LMFDB - Embedded newform 1575.4.a.bk.1.4

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 = [4,0,0,16,0,0,28,-9,0,0,-21] 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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} - 3x + 46 $ x^4 - 24x^2 - 3x + 46
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.60171$ 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)

$f(q)$	$=$	$q+4.60171 q^{2} +13.1758 q^{4} +7.00000 q^{7} +23.8174 q^{8} +52.9465 q^{11} +19.6024 q^{13} +32.2120 q^{14} +4.19486 q^{16} -61.5076 q^{17} +27.0928 q^{19} +243.645 q^{22} -19.2163 q^{23} +90.2046 q^{26} +92.2304 q^{28} +167.409 q^{29} +225.638 q^{31} -171.236 q^{32} -283.040 q^{34} +311.705 q^{37} +124.673 q^{38} -12.8071 q^{41} -114.311 q^{43} +697.611 q^{44} -88.4281 q^{46} +207.919 q^{47} +49.0000 q^{49} +258.277 q^{52} -227.402 q^{53} +166.722 q^{56} +770.369 q^{58} +605.531 q^{59} -315.981 q^{61} +1038.32 q^{62} -821.538 q^{64} +720.254 q^{67} -810.410 q^{68} +56.2766 q^{71} -1154.36 q^{73} +1434.38 q^{74} +356.969 q^{76} +370.625 q^{77} +1152.80 q^{79} -58.9347 q^{82} -692.581 q^{83} -526.029 q^{86} +1261.05 q^{88} -1417.70 q^{89} +137.217 q^{91} -253.190 q^{92} +956.783 q^{94} +661.229 q^{97} +225.484 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{4} + 28 q^{7} - 9 q^{8} - 21 q^{11} + 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 221 q^{22} - 102 q^{23} - 129 q^{26} + 112 q^{28} + 240 q^{29} + 351 q^{31} - 72 q^{32} - 285 q^{34} + 399 q^{37}+ \cdots - 372 q^{97}+O(q^{100}) $ 4 * q + 16 * q^4 + 28 * q^7 - 9 * q^8 - 21 * q^11 + 5 * q^13 + 72 * q^16 - 99 * q^17 + 72 * q^19 + 221 * q^22 - 102 * q^23 - 129 * q^26 + 112 * q^28 + 240 * q^29 + 351 * q^31 - 72 * q^32 - 285 * q^34 + 399 * q^37 - 324 * q^38 - 381 * q^41 + 460 * q^43 + 975 * q^44 + 550 * q^46 - 60 * q^47 + 196 * q^49 + 223 * q^52 - 873 * q^53 - 63 * q^56 + 1408 * q^58 + 855 * q^59 + 687 * q^61 + 477 * q^62 - 1285 * q^64 - 503 * q^67 - 1725 * q^68 + 681 * q^71 + 1228 * q^73 + 3369 * q^74 + 1910 * q^76 - 147 * q^77 + 345 * q^79 - 495 * q^82 - 1509 * q^83 + 540 * q^86 + 2266 * q^88 + 198 * q^89 + 35 * q^91 - 2994 * q^92 - 1732 * q^94 - 372 * q^97

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 $

$2$	4.60171	1.62695	0.813476	−	0.581599i	$-0.197573\pi$
$2$	4.60171	1.62695	0.813476	+	0.581599i	$0.197573\pi$
$3$	0	0
$3$	0	0
$4$	13.1758	1.64697
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	23.8174	1.05259
$9$	0	0
$10$	0	0
$11$	52.9465	1.45127	0.725635	−	0.688080i	$-0.241546\pi$
$11$	52.9465	1.45127	0.725635	+	0.688080i	$0.241546\pi$
$12$	0	0
$13$	19.6024	0.418209	0.209105	−	0.977893i	$-0.432945\pi$
$13$	19.6024	0.418209	0.209105	+	0.977893i	$0.432945\pi$
$14$	32.2120	0.614930
$15$	0	0
$16$	4.19486	0.0655446
$17$	−61.5076	−0.877516	−0.438758	−	0.898605i	$-0.644582\pi$
$17$	−61.5076	−0.877516	−0.438758	+	0.898605i	$0.644582\pi$
$18$	0	0
$19$	27.0928	0.327133	0.163566	−	0.986532i	$-0.447700\pi$
$19$	27.0928	0.327133	0.163566	+	0.986532i	$0.447700\pi$
$20$	0	0
$21$	0	0
$22$	243.645	2.36115
$23$	−19.2163	−0.174212	−0.0871062	−	0.996199i	$-0.527762\pi$
$23$	−19.2163	−0.174212	−0.0871062	+	0.996199i	$0.527762\pi$
$24$	0	0
$25$	0	0
$26$	90.2046	0.680407
$27$	0	0
$28$	92.2304	0.622497
$29$	167.409	1.07197	0.535984	−	0.844228i	$-0.319941\pi$
$29$	167.409	1.07197	0.535984	+	0.844228i	$0.319941\pi$
$30$	0	0
$31$	225.638	1.30728	0.653642	−	0.756804i	$-0.273240\pi$
$31$	225.638	1.30728	0.653642	+	0.756804i	$0.273240\pi$
$32$	−171.236	−0.945954
$33$	0	0
$34$	−283.040	−1.42768
$35$	0	0
$36$	0	0
$37$	311.705	1.38497	0.692485	−	0.721432i	$-0.256516\pi$
$37$	311.705	1.38497	0.692485	+	0.721432i	$0.256516\pi$
$38$	124.673	0.532229
$39$	0	0
$40$	0	0
$41$	−12.8071	−0.0487838	−0.0243919	−	0.999702i	$-0.507765\pi$
$41$	−12.8071	−0.0487838	−0.0243919	+	0.999702i	$0.507765\pi$
$42$	0	0
$43$	−114.311	−0.405403	−0.202702	−	0.979241i	$-0.564972\pi$
$43$	−114.311	−0.405403	−0.202702	+	0.979241i	$0.564972\pi$
$44$	697.611	2.39020
$45$	0	0
$46$	−88.4281	−0.283435
$47$	207.919	0.645278	0.322639	−	0.946522i	$-0.395430\pi$
$47$	207.919	0.645278	0.322639	+	0.946522i	$0.395430\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	258.277	0.688779
$53$	−227.402	−0.589360	−0.294680	−	0.955596i	$-0.595213\pi$
$53$	−227.402	−0.589360	−0.294680	+	0.955596i	$0.595213\pi$
$54$	0	0
$55$	0	0
$56$	166.722	0.397842
$57$	0	0
$58$	770.369	1.74404
$59$	605.531	1.33616	0.668080	−	0.744090i	$-0.267116\pi$
$59$	605.531	1.33616	0.668080	+	0.744090i	$0.267116\pi$
$60$	0	0
$61$	−315.981	−0.663232	−0.331616	−	0.943414i	$-0.607594\pi$
$61$	−315.981	−0.663232	−0.331616	+	0.943414i	$0.607594\pi$
$62$	1038.32	2.12689
$63$	0	0
$64$	−821.538	−1.60457
$65$	0	0
$66$	0	0
$67$	720.254	1.31333	0.656664	−	0.754183i	$-0.271967\pi$
$67$	720.254	1.31333	0.656664	+	0.754183i	$0.271967\pi$
$68$	−810.410	−1.44524
$69$	0	0
$70$	0	0
$71$	56.2766	0.0940676	0.0470338	−	0.998893i	$-0.485023\pi$
$71$	56.2766	0.0940676	0.0470338	+	0.998893i	$0.485023\pi$
$72$	0	0
$73$	−1154.36	−1.85079	−0.925393	−	0.379010i	$-0.876265\pi$
$73$	−1154.36	−1.85079	−0.925393	+	0.379010i	$0.876265\pi$
$74$	1434.38	2.25328
$75$	0	0
$76$	356.969	0.538778
$77$	370.625	0.548528
$78$	0	0
$79$	1152.80	1.64177	0.820885	−	0.571093i	$-0.193480\pi$
$79$	1152.80	1.64177	0.820885	+	0.571093i	$0.193480\pi$
$80$	0	0
$81$	0	0
$82$	−58.9347	−0.0793689
$83$	−692.581	−0.915911	−0.457956	−	0.888975i	$-0.651418\pi$
$83$	−692.581	−0.915911	−0.457956	+	0.888975i	$0.651418\pi$
$84$	0	0
$85$	0	0
$86$	−526.029	−0.659571
$87$	0	0
$88$	1261.05	1.52759
$89$	−1417.70	−1.68849	−0.844246	−	0.535956i	$-0.819951\pi$
$89$	−1417.70	−1.68849	−0.844246	+	0.535956i	$0.819951\pi$
$90$	0	0
$91$	137.217	0.158068
$92$	−253.190	−0.286923
$93$	0	0
$94$	956.783	1.04984
$95$	0	0
$96$	0	0
$97$	661.229	0.692140	0.346070	−	0.938209i	$-0.387516\pi$
$97$	661.229	0.692140	0.346070	+	0.938209i	$0.387516\pi$
$98$	225.484	0.232422
$99$	0	0
$100$	0	0
$101$	−187.231	−0.184458	−0.0922289	−	0.995738i	$-0.529399\pi$
$101$	−187.231	−0.184458	−0.0922289	+	0.995738i	$0.529399\pi$
$102$	0	0
$103$	1076.28	1.02961	0.514803	−	0.857308i	$-0.327865\pi$
$103$	1076.28	1.02961	0.514803	+	0.857308i	$0.327865\pi$
$104$	466.879	0.440204
$105$	0	0
$106$	−1046.44	−0.958861
$107$	−1591.65	−1.43805	−0.719024	−	0.694986i	$-0.755411\pi$
$107$	−1591.65	−1.43805	−0.719024	+	0.694986i	$0.755411\pi$
$108$	0	0
$109$	1706.29	1.49938	0.749692	−	0.661786i	$-0.230201\pi$
$109$	1706.29	1.49938	0.749692	+	0.661786i	$0.230201\pi$
$110$	0	0
$111$	0	0
$112$	29.3640	0.0247735
$113$	560.000	0.466198	0.233099	−	0.972453i	$-0.425113\pi$
$113$	560.000	0.466198	0.233099	+	0.972453i	$0.425113\pi$
$114$	0	0
$115$	0	0
$116$	2205.74	1.76550
$117$	0	0
$118$	2786.48	2.17387
$119$	−430.553	−0.331670
$120$	0	0
$121$	1472.33	1.10618
$122$	−1454.05	−1.07905
$123$	0	0
$124$	2972.96	2.15306
$125$	0	0
$126$	0	0
$127$	2098.96	1.46655	0.733276	−	0.679931i	$-0.237990\pi$
$127$	2098.96	1.46655	0.733276	+	0.679931i	$0.237990\pi$
$128$	−2410.60	−1.66460
$129$	0	0
$130$	0	0
$131$	−1675.28	−1.11733	−0.558664	−	0.829394i	$-0.688686\pi$
$131$	−1675.28	−1.11733	−0.558664	+	0.829394i	$0.688686\pi$
$132$	0	0
$133$	189.650	0.123644
$134$	3314.40	2.13672
$135$	0	0
$136$	−1464.95	−0.923667
$137$	2678.78	1.67054	0.835269	−	0.549841i	$-0.185312\pi$
$137$	2678.78	1.67054	0.835269	+	0.549841i	$0.185312\pi$
$138$	0	0
$139$	1032.11	0.629800	0.314900	−	0.949125i	$-0.398029\pi$
$139$	1032.11	0.629800	0.314900	+	0.949125i	$0.398029\pi$
$140$	0	0
$141$	0	0
$142$	258.969	0.153043
$143$	1037.88	0.606935
$144$	0	0
$145$	0	0
$146$	−5312.02	−3.01114
$147$	0	0
$148$	4106.95	2.28101
$149$	−1187.33	−0.652815	−0.326408	−	0.945229i	$-0.605838\pi$
$149$	−1187.33	−0.652815	−0.326408	+	0.945229i	$0.605838\pi$
$150$	0	0
$151$	−138.484	−0.0746338	−0.0373169	−	0.999303i	$-0.511881\pi$
$151$	−138.484	−0.0746338	−0.0373169	+	0.999303i	$0.511881\pi$
$152$	645.282	0.344337
$153$	0	0
$154$	1705.51	0.892429
$155$	0	0
$156$	0	0
$157$	441.575	0.224469	0.112234	−	0.993682i	$-0.464199\pi$
$157$	441.575	0.224469	0.112234	+	0.993682i	$0.464199\pi$
$158$	5304.85	2.67108
$159$	0	0
$160$	0	0
$161$	−134.514	−0.0658461
$162$	0	0
$163$	2116.62	1.01710	0.508548	−	0.861034i	$-0.330183\pi$
$163$	2116.62	1.01710	0.508548	+	0.861034i	$0.330183\pi$
$164$	−168.744	−0.0803455
$165$	0	0
$166$	−3187.06	−1.49014
$167$	−843.566	−0.390881	−0.195440	−	0.980716i	$-0.562614\pi$
$167$	−843.566	−0.390881	−0.195440	+	0.980716i	$0.562614\pi$
$168$	0	0
$169$	−1812.75	−0.825101
$170$	0	0
$171$	0	0
$172$	−1506.14	−0.667687
$173$	4319.01	1.89808	0.949042	−	0.315150i	$-0.102055\pi$
$173$	4319.01	1.89808	0.949042	+	0.315150i	$0.102055\pi$
$174$	0	0
$175$	0	0
$176$	222.103	0.0951229
$177$	0	0
$178$	−6523.84	−2.74709
$179$	421.744	0.176104	0.0880520	−	0.996116i	$-0.471936\pi$
$179$	421.744	0.176104	0.0880520	+	0.996116i	$0.471936\pi$
$180$	0	0
$181$	791.205	0.324916	0.162458	−	0.986715i	$-0.448058\pi$
$181$	791.205	0.324916	0.162458	+	0.986715i	$0.448058\pi$
$182$	631.432	0.257170
$183$	0	0
$184$	−457.684	−0.183375
$185$	0	0
$186$	0	0
$187$	−3256.61	−1.27351
$188$	2739.49	1.06276
$189$	0	0
$190$	0	0
$191$	−1790.22	−0.678199	−0.339100	−	0.940750i	$-0.610122\pi$
$191$	−1790.22	−0.678199	−0.339100	+	0.940750i	$0.610122\pi$
$192$	0	0
$193$	−2301.10	−0.858221	−0.429111	−	0.903252i	$-0.641173\pi$
$193$	−2301.10	−0.858221	−0.429111	+	0.903252i	$0.641173\pi$
$194$	3042.79	1.12608
$195$	0	0
$196$	645.613	0.235282
$197$	−3490.59	−1.26241	−0.631203	−	0.775617i	$-0.717439\pi$
$197$	−3490.59	−1.26241	−0.631203	+	0.775617i	$0.717439\pi$
$198$	0	0
$199$	822.357	0.292942	0.146471	−	0.989215i	$-0.453209\pi$
$199$	822.357	0.292942	0.146471	+	0.989215i	$0.453209\pi$
$200$	0	0
$201$	0	0
$202$	−861.586	−0.300104
$203$	1171.86	0.405166
$204$	0	0
$205$	0	0
$206$	4952.75	1.67512
$207$	0	0
$208$	82.2292	0.0274114
$209$	1434.47	0.474757
$210$	0	0
$211$	−2323.79	−0.758181	−0.379091	−	0.925360i	$-0.623763\pi$
$211$	−2323.79	−0.758181	−0.379091	+	0.925360i	$0.623763\pi$
$212$	−2996.20	−0.970660
$213$	0	0
$214$	−7324.34	−2.33963
$215$	0	0
$216$	0	0
$217$	1579.47	0.494107
$218$	7851.86	2.43943
$219$	0	0
$220$	0	0
$221$	−1205.70	−0.366986
$222$	0	0
$223$	−2526.13	−0.758574	−0.379287	−	0.925279i	$-0.623831\pi$
$223$	−2526.13	−0.758574	−0.379287	+	0.925279i	$0.623831\pi$
$224$	−1198.65	−0.357537
$225$	0	0
$226$	2576.96	0.758482
$227$	−3924.06	−1.14735	−0.573676	−	0.819082i	$-0.694483\pi$
$227$	−3924.06	−1.14735	−0.573676	+	0.819082i	$0.694483\pi$
$228$	0	0
$229$	3591.55	1.03640	0.518201	−	0.855259i	$-0.326602\pi$
$229$	3591.55	1.03640	0.518201	+	0.855259i	$0.326602\pi$
$230$	0	0
$231$	0	0
$232$	3987.26	1.12835
$233$	−4957.57	−1.39391	−0.696955	−	0.717115i	$-0.745462\pi$
$233$	−4957.57	−1.39391	−0.696955	+	0.717115i	$0.745462\pi$
$234$	0	0
$235$	0	0
$236$	7978.34	2.20062
$237$	0	0
$238$	−1981.28	−0.539611
$239$	5663.57	1.53283	0.766414	−	0.642347i	$-0.222039\pi$
$239$	5663.57	1.53283	0.766414	+	0.642347i	$0.222039\pi$
$240$	0	0
$241$	−2246.64	−0.600493	−0.300246	−	0.953862i	$-0.597069\pi$
$241$	−2246.64	−0.600493	−0.300246	+	0.953862i	$0.597069\pi$
$242$	6775.24	1.79971
$243$	0	0
$244$	−4163.29	−1.09232
$245$	0	0
$246$	0	0
$247$	531.084	0.136810
$248$	5374.12	1.37604
$249$	0	0
$250$	0	0
$251$	3382.51	0.850606	0.425303	−	0.905051i	$-0.360167\pi$
$251$	3382.51	0.850606	0.425303	+	0.905051i	$0.360167\pi$
$252$	0	0
$253$	−1017.44	−0.252829
$254$	9658.79	2.38601
$255$	0	0
$256$	−4520.57	−1.10365
$257$	−8116.97	−1.97013	−0.985064	−	0.172190i	$-0.944916\pi$
$257$	−8116.97	−1.97013	−0.985064	+	0.172190i	$0.944916\pi$
$258$	0	0
$259$	2181.93	0.523470
$260$	0	0
$261$	0	0
$262$	−7709.17	−1.81784
$263$	−3643.53	−0.854257	−0.427129	−	0.904191i	$-0.640475\pi$
$263$	−3643.53	−0.854257	−0.427129	+	0.904191i	$0.640475\pi$
$264$	0	0
$265$	0	0
$266$	872.714	0.201164
$267$	0	0
$268$	9489.90	2.16302
$269$	−7509.24	−1.70203	−0.851015	−	0.525141i	$-0.824013\pi$
$269$	−7509.24	−1.70203	−0.851015	+	0.525141i	$0.824013\pi$
$270$	0	0
$271$	−8350.12	−1.87171	−0.935855	−	0.352385i	$-0.885371\pi$
$271$	−8350.12	−1.87171	−0.935855	+	0.352385i	$0.885371\pi$
$272$	−258.015	−0.0575165
$273$	0	0
$274$	12327.0	2.71788
$275$	0	0
$276$	0	0
$277$	3359.70	0.728753	0.364377	−	0.931252i	$-0.381282\pi$
$277$	3359.70	0.728753	0.364377	+	0.931252i	$0.381282\pi$
$278$	4749.47	1.02465
$279$	0	0
$280$	0	0
$281$	−6403.33	−1.35940	−0.679699	−	0.733491i	$-0.737890\pi$
$281$	−6403.33	−1.35940	−0.679699	+	0.733491i	$0.737890\pi$
$282$	0	0
$283$	−5862.59	−1.23143	−0.615715	−	0.787969i	$-0.711133\pi$
$283$	−5862.59	−1.23143	−0.615715	+	0.787969i	$0.711133\pi$
$284$	741.487	0.154927
$285$	0	0
$286$	4776.01	0.987453
$287$	−89.6498	−0.0184385
$288$	0	0
$289$	−1129.82	−0.229965
$290$	0	0
$291$	0	0
$292$	−15209.6	−3.04819
$293$	−3866.04	−0.770840	−0.385420	−	0.922741i	$-0.625944\pi$
$293$	−3866.04	−0.770840	−0.385420	+	0.922741i	$0.625944\pi$
$294$	0	0
$295$	0	0
$296$	7424.01	1.45781
$297$	0	0
$298$	−5463.73	−1.06210
$299$	−376.686	−0.0728573
$300$	0	0
$301$	−800.180	−0.153228
$302$	−637.266	−0.121426
$303$	0	0
$304$	113.650	0.0214418
$305$	0	0
$306$	0	0
$307$	7084.81	1.31711	0.658553	−	0.752535i	$-0.271169\pi$
$307$	7084.81	1.31711	0.658553	+	0.752535i	$0.271169\pi$
$308$	4883.28	0.903411
$309$	0	0
$310$	0	0
$311$	10616.6	1.93572	0.967862	−	0.251483i	$-0.0809184\pi$
$311$	10616.6	1.93572	0.967862	+	0.251483i	$0.0809184\pi$
$312$	0	0
$313$	−7247.78	−1.30885	−0.654423	−	0.756128i	$-0.727089\pi$
$313$	−7247.78	−1.30885	−0.654423	+	0.756128i	$0.727089\pi$
$314$	2032.00	0.365199
$315$	0	0
$316$	15189.0	2.70395
$317$	−6308.24	−1.11768	−0.558842	−	0.829274i	$-0.688754\pi$
$317$	−6308.24	−1.11768	−0.558842	+	0.829274i	$0.688754\pi$
$318$	0	0
$319$	8863.72	1.55572
$320$	0	0
$321$	0	0
$322$	−618.997	−0.107128
$323$	−1666.41	−0.287064
$324$	0	0
$325$	0	0
$326$	9740.09	1.65477
$327$	0	0
$328$	−305.033	−0.0513494
$329$	1455.43	0.243892
$330$	0	0
$331$	6656.25	1.10532	0.552660	−	0.833407i	$-0.313613\pi$
$331$	6656.25	1.10532	0.552660	+	0.833407i	$0.313613\pi$
$332$	−9125.29	−1.50848
$333$	0	0
$334$	−3881.85	−0.635944
$335$	0	0
$336$	0	0
$337$	−11941.8	−1.93030	−0.965152	−	0.261691i	$-0.915720\pi$
$337$	−11941.8	−1.93030	−0.965152	+	0.261691i	$0.915720\pi$
$338$	−8341.74	−1.34240
$339$	0	0
$340$	0	0
$341$	11946.7	1.89722
$342$	0	0
$343$	343.000	0.0539949
$344$	−2722.61	−0.426724
$345$	0	0
$346$	19874.9	3.08809
$347$	6390.65	0.988669	0.494335	−	0.869272i	$-0.335412\pi$
$347$	6390.65	0.988669	0.494335	+	0.869272i	$0.335412\pi$
$348$	0	0
$349$	−7760.54	−1.19029	−0.595147	−	0.803617i	$-0.702906\pi$
$349$	−7760.54	−1.19029	−0.595147	+	0.803617i	$0.702906\pi$
$350$	0	0
$351$	0	0
$352$	−9066.34	−1.37283
$353$	−5416.46	−0.816684	−0.408342	−	0.912829i	$-0.633893\pi$
$353$	−5416.46	−0.816684	−0.408342	+	0.912829i	$0.633893\pi$
$354$	0	0
$355$	0	0
$356$	−18679.3	−2.78090
$357$	0	0
$358$	1940.74	0.286513
$359$	2563.26	0.376834	0.188417	−	0.982089i	$-0.439664\pi$
$359$	2563.26	0.376834	0.188417	+	0.982089i	$0.439664\pi$
$360$	0	0
$361$	−6124.98	−0.892984
$362$	3640.90	0.528623
$363$	0	0
$364$	1807.94	0.260334
$365$	0	0
$366$	0	0
$367$	−10118.8	−1.43923	−0.719614	−	0.694374i	$-0.755681\pi$
$367$	−10118.8	−1.43923	−0.719614	+	0.694374i	$0.755681\pi$
$368$	−80.6098	−0.0114187
$369$	0	0
$370$	0	0
$371$	−1591.82	−0.222757
$372$	0	0
$373$	1870.64	0.259673	0.129837	−	0.991535i	$-0.458555\pi$
$373$	1870.64	0.259673	0.129837	+	0.991535i	$0.458555\pi$
$374$	−14986.0	−2.07194
$375$	0	0
$376$	4952.09	0.679215
$377$	3281.62	0.448307
$378$	0	0
$379$	3804.87	0.515681	0.257841	−	0.966187i	$-0.416989\pi$
$379$	3804.87	0.515681	0.257841	+	0.966187i	$0.416989\pi$
$380$	0	0
$381$	0	0
$382$	−8238.10	−1.10340
$383$	−11709.3	−1.56219	−0.781094	−	0.624414i	$-0.785338\pi$
$383$	−11709.3	−1.56219	−0.781094	+	0.624414i	$0.785338\pi$
$384$	0	0
$385$	0	0
$386$	−10589.0	−1.39628
$387$	0	0
$388$	8712.20	1.13994
$389$	1278.24	0.166605	0.0833023	−	0.996524i	$-0.473453\pi$
$389$	1278.24	0.166605	0.0833023	+	0.996524i	$0.473453\pi$
$390$	0	0
$391$	1181.95	0.152874
$392$	1167.05	0.150370
$393$	0	0
$394$	−16062.7	−2.05387
$395$	0	0
$396$	0	0
$397$	2101.91	0.265723	0.132862	−	0.991135i	$-0.457583\pi$
$397$	2101.91	0.265723	0.132862	+	0.991135i	$0.457583\pi$
$398$	3784.25	0.476602
$399$	0	0
$400$	0	0
$401$	11239.5	1.39969	0.699846	−	0.714294i	$-0.253252\pi$
$401$	11239.5	1.39969	0.699846	+	0.714294i	$0.253252\pi$
$402$	0	0
$403$	4423.04	0.546718
$404$	−2466.92	−0.303797
$405$	0	0
$406$	5392.58	0.659186
$407$	16503.7	2.00997
$408$	0	0
$409$	6751.66	0.816254	0.408127	−	0.912925i	$-0.366182\pi$
$409$	6751.66	0.816254	0.408127	+	0.912925i	$0.366182\pi$
$410$	0	0
$411$	0	0
$412$	14180.9	1.69573
$413$	4238.72	0.505021
$414$	0	0
$415$	0	0
$416$	−3356.63	−0.395607
$417$	0	0
$418$	6601.02	0.772408
$419$	8449.93	0.985218	0.492609	−	0.870251i	$-0.336043\pi$
$419$	8449.93	0.985218	0.492609	+	0.870251i	$0.336043\pi$
$420$	0	0
$421$	2267.58	0.262506	0.131253	−	0.991349i	$-0.458100\pi$
$421$	2267.58	0.262506	0.131253	+	0.991349i	$0.458100\pi$
$422$	−10693.4	−1.23352
$423$	0	0
$424$	−5416.14	−0.620356
$425$	0	0
$426$	0	0
$427$	−2211.86	−0.250678
$428$	−20971.3	−2.36842
$429$	0	0
$430$	0	0
$431$	3172.81	0.354591	0.177296	−	0.984158i	$-0.443265\pi$
$431$	3172.81	0.354591	0.177296	+	0.984158i	$0.443265\pi$
$432$	0	0
$433$	−13569.4	−1.50601	−0.753006	−	0.658013i	$-0.771397\pi$
$433$	−13569.4	−1.50601	−0.753006	+	0.658013i	$0.771397\pi$
$434$	7268.25	0.803888
$435$	0	0
$436$	22481.7	2.46944
$437$	−520.625	−0.0569905
$438$	0	0
$439$	−12021.7	−1.30698	−0.653488	−	0.756936i	$-0.726695\pi$
$439$	−12021.7	−1.30698	−0.653488	+	0.756936i	$0.726695\pi$
$440$	0	0
$441$	0	0
$442$	−5548.26	−0.597068
$443$	−2072.36	−0.222259	−0.111129	−	0.993806i	$-0.535447\pi$
$443$	−2072.36	−0.222259	−0.111129	+	0.993806i	$0.535447\pi$
$444$	0	0
$445$	0	0
$446$	−11624.5	−1.23416
$447$	0	0
$448$	−5750.77	−0.606469
$449$	−154.727	−0.0162628	−0.00813142	−	0.999967i	$-0.502588\pi$
$449$	−154.727	−0.0162628	−0.00813142	+	0.999967i	$0.502588\pi$
$450$	0	0
$451$	−678.092	−0.0707984
$452$	7378.44	0.767815
$453$	0	0
$454$	−18057.4	−1.86669
$455$	0	0
$456$	0	0
$457$	4347.26	0.444981	0.222491	−	0.974935i	$-0.428581\pi$
$457$	4347.26	0.444981	0.222491	+	0.974935i	$0.428581\pi$
$458$	16527.3	1.68618
$459$	0	0
$460$	0	0
$461$	5510.76	0.556750	0.278375	−	0.960473i	$-0.410204\pi$
$461$	5510.76	0.556750	0.278375	+	0.960473i	$0.410204\pi$
$462$	0	0
$463$	9166.14	0.920058	0.460029	−	0.887904i	$-0.347839\pi$
$463$	9166.14	0.920058	0.460029	+	0.887904i	$0.347839\pi$
$464$	702.257	0.0702618
$465$	0	0
$466$	−22813.3	−2.26783
$467$	−9008.92	−0.892683	−0.446342	−	0.894863i	$-0.647273\pi$
$467$	−9008.92	−0.892683	−0.446342	+	0.894863i	$0.647273\pi$
$468$	0	0
$469$	5041.78	0.496392
$470$	0	0
$471$	0	0
$472$	14422.2	1.40643
$473$	−6052.39	−0.588349
$474$	0	0
$475$	0	0
$476$	−5672.87	−0.546251
$477$	0	0
$478$	26062.1	2.49384
$479$	−10128.1	−0.966108	−0.483054	−	0.875590i	$-0.660473\pi$
$479$	−10128.1	−0.966108	−0.483054	+	0.875590i	$0.660473\pi$
$480$	0	0
$481$	6110.15	0.579208
$482$	−10338.4	−0.976973
$483$	0	0
$484$	19399.1	1.82185
$485$	0	0
$486$	0	0
$487$	−954.443	−0.0888089	−0.0444045	−	0.999014i	$-0.514139\pi$
$487$	−954.443	−0.0888089	−0.0444045	+	0.999014i	$0.514139\pi$
$488$	−7525.85	−0.698113
$489$	0	0
$490$	0	0
$491$	−7336.23	−0.674296	−0.337148	−	0.941452i	$-0.609462\pi$
$491$	−7336.23	−0.674296	−0.337148	+	0.941452i	$0.609462\pi$
$492$	0	0
$493$	−10296.9	−0.940670
$494$	2443.90	0.222583
$495$	0	0
$496$	946.519	0.0856854
$497$	393.936	0.0355542
$498$	0	0
$499$	16572.9	1.48679	0.743393	−	0.668855i	$-0.233215\pi$
$499$	16572.9	1.48679	0.743393	+	0.668855i	$0.233215\pi$
$500$	0	0
$501$	0	0
$502$	15565.3	1.38390
$503$	−13628.8	−1.20811	−0.604053	−	0.796944i	$-0.706448\pi$
$503$	−13628.8	−1.20811	−0.604053	+	0.796944i	$0.706448\pi$
$504$	0	0
$505$	0	0
$506$	−4681.96	−0.411341
$507$	0	0
$508$	27655.4	2.41537
$509$	8370.77	0.728935	0.364467	−	0.931216i	$-0.381251\pi$
$509$	8370.77	0.728935	0.364467	+	0.931216i	$0.381251\pi$
$510$	0	0
$511$	−8080.50	−0.699531
$512$	−1517.60	−0.130994
$513$	0	0
$514$	−37352.0	−3.20530
$515$	0	0
$516$	0	0
$517$	11008.6	0.936473
$518$	10040.6	0.851660
$519$	0	0
$520$	0	0
$521$	705.491	0.0593246	0.0296623	−	0.999560i	$-0.490557\pi$
$521$	705.491	0.0593246	0.0296623	+	0.999560i	$0.490557\pi$
$522$	0	0
$523$	−4556.70	−0.380977	−0.190488	−	0.981689i	$-0.561007\pi$
$523$	−4556.70	−0.380977	−0.190488	+	0.981689i	$0.561007\pi$
$524$	−22073.1	−1.84021
$525$	0	0
$526$	−16766.5	−1.38984
$527$	−13878.4	−1.14716
$528$	0	0
$529$	−11797.7	−0.969650
$530$	0	0
$531$	0	0
$532$	2498.78	0.203639
$533$	−251.050	−0.0204018
$534$	0	0
$535$	0	0
$536$	17154.6	1.38240
$537$	0	0
$538$	−34555.4	−2.76912
$539$	2594.38	0.207324
$540$	0	0
$541$	−21336.7	−1.69563	−0.847817	−	0.530289i	$-0.822084\pi$
$541$	−21336.7	−1.69563	−0.847817	+	0.530289i	$0.822084\pi$
$542$	−38424.9	−3.04518
$543$	0	0
$544$	10532.3	0.830090
$545$	0	0
$546$	0	0
$547$	−957.875	−0.0748734	−0.0374367	−	0.999299i	$-0.511919\pi$
$547$	−957.875	−0.0748734	−0.0374367	+	0.999299i	$0.511919\pi$
$548$	35295.0	2.75133
$549$	0	0
$550$	0	0
$551$	4535.58	0.350676
$552$	0	0
$553$	8069.58	0.620531
$554$	15460.4	1.18565
$555$	0	0
$556$	13598.8	1.03726
$557$	13357.8	1.01614	0.508069	−	0.861316i	$-0.330359\pi$
$557$	13357.8	1.01614	0.508069	+	0.861316i	$0.330359\pi$
$558$	0	0
$559$	−2240.78	−0.169543
$560$	0	0
$561$	0	0
$562$	−29466.3	−2.21167
$563$	−14988.5	−1.12201	−0.561003	−	0.827814i	$-0.689584\pi$
$563$	−14988.5	−1.12201	−0.561003	+	0.827814i	$0.689584\pi$
$564$	0	0
$565$	0	0
$566$	−26978.0	−2.00348
$567$	0	0
$568$	1340.36	0.0990148
$569$	12901.9	0.950572	0.475286	−	0.879831i	$-0.342345\pi$
$569$	12901.9	0.950572	0.475286	+	0.879831i	$0.342345\pi$
$570$	0	0
$571$	−19768.4	−1.44883	−0.724414	−	0.689365i	$-0.757890\pi$
$571$	−19768.4	−1.44883	−0.724414	+	0.689365i	$0.757890\pi$
$572$	13674.8	0.999604
$573$	0	0
$574$	−412.543	−0.0299986
$575$	0	0
$576$	0	0
$577$	−5598.68	−0.403945	−0.201972	−	0.979391i	$-0.564735\pi$
$577$	−5598.68	−0.403945	−0.201972	+	0.979391i	$0.564735\pi$
$578$	−5199.10	−0.374142
$579$	0	0
$580$	0	0
$581$	−4848.07	−0.346182
$582$	0	0
$583$	−12040.2	−0.855321
$584$	−27493.8	−1.94812
$585$	0	0
$586$	−17790.4	−1.25412
$587$	−2915.30	−0.204987	−0.102494	−	0.994734i	$-0.532682\pi$
$587$	−2915.30	−0.204987	−0.102494	+	0.994734i	$0.532682\pi$
$588$	0	0
$589$	6113.17	0.427655
$590$	0	0
$591$	0	0
$592$	1307.56	0.0907774
$593$	6126.89	0.424285	0.212143	−	0.977239i	$-0.431956\pi$
$593$	6126.89	0.424285	0.212143	+	0.977239i	$0.431956\pi$
$594$	0	0
$595$	0	0
$596$	−15643.9	−1.07517
$597$	0	0
$598$	−1733.40	−0.118535
$599$	13503.6	0.921104	0.460552	−	0.887633i	$-0.347652\pi$
$599$	13503.6	0.921104	0.460552	+	0.887633i	$0.347652\pi$
$600$	0	0
$601$	−28199.1	−1.91392	−0.956960	−	0.290220i	$-0.906272\pi$
$601$	−28199.1	−1.91392	−0.956960	+	0.290220i	$0.906272\pi$
$602$	−3682.20	−0.249294
$603$	0	0
$604$	−1824.64	−0.122920
$605$	0	0
$606$	0	0
$607$	−13557.2	−0.906543	−0.453272	−	0.891372i	$-0.649743\pi$
$607$	−13557.2	−0.906543	−0.453272	+	0.891372i	$0.649743\pi$
$608$	−4639.27	−0.309452
$609$	0	0
$610$	0	0
$611$	4075.70	0.269861
$612$	0	0
$613$	−10357.7	−0.682452	−0.341226	−	0.939981i	$-0.610842\pi$
$613$	−10357.7	−0.682452	−0.341226	+	0.939981i	$0.610842\pi$
$614$	32602.3	2.14287
$615$	0	0
$616$	8827.35	0.577377
$617$	9433.74	0.615540	0.307770	−	0.951461i	$-0.400417\pi$
$617$	9433.74	0.615540	0.307770	+	0.951461i	$0.400417\pi$
$618$	0	0
$619$	−5686.15	−0.369217	−0.184609	−	0.982812i	$-0.559102\pi$
$619$	−5686.15	−0.369217	−0.184609	+	0.982812i	$0.559102\pi$
$620$	0	0
$621$	0	0
$622$	48854.4	3.14933
$623$	−9923.89	−0.638190
$624$	0	0
$625$	0	0
$626$	−33352.2	−2.12943
$627$	0	0
$628$	5818.10	0.369693
$629$	−19172.2	−1.21533
$630$	0	0
$631$	24059.8	1.51792	0.758959	−	0.651138i	$-0.225708\pi$
$631$	24059.8	1.51792	0.758959	+	0.651138i	$0.225708\pi$
$632$	27456.7	1.72811
$633$	0	0
$634$	−29028.7	−1.81842
$635$	0	0
$636$	0	0
$637$	960.517	0.0597442
$638$	40788.3	2.53107
$639$	0	0
$640$	0	0
$641$	−18543.7	−1.14264	−0.571319	−	0.820728i	$-0.693568\pi$
$641$	−18543.7	−1.14264	−0.571319	+	0.820728i	$0.693568\pi$
$642$	0	0
$643$	6491.72	0.398147	0.199073	−	0.979985i	$-0.436207\pi$
$643$	6491.72	0.398147	0.199073	+	0.979985i	$0.436207\pi$
$644$	−1772.33	−0.108447
$645$	0	0
$646$	−7668.36	−0.467040
$647$	12238.1	0.743630	0.371815	−	0.928307i	$-0.378736\pi$
$647$	12238.1	0.743630	0.371815	+	0.928307i	$0.378736\pi$
$648$	0	0
$649$	32060.7	1.93913
$650$	0	0
$651$	0	0
$652$	27888.1	1.67513
$653$	2507.00	0.150239	0.0751197	−	0.997175i	$-0.476066\pi$
$653$	2507.00	0.150239	0.0751197	+	0.997175i	$0.476066\pi$
$654$	0	0
$655$	0	0
$656$	−53.7240	−0.00319752
$657$	0	0
$658$	6697.48	0.396801
$659$	16501.0	0.975398	0.487699	−	0.873012i	$-0.337836\pi$
$659$	16501.0	0.975398	0.487699	+	0.873012i	$0.337836\pi$
$660$	0	0
$661$	19086.4	1.12311	0.561554	−	0.827440i	$-0.310204\pi$
$661$	19086.4	1.12311	0.561554	+	0.827440i	$0.310204\pi$
$662$	30630.2	1.79830
$663$	0	0
$664$	−16495.5	−0.964081
$665$	0	0
$666$	0	0
$667$	−3216.99	−0.186750
$668$	−11114.6	−0.643770
$669$	0	0
$670$	0	0
$671$	−16730.1	−0.962529
$672$	0	0
$673$	−28326.1	−1.62242	−0.811211	−	0.584754i	$-0.801191\pi$
$673$	−28326.1	−1.62242	−0.811211	+	0.584754i	$0.801191\pi$
$674$	−54952.8	−3.14051
$675$	0	0
$676$	−23884.3	−1.35892
$677$	−9012.99	−0.511665	−0.255833	−	0.966721i	$-0.582350\pi$
$677$	−9012.99	−0.511665	−0.255833	+	0.966721i	$0.582350\pi$
$678$	0	0
$679$	4628.60	0.261604
$680$	0	0
$681$	0	0
$682$	54975.5	3.08669
$683$	3111.96	0.174343	0.0871713	−	0.996193i	$-0.472217\pi$
$683$	3111.96	0.174343	0.0871713	+	0.996193i	$0.472217\pi$
$684$	0	0
$685$	0	0
$686$	1578.39	0.0878471
$687$	0	0
$688$	−479.520	−0.0265720
$689$	−4457.63	−0.246476
$690$	0	0
$691$	21905.5	1.20597	0.602984	−	0.797753i	$-0.293978\pi$
$691$	21905.5	1.20597	0.602984	+	0.797753i	$0.293978\pi$
$692$	56906.3	3.12609
$693$	0	0
$694$	29408.0	1.60852
$695$	0	0
$696$	0	0
$697$	787.735	0.0428086
$698$	−35711.8	−1.93655
$699$	0	0
$700$	0	0
$701$	13337.0	0.718592	0.359296	−	0.933224i	$-0.383017\pi$
$701$	13337.0	0.718592	0.359296	+	0.933224i	$0.383017\pi$
$702$	0	0
$703$	8444.95	0.453069
$704$	−43497.5	−2.32866
$705$	0	0
$706$	−24925.0	−1.32870
$707$	−1310.62	−0.0697185
$708$	0	0
$709$	31571.2	1.67233	0.836164	−	0.548480i	$-0.184793\pi$
$709$	31571.2	1.67233	0.836164	+	0.548480i	$0.184793\pi$
$710$	0	0
$711$	0	0
$712$	−33766.0	−1.77729
$713$	−4335.94	−0.227745
$714$	0	0
$715$	0	0
$716$	5556.80	0.290038
$717$	0	0
$718$	11795.4	0.613091
$719$	1983.80	0.102897	0.0514486	−	0.998676i	$-0.483616\pi$
$719$	1983.80	0.102897	0.0514486	+	0.998676i	$0.483616\pi$
$720$	0	0
$721$	7533.99	0.389155
$722$	−28185.4	−1.45284
$723$	0	0
$724$	10424.7	0.535127
$725$	0	0
$726$	0	0
$727$	28855.0	1.47204	0.736020	−	0.676960i	$-0.236703\pi$
$727$	28855.0	1.47204	0.736020	+	0.676960i	$0.236703\pi$
$728$	3268.15	0.166381
$729$	0	0
$730$	0	0
$731$	7031.02	0.355748
$732$	0	0
$733$	2709.63	0.136538	0.0682691	−	0.997667i	$-0.478252\pi$
$733$	2709.63	0.136538	0.0682691	+	0.997667i	$0.478252\pi$
$734$	−46563.8	−2.34156
$735$	0	0
$736$	3290.53	0.164797
$737$	38134.9	1.90599
$738$	0	0
$739$	−24900.6	−1.23949	−0.619746	−	0.784802i	$-0.712764\pi$
$739$	−24900.6	−1.23949	−0.619746	+	0.784802i	$0.712764\pi$
$740$	0	0
$741$	0	0
$742$	−7325.08	−0.362415
$743$	14522.6	0.717070	0.358535	−	0.933516i	$-0.383276\pi$
$743$	14522.6	0.717070	0.358535	+	0.933516i	$0.383276\pi$
$744$	0	0
$745$	0	0
$746$	8608.16	0.422476
$747$	0	0
$748$	−42908.4	−2.09744
$749$	−11141.6	−0.543531
$750$	0	0
$751$	−14360.6	−0.697768	−0.348884	−	0.937166i	$-0.613439\pi$
$751$	−14360.6	−0.697768	−0.348884	+	0.937166i	$0.613439\pi$
$752$	872.190	0.0422945
$753$	0	0
$754$	15101.1	0.729374
$755$	0	0
$756$	0	0
$757$	−6200.35	−0.297695	−0.148848	−	0.988860i	$-0.547556\pi$
$757$	−6200.35	−0.297695	−0.148848	+	0.988860i	$0.547556\pi$
$758$	17508.9	0.838988
$759$	0	0
$760$	0	0
$761$	22218.6	1.05838	0.529188	−	0.848505i	$-0.322497\pi$
$761$	22218.6	1.05838	0.529188	+	0.848505i	$0.322497\pi$
$762$	0	0
$763$	11944.0	0.566714
$764$	−23587.6	−1.11698
$765$	0	0
$766$	−53882.9	−2.54160
$767$	11869.8	0.558795
$768$	0	0
$769$	9799.16	0.459515	0.229757	−	0.973248i	$-0.426207\pi$
$769$	9799.16	0.459515	0.229757	+	0.973248i	$0.426207\pi$
$770$	0	0
$771$	0	0
$772$	−30318.8	−1.41347
$773$	23174.1	1.07828	0.539141	−	0.842215i	$-0.318749\pi$
$773$	23174.1	1.07828	0.539141	+	0.842215i	$0.318749\pi$
$774$	0	0
$775$	0	0
$776$	15748.8	0.728541
$777$	0	0
$778$	5882.08	0.271058
$779$	−346.981	−0.0159588
$780$	0	0
$781$	2979.65	0.136517
$782$	5439.00	0.248719
$783$	0	0
$784$	205.548	0.00936352
$785$	0	0
$786$	0	0
$787$	13218.3	0.598707	0.299353	−	0.954142i	$-0.403229\pi$
$787$	13218.3	0.598707	0.299353	+	0.954142i	$0.403229\pi$
$788$	−45991.2	−2.07915
$789$	0	0
$790$	0	0
$791$	3920.00	0.176206
$792$	0	0
$793$	−6193.97	−0.277370
$794$	9672.41	0.432319
$795$	0	0
$796$	10835.2	0.482466
$797$	−1421.60	−0.0631815	−0.0315908	−	0.999501i	$-0.510057\pi$
$797$	−1421.60	−0.0631815	−0.0315908	+	0.999501i	$0.510057\pi$
$798$	0	0
$799$	−12788.6	−0.566242
$800$	0	0
$801$	0	0
$802$	51721.2	2.27723
$803$	−61119.2	−2.68599
$804$	0	0
$805$	0	0
$806$	20353.6	0.889484
$807$	0	0
$808$	−4459.37	−0.194159
$809$	−11669.3	−0.507131	−0.253566	−	0.967318i	$-0.581603\pi$
$809$	−11669.3	−0.507131	−0.253566	+	0.967318i	$0.581603\pi$
$810$	0	0
$811$	2165.42	0.0937587	0.0468793	−	0.998901i	$-0.485072\pi$
$811$	2165.42	0.0937587	0.0468793	+	0.998901i	$0.485072\pi$
$812$	15440.2	0.667297
$813$	0	0
$814$	75945.1	3.27012
$815$	0	0
$816$	0	0
$817$	−3097.02	−0.132621
$818$	31069.2	1.32801
$819$	0	0
$820$	0	0
$821$	37709.7	1.60302	0.801509	−	0.597982i	$-0.204031\pi$
$821$	37709.7	1.60302	0.801509	+	0.597982i	$0.204031\pi$
$822$	0	0
$823$	20883.1	0.884497	0.442248	−	0.896893i	$-0.354181\pi$
$823$	20883.1	0.884497	0.442248	+	0.896893i	$0.354181\pi$
$824$	25634.3	1.08376
$825$	0	0
$826$	19505.4	0.821645
$827$	13018.1	0.547379	0.273690	−	0.961818i	$-0.411756\pi$
$827$	13018.1	0.547379	0.273690	+	0.961818i	$0.411756\pi$
$828$	0	0
$829$	1309.94	0.0548807	0.0274404	−	0.999623i	$-0.491264\pi$
$829$	1309.94	0.0548807	0.0274404	+	0.999623i	$0.491264\pi$
$830$	0	0
$831$	0	0
$832$	−16104.1	−0.671045
$833$	−3013.87	−0.125359
$834$	0	0
$835$	0	0
$836$	18900.2	0.781912
$837$	0	0
$838$	38884.2	1.60290
$839$	−35223.2	−1.44939	−0.724696	−	0.689069i	$-0.758020\pi$
$839$	−35223.2	−1.44939	−0.724696	+	0.689069i	$0.758020\pi$
$840$	0	0
$841$	3636.81	0.149117
$842$	10434.8	0.427085
$843$	0	0
$844$	−30617.7	−1.24870
$845$	0	0
$846$	0	0
$847$	10306.3	0.418098
$848$	−953.920	−0.0386294
$849$	0	0
$850$	0	0
$851$	−5989.82	−0.241279
$852$	0	0
$853$	−14907.6	−0.598390	−0.299195	−	0.954192i	$-0.596718\pi$
$853$	−14907.6	−0.598390	−0.299195	+	0.954192i	$0.596718\pi$
$854$	−10178.4	−0.407841
$855$	0	0
$856$	−37909.1	−1.51368
$857$	30377.1	1.21081	0.605404	−	0.795919i	$-0.293012\pi$
$857$	30377.1	1.21081	0.605404	+	0.795919i	$0.293012\pi$
$858$	0	0
$859$	24031.5	0.954535	0.477267	−	0.878758i	$-0.341627\pi$
$859$	24031.5	0.954535	0.477267	+	0.878758i	$0.341627\pi$
$860$	0	0
$861$	0	0
$862$	14600.4	0.576903
$863$	33363.7	1.31601	0.658004	−	0.753015i	$-0.271401\pi$
$863$	33363.7	1.31601	0.658004	+	0.753015i	$0.271401\pi$
$864$	0	0
$865$	0	0
$866$	−62442.5	−2.45021
$867$	0	0
$868$	20810.7	0.813780
$869$	61036.6	2.38265
$870$	0	0
$871$	14118.7	0.549246
$872$	40639.5	1.57824
$873$	0	0
$874$	−2395.77	−0.0927208
$875$	0	0
$876$	0	0
$877$	4392.19	0.169115	0.0845574	−	0.996419i	$-0.473052\pi$
$877$	4392.19	0.169115	0.0845574	+	0.996419i	$0.473052\pi$
$878$	−55320.3	−2.12639
$879$	0	0
$880$	0	0
$881$	8132.72	0.311008	0.155504	−	0.987835i	$-0.450300\pi$
$881$	8132.72	0.311008	0.155504	+	0.987835i	$0.450300\pi$
$882$	0	0
$883$	−16582.9	−0.632003	−0.316002	−	0.948759i	$-0.602341\pi$
$883$	−16582.9	−0.632003	−0.316002	+	0.948759i	$0.602341\pi$
$884$	−15886.0	−0.604415
$885$	0	0
$886$	−9536.40	−0.361605
$887$	9140.65	0.346012	0.173006	−	0.984921i	$-0.444652\pi$
$887$	9140.65	0.346012	0.173006	+	0.984921i	$0.444652\pi$
$888$	0	0
$889$	14692.7	0.554305
$890$	0	0
$891$	0	0
$892$	−33283.7	−1.24935
$893$	5633.11	0.211092
$894$	0	0
$895$	0	0
$896$	−16874.2	−0.629159
$897$	0	0
$898$	−712.009	−0.0264589
$899$	37773.9	1.40137
$900$	0	0
$901$	13987.0	0.517173
$902$	−3120.38	−0.115186
$903$	0	0
$904$	13337.8	0.490717
$905$	0	0
$906$	0	0
$907$	25359.3	0.928382	0.464191	−	0.885735i	$-0.346345\pi$
$907$	25359.3	0.928382	0.464191	+	0.885735i	$0.346345\pi$
$908$	−51702.5	−1.88966
$909$	0	0
$910$	0	0
$911$	−33308.6	−1.21138	−0.605688	−	0.795702i	$-0.707102\pi$
$911$	−33308.6	−1.21138	−0.605688	+	0.795702i	$0.707102\pi$
$912$	0	0
$913$	−36669.7	−1.32923
$914$	20004.9	0.723963
$915$	0	0
$916$	47321.4	1.70692
$917$	−11727.0	−0.422311
$918$	0	0
$919$	−5827.47	−0.209174	−0.104587	−	0.994516i	$-0.533352\pi$
$919$	−5827.47	−0.209174	−0.104587	+	0.994516i	$0.533352\pi$
$920$	0	0
$921$	0	0
$922$	25358.9	0.905805
$923$	1103.15	0.0393400
$924$	0	0
$925$	0	0
$926$	42180.0	1.49689
$927$	0	0
$928$	−28666.5	−1.01403
$929$	−19224.3	−0.678934	−0.339467	−	0.940618i	$-0.610247\pi$
$929$	−19224.3	−0.678934	−0.339467	+	0.940618i	$0.610247\pi$
$930$	0	0
$931$	1327.55	0.0467332
$932$	−65319.8	−2.29573
$933$	0	0
$934$	−41456.5	−1.45235
$935$	0	0
$936$	0	0
$937$	−37202.8	−1.29708	−0.648539	−	0.761181i	$-0.724620\pi$
$937$	−37202.8	−1.29708	−0.648539	+	0.761181i	$0.724620\pi$
$938$	23200.8	0.807605
$939$	0	0
$940$	0	0
$941$	−43155.8	−1.49505	−0.747523	−	0.664236i	$-0.768757\pi$
$941$	−43155.8	−1.49505	−0.747523	+	0.664236i	$0.768757\pi$
$942$	0	0
$943$	246.106	0.00849874
$944$	2540.12	0.0875781
$945$	0	0
$946$	−27851.4	−0.957215
$947$	1442.59	0.0495014	0.0247507	−	0.999694i	$-0.492121\pi$
$947$	1442.59	0.0495014	0.0247507	+	0.999694i	$0.492121\pi$
$948$	0	0
$949$	−22628.2	−0.774016
$950$	0	0
$951$	0	0
$952$	−10254.7	−0.349113
$953$	−2964.70	−0.100773	−0.0503863	−	0.998730i	$-0.516045\pi$
$953$	−2964.70	−0.100773	−0.0503863	+	0.998730i	$0.516045\pi$
$954$	0	0
$955$	0	0
$956$	74621.9	2.52452
$957$	0	0
$958$	−46606.7	−1.57181
$959$	18751.5	0.631404
$960$	0	0
$961$	21121.5	0.708990
$962$	28117.2	0.942343
$963$	0	0
$964$	−29601.2	−0.988994
$965$	0	0
$966$	0	0
$967$	−59088.1	−1.96499	−0.982494	−	0.186292i	$-0.940353\pi$
$967$	−59088.1	−1.96499	−0.982494	+	0.186292i	$0.940353\pi$
$968$	35067.1	1.16436
$969$	0	0
$970$	0	0
$971$	6678.75	0.220732	0.110366	−	0.993891i	$-0.464798\pi$
$971$	6678.75	0.220732	0.110366	+	0.993891i	$0.464798\pi$
$972$	0	0
$973$	7224.76	0.238042
$974$	−4392.07	−0.144488
$975$	0	0
$976$	−1325.49	−0.0434713
$977$	−46454.6	−1.52120	−0.760600	−	0.649220i	$-0.775095\pi$
$977$	−46454.6	−1.52120	−0.760600	+	0.649220i	$0.775095\pi$
$978$	0	0
$979$	−75062.2	−2.45046
$980$	0	0
$981$	0	0
$982$	−33759.2	−1.09705
$983$	3154.80	0.102363	0.0511814	−	0.998689i	$-0.483701\pi$
$983$	3154.80	0.102363	0.0511814	+	0.998689i	$0.483701\pi$
$984$	0	0
$985$	0	0
$986$	−47383.5	−1.53042
$987$	0	0
$988$	6997.44	0.225322
$989$	2196.65	0.0706262
$990$	0	0
$991$	36207.7	1.16062	0.580311	−	0.814395i	$-0.302931\pi$
$991$	36207.7	1.16062	0.580311	+	0.814395i	$0.302931\pi$
$992$	−38637.3	−1.23663
$993$	0	0
$994$	1812.78	0.0578450
$995$	0	0
$996$	0	0
$997$	18452.9	0.586166	0.293083	−	0.956087i	$-0.405319\pi$
$997$	18452.9	0.586166	0.293083	+	0.956087i	$0.405319\pi$
$998$	76263.9	2.41893
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bk.1.4		4
3.2	odd	2		525.4.a.u.1.1	yes	4
5.4	even	2		1575.4.a.bj.1.1		4
15.2	even	4		525.4.d.n.274.2		8
15.8	even	4		525.4.d.n.274.7		8
15.14	odd	2		525.4.a.t.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.t.1.4	✓	4	15.14	odd	2
525.4.a.u.1.1	yes	4	3.2	odd	2
525.4.d.n.274.2		8	15.2	even	4
525.4.d.n.274.7		8	15.8	even	4
1575.4.a.bj.1.1		4	5.4	even	2
1575.4.a.bk.1.4		4	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+4.60171 q^{2} +13.1758 q^{4} +7.00000 q^{7} +23.8174 q^{8} +52.9465 q^{11} +19.6024 q^{13} +32.2120 q^{14} +4.19486 q^{16} -61.5076 q^{17} +27.0928 q^{19} +243.645 q^{22} -19.2163 q^{23} +90.2046 q^{26} +92.2304 q^{28} +167.409 q^{29} +225.638 q^{31} -171.236 q^{32} -283.040 q^{34} +311.705 q^{37} +124.673 q^{38} -12.8071 q^{41} -114.311 q^{43} +697.611 q^{44} -88.4281 q^{46} +207.919 q^{47} +49.0000 q^{49} +258.277 q^{52} -227.402 q^{53} +166.722 q^{56} +770.369 q^{58} +605.531 q^{59} -315.981 q^{61} +1038.32 q^{62} -821.538 q^{64} +720.254 q^{67} -810.410 q^{68} +56.2766 q^{71} -1154.36 q^{73} +1434.38 q^{74} +356.969 q^{76} +370.625 q^{77} +1152.80 q^{79} -58.9347 q^{82} -692.581 q^{83} -526.029 q^{86} +1261.05 q^{88} -1417.70 q^{89} +137.217 q^{91} -253.190 q^{92} +956.783 q^{94} +661.229 q^{97} +225.484 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{4} + 28 q^{7} - 9 q^{8} - 21 q^{11} + 5 q^{13} + 72 q^{16} - 99 q^{17} + 72 q^{19} + 221 q^{22} - 102 q^{23} - 129 q^{26} + 112 q^{28} + 240 q^{29} + 351 q^{31} - 72 q^{32} - 285 q^{34} + 399 q^{37}+ \cdots - 372 q^{97}+O(q^{100}) \) 4 * q + 16 * q^4 + 28 * q^7 - 9 * q^8 - 21 * q^11 + 5 * q^13 + 72 * q^16 - 99 * q^17 + 72 * q^19 + 221 * q^22 - 102 * q^23 - 129 * q^26 + 112 * q^28 + 240 * q^29 + 351 * q^31 - 72 * q^32 - 285 * q^34 + 399 * q^37 - 324 * q^38 - 381 * q^41 + 460 * q^43 + 975 * q^44 + 550 * q^46 - 60 * q^47 + 196 * q^49 + 223 * q^52 - 873 * q^53 - 63 * q^56 + 1408 * q^58 + 855 * q^59 + 687 * q^61 + 477 * q^62 - 1285 * q^64 - 503 * q^67 - 1725 * q^68 + 681 * q^71 + 1228 * q^73 + 3369 * q^74 + 1910 * q^76 - 147 * q^77 + 345 * q^79 - 495 * q^82 - 1509 * q^83 + 540 * q^86 + 2266 * q^88 + 198 * q^89 + 35 * q^91 - 2994 * q^92 - 1732 * q^94 - 372 * q^97

Introduction

L-functions

Modular forms

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Database

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