LMFDB - Embedded newform 1900.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.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:	$15.1715763840$
Analytic rank:	$0$
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			1.1
Root			$-0.608430$ of defining polynomial
Character	$\chi$	$=$	1900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.28715 q^{3} +1.93210 q^{7} +7.80536 q^{9} +O(q^{10})$	$q-3.28715 q^{3} +1.93210 q^{7} +7.80536 q^{9} +5.62981 q^{11} +2.07029 q^{13} -3.42535 q^{17} +1.00000 q^{19} -6.35109 q^{21} +5.35744 q^{23} -15.7959 q^{27} +1.09146 q^{29} +3.09146 q^{31} -18.5060 q^{33} -3.28715 q^{37} -6.80536 q^{39} -11.6107 q^{41} +0.501623 q^{43} +12.6470 q^{47} -3.26701 q^{49} +11.2596 q^{51} -3.01076 q^{53} -3.28715 q^{57} -2.35109 q^{59} +7.62981 q^{61} +15.0807 q^{63} -12.2324 q^{67} -17.6107 q^{69} +14.3511 q^{71} +2.87256 q^{73} +10.8773 q^{77} +4.00000 q^{79} +28.5075 q^{81} -5.35744 q^{83} -3.58780 q^{87} +8.35109 q^{89} +4.00000 q^{91} -10.1621 q^{93} -3.01076 q^{97} +43.9427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.28715	−1.89784	−0.948919	−	0.315521i	$-0.897821\pi$
$3$	−3.28715	−1.89784	−0.948919	+	0.315521i	$0.897821\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.93210	0.730263	0.365132	−	0.930956i	$-0.381024\pi$
$7$	1.93210	0.730263	0.365132	+	0.930956i	$0.381024\pi$
$8$	0	0
$9$	7.80536	2.60179
$10$	0	0
$11$	5.62981	1.69745	0.848726	−	0.528832i	$-0.177370\pi$
$11$	5.62981	1.69745	0.848726	+	0.528832i	$0.177370\pi$
$12$	0	0
$13$	2.07029	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$13$	2.07029	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.42535	−0.830768	−0.415384	−	0.909646i	$-0.636353\pi$
$17$	−3.42535	−0.830768	−0.415384	+	0.909646i	$0.636353\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.35109	−1.38592
$22$	0	0
$23$	5.35744	1.11710	0.558552	−	0.829470i	$-0.311357\pi$
$23$	5.35744	1.11710	0.558552	+	0.829470i	$0.311357\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−15.7959	−3.03993
$28$	0	0
$29$	1.09146	0.202679	0.101340	−	0.994852i	$-0.467687\pi$
$29$	1.09146	0.202679	0.101340	+	0.994852i	$0.467687\pi$
$30$	0	0
$31$	3.09146	0.555243	0.277622	−	0.960691i	$-0.410454\pi$
$31$	3.09146	0.555243	0.277622	+	0.960691i	$0.410454\pi$
$32$	0	0
$33$	−18.5060	−3.22149
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.28715	−0.540404	−0.270202	−	0.962804i	$-0.587091\pi$
$37$	−3.28715	−0.540404	−0.270202	+	0.962804i	$0.587091\pi$
$38$	0	0
$39$	−6.80536	−1.08973
$40$	0	0
$41$	−11.6107	−1.81329	−0.906645	−	0.421895i	$-0.861365\pi$
$41$	−11.6107	−1.81329	−0.906645	+	0.421895i	$0.861365\pi$
$42$	0	0
$43$	0.501623	0.0764968	0.0382484	−	0.999268i	$-0.487822\pi$
$43$	0.501623	0.0764968	0.0382484	+	0.999268i	$0.487822\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.6470	1.84475	0.922376	−	0.386294i	$-0.126245\pi$
$47$	12.6470	1.84475	0.922376	+	0.386294i	$0.126245\pi$
$48$	0	0
$49$	−3.26701	−0.466715
$50$	0	0
$51$	11.2596	1.57666
$52$	0	0
$53$	−3.01076	−0.413560	−0.206780	−	0.978388i	$-0.566298\pi$
$53$	−3.01076	−0.413560	−0.206780	+	0.978388i	$0.566298\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.28715	−0.435394
$58$	0	0
$59$	−2.35109	−0.306086	−0.153043	−	0.988220i	$-0.548907\pi$
$59$	−2.35109	−0.306086	−0.153043	+	0.988220i	$0.548907\pi$
$60$	0	0
$61$	7.62981	0.976897	0.488449	−	0.872593i	$-0.337563\pi$
$61$	7.62981	0.976897	0.488449	+	0.872593i	$0.337563\pi$
$62$	0	0
$63$	15.0807	1.89999
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.2324	−1.49442	−0.747212	−	0.664585i	$-0.768608\pi$
$67$	−12.2324	−1.49442	−0.747212	+	0.664585i	$0.768608\pi$
$68$	0	0
$69$	−17.6107	−2.12008
$70$	0	0
$71$	14.3511	1.70316	0.851580	−	0.524224i	$-0.175645\pi$
$71$	14.3511	1.70316	0.851580	+	0.524224i	$0.175645\pi$
$72$	0	0
$73$	2.87256	0.336208	0.168104	−	0.985769i	$-0.446236\pi$
$73$	2.87256	0.336208	0.168104	+	0.985769i	$0.446236\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.8773	1.23959
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	28.5075	3.16750
$82$	0	0
$83$	−5.35744	−0.588056	−0.294028	−	0.955797i	$-0.594996\pi$
$83$	−5.35744	−0.588056	−0.294028	+	0.955797i	$0.594996\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.58780	−0.384653
$88$	0	0
$89$	8.35109	0.885214	0.442607	−	0.896716i	$-0.354054\pi$
$89$	8.35109	0.885214	0.442607	+	0.896716i	$0.354054\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	−10.1621	−1.05376
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.01076	−0.305696	−0.152848	−	0.988250i	$-0.548845\pi$
$97$	−3.01076	−0.305696	−0.152848	+	0.988250i	$0.548845\pi$
$98$	0	0
$99$	43.9427	4.41641
$100$	0	0
$101$	−4.80536	−0.478151	−0.239075	−	0.971001i	$-0.576844\pi$
$101$	−4.80536	−0.478151	−0.239075	+	0.971001i	$0.576844\pi$
$102$	0	0
$103$	4.22762	0.416560	0.208280	−	0.978069i	$-0.433214\pi$
$103$	4.22762	0.416560	0.208280	+	0.978069i	$0.433214\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.92098	0.862424	0.431212	−	0.902251i	$-0.358086\pi$
$107$	8.92098	0.862424	0.431212	+	0.902251i	$0.358086\pi$
$108$	0	0
$109$	5.25963	0.503781	0.251890	−	0.967756i	$-0.418948\pi$
$109$	5.25963	0.503781	0.251890	+	0.967756i	$0.418948\pi$
$110$	0	0
$111$	10.8054	1.02560
$112$	0	0
$113$	12.5088	1.17673	0.588364	−	0.808596i	$-0.299772\pi$
$113$	12.5088	1.17673	0.588364	+	0.808596i	$0.299772\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.1594	1.49393
$118$	0	0
$119$	−6.61810	−0.606680
$120$	0	0
$121$	20.6948	1.88135
$122$	0	0
$123$	38.1662	3.44133
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.2952	1.09102	0.545510	−	0.838104i	$-0.316336\pi$
$127$	12.2952	1.09102	0.545510	+	0.838104i	$0.316336\pi$
$128$	0	0
$129$	−1.64891	−0.145179
$130$	0	0
$131$	−10.7863	−0.942400	−0.471200	−	0.882026i	$-0.656179\pi$
$131$	−10.7863	−0.942400	−0.471200	+	0.882026i	$0.656179\pi$
$132$	0	0
$133$	1.93210	0.167534
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.85582	−0.414861	−0.207430	−	0.978250i	$-0.566510\pi$
$137$	−4.85582	−0.414861	−0.207430	+	0.978250i	$0.566510\pi$
$138$	0	0
$139$	−11.2405	−0.953409	−0.476705	−	0.879064i	$-0.658169\pi$
$139$	−11.2405	−0.953409	−0.476705	+	0.879064i	$0.658169\pi$
$140$	0	0
$141$	−41.5725	−3.50104
$142$	0	0
$143$	11.6554	0.974669
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.7391	0.885750
$148$	0	0
$149$	3.62981	0.297366	0.148683	−	0.988885i	$-0.452497\pi$
$149$	3.62981	0.297366	0.148683	+	0.988885i	$0.452497\pi$
$150$	0	0
$151$	−10.3511	−0.842360	−0.421180	−	0.906977i	$-0.638384\pi$
$151$	−10.3511	−0.842360	−0.421180	+	0.906977i	$0.638384\pi$
$152$	0	0
$153$	−26.7360	−2.16148
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.12708	0.568803	0.284402	−	0.958705i	$-0.408205\pi$
$157$	7.12708	0.568803	0.284402	+	0.958705i	$0.408205\pi$
$158$	0	0
$159$	9.89682	0.784869
$160$	0	0
$161$	10.3511	0.815780
$162$	0	0
$163$	1.21686	0.0953118	0.0476559	−	0.998864i	$-0.484825\pi$
$163$	1.21686	0.0953118	0.0476559	+	0.998864i	$0.484825\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.8830	1.22906	0.614531	−	0.788893i	$-0.289345\pi$
$167$	15.8830	1.22906	0.614531	+	0.788893i	$0.289345\pi$
$168$	0	0
$169$	−8.71390	−0.670300
$170$	0	0
$171$	7.80536	0.596891
$172$	0	0
$173$	−13.7884	−1.04831	−0.524157	−	0.851622i	$-0.675620\pi$
$173$	−13.7884	−1.04831	−0.524157	+	0.851622i	$0.675620\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.72838	0.580901
$178$	0	0
$179$	10.3511	0.773677	0.386838	−	0.922148i	$-0.373567\pi$
$179$	10.3511	0.773677	0.386838	+	0.922148i	$0.373567\pi$
$180$	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
$183$	−25.0803	−1.85399
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−19.2841	−1.41019
$188$	0	0
$189$	−30.5193	−2.21995
$190$	0	0
$191$	−8.43517	−0.610348	−0.305174	−	0.952297i	$-0.598715\pi$
$191$	−8.43517	−0.610348	−0.305174	+	0.952297i	$0.598715\pi$
$192$	0	0
$193$	23.2864	1.67619	0.838097	−	0.545521i	$-0.183668\pi$
$193$	23.2864	1.67619	0.838097	+	0.545521i	$0.183668\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.28116	−0.590008	−0.295004	−	0.955496i	$-0.595321\pi$
$197$	−8.28116	−0.590008	−0.295004	+	0.955496i	$0.595321\pi$
$198$	0	0
$199$	−18.7863	−1.33172	−0.665861	−	0.746076i	$-0.731936\pi$
$199$	−18.7863	−1.33172	−0.665861	+	0.746076i	$0.731936\pi$
$200$	0	0
$201$	40.2097	2.83617
$202$	0	0
$203$	2.10881	0.148009
$204$	0	0
$205$	0	0
$206$	0	0
$207$	41.8167	2.90646
$208$	0	0
$209$	5.62981	0.389422
$210$	0	0
$211$	−21.7789	−1.49932	−0.749660	−	0.661823i	$-0.769783\pi$
$211$	−21.7789	−1.49932	−0.749660	+	0.661823i	$0.769783\pi$
$212$	0	0
$213$	−47.1742	−3.23232
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.97300	0.405474
$218$	0	0
$219$	−9.44255	−0.638068
$220$	0	0
$221$	−7.09146	−0.477023
$222$	0	0
$223$	14.5548	0.974662	0.487331	−	0.873217i	$-0.337970\pi$
$223$	14.5548	0.974662	0.487331	+	0.873217i	$0.337970\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.07029	0.137410	0.0687050	−	0.997637i	$-0.478113\pi$
$227$	2.07029	0.137410	0.0687050	+	0.997637i	$0.478113\pi$
$228$	0	0
$229$	−15.4616	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$229$	−15.4616	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$230$	0	0
$231$	−35.7554	−2.35254
$232$	0	0
$233$	28.8934	1.89287	0.946434	−	0.322897i	$-0.104657\pi$
$233$	28.8934	1.89287	0.946434	+	0.322897i	$0.104657\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−13.1486	−0.854093
$238$	0	0
$239$	26.6181	1.72178	0.860891	−	0.508790i	$-0.169907\pi$
$239$	26.6181	1.72178	0.860891	+	0.508790i	$0.169907\pi$
$240$	0	0
$241$	−19.6107	−1.26324	−0.631619	−	0.775279i	$-0.717609\pi$
$241$	−19.6107	−1.26324	−0.631619	+	0.775279i	$0.717609\pi$
$242$	0	0
$243$	−46.3208	−2.97148
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.07029	0.131729
$248$	0	0
$249$	17.6107	1.11603
$250$	0	0
$251$	14.2522	0.899594	0.449797	−	0.893131i	$-0.351496\pi$
$251$	14.2522	0.899594	0.449797	+	0.893131i	$0.351496\pi$
$252$	0	0
$253$	30.1614	1.89623
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.2919	0.704371	0.352185	−	0.935930i	$-0.385439\pi$
$257$	11.2919	0.704371	0.352185	+	0.935930i	$0.385439\pi$
$258$	0	0
$259$	−6.35109	−0.394637
$260$	0	0
$261$	8.51925	0.527329
$262$	0	0
$263$	−15.3571	−0.946959	−0.473479	−	0.880805i	$-0.657002\pi$
$263$	−15.3571	−0.946959	−0.473479	+	0.880805i	$0.657002\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−27.4513	−1.67999
$268$	0	0
$269$	11.4426	0.697665	0.348832	−	0.937185i	$-0.386578\pi$
$269$	11.4426	0.697665	0.348832	+	0.937185i	$0.386578\pi$
$270$	0	0
$271$	12.4161	0.754223	0.377111	−	0.926168i	$-0.376917\pi$
$271$	12.4161	0.754223	0.377111	+	0.926168i	$0.376917\pi$
$272$	0	0
$273$	−13.1486	−0.795790
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−16.0212	−0.962619	−0.481309	−	0.876551i	$-0.659839\pi$
$277$	−16.0212	−0.962619	−0.481309	+	0.876551i	$0.659839\pi$
$278$	0	0
$279$	24.1300	1.44462
$280$	0	0
$281$	−24.8851	−1.48452	−0.742260	−	0.670112i	$-0.766246\pi$
$281$	−24.8851	−1.48452	−0.742260	+	0.670112i	$0.766246\pi$
$282$	0	0
$283$	16.2348	0.965057	0.482529	−	0.875880i	$-0.339718\pi$
$283$	16.2348	0.965057	0.482529	+	0.875880i	$0.339718\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−22.4330	−1.32418
$288$	0	0
$289$	−5.26701	−0.309824
$290$	0	0
$291$	9.89682	0.580162
$292$	0	0
$293$	23.2864	1.36041	0.680204	−	0.733023i	$-0.261891\pi$
$293$	23.2864	1.36041	0.680204	+	0.733023i	$0.261891\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−88.9282	−5.16013
$298$	0	0
$299$	11.0915	0.641436
$300$	0	0
$301$	0.969184	0.0558628
$302$	0	0
$303$	15.7959	0.907453
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.964728	0.0550599	0.0275300	−	0.999621i	$-0.491236\pi$
$307$	0.964728	0.0550599	0.0275300	+	0.999621i	$0.491236\pi$
$308$	0	0
$309$	−13.8968	−0.790562
$310$	0	0
$311$	18.1638	1.02998	0.514988	−	0.857197i	$-0.327796\pi$
$311$	18.1638	1.02998	0.514988	+	0.857197i	$0.327796\pi$
$312$	0	0
$313$	7.30116	0.412686	0.206343	−	0.978480i	$-0.433844\pi$
$313$	7.30116	0.412686	0.206343	+	0.978480i	$0.433844\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.64135	−0.429181	−0.214590	−	0.976704i	$-0.568842\pi$
$317$	−7.64135	−0.429181	−0.214590	+	0.976704i	$0.568842\pi$
$318$	0	0
$319$	6.14473	0.344039
$320$	0	0
$321$	−29.3246	−1.63674
$322$	0	0
$323$	−3.42535	−0.190591
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−17.2892	−0.956094
$328$	0	0
$329$	24.4352	1.34715
$330$	0	0
$331$	22.3129	1.22643	0.613214	−	0.789917i	$-0.289876\pi$
$331$	22.3129	1.22643	0.613214	+	0.789917i	$0.289876\pi$
$332$	0	0
$333$	−25.6574	−1.40602
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0.790654	0.0430696	0.0215348	−	0.999768i	$-0.493145\pi$
$337$	0.790654	0.0430696	0.0215348	+	0.999768i	$0.493145\pi$
$338$	0	0
$339$	−41.1183	−2.23324
$340$	0	0
$341$	17.4044	0.942499
$342$	0	0
$343$	−19.8368	−1.07109
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.95361	−0.426972	−0.213486	−	0.976946i	$-0.568482\pi$
$347$	−7.95361	−0.426972	−0.213486	+	0.976946i	$0.568482\pi$
$348$	0	0
$349$	15.8777	0.849915	0.424957	−	0.905213i	$-0.360289\pi$
$349$	15.8777	0.849915	0.424957	+	0.905213i	$0.360289\pi$
$350$	0	0
$351$	−32.7022	−1.74551
$352$	0	0
$353$	8.45524	0.450027	0.225013	−	0.974356i	$-0.427757\pi$
$353$	8.45524	0.450027	0.225013	+	0.974356i	$0.427757\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	21.7547	1.15138
$358$	0	0
$359$	15.2787	0.806380	0.403190	−	0.915116i	$-0.367901\pi$
$359$	15.2787	0.806380	0.403190	+	0.915116i	$0.367901\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−68.0269	−3.57049
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.6331	1.33804	0.669019	−	0.743245i	$-0.266714\pi$
$367$	25.6331	1.33804	0.669019	+	0.743245i	$0.266714\pi$
$368$	0	0
$369$	−90.6258	−4.71779
$370$	0	0
$371$	−5.81708	−0.302008
$372$	0	0
$373$	1.30390	0.0675132	0.0337566	−	0.999430i	$-0.489253\pi$
$373$	1.30390	0.0675132	0.0337566	+	0.999430i	$0.489253\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.25964	0.116378
$378$	0	0
$379$	4.87034	0.250173	0.125086	−	0.992146i	$-0.460079\pi$
$379$	4.87034	0.250173	0.125086	+	0.992146i	$0.460079\pi$
$380$	0	0
$381$	−40.4161	−2.07058
$382$	0	0
$383$	−20.6876	−1.05709	−0.528544	−	0.848906i	$-0.677262\pi$
$383$	−20.6876	−1.05709	−0.528544	+	0.848906i	$0.677262\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.91535	0.199028
$388$	0	0
$389$	0.252246	0.0127894	0.00639470	−	0.999980i	$-0.497964\pi$
$389$	0.252246	0.0127894	0.00639470	+	0.999980i	$0.497964\pi$
$390$	0	0
$391$	−18.3511	−0.928054
$392$	0	0
$393$	35.4561	1.78852
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−28.1665	−1.41364	−0.706819	−	0.707395i	$-0.749870\pi$
$397$	−28.1665	−1.41364	−0.706819	+	0.707395i	$0.749870\pi$
$398$	0	0
$399$	−6.35109	−0.317952
$400$	0	0
$401$	5.25963	0.262653	0.131327	−	0.991339i	$-0.458076\pi$
$401$	5.25963	0.262653	0.131327	+	0.991339i	$0.458076\pi$
$402$	0	0
$403$	6.40023	0.318818
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.5060	−0.917310
$408$	0	0
$409$	−25.0533	−1.23880	−0.619402	−	0.785074i	$-0.712625\pi$
$409$	−25.0533	−1.23880	−0.619402	+	0.785074i	$0.712625\pi$
$410$	0	0
$411$	15.9618	0.787338
$412$	0	0
$413$	−4.54253	−0.223523
$414$	0	0
$415$	0	0
$416$	0	0
$417$	36.9493	1.80942
$418$	0	0
$419$	12.7022	0.620542	0.310271	−	0.950648i	$-0.399580\pi$
$419$	12.7022	0.620542	0.310271	+	0.950648i	$0.399580\pi$
$420$	0	0
$421$	13.9618	0.680457	0.340228	−	0.940343i	$-0.389496\pi$
$421$	13.9618	0.680457	0.340228	+	0.940343i	$0.389496\pi$
$422$	0	0
$423$	98.7142	4.79965
$424$	0	0
$425$	0	0
$426$	0	0
$427$	14.7415	0.713393
$428$	0	0
$429$	−38.3129	−1.84976
$430$	0	0
$431$	36.1300	1.74032	0.870160	−	0.492770i	$-0.164016\pi$
$431$	36.1300	1.74032	0.870160	+	0.492770i	$0.164016\pi$
$432$	0	0
$433$	−5.50726	−0.264662	−0.132331	−	0.991206i	$-0.542246\pi$
$433$	−5.50726	−0.264662	−0.132331	+	0.991206i	$0.542246\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.35744	0.256281
$438$	0	0
$439$	−34.4811	−1.64569	−0.822846	−	0.568265i	$-0.807615\pi$
$439$	−34.4811	−1.64569	−0.822846	+	0.568265i	$0.807615\pi$
$440$	0	0
$441$	−25.5002	−1.21429
$442$	0	0
$443$	28.9562	1.37575	0.687874	−	0.725830i	$-0.258544\pi$
$443$	28.9562	1.37575	0.687874	+	0.725830i	$0.258544\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−11.9317	−0.564352
$448$	0	0
$449$	−15.5725	−0.734913	−0.367456	−	0.930041i	$-0.619771\pi$
$449$	−15.5725	−0.734913	−0.367456	+	0.930041i	$0.619771\pi$
$450$	0	0
$451$	−65.3662	−3.07797
$452$	0	0
$453$	34.0256	1.59866
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.715236	−0.0334573	−0.0167287	−	0.999860i	$-0.505325\pi$
$457$	−0.715236	−0.0334573	−0.0167287	+	0.999860i	$0.505325\pi$
$458$	0	0
$459$	54.1065	2.52548
$460$	0	0
$461$	−28.7863	−1.34071	−0.670355	−	0.742041i	$-0.733858\pi$
$461$	−28.7863	−1.34071	−0.670355	+	0.742041i	$0.733858\pi$
$462$	0	0
$463$	−22.1055	−1.02733	−0.513664	−	0.857991i	$-0.671712\pi$
$463$	−22.1055	−1.02733	−0.513664	+	0.857991i	$0.671712\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−37.6645	−1.74291	−0.871454	−	0.490478i	$-0.836822\pi$
$467$	−37.6645	−1.74291	−0.871454	+	0.490478i	$0.836822\pi$
$468$	0	0
$469$	−23.6342	−1.09132
$470$	0	0
$471$	−23.4278	−1.07950
$472$	0	0
$473$	2.82405	0.129850
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−23.5001	−1.07599
$478$	0	0
$479$	−5.32461	−0.243288	−0.121644	−	0.992574i	$-0.538817\pi$
$479$	−5.32461	−0.243288	−0.121644	+	0.992574i	$0.538817\pi$
$480$	0	0
$481$	−6.80536	−0.310298
$482$	0	0
$483$	−34.0256	−1.54822
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.9425	0.677109	0.338555	−	0.940947i	$-0.390062\pi$
$487$	14.9425	0.677109	0.338555	+	0.940947i	$0.390062\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	5.81708	0.262521	0.131260	−	0.991348i	$-0.458098\pi$
$491$	5.81708	0.262521	0.131260	+	0.991348i	$0.458098\pi$
$492$	0	0
$493$	−3.73864	−0.168380
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.7277	1.24376
$498$	0	0
$499$	17.4235	0.779981	0.389990	−	0.920819i	$-0.372478\pi$
$499$	17.4235	0.779981	0.389990	+	0.920819i	$0.372478\pi$
$500$	0	0
$501$	−52.2097	−2.33256
$502$	0	0
$503$	−0.490004	−0.0218482	−0.0109241	−	0.999940i	$-0.503477\pi$
$503$	−0.490004	−0.0218482	−0.0109241	+	0.999940i	$0.503477\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	28.6439	1.27212
$508$	0	0
$509$	16.4811	0.730510	0.365255	−	0.930908i	$-0.380982\pi$
$509$	16.4811	0.730510	0.365255	+	0.930908i	$0.380982\pi$
$510$	0	0
$511$	5.55007	0.245521
$512$	0	0
$513$	−15.7959	−0.697407
$514$	0	0
$515$	0	0
$516$	0	0
$517$	71.2001	3.13138
$518$	0	0
$519$	45.3246	1.98953
$520$	0	0
$521$	17.0533	0.747117	0.373559	−	0.927607i	$-0.378137\pi$
$521$	17.0533	0.747117	0.373559	+	0.927607i	$0.378137\pi$
$522$	0	0
$523$	−34.2777	−1.49886	−0.749430	−	0.662084i	$-0.769672\pi$
$523$	−34.2777	−1.49886	−0.749430	+	0.662084i	$0.769672\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.5893	−0.461278
$528$	0	0
$529$	5.70218	0.247921
$530$	0	0
$531$	−18.3511	−0.796369
$532$	0	0
$533$	−24.0376	−1.04118
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−34.0256	−1.46831
$538$	0	0
$539$	−18.3926	−0.792227
$540$	0	0
$541$	29.9427	1.28734	0.643669	−	0.765304i	$-0.277411\pi$
$541$	29.9427	1.28734	0.643669	+	0.765304i	$0.277411\pi$
$542$	0	0
$543$	6.57430	0.282130
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−29.1339	−1.24568	−0.622838	−	0.782351i	$-0.714020\pi$
$547$	−29.1339	−1.24568	−0.622838	+	0.782351i	$0.714020\pi$
$548$	0	0
$549$	59.5534	2.54168
$550$	0	0
$551$	1.09146	0.0464979
$552$	0	0
$553$	7.72838	0.328644
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.4542	1.62936	0.814678	−	0.579914i	$-0.196914\pi$
$557$	38.4542	1.62936	0.814678	+	0.579914i	$0.196914\pi$
$558$	0	0
$559$	1.03851	0.0439241
$560$	0	0
$561$	63.3896	2.67631
$562$	0	0
$563$	8.58181	0.361680	0.180840	−	0.983512i	$-0.442118\pi$
$563$	8.58181	0.361680	0.180840	+	0.983512i	$0.442118\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	55.0793	2.31311
$568$	0	0
$569$	−0.221120	−0.00926985	−0.00463492	−	0.999989i	$-0.501475\pi$
$569$	−0.221120	−0.00926985	−0.00463492	+	0.999989i	$0.501475\pi$
$570$	0	0
$571$	2.23315	0.0934544	0.0467272	−	0.998908i	$-0.485121\pi$
$571$	2.23315	0.0934544	0.0467272	+	0.998908i	$0.485121\pi$
$572$	0	0
$573$	27.7277	1.15834
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.6225	0.692002	0.346001	−	0.938234i	$-0.387539\pi$
$577$	16.6225	0.692002	0.346001	+	0.938234i	$0.387539\pi$
$578$	0	0
$579$	−76.5460	−3.18114
$580$	0	0
$581$	−10.3511	−0.429436
$582$	0	0
$583$	−16.9500	−0.701998
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.2213	−0.793347	−0.396674	−	0.917960i	$-0.629835\pi$
$587$	−19.2213	−0.793347	−0.396674	+	0.917960i	$0.629835\pi$
$588$	0	0
$589$	3.09146	0.127381
$590$	0	0
$591$	27.2214	1.11974
$592$	0	0
$593$	−13.0230	−0.534792	−0.267396	−	0.963587i	$-0.586163\pi$
$593$	−13.0230	−0.534792	−0.267396	+	0.963587i	$0.586163\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	61.7533	2.52739
$598$	0	0
$599$	27.0915	1.10693	0.553464	−	0.832873i	$-0.313306\pi$
$599$	27.0915	1.10693	0.553464	+	0.832873i	$0.313306\pi$
$600$	0	0
$601$	−5.42779	−0.221404	−0.110702	−	0.993854i	$-0.535310\pi$
$601$	−5.42779	−0.221404	−0.110702	+	0.993854i	$0.535310\pi$
$602$	0	0
$603$	−95.4782	−3.88817
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−21.0663	−0.855056	−0.427528	−	0.904002i	$-0.640616\pi$
$607$	−21.0663	−0.855056	−0.427528	+	0.904002i	$0.640616\pi$
$608$	0	0
$609$	−6.93197	−0.280898
$610$	0	0
$611$	26.1829	1.05925
$612$	0	0
$613$	30.2753	1.22281	0.611405	−	0.791318i	$-0.290605\pi$
$613$	30.2753	1.22281	0.611405	+	0.791318i	$0.290605\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.3049	1.82391	0.911953	−	0.410295i	$-0.134574\pi$
$617$	45.3049	1.82391	0.911953	+	0.410295i	$0.134574\pi$
$618$	0	0
$619$	18.2331	0.732852	0.366426	−	0.930447i	$-0.380581\pi$
$619$	18.2331	0.732852	0.366426	+	0.930447i	$0.380581\pi$
$620$	0	0
$621$	−84.6258	−3.39592
$622$	0	0
$623$	16.1351	0.646439
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−18.5060	−0.739060
$628$	0	0
$629$	11.2596	0.448951
$630$	0	0
$631$	−11.4087	−0.454173	−0.227086	−	0.973875i	$-0.572920\pi$
$631$	−11.4087	−0.454173	−0.227086	+	0.973875i	$0.572920\pi$
$632$	0	0
$633$	71.5905	2.84547
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.76365	−0.267986
$638$	0	0
$639$	112.015	4.43126
$640$	0	0
$641$	4.92330	0.194459	0.0972293	−	0.995262i	$-0.469002\pi$
$641$	4.92330	0.194459	0.0972293	+	0.995262i	$0.469002\pi$
$642$	0	0
$643$	−24.4136	−0.962779	−0.481390	−	0.876507i	$-0.659868\pi$
$643$	−24.4136	−0.962779	−0.481390	+	0.876507i	$0.659868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.0424	0.434123	0.217061	−	0.976158i	$-0.430353\pi$
$647$	11.0424	0.434123	0.217061	+	0.976158i	$0.430353\pi$
$648$	0	0
$649$	−13.2362	−0.519566
$650$	0	0
$651$	−19.6342	−0.769523
$652$	0	0
$653$	−28.1665	−1.10224	−0.551121	−	0.834426i	$-0.685800\pi$
$653$	−28.1665	−1.10224	−0.551121	+	0.834426i	$0.685800\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	22.4214	0.874742
$658$	0	0
$659$	4.74037	0.184659	0.0923294	−	0.995729i	$-0.470569\pi$
$659$	4.74037	0.184659	0.0923294	+	0.995729i	$0.470569\pi$
$660$	0	0
$661$	−19.4426	−0.756228	−0.378114	−	0.925759i	$-0.623427\pi$
$661$	−19.4426	−0.756228	−0.378114	+	0.925759i	$0.623427\pi$
$662$	0	0
$663$	23.3107	0.905313
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.84744	0.226414
$668$	0	0
$669$	−47.8439	−1.84975
$670$	0	0
$671$	42.9544	1.65824
$672$	0	0
$673$	−12.5088	−0.482178	−0.241089	−	0.970503i	$-0.577505\pi$
$673$	−12.5088	−0.482178	−0.241089	+	0.970503i	$0.577505\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.7265	1.56525	0.782623	−	0.622497i	$-0.213882\pi$
$677$	40.7265	1.56525	0.782623	+	0.622497i	$0.213882\pi$
$678$	0	0
$679$	−5.81708	−0.223239
$680$	0	0
$681$	−6.80536	−0.260782
$682$	0	0
$683$	−14.8312	−0.567500	−0.283750	−	0.958898i	$-0.591579\pi$
$683$	−14.8312	−0.567500	−0.283750	+	0.958898i	$0.591579\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	50.8248	1.93909
$688$	0	0
$689$	−6.23315	−0.237464
$690$	0	0
$691$	13.0958	0.498188	0.249094	−	0.968479i	$-0.419867\pi$
$691$	13.0958	0.498188	0.249094	+	0.968479i	$0.419867\pi$
$692$	0	0
$693$	84.9015	3.22514
$694$	0	0
$695$	0	0
$696$	0	0
$697$	39.7707	1.50642
$698$	0	0
$699$	−94.9769	−3.59236
$700$	0	0
$701$	−14.0797	−0.531785	−0.265892	−	0.964003i	$-0.585667\pi$
$701$	−14.0797	−0.531785	−0.265892	+	0.964003i	$0.585667\pi$
$702$	0	0
$703$	−3.28715	−0.123977
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.28441	−0.349176
$708$	0	0
$709$	8.38928	0.315066	0.157533	−	0.987514i	$-0.449646\pi$
$709$	8.38928	0.315066	0.157533	+	0.987514i	$0.449646\pi$
$710$	0	0
$711$	31.2214	1.17090
$712$	0	0
$713$	16.5623	0.620264
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−87.4977	−3.26766
$718$	0	0
$719$	6.70652	0.250111	0.125055	−	0.992150i	$-0.460089\pi$
$719$	6.70652	0.250111	0.125055	+	0.992150i	$0.460089\pi$
$720$	0	0
$721$	8.16816	0.304198
$722$	0	0
$723$	64.4634	2.39742
$724$	0	0
$725$	0	0
$726$	0	0
$727$	20.5009	0.760337	0.380168	−	0.924917i	$-0.375866\pi$
$727$	20.5009	0.760337	0.380168	+	0.924917i	$0.375866\pi$
$728$	0	0
$729$	66.7407	2.47188
$730$	0	0
$731$	−1.71823	−0.0635512
$732$	0	0
$733$	−42.4323	−1.56727	−0.783636	−	0.621220i	$-0.786637\pi$
$733$	−42.4323	−1.56727	−0.783636	+	0.621220i	$0.786637\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−68.8661	−2.53671
$738$	0	0
$739$	44.4352	1.63457	0.817287	−	0.576231i	$-0.195477\pi$
$739$	44.4352	1.63457	0.817287	+	0.576231i	$0.195477\pi$
$740$	0	0
$741$	−6.80536	−0.250001
$742$	0	0
$743$	−36.4350	−1.33667	−0.668336	−	0.743859i	$-0.732993\pi$
$743$	−36.4350	−1.33667	−0.668336	+	0.743859i	$0.732993\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−41.8167	−1.52999
$748$	0	0
$749$	17.2362	0.629797
$750$	0	0
$751$	−29.6107	−1.08051	−0.540255	−	0.841501i	$-0.681672\pi$
$751$	−29.6107	−1.08051	−0.540255	+	0.841501i	$0.681672\pi$
$752$	0	0
$753$	−46.8493	−1.70728
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−44.0252	−1.60012	−0.800062	−	0.599917i	$-0.795200\pi$
$757$	−44.0252	−1.60012	−0.800062	+	0.599917i	$0.795200\pi$
$758$	0	0
$759$	−99.1450	−3.59874
$760$	0	0
$761$	25.9809	0.941807	0.470903	−	0.882185i	$-0.343928\pi$
$761$	25.9809	0.941807	0.470903	+	0.882185i	$0.343928\pi$
$762$	0	0
$763$	10.1621	0.367893
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.86744	−0.175753
$768$	0	0
$769$	2.68308	0.0967543	0.0483772	−	0.998829i	$-0.484595\pi$
$769$	2.68308	0.0967543	0.0483772	+	0.998829i	$0.484595\pi$
$770$	0	0
$771$	−37.1183	−1.33678
$772$	0	0
$773$	39.5328	1.42190	0.710949	−	0.703244i	$-0.248266\pi$
$773$	39.5328	1.42190	0.710949	+	0.703244i	$0.248266\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.8770	0.748958
$778$	0	0
$779$	−11.6107	−0.415997
$780$	0	0
$781$	80.7940	2.89103
$782$	0	0
$783$	−17.2407	−0.616131
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9.15783	−0.326442	−0.163221	−	0.986590i	$-0.552188\pi$
$787$	−9.15783	−0.326442	−0.163221	+	0.986590i	$0.552188\pi$
$788$	0	0
$789$	50.4811	1.79717
$790$	0	0
$791$	24.1682	0.859321
$792$	0	0
$793$	15.7959	0.560930
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.5581	0.551095	0.275547	−	0.961287i	$-0.411141\pi$
$797$	15.5581	0.551095	0.275547	+	0.961287i	$0.411141\pi$
$798$	0	0
$799$	−43.3203	−1.53256
$800$	0	0
$801$	65.1832	2.30314
$802$	0	0
$803$	16.1720	0.570698
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−37.6134	−1.32405
$808$	0	0
$809$	−41.6184	−1.46323	−0.731613	−	0.681721i	$-0.761232\pi$
$809$	−41.6184	−1.46323	−0.731613	+	0.681721i	$0.761232\pi$
$810$	0	0
$811$	−49.1685	−1.72654	−0.863269	−	0.504744i	$-0.831587\pi$
$811$	−49.1685	−1.72654	−0.863269	+	0.504744i	$0.831587\pi$
$812$	0	0
$813$	−40.8135	−1.43139
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.501623	0.0175496
$818$	0	0
$819$	31.2214	1.09097
$820$	0	0
$821$	−13.9012	−0.485154	−0.242577	−	0.970132i	$-0.577993\pi$
$821$	−13.9012	−0.485154	−0.242577	+	0.970132i	$0.577993\pi$
$822$	0	0
$823$	19.0704	0.664754	0.332377	−	0.943147i	$-0.392149\pi$
$823$	19.0704	0.664754	0.332377	+	0.943147i	$0.392149\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.7712	−1.62639	−0.813197	−	0.581988i	$-0.802275\pi$
$827$	−46.7712	−1.62639	−0.813197	+	0.581988i	$0.802275\pi$
$828$	0	0
$829$	2.37452	0.0824706	0.0412353	−	0.999149i	$-0.486871\pi$
$829$	2.37452	0.0824706	0.0412353	+	0.999149i	$0.486871\pi$
$830$	0	0
$831$	52.6640	1.82689
$832$	0	0
$833$	11.1906	0.387732
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−48.8325	−1.68790
$838$	0	0
$839$	−1.57252	−0.0542894	−0.0271447	−	0.999632i	$-0.508641\pi$
$839$	−1.57252	−0.0542894	−0.0271447	+	0.999632i	$0.508641\pi$
$840$	0	0
$841$	−27.8087	−0.958921
$842$	0	0
$843$	81.8011	2.81738
$844$	0	0
$845$	0	0
$846$	0	0
$847$	39.9843	1.37388
$848$	0	0
$849$	−53.3662	−1.83152
$850$	0	0
$851$	−17.6107	−0.603688
$852$	0	0
$853$	1.88094	0.0644021	0.0322010	−	0.999481i	$-0.489748\pi$
$853$	1.88094	0.0644021	0.0322010	+	0.999481i	$0.489748\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17.0371	0.581975	0.290987	−	0.956727i	$-0.406016\pi$
$857$	17.0371	0.581975	0.290987	+	0.956727i	$0.406016\pi$
$858$	0	0
$859$	49.8395	1.70050	0.850251	−	0.526377i	$-0.176450\pi$
$859$	49.8395	1.70050	0.850251	+	0.526377i	$0.176450\pi$
$860$	0	0
$861$	73.7407	2.51308
$862$	0	0
$863$	12.1696	0.414258	0.207129	−	0.978314i	$-0.433588\pi$
$863$	12.1696	0.414258	0.207129	+	0.978314i	$0.433588\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.3134	0.587995
$868$	0	0
$869$	22.5193	0.763913
$870$	0	0
$871$	−25.3246	−0.858092
$872$	0	0
$873$	−23.5001	−0.795356
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−42.2287	−1.42596	−0.712981	−	0.701184i	$-0.752655\pi$
$877$	−42.2287	−1.42596	−0.712981	+	0.701184i	$0.752655\pi$
$878$	0	0
$879$	−76.5460	−2.58183
$880$	0	0
$881$	−12.6683	−0.426807	−0.213403	−	0.976964i	$-0.568455\pi$
$881$	−12.6683	−0.426807	−0.213403	+	0.976964i	$0.568455\pi$
$882$	0	0
$883$	−11.5414	−0.388400	−0.194200	−	0.980962i	$-0.562211\pi$
$883$	−11.5414	−0.388400	−0.194200	+	0.980962i	$0.562211\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.36171	0.213605	0.106803	−	0.994280i	$-0.465939\pi$
$887$	6.36171	0.213605	0.106803	+	0.994280i	$0.465939\pi$
$888$	0	0
$889$	23.7554	0.796732
$890$	0	0
$891$	160.492	5.37669
$892$	0	0
$893$	12.6470	0.423215
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−36.4593	−1.21734
$898$	0	0
$899$	3.37421	0.112536
$900$	0	0
$901$	10.3129	0.343572
$902$	0	0
$903$	−3.18585	−0.106019
$904$	0	0
$905$	0	0
$906$	0	0
$907$	24.9538	0.828576	0.414288	−	0.910146i	$-0.364031\pi$
$907$	24.9538	0.828576	0.414288	+	0.910146i	$0.364031\pi$
$908$	0	0
$909$	−37.5075	−1.24405
$910$	0	0
$911$	−28.6640	−0.949680	−0.474840	−	0.880072i	$-0.657494\pi$
$911$	−28.6640	−0.949680	−0.474840	+	0.880072i	$0.657494\pi$
$912$	0	0
$913$	−30.1614	−0.998196
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20.8401	−0.688200
$918$	0	0
$919$	−25.0385	−0.825944	−0.412972	−	0.910744i	$-0.635509\pi$
$919$	−25.0385	−0.825944	−0.412972	+	0.910744i	$0.635509\pi$
$920$	0	0
$921$	−3.17121	−0.104495
$922$	0	0
$923$	29.7109	0.977947
$924$	0	0
$925$	0	0
$926$	0	0
$927$	32.9981	1.08380
$928$	0	0
$929$	27.2744	0.894844	0.447422	−	0.894323i	$-0.352342\pi$
$929$	27.2744	0.894844	0.447422	+	0.894323i	$0.352342\pi$
$930$	0	0
$931$	−3.26701	−0.107072
$932$	0	0
$933$	−59.7072	−1.95473
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−28.7193	−0.938219	−0.469109	−	0.883140i	$-0.655425\pi$
$937$	−28.7193	−0.938219	−0.469109	+	0.883140i	$0.655425\pi$
$938$	0	0
$939$	−24.0000	−0.783210
$940$	0	0
$941$	7.81708	0.254829	0.127415	−	0.991850i	$-0.459332\pi$
$941$	7.81708	0.254829	0.127415	+	0.991850i	$0.459332\pi$
$942$	0	0
$943$	−62.2037	−2.02563
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.4886	−1.31570	−0.657851	−	0.753148i	$-0.728534\pi$
$947$	−40.4886	−1.31570	−0.657851	+	0.753148i	$0.728534\pi$
$948$	0	0
$949$	5.94704	0.193049
$950$	0	0
$951$	25.1183	0.814515
$952$	0	0
$953$	−7.57857	−0.245494	−0.122747	−	0.992438i	$-0.539170\pi$
$953$	−7.57857	−0.245494	−0.122747	+	0.992438i	$0.539170\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−20.1986	−0.652930
$958$	0	0
$959$	−9.38190	−0.302958
$960$	0	0
$961$	−21.4429	−0.691705
$962$	0	0
$963$	69.6315	2.24384
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−15.5195	−0.499075	−0.249537	−	0.968365i	$-0.580279\pi$
$967$	−15.5195	−0.499075	−0.249537	+	0.968365i	$0.580279\pi$
$968$	0	0
$969$	11.2596	0.361711
$970$	0	0
$971$	1.81708	0.0583127	0.0291564	−	0.999575i	$-0.490718\pi$
$971$	1.81708	0.0583127	0.0291564	+	0.999575i	$0.490718\pi$
$972$	0	0
$973$	−21.7178	−0.696240
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.0178	−0.672420	−0.336210	−	0.941787i	$-0.609145\pi$
$977$	−21.0178	−0.672420	−0.336210	+	0.941787i	$0.609145\pi$
$978$	0	0
$979$	47.0151	1.50261
$980$	0	0
$981$	41.0533	1.31073
$982$	0	0
$983$	−19.2339	−0.613467	−0.306733	−	0.951795i	$-0.599236\pi$
$983$	−19.2339	−0.613467	−0.306733	+	0.951795i	$0.599236\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−80.3221	−2.55668
$988$	0	0
$989$	2.68742	0.0854549
$990$	0	0
$991$	36.1682	1.14892	0.574460	−	0.818533i	$-0.305212\pi$
$991$	36.1682	1.14892	0.574460	+	0.818533i	$0.305212\pi$
$992$	0	0
$993$	−73.3458	−2.32756
$994$	0	0
$995$	0	0
$996$	0	0
$997$	35.6955	1.13049	0.565245	−	0.824923i	$-0.308782\pi$
$997$	35.6955	1.13049	0.565245	+	0.824923i	$0.308782\pi$
$998$	0	0
$999$	51.9236	1.64279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.a.k.1.1		6
4.3	odd	2		7600.2.a.cj.1.6		6
5.2	odd	4		380.2.c.b.229.6	yes	6
5.3	odd	4		380.2.c.b.229.1	✓	6
5.4	even	2	inner	1900.2.a.k.1.6		6
15.2	even	4		3420.2.f.c.1369.1		6
15.8	even	4		3420.2.f.c.1369.2		6
20.3	even	4		1520.2.d.i.609.6		6
20.7	even	4		1520.2.d.i.609.1		6
20.19	odd	2		7600.2.a.cj.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.c.b.229.1	✓	6	5.3	odd	4
380.2.c.b.229.6	yes	6	5.2	odd	4
1520.2.d.i.609.1		6	20.7	even	4
1520.2.d.i.609.6		6	20.3	even	4
1900.2.a.k.1.1		6	1.1	even	1	trivial
1900.2.a.k.1.6		6	5.4	even	2	inner
3420.2.f.c.1369.1		6	15.2	even	4
3420.2.f.c.1369.2		6	15.8	even	4
7600.2.a.cj.1.1		6	20.19	odd	2
7600.2.a.cj.1.6		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.a (trivial)

\(f(q)\)	\(=\)	\(q-3.28715 q^{3} +1.93210 q^{7} +7.80536 q^{9} +O(q^{10})\)	\(q-3.28715 q^{3} +1.93210 q^{7} +7.80536 q^{9} +5.62981 q^{11} +2.07029 q^{13} -3.42535 q^{17} +1.00000 q^{19} -6.35109 q^{21} +5.35744 q^{23} -15.7959 q^{27} +1.09146 q^{29} +3.09146 q^{31} -18.5060 q^{33} -3.28715 q^{37} -6.80536 q^{39} -11.6107 q^{41} +0.501623 q^{43} +12.6470 q^{47} -3.26701 q^{49} +11.2596 q^{51} -3.01076 q^{53} -3.28715 q^{57} -2.35109 q^{59} +7.62981 q^{61} +15.0807 q^{63} -12.2324 q^{67} -17.6107 q^{69} +14.3511 q^{71} +2.87256 q^{73} +10.8773 q^{77} +4.00000 q^{79} +28.5075 q^{81} -5.35744 q^{83} -3.58780 q^{87} +8.35109 q^{89} +4.00000 q^{91} -10.1621 q^{93} -3.01076 q^{97} +43.9427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{9}+O(q^{10}) \) 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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