LMFDB - Embedded newform 1148.4.a.c.1.8

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} - 5 x^{14} - 270 x^{13} + 1158 x^{12} + 28413 x^{11} - 102669 x^{10} - 1445580 x^{9} + \cdots + 1052740152 $ x^15 - 5x^14 - 270x^13 + 1158x^12 + 28413x^11 - 102669x^10 - 1445580x^9 + 4412670x^8 + 35401830x^7 - 96527282x^6 - 348573059x^5 + 982943895x^4 + 672972471x^3 - 2124774315x^2 - 569503386x + 1052740152
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.707687$ 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)

$f(q)$	$=$	$q+0.292313 q^{3} -17.4774 q^{5} +7.00000 q^{7} -26.9146 q^{9} +O(q^{10})$	$q+0.292313 q^{3} -17.4774 q^{5} +7.00000 q^{7} -26.9146 q^{9} +28.3731 q^{11} -16.1056 q^{13} -5.10888 q^{15} -96.8854 q^{17} -9.67173 q^{19} +2.04619 q^{21} -123.483 q^{23} +180.460 q^{25} -15.7599 q^{27} +83.1360 q^{29} -44.3239 q^{31} +8.29383 q^{33} -122.342 q^{35} -128.528 q^{37} -4.70789 q^{39} -41.0000 q^{41} -384.274 q^{43} +470.397 q^{45} +538.019 q^{47} +49.0000 q^{49} -28.3209 q^{51} -436.867 q^{53} -495.888 q^{55} -2.82717 q^{57} -390.109 q^{59} +626.187 q^{61} -188.402 q^{63} +281.485 q^{65} +648.916 q^{67} -36.0958 q^{69} -925.584 q^{71} +438.692 q^{73} +52.7508 q^{75} +198.612 q^{77} -433.100 q^{79} +722.086 q^{81} -55.2788 q^{83} +1693.31 q^{85} +24.3017 q^{87} +1611.88 q^{89} -112.740 q^{91} -12.9565 q^{93} +169.037 q^{95} +1151.47 q^{97} -763.649 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9}+O(q^{10}) $ 15 * q + 10 * q^3 + 6 * q^5 + 105 * q^7 + 165 * q^9	$ 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9} + 62 q^{11} + 148 q^{13} + 152 q^{15} + 132 q^{17} + 260 q^{19} + 70 q^{21} - 26 q^{23} + 453 q^{25} + 454 q^{27} + 12 q^{29} + 144 q^{31} + 336 q^{33} + 42 q^{35} + 274 q^{37} + 218 q^{39} - 615 q^{41} + 986 q^{43} + 648 q^{45} + 44 q^{47} + 735 q^{49} + 202 q^{51} - 366 q^{53} + 928 q^{55} - 294 q^{57} - 8 q^{59} + 1396 q^{61} + 1155 q^{63} + 156 q^{65} - 24 q^{67} + 892 q^{69} + 1464 q^{71} + 1174 q^{73} + 318 q^{75} + 434 q^{77} + 1590 q^{79} + 1959 q^{81} + 1970 q^{83} + 2348 q^{85} + 2544 q^{87} + 1972 q^{89} + 1036 q^{91} + 4034 q^{93} + 1070 q^{95} + 954 q^{97} + 3398 q^{99}+O(q^{100}) $ 15 * q + 10 * q^3 + 6 * q^5 + 105 * q^7 + 165 * q^9 + 62 * q^11 + 148 * q^13 + 152 * q^15 + 132 * q^17 + 260 * q^19 + 70 * q^21 - 26 * q^23 + 453 * q^25 + 454 * q^27 + 12 * q^29 + 144 * q^31 + 336 * q^33 + 42 * q^35 + 274 * q^37 + 218 * q^39 - 615 * q^41 + 986 * q^43 + 648 * q^45 + 44 * q^47 + 735 * q^49 + 202 * q^51 - 366 * q^53 + 928 * q^55 - 294 * q^57 - 8 * q^59 + 1396 * q^61 + 1155 * q^63 + 156 * q^65 - 24 * q^67 + 892 * q^69 + 1464 * q^71 + 1174 * q^73 + 318 * q^75 + 434 * q^77 + 1590 * q^79 + 1959 * q^81 + 1970 * q^83 + 2348 * q^85 + 2544 * q^87 + 1972 * q^89 + 1036 * q^91 + 4034 * q^93 + 1070 * q^95 + 954 * q^97 + 3398 * 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
$3$	0.292313	0.0562557	0.0281279	−	0.999604i	$-0.491045\pi$
$3$	0.292313	0.0562557	0.0281279	+	0.999604i	$0.491045\pi$
$4$	0	0
$5$	−17.4774	−1.56323	−0.781614	−	0.623763i	$-0.785603\pi$
$5$	−17.4774	−1.56323	−0.781614	+	0.623763i	$0.785603\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−26.9146	−0.996835
$10$	0	0
$11$	28.3731	0.777710	0.388855	−	0.921299i	$-0.372871\pi$
$11$	28.3731	0.777710	0.388855	+	0.921299i	$0.372871\pi$
$12$	0	0
$13$	−16.1056	−0.343608	−0.171804	−	0.985131i	$-0.554960\pi$
$13$	−16.1056	−0.343608	−0.171804	+	0.985131i	$0.554960\pi$
$14$	0	0
$15$	−5.10888	−0.0879405
$16$	0	0
$17$	−96.8854	−1.38224	−0.691122	−	0.722738i	$-0.742883\pi$
$17$	−96.8854	−1.38224	−0.691122	+	0.722738i	$0.742883\pi$
$18$	0	0
$19$	−9.67173	−0.116781	−0.0583907	−	0.998294i	$-0.518597\pi$
$19$	−9.67173	−0.116781	−0.0583907	+	0.998294i	$0.518597\pi$
$20$	0	0
$21$	2.04619	0.0212627
$22$	0	0
$23$	−123.483	−1.11948	−0.559740	−	0.828668i	$-0.689099\pi$
$23$	−123.483	−1.11948	−0.559740	+	0.828668i	$0.689099\pi$
$24$	0	0
$25$	180.460	1.44368
$26$	0	0
$27$	−15.7599	−0.112333
$28$	0	0
$29$	83.1360	0.532344	0.266172	−	0.963926i	$-0.414241\pi$
$29$	83.1360	0.532344	0.266172	+	0.963926i	$0.414241\pi$
$30$	0	0
$31$	−44.3239	−0.256800	−0.128400	−	0.991722i	$-0.540984\pi$
$31$	−44.3239	−0.256800	−0.128400	+	0.991722i	$0.540984\pi$
$32$	0	0
$33$	8.29383	0.0437506
$34$	0	0
$35$	−122.342	−0.590844
$36$	0	0
$37$	−128.528	−0.571076	−0.285538	−	0.958367i	$-0.592172\pi$
$37$	−128.528	−0.571076	−0.285538	+	0.958367i	$0.592172\pi$
$38$	0	0
$39$	−4.70789	−0.0193299
$40$	0	0
$41$	−41.0000	−0.156174
$41$	−41.0000	−0.156174
$42$	0	0
$43$	−384.274	−1.36282	−0.681409	−	0.731902i	$-0.738633\pi$
$43$	−384.274	−1.36282	−0.681409	+	0.731902i	$0.738633\pi$
$44$	0	0
$45$	470.397	1.55828
$46$	0	0
$47$	538.019	1.66975	0.834874	−	0.550441i	$-0.185540\pi$
$47$	538.019	1.66975	0.834874	+	0.550441i	$0.185540\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−28.3209	−0.0777591
$52$	0	0
$53$	−436.867	−1.13223	−0.566116	−	0.824325i	$-0.691555\pi$
$53$	−436.867	−1.13223	−0.566116	+	0.824325i	$0.691555\pi$
$54$	0	0
$55$	−495.888	−1.21574
$56$	0	0
$57$	−2.82717	−0.00656962
$58$	0	0
$59$	−390.109	−0.860812	−0.430406	−	0.902635i	$-0.641630\pi$
$59$	−390.109	−0.860812	−0.430406	+	0.902635i	$0.641630\pi$
$60$	0	0
$61$	626.187	1.31435	0.657173	−	0.753740i	$-0.271752\pi$
$61$	626.187	1.31435	0.657173	+	0.753740i	$0.271752\pi$
$62$	0	0
$63$	−188.402	−0.376768
$64$	0	0
$65$	281.485	0.537137
$66$	0	0
$67$	648.916	1.18325	0.591625	−	0.806213i	$-0.298487\pi$
$67$	648.916	1.18325	0.591625	+	0.806213i	$0.298487\pi$
$68$	0	0
$69$	−36.0958	−0.0629771
$70$	0	0
$71$	−925.584	−1.54714	−0.773568	−	0.633713i	$-0.781530\pi$
$71$	−925.584	−1.54714	−0.773568	+	0.633713i	$0.781530\pi$
$72$	0	0
$73$	438.692	0.703357	0.351679	−	0.936121i	$-0.385611\pi$
$73$	438.692	0.703357	0.351679	+	0.936121i	$0.385611\pi$
$74$	0	0
$75$	52.7508	0.0812152
$76$	0	0
$77$	198.612	0.293947
$78$	0	0
$79$	−433.100	−0.616804	−0.308402	−	0.951256i	$-0.599794\pi$
$79$	−433.100	−0.616804	−0.308402	+	0.951256i	$0.599794\pi$
$80$	0	0
$81$	722.086	0.990516
$82$	0	0
$83$	−55.2788	−0.0731040	−0.0365520	−	0.999332i	$-0.511637\pi$
$83$	−55.2788	−0.0731040	−0.0365520	+	0.999332i	$0.511637\pi$
$84$	0	0
$85$	1693.31	2.16076
$86$	0	0
$87$	24.3017	0.0299474
$88$	0	0
$89$	1611.88	1.91977	0.959883	−	0.280400i	$-0.0904671\pi$
$89$	1611.88	1.91977	0.959883	+	0.280400i	$0.0904671\pi$
$90$	0	0
$91$	−112.740	−0.129872
$92$	0	0
$93$	−12.9565	−0.0144465
$94$	0	0
$95$	169.037	0.182556
$96$	0	0
$97$	1151.47	1.20530	0.602649	−	0.798007i	$-0.294112\pi$
$97$	1151.47	1.20530	0.602649	+	0.798007i	$0.294112\pi$
$98$	0	0
$99$	−763.649	−0.775249
$100$	0	0
$101$	1165.17	1.14791	0.573954	−	0.818887i	$-0.305409\pi$
$101$	1165.17	1.14791	0.573954	+	0.818887i	$0.305409\pi$
$102$	0	0
$103$	1791.35	1.71366	0.856832	−	0.515596i	$-0.172429\pi$
$103$	1791.35	1.71366	0.856832	+	0.515596i	$0.172429\pi$
$104$	0	0
$105$	−35.7622	−0.0332384
$106$	0	0
$107$	678.859	0.613344	0.306672	−	0.951815i	$-0.400785\pi$
$107$	678.859	0.613344	0.306672	+	0.951815i	$0.400785\pi$
$108$	0	0
$109$	−1280.95	−1.12562	−0.562809	−	0.826587i	$-0.690280\pi$
$109$	−1280.95	−1.12562	−0.562809	+	0.826587i	$0.690280\pi$
$110$	0	0
$111$	−37.5703	−0.0321263
$112$	0	0
$113$	−783.787	−0.652500	−0.326250	−	0.945283i	$-0.605785\pi$
$113$	−783.787	−0.652500	−0.326250	+	0.945283i	$0.605785\pi$
$114$	0	0
$115$	2158.17	1.75000
$116$	0	0
$117$	433.476	0.342521
$118$	0	0
$119$	−678.198	−0.522439
$120$	0	0
$121$	−525.967	−0.395167
$122$	0	0
$123$	−11.9848	−0.00878567
$124$	0	0
$125$	−969.297	−0.693572
$126$	0	0
$127$	1094.83	0.764967	0.382483	−	0.923962i	$-0.375069\pi$
$127$	1094.83	0.764967	0.382483	+	0.923962i	$0.375069\pi$
$128$	0	0
$129$	−112.328	−0.0766663
$130$	0	0
$131$	−2709.07	−1.80681	−0.903407	−	0.428785i	$-0.858942\pi$
$131$	−2709.07	−1.80681	−0.903407	+	0.428785i	$0.858942\pi$
$132$	0	0
$133$	−67.7021	−0.0441392
$134$	0	0
$135$	275.443	0.175603
$136$	0	0
$137$	1389.74	0.866666	0.433333	−	0.901234i	$-0.357337\pi$
$137$	1389.74	0.866666	0.433333	+	0.901234i	$0.357337\pi$
$138$	0	0
$139$	710.741	0.433700	0.216850	−	0.976205i	$-0.430422\pi$
$139$	710.741	0.433700	0.216850	+	0.976205i	$0.430422\pi$
$140$	0	0
$141$	157.270	0.0939329
$142$	0	0
$143$	−456.967	−0.267227
$144$	0	0
$145$	−1453.00	−0.832174
$146$	0	0
$147$	14.3233	0.00803653
$148$	0	0
$149$	3184.29	1.75078	0.875392	−	0.483414i	$-0.160603\pi$
$149$	3184.29	1.75078	0.875392	+	0.483414i	$0.160603\pi$
$150$	0	0
$151$	2603.81	1.40328	0.701639	−	0.712532i	$-0.252452\pi$
$151$	2603.81	1.40328	0.701639	+	0.712532i	$0.252452\pi$
$152$	0	0
$153$	2607.63	1.37787
$154$	0	0
$155$	774.668	0.401438
$156$	0	0
$157$	112.253	0.0570624	0.0285312	−	0.999593i	$-0.490917\pi$
$157$	112.253	0.0570624	0.0285312	+	0.999593i	$0.490917\pi$
$158$	0	0
$159$	−127.702	−0.0636945
$160$	0	0
$161$	−864.383	−0.423124
$162$	0	0
$163$	3053.47	1.46728	0.733638	−	0.679540i	$-0.237821\pi$
$163$	3053.47	1.46728	0.733638	+	0.679540i	$0.237821\pi$
$164$	0	0
$165$	−144.955	−0.0683922
$166$	0	0
$167$	−1237.30	−0.573322	−0.286661	−	0.958032i	$-0.592545\pi$
$167$	−1237.30	−0.573322	−0.286661	+	0.958032i	$0.592545\pi$
$168$	0	0
$169$	−1937.61	−0.881934
$170$	0	0
$171$	260.310	0.116412
$172$	0	0
$173$	1136.74	0.499566	0.249783	−	0.968302i	$-0.419641\pi$
$173$	1136.74	0.499566	0.249783	+	0.968302i	$0.419641\pi$
$174$	0	0
$175$	1263.22	0.545660
$176$	0	0
$177$	−114.034	−0.0484256
$178$	0	0
$179$	−2071.02	−0.864778	−0.432389	−	0.901687i	$-0.642329\pi$
$179$	−2071.02	−0.864778	−0.432389	+	0.901687i	$0.642329\pi$
$180$	0	0
$181$	−3160.71	−1.29798	−0.648988	−	0.760799i	$-0.724807\pi$
$181$	−3160.71	−1.29798	−0.648988	+	0.760799i	$0.724807\pi$
$182$	0	0
$183$	183.043	0.0739394
$184$	0	0
$185$	2246.33	0.892722
$186$	0	0
$187$	−2748.94	−1.07499
$188$	0	0
$189$	−110.320	−0.0424580
$190$	0	0
$191$	1223.75	0.463599	0.231800	−	0.972764i	$-0.425539\pi$
$191$	1223.75	0.463599	0.231800	+	0.972764i	$0.425539\pi$
$192$	0	0
$193$	−644.851	−0.240505	−0.120252	−	0.992743i	$-0.538370\pi$
$193$	−644.851	−0.240505	−0.120252	+	0.992743i	$0.538370\pi$
$194$	0	0
$195$	82.2818	0.0302170
$196$	0	0
$197$	−2114.44	−0.764709	−0.382354	−	0.924016i	$-0.624887\pi$
$197$	−2114.44	−0.764709	−0.382354	+	0.924016i	$0.624887\pi$
$198$	0	0
$199$	1927.53	0.686627	0.343313	−	0.939221i	$-0.388451\pi$
$199$	1927.53	0.686627	0.343313	+	0.939221i	$0.388451\pi$
$200$	0	0
$201$	189.687	0.0665646
$202$	0	0
$203$	581.952	0.201207
$204$	0	0
$205$	716.574	0.244135
$206$	0	0
$207$	3323.50	1.11594
$208$	0	0
$209$	−274.417	−0.0908221
$210$	0	0
$211$	−274.308	−0.0894982	−0.0447491	−	0.998998i	$-0.514249\pi$
$211$	−274.308	−0.0894982	−0.0447491	+	0.998998i	$0.514249\pi$
$212$	0	0
$213$	−270.561	−0.0870352
$214$	0	0
$215$	6716.11	2.13040
$216$	0	0
$217$	−310.268	−0.0970615
$218$	0	0
$219$	128.236	0.0395678
$220$	0	0
$221$	1560.40	0.474950
$222$	0	0
$223$	−2291.92	−0.688243	−0.344121	−	0.938925i	$-0.611823\pi$
$223$	−2291.92	−0.688243	−0.344121	+	0.938925i	$0.611823\pi$
$224$	0	0
$225$	−4857.00	−1.43911
$226$	0	0
$227$	5609.43	1.64014	0.820068	−	0.572266i	$-0.193936\pi$
$227$	5609.43	1.64014	0.820068	+	0.572266i	$0.193936\pi$
$228$	0	0
$229$	5386.65	1.55441	0.777204	−	0.629248i	$-0.216637\pi$
$229$	5386.65	1.55441	0.777204	+	0.629248i	$0.216637\pi$
$230$	0	0
$231$	58.0568	0.0165362
$232$	0	0
$233$	1097.18	0.308491	0.154246	−	0.988033i	$-0.450705\pi$
$233$	1097.18	0.308491	0.154246	+	0.988033i	$0.450705\pi$
$234$	0	0
$235$	−9403.18	−2.61020
$236$	0	0
$237$	−126.601	−0.0346987
$238$	0	0
$239$	−4552.04	−1.23200	−0.615998	−	0.787748i	$-0.711247\pi$
$239$	−4552.04	−1.23200	−0.615998	+	0.787748i	$0.711247\pi$
$240$	0	0
$241$	3712.59	0.992319	0.496159	−	0.868232i	$-0.334743\pi$
$241$	3712.59	0.992319	0.496159	+	0.868232i	$0.334743\pi$
$242$	0	0
$243$	636.594	0.168056
$244$	0	0
$245$	−856.393	−0.223318
$246$	0	0
$247$	155.769	0.0401270
$248$	0	0
$249$	−16.1587	−0.00411252
$250$	0	0
$251$	−7857.65	−1.97598	−0.987989	−	0.154524i	$-0.950616\pi$
$251$	−7857.65	−1.97598	−0.987989	+	0.154524i	$0.950616\pi$
$252$	0	0
$253$	−3503.60	−0.870631
$254$	0	0
$255$	494.976	0.121555
$256$	0	0
$257$	−3875.81	−0.940725	−0.470363	−	0.882473i	$-0.655877\pi$
$257$	−3875.81	−0.940725	−0.470363	+	0.882473i	$0.655877\pi$
$258$	0	0
$259$	−899.693	−0.215846
$260$	0	0
$261$	−2237.57	−0.530659
$262$	0	0
$263$	4066.20	0.953355	0.476678	−	0.879078i	$-0.341841\pi$
$263$	4066.20	0.953355	0.476678	+	0.879078i	$0.341841\pi$
$264$	0	0
$265$	7635.31	1.76994
$266$	0	0
$267$	471.175	0.107998
$268$	0	0
$269$	2140.26	0.485107	0.242554	−	0.970138i	$-0.422015\pi$
$269$	2140.26	0.485107	0.242554	+	0.970138i	$0.422015\pi$
$270$	0	0
$271$	−2080.27	−0.466301	−0.233151	−	0.972441i	$-0.574904\pi$
$271$	−2080.27	−0.466301	−0.233151	+	0.972441i	$0.574904\pi$
$272$	0	0
$273$	−32.9553	−0.00730602
$274$	0	0
$275$	5120.21	1.12276
$276$	0	0
$277$	−18.1765	−0.00394268	−0.00197134	−	0.999998i	$-0.500627\pi$
$277$	−18.1765	−0.00394268	−0.00197134	+	0.999998i	$0.500627\pi$
$278$	0	0
$279$	1192.96	0.255988
$280$	0	0
$281$	−1577.23	−0.334838	−0.167419	−	0.985886i	$-0.553543\pi$
$281$	−1577.23	−0.334838	−0.167419	+	0.985886i	$0.553543\pi$
$282$	0	0
$283$	94.2151	0.0197898	0.00989488	−	0.999951i	$-0.496850\pi$
$283$	94.2151	0.0197898	0.00989488	+	0.999951i	$0.496850\pi$
$284$	0	0
$285$	49.4117	0.0102698
$286$	0	0
$287$	−287.000	−0.0590281
$288$	0	0
$289$	4473.77	0.910599
$290$	0	0
$291$	336.589	0.0678049
$292$	0	0
$293$	−899.532	−0.179356	−0.0896779	−	0.995971i	$-0.528584\pi$
$293$	−899.532	−0.179356	−0.0896779	+	0.995971i	$0.528584\pi$
$294$	0	0
$295$	6818.10	1.34564
$296$	0	0
$297$	−447.158	−0.0873628
$298$	0	0
$299$	1988.78	0.384662
$300$	0	0
$301$	−2689.92	−0.515097
$302$	0	0
$303$	340.595	0.0645764
$304$	0	0
$305$	−10944.1	−2.05462
$306$	0	0
$307$	9652.68	1.79449	0.897244	−	0.441536i	$-0.145566\pi$
$307$	9652.68	1.79449	0.897244	+	0.441536i	$0.145566\pi$
$308$	0	0
$309$	523.637	0.0964034
$310$	0	0
$311$	−4839.66	−0.882418	−0.441209	−	0.897404i	$-0.645450\pi$
$311$	−4839.66	−0.882418	−0.441209	+	0.897404i	$0.645450\pi$
$312$	0	0
$313$	2379.35	0.429677	0.214838	−	0.976650i	$-0.431077\pi$
$313$	2379.35	0.429677	0.214838	+	0.976650i	$0.431077\pi$
$314$	0	0
$315$	3292.78	0.588975
$316$	0	0
$317$	−5320.51	−0.942679	−0.471340	−	0.881952i	$-0.656229\pi$
$317$	−5320.51	−0.942679	−0.471340	+	0.881952i	$0.656229\pi$
$318$	0	0
$319$	2358.83	0.414009
$320$	0	0
$321$	198.439	0.0345041
$322$	0	0
$323$	937.049	0.161420
$324$	0	0
$325$	−2906.42	−0.496060
$326$	0	0
$327$	−374.438	−0.0633225
$328$	0	0
$329$	3766.13	0.631105
$330$	0	0
$331$	6667.28	1.10715	0.553576	−	0.832799i	$-0.313263\pi$
$331$	6667.28	1.10715	0.553576	+	0.832799i	$0.313263\pi$
$332$	0	0
$333$	3459.26	0.569269
$334$	0	0
$335$	−11341.4	−1.84969
$336$	0	0
$337$	−1963.26	−0.317345	−0.158673	−	0.987331i	$-0.550721\pi$
$337$	−1963.26	−0.317345	−0.158673	+	0.987331i	$0.550721\pi$
$338$	0	0
$339$	−229.111	−0.0367069
$340$	0	0
$341$	−1257.61	−0.199716
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	630.861	0.0984476
$346$	0	0
$347$	−2425.34	−0.375213	−0.187607	−	0.982244i	$-0.560073\pi$
$347$	−2425.34	−0.375213	−0.187607	+	0.982244i	$0.560073\pi$
$348$	0	0
$349$	10783.4	1.65393	0.826966	−	0.562251i	$-0.190065\pi$
$349$	10783.4	1.65393	0.826966	+	0.562251i	$0.190065\pi$
$350$	0	0
$351$	253.824	0.0385986
$352$	0	0
$353$	4394.66	0.662618	0.331309	−	0.943522i	$-0.392510\pi$
$353$	4394.66	0.662618	0.331309	+	0.943522i	$0.392510\pi$
$354$	0	0
$355$	16176.8	2.41853
$356$	0	0
$357$	−198.246	−0.0293902
$358$	0	0
$359$	−226.372	−0.0332798	−0.0166399	−	0.999862i	$-0.505297\pi$
$359$	−226.372	−0.0332798	−0.0166399	+	0.999862i	$0.505297\pi$
$360$	0	0
$361$	−6765.46	−0.986362
$362$	0	0
$363$	−153.747	−0.0222304
$364$	0	0
$365$	−7667.21	−1.09951
$366$	0	0
$367$	8193.42	1.16538	0.582688	−	0.812696i	$-0.302001\pi$
$367$	8193.42	1.16538	0.582688	+	0.812696i	$0.302001\pi$
$368$	0	0
$369$	1103.50	0.155680
$370$	0	0
$371$	−3058.07	−0.427944
$372$	0	0
$373$	8448.68	1.17280	0.586402	−	0.810020i	$-0.300544\pi$
$373$	8448.68	1.17280	0.586402	+	0.810020i	$0.300544\pi$
$374$	0	0
$375$	−283.338	−0.0390174
$376$	0	0
$377$	−1338.96	−0.182917
$378$	0	0
$379$	−4200.66	−0.569323	−0.284662	−	0.958628i	$-0.591881\pi$
$379$	−4200.66	−0.569323	−0.284662	+	0.958628i	$0.591881\pi$
$380$	0	0
$381$	320.034	0.0430338
$382$	0	0
$383$	−2495.16	−0.332889	−0.166445	−	0.986051i	$-0.553229\pi$
$383$	−2495.16	−0.332889	−0.166445	+	0.986051i	$0.553229\pi$
$384$	0	0
$385$	−3471.22	−0.459506
$386$	0	0
$387$	10342.6	1.35851
$388$	0	0
$389$	7854.34	1.02373	0.511865	−	0.859066i	$-0.328955\pi$
$389$	7854.34	1.02373	0.511865	+	0.859066i	$0.328955\pi$
$390$	0	0
$391$	11963.7	1.54739
$392$	0	0
$393$	−791.897	−0.101644
$394$	0	0
$395$	7569.46	0.964205
$396$	0	0
$397$	−9190.86	−1.16190	−0.580952	−	0.813938i	$-0.697320\pi$
$397$	−9190.86	−1.16190	−0.580952	+	0.813938i	$0.697320\pi$
$398$	0	0
$399$	−19.7902	−0.00248308
$400$	0	0
$401$	−9624.39	−1.19855	−0.599276	−	0.800543i	$-0.704545\pi$
$401$	−9624.39	−1.19855	−0.599276	+	0.800543i	$0.704545\pi$
$402$	0	0
$403$	713.866	0.0882387
$404$	0	0
$405$	−12620.2	−1.54840
$406$	0	0
$407$	−3646.73	−0.444132
$408$	0	0
$409$	−5243.74	−0.633951	−0.316976	−	0.948434i	$-0.602667\pi$
$409$	−5243.74	−0.633951	−0.316976	+	0.948434i	$0.602667\pi$
$410$	0	0
$411$	406.239	0.0487549
$412$	0	0
$413$	−2730.77	−0.325356
$414$	0	0
$415$	966.130	0.114278
$416$	0	0
$417$	207.759	0.0243981
$418$	0	0
$419$	8647.05	1.00820	0.504100	−	0.863645i	$-0.331824\pi$
$419$	8647.05	1.00820	0.504100	+	0.863645i	$0.331824\pi$
$420$	0	0
$421$	12999.9	1.50493	0.752466	−	0.658632i	$-0.228864\pi$
$421$	12999.9	1.50493	0.752466	+	0.658632i	$0.228864\pi$
$422$	0	0
$423$	−14480.5	−1.66446
$424$	0	0
$425$	−17483.9	−1.99552
$426$	0	0
$427$	4383.31	0.496776
$428$	0	0
$429$	−133.578	−0.0150331
$430$	0	0
$431$	−7411.69	−0.828326	−0.414163	−	0.910203i	$-0.635926\pi$
$431$	−7411.69	−0.828326	−0.414163	+	0.910203i	$0.635926\pi$
$432$	0	0
$433$	−11346.2	−1.25927	−0.629636	−	0.776890i	$-0.716796\pi$
$433$	−11346.2	−1.25927	−0.629636	+	0.776890i	$0.716796\pi$
$434$	0	0
$435$	−424.732	−0.0468145
$436$	0	0
$437$	1194.30	0.130734
$438$	0	0
$439$	−13419.3	−1.45893	−0.729464	−	0.684019i	$-0.760230\pi$
$439$	−13419.3	−1.45893	−0.729464	+	0.684019i	$0.760230\pi$
$440$	0	0
$441$	−1318.81	−0.142405
$442$	0	0
$443$	−10803.8	−1.15870	−0.579351	−	0.815078i	$-0.696694\pi$
$443$	−10803.8	−1.15870	−0.579351	+	0.815078i	$0.696694\pi$
$444$	0	0
$445$	−28171.5	−3.00103
$446$	0	0
$447$	930.809	0.0984916
$448$	0	0
$449$	−777.751	−0.0817469	−0.0408734	−	0.999164i	$-0.513014\pi$
$449$	−777.751	−0.0817469	−0.0408734	+	0.999164i	$0.513014\pi$
$450$	0	0
$451$	−1163.30	−0.121458
$452$	0	0
$453$	761.128	0.0789424
$454$	0	0
$455$	1970.40	0.203019
$456$	0	0
$457$	−3625.99	−0.371152	−0.185576	−	0.982630i	$-0.559415\pi$
$457$	−3625.99	−0.371152	−0.185576	+	0.982630i	$0.559415\pi$
$458$	0	0
$459$	1526.91	0.155272
$460$	0	0
$461$	−3852.21	−0.389187	−0.194594	−	0.980884i	$-0.562339\pi$
$461$	−3852.21	−0.389187	−0.194594	+	0.980884i	$0.562339\pi$
$462$	0	0
$463$	2784.80	0.279526	0.139763	−	0.990185i	$-0.455366\pi$
$463$	2784.80	0.279526	0.139763	+	0.990185i	$0.455366\pi$
$464$	0	0
$465$	226.446	0.0225832
$466$	0	0
$467$	7408.13	0.734063	0.367032	−	0.930208i	$-0.380374\pi$
$467$	7408.13	0.734063	0.367032	+	0.930208i	$0.380374\pi$
$468$	0	0
$469$	4542.41	0.447226
$470$	0	0
$471$	32.8131	0.00321008
$472$	0	0
$473$	−10903.0	−1.05988
$474$	0	0
$475$	−1745.36	−0.168595
$476$	0	0
$477$	11758.1	1.12865
$478$	0	0
$479$	4477.72	0.427124	0.213562	−	0.976930i	$-0.431493\pi$
$479$	4477.72	0.427124	0.213562	+	0.976930i	$0.431493\pi$
$480$	0	0
$481$	2070.02	0.196226
$482$	0	0
$483$	−252.671	−0.0238031
$484$	0	0
$485$	−20124.7	−1.88415
$486$	0	0
$487$	−1411.88	−0.131372	−0.0656862	−	0.997840i	$-0.520924\pi$
$487$	−1411.88	−0.131372	−0.0656862	+	0.997840i	$0.520924\pi$
$488$	0	0
$489$	892.569	0.0825427
$490$	0	0
$491$	−14647.9	−1.34634	−0.673168	−	0.739490i	$-0.735067\pi$
$491$	−14647.9	−1.34634	−0.673168	+	0.739490i	$0.735067\pi$
$492$	0	0
$493$	−8054.66	−0.735829
$494$	0	0
$495$	13346.6	1.21189
$496$	0	0
$497$	−6479.09	−0.584762
$498$	0	0
$499$	−4499.54	−0.403662	−0.201831	−	0.979420i	$-0.564689\pi$
$499$	−4499.54	−0.403662	−0.201831	+	0.979420i	$0.564689\pi$
$500$	0	0
$501$	−361.678	−0.0322527
$502$	0	0
$503$	−10117.4	−0.896841	−0.448421	−	0.893823i	$-0.648013\pi$
$503$	−10117.4	−0.896841	−0.448421	+	0.893823i	$0.648013\pi$
$504$	0	0
$505$	−20364.2	−1.79444
$506$	0	0
$507$	−566.389	−0.0496138
$508$	0	0
$509$	8635.87	0.752020	0.376010	−	0.926616i	$-0.377296\pi$
$509$	8635.87	0.752020	0.376010	+	0.926616i	$0.377296\pi$
$510$	0	0
$511$	3070.85	0.265844
$512$	0	0
$513$	152.426	0.0131184
$514$	0	0
$515$	−31308.2	−2.67885
$516$	0	0
$517$	15265.3	1.29858
$518$	0	0
$519$	332.285	0.0281034
$520$	0	0
$521$	−19012.7	−1.59878	−0.799389	−	0.600814i	$-0.794843\pi$
$521$	−19012.7	−1.59878	−0.799389	+	0.600814i	$0.794843\pi$
$522$	0	0
$523$	−15582.7	−1.30284	−0.651418	−	0.758719i	$-0.725825\pi$
$523$	−15582.7	−1.30284	−0.651418	+	0.758719i	$0.725825\pi$
$524$	0	0
$525$	369.256	0.0306965
$526$	0	0
$527$	4294.34	0.354961
$528$	0	0
$529$	3081.12	0.253236
$530$	0	0
$531$	10499.6	0.858088
$532$	0	0
$533$	660.332	0.0536625
$534$	0	0
$535$	−11864.7	−0.958796
$536$	0	0
$537$	−605.386	−0.0486487
$538$	0	0
$539$	1390.28	0.111101
$540$	0	0
$541$	12098.1	0.961437	0.480718	−	0.876875i	$-0.340376\pi$
$541$	12098.1	0.961437	0.480718	+	0.876875i	$0.340376\pi$
$542$	0	0
$543$	−923.917	−0.0730186
$544$	0	0
$545$	22387.6	1.75960
$546$	0	0
$547$	−17861.7	−1.39618	−0.698090	−	0.716010i	$-0.745966\pi$
$547$	−17861.7	−1.39618	−0.698090	+	0.716010i	$0.745966\pi$
$548$	0	0
$549$	−16853.6	−1.31019
$550$	0	0
$551$	−804.068	−0.0621678
$552$	0	0
$553$	−3031.70	−0.233130
$554$	0	0
$555$	656.632	0.0502207
$556$	0	0
$557$	108.840	0.00827955	0.00413977	−	0.999991i	$-0.498682\pi$
$557$	108.840	0.00827955	0.00413977	+	0.999991i	$0.498682\pi$
$558$	0	0
$559$	6188.98	0.468275
$560$	0	0
$561$	−803.551	−0.0604741
$562$	0	0
$563$	−5231.04	−0.391584	−0.195792	−	0.980645i	$-0.562728\pi$
$563$	−5231.04	−0.391584	−0.195792	+	0.980645i	$0.562728\pi$
$564$	0	0
$565$	13698.6	1.02001
$566$	0	0
$567$	5054.60	0.374380
$568$	0	0
$569$	−14822.2	−1.09205	−0.546027	−	0.837768i	$-0.683860\pi$
$569$	−14822.2	−1.09205	−0.546027	+	0.837768i	$0.683860\pi$
$570$	0	0
$571$	18386.5	1.34755	0.673773	−	0.738938i	$-0.264672\pi$
$571$	18386.5	1.34755	0.673773	+	0.738938i	$0.264672\pi$
$572$	0	0
$573$	357.718	0.0260801
$574$	0	0
$575$	−22283.8	−1.61617
$576$	0	0
$577$	−21812.5	−1.57377	−0.786885	−	0.617100i	$-0.788307\pi$
$577$	−21812.5	−1.57377	−0.786885	+	0.617100i	$0.788307\pi$
$578$	0	0
$579$	−188.498	−0.0135298
$580$	0	0
$581$	−386.952	−0.0276307
$582$	0	0
$583$	−12395.3	−0.880549
$584$	0	0
$585$	−7576.04	−0.535437
$586$	0	0
$587$	23651.7	1.66305	0.831524	−	0.555489i	$-0.187469\pi$
$587$	23651.7	1.66305	0.831524	+	0.555489i	$0.187469\pi$
$588$	0	0
$589$	428.689	0.0299895
$590$	0	0
$591$	−618.079	−0.0430192
$592$	0	0
$593$	2531.01	0.175272	0.0876359	−	0.996153i	$-0.472069\pi$
$593$	2531.01	0.175272	0.0876359	+	0.996153i	$0.472069\pi$
$594$	0	0
$595$	11853.1	0.816691
$596$	0	0
$597$	563.441	0.0386267
$598$	0	0
$599$	20858.7	1.42281	0.711406	−	0.702781i	$-0.248059\pi$
$599$	20858.7	1.42281	0.711406	+	0.702781i	$0.248059\pi$
$600$	0	0
$601$	−20373.3	−1.38277	−0.691384	−	0.722487i	$-0.742999\pi$
$601$	−20373.3	−1.38277	−0.691384	+	0.722487i	$0.742999\pi$
$602$	0	0
$603$	−17465.3	−1.17951
$604$	0	0
$605$	9192.54	0.617735
$606$	0	0
$607$	17966.3	1.20136	0.600682	−	0.799488i	$-0.294896\pi$
$607$	17966.3	1.20136	0.600682	+	0.799488i	$0.294896\pi$
$608$	0	0
$609$	170.112	0.0113190
$610$	0	0
$611$	−8665.15	−0.573739
$612$	0	0
$613$	19179.7	1.26372	0.631861	−	0.775082i	$-0.282292\pi$
$613$	19179.7	1.26372	0.631861	+	0.775082i	$0.282292\pi$
$614$	0	0
$615$	209.464	0.0137340
$616$	0	0
$617$	13626.6	0.889119	0.444560	−	0.895749i	$-0.353360\pi$
$617$	13626.6	0.889119	0.444560	+	0.895749i	$0.353360\pi$
$618$	0	0
$619$	18410.0	1.19541	0.597707	−	0.801714i	$-0.296078\pi$
$619$	18410.0	1.19541	0.597707	+	0.801714i	$0.296078\pi$
$620$	0	0
$621$	1946.09	0.125755
$622$	0	0
$623$	11283.2	0.725604
$624$	0	0
$625$	−5616.69	−0.359468
$626$	0	0
$627$	−80.2157	−0.00510926
$628$	0	0
$629$	12452.4	0.789366
$630$	0	0
$631$	−7383.91	−0.465846	−0.232923	−	0.972495i	$-0.574829\pi$
$631$	−7383.91	−0.465846	−0.232923	+	0.972495i	$0.574829\pi$
$632$	0	0
$633$	−80.1838	−0.00503478
$634$	0	0
$635$	−19134.9	−1.19582
$636$	0	0
$637$	−789.177	−0.0490868
$638$	0	0
$639$	24911.7	1.54224
$640$	0	0
$641$	−20042.1	−1.23497	−0.617486	−	0.786582i	$-0.711849\pi$
$641$	−20042.1	−1.23497	−0.617486	+	0.786582i	$0.711849\pi$
$642$	0	0
$643$	−17547.9	−1.07624	−0.538121	−	0.842868i	$-0.680866\pi$
$643$	−17547.9	−1.07624	−0.538121	+	0.842868i	$0.680866\pi$
$644$	0	0
$645$	1963.21	0.119847
$646$	0	0
$647$	−10466.6	−0.635991	−0.317995	−	0.948092i	$-0.603010\pi$
$647$	−10466.6	−0.635991	−0.317995	+	0.948092i	$0.603010\pi$
$648$	0	0
$649$	−11068.6	−0.669462
$650$	0	0
$651$	−90.6953	−0.00546026
$652$	0	0
$653$	25566.0	1.53212	0.766059	−	0.642770i	$-0.222215\pi$
$653$	25566.0	1.53212	0.766059	+	0.642770i	$0.222215\pi$
$654$	0	0
$655$	47347.5	2.82446
$656$	0	0
$657$	−11807.2	−0.701131
$658$	0	0
$659$	4174.04	0.246734	0.123367	−	0.992361i	$-0.460631\pi$
$659$	4174.04	0.246734	0.123367	+	0.992361i	$0.460631\pi$
$660$	0	0
$661$	−388.996	−0.0228898	−0.0114449	−	0.999935i	$-0.503643\pi$
$661$	−388.996	−0.0228898	−0.0114449	+	0.999935i	$0.503643\pi$
$662$	0	0
$663$	456.126	0.0267187
$664$	0	0
$665$	1183.26	0.0689996
$666$	0	0
$667$	−10265.9	−0.595948
$668$	0	0
$669$	−669.958	−0.0387176
$670$	0	0
$671$	17766.9	1.02218
$672$	0	0
$673$	3874.95	0.221944	0.110972	−	0.993824i	$-0.464604\pi$
$673$	3874.95	0.221944	0.110972	+	0.993824i	$0.464604\pi$
$674$	0	0
$675$	−2844.04	−0.162173
$676$	0	0
$677$	−27222.6	−1.54542	−0.772709	−	0.634760i	$-0.781099\pi$
$677$	−27222.6	−1.54542	−0.772709	+	0.634760i	$0.781099\pi$
$678$	0	0
$679$	8060.28	0.455560
$680$	0	0
$681$	1639.71	0.0922670
$682$	0	0
$683$	35266.9	1.97577	0.987883	−	0.155198i	$-0.0496014\pi$
$683$	35266.9	1.97577	0.987883	+	0.155198i	$0.0496014\pi$
$684$	0	0
$685$	−24289.0	−1.35480
$686$	0	0
$687$	1574.59	0.0874444
$688$	0	0
$689$	7036.03	0.389044
$690$	0	0
$691$	14368.1	0.791010	0.395505	−	0.918464i	$-0.370570\pi$
$691$	14368.1	0.791010	0.395505	+	0.918464i	$0.370570\pi$
$692$	0	0
$693$	−5345.55	−0.293017
$694$	0	0
$695$	−12421.9	−0.677972
$696$	0	0
$697$	3972.30	0.215870
$698$	0	0
$699$	320.719	0.0173544
$700$	0	0
$701$	23636.1	1.27350	0.636750	−	0.771070i	$-0.280278\pi$
$701$	23636.1	1.27350	0.636750	+	0.771070i	$0.280278\pi$
$702$	0	0
$703$	1243.08	0.0666910
$704$	0	0
$705$	−2748.67	−0.146838
$706$	0	0
$707$	8156.19	0.433869
$708$	0	0
$709$	−12755.9	−0.675681	−0.337841	−	0.941203i	$-0.609697\pi$
$709$	−12755.9	−0.675681	−0.337841	+	0.941203i	$0.609697\pi$
$710$	0	0
$711$	11656.7	0.614852
$712$	0	0
$713$	5473.27	0.287483
$714$	0	0
$715$	7986.61	0.417737
$716$	0	0
$717$	−1330.62	−0.0693068
$718$	0	0
$719$	−14060.4	−0.729299	−0.364649	−	0.931145i	$-0.618811\pi$
$719$	−14060.4	−0.729299	−0.364649	+	0.931145i	$0.618811\pi$
$720$	0	0
$721$	12539.5	0.647704
$722$	0	0
$723$	1085.24	0.0558236
$724$	0	0
$725$	15002.7	0.768534
$726$	0	0
$727$	−711.908	−0.0363180	−0.0181590	−	0.999835i	$-0.505781\pi$
$727$	−711.908	−0.0363180	−0.0181590	+	0.999835i	$0.505781\pi$
$728$	0	0
$729$	−19310.2	−0.981062
$730$	0	0
$731$	37230.5	1.88375
$732$	0	0
$733$	38720.0	1.95110	0.975549	−	0.219784i	$-0.0705353\pi$
$733$	38720.0	1.95110	0.975549	+	0.219784i	$0.0705353\pi$
$734$	0	0
$735$	−250.335	−0.0125629
$736$	0	0
$737$	18411.8	0.920226
$738$	0	0
$739$	28977.6	1.44243	0.721217	−	0.692709i	$-0.243583\pi$
$739$	28977.6	1.44243	0.721217	+	0.692709i	$0.243583\pi$
$740$	0	0
$741$	45.5335	0.00225737
$742$	0	0
$743$	−14734.3	−0.727523	−0.363762	−	0.931492i	$-0.618508\pi$
$743$	−14734.3	−0.727523	−0.363762	+	0.931492i	$0.618508\pi$
$744$	0	0
$745$	−55653.1	−2.73687
$746$	0	0
$747$	1487.80	0.0728727
$748$	0	0
$749$	4752.01	0.231822
$750$	0	0
$751$	−12104.0	−0.588126	−0.294063	−	0.955786i	$-0.595008\pi$
$751$	−12104.0	−0.588126	−0.294063	+	0.955786i	$0.595008\pi$
$752$	0	0
$753$	−2296.90	−0.111160
$754$	0	0
$755$	−45507.9	−2.19364
$756$	0	0
$757$	4317.29	0.207285	0.103642	−	0.994615i	$-0.466950\pi$
$757$	4317.29	0.207285	0.103642	+	0.994615i	$0.466950\pi$
$758$	0	0
$759$	−1024.15	−0.0489780
$760$	0	0
$761$	22799.9	1.08607	0.543034	−	0.839711i	$-0.317276\pi$
$761$	22799.9	1.08607	0.543034	+	0.839711i	$0.317276\pi$
$762$	0	0
$763$	−8966.63	−0.425444
$764$	0	0
$765$	−45574.6	−2.15392
$766$	0	0
$767$	6282.96	0.295782
$768$	0	0
$769$	9902.19	0.464346	0.232173	−	0.972674i	$-0.425416\pi$
$769$	9902.19	0.464346	0.232173	+	0.972674i	$0.425416\pi$
$770$	0	0
$771$	−1132.95	−0.0529212
$772$	0	0
$773$	−25926.1	−1.20634	−0.603168	−	0.797614i	$-0.706095\pi$
$773$	−25926.1	−1.20634	−0.603168	+	0.797614i	$0.706095\pi$
$774$	0	0
$775$	−7998.70	−0.370738
$776$	0	0
$777$	−262.992	−0.0121426
$778$	0	0
$779$	396.541	0.0182382
$780$	0	0
$781$	−26261.7	−1.20322
$782$	0	0
$783$	−1310.22	−0.0598000
$784$	0	0
$785$	−1961.90	−0.0892014
$786$	0	0
$787$	−7394.83	−0.334939	−0.167470	−	0.985877i	$-0.553560\pi$
$787$	−7394.83	−0.334939	−0.167470	+	0.985877i	$0.553560\pi$
$788$	0	0
$789$	1188.60	0.0536317
$790$	0	0
$791$	−5486.51	−0.246622
$792$	0	0
$793$	−10085.2	−0.451620
$794$	0	0
$795$	2231.90	0.0995690
$796$	0	0
$797$	14933.8	0.663718	0.331859	−	0.943329i	$-0.392324\pi$
$797$	14933.8	0.663718	0.331859	+	0.943329i	$0.392324\pi$
$798$	0	0
$799$	−52126.2	−2.30800
$800$	0	0
$801$	−43383.1	−1.91369
$802$	0	0
$803$	12447.1	0.547008
$804$	0	0
$805$	15107.2	0.661439
$806$	0	0
$807$	625.626	0.0272901
$808$	0	0
$809$	−12322.6	−0.535525	−0.267763	−	0.963485i	$-0.586284\pi$
$809$	−12322.6	−0.535525	−0.267763	+	0.963485i	$0.586284\pi$
$810$	0	0
$811$	−67.8918	−0.00293959	−0.00146979	−	0.999999i	$-0.500468\pi$
$811$	−67.8918	−0.00293959	−0.00146979	+	0.999999i	$0.500468\pi$
$812$	0	0
$813$	−608.092	−0.0262321
$814$	0	0
$815$	−53366.7	−2.29369
$816$	0	0
$817$	3716.59	0.159152
$818$	0	0
$819$	3034.33	0.129461
$820$	0	0
$821$	31199.5	1.32627	0.663136	−	0.748499i	$-0.269225\pi$
$821$	31199.5	1.32627	0.663136	+	0.748499i	$0.269225\pi$
$822$	0	0
$823$	−11139.7	−0.471818	−0.235909	−	0.971775i	$-0.575807\pi$
$823$	−11139.7	−0.471818	−0.235909	+	0.971775i	$0.575807\pi$
$824$	0	0
$825$	1496.71	0.0631619
$826$	0	0
$827$	−18098.2	−0.760987	−0.380493	−	0.924784i	$-0.624246\pi$
$827$	−18098.2	−0.760987	−0.380493	+	0.924784i	$0.624246\pi$
$828$	0	0
$829$	7418.90	0.310819	0.155410	−	0.987850i	$-0.450330\pi$
$829$	7418.90	0.310819	0.155410	+	0.987850i	$0.450330\pi$
$830$	0	0
$831$	−5.31324	−0.000221798 0
$832$	0	0
$833$	−4747.38	−0.197463
$834$	0	0
$835$	21624.7	0.896233
$836$	0	0
$837$	698.543	0.0288473
$838$	0	0
$839$	17626.7	0.725315	0.362658	−	0.931922i	$-0.381869\pi$
$839$	17626.7	0.725315	0.362658	+	0.931922i	$0.381869\pi$
$840$	0	0
$841$	−17477.4	−0.716610
$842$	0	0
$843$	−461.045	−0.0188366
$844$	0	0
$845$	33864.4	1.37866
$846$	0	0
$847$	−3681.77	−0.149359
$848$	0	0
$849$	27.5403	0.00111329
$850$	0	0
$851$	15871.0	0.639308
$852$	0	0
$853$	26268.0	1.05439	0.527197	−	0.849743i	$-0.323243\pi$
$853$	26268.0	1.05439	0.527197	+	0.849743i	$0.323243\pi$
$854$	0	0
$855$	−4549.55	−0.181978
$856$	0	0
$857$	1757.44	0.0700503	0.0350252	−	0.999386i	$-0.488849\pi$
$857$	1757.44	0.0700503	0.0350252	+	0.999386i	$0.488849\pi$
$858$	0	0
$859$	8831.43	0.350785	0.175393	−	0.984499i	$-0.443880\pi$
$859$	8831.43	0.350785	0.175393	+	0.984499i	$0.443880\pi$
$860$	0	0
$861$	−83.8939	−0.00332067
$862$	0	0
$863$	3432.14	0.135378	0.0676892	−	0.997706i	$-0.478437\pi$
$863$	3432.14	0.135378	0.0676892	+	0.997706i	$0.478437\pi$
$864$	0	0
$865$	−19867.3	−0.780935
$866$	0	0
$867$	1307.74	0.0512264
$868$	0	0
$869$	−12288.4	−0.479695
$870$	0	0
$871$	−10451.2	−0.406574
$872$	0	0
$873$	−30991.2	−1.20148
$874$	0	0
$875$	−6785.08	−0.262146
$876$	0	0
$877$	−158.685	−0.00610993	−0.00305496	−	0.999995i	$-0.500972\pi$
$877$	−158.685	−0.00610993	−0.00305496	+	0.999995i	$0.500972\pi$
$878$	0	0
$879$	−262.945	−0.0100898
$880$	0	0
$881$	41079.5	1.57095	0.785473	−	0.618896i	$-0.212420\pi$
$881$	41079.5	1.57095	0.785473	+	0.618896i	$0.212420\pi$
$882$	0	0
$883$	30256.7	1.15314	0.576568	−	0.817049i	$-0.304392\pi$
$883$	30256.7	1.15314	0.576568	+	0.817049i	$0.304392\pi$
$884$	0	0
$885$	1993.02	0.0757002
$886$	0	0
$887$	24071.9	0.911222	0.455611	−	0.890179i	$-0.349421\pi$
$887$	24071.9	0.911222	0.455611	+	0.890179i	$0.349421\pi$
$888$	0	0
$889$	7663.84	0.289130
$890$	0	0
$891$	20487.8	0.770334
$892$	0	0
$893$	−5203.57	−0.194995
$894$	0	0
$895$	36196.1	1.35184
$896$	0	0
$897$	581.346	0.0216394
$898$	0	0
$899$	−3684.91	−0.136706
$900$	0	0
$901$	42326.0	1.56502
$902$	0	0
$903$	−786.298	−0.0289772
$904$	0	0
$905$	55241.0	2.02903
$906$	0	0
$907$	−10918.9	−0.399732	−0.199866	−	0.979823i	$-0.564051\pi$
$907$	−10918.9	−0.399732	−0.199866	+	0.979823i	$0.564051\pi$
$908$	0	0
$909$	−31360.0	−1.14428
$910$	0	0
$911$	33104.5	1.20395	0.601977	−	0.798513i	$-0.294380\pi$
$911$	33104.5	1.20395	0.601977	+	0.798513i	$0.294380\pi$
$912$	0	0
$913$	−1568.43	−0.0568538
$914$	0	0
$915$	−3199.12	−0.115584
$916$	0	0
$917$	−18963.5	−0.682911
$918$	0	0
$919$	18888.7	0.677999	0.339000	−	0.940787i	$-0.389911\pi$
$919$	18888.7	0.677999	0.339000	+	0.940787i	$0.389911\pi$
$920$	0	0
$921$	2821.61	0.100950
$922$	0	0
$923$	14907.1	0.531608
$924$	0	0
$925$	−23194.1	−0.824451
$926$	0	0
$927$	−48213.5	−1.70824
$928$	0	0
$929$	44487.9	1.57115	0.785575	−	0.618766i	$-0.212367\pi$
$929$	44487.9	1.57115	0.785575	+	0.618766i	$0.212367\pi$
$930$	0	0
$931$	−473.915	−0.0166831
$932$	0	0
$933$	−1414.70	−0.0496410
$934$	0	0
$935$	48044.3	1.68045
$936$	0	0
$937$	18494.1	0.644800	0.322400	−	0.946604i	$-0.395510\pi$
$937$	18494.1	0.644800	0.322400	+	0.946604i	$0.395510\pi$
$938$	0	0
$939$	695.516	0.0241718
$940$	0	0
$941$	13084.3	0.453280	0.226640	−	0.973979i	$-0.427226\pi$
$941$	13084.3	0.453280	0.226640	+	0.973979i	$0.427226\pi$
$942$	0	0
$943$	5062.81	0.174833
$944$	0	0
$945$	1928.10	0.0663716
$946$	0	0
$947$	8087.01	0.277500	0.138750	−	0.990327i	$-0.455692\pi$
$947$	8087.01	0.277500	0.138750	+	0.990327i	$0.455692\pi$
$948$	0	0
$949$	−7065.43	−0.241679
$950$	0	0
$951$	−1555.25	−0.0530311
$952$	0	0
$953$	−37291.6	−1.26757	−0.633785	−	0.773509i	$-0.718500\pi$
$953$	−37291.6	−1.26757	−0.633785	+	0.773509i	$0.718500\pi$
$954$	0	0
$955$	−21388.0	−0.724711
$956$	0	0
$957$	689.516	0.0232904
$958$	0	0
$959$	9728.16	0.327569
$960$	0	0
$961$	−27826.4	−0.934054
$962$	0	0
$963$	−18271.2	−0.611403
$964$	0	0
$965$	11270.3	0.375963
$966$	0	0
$967$	43860.8	1.45860	0.729301	−	0.684193i	$-0.239846\pi$
$967$	43860.8	1.45860	0.729301	+	0.684193i	$0.239846\pi$
$968$	0	0
$969$	273.912	0.00908082
$970$	0	0
$971$	−30622.0	−1.01206	−0.506029	−	0.862517i	$-0.668887\pi$
$971$	−30622.0	−1.01206	−0.506029	+	0.862517i	$0.668887\pi$
$972$	0	0
$973$	4975.19	0.163923
$974$	0	0
$975$	−849.586	−0.0279062
$976$	0	0
$977$	−40426.3	−1.32380	−0.661900	−	0.749592i	$-0.730250\pi$
$977$	−40426.3	−1.32380	−0.661900	+	0.749592i	$0.730250\pi$
$978$	0	0
$979$	45734.1	1.49302
$980$	0	0
$981$	34476.1	1.12206
$982$	0	0
$983$	25390.0	0.823819	0.411910	−	0.911225i	$-0.364862\pi$
$983$	25390.0	0.823819	0.411910	+	0.911225i	$0.364862\pi$
$984$	0	0
$985$	36954.9	1.19541
$986$	0	0
$987$	1100.89	0.0355033
$988$	0	0
$989$	47451.4	1.52565
$990$	0	0
$991$	−17971.9	−0.576081	−0.288041	−	0.957618i	$-0.593004\pi$
$991$	−17971.9	−0.576081	−0.288041	+	0.957618i	$0.593004\pi$
$992$	0	0
$993$	1948.93	0.0622836
$994$	0	0
$995$	−33688.2	−1.07335
$996$	0	0
$997$	31754.9	1.00872	0.504358	−	0.863495i	$-0.331729\pi$
$997$	31754.9	1.00872	0.504358	+	0.863495i	$0.331729\pi$
$998$	0	0
$999$	2025.59	0.0641509

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.c.1.8	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.c.1.8	✓	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)

\(f(q)\)	\(=\)	\(q+0.292313 q^{3} -17.4774 q^{5} +7.00000 q^{7} -26.9146 q^{9} +O(q^{10})\)	\(q+0.292313 q^{3} -17.4774 q^{5} +7.00000 q^{7} -26.9146 q^{9} +28.3731 q^{11} -16.1056 q^{13} -5.10888 q^{15} -96.8854 q^{17} -9.67173 q^{19} +2.04619 q^{21} -123.483 q^{23} +180.460 q^{25} -15.7599 q^{27} +83.1360 q^{29} -44.3239 q^{31} +8.29383 q^{33} -122.342 q^{35} -128.528 q^{37} -4.70789 q^{39} -41.0000 q^{41} -384.274 q^{43} +470.397 q^{45} +538.019 q^{47} +49.0000 q^{49} -28.3209 q^{51} -436.867 q^{53} -495.888 q^{55} -2.82717 q^{57} -390.109 q^{59} +626.187 q^{61} -188.402 q^{63} +281.485 q^{65} +648.916 q^{67} -36.0958 q^{69} -925.584 q^{71} +438.692 q^{73} +52.7508 q^{75} +198.612 q^{77} -433.100 q^{79} +722.086 q^{81} -55.2788 q^{83} +1693.31 q^{85} +24.3017 q^{87} +1611.88 q^{89} -112.740 q^{91} -12.9565 q^{93} +169.037 q^{95} +1151.47 q^{97} -763.649 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9}+O(q^{10}) \) 15 * q + 10 * q^3 + 6 * q^5 + 105 * q^7 + 165 * q^9	\( 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9} + 62 q^{11} + 148 q^{13} + 152 q^{15} + 132 q^{17} + 260 q^{19} + 70 q^{21} - 26 q^{23} + 453 q^{25} + 454 q^{27} + 12 q^{29} + 144 q^{31} + 336 q^{33} + 42 q^{35} + 274 q^{37} + 218 q^{39} - 615 q^{41} + 986 q^{43} + 648 q^{45} + 44 q^{47} + 735 q^{49} + 202 q^{51} - 366 q^{53} + 928 q^{55} - 294 q^{57} - 8 q^{59} + 1396 q^{61} + 1155 q^{63} + 156 q^{65} - 24 q^{67} + 892 q^{69} + 1464 q^{71} + 1174 q^{73} + 318 q^{75} + 434 q^{77} + 1590 q^{79} + 1959 q^{81} + 1970 q^{83} + 2348 q^{85} + 2544 q^{87} + 1972 q^{89} + 1036 q^{91} + 4034 q^{93} + 1070 q^{95} + 954 q^{97} + 3398 q^{99}+O(q^{100}) \) 15 * q + 10 * q^3 + 6 * q^5 + 105 * q^7 + 165 * q^9 + 62 * q^11 + 148 * q^13 + 152 * q^15 + 132 * q^17 + 260 * q^19 + 70 * q^21 - 26 * q^23 + 453 * q^25 + 454 * q^27 + 12 * q^29 + 144 * q^31 + 336 * q^33 + 42 * q^35 + 274 * q^37 + 218 * q^39 - 615 * q^41 + 986 * q^43 + 648 * q^45 + 44 * q^47 + 735 * q^49 + 202 * q^51 - 366 * q^53 + 928 * q^55 - 294 * q^57 - 8 * q^59 + 1396 * q^61 + 1155 * q^63 + 156 * q^65 - 24 * q^67 + 892 * q^69 + 1464 * q^71 + 1174 * q^73 + 318 * q^75 + 434 * q^77 + 1590 * q^79 + 1959 * q^81 + 1970 * q^83 + 2348 * q^85 + 2544 * q^87 + 1972 * q^89 + 1036 * q^91 + 4034 * q^93 + 1070 * q^95 + 954 * q^97 + 3398 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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