LMFDB - Embedded newform 1575.4.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.26729725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 18x^{2} + 19x + 44 $ x^4 - 2x^3 - 18x^2 + 19*x + 44
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			1.1
Root			$4.56826$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.56826 q^{2} +23.0055 q^{4} -7.00000 q^{7} -83.5546 q^{8} -23.2885 q^{11} -46.5366 q^{13} +38.9778 q^{14} +281.210 q^{16} -76.0316 q^{17} -114.605 q^{19} +129.677 q^{22} -113.880 q^{23} +259.128 q^{26} -161.039 q^{28} -120.470 q^{29} -182.048 q^{31} -897.411 q^{32} +423.363 q^{34} -322.477 q^{37} +638.151 q^{38} +93.0481 q^{41} -452.579 q^{43} -535.765 q^{44} +634.113 q^{46} -402.565 q^{47} +49.0000 q^{49} -1070.60 q^{52} +495.787 q^{53} +584.882 q^{56} +670.808 q^{58} -496.707 q^{59} -265.742 q^{61} +1013.69 q^{62} +2747.34 q^{64} +594.240 q^{67} -1749.14 q^{68} +510.099 q^{71} +470.181 q^{73} +1795.63 q^{74} -2636.55 q^{76} +163.020 q^{77} -487.916 q^{79} -518.116 q^{82} +1250.53 q^{83} +2520.08 q^{86} +1945.86 q^{88} +1561.98 q^{89} +325.756 q^{91} -2619.87 q^{92} +2241.59 q^{94} +21.5118 q^{97} -272.845 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} + 16 q^{4} - 28 q^{7} - 93 q^{8} - 57 q^{11} + 43 q^{13} + 42 q^{14} + 216 q^{16} - 99 q^{17} - 12 q^{19} - 41 q^{22} - 156 q^{23} + 81 q^{26} - 112 q^{28} - 378 q^{29} - 93 q^{31} - 690 q^{32}+ \cdots - 294 q^{98}+O(q^{100}) $ 4 * q - 6 * q^2 + 16 * q^4 - 28 * q^7 - 93 * q^8 - 57 * q^11 + 43 * q^13 + 42 * q^14 + 216 * q^16 - 99 * q^17 - 12 * q^19 - 41 * q^22 - 156 * q^23 + 81 * q^26 - 112 * q^28 - 378 * q^29 - 93 * q^31 - 690 * q^32 + 783 * q^34 + 81 * q^37 - 216 * q^38 + 465 * q^41 - 64 * q^43 - 681 * q^44 + 310 * q^46 - 744 * q^47 + 196 * q^49 - 727 * q^52 - 729 * q^53 + 651 * q^56 + 1172 * q^58 - 231 * q^59 - 1353 * q^61 + 165 * q^62 + 3107 * q^64 + 1487 * q^67 - 2577 * q^68 + 1725 * q^71 + 512 * q^73 + 1953 * q^74 - 3046 * q^76 + 399 * q^77 + 1629 * q^79 - 693 * q^82 - 321 * q^83 + 4542 * q^86 + 3482 * q^88 + 978 * q^89 - 301 * q^91 - 852 * q^92 + 2480 * q^94 + 2616 * q^97 - 294 * 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 $

$2$	−5.56826	−1.96868	−0.984339	−	0.176289i	$-0.943591\pi$
$2$	−5.56826	−1.96868	−0.984339	+	0.176289i	$0.943591\pi$
$3$	0	0
$3$	0	0
$4$	23.0055	2.87569
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−83.5546	−3.69263
$9$	0	0
$10$	0	0
$11$	−23.2885	−0.638342	−0.319171	−	0.947697i	$-0.603404\pi$
$11$	−23.2885	−0.638342	−0.319171	+	0.947697i	$0.603404\pi$
$12$	0	0
$13$	−46.5366	−0.992840	−0.496420	−	0.868082i	$-0.665352\pi$
$13$	−46.5366	−0.992840	−0.496420	+	0.868082i	$0.665352\pi$
$14$	38.9778	0.744090
$15$	0	0
$16$	281.210	4.39390
$17$	−76.0316	−1.08473	−0.542364	−	0.840144i	$-0.682470\pi$
$17$	−76.0316	−1.08473	−0.542364	+	0.840144i	$0.682470\pi$
$18$	0	0
$19$	−114.605	−1.38380	−0.691901	−	0.721993i	$-0.743226\pi$
$19$	−114.605	−1.38380	−0.691901	+	0.721993i	$0.743226\pi$
$20$	0	0
$21$	0	0
$22$	129.677	1.25669
$23$	−113.880	−1.03242	−0.516209	−	0.856463i	$-0.672657\pi$
$23$	−113.880	−1.03242	−0.516209	+	0.856463i	$0.672657\pi$
$24$	0	0
$25$	0	0
$26$	259.128	1.95458
$27$	0	0
$28$	−161.039	−1.08691
$29$	−120.470	−0.771404	−0.385702	−	0.922623i	$-0.626041\pi$
$29$	−120.470	−0.771404	−0.385702	+	0.922623i	$0.626041\pi$
$30$	0	0
$31$	−182.048	−1.05474	−0.527369	−	0.849637i	$-0.676821\pi$
$31$	−182.048	−1.05474	−0.527369	+	0.849637i	$0.676821\pi$
$32$	−897.411	−4.95754
$33$	0	0
$34$	423.363	2.13548
$35$	0	0
$36$	0	0
$37$	−322.477	−1.43283	−0.716417	−	0.697673i	$-0.754219\pi$
$37$	−322.477	−1.43283	−0.716417	+	0.697673i	$0.754219\pi$
$38$	638.151	2.72426
$39$	0	0
$40$	0	0
$41$	93.0481	0.354431	0.177215	−	0.984172i	$-0.443291\pi$
$41$	93.0481	0.354431	0.177215	+	0.984172i	$0.443291\pi$
$42$	0	0
$43$	−452.579	−1.60506	−0.802530	−	0.596612i	$-0.796513\pi$
$43$	−452.579	−1.60506	−0.802530	+	0.596612i	$0.796513\pi$
$44$	−535.765	−1.83567
$45$	0	0
$46$	634.113	2.03250
$47$	−402.565	−1.24937	−0.624683	−	0.780879i	$-0.714772\pi$
$47$	−402.565	−1.24937	−0.624683	+	0.780879i	$0.714772\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−1070.60	−2.85510
$53$	495.787	1.28494	0.642468	−	0.766313i	$-0.277911\pi$
$53$	495.787	1.28494	0.642468	+	0.766313i	$0.277911\pi$
$54$	0	0
$55$	0	0
$56$	584.882	1.39568
$57$	0	0
$58$	670.808	1.51865
$59$	−496.707	−1.09603	−0.548015	−	0.836468i	$-0.684616\pi$
$59$	−496.707	−1.09603	−0.548015	+	0.836468i	$0.684616\pi$
$60$	0	0
$61$	−265.742	−0.557783	−0.278891	−	0.960323i	$-0.589967\pi$
$61$	−265.742	−0.557783	−0.278891	+	0.960323i	$0.589967\pi$
$62$	1013.69	2.07644
$63$	0	0
$64$	2747.34	5.36590
$65$	0	0
$66$	0	0
$67$	594.240	1.08355	0.541776	−	0.840523i	$-0.317752\pi$
$67$	594.240	1.08355	0.541776	+	0.840523i	$0.317752\pi$
$68$	−1749.14	−3.11934
$69$	0	0
$70$	0	0
$71$	510.099	0.852642	0.426321	−	0.904572i	$-0.359809\pi$
$71$	510.099	0.852642	0.426321	+	0.904572i	$0.359809\pi$
$72$	0	0
$73$	470.181	0.753843	0.376921	−	0.926245i	$-0.376983\pi$
$73$	470.181	0.753843	0.376921	+	0.926245i	$0.376983\pi$
$74$	1795.63	2.82079
$75$	0	0
$76$	−2636.55	−3.97938
$77$	163.020	0.241271
$78$	0	0
$79$	−487.916	−0.694871	−0.347435	−	0.937704i	$-0.612947\pi$
$79$	−487.916	−0.694871	−0.347435	+	0.937704i	$0.612947\pi$
$80$	0	0
$81$	0	0
$82$	−518.116	−0.697760
$83$	1250.53	1.65378	0.826891	−	0.562362i	$-0.190107\pi$
$83$	1250.53	1.65378	0.826891	+	0.562362i	$0.190107\pi$
$84$	0	0
$85$	0	0
$86$	2520.08	3.15985
$87$	0	0
$88$	1945.86	2.35716
$89$	1561.98	1.86033	0.930164	−	0.367145i	$-0.119665\pi$
$89$	1561.98	1.86033	0.930164	+	0.367145i	$0.119665\pi$
$90$	0	0
$91$	325.756	0.375258
$92$	−2619.87	−2.96891
$93$	0	0
$94$	2241.59	2.45960
$95$	0	0
$96$	0	0
$97$	21.5118	0.0225175	0.0112587	−	0.999937i	$-0.496416\pi$
$97$	21.5118	0.0225175	0.0112587	+	0.999937i	$0.496416\pi$
$98$	−272.845	−0.281240
$99$	0	0
$100$	0	0
$101$	−211.353	−0.208222	−0.104111	−	0.994566i	$-0.533200\pi$
$101$	−211.353	−0.208222	−0.104111	+	0.994566i	$0.533200\pi$
$102$	0	0
$103$	−1180.82	−1.12961	−0.564806	−	0.825224i	$-0.691049\pi$
$103$	−1180.82	−1.12961	−0.564806	+	0.825224i	$0.691049\pi$
$104$	3888.35	3.66619
$105$	0	0
$106$	−2760.67	−2.52962
$107$	−1114.80	−1.00721	−0.503607	−	0.863933i	$-0.667994\pi$
$107$	−1114.80	−1.00721	−0.503607	+	0.863933i	$0.667994\pi$
$108$	0	0
$109$	970.519	0.852834	0.426417	−	0.904527i	$-0.359776\pi$
$109$	970.519	0.852834	0.426417	+	0.904527i	$0.359776\pi$
$110$	0	0
$111$	0	0
$112$	−1968.47	−1.66074
$113$	−800.030	−0.666022	−0.333011	−	0.942923i	$-0.608065\pi$
$113$	−800.030	−0.666022	−0.333011	+	0.942923i	$0.608065\pi$
$114$	0	0
$115$	0	0
$116$	−2771.47	−2.21832
$117$	0	0
$118$	2765.79	2.15773
$119$	532.221	0.409988
$120$	0	0
$121$	−788.644	−0.592520
$122$	1479.72	1.09809
$123$	0	0
$124$	−4188.12	−3.03310
$125$	0	0
$126$	0	0
$127$	526.685	0.367998	0.183999	−	0.982926i	$-0.441096\pi$
$127$	526.685	0.367998	0.183999	+	0.982926i	$0.441096\pi$
$128$	−8118.62	−5.60618
$129$	0	0
$130$	0	0
$131$	−827.813	−0.552110	−0.276055	−	0.961142i	$-0.589027\pi$
$131$	−827.813	−0.552110	−0.276055	+	0.961142i	$0.589027\pi$
$132$	0	0
$133$	802.236	0.523028
$134$	−3308.88	−2.13316
$135$	0	0
$136$	6352.79	4.00549
$137$	−1636.01	−1.02025	−0.510124	−	0.860101i	$-0.670401\pi$
$137$	−1636.01	−1.02025	−0.510124	+	0.860101i	$0.670401\pi$
$138$	0	0
$139$	−1463.08	−0.892783	−0.446391	−	0.894838i	$-0.647291\pi$
$139$	−1463.08	−0.892783	−0.446391	+	0.894838i	$0.647291\pi$
$140$	0	0
$141$	0	0
$142$	−2840.36	−1.67858
$143$	1083.77	0.633772
$144$	0	0
$145$	0	0
$146$	−2618.09	−1.48407
$147$	0	0
$148$	−7418.74	−4.12038
$149$	−330.833	−0.181898	−0.0909492	−	0.995856i	$-0.528990\pi$
$149$	−330.833	−0.181898	−0.0909492	+	0.995856i	$0.528990\pi$
$150$	0	0
$151$	−1139.90	−0.614327	−0.307164	−	0.951657i	$-0.599380\pi$
$151$	−1139.90	−0.614327	−0.307164	+	0.951657i	$0.599380\pi$
$152$	9575.79	5.10986
$153$	0	0
$154$	−907.737	−0.474984
$155$	0	0
$156$	0	0
$157$	3297.93	1.67645	0.838227	−	0.545322i	$-0.183592\pi$
$157$	3297.93	1.67645	0.838227	+	0.545322i	$0.183592\pi$
$158$	2716.84	1.36798
$159$	0	0
$160$	0	0
$161$	797.159	0.390217
$162$	0	0
$163$	2569.02	1.23449	0.617243	−	0.786773i	$-0.288250\pi$
$163$	2569.02	1.23449	0.617243	+	0.786773i	$0.288250\pi$
$164$	2140.62	1.01923
$165$	0	0
$166$	−6963.30	−3.25576
$167$	−1034.90	−0.479540	−0.239770	−	0.970830i	$-0.577072\pi$
$167$	−1034.90	−0.479540	−0.239770	+	0.970830i	$0.577072\pi$
$168$	0	0
$169$	−31.3467	−0.0142679
$170$	0	0
$171$	0	0
$172$	−10411.8	−4.61565
$173$	−244.188	−0.107314	−0.0536568	−	0.998559i	$-0.517088\pi$
$173$	−244.188	−0.107314	−0.0536568	+	0.998559i	$0.517088\pi$
$174$	0	0
$175$	0	0
$176$	−6548.96	−2.80481
$177$	0	0
$178$	−8697.49	−3.66238
$179$	−1523.83	−0.636294	−0.318147	−	0.948041i	$-0.603061\pi$
$179$	−1523.83	−0.636294	−0.318147	+	0.948041i	$0.603061\pi$
$180$	0	0
$181$	−3144.63	−1.29137	−0.645687	−	0.763602i	$-0.723429\pi$
$181$	−3144.63	−1.29137	−0.645687	+	0.763602i	$0.723429\pi$
$182$	−1813.89	−0.738763
$183$	0	0
$184$	9515.19	3.81233
$185$	0	0
$186$	0	0
$187$	1770.66	0.692427
$188$	−9261.22	−3.59279
$189$	0	0
$190$	0	0
$191$	734.675	0.278320	0.139160	−	0.990270i	$-0.455560\pi$
$191$	734.675	0.278320	0.139160	+	0.990270i	$0.455560\pi$
$192$	0	0
$193$	−3248.41	−1.21153	−0.605765	−	0.795643i	$-0.707133\pi$
$193$	−3248.41	−1.21153	−0.605765	+	0.795643i	$0.707133\pi$
$194$	−119.783	−0.0443297
$195$	0	0
$196$	1127.27	0.410813
$197$	−2915.78	−1.05452	−0.527261	−	0.849703i	$-0.676781\pi$
$197$	−2915.78	−1.05452	−0.527261	+	0.849703i	$0.676781\pi$
$198$	0	0
$199$	−1397.88	−0.497955	−0.248977	−	0.968509i	$-0.580095\pi$
$199$	−1397.88	−0.497955	−0.248977	+	0.968509i	$0.580095\pi$
$200$	0	0
$201$	0	0
$202$	1176.87	0.409923
$203$	843.290	0.291563
$204$	0	0
$205$	0	0
$206$	6575.14	2.22384
$207$	0	0
$208$	−13086.5	−4.36244
$209$	2668.99	0.883338
$210$	0	0
$211$	−998.781	−0.325872	−0.162936	−	0.986637i	$-0.552096\pi$
$211$	−998.781	−0.325872	−0.162936	+	0.986637i	$0.552096\pi$
$212$	11405.8	3.69508
$213$	0	0
$214$	6207.50	1.98288
$215$	0	0
$216$	0	0
$217$	1274.34	0.398653
$218$	−5404.10	−1.67895
$219$	0	0
$220$	0	0
$221$	3538.25	1.07696
$222$	0	0
$223$	2820.52	0.846979	0.423489	−	0.905901i	$-0.360805\pi$
$223$	2820.52	0.846979	0.423489	+	0.905901i	$0.360805\pi$
$224$	6281.88	1.87377
$225$	0	0
$226$	4454.78	1.31118
$227$	1420.92	0.415460	0.207730	−	0.978186i	$-0.433392\pi$
$227$	1420.92	0.415460	0.207730	+	0.978186i	$0.433392\pi$
$228$	0	0
$229$	−87.4506	−0.0252354	−0.0126177	−	0.999920i	$-0.504016\pi$
$229$	−87.4506	−0.0252354	−0.0126177	+	0.999920i	$0.504016\pi$
$230$	0	0
$231$	0	0
$232$	10065.8	2.84851
$233$	4719.66	1.32702	0.663509	−	0.748169i	$-0.269067\pi$
$233$	4719.66	1.32702	0.663509	+	0.748169i	$0.269067\pi$
$234$	0	0
$235$	0	0
$236$	−11427.0	−3.15184
$237$	0	0
$238$	−2963.54	−0.807135
$239$	850.656	0.230227	0.115114	−	0.993352i	$-0.463277\pi$
$239$	850.656	0.230227	0.115114	+	0.993352i	$0.463277\pi$
$240$	0	0
$241$	4036.74	1.07896	0.539480	−	0.841999i	$-0.318621\pi$
$241$	4036.74	1.07896	0.539480	+	0.841999i	$0.318621\pi$
$242$	4391.37	1.16648
$243$	0	0
$244$	−6113.52	−1.60401
$245$	0	0
$246$	0	0
$247$	5333.33	1.37389
$248$	15211.0	3.89475
$249$	0	0
$250$	0	0
$251$	−4659.33	−1.17169	−0.585845	−	0.810423i	$-0.699237\pi$
$251$	−4659.33	−1.17169	−0.585845	+	0.810423i	$0.699237\pi$
$252$	0	0
$253$	2652.10	0.659035
$254$	−2932.72	−0.724469
$255$	0	0
$256$	23227.9	5.67086
$257$	1160.22	0.281605	0.140802	−	0.990038i	$-0.455032\pi$
$257$	1160.22	0.281605	0.140802	+	0.990038i	$0.455032\pi$
$258$	0	0
$259$	2257.34	0.541560
$260$	0	0
$261$	0	0
$262$	4609.48	1.08693
$263$	−1589.53	−0.372680	−0.186340	−	0.982485i	$-0.559663\pi$
$263$	−1589.53	−0.372680	−0.186340	+	0.982485i	$0.559663\pi$
$264$	0	0
$265$	0	0
$266$	−4467.06	−1.02967
$267$	0	0
$268$	13670.8	3.11596
$269$	−4568.96	−1.03559	−0.517796	−	0.855504i	$-0.673247\pi$
$269$	−4568.96	−1.03559	−0.517796	+	0.855504i	$0.673247\pi$
$270$	0	0
$271$	−6520.72	−1.46164	−0.730822	−	0.682569i	$-0.760863\pi$
$271$	−6520.72	−1.46164	−0.730822	+	0.682569i	$0.760863\pi$
$272$	−21380.8	−4.76618
$273$	0	0
$274$	9109.75	2.00854
$275$	0	0
$276$	0	0
$277$	−4921.65	−1.06756	−0.533779	−	0.845624i	$-0.679229\pi$
$277$	−4921.65	−1.06756	−0.533779	+	0.845624i	$0.679229\pi$
$278$	8146.81	1.75760
$279$	0	0
$280$	0	0
$281$	−2378.08	−0.504856	−0.252428	−	0.967616i	$-0.581229\pi$
$281$	−2378.08	−0.504856	−0.252428	+	0.967616i	$0.581229\pi$
$282$	0	0
$283$	8673.45	1.82185	0.910924	−	0.412574i	$-0.135370\pi$
$283$	8673.45	1.82185	0.910924	+	0.412574i	$0.135370\pi$
$284$	11735.1	2.45193
$285$	0	0
$286$	−6034.71	−1.24769
$287$	−651.337	−0.133962
$288$	0	0
$289$	867.797	0.176633
$290$	0	0
$291$	0	0
$292$	10816.8	2.16782
$293$	−4334.99	−0.864344	−0.432172	−	0.901791i	$-0.642253\pi$
$293$	−4334.99	−0.864344	−0.432172	+	0.901791i	$0.642253\pi$
$294$	0	0
$295$	0	0
$296$	26944.4	5.29092
$297$	0	0
$298$	1842.16	0.358099
$299$	5299.58	1.02503
$300$	0	0
$301$	3168.05	0.606656
$302$	6347.24	1.20941
$303$	0	0
$304$	−32228.1	−6.08028
$305$	0	0
$306$	0	0
$307$	5849.36	1.08743	0.543715	−	0.839270i	$-0.317017\pi$
$307$	5849.36	1.08743	0.543715	+	0.839270i	$0.317017\pi$
$308$	3750.35	0.693819
$309$	0	0
$310$	0	0
$311$	5005.00	0.912564	0.456282	−	0.889835i	$-0.349181\pi$
$311$	5005.00	0.912564	0.456282	+	0.889835i	$0.349181\pi$
$312$	0	0
$313$	7362.05	1.32948	0.664741	−	0.747074i	$-0.268542\pi$
$313$	7362.05	1.32948	0.664741	+	0.747074i	$0.268542\pi$
$314$	−18363.7	−3.30040
$315$	0	0
$316$	−11224.7	−1.99823
$317$	−7869.89	−1.39438	−0.697188	−	0.716889i	$-0.745566\pi$
$317$	−7869.89	−1.39438	−0.697188	+	0.716889i	$0.745566\pi$
$318$	0	0
$319$	2805.57	0.492419
$320$	0	0
$321$	0	0
$322$	−4438.79	−0.768211
$323$	8713.61	1.50105
$324$	0	0
$325$	0	0
$326$	−14305.0	−2.43030
$327$	0	0
$328$	−7774.60	−1.30878
$329$	2817.96	0.472216
$330$	0	0
$331$	−2608.54	−0.433167	−0.216584	−	0.976264i	$-0.569491\pi$
$331$	−2608.54	−0.433167	−0.216584	+	0.976264i	$0.569491\pi$
$332$	28769.2	4.75576
$333$	0	0
$334$	5762.61	0.944059
$335$	0	0
$336$	0	0
$337$	−5355.89	−0.865739	−0.432870	−	0.901457i	$-0.642499\pi$
$337$	−5355.89	−0.865739	−0.432870	+	0.901457i	$0.642499\pi$
$338$	174.546	0.0280890
$339$	0	0
$340$	0	0
$341$	4239.64	0.673283
$342$	0	0
$343$	−343.000	−0.0539949
$344$	37815.0	5.92689
$345$	0	0
$346$	1359.70	0.211266
$347$	−10607.0	−1.64097	−0.820484	−	0.571670i	$-0.806296\pi$
$347$	−10607.0	−1.64097	−0.820484	+	0.571670i	$0.806296\pi$
$348$	0	0
$349$	−1896.78	−0.290923	−0.145462	−	0.989364i	$-0.546467\pi$
$349$	−1896.78	−0.290923	−0.145462	+	0.989364i	$0.546467\pi$
$350$	0	0
$351$	0	0
$352$	20899.4	3.16461
$353$	4040.68	0.609245	0.304622	−	0.952473i	$-0.401470\pi$
$353$	4040.68	0.609245	0.304622	+	0.952473i	$0.401470\pi$
$354$	0	0
$355$	0	0
$356$	35934.1	5.34972
$357$	0	0
$358$	8485.09	1.25266
$359$	2054.91	0.302100	0.151050	−	0.988526i	$-0.451734\pi$
$359$	2054.91	0.302100	0.151050	+	0.988526i	$0.451734\pi$
$360$	0	0
$361$	6275.34	0.914906
$362$	17510.1	2.54230
$363$	0	0
$364$	7494.19	1.07913
$365$	0	0
$366$	0	0
$367$	−9866.34	−1.40332	−0.701661	−	0.712511i	$-0.747558\pi$
$367$	−9866.34	−1.40332	−0.701661	+	0.712511i	$0.747558\pi$
$368$	−32024.1	−4.53634
$369$	0	0
$370$	0	0
$371$	−3470.51	−0.485660
$372$	0	0
$373$	4420.36	0.613613	0.306806	−	0.951772i	$-0.400740\pi$
$373$	4420.36	0.613613	0.306806	+	0.951772i	$0.400740\pi$
$374$	−9859.52	−1.36316
$375$	0	0
$376$	33636.2	4.61344
$377$	5606.26	0.765881
$378$	0	0
$379$	−13.7935	−0.00186945	−0.000934727	−	1.00000i	$-0.500298\pi$
$379$	−13.7935	−0.00186945	−0.000934727		1.00000i	$0.500298\pi$
$380$	0	0
$381$	0	0
$382$	−4090.86	−0.547923
$383$	1229.76	0.164067	0.0820335	−	0.996630i	$-0.473859\pi$
$383$	1229.76	0.164067	0.0820335	+	0.996630i	$0.473859\pi$
$384$	0	0
$385$	0	0
$386$	18088.0	2.38511
$387$	0	0
$388$	494.891	0.0647533
$389$	2827.48	0.368532	0.184266	−	0.982876i	$-0.441009\pi$
$389$	2827.48	0.368532	0.184266	+	0.982876i	$0.441009\pi$
$390$	0	0
$391$	8658.47	1.11989
$392$	−4094.18	−0.527518
$393$	0	0
$394$	16235.8	2.07601
$395$	0	0
$396$	0	0
$397$	1322.12	0.167142	0.0835711	−	0.996502i	$-0.473367\pi$
$397$	1322.12	0.167142	0.0835711	+	0.996502i	$0.473367\pi$
$398$	7783.76	0.980313
$399$	0	0
$400$	0	0
$401$	−7112.69	−0.885762	−0.442881	−	0.896580i	$-0.646044\pi$
$401$	−7112.69	−0.885762	−0.442881	+	0.896580i	$0.646044\pi$
$402$	0	0
$403$	8471.91	1.04719
$404$	−4862.30	−0.598783
$405$	0	0
$406$	−4695.66	−0.573994
$407$	7510.01	0.914637
$408$	0	0
$409$	5986.22	0.723715	0.361858	−	0.932233i	$-0.382143\pi$
$409$	5986.22	0.723715	0.361858	+	0.932233i	$0.382143\pi$
$410$	0	0
$411$	0	0
$412$	−27165.5	−3.24841
$413$	3476.95	0.414260
$414$	0	0
$415$	0	0
$416$	41762.4	4.92205
$417$	0	0
$418$	−14861.6	−1.73901
$419$	64.5704	0.00752857	0.00376428	−	0.999993i	$-0.498802\pi$
$419$	64.5704	0.00752857	0.00376428	+	0.999993i	$0.498802\pi$
$420$	0	0
$421$	−4482.13	−0.518874	−0.259437	−	0.965760i	$-0.583537\pi$
$421$	−4482.13	−0.518874	−0.259437	+	0.965760i	$0.583537\pi$
$422$	5561.47	0.641536
$423$	0	0
$424$	−41425.3	−4.74479
$425$	0	0
$426$	0	0
$427$	1860.19	0.210822
$428$	−25646.6	−2.89643
$429$	0	0
$430$	0	0
$431$	−10161.5	−1.13564	−0.567821	−	0.823152i	$-0.692213\pi$
$431$	−10161.5	−1.13564	−0.567821	+	0.823152i	$0.692213\pi$
$432$	0	0
$433$	2027.12	0.224982	0.112491	−	0.993653i	$-0.464117\pi$
$433$	2027.12	0.224982	0.112491	+	0.993653i	$0.464117\pi$
$434$	−7095.85	−0.784819
$435$	0	0
$436$	22327.3	2.45249
$437$	13051.2	1.42866
$438$	0	0
$439$	−9366.44	−1.01830	−0.509152	−	0.860676i	$-0.670041\pi$
$439$	−9366.44	−1.01830	−0.509152	+	0.860676i	$0.670041\pi$
$440$	0	0
$441$	0	0
$442$	−19701.9	−2.12019
$443$	−7341.52	−0.787373	−0.393686	−	0.919245i	$-0.628800\pi$
$443$	−7341.52	−0.787373	−0.393686	+	0.919245i	$0.628800\pi$
$444$	0	0
$445$	0	0
$446$	−15705.4	−1.66743
$447$	0	0
$448$	−19231.4	−2.02812
$449$	4667.02	0.490535	0.245267	−	0.969455i	$-0.421124\pi$
$449$	4667.02	0.490535	0.245267	+	0.969455i	$0.421124\pi$
$450$	0	0
$451$	−2166.95	−0.226248
$452$	−18405.1	−1.91527
$453$	0	0
$454$	−7912.03	−0.817907
$455$	0	0
$456$	0	0
$457$	9648.38	0.987598	0.493799	−	0.869576i	$-0.335608\pi$
$457$	9648.38	0.987598	0.493799	+	0.869576i	$0.335608\pi$
$458$	486.948	0.0496803
$459$	0	0
$460$	0	0
$461$	−16097.6	−1.62634	−0.813168	−	0.582028i	$-0.802259\pi$
$461$	−16097.6	−1.62634	−0.813168	+	0.582028i	$0.802259\pi$
$462$	0	0
$463$	11024.6	1.10661	0.553303	−	0.832980i	$-0.313367\pi$
$463$	11024.6	1.10661	0.553303	+	0.832980i	$0.313367\pi$
$464$	−33877.3	−3.38947
$465$	0	0
$466$	−26280.3	−2.61247
$467$	−15275.8	−1.51366	−0.756829	−	0.653613i	$-0.773252\pi$
$467$	−15275.8	−1.51366	−0.756829	+	0.653613i	$0.773252\pi$
$468$	0	0
$469$	−4159.68	−0.409544
$470$	0	0
$471$	0	0
$472$	41502.2	4.04723
$473$	10539.9	1.02458
$474$	0	0
$475$	0	0
$476$	12244.0	1.17900
$477$	0	0
$478$	−4736.67	−0.453243
$479$	6247.49	0.595939	0.297970	−	0.954575i	$-0.403691\pi$
$479$	6247.49	0.595939	0.297970	+	0.954575i	$0.403691\pi$
$480$	0	0
$481$	15007.0	1.42257
$482$	−22477.6	−2.12412
$483$	0	0
$484$	−18143.2	−1.70390
$485$	0	0
$486$	0	0
$487$	15579.8	1.44967	0.724835	−	0.688922i	$-0.241916\pi$
$487$	15579.8	1.44967	0.724835	+	0.688922i	$0.241916\pi$
$488$	22203.9	2.05968
$489$	0	0
$490$	0	0
$491$	−5604.60	−0.515137	−0.257568	−	0.966260i	$-0.582921\pi$
$491$	−5604.60	−0.515137	−0.257568	+	0.966260i	$0.582921\pi$
$492$	0	0
$493$	9159.52	0.836763
$494$	−29697.4	−2.70475
$495$	0	0
$496$	−51193.7	−4.63441
$497$	−3570.69	−0.322268
$498$	0	0
$499$	18016.9	1.61633	0.808163	−	0.588958i	$-0.200462\pi$
$499$	18016.9	1.61633	0.808163	+	0.588958i	$0.200462\pi$
$500$	0	0
$501$	0	0
$502$	25944.4	2.30668
$503$	−3405.38	−0.301865	−0.150933	−	0.988544i	$-0.548228\pi$
$503$	−3405.38	−0.301865	−0.150933	+	0.988544i	$0.548228\pi$
$504$	0	0
$505$	0	0
$506$	−14767.6	−1.29743
$507$	0	0
$508$	12116.7	1.05825
$509$	3669.88	0.319577	0.159788	−	0.987151i	$-0.448919\pi$
$509$	3669.88	0.319577	0.159788	+	0.987151i	$0.448919\pi$
$510$	0	0
$511$	−3291.27	−0.284926
$512$	−64389.7	−5.55791
$513$	0	0
$514$	−6460.40	−0.554389
$515$	0	0
$516$	0	0
$517$	9375.16	0.797522
$518$	−12569.4	−1.06616
$519$	0	0
$520$	0	0
$521$	−12171.2	−1.02347	−0.511736	−	0.859143i	$-0.670997\pi$
$521$	−12171.2	−1.02347	−0.511736	+	0.859143i	$0.670997\pi$
$522$	0	0
$523$	15624.1	1.30630	0.653148	−	0.757230i	$-0.273448\pi$
$523$	15624.1	1.30630	0.653148	+	0.757230i	$0.273448\pi$
$524$	−19044.3	−1.58770
$525$	0	0
$526$	8850.93	0.733686
$527$	13841.4	1.14410
$528$	0	0
$529$	801.633	0.0658858
$530$	0	0
$531$	0	0
$532$	18455.9	1.50407
$533$	−4330.14	−0.351893
$534$	0	0
$535$	0	0
$536$	−49651.5	−4.00115
$537$	0	0
$538$	25441.2	2.03875
$539$	−1141.14	−0.0911917
$540$	0	0
$541$	10905.8	0.866688	0.433344	−	0.901229i	$-0.357334\pi$
$541$	10905.8	0.866688	0.433344	+	0.901229i	$0.357334\pi$
$542$	36309.0	2.87750
$543$	0	0
$544$	68231.5	5.37758
$545$	0	0
$546$	0	0
$547$	−9430.33	−0.737133	−0.368566	−	0.929601i	$-0.620151\pi$
$547$	−9430.33	−0.737133	−0.368566	+	0.929601i	$0.620151\pi$
$548$	−37637.3	−2.93392
$549$	0	0
$550$	0	0
$551$	13806.5	1.06747
$552$	0	0
$553$	3415.41	0.262636
$554$	27405.0	2.10168
$555$	0	0
$556$	−33658.9	−2.56737
$557$	12304.7	0.936029	0.468015	−	0.883721i	$-0.344969\pi$
$557$	12304.7	0.936029	0.468015	+	0.883721i	$0.344969\pi$
$558$	0	0
$559$	21061.5	1.59357
$560$	0	0
$561$	0	0
$562$	13241.8	0.993899
$563$	15768.1	1.18037	0.590184	−	0.807268i	$-0.299055\pi$
$563$	15768.1	1.18037	0.590184	+	0.807268i	$0.299055\pi$
$564$	0	0
$565$	0	0
$566$	−48296.0	−3.58663
$567$	0	0
$568$	−42621.1	−3.14849
$569$	−243.490	−0.0179396	−0.00896981	−	0.999960i	$-0.502855\pi$
$569$	−243.490	−0.0179396	−0.00896981	+	0.999960i	$0.502855\pi$
$570$	0	0
$571$	−26619.7	−1.95096	−0.975480	−	0.220090i	$-0.929365\pi$
$571$	−26619.7	−1.95096	−0.975480	+	0.220090i	$0.929365\pi$
$572$	24932.7	1.82253
$573$	0	0
$574$	3626.81	0.263729
$575$	0	0
$576$	0	0
$577$	4868.88	0.351290	0.175645	−	0.984454i	$-0.443799\pi$
$577$	4868.88	0.351290	0.175645	+	0.984454i	$0.443799\pi$
$578$	−4832.12	−0.347733
$579$	0	0
$580$	0	0
$581$	−8753.74	−0.625071
$582$	0	0
$583$	−11546.2	−0.820228
$584$	−39285.8	−2.78366
$585$	0	0
$586$	24138.4	1.70161
$587$	−12698.7	−0.892898	−0.446449	−	0.894809i	$-0.647312\pi$
$587$	−12698.7	−0.892898	−0.446449	+	0.894809i	$0.647312\pi$
$588$	0	0
$589$	20863.7	1.45955
$590$	0	0
$591$	0	0
$592$	−90683.5	−6.29573
$593$	−22043.0	−1.52647	−0.763237	−	0.646119i	$-0.776391\pi$
$593$	−22043.0	−1.52647	−0.763237	+	0.646119i	$0.776391\pi$
$594$	0	0
$595$	0	0
$596$	−7610.97	−0.523083
$597$	0	0
$598$	−29509.4	−2.01794
$599$	15180.6	1.03550	0.517750	−	0.855532i	$-0.326770\pi$
$599$	15180.6	1.03550	0.517750	+	0.855532i	$0.326770\pi$
$600$	0	0
$601$	415.176	0.0281787	0.0140893	−	0.999901i	$-0.495515\pi$
$601$	415.176	0.0281787	0.0140893	+	0.999901i	$0.495515\pi$
$602$	−17640.5	−1.19431
$603$	0	0
$604$	−26223.9	−1.76661
$605$	0	0
$606$	0	0
$607$	−23973.2	−1.60303	−0.801516	−	0.597973i	$-0.795973\pi$
$607$	−23973.2	−1.60303	−0.801516	+	0.597973i	$0.795973\pi$
$608$	102848.	6.86025
$609$	0	0
$610$	0	0
$611$	18734.0	1.24042
$612$	0	0
$613$	−5616.87	−0.370087	−0.185043	−	0.982730i	$-0.559243\pi$
$613$	−5616.87	−0.370087	−0.185043	+	0.982730i	$0.559243\pi$
$614$	−32570.8	−2.14080
$615$	0	0
$616$	−13621.1	−0.890922
$617$	15690.3	1.02377	0.511886	−	0.859053i	$-0.328947\pi$
$617$	15690.3	1.02377	0.511886	+	0.859053i	$0.328947\pi$
$618$	0	0
$619$	7832.79	0.508605	0.254302	−	0.967125i	$-0.418154\pi$
$619$	7832.79	0.508605	0.254302	+	0.967125i	$0.418154\pi$
$620$	0	0
$621$	0	0
$622$	−27869.1	−1.79654
$623$	−10933.8	−0.703138
$624$	0	0
$625$	0	0
$626$	−40993.8	−2.61732
$627$	0	0
$628$	75870.5	4.82096
$629$	24518.4	1.55423
$630$	0	0
$631$	−16365.7	−1.03250	−0.516250	−	0.856438i	$-0.672673\pi$
$631$	−16365.7	−1.03250	−0.516250	+	0.856438i	$0.672673\pi$
$632$	40767.6	2.56590
$633$	0	0
$634$	43821.6	2.74507
$635$	0	0
$636$	0	0
$637$	−2280.29	−0.141834
$638$	−15622.1	−0.969415
$639$	0	0
$640$	0	0
$641$	8208.10	0.505773	0.252886	−	0.967496i	$-0.418620\pi$
$641$	8208.10	0.505773	0.252886	+	0.967496i	$0.418620\pi$
$642$	0	0
$643$	13351.5	0.818868	0.409434	−	0.912340i	$-0.365726\pi$
$643$	13351.5	0.818868	0.409434	+	0.912340i	$0.365726\pi$
$644$	18339.1	1.12214
$645$	0	0
$646$	−48519.6	−2.95508
$647$	−23313.9	−1.41663	−0.708317	−	0.705894i	$-0.750545\pi$
$647$	−23313.9	−1.41663	−0.708317	+	0.705894i	$0.750545\pi$
$648$	0	0
$649$	11567.6	0.699642
$650$	0	0
$651$	0	0
$652$	59101.6	3.55000
$653$	21077.4	1.26313	0.631565	−	0.775323i	$-0.282413\pi$
$653$	21077.4	1.26313	0.631565	+	0.775323i	$0.282413\pi$
$654$	0	0
$655$	0	0
$656$	26166.0	1.55733
$657$	0	0
$658$	−15691.1	−0.929640
$659$	−31971.5	−1.88988	−0.944940	−	0.327243i	$-0.893881\pi$
$659$	−31971.5	−1.88988	−0.944940	+	0.327243i	$0.893881\pi$
$660$	0	0
$661$	482.048	0.0283654	0.0141827	−	0.999899i	$-0.495485\pi$
$661$	482.048	0.0283654	0.0141827	+	0.999899i	$0.495485\pi$
$662$	14525.0	0.852766
$663$	0	0
$664$	−104488.	−6.10680
$665$	0	0
$666$	0	0
$667$	13719.1	0.796411
$668$	−23808.5	−1.37901
$669$	0	0
$670$	0	0
$671$	6188.74	0.356056
$672$	0	0
$673$	−22561.8	−1.29226	−0.646132	−	0.763225i	$-0.723615\pi$
$673$	−22561.8	−1.29226	−0.646132	+	0.763225i	$0.723615\pi$
$674$	29823.0	1.70436
$675$	0	0
$676$	−721.146	−0.0410302
$677$	−1449.65	−0.0822962	−0.0411481	−	0.999153i	$-0.513102\pi$
$677$	−1449.65	−0.0822962	−0.0411481	+	0.999153i	$0.513102\pi$
$678$	0	0
$679$	−150.583	−0.00851081
$680$	0	0
$681$	0	0
$682$	−23607.4	−1.32548
$683$	10176.4	0.570114	0.285057	−	0.958511i	$-0.407987\pi$
$683$	10176.4	0.570114	0.285057	+	0.958511i	$0.407987\pi$
$684$	0	0
$685$	0	0
$686$	1909.91	0.106299
$687$	0	0
$688$	−127269.	−7.05247
$689$	−23072.2	−1.27574
$690$	0	0
$691$	23323.3	1.28402	0.642011	−	0.766695i	$-0.278100\pi$
$691$	23323.3	1.28402	0.642011	+	0.766695i	$0.278100\pi$
$692$	−5617.67	−0.308601
$693$	0	0
$694$	59062.7	3.23053
$695$	0	0
$696$	0	0
$697$	−7074.59	−0.384461
$698$	10561.8	0.572734
$699$	0	0
$700$	0	0
$701$	15221.7	0.820138	0.410069	−	0.912055i	$-0.365505\pi$
$701$	15221.7	0.820138	0.410069	+	0.912055i	$0.365505\pi$
$702$	0	0
$703$	36957.5	1.98276
$704$	−63981.6	−3.42528
$705$	0	0
$706$	−22499.5	−1.19941
$707$	1479.47	0.0787006
$708$	0	0
$709$	−13088.0	−0.693274	−0.346637	−	0.937999i	$-0.612676\pi$
$709$	−13088.0	−0.693274	−0.346637	+	0.937999i	$0.612676\pi$
$710$	0	0
$711$	0	0
$712$	−130510.	−6.86949
$713$	20731.6	1.08893
$714$	0	0
$715$	0	0
$716$	−35056.5	−1.82978
$717$	0	0
$718$	−11442.3	−0.594738
$719$	13845.7	0.718160	0.359080	−	0.933307i	$-0.383091\pi$
$719$	13845.7	0.718160	0.359080	+	0.933307i	$0.383091\pi$
$720$	0	0
$721$	8265.77	0.426953
$722$	−34942.7	−1.80115
$723$	0	0
$724$	−72343.9	−3.71359
$725$	0	0
$726$	0	0
$727$	2691.12	0.137287	0.0686437	−	0.997641i	$-0.478133\pi$
$727$	2691.12	0.137287	0.0686437	+	0.997641i	$0.478133\pi$
$728$	−27218.4	−1.38569
$729$	0	0
$730$	0	0
$731$	34410.3	1.74105
$732$	0	0
$733$	765.975	0.0385975	0.0192987	−	0.999814i	$-0.493857\pi$
$733$	765.975	0.0385975	0.0192987	+	0.999814i	$0.493857\pi$
$734$	54938.4	2.76269
$735$	0	0
$736$	102197.	5.11825
$737$	−13839.0	−0.691676
$738$	0	0
$739$	−20961.2	−1.04340	−0.521698	−	0.853130i	$-0.674701\pi$
$739$	−20961.2	−1.04340	−0.521698	+	0.853130i	$0.674701\pi$
$740$	0	0
$741$	0	0
$742$	19324.7	0.956108
$743$	28175.2	1.39118	0.695590	−	0.718439i	$-0.255143\pi$
$743$	28175.2	1.39118	0.695590	+	0.718439i	$0.255143\pi$
$744$	0	0
$745$	0	0
$746$	−24613.7	−1.20801
$747$	0	0
$748$	40735.0	1.99120
$749$	7803.60	0.380691
$750$	0	0
$751$	10303.4	0.500636	0.250318	−	0.968164i	$-0.419465\pi$
$751$	10303.4	0.500636	0.250318	+	0.968164i	$0.419465\pi$
$752$	−113205.	−5.48959
$753$	0	0
$754$	−31217.1	−1.50777
$755$	0	0
$756$	0	0
$757$	−38444.4	−1.84582	−0.922909	−	0.385019i	$-0.874195\pi$
$757$	−38444.4	−1.84582	−0.922909	+	0.385019i	$0.874195\pi$
$758$	76.8057	0.00368035
$759$	0	0
$760$	0	0
$761$	7765.25	0.369895	0.184947	−	0.982748i	$-0.440789\pi$
$761$	7765.25	0.369895	0.184947	+	0.982748i	$0.440789\pi$
$762$	0	0
$763$	−6793.64	−0.322341
$764$	16901.6	0.800363
$765$	0	0
$766$	−6847.61	−0.322995
$767$	23115.1	1.08818
$768$	0	0
$769$	−10320.1	−0.483945	−0.241972	−	0.970283i	$-0.577794\pi$
$769$	−10320.1	−0.483945	−0.241972	+	0.970283i	$0.577794\pi$
$770$	0	0
$771$	0	0
$772$	−74731.3	−3.48399
$773$	167.975	0.00781584	0.00390792	−	0.999992i	$-0.498756\pi$
$773$	167.975	0.00781584	0.00390792	+	0.999992i	$0.498756\pi$
$774$	0	0
$775$	0	0
$776$	−1797.41	−0.0831487
$777$	0	0
$778$	−15744.1	−0.725520
$779$	−10663.8	−0.490462
$780$	0	0
$781$	−11879.5	−0.544277
$782$	−48212.6	−2.20470
$783$	0	0
$784$	13779.3	0.627700
$785$	0	0
$786$	0	0
$787$	−12700.6	−0.575256	−0.287628	−	0.957742i	$-0.592867\pi$
$787$	−12700.6	−0.575256	−0.287628	+	0.957742i	$0.592867\pi$
$788$	−67079.1	−3.03248
$789$	0	0
$790$	0	0
$791$	5600.21	0.251733
$792$	0	0
$793$	12366.7	0.553789
$794$	−7361.93	−0.329049
$795$	0	0
$796$	−32158.9	−1.43196
$797$	−7995.40	−0.355347	−0.177674	−	0.984089i	$-0.556857\pi$
$797$	−7995.40	−0.355347	−0.177674	+	0.984089i	$0.556857\pi$
$798$	0	0
$799$	30607.7	1.35522
$800$	0	0
$801$	0	0
$802$	39605.3	1.74378
$803$	−10949.8	−0.481209
$804$	0	0
$805$	0	0
$806$	−47173.8	−2.06157
$807$	0	0
$808$	17659.6	0.768887
$809$	−7248.83	−0.315025	−0.157513	−	0.987517i	$-0.550348\pi$
$809$	−7248.83	−0.315025	−0.157513	+	0.987517i	$0.550348\pi$
$810$	0	0
$811$	12097.4	0.523793	0.261896	−	0.965096i	$-0.415652\pi$
$811$	12097.4	0.523793	0.261896	+	0.965096i	$0.415652\pi$
$812$	19400.3	0.838445
$813$	0	0
$814$	−41817.7	−1.80063
$815$	0	0
$816$	0	0
$817$	51867.8	2.22108
$818$	−33332.8	−1.42476
$819$	0	0
$820$	0	0
$821$	−4821.38	−0.204954	−0.102477	−	0.994735i	$-0.532677\pi$
$821$	−4821.38	−0.204954	−0.102477	+	0.994735i	$0.532677\pi$
$822$	0	0
$823$	−3733.51	−0.158131	−0.0790657	−	0.996869i	$-0.525194\pi$
$823$	−3733.51	−0.158131	−0.0790657	+	0.996869i	$0.525194\pi$
$824$	98663.3	4.17124
$825$	0	0
$826$	−19360.6	−0.815545
$827$	1160.78	0.0488079	0.0244040	−	0.999702i	$-0.492231\pi$
$827$	1160.78	0.0488079	0.0244040	+	0.999702i	$0.492231\pi$
$828$	0	0
$829$	18627.0	0.780389	0.390195	−	0.920732i	$-0.372408\pi$
$829$	18627.0	0.780389	0.390195	+	0.920732i	$0.372408\pi$
$830$	0	0
$831$	0	0
$832$	−127852.	−5.32748
$833$	−3725.55	−0.154961
$834$	0	0
$835$	0	0
$836$	61401.4	2.54021
$837$	0	0
$838$	−359.545	−0.0148213
$839$	−4213.40	−0.173376	−0.0866881	−	0.996236i	$-0.527628\pi$
$839$	−4213.40	−0.173376	−0.0866881	+	0.996236i	$0.527628\pi$
$840$	0	0
$841$	−9875.99	−0.404936
$842$	24957.7	1.02150
$843$	0	0
$844$	−22977.5	−0.937106
$845$	0	0
$846$	0	0
$847$	5520.51	0.223951
$848$	139420.	5.64588
$849$	0	0
$850$	0	0
$851$	36723.6	1.47928
$852$	0	0
$853$	−3940.85	−0.158185	−0.0790926	−	0.996867i	$-0.525202\pi$
$853$	−3940.85	−0.158185	−0.0790926	+	0.996867i	$0.525202\pi$
$854$	−10358.0	−0.415040
$855$	0	0
$856$	93146.7	3.71926
$857$	38267.2	1.52530	0.762650	−	0.646811i	$-0.223898\pi$
$857$	38267.2	1.52530	0.762650	+	0.646811i	$0.223898\pi$
$858$	0	0
$859$	10145.2	0.402969	0.201484	−	0.979492i	$-0.435423\pi$
$859$	10145.2	0.402969	0.201484	+	0.979492i	$0.435423\pi$
$860$	0	0
$861$	0	0
$862$	56581.8	2.23571
$863$	−17608.5	−0.694553	−0.347277	−	0.937763i	$-0.612894\pi$
$863$	−17608.5	−0.694553	−0.347277	+	0.937763i	$0.612894\pi$
$864$	0	0
$865$	0	0
$866$	−11287.5	−0.442917
$867$	0	0
$868$	29316.8	1.14640
$869$	11362.8	0.443565
$870$	0	0
$871$	−27653.9	−1.07579
$872$	−81091.3	−3.14920
$873$	0	0
$874$	−72672.6	−2.81257
$875$	0	0
$876$	0	0
$877$	23555.5	0.906972	0.453486	−	0.891263i	$-0.350180\pi$
$877$	23555.5	0.906972	0.453486	+	0.891263i	$0.350180\pi$
$878$	52154.8	2.00471
$879$	0	0
$880$	0	0
$881$	43719.7	1.67191	0.835955	−	0.548797i	$-0.184914\pi$
$881$	43719.7	1.67191	0.835955	+	0.548797i	$0.184914\pi$
$882$	0	0
$883$	−31393.4	−1.19646	−0.598229	−	0.801325i	$-0.704129\pi$
$883$	−31393.4	−1.19646	−0.598229	+	0.801325i	$0.704129\pi$
$884$	81399.2	3.09700
$885$	0	0
$886$	40879.5	1.55008
$887$	−40021.9	−1.51500	−0.757499	−	0.652836i	$-0.773579\pi$
$887$	−40021.9	−1.51500	−0.757499	+	0.652836i	$0.773579\pi$
$888$	0	0
$889$	−3686.80	−0.139090
$890$	0	0
$891$	0	0
$892$	64887.6	2.43565
$893$	46136.0	1.72887
$894$	0	0
$895$	0	0
$896$	56830.3	2.11894
$897$	0	0
$898$	−25987.2	−0.965705
$899$	21931.4	0.813628
$900$	0	0
$901$	−37695.5	−1.39380
$902$	12066.2	0.445409
$903$	0	0
$904$	66846.2	2.45937
$905$	0	0
$906$	0	0
$907$	2729.28	0.0999165	0.0499583	−	0.998751i	$-0.484091\pi$
$907$	2729.28	0.0999165	0.0499583	+	0.998751i	$0.484091\pi$
$908$	32688.9	1.19473
$909$	0	0
$910$	0	0
$911$	20874.8	0.759181	0.379590	−	0.925155i	$-0.376065\pi$
$911$	20874.8	0.759181	0.379590	+	0.925155i	$0.376065\pi$
$912$	0	0
$913$	−29123.1	−1.05568
$914$	−53724.7	−1.94426
$915$	0	0
$916$	−2011.85	−0.0725691
$917$	5794.69	0.208678
$918$	0	0
$919$	12339.8	0.442931	0.221466	−	0.975168i	$-0.428916\pi$
$919$	12339.8	0.442931	0.221466	+	0.975168i	$0.428916\pi$
$920$	0	0
$921$	0	0
$922$	89635.8	3.20173
$923$	−23738.2	−0.846537
$924$	0	0
$925$	0	0
$926$	−61388.1	−2.17855
$927$	0	0
$928$	108111.	3.82427
$929$	51868.3	1.83180	0.915900	−	0.401407i	$-0.131479\pi$
$929$	51868.3	1.83180	0.915900	+	0.401407i	$0.131479\pi$
$930$	0	0
$931$	−5615.65	−0.197686
$932$	108578.	3.81609
$933$	0	0
$934$	85059.5	2.97990
$935$	0	0
$936$	0	0
$937$	−47757.7	−1.66508	−0.832538	−	0.553967i	$-0.813113\pi$
$937$	−47757.7	−1.66508	−0.832538	+	0.553967i	$0.813113\pi$
$938$	23162.2	0.806260
$939$	0	0
$940$	0	0
$941$	41845.5	1.44966	0.724828	−	0.688930i	$-0.241919\pi$
$941$	41845.5	1.44966	0.724828	+	0.688930i	$0.241919\pi$
$942$	0	0
$943$	−10596.3	−0.365921
$944$	−139679.	−4.81585
$945$	0	0
$946$	−58688.9	−2.01706
$947$	−14121.8	−0.484580	−0.242290	−	0.970204i	$-0.577899\pi$
$947$	−14121.8	−0.484580	−0.242290	+	0.970204i	$0.577899\pi$
$948$	0	0
$949$	−21880.6	−0.748445
$950$	0	0
$951$	0	0
$952$	−44469.5	−1.51393
$953$	31178.0	1.05976	0.529882	−	0.848071i	$-0.322236\pi$
$953$	31178.0	1.05976	0.529882	+	0.848071i	$0.322236\pi$
$954$	0	0
$955$	0	0
$956$	19569.8	0.662062
$957$	0	0
$958$	−34787.6	−1.17321
$959$	11452.1	0.385618
$960$	0	0
$961$	3350.60	0.112470
$962$	−83562.6	−2.80059
$963$	0	0
$964$	92867.2	3.10275
$965$	0	0
$966$	0	0
$967$	−34789.7	−1.15694	−0.578470	−	0.815704i	$-0.696350\pi$
$967$	−34789.7	−1.15694	−0.578470	+	0.815704i	$0.696350\pi$
$968$	65894.8	2.18795
$969$	0	0
$970$	0	0
$971$	−20481.9	−0.676926	−0.338463	−	0.940980i	$-0.609907\pi$
$971$	−20481.9	−0.676926	−0.338463	+	0.940980i	$0.609907\pi$
$972$	0	0
$973$	10241.6	0.337440
$974$	−86752.6	−2.85393
$975$	0	0
$976$	−74729.1	−2.45084
$977$	−5605.71	−0.183565	−0.0917823	−	0.995779i	$-0.529256\pi$
$977$	−5605.71	−0.183565	−0.0917823	+	0.995779i	$0.529256\pi$
$978$	0	0
$979$	−36376.1	−1.18752
$980$	0	0
$981$	0	0
$982$	31207.9	1.01414
$983$	−34728.1	−1.12681	−0.563406	−	0.826180i	$-0.690509\pi$
$983$	−34728.1	−1.12681	−0.563406	+	0.826180i	$0.690509\pi$
$984$	0	0
$985$	0	0
$986$	−51002.6	−1.64732
$987$	0	0
$988$	122696.	3.95089
$989$	51539.6	1.65709
$990$	0	0
$991$	−2145.57	−0.0687753	−0.0343876	−	0.999409i	$-0.510948\pi$
$991$	−2145.57	−0.0687753	−0.0343876	+	0.999409i	$0.510948\pi$
$992$	163372.	5.22890
$993$	0	0
$994$	19882.5	0.634442
$995$	0	0
$996$	0	0
$997$	−59620.3	−1.89388	−0.946938	−	0.321417i	$-0.895841\pi$
$997$	−59620.3	−1.89388	−0.946938	+	0.321417i	$0.895841\pi$
$998$	−100323.	−3.18203
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bf.1.1		4
3.2	odd	2		525.4.a.v.1.4	yes	4
5.4	even	2		1575.4.a.bm.1.4		4
15.2	even	4		525.4.d.o.274.8		8
15.8	even	4		525.4.d.o.274.1		8
15.14	odd	2		525.4.a.s.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.s.1.1	✓	4	15.14	odd	2
525.4.a.v.1.4	yes	4	3.2	odd	2
525.4.d.o.274.1		8	15.8	even	4
525.4.d.o.274.8		8	15.2	even	4
1575.4.a.bf.1.1		4	1.1	even	1	trivial
1575.4.a.bm.1.4		4	5.4	even	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)

\(f(q)\)	\(=\)	\(q-5.56826 q^{2} +23.0055 q^{4} -7.00000 q^{7} -83.5546 q^{8} -23.2885 q^{11} -46.5366 q^{13} +38.9778 q^{14} +281.210 q^{16} -76.0316 q^{17} -114.605 q^{19} +129.677 q^{22} -113.880 q^{23} +259.128 q^{26} -161.039 q^{28} -120.470 q^{29} -182.048 q^{31} -897.411 q^{32} +423.363 q^{34} -322.477 q^{37} +638.151 q^{38} +93.0481 q^{41} -452.579 q^{43} -535.765 q^{44} +634.113 q^{46} -402.565 q^{47} +49.0000 q^{49} -1070.60 q^{52} +495.787 q^{53} +584.882 q^{56} +670.808 q^{58} -496.707 q^{59} -265.742 q^{61} +1013.69 q^{62} +2747.34 q^{64} +594.240 q^{67} -1749.14 q^{68} +510.099 q^{71} +470.181 q^{73} +1795.63 q^{74} -2636.55 q^{76} +163.020 q^{77} -487.916 q^{79} -518.116 q^{82} +1250.53 q^{83} +2520.08 q^{86} +1945.86 q^{88} +1561.98 q^{89} +325.756 q^{91} -2619.87 q^{92} +2241.59 q^{94} +21.5118 q^{97} -272.845 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} + 16 q^{4} - 28 q^{7} - 93 q^{8} - 57 q^{11} + 43 q^{13} + 42 q^{14} + 216 q^{16} - 99 q^{17} - 12 q^{19} - 41 q^{22} - 156 q^{23} + 81 q^{26} - 112 q^{28} - 378 q^{29} - 93 q^{31} - 690 q^{32}+ \cdots - 294 q^{98}+O(q^{100}) \) 4 * q - 6 * q^2 + 16 * q^4 - 28 * q^7 - 93 * q^8 - 57 * q^11 + 43 * q^13 + 42 * q^14 + 216 * q^16 - 99 * q^17 - 12 * q^19 - 41 * q^22 - 156 * q^23 + 81 * q^26 - 112 * q^28 - 378 * q^29 - 93 * q^31 - 690 * q^32 + 783 * q^34 + 81 * q^37 - 216 * q^38 + 465 * q^41 - 64 * q^43 - 681 * q^44 + 310 * q^46 - 744 * q^47 + 196 * q^49 - 727 * q^52 - 729 * q^53 + 651 * q^56 + 1172 * q^58 - 231 * q^59 - 1353 * q^61 + 165 * q^62 + 3107 * q^64 + 1487 * q^67 - 2577 * q^68 + 1725 * q^71 + 512 * q^73 + 1953 * q^74 - 3046 * q^76 + 399 * q^77 + 1629 * q^79 - 693 * q^82 - 321 * q^83 + 4542 * q^86 + 3482 * q^88 + 978 * q^89 - 301 * q^91 - 852 * q^92 + 2480 * q^94 + 2616 * q^97 - 294 * q^98

Introduction

L-functions

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