LMFDB - Embedded newform 2303.4.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2303,4,Mod(1,2303)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2303, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2303.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2303 = 7^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2303.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:	$135.881398743$
Analytic rank:	$0$
Dimension:	$68$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	2303.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.98878 q^{2} -0.891652 q^{3} +16.8879 q^{4} -13.0126 q^{5} +4.44826 q^{6} -44.3397 q^{8} -26.2050 q^{9} +O(q^{10})$	\(q-4.98878 q^{2} -0.891652 q^{3} +16.8879 q^{4} -13.0126 q^{5} +4.44826 q^{6} -44.3397 q^{8} -26.2050 q^{9} +64.9169 q^{10} +28.2050 q^{11} -15.0581 q^{12} -11.9867 q^{13} +11.6027 q^{15} +86.0979 q^{16} +39.6166 q^{17} +130.731 q^{18} -46.9639 q^{19} -219.755 q^{20} -140.708 q^{22} -219.832 q^{23} +39.5356 q^{24} +44.3276 q^{25} +59.7988 q^{26} +47.4403 q^{27} +142.749 q^{29} -57.8833 q^{30} +97.5808 q^{31} -74.8053 q^{32} -25.1490 q^{33} -197.638 q^{34} -442.547 q^{36} -147.706 q^{37} +234.292 q^{38} +10.6879 q^{39} +576.975 q^{40} -257.955 q^{41} +443.431 q^{43} +476.323 q^{44} +340.994 q^{45} +1096.69 q^{46} -47.0000 q^{47} -76.7694 q^{48} -221.140 q^{50} -35.3242 q^{51} -202.430 q^{52} -196.229 q^{53} -236.669 q^{54} -367.020 q^{55} +41.8755 q^{57} -712.142 q^{58} -805.515 q^{59} +195.945 q^{60} -82.3391 q^{61} -486.809 q^{62} -315.596 q^{64} +155.978 q^{65} +125.463 q^{66} -437.934 q^{67} +669.040 q^{68} +196.014 q^{69} -691.951 q^{71} +1161.92 q^{72} -294.446 q^{73} +736.873 q^{74} -39.5248 q^{75} -793.121 q^{76} -53.3198 q^{78} +65.8073 q^{79} -1120.36 q^{80} +665.234 q^{81} +1286.88 q^{82} +382.658 q^{83} -515.514 q^{85} -2212.18 q^{86} -127.282 q^{87} -1250.60 q^{88} +388.053 q^{89} -1701.15 q^{90} -3712.51 q^{92} -87.0081 q^{93} +234.473 q^{94} +611.122 q^{95} +66.7004 q^{96} +1668.17 q^{97} -739.110 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 68 q - 2 q^{2} + 24 q^{3} + 254 q^{4} + 40 q^{5} + 48 q^{6} - 66 q^{8} + 576 q^{9}+O(q^{10}) $ 68 * q - 2 * q^2 + 24 * q^3 + 254 * q^4 + 40 * q^5 + 48 * q^6 - 66 * q^8 + 576 * q^9	\( 68 q - 2 q^{2} + 24 q^{3} + 254 q^{4} + 40 q^{5} + 48 q^{6} - 66 q^{8} + 576 q^{9} + 200 q^{10} - 20 q^{11} + 288 q^{12} + 520 q^{13} + 88 q^{15} + 1062 q^{16} + 784 q^{17} - 2 q^{18} + 532 q^{19} + 400 q^{20} - 4 q^{22} - 268 q^{23} + 576 q^{24} + 1864 q^{25} + 312 q^{26} + 864 q^{27} + 200 q^{29} + 792 q^{30} + 936 q^{31} + 30 q^{32} + 2112 q^{33} + 1088 q^{34} + 2130 q^{36} - 356 q^{37} + 1192 q^{38} - 488 q^{39} + 2400 q^{40} + 1476 q^{41} - 92 q^{43} + 192 q^{44} + 1848 q^{45} - 424 q^{46} - 3196 q^{47} + 2688 q^{48} - 1338 q^{50} - 148 q^{51} + 4980 q^{52} - 80 q^{53} + 4944 q^{54} + 2200 q^{55} + 2244 q^{57} - 356 q^{58} + 560 q^{59} - 736 q^{60} + 3944 q^{61} + 1488 q^{62} + 3778 q^{64} + 2004 q^{65} - 1000 q^{66} + 2768 q^{67} + 8192 q^{68} + 2208 q^{69} - 2448 q^{71} - 5234 q^{72} + 9532 q^{73} - 2000 q^{74} + 11136 q^{75} + 6384 q^{76} - 3460 q^{78} - 1520 q^{79} + 616 q^{80} + 6976 q^{81} + 4976 q^{82} + 3320 q^{83} + 3244 q^{85} - 2892 q^{86} + 2360 q^{87} - 2868 q^{88} + 8152 q^{89} + 5400 q^{90} - 4684 q^{92} + 2840 q^{93} + 94 q^{94} - 4256 q^{95} + 5376 q^{96} + 13968 q^{97} + 2380 q^{99}+O(q^{100}) \) 68 * q - 2 * q^2 + 24 * q^3 + 254 * q^4 + 40 * q^5 + 48 * q^6 - 66 * q^8 + 576 * q^9 + 200 * q^10 - 20 * q^11 + 288 * q^12 + 520 * q^13 + 88 * q^15 + 1062 * q^16 + 784 * q^17 - 2 * q^18 + 532 * q^19 + 400 * q^20 - 4 * q^22 - 268 * q^23 + 576 * q^24 + 1864 * q^25 + 312 * q^26 + 864 * q^27 + 200 * q^29 + 792 * q^30 + 936 * q^31 + 30 * q^32 + 2112 * q^33 + 1088 * q^34 + 2130 * q^36 - 356 * q^37 + 1192 * q^38 - 488 * q^39 + 2400 * q^40 + 1476 * q^41 - 92 * q^43 + 192 * q^44 + 1848 * q^45 - 424 * q^46 - 3196 * q^47 + 2688 * q^48 - 1338 * q^50 - 148 * q^51 + 4980 * q^52 - 80 * q^53 + 4944 * q^54 + 2200 * q^55 + 2244 * q^57 - 356 * q^58 + 560 * q^59 - 736 * q^60 + 3944 * q^61 + 1488 * q^62 + 3778 * q^64 + 2004 * q^65 - 1000 * q^66 + 2768 * q^67 + 8192 * q^68 + 2208 * q^69 - 2448 * q^71 - 5234 * q^72 + 9532 * q^73 - 2000 * q^74 + 11136 * q^75 + 6384 * q^76 - 3460 * q^78 - 1520 * q^79 + 616 * q^80 + 6976 * q^81 + 4976 * q^82 + 3320 * q^83 + 3244 * q^85 - 2892 * q^86 + 2360 * q^87 - 2868 * q^88 + 8152 * q^89 + 5400 * q^90 - 4684 * q^92 + 2840 * q^93 + 94 * q^94 - 4256 * q^95 + 5376 * q^96 + 13968 * q^97 + 2380 * 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 $

$2$	−4.98878	−1.76380	−0.881900	−	0.471437i	$-0.843735\pi$
$2$	−4.98878	−1.76380	−0.881900	+	0.471437i	$0.843735\pi$
$3$	−0.891652	−0.171599	−0.0857993	−	0.996312i	$-0.527344\pi$
$3$	−0.891652	−0.171599	−0.0857993	+	0.996312i	$0.527344\pi$
$4$	16.8879	2.11099
$5$	−13.0126	−1.16388	−0.581941	−	0.813231i	$-0.697706\pi$
$5$	−13.0126	−1.16388	−0.581941	+	0.813231i	$0.697706\pi$
$6$	4.44826	0.302665
$7$	0	0
$7$	0	0
$8$	−44.3397	−1.95956
$9$	−26.2050	−0.970554
$10$	64.9169	2.05285
$11$	28.2050	0.773102	0.386551	−	0.922268i	$-0.373666\pi$
$11$	28.2050	0.773102	0.386551	+	0.922268i	$0.373666\pi$
$12$	−15.0581	−0.362242
$13$	−11.9867	−0.255731	−0.127866	−	0.991792i	$-0.540813\pi$
$13$	−11.9867	−0.255731	−0.127866	+	0.991792i	$0.540813\pi$
$14$	0	0
$15$	11.6027	0.199720
$16$	86.0979	1.34528
$17$	39.6166	0.565202	0.282601	−	0.959238i	$-0.408803\pi$
$17$	39.6166	0.565202	0.282601	+	0.959238i	$0.408803\pi$
$18$	130.731	1.71186
$19$	−46.9639	−0.567066	−0.283533	−	0.958962i	$-0.591507\pi$
$19$	−46.9639	−0.567066	−0.283533	+	0.958962i	$0.591507\pi$
$20$	−219.755	−2.45694
$21$	0	0
$22$	−140.708	−1.36360
$23$	−219.832	−1.99297	−0.996483	−	0.0837963i	$-0.973295\pi$
$23$	−219.832	−1.99297	−0.996483	+	0.0837963i	$0.973295\pi$
$24$	39.5356	0.336257
$25$	44.3276	0.354621
$26$	59.7988	0.451058
$27$	47.4403	0.338144
$28$	0	0
$29$	142.749	0.914061	0.457031	−	0.889451i	$-0.348913\pi$
$29$	142.749	0.914061	0.457031	+	0.889451i	$0.348913\pi$
$30$	−57.8833	−0.352267
$31$	97.5808	0.565356	0.282678	−	0.959215i	$-0.408777\pi$
$31$	97.5808	0.565356	0.282678	+	0.959215i	$0.408777\pi$
$32$	−74.8053	−0.413245
$33$	−25.1490	−0.132663
$34$	−197.638	−0.996902
$35$	0	0
$36$	−442.547	−2.04883
$37$	−147.706	−0.656291	−0.328145	−	0.944627i	$-0.606424\pi$
$37$	−147.706	−0.656291	−0.328145	+	0.944627i	$0.606424\pi$
$38$	234.292	1.00019
$39$	10.6879	0.0438831
$40$	576.975	2.28069
$41$	−257.955	−0.982582	−0.491291	−	0.870996i	$-0.663475\pi$
$41$	−257.955	−0.982582	−0.491291	+	0.870996i	$0.663475\pi$
$42$	0	0
$43$	443.431	1.57262	0.786309	−	0.617833i	$-0.211989\pi$
$43$	443.431	1.57262	0.786309	+	0.617833i	$0.211989\pi$
$44$	476.323	1.63201
$45$	340.994	1.12961
$46$	1096.69	3.51519
$47$	−47.0000	−0.145865
$47$	−47.0000	−0.145865
$48$	−76.7694	−0.230848
$49$	0	0
$50$	−221.140	−0.625479
$51$	−35.3242	−0.0969878
$52$	−202.430	−0.539845
$53$	−196.229	−0.508569	−0.254284	−	0.967130i	$-0.581840\pi$
$53$	−196.229	−0.508569	−0.254284	+	0.967130i	$0.581840\pi$
$54$	−236.669	−0.596419
$55$	−367.020	−0.899799
$56$	0	0
$57$	41.8755	0.0973077
$58$	−712.142	−1.61222
$59$	−805.515	−1.77744	−0.888721	−	0.458449i	$-0.848405\pi$
$59$	−805.515	−1.77744	−0.888721	+	0.458449i	$0.848405\pi$
$60$	195.945	0.421607
$61$	−82.3391	−0.172827	−0.0864134	−	0.996259i	$-0.527541\pi$
$61$	−82.3391	−0.172827	−0.0864134	+	0.996259i	$0.527541\pi$
$62$	−486.809	−0.997174
$63$	0	0
$64$	−315.596	−0.616399
$65$	155.978	0.297641
$66$	125.463	0.233991
$67$	−437.934	−0.798540	−0.399270	−	0.916833i	$-0.630736\pi$
$67$	−437.934	−0.798540	−0.399270	+	0.916833i	$0.630736\pi$
$68$	669.040	1.19313
$69$	196.014	0.341990
$70$	0	0
$71$	−691.951	−1.15661	−0.578306	−	0.815820i	$-0.696286\pi$
$71$	−691.951	−1.15661	−0.578306	+	0.815820i	$0.696286\pi$
$72$	1161.92	1.90186
$73$	−294.446	−0.472087	−0.236043	−	0.971743i	$-0.575851\pi$
$73$	−294.446	−0.472087	−0.236043	+	0.971743i	$0.575851\pi$
$74$	736.873	1.15756
$75$	−39.5248	−0.0608524
$76$	−793.121	−1.19707
$77$	0	0
$78$	−53.3198	−0.0774010
$79$	65.8073	0.0937203	0.0468601	−	0.998901i	$-0.485078\pi$
$79$	65.8073	0.0937203	0.0468601	+	0.998901i	$0.485078\pi$
$80$	−1120.36	−1.56575
$81$	665.234	0.912529
$82$	1286.88	1.73308
$83$	382.658	0.506050	0.253025	−	0.967460i	$-0.418574\pi$
$83$	382.658	0.506050	0.253025	+	0.967460i	$0.418574\pi$
$84$	0	0
$85$	−515.514	−0.657828
$86$	−2212.18	−2.77378
$87$	−127.282	−0.156852
$88$	−1250.60	−1.51494
$89$	388.053	0.462175	0.231088	−	0.972933i	$-0.425772\pi$
$89$	388.053	0.462175	0.231088	+	0.972933i	$0.425772\pi$
$90$	−1701.15	−1.99240
$91$	0	0
$92$	−3712.51	−4.20713
$93$	−87.0081	−0.0970142
$94$	234.473	0.257277
$95$	611.122	0.659998
$96$	66.7004	0.0709123
$97$	1668.17	1.74615	0.873076	−	0.487584i	$-0.162122\pi$
$97$	1668.17	1.74615	0.873076	+	0.487584i	$0.162122\pi$
$98$	0	0
$99$	−739.110	−0.750337
$100$	748.600	0.748600
$101$	−751.368	−0.740237	−0.370119	−	0.928985i	$-0.620683\pi$
$101$	−751.368	−0.740237	−0.370119	+	0.928985i	$0.620683\pi$
$102$	176.225	0.171067
$103$	520.625	0.498045	0.249023	−	0.968498i	$-0.419891\pi$
$103$	520.625	0.498045	0.249023	+	0.968498i	$0.419891\pi$
$104$	531.486	0.501120
$105$	0	0
$106$	978.943	0.897013
$107$	−1017.20	−0.919032	−0.459516	−	0.888169i	$-0.651977\pi$
$107$	−1017.20	−0.919032	−0.459516	+	0.888169i	$0.651977\pi$
$108$	801.167	0.713818
$109$	99.2035	0.0871741	0.0435870	−	0.999050i	$-0.486121\pi$
$109$	99.2035	0.0871741	0.0435870	+	0.999050i	$0.486121\pi$
$110$	1830.98	1.58706
$111$	131.703	0.112619
$112$	0	0
$113$	−811.319	−0.675420	−0.337710	−	0.941250i	$-0.609652\pi$
$113$	−811.319	−0.675420	−0.337710	+	0.941250i	$0.609652\pi$
$114$	−208.907	−0.171631
$115$	2860.59	2.31958
$116$	2410.73	1.92957
$117$	314.110	0.248201
$118$	4018.53	3.13505
$119$	0	0
$120$	−514.461	−0.391364
$121$	−535.480	−0.402314
$122$	410.771	0.304832
$123$	230.006	0.168610
$124$	1647.93	1.19346
$125$	1049.76	0.751145
$126$	0	0
$127$	−1961.48	−1.37050	−0.685250	−	0.728308i	$-0.740307\pi$
$127$	−1961.48	−1.37050	−0.685250	+	0.728308i	$0.740307\pi$
$128$	2172.88	1.50045
$129$	−395.386	−0.269859
$130$	−778.138	−0.524979
$131$	−146.910	−0.0979813	−0.0489907	−	0.998799i	$-0.515600\pi$
$131$	−146.910	−0.0979813	−0.0489907	+	0.998799i	$0.515600\pi$
$132$	−424.714	−0.280050
$133$	0	0
$134$	2184.76	1.40846
$135$	−617.322	−0.393560
$136$	−1756.59	−1.10755
$137$	−800.723	−0.499346	−0.249673	−	0.968330i	$-0.580323\pi$
$137$	−800.723	−0.499346	−0.249673	+	0.968330i	$0.580323\pi$
$138$	−977.870	−0.603202
$139$	−1576.33	−0.961890	−0.480945	−	0.876751i	$-0.659706\pi$
$139$	−1576.33	−0.961890	−0.480945	+	0.876751i	$0.659706\pi$
$140$	0	0
$141$	41.9077	0.0250302
$142$	3451.99	2.04003
$143$	−338.084	−0.197706
$144$	−2256.19	−1.30567
$145$	−1857.53	−1.06386
$146$	1468.93	0.832666
$147$	0	0
$148$	−2494.45	−1.38542
$149$	−305.785	−0.168127	−0.0840635	−	0.996460i	$-0.526790\pi$
$149$	−305.785	−0.168127	−0.0840635	+	0.996460i	$0.526790\pi$
$150$	197.180	0.107331
$151$	−1301.82	−0.701591	−0.350795	−	0.936452i	$-0.614089\pi$
$151$	−1301.82	−0.701591	−0.350795	+	0.936452i	$0.614089\pi$
$152$	2082.37	1.11120
$153$	−1038.15	−0.548559
$154$	0	0
$155$	−1269.78	−0.658007
$156$	180.497	0.0926367
$157$	−3062.76	−1.55691	−0.778454	−	0.627701i	$-0.783996\pi$
$157$	−3062.76	−1.55691	−0.778454	+	0.627701i	$0.783996\pi$
$158$	−328.298	−0.165304
$159$	174.968	0.0872697
$160$	973.411	0.480968
$161$	0	0
$162$	−3318.70	−1.60952
$163$	3183.97	1.52999	0.764994	−	0.644037i	$-0.222742\pi$
$163$	3183.97	1.52999	0.764994	+	0.644037i	$0.222742\pi$
$164$	−4356.32	−2.07422
$165$	327.254	0.154404
$166$	−1909.00	−0.892571
$167$	−2282.38	−1.05758	−0.528790	−	0.848753i	$-0.677354\pi$
$167$	−2282.38	−1.05758	−0.528790	+	0.848753i	$0.677354\pi$
$168$	0	0
$169$	−2053.32	−0.934602
$170$	2571.78	1.16028
$171$	1230.69	0.550368
$172$	7488.62	3.31978
$173$	1831.63	0.804950	0.402475	−	0.915431i	$-0.368150\pi$
$173$	1831.63	0.804950	0.402475	+	0.915431i	$0.368150\pi$
$174$	634.983	0.276655
$175$	0	0
$176$	2428.39	1.04004
$177$	718.239	0.305007
$178$	−1935.91	−0.815184
$179$	−2307.24	−0.963414	−0.481707	−	0.876332i	$-0.659983\pi$
$179$	−2307.24	−0.963414	−0.481707	+	0.876332i	$0.659983\pi$
$180$	5758.68	2.38459
$181$	1536.75	0.631082	0.315541	−	0.948912i	$-0.397814\pi$
$181$	1536.75	0.631082	0.315541	+	0.948912i	$0.397814\pi$
$182$	0	0
$183$	73.4178	0.0296568
$184$	9747.31	3.90533
$185$	1922.04	0.763845
$186$	434.064	0.171114
$187$	1117.38	0.436958
$188$	−793.731	−0.307919
$189$	0	0
$190$	−3048.75	−1.16410
$191$	−4013.88	−1.52060	−0.760299	−	0.649573i	$-0.774948\pi$
$191$	−4013.88	−1.52060	−0.760299	+	0.649573i	$0.774948\pi$
$192$	281.402	0.105773
$193$	291.554	0.108738	0.0543692	−	0.998521i	$-0.482685\pi$
$193$	291.554	0.108738	0.0543692	+	0.998521i	$0.482685\pi$
$194$	−8322.12	−3.07986
$195$	−139.078	−0.0510747
$196$	0	0
$197$	−96.9647	−0.0350683	−0.0175341	−	0.999846i	$-0.505582\pi$
$197$	−96.9647	−0.0350683	−0.0175341	+	0.999846i	$0.505582\pi$
$198$	3687.25	1.32344
$199$	588.904	0.209781	0.104890	−	0.994484i	$-0.466551\pi$
$199$	588.904	0.209781	0.104890	+	0.994484i	$0.466551\pi$
$200$	−1965.47	−0.694900
$201$	390.485	0.137028
$202$	3748.41	1.30563
$203$	0	0
$204$	−596.551	−0.204740
$205$	3356.67	1.14361
$206$	−2597.28	−0.878452
$207$	5760.70	1.93428
$208$	−1032.03	−0.344030
$209$	−1324.61	−0.438400
$210$	0	0
$211$	1313.03	0.428401	0.214201	−	0.976790i	$-0.431285\pi$
$211$	1313.03	0.428401	0.214201	+	0.976790i	$0.431285\pi$
$212$	−3313.90	−1.07358
$213$	616.980	0.198473
$214$	5074.59	1.62099
$215$	−5770.19	−1.83034
$216$	−2103.49	−0.662613
$217$	0	0
$218$	−494.904	−0.153758
$219$	262.544	0.0810094
$220$	−6198.19	−1.89946
$221$	−474.871	−0.144540
$222$	−657.035	−0.198636
$223$	2336.33	0.701579	0.350790	−	0.936454i	$-0.385913\pi$
$223$	2336.33	0.701579	0.350790	+	0.936454i	$0.385913\pi$
$224$	0	0
$225$	−1161.60	−0.344178
$226$	4047.49	1.19131
$227$	−6323.09	−1.84880	−0.924401	−	0.381422i	$-0.875435\pi$
$227$	−6323.09	−1.84880	−0.924401	+	0.381422i	$0.875435\pi$
$228$	707.188	0.205415
$229$	−3704.81	−1.06909	−0.534544	−	0.845141i	$-0.679517\pi$
$229$	−3704.81	−1.06909	−0.534544	+	0.845141i	$0.679517\pi$
$230$	−14270.8	−4.09127
$231$	0	0
$232$	−6329.44	−1.79116
$233$	−3346.63	−0.940965	−0.470482	−	0.882409i	$-0.655920\pi$
$233$	−3346.63	−0.940965	−0.470482	+	0.882409i	$0.655920\pi$
$234$	−1567.03	−0.437776
$235$	611.592	0.169770
$236$	−13603.4	−3.75216
$237$	−58.6772	−0.0160823
$238$	0	0
$239$	−922.380	−0.249639	−0.124820	−	0.992179i	$-0.539835\pi$
$239$	−922.380	−0.249639	−0.124820	+	0.992179i	$0.539835\pi$
$240$	998.969	0.268680
$241$	−2712.07	−0.724894	−0.362447	−	0.932004i	$-0.618059\pi$
$241$	−2712.07	−0.724894	−0.362447	+	0.932004i	$0.618059\pi$
$242$	2671.39	0.709601
$243$	−1874.05	−0.494733
$244$	−1390.53	−0.364835
$245$	0	0
$246$	−1147.45	−0.297394
$247$	562.941	0.145016
$248$	−4326.71	−1.10785
$249$	−341.198	−0.0868375
$250$	−5237.01	−1.32487
$251$	3245.42	0.816133	0.408066	−	0.912952i	$-0.366203\pi$
$251$	3245.42	0.816133	0.408066	+	0.912952i	$0.366203\pi$
$252$	0	0
$253$	−6200.36	−1.54077
$254$	9785.41	2.41729
$255$	459.659	0.112882
$256$	−8315.25	−2.03009
$257$	3981.97	0.966492	0.483246	−	0.875485i	$-0.339458\pi$
$257$	3981.97	0.966492	0.483246	+	0.875485i	$0.339458\pi$
$258$	1972.49	0.475977
$259$	0	0
$260$	2634.14	0.628316
$261$	−3740.73	−0.887146
$262$	732.899	0.172819
$263$	−3519.25	−0.825119	−0.412560	−	0.910931i	$-0.635365\pi$
$263$	−3519.25	−0.825119	−0.412560	+	0.910931i	$0.635365\pi$
$264$	1115.10	0.259961
$265$	2553.45	0.591914
$266$	0	0
$267$	−346.009	−0.0793086
$268$	−7395.79	−1.68571
$269$	−1019.62	−0.231106	−0.115553	−	0.993301i	$-0.536864\pi$
$269$	−1019.62	−0.231106	−0.115553	+	0.993301i	$0.536864\pi$
$270$	3079.68	0.694161
$271$	2338.90	0.524272	0.262136	−	0.965031i	$-0.415573\pi$
$271$	2338.90	0.524272	0.262136	+	0.965031i	$0.415573\pi$
$272$	3410.90	0.760354
$273$	0	0
$274$	3994.63	0.880745
$275$	1250.26	0.274158
$276$	3310.27	0.721937
$277$	−5941.66	−1.28881	−0.644403	−	0.764686i	$-0.722894\pi$
$277$	−5941.66	−1.28881	−0.644403	+	0.764686i	$0.722894\pi$
$278$	7863.96	1.69658
$279$	−2557.10	−0.548708
$280$	0	0
$281$	734.138	0.155854	0.0779271	−	0.996959i	$-0.475170\pi$
$281$	734.138	0.155854	0.0779271	+	0.996959i	$0.475170\pi$
$282$	−209.068	−0.0441483
$283$	2311.26	0.485477	0.242738	−	0.970092i	$-0.421954\pi$
$283$	2311.26	0.485477	0.242738	+	0.970092i	$0.421954\pi$
$284$	−11685.6	−2.44159
$285$	−544.908	−0.113255
$286$	1686.62	0.348714
$287$	0	0
$288$	1960.27	0.401077
$289$	−3343.53	−0.680547
$290$	9266.81	1.87643
$291$	−1487.43	−0.299637
$292$	−4972.58	−0.996569
$293$	−7107.24	−1.41710	−0.708549	−	0.705662i	$-0.750650\pi$
$293$	−7107.24	−1.41710	−0.708549	+	0.705662i	$0.750650\pi$
$294$	0	0
$295$	10481.8	2.06873
$296$	6549.26	1.28604
$297$	1338.05	0.261420
$298$	1525.50	0.296542
$299$	2635.06	0.509663
$300$	−667.491	−0.128459
$301$	0	0
$302$	6494.47	1.23747
$303$	669.959	0.127024
$304$	−4043.49	−0.762862
$305$	1071.44	0.201150
$306$	5179.10	0.967547
$307$	3209.17	0.596602	0.298301	−	0.954472i	$-0.403580\pi$
$307$	3209.17	0.596602	0.298301	+	0.954472i	$0.403580\pi$
$308$	0	0
$309$	−464.216	−0.0854639
$310$	6334.64	1.16059
$311$	−4108.02	−0.749017	−0.374508	−	0.927224i	$-0.622189\pi$
$311$	−4108.02	−0.749017	−0.374508	+	0.927224i	$0.622189\pi$
$312$	−473.901	−0.0859915
$313$	2728.82	0.492786	0.246393	−	0.969170i	$-0.420755\pi$
$313$	2728.82	0.492786	0.246393	+	0.969170i	$0.420755\pi$
$314$	15279.4	2.74607
$315$	0	0
$316$	1111.35	0.197842
$317$	6677.94	1.18319	0.591594	−	0.806236i	$-0.298499\pi$
$317$	6677.94	1.18319	0.591594	+	0.806236i	$0.298499\pi$
$318$	−872.877	−0.153926
$319$	4026.22	0.706662
$320$	4106.72	0.717415
$321$	906.989	0.157705
$322$	0	0
$323$	−1860.55	−0.320507
$324$	11234.4	1.92634
$325$	−531.340	−0.0906875
$326$	−15884.1	−2.69859
$327$	−88.4550	−0.0149589
$328$	11437.7	1.92543
$329$	0	0
$330$	−1632.60	−0.272338
$331$	−1293.30	−0.214762	−0.107381	−	0.994218i	$-0.534246\pi$
$331$	−1293.30	−0.214762	−0.107381	+	0.994218i	$0.534246\pi$
$332$	6462.29	1.06827
$333$	3870.64	0.636965
$334$	11386.3	1.86536
$335$	5698.66	0.929406
$336$	0	0
$337$	3556.90	0.574946	0.287473	−	0.957789i	$-0.407185\pi$
$337$	3556.90	0.574946	0.287473	+	0.957789i	$0.407185\pi$
$338$	10243.6	1.64845
$339$	723.415	0.115901
$340$	−8705.95	−1.38867
$341$	2752.26	0.437077
$342$	−6139.62	−0.970739
$343$	0	0
$344$	−19661.6	−3.08164
$345$	−2550.65	−0.398036
$346$	−9137.60	−1.41977
$347$	−2872.61	−0.444409	−0.222205	−	0.975000i	$-0.571325\pi$
$347$	−2872.61	−0.444409	−0.222205	+	0.975000i	$0.571325\pi$
$348$	−2149.53	−0.331112
$349$	9120.82	1.39893	0.699465	−	0.714667i	$-0.253422\pi$
$349$	9120.82	1.39893	0.699465	+	0.714667i	$0.253422\pi$
$350$	0	0
$351$	−568.652	−0.0864740
$352$	−2109.88	−0.319480
$353$	−4378.86	−0.660236	−0.330118	−	0.943940i	$-0.607089\pi$
$353$	−4378.86	−0.660236	−0.330118	+	0.943940i	$0.607089\pi$
$354$	−3583.13	−0.537970
$355$	9004.07	1.34616
$356$	6553.41	0.975646
$357$	0	0
$358$	11510.3	1.69927
$359$	−5881.38	−0.864644	−0.432322	−	0.901719i	$-0.642306\pi$
$359$	−5881.38	−0.864644	−0.432322	+	0.901719i	$0.642306\pi$
$360$	−15119.6	−2.21354
$361$	−4653.39	−0.678436
$362$	−7666.52	−1.11310
$363$	477.462	0.0690365
$364$	0	0
$365$	3831.51	0.549453
$366$	−366.265	−0.0523087
$367$	5889.08	0.837623	0.418812	−	0.908073i	$-0.362447\pi$
$367$	5889.08	0.837623	0.418812	+	0.908073i	$0.362447\pi$
$368$	−18927.1	−2.68110
$369$	6759.71	0.953649
$370$	−9588.63	−1.34727
$371$	0	0
$372$	−1469.38	−0.204796
$373$	−2101.81	−0.291764	−0.145882	−	0.989302i	$-0.546602\pi$
$373$	−2101.81	−0.291764	−0.145882	+	0.989302i	$0.546602\pi$
$374$	−5574.38	−0.770706
$375$	−936.019	−0.128895
$376$	2083.97	0.285831
$377$	−1711.08	−0.233754
$378$	0	0
$379$	8431.85	1.14278	0.571392	−	0.820677i	$-0.306404\pi$
$379$	8431.85	1.14278	0.571392	+	0.820677i	$0.306404\pi$
$380$	10320.6	1.39325
$381$	1748.96	0.235176
$382$	20024.4	2.68203
$383$	−11731.4	−1.56514	−0.782568	−	0.622565i	$-0.786091\pi$
$383$	−11731.4	−1.56514	−0.782568	+	0.622565i	$0.786091\pi$
$384$	−1937.45	−0.257475
$385$	0	0
$386$	−1454.50	−0.191793
$387$	−11620.1	−1.52631
$388$	28171.8	3.68611
$389$	11270.8	1.46903	0.734516	−	0.678591i	$-0.237409\pi$
$389$	11270.8	1.46903	0.734516	+	0.678591i	$0.237409\pi$
$390$	693.829	0.0900856
$391$	−8709.00	−1.12643
$392$	0	0
$393$	130.992	0.0168135
$394$	483.735	0.0618534
$395$	−856.324	−0.109079
$396$	−12482.0	−1.58395
$397$	8117.08	1.02616	0.513079	−	0.858341i	$-0.328505\pi$
$397$	8117.08	1.02616	0.513079	+	0.858341i	$0.328505\pi$
$398$	−2937.91	−0.370011
$399$	0	0
$400$	3816.51	0.477064
$401$	2589.72	0.322505	0.161252	−	0.986913i	$-0.448447\pi$
$401$	2589.72	0.322505	0.161252	+	0.986913i	$0.448447\pi$
$402$	−1948.04	−0.241690
$403$	−1169.67	−0.144579
$404$	−12689.0	−1.56263
$405$	−8656.41	−1.06208
$406$	0	0
$407$	−4166.05	−0.507379
$408$	1566.27	0.190053
$409$	4127.30	0.498977	0.249489	−	0.968378i	$-0.419737\pi$
$409$	4127.30	0.498977	0.249489	+	0.968378i	$0.419737\pi$
$410$	−16745.7	−2.01710
$411$	713.966	0.0856870
$412$	8792.26	1.05137
$413$	0	0
$414$	−28738.8	−3.41168
$415$	−4979.37	−0.588983
$416$	896.667	0.105680
$417$	1405.54	0.165059
$418$	6608.21	0.773249
$419$	−6711.47	−0.782522	−0.391261	−	0.920280i	$-0.627961\pi$
$419$	−6711.47	−0.782522	−0.391261	+	0.920280i	$0.627961\pi$
$420$	0	0
$421$	1485.21	0.171935	0.0859673	−	0.996298i	$-0.472602\pi$
$421$	1485.21	0.171935	0.0859673	+	0.996298i	$0.472602\pi$
$422$	−6550.41	−0.755613
$423$	1231.63	0.141570
$424$	8700.75	0.996570
$425$	1756.11	0.200432
$426$	−3077.97	−0.350066
$427$	0	0
$428$	−17178.4	−1.94007
$429$	301.453	0.0339261
$430$	28786.2	3.22836
$431$	4984.95	0.557115	0.278558	−	0.960420i	$-0.410144\pi$
$431$	4984.95	0.557115	0.278558	+	0.960420i	$0.410144\pi$
$432$	4084.51	0.454899
$433$	16424.2	1.82286	0.911431	−	0.411454i	$-0.134979\pi$
$433$	16424.2	1.82286	0.911431	+	0.411454i	$0.134979\pi$
$434$	0	0
$435$	1656.27	0.182557
$436$	1675.34	0.184023
$437$	10324.2	1.13014
$438$	−1309.77	−0.142884
$439$	13769.3	1.49697	0.748486	−	0.663150i	$-0.230781\pi$
$439$	13769.3	1.49697	0.748486	+	0.663150i	$0.230781\pi$
$440$	16273.6	1.76321
$441$	0	0
$442$	2369.02	0.254939
$443$	14674.1	1.57379	0.786893	−	0.617089i	$-0.211688\pi$
$443$	14674.1	1.57379	0.786893	+	0.617089i	$0.211688\pi$
$444$	2224.18	0.237736
$445$	−5049.58	−0.537917
$446$	−11655.4	−1.23745
$447$	272.654	0.0288504
$448$	0	0
$449$	−5541.32	−0.582430	−0.291215	−	0.956658i	$-0.594060\pi$
$449$	−5541.32	−0.582430	−0.291215	+	0.956658i	$0.594060\pi$
$450$	5794.97	0.607062
$451$	−7275.62	−0.759636
$452$	−13701.5	−1.42580
$453$	1160.77	0.120392
$454$	31544.5	3.26092
$455$	0	0
$456$	−1856.75	−0.190680
$457$	−15963.4	−1.63399	−0.816996	−	0.576643i	$-0.804362\pi$
$457$	−15963.4	−1.63399	−0.816996	+	0.576643i	$0.804362\pi$
$458$	18482.5	1.88566
$459$	1879.42	0.191120
$460$	48309.3	4.89660
$461$	−7017.86	−0.709011	−0.354506	−	0.935054i	$-0.615351\pi$
$461$	−7017.86	−0.709011	−0.354506	+	0.935054i	$0.615351\pi$
$462$	0	0
$463$	3839.60	0.385402	0.192701	−	0.981257i	$-0.438275\pi$
$463$	3839.60	0.385402	0.192701	+	0.981257i	$0.438275\pi$
$464$	12290.4	1.22967
$465$	1132.20	0.112913
$466$	16695.6	1.65967
$467$	18626.2	1.84564	0.922822	−	0.385227i	$-0.125877\pi$
$467$	18626.2	1.84564	0.922822	+	0.385227i	$0.125877\pi$
$468$	5304.66	0.523949
$469$	0	0
$470$	−3051.10	−0.299439
$471$	2730.92	0.267163
$472$	35716.3	3.48300
$473$	12507.0	1.21579
$474$	292.728	0.0283659
$475$	−2081.79	−0.201093
$476$	0	0
$477$	5142.17	0.493593
$478$	4601.55	0.440314
$479$	16919.9	1.61396	0.806982	−	0.590576i	$-0.201099\pi$
$479$	16919.9	1.61396	0.806982	+	0.590576i	$0.201099\pi$
$480$	−867.945	−0.0825335
$481$	1770.51	0.167834
$482$	13529.9	1.27857
$483$	0	0
$484$	−9043.13	−0.849280
$485$	−21707.2	−2.03231
$486$	9349.20	0.872609
$487$	6090.66	0.566723	0.283362	−	0.959013i	$-0.408550\pi$
$487$	6090.66	0.566723	0.283362	+	0.959013i	$0.408550\pi$
$488$	3650.89	0.338664
$489$	−2839.00	−0.262544
$490$	0	0
$491$	−6399.42	−0.588191	−0.294095	−	0.955776i	$-0.595018\pi$
$491$	−6399.42	−0.588191	−0.294095	+	0.955776i	$0.595018\pi$
$492$	3884.33	0.355933
$493$	5655.21	0.516629
$494$	−2808.39	−0.255780
$495$	9617.74	0.873303
$496$	8401.50	0.760562
$497$	0	0
$498$	1702.16	0.153164
$499$	−13912.9	−1.24815	−0.624075	−	0.781364i	$-0.714524\pi$
$499$	−13912.9	−1.24815	−0.624075	+	0.781364i	$0.714524\pi$
$500$	17728.2	1.58566
$501$	2035.09	0.181479
$502$	−16190.7	−1.43949
$503$	1334.18	0.118267	0.0591334	−	0.998250i	$-0.481166\pi$
$503$	1334.18	0.118267	0.0591334	+	0.998250i	$0.481166\pi$
$504$	0	0
$505$	9777.25	0.861548
$506$	30932.2	2.71760
$507$	1830.85	0.160376
$508$	−33125.4	−2.89311
$509$	−18303.2	−1.59386	−0.796928	−	0.604074i	$-0.793543\pi$
$509$	−18303.2	−1.59386	−0.796928	+	0.604074i	$0.793543\pi$
$510$	−2293.14	−0.199102
$511$	0	0
$512$	24099.9	2.08022
$513$	−2227.98	−0.191750
$514$	−19865.2	−1.70470
$515$	−6774.68	−0.579666
$516$	−6677.24	−0.569669
$517$	−1325.63	−0.112768
$518$	0	0
$519$	−1633.18	−0.138128
$520$	−6916.01	−0.583244
$521$	−2201.52	−0.185125	−0.0925627	−	0.995707i	$-0.529506\pi$
$521$	−2201.52	−0.185125	−0.0925627	+	0.995707i	$0.529506\pi$
$522$	18661.6	1.56475
$523$	22705.5	1.89836	0.949178	−	0.314739i	$-0.101917\pi$
$523$	22705.5	1.89836	0.949178	+	0.314739i	$0.101917\pi$
$524$	−2480.99	−0.206837
$525$	0	0
$526$	17556.8	1.45534
$527$	3865.81	0.319540
$528$	−2165.28	−0.178469
$529$	36159.3	2.97191
$530$	−12738.6	−1.04402
$531$	21108.5	1.72510
$532$	0	0
$533$	3092.03	0.251277
$534$	1726.16	0.139884
$535$	13236.4	1.06964
$536$	19417.9	1.56479
$537$	2057.25	0.165320
$538$	5086.67	0.407624
$539$	0	0
$540$	−10425.3	−0.830800
$541$	13694.1	1.08827	0.544137	−	0.838997i	$-0.316857\pi$
$541$	13694.1	1.08827	0.544137	+	0.838997i	$0.316857\pi$
$542$	−11668.2	−0.924711
$543$	−1370.25	−0.108293
$544$	−2963.53	−0.233567
$545$	−1290.89	−0.101460
$546$	0	0
$547$	7078.62	0.553309	0.276654	−	0.960970i	$-0.410774\pi$
$547$	7078.62	0.553309	0.276654	+	0.960970i	$0.410774\pi$
$548$	−13522.5	−1.05411
$549$	2157.69	0.167738
$550$	−6237.26	−0.483559
$551$	−6704.04	−0.518333
$552$	−8691.21	−0.670150
$553$	0	0
$554$	29641.6	2.27320
$555$	−1713.79	−0.131075
$556$	−26620.9	−2.03054
$557$	2577.49	0.196071	0.0980355	−	0.995183i	$-0.468744\pi$
$557$	2577.49	0.196071	0.0980355	+	0.995183i	$0.468744\pi$
$558$	12756.8	0.967811
$559$	−5315.26	−0.402168
$560$	0	0
$561$	−996.318	−0.0749814
$562$	−3662.45	−0.274895
$563$	6121.91	0.458273	0.229137	−	0.973394i	$-0.426410\pi$
$563$	6121.91	0.458273	0.229137	+	0.973394i	$0.426410\pi$
$564$	707.732	0.0528385
$565$	10557.4	0.786109
$566$	−11530.3	−0.856284
$567$	0	0
$568$	30680.9	2.26645
$569$	−15681.3	−1.15535	−0.577676	−	0.816266i	$-0.696040\pi$
$569$	−15681.3	−1.15535	−0.577676	+	0.816266i	$0.696040\pi$
$570$	2718.43	0.199758
$571$	−3818.54	−0.279861	−0.139931	−	0.990161i	$-0.544688\pi$
$571$	−3818.54	−0.279861	−0.139931	+	0.990161i	$0.544688\pi$
$572$	−5709.52	−0.417355
$573$	3578.99	0.260933
$574$	0	0
$575$	−9744.63	−0.706747
$576$	8270.18	0.598248
$577$	14628.8	1.05547	0.527733	−	0.849410i	$-0.323042\pi$
$577$	14628.8	1.05547	0.527733	+	0.849410i	$0.323042\pi$
$578$	16680.1	1.20035
$579$	−259.965	−0.0186594
$580$	−31369.8	−2.24579
$581$	0	0
$582$	7420.44	0.528500
$583$	−5534.63	−0.393175
$584$	13055.7	0.925082
$585$	−4087.39	−0.288876
$586$	35456.4	2.49948
$587$	2336.85	0.164314	0.0821569	−	0.996619i	$-0.473819\pi$
$587$	2336.85	0.164314	0.0821569	+	0.996619i	$0.473819\pi$
$588$	0	0
$589$	−4582.77	−0.320594
$590$	−52291.5	−3.64883
$591$	86.4588	0.00601766
$592$	−12717.2	−0.882895
$593$	603.191	0.0417708	0.0208854	−	0.999782i	$-0.493351\pi$
$593$	603.191	0.0417708	0.0208854	+	0.999782i	$0.493351\pi$
$594$	−6675.25	−0.461092
$595$	0	0
$596$	−5164.07	−0.354914
$597$	−525.098	−0.0359980
$598$	−13145.7	−0.898944
$599$	26046.4	1.77668	0.888338	−	0.459190i	$-0.151860\pi$
$599$	26046.4	1.77668	0.888338	+	0.459190i	$0.151860\pi$
$600$	1752.52	0.119244
$601$	15602.0	1.05893	0.529467	−	0.848331i	$-0.322392\pi$
$601$	15602.0	1.05893	0.529467	+	0.848331i	$0.322392\pi$
$602$	0	0
$603$	11476.0	0.775026
$604$	−21984.9	−1.48105
$605$	6967.98	0.468246
$606$	−3342.28	−0.224044
$607$	348.895	0.0233298	0.0116649	−	0.999932i	$-0.496287\pi$
$607$	348.895	0.0233298	0.0116649	+	0.999932i	$0.496287\pi$
$608$	3513.15	0.234337
$609$	0	0
$610$	−5345.20	−0.354788
$611$	563.374	0.0373022
$612$	−17532.2	−1.15800
$613$	18175.2	1.19754	0.598769	−	0.800922i	$-0.295657\pi$
$613$	18175.2	1.19754	0.598769	+	0.800922i	$0.295657\pi$
$614$	−16009.8	−1.05229
$615$	−2992.98	−0.196242
$616$	0	0
$617$	−6597.25	−0.430462	−0.215231	−	0.976563i	$-0.569051\pi$
$617$	−6597.25	−0.430462	−0.215231	+	0.976563i	$0.569051\pi$
$618$	2315.87	0.150741
$619$	21808.0	1.41605	0.708026	−	0.706186i	$-0.249586\pi$
$619$	21808.0	1.41605	0.708026	+	0.706186i	$0.249586\pi$
$620$	−21443.9	−1.38904
$621$	−10428.9	−0.673910
$622$	20494.0	1.32112
$623$	0	0
$624$	920.210	0.0590350
$625$	−19201.0	−1.22886
$626$	−13613.5	−0.869175
$627$	1181.10	0.0752287
$628$	−51723.5	−3.28661
$629$	−5851.61	−0.370936
$630$	0	0
$631$	−12252.6	−0.773007	−0.386504	−	0.922288i	$-0.626317\pi$
$631$	−12252.6	−0.773007	−0.386504	+	0.922288i	$0.626317\pi$
$632$	−2917.88	−0.183650
$633$	−1170.77	−0.0735130
$634$	−33314.7	−2.08690
$635$	25524.0	1.59510
$636$	2954.84	0.184225
$637$	0	0
$638$	−20085.9	−1.24641
$639$	18132.5	1.12255
$640$	−28274.8	−1.74634
$641$	11279.6	0.695035	0.347517	−	0.937674i	$-0.387025\pi$
$641$	11279.6	0.695035	0.347517	+	0.937674i	$0.387025\pi$
$642$	−4524.77	−0.278159
$643$	29516.3	1.81028	0.905140	−	0.425113i	$-0.139766\pi$
$643$	29516.3	1.81028	0.905140	+	0.425113i	$0.139766\pi$
$644$	0	0
$645$	5145.00	0.314084
$646$	9281.85	0.565309
$647$	8983.50	0.545870	0.272935	−	0.962032i	$-0.412006\pi$
$647$	8983.50	0.545870	0.272935	+	0.962032i	$0.412006\pi$
$648$	−29496.3	−1.78815
$649$	−22719.5	−1.37414
$650$	2650.74	0.159955
$651$	0	0
$652$	53770.6	3.22979
$653$	1791.31	0.107350	0.0536750	−	0.998558i	$-0.482907\pi$
$653$	1791.31	0.107350	0.0536750	+	0.998558i	$0.482907\pi$
$654$	441.282	0.0263846
$655$	1911.67	0.114039
$656$	−22209.4	−1.32185
$657$	7715.95	0.458186
$658$	0	0
$659$	12207.0	0.721572	0.360786	−	0.932649i	$-0.382508\pi$
$659$	12207.0	0.721572	0.360786	+	0.932649i	$0.382508\pi$
$660$	5526.63	0.325945
$661$	−19308.8	−1.13619	−0.568097	−	0.822961i	$-0.692320\pi$
$661$	−19308.8	−1.13619	−0.568097	+	0.822961i	$0.692320\pi$
$662$	6451.98	0.378797
$663$	423.420	0.0248028
$664$	−16967.0	−0.991635
$665$	0	0
$666$	−19309.7	−1.12348
$667$	−31380.8	−1.82169
$668$	−38544.6	−2.23254
$669$	−2083.19	−0.120390
$670$	−28429.3	−1.63929
$671$	−2322.37	−0.133613
$672$	0	0
$673$	−20328.8	−1.16437	−0.582184	−	0.813057i	$-0.697802\pi$
$673$	−20328.8	−1.16437	−0.582184	+	0.813057i	$0.697802\pi$
$674$	−17744.6	−1.01409
$675$	2102.91	0.119913
$676$	−34676.3	−1.97293
$677$	25203.2	1.43078	0.715390	−	0.698725i	$-0.246249\pi$
$677$	25203.2	1.43078	0.715390	+	0.698725i	$0.246249\pi$
$678$	−3608.96	−0.204426
$679$	0	0
$680$	22857.8	1.28905
$681$	5638.00	0.317252
$682$	−13730.4	−0.770917
$683$	−21263.3	−1.19124	−0.595621	−	0.803266i	$-0.703094\pi$
$683$	−21263.3	−1.19124	−0.595621	+	0.803266i	$0.703094\pi$
$684$	20783.7	1.16182
$685$	10419.5	0.581179
$686$	0	0
$687$	3303.41	0.183454
$688$	38178.5	2.11561
$689$	2352.13	0.130057
$690$	12724.6	0.702056
$691$	28310.7	1.55860	0.779299	−	0.626652i	$-0.215575\pi$
$691$	28310.7	1.55860	0.779299	+	0.626652i	$0.215575\pi$
$692$	30932.4	1.69924
$693$	0	0
$694$	14330.8	0.783849
$695$	20512.2	1.11953
$696$	5643.66	0.307360
$697$	−10219.3	−0.555357
$698$	−45501.7	−2.46743
$699$	2984.03	0.161468
$700$	0	0
$701$	−20004.0	−1.07780	−0.538902	−	0.842369i	$-0.681161\pi$
$701$	−20004.0	−1.07780	−0.538902	+	0.842369i	$0.681161\pi$
$702$	2836.88	0.152523
$703$	6936.86	0.372160
$704$	−8901.38	−0.476539
$705$	−545.327	−0.0291322
$706$	21845.2	1.16452
$707$	0	0
$708$	12129.5	0.643865
$709$	−2192.56	−0.116140	−0.0580700	−	0.998313i	$-0.518495\pi$
$709$	−2192.56	−0.116140	−0.0580700	+	0.998313i	$0.518495\pi$
$710$	−44919.3	−2.37435
$711$	−1724.48	−0.0909606
$712$	−17206.2	−0.905659
$713$	−21451.4	−1.12673
$714$	0	0
$715$	4399.35	0.230107
$716$	−38964.4	−2.03375
$717$	822.442	0.0428378
$718$	29340.9	1.52506
$719$	13745.9	0.712985	0.356492	−	0.934298i	$-0.383973\pi$
$719$	13745.9	0.712985	0.356492	+	0.934298i	$0.383973\pi$
$720$	29358.9	1.51964
$721$	0	0
$722$	23214.7	1.19663
$723$	2418.22	0.124391
$724$	25952.5	1.33221
$725$	6327.71	0.324145
$726$	−2381.95	−0.121767
$727$	−17399.1	−0.887617	−0.443809	−	0.896122i	$-0.646373\pi$
$727$	−17399.1	−0.887617	−0.443809	+	0.896122i	$0.646373\pi$
$728$	0	0
$729$	−16290.3	−0.827633
$730$	−19114.6	−0.969125
$731$	17567.2	0.888846
$732$	1239.87	0.0626052
$733$	−30770.4	−1.55052	−0.775260	−	0.631643i	$-0.782381\pi$
$733$	−30770.4	−1.55052	−0.775260	+	0.631643i	$0.782381\pi$
$734$	−29379.3	−1.47740
$735$	0	0
$736$	16444.6	0.823583
$737$	−12351.9	−0.617353
$738$	−33722.7	−1.68204
$739$	−13120.0	−0.653083	−0.326542	−	0.945183i	$-0.605883\pi$
$739$	−13120.0	−0.653083	−0.326542	+	0.945183i	$0.605883\pi$
$740$	32459.2	1.61247
$741$	−501.947	−0.0248846
$742$	0	0
$743$	−1097.41	−0.0541857	−0.0270928	−	0.999633i	$-0.508625\pi$
$743$	−1097.41	−0.0541857	−0.0270928	+	0.999633i	$0.508625\pi$
$744$	3857.92	0.190105
$745$	3979.06	0.195680
$746$	10485.5	0.514612
$747$	−10027.5	−0.491149
$748$	18870.3	0.922413
$749$	0	0
$750$	4669.59	0.227346
$751$	−5144.32	−0.249959	−0.124979	−	0.992159i	$-0.539886\pi$
$751$	−5144.32	−0.249959	−0.124979	+	0.992159i	$0.539886\pi$
$752$	−4046.60	−0.196229
$753$	−2893.79	−0.140047
$754$	8536.21	0.412295
$755$	16940.0	0.816569
$756$	0	0
$757$	13188.5	0.633218	0.316609	−	0.948556i	$-0.397456\pi$
$757$	13188.5	0.633218	0.316609	+	0.948556i	$0.397456\pi$
$758$	−42064.6	−2.01564
$759$	5528.57	0.264393
$760$	−27097.0	−1.29330
$761$	−39177.9	−1.86623	−0.933113	−	0.359584i	$-0.882919\pi$
$761$	−39177.9	−1.86623	−0.933113	+	0.359584i	$0.882919\pi$
$762$	−8725.18	−0.414803
$763$	0	0
$764$	−67786.0	−3.20996
$765$	13509.0	0.638457
$766$	58525.4	2.76059
$767$	9655.44	0.454547
$768$	7414.31	0.348361
$769$	−21721.5	−1.01859	−0.509295	−	0.860592i	$-0.670094\pi$
$769$	−21721.5	−1.01859	−0.509295	+	0.860592i	$0.670094\pi$
$770$	0	0
$771$	−3550.53	−0.165849
$772$	4923.74	0.229545
$773$	−1146.57	−0.0533496	−0.0266748	−	0.999644i	$-0.508492\pi$
$773$	−1146.57	−0.0533496	−0.0266748	+	0.999644i	$0.508492\pi$
$774$	57970.0	2.69211
$775$	4325.52	0.200487
$776$	−73966.1	−3.42169
$777$	0	0
$778$	−56227.6	−2.59108
$779$	12114.6	0.557189
$780$	−2348.73	−0.107818
$781$	−19516.4	−0.894178
$782$	43447.3	1.98679
$783$	6772.05	0.309085
$784$	0	0
$785$	39854.4	1.81206
$786$	−653.491	−0.0296556
$787$	31809.4	1.44076	0.720382	−	0.693578i	$-0.243967\pi$
$787$	31809.4	1.44076	0.720382	+	0.693578i	$0.243967\pi$
$788$	−1637.53	−0.0740287
$789$	3137.95	0.141589
$790$	4272.01	0.192394
$791$	0	0
$792$	32771.9	1.47033
$793$	986.971	0.0441972
$794$	−40494.3	−1.80994
$795$	−2276.79	−0.101572
$796$	9945.36	0.442844
$797$	13085.5	0.581573	0.290786	−	0.956788i	$-0.406083\pi$
$797$	13085.5	0.581573	0.290786	+	0.956788i	$0.406083\pi$
$798$	0	0
$799$	−1861.98	−0.0824431
$800$	−3315.94	−0.146545
$801$	−10168.9	−0.448566
$802$	−12919.5	−0.568833
$803$	−8304.85	−0.364971
$804$	6594.47	0.289265
$805$	0	0
$806$	5835.22	0.255008
$807$	909.149	0.0396575
$808$	33315.5	1.45054
$809$	−30870.9	−1.34161	−0.670805	−	0.741634i	$-0.734051\pi$
$809$	−30870.9	−1.34161	−0.670805	+	0.741634i	$0.734051\pi$
$810$	43184.9	1.87329
$811$	22207.2	0.961528	0.480764	−	0.876850i	$-0.340359\pi$
$811$	22207.2	0.961528	0.480764	+	0.876850i	$0.340359\pi$
$812$	0	0
$813$	−2085.48	−0.0899644
$814$	20783.5	0.894915
$815$	−41431.8	−1.78073
$816$	−3041.34	−0.130476
$817$	−20825.2	−0.891778
$818$	−20590.2	−0.880096
$819$	0	0
$820$	56687.1	2.41414
$821$	12368.9	0.525796	0.262898	−	0.964824i	$-0.415322\pi$
$821$	12368.9	0.525796	0.262898	+	0.964824i	$0.415322\pi$
$822$	−3561.82	−0.151135
$823$	−12937.7	−0.547970	−0.273985	−	0.961734i	$-0.588342\pi$
$823$	−12937.7	−0.547970	−0.273985	+	0.961734i	$0.588342\pi$
$824$	−23084.4	−0.975949
$825$	−1114.80	−0.0470451
$826$	0	0
$827$	11840.2	0.497854	0.248927	−	0.968522i	$-0.419922\pi$
$827$	11840.2	0.497854	0.248927	+	0.968522i	$0.419922\pi$
$828$	97286.1	4.08324
$829$	15077.7	0.631690	0.315845	−	0.948811i	$-0.397712\pi$
$829$	15077.7	0.631690	0.315845	+	0.948811i	$0.397712\pi$
$830$	24841.0	1.03885
$831$	5297.89	0.221157
$832$	3782.95	0.157632
$833$	0	0
$834$	−7011.92	−0.291131
$835$	29699.7	1.23090
$836$	−22370.0	−0.925456
$837$	4629.26	0.191172
$838$	33482.0	1.38021
$839$	16663.1	0.685668	0.342834	−	0.939396i	$-0.388613\pi$
$839$	16663.1	0.685668	0.342834	+	0.939396i	$0.388613\pi$
$840$	0	0
$841$	−4011.79	−0.164492
$842$	−7409.36	−0.303258
$843$	−654.596	−0.0267444
$844$	22174.3	0.904349
$845$	26719.0	1.08777
$846$	−6144.34	−0.249701
$847$	0	0
$848$	−16894.9	−0.684167
$849$	−2060.84	−0.0833071
$850$	−8760.82	−0.353522
$851$	32470.6	1.30796
$852$	10419.5	0.418974
$853$	22002.2	0.883166	0.441583	−	0.897220i	$-0.354417\pi$
$853$	22002.2	0.883166	0.441583	+	0.897220i	$0.354417\pi$
$854$	0	0
$855$	−16014.4	−0.640563
$856$	45102.4	1.80090
$857$	47709.4	1.90166	0.950829	−	0.309716i	$-0.100234\pi$
$857$	47709.4	1.90166	0.950829	+	0.309716i	$0.100234\pi$
$858$	−1503.88	−0.0598388
$859$	18513.0	0.735337	0.367668	−	0.929957i	$-0.380156\pi$
$859$	18513.0	0.735337	0.367668	+	0.929957i	$0.380156\pi$
$860$	−97446.3	−3.86383
$861$	0	0
$862$	−24868.8	−0.982639
$863$	32289.8	1.27365	0.636823	−	0.771010i	$-0.280248\pi$
$863$	32289.8	1.27365	0.636823	+	0.771010i	$0.280248\pi$
$864$	−3548.79	−0.139736
$865$	−23834.3	−0.936866
$866$	−81936.9	−3.21516
$867$	2981.27	0.116781
$868$	0	0
$869$	1856.09	0.0724553
$870$	−8262.77	−0.321993
$871$	5249.37	0.204212
$872$	−4398.66	−0.170823
$873$	−43714.3	−1.69474
$874$	−51505.0	−1.99335
$875$	0	0
$876$	4433.81	0.171010
$877$	−30780.9	−1.18517	−0.592586	−	0.805507i	$-0.701893\pi$
$877$	−30780.9	−1.18517	−0.592586	+	0.805507i	$0.701893\pi$
$878$	−68691.8	−2.64036
$879$	6337.19	0.243172
$880$	−31599.6	−1.21048
$881$	−24135.9	−0.922995	−0.461497	−	0.887142i	$-0.652688\pi$
$881$	−24135.9	−0.922995	−0.461497	+	0.887142i	$0.652688\pi$
$882$	0	0
$883$	16684.8	0.635889	0.317944	−	0.948109i	$-0.397008\pi$
$883$	16684.8	0.635889	0.317944	+	0.948109i	$0.397008\pi$
$884$	−8019.57	−0.305121
$885$	−9346.15	−0.354991
$886$	−73205.8	−2.77584
$887$	−45205.0	−1.71120	−0.855600	−	0.517638i	$-0.826811\pi$
$887$	−45205.0	−1.71120	−0.855600	+	0.517638i	$0.826811\pi$
$888$	−5839.66	−0.220683
$889$	0	0
$890$	25191.2	0.948778
$891$	18762.9	0.705477
$892$	39455.7	1.48103
$893$	2207.30	0.0827151
$894$	−1360.21	−0.0508862
$895$	30023.2	1.12130
$896$	0	0
$897$	−2349.56	−0.0874575
$898$	27644.4	1.02729
$899$	13929.5	0.516770
$900$	−19617.0	−0.726556
$901$	−7773.92	−0.287444
$902$	36296.5	1.33984
$903$	0	0
$904$	35973.7	1.32353
$905$	−19997.1	−0.734505
$906$	−5790.81	−0.212347
$907$	−22804.9	−0.834867	−0.417433	−	0.908708i	$-0.637070\pi$
$907$	−22804.9	−0.834867	−0.417433	+	0.908708i	$0.637070\pi$
$908$	−106784.	−3.90280
$909$	19689.6	0.718440
$910$	0	0
$911$	−3547.71	−0.129024	−0.0645121	−	0.997917i	$-0.520549\pi$
$911$	−3547.71	−0.129024	−0.0645121	+	0.997917i	$0.520549\pi$
$912$	3605.39	0.130906
$913$	10792.9	0.391228
$914$	79637.6	2.88203
$915$	−955.356	−0.0345171
$916$	−62566.5	−2.25683
$917$	0	0
$918$	−9376.02	−0.337097
$919$	44157.5	1.58501	0.792504	−	0.609867i	$-0.208777\pi$
$919$	44157.5	1.58501	0.792504	+	0.609867i	$0.208777\pi$
$920$	−126838.	−4.54535
$921$	−2861.46	−0.102376
$922$	35010.5	1.25055
$923$	8294.19	0.295782
$924$	0	0
$925$	−6547.46	−0.232734
$926$	−19154.9	−0.679773
$927$	−13642.9	−0.483380
$928$	−10678.4	−0.377731
$929$	−3868.19	−0.136611	−0.0683053	−	0.997664i	$-0.521759\pi$
$929$	−3868.19	−0.136611	−0.0683053	+	0.997664i	$0.521759\pi$
$930$	−5648.30	−0.199156
$931$	0	0
$932$	−56517.5	−1.98636
$933$	3662.92	0.128530
$934$	−92921.7	−3.25534
$935$	−14540.1	−0.508568
$936$	−13927.6	−0.486364
$937$	−6819.48	−0.237762	−0.118881	−	0.992909i	$-0.537931\pi$
$937$	−6819.48	−0.237762	−0.118881	+	0.992909i	$0.537931\pi$
$938$	0	0
$939$	−2433.16	−0.0845613
$940$	10328.5	0.358381
$941$	40220.0	1.39334	0.696671	−	0.717391i	$-0.254664\pi$
$941$	40220.0	1.39334	0.696671	+	0.717391i	$0.254664\pi$
$942$	−13623.9	−0.471222
$943$	56706.9	1.95825
$944$	−69353.1	−2.39116
$945$	0	0
$946$	−62394.4	−2.14442
$947$	34688.6	1.19031	0.595157	−	0.803609i	$-0.297090\pi$
$947$	34688.6	1.19031	0.595157	+	0.803609i	$0.297090\pi$
$948$	−990.935	−0.0339495
$949$	3529.43	0.120727
$950$	10385.6	0.354688
$951$	−5954.40	−0.203033
$952$	0	0
$953$	−38916.3	−1.32279	−0.661397	−	0.750036i	$-0.730036\pi$
$953$	−38916.3	−1.32279	−0.661397	+	0.750036i	$0.730036\pi$
$954$	−25653.2	−0.870599
$955$	52231.0	1.76980
$956$	−15577.1	−0.526985
$957$	−3589.99	−0.121262
$958$	−84409.5	−2.84671
$959$	0	0
$960$	−3661.77	−0.123107
$961$	−20269.0	−0.680373
$962$	−8832.66	−0.296025
$963$	26655.7	0.891970
$964$	−45801.1	−1.53024
$965$	−3793.87	−0.126559
$966$	0	0
$967$	12979.3	0.431629	0.215815	−	0.976434i	$-0.430759\pi$
$967$	12979.3	0.431629	0.215815	+	0.976434i	$0.430759\pi$
$968$	23743.0	0.788358
$969$	1658.96	0.0549985
$970$	108292.	3.58460
$971$	−20267.4	−0.669838	−0.334919	−	0.942247i	$-0.608709\pi$
$971$	−20267.4	−0.669838	−0.334919	+	0.942247i	$0.608709\pi$
$972$	−31648.7	−1.04437
$973$	0	0
$974$	−30385.0	−0.999586
$975$	473.771	0.0155619
$976$	−7089.22	−0.232500
$977$	46934.4	1.53691	0.768457	−	0.639902i	$-0.221025\pi$
$977$	46934.4	1.53691	0.768457	+	0.639902i	$0.221025\pi$
$978$	14163.1	0.463075
$979$	10945.0	0.357308
$980$	0	0
$981$	−2599.62	−0.0846071
$982$	31925.3	1.03745
$983$	−38610.5	−1.25278	−0.626390	−	0.779510i	$-0.715468\pi$
$983$	−38610.5	−1.25278	−0.626390	+	0.779510i	$0.715468\pi$
$984$	−10198.4	−0.330400
$985$	1261.76	0.0408153
$986$	−28212.6	−0.911230
$987$	0	0
$988$	9506.88	0.306128
$989$	−97480.5	−3.13418
$990$	−47980.7	−1.54033
$991$	−45552.6	−1.46017	−0.730084	−	0.683357i	$-0.760519\pi$
$991$	−45552.6	−1.46017	−0.730084	+	0.683357i	$0.760519\pi$
$992$	−7299.56	−0.233630
$993$	1153.17	0.0368528
$994$	0	0
$995$	−7663.17	−0.244160
$996$	−5762.11	−0.183313
$997$	−31215.6	−0.991582	−0.495791	−	0.868442i	$-0.665122\pi$
$997$	−31215.6	−0.991582	−0.495791	+	0.868442i	$0.665122\pi$
$998$	69408.4	2.20149
$999$	−7007.23	−0.221921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2303.4.a.n.1.5	yes	68
7.6	odd	2		2303.4.a.m.1.5	✓	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2303.4.a.m.1.5	✓	68	7.6	odd	2
2303.4.a.n.1.5	yes	68	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:	\( N \)	\(=\)	\( 2303 = 7^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2303.a (trivial)

\(f(q)\)	\(=\)	\(q-4.98878 q^{2} -0.891652 q^{3} +16.8879 q^{4} -13.0126 q^{5} +4.44826 q^{6} -44.3397 q^{8} -26.2050 q^{9} +O(q^{10})\)	\(q-4.98878 q^{2} -0.891652 q^{3} +16.8879 q^{4} -13.0126 q^{5} +4.44826 q^{6} -44.3397 q^{8} -26.2050 q^{9} +64.9169 q^{10} +28.2050 q^{11} -15.0581 q^{12} -11.9867 q^{13} +11.6027 q^{15} +86.0979 q^{16} +39.6166 q^{17} +130.731 q^{18} -46.9639 q^{19} -219.755 q^{20} -140.708 q^{22} -219.832 q^{23} +39.5356 q^{24} +44.3276 q^{25} +59.7988 q^{26} +47.4403 q^{27} +142.749 q^{29} -57.8833 q^{30} +97.5808 q^{31} -74.8053 q^{32} -25.1490 q^{33} -197.638 q^{34} -442.547 q^{36} -147.706 q^{37} +234.292 q^{38} +10.6879 q^{39} +576.975 q^{40} -257.955 q^{41} +443.431 q^{43} +476.323 q^{44} +340.994 q^{45} +1096.69 q^{46} -47.0000 q^{47} -76.7694 q^{48} -221.140 q^{50} -35.3242 q^{51} -202.430 q^{52} -196.229 q^{53} -236.669 q^{54} -367.020 q^{55} +41.8755 q^{57} -712.142 q^{58} -805.515 q^{59} +195.945 q^{60} -82.3391 q^{61} -486.809 q^{62} -315.596 q^{64} +155.978 q^{65} +125.463 q^{66} -437.934 q^{67} +669.040 q^{68} +196.014 q^{69} -691.951 q^{71} +1161.92 q^{72} -294.446 q^{73} +736.873 q^{74} -39.5248 q^{75} -793.121 q^{76} -53.3198 q^{78} +65.8073 q^{79} -1120.36 q^{80} +665.234 q^{81} +1286.88 q^{82} +382.658 q^{83} -515.514 q^{85} -2212.18 q^{86} -127.282 q^{87} -1250.60 q^{88} +388.053 q^{89} -1701.15 q^{90} -3712.51 q^{92} -87.0081 q^{93} +234.473 q^{94} +611.122 q^{95} +66.7004 q^{96} +1668.17 q^{97} -739.110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 2 q^{2} + 24 q^{3} + 254 q^{4} + 40 q^{5} + 48 q^{6} - 66 q^{8} + 576 q^{9}+O(q^{10}) \) 68 * q - 2 * q^2 + 24 * q^3 + 254 * q^4 + 40 * q^5 + 48 * q^6 - 66 * q^8 + 576 * q^9	\( 68 q - 2 q^{2} + 24 q^{3} + 254 q^{4} + 40 q^{5} + 48 q^{6} - 66 q^{8} + 576 q^{9} + 200 q^{10} - 20 q^{11} + 288 q^{12} + 520 q^{13} + 88 q^{15} + 1062 q^{16} + 784 q^{17} - 2 q^{18} + 532 q^{19} + 400 q^{20} - 4 q^{22} - 268 q^{23} + 576 q^{24} + 1864 q^{25} + 312 q^{26} + 864 q^{27} + 200 q^{29} + 792 q^{30} + 936 q^{31} + 30 q^{32} + 2112 q^{33} + 1088 q^{34} + 2130 q^{36} - 356 q^{37} + 1192 q^{38} - 488 q^{39} + 2400 q^{40} + 1476 q^{41} - 92 q^{43} + 192 q^{44} + 1848 q^{45} - 424 q^{46} - 3196 q^{47} + 2688 q^{48} - 1338 q^{50} - 148 q^{51} + 4980 q^{52} - 80 q^{53} + 4944 q^{54} + 2200 q^{55} + 2244 q^{57} - 356 q^{58} + 560 q^{59} - 736 q^{60} + 3944 q^{61} + 1488 q^{62} + 3778 q^{64} + 2004 q^{65} - 1000 q^{66} + 2768 q^{67} + 8192 q^{68} + 2208 q^{69} - 2448 q^{71} - 5234 q^{72} + 9532 q^{73} - 2000 q^{74} + 11136 q^{75} + 6384 q^{76} - 3460 q^{78} - 1520 q^{79} + 616 q^{80} + 6976 q^{81} + 4976 q^{82} + 3320 q^{83} + 3244 q^{85} - 2892 q^{86} + 2360 q^{87} - 2868 q^{88} + 8152 q^{89} + 5400 q^{90} - 4684 q^{92} + 2840 q^{93} + 94 q^{94} - 4256 q^{95} + 5376 q^{96} + 13968 q^{97} + 2380 q^{99}+O(q^{100}) \) 68 * q - 2 * q^2 + 24 * q^3 + 254 * q^4 + 40 * q^5 + 48 * q^6 - 66 * q^8 + 576 * q^9 + 200 * q^10 - 20 * q^11 + 288 * q^12 + 520 * q^13 + 88 * q^15 + 1062 * q^16 + 784 * q^17 - 2 * q^18 + 532 * q^19 + 400 * q^20 - 4 * q^22 - 268 * q^23 + 576 * q^24 + 1864 * q^25 + 312 * q^26 + 864 * q^27 + 200 * q^29 + 792 * q^30 + 936 * q^31 + 30 * q^32 + 2112 * q^33 + 1088 * q^34 + 2130 * q^36 - 356 * q^37 + 1192 * q^38 - 488 * q^39 + 2400 * q^40 + 1476 * q^41 - 92 * q^43 + 192 * q^44 + 1848 * q^45 - 424 * q^46 - 3196 * q^47 + 2688 * q^48 - 1338 * q^50 - 148 * q^51 + 4980 * q^52 - 80 * q^53 + 4944 * q^54 + 2200 * q^55 + 2244 * q^57 - 356 * q^58 + 560 * q^59 - 736 * q^60 + 3944 * q^61 + 1488 * q^62 + 3778 * q^64 + 2004 * q^65 - 1000 * q^66 + 2768 * q^67 + 8192 * q^68 + 2208 * q^69 - 2448 * q^71 - 5234 * q^72 + 9532 * q^73 - 2000 * q^74 + 11136 * q^75 + 6384 * q^76 - 3460 * q^78 - 1520 * q^79 + 616 * q^80 + 6976 * q^81 + 4976 * q^82 + 3320 * q^83 + 3244 * q^85 - 2892 * q^86 + 2360 * q^87 - 2868 * q^88 + 8152 * q^89 + 5400 * q^90 - 4684 * q^92 + 2840 * q^93 + 94 * q^94 - 4256 * q^95 + 5376 * q^96 + 13968 * q^97 + 2380 * q^99

Introduction

L-functions

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