LMFDB - Embedded newform 2312.4.a.q.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, 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("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.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:	$136.412415933$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 294 x^{16} - 14 x^{15} + 34371 x^{14} + 2670 x^{13} - 2054705 x^{12} - 160284 x^{11} + \cdots - 176969301147 $ x^18 - 294x^16 - 14x^15 + 34371x^14 + 2670x^13 - 2054705x^12 - 160284x^11 + 67981059x^10 + 2824200x^9 - 1279285428x^8 + 12108606x^7 + 13538185208x^6 - 853720902x^5 - 75816227907x^4 + 7747530606x^3 + 195817930515x^2 - 12873873186x - 176969301147
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 17^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-7.45229$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.45229 q^{3} +17.0781 q^{5} -29.7278 q^{7} +44.4413 q^{9} +O(q^{10})$	$q+8.45229 q^{3} +17.0781 q^{5} -29.7278 q^{7} +44.4413 q^{9} +49.6400 q^{11} -67.1970 q^{13} +144.349 q^{15} +61.2676 q^{19} -251.268 q^{21} -93.9175 q^{23} +166.662 q^{25} +147.419 q^{27} -10.2954 q^{29} +288.019 q^{31} +419.572 q^{33} -507.696 q^{35} +345.264 q^{37} -567.969 q^{39} +50.3858 q^{41} -83.3053 q^{43} +758.973 q^{45} +201.195 q^{47} +540.744 q^{49} +398.772 q^{53} +847.759 q^{55} +517.852 q^{57} +69.6651 q^{59} +310.111 q^{61} -1321.14 q^{63} -1147.60 q^{65} +819.176 q^{67} -793.818 q^{69} -1042.32 q^{71} +728.809 q^{73} +1408.68 q^{75} -1475.69 q^{77} -351.515 q^{79} +46.1114 q^{81} +1134.30 q^{83} -87.0198 q^{87} -137.678 q^{89} +1997.62 q^{91} +2434.42 q^{93} +1046.34 q^{95} -636.474 q^{97} +2206.07 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 51 q^{7} + 120 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 51 * q^7 + 120 * q^9	$ 18 q + 18 q^{3} + 51 q^{7} + 120 q^{9} + 132 q^{11} + 30 q^{13} + 102 q^{15} + 66 q^{19} + 144 q^{21} + 153 q^{23} + 306 q^{25} + 768 q^{27} + 51 q^{29} + 303 q^{31} + 525 q^{33} - 255 q^{35} + 717 q^{37} - 216 q^{39} - 393 q^{41} - 390 q^{43} + 558 q^{45} - 633 q^{47} + 1443 q^{49} + 1275 q^{53} + 1539 q^{55} + 810 q^{57} - 204 q^{59} + 534 q^{61} + 2556 q^{63} - 2127 q^{65} - 405 q^{67} + 2547 q^{69} - 426 q^{71} + 1149 q^{73} + 2226 q^{75} - 357 q^{77} + 1053 q^{79} + 2802 q^{81} + 66 q^{83} + 2487 q^{87} - 4119 q^{89} + 6090 q^{91} + 606 q^{93} + 2109 q^{95} + 2349 q^{97} + 1428 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 51 * q^7 + 120 * q^9 + 132 * q^11 + 30 * q^13 + 102 * q^15 + 66 * q^19 + 144 * q^21 + 153 * q^23 + 306 * q^25 + 768 * q^27 + 51 * q^29 + 303 * q^31 + 525 * q^33 - 255 * q^35 + 717 * q^37 - 216 * q^39 - 393 * q^41 - 390 * q^43 + 558 * q^45 - 633 * q^47 + 1443 * q^49 + 1275 * q^53 + 1539 * q^55 + 810 * q^57 - 204 * q^59 + 534 * q^61 + 2556 * q^63 - 2127 * q^65 - 405 * q^67 + 2547 * q^69 - 426 * q^71 + 1149 * q^73 + 2226 * q^75 - 357 * q^77 + 1053 * q^79 + 2802 * q^81 + 66 * q^83 + 2487 * q^87 - 4119 * q^89 + 6090 * q^91 + 606 * q^93 + 2109 * q^95 + 2349 * q^97 + 1428 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	8.45229	1.62664	0.813322	−	0.581813i	$-0.197657\pi$
$3$	8.45229	1.62664	0.813322	+	0.581813i	$0.197657\pi$
$4$	0	0
$5$	17.0781	1.52751	0.763757	−	0.645504i	$-0.223353\pi$
$5$	17.0781	1.52751	0.763757	+	0.645504i	$0.223353\pi$
$6$	0	0
$7$	−29.7278	−1.60515	−0.802576	−	0.596550i	$-0.796538\pi$
$7$	−29.7278	−1.60515	−0.802576	+	0.596550i	$0.796538\pi$
$8$	0	0
$9$	44.4413	1.64597
$10$	0	0
$11$	49.6400	1.36064	0.680320	−	0.732915i	$-0.261841\pi$
$11$	49.6400	1.36064	0.680320	+	0.732915i	$0.261841\pi$
$12$	0	0
$13$	−67.1970	−1.43362	−0.716812	−	0.697267i	$-0.754399\pi$
$13$	−67.1970	−1.43362	−0.716812	+	0.697267i	$0.754399\pi$
$14$	0	0
$15$	144.349	2.48472
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	61.2676	0.739777	0.369888	−	0.929076i	$-0.379396\pi$
$19$	61.2676	0.739777	0.369888	+	0.929076i	$0.379396\pi$
$20$	0	0
$21$	−251.268	−2.61101
$22$	0	0
$23$	−93.9175	−0.851441	−0.425721	−	0.904855i	$-0.639979\pi$
$23$	−93.9175	−0.851441	−0.425721	+	0.904855i	$0.639979\pi$
$24$	0	0
$25$	166.662	1.33330
$26$	0	0
$27$	147.419	1.05077
$28$	0	0
$29$	−10.2954	−0.0659244	−0.0329622	−	0.999457i	$-0.510494\pi$
$29$	−10.2954	−0.0659244	−0.0329622	+	0.999457i	$0.510494\pi$
$30$	0	0
$31$	288.019	1.66870	0.834350	−	0.551235i	$-0.185843\pi$
$31$	288.019	1.66870	0.834350	+	0.551235i	$0.185843\pi$
$32$	0	0
$33$	419.572	2.21328
$34$	0	0
$35$	−507.696	−2.45189
$36$	0	0
$37$	345.264	1.53408	0.767041	−	0.641598i	$-0.221728\pi$
$37$	345.264	1.53408	0.767041	+	0.641598i	$0.221728\pi$
$38$	0	0
$39$	−567.969	−2.33200
$40$	0	0
$41$	50.3858	0.191925	0.0959627	−	0.995385i	$-0.469407\pi$
$41$	50.3858	0.191925	0.0959627	+	0.995385i	$0.469407\pi$
$42$	0	0
$43$	−83.3053	−0.295440	−0.147720	−	0.989029i	$-0.547193\pi$
$43$	−83.3053	−0.295440	−0.147720	+	0.989029i	$0.547193\pi$
$44$	0	0
$45$	758.973	2.51425
$46$	0	0
$47$	201.195	0.624411	0.312205	−	0.950015i	$-0.398932\pi$
$47$	201.195	0.624411	0.312205	+	0.950015i	$0.398932\pi$
$48$	0	0
$49$	540.744	1.57651
$50$	0	0
$51$	0	0
$52$	0	0
$53$	398.772	1.03350	0.516751	−	0.856136i	$-0.327141\pi$
$53$	398.772	1.03350	0.516751	+	0.856136i	$0.327141\pi$
$54$	0	0
$55$	847.759	2.07840
$56$	0	0
$57$	517.852	1.20335
$58$	0	0
$59$	69.6651	0.153722	0.0768612	−	0.997042i	$-0.475510\pi$
$59$	69.6651	0.153722	0.0768612	+	0.997042i	$0.475510\pi$
$60$	0	0
$61$	310.111	0.650913	0.325457	−	0.945557i	$-0.394482\pi$
$61$	310.111	0.650913	0.325457	+	0.945557i	$0.394482\pi$
$62$	0	0
$63$	−1321.14	−2.64204
$64$	0	0
$65$	−1147.60	−2.18988
$66$	0	0
$67$	819.176	1.49371	0.746853	−	0.664990i	$-0.231564\pi$
$67$	819.176	1.49371	0.746853	+	0.664990i	$0.231564\pi$
$68$	0	0
$69$	−793.818	−1.38499
$70$	0	0
$71$	−1042.32	−1.74226	−0.871129	−	0.491053i	$-0.836612\pi$
$71$	−1042.32	−1.74226	−0.871129	+	0.491053i	$0.836612\pi$
$72$	0	0
$73$	728.809	1.16850	0.584251	−	0.811573i	$-0.301388\pi$
$73$	728.809	1.16850	0.584251	+	0.811573i	$0.301388\pi$
$74$	0	0
$75$	1408.68	2.16880
$76$	0	0
$77$	−1475.69	−2.18403
$78$	0	0
$79$	−351.515	−0.500615	−0.250307	−	0.968166i	$-0.580532\pi$
$79$	−351.515	−0.500615	−0.250307	+	0.968166i	$0.580532\pi$
$80$	0	0
$81$	46.1114	0.0632530
$82$	0	0
$83$	1134.30	1.50007	0.750033	−	0.661400i	$-0.230037\pi$
$83$	1134.30	1.50007	0.750033	+	0.661400i	$0.230037\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−87.0198	−0.107236
$88$	0	0
$89$	−137.678	−0.163976	−0.0819878	−	0.996633i	$-0.526127\pi$
$89$	−137.678	−0.163976	−0.0819878	+	0.996633i	$0.526127\pi$
$90$	0	0
$91$	1997.62	2.30118
$92$	0	0
$93$	2434.42	2.71438
$94$	0	0
$95$	1046.34	1.13002
$96$	0	0
$97$	−636.474	−0.666228	−0.333114	−	0.942887i	$-0.608099\pi$
$97$	−636.474	−0.666228	−0.333114	+	0.942887i	$0.608099\pi$
$98$	0	0
$99$	2206.07	2.23958
$100$	0	0
$101$	1333.73	1.31397	0.656986	−	0.753903i	$-0.271831\pi$
$101$	1333.73	1.31397	0.656986	+	0.753903i	$0.271831\pi$
$102$	0	0
$103$	826.016	0.790192	0.395096	−	0.918640i	$-0.370711\pi$
$103$	826.016	0.790192	0.395096	+	0.918640i	$0.370711\pi$
$104$	0	0
$105$	−4291.19	−3.98836
$106$	0	0
$107$	62.8955	0.0568256	0.0284128	−	0.999596i	$-0.490955\pi$
$107$	62.8955	0.0568256	0.0284128	+	0.999596i	$0.490955\pi$
$108$	0	0
$109$	−1807.02	−1.58790	−0.793950	−	0.607984i	$-0.791979\pi$
$109$	−1807.02	−1.58790	−0.793950	+	0.607984i	$0.791979\pi$
$110$	0	0
$111$	2918.27	2.49541
$112$	0	0
$113$	−249.241	−0.207493	−0.103746	−	0.994604i	$-0.533083\pi$
$113$	−249.241	−0.207493	−0.103746	+	0.994604i	$0.533083\pi$
$114$	0	0
$115$	−1603.93	−1.30059
$116$	0	0
$117$	−2986.32	−2.35971
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1133.13	0.851341
$122$	0	0
$123$	425.875	0.312194
$124$	0	0
$125$	711.515	0.509119
$126$	0	0
$127$	−429.655	−0.300202	−0.150101	−	0.988671i	$-0.547960\pi$
$127$	−429.655	−0.300202	−0.150101	+	0.988671i	$0.547960\pi$
$128$	0	0
$129$	−704.120	−0.480576
$130$	0	0
$131$	980.514	0.653953	0.326977	−	0.945032i	$-0.393970\pi$
$131$	980.514	0.653953	0.326977	+	0.945032i	$0.393970\pi$
$132$	0	0
$133$	−1821.35	−1.18745
$134$	0	0
$135$	2517.63	1.60506
$136$	0	0
$137$	440.529	0.274722	0.137361	−	0.990521i	$-0.456138\pi$
$137$	440.529	0.274722	0.137361	+	0.990521i	$0.456138\pi$
$138$	0	0
$139$	851.800	0.519775	0.259888	−	0.965639i	$-0.416314\pi$
$139$	851.800	0.519775	0.259888	+	0.965639i	$0.416314\pi$
$140$	0	0
$141$	1700.56	1.01569
$142$	0	0
$143$	−3335.66	−1.95065
$144$	0	0
$145$	−175.826	−0.100700
$146$	0	0
$147$	4570.53	2.56443
$148$	0	0
$149$	1174.43	0.645728	0.322864	−	0.946445i	$-0.395354\pi$
$149$	1174.43	0.645728	0.322864	+	0.946445i	$0.395354\pi$
$150$	0	0
$151$	−3380.75	−1.82200	−0.910999	−	0.412408i	$-0.864688\pi$
$151$	−3380.75	−1.82200	−0.910999	+	0.412408i	$0.864688\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4918.82	2.54896
$156$	0	0
$157$	2565.63	1.30420	0.652101	−	0.758132i	$-0.273888\pi$
$157$	2565.63	1.30420	0.652101	+	0.758132i	$0.273888\pi$
$158$	0	0
$159$	3370.54	1.68114
$160$	0	0
$161$	2791.96	1.36669
$162$	0	0
$163$	1843.88	0.886034	0.443017	−	0.896513i	$-0.353908\pi$
$163$	1843.88	0.886034	0.443017	+	0.896513i	$0.353908\pi$
$164$	0	0
$165$	7165.51	3.38081
$166$	0	0
$167$	1315.34	0.609484	0.304742	−	0.952435i	$-0.401430\pi$
$167$	1315.34	0.609484	0.304742	+	0.952435i	$0.401430\pi$
$168$	0	0
$169$	2318.44	1.05528
$170$	0	0
$171$	2722.81	1.21765
$172$	0	0
$173$	−824.943	−0.362539	−0.181269	−	0.983433i	$-0.558021\pi$
$173$	−824.943	−0.362539	−0.181269	+	0.983433i	$0.558021\pi$
$174$	0	0
$175$	−4954.51	−2.14015
$176$	0	0
$177$	588.830	0.250052
$178$	0	0
$179$	828.515	0.345956	0.172978	−	0.984926i	$-0.444661\pi$
$179$	828.515	0.345956	0.172978	+	0.984926i	$0.444661\pi$
$180$	0	0
$181$	−3120.70	−1.28155	−0.640774	−	0.767730i	$-0.721386\pi$
$181$	−3120.70	−1.28155	−0.640774	+	0.767730i	$0.721386\pi$
$182$	0	0
$183$	2621.15	1.05880
$184$	0	0
$185$	5896.46	2.34333
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4382.44	−1.68664
$190$	0	0
$191$	−2360.26	−0.894148	−0.447074	−	0.894497i	$-0.647534\pi$
$191$	−2360.26	−0.894148	−0.447074	+	0.894497i	$0.647534\pi$
$192$	0	0
$193$	−686.984	−0.256219	−0.128109	−	0.991760i	$-0.540891\pi$
$193$	−686.984	−0.256219	−0.128109	+	0.991760i	$0.540891\pi$
$194$	0	0
$195$	−9699.85	−3.56216
$196$	0	0
$197$	−1435.15	−0.519037	−0.259518	−	0.965738i	$-0.583564\pi$
$197$	−1435.15	−0.519037	−0.259518	+	0.965738i	$0.583564\pi$
$198$	0	0
$199$	4829.56	1.72039	0.860197	−	0.509963i	$-0.170341\pi$
$199$	4829.56	1.72039	0.860197	+	0.509963i	$0.170341\pi$
$200$	0	0
$201$	6923.91	2.42973
$202$	0	0
$203$	306.060	0.105819
$204$	0	0
$205$	860.495	0.293169
$206$	0	0
$207$	−4173.81	−1.40145
$208$	0	0
$209$	3041.33	1.00657
$210$	0	0
$211$	−1975.10	−0.644414	−0.322207	−	0.946669i	$-0.604425\pi$
$211$	−1975.10	−0.644414	−0.322207	+	0.946669i	$0.604425\pi$
$212$	0	0
$213$	−8809.98	−2.83404
$214$	0	0
$215$	−1422.70	−0.451289
$216$	0	0
$217$	−8562.18	−2.67852
$218$	0	0
$219$	6160.11	1.90074
$220$	0	0
$221$	0	0
$222$	0	0
$223$	839.948	0.252229	0.126115	−	0.992016i	$-0.459749\pi$
$223$	839.948	0.252229	0.126115	+	0.992016i	$0.459749\pi$
$224$	0	0
$225$	7406.69	2.19457
$226$	0	0
$227$	1984.22	0.580166	0.290083	−	0.957002i	$-0.406317\pi$
$227$	1984.22	0.580166	0.290083	+	0.957002i	$0.406317\pi$
$228$	0	0
$229$	−4631.38	−1.33646	−0.668231	−	0.743953i	$-0.732948\pi$
$229$	−4631.38	−1.33646	−0.668231	+	0.743953i	$0.732948\pi$
$230$	0	0
$231$	−12473.0	−3.55265
$232$	0	0
$233$	583.910	0.164177	0.0820885	−	0.996625i	$-0.473841\pi$
$233$	583.910	0.164177	0.0820885	+	0.996625i	$0.473841\pi$
$234$	0	0
$235$	3436.03	0.953796
$236$	0	0
$237$	−2971.11	−0.814322
$238$	0	0
$239$	−5190.18	−1.40471	−0.702353	−	0.711828i	$-0.747867\pi$
$239$	−5190.18	−1.40471	−0.702353	+	0.711828i	$0.747867\pi$
$240$	0	0
$241$	−3352.95	−0.896194	−0.448097	−	0.893985i	$-0.647898\pi$
$241$	−3352.95	−0.896194	−0.448097	+	0.893985i	$0.647898\pi$
$242$	0	0
$243$	−3590.56	−0.947878
$244$	0	0
$245$	9234.90	2.40815
$246$	0	0
$247$	−4117.00	−1.06056
$248$	0	0
$249$	9587.43	2.44007
$250$	0	0
$251$	−2957.51	−0.743729	−0.371865	−	0.928287i	$-0.621281\pi$
$251$	−2957.51	−0.743729	−0.371865	+	0.928287i	$0.621281\pi$
$252$	0	0
$253$	−4662.07	−1.15850
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7327.39	−1.77848	−0.889241	−	0.457439i	$-0.848767\pi$
$257$	−7327.39	−1.77848	−0.889241	+	0.457439i	$0.848767\pi$
$258$	0	0
$259$	−10263.9	−2.46244
$260$	0	0
$261$	−457.541	−0.108510
$262$	0	0
$263$	1770.01	0.414995	0.207498	−	0.978236i	$-0.433468\pi$
$263$	1770.01	0.414995	0.207498	+	0.978236i	$0.433468\pi$
$264$	0	0
$265$	6810.28	1.57869
$266$	0	0
$267$	−1163.69	−0.266730
$268$	0	0
$269$	−4962.67	−1.12483	−0.562415	−	0.826855i	$-0.690128\pi$
$269$	−4962.67	−1.12483	−0.562415	+	0.826855i	$0.690128\pi$
$270$	0	0
$271$	−5101.29	−1.14347	−0.571736	−	0.820437i	$-0.693730\pi$
$271$	−5101.29	−1.14347	−0.571736	+	0.820437i	$0.693730\pi$
$272$	0	0
$273$	16884.5	3.74321
$274$	0	0
$275$	8273.13	1.81414
$276$	0	0
$277$	395.095	0.0857002	0.0428501	−	0.999082i	$-0.486356\pi$
$277$	395.095	0.0857002	0.0428501	+	0.999082i	$0.486356\pi$
$278$	0	0
$279$	12799.9	2.74663
$280$	0	0
$281$	843.542	0.179080	0.0895401	−	0.995983i	$-0.471460\pi$
$281$	843.542	0.179080	0.0895401	+	0.995983i	$0.471460\pi$
$282$	0	0
$283$	1850.61	0.388719	0.194359	−	0.980930i	$-0.437737\pi$
$283$	1850.61	0.388719	0.194359	+	0.980930i	$0.437737\pi$
$284$	0	0
$285$	8843.94	1.83814
$286$	0	0
$287$	−1497.86	−0.308069
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−5379.66	−1.08372
$292$	0	0
$293$	−8340.67	−1.66303	−0.831514	−	0.555504i	$-0.812525\pi$
$293$	−8340.67	−1.66303	−0.831514	+	0.555504i	$0.812525\pi$
$294$	0	0
$295$	1189.75	0.234813
$296$	0	0
$297$	7317.87	1.42972
$298$	0	0
$299$	6310.98	1.22065
$300$	0	0
$301$	2476.49	0.474227
$302$	0	0
$303$	11273.1	2.13737
$304$	0	0
$305$	5296.12	0.994279
$306$	0	0
$307$	−5115.33	−0.950968	−0.475484	−	0.879724i	$-0.657727\pi$
$307$	−5115.33	−0.950968	−0.475484	+	0.879724i	$0.657727\pi$
$308$	0	0
$309$	6981.73	1.28536
$310$	0	0
$311$	−4257.49	−0.776270	−0.388135	−	0.921602i	$-0.626881\pi$
$311$	−4257.49	−0.776270	−0.388135	+	0.921602i	$0.626881\pi$
$312$	0	0
$313$	−9653.21	−1.74323	−0.871616	−	0.490189i	$-0.836928\pi$
$313$	−9653.21	−1.74323	−0.871616	+	0.490189i	$0.836928\pi$
$314$	0	0
$315$	−22562.6	−4.03575
$316$	0	0
$317$	758.920	0.134464	0.0672321	−	0.997737i	$-0.478583\pi$
$317$	758.920	0.134464	0.0672321	+	0.997737i	$0.478583\pi$
$318$	0	0
$319$	−511.064	−0.0896994
$320$	0	0
$321$	531.611	0.0924350
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−11199.2	−1.91145
$326$	0	0
$327$	−15273.5	−2.58295
$328$	0	0
$329$	−5981.09	−1.00227
$330$	0	0
$331$	4474.29	0.742988	0.371494	−	0.928435i	$-0.378846\pi$
$331$	4474.29	0.742988	0.371494	+	0.928435i	$0.378846\pi$
$332$	0	0
$333$	15344.0	2.52506
$334$	0	0
$335$	13990.0	2.28166
$336$	0	0
$337$	4115.13	0.665179	0.332590	−	0.943072i	$-0.392078\pi$
$337$	4115.13	0.665179	0.332590	+	0.943072i	$0.392078\pi$
$338$	0	0
$339$	−2106.66	−0.337517
$340$	0	0
$341$	14297.3	2.27050
$342$	0	0
$343$	−5878.51	−0.925392
$344$	0	0
$345$	−13556.9	−2.11560
$346$	0	0
$347$	3755.73	0.581032	0.290516	−	0.956870i	$-0.406173\pi$
$347$	3755.73	0.581032	0.290516	+	0.956870i	$0.406173\pi$
$348$	0	0
$349$	−4192.69	−0.643065	−0.321532	−	0.946899i	$-0.604198\pi$
$349$	−4192.69	−0.643065	−0.321532	+	0.946899i	$0.604198\pi$
$350$	0	0
$351$	−9906.10	−1.50641
$352$	0	0
$353$	5372.21	0.810011	0.405005	−	0.914314i	$-0.367270\pi$
$353$	5372.21	0.810011	0.405005	+	0.914314i	$0.367270\pi$
$354$	0	0
$355$	−17800.8	−2.66132
$356$	0	0
$357$	0	0
$358$	0	0
$359$	811.576	0.119313	0.0596565	−	0.998219i	$-0.480999\pi$
$359$	811.576	0.119313	0.0596565	+	0.998219i	$0.480999\pi$
$360$	0	0
$361$	−3105.28	−0.452730
$362$	0	0
$363$	9577.58	1.38483
$364$	0	0
$365$	12446.7	1.78490
$366$	0	0
$367$	4773.43	0.678940	0.339470	−	0.940617i	$-0.389752\pi$
$367$	4773.43	0.678940	0.339470	+	0.940617i	$0.389752\pi$
$368$	0	0
$369$	2239.21	0.315904
$370$	0	0
$371$	−11854.6	−1.65893
$372$	0	0
$373$	−1309.93	−0.181838	−0.0909192	−	0.995858i	$-0.528980\pi$
$373$	−1309.93	−0.181838	−0.0909192	+	0.995858i	$0.528980\pi$
$374$	0	0
$375$	6013.94	0.828156
$376$	0	0
$377$	691.821	0.0945108
$378$	0	0
$379$	5099.43	0.691135	0.345568	−	0.938394i	$-0.387686\pi$
$379$	5099.43	0.691135	0.345568	+	0.938394i	$0.387686\pi$
$380$	0	0
$381$	−3631.57	−0.488322
$382$	0	0
$383$	2429.08	0.324074	0.162037	−	0.986785i	$-0.448194\pi$
$383$	2429.08	0.324074	0.162037	+	0.986785i	$0.448194\pi$
$384$	0	0
$385$	−25202.0	−3.33614
$386$	0	0
$387$	−3702.19	−0.486287
$388$	0	0
$389$	14671.3	1.91224	0.956122	−	0.292970i	$-0.0946435\pi$
$389$	14671.3	1.91224	0.956122	+	0.292970i	$0.0946435\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	8287.59	1.06375
$394$	0	0
$395$	−6003.22	−0.764696
$396$	0	0
$397$	−2415.58	−0.305377	−0.152688	−	0.988274i	$-0.548793\pi$
$397$	−2415.58	−0.305377	−0.152688	+	0.988274i	$0.548793\pi$
$398$	0	0
$399$	−15394.6	−1.93157
$400$	0	0
$401$	388.973	0.0484399	0.0242200	−	0.999707i	$-0.492290\pi$
$401$	388.973	0.0484399	0.0242200	+	0.999707i	$0.492290\pi$
$402$	0	0
$403$	−19354.0	−2.39229
$404$	0	0
$405$	787.496	0.0966198
$406$	0	0
$407$	17138.9	2.08733
$408$	0	0
$409$	1325.56	0.160256	0.0801279	−	0.996785i	$-0.474467\pi$
$409$	1325.56	0.160256	0.0801279	+	0.996785i	$0.474467\pi$
$410$	0	0
$411$	3723.48	0.446875
$412$	0	0
$413$	−2070.99	−0.246748
$414$	0	0
$415$	19371.7	2.29137
$416$	0	0
$417$	7199.67	0.845490
$418$	0	0
$419$	4567.74	0.532575	0.266287	−	0.963894i	$-0.414203\pi$
$419$	4567.74	0.532575	0.266287	+	0.963894i	$0.414203\pi$
$420$	0	0
$421$	−3496.17	−0.404734	−0.202367	−	0.979310i	$-0.564863\pi$
$421$	−3496.17	−0.404734	−0.202367	+	0.979310i	$0.564863\pi$
$422$	0	0
$423$	8941.36	1.02776
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−9218.94	−1.04481
$428$	0	0
$429$	−28194.0	−3.17301
$430$	0	0
$431$	−13614.7	−1.52158	−0.760788	−	0.649001i	$-0.775187\pi$
$431$	−13614.7	−1.52158	−0.760788	+	0.649001i	$0.775187\pi$
$432$	0	0
$433$	−13901.8	−1.54290	−0.771450	−	0.636290i	$-0.780468\pi$
$433$	−13901.8	−1.54290	−0.771450	+	0.636290i	$0.780468\pi$
$434$	0	0
$435$	−1486.13	−0.163804
$436$	0	0
$437$	−5754.10	−0.629877
$438$	0	0
$439$	11575.6	1.25848	0.629240	−	0.777211i	$-0.283366\pi$
$439$	11575.6	1.25848	0.629240	+	0.777211i	$0.283366\pi$
$440$	0	0
$441$	24031.4	2.59490
$442$	0	0
$443$	16252.9	1.74311	0.871557	−	0.490294i	$-0.163111\pi$
$443$	16252.9	1.74311	0.871557	+	0.490294i	$0.163111\pi$
$444$	0	0
$445$	−2351.28	−0.250475
$446$	0	0
$447$	9926.67	1.05037
$448$	0	0
$449$	−2793.63	−0.293629	−0.146814	−	0.989164i	$-0.546902\pi$
$449$	−2793.63	−0.293629	−0.146814	+	0.989164i	$0.546902\pi$
$450$	0	0
$451$	2501.15	0.261141
$452$	0	0
$453$	−28575.1	−2.96374
$454$	0	0
$455$	34115.7	3.51509
$456$	0	0
$457$	6554.55	0.670916	0.335458	−	0.942055i	$-0.391109\pi$
$457$	6554.55	0.670916	0.335458	+	0.942055i	$0.391109\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18115.9	−1.83024	−0.915121	−	0.403178i	$-0.867905\pi$
$461$	−18115.9	−1.83024	−0.915121	+	0.403178i	$0.867905\pi$
$462$	0	0
$463$	9668.53	0.970485	0.485242	−	0.874380i	$-0.338731\pi$
$463$	9668.53	0.970485	0.485242	+	0.874380i	$0.338731\pi$
$464$	0	0
$465$	41575.3	4.14626
$466$	0	0
$467$	−1777.43	−0.176124	−0.0880619	−	0.996115i	$-0.528067\pi$
$467$	−1777.43	−0.176124	−0.0880619	+	0.996115i	$0.528067\pi$
$468$	0	0
$469$	−24352.3	−2.39762
$470$	0	0
$471$	21685.5	2.12147
$472$	0	0
$473$	−4135.28	−0.401988
$474$	0	0
$475$	10211.0	0.986344
$476$	0	0
$477$	17721.9	1.70112
$478$	0	0
$479$	−14388.9	−1.37254	−0.686268	−	0.727349i	$-0.740752\pi$
$479$	−14388.9	−1.37254	−0.686268	+	0.727349i	$0.740752\pi$
$480$	0	0
$481$	−23200.7	−2.19930
$482$	0	0
$483$	23598.5	2.22312
$484$	0	0
$485$	−10869.8	−1.01767
$486$	0	0
$487$	1901.73	0.176952	0.0884761	−	0.996078i	$-0.471800\pi$
$487$	1901.73	0.176952	0.0884761	+	0.996078i	$0.471800\pi$
$488$	0	0
$489$	15585.0	1.44126
$490$	0	0
$491$	17567.4	1.61468	0.807338	−	0.590089i	$-0.200907\pi$
$491$	17567.4	1.61468	0.807338	+	0.590089i	$0.200907\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	37675.5	3.42098
$496$	0	0
$497$	30985.9	2.79659
$498$	0	0
$499$	−15183.1	−1.36210	−0.681052	−	0.732235i	$-0.738477\pi$
$499$	−15183.1	−1.36210	−0.681052	+	0.732235i	$0.738477\pi$
$500$	0	0
$501$	11117.6	0.991414
$502$	0	0
$503$	11760.1	1.04246	0.521228	−	0.853418i	$-0.325474\pi$
$503$	11760.1	1.04246	0.521228	+	0.853418i	$0.325474\pi$
$504$	0	0
$505$	22777.6	2.00711
$506$	0	0
$507$	19596.2	1.71656
$508$	0	0
$509$	−7919.19	−0.689611	−0.344806	−	0.938674i	$-0.612055\pi$
$509$	−7919.19	−0.689611	−0.344806	+	0.938674i	$0.612055\pi$
$510$	0	0
$511$	−21665.9	−1.87562
$512$	0	0
$513$	9031.99	0.777333
$514$	0	0
$515$	14106.8	1.20703
$516$	0	0
$517$	9987.33	0.849598
$518$	0	0
$519$	−6972.66	−0.589722
$520$	0	0
$521$	9794.45	0.823614	0.411807	−	0.911271i	$-0.364898\pi$
$521$	9794.45	0.823614	0.411807	+	0.911271i	$0.364898\pi$
$522$	0	0
$523$	−14470.2	−1.20983	−0.604914	−	0.796291i	$-0.706792\pi$
$523$	−14470.2	−1.20983	−0.604914	+	0.796291i	$0.706792\pi$
$524$	0	0
$525$	−41877.0	−3.48126
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3346.50	−0.275048
$530$	0	0
$531$	3096.00	0.253023
$532$	0	0
$533$	−3385.78	−0.275149
$534$	0	0
$535$	1074.14	0.0868018
$536$	0	0
$537$	7002.85	0.562747
$538$	0	0
$539$	26842.6	2.14507
$540$	0	0
$541$	21316.2	1.69400	0.847002	−	0.531589i	$-0.178405\pi$
$541$	21316.2	1.69400	0.847002	+	0.531589i	$0.178405\pi$
$542$	0	0
$543$	−26377.1	−2.08462
$544$	0	0
$545$	−30860.5	−2.42554
$546$	0	0
$547$	−4404.84	−0.344309	−0.172155	−	0.985070i	$-0.555073\pi$
$547$	−4404.84	−0.344309	−0.172155	+	0.985070i	$0.555073\pi$
$548$	0	0
$549$	13781.7	1.07138
$550$	0	0
$551$	−630.775	−0.0487694
$552$	0	0
$553$	10449.8	0.803563
$554$	0	0
$555$	49838.6	3.81177
$556$	0	0
$557$	−8518.80	−0.648031	−0.324016	−	0.946052i	$-0.605033\pi$
$557$	−8518.80	−0.648031	−0.324016	+	0.946052i	$0.605033\pi$
$558$	0	0
$559$	5597.87	0.423550
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10450.3	−0.782289	−0.391144	−	0.920329i	$-0.627921\pi$
$563$	−10450.3	−0.782289	−0.391144	+	0.920329i	$0.627921\pi$
$564$	0	0
$565$	−4256.58	−0.316948
$566$	0	0
$567$	−1370.79	−0.101531
$568$	0	0
$569$	−15139.1	−1.11541	−0.557703	−	0.830041i	$-0.688317\pi$
$569$	−15139.1	−1.11541	−0.557703	+	0.830041i	$0.688317\pi$
$570$	0	0
$571$	13756.5	1.00821	0.504107	−	0.863641i	$-0.331822\pi$
$571$	13756.5	1.00821	0.504107	+	0.863641i	$0.331822\pi$
$572$	0	0
$573$	−19949.6	−1.45446
$574$	0	0
$575$	−15652.5	−1.13523
$576$	0	0
$577$	24916.2	1.79770	0.898850	−	0.438256i	$-0.144404\pi$
$577$	24916.2	1.79770	0.898850	+	0.438256i	$0.144404\pi$
$578$	0	0
$579$	−5806.59	−0.416776
$580$	0	0
$581$	−33720.3	−2.40783
$582$	0	0
$583$	19795.1	1.40622
$584$	0	0
$585$	−51000.8	−3.60448
$586$	0	0
$587$	11026.2	0.775296	0.387648	−	0.921808i	$-0.373288\pi$
$587$	11026.2	0.775296	0.387648	+	0.921808i	$0.373288\pi$
$588$	0	0
$589$	17646.2	1.23447
$590$	0	0
$591$	−12130.3	−0.844288
$592$	0	0
$593$	−5204.96	−0.360442	−0.180221	−	0.983626i	$-0.557681\pi$
$593$	−5204.96	−0.360442	−0.180221	+	0.983626i	$0.557681\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	40820.8	2.79847
$598$	0	0
$599$	8309.34	0.566795	0.283398	−	0.959002i	$-0.408538\pi$
$599$	8309.34	0.566795	0.283398	+	0.959002i	$0.408538\pi$
$600$	0	0
$601$	15392.3	1.04470	0.522349	−	0.852732i	$-0.325056\pi$
$601$	15392.3	1.04470	0.522349	+	0.852732i	$0.325056\pi$
$602$	0	0
$603$	36405.2	2.45860
$604$	0	0
$605$	19351.8	1.30043
$606$	0	0
$607$	4303.81	0.287786	0.143893	−	0.989593i	$-0.454038\pi$
$607$	4303.81	0.287786	0.143893	+	0.989593i	$0.454038\pi$
$608$	0	0
$609$	2586.91	0.172130
$610$	0	0
$611$	−13519.7	−0.895170
$612$	0	0
$613$	−597.249	−0.0393518	−0.0196759	−	0.999806i	$-0.506263\pi$
$613$	−597.249	−0.0393518	−0.0196759	+	0.999806i	$0.506263\pi$
$614$	0	0
$615$	7273.15	0.476881
$616$	0	0
$617$	16387.1	1.06924	0.534620	−	0.845093i	$-0.320455\pi$
$617$	16387.1	1.06924	0.534620	+	0.845093i	$0.320455\pi$
$618$	0	0
$619$	10244.0	0.665169	0.332585	−	0.943073i	$-0.392079\pi$
$619$	10244.0	0.665169	0.332585	+	0.943073i	$0.392079\pi$
$620$	0	0
$621$	−13845.2	−0.894667
$622$	0	0
$623$	4092.87	0.263206
$624$	0	0
$625$	−8681.45	−0.555613
$626$	0	0
$627$	25706.2	1.63733
$628$	0	0
$629$	0	0
$630$	0	0
$631$	3847.21	0.242718	0.121359	−	0.992609i	$-0.461275\pi$
$631$	3847.21	0.242718	0.121359	+	0.992609i	$0.461275\pi$
$632$	0	0
$633$	−16694.1	−1.04823
$634$	0	0
$635$	−7337.69	−0.458563
$636$	0	0
$637$	−36336.4	−2.26013
$638$	0	0
$639$	−46321.9	−2.86771
$640$	0	0
$641$	−27670.3	−1.70501	−0.852505	−	0.522718i	$-0.824918\pi$
$641$	−27670.3	−1.70501	−0.852505	+	0.522718i	$0.824918\pi$
$642$	0	0
$643$	−56.8825	−0.00348869	−0.00174434	−	0.999998i	$-0.500555\pi$
$643$	−56.8825	−0.00348869	−0.00174434	+	0.999998i	$0.500555\pi$
$644$	0	0
$645$	−12025.1	−0.734087
$646$	0	0
$647$	26907.9	1.63502	0.817512	−	0.575912i	$-0.195353\pi$
$647$	26907.9	1.63502	0.817512	+	0.575912i	$0.195353\pi$
$648$	0	0
$649$	3458.18	0.209161
$650$	0	0
$651$	−72370.0	−4.35700
$652$	0	0
$653$	3952.95	0.236893	0.118446	−	0.992960i	$-0.462209\pi$
$653$	3952.95	0.236893	0.118446	+	0.992960i	$0.462209\pi$
$654$	0	0
$655$	16745.3	0.998923
$656$	0	0
$657$	32389.2	1.92332
$658$	0	0
$659$	−25227.3	−1.49122	−0.745612	−	0.666380i	$-0.767843\pi$
$659$	−25227.3	−1.49122	−0.745612	+	0.666380i	$0.767843\pi$
$660$	0	0
$661$	−19797.9	−1.16498	−0.582489	−	0.812838i	$-0.697921\pi$
$661$	−19797.9	−1.16498	−0.582489	+	0.812838i	$0.697921\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−31105.3	−1.81385
$666$	0	0
$667$	966.919	0.0561308
$668$	0	0
$669$	7099.49	0.410287
$670$	0	0
$671$	15393.9	0.885658
$672$	0	0
$673$	1736.56	0.0994642	0.0497321	−	0.998763i	$-0.484163\pi$
$673$	1736.56	0.0994642	0.0497321	+	0.998763i	$0.484163\pi$
$674$	0	0
$675$	24569.1	1.40099
$676$	0	0
$677$	−29129.0	−1.65364	−0.826822	−	0.562464i	$-0.809853\pi$
$677$	−29129.0	−1.65364	−0.826822	+	0.562464i	$0.809853\pi$
$678$	0	0
$679$	18921.0	1.06940
$680$	0	0
$681$	16771.3	0.943724
$682$	0	0
$683$	−28408.9	−1.59156	−0.795780	−	0.605586i	$-0.792939\pi$
$683$	−28408.9	−1.59156	−0.795780	+	0.605586i	$0.792939\pi$
$684$	0	0
$685$	7523.41	0.419642
$686$	0	0
$687$	−39145.8	−2.17395
$688$	0	0
$689$	−26796.3	−1.48165
$690$	0	0
$691$	12784.1	0.703807	0.351903	−	0.936036i	$-0.385535\pi$
$691$	12784.1	0.703807	0.351903	+	0.936036i	$0.385535\pi$
$692$	0	0
$693$	−65581.6	−3.59486
$694$	0	0
$695$	14547.2	0.793964
$696$	0	0
$697$	0	0
$698$	0	0
$699$	4935.38	0.267057
$700$	0	0
$701$	15017.9	0.809155	0.404577	−	0.914504i	$-0.367419\pi$
$701$	15017.9	0.809155	0.404577	+	0.914504i	$0.367419\pi$
$702$	0	0
$703$	21153.5	1.13488
$704$	0	0
$705$	29042.4	1.55149
$706$	0	0
$707$	−39648.9	−2.10913
$708$	0	0
$709$	−31017.1	−1.64298	−0.821489	−	0.570224i	$-0.806856\pi$
$709$	−31017.1	−1.64298	−0.821489	+	0.570224i	$0.806856\pi$
$710$	0	0
$711$	−15621.8	−0.823998
$712$	0	0
$713$	−27050.0	−1.42080
$714$	0	0
$715$	−56966.9	−2.97964
$716$	0	0
$717$	−43868.9	−2.28496
$718$	0	0
$719$	17617.6	0.913804	0.456902	−	0.889517i	$-0.348959\pi$
$719$	17617.6	0.913804	0.456902	+	0.889517i	$0.348959\pi$
$720$	0	0
$721$	−24555.7	−1.26838
$722$	0	0
$723$	−28340.1	−1.45779
$724$	0	0
$725$	−1715.86	−0.0878970
$726$	0	0
$727$	−26891.1	−1.37185	−0.685924	−	0.727673i	$-0.740602\pi$
$727$	−26891.1	−1.37185	−0.685924	+	0.727673i	$0.740602\pi$
$728$	0	0
$729$	−31593.4	−1.60511
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−2397.78	−0.120824	−0.0604121	−	0.998174i	$-0.519241\pi$
$733$	−2397.78	−0.120824	−0.0604121	+	0.998174i	$0.519241\pi$
$734$	0	0
$735$	78056.1	3.91720
$736$	0	0
$737$	40663.9	2.03239
$738$	0	0
$739$	−10704.0	−0.532820	−0.266410	−	0.963860i	$-0.585837\pi$
$739$	−10704.0	−0.532820	−0.266410	+	0.963860i	$0.585837\pi$
$740$	0	0
$741$	−34798.1	−1.72516
$742$	0	0
$743$	31255.1	1.54326	0.771628	−	0.636074i	$-0.219443\pi$
$743$	31255.1	1.54326	0.771628	+	0.636074i	$0.219443\pi$
$744$	0	0
$745$	20057.1	0.986358
$746$	0	0
$747$	50409.7	2.46907
$748$	0	0
$749$	−1869.75	−0.0912137
$750$	0	0
$751$	9614.29	0.467151	0.233576	−	0.972339i	$-0.424957\pi$
$751$	9614.29	0.467151	0.233576	+	0.972339i	$0.424957\pi$
$752$	0	0
$753$	−24997.7	−1.20978
$754$	0	0
$755$	−57736.9	−2.78313
$756$	0	0
$757$	18829.3	0.904046	0.452023	−	0.892006i	$-0.350703\pi$
$757$	18829.3	0.904046	0.452023	+	0.892006i	$0.350703\pi$
$758$	0	0
$759$	−39405.2	−1.88448
$760$	0	0
$761$	16958.3	0.807804	0.403902	−	0.914802i	$-0.367654\pi$
$761$	16958.3	0.807804	0.403902	+	0.914802i	$0.367654\pi$
$762$	0	0
$763$	53718.8	2.54882
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4681.29	−0.220380
$768$	0	0
$769$	19770.1	0.927085	0.463542	−	0.886075i	$-0.346578\pi$
$769$	19770.1	0.927085	0.463542	+	0.886075i	$0.346578\pi$
$770$	0	0
$771$	−61933.2	−2.89296
$772$	0	0
$773$	35459.5	1.64992	0.824962	−	0.565188i	$-0.191196\pi$
$773$	35459.5	1.64992	0.824962	+	0.565188i	$0.191196\pi$
$774$	0	0
$775$	48001.9	2.22488
$776$	0	0
$777$	−86753.9	−4.00551
$778$	0	0
$779$	3087.02	0.141982
$780$	0	0
$781$	−51740.7	−2.37059
$782$	0	0
$783$	−1517.73	−0.0692713
$784$	0	0
$785$	43816.2	1.99219
$786$	0	0
$787$	2344.16	0.106176	0.0530879	−	0.998590i	$-0.483094\pi$
$787$	2344.16	0.106176	0.0530879	+	0.998590i	$0.483094\pi$
$788$	0	0
$789$	14960.7	0.675050
$790$	0	0
$791$	7409.41	0.333057
$792$	0	0
$793$	−20838.6	−0.933164
$794$	0	0
$795$	57562.5	2.56796
$796$	0	0
$797$	−19472.1	−0.865418	−0.432709	−	0.901534i	$-0.642442\pi$
$797$	−19472.1	−0.865418	−0.432709	+	0.901534i	$0.642442\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6118.58	−0.269899
$802$	0	0
$803$	36178.1	1.58991
$804$	0	0
$805$	47681.5	2.08764
$806$	0	0
$807$	−41945.9	−1.82970
$808$	0	0
$809$	23421.1	1.01785	0.508926	−	0.860810i	$-0.330043\pi$
$809$	23421.1	1.01785	0.508926	+	0.860810i	$0.330043\pi$
$810$	0	0
$811$	−23146.2	−1.00218	−0.501092	−	0.865394i	$-0.667068\pi$
$811$	−23146.2	−1.00218	−0.501092	+	0.865394i	$0.667068\pi$
$812$	0	0
$813$	−43117.6	−1.86002
$814$	0	0
$815$	31490.0	1.35343
$816$	0	0
$817$	−5103.92	−0.218560
$818$	0	0
$819$	88776.9	3.78769
$820$	0	0
$821$	−19984.8	−0.849542	−0.424771	−	0.905301i	$-0.639645\pi$
$821$	−19984.8	−0.849542	−0.424771	+	0.905301i	$0.639645\pi$
$822$	0	0
$823$	37756.9	1.59918	0.799588	−	0.600549i	$-0.205051\pi$
$823$	37756.9	1.59918	0.799588	+	0.600549i	$0.205051\pi$
$824$	0	0
$825$	69926.9	2.95096
$826$	0	0
$827$	27594.1	1.16027	0.580134	−	0.814521i	$-0.303000\pi$
$827$	27594.1	1.16027	0.580134	+	0.814521i	$0.303000\pi$
$828$	0	0
$829$	15631.5	0.654890	0.327445	−	0.944870i	$-0.393812\pi$
$829$	15631.5	0.654890	0.327445	+	0.944870i	$0.393812\pi$
$830$	0	0
$831$	3339.46	0.139404
$832$	0	0
$833$	0	0
$834$	0	0
$835$	22463.5	0.930995
$836$	0	0
$837$	42459.3	1.75342
$838$	0	0
$839$	7050.42	0.290116	0.145058	−	0.989423i	$-0.453663\pi$
$839$	7050.42	0.290116	0.145058	+	0.989423i	$0.453663\pi$
$840$	0	0
$841$	−24283.0	−0.995654
$842$	0	0
$843$	7129.87	0.291300
$844$	0	0
$845$	39594.7	1.61195
$846$	0	0
$847$	−33685.6	−1.36653
$848$	0	0
$849$	15641.9	0.632307
$850$	0	0
$851$	−32426.3	−1.30618
$852$	0	0
$853$	−21750.6	−0.873066	−0.436533	−	0.899688i	$-0.643794\pi$
$853$	−21750.6	−0.873066	−0.436533	+	0.899688i	$0.643794\pi$
$854$	0	0
$855$	46500.5	1.85998
$856$	0	0
$857$	−19592.1	−0.780925	−0.390462	−	0.920619i	$-0.627685\pi$
$857$	−19592.1	−0.780925	−0.390462	+	0.920619i	$0.627685\pi$
$858$	0	0
$859$	−1538.80	−0.0611214	−0.0305607	−	0.999533i	$-0.509729\pi$
$859$	−1538.80	−0.0611214	−0.0305607	+	0.999533i	$0.509729\pi$
$860$	0	0
$861$	−12660.4	−0.501119
$862$	0	0
$863$	−9161.98	−0.361387	−0.180694	−	0.983539i	$-0.557834\pi$
$863$	−9161.98	−0.361387	−0.180694	+	0.983539i	$0.557834\pi$
$864$	0	0
$865$	−14088.5	−0.553783
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−17449.2	−0.681156
$870$	0	0
$871$	−55046.2	−2.14141
$872$	0	0
$873$	−28285.7	−1.09659
$874$	0	0
$875$	−21151.8	−0.817213
$876$	0	0
$877$	46897.3	1.80571	0.902855	−	0.429945i	$-0.141467\pi$
$877$	46897.3	1.80571	0.902855	+	0.429945i	$0.141467\pi$
$878$	0	0
$879$	−70497.8	−2.70516
$880$	0	0
$881$	43003.6	1.64453	0.822263	−	0.569108i	$-0.192711\pi$
$881$	43003.6	1.64453	0.822263	+	0.569108i	$0.192711\pi$
$882$	0	0
$883$	−41978.0	−1.59986	−0.799928	−	0.600097i	$-0.795129\pi$
$883$	−41978.0	−1.59986	−0.799928	+	0.600097i	$0.795129\pi$
$884$	0	0
$885$	10056.1	0.381957
$886$	0	0
$887$	−7899.61	−0.299034	−0.149517	−	0.988759i	$-0.547772\pi$
$887$	−7899.61	−0.299034	−0.149517	+	0.988759i	$0.547772\pi$
$888$	0	0
$889$	12772.7	0.481870
$890$	0	0
$891$	2288.97	0.0860645
$892$	0	0
$893$	12326.7	0.461925
$894$	0	0
$895$	14149.5	0.528453
$896$	0	0
$897$	53342.2	1.98556
$898$	0	0
$899$	−2965.27	−0.110008
$900$	0	0
$901$	0	0
$902$	0	0
$903$	20932.0	0.771398
$904$	0	0
$905$	−53295.8	−1.95758
$906$	0	0
$907$	−7515.34	−0.275130	−0.137565	−	0.990493i	$-0.543928\pi$
$907$	−7515.34	−0.275130	−0.137565	+	0.990493i	$0.543928\pi$
$908$	0	0
$909$	59272.7	2.16276
$910$	0	0
$911$	−13089.6	−0.476047	−0.238023	−	0.971259i	$-0.576499\pi$
$911$	−13089.6	−0.476047	−0.238023	+	0.971259i	$0.576499\pi$
$912$	0	0
$913$	56306.7	2.04105
$914$	0	0
$915$	44764.4	1.61734
$916$	0	0
$917$	−29148.6	−1.04969
$918$	0	0
$919$	6706.94	0.240742	0.120371	−	0.992729i	$-0.461592\pi$
$919$	6706.94	0.240742	0.120371	+	0.992729i	$0.461592\pi$
$920$	0	0
$921$	−43236.2	−1.54689
$922$	0	0
$923$	70040.7	2.49774
$924$	0	0
$925$	57542.5	2.04539
$926$	0	0
$927$	36709.2	1.30063
$928$	0	0
$929$	−36444.9	−1.28710	−0.643550	−	0.765404i	$-0.722539\pi$
$929$	−36444.9	−1.28710	−0.643550	+	0.765404i	$0.722539\pi$
$930$	0	0
$931$	33130.1	1.16627
$932$	0	0
$933$	−35985.5	−1.26272
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15266.0	0.532250	0.266125	−	0.963939i	$-0.414257\pi$
$937$	15266.0	0.532250	0.266125	+	0.963939i	$0.414257\pi$
$938$	0	0
$939$	−81591.8	−2.83562
$940$	0	0
$941$	−53762.6	−1.86250	−0.931249	−	0.364385i	$-0.881279\pi$
$941$	−53762.6	−1.86250	−0.931249	+	0.364385i	$0.881279\pi$
$942$	0	0
$943$	−4732.11	−0.163413
$944$	0	0
$945$	−74843.8	−2.57637
$946$	0	0
$947$	26312.0	0.902876	0.451438	−	0.892302i	$-0.350911\pi$
$947$	26312.0	0.902876	0.451438	+	0.892302i	$0.350911\pi$
$948$	0	0
$949$	−48973.8	−1.67519
$950$	0	0
$951$	6414.61	0.218726
$952$	0	0
$953$	13453.8	0.457304	0.228652	−	0.973508i	$-0.426568\pi$
$953$	13453.8	0.457304	0.228652	+	0.973508i	$0.426568\pi$
$954$	0	0
$955$	−40308.8	−1.36582
$956$	0	0
$957$	−4319.67	−0.145909
$958$	0	0
$959$	−13096.0	−0.440971
$960$	0	0
$961$	53163.9	1.78456
$962$	0	0
$963$	2795.15	0.0935333
$964$	0	0
$965$	−11732.4	−0.391377
$966$	0	0
$967$	15356.9	0.510697	0.255348	−	0.966849i	$-0.417810\pi$
$967$	15356.9	0.510697	0.255348	+	0.966849i	$0.417810\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14810.7	−0.489493	−0.244747	−	0.969587i	$-0.578705\pi$
$971$	−14810.7	−0.489493	−0.244747	+	0.969587i	$0.578705\pi$
$972$	0	0
$973$	−25322.2	−0.834318
$974$	0	0
$975$	−94659.1	−3.10925
$976$	0	0
$977$	−12726.3	−0.416737	−0.208368	−	0.978050i	$-0.566815\pi$
$977$	−12726.3	−0.416737	−0.208368	+	0.978050i	$0.566815\pi$
$978$	0	0
$979$	−6834.34	−0.223112
$980$	0	0
$981$	−80306.2	−2.61364
$982$	0	0
$983$	11520.5	0.373803	0.186901	−	0.982379i	$-0.440156\pi$
$983$	11520.5	0.373803	0.186901	+	0.982379i	$0.440156\pi$
$984$	0	0
$985$	−24509.7	−0.792836
$986$	0	0
$987$	−50554.0	−1.63034
$988$	0	0
$989$	7823.82	0.251550
$990$	0	0
$991$	6319.55	0.202570	0.101285	−	0.994857i	$-0.467705\pi$
$991$	6319.55	0.202570	0.101285	+	0.994857i	$0.467705\pi$
$992$	0	0
$993$	37818.0	1.20858
$994$	0	0
$995$	82479.8	2.62792
$996$	0	0
$997$	19548.7	0.620977	0.310488	−	0.950577i	$-0.399507\pi$
$997$	19548.7	0.620977	0.310488	+	0.950577i	$0.399507\pi$
$998$	0	0
$999$	50898.3	1.61196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.q.1.16	yes	18
17.16	even	2		2312.4.a.n.1.3	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.n.1.3	✓	18	17.16	even	2
2312.4.a.q.1.16	yes	18	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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q+8.45229 q^{3} +17.0781 q^{5} -29.7278 q^{7} +44.4413 q^{9} +O(q^{10})\)	\(q+8.45229 q^{3} +17.0781 q^{5} -29.7278 q^{7} +44.4413 q^{9} +49.6400 q^{11} -67.1970 q^{13} +144.349 q^{15} +61.2676 q^{19} -251.268 q^{21} -93.9175 q^{23} +166.662 q^{25} +147.419 q^{27} -10.2954 q^{29} +288.019 q^{31} +419.572 q^{33} -507.696 q^{35} +345.264 q^{37} -567.969 q^{39} +50.3858 q^{41} -83.3053 q^{43} +758.973 q^{45} +201.195 q^{47} +540.744 q^{49} +398.772 q^{53} +847.759 q^{55} +517.852 q^{57} +69.6651 q^{59} +310.111 q^{61} -1321.14 q^{63} -1147.60 q^{65} +819.176 q^{67} -793.818 q^{69} -1042.32 q^{71} +728.809 q^{73} +1408.68 q^{75} -1475.69 q^{77} -351.515 q^{79} +46.1114 q^{81} +1134.30 q^{83} -87.0198 q^{87} -137.678 q^{89} +1997.62 q^{91} +2434.42 q^{93} +1046.34 q^{95} -636.474 q^{97} +2206.07 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 51 q^{7} + 120 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 51 * q^7 + 120 * q^9	\( 18 q + 18 q^{3} + 51 q^{7} + 120 q^{9} + 132 q^{11} + 30 q^{13} + 102 q^{15} + 66 q^{19} + 144 q^{21} + 153 q^{23} + 306 q^{25} + 768 q^{27} + 51 q^{29} + 303 q^{31} + 525 q^{33} - 255 q^{35} + 717 q^{37} - 216 q^{39} - 393 q^{41} - 390 q^{43} + 558 q^{45} - 633 q^{47} + 1443 q^{49} + 1275 q^{53} + 1539 q^{55} + 810 q^{57} - 204 q^{59} + 534 q^{61} + 2556 q^{63} - 2127 q^{65} - 405 q^{67} + 2547 q^{69} - 426 q^{71} + 1149 q^{73} + 2226 q^{75} - 357 q^{77} + 1053 q^{79} + 2802 q^{81} + 66 q^{83} + 2487 q^{87} - 4119 q^{89} + 6090 q^{91} + 606 q^{93} + 2109 q^{95} + 2349 q^{97} + 1428 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 51 * q^7 + 120 * q^9 + 132 * q^11 + 30 * q^13 + 102 * q^15 + 66 * q^19 + 144 * q^21 + 153 * q^23 + 306 * q^25 + 768 * q^27 + 51 * q^29 + 303 * q^31 + 525 * q^33 - 255 * q^35 + 717 * q^37 - 216 * q^39 - 393 * q^41 - 390 * q^43 + 558 * q^45 - 633 * q^47 + 1443 * q^49 + 1275 * q^53 + 1539 * q^55 + 810 * q^57 - 204 * q^59 + 534 * q^61 + 2556 * q^63 - 2127 * q^65 - 405 * q^67 + 2547 * q^69 - 426 * q^71 + 1149 * q^73 + 2226 * q^75 - 357 * q^77 + 1053 * q^79 + 2802 * q^81 + 66 * q^83 + 2487 * q^87 - 4119 * q^89 + 6090 * q^91 + 606 * q^93 + 2109 * q^95 + 2349 * q^97 + 1428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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