LMFDB - Embedded newform 2475.4.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43845 q^{2} -5.93087 q^{4} +31.0540 q^{7} +20.0388 q^{8} +11.0000 q^{11} +45.6155 q^{13} -44.6695 q^{14} +18.6222 q^{16} -40.4536 q^{17} +91.2699 q^{19} -15.8229 q^{22} +32.2462 q^{23} -65.6155 q^{26} -184.177 q^{28} -35.8702 q^{29} +311.702 q^{31} -187.098 q^{32} +58.1904 q^{34} +368.147 q^{37} -131.287 q^{38} +393.602 q^{41} +351.602 q^{43} -65.2396 q^{44} -46.3845 q^{46} -230.155 q^{47} +621.349 q^{49} -270.540 q^{52} +406.902 q^{53} +622.285 q^{56} +51.5975 q^{58} +368.725 q^{59} -322.825 q^{61} -448.366 q^{62} +120.153 q^{64} -442.233 q^{67} +239.925 q^{68} -667.745 q^{71} +84.5701 q^{73} -529.560 q^{74} -541.310 q^{76} +341.594 q^{77} -411.983 q^{79} -566.176 q^{82} -835.619 q^{83} -505.761 q^{86} +220.427 q^{88} +799.853 q^{89} +1416.54 q^{91} -191.248 q^{92} +331.066 q^{94} -768.945 q^{97} -893.778 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 7 q^{2} + 17 q^{4} + 25 q^{7} - 63 q^{8} + 22 q^{11} + 50 q^{13} - 11 q^{14} + 297 q^{16} - 151 q^{17} - 3 q^{19} - 77 q^{22} + 48 q^{23} - 90 q^{26} - 323 q^{28} + 221 q^{29} + 141 q^{31} - 1071 q^{32}+ \cdots + 810 q^{98}+O(q^{100}) $ 2 * q - 7 * q^2 + 17 * q^4 + 25 * q^7 - 63 * q^8 + 22 * q^11 + 50 * q^13 - 11 * q^14 + 297 * q^16 - 151 * q^17 - 3 * q^19 - 77 * q^22 + 48 * q^23 - 90 * q^26 - 323 * q^28 + 221 * q^29 + 141 * q^31 - 1071 * q^32 + 673 * q^34 + 559 * q^37 + 393 * q^38 + 144 * q^41 + 60 * q^43 + 187 * q^44 - 134 * q^46 - 48 * q^47 + 315 * q^49 - 170 * q^52 + 117 * q^53 + 1125 * q^56 - 1377 * q^58 + 86 * q^59 - 155 * q^61 + 501 * q^62 + 2809 * q^64 - 266 * q^67 - 2295 * q^68 - 1587 * q^71 - 70 * q^73 - 1591 * q^74 - 2703 * q^76 + 275 * q^77 - 1294 * q^79 + 822 * q^82 - 558 * q^83 + 1116 * q^86 - 693 * q^88 + 1777 * q^89 + 1390 * q^91 + 170 * q^92 - 682 * q^94 + 334 * q^97 + 810 * q^98

Coefficient data

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

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

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

$2$	−1.43845	−0.508568	−0.254284	−	0.967130i	$-0.581840\pi$
$2$	−1.43845	−0.508568	−0.254284	+	0.967130i	$0.581840\pi$
$3$	0	0
$3$	0	0
$4$	−5.93087	−0.741359
$5$	0	0
$5$	0	0
$6$	0	0
$7$	31.0540	1.67676	0.838379	−	0.545088i	$-0.183504\pi$
$7$	31.0540	1.67676	0.838379	+	0.545088i	$0.183504\pi$
$8$	20.0388	0.885599
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	45.6155	0.973190	0.486595	−	0.873628i	$-0.338239\pi$
$13$	45.6155	0.973190	0.486595	+	0.873628i	$0.338239\pi$
$14$	−44.6695	−0.852745
$15$	0	0
$16$	18.6222	0.290971
$17$	−40.4536	−0.577144	−0.288572	−	0.957458i	$-0.593180\pi$
$17$	−40.4536	−0.577144	−0.288572	+	0.957458i	$0.593180\pi$
$18$	0	0
$19$	91.2699	1.10204	0.551020	−	0.834492i	$-0.314239\pi$
$19$	91.2699	1.10204	0.551020	+	0.834492i	$0.314239\pi$
$20$	0	0
$21$	0	0
$22$	−15.8229	−0.153339
$23$	32.2462	0.292339	0.146170	−	0.989260i	$-0.453305\pi$
$23$	32.2462	0.292339	0.146170	+	0.989260i	$0.453305\pi$
$24$	0	0
$25$	0	0
$26$	−65.6155	−0.494933
$27$	0	0
$28$	−184.177	−1.24308
$29$	−35.8702	−0.229688	−0.114844	−	0.993384i	$-0.536637\pi$
$29$	−35.8702	−0.229688	−0.114844	+	0.993384i	$0.536637\pi$
$30$	0	0
$31$	311.702	1.80591	0.902956	−	0.429733i	$-0.141392\pi$
$31$	311.702	1.80591	0.902956	+	0.429733i	$0.141392\pi$
$32$	−187.098	−1.03358
$33$	0	0
$34$	58.1904	0.293517
$35$	0	0
$36$	0	0
$37$	368.147	1.63576	0.817878	−	0.575392i	$-0.195150\pi$
$37$	368.147	1.63576	0.817878	+	0.575392i	$0.195150\pi$
$38$	−131.287	−0.560462
$39$	0	0
$40$	0	0
$41$	393.602	1.49928	0.749638	−	0.661848i	$-0.230227\pi$
$41$	393.602	1.49928	0.749638	+	0.661848i	$0.230227\pi$
$42$	0	0
$43$	351.602	1.24695	0.623475	−	0.781843i	$-0.285720\pi$
$43$	351.602	1.24695	0.623475	+	0.781843i	$0.285720\pi$
$44$	−65.2396	−0.223528
$45$	0	0
$46$	−46.3845	−0.148674
$47$	−230.155	−0.714289	−0.357145	−	0.934049i	$-0.616250\pi$
$47$	−230.155	−0.714289	−0.357145	+	0.934049i	$0.616250\pi$
$48$	0	0
$49$	621.349	1.81151
$50$	0	0
$51$	0	0
$52$	−270.540	−0.721483
$53$	406.902	1.05457	0.527286	−	0.849688i	$-0.323210\pi$
$53$	406.902	1.05457	0.527286	+	0.849688i	$0.323210\pi$
$54$	0	0
$55$	0	0
$56$	622.285	1.48493
$57$	0	0
$58$	51.5975	0.116812
$59$	368.725	0.813626	0.406813	−	0.913511i	$-0.366640\pi$
$59$	368.725	0.813626	0.406813	+	0.913511i	$0.366640\pi$
$60$	0	0
$61$	−322.825	−0.677598	−0.338799	−	0.940859i	$-0.610021\pi$
$61$	−322.825	−0.677598	−0.338799	+	0.940859i	$0.610021\pi$
$62$	−448.366	−0.918429
$63$	0	0
$64$	120.153	0.234673
$65$	0	0
$66$	0	0
$67$	−442.233	−0.806378	−0.403189	−	0.915117i	$-0.632098\pi$
$67$	−442.233	−0.806378	−0.403189	+	0.915117i	$0.632098\pi$
$68$	239.925	0.427870
$69$	0	0
$70$	0	0
$71$	−667.745	−1.11615	−0.558076	−	0.829790i	$-0.688460\pi$
$71$	−667.745	−1.11615	−0.558076	+	0.829790i	$0.688460\pi$
$72$	0	0
$73$	84.5701	0.135591	0.0677957	−	0.997699i	$-0.478403\pi$
$73$	84.5701	0.135591	0.0677957	+	0.997699i	$0.478403\pi$
$74$	−529.560	−0.831893
$75$	0	0
$76$	−541.310	−0.817006
$77$	341.594	0.505561
$78$	0	0
$79$	−411.983	−0.586730	−0.293365	−	0.956000i	$-0.594775\pi$
$79$	−411.983	−0.586730	−0.293365	+	0.956000i	$0.594775\pi$
$80$	0	0
$81$	0	0
$82$	−566.176	−0.762484
$83$	−835.619	−1.10507	−0.552537	−	0.833488i	$-0.686340\pi$
$83$	−835.619	−1.10507	−0.552537	+	0.833488i	$0.686340\pi$
$84$	0	0
$85$	0	0
$86$	−505.761	−0.634159
$87$	0	0
$88$	220.427	0.267018
$89$	799.853	0.952632	0.476316	−	0.879274i	$-0.341972\pi$
$89$	799.853	0.952632	0.476316	+	0.879274i	$0.341972\pi$
$90$	0	0
$91$	1416.54	1.63180
$92$	−191.248	−0.216728
$93$	0	0
$94$	331.066	0.363265
$95$	0	0
$96$	0	0
$97$	−768.945	−0.804892	−0.402446	−	0.915444i	$-0.631840\pi$
$97$	−768.945	−0.804892	−0.402446	+	0.915444i	$0.631840\pi$
$98$	−893.778	−0.921278
$99$	0	0
$100$	0	0
$101$	1412.31	1.39139	0.695696	−	0.718337i	$-0.255096\pi$
$101$	1412.31	1.39139	0.695696	+	0.718337i	$0.255096\pi$
$102$	0	0
$103$	−24.4185	−0.0233595	−0.0116797	−	0.999932i	$-0.503718\pi$
$103$	−24.4185	−0.0233595	−0.0116797	+	0.999932i	$0.503718\pi$
$104$	914.081	0.861856
$105$	0	0
$106$	−585.308	−0.536322
$107$	532.155	0.480798	0.240399	−	0.970674i	$-0.422722\pi$
$107$	532.155	0.480798	0.240399	+	0.970674i	$0.422722\pi$
$108$	0	0
$109$	−1234.62	−1.08491	−0.542455	−	0.840085i	$-0.682505\pi$
$109$	−1234.62	−1.08491	−0.542455	+	0.840085i	$0.682505\pi$
$110$	0	0
$111$	0	0
$112$	578.292	0.487888
$113$	−1304.45	−1.08595	−0.542976	−	0.839748i	$-0.682703\pi$
$113$	−1304.45	−1.08595	−0.542976	+	0.839748i	$0.682703\pi$
$114$	0	0
$115$	0	0
$116$	212.742	0.170281
$117$	0	0
$118$	−530.392	−0.413784
$119$	−1256.25	−0.967729
$120$	0	0
$121$	121.000	0.0909091
$122$	464.366	0.344605
$123$	0	0
$124$	−1848.66	−1.33883
$125$	0	0
$126$	0	0
$127$	−1167.88	−0.816004	−0.408002	−	0.912981i	$-0.633774\pi$
$127$	−1167.88	−0.816004	−0.408002	+	0.912981i	$0.633774\pi$
$128$	1323.95	0.914231
$129$	0	0
$130$	0	0
$131$	−1549.95	−1.03374	−0.516869	−	0.856065i	$-0.672902\pi$
$131$	−1549.95	−1.03374	−0.516869	+	0.856065i	$0.672902\pi$
$132$	0	0
$133$	2834.29	1.84785
$134$	636.129	0.410098
$135$	0	0
$136$	−810.642	−0.511118
$137$	−2440.68	−1.52205	−0.761026	−	0.648722i	$-0.775304\pi$
$137$	−2440.68	−1.52205	−0.761026	+	0.648722i	$0.775304\pi$
$138$	0	0
$139$	−1861.43	−1.13586	−0.567930	−	0.823077i	$-0.692256\pi$
$139$	−1861.43	−1.13586	−0.567930	+	0.823077i	$0.692256\pi$
$140$	0	0
$141$	0	0
$142$	960.516	0.567639
$143$	501.771	0.293428
$144$	0	0
$145$	0	0
$146$	−121.650	−0.0689575
$147$	0	0
$148$	−2183.43	−1.21268
$149$	−814.376	−0.447760	−0.223880	−	0.974617i	$-0.571872\pi$
$149$	−814.376	−0.447760	−0.223880	+	0.974617i	$0.571872\pi$
$150$	0	0
$151$	3666.43	1.97596	0.987979	−	0.154591i	$-0.0494060\pi$
$151$	3666.43	1.97596	0.987979	+	0.154591i	$0.0494060\pi$
$152$	1828.94	0.975965
$153$	0	0
$154$	−491.365	−0.257112
$155$	0	0
$156$	0	0
$157$	−2671.96	−1.35825	−0.679127	−	0.734021i	$-0.737641\pi$
$157$	−2671.96	−1.35825	−0.679127	+	0.734021i	$0.737641\pi$
$158$	592.616	0.298392
$159$	0	0
$160$	0	0
$161$	1001.37	0.490182
$162$	0	0
$163$	1728.53	0.830605	0.415302	−	0.909683i	$-0.363676\pi$
$163$	1728.53	0.830605	0.415302	+	0.909683i	$0.363676\pi$
$164$	−2334.40	−1.11150
$165$	0	0
$166$	1201.99	0.562005
$167$	−0.0653990	−3.03037e−5 0	−1.51519e−5	−	1.00000i	$-0.500005\pi$
$167$	−0.0653990	−3.03037e−5 0	−1.51519e−5		1.00000i	$0.500005\pi$
$168$	0	0
$169$	−116.224	−0.0529010
$170$	0	0
$171$	0	0
$172$	−2085.31	−0.924437
$173$	−1816.75	−0.798411	−0.399205	−	0.916861i	$-0.630714\pi$
$173$	−1816.75	−0.798411	−0.399205	+	0.916861i	$0.630714\pi$
$174$	0	0
$175$	0	0
$176$	204.844	0.0877312
$177$	0	0
$178$	−1150.55	−0.484478
$179$	−2381.23	−0.994312	−0.497156	−	0.867661i	$-0.665622\pi$
$179$	−2381.23	−0.994312	−0.497156	+	0.867661i	$0.665622\pi$
$180$	0	0
$181$	1275.61	0.523840	0.261920	−	0.965090i	$-0.415644\pi$
$181$	1275.61	0.523840	0.261920	+	0.965090i	$0.415644\pi$
$182$	−2037.62	−0.829883
$183$	0	0
$184$	646.176	0.258895
$185$	0	0
$186$	0	0
$187$	−444.990	−0.174015
$188$	1365.02	0.529545
$189$	0	0
$190$	0	0
$191$	3699.04	1.40133	0.700663	−	0.713492i	$-0.252888\pi$
$191$	3699.04	1.40133	0.700663	+	0.713492i	$0.252888\pi$
$192$	0	0
$193$	−1231.22	−0.459196	−0.229598	−	0.973285i	$-0.573741\pi$
$193$	−1231.22	−0.459196	−0.229598	+	0.973285i	$0.573741\pi$
$194$	1106.09	0.409342
$195$	0	0
$196$	−3685.14	−1.34298
$197$	728.087	0.263320	0.131660	−	0.991295i	$-0.457969\pi$
$197$	728.087	0.263320	0.131660	+	0.991295i	$0.457969\pi$
$198$	0	0
$199$	−1650.81	−0.588055	−0.294027	−	0.955797i	$-0.594996\pi$
$199$	−1650.81	−0.588055	−0.294027	+	0.955797i	$0.594996\pi$
$200$	0	0
$201$	0	0
$202$	−2031.54	−0.707617
$203$	−1113.91	−0.385130
$204$	0	0
$205$	0	0
$206$	35.1247	0.0118799
$207$	0	0
$208$	849.460	0.283171
$209$	1003.97	0.332277
$210$	0	0
$211$	1587.01	0.517792	0.258896	−	0.965905i	$-0.416641\pi$
$211$	1587.01	0.517792	0.258896	+	0.965905i	$0.416641\pi$
$212$	−2413.29	−0.781817
$213$	0	0
$214$	−765.477	−0.244518
$215$	0	0
$216$	0	0
$217$	9679.58	3.02808
$218$	1775.94	0.551751
$219$	0	0
$220$	0	0
$221$	−1845.31	−0.561670
$222$	0	0
$223$	1998.59	0.600160	0.300080	−	0.953914i	$-0.402987\pi$
$223$	1998.59	0.600160	0.300080	+	0.953914i	$0.402987\pi$
$224$	−5810.12	−1.73306
$225$	0	0
$226$	1876.39	0.552280
$227$	4212.00	1.23154	0.615771	−	0.787925i	$-0.288845\pi$
$227$	4212.00	1.23154	0.615771	+	0.787925i	$0.288845\pi$
$228$	0	0
$229$	1371.82	0.395863	0.197931	−	0.980216i	$-0.436578\pi$
$229$	1371.82	0.395863	0.197931	+	0.980216i	$0.436578\pi$
$230$	0	0
$231$	0	0
$232$	−718.797	−0.203411
$233$	−1718.37	−0.483152	−0.241576	−	0.970382i	$-0.577664\pi$
$233$	−1718.37	−0.483152	−0.241576	+	0.970382i	$0.577664\pi$
$234$	0	0
$235$	0	0
$236$	−2186.86	−0.603189
$237$	0	0
$238$	1807.04	0.492156
$239$	1794.77	0.485749	0.242875	−	0.970058i	$-0.421910\pi$
$239$	1794.77	0.485749	0.242875	+	0.970058i	$0.421910\pi$
$240$	0	0
$241$	5185.02	1.38588	0.692939	−	0.720996i	$-0.256315\pi$
$241$	5185.02	1.38588	0.692939	+	0.720996i	$0.256315\pi$
$242$	−174.052	−0.0462334
$243$	0	0
$244$	1914.63	0.502343
$245$	0	0
$246$	0	0
$247$	4163.32	1.07249
$248$	6246.13	1.59931
$249$	0	0
$250$	0	0
$251$	3827.69	0.962556	0.481278	−	0.876568i	$-0.340173\pi$
$251$	3827.69	0.962556	0.481278	+	0.876568i	$0.340173\pi$
$252$	0	0
$253$	354.708	0.0881436
$254$	1679.93	0.414993
$255$	0	0
$256$	−2865.65	−0.699621
$257$	−2671.37	−0.648387	−0.324193	−	0.945991i	$-0.605093\pi$
$257$	−2671.37	−0.648387	−0.324193	+	0.945991i	$0.605093\pi$
$258$	0	0
$259$	11432.4	2.74276
$260$	0	0
$261$	0	0
$262$	2229.52	0.525726
$263$	6659.10	1.56128	0.780642	−	0.624979i	$-0.214892\pi$
$263$	6659.10	1.56128	0.780642	+	0.624979i	$0.214892\pi$
$264$	0	0
$265$	0	0
$266$	−4076.98	−0.939758
$267$	0	0
$268$	2622.83	0.597816
$269$	−8473.05	−1.92049	−0.960244	−	0.279162i	$-0.909943\pi$
$269$	−8473.05	−1.92049	−0.960244	+	0.279162i	$0.909943\pi$
$270$	0	0
$271$	−2643.21	−0.592486	−0.296243	−	0.955113i	$-0.595734\pi$
$271$	−2643.21	−0.592486	−0.296243	+	0.955113i	$0.595734\pi$
$272$	−753.334	−0.167932
$273$	0	0
$274$	3510.78	0.774066
$275$	0	0
$276$	0	0
$277$	386.962	0.0839361	0.0419680	−	0.999119i	$-0.486637\pi$
$277$	386.962	0.0839361	0.0419680	+	0.999119i	$0.486637\pi$
$278$	2677.57	0.577662
$279$	0	0
$280$	0	0
$281$	1106.42	0.234887	0.117444	−	0.993080i	$-0.462530\pi$
$281$	1106.42	0.234887	0.117444	+	0.993080i	$0.462530\pi$
$282$	0	0
$283$	−1165.71	−0.244855	−0.122428	−	0.992477i	$-0.539068\pi$
$283$	−1165.71	−0.244855	−0.122428	+	0.992477i	$0.539068\pi$
$284$	3960.31	0.827469
$285$	0	0
$286$	−721.771	−0.149228
$287$	12222.9	2.51392
$288$	0	0
$289$	−3276.51	−0.666905
$290$	0	0
$291$	0	0
$292$	−501.574	−0.100522
$293$	−3646.82	−0.727131	−0.363566	−	0.931569i	$-0.618441\pi$
$293$	−3646.82	−0.727131	−0.363566	+	0.931569i	$0.618441\pi$
$294$	0	0
$295$	0	0
$296$	7377.23	1.44862
$297$	0	0
$298$	1171.44	0.227716
$299$	1470.93	0.284502
$300$	0	0
$301$	10918.6	2.09083
$302$	−5273.96	−1.00491
$303$	0	0
$304$	1699.64	0.320662
$305$	0	0
$306$	0	0
$307$	816.066	0.151711	0.0758556	−	0.997119i	$-0.475831\pi$
$307$	816.066	0.151711	0.0758556	+	0.997119i	$0.475831\pi$
$308$	−2025.95	−0.374802
$309$	0	0
$310$	0	0
$311$	−3146.99	−0.573793	−0.286896	−	0.957962i	$-0.592624\pi$
$311$	−3146.99	−0.573793	−0.286896	+	0.957962i	$0.592624\pi$
$312$	0	0
$313$	5085.49	0.918367	0.459184	−	0.888341i	$-0.348142\pi$
$313$	5085.49	0.918367	0.459184	+	0.888341i	$0.348142\pi$
$314$	3843.48	0.690764
$315$	0	0
$316$	2443.42	0.434978
$317$	−4801.89	−0.850791	−0.425396	−	0.905007i	$-0.639865\pi$
$317$	−4801.89	−0.850791	−0.425396	+	0.905007i	$0.639865\pi$
$318$	0	0
$319$	−394.573	−0.0692534
$320$	0	0
$321$	0	0
$322$	−1440.42	−0.249291
$323$	−3692.20	−0.636035
$324$	0	0
$325$	0	0
$326$	−2486.39	−0.422419
$327$	0	0
$328$	7887.32	1.32776
$329$	−7147.24	−1.19769
$330$	0	0
$331$	−2597.31	−0.431302	−0.215651	−	0.976470i	$-0.569187\pi$
$331$	−2597.31	−0.431302	−0.215651	+	0.976470i	$0.569187\pi$
$332$	4955.95	0.819256
$333$	0	0
$334$	0.0940730	1.54115e−5 0
$335$	0	0
$336$	0	0
$337$	−2695.45	−0.435698	−0.217849	−	0.975982i	$-0.569904\pi$
$337$	−2695.45	−0.435698	−0.217849	+	0.975982i	$0.569904\pi$
$338$	167.182	0.0269038
$339$	0	0
$340$	0	0
$341$	3428.72	0.544503
$342$	0	0
$343$	8643.85	1.36071
$344$	7045.69	1.10430
$345$	0	0
$346$	2613.30	0.406046
$347$	−9239.79	−1.42945	−0.714723	−	0.699407i	$-0.753447\pi$
$347$	−9239.79	−1.42945	−0.714723	+	0.699407i	$0.753447\pi$
$348$	0	0
$349$	3857.67	0.591680	0.295840	−	0.955237i	$-0.404400\pi$
$349$	3857.67	0.591680	0.295840	+	0.955237i	$0.404400\pi$
$350$	0	0
$351$	0	0
$352$	−2058.07	−0.311635
$353$	−3378.21	−0.509360	−0.254680	−	0.967025i	$-0.581970\pi$
$353$	−3378.21	−0.509360	−0.254680	+	0.967025i	$0.581970\pi$
$354$	0	0
$355$	0	0
$356$	−4743.83	−0.706242
$357$	0	0
$358$	3425.28	0.505675
$359$	9892.96	1.45440	0.727201	−	0.686425i	$-0.240821\pi$
$359$	9892.96	1.45440	0.727201	+	0.686425i	$0.240821\pi$
$360$	0	0
$361$	1471.19	0.214491
$362$	−1834.89	−0.266408
$363$	0	0
$364$	−8401.33	−1.20975
$365$	0	0
$366$	0	0
$367$	−5295.58	−0.753208	−0.376604	−	0.926374i	$-0.622908\pi$
$367$	−5295.58	−0.753208	−0.376604	+	0.926374i	$0.622908\pi$
$368$	600.495	0.0850623
$369$	0	0
$370$	0	0
$371$	12635.9	1.76826
$372$	0	0
$373$	−7786.10	−1.08083	−0.540414	−	0.841399i	$-0.681732\pi$
$373$	−7786.10	−1.08083	−0.540414	+	0.841399i	$0.681732\pi$
$374$	640.094	0.0884986
$375$	0	0
$376$	−4612.04	−0.632574
$377$	−1636.24	−0.223530
$378$	0	0
$379$	−1940.17	−0.262955	−0.131477	−	0.991319i	$-0.541972\pi$
$379$	−1940.17	−0.262955	−0.131477	+	0.991319i	$0.541972\pi$
$380$	0	0
$381$	0	0
$382$	−5320.88	−0.712669
$383$	10805.6	1.44161	0.720807	−	0.693136i	$-0.243772\pi$
$383$	10805.6	1.44161	0.720807	+	0.693136i	$0.243772\pi$
$384$	0	0
$385$	0	0
$386$	1771.04	0.233533
$387$	0	0
$388$	4560.51	0.596714
$389$	9225.11	1.20239	0.601197	−	0.799101i	$-0.294691\pi$
$389$	9225.11	1.20239	0.601197	+	0.799101i	$0.294691\pi$
$390$	0	0
$391$	−1304.48	−0.168722
$392$	12451.1	1.60428
$393$	0	0
$394$	−1047.31	−0.133916
$395$	0	0
$396$	0	0
$397$	−13364.3	−1.68951	−0.844753	−	0.535156i	$-0.820253\pi$
$397$	−13364.3	−1.68951	−0.844753	+	0.535156i	$0.820253\pi$
$398$	2374.61	0.299066
$399$	0	0
$400$	0	0
$401$	6030.88	0.751042	0.375521	−	0.926814i	$-0.377464\pi$
$401$	6030.88	0.751042	0.375521	+	0.926814i	$0.377464\pi$
$402$	0	0
$403$	14218.4	1.75750
$404$	−8376.25	−1.03152
$405$	0	0
$406$	1602.31	0.195865
$407$	4049.61	0.493199
$408$	0	0
$409$	931.381	0.112601	0.0563005	−	0.998414i	$-0.482069\pi$
$409$	931.381	0.112601	0.0563005	+	0.998414i	$0.482069\pi$
$410$	0	0
$411$	0	0
$412$	144.823	0.0173178
$413$	11450.4	1.36425
$414$	0	0
$415$	0	0
$416$	−8534.55	−1.00587
$417$	0	0
$418$	−1444.16	−0.168986
$419$	13161.8	1.53460	0.767300	−	0.641289i	$-0.221600\pi$
$419$	13161.8	1.53460	0.767300	+	0.641289i	$0.221600\pi$
$420$	0	0
$421$	−1127.05	−0.130473	−0.0652365	−	0.997870i	$-0.520780\pi$
$421$	−1127.05	−0.130473	−0.0652365	+	0.997870i	$0.520780\pi$
$422$	−2282.83	−0.263332
$423$	0	0
$424$	8153.84	0.933929
$425$	0	0
$426$	0	0
$427$	−10025.0	−1.13617
$428$	−3156.14	−0.356444
$429$	0	0
$430$	0	0
$431$	4386.13	0.490191	0.245096	−	0.969499i	$-0.421181\pi$
$431$	4386.13	0.490191	0.245096	+	0.969499i	$0.421181\pi$
$432$	0	0
$433$	−10905.5	−1.21035	−0.605177	−	0.796091i	$-0.706898\pi$
$433$	−10905.5	−1.21035	−0.605177	+	0.796091i	$0.706898\pi$
$434$	−13923.6	−1.53998
$435$	0	0
$436$	7322.38	0.804308
$437$	2943.11	0.322169
$438$	0	0
$439$	−3900.94	−0.424104	−0.212052	−	0.977258i	$-0.568015\pi$
$439$	−3900.94	−0.424104	−0.212052	+	0.977258i	$0.568015\pi$
$440$	0	0
$441$	0	0
$442$	2654.38	0.285647
$443$	−15537.5	−1.66638	−0.833192	−	0.552985i	$-0.813489\pi$
$443$	−15537.5	−1.66638	−0.833192	+	0.552985i	$0.813489\pi$
$444$	0	0
$445$	0	0
$446$	−2874.87	−0.305222
$447$	0	0
$448$	3731.22	0.393490
$449$	3464.47	0.364139	0.182070	−	0.983286i	$-0.441720\pi$
$449$	3464.47	0.364139	0.182070	+	0.983286i	$0.441720\pi$
$450$	0	0
$451$	4329.62	0.452049
$452$	7736.54	0.805080
$453$	0	0
$454$	−6058.74	−0.626323
$455$	0	0
$456$	0	0
$457$	14410.6	1.47506	0.737528	−	0.675317i	$-0.235993\pi$
$457$	14410.6	1.47506	0.737528	+	0.675317i	$0.235993\pi$
$458$	−1973.29	−0.201323
$459$	0	0
$460$	0	0
$461$	3219.12	0.325226	0.162613	−	0.986690i	$-0.448008\pi$
$461$	3219.12	0.325226	0.162613	+	0.986690i	$0.448008\pi$
$462$	0	0
$463$	1338.77	0.134380	0.0671899	−	0.997740i	$-0.478597\pi$
$463$	1338.77	0.134380	0.0671899	+	0.997740i	$0.478597\pi$
$464$	−667.982	−0.0668325
$465$	0	0
$466$	2471.79	0.245716
$467$	−12221.6	−1.21102	−0.605512	−	0.795836i	$-0.707032\pi$
$467$	−12221.6	−1.21102	−0.605512	+	0.795836i	$0.707032\pi$
$468$	0	0
$469$	−13733.1	−1.35210
$470$	0	0
$471$	0	0
$472$	7388.82	0.720547
$473$	3867.62	0.375969
$474$	0	0
$475$	0	0
$476$	7450.63	0.717435
$477$	0	0
$478$	−2581.68	−0.247036
$479$	20290.0	1.93544	0.967718	−	0.252036i	$-0.0811001\pi$
$479$	20290.0	1.93544	0.967718	+	0.252036i	$0.0811001\pi$
$480$	0	0
$481$	16793.2	1.59190
$482$	−7458.38	−0.704813
$483$	0	0
$484$	−717.635	−0.0673962
$485$	0	0
$486$	0	0
$487$	−17360.7	−1.61538	−0.807688	−	0.589610i	$-0.799282\pi$
$487$	−17360.7	−1.61538	−0.807688	+	0.589610i	$0.799282\pi$
$488$	−6469.03	−0.600080
$489$	0	0
$490$	0	0
$491$	−5129.01	−0.471424	−0.235712	−	0.971823i	$-0.575742\pi$
$491$	−5129.01	−0.471424	−0.235712	+	0.971823i	$0.575742\pi$
$492$	0	0
$493$	1451.08	0.132563
$494$	−5988.72	−0.545436
$495$	0	0
$496$	5804.56	0.525469
$497$	−20736.1	−1.87152
$498$	0	0
$499$	13009.6	1.16712	0.583558	−	0.812072i	$-0.301660\pi$
$499$	13009.6	1.16712	0.583558	+	0.812072i	$0.301660\pi$
$500$	0	0
$501$	0	0
$502$	−5505.93	−0.489525
$503$	3972.02	0.352095	0.176047	−	0.984382i	$-0.443669\pi$
$503$	3972.02	0.352095	0.176047	+	0.984382i	$0.443669\pi$
$504$	0	0
$505$	0	0
$506$	−510.229	−0.0448270
$507$	0	0
$508$	6926.54	0.604952
$509$	−11174.1	−0.973048	−0.486524	−	0.873667i	$-0.661735\pi$
$509$	−11174.1	−0.973048	−0.486524	+	0.873667i	$0.661735\pi$
$510$	0	0
$511$	2626.24	0.227354
$512$	−6469.49	−0.558426
$513$	0	0
$514$	3842.62	0.329749
$515$	0	0
$516$	0	0
$517$	−2531.71	−0.215366
$518$	−16444.9	−1.39488
$519$	0	0
$520$	0	0
$521$	18614.6	1.56530	0.782648	−	0.622464i	$-0.213868\pi$
$521$	18614.6	1.56530	0.782648	+	0.622464i	$0.213868\pi$
$522$	0	0
$523$	−3792.30	−0.317066	−0.158533	−	0.987354i	$-0.550676\pi$
$523$	−3792.30	−0.317066	−0.158533	+	0.987354i	$0.550676\pi$
$524$	9192.54	0.766370
$525$	0	0
$526$	−9578.76	−0.794019
$527$	−12609.5	−1.04227
$528$	0	0
$529$	−11127.2	−0.914538
$530$	0	0
$531$	0	0
$532$	−16809.8	−1.36992
$533$	17954.4	1.45908
$534$	0	0
$535$	0	0
$536$	−8861.83	−0.714128
$537$	0	0
$538$	12188.0	0.976698
$539$	6834.84	0.546192
$540$	0	0
$541$	5187.61	0.412260	0.206130	−	0.978525i	$-0.433913\pi$
$541$	5187.61	0.412260	0.206130	+	0.978525i	$0.433913\pi$
$542$	3802.12	0.301319
$543$	0	0
$544$	7568.77	0.596523
$545$	0	0
$546$	0	0
$547$	15642.1	1.22268	0.611342	−	0.791366i	$-0.290630\pi$
$547$	15642.1	1.22268	0.611342	+	0.791366i	$0.290630\pi$
$548$	14475.3	1.12839
$549$	0	0
$550$	0	0
$551$	−3273.87	−0.253125
$552$	0	0
$553$	−12793.7	−0.983804
$554$	−556.624	−0.0426872
$555$	0	0
$556$	11039.9	0.842080
$557$	15798.8	1.20182	0.600912	−	0.799315i	$-0.294804\pi$
$557$	15798.8	1.20182	0.600912	+	0.799315i	$0.294804\pi$
$558$	0	0
$559$	16038.5	1.21352
$560$	0	0
$561$	0	0
$562$	−1591.52	−0.119456
$563$	5028.68	0.376436	0.188218	−	0.982127i	$-0.439729\pi$
$563$	5028.68	0.376436	0.188218	+	0.982127i	$0.439729\pi$
$564$	0	0
$565$	0	0
$566$	1676.81	0.124526
$567$	0	0
$568$	−13380.8	−0.988463
$569$	−24194.1	−1.78255	−0.891273	−	0.453467i	$-0.850187\pi$
$569$	−24194.1	−1.78255	−0.891273	+	0.453467i	$0.850187\pi$
$570$	0	0
$571$	−8184.61	−0.599852	−0.299926	−	0.953963i	$-0.596962\pi$
$571$	−8184.61	−0.599852	−0.299926	+	0.953963i	$0.596962\pi$
$572$	−2975.94	−0.217535
$573$	0	0
$574$	−17582.0	−1.27850
$575$	0	0
$576$	0	0
$577$	−550.908	−0.0397480	−0.0198740	−	0.999802i	$-0.506327\pi$
$577$	−550.908	−0.0397480	−0.0198740	+	0.999802i	$0.506327\pi$
$578$	4713.08	0.339167
$579$	0	0
$580$	0	0
$581$	−25949.3	−1.85294
$582$	0	0
$583$	4475.93	0.317966
$584$	1694.68	0.120080
$585$	0	0
$586$	5245.76	0.369796
$587$	3080.03	0.216570	0.108285	−	0.994120i	$-0.465464\pi$
$587$	3080.03	0.216570	0.108285	+	0.994120i	$0.465464\pi$
$588$	0	0
$589$	28449.0	1.99019
$590$	0	0
$591$	0	0
$592$	6855.69	0.475958
$593$	5257.89	0.364108	0.182054	−	0.983289i	$-0.441726\pi$
$593$	5257.89	0.364108	0.182054	+	0.983289i	$0.441726\pi$
$594$	0	0
$595$	0	0
$596$	4829.96	0.331951
$597$	0	0
$598$	−2115.85	−0.144688
$599$	1181.80	0.0806125	0.0403062	−	0.999187i	$-0.487167\pi$
$599$	1181.80	0.0806125	0.0403062	+	0.999187i	$0.487167\pi$
$600$	0	0
$601$	−27879.8	−1.89225	−0.946123	−	0.323807i	$-0.895037\pi$
$601$	−27879.8	−1.89225	−0.946123	+	0.323807i	$0.895037\pi$
$602$	−15705.9	−1.06333
$603$	0	0
$604$	−21745.1	−1.46489
$605$	0	0
$606$	0	0
$607$	4500.25	0.300922	0.150461	−	0.988616i	$-0.451924\pi$
$607$	4500.25	0.300922	0.150461	+	0.988616i	$0.451924\pi$
$608$	−17076.4	−1.13904
$609$	0	0
$610$	0	0
$611$	−10498.7	−0.695139
$612$	0	0
$613$	−8963.16	−0.590569	−0.295284	−	0.955409i	$-0.595414\pi$
$613$	−8963.16	−0.590569	−0.295284	+	0.955409i	$0.595414\pi$
$614$	−1173.87	−0.0771555
$615$	0	0
$616$	6845.14	0.447725
$617$	−9394.42	−0.612974	−0.306487	−	0.951875i	$-0.599154\pi$
$617$	−9394.42	−0.612974	−0.306487	+	0.951875i	$0.599154\pi$
$618$	0	0
$619$	4816.38	0.312741	0.156371	−	0.987698i	$-0.450021\pi$
$619$	4816.38	0.312741	0.156371	+	0.987698i	$0.450021\pi$
$620$	0	0
$621$	0	0
$622$	4526.78	0.291813
$623$	24838.6	1.59733
$624$	0	0
$625$	0	0
$626$	−7315.21	−0.467052
$627$	0	0
$628$	15847.1	1.00695
$629$	−14892.9	−0.944066
$630$	0	0
$631$	−24847.7	−1.56763	−0.783814	−	0.620996i	$-0.786728\pi$
$631$	−24847.7	−1.56763	−0.783814	+	0.620996i	$0.786728\pi$
$632$	−8255.65	−0.519608
$633$	0	0
$634$	6907.26	0.432685
$635$	0	0
$636$	0	0
$637$	28343.2	1.76295
$638$	567.572	0.0352201
$639$	0	0
$640$	0	0
$641$	30497.0	1.87919	0.939594	−	0.342290i	$-0.111202\pi$
$641$	30497.0	1.87919	0.939594	+	0.342290i	$0.111202\pi$
$642$	0	0
$643$	−5407.75	−0.331665	−0.165833	−	0.986154i	$-0.553031\pi$
$643$	−5407.75	−0.331665	−0.165833	+	0.986154i	$0.553031\pi$
$644$	−5939.01	−0.363400
$645$	0	0
$646$	5311.03	0.323467
$647$	22118.8	1.34402	0.672011	−	0.740542i	$-0.265431\pi$
$647$	22118.8	1.34402	0.672011	+	0.740542i	$0.265431\pi$
$648$	0	0
$649$	4055.98	0.245318
$650$	0	0
$651$	0	0
$652$	−10251.7	−0.615776
$653$	−12862.4	−0.770817	−0.385409	−	0.922746i	$-0.625940\pi$
$653$	−12862.4	−0.770817	−0.385409	+	0.922746i	$0.625940\pi$
$654$	0	0
$655$	0	0
$656$	7329.73	0.436247
$657$	0	0
$658$	10280.9	0.609106
$659$	−13216.0	−0.781218	−0.390609	−	0.920557i	$-0.627736\pi$
$659$	−13216.0	−0.781218	−0.390609	+	0.920557i	$0.627736\pi$
$660$	0	0
$661$	32098.3	1.88878	0.944388	−	0.328833i	$-0.106655\pi$
$661$	32098.3	1.88878	0.944388	+	0.328833i	$0.106655\pi$
$662$	3736.09	0.219347
$663$	0	0
$664$	−16744.8	−0.978652
$665$	0	0
$666$	0	0
$667$	−1156.68	−0.0671466
$668$	0.387873	2.24659e−5 0
$669$	0	0
$670$	0	0
$671$	−3551.07	−0.204303
$672$	0	0
$673$	19956.4	1.14304	0.571518	−	0.820590i	$-0.306355\pi$
$673$	19956.4	1.14304	0.571518	+	0.820590i	$0.306355\pi$
$674$	3877.26	0.221582
$675$	0	0
$676$	689.307	0.0392186
$677$	−1483.58	−0.0842223	−0.0421111	−	0.999113i	$-0.513408\pi$
$677$	−1483.58	−0.0842223	−0.0421111	+	0.999113i	$0.513408\pi$
$678$	0	0
$679$	−23878.8	−1.34961
$680$	0	0
$681$	0	0
$682$	−4932.03	−0.276917
$683$	23228.2	1.30132	0.650659	−	0.759370i	$-0.274493\pi$
$683$	23228.2	1.30132	0.650659	+	0.759370i	$0.274493\pi$
$684$	0	0
$685$	0	0
$686$	−12433.7	−0.692015
$687$	0	0
$688$	6547.60	0.362827
$689$	18561.1	1.02630
$690$	0	0
$691$	−7333.20	−0.403717	−0.201858	−	0.979415i	$-0.564698\pi$
$691$	−7333.20	−0.403717	−0.201858	+	0.979415i	$0.564698\pi$
$692$	10774.9	0.591909
$693$	0	0
$694$	13290.9	0.726971
$695$	0	0
$696$	0	0
$697$	−15922.6	−0.865298
$698$	−5549.06	−0.300909
$699$	0	0
$700$	0	0
$701$	19481.6	1.04966	0.524829	−	0.851208i	$-0.324129\pi$
$701$	19481.6	1.04966	0.524829	+	0.851208i	$0.324129\pi$
$702$	0	0
$703$	33600.7	1.80267
$704$	1321.68	0.0707566
$705$	0	0
$706$	4859.38	0.259044
$707$	43858.0	2.33303
$708$	0	0
$709$	14807.5	0.784353	0.392177	−	0.919890i	$-0.371722\pi$
$709$	14807.5	0.784353	0.392177	+	0.919890i	$0.371722\pi$
$710$	0	0
$711$	0	0
$712$	16028.1	0.843650
$713$	10051.2	0.527939
$714$	0	0
$715$	0	0
$716$	14122.8	0.737142
$717$	0	0
$718$	−14230.5	−0.739662
$719$	−22891.1	−1.18734	−0.593669	−	0.804709i	$-0.702321\pi$
$719$	−22891.1	−1.18734	−0.593669	+	0.804709i	$0.702321\pi$
$720$	0	0
$721$	−758.292	−0.0391682
$722$	−2116.23	−0.109083
$723$	0	0
$724$	−7565.45	−0.388353
$725$	0	0
$726$	0	0
$727$	−24318.7	−1.24062	−0.620309	−	0.784357i	$-0.712993\pi$
$727$	−24318.7	−1.24062	−0.620309	+	0.784357i	$0.712993\pi$
$728$	28385.9	1.44512
$729$	0	0
$730$	0	0
$731$	−14223.6	−0.719669
$732$	0	0
$733$	34939.0	1.76057	0.880287	−	0.474441i	$-0.157350\pi$
$733$	34939.0	1.76057	0.880287	+	0.474441i	$0.157350\pi$
$734$	7617.42	0.383057
$735$	0	0
$736$	−6033.19	−0.302155
$737$	−4864.56	−0.243132
$738$	0	0
$739$	−12663.1	−0.630337	−0.315168	−	0.949036i	$-0.602061\pi$
$739$	−12663.1	−0.630337	−0.315168	+	0.949036i	$0.602061\pi$
$740$	0	0
$741$	0	0
$742$	−18176.1	−0.899281
$743$	−1711.32	−0.0844985	−0.0422492	−	0.999107i	$-0.513452\pi$
$743$	−1711.32	−0.0844985	−0.0422492	+	0.999107i	$0.513452\pi$
$744$	0	0
$745$	0	0
$746$	11199.9	0.549675
$747$	0	0
$748$	2639.18	0.129008
$749$	16525.5	0.806182
$750$	0	0
$751$	−5165.32	−0.250979	−0.125489	−	0.992095i	$-0.540050\pi$
$751$	−5165.32	−0.250979	−0.125489	+	0.992095i	$0.540050\pi$
$752$	−4285.99	−0.207838
$753$	0	0
$754$	2353.65	0.113680
$755$	0	0
$756$	0	0
$757$	40684.0	1.95335	0.976675	−	0.214722i	$-0.0688847\pi$
$757$	40684.0	1.95335	0.976675	+	0.214722i	$0.0688847\pi$
$758$	2790.83	0.133730
$759$	0	0
$760$	0	0
$761$	−26564.1	−1.26537	−0.632687	−	0.774408i	$-0.718048\pi$
$761$	−26564.1	−1.26537	−0.632687	+	0.774408i	$0.718048\pi$
$762$	0	0
$763$	−38339.9	−1.81913
$764$	−21938.5	−1.03889
$765$	0	0
$766$	−15543.2	−0.733158
$767$	16819.6	0.791813
$768$	0	0
$769$	24381.9	1.14335	0.571674	−	0.820481i	$-0.306294\pi$
$769$	24381.9	1.14335	0.571674	+	0.820481i	$0.306294\pi$
$770$	0	0
$771$	0	0
$772$	7302.19	0.340429
$773$	−18945.0	−0.881508	−0.440754	−	0.897628i	$-0.645289\pi$
$773$	−18945.0	−0.881508	−0.440754	+	0.897628i	$0.645289\pi$
$774$	0	0
$775$	0	0
$776$	−15408.8	−0.712812
$777$	0	0
$778$	−13269.8	−0.611499
$779$	35924.0	1.65226
$780$	0	0
$781$	−7345.20	−0.336532
$782$	1876.42	0.0858064
$783$	0	0
$784$	11570.9	0.527099
$785$	0	0
$786$	0	0
$787$	7610.70	0.344717	0.172359	−	0.985034i	$-0.444861\pi$
$787$	7610.70	0.344717	0.172359	+	0.985034i	$0.444861\pi$
$788$	−4318.19	−0.195215
$789$	0	0
$790$	0	0
$791$	−40508.4	−1.82088
$792$	0	0
$793$	−14725.8	−0.659432
$794$	19223.8	0.859229
$795$	0	0
$796$	9790.75	0.435960
$797$	3732.50	0.165887	0.0829435	−	0.996554i	$-0.473568\pi$
$797$	3732.50	0.165887	0.0829435	+	0.996554i	$0.473568\pi$
$798$	0	0
$799$	9310.61	0.412247
$800$	0	0
$801$	0	0
$802$	−8675.11	−0.381956
$803$	930.271	0.0408824
$804$	0	0
$805$	0	0
$806$	−20452.5	−0.893806
$807$	0	0
$808$	28301.1	1.23221
$809$	27621.8	1.20041	0.600204	−	0.799847i	$-0.295086\pi$
$809$	27621.8	1.20041	0.600204	+	0.799847i	$0.295086\pi$
$810$	0	0
$811$	−29996.0	−1.29877	−0.649384	−	0.760461i	$-0.724973\pi$
$811$	−29996.0	−1.29877	−0.649384	+	0.760461i	$0.724973\pi$
$812$	6606.48	0.285520
$813$	0	0
$814$	−5825.16	−0.250825
$815$	0	0
$816$	0	0
$817$	32090.7	1.37419
$818$	−1339.74	−0.0572653
$819$	0	0
$820$	0	0
$821$	−18075.1	−0.768362	−0.384181	−	0.923258i	$-0.625516\pi$
$821$	−18075.1	−0.768362	−0.384181	+	0.923258i	$0.625516\pi$
$822$	0	0
$823$	26723.7	1.13187	0.565935	−	0.824450i	$-0.308515\pi$
$823$	26723.7	1.13187	0.565935	+	0.824450i	$0.308515\pi$
$824$	−489.318	−0.0206871
$825$	0	0
$826$	−16470.8	−0.693816
$827$	36788.1	1.54685	0.773427	−	0.633885i	$-0.218541\pi$
$827$	36788.1	1.54685	0.773427	+	0.633885i	$0.218541\pi$
$828$	0	0
$829$	−1260.15	−0.0527948	−0.0263974	−	0.999652i	$-0.508404\pi$
$829$	−1260.15	−0.0527948	−0.0263974	+	0.999652i	$0.508404\pi$
$830$	0	0
$831$	0	0
$832$	5480.82	0.228381
$833$	−25135.8	−1.04550
$834$	0	0
$835$	0	0
$836$	−5954.41	−0.246337
$837$	0	0
$838$	−18932.6	−0.780448
$839$	4023.16	0.165548	0.0827742	−	0.996568i	$-0.473622\pi$
$839$	4023.16	0.165548	0.0827742	+	0.996568i	$0.473622\pi$
$840$	0	0
$841$	−23102.3	−0.947244
$842$	1621.20	0.0663543
$843$	0	0
$844$	−9412.34	−0.383870
$845$	0	0
$846$	0	0
$847$	3757.53	0.152432
$848$	7577.41	0.306851
$849$	0	0
$850$	0	0
$851$	11871.3	0.478195
$852$	0	0
$853$	−40707.7	−1.63400	−0.817001	−	0.576636i	$-0.804365\pi$
$853$	−40707.7	−1.63400	−0.817001	+	0.576636i	$0.804365\pi$
$854$	14420.4	0.577818
$855$	0	0
$856$	10663.8	0.425794
$857$	24321.4	0.969431	0.484715	−	0.874672i	$-0.338923\pi$
$857$	24321.4	0.969431	0.484715	+	0.874672i	$0.338923\pi$
$858$	0	0
$859$	16912.7	0.671776	0.335888	−	0.941902i	$-0.390964\pi$
$859$	16912.7	0.671776	0.335888	+	0.941902i	$0.390964\pi$
$860$	0	0
$861$	0	0
$862$	−6309.22	−0.249295
$863$	−25793.8	−1.01742	−0.508708	−	0.860939i	$-0.669877\pi$
$863$	−25793.8	−1.01742	−0.508708	+	0.860939i	$0.669877\pi$
$864$	0	0
$865$	0	0
$866$	15686.9	0.615547
$867$	0	0
$868$	−57408.3	−2.24489
$869$	−4531.81	−0.176906
$870$	0	0
$871$	−20172.7	−0.784759
$872$	−24740.4	−0.960796
$873$	0	0
$874$	−4233.51	−0.163845
$875$	0	0
$876$	0	0
$877$	−20143.8	−0.775607	−0.387803	−	0.921742i	$-0.626766\pi$
$877$	−20143.8	−0.775607	−0.387803	+	0.921742i	$0.626766\pi$
$878$	5611.30	0.215686
$879$	0	0
$880$	0	0
$881$	18231.3	0.697196	0.348598	−	0.937272i	$-0.386658\pi$
$881$	18231.3	0.697196	0.348598	+	0.937272i	$0.386658\pi$
$882$	0	0
$883$	34487.5	1.31438	0.657189	−	0.753725i	$-0.271745\pi$
$883$	34487.5	1.31438	0.657189	+	0.753725i	$0.271745\pi$
$884$	10944.3	0.416399
$885$	0	0
$886$	22349.8	0.847469
$887$	−16039.9	−0.607180	−0.303590	−	0.952803i	$-0.598185\pi$
$887$	−16039.9	−0.607180	−0.303590	+	0.952803i	$0.598185\pi$
$888$	0	0
$889$	−36267.3	−1.36824
$890$	0	0
$891$	0	0
$892$	−11853.4	−0.444934
$893$	−21006.2	−0.787175
$894$	0	0
$895$	0	0
$896$	41113.8	1.53294
$897$	0	0
$898$	−4983.46	−0.185189
$899$	−11180.8	−0.414795
$900$	0	0
$901$	−16460.7	−0.608640
$902$	−6227.94	−0.229898
$903$	0	0
$904$	−26139.7	−0.961718
$905$	0	0
$906$	0	0
$907$	23029.2	0.843077	0.421539	−	0.906810i	$-0.361490\pi$
$907$	23029.2	0.843077	0.421539	+	0.906810i	$0.361490\pi$
$908$	−24980.8	−0.913015
$909$	0	0
$910$	0	0
$911$	9206.00	0.334806	0.167403	−	0.985889i	$-0.446462\pi$
$911$	9206.00	0.334806	0.167403	+	0.985889i	$0.446462\pi$
$912$	0	0
$913$	−9191.81	−0.333192
$914$	−20728.9	−0.750166
$915$	0	0
$916$	−8136.10	−0.293476
$917$	−48132.0	−1.73333
$918$	0	0
$919$	13329.5	0.478456	0.239228	−	0.970963i	$-0.423106\pi$
$919$	13329.5	0.478456	0.239228	+	0.970963i	$0.423106\pi$
$920$	0	0
$921$	0	0
$922$	−4630.53	−0.165400
$923$	−30459.6	−1.08623
$924$	0	0
$925$	0	0
$926$	−1925.75	−0.0683412
$927$	0	0
$928$	6711.24	0.237400
$929$	−1819.10	−0.0642442	−0.0321221	−	0.999484i	$-0.510227\pi$
$929$	−1819.10	−0.0642442	−0.0321221	+	0.999484i	$0.510227\pi$
$930$	0	0
$931$	56710.5	1.99636
$932$	10191.5	0.358189
$933$	0	0
$934$	17580.1	0.615888
$935$	0	0
$936$	0	0
$937$	−4012.08	−0.139882	−0.0699408	−	0.997551i	$-0.522281\pi$
$937$	−4012.08	−0.139882	−0.0699408	+	0.997551i	$0.522281\pi$
$938$	19754.3	0.687635
$939$	0	0
$940$	0	0
$941$	−8444.53	−0.292544	−0.146272	−	0.989244i	$-0.546727\pi$
$941$	−8444.53	−0.292544	−0.146272	+	0.989244i	$0.546727\pi$
$942$	0	0
$943$	12692.2	0.438297
$944$	6866.47	0.236742
$945$	0	0
$946$	−5563.37	−0.191206
$947$	45316.0	1.55499	0.777493	−	0.628892i	$-0.216491\pi$
$947$	45316.0	1.55499	0.777493	+	0.628892i	$0.216491\pi$
$948$	0	0
$949$	3857.71	0.131956
$950$	0	0
$951$	0	0
$952$	−25173.7	−0.857020
$953$	96.7143	0.00328739	0.00164370	−	0.999999i	$-0.499477\pi$
$953$	96.7143	0.00328739	0.00164370	+	0.999999i	$0.499477\pi$
$954$	0	0
$955$	0	0
$956$	−10644.6	−0.360114
$957$	0	0
$958$	−29186.1	−0.984300
$959$	−75792.7	−2.55211
$960$	0	0
$961$	67366.9	2.26132
$962$	−24156.1	−0.809590
$963$	0	0
$964$	−30751.7	−1.02743
$965$	0	0
$966$	0	0
$967$	−7666.63	−0.254956	−0.127478	−	0.991841i	$-0.540688\pi$
$967$	−7666.63	−0.254956	−0.127478	+	0.991841i	$0.540688\pi$
$968$	2424.70	0.0805090
$969$	0	0
$970$	0	0
$971$	5221.18	0.172560	0.0862799	−	0.996271i	$-0.472502\pi$
$971$	5221.18	0.172560	0.0862799	+	0.996271i	$0.472502\pi$
$972$	0	0
$973$	−57804.9	−1.90456
$974$	24972.4	0.821529
$975$	0	0
$976$	−6011.70	−0.197162
$977$	−45769.9	−1.49878	−0.749390	−	0.662129i	$-0.769653\pi$
$977$	−45769.9	−1.49878	−0.749390	+	0.662129i	$0.769653\pi$
$978$	0	0
$979$	8798.39	0.287229
$980$	0	0
$981$	0	0
$982$	7377.81	0.239751
$983$	−14777.8	−0.479488	−0.239744	−	0.970836i	$-0.577064\pi$
$983$	−14777.8	−0.479488	−0.239744	+	0.970836i	$0.577064\pi$
$984$	0	0
$985$	0	0
$986$	−2087.30	−0.0674171
$987$	0	0
$988$	−24692.1	−0.795103
$989$	11337.8	0.364532
$990$	0	0
$991$	−16783.9	−0.538001	−0.269001	−	0.963140i	$-0.586693\pi$
$991$	−16783.9	−0.538001	−0.269001	+	0.963140i	$0.586693\pi$
$992$	−58318.6	−1.86655
$993$	0	0
$994$	29827.9	0.951793
$995$	0	0
$996$	0	0
$997$	8720.36	0.277008	0.138504	−	0.990362i	$-0.455771\pi$
$997$	8720.36	0.277008	0.138504	+	0.990362i	$0.455771\pi$
$998$	−18713.7	−0.593557
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.l.1.2		2
3.2	odd	2		275.4.a.c.1.1		2
5.4	even	2		495.4.a.e.1.1		2
15.2	even	4		275.4.b.b.199.3		4
15.8	even	4		275.4.b.b.199.2		4
15.14	odd	2		55.4.a.b.1.2	✓	2
60.59	even	2		880.4.a.r.1.1		2
165.164	even	2		605.4.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.a.b.1.2	✓	2	15.14	odd	2
275.4.a.c.1.1		2	3.2	odd	2
275.4.b.b.199.2		4	15.8	even	4
275.4.b.b.199.3		4	15.2	even	4
495.4.a.e.1.1		2	5.4	even	2
605.4.a.g.1.1		2	165.164	even	2
880.4.a.r.1.1		2	60.59	even	2
2475.4.a.l.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43845 q^{2} -5.93087 q^{4} +31.0540 q^{7} +20.0388 q^{8} +11.0000 q^{11} +45.6155 q^{13} -44.6695 q^{14} +18.6222 q^{16} -40.4536 q^{17} +91.2699 q^{19} -15.8229 q^{22} +32.2462 q^{23} -65.6155 q^{26} -184.177 q^{28} -35.8702 q^{29} +311.702 q^{31} -187.098 q^{32} +58.1904 q^{34} +368.147 q^{37} -131.287 q^{38} +393.602 q^{41} +351.602 q^{43} -65.2396 q^{44} -46.3845 q^{46} -230.155 q^{47} +621.349 q^{49} -270.540 q^{52} +406.902 q^{53} +622.285 q^{56} +51.5975 q^{58} +368.725 q^{59} -322.825 q^{61} -448.366 q^{62} +120.153 q^{64} -442.233 q^{67} +239.925 q^{68} -667.745 q^{71} +84.5701 q^{73} -529.560 q^{74} -541.310 q^{76} +341.594 q^{77} -411.983 q^{79} -566.176 q^{82} -835.619 q^{83} -505.761 q^{86} +220.427 q^{88} +799.853 q^{89} +1416.54 q^{91} -191.248 q^{92} +331.066 q^{94} -768.945 q^{97} -893.778 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 7 q^{2} + 17 q^{4} + 25 q^{7} - 63 q^{8} + 22 q^{11} + 50 q^{13} - 11 q^{14} + 297 q^{16} - 151 q^{17} - 3 q^{19} - 77 q^{22} + 48 q^{23} - 90 q^{26} - 323 q^{28} + 221 q^{29} + 141 q^{31} - 1071 q^{32}+ \cdots + 810 q^{98}+O(q^{100}) \) 2 * q - 7 * q^2 + 17 * q^4 + 25 * q^7 - 63 * q^8 + 22 * q^11 + 50 * q^13 - 11 * q^14 + 297 * q^16 - 151 * q^17 - 3 * q^19 - 77 * q^22 + 48 * q^23 - 90 * q^26 - 323 * q^28 + 221 * q^29 + 141 * q^31 - 1071 * q^32 + 673 * q^34 + 559 * q^37 + 393 * q^38 + 144 * q^41 + 60 * q^43 + 187 * q^44 - 134 * q^46 - 48 * q^47 + 315 * q^49 - 170 * q^52 + 117 * q^53 + 1125 * q^56 - 1377 * q^58 + 86 * q^59 - 155 * q^61 + 501 * q^62 + 2809 * q^64 - 266 * q^67 - 2295 * q^68 - 1587 * q^71 - 70 * q^73 - 1591 * q^74 - 2703 * q^76 + 275 * q^77 - 1294 * q^79 + 822 * q^82 - 558 * q^83 + 1116 * q^86 - 693 * q^88 + 1777 * q^89 + 1390 * q^91 + 170 * q^92 - 682 * q^94 + 334 * q^97 + 810 * q^98

Introduction

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