LMFDB - Embedded newform 1575.4.a.bu.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 67x^{8} + 1523x^{6} - 13569x^{4} + 36944x^{2} - 256 $ x^10 - 67x^8 + 1523x^6 - 13569x^4 + 36944x^2 - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.44617$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.44617 q^{2} +21.6607 q^{4} +7.00000 q^{7} -74.3987 q^{8} +O(q^{10})$	$q-5.44617 q^{2} +21.6607 q^{4} +7.00000 q^{7} -74.3987 q^{8} -60.4870 q^{11} +64.0698 q^{13} -38.1232 q^{14} +231.902 q^{16} +37.7444 q^{17} -134.779 q^{19} +329.422 q^{22} +52.6223 q^{23} -348.935 q^{26} +151.625 q^{28} -165.252 q^{29} -71.9304 q^{31} -667.787 q^{32} -205.563 q^{34} +48.7993 q^{37} +734.031 q^{38} +10.5566 q^{41} -425.906 q^{43} -1310.19 q^{44} -286.590 q^{46} +249.258 q^{47} +49.0000 q^{49} +1387.80 q^{52} -544.007 q^{53} -520.791 q^{56} +899.988 q^{58} +567.390 q^{59} +614.549 q^{61} +391.745 q^{62} +1781.66 q^{64} +201.820 q^{67} +817.573 q^{68} +525.114 q^{71} +137.016 q^{73} -265.769 q^{74} -2919.42 q^{76} -423.409 q^{77} -234.355 q^{79} -57.4929 q^{82} -10.3089 q^{83} +2319.56 q^{86} +4500.15 q^{88} +20.9202 q^{89} +448.488 q^{91} +1139.84 q^{92} -1357.50 q^{94} +1388.80 q^{97} -266.862 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 54 q^{4} + 70 q^{7}+O(q^{10}) $ 10 * q + 54 * q^4 + 70 * q^7	$ 10 q + 54 q^{4} + 70 q^{7} + 104 q^{13} + 310 q^{16} + 36 q^{19} + 644 q^{22} + 378 q^{28} - 24 q^{31} + 116 q^{34} + 732 q^{37} + 212 q^{43} - 184 q^{46} + 490 q^{49} + 2692 q^{52} + 3196 q^{58} + 1024 q^{61} + 1590 q^{64} + 2388 q^{67} + 3432 q^{73} - 4184 q^{76} - 776 q^{79} + 1860 q^{82} + 10164 q^{88} + 728 q^{91} - 4028 q^{94} + 4024 q^{97}+O(q^{100}) $ 10 * q + 54 * q^4 + 70 * q^7 + 104 * q^13 + 310 * q^16 + 36 * q^19 + 644 * q^22 + 378 * q^28 - 24 * q^31 + 116 * q^34 + 732 * q^37 + 212 * q^43 - 184 * q^46 + 490 * q^49 + 2692 * q^52 + 3196 * q^58 + 1024 * q^61 + 1590 * q^64 + 2388 * q^67 + 3432 * q^73 - 4184 * q^76 - 776 * q^79 + 1860 * q^82 + 10164 * q^88 + 728 * q^91 - 4028 * q^94 + 4024 * q^97

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 $

$2$	−5.44617	−1.92551	−0.962756	−	0.270373i	$-0.912853\pi$
$2$	−5.44617	−1.92551	−0.962756	+	0.270373i	$0.912853\pi$
$3$	0	0
$3$	0	0
$4$	21.6607	2.70759
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−74.3987	−3.28799
$9$	0	0
$10$	0	0
$11$	−60.4870	−1.65796	−0.828978	−	0.559281i	$-0.811077\pi$
$11$	−60.4870	−1.65796	−0.828978	+	0.559281i	$0.811077\pi$
$12$	0	0
$13$	64.0698	1.36690	0.683452	−	0.729995i	$-0.260478\pi$
$13$	64.0698	1.36690	0.683452	+	0.729995i	$0.260478\pi$
$14$	−38.1232	−0.727775
$15$	0	0
$16$	231.902	3.62347
$17$	37.7444	0.538492	0.269246	−	0.963071i	$-0.413225\pi$
$17$	37.7444	0.538492	0.269246	+	0.963071i	$0.413225\pi$
$18$	0	0
$19$	−134.779	−1.62739	−0.813697	−	0.581289i	$-0.802549\pi$
$19$	−134.779	−1.62739	−0.813697	+	0.581289i	$0.802549\pi$
$20$	0	0
$21$	0	0
$22$	329.422	3.19241
$23$	52.6223	0.477065	0.238533	−	0.971134i	$-0.423334\pi$
$23$	52.6223	0.477065	0.238533	+	0.971134i	$0.423334\pi$
$24$	0	0
$25$	0	0
$26$	−348.935	−2.63199
$27$	0	0
$28$	151.625	1.02337
$29$	−165.252	−1.05815	−0.529077	−	0.848574i	$-0.677462\pi$
$29$	−165.252	−1.05815	−0.529077	+	0.848574i	$0.677462\pi$
$30$	0	0
$31$	−71.9304	−0.416745	−0.208372	−	0.978050i	$-0.566817\pi$
$31$	−71.9304	−0.416745	−0.208372	+	0.978050i	$0.566817\pi$
$32$	−667.787	−3.68904
$33$	0	0
$34$	−205.563	−1.03687
$35$	0	0
$36$	0	0
$37$	48.7993	0.216826	0.108413	−	0.994106i	$-0.465423\pi$
$37$	48.7993	0.216826	0.108413	+	0.994106i	$0.465423\pi$
$38$	734.031	3.13357
$39$	0	0
$40$	0	0
$41$	10.5566	0.0402113	0.0201056	−	0.999798i	$-0.493600\pi$
$41$	10.5566	0.0402113	0.0201056	+	0.999798i	$0.493600\pi$
$42$	0	0
$43$	−425.906	−1.51047	−0.755233	−	0.655456i	$-0.772476\pi$
$43$	−425.906	−1.51047	−0.755233	+	0.655456i	$0.772476\pi$
$44$	−1310.19	−4.48907
$45$	0	0
$46$	−286.590	−0.918594
$47$	249.258	0.773575	0.386787	−	0.922169i	$-0.373585\pi$
$47$	249.258	0.773575	0.386787	+	0.922169i	$0.373585\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	1387.80	3.70102
$53$	−544.007	−1.40991	−0.704953	−	0.709254i	$-0.749032\pi$
$53$	−544.007	−1.40991	−0.704953	+	0.709254i	$0.749032\pi$
$54$	0	0
$55$	0	0
$56$	−520.791	−1.24274
$57$	0	0
$58$	899.988	2.03749
$59$	567.390	1.25200	0.625999	−	0.779824i	$-0.284691\pi$
$59$	567.390	1.25200	0.625999	+	0.779824i	$0.284691\pi$
$60$	0	0
$61$	614.549	1.28992	0.644958	−	0.764218i	$-0.276875\pi$
$61$	614.549	1.28992	0.644958	+	0.764218i	$0.276875\pi$
$62$	391.745	0.802447
$63$	0	0
$64$	1781.66	3.47981
$65$	0	0
$66$	0	0
$67$	201.820	0.368004	0.184002	−	0.982926i	$-0.441095\pi$
$67$	201.820	0.368004	0.184002	+	0.982926i	$0.441095\pi$
$68$	817.573	1.45802
$69$	0	0
$70$	0	0
$71$	525.114	0.877741	0.438871	−	0.898550i	$-0.355379\pi$
$71$	525.114	0.877741	0.438871	+	0.898550i	$0.355379\pi$
$72$	0	0
$73$	137.016	0.219679	0.109839	−	0.993949i	$-0.464966\pi$
$73$	137.016	0.219679	0.109839	+	0.993949i	$0.464966\pi$
$74$	−265.769	−0.417500
$75$	0	0
$76$	−2919.42	−4.40632
$77$	−423.409	−0.626649
$78$	0	0
$79$	−234.355	−0.333759	−0.166880	−	0.985977i	$-0.553369\pi$
$79$	−234.355	−0.333759	−0.166880	+	0.985977i	$0.553369\pi$
$80$	0	0
$81$	0	0
$82$	−57.4929	−0.0774272
$83$	−10.3089	−0.0136332	−0.00681659	−	0.999977i	$-0.502170\pi$
$83$	−10.3089	−0.0136332	−0.00681659	+	0.999977i	$0.502170\pi$
$84$	0	0
$85$	0	0
$86$	2319.56	2.90842
$87$	0	0
$88$	4500.15	5.45134
$89$	20.9202	0.0249162	0.0124581	−	0.999922i	$-0.496034\pi$
$89$	20.9202	0.0249162	0.0124581	+	0.999922i	$0.496034\pi$
$90$	0	0
$91$	448.488	0.516641
$92$	1139.84	1.29170
$93$	0	0
$94$	−1357.50	−1.48953
$95$	0	0
$96$	0	0
$97$	1388.80	1.45373	0.726864	−	0.686781i	$-0.240977\pi$
$97$	1388.80	1.45373	0.726864	+	0.686781i	$0.240977\pi$
$98$	−266.862	−0.275073
$99$	0	0
$100$	0	0
$101$	−95.4423	−0.0940283	−0.0470142	−	0.998894i	$-0.514971\pi$
$101$	−95.4423	−0.0940283	−0.0470142	+	0.998894i	$0.514971\pi$
$102$	0	0
$103$	−368.235	−0.352265	−0.176132	−	0.984366i	$-0.556359\pi$
$103$	−368.235	−0.352265	−0.176132	+	0.984366i	$0.556359\pi$
$104$	−4766.71	−4.49437
$105$	0	0
$106$	2962.75	2.71479
$107$	−623.980	−0.563761	−0.281881	−	0.959449i	$-0.590958\pi$
$107$	−623.980	−0.563761	−0.281881	+	0.959449i	$0.590958\pi$
$108$	0	0
$109$	−381.664	−0.335384	−0.167692	−	0.985839i	$-0.553631\pi$
$109$	−381.664	−0.335384	−0.167692	+	0.985839i	$0.553631\pi$
$110$	0	0
$111$	0	0
$112$	1623.31	1.36954
$113$	1189.96	0.990637	0.495318	−	0.868711i	$-0.335051\pi$
$113$	1189.96	0.990637	0.495318	+	0.868711i	$0.335051\pi$
$114$	0	0
$115$	0	0
$116$	−3579.47	−2.86505
$117$	0	0
$118$	−3090.10	−2.41074
$119$	264.211	0.203531
$120$	0	0
$121$	2327.68	1.74882
$122$	−3346.94	−2.48375
$123$	0	0
$124$	−1558.07	−1.12838
$125$	0	0
$126$	0	0
$127$	−1681.29	−1.17472	−0.587362	−	0.809324i	$-0.699834\pi$
$127$	−1681.29	−1.17472	−0.587362	+	0.809324i	$0.699834\pi$
$128$	−4360.95	−3.01138
$129$	0	0
$130$	0	0
$131$	−41.0906	−0.0274053	−0.0137027	−	0.999906i	$-0.504362\pi$
$131$	−41.0906	−0.0274053	−0.0137027	+	0.999906i	$0.504362\pi$
$132$	0	0
$133$	−943.455	−0.615097
$134$	−1099.15	−0.708596
$135$	0	0
$136$	−2808.14	−1.77056
$137$	−1265.83	−0.789394	−0.394697	−	0.918811i	$-0.629150\pi$
$137$	−1265.83	−0.789394	−0.394697	+	0.918811i	$0.629150\pi$
$138$	0	0
$139$	−576.135	−0.351562	−0.175781	−	0.984429i	$-0.556245\pi$
$139$	−576.135	−0.351562	−0.175781	+	0.984429i	$0.556245\pi$
$140$	0	0
$141$	0	0
$142$	−2859.86	−1.69010
$143$	−3875.39	−2.26627
$144$	0	0
$145$	0	0
$146$	−746.214	−0.422994
$147$	0	0
$148$	1057.03	0.587076
$149$	−1588.69	−0.873496	−0.436748	−	0.899584i	$-0.643870\pi$
$149$	−1588.69	−0.873496	−0.436748	+	0.899584i	$0.643870\pi$
$150$	0	0
$151$	1317.30	0.709935	0.354968	−	0.934879i	$-0.384492\pi$
$151$	1317.30	0.709935	0.354968	+	0.934879i	$0.384492\pi$
$152$	10027.4	5.35086
$153$	0	0
$154$	2305.96	1.20662
$155$	0	0
$156$	0	0
$157$	1803.58	0.916826	0.458413	−	0.888739i	$-0.348418\pi$
$157$	1803.58	0.916826	0.458413	+	0.888739i	$0.348418\pi$
$158$	1276.34	0.642657
$159$	0	0
$160$	0	0
$161$	368.356	0.180314
$162$	0	0
$163$	−324.426	−0.155896	−0.0779479	−	0.996957i	$-0.524837\pi$
$163$	−324.426	−0.155896	−0.0779479	+	0.996957i	$0.524837\pi$
$164$	228.663	0.108876
$165$	0	0
$166$	56.1443	0.0262508
$167$	3644.85	1.68891	0.844453	−	0.535630i	$-0.179926\pi$
$167$	3644.85	1.68891	0.844453	+	0.535630i	$0.179926\pi$
$168$	0	0
$169$	1907.94	0.868428
$170$	0	0
$171$	0	0
$172$	−9225.44	−4.08973
$173$	−193.361	−0.0849765	−0.0424882	−	0.999097i	$-0.513529\pi$
$173$	−193.361	−0.0849765	−0.0424882	+	0.999097i	$0.513529\pi$
$174$	0	0
$175$	0	0
$176$	−14027.0	−6.00755
$177$	0	0
$178$	−113.935	−0.0479764
$179$	−3498.06	−1.46066	−0.730328	−	0.683096i	$-0.760633\pi$
$179$	−3498.06	−1.46066	−0.730328	+	0.683096i	$0.760633\pi$
$180$	0	0
$181$	−590.623	−0.242545	−0.121273	−	0.992619i	$-0.538698\pi$
$181$	−590.623	−0.242545	−0.121273	+	0.992619i	$0.538698\pi$
$182$	−2442.54	−0.994799
$183$	0	0
$184$	−3915.03	−1.56858
$185$	0	0
$186$	0	0
$187$	−2283.05	−0.892797
$188$	5399.11	2.09452
$189$	0	0
$190$	0	0
$191$	−1398.87	−0.529940	−0.264970	−	0.964257i	$-0.585362\pi$
$191$	−1398.87	−0.529940	−0.264970	+	0.964257i	$0.585362\pi$
$192$	0	0
$193$	−529.184	−0.197365	−0.0986826	−	0.995119i	$-0.531463\pi$
$193$	−529.184	−0.197365	−0.0986826	+	0.995119i	$0.531463\pi$
$194$	−7563.66	−2.79917
$195$	0	0
$196$	1061.38	0.386799
$197$	3841.24	1.38922	0.694612	−	0.719385i	$-0.255576\pi$
$197$	3841.24	1.38922	0.694612	+	0.719385i	$0.255576\pi$
$198$	0	0
$199$	−3052.58	−1.08740	−0.543698	−	0.839281i	$-0.682976\pi$
$199$	−3052.58	−1.08740	−0.543698	+	0.839281i	$0.682976\pi$
$200$	0	0
$201$	0	0
$202$	519.795	0.181053
$203$	−1156.76	−0.399945
$204$	0	0
$205$	0	0
$206$	2005.47	0.678289
$207$	0	0
$208$	14857.9	4.95293
$209$	8152.40	2.69815
$210$	0	0
$211$	−3340.97	−1.09006	−0.545028	−	0.838418i	$-0.683481\pi$
$211$	−3340.97	−1.09006	−0.545028	+	0.838418i	$0.683481\pi$
$212$	−11783.6	−3.81745
$213$	0	0
$214$	3398.30	1.08553
$215$	0	0
$216$	0	0
$217$	−503.513	−0.157515
$218$	2078.61	0.645785
$219$	0	0
$220$	0	0
$221$	2418.28	0.736068
$222$	0	0
$223$	5475.60	1.64428	0.822138	−	0.569289i	$-0.192781\pi$
$223$	5475.60	1.64428	0.822138	+	0.569289i	$0.192781\pi$
$224$	−4674.51	−1.39432
$225$	0	0
$226$	−6480.72	−1.90748
$227$	−527.894	−0.154350	−0.0771752	−	0.997018i	$-0.524590\pi$
$227$	−527.894	−0.154350	−0.0771752	+	0.997018i	$0.524590\pi$
$228$	0	0
$229$	1038.38	0.299642	0.149821	−	0.988713i	$-0.452130\pi$
$229$	1038.38	0.299642	0.149821	+	0.988713i	$0.452130\pi$
$230$	0	0
$231$	0	0
$232$	12294.5	3.47920
$233$	1854.95	0.521554	0.260777	−	0.965399i	$-0.416021\pi$
$233$	1854.95	0.521554	0.260777	+	0.965399i	$0.416021\pi$
$234$	0	0
$235$	0	0
$236$	12290.1	3.38990
$237$	0	0
$238$	−1438.94	−0.391901
$239$	4071.39	1.10191	0.550955	−	0.834535i	$-0.314264\pi$
$239$	4071.39	1.10191	0.550955	+	0.834535i	$0.314264\pi$
$240$	0	0
$241$	−5734.52	−1.53275	−0.766375	−	0.642394i	$-0.777941\pi$
$241$	−5734.52	−1.53275	−0.766375	+	0.642394i	$0.777941\pi$
$242$	−12676.9	−3.36737
$243$	0	0
$244$	13311.6	3.49257
$245$	0	0
$246$	0	0
$247$	−8635.28	−2.22449
$248$	5351.53	1.37025
$249$	0	0
$250$	0	0
$251$	4371.18	1.09923	0.549614	−	0.835419i	$-0.314775\pi$
$251$	4371.18	1.09923	0.549614	+	0.835419i	$0.314775\pi$
$252$	0	0
$253$	−3182.96	−0.790953
$254$	9156.57	2.26195
$255$	0	0
$256$	9497.13	2.31864
$257$	6667.99	1.61844	0.809218	−	0.587509i	$-0.199891\pi$
$257$	6667.99	1.61844	0.809218	+	0.587509i	$0.199891\pi$
$258$	0	0
$259$	341.595	0.0819524
$260$	0	0
$261$	0	0
$262$	223.786	0.0527693
$263$	−4668.18	−1.09450	−0.547248	−	0.836970i	$-0.684325\pi$
$263$	−4668.18	−1.09450	−0.547248	+	0.836970i	$0.684325\pi$
$264$	0	0
$265$	0	0
$266$	5138.22	1.18438
$267$	0	0
$268$	4371.58	0.996406
$269$	−153.845	−0.0348701	−0.0174351	−	0.999848i	$-0.505550\pi$
$269$	−153.845	−0.0348701	−0.0174351	+	0.999848i	$0.505550\pi$
$270$	0	0
$271$	−3774.11	−0.845982	−0.422991	−	0.906134i	$-0.639020\pi$
$271$	−3774.11	−0.845982	−0.422991	+	0.906134i	$0.639020\pi$
$272$	8753.01	1.95121
$273$	0	0
$274$	6893.91	1.51999
$275$	0	0
$276$	0	0
$277$	7464.31	1.61909	0.809544	−	0.587060i	$-0.199715\pi$
$277$	7464.31	1.61909	0.809544	+	0.587060i	$0.199715\pi$
$278$	3137.73	0.676937
$279$	0	0
$280$	0	0
$281$	5596.80	1.18818	0.594088	−	0.804400i	$-0.297513\pi$
$281$	5596.80	1.18818	0.594088	+	0.804400i	$0.297513\pi$
$282$	0	0
$283$	−8132.88	−1.70830	−0.854151	−	0.520025i	$-0.825923\pi$
$283$	−8132.88	−1.70830	−0.854151	+	0.520025i	$0.825923\pi$
$284$	11374.4	2.37657
$285$	0	0
$286$	21106.0	4.36372
$287$	73.8961	0.0151984
$288$	0	0
$289$	−3488.36	−0.710026
$290$	0	0
$291$	0	0
$292$	2967.88	0.594801
$293$	4134.83	0.824434	0.412217	−	0.911086i	$-0.364755\pi$
$293$	4134.83	0.824434	0.412217	+	0.911086i	$0.364755\pi$
$294$	0	0
$295$	0	0
$296$	−3630.60	−0.712921
$297$	0	0
$298$	8652.30	1.68193
$299$	3371.50	0.652102
$300$	0	0
$301$	−2981.34	−0.570903
$302$	−7174.23	−1.36699
$303$	0	0
$304$	−31255.6	−5.89681
$305$	0	0
$306$	0	0
$307$	9663.06	1.79642	0.898209	−	0.439569i	$-0.144869\pi$
$307$	9663.06	1.79642	0.898209	+	0.439569i	$0.144869\pi$
$308$	−9171.36	−1.69671
$309$	0	0
$310$	0	0
$311$	7930.53	1.44598	0.722989	−	0.690860i	$-0.242768\pi$
$311$	7930.53	1.44598	0.722989	+	0.690860i	$0.242768\pi$
$312$	0	0
$313$	1247.21	0.225228	0.112614	−	0.993639i	$-0.464078\pi$
$313$	1247.21	0.225228	0.112614	+	0.993639i	$0.464078\pi$
$314$	−9822.63	−1.76536
$315$	0	0
$316$	−5076.30	−0.903684
$317$	−2884.20	−0.511019	−0.255510	−	0.966807i	$-0.582243\pi$
$317$	−2884.20	−0.511019	−0.255510	+	0.966807i	$0.582243\pi$
$318$	0	0
$319$	9995.58	1.75437
$320$	0	0
$321$	0	0
$322$	−2006.13	−0.347196
$323$	−5087.17	−0.876340
$324$	0	0
$325$	0	0
$326$	1766.88	0.300179
$327$	0	0
$328$	−785.396	−0.132214
$329$	1744.81	0.292384
$330$	0	0
$331$	−5668.46	−0.941289	−0.470645	−	0.882323i	$-0.655979\pi$
$331$	−5668.46	−0.941289	−0.470645	+	0.882323i	$0.655979\pi$
$332$	−223.299	−0.0369131
$333$	0	0
$334$	−19850.5	−3.25201
$335$	0	0
$336$	0	0
$337$	−8556.34	−1.38307	−0.691534	−	0.722344i	$-0.743065\pi$
$337$	−8556.34	−1.38307	−0.691534	+	0.722344i	$0.743065\pi$
$338$	−10390.9	−1.67217
$339$	0	0
$340$	0	0
$341$	4350.86	0.690945
$342$	0	0
$343$	343.000	0.0539949
$344$	31686.9	4.96640
$345$	0	0
$346$	1053.07	0.163623
$347$	6794.69	1.05118	0.525588	−	0.850739i	$-0.323845\pi$
$347$	6794.69	1.05118	0.525588	+	0.850739i	$0.323845\pi$
$348$	0	0
$349$	8922.38	1.36849	0.684246	−	0.729251i	$-0.260131\pi$
$349$	8922.38	1.36849	0.684246	+	0.729251i	$0.260131\pi$
$350$	0	0
$351$	0	0
$352$	40392.4	6.11626
$353$	8668.14	1.30696	0.653482	−	0.756942i	$-0.273307\pi$
$353$	8668.14	1.30696	0.653482	+	0.756942i	$0.273307\pi$
$354$	0	0
$355$	0	0
$356$	453.148	0.0674629
$357$	0	0
$358$	19051.0	2.81251
$359$	3084.48	0.453462	0.226731	−	0.973957i	$-0.427196\pi$
$359$	3084.48	0.453462	0.226731	+	0.973957i	$0.427196\pi$
$360$	0	0
$361$	11306.5	1.64841
$362$	3216.63	0.467023
$363$	0	0
$364$	9714.59	1.39885
$365$	0	0
$366$	0	0
$367$	7926.72	1.12744	0.563721	−	0.825965i	$-0.309369\pi$
$367$	7926.72	1.12744	0.563721	+	0.825965i	$0.309369\pi$
$368$	12203.2	1.72863
$369$	0	0
$370$	0	0
$371$	−3808.05	−0.532895
$372$	0	0
$373$	9724.91	1.34996	0.674982	−	0.737834i	$-0.264151\pi$
$373$	9724.91	1.34996	0.674982	+	0.737834i	$0.264151\pi$
$374$	12433.9	1.71909
$375$	0	0
$376$	−18544.5	−2.54350
$377$	−10587.6	−1.44640
$378$	0	0
$379$	−793.694	−0.107571	−0.0537854	−	0.998553i	$-0.517129\pi$
$379$	−793.694	−0.107571	−0.0537854	+	0.998553i	$0.517129\pi$
$380$	0	0
$381$	0	0
$382$	7618.46	1.02040
$383$	−2732.63	−0.364571	−0.182286	−	0.983246i	$-0.558350\pi$
$383$	−2732.63	−0.364571	−0.182286	+	0.983246i	$0.558350\pi$
$384$	0	0
$385$	0	0
$386$	2882.02	0.380029
$387$	0	0
$388$	30082.5	3.93610
$389$	−11011.1	−1.43518	−0.717588	−	0.696468i	$-0.754754\pi$
$389$	−11011.1	−1.43518	−0.717588	+	0.696468i	$0.754754\pi$
$390$	0	0
$391$	1986.20	0.256896
$392$	−3645.54	−0.469713
$393$	0	0
$394$	−20920.0	−2.67497
$395$	0	0
$396$	0	0
$397$	9688.40	1.22480	0.612402	−	0.790547i	$-0.290204\pi$
$397$	9688.40	1.22480	0.612402	+	0.790547i	$0.290204\pi$
$398$	16624.9	2.09379
$399$	0	0
$400$	0	0
$401$	7442.12	0.926788	0.463394	−	0.886152i	$-0.346632\pi$
$401$	7442.12	0.926788	0.463394	+	0.886152i	$0.346632\pi$
$402$	0	0
$403$	−4608.57	−0.569650
$404$	−2067.35	−0.254590
$405$	0	0
$406$	6299.92	0.770098
$407$	−2951.72	−0.359488
$408$	0	0
$409$	−10839.1	−1.31042	−0.655208	−	0.755449i	$-0.727419\pi$
$409$	−10839.1	−1.31042	−0.655208	+	0.755449i	$0.727419\pi$
$410$	0	0
$411$	0	0
$412$	−7976.24	−0.953789
$413$	3971.73	0.473211
$414$	0	0
$415$	0	0
$416$	−42784.9	−5.04256
$417$	0	0
$418$	−44399.3	−5.19532
$419$	475.629	0.0554558	0.0277279	−	0.999616i	$-0.491173\pi$
$419$	475.629	0.0554558	0.0277279	+	0.999616i	$0.491173\pi$
$420$	0	0
$421$	6879.42	0.796396	0.398198	−	0.917300i	$-0.369636\pi$
$421$	6879.42	0.796396	0.398198	+	0.917300i	$0.369636\pi$
$422$	18195.5	2.09891
$423$	0	0
$424$	40473.4	4.63576
$425$	0	0
$426$	0	0
$427$	4301.84	0.487543
$428$	−13515.9	−1.52644
$429$	0	0
$430$	0	0
$431$	3993.61	0.446324	0.223162	−	0.974781i	$-0.428362\pi$
$431$	3993.61	0.446324	0.223162	+	0.974781i	$0.428362\pi$
$432$	0	0
$433$	13362.5	1.48306	0.741528	−	0.670922i	$-0.234102\pi$
$433$	13362.5	1.48306	0.741528	+	0.670922i	$0.234102\pi$
$434$	2742.22	0.303296
$435$	0	0
$436$	−8267.13	−0.908082
$437$	−7092.39	−0.776373
$438$	0	0
$439$	16047.5	1.74466	0.872330	−	0.488918i	$-0.162608\pi$
$439$	16047.5	1.74466	0.872330	+	0.488918i	$0.162608\pi$
$440$	0	0
$441$	0	0
$442$	−13170.3	−1.41731
$443$	−16538.8	−1.77378	−0.886888	−	0.461985i	$-0.847137\pi$
$443$	−16538.8	−1.77378	−0.886888	+	0.461985i	$0.847137\pi$
$444$	0	0
$445$	0	0
$446$	−29821.0	−3.16607
$447$	0	0
$448$	12471.6	1.31525
$449$	2790.61	0.293312	0.146656	−	0.989188i	$-0.453149\pi$
$449$	2790.61	0.293312	0.146656	+	0.989188i	$0.453149\pi$
$450$	0	0
$451$	−638.536	−0.0666685
$452$	25775.4	2.68224
$453$	0	0
$454$	2875.00	0.297204
$455$	0	0
$456$	0	0
$457$	−8003.62	−0.819242	−0.409621	−	0.912256i	$-0.634339\pi$
$457$	−8003.62	−0.819242	−0.409621	+	0.912256i	$0.634339\pi$
$458$	−5655.18	−0.576964
$459$	0	0
$460$	0	0
$461$	11025.5	1.11390	0.556948	−	0.830547i	$-0.311972\pi$
$461$	11025.5	1.11390	0.556948	+	0.830547i	$0.311972\pi$
$462$	0	0
$463$	−539.353	−0.0541379	−0.0270689	−	0.999634i	$-0.508617\pi$
$463$	−539.353	−0.0541379	−0.0270689	+	0.999634i	$0.508617\pi$
$464$	−38322.2	−3.83418
$465$	0	0
$466$	−10102.4	−1.00426
$467$	−10642.7	−1.05457	−0.527286	−	0.849688i	$-0.676790\pi$
$467$	−10642.7	−1.05457	−0.527286	+	0.849688i	$0.676790\pi$
$468$	0	0
$469$	1412.74	0.139093
$470$	0	0
$471$	0	0
$472$	−42213.1	−4.11656
$473$	25761.8	2.50429
$474$	0	0
$475$	0	0
$476$	5723.01	0.551079
$477$	0	0
$478$	−22173.5	−2.12174
$479$	12976.4	1.23780	0.618899	−	0.785471i	$-0.287579\pi$
$479$	12976.4	1.23780	0.618899	+	0.785471i	$0.287579\pi$
$480$	0	0
$481$	3126.56	0.296380
$482$	31231.1	2.95133
$483$	0	0
$484$	50419.3	4.73509
$485$	0	0
$486$	0	0
$487$	12739.8	1.18541	0.592705	−	0.805419i	$-0.298060\pi$
$487$	12739.8	1.18541	0.592705	+	0.805419i	$0.298060\pi$
$488$	−45721.6	−4.24123
$489$	0	0
$490$	0	0
$491$	7951.24	0.730824	0.365412	−	0.930846i	$-0.380928\pi$
$491$	7951.24	0.730824	0.365412	+	0.930846i	$0.380928\pi$
$492$	0	0
$493$	−6237.33	−0.569808
$494$	47029.2	4.28329
$495$	0	0
$496$	−16680.8	−1.51006
$497$	3675.80	0.331755
$498$	0	0
$499$	−13883.2	−1.24549	−0.622744	−	0.782426i	$-0.713982\pi$
$499$	−13883.2	−1.24549	−0.622744	+	0.782426i	$0.713982\pi$
$500$	0	0
$501$	0	0
$502$	−23806.2	−2.11658
$503$	3388.74	0.300391	0.150195	−	0.988656i	$-0.452010\pi$
$503$	3388.74	0.300391	0.150195	+	0.988656i	$0.452010\pi$
$504$	0	0
$505$	0	0
$506$	17335.0	1.52299
$507$	0	0
$508$	−36417.9	−3.18068
$509$	20440.9	1.78001	0.890005	−	0.455950i	$-0.150700\pi$
$509$	20440.9	1.78001	0.890005	+	0.455950i	$0.150700\pi$
$510$	0	0
$511$	959.114	0.0830308
$512$	−16835.4	−1.45318
$513$	0	0
$514$	−36315.0	−3.11632
$515$	0	0
$516$	0	0
$517$	−15076.9	−1.28255
$518$	−1860.38	−0.157800
$519$	0	0
$520$	0	0
$521$	2391.88	0.201133	0.100566	−	0.994930i	$-0.467935\pi$
$521$	2391.88	0.201133	0.100566	+	0.994930i	$0.467935\pi$
$522$	0	0
$523$	−4839.38	−0.404611	−0.202305	−	0.979322i	$-0.564843\pi$
$523$	−4839.38	−0.404611	−0.202305	+	0.979322i	$0.564843\pi$
$524$	−890.052	−0.0742025
$525$	0	0
$526$	25423.7	2.10747
$527$	−2714.97	−0.224414
$528$	0	0
$529$	−9397.90	−0.772409
$530$	0	0
$531$	0	0
$532$	−20435.9	−1.66543
$533$	676.358	0.0549649
$534$	0	0
$535$	0	0
$536$	−15015.2	−1.20999
$537$	0	0
$538$	837.863	0.0671429
$539$	−2963.86	−0.236851
$540$	0	0
$541$	21183.6	1.68347	0.841733	−	0.539894i	$-0.181536\pi$
$541$	21183.6	1.68347	0.841733	+	0.539894i	$0.181536\pi$
$542$	20554.4	1.62895
$543$	0	0
$544$	−25205.2	−1.98652
$545$	0	0
$546$	0	0
$547$	3180.06	0.248574	0.124287	−	0.992246i	$-0.460336\pi$
$547$	3180.06	0.248574	0.124287	+	0.992246i	$0.460336\pi$
$548$	−27418.8	−2.13736
$549$	0	0
$550$	0	0
$551$	22272.5	1.72203
$552$	0	0
$553$	−1640.48	−0.126149
$554$	−40651.9	−3.11757
$555$	0	0
$556$	−12479.5	−0.951888
$557$	−1196.88	−0.0910476	−0.0455238	−	0.998963i	$-0.514496\pi$
$557$	−1196.88	−0.0910476	−0.0455238	+	0.998963i	$0.514496\pi$
$558$	0	0
$559$	−27287.7	−2.06466
$560$	0	0
$561$	0	0
$562$	−30481.1	−2.28784
$563$	−2800.80	−0.209662	−0.104831	−	0.994490i	$-0.533430\pi$
$563$	−2800.80	−0.209662	−0.104831	+	0.994490i	$0.533430\pi$
$564$	0	0
$565$	0	0
$566$	44293.0	3.28935
$567$	0	0
$568$	−39067.8	−2.88600
$569$	−12783.5	−0.941849	−0.470924	−	0.882174i	$-0.656080\pi$
$569$	−12783.5	−0.941849	−0.470924	+	0.882174i	$0.656080\pi$
$570$	0	0
$571$	−12490.6	−0.915439	−0.457719	−	0.889097i	$-0.651334\pi$
$571$	−12490.6	−0.915439	−0.457719	+	0.889097i	$0.651334\pi$
$572$	−83943.8	−6.13613
$573$	0	0
$574$	−402.450	−0.0292647
$575$	0	0
$576$	0	0
$577$	19884.0	1.43463	0.717317	−	0.696747i	$-0.245370\pi$
$577$	19884.0	1.43463	0.717317	+	0.696747i	$0.245370\pi$
$578$	18998.2	1.36716
$579$	0	0
$580$	0	0
$581$	−72.1626	−0.00515286
$582$	0	0
$583$	32905.3	2.33756
$584$	−10193.8	−0.722301
$585$	0	0
$586$	−22519.0	−1.58746
$587$	−586.249	−0.0412216	−0.0206108	−	0.999788i	$-0.506561\pi$
$587$	−586.249	−0.0412216	−0.0206108	+	0.999788i	$0.506561\pi$
$588$	0	0
$589$	9694.74	0.678208
$590$	0	0
$591$	0	0
$592$	11316.6	0.785661
$593$	7891.38	0.546476	0.273238	−	0.961946i	$-0.411905\pi$
$593$	7891.38	0.546476	0.273238	+	0.961946i	$0.411905\pi$
$594$	0	0
$595$	0	0
$596$	−34412.3	−2.36507
$597$	0	0
$598$	−18361.7	−1.25563
$599$	−8924.75	−0.608774	−0.304387	−	0.952549i	$-0.598452\pi$
$599$	−8924.75	−0.608774	−0.304387	+	0.952549i	$0.598452\pi$
$600$	0	0
$601$	−14545.7	−0.987241	−0.493620	−	0.869677i	$-0.664327\pi$
$601$	−14545.7	−0.987241	−0.493620	+	0.869677i	$0.664327\pi$
$602$	16236.9	1.09928
$603$	0	0
$604$	28533.7	1.92222
$605$	0	0
$606$	0	0
$607$	20253.5	1.35431	0.677154	−	0.735841i	$-0.263213\pi$
$607$	20253.5	1.35431	0.677154	+	0.735841i	$0.263213\pi$
$608$	90003.9	6.00352
$609$	0	0
$610$	0	0
$611$	15969.9	1.05740
$612$	0	0
$613$	−14610.9	−0.962686	−0.481343	−	0.876532i	$-0.659851\pi$
$613$	−14610.9	−0.962686	−0.481343	+	0.876532i	$0.659851\pi$
$614$	−52626.7	−3.45902
$615$	0	0
$616$	31501.1	2.06041
$617$	−15791.8	−1.03039	−0.515197	−	0.857072i	$-0.672281\pi$
$617$	−15791.8	−1.03039	−0.515197	+	0.857072i	$0.672281\pi$
$618$	0	0
$619$	561.212	0.0364410	0.0182205	−	0.999834i	$-0.494200\pi$
$619$	561.212	0.0364410	0.0182205	+	0.999834i	$0.494200\pi$
$620$	0	0
$621$	0	0
$622$	−43191.0	−2.78425
$623$	146.442	0.00941743
$624$	0	0
$625$	0	0
$626$	−6792.50	−0.433679
$627$	0	0
$628$	39067.0	2.48239
$629$	1841.90	0.116759
$630$	0	0
$631$	16360.8	1.03219	0.516096	−	0.856531i	$-0.327385\pi$
$631$	16360.8	1.03219	0.516096	+	0.856531i	$0.327385\pi$
$632$	17435.7	1.09740
$633$	0	0
$634$	15707.9	0.983973
$635$	0	0
$636$	0	0
$637$	3139.42	0.195272
$638$	−54437.6	−3.37806
$639$	0	0
$640$	0	0
$641$	30298.1	1.86693	0.933465	−	0.358667i	$-0.116769\pi$
$641$	30298.1	1.86693	0.933465	+	0.358667i	$0.116769\pi$
$642$	0	0
$643$	121.613	0.00745871	0.00372935	−	0.999993i	$-0.498813\pi$
$643$	121.613	0.00745871	0.00372935	+	0.999993i	$0.498813\pi$
$644$	7978.86	0.488216
$645$	0	0
$646$	27705.6	1.68740
$647$	17068.6	1.03715	0.518574	−	0.855033i	$-0.326463\pi$
$647$	17068.6	1.03715	0.518574	+	0.855033i	$0.326463\pi$
$648$	0	0
$649$	−34319.7	−2.07576
$650$	0	0
$651$	0	0
$652$	−7027.31	−0.422102
$653$	−2667.60	−0.159864	−0.0799319	−	0.996800i	$-0.525470\pi$
$653$	−2667.60	−0.159864	−0.0799319	+	0.996800i	$0.525470\pi$
$654$	0	0
$655$	0	0
$656$	2448.09	0.145704
$657$	0	0
$658$	−9502.50	−0.562988
$659$	11943.1	0.705976	0.352988	−	0.935628i	$-0.385166\pi$
$659$	11943.1	0.705976	0.352988	+	0.935628i	$0.385166\pi$
$660$	0	0
$661$	−21867.4	−1.28675	−0.643375	−	0.765551i	$-0.722466\pi$
$661$	−21867.4	−1.28675	−0.643375	+	0.765551i	$0.722466\pi$
$662$	30871.4	1.81246
$663$	0	0
$664$	766.972	0.0448258
$665$	0	0
$666$	0	0
$667$	−8695.91	−0.504808
$668$	78950.2	4.57287
$669$	0	0
$670$	0	0
$671$	−37172.2	−2.13863
$672$	0	0
$673$	480.080	0.0274974	0.0137487	−	0.999905i	$-0.495624\pi$
$673$	480.080	0.0274974	0.0137487	+	0.999905i	$0.495624\pi$
$674$	46599.3	2.66311
$675$	0	0
$676$	41327.3	2.35135
$677$	−28054.8	−1.59267	−0.796333	−	0.604858i	$-0.793230\pi$
$677$	−28054.8	−1.59267	−0.796333	+	0.604858i	$0.793230\pi$
$678$	0	0
$679$	9721.62	0.549458
$680$	0	0
$681$	0	0
$682$	−23695.5	−1.33042
$683$	−15552.3	−0.871291	−0.435645	−	0.900118i	$-0.643480\pi$
$683$	−15552.3	−0.871291	−0.435645	+	0.900118i	$0.643480\pi$
$684$	0	0
$685$	0	0
$686$	−1868.04	−0.103968
$687$	0	0
$688$	−98768.4	−5.47312
$689$	−34854.4	−1.92721
$690$	0	0
$691$	20777.6	1.14387	0.571936	−	0.820298i	$-0.306193\pi$
$691$	20777.6	1.14387	0.571936	+	0.820298i	$0.306193\pi$
$692$	−4188.33	−0.230082
$693$	0	0
$694$	−37005.0	−2.02405
$695$	0	0
$696$	0	0
$697$	398.452	0.0216535
$698$	−48592.8	−2.63505
$699$	0	0
$700$	0	0
$701$	2132.81	0.114915	0.0574573	−	0.998348i	$-0.481701\pi$
$701$	2132.81	0.114915	0.0574573	+	0.998348i	$0.481701\pi$
$702$	0	0
$703$	−6577.14	−0.352861
$704$	−107768.	−5.76938
$705$	0	0
$706$	−47208.1	−2.51657
$707$	−668.096	−0.0355394
$708$	0	0
$709$	−2908.28	−0.154052	−0.0770260	−	0.997029i	$-0.524542\pi$
$709$	−2908.28	−0.154052	−0.0770260	+	0.997029i	$0.524542\pi$
$710$	0	0
$711$	0	0
$712$	−1556.44	−0.0819241
$713$	−3785.14	−0.198814
$714$	0	0
$715$	0	0
$716$	−75770.6	−3.95486
$717$	0	0
$718$	−16798.6	−0.873145
$719$	12241.1	0.634931	0.317466	−	0.948270i	$-0.397168\pi$
$719$	12241.1	0.634931	0.317466	+	0.948270i	$0.397168\pi$
$720$	0	0
$721$	−2577.64	−0.133144
$722$	−61576.9	−3.17404
$723$	0	0
$724$	−12793.3	−0.656713
$725$	0	0
$726$	0	0
$727$	8776.97	0.447758	0.223879	−	0.974617i	$-0.428128\pi$
$727$	8776.97	0.447758	0.223879	+	0.974617i	$0.428128\pi$
$728$	−33367.0	−1.69871
$729$	0	0
$730$	0	0
$731$	−16075.6	−0.813375
$732$	0	0
$733$	−3093.56	−0.155884	−0.0779421	−	0.996958i	$-0.524835\pi$
$733$	−3093.56	−0.155884	−0.0779421	+	0.996958i	$0.524835\pi$
$734$	−43170.3	−2.17090
$735$	0	0
$736$	−35140.4	−1.75991
$737$	−12207.5	−0.610135
$738$	0	0
$739$	18338.3	0.912837	0.456419	−	0.889765i	$-0.349132\pi$
$739$	18338.3	0.912837	0.456419	+	0.889765i	$0.349132\pi$
$740$	0	0
$741$	0	0
$742$	20739.3	1.02609
$743$	3193.45	0.157680	0.0788401	−	0.996887i	$-0.474878\pi$
$743$	3193.45	0.157680	0.0788401	+	0.996887i	$0.474878\pi$
$744$	0	0
$745$	0	0
$746$	−52963.5	−2.59937
$747$	0	0
$748$	−49452.5	−2.41733
$749$	−4367.86	−0.213082
$750$	0	0
$751$	−13439.8	−0.653029	−0.326515	−	0.945192i	$-0.605874\pi$
$751$	−13439.8	−0.653029	−0.326515	+	0.945192i	$0.605874\pi$
$752$	57803.4	2.80302
$753$	0	0
$754$	57662.0	2.78505
$755$	0	0
$756$	0	0
$757$	8314.25	0.399190	0.199595	−	0.979879i	$-0.436037\pi$
$757$	8314.25	0.399190	0.199595	+	0.979879i	$0.436037\pi$
$758$	4322.59	0.207129
$759$	0	0
$760$	0	0
$761$	22244.7	1.05962	0.529809	−	0.848117i	$-0.322264\pi$
$761$	22244.7	1.05962	0.529809	+	0.848117i	$0.322264\pi$
$762$	0	0
$763$	−2671.65	−0.126763
$764$	−30300.5	−1.43486
$765$	0	0
$766$	14882.3	0.701986
$767$	36352.5	1.71136
$768$	0	0
$769$	14250.1	0.668236	0.334118	−	0.942531i	$-0.391562\pi$
$769$	14250.1	0.668236	0.334118	+	0.942531i	$0.391562\pi$
$770$	0	0
$771$	0	0
$772$	−11462.5	−0.534385
$773$	20201.9	0.939988	0.469994	−	0.882670i	$-0.344256\pi$
$773$	20201.9	0.939988	0.469994	+	0.882670i	$0.344256\pi$
$774$	0	0
$775$	0	0
$776$	−103325.	−4.77984
$777$	0	0
$778$	59968.1	2.76345
$779$	−1422.81	−0.0654396
$780$	0	0
$781$	−31762.6	−1.45526
$782$	−10817.2	−0.494656
$783$	0	0
$784$	11363.2	0.517638
$785$	0	0
$786$	0	0
$787$	3277.11	0.148433	0.0742163	−	0.997242i	$-0.476354\pi$
$787$	3277.11	0.148433	0.0742163	+	0.997242i	$0.476354\pi$
$788$	83204.1	3.76145
$789$	0	0
$790$	0	0
$791$	8329.72	0.374426
$792$	0	0
$793$	39374.0	1.76319
$794$	−52764.6	−2.35837
$795$	0	0
$796$	−66121.2	−2.94422
$797$	3762.85	0.167236	0.0836180	−	0.996498i	$-0.473352\pi$
$797$	3762.85	0.167236	0.0836180	+	0.996498i	$0.473352\pi$
$798$	0	0
$799$	9408.10	0.416564
$800$	0	0
$801$	0	0
$802$	−40531.0	−1.78454
$803$	−8287.71	−0.364218
$804$	0	0
$805$	0	0
$806$	25099.0	1.09687
$807$	0	0
$808$	7100.78	0.309164
$809$	19453.8	0.845438	0.422719	−	0.906261i	$-0.361076\pi$
$809$	19453.8	0.845438	0.422719	+	0.906261i	$0.361076\pi$
$810$	0	0
$811$	45456.3	1.96817	0.984085	−	0.177700i	$-0.0568655\pi$
$811$	45456.3	1.96817	0.984085	+	0.177700i	$0.0568655\pi$
$812$	−25056.3	−1.08289
$813$	0	0
$814$	16075.6	0.692198
$815$	0	0
$816$	0	0
$817$	57403.3	2.45813
$818$	59031.7	2.52322
$819$	0	0
$820$	0	0
$821$	−14653.1	−0.622897	−0.311448	−	0.950263i	$-0.600814\pi$
$821$	−14653.1	−0.622897	−0.311448	+	0.950263i	$0.600814\pi$
$822$	0	0
$823$	5569.57	0.235897	0.117948	−	0.993020i	$-0.462368\pi$
$823$	5569.57	0.235897	0.117948	+	0.993020i	$0.462368\pi$
$824$	27396.2	1.15824
$825$	0	0
$826$	−21630.7	−0.911173
$827$	11578.6	0.486852	0.243426	−	0.969920i	$-0.421729\pi$
$827$	11578.6	0.486852	0.243426	+	0.969920i	$0.421729\pi$
$828$	0	0
$829$	31832.6	1.33365	0.666823	−	0.745216i	$-0.267654\pi$
$829$	31832.6	1.33365	0.666823	+	0.745216i	$0.267654\pi$
$830$	0	0
$831$	0	0
$832$	114151.	4.75657
$833$	1849.48	0.0769275
$834$	0	0
$835$	0	0
$836$	176587.	7.30549
$837$	0	0
$838$	−2590.35	−0.106781
$839$	16909.4	0.695802	0.347901	−	0.937531i	$-0.386895\pi$
$839$	16909.4	0.695802	0.347901	+	0.937531i	$0.386895\pi$
$840$	0	0
$841$	2919.11	0.119690
$842$	−37466.5	−1.53347
$843$	0	0
$844$	−72367.8	−2.95143
$845$	0	0
$846$	0	0
$847$	16293.8	0.660992
$848$	−126156.	−5.10875
$849$	0	0
$850$	0	0
$851$	2567.93	0.103440
$852$	0	0
$853$	34133.0	1.37010	0.685048	−	0.728498i	$-0.259781\pi$
$853$	34133.0	1.37010	0.685048	+	0.728498i	$0.259781\pi$
$854$	−23428.6	−0.938769
$855$	0	0
$856$	46423.3	1.85364
$857$	−27751.3	−1.10614	−0.553072	−	0.833134i	$-0.686544\pi$
$857$	−27751.3	−1.10614	−0.553072	+	0.833134i	$0.686544\pi$
$858$	0	0
$859$	23061.8	0.916016	0.458008	−	0.888948i	$-0.348563\pi$
$859$	23061.8	0.916016	0.458008	+	0.888948i	$0.348563\pi$
$860$	0	0
$861$	0	0
$862$	−21749.9	−0.859402
$863$	26058.3	1.02785	0.513924	−	0.857835i	$-0.328191\pi$
$863$	26058.3	1.02785	0.513924	+	0.857835i	$0.328191\pi$
$864$	0	0
$865$	0	0
$866$	−72774.6	−2.85564
$867$	0	0
$868$	−10906.5	−0.426486
$869$	14175.4	0.553358
$870$	0	0
$871$	12930.6	0.503027
$872$	28395.3	1.10274
$873$	0	0
$874$	38626.4	1.49492
$875$	0	0
$876$	0	0
$877$	32589.3	1.25480	0.627402	−	0.778696i	$-0.284118\pi$
$877$	32589.3	1.25480	0.627402	+	0.778696i	$0.284118\pi$
$878$	−87397.4	−3.35936
$879$	0	0
$880$	0	0
$881$	17236.6	0.659154	0.329577	−	0.944129i	$-0.393094\pi$
$881$	17236.6	0.659154	0.329577	+	0.944129i	$0.393094\pi$
$882$	0	0
$883$	12400.1	0.472591	0.236296	−	0.971681i	$-0.424067\pi$
$883$	12400.1	0.472591	0.236296	+	0.971681i	$0.424067\pi$
$884$	52381.7	1.99297
$885$	0	0
$886$	90073.1	3.41542
$887$	43060.4	1.63002	0.815010	−	0.579447i	$-0.196731\pi$
$887$	43060.4	1.63002	0.815010	+	0.579447i	$0.196731\pi$
$888$	0	0
$889$	−11769.0	−0.444004
$890$	0	0
$891$	0	0
$892$	118606.	4.45203
$893$	−33594.8	−1.25891
$894$	0	0
$895$	0	0
$896$	−30526.6	−1.13820
$897$	0	0
$898$	−15198.1	−0.564775
$899$	11886.6	0.440980
$900$	0	0
$901$	−20533.2	−0.759224
$902$	3477.58	0.128371
$903$	0	0
$904$	−88531.4	−3.25720
$905$	0	0
$906$	0	0
$907$	−13578.2	−0.497087	−0.248544	−	0.968621i	$-0.579952\pi$
$907$	−13578.2	−0.497087	−0.248544	+	0.968621i	$0.579952\pi$
$908$	−11434.6	−0.417918
$909$	0	0
$910$	0	0
$911$	−977.222	−0.0355399	−0.0177699	−	0.999842i	$-0.505657\pi$
$911$	−977.222	−0.0355399	−0.0177699	+	0.999842i	$0.505657\pi$
$912$	0	0
$913$	623.558	0.0226032
$914$	43589.1	1.57746
$915$	0	0
$916$	22492.1	0.811308
$917$	−287.634	−0.0103582
$918$	0	0
$919$	−3695.14	−0.132635	−0.0663174	−	0.997799i	$-0.521125\pi$
$919$	−3695.14	−0.132635	−0.0663174	+	0.997799i	$0.521125\pi$
$920$	0	0
$921$	0	0
$922$	−60046.5	−2.14482
$923$	33644.0	1.19979
$924$	0	0
$925$	0	0
$926$	2937.41	0.104243
$927$	0	0
$928$	110353.	3.90357
$929$	−28459.7	−1.00510	−0.502548	−	0.864549i	$-0.667604\pi$
$929$	−28459.7	−1.00510	−0.502548	+	0.864549i	$0.667604\pi$
$930$	0	0
$931$	−6604.19	−0.232485
$932$	40179.7	1.41215
$933$	0	0
$934$	57961.9	2.03059
$935$	0	0
$936$	0	0
$937$	−48937.4	−1.70621	−0.853103	−	0.521743i	$-0.825282\pi$
$937$	−48937.4	−1.70621	−0.853103	+	0.521743i	$0.825282\pi$
$938$	−7694.03	−0.267824
$939$	0	0
$940$	0	0
$941$	−32372.5	−1.12148	−0.560741	−	0.827991i	$-0.689484\pi$
$941$	−32372.5	−1.12148	−0.560741	+	0.827991i	$0.689484\pi$
$942$	0	0
$943$	555.511	0.0191834
$944$	131579.	4.53657
$945$	0	0
$946$	−140303.	−4.82203
$947$	20328.9	0.697572	0.348786	−	0.937202i	$-0.386594\pi$
$947$	20328.9	0.697572	0.348786	+	0.937202i	$0.386594\pi$
$948$	0	0
$949$	8778.61	0.300280
$950$	0	0
$951$	0	0
$952$	−19657.0	−0.669208
$953$	31183.7	1.05996	0.529979	−	0.848011i	$-0.322200\pi$
$953$	31183.7	1.05996	0.529979	+	0.848011i	$0.322200\pi$
$954$	0	0
$955$	0	0
$956$	88189.3	2.98352
$957$	0	0
$958$	−70671.5	−2.38339
$959$	−8860.79	−0.298363
$960$	0	0
$961$	−24617.0	−0.826324
$962$	−17027.8	−0.570683
$963$	0	0
$964$	−124214.	−4.15006
$965$	0	0
$966$	0	0
$967$	36948.1	1.22872	0.614359	−	0.789027i	$-0.289415\pi$
$967$	36948.1	1.22872	0.614359	+	0.789027i	$0.289415\pi$
$968$	−173176.	−5.75010
$969$	0	0
$970$	0	0
$971$	22035.3	0.728266	0.364133	−	0.931347i	$-0.381365\pi$
$971$	22035.3	0.728266	0.364133	+	0.931347i	$0.381365\pi$
$972$	0	0
$973$	−4032.95	−0.132878
$974$	−69383.0	−2.28252
$975$	0	0
$976$	142515.	4.67397
$977$	13220.8	0.432929	0.216465	−	0.976290i	$-0.430547\pi$
$977$	13220.8	0.432929	0.216465	+	0.976290i	$0.430547\pi$
$978$	0	0
$979$	−1265.40	−0.0413099
$980$	0	0
$981$	0	0
$982$	−43303.8	−1.40721
$983$	−28323.8	−0.919012	−0.459506	−	0.888175i	$-0.651973\pi$
$983$	−28323.8	−0.919012	−0.459506	+	0.888175i	$0.651973\pi$
$984$	0	0
$985$	0	0
$986$	33969.6	1.09717
$987$	0	0
$988$	−187047.	−6.02302
$989$	−22412.1	−0.720591
$990$	0	0
$991$	7605.32	0.243785	0.121893	−	0.992543i	$-0.461104\pi$
$991$	7605.32	0.243785	0.121893	+	0.992543i	$0.461104\pi$
$992$	48034.2	1.53739
$993$	0	0
$994$	−20019.0	−0.638798
$995$	0	0
$996$	0	0
$997$	14387.3	0.457021	0.228510	−	0.973541i	$-0.426615\pi$
$997$	14387.3	0.457021	0.228510	+	0.973541i	$0.426615\pi$
$998$	75610.4	2.39820
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bu.1.1		10
3.2	odd	2	inner	1575.4.a.bu.1.10		10
5.2	odd	4		315.4.d.d.64.1	✓	20
5.3	odd	4		315.4.d.d.64.19	yes	20
5.4	even	2		1575.4.a.bt.1.10		10
15.2	even	4		315.4.d.d.64.20	yes	20
15.8	even	4		315.4.d.d.64.2	yes	20
15.14	odd	2		1575.4.a.bt.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.d.d.64.1	✓	20	5.2	odd	4
315.4.d.d.64.2	yes	20	15.8	even	4
315.4.d.d.64.19	yes	20	5.3	odd	4
315.4.d.d.64.20	yes	20	15.2	even	4
1575.4.a.bt.1.1		10	15.14	odd	2
1575.4.a.bt.1.10		10	5.4	even	2
1575.4.a.bu.1.1		10	1.1	even	1	trivial
1575.4.a.bu.1.10		10	3.2	odd	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-5.44617 q^{2} +21.6607 q^{4} +7.00000 q^{7} -74.3987 q^{8} +O(q^{10})\)	\(q-5.44617 q^{2} +21.6607 q^{4} +7.00000 q^{7} -74.3987 q^{8} -60.4870 q^{11} +64.0698 q^{13} -38.1232 q^{14} +231.902 q^{16} +37.7444 q^{17} -134.779 q^{19} +329.422 q^{22} +52.6223 q^{23} -348.935 q^{26} +151.625 q^{28} -165.252 q^{29} -71.9304 q^{31} -667.787 q^{32} -205.563 q^{34} +48.7993 q^{37} +734.031 q^{38} +10.5566 q^{41} -425.906 q^{43} -1310.19 q^{44} -286.590 q^{46} +249.258 q^{47} +49.0000 q^{49} +1387.80 q^{52} -544.007 q^{53} -520.791 q^{56} +899.988 q^{58} +567.390 q^{59} +614.549 q^{61} +391.745 q^{62} +1781.66 q^{64} +201.820 q^{67} +817.573 q^{68} +525.114 q^{71} +137.016 q^{73} -265.769 q^{74} -2919.42 q^{76} -423.409 q^{77} -234.355 q^{79} -57.4929 q^{82} -10.3089 q^{83} +2319.56 q^{86} +4500.15 q^{88} +20.9202 q^{89} +448.488 q^{91} +1139.84 q^{92} -1357.50 q^{94} +1388.80 q^{97} -266.862 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 54 q^{4} + 70 q^{7}+O(q^{10}) \) 10 * q + 54 * q^4 + 70 * q^7	\( 10 q + 54 q^{4} + 70 q^{7} + 104 q^{13} + 310 q^{16} + 36 q^{19} + 644 q^{22} + 378 q^{28} - 24 q^{31} + 116 q^{34} + 732 q^{37} + 212 q^{43} - 184 q^{46} + 490 q^{49} + 2692 q^{52} + 3196 q^{58} + 1024 q^{61} + 1590 q^{64} + 2388 q^{67} + 3432 q^{73} - 4184 q^{76} - 776 q^{79} + 1860 q^{82} + 10164 q^{88} + 728 q^{91} - 4028 q^{94} + 4024 q^{97}+O(q^{100}) \) 10 * q + 54 * q^4 + 70 * q^7 + 104 * q^13 + 310 * q^16 + 36 * q^19 + 644 * q^22 + 378 * q^28 - 24 * q^31 + 116 * q^34 + 732 * q^37 + 212 * q^43 - 184 * q^46 + 490 * q^49 + 2692 * q^52 + 3196 * q^58 + 1024 * q^61 + 1590 * q^64 + 2388 * q^67 + 3432 * q^73 - 4184 * q^76 - 776 * q^79 + 1860 * q^82 + 10164 * q^88 + 728 * q^91 - 4028 * q^94 + 4024 * q^97

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