LMFDB - Embedded newform 675.4.a.v.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-1,0,19,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.183945.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 3x + 18 $ x^4 - x^3 - 12x^2 + 3x + 18
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.39127$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53008 q^{2} -1.59867 q^{4} +27.8841 q^{7} +24.2855 q^{8} +35.8155 q^{11} -27.3694 q^{13} -70.5492 q^{14} -48.6549 q^{16} +93.6094 q^{17} +135.589 q^{19} -90.6163 q^{22} -0.407389 q^{23} +69.2470 q^{26} -44.5775 q^{28} +194.946 q^{29} -96.7790 q^{31} -71.1826 q^{32} -236.840 q^{34} -186.968 q^{37} -343.051 q^{38} -53.6700 q^{41} -519.172 q^{43} -57.2573 q^{44} +1.03073 q^{46} +190.431 q^{47} +434.524 q^{49} +43.7547 q^{52} -533.255 q^{53} +677.178 q^{56} -493.229 q^{58} +472.800 q^{59} -327.057 q^{61} +244.859 q^{62} +569.337 q^{64} -78.0010 q^{67} -149.651 q^{68} +707.467 q^{71} +344.812 q^{73} +473.044 q^{74} -216.761 q^{76} +998.685 q^{77} +98.3522 q^{79} +135.790 q^{82} +1248.37 q^{83} +1313.55 q^{86} +869.796 q^{88} -448.920 q^{89} -763.173 q^{91} +0.651281 q^{92} -481.808 q^{94} +472.081 q^{97} -1099.38 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 19 q^{4} + 4 q^{7} + 15 q^{8} + 52 q^{11} - 2 q^{13} + 138 q^{14} - 5 q^{16} - 64 q^{17} - 46 q^{19} + 87 q^{22} + 90 q^{23} + 469 q^{26} - 110 q^{28} + 470 q^{29} - 262 q^{31} + 199 q^{32}+ \cdots - 4301 q^{98}+O(q^{100}) $ 4 * q - q^2 + 19 * q^4 + 4 * q^7 + 15 * q^8 + 52 * q^11 - 2 * q^13 + 138 * q^14 - 5 * q^16 - 64 * q^17 - 46 * q^19 + 87 * q^22 + 90 * q^23 + 469 * q^26 - 110 * q^28 + 470 * q^29 - 262 * q^31 + 199 * q^32 - 42 * q^34 - 542 * q^37 - 532 * q^38 + 698 * q^41 + 142 * q^43 + 419 * q^44 + 537 * q^46 + 542 * q^47 + 780 * q^49 + 409 * q^52 - 910 * q^53 + 2034 * q^56 - 576 * q^58 + 100 * q^59 + 74 * q^61 + 2406 * q^62 - 965 * q^64 + 928 * q^67 - 2810 * q^68 + 1622 * q^71 - 536 * q^73 - 253 * q^74 - 2068 * q^76 + 1932 * q^77 - 508 * q^79 - 1782 * q^82 - 1524 * q^83 + 3940 * q^86 + 2247 * q^88 + 756 * q^89 + 1120 * q^91 + 3645 * q^92 + 2847 * q^94 + 892 * q^97 - 4301 * 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 $

$2$	−2.53008	−0.894520	−0.447260	−	0.894404i	$-0.647600\pi$
$2$	−2.53008	−0.894520	−0.447260	+	0.894404i	$0.647600\pi$
$3$	0	0
$3$	0	0
$4$	−1.59867	−0.199834
$5$	0	0
$5$	0	0
$6$	0	0
$7$	27.8841	1.50560	0.752801	−	0.658249i	$-0.228702\pi$
$7$	27.8841	1.50560	0.752801	+	0.658249i	$0.228702\pi$
$8$	24.2855	1.07328
$9$	0	0
$10$	0	0
$11$	35.8155	0.981708	0.490854	−	0.871242i	$-0.336685\pi$
$11$	35.8155	0.981708	0.490854	+	0.871242i	$0.336685\pi$
$12$	0	0
$13$	−27.3694	−0.583917	−0.291958	−	0.956431i	$-0.594307\pi$
$13$	−27.3694	−0.583917	−0.291958	+	0.956431i	$0.594307\pi$
$14$	−70.5492	−1.34679
$15$	0	0
$16$	−48.6549	−0.760233
$17$	93.6094	1.33551	0.667753	−	0.744383i	$-0.267256\pi$
$17$	93.6094	1.33551	0.667753	+	0.744383i	$0.267256\pi$
$18$	0	0
$19$	135.589	1.63717	0.818583	−	0.574388i	$-0.194760\pi$
$19$	135.589	1.63717	0.818583	+	0.574388i	$0.194760\pi$
$20$	0	0
$21$	0	0
$22$	−90.6163	−0.878158
$23$	−0.407389	−0.00369332	−0.00184666	−	0.999998i	$-0.500588\pi$
$23$	−0.407389	−0.00369332	−0.00184666	+	0.999998i	$0.500588\pi$
$24$	0	0
$25$	0	0
$26$	69.2470	0.522325
$27$	0	0
$28$	−44.5775	−0.300870
$29$	194.946	1.24829	0.624147	−	0.781307i	$-0.285447\pi$
$29$	194.946	1.24829	0.624147	+	0.781307i	$0.285447\pi$
$30$	0	0
$31$	−96.7790	−0.560710	−0.280355	−	0.959896i	$-0.590452\pi$
$31$	−96.7790	−0.560710	−0.280355	+	0.959896i	$0.590452\pi$
$32$	−71.1826	−0.393232
$33$	0	0
$34$	−236.840	−1.19464
$35$	0	0
$36$	0	0
$37$	−186.968	−0.830737	−0.415369	−	0.909653i	$-0.636347\pi$
$37$	−186.968	−0.830737	−0.415369	+	0.909653i	$0.636347\pi$
$38$	−343.051	−1.46448
$39$	0	0
$40$	0	0
$41$	−53.6700	−0.204435	−0.102218	−	0.994762i	$-0.532594\pi$
$41$	−53.6700	−0.204435	−0.102218	+	0.994762i	$0.532594\pi$
$42$	0	0
$43$	−519.172	−1.84123	−0.920616	−	0.390470i	$-0.872313\pi$
$43$	−519.172	−1.84123	−0.920616	+	0.390470i	$0.872313\pi$
$44$	−57.2573	−0.196179
$45$	0	0
$46$	1.03073	0.00330375
$47$	190.431	0.591006	0.295503	−	0.955342i	$-0.404513\pi$
$47$	190.431	0.591006	0.295503	+	0.955342i	$0.404513\pi$
$48$	0	0
$49$	434.524	1.26683
$50$	0	0
$51$	0	0
$52$	43.7547	0.116686
$53$	−533.255	−1.38204	−0.691021	−	0.722834i	$-0.742839\pi$
$53$	−533.255	−1.38204	−0.691021	+	0.722834i	$0.742839\pi$
$54$	0	0
$55$	0	0
$56$	677.178	1.61592
$57$	0	0
$58$	−493.229	−1.11662
$59$	472.800	1.04328	0.521639	−	0.853167i	$-0.325321\pi$
$59$	472.800	1.04328	0.521639	+	0.853167i	$0.325321\pi$
$60$	0	0
$61$	−327.057	−0.686480	−0.343240	−	0.939248i	$-0.611525\pi$
$61$	−327.057	−0.686480	−0.343240	+	0.939248i	$0.611525\pi$
$62$	244.859	0.501567
$63$	0	0
$64$	569.337	1.11199
$65$	0	0
$66$	0	0
$67$	−78.0010	−0.142229	−0.0711145	−	0.997468i	$-0.522656\pi$
$67$	−78.0010	−0.142229	−0.0711145	+	0.997468i	$0.522656\pi$
$68$	−149.651	−0.266879
$69$	0	0
$70$	0	0
$71$	707.467	1.18255	0.591274	−	0.806471i	$-0.298625\pi$
$71$	707.467	1.18255	0.591274	+	0.806471i	$0.298625\pi$
$72$	0	0
$73$	344.812	0.552838	0.276419	−	0.961037i	$-0.410852\pi$
$73$	344.812	0.552838	0.276419	+	0.961037i	$0.410852\pi$
$74$	473.044	0.743111
$75$	0	0
$76$	−216.761	−0.327161
$77$	998.685	1.47806
$78$	0	0
$79$	98.3522	0.140069	0.0700347	−	0.997545i	$-0.477689\pi$
$79$	98.3522	0.140069	0.0700347	+	0.997545i	$0.477689\pi$
$80$	0	0
$81$	0	0
$82$	135.790	0.182872
$83$	1248.37	1.65092	0.825460	−	0.564461i	$-0.190916\pi$
$83$	1248.37	1.65092	0.825460	+	0.564461i	$0.190916\pi$
$84$	0	0
$85$	0	0
$86$	1313.55	1.64702
$87$	0	0
$88$	869.796	1.05364
$89$	−448.920	−0.534668	−0.267334	−	0.963604i	$-0.586143\pi$
$89$	−448.920	−0.534668	−0.267334	+	0.963604i	$0.586143\pi$
$90$	0	0
$91$	−763.173	−0.879145
$92$	0.651281	0.000738051 0
$93$	0	0
$94$	−481.808	−0.528667
$95$	0	0
$96$	0	0
$97$	472.081	0.494150	0.247075	−	0.968996i	$-0.420531\pi$
$97$	472.081	0.494150	0.247075	+	0.968996i	$0.420531\pi$
$98$	−1099.38	−1.13321
$99$	0	0
$100$	0	0
$101$	1474.17	1.45233	0.726166	−	0.687519i	$-0.241300\pi$
$101$	1474.17	1.45233	0.726166	+	0.687519i	$0.241300\pi$
$102$	0	0
$103$	−967.934	−0.925955	−0.462977	−	0.886370i	$-0.653219\pi$
$103$	−967.934	−0.925955	−0.462977	+	0.886370i	$0.653219\pi$
$104$	−664.679	−0.626703
$105$	0	0
$106$	1349.18	1.23626
$107$	1280.19	1.15664	0.578321	−	0.815809i	$-0.303708\pi$
$107$	1280.19	1.15664	0.578321	+	0.815809i	$0.303708\pi$
$108$	0	0
$109$	−1445.89	−1.27056	−0.635281	−	0.772281i	$-0.719116\pi$
$109$	−1445.89	−1.27056	−0.635281	+	0.772281i	$0.719116\pi$
$110$	0	0
$111$	0	0
$112$	−1356.70	−1.14461
$113$	−2311.88	−1.92463	−0.962315	−	0.271938i	$-0.912335\pi$
$113$	−2311.88	−1.92463	−0.962315	+	0.271938i	$0.912335\pi$
$114$	0	0
$115$	0	0
$116$	−311.654	−0.249451
$117$	0	0
$118$	−1196.22	−0.933232
$119$	2610.21	2.01074
$120$	0	0
$121$	−48.2474	−0.0362490
$122$	827.481	0.614071
$123$	0	0
$124$	154.718	0.112049
$125$	0	0
$126$	0	0
$127$	1483.99	1.03687	0.518435	−	0.855117i	$-0.326515\pi$
$127$	1483.99	1.03687	0.518435	+	0.855117i	$0.326515\pi$
$128$	−871.010	−0.601462
$129$	0	0
$130$	0	0
$131$	1950.53	1.30090	0.650452	−	0.759547i	$-0.274579\pi$
$131$	1950.53	1.30090	0.650452	+	0.759547i	$0.274579\pi$
$132$	0	0
$133$	3780.77	2.46492
$134$	197.349	0.127227
$135$	0	0
$136$	2273.35	1.43337
$137$	−2318.92	−1.44612	−0.723060	−	0.690785i	$-0.757265\pi$
$137$	−2318.92	−1.44612	−0.723060	+	0.690785i	$0.757265\pi$
$138$	0	0
$139$	1461.53	0.891836	0.445918	−	0.895074i	$-0.352877\pi$
$139$	1461.53	0.891836	0.445918	+	0.895074i	$0.352877\pi$
$140$	0	0
$141$	0	0
$142$	−1789.95	−1.05781
$143$	−980.251	−0.573236
$144$	0	0
$145$	0	0
$146$	−872.403	−0.494524
$147$	0	0
$148$	298.900	0.166009
$149$	1404.10	0.772003	0.386002	−	0.922498i	$-0.373856\pi$
$149$	1404.10	0.772003	0.386002	+	0.922498i	$0.373856\pi$
$150$	0	0
$151$	2722.74	1.46737	0.733687	−	0.679488i	$-0.237798\pi$
$151$	2722.74	1.46737	0.733687	+	0.679488i	$0.237798\pi$
$152$	3292.83	1.75713
$153$	0	0
$154$	−2526.76	−1.32216
$155$	0	0
$156$	0	0
$157$	2731.98	1.38876	0.694382	−	0.719607i	$-0.255678\pi$
$157$	2731.98	1.38876	0.694382	+	0.719607i	$0.255678\pi$
$158$	−248.839	−0.125295
$159$	0	0
$160$	0	0
$161$	−11.3597	−0.00556067
$162$	0	0
$163$	2886.65	1.38712	0.693558	−	0.720401i	$-0.256042\pi$
$163$	2886.65	1.38712	0.693558	+	0.720401i	$0.256042\pi$
$164$	85.8007	0.0408531
$165$	0	0
$166$	−3158.48	−1.47678
$167$	−636.737	−0.295043	−0.147522	−	0.989059i	$-0.547130\pi$
$167$	−636.737	−0.295043	−0.147522	+	0.989059i	$0.547130\pi$
$168$	0	0
$169$	−1447.91	−0.659041
$170$	0	0
$171$	0	0
$172$	829.985	0.367940
$173$	−183.943	−0.0808377	−0.0404189	−	0.999183i	$-0.512869\pi$
$173$	−183.943	−0.0808377	−0.0404189	+	0.999183i	$0.512869\pi$
$174$	0	0
$175$	0	0
$176$	−1742.60	−0.746327
$177$	0	0
$178$	1135.81	0.478271
$179$	−4639.38	−1.93723	−0.968614	−	0.248570i	$-0.920039\pi$
$179$	−4639.38	−1.93723	−0.968614	+	0.248570i	$0.920039\pi$
$180$	0	0
$181$	−2196.49	−0.902010	−0.451005	−	0.892522i	$-0.648934\pi$
$181$	−2196.49	−0.902010	−0.451005	+	0.892522i	$0.648934\pi$
$182$	1930.89	0.786413
$183$	0	0
$184$	−9.89362	−0.00396395
$185$	0	0
$186$	0	0
$187$	3352.67	1.31108
$188$	−304.437	−0.118103
$189$	0	0
$190$	0	0
$191$	3135.80	1.18795	0.593975	−	0.804484i	$-0.297558\pi$
$191$	3135.80	1.18795	0.593975	+	0.804484i	$0.297558\pi$
$192$	0	0
$193$	−3516.99	−1.31170	−0.655851	−	0.754890i	$-0.727690\pi$
$193$	−3516.99	−1.31170	−0.655851	+	0.754890i	$0.727690\pi$
$194$	−1194.41	−0.442027
$195$	0	0
$196$	−694.661	−0.253156
$197$	−3453.43	−1.24897	−0.624483	−	0.781038i	$-0.714690\pi$
$197$	−3453.43	−1.24897	−0.624483	+	0.781038i	$0.714690\pi$
$198$	0	0
$199$	2258.23	0.804429	0.402215	−	0.915545i	$-0.368241\pi$
$199$	2258.23	0.804429	0.402215	+	0.915545i	$0.368241\pi$
$200$	0	0
$201$	0	0
$202$	−3729.78	−1.29914
$203$	5435.89	1.87943
$204$	0	0
$205$	0	0
$206$	2448.95	0.828285
$207$	0	0
$208$	1331.66	0.443912
$209$	4856.18	1.60722
$210$	0	0
$211$	795.126	0.259425	0.129713	−	0.991552i	$-0.458595\pi$
$211$	795.126	0.259425	0.129713	+	0.991552i	$0.458595\pi$
$212$	852.500	0.276179
$213$	0	0
$214$	−3238.99	−1.03464
$215$	0	0
$216$	0	0
$217$	−2698.60	−0.844206
$218$	3658.23	1.13654
$219$	0	0
$220$	0	0
$221$	−2562.04	−0.779824
$222$	0	0
$223$	2719.12	0.816529	0.408265	−	0.912864i	$-0.366134\pi$
$223$	2719.12	0.816529	0.408265	+	0.912864i	$0.366134\pi$
$224$	−1984.86	−0.592051
$225$	0	0
$226$	5849.25	1.72162
$227$	−1935.94	−0.566047	−0.283024	−	0.959113i	$-0.591337\pi$
$227$	−1935.94	−0.566047	−0.283024	+	0.959113i	$0.591337\pi$
$228$	0	0
$229$	4781.62	1.37982	0.689909	−	0.723896i	$-0.257651\pi$
$229$	4781.62	1.37982	0.689909	+	0.723896i	$0.257651\pi$
$230$	0	0
$231$	0	0
$232$	4734.34	1.33976
$233$	794.207	0.223306	0.111653	−	0.993747i	$-0.464386\pi$
$233$	794.207	0.223306	0.111653	+	0.993747i	$0.464386\pi$
$234$	0	0
$235$	0	0
$236$	−755.852	−0.208482
$237$	0	0
$238$	−6604.07	−1.79865
$239$	−3253.17	−0.880461	−0.440230	−	0.897885i	$-0.645103\pi$
$239$	−3253.17	−0.880461	−0.440230	+	0.897885i	$0.645103\pi$
$240$	0	0
$241$	1428.53	0.381826	0.190913	−	0.981607i	$-0.438855\pi$
$241$	1428.53	0.381826	0.190913	+	0.981607i	$0.438855\pi$
$242$	122.070	0.0324254
$243$	0	0
$244$	522.856	0.137182
$245$	0	0
$246$	0	0
$247$	−3710.98	−0.955968
$248$	−2350.32	−0.601797
$249$	0	0
$250$	0	0
$251$	−7809.50	−1.96387	−0.981935	−	0.189218i	$-0.939405\pi$
$251$	−7809.50	−1.96387	−0.981935	+	0.189218i	$0.939405\pi$
$252$	0	0
$253$	−14.5909	−0.00362577
$254$	−3754.61	−0.927501
$255$	0	0
$256$	−2350.97	−0.573967
$257$	−2790.73	−0.677359	−0.338679	−	0.940902i	$-0.609980\pi$
$257$	−2790.73	−0.677359	−0.338679	+	0.940902i	$0.609980\pi$
$258$	0	0
$259$	−5213.43	−1.25076
$260$	0	0
$261$	0	0
$262$	−4935.00	−1.16368
$263$	3146.75	0.737782	0.368891	−	0.929473i	$-0.379737\pi$
$263$	3146.75	0.737782	0.368891	+	0.929473i	$0.379737\pi$
$264$	0	0
$265$	0	0
$266$	−9565.66	−2.20492
$267$	0	0
$268$	124.698	0.0284222
$269$	8006.02	1.81463	0.907315	−	0.420452i	$-0.138128\pi$
$269$	8006.02	1.81463	0.907315	+	0.420452i	$0.138128\pi$
$270$	0	0
$271$	−5598.74	−1.25498	−0.627489	−	0.778625i	$-0.715917\pi$
$271$	−5598.74	−1.25498	−0.627489	+	0.778625i	$0.715917\pi$
$272$	−4554.55	−1.01530
$273$	0	0
$274$	5867.06	1.29358
$275$	0	0
$276$	0	0
$277$	−759.525	−0.164749	−0.0823744	−	0.996601i	$-0.526250\pi$
$277$	−759.525	−0.164749	−0.0823744	+	0.996601i	$0.526250\pi$
$278$	−3697.79	−0.797765
$279$	0	0
$280$	0	0
$281$	−2516.27	−0.534193	−0.267097	−	0.963670i	$-0.586064\pi$
$281$	−2516.27	−0.534193	−0.267097	+	0.963670i	$0.586064\pi$
$282$	0	0
$283$	−7742.08	−1.62622	−0.813108	−	0.582113i	$-0.802226\pi$
$283$	−7742.08	−1.62622	−0.813108	+	0.582113i	$0.802226\pi$
$284$	−1131.01	−0.236313
$285$	0	0
$286$	2480.12	0.512771
$287$	−1496.54	−0.307798
$288$	0	0
$289$	3849.71	0.783577
$290$	0	0
$291$	0	0
$292$	−551.240	−0.110476
$293$	−577.798	−0.115206	−0.0576029	−	0.998340i	$-0.518346\pi$
$293$	−577.798	−0.115206	−0.0576029	+	0.998340i	$0.518346\pi$
$294$	0	0
$295$	0	0
$296$	−4540.59	−0.891610
$297$	0	0
$298$	−3552.50	−0.690572
$299$	11.1500	0.00215659
$300$	0	0
$301$	−14476.7	−2.77216
$302$	−6888.76	−1.31259
$303$	0	0
$304$	−6597.05	−1.24463
$305$	0	0
$306$	0	0
$307$	5530.03	1.02806	0.514031	−	0.857771i	$-0.328151\pi$
$307$	5530.03	1.02806	0.514031	+	0.857771i	$0.328151\pi$
$308$	−1596.57	−0.295367
$309$	0	0
$310$	0	0
$311$	7071.80	1.28941	0.644703	−	0.764433i	$-0.276981\pi$
$311$	7071.80	1.28941	0.644703	+	0.764433i	$0.276981\pi$
$312$	0	0
$313$	−54.0308	−0.00975719	−0.00487860	−	0.999988i	$-0.501553\pi$
$313$	−54.0308	−0.00975719	−0.00487860	+	0.999988i	$0.501553\pi$
$314$	−6912.14	−1.24228
$315$	0	0
$316$	−157.233	−0.0279906
$317$	158.754	0.0281279	0.0140639	−	0.999901i	$-0.495523\pi$
$317$	158.754	0.0281279	0.0140639	+	0.999901i	$0.495523\pi$
$318$	0	0
$319$	6982.09	1.22546
$320$	0	0
$321$	0	0
$322$	28.7410	0.00497413
$323$	12692.4	2.18645
$324$	0	0
$325$	0	0
$326$	−7303.47	−1.24080
$327$	0	0
$328$	−1303.40	−0.219416
$329$	5310.01	0.889819
$330$	0	0
$331$	−914.263	−0.151820	−0.0759101	−	0.997115i	$-0.524186\pi$
$331$	−914.263	−0.151820	−0.0759101	+	0.997115i	$0.524186\pi$
$332$	−1995.73	−0.329910
$333$	0	0
$334$	1611.00	0.263922
$335$	0	0
$336$	0	0
$337$	1258.22	0.203381	0.101691	−	0.994816i	$-0.467575\pi$
$337$	1258.22	0.203381	0.101691	+	0.994816i	$0.467575\pi$
$338$	3663.35	0.589526
$339$	0	0
$340$	0	0
$341$	−3466.19	−0.550454
$342$	0	0
$343$	2552.07	0.401746
$344$	−12608.3	−1.97615
$345$	0	0
$346$	465.392	0.0723110
$347$	−591.507	−0.0915094	−0.0457547	−	0.998953i	$-0.514569\pi$
$347$	−591.507	−0.0915094	−0.0457547	+	0.998953i	$0.514569\pi$
$348$	0	0
$349$	6362.97	0.975937	0.487969	−	0.872861i	$-0.337738\pi$
$349$	6362.97	0.975937	0.487969	+	0.872861i	$0.337738\pi$
$350$	0	0
$351$	0	0
$352$	−2549.44	−0.386039
$353$	2735.16	0.412402	0.206201	−	0.978510i	$-0.433890\pi$
$353$	2735.16	0.412402	0.206201	+	0.978510i	$0.433890\pi$
$354$	0	0
$355$	0	0
$356$	717.676	0.106845
$357$	0	0
$358$	11738.0	1.73289
$359$	2775.11	0.407980	0.203990	−	0.978973i	$-0.434609\pi$
$359$	2775.11	0.407980	0.203990	+	0.978973i	$0.434609\pi$
$360$	0	0
$361$	11525.3	1.68031
$362$	5557.30	0.806866
$363$	0	0
$364$	1220.06	0.175683
$365$	0	0
$366$	0	0
$367$	716.105	0.101854	0.0509269	−	0.998702i	$-0.483782\pi$
$367$	716.105	0.101854	0.0509269	+	0.998702i	$0.483782\pi$
$368$	19.8215	0.00280779
$369$	0	0
$370$	0	0
$371$	−14869.4	−2.08080
$372$	0	0
$373$	−11605.3	−1.61099	−0.805494	−	0.592603i	$-0.798100\pi$
$373$	−11605.3	−1.61099	−0.805494	+	0.592603i	$0.798100\pi$
$374$	−8482.54	−1.17279
$375$	0	0
$376$	4624.71	0.634312
$377$	−5335.55	−0.728899
$378$	0	0
$379$	−7893.11	−1.06977	−0.534884	−	0.844926i	$-0.679645\pi$
$379$	−7893.11	−1.06977	−0.534884	+	0.844926i	$0.679645\pi$
$380$	0	0
$381$	0	0
$382$	−7933.84	−1.06264
$383$	6965.23	0.929261	0.464630	−	0.885505i	$-0.346187\pi$
$383$	6965.23	0.929261	0.464630	+	0.885505i	$0.346187\pi$
$384$	0	0
$385$	0	0
$386$	8898.29	1.17334
$387$	0	0
$388$	−754.702	−0.0987480
$389$	−6906.36	−0.900171	−0.450085	−	0.892986i	$-0.648606\pi$
$389$	−6906.36	−0.900171	−0.450085	+	0.892986i	$0.648606\pi$
$390$	0	0
$391$	−38.1354	−0.00493246
$392$	10552.6	1.35966
$393$	0	0
$394$	8737.46	1.11723
$395$	0	0
$396$	0	0
$397$	8237.28	1.04135	0.520676	−	0.853754i	$-0.325680\pi$
$397$	8237.28	1.04135	0.520676	+	0.853754i	$0.325680\pi$
$398$	−5713.50	−0.719578
$399$	0	0
$400$	0	0
$401$	1460.02	0.181820	0.0909101	−	0.995859i	$-0.471022\pi$
$401$	1460.02	0.181820	0.0909101	+	0.995859i	$0.471022\pi$
$402$	0	0
$403$	2648.79	0.327408
$404$	−2356.72	−0.290225
$405$	0	0
$406$	−13753.3	−1.68119
$407$	−6696.34	−0.815542
$408$	0	0
$409$	−9821.99	−1.18745	−0.593724	−	0.804669i	$-0.702343\pi$
$409$	−9821.99	−1.18745	−0.593724	+	0.804669i	$0.702343\pi$
$410$	0	0
$411$	0	0
$412$	1547.41	0.185037
$413$	13183.6	1.57076
$414$	0	0
$415$	0	0
$416$	1948.23	0.229615
$417$	0	0
$418$	−12286.5	−1.43769
$419$	14970.5	1.74549	0.872743	−	0.488181i	$-0.162339\pi$
$419$	14970.5	1.74549	0.872743	+	0.488181i	$0.162339\pi$
$420$	0	0
$421$	12909.9	1.49452	0.747259	−	0.664533i	$-0.231369\pi$
$421$	12909.9	1.49452	0.747259	+	0.664533i	$0.231369\pi$
$422$	−2011.74	−0.232061
$423$	0	0
$424$	−12950.3	−1.48331
$425$	0	0
$426$	0	0
$427$	−9119.69	−1.03357
$428$	−2046.60	−0.231136
$429$	0	0
$430$	0	0
$431$	5841.94	0.652892	0.326446	−	0.945216i	$-0.394149\pi$
$431$	5841.94	0.652892	0.326446	+	0.945216i	$0.394149\pi$
$432$	0	0
$433$	1317.13	0.146182	0.0730912	−	0.997325i	$-0.476714\pi$
$433$	1317.13	0.146182	0.0730912	+	0.997325i	$0.476714\pi$
$434$	6827.68	0.755159
$435$	0	0
$436$	2311.50	0.253901
$437$	−55.2373	−0.00604659
$438$	0	0
$439$	−10825.6	−1.17694	−0.588469	−	0.808520i	$-0.700269\pi$
$439$	−10825.6	−1.17694	−0.588469	+	0.808520i	$0.700269\pi$
$440$	0	0
$441$	0	0
$442$	6482.17	0.697568
$443$	−7857.87	−0.842751	−0.421375	−	0.906886i	$-0.638452\pi$
$443$	−7857.87	−0.842751	−0.421375	+	0.906886i	$0.638452\pi$
$444$	0	0
$445$	0	0
$446$	−6879.61	−0.730402
$447$	0	0
$448$	15875.5	1.67421
$449$	12512.7	1.31517	0.657587	−	0.753379i	$-0.271577\pi$
$449$	12512.7	1.31517	0.657587	+	0.753379i	$0.271577\pi$
$450$	0	0
$451$	−1922.22	−0.200696
$452$	3695.93	0.384606
$453$	0	0
$454$	4898.09	0.506341
$455$	0	0
$456$	0	0
$457$	−11041.9	−1.13024	−0.565118	−	0.825010i	$-0.691170\pi$
$457$	−11041.9	−1.13024	−0.565118	+	0.825010i	$0.691170\pi$
$458$	−12097.9	−1.23427
$459$	0	0
$460$	0	0
$461$	9204.91	0.929968	0.464984	−	0.885319i	$-0.346060\pi$
$461$	9204.91	0.929968	0.464984	+	0.885319i	$0.346060\pi$
$462$	0	0
$463$	17536.1	1.76020	0.880101	−	0.474786i	$-0.157475\pi$
$463$	17536.1	1.76020	0.880101	+	0.474786i	$0.157475\pi$
$464$	−9485.06	−0.948993
$465$	0	0
$466$	−2009.41	−0.199752
$467$	−13734.5	−1.36093	−0.680467	−	0.732779i	$-0.738223\pi$
$467$	−13734.5	−1.36093	−0.680467	+	0.732779i	$0.738223\pi$
$468$	0	0
$469$	−2174.99	−0.214140
$470$	0	0
$471$	0	0
$472$	11482.2	1.11972
$473$	−18594.4	−1.80755
$474$	0	0
$475$	0	0
$476$	−4172.87	−0.401814
$477$	0	0
$478$	8230.80	0.787590
$479$	−5272.93	−0.502977	−0.251489	−	0.967860i	$-0.580920\pi$
$479$	−5272.93	−0.502977	−0.251489	+	0.967860i	$0.580920\pi$
$480$	0	0
$481$	5117.20	0.485081
$482$	−3614.31	−0.341551
$483$	0	0
$484$	77.1317	0.00724377
$485$	0	0
$486$	0	0
$487$	−918.490	−0.0854636	−0.0427318	−	0.999087i	$-0.513606\pi$
$487$	−918.490	−0.0854636	−0.0427318	+	0.999087i	$0.513606\pi$
$488$	−7942.72	−0.736783
$489$	0	0
$490$	0	0
$491$	1746.00	0.160480	0.0802400	−	0.996776i	$-0.474431\pi$
$491$	1746.00	0.160480	0.0802400	+	0.996776i	$0.474431\pi$
$492$	0	0
$493$	18248.7	1.66710
$494$	9389.10	0.855133
$495$	0	0
$496$	4708.77	0.426270
$497$	19727.1	1.78044
$498$	0	0
$499$	10841.2	0.972585	0.486292	−	0.873796i	$-0.338349\pi$
$499$	10841.2	0.972585	0.486292	+	0.873796i	$0.338349\pi$
$500$	0	0
$501$	0	0
$502$	19758.7	1.75672
$503$	−6427.44	−0.569752	−0.284876	−	0.958564i	$-0.591952\pi$
$503$	−6427.44	−0.569752	−0.284876	+	0.958564i	$0.591952\pi$
$504$	0	0
$505$	0	0
$506$	36.9161	0.00324332
$507$	0	0
$508$	−2372.41	−0.207202
$509$	5482.99	0.477464	0.238732	−	0.971085i	$-0.423268\pi$
$509$	5482.99	0.477464	0.238732	+	0.971085i	$0.423268\pi$
$510$	0	0
$511$	9614.77	0.832353
$512$	12916.2	1.11489
$513$	0	0
$514$	7060.79	0.605911
$515$	0	0
$516$	0	0
$517$	6820.40	0.580195
$518$	13190.4	1.11883
$519$	0	0
$520$	0	0
$521$	8507.07	0.715358	0.357679	−	0.933845i	$-0.383568\pi$
$521$	8507.07	0.715358	0.357679	+	0.933845i	$0.383568\pi$
$522$	0	0
$523$	−5908.55	−0.494002	−0.247001	−	0.969015i	$-0.579445\pi$
$523$	−5908.55	−0.494002	−0.247001	+	0.969015i	$0.579445\pi$
$524$	−3118.25	−0.259965
$525$	0	0
$526$	−7961.54	−0.659961
$527$	−9059.42	−0.748832
$528$	0	0
$529$	−12166.8	−0.999986
$530$	0	0
$531$	0	0
$532$	−6044.20	−0.492574
$533$	1468.92	0.119373
$534$	0	0
$535$	0	0
$536$	−1894.29	−0.152651
$537$	0	0
$538$	−20255.9	−1.62322
$539$	15562.7	1.24366
$540$	0	0
$541$	10557.4	0.838995	0.419497	−	0.907757i	$-0.362206\pi$
$541$	10557.4	0.838995	0.419497	+	0.907757i	$0.362206\pi$
$542$	14165.3	1.12260
$543$	0	0
$544$	−6663.36	−0.525164
$545$	0	0
$546$	0	0
$547$	−6056.60	−0.473421	−0.236711	−	0.971580i	$-0.576069\pi$
$547$	−6056.60	−0.473421	−0.236711	+	0.971580i	$0.576069\pi$
$548$	3707.18	0.288984
$549$	0	0
$550$	0	0
$551$	26432.4	2.04366
$552$	0	0
$553$	2742.46	0.210889
$554$	1921.66	0.147371
$555$	0	0
$556$	−2336.50	−0.178219
$557$	816.783	0.0621332	0.0310666	−	0.999517i	$-0.490110\pi$
$557$	816.783	0.0621332	0.0310666	+	0.999517i	$0.490110\pi$
$558$	0	0
$559$	14209.4	1.07513
$560$	0	0
$561$	0	0
$562$	6366.39	0.477847
$563$	6891.09	0.515852	0.257926	−	0.966165i	$-0.416961\pi$
$563$	6891.09	0.515852	0.257926	+	0.966165i	$0.416961\pi$
$564$	0	0
$565$	0	0
$566$	19588.1	1.45468
$567$	0	0
$568$	17181.1	1.26920
$569$	−9267.91	−0.682831	−0.341416	−	0.939912i	$-0.610906\pi$
$569$	−9267.91	−0.682831	−0.341416	+	0.939912i	$0.610906\pi$
$570$	0	0
$571$	−10142.9	−0.743378	−0.371689	−	0.928357i	$-0.621221\pi$
$571$	−10142.9	−0.743378	−0.371689	+	0.928357i	$0.621221\pi$
$572$	1567.10	0.114552
$573$	0	0
$574$	3786.38	0.275332
$575$	0	0
$576$	0	0
$577$	4905.22	0.353912	0.176956	−	0.984219i	$-0.443375\pi$
$577$	4905.22	0.353912	0.176956	+	0.984219i	$0.443375\pi$
$578$	−9740.10	−0.700925
$579$	0	0
$580$	0	0
$581$	34809.7	2.48563
$582$	0	0
$583$	−19098.8	−1.35676
$584$	8373.91	0.593347
$585$	0	0
$586$	1461.88	0.103054
$587$	2570.78	0.180762	0.0903810	−	0.995907i	$-0.471192\pi$
$587$	2570.78	0.180762	0.0903810	+	0.995907i	$0.471192\pi$
$588$	0	0
$589$	−13122.1	−0.917976
$590$	0	0
$591$	0	0
$592$	9096.89	0.631554
$593$	4740.85	0.328303	0.164151	−	0.986435i	$-0.447511\pi$
$593$	4740.85	0.328303	0.164151	+	0.986435i	$0.447511\pi$
$594$	0	0
$595$	0	0
$596$	−2244.70	−0.154272
$597$	0	0
$598$	−28.2105	−0.00192912
$599$	1784.53	0.121726	0.0608632	−	0.998146i	$-0.480615\pi$
$599$	1784.53	0.121726	0.0608632	+	0.998146i	$0.480615\pi$
$600$	0	0
$601$	−11189.9	−0.759479	−0.379739	−	0.925094i	$-0.623986\pi$
$601$	−11189.9	−0.759479	−0.379739	+	0.925094i	$0.623986\pi$
$602$	36627.2	2.47975
$603$	0	0
$604$	−4352.76	−0.293231
$605$	0	0
$606$	0	0
$607$	−21524.7	−1.43931	−0.719653	−	0.694334i	$-0.755699\pi$
$607$	−21524.7	−1.43931	−0.719653	+	0.694334i	$0.755699\pi$
$608$	−9651.55	−0.643786
$609$	0	0
$610$	0	0
$611$	−5212.00	−0.345098
$612$	0	0
$613$	2787.57	0.183668	0.0918342	−	0.995774i	$-0.470727\pi$
$613$	2787.57	0.183668	0.0918342	+	0.995774i	$0.470727\pi$
$614$	−13991.4	−0.919623
$615$	0	0
$616$	24253.5	1.58637
$617$	8076.49	0.526981	0.263490	−	0.964662i	$-0.415126\pi$
$617$	8076.49	0.526981	0.263490	+	0.964662i	$0.415126\pi$
$618$	0	0
$619$	−28161.9	−1.82863	−0.914315	−	0.405005i	$-0.867270\pi$
$619$	−28161.9	−1.82863	−0.914315	+	0.405005i	$0.867270\pi$
$620$	0	0
$621$	0	0
$622$	−17892.3	−1.15340
$623$	−12517.8	−0.804997
$624$	0	0
$625$	0	0
$626$	136.703	0.00872801
$627$	0	0
$628$	−4367.54	−0.277522
$629$	−17501.9	−1.10945
$630$	0	0
$631$	3519.36	0.222034	0.111017	−	0.993818i	$-0.464589\pi$
$631$	3519.36	0.222034	0.111017	+	0.993818i	$0.464589\pi$
$632$	2388.53	0.150333
$633$	0	0
$634$	−401.662	−0.0251609
$635$	0	0
$636$	0	0
$637$	−11892.7	−0.739726
$638$	−17665.3	−1.09620
$639$	0	0
$640$	0	0
$641$	17397.1	1.07199	0.535995	−	0.844221i	$-0.319937\pi$
$641$	17397.1	1.07199	0.535995	+	0.844221i	$0.319937\pi$
$642$	0	0
$643$	5773.23	0.354081	0.177040	−	0.984204i	$-0.443348\pi$
$643$	5773.23	0.354081	0.177040	+	0.984204i	$0.443348\pi$
$644$	18.1604	0.00111121
$645$	0	0
$646$	−32112.7	−1.95582
$647$	−27954.2	−1.69860	−0.849299	−	0.527913i	$-0.822975\pi$
$647$	−27954.2	−1.69860	−0.849299	+	0.527913i	$0.822975\pi$
$648$	0	0
$649$	16933.6	1.02419
$650$	0	0
$651$	0	0
$652$	−4614.80	−0.277193
$653$	−3933.82	−0.235746	−0.117873	−	0.993029i	$-0.537608\pi$
$653$	−3933.82	−0.235746	−0.117873	+	0.993029i	$0.537608\pi$
$654$	0	0
$655$	0	0
$656$	2611.31	0.155418
$657$	0	0
$658$	−13434.8	−0.795961
$659$	24920.9	1.47311	0.736555	−	0.676377i	$-0.236451\pi$
$659$	24920.9	1.47311	0.736555	+	0.676377i	$0.236451\pi$
$660$	0	0
$661$	4021.88	0.236661	0.118331	−	0.992974i	$-0.462246\pi$
$661$	4021.88	0.236661	0.118331	+	0.992974i	$0.462246\pi$
$662$	2313.16	0.135806
$663$	0	0
$664$	30317.2	1.77189
$665$	0	0
$666$	0	0
$667$	−79.4187	−0.00461035
$668$	1017.93	0.0589596
$669$	0	0
$670$	0	0
$671$	−11713.7	−0.673924
$672$	0	0
$673$	−20002.2	−1.14566	−0.572828	−	0.819675i	$-0.694154\pi$
$673$	−20002.2	−1.14566	−0.572828	+	0.819675i	$0.694154\pi$
$674$	−3183.40	−0.181929
$675$	0	0
$676$	2314.74	0.131699
$677$	1955.82	0.111031	0.0555156	−	0.998458i	$-0.482320\pi$
$677$	1955.82	0.111031	0.0555156	+	0.998458i	$0.482320\pi$
$678$	0	0
$679$	13163.6	0.743993
$680$	0	0
$681$	0	0
$682$	8769.76	0.492392
$683$	7805.42	0.437285	0.218643	−	0.975805i	$-0.429837\pi$
$683$	7805.42	0.437285	0.218643	+	0.975805i	$0.429837\pi$
$684$	0	0
$685$	0	0
$686$	−6456.96	−0.359370
$687$	0	0
$688$	25260.2	1.39976
$689$	14594.9	0.806998
$690$	0	0
$691$	−4435.06	−0.244165	−0.122082	−	0.992520i	$-0.538957\pi$
$691$	−4435.06	−0.244165	−0.122082	+	0.992520i	$0.538957\pi$
$692$	294.064	0.0161541
$693$	0	0
$694$	1496.56	0.0818570
$695$	0	0
$696$	0	0
$697$	−5024.02	−0.273025
$698$	−16098.9	−0.872996
$699$	0	0
$700$	0	0
$701$	12579.2	0.677762	0.338881	−	0.940829i	$-0.389952\pi$
$701$	12579.2	0.677762	0.338881	+	0.940829i	$0.389952\pi$
$702$	0	0
$703$	−25350.7	−1.36005
$704$	20391.1	1.09165
$705$	0	0
$706$	−6920.19	−0.368902
$707$	41106.0	2.18663
$708$	0	0
$709$	18736.3	0.992461	0.496231	−	0.868191i	$-0.334717\pi$
$709$	18736.3	0.992461	0.496231	+	0.868191i	$0.334717\pi$
$710$	0	0
$711$	0	0
$712$	−10902.2	−0.573846
$713$	39.4267	0.00207089
$714$	0	0
$715$	0	0
$716$	7416.85	0.387124
$717$	0	0
$718$	−7021.27	−0.364946
$719$	−6632.57	−0.344024	−0.172012	−	0.985095i	$-0.555027\pi$
$719$	−6632.57	−0.344024	−0.172012	+	0.985095i	$0.555027\pi$
$720$	0	0
$721$	−26990.0	−1.39412
$722$	−29159.9	−1.50307
$723$	0	0
$724$	3511.46	0.180252
$725$	0	0
$726$	0	0
$727$	−5599.91	−0.285680	−0.142840	−	0.989746i	$-0.545623\pi$
$727$	−5599.91	−0.285680	−0.142840	+	0.989746i	$0.545623\pi$
$728$	−18534.0	−0.943565
$729$	0	0
$730$	0	0
$731$	−48599.3	−2.45898
$732$	0	0
$733$	−25381.4	−1.27897	−0.639483	−	0.768805i	$-0.720852\pi$
$733$	−25381.4	−1.27897	−0.639483	+	0.768805i	$0.720852\pi$
$734$	−1811.81	−0.0911103
$735$	0	0
$736$	28.9990	0.00145233
$737$	−2793.65	−0.139627
$738$	0	0
$739$	3153.26	0.156962	0.0784808	−	0.996916i	$-0.474993\pi$
$739$	3153.26	0.156962	0.0784808	+	0.996916i	$0.474993\pi$
$740$	0	0
$741$	0	0
$742$	37620.7	1.86132
$743$	−23892.5	−1.17972	−0.589859	−	0.807506i	$-0.700817\pi$
$743$	−23892.5	−1.17972	−0.589859	+	0.807506i	$0.700817\pi$
$744$	0	0
$745$	0	0
$746$	29362.4	1.44106
$747$	0	0
$748$	−5359.81	−0.261998
$749$	35697.0	1.74144
$750$	0	0
$751$	7232.32	0.351413	0.175707	−	0.984443i	$-0.443779\pi$
$751$	7232.32	0.351413	0.175707	+	0.984443i	$0.443779\pi$
$752$	−9265.42	−0.449302
$753$	0	0
$754$	13499.4	0.652015
$755$	0	0
$756$	0	0
$757$	9385.12	0.450605	0.225302	−	0.974289i	$-0.427663\pi$
$757$	9385.12	0.450605	0.225302	+	0.974289i	$0.427663\pi$
$758$	19970.2	0.956929
$759$	0	0
$760$	0	0
$761$	13138.3	0.625840	0.312920	−	0.949779i	$-0.398693\pi$
$761$	13138.3	0.625840	0.312920	+	0.949779i	$0.398693\pi$
$762$	0	0
$763$	−40317.4	−1.91296
$764$	−5013.11	−0.237393
$765$	0	0
$766$	−17622.6	−0.831242
$767$	−12940.3	−0.609187
$768$	0	0
$769$	−34393.8	−1.61284	−0.806419	−	0.591344i	$-0.798597\pi$
$769$	−34393.8	−1.61284	−0.806419	+	0.591344i	$0.798597\pi$
$770$	0	0
$771$	0	0
$772$	5622.51	0.262123
$773$	−5527.75	−0.257205	−0.128602	−	0.991696i	$-0.541049\pi$
$773$	−5527.75	−0.257205	−0.128602	+	0.991696i	$0.541049\pi$
$774$	0	0
$775$	0	0
$776$	11464.7	0.530359
$777$	0	0
$778$	17473.7	0.805221
$779$	−7277.05	−0.334695
$780$	0	0
$781$	25338.3	1.16092
$782$	96.4858	0.00441218
$783$	0	0
$784$	−21141.7	−0.963089
$785$	0	0
$786$	0	0
$787$	39668.3	1.79673	0.898363	−	0.439254i	$-0.144757\pi$
$787$	39668.3	1.79673	0.898363	+	0.439254i	$0.144757\pi$
$788$	5520.89	0.249586
$789$	0	0
$790$	0	0
$791$	−64464.7	−2.89772
$792$	0	0
$793$	8951.35	0.400847
$794$	−20841.0	−0.931511
$795$	0	0
$796$	−3610.16	−0.160752
$797$	11474.1	0.509956	0.254978	−	0.966947i	$-0.417932\pi$
$797$	11474.1	0.509956	0.254978	+	0.966947i	$0.417932\pi$
$798$	0	0
$799$	17826.2	0.789292
$800$	0	0
$801$	0	0
$802$	−3693.97	−0.162642
$803$	12349.6	0.542725
$804$	0	0
$805$	0	0
$806$	−6701.65	−0.292873
$807$	0	0
$808$	35800.9	1.55875
$809$	−38806.9	−1.68650	−0.843249	−	0.537524i	$-0.819360\pi$
$809$	−38806.9	−1.68650	−0.843249	+	0.537524i	$0.819360\pi$
$810$	0	0
$811$	1873.12	0.0811027	0.0405513	−	0.999177i	$-0.487089\pi$
$811$	1873.12	0.0811027	0.0405513	+	0.999177i	$0.487089\pi$
$812$	−8690.20	−0.375574
$813$	0	0
$814$	16942.3	0.729518
$815$	0	0
$816$	0	0
$817$	−70393.8	−3.01440
$818$	24850.5	1.06220
$819$	0	0
$820$	0	0
$821$	39300.8	1.67066	0.835328	−	0.549752i	$-0.185278\pi$
$821$	39300.8	1.67066	0.835328	+	0.549752i	$0.185278\pi$
$822$	0	0
$823$	44666.5	1.89183	0.945916	−	0.324413i	$-0.105167\pi$
$823$	44666.5	1.89183	0.945916	+	0.324413i	$0.105167\pi$
$824$	−23506.7	−0.993805
$825$	0	0
$826$	−33355.7	−1.40508
$827$	39263.8	1.65095	0.825475	−	0.564438i	$-0.190907\pi$
$827$	39263.8	1.65095	0.825475	+	0.564438i	$0.190907\pi$
$828$	0	0
$829$	1566.69	0.0656372	0.0328186	−	0.999461i	$-0.489552\pi$
$829$	1566.69	0.0656372	0.0328186	+	0.999461i	$0.489552\pi$
$830$	0	0
$831$	0	0
$832$	−15582.4	−0.649307
$833$	40675.5	1.69187
$834$	0	0
$835$	0	0
$836$	−7763.43	−0.321177
$837$	0	0
$838$	−37876.7	−1.56137
$839$	−15525.9	−0.638870	−0.319435	−	0.947608i	$-0.603493\pi$
$839$	−15525.9	−0.638870	−0.319435	+	0.947608i	$0.603493\pi$
$840$	0	0
$841$	13614.8	0.558237
$842$	−32663.2	−1.33688
$843$	0	0
$844$	−1271.14	−0.0518420
$845$	0	0
$846$	0	0
$847$	−1345.34	−0.0545765
$848$	25945.5	1.05067
$849$	0	0
$850$	0	0
$851$	76.1685	0.00306818
$852$	0	0
$853$	−9716.17	−0.390006	−0.195003	−	0.980803i	$-0.562472\pi$
$853$	−9716.17	−0.390006	−0.195003	+	0.980803i	$0.562472\pi$
$854$	23073.6	0.924545
$855$	0	0
$856$	31090.0	1.24140
$857$	−12663.6	−0.504762	−0.252381	−	0.967628i	$-0.581214\pi$
$857$	−12663.6	−0.504762	−0.252381	+	0.967628i	$0.581214\pi$
$858$	0	0
$859$	−22605.5	−0.897892	−0.448946	−	0.893559i	$-0.648200\pi$
$859$	−22605.5	−0.897892	−0.448946	+	0.893559i	$0.648200\pi$
$860$	0	0
$861$	0	0
$862$	−14780.6	−0.584025
$863$	1672.68	0.0659778	0.0329889	−	0.999456i	$-0.489497\pi$
$863$	1672.68	0.0659778	0.0329889	+	0.999456i	$0.489497\pi$
$864$	0	0
$865$	0	0
$866$	−3332.44	−0.130763
$867$	0	0
$868$	4314.17	0.168701
$869$	3522.54	0.137507
$870$	0	0
$871$	2134.84	0.0830499
$872$	−35114.1	−1.36366
$873$	0	0
$874$	139.755	0.00540879
$875$	0	0
$876$	0	0
$877$	−3237.58	−0.124658	−0.0623291	−	0.998056i	$-0.519853\pi$
$877$	−3237.58	−0.124658	−0.0623291	+	0.998056i	$0.519853\pi$
$878$	27389.6	1.05279
$879$	0	0
$880$	0	0
$881$	−8650.65	−0.330815	−0.165407	−	0.986225i	$-0.552894\pi$
$881$	−8650.65	−0.330815	−0.165407	+	0.986225i	$0.552894\pi$
$882$	0	0
$883$	16846.2	0.642037	0.321018	−	0.947073i	$-0.395975\pi$
$883$	16846.2	0.642037	0.321018	+	0.947073i	$0.395975\pi$
$884$	4095.85	0.155835
$885$	0	0
$886$	19881.1	0.753857
$887$	−47809.6	−1.80980	−0.904899	−	0.425627i	$-0.860053\pi$
$887$	−47809.6	−1.80980	−0.904899	+	0.425627i	$0.860053\pi$
$888$	0	0
$889$	41379.7	1.56111
$890$	0	0
$891$	0	0
$892$	−4346.98	−0.163170
$893$	25820.3	0.967575
$894$	0	0
$895$	0	0
$896$	−24287.4	−0.905562
$897$	0	0
$898$	−31658.3	−1.17645
$899$	−18866.7	−0.699931
$900$	0	0
$901$	−49917.7	−1.84573
$902$	4863.38	0.179527
$903$	0	0
$904$	−56145.0	−2.06566
$905$	0	0
$906$	0	0
$907$	1628.39	0.0596140	0.0298070	−	0.999556i	$-0.490511\pi$
$907$	1628.39	0.0596140	0.0298070	+	0.999556i	$0.490511\pi$
$908$	3094.93	0.113115
$909$	0	0
$910$	0	0
$911$	−17753.9	−0.645677	−0.322839	−	0.946454i	$-0.604637\pi$
$911$	−17753.9	−0.645677	−0.322839	+	0.946454i	$0.604637\pi$
$912$	0	0
$913$	44711.0	1.62072
$914$	27936.9	1.01102
$915$	0	0
$916$	−7644.24	−0.275734
$917$	54388.8	1.95864
$918$	0	0
$919$	−42025.9	−1.50849	−0.754247	−	0.656591i	$-0.771998\pi$
$919$	−42025.9	−1.50849	−0.754247	+	0.656591i	$0.771998\pi$
$920$	0	0
$921$	0	0
$922$	−23289.2	−0.831875
$923$	−19363.0	−0.690509
$924$	0	0
$925$	0	0
$926$	−44367.9	−1.57454
$927$	0	0
$928$	−13876.7	−0.490869
$929$	24361.1	0.860348	0.430174	−	0.902746i	$-0.358452\pi$
$929$	24361.1	0.860348	0.430174	+	0.902746i	$0.358452\pi$
$930$	0	0
$931$	58916.5	2.07402
$932$	−1269.68	−0.0446241
$933$	0	0
$934$	34749.4	1.21738
$935$	0	0
$936$	0	0
$937$	51718.7	1.80318	0.901589	−	0.432595i	$-0.142402\pi$
$937$	51718.7	1.80318	0.901589	+	0.432595i	$0.142402\pi$
$938$	5502.91	0.191553
$939$	0	0
$940$	0	0
$941$	−24630.2	−0.853266	−0.426633	−	0.904425i	$-0.640300\pi$
$941$	−24630.2	−0.853266	−0.426633	+	0.904425i	$0.640300\pi$
$942$	0	0
$943$	21.8646	0.000755046 0
$944$	−23004.0	−0.793133
$945$	0	0
$946$	47045.5	1.61689
$947$	−19674.2	−0.675105	−0.337552	−	0.941307i	$-0.609599\pi$
$947$	−19674.2	−0.675105	−0.337552	+	0.941307i	$0.609599\pi$
$948$	0	0
$949$	−9437.30	−0.322811
$950$	0	0
$951$	0	0
$952$	63390.2	2.15808
$953$	−40472.3	−1.37568	−0.687841	−	0.725861i	$-0.741442\pi$
$953$	−40472.3	−1.37568	−0.687841	+	0.725861i	$0.741442\pi$
$954$	0	0
$955$	0	0
$956$	5200.75	0.175946
$957$	0	0
$958$	13341.0	0.449923
$959$	−64661.0	−2.17728
$960$	0	0
$961$	−20424.8	−0.685604
$962$	−12946.9	−0.433915
$963$	0	0
$964$	−2283.76	−0.0763017
$965$	0	0
$966$	0	0
$967$	29381.6	0.977093	0.488547	−	0.872538i	$-0.337527\pi$
$967$	29381.6	0.977093	0.488547	+	0.872538i	$0.337527\pi$
$968$	−1171.71	−0.0389051
$969$	0	0
$970$	0	0
$971$	−22752.2	−0.751961	−0.375981	−	0.926628i	$-0.622694\pi$
$971$	−22752.2	−0.751961	−0.375981	+	0.926628i	$0.622694\pi$
$972$	0	0
$973$	40753.4	1.34275
$974$	2323.86	0.0764489
$975$	0	0
$976$	15912.9	0.521885
$977$	10823.1	0.354413	0.177206	−	0.984174i	$-0.443294\pi$
$977$	10823.1	0.354413	0.177206	+	0.984174i	$0.443294\pi$
$978$	0	0
$979$	−16078.3	−0.524888
$980$	0	0
$981$	0	0
$982$	−4417.52	−0.143553
$983$	17495.4	0.567666	0.283833	−	0.958874i	$-0.408394\pi$
$983$	17495.4	0.567666	0.283833	+	0.958874i	$0.408394\pi$
$984$	0	0
$985$	0	0
$986$	−46170.9	−1.49126
$987$	0	0
$988$	5932.64	0.191035
$989$	211.505	0.00680027
$990$	0	0
$991$	300.294	0.00962578	0.00481289	−	0.999988i	$-0.498468\pi$
$991$	300.294	0.00962578	0.00481289	+	0.999988i	$0.498468\pi$
$992$	6888.98	0.220489
$993$	0	0
$994$	−49911.2	−1.59264
$995$	0	0
$996$	0	0
$997$	−6360.83	−0.202056	−0.101028	−	0.994884i	$-0.532213\pi$
$997$	−6360.83	−0.202056	−0.101028	+	0.994884i	$0.532213\pi$
$998$	−27429.2	−0.869997
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.v.1.2	yes	4
3.2	odd	2		675.4.a.z.1.3	yes	4
5.2	odd	4		675.4.b.q.649.3		8
5.3	odd	4		675.4.b.q.649.6		8
5.4	even	2		675.4.a.y.1.3	yes	4
15.2	even	4		675.4.b.p.649.6		8
15.8	even	4		675.4.b.p.649.3		8
15.14	odd	2		675.4.a.u.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.4.a.u.1.2	✓	4	15.14	odd	2
675.4.a.v.1.2	yes	4	1.1	even	1	trivial
675.4.a.y.1.3	yes	4	5.4	even	2
675.4.a.z.1.3	yes	4	3.2	odd	2
675.4.b.p.649.3		8	15.8	even	4
675.4.b.p.649.6		8	15.2	even	4
675.4.b.q.649.3		8	5.2	odd	4
675.4.b.q.649.6		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53008 q^{2} -1.59867 q^{4} +27.8841 q^{7} +24.2855 q^{8} +35.8155 q^{11} -27.3694 q^{13} -70.5492 q^{14} -48.6549 q^{16} +93.6094 q^{17} +135.589 q^{19} -90.6163 q^{22} -0.407389 q^{23} +69.2470 q^{26} -44.5775 q^{28} +194.946 q^{29} -96.7790 q^{31} -71.1826 q^{32} -236.840 q^{34} -186.968 q^{37} -343.051 q^{38} -53.6700 q^{41} -519.172 q^{43} -57.2573 q^{44} +1.03073 q^{46} +190.431 q^{47} +434.524 q^{49} +43.7547 q^{52} -533.255 q^{53} +677.178 q^{56} -493.229 q^{58} +472.800 q^{59} -327.057 q^{61} +244.859 q^{62} +569.337 q^{64} -78.0010 q^{67} -149.651 q^{68} +707.467 q^{71} +344.812 q^{73} +473.044 q^{74} -216.761 q^{76} +998.685 q^{77} +98.3522 q^{79} +135.790 q^{82} +1248.37 q^{83} +1313.55 q^{86} +869.796 q^{88} -448.920 q^{89} -763.173 q^{91} +0.651281 q^{92} -481.808 q^{94} +472.081 q^{97} -1099.38 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 19 q^{4} + 4 q^{7} + 15 q^{8} + 52 q^{11} - 2 q^{13} + 138 q^{14} - 5 q^{16} - 64 q^{17} - 46 q^{19} + 87 q^{22} + 90 q^{23} + 469 q^{26} - 110 q^{28} + 470 q^{29} - 262 q^{31} + 199 q^{32}+ \cdots - 4301 q^{98}+O(q^{100}) \) 4 * q - q^2 + 19 * q^4 + 4 * q^7 + 15 * q^8 + 52 * q^11 - 2 * q^13 + 138 * q^14 - 5 * q^16 - 64 * q^17 - 46 * q^19 + 87 * q^22 + 90 * q^23 + 469 * q^26 - 110 * q^28 + 470 * q^29 - 262 * q^31 + 199 * q^32 - 42 * q^34 - 542 * q^37 - 532 * q^38 + 698 * q^41 + 142 * q^43 + 419 * q^44 + 537 * q^46 + 542 * q^47 + 780 * q^49 + 409 * q^52 - 910 * q^53 + 2034 * q^56 - 576 * q^58 + 100 * q^59 + 74 * q^61 + 2406 * q^62 - 965 * q^64 + 928 * q^67 - 2810 * q^68 + 1622 * q^71 - 536 * q^73 - 253 * q^74 - 2068 * q^76 + 1932 * q^77 - 508 * q^79 - 1782 * q^82 - 1524 * q^83 + 3940 * q^86 + 2247 * q^88 + 756 * q^89 + 1120 * q^91 + 3645 * q^92 + 2847 * q^94 + 892 * q^97 - 4301 * q^98

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