LMFDB - Embedded newform 7600.2.a.cj.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.56310016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 14x^{2} - 4 $ x^6 - 9x^4 + 14x^2 - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.23277$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.62236 q^{3} +4.74397 q^{7} -0.367938 q^{9} +O(q^{10})$	$q+1.62236 q^{3} +4.74397 q^{7} -0.367938 q^{9} -4.48028 q^{11} -0.843176 q^{13} -5.52315 q^{17} -1.00000 q^{19} +7.69643 q^{21} -0.779187 q^{23} -5.46402 q^{27} -10.6570 q^{29} +8.65699 q^{31} -7.26864 q^{33} -1.62236 q^{37} -1.36794 q^{39} +4.73588 q^{41} -9.67504 q^{43} +3.18559 q^{47} +15.5052 q^{49} -8.96056 q^{51} +6.17922 q^{53} -1.62236 q^{57} -11.6964 q^{59} +6.48028 q^{61} -1.74549 q^{63} -14.8880 q^{67} -1.26412 q^{69} -0.303566 q^{71} -10.0800 q^{73} -21.2543 q^{77} -4.00000 q^{79} -7.76081 q^{81} +0.779187 q^{83} -17.2895 q^{87} -5.69643 q^{89} -4.00000 q^{91} +14.0448 q^{93} +6.17922 q^{97} +1.64847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{9}+O(q^{10}) $ 6 * q + 10 * q^9	$ 6 q + 10 q^{9} - 18 q^{11} - 6 q^{19} + 4 q^{21} - 4 q^{29} - 8 q^{31} + 4 q^{39} + 4 q^{41} + 12 q^{49} - 36 q^{51} - 28 q^{59} + 30 q^{61} - 32 q^{69} - 44 q^{71} - 24 q^{79} + 50 q^{81} + 8 q^{89} - 24 q^{91} - 90 q^{99}+O(q^{100}) $ 6 * q + 10 * q^9 - 18 * q^11 - 6 * q^19 + 4 * q^21 - 4 * q^29 - 8 * q^31 + 4 * q^39 + 4 * q^41 + 12 * q^49 - 36 * q^51 - 28 * q^59 + 30 * q^61 - 32 * q^69 - 44 * q^71 - 24 * q^79 + 50 * q^81 + 8 * q^89 - 24 * q^91 - 90 * 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
$3$	1.62236	0.936672	0.468336	−	0.883551i	$-0.344854\pi$
$3$	1.62236	0.936672	0.468336	+	0.883551i	$0.344854\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.74397	1.79305	0.896525	−	0.442993i	$-0.146083\pi$
$7$	4.74397	1.79305	0.896525	+	0.442993i	$0.146083\pi$
$8$	0	0
$9$	−0.367938	−0.122646
$10$	0	0
$11$	−4.48028	−1.35085	−0.675427	−	0.737426i	$-0.736041\pi$
$11$	−4.48028	−1.35085	−0.675427	+	0.737426i	$0.736041\pi$
$12$	0	0
$13$	−0.843176	−0.233855	−0.116928	−	0.993140i	$-0.537305\pi$
$13$	−0.843176	−0.233855	−0.116928	+	0.993140i	$0.537305\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.52315	−1.33956	−0.669781	−	0.742559i	$-0.733612\pi$
$17$	−5.52315	−1.33956	−0.669781	+	0.742559i	$0.733612\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	7.69643	1.67950
$22$	0	0
$23$	−0.779187	−0.162472	−0.0812358	−	0.996695i	$-0.525887\pi$
$23$	−0.779187	−0.162472	−0.0812358	+	0.996695i	$0.525887\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.46402	−1.05155
$28$	0	0
$29$	−10.6570	−1.97895	−0.989477	−	0.144691i	$-0.953781\pi$
$29$	−10.6570	−1.97895	−0.989477	+	0.144691i	$0.953781\pi$
$30$	0	0
$31$	8.65699	1.55484	0.777421	−	0.628981i	$-0.216528\pi$
$31$	8.65699	1.55484	0.777421	+	0.628981i	$0.216528\pi$
$32$	0	0
$33$	−7.26864	−1.26531
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.62236	−0.266715	−0.133357	−	0.991068i	$-0.542576\pi$
$37$	−1.62236	−0.266715	−0.133357	+	0.991068i	$0.542576\pi$
$38$	0	0
$39$	−1.36794	−0.219045
$40$	0	0
$41$	4.73588	0.739620	0.369810	−	0.929107i	$-0.379423\pi$
$41$	4.73588	0.739620	0.369810	+	0.929107i	$0.379423\pi$
$42$	0	0
$43$	−9.67504	−1.47543	−0.737715	−	0.675112i	$-0.764095\pi$
$43$	−9.67504	−1.47543	−0.737715	+	0.675112i	$0.764095\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.18559	0.464666	0.232333	−	0.972636i	$-0.425364\pi$
$47$	3.18559	0.464666	0.232333	+	0.972636i	$0.425364\pi$
$48$	0	0
$49$	15.5052	2.21503
$50$	0	0
$51$	−8.96056	−1.25473
$52$	0	0
$53$	6.17922	0.848780	0.424390	−	0.905479i	$-0.360488\pi$
$53$	6.17922	0.848780	0.424390	+	0.905479i	$0.360488\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.62236	−0.214887
$58$	0	0
$59$	−11.6964	−1.52275	−0.761373	−	0.648314i	$-0.775474\pi$
$59$	−11.6964	−1.52275	−0.761373	+	0.648314i	$0.775474\pi$
$60$	0	0
$61$	6.48028	0.829715	0.414857	−	0.909886i	$-0.363831\pi$
$61$	6.48028	0.829715	0.414857	+	0.909886i	$0.363831\pi$
$62$	0	0
$63$	−1.74549	−0.219911
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−14.8880	−1.81885	−0.909427	−	0.415864i	$-0.863479\pi$
$67$	−14.8880	−1.81885	−0.909427	+	0.415864i	$0.863479\pi$
$68$	0	0
$69$	−1.26412	−0.152183
$70$	0	0
$71$	−0.303566	−0.0360266	−0.0180133	−	0.999838i	$-0.505734\pi$
$71$	−0.303566	−0.0360266	−0.0180133	+	0.999838i	$0.505734\pi$
$72$	0	0
$73$	−10.0800	−1.17978	−0.589888	−	0.807485i	$-0.700828\pi$
$73$	−10.0800	−1.17978	−0.589888	+	0.807485i	$0.700828\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−21.2543	−2.42215
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−7.76081	−0.862312
$82$	0	0
$83$	0.779187	0.0855268	0.0427634	−	0.999085i	$-0.486384\pi$
$83$	0.779187	0.0855268	0.0427634	+	0.999085i	$0.486384\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−17.2895	−1.85363
$88$	0	0
$89$	−5.69643	−0.603821	−0.301910	−	0.953336i	$-0.597624\pi$
$89$	−5.69643	−0.603821	−0.301910	+	0.953336i	$0.597624\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	14.0448	1.45638
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.17922	0.627404	0.313702	−	0.949521i	$-0.398431\pi$
$97$	6.17922	0.627404	0.313702	+	0.949521i	$0.398431\pi$
$98$	0	0
$99$	1.64847	0.165677
$100$	0	0
$101$	3.36794	0.335122	0.167561	−	0.985862i	$-0.446411\pi$
$101$	3.36794	0.335122	0.167561	+	0.985862i	$0.446411\pi$
$102$	0	0
$103$	3.71368	0.365919	0.182960	−	0.983120i	$-0.441432\pi$
$103$	3.71368	0.365919	0.182960	+	0.983120i	$0.441432\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.2031	−0.986374	−0.493187	−	0.869923i	$-0.664168\pi$
$107$	−10.2031	−0.986374	−0.493187	+	0.869923i	$0.664168\pi$
$108$	0	0
$109$	2.96056	0.283570	0.141785	−	0.989897i	$-0.454716\pi$
$109$	2.96056	0.283570	0.141785	+	0.989897i	$0.454716\pi$
$110$	0	0
$111$	−2.63206	−0.249824
$112$	0	0
$113$	−7.08638	−0.666631	−0.333315	−	0.942815i	$-0.608167\pi$
$113$	−7.08638	−0.666631	−0.333315	+	0.942815i	$0.608167\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.310237	0.0286814
$118$	0	0
$119$	−26.2016	−2.40190
$120$	0	0
$121$	9.07290	0.824809
$122$	0	0
$123$	7.68331	0.692781
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−9.79817	−0.869447	−0.434723	−	0.900564i	$-0.643154\pi$
$127$	−9.79817	−0.869447	−0.434723	+	0.900564i	$0.643154\pi$
$128$	0	0
$129$	−15.6964	−1.38199
$130$	0	0
$131$	−12.5841	−1.09948	−0.549739	−	0.835337i	$-0.685273\pi$
$131$	−12.5841	−1.09948	−0.549739	+	0.835337i	$0.685273\pi$
$132$	0	0
$133$	−4.74397	−0.411354
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.89586	0.760024	0.380012	−	0.924981i	$-0.375920\pi$
$137$	8.89586	0.760024	0.380012	+	0.924981i	$0.375920\pi$
$138$	0	0
$139$	−6.25560	−0.530593	−0.265296	−	0.964167i	$-0.585470\pi$
$139$	−6.25560	−0.530593	−0.265296	+	0.964167i	$0.585470\pi$
$140$	0	0
$141$	5.16819	0.435240
$142$	0	0
$143$	3.77767	0.315904
$144$	0	0
$145$	0	0
$146$	0	0
$147$	25.1551	2.07476
$148$	0	0
$149$	2.48028	0.203192	0.101596	−	0.994826i	$-0.467605\pi$
$149$	2.48028	0.203192	0.101596	+	0.994826i	$0.467605\pi$
$150$	0	0
$151$	−3.69643	−0.300812	−0.150406	−	0.988624i	$-0.548058\pi$
$151$	−3.69643	−0.300812	−0.150406	+	0.988624i	$0.548058\pi$
$152$	0	0
$153$	2.03218	0.164292
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.8479	1.50422	0.752112	−	0.659035i	$-0.229035\pi$
$157$	18.8479	1.50422	0.752112	+	0.659035i	$0.229035\pi$
$158$	0	0
$159$	10.0249	0.795029
$160$	0	0
$161$	−3.69643	−0.291320
$162$	0	0
$163$	−2.46554	−0.193116	−0.0965580	−	0.995327i	$-0.530783\pi$
$163$	−2.46554	−0.193116	−0.0965580	+	0.995327i	$0.530783\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.49134	0.579697	0.289849	−	0.957072i	$-0.406395\pi$
$167$	7.49134	0.579697	0.289849	+	0.957072i	$0.406395\pi$
$168$	0	0
$169$	−12.2891	−0.945312
$170$	0	0
$171$	0.367938	0.0281369
$172$	0	0
$173$	−20.0653	−1.52554	−0.762768	−	0.646673i	$-0.776160\pi$
$173$	−20.0653	−1.52554	−0.762768	+	0.646673i	$0.776160\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.9759	−1.42631
$178$	0	0
$179$	3.69643	0.276284	0.138142	−	0.990412i	$-0.455887\pi$
$179$	3.69643	0.276284	0.138142	+	0.990412i	$0.455887\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	10.5134	0.777170
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.7453	1.80955
$188$	0	0
$189$	−25.9211	−1.88548
$190$	0	0
$191$	−0.887659	−0.0642288	−0.0321144	−	0.999484i	$-0.510224\pi$
$191$	−0.887659	−0.0642288	−0.0321144	+	0.999484i	$0.510224\pi$
$192$	0	0
$193$	19.1581	1.37903	0.689516	−	0.724271i	$-0.257823\pi$
$193$	19.1581	1.37903	0.689516	+	0.724271i	$0.257823\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.37271	0.240295	0.120148	−	0.992756i	$-0.461663\pi$
$197$	3.37271	0.240295	0.120148	+	0.992756i	$0.461663\pi$
$198$	0	0
$199$	−4.58409	−0.324958	−0.162479	−	0.986712i	$-0.551949\pi$
$199$	−4.58409	−0.324958	−0.162479	+	0.986712i	$0.551949\pi$
$200$	0	0
$201$	−24.1537	−1.70367
$202$	0	0
$203$	−50.5564	−3.54836
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.286693	0.0199265
$208$	0	0
$209$	4.48028	0.309907
$210$	0	0
$211$	14.8817	1.02450	0.512248	−	0.858837i	$-0.328813\pi$
$211$	14.8817	1.02450	0.512248	+	0.858837i	$0.328813\pi$
$212$	0	0
$213$	−0.492494	−0.0337451
$214$	0	0
$215$	0	0
$216$	0	0
$217$	41.0685	2.78791
$218$	0	0
$219$	−16.3534	−1.10506
$220$	0	0
$221$	4.65699	0.313263
$222$	0	0
$223$	−18.7839	−1.25786	−0.628931	−	0.777461i	$-0.716507\pi$
$223$	−18.7839	−1.25786	−0.628931	+	0.777461i	$0.716507\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.843176	0.0559636	0.0279818	−	0.999608i	$-0.491092\pi$
$227$	0.843176	0.0559636	0.0279818	+	0.999608i	$0.491092\pi$
$228$	0	0
$229$	−4.86273	−0.321338	−0.160669	−	0.987008i	$-0.551365\pi$
$229$	−4.86273	−0.321338	−0.160669	+	0.987008i	$0.551365\pi$
$230$	0	0
$231$	−34.4822	−2.26876
$232$	0	0
$233$	−4.90268	−0.321185	−0.160593	−	0.987021i	$-0.551341\pi$
$233$	−4.90268	−0.321185	−0.160593	+	0.987021i	$0.551341\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.48945	−0.421535
$238$	0	0
$239$	6.20164	0.401151	0.200575	−	0.979678i	$-0.435719\pi$
$239$	6.20164	0.401151	0.200575	+	0.979678i	$0.435719\pi$
$240$	0	0
$241$	−3.26412	−0.210261	−0.105130	−	0.994458i	$-0.533526\pi$
$241$	−3.26412	−0.210261	−0.105130	+	0.994458i	$0.533526\pi$
$242$	0	0
$243$	3.80121	0.243848
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.843176	0.0536500
$248$	0	0
$249$	1.26412	0.0801106
$250$	0	0
$251$	−28.4263	−1.79425	−0.897127	−	0.441773i	$-0.854350\pi$
$251$	−28.4263	−1.79425	−0.897127	+	0.441773i	$0.854350\pi$
$252$	0	0
$253$	3.49097	0.219476
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.55192	−0.595832	−0.297916	−	0.954592i	$-0.596292\pi$
$257$	−9.55192	−0.595832	−0.297916	+	0.954592i	$0.596292\pi$
$258$	0	0
$259$	−7.69643	−0.478233
$260$	0	0
$261$	3.92112	0.242711
$262$	0	0
$263$	9.54706	0.588697	0.294349	−	0.955698i	$-0.404897\pi$
$263$	9.54706	0.588697	0.294349	+	0.955698i	$0.404897\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.24168	−0.565582
$268$	0	0
$269$	−14.3534	−0.875144	−0.437572	−	0.899183i	$-0.644161\pi$
$269$	−14.3534	−0.875144	−0.437572	+	0.899183i	$0.644161\pi$
$270$	0	0
$271$	12.1038	0.735254	0.367627	−	0.929973i	$-0.380170\pi$
$271$	12.1038	0.735254	0.367627	+	0.929973i	$0.380170\pi$
$272$	0	0
$273$	−6.48945	−0.392760
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.59055	0.215735	0.107868	−	0.994165i	$-0.465598\pi$
$277$	3.59055	0.215735	0.107868	+	0.994165i	$0.465598\pi$
$278$	0	0
$279$	−3.18524	−0.190695
$280$	0	0
$281$	26.7069	1.59320	0.796599	−	0.604509i	$-0.206631\pi$
$281$	26.7069	1.59320	0.796599	+	0.604509i	$0.206631\pi$
$282$	0	0
$283$	20.4751	1.21712	0.608559	−	0.793508i	$-0.291748\pi$
$283$	20.4751	1.21712	0.608559	+	0.793508i	$0.291748\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	22.4668	1.32618
$288$	0	0
$289$	13.5052	0.794424
$290$	0	0
$291$	10.0249	0.587672
$292$	0	0
$293$	19.1581	1.11923	0.559615	−	0.828753i	$-0.310949\pi$
$293$	19.1581	1.11923	0.559615	+	0.828753i	$0.310949\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	24.4803	1.42049
$298$	0	0
$299$	0.656992	0.0379948
$300$	0	0
$301$	−45.8981	−2.64552
$302$	0	0
$303$	5.46402	0.313900
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.0495	1.82916	0.914580	−	0.404404i	$-0.132521\pi$
$307$	32.0495	1.82916	0.914580	+	0.404404i	$0.132521\pi$
$308$	0	0
$309$	6.02493	0.342746
$310$	0	0
$311$	20.5301	1.16416	0.582079	−	0.813132i	$-0.302240\pi$
$311$	20.5301	1.16416	0.582079	+	0.813132i	$0.302240\pi$
$312$	0	0
$313$	14.7932	0.836163	0.418082	−	0.908409i	$-0.362703\pi$
$313$	14.7932	0.836163	0.418082	+	0.908409i	$0.362703\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.9485	0.951925	0.475962	−	0.879466i	$-0.342100\pi$
$317$	16.9485	0.951925	0.475962	+	0.879466i	$0.342100\pi$
$318$	0	0
$319$	47.7463	2.67328
$320$	0	0
$321$	−16.5532	−0.923908
$322$	0	0
$323$	5.52315	0.307316
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.80310	0.265612
$328$	0	0
$329$	15.1123	0.833170
$330$	0	0
$331$	22.1287	1.21631	0.608153	−	0.793820i	$-0.291911\pi$
$331$	22.1287	1.21631	0.608153	+	0.793820i	$0.291911\pi$
$332$	0	0
$333$	0.596929	0.0327115
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−27.9948	−1.52498	−0.762488	−	0.647002i	$-0.776022\pi$
$337$	−27.9948	−1.52498	−0.762488	+	0.647002i	$0.776022\pi$
$338$	0	0
$339$	−11.4967	−0.624414
$340$	0	0
$341$	−38.7857	−2.10037
$342$	0	0
$343$	40.3484	2.17861
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.1024	−0.918105	−0.459052	−	0.888409i	$-0.651811\pi$
$347$	−17.1024	−0.918105	−0.459052	+	0.888409i	$0.651811\pi$
$348$	0	0
$349$	−19.2411	−1.02995	−0.514976	−	0.857205i	$-0.672199\pi$
$349$	−19.2411	−1.02995	−0.514976	+	0.857205i	$0.672199\pi$
$350$	0	0
$351$	4.60713	0.245910
$352$	0	0
$353$	−7.42735	−0.395318	−0.197659	−	0.980271i	$-0.563334\pi$
$353$	−7.42735	−0.395318	−0.197659	+	0.980271i	$0.563334\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−42.5086	−2.24979
$358$	0	0
$359$	−28.1767	−1.48711	−0.743555	−	0.668675i	$-0.766862\pi$
$359$	−28.1767	−1.48711	−0.743555	+	0.668675i	$0.766862\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	14.7195	0.772575
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−26.1165	−1.36327	−0.681636	−	0.731692i	$-0.738731\pi$
$367$	−26.1165	−1.36327	−0.681636	+	0.731692i	$0.738731\pi$
$368$	0	0
$369$	−1.74251	−0.0907115
$370$	0	0
$371$	29.3140	1.52191
$372$	0	0
$373$	0.438217	0.0226900	0.0113450	−	0.999936i	$-0.496389\pi$
$373$	0.438217	0.0226900	0.0113450	+	0.999936i	$0.496389\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.98572	0.462788
$378$	0	0
$379$	13.7753	0.707591	0.353795	−	0.935323i	$-0.384891\pi$
$379$	13.7753	0.707591	0.353795	+	0.935323i	$0.384891\pi$
$380$	0	0
$381$	−15.8962	−0.814386
$382$	0	0
$383$	−22.3153	−1.14026	−0.570130	−	0.821555i	$-0.693107\pi$
$383$	−22.3153	−1.14026	−0.570130	+	0.821555i	$0.693107\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.55982	0.180956
$388$	0	0
$389$	14.4263	0.731444	0.365722	−	0.930724i	$-0.380822\pi$
$389$	14.4263	0.731444	0.365722	+	0.930724i	$0.380822\pi$
$390$	0	0
$391$	4.30357	0.217641
$392$	0	0
$393$	−20.4160	−1.02985
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.4512	0.825662	0.412831	−	0.910808i	$-0.364540\pi$
$397$	16.4512	0.825662	0.412831	+	0.910808i	$0.364540\pi$
$398$	0	0
$399$	−7.69643	−0.385304
$400$	0	0
$401$	2.96056	0.147843	0.0739216	−	0.997264i	$-0.476449\pi$
$401$	2.96056	0.147843	0.0739216	+	0.997264i	$0.476449\pi$
$402$	0	0
$403$	−7.29937	−0.363608
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.26864	0.360293
$408$	0	0
$409$	17.0893	0.845012	0.422506	−	0.906360i	$-0.361151\pi$
$409$	17.0893	0.845012	0.422506	+	0.906360i	$0.361151\pi$
$410$	0	0
$411$	14.4323	0.711893
$412$	0	0
$413$	−55.4875	−2.73036
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.1489	−0.496991
$418$	0	0
$419$	15.3929	0.751991	0.375995	−	0.926622i	$-0.377301\pi$
$419$	15.3929	0.751991	0.375995	+	0.926622i	$0.377301\pi$
$420$	0	0
$421$	−16.4323	−0.800862	−0.400431	−	0.916327i	$-0.631140\pi$
$421$	−16.4323	−0.800862	−0.400431	+	0.916327i	$0.631140\pi$
$422$	0	0
$423$	−1.17210	−0.0569895
$424$	0	0
$425$	0	0
$426$	0	0
$427$	30.7422	1.48772
$428$	0	0
$429$	6.12874	0.295899
$430$	0	0
$431$	−15.1852	−0.731447	−0.365724	−	0.930724i	$-0.619178\pi$
$431$	−15.1852	−0.731447	−0.365724	+	0.930724i	$0.619178\pi$
$432$	0	0
$433$	−23.4380	−1.12636	−0.563179	−	0.826335i	$-0.690422\pi$
$433$	−23.4380	−1.12636	−0.563179	+	0.826335i	$0.690422\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.779187	0.0372735
$438$	0	0
$439$	−0.511196	−0.0243980	−0.0121990	−	0.999926i	$-0.503883\pi$
$439$	−0.511196	−0.0243980	−0.0121990	+	0.999926i	$0.503883\pi$
$440$	0	0
$441$	−5.70496	−0.271665
$442$	0	0
$443$	−19.7834	−0.939940	−0.469970	−	0.882682i	$-0.655735\pi$
$443$	−19.7834	−0.939940	−0.469970	+	0.882682i	$0.655735\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.02391	0.190325
$448$	0	0
$449$	31.1682	1.47092	0.735459	−	0.677569i	$-0.236967\pi$
$449$	31.1682	1.47092	0.735459	+	0.677569i	$0.236967\pi$
$450$	0	0
$451$	−21.2180	−0.999119
$452$	0	0
$453$	−5.99696	−0.281762
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.20950	0.337246	0.168623	−	0.985681i	$-0.446068\pi$
$457$	7.20950	0.337246	0.168623	+	0.985681i	$0.446068\pi$
$458$	0	0
$459$	30.1786	1.40862
$460$	0	0
$461$	−5.41591	−0.252244	−0.126122	−	0.992015i	$-0.540253\pi$
$461$	−5.41591	−0.252244	−0.126122	+	0.992015i	$0.540253\pi$
$462$	0	0
$463$	8.73715	0.406050	0.203025	−	0.979174i	$-0.434923\pi$
$463$	8.73715	0.406050	0.203025	+	0.979174i	$0.434923\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.3584	−0.803249	−0.401624	−	0.915804i	$-0.631554\pi$
$467$	−17.3584	−0.803249	−0.401624	+	0.915804i	$0.631554\pi$
$468$	0	0
$469$	−70.6280	−3.26130
$470$	0	0
$471$	30.5781	1.40896
$472$	0	0
$473$	43.3469	1.99309
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.27357	−0.104100
$478$	0	0
$479$	−7.44682	−0.340254	−0.170127	−	0.985422i	$-0.554418\pi$
$479$	−7.44682	−0.340254	−0.170127	+	0.985422i	$0.554418\pi$
$480$	0	0
$481$	1.36794	0.0623726
$482$	0	0
$483$	−5.99696	−0.272871
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.15530	0.0976661	0.0488330	−	0.998807i	$-0.484450\pi$
$487$	2.15530	0.0976661	0.0488330	+	0.998807i	$0.484450\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−29.3140	−1.32292	−0.661461	−	0.749980i	$-0.730063\pi$
$491$	−29.3140	−1.32292	−0.661461	+	0.749980i	$0.730063\pi$
$492$	0	0
$493$	58.8602	2.65093
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.44011	−0.0645976
$498$	0	0
$499$	23.5696	1.05512	0.527560	−	0.849518i	$-0.323107\pi$
$499$	23.5696	1.05512	0.527560	+	0.849518i	$0.323107\pi$
$500$	0	0
$501$	12.1537	0.542986
$502$	0	0
$503$	−9.08297	−0.404990	−0.202495	−	0.979283i	$-0.564905\pi$
$503$	−9.08297	−0.404990	−0.202495	+	0.979283i	$0.564905\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−19.9373	−0.885447
$508$	0	0
$509$	−18.5112	−0.820494	−0.410247	−	0.911974i	$-0.634558\pi$
$509$	−18.5112	−0.820494	−0.410247	+	0.911974i	$0.634558\pi$
$510$	0	0
$511$	−47.8192	−2.11540
$512$	0	0
$513$	5.46402	0.241242
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−14.2723	−0.627697
$518$	0	0
$519$	−32.5532	−1.42893
$520$	0	0
$521$	−25.0893	−1.09918	−0.549591	−	0.835434i	$-0.685216\pi$
$521$	−25.0893	−1.09918	−0.549591	+	0.835434i	$0.685216\pi$
$522$	0	0
$523$	28.5181	1.24701	0.623504	−	0.781820i	$-0.285708\pi$
$523$	28.5181	1.24701	0.623504	+	0.781820i	$0.285708\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−47.8139	−2.08281
$528$	0	0
$529$	−22.3929	−0.973603
$530$	0	0
$531$	4.30357	0.186759
$532$	0	0
$533$	−3.99318	−0.172964
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.99696	0.258788
$538$	0	0
$539$	−69.4677	−2.99218
$540$	0	0
$541$	−15.6485	−0.672780	−0.336390	−	0.941723i	$-0.609206\pi$
$541$	−15.6485	−0.672780	−0.336390	+	0.941723i	$0.609206\pi$
$542$	0	0
$543$	−3.24473	−0.139245
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10.8543	0.464098	0.232049	−	0.972704i	$-0.425457\pi$
$547$	10.8543	0.464098	0.232049	+	0.972704i	$0.425457\pi$
$548$	0	0
$549$	−2.38434	−0.101761
$550$	0	0
$551$	10.6570	0.454003
$552$	0	0
$553$	−18.9759	−0.806936
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.8763	0.799814	0.399907	−	0.916556i	$-0.369042\pi$
$557$	18.8763	0.799814	0.399907	+	0.916556i	$0.369042\pi$
$558$	0	0
$559$	8.15777	0.345037
$560$	0	0
$561$	40.1458	1.69496
$562$	0	0
$563$	22.2846	0.939183	0.469591	−	0.882884i	$-0.344401\pi$
$563$	22.2846	0.939183	0.469591	+	0.882884i	$0.344401\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−36.8170	−1.54617
$568$	0	0
$569$	−7.11833	−0.298416	−0.149208	−	0.988806i	$-0.547672\pi$
$569$	−7.11833	−0.298416	−0.149208	+	0.988806i	$0.547672\pi$
$570$	0	0
$571$	−1.21017	−0.0506440	−0.0253220	−	0.999679i	$-0.508061\pi$
$571$	−1.21017	−0.0506440	−0.0253220	+	0.999679i	$0.508061\pi$
$572$	0	0
$573$	−1.44011	−0.0601613
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−41.4143	−1.72410	−0.862050	−	0.506823i	$-0.830820\pi$
$577$	−41.4143	−1.72410	−0.862050	+	0.506823i	$0.830820\pi$
$578$	0	0
$579$	31.0814	1.29170
$580$	0	0
$581$	3.69643	0.153354
$582$	0	0
$583$	−27.6846	−1.14658
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.0591336	0.00244071	0.00122035	−	0.999999i	$-0.499612\pi$
$587$	0.0591336	0.00244071	0.00122035	+	0.999999i	$0.499612\pi$
$588$	0	0
$589$	−8.65699	−0.356705
$590$	0	0
$591$	5.47175	0.225078
$592$	0	0
$593$	42.8828	1.76099	0.880493	−	0.474059i	$-0.157212\pi$
$593$	42.8828	1.76099	0.880493	+	0.474059i	$0.157212\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.43706	−0.304379
$598$	0	0
$599$	−15.3430	−0.626898	−0.313449	−	0.949605i	$-0.601485\pi$
$599$	−15.3430	−0.626898	−0.313449	+	0.949605i	$0.601485\pi$
$600$	0	0
$601$	−12.5781	−0.513072	−0.256536	−	0.966535i	$-0.582581\pi$
$601$	−12.5781	−0.513072	−0.256536	+	0.966535i	$0.582581\pi$
$602$	0	0
$603$	5.47785	0.223075
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2.65751	−0.107865	−0.0539325	−	0.998545i	$-0.517176\pi$
$607$	−2.65751	−0.107865	−0.0539325	+	0.998545i	$0.517176\pi$
$608$	0	0
$609$	−82.0208	−3.32365
$610$	0	0
$611$	−2.68602	−0.108665
$612$	0	0
$613$	34.1052	1.37750	0.688748	−	0.725001i	$-0.258161\pi$
$613$	34.1052	1.37750	0.688748	+	0.725001i	$0.258161\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.9226	1.20464	0.602319	−	0.798255i	$-0.294244\pi$
$617$	29.9226	1.20464	0.602319	+	0.798255i	$0.294244\pi$
$618$	0	0
$619$	−17.2102	−0.691735	−0.345868	−	0.938283i	$-0.612415\pi$
$619$	−17.2102	−0.691735	−0.345868	+	0.938283i	$0.612415\pi$
$620$	0	0
$621$	4.25749	0.170847
$622$	0	0
$623$	−27.0237	−1.08268
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.26864	0.290281
$628$	0	0
$629$	8.96056	0.357281
$630$	0	0
$631$	3.36195	0.133837	0.0669186	−	0.997758i	$-0.478683\pi$
$631$	3.36195	0.133837	0.0669186	+	0.997758i	$0.478683\pi$
$632$	0	0
$633$	24.1435	0.959617
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−13.0736	−0.517996
$638$	0	0
$639$	0.111693	0.00441853
$640$	0	0
$641$	−16.2745	−0.642806	−0.321403	−	0.946943i	$-0.604154\pi$
$641$	−16.2745	−0.642806	−0.321403	+	0.946943i	$0.604154\pi$
$642$	0	0
$643$	−35.7040	−1.40803	−0.704015	−	0.710185i	$-0.748611\pi$
$643$	−35.7040	−1.40803	−0.704015	+	0.710185i	$0.748611\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.2881	−0.601036	−0.300518	−	0.953776i	$-0.597160\pi$
$647$	−15.2881	−0.601036	−0.300518	+	0.953776i	$0.597160\pi$
$648$	0	0
$649$	52.4033	2.05701
$650$	0	0
$651$	66.6280	2.61136
$652$	0	0
$653$	16.4512	0.643785	0.321892	−	0.946776i	$-0.395681\pi$
$653$	16.4512	0.643785	0.321892	+	0.946776i	$0.395681\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.70882	0.144695
$658$	0	0
$659$	−7.03944	−0.274218	−0.137109	−	0.990556i	$-0.543781\pi$
$659$	−7.03944	−0.274218	−0.137109	+	0.990556i	$0.543781\pi$
$660$	0	0
$661$	6.35343	0.247120	0.123560	−	0.992337i	$-0.460569\pi$
$661$	6.35343	0.247120	0.123560	+	0.992337i	$0.460569\pi$
$662$	0	0
$663$	7.55533	0.293425
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.30378	0.321524
$668$	0	0
$669$	−30.4743	−1.17820
$670$	0	0
$671$	−29.0335	−1.12082
$672$	0	0
$673$	7.08638	0.273160	0.136580	−	0.990629i	$-0.456389\pi$
$673$	7.08638	0.273160	0.136580	+	0.990629i	$0.456389\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.6095	−0.676786	−0.338393	−	0.941005i	$-0.609883\pi$
$677$	−17.6095	−0.676786	−0.338393	+	0.941005i	$0.609883\pi$
$678$	0	0
$679$	29.3140	1.12497
$680$	0	0
$681$	1.36794	0.0524195
$682$	0	0
$683$	26.5855	1.01726	0.508632	−	0.860984i	$-0.330151\pi$
$683$	26.5855	1.01726	0.508632	+	0.860984i	$0.330151\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−7.88911	−0.300988
$688$	0	0
$689$	−5.21017	−0.198492
$690$	0	0
$691$	−49.4907	−1.88271	−0.941357	−	0.337411i	$-0.890449\pi$
$691$	−49.4907	−1.88271	−0.941357	+	0.337411i	$0.890449\pi$
$692$	0	0
$693$	7.82027	0.297067
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−26.1570	−0.990766
$698$	0	0
$699$	−7.95392	−0.300845
$700$	0	0
$701$	29.3389	1.10812	0.554058	−	0.832478i	$-0.313079\pi$
$701$	29.3389	1.10812	0.554058	+	0.832478i	$0.313079\pi$
$702$	0	0
$703$	1.62236	0.0611886
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.9774	0.600891
$708$	0	0
$709$	24.7359	0.928975	0.464488	−	0.885580i	$-0.346238\pi$
$709$	24.7359	0.928975	0.464488	+	0.885580i	$0.346238\pi$
$710$	0	0
$711$	1.47175	0.0551951
$712$	0	0
$713$	−6.74541	−0.252618
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.0613	0.375747
$718$	0	0
$719$	−26.7548	−0.997786	−0.498893	−	0.866663i	$-0.666260\pi$
$719$	−26.7548	−0.997786	−0.498893	+	0.866663i	$0.666260\pi$
$720$	0	0
$721$	17.6175	0.656112
$722$	0	0
$723$	−5.29559	−0.196945
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−27.2108	−1.00919	−0.504596	−	0.863355i	$-0.668359\pi$
$727$	−27.2108	−1.00919	−0.504596	+	0.863355i	$0.668359\pi$
$728$	0	0
$729$	29.4494	1.09072
$730$	0	0
$731$	53.4367	1.97643
$732$	0	0
$733$	−40.0026	−1.47753	−0.738765	−	0.673963i	$-0.764591\pi$
$733$	−40.0026	−1.47753	−0.738765	+	0.673963i	$0.764591\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	66.7022	2.45701
$738$	0	0
$739$	−35.1123	−1.29163	−0.645814	−	0.763495i	$-0.723482\pi$
$739$	−35.1123	−1.29163	−0.645814	+	0.763495i	$0.723482\pi$
$740$	0	0
$741$	1.36794	0.0502525
$742$	0	0
$743$	25.6476	0.940918	0.470459	−	0.882422i	$-0.344088\pi$
$743$	25.6476	0.940918	0.470459	+	0.882422i	$0.344088\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.286693	−0.0104895
$748$	0	0
$749$	−48.4033	−1.76862
$750$	0	0
$751$	13.2641	0.484015	0.242007	−	0.970274i	$-0.422194\pi$
$751$	13.2641	0.484015	0.242007	+	0.970274i	$0.422194\pi$
$752$	0	0
$753$	−46.1178	−1.68063
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.77092	−0.100711	−0.0503554	−	0.998731i	$-0.516035\pi$
$757$	−2.77092	−0.100711	−0.0503554	+	0.998731i	$0.516035\pi$
$758$	0	0
$759$	5.66363	0.205577
$760$	0	0
$761$	10.7838	0.390914	0.195457	−	0.980712i	$-0.437381\pi$
$761$	10.7838	0.390914	0.195457	+	0.980712i	$0.437381\pi$
$762$	0	0
$763$	14.0448	0.508455
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.86216	0.356102
$768$	0	0
$769$	−40.6090	−1.46440	−0.732199	−	0.681090i	$-0.761506\pi$
$769$	−40.6090	−1.46440	−0.732199	+	0.681090i	$0.761506\pi$
$770$	0	0
$771$	−15.4967	−0.558099
$772$	0	0
$773$	17.4410	0.627310	0.313655	−	0.949537i	$-0.398446\pi$
$773$	17.4410	0.627310	0.313655	+	0.949537i	$0.398446\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−12.4864	−0.447947
$778$	0	0
$779$	−4.73588	−0.169680
$780$	0	0
$781$	1.36006	0.0486668
$782$	0	0
$783$	58.2300	2.08097
$784$	0	0
$785$	0	0
$786$	0	0
$787$	30.8346	1.09914	0.549568	−	0.835449i	$-0.314792\pi$
$787$	30.8346	1.09914	0.549568	+	0.835449i	$0.314792\pi$
$788$	0	0
$789$	15.4888	0.551416
$790$	0	0
$791$	−33.6175	−1.19530
$792$	0	0
$793$	−5.46402	−0.194033
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.1340	1.35077	0.675387	−	0.737463i	$-0.263976\pi$
$797$	38.1340	1.35077	0.675387	+	0.737463i	$0.263976\pi$
$798$	0	0
$799$	−17.5945	−0.622449
$800$	0	0
$801$	2.09594	0.0740563
$802$	0	0
$803$	45.1612	1.59371
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−23.2865	−0.819722
$808$	0	0
$809$	30.7917	1.08258	0.541290	−	0.840836i	$-0.317936\pi$
$809$	30.7917	1.08258	0.541290	+	0.840836i	$0.317936\pi$
$810$	0	0
$811$	19.0275	0.668145	0.334072	−	0.942547i	$-0.391577\pi$
$811$	19.0275	0.668145	0.334072	+	0.942547i	$0.391577\pi$
$812$	0	0
$813$	19.6368	0.688692
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.67504	0.338487
$818$	0	0
$819$	1.47175	0.0514272
$820$	0	0
$821$	−42.1228	−1.47009	−0.735047	−	0.678016i	$-0.762840\pi$
$821$	−42.1228	−1.47009	−0.735047	+	0.678016i	$0.762840\pi$
$822$	0	0
$823$	−41.6298	−1.45112	−0.725562	−	0.688157i	$-0.758420\pi$
$823$	−41.6298	−1.45112	−0.725562	+	0.688157i	$0.758420\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.54814	0.262475	0.131237	−	0.991351i	$-0.458105\pi$
$827$	7.54814	0.262475	0.131237	+	0.991351i	$0.458105\pi$
$828$	0	0
$829$	51.6674	1.79448	0.897242	−	0.441540i	$-0.145568\pi$
$829$	51.6674	1.79448	0.897242	+	0.441540i	$0.145568\pi$
$830$	0	0
$831$	5.82518	0.202073
$832$	0	0
$833$	−85.6376	−2.96717
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−47.3020	−1.63500
$838$	0	0
$839$	−45.1682	−1.55938	−0.779689	−	0.626166i	$-0.784623\pi$
$839$	−45.1682	−1.55938	−0.779689	+	0.626166i	$0.784623\pi$
$840$	0	0
$841$	84.5715	2.91626
$842$	0	0
$843$	43.3282	1.49230
$844$	0	0
$845$	0	0
$846$	0	0
$847$	43.0415	1.47892
$848$	0	0
$849$	33.2180	1.14004
$850$	0	0
$851$	1.26412	0.0433336
$852$	0	0
$853$	−10.6721	−0.365405	−0.182702	−	0.983168i	$-0.558485\pi$
$853$	−10.6721	−0.365405	−0.182702	+	0.983168i	$0.558485\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29.7119	−1.01494	−0.507470	−	0.861669i	$-0.669419\pi$
$857$	−29.7119	−1.01494	−0.507470	+	0.861669i	$0.669419\pi$
$858$	0	0
$859$	15.6734	0.534769	0.267385	−	0.963590i	$-0.413841\pi$
$859$	15.6734	0.534769	0.267385	+	0.963590i	$0.413841\pi$
$860$	0	0
$861$	36.4494	1.24219
$862$	0	0
$863$	39.5741	1.34712	0.673559	−	0.739134i	$-0.264765\pi$
$863$	39.5741	1.34712	0.673559	+	0.739134i	$0.264765\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	21.9104	0.744115
$868$	0	0
$869$	17.9211	0.607932
$870$	0	0
$871$	12.5532	0.425348
$872$	0	0
$873$	−2.27357	−0.0769487
$874$	0	0
$875$	0	0
$876$	0	0
$877$	47.9393	1.61880	0.809398	−	0.587260i	$-0.199793\pi$
$877$	47.9393	1.61880	0.809398	+	0.587260i	$0.199793\pi$
$878$	0	0
$879$	31.0814	1.04835
$880$	0	0
$881$	−2.32251	−0.0782473	−0.0391237	−	0.999234i	$-0.512457\pi$
$881$	−2.32251	−0.0782473	−0.0391237	+	0.999234i	$0.512457\pi$
$882$	0	0
$883$	−34.3919	−1.15738	−0.578690	−	0.815548i	$-0.696436\pi$
$883$	−34.3919	−1.15738	−0.578690	+	0.815548i	$0.696436\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44.1002	1.48074	0.740370	−	0.672200i	$-0.234650\pi$
$887$	44.1002	1.48074	0.740370	+	0.672200i	$0.234650\pi$
$888$	0	0
$889$	−46.4822	−1.55896
$890$	0	0
$891$	34.7706	1.16486
$892$	0	0
$893$	−3.18559	−0.106602
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.06588	0.0355887
$898$	0	0
$899$	−92.2575	−3.07696
$900$	0	0
$901$	−34.1287	−1.13699
$902$	0	0
$903$	−74.4633	−2.47798
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−25.3706	−0.842417	−0.421208	−	0.906964i	$-0.638394\pi$
$907$	−25.3706	−0.842417	−0.421208	+	0.906964i	$0.638394\pi$
$908$	0	0
$909$	−1.23919	−0.0411015
$910$	0	0
$911$	−29.8252	−0.988152	−0.494076	−	0.869419i	$-0.664494\pi$
$911$	−29.8252	−0.988152	−0.494076	+	0.869419i	$0.664494\pi$
$912$	0	0
$913$	−3.49097	−0.115534
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−59.6985	−1.97142
$918$	0	0
$919$	15.8422	0.522587	0.261293	−	0.965259i	$-0.415851\pi$
$919$	15.8422	0.522587	0.261293	+	0.965259i	$0.415851\pi$
$920$	0	0
$921$	51.9959	1.71332
$922$	0	0
$923$	0.255960	0.00842501
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.36640	−0.0448786
$928$	0	0
$929$	−7.97098	−0.261519	−0.130760	−	0.991414i	$-0.541742\pi$
$929$	−7.97098	−0.261519	−0.130760	+	0.991414i	$0.541742\pi$
$930$	0	0
$931$	−15.5052	−0.508163
$932$	0	0
$933$	33.3073	1.09043
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.848033	0.0277040	0.0138520	−	0.999904i	$-0.495591\pi$
$937$	0.848033	0.0277040	0.0138520	+	0.999904i	$0.495591\pi$
$938$	0	0
$939$	24.0000	0.783210
$940$	0	0
$941$	31.3140	1.02081	0.510403	−	0.859935i	$-0.329496\pi$
$941$	31.3140	1.02081	0.510403	+	0.859935i	$0.329496\pi$
$942$	0	0
$943$	−3.69013	−0.120167
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.9885	0.844514	0.422257	−	0.906476i	$-0.361238\pi$
$947$	25.9885	0.844514	0.422257	+	0.906476i	$0.361238\pi$
$948$	0	0
$949$	8.49922	0.275896
$950$	0	0
$951$	27.4967	0.891641
$952$	0	0
$953$	41.6347	1.34868	0.674340	−	0.738421i	$-0.264428\pi$
$953$	41.6347	1.34868	0.674340	+	0.738421i	$0.264428\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	77.4618	2.50399
$958$	0	0
$959$	42.2016	1.36276
$960$	0	0
$961$	43.9435	1.41753
$962$	0	0
$963$	3.75412	0.120975
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−13.2656	−0.426593	−0.213296	−	0.976988i	$-0.568420\pi$
$967$	−13.2656	−0.426593	−0.213296	+	0.976988i	$0.568420\pi$
$968$	0	0
$969$	8.96056	0.287855
$970$	0	0
$971$	−25.3140	−0.812364	−0.406182	−	0.913792i	$-0.633140\pi$
$971$	−25.3140	−0.812364	−0.406182	+	0.913792i	$0.633140\pi$
$972$	0	0
$973$	−29.6763	−0.951380
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.7694	−1.62426	−0.812128	−	0.583479i	$-0.801691\pi$
$977$	−50.7694	−1.62426	−0.812128	+	0.583479i	$0.801691\pi$
$978$	0	0
$979$	25.5216	0.815674
$980$	0	0
$981$	−1.08930	−0.0347788
$982$	0	0
$983$	−45.4123	−1.44843	−0.724214	−	0.689575i	$-0.757797\pi$
$983$	−45.4123	−1.44843	−0.724214	+	0.689575i	$0.757797\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.5177	0.780407
$988$	0	0
$989$	7.53866	0.239716
$990$	0	0
$991$	−45.6175	−1.44909	−0.724545	−	0.689228i	$-0.757950\pi$
$991$	−45.6175	−1.44909	−0.724545	+	0.689228i	$0.757950\pi$
$992$	0	0
$993$	35.9009	1.13928
$994$	0	0
$995$	0	0
$996$	0	0
$997$	59.5705	1.88662	0.943309	−	0.331917i	$-0.107695\pi$
$997$	59.5705	1.88662	0.943309	+	0.331917i	$0.107695\pi$
$998$	0	0
$999$	8.86462	0.280464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cj.1.5		6
4.3	odd	2		1900.2.a.k.1.2		6
5.2	odd	4		1520.2.d.i.609.2		6
5.3	odd	4		1520.2.d.i.609.5		6
5.4	even	2	inner	7600.2.a.cj.1.2		6
20.3	even	4		380.2.c.b.229.2	✓	6
20.7	even	4		380.2.c.b.229.5	yes	6
20.19	odd	2		1900.2.a.k.1.5		6
60.23	odd	4		3420.2.f.c.1369.5		6
60.47	odd	4		3420.2.f.c.1369.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.c.b.229.2	✓	6	20.3	even	4
380.2.c.b.229.5	yes	6	20.7	even	4
1520.2.d.i.609.2		6	5.2	odd	4
1520.2.d.i.609.5		6	5.3	odd	4
1900.2.a.k.1.2		6	4.3	odd	2
1900.2.a.k.1.5		6	20.19	odd	2
3420.2.f.c.1369.5		6	60.23	odd	4
3420.2.f.c.1369.6		6	60.47	odd	4
7600.2.a.cj.1.2		6	5.4	even	2	inner
7600.2.a.cj.1.5		6	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.62236 q^{3} +4.74397 q^{7} -0.367938 q^{9} +O(q^{10})\)	\(q+1.62236 q^{3} +4.74397 q^{7} -0.367938 q^{9} -4.48028 q^{11} -0.843176 q^{13} -5.52315 q^{17} -1.00000 q^{19} +7.69643 q^{21} -0.779187 q^{23} -5.46402 q^{27} -10.6570 q^{29} +8.65699 q^{31} -7.26864 q^{33} -1.62236 q^{37} -1.36794 q^{39} +4.73588 q^{41} -9.67504 q^{43} +3.18559 q^{47} +15.5052 q^{49} -8.96056 q^{51} +6.17922 q^{53} -1.62236 q^{57} -11.6964 q^{59} +6.48028 q^{61} -1.74549 q^{63} -14.8880 q^{67} -1.26412 q^{69} -0.303566 q^{71} -10.0800 q^{73} -21.2543 q^{77} -4.00000 q^{79} -7.76081 q^{81} +0.779187 q^{83} -17.2895 q^{87} -5.69643 q^{89} -4.00000 q^{91} +14.0448 q^{93} +6.17922 q^{97} +1.64847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{9}+O(q^{10}) \) 6 * q + 10 * q^9	\( 6 q + 10 q^{9} - 18 q^{11} - 6 q^{19} + 4 q^{21} - 4 q^{29} - 8 q^{31} + 4 q^{39} + 4 q^{41} + 12 q^{49} - 36 q^{51} - 28 q^{59} + 30 q^{61} - 32 q^{69} - 44 q^{71} - 24 q^{79} + 50 q^{81} + 8 q^{89} - 24 q^{91} - 90 q^{99}+O(q^{100}) \) 6 * q + 10 * q^9 - 18 * q^11 - 6 * q^19 + 4 * q^21 - 4 * q^29 - 8 * q^31 + 4 * q^39 + 4 * q^41 + 12 * q^49 - 36 * q^51 - 28 * q^59 + 30 * q^61 - 32 * q^69 - 44 * q^71 - 24 * q^79 + 50 * q^81 + 8 * q^89 - 24 * q^91 - 90 * q^99

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