2.31613e6	−2.41558
$248$	2.87972e6	2.97318
$249$	19465.9	0.0198965
$250$	0	0
$251$	−873829.	−0.875472	−0.437736	−	0.899104i	$-0.644220\pi$
$251$	−873829.	−0.875472	−0.437736	+	0.899104i	$0.644220\pi$
$252$	891165.	0.884009
$253$	508333.	0.499283
$254$	−2.25322e6	−2.19139
$255$	0	0
$256$	−1.34948e6	−1.28696
$257$	1.69731e6	1.60298	0.801491	−	0.598007i	$-0.204040\pi$
$257$	1.69731e6	1.60298	0.801491	+	0.598007i	$0.204040\pi$
$258$	60554.5	0.0566366
$259$	−61342.9	−0.0568218
$260$	0	0
$261$	−358145.	−0.325430
$262$	1.94626e6	1.75165
$263$	793063.	0.706998	0.353499	−	0.935435i	$-0.384992\pi$
$263$	793063.	0.706998	0.353499	+	0.935435i	$0.384992\pi$
$264$	−178965.	−0.158037
$265$	0	0
$266$	1.22857e6	1.06462
$267$	−157817.	−0.135480
$268$	−368459.	−0.313366
$269$	641402.	0.540442	0.270221	−	0.962798i	$-0.412903\pi$
$269$	641402.	0.540442	0.270221	+	0.962798i	$0.412903\pi$
$270$	0	0
$271$	678036.	0.560828	0.280414	−	0.959879i	$-0.409528\pi$
$271$	678036.	0.560828	0.280414	+	0.959879i	$0.409528\pi$
$272$	−980477.	−0.803554
$273$	−163053.	−0.132411
$274$	−1.82221e6	−1.46630
$275$	0	0
$276$	1.03597e6	0.818602
$277$	1.79159e6	1.40294	0.701469	−	0.712700i	$-0.252528\pi$
$277$	1.79159e6	1.40294	0.701469	+	0.712700i	$0.252528\pi$
$278$	−3.56300e6	−2.76506
$279$	1.47247e6	1.13249
$280$	0	0
$281$	−199237.	−0.150524	−0.0752619	−	0.997164i	$-0.523979\pi$
$281$	−199237.	−0.150524	−0.0752619	+	0.997164i	$0.523979\pi$
$282$	56408.4	0.0422397
$283$	−8539.72	−0.00633837	−0.00316918	−	0.999995i	$-0.501009\pi$
$283$	−8539.72	−0.00633837	−0.00316918	+	0.999995i	$0.501009\pi$
$284$	−1.42680e6	−1.04971
$285$	0	0
$286$	−1.24403e6	−0.899321
$287$	757868.	0.543111
$288$	2.15424e6	1.53043
$289$	−1.23715e6	−0.871321
$290$	0	0
$291$	355360.	0.246000
$292$	5.01630e6	3.44292
$293$	1.35528e6	0.922277	0.461139	−	0.887328i	$-0.347441\pi$
$293$	1.35528e6	0.922277	0.461139	+	0.887328i	$0.347441\pi$
$294$	−481293.	−0.324744
$295$	0	0
$296$	−551049.	−0.365562
$297$	−187187.	−0.123136
$298$	3.19309e6	2.08291
$299$	4.16040e6	2.69127
$300$	0	0
$301$	−90696.7	−0.0576999
$302$	−3.68839e6	−2.32712
$303$	268267.	0.167865
$304$	5.36478e6	3.32942
$305$	0	0
$306$	−1.03135e6	−0.629654
$307$	−1.04893e6	−0.635183	−0.317592	−	0.948228i	$-0.602874\pi$
$307$	−1.04893e6	−0.635183	−0.317592	+	0.948228i	$0.602874\pi$
$308$	463966.	0.278682
$309$	−541887.	−0.322859
$310$	0	0
$311$	540559.	0.316915	0.158457	−	0.987366i	$-0.449348\pi$
$311$	540559.	0.316915	0.158457	+	0.987366i	$0.449348\pi$
$312$	−1.46472e6	−0.851861
$313$	−1.13973e6	−0.657569	−0.328784	−	0.944405i	$-0.606639\pi$
$313$	−1.13973e6	−0.657569	−0.328784	+	0.944405i	$0.606639\pi$
$314$	−1.71580e6	−0.982071
$315$	0	0
$316$	6.29545e6	3.54658
$317$	1.80634e6	1.00961	0.504803	−	0.863235i	$-0.331565\pi$
$317$	1.80634e6	1.00961	0.504803	+	0.863235i	$0.331565\pi$
$318$	−403249.	−0.223617
$319$	−186461.	−0.102591
$320$	0	0
$321$	−473768.	−0.256627
$322$	−2.20685e6	−1.18613
$323$	−999694.	−0.533164
$324$	3.89835e6	2.06309
$325$	0	0
$326$	3.93775e6	2.05213
$327$	738875.	0.382122
$328$	6.80799e6	3.49409
$329$	−84486.9	−0.0430328
$330$	0	0
$331$	−3.76943e6	−1.89106	−0.945531	−	0.325533i	$-0.894456\pi$
$331$	−3.76943e6	−1.89106	−0.945531	+	0.325533i	$0.894456\pi$
$332$	−453333.	−0.225721
$333$	−281763.	−0.139243
$334$	−336120.	−0.164865
$335$	0	0
$336$	377675.	0.182503
$337$	−4.09543e6	−1.96438	−0.982189	−	0.187898i	$-0.939833\pi$
$337$	−4.09543e6	−1.96438	−0.982189	+	0.187898i	$0.939833\pi$
$338$	−6.32694e6	−3.01233
$339$	584313.	0.276151
$340$	0	0
$341$	766608.	0.357016
$342$	5.64313e6	2.60889
$343$	1.57127e6	0.721135
$344$	−814736.	−0.371211
$345$	0	0
$346$	521774.	0.234311
$347$	−590909.	−0.263449	−0.131725	−	0.991286i	$-0.542051\pi$
$347$	−590909.	−0.263449	−0.131725	+	0.991286i	$0.542051\pi$
$348$	−380001.	−0.168204
$349$	3.61334e6	1.58798	0.793991	−	0.607930i	$-0.208000\pi$
$349$	3.61334e6	1.58798	0.793991	+	0.607930i	$0.208000\pi$
$350$	0	0
$351$	−1.53201e6	−0.663734
$352$	1.12156e6	0.482464
$353$	−3.14648e6	−1.34396	−0.671982	−	0.740567i	$-0.734557\pi$
$353$	−3.14648e6	−1.34396	−0.671982	+	0.740567i	$0.734557\pi$
$354$	−553694.	−0.234834
$355$	0	0
$356$	3.67533e6	1.53699
$357$	−70377.4	−0.0292256
$358$	3.99637e6	1.64800
$359$	−1.08362e6	−0.443754	−0.221877	−	0.975075i	$-0.571218\pi$
$359$	−1.08362e6	−0.443754	−0.221877	+	0.975075i	$0.571218\pi$
$360$	0	0
$361$	2.99383e6	1.20909
$362$	−7.52374e6	−3.01760
$363$	−47642.1	−0.0189768
$364$	3.79728e6	1.50217
$365$	0	0
$366$	−234374.	−0.0914548
$367$	−3.15625e6	−1.22322	−0.611612	−	0.791158i	$-0.709479\pi$
$367$	−3.15625e6	−1.22322	−0.611612	+	0.791158i	$0.709479\pi$
$368$	−9.63660e6	−3.70940
$369$	3.48108e6	1.33091
$370$	0	0
$371$	603974.	0.227816
$372$	1.56232e6	0.585347
$373$	−2.03193e6	−0.756202	−0.378101	−	0.925764i	$-0.623423\pi$
$373$	−2.03193e6	−0.756202	−0.378101	+	0.925764i	$0.623423\pi$
$374$	−536950.	−0.198497
$375$	0	0
$376$	−758953.	−0.276850
$377$	−1.52607e6	−0.552995
$378$	812641.	0.292529
$379$	−998682.	−0.357132	−0.178566	−	0.983928i	$-0.557146\pi$
$379$	−998682.	−0.357132	−0.178566	+	0.983928i	$0.557146\pi$
$380$	0	0
$381$	−706240.	−0.249253
$382$	165353.	0.0579767
$383$	−1.89593e6	−0.660427	−0.330214	−	0.943906i	$-0.607121\pi$
$383$	−1.89593e6	−0.660427	−0.330214	+	0.943906i	$0.607121\pi$
$384$	−194020.	−0.0671458
$385$	0	0
$386$	−5.11026e6	−1.74572
$387$	−416593.	−0.141395
$388$	−8.27583e6	−2.79082
$389$	−1.84597e6	−0.618514	−0.309257	−	0.950978i	$-0.600080\pi$
$389$	−1.84597e6	−0.618514	−0.309257	+	0.950978i	$0.600080\pi$
$390$	0	0
$391$	1.79572e6	0.594014
$392$	6.47560e6	2.12846
$393$	610026.	0.199236
$394$	−4.52132e6	−1.46732
$395$	0	0
$396$	2.13111e6	0.682917
$397$	−180968.	−0.0576270	−0.0288135	−	0.999585i	$-0.509173\pi$
$397$	−180968.	−0.0576270	−0.0288135	+	0.999585i	$0.509173\pi$
$398$	−6.12677e6	−1.93876
$399$	385077.	0.121092
$400$	0	0
$401$	3.73198e6	1.15899	0.579493	−	0.814978i	$-0.303251\pi$
$401$	3.73198e6	1.15899	0.579493	+	0.814978i	$0.303251\pi$
$402$	−164255.	−0.0506936
$403$	6.27423e6	1.92441
$404$	−6.24756e6	−1.90440
$405$	0	0
$406$	809490.	0.243723
$407$	−146694.	−0.0438961
$408$	−632206.	−0.188022
$409$	−2.93967e6	−0.868940	−0.434470	−	0.900686i	$-0.643064\pi$
$409$	−2.93967e6	−0.868940	−0.434470	+	0.900686i	$0.643064\pi$
$410$	0	0
$411$	−571144.	−0.166779
$412$	1.26198e7	3.66277
$413$	829307.	0.239243
$414$	−1.01366e7	−2.90664
$415$	0	0
$416$	9.17929e6	2.60061
$417$	−1.11677e6	−0.314502
$418$	2.93798e6	0.822446
$419$	−2.38197e6	−0.662830	−0.331415	−	0.943485i	$-0.607526\pi$
$419$	−2.38197e6	−0.662830	−0.331415	+	0.943485i	$0.607526\pi$
$420$	0	0
$421$	−663533.	−0.182456	−0.0912279	−	0.995830i	$-0.529079\pi$
$421$	−663533.	−0.182456	−0.0912279	+	0.995830i	$0.529079\pi$
$422$	−2.26338e6	−0.618694
$423$	−388069.	−0.105453
$424$	5.42555e6	1.46565
$425$	0	0
$426$	−636051.	−0.169812
$427$	351038.	0.0931719
$428$	1.10334e7	2.91138
$429$	−389922.	−0.102290
$430$	0	0
$431$	−1.80832e6	−0.468903	−0.234451	−	0.972128i	$-0.575329\pi$
$431$	−1.80832e6	−0.468903	−0.234451	+	0.972128i	$0.575329\pi$
$432$	3.54855e6	0.914832
$433$	−6.02055e6	−1.54318	−0.771590	−	0.636121i	$-0.780538\pi$
$433$	−6.02055e6	−1.54318	−0.771590	+	0.636121i	$0.780538\pi$
$434$	−3.32811e6	−0.848151
$435$	0	0
$436$	−1.72074e7	−4.33509
$437$	−9.82547e6	−2.46122
$438$	2.23621e6	0.556964
$439$	−3.14357e6	−0.778507	−0.389253	−	0.921131i	$-0.627267\pi$
$439$	−3.14357e6	−0.778507	−0.389253	+	0.921131i	$0.627267\pi$
$440$	0	0
$441$	3.31112e6	0.810733
$442$	−4.39461e6	−1.06995
$443$	−5.58480e6	−1.35207	−0.676034	−	0.736871i	$-0.736303\pi$
$443$	−5.58480e6	−1.35207	−0.676034	+	0.736871i	$0.736303\pi$
$444$	−298958.	−0.0719702
$445$	0	0
$446$	−6.38234e6	−1.51930
$447$	1.00083e6	0.236913
$448$	−1.15501e6	−0.271889
$449$	7.04017e6	1.64804	0.824019	−	0.566562i	$-0.191727\pi$
$449$	7.04017e6	1.64804	0.824019	+	0.566562i	$0.191727\pi$
$450$	0	0
$451$	1.81235e6	0.419566
$452$	−1.36078e7	−3.13287
$453$	−1.15607e6	−0.264691
$454$	1.11176e7	2.53146
$455$	0	0
$456$	3.45918e6	0.779043
$457$	249299.	0.0558379	0.0279190	−	0.999610i	$-0.491112\pi$
$457$	249299.	0.0558379	0.0279190	+	0.999610i	$0.491112\pi$
$458$	1.39858e6	0.311548
$459$	−661250.	−0.146499
$460$	0	0
$461$	7.02575e6	1.53971	0.769857	−	0.638216i	$-0.220327\pi$
$461$	7.02575e6	1.53971	0.769857	+	0.638216i	$0.220327\pi$
$462$	206831.	0.0450827
$463$	200249.	0.0434127	0.0217064	−	0.999764i	$-0.493090\pi$
$463$	200249.	0.0434127	0.0217064	+	0.999764i	$0.493090\pi$
$464$	3.53479e6	0.762199
$465$	0	0
$466$	−186001.	−0.0396780
$467$	−2.10125e6	−0.445847	−0.222923	−	0.974836i	$-0.571560\pi$
$467$	−2.10125e6	−0.445847	−0.222923	+	0.974836i	$0.571560\pi$
$468$	1.74419e7	3.68111
$469$	246016.	0.0516453
$470$	0	0
$471$	−537792.	−0.111702
$472$	7.44973e6	1.53917
$473$	−216890.	−0.0445745
$474$	2.80644e6	0.573733
$475$	0	0
$476$	1.63899e6	0.331558
$477$	2.77420e6	0.558268
$478$	−9.11808e6	−1.82530
$479$	−303525.	−0.0604443	−0.0302222	−	0.999543i	$-0.509621\pi$
$479$	−303525.	−0.0604443	−0.0302222	+	0.999543i	$0.509621\pi$
$480$	0	0
$481$	−1.20060e6	−0.236612
$482$	−4.98839e6	−0.978008
$483$	−691703.	−0.134912
$484$	1.10952e6	0.215288
$485$	0	0
$486$	5.64056e6	1.08326
$487$	847330.	0.161894	0.0809469	−	0.996718i	$-0.474206\pi$
$487$	847330.	0.161894	0.0809469	+	0.996718i	$0.474206\pi$
$488$	3.15341e6	0.599419
$489$	1.23423e6	0.233413
$490$	0	0
$491$	3.46005e6	0.647708	0.323854	−	0.946107i	$-0.395021\pi$
$491$	3.46005e6	0.647708	0.323854	+	0.946107i	$0.395021\pi$
$492$	3.69351e6	0.687902
$493$	−658686.	−0.122057
$494$	2.40456e7	4.43321
$495$	0	0
$496$	−1.45328e7	−2.65244
$497$	952659.	0.173000
$498$	−202091.	−0.0365151
$499$	3.44516e6	0.619381	0.309691	−	0.950837i	$-0.399775\pi$
$499$	3.44516e6	0.619381	0.309691	+	0.950837i	$0.399775\pi$
$500$	0	0
$501$	−105352.	−0.0187520
$502$	9.07191e6	1.60672
$503$	−5.30250e6	−0.934459	−0.467230	−	0.884136i	$-0.654748\pi$
$503$	−5.30250e6	−0.934459	−0.467230	+	0.884136i	$0.654748\pi$
$504$	−5.34512e6	−0.937306
$505$	0	0
$506$	−5.27740e6	−0.916313
$507$	−1.98309e6	−0.342627
$508$	1.64473e7	2.82772
$509$	−1.18529e6	−0.202783	−0.101391	−	0.994847i	$-0.532329\pi$
$509$	−1.18529e6	−0.202783	−0.101391	+	0.994847i	$0.532329\pi$
$510$	0	0
$511$	−3.34933e6	−0.567421
$512$	1.21020e7	2.04024
$513$	3.61810e6	0.606997
$514$	−1.76211e7	−2.94189
$515$	0	0
$516$	−442015.	−0.0730824
$517$	−202040.	−0.0332438
$518$	636849.	0.104283
$519$	163542.	0.0266509
$520$	0	0
$521$	−9.10974e6	−1.47032	−0.735159	−	0.677894i	$-0.762893\pi$
$521$	−9.10974e6	−1.47032	−0.735159	+	0.677894i	$0.762893\pi$
$522$	3.71819e6	0.597249
$523$	6.68845e6	1.06923	0.534615	−	0.845096i	$-0.320457\pi$
$523$	6.68845e6	1.06923	0.534615	+	0.845096i	$0.320457\pi$
$524$	−1.42067e7	−2.26029
$525$	0	0
$526$	−8.23341e6	−1.29752
$527$	2.70810e6	0.424754
$528$	903164.	0.140988
$529$	1.12129e7	1.74212
$530$	0	0
$531$	3.80921e6	0.586272
$532$	−8.96791e6	−1.37376
$533$	1.48330e7	2.26157
$534$	1.63842e6	0.248640
$535$	0	0
$536$	2.20998e6	0.332259
$537$	1.25260e6	0.187446
$538$	−6.65890e6	−0.991851
$539$	1.72386e6	0.255582
$540$	0	0
$541$	−5.79737e6	−0.851605	−0.425802	−	0.904816i	$-0.640008\pi$
$541$	−5.79737e6	−0.851605	−0.425802	+	0.904816i	$0.640008\pi$
$542$	−7.03923e6	−1.02926
$543$	−2.35820e6	−0.343227
$544$	3.96198e6	0.574005
$545$	0	0
$546$	1.69279e6	0.243008
$547$	1.31793e7	1.88332	0.941660	−	0.336566i	$-0.109266\pi$
$547$	1.31793e7	1.88332	0.941660	+	0.336566i	$0.109266\pi$
$548$	1.33011e7	1.89207
$549$	1.61241e6	0.228320
$550$	0	0
$551$	3.60407e6	0.505724
$552$	−6.21363e6	−0.867956
$553$	−4.20340e6	−0.584505
$554$	−1.85999e7	−2.57476
$555$	0	0
$556$	2.60080e7	3.56796
$557$	3.17990e6	0.434286	0.217143	−	0.976140i	$-0.430326\pi$
$557$	3.17990e6	0.434286	0.217143	+	0.976140i	$0.430326\pi$
$558$	−1.52868e7	−2.07841
$559$	−1.77512e6	−0.240269
$560$	0	0
$561$	−168299.	−0.0225774
$562$	2.06844e6	0.276250
$563$	−5.49194e6	−0.730222	−0.365111	−	0.930964i	$-0.618969\pi$
$563$	−5.49194e6	−0.730222	−0.365111	+	0.930964i	$0.618969\pi$
$564$	−411751.	−0.0545051
$565$	0	0
$566$	88657.6	0.0116325
$567$	−2.60289e6	−0.340015
$568$	8.55782e6	1.11299
$569$	−2.24494e6	−0.290686	−0.145343	−	0.989381i	$-0.546429\pi$
$569$	−2.24494e6	−0.290686	−0.145343	+	0.989381i	$0.546429\pi$
$570$	0	0
$571$	−5.92316e6	−0.760262	−0.380131	−	0.924933i	$-0.624121\pi$
$571$	−5.92316e6	−0.760262	−0.380131	+	0.924933i	$0.624121\pi$
$572$	9.08074e6	1.16046
$573$	51827.4	0.00659436
$574$	−7.86802e6	−0.996749
$575$	0	0
$576$	−5.30527e6	−0.666272
$577$	−1.72233e6	−0.215366	−0.107683	−	0.994185i	$-0.534343\pi$
$577$	−1.72233e6	−0.215366	−0.107683	+	0.994185i	$0.534343\pi$
$578$	1.28438e7	1.59910
$579$	−1.60173e6	−0.198561
$580$	0	0
$581$	302686.	0.0372007
$582$	−3.68927e6	−0.451474
$583$	1.44433e6	0.175993
$584$	−3.00873e7	−3.65049
$585$	0	0
$586$	−1.40703e7	−1.69262
$587$	−4.01191e6	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$587$	−4.01191e6	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$588$	3.51318e6	0.419041
$589$	−1.48176e7	−1.75991
$590$	0	0
$591$	−1.41714e6	−0.166895
$592$	2.78092e6	0.326125
$593$	−1.81663e6	−0.212143	−0.106072	−	0.994358i	$-0.533827\pi$
$593$	−1.81663e6	−0.212143	−0.106072	+	0.994358i	$0.533827\pi$
$594$	1.94333e6	0.225986
$595$	0	0
$596$	−2.33078e7	−2.68773
$597$	−1.92035e6	−0.220518
$598$	−4.31924e7	−4.93917
$599$	−7.43101e6	−0.846215	−0.423108	−	0.906079i	$-0.639061\pi$
$599$	−7.43101e6	−0.846215	−0.423108	+	0.906079i	$0.639061\pi$
$600$	0	0
$601$	−2.83570e6	−0.320239	−0.160119	−	0.987098i	$-0.551188\pi$
$601$	−2.83570e6	−0.320239	−0.160119	+	0.987098i	$0.551188\pi$
$602$	941594.	0.105894
$603$	1.13001e6	0.126558
$604$	2.69232e7	3.00286
$605$	0	0
$606$	−2.78509e6	−0.308076
$607$	−1.58704e7	−1.74830	−0.874151	−	0.485654i	$-0.838581\pi$
$607$	−1.58704e7	−1.74830	−0.874151	+	0.485654i	$0.838581\pi$
$608$	−2.16784e7	−2.37831
$609$	253723.	0.0277215
$610$	0	0
$611$	−1.65358e6	−0.179193
$612$	7.52830e6	0.812490
$613$	−1.07890e6	−0.115965	−0.0579827	−	0.998318i	$-0.518467\pi$
$613$	−1.07890e6	−0.115965	−0.0579827	+	0.998318i	$0.518467\pi$
$614$	1.08897e7	1.16573
$615$	0	0
$616$	−2.78282e6	−0.295484
$617$	4.90715e6	0.518940	0.259470	−	0.965751i	$-0.416452\pi$
$617$	4.90715e6	0.518940	0.259470	+	0.965751i	$0.416452\pi$
$618$	5.62576e6	0.592529
$619$	6.37999e6	0.669258	0.334629	−	0.942350i	$-0.391389\pi$
$619$	6.37999e6	0.669258	0.334629	+	0.942350i	$0.391389\pi$
$620$	0	0
$621$	−6.49908e6	−0.676274
$622$	−5.61197e6	−0.581620
$623$	−2.45397e6	−0.253308
$624$	7.39186e6	0.759962
$625$	0	0
$626$	1.18324e7	1.20681
$627$	920865.	0.0935464
$628$	1.25244e7	1.26724
$629$	−518207.	−0.0522248
$630$	0	0
$631$	−4.34499e6	−0.434426	−0.217213	−	0.976124i	$-0.569697\pi$
$631$	−4.34499e6	−0.434426	−0.217213	+	0.976124i	$0.569697\pi$
$632$	−3.77595e7	−3.76040
$633$	−709422.	−0.0703713
$634$	−1.87530e7	−1.85289
$635$	0	0
$636$	2.94350e6	0.288550
$637$	1.41088e7	1.37766
$638$	1.93580e6	0.188282
$639$	4.37580e6	0.423941
$640$	0	0
$641$	1.84166e7	1.77037	0.885184	−	0.465241i	$-0.154032\pi$
$641$	1.84166e7	1.77037	0.885184	+	0.465241i	$0.154032\pi$
$642$	4.91856e6	0.470977
$643$	−8.94487e6	−0.853192	−0.426596	−	0.904442i	$-0.640287\pi$
$643$	−8.94487e6	−0.853192	−0.426596	+	0.904442i	$0.640287\pi$
$644$	1.61088e7	1.53055
$645$	0	0
$646$	1.03786e7	0.978493
$647$	7.92058e6	0.743869	0.371934	−	0.928259i	$-0.378695\pi$
$647$	7.92058e6	0.743869	0.371934	+	0.928259i	$0.378695\pi$
$648$	−2.33820e7	−2.18748
$649$	1.98319e6	0.184821
$650$	0	0
$651$	−1.04315e6	−0.0964701
$652$	−2.87435e7	−2.64802
$653$	1.28235e7	1.17685	0.588426	−	0.808551i	$-0.299748\pi$
$653$	1.28235e7	1.17685	0.588426	+	0.808551i	$0.299748\pi$
$654$	−7.67084e6	−0.701292
$655$	0	0
$656$	−3.43572e7	−3.11715
$657$	−1.53843e7	−1.39048
$658$	877125.	0.0789763
$659$	−4.40034e6	−0.394705	−0.197353	−	0.980333i	$-0.563234\pi$
$659$	−4.40034e6	−0.394705	−0.197353	+	0.980333i	$0.563234\pi$
$660$	0	0
$661$	1.25957e6	0.112129	0.0560647	−	0.998427i	$-0.482145\pi$
$661$	1.25957e6	0.112129	0.0560647	+	0.998427i	$0.482145\pi$
$662$	3.91334e7	3.47059
$663$	−1.37743e6	−0.121698
$664$	2.71905e6	0.239330
$665$	0	0
$666$	2.92521e6	0.255547
$667$	−6.47388e6	−0.563443
$668$	2.45349e6	0.212737
$669$	−2.00045e6	−0.172807
$670$	0	0
$671$	839465.	0.0719774
$672$	−1.52614e6	−0.130368
$673$	304612.	0.0259244	0.0129622	−	0.999916i	$-0.495874\pi$
$673$	304612.	0.0259244	0.0129622	+	0.999916i	$0.495874\pi$
$674$	4.25179e7	3.60514
$675$	0	0
$676$	4.61833e7	3.88703
$677$	3.54969e6	0.297659	0.148829	−	0.988863i	$-0.452449\pi$
$677$	3.54969e6	0.297659	0.148829	+	0.988863i	$0.452449\pi$
$678$	−6.06621e6	−0.506808
$679$	5.52568e6	0.459951
$680$	0	0
$681$	3.48465e6	0.287933
$682$	−7.95876e6	−0.655216
$683$	1.80415e7	1.47986	0.739932	−	0.672681i	$-0.234858\pi$
$683$	1.80415e7	1.47986	0.739932	+	0.672681i	$0.234858\pi$
$684$	−4.11919e7	−3.36644
$685$	0	0
$686$	−1.63126e7	−1.32347
$687$	438365.	0.0354359
$688$	4.11164e6	0.331165
$689$	1.18210e7	0.948649
$690$	0	0
$691$	1.76063e7	1.40273	0.701365	−	0.712802i	$-0.252574\pi$
$691$	1.76063e7	1.40273	0.701365	+	0.712802i	$0.252574\pi$
$692$	−3.80868e6	−0.302349
$693$	−1.42292e6	−0.112550
$694$	6.13469e6	0.483497
$695$	0	0
$696$	2.27921e6	0.178345
$697$	6.40224e6	0.499172
$698$	−3.75129e7	−2.91436
$699$	−58299.2	−0.00451304
$700$	0	0
$701$	−5.97597e6	−0.459318	−0.229659	−	0.973271i	$-0.573761\pi$
$701$	−5.97597e6	−0.459318	−0.229659	+	0.973271i	$0.573761\pi$
$702$	1.59050e7	1.21812
$703$	2.83542e6	0.216386
$704$	−2.76208e6	−0.210041
$705$	0	0
$706$	3.26660e7	2.46652
$707$	4.17143e6	0.313860
$708$	4.04167e6	0.303025
$709$	4.89639e6	0.365814	0.182907	−	0.983130i	$-0.441449\pi$
$709$	4.89639e6	0.365814	0.182907	+	0.983130i	$0.441449\pi$
$710$	0	0
$711$	−1.93073e7	−1.43234
$712$	−2.20443e7	−1.62965
$713$	2.66165e7	1.96077
$714$	730643.	0.0536364
$715$	0	0
$716$	−2.91713e7	−2.12654
$717$	−2.85793e6	−0.207612
$718$	1.12500e7	0.814403
$719$	−2.14209e7	−1.54531	−0.772653	−	0.634828i	$-0.781071\pi$
$719$	−2.14209e7	−1.54531	−0.772653	+	0.634828i	$0.781071\pi$
$720$	0	0
$721$	−8.42610e6	−0.603654
$722$	−3.10813e7	−2.21899
$723$	−1.56354e6	−0.111240
$724$	5.49193e7	3.89384
$725$	0	0
$726$	494610.	0.0348274
$727$	2.80489e7	1.96825	0.984125	−	0.177479i	$-0.0567941\pi$
$727$	2.80489e7	1.96825	0.984125	+	0.177479i	$0.0567941\pi$
$728$	−2.27758e7	−1.59274
$729$	−1.07325e7	−0.747963
$730$	0	0
$731$	−766179.	−0.0530319
$732$	1.71080e6	0.118011
$733$	−1.26781e7	−0.871553	−0.435776	−	0.900055i	$-0.643526\pi$
$733$	−1.26781e7	−0.871553	−0.435776	+	0.900055i	$0.643526\pi$
$734$	3.27675e7	2.24493
$735$	0	0
$736$	3.89403e7	2.64975
$737$	588317.	0.0398972
$738$	−3.61398e7	−2.44256
$739$	1.14208e7	0.769284	0.384642	−	0.923066i	$-0.374325\pi$
$739$	1.14208e7	0.769284	0.384642	+	0.923066i	$0.374325\pi$
$740$	0	0
$741$	7.53673e6	0.504240
$742$	−6.27033e6	−0.418101
$743$	−2.69184e6	−0.178887	−0.0894433	−	0.995992i	$-0.528509\pi$
$743$	−2.69184e6	−0.178887	−0.0894433	+	0.995992i	$0.528509\pi$
$744$	−9.37067e6	−0.620638
$745$	0	0
$746$	2.10951e7	1.38783
$747$	1.39031e6	0.0911612
$748$	3.91945e6	0.256136
$749$	−7.36687e6	−0.479820
$750$	0	0
$751$	2.39859e6	0.155187	0.0775935	−	0.996985i	$-0.475276\pi$
$751$	2.39859e6	0.155187	0.0775935	+	0.996985i	$0.475276\pi$
$752$	3.83013e6	0.246984
$753$	2.84346e6	0.182751
$754$	1.58433e7	1.01489
$755$	0	0
$756$	−5.93185e6	−0.377473
$757$	1.21139e7	0.768324	0.384162	−	0.923266i	$-0.374490\pi$
$757$	1.21139e7	0.768324	0.384162	+	0.923266i	$0.374490\pi$
$758$	1.03681e7	0.655430
$759$	−1.65412e6	−0.104223
$760$	0	0
$761$	−8.03382e6	−0.502875	−0.251438	−	0.967873i	$-0.580903\pi$
$761$	−8.03382e6	−0.502875	−0.251438	+	0.967873i	$0.580903\pi$
$762$	7.33203e6	0.457443
$763$	1.14892e7	0.714459
$764$	−1.20699e6	−0.0748117
$765$	0	0
$766$	1.96831e7	1.21205
$767$	1.62312e7	0.996235
$768$	4.39122e6	0.268647
$769$	−4.58340e6	−0.279494	−0.139747	−	0.990187i	$-0.544629\pi$
$769$	−4.58340e6	−0.279494	−0.139747	+	0.990187i	$0.544629\pi$
$770$	0	0
$771$	−5.52308e6	−0.334615
$772$	3.73022e7	2.25263
$773$	−1.69473e7	−1.02012	−0.510062	−	0.860138i	$-0.670377\pi$
$773$	−1.69473e7	−1.02012	−0.510062	+	0.860138i	$0.670377\pi$
$774$	4.32498e6	0.259496
$775$	0	0
$776$	4.96377e7	2.95908
$777$	199611.	0.0118613
$778$	1.91644e7	1.13513
$779$	−3.50305e7	−2.06825
$780$	0	0
$781$	2.27817e6	0.133647
$782$	−1.86428e7	−1.09017
$783$	2.38392e6	0.138959
$784$	−3.26797e7	−1.89884
$785$	0	0
$786$	−6.33316e6	−0.365649
$787$	4.01619e6	0.231141	0.115571	−	0.993299i	$-0.463130\pi$
$787$	4.01619e6	0.231141	0.115571	+	0.993299i	$0.463130\pi$
$788$	3.30032e7	1.89339
$789$	−2.58064e6	−0.147583
$790$	0	0
$791$	9.08579e6	0.516323
$792$	−1.27822e7	−0.724090
$793$	6.87052e6	0.387978
$794$	1.87877e6	0.105760
$795$	0	0
$796$	4.47222e7	2.50173
$797$	2.36467e7	1.31863	0.659317	−	0.751865i	$-0.270845\pi$
$797$	2.36467e7	1.31863	0.659317	+	0.751865i	$0.270845\pi$
$798$	−3.99779e6	−0.222235
$799$	−713720.	−0.0395513
$800$	0	0
$801$	−1.12717e7	−0.620738
$802$	−3.87446e7	−2.12704
$803$	−8.00951e6	−0.438346
$804$	1.19897e6	0.0654137
$805$	0	0
$806$	−6.51377e7	−3.53179
$807$	−2.08713e6	−0.112815
$808$	3.74723e7	2.01921
$809$	1.06140e7	0.570174	0.285087	−	0.958502i	$-0.407978\pi$
$809$	1.06140e7	0.570174	0.285087	+	0.958502i	$0.407978\pi$
$810$	0	0
$811$	3.28249e6	0.175247	0.0876236	−	0.996154i	$-0.472073\pi$
$811$	3.28249e6	0.175247	0.0876236	+	0.996154i	$0.472073\pi$
$812$	−5.90885e6	−0.314494
$813$	−2.20634e6	−0.117070
$814$	1.52295e6	0.0805607
$815$	0	0
$816$	3.19049e6	0.167738
$817$	4.19223e6	0.219730
$818$	3.05190e7	1.59473
$819$	−1.16457e7	−0.606677
$820$	0	0
$821$	8.61250e6	0.445935	0.222967	−	0.974826i	$-0.428426\pi$
$821$	8.61250e6	0.445935	0.222967	+	0.974826i	$0.428426\pi$
$822$	5.92949e6	0.306082
$823$	−1.41290e7	−0.727131	−0.363565	−	0.931569i	$-0.618441\pi$
$823$	−1.41290e7	−0.727131	−0.363565	+	0.931569i	$0.618441\pi$
$824$	−7.56924e7	−3.88360
$825$	0	0
$826$	−8.60969e6	−0.439074
$827$	597955.	0.0304022	0.0152011	−	0.999884i	$-0.495161\pi$
$827$	597955.	0.0304022	0.0152011	+	0.999884i	$0.495161\pi$
$828$	7.39917e7	3.75066
$829$	1.66755e7	0.842736	0.421368	−	0.906890i	$-0.361550\pi$
$829$	1.66755e7	0.842736	0.421368	+	0.906890i	$0.361550\pi$
$830$	0	0
$831$	−5.82986e6	−0.292857
$832$	−2.26059e7	−1.13218
$833$	6.08966e6	0.304075
$834$	1.15941e7	0.577193
$835$	0	0
$836$	−2.14457e7	−1.06126
$837$	−9.80116e6	−0.483575
$838$	2.47291e7	1.21646
$839$	−1.15062e7	−0.564322	−0.282161	−	0.959367i	$-0.591051\pi$
$839$	−1.15062e7	−0.564322	−0.282161	+	0.959367i	$0.591051\pi$
$840$	0	0
$841$	−1.81365e7	−0.884225
$842$	6.88866e6	0.334853
$843$	648322.	0.0314211
$844$	1.65215e7	0.798348
$845$	0	0
$846$	4.02885e6	0.193533
$847$	−740812.	−0.0354813
$848$	−2.73806e7	−1.30753
$849$	27788.4	0.00132310
$850$	0	0
$851$	−5.09319e6	−0.241082
$852$	4.64283e6	0.219121
$853$	−1.62039e7	−0.762513	−0.381257	−	0.924469i	$-0.624509\pi$
$853$	−1.62039e7	−0.762513	−0.381257	+	0.924469i	$0.624509\pi$
$854$	−3.64440e6	−0.170994
$855$	0	0
$856$	−6.61772e7	−3.08691
$857$	9.97322e6	0.463856	0.231928	−	0.972733i	$-0.425497\pi$
$857$	9.97322e6	0.463856	0.231928	+	0.972733i	$0.425497\pi$
$858$	4.04809e6	0.187729
$859$	−3.22135e7	−1.48955	−0.744774	−	0.667316i	$-0.767443\pi$
$859$	−3.22135e7	−1.48955	−0.744774	+	0.667316i	$0.767443\pi$
$860$	0	0
$861$	−2.46611e6	−0.113372
$862$	1.87736e7	0.860558
$863$	−1.41832e7	−0.648257	−0.324128	−	0.946013i	$-0.605071\pi$
$863$	−1.41832e7	−0.648257	−0.324128	+	0.946013i	$0.605071\pi$
$864$	−1.43392e7	−0.653494
$865$	0	0
$866$	6.25041e7	2.83213
$867$	4.02571e6	0.181884
$868$	2.42934e7	1.09443
$869$	−1.00519e7	−0.451544
$870$	0	0
$871$	4.81502e6	0.215057
$872$	1.03208e8	4.59645
$873$	2.53808e7	1.12712
$874$	1.02006e8	4.51697
$875$	0	0
$876$	−1.63231e7	−0.718693
$877$	−1.87883e7	−0.824875	−0.412437	−	0.910986i	$-0.635322\pi$
$877$	−1.87883e7	−0.824875	−0.412437	+	0.910986i	$0.635322\pi$
$878$	3.26359e7	1.42876
$879$	−4.41012e6	−0.192521
$880$	0	0
$881$	3.00458e7	1.30420	0.652099	−	0.758134i	$-0.273889\pi$
$881$	3.00458e7	1.30420	0.652099	+	0.758134i	$0.273889\pi$
$882$	−3.43753e7	−1.48791
$883$	−4.10805e7	−1.77310	−0.886552	−	0.462629i	$-0.846906\pi$
$883$	−4.10805e7	−1.77310	−0.886552	+	0.462629i	$0.846906\pi$
$884$	3.20783e7	1.38064
$885$	0	0
$886$	5.79802e7	2.48139
$887$	1.64567e7	0.702316	0.351158	−	0.936316i	$-0.385788\pi$
$887$	1.64567e7	0.702316	0.351158	+	0.936316i	$0.385788\pi$
$888$	1.79312e6	0.0763093
$889$	−1.09817e7	−0.466031
$890$	0	0
$891$	−6.22448e6	−0.262669
$892$	4.65877e7	1.96046
$893$	3.90519e6	0.163875
$894$	−1.03904e7	−0.434797
$895$	0	0
$896$	−3.01692e6	−0.125543
$897$	−1.35380e7	−0.561790
$898$	−7.30896e7	−3.02458
$899$	−9.76315e6	−0.402894
$900$	0	0
$901$	5.10220e6	0.209385
$902$	−1.88154e7	−0.770012
$903$	295129.	0.0120446
$904$	8.16185e7	3.32175
$905$	0	0
$906$	1.20021e7	0.485776
$907$	−1.72342e6	−0.0695621	−0.0347810	−	0.999395i	$-0.511073\pi$
$907$	−1.72342e6	−0.0695621	−0.0347810	+	0.999395i	$0.511073\pi$
$908$	−8.11526e7	−3.26654
$909$	1.91604e7	0.769122
$910$	0	0
$911$	−3.98387e7	−1.59041	−0.795206	−	0.606340i	$-0.792637\pi$
$911$	−3.98387e7	−1.59041	−0.795206	+	0.606340i	$0.792637\pi$
$912$	−1.74571e7	−0.695000
$913$	723835.	0.0287384
$914$	−2.58816e6	−0.102477
$915$	0	0
$916$	−1.02089e7	−0.402013
$917$	9.48563e6	0.372514
$918$	6.86496e6	0.268863
$919$	309349.	0.0120826	0.00604129	−	0.999982i	$-0.498077\pi$
$919$	309349.	0.0120826	0.00604129	+	0.999982i	$0.498077\pi$
$920$	0	0
$921$	3.41322e6	0.132592
$922$	−7.29398e7	−2.82577
$923$	1.86454e7	0.720391
$924$	−1.50975e6	−0.0581736
$925$	0	0
$926$	−2.07894e6	−0.0796736
$927$	−3.87032e7	−1.47927
$928$	−1.42836e7	−0.544463
$929$	3.31582e7	1.26053	0.630263	−	0.776382i	$-0.282947\pi$
$929$	3.31582e7	1.26053	0.630263	+	0.776382i	$0.282947\pi$
$930$	0	0
$931$	−3.33202e7	−1.25989
$932$	1.35771e6	0.0511995
$933$	−1.75899e6	−0.0661545
$934$	2.18147e7	0.818244
$935$	0	0
$936$	−1.04615e8	−3.90304
$937$	−5.04047e7	−1.87552	−0.937760	−	0.347283i	$-0.887104\pi$
$937$	−5.04047e7	−1.87552	−0.937760	+	0.347283i	$0.887104\pi$
$938$	−2.55409e6	−0.0947825
$939$	3.70870e6	0.137264
$940$	0	0
$941$	1.93209e7	0.711299	0.355649	−	0.934619i	$-0.384260\pi$
$941$	1.93209e7	0.711299	0.355649	+	0.934619i	$0.384260\pi$
$942$	5.58325e6	0.205003
$943$	6.29243e7	2.30430
$944$	−3.75958e7	−1.37312
$945$	0	0
$946$	2.25171e6	0.0818057
$947$	−6.05379e6	−0.219358	−0.109679	−	0.993967i	$-0.534982\pi$
$947$	−6.05379e6	−0.219358	−0.109679	+	0.993967i	$0.534982\pi$
$948$	−2.04855e7	−0.740331
$949$	−6.55530e7	−2.36280
$950$	0	0
$951$	−5.87786e6	−0.210750
$952$	−9.83052e6	−0.351548
$953$	2.45689e7	0.876302	0.438151	−	0.898901i	$-0.355634\pi$
$953$	2.45689e7	0.876302	0.438151	+	0.898901i	$0.355634\pi$
$954$	−2.88012e7	−1.02457
$955$	0	0
$956$	6.65571e7	2.35532
$957$	606747.	0.0214155
$958$	3.15113e6	0.110931
$959$	−8.88102e6	−0.311829
$960$	0	0
$961$	1.15107e7	0.402062
$962$	1.24644e7	0.434244
$963$	−3.38379e7	−1.17581
$964$	3.64126e7	1.26200
$965$	0	0
$966$	7.18111e6	0.247599
$967$	4.76945e7	1.64022	0.820110	−	0.572206i	$-0.193912\pi$
$967$	4.76945e7	1.64022	0.820110	+	0.572206i	$0.193912\pi$
$968$	−6.65478e6	−0.228268
$969$	3.25302e6	0.111295
$970$	0	0
$971$	−39483.8	−0.00134391	−0.000671957	−	1.00000i	$-0.500214\pi$
$971$	−39483.8	−0.00134391	−0.000671957		1.00000i	$0.500214\pi$
$972$	−4.11731e7	−1.39781
$973$	−1.73653e7	−0.588030
$974$	−8.79680e6	−0.297117
$975$	0	0
$976$	−1.59140e7	−0.534754
$977$	−3.06451e7	−1.02713	−0.513564	−	0.858051i	$-0.671675\pi$
$977$	−3.06451e7	−1.02713	−0.513564	+	0.858051i	$0.671675\pi$
$978$	−1.28135e7	−0.428372
$979$	−5.86838e6	−0.195687
$980$	0	0
$981$	5.27726e7	1.75080
$982$	−3.59215e7	−1.18871
$983$	2.19736e7	0.725300	0.362650	−	0.931925i	$-0.381872\pi$
$983$	2.19736e7	0.725300	0.362650	+	0.931925i	$0.381872\pi$
$984$	−2.21533e7	−0.729375
$985$	0	0
$986$	6.83833e6	0.224005
$987$	274922.	0.00898289
$988$	−1.75520e8	−5.72050
$989$	−7.53038e6	−0.244808
$990$	0	0
$991$	−5.06580e6	−0.163857	−0.0819284	−	0.996638i	$-0.526108\pi$
$991$	−5.06580e6	−0.163857	−0.0819284	+	0.996638i	$0.526108\pi$
$992$	5.87252e7	1.89472
$993$	1.22658e7	0.394750
$994$	−9.89030e6	−0.317500
$995$	0	0
$996$	1.47515e6	0.0471183
$997$	1.77819e7	0.566552	0.283276	−	0.959038i	$-0.408579\pi$
$997$	1.77819e7	0.566552	0.283276	+	0.959038i	$0.408579\pi$
$998$	−3.57669e7	−1.13672
$999$	1.87550e6	0.0594570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.6.a.l.1.1		14
5.2	odd	4		55.6.b.b.34.1	✓	14
5.3	odd	4		55.6.b.b.34.14	yes	14
5.4	even	2	inner	275.6.a.l.1.14		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.b.b.34.1	✓	14	5.2	odd	4
55.6.b.b.34.14	yes	14	5.3	odd	4
275.6.a.l.1.1		14	1.1	even	1	trivial
275.6.a.l.1.14		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)

\(f(q)\)	\(=\)	\(q-10.3818 q^{2} -3.25402 q^{3} +75.7815 q^{4} +33.7825 q^{6} -50.5985 q^{7} -454.530 q^{8} -232.411 q^{9} +O(q^{10})\)	\(q-10.3818 q^{2} -3.25402 q^{3} +75.7815 q^{4} +33.7825 q^{6} -50.5985 q^{7} -454.530 q^{8} -232.411 q^{9} -121.000 q^{11} -246.594 q^{12} -990.313 q^{13} +525.303 q^{14} +2293.83 q^{16} -427.441 q^{17} +2412.85 q^{18} +2338.79 q^{19} +164.648 q^{21} +1256.20 q^{22} -4201.10 q^{23} +1479.05 q^{24} +10281.2 q^{26} +1547.00 q^{27} -3834.43 q^{28} +1541.00 q^{29} -6335.60 q^{31} -9269.08 q^{32} +393.736 q^{33} +4437.60 q^{34} -17612.5 q^{36} +1212.35 q^{37} -24280.8 q^{38} +3222.50 q^{39} -14978.1 q^{41} -1709.34 q^{42} +1792.48 q^{43} -9169.56 q^{44} +43614.9 q^{46} +1669.75 q^{47} -7464.16 q^{48} -14246.8 q^{49} +1390.90 q^{51} -75047.4 q^{52} -11936.6 q^{53} -16060.6 q^{54} +22998.5 q^{56} -7610.45 q^{57} -15998.3 q^{58} -16390.0 q^{59} -6937.73 q^{61} +65774.9 q^{62} +11759.7 q^{63} +22827.1 q^{64} -4087.68 q^{66} -4862.12 q^{67} -32392.1 q^{68} +13670.4 q^{69} -18827.8 q^{71} +105638. q^{72} +66194.3 q^{73} -12586.3 q^{74} +177237. q^{76} +6122.41 q^{77} -33455.3 q^{78} +83073.7 q^{79} +51442.0 q^{81} +155499. q^{82} -5982.11 q^{83} +12477.3 q^{84} -18609.1 q^{86} -5014.43 q^{87} +54998.2 q^{88} +48499.0 q^{89} +50108.3 q^{91} -318365. q^{92} +20616.2 q^{93} -17335.0 q^{94} +30161.7 q^{96} -109206. q^{97} +147907. q^{98} +28121.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 256 q^{4} + 14 q^{6} + 1550 q^{9}+O(q^{10}) \) 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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