LMFDB - Embedded newform 7488.2.a.ch.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7488,2,Mod(1,7488)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7488, 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("7488.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7488 = 2^{6} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7488.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:	$59.7919810335$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1248)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7488.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607 q^{5} +0.763932 q^{7} -4.47214 q^{11} +1.00000 q^{13} +4.47214 q^{17} -3.23607 q^{19} -6.47214 q^{23} +5.47214 q^{25} +0.472136 q^{29} +4.76393 q^{31} -2.47214 q^{35} -4.47214 q^{37} +4.76393 q^{41} +2.47214 q^{43} -8.47214 q^{47} -6.41641 q^{49} -8.47214 q^{53} +14.4721 q^{55} +10.9443 q^{59} -12.4721 q^{61} -3.23607 q^{65} -5.70820 q^{67} -10.0000 q^{71} +4.47214 q^{73} -3.41641 q^{77} -8.94427 q^{79} +14.9443 q^{83} -14.4721 q^{85} +2.29180 q^{89} +0.763932 q^{91} +10.4721 q^{95} +7.52786 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 6 q^{7} + 2 q^{13} - 2 q^{19} - 4 q^{23} + 2 q^{25} - 8 q^{29} + 14 q^{31} + 4 q^{35} + 14 q^{41} - 4 q^{43} - 8 q^{47} + 14 q^{49} - 8 q^{53} + 20 q^{55} + 4 q^{59} - 16 q^{61} - 2 q^{65}+ \cdots + 24 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 6 * q^7 + 2 * q^13 - 2 * q^19 - 4 * q^23 + 2 * q^25 - 8 * q^29 + 14 * q^31 + 4 * q^35 + 14 * q^41 - 4 * q^43 - 8 * q^47 + 14 * q^49 - 8 * q^53 + 20 * q^55 + 4 * q^59 - 16 * q^61 - 2 * q^65 + 2 * q^67 - 20 * q^71 + 20 * q^77 + 12 * q^83 - 20 * q^85 + 18 * q^89 + 6 * q^91 + 12 * q^95 + 24 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	0.763932	0.288739	0.144370	−	0.989524i	$-0.453885\pi$
$7$	0.763932	0.288739	0.144370	+	0.989524i	$0.453885\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	−3.23607	−0.742405	−0.371202	−	0.928552i	$-0.621054\pi$
$19$	−3.23607	−0.742405	−0.371202	+	0.928552i	$0.621054\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.47214	−1.34953	−0.674767	−	0.738031i	$-0.735756\pi$
$23$	−6.47214	−1.34953	−0.674767	+	0.738031i	$0.735756\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	4.76393	0.855627	0.427814	−	0.903867i	$-0.359284\pi$
$31$	4.76393	0.855627	0.427814	+	0.903867i	$0.359284\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.47214	−0.417867
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.76393	0.744001	0.372001	−	0.928232i	$-0.378672\pi$
$41$	4.76393	0.744001	0.372001	+	0.928232i	$0.378672\pi$
$42$	0	0
$43$	2.47214	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$43$	2.47214	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.47214	−1.23579	−0.617894	−	0.786261i	$-0.712014\pi$
$47$	−8.47214	−1.23579	−0.617894	+	0.786261i	$0.712014\pi$
$48$	0	0
$49$	−6.41641	−0.916630
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	14.4721	1.95142
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.9443	1.42482	0.712411	−	0.701762i	$-0.247603\pi$
$59$	10.9443	1.42482	0.712411	+	0.701762i	$0.247603\pi$
$60$	0	0
$61$	−12.4721	−1.59689	−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721	−1.59689	−0.798447	+	0.602066i	$0.794345\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.23607	−0.401385
$66$	0	0
$67$	−5.70820	−0.697368	−0.348684	−	0.937240i	$-0.613371\pi$
$67$	−5.70820	−0.697368	−0.348684	+	0.937240i	$0.613371\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	4.47214	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$73$	4.47214	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.41641	−0.389336
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.9443	1.64035	0.820173	−	0.572115i	$-0.193877\pi$
$83$	14.9443	1.64035	0.820173	+	0.572115i	$0.193877\pi$
$84$	0	0
$85$	−14.4721	−1.56972
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.29180	0.242930	0.121465	−	0.992596i	$-0.461241\pi$
$89$	2.29180	0.242930	0.121465	+	0.992596i	$0.461241\pi$
$90$	0	0
$91$	0.763932	0.0800818
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.4721	1.07442
$96$	0	0
$97$	7.52786	0.764339	0.382169	−	0.924092i	$-0.375177\pi$
$97$	7.52786	0.764339	0.382169	+	0.924092i	$0.375177\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	15.4164	1.51902	0.759512	−	0.650493i	$-0.225438\pi$
$103$	15.4164	1.51902	0.759512	+	0.650493i	$0.225438\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.47214	−0.238990	−0.119495	−	0.992835i	$-0.538128\pi$
$107$	−2.47214	−0.238990	−0.119495	+	0.992835i	$0.538128\pi$
$108$	0	0
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.9443	−1.02955	−0.514775	−	0.857325i	$-0.672125\pi$
$113$	−10.9443	−1.02955	−0.514775	+	0.857325i	$0.672125\pi$
$114$	0	0
$115$	20.9443	1.95306
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.41641	0.313182
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−15.4164	−1.36798	−0.683992	−	0.729489i	$-0.739758\pi$
$127$	−15.4164	−1.36798	−0.683992	+	0.729489i	$0.739758\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.4721	−1.61392	−0.806959	−	0.590607i	$-0.798888\pi$
$131$	−18.4721	−1.61392	−0.806959	+	0.590607i	$0.798888\pi$
$132$	0	0
$133$	−2.47214	−0.214361
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.6525	1.93533	0.967666	−	0.252236i	$-0.0811658\pi$
$137$	22.6525	1.93533	0.967666	+	0.252236i	$0.0811658\pi$
$138$	0	0
$139$	−0.944272	−0.0800921	−0.0400460	−	0.999198i	$-0.512750\pi$
$139$	−0.944272	−0.0800921	−0.0400460	+	0.999198i	$0.512750\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.47214	−0.373979
$144$	0	0
$145$	−1.52786	−0.126882
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.1803	0.997852	0.498926	−	0.866644i	$-0.333728\pi$
$149$	12.1803	0.997852	0.498926	+	0.866644i	$0.333728\pi$
$150$	0	0
$151$	8.18034	0.665707	0.332853	−	0.942979i	$-0.391989\pi$
$151$	8.18034	0.665707	0.332853	+	0.942979i	$0.391989\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−15.4164	−1.23828
$156$	0	0
$157$	12.4721	0.995385	0.497692	−	0.867354i	$-0.334181\pi$
$157$	12.4721	0.995385	0.497692	+	0.867354i	$0.334181\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.94427	−0.389663
$162$	0	0
$163$	11.2361	0.880077	0.440038	−	0.897979i	$-0.354965\pi$
$163$	11.2361	0.880077	0.440038	+	0.897979i	$0.354965\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.52786	−0.272994	−0.136497	−	0.990640i	$-0.543584\pi$
$167$	−3.52786	−0.272994	−0.136497	+	0.990640i	$0.543584\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.94427	0.527963	0.263982	−	0.964528i	$-0.414964\pi$
$173$	6.94427	0.527963	0.263982	+	0.964528i	$0.414964\pi$
$174$	0	0
$175$	4.18034	0.316004
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.4721	−1.08170	−0.540849	−	0.841120i	$-0.681897\pi$
$179$	−14.4721	−1.08170	−0.540849	+	0.841120i	$0.681897\pi$
$180$	0	0
$181$	1.41641	0.105281	0.0526404	−	0.998614i	$-0.483236\pi$
$181$	1.41641	0.105281	0.0526404	+	0.998614i	$0.483236\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	14.4721	1.06401
$186$	0	0
$187$	−20.0000	−1.46254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.4164	1.69435	0.847176	−	0.531313i	$-0.178301\pi$
$191$	23.4164	1.69435	0.847176	+	0.531313i	$0.178301\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.6525	1.04395	0.521973	−	0.852962i	$-0.325196\pi$
$197$	14.6525	1.04395	0.521973	+	0.852962i	$0.325196\pi$
$198$	0	0
$199$	10.4721	0.742350	0.371175	−	0.928563i	$-0.378955\pi$
$199$	10.4721	0.742350	0.371175	+	0.928563i	$0.378955\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.360680	0.0253148
$204$	0	0
$205$	−15.4164	−1.07673
$206$	0	0
$207$	0	0
$208$	0	0
$209$	14.4721	1.00106
$210$	0	0
$211$	8.94427	0.615749	0.307875	−	0.951427i	$-0.400382\pi$
$211$	8.94427	0.615749	0.307875	+	0.951427i	$0.400382\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	3.63932	0.247053
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.47214	0.300828
$222$	0	0
$223$	−16.1803	−1.08352	−0.541758	−	0.840535i	$-0.682241\pi$
$223$	−16.1803	−1.08352	−0.541758	+	0.840535i	$0.682241\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.4164	1.68695	0.843473	−	0.537171i	$-0.180507\pi$
$227$	25.4164	1.68695	0.843473	+	0.537171i	$0.180507\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.5279	1.27931	0.639656	−	0.768661i	$-0.279077\pi$
$233$	19.5279	1.27931	0.639656	+	0.768661i	$0.279077\pi$
$234$	0	0
$235$	27.4164	1.78845
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.8885	−0.769006	−0.384503	−	0.923124i	$-0.625627\pi$
$239$	−11.8885	−0.769006	−0.384503	+	0.923124i	$0.625627\pi$
$240$	0	0
$241$	25.4164	1.63721	0.818607	−	0.574354i	$-0.194747\pi$
$241$	25.4164	1.63721	0.818607	+	0.574354i	$0.194747\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.7639	1.32656
$246$	0	0
$247$	−3.23607	−0.205906
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	0	0
$253$	28.9443	1.81971
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.4164	−1.58543	−0.792716	−	0.609591i	$-0.791334\pi$
$257$	−25.4164	−1.58543	−0.792716	+	0.609591i	$0.791334\pi$
$258$	0	0
$259$	−3.41641	−0.212285
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	27.4164	1.68418
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.41641	−0.574129	−0.287064	−	0.957911i	$-0.592679\pi$
$269$	−9.41641	−0.574129	−0.287064	+	0.957911i	$0.592679\pi$
$270$	0	0
$271$	18.6525	1.13306	0.566529	−	0.824042i	$-0.308286\pi$
$271$	18.6525	1.13306	0.566529	+	0.824042i	$0.308286\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−24.4721	−1.47573
$276$	0	0
$277$	−18.9443	−1.13825	−0.569125	−	0.822251i	$-0.692718\pi$
$277$	−18.9443	−1.13825	−0.569125	+	0.822251i	$0.692718\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.7082	1.05638	0.528191	−	0.849125i	$-0.322870\pi$
$281$	17.7082	1.05638	0.528191	+	0.849125i	$0.322870\pi$
$282$	0	0
$283$	−21.5279	−1.27970	−0.639849	−	0.768500i	$-0.721003\pi$
$283$	−21.5279	−1.27970	−0.639849	+	0.768500i	$0.721003\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.63932	0.214822
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−15.5967	−0.911172	−0.455586	−	0.890192i	$-0.650570\pi$
$293$	−15.5967	−0.911172	−0.455586	+	0.890192i	$0.650570\pi$
$294$	0	0
$295$	−35.4164	−2.06202
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.47214	−0.374293
$300$	0	0
$301$	1.88854	0.108854
$302$	0	0
$303$	0	0
$304$	0	0
$305$	40.3607	2.31105
$306$	0	0
$307$	21.7082	1.23895	0.619476	−	0.785015i	$-0.287345\pi$
$307$	21.7082	1.23895	0.619476	+	0.785015i	$0.287345\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	17.5279	0.993914	0.496957	−	0.867775i	$-0.334451\pi$
$311$	17.5279	0.993914	0.496957	+	0.867775i	$0.334451\pi$
$312$	0	0
$313$	31.8885	1.80245	0.901224	−	0.433355i	$-0.142670\pi$
$313$	31.8885	1.80245	0.901224	+	0.433355i	$0.142670\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.6525	0.822965	0.411483	−	0.911418i	$-0.365011\pi$
$317$	14.6525	0.822965	0.411483	+	0.911418i	$0.365011\pi$
$318$	0	0
$319$	−2.11146	−0.118219
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14.4721	−0.805251
$324$	0	0
$325$	5.47214	0.303539
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.47214	−0.356820
$330$	0	0
$331$	−8.76393	−0.481709	−0.240855	−	0.970561i	$-0.577428\pi$
$331$	−8.76393	−0.481709	−0.240855	+	0.970561i	$0.577428\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	18.4721	1.00924
$336$	0	0
$337$	−5.05573	−0.275403	−0.137702	−	0.990474i	$-0.543971\pi$
$337$	−5.05573	−0.275403	−0.137702	+	0.990474i	$0.543971\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−21.3050	−1.15373
$342$	0	0
$343$	−10.2492	−0.553406
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	34.3607	1.83929	0.919643	−	0.392756i	$-0.128478\pi$
$349$	34.3607	1.83929	0.919643	+	0.392756i	$0.128478\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.7082	1.36831	0.684155	−	0.729337i	$-0.260171\pi$
$353$	25.7082	1.36831	0.684155	+	0.729337i	$0.260171\pi$
$354$	0	0
$355$	32.3607	1.71753
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.41641	0.496979	0.248489	−	0.968635i	$-0.420066\pi$
$359$	9.41641	0.496979	0.248489	+	0.968635i	$0.420066\pi$
$360$	0	0
$361$	−8.52786	−0.448835
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14.4721	−0.757506
$366$	0	0
$367$	32.9443	1.71968	0.859838	−	0.510566i	$-0.170564\pi$
$367$	32.9443	1.71968	0.859838	+	0.510566i	$0.170564\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.47214	−0.336017
$372$	0	0
$373$	−6.94427	−0.359561	−0.179780	−	0.983707i	$-0.557539\pi$
$373$	−6.94427	−0.359561	−0.179780	+	0.983707i	$0.557539\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.472136	0.0243162
$378$	0	0
$379$	−26.0689	−1.33907	−0.669534	−	0.742781i	$-0.733506\pi$
$379$	−26.0689	−1.33907	−0.669534	+	0.742781i	$0.733506\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.4164	1.09433	0.547164	−	0.837026i	$-0.315708\pi$
$383$	21.4164	1.09433	0.547164	+	0.837026i	$0.315708\pi$
$384$	0	0
$385$	11.0557	0.563452
$386$	0	0
$387$	0	0
$388$	0	0
$389$	31.8885	1.61681	0.808407	−	0.588624i	$-0.200330\pi$
$389$	31.8885	1.61681	0.808407	+	0.588624i	$0.200330\pi$
$390$	0	0
$391$	−28.9443	−1.46377
$392$	0	0
$393$	0	0
$394$	0	0
$395$	28.9443	1.45634
$396$	0	0
$397$	−12.4721	−0.625959	−0.312979	−	0.949760i	$-0.601327\pi$
$397$	−12.4721	−0.625959	−0.312979	+	0.949760i	$0.601327\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.7082	−1.08406	−0.542028	−	0.840360i	$-0.682343\pi$
$401$	−21.7082	−1.08406	−0.542028	+	0.840360i	$0.682343\pi$
$402$	0	0
$403$	4.76393	0.237308
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.0000	0.991363
$408$	0	0
$409$	22.3607	1.10566	0.552832	−	0.833293i	$-0.313547\pi$
$409$	22.3607	1.10566	0.552832	+	0.833293i	$0.313547\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.36068	0.411402
$414$	0	0
$415$	−48.3607	−2.37393
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.52786	0.270054	0.135027	−	0.990842i	$-0.456888\pi$
$419$	5.52786	0.270054	0.135027	+	0.990842i	$0.456888\pi$
$420$	0	0
$421$	−6.94427	−0.338443	−0.169222	−	0.985578i	$-0.554125\pi$
$421$	−6.94427	−0.338443	−0.169222	+	0.985578i	$0.554125\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	24.4721	1.18707
$426$	0	0
$427$	−9.52786	−0.461086
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.52786	0.169931	0.0849656	−	0.996384i	$-0.472922\pi$
$431$	3.52786	0.169931	0.0849656	+	0.996384i	$0.472922\pi$
$432$	0	0
$433$	−25.4164	−1.22143	−0.610717	−	0.791849i	$-0.709119\pi$
$433$	−25.4164	−1.22143	−0.610717	+	0.791849i	$0.709119\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.9443	1.00190
$438$	0	0
$439$	7.41641	0.353966	0.176983	−	0.984214i	$-0.443366\pi$
$439$	7.41641	0.353966	0.176983	+	0.984214i	$0.443366\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.4721	0.877638	0.438819	−	0.898576i	$-0.355397\pi$
$443$	18.4721	0.877638	0.438819	+	0.898576i	$0.355397\pi$
$444$	0	0
$445$	−7.41641	−0.351571
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.34752	0.252365	0.126183	−	0.992007i	$-0.459727\pi$
$449$	5.34752	0.252365	0.126183	+	0.992007i	$0.459727\pi$
$450$	0	0
$451$	−21.3050	−1.00321
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.47214	−0.115896
$456$	0	0
$457$	36.4721	1.70609	0.853047	−	0.521834i	$-0.174752\pi$
$457$	36.4721	1.70609	0.853047	+	0.521834i	$0.174752\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.23607	−0.150719	−0.0753594	−	0.997156i	$-0.524010\pi$
$461$	−3.23607	−0.150719	−0.0753594	+	0.997156i	$0.524010\pi$
$462$	0	0
$463$	−36.1803	−1.68144	−0.840721	−	0.541468i	$-0.817869\pi$
$463$	−36.1803	−1.68144	−0.840721	+	0.541468i	$0.817869\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.0557	0.881794	0.440897	−	0.897558i	$-0.354660\pi$
$467$	19.0557	0.881794	0.440897	+	0.897558i	$0.354660\pi$
$468$	0	0
$469$	−4.36068	−0.201357
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.0557	−0.508343
$474$	0	0
$475$	−17.7082	−0.812508
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	−4.47214	−0.203912
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−24.3607	−1.10616
$486$	0	0
$487$	14.6525	0.663967	0.331984	−	0.943285i	$-0.392282\pi$
$487$	14.6525	0.663967	0.331984	+	0.943285i	$0.392282\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.8885	1.16833	0.584167	−	0.811634i	$-0.301421\pi$
$491$	25.8885	1.16833	0.584167	+	0.811634i	$0.301421\pi$
$492$	0	0
$493$	2.11146	0.0950952
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.63932	−0.342670
$498$	0	0
$499$	−24.7639	−1.10859	−0.554293	−	0.832322i	$-0.687011\pi$
$499$	−24.7639	−1.10859	−0.554293	+	0.832322i	$0.687011\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.05573	0.314599	0.157300	−	0.987551i	$-0.449721\pi$
$503$	7.05573	0.314599	0.157300	+	0.987551i	$0.449721\pi$
$504$	0	0
$505$	−32.3607	−1.44003
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.8197	0.701194	0.350597	−	0.936526i	$-0.385979\pi$
$509$	15.8197	0.701194	0.350597	+	0.936526i	$0.385979\pi$
$510$	0	0
$511$	3.41641	0.151133
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−49.8885	−2.19835
$516$	0	0
$517$	37.8885	1.66634
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.8885	1.57231	0.786153	−	0.618032i	$-0.212070\pi$
$521$	35.8885	1.57231	0.786153	+	0.618032i	$0.212070\pi$
$522$	0	0
$523$	−37.8885	−1.65675	−0.828375	−	0.560174i	$-0.810734\pi$
$523$	−37.8885	−1.65675	−0.828375	+	0.560174i	$0.810734\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.3050	0.928058
$528$	0	0
$529$	18.8885	0.821241
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.76393	0.206349
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	0	0
$538$	0	0
$539$	28.6950	1.23598
$540$	0	0
$541$	33.7771	1.45219	0.726095	−	0.687594i	$-0.241333\pi$
$541$	33.7771	1.45219	0.726095	+	0.687594i	$0.241333\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.52786	−0.408129
$546$	0	0
$547$	−31.7771	−1.35869	−0.679345	−	0.733819i	$-0.737736\pi$
$547$	−31.7771	−1.35869	−0.679345	+	0.733819i	$0.737736\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.52786	−0.0650892
$552$	0	0
$553$	−6.83282	−0.290561
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0689	1.27406	0.637030	−	0.770839i	$-0.280163\pi$
$557$	30.0689	1.27406	0.637030	+	0.770839i	$0.280163\pi$
$558$	0	0
$559$	2.47214	0.104560
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.47214	0.272768	0.136384	−	0.990656i	$-0.456452\pi$
$563$	6.47214	0.272768	0.136384	+	0.990656i	$0.456452\pi$
$564$	0	0
$565$	35.4164	1.48998
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.4164	−0.730134	−0.365067	−	0.930981i	$-0.618954\pi$
$569$	−17.4164	−0.730134	−0.365067	+	0.930981i	$0.618954\pi$
$570$	0	0
$571$	15.4164	0.645157	0.322578	−	0.946543i	$-0.395450\pi$
$571$	15.4164	0.645157	0.322578	+	0.946543i	$0.395450\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−35.4164	−1.47697
$576$	0	0
$577$	−28.8328	−1.20033	−0.600163	−	0.799878i	$-0.704898\pi$
$577$	−28.8328	−1.20033	−0.600163	+	0.799878i	$0.704898\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.4164	0.473632
$582$	0	0
$583$	37.8885	1.56918
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.05573	0.208672	0.104336	−	0.994542i	$-0.466728\pi$
$587$	5.05573	0.208672	0.104336	+	0.994542i	$0.466728\pi$
$588$	0	0
$589$	−15.4164	−0.635222
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26.6525	−1.09449	−0.547243	−	0.836974i	$-0.684323\pi$
$593$	−26.6525	−1.09449	−0.547243	+	0.836974i	$0.684323\pi$
$594$	0	0
$595$	−11.0557	−0.453241
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−0.111456	−0.00454639	−0.00227320	−	0.999997i	$-0.500724\pi$
$601$	−0.111456	−0.00454639	−0.00227320	+	0.999997i	$0.500724\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−29.1246	−1.18408
$606$	0	0
$607$	0.944272	0.0383268	0.0191634	−	0.999816i	$-0.493900\pi$
$607$	0.944272	0.0383268	0.0191634	+	0.999816i	$0.493900\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.47214	−0.342746
$612$	0	0
$613$	8.47214	0.342186	0.171093	−	0.985255i	$-0.445270\pi$
$613$	8.47214	0.342186	0.171093	+	0.985255i	$0.445270\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.0689	−0.727426	−0.363713	−	0.931511i	$-0.618491\pi$
$617$	−18.0689	−0.727426	−0.363713	+	0.931511i	$0.618491\pi$
$618$	0	0
$619$	39.5967	1.59153	0.795764	−	0.605607i	$-0.207070\pi$
$619$	39.5967	1.59153	0.795764	+	0.605607i	$0.207070\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.75078	0.0701434
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−1.70820	−0.0680025	−0.0340013	−	0.999422i	$-0.510825\pi$
$631$	−1.70820	−0.0680025	−0.0340013	+	0.999422i	$0.510825\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	49.8885	1.97977
$636$	0	0
$637$	−6.41641	−0.254227
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.47214	0.176639	0.0883194	−	0.996092i	$-0.471850\pi$
$641$	4.47214	0.176639	0.0883194	+	0.996092i	$0.471850\pi$
$642$	0	0
$643$	−19.8197	−0.781611	−0.390806	−	0.920473i	$-0.627804\pi$
$643$	−19.8197	−0.781611	−0.390806	+	0.920473i	$0.627804\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	−48.9443	−1.92123
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.5279	0.451120	0.225560	−	0.974229i	$-0.427579\pi$
$653$	11.5279	0.451120	0.225560	+	0.974229i	$0.427579\pi$
$654$	0	0
$655$	59.7771	2.33568
$656$	0	0
$657$	0	0
$658$	0	0
$659$	42.8328	1.66853	0.834265	−	0.551364i	$-0.185892\pi$
$659$	42.8328	1.66853	0.834265	+	0.551364i	$0.185892\pi$
$660$	0	0
$661$	19.5279	0.759546	0.379773	−	0.925080i	$-0.376002\pi$
$661$	19.5279	0.759546	0.379773	+	0.925080i	$0.376002\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	−3.05573	−0.118318
$668$	0	0
$669$	0	0
$670$	0	0
$671$	55.7771	2.15325
$672$	0	0
$673$	−35.5279	−1.36950	−0.684749	−	0.728779i	$-0.740088\pi$
$673$	−35.5279	−1.36950	−0.684749	+	0.728779i	$0.740088\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25.4164	0.976832	0.488416	−	0.872611i	$-0.337575\pi$
$677$	25.4164	0.976832	0.488416	+	0.872611i	$0.337575\pi$
$678$	0	0
$679$	5.75078	0.220695
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2.58359	−0.0988584	−0.0494292	−	0.998778i	$-0.515740\pi$
$683$	−2.58359	−0.0988584	−0.0494292	+	0.998778i	$0.515740\pi$
$684$	0	0
$685$	−73.3050	−2.80084
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.47214	−0.322763
$690$	0	0
$691$	33.1246	1.26012	0.630060	−	0.776547i	$-0.283030\pi$
$691$	33.1246	1.26012	0.630060	+	0.776547i	$0.283030\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.05573	0.115910
$696$	0	0
$697$	21.3050	0.806983
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.88854	0.297946	0.148973	−	0.988841i	$-0.452403\pi$
$701$	7.88854	0.297946	0.148973	+	0.988841i	$0.452403\pi$
$702$	0	0
$703$	14.4721	0.545827
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.63932	0.287306
$708$	0	0
$709$	4.11146	0.154409	0.0772045	−	0.997015i	$-0.475401\pi$
$709$	4.11146	0.154409	0.0772045	+	0.997015i	$0.475401\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−30.8328	−1.15470
$714$	0	0
$715$	14.4721	0.541227
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.47214	−0.241370	−0.120685	−	0.992691i	$-0.538509\pi$
$719$	−6.47214	−0.241370	−0.120685	+	0.992691i	$0.538509\pi$
$720$	0	0
$721$	11.7771	0.438602
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.58359	0.0959522
$726$	0	0
$727$	−28.3607	−1.05184	−0.525920	−	0.850534i	$-0.676279\pi$
$727$	−28.3607	−1.05184	−0.525920	+	0.850534i	$0.676279\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.0557	0.408911
$732$	0	0
$733$	7.88854	0.291370	0.145685	−	0.989331i	$-0.453461\pi$
$733$	7.88854	0.291370	0.145685	+	0.989331i	$0.453461\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.5279	0.940331
$738$	0	0
$739$	8.76393	0.322386	0.161193	−	0.986923i	$-0.448466\pi$
$739$	8.76393	0.322386	0.161193	+	0.986923i	$0.448466\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.9443	−1.42873	−0.714363	−	0.699775i	$-0.753284\pi$
$743$	−38.9443	−1.42873	−0.714363	+	0.699775i	$0.753284\pi$
$744$	0	0
$745$	−39.4164	−1.44411
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.88854	−0.0690059
$750$	0	0
$751$	−34.4721	−1.25791	−0.628953	−	0.777443i	$-0.716516\pi$
$751$	−34.4721	−1.25791	−0.628953	+	0.777443i	$0.716516\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−26.4721	−0.963420
$756$	0	0
$757$	21.4164	0.778393	0.389196	−	0.921155i	$-0.372753\pi$
$757$	21.4164	0.778393	0.389196	+	0.921155i	$0.372753\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24.1803	−0.876537	−0.438268	−	0.898844i	$-0.644408\pi$
$761$	−24.1803	−0.876537	−0.438268	+	0.898844i	$0.644408\pi$
$762$	0	0
$763$	2.24922	0.0814274
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.9443	0.395175
$768$	0	0
$769$	−2.94427	−0.106173	−0.0530866	−	0.998590i	$-0.516906\pi$
$769$	−2.94427	−0.106173	−0.0530866	+	0.998590i	$0.516906\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.1803	1.58906	0.794528	−	0.607227i	$-0.207718\pi$
$773$	44.1803	1.58906	0.794528	+	0.607227i	$0.207718\pi$
$774$	0	0
$775$	26.0689	0.936422
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−15.4164	−0.552350
$780$	0	0
$781$	44.7214	1.60026
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−40.3607	−1.44053
$786$	0	0
$787$	−39.2361	−1.39861	−0.699307	−	0.714821i	$-0.746508\pi$
$787$	−39.2361	−1.39861	−0.699307	+	0.714821i	$0.746508\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.36068	−0.297272
$792$	0	0
$793$	−12.4721	−0.442899
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.7771	−1.05476	−0.527379	−	0.849630i	$-0.676825\pi$
$797$	−29.7771	−1.05476	−0.527379	+	0.849630i	$0.676825\pi$
$798$	0	0
$799$	−37.8885	−1.34040
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	16.0000	0.563926
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	51.5967	1.81181	0.905903	−	0.423484i	$-0.139193\pi$
$811$	51.5967	1.81181	0.905903	+	0.423484i	$0.139193\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−36.3607	−1.27366
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.1246	1.15606	0.578028	−	0.816017i	$-0.303822\pi$
$821$	33.1246	1.15606	0.578028	+	0.816017i	$0.303822\pi$
$822$	0	0
$823$	−2.47214	−0.0861732	−0.0430866	−	0.999071i	$-0.513719\pi$
$823$	−2.47214	−0.0861732	−0.0430866	+	0.999071i	$0.513719\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.5836	−0.368028	−0.184014	−	0.982924i	$-0.558909\pi$
$827$	−10.5836	−0.368028	−0.184014	+	0.982924i	$0.558909\pi$
$828$	0	0
$829$	−7.52786	−0.261454	−0.130727	−	0.991418i	$-0.541731\pi$
$829$	−7.52786	−0.261454	−0.130727	+	0.991418i	$0.541731\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−28.6950	−0.994224
$834$	0	0
$835$	11.4164	0.395081
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.8885	−1.10091	−0.550457	−	0.834863i	$-0.685547\pi$
$839$	−31.8885	−1.10091	−0.550457	+	0.834863i	$0.685547\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.23607	−0.111324
$846$	0	0
$847$	6.87539	0.236241
$848$	0	0
$849$	0	0
$850$	0	0
$851$	28.9443	0.992197
$852$	0	0
$853$	14.5836	0.499333	0.249666	−	0.968332i	$-0.419679\pi$
$853$	14.5836	0.499333	0.249666	+	0.968332i	$0.419679\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.52786	−0.257147	−0.128573	−	0.991700i	$-0.541040\pi$
$857$	−7.52786	−0.257147	−0.128573	+	0.991700i	$0.541040\pi$
$858$	0	0
$859$	−0.944272	−0.0322181	−0.0161091	−	0.999870i	$-0.505128\pi$
$859$	−0.944272	−0.0322181	−0.0161091	+	0.999870i	$0.505128\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.30495	0.248663	0.124332	−	0.992241i	$-0.460321\pi$
$863$	7.30495	0.248663	0.124332	+	0.992241i	$0.460321\pi$
$864$	0	0
$865$	−22.4721	−0.764076
$866$	0	0
$867$	0	0
$868$	0	0
$869$	40.0000	1.35691
$870$	0	0
$871$	−5.70820	−0.193415
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.16718	−0.0394580
$876$	0	0
$877$	−20.4721	−0.691295	−0.345647	−	0.938364i	$-0.612341\pi$
$877$	−20.4721	−0.691295	−0.345647	+	0.938364i	$0.612341\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−28.4721	−0.959251	−0.479625	−	0.877473i	$-0.659227\pi$
$881$	−28.4721	−0.959251	−0.479625	+	0.877473i	$0.659227\pi$
$882$	0	0
$883$	−5.52786	−0.186027	−0.0930137	−	0.995665i	$-0.529650\pi$
$883$	−5.52786	−0.186027	−0.0930137	+	0.995665i	$0.529650\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.4164	−1.59209	−0.796044	−	0.605239i	$-0.793077\pi$
$887$	−47.4164	−1.59209	−0.796044	+	0.605239i	$0.793077\pi$
$888$	0	0
$889$	−11.7771	−0.394991
$890$	0	0
$891$	0	0
$892$	0	0
$893$	27.4164	0.917455
$894$	0	0
$895$	46.8328	1.56545
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.24922	0.0750158
$900$	0	0
$901$	−37.8885	−1.26225
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.58359	−0.152364
$906$	0	0
$907$	−49.3050	−1.63714	−0.818572	−	0.574404i	$-0.805234\pi$
$907$	−49.3050	−1.63714	−0.818572	+	0.574404i	$0.805234\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	42.4721	1.40716	0.703582	−	0.710614i	$-0.251583\pi$
$911$	42.4721	1.40716	0.703582	+	0.710614i	$0.251583\pi$
$912$	0	0
$913$	−66.8328	−2.21184
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.1115	−0.466001
$918$	0	0
$919$	29.8885	0.985932	0.492966	−	0.870049i	$-0.335913\pi$
$919$	29.8885	0.985932	0.492966	+	0.870049i	$0.335913\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.0000	−0.329154
$924$	0	0
$925$	−24.4721	−0.804639
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.65248	0.218261	0.109130	−	0.994027i	$-0.465193\pi$
$929$	6.65248	0.218261	0.109130	+	0.994027i	$0.465193\pi$
$930$	0	0
$931$	20.7639	0.680510
$932$	0	0
$933$	0	0
$934$	0	0
$935$	64.7214	2.11661
$936$	0	0
$937$	57.1935	1.86843	0.934215	−	0.356710i	$-0.116102\pi$
$937$	57.1935	1.86843	0.934215	+	0.356710i	$0.116102\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−40.7639	−1.32887	−0.664433	−	0.747348i	$-0.731327\pi$
$941$	−40.7639	−1.32887	−0.664433	+	0.747348i	$0.731327\pi$
$942$	0	0
$943$	−30.8328	−1.00405
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.9443	0.355641	0.177821	−	0.984063i	$-0.443095\pi$
$947$	10.9443	0.355641	0.177821	+	0.984063i	$0.443095\pi$
$948$	0	0
$949$	4.47214	0.145172
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13.4164	0.434600	0.217300	−	0.976105i	$-0.430275\pi$
$953$	13.4164	0.434600	0.217300	+	0.976105i	$0.430275\pi$
$954$	0	0
$955$	−75.7771	−2.45209
$956$	0	0
$957$	0	0
$958$	0	0
$959$	17.3050	0.558806
$960$	0	0
$961$	−8.30495	−0.267902
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19.4164	0.625036
$966$	0	0
$967$	9.70820	0.312195	0.156097	−	0.987742i	$-0.450109\pi$
$967$	9.70820	0.312195	0.156097	+	0.987742i	$0.450109\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.8328	−1.37457	−0.687285	−	0.726388i	$-0.741198\pi$
$971$	−42.8328	−1.37457	−0.687285	+	0.726388i	$0.741198\pi$
$972$	0	0
$973$	−0.721360	−0.0231257
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−33.4853	−1.07129	−0.535645	−	0.844443i	$-0.679931\pi$
$977$	−33.4853	−1.07129	−0.535645	+	0.844443i	$0.679931\pi$
$978$	0	0
$979$	−10.2492	−0.327567
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−42.9443	−1.36971	−0.684855	−	0.728680i	$-0.740134\pi$
$983$	−42.9443	−1.36971	−0.684855	+	0.728680i	$0.740134\pi$
$984$	0	0
$985$	−47.4164	−1.51081
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−33.8885	−1.07434
$996$	0	0
$997$	27.8885	0.883239	0.441620	−	0.897202i	$-0.354404\pi$
$997$	27.8885	0.883239	0.441620	+	0.897202i	$0.354404\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7488.2.a.ch.1.1		2
3.2	odd	2		2496.2.a.bj.1.2		2
4.3	odd	2		7488.2.a.cg.1.1		2
8.3	odd	2		3744.2.a.v.1.2		2
8.5	even	2		3744.2.a.w.1.2		2
12.11	even	2		2496.2.a.bg.1.2		2
24.5	odd	2		1248.2.a.k.1.1	✓	2
24.11	even	2		1248.2.a.m.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.a.k.1.1	✓	2	24.5	odd	2
1248.2.a.m.1.1	yes	2	24.11	even	2
2496.2.a.bg.1.2		2	12.11	even	2
2496.2.a.bj.1.2		2	3.2	odd	2
3744.2.a.v.1.2		2	8.3	odd	2
3744.2.a.w.1.2		2	8.5	even	2
7488.2.a.cg.1.1		2	4.3	odd	2
7488.2.a.ch.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:	\( N \)	\(=\)	\( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7488.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23607 q^{5} +0.763932 q^{7} -4.47214 q^{11} +1.00000 q^{13} +4.47214 q^{17} -3.23607 q^{19} -6.47214 q^{23} +5.47214 q^{25} +0.472136 q^{29} +4.76393 q^{31} -2.47214 q^{35} -4.47214 q^{37} +4.76393 q^{41} +2.47214 q^{43} -8.47214 q^{47} -6.41641 q^{49} -8.47214 q^{53} +14.4721 q^{55} +10.9443 q^{59} -12.4721 q^{61} -3.23607 q^{65} -5.70820 q^{67} -10.0000 q^{71} +4.47214 q^{73} -3.41641 q^{77} -8.94427 q^{79} +14.9443 q^{83} -14.4721 q^{85} +2.29180 q^{89} +0.763932 q^{91} +10.4721 q^{95} +7.52786 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 6 q^{7} + 2 q^{13} - 2 q^{19} - 4 q^{23} + 2 q^{25} - 8 q^{29} + 14 q^{31} + 4 q^{35} + 14 q^{41} - 4 q^{43} - 8 q^{47} + 14 q^{49} - 8 q^{53} + 20 q^{55} + 4 q^{59} - 16 q^{61} - 2 q^{65}+ \cdots + 24 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 6 * q^7 + 2 * q^13 - 2 * q^19 - 4 * q^23 + 2 * q^25 - 8 * q^29 + 14 * q^31 + 4 * q^35 + 14 * q^41 - 4 * q^43 - 8 * q^47 + 14 * q^49 - 8 * q^53 + 20 * q^55 + 4 * q^59 - 16 * q^61 - 2 * q^65 + 2 * q^67 - 20 * q^71 + 20 * q^77 + 12 * q^83 - 20 * q^85 + 18 * q^89 + 6 * q^91 + 12 * q^95 + 24 * q^97

Introduction

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