LMFDB - Embedded newform 975.6.a.u.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [975,6,Mod(1,975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("975.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,2,-99,224,0,-18,-55] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$156.374224318$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 286 x^{9} + 442 x^{8} + 28715 x^{7} - 29138 x^{6} - 1208172 x^{5} + 509768 x^{4} + \cdots + 55036800 $ x^11 - 2x^10 - 286x^9 + 442x^8 + 28715x^7 - 29138x^6 - 1208172x^5 + 509768x^4 + 20144528x^3 - 3093984x^2 - 98068800x + 55036800
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3\cdot 5^{3}\cdot 11 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$8.18506$ of defining polynomial
Character	$\chi$	$=$	975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.18506 q^{2} -9.00000 q^{3} +34.9953 q^{4} -73.6656 q^{6} -143.020 q^{7} +24.5164 q^{8} +81.0000 q^{9} +156.357 q^{11} -314.957 q^{12} +169.000 q^{13} -1170.63 q^{14} -919.180 q^{16} +1172.37 q^{17} +662.990 q^{18} +1536.81 q^{19} +1287.18 q^{21} +1279.79 q^{22} +2933.88 q^{23} -220.648 q^{24} +1383.28 q^{26} -729.000 q^{27} -5005.03 q^{28} -6994.66 q^{29} +654.590 q^{31} -8308.07 q^{32} -1407.21 q^{33} +9595.96 q^{34} +2834.62 q^{36} -1167.29 q^{37} +12578.9 q^{38} -1521.00 q^{39} +18120.5 q^{41} +10535.7 q^{42} -6173.92 q^{43} +5471.75 q^{44} +24014.0 q^{46} -1002.86 q^{47} +8272.62 q^{48} +3647.81 q^{49} -10551.4 q^{51} +5914.20 q^{52} -30117.1 q^{53} -5966.91 q^{54} -3506.34 q^{56} -13831.3 q^{57} -57251.7 q^{58} -26017.9 q^{59} -11346.4 q^{61} +5357.86 q^{62} -11584.6 q^{63} -38588.3 q^{64} -11518.1 q^{66} +3312.08 q^{67} +41027.5 q^{68} -26404.9 q^{69} -23802.5 q^{71} +1985.83 q^{72} -72289.8 q^{73} -9554.37 q^{74} +53781.2 q^{76} -22362.2 q^{77} -12449.5 q^{78} -39681.2 q^{79} +6561.00 q^{81} +148318. q^{82} -56017.1 q^{83} +45045.3 q^{84} -50533.9 q^{86} +62951.9 q^{87} +3833.31 q^{88} +133295. q^{89} -24170.4 q^{91} +102672. q^{92} -5891.31 q^{93} -8208.46 q^{94} +74772.6 q^{96} -114564. q^{97} +29857.6 q^{98} +12664.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} - 99 q^{3} + 224 q^{4} - 18 q^{6} - 55 q^{7} + 270 q^{8} + 891 q^{9} - 125 q^{11} - 2016 q^{12} + 1859 q^{13} - 1311 q^{14} + 5756 q^{16} - 4507 q^{17} + 162 q^{18} + 142 q^{19} + 495 q^{21}+ \cdots - 10125 q^{99}+O(q^{100}) $ 11 * q + 2 * q^2 - 99 * q^3 + 224 * q^4 - 18 * q^6 - 55 * q^7 + 270 * q^8 + 891 * q^9 - 125 * q^11 - 2016 * q^12 + 1859 * q^13 - 1311 * q^14 + 5756 * q^16 - 4507 * q^17 + 162 * q^18 + 142 * q^19 + 495 * q^21 - 5975 * q^22 - 4672 * q^23 - 2430 * q^24 + 338 * q^26 - 8019 * q^27 - 4149 * q^28 + 5197 * q^29 - 277 * q^31 + 2272 * q^32 + 1125 * q^33 - 21225 * q^34 + 18144 * q^36 - 1696 * q^37 - 27485 * q^38 - 16731 * q^39 + 2040 * q^41 + 11799 * q^42 - 31436 * q^43 - 39883 * q^44 + 67530 * q^46 + 15499 * q^47 - 51804 * q^48 + 29212 * q^49 + 40563 * q^51 + 37856 * q^52 - 119577 * q^53 - 1458 * q^54 + 1677 * q^56 - 1278 * q^57 - 91154 * q^58 + 17115 * q^59 + 26273 * q^61 - 45211 * q^62 - 4455 * q^63 + 137480 * q^64 + 53775 * q^66 + 26313 * q^67 - 191297 * q^68 + 42048 * q^69 - 105756 * q^71 + 21870 * q^72 - 103230 * q^73 - 6501 * q^74 + 124495 * q^76 - 104415 * q^77 - 3042 * q^78 + 210732 * q^79 + 72171 * q^81 - 60283 * q^82 - 144247 * q^83 + 37341 * q^84 - 42260 * q^86 - 46773 * q^87 - 415403 * q^88 + 117088 * q^89 - 9295 * q^91 - 70700 * q^92 + 2493 * q^93 + 383414 * q^94 - 20448 * q^96 - 112934 * q^97 - 159305 * q^98 - 10125 * q^99

Coefficient data

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

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

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

$2$	8.18506	1.44693	0.723464	−	0.690362i	$-0.242549\pi$
$2$	8.18506	1.44693	0.723464	+	0.690362i	$0.242549\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	34.9953	1.09360
$5$	0	0
$5$	0	0
$6$	−73.6656	−0.835385
$7$	−143.020	−1.10320	−0.551598	−	0.834110i	$-0.685982\pi$
$7$	−143.020	−1.10320	−0.551598	+	0.834110i	$0.685982\pi$
$8$	24.5164	0.135435
$9$	81.0000	0.333333
$10$	0	0
$11$	156.357	0.389615	0.194807	−	0.980842i	$-0.437592\pi$
$11$	156.357	0.389615	0.194807	+	0.980842i	$0.437592\pi$
$12$	−314.957	−0.631391
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−1170.63	−1.59625
$15$	0	0
$16$	−919.180	−0.897637
$17$	1172.37	0.983884	0.491942	−	0.870628i	$-0.336287\pi$
$17$	1172.37	0.983884	0.491942	+	0.870628i	$0.336287\pi$
$18$	662.990	0.482309
$19$	1536.81	0.976646	0.488323	−	0.872663i	$-0.337609\pi$
$19$	1536.81	0.976646	0.488323	+	0.872663i	$0.337609\pi$
$20$	0	0
$21$	1287.18	0.636930
$22$	1279.79	0.563745
$23$	2933.88	1.15644	0.578220	−	0.815881i	$-0.303748\pi$
$23$	2933.88	1.15644	0.578220	+	0.815881i	$0.303748\pi$
$24$	−220.648	−0.0781935
$25$	0	0
$26$	1383.28	0.401306
$27$	−729.000	−0.192450
$28$	−5005.03	−1.20646
$29$	−6994.66	−1.54444	−0.772221	−	0.635354i	$-0.780854\pi$
$29$	−6994.66	−1.54444	−0.772221	+	0.635354i	$0.780854\pi$
$30$	0	0
$31$	654.590	0.122339	0.0611695	−	0.998127i	$-0.480517\pi$
$31$	654.590	0.122339	0.0611695	+	0.998127i	$0.480517\pi$
$32$	−8308.07	−1.43425
$33$	−1407.21	−0.224944
$34$	9595.96	1.42361
$35$	0	0
$36$	2834.62	0.364534
$37$	−1167.29	−0.140177	−0.0700883	−	0.997541i	$-0.522328\pi$
$37$	−1167.29	−0.140177	−0.0700883	+	0.997541i	$0.522328\pi$
$38$	12578.9	1.41314
$39$	−1521.00	−0.160128
$40$	0	0
$41$	18120.5	1.68349	0.841746	−	0.539874i	$-0.181528\pi$
$41$	18120.5	1.68349	0.841746	+	0.539874i	$0.181528\pi$
$42$	10535.7	0.921593
$43$	−6173.92	−0.509202	−0.254601	−	0.967046i	$-0.581944\pi$
$43$	−6173.92	−0.509202	−0.254601	+	0.967046i	$0.581944\pi$
$44$	5471.75	0.426083
$45$	0	0
$46$	24014.0	1.67328
$47$	−1002.86	−0.0662209	−0.0331104	−	0.999452i	$-0.510541\pi$
$47$	−1002.86	−0.0662209	−0.0331104	+	0.999452i	$0.510541\pi$
$48$	8272.62	0.518251
$49$	3647.81	0.217041
$50$	0	0
$51$	−10551.4	−0.568046
$52$	5914.20	0.303311
$53$	−30117.1	−1.47273	−0.736366	−	0.676584i	$-0.763460\pi$
$53$	−30117.1	−1.47273	−0.736366	+	0.676584i	$0.763460\pi$
$54$	−5966.91	−0.278462
$55$	0	0
$56$	−3506.34	−0.149412
$57$	−13831.3	−0.563867
$58$	−57251.7	−2.23470
$59$	−26017.9	−0.973067	−0.486533	−	0.873662i	$-0.661739\pi$
$59$	−26017.9	−0.973067	−0.486533	+	0.873662i	$0.661739\pi$
$60$	0	0
$61$	−11346.4	−0.390422	−0.195211	−	0.980761i	$-0.562539\pi$
$61$	−11346.4	−0.390422	−0.195211	+	0.980761i	$0.562539\pi$
$62$	5357.86	0.177016
$63$	−11584.6	−0.367732
$64$	−38588.3	−1.17762
$65$	0	0
$66$	−11518.1	−0.325478
$67$	3312.08	0.0901392	0.0450696	−	0.998984i	$-0.485649\pi$
$67$	3312.08	0.0901392	0.0450696	+	0.998984i	$0.485649\pi$
$68$	41027.5	1.07598
$69$	−26404.9	−0.667671
$70$	0	0
$71$	−23802.5	−0.560373	−0.280186	−	0.959946i	$-0.590396\pi$
$71$	−23802.5	−0.560373	−0.280186	+	0.959946i	$0.590396\pi$
$72$	1985.83	0.0451451
$73$	−72289.8	−1.58770	−0.793852	−	0.608111i	$-0.791928\pi$
$73$	−72289.8	−1.58770	−0.793852	+	0.608111i	$0.791928\pi$
$74$	−9554.37	−0.202826
$75$	0	0
$76$	53781.2	1.06806
$77$	−22362.2	−0.429821
$78$	−12449.5	−0.231694
$79$	−39681.2	−0.715348	−0.357674	−	0.933847i	$-0.616430\pi$
$79$	−39681.2	−0.715348	−0.357674	+	0.933847i	$0.616430\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	148318.	2.43589
$83$	−56017.1	−0.892535	−0.446268	−	0.894900i	$-0.647247\pi$
$83$	−56017.1	−0.892535	−0.446268	+	0.894900i	$0.647247\pi$
$84$	45045.3	0.696548
$85$	0	0
$86$	−50533.9	−0.736779
$87$	62951.9	0.891684
$88$	3833.31	0.0527675
$89$	133295.	1.78377	0.891884	−	0.452265i	$-0.149384\pi$
$89$	133295.	1.78377	0.891884	+	0.452265i	$0.149384\pi$
$90$	0	0
$91$	−24170.4	−0.305971
$92$	102672.	1.26468
$93$	−5891.31	−0.0706325
$94$	−8208.46	−0.0958168
$95$	0	0
$96$	74772.6	0.828065
$97$	−114564.	−1.23629	−0.618145	−	0.786064i	$-0.712115\pi$
$97$	−114564.	−1.23629	−0.618145	+	0.786064i	$0.712115\pi$
$98$	29857.6	0.314043
$99$	12664.9	0.129872
$100$	0	0
$101$	−158248.	−1.54360	−0.771800	−	0.635865i	$-0.780643\pi$
$101$	−158248.	−1.54360	−0.771800	+	0.635865i	$0.780643\pi$
$102$	−86363.6	−0.821922
$103$	−62879.2	−0.584001	−0.292001	−	0.956418i	$-0.594321\pi$
$103$	−62879.2	−0.584001	−0.292001	+	0.956418i	$0.594321\pi$
$104$	4143.27	0.0375630
$105$	0	0
$106$	−246510.	−2.13094
$107$	−37317.7	−0.315105	−0.157553	−	0.987511i	$-0.550360\pi$
$107$	−37317.7	−0.315105	−0.157553	+	0.987511i	$0.550360\pi$
$108$	−25511.5	−0.210464
$109$	193608.	1.56084	0.780418	−	0.625258i	$-0.215006\pi$
$109$	193608.	1.56084	0.780418	+	0.625258i	$0.215006\pi$
$110$	0	0
$111$	10505.6	0.0809310
$112$	131461.	0.990269
$113$	−86575.3	−0.637820	−0.318910	−	0.947785i	$-0.603317\pi$
$113$	−86575.3	−0.637820	−0.318910	+	0.947785i	$0.603317\pi$
$114$	−113210.	−0.815875
$115$	0	0
$116$	−244780.	−1.68900
$117$	13689.0	0.0924500
$118$	−212958.	−1.40796
$119$	−167673.	−1.08542
$120$	0	0
$121$	−136604.	−0.848200
$122$	−92871.1	−0.564913
$123$	−163085.	−0.971964
$124$	22907.5	0.133790
$125$	0	0
$126$	−94821.1	−0.532082
$127$	−231264.	−1.27233	−0.636163	−	0.771554i	$-0.719480\pi$
$127$	−231264.	−1.27233	−0.636163	+	0.771554i	$0.719480\pi$
$128$	−49989.6	−0.269684
$129$	55565.3	0.293988
$130$	0	0
$131$	251033.	1.27806	0.639032	−	0.769180i	$-0.279335\pi$
$131$	251033.	1.27806	0.639032	+	0.769180i	$0.279335\pi$
$132$	−49245.8	−0.245999
$133$	−219796.	−1.07743
$134$	27109.6	0.130425
$135$	0	0
$136$	28742.4	0.133253
$137$	185151.	0.842802	0.421401	−	0.906874i	$-0.361538\pi$
$137$	185151.	0.842802	0.421401	+	0.906874i	$0.361538\pi$
$138$	−216126.	−0.966071
$139$	−58662.0	−0.257525	−0.128763	−	0.991675i	$-0.541101\pi$
$139$	−58662.0	−0.257525	−0.128763	+	0.991675i	$0.541101\pi$
$140$	0	0
$141$	9025.72	0.0382326
$142$	−194825.	−0.810819
$143$	26424.3	0.108060
$144$	−74453.6	−0.299212
$145$	0	0
$146$	−591696.	−2.29729
$147$	−32830.3	−0.125309
$148$	−40849.7	−0.153297
$149$	−218175.	−0.805081	−0.402540	−	0.915402i	$-0.631873\pi$
$149$	−218175.	−0.805081	−0.402540	+	0.915402i	$0.631873\pi$
$150$	0	0
$151$	151474.	0.540626	0.270313	−	0.962773i	$-0.412873\pi$
$151$	151474.	0.540626	0.270313	+	0.962773i	$0.412873\pi$
$152$	37677.1	0.132272
$153$	94962.3	0.327961
$154$	−183036.	−0.621921
$155$	0	0
$156$	−53227.8	−0.175116
$157$	453507.	1.46837	0.734184	−	0.678950i	$-0.237565\pi$
$157$	453507.	1.46837	0.734184	+	0.678950i	$0.237565\pi$
$158$	−324793.	−1.03506
$159$	271054.	0.850282
$160$	0	0
$161$	−419605.	−1.27578
$162$	53702.2	0.160770
$163$	−383364.	−1.13017	−0.565083	−	0.825034i	$-0.691156\pi$
$163$	−383364.	−1.13017	−0.565083	+	0.825034i	$0.691156\pi$
$164$	634132.	1.84107
$165$	0	0
$166$	−458504.	−1.29143
$167$	452963.	1.25682	0.628408	−	0.777884i	$-0.283707\pi$
$167$	452963.	1.25682	0.628408	+	0.777884i	$0.283707\pi$
$168$	31557.1	0.0862628
$169$	28561.0	0.0769231
$170$	0	0
$171$	124482.	0.325549
$172$	−216058.	−0.556864
$173$	466668.	1.18548	0.592738	−	0.805396i	$-0.298047\pi$
$173$	466668.	1.18548	0.592738	+	0.805396i	$0.298047\pi$
$174$	515265.	1.29020
$175$	0	0
$176$	−143720.	−0.349733
$177$	234161.	0.561800
$178$	1.09103e6	2.58098
$179$	−131351.	−0.306409	−0.153204	−	0.988195i	$-0.548959\pi$
$179$	−131351.	−0.306409	−0.153204	+	0.988195i	$0.548959\pi$
$180$	0	0
$181$	−410272.	−0.930841	−0.465420	−	0.885090i	$-0.654097\pi$
$181$	−410272.	−0.930841	−0.465420	+	0.885090i	$0.654097\pi$
$182$	−197837.	−0.442719
$183$	102118.	0.225410
$184$	71928.2	0.156623
$185$	0	0
$186$	−48220.7	−0.102200
$187$	183309.	0.383336
$188$	−35095.3	−0.0724193
$189$	104262.	0.212310
$190$	0	0
$191$	−627101.	−1.24381	−0.621905	−	0.783093i	$-0.713641\pi$
$191$	−627101.	−1.24381	−0.621905	+	0.783093i	$0.713641\pi$
$192$	347295.	0.679901
$193$	−611648.	−1.18197	−0.590987	−	0.806681i	$-0.701262\pi$
$193$	−611648.	−1.18197	−0.590987	+	0.806681i	$0.701262\pi$
$194$	−937716.	−1.78882
$195$	0	0
$196$	127656.	0.237357
$197$	363322.	0.667001	0.333501	−	0.942750i	$-0.391770\pi$
$197$	363322.	0.667001	0.333501	+	0.942750i	$0.391770\pi$
$198$	103663.	0.187915
$199$	−222387.	−0.398086	−0.199043	−	0.979991i	$-0.563783\pi$
$199$	−222387.	−0.398086	−0.199043	+	0.979991i	$0.563783\pi$
$200$	0	0
$201$	−29808.7	−0.0520419
$202$	−1.29527e6	−2.23348
$203$	1.00038e6	1.70382
$204$	−369248.	−0.621216
$205$	0	0
$206$	−514670.	−0.845008
$207$	237644.	0.385480
$208$	−155341.	−0.248960
$209$	240291.	0.380516
$210$	0	0
$211$	168183.	0.260061	0.130031	−	0.991510i	$-0.458492\pi$
$211$	168183.	0.260061	0.130031	+	0.991510i	$0.458492\pi$
$212$	−1.05396e6	−1.61058
$213$	214223.	0.323531
$214$	−305448.	−0.455935
$215$	0	0
$216$	−17872.4	−0.0260645
$217$	−93619.6	−0.134964
$218$	1.58469e6	2.25842
$219$	650608.	0.916662
$220$	0	0
$221$	198131.	0.272880
$222$	85989.3	0.117101
$223$	−1.03260e6	−1.39049	−0.695246	−	0.718771i	$-0.744705\pi$
$223$	−1.03260e6	−1.39049	−0.695246	+	0.718771i	$0.744705\pi$
$224$	1.18822e6	1.58226
$225$	0	0
$226$	−708625.	−0.922880
$227$	−598593.	−0.771023	−0.385511	−	0.922703i	$-0.625975\pi$
$227$	−598593.	−0.771023	−0.385511	+	0.922703i	$0.625975\pi$
$228$	−484031.	−0.616646
$229$	1.19141e6	1.50132	0.750662	−	0.660687i	$-0.229735\pi$
$229$	1.19141e6	1.50132	0.750662	+	0.660687i	$0.229735\pi$
$230$	0	0
$231$	201260.	0.248157
$232$	−171484.	−0.209172
$233$	−606629.	−0.732037	−0.366018	−	0.930608i	$-0.619279\pi$
$233$	−606629.	−0.732037	−0.366018	+	0.930608i	$0.619279\pi$
$234$	112045.	0.133769
$235$	0	0
$236$	−910504.	−1.06415
$237$	357131.	0.413006
$238$	−1.37242e6	−1.57052
$239$	−143243.	−0.162211	−0.0811053	−	0.996706i	$-0.525845\pi$
$239$	−143243.	−0.162211	−0.0811053	+	0.996706i	$0.525845\pi$
$240$	0	0
$241$	721352.	0.800027	0.400013	−	0.916509i	$-0.369005\pi$
$241$	721352.	0.800027	0.400013	+	0.916509i	$0.369005\pi$
$242$	−1.11811e6	−1.22729
$243$	−59049.0	−0.0641500
$244$	−397071.	−0.426966
$245$	0	0
$246$	−1.33486e6	−1.40636
$247$	259721.	0.270873
$248$	16048.2	0.0165690
$249$	504154.	0.515306
$250$	0	0
$251$	495824.	0.496756	0.248378	−	0.968663i	$-0.420102\pi$
$251$	495824.	0.496756	0.248378	+	0.968663i	$0.420102\pi$
$252$	−405408.	−0.402152
$253$	458732.	0.450566
$254$	−1.89291e6	−1.84097
$255$	0	0
$256$	825658.	0.787409
$257$	−543343.	−0.513146	−0.256573	−	0.966525i	$-0.582593\pi$
$257$	−543343.	−0.513146	−0.256573	+	0.966525i	$0.582593\pi$
$258$	454806.	0.425379
$259$	166947.	0.154642
$260$	0	0
$261$	−566567.	−0.514814
$262$	2.05472e6	1.84927
$263$	−1.66097e6	−1.48071	−0.740357	−	0.672214i	$-0.765344\pi$
$263$	−1.66097e6	−1.48071	−0.740357	+	0.672214i	$0.765344\pi$
$264$	−34499.8	−0.0304654
$265$	0	0
$266$	−1.79904e6	−1.55897
$267$	−1.19965e6	−1.02986
$268$	115907.	0.0985764
$269$	−1.66042e6	−1.39907	−0.699533	−	0.714601i	$-0.746608\pi$
$269$	−1.66042e6	−1.39907	−0.699533	+	0.714601i	$0.746608\pi$
$270$	0	0
$271$	1.08347e6	0.896179	0.448089	−	0.893989i	$-0.352105\pi$
$271$	1.08347e6	0.896179	0.448089	+	0.893989i	$0.352105\pi$
$272$	−1.07762e6	−0.883171
$273$	217534.	0.176653
$274$	1.51548e6	1.21947
$275$	0	0
$276$	−924047.	−0.730166
$277$	−538273.	−0.421505	−0.210753	−	0.977539i	$-0.567591\pi$
$277$	−538273.	−0.421505	−0.210753	+	0.977539i	$0.567591\pi$
$278$	−480153.	−0.372621
$279$	53021.8	0.0407797
$280$	0	0
$281$	−948034.	−0.716239	−0.358120	−	0.933676i	$-0.616582\pi$
$281$	−948034.	−0.716239	−0.358120	+	0.933676i	$0.616582\pi$
$282$	73876.1	0.0553199
$283$	−1.88148e6	−1.39648	−0.698238	−	0.715865i	$-0.746032\pi$
$283$	−1.88148e6	−1.39648	−0.698238	+	0.715865i	$0.746032\pi$
$284$	−832975.	−0.612825
$285$	0	0
$286$	216285.	0.156355
$287$	−2.59160e6	−1.85722
$288$	−672954.	−0.478084
$289$	−45395.1	−0.0319716
$290$	0	0
$291$	1.03108e6	0.713772
$292$	−2.52980e6	−1.73632
$293$	−1805.69	−0.00122878	−0.000614391	−	1.00000i	$-0.500196\pi$
$293$	−1805.69	−0.00122878	−0.000614391		1.00000i	$0.500196\pi$
$294$	−268718.	−0.181313
$295$	0	0
$296$	−28617.8	−0.0189848
$297$	−113984.	−0.0749814
$298$	−1.78578e6	−1.16489
$299$	495826.	0.320739
$300$	0	0
$301$	882996.	0.561749
$302$	1.23983e6	0.782247
$303$	1.42423e6	0.891198
$304$	−1.41261e6	−0.876673
$305$	0	0
$306$	777273.	0.474537
$307$	−33716.5	−0.0204172	−0.0102086	−	0.999948i	$-0.503250\pi$
$307$	−33716.5	−0.0204172	−0.0102086	+	0.999948i	$0.503250\pi$
$308$	−782571.	−0.470053
$309$	565913.	0.337173
$310$	0	0
$311$	2.05265e6	1.20341	0.601706	−	0.798718i	$-0.294488\pi$
$311$	2.05265e6	1.20341	0.601706	+	0.798718i	$0.294488\pi$
$312$	−37289.4	−0.0216870
$313$	−1.86632e6	−1.07678	−0.538389	−	0.842697i	$-0.680967\pi$
$313$	−1.86632e6	−1.07678	−0.538389	+	0.842697i	$0.680967\pi$
$314$	3.71199e6	2.12462
$315$	0	0
$316$	−1.38865e6	−0.782305
$317$	2.01239e6	1.12477	0.562385	−	0.826876i	$-0.309884\pi$
$317$	2.01239e6	1.12477	0.562385	+	0.826876i	$0.309884\pi$
$318$	2.21859e6	1.23030
$319$	−1.09366e6	−0.601737
$320$	0	0
$321$	335859.	0.181926
$322$	−3.43449e6	−1.84596
$323$	1.80172e6	0.960907
$324$	229604.	0.121511
$325$	0	0
$326$	−3.13786e6	−1.63527
$327$	−1.74247e6	−0.901149
$328$	444250.	0.228004
$329$	143429.	0.0730546
$330$	0	0
$331$	−1.99627e6	−1.00150	−0.500748	−	0.865593i	$-0.666942\pi$
$331$	−1.99627e6	−1.00150	−0.500748	+	0.865593i	$0.666942\pi$
$332$	−1.96033e6	−0.976078
$333$	−94550.7	−0.0467255
$334$	3.70753e6	1.81852
$335$	0	0
$336$	−1.18315e6	−0.571732
$337$	−3.13833e6	−1.50530	−0.752652	−	0.658419i	$-0.771226\pi$
$337$	−3.13833e6	−1.50530	−0.752652	+	0.658419i	$0.771226\pi$
$338$	233774.	0.111302
$339$	779178.	0.368245
$340$	0	0
$341$	102350.	0.0476651
$342$	1.01889e6	0.471046
$343$	1.88203e6	0.863757
$344$	−151362.	−0.0689639
$345$	0	0
$346$	3.81970e6	1.71530
$347$	−2.20770e6	−0.984276	−0.492138	−	0.870517i	$-0.663784\pi$
$347$	−2.20770e6	−0.984276	−0.492138	+	0.870517i	$0.663784\pi$
$348$	2.20302e6	0.975147
$349$	311348.	0.136830	0.0684152	−	0.997657i	$-0.478206\pi$
$349$	311348.	0.136830	0.0684152	+	0.997657i	$0.478206\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	−1.29902e6	−0.558805
$353$	−1.58411e6	−0.676627	−0.338313	−	0.941034i	$-0.609856\pi$
$353$	−1.58411e6	−0.676627	−0.338313	+	0.941034i	$0.609856\pi$
$354$	1.91662e6	0.812885
$355$	0	0
$356$	4.66469e6	1.95073
$357$	1.50906e6	0.626666
$358$	−1.07512e6	−0.443352
$359$	2.08298e6	0.852999	0.426500	−	0.904488i	$-0.359747\pi$
$359$	2.08298e6	0.852999	0.426500	+	0.904488i	$0.359747\pi$
$360$	0	0
$361$	−114304.	−0.0461628
$362$	−3.35810e6	−1.34686
$363$	1.22943e6	0.489709
$364$	−845851.	−0.334611
$365$	0	0
$366$	835840.	0.326152
$367$	−870074.	−0.337203	−0.168601	−	0.985684i	$-0.553925\pi$
$367$	−870074.	−0.337203	−0.168601	+	0.985684i	$0.553925\pi$
$368$	−2.69676e6	−1.03806
$369$	1.46776e6	0.561164
$370$	0	0
$371$	4.30736e6	1.62471
$372$	−206168.	−0.0772438
$373$	258867.	0.0963395	0.0481697	−	0.998839i	$-0.484661\pi$
$373$	258867.	0.0963395	0.0481697	+	0.998839i	$0.484661\pi$
$374$	1.50039e6	0.554659
$375$	0	0
$376$	−24586.5	−0.00896863
$377$	−1.18210e6	−0.428351
$378$	853389.	0.307198
$379$	166708.	0.0596153	0.0298076	−	0.999556i	$-0.490511\pi$
$379$	166708.	0.0596153	0.0298076	+	0.999556i	$0.490511\pi$
$380$	0	0
$381$	2.08138e6	0.734578
$382$	−5.13286e6	−1.79970
$383$	2.76519e6	0.963227	0.481613	−	0.876384i	$-0.340051\pi$
$383$	2.76519e6	0.963227	0.481613	+	0.876384i	$0.340051\pi$
$384$	449907.	0.155702
$385$	0	0
$386$	−5.00638e6	−1.71023
$387$	−500088.	−0.169734
$388$	−4.00921e6	−1.35201
$389$	1.74605e6	0.585037	0.292518	−	0.956260i	$-0.405507\pi$
$389$	1.74605e6	0.585037	0.292518	+	0.956260i	$0.405507\pi$
$390$	0	0
$391$	3.43961e6	1.13780
$392$	89431.1	0.0293950
$393$	−2.25930e6	−0.737890
$394$	2.97382e6	0.965103
$395$	0	0
$396$	443212.	0.142028
$397$	−2.69673e6	−0.858740	−0.429370	−	0.903129i	$-0.641264\pi$
$397$	−2.69673e6	−0.858740	−0.429370	+	0.903129i	$0.641264\pi$
$398$	−1.82025e6	−0.576002
$399$	1.97816e6	0.622056
$400$	0	0
$401$	3.57209e6	1.10933	0.554666	−	0.832073i	$-0.312846\pi$
$401$	3.57209e6	1.10933	0.554666	+	0.832073i	$0.312846\pi$
$402$	−243986.	−0.0753009
$403$	110626.	0.0339307
$404$	−5.53793e6	−1.68808
$405$	0	0
$406$	8.18816e6	2.46531
$407$	−182514.	−0.0546149
$408$	−258682.	−0.0769334
$409$	−6.35701e6	−1.87908	−0.939538	−	0.342444i	$-0.888745\pi$
$409$	−6.35701e6	−1.87908	−0.939538	+	0.342444i	$0.888745\pi$
$410$	0	0
$411$	−1.66636e6	−0.486592
$412$	−2.20047e6	−0.638665
$413$	3.72109e6	1.07348
$414$	1.94513e6	0.557762
$415$	0	0
$416$	−1.40406e6	−0.397790
$417$	527958.	0.148682
$418$	1.96680e6	0.550579
$419$	6.67598e6	1.85772	0.928859	−	0.370432i	$-0.120790\pi$
$419$	6.67598e6	1.85772	0.928859	+	0.370432i	$0.120790\pi$
$420$	0	0
$421$	−2.03061e6	−0.558368	−0.279184	−	0.960238i	$-0.590064\pi$
$421$	−2.03061e6	−0.558368	−0.279184	+	0.960238i	$0.590064\pi$
$422$	1.37659e6	0.376290
$423$	−81231.5	−0.0220736
$424$	−738363.	−0.199460
$425$	0	0
$426$	1.75343e6	0.468127
$427$	1.62277e6	0.430712
$428$	−1.30594e6	−0.344600
$429$	−237819.	−0.0623883
$430$	0	0
$431$	−4.89190e6	−1.26848	−0.634240	−	0.773136i	$-0.718687\pi$
$431$	−4.89190e6	−1.26848	−0.634240	+	0.773136i	$0.718687\pi$
$432$	670082.	0.172750
$433$	1.43250e6	0.367176	0.183588	−	0.983003i	$-0.441229\pi$
$433$	1.43250e6	0.367176	0.183588	+	0.983003i	$0.441229\pi$
$434$	−766283.	−0.195283
$435$	0	0
$436$	6.77537e6	1.70693
$437$	4.50883e6	1.12943
$438$	5.32527e6	1.32634
$439$	4.36884e6	1.08194	0.540972	−	0.841041i	$-0.318056\pi$
$439$	4.36884e6	1.08194	0.540972	+	0.841041i	$0.318056\pi$
$440$	0	0
$441$	295473.	0.0723470
$442$	1.62172e6	0.394838
$443$	−1.45431e6	−0.352084	−0.176042	−	0.984383i	$-0.556329\pi$
$443$	−1.45431e6	−0.352084	−0.176042	+	0.984383i	$0.556329\pi$
$444$	367648.	0.0885063
$445$	0	0
$446$	−8.45187e6	−2.01194
$447$	1.96358e6	0.464814
$448$	5.51891e6	1.29915
$449$	1.53724e6	0.359854	0.179927	−	0.983680i	$-0.442414\pi$
$449$	1.53724e6	0.359854	0.179927	+	0.983680i	$0.442414\pi$
$450$	0	0
$451$	2.83327e6	0.655913
$452$	−3.02973e6	−0.697521
$453$	−1.36327e6	−0.312130
$454$	−4.89953e6	−1.11562
$455$	0	0
$456$	−339094.	−0.0763674
$457$	−3.68988e6	−0.826461	−0.413230	−	0.910627i	$-0.635600\pi$
$457$	−3.68988e6	−0.826461	−0.413230	+	0.910627i	$0.635600\pi$
$458$	9.75180e6	2.17231
$459$	−854661.	−0.189349
$460$	0	0
$461$	1.91355e6	0.419360	0.209680	−	0.977770i	$-0.432758\pi$
$461$	1.91355e6	0.419360	0.209680	+	0.977770i	$0.432758\pi$
$462$	1.64733e6	0.359066
$463$	−4.78828e6	−1.03807	−0.519036	−	0.854753i	$-0.673709\pi$
$463$	−4.78828e6	−1.03807	−0.519036	+	0.854753i	$0.673709\pi$
$464$	6.42935e6	1.38635
$465$	0	0
$466$	−4.96529e6	−1.05921
$467$	8.12853e6	1.72472	0.862362	−	0.506291i	$-0.168984\pi$
$467$	8.12853e6	1.72472	0.862362	+	0.506291i	$0.168984\pi$
$468$	479050.	0.101104
$469$	−473694.	−0.0994412
$470$	0	0
$471$	−4.08157e6	−0.847763
$472$	−637865.	−0.131787
$473$	−965335.	−0.198393
$474$	2.92314e6	0.597590
$475$	0	0
$476$	−5.86777e6	−1.18701
$477$	−2.43949e6	−0.490911
$478$	−1.17246e6	−0.234707
$479$	6.07022e6	1.20883	0.604415	−	0.796669i	$-0.293407\pi$
$479$	6.07022e6	1.20883	0.604415	+	0.796669i	$0.293407\pi$
$480$	0	0
$481$	−197273.	−0.0388780
$482$	5.90431e6	1.15758
$483$	3.77644e6	0.736571
$484$	−4.78048e6	−0.927594
$485$	0	0
$486$	−483320.	−0.0928205
$487$	5.88897e6	1.12517	0.562583	−	0.826741i	$-0.309808\pi$
$487$	5.88897e6	1.12517	0.562583	+	0.826741i	$0.309808\pi$
$488$	−278173.	−0.0528769
$489$	3.45027e6	0.652501
$490$	0	0
$491$	5.50454e6	1.03043	0.515214	−	0.857062i	$-0.327713\pi$
$491$	5.50454e6	1.03043	0.515214	+	0.857062i	$0.327713\pi$
$492$	−5.70719e6	−1.06294
$493$	−8.20036e6	−1.51955
$494$	2.12584e6	0.391934
$495$	0	0
$496$	−601686.	−0.109816
$497$	3.40424e6	0.618201
$498$	4.12653e6	0.745610
$499$	5.84569e6	1.05096	0.525478	−	0.850807i	$-0.323886\pi$
$499$	5.84569e6	1.05096	0.525478	+	0.850807i	$0.323886\pi$
$500$	0	0
$501$	−4.07667e6	−0.725623
$502$	4.05835e6	0.718771
$503$	−2.08903e6	−0.368149	−0.184075	−	0.982912i	$-0.558929\pi$
$503$	−2.08903e6	−0.368149	−0.184075	+	0.982912i	$0.558929\pi$
$504$	−284014.	−0.0498039
$505$	0	0
$506$	3.75475e6	0.651936
$507$	−257049.	−0.0444116
$508$	−8.09314e6	−1.39142
$509$	−3.78119e6	−0.646896	−0.323448	−	0.946246i	$-0.604842\pi$
$509$	−3.78119e6	−0.646896	−0.323448	+	0.946246i	$0.604842\pi$
$510$	0	0
$511$	1.03389e7	1.75155
$512$	8.35773e6	1.40901
$513$	−1.12034e6	−0.187956
$514$	−4.44730e6	−0.742486
$515$	0	0
$516$	1.94452e6	0.321506
$517$	−156804.	−0.0258006
$518$	1.36647e6	0.223756
$519$	−4.20001e6	−0.684434
$520$	0	0
$521$	−8.90077e6	−1.43659	−0.718296	−	0.695738i	$-0.755078\pi$
$521$	−8.90077e6	−1.43659	−0.718296	+	0.695738i	$0.755078\pi$
$522$	−4.63739e6	−0.744899
$523$	−879413.	−0.140585	−0.0702924	−	0.997526i	$-0.522393\pi$
$523$	−879413.	−0.140585	−0.0702924	+	0.997526i	$0.522393\pi$
$524$	8.78496e6	1.39769
$525$	0	0
$526$	−1.35951e7	−2.14249
$527$	767424.	0.120367
$528$	1.29348e6	0.201918
$529$	2.17131e6	0.337352
$530$	0	0
$531$	−2.10745e6	−0.324356
$532$	−7.69180e6	−1.17828
$533$	3.06237e6	0.466917
$534$	−9.81924e6	−1.49013
$535$	0	0
$536$	81200.2	0.0122080
$537$	1.18216e6	0.176905
$538$	−1.35907e7	−2.02435
$539$	570360.	0.0845624
$540$	0	0
$541$	8.75296e6	1.28577	0.642883	−	0.765965i	$-0.277738\pi$
$541$	8.75296e6	1.28577	0.642883	+	0.765965i	$0.277738\pi$
$542$	8.86829e6	1.29671
$543$	3.69245e6	0.537421
$544$	−9.74017e6	−1.41114
$545$	0	0
$546$	1.78053e6	0.255604
$547$	−4.44149e6	−0.634688	−0.317344	−	0.948310i	$-0.602791\pi$
$547$	−4.44149e6	−0.634688	−0.317344	+	0.948310i	$0.602791\pi$
$548$	6.47942e6	0.921690
$549$	−919060.	−0.130141
$550$	0	0
$551$	−1.07495e7	−1.50837
$552$	−647353.	−0.0904261
$553$	5.67522e6	0.789169
$554$	−4.40580e6	−0.609888
$555$	0	0
$556$	−2.05289e6	−0.281630
$557$	5.48849e6	0.749574	0.374787	−	0.927111i	$-0.377716\pi$
$557$	5.48849e6	0.749574	0.374787	+	0.927111i	$0.377716\pi$
$558$	433987.	0.0590053
$559$	−1.04339e6	−0.141227
$560$	0	0
$561$	−1.64978e6	−0.221319
$562$	−7.75972e6	−1.03635
$563$	1.15800e7	1.53971	0.769855	−	0.638219i	$-0.220329\pi$
$563$	1.15800e7	1.53971	0.769855	+	0.638219i	$0.220329\pi$
$564$	315857.	0.0418113
$565$	0	0
$566$	−1.54000e7	−2.02060
$567$	−938356.	−0.122577
$568$	−583552.	−0.0758942
$569$	−2.45605e6	−0.318022	−0.159011	−	0.987277i	$-0.550831\pi$
$569$	−2.45605e6	−0.318022	−0.159011	+	0.987277i	$0.550831\pi$
$570$	0	0
$571$	7.26803e6	0.932881	0.466441	−	0.884553i	$-0.345536\pi$
$571$	7.26803e6	0.932881	0.466441	+	0.884553i	$0.345536\pi$
$572$	924726.	0.118174
$573$	5.64391e6	0.718114
$574$	−2.12124e7	−2.68727
$575$	0	0
$576$	−3.12565e6	−0.392541
$577$	−3.53959e6	−0.442602	−0.221301	−	0.975206i	$-0.571030\pi$
$577$	−3.53959e6	−0.442602	−0.221301	+	0.975206i	$0.571030\pi$
$578$	−371562.	−0.0462606
$579$	5.50483e6	0.682413
$580$	0	0
$581$	8.01158e6	0.984641
$582$	8.43945e6	1.03278
$583$	−4.70902e6	−0.573798
$584$	−1.77228e6	−0.215031
$585$	0	0
$586$	−14779.7	−0.00177796
$587$	−6.77644e6	−0.811721	−0.405860	−	0.913935i	$-0.633028\pi$
$587$	−6.77644e6	−0.811721	−0.405860	+	0.913935i	$0.633028\pi$
$588$	−1.14890e6	−0.137038
$589$	1.00598e6	0.119482
$590$	0	0
$591$	−3.26990e6	−0.385093
$592$	1.07295e6	0.125828
$593$	−1.55084e7	−1.81105	−0.905523	−	0.424297i	$-0.860521\pi$
$593$	−1.55084e7	−1.81105	−0.905523	+	0.424297i	$0.860521\pi$
$594$	−932968.	−0.108493
$595$	0	0
$596$	−7.63509e6	−0.880438
$597$	2.00148e6	0.229835
$598$	4.05837e6	0.464086
$599$	4.07861e6	0.464457	0.232228	−	0.972661i	$-0.425398\pi$
$599$	4.07861e6	0.464457	0.232228	+	0.972661i	$0.425398\pi$
$600$	0	0
$601$	−1.55110e7	−1.75168	−0.875838	−	0.482606i	$-0.839690\pi$
$601$	−1.55110e7	−1.75168	−0.875838	+	0.482606i	$0.839690\pi$
$602$	7.22738e6	0.812811
$603$	268278.	0.0300464
$604$	5.30089e6	0.591229
$605$	0	0
$606$	1.16574e7	1.28950
$607$	1.02435e7	1.12844	0.564220	−	0.825624i	$-0.309177\pi$
$607$	1.02435e7	1.12844	0.564220	+	0.825624i	$0.309177\pi$
$608$	−1.27680e7	−1.40076
$609$	−9.00340e6	−0.983702
$610$	0	0
$611$	−169483.	−0.0183664
$612$	3.32323e6	0.358659
$613$	−8.82712e6	−0.948785	−0.474392	−	0.880314i	$-0.657332\pi$
$613$	−8.82712e6	−0.948785	−0.474392	+	0.880314i	$0.657332\pi$
$614$	−275972.	−0.0295423
$615$	0	0
$616$	−548241.	−0.0582129
$617$	4.01650e6	0.424751	0.212376	−	0.977188i	$-0.431880\pi$
$617$	4.01650e6	0.424751	0.212376	+	0.977188i	$0.431880\pi$
$618$	4.63203e6	0.487866
$619$	−1.28449e7	−1.34743	−0.673714	−	0.738992i	$-0.735302\pi$
$619$	−1.28449e7	−1.34743	−0.673714	+	0.738992i	$0.735302\pi$
$620$	0	0
$621$	−2.13880e6	−0.222557
$622$	1.68011e7	1.74125
$623$	−1.90639e7	−1.96784
$624$	1.39807e6	0.143737
$625$	0	0
$626$	−1.52760e7	−1.55802
$627$	−2.16262e6	−0.219691
$628$	1.58706e7	1.60581
$629$	−1.36850e6	−0.137918
$630$	0	0
$631$	1.29871e7	1.29849	0.649247	−	0.760578i	$-0.275084\pi$
$631$	1.29871e7	1.29849	0.649247	+	0.760578i	$0.275084\pi$
$632$	−972840.	−0.0968832
$633$	−1.51365e6	−0.150146
$634$	1.64715e7	1.62746
$635$	0	0
$636$	9.48560e6	0.929870
$637$	616480.	0.0601964
$638$	−8.95170e6	−0.870671
$639$	−1.92800e6	−0.186791
$640$	0	0
$641$	2.73171e6	0.262597	0.131298	−	0.991343i	$-0.458085\pi$
$641$	2.73171e6	0.262597	0.131298	+	0.991343i	$0.458085\pi$
$642$	2.74903e6	0.263234
$643$	−4.94400e6	−0.471575	−0.235788	−	0.971805i	$-0.575767\pi$
$643$	−4.94400e6	−0.471575	−0.235788	+	0.971805i	$0.575767\pi$
$644$	−1.46842e7	−1.39519
$645$	0	0
$646$	1.47472e7	1.39036
$647$	1.65285e7	1.55229	0.776144	−	0.630555i	$-0.217173\pi$
$647$	1.65285e7	1.55229	0.776144	+	0.630555i	$0.217173\pi$
$648$	160852.	0.0150484
$649$	−4.06808e6	−0.379121
$650$	0	0
$651$	842577.	0.0779214
$652$	−1.34159e7	−1.23595
$653$	−4.86572e6	−0.446544	−0.223272	−	0.974756i	$-0.571674\pi$
$653$	−4.86572e6	−0.446544	−0.223272	+	0.974756i	$0.571674\pi$
$654$	−1.42623e7	−1.30390
$655$	0	0
$656$	−1.66560e7	−1.51116
$657$	−5.85547e6	−0.529235
$658$	1.17398e6	0.105705
$659$	1.29753e7	1.16387	0.581934	−	0.813236i	$-0.302296\pi$
$659$	1.29753e7	1.16387	0.581934	+	0.813236i	$0.302296\pi$
$660$	0	0
$661$	1.33566e7	1.18902	0.594512	−	0.804086i	$-0.297345\pi$
$661$	1.33566e7	1.18902	0.594512	+	0.804086i	$0.297345\pi$
$662$	−1.63396e7	−1.44909
$663$	−1.78318e6	−0.157548
$664$	−1.37334e6	−0.120881
$665$	0	0
$666$	−773904.	−0.0676085
$667$	−2.05215e7	−1.78605
$668$	1.58516e7	1.37446
$669$	9.29337e6	0.802801
$670$	0	0
$671$	−1.77409e6	−0.152114
$672$	−1.06940e7	−0.913518
$673$	−3.79552e6	−0.323023	−0.161511	−	0.986871i	$-0.551637\pi$
$673$	−3.79552e6	−0.323023	−0.161511	+	0.986871i	$0.551637\pi$
$674$	−2.56874e7	−2.17807
$675$	0	0
$676$	999500.	0.0841232
$677$	1.31239e7	1.10051	0.550253	−	0.834998i	$-0.314531\pi$
$677$	1.31239e7	1.10051	0.550253	+	0.834998i	$0.314531\pi$
$678$	6.37762e6	0.532825
$679$	1.63850e7	1.36387
$680$	0	0
$681$	5.38734e6	0.445150
$682$	837738.	0.0689680
$683$	−1.07825e7	−0.884437	−0.442219	−	0.896907i	$-0.645808\pi$
$683$	−1.07825e7	−0.884437	−0.442219	+	0.896907i	$0.645808\pi$
$684$	4.35628e6	0.356021
$685$	0	0
$686$	1.54045e7	1.24979
$687$	−1.07227e7	−0.866789
$688$	5.67495e6	0.457078
$689$	−5.08979e6	−0.408462
$690$	0	0
$691$	1.62237e6	0.129257	0.0646285	−	0.997909i	$-0.479414\pi$
$691$	1.62237e6	0.129257	0.0646285	+	0.997909i	$0.479414\pi$
$692$	1.63312e7	1.29644
$693$	−1.81134e6	−0.143274
$694$	−1.80702e7	−1.42418
$695$	0	0
$696$	1.54335e6	0.120765
$697$	2.12440e7	1.65636
$698$	2.54840e6	0.197984
$699$	5.45966e6	0.422642
$700$	0	0
$701$	−5.04999e6	−0.388147	−0.194073	−	0.980987i	$-0.562170\pi$
$701$	−5.04999e6	−0.388147	−0.194073	+	0.980987i	$0.562170\pi$
$702$	−1.00841e6	−0.0772313
$703$	−1.79391e6	−0.136903
$704$	−6.03355e6	−0.458819
$705$	0	0
$706$	−1.29661e7	−0.979030
$707$	2.26327e7	1.70289
$708$	8.19453e6	0.614386
$709$	−9.81119e6	−0.733003	−0.366502	−	0.930417i	$-0.619445\pi$
$709$	−9.81119e6	−0.733003	−0.366502	+	0.930417i	$0.619445\pi$
$710$	0	0
$711$	−3.21418e6	−0.238449
$712$	3.26791e6	0.241585
$713$	1.92049e6	0.141478
$714$	1.23518e7	0.906741
$715$	0	0
$716$	−4.59667e6	−0.335089
$717$	1.28919e6	0.0936524
$718$	1.70493e7	1.23423
$719$	−1.90640e7	−1.37528	−0.687641	−	0.726051i	$-0.741354\pi$
$719$	−1.90640e7	−1.37528	−0.687641	+	0.726051i	$0.741354\pi$
$720$	0	0
$721$	8.99300e6	0.644268
$722$	−935582.	−0.0667942
$723$	−6.49217e6	−0.461896
$724$	−1.43576e7	−1.01797
$725$	0	0
$726$	1.00630e7	0.708573
$727$	2.87144e6	0.201495	0.100747	−	0.994912i	$-0.467877\pi$
$727$	2.87144e6	0.201495	0.100747	+	0.994912i	$0.467877\pi$
$728$	−592572.	−0.0414393
$729$	531441.	0.0370370
$730$	0	0
$731$	−7.23815e6	−0.500996
$732$	3.57364e6	0.246509
$733$	−2.15097e6	−0.147868	−0.0739339	−	0.997263i	$-0.523555\pi$
$733$	−2.15097e6	−0.147868	−0.0739339	+	0.997263i	$0.523555\pi$
$734$	−7.12161e6	−0.487908
$735$	0	0
$736$	−2.43749e7	−1.65862
$737$	517866.	0.0351195
$738$	1.20137e7	0.811964
$739$	−9.55920e6	−0.643888	−0.321944	−	0.946759i	$-0.604336\pi$
$739$	−9.55920e6	−0.643888	−0.321944	+	0.946759i	$0.604336\pi$
$740$	0	0
$741$	−2.33749e6	−0.156389
$742$	3.52560e7	2.35084
$743$	−680775.	−0.0452409	−0.0226205	−	0.999744i	$-0.507201\pi$
$743$	−680775.	−0.0452409	−0.0226205	+	0.999744i	$0.507201\pi$
$744$	−144434.	−0.00956612
$745$	0	0
$746$	2.11884e6	0.139396
$747$	−4.53739e6	−0.297512
$748$	6.41494e6	0.419217
$749$	5.33719e6	0.347623
$750$	0	0
$751$	7.29838e6	0.472200	0.236100	−	0.971729i	$-0.424131\pi$
$751$	7.29838e6	0.472200	0.236100	+	0.971729i	$0.424131\pi$
$752$	921807.	0.0594423
$753$	−4.46242e6	−0.286802
$754$	−9.67554e6	−0.619793
$755$	0	0
$756$	3.64867e6	0.232183
$757$	1.74116e7	1.10433	0.552165	−	0.833735i	$-0.313802\pi$
$757$	1.74116e7	1.10433	0.552165	+	0.833735i	$0.313802\pi$
$758$	1.36451e6	0.0862590
$759$	−4.12859e6	−0.260134
$760$	0	0
$761$	−2.59033e6	−0.162141	−0.0810707	−	0.996708i	$-0.525834\pi$
$761$	−2.59033e6	−0.162141	−0.0810707	+	0.996708i	$0.525834\pi$
$762$	1.70362e7	1.06288
$763$	−2.76899e7	−1.72191
$764$	−2.19456e7	−1.36023
$765$	0	0
$766$	2.26333e7	1.39372
$767$	−4.39703e6	−0.269880
$768$	−7.43093e6	−0.454611
$769$	−834051.	−0.0508600	−0.0254300	−	0.999677i	$-0.508095\pi$
$769$	−834051.	−0.0508600	−0.0254300	+	0.999677i	$0.508095\pi$
$770$	0	0
$771$	4.89009e6	0.296265
$772$	−2.14048e7	−1.29261
$773$	−1.63773e7	−0.985808	−0.492904	−	0.870084i	$-0.664065\pi$
$773$	−1.63773e7	−0.985808	−0.492904	+	0.870084i	$0.664065\pi$
$774$	−4.09325e6	−0.245593
$775$	0	0
$776$	−2.80870e6	−0.167437
$777$	−1.50252e6	−0.0892828
$778$	1.42915e7	0.846506
$779$	2.78479e7	1.64418
$780$	0	0
$781$	−3.72169e6	−0.218329
$782$	2.81534e7	1.64632
$783$	5.09911e6	0.297228
$784$	−3.35299e6	−0.194824
$785$	0	0
$786$	−1.84925e7	−1.06767
$787$	−1.34299e7	−0.772924	−0.386462	−	0.922305i	$-0.626303\pi$
$787$	−1.34299e7	−0.772924	−0.386462	+	0.922305i	$0.626303\pi$
$788$	1.27146e7	0.729434
$789$	1.49487e7	0.854891
$790$	0	0
$791$	1.23820e7	0.703640
$792$	310498.	0.0175892
$793$	−1.91754e6	−0.108284
$794$	−2.20729e7	−1.24253
$795$	0	0
$796$	−7.78250e6	−0.435348
$797$	−43136.0	−0.00240544	−0.00120272	−	0.999999i	$-0.500383\pi$
$797$	−43136.0	−0.00240544	−0.00120272	+	0.999999i	$0.500383\pi$
$798$	1.61914e7	0.900070
$799$	−1.17573e6	−0.0651537
$800$	0	0
$801$	1.07969e7	0.594589
$802$	2.92378e7	1.60512
$803$	−1.13030e7	−0.618593
$804$	−1.04316e6	−0.0569131
$805$	0	0
$806$	905478.	0.0490953
$807$	1.49438e7	0.807751
$808$	−3.87967e6	−0.209058
$809$	−433364.	−0.0232799	−0.0116400	−	0.999932i	$-0.503705\pi$
$809$	−433364.	−0.0232799	−0.0116400	+	0.999932i	$0.503705\pi$
$810$	0	0
$811$	572080.	0.0305425	0.0152712	−	0.999883i	$-0.495139\pi$
$811$	572080.	0.0305425	0.0152712	+	0.999883i	$0.495139\pi$
$812$	3.50085e7	1.86330
$813$	−9.75125e6	−0.517409
$814$	−1.49389e6	−0.0790238
$815$	0	0
$816$	9.69861e6	0.509899
$817$	−9.48817e6	−0.497310
$818$	−5.20325e7	−2.71889
$819$	−1.95781e6	−0.101990
$820$	0	0
$821$	−1.28251e7	−0.664055	−0.332028	−	0.943270i	$-0.607733\pi$
$821$	−1.28251e7	−0.664055	−0.332028	+	0.943270i	$0.607733\pi$
$822$	−1.36393e7	−0.704064
$823$	1.19649e7	0.615755	0.307878	−	0.951426i	$-0.400381\pi$
$823$	1.19649e7	0.615755	0.307878	+	0.951426i	$0.400381\pi$
$824$	−1.54157e6	−0.0790943
$825$	0	0
$826$	3.04574e7	1.55325
$827$	−1.44786e7	−0.736142	−0.368071	−	0.929798i	$-0.619982\pi$
$827$	−1.44786e7	−0.736142	−0.368071	+	0.929798i	$0.619982\pi$
$828$	8.31643e6	0.421561
$829$	−3.92219e6	−0.198218	−0.0991088	−	0.995077i	$-0.531599\pi$
$829$	−3.92219e6	−0.198218	−0.0991088	+	0.995077i	$0.531599\pi$
$830$	0	0
$831$	4.84445e6	0.243356
$832$	−6.52143e6	−0.326614
$833$	4.27660e6	0.213543
$834$	4.32137e6	0.215133
$835$	0	0
$836$	8.40906e6	0.416133
$837$	−477196.	−0.0235442
$838$	5.46433e7	2.68799
$839$	−3.30984e7	−1.62331	−0.811656	−	0.584136i	$-0.801433\pi$
$839$	−3.30984e7	−1.62331	−0.811656	+	0.584136i	$0.801433\pi$
$840$	0	0
$841$	2.84141e7	1.38530
$842$	−1.66206e7	−0.807919
$843$	8.53230e6	0.413521
$844$	5.88561e6	0.284404
$845$	0	0
$846$	−664885.	−0.0319389
$847$	1.95371e7	0.935731
$848$	2.76830e7	1.32198
$849$	1.69333e7	0.806256
$850$	0	0
$851$	−3.42470e6	−0.162106
$852$	7.49678e6	0.353814
$853$	−2.61505e7	−1.23057	−0.615287	−	0.788303i	$-0.710960\pi$
$853$	−2.61505e7	−1.23057	−0.615287	+	0.788303i	$0.710960\pi$
$854$	1.32825e7	0.623209
$855$	0	0
$856$	−914896.	−0.0426763
$857$	5.69689e6	0.264963	0.132482	−	0.991185i	$-0.457705\pi$
$857$	5.69689e6	0.264963	0.132482	+	0.991185i	$0.457705\pi$
$858$	−1.94656e6	−0.0902714
$859$	−6.71921e6	−0.310696	−0.155348	−	0.987860i	$-0.549650\pi$
$859$	−6.71921e6	−0.310696	−0.155348	+	0.987860i	$0.549650\pi$
$860$	0	0
$861$	2.33244e7	1.07227
$862$	−4.00405e7	−1.83540
$863$	−3.34678e6	−0.152968	−0.0764839	−	0.997071i	$-0.524369\pi$
$863$	−3.34678e6	−0.152968	−0.0764839	+	0.997071i	$0.524369\pi$
$864$	6.05658e6	0.276022
$865$	0	0
$866$	1.17251e7	0.531278
$867$	408556.	0.0184588
$868$	−3.27624e6	−0.147597
$869$	−6.20443e6	−0.278710
$870$	0	0
$871$	559741.	0.0250001
$872$	4.74657e6	0.211392
$873$	−9.27971e6	−0.412096
$874$	3.69050e7	1.63421
$875$	0	0
$876$	2.27682e7	1.00246
$877$	4.49406e7	1.97306	0.986530	−	0.163581i	$-0.0523044\pi$
$877$	4.49406e7	1.97306	0.986530	+	0.163581i	$0.0523044\pi$
$878$	3.57592e7	1.56550
$879$	16251.2	0.000709438 0
$880$	0	0
$881$	−1.50813e7	−0.654633	−0.327317	−	0.944915i	$-0.606144\pi$
$881$	−1.50813e7	−0.654633	−0.327317	+	0.944915i	$0.606144\pi$
$882$	2.41846e6	0.104681
$883$	2.76226e7	1.19224	0.596120	−	0.802896i	$-0.296708\pi$
$883$	2.76226e7	1.19224	0.596120	+	0.802896i	$0.296708\pi$
$884$	6.93366e6	0.298423
$885$	0	0
$886$	−1.19036e7	−0.509440
$887$	3.02405e7	1.29056	0.645282	−	0.763945i	$-0.276740\pi$
$887$	3.02405e7	1.29056	0.645282	+	0.763945i	$0.276740\pi$
$888$	257560.	0.0109609
$889$	3.30754e7	1.40363
$890$	0	0
$891$	1.02586e6	0.0432905
$892$	−3.61360e7	−1.52065
$893$	−1.54121e6	−0.0646743
$894$	1.60720e7	0.672552
$895$	0	0
$896$	7.14953e6	0.297514
$897$	−4.46243e6	−0.185178
$898$	1.25824e7	0.520683
$899$	−4.57863e6	−0.188945
$900$	0	0
$901$	−3.53085e7	−1.44900
$902$	2.31905e7	0.949059
$903$	−7.94697e6	−0.324326
$904$	−2.12251e6	−0.0863833
$905$	0	0
$906$	−1.11584e7	−0.451630
$907$	1.03037e7	0.415885	0.207943	−	0.978141i	$-0.433323\pi$
$907$	1.03037e7	0.415885	0.207943	+	0.978141i	$0.433323\pi$
$908$	−2.09479e7	−0.843192
$909$	−1.28181e7	−0.514534
$910$	0	0
$911$	1.45719e7	0.581728	0.290864	−	0.956764i	$-0.406057\pi$
$911$	1.45719e7	0.581728	0.290864	+	0.956764i	$0.406057\pi$
$912$	1.27135e7	0.506148
$913$	−8.75866e6	−0.347745
$914$	−3.02019e7	−1.19583
$915$	0	0
$916$	4.16939e7	1.64185
$917$	−3.59028e7	−1.40995
$918$	−6.99545e6	−0.273974
$919$	4.11548e7	1.60743	0.803715	−	0.595014i	$-0.202854\pi$
$919$	4.11548e7	1.60743	0.803715	+	0.595014i	$0.202854\pi$
$920$	0	0
$921$	303449.	0.0117879
$922$	1.56625e7	0.606785
$923$	−4.02263e6	−0.155419
$924$	7.04314e6	0.271385
$925$	0	0
$926$	−3.91924e7	−1.50201
$927$	−5.09321e6	−0.194667
$928$	5.81121e7	2.21512
$929$	−1.32533e7	−0.503830	−0.251915	−	0.967749i	$-0.581060\pi$
$929$	−1.32533e7	−0.503830	−0.251915	+	0.967749i	$0.581060\pi$
$930$	0	0
$931$	5.60600e6	0.211972
$932$	−2.12291e7	−0.800557
$933$	−1.84739e7	−0.694790
$934$	6.65325e7	2.49555
$935$	0	0
$936$	335605.	0.0125210
$937$	−3.67751e7	−1.36837	−0.684186	−	0.729307i	$-0.739842\pi$
$937$	−3.67751e7	−1.36837	−0.684186	+	0.729307i	$0.739842\pi$
$938$	−3.87722e6	−0.143884
$939$	1.67969e7	0.621678
$940$	0	0
$941$	2.09808e7	0.772409	0.386205	−	0.922413i	$-0.373786\pi$
$941$	2.09808e7	0.772409	0.386205	+	0.922413i	$0.373786\pi$
$942$	−3.34079e7	−1.22665
$943$	5.31634e7	1.94686
$944$	2.39152e7	0.873460
$945$	0	0
$946$	−7.90133e6	−0.287060
$947$	−1.94868e7	−0.706097	−0.353049	−	0.935605i	$-0.614855\pi$
$947$	−1.94868e7	−0.706097	−0.353049	+	0.935605i	$0.614855\pi$
$948$	1.24979e7	0.451664
$949$	−1.22170e7	−0.440350
$950$	0	0
$951$	−1.81115e7	−0.649386
$952$	−4.11075e6	−0.147004
$953$	1.15965e6	0.0413612	0.0206806	−	0.999786i	$-0.493417\pi$
$953$	1.15965e6	0.0413612	0.0206806	+	0.999786i	$0.493417\pi$
$954$	−1.99673e7	−0.710312
$955$	0	0
$956$	−5.01284e6	−0.177394
$957$	9.84297e6	0.347413
$958$	4.96851e7	1.74909
$959$	−2.64804e7	−0.929776
$960$	0	0
$961$	−2.82007e7	−0.985033
$962$	−1.61469e6	−0.0562537
$963$	−3.02274e6	−0.105035
$964$	2.52439e7	0.874911
$965$	0	0
$966$	3.09104e7	1.06577
$967$	−432431.	−0.0148714	−0.00743569	−	0.999972i	$-0.502367\pi$
$967$	−432431.	−0.0148714	−0.00743569	+	0.999972i	$0.502367\pi$
$968$	−3.34903e6	−0.114876
$969$	−1.62155e7	−0.554780
$970$	0	0
$971$	4.89492e7	1.66609	0.833043	−	0.553208i	$-0.186597\pi$
$971$	4.89492e7	1.66609	0.833043	+	0.553208i	$0.186597\pi$
$972$	−2.06644e6	−0.0701546
$973$	8.38986e6	0.284101
$974$	4.82016e7	1.62804
$975$	0	0
$976$	1.04294e7	0.350457
$977$	−5.20849e7	−1.74572	−0.872861	−	0.487969i	$-0.837738\pi$
$977$	−5.20849e7	−1.74572	−0.872861	+	0.487969i	$0.837738\pi$
$978$	2.82407e7	0.944123
$979$	2.08416e7	0.694982
$980$	0	0
$981$	1.56823e7	0.520279
$982$	4.50550e7	1.49095
$983$	−5.63533e7	−1.86010	−0.930049	−	0.367435i	$-0.880236\pi$
$983$	−5.63533e7	−1.86010	−0.930049	+	0.367435i	$0.880236\pi$
$984$	−3.99825e6	−0.131638
$985$	0	0
$986$	−6.71204e7	−2.19868
$987$	−1.29086e6	−0.0421781
$988$	9.08902e6	0.296227
$989$	−1.81136e7	−0.588861
$990$	0	0
$991$	4.29172e7	1.38818	0.694092	−	0.719886i	$-0.255806\pi$
$991$	4.29172e7	1.38818	0.694092	+	0.719886i	$0.255806\pi$
$992$	−5.43838e6	−0.175465
$993$	1.79664e7	0.578214
$994$	2.78639e7	0.894492
$995$	0	0
$996$	1.76430e7	0.563539
$997$	2.54188e7	0.809874	0.404937	−	0.914345i	$-0.367293\pi$
$997$	2.54188e7	0.809874	0.404937	+	0.914345i	$0.367293\pi$
$998$	4.78474e7	1.52066
$999$	850957.	0.0269770

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.6.a.u.1.9	yes	11
5.4	even	2		975.6.a.r.1.3	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.6.a.r.1.3	✓	11	5.4	even	2
975.6.a.u.1.9	yes	11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

\(f(q)\)	\(=\)	\(q+8.18506 q^{2} -9.00000 q^{3} +34.9953 q^{4} -73.6656 q^{6} -143.020 q^{7} +24.5164 q^{8} +81.0000 q^{9} +156.357 q^{11} -314.957 q^{12} +169.000 q^{13} -1170.63 q^{14} -919.180 q^{16} +1172.37 q^{17} +662.990 q^{18} +1536.81 q^{19} +1287.18 q^{21} +1279.79 q^{22} +2933.88 q^{23} -220.648 q^{24} +1383.28 q^{26} -729.000 q^{27} -5005.03 q^{28} -6994.66 q^{29} +654.590 q^{31} -8308.07 q^{32} -1407.21 q^{33} +9595.96 q^{34} +2834.62 q^{36} -1167.29 q^{37} +12578.9 q^{38} -1521.00 q^{39} +18120.5 q^{41} +10535.7 q^{42} -6173.92 q^{43} +5471.75 q^{44} +24014.0 q^{46} -1002.86 q^{47} +8272.62 q^{48} +3647.81 q^{49} -10551.4 q^{51} +5914.20 q^{52} -30117.1 q^{53} -5966.91 q^{54} -3506.34 q^{56} -13831.3 q^{57} -57251.7 q^{58} -26017.9 q^{59} -11346.4 q^{61} +5357.86 q^{62} -11584.6 q^{63} -38588.3 q^{64} -11518.1 q^{66} +3312.08 q^{67} +41027.5 q^{68} -26404.9 q^{69} -23802.5 q^{71} +1985.83 q^{72} -72289.8 q^{73} -9554.37 q^{74} +53781.2 q^{76} -22362.2 q^{77} -12449.5 q^{78} -39681.2 q^{79} +6561.00 q^{81} +148318. q^{82} -56017.1 q^{83} +45045.3 q^{84} -50533.9 q^{86} +62951.9 q^{87} +3833.31 q^{88} +133295. q^{89} -24170.4 q^{91} +102672. q^{92} -5891.31 q^{93} -8208.46 q^{94} +74772.6 q^{96} -114564. q^{97} +29857.6 q^{98} +12664.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} - 99 q^{3} + 224 q^{4} - 18 q^{6} - 55 q^{7} + 270 q^{8} + 891 q^{9} - 125 q^{11} - 2016 q^{12} + 1859 q^{13} - 1311 q^{14} + 5756 q^{16} - 4507 q^{17} + 162 q^{18} + 142 q^{19} + 495 q^{21}+ \cdots - 10125 q^{99}+O(q^{100}) \) 11 * q + 2 * q^2 - 99 * q^3 + 224 * q^4 - 18 * q^6 - 55 * q^7 + 270 * q^8 + 891 * q^9 - 125 * q^11 - 2016 * q^12 + 1859 * q^13 - 1311 * q^14 + 5756 * q^16 - 4507 * q^17 + 162 * q^18 + 142 * q^19 + 495 * q^21 - 5975 * q^22 - 4672 * q^23 - 2430 * q^24 + 338 * q^26 - 8019 * q^27 - 4149 * q^28 + 5197 * q^29 - 277 * q^31 + 2272 * q^32 + 1125 * q^33 - 21225 * q^34 + 18144 * q^36 - 1696 * q^37 - 27485 * q^38 - 16731 * q^39 + 2040 * q^41 + 11799 * q^42 - 31436 * q^43 - 39883 * q^44 + 67530 * q^46 + 15499 * q^47 - 51804 * q^48 + 29212 * q^49 + 40563 * q^51 + 37856 * q^52 - 119577 * q^53 - 1458 * q^54 + 1677 * q^56 - 1278 * q^57 - 91154 * q^58 + 17115 * q^59 + 26273 * q^61 - 45211 * q^62 - 4455 * q^63 + 137480 * q^64 + 53775 * q^66 + 26313 * q^67 - 191297 * q^68 + 42048 * q^69 - 105756 * q^71 + 21870 * q^72 - 103230 * q^73 - 6501 * q^74 + 124495 * q^76 - 104415 * q^77 - 3042 * q^78 + 210732 * q^79 + 72171 * q^81 - 60283 * q^82 - 144247 * q^83 + 37341 * q^84 - 42260 * q^86 - 46773 * q^87 - 415403 * q^88 + 117088 * q^89 - 9295 * q^91 - 70700 * q^92 + 2493 * q^93 + 383414 * q^94 - 20448 * q^96 - 112934 * q^97 - 159305 * q^98 - 10125 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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