LMFDB - Embedded newform 448.4.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, 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("448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.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:	$26.4328556826$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.20293 q^{3} -5.37748 q^{5} -7.00000 q^{7} -25.5530 q^{9} +O(q^{10})$	$q-1.20293 q^{3} -5.37748 q^{5} -7.00000 q^{7} -25.5530 q^{9} +64.8746 q^{11} +40.9735 q^{13} +6.46874 q^{15} +114.967 q^{17} -77.3580 q^{19} +8.42052 q^{21} -193.716 q^{23} -96.0827 q^{25} +63.2176 q^{27} -15.2450 q^{29} -285.720 q^{31} -78.0397 q^{33} +37.6424 q^{35} -267.265 q^{37} -49.2883 q^{39} +107.093 q^{41} +204.247 q^{43} +137.411 q^{45} -367.352 q^{47} +49.0000 q^{49} -138.297 q^{51} +474.907 q^{53} -348.862 q^{55} +93.0563 q^{57} -801.275 q^{59} -295.589 q^{61} +178.871 q^{63} -220.334 q^{65} -379.222 q^{67} +233.027 q^{69} -374.737 q^{71} -242.238 q^{73} +115.581 q^{75} -454.122 q^{77} +44.7115 q^{79} +613.883 q^{81} -631.985 q^{83} -618.232 q^{85} +18.3387 q^{87} -248.867 q^{89} -286.814 q^{91} +343.701 q^{93} +415.991 q^{95} -1567.25 q^{97} -1657.74 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{5} - 21 q^{7} + 15 q^{9}+O(q^{10}) $ 3 * q + 6 * q^5 - 21 * q^7 + 15 * q^9	$ 3 q + 6 q^{5} - 21 q^{7} + 15 q^{9} + 6 q^{13} - 168 q^{15} - 66 q^{17} + 168 q^{19} - 336 q^{23} + 69 q^{25} + 168 q^{27} - 90 q^{29} - 504 q^{31} + 120 q^{33} - 42 q^{35} - 18 q^{37} - 840 q^{39} - 450 q^{41} + 150 q^{45} - 504 q^{47} + 147 q^{49} - 336 q^{51} + 78 q^{53} - 1176 q^{55} + 48 q^{57} - 504 q^{59} - 498 q^{61} - 105 q^{63} + 1068 q^{65} - 1008 q^{67} + 1224 q^{69} - 504 q^{71} - 234 q^{73} - 1848 q^{75} - 168 q^{79} - 981 q^{81} - 3024 q^{83} - 1476 q^{85} + 336 q^{87} + 246 q^{89} - 42 q^{91} - 1200 q^{93} + 2184 q^{95} - 2514 q^{97} - 2688 q^{99}+O(q^{100}) $ 3 * q + 6 * q^5 - 21 * q^7 + 15 * q^9 + 6 * q^13 - 168 * q^15 - 66 * q^17 + 168 * q^19 - 336 * q^23 + 69 * q^25 + 168 * q^27 - 90 * q^29 - 504 * q^31 + 120 * q^33 - 42 * q^35 - 18 * q^37 - 840 * q^39 - 450 * q^41 + 150 * q^45 - 504 * q^47 + 147 * q^49 - 336 * q^51 + 78 * q^53 - 1176 * q^55 + 48 * q^57 - 504 * q^59 - 498 * q^61 - 105 * q^63 + 1068 * q^65 - 1008 * q^67 + 1224 * q^69 - 504 * q^71 - 234 * q^73 - 1848 * q^75 - 168 * q^79 - 981 * q^81 - 3024 * q^83 - 1476 * q^85 + 336 * q^87 + 246 * q^89 - 42 * q^91 - 1200 * q^93 + 2184 * q^95 - 2514 * q^97 - 2688 * 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
$3$	−1.20293	−0.231504	−0.115752	−	0.993278i	$-0.536928\pi$
$3$	−1.20293	−0.231504	−0.115752	+	0.993278i	$0.536928\pi$
$4$	0	0
$5$	−5.37748	−0.480976	−0.240488	−	0.970652i	$-0.577308\pi$
$5$	−5.37748	−0.480976	−0.240488	+	0.970652i	$0.577308\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−25.5530	−0.946406
$10$	0	0
$11$	64.8746	1.77822	0.889110	−	0.457693i	$-0.151324\pi$
$11$	64.8746	1.77822	0.889110	+	0.457693i	$0.151324\pi$
$12$	0	0
$13$	40.9735	0.874154	0.437077	−	0.899424i	$-0.356014\pi$
$13$	40.9735	0.874154	0.437077	+	0.899424i	$0.356014\pi$
$14$	0	0
$15$	6.46874	0.111348
$16$	0	0
$17$	114.967	1.64021	0.820104	−	0.572214i	$-0.193915\pi$
$17$	114.967	1.64021	0.820104	+	0.572214i	$0.193915\pi$
$18$	0	0
$19$	−77.3580	−0.934060	−0.467030	−	0.884241i	$-0.654676\pi$
$19$	−77.3580	−0.934060	−0.467030	+	0.884241i	$0.654676\pi$
$20$	0	0
$21$	8.42052	0.0875004
$22$	0	0
$23$	−193.716	−1.75620	−0.878098	−	0.478481i	$-0.841188\pi$
$23$	−193.716	−1.75620	−0.878098	+	0.478481i	$0.841188\pi$
$24$	0	0
$25$	−96.0827	−0.768662
$26$	0	0
$27$	63.2176	0.450601
$28$	0	0
$29$	−15.2450	−0.0976184	−0.0488092	−	0.998808i	$-0.515543\pi$
$29$	−15.2450	−0.0976184	−0.0488092	+	0.998808i	$0.515543\pi$
$30$	0	0
$31$	−285.720	−1.65538	−0.827690	−	0.561185i	$-0.810346\pi$
$31$	−285.720	−1.65538	−0.827690	+	0.561185i	$0.810346\pi$
$32$	0	0
$33$	−78.0397	−0.411666
$34$	0	0
$35$	37.6424	0.181792
$36$	0	0
$37$	−267.265	−1.18751	−0.593757	−	0.804644i	$-0.702356\pi$
$37$	−267.265	−1.18751	−0.593757	+	0.804644i	$0.702356\pi$
$38$	0	0
$39$	−49.2883	−0.202370
$40$	0	0
$41$	107.093	0.407928	0.203964	−	0.978978i	$-0.434617\pi$
$41$	107.093	0.407928	0.203964	+	0.978978i	$0.434617\pi$
$42$	0	0
$43$	204.247	0.724358	0.362179	−	0.932108i	$-0.382033\pi$
$43$	204.247	0.724358	0.362179	+	0.932108i	$0.382033\pi$
$44$	0	0
$45$	137.411	0.455199
$46$	0	0
$47$	−367.352	−1.14008	−0.570040	−	0.821617i	$-0.693072\pi$
$47$	−367.352	−1.14008	−0.570040	+	0.821617i	$0.693072\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−138.297	−0.379715
$52$	0	0
$53$	474.907	1.23082	0.615410	−	0.788207i	$-0.288990\pi$
$53$	474.907	1.23082	0.615410	+	0.788207i	$0.288990\pi$
$54$	0	0
$55$	−348.862	−0.855282
$56$	0	0
$57$	93.0563	0.216239
$58$	0	0
$59$	−801.275	−1.76809	−0.884044	−	0.467404i	$-0.845189\pi$
$59$	−801.275	−1.76809	−0.884044	+	0.467404i	$0.845189\pi$
$60$	0	0
$61$	−295.589	−0.620432	−0.310216	−	0.950666i	$-0.600401\pi$
$61$	−295.589	−0.620432	−0.310216	+	0.950666i	$0.600401\pi$
$62$	0	0
$63$	178.871	0.357708
$64$	0	0
$65$	−220.334	−0.420447
$66$	0	0
$67$	−379.222	−0.691482	−0.345741	−	0.938330i	$-0.612372\pi$
$67$	−379.222	−0.691482	−0.345741	+	0.938330i	$0.612372\pi$
$68$	0	0
$69$	233.027	0.406567
$70$	0	0
$71$	−374.737	−0.626381	−0.313191	−	0.949690i	$-0.601398\pi$
$71$	−374.737	−0.626381	−0.313191	+	0.949690i	$0.601398\pi$
$72$	0	0
$73$	−242.238	−0.388381	−0.194191	−	0.980964i	$-0.562208\pi$
$73$	−242.238	−0.388381	−0.194191	+	0.980964i	$0.562208\pi$
$74$	0	0
$75$	115.581	0.177948
$76$	0	0
$77$	−454.122	−0.672104
$78$	0	0
$79$	44.7115	0.0636764	0.0318382	−	0.999493i	$-0.489864\pi$
$79$	44.7115	0.0636764	0.0318382	+	0.999493i	$0.489864\pi$
$80$	0	0
$81$	613.883	0.842090
$82$	0	0
$83$	−631.985	−0.835775	−0.417888	−	0.908499i	$-0.637229\pi$
$83$	−631.985	−0.835775	−0.417888	+	0.908499i	$0.637229\pi$
$84$	0	0
$85$	−618.232	−0.788901
$86$	0	0
$87$	18.3387	0.0225991
$88$	0	0
$89$	−248.867	−0.296403	−0.148202	−	0.988957i	$-0.547348\pi$
$89$	−248.867	−0.296403	−0.148202	+	0.988957i	$0.547348\pi$
$90$	0	0
$91$	−286.814	−0.330399
$92$	0	0
$93$	343.701	0.383228
$94$	0	0
$95$	415.991	0.449261
$96$	0	0
$97$	−1567.25	−1.64052	−0.820259	−	0.571992i	$-0.806171\pi$
$97$	−1567.25	−1.64052	−0.820259	+	0.571992i	$0.806171\pi$
$98$	0	0
$99$	−1657.74	−1.68292
$100$	0	0
$101$	305.715	0.301186	0.150593	−	0.988596i	$-0.451882\pi$
$101$	305.715	0.301186	0.150593	+	0.988596i	$0.451882\pi$
$102$	0	0
$103$	1262.06	1.20732	0.603661	−	0.797241i	$-0.293708\pi$
$103$	1262.06	1.20732	0.603661	+	0.797241i	$0.293708\pi$
$104$	0	0
$105$	−45.2812	−0.0420856
$106$	0	0
$107$	−127.196	−0.114920	−0.0574602	−	0.998348i	$-0.518300\pi$
$107$	−127.196	−0.114920	−0.0574602	+	0.998348i	$0.518300\pi$
$108$	0	0
$109$	355.146	0.312081	0.156040	−	0.987751i	$-0.450127\pi$
$109$	355.146	0.312081	0.156040	+	0.987751i	$0.450127\pi$
$110$	0	0
$111$	321.501	0.274915
$112$	0	0
$113$	1436.68	1.19603	0.598015	−	0.801485i	$-0.295956\pi$
$113$	1436.68	1.19603	0.598015	+	0.801485i	$0.295956\pi$
$114$	0	0
$115$	1041.70	0.844689
$116$	0	0
$117$	−1046.99	−0.827304
$118$	0	0
$119$	−804.768	−0.619940
$120$	0	0
$121$	2877.71	2.16207
$122$	0	0
$123$	−128.825	−0.0944370
$124$	0	0
$125$	1188.87	0.850685
$126$	0	0
$127$	12.3678	0.00864146	0.00432073	−	0.999991i	$-0.498625\pi$
$127$	12.3678	0.00864146	0.00432073	+	0.999991i	$0.498625\pi$
$128$	0	0
$129$	−245.695	−0.167692
$130$	0	0
$131$	−2155.93	−1.43790	−0.718948	−	0.695063i	$-0.755376\pi$
$131$	−2155.93	−1.43790	−0.718948	+	0.695063i	$0.755376\pi$
$132$	0	0
$133$	541.506	0.353042
$134$	0	0
$135$	−339.951	−0.216729
$136$	0	0
$137$	−1959.34	−1.22188	−0.610939	−	0.791677i	$-0.709208\pi$
$137$	−1959.34	−1.22188	−0.610939	+	0.791677i	$0.709208\pi$
$138$	0	0
$139$	49.3878	0.0301368	0.0150684	−	0.999886i	$-0.495203\pi$
$139$	49.3878	0.0301368	0.0150684	+	0.999886i	$0.495203\pi$
$140$	0	0
$141$	441.899	0.263933
$142$	0	0
$143$	2658.14	1.55444
$144$	0	0
$145$	81.9799	0.0469521
$146$	0	0
$147$	−58.9436	−0.0330720
$148$	0	0
$149$	−1813.83	−0.997278	−0.498639	−	0.866810i	$-0.666167\pi$
$149$	−1813.83	−0.997278	−0.498639	+	0.866810i	$0.666167\pi$
$150$	0	0
$151$	−1542.51	−0.831310	−0.415655	−	0.909522i	$-0.636448\pi$
$151$	−1542.51	−0.831310	−0.415655	+	0.909522i	$0.636448\pi$
$152$	0	0
$153$	−2937.74	−1.55230
$154$	0	0
$155$	1536.45	0.796199
$156$	0	0
$157$	1396.93	0.710111	0.355055	−	0.934845i	$-0.384462\pi$
$157$	1396.93	0.710111	0.355055	+	0.934845i	$0.384462\pi$
$158$	0	0
$159$	−571.280	−0.284940
$160$	0	0
$161$	1356.01	0.663780
$162$	0	0
$163$	1971.16	0.947198	0.473599	−	0.880741i	$-0.342955\pi$
$163$	1971.16	0.947198	0.473599	+	0.880741i	$0.342955\pi$
$164$	0	0
$165$	419.657	0.198001
$166$	0	0
$167$	−744.765	−0.345100	−0.172550	−	0.985001i	$-0.555201\pi$
$167$	−744.765	−0.345100	−0.172550	+	0.985001i	$0.555201\pi$
$168$	0	0
$169$	−518.174	−0.235855
$170$	0	0
$171$	1976.73	0.884000
$172$	0	0
$173$	−3753.04	−1.64935	−0.824677	−	0.565604i	$-0.808643\pi$
$173$	−3753.04	−1.64935	−0.824677	+	0.565604i	$0.808643\pi$
$174$	0	0
$175$	672.579	0.290527
$176$	0	0
$177$	963.879	0.409320
$178$	0	0
$179$	1951.64	0.814928	0.407464	−	0.913221i	$-0.366413\pi$
$179$	1951.64	0.814928	0.407464	+	0.913221i	$0.366413\pi$
$180$	0	0
$181$	3370.13	1.38398	0.691989	−	0.721908i	$-0.256735\pi$
$181$	3370.13	1.38398	0.691989	+	0.721908i	$0.256735\pi$
$182$	0	0
$183$	355.574	0.143633
$184$	0	0
$185$	1437.21	0.571167
$186$	0	0
$187$	7458.42	2.91665
$188$	0	0
$189$	−442.523	−0.170311
$190$	0	0
$191$	3566.04	1.35094	0.675470	−	0.737387i	$-0.263941\pi$
$191$	3566.04	1.35094	0.675470	+	0.737387i	$0.263941\pi$
$192$	0	0
$193$	424.407	0.158287	0.0791437	−	0.996863i	$-0.474781\pi$
$193$	424.407	0.158287	0.0791437	+	0.996863i	$0.474781\pi$
$194$	0	0
$195$	265.047	0.0973353
$196$	0	0
$197$	−1812.40	−0.655475	−0.327737	−	0.944769i	$-0.606286\pi$
$197$	−1812.40	−0.655475	−0.327737	+	0.944769i	$0.606286\pi$
$198$	0	0
$199$	−4674.96	−1.66532	−0.832662	−	0.553782i	$-0.813184\pi$
$199$	−4674.96	−1.66532	−0.832662	+	0.553782i	$0.813184\pi$
$200$	0	0
$201$	456.178	0.160081
$202$	0	0
$203$	106.715	0.0368963
$204$	0	0
$205$	−575.888	−0.196204
$206$	0	0
$207$	4950.01	1.66207
$208$	0	0
$209$	−5018.57	−1.66096
$210$	0	0
$211$	2528.32	0.824915	0.412457	−	0.910977i	$-0.364671\pi$
$211$	2528.32	0.824915	0.412457	+	0.910977i	$0.364671\pi$
$212$	0	0
$213$	450.783	0.145010
$214$	0	0
$215$	−1098.34	−0.348399
$216$	0	0
$217$	2000.04	0.625675
$218$	0	0
$219$	291.396	0.0899119
$220$	0	0
$221$	4710.59	1.43379
$222$	0	0
$223$	−752.732	−0.226039	−0.113019	−	0.993593i	$-0.536052\pi$
$223$	−752.732	−0.226039	−0.113019	+	0.993593i	$0.536052\pi$
$224$	0	0
$225$	2455.20	0.727466
$226$	0	0
$227$	−4070.90	−1.19029	−0.595144	−	0.803619i	$-0.702905\pi$
$227$	−4070.90	−1.19029	−0.595144	+	0.803619i	$0.702905\pi$
$228$	0	0
$229$	−982.525	−0.283524	−0.141762	−	0.989901i	$-0.545277\pi$
$229$	−982.525	−0.283524	−0.141762	+	0.989901i	$0.545277\pi$
$230$	0	0
$231$	546.278	0.155595
$232$	0	0
$233$	2613.56	0.734851	0.367425	−	0.930053i	$-0.380239\pi$
$233$	2613.56	0.734851	0.367425	+	0.930053i	$0.380239\pi$
$234$	0	0
$235$	1975.43	0.548352
$236$	0	0
$237$	−53.7848	−0.0147414
$238$	0	0
$239$	−141.870	−0.0383967	−0.0191984	−	0.999816i	$-0.506111\pi$
$239$	−141.870	−0.0383967	−0.0191984	+	0.999816i	$0.506111\pi$
$240$	0	0
$241$	−4180.11	−1.11728	−0.558640	−	0.829410i	$-0.688677\pi$
$241$	−4180.11	−1.11728	−0.558640	+	0.829410i	$0.688677\pi$
$242$	0	0
$243$	−2445.33	−0.645548
$244$	0	0
$245$	−263.497	−0.0687109
$246$	0	0
$247$	−3169.63	−0.816512
$248$	0	0
$249$	760.234	0.193485
$250$	0	0
$251$	4988.02	1.25435	0.627173	−	0.778880i	$-0.284212\pi$
$251$	4988.02	1.25435	0.627173	+	0.778880i	$0.284212\pi$
$252$	0	0
$253$	−12567.2	−3.12290
$254$	0	0
$255$	743.690	0.182634
$256$	0	0
$257$	−2496.21	−0.605873	−0.302936	−	0.953011i	$-0.597967\pi$
$257$	−2496.21	−0.605873	−0.302936	+	0.953011i	$0.597967\pi$
$258$	0	0
$259$	1870.85	0.448838
$260$	0	0
$261$	389.556	0.0923866
$262$	0	0
$263$	−1718.47	−0.402911	−0.201456	−	0.979498i	$-0.564567\pi$
$263$	−1718.47	−0.402911	−0.201456	+	0.979498i	$0.564567\pi$
$264$	0	0
$265$	−2553.80	−0.591996
$266$	0	0
$267$	299.370	0.0686186
$268$	0	0
$269$	5653.67	1.28145	0.640726	−	0.767770i	$-0.278634\pi$
$269$	5653.67	1.28145	0.640726	+	0.767770i	$0.278634\pi$
$270$	0	0
$271$	2809.53	0.629766	0.314883	−	0.949130i	$-0.398035\pi$
$271$	2809.53	0.629766	0.314883	+	0.949130i	$0.398035\pi$
$272$	0	0
$273$	345.018	0.0764888
$274$	0	0
$275$	−6233.33	−1.36685
$276$	0	0
$277$	6707.40	1.45490	0.727452	−	0.686158i	$-0.240704\pi$
$277$	6707.40	1.45490	0.727452	+	0.686158i	$0.240704\pi$
$278$	0	0
$279$	7300.99	1.56666
$280$	0	0
$281$	−2599.96	−0.551959	−0.275980	−	0.961164i	$-0.589002\pi$
$281$	−2599.96	−0.551959	−0.275980	+	0.961164i	$0.589002\pi$
$282$	0	0
$283$	1576.37	0.331115	0.165558	−	0.986200i	$-0.447058\pi$
$283$	1576.37	0.331115	0.165558	+	0.986200i	$0.447058\pi$
$284$	0	0
$285$	−500.409	−0.104006
$286$	0	0
$287$	−749.648	−0.154182
$288$	0	0
$289$	8304.36	1.69028
$290$	0	0
$291$	1885.30	0.379787
$292$	0	0
$293$	6393.39	1.27476	0.637382	−	0.770548i	$-0.280017\pi$
$293$	6393.39	1.27476	0.637382	+	0.770548i	$0.280017\pi$
$294$	0	0
$295$	4308.84	0.850408
$296$	0	0
$297$	4101.22	0.801268
$298$	0	0
$299$	−7937.20	−1.53518
$300$	0	0
$301$	−1429.73	−0.273782
$302$	0	0
$303$	−367.755	−0.0697259
$304$	0	0
$305$	1589.53	0.298413
$306$	0	0
$307$	−6139.29	−1.14133	−0.570664	−	0.821184i	$-0.693314\pi$
$307$	−6139.29	−1.14133	−0.570664	+	0.821184i	$0.693314\pi$
$308$	0	0
$309$	−1518.17	−0.279500
$310$	0	0
$311$	−1230.29	−0.224320	−0.112160	−	0.993690i	$-0.535777\pi$
$311$	−1230.29	−0.224320	−0.112160	+	0.993690i	$0.535777\pi$
$312$	0	0
$313$	−5046.27	−0.911285	−0.455642	−	0.890163i	$-0.650590\pi$
$313$	−5046.27	−0.911285	−0.455642	+	0.890163i	$0.650590\pi$
$314$	0	0
$315$	−961.874	−0.172049
$316$	0	0
$317$	−1209.75	−0.214342	−0.107171	−	0.994241i	$-0.534179\pi$
$317$	−1209.75	−0.214342	−0.107171	+	0.994241i	$0.534179\pi$
$318$	0	0
$319$	−989.016	−0.173587
$320$	0	0
$321$	153.008	0.0266046
$322$	0	0
$323$	−8893.60	−1.53205
$324$	0	0
$325$	−3936.84	−0.671928
$326$	0	0
$327$	−427.216	−0.0722480
$328$	0	0
$329$	2571.46	0.430910
$330$	0	0
$331$	4940.85	0.820465	0.410232	−	0.911981i	$-0.365448\pi$
$331$	4940.85	0.820465	0.410232	+	0.911981i	$0.365448\pi$
$332$	0	0
$333$	6829.40	1.12387
$334$	0	0
$335$	2039.26	0.332587
$336$	0	0
$337$	−5554.70	−0.897875	−0.448937	−	0.893563i	$-0.648197\pi$
$337$	−5554.70	−0.897875	−0.448937	+	0.893563i	$0.648197\pi$
$338$	0	0
$339$	−1728.22	−0.276886
$340$	0	0
$341$	−18536.0	−2.94363
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1253.10	−0.195549
$346$	0	0
$347$	−6793.58	−1.05100	−0.525502	−	0.850792i	$-0.676123\pi$
$347$	−6793.58	−1.05100	−0.525502	+	0.850792i	$0.676123\pi$
$348$	0	0
$349$	8080.59	1.23938	0.619691	−	0.784846i	$-0.287258\pi$
$349$	8080.59	1.23938	0.619691	+	0.784846i	$0.287258\pi$
$350$	0	0
$351$	2590.24	0.393895
$352$	0	0
$353$	−275.755	−0.0415777	−0.0207889	−	0.999784i	$-0.506618\pi$
$353$	−275.755	−0.0415777	−0.0207889	+	0.999784i	$0.506618\pi$
$354$	0	0
$355$	2015.14	0.301275
$356$	0	0
$357$	968.080	0.143519
$358$	0	0
$359$	525.031	0.0771869	0.0385935	−	0.999255i	$-0.487712\pi$
$359$	525.031	0.0771869	0.0385935	+	0.999255i	$0.487712\pi$
$360$	0	0
$361$	−874.741	−0.127532
$362$	0	0
$363$	−3461.69	−0.500528
$364$	0	0
$365$	1302.63	0.186802
$366$	0	0
$367$	3243.27	0.461300	0.230650	−	0.973037i	$-0.425915\pi$
$367$	3243.27	0.461300	0.230650	+	0.973037i	$0.425915\pi$
$368$	0	0
$369$	−2736.53	−0.386065
$370$	0	0
$371$	−3324.35	−0.465206
$372$	0	0
$373$	6731.12	0.934381	0.467191	−	0.884157i	$-0.345266\pi$
$373$	6731.12	0.934381	0.467191	+	0.884157i	$0.345266\pi$
$374$	0	0
$375$	−1430.13	−0.196937
$376$	0	0
$377$	−624.642	−0.0853335
$378$	0	0
$379$	6185.38	0.838315	0.419158	−	0.907913i	$-0.362325\pi$
$379$	6185.38	0.838315	0.419158	+	0.907913i	$0.362325\pi$
$380$	0	0
$381$	−14.8776	−0.00200053
$382$	0	0
$383$	−184.777	−0.0246518	−0.0123259	−	0.999924i	$-0.503924\pi$
$383$	−184.777	−0.0246518	−0.0123259	+	0.999924i	$0.503924\pi$
$384$	0	0
$385$	2442.03	0.323266
$386$	0	0
$387$	−5219.12	−0.685537
$388$	0	0
$389$	−4583.77	−0.597446	−0.298723	−	0.954340i	$-0.596561\pi$
$389$	−4583.77	−0.597446	−0.298723	+	0.954340i	$0.596561\pi$
$390$	0	0
$391$	−22270.9	−2.88053
$392$	0	0
$393$	2593.44	0.332879
$394$	0	0
$395$	−240.435	−0.0306268
$396$	0	0
$397$	−5957.42	−0.753134	−0.376567	−	0.926389i	$-0.622896\pi$
$397$	−5957.42	−0.753134	−0.376567	+	0.926389i	$0.622896\pi$
$398$	0	0
$399$	−651.394	−0.0817306
$400$	0	0
$401$	12036.0	1.49888	0.749438	−	0.662075i	$-0.230324\pi$
$401$	12036.0	1.49888	0.749438	+	0.662075i	$0.230324\pi$
$402$	0	0
$403$	−11706.9	−1.44706
$404$	0	0
$405$	−3301.15	−0.405025
$406$	0	0
$407$	−17338.7	−2.11166
$408$	0	0
$409$	7437.23	0.899138	0.449569	−	0.893246i	$-0.351578\pi$
$409$	7437.23	0.899138	0.449569	+	0.893246i	$0.351578\pi$
$410$	0	0
$411$	2356.95	0.282870
$412$	0	0
$413$	5608.93	0.668274
$414$	0	0
$415$	3398.48	0.401988
$416$	0	0
$417$	−59.4101	−0.00697680
$418$	0	0
$419$	−3800.96	−0.443172	−0.221586	−	0.975141i	$-0.571123\pi$
$419$	−3800.96	−0.443172	−0.221586	+	0.975141i	$0.571123\pi$
$420$	0	0
$421$	787.098	0.0911184	0.0455592	−	0.998962i	$-0.485493\pi$
$421$	787.098	0.0911184	0.0455592	+	0.998962i	$0.485493\pi$
$422$	0	0
$423$	9386.93	1.07898
$424$	0	0
$425$	−11046.3	−1.26077
$426$	0	0
$427$	2069.13	0.234501
$428$	0	0
$429$	−3197.56	−0.359859
$430$	0	0
$431$	−5732.42	−0.640652	−0.320326	−	0.947307i	$-0.603792\pi$
$431$	−5732.42	−0.640652	−0.320326	+	0.947307i	$0.603792\pi$
$432$	0	0
$433$	−9816.02	−1.08944	−0.544720	−	0.838618i	$-0.683364\pi$
$433$	−9816.02	−1.08944	−0.544720	+	0.838618i	$0.683364\pi$
$434$	0	0
$435$	−98.6162	−0.0108696
$436$	0	0
$437$	14985.5	1.64039
$438$	0	0
$439$	−13729.9	−1.49270	−0.746349	−	0.665555i	$-0.768195\pi$
$439$	−13729.9	−1.49270	−0.746349	+	0.665555i	$0.768195\pi$
$440$	0	0
$441$	−1252.09	−0.135201
$442$	0	0
$443$	3063.57	0.328565	0.164283	−	0.986413i	$-0.447469\pi$
$443$	3063.57	0.328565	0.164283	+	0.986413i	$0.447469\pi$
$444$	0	0
$445$	1338.28	0.142563
$446$	0	0
$447$	2181.91	0.230874
$448$	0	0
$449$	−4005.72	−0.421028	−0.210514	−	0.977591i	$-0.567514\pi$
$449$	−4005.72	−0.421028	−0.210514	+	0.977591i	$0.567514\pi$
$450$	0	0
$451$	6947.59	0.725386
$452$	0	0
$453$	1855.53	0.192452
$454$	0	0
$455$	1542.34	0.158914
$456$	0	0
$457$	287.151	0.0293925	0.0146962	−	0.999892i	$-0.495322\pi$
$457$	287.151	0.0293925	0.0146962	+	0.999892i	$0.495322\pi$
$458$	0	0
$459$	7267.92	0.739080
$460$	0	0
$461$	−11013.0	−1.11263	−0.556317	−	0.830970i	$-0.687786\pi$
$461$	−11013.0	−1.11263	−0.556317	+	0.830970i	$0.687786\pi$
$462$	0	0
$463$	−3512.06	−0.352525	−0.176263	−	0.984343i	$-0.556401\pi$
$463$	−3512.06	−0.352525	−0.176263	+	0.984343i	$0.556401\pi$
$464$	0	0
$465$	−1848.25	−0.184323
$466$	0	0
$467$	9305.20	0.922041	0.461021	−	0.887389i	$-0.347483\pi$
$467$	9305.20	0.922041	0.461021	+	0.887389i	$0.347483\pi$
$468$	0	0
$469$	2654.55	0.261356
$470$	0	0
$471$	−1680.41	−0.164394
$472$	0	0
$473$	13250.5	1.28807
$474$	0	0
$475$	7432.77	0.717976
$476$	0	0
$477$	−12135.3	−1.16486
$478$	0	0
$479$	−12393.9	−1.18223	−0.591117	−	0.806586i	$-0.701313\pi$
$479$	−12393.9	−1.18223	−0.591117	+	0.806586i	$0.701313\pi$
$480$	0	0
$481$	−10950.8	−1.03807
$482$	0	0
$483$	−1631.19	−0.153668
$484$	0	0
$485$	8427.86	0.789050
$486$	0	0
$487$	−7406.91	−0.689197	−0.344599	−	0.938750i	$-0.611985\pi$
$487$	−7406.91	−0.689197	−0.344599	+	0.938750i	$0.611985\pi$
$488$	0	0
$489$	−2371.17	−0.219280
$490$	0	0
$491$	−13517.4	−1.24242	−0.621212	−	0.783643i	$-0.713359\pi$
$491$	−13517.4	−1.24242	−0.621212	+	0.783643i	$0.713359\pi$
$492$	0	0
$493$	−1752.67	−0.160114
$494$	0	0
$495$	8914.45	0.809444
$496$	0	0
$497$	2623.16	0.236750
$498$	0	0
$499$	10089.0	0.905102	0.452551	−	0.891739i	$-0.350514\pi$
$499$	10089.0	0.905102	0.452551	+	0.891739i	$0.350514\pi$
$500$	0	0
$501$	895.901	0.0798921
$502$	0	0
$503$	14624.9	1.29640	0.648201	−	0.761469i	$-0.275522\pi$
$503$	14624.9	1.29640	0.648201	+	0.761469i	$0.275522\pi$
$504$	0	0
$505$	−1643.98	−0.144863
$506$	0	0
$507$	623.328	0.0546015
$508$	0	0
$509$	−5366.07	−0.467283	−0.233641	−	0.972323i	$-0.575064\pi$
$509$	−5366.07	−0.467283	−0.233641	+	0.972323i	$0.575064\pi$
$510$	0	0
$511$	1695.67	0.146794
$512$	0	0
$513$	−4890.39	−0.420888
$514$	0	0
$515$	−6786.69	−0.580694
$516$	0	0
$517$	−23831.8	−2.02731
$518$	0	0
$519$	4514.65	0.381832
$520$	0	0
$521$	−2940.22	−0.247242	−0.123621	−	0.992329i	$-0.539451\pi$
$521$	−2940.22	−0.247242	−0.123621	+	0.992329i	$0.539451\pi$
$522$	0	0
$523$	−18921.2	−1.58196	−0.790982	−	0.611839i	$-0.790430\pi$
$523$	−18921.2	−1.58196	−0.790982	+	0.611839i	$0.790430\pi$
$524$	0	0
$525$	−809.066	−0.0672582
$526$	0	0
$527$	−32848.3	−2.71517
$528$	0	0
$529$	25358.7	2.08422
$530$	0	0
$531$	20475.0	1.67333
$532$	0	0
$533$	4387.95	0.356592
$534$	0	0
$535$	683.993	0.0552740
$536$	0	0
$537$	−2347.68	−0.188659
$538$	0	0
$539$	3178.86	0.254032
$540$	0	0
$541$	20592.8	1.63651	0.818255	−	0.574856i	$-0.194942\pi$
$541$	20592.8	1.63651	0.818255	+	0.574856i	$0.194942\pi$
$542$	0	0
$543$	−4054.04	−0.320397
$544$	0	0
$545$	−1909.79	−0.150104
$546$	0	0
$547$	15822.9	1.23682	0.618408	−	0.785857i	$-0.287778\pi$
$547$	15822.9	1.23682	0.618408	+	0.785857i	$0.287778\pi$
$548$	0	0
$549$	7553.18	0.587180
$550$	0	0
$551$	1179.33	0.0911814
$552$	0	0
$553$	−312.980	−0.0240674
$554$	0	0
$555$	−1728.87	−0.132227
$556$	0	0
$557$	16750.8	1.27424	0.637121	−	0.770764i	$-0.280125\pi$
$557$	16750.8	1.27424	0.637121	+	0.770764i	$0.280125\pi$
$558$	0	0
$559$	8368.72	0.633201
$560$	0	0
$561$	−8971.97	−0.675217
$562$	0	0
$563$	835.723	0.0625604	0.0312802	−	0.999511i	$-0.490042\pi$
$563$	835.723	0.0625604	0.0312802	+	0.999511i	$0.490042\pi$
$564$	0	0
$565$	−7725.71	−0.575262
$566$	0	0
$567$	−4297.18	−0.318280
$568$	0	0
$569$	−7960.01	−0.586469	−0.293235	−	0.956041i	$-0.594732\pi$
$569$	−7960.01	−0.586469	−0.293235	+	0.956041i	$0.594732\pi$
$570$	0	0
$571$	24987.4	1.83133	0.915664	−	0.401944i	$-0.131665\pi$
$571$	24987.4	1.83133	0.915664	+	0.401944i	$0.131665\pi$
$572$	0	0
$573$	−4289.70	−0.312748
$574$	0	0
$575$	18612.7	1.34992
$576$	0	0
$577$	20271.3	1.46258	0.731289	−	0.682068i	$-0.238919\pi$
$577$	20271.3	1.46258	0.731289	+	0.682068i	$0.238919\pi$
$578$	0	0
$579$	−510.532	−0.0366442
$580$	0	0
$581$	4423.89	0.315893
$582$	0	0
$583$	30809.4	2.18867
$584$	0	0
$585$	5630.19	0.397914
$586$	0	0
$587$	−13258.8	−0.932282	−0.466141	−	0.884710i	$-0.654356\pi$
$587$	−13258.8	−0.932282	−0.466141	+	0.884710i	$0.654356\pi$
$588$	0	0
$589$	22102.7	1.54623
$590$	0	0
$591$	2180.20	0.151745
$592$	0	0
$593$	1079.82	0.0747772	0.0373886	−	0.999301i	$-0.488096\pi$
$593$	1079.82	0.0747772	0.0373886	+	0.999301i	$0.488096\pi$
$594$	0	0
$595$	4327.62	0.298177
$596$	0	0
$597$	5623.66	0.385529
$598$	0	0
$599$	29080.1	1.98361	0.991805	−	0.127764i	$-0.0407799\pi$
$599$	29080.1	1.98361	0.991805	+	0.127764i	$0.0407799\pi$
$600$	0	0
$601$	19868.1	1.34848	0.674240	−	0.738512i	$-0.264471\pi$
$601$	19868.1	1.34848	0.674240	+	0.738512i	$0.264471\pi$
$602$	0	0
$603$	9690.24	0.654423
$604$	0	0
$605$	−15474.8	−1.03990
$606$	0	0
$607$	16114.9	1.07757	0.538785	−	0.842444i	$-0.318884\pi$
$607$	16114.9	1.07757	0.538785	+	0.842444i	$0.318884\pi$
$608$	0	0
$609$	−128.371	−0.00854164
$610$	0	0
$611$	−15051.7	−0.996605
$612$	0	0
$613$	−7248.39	−0.477585	−0.238792	−	0.971071i	$-0.576752\pi$
$613$	−7248.39	−0.477585	−0.238792	+	0.971071i	$0.576752\pi$
$614$	0	0
$615$	692.754	0.0454220
$616$	0	0
$617$	−10608.8	−0.692208	−0.346104	−	0.938196i	$-0.612496\pi$
$617$	−10608.8	−0.692208	−0.346104	+	0.938196i	$0.612496\pi$
$618$	0	0
$619$	−19228.9	−1.24858	−0.624291	−	0.781192i	$-0.714612\pi$
$619$	−19228.9	−1.24858	−0.624291	+	0.781192i	$0.714612\pi$
$620$	0	0
$621$	−12246.2	−0.791344
$622$	0	0
$623$	1742.07	0.112030
$624$	0	0
$625$	5617.23	0.359503
$626$	0	0
$627$	6036.99	0.384520
$628$	0	0
$629$	−30726.6	−1.94777
$630$	0	0
$631$	13310.3	0.839740	0.419870	−	0.907584i	$-0.362076\pi$
$631$	13310.3	0.839740	0.419870	+	0.907584i	$0.362076\pi$
$632$	0	0
$633$	−3041.40	−0.190971
$634$	0	0
$635$	−66.5076	−0.00415634
$636$	0	0
$637$	2007.70	0.124879
$638$	0	0
$639$	9575.63	0.592811
$640$	0	0
$641$	−15434.7	−0.951066	−0.475533	−	0.879698i	$-0.657745\pi$
$641$	−15434.7	−0.951066	−0.475533	+	0.879698i	$0.657745\pi$
$642$	0	0
$643$	−5672.98	−0.347932	−0.173966	−	0.984752i	$-0.555658\pi$
$643$	−5672.98	−0.347932	−0.173966	+	0.984752i	$0.555658\pi$
$644$	0	0
$645$	1321.22	0.0806559
$646$	0	0
$647$	−11473.0	−0.697142	−0.348571	−	0.937282i	$-0.613333\pi$
$647$	−11473.0	−0.697142	−0.348571	+	0.937282i	$0.613333\pi$
$648$	0	0
$649$	−51982.4	−3.14405
$650$	0	0
$651$	−2405.91	−0.144846
$652$	0	0
$653$	3622.81	0.217108	0.108554	−	0.994091i	$-0.465378\pi$
$653$	3622.81	0.217108	0.108554	+	0.994091i	$0.465378\pi$
$654$	0	0
$655$	11593.5	0.691594
$656$	0	0
$657$	6189.90	0.367566
$658$	0	0
$659$	9805.78	0.579634	0.289817	−	0.957082i	$-0.406405\pi$
$659$	9805.78	0.579634	0.289817	+	0.957082i	$0.406405\pi$
$660$	0	0
$661$	582.290	0.0342639	0.0171320	−	0.999853i	$-0.494546\pi$
$661$	582.290	0.0342639	0.0171320	+	0.999853i	$0.494546\pi$
$662$	0	0
$663$	−5666.51	−0.331929
$664$	0	0
$665$	−2911.94	−0.169805
$666$	0	0
$667$	2953.20	0.171437
$668$	0	0
$669$	905.484	0.0523289
$670$	0	0
$671$	−19176.2	−1.10326
$672$	0	0
$673$	−5536.24	−0.317097	−0.158549	−	0.987351i	$-0.550681\pi$
$673$	−5536.24	−0.317097	−0.158549	+	0.987351i	$0.550681\pi$
$674$	0	0
$675$	−6074.12	−0.346360
$676$	0	0
$677$	14403.6	0.817691	0.408846	−	0.912604i	$-0.365931\pi$
$677$	14403.6	0.817691	0.408846	+	0.912604i	$0.365931\pi$
$678$	0	0
$679$	10970.8	0.620058
$680$	0	0
$681$	4897.02	0.275557
$682$	0	0
$683$	−31790.2	−1.78100	−0.890498	−	0.454988i	$-0.849644\pi$
$683$	−31790.2	−1.78100	−0.890498	+	0.454988i	$0.849644\pi$
$684$	0	0
$685$	10536.3	0.587695
$686$	0	0
$687$	1181.91	0.0656371
$688$	0	0
$689$	19458.6	1.07593
$690$	0	0
$691$	21738.4	1.19677	0.598386	−	0.801208i	$-0.295809\pi$
$691$	21738.4	1.19677	0.598386	+	0.801208i	$0.295809\pi$
$692$	0	0
$693$	11604.2	0.636083
$694$	0	0
$695$	−265.582	−0.0144951
$696$	0	0
$697$	12312.1	0.669087
$698$	0	0
$699$	−3143.94	−0.170121
$700$	0	0
$701$	−8810.35	−0.474697	−0.237348	−	0.971425i	$-0.576278\pi$
$701$	−8810.35	−0.474697	−0.237348	+	0.971425i	$0.576278\pi$
$702$	0	0
$703$	20675.1	1.10921
$704$	0	0
$705$	−2376.30	−0.126946
$706$	0	0
$707$	−2140.01	−0.113838
$708$	0	0
$709$	3214.64	0.170280	0.0851398	−	0.996369i	$-0.472866\pi$
$709$	3214.64	0.170280	0.0851398	+	0.996369i	$0.472866\pi$
$710$	0	0
$711$	−1142.51	−0.0602637
$712$	0	0
$713$	55348.4	2.90717
$714$	0	0
$715$	−14294.1	−0.747648
$716$	0	0
$717$	170.660	0.00888901
$718$	0	0
$719$	−25591.7	−1.32741	−0.663706	−	0.747994i	$-0.731017\pi$
$719$	−25591.7	−1.32741	−0.663706	+	0.747994i	$0.731017\pi$
$720$	0	0
$721$	−8834.40	−0.456325
$722$	0	0
$723$	5028.38	0.258655
$724$	0	0
$725$	1464.78	0.0750355
$726$	0	0
$727$	33528.3	1.71045	0.855225	−	0.518257i	$-0.173419\pi$
$727$	33528.3	1.71045	0.855225	+	0.518257i	$0.173419\pi$
$728$	0	0
$729$	−13633.3	−0.692643
$730$	0	0
$731$	23481.6	1.18810
$732$	0	0
$733$	−17822.6	−0.898082	−0.449041	−	0.893511i	$-0.648234\pi$
$733$	−17822.6	−0.898082	−0.449041	+	0.893511i	$0.648234\pi$
$734$	0	0
$735$	316.968	0.0159069
$736$	0	0
$737$	−24601.9	−1.22961
$738$	0	0
$739$	3928.80	0.195566	0.0977830	−	0.995208i	$-0.468825\pi$
$739$	3928.80	0.195566	0.0977830	+	0.995208i	$0.468825\pi$
$740$	0	0
$741$	3812.84	0.189026
$742$	0	0
$743$	4426.55	0.218566	0.109283	−	0.994011i	$-0.465145\pi$
$743$	4426.55	0.218566	0.109283	+	0.994011i	$0.465145\pi$
$744$	0	0
$745$	9753.82	0.479667
$746$	0	0
$747$	16149.1	0.790982
$748$	0	0
$749$	890.371	0.0434358
$750$	0	0
$751$	12957.4	0.629592	0.314796	−	0.949159i	$-0.398064\pi$
$751$	12957.4	0.629592	0.314796	+	0.949159i	$0.398064\pi$
$752$	0	0
$753$	−6000.24	−0.290386
$754$	0	0
$755$	8294.82	0.399840
$756$	0	0
$757$	990.820	0.0475719	0.0237860	−	0.999717i	$-0.492428\pi$
$757$	990.820	0.0475719	0.0237860	+	0.999717i	$0.492428\pi$
$758$	0	0
$759$	15117.5	0.722965
$760$	0	0
$761$	−3305.77	−0.157469	−0.0787346	−	0.996896i	$-0.525088\pi$
$761$	−3305.77	−0.157469	−0.0787346	+	0.996896i	$0.525088\pi$
$762$	0	0
$763$	−2486.02	−0.117956
$764$	0	0
$765$	15797.6	0.746621
$766$	0	0
$767$	−32831.0	−1.54558
$768$	0	0
$769$	1146.04	0.0537418	0.0268709	−	0.999639i	$-0.491446\pi$
$769$	1146.04	0.0537418	0.0268709	+	0.999639i	$0.491446\pi$
$770$	0	0
$771$	3002.77	0.140262
$772$	0	0
$773$	−12184.6	−0.566947	−0.283474	−	0.958980i	$-0.591487\pi$
$773$	−12184.6	−0.566947	−0.283474	+	0.958980i	$0.591487\pi$
$774$	0	0
$775$	27452.7	1.27243
$776$	0	0
$777$	−2250.51	−0.103908
$778$	0	0
$779$	−8284.46	−0.381029
$780$	0	0
$781$	−24310.9	−1.11384
$782$	0	0
$783$	−963.755	−0.0439870
$784$	0	0
$785$	−7511.98	−0.341547
$786$	0	0
$787$	17720.1	0.802611	0.401305	−	0.915944i	$-0.368557\pi$
$787$	17720.1	0.802611	0.401305	+	0.915944i	$0.368557\pi$
$788$	0	0
$789$	2067.21	0.0932756
$790$	0	0
$791$	−10056.7	−0.452057
$792$	0	0
$793$	−12111.3	−0.542353
$794$	0	0
$795$	3072.05	0.137049
$796$	0	0
$797$	−7867.90	−0.349680	−0.174840	−	0.984597i	$-0.555941\pi$
$797$	−7867.90	−0.349680	−0.174840	+	0.984597i	$0.555941\pi$
$798$	0	0
$799$	−42233.3	−1.86997
$800$	0	0
$801$	6359.30	0.280518
$802$	0	0
$803$	−15715.1	−0.690628
$804$	0	0
$805$	−7291.91	−0.319262
$806$	0	0
$807$	−6800.98	−0.296661
$808$	0	0
$809$	30693.9	1.33392	0.666959	−	0.745094i	$-0.267595\pi$
$809$	30693.9	1.33392	0.666959	+	0.745094i	$0.267595\pi$
$810$	0	0
$811$	8432.57	0.365114	0.182557	−	0.983195i	$-0.441563\pi$
$811$	8432.57	0.365114	0.182557	+	0.983195i	$0.441563\pi$
$812$	0	0
$813$	−3379.67	−0.145794
$814$	0	0
$815$	−10599.9	−0.455580
$816$	0	0
$817$	−15800.2	−0.676594
$818$	0	0
$819$	7328.95	0.312692
$820$	0	0
$821$	41883.3	1.78043	0.890217	−	0.455537i	$-0.150553\pi$
$821$	41883.3	1.78043	0.890217	+	0.455537i	$0.150553\pi$
$822$	0	0
$823$	9670.14	0.409575	0.204787	−	0.978806i	$-0.434350\pi$
$823$	9670.14	0.409575	0.204787	+	0.978806i	$0.434350\pi$
$824$	0	0
$825$	7498.26	0.316432
$826$	0	0
$827$	−13903.8	−0.584621	−0.292311	−	0.956323i	$-0.594424\pi$
$827$	−13903.8	−0.584621	−0.292311	+	0.956323i	$0.594424\pi$
$828$	0	0
$829$	21460.5	0.899101	0.449551	−	0.893255i	$-0.351584\pi$
$829$	21460.5	0.899101	0.449551	+	0.893255i	$0.351584\pi$
$830$	0	0
$831$	−8068.54	−0.336817
$832$	0	0
$833$	5633.37	0.234315
$834$	0	0
$835$	4004.96	0.165985
$836$	0	0
$837$	−18062.5	−0.745916
$838$	0	0
$839$	−33587.6	−1.38209	−0.691045	−	0.722812i	$-0.742849\pi$
$839$	−33587.6	−1.38209	−0.691045	+	0.722812i	$0.742849\pi$
$840$	0	0
$841$	−24156.6	−0.990471
$842$	0	0
$843$	3127.57	0.127781
$844$	0	0
$845$	2786.47	0.113441
$846$	0	0
$847$	−20144.0	−0.817185
$848$	0	0
$849$	−1896.27	−0.0766546
$850$	0	0
$851$	51773.3	2.08551
$852$	0	0
$853$	−37270.7	−1.49604	−0.748022	−	0.663674i	$-0.768996\pi$
$853$	−37270.7	−1.49604	−0.748022	+	0.663674i	$0.768996\pi$
$854$	0	0
$855$	−10629.8	−0.425183
$856$	0	0
$857$	−22549.5	−0.898805	−0.449403	−	0.893329i	$-0.648363\pi$
$857$	−22549.5	−0.898805	−0.449403	+	0.893329i	$0.648363\pi$
$858$	0	0
$859$	−16891.2	−0.670922	−0.335461	−	0.942054i	$-0.608892\pi$
$859$	−16891.2	−0.670922	−0.335461	+	0.942054i	$0.608892\pi$
$860$	0	0
$861$	901.775	0.0356938
$862$	0	0
$863$	−3765.10	−0.148511	−0.0742557	−	0.997239i	$-0.523658\pi$
$863$	−3765.10	−0.148511	−0.0742557	+	0.997239i	$0.523658\pi$
$864$	0	0
$865$	20181.9	0.793300
$866$	0	0
$867$	−9989.58	−0.391308
$868$	0	0
$869$	2900.64	0.113231
$870$	0	0
$871$	−15538.0	−0.604462
$872$	0	0
$873$	40047.9	1.55260
$874$	0	0
$875$	−8322.07	−0.321529
$876$	0	0
$877$	−33576.5	−1.29281	−0.646406	−	0.762994i	$-0.723729\pi$
$877$	−33576.5	−1.29281	−0.646406	+	0.762994i	$0.723729\pi$
$878$	0	0
$879$	−7690.81	−0.295113
$880$	0	0
$881$	−13514.0	−0.516797	−0.258398	−	0.966038i	$-0.583195\pi$
$881$	−13514.0	−0.516797	−0.258398	+	0.966038i	$0.583195\pi$
$882$	0	0
$883$	−40066.2	−1.52699	−0.763497	−	0.645811i	$-0.776519\pi$
$883$	−40066.2	−1.52699	−0.763497	+	0.645811i	$0.776519\pi$
$884$	0	0
$885$	−5183.24	−0.196873
$886$	0	0
$887$	15292.3	0.578880	0.289440	−	0.957196i	$-0.406531\pi$
$887$	15292.3	0.578880	0.289440	+	0.957196i	$0.406531\pi$
$888$	0	0
$889$	−86.5746	−0.00326616
$890$	0	0
$891$	39825.4	1.49742
$892$	0	0
$893$	28417.6	1.06490
$894$	0	0
$895$	−10494.9	−0.391961
$896$	0	0
$897$	9547.91	0.355402
$898$	0	0
$899$	4355.81	0.161596
$900$	0	0
$901$	54598.5	2.01880
$902$	0	0
$903$	1719.87	0.0633816
$904$	0	0
$905$	−18122.8	−0.665661
$906$	0	0
$907$	−50444.7	−1.84674	−0.923368	−	0.383915i	$-0.874575\pi$
$907$	−50444.7	−1.84674	−0.923368	+	0.383915i	$0.874575\pi$
$908$	0	0
$909$	−7811.93	−0.285044
$910$	0	0
$911$	−34326.0	−1.24838	−0.624188	−	0.781274i	$-0.714570\pi$
$911$	−34326.0	−1.24838	−0.624188	+	0.781274i	$0.714570\pi$
$912$	0	0
$913$	−40999.8	−1.48619
$914$	0	0
$915$	−1912.09	−0.0690839
$916$	0	0
$917$	15091.5	0.543474
$918$	0	0
$919$	−37001.3	−1.32814	−0.664071	−	0.747670i	$-0.731173\pi$
$919$	−37001.3	−1.32814	−0.664071	+	0.747670i	$0.731173\pi$
$920$	0	0
$921$	7385.14	0.264222
$922$	0	0
$923$	−15354.3	−0.547554
$924$	0	0
$925$	25679.5	0.912797
$926$	0	0
$927$	−32249.3	−1.14262
$928$	0	0
$929$	19715.0	0.696263	0.348132	−	0.937446i	$-0.386816\pi$
$929$	19715.0	0.696263	0.348132	+	0.937446i	$0.386816\pi$
$930$	0	0
$931$	−3790.54	−0.133437
$932$	0	0
$933$	1479.96	0.0519310
$934$	0	0
$935$	−40107.5	−1.40284
$936$	0	0
$937$	8382.89	0.292270	0.146135	−	0.989265i	$-0.453317\pi$
$937$	8382.89	0.292270	0.146135	+	0.989265i	$0.453317\pi$
$938$	0	0
$939$	6070.32	0.210966
$940$	0	0
$941$	−6731.06	−0.233184	−0.116592	−	0.993180i	$-0.537197\pi$
$941$	−6731.06	−0.233184	−0.116592	+	0.993180i	$0.537197\pi$
$942$	0	0
$943$	−20745.5	−0.716401
$944$	0	0
$945$	2379.66	0.0819157
$946$	0	0
$947$	12109.8	0.415540	0.207770	−	0.978178i	$-0.433379\pi$
$947$	12109.8	0.415540	0.207770	+	0.978178i	$0.433379\pi$
$948$	0	0
$949$	−9925.34	−0.339505
$950$	0	0
$951$	1455.25	0.0496212
$952$	0	0
$953$	−22423.5	−0.762191	−0.381095	−	0.924536i	$-0.624453\pi$
$953$	−22423.5	−0.762191	−0.381095	+	0.924536i	$0.624453\pi$
$954$	0	0
$955$	−19176.3	−0.649770
$956$	0	0
$957$	1189.72	0.0401861
$958$	0	0
$959$	13715.4	0.461827
$960$	0	0
$961$	51844.9	1.74029
$962$	0	0
$963$	3250.23	0.108761
$964$	0	0
$965$	−2282.24	−0.0761325
$966$	0	0
$967$	52331.8	1.74031	0.870154	−	0.492780i	$-0.164019\pi$
$967$	52331.8	1.74031	0.870154	+	0.492780i	$0.164019\pi$
$968$	0	0
$969$	10698.4	0.354677
$970$	0	0
$971$	11027.1	0.364444	0.182222	−	0.983257i	$-0.441671\pi$
$971$	11027.1	0.364444	0.182222	+	0.983257i	$0.441671\pi$
$972$	0	0
$973$	−345.715	−0.0113907
$974$	0	0
$975$	4735.75	0.155554
$976$	0	0
$977$	−6912.76	−0.226365	−0.113183	−	0.993574i	$-0.536105\pi$
$977$	−6912.76	−0.226365	−0.113183	+	0.993574i	$0.536105\pi$
$978$	0	0
$979$	−16145.2	−0.527070
$980$	0	0
$981$	−9075.03	−0.295355
$982$	0	0
$983$	16989.0	0.551235	0.275617	−	0.961267i	$-0.411118\pi$
$983$	16989.0	0.551235	0.275617	+	0.961267i	$0.411118\pi$
$984$	0	0
$985$	9746.17	0.315268
$986$	0	0
$987$	−3093.29	−0.0997574
$988$	0	0
$989$	−39565.9	−1.27212
$990$	0	0
$991$	44939.3	1.44051	0.720254	−	0.693710i	$-0.244025\pi$
$991$	44939.3	1.44051	0.720254	+	0.693710i	$0.244025\pi$
$992$	0	0
$993$	−5943.51	−0.189941
$994$	0	0
$995$	25139.5	0.800981
$996$	0	0
$997$	−27719.4	−0.880525	−0.440263	−	0.897869i	$-0.645115\pi$
$997$	−27719.4	−0.880525	−0.440263	+	0.897869i	$0.645115\pi$
$998$	0	0
$999$	−16895.8	−0.535095

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.v.1.2		3
4.3	odd	2		448.4.a.w.1.2		3
8.3	odd	2		224.4.a.g.1.2	yes	3
8.5	even	2		224.4.a.f.1.2	✓	3
24.5	odd	2		2016.4.a.w.1.2		3
24.11	even	2		2016.4.a.x.1.2		3
56.13	odd	2		1568.4.a.w.1.2		3
56.27	even	2		1568.4.a.x.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.4.a.f.1.2	✓	3	8.5	even	2
224.4.a.g.1.2	yes	3	8.3	odd	2
448.4.a.v.1.2		3	1.1	even	1	trivial
448.4.a.w.1.2		3	4.3	odd	2
1568.4.a.w.1.2		3	56.13	odd	2
1568.4.a.x.1.2		3	56.27	even	2
2016.4.a.w.1.2		3	24.5	odd	2
2016.4.a.x.1.2		3	24.11	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q-1.20293 q^{3} -5.37748 q^{5} -7.00000 q^{7} -25.5530 q^{9} +O(q^{10})\)	\(q-1.20293 q^{3} -5.37748 q^{5} -7.00000 q^{7} -25.5530 q^{9} +64.8746 q^{11} +40.9735 q^{13} +6.46874 q^{15} +114.967 q^{17} -77.3580 q^{19} +8.42052 q^{21} -193.716 q^{23} -96.0827 q^{25} +63.2176 q^{27} -15.2450 q^{29} -285.720 q^{31} -78.0397 q^{33} +37.6424 q^{35} -267.265 q^{37} -49.2883 q^{39} +107.093 q^{41} +204.247 q^{43} +137.411 q^{45} -367.352 q^{47} +49.0000 q^{49} -138.297 q^{51} +474.907 q^{53} -348.862 q^{55} +93.0563 q^{57} -801.275 q^{59} -295.589 q^{61} +178.871 q^{63} -220.334 q^{65} -379.222 q^{67} +233.027 q^{69} -374.737 q^{71} -242.238 q^{73} +115.581 q^{75} -454.122 q^{77} +44.7115 q^{79} +613.883 q^{81} -631.985 q^{83} -618.232 q^{85} +18.3387 q^{87} -248.867 q^{89} -286.814 q^{91} +343.701 q^{93} +415.991 q^{95} -1567.25 q^{97} -1657.74 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{5} - 21 q^{7} + 15 q^{9}+O(q^{10}) \) 3 * q + 6 * q^5 - 21 * q^7 + 15 * q^9	\( 3 q + 6 q^{5} - 21 q^{7} + 15 q^{9} + 6 q^{13} - 168 q^{15} - 66 q^{17} + 168 q^{19} - 336 q^{23} + 69 q^{25} + 168 q^{27} - 90 q^{29} - 504 q^{31} + 120 q^{33} - 42 q^{35} - 18 q^{37} - 840 q^{39} - 450 q^{41} + 150 q^{45} - 504 q^{47} + 147 q^{49} - 336 q^{51} + 78 q^{53} - 1176 q^{55} + 48 q^{57} - 504 q^{59} - 498 q^{61} - 105 q^{63} + 1068 q^{65} - 1008 q^{67} + 1224 q^{69} - 504 q^{71} - 234 q^{73} - 1848 q^{75} - 168 q^{79} - 981 q^{81} - 3024 q^{83} - 1476 q^{85} + 336 q^{87} + 246 q^{89} - 42 q^{91} - 1200 q^{93} + 2184 q^{95} - 2514 q^{97} - 2688 q^{99}+O(q^{100}) \) 3 * q + 6 * q^5 - 21 * q^7 + 15 * q^9 + 6 * q^13 - 168 * q^15 - 66 * q^17 + 168 * q^19 - 336 * q^23 + 69 * q^25 + 168 * q^27 - 90 * q^29 - 504 * q^31 + 120 * q^33 - 42 * q^35 - 18 * q^37 - 840 * q^39 - 450 * q^41 + 150 * q^45 - 504 * q^47 + 147 * q^49 - 336 * q^51 + 78 * q^53 - 1176 * q^55 + 48 * q^57 - 504 * q^59 - 498 * q^61 - 105 * q^63 + 1068 * q^65 - 1008 * q^67 + 1224 * q^69 - 504 * q^71 - 234 * q^73 - 1848 * q^75 - 168 * q^79 - 981 * q^81 - 3024 * q^83 - 1476 * q^85 + 336 * q^87 + 246 * q^89 - 42 * q^91 - 1200 * q^93 + 2184 * q^95 - 2514 * q^97 - 2688 * q^99

Introduction

L-functions

Modular forms

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