LMFDB - Embedded newform 7248.2.a.bm.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7248,2,Mod(1,7248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7248, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7248 = 2^{4} \cdot 3 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7248.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:	$57.8755713850$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 27x^{8} + 45x^{7} + 258x^{6} - 289x^{5} - 1133x^{4} + 510x^{3} + 2070x^{2} + 341x - 500 $ x^10 - 2x^9 - 27x^8 + 45x^7 + 258x^6 - 289x^5 - 1133x^4 + 510x^3 + 2070x^2 + 341*x - 500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3624)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.35125$ of defining polynomial
Character	$\chi$	$=$	7248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.35125 q^{5} -3.88084 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.35125 q^{5} -3.88084 q^{7} +1.00000 q^{9} +3.62475 q^{11} +1.11544 q^{13} -2.35125 q^{15} -3.39276 q^{17} -2.95531 q^{19} +3.88084 q^{21} -4.91731 q^{23} +0.528370 q^{25} -1.00000 q^{27} +6.13816 q^{29} +6.20359 q^{31} -3.62475 q^{33} -9.12481 q^{35} -6.08861 q^{37} -1.11544 q^{39} +11.9689 q^{41} -0.552994 q^{43} +2.35125 q^{45} -1.47835 q^{47} +8.06089 q^{49} +3.39276 q^{51} +0.529419 q^{53} +8.52268 q^{55} +2.95531 q^{57} -15.1543 q^{59} -0.593677 q^{61} -3.88084 q^{63} +2.62269 q^{65} -6.27209 q^{67} +4.91731 q^{69} +10.2326 q^{71} -13.6241 q^{73} -0.528370 q^{75} -14.0671 q^{77} -12.2683 q^{79} +1.00000 q^{81} -13.7919 q^{83} -7.97721 q^{85} -6.13816 q^{87} +14.1095 q^{89} -4.32886 q^{91} -6.20359 q^{93} -6.94866 q^{95} -10.8690 q^{97} +3.62475 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9	$ 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9 - 7 * q^11 + 6 * q^13 - 2 * q^15 + 7 * q^17 + 8 * q^21 - 25 * q^23 + 8 * q^25 - 10 * q^27 + 12 * q^29 - 11 * q^31 + 7 * q^33 - 9 * q^35 - 3 * q^37 - 6 * q^39 + 12 * q^41 + 2 * q^45 - 31 * q^47 + 14 * q^49 - 7 * q^51 + q^53 - 9 * q^55 - 19 * q^59 + 24 * q^61 - 8 * q^63 + 20 * q^65 + q^67 + 25 * q^69 - 34 * q^71 - 18 * q^73 - 8 * q^75 + 27 * q^77 - 25 * q^79 + 10 * q^81 - 14 * q^83 - 3 * q^85 - 12 * q^87 + 20 * q^89 + 12 * q^91 + 11 * q^93 - 48 * q^95 - 15 * q^97 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.35125	1.05151	0.525755	−	0.850636i	$-0.323783\pi$
$5$	2.35125	1.05151	0.525755	+	0.850636i	$0.323783\pi$
$6$	0	0
$7$	−3.88084	−1.46682	−0.733409	−	0.679788i	$-0.762072\pi$
$7$	−3.88084	−1.46682	−0.733409	+	0.679788i	$0.762072\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.62475	1.09290	0.546451	−	0.837491i	$-0.315978\pi$
$11$	3.62475	1.09290	0.546451	+	0.837491i	$0.315978\pi$
$12$	0	0
$13$	1.11544	0.309368	0.154684	−	0.987964i	$-0.450564\pi$
$13$	1.11544	0.309368	0.154684	+	0.987964i	$0.450564\pi$
$14$	0	0
$15$	−2.35125	−0.607090
$16$	0	0
$17$	−3.39276	−0.822864	−0.411432	−	0.911440i	$-0.634971\pi$
$17$	−3.39276	−0.822864	−0.411432	+	0.911440i	$0.634971\pi$
$18$	0	0
$19$	−2.95531	−0.677994	−0.338997	−	0.940787i	$-0.610088\pi$
$19$	−2.95531	−0.677994	−0.338997	+	0.940787i	$0.610088\pi$
$20$	0	0
$21$	3.88084	0.846868
$22$	0	0
$23$	−4.91731	−1.02533	−0.512665	−	0.858589i	$-0.671342\pi$
$23$	−4.91731	−1.02533	−0.512665	+	0.858589i	$0.671342\pi$
$24$	0	0
$25$	0.528370	0.105674
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.13816	1.13983	0.569914	−	0.821704i	$-0.306976\pi$
$29$	6.13816	1.13983	0.569914	+	0.821704i	$0.306976\pi$
$30$	0	0
$31$	6.20359	1.11420	0.557099	−	0.830446i	$-0.311915\pi$
$31$	6.20359	1.11420	0.557099	+	0.830446i	$0.311915\pi$
$32$	0	0
$33$	−3.62475	−0.630988
$34$	0	0
$35$	−9.12481	−1.54237
$36$	0	0
$37$	−6.08861	−1.00096	−0.500480	−	0.865748i	$-0.666843\pi$
$37$	−6.08861	−1.00096	−0.500480	+	0.865748i	$0.666843\pi$
$38$	0	0
$39$	−1.11544	−0.178614
$40$	0	0
$41$	11.9689	1.86922	0.934611	−	0.355671i	$-0.115748\pi$
$41$	11.9689	1.86922	0.934611	+	0.355671i	$0.115748\pi$
$42$	0	0
$43$	−0.552994	−0.0843307	−0.0421654	−	0.999111i	$-0.513426\pi$
$43$	−0.552994	−0.0843307	−0.0421654	+	0.999111i	$0.513426\pi$
$44$	0	0
$45$	2.35125	0.350503
$46$	0	0
$47$	−1.47835	−0.215640	−0.107820	−	0.994170i	$-0.534387\pi$
$47$	−1.47835	−0.215640	−0.107820	+	0.994170i	$0.534387\pi$
$48$	0	0
$49$	8.06089	1.15156
$50$	0	0
$51$	3.39276	0.475081
$52$	0	0
$53$	0.529419	0.0727213	0.0363606	−	0.999339i	$-0.488423\pi$
$53$	0.529419	0.0727213	0.0363606	+	0.999339i	$0.488423\pi$
$54$	0	0
$55$	8.52268	1.14920
$56$	0	0
$57$	2.95531	0.391440
$58$	0	0
$59$	−15.1543	−1.97293	−0.986464	−	0.163978i	$-0.947567\pi$
$59$	−15.1543	−1.97293	−0.986464	+	0.163978i	$0.947567\pi$
$60$	0	0
$61$	−0.593677	−0.0760125	−0.0380063	−	0.999278i	$-0.512101\pi$
$61$	−0.593677	−0.0760125	−0.0380063	+	0.999278i	$0.512101\pi$
$62$	0	0
$63$	−3.88084	−0.488939
$64$	0	0
$65$	2.62269	0.325304
$66$	0	0
$67$	−6.27209	−0.766258	−0.383129	−	0.923695i	$-0.625153\pi$
$67$	−6.27209	−0.766258	−0.383129	+	0.923695i	$0.625153\pi$
$68$	0	0
$69$	4.91731	0.591975
$70$	0	0
$71$	10.2326	1.21439	0.607193	−	0.794554i	$-0.292295\pi$
$71$	10.2326	1.21439	0.607193	+	0.794554i	$0.292295\pi$
$72$	0	0
$73$	−13.6241	−1.59458	−0.797288	−	0.603599i	$-0.793733\pi$
$73$	−13.6241	−1.59458	−0.797288	+	0.603599i	$0.793733\pi$
$74$	0	0
$75$	−0.528370	−0.0610109
$76$	0	0
$77$	−14.0671	−1.60309
$78$	0	0
$79$	−12.2683	−1.38029	−0.690147	−	0.723669i	$-0.742454\pi$
$79$	−12.2683	−1.38029	−0.690147	+	0.723669i	$0.742454\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.7919	−1.51385	−0.756927	−	0.653499i	$-0.773300\pi$
$83$	−13.7919	−1.51385	−0.756927	+	0.653499i	$0.773300\pi$
$84$	0	0
$85$	−7.97721	−0.865250
$86$	0	0
$87$	−6.13816	−0.658080
$88$	0	0
$89$	14.1095	1.49561	0.747804	−	0.663920i	$-0.231108\pi$
$89$	14.1095	1.49561	0.747804	+	0.663920i	$0.231108\pi$
$90$	0	0
$91$	−4.32886	−0.453787
$92$	0	0
$93$	−6.20359	−0.643282
$94$	0	0
$95$	−6.94866	−0.712917
$96$	0	0
$97$	−10.8690	−1.10358	−0.551791	−	0.833982i	$-0.686055\pi$
$97$	−10.8690	−1.10358	−0.551791	+	0.833982i	$0.686055\pi$
$98$	0	0
$99$	3.62475	0.364301
$100$	0	0
$101$	9.83708	0.978826	0.489413	−	0.872052i	$-0.337211\pi$
$101$	9.83708	0.978826	0.489413	+	0.872052i	$0.337211\pi$
$102$	0	0
$103$	−0.135266	−0.0133282	−0.00666408	−	0.999978i	$-0.502121\pi$
$103$	−0.135266	−0.0133282	−0.00666408	+	0.999978i	$0.502121\pi$
$104$	0	0
$105$	9.12481	0.890490
$106$	0	0
$107$	13.4430	1.29959	0.649793	−	0.760112i	$-0.274856\pi$
$107$	13.4430	1.29959	0.649793	+	0.760112i	$0.274856\pi$
$108$	0	0
$109$	−14.7691	−1.41463	−0.707314	−	0.706900i	$-0.750093\pi$
$109$	−14.7691	−1.41463	−0.707314	+	0.706900i	$0.750093\pi$
$110$	0	0
$111$	6.08861	0.577905
$112$	0	0
$113$	18.7300	1.76197	0.880986	−	0.473143i	$-0.156881\pi$
$113$	18.7300	1.76197	0.880986	+	0.473143i	$0.156881\pi$
$114$	0	0
$115$	−11.5618	−1.07815
$116$	0	0
$117$	1.11544	0.103123
$118$	0	0
$119$	13.1667	1.20699
$120$	0	0
$121$	2.13879	0.194436
$122$	0	0
$123$	−11.9689	−1.07920
$124$	0	0
$125$	−10.5139	−0.940393
$126$	0	0
$127$	1.13792	0.100974	0.0504871	−	0.998725i	$-0.483923\pi$
$127$	1.13792	0.100974	0.0504871	+	0.998725i	$0.483923\pi$
$128$	0	0
$129$	0.552994	0.0486884
$130$	0	0
$131$	17.7068	1.54705	0.773525	−	0.633766i	$-0.218492\pi$
$131$	17.7068	1.54705	0.773525	+	0.633766i	$0.218492\pi$
$132$	0	0
$133$	11.4691	0.994494
$134$	0	0
$135$	−2.35125	−0.202363
$136$	0	0
$137$	−16.7899	−1.43445	−0.717227	−	0.696839i	$-0.754589\pi$
$137$	−16.7899	−1.43445	−0.717227	+	0.696839i	$0.754589\pi$
$138$	0	0
$139$	5.14673	0.436540	0.218270	−	0.975888i	$-0.429959\pi$
$139$	5.14673	0.436540	0.218270	+	0.975888i	$0.429959\pi$
$140$	0	0
$141$	1.47835	0.124500
$142$	0	0
$143$	4.04320	0.338110
$144$	0	0
$145$	14.4323	1.19854
$146$	0	0
$147$	−8.06089	−0.664851
$148$	0	0
$149$	−11.4196	−0.935530	−0.467765	−	0.883853i	$-0.654941\pi$
$149$	−11.4196	−0.935530	−0.467765	+	0.883853i	$0.654941\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−3.39276	−0.274288
$154$	0	0
$155$	14.5862	1.17159
$156$	0	0
$157$	−12.8970	−1.02929	−0.514644	−	0.857404i	$-0.672076\pi$
$157$	−12.8970	−1.02929	−0.514644	+	0.857404i	$0.672076\pi$
$158$	0	0
$159$	−0.529419	−0.0419857
$160$	0	0
$161$	19.0833	1.50397
$162$	0	0
$163$	8.39667	0.657678	0.328839	−	0.944386i	$-0.393343\pi$
$163$	8.39667	0.657678	0.328839	+	0.944386i	$0.393343\pi$
$164$	0	0
$165$	−8.52268	−0.663490
$166$	0	0
$167$	13.8997	1.07559	0.537795	−	0.843075i	$-0.319257\pi$
$167$	13.8997	1.07559	0.537795	+	0.843075i	$0.319257\pi$
$168$	0	0
$169$	−11.7558	−0.904291
$170$	0	0
$171$	−2.95531	−0.225998
$172$	0	0
$173$	−13.4580	−1.02319	−0.511595	−	0.859227i	$-0.670945\pi$
$173$	−13.4580	−1.02319	−0.511595	+	0.859227i	$0.670945\pi$
$174$	0	0
$175$	−2.05052	−0.155005
$176$	0	0
$177$	15.1543	1.13907
$178$	0	0
$179$	−5.14776	−0.384762	−0.192381	−	0.981320i	$-0.561621\pi$
$179$	−5.14776	−0.384762	−0.192381	+	0.981320i	$0.561621\pi$
$180$	0	0
$181$	5.67759	0.422012	0.211006	−	0.977485i	$-0.432326\pi$
$181$	5.67759	0.422012	0.211006	+	0.977485i	$0.432326\pi$
$182$	0	0
$183$	0.593677	0.0438859
$184$	0	0
$185$	−14.3158	−1.05252
$186$	0	0
$187$	−12.2979	−0.899310
$188$	0	0
$189$	3.88084	0.282289
$190$	0	0
$191$	−4.11362	−0.297651	−0.148825	−	0.988863i	$-0.547549\pi$
$191$	−4.11362	−0.297651	−0.148825	+	0.988863i	$0.547549\pi$
$192$	0	0
$193$	8.14674	0.586415	0.293208	−	0.956049i	$-0.405277\pi$
$193$	8.14674	0.586415	0.293208	+	0.956049i	$0.405277\pi$
$194$	0	0
$195$	−2.62269	−0.187814
$196$	0	0
$197$	−22.6105	−1.61093	−0.805464	−	0.592645i	$-0.798084\pi$
$197$	−22.6105	−1.61093	−0.805464	+	0.592645i	$0.798084\pi$
$198$	0	0
$199$	−12.3989	−0.878936	−0.439468	−	0.898258i	$-0.644833\pi$
$199$	−12.3989	−0.878936	−0.439468	+	0.898258i	$0.644833\pi$
$200$	0	0
$201$	6.27209	0.442399
$202$	0	0
$203$	−23.8212	−1.67192
$204$	0	0
$205$	28.1418	1.96551
$206$	0	0
$207$	−4.91731	−0.341777
$208$	0	0
$209$	−10.7122	−0.740981
$210$	0	0
$211$	−0.0118489	−0.000815709 0	−0.000407854	−	1.00000i	$-0.500130\pi$
$211$	−0.0118489	−0.000815709 0	−0.000407854		1.00000i	$0.500130\pi$
$212$	0	0
$213$	−10.2326	−0.701127
$214$	0	0
$215$	−1.30023	−0.0886746
$216$	0	0
$217$	−24.0751	−1.63432
$218$	0	0
$219$	13.6241	0.920629
$220$	0	0
$221$	−3.78443	−0.254568
$222$	0	0
$223$	−15.2626	−1.02206	−0.511030	−	0.859563i	$-0.670736\pi$
$223$	−15.2626	−1.02206	−0.511030	+	0.859563i	$0.670736\pi$
$224$	0	0
$225$	0.528370	0.0352247
$226$	0	0
$227$	−16.6606	−1.10581	−0.552903	−	0.833246i	$-0.686480\pi$
$227$	−16.6606	−1.10581	−0.552903	+	0.833246i	$0.686480\pi$
$228$	0	0
$229$	0.0631747	0.00417470	0.00208735	−	0.999998i	$-0.499336\pi$
$229$	0.0631747	0.00417470	0.00208735	+	0.999998i	$0.499336\pi$
$230$	0	0
$231$	14.0671	0.925544
$232$	0	0
$233$	15.2593	0.999667	0.499834	−	0.866121i	$-0.333395\pi$
$233$	15.2593	0.999667	0.499834	+	0.866121i	$0.333395\pi$
$234$	0	0
$235$	−3.47598	−0.226748
$236$	0	0
$237$	12.2683	0.796913
$238$	0	0
$239$	−25.1314	−1.62561	−0.812807	−	0.582533i	$-0.802062\pi$
$239$	−25.1314	−1.62561	−0.812807	+	0.582533i	$0.802062\pi$
$240$	0	0
$241$	−18.0615	−1.16345	−0.581723	−	0.813387i	$-0.697621\pi$
$241$	−18.0615	−1.16345	−0.581723	+	0.813387i	$0.697621\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	18.9532	1.21087
$246$	0	0
$247$	−3.29648	−0.209750
$248$	0	0
$249$	13.7919	0.874025
$250$	0	0
$251$	11.3978	0.719420	0.359710	−	0.933064i	$-0.382876\pi$
$251$	11.3978	0.719420	0.359710	+	0.933064i	$0.382876\pi$
$252$	0	0
$253$	−17.8240	−1.12059
$254$	0	0
$255$	7.97721	0.499553
$256$	0	0
$257$	22.1981	1.38468	0.692340	−	0.721571i	$-0.256580\pi$
$257$	22.1981	1.38468	0.692340	+	0.721571i	$0.256580\pi$
$258$	0	0
$259$	23.6289	1.46823
$260$	0	0
$261$	6.13816	0.379943
$262$	0	0
$263$	−2.76605	−0.170562	−0.0852810	−	0.996357i	$-0.527179\pi$
$263$	−2.76605	−0.170562	−0.0852810	+	0.996357i	$0.527179\pi$
$264$	0	0
$265$	1.24480	0.0764672
$266$	0	0
$267$	−14.1095	−0.863490
$268$	0	0
$269$	27.3100	1.66512	0.832561	−	0.553933i	$-0.186874\pi$
$269$	27.3100	1.66512	0.832561	+	0.553933i	$0.186874\pi$
$270$	0	0
$271$	−12.0453	−0.731703	−0.365851	−	0.930673i	$-0.619222\pi$
$271$	−12.0453	−0.731703	−0.365851	+	0.930673i	$0.619222\pi$
$272$	0	0
$273$	4.32886	0.261994
$274$	0	0
$275$	1.91521	0.115491
$276$	0	0
$277$	12.7927	0.768639	0.384320	−	0.923200i	$-0.374436\pi$
$277$	12.7927	0.768639	0.384320	+	0.923200i	$0.374436\pi$
$278$	0	0
$279$	6.20359	0.371399
$280$	0	0
$281$	−10.7664	−0.642268	−0.321134	−	0.947034i	$-0.604064\pi$
$281$	−10.7664	−0.642268	−0.321134	+	0.947034i	$0.604064\pi$
$282$	0	0
$283$	14.8003	0.879786	0.439893	−	0.898050i	$-0.355016\pi$
$283$	14.8003	0.879786	0.439893	+	0.898050i	$0.355016\pi$
$284$	0	0
$285$	6.94866	0.411603
$286$	0	0
$287$	−46.4492	−2.74181
$288$	0	0
$289$	−5.48920	−0.322894
$290$	0	0
$291$	10.8690	0.637154
$292$	0	0
$293$	−17.2501	−1.00776	−0.503880	−	0.863774i	$-0.668095\pi$
$293$	−17.2501	−1.00776	−0.503880	+	0.863774i	$0.668095\pi$
$294$	0	0
$295$	−35.6316	−2.07455
$296$	0	0
$297$	−3.62475	−0.210329
$298$	0	0
$299$	−5.48499	−0.317205
$300$	0	0
$301$	2.14608	0.123698
$302$	0	0
$303$	−9.83708	−0.565126
$304$	0	0
$305$	−1.39588	−0.0799280
$306$	0	0
$307$	−11.4676	−0.654491	−0.327246	−	0.944939i	$-0.606120\pi$
$307$	−11.4676	−0.654491	−0.327246	+	0.944939i	$0.606120\pi$
$308$	0	0
$309$	0.135266	0.00769502
$310$	0	0
$311$	15.8947	0.901307	0.450654	−	0.892699i	$-0.351191\pi$
$311$	15.8947	0.901307	0.450654	+	0.892699i	$0.351191\pi$
$312$	0	0
$313$	−2.85491	−0.161369	−0.0806846	−	0.996740i	$-0.525711\pi$
$313$	−2.85491	−0.161369	−0.0806846	+	0.996740i	$0.525711\pi$
$314$	0	0
$315$	−9.12481	−0.514125
$316$	0	0
$317$	−3.11186	−0.174780	−0.0873899	−	0.996174i	$-0.527853\pi$
$317$	−3.11186	−0.174780	−0.0873899	+	0.996174i	$0.527853\pi$
$318$	0	0
$319$	22.2493	1.24572
$320$	0	0
$321$	−13.4430	−0.750316
$322$	0	0
$323$	10.0266	0.557897
$324$	0	0
$325$	0.589367	0.0326922
$326$	0	0
$327$	14.7691	0.816735
$328$	0	0
$329$	5.73725	0.316305
$330$	0	0
$331$	−22.1198	−1.21581	−0.607906	−	0.794009i	$-0.707990\pi$
$331$	−22.1198	−1.21581	−0.607906	+	0.794009i	$0.707990\pi$
$332$	0	0
$333$	−6.08861	−0.333654
$334$	0	0
$335$	−14.7472	−0.805728
$336$	0	0
$337$	4.16733	0.227009	0.113504	−	0.993537i	$-0.463792\pi$
$337$	4.16733	0.227009	0.113504	+	0.993537i	$0.463792\pi$
$338$	0	0
$339$	−18.7300	−1.01727
$340$	0	0
$341$	22.4864	1.21771
$342$	0	0
$343$	−4.11714	−0.222305
$344$	0	0
$345$	11.5618	0.622468
$346$	0	0
$347$	−1.29457	−0.0694961	−0.0347481	−	0.999396i	$-0.511063\pi$
$347$	−1.29457	−0.0694961	−0.0347481	+	0.999396i	$0.511063\pi$
$348$	0	0
$349$	−16.9311	−0.906303	−0.453151	−	0.891434i	$-0.649700\pi$
$349$	−16.9311	−0.906303	−0.453151	+	0.891434i	$0.649700\pi$
$350$	0	0
$351$	−1.11544	−0.0595380
$352$	0	0
$353$	−28.1605	−1.49883	−0.749415	−	0.662100i	$-0.769665\pi$
$353$	−28.1605	−1.49883	−0.749415	+	0.662100i	$0.769665\pi$
$354$	0	0
$355$	24.0594	1.27694
$356$	0	0
$357$	−13.1667	−0.696857
$358$	0	0
$359$	13.1074	0.691780	0.345890	−	0.938275i	$-0.387577\pi$
$359$	13.1074	0.691780	0.345890	+	0.938275i	$0.387577\pi$
$360$	0	0
$361$	−10.2662	−0.540324
$362$	0	0
$363$	−2.13879	−0.112258
$364$	0	0
$365$	−32.0336	−1.67671
$366$	0	0
$367$	−19.7133	−1.02902	−0.514512	−	0.857483i	$-0.672027\pi$
$367$	−19.7133	−1.02902	−0.514512	+	0.857483i	$0.672027\pi$
$368$	0	0
$369$	11.9689	0.623074
$370$	0	0
$371$	−2.05459	−0.106669
$372$	0	0
$373$	−17.9188	−0.927799	−0.463900	−	0.885888i	$-0.653550\pi$
$373$	−17.9188	−0.927799	−0.463900	+	0.885888i	$0.653550\pi$
$374$	0	0
$375$	10.5139	0.542936
$376$	0	0
$377$	6.84678	0.352627
$378$	0	0
$379$	35.9206	1.84512	0.922559	−	0.385857i	$-0.126094\pi$
$379$	35.9206	1.84512	0.922559	+	0.385857i	$0.126094\pi$
$380$	0	0
$381$	−1.13792	−0.0582975
$382$	0	0
$383$	−32.0882	−1.63963	−0.819816	−	0.572627i	$-0.805924\pi$
$383$	−32.0882	−1.63963	−0.819816	+	0.572627i	$0.805924\pi$
$384$	0	0
$385$	−33.0751	−1.68566
$386$	0	0
$387$	−0.552994	−0.0281102
$388$	0	0
$389$	−21.5318	−1.09171	−0.545854	−	0.837880i	$-0.683795\pi$
$389$	−21.5318	−1.09171	−0.545854	+	0.837880i	$0.683795\pi$
$390$	0	0
$391$	16.6832	0.843708
$392$	0	0
$393$	−17.7068	−0.893189
$394$	0	0
$395$	−28.8459	−1.45139
$396$	0	0
$397$	26.8254	1.34633	0.673163	−	0.739494i	$-0.264935\pi$
$397$	26.8254	1.34633	0.673163	+	0.739494i	$0.264935\pi$
$398$	0	0
$399$	−11.4691	−0.574171
$400$	0	0
$401$	26.9734	1.34699	0.673493	−	0.739194i	$-0.264793\pi$
$401$	26.9734	1.34699	0.673493	+	0.739194i	$0.264793\pi$
$402$	0	0
$403$	6.91975	0.344697
$404$	0	0
$405$	2.35125	0.116834
$406$	0	0
$407$	−22.0697	−1.09395
$408$	0	0
$409$	−27.9384	−1.38147	−0.690733	−	0.723110i	$-0.742712\pi$
$409$	−27.9384	−1.38147	−0.690733	+	0.723110i	$0.742712\pi$
$410$	0	0
$411$	16.7899	0.828183
$412$	0	0
$413$	58.8115	2.89393
$414$	0	0
$415$	−32.4281	−1.59183
$416$	0	0
$417$	−5.14673	−0.252037
$418$	0	0
$419$	−12.6216	−0.616604	−0.308302	−	0.951288i	$-0.599761\pi$
$419$	−12.6216	−0.616604	−0.308302	+	0.951288i	$0.599761\pi$
$420$	0	0
$421$	8.52131	0.415303	0.207652	−	0.978203i	$-0.433418\pi$
$421$	8.52131	0.415303	0.207652	+	0.978203i	$0.433418\pi$
$422$	0	0
$423$	−1.47835	−0.0718800
$424$	0	0
$425$	−1.79263	−0.0869553
$426$	0	0
$427$	2.30396	0.111497
$428$	0	0
$429$	−4.04320	−0.195208
$430$	0	0
$431$	−21.1997	−1.02116	−0.510578	−	0.859831i	$-0.670569\pi$
$431$	−21.1997	−1.02116	−0.510578	+	0.859831i	$0.670569\pi$
$432$	0	0
$433$	−26.4294	−1.27011	−0.635057	−	0.772465i	$-0.719023\pi$
$433$	−26.4294	−1.27011	−0.635057	+	0.772465i	$0.719023\pi$
$434$	0	0
$435$	−14.4323	−0.691978
$436$	0	0
$437$	14.5322	0.695168
$438$	0	0
$439$	−6.38549	−0.304763	−0.152381	−	0.988322i	$-0.548694\pi$
$439$	−6.38549	−0.304763	−0.152381	+	0.988322i	$0.548694\pi$
$440$	0	0
$441$	8.06089	0.383852
$442$	0	0
$443$	−3.58684	−0.170416	−0.0852080	−	0.996363i	$-0.527155\pi$
$443$	−3.58684	−0.170416	−0.0852080	+	0.996363i	$0.527155\pi$
$444$	0	0
$445$	33.1750	1.57265
$446$	0	0
$447$	11.4196	0.540128
$448$	0	0
$449$	−16.0200	−0.756029	−0.378015	−	0.925800i	$-0.623393\pi$
$449$	−16.0200	−0.756029	−0.378015	+	0.925800i	$0.623393\pi$
$450$	0	0
$451$	43.3841	2.04288
$452$	0	0
$453$	1.00000	0.0469841
$454$	0	0
$455$	−10.1782	−0.477162
$456$	0	0
$457$	11.6281	0.543938	0.271969	−	0.962306i	$-0.412325\pi$
$457$	11.6281	0.543938	0.271969	+	0.962306i	$0.412325\pi$
$458$	0	0
$459$	3.39276	0.158360
$460$	0	0
$461$	−41.2767	−1.92245	−0.961223	−	0.275771i	$-0.911067\pi$
$461$	−41.2767	−1.92245	−0.961223	+	0.275771i	$0.911067\pi$
$462$	0	0
$463$	−8.85224	−0.411398	−0.205699	−	0.978615i	$-0.565947\pi$
$463$	−8.85224	−0.411398	−0.205699	+	0.978615i	$0.565947\pi$
$464$	0	0
$465$	−14.5862	−0.676418
$466$	0	0
$467$	2.12101	0.0981488	0.0490744	−	0.998795i	$-0.484373\pi$
$467$	2.12101	0.0981488	0.0490744	+	0.998795i	$0.484373\pi$
$468$	0	0
$469$	24.3409	1.12396
$470$	0	0
$471$	12.8970	0.594260
$472$	0	0
$473$	−2.00446	−0.0921653
$474$	0	0
$475$	−1.56149	−0.0716463
$476$	0	0
$477$	0.529419	0.0242404
$478$	0	0
$479$	−1.97936	−0.0904390	−0.0452195	−	0.998977i	$-0.514399\pi$
$479$	−1.97936	−0.0904390	−0.0452195	+	0.998977i	$0.514399\pi$
$480$	0	0
$481$	−6.79150	−0.309666
$482$	0	0
$483$	−19.0833	−0.868320
$484$	0	0
$485$	−25.5558	−1.16043
$486$	0	0
$487$	17.9650	0.814073	0.407037	−	0.913412i	$-0.366562\pi$
$487$	17.9650	0.814073	0.407037	+	0.913412i	$0.366562\pi$
$488$	0	0
$489$	−8.39667	−0.379711
$490$	0	0
$491$	−2.70638	−0.122137	−0.0610686	−	0.998134i	$-0.519451\pi$
$491$	−2.70638	−0.122137	−0.0610686	+	0.998134i	$0.519451\pi$
$492$	0	0
$493$	−20.8253	−0.937924
$494$	0	0
$495$	8.52268	0.383066
$496$	0	0
$497$	−39.7111	−1.78128
$498$	0	0
$499$	−25.8660	−1.15792	−0.578961	−	0.815355i	$-0.696542\pi$
$499$	−25.8660	−1.15792	−0.578961	+	0.815355i	$0.696542\pi$
$500$	0	0
$501$	−13.8997	−0.620993
$502$	0	0
$503$	6.75191	0.301053	0.150526	−	0.988606i	$-0.451903\pi$
$503$	6.75191	0.301053	0.150526	+	0.988606i	$0.451903\pi$
$504$	0	0
$505$	23.1294	1.02925
$506$	0	0
$507$	11.7558	0.522093
$508$	0	0
$509$	−31.7805	−1.40865	−0.704324	−	0.709879i	$-0.748750\pi$
$509$	−31.7805	−1.40865	−0.704324	+	0.709879i	$0.748750\pi$
$510$	0	0
$511$	52.8728	2.33895
$512$	0	0
$513$	2.95531	0.130480
$514$	0	0
$515$	−0.318044	−0.0140147
$516$	0	0
$517$	−5.35866	−0.235673
$518$	0	0
$519$	13.4580	0.590739
$520$	0	0
$521$	5.63807	0.247008	0.123504	−	0.992344i	$-0.460587\pi$
$521$	5.63807	0.247008	0.123504	+	0.992344i	$0.460587\pi$
$522$	0	0
$523$	12.5801	0.550091	0.275046	−	0.961431i	$-0.411307\pi$
$523$	12.5801	0.550091	0.275046	+	0.961431i	$0.411307\pi$
$524$	0	0
$525$	2.05052	0.0894919
$526$	0	0
$527$	−21.0473	−0.916833
$528$	0	0
$529$	1.17996	0.0513027
$530$	0	0
$531$	−15.1543	−0.657643
$532$	0	0
$533$	13.3506	0.578278
$534$	0	0
$535$	31.6079	1.36653
$536$	0	0
$537$	5.14776	0.222142
$538$	0	0
$539$	29.2187	1.25854
$540$	0	0
$541$	−31.3894	−1.34953	−0.674767	−	0.738031i	$-0.735756\pi$
$541$	−31.3894	−1.34953	−0.674767	+	0.738031i	$0.735756\pi$
$542$	0	0
$543$	−5.67759	−0.243649
$544$	0	0
$545$	−34.7259	−1.48750
$546$	0	0
$547$	41.2763	1.76485	0.882424	−	0.470454i	$-0.155910\pi$
$547$	41.2763	1.76485	0.882424	+	0.470454i	$0.155910\pi$
$548$	0	0
$549$	−0.593677	−0.0253375
$550$	0	0
$551$	−18.1401	−0.772796
$552$	0	0
$553$	47.6113	2.02464
$554$	0	0
$555$	14.3158	0.607673
$556$	0	0
$557$	−5.02543	−0.212934	−0.106467	−	0.994316i	$-0.533954\pi$
$557$	−5.02543	−0.212934	−0.106467	+	0.994316i	$0.533954\pi$
$558$	0	0
$559$	−0.616833	−0.0260893
$560$	0	0
$561$	12.2979	0.519217
$562$	0	0
$563$	17.5117	0.738031	0.369015	−	0.929423i	$-0.379695\pi$
$563$	17.5117	0.738031	0.369015	+	0.929423i	$0.379695\pi$
$564$	0	0
$565$	44.0389	1.85273
$566$	0	0
$567$	−3.88084	−0.162980
$568$	0	0
$569$	−6.22964	−0.261160	−0.130580	−	0.991438i	$-0.541684\pi$
$569$	−6.22964	−0.261160	−0.130580	+	0.991438i	$0.541684\pi$
$570$	0	0
$571$	−18.6297	−0.779631	−0.389815	−	0.920893i	$-0.627461\pi$
$571$	−18.6297	−0.779631	−0.389815	+	0.920893i	$0.627461\pi$
$572$	0	0
$573$	4.11362	0.171849
$574$	0	0
$575$	−2.59816	−0.108351
$576$	0	0
$577$	−22.8196	−0.949993	−0.474997	−	0.879988i	$-0.657551\pi$
$577$	−22.8196	−0.949993	−0.474997	+	0.879988i	$0.657551\pi$
$578$	0	0
$579$	−8.14674	−0.338567
$580$	0	0
$581$	53.5240	2.22055
$582$	0	0
$583$	1.91901	0.0794773
$584$	0	0
$585$	2.62269	0.108435
$586$	0	0
$587$	−4.69083	−0.193611	−0.0968057	−	0.995303i	$-0.530863\pi$
$587$	−4.69083	−0.193611	−0.0968057	+	0.995303i	$0.530863\pi$
$588$	0	0
$589$	−18.3335	−0.755419
$590$	0	0
$591$	22.6105	0.930070
$592$	0	0
$593$	−25.4293	−1.04425	−0.522127	−	0.852868i	$-0.674861\pi$
$593$	−25.4293	−1.04425	−0.522127	+	0.852868i	$0.674861\pi$
$594$	0	0
$595$	30.9583	1.26916
$596$	0	0
$597$	12.3989	0.507454
$598$	0	0
$599$	7.64008	0.312165	0.156083	−	0.987744i	$-0.450113\pi$
$599$	7.64008	0.312165	0.156083	+	0.987744i	$0.450113\pi$
$600$	0	0
$601$	8.80835	0.359300	0.179650	−	0.983731i	$-0.442503\pi$
$601$	8.80835	0.359300	0.179650	+	0.983731i	$0.442503\pi$
$602$	0	0
$603$	−6.27209	−0.255419
$604$	0	0
$605$	5.02883	0.204451
$606$	0	0
$607$	36.7962	1.49351	0.746756	−	0.665098i	$-0.231610\pi$
$607$	36.7962	1.49351	0.746756	+	0.665098i	$0.231610\pi$
$608$	0	0
$609$	23.8212	0.965284
$610$	0	0
$611$	−1.64902	−0.0667122
$612$	0	0
$613$	19.6284	0.792782	0.396391	−	0.918082i	$-0.370263\pi$
$613$	19.6284	0.792782	0.396391	+	0.918082i	$0.370263\pi$
$614$	0	0
$615$	−28.1418	−1.13479
$616$	0	0
$617$	−19.9178	−0.801861	−0.400931	−	0.916108i	$-0.631313\pi$
$617$	−19.9178	−0.801861	−0.400931	+	0.916108i	$0.631313\pi$
$618$	0	0
$619$	−3.68189	−0.147988	−0.0739938	−	0.997259i	$-0.523575\pi$
$619$	−3.68189	−0.147988	−0.0739938	+	0.997259i	$0.523575\pi$
$620$	0	0
$621$	4.91731	0.197325
$622$	0	0
$623$	−54.7568	−2.19378
$624$	0	0
$625$	−27.3627	−1.09451
$626$	0	0
$627$	10.7122	0.427806
$628$	0	0
$629$	20.6572	0.823655
$630$	0	0
$631$	40.7860	1.62367	0.811833	−	0.583889i	$-0.198470\pi$
$631$	40.7860	1.62367	0.811833	+	0.583889i	$0.198470\pi$
$632$	0	0
$633$	0.0118489	0.000470950 0
$634$	0	0
$635$	2.67554	0.106175
$636$	0	0
$637$	8.99147	0.356255
$638$	0	0
$639$	10.2326	0.404796
$640$	0	0
$641$	−24.4845	−0.967078	−0.483539	−	0.875323i	$-0.660649\pi$
$641$	−24.4845	−0.967078	−0.483539	+	0.875323i	$0.660649\pi$
$642$	0	0
$643$	−27.7844	−1.09571	−0.547854	−	0.836574i	$-0.684555\pi$
$643$	−27.7844	−1.09571	−0.547854	+	0.836574i	$0.684555\pi$
$644$	0	0
$645$	1.30023	0.0511963
$646$	0	0
$647$	30.8777	1.21393	0.606963	−	0.794730i	$-0.292388\pi$
$647$	30.8777	1.21393	0.606963	+	0.794730i	$0.292388\pi$
$648$	0	0
$649$	−54.9307	−2.15622
$650$	0	0
$651$	24.0751	0.943578
$652$	0	0
$653$	2.24876	0.0880009	0.0440004	−	0.999032i	$-0.485990\pi$
$653$	2.24876	0.0880009	0.0440004	+	0.999032i	$0.485990\pi$
$654$	0	0
$655$	41.6331	1.62674
$656$	0	0
$657$	−13.6241	−0.531526
$658$	0	0
$659$	−13.4795	−0.525088	−0.262544	−	0.964920i	$-0.584561\pi$
$659$	−13.4795	−0.525088	−0.262544	+	0.964920i	$0.584561\pi$
$660$	0	0
$661$	15.3425	0.596755	0.298378	−	0.954448i	$-0.403554\pi$
$661$	15.3425	0.596755	0.298378	+	0.954448i	$0.403554\pi$
$662$	0	0
$663$	3.78443	0.146975
$664$	0	0
$665$	26.9666	1.04572
$666$	0	0
$667$	−30.1833	−1.16870
$668$	0	0
$669$	15.2626	0.590087
$670$	0	0
$671$	−2.15193	−0.0830743
$672$	0	0
$673$	31.9217	1.23049	0.615245	−	0.788336i	$-0.289057\pi$
$673$	31.9217	1.23049	0.615245	+	0.788336i	$0.289057\pi$
$674$	0	0
$675$	−0.528370	−0.0203370
$676$	0	0
$677$	44.2592	1.70102	0.850510	−	0.525959i	$-0.176294\pi$
$677$	44.2592	1.70102	0.850510	+	0.525959i	$0.176294\pi$
$678$	0	0
$679$	42.1809	1.61875
$680$	0	0
$681$	16.6606	0.638437
$682$	0	0
$683$	19.7595	0.756077	0.378039	−	0.925790i	$-0.376599\pi$
$683$	19.7595	0.756077	0.378039	+	0.925790i	$0.376599\pi$
$684$	0	0
$685$	−39.4771	−1.50834
$686$	0	0
$687$	−0.0631747	−0.00241027
$688$	0	0
$689$	0.590537	0.0224977
$690$	0	0
$691$	−21.4592	−0.816347	−0.408173	−	0.912904i	$-0.633834\pi$
$691$	−21.4592	−0.816347	−0.408173	+	0.912904i	$0.633834\pi$
$692$	0	0
$693$	−14.0671	−0.534363
$694$	0	0
$695$	12.1012	0.459026
$696$	0	0
$697$	−40.6074	−1.53812
$698$	0	0
$699$	−15.2593	−0.577158
$700$	0	0
$701$	−26.8987	−1.01595	−0.507975	−	0.861372i	$-0.669606\pi$
$701$	−26.8987	−1.01595	−0.507975	+	0.861372i	$0.669606\pi$
$702$	0	0
$703$	17.9937	0.678645
$704$	0	0
$705$	3.47598	0.130913
$706$	0	0
$707$	−38.1761	−1.43576
$708$	0	0
$709$	−14.7077	−0.552359	−0.276179	−	0.961106i	$-0.589068\pi$
$709$	−14.7077	−0.552359	−0.276179	+	0.961106i	$0.589068\pi$
$710$	0	0
$711$	−12.2683	−0.460098
$712$	0	0
$713$	−30.5050	−1.14242
$714$	0	0
$715$	9.50657	0.355526
$716$	0	0
$717$	25.1314	0.938548
$718$	0	0
$719$	−49.4596	−1.84453	−0.922266	−	0.386556i	$-0.873665\pi$
$719$	−49.4596	−1.84453	−0.922266	+	0.386556i	$0.873665\pi$
$720$	0	0
$721$	0.524945	0.0195500
$722$	0	0
$723$	18.0615	0.671716
$724$	0	0
$725$	3.24322	0.120450
$726$	0	0
$727$	−23.9210	−0.887181	−0.443590	−	0.896230i	$-0.646295\pi$
$727$	−23.9210	−0.887181	−0.443590	+	0.896230i	$0.646295\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.87617	0.0693928
$732$	0	0
$733$	14.3981	0.531805	0.265902	−	0.964000i	$-0.414330\pi$
$733$	14.3981	0.531805	0.265902	+	0.964000i	$0.414330\pi$
$734$	0	0
$735$	−18.9532	−0.699098
$736$	0	0
$737$	−22.7347	−0.837445
$738$	0	0
$739$	−22.5354	−0.828979	−0.414490	−	0.910054i	$-0.636040\pi$
$739$	−22.5354	−0.828979	−0.414490	+	0.910054i	$0.636040\pi$
$740$	0	0
$741$	3.29648	0.121099
$742$	0	0
$743$	−18.0877	−0.663575	−0.331788	−	0.943354i	$-0.607652\pi$
$743$	−18.0877	−0.663575	−0.331788	+	0.943354i	$0.607652\pi$
$744$	0	0
$745$	−26.8503	−0.983719
$746$	0	0
$747$	−13.7919	−0.504618
$748$	0	0
$749$	−52.1701	−1.90626
$750$	0	0
$751$	11.9054	0.434436	0.217218	−	0.976123i	$-0.430302\pi$
$751$	11.9054	0.434436	0.217218	+	0.976123i	$0.430302\pi$
$752$	0	0
$753$	−11.3978	−0.415357
$754$	0	0
$755$	−2.35125	−0.0855707
$756$	0	0
$757$	−27.1282	−0.985990	−0.492995	−	0.870032i	$-0.664098\pi$
$757$	−27.1282	−0.985990	−0.492995	+	0.870032i	$0.664098\pi$
$758$	0	0
$759$	17.8240	0.646971
$760$	0	0
$761$	36.4257	1.32043	0.660215	−	0.751077i	$-0.270465\pi$
$761$	36.4257	1.32043	0.660215	+	0.751077i	$0.270465\pi$
$762$	0	0
$763$	57.3166	2.07500
$764$	0	0
$765$	−7.97721	−0.288417
$766$	0	0
$767$	−16.9038	−0.610362
$768$	0	0
$769$	14.4035	0.519402	0.259701	−	0.965689i	$-0.416376\pi$
$769$	14.4035	0.519402	0.259701	+	0.965689i	$0.416376\pi$
$770$	0	0
$771$	−22.1981	−0.799445
$772$	0	0
$773$	−27.7819	−0.999246	−0.499623	−	0.866243i	$-0.666528\pi$
$773$	−27.7819	−0.999246	−0.499623	+	0.866243i	$0.666528\pi$
$774$	0	0
$775$	3.27779	0.117742
$776$	0	0
$777$	−23.6289	−0.847681
$778$	0	0
$779$	−35.3717	−1.26732
$780$	0	0
$781$	37.0906	1.32721
$782$	0	0
$783$	−6.13816	−0.219360
$784$	0	0
$785$	−30.3239	−1.08231
$786$	0	0
$787$	43.2395	1.54132	0.770661	−	0.637245i	$-0.219926\pi$
$787$	43.2395	1.54132	0.770661	+	0.637245i	$0.219926\pi$
$788$	0	0
$789$	2.76605	0.0984740
$790$	0	0
$791$	−72.6881	−2.58449
$792$	0	0
$793$	−0.662213	−0.0235159
$794$	0	0
$795$	−1.24480	−0.0441483
$796$	0	0
$797$	35.1908	1.24652	0.623261	−	0.782014i	$-0.285807\pi$
$797$	35.1908	1.24652	0.623261	+	0.782014i	$0.285807\pi$
$798$	0	0
$799$	5.01569	0.177442
$800$	0	0
$801$	14.1095	0.498536
$802$	0	0
$803$	−49.3838	−1.74272
$804$	0	0
$805$	44.8695	1.58144
$806$	0	0
$807$	−27.3100	−0.961359
$808$	0	0
$809$	22.9348	0.806343	0.403172	−	0.915124i	$-0.367908\pi$
$809$	22.9348	0.806343	0.403172	+	0.915124i	$0.367908\pi$
$810$	0	0
$811$	−42.8809	−1.50575	−0.752875	−	0.658163i	$-0.771334\pi$
$811$	−42.8809	−1.50575	−0.752875	+	0.658163i	$0.771334\pi$
$812$	0	0
$813$	12.0453	0.422449
$814$	0	0
$815$	19.7427	0.691555
$816$	0	0
$817$	1.63427	0.0571757
$818$	0	0
$819$	−4.32886	−0.151262
$820$	0	0
$821$	−44.9376	−1.56833	−0.784166	−	0.620551i	$-0.786909\pi$
$821$	−44.9376	−1.56833	−0.784166	+	0.620551i	$0.786909\pi$
$822$	0	0
$823$	−39.4716	−1.37590	−0.687948	−	0.725760i	$-0.741488\pi$
$823$	−39.4716	−1.37590	−0.687948	+	0.725760i	$0.741488\pi$
$824$	0	0
$825$	−1.91521	−0.0666790
$826$	0	0
$827$	56.9847	1.98155	0.990776	−	0.135509i	$-0.0432670\pi$
$827$	56.9847	1.98155	0.990776	+	0.135509i	$0.0432670\pi$
$828$	0	0
$829$	2.60062	0.0903232	0.0451616	−	0.998980i	$-0.485620\pi$
$829$	2.60062	0.0903232	0.0451616	+	0.998980i	$0.485620\pi$
$830$	0	0
$831$	−12.7927	−0.443774
$832$	0	0
$833$	−27.3486	−0.947574
$834$	0	0
$835$	32.6816	1.13099
$836$	0	0
$837$	−6.20359	−0.214427
$838$	0	0
$839$	−20.4437	−0.705796	−0.352898	−	0.935662i	$-0.614804\pi$
$839$	−20.4437	−0.705796	−0.352898	+	0.935662i	$0.614804\pi$
$840$	0	0
$841$	8.67703	0.299208
$842$	0	0
$843$	10.7664	0.370814
$844$	0	0
$845$	−27.6408	−0.950871
$846$	0	0
$847$	−8.30031	−0.285202
$848$	0	0
$849$	−14.8003	−0.507944
$850$	0	0
$851$	29.9396	1.02632
$852$	0	0
$853$	−45.8515	−1.56992	−0.784962	−	0.619543i	$-0.787318\pi$
$853$	−45.8515	−1.56992	−0.784962	+	0.619543i	$0.787318\pi$
$854$	0	0
$855$	−6.94866	−0.237639
$856$	0	0
$857$	26.7134	0.912512	0.456256	−	0.889849i	$-0.349190\pi$
$857$	26.7134	0.912512	0.456256	+	0.889849i	$0.349190\pi$
$858$	0	0
$859$	56.7583	1.93657	0.968284	−	0.249852i	$-0.0803820\pi$
$859$	56.7583	1.93657	0.968284	+	0.249852i	$0.0803820\pi$
$860$	0	0
$861$	46.4492	1.58298
$862$	0	0
$863$	−28.8774	−0.982999	−0.491499	−	0.870878i	$-0.663551\pi$
$863$	−28.8774	−0.982999	−0.491499	+	0.870878i	$0.663551\pi$
$864$	0	0
$865$	−31.6430	−1.07589
$866$	0	0
$867$	5.48920	0.186423
$868$	0	0
$869$	−44.4696	−1.50853
$870$	0	0
$871$	−6.99616	−0.237056
$872$	0	0
$873$	−10.8690	−0.367861
$874$	0	0
$875$	40.8028	1.37939
$876$	0	0
$877$	2.31611	0.0782094	0.0391047	−	0.999235i	$-0.487549\pi$
$877$	2.31611	0.0782094	0.0391047	+	0.999235i	$0.487549\pi$
$878$	0	0
$879$	17.2501	0.581830
$880$	0	0
$881$	−7.86431	−0.264955	−0.132478	−	0.991186i	$-0.542293\pi$
$881$	−7.86431	−0.264955	−0.132478	+	0.991186i	$0.542293\pi$
$882$	0	0
$883$	−22.0864	−0.743267	−0.371633	−	0.928380i	$-0.621202\pi$
$883$	−22.0864	−0.743267	−0.371633	+	0.928380i	$0.621202\pi$
$884$	0	0
$885$	35.6316	1.19774
$886$	0	0
$887$	14.5186	0.487488	0.243744	−	0.969840i	$-0.421624\pi$
$887$	14.5186	0.487488	0.243744	+	0.969840i	$0.421624\pi$
$888$	0	0
$889$	−4.41609	−0.148111
$890$	0	0
$891$	3.62475	0.121434
$892$	0	0
$893$	4.36899	0.146203
$894$	0	0
$895$	−12.1037	−0.404581
$896$	0	0
$897$	5.48499	0.183138
$898$	0	0
$899$	38.0786	1.26999
$900$	0	0
$901$	−1.79619	−0.0598397
$902$	0	0
$903$	−2.14608	−0.0714170
$904$	0	0
$905$	13.3494	0.443750
$906$	0	0
$907$	−34.4847	−1.14504	−0.572522	−	0.819889i	$-0.694035\pi$
$907$	−34.4847	−1.14504	−0.572522	+	0.819889i	$0.694035\pi$
$908$	0	0
$909$	9.83708	0.326275
$910$	0	0
$911$	32.9237	1.09081	0.545405	−	0.838173i	$-0.316376\pi$
$911$	32.9237	1.09081	0.545405	+	0.838173i	$0.316376\pi$
$912$	0	0
$913$	−49.9921	−1.65450
$914$	0	0
$915$	1.39588	0.0461464
$916$	0	0
$917$	−68.7172	−2.26924
$918$	0	0
$919$	−8.65283	−0.285431	−0.142715	−	0.989764i	$-0.545583\pi$
$919$	−8.65283	−0.285431	−0.142715	+	0.989764i	$0.545583\pi$
$920$	0	0
$921$	11.4676	0.377871
$922$	0	0
$923$	11.4139	0.375693
$924$	0	0
$925$	−3.21704	−0.105775
$926$	0	0
$927$	−0.135266	−0.00444272
$928$	0	0
$929$	41.5725	1.36395	0.681974	−	0.731376i	$-0.261122\pi$
$929$	41.5725	1.36395	0.681974	+	0.731376i	$0.261122\pi$
$930$	0	0
$931$	−23.8224	−0.780748
$932$	0	0
$933$	−15.8947	−0.520370
$934$	0	0
$935$	−28.9154	−0.945634
$936$	0	0
$937$	49.3131	1.61099	0.805495	−	0.592603i	$-0.201900\pi$
$937$	49.3131	1.61099	0.805495	+	0.592603i	$0.201900\pi$
$938$	0	0
$939$	2.85491	0.0931666
$940$	0	0
$941$	−17.4601	−0.569184	−0.284592	−	0.958649i	$-0.591858\pi$
$941$	−17.4601	−0.569184	−0.284592	+	0.958649i	$0.591858\pi$
$942$	0	0
$943$	−58.8546	−1.91657
$944$	0	0
$945$	9.12481	0.296830
$946$	0	0
$947$	34.3319	1.11564	0.557818	−	0.829963i	$-0.311639\pi$
$947$	34.3319	1.11564	0.557818	+	0.829963i	$0.311639\pi$
$948$	0	0
$949$	−15.1969	−0.493312
$950$	0	0
$951$	3.11186	0.100909
$952$	0	0
$953$	42.7429	1.38458	0.692289	−	0.721621i	$-0.256602\pi$
$953$	42.7429	1.38458	0.692289	+	0.721621i	$0.256602\pi$
$954$	0	0
$955$	−9.67214	−0.312983
$956$	0	0
$957$	−22.2493	−0.719217
$958$	0	0
$959$	65.1587	2.10408
$960$	0	0
$961$	7.48449	0.241435
$962$	0	0
$963$	13.4430	0.433195
$964$	0	0
$965$	19.1550	0.616622
$966$	0	0
$967$	36.4676	1.17272	0.586359	−	0.810051i	$-0.300561\pi$
$967$	36.4676	1.17272	0.586359	+	0.810051i	$0.300561\pi$
$968$	0	0
$969$	−10.0266	−0.322102
$970$	0	0
$971$	20.1056	0.645219	0.322609	−	0.946532i	$-0.395440\pi$
$971$	20.1056	0.645219	0.322609	+	0.946532i	$0.395440\pi$
$972$	0	0
$973$	−19.9736	−0.640325
$974$	0	0
$975$	−0.589367	−0.0188748
$976$	0	0
$977$	−52.4746	−1.67881	−0.839406	−	0.543504i	$-0.817097\pi$
$977$	−52.4746	−1.67881	−0.839406	+	0.543504i	$0.817097\pi$
$978$	0	0
$979$	51.1435	1.63455
$980$	0	0
$981$	−14.7691	−0.471542
$982$	0	0
$983$	−0.619139	−0.0197475	−0.00987373	−	0.999951i	$-0.503143\pi$
$983$	−0.619139	−0.0197475	−0.00987373	+	0.999951i	$0.503143\pi$
$984$	0	0
$985$	−53.1628	−1.69391
$986$	0	0
$987$	−5.73725	−0.182619
$988$	0	0
$989$	2.71924	0.0864669
$990$	0	0
$991$	20.6423	0.655726	0.327863	−	0.944725i	$-0.393672\pi$
$991$	20.6423	0.655726	0.327863	+	0.944725i	$0.393672\pi$
$992$	0	0
$993$	22.1198	0.701949
$994$	0	0
$995$	−29.1529	−0.924210
$996$	0	0
$997$	35.8162	1.13431	0.567155	−	0.823611i	$-0.308044\pi$
$997$	35.8162	1.13431	0.567155	+	0.823611i	$0.308044\pi$
$998$	0	0
$999$	6.08861	0.192635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7248.2.a.bm.1.7		10
4.3	odd	2		3624.2.a.l.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3624.2.a.l.1.7	✓	10	4.3	odd	2
7248.2.a.bm.1.7		10	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:	\( N \)	\(=\)	\( 7248 = 2^{4} \cdot 3 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7248.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.35125 q^{5} -3.88084 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.35125 q^{5} -3.88084 q^{7} +1.00000 q^{9} +3.62475 q^{11} +1.11544 q^{13} -2.35125 q^{15} -3.39276 q^{17} -2.95531 q^{19} +3.88084 q^{21} -4.91731 q^{23} +0.528370 q^{25} -1.00000 q^{27} +6.13816 q^{29} +6.20359 q^{31} -3.62475 q^{33} -9.12481 q^{35} -6.08861 q^{37} -1.11544 q^{39} +11.9689 q^{41} -0.552994 q^{43} +2.35125 q^{45} -1.47835 q^{47} +8.06089 q^{49} +3.39276 q^{51} +0.529419 q^{53} +8.52268 q^{55} +2.95531 q^{57} -15.1543 q^{59} -0.593677 q^{61} -3.88084 q^{63} +2.62269 q^{65} -6.27209 q^{67} +4.91731 q^{69} +10.2326 q^{71} -13.6241 q^{73} -0.528370 q^{75} -14.0671 q^{77} -12.2683 q^{79} +1.00000 q^{81} -13.7919 q^{83} -7.97721 q^{85} -6.13816 q^{87} +14.1095 q^{89} -4.32886 q^{91} -6.20359 q^{93} -6.94866 q^{95} -10.8690 q^{97} +3.62475 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9 - 7 * q^11 + 6 * q^13 - 2 * q^15 + 7 * q^17 + 8 * q^21 - 25 * q^23 + 8 * q^25 - 10 * q^27 + 12 * q^29 - 11 * q^31 + 7 * q^33 - 9 * q^35 - 3 * q^37 - 6 * q^39 + 12 * q^41 + 2 * q^45 - 31 * q^47 + 14 * q^49 - 7 * q^51 + q^53 - 9 * q^55 - 19 * q^59 + 24 * q^61 - 8 * q^63 + 20 * q^65 + q^67 + 25 * q^69 - 34 * q^71 - 18 * q^73 - 8 * q^75 + 27 * q^77 - 25 * q^79 + 10 * q^81 - 14 * q^83 - 3 * q^85 - 12 * q^87 + 20 * q^89 + 12 * q^91 + 11 * q^93 - 48 * q^95 - 15 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

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