LMFDB - Embedded newform 1575.4.a.bt.1.9

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:	$1$
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			1.9
Root			$4.31226$ 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)

$f(q)$	$=$	$q+4.31226 q^{2} +10.5956 q^{4} -7.00000 q^{7} +11.1930 q^{8} +O(q^{10})$	$q+4.31226 q^{2} +10.5956 q^{4} -7.00000 q^{7} +11.1930 q^{8} +12.8336 q^{11} -57.3520 q^{13} -30.1858 q^{14} -36.4978 q^{16} +17.1377 q^{17} +107.753 q^{19} +55.3419 q^{22} -67.1525 q^{23} -247.317 q^{26} -74.1693 q^{28} +52.7411 q^{29} +117.294 q^{31} -246.932 q^{32} +73.9022 q^{34} -150.611 q^{37} +464.661 q^{38} -396.846 q^{41} -431.999 q^{43} +135.980 q^{44} -289.579 q^{46} -389.297 q^{47} +49.0000 q^{49} -607.680 q^{52} -16.7752 q^{53} -78.3509 q^{56} +227.434 q^{58} -537.732 q^{59} -287.617 q^{61} +505.802 q^{62} -772.854 q^{64} -826.916 q^{67} +181.584 q^{68} +614.661 q^{71} -234.183 q^{73} -649.476 q^{74} +1141.71 q^{76} -89.8353 q^{77} +235.764 q^{79} -1711.30 q^{82} -516.982 q^{83} -1862.89 q^{86} +143.647 q^{88} -1593.49 q^{89} +401.464 q^{91} -711.522 q^{92} -1678.75 q^{94} +1113.09 q^{97} +211.301 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.31226	1.52462	0.762308	−	0.647215i	$-0.224066\pi$
$2$	4.31226	1.52462	0.762308	+	0.647215i	$0.224066\pi$
$3$	0	0
$3$	0	0
$4$	10.5956	1.32445
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	11.1930	0.494665
$9$	0	0
$10$	0	0
$11$	12.8336	0.351771	0.175886	−	0.984411i	$-0.443721\pi$
$11$	12.8336	0.351771	0.175886	+	0.984411i	$0.443721\pi$
$12$	0	0
$13$	−57.3520	−1.22358	−0.611791	−	0.791019i	$-0.709551\pi$
$13$	−57.3520	−1.22358	−0.611791	+	0.791019i	$0.709551\pi$
$14$	−30.1858	−0.576250
$15$	0	0
$16$	−36.4978	−0.570279
$17$	17.1377	0.244500	0.122250	−	0.992499i	$-0.460989\pi$
$17$	17.1377	0.244500	0.122250	+	0.992499i	$0.460989\pi$
$18$	0	0
$19$	107.753	1.30107	0.650535	−	0.759476i	$-0.274545\pi$
$19$	107.753	1.30107	0.650535	+	0.759476i	$0.274545\pi$
$20$	0	0
$21$	0	0
$22$	55.3419	0.536316
$23$	−67.1525	−0.608794	−0.304397	−	0.952545i	$-0.598455\pi$
$23$	−67.1525	−0.608794	−0.304397	+	0.952545i	$0.598455\pi$
$24$	0	0
$25$	0	0
$26$	−247.317	−1.86549
$27$	0	0
$28$	−74.1693	−0.500596
$29$	52.7411	0.337717	0.168858	−	0.985640i	$-0.445992\pi$
$29$	52.7411	0.337717	0.168858	+	0.985640i	$0.445992\pi$
$30$	0	0
$31$	117.294	0.679568	0.339784	−	0.940504i	$-0.389646\pi$
$31$	117.294	0.679568	0.339784	+	0.940504i	$0.389646\pi$
$32$	−246.932	−1.36412
$33$	0	0
$34$	73.9022	0.372768
$35$	0	0
$36$	0	0
$37$	−150.611	−0.669199	−0.334599	−	0.942360i	$-0.608601\pi$
$37$	−150.611	−0.669199	−0.334599	+	0.942360i	$0.608601\pi$
$38$	464.661	1.98363
$39$	0	0
$40$	0	0
$41$	−396.846	−1.51163	−0.755816	−	0.654784i	$-0.772760\pi$
$41$	−396.846	−1.51163	−0.755816	+	0.654784i	$0.772760\pi$
$42$	0	0
$43$	−431.999	−1.53208	−0.766038	−	0.642795i	$-0.777775\pi$
$43$	−431.999	−1.53208	−0.766038	+	0.642795i	$0.777775\pi$
$44$	135.980	0.465904
$45$	0	0
$46$	−289.579	−0.928177
$47$	−389.297	−1.20819	−0.604094	−	0.796913i	$-0.706465\pi$
$47$	−389.297	−1.20819	−0.604094	+	0.796913i	$0.706465\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−607.680	−1.62058
$53$	−16.7752	−0.0434765	−0.0217382	−	0.999764i	$-0.506920\pi$
$53$	−16.7752	−0.0434765	−0.0217382	+	0.999764i	$0.506920\pi$
$54$	0	0
$55$	0	0
$56$	−78.3509	−0.186966
$57$	0	0
$58$	227.434	0.514888
$59$	−537.732	−1.18655	−0.593277	−	0.804998i	$-0.702166\pi$
$59$	−537.732	−1.18655	−0.593277	+	0.804998i	$0.702166\pi$
$60$	0	0
$61$	−287.617	−0.603697	−0.301849	−	0.953356i	$-0.597604\pi$
$61$	−287.617	−0.603697	−0.301849	+	0.953356i	$0.597604\pi$
$62$	505.802	1.03608
$63$	0	0
$64$	−772.854	−1.50948
$65$	0	0
$66$	0	0
$67$	−826.916	−1.50782	−0.753909	−	0.656979i	$-0.771834\pi$
$67$	−826.916	−1.50782	−0.753909	+	0.656979i	$0.771834\pi$
$68$	181.584	0.323828
$69$	0	0
$70$	0	0
$71$	614.661	1.02742	0.513711	−	0.857964i	$-0.328271\pi$
$71$	614.661	1.02742	0.513711	+	0.857964i	$0.328271\pi$
$72$	0	0
$73$	−234.183	−0.375467	−0.187733	−	0.982220i	$-0.560114\pi$
$73$	−234.183	−0.375467	−0.187733	+	0.982220i	$0.560114\pi$
$74$	−649.476	−1.02027
$75$	0	0
$76$	1141.71	1.72321
$77$	−89.8353	−0.132957
$78$	0	0
$79$	235.764	0.335767	0.167883	−	0.985807i	$-0.446307\pi$
$79$	235.764	0.335767	0.167883	+	0.985807i	$0.446307\pi$
$80$	0	0
$81$	0	0
$82$	−1711.30	−2.30466
$83$	−516.982	−0.683689	−0.341844	−	0.939757i	$-0.611052\pi$
$83$	−516.982	−0.683689	−0.341844	+	0.939757i	$0.611052\pi$
$84$	0	0
$85$	0	0
$86$	−1862.89	−2.33583
$87$	0	0
$88$	143.647	0.174009
$89$	−1593.49	−1.89786	−0.948932	−	0.315480i	$-0.897834\pi$
$89$	−1593.49	−1.89786	−0.948932	+	0.315480i	$0.897834\pi$
$90$	0	0
$91$	401.464	0.462471
$92$	−711.522	−0.806319
$93$	0	0
$94$	−1678.75	−1.84202
$95$	0	0
$96$	0	0
$97$	1113.09	1.16512	0.582562	−	0.812786i	$-0.302050\pi$
$97$	1113.09	1.16512	0.582562	+	0.812786i	$0.302050\pi$
$98$	211.301	0.217802
$99$	0	0
$100$	0	0
$101$	502.736	0.495288	0.247644	−	0.968851i	$-0.420344\pi$
$101$	502.736	0.495288	0.247644	+	0.968851i	$0.420344\pi$
$102$	0	0
$103$	−863.314	−0.825872	−0.412936	−	0.910760i	$-0.635497\pi$
$103$	−863.314	−0.825872	−0.412936	+	0.910760i	$0.635497\pi$
$104$	−641.940	−0.605263
$105$	0	0
$106$	−72.3392	−0.0662849
$107$	1706.34	1.54166	0.770832	−	0.637038i	$-0.219841\pi$
$107$	1706.34	1.54166	0.770832	+	0.637038i	$0.219841\pi$
$108$	0	0
$109$	2131.41	1.87296	0.936479	−	0.350723i	$-0.114064\pi$
$109$	2131.41	1.87296	0.936479	+	0.350723i	$0.114064\pi$
$110$	0	0
$111$	0	0
$112$	255.485	0.215545
$113$	320.895	0.267144	0.133572	−	0.991039i	$-0.457355\pi$
$113$	320.895	0.267144	0.133572	+	0.991039i	$0.457355\pi$
$114$	0	0
$115$	0	0
$116$	558.825	0.447290
$117$	0	0
$118$	−2318.84	−1.80904
$119$	−119.964	−0.0924123
$120$	0	0
$121$	−1166.30	−0.876257
$122$	−1240.28	−0.920406
$123$	0	0
$124$	1242.80	0.900055
$125$	0	0
$126$	0	0
$127$	775.087	0.541558	0.270779	−	0.962642i	$-0.412719\pi$
$127$	775.087	0.541558	0.270779	+	0.962642i	$0.412719\pi$
$128$	−1357.29	−0.937257
$129$	0	0
$130$	0	0
$131$	178.450	0.119017	0.0595085	−	0.998228i	$-0.481047\pi$
$131$	178.450	0.119017	0.0595085	+	0.998228i	$0.481047\pi$
$132$	0	0
$133$	−754.274	−0.491758
$134$	−3565.88	−2.29884
$135$	0	0
$136$	191.822	0.120946
$137$	1618.85	1.00954	0.504772	−	0.863252i	$-0.331576\pi$
$137$	1618.85	1.00954	0.504772	+	0.863252i	$0.331576\pi$
$138$	0	0
$139$	−335.854	−0.204941	−0.102470	−	0.994736i	$-0.532675\pi$
$139$	−335.854	−0.204941	−0.102470	+	0.994736i	$0.532675\pi$
$140$	0	0
$141$	0	0
$142$	2650.58	1.56642
$143$	−736.033	−0.430421
$144$	0	0
$145$	0	0
$146$	−1009.86	−0.572442
$147$	0	0
$148$	−1595.82	−0.886322
$149$	−1953.30	−1.07396	−0.536981	−	0.843594i	$-0.680435\pi$
$149$	−1953.30	−1.07396	−0.536981	+	0.843594i	$0.680435\pi$
$150$	0	0
$151$	−2040.94	−1.09993	−0.549965	−	0.835188i	$-0.685359\pi$
$151$	−2040.94	−1.09993	−0.549965	+	0.835188i	$0.685359\pi$
$152$	1206.08	0.643594
$153$	0	0
$154$	−387.394	−0.202708
$155$	0	0
$156$	0	0
$157$	1793.44	0.911670	0.455835	−	0.890064i	$-0.349341\pi$
$157$	1793.44	0.911670	0.455835	+	0.890064i	$0.349341\pi$
$158$	1016.68	0.511915
$159$	0	0
$160$	0	0
$161$	470.068	0.230103
$162$	0	0
$163$	273.817	0.131577	0.0657884	−	0.997834i	$-0.479044\pi$
$163$	273.817	0.131577	0.0657884	+	0.997834i	$0.479044\pi$
$164$	−4204.83	−2.00209
$165$	0	0
$166$	−2229.36	−1.04236
$167$	−895.708	−0.415042	−0.207521	−	0.978231i	$-0.566539\pi$
$167$	−895.708	−0.415042	−0.207521	+	0.978231i	$0.566539\pi$
$168$	0	0
$169$	1092.25	0.497155
$170$	0	0
$171$	0	0
$172$	−4577.30	−2.02916
$173$	−2708.36	−1.19025	−0.595124	−	0.803634i	$-0.702897\pi$
$173$	−2708.36	−1.19025	−0.595124	+	0.803634i	$0.702897\pi$
$174$	0	0
$175$	0	0
$176$	−468.399	−0.200607
$177$	0	0
$178$	−6871.56	−2.89351
$179$	−2759.29	−1.15217	−0.576087	−	0.817388i	$-0.695421\pi$
$179$	−2759.29	−1.15217	−0.576087	+	0.817388i	$0.695421\pi$
$180$	0	0
$181$	2100.48	0.862582	0.431291	−	0.902213i	$-0.358058\pi$
$181$	2100.48	0.862582	0.431291	+	0.902213i	$0.358058\pi$
$182$	1731.22	0.705090
$183$	0	0
$184$	−751.637	−0.301149
$185$	0	0
$186$	0	0
$187$	219.938	0.0860080
$188$	−4124.84	−1.60019
$189$	0	0
$190$	0	0
$191$	3102.45	1.17532	0.587658	−	0.809110i	$-0.300050\pi$
$191$	3102.45	1.17532	0.587658	+	0.809110i	$0.300050\pi$
$192$	0	0
$193$	3242.25	1.20924	0.604618	−	0.796515i	$-0.293326\pi$
$193$	3242.25	1.20924	0.604618	+	0.796515i	$0.293326\pi$
$194$	4799.93	1.77637
$195$	0	0
$196$	519.185	0.189207
$197$	−2066.28	−0.747290	−0.373645	−	0.927572i	$-0.621892\pi$
$197$	−2066.28	−0.747290	−0.373645	+	0.927572i	$0.621892\pi$
$198$	0	0
$199$	−5504.07	−1.96067	−0.980335	−	0.197338i	$-0.936770\pi$
$199$	−5504.07	−1.96067	−0.980335	+	0.197338i	$0.936770\pi$
$200$	0	0
$201$	0	0
$202$	2167.93	0.755124
$203$	−369.188	−0.127645
$204$	0	0
$205$	0	0
$206$	−3722.84	−1.25914
$207$	0	0
$208$	2093.22	0.697783
$209$	1382.87	0.457679
$210$	0	0
$211$	3461.88	1.12950	0.564752	−	0.825261i	$-0.308972\pi$
$211$	3461.88	1.12950	0.564752	+	0.825261i	$0.308972\pi$
$212$	−177.744	−0.0575825
$213$	0	0
$214$	7358.19	2.35045
$215$	0	0
$216$	0	0
$217$	−821.057	−0.256853
$218$	9191.22	2.85554
$219$	0	0
$220$	0	0
$221$	−982.880	−0.299166
$222$	0	0
$223$	−1197.98	−0.359744	−0.179872	−	0.983690i	$-0.557568\pi$
$223$	−1197.98	−0.359744	−0.179872	+	0.983690i	$0.557568\pi$
$224$	1728.53	0.515589
$225$	0	0
$226$	1383.78	0.407291
$227$	2020.12	0.590660	0.295330	−	0.955395i	$-0.404570\pi$
$227$	2020.12	0.590660	0.295330	+	0.955395i	$0.404570\pi$
$228$	0	0
$229$	−3108.53	−0.897019	−0.448509	−	0.893778i	$-0.648045\pi$
$229$	−3108.53	−0.897019	−0.448509	+	0.893778i	$0.648045\pi$
$230$	0	0
$231$	0	0
$232$	590.331	0.167057
$233$	6840.90	1.92344	0.961722	−	0.274027i	$-0.0883558\pi$
$233$	6840.90	1.92344	0.961722	+	0.274027i	$0.0883558\pi$
$234$	0	0
$235$	0	0
$236$	−5697.60	−1.57154
$237$	0	0
$238$	−517.315	−0.140893
$239$	−569.240	−0.154063	−0.0770316	−	0.997029i	$-0.524544\pi$
$239$	−569.240	−0.154063	−0.0770316	+	0.997029i	$0.524544\pi$
$240$	0	0
$241$	2731.59	0.730112	0.365056	−	0.930986i	$-0.381050\pi$
$241$	2731.59	0.730112	0.365056	+	0.930986i	$0.381050\pi$
$242$	−5029.39	−1.33596
$243$	0	0
$244$	−3047.48	−0.799568
$245$	0	0
$246$	0	0
$247$	−6179.87	−1.59197
$248$	1312.87	0.336158
$249$	0	0
$250$	0	0
$251$	4339.51	1.09126	0.545632	−	0.838025i	$-0.316290\pi$
$251$	4339.51	1.09126	0.545632	+	0.838025i	$0.316290\pi$
$252$	0	0
$253$	−861.809	−0.214156
$254$	3342.38	0.825667
$255$	0	0
$256$	329.827	0.0805242
$257$	519.419	0.126072	0.0630360	−	0.998011i	$-0.479922\pi$
$257$	519.419	0.126072	0.0630360	+	0.998011i	$0.479922\pi$
$258$	0	0
$259$	1054.28	0.252933
$260$	0	0
$261$	0	0
$262$	769.522	0.181455
$263$	−4887.46	−1.14591	−0.572954	−	0.819588i	$-0.694203\pi$
$263$	−4887.46	−1.14591	−0.572954	+	0.819588i	$0.694203\pi$
$264$	0	0
$265$	0	0
$266$	−3252.63	−0.749742
$267$	0	0
$268$	−8761.68	−1.99703
$269$	7186.66	1.62892	0.814458	−	0.580222i	$-0.197034\pi$
$269$	7186.66	1.62892	0.814458	+	0.580222i	$0.197034\pi$
$270$	0	0
$271$	−2808.71	−0.629582	−0.314791	−	0.949161i	$-0.601934\pi$
$271$	−2808.71	−0.629582	−0.314791	+	0.949161i	$0.601934\pi$
$272$	−625.488	−0.139433
$273$	0	0
$274$	6980.90	1.53917
$275$	0	0
$276$	0	0
$277$	−2515.35	−0.545605	−0.272803	−	0.962070i	$-0.587951\pi$
$277$	−2515.35	−0.545605	−0.272803	+	0.962070i	$0.587951\pi$
$278$	−1448.29	−0.312456
$279$	0	0
$280$	0	0
$281$	8726.90	1.85268	0.926341	−	0.376687i	$-0.122937\pi$
$281$	8726.90	1.85268	0.926341	+	0.376687i	$0.122937\pi$
$282$	0	0
$283$	−382.046	−0.0802483	−0.0401241	−	0.999195i	$-0.512775\pi$
$283$	−382.046	−0.0802483	−0.0401241	+	0.999195i	$0.512775\pi$
$284$	6512.72	1.36077
$285$	0	0
$286$	−3173.97	−0.656226
$287$	2777.92	0.571343
$288$	0	0
$289$	−4619.30	−0.940220
$290$	0	0
$291$	0	0
$292$	−2481.32	−0.497288
$293$	−5688.87	−1.13429	−0.567145	−	0.823618i	$-0.691952\pi$
$293$	−5688.87	−1.13429	−0.567145	+	0.823618i	$0.691952\pi$
$294$	0	0
$295$	0	0
$296$	−1685.79	−0.331029
$297$	0	0
$298$	−8423.14	−1.63738
$299$	3851.33	0.744910
$300$	0	0
$301$	3023.99	0.579070
$302$	−8801.07	−1.67697
$303$	0	0
$304$	−3932.77	−0.741972
$305$	0	0
$306$	0	0
$307$	−5746.25	−1.06826	−0.534130	−	0.845403i	$-0.679361\pi$
$307$	−5746.25	−1.06826	−0.534130	+	0.845403i	$0.679361\pi$
$308$	−951.861	−0.176095
$309$	0	0
$310$	0	0
$311$	7445.70	1.35758	0.678789	−	0.734333i	$-0.262505\pi$
$311$	7445.70	1.35758	0.678789	+	0.734333i	$0.262505\pi$
$312$	0	0
$313$	−9252.13	−1.67080	−0.835401	−	0.549641i	$-0.814765\pi$
$313$	−9252.13	−1.67080	−0.835401	+	0.549641i	$0.814765\pi$
$314$	7733.79	1.38995
$315$	0	0
$316$	2498.07	0.444707
$317$	2211.49	0.391829	0.195915	−	0.980621i	$-0.437232\pi$
$317$	2211.49	0.391829	0.195915	+	0.980621i	$0.437232\pi$
$318$	0	0
$319$	676.860	0.118799
$320$	0	0
$321$	0	0
$322$	2027.06	0.350818
$323$	1846.64	0.318112
$324$	0	0
$325$	0	0
$326$	1180.77	0.200604
$327$	0	0
$328$	−4441.89	−0.747752
$329$	2725.08	0.456652
$330$	0	0
$331$	−7484.30	−1.24282	−0.621412	−	0.783484i	$-0.713441\pi$
$331$	−7484.30	−1.24282	−0.621412	+	0.783484i	$0.713441\pi$
$332$	−5477.75	−0.905513
$333$	0	0
$334$	−3862.53	−0.632779
$335$	0	0
$336$	0	0
$337$	−6661.17	−1.07673	−0.538364	−	0.842713i	$-0.680957\pi$
$337$	−6661.17	−1.07673	−0.538364	+	0.842713i	$0.680957\pi$
$338$	4710.07	0.757970
$339$	0	0
$340$	0	0
$341$	1505.30	0.239052
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−4835.36	−0.757864
$345$	0	0
$346$	−11679.2	−1.81467
$347$	885.113	0.136932	0.0684659	−	0.997653i	$-0.478190\pi$
$347$	885.113	0.136932	0.0684659	+	0.997653i	$0.478190\pi$
$348$	0	0
$349$	8672.23	1.33013	0.665063	−	0.746787i	$-0.268405\pi$
$349$	8672.23	1.33013	0.665063	+	0.746787i	$0.268405\pi$
$350$	0	0
$351$	0	0
$352$	−3169.03	−0.479858
$353$	4691.67	0.707400	0.353700	−	0.935359i	$-0.384923\pi$
$353$	4691.67	0.707400	0.353700	+	0.935359i	$0.384923\pi$
$354$	0	0
$355$	0	0
$356$	−16884.0	−2.51363
$357$	0	0
$358$	−11898.8	−1.75662
$359$	8421.84	1.23813	0.619063	−	0.785341i	$-0.287512\pi$
$359$	8421.84	1.23813	0.619063	+	0.785341i	$0.287512\pi$
$360$	0	0
$361$	4751.81	0.692784
$362$	9057.81	1.31511
$363$	0	0
$364$	4253.76	0.612520
$365$	0	0
$366$	0	0
$367$	−5968.53	−0.848923	−0.424461	−	0.905446i	$-0.639537\pi$
$367$	−5968.53	−0.848923	−0.424461	+	0.905446i	$0.639537\pi$
$368$	2450.92	0.347182
$369$	0	0
$370$	0	0
$371$	117.427	0.0164326
$372$	0	0
$373$	4620.46	0.641390	0.320695	−	0.947183i	$-0.396084\pi$
$373$	4620.46	0.641390	0.320695	+	0.947183i	$0.396084\pi$
$374$	948.433	0.131129
$375$	0	0
$376$	−4357.40	−0.597648
$377$	−3024.81	−0.413224
$378$	0	0
$379$	13447.3	1.82254	0.911271	−	0.411808i	$-0.135103\pi$
$379$	13447.3	1.82254	0.911271	+	0.411808i	$0.135103\pi$
$380$	0	0
$381$	0	0
$382$	13378.6	1.79190
$383$	5930.11	0.791161	0.395580	−	0.918431i	$-0.370544\pi$
$383$	5930.11	0.791161	0.395580	+	0.918431i	$0.370544\pi$
$384$	0	0
$385$	0	0
$386$	13981.5	1.84362
$387$	0	0
$388$	11793.9	1.54315
$389$	6096.88	0.794664	0.397332	−	0.917675i	$-0.369936\pi$
$389$	6096.88	0.794664	0.397332	+	0.917675i	$0.369936\pi$
$390$	0	0
$391$	−1150.84	−0.148850
$392$	548.456	0.0706664
$393$	0	0
$394$	−8910.33	−1.13933
$395$	0	0
$396$	0	0
$397$	667.530	0.0843888	0.0421944	−	0.999109i	$-0.486565\pi$
$397$	667.530	0.0843888	0.0421944	+	0.999109i	$0.486565\pi$
$398$	−23735.0	−2.98927
$399$	0	0
$400$	0	0
$401$	−6799.07	−0.846706	−0.423353	−	0.905965i	$-0.639147\pi$
$401$	−6799.07	−0.846706	−0.423353	+	0.905965i	$0.639147\pi$
$402$	0	0
$403$	−6727.04	−0.831508
$404$	5326.80	0.655985
$405$	0	0
$406$	−1592.04	−0.194609
$407$	−1932.89	−0.235405
$408$	0	0
$409$	−15002.7	−1.81378	−0.906892	−	0.421364i	$-0.861552\pi$
$409$	−15002.7	−1.81378	−0.906892	+	0.421364i	$0.861552\pi$
$410$	0	0
$411$	0	0
$412$	−9147.34	−1.09383
$413$	3764.12	0.448476
$414$	0	0
$415$	0	0
$416$	14162.0	1.66911
$417$	0	0
$418$	5963.29	0.697784
$419$	3361.16	0.391893	0.195947	−	0.980615i	$-0.437222\pi$
$419$	3361.16	0.391893	0.195947	+	0.980615i	$0.437222\pi$
$420$	0	0
$421$	−2585.35	−0.299293	−0.149647	−	0.988740i	$-0.547814\pi$
$421$	−2585.35	−0.299293	−0.149647	+	0.988740i	$0.547814\pi$
$422$	14928.5	1.72206
$423$	0	0
$424$	−187.765	−0.0215063
$425$	0	0
$426$	0	0
$427$	2013.32	0.228176
$428$	18079.7	2.04186
$429$	0	0
$430$	0	0
$431$	15105.3	1.68816	0.844080	−	0.536217i	$-0.180147\pi$
$431$	15105.3	1.68816	0.844080	+	0.536217i	$0.180147\pi$
$432$	0	0
$433$	−4538.96	−0.503761	−0.251881	−	0.967758i	$-0.581049\pi$
$433$	−4538.96	−0.503761	−0.251881	+	0.967758i	$0.581049\pi$
$434$	−3540.62	−0.391601
$435$	0	0
$436$	22583.7	2.48064
$437$	−7235.91	−0.792084
$438$	0	0
$439$	−10049.4	−1.09255	−0.546275	−	0.837606i	$-0.683955\pi$
$439$	−10049.4	−1.09255	−0.546275	+	0.837606i	$0.683955\pi$
$440$	0	0
$441$	0	0
$442$	−4238.44	−0.456113
$443$	−10637.4	−1.14085	−0.570424	−	0.821350i	$-0.693221\pi$
$443$	−10637.4	−1.14085	−0.570424	+	0.821350i	$0.693221\pi$
$444$	0	0
$445$	0	0
$446$	−5166.02	−0.548471
$447$	0	0
$448$	5409.98	0.570530
$449$	6126.87	0.643975	0.321987	−	0.946744i	$-0.395649\pi$
$449$	6126.87	0.643975	0.321987	+	0.946744i	$0.395649\pi$
$450$	0	0
$451$	−5092.97	−0.531749
$452$	3400.08	0.353819
$453$	0	0
$454$	8711.28	0.900530
$455$	0	0
$456$	0	0
$457$	−13342.3	−1.36571	−0.682853	−	0.730556i	$-0.739261\pi$
$457$	−13342.3	−1.36571	−0.682853	+	0.730556i	$0.739261\pi$
$458$	−13404.8	−1.36761
$459$	0	0
$460$	0	0
$461$	8558.83	0.864695	0.432348	−	0.901707i	$-0.357685\pi$
$461$	8558.83	0.864695	0.432348	+	0.901707i	$0.357685\pi$
$462$	0	0
$463$	18213.7	1.82822	0.914108	−	0.405471i	$-0.132893\pi$
$463$	18213.7	1.82822	0.914108	+	0.405471i	$0.132893\pi$
$464$	−1924.94	−0.192593
$465$	0	0
$466$	29499.8	2.93251
$467$	5928.32	0.587430	0.293715	−	0.955893i	$-0.405108\pi$
$467$	5928.32	0.587430	0.293715	+	0.955893i	$0.405108\pi$
$468$	0	0
$469$	5788.41	0.569902
$470$	0	0
$471$	0	0
$472$	−6018.83	−0.586947
$473$	−5544.11	−0.538940
$474$	0	0
$475$	0	0
$476$	−1271.09	−0.122396
$477$	0	0
$478$	−2454.71	−0.234887
$479$	−13360.2	−1.27441	−0.637207	−	0.770693i	$-0.719910\pi$
$479$	−13360.2	−1.27441	−0.637207	+	0.770693i	$0.719910\pi$
$480$	0	0
$481$	8637.86	0.818820
$482$	11779.3	1.11314
$483$	0	0
$484$	−12357.7	−1.16056
$485$	0	0
$486$	0	0
$487$	13063.5	1.21553	0.607766	−	0.794116i	$-0.292066\pi$
$487$	13063.5	1.21553	0.607766	+	0.794116i	$0.292066\pi$
$488$	−3219.29	−0.298628
$489$	0	0
$490$	0	0
$491$	−6239.42	−0.573485	−0.286742	−	0.958008i	$-0.592572\pi$
$491$	−6239.42	−0.573485	−0.286742	+	0.958008i	$0.592572\pi$
$492$	0	0
$493$	903.861	0.0825717
$494$	−26649.2	−2.42714
$495$	0	0
$496$	−4280.97	−0.387543
$497$	−4302.63	−0.388329
$498$	0	0
$499$	9618.30	0.862874	0.431437	−	0.902143i	$-0.358007\pi$
$499$	9618.30	0.862874	0.431437	+	0.902143i	$0.358007\pi$
$500$	0	0
$501$	0	0
$502$	18713.1	1.66376
$503$	−20789.1	−1.84282	−0.921411	−	0.388588i	$-0.872963\pi$
$503$	−20789.1	−1.84282	−0.921411	+	0.388588i	$0.872963\pi$
$504$	0	0
$505$	0	0
$506$	−3716.35	−0.326506
$507$	0	0
$508$	8212.52	0.717267
$509$	5511.08	0.479911	0.239955	−	0.970784i	$-0.422867\pi$
$509$	5511.08	0.479911	0.239955	+	0.970784i	$0.422867\pi$
$510$	0	0
$511$	1639.28	0.141913
$512$	12280.6	1.06003
$513$	0	0
$514$	2239.87	0.192211
$515$	0	0
$516$	0	0
$517$	−4996.09	−0.425005
$518$	4546.33	0.385626
$519$	0	0
$520$	0	0
$521$	7904.05	0.664651	0.332325	−	0.943165i	$-0.392167\pi$
$521$	7904.05	0.664651	0.332325	+	0.943165i	$0.392167\pi$
$522$	0	0
$523$	−10740.6	−0.898000	−0.449000	−	0.893532i	$-0.648220\pi$
$523$	−10740.6	−0.898000	−0.449000	+	0.893532i	$0.648220\pi$
$524$	1890.78	0.157632
$525$	0	0
$526$	−21076.0	−1.74707
$527$	2010.15	0.166154
$528$	0	0
$529$	−7657.54	−0.629370
$530$	0	0
$531$	0	0
$532$	−7992.00	−0.651310
$533$	22759.9	1.84961
$534$	0	0
$535$	0	0
$536$	−9255.66	−0.745865
$537$	0	0
$538$	30990.8	2.48347
$539$	628.847	0.0502530
$540$	0	0
$541$	−14259.7	−1.13322	−0.566611	−	0.823985i	$-0.691746\pi$
$541$	−14259.7	−1.13322	−0.566611	+	0.823985i	$0.691746\pi$
$542$	−12111.9	−0.959871
$543$	0	0
$544$	−4231.84	−0.333527
$545$	0	0
$546$	0	0
$547$	11740.0	0.917673	0.458837	−	0.888521i	$-0.348266\pi$
$547$	11740.0	0.917673	0.458837	+	0.888521i	$0.348266\pi$
$548$	17152.7	1.33709
$549$	0	0
$550$	0	0
$551$	5683.04	0.439393
$552$	0	0
$553$	−1650.35	−0.126908
$554$	−10846.9	−0.831838
$555$	0	0
$556$	−3558.58	−0.271434
$557$	−23038.8	−1.75258	−0.876290	−	0.481784i	$-0.839989\pi$
$557$	−23038.8	−1.75258	−0.876290	+	0.481784i	$0.839989\pi$
$558$	0	0
$559$	24776.0	1.87462
$560$	0	0
$561$	0	0
$562$	37632.7	2.82463
$563$	7093.18	0.530980	0.265490	−	0.964114i	$-0.414466\pi$
$563$	7093.18	0.530980	0.265490	+	0.964114i	$0.414466\pi$
$564$	0	0
$565$	0	0
$566$	−1647.48	−0.122348
$567$	0	0
$568$	6879.90	0.508229
$569$	18680.3	1.37631	0.688153	−	0.725566i	$-0.258422\pi$
$569$	18680.3	1.37631	0.688153	+	0.725566i	$0.258422\pi$
$570$	0	0
$571$	10980.5	0.804762	0.402381	−	0.915472i	$-0.368183\pi$
$571$	10980.5	0.804762	0.402381	+	0.915472i	$0.368183\pi$
$572$	−7798.73	−0.570072
$573$	0	0
$574$	11979.1	0.871079
$575$	0	0
$576$	0	0
$577$	17043.7	1.22970	0.614850	−	0.788644i	$-0.289216\pi$
$577$	17043.7	1.22970	0.614850	+	0.788644i	$0.289216\pi$
$578$	−19919.6	−1.43347
$579$	0	0
$580$	0	0
$581$	3618.88	0.258410
$582$	0	0
$583$	−215.287	−0.0152938
$584$	−2621.21	−0.185730
$585$	0	0
$586$	−24531.9	−1.72936
$587$	−16599.1	−1.16715	−0.583575	−	0.812059i	$-0.698347\pi$
$587$	−16599.1	−1.16715	−0.583575	+	0.812059i	$0.698347\pi$
$588$	0	0
$589$	12638.8	0.884166
$590$	0	0
$591$	0	0
$592$	5496.99	0.381630
$593$	17589.7	1.21808	0.609042	−	0.793138i	$-0.291554\pi$
$593$	17589.7	1.21808	0.609042	+	0.793138i	$0.291554\pi$
$594$	0	0
$595$	0	0
$596$	−20696.4	−1.42241
$597$	0	0
$598$	16607.9	1.13570
$599$	12914.6	0.880931	0.440466	−	0.897769i	$-0.354813\pi$
$599$	12914.6	0.880931	0.440466	+	0.897769i	$0.354813\pi$
$600$	0	0
$601$	−12810.8	−0.869488	−0.434744	−	0.900554i	$-0.643161\pi$
$601$	−12810.8	−0.869488	−0.434744	+	0.900554i	$0.643161\pi$
$602$	13040.3	0.882859
$603$	0	0
$604$	−21625.0	−1.45680
$605$	0	0
$606$	0	0
$607$	10549.6	0.705429	0.352714	−	0.935731i	$-0.385259\pi$
$607$	10549.6	0.705429	0.352714	+	0.935731i	$0.385259\pi$
$608$	−26607.8	−1.77482
$609$	0	0
$610$	0	0
$611$	22327.0	1.47832
$612$	0	0
$613$	−23485.3	−1.54741	−0.773706	−	0.633545i	$-0.781599\pi$
$613$	−23485.3	−1.54741	−0.773706	+	0.633545i	$0.781599\pi$
$614$	−24779.3	−1.62868
$615$	0	0
$616$	−1005.53	−0.0657691
$617$	6831.26	0.445731	0.222866	−	0.974849i	$-0.428459\pi$
$617$	6831.26	0.445731	0.222866	+	0.974849i	$0.428459\pi$
$618$	0	0
$619$	−21443.3	−1.39237	−0.696186	−	0.717861i	$-0.745121\pi$
$619$	−21443.3	−1.39237	−0.696186	+	0.717861i	$0.745121\pi$
$620$	0	0
$621$	0	0
$622$	32107.8	2.06979
$623$	11154.5	0.717325
$624$	0	0
$625$	0	0
$626$	−39897.6	−2.54733
$627$	0	0
$628$	19002.6	1.20746
$629$	−2581.13	−0.163619
$630$	0	0
$631$	−3100.35	−0.195599	−0.0977995	−	0.995206i	$-0.531180\pi$
$631$	−3100.35	−0.195599	−0.0977995	+	0.995206i	$0.531180\pi$
$632$	2638.91	0.166092
$633$	0	0
$634$	9536.55	0.597389
$635$	0	0
$636$	0	0
$637$	−2810.25	−0.174798
$638$	2918.80	0.181123
$639$	0	0
$640$	0	0
$641$	−31019.8	−1.91140	−0.955702	−	0.294336i	$-0.904902\pi$
$641$	−31019.8	−1.91140	−0.955702	+	0.294336i	$0.904902\pi$
$642$	0	0
$643$	14636.9	0.897703	0.448852	−	0.893606i	$-0.351833\pi$
$643$	14636.9	0.897703	0.448852	+	0.893606i	$0.351833\pi$
$644$	4980.66	0.304760
$645$	0	0
$646$	7963.22	0.484998
$647$	8902.74	0.540963	0.270481	−	0.962725i	$-0.412817\pi$
$647$	8902.74	0.540963	0.270481	+	0.962725i	$0.412817\pi$
$648$	0	0
$649$	−6901.05	−0.417396
$650$	0	0
$651$	0	0
$652$	2901.26	0.174267
$653$	21809.9	1.30703	0.653513	−	0.756915i	$-0.273294\pi$
$653$	21809.9	1.30703	0.653513	+	0.756915i	$0.273294\pi$
$654$	0	0
$655$	0	0
$656$	14484.0	0.862052
$657$	0	0
$658$	11751.3	0.696219
$659$	−26206.6	−1.54911	−0.774556	−	0.632505i	$-0.782027\pi$
$659$	−26206.6	−1.54911	−0.774556	+	0.632505i	$0.782027\pi$
$660$	0	0
$661$	−4008.03	−0.235846	−0.117923	−	0.993023i	$-0.537624\pi$
$661$	−4008.03	−0.235846	−0.117923	+	0.993023i	$0.537624\pi$
$662$	−32274.3	−1.89483
$663$	0	0
$664$	−5786.58	−0.338197
$665$	0	0
$666$	0	0
$667$	−3541.70	−0.205600
$668$	−9490.58	−0.549703
$669$	0	0
$670$	0	0
$671$	−3691.16	−0.212363
$672$	0	0
$673$	8924.47	0.511164	0.255582	−	0.966787i	$-0.417733\pi$
$673$	8924.47	0.511164	0.255582	+	0.966787i	$0.417733\pi$
$674$	−28724.7	−1.64159
$675$	0	0
$676$	11573.1	0.658458
$677$	28920.9	1.64183	0.820917	−	0.571048i	$-0.193463\pi$
$677$	28920.9	1.64183	0.820917	+	0.571048i	$0.193463\pi$
$678$	0	0
$679$	−7791.62	−0.440376
$680$	0	0
$681$	0	0
$682$	6491.27	0.364463
$683$	−23258.8	−1.30304	−0.651518	−	0.758633i	$-0.725868\pi$
$683$	−23258.8	−1.30304	−0.651518	+	0.758633i	$0.725868\pi$
$684$	0	0
$685$	0	0
$686$	−1479.11	−0.0823215
$687$	0	0
$688$	15767.0	0.873710
$689$	962.092	0.0531971
$690$	0	0
$691$	−7735.27	−0.425852	−0.212926	−	0.977068i	$-0.568299\pi$
$691$	−7735.27	−0.425852	−0.212926	+	0.977068i	$0.568299\pi$
$692$	−28696.8	−1.57643
$693$	0	0
$694$	3816.84	0.208768
$695$	0	0
$696$	0	0
$697$	−6801.02	−0.369594
$698$	37397.0	2.02793
$699$	0	0
$700$	0	0
$701$	899.599	0.0484699	0.0242349	−	0.999706i	$-0.492285\pi$
$701$	899.599	0.0484699	0.0242349	+	0.999706i	$0.492285\pi$
$702$	0	0
$703$	−16228.9	−0.870675
$704$	−9918.51	−0.530991
$705$	0	0
$706$	20231.7	1.07851
$707$	−3519.15	−0.187201
$708$	0	0
$709$	11448.3	0.606418	0.303209	−	0.952924i	$-0.401942\pi$
$709$	11448.3	0.606418	0.303209	+	0.952924i	$0.401942\pi$
$710$	0	0
$711$	0	0
$712$	−17835.9	−0.938807
$713$	−7876.58	−0.413717
$714$	0	0
$715$	0	0
$716$	−29236.4	−1.52600
$717$	0	0
$718$	36317.2	1.88767
$719$	2408.92	0.124948	0.0624740	−	0.998047i	$-0.480101\pi$
$719$	2408.92	0.124948	0.0624740	+	0.998047i	$0.480101\pi$
$720$	0	0
$721$	6043.20	0.312150
$722$	20491.0	1.05623
$723$	0	0
$724$	22255.9	1.14245
$725$	0	0
$726$	0	0
$727$	−35748.0	−1.82369	−0.911844	−	0.410537i	$-0.865341\pi$
$727$	−35748.0	−1.82369	−0.911844	+	0.410537i	$0.865341\pi$
$728$	4493.58	0.228768
$729$	0	0
$730$	0	0
$731$	−7403.46	−0.374592
$732$	0	0
$733$	127.891	0.00644443	0.00322221	−	0.999995i	$-0.498974\pi$
$733$	127.891	0.00644443	0.00322221	+	0.999995i	$0.498974\pi$
$734$	−25737.9	−1.29428
$735$	0	0
$736$	16582.1	0.830468
$737$	−10612.3	−0.530407
$738$	0	0
$739$	5766.59	0.287047	0.143523	−	0.989647i	$-0.454157\pi$
$739$	5766.59	0.287047	0.143523	+	0.989647i	$0.454157\pi$
$740$	0	0
$741$	0	0
$742$	506.374	0.0250533
$743$	9227.18	0.455602	0.227801	−	0.973708i	$-0.426846\pi$
$743$	9227.18	0.455602	0.227801	+	0.973708i	$0.426846\pi$
$744$	0	0
$745$	0	0
$746$	19924.6	0.977873
$747$	0	0
$748$	2330.38	0.113913
$749$	−11944.4	−0.582695
$750$	0	0
$751$	14654.8	0.712068	0.356034	−	0.934473i	$-0.384129\pi$
$751$	14654.8	0.712068	0.356034	+	0.934473i	$0.384129\pi$
$752$	14208.5	0.689003
$753$	0	0
$754$	−13043.8	−0.630008
$755$	0	0
$756$	0	0
$757$	−30075.8	−1.44402	−0.722010	−	0.691882i	$-0.756782\pi$
$757$	−30075.8	−1.44402	−0.722010	+	0.691882i	$0.756782\pi$
$758$	57988.4	2.77867
$759$	0	0
$760$	0	0
$761$	−23680.3	−1.12800	−0.564002	−	0.825774i	$-0.690739\pi$
$761$	−23680.3	−1.12800	−0.564002	+	0.825774i	$0.690739\pi$
$762$	0	0
$763$	−14919.9	−0.707912
$764$	32872.4	1.55665
$765$	0	0
$766$	25572.2	1.20622
$767$	30840.0	1.45185
$768$	0	0
$769$	−28376.3	−1.33066	−0.665328	−	0.746551i	$-0.731708\pi$
$769$	−28376.3	−1.33066	−0.665328	+	0.746551i	$0.731708\pi$
$770$	0	0
$771$	0	0
$772$	34353.7	1.60158
$773$	−19138.8	−0.890525	−0.445263	−	0.895400i	$-0.646890\pi$
$773$	−19138.8	−0.890525	−0.445263	+	0.895400i	$0.646890\pi$
$774$	0	0
$775$	0	0
$776$	12458.8	0.576346
$777$	0	0
$778$	26291.4	1.21156
$779$	−42761.5	−1.96674
$780$	0	0
$781$	7888.33	0.361417
$782$	−4962.72	−0.226939
$783$	0	0
$784$	−1788.39	−0.0814684
$785$	0	0
$786$	0	0
$787$	−32890.9	−1.48975	−0.744876	−	0.667203i	$-0.767491\pi$
$787$	−32890.9	−1.48975	−0.744876	+	0.667203i	$0.767491\pi$
$788$	−21893.5	−0.989750
$789$	0	0
$790$	0	0
$791$	−2246.26	−0.100971
$792$	0	0
$793$	16495.4	0.738674
$794$	2878.57	0.128661
$795$	0	0
$796$	−58319.1	−2.59682
$797$	−37925.0	−1.68554	−0.842769	−	0.538276i	$-0.819076\pi$
$797$	−37925.0	−1.68554	−0.842769	+	0.538276i	$0.819076\pi$
$798$	0	0
$799$	−6671.65	−0.295402
$800$	0	0
$801$	0	0
$802$	−29319.4	−1.29090
$803$	−3005.42	−0.132078
$804$	0	0
$805$	0	0
$806$	−29008.8	−1.26773
$807$	0	0
$808$	5627.12	0.245002
$809$	29799.1	1.29503	0.647515	−	0.762052i	$-0.275808\pi$
$809$	29799.1	1.29503	0.647515	+	0.762052i	$0.275808\pi$
$810$	0	0
$811$	65.6762	0.00284365	0.00142183	−	0.999999i	$-0.499547\pi$
$811$	65.6762	0.00284365	0.00142183	+	0.999999i	$0.499547\pi$
$812$	−3911.77	−0.169060
$813$	0	0
$814$	−8335.12	−0.358902
$815$	0	0
$816$	0	0
$817$	−46549.4	−1.99334
$818$	−64695.7	−2.76532
$819$	0	0
$820$	0	0
$821$	−2199.22	−0.0934877	−0.0467438	−	0.998907i	$-0.514884\pi$
$821$	−2199.22	−0.0934877	−0.0467438	+	0.998907i	$0.514884\pi$
$822$	0	0
$823$	33734.0	1.42879	0.714395	−	0.699743i	$-0.246702\pi$
$823$	33734.0	1.42879	0.714395	+	0.699743i	$0.246702\pi$
$824$	−9663.06	−0.408530
$825$	0	0
$826$	16231.9	0.683753
$827$	−1502.97	−0.0631964	−0.0315982	−	0.999501i	$-0.510060\pi$
$827$	−1502.97	−0.0631964	−0.0315982	+	0.999501i	$0.510060\pi$
$828$	0	0
$829$	8866.61	0.371472	0.185736	−	0.982600i	$-0.440533\pi$
$829$	8866.61	0.371472	0.185736	+	0.982600i	$0.440533\pi$
$830$	0	0
$831$	0	0
$832$	44324.7	1.84697
$833$	839.746	0.0349286
$834$	0	0
$835$	0	0
$836$	14652.3	0.606174
$837$	0	0
$838$	14494.2	0.597487
$839$	−34958.5	−1.43850	−0.719250	−	0.694751i	$-0.755514\pi$
$839$	−34958.5	−1.43850	−0.719250	+	0.694751i	$0.755514\pi$
$840$	0	0
$841$	−21607.4	−0.885947
$842$	−11148.7	−0.456307
$843$	0	0
$844$	36680.7	1.49597
$845$	0	0
$846$	0	0
$847$	8164.09	0.331194
$848$	612.259	0.0247937
$849$	0	0
$850$	0	0
$851$	10113.9	0.407404
$852$	0	0
$853$	−17117.8	−0.687105	−0.343553	−	0.939133i	$-0.611630\pi$
$853$	−17117.8	−0.687105	−0.343553	+	0.939133i	$0.611630\pi$
$854$	8681.95	0.347881
$855$	0	0
$856$	19099.0	0.762608
$857$	−22884.8	−0.912169	−0.456085	−	0.889936i	$-0.650749\pi$
$857$	−22884.8	−0.912169	−0.456085	+	0.889936i	$0.650749\pi$
$858$	0	0
$859$	−26159.2	−1.03905	−0.519523	−	0.854456i	$-0.673890\pi$
$859$	−26159.2	−1.03905	−0.519523	+	0.854456i	$0.673890\pi$
$860$	0	0
$861$	0	0
$862$	65138.1	2.57380
$863$	47327.4	1.86680	0.933398	−	0.358844i	$-0.116829\pi$
$863$	47327.4	1.86680	0.933398	+	0.358844i	$0.116829\pi$
$864$	0	0
$865$	0	0
$866$	−19573.2	−0.768042
$867$	0	0
$868$	−8699.61	−0.340189
$869$	3025.71	0.118113
$870$	0	0
$871$	47425.2	1.84494
$872$	23856.9	0.926487
$873$	0	0
$874$	−31203.2	−1.20762
$875$	0	0
$876$	0	0
$877$	−33296.3	−1.28203	−0.641013	−	0.767530i	$-0.721486\pi$
$877$	−33296.3	−1.28203	−0.641013	+	0.767530i	$0.721486\pi$
$878$	−43335.5	−1.66572
$879$	0	0
$880$	0	0
$881$	7158.46	0.273751	0.136875	−	0.990588i	$-0.456294\pi$
$881$	7158.46	0.273751	0.136875	+	0.990588i	$0.456294\pi$
$882$	0	0
$883$	−16972.4	−0.646847	−0.323423	−	0.946254i	$-0.604834\pi$
$883$	−16972.4	−0.646847	−0.323423	+	0.946254i	$0.604834\pi$
$884$	−10414.2	−0.396231
$885$	0	0
$886$	−45871.1	−1.73936
$887$	14696.9	0.556339	0.278169	−	0.960532i	$-0.410272\pi$
$887$	14696.9	0.556339	0.278169	+	0.960532i	$0.410272\pi$
$888$	0	0
$889$	−5425.61	−0.204690
$890$	0	0
$891$	0	0
$892$	−12693.4	−0.476464
$893$	−41948.1	−1.57194
$894$	0	0
$895$	0	0
$896$	9501.05	0.354250
$897$	0	0
$898$	26420.7	0.981814
$899$	6186.21	0.229501
$900$	0	0
$901$	−287.488	−0.0106300
$902$	−21962.2	−0.810712
$903$	0	0
$904$	3591.77	0.132147
$905$	0	0
$906$	0	0
$907$	−8554.10	−0.313158	−0.156579	−	0.987665i	$-0.550047\pi$
$907$	−8554.10	−0.313158	−0.156579	+	0.987665i	$0.550047\pi$
$908$	21404.4	0.782302
$909$	0	0
$910$	0	0
$911$	43734.7	1.59055	0.795277	−	0.606246i	$-0.207325\pi$
$911$	43734.7	1.59055	0.795277	+	0.606246i	$0.207325\pi$
$912$	0	0
$913$	−6634.75	−0.240502
$914$	−57535.6	−2.08218
$915$	0	0
$916$	−32936.8	−1.18806
$917$	−1249.15	−0.0449842
$918$	0	0
$919$	50797.6	1.82335	0.911674	−	0.410914i	$-0.134790\pi$
$919$	50797.6	1.82335	0.911674	+	0.410914i	$0.134790\pi$
$920$	0	0
$921$	0	0
$922$	36907.9	1.31833
$923$	−35252.0	−1.25713
$924$	0	0
$925$	0	0
$926$	78542.4	2.78733
$927$	0	0
$928$	−13023.5	−0.460686
$929$	−7003.67	−0.247344	−0.123672	−	0.992323i	$-0.539467\pi$
$929$	−7003.67	−0.247344	−0.123672	+	0.992323i	$0.539467\pi$
$930$	0	0
$931$	5279.92	0.185867
$932$	72483.6	2.54751
$933$	0	0
$934$	25564.5	0.895606
$935$	0	0
$936$	0	0
$937$	8481.46	0.295707	0.147853	−	0.989009i	$-0.452764\pi$
$937$	8481.46	0.295707	0.147853	+	0.989009i	$0.452764\pi$
$938$	24961.1	0.868881
$939$	0	0
$940$	0	0
$941$	4608.08	0.159638	0.0798188	−	0.996809i	$-0.474566\pi$
$941$	4608.08	0.159638	0.0798188	+	0.996809i	$0.474566\pi$
$942$	0	0
$943$	26649.2	0.920273
$944$	19626.0	0.676667
$945$	0	0
$946$	−23907.7	−0.821676
$947$	43416.5	1.48981	0.744903	−	0.667172i	$-0.232496\pi$
$947$	43416.5	1.48981	0.744903	+	0.667172i	$0.232496\pi$
$948$	0	0
$949$	13430.9	0.459415
$950$	0	0
$951$	0	0
$952$	−1342.75	−0.0457131
$953$	24453.1	0.831178	0.415589	−	0.909553i	$-0.363576\pi$
$953$	24453.1	0.831178	0.415589	+	0.909553i	$0.363576\pi$
$954$	0	0
$955$	0	0
$956$	−6031.45	−0.204049
$957$	0	0
$958$	−57612.8	−1.94299
$959$	−11331.9	−0.381572
$960$	0	0
$961$	−16033.1	−0.538187
$962$	37248.7	1.24839
$963$	0	0
$964$	28942.8	0.966998
$965$	0	0
$966$	0	0
$967$	−23085.2	−0.767703	−0.383851	−	0.923395i	$-0.625403\pi$
$967$	−23085.2	−0.767703	−0.383851	+	0.923395i	$0.625403\pi$
$968$	−13054.4	−0.433454
$969$	0	0
$970$	0	0
$971$	26971.5	0.891406	0.445703	−	0.895181i	$-0.352954\pi$
$971$	26971.5	0.891406	0.445703	+	0.895181i	$0.352954\pi$
$972$	0	0
$973$	2350.98	0.0774603
$974$	56333.3	1.85322
$975$	0	0
$976$	10497.4	0.344276
$977$	9880.27	0.323539	0.161770	−	0.986829i	$-0.448280\pi$
$977$	9880.27	0.323539	0.161770	+	0.986829i	$0.448280\pi$
$978$	0	0
$979$	−20450.3	−0.667614
$980$	0	0
$981$	0	0
$982$	−26906.0	−0.874344
$983$	−32148.0	−1.04310	−0.521548	−	0.853222i	$-0.674645\pi$
$983$	−32148.0	−1.04310	−0.521548	+	0.853222i	$0.674645\pi$
$984$	0	0
$985$	0	0
$986$	3897.69	0.125890
$987$	0	0
$988$	−65479.6	−2.10848
$989$	29009.8	0.932719
$990$	0	0
$991$	16109.8	0.516391	0.258196	−	0.966093i	$-0.416872\pi$
$991$	16109.8	0.516391	0.258196	+	0.966093i	$0.416872\pi$
$992$	−28963.6	−0.927012
$993$	0	0
$994$	−18554.1	−0.592052
$995$	0	0
$996$	0	0
$997$	46401.1	1.47396	0.736979	−	0.675915i	$-0.236251\pi$
$997$	46401.1	1.47396	0.736979	+	0.675915i	$0.236251\pi$
$998$	41476.7	1.31555
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bt.1.9		10
3.2	odd	2	inner	1575.4.a.bt.1.2		10
5.2	odd	4		315.4.d.d.64.18	yes	20
5.3	odd	4		315.4.d.d.64.4	yes	20
5.4	even	2		1575.4.a.bu.1.2		10
15.2	even	4		315.4.d.d.64.3	✓	20
15.8	even	4		315.4.d.d.64.17	yes	20
15.14	odd	2		1575.4.a.bu.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.d.d.64.3	✓	20	15.2	even	4
315.4.d.d.64.4	yes	20	5.3	odd	4
315.4.d.d.64.17	yes	20	15.8	even	4
315.4.d.d.64.18	yes	20	5.2	odd	4
1575.4.a.bt.1.2		10	3.2	odd	2	inner
1575.4.a.bt.1.9		10	1.1	even	1	trivial
1575.4.a.bu.1.2		10	5.4	even	2
1575.4.a.bu.1.9		10	15.14	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+4.31226 q^{2} +10.5956 q^{4} -7.00000 q^{7} +11.1930 q^{8} +O(q^{10})\)	\(q+4.31226 q^{2} +10.5956 q^{4} -7.00000 q^{7} +11.1930 q^{8} +12.8336 q^{11} -57.3520 q^{13} -30.1858 q^{14} -36.4978 q^{16} +17.1377 q^{17} +107.753 q^{19} +55.3419 q^{22} -67.1525 q^{23} -247.317 q^{26} -74.1693 q^{28} +52.7411 q^{29} +117.294 q^{31} -246.932 q^{32} +73.9022 q^{34} -150.611 q^{37} +464.661 q^{38} -396.846 q^{41} -431.999 q^{43} +135.980 q^{44} -289.579 q^{46} -389.297 q^{47} +49.0000 q^{49} -607.680 q^{52} -16.7752 q^{53} -78.3509 q^{56} +227.434 q^{58} -537.732 q^{59} -287.617 q^{61} +505.802 q^{62} -772.854 q^{64} -826.916 q^{67} +181.584 q^{68} +614.661 q^{71} -234.183 q^{73} -649.476 q^{74} +1141.71 q^{76} -89.8353 q^{77} +235.764 q^{79} -1711.30 q^{82} -516.982 q^{83} -1862.89 q^{86} +143.647 q^{88} -1593.49 q^{89} +401.464 q^{91} -711.522 q^{92} -1678.75 q^{94} +1113.09 q^{97} +211.301 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 54 q^{4} - 70 q^{7}+O(q^{10}) \) 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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