LMFDB - Embedded newform 1148.4.a.d.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, 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("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.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:	$67.7341926866$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 238 x^{13} + 602 x^{12} + 21013 x^{11} - 44923 x^{10} - 876344 x^{9} + \cdots - 45134496 $ x^15 - 3x^14 - 238x^13 + 602x^12 + 21013x^11 - 44923x^10 - 876344x^9 + 1512470x^8 + 18440602x^7 - 23548938x^6 - 192795335x^5 + 157072885x^4 + 930883767x^3 - 339053445x^2 - 1549251522x - 45134496
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.79210$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.79210 q^{3} +13.9605 q^{5} -7.00000 q^{7} -12.6200 q^{9} +O(q^{10})$	$q+3.79210 q^{3} +13.9605 q^{5} -7.00000 q^{7} -12.6200 q^{9} +11.0728 q^{11} +66.5295 q^{13} +52.9397 q^{15} -68.5177 q^{17} +15.3165 q^{19} -26.5447 q^{21} +62.8361 q^{23} +69.8960 q^{25} -150.243 q^{27} +125.214 q^{29} +216.905 q^{31} +41.9890 q^{33} -97.7236 q^{35} +234.751 q^{37} +252.287 q^{39} +41.0000 q^{41} +288.642 q^{43} -176.181 q^{45} -543.451 q^{47} +49.0000 q^{49} -259.826 q^{51} -396.165 q^{53} +154.581 q^{55} +58.0819 q^{57} +675.069 q^{59} +505.845 q^{61} +88.3398 q^{63} +928.786 q^{65} +1025.36 q^{67} +238.281 q^{69} +87.8969 q^{71} +622.842 q^{73} +265.053 q^{75} -77.5093 q^{77} -826.184 q^{79} -228.997 q^{81} +494.349 q^{83} -956.543 q^{85} +474.825 q^{87} -658.031 q^{89} -465.707 q^{91} +822.527 q^{93} +213.827 q^{95} +379.367 q^{97} -139.738 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * 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
$3$	3.79210	0.729790	0.364895	−	0.931049i	$-0.381105\pi$
$3$	3.79210	0.729790	0.364895	+	0.931049i	$0.381105\pi$
$4$	0	0
$5$	13.9605	1.24867	0.624333	−	0.781158i	$-0.285371\pi$
$5$	13.9605	1.24867	0.624333	+	0.781158i	$0.285371\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−12.6200	−0.467407
$10$	0	0
$11$	11.0728	0.303506	0.151753	−	0.988418i	$-0.451508\pi$
$11$	11.0728	0.303506	0.151753	+	0.988418i	$0.451508\pi$
$12$	0	0
$13$	66.5295	1.41938	0.709691	−	0.704513i	$-0.248835\pi$
$13$	66.5295	1.41938	0.709691	+	0.704513i	$0.248835\pi$
$14$	0	0
$15$	52.9397	0.911264
$16$	0	0
$17$	−68.5177	−0.977529	−0.488765	−	0.872416i	$-0.662552\pi$
$17$	−68.5177	−0.977529	−0.488765	+	0.872416i	$0.662552\pi$
$18$	0	0
$19$	15.3165	0.184940	0.0924699	−	0.995715i	$-0.470524\pi$
$19$	15.3165	0.184940	0.0924699	+	0.995715i	$0.470524\pi$
$20$	0	0
$21$	−26.5447	−0.275835
$22$	0	0
$23$	62.8361	0.569662	0.284831	−	0.958578i	$-0.408063\pi$
$23$	62.8361	0.569662	0.284831	+	0.958578i	$0.408063\pi$
$24$	0	0
$25$	69.8960	0.559168
$26$	0	0
$27$	−150.243	−1.07090
$28$	0	0
$29$	125.214	0.801783	0.400892	−	0.916126i	$-0.368700\pi$
$29$	125.214	0.801783	0.400892	+	0.916126i	$0.368700\pi$
$30$	0	0
$31$	216.905	1.25669	0.628344	−	0.777935i	$-0.283733\pi$
$31$	216.905	1.25669	0.628344	+	0.777935i	$0.283733\pi$
$32$	0	0
$33$	41.9890	0.221495
$34$	0	0
$35$	−97.7236	−0.471952
$36$	0	0
$37$	234.751	1.04305	0.521524	−	0.853237i	$-0.325364\pi$
$37$	234.751	1.04305	0.521524	+	0.853237i	$0.325364\pi$
$38$	0	0
$39$	252.287	1.03585
$40$	0	0
$41$	41.0000	0.156174
$41$	41.0000	0.156174
$42$	0	0
$43$	288.642	1.02366	0.511831	−	0.859086i	$-0.328967\pi$
$43$	288.642	1.02366	0.511831	+	0.859086i	$0.328967\pi$
$44$	0	0
$45$	−176.181	−0.583635
$46$	0	0
$47$	−543.451	−1.68661	−0.843303	−	0.537439i	$-0.819392\pi$
$47$	−543.451	−1.68661	−0.843303	+	0.537439i	$0.819392\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−259.826	−0.713391
$52$	0	0
$53$	−396.165	−1.02674	−0.513372	−	0.858166i	$-0.671604\pi$
$53$	−396.165	−1.02674	−0.513372	+	0.858166i	$0.671604\pi$
$54$	0	0
$55$	154.581	0.378977
$56$	0	0
$57$	58.0819	0.134967
$58$	0	0
$59$	675.069	1.48960	0.744801	−	0.667286i	$-0.232544\pi$
$59$	675.069	1.48960	0.744801	+	0.667286i	$0.232544\pi$
$60$	0	0
$61$	505.845	1.06175	0.530876	−	0.847449i	$-0.321863\pi$
$61$	505.845	1.06175	0.530876	+	0.847449i	$0.321863\pi$
$62$	0	0
$63$	88.3398	0.176663
$64$	0	0
$65$	928.786	1.77234
$66$	0	0
$67$	1025.36	1.86966	0.934829	−	0.355097i	$-0.115552\pi$
$67$	1025.36	1.86966	0.934829	+	0.355097i	$0.115552\pi$
$68$	0	0
$69$	238.281	0.415734
$70$	0	0
$71$	87.8969	0.146922	0.0734608	−	0.997298i	$-0.476596\pi$
$71$	87.8969	0.146922	0.0734608	+	0.997298i	$0.476596\pi$
$72$	0	0
$73$	622.842	0.998605	0.499302	−	0.866428i	$-0.333590\pi$
$73$	622.842	0.998605	0.499302	+	0.866428i	$0.333590\pi$
$74$	0	0
$75$	265.053	0.408075
$76$	0	0
$77$	−77.5093	−0.114714
$78$	0	0
$79$	−826.184	−1.17662	−0.588310	−	0.808636i	$-0.700206\pi$
$79$	−826.184	−1.17662	−0.588310	+	0.808636i	$0.700206\pi$
$80$	0	0
$81$	−228.997	−0.314124
$82$	0	0
$83$	494.349	0.653757	0.326879	−	0.945066i	$-0.394003\pi$
$83$	494.349	0.653757	0.326879	+	0.945066i	$0.394003\pi$
$84$	0	0
$85$	−956.543	−1.22061
$86$	0	0
$87$	474.825	0.585133
$88$	0	0
$89$	−658.031	−0.783721	−0.391860	−	0.920025i	$-0.628168\pi$
$89$	−658.031	−0.783721	−0.391860	+	0.920025i	$0.628168\pi$
$90$	0	0
$91$	−465.707	−0.536476
$92$	0	0
$93$	822.527	0.917119
$94$	0	0
$95$	213.827	0.230928
$96$	0	0
$97$	379.367	0.397101	0.198551	−	0.980091i	$-0.436377\pi$
$97$	379.367	0.397101	0.198551	+	0.980091i	$0.436377\pi$
$98$	0	0
$99$	−139.738	−0.141861
$100$	0	0
$101$	185.586	0.182836	0.0914182	−	0.995813i	$-0.470860\pi$
$101$	185.586	0.182836	0.0914182	+	0.995813i	$0.470860\pi$
$102$	0	0
$103$	−4.29642	−0.00411009	−0.00205504	−	0.999998i	$-0.500654\pi$
$103$	−4.29642	−0.00411009	−0.00205504	+	0.999998i	$0.500654\pi$
$104$	0	0
$105$	−370.578	−0.344426
$106$	0	0
$107$	696.718	0.629479	0.314740	−	0.949178i	$-0.398083\pi$
$107$	696.718	0.629479	0.314740	+	0.949178i	$0.398083\pi$
$108$	0	0
$109$	−1598.36	−1.40454	−0.702271	−	0.711910i	$-0.747831\pi$
$109$	−1598.36	−1.40454	−0.702271	+	0.711910i	$0.747831\pi$
$110$	0	0
$111$	890.198	0.761206
$112$	0	0
$113$	1396.59	1.16266	0.581328	−	0.813669i	$-0.302533\pi$
$113$	1396.59	1.16266	0.581328	+	0.813669i	$0.302533\pi$
$114$	0	0
$115$	877.224	0.711318
$116$	0	0
$117$	−839.601	−0.663429
$118$	0	0
$119$	479.624	0.369471
$120$	0	0
$121$	−1208.39	−0.907884
$122$	0	0
$123$	155.476	0.113974
$124$	0	0
$125$	−769.280	−0.550452
$126$	0	0
$127$	−2438.79	−1.70400	−0.851999	−	0.523543i	$-0.824610\pi$
$127$	−2438.79	−1.70400	−0.851999	+	0.523543i	$0.824610\pi$
$128$	0	0
$129$	1094.56	0.747058
$130$	0	0
$131$	−774.648	−0.516651	−0.258325	−	0.966058i	$-0.583171\pi$
$131$	−774.648	−0.516651	−0.258325	+	0.966058i	$0.583171\pi$
$132$	0	0
$133$	−107.216	−0.0699007
$134$	0	0
$135$	−2097.47	−1.33720
$136$	0	0
$137$	−1825.17	−1.13821	−0.569107	−	0.822264i	$-0.692711\pi$
$137$	−1825.17	−1.13821	−0.569107	+	0.822264i	$0.692711\pi$
$138$	0	0
$139$	1879.67	1.14699	0.573495	−	0.819209i	$-0.305587\pi$
$139$	1879.67	1.14699	0.573495	+	0.819209i	$0.305587\pi$
$140$	0	0
$141$	−2060.82	−1.23087
$142$	0	0
$143$	736.665	0.430791
$144$	0	0
$145$	1748.06	1.00116
$146$	0	0
$147$	185.813	0.104256
$148$	0	0
$149$	−2155.42	−1.18510	−0.592548	−	0.805535i	$-0.701878\pi$
$149$	−2155.42	−1.18510	−0.592548	+	0.805535i	$0.701878\pi$
$150$	0	0
$151$	2697.69	1.45387	0.726936	−	0.686705i	$-0.240944\pi$
$151$	2697.69	1.45387	0.726936	+	0.686705i	$0.240944\pi$
$152$	0	0
$153$	864.692	0.456904
$154$	0	0
$155$	3028.11	1.56918
$156$	0	0
$157$	258.548	0.131429	0.0657147	−	0.997838i	$-0.479067\pi$
$157$	258.548	0.131429	0.0657147	+	0.997838i	$0.479067\pi$
$158$	0	0
$159$	−1502.30	−0.749308
$160$	0	0
$161$	−439.853	−0.215312
$162$	0	0
$163$	2485.74	1.19447	0.597233	−	0.802068i	$-0.296267\pi$
$163$	2485.74	1.19447	0.597233	+	0.802068i	$0.296267\pi$
$164$	0	0
$165$	586.188	0.276574
$166$	0	0
$167$	−454.402	−0.210555	−0.105278	−	0.994443i	$-0.533573\pi$
$167$	−454.402	−0.210555	−0.105278	+	0.994443i	$0.533573\pi$
$168$	0	0
$169$	2229.18	1.01465
$170$	0	0
$171$	−193.294	−0.0864421
$172$	0	0
$173$	2861.06	1.25736	0.628678	−	0.777666i	$-0.283596\pi$
$173$	2861.06	1.25736	0.628678	+	0.777666i	$0.283596\pi$
$174$	0	0
$175$	−489.272	−0.211346
$176$	0	0
$177$	2559.93	1.08710
$178$	0	0
$179$	250.958	0.104790	0.0523952	−	0.998626i	$-0.483314\pi$
$179$	250.958	0.104790	0.0523952	+	0.998626i	$0.483314\pi$
$180$	0	0
$181$	−2640.40	−1.08430	−0.542152	−	0.840280i	$-0.682390\pi$
$181$	−2640.40	−1.08430	−0.542152	+	0.840280i	$0.682390\pi$
$182$	0	0
$183$	1918.22	0.774856
$184$	0	0
$185$	3277.24	1.30242
$186$	0	0
$187$	−758.680	−0.296686
$188$	0	0
$189$	1051.70	0.404762
$190$	0	0
$191$	458.166	0.173569	0.0867847	−	0.996227i	$-0.472341\pi$
$191$	458.166	0.173569	0.0867847	+	0.996227i	$0.472341\pi$
$192$	0	0
$193$	−743.907	−0.277449	−0.138724	−	0.990331i	$-0.544300\pi$
$193$	−743.907	−0.277449	−0.138724	+	0.990331i	$0.544300\pi$
$194$	0	0
$195$	3522.05	1.29343
$196$	0	0
$197$	1757.84	0.635741	0.317871	−	0.948134i	$-0.397032\pi$
$197$	1757.84	0.635741	0.317871	+	0.948134i	$0.397032\pi$
$198$	0	0
$199$	334.691	0.119224	0.0596121	−	0.998222i	$-0.481014\pi$
$199$	334.691	0.119224	0.0596121	+	0.998222i	$0.481014\pi$
$200$	0	0
$201$	3888.25	1.36446
$202$	0	0
$203$	−876.500	−0.303046
$204$	0	0
$205$	572.381	0.195009
$206$	0	0
$207$	−792.990	−0.266264
$208$	0	0
$209$	169.596	0.0561303
$210$	0	0
$211$	−3784.89	−1.23489	−0.617446	−	0.786613i	$-0.711833\pi$
$211$	−3784.89	−1.23489	−0.617446	+	0.786613i	$0.711833\pi$
$212$	0	0
$213$	333.314	0.107222
$214$	0	0
$215$	4029.59	1.27821
$216$	0	0
$217$	−1518.34	−0.474984
$218$	0	0
$219$	2361.88	0.728772
$220$	0	0
$221$	−4558.45	−1.38749
$222$	0	0
$223$	5011.41	1.50488	0.752441	−	0.658660i	$-0.228876\pi$
$223$	5011.41	1.50488	0.752441	+	0.658660i	$0.228876\pi$
$224$	0	0
$225$	−882.086	−0.261359
$226$	0	0
$227$	−1698.90	−0.496739	−0.248369	−	0.968665i	$-0.579895\pi$
$227$	−1698.90	−0.496739	−0.248369	+	0.968665i	$0.579895\pi$
$228$	0	0
$229$	3323.23	0.958976	0.479488	−	0.877548i	$-0.340822\pi$
$229$	3323.23	0.958976	0.479488	+	0.877548i	$0.340822\pi$
$230$	0	0
$231$	−293.923	−0.0837174
$232$	0	0
$233$	3474.71	0.976979	0.488489	−	0.872570i	$-0.337548\pi$
$233$	3474.71	0.976979	0.488489	+	0.872570i	$0.337548\pi$
$234$	0	0
$235$	−7586.86	−2.10601
$236$	0	0
$237$	−3132.97	−0.858685
$238$	0	0
$239$	4458.67	1.20673	0.603363	−	0.797467i	$-0.293827\pi$
$239$	4458.67	1.20673	0.603363	+	0.797467i	$0.293827\pi$
$240$	0	0
$241$	4325.72	1.15620	0.578099	−	0.815966i	$-0.303795\pi$
$241$	4325.72	1.15620	0.578099	+	0.815966i	$0.303795\pi$
$242$	0	0
$243$	3188.18	0.841654
$244$	0	0
$245$	684.065	0.178381
$246$	0	0
$247$	1019.00	0.262500
$248$	0	0
$249$	1874.62	0.477106
$250$	0	0
$251$	−5292.07	−1.33081	−0.665404	−	0.746484i	$-0.731740\pi$
$251$	−5292.07	−1.33081	−0.665404	+	0.746484i	$0.731740\pi$
$252$	0	0
$253$	695.769	0.172896
$254$	0	0
$255$	−3627.31	−0.890787
$256$	0	0
$257$	1461.49	0.354729	0.177364	−	0.984145i	$-0.443243\pi$
$257$	1461.49	0.354729	0.177364	+	0.984145i	$0.443243\pi$
$258$	0	0
$259$	−1643.25	−0.394235
$260$	0	0
$261$	−1580.20	−0.374759
$262$	0	0
$263$	−3997.28	−0.937197	−0.468598	−	0.883411i	$-0.655241\pi$
$263$	−3997.28	−0.937197	−0.468598	+	0.883411i	$0.655241\pi$
$264$	0	0
$265$	−5530.67	−1.28206
$266$	0	0
$267$	−2495.32	−0.571952
$268$	0	0
$269$	7093.07	1.60770	0.803852	−	0.594830i	$-0.202781\pi$
$269$	7093.07	1.60770	0.803852	+	0.594830i	$0.202781\pi$
$270$	0	0
$271$	7550.49	1.69247	0.846236	−	0.532809i	$-0.178864\pi$
$271$	7550.49	1.69247	0.846236	+	0.532809i	$0.178864\pi$
$272$	0	0
$273$	−1766.01	−0.391515
$274$	0	0
$275$	773.942	0.169711
$276$	0	0
$277$	−3752.62	−0.813982	−0.406991	−	0.913432i	$-0.633422\pi$
$277$	−3752.62	−0.813982	−0.406991	+	0.913432i	$0.633422\pi$
$278$	0	0
$279$	−2737.34	−0.587385
$280$	0	0
$281$	−659.862	−0.140086	−0.0700428	−	0.997544i	$-0.522314\pi$
$281$	−659.862	−0.140086	−0.0700428	+	0.997544i	$0.522314\pi$
$282$	0	0
$283$	−6223.83	−1.30731	−0.653655	−	0.756793i	$-0.726765\pi$
$283$	−6223.83	−1.30731	−0.653655	+	0.756793i	$0.726765\pi$
$284$	0	0
$285$	810.853	0.168529
$286$	0	0
$287$	−287.000	−0.0590281
$288$	0	0
$289$	−218.319	−0.0444370
$290$	0	0
$291$	1438.60	0.289801
$292$	0	0
$293$	−6211.53	−1.23850	−0.619252	−	0.785192i	$-0.712564\pi$
$293$	−6211.53	−1.23850	−0.619252	+	0.785192i	$0.712564\pi$
$294$	0	0
$295$	9424.32	1.86002
$296$	0	0
$297$	−1663.60	−0.325024
$298$	0	0
$299$	4180.45	0.808568
$300$	0	0
$301$	−2020.49	−0.386908
$302$	0	0
$303$	703.760	0.133432
$304$	0	0
$305$	7061.86	1.32577
$306$	0	0
$307$	−5941.30	−1.10452	−0.552260	−	0.833672i	$-0.686235\pi$
$307$	−5941.30	−1.10452	−0.552260	+	0.833672i	$0.686235\pi$
$308$	0	0
$309$	−16.2925	−0.00299950
$310$	0	0
$311$	4974.67	0.907034	0.453517	−	0.891248i	$-0.350169\pi$
$311$	4974.67	0.907034	0.453517	+	0.891248i	$0.350169\pi$
$312$	0	0
$313$	−2510.88	−0.453430	−0.226715	−	0.973961i	$-0.572799\pi$
$313$	−2510.88	−0.453430	−0.226715	+	0.973961i	$0.572799\pi$
$314$	0	0
$315$	1233.27	0.220593
$316$	0	0
$317$	−445.903	−0.0790044	−0.0395022	−	0.999219i	$-0.512577\pi$
$317$	−445.903	−0.0790044	−0.0395022	+	0.999219i	$0.512577\pi$
$318$	0	0
$319$	1386.47	0.243346
$320$	0	0
$321$	2642.02	0.459388
$322$	0	0
$323$	−1049.46	−0.180784
$324$	0	0
$325$	4650.15	0.793673
$326$	0	0
$327$	−6061.14	−1.02502
$328$	0	0
$329$	3804.16	0.637477
$330$	0	0
$331$	−3788.75	−0.629149	−0.314575	−	0.949233i	$-0.601862\pi$
$331$	−3788.75	−0.629149	−0.314575	+	0.949233i	$0.601862\pi$
$332$	0	0
$333$	−2962.55	−0.487527
$334$	0	0
$335$	14314.5	2.33458
$336$	0	0
$337$	−5438.68	−0.879122	−0.439561	−	0.898213i	$-0.644866\pi$
$337$	−5438.68	−0.879122	−0.439561	+	0.898213i	$0.644866\pi$
$338$	0	0
$339$	5296.01	0.848495
$340$	0	0
$341$	2401.74	0.381412
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	3326.52	0.519113
$346$	0	0
$347$	2651.06	0.410133	0.205067	−	0.978748i	$-0.434259\pi$
$347$	2651.06	0.410133	0.205067	+	0.978748i	$0.434259\pi$
$348$	0	0
$349$	−6889.11	−1.05663	−0.528317	−	0.849047i	$-0.677177\pi$
$349$	−6889.11	−1.05663	−0.528317	+	0.849047i	$0.677177\pi$
$350$	0	0
$351$	−9995.59	−1.52001
$352$	0	0
$353$	243.216	0.0366716	0.0183358	−	0.999832i	$-0.494163\pi$
$353$	243.216	0.0366716	0.0183358	+	0.999832i	$0.494163\pi$
$354$	0	0
$355$	1227.09	0.183456
$356$	0	0
$357$	1818.78	0.269636
$358$	0	0
$359$	−2689.99	−0.395466	−0.197733	−	0.980256i	$-0.563358\pi$
$359$	−2689.99	−0.395466	−0.197733	+	0.980256i	$0.563358\pi$
$360$	0	0
$361$	−6624.40	−0.965797
$362$	0	0
$363$	−4582.35	−0.662565
$364$	0	0
$365$	8695.20	1.24692
$366$	0	0
$367$	628.015	0.0893245	0.0446623	−	0.999002i	$-0.485779\pi$
$367$	628.015	0.0893245	0.0446623	+	0.999002i	$0.485779\pi$
$368$	0	0
$369$	−517.419	−0.0729966
$370$	0	0
$371$	2773.16	0.388073
$372$	0	0
$373$	−5895.51	−0.818386	−0.409193	−	0.912448i	$-0.634190\pi$
$373$	−5895.51	−0.818386	−0.409193	+	0.912448i	$0.634190\pi$
$374$	0	0
$375$	−2917.19	−0.401714
$376$	0	0
$377$	8330.45	1.13804
$378$	0	0
$379$	4808.33	0.651681	0.325841	−	0.945425i	$-0.394353\pi$
$379$	4808.33	0.651681	0.325841	+	0.945425i	$0.394353\pi$
$380$	0	0
$381$	−9248.14	−1.24356
$382$	0	0
$383$	−2587.97	−0.345272	−0.172636	−	0.984986i	$-0.555228\pi$
$383$	−2587.97	−0.345272	−0.172636	+	0.984986i	$0.555228\pi$
$384$	0	0
$385$	−1082.07	−0.143240
$386$	0	0
$387$	−3642.65	−0.478466
$388$	0	0
$389$	−9071.61	−1.18239	−0.591194	−	0.806529i	$-0.701343\pi$
$389$	−9071.61	−1.18239	−0.591194	+	0.806529i	$0.701343\pi$
$390$	0	0
$391$	−4305.39	−0.556861
$392$	0	0
$393$	−2937.54	−0.377047
$394$	0	0
$395$	−11534.0	−1.46921
$396$	0	0
$397$	−11072.4	−1.39977	−0.699884	−	0.714257i	$-0.746765\pi$
$397$	−11072.4	−1.39977	−0.699884	+	0.714257i	$0.746765\pi$
$398$	0	0
$399$	−406.573	−0.0510128
$400$	0	0
$401$	−6167.27	−0.768026	−0.384013	−	0.923328i	$-0.625458\pi$
$401$	−6167.27	−0.768026	−0.384013	+	0.923328i	$0.625458\pi$
$402$	0	0
$403$	14430.6	1.78372
$404$	0	0
$405$	−3196.91	−0.392237
$406$	0	0
$407$	2599.34	0.316571
$408$	0	0
$409$	9683.70	1.17073	0.585365	−	0.810770i	$-0.300951\pi$
$409$	9683.70	1.17073	0.585365	+	0.810770i	$0.300951\pi$
$410$	0	0
$411$	−6921.24	−0.830657
$412$	0	0
$413$	−4725.49	−0.563017
$414$	0	0
$415$	6901.37	0.816325
$416$	0	0
$417$	7127.91	0.837062
$418$	0	0
$419$	−8030.33	−0.936294	−0.468147	−	0.883651i	$-0.655078\pi$
$419$	−8030.33	−0.936294	−0.468147	+	0.883651i	$0.655078\pi$
$420$	0	0
$421$	3990.61	0.461972	0.230986	−	0.972957i	$-0.425805\pi$
$421$	3990.61	0.461972	0.230986	+	0.972957i	$0.425805\pi$
$422$	0	0
$423$	6858.34	0.788331
$424$	0	0
$425$	−4789.12	−0.546603
$426$	0	0
$427$	−3540.92	−0.401305
$428$	0	0
$429$	2793.51	0.314387
$430$	0	0
$431$	11219.7	1.25391	0.626953	−	0.779057i	$-0.284302\pi$
$431$	11219.7	1.25391	0.626953	+	0.779057i	$0.284302\pi$
$432$	0	0
$433$	−3331.86	−0.369789	−0.184895	−	0.982758i	$-0.559194\pi$
$433$	−3331.86	−0.369789	−0.184895	+	0.982758i	$0.559194\pi$
$434$	0	0
$435$	6628.80	0.730636
$436$	0	0
$437$	962.432	0.105353
$438$	0	0
$439$	−5854.69	−0.636513	−0.318256	−	0.948005i	$-0.603097\pi$
$439$	−5854.69	−0.636513	−0.318256	+	0.948005i	$0.603097\pi$
$440$	0	0
$441$	−618.379	−0.0667724
$442$	0	0
$443$	2197.40	0.235669	0.117835	−	0.993033i	$-0.462405\pi$
$443$	2197.40	0.235669	0.117835	+	0.993033i	$0.462405\pi$
$444$	0	0
$445$	−9186.45	−0.978606
$446$	0	0
$447$	−8173.58	−0.864871
$448$	0	0
$449$	−8007.96	−0.841690	−0.420845	−	0.907132i	$-0.638266\pi$
$449$	−8007.96	−0.841690	−0.420845	+	0.907132i	$0.638266\pi$
$450$	0	0
$451$	453.983	0.0473996
$452$	0	0
$453$	10229.9	1.06102
$454$	0	0
$455$	−6501.51	−0.669880
$456$	0	0
$457$	−6771.88	−0.693162	−0.346581	−	0.938020i	$-0.612657\pi$
$457$	−6771.88	−0.693162	−0.346581	+	0.938020i	$0.612657\pi$
$458$	0	0
$459$	10294.3	1.04683
$460$	0	0
$461$	11217.5	1.13330	0.566651	−	0.823958i	$-0.308239\pi$
$461$	11217.5	1.13330	0.566651	+	0.823958i	$0.308239\pi$
$462$	0	0
$463$	3970.54	0.398546	0.199273	−	0.979944i	$-0.436142\pi$
$463$	3970.54	0.398546	0.199273	+	0.979944i	$0.436142\pi$
$464$	0	0
$465$	11482.9	1.14518
$466$	0	0
$467$	−11975.2	−1.18661	−0.593303	−	0.804979i	$-0.702176\pi$
$467$	−11975.2	−1.18661	−0.593303	+	0.804979i	$0.702176\pi$
$468$	0	0
$469$	−7177.49	−0.706665
$470$	0	0
$471$	980.442	0.0959158
$472$	0	0
$473$	3196.06	0.310687
$474$	0	0
$475$	1070.57	0.103412
$476$	0	0
$477$	4999.60	0.479907
$478$	0	0
$479$	−13659.5	−1.30296	−0.651479	−	0.758667i	$-0.725851\pi$
$479$	−13659.5	−1.30296	−0.651479	+	0.758667i	$0.725851\pi$
$480$	0	0
$481$	15617.8	1.48048
$482$	0	0
$483$	−1667.96	−0.157133
$484$	0	0
$485$	5296.15	0.495847
$486$	0	0
$487$	−7300.97	−0.679340	−0.339670	−	0.940545i	$-0.610315\pi$
$487$	−7300.97	−0.679340	−0.339670	+	0.940545i	$0.610315\pi$
$488$	0	0
$489$	9426.16	0.871709
$490$	0	0
$491$	−16314.0	−1.49948	−0.749738	−	0.661735i	$-0.769820\pi$
$491$	−16314.0	−1.49948	−0.749738	+	0.661735i	$0.769820\pi$
$492$	0	0
$493$	−8579.40	−0.783766
$494$	0	0
$495$	−1950.81	−0.177137
$496$	0	0
$497$	−615.278	−0.0555312
$498$	0	0
$499$	−12123.6	−1.08763	−0.543813	−	0.839207i	$-0.683020\pi$
$499$	−12123.6	−1.08763	−0.543813	+	0.839207i	$0.683020\pi$
$500$	0	0
$501$	−1723.14	−0.153661
$502$	0	0
$503$	3316.45	0.293982	0.146991	−	0.989138i	$-0.453041\pi$
$503$	3316.45	0.293982	0.146991	+	0.989138i	$0.453041\pi$
$504$	0	0
$505$	2590.87	0.228302
$506$	0	0
$507$	8453.26	0.740478
$508$	0	0
$509$	−3570.71	−0.310941	−0.155470	−	0.987841i	$-0.549689\pi$
$509$	−3570.71	−0.310941	−0.155470	+	0.987841i	$0.549689\pi$
$510$	0	0
$511$	−4359.90	−0.377437
$512$	0	0
$513$	−2301.20	−0.198052
$514$	0	0
$515$	−59.9803	−0.00513213
$516$	0	0
$517$	−6017.50	−0.511894
$518$	0	0
$519$	10849.4	0.917606
$520$	0	0
$521$	−11938.2	−1.00388	−0.501942	−	0.864901i	$-0.667381\pi$
$521$	−11938.2	−1.00388	−0.501942	+	0.864901i	$0.667381\pi$
$522$	0	0
$523$	2148.31	0.179616	0.0898079	−	0.995959i	$-0.471375\pi$
$523$	2148.31	0.179616	0.0898079	+	0.995959i	$0.471375\pi$
$524$	0	0
$525$	−1855.37	−0.154238
$526$	0	0
$527$	−14861.9	−1.22845
$528$	0	0
$529$	−8218.63	−0.675485
$530$	0	0
$531$	−8519.36	−0.696250
$532$	0	0
$533$	2727.71	0.221670
$534$	0	0
$535$	9726.54	0.786009
$536$	0	0
$537$	951.657	0.0764749
$538$	0	0
$539$	542.565	0.0433580
$540$	0	0
$541$	−22994.2	−1.82735	−0.913675	−	0.406445i	$-0.866768\pi$
$541$	−22994.2	−1.82735	−0.913675	+	0.406445i	$0.866768\pi$
$542$	0	0
$543$	−10012.6	−0.791315
$544$	0	0
$545$	−22313.9	−1.75381
$546$	0	0
$547$	11684.1	0.913305	0.456653	−	0.889645i	$-0.349048\pi$
$547$	11684.1	0.913305	0.456653	+	0.889645i	$0.349048\pi$
$548$	0	0
$549$	−6383.76	−0.496270
$550$	0	0
$551$	1917.85	0.148282
$552$	0	0
$553$	5783.29	0.444720
$554$	0	0
$555$	12427.6	0.950492
$556$	0	0
$557$	8163.62	0.621012	0.310506	−	0.950571i	$-0.399502\pi$
$557$	8163.62	0.621012	0.310506	+	0.950571i	$0.399502\pi$
$558$	0	0
$559$	19203.2	1.45297
$560$	0	0
$561$	−2876.99	−0.216518
$562$	0	0
$563$	4415.81	0.330558	0.165279	−	0.986247i	$-0.447148\pi$
$563$	4415.81	0.330558	0.165279	+	0.986247i	$0.447148\pi$
$564$	0	0
$565$	19497.1	1.45177
$566$	0	0
$567$	1602.98	0.118728
$568$	0	0
$569$	−25237.7	−1.85943	−0.929717	−	0.368275i	$-0.879949\pi$
$569$	−25237.7	−1.85943	−0.929717	+	0.368275i	$0.879949\pi$
$570$	0	0
$571$	−2372.04	−0.173847	−0.0869237	−	0.996215i	$-0.527704\pi$
$571$	−2372.04	−0.173847	−0.0869237	+	0.996215i	$0.527704\pi$
$572$	0	0
$573$	1737.41	0.126669
$574$	0	0
$575$	4391.99	0.318537
$576$	0	0
$577$	−18059.7	−1.30301	−0.651504	−	0.758645i	$-0.725862\pi$
$577$	−18059.7	−1.30301	−0.651504	+	0.758645i	$0.725862\pi$
$578$	0	0
$579$	−2820.97	−0.202479
$580$	0	0
$581$	−3460.44	−0.247097
$582$	0	0
$583$	−4386.64	−0.311623
$584$	0	0
$585$	−11721.3	−0.828401
$586$	0	0
$587$	−430.010	−0.0302358	−0.0151179	−	0.999886i	$-0.504812\pi$
$587$	−430.010	−0.0302358	−0.0151179	+	0.999886i	$0.504812\pi$
$588$	0	0
$589$	3322.24	0.232412
$590$	0	0
$591$	6665.91	0.463958
$592$	0	0
$593$	−9904.91	−0.685912	−0.342956	−	0.939351i	$-0.611428\pi$
$593$	−9904.91	−0.685912	−0.342956	+	0.939351i	$0.611428\pi$
$594$	0	0
$595$	6695.80	0.461346
$596$	0	0
$597$	1269.18	0.0870086
$598$	0	0
$599$	13551.7	0.924388	0.462194	−	0.886779i	$-0.347062\pi$
$599$	13551.7	0.924388	0.462194	+	0.886779i	$0.347062\pi$
$600$	0	0
$601$	22246.9	1.50994	0.754968	−	0.655762i	$-0.227652\pi$
$601$	22246.9	1.50994	0.754968	+	0.655762i	$0.227652\pi$
$602$	0	0
$603$	−12940.0	−0.873891
$604$	0	0
$605$	−16869.8	−1.13364
$606$	0	0
$607$	−17935.3	−1.19929	−0.599647	−	0.800265i	$-0.704692\pi$
$607$	−17935.3	−1.19929	−0.599647	+	0.800265i	$0.704692\pi$
$608$	0	0
$609$	−3323.78	−0.221160
$610$	0	0
$611$	−36155.5	−2.39394
$612$	0	0
$613$	−22506.9	−1.48294	−0.741472	−	0.670984i	$-0.765872\pi$
$613$	−22506.9	−1.48294	−0.741472	+	0.670984i	$0.765872\pi$
$614$	0	0
$615$	2170.53	0.142316
$616$	0	0
$617$	29710.3	1.93856	0.969280	−	0.245961i	$-0.0791035\pi$
$617$	29710.3	1.93856	0.969280	+	0.245961i	$0.0791035\pi$
$618$	0	0
$619$	25289.3	1.64211	0.821054	−	0.570851i	$-0.193386\pi$
$619$	25289.3	1.64211	0.821054	+	0.570851i	$0.193386\pi$
$620$	0	0
$621$	−9440.68	−0.610050
$622$	0	0
$623$	4606.22	0.296219
$624$	0	0
$625$	−19476.5	−1.24650
$626$	0	0
$627$	643.126	0.0409633
$628$	0	0
$629$	−16084.6	−1.01961
$630$	0	0
$631$	25765.1	1.62550	0.812752	−	0.582610i	$-0.197968\pi$
$631$	25765.1	1.62550	0.812752	+	0.582610i	$0.197968\pi$
$632$	0	0
$633$	−14352.7	−0.901212
$634$	0	0
$635$	−34046.8	−2.12773
$636$	0	0
$637$	3259.95	0.202769
$638$	0	0
$639$	−1109.26	−0.0686722
$640$	0	0
$641$	−16452.1	−1.01376	−0.506878	−	0.862018i	$-0.669201\pi$
$641$	−16452.1	−1.01376	−0.506878	+	0.862018i	$0.669201\pi$
$642$	0	0
$643$	−4328.58	−0.265478	−0.132739	−	0.991151i	$-0.542377\pi$
$643$	−4328.58	−0.265478	−0.132739	+	0.991151i	$0.542377\pi$
$644$	0	0
$645$	15280.6	0.932827
$646$	0	0
$647$	−21986.2	−1.33596	−0.667981	−	0.744179i	$-0.732841\pi$
$647$	−21986.2	−1.33596	−0.667981	+	0.744179i	$0.732841\pi$
$648$	0	0
$649$	7474.88	0.452103
$650$	0	0
$651$	−5757.69	−0.346638
$652$	0	0
$653$	−12227.0	−0.732740	−0.366370	−	0.930469i	$-0.619400\pi$
$653$	−12227.0	−0.732740	−0.366370	+	0.930469i	$0.619400\pi$
$654$	0	0
$655$	−10814.5	−0.645125
$656$	0	0
$657$	−7860.25	−0.466755
$658$	0	0
$659$	23121.0	1.36672	0.683359	−	0.730082i	$-0.260518\pi$
$659$	23121.0	1.36672	0.683359	+	0.730082i	$0.260518\pi$
$660$	0	0
$661$	12600.0	0.741425	0.370712	−	0.928748i	$-0.379114\pi$
$661$	12600.0	0.741425	0.370712	+	0.928748i	$0.379114\pi$
$662$	0	0
$663$	−17286.1	−1.01257
$664$	0	0
$665$	−1496.79	−0.0872826
$666$	0	0
$667$	7867.98	0.456745
$668$	0	0
$669$	19003.8	1.09825
$670$	0	0
$671$	5601.10	0.322248
$672$	0	0
$673$	−22288.1	−1.27659	−0.638293	−	0.769794i	$-0.720359\pi$
$673$	−22288.1	−1.27659	−0.638293	+	0.769794i	$0.720359\pi$
$674$	0	0
$675$	−10501.4	−0.598813
$676$	0	0
$677$	−26297.6	−1.49291	−0.746455	−	0.665436i	$-0.768246\pi$
$677$	−26297.6	−1.49291	−0.746455	+	0.665436i	$0.768246\pi$
$678$	0	0
$679$	−2655.57	−0.150090
$680$	0	0
$681$	−6442.38	−0.362515
$682$	0	0
$683$	15825.0	0.886569	0.443285	−	0.896381i	$-0.353813\pi$
$683$	15825.0	0.886569	0.443285	+	0.896381i	$0.353813\pi$
$684$	0	0
$685$	−25480.4	−1.42125
$686$	0	0
$687$	12602.0	0.699851
$688$	0	0
$689$	−26356.7	−1.45734
$690$	0	0
$691$	28078.8	1.54583	0.772914	−	0.634511i	$-0.218798\pi$
$691$	28078.8	1.54583	0.772914	+	0.634511i	$0.218798\pi$
$692$	0	0
$693$	978.166	0.0536182
$694$	0	0
$695$	26241.2	1.43221
$696$	0	0
$697$	−2809.23	−0.152664
$698$	0	0
$699$	13176.5	0.712989
$700$	0	0
$701$	−13308.6	−0.717057	−0.358529	−	0.933519i	$-0.616721\pi$
$701$	−13308.6	−0.717057	−0.358529	+	0.933519i	$0.616721\pi$
$702$	0	0
$703$	3595.57	0.192901
$704$	0	0
$705$	−28770.1	−1.53694
$706$	0	0
$707$	−1299.10	−0.0691057
$708$	0	0
$709$	−13259.2	−0.702340	−0.351170	−	0.936312i	$-0.614216\pi$
$709$	−13259.2	−0.702340	−0.351170	+	0.936312i	$0.614216\pi$
$710$	0	0
$711$	10426.4	0.549960
$712$	0	0
$713$	13629.5	0.715888
$714$	0	0
$715$	10284.2	0.537914
$716$	0	0
$717$	16907.7	0.880656
$718$	0	0
$719$	13787.0	0.715117	0.357559	−	0.933891i	$-0.383609\pi$
$719$	13787.0	0.715117	0.357559	+	0.933891i	$0.383609\pi$
$720$	0	0
$721$	30.0750	0.00155347
$722$	0	0
$723$	16403.5	0.843782
$724$	0	0
$725$	8751.98	0.448332
$726$	0	0
$727$	−22346.5	−1.14001	−0.570004	−	0.821642i	$-0.693058\pi$
$727$	−22346.5	−1.14001	−0.570004	+	0.821642i	$0.693058\pi$
$728$	0	0
$729$	18272.8	0.928355
$730$	0	0
$731$	−19777.1	−1.00066
$732$	0	0
$733$	34032.6	1.71490	0.857451	−	0.514566i	$-0.172047\pi$
$733$	34032.6	1.71490	0.857451	+	0.514566i	$0.172047\pi$
$734$	0	0
$735$	2594.04	0.130181
$736$	0	0
$737$	11353.5	0.567452
$738$	0	0
$739$	25031.7	1.24602	0.623009	−	0.782215i	$-0.285910\pi$
$739$	25031.7	1.24602	0.623009	+	0.782215i	$0.285910\pi$
$740$	0	0
$741$	3864.16	0.191570
$742$	0	0
$743$	−17954.8	−0.886539	−0.443269	−	0.896389i	$-0.646181\pi$
$743$	−17954.8	−0.886539	−0.443269	+	0.896389i	$0.646181\pi$
$744$	0	0
$745$	−30090.8	−1.47979
$746$	0	0
$747$	−6238.67	−0.305571
$748$	0	0
$749$	−4877.03	−0.237921
$750$	0	0
$751$	33796.8	1.64216	0.821081	−	0.570812i	$-0.193371\pi$
$751$	33796.8	1.64216	0.821081	+	0.570812i	$0.193371\pi$
$752$	0	0
$753$	−20068.1	−0.971210
$754$	0	0
$755$	37661.1	1.81540
$756$	0	0
$757$	4987.84	0.239480	0.119740	−	0.992805i	$-0.461794\pi$
$757$	4987.84	0.239480	0.119740	+	0.992805i	$0.461794\pi$
$758$	0	0
$759$	2638.42	0.126178
$760$	0	0
$761$	−27320.1	−1.30138	−0.650691	−	0.759343i	$-0.725521\pi$
$761$	−27320.1	−1.30138	−0.650691	+	0.759343i	$0.725521\pi$
$762$	0	0
$763$	11188.5	0.530867
$764$	0	0
$765$	12071.6	0.570520
$766$	0	0
$767$	44912.0	2.11432
$768$	0	0
$769$	3724.67	0.174662	0.0873309	−	0.996179i	$-0.472166\pi$
$769$	3724.67	0.174662	0.0873309	+	0.996179i	$0.472166\pi$
$770$	0	0
$771$	5542.12	0.258878
$772$	0	0
$773$	−13824.4	−0.643248	−0.321624	−	0.946868i	$-0.604229\pi$
$773$	−13824.4	−0.643248	−0.321624	+	0.946868i	$0.604229\pi$
$774$	0	0
$775$	15160.8	0.702700
$776$	0	0
$777$	−6231.38	−0.287709
$778$	0	0
$779$	627.978	0.0288827
$780$	0	0
$781$	973.261	0.0445916
$782$	0	0
$783$	−18812.6	−0.858629
$784$	0	0
$785$	3609.47	0.164111
$786$	0	0
$787$	−8441.71	−0.382356	−0.191178	−	0.981555i	$-0.561231\pi$
$787$	−8441.71	−0.382356	−0.191178	+	0.981555i	$0.561231\pi$
$788$	0	0
$789$	−15158.1	−0.683957
$790$	0	0
$791$	−9776.13	−0.439443
$792$	0	0
$793$	33653.7	1.50703
$794$	0	0
$795$	−20972.9	−0.935636
$796$	0	0
$797$	−38305.8	−1.70246	−0.851230	−	0.524793i	$-0.824143\pi$
$797$	−38305.8	−1.70246	−0.851230	+	0.524793i	$0.824143\pi$
$798$	0	0
$799$	37236.0	1.64871
$800$	0	0
$801$	8304.34	0.366316
$802$	0	0
$803$	6896.58	0.303082
$804$	0	0
$805$	−6140.57	−0.268853
$806$	0	0
$807$	26897.6	1.17329
$808$	0	0
$809$	6226.56	0.270598	0.135299	−	0.990805i	$-0.456800\pi$
$809$	6226.56	0.270598	0.135299	+	0.990805i	$0.456800\pi$
$810$	0	0
$811$	−6011.22	−0.260274	−0.130137	−	0.991496i	$-0.541542\pi$
$811$	−6011.22	−0.260274	−0.130137	+	0.991496i	$0.541542\pi$
$812$	0	0
$813$	28632.2	1.23515
$814$	0	0
$815$	34702.2	1.49149
$816$	0	0
$817$	4420.99	0.189316
$818$	0	0
$819$	5877.21	0.250752
$820$	0	0
$821$	−24955.9	−1.06086	−0.530429	−	0.847729i	$-0.677969\pi$
$821$	−24955.9	−1.06086	−0.530429	+	0.847729i	$0.677969\pi$
$822$	0	0
$823$	20339.7	0.861478	0.430739	−	0.902477i	$-0.358253\pi$
$823$	20339.7	0.861478	0.430739	+	0.902477i	$0.358253\pi$
$824$	0	0
$825$	2934.87	0.123853
$826$	0	0
$827$	6434.94	0.270574	0.135287	−	0.990806i	$-0.456804\pi$
$827$	6434.94	0.270574	0.135287	+	0.990806i	$0.456804\pi$
$828$	0	0
$829$	−38186.6	−1.59985	−0.799926	−	0.600099i	$-0.795128\pi$
$829$	−38186.6	−1.59985	−0.799926	+	0.600099i	$0.795128\pi$
$830$	0	0
$831$	−14230.3	−0.594036
$832$	0	0
$833$	−3357.37	−0.139647
$834$	0	0
$835$	−6343.69	−0.262913
$836$	0	0
$837$	−32588.5	−1.34579
$838$	0	0
$839$	16761.3	0.689706	0.344853	−	0.938657i	$-0.387929\pi$
$839$	16761.3	0.689706	0.344853	+	0.938657i	$0.387929\pi$
$840$	0	0
$841$	−8710.38	−0.357144
$842$	0	0
$843$	−2502.26	−0.102233
$844$	0	0
$845$	31120.5	1.26695
$846$	0	0
$847$	8458.76	0.343148
$848$	0	0
$849$	−23601.4	−0.954061
$850$	0	0
$851$	14750.8	0.594185
$852$	0	0
$853$	46470.0	1.86530	0.932650	−	0.360781i	$-0.117490\pi$
$853$	46470.0	1.86530	0.932650	+	0.360781i	$0.117490\pi$
$854$	0	0
$855$	−2698.49	−0.107937
$856$	0	0
$857$	25353.0	1.01055	0.505275	−	0.862959i	$-0.331391\pi$
$857$	25353.0	1.01055	0.505275	+	0.862959i	$0.331391\pi$
$858$	0	0
$859$	−34441.7	−1.36803	−0.684014	−	0.729469i	$-0.739767\pi$
$859$	−34441.7	−1.36803	−0.684014	+	0.729469i	$0.739767\pi$
$860$	0	0
$861$	−1088.33	−0.0430781
$862$	0	0
$863$	−48569.1	−1.91577	−0.957885	−	0.287151i	$-0.907292\pi$
$863$	−48569.1	−1.91577	−0.957885	+	0.287151i	$0.907292\pi$
$864$	0	0
$865$	39941.9	1.57002
$866$	0	0
$867$	−827.887	−0.0324297
$868$	0	0
$869$	−9148.13	−0.357111
$870$	0	0
$871$	68216.4	2.65376
$872$	0	0
$873$	−4787.60	−0.185608
$874$	0	0
$875$	5384.96	0.208051
$876$	0	0
$877$	8497.93	0.327200	0.163600	−	0.986527i	$-0.447689\pi$
$877$	8497.93	0.327200	0.163600	+	0.986527i	$0.447689\pi$
$878$	0	0
$879$	−23554.8	−0.903848
$880$	0	0
$881$	−38132.9	−1.45826	−0.729132	−	0.684373i	$-0.760076\pi$
$881$	−38132.9	−1.45826	−0.729132	+	0.684373i	$0.760076\pi$
$882$	0	0
$883$	−13604.5	−0.518491	−0.259245	−	0.965811i	$-0.583474\pi$
$883$	−13604.5	−0.518491	−0.259245	+	0.965811i	$0.583474\pi$
$884$	0	0
$885$	35738.0	1.35742
$886$	0	0
$887$	37239.0	1.40965	0.704826	−	0.709380i	$-0.251025\pi$
$887$	37239.0	1.40965	0.704826	+	0.709380i	$0.251025\pi$
$888$	0	0
$889$	17071.5	0.644051
$890$	0	0
$891$	−2535.63	−0.0953386
$892$	0	0
$893$	−8323.79	−0.311921
$894$	0	0
$895$	3503.50	0.130848
$896$	0	0
$897$	15852.7	0.590085
$898$	0	0
$899$	27159.6	1.00759
$900$	0	0
$901$	27144.3	1.00367
$902$	0	0
$903$	−7661.91	−0.282361
$904$	0	0
$905$	−36861.3	−1.35394
$906$	0	0
$907$	30240.5	1.10708	0.553539	−	0.832823i	$-0.313277\pi$
$907$	30240.5	1.10708	0.553539	+	0.832823i	$0.313277\pi$
$908$	0	0
$909$	−2342.09	−0.0854590
$910$	0	0
$911$	25898.4	0.941880	0.470940	−	0.882165i	$-0.343915\pi$
$911$	25898.4	0.941880	0.470940	+	0.882165i	$0.343915\pi$
$912$	0	0
$913$	5473.81	0.198419
$914$	0	0
$915$	26779.3	0.967537
$916$	0	0
$917$	5422.53	0.195276
$918$	0	0
$919$	1303.01	0.0467708	0.0233854	−	0.999727i	$-0.492556\pi$
$919$	1303.01	0.0467708	0.0233854	+	0.999727i	$0.492556\pi$
$920$	0	0
$921$	−22530.0	−0.806068
$922$	0	0
$923$	5847.74	0.208538
$924$	0	0
$925$	16408.1	0.583239
$926$	0	0
$927$	54.2208	0.00192108
$928$	0	0
$929$	−33763.6	−1.19241	−0.596204	−	0.802833i	$-0.703325\pi$
$929$	−33763.6	−1.19241	−0.596204	+	0.802833i	$0.703325\pi$
$930$	0	0
$931$	750.511	0.0264200
$932$	0	0
$933$	18864.4	0.661944
$934$	0	0
$935$	−10591.6	−0.370461
$936$	0	0
$937$	2250.74	0.0784723	0.0392362	−	0.999230i	$-0.487508\pi$
$937$	2250.74	0.0784723	0.0392362	+	0.999230i	$0.487508\pi$
$938$	0	0
$939$	−9521.52	−0.330909
$940$	0	0
$941$	27082.0	0.938203	0.469102	−	0.883144i	$-0.344578\pi$
$941$	27082.0	0.938203	0.469102	+	0.883144i	$0.344578\pi$
$942$	0	0
$943$	2576.28	0.0889663
$944$	0	0
$945$	14682.3	0.505412
$946$	0	0
$947$	31149.3	1.06886	0.534432	−	0.845211i	$-0.320525\pi$
$947$	31149.3	1.06886	0.534432	+	0.845211i	$0.320525\pi$
$948$	0	0
$949$	41437.4	1.41740
$950$	0	0
$951$	−1690.91	−0.0576566
$952$	0	0
$953$	29269.3	0.994884	0.497442	−	0.867497i	$-0.334273\pi$
$953$	29269.3	0.994884	0.497442	+	0.867497i	$0.334273\pi$
$954$	0	0
$955$	6396.24	0.216730
$956$	0	0
$957$	5257.62	0.177591
$958$	0	0
$959$	12776.2	0.430204
$960$	0	0
$961$	17256.9	0.579266
$962$	0	0
$963$	−8792.56	−0.294223
$964$	0	0
$965$	−10385.3	−0.346441
$966$	0	0
$967$	44952.0	1.49489	0.747445	−	0.664324i	$-0.231280\pi$
$967$	44952.0	1.49489	0.747445	+	0.664324i	$0.231280\pi$
$968$	0	0
$969$	−3979.64	−0.131934
$970$	0	0
$971$	−43065.9	−1.42333	−0.711664	−	0.702520i	$-0.752058\pi$
$971$	−43065.9	−1.42333	−0.711664	+	0.702520i	$0.752058\pi$
$972$	0	0
$973$	−13157.7	−0.433522
$974$	0	0
$975$	17633.8	0.579215
$976$	0	0
$977$	34642.8	1.13441	0.567206	−	0.823576i	$-0.308024\pi$
$977$	34642.8	1.13441	0.567206	+	0.823576i	$0.308024\pi$
$978$	0	0
$979$	−7286.22	−0.237864
$980$	0	0
$981$	20171.3	0.656492
$982$	0	0
$983$	57038.2	1.85070	0.925349	−	0.379116i	$-0.123772\pi$
$983$	57038.2	1.85070	0.925349	+	0.379116i	$0.123772\pi$
$984$	0	0
$985$	24540.4	0.793829
$986$	0	0
$987$	14425.7	0.465224
$988$	0	0
$989$	18137.1	0.583141
$990$	0	0
$991$	−28952.3	−0.928051	−0.464026	−	0.885822i	$-0.653595\pi$
$991$	−28952.3	−0.928051	−0.464026	+	0.885822i	$0.653595\pi$
$992$	0	0
$993$	−14367.3	−0.459147
$994$	0	0
$995$	4672.46	0.148871
$996$	0	0
$997$	−45193.8	−1.43561	−0.717805	−	0.696245i	$-0.754853\pi$
$997$	−45193.8	−1.43561	−0.717805	+	0.696245i	$0.754853\pi$
$998$	0	0
$999$	−35269.6	−1.11700

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.d.1.10	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.d.1.10	✓	15	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q+3.79210 q^{3} +13.9605 q^{5} -7.00000 q^{7} -12.6200 q^{9} +O(q^{10})\)	\(q+3.79210 q^{3} +13.9605 q^{5} -7.00000 q^{7} -12.6200 q^{9} +11.0728 q^{11} +66.5295 q^{13} +52.9397 q^{15} -68.5177 q^{17} +15.3165 q^{19} -26.5447 q^{21} +62.8361 q^{23} +69.8960 q^{25} -150.243 q^{27} +125.214 q^{29} +216.905 q^{31} +41.9890 q^{33} -97.7236 q^{35} +234.751 q^{37} +252.287 q^{39} +41.0000 q^{41} +288.642 q^{43} -176.181 q^{45} -543.451 q^{47} +49.0000 q^{49} -259.826 q^{51} -396.165 q^{53} +154.581 q^{55} +58.0819 q^{57} +675.069 q^{59} +505.845 q^{61} +88.3398 q^{63} +928.786 q^{65} +1025.36 q^{67} +238.281 q^{69} +87.8969 q^{71} +622.842 q^{73} +265.053 q^{75} -77.5093 q^{77} -826.184 q^{79} -228.997 q^{81} +494.349 q^{83} -956.543 q^{85} +474.825 q^{87} -658.031 q^{89} -465.707 q^{91} +822.527 q^{93} +213.827 q^{95} +379.367 q^{97} -139.738 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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