LMFDB - Embedded newform 1575.4.a.bn.1.4

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 7x^{2} + 120x + 60 $ x^5 - x^4 - 27x^3 + 7x^2 + 120*x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.67516$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67516 q^{2} -5.19383 q^{4} +7.00000 q^{7} -22.1018 q^{8} +57.5880 q^{11} -45.5159 q^{13} +11.7261 q^{14} +4.52655 q^{16} -92.0051 q^{17} +125.177 q^{19} +96.4692 q^{22} -158.496 q^{23} -76.2466 q^{26} -36.3568 q^{28} +40.1708 q^{29} +49.5590 q^{31} +184.397 q^{32} -154.123 q^{34} +231.307 q^{37} +209.692 q^{38} -169.556 q^{41} -147.428 q^{43} -299.102 q^{44} -265.507 q^{46} +67.0327 q^{47} +49.0000 q^{49} +236.402 q^{52} -268.647 q^{53} -154.713 q^{56} +67.2926 q^{58} +240.843 q^{59} +90.4579 q^{61} +83.0194 q^{62} +272.683 q^{64} -406.498 q^{67} +477.859 q^{68} -330.782 q^{71} +546.255 q^{73} +387.477 q^{74} -650.149 q^{76} +403.116 q^{77} -25.3087 q^{79} -284.034 q^{82} -376.255 q^{83} -246.965 q^{86} -1272.80 q^{88} -1026.44 q^{89} -318.612 q^{91} +823.203 q^{92} +112.291 q^{94} -942.660 q^{97} +82.0829 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} + 18 q^{4} + 35 q^{7} - 42 q^{8} - 42 q^{11} + 34 q^{13} - 28 q^{14} + 74 q^{16} - 238 q^{17} - 36 q^{19} + 358 q^{22} - 152 q^{23} + 310 q^{26} + 126 q^{28} + 44 q^{29} + 60 q^{31} - 710 q^{32}+ \cdots - 196 q^{98}+O(q^{100}) $ 5 * q - 4 * q^2 + 18 * q^4 + 35 * q^7 - 42 * q^8 - 42 * q^11 + 34 * q^13 - 28 * q^14 + 74 * q^16 - 238 * q^17 - 36 * q^19 + 358 * q^22 - 152 * q^23 + 310 * q^26 + 126 * q^28 + 44 * q^29 + 60 * q^31 - 710 * q^32 - 482 * q^34 + 312 * q^37 - 280 * q^38 + 426 * q^41 + 304 * q^43 - 712 * q^44 + 88 * q^46 - 370 * q^47 + 245 * q^49 - 1156 * q^52 - 976 * q^53 - 294 * q^56 - 2722 * q^58 + 432 * q^59 - 442 * q^61 + 956 * q^62 + 1362 * q^64 - 804 * q^67 + 420 * q^68 - 440 * q^71 + 564 * q^73 + 1512 * q^74 - 1336 * q^76 - 294 * q^77 + 1790 * q^79 - 3480 * q^82 - 1656 * q^83 - 1216 * q^86 - 1092 * q^88 - 746 * q^89 + 238 * q^91 - 572 * q^92 - 826 * q^94 + 518 * q^97 - 196 * q^98

Coefficient data

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

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

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

$2$	1.67516	0.592259	0.296130	−	0.955148i	$-0.404304\pi$
$2$	1.67516	0.592259	0.296130	+	0.955148i	$0.404304\pi$
$3$	0	0
$3$	0	0
$4$	−5.19383	−0.649229
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−22.1018	−0.976771
$9$	0	0
$10$	0	0
$11$	57.5880	1.57849	0.789247	−	0.614076i	$-0.210471\pi$
$11$	57.5880	1.57849	0.789247	+	0.614076i	$0.210471\pi$
$12$	0	0
$13$	−45.5159	−0.971066	−0.485533	−	0.874218i	$-0.661374\pi$
$13$	−45.5159	−0.971066	−0.485533	+	0.874218i	$0.661374\pi$
$14$	11.7261	0.223853
$15$	0	0
$16$	4.52655	0.0707273
$17$	−92.0051	−1.31262	−0.656309	−	0.754492i	$-0.727883\pi$
$17$	−92.0051	−1.31262	−0.656309	+	0.754492i	$0.727883\pi$
$18$	0	0
$19$	125.177	1.51145	0.755726	−	0.654888i	$-0.227284\pi$
$19$	125.177	1.51145	0.755726	+	0.654888i	$0.227284\pi$
$20$	0	0
$21$	0	0
$22$	96.4692	0.934878
$23$	−158.496	−1.43690	−0.718451	−	0.695578i	$-0.755149\pi$
$23$	−158.496	−1.43690	−0.718451	+	0.695578i	$0.755149\pi$
$24$	0	0
$25$	0	0
$26$	−76.2466	−0.575123
$27$	0	0
$28$	−36.3568	−0.245386
$29$	40.1708	0.257225	0.128613	−	0.991695i	$-0.458948\pi$
$29$	40.1708	0.257225	0.128613	+	0.991695i	$0.458948\pi$
$30$	0	0
$31$	49.5590	0.287131	0.143566	−	0.989641i	$-0.454143\pi$
$31$	49.5590	0.287131	0.143566	+	0.989641i	$0.454143\pi$
$32$	184.397	1.01866
$33$	0	0
$34$	−154.123	−0.777410
$35$	0	0
$36$	0	0
$37$	231.307	1.02775	0.513874	−	0.857866i	$-0.328210\pi$
$37$	231.307	1.02775	0.513874	+	0.857866i	$0.328210\pi$
$38$	209.692	0.895171
$39$	0	0
$40$	0	0
$41$	−169.556	−0.645859	−0.322929	−	0.946423i	$-0.604668\pi$
$41$	−169.556	−0.645859	−0.322929	+	0.946423i	$0.604668\pi$
$42$	0	0
$43$	−147.428	−0.522849	−0.261425	−	0.965224i	$-0.584192\pi$
$43$	−147.428	−0.522849	−0.261425	+	0.965224i	$0.584192\pi$
$44$	−299.102	−1.02480
$45$	0	0
$46$	−265.507	−0.851018
$47$	67.0327	0.208037	0.104018	−	0.994575i	$-0.466830\pi$
$47$	67.0327	0.208037	0.104018	+	0.994575i	$0.466830\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	236.402	0.630444
$53$	−268.647	−0.696254	−0.348127	−	0.937447i	$-0.613182\pi$
$53$	−268.647	−0.696254	−0.348127	+	0.937447i	$0.613182\pi$
$54$	0	0
$55$	0	0
$56$	−154.713	−0.369185
$57$	0	0
$58$	67.2926	0.152344
$59$	240.843	0.531442	0.265721	−	0.964050i	$-0.414390\pi$
$59$	240.843	0.531442	0.265721	+	0.964050i	$0.414390\pi$
$60$	0	0
$61$	90.4579	0.189868	0.0949340	−	0.995484i	$-0.469736\pi$
$61$	90.4579	0.189868	0.0949340	+	0.995484i	$0.469736\pi$
$62$	83.0194	0.170056
$63$	0	0
$64$	272.683	0.532583
$65$	0	0
$66$	0	0
$67$	−406.498	−0.741218	−0.370609	−	0.928789i	$-0.620851\pi$
$67$	−406.498	−0.741218	−0.370609	+	0.928789i	$0.620851\pi$
$68$	477.859	0.852190
$69$	0	0
$70$	0	0
$71$	−330.782	−0.552910	−0.276455	−	0.961027i	$-0.589160\pi$
$71$	−330.782	−0.552910	−0.276455	+	0.961027i	$0.589160\pi$
$72$	0	0
$73$	546.255	0.875812	0.437906	−	0.899021i	$-0.355720\pi$
$73$	546.255	0.875812	0.437906	+	0.899021i	$0.355720\pi$
$74$	387.477	0.608693
$75$	0	0
$76$	−650.149	−0.981279
$77$	403.116	0.596615
$78$	0	0
$79$	−25.3087	−0.0360436	−0.0180218	−	0.999838i	$-0.505737\pi$
$79$	−25.3087	−0.0360436	−0.0180218	+	0.999838i	$0.505737\pi$
$80$	0	0
$81$	0	0
$82$	−284.034	−0.382516
$83$	−376.255	−0.497582	−0.248791	−	0.968557i	$-0.580033\pi$
$83$	−376.255	−0.497582	−0.248791	+	0.968557i	$0.580033\pi$
$84$	0	0
$85$	0	0
$86$	−246.965	−0.309662
$87$	0	0
$88$	−1272.80	−1.54183
$89$	−1026.44	−1.22250	−0.611248	−	0.791439i	$-0.709332\pi$
$89$	−1026.44	−1.22250	−0.611248	+	0.791439i	$0.709332\pi$
$90$	0	0
$91$	−318.612	−0.367028
$92$	823.203	0.932878
$93$	0	0
$94$	112.291	0.123212
$95$	0	0
$96$	0	0
$97$	−942.660	−0.986728	−0.493364	−	0.869823i	$-0.664233\pi$
$97$	−942.660	−0.986728	−0.493364	+	0.869823i	$0.664233\pi$
$98$	82.0829	0.0846085
$99$	0	0
$100$	0	0
$101$	604.617	0.595659	0.297830	−	0.954619i	$-0.403737\pi$
$101$	604.617	0.595659	0.297830	+	0.954619i	$0.403737\pi$
$102$	0	0
$103$	−300.967	−0.287914	−0.143957	−	0.989584i	$-0.545983\pi$
$103$	−300.967	−0.287914	−0.143957	+	0.989584i	$0.545983\pi$
$104$	1005.98	0.948509
$105$	0	0
$106$	−450.027	−0.412363
$107$	−1511.66	−1.36577	−0.682886	−	0.730525i	$-0.739276\pi$
$107$	−1511.66	−1.36577	−0.682886	+	0.730525i	$0.739276\pi$
$108$	0	0
$109$	−1767.09	−1.55281	−0.776406	−	0.630233i	$-0.782959\pi$
$109$	−1767.09	−1.55281	−0.776406	+	0.630233i	$0.782959\pi$
$110$	0	0
$111$	0	0
$112$	31.6858	0.0267324
$113$	−1045.27	−0.870182	−0.435091	−	0.900387i	$-0.643284\pi$
$113$	−1045.27	−0.870182	−0.435091	+	0.900387i	$0.643284\pi$
$114$	0	0
$115$	0	0
$116$	−208.640	−0.166998
$117$	0	0
$118$	403.451	0.314751
$119$	−644.036	−0.496123
$120$	0	0
$121$	1985.38	1.49164
$122$	151.532	0.112451
$123$	0	0
$124$	−257.401	−0.186414
$125$	0	0
$126$	0	0
$127$	260.727	0.182171	0.0910857	−	0.995843i	$-0.470966\pi$
$127$	260.727	0.182171	0.0910857	+	0.995843i	$0.470966\pi$
$128$	−1018.39	−0.703233
$129$	0	0
$130$	0	0
$131$	723.522	0.482553	0.241276	−	0.970456i	$-0.422434\pi$
$131$	723.522	0.482553	0.241276	+	0.970456i	$0.422434\pi$
$132$	0	0
$133$	876.240	0.571275
$134$	−680.950	−0.438993
$135$	0	0
$136$	2033.48	1.28213
$137$	−773.693	−0.482490	−0.241245	−	0.970464i	$-0.577556\pi$
$137$	−773.693	−0.482490	−0.241245	+	0.970464i	$0.577556\pi$
$138$	0	0
$139$	−2952.97	−1.80192	−0.900961	−	0.433899i	$-0.857137\pi$
$139$	−2952.97	−1.80192	−0.900961	+	0.433899i	$0.857137\pi$
$140$	0	0
$141$	0	0
$142$	−554.114	−0.327466
$143$	−2621.17	−1.53282
$144$	0	0
$145$	0	0
$146$	915.066	0.518708
$147$	0	0
$148$	−1201.37	−0.667244
$149$	−2514.00	−1.38225	−0.691124	−	0.722736i	$-0.742884\pi$
$149$	−2514.00	−1.38225	−0.691124	+	0.722736i	$0.742884\pi$
$150$	0	0
$151$	101.052	0.0544605	0.0272302	−	0.999629i	$-0.491331\pi$
$151$	101.052	0.0544605	0.0272302	+	0.999629i	$0.491331\pi$
$152$	−2766.64	−1.47634
$153$	0	0
$154$	675.285	0.353351
$155$	0	0
$156$	0	0
$157$	2338.35	1.18867	0.594333	−	0.804219i	$-0.297416\pi$
$157$	2338.35	1.18867	0.594333	+	0.804219i	$0.297416\pi$
$158$	−42.3961	−0.0213472
$159$	0	0
$160$	0	0
$161$	−1109.47	−0.543098
$162$	0	0
$163$	−1325.20	−0.636798	−0.318399	−	0.947957i	$-0.603145\pi$
$163$	−1325.20	−0.636798	−0.318399	+	0.947957i	$0.603145\pi$
$164$	880.646	0.419310
$165$	0	0
$166$	−630.288	−0.294698
$167$	−2086.20	−0.966675	−0.483338	−	0.875434i	$-0.660576\pi$
$167$	−2086.20	−0.966675	−0.483338	+	0.875434i	$0.660576\pi$
$168$	0	0
$169$	−125.299	−0.0570317
$170$	0	0
$171$	0	0
$172$	765.715	0.339449
$173$	−1918.19	−0.842990	−0.421495	−	0.906831i	$-0.638495\pi$
$173$	−1918.19	−0.842990	−0.421495	+	0.906831i	$0.638495\pi$
$174$	0	0
$175$	0	0
$176$	260.675	0.111643
$177$	0	0
$178$	−1719.45	−0.724035
$179$	−629.046	−0.262665	−0.131333	−	0.991338i	$-0.541926\pi$
$179$	−629.046	−0.262665	−0.131333	+	0.991338i	$0.541926\pi$
$180$	0	0
$181$	−2800.85	−1.15020	−0.575099	−	0.818084i	$-0.695036\pi$
$181$	−2800.85	−1.15020	−0.575099	+	0.818084i	$0.695036\pi$
$182$	−533.726	−0.217376
$183$	0	0
$184$	3503.05	1.40352
$185$	0	0
$186$	0	0
$187$	−5298.39	−2.07196
$188$	−348.157	−0.135063
$189$	0	0
$190$	0	0
$191$	−740.255	−0.280434	−0.140217	−	0.990121i	$-0.544780\pi$
$191$	−740.255	−0.280434	−0.140217	+	0.990121i	$0.544780\pi$
$192$	0	0
$193$	4082.57	1.52264	0.761321	−	0.648375i	$-0.224551\pi$
$193$	4082.57	1.52264	0.761321	+	0.648375i	$0.224551\pi$
$194$	−1579.11	−0.584399
$195$	0	0
$196$	−254.498	−0.0927470
$197$	−3414.89	−1.23503	−0.617515	−	0.786559i	$-0.711860\pi$
$197$	−3414.89	−1.23503	−0.617515	+	0.786559i	$0.711860\pi$
$198$	0	0
$199$	−3392.44	−1.20846	−0.604231	−	0.796809i	$-0.706520\pi$
$199$	−3392.44	−1.20846	−0.604231	+	0.796809i	$0.706520\pi$
$200$	0	0
$201$	0	0
$202$	1012.83	0.352785
$203$	281.196	0.0972220
$204$	0	0
$205$	0	0
$206$	−504.169	−0.170520
$207$	0	0
$208$	−206.030	−0.0686809
$209$	7208.70	2.38582
$210$	0	0
$211$	3398.04	1.10867	0.554337	−	0.832292i	$-0.312972\pi$
$211$	3398.04	1.10867	0.554337	+	0.832292i	$0.312972\pi$
$212$	1395.31	0.452029
$213$	0	0
$214$	−2532.28	−0.808892
$215$	0	0
$216$	0	0
$217$	346.913	0.108525
$218$	−2960.16	−0.919667
$219$	0	0
$220$	0	0
$221$	4187.70	1.27464
$222$	0	0
$223$	182.611	0.0548365	0.0274183	−	0.999624i	$-0.491271\pi$
$223$	182.611	0.0548365	0.0274183	+	0.999624i	$0.491271\pi$
$224$	1290.78	0.385017
$225$	0	0
$226$	−1750.99	−0.515373
$227$	3152.33	0.921707	0.460854	−	0.887476i	$-0.347543\pi$
$227$	3152.33	0.921707	0.460854	+	0.887476i	$0.347543\pi$
$228$	0	0
$229$	6012.35	1.73497	0.867483	−	0.497466i	$-0.165736\pi$
$229$	6012.35	1.73497	0.867483	+	0.497466i	$0.165736\pi$
$230$	0	0
$231$	0	0
$232$	−887.848	−0.251250
$233$	940.660	0.264484	0.132242	−	0.991217i	$-0.457782\pi$
$233$	940.660	0.264484	0.132242	+	0.991217i	$0.457782\pi$
$234$	0	0
$235$	0	0
$236$	−1250.90	−0.345027
$237$	0	0
$238$	−1078.86	−0.293833
$239$	−5158.82	−1.39622	−0.698109	−	0.715991i	$-0.745975\pi$
$239$	−5158.82	−1.39622	−0.698109	+	0.715991i	$0.745975\pi$
$240$	0	0
$241$	−463.836	−0.123976	−0.0619882	−	0.998077i	$-0.519744\pi$
$241$	−463.836	−0.123976	−0.0619882	+	0.998077i	$0.519744\pi$
$242$	3325.83	0.883440
$243$	0	0
$244$	−469.823	−0.123268
$245$	0	0
$246$	0	0
$247$	−5697.55	−1.46772
$248$	−1095.34	−0.280461
$249$	0	0
$250$	0	0
$251$	−2290.25	−0.575934	−0.287967	−	0.957640i	$-0.592979\pi$
$251$	−2290.25	−0.575934	−0.287967	+	0.957640i	$0.592979\pi$
$252$	0	0
$253$	−9127.48	−2.26814
$254$	436.760	0.107893
$255$	0	0
$256$	−3887.43	−0.949079
$257$	802.202	0.194708	0.0973541	−	0.995250i	$-0.468962\pi$
$257$	802.202	0.194708	0.0973541	+	0.995250i	$0.468962\pi$
$258$	0	0
$259$	1619.15	0.388452
$260$	0	0
$261$	0	0
$262$	1212.02	0.285796
$263$	286.978	0.0672845	0.0336423	−	0.999434i	$-0.489289\pi$
$263$	286.978	0.0672845	0.0336423	+	0.999434i	$0.489289\pi$
$264$	0	0
$265$	0	0
$266$	1467.84	0.338343
$267$	0	0
$268$	2111.28	0.481221
$269$	−3561.22	−0.807180	−0.403590	−	0.914940i	$-0.632238\pi$
$269$	−3561.22	−0.807180	−0.403590	+	0.914940i	$0.632238\pi$
$270$	0	0
$271$	1928.81	0.432349	0.216175	−	0.976355i	$-0.430642\pi$
$271$	1928.81	0.432349	0.216175	+	0.976355i	$0.430642\pi$
$272$	−416.466	−0.0928380
$273$	0	0
$274$	−1296.06	−0.285759
$275$	0	0
$276$	0	0
$277$	6588.69	1.42916	0.714578	−	0.699556i	$-0.246619\pi$
$277$	6588.69	1.42916	0.714578	+	0.699556i	$0.246619\pi$
$278$	−4946.70	−1.06721
$279$	0	0
$280$	0	0
$281$	−815.552	−0.173138	−0.0865689	−	0.996246i	$-0.527590\pi$
$281$	−815.552	−0.173138	−0.0865689	+	0.996246i	$0.527590\pi$
$282$	0	0
$283$	−6513.49	−1.36815	−0.684076	−	0.729411i	$-0.739794\pi$
$283$	−6513.49	−1.36815	−0.684076	+	0.729411i	$0.739794\pi$
$284$	1718.03	0.358965
$285$	0	0
$286$	−4390.89	−0.907828
$287$	−1186.89	−0.244112
$288$	0	0
$289$	3551.94	0.722967
$290$	0	0
$291$	0	0
$292$	−2837.16	−0.568603
$293$	−435.520	−0.0868373	−0.0434186	−	0.999057i	$-0.513825\pi$
$293$	−435.520	−0.0868373	−0.0434186	+	0.999057i	$0.513825\pi$
$294$	0	0
$295$	0	0
$296$	−5112.31	−1.00387
$297$	0	0
$298$	−4211.36	−0.818650
$299$	7214.10	1.39533
$300$	0	0
$301$	−1031.99	−0.197618
$302$	169.279	0.0322547
$303$	0	0
$304$	566.620	0.106901
$305$	0	0
$306$	0	0
$307$	−4915.99	−0.913910	−0.456955	−	0.889490i	$-0.651060\pi$
$307$	−4915.99	−0.913910	−0.456955	+	0.889490i	$0.651060\pi$
$308$	−2093.72	−0.387340
$309$	0	0
$310$	0	0
$311$	1831.11	0.333868	0.166934	−	0.985968i	$-0.446613\pi$
$311$	1831.11	0.333868	0.166934	+	0.985968i	$0.446613\pi$
$312$	0	0
$313$	−2442.96	−0.441163	−0.220582	−	0.975369i	$-0.570796\pi$
$313$	−2442.96	−0.441163	−0.220582	+	0.975369i	$0.570796\pi$
$314$	3917.11	0.703998
$315$	0	0
$316$	131.449	0.0234006
$317$	−1666.19	−0.295214	−0.147607	−	0.989046i	$-0.547157\pi$
$317$	−1666.19	−0.295214	−0.147607	+	0.989046i	$0.547157\pi$
$318$	0	0
$319$	2313.36	0.406029
$320$	0	0
$321$	0	0
$322$	−1858.55	−0.321655
$323$	−11516.9	−1.98396
$324$	0	0
$325$	0	0
$326$	−2219.93	−0.377149
$327$	0	0
$328$	3747.50	0.630856
$329$	469.229	0.0786305
$330$	0	0
$331$	−5466.38	−0.907732	−0.453866	−	0.891070i	$-0.649956\pi$
$331$	−5466.38	−0.907732	−0.453866	+	0.891070i	$0.649956\pi$
$332$	1954.20	0.323045
$333$	0	0
$334$	−3494.72	−0.572522
$335$	0	0
$336$	0	0
$337$	10650.5	1.72157	0.860784	−	0.508970i	$-0.169973\pi$
$337$	10650.5	1.72157	0.860784	+	0.508970i	$0.169973\pi$
$338$	−209.896	−0.0337776
$339$	0	0
$340$	0	0
$341$	2854.01	0.453235
$342$	0	0
$343$	343.000	0.0539949
$344$	3258.42	0.510704
$345$	0	0
$346$	−3213.28	−0.499269
$347$	−4019.13	−0.621782	−0.310891	−	0.950446i	$-0.600627\pi$
$347$	−4019.13	−0.621782	−0.310891	+	0.950446i	$0.600627\pi$
$348$	0	0
$349$	10544.9	1.61735	0.808674	−	0.588256i	$-0.200185\pi$
$349$	10544.9	1.61735	0.808674	+	0.588256i	$0.200185\pi$
$350$	0	0
$351$	0	0
$352$	10619.1	1.60795
$353$	−2959.98	−0.446300	−0.223150	−	0.974784i	$-0.571634\pi$
$353$	−2959.98	−0.446300	−0.223150	+	0.974784i	$0.571634\pi$
$354$	0	0
$355$	0	0
$356$	5331.14	0.793680
$357$	0	0
$358$	−1053.75	−0.155566
$359$	−2170.17	−0.319045	−0.159523	−	0.987194i	$-0.550995\pi$
$359$	−2170.17	−0.319045	−0.159523	+	0.987194i	$0.550995\pi$
$360$	0	0
$361$	8810.30	1.28449
$362$	−4691.88	−0.681215
$363$	0	0
$364$	1654.82	0.238285
$365$	0	0
$366$	0	0
$367$	−1252.20	−0.178105	−0.0890523	−	0.996027i	$-0.528384\pi$
$367$	−1252.20	−0.178105	−0.0890523	+	0.996027i	$0.528384\pi$
$368$	−717.441	−0.101628
$369$	0	0
$370$	0	0
$371$	−1880.53	−0.263159
$372$	0	0
$373$	4646.02	0.644938	0.322469	−	0.946580i	$-0.395487\pi$
$373$	4646.02	0.644938	0.322469	+	0.946580i	$0.395487\pi$
$374$	−8875.66	−1.22714
$375$	0	0
$376$	−1481.54	−0.203204
$377$	−1828.41	−0.249783
$378$	0	0
$379$	1434.84	0.194466	0.0972331	−	0.995262i	$-0.469001\pi$
$379$	1434.84	0.194466	0.0972331	+	0.995262i	$0.469001\pi$
$380$	0	0
$381$	0	0
$382$	−1240.05	−0.166090
$383$	13216.7	1.76330	0.881649	−	0.471905i	$-0.156434\pi$
$383$	13216.7	1.76330	0.881649	+	0.471905i	$0.156434\pi$
$384$	0	0
$385$	0	0
$386$	6838.97	0.901799
$387$	0	0
$388$	4896.02	0.640612
$389$	7755.01	1.01078	0.505391	−	0.862890i	$-0.331348\pi$
$389$	7755.01	1.01078	0.505391	+	0.862890i	$0.331348\pi$
$390$	0	0
$391$	14582.5	1.88610
$392$	−1082.99	−0.139539
$393$	0	0
$394$	−5720.49	−0.731457
$395$	0	0
$396$	0	0
$397$	−3560.83	−0.450158	−0.225079	−	0.974341i	$-0.572264\pi$
$397$	−3560.83	−0.450158	−0.225079	+	0.974341i	$0.572264\pi$
$398$	−5682.89	−0.715723
$399$	0	0
$400$	0	0
$401$	5430.61	0.676288	0.338144	−	0.941094i	$-0.390201\pi$
$401$	5430.61	0.676288	0.338144	+	0.941094i	$0.390201\pi$
$402$	0	0
$403$	−2255.73	−0.278823
$404$	−3140.28	−0.386719
$405$	0	0
$406$	471.048	0.0575806
$407$	13320.5	1.62229
$408$	0	0
$409$	−9698.79	−1.17255	−0.586277	−	0.810111i	$-0.699407\pi$
$409$	−9698.79	−1.17255	−0.586277	+	0.810111i	$0.699407\pi$
$410$	0	0
$411$	0	0
$412$	1563.17	0.186922
$413$	1685.90	0.200866
$414$	0	0
$415$	0	0
$416$	−8393.01	−0.989186
$417$	0	0
$418$	12075.7	1.41302
$419$	−13830.9	−1.61261	−0.806307	−	0.591498i	$-0.798537\pi$
$419$	−13830.9	−1.61261	−0.806307	+	0.591498i	$0.798537\pi$
$420$	0	0
$421$	16703.0	1.93362	0.966810	−	0.255498i	$-0.0822393\pi$
$421$	16703.0	1.93362	0.966810	+	0.255498i	$0.0822393\pi$
$422$	5692.26	0.656623
$423$	0	0
$424$	5937.58	0.680081
$425$	0	0
$426$	0	0
$427$	633.205	0.0717634
$428$	7851.31	0.886699
$429$	0	0
$430$	0	0
$431$	−8174.07	−0.913530	−0.456765	−	0.889588i	$-0.650992\pi$
$431$	−8174.07	−0.913530	−0.456765	+	0.889588i	$0.650992\pi$
$432$	0	0
$433$	14222.8	1.57853	0.789267	−	0.614051i	$-0.210461\pi$
$433$	14222.8	1.57853	0.789267	+	0.614051i	$0.210461\pi$
$434$	581.136	0.0642752
$435$	0	0
$436$	9177.96	1.00813
$437$	−19840.1	−2.17181
$438$	0	0
$439$	5537.38	0.602016	0.301008	−	0.953622i	$-0.402677\pi$
$439$	5537.38	0.602016	0.301008	+	0.953622i	$0.402677\pi$
$440$	0	0
$441$	0	0
$442$	7015.07	0.754916
$443$	−3974.09	−0.426218	−0.213109	−	0.977028i	$-0.568359\pi$
$443$	−3974.09	−0.426218	−0.213109	+	0.977028i	$0.568359\pi$
$444$	0	0
$445$	0	0
$446$	305.903	0.0324774
$447$	0	0
$448$	1908.78	0.201298
$449$	15243.1	1.60216	0.801078	−	0.598559i	$-0.204260\pi$
$449$	15243.1	1.60216	0.801078	+	0.598559i	$0.204260\pi$
$450$	0	0
$451$	−9764.40	−1.01948
$452$	5428.95	0.564947
$453$	0	0
$454$	5280.67	0.545890
$455$	0	0
$456$	0	0
$457$	−10768.9	−1.10229	−0.551145	−	0.834410i	$-0.685809\pi$
$457$	−10768.9	−1.10229	−0.551145	+	0.834410i	$0.685809\pi$
$458$	10071.7	1.02755
$459$	0	0
$460$	0	0
$461$	332.605	0.0336029	0.0168015	−	0.999859i	$-0.494652\pi$
$461$	332.605	0.0336029	0.0168015	+	0.999859i	$0.494652\pi$
$462$	0	0
$463$	−8205.35	−0.823618	−0.411809	−	0.911270i	$-0.635103\pi$
$463$	−8205.35	−0.823618	−0.411809	+	0.911270i	$0.635103\pi$
$464$	181.835	0.0181929
$465$	0	0
$466$	1575.76	0.156643
$467$	167.628	0.0166100	0.00830501	−	0.999966i	$-0.497356\pi$
$467$	167.628	0.0166100	0.00830501	+	0.999966i	$0.497356\pi$
$468$	0	0
$469$	−2845.49	−0.280154
$470$	0	0
$471$	0	0
$472$	−5323.06	−0.519097
$473$	−8490.07	−0.825315
$474$	0	0
$475$	0	0
$476$	3345.01	0.322098
$477$	0	0
$478$	−8641.86	−0.826924
$479$	−6628.58	−0.632292	−0.316146	−	0.948711i	$-0.602389\pi$
$479$	−6628.58	−0.632292	−0.316146	+	0.948711i	$0.602389\pi$
$480$	0	0
$481$	−10528.2	−0.998010
$482$	−777.001	−0.0734262
$483$	0	0
$484$	−10311.7	−0.968419
$485$	0	0
$486$	0	0
$487$	−20641.6	−1.92065	−0.960327	−	0.278875i	$-0.910039\pi$
$487$	−20641.6	−1.92065	−0.960327	+	0.278875i	$0.910039\pi$
$488$	−1999.28	−0.185458
$489$	0	0
$490$	0	0
$491$	16710.8	1.53594	0.767972	−	0.640484i	$-0.221266\pi$
$491$	16710.8	1.53594	0.767972	+	0.640484i	$0.221266\pi$
$492$	0	0
$493$	−3695.92	−0.337639
$494$	−9544.32	−0.869270
$495$	0	0
$496$	224.331	0.0203080
$497$	−2315.47	−0.208980
$498$	0	0
$499$	−13728.7	−1.23162	−0.615812	−	0.787893i	$-0.711172\pi$
$499$	−13728.7	−1.23162	−0.615812	+	0.787893i	$0.711172\pi$
$500$	0	0
$501$	0	0
$502$	−3836.54	−0.341102
$503$	19523.7	1.73065	0.865326	−	0.501209i	$-0.167111\pi$
$503$	19523.7	1.73065	0.865326	+	0.501209i	$0.167111\pi$
$504$	0	0
$505$	0	0
$506$	−15290.0	−1.34333
$507$	0	0
$508$	−1354.17	−0.118271
$509$	8688.17	0.756574	0.378287	−	0.925688i	$-0.376513\pi$
$509$	8688.17	0.756574	0.378287	+	0.925688i	$0.376513\pi$
$510$	0	0
$511$	3823.78	0.331026
$512$	1635.04	0.141131
$513$	0	0
$514$	1343.82	0.115318
$515$	0	0
$516$	0	0
$517$	3860.28	0.328385
$518$	2712.34	0.230064
$519$	0	0
$520$	0	0
$521$	−6771.36	−0.569402	−0.284701	−	0.958616i	$-0.591894\pi$
$521$	−6771.36	−0.569402	−0.284701	+	0.958616i	$0.591894\pi$
$522$	0	0
$523$	−1365.89	−0.114200	−0.0570998	−	0.998368i	$-0.518185\pi$
$523$	−1365.89	−0.114200	−0.0570998	+	0.998368i	$0.518185\pi$
$524$	−3757.85	−0.313287
$525$	0	0
$526$	480.735	0.0398499
$527$	−4559.68	−0.376894
$528$	0	0
$529$	12954.0	1.06469
$530$	0	0
$531$	0	0
$532$	−4551.04	−0.370888
$533$	7717.51	0.627171
$534$	0	0
$535$	0	0
$536$	8984.34	0.724001
$537$	0	0
$538$	−5965.62	−0.478060
$539$	2821.81	0.225499
$540$	0	0
$541$	−23250.1	−1.84769	−0.923844	−	0.382770i	$-0.874970\pi$
$541$	−23250.1	−1.84769	−0.923844	+	0.382770i	$0.874970\pi$
$542$	3231.06	0.256063
$543$	0	0
$544$	−16965.5	−1.33711
$545$	0	0
$546$	0	0
$547$	−11552.7	−0.903033	−0.451516	−	0.892263i	$-0.649117\pi$
$547$	−11552.7	−0.903033	−0.451516	+	0.892263i	$0.649117\pi$
$548$	4018.43	0.313246
$549$	0	0
$550$	0	0
$551$	5028.47	0.388784
$552$	0	0
$553$	−177.161	−0.0136232
$554$	11037.1	0.846430
$555$	0	0
$556$	15337.2	1.16986
$557$	16406.2	1.24803	0.624014	−	0.781413i	$-0.285501\pi$
$557$	16406.2	1.24803	0.624014	+	0.781413i	$0.285501\pi$
$558$	0	0
$559$	6710.31	0.507721
$560$	0	0
$561$	0	0
$562$	−1366.18	−0.102542
$563$	−13631.9	−1.02045	−0.510227	−	0.860040i	$-0.670439\pi$
$563$	−13631.9	−1.02045	−0.510227	+	0.860040i	$0.670439\pi$
$564$	0	0
$565$	0	0
$566$	−10911.2	−0.810300
$567$	0	0
$568$	7310.88	0.540067
$569$	3086.83	0.227428	0.113714	−	0.993514i	$-0.463725\pi$
$569$	3086.83	0.227428	0.113714	+	0.993514i	$0.463725\pi$
$570$	0	0
$571$	−3258.06	−0.238784	−0.119392	−	0.992847i	$-0.538095\pi$
$571$	−3258.06	−0.238784	−0.119392	+	0.992847i	$0.538095\pi$
$572$	13613.9	0.995152
$573$	0	0
$574$	−1988.24	−0.144577
$575$	0	0
$576$	0	0
$577$	23758.4	1.71417	0.857083	−	0.515178i	$-0.172274\pi$
$577$	23758.4	1.71417	0.857083	+	0.515178i	$0.172274\pi$
$578$	5950.07	0.428184
$579$	0	0
$580$	0	0
$581$	−2633.78	−0.188068
$582$	0	0
$583$	−15470.8	−1.09903
$584$	−12073.2	−0.855468
$585$	0	0
$586$	−729.566	−0.0514302
$587$	−596.893	−0.0419701	−0.0209850	−	0.999780i	$-0.506680\pi$
$587$	−596.893	−0.0419701	−0.0209850	+	0.999780i	$0.506680\pi$
$588$	0	0
$589$	6203.66	0.433985
$590$	0	0
$591$	0	0
$592$	1047.02	0.0726898
$593$	−19496.3	−1.35012	−0.675058	−	0.737765i	$-0.735881\pi$
$593$	−19496.3	−1.35012	−0.675058	+	0.737765i	$0.735881\pi$
$594$	0	0
$595$	0	0
$596$	13057.3	0.897396
$597$	0	0
$598$	12084.8	0.826395
$599$	−3797.02	−0.259001	−0.129501	−	0.991579i	$-0.541337\pi$
$599$	−3797.02	−0.259001	−0.129501	+	0.991579i	$0.541337\pi$
$600$	0	0
$601$	5789.33	0.392931	0.196466	−	0.980511i	$-0.437054\pi$
$601$	5789.33	0.392931	0.196466	+	0.980511i	$0.437054\pi$
$602$	−1728.76	−0.117041
$603$	0	0
$604$	−524.850	−0.0353573
$605$	0	0
$606$	0	0
$607$	18536.4	1.23949	0.619745	−	0.784803i	$-0.287236\pi$
$607$	18536.4	1.23949	0.619745	+	0.784803i	$0.287236\pi$
$608$	23082.3	1.53966
$609$	0	0
$610$	0	0
$611$	−3051.06	−0.202017
$612$	0	0
$613$	2163.47	0.142548	0.0712738	−	0.997457i	$-0.477294\pi$
$613$	2163.47	0.142548	0.0712738	+	0.997457i	$0.477294\pi$
$614$	−8235.08	−0.541272
$615$	0	0
$616$	−8909.59	−0.582756
$617$	−22964.9	−1.49843	−0.749215	−	0.662327i	$-0.769569\pi$
$617$	−22964.9	−1.49843	−0.749215	+	0.662327i	$0.769569\pi$
$618$	0	0
$619$	−1386.67	−0.0900401	−0.0450200	−	0.998986i	$-0.514335\pi$
$619$	−1386.67	−0.0900401	−0.0450200	+	0.998986i	$0.514335\pi$
$620$	0	0
$621$	0	0
$622$	3067.41	0.197736
$623$	−7185.06	−0.462060
$624$	0	0
$625$	0	0
$626$	−4092.35	−0.261283
$627$	0	0
$628$	−12145.0	−0.771716
$629$	−21281.4	−1.34904
$630$	0	0
$631$	5969.39	0.376605	0.188303	−	0.982111i	$-0.439701\pi$
$631$	5969.39	0.376605	0.188303	+	0.982111i	$0.439701\pi$
$632$	559.367	0.0352064
$633$	0	0
$634$	−2791.14	−0.174843
$635$	0	0
$636$	0	0
$637$	−2230.28	−0.138724
$638$	3875.25	0.240474
$639$	0	0
$640$	0	0
$641$	−30367.1	−1.87118	−0.935592	−	0.353084i	$-0.885133\pi$
$641$	−30367.1	−1.87118	−0.935592	+	0.353084i	$0.885133\pi$
$642$	0	0
$643$	28592.2	1.75360	0.876802	−	0.480851i	$-0.159672\pi$
$643$	28592.2	1.75360	0.876802	+	0.480851i	$0.159672\pi$
$644$	5762.42	0.352595
$645$	0	0
$646$	−19292.7	−1.17502
$647$	14507.9	0.881555	0.440778	−	0.897616i	$-0.354703\pi$
$647$	14507.9	0.881555	0.440778	+	0.897616i	$0.354703\pi$
$648$	0	0
$649$	13869.7	0.838877
$650$	0	0
$651$	0	0
$652$	6882.89	0.413428
$653$	−6999.85	−0.419488	−0.209744	−	0.977756i	$-0.567263\pi$
$653$	−6999.85	−0.419488	−0.209744	+	0.977756i	$0.567263\pi$
$654$	0	0
$655$	0	0
$656$	−767.504	−0.0456799
$657$	0	0
$658$	786.035	0.0465696
$659$	−7308.92	−0.432041	−0.216020	−	0.976389i	$-0.569308\pi$
$659$	−7308.92	−0.432041	−0.216020	+	0.976389i	$0.569308\pi$
$660$	0	0
$661$	−30097.2	−1.77102	−0.885512	−	0.464617i	$-0.846192\pi$
$661$	−30097.2	−1.77102	−0.885512	+	0.464617i	$0.846192\pi$
$662$	−9157.07	−0.537613
$663$	0	0
$664$	8315.91	0.486024
$665$	0	0
$666$	0	0
$667$	−6366.92	−0.369608
$668$	10835.4	0.627594
$669$	0	0
$670$	0	0
$671$	5209.29	0.299706
$672$	0	0
$673$	−5400.26	−0.309309	−0.154654	−	0.987969i	$-0.549426\pi$
$673$	−5400.26	−0.309309	−0.154654	+	0.987969i	$0.549426\pi$
$674$	17841.3	1.01962
$675$	0	0
$676$	650.781	0.0370267
$677$	−6431.09	−0.365091	−0.182546	−	0.983197i	$-0.558434\pi$
$677$	−6431.09	−0.365091	−0.182546	+	0.983197i	$0.558434\pi$
$678$	0	0
$679$	−6598.62	−0.372948
$680$	0	0
$681$	0	0
$682$	4780.92	0.268433
$683$	20865.8	1.16897	0.584486	−	0.811404i	$-0.301296\pi$
$683$	20865.8	1.16897	0.584486	+	0.811404i	$0.301296\pi$
$684$	0	0
$685$	0	0
$686$	574.581	0.0319790
$687$	0	0
$688$	−667.339	−0.0369797
$689$	12227.7	0.676109
$690$	0	0
$691$	18450.3	1.01575	0.507873	−	0.861432i	$-0.330432\pi$
$691$	18450.3	1.01575	0.507873	+	0.861432i	$0.330432\pi$
$692$	9962.76	0.547294
$693$	0	0
$694$	−6732.70	−0.368256
$695$	0	0
$696$	0	0
$697$	15600.0	0.847766
$698$	17664.4	0.957890
$699$	0	0
$700$	0	0
$701$	−12639.3	−0.680996	−0.340498	−	0.940245i	$-0.610596\pi$
$701$	−12639.3	−0.680996	−0.340498	+	0.940245i	$0.610596\pi$
$702$	0	0
$703$	28954.4	1.55339
$704$	15703.3	0.840680
$705$	0	0
$706$	−4958.45	−0.264325
$707$	4232.32	0.225138
$708$	0	0
$709$	−23126.8	−1.22503	−0.612514	−	0.790460i	$-0.709842\pi$
$709$	−23126.8	−1.22503	−0.612514	+	0.790460i	$0.709842\pi$
$710$	0	0
$711$	0	0
$712$	22686.1	1.19410
$713$	−7854.92	−0.412579
$714$	0	0
$715$	0	0
$716$	3267.16	0.170530
$717$	0	0
$718$	−3635.39	−0.188957
$719$	24093.1	1.24968	0.624841	−	0.780752i	$-0.285164\pi$
$719$	24093.1	1.24968	0.624841	+	0.780752i	$0.285164\pi$
$720$	0	0
$721$	−2106.77	−0.108821
$722$	14758.7	0.760750
$723$	0	0
$724$	14547.2	0.746741
$725$	0	0
$726$	0	0
$727$	35983.4	1.83570	0.917849	−	0.396931i	$-0.129925\pi$
$727$	35983.4	1.83570	0.917849	+	0.396931i	$0.129925\pi$
$728$	7041.89	0.358503
$729$	0	0
$730$	0	0
$731$	13564.1	0.686302
$732$	0	0
$733$	−1451.50	−0.0731413	−0.0365706	−	0.999331i	$-0.511643\pi$
$733$	−1451.50	−0.0731413	−0.0365706	+	0.999331i	$0.511643\pi$
$734$	−2097.64	−0.105484
$735$	0	0
$736$	−29226.3	−1.46371
$737$	−23409.4	−1.17001
$738$	0	0
$739$	−5891.67	−0.293273	−0.146636	−	0.989190i	$-0.546845\pi$
$739$	−5891.67	−0.293273	−0.146636	+	0.989190i	$0.546845\pi$
$740$	0	0
$741$	0	0
$742$	−3150.19	−0.155859
$743$	−7438.65	−0.367292	−0.183646	−	0.982992i	$-0.558790\pi$
$743$	−7438.65	−0.367292	−0.183646	+	0.982992i	$0.558790\pi$
$744$	0	0
$745$	0	0
$746$	7782.84	0.381970
$747$	0	0
$748$	27518.9	1.34518
$749$	−10581.6	−0.516214
$750$	0	0
$751$	20272.4	0.985018	0.492509	−	0.870307i	$-0.336080\pi$
$751$	20272.4	0.985018	0.492509	+	0.870307i	$0.336080\pi$
$752$	303.427	0.0147139
$753$	0	0
$754$	−3062.89	−0.147936
$755$	0	0
$756$	0	0
$757$	−10193.8	−0.489432	−0.244716	−	0.969595i	$-0.578695\pi$
$757$	−10193.8	−0.489432	−0.244716	+	0.969595i	$0.578695\pi$
$758$	2403.59	0.115174
$759$	0	0
$760$	0	0
$761$	41117.6	1.95862	0.979311	−	0.202362i	$-0.0648618\pi$
$761$	41117.6	1.95862	0.979311	+	0.202362i	$0.0648618\pi$
$762$	0	0
$763$	−12369.6	−0.586908
$764$	3844.76	0.182066
$765$	0	0
$766$	22140.2	1.04433
$767$	−10962.2	−0.516065
$768$	0	0
$769$	11486.6	0.538642	0.269321	−	0.963050i	$-0.413201\pi$
$769$	11486.6	0.538642	0.269321	+	0.963050i	$0.413201\pi$
$770$	0	0
$771$	0	0
$772$	−21204.2	−0.988544
$773$	21799.2	1.01431	0.507156	−	0.861854i	$-0.330697\pi$
$773$	21799.2	1.01431	0.507156	+	0.861854i	$0.330697\pi$
$774$	0	0
$775$	0	0
$776$	20834.5	0.963807
$777$	0	0
$778$	12990.9	0.598645
$779$	−21224.5	−0.976185
$780$	0	0
$781$	−19049.1	−0.872765
$782$	24428.0	1.11706
$783$	0	0
$784$	221.801	0.0101039
$785$	0	0
$786$	0	0
$787$	−24080.3	−1.09068	−0.545342	−	0.838214i	$-0.683600\pi$
$787$	−24080.3	−1.09068	−0.545342	+	0.838214i	$0.683600\pi$
$788$	17736.4	0.801817
$789$	0	0
$790$	0	0
$791$	−7316.87	−0.328898
$792$	0	0
$793$	−4117.28	−0.184374
$794$	−5964.96	−0.266610
$795$	0	0
$796$	17619.8	0.784569
$797$	17194.3	0.764184	0.382092	−	0.924124i	$-0.375204\pi$
$797$	17194.3	0.764184	0.382092	+	0.924124i	$0.375204\pi$
$798$	0	0
$799$	−6167.35	−0.273073
$800$	0	0
$801$	0	0
$802$	9097.15	0.400538
$803$	31457.7	1.38246
$804$	0	0
$805$	0	0
$806$	−3778.71	−0.165136
$807$	0	0
$808$	−13363.1	−0.581823
$809$	−33349.1	−1.44931	−0.724656	−	0.689111i	$-0.758001\pi$
$809$	−33349.1	−1.44931	−0.724656	+	0.689111i	$0.758001\pi$
$810$	0	0
$811$	4577.87	0.198213	0.0991066	−	0.995077i	$-0.468402\pi$
$811$	4577.87	0.198213	0.0991066	+	0.995077i	$0.468402\pi$
$812$	−1460.48	−0.0631194
$813$	0	0
$814$	22314.0	0.960819
$815$	0	0
$816$	0	0
$817$	−18454.6	−0.790262
$818$	−16247.0	−0.694455
$819$	0	0
$820$	0	0
$821$	5832.52	0.247937	0.123969	−	0.992286i	$-0.460438\pi$
$821$	5832.52	0.247937	0.123969	+	0.992286i	$0.460438\pi$
$822$	0	0
$823$	37974.3	1.60838	0.804192	−	0.594369i	$-0.202598\pi$
$823$	37974.3	1.60838	0.804192	+	0.594369i	$0.202598\pi$
$824$	6651.92	0.281226
$825$	0	0
$826$	2824.15	0.118965
$827$	−15796.2	−0.664193	−0.332096	−	0.943245i	$-0.607756\pi$
$827$	−15796.2	−0.664193	−0.332096	+	0.943245i	$0.607756\pi$
$828$	0	0
$829$	−12714.1	−0.532666	−0.266333	−	0.963881i	$-0.585812\pi$
$829$	−12714.1	−0.532666	−0.266333	+	0.963881i	$0.585812\pi$
$830$	0	0
$831$	0	0
$832$	−12411.4	−0.517173
$833$	−4508.25	−0.187517
$834$	0	0
$835$	0	0
$836$	−37440.8	−1.54894
$837$	0	0
$838$	−23169.0	−0.955085
$839$	42924.2	1.76628	0.883140	−	0.469109i	$-0.155425\pi$
$839$	42924.2	1.76628	0.883140	+	0.469109i	$0.155425\pi$
$840$	0	0
$841$	−22775.3	−0.933835
$842$	27980.2	1.14520
$843$	0	0
$844$	−17648.8	−0.719784
$845$	0	0
$846$	0	0
$847$	13897.6	0.563788
$848$	−1216.04	−0.0492442
$849$	0	0
$850$	0	0
$851$	−36661.3	−1.47677
$852$	0	0
$853$	−36172.5	−1.45196	−0.725980	−	0.687716i	$-0.758613\pi$
$853$	−36172.5	−1.45196	−0.725980	+	0.687716i	$0.758613\pi$
$854$	1060.72	0.0425025
$855$	0	0
$856$	33410.4	1.33405
$857$	32039.9	1.27709	0.638544	−	0.769586i	$-0.279537\pi$
$857$	32039.9	1.27709	0.638544	+	0.769586i	$0.279537\pi$
$858$	0	0
$859$	−6798.07	−0.270020	−0.135010	−	0.990844i	$-0.543107\pi$
$859$	−6798.07	−0.270020	−0.135010	+	0.990844i	$0.543107\pi$
$860$	0	0
$861$	0	0
$862$	−13692.9	−0.541046
$863$	30179.3	1.19040	0.595200	−	0.803577i	$-0.297073\pi$
$863$	30179.3	1.19040	0.595200	+	0.803577i	$0.297073\pi$
$864$	0	0
$865$	0	0
$866$	23825.5	0.934901
$867$	0	0
$868$	−1801.81	−0.0704578
$869$	−1457.47	−0.0568946
$870$	0	0
$871$	18502.1	0.719772
$872$	39055.9	1.51674
$873$	0	0
$874$	−33235.4	−1.28627
$875$	0	0
$876$	0	0
$877$	−1700.51	−0.0654758	−0.0327379	−	0.999464i	$-0.510423\pi$
$877$	−1700.51	−0.0654758	−0.0327379	+	0.999464i	$0.510423\pi$
$878$	9276.02	0.356549
$879$	0	0
$880$	0	0
$881$	−1678.46	−0.0641869	−0.0320935	−	0.999485i	$-0.510217\pi$
$881$	−1678.46	−0.0641869	−0.0320935	+	0.999485i	$0.510217\pi$
$882$	0	0
$883$	10285.8	0.392009	0.196005	−	0.980603i	$-0.437203\pi$
$883$	10285.8	0.392009	0.196005	+	0.980603i	$0.437203\pi$
$884$	−21750.2	−0.827532
$885$	0	0
$886$	−6657.24	−0.252432
$887$	12167.5	0.460593	0.230296	−	0.973121i	$-0.426030\pi$
$887$	12167.5	0.460593	0.230296	+	0.973121i	$0.426030\pi$
$888$	0	0
$889$	1825.09	0.0688543
$890$	0	0
$891$	0	0
$892$	−948.452	−0.0356015
$893$	8390.96	0.314438
$894$	0	0
$895$	0	0
$896$	−7128.73	−0.265797
$897$	0	0
$898$	25534.7	0.948892
$899$	1990.83	0.0738574
$900$	0	0
$901$	24716.9	0.913916
$902$	−16356.9	−0.603799
$903$	0	0
$904$	23102.3	0.849968
$905$	0	0
$906$	0	0
$907$	−50766.0	−1.85850	−0.929250	−	0.369452i	$-0.879545\pi$
$907$	−50766.0	−1.85850	−0.929250	+	0.369452i	$0.879545\pi$
$908$	−16372.7	−0.598399
$909$	0	0
$910$	0	0
$911$	−18451.1	−0.671033	−0.335517	−	0.942034i	$-0.608911\pi$
$911$	−18451.1	−0.671033	−0.335517	+	0.942034i	$0.608911\pi$
$912$	0	0
$913$	−21667.8	−0.785431
$914$	−18039.6	−0.652841
$915$	0	0
$916$	−31227.1	−1.12639
$917$	5064.65	0.182388
$918$	0	0
$919$	393.861	0.0141374	0.00706870	−	0.999975i	$-0.497750\pi$
$919$	393.861	0.0141374	0.00706870	+	0.999975i	$0.497750\pi$
$920$	0	0
$921$	0	0
$922$	557.167	0.0199016
$923$	15055.9	0.536912
$924$	0	0
$925$	0	0
$926$	−13745.3	−0.487795
$927$	0	0
$928$	7407.39	0.262025
$929$	−27452.3	−0.969517	−0.484759	−	0.874648i	$-0.661093\pi$
$929$	−27452.3	−0.969517	−0.484759	+	0.874648i	$0.661093\pi$
$930$	0	0
$931$	6133.68	0.215922
$932$	−4885.63	−0.171710
$933$	0	0
$934$	280.803	0.00983744
$935$	0	0
$936$	0	0
$937$	21608.3	0.753375	0.376688	−	0.926340i	$-0.377063\pi$
$937$	21608.3	0.753375	0.376688	+	0.926340i	$0.377063\pi$
$938$	−4766.65	−0.165924
$939$	0	0
$940$	0	0
$941$	16708.5	0.578832	0.289416	−	0.957203i	$-0.406539\pi$
$941$	16708.5	0.578832	0.289416	+	0.957203i	$0.406539\pi$
$942$	0	0
$943$	26874.0	0.928036
$944$	1090.19	0.0375874
$945$	0	0
$946$	−14222.2	−0.488800
$947$	−13972.8	−0.479466	−0.239733	−	0.970839i	$-0.577060\pi$
$947$	−13972.8	−0.479466	−0.239733	+	0.970839i	$0.577060\pi$
$948$	0	0
$949$	−24863.3	−0.850471
$950$	0	0
$951$	0	0
$952$	14234.4	0.484599
$953$	11155.9	0.379197	0.189598	−	0.981862i	$-0.439281\pi$
$953$	11155.9	0.379197	0.189598	+	0.981862i	$0.439281\pi$
$954$	0	0
$955$	0	0
$956$	26794.0	0.906466
$957$	0	0
$958$	−11104.0	−0.374481
$959$	−5415.85	−0.182364
$960$	0	0
$961$	−27334.9	−0.917556
$962$	−17636.4	−0.591081
$963$	0	0
$964$	2409.09	0.0804891
$965$	0	0
$966$	0	0
$967$	−1212.55	−0.0403238	−0.0201619	−	0.999797i	$-0.506418\pi$
$967$	−1212.55	−0.0403238	−0.0201619	+	0.999797i	$0.506418\pi$
$968$	−43880.4	−1.45699
$969$	0	0
$970$	0	0
$971$	48832.2	1.61390	0.806952	−	0.590617i	$-0.201115\pi$
$971$	48832.2	1.61390	0.806952	+	0.590617i	$0.201115\pi$
$972$	0	0
$973$	−20670.8	−0.681063
$974$	−34578.0	−1.13753
$975$	0	0
$976$	409.462	0.0134289
$977$	−5939.19	−0.194485	−0.0972424	−	0.995261i	$-0.531002\pi$
$977$	−5939.19	−0.194485	−0.0972424	+	0.995261i	$0.531002\pi$
$978$	0	0
$979$	−59110.5	−1.92970
$980$	0	0
$981$	0	0
$982$	27993.3	0.909677
$983$	21863.5	0.709398	0.354699	−	0.934980i	$-0.384583\pi$
$983$	21863.5	0.709398	0.354699	+	0.934980i	$0.384583\pi$
$984$	0	0
$985$	0	0
$986$	−6191.26	−0.199970
$987$	0	0
$988$	29592.1	0.952886
$989$	23366.7	0.751283
$990$	0	0
$991$	−44618.5	−1.43023	−0.715113	−	0.699009i	$-0.753625\pi$
$991$	−44618.5	−1.43023	−0.715113	+	0.699009i	$0.753625\pi$
$992$	9138.55	0.292489
$993$	0	0
$994$	−3878.80	−0.123771
$995$	0	0
$996$	0	0
$997$	−37923.9	−1.20468	−0.602338	−	0.798241i	$-0.705764\pi$
$997$	−37923.9	−1.20468	−0.602338	+	0.798241i	$0.705764\pi$
$998$	−22997.8	−0.729441
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bn.1.4		5
3.2	odd	2		175.4.a.j.1.2		5
5.2	odd	4		315.4.d.c.64.7		10
5.3	odd	4		315.4.d.c.64.4		10
5.4	even	2		1575.4.a.bq.1.2		5
15.2	even	4		35.4.b.a.29.4	✓	10
15.8	even	4		35.4.b.a.29.7	yes	10
15.14	odd	2		175.4.a.i.1.4		5
21.20	even	2		1225.4.a.bh.1.2		5
60.23	odd	4		560.4.g.f.449.7		10
60.47	odd	4		560.4.g.f.449.4		10
105.2	even	12		245.4.j.e.214.4		20
105.17	odd	12		245.4.j.f.79.7		20
105.23	even	12		245.4.j.e.214.7		20
105.32	even	12		245.4.j.e.79.7		20
105.38	odd	12		245.4.j.f.79.4		20
105.47	odd	12		245.4.j.f.214.4		20
105.53	even	12		245.4.j.e.79.4		20
105.62	odd	4		245.4.b.d.99.4		10
105.68	odd	12		245.4.j.f.214.7		20
105.83	odd	4		245.4.b.d.99.7		10
105.104	even	2		1225.4.a.be.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.4	✓	10	15.2	even	4
35.4.b.a.29.7	yes	10	15.8	even	4
175.4.a.i.1.4		5	15.14	odd	2
175.4.a.j.1.2		5	3.2	odd	2
245.4.b.d.99.4		10	105.62	odd	4
245.4.b.d.99.7		10	105.83	odd	4
245.4.j.e.79.4		20	105.53	even	12
245.4.j.e.79.7		20	105.32	even	12
245.4.j.e.214.4		20	105.2	even	12
245.4.j.e.214.7		20	105.23	even	12
245.4.j.f.79.4		20	105.38	odd	12
245.4.j.f.79.7		20	105.17	odd	12
245.4.j.f.214.4		20	105.47	odd	12
245.4.j.f.214.7		20	105.68	odd	12
315.4.d.c.64.4		10	5.3	odd	4
315.4.d.c.64.7		10	5.2	odd	4
560.4.g.f.449.4		10	60.47	odd	4
560.4.g.f.449.7		10	60.23	odd	4
1225.4.a.be.1.4		5	105.104	even	2
1225.4.a.bh.1.2		5	21.20	even	2
1575.4.a.bn.1.4		5	1.1	even	1	trivial
1575.4.a.bq.1.2		5	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)

\(f(q)\)	\(=\)	\(q+1.67516 q^{2} -5.19383 q^{4} +7.00000 q^{7} -22.1018 q^{8} +57.5880 q^{11} -45.5159 q^{13} +11.7261 q^{14} +4.52655 q^{16} -92.0051 q^{17} +125.177 q^{19} +96.4692 q^{22} -158.496 q^{23} -76.2466 q^{26} -36.3568 q^{28} +40.1708 q^{29} +49.5590 q^{31} +184.397 q^{32} -154.123 q^{34} +231.307 q^{37} +209.692 q^{38} -169.556 q^{41} -147.428 q^{43} -299.102 q^{44} -265.507 q^{46} +67.0327 q^{47} +49.0000 q^{49} +236.402 q^{52} -268.647 q^{53} -154.713 q^{56} +67.2926 q^{58} +240.843 q^{59} +90.4579 q^{61} +83.0194 q^{62} +272.683 q^{64} -406.498 q^{67} +477.859 q^{68} -330.782 q^{71} +546.255 q^{73} +387.477 q^{74} -650.149 q^{76} +403.116 q^{77} -25.3087 q^{79} -284.034 q^{82} -376.255 q^{83} -246.965 q^{86} -1272.80 q^{88} -1026.44 q^{89} -318.612 q^{91} +823.203 q^{92} +112.291 q^{94} -942.660 q^{97} +82.0829 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} + 18 q^{4} + 35 q^{7} - 42 q^{8} - 42 q^{11} + 34 q^{13} - 28 q^{14} + 74 q^{16} - 238 q^{17} - 36 q^{19} + 358 q^{22} - 152 q^{23} + 310 q^{26} + 126 q^{28} + 44 q^{29} + 60 q^{31} - 710 q^{32}+ \cdots - 196 q^{98}+O(q^{100}) \) 5 * q - 4 * q^2 + 18 * q^4 + 35 * q^7 - 42 * q^8 - 42 * q^11 + 34 * q^13 - 28 * q^14 + 74 * q^16 - 238 * q^17 - 36 * q^19 + 358 * q^22 - 152 * q^23 + 310 * q^26 + 126 * q^28 + 44 * q^29 + 60 * q^31 - 710 * q^32 - 482 * q^34 + 312 * q^37 - 280 * q^38 + 426 * q^41 + 304 * q^43 - 712 * q^44 + 88 * q^46 - 370 * q^47 + 245 * q^49 - 1156 * q^52 - 976 * q^53 - 294 * q^56 - 2722 * q^58 + 432 * q^59 - 442 * q^61 + 956 * q^62 + 1362 * q^64 - 804 * q^67 + 420 * q^68 - 440 * q^71 + 564 * q^73 + 1512 * q^74 - 1336 * q^76 - 294 * q^77 + 1790 * q^79 - 3480 * q^82 - 1656 * q^83 - 1216 * q^86 - 1092 * q^88 - 746 * q^89 + 238 * q^91 - 572 * q^92 - 826 * q^94 + 518 * q^97 - 196 * q^98

Introduction

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