LMFDB - Embedded newform 2312.4.a.n.1.4

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:	$1$
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			1.4
Root			$-5.63767$ 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)

$f(q)$	$=$	$q-6.63767 q^{3} +3.97104 q^{5} -24.2054 q^{7} +17.0586 q^{9} +O(q^{10})$	$q-6.63767 q^{3} +3.97104 q^{5} -24.2054 q^{7} +17.0586 q^{9} -14.5546 q^{11} +62.7717 q^{13} -26.3585 q^{15} +12.7935 q^{19} +160.667 q^{21} -67.4068 q^{23} -109.231 q^{25} +65.9875 q^{27} -146.320 q^{29} -211.066 q^{31} +96.6086 q^{33} -96.1206 q^{35} +313.135 q^{37} -416.658 q^{39} +12.3943 q^{41} +300.684 q^{43} +67.7406 q^{45} +65.4144 q^{47} +242.899 q^{49} +610.708 q^{53} -57.7970 q^{55} -84.9188 q^{57} -152.734 q^{59} +174.780 q^{61} -412.910 q^{63} +249.269 q^{65} +525.734 q^{67} +447.424 q^{69} +837.745 q^{71} +494.089 q^{73} +725.038 q^{75} +352.299 q^{77} -885.107 q^{79} -898.586 q^{81} -1207.08 q^{83} +971.226 q^{87} +1295.94 q^{89} -1519.41 q^{91} +1400.98 q^{93} +50.8034 q^{95} -850.332 q^{97} -248.282 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−6.63767	−1.27742	−0.638710	−	0.769448i	$-0.720532\pi$
$3$	−6.63767	−1.27742	−0.638710	+	0.769448i	$0.720532\pi$
$4$	0	0
$5$	3.97104	0.355181	0.177591	−	0.984104i	$-0.443170\pi$
$5$	3.97104	0.355181	0.177591	+	0.984104i	$0.443170\pi$
$6$	0	0
$7$	−24.2054	−1.30697	−0.653483	−	0.756941i	$-0.726693\pi$
$7$	−24.2054	−1.30697	−0.653483	+	0.756941i	$0.726693\pi$
$8$	0	0
$9$	17.0586	0.631801
$10$	0	0
$11$	−14.5546	−0.398943	−0.199472	−	0.979904i	$-0.563923\pi$
$11$	−14.5546	−0.398943	−0.199472	+	0.979904i	$0.563923\pi$
$12$	0	0
$13$	62.7717	1.33921	0.669605	−	0.742717i	$-0.266463\pi$
$13$	62.7717	1.33921	0.669605	+	0.742717i	$0.266463\pi$
$14$	0	0
$15$	−26.3585	−0.453715
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	12.7935	0.154475	0.0772374	−	0.997013i	$-0.475390\pi$
$19$	12.7935	0.154475	0.0772374	+	0.997013i	$0.475390\pi$
$20$	0	0
$21$	160.667	1.66954
$22$	0	0
$23$	−67.4068	−0.611099	−0.305550	−	0.952176i	$-0.598840\pi$
$23$	−67.4068	−0.611099	−0.305550	+	0.952176i	$0.598840\pi$
$24$	0	0
$25$	−109.231	−0.873846
$26$	0	0
$27$	65.9875	0.470344
$28$	0	0
$29$	−146.320	−0.936931	−0.468466	−	0.883482i	$-0.655193\pi$
$29$	−146.320	−0.936931	−0.468466	+	0.883482i	$0.655193\pi$
$30$	0	0
$31$	−211.066	−1.22286	−0.611428	−	0.791300i	$-0.709405\pi$
$31$	−211.066	−1.22286	−0.611428	+	0.791300i	$0.709405\pi$
$32$	0	0
$33$	96.6086	0.509618
$34$	0	0
$35$	−96.1206	−0.464210
$36$	0	0
$37$	313.135	1.39133	0.695663	−	0.718368i	$-0.255111\pi$
$37$	313.135	1.39133	0.695663	+	0.718368i	$0.255111\pi$
$38$	0	0
$39$	−416.658	−1.71073
$40$	0	0
$41$	12.3943	0.0472112	0.0236056	−	0.999721i	$-0.492485\pi$
$41$	12.3943	0.0472112	0.0236056	+	0.999721i	$0.492485\pi$
$42$	0	0
$43$	300.684	1.06637	0.533186	−	0.845998i	$-0.320995\pi$
$43$	300.684	1.06637	0.533186	+	0.845998i	$0.320995\pi$
$44$	0	0
$45$	67.7406	0.224404
$46$	0	0
$47$	65.4144	0.203014	0.101507	−	0.994835i	$-0.467634\pi$
$47$	65.4144	0.203014	0.101507	+	0.994835i	$0.467634\pi$
$48$	0	0
$49$	242.899	0.708161
$50$	0	0
$51$	0	0
$52$	0	0
$53$	610.708	1.58278	0.791388	−	0.611314i	$-0.209359\pi$
$53$	610.708	1.58278	0.791388	+	0.611314i	$0.209359\pi$
$54$	0	0
$55$	−57.7970	−0.141697
$56$	0	0
$57$	−84.9188	−0.197329
$58$	0	0
$59$	−152.734	−0.337021	−0.168510	−	0.985700i	$-0.553896\pi$
$59$	−152.734	−0.337021	−0.168510	+	0.985700i	$0.553896\pi$
$60$	0	0
$61$	174.780	0.366857	0.183429	−	0.983033i	$-0.441280\pi$
$61$	174.780	0.366857	0.183429	+	0.983033i	$0.441280\pi$
$62$	0	0
$63$	−412.910	−0.825743
$64$	0	0
$65$	249.269	0.475662
$66$	0	0
$67$	525.734	0.958637	0.479318	−	0.877641i	$-0.340884\pi$
$67$	525.734	0.958637	0.479318	+	0.877641i	$0.340884\pi$
$68$	0	0
$69$	447.424	0.780630
$70$	0	0
$71$	837.745	1.40031	0.700155	−	0.713990i	$-0.253114\pi$
$71$	837.745	1.40031	0.700155	+	0.713990i	$0.253114\pi$
$72$	0	0
$73$	494.089	0.792175	0.396087	−	0.918213i	$-0.370368\pi$
$73$	494.089	0.792175	0.396087	+	0.918213i	$0.370368\pi$
$74$	0	0
$75$	725.038	1.11627
$76$	0	0
$77$	352.299	0.521406
$78$	0	0
$79$	−885.107	−1.26054	−0.630268	−	0.776378i	$-0.717055\pi$
$79$	−885.107	−1.26054	−0.630268	+	0.776378i	$0.717055\pi$
$80$	0	0
$81$	−898.586	−1.23263
$82$	0	0
$83$	−1207.08	−1.59632	−0.798161	−	0.602444i	$-0.794194\pi$
$83$	−1207.08	−1.59632	−0.798161	+	0.602444i	$0.794194\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	971.226	1.19685
$88$	0	0
$89$	1295.94	1.54347	0.771737	−	0.635942i	$-0.219388\pi$
$89$	1295.94	1.54347	0.771737	+	0.635942i	$0.219388\pi$
$90$	0	0
$91$	−1519.41	−1.75030
$92$	0	0
$93$	1400.98	1.56210
$94$	0	0
$95$	50.8034	0.0548665
$96$	0	0
$97$	−850.332	−0.890083	−0.445042	−	0.895510i	$-0.646811\pi$
$97$	−850.332	−0.890083	−0.445042	+	0.895510i	$0.646811\pi$
$98$	0	0
$99$	−248.282	−0.252053
$100$	0	0
$101$	1488.88	1.46682	0.733410	−	0.679786i	$-0.237928\pi$
$101$	1488.88	1.46682	0.733410	+	0.679786i	$0.237928\pi$
$102$	0	0
$103$	−1581.78	−1.51318	−0.756588	−	0.653892i	$-0.773135\pi$
$103$	−1581.78	−1.51318	−0.756588	+	0.653892i	$0.773135\pi$
$104$	0	0
$105$	638.016	0.592991
$106$	0	0
$107$	398.908	0.360410	0.180205	−	0.983629i	$-0.442324\pi$
$107$	398.908	0.360410	0.180205	+	0.983629i	$0.442324\pi$
$108$	0	0
$109$	687.592	0.604215	0.302107	−	0.953274i	$-0.402310\pi$
$109$	687.592	0.604215	0.302107	+	0.953274i	$0.402310\pi$
$110$	0	0
$111$	−2078.49	−1.77731
$112$	0	0
$113$	−844.900	−0.703376	−0.351688	−	0.936117i	$-0.614392\pi$
$113$	−844.900	−0.703376	−0.351688	+	0.936117i	$0.614392\pi$
$114$	0	0
$115$	−267.675	−0.217051
$116$	0	0
$117$	1070.80	0.846115
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1119.16	−0.840844
$122$	0	0
$123$	−82.2691	−0.0603085
$124$	0	0
$125$	−930.141	−0.665555
$126$	0	0
$127$	−257.267	−0.179754	−0.0898770	−	0.995953i	$-0.528647\pi$
$127$	−257.267	−0.179754	−0.0898770	+	0.995953i	$0.528647\pi$
$128$	0	0
$129$	−1995.84	−1.36220
$130$	0	0
$131$	2412.41	1.60895	0.804477	−	0.593984i	$-0.202446\pi$
$131$	2412.41	1.60895	0.804477	+	0.593984i	$0.202446\pi$
$132$	0	0
$133$	−309.670	−0.201893
$134$	0	0
$135$	262.039	0.167057
$136$	0	0
$137$	1436.44	0.895794	0.447897	−	0.894085i	$-0.352173\pi$
$137$	1436.44	0.895794	0.447897	+	0.894085i	$0.352173\pi$
$138$	0	0
$139$	1947.71	1.18851	0.594253	−	0.804278i	$-0.297448\pi$
$139$	1947.71	1.18851	0.594253	+	0.804278i	$0.297448\pi$
$140$	0	0
$141$	−434.199	−0.259335
$142$	0	0
$143$	−913.617	−0.534269
$144$	0	0
$145$	−581.045	−0.332780
$146$	0	0
$147$	−1612.29	−0.904619
$148$	0	0
$149$	−1196.71	−0.657976	−0.328988	−	0.944334i	$-0.606708\pi$
$149$	−1196.71	−0.657976	−0.328988	+	0.944334i	$0.606708\pi$
$150$	0	0
$151$	2612.21	1.40780	0.703902	−	0.710297i	$-0.251439\pi$
$151$	2612.21	1.40780	0.703902	+	0.710297i	$0.251439\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−838.152	−0.434335
$156$	0	0
$157$	1575.35	0.800804	0.400402	−	0.916340i	$-0.368870\pi$
$157$	1575.35	0.800804	0.400402	+	0.916340i	$0.368870\pi$
$158$	0	0
$159$	−4053.68	−2.02187
$160$	0	0
$161$	1631.61	0.798686
$162$	0	0
$163$	1692.30	0.813199	0.406599	−	0.913607i	$-0.366715\pi$
$163$	1692.30	0.813199	0.406599	+	0.913607i	$0.366715\pi$
$164$	0	0
$165$	383.637	0.181007
$166$	0	0
$167$	−2784.41	−1.29020	−0.645102	−	0.764097i	$-0.723185\pi$
$167$	−2784.41	−1.29020	−0.645102	+	0.764097i	$0.723185\pi$
$168$	0	0
$169$	1743.29	0.793485
$170$	0	0
$171$	218.239	0.0975974
$172$	0	0
$173$	2948.71	1.29588	0.647938	−	0.761693i	$-0.275632\pi$
$173$	2948.71	1.29588	0.647938	+	0.761693i	$0.275632\pi$
$174$	0	0
$175$	2643.97	1.14209
$176$	0	0
$177$	1013.79	0.430517
$178$	0	0
$179$	−278.519	−0.116299	−0.0581494	−	0.998308i	$-0.518520\pi$
$179$	−278.519	−0.116299	−0.0581494	+	0.998308i	$0.518520\pi$
$180$	0	0
$181$	−4027.33	−1.65386	−0.826931	−	0.562303i	$-0.809915\pi$
$181$	−4027.33	−1.65386	−0.826931	+	0.562303i	$0.809915\pi$
$182$	0	0
$183$	−1160.13	−0.468631
$184$	0	0
$185$	1243.47	0.494173
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1597.25	−0.614724
$190$	0	0
$191$	−2597.96	−0.984198	−0.492099	−	0.870539i	$-0.663770\pi$
$191$	−2597.96	−0.984198	−0.492099	+	0.870539i	$0.663770\pi$
$192$	0	0
$193$	1443.76	0.538468	0.269234	−	0.963075i	$-0.413229\pi$
$193$	1443.76	0.538468	0.269234	+	0.963075i	$0.413229\pi$
$194$	0	0
$195$	−1654.57	−0.607620
$196$	0	0
$197$	−2638.59	−0.954273	−0.477137	−	0.878829i	$-0.658325\pi$
$197$	−2638.59	−0.954273	−0.477137	+	0.878829i	$0.658325\pi$
$198$	0	0
$199$	−2817.60	−1.00369	−0.501845	−	0.864958i	$-0.667345\pi$
$199$	−2817.60	−1.00369	−0.501845	+	0.864958i	$0.667345\pi$
$200$	0	0
$201$	−3489.65	−1.22458
$202$	0	0
$203$	3541.74	1.22454
$204$	0	0
$205$	49.2182	0.0167685
$206$	0	0
$207$	−1149.87	−0.386093
$208$	0	0
$209$	−186.204	−0.0616267
$210$	0	0
$211$	−5144.22	−1.67840	−0.839201	−	0.543821i	$-0.816977\pi$
$211$	−5144.22	−1.67840	−0.839201	+	0.543821i	$0.816977\pi$
$212$	0	0
$213$	−5560.67	−1.78878
$214$	0	0
$215$	1194.03	0.378755
$216$	0	0
$217$	5108.92	1.59823
$218$	0	0
$219$	−3279.60	−1.01194
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5689.18	1.70841	0.854206	−	0.519934i	$-0.174044\pi$
$223$	5689.18	1.70841	0.854206	+	0.519934i	$0.174044\pi$
$224$	0	0
$225$	−1863.33	−0.552097
$226$	0	0
$227$	−567.753	−0.166005	−0.0830024	−	0.996549i	$-0.526451\pi$
$227$	−567.753	−0.166005	−0.0830024	+	0.996549i	$0.526451\pi$
$228$	0	0
$229$	3174.35	0.916013	0.458007	−	0.888949i	$-0.348564\pi$
$229$	3174.35	0.916013	0.458007	+	0.888949i	$0.348564\pi$
$230$	0	0
$231$	−2338.45	−0.666054
$232$	0	0
$233$	−5681.36	−1.59742	−0.798709	−	0.601717i	$-0.794483\pi$
$233$	−5681.36	−1.59742	−0.798709	+	0.601717i	$0.794483\pi$
$234$	0	0
$235$	259.764	0.0721069
$236$	0	0
$237$	5875.04	1.61023
$238$	0	0
$239$	85.0026	0.0230057	0.0115028	−	0.999934i	$-0.496338\pi$
$239$	85.0026	0.0230057	0.0115028	+	0.999934i	$0.496338\pi$
$240$	0	0
$241$	−3246.38	−0.867708	−0.433854	−	0.900983i	$-0.642847\pi$
$241$	−3246.38	−0.867708	−0.433854	+	0.900983i	$0.642847\pi$
$242$	0	0
$243$	4182.85	1.10424
$244$	0	0
$245$	964.564	0.251525
$246$	0	0
$247$	803.067	0.206874
$248$	0	0
$249$	8012.23	2.03917
$250$	0	0
$251$	−544.737	−0.136986	−0.0684930	−	0.997652i	$-0.521819\pi$
$251$	−544.737	−0.136986	−0.0684930	+	0.997652i	$0.521819\pi$
$252$	0	0
$253$	981.079	0.243794
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5243.44	−1.27267	−0.636336	−	0.771412i	$-0.719551\pi$
$257$	−5243.44	−1.27267	−0.636336	+	0.771412i	$0.719551\pi$
$258$	0	0
$259$	−7579.54	−1.81842
$260$	0	0
$261$	−2496.03	−0.591954
$262$	0	0
$263$	−4004.37	−0.938860	−0.469430	−	0.882970i	$-0.655541\pi$
$263$	−4004.37	−0.938860	−0.469430	+	0.882970i	$0.655541\pi$
$264$	0	0
$265$	2425.15	0.562172
$266$	0	0
$267$	−8602.00	−1.97166
$268$	0	0
$269$	3257.10	0.738249	0.369125	−	0.929380i	$-0.379658\pi$
$269$	3257.10	0.738249	0.369125	+	0.929380i	$0.379658\pi$
$270$	0	0
$271$	−4165.16	−0.933637	−0.466818	−	0.884353i	$-0.654600\pi$
$271$	−4165.16	−0.933637	−0.466818	+	0.884353i	$0.654600\pi$
$272$	0	0
$273$	10085.3	2.23587
$274$	0	0
$275$	1589.81	0.348615
$276$	0	0
$277$	−6317.39	−1.37031	−0.685154	−	0.728399i	$-0.740265\pi$
$277$	−6317.39	−1.37031	−0.685154	+	0.728399i	$0.740265\pi$
$278$	0	0
$279$	−3600.49	−0.772602
$280$	0	0
$281$	−3809.23	−0.808681	−0.404340	−	0.914609i	$-0.632499\pi$
$281$	−3809.23	−0.808681	−0.404340	+	0.914609i	$0.632499\pi$
$282$	0	0
$283$	−3144.81	−0.660564	−0.330282	−	0.943882i	$-0.607144\pi$
$283$	−3144.81	−0.660564	−0.330282	+	0.943882i	$0.607144\pi$
$284$	0	0
$285$	−337.216	−0.0700876
$286$	0	0
$287$	−300.008	−0.0617035
$288$	0	0
$289$	0	0
$290$	0	0
$291$	5644.22	1.13701
$292$	0	0
$293$	−3106.80	−0.619458	−0.309729	−	0.950825i	$-0.600238\pi$
$293$	−3106.80	−0.619458	−0.309729	+	0.950825i	$0.600238\pi$
$294$	0	0
$295$	−606.512	−0.119703
$296$	0	0
$297$	−960.422	−0.187641
$298$	0	0
$299$	−4231.24	−0.818391
$300$	0	0
$301$	−7278.18	−1.39371
$302$	0	0
$303$	−9882.67	−1.87375
$304$	0	0
$305$	694.060	0.130301
$306$	0	0
$307$	9783.03	1.81872	0.909360	−	0.416011i	$-0.136572\pi$
$307$	9783.03	1.81872	0.909360	+	0.416011i	$0.136572\pi$
$308$	0	0
$309$	10499.3	1.93296
$310$	0	0
$311$	4872.82	0.888463	0.444232	−	0.895912i	$-0.353477\pi$
$311$	4872.82	0.888463	0.444232	+	0.895912i	$0.353477\pi$
$312$	0	0
$313$	−5110.76	−0.922931	−0.461465	−	0.887158i	$-0.652676\pi$
$313$	−5110.76	−0.922931	−0.461465	+	0.887158i	$0.652676\pi$
$314$	0	0
$315$	−1639.68	−0.293288
$316$	0	0
$317$	9671.62	1.71360	0.856802	−	0.515646i	$-0.172448\pi$
$317$	9671.62	1.71360	0.856802	+	0.515646i	$0.172448\pi$
$318$	0	0
$319$	2129.63	0.373783
$320$	0	0
$321$	−2647.82	−0.460395
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−6856.60	−1.17026
$326$	0	0
$327$	−4564.01	−0.771836
$328$	0	0
$329$	−1583.38	−0.265333
$330$	0	0
$331$	−7743.77	−1.28591	−0.642955	−	0.765904i	$-0.722292\pi$
$331$	−7743.77	−1.28591	−0.642955	+	0.765904i	$0.722292\pi$
$332$	0	0
$333$	5341.65	0.879041
$334$	0	0
$335$	2087.71	0.340489
$336$	0	0
$337$	1725.44	0.278903	0.139452	−	0.990229i	$-0.455466\pi$
$337$	1725.44	0.278903	0.139452	+	0.990229i	$0.455466\pi$
$338$	0	0
$339$	5608.16	0.898506
$340$	0	0
$341$	3071.98	0.487850
$342$	0	0
$343$	2422.97	0.381423
$344$	0	0
$345$	1776.74	0.277265
$346$	0	0
$347$	−8712.15	−1.34782	−0.673909	−	0.738815i	$-0.735386\pi$
$347$	−8712.15	−1.34782	−0.673909	+	0.738815i	$0.735386\pi$
$348$	0	0
$349$	−7490.89	−1.14893	−0.574467	−	0.818527i	$-0.694791\pi$
$349$	−7490.89	−1.14893	−0.574467	+	0.818527i	$0.694791\pi$
$350$	0	0
$351$	4142.15	0.629890
$352$	0	0
$353$	4734.52	0.713862	0.356931	−	0.934131i	$-0.383823\pi$
$353$	4734.52	0.713862	0.356931	+	0.934131i	$0.383823\pi$
$354$	0	0
$355$	3326.72	0.497364
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9814.52	−1.44287	−0.721435	−	0.692482i	$-0.756517\pi$
$359$	−9814.52	−1.44287	−0.721435	+	0.692482i	$0.756517\pi$
$360$	0	0
$361$	−6695.33	−0.976138
$362$	0	0
$363$	7428.64	1.07411
$364$	0	0
$365$	1962.05	0.281365
$366$	0	0
$367$	−378.501	−0.0538354	−0.0269177	−	0.999638i	$-0.508569\pi$
$367$	−378.501	−0.0538354	−0.0269177	+	0.999638i	$0.508569\pi$
$368$	0	0
$369$	211.429	0.0298281
$370$	0	0
$371$	−14782.4	−2.06864
$372$	0	0
$373$	−3603.72	−0.500251	−0.250126	−	0.968213i	$-0.580472\pi$
$373$	−3603.72	−0.500251	−0.250126	+	0.968213i	$0.580472\pi$
$374$	0	0
$375$	6173.97	0.850193
$376$	0	0
$377$	−9184.78	−1.25475
$378$	0	0
$379$	−2074.13	−0.281111	−0.140556	−	0.990073i	$-0.544889\pi$
$379$	−2074.13	−0.281111	−0.140556	+	0.990073i	$0.544889\pi$
$380$	0	0
$381$	1707.65	0.229621
$382$	0	0
$383$	−7957.96	−1.06170	−0.530852	−	0.847464i	$-0.678128\pi$
$383$	−7957.96	−1.06170	−0.530852	+	0.847464i	$0.678128\pi$
$384$	0	0
$385$	1399.00	0.185193
$386$	0	0
$387$	5129.27	0.673734
$388$	0	0
$389$	−13060.2	−1.70226	−0.851129	−	0.524957i	$-0.824081\pi$
$389$	−13060.2	−1.70226	−0.851129	+	0.524957i	$0.824081\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−16012.8	−2.05531
$394$	0	0
$395$	−3514.80	−0.447718
$396$	0	0
$397$	−12650.4	−1.59926	−0.799628	−	0.600496i	$-0.794970\pi$
$397$	−12650.4	−1.59926	−0.799628	+	0.600496i	$0.794970\pi$
$398$	0	0
$399$	2055.49	0.257903
$400$	0	0
$401$	−10828.2	−1.34846	−0.674231	−	0.738521i	$-0.735525\pi$
$401$	−10828.2	−1.34846	−0.674231	+	0.738521i	$0.735525\pi$
$402$	0	0
$403$	−13249.0	−1.63766
$404$	0	0
$405$	−3568.33	−0.437806
$406$	0	0
$407$	−4557.55	−0.555060
$408$	0	0
$409$	15232.9	1.84160	0.920802	−	0.390030i	$-0.127535\pi$
$409$	15232.9	1.84160	0.920802	+	0.390030i	$0.127535\pi$
$410$	0	0
$411$	−9534.64	−1.14430
$412$	0	0
$413$	3696.97	0.440475
$414$	0	0
$415$	−4793.39	−0.566983
$416$	0	0
$417$	−12928.2	−1.51822
$418$	0	0
$419$	−5603.72	−0.653364	−0.326682	−	0.945134i	$-0.605931\pi$
$419$	−5603.72	−0.653364	−0.326682	+	0.945134i	$0.605931\pi$
$420$	0	0
$421$	−1390.33	−0.160952	−0.0804758	−	0.996757i	$-0.525644\pi$
$421$	−1390.33	−0.160952	−0.0804758	+	0.996757i	$0.525644\pi$
$422$	0	0
$423$	1115.88	0.128265
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4230.62	−0.479470
$428$	0	0
$429$	6064.29	0.682486
$430$	0	0
$431$	−4512.44	−0.504308	−0.252154	−	0.967687i	$-0.581139\pi$
$431$	−4512.44	−0.504308	−0.252154	+	0.967687i	$0.581139\pi$
$432$	0	0
$433$	−1370.44	−0.152100	−0.0760498	−	0.997104i	$-0.524231\pi$
$433$	−1370.44	−0.152100	−0.0760498	+	0.997104i	$0.524231\pi$
$434$	0	0
$435$	3856.78	0.425100
$436$	0	0
$437$	−862.366	−0.0943995
$438$	0	0
$439$	5042.52	0.548215	0.274107	−	0.961699i	$-0.411618\pi$
$439$	5042.52	0.548215	0.274107	+	0.961699i	$0.411618\pi$
$440$	0	0
$441$	4143.53	0.447417
$442$	0	0
$443$	−11234.9	−1.20493	−0.602467	−	0.798144i	$-0.705815\pi$
$443$	−11234.9	−1.20493	−0.602467	+	0.798144i	$0.705815\pi$
$444$	0	0
$445$	5146.23	0.548212
$446$	0	0
$447$	7943.38	0.840512
$448$	0	0
$449$	13470.6	1.41586	0.707928	−	0.706285i	$-0.249630\pi$
$449$	13470.6	1.41586	0.707928	+	0.706285i	$0.249630\pi$
$450$	0	0
$451$	−180.394	−0.0188346
$452$	0	0
$453$	−17339.0	−1.79836
$454$	0	0
$455$	−6033.65	−0.621674
$456$	0	0
$457$	13318.0	1.36321	0.681606	−	0.731720i	$-0.261282\pi$
$457$	13318.0	1.36321	0.681606	+	0.731720i	$0.261282\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11201.7	1.13170	0.565852	−	0.824507i	$-0.308547\pi$
$461$	11201.7	1.13170	0.565852	+	0.824507i	$0.308547\pi$
$462$	0	0
$463$	−3221.90	−0.323400	−0.161700	−	0.986840i	$-0.551698\pi$
$463$	−3221.90	−0.323400	−0.161700	+	0.986840i	$0.551698\pi$
$464$	0	0
$465$	5563.37	0.554828
$466$	0	0
$467$	−8759.46	−0.867964	−0.433982	−	0.900921i	$-0.642892\pi$
$467$	−8759.46	−0.867964	−0.433982	+	0.900921i	$0.642892\pi$
$468$	0	0
$469$	−12725.6	−1.25291
$470$	0	0
$471$	−10456.6	−1.02296
$472$	0	0
$473$	−4376.34	−0.425422
$474$	0	0
$475$	−1397.44	−0.134987
$476$	0	0
$477$	10417.8	1.00000
$478$	0	0
$479$	2003.89	0.191148	0.0955741	−	0.995422i	$-0.469531\pi$
$479$	2003.89	0.191148	0.0955741	+	0.995422i	$0.469531\pi$
$480$	0	0
$481$	19656.0	1.86328
$482$	0	0
$483$	−10830.1	−1.02026
$484$	0	0
$485$	−3376.70	−0.316141
$486$	0	0
$487$	−6331.03	−0.589089	−0.294544	−	0.955638i	$-0.595168\pi$
$487$	−6331.03	−0.589089	−0.294544	+	0.955638i	$0.595168\pi$
$488$	0	0
$489$	−11232.9	−1.03880
$490$	0	0
$491$	19797.5	1.81965	0.909824	−	0.414995i	$-0.136217\pi$
$491$	19797.5	1.81965	0.909824	+	0.414995i	$0.136217\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−985.937	−0.0895244
$496$	0	0
$497$	−20277.9	−1.83016
$498$	0	0
$499$	−11191.1	−1.00397	−0.501985	−	0.864876i	$-0.667397\pi$
$499$	−11191.1	−1.00397	−0.501985	+	0.864876i	$0.667397\pi$
$500$	0	0
$501$	18482.0	1.64813
$502$	0	0
$503$	7953.69	0.705044	0.352522	−	0.935803i	$-0.385324\pi$
$503$	7953.69	0.705044	0.352522	+	0.935803i	$0.385324\pi$
$504$	0	0
$505$	5912.40	0.520987
$506$	0	0
$507$	−11571.4	−1.01361
$508$	0	0
$509$	4573.74	0.398286	0.199143	−	0.979970i	$-0.436184\pi$
$509$	4573.74	0.398286	0.199143	+	0.979970i	$0.436184\pi$
$510$	0	0
$511$	−11959.6	−1.03535
$512$	0	0
$513$	844.209	0.0726564
$514$	0	0
$515$	−6281.31	−0.537451
$516$	0	0
$517$	−952.081	−0.0809912
$518$	0	0
$519$	−19572.6	−1.65538
$520$	0	0
$521$	18335.1	1.54179	0.770897	−	0.636960i	$-0.219808\pi$
$521$	18335.1	1.54179	0.770897	+	0.636960i	$0.219808\pi$
$522$	0	0
$523$	−4043.38	−0.338058	−0.169029	−	0.985611i	$-0.554063\pi$
$523$	−4043.38	−0.338058	−0.169029	+	0.985611i	$0.554063\pi$
$524$	0	0
$525$	−17549.8	−1.45893
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7623.33	−0.626558
$530$	0	0
$531$	−2605.42	−0.212930
$532$	0	0
$533$	778.010	0.0632258
$534$	0	0
$535$	1584.08	0.128011
$536$	0	0
$537$	1848.71	0.148562
$538$	0	0
$539$	−3535.30	−0.282516
$540$	0	0
$541$	10179.8	0.808991	0.404496	−	0.914540i	$-0.367447\pi$
$541$	10179.8	0.808991	0.404496	+	0.914540i	$0.367447\pi$
$542$	0	0
$543$	26732.1	2.11268
$544$	0	0
$545$	2730.46	0.214606
$546$	0	0
$547$	144.970	0.0113318	0.00566589	−	0.999984i	$-0.498196\pi$
$547$	144.970	0.0113318	0.00566589	+	0.999984i	$0.498196\pi$
$548$	0	0
$549$	2981.51	0.231781
$550$	0	0
$551$	−1871.94	−0.144732
$552$	0	0
$553$	21424.3	1.64748
$554$	0	0
$555$	−8253.76	−0.631266
$556$	0	0
$557$	−19694.0	−1.49814	−0.749068	−	0.662493i	$-0.769498\pi$
$557$	−19694.0	−1.49814	−0.749068	+	0.662493i	$0.769498\pi$
$558$	0	0
$559$	18874.5	1.42810
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12954.2	0.969723	0.484861	−	0.874591i	$-0.338870\pi$
$563$	12954.2	0.969723	0.484861	+	0.874591i	$0.338870\pi$
$564$	0	0
$565$	−3355.13	−0.249826
$566$	0	0
$567$	21750.6	1.61100
$568$	0	0
$569$	−16463.2	−1.21296	−0.606480	−	0.795099i	$-0.707419\pi$
$569$	−16463.2	−1.21296	−0.606480	+	0.795099i	$0.707419\pi$
$570$	0	0
$571$	6518.66	0.477754	0.238877	−	0.971050i	$-0.423221\pi$
$571$	6518.66	0.477754	0.238877	+	0.971050i	$0.423221\pi$
$572$	0	0
$573$	17244.4	1.25723
$574$	0	0
$575$	7362.90	0.534007
$576$	0	0
$577$	−763.092	−0.0550571	−0.0275285	−	0.999621i	$-0.508764\pi$
$577$	−763.092	−0.0550571	−0.0275285	+	0.999621i	$0.508764\pi$
$578$	0	0
$579$	−9583.22	−0.687850
$580$	0	0
$581$	29217.9	2.08634
$582$	0	0
$583$	−8888.61	−0.631438
$584$	0	0
$585$	4252.19	0.300524
$586$	0	0
$587$	−9258.30	−0.650989	−0.325495	−	0.945544i	$-0.605531\pi$
$587$	−9258.30	−0.650989	−0.325495	+	0.945544i	$0.605531\pi$
$588$	0	0
$589$	−2700.26	−0.188901
$590$	0	0
$591$	17514.1	1.21901
$592$	0	0
$593$	−15031.9	−1.04096	−0.520478	−	0.853875i	$-0.674246\pi$
$593$	−15031.9	−1.04096	−0.520478	+	0.853875i	$0.674246\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18702.3	1.28213
$598$	0	0
$599$	17976.1	1.22618	0.613091	−	0.790012i	$-0.289926\pi$
$599$	17976.1	1.22618	0.613091	+	0.790012i	$0.289926\pi$
$600$	0	0
$601$	−28097.4	−1.90702	−0.953508	−	0.301369i	$-0.902556\pi$
$601$	−28097.4	−1.90702	−0.953508	+	0.301369i	$0.902556\pi$
$602$	0	0
$603$	8968.30	0.605668
$604$	0	0
$605$	−4444.25	−0.298652
$606$	0	0
$607$	−11850.5	−0.792418	−0.396209	−	0.918160i	$-0.629674\pi$
$607$	−11850.5	−0.792418	−0.396209	+	0.918160i	$0.629674\pi$
$608$	0	0
$609$	−23508.9	−1.56425
$610$	0	0
$611$	4106.18	0.271879
$612$	0	0
$613$	961.994	0.0633843	0.0316921	−	0.999498i	$-0.489910\pi$
$613$	961.994	0.0633843	0.0316921	+	0.999498i	$0.489910\pi$
$614$	0	0
$615$	−326.694	−0.0214205
$616$	0	0
$617$	2713.80	0.177072	0.0885359	−	0.996073i	$-0.471781\pi$
$617$	2713.80	0.177072	0.0885359	+	0.996073i	$0.471781\pi$
$618$	0	0
$619$	19997.4	1.29848	0.649242	−	0.760582i	$-0.275086\pi$
$619$	19997.4	1.29848	0.649242	+	0.760582i	$0.275086\pi$
$620$	0	0
$621$	−4448.01	−0.287427
$622$	0	0
$623$	−31368.6	−2.01727
$624$	0	0
$625$	9960.22	0.637454
$626$	0	0
$627$	1235.96	0.0787232
$628$	0	0
$629$	0	0
$630$	0	0
$631$	6160.17	0.388641	0.194320	−	0.980938i	$-0.437750\pi$
$631$	6160.17	0.388641	0.194320	+	0.980938i	$0.437750\pi$
$632$	0	0
$633$	34145.6	2.14402
$634$	0	0
$635$	−1021.62	−0.0638452
$636$	0	0
$637$	15247.2	0.948377
$638$	0	0
$639$	14290.8	0.884718
$640$	0	0
$641$	−4474.48	−0.275712	−0.137856	−	0.990452i	$-0.544021\pi$
$641$	−4474.48	−0.275712	−0.137856	+	0.990452i	$0.544021\pi$
$642$	0	0
$643$	20529.9	1.25913	0.629564	−	0.776949i	$-0.283234\pi$
$643$	20529.9	1.25913	0.629564	+	0.776949i	$0.283234\pi$
$644$	0	0
$645$	−7925.58	−0.483829
$646$	0	0
$647$	−15233.2	−0.925622	−0.462811	−	0.886457i	$-0.653159\pi$
$647$	−15233.2	−0.925622	−0.462811	+	0.886457i	$0.653159\pi$
$648$	0	0
$649$	2222.98	0.134452
$650$	0	0
$651$	−33911.3	−2.04161
$652$	0	0
$653$	−23770.8	−1.42454	−0.712269	−	0.701906i	$-0.752333\pi$
$653$	−23770.8	−1.42454	−0.712269	+	0.701906i	$0.752333\pi$
$654$	0	0
$655$	9579.78	0.571470
$656$	0	0
$657$	8428.49	0.500497
$658$	0	0
$659$	14722.8	0.870285	0.435143	−	0.900362i	$-0.356698\pi$
$659$	14722.8	0.870285	0.435143	+	0.900362i	$0.356698\pi$
$660$	0	0
$661$	−11881.1	−0.699122	−0.349561	−	0.936914i	$-0.613669\pi$
$661$	−11881.1	−0.699122	−0.349561	+	0.936914i	$0.613669\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1229.71	−0.0717087
$666$	0	0
$667$	9862.99	0.572558
$668$	0	0
$669$	−37762.9	−2.18236
$670$	0	0
$671$	−2543.86	−0.146355
$672$	0	0
$673$	−11159.3	−0.639167	−0.319583	−	0.947558i	$-0.603543\pi$
$673$	−11159.3	−0.639167	−0.319583	+	0.947558i	$0.603543\pi$
$674$	0	0
$675$	−7207.87	−0.411009
$676$	0	0
$677$	11031.6	0.626261	0.313131	−	0.949710i	$-0.398622\pi$
$677$	11031.6	0.626261	0.313131	+	0.949710i	$0.398622\pi$
$678$	0	0
$679$	20582.6	1.16331
$680$	0	0
$681$	3768.56	0.212058
$682$	0	0
$683$	22233.7	1.24561	0.622803	−	0.782379i	$-0.285994\pi$
$683$	22233.7	1.24561	0.622803	+	0.782379i	$0.285994\pi$
$684$	0	0
$685$	5704.19	0.318169
$686$	0	0
$687$	−21070.3	−1.17013
$688$	0	0
$689$	38335.2	2.11967
$690$	0	0
$691$	10971.3	0.604007	0.302003	−	0.953307i	$-0.402345\pi$
$691$	10971.3	0.604007	0.302003	+	0.953307i	$0.402345\pi$
$692$	0	0
$693$	6009.74	0.329425
$694$	0	0
$695$	7734.43	0.422135
$696$	0	0
$697$	0	0
$698$	0	0
$699$	37711.0	2.04057
$700$	0	0
$701$	20973.0	1.13001	0.565007	−	0.825086i	$-0.308873\pi$
$701$	20973.0	1.13001	0.565007	+	0.825086i	$0.308873\pi$
$702$	0	0
$703$	4006.08	0.214925
$704$	0	0
$705$	−1724.22	−0.0921107
$706$	0	0
$707$	−36038.8	−1.91709
$708$	0	0
$709$	31393.7	1.66293	0.831464	−	0.555578i	$-0.187503\pi$
$709$	31393.7	1.66293	0.831464	+	0.555578i	$0.187503\pi$
$710$	0	0
$711$	−15098.7	−0.796408
$712$	0	0
$713$	14227.3	0.747287
$714$	0	0
$715$	−3628.01	−0.189762
$716$	0	0
$717$	−564.219	−0.0293879
$718$	0	0
$719$	−8700.65	−0.451293	−0.225646	−	0.974209i	$-0.572449\pi$
$719$	−8700.65	−0.451293	−0.225646	+	0.974209i	$0.572449\pi$
$720$	0	0
$721$	38287.5	1.97767
$722$	0	0
$723$	21548.4	1.10843
$724$	0	0
$725$	15982.7	0.818734
$726$	0	0
$727$	−2902.76	−0.148085	−0.0740423	−	0.997255i	$-0.523590\pi$
$727$	−2902.76	−0.148085	−0.0740423	+	0.997255i	$0.523590\pi$
$728$	0	0
$729$	−3502.56	−0.177949
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−19860.8	−1.00078	−0.500392	−	0.865799i	$-0.666811\pi$
$733$	−19860.8	−1.00078	−0.500392	+	0.865799i	$0.666811\pi$
$734$	0	0
$735$	−6402.46	−0.321304
$736$	0	0
$737$	−7651.85	−0.382442
$738$	0	0
$739$	31242.8	1.55519	0.777594	−	0.628767i	$-0.216440\pi$
$739$	31242.8	1.55519	0.777594	+	0.628767i	$0.216440\pi$
$740$	0	0
$741$	−5330.49	−0.264265
$742$	0	0
$743$	−10749.3	−0.530759	−0.265379	−	0.964144i	$-0.585497\pi$
$743$	−10749.3	−0.530759	−0.265379	+	0.964144i	$0.585497\pi$
$744$	0	0
$745$	−4752.20	−0.233701
$746$	0	0
$747$	−20591.2	−1.00856
$748$	0	0
$749$	−9655.70	−0.471044
$750$	0	0
$751$	11931.8	0.579755	0.289877	−	0.957064i	$-0.406385\pi$
$751$	11931.8	0.579755	0.289877	+	0.957064i	$0.406385\pi$
$752$	0	0
$753$	3615.78	0.174989
$754$	0	0
$755$	10373.2	0.500025
$756$	0	0
$757$	5667.85	0.272129	0.136064	−	0.990700i	$-0.456555\pi$
$757$	5667.85	0.272129	0.136064	+	0.990700i	$0.456555\pi$
$758$	0	0
$759$	−6512.08	−0.311427
$760$	0	0
$761$	6308.12	0.300485	0.150243	−	0.988649i	$-0.451995\pi$
$761$	6308.12	0.300485	0.150243	+	0.988649i	$0.451995\pi$
$762$	0	0
$763$	−16643.4	−0.789688
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9587.34	−0.451341
$768$	0	0
$769$	10642.8	0.499075	0.249538	−	0.968365i	$-0.419721\pi$
$769$	10642.8	0.499075	0.249538	+	0.968365i	$0.419721\pi$
$770$	0	0
$771$	34804.2	1.62574
$772$	0	0
$773$	−18898.8	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$773$	−18898.8	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$774$	0	0
$775$	23054.9	1.06859
$776$	0	0
$777$	50310.5	2.32288
$778$	0	0
$779$	158.566	0.00729295
$780$	0	0
$781$	−12193.0	−0.558645
$782$	0	0
$783$	−9655.32	−0.440681
$784$	0	0
$785$	6255.77	0.284430
$786$	0	0
$787$	−17642.0	−0.799074	−0.399537	−	0.916717i	$-0.630829\pi$
$787$	−17642.0	−0.799074	−0.399537	+	0.916717i	$0.630829\pi$
$788$	0	0
$789$	26579.7	1.19932
$790$	0	0
$791$	20451.1	0.919289
$792$	0	0
$793$	10971.2	0.491299
$794$	0	0
$795$	−16097.3	−0.718130
$796$	0	0
$797$	−26611.1	−1.18270	−0.591350	−	0.806415i	$-0.701405\pi$
$797$	−26611.1	−1.18270	−0.591350	+	0.806415i	$0.701405\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	22106.9	0.975168
$802$	0	0
$803$	−7191.27	−0.316033
$804$	0	0
$805$	6479.18	0.283678
$806$	0	0
$807$	−21619.6	−0.943054
$808$	0	0
$809$	33990.6	1.47719	0.738595	−	0.674149i	$-0.235490\pi$
$809$	33990.6	1.47719	0.738595	+	0.674149i	$0.235490\pi$
$810$	0	0
$811$	−20253.4	−0.876932	−0.438466	−	0.898748i	$-0.644478\pi$
$811$	−20253.4	−0.876932	−0.438466	+	0.898748i	$0.644478\pi$
$812$	0	0
$813$	27647.0	1.19265
$814$	0	0
$815$	6720.21	0.288833
$816$	0	0
$817$	3846.80	0.164728
$818$	0	0
$819$	−25919.1	−1.10584
$820$	0	0
$821$	−3978.77	−0.169135	−0.0845675	−	0.996418i	$-0.526951\pi$
$821$	−3978.77	−0.169135	−0.0845675	+	0.996418i	$0.526951\pi$
$822$	0	0
$823$	−11402.6	−0.482952	−0.241476	−	0.970407i	$-0.577631\pi$
$823$	−11402.6	−0.482952	−0.241476	+	0.970407i	$0.577631\pi$
$824$	0	0
$825$	−10552.6	−0.445328
$826$	0	0
$827$	−19098.8	−0.803061	−0.401531	−	0.915846i	$-0.631522\pi$
$827$	−19098.8	−0.803061	−0.401531	+	0.915846i	$0.631522\pi$
$828$	0	0
$829$	−21222.5	−0.889129	−0.444564	−	0.895747i	$-0.646642\pi$
$829$	−21222.5	−0.889129	−0.444564	+	0.895747i	$0.646642\pi$
$830$	0	0
$831$	41932.7	1.75046
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−11057.0	−0.458256
$836$	0	0
$837$	−13927.7	−0.575164
$838$	0	0
$839$	15057.1	0.619582	0.309791	−	0.950805i	$-0.399741\pi$
$839$	15057.1	0.619582	0.309791	+	0.950805i	$0.399741\pi$
$840$	0	0
$841$	−2979.35	−0.122160
$842$	0	0
$843$	25284.4	1.03303
$844$	0	0
$845$	6922.67	0.281831
$846$	0	0
$847$	27089.8	1.09896
$848$	0	0
$849$	20874.2	0.843818
$850$	0	0
$851$	−21107.4	−0.850239
$852$	0	0
$853$	22060.4	0.885502	0.442751	−	0.896645i	$-0.354003\pi$
$853$	22060.4	0.885502	0.442751	+	0.896645i	$0.354003\pi$
$854$	0	0
$855$	866.637	0.0346647
$856$	0	0
$857$	45460.6	1.81203	0.906013	−	0.423251i	$-0.139111\pi$
$857$	45460.6	1.81203	0.906013	+	0.423251i	$0.139111\pi$
$858$	0	0
$859$	18788.1	0.746264	0.373132	−	0.927778i	$-0.378284\pi$
$859$	18788.1	0.746264	0.373132	+	0.927778i	$0.378284\pi$
$860$	0	0
$861$	1991.35	0.0788213
$862$	0	0
$863$	−39141.1	−1.54389	−0.771945	−	0.635689i	$-0.780716\pi$
$863$	−39141.1	−1.54389	−0.771945	+	0.635689i	$0.780716\pi$
$864$	0	0
$865$	11709.5	0.460270
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12882.4	0.502882
$870$	0	0
$871$	33001.2	1.28382
$872$	0	0
$873$	−14505.5	−0.562356
$874$	0	0
$875$	22514.4	0.869858
$876$	0	0
$877$	916.994	0.0353075	0.0176538	−	0.999844i	$-0.494380\pi$
$877$	916.994	0.0353075	0.0176538	+	0.999844i	$0.494380\pi$
$878$	0	0
$879$	20621.9	0.791308
$880$	0	0
$881$	12596.7	0.481718	0.240859	−	0.970560i	$-0.422571\pi$
$881$	12596.7	0.481718	0.240859	+	0.970560i	$0.422571\pi$
$882$	0	0
$883$	−6307.73	−0.240399	−0.120199	−	0.992750i	$-0.538353\pi$
$883$	−6307.73	−0.240399	−0.120199	+	0.992750i	$0.538353\pi$
$884$	0	0
$885$	4025.82	0.152911
$886$	0	0
$887$	−49545.5	−1.87551	−0.937753	−	0.347303i	$-0.887097\pi$
$887$	−49545.5	−1.87551	−0.937753	+	0.347303i	$0.887097\pi$
$888$	0	0
$889$	6227.24	0.234932
$890$	0	0
$891$	13078.6	0.491749
$892$	0	0
$893$	836.877	0.0313606
$894$	0	0
$895$	−1106.01	−0.0413071
$896$	0	0
$897$	28085.6	1.04543
$898$	0	0
$899$	30883.2	1.14573
$900$	0	0
$901$	0	0
$902$	0	0
$903$	48310.1	1.78035
$904$	0	0
$905$	−15992.7	−0.587421
$906$	0	0
$907$	24422.2	0.894074	0.447037	−	0.894515i	$-0.352479\pi$
$907$	24422.2	0.894074	0.447037	+	0.894515i	$0.352479\pi$
$908$	0	0
$909$	25398.2	0.926739
$910$	0	0
$911$	−17953.7	−0.652945	−0.326473	−	0.945207i	$-0.605860\pi$
$911$	−17953.7	−0.652945	−0.326473	+	0.945207i	$0.605860\pi$
$912$	0	0
$913$	17568.6	0.636842
$914$	0	0
$915$	−4606.94	−0.166449
$916$	0	0
$917$	−58393.2	−2.10285
$918$	0	0
$919$	−31677.4	−1.13704	−0.568520	−	0.822669i	$-0.692484\pi$
$919$	−31677.4	−1.13704	−0.568520	+	0.822669i	$0.692484\pi$
$920$	0	0
$921$	−64936.5	−2.32327
$922$	0	0
$923$	52586.7	1.87531
$924$	0	0
$925$	−34204.0	−1.21581
$926$	0	0
$927$	−26983.0	−0.956026
$928$	0	0
$929$	−22251.5	−0.785844	−0.392922	−	0.919572i	$-0.628536\pi$
$929$	−22251.5	−0.785844	−0.392922	+	0.919572i	$0.628536\pi$
$930$	0	0
$931$	3107.52	0.109393
$932$	0	0
$933$	−32344.1	−1.13494
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−48139.3	−1.67838	−0.839190	−	0.543838i	$-0.816971\pi$
$937$	−48139.3	−1.67838	−0.839190	+	0.543838i	$0.816971\pi$
$938$	0	0
$939$	33923.5	1.17897
$940$	0	0
$941$	27519.9	0.953373	0.476686	−	0.879073i	$-0.341838\pi$
$941$	27519.9	0.953373	0.476686	+	0.879073i	$0.341838\pi$
$942$	0	0
$943$	−835.458	−0.0288508
$944$	0	0
$945$	−6342.76	−0.218338
$946$	0	0
$947$	2914.79	0.100019	0.0500094	−	0.998749i	$-0.484075\pi$
$947$	2914.79	0.100019	0.0500094	+	0.998749i	$0.484075\pi$
$948$	0	0
$949$	31014.8	1.06089
$950$	0	0
$951$	−64197.0	−2.18899
$952$	0	0
$953$	−13351.6	−0.453832	−0.226916	−	0.973914i	$-0.572864\pi$
$953$	−13351.6	−0.453832	−0.226916	+	0.973914i	$0.572864\pi$
$954$	0	0
$955$	−10316.6	−0.349568
$956$	0	0
$957$	−14135.8	−0.477477
$958$	0	0
$959$	−34769.7	−1.17077
$960$	0	0
$961$	14757.8	0.495377
$962$	0	0
$963$	6804.82	0.227707
$964$	0	0
$965$	5733.25	0.191254
$966$	0	0
$967$	−21177.3	−0.704257	−0.352129	−	0.935952i	$-0.614542\pi$
$967$	−21177.3	−0.704257	−0.352129	+	0.935952i	$0.614542\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−31599.2	−1.04435	−0.522177	−	0.852837i	$-0.674880\pi$
$971$	−31599.2	−1.04435	−0.522177	+	0.852837i	$0.674880\pi$
$972$	0	0
$973$	−47144.9	−1.55334
$974$	0	0
$975$	45511.9	1.49492
$976$	0	0
$977$	−46904.5	−1.53594	−0.767968	−	0.640489i	$-0.778732\pi$
$977$	−46904.5	−1.53594	−0.767968	+	0.640489i	$0.778732\pi$
$978$	0	0
$979$	−18861.9	−0.615758
$980$	0	0
$981$	11729.4	0.381744
$982$	0	0
$983$	38324.0	1.24348	0.621742	−	0.783222i	$-0.286425\pi$
$983$	38324.0	1.24348	0.621742	+	0.783222i	$0.286425\pi$
$984$	0	0
$985$	−10478.0	−0.338940
$986$	0	0
$987$	10509.9	0.338942
$988$	0	0
$989$	−20268.2	−0.651659
$990$	0	0
$991$	2292.44	0.0734831	0.0367416	−	0.999325i	$-0.488302\pi$
$991$	2292.44	0.0734831	0.0367416	+	0.999325i	$0.488302\pi$
$992$	0	0
$993$	51400.6	1.64265
$994$	0	0
$995$	−11188.8	−0.356491
$996$	0	0
$997$	43002.8	1.36601	0.683005	−	0.730414i	$-0.260673\pi$
$997$	43002.8	1.36601	0.683005	+	0.730414i	$0.260673\pi$
$998$	0	0
$999$	20663.0	0.654403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.n.1.4	✓	18
17.16	even	2		2312.4.a.q.1.15	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.n.1.4	✓	18	1.1	even	1	trivial
2312.4.a.q.1.15	yes	18	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-6.63767 q^{3} +3.97104 q^{5} -24.2054 q^{7} +17.0586 q^{9} +O(q^{10})\)	\(q-6.63767 q^{3} +3.97104 q^{5} -24.2054 q^{7} +17.0586 q^{9} -14.5546 q^{11} +62.7717 q^{13} -26.3585 q^{15} +12.7935 q^{19} +160.667 q^{21} -67.4068 q^{23} -109.231 q^{25} +65.9875 q^{27} -146.320 q^{29} -211.066 q^{31} +96.6086 q^{33} -96.1206 q^{35} +313.135 q^{37} -416.658 q^{39} +12.3943 q^{41} +300.684 q^{43} +67.7406 q^{45} +65.4144 q^{47} +242.899 q^{49} +610.708 q^{53} -57.7970 q^{55} -84.9188 q^{57} -152.734 q^{59} +174.780 q^{61} -412.910 q^{63} +249.269 q^{65} +525.734 q^{67} +447.424 q^{69} +837.745 q^{71} +494.089 q^{73} +725.038 q^{75} +352.299 q^{77} -885.107 q^{79} -898.586 q^{81} -1207.08 q^{83} +971.226 q^{87} +1295.94 q^{89} -1519.41 q^{91} +1400.98 q^{93} +50.8034 q^{95} -850.332 q^{97} -248.282 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 18 q^{3} - 51 q^{7} + 120 q^{9}+O(q^{10}) \) 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

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