LMFDB - Embedded newform 240.12.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,12,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("240.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	240.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:	$184.402363334$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1801}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 450 $ x^2 - x - 450
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$21.7191$ of defining polynomial
Character	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+243.000 q^{3} -3125.00 q^{5} -79941.2 q^{7} +59049.0 q^{9} -805067. q^{11} -1.19767e6 q^{13} -759375. q^{15} +2.63319e6 q^{17} -1.16061e7 q^{19} -1.94257e7 q^{21} -1.84216e7 q^{23} +9.76562e6 q^{25} +1.43489e7 q^{27} -1.90527e8 q^{29} -1.01127e8 q^{31} -1.95631e8 q^{33} +2.49816e8 q^{35} +8.06675e7 q^{37} -2.91034e8 q^{39} +2.26316e8 q^{41} -1.67149e9 q^{43} -1.84528e8 q^{45} -8.58507e8 q^{47} +4.41327e9 q^{49} +6.39864e8 q^{51} -3.52750e9 q^{53} +2.51583e9 q^{55} -2.82028e9 q^{57} -4.35760e9 q^{59} -1.65393e9 q^{61} -4.72045e9 q^{63} +3.74272e9 q^{65} -7.58610e9 q^{67} -4.47645e9 q^{69} +2.75809e10 q^{71} +3.22368e10 q^{73} +2.37305e9 q^{75} +6.43580e10 q^{77} +2.43149e9 q^{79} +3.48678e9 q^{81} -1.20729e10 q^{83} -8.22871e9 q^{85} -4.62981e10 q^{87} +4.44073e9 q^{89} +9.57433e10 q^{91} -2.45737e10 q^{93} +3.62690e10 q^{95} -2.04453e10 q^{97} -4.75384e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 486 q^{3} - 6250 q^{5} - 7784 q^{7} + 118098 q^{9} - 295568 q^{11} + 657492 q^{13} - 1518750 q^{15} + 8579948 q^{17} - 17627976 q^{19} - 1891512 q^{21} + 29841072 q^{23} + 19531250 q^{25} + 28697814 q^{27}+ \cdots - 17452994832 q^{99}+O(q^{100}) $ 2 * q + 486 * q^3 - 6250 * q^5 - 7784 * q^7 + 118098 * q^9 - 295568 * q^11 + 657492 * q^13 - 1518750 * q^15 + 8579948 * q^17 - 17627976 * q^19 - 1891512 * q^21 + 29841072 * q^23 + 19531250 * q^25 + 28697814 * q^27 - 201881948 * q^29 + 71057008 * q^31 - 71823024 * q^33 + 24325000 * q^35 + 705858484 * q^37 + 159770556 * q^39 - 327655148 * q^41 - 3192552120 * q^43 - 369056250 * q^45 - 2053064720 * q^47 + 7642614770 * q^49 + 2084927364 * q^51 - 2304299452 * q^53 + 923650000 * q^55 - 4283598168 * q^57 + 1478770576 * q^59 - 8264891460 * q^61 - 459637416 * q^63 - 2054662500 * q^65 - 24212177528 * q^67 + 7251380496 * q^69 + 20218888256 * q^71 + 25879583268 * q^73 + 4746093750 * q^75 + 101122045248 * q^77 - 22324995440 * q^79 + 6973568802 * q^81 - 48014508984 * q^83 - 26812337500 * q^85 - 49057313364 * q^87 + 79209683076 * q^89 + 229606712048 * q^91 + 17266852944 * q^93 + 55087425000 * q^95 - 37075227452 * q^97 - 17452994832 * q^99

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
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	0	0
$5$	−3125.00	−0.447214
$5$	−3125.00	−0.447214
$6$	0	0
$7$	−79941.2	−1.79776	−0.898880	−	0.438195i	$-0.855618\pi$
$7$	−79941.2	−1.79776	−0.898880	+	0.438195i	$0.855618\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	−805067.	−1.50720	−0.753602	−	0.657331i	$-0.771685\pi$
$11$	−805067.	−1.50720	−0.753602	+	0.657331i	$0.771685\pi$
$12$	0	0
$13$	−1.19767e6	−0.894641	−0.447321	−	0.894374i	$-0.647622\pi$
$13$	−1.19767e6	−0.894641	−0.447321	+	0.894374i	$0.647622\pi$
$14$	0	0
$15$	−759375.	−0.258199
$16$	0	0
$17$	2.63319e6	0.449793	0.224896	−	0.974383i	$-0.427796\pi$
$17$	2.63319e6	0.449793	0.224896	+	0.974383i	$0.427796\pi$
$18$	0	0
$19$	−1.16061e7	−1.07533	−0.537664	−	0.843159i	$-0.680693\pi$
$19$	−1.16061e7	−1.07533	−0.537664	+	0.843159i	$0.680693\pi$
$20$	0	0
$21$	−1.94257e7	−1.03794
$22$	0	0
$23$	−1.84216e7	−0.596794	−0.298397	−	0.954442i	$-0.596452\pi$
$23$	−1.84216e7	−0.596794	−0.298397	+	0.954442i	$0.596452\pi$
$24$	0	0
$25$	9.76562e6	0.200000
$26$	0	0
$27$	1.43489e7	0.192450
$28$	0	0
$29$	−1.90527e8	−1.72491	−0.862457	−	0.506130i	$-0.831076\pi$
$29$	−1.90527e8	−1.72491	−0.862457	+	0.506130i	$0.831076\pi$
$30$	0	0
$31$	−1.01127e8	−0.634419	−0.317209	−	0.948356i	$-0.602746\pi$
$31$	−1.01127e8	−0.634419	−0.317209	+	0.948356i	$0.602746\pi$
$32$	0	0
$33$	−1.95631e8	−0.870185
$34$	0	0
$35$	2.49816e8	0.803983
$36$	0	0
$37$	8.06675e7	0.191245	0.0956223	−	0.995418i	$-0.469516\pi$
$37$	8.06675e7	0.191245	0.0956223	+	0.995418i	$0.469516\pi$
$38$	0	0
$39$	−2.91034e8	−0.516522
$40$	0	0
$41$	2.26316e8	0.305073	0.152537	−	0.988298i	$-0.451256\pi$
$41$	2.26316e8	0.305073	0.152537	+	0.988298i	$0.451256\pi$
$42$	0	0
$43$	−1.67149e9	−1.73391	−0.866956	−	0.498385i	$-0.833927\pi$
$43$	−1.67149e9	−1.73391	−0.866956	+	0.498385i	$0.833927\pi$
$44$	0	0
$45$	−1.84528e8	−0.149071
$46$	0	0
$47$	−8.58507e8	−0.546016	−0.273008	−	0.962012i	$-0.588019\pi$
$47$	−8.58507e8	−0.546016	−0.273008	+	0.962012i	$0.588019\pi$
$48$	0	0
$49$	4.41327e9	2.23194
$50$	0	0
$51$	6.39864e8	0.259688
$52$	0	0
$53$	−3.52750e9	−1.15864	−0.579322	−	0.815099i	$-0.696683\pi$
$53$	−3.52750e9	−1.15864	−0.579322	+	0.815099i	$0.696683\pi$
$54$	0	0
$55$	2.51583e9	0.674042
$56$	0	0
$57$	−2.82028e9	−0.620841
$58$	0	0
$59$	−4.35760e9	−0.793527	−0.396763	−	0.917921i	$-0.629867\pi$
$59$	−4.35760e9	−0.793527	−0.396763	+	0.917921i	$0.629867\pi$
$60$	0	0
$61$	−1.65393e9	−0.250727	−0.125364	−	0.992111i	$-0.540010\pi$
$61$	−1.65393e9	−0.250727	−0.125364	+	0.992111i	$0.540010\pi$
$62$	0	0
$63$	−4.72045e9	−0.599253
$64$	0	0
$65$	3.74272e9	0.400096
$66$	0	0
$67$	−7.58610e9	−0.686447	−0.343224	−	0.939254i	$-0.611519\pi$
$67$	−7.58610e9	−0.686447	−0.343224	+	0.939254i	$0.611519\pi$
$68$	0	0
$69$	−4.47645e9	−0.344559
$70$	0	0
$71$	2.75809e10	1.81421	0.907104	−	0.420905i	$-0.138288\pi$
$71$	2.75809e10	1.81421	0.907104	+	0.420905i	$0.138288\pi$
$72$	0	0
$73$	3.22368e10	1.82002	0.910011	−	0.414584i	$-0.136073\pi$
$73$	3.22368e10	1.82002	0.910011	+	0.414584i	$0.136073\pi$
$74$	0	0
$75$	2.37305e9	0.115470
$76$	0	0
$77$	6.43580e10	2.70959
$78$	0	0
$79$	2.43149e9	0.0889046	0.0444523	−	0.999012i	$-0.485846\pi$
$79$	2.43149e9	0.0889046	0.0444523	+	0.999012i	$0.485846\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	−1.20729e10	−0.336421	−0.168210	−	0.985751i	$-0.553799\pi$
$83$	−1.20729e10	−0.336421	−0.168210	+	0.985751i	$0.553799\pi$
$84$	0	0
$85$	−8.22871e9	−0.201154
$86$	0	0
$87$	−4.62981e10	−0.995880
$88$	0	0
$89$	4.44073e9	0.0842964	0.0421482	−	0.999111i	$-0.486580\pi$
$89$	4.44073e9	0.0842964	0.0421482	+	0.999111i	$0.486580\pi$
$90$	0	0
$91$	9.57433e10	1.60835
$92$	0	0
$93$	−2.45737e10	−0.366282
$94$	0	0
$95$	3.62690e10	0.480901
$96$	0	0
$97$	−2.04453e10	−0.241740	−0.120870	−	0.992668i	$-0.538568\pi$
$97$	−2.04453e10	−0.241740	−0.120870	+	0.992668i	$0.538568\pi$
$98$	0	0
$99$	−4.75384e10	−0.502401
$100$	0	0
$101$	−1.55947e11	−1.47642	−0.738209	−	0.674572i	$-0.764328\pi$
$101$	−1.55947e11	−1.47642	−0.738209	+	0.674572i	$0.764328\pi$
$102$	0	0
$103$	5.10325e10	0.433752	0.216876	−	0.976199i	$-0.430413\pi$
$103$	5.10325e10	0.433752	0.216876	+	0.976199i	$0.430413\pi$
$104$	0	0
$105$	6.07054e10	0.464180
$106$	0	0
$107$	−1.06665e10	−0.0735207	−0.0367603	−	0.999324i	$-0.511704\pi$
$107$	−1.06665e10	−0.0735207	−0.0367603	+	0.999324i	$0.511704\pi$
$108$	0	0
$109$	−3.12513e10	−0.194546	−0.0972729	−	0.995258i	$-0.531012\pi$
$109$	−3.12513e10	−0.194546	−0.0972729	+	0.995258i	$0.531012\pi$
$110$	0	0
$111$	1.96022e10	0.110415
$112$	0	0
$113$	−3.96346e10	−0.202369	−0.101184	−	0.994868i	$-0.532263\pi$
$113$	−3.96346e10	−0.202369	−0.101184	+	0.994868i	$0.532263\pi$
$114$	0	0
$115$	5.75676e10	0.266895
$116$	0	0
$117$	−7.07213e10	−0.298214
$118$	0	0
$119$	−2.10500e11	−0.808620
$120$	0	0
$121$	3.62821e11	1.27166
$122$	0	0
$123$	5.49948e10	0.176134
$124$	0	0
$125$	−3.05176e10	−0.0894427
$126$	0	0
$127$	−2.63460e10	−0.0707611	−0.0353806	−	0.999374i	$-0.511264\pi$
$127$	−2.63460e10	−0.0707611	−0.0353806	+	0.999374i	$0.511264\pi$
$128$	0	0
$129$	−4.06172e11	−1.00107
$130$	0	0
$131$	2.19917e11	0.498044	0.249022	−	0.968498i	$-0.419891\pi$
$131$	2.19917e11	0.498044	0.249022	+	0.968498i	$0.419891\pi$
$132$	0	0
$133$	9.27805e11	1.93318
$134$	0	0
$135$	−4.48403e10	−0.0860663
$136$	0	0
$137$	−5.54041e11	−0.980795	−0.490398	−	0.871499i	$-0.663148\pi$
$137$	−5.54041e11	−0.980795	−0.490398	+	0.871499i	$0.663148\pi$
$138$	0	0
$139$	6.17540e11	1.00945	0.504723	−	0.863281i	$-0.331594\pi$
$139$	6.17540e11	1.00945	0.504723	+	0.863281i	$0.331594\pi$
$140$	0	0
$141$	−2.08617e11	−0.315243
$142$	0	0
$143$	9.64205e11	1.34841
$144$	0	0
$145$	5.95397e11	0.771405
$146$	0	0
$147$	1.07243e12	1.28861
$148$	0	0
$149$	−6.68742e11	−0.745992	−0.372996	−	0.927833i	$-0.621670\pi$
$149$	−6.68742e11	−0.745992	−0.372996	+	0.927833i	$0.621670\pi$
$150$	0	0
$151$	−1.38243e12	−1.43308	−0.716541	−	0.697545i	$-0.754276\pi$
$151$	−1.38243e12	−1.43308	−0.716541	+	0.697545i	$0.754276\pi$
$152$	0	0
$153$	1.55487e11	0.149931
$154$	0	0
$155$	3.16020e11	0.283721
$156$	0	0
$157$	−7.93661e11	−0.664029	−0.332014	−	0.943274i	$-0.607728\pi$
$157$	−7.93661e11	−0.664029	−0.332014	+	0.943274i	$0.607728\pi$
$158$	0	0
$159$	−8.57184e11	−0.668944
$160$	0	0
$161$	1.47265e12	1.07289
$162$	0	0
$163$	4.67399e11	0.318168	0.159084	−	0.987265i	$-0.449146\pi$
$163$	4.67399e11	0.318168	0.159084	+	0.987265i	$0.449146\pi$
$164$	0	0
$165$	6.11348e11	0.389158
$166$	0	0
$167$	−2.87482e12	−1.71266	−0.856328	−	0.516432i	$-0.827260\pi$
$167$	−2.87482e12	−1.71266	−0.856328	+	0.516432i	$0.827260\pi$
$168$	0	0
$169$	−3.57745e11	−0.199617
$170$	0	0
$171$	−6.85328e11	−0.358443
$172$	0	0
$173$	1.53085e12	0.751068	0.375534	−	0.926809i	$-0.377459\pi$
$173$	1.53085e12	0.751068	0.375534	+	0.926809i	$0.377459\pi$
$174$	0	0
$175$	−7.80676e11	−0.359552
$176$	0	0
$177$	−1.05890e12	−0.458143
$178$	0	0
$179$	1.91337e12	0.778228	0.389114	−	0.921190i	$-0.372781\pi$
$179$	1.91337e12	0.778228	0.389114	+	0.921190i	$0.372781\pi$
$180$	0	0
$181$	−1.70819e12	−0.653587	−0.326794	−	0.945096i	$-0.605968\pi$
$181$	−1.70819e12	−0.653587	−0.326794	+	0.945096i	$0.605968\pi$
$182$	0	0
$183$	−4.01904e11	−0.144758
$184$	0	0
$185$	−2.52086e11	−0.0855272
$186$	0	0
$187$	−2.11989e12	−0.677930
$188$	0	0
$189$	−1.14707e12	−0.345979
$190$	0	0
$191$	−3.16020e12	−0.899562	−0.449781	−	0.893139i	$-0.648498\pi$
$191$	−3.16020e12	−0.899562	−0.449781	+	0.893139i	$0.648498\pi$
$192$	0	0
$193$	2.77296e12	0.745382	0.372691	−	0.927956i	$-0.378435\pi$
$193$	2.77296e12	0.745382	0.372691	+	0.927956i	$0.378435\pi$
$194$	0	0
$195$	9.09481e11	0.230995
$196$	0	0
$197$	3.86504e12	0.928089	0.464044	−	0.885812i	$-0.346398\pi$
$197$	3.86504e12	0.928089	0.464044	+	0.885812i	$0.346398\pi$
$198$	0	0
$199$	1.01839e12	0.231325	0.115663	−	0.993289i	$-0.463101\pi$
$199$	1.01839e12	0.231325	0.115663	+	0.993289i	$0.463101\pi$
$200$	0	0
$201$	−1.84342e12	−0.396321
$202$	0	0
$203$	1.52310e13	3.10098
$204$	0	0
$205$	−7.07237e11	−0.136433
$206$	0	0
$207$	−1.08778e12	−0.198931
$208$	0	0
$209$	9.34367e12	1.62074
$210$	0	0
$211$	−2.05668e12	−0.338543	−0.169271	−	0.985569i	$-0.554141\pi$
$211$	−2.05668e12	−0.338543	−0.169271	+	0.985569i	$0.554141\pi$
$212$	0	0
$213$	6.70216e12	1.04743
$214$	0	0
$215$	5.22340e12	0.775429
$216$	0	0
$217$	8.08418e12	1.14053
$218$	0	0
$219$	7.83355e12	1.05079
$220$	0	0
$221$	−3.15369e12	−0.402403
$222$	0	0
$223$	−9.99986e12	−1.21428	−0.607138	−	0.794597i	$-0.707682\pi$
$223$	−9.99986e12	−1.21428	−0.607138	+	0.794597i	$0.707682\pi$
$224$	0	0
$225$	5.76650e11	0.0666667
$226$	0	0
$227$	6.77123e12	0.745634	0.372817	−	0.927905i	$-0.378392\pi$
$227$	6.77123e12	0.745634	0.372817	+	0.927905i	$0.378392\pi$
$228$	0	0
$229$	1.01933e13	1.06959	0.534796	−	0.844981i	$-0.320389\pi$
$229$	1.01933e13	1.06959	0.534796	+	0.844981i	$0.320389\pi$
$230$	0	0
$231$	1.56390e13	1.56438
$232$	0	0
$233$	6.74446e12	0.643412	0.321706	−	0.946840i	$-0.395744\pi$
$233$	6.74446e12	0.643412	0.321706	+	0.946840i	$0.395744\pi$
$234$	0	0
$235$	2.68283e12	0.244186
$236$	0	0
$237$	5.90853e11	0.0513291
$238$	0	0
$239$	1.76501e13	1.46406	0.732030	−	0.681272i	$-0.238573\pi$
$239$	1.76501e13	1.46406	0.732030	+	0.681272i	$0.238573\pi$
$240$	0	0
$241$	−1.37662e13	−1.09074	−0.545369	−	0.838196i	$-0.683611\pi$
$241$	−1.37662e13	−1.09074	−0.545369	+	0.838196i	$0.683611\pi$
$242$	0	0
$243$	8.47289e11	0.0641500
$244$	0	0
$245$	−1.37915e13	−0.998154
$246$	0	0
$247$	1.39003e13	0.962033
$248$	0	0
$249$	−2.93372e12	−0.194233
$250$	0	0
$251$	−3.02252e13	−1.91498	−0.957490	−	0.288467i	$-0.906855\pi$
$251$	−3.02252e13	−1.91498	−0.957490	+	0.288467i	$0.906855\pi$
$252$	0	0
$253$	1.48306e13	0.899491
$254$	0	0
$255$	−1.99958e12	−0.116136
$256$	0	0
$257$	6.88263e12	0.382933	0.191466	−	0.981499i	$-0.438676\pi$
$257$	6.88263e12	0.382933	0.191466	+	0.981499i	$0.438676\pi$
$258$	0	0
$259$	−6.44866e12	−0.343812
$260$	0	0
$261$	−1.12504e13	−0.574971
$262$	0	0
$263$	3.29948e13	1.61692	0.808462	−	0.588549i	$-0.200301\pi$
$263$	3.29948e13	1.61692	0.808462	+	0.588549i	$0.200301\pi$
$264$	0	0
$265$	1.10235e13	0.518162
$266$	0	0
$267$	1.07910e12	0.0486686
$268$	0	0
$269$	−2.97478e13	−1.28771	−0.643855	−	0.765148i	$-0.722666\pi$
$269$	−2.97478e13	−1.28771	−0.643855	+	0.765148i	$0.722666\pi$
$270$	0	0
$271$	4.30410e13	1.78876	0.894378	−	0.447312i	$-0.147619\pi$
$271$	4.30410e13	1.78876	0.894378	+	0.447312i	$0.147619\pi$
$272$	0	0
$273$	2.32656e13	0.928582
$274$	0	0
$275$	−7.86198e12	−0.301441
$276$	0	0
$277$	1.46706e13	0.540518	0.270259	−	0.962788i	$-0.412891\pi$
$277$	1.46706e13	0.540518	0.270259	+	0.962788i	$0.412891\pi$
$278$	0	0
$279$	−5.97142e12	−0.211473
$280$	0	0
$281$	4.18378e13	1.42457	0.712286	−	0.701890i	$-0.247660\pi$
$281$	4.18378e13	1.42457	0.712286	+	0.701890i	$0.247660\pi$
$282$	0	0
$283$	−2.34242e13	−0.767078	−0.383539	−	0.923525i	$-0.625295\pi$
$283$	−2.34242e13	−0.767078	−0.383539	+	0.923525i	$0.625295\pi$
$284$	0	0
$285$	8.81337e12	0.277648
$286$	0	0
$287$	−1.80920e13	−0.548448
$288$	0	0
$289$	−2.73382e13	−0.797686
$290$	0	0
$291$	−4.96820e12	−0.139568
$292$	0	0
$293$	−7.31258e12	−0.197833	−0.0989166	−	0.995096i	$-0.531538\pi$
$293$	−7.31258e12	−0.197833	−0.0989166	+	0.995096i	$0.531538\pi$
$294$	0	0
$295$	1.36175e13	0.354876
$296$	0	0
$297$	−1.15518e13	−0.290062
$298$	0	0
$299$	2.20630e13	0.533917
$300$	0	0
$301$	1.33621e14	3.11716
$302$	0	0
$303$	−3.78951e13	−0.852410
$304$	0	0
$305$	5.16852e12	0.112129
$306$	0	0
$307$	5.37239e13	1.12436	0.562181	−	0.827014i	$-0.309962\pi$
$307$	5.37239e13	1.12436	0.562181	+	0.827014i	$0.309962\pi$
$308$	0	0
$309$	1.24009e13	0.250427
$310$	0	0
$311$	−5.41406e13	−1.05521	−0.527607	−	0.849489i	$-0.676911\pi$
$311$	−5.41406e13	−1.05521	−0.527607	+	0.849489i	$0.676911\pi$
$312$	0	0
$313$	−4.29721e12	−0.0808524	−0.0404262	−	0.999183i	$-0.512872\pi$
$313$	−4.29721e12	−0.0808524	−0.0404262	+	0.999183i	$0.512872\pi$
$314$	0	0
$315$	1.47514e13	0.267994
$316$	0	0
$317$	−2.81928e13	−0.494666	−0.247333	−	0.968931i	$-0.579554\pi$
$317$	−2.81928e13	−0.494666	−0.247333	+	0.968931i	$0.579554\pi$
$318$	0	0
$319$	1.53387e14	2.59980
$320$	0	0
$321$	−2.59195e12	−0.0424472
$322$	0	0
$323$	−3.05610e13	−0.483675
$324$	0	0
$325$	−1.16960e13	−0.178928
$326$	0	0
$327$	−7.59406e12	−0.112321
$328$	0	0
$329$	6.86301e13	0.981606
$330$	0	0
$331$	9.95469e13	1.37713	0.688563	−	0.725176i	$-0.258242\pi$
$331$	9.95469e13	1.37713	0.688563	+	0.725176i	$0.258242\pi$
$332$	0	0
$333$	4.76334e12	0.0637482
$334$	0	0
$335$	2.37066e13	0.306989
$336$	0	0
$337$	−9.05175e13	−1.13440	−0.567202	−	0.823579i	$-0.691974\pi$
$337$	−9.05175e13	−1.13440	−0.567202	+	0.823579i	$0.691974\pi$
$338$	0	0
$339$	−9.63121e12	−0.116838
$340$	0	0
$341$	8.14136e13	0.956198
$342$	0	0
$343$	−1.94733e14	−2.21473
$344$	0	0
$345$	1.39889e13	0.154092
$346$	0	0
$347$	6.36892e13	0.679601	0.339801	−	0.940498i	$-0.389640\pi$
$347$	6.36892e13	0.679601	0.339801	+	0.940498i	$0.389640\pi$
$348$	0	0
$349$	7.08598e13	0.732588	0.366294	−	0.930499i	$-0.380626\pi$
$349$	7.08598e13	0.732588	0.366294	+	0.930499i	$0.380626\pi$
$350$	0	0
$351$	−1.71853e13	−0.172174
$352$	0	0
$353$	6.62893e13	0.643699	0.321849	−	0.946791i	$-0.395696\pi$
$353$	6.62893e13	0.643699	0.321849	+	0.946791i	$0.395696\pi$
$354$	0	0
$355$	−8.61903e13	−0.811339
$356$	0	0
$357$	−5.11516e13	−0.466857
$358$	0	0
$359$	−6.80217e13	−0.602044	−0.301022	−	0.953617i	$-0.597328\pi$
$359$	−6.80217e13	−0.602044	−0.301022	+	0.953617i	$0.597328\pi$
$360$	0	0
$361$	1.82109e13	0.156330
$362$	0	0
$363$	8.81654e13	0.734196
$364$	0	0
$365$	−1.00740e14	−0.813939
$366$	0	0
$367$	−2.19771e14	−1.72309	−0.861543	−	0.507685i	$-0.830501\pi$
$367$	−2.19771e14	−1.72309	−0.861543	+	0.507685i	$0.830501\pi$
$368$	0	0
$369$	1.33637e13	0.101691
$370$	0	0
$371$	2.81993e14	2.08296
$372$	0	0
$373$	−1.67232e14	−1.19928	−0.599639	−	0.800271i	$-0.704689\pi$
$373$	−1.67232e14	−1.19928	−0.599639	+	0.800271i	$0.704689\pi$
$374$	0	0
$375$	−7.41577e12	−0.0516398
$376$	0	0
$377$	2.28189e14	1.54318
$378$	0	0
$379$	1.57778e14	1.03641	0.518205	−	0.855256i	$-0.326600\pi$
$379$	1.57778e14	1.03641	0.518205	+	0.855256i	$0.326600\pi$
$380$	0	0
$381$	−6.40209e12	−0.0408540
$382$	0	0
$383$	−5.49255e13	−0.340550	−0.170275	−	0.985397i	$-0.554466\pi$
$383$	−5.49255e13	−0.340550	−0.170275	+	0.985397i	$0.554466\pi$
$384$	0	0
$385$	−2.01119e14	−1.21177
$386$	0	0
$387$	−9.86997e13	−0.577971
$388$	0	0
$389$	−2.32940e14	−1.32593	−0.662965	−	0.748650i	$-0.730702\pi$
$389$	−2.32940e14	−1.32593	−0.662965	+	0.748650i	$0.730702\pi$
$390$	0	0
$391$	−4.85076e13	−0.268434
$392$	0	0
$393$	5.34399e13	0.287546
$394$	0	0
$395$	−7.59842e12	−0.0397593
$396$	0	0
$397$	−4.76134e13	−0.242316	−0.121158	−	0.992633i	$-0.538661\pi$
$397$	−4.76134e13	−0.242316	−0.121158	+	0.992633i	$0.538661\pi$
$398$	0	0
$399$	2.25456e14	1.11612
$400$	0	0
$401$	−2.99074e14	−1.44040	−0.720202	−	0.693764i	$-0.755951\pi$
$401$	−2.99074e14	−1.44040	−0.720202	+	0.693764i	$0.755951\pi$
$402$	0	0
$403$	1.21116e14	0.567577
$404$	0	0
$405$	−1.08962e13	−0.0496904
$406$	0	0
$407$	−6.49428e13	−0.288245
$408$	0	0
$409$	−2.55879e14	−1.10550	−0.552748	−	0.833349i	$-0.686421\pi$
$409$	−2.55879e14	−1.10550	−0.552748	+	0.833349i	$0.686421\pi$
$410$	0	0
$411$	−1.34632e14	−0.566262
$412$	0	0
$413$	3.48352e14	1.42657
$414$	0	0
$415$	3.77279e13	0.150452
$416$	0	0
$417$	1.50062e14	0.582804
$418$	0	0
$419$	−2.21825e14	−0.839139	−0.419570	−	0.907723i	$-0.637819\pi$
$419$	−2.21825e14	−0.839139	−0.419570	+	0.907723i	$0.637819\pi$
$420$	0	0
$421$	9.78949e13	0.360752	0.180376	−	0.983598i	$-0.442269\pi$
$421$	9.78949e13	0.360752	0.180376	+	0.983598i	$0.442269\pi$
$422$	0	0
$423$	−5.06940e13	−0.182005
$424$	0	0
$425$	2.57147e13	0.0899586
$426$	0	0
$427$	1.32217e14	0.450748
$428$	0	0
$429$	2.34302e14	0.778503
$430$	0	0
$431$	−1.11415e14	−0.360842	−0.180421	−	0.983589i	$-0.557746\pi$
$431$	−1.11415e14	−0.360842	−0.180421	+	0.983589i	$0.557746\pi$
$432$	0	0
$433$	−3.42045e14	−1.07994	−0.539970	−	0.841684i	$-0.681564\pi$
$433$	−3.42045e14	−1.07994	−0.539970	+	0.841684i	$0.681564\pi$
$434$	0	0
$435$	1.44681e14	0.445371
$436$	0	0
$437$	2.13803e14	0.641750
$438$	0	0
$439$	−1.20947e14	−0.354029	−0.177015	−	0.984208i	$-0.556644\pi$
$439$	−1.20947e14	−0.354029	−0.177015	+	0.984208i	$0.556644\pi$
$440$	0	0
$441$	2.60599e14	0.743980
$442$	0	0
$443$	1.27316e14	0.354539	0.177270	−	0.984162i	$-0.443274\pi$
$443$	1.27316e14	0.354539	0.177270	+	0.984162i	$0.443274\pi$
$444$	0	0
$445$	−1.38773e13	−0.0376985
$446$	0	0
$447$	−1.62504e14	−0.430699
$448$	0	0
$449$	−1.61837e14	−0.418526	−0.209263	−	0.977859i	$-0.567106\pi$
$449$	−1.61837e14	−0.418526	−0.209263	+	0.977859i	$0.567106\pi$
$450$	0	0
$451$	−1.82199e14	−0.459807
$452$	0	0
$453$	−3.35931e14	−0.827390
$454$	0	0
$455$	−2.99198e14	−0.719276
$456$	0	0
$457$	−3.49165e14	−0.819391	−0.409696	−	0.912222i	$-0.634365\pi$
$457$	−3.49165e14	−0.819391	−0.409696	+	0.912222i	$0.634365\pi$
$458$	0	0
$459$	3.77834e13	0.0865627
$460$	0	0
$461$	−4.49619e14	−1.00575	−0.502875	−	0.864359i	$-0.667724\pi$
$461$	−4.49619e14	−1.00575	−0.502875	+	0.864359i	$0.667724\pi$
$462$	0	0
$463$	−4.21098e13	−0.0919787	−0.0459894	−	0.998942i	$-0.514644\pi$
$463$	−4.21098e13	−0.0919787	−0.0459894	+	0.998942i	$0.514644\pi$
$464$	0	0
$465$	7.67930e13	0.163806
$466$	0	0
$467$	7.91358e13	0.164866	0.0824328	−	0.996597i	$-0.473731\pi$
$467$	7.91358e13	0.164866	0.0824328	+	0.996597i	$0.473731\pi$
$468$	0	0
$469$	6.06442e14	1.23407
$470$	0	0
$471$	−1.92860e14	−0.383377
$472$	0	0
$473$	1.34566e15	2.61336
$474$	0	0
$475$	−1.13341e14	−0.215066
$476$	0	0
$477$	−2.08296e14	−0.386215
$478$	0	0
$479$	9.62318e14	1.74371	0.871853	−	0.489768i	$-0.162919\pi$
$479$	9.62318e14	1.74371	0.871853	+	0.489768i	$0.162919\pi$
$480$	0	0
$481$	−9.66132e13	−0.171095
$482$	0	0
$483$	3.57853e14	0.619435
$484$	0	0
$485$	6.38914e13	0.108109
$486$	0	0
$487$	−4.57117e14	−0.756168	−0.378084	−	0.925771i	$-0.623417\pi$
$487$	−4.57117e14	−0.756168	−0.378084	+	0.925771i	$0.623417\pi$
$488$	0	0
$489$	1.13578e14	0.183694
$490$	0	0
$491$	−4.40122e14	−0.696025	−0.348013	−	0.937490i	$-0.613143\pi$
$491$	−4.40122e14	−0.696025	−0.348013	+	0.937490i	$0.613143\pi$
$492$	0	0
$493$	−5.01693e14	−0.775854
$494$	0	0
$495$	1.48557e14	0.224681
$496$	0	0
$497$	−2.20485e15	−3.26151
$498$	0	0
$499$	−6.36085e14	−0.920369	−0.460185	−	0.887823i	$-0.652217\pi$
$499$	−6.36085e14	−0.920369	−0.460185	+	0.887823i	$0.652217\pi$
$500$	0	0
$501$	−6.98581e14	−0.988802
$502$	0	0
$503$	4.81639e12	0.00666957	0.00333478	−	0.999994i	$-0.498939\pi$
$503$	4.81639e12	0.00666957	0.00333478	+	0.999994i	$0.498939\pi$
$504$	0	0
$505$	4.87334e14	0.660274
$506$	0	0
$507$	−8.69320e13	−0.115249
$508$	0	0
$509$	−5.73829e14	−0.744449	−0.372224	−	0.928143i	$-0.621405\pi$
$509$	−5.73829e14	−0.744449	−0.372224	+	0.928143i	$0.621405\pi$
$510$	0	0
$511$	−2.57705e15	−3.27196
$512$	0	0
$513$	−1.66535e14	−0.206947
$514$	0	0
$515$	−1.59476e14	−0.193980
$516$	0	0
$517$	6.91155e14	0.822958
$518$	0	0
$519$	3.71997e14	0.433630
$520$	0	0
$521$	5.14705e14	0.587422	0.293711	−	0.955894i	$-0.405110\pi$
$521$	5.14705e14	0.587422	0.293711	+	0.955894i	$0.405110\pi$
$522$	0	0
$523$	−1.20146e15	−1.34261	−0.671304	−	0.741182i	$-0.734265\pi$
$523$	−1.20146e15	−1.34261	−0.671304	+	0.741182i	$0.734265\pi$
$524$	0	0
$525$	−1.89704e14	−0.207587
$526$	0	0
$527$	−2.66285e14	−0.285357
$528$	0	0
$529$	−6.13454e14	−0.643836
$530$	0	0
$531$	−2.57312e14	−0.264509
$532$	0	0
$533$	−2.71052e14	−0.272931
$534$	0	0
$535$	3.33327e13	0.0328795
$536$	0	0
$537$	4.64948e14	0.449310
$538$	0	0
$539$	−3.55298e15	−3.36399
$540$	0	0
$541$	−1.89107e15	−1.75438	−0.877190	−	0.480143i	$-0.840585\pi$
$541$	−1.89107e15	−1.75438	−0.877190	+	0.480143i	$0.840585\pi$
$542$	0	0
$543$	−4.15090e14	−0.377349
$544$	0	0
$545$	9.76602e13	0.0870035
$546$	0	0
$547$	−1.37400e14	−0.119966	−0.0599830	−	0.998199i	$-0.519105\pi$
$547$	−1.37400e14	−0.119966	−0.0599830	+	0.998199i	$0.519105\pi$
$548$	0	0
$549$	−9.76626e13	−0.0835758
$550$	0	0
$551$	2.21127e15	1.85485
$552$	0	0
$553$	−1.94377e14	−0.159829
$554$	0	0
$555$	−6.12569e13	−0.0493792
$556$	0	0
$557$	−1.18622e15	−0.937483	−0.468742	−	0.883335i	$-0.655292\pi$
$557$	−1.18622e15	−0.937483	−0.468742	+	0.883335i	$0.655292\pi$
$558$	0	0
$559$	2.00189e15	1.55123
$560$	0	0
$561$	−5.15134e14	−0.391403
$562$	0	0
$563$	2.58208e15	1.92386	0.961929	−	0.273298i	$-0.0881146\pi$
$563$	2.58208e15	1.92386	0.961929	+	0.273298i	$0.0881146\pi$
$564$	0	0
$565$	1.23858e14	0.0905020
$566$	0	0
$567$	−2.78738e14	−0.199751
$568$	0	0
$569$	6.86455e13	0.0482497	0.0241249	−	0.999709i	$-0.492320\pi$
$569$	6.86455e13	0.0482497	0.0241249	+	0.999709i	$0.492320\pi$
$570$	0	0
$571$	1.48521e15	1.02397	0.511986	−	0.858994i	$-0.328910\pi$
$571$	1.48521e15	1.02397	0.511986	+	0.858994i	$0.328910\pi$
$572$	0	0
$573$	−7.67929e14	−0.519363
$574$	0	0
$575$	−1.79899e14	−0.119359
$576$	0	0
$577$	−1.11682e14	−0.0726966	−0.0363483	−	0.999339i	$-0.511573\pi$
$577$	−1.11682e14	−0.0726966	−0.0363483	+	0.999339i	$0.511573\pi$
$578$	0	0
$579$	6.73830e14	0.430346
$580$	0	0
$581$	9.65124e14	0.604803
$582$	0	0
$583$	2.83988e15	1.74631
$584$	0	0
$585$	2.21004e14	0.133365
$586$	0	0
$587$	−1.03228e15	−0.611347	−0.305673	−	0.952136i	$-0.598882\pi$
$587$	−1.03228e15	−0.611347	−0.305673	+	0.952136i	$0.598882\pi$
$588$	0	0
$589$	1.17368e15	0.682208
$590$	0	0
$591$	9.39204e14	0.535832
$592$	0	0
$593$	5.05946e14	0.283337	0.141668	−	0.989914i	$-0.454753\pi$
$593$	5.05946e14	0.283337	0.141668	+	0.989914i	$0.454753\pi$
$594$	0	0
$595$	6.57813e14	0.361626
$596$	0	0
$597$	2.47469e14	0.133556
$598$	0	0
$599$	1.86439e14	0.0987846	0.0493923	−	0.998779i	$-0.484272\pi$
$599$	1.86439e14	0.0987846	0.0493923	+	0.998779i	$0.484272\pi$
$600$	0	0
$601$	−6.39018e14	−0.332433	−0.166216	−	0.986089i	$-0.553155\pi$
$601$	−6.39018e14	−0.332433	−0.166216	+	0.986089i	$0.553155\pi$
$602$	0	0
$603$	−4.47952e14	−0.228816
$604$	0	0
$605$	−1.13381e15	−0.568706
$606$	0	0
$607$	−3.07440e15	−1.51434	−0.757169	−	0.653219i	$-0.773418\pi$
$607$	−3.07440e15	−1.51434	−0.757169	+	0.653219i	$0.773418\pi$
$608$	0	0
$609$	3.70112e15	1.79035
$610$	0	0
$611$	1.02821e15	0.488489
$612$	0	0
$613$	1.45823e15	0.680447	0.340223	−	0.940345i	$-0.389497\pi$
$613$	1.45823e15	0.680447	0.340223	+	0.940345i	$0.389497\pi$
$614$	0	0
$615$	−1.71859e14	−0.0787695
$616$	0	0
$617$	2.52038e15	1.13474	0.567372	−	0.823462i	$-0.307960\pi$
$617$	2.52038e15	1.13474	0.567372	+	0.823462i	$0.307960\pi$
$618$	0	0
$619$	−2.76117e15	−1.22122	−0.610612	−	0.791930i	$-0.709076\pi$
$619$	−2.76117e15	−1.22122	−0.610612	+	0.791930i	$0.709076\pi$
$620$	0	0
$621$	−2.64330e14	−0.114853
$622$	0	0
$623$	−3.54997e14	−0.151545
$624$	0	0
$625$	9.53674e13	0.0400000
$626$	0	0
$627$	2.27051e15	0.935734
$628$	0	0
$629$	2.12413e14	0.0860205
$630$	0	0
$631$	5.89126e14	0.234448	0.117224	−	0.993105i	$-0.462600\pi$
$631$	5.89126e14	0.234448	0.117224	+	0.993105i	$0.462600\pi$
$632$	0	0
$633$	−4.99774e14	−0.195458
$634$	0	0
$635$	8.23314e13	0.0316453
$636$	0	0
$637$	−5.28565e15	−1.99679
$638$	0	0
$639$	1.62862e15	0.604736
$640$	0	0
$641$	1.59787e15	0.583208	0.291604	−	0.956539i	$-0.405811\pi$
$641$	1.59787e15	0.583208	0.291604	+	0.956539i	$0.405811\pi$
$642$	0	0
$643$	−4.44445e15	−1.59462	−0.797310	−	0.603569i	$-0.793745\pi$
$643$	−4.44445e15	−1.59462	−0.797310	+	0.603569i	$0.793745\pi$
$644$	0	0
$645$	1.26929e15	0.447694
$646$	0	0
$647$	8.54247e14	0.296217	0.148108	−	0.988971i	$-0.452682\pi$
$647$	8.54247e14	0.296217	0.148108	+	0.988971i	$0.452682\pi$
$648$	0	0
$649$	3.50816e15	1.19601
$650$	0	0
$651$	1.96446e15	0.658486
$652$	0	0
$653$	−2.86428e15	−0.944045	−0.472022	−	0.881587i	$-0.656476\pi$
$653$	−2.86428e15	−0.944045	−0.472022	+	0.881587i	$0.656476\pi$
$654$	0	0
$655$	−6.87242e14	−0.222732
$656$	0	0
$657$	1.90355e15	0.606674
$658$	0	0
$659$	−4.78135e15	−1.49858	−0.749291	−	0.662241i	$-0.769606\pi$
$659$	−4.78135e15	−1.49858	−0.749291	+	0.662241i	$0.769606\pi$
$660$	0	0
$661$	−5.52275e14	−0.170234	−0.0851172	−	0.996371i	$-0.527126\pi$
$661$	−5.52275e14	−0.170234	−0.0851172	+	0.996371i	$0.527126\pi$
$662$	0	0
$663$	−7.66347e14	−0.232328
$664$	0	0
$665$	−2.89939e15	−0.864545
$666$	0	0
$667$	3.50982e15	1.02942
$668$	0	0
$669$	−2.42997e15	−0.701062
$670$	0	0
$671$	1.33152e15	0.377897
$672$	0	0
$673$	−3.31175e15	−0.924645	−0.462323	−	0.886712i	$-0.652984\pi$
$673$	−3.31175e15	−0.924645	−0.462323	+	0.886712i	$0.652984\pi$
$674$	0	0
$675$	1.40126e14	0.0384900
$676$	0	0
$677$	6.69181e15	1.80845	0.904223	−	0.427060i	$-0.140451\pi$
$677$	6.69181e15	1.80845	0.904223	+	0.427060i	$0.140451\pi$
$678$	0	0
$679$	1.63442e15	0.434590
$680$	0	0
$681$	1.64541e15	0.430492
$682$	0	0
$683$	7.20810e15	1.85570	0.927848	−	0.372959i	$-0.121657\pi$
$683$	7.20810e15	1.85570	0.927848	+	0.372959i	$0.121657\pi$
$684$	0	0
$685$	1.73138e15	0.438625
$686$	0	0
$687$	2.47696e15	0.617529
$688$	0	0
$689$	4.22479e15	1.03657
$690$	0	0
$691$	−7.79467e15	−1.88221	−0.941106	−	0.338112i	$-0.890212\pi$
$691$	−7.79467e15	−1.88221	−0.941106	+	0.338112i	$0.890212\pi$
$692$	0	0
$693$	3.80028e15	0.903197
$694$	0	0
$695$	−1.92981e15	−0.451438
$696$	0	0
$697$	5.95932e14	0.137220
$698$	0	0
$699$	1.63890e15	0.371474
$700$	0	0
$701$	6.71290e15	1.49782	0.748912	−	0.662669i	$-0.230577\pi$
$701$	6.71290e15	1.49782	0.748912	+	0.662669i	$0.230577\pi$
$702$	0	0
$703$	−9.36234e14	−0.205651
$704$	0	0
$705$	6.51929e14	0.140981
$706$	0	0
$707$	1.24666e16	2.65424
$708$	0	0
$709$	−4.10214e15	−0.859917	−0.429959	−	0.902849i	$-0.641472\pi$
$709$	−4.10214e15	−0.859917	−0.429959	+	0.902849i	$0.641472\pi$
$710$	0	0
$711$	1.43577e14	0.0296349
$712$	0	0
$713$	1.86291e15	0.378617
$714$	0	0
$715$	−3.01314e15	−0.603026
$716$	0	0
$717$	4.28898e15	0.845276
$718$	0	0
$719$	3.31818e15	0.644008	0.322004	−	0.946738i	$-0.395644\pi$
$719$	3.31818e15	0.644008	0.322004	+	0.946738i	$0.395644\pi$
$720$	0	0
$721$	−4.07960e15	−0.779783
$722$	0	0
$723$	−3.34519e15	−0.629738
$724$	0	0
$725$	−1.86061e15	−0.344983
$726$	0	0
$727$	−3.34273e15	−0.610467	−0.305233	−	0.952278i	$-0.598734\pi$
$727$	−3.34273e15	−0.610467	−0.305233	+	0.952278i	$0.598734\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−4.40134e15	−0.779901
$732$	0	0
$733$	−6.14966e15	−1.07344	−0.536722	−	0.843759i	$-0.680338\pi$
$733$	−6.14966e15	−1.07344	−0.536722	+	0.843759i	$0.680338\pi$
$734$	0	0
$735$	−3.35133e15	−0.576284
$736$	0	0
$737$	6.10732e15	1.03462
$738$	0	0
$739$	4.15701e15	0.693804	0.346902	−	0.937901i	$-0.387234\pi$
$739$	4.15701e15	0.693804	0.346902	+	0.937901i	$0.387234\pi$
$740$	0	0
$741$	3.37776e15	0.555430
$742$	0	0
$743$	4.90364e15	0.794474	0.397237	−	0.917716i	$-0.369969\pi$
$743$	4.90364e15	0.794474	0.397237	+	0.917716i	$0.369969\pi$
$744$	0	0
$745$	2.08982e15	0.333618
$746$	0	0
$747$	−7.12893e14	−0.112140
$748$	0	0
$749$	8.52690e14	0.132173
$750$	0	0
$751$	2.00580e15	0.306385	0.153192	−	0.988196i	$-0.451045\pi$
$751$	2.00580e15	0.306385	0.153192	+	0.988196i	$0.451045\pi$
$752$	0	0
$753$	−7.34473e15	−1.10561
$754$	0	0
$755$	4.32010e15	0.640894
$756$	0	0
$757$	2.85016e15	0.416718	0.208359	−	0.978052i	$-0.433188\pi$
$757$	2.85016e15	0.416718	0.208359	+	0.978052i	$0.433188\pi$
$758$	0	0
$759$	3.60384e15	0.519321
$760$	0	0
$761$	8.65962e15	1.22994	0.614969	−	0.788551i	$-0.289168\pi$
$761$	8.65962e15	1.22994	0.614969	+	0.788551i	$0.289168\pi$
$762$	0	0
$763$	2.49827e15	0.349747
$764$	0	0
$765$	−4.85897e14	−0.0670512
$766$	0	0
$767$	5.21897e15	0.709922
$768$	0	0
$769$	−7.49727e15	−1.00533	−0.502665	−	0.864481i	$-0.667647\pi$
$769$	−7.49727e15	−1.00533	−0.502665	+	0.864481i	$0.667647\pi$
$770$	0	0
$771$	1.67248e15	0.221086
$772$	0	0
$773$	2.07867e15	0.270894	0.135447	−	0.990785i	$-0.456753\pi$
$773$	2.07867e15	0.270894	0.135447	+	0.990785i	$0.456753\pi$
$774$	0	0
$775$	−9.87564e14	−0.126884
$776$	0	0
$777$	−1.56703e15	−0.198500
$778$	0	0
$779$	−2.62664e15	−0.328053
$780$	0	0
$781$	−2.22045e16	−2.73438
$782$	0	0
$783$	−2.73385e15	−0.331960
$784$	0	0
$785$	2.48019e15	0.296963
$786$	0	0
$787$	1.58421e16	1.87047	0.935235	−	0.354026i	$-0.115188\pi$
$787$	1.58421e16	1.87047	0.935235	+	0.354026i	$0.115188\pi$
$788$	0	0
$789$	8.01774e15	0.933531
$790$	0	0
$791$	3.16844e15	0.363810
$792$	0	0
$793$	1.98086e15	0.224311
$794$	0	0
$795$	2.67870e15	0.299161
$796$	0	0
$797$	−4.58469e15	−0.504997	−0.252499	−	0.967597i	$-0.581252\pi$
$797$	−4.58469e15	−0.504997	−0.252499	+	0.967597i	$0.581252\pi$
$798$	0	0
$799$	−2.26061e15	−0.245594
$800$	0	0
$801$	2.62221e14	0.0280988
$802$	0	0
$803$	−2.59528e16	−2.74315
$804$	0	0
$805$	−4.60202e15	−0.479812
$806$	0	0
$807$	−7.22872e15	−0.743459
$808$	0	0
$809$	−1.72813e15	−0.175331	−0.0876656	−	0.996150i	$-0.527941\pi$
$809$	−1.72813e15	−0.175331	−0.0876656	+	0.996150i	$0.527941\pi$
$810$	0	0
$811$	−4.24638e14	−0.0425015	−0.0212507	−	0.999774i	$-0.506765\pi$
$811$	−4.24638e14	−0.0425015	−0.0212507	+	0.999774i	$0.506765\pi$
$812$	0	0
$813$	1.04590e16	1.03274
$814$	0	0
$815$	−1.46062e15	−0.142289
$816$	0	0
$817$	1.93994e16	1.86452
$818$	0	0
$819$	5.65355e15	0.536117
$820$	0	0
$821$	−1.96517e16	−1.83871	−0.919354	−	0.393431i	$-0.871288\pi$
$821$	−1.96517e16	−1.83871	−0.919354	+	0.393431i	$0.871288\pi$
$822$	0	0
$823$	−5.95701e14	−0.0549957	−0.0274979	−	0.999622i	$-0.508754\pi$
$823$	−5.95701e14	−0.0549957	−0.0274979	+	0.999622i	$0.508754\pi$
$824$	0	0
$825$	−1.91046e15	−0.174037
$826$	0	0
$827$	1.40975e16	1.26725	0.633626	−	0.773639i	$-0.281566\pi$
$827$	1.40975e16	1.26725	0.633626	+	0.773639i	$0.281566\pi$
$828$	0	0
$829$	−7.87187e15	−0.698277	−0.349139	−	0.937071i	$-0.613526\pi$
$829$	−7.87187e15	−0.698277	−0.349139	+	0.937071i	$0.613526\pi$
$830$	0	0
$831$	3.56497e15	0.312068
$832$	0	0
$833$	1.16210e16	1.00391
$834$	0	0
$835$	8.98381e15	0.765923
$836$	0	0
$837$	−1.45106e15	−0.122094
$838$	0	0
$839$	−6.12117e15	−0.508327	−0.254164	−	0.967161i	$-0.581800\pi$
$839$	−6.12117e15	−0.508327	−0.254164	+	0.967161i	$0.581800\pi$
$840$	0	0
$841$	2.41000e16	1.97533
$842$	0	0
$843$	1.01666e16	0.822476
$844$	0	0
$845$	1.11795e15	0.0892713
$846$	0	0
$847$	−2.90043e16	−2.28615
$848$	0	0
$849$	−5.69208e15	−0.442873
$850$	0	0
$851$	−1.48603e15	−0.114134
$852$	0	0
$853$	2.83032e15	0.214594	0.107297	−	0.994227i	$-0.465781\pi$
$853$	2.83032e15	0.214594	0.107297	+	0.994227i	$0.465781\pi$
$854$	0	0
$855$	2.14165e15	0.160300
$856$	0	0
$857$	−4.83193e14	−0.0357048	−0.0178524	−	0.999841i	$-0.505683\pi$
$857$	−4.83193e14	−0.0357048	−0.0178524	+	0.999841i	$0.505683\pi$
$858$	0	0
$859$	−1.23886e16	−0.903775	−0.451888	−	0.892075i	$-0.649249\pi$
$859$	−1.23886e16	−0.903775	−0.451888	+	0.892075i	$0.649249\pi$
$860$	0	0
$861$	−4.39635e15	−0.316647
$862$	0	0
$863$	2.26458e16	1.61038	0.805191	−	0.593016i	$-0.202063\pi$
$863$	2.26458e16	1.61038	0.805191	+	0.593016i	$0.202063\pi$
$864$	0	0
$865$	−4.78391e15	−0.335888
$866$	0	0
$867$	−6.64319e15	−0.460544
$868$	0	0
$869$	−1.95751e15	−0.133997
$870$	0	0
$871$	9.08565e15	0.614124
$872$	0	0
$873$	−1.20727e15	−0.0805799
$874$	0	0
$875$	2.43961e15	0.160797
$876$	0	0
$877$	−8.89622e15	−0.579039	−0.289519	−	0.957172i	$-0.593495\pi$
$877$	−8.89622e15	−0.579039	−0.289519	+	0.957172i	$0.593495\pi$
$878$	0	0
$879$	−1.77696e15	−0.114219
$880$	0	0
$881$	−1.82720e16	−1.15989	−0.579946	−	0.814655i	$-0.696927\pi$
$881$	−1.82720e16	−1.15989	−0.579946	+	0.814655i	$0.696927\pi$
$882$	0	0
$883$	−8.75758e15	−0.549035	−0.274518	−	0.961582i	$-0.588518\pi$
$883$	−8.75758e15	−0.549035	−0.274518	+	0.961582i	$0.588518\pi$
$884$	0	0
$885$	3.30905e15	0.204888
$886$	0	0
$887$	−8.52159e15	−0.521124	−0.260562	−	0.965457i	$-0.583908\pi$
$887$	−8.52159e15	−0.521124	−0.260562	+	0.965457i	$0.583908\pi$
$888$	0	0
$889$	2.10613e15	0.127212
$890$	0	0
$891$	−2.80709e15	−0.167467
$892$	0	0
$893$	9.96390e15	0.587146
$894$	0	0
$895$	−5.97928e15	−0.348034
$896$	0	0
$897$	5.36132e15	0.308257
$898$	0	0
$899$	1.92673e16	1.09432
$900$	0	0
$901$	−9.28858e15	−0.521150
$902$	0	0
$903$	3.24699e16	1.79969
$904$	0	0
$905$	5.33809e15	0.292293
$906$	0	0
$907$	−2.10550e16	−1.13898	−0.569490	−	0.821998i	$-0.692859\pi$
$907$	−2.10550e16	−1.13898	−0.569490	+	0.821998i	$0.692859\pi$
$908$	0	0
$909$	−9.20851e15	−0.492139
$910$	0	0
$911$	−3.04479e16	−1.60771	−0.803854	−	0.594827i	$-0.797220\pi$
$911$	−3.04479e16	−1.60771	−0.803854	+	0.594827i	$0.797220\pi$
$912$	0	0
$913$	9.71950e15	0.507055
$914$	0	0
$915$	1.25595e15	0.0647375
$916$	0	0
$917$	−1.75805e16	−0.895363
$918$	0	0
$919$	4.47695e15	0.225293	0.112646	−	0.993635i	$-0.464067\pi$
$919$	4.47695e15	0.225293	0.112646	+	0.993635i	$0.464067\pi$
$920$	0	0
$921$	1.30549e16	0.649151
$922$	0	0
$923$	−3.30328e16	−1.62307
$924$	0	0
$925$	7.87769e14	0.0382489
$926$	0	0
$927$	3.01342e15	0.144584
$928$	0	0
$929$	2.63310e16	1.24848	0.624238	−	0.781234i	$-0.285409\pi$
$929$	2.63310e16	1.24848	0.624238	+	0.781234i	$0.285409\pi$
$930$	0	0
$931$	−5.12208e16	−2.40007
$932$	0	0
$933$	−1.31562e16	−0.609228
$934$	0	0
$935$	6.62466e15	0.303179
$936$	0	0
$937$	1.18259e16	0.534894	0.267447	−	0.963573i	$-0.413820\pi$
$937$	1.18259e16	0.534894	0.267447	+	0.963573i	$0.413820\pi$
$938$	0	0
$939$	−1.04422e15	−0.0466802
$940$	0	0
$941$	6.96003e15	0.307517	0.153758	−	0.988108i	$-0.450862\pi$
$941$	6.96003e15	0.307517	0.153758	+	0.988108i	$0.450862\pi$
$942$	0	0
$943$	−4.16911e15	−0.182066
$944$	0	0
$945$	3.58459e15	0.154727
$946$	0	0
$947$	3.38501e16	1.44423	0.722113	−	0.691775i	$-0.243171\pi$
$947$	3.38501e16	1.44423	0.722113	+	0.691775i	$0.243171\pi$
$948$	0	0
$949$	−3.86091e16	−1.62827
$950$	0	0
$951$	−6.85084e15	−0.285595
$952$	0	0
$953$	4.50062e16	1.85465	0.927323	−	0.374262i	$-0.122104\pi$
$953$	4.50062e16	1.85465	0.927323	+	0.374262i	$0.122104\pi$
$954$	0	0
$955$	9.87563e15	0.402297
$956$	0	0
$957$	3.72730e16	1.50099
$958$	0	0
$959$	4.42907e16	1.76323
$960$	0	0
$961$	−1.51819e16	−0.597513
$962$	0	0
$963$	−6.29844e14	−0.0245069
$964$	0	0
$965$	−8.66551e15	−0.333345
$966$	0	0
$967$	1.48359e16	0.564246	0.282123	−	0.959378i	$-0.408961\pi$
$967$	1.48359e16	0.564246	0.282123	+	0.959378i	$0.408961\pi$
$968$	0	0
$969$	−7.42632e15	−0.279250
$970$	0	0
$971$	−1.14414e16	−0.425378	−0.212689	−	0.977120i	$-0.568222\pi$
$971$	−1.14414e16	−0.425378	−0.212689	+	0.977120i	$0.568222\pi$
$972$	0	0
$973$	−4.93669e16	−1.81474
$974$	0	0
$975$	−2.84213e15	−0.103304
$976$	0	0
$977$	−2.64437e16	−0.950390	−0.475195	−	0.879881i	$-0.657622\pi$
$977$	−2.64437e16	−0.950390	−0.475195	+	0.879881i	$0.657622\pi$
$978$	0	0
$979$	−3.57508e15	−0.127052
$980$	0	0
$981$	−1.84536e15	−0.0648486
$982$	0	0
$983$	−3.63656e16	−1.26371	−0.631854	−	0.775088i	$-0.717706\pi$
$983$	−3.63656e16	−1.26371	−0.631854	+	0.775088i	$0.717706\pi$
$984$	0	0
$985$	−1.20782e16	−0.415054
$986$	0	0
$987$	1.66771e16	0.566731
$988$	0	0
$989$	3.07915e16	1.03479
$990$	0	0
$991$	−5.09067e16	−1.69188	−0.845941	−	0.533276i	$-0.820961\pi$
$991$	−5.09067e16	−1.69188	−0.845941	+	0.533276i	$0.820961\pi$
$992$	0	0
$993$	2.41899e16	0.795084
$994$	0	0
$995$	−3.18247e15	−0.103452
$996$	0	0
$997$	−4.28291e16	−1.37694	−0.688471	−	0.725264i	$-0.741718\pi$
$997$	−4.28291e16	−1.37694	−0.688471	+	0.725264i	$0.741718\pi$
$998$	0	0
$999$	1.15749e15	0.0368051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.12.a.m.1.1		2
4.3	odd	2		15.12.a.c.1.2	✓	2
12.11	even	2		45.12.a.c.1.1		2
20.3	even	4		75.12.b.d.49.2		4
20.7	even	4		75.12.b.d.49.3		4
20.19	odd	2		75.12.a.c.1.1		2
60.23	odd	4		225.12.b.i.199.3		4
60.47	odd	4		225.12.b.i.199.2		4
60.59	even	2		225.12.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.c.1.2	✓	2	4.3	odd	2
45.12.a.c.1.1		2	12.11	even	2
75.12.a.c.1.1		2	20.19	odd	2
75.12.b.d.49.2		4	20.3	even	4
75.12.b.d.49.3		4	20.7	even	4
225.12.a.i.1.2		2	60.59	even	2
225.12.b.i.199.2		4	60.47	odd	4
225.12.b.i.199.3		4	60.23	odd	4
240.12.a.m.1.1		2	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

\(f(q)\)	\(=\)	\(q+243.000 q^{3} -3125.00 q^{5} -79941.2 q^{7} +59049.0 q^{9} -805067. q^{11} -1.19767e6 q^{13} -759375. q^{15} +2.63319e6 q^{17} -1.16061e7 q^{19} -1.94257e7 q^{21} -1.84216e7 q^{23} +9.76562e6 q^{25} +1.43489e7 q^{27} -1.90527e8 q^{29} -1.01127e8 q^{31} -1.95631e8 q^{33} +2.49816e8 q^{35} +8.06675e7 q^{37} -2.91034e8 q^{39} +2.26316e8 q^{41} -1.67149e9 q^{43} -1.84528e8 q^{45} -8.58507e8 q^{47} +4.41327e9 q^{49} +6.39864e8 q^{51} -3.52750e9 q^{53} +2.51583e9 q^{55} -2.82028e9 q^{57} -4.35760e9 q^{59} -1.65393e9 q^{61} -4.72045e9 q^{63} +3.74272e9 q^{65} -7.58610e9 q^{67} -4.47645e9 q^{69} +2.75809e10 q^{71} +3.22368e10 q^{73} +2.37305e9 q^{75} +6.43580e10 q^{77} +2.43149e9 q^{79} +3.48678e9 q^{81} -1.20729e10 q^{83} -8.22871e9 q^{85} -4.62981e10 q^{87} +4.44073e9 q^{89} +9.57433e10 q^{91} -2.45737e10 q^{93} +3.62690e10 q^{95} -2.04453e10 q^{97} -4.75384e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 486 q^{3} - 6250 q^{5} - 7784 q^{7} + 118098 q^{9} - 295568 q^{11} + 657492 q^{13} - 1518750 q^{15} + 8579948 q^{17} - 17627976 q^{19} - 1891512 q^{21} + 29841072 q^{23} + 19531250 q^{25} + 28697814 q^{27}+ \cdots - 17452994832 q^{99}+O(q^{100}) \) 2 * q + 486 * q^3 - 6250 * q^5 - 7784 * q^7 + 118098 * q^9 - 295568 * q^11 + 657492 * q^13 - 1518750 * q^15 + 8579948 * q^17 - 17627976 * q^19 - 1891512 * q^21 + 29841072 * q^23 + 19531250 * q^25 + 28697814 * q^27 - 201881948 * q^29 + 71057008 * q^31 - 71823024 * q^33 + 24325000 * q^35 + 705858484 * q^37 + 159770556 * q^39 - 327655148 * q^41 - 3192552120 * q^43 - 369056250 * q^45 - 2053064720 * q^47 + 7642614770 * q^49 + 2084927364 * q^51 - 2304299452 * q^53 + 923650000 * q^55 - 4283598168 * q^57 + 1478770576 * q^59 - 8264891460 * q^61 - 459637416 * q^63 - 2054662500 * q^65 - 24212177528 * q^67 + 7251380496 * q^69 + 20218888256 * q^71 + 25879583268 * q^73 + 4746093750 * q^75 + 101122045248 * q^77 - 22324995440 * q^79 + 6973568802 * q^81 - 48014508984 * q^83 - 26812337500 * q^85 - 49057313364 * q^87 + 79209683076 * q^89 + 229606712048 * q^91 + 17266852944 * q^93 + 55087425000 * q^95 - 37075227452 * q^97 - 17452994832 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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