LMFDB - Embedded newform 1575.4.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] 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("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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:traces = [2,-7,0,17,0,0,14,-63,0,0,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.43845 q^{2} -5.93087 q^{4} +7.00000 q^{7} +20.0388 q^{8} -7.61553 q^{11} -52.3542 q^{13} -10.0691 q^{14} +18.6222 q^{16} -49.7235 q^{17} +140.600 q^{19} +10.9545 q^{22} -23.4470 q^{23} +75.3087 q^{26} -41.5161 q^{28} -157.170 q^{29} +127.892 q^{31} -187.098 q^{32} +71.5246 q^{34} +115.477 q^{37} -202.246 q^{38} +188.617 q^{41} -322.186 q^{43} +45.1667 q^{44} +33.7272 q^{46} -76.6477 q^{47} +49.0000 q^{49} +310.506 q^{52} -424.172 q^{53} +140.272 q^{56} +226.081 q^{58} -107.784 q^{59} +915.511 q^{61} -183.966 q^{62} +120.153 q^{64} +451.723 q^{67} +294.903 q^{68} -907.312 q^{71} -755.956 q^{73} -166.108 q^{74} -833.882 q^{76} -53.3087 q^{77} +22.5834 q^{79} -271.316 q^{82} +1112.25 q^{83} +463.447 q^{86} -152.606 q^{88} -1518.81 q^{89} -366.479 q^{91} +139.061 q^{92} +110.254 q^{94} +549.987 q^{97} -70.4839 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + 26 q^{11} - 14 q^{13} - 49 q^{14} + 297 q^{16} + 16 q^{17} + 174 q^{19} - 176 q^{22} + 184 q^{23} - 138 q^{26} + 119 q^{28} + 32 q^{29} + 330 q^{31} - 1071 q^{32}+ \cdots - 343 q^{98}+O(q^{100}) $ 2 * q - 7 * q^2 + 17 * q^4 + 14 * q^7 - 63 * q^8 + 26 * q^11 - 14 * q^13 - 49 * q^14 + 297 * q^16 + 16 * q^17 + 174 * q^19 - 176 * q^22 + 184 * q^23 - 138 * q^26 + 119 * q^28 + 32 * q^29 + 330 * q^31 - 1071 * q^32 - 294 * q^34 + 132 * q^37 - 388 * q^38 - 200 * q^41 - 364 * q^43 + 816 * q^44 - 1120 * q^46 + 292 * q^47 + 98 * q^49 + 1190 * q^52 + 34 * q^53 - 441 * q^56 - 826 * q^58 - 364 * q^59 + 792 * q^61 - 1308 * q^62 + 2809 * q^64 + 788 * q^67 + 1802 * q^68 - 454 * q^71 - 778 * q^73 - 258 * q^74 - 68 * q^76 + 182 * q^77 + 408 * q^79 + 1890 * q^82 + 1136 * q^83 + 696 * q^86 - 2944 * q^88 - 36 * q^89 - 98 * q^91 + 4896 * q^92 - 1940 * q^94 + 498 * q^97 - 343 * q^98

Coefficient data

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

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

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

$2$	−1.43845	−0.508568	−0.254284	−	0.967130i	$-0.581840\pi$
$2$	−1.43845	−0.508568	−0.254284	+	0.967130i	$0.581840\pi$
$3$	0	0
$3$	0	0
$4$	−5.93087	−0.741359
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	20.0388	0.885599
$9$	0	0
$10$	0	0
$11$	−7.61553	−0.208743	−0.104371	−	0.994538i	$-0.533283\pi$
$11$	−7.61553	−0.208743	−0.104371	+	0.994538i	$0.533283\pi$
$12$	0	0
$13$	−52.3542	−1.11696	−0.558478	−	0.829519i	$-0.688615\pi$
$13$	−52.3542	−1.11696	−0.558478	+	0.829519i	$0.688615\pi$
$14$	−10.0691	−0.192221
$15$	0	0
$16$	18.6222	0.290971
$17$	−49.7235	−0.709395	−0.354697	−	0.934981i	$-0.615416\pi$
$17$	−49.7235	−0.709395	−0.354697	+	0.934981i	$0.615416\pi$
$18$	0	0
$19$	140.600	1.69768	0.848840	−	0.528649i	$-0.177301\pi$
$19$	140.600	1.69768	0.848840	+	0.528649i	$0.177301\pi$
$20$	0	0
$21$	0	0
$22$	10.9545	0.106160
$23$	−23.4470	−0.212566	−0.106283	−	0.994336i	$-0.533895\pi$
$23$	−23.4470	−0.212566	−0.106283	+	0.994336i	$0.533895\pi$
$24$	0	0
$25$	0	0
$26$	75.3087	0.568048
$27$	0	0
$28$	−41.5161	−0.280207
$29$	−157.170	−1.00641	−0.503204	−	0.864168i	$-0.667845\pi$
$29$	−157.170	−1.00641	−0.503204	+	0.864168i	$0.667845\pi$
$30$	0	0
$31$	127.892	0.740971	0.370485	−	0.928838i	$-0.379191\pi$
$31$	127.892	0.740971	0.370485	+	0.928838i	$0.379191\pi$
$32$	−187.098	−1.03358
$33$	0	0
$34$	71.5246	0.360776
$35$	0	0
$36$	0	0
$37$	115.477	0.513090	0.256545	−	0.966532i	$-0.417416\pi$
$37$	115.477	0.513090	0.256545	+	0.966532i	$0.417416\pi$
$38$	−202.246	−0.863386
$39$	0	0
$40$	0	0
$41$	188.617	0.718466	0.359233	−	0.933248i	$-0.383038\pi$
$41$	188.617	0.718466	0.359233	+	0.933248i	$0.383038\pi$
$42$	0	0
$43$	−322.186	−1.14262	−0.571312	−	0.820733i	$-0.693565\pi$
$43$	−322.186	−1.14262	−0.571312	+	0.820733i	$0.693565\pi$
$44$	45.1667	0.154753
$45$	0	0
$46$	33.7272	0.108104
$47$	−76.6477	−0.237877	−0.118938	−	0.992902i	$-0.537949\pi$
$47$	−76.6477	−0.237877	−0.118938	+	0.992902i	$0.537949\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	310.506	0.828065
$53$	−424.172	−1.09933	−0.549666	−	0.835385i	$-0.685245\pi$
$53$	−424.172	−1.09933	−0.549666	+	0.835385i	$0.685245\pi$
$54$	0	0
$55$	0	0
$56$	140.272	0.334725
$57$	0	0
$58$	226.081	0.511827
$59$	−107.784	−0.237835	−0.118918	−	0.992904i	$-0.537942\pi$
$59$	−107.784	−0.237835	−0.118918	+	0.992904i	$0.537942\pi$
$60$	0	0
$61$	915.511	1.92163	0.960813	−	0.277197i	$-0.0894053\pi$
$61$	915.511	1.92163	0.960813	+	0.277197i	$0.0894053\pi$
$62$	−183.966	−0.376834
$63$	0	0
$64$	120.153	0.234673
$65$	0	0
$66$	0	0
$67$	451.723	0.823684	0.411842	−	0.911255i	$-0.364886\pi$
$67$	451.723	0.823684	0.411842	+	0.911255i	$0.364886\pi$
$68$	294.903	0.525916
$69$	0	0
$70$	0	0
$71$	−907.312	−1.51659	−0.758297	−	0.651909i	$-0.773968\pi$
$71$	−907.312	−1.51659	−0.758297	+	0.651909i	$0.773968\pi$
$72$	0	0
$73$	−755.956	−1.21203	−0.606014	−	0.795454i	$-0.707232\pi$
$73$	−755.956	−1.21203	−0.606014	+	0.795454i	$0.707232\pi$
$74$	−166.108	−0.260941
$75$	0	0
$76$	−833.882	−1.25859
$77$	−53.3087	−0.0788973
$78$	0	0
$79$	22.5834	0.0321624	0.0160812	−	0.999871i	$-0.494881\pi$
$79$	22.5834	0.0321624	0.0160812	+	0.999871i	$0.494881\pi$
$80$	0	0
$81$	0	0
$82$	−271.316	−0.365389
$83$	1112.25	1.47091	0.735454	−	0.677575i	$-0.236969\pi$
$83$	1112.25	1.47091	0.735454	+	0.677575i	$0.236969\pi$
$84$	0	0
$85$	0	0
$86$	463.447	0.581102
$87$	0	0
$88$	−152.606	−0.184862
$89$	−1518.81	−1.80892	−0.904458	−	0.426562i	$-0.859725\pi$
$89$	−1518.81	−1.80892	−0.904458	+	0.426562i	$0.859725\pi$
$90$	0	0
$91$	−366.479	−0.422170
$92$	139.061	0.157588
$93$	0	0
$94$	110.254	0.120977
$95$	0	0
$96$	0	0
$97$	549.987	0.575698	0.287849	−	0.957676i	$-0.407060\pi$
$97$	549.987	0.575698	0.287849	+	0.957676i	$0.407060\pi$
$98$	−70.4839	−0.0726526
$99$	0	0
$100$	0	0
$101$	533.299	0.525398	0.262699	−	0.964878i	$-0.415387\pi$
$101$	533.299	0.525398	0.262699	+	0.964878i	$0.415387\pi$
$102$	0	0
$103$	−1357.79	−1.29890	−0.649451	−	0.760404i	$-0.725001\pi$
$103$	−1357.79	−1.29890	−0.649451	+	0.760404i	$0.725001\pi$
$104$	−1049.12	−0.989176
$105$	0	0
$106$	610.149	0.559084
$107$	913.693	0.825515	0.412757	−	0.910841i	$-0.364566\pi$
$107$	913.693	0.825515	0.412757	+	0.910841i	$0.364566\pi$
$108$	0	0
$109$	−160.489	−0.141028	−0.0705139	−	0.997511i	$-0.522464\pi$
$109$	−160.489	−0.141028	−0.0705139	+	0.997511i	$0.522464\pi$
$110$	0	0
$111$	0	0
$112$	130.355	0.109977
$113$	−1788.48	−1.48891	−0.744453	−	0.667675i	$-0.767290\pi$
$113$	−1788.48	−1.48891	−0.744453	+	0.667675i	$0.767290\pi$
$114$	0	0
$115$	0	0
$116$	932.157	0.746109
$117$	0	0
$118$	155.042	0.120955
$119$	−348.064	−0.268126
$120$	0	0
$121$	−1273.00	−0.956427
$122$	−1316.91	−0.977277
$123$	0	0
$124$	−758.511	−0.549325
$125$	0	0
$126$	0	0
$127$	271.015	0.189360	0.0946799	−	0.995508i	$-0.469817\pi$
$127$	271.015	0.189360	0.0946799	+	0.995508i	$0.469817\pi$
$128$	1323.95	0.914231
$129$	0	0
$130$	0	0
$131$	763.151	0.508984	0.254492	−	0.967075i	$-0.418092\pi$
$131$	763.151	0.508984	0.254492	+	0.967075i	$0.418092\pi$
$132$	0	0
$133$	984.203	0.641663
$134$	−649.780	−0.418899
$135$	0	0
$136$	−996.400	−0.628240
$137$	−240.934	−0.150251	−0.0751254	−	0.997174i	$-0.523936\pi$
$137$	−240.934	−0.150251	−0.0751254	+	0.997174i	$0.523936\pi$
$138$	0	0
$139$	−103.150	−0.0629427	−0.0314714	−	0.999505i	$-0.510019\pi$
$139$	−103.150	−0.0629427	−0.0314714	+	0.999505i	$0.510019\pi$
$140$	0	0
$141$	0	0
$142$	1305.12	0.771291
$143$	398.705	0.233156
$144$	0	0
$145$	0	0
$146$	1087.40	0.616398
$147$	0	0
$148$	−684.881	−0.380384
$149$	3256.71	1.79061	0.895303	−	0.445458i	$-0.146959\pi$
$149$	3256.71	1.79061	0.895303	+	0.445458i	$0.146959\pi$
$150$	0	0
$151$	1471.04	0.792793	0.396396	−	0.918080i	$-0.370261\pi$
$151$	1471.04	0.792793	0.396396	+	0.918080i	$0.370261\pi$
$152$	2817.47	1.50346
$153$	0	0
$154$	76.6817	0.0401246
$155$	0	0
$156$	0	0
$157$	1394.22	0.708730	0.354365	−	0.935107i	$-0.384697\pi$
$157$	1394.22	0.708730	0.354365	+	0.935107i	$0.384697\pi$
$158$	−32.4850	−0.0163567
$159$	0	0
$160$	0	0
$161$	−164.129	−0.0803426
$162$	0	0
$163$	3674.27	1.76559	0.882794	−	0.469760i	$-0.155659\pi$
$163$	3674.27	1.76559	0.882794	+	0.469760i	$0.155659\pi$
$164$	−1118.67	−0.532641
$165$	0	0
$166$	−1599.91	−0.748056
$167$	4041.09	1.87251	0.936254	−	0.351325i	$-0.114269\pi$
$167$	4041.09	1.87251	0.936254	+	0.351325i	$0.114269\pi$
$168$	0	0
$169$	543.958	0.247591
$170$	0	0
$171$	0	0
$172$	1910.84	0.847094
$173$	59.4582	0.0261302	0.0130651	−	0.999915i	$-0.495841\pi$
$173$	59.4582	0.0261302	0.0130651	+	0.999915i	$0.495841\pi$
$174$	0	0
$175$	0	0
$176$	−141.818	−0.0607381
$177$	0	0
$178$	2184.73	0.919957
$179$	2973.32	1.24155	0.620773	−	0.783991i	$-0.286819\pi$
$179$	2973.32	1.24155	0.620773	+	0.783991i	$0.286819\pi$
$180$	0	0
$181$	−676.220	−0.277696	−0.138848	−	0.990314i	$-0.544340\pi$
$181$	−676.220	−0.277696	−0.138848	+	0.990314i	$0.544340\pi$
$182$	527.161	0.214702
$183$	0	0
$184$	−469.849	−0.188249
$185$	0	0
$186$	0	0
$187$	378.671	0.148081
$188$	454.588	0.176352
$189$	0	0
$190$	0	0
$191$	−16.5589	−0.00627308	−0.00313654	−	0.999995i	$-0.500998\pi$
$191$	−16.5589	−0.00627308	−0.00313654	+	0.999995i	$0.500998\pi$
$192$	0	0
$193$	−2694.68	−1.00501	−0.502506	−	0.864574i	$-0.667589\pi$
$193$	−2694.68	−1.00501	−0.502506	+	0.864574i	$0.667589\pi$
$194$	−791.127	−0.292781
$195$	0	0
$196$	−290.613	−0.105908
$197$	1027.38	0.371561	0.185781	−	0.982591i	$-0.440519\pi$
$197$	1027.38	0.371561	0.185781	+	0.982591i	$0.440519\pi$
$198$	0	0
$199$	2823.77	1.00589	0.502944	−	0.864319i	$-0.332250\pi$
$199$	2823.77	1.00589	0.502944	+	0.864319i	$0.332250\pi$
$200$	0	0
$201$	0	0
$202$	−767.123	−0.267201
$203$	−1100.19	−0.380386
$204$	0	0
$205$	0	0
$206$	1953.11	0.660579
$207$	0	0
$208$	−974.948	−0.325002
$209$	−1070.75	−0.354378
$210$	0	0
$211$	5151.16	1.68067	0.840333	−	0.542071i	$-0.182360\pi$
$211$	5151.16	1.68067	0.840333	+	0.542071i	$0.182360\pi$
$212$	2515.71	0.814999
$213$	0	0
$214$	−1314.30	−0.419830
$215$	0	0
$216$	0	0
$217$	895.244	0.280061
$218$	230.855	0.0717222
$219$	0	0
$220$	0	0
$221$	2603.23	0.792363
$222$	0	0
$223$	114.496	0.0343822	0.0171911	−	0.999852i	$-0.494528\pi$
$223$	114.496	0.0343822	0.0171911	+	0.999852i	$0.494528\pi$
$224$	−1309.68	−0.390656
$225$	0	0
$226$	2572.64	0.757209
$227$	4744.75	1.38731	0.693657	−	0.720306i	$-0.255999\pi$
$227$	4744.75	1.38731	0.693657	+	0.720306i	$0.255999\pi$
$228$	0	0
$229$	−5384.47	−1.55378	−0.776891	−	0.629635i	$-0.783204\pi$
$229$	−5384.47	−1.55378	−0.776891	+	0.629635i	$0.783204\pi$
$230$	0	0
$231$	0	0
$232$	−3149.51	−0.891274
$233$	−1608.91	−0.452373	−0.226187	−	0.974084i	$-0.572626\pi$
$233$	−1608.91	−0.452373	−0.226187	+	0.974084i	$0.572626\pi$
$234$	0	0
$235$	0	0
$236$	639.253	0.176321
$237$	0	0
$238$	500.672	0.136360
$239$	3113.11	0.842554	0.421277	−	0.906932i	$-0.361582\pi$
$239$	3113.11	0.842554	0.421277	+	0.906932i	$0.361582\pi$
$240$	0	0
$241$	7136.38	1.90745	0.953724	−	0.300685i	$-0.0972151\pi$
$241$	7136.38	1.90745	0.953724	+	0.300685i	$0.0972151\pi$
$242$	1831.15	0.486408
$243$	0	0
$244$	−5429.78	−1.42461
$245$	0	0
$246$	0	0
$247$	−7361.01	−1.89624
$248$	2562.81	0.656203
$249$	0	0
$250$	0	0
$251$	225.504	0.0567079	0.0283539	−	0.999598i	$-0.490973\pi$
$251$	225.504	0.0567079	0.0283539	+	0.999598i	$0.490973\pi$
$252$	0	0
$253$	178.561	0.0443717
$254$	−389.841	−0.0963024
$255$	0	0
$256$	−2865.65	−0.699621
$257$	−4423.05	−1.07355	−0.536775	−	0.843725i	$-0.680358\pi$
$257$	−4423.05	−1.07355	−0.536775	+	0.843725i	$0.680358\pi$
$258$	0	0
$259$	808.341	0.193930
$260$	0	0
$261$	0	0
$262$	−1097.75	−0.258853
$263$	6540.56	1.53349	0.766746	−	0.641950i	$-0.221875\pi$
$263$	6540.56	1.53349	0.766746	+	0.641950i	$0.221875\pi$
$264$	0	0
$265$	0	0
$266$	−1415.72	−0.326329
$267$	0	0
$268$	−2679.11	−0.610645
$269$	−2262.97	−0.512920	−0.256460	−	0.966555i	$-0.582556\pi$
$269$	−2262.97	−0.512920	−0.256460	+	0.966555i	$0.582556\pi$
$270$	0	0
$271$	1615.68	0.362160	0.181080	−	0.983468i	$-0.442041\pi$
$271$	1615.68	0.362160	0.181080	+	0.983468i	$0.442041\pi$
$272$	−925.959	−0.206414
$273$	0	0
$274$	346.571	0.0764127
$275$	0	0
$276$	0	0
$277$	−4691.55	−1.01765	−0.508823	−	0.860871i	$-0.669919\pi$
$277$	−4691.55	−1.01765	−0.508823	+	0.860871i	$0.669919\pi$
$278$	148.375	0.0320107
$279$	0	0
$280$	0	0
$281$	8119.00	1.72363	0.861813	−	0.507226i	$-0.169329\pi$
$281$	8119.00	1.72363	0.861813	+	0.507226i	$0.169329\pi$
$282$	0	0
$283$	3633.75	0.763265	0.381632	−	0.924314i	$-0.375362\pi$
$283$	3633.75	0.763265	0.381632	+	0.924314i	$0.375362\pi$
$284$	5381.15	1.12434
$285$	0	0
$286$	−573.515	−0.118576
$287$	1320.32	0.271554
$288$	0	0
$289$	−2440.58	−0.496759
$290$	0	0
$291$	0	0
$292$	4483.48	0.898547
$293$	−7981.99	−1.59151	−0.795756	−	0.605618i	$-0.792926\pi$
$293$	−7981.99	−1.59151	−0.795756	+	0.605618i	$0.792926\pi$
$294$	0	0
$295$	0	0
$296$	2314.03	0.454392
$297$	0	0
$298$	−4684.61	−0.910645
$299$	1227.55	0.237427
$300$	0	0
$301$	−2255.30	−0.431871
$302$	−2116.02	−0.403189
$303$	0	0
$304$	2618.28	0.493977
$305$	0	0
$306$	0	0
$307$	7118.15	1.32330	0.661652	−	0.749811i	$-0.269856\pi$
$307$	7118.15	1.32330	0.661652	+	0.749811i	$0.269856\pi$
$308$	316.167	0.0584912
$309$	0	0
$310$	0	0
$311$	9155.92	1.66940	0.834702	−	0.550703i	$-0.185640\pi$
$311$	9155.92	1.66940	0.834702	+	0.550703i	$0.185640\pi$
$312$	0	0
$313$	6163.44	1.11303	0.556515	−	0.830838i	$-0.312138\pi$
$313$	6163.44	1.11303	0.556515	+	0.830838i	$0.312138\pi$
$314$	−2005.51	−0.360438
$315$	0	0
$316$	−133.939	−0.0238438
$317$	8658.37	1.53408	0.767038	−	0.641601i	$-0.221730\pi$
$317$	8658.37	1.53408	0.767038	+	0.641601i	$0.221730\pi$
$318$	0	0
$319$	1196.94	0.210080
$320$	0	0
$321$	0	0
$322$	236.090	0.0408597
$323$	−6991.14	−1.20433
$324$	0	0
$325$	0	0
$326$	−5285.24	−0.897922
$327$	0	0
$328$	3779.67	0.636272
$329$	−536.534	−0.0899090
$330$	0	0
$331$	−128.477	−0.0213346	−0.0106673	−	0.999943i	$-0.503396\pi$
$331$	−128.477	−0.0213346	−0.0106673	+	0.999943i	$0.503396\pi$
$332$	−6596.61	−1.09047
$333$	0	0
$334$	−5812.89	−0.952297
$335$	0	0
$336$	0	0
$337$	−7784.57	−1.25832	−0.629158	−	0.777277i	$-0.716600\pi$
$337$	−7784.57	−1.25832	−0.629158	+	0.777277i	$0.716600\pi$
$338$	−782.455	−0.125917
$339$	0	0
$340$	0	0
$341$	−973.965	−0.154672
$342$	0	0
$343$	343.000	0.0539949
$344$	−6456.22	−1.01191
$345$	0	0
$346$	−85.5274	−0.0132890
$347$	3740.26	0.578638	0.289319	−	0.957233i	$-0.406571\pi$
$347$	3740.26	0.578638	0.289319	+	0.957233i	$0.406571\pi$
$348$	0	0
$349$	5676.86	0.870703	0.435351	−	0.900261i	$-0.356624\pi$
$349$	5676.86	0.870703	0.435351	+	0.900261i	$0.356624\pi$
$350$	0	0
$351$	0	0
$352$	1424.85	0.215752
$353$	909.564	0.137142	0.0685711	−	0.997646i	$-0.478156\pi$
$353$	909.564	0.137142	0.0685711	+	0.997646i	$0.478156\pi$
$354$	0	0
$355$	0	0
$356$	9007.87	1.34106
$357$	0	0
$358$	−4276.97	−0.631410
$359$	−2678.57	−0.393788	−0.196894	−	0.980425i	$-0.563085\pi$
$359$	−2678.57	−0.393788	−0.196894	+	0.980425i	$0.563085\pi$
$360$	0	0
$361$	12909.5	1.88212
$362$	972.706	0.141227
$363$	0	0
$364$	2173.54	0.312979
$365$	0	0
$366$	0	0
$367$	716.898	0.101967	0.0509833	−	0.998700i	$-0.483764\pi$
$367$	716.898	0.101967	0.0509833	+	0.998700i	$0.483764\pi$
$368$	−436.633	−0.0618508
$369$	0	0
$370$	0	0
$371$	−2969.21	−0.415508
$372$	0	0
$373$	2006.15	0.278484	0.139242	−	0.990258i	$-0.455533\pi$
$373$	2006.15	0.278484	0.139242	+	0.990258i	$0.455533\pi$
$374$	−544.698	−0.0753092
$375$	0	0
$376$	−1535.93	−0.210664
$377$	8228.53	1.12411
$378$	0	0
$379$	7277.53	0.986336	0.493168	−	0.869934i	$-0.335839\pi$
$379$	7277.53	0.986336	0.493168	+	0.869934i	$0.335839\pi$
$380$	0	0
$381$	0	0
$382$	23.8191	0.00319029
$383$	5953.94	0.794339	0.397170	−	0.917745i	$-0.369992\pi$
$383$	5953.94	0.794339	0.397170	+	0.917745i	$0.369992\pi$
$384$	0	0
$385$	0	0
$386$	3876.16	0.511117
$387$	0	0
$388$	−3261.90	−0.426799
$389$	−1867.98	−0.243472	−0.121736	−	0.992563i	$-0.538846\pi$
$389$	−1867.98	−0.243472	−0.121736	+	0.992563i	$0.538846\pi$
$390$	0	0
$391$	1165.86	0.150794
$392$	981.902	0.126514
$393$	0	0
$394$	−1477.83	−0.188964
$395$	0	0
$396$	0	0
$397$	2160.07	0.273075	0.136538	−	0.990635i	$-0.456403\pi$
$397$	2160.07	0.273075	0.136538	+	0.990635i	$0.456403\pi$
$398$	−4061.85	−0.511563
$399$	0	0
$400$	0	0
$401$	−1954.81	−0.243438	−0.121719	−	0.992565i	$-0.538841\pi$
$401$	−1954.81	−0.243438	−0.121719	+	0.992565i	$0.538841\pi$
$402$	0	0
$403$	−6695.68	−0.827632
$404$	−3162.93	−0.389509
$405$	0	0
$406$	1582.57	0.193452
$407$	−879.420	−0.107104
$408$	0	0
$409$	14895.1	1.80077	0.900384	−	0.435096i	$-0.143286\pi$
$409$	14895.1	1.80077	0.900384	+	0.435096i	$0.143286\pi$
$410$	0	0
$411$	0	0
$412$	8052.86	0.962952
$413$	−754.489	−0.0898934
$414$	0	0
$415$	0	0
$416$	9795.34	1.15446
$417$	0	0
$418$	1540.21	0.180225
$419$	−12608.9	−1.47013	−0.735067	−	0.677994i	$-0.762849\pi$
$419$	−12608.9	−1.47013	−0.735067	+	0.677994i	$0.762849\pi$
$420$	0	0
$421$	−7862.86	−0.910243	−0.455122	−	0.890429i	$-0.650404\pi$
$421$	−7862.86	−0.910243	−0.455122	+	0.890429i	$0.650404\pi$
$422$	−7409.67	−0.854732
$423$	0	0
$424$	−8499.91	−0.973567
$425$	0	0
$426$	0	0
$427$	6408.58	0.726307
$428$	−5419.00	−0.612002
$429$	0	0
$430$	0	0
$431$	−14291.0	−1.59715	−0.798575	−	0.601896i	$-0.794412\pi$
$431$	−14291.0	−1.59715	−0.798575	+	0.601896i	$0.794412\pi$
$432$	0	0
$433$	13759.0	1.52705	0.763527	−	0.645776i	$-0.223466\pi$
$433$	13759.0	1.52705	0.763527	+	0.645776i	$0.223466\pi$
$434$	−1287.76	−0.142430
$435$	0	0
$436$	951.838	0.104552
$437$	−3296.65	−0.360870
$438$	0	0
$439$	−6093.13	−0.662436	−0.331218	−	0.943554i	$-0.607459\pi$
$439$	−6093.13	−0.662436	−0.331218	+	0.943554i	$0.607459\pi$
$440$	0	0
$441$	0	0
$442$	−3744.61	−0.402970
$443$	13449.5	1.44244	0.721222	−	0.692704i	$-0.243581\pi$
$443$	13449.5	1.44244	0.721222	+	0.692704i	$0.243581\pi$
$444$	0	0
$445$	0	0
$446$	−164.697	−0.0174857
$447$	0	0
$448$	841.068	0.0886981
$449$	−6893.83	−0.724588	−0.362294	−	0.932064i	$-0.618006\pi$
$449$	−6893.83	−0.724588	−0.362294	+	0.932064i	$0.618006\pi$
$450$	0	0
$451$	−1436.42	−0.149974
$452$	10607.3	1.10381
$453$	0	0
$454$	−6825.07	−0.705543
$455$	0	0
$456$	0	0
$457$	11820.1	1.20990	0.604948	−	0.796265i	$-0.293194\pi$
$457$	11820.1	1.20990	0.604948	+	0.796265i	$0.293194\pi$
$458$	7745.28	0.790203
$459$	0	0
$460$	0	0
$461$	−8443.38	−0.853031	−0.426516	−	0.904480i	$-0.640259\pi$
$461$	−8443.38	−0.853031	−0.426516	+	0.904480i	$0.640259\pi$
$462$	0	0
$463$	1269.74	0.127451	0.0637257	−	0.997967i	$-0.479702\pi$
$463$	1269.74	0.127451	0.0637257	+	0.997967i	$0.479702\pi$
$464$	−2926.86	−0.292836
$465$	0	0
$466$	2314.33	0.230063
$467$	−16481.8	−1.63316	−0.816579	−	0.577233i	$-0.804132\pi$
$467$	−16481.8	−1.63316	−0.816579	+	0.577233i	$0.804132\pi$
$468$	0	0
$469$	3162.06	0.311323
$470$	0	0
$471$	0	0
$472$	−2159.87	−0.210627
$473$	2453.61	0.238514
$474$	0	0
$475$	0	0
$476$	2064.32	0.198778
$477$	0	0
$478$	−4478.05	−0.428496
$479$	1400.21	0.133564	0.0667822	−	0.997768i	$-0.478727\pi$
$479$	1400.21	0.133564	0.0667822	+	0.997768i	$0.478727\pi$
$480$	0	0
$481$	−6045.72	−0.573100
$482$	−10265.3	−0.970066
$483$	0	0
$484$	7550.02	0.709055
$485$	0	0
$486$	0	0
$487$	14165.9	1.31811	0.659055	−	0.752094i	$-0.270956\pi$
$487$	14165.9	1.31811	0.659055	+	0.752094i	$0.270956\pi$
$488$	18345.8	1.70179
$489$	0	0
$490$	0	0
$491$	−4739.28	−0.435603	−0.217801	−	0.975993i	$-0.569888\pi$
$491$	−4739.28	−0.435603	−0.217801	+	0.975993i	$0.569888\pi$
$492$	0	0
$493$	7815.06	0.713940
$494$	10588.4	0.964364
$495$	0	0
$496$	2381.63	0.215601
$497$	−6351.19	−0.573219
$498$	0	0
$499$	−11370.0	−1.02003	−0.510013	−	0.860167i	$-0.670360\pi$
$499$	−11370.0	−1.02003	−0.510013	+	0.860167i	$0.670360\pi$
$500$	0	0
$501$	0	0
$502$	−324.375	−0.0288398
$503$	9212.48	0.816629	0.408314	−	0.912841i	$-0.366117\pi$
$503$	9212.48	0.816629	0.408314	+	0.912841i	$0.366117\pi$
$504$	0	0
$505$	0	0
$506$	−256.851	−0.0225660
$507$	0	0
$508$	−1607.36	−0.140384
$509$	15938.3	1.38792	0.693960	−	0.720014i	$-0.255864\pi$
$509$	15938.3	1.38792	0.693960	+	0.720014i	$0.255864\pi$
$510$	0	0
$511$	−5291.69	−0.458103
$512$	−6469.49	−0.558426
$513$	0	0
$514$	6362.33	0.545973
$515$	0	0
$516$	0	0
$517$	583.713	0.0496550
$518$	−1162.76	−0.0986265
$519$	0	0
$520$	0	0
$521$	−6442.99	−0.541790	−0.270895	−	0.962609i	$-0.587320\pi$
$521$	−6442.99	−0.541790	−0.270895	+	0.962609i	$0.587320\pi$
$522$	0	0
$523$	−986.655	−0.0824922	−0.0412461	−	0.999149i	$-0.513133\pi$
$523$	−986.655	−0.0824922	−0.0412461	+	0.999149i	$0.513133\pi$
$524$	−4526.15	−0.377339
$525$	0	0
$526$	−9408.26	−0.779885
$527$	−6359.24	−0.525641
$528$	0	0
$529$	−11617.2	−0.954815
$530$	0	0
$531$	0	0
$532$	−5837.18	−0.475703
$533$	−9874.91	−0.802495
$534$	0	0
$535$	0	0
$536$	9052.01	0.729454
$537$	0	0
$538$	3255.16	0.260855
$539$	−373.161	−0.0298204
$540$	0	0
$541$	−12681.1	−1.00777	−0.503885	−	0.863771i	$-0.668097\pi$
$541$	−12681.1	−1.00777	−0.503885	+	0.863771i	$0.668097\pi$
$542$	−2324.06	−0.184183
$543$	0	0
$544$	9303.14	0.733215
$545$	0	0
$546$	0	0
$547$	14826.4	1.15892	0.579462	−	0.814999i	$-0.303263\pi$
$547$	14826.4	1.15892	0.579462	+	0.814999i	$0.303263\pi$
$548$	1428.95	0.111390
$549$	0	0
$550$	0	0
$551$	−22098.2	−1.70856
$552$	0	0
$553$	158.083	0.0121562
$554$	6748.55	0.517542
$555$	0	0
$556$	611.767	0.0466632
$557$	1926.46	0.146547	0.0732737	−	0.997312i	$-0.476655\pi$
$557$	1926.46	0.146547	0.0732737	+	0.997312i	$0.476655\pi$
$558$	0	0
$559$	16867.8	1.27626
$560$	0	0
$561$	0	0
$562$	−11678.8	−0.876581
$563$	−18624.8	−1.39422	−0.697108	−	0.716966i	$-0.745530\pi$
$563$	−18624.8	−1.39422	−0.697108	+	0.716966i	$0.745530\pi$
$564$	0	0
$565$	0	0
$566$	−5226.96	−0.388172
$567$	0	0
$568$	−18181.5	−1.34309
$569$	20093.9	1.48045	0.740227	−	0.672357i	$-0.234718\pi$
$569$	20093.9	1.48045	0.740227	+	0.672357i	$0.234718\pi$
$570$	0	0
$571$	4535.25	0.332389	0.166195	−	0.986093i	$-0.446852\pi$
$571$	4535.25	0.332389	0.166195	+	0.986093i	$0.446852\pi$
$572$	−2364.66	−0.172852
$573$	0	0
$574$	−1899.21	−0.138104
$575$	0	0
$576$	0	0
$577$	−10034.6	−0.723994	−0.361997	−	0.932179i	$-0.617905\pi$
$577$	−10034.6	−0.723994	−0.361997	+	0.932179i	$0.617905\pi$
$578$	3510.64	0.252636
$579$	0	0
$580$	0	0
$581$	7785.75	0.555951
$582$	0	0
$583$	3230.30	0.229477
$584$	−15148.5	−1.07337
$585$	0	0
$586$	11481.7	0.809391
$587$	−11192.6	−0.786999	−0.393499	−	0.919325i	$-0.628736\pi$
$587$	−11192.6	−0.786999	−0.393499	+	0.919325i	$0.628736\pi$
$588$	0	0
$589$	17981.7	1.25793
$590$	0	0
$591$	0	0
$592$	2150.44	0.149295
$593$	20317.6	1.40699	0.703493	−	0.710703i	$-0.251623\pi$
$593$	20317.6	1.40699	0.703493	+	0.710703i	$0.251623\pi$
$594$	0	0
$595$	0	0
$596$	−19315.1	−1.32748
$597$	0	0
$598$	−1765.76	−0.120748
$599$	26376.5	1.79919	0.899594	−	0.436727i	$-0.143862\pi$
$599$	26376.5	1.79919	0.899594	+	0.436727i	$0.143862\pi$
$600$	0	0
$601$	9266.13	0.628907	0.314454	−	0.949273i	$-0.398179\pi$
$601$	9266.13	0.628907	0.314454	+	0.949273i	$0.398179\pi$
$602$	3244.13	0.219636
$603$	0	0
$604$	−8724.56	−0.587744
$605$	0	0
$606$	0	0
$607$	−11338.0	−0.758149	−0.379075	−	0.925366i	$-0.623758\pi$
$607$	−11338.0	−0.758149	−0.379075	+	0.925366i	$0.623758\pi$
$608$	−26306.0	−1.75469
$609$	0	0
$610$	0	0
$611$	4012.83	0.265698
$612$	0	0
$613$	−25712.5	−1.69416	−0.847078	−	0.531469i	$-0.821640\pi$
$613$	−25712.5	−1.69416	−0.847078	+	0.531469i	$0.821640\pi$
$614$	−10239.1	−0.672990
$615$	0	0
$616$	−1068.24	−0.0698714
$617$	663.465	0.0432903	0.0216451	−	0.999766i	$-0.493110\pi$
$617$	663.465	0.0432903	0.0216451	+	0.999766i	$0.493110\pi$
$618$	0	0
$619$	−12768.7	−0.829108	−0.414554	−	0.910025i	$-0.636062\pi$
$619$	−12768.7	−0.829108	−0.414554	+	0.910025i	$0.636062\pi$
$620$	0	0
$621$	0	0
$622$	−13170.3	−0.849005
$623$	−10631.7	−0.683706
$624$	0	0
$625$	0	0
$626$	−8865.78	−0.566051
$627$	0	0
$628$	−8268.92	−0.525423
$629$	−5741.93	−0.363984
$630$	0	0
$631$	−14937.6	−0.942400	−0.471200	−	0.882026i	$-0.656179\pi$
$631$	−14937.6	−0.942400	−0.471200	+	0.882026i	$0.656179\pi$
$632$	452.544	0.0284829
$633$	0	0
$634$	−12454.6	−0.780182
$635$	0	0
$636$	0	0
$637$	−2565.35	−0.159565
$638$	−1721.73	−0.106840
$639$	0	0
$640$	0	0
$641$	−10903.0	−0.671829	−0.335914	−	0.941893i	$-0.609045\pi$
$641$	−10903.0	−0.671829	−0.335914	+	0.941893i	$0.609045\pi$
$642$	0	0
$643$	7623.47	0.467559	0.233779	−	0.972290i	$-0.424891\pi$
$643$	7623.47	0.467559	0.233779	+	0.972290i	$0.424891\pi$
$644$	973.426	0.0595627
$645$	0	0
$646$	10056.4	0.612482
$647$	5384.84	0.327202	0.163601	−	0.986527i	$-0.447689\pi$
$647$	5384.84	0.327202	0.163601	+	0.986527i	$0.447689\pi$
$648$	0	0
$649$	820.833	0.0496464
$650$	0	0
$651$	0	0
$652$	−21791.6	−1.30893
$653$	297.318	0.0178177	0.00890883	−	0.999960i	$-0.497164\pi$
$653$	297.318	0.0178177	0.00890883	+	0.999960i	$0.497164\pi$
$654$	0	0
$655$	0	0
$656$	3512.47	0.209053
$657$	0	0
$658$	771.776	0.0457248
$659$	−10324.7	−0.610309	−0.305155	−	0.952303i	$-0.598708\pi$
$659$	−10324.7	−0.610309	−0.305155	+	0.952303i	$0.598708\pi$
$660$	0	0
$661$	4272.98	0.251437	0.125718	−	0.992066i	$-0.459876\pi$
$661$	4272.98	0.251437	0.125718	+	0.992066i	$0.459876\pi$
$662$	184.808	0.0108501
$663$	0	0
$664$	22288.2	1.30263
$665$	0	0
$666$	0	0
$667$	3685.17	0.213928
$668$	−23967.2	−1.38820
$669$	0	0
$670$	0	0
$671$	−6972.10	−0.401125
$672$	0	0
$673$	23033.4	1.31927	0.659637	−	0.751584i	$-0.270710\pi$
$673$	23033.4	1.31927	0.659637	+	0.751584i	$0.270710\pi$
$674$	11197.7	0.639940
$675$	0	0
$676$	−3226.15	−0.183554
$677$	5113.50	0.290292	0.145146	−	0.989410i	$-0.453635\pi$
$677$	5113.50	0.290292	0.145146	+	0.989410i	$0.453635\pi$
$678$	0	0
$679$	3849.91	0.217593
$680$	0	0
$681$	0	0
$682$	1401.00	0.0786613
$683$	1341.02	0.0751286	0.0375643	−	0.999294i	$-0.488040\pi$
$683$	1341.02	0.0751286	0.0375643	+	0.999294i	$0.488040\pi$
$684$	0	0
$685$	0	0
$686$	−493.387	−0.0274601
$687$	0	0
$688$	−5999.80	−0.332471
$689$	22207.2	1.22790
$690$	0	0
$691$	−16809.3	−0.925404	−0.462702	−	0.886514i	$-0.653120\pi$
$691$	−16809.3	−0.925404	−0.462702	+	0.886514i	$0.653120\pi$
$692$	−352.639	−0.0193718
$693$	0	0
$694$	−5380.16	−0.294277
$695$	0	0
$696$	0	0
$697$	−9378.71	−0.509676
$698$	−8165.86	−0.442812
$699$	0	0
$700$	0	0
$701$	−13467.0	−0.725592	−0.362796	−	0.931869i	$-0.618178\pi$
$701$	−13467.0	−0.725592	−0.362796	+	0.931869i	$0.618178\pi$
$702$	0	0
$703$	16236.1	0.871064
$704$	−915.025	−0.0489863
$705$	0	0
$706$	−1308.36	−0.0697461
$707$	3733.09	0.198582
$708$	0	0
$709$	35514.3	1.88119	0.940597	−	0.339525i	$-0.110266\pi$
$709$	35514.3	1.88119	0.940597	+	0.339525i	$0.110266\pi$
$710$	0	0
$711$	0	0
$712$	−30435.2	−1.60198
$713$	−2998.68	−0.157506
$714$	0	0
$715$	0	0
$716$	−17634.4	−0.920431
$717$	0	0
$718$	3852.99	0.200268
$719$	−4993.61	−0.259013	−0.129506	−	0.991579i	$-0.541339\pi$
$719$	−4993.61	−0.259013	−0.129506	+	0.991579i	$0.541339\pi$
$720$	0	0
$721$	−9504.51	−0.490938
$722$	−18569.6	−0.957186
$723$	0	0
$724$	4010.57	0.205872
$725$	0	0
$726$	0	0
$727$	4223.35	0.215454	0.107727	−	0.994181i	$-0.465643\pi$
$727$	4223.35	0.215454	0.107727	+	0.994181i	$0.465643\pi$
$728$	−7343.81	−0.373873
$729$	0	0
$730$	0	0
$731$	16020.2	0.810572
$732$	0	0
$733$	−19030.9	−0.958968	−0.479484	−	0.877551i	$-0.659176\pi$
$733$	−19030.9	−0.958968	−0.479484	+	0.877551i	$0.659176\pi$
$734$	−1031.22	−0.0518570
$735$	0	0
$736$	4386.87	0.219704
$737$	−3440.11	−0.171938
$738$	0	0
$739$	−27772.5	−1.38245	−0.691224	−	0.722641i	$-0.742928\pi$
$739$	−27772.5	−1.38245	−0.691224	+	0.722641i	$0.742928\pi$
$740$	0	0
$741$	0	0
$742$	4271.05	0.211314
$743$	26880.7	1.32726	0.663631	−	0.748060i	$-0.269015\pi$
$743$	26880.7	1.32726	0.663631	+	0.748060i	$0.269015\pi$
$744$	0	0
$745$	0	0
$746$	−2885.74	−0.141628
$747$	0	0
$748$	−2245.85	−0.109781
$749$	6395.85	0.312015
$750$	0	0
$751$	−35166.6	−1.70872	−0.854360	−	0.519681i	$-0.826051\pi$
$751$	−35166.6	−1.70872	−0.854360	+	0.519681i	$0.826051\pi$
$752$	−1427.35	−0.0692154
$753$	0	0
$754$	−11836.3	−0.571688
$755$	0	0
$756$	0	0
$757$	−14589.2	−0.700467	−0.350233	−	0.936662i	$-0.613898\pi$
$757$	−14589.2	−0.700467	−0.350233	+	0.936662i	$0.613898\pi$
$758$	−10468.3	−0.501619
$759$	0	0
$760$	0	0
$761$	782.826	0.0372897	0.0186448	−	0.999826i	$-0.494065\pi$
$761$	782.826	0.0372897	0.0186448	+	0.999826i	$0.494065\pi$
$762$	0	0
$763$	−1123.42	−0.0533035
$764$	98.2085	0.00465060
$765$	0	0
$766$	−8564.42	−0.403975
$767$	5642.95	0.265652
$768$	0	0
$769$	−16548.4	−0.776008	−0.388004	−	0.921658i	$-0.626835\pi$
$769$	−16548.4	−0.776008	−0.388004	+	0.921658i	$0.626835\pi$
$770$	0	0
$771$	0	0
$772$	15981.8	0.745075
$773$	5744.94	0.267310	0.133655	−	0.991028i	$-0.457329\pi$
$773$	5744.94	0.267310	0.133655	+	0.991028i	$0.457329\pi$
$774$	0	0
$775$	0	0
$776$	11021.1	0.509837
$777$	0	0
$778$	2687.00	0.123822
$779$	26519.7	1.21973
$780$	0	0
$781$	6909.66	0.316578
$782$	−1677.03	−0.0766888
$783$	0	0
$784$	912.486	0.0415673
$785$	0	0
$786$	0	0
$787$	34744.3	1.57370	0.786848	−	0.617146i	$-0.211711\pi$
$787$	34744.3	1.57370	0.786848	+	0.617146i	$0.211711\pi$
$788$	−6093.24	−0.275460
$789$	0	0
$790$	0	0
$791$	−12519.4	−0.562753
$792$	0	0
$793$	−47930.8	−2.14637
$794$	−3107.15	−0.138877
$795$	0	0
$796$	−16747.4	−0.745724
$797$	21748.7	0.966600	0.483300	−	0.875455i	$-0.339438\pi$
$797$	21748.7	0.966600	0.483300	+	0.875455i	$0.339438\pi$
$798$	0	0
$799$	3811.19	0.168749
$800$	0	0
$801$	0	0
$802$	2811.89	0.123805
$803$	5757.01	0.253002
$804$	0	0
$805$	0	0
$806$	9631.38	0.420907
$807$	0	0
$808$	10686.7	0.465292
$809$	42350.6	1.84050	0.920252	−	0.391325i	$-0.127983\pi$
$809$	42350.6	1.84050	0.920252	+	0.391325i	$0.127983\pi$
$810$	0	0
$811$	−18910.7	−0.818796	−0.409398	−	0.912356i	$-0.634261\pi$
$811$	−18910.7	−0.818796	−0.409398	+	0.912356i	$0.634261\pi$
$812$	6525.10	0.282003
$813$	0	0
$814$	1265.00	0.0544696
$815$	0	0
$816$	0	0
$817$	−45299.4	−1.93981
$818$	−21425.8	−0.915813
$819$	0	0
$820$	0	0
$821$	−6593.42	−0.280283	−0.140141	−	0.990132i	$-0.544756\pi$
$821$	−6593.42	−0.280283	−0.140141	+	0.990132i	$0.544756\pi$
$822$	0	0
$823$	−26762.4	−1.13351	−0.566755	−	0.823886i	$-0.691801\pi$
$823$	−26762.4	−1.13351	−0.566755	+	0.823886i	$0.691801\pi$
$824$	−27208.5	−1.15031
$825$	0	0
$826$	1085.29	0.0457169
$827$	−24016.7	−1.00985	−0.504924	−	0.863164i	$-0.668479\pi$
$827$	−24016.7	−1.00985	−0.504924	+	0.863164i	$0.668479\pi$
$828$	0	0
$829$	−28422.3	−1.19077	−0.595383	−	0.803442i	$-0.703000\pi$
$829$	−28422.3	−1.19077	−0.595383	+	0.803442i	$0.703000\pi$
$830$	0	0
$831$	0	0
$832$	−6290.49	−0.262120
$833$	−2436.45	−0.101342
$834$	0	0
$835$	0	0
$836$	6350.46	0.262721
$837$	0	0
$838$	18137.3	0.747663
$839$	−6637.09	−0.273108	−0.136554	−	0.990633i	$-0.543603\pi$
$839$	−6637.09	−0.273108	−0.136554	+	0.990633i	$0.543603\pi$
$840$	0	0
$841$	313.546	0.0128560
$842$	11310.3	0.462921
$843$	0	0
$844$	−30550.9	−1.24598
$845$	0	0
$846$	0	0
$847$	−8911.03	−0.361495
$848$	−7899.01	−0.319874
$849$	0	0
$850$	0	0
$851$	−2707.59	−0.109066
$852$	0	0
$853$	1406.88	0.0564720	0.0282360	−	0.999601i	$-0.491011\pi$
$853$	1406.88	0.0564720	0.0282360	+	0.999601i	$0.491011\pi$
$854$	−9218.40	−0.369376
$855$	0	0
$856$	18309.3	0.731075
$857$	27943.4	1.11380	0.556901	−	0.830579i	$-0.311990\pi$
$857$	27943.4	1.11380	0.556901	+	0.830579i	$0.311990\pi$
$858$	0	0
$859$	1936.63	0.0769233	0.0384616	−	0.999260i	$-0.487754\pi$
$859$	1936.63	0.0769233	0.0384616	+	0.999260i	$0.487754\pi$
$860$	0	0
$861$	0	0
$862$	20556.8	0.812259
$863$	7947.70	0.313491	0.156746	−	0.987639i	$-0.449900\pi$
$863$	7947.70	0.313491	0.156746	+	0.987639i	$0.449900\pi$
$864$	0	0
$865$	0	0
$866$	−19791.6	−0.776611
$867$	0	0
$868$	−5309.58	−0.207625
$869$	−171.984	−0.00671365
$870$	0	0
$871$	−23649.6	−0.920019
$872$	−3216.00	−0.124894
$873$	0	0
$874$	4742.06	0.183527
$875$	0	0
$876$	0	0
$877$	−38655.0	−1.48835	−0.744177	−	0.667983i	$-0.767158\pi$
$877$	−38655.0	−1.48835	−0.744177	+	0.667983i	$0.767158\pi$
$878$	8764.65	0.336893
$879$	0	0
$880$	0	0
$881$	18879.4	0.721978	0.360989	−	0.932570i	$-0.382439\pi$
$881$	18879.4	0.721978	0.360989	+	0.932570i	$0.382439\pi$
$882$	0	0
$883$	35098.1	1.33765	0.668825	−	0.743420i	$-0.266798\pi$
$883$	35098.1	1.33765	0.668825	+	0.743420i	$0.266798\pi$
$884$	−15439.4	−0.587425
$885$	0	0
$886$	−19346.3	−0.733581
$887$	−48816.4	−1.84791	−0.923954	−	0.382504i	$-0.875062\pi$
$887$	−48816.4	−1.84791	−0.923954	+	0.382504i	$0.875062\pi$
$888$	0	0
$889$	1897.11	0.0715713
$890$	0	0
$891$	0	0
$892$	−679.062	−0.0254895
$893$	−10776.7	−0.403839
$894$	0	0
$895$	0	0
$896$	9267.63	0.345547
$897$	0	0
$898$	9916.41	0.368502
$899$	−20100.8	−0.745718
$900$	0	0
$901$	21091.3	0.779860
$902$	2066.22	0.0762721
$903$	0	0
$904$	−35839.1	−1.31857
$905$	0	0
$906$	0	0
$907$	6010.83	0.220051	0.110026	−	0.993929i	$-0.464907\pi$
$907$	6010.83	0.220051	0.110026	+	0.993929i	$0.464907\pi$
$908$	−28140.5	−1.02850
$909$	0	0
$910$	0	0
$911$	−25780.1	−0.937576	−0.468788	−	0.883311i	$-0.655309\pi$
$911$	−25780.1	−0.937576	−0.468788	+	0.883311i	$0.655309\pi$
$912$	0	0
$913$	−8470.37	−0.307041
$914$	−17002.6	−0.615314
$915$	0	0
$916$	31934.6	1.15191
$917$	5342.06	0.192378
$918$	0	0
$919$	26731.8	0.959522	0.479761	−	0.877399i	$-0.340723\pi$
$919$	26731.8	0.959522	0.479761	+	0.877399i	$0.340723\pi$
$920$	0	0
$921$	0	0
$922$	12145.4	0.433824
$923$	47501.6	1.69397
$924$	0	0
$925$	0	0
$926$	−1826.46	−0.0648177
$927$	0	0
$928$	29406.2	1.04020
$929$	30464.6	1.07590	0.537949	−	0.842977i	$-0.319199\pi$
$929$	30464.6	1.07590	0.537949	+	0.842977i	$0.319199\pi$
$930$	0	0
$931$	6889.42	0.242526
$932$	9542.22	0.335371
$933$	0	0
$934$	23708.1	0.830572
$935$	0	0
$936$	0	0
$937$	28533.4	0.994819	0.497409	−	0.867516i	$-0.334285\pi$
$937$	28533.4	0.994819	0.497409	+	0.867516i	$0.334285\pi$
$938$	−4548.46	−0.158329
$939$	0	0
$940$	0	0
$941$	−34455.8	−1.19365	−0.596827	−	0.802370i	$-0.703572\pi$
$941$	−34455.8	−1.19365	−0.596827	+	0.802370i	$0.703572\pi$
$942$	0	0
$943$	−4422.50	−0.152722
$944$	−2007.17	−0.0692033
$945$	0	0
$946$	−3529.39	−0.121301
$947$	2477.68	0.0850198	0.0425099	−	0.999096i	$-0.486465\pi$
$947$	2477.68	0.0850198	0.0425099	+	0.999096i	$0.486465\pi$
$948$	0	0
$949$	39577.5	1.35378
$950$	0	0
$951$	0	0
$952$	−6974.80	−0.237452
$953$	41690.6	1.41709	0.708547	−	0.705663i	$-0.249351\pi$
$953$	41690.6	1.41709	0.708547	+	0.705663i	$0.249351\pi$
$954$	0	0
$955$	0	0
$956$	−18463.5	−0.624635
$957$	0	0
$958$	−2014.13	−0.0679266
$959$	−1686.54	−0.0567895
$960$	0	0
$961$	−13434.6	−0.450962
$962$	8696.44	0.291460
$963$	0	0
$964$	−42325.0	−1.41410
$965$	0	0
$966$	0	0
$967$	20641.3	0.686430	0.343215	−	0.939257i	$-0.388484\pi$
$967$	20641.3	0.686430	0.343215	+	0.939257i	$0.388484\pi$
$968$	−25509.5	−0.847010
$969$	0	0
$970$	0	0
$971$	−6626.17	−0.218995	−0.109497	−	0.993987i	$-0.534924\pi$
$971$	−6626.17	−0.218995	−0.109497	+	0.993987i	$0.534924\pi$
$972$	0	0
$973$	−722.048	−0.0237901
$974$	−20377.0	−0.670349
$975$	0	0
$976$	17048.8	0.559138
$977$	41961.0	1.37405	0.687027	−	0.726632i	$-0.258915\pi$
$977$	41961.0	1.37405	0.687027	+	0.726632i	$0.258915\pi$
$978$	0	0
$979$	11566.5	0.377598
$980$	0	0
$981$	0	0
$982$	6817.21	0.221533
$983$	−16781.7	−0.544510	−0.272255	−	0.962225i	$-0.587769\pi$
$983$	−16781.7	−0.544510	−0.272255	+	0.962225i	$0.587769\pi$
$984$	0	0
$985$	0	0
$986$	−11241.6	−0.363087
$987$	0	0
$988$	43657.2	1.40579
$989$	7554.27	0.242884
$990$	0	0
$991$	50319.8	1.61298	0.806489	−	0.591250i	$-0.201365\pi$
$991$	50319.8	1.61298	0.806489	+	0.591250i	$0.201365\pi$
$992$	−23928.3	−0.765851
$993$	0	0
$994$	9135.85	0.291521
$995$	0	0
$996$	0	0
$997$	−12949.5	−0.411348	−0.205674	−	0.978621i	$-0.565939\pi$
$997$	−12949.5	−0.411348	−0.205674	+	0.978621i	$0.565939\pi$
$998$	16355.2	0.518753
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.m.1.2		2
3.2	odd	2		525.4.a.p.1.1		2
5.4	even	2		315.4.a.m.1.1		2
15.2	even	4		525.4.d.i.274.3		4
15.8	even	4		525.4.d.i.274.2		4
15.14	odd	2		105.4.a.c.1.2	✓	2
35.34	odd	2		2205.4.a.bh.1.1		2
60.59	even	2		1680.4.a.bk.1.1		2
105.104	even	2		735.4.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.c.1.2	✓	2	15.14	odd	2
315.4.a.m.1.1		2	5.4	even	2
525.4.a.p.1.1		2	3.2	odd	2
525.4.d.i.274.2		4	15.8	even	4
525.4.d.i.274.3		4	15.2	even	4
735.4.a.k.1.2		2	105.104	even	2
1575.4.a.m.1.2		2	1.1	even	1	trivial
1680.4.a.bk.1.1		2	60.59	even	2
2205.4.a.bh.1.1		2	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43845 q^{2} -5.93087 q^{4} +7.00000 q^{7} +20.0388 q^{8} -7.61553 q^{11} -52.3542 q^{13} -10.0691 q^{14} +18.6222 q^{16} -49.7235 q^{17} +140.600 q^{19} +10.9545 q^{22} -23.4470 q^{23} +75.3087 q^{26} -41.5161 q^{28} -157.170 q^{29} +127.892 q^{31} -187.098 q^{32} +71.5246 q^{34} +115.477 q^{37} -202.246 q^{38} +188.617 q^{41} -322.186 q^{43} +45.1667 q^{44} +33.7272 q^{46} -76.6477 q^{47} +49.0000 q^{49} +310.506 q^{52} -424.172 q^{53} +140.272 q^{56} +226.081 q^{58} -107.784 q^{59} +915.511 q^{61} -183.966 q^{62} +120.153 q^{64} +451.723 q^{67} +294.903 q^{68} -907.312 q^{71} -755.956 q^{73} -166.108 q^{74} -833.882 q^{76} -53.3087 q^{77} +22.5834 q^{79} -271.316 q^{82} +1112.25 q^{83} +463.447 q^{86} -152.606 q^{88} -1518.81 q^{89} -366.479 q^{91} +139.061 q^{92} +110.254 q^{94} +549.987 q^{97} -70.4839 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + 26 q^{11} - 14 q^{13} - 49 q^{14} + 297 q^{16} + 16 q^{17} + 174 q^{19} - 176 q^{22} + 184 q^{23} - 138 q^{26} + 119 q^{28} + 32 q^{29} + 330 q^{31} - 1071 q^{32}+ \cdots - 343 q^{98}+O(q^{100}) \) 2 * q - 7 * q^2 + 17 * q^4 + 14 * q^7 - 63 * q^8 + 26 * q^11 - 14 * q^13 - 49 * q^14 + 297 * q^16 + 16 * q^17 + 174 * q^19 - 176 * q^22 + 184 * q^23 - 138 * q^26 + 119 * q^28 + 32 * q^29 + 330 * q^31 - 1071 * q^32 - 294 * q^34 + 132 * q^37 - 388 * q^38 - 200 * q^41 - 364 * q^43 + 816 * q^44 - 1120 * q^46 + 292 * q^47 + 98 * q^49 + 1190 * q^52 + 34 * q^53 - 441 * q^56 - 826 * q^58 - 364 * q^59 + 792 * q^61 - 1308 * q^62 + 2809 * q^64 + 788 * q^67 + 1802 * q^68 - 454 * q^71 - 778 * q^73 - 258 * q^74 - 68 * q^76 + 182 * q^77 + 408 * q^79 + 1890 * q^82 + 1136 * q^83 + 696 * q^86 - 2944 * q^88 - 36 * q^89 - 98 * q^91 + 4896 * q^92 - 1940 * q^94 + 498 * q^97 - 343 * q^98

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