LMFDB - Embedded newform 200.14.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,14,Mod(1,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("200.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	200.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:	$214.461857904$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7121x^{2} - 128406x + 3057138 $ x^4 - x^3 - 7121x^2 - 128406x + 3057138
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{2}\cdot 5^{2}\cdot 17 $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-66.6188$ of defining polynomial
Character	$\chi$	$=$	200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+341.481 q^{3} +519291. q^{7} -1.47771e6 q^{9} +O(q^{10})$	$q+341.481 q^{3} +519291. q^{7} -1.47771e6 q^{9} +4.14757e6 q^{11} -1.64761e7 q^{13} -2.49229e7 q^{17} -3.23445e8 q^{19} +1.77328e8 q^{21} +4.72725e8 q^{23} -1.04904e9 q^{27} -3.21281e8 q^{29} +8.21228e9 q^{31} +1.41632e9 q^{33} -3.84589e9 q^{37} -5.62627e9 q^{39} +5.13789e10 q^{41} +6.55191e9 q^{43} -1.08909e11 q^{47} +1.72774e11 q^{49} -8.51071e9 q^{51} +1.80566e11 q^{53} -1.10450e11 q^{57} +1.58372e11 q^{59} +6.54853e11 q^{61} -7.67363e11 q^{63} -1.96867e11 q^{67} +1.61427e11 q^{69} -1.46786e12 q^{71} -2.18020e12 q^{73} +2.15380e12 q^{77} +1.48519e12 q^{79} +1.99772e12 q^{81} -5.36718e11 q^{83} -1.09712e11 q^{87} -4.39392e12 q^{89} -8.55588e12 q^{91} +2.80434e12 q^{93} -3.00844e12 q^{97} -6.12893e12 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 936 q^{3} - 112424 q^{7} + 2737236 q^{9}+O(q^{10}) $ 4 * q - 936 * q^3 - 112424 * q^7 + 2737236 * q^9	$ 4 q - 936 q^{3} - 112424 q^{7} + 2737236 q^{9} - 85280 q^{11} - 22276696 q^{13} - 92966696 q^{17} - 52003280 q^{19} - 124736832 q^{21} + 498172904 q^{23} - 1984218768 q^{27} + 5868734200 q^{29} + 1204156144 q^{31} + 2413789344 q^{33} + 1227431400 q^{37} + 47183929776 q^{39} + 17822906952 q^{41} - 92192384488 q^{43} - 187731312904 q^{47} + 218812853892 q^{49} + 436920457680 q^{51} - 617748697656 q^{53} - 962043447456 q^{57} + 155305995248 q^{59} + 918726826936 q^{61} - 453684626952 q^{63} - 1172014952696 q^{67} + 2007735561792 q^{69} + 1499341271376 q^{71} - 3669705031176 q^{73} - 2543134824608 q^{77} + 1474589017632 q^{79} + 1358685231876 q^{81} + 10007128523992 q^{83} + 6056521948368 q^{87} - 2840927507160 q^{89} - 16479691138768 q^{91} + 4603403842560 q^{93} - 7763871939496 q^{97} - 50298890417952 q^{99}+O(q^{100}) $ 4 * q - 936 * q^3 - 112424 * q^7 + 2737236 * q^9 - 85280 * q^11 - 22276696 * q^13 - 92966696 * q^17 - 52003280 * q^19 - 124736832 * q^21 + 498172904 * q^23 - 1984218768 * q^27 + 5868734200 * q^29 + 1204156144 * q^31 + 2413789344 * q^33 + 1227431400 * q^37 + 47183929776 * q^39 + 17822906952 * q^41 - 92192384488 * q^43 - 187731312904 * q^47 + 218812853892 * q^49 + 436920457680 * q^51 - 617748697656 * q^53 - 962043447456 * q^57 + 155305995248 * q^59 + 918726826936 * q^61 - 453684626952 * q^63 - 1172014952696 * q^67 + 2007735561792 * q^69 + 1499341271376 * q^71 - 3669705031176 * q^73 - 2543134824608 * q^77 + 1474589017632 * q^79 + 1358685231876 * q^81 + 10007128523992 * q^83 + 6056521948368 * q^87 - 2840927507160 * q^89 - 16479691138768 * q^91 + 4603403842560 * q^93 - 7763871939496 * q^97 - 50298890417952 * 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
$3$	341.481	0.270445	0.135222	−	0.990815i	$-0.456825\pi$
$3$	341.481	0.270445	0.135222	+	0.990815i	$0.456825\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	519291.	1.66830	0.834149	−	0.551539i	$-0.185959\pi$
$7$	519291.	1.66830	0.834149	+	0.551539i	$0.185959\pi$
$8$	0	0
$9$	−1.47771e6	−0.926860
$10$	0	0
$11$	4.14757e6	0.705898	0.352949	−	0.935643i	$-0.385179\pi$
$11$	4.14757e6	0.705898	0.352949	+	0.935643i	$0.385179\pi$
$12$	0	0
$13$	−1.64761e7	−0.946721	−0.473361	−	0.880869i	$-0.656959\pi$
$13$	−1.64761e7	−0.946721	−0.473361	+	0.880869i	$0.656959\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.49229e7	−0.250427	−0.125213	−	0.992130i	$-0.539962\pi$
$17$	−2.49229e7	−0.250427	−0.125213	+	0.992130i	$0.539962\pi$
$18$	0	0
$19$	−3.23445e8	−1.57726	−0.788628	−	0.614871i	$-0.789208\pi$
$19$	−3.23445e8	−1.57726	−0.788628	+	0.614871i	$0.789208\pi$
$20$	0	0
$21$	1.77328e8	0.451183
$22$	0	0
$23$	4.72725e8	0.665852	0.332926	−	0.942953i	$-0.391964\pi$
$23$	4.72725e8	0.665852	0.332926	+	0.942953i	$0.391964\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.04904e9	−0.521109
$28$	0	0
$29$	−3.21281e8	−0.100299	−0.0501497	−	0.998742i	$-0.515970\pi$
$29$	−3.21281e8	−0.100299	−0.0501497	+	0.998742i	$0.515970\pi$
$30$	0	0
$31$	8.21228e9	1.66193	0.830965	−	0.556325i	$-0.187789\pi$
$31$	8.21228e9	1.66193	0.830965	+	0.556325i	$0.187789\pi$
$32$	0	0
$33$	1.41632e9	0.190906
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.84589e9	−0.246425	−0.123213	−	0.992380i	$-0.539320\pi$
$37$	−3.84589e9	−0.246425	−0.123213	+	0.992380i	$0.539320\pi$
$38$	0	0
$39$	−5.62627e9	−0.256036
$40$	0	0
$41$	5.13789e10	1.68923	0.844617	−	0.535371i	$-0.179828\pi$
$41$	5.13789e10	1.68923	0.844617	+	0.535371i	$0.179828\pi$
$42$	0	0
$43$	6.55191e9	0.158060	0.0790302	−	0.996872i	$-0.474818\pi$
$43$	6.55191e9	0.158060	0.0790302	+	0.996872i	$0.474818\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.08909e11	−1.47377	−0.736883	−	0.676020i	$-0.763703\pi$
$47$	−1.08909e11	−1.47377	−0.736883	+	0.676020i	$0.763703\pi$
$48$	0	0
$49$	1.72774e11	1.78322
$50$	0	0
$51$	−8.51071e9	−0.0677267
$52$	0	0
$53$	1.80566e11	1.11903	0.559517	−	0.828819i	$-0.310987\pi$
$53$	1.80566e11	1.11903	0.559517	+	0.828819i	$0.310987\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.10450e11	−0.426561
$58$	0	0
$59$	1.58372e11	0.488810	0.244405	−	0.969673i	$-0.421407\pi$
$59$	1.58372e11	0.488810	0.244405	+	0.969673i	$0.421407\pi$
$60$	0	0
$61$	6.54853e11	1.62742	0.813710	−	0.581271i	$-0.197444\pi$
$61$	6.54853e11	1.62742	0.813710	+	0.581271i	$0.197444\pi$
$62$	0	0
$63$	−7.67363e11	−1.54628
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.96867e11	−0.265881	−0.132940	−	0.991124i	$-0.542442\pi$
$67$	−1.96867e11	−0.265881	−0.132940	+	0.991124i	$0.542442\pi$
$68$	0	0
$69$	1.61427e11	0.180076
$70$	0	0
$71$	−1.46786e12	−1.35990	−0.679948	−	0.733261i	$-0.737998\pi$
$71$	−1.46786e12	−1.35990	−0.679948	+	0.733261i	$0.737998\pi$
$72$	0	0
$73$	−2.18020e12	−1.68615	−0.843077	−	0.537793i	$-0.819258\pi$
$73$	−2.18020e12	−1.68615	−0.843077	+	0.537793i	$0.819258\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.15380e12	1.17765
$78$	0	0
$79$	1.48519e12	0.687396	0.343698	−	0.939080i	$-0.388320\pi$
$79$	1.48519e12	0.687396	0.343698	+	0.939080i	$0.388320\pi$
$80$	0	0
$81$	1.99772e12	0.785928
$82$	0	0
$83$	−5.36718e11	−0.180193	−0.0900967	−	0.995933i	$-0.528718\pi$
$83$	−5.36718e11	−0.180193	−0.0900967	+	0.995933i	$0.528718\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.09712e11	−0.0271255
$88$	0	0
$89$	−4.39392e12	−0.937167	−0.468583	−	0.883419i	$-0.655235\pi$
$89$	−4.39392e12	−0.937167	−0.468583	+	0.883419i	$0.655235\pi$
$90$	0	0
$91$	−8.55588e12	−1.57941
$92$	0	0
$93$	2.80434e12	0.449461
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.00844e12	−0.366712	−0.183356	−	0.983047i	$-0.558696\pi$
$97$	−3.00844e12	−0.366712	−0.183356	+	0.983047i	$0.558696\pi$
$98$	0	0
$99$	−6.12893e12	−0.654268
$100$	0	0
$101$	1.40811e13	1.31992	0.659958	−	0.751302i	$-0.270574\pi$
$101$	1.40811e13	1.31992	0.659958	+	0.751302i	$0.270574\pi$
$102$	0	0
$103$	1.94525e13	1.60522	0.802609	−	0.596506i	$-0.203445\pi$
$103$	1.94525e13	1.60522	0.802609	+	0.596506i	$0.203445\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.59543e13	1.02774	0.513871	−	0.857867i	$-0.328211\pi$
$107$	1.59543e13	1.02774	0.513871	+	0.857867i	$0.328211\pi$
$108$	0	0
$109$	2.40526e13	1.37369	0.686846	−	0.726803i	$-0.258995\pi$
$109$	2.40526e13	1.37369	0.686846	+	0.726803i	$0.258995\pi$
$110$	0	0
$111$	−1.31330e12	−0.0666445
$112$	0	0
$113$	1.82576e13	0.824962	0.412481	−	0.910966i	$-0.364662\pi$
$113$	1.82576e13	0.824962	0.412481	+	0.910966i	$0.364662\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.43469e13	0.877477
$118$	0	0
$119$	−1.29422e13	−0.417786
$120$	0	0
$121$	−1.73203e13	−0.501709
$122$	0	0
$123$	1.75449e13	0.456845
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.04435e13	0.220863	0.110431	−	0.993884i	$-0.464777\pi$
$127$	1.04435e13	0.220863	0.110431	+	0.993884i	$0.464777\pi$
$128$	0	0
$129$	2.23736e12	0.0427466
$130$	0	0
$131$	3.88754e13	0.672065	0.336033	−	0.941850i	$-0.390915\pi$
$131$	3.88754e13	0.672065	0.336033	+	0.941850i	$0.390915\pi$
$132$	0	0
$133$	−1.67962e14	−2.63133
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.10112e14	−1.42283	−0.711413	−	0.702774i	$-0.751944\pi$
$137$	−1.10112e14	−1.42283	−0.711413	+	0.702774i	$0.751944\pi$
$138$	0	0
$139$	−8.21149e13	−0.965663	−0.482831	−	0.875713i	$-0.660392\pi$
$139$	−8.21149e13	−0.965663	−0.482831	+	0.875713i	$0.660392\pi$
$140$	0	0
$141$	−3.71905e13	−0.398573
$142$	0	0
$143$	−6.83358e13	−0.668288
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.89992e13	0.482262
$148$	0	0
$149$	1.33385e14	0.998610	0.499305	−	0.866426i	$-0.333589\pi$
$149$	1.33385e14	0.998610	0.499305	+	0.866426i	$0.333589\pi$
$150$	0	0
$151$	−4.09054e13	−0.280821	−0.140411	−	0.990093i	$-0.544842\pi$
$151$	−4.09054e13	−0.280821	−0.140411	+	0.990093i	$0.544842\pi$
$152$	0	0
$153$	3.68289e13	0.232110
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−9.86234e13	−0.525572	−0.262786	−	0.964854i	$-0.584641\pi$
$157$	−9.86234e13	−0.525572	−0.262786	+	0.964854i	$0.584641\pi$
$158$	0	0
$159$	6.16600e13	0.302637
$160$	0	0
$161$	2.45482e14	1.11084
$162$	0	0
$163$	−5.38507e13	−0.224891	−0.112445	−	0.993658i	$-0.535868\pi$
$163$	−5.38507e13	−0.224891	−0.112445	+	0.993658i	$0.535868\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.77101e14	0.631776	0.315888	−	0.948796i	$-0.397698\pi$
$167$	1.77101e14	0.631776	0.315888	+	0.948796i	$0.397698\pi$
$168$	0	0
$169$	−3.14139e13	−0.103719
$170$	0	0
$171$	4.77959e14	1.46189
$172$	0	0
$173$	6.62187e14	1.87794	0.938968	−	0.344004i	$-0.111783\pi$
$173$	6.62187e14	1.87794	0.938968	+	0.344004i	$0.111783\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.40811e13	0.132196
$178$	0	0
$179$	−1.75476e14	−0.398726	−0.199363	−	0.979926i	$-0.563887\pi$
$179$	−1.75476e14	−0.398726	−0.199363	+	0.979926i	$0.563887\pi$
$180$	0	0
$181$	−1.26205e14	−0.266787	−0.133394	−	0.991063i	$-0.542587\pi$
$181$	−1.26205e14	−0.266787	−0.133394	+	0.991063i	$0.542587\pi$
$182$	0	0
$183$	2.23620e14	0.440128
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.03370e14	−0.176776
$188$	0	0
$189$	−5.44759e14	−0.869366
$190$	0	0
$191$	−4.87160e13	−0.0726030	−0.0363015	−	0.999341i	$-0.511558\pi$
$191$	−4.87160e13	−0.0726030	−0.0363015	+	0.999341i	$0.511558\pi$
$192$	0	0
$193$	3.51570e14	0.489654	0.244827	−	0.969567i	$-0.421269\pi$
$193$	3.51570e14	0.489654	0.244827	+	0.969567i	$0.421269\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.75069e14	0.944735	0.472367	−	0.881402i	$-0.343400\pi$
$197$	7.75069e14	0.944735	0.472367	+	0.881402i	$0.343400\pi$
$198$	0	0
$199$	−1.53701e14	−0.175441	−0.0877207	−	0.996145i	$-0.527958\pi$
$199$	−1.53701e14	−0.175441	−0.0877207	+	0.996145i	$0.527958\pi$
$200$	0	0
$201$	−6.72265e13	−0.0719062
$202$	0	0
$203$	−1.66838e14	−0.167329
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.98552e14	−0.617151
$208$	0	0
$209$	−1.34151e15	−1.11338
$210$	0	0
$211$	3.33974e14	0.260541	0.130271	−	0.991478i	$-0.458415\pi$
$211$	3.33974e14	0.260541	0.130271	+	0.991478i	$0.458415\pi$
$212$	0	0
$213$	−5.01247e14	−0.367777
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.26456e15	2.77259
$218$	0	0
$219$	−7.44497e14	−0.456012
$220$	0	0
$221$	4.10632e14	0.237084
$222$	0	0
$223$	−2.67761e15	−1.45803	−0.729015	−	0.684498i	$-0.760021\pi$
$223$	−2.67761e15	−1.45803	−0.729015	+	0.684498i	$0.760021\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.08407e14	0.0525884	0.0262942	−	0.999654i	$-0.491629\pi$
$227$	1.08407e14	0.0525884	0.0262942	+	0.999654i	$0.491629\pi$
$228$	0	0
$229$	6.80147e14	0.311653	0.155827	−	0.987784i	$-0.450196\pi$
$229$	6.80147e14	0.311653	0.155827	+	0.987784i	$0.450196\pi$
$230$	0	0
$231$	7.35482e14	0.318489
$232$	0	0
$233$	−1.97048e14	−0.0806788	−0.0403394	−	0.999186i	$-0.512844\pi$
$233$	−1.97048e14	−0.0806788	−0.0403394	+	0.999186i	$0.512844\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.07166e14	0.185903
$238$	0	0
$239$	3.63126e15	1.26029	0.630146	−	0.776477i	$-0.282995\pi$
$239$	3.63126e15	1.26029	0.630146	+	0.776477i	$0.282995\pi$
$240$	0	0
$241$	5.71228e15	1.87801	0.939006	−	0.343901i	$-0.111748\pi$
$241$	5.71228e15	1.87801	0.939006	+	0.343901i	$0.111748\pi$
$242$	0	0
$243$	2.35470e15	0.733660
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.32911e15	1.49322
$248$	0	0
$249$	−1.83279e14	−0.0487324
$250$	0	0
$251$	6.32445e15	1.59641	0.798204	−	0.602387i	$-0.205784\pi$
$251$	6.32445e15	1.59641	0.798204	+	0.602387i	$0.205784\pi$
$252$	0	0
$253$	1.96066e15	0.470023
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.66205e15	−1.44226	−0.721129	−	0.692801i	$-0.756376\pi$
$257$	−6.66205e15	−1.44226	−0.721129	+	0.692801i	$0.756376\pi$
$258$	0	0
$259$	−1.99713e15	−0.411111
$260$	0	0
$261$	4.74762e14	0.0929635
$262$	0	0
$263$	3.61047e15	0.672746	0.336373	−	0.941729i	$-0.390800\pi$
$263$	3.61047e15	0.672746	0.336373	+	0.941729i	$0.390800\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.50044e15	−0.253452
$268$	0	0
$269$	3.64277e15	0.586196	0.293098	−	0.956082i	$-0.405314\pi$
$269$	3.64277e15	0.586196	0.293098	+	0.956082i	$0.405314\pi$
$270$	0	0
$271$	1.17574e16	1.80306	0.901529	−	0.432718i	$-0.142445\pi$
$271$	1.17574e16	1.80306	0.901529	+	0.432718i	$0.142445\pi$
$272$	0	0
$273$	−2.92167e15	−0.427144
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.57993e15	1.14121	0.570605	−	0.821225i	$-0.306709\pi$
$277$	8.57993e15	1.14121	0.570605	+	0.821225i	$0.306709\pi$
$278$	0	0
$279$	−1.21354e16	−1.54038
$280$	0	0
$281$	9.94598e15	1.20519	0.602596	−	0.798046i	$-0.294133\pi$
$281$	9.94598e15	1.20519	0.602596	+	0.798046i	$0.294133\pi$
$282$	0	0
$283$	−5.56413e15	−0.643851	−0.321925	−	0.946765i	$-0.604330\pi$
$283$	−5.56413e15	−0.643851	−0.321925	+	0.946765i	$0.604330\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.66806e16	2.81815
$288$	0	0
$289$	−9.28343e15	−0.937286
$290$	0	0
$291$	−1.02733e15	−0.0991755
$292$	0	0
$293$	1.14644e16	1.05855	0.529275	−	0.848451i	$-0.322464\pi$
$293$	1.14644e16	1.05855	0.529275	+	0.848451i	$0.322464\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.35098e15	−0.367850
$298$	0	0
$299$	−7.78866e15	−0.630376
$300$	0	0
$301$	3.40235e15	0.263692
$302$	0	0
$303$	4.80842e15	0.356965
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.21107e16	1.50731	0.753655	−	0.657270i	$-0.228289\pi$
$307$	2.21107e16	1.50731	0.753655	+	0.657270i	$0.228289\pi$
$308$	0	0
$309$	6.64267e15	0.434123
$310$	0	0
$311$	−1.62272e15	−0.101695	−0.0508476	−	0.998706i	$-0.516192\pi$
$311$	−1.62272e15	−0.101695	−0.0508476	+	0.998706i	$0.516192\pi$
$312$	0	0
$313$	−1.65696e16	−0.996031	−0.498015	−	0.867168i	$-0.665938\pi$
$313$	−1.65696e16	−0.996031	−0.498015	+	0.867168i	$0.665938\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.01064e16	0.559383	0.279692	−	0.960090i	$-0.409768\pi$
$317$	1.01064e16	0.559383	0.279692	+	0.960090i	$0.409768\pi$
$318$	0	0
$319$	−1.33254e15	−0.0708011
$320$	0	0
$321$	5.44811e15	0.277948
$322$	0	0
$323$	8.06119e15	0.394987
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.21350e15	0.371508
$328$	0	0
$329$	−5.65556e16	−2.45868
$330$	0	0
$331$	5.35986e15	0.224012	0.112006	−	0.993708i	$-0.464272\pi$
$331$	5.35986e15	0.224012	0.112006	+	0.993708i	$0.464272\pi$
$332$	0	0
$333$	5.68312e15	0.228402
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.16047e16	1.17532	0.587661	−	0.809107i	$-0.300049\pi$
$337$	3.16047e16	1.17532	0.587661	+	0.809107i	$0.300049\pi$
$338$	0	0
$339$	6.23464e15	0.223107
$340$	0	0
$341$	3.40610e16	1.17315
$342$	0	0
$343$	3.94065e16	1.30664
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.98391e16	−1.84011	−0.920054	−	0.391791i	$-0.871856\pi$
$347$	−5.98391e16	−1.84011	−0.920054	+	0.391791i	$0.871856\pi$
$348$	0	0
$349$	4.73910e16	1.40388	0.701942	−	0.712234i	$-0.252317\pi$
$349$	4.73910e16	1.40388	0.701942	+	0.712234i	$0.252317\pi$
$350$	0	0
$351$	1.72841e16	0.493345
$352$	0	0
$353$	−3.88639e16	−1.06908	−0.534541	−	0.845143i	$-0.679515\pi$
$353$	−3.88639e16	−1.06908	−0.534541	+	0.845143i	$0.679515\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.41953e15	−0.112988
$358$	0	0
$359$	2.03162e16	0.500873	0.250437	−	0.968133i	$-0.419426\pi$
$359$	2.03162e16	0.500873	0.250437	+	0.968133i	$0.419426\pi$
$360$	0	0
$361$	6.25638e16	1.48774
$362$	0	0
$363$	−5.91457e15	−0.135685
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.15372e16	1.31464	0.657322	−	0.753610i	$-0.271689\pi$
$367$	6.15372e16	1.31464	0.657322	+	0.753610i	$0.271689\pi$
$368$	0	0
$369$	−7.59233e16	−1.56568
$370$	0	0
$371$	9.37664e16	1.86688
$372$	0	0
$373$	8.61573e16	1.65647	0.828237	−	0.560378i	$-0.189344\pi$
$373$	8.61573e16	1.65647	0.828237	+	0.560378i	$0.189344\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.29346e15	0.0949556
$378$	0	0
$379$	8.34144e16	1.44573	0.722863	−	0.690991i	$-0.242826\pi$
$379$	8.34144e16	1.44573	0.722863	+	0.690991i	$0.242826\pi$
$380$	0	0
$381$	3.56626e15	0.0597312
$382$	0	0
$383$	−1.01726e17	−1.64679	−0.823395	−	0.567469i	$-0.807923\pi$
$383$	−1.01726e17	−1.64679	−0.823395	+	0.567469i	$0.807923\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.68185e15	−0.146500
$388$	0	0
$389$	−3.98513e16	−0.583136	−0.291568	−	0.956550i	$-0.594177\pi$
$389$	−3.98513e16	−0.583136	−0.291568	+	0.956550i	$0.594177\pi$
$390$	0	0
$391$	−1.17817e16	−0.166747
$392$	0	0
$393$	1.32752e16	0.181757
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9.10381e15	−0.116704	−0.0583519	−	0.998296i	$-0.518585\pi$
$397$	−9.10381e15	−0.116704	−0.0583519	+	0.998296i	$0.518585\pi$
$398$	0	0
$399$	−5.73559e16	−0.711631
$400$	0	0
$401$	−3.12232e15	−0.0375007	−0.0187503	−	0.999824i	$-0.505969\pi$
$401$	−3.12232e15	−0.0375007	−0.0187503	+	0.999824i	$0.505969\pi$
$402$	0	0
$403$	−1.35306e17	−1.57338
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.59511e16	−0.173951
$408$	0	0
$409$	−1.14106e15	−0.0120534	−0.00602669	−	0.999982i	$-0.501918\pi$
$409$	−1.14106e15	−0.0120534	−0.00602669	+	0.999982i	$0.501918\pi$
$410$	0	0
$411$	−3.76013e16	−0.384796
$412$	0	0
$413$	8.22411e16	0.815481
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.80407e16	−0.261159
$418$	0	0
$419$	1.12123e17	1.01229	0.506145	−	0.862448i	$-0.331070\pi$
$419$	1.12123e17	1.01229	0.506145	+	0.862448i	$0.331070\pi$
$420$	0	0
$421$	−1.91630e17	−1.67737	−0.838686	−	0.544616i	$-0.816676\pi$
$421$	−1.91630e17	−1.67737	−0.838686	+	0.544616i	$0.816676\pi$
$422$	0	0
$423$	1.60937e17	1.36597
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.40059e17	2.71502
$428$	0	0
$429$	−2.33354e16	−0.180735
$430$	0	0
$431$	−1.58423e17	−1.19046	−0.595231	−	0.803555i	$-0.702939\pi$
$431$	−1.58423e17	−1.19046	−0.595231	+	0.803555i	$0.702939\pi$
$432$	0	0
$433$	−7.89897e16	−0.575969	−0.287984	−	0.957635i	$-0.592985\pi$
$433$	−7.89897e16	−0.575969	−0.287984	+	0.957635i	$0.592985\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.52901e17	−1.05022
$438$	0	0
$439$	1.94342e16	0.129583	0.0647915	−	0.997899i	$-0.479362\pi$
$439$	1.94342e16	0.129583	0.0647915	+	0.997899i	$0.479362\pi$
$440$	0	0
$441$	−2.55311e17	−1.65279
$442$	0	0
$443$	−1.04083e17	−0.654265	−0.327132	−	0.944979i	$-0.606082\pi$
$443$	−1.04083e17	−0.654265	−0.327132	+	0.944979i	$0.606082\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.55485e16	0.270069
$448$	0	0
$449$	2.83397e16	0.163228	0.0816140	−	0.996664i	$-0.473993\pi$
$449$	2.83397e16	0.163228	0.0816140	+	0.996664i	$0.473993\pi$
$450$	0	0
$451$	2.13098e17	1.19243
$452$	0	0
$453$	−1.39684e16	−0.0759467
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9.04138e16	−0.464279	−0.232140	−	0.972682i	$-0.574573\pi$
$457$	−9.04138e16	−0.464279	−0.232140	+	0.972682i	$0.574573\pi$
$458$	0	0
$459$	2.61452e16	0.130500
$460$	0	0
$461$	4.40414e16	0.213700	0.106850	−	0.994275i	$-0.465924\pi$
$461$	4.40414e16	0.213700	0.106850	+	0.994275i	$0.465924\pi$
$462$	0	0
$463$	1.56397e17	0.737823	0.368912	−	0.929464i	$-0.379730\pi$
$463$	1.56397e17	0.737823	0.368912	+	0.929464i	$0.379730\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.36099e17	−1.49936	−0.749682	−	0.661798i	$-0.769794\pi$
$467$	−3.36099e17	−1.49936	−0.749682	+	0.661798i	$0.769794\pi$
$468$	0	0
$469$	−1.02231e17	−0.443569
$470$	0	0
$471$	−3.36780e16	−0.142138
$472$	0	0
$473$	2.71745e16	0.111574
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.66825e17	−1.03719
$478$	0	0
$479$	−2.48069e17	−0.938408	−0.469204	−	0.883090i	$-0.655459\pi$
$479$	−2.48069e17	−0.938408	−0.469204	+	0.883090i	$0.655459\pi$
$480$	0	0
$481$	6.33651e16	0.233296
$482$	0	0
$483$	8.38275e16	0.300421
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.45229e17	0.493305	0.246653	−	0.969104i	$-0.420669\pi$
$487$	1.45229e17	0.493305	0.246653	+	0.969104i	$0.420669\pi$
$488$	0	0
$489$	−1.83890e16	−0.0608206
$490$	0	0
$491$	1.37365e16	0.0442433	0.0221217	−	0.999755i	$-0.492958\pi$
$491$	1.37365e16	0.0442433	0.0221217	+	0.999755i	$0.492958\pi$
$492$	0	0
$493$	8.00726e15	0.0251177
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.62247e17	−2.26871
$498$	0	0
$499$	6.40351e17	1.85680	0.928399	−	0.371585i	$-0.121186\pi$
$499$	6.40351e17	1.85680	0.928399	+	0.371585i	$0.121186\pi$
$500$	0	0
$501$	6.04766e16	0.170861
$502$	0	0
$503$	−8.60860e16	−0.236996	−0.118498	−	0.992954i	$-0.537808\pi$
$503$	−8.60860e16	−0.236996	−0.118498	+	0.992954i	$0.537808\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.07273e16	−0.0280503
$508$	0	0
$509$	5.21435e17	1.32903	0.664515	−	0.747275i	$-0.268638\pi$
$509$	5.21435e17	1.32903	0.664515	+	0.747275i	$0.268638\pi$
$510$	0	0
$511$	−1.13216e18	−2.81301
$512$	0	0
$513$	3.39308e17	0.821923
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.51709e17	−1.04033
$518$	0	0
$519$	2.26125e17	0.507878
$520$	0	0
$521$	−9.29755e16	−0.203668	−0.101834	−	0.994801i	$-0.532471\pi$
$521$	−9.29755e16	−0.203668	−0.101834	+	0.994801i	$0.532471\pi$
$522$	0	0
$523$	−8.41876e17	−1.79882	−0.899409	−	0.437108i	$-0.856003\pi$
$523$	−8.41876e17	−1.79882	−0.899409	+	0.437108i	$0.856003\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.04674e17	−0.416192
$528$	0	0
$529$	−2.80567e17	−0.556641
$530$	0	0
$531$	−2.34028e17	−0.453058
$532$	0	0
$533$	−8.46523e17	−1.59923
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.99219e16	−0.107833
$538$	0	0
$539$	7.16594e17	1.25877
$540$	0	0
$541$	−9.15797e17	−1.57042	−0.785212	−	0.619227i	$-0.787446\pi$
$541$	−9.15797e17	−1.57042	−0.785212	+	0.619227i	$0.787446\pi$
$542$	0	0
$543$	−4.30966e16	−0.0721512
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−4.67586e16	−0.0746353	−0.0373177	−	0.999303i	$-0.511881\pi$
$547$	−4.67586e16	−0.0746353	−0.0373177	+	0.999303i	$0.511881\pi$
$548$	0	0
$549$	−9.67685e17	−1.50839
$550$	0	0
$551$	1.03917e17	0.158198
$552$	0	0
$553$	7.71248e17	1.14678
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.80559e17	−0.398075	−0.199038	−	0.979992i	$-0.563782\pi$
$557$	−2.80559e17	−0.398075	−0.199038	+	0.979992i	$0.563782\pi$
$558$	0	0
$559$	−1.07950e17	−0.149639
$560$	0	0
$561$	−3.52988e16	−0.0478081
$562$	0	0
$563$	−4.99828e17	−0.661479	−0.330739	−	0.943722i	$-0.607298\pi$
$563$	−4.99828e17	−0.661479	−0.330739	+	0.943722i	$0.607298\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.03740e18	1.31116
$568$	0	0
$569$	7.81630e17	0.965543	0.482771	−	0.875746i	$-0.339630\pi$
$569$	7.81630e17	0.965543	0.482771	+	0.875746i	$0.339630\pi$
$570$	0	0
$571$	−1.05601e18	−1.27506	−0.637532	−	0.770424i	$-0.720045\pi$
$571$	−1.05601e18	−1.27506	−0.637532	+	0.770424i	$0.720045\pi$
$572$	0	0
$573$	−1.66356e16	−0.0196351
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.26116e18	−1.42275	−0.711375	−	0.702812i	$-0.751927\pi$
$577$	−1.26116e18	−1.42275	−0.711375	+	0.702812i	$0.751927\pi$
$578$	0	0
$579$	1.20055e17	0.132424
$580$	0	0
$581$	−2.78713e17	−0.300616
$582$	0	0
$583$	7.48912e17	0.789923
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.10590e16	0.0616030	0.0308015	−	0.999526i	$-0.490194\pi$
$587$	6.10590e16	0.0616030	0.0308015	+	0.999526i	$0.490194\pi$
$588$	0	0
$589$	−2.65622e18	−2.62129
$590$	0	0
$591$	2.64672e17	0.255499
$592$	0	0
$593$	1.57661e18	1.48891	0.744454	−	0.667674i	$-0.232710\pi$
$593$	1.57661e18	1.48891	0.744454	+	0.667674i	$0.232710\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.24861e16	−0.0474473
$598$	0	0
$599$	1.25519e18	1.11029	0.555144	−	0.831754i	$-0.312663\pi$
$599$	1.25519e18	1.11029	0.555144	+	0.831754i	$0.312663\pi$
$600$	0	0
$601$	−7.69850e16	−0.0666380	−0.0333190	−	0.999445i	$-0.510608\pi$
$601$	−7.69850e16	−0.0666380	−0.0333190	+	0.999445i	$0.510608\pi$
$602$	0	0
$603$	2.90913e17	0.246434
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.30309e18	1.05742	0.528709	−	0.848803i	$-0.322676\pi$
$607$	1.30309e18	1.05742	0.528709	+	0.848803i	$0.322676\pi$
$608$	0	0
$609$	−5.69722e16	−0.0452534
$610$	0	0
$611$	1.79440e18	1.39525
$612$	0	0
$613$	4.77302e17	0.363329	0.181664	−	0.983361i	$-0.441852\pi$
$613$	4.77302e17	0.363329	0.181664	+	0.983361i	$0.441852\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.58149e17	−0.115402	−0.0577008	−	0.998334i	$-0.518377\pi$
$617$	−1.58149e17	−0.115402	−0.0577008	+	0.998334i	$0.518377\pi$
$618$	0	0
$619$	5.89078e17	0.420904	0.210452	−	0.977604i	$-0.432506\pi$
$619$	5.89078e17	0.420904	0.210452	+	0.977604i	$0.432506\pi$
$620$	0	0
$621$	−4.95909e17	−0.346982
$622$	0	0
$623$	−2.28172e18	−1.56347
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.58102e17	−0.301108
$628$	0	0
$629$	9.58507e16	0.0617115
$630$	0	0
$631$	1.01185e18	0.638155	0.319077	−	0.947729i	$-0.396627\pi$
$631$	1.01185e18	0.638155	0.319077	+	0.947729i	$0.396627\pi$
$632$	0	0
$633$	1.14046e17	0.0704621
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.84664e18	−1.68821
$638$	0	0
$639$	2.16908e18	1.26043
$640$	0	0
$641$	−3.07604e18	−1.75152	−0.875759	−	0.482748i	$-0.839639\pi$
$641$	−3.07604e18	−1.75152	−0.875759	+	0.482748i	$0.839639\pi$
$642$	0	0
$643$	1.10506e18	0.616615	0.308308	−	0.951287i	$-0.400237\pi$
$643$	1.10506e18	0.616615	0.308308	+	0.951287i	$0.400237\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.10565e18	1.66447	0.832233	−	0.554426i	$-0.187062\pi$
$647$	3.10565e18	1.66447	0.832233	+	0.554426i	$0.187062\pi$
$648$	0	0
$649$	6.56860e17	0.345050
$650$	0	0
$651$	1.45627e18	0.749834
$652$	0	0
$653$	2.53908e17	0.128156	0.0640782	−	0.997945i	$-0.479589\pi$
$653$	2.53908e17	0.128156	0.0640782	+	0.997945i	$0.479589\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.22171e18	1.56283
$658$	0	0
$659$	9.41778e17	0.447912	0.223956	−	0.974599i	$-0.428103\pi$
$659$	9.41778e17	0.447912	0.223956	+	0.974599i	$0.428103\pi$
$660$	0	0
$661$	1.94057e18	0.904941	0.452471	−	0.891779i	$-0.350543\pi$
$661$	1.94057e18	0.904941	0.452471	+	0.891779i	$0.350543\pi$
$662$	0	0
$663$	1.40223e17	0.0641183
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.51878e17	−0.0667846
$668$	0	0
$669$	−9.14355e17	−0.394317
$670$	0	0
$671$	2.71605e18	1.14879
$672$	0	0
$673$	−2.86620e18	−1.18907	−0.594537	−	0.804068i	$-0.702665\pi$
$673$	−2.86620e18	−1.18907	−0.594537	+	0.804068i	$0.702665\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.97425e18	−1.18727	−0.593637	−	0.804733i	$-0.702309\pi$
$677$	−2.97425e18	−1.18727	−0.593637	+	0.804733i	$0.702309\pi$
$678$	0	0
$679$	−1.56226e18	−0.611785
$680$	0	0
$681$	3.70190e16	0.0142223
$682$	0	0
$683$	−3.71376e18	−1.39984	−0.699922	−	0.714219i	$-0.746782\pi$
$683$	−3.71376e18	−1.39984	−0.699922	+	0.714219i	$0.746782\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.32257e17	0.0842851
$688$	0	0
$689$	−2.97502e18	−1.05941
$690$	0	0
$691$	−1.44431e18	−0.504723	−0.252361	−	0.967633i	$-0.581207\pi$
$691$	−1.44431e18	−0.504723	−0.252361	+	0.967633i	$0.581207\pi$
$692$	0	0
$693$	−3.18270e18	−1.09151
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.28051e18	−0.423030
$698$	0	0
$699$	−6.72884e16	−0.0218192
$700$	0	0
$701$	4.20739e18	1.33920	0.669600	−	0.742722i	$-0.266466\pi$
$701$	4.20739e18	1.33920	0.669600	+	0.742722i	$0.266466\pi$
$702$	0	0
$703$	1.24393e18	0.388676
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.31217e18	2.20201
$708$	0	0
$709$	−6.28788e18	−1.85910	−0.929552	−	0.368692i	$-0.879806\pi$
$709$	−6.28788e18	−1.85910	−0.929552	+	0.368692i	$0.879806\pi$
$710$	0	0
$711$	−2.19469e18	−0.637120
$712$	0	0
$713$	3.88215e18	1.10660
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.24001e18	0.340840
$718$	0	0
$719$	2.77418e18	0.748855	0.374427	−	0.927256i	$-0.377839\pi$
$719$	2.77418e18	0.748855	0.374427	+	0.927256i	$0.377839\pi$
$720$	0	0
$721$	1.01015e19	2.67798
$722$	0	0
$723$	1.95064e18	0.507899
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.10980e18	0.529991	0.264995	−	0.964250i	$-0.414630\pi$
$727$	2.10980e18	0.529991	0.264995	+	0.964250i	$0.414630\pi$
$728$	0	0
$729$	−2.38093e18	−0.587514
$730$	0	0
$731$	−1.63293e17	−0.0395825
$732$	0	0
$733$	5.91497e18	1.40856	0.704282	−	0.709921i	$-0.251269\pi$
$733$	5.91497e18	1.40856	0.704282	+	0.709921i	$0.251269\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.16521e17	−0.187685
$738$	0	0
$739$	−7.27738e17	−0.164356	−0.0821782	−	0.996618i	$-0.526188\pi$
$739$	−7.27738e17	−0.164356	−0.0821782	+	0.996618i	$0.526188\pi$
$740$	0	0
$741$	1.81979e18	0.403834
$742$	0	0
$743$	−2.33651e18	−0.509496	−0.254748	−	0.967007i	$-0.581993\pi$
$743$	−2.33651e18	−0.509496	−0.254748	+	0.967007i	$0.581993\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.93116e17	0.167014
$748$	0	0
$749$	8.28494e18	1.71458
$750$	0	0
$751$	1.45486e17	0.0295911	0.0147955	−	0.999891i	$-0.495290\pi$
$751$	1.45486e17	0.0295911	0.0147955	+	0.999891i	$0.495290\pi$
$752$	0	0
$753$	2.15968e18	0.431740
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.29556e18	1.02279	0.511397	−	0.859344i	$-0.329128\pi$
$757$	5.29556e18	1.02279	0.511397	+	0.859344i	$0.329128\pi$
$758$	0	0
$759$	6.69530e17	0.127115
$760$	0	0
$761$	3.01218e18	0.562187	0.281094	−	0.959680i	$-0.409303\pi$
$761$	3.01218e18	0.562187	0.281094	+	0.959680i	$0.409303\pi$
$762$	0	0
$763$	1.24903e19	2.29173
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.60935e18	−0.462767
$768$	0	0
$769$	−5.11313e18	−0.891591	−0.445795	−	0.895135i	$-0.647079\pi$
$769$	−5.11313e18	−0.891591	−0.445795	+	0.895135i	$0.647079\pi$
$770$	0	0
$771$	−2.27497e18	−0.390051
$772$	0	0
$773$	4.35596e18	0.734375	0.367187	−	0.930147i	$-0.380321\pi$
$773$	4.35596e18	0.734375	0.367187	+	0.930147i	$0.380321\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6.81984e17	−0.111183
$778$	0	0
$779$	−1.66183e19	−2.66436
$780$	0	0
$781$	−6.08806e18	−0.959947
$782$	0	0
$783$	3.37038e17	0.0522670
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.74931e18	1.31262	0.656308	−	0.754493i	$-0.272117\pi$
$787$	8.74931e18	1.31262	0.656308	+	0.754493i	$0.272117\pi$
$788$	0	0
$789$	1.23291e18	0.181941
$790$	0	0
$791$	9.48102e18	1.37628
$792$	0	0
$793$	−1.07894e19	−1.54071
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.68001e18	0.785000	0.392500	−	0.919752i	$-0.371610\pi$
$797$	5.68001e18	0.785000	0.392500	+	0.919752i	$0.371610\pi$
$798$	0	0
$799$	2.71433e18	0.369070
$800$	0	0
$801$	6.49295e18	0.868622
$802$	0	0
$803$	−9.04253e18	−1.19025
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.24394e18	0.158534
$808$	0	0
$809$	−7.72134e18	−0.968338	−0.484169	−	0.874974i	$-0.660878\pi$
$809$	−7.72134e18	−0.968338	−0.484169	+	0.874974i	$0.660878\pi$
$810$	0	0
$811$	1.12177e19	1.38442	0.692208	−	0.721698i	$-0.256638\pi$
$811$	1.12177e19	1.38442	0.692208	+	0.721698i	$0.256638\pi$
$812$	0	0
$813$	4.01492e18	0.487628
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.11918e18	−0.249302
$818$	0	0
$819$	1.26431e19	1.46389
$820$	0	0
$821$	−1.16499e19	−1.32767	−0.663835	−	0.747879i	$-0.731072\pi$
$821$	−1.16499e19	−1.32767	−0.663835	+	0.747879i	$0.731072\pi$
$822$	0	0
$823$	−1.33989e19	−1.50304	−0.751520	−	0.659710i	$-0.770679\pi$
$823$	−1.33989e19	−1.50304	−0.751520	+	0.659710i	$0.770679\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.57639e18	0.932221	0.466110	−	0.884727i	$-0.345655\pi$
$827$	8.57639e18	0.932221	0.466110	+	0.884727i	$0.345655\pi$
$828$	0	0
$829$	1.46275e19	1.56519	0.782595	−	0.622532i	$-0.213896\pi$
$829$	1.46275e19	1.56519	0.782595	+	0.622532i	$0.213896\pi$
$830$	0	0
$831$	2.92989e18	0.308634
$832$	0	0
$833$	−4.30603e18	−0.446565
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.61504e18	−0.866047
$838$	0	0
$839$	−5.70394e18	−0.564576	−0.282288	−	0.959330i	$-0.591093\pi$
$839$	−5.70394e18	−0.564576	−0.282288	+	0.959330i	$0.591093\pi$
$840$	0	0
$841$	−1.01574e19	−0.989940
$842$	0	0
$843$	3.39637e18	0.325938
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8.99430e18	−0.836999
$848$	0	0
$849$	−1.90005e18	−0.174126
$850$	0	0
$851$	−1.81805e18	−0.164083
$852$	0	0
$853$	−1.75020e19	−1.55568	−0.777839	−	0.628464i	$-0.783684\pi$
$853$	−1.75020e19	−1.55568	−0.777839	+	0.628464i	$0.783684\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.13838e19	0.981550	0.490775	−	0.871286i	$-0.336714\pi$
$857$	1.13838e19	0.981550	0.490775	+	0.871286i	$0.336714\pi$
$858$	0	0
$859$	4.32333e17	0.0367166	0.0183583	−	0.999831i	$-0.494156\pi$
$859$	4.32333e17	0.0367166	0.0183583	+	0.999831i	$0.494156\pi$
$860$	0	0
$861$	9.11093e18	0.762153
$862$	0	0
$863$	2.54755e17	0.0209920	0.0104960	−	0.999945i	$-0.496659\pi$
$863$	2.54755e17	0.0209920	0.0104960	+	0.999945i	$0.496659\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.17012e18	−0.253484
$868$	0	0
$869$	6.15995e18	0.485231
$870$	0	0
$871$	3.24360e18	0.251715
$872$	0	0
$873$	4.44562e18	0.339891
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9.46060e18	−0.702136	−0.351068	−	0.936350i	$-0.614181\pi$
$877$	−9.46060e18	−0.702136	−0.351068	+	0.936350i	$0.614181\pi$
$878$	0	0
$879$	3.91487e18	0.286279
$880$	0	0
$881$	−7.69724e18	−0.554615	−0.277308	−	0.960781i	$-0.589442\pi$
$881$	−7.69724e18	−0.554615	−0.277308	+	0.960781i	$0.589442\pi$
$882$	0	0
$883$	−3.90855e18	−0.277505	−0.138752	−	0.990327i	$-0.544309\pi$
$883$	−3.90855e18	−0.277505	−0.138752	+	0.990327i	$0.544309\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.90686e19	−1.31467	−0.657333	−	0.753600i	$-0.728316\pi$
$887$	−1.90686e19	−1.31467	−0.657333	+	0.753600i	$0.728316\pi$
$888$	0	0
$889$	5.42322e18	0.368464
$890$	0	0
$891$	8.28571e18	0.554785
$892$	0	0
$893$	3.52262e19	2.32451
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.65968e18	−0.170482
$898$	0	0
$899$	−2.63845e18	−0.166691
$900$	0	0
$901$	−4.50023e18	−0.280236
$902$	0	0
$903$	1.16184e18	0.0713141
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.23169e18	−0.133102	−0.0665512	−	0.997783i	$-0.521200\pi$
$907$	−2.23169e18	−0.133102	−0.0665512	+	0.997783i	$0.521200\pi$
$908$	0	0
$909$	−2.08078e19	−1.22338
$910$	0	0
$911$	−1.09912e18	−0.0637053	−0.0318527	−	0.999493i	$-0.510141\pi$
$911$	−1.09912e18	−0.0637053	−0.0318527	+	0.999493i	$0.510141\pi$
$912$	0	0
$913$	−2.22608e18	−0.127198
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.01877e19	1.12121
$918$	0	0
$919$	2.39854e19	1.31340	0.656699	−	0.754153i	$-0.271952\pi$
$919$	2.39854e19	1.31340	0.656699	+	0.754153i	$0.271952\pi$
$920$	0	0
$921$	7.55039e18	0.407645
$922$	0	0
$923$	2.41846e19	1.28744
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2.87453e19	−1.48781
$928$	0	0
$929$	7.03673e18	0.359144	0.179572	−	0.983745i	$-0.442529\pi$
$929$	7.03673e18	0.359144	0.179572	+	0.983745i	$0.442529\pi$
$930$	0	0
$931$	−5.58829e19	−2.81259
$932$	0	0
$933$	−5.54128e17	−0.0275030
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.82749e19	0.882163	0.441081	−	0.897467i	$-0.354595\pi$
$937$	1.82749e19	0.882163	0.441081	+	0.897467i	$0.354595\pi$
$938$	0	0
$939$	−5.65820e18	−0.269371
$940$	0	0
$941$	−2.84709e19	−1.33681	−0.668404	−	0.743798i	$-0.733022\pi$
$941$	−2.84709e19	−1.33681	−0.668404	+	0.743798i	$0.733022\pi$
$942$	0	0
$943$	2.42881e19	1.12478
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.84313e19	−1.73145	−0.865725	−	0.500520i	$-0.833142\pi$
$947$	−3.84313e19	−1.73145	−0.865725	+	0.500520i	$0.833142\pi$
$948$	0	0
$949$	3.59211e19	1.59632
$950$	0	0
$951$	3.45113e18	0.151282
$952$	0	0
$953$	−5.75110e18	−0.248684	−0.124342	−	0.992239i	$-0.539682\pi$
$953$	−5.75110e18	−0.248684	−0.124342	+	0.992239i	$0.539682\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.55037e17	−0.0191478
$958$	0	0
$959$	−5.71803e19	−2.37370
$960$	0	0
$961$	4.30240e19	1.76201
$962$	0	0
$963$	−2.35759e19	−0.952573
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.78997e19	1.09731	0.548653	−	0.836050i	$-0.315141\pi$
$967$	2.78997e19	1.09731	0.548653	+	0.836050i	$0.315141\pi$
$968$	0	0
$969$	2.75275e18	0.106822
$970$	0	0
$971$	−4.64605e19	−1.77893	−0.889466	−	0.457002i	$-0.848923\pi$
$971$	−4.64605e19	−1.77893	−0.889466	+	0.457002i	$0.848923\pi$
$972$	0	0
$973$	−4.26415e19	−1.61101
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.26224e18	0.193578	0.0967890	−	0.995305i	$-0.469143\pi$
$977$	5.26224e18	0.193578	0.0967890	+	0.995305i	$0.469143\pi$
$978$	0	0
$979$	−1.82241e19	−0.661544
$980$	0	0
$981$	−3.55428e19	−1.27322
$982$	0	0
$983$	−3.27781e19	−1.15874	−0.579370	−	0.815064i	$-0.696702\pi$
$983$	−3.27781e19	−1.15874	−0.579370	+	0.815064i	$0.696702\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.93127e19	−0.664938
$988$	0	0
$989$	3.09725e18	0.105245
$990$	0	0
$991$	−5.21648e19	−1.74944	−0.874719	−	0.484630i	$-0.838954\pi$
$991$	−5.21648e19	−1.74944	−0.874719	+	0.484630i	$0.838954\pi$
$992$	0	0
$993$	1.83029e18	0.0605830
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−8.76768e18	−0.282726	−0.141363	−	0.989958i	$-0.545149\pi$
$997$	−8.76768e18	−0.282726	−0.141363	+	0.989958i	$0.545149\pi$
$998$	0	0
$999$	4.03450e18	0.128415

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.14.a.f.1.3		4
5.2	odd	4		200.14.c.f.49.4		8
5.3	odd	4		200.14.c.f.49.5		8
5.4	even	2		40.14.a.d.1.2	✓	4
20.19	odd	2		80.14.a.k.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.14.a.d.1.2	✓	4	5.4	even	2
80.14.a.k.1.3		4	20.19	odd	2
200.14.a.f.1.3		4	1.1	even	1	trivial
200.14.c.f.49.4		8	5.2	odd	4
200.14.c.f.49.5		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)

\(f(q)\)	\(=\)	\(q+341.481 q^{3} +519291. q^{7} -1.47771e6 q^{9} +O(q^{10})\)	\(q+341.481 q^{3} +519291. q^{7} -1.47771e6 q^{9} +4.14757e6 q^{11} -1.64761e7 q^{13} -2.49229e7 q^{17} -3.23445e8 q^{19} +1.77328e8 q^{21} +4.72725e8 q^{23} -1.04904e9 q^{27} -3.21281e8 q^{29} +8.21228e9 q^{31} +1.41632e9 q^{33} -3.84589e9 q^{37} -5.62627e9 q^{39} +5.13789e10 q^{41} +6.55191e9 q^{43} -1.08909e11 q^{47} +1.72774e11 q^{49} -8.51071e9 q^{51} +1.80566e11 q^{53} -1.10450e11 q^{57} +1.58372e11 q^{59} +6.54853e11 q^{61} -7.67363e11 q^{63} -1.96867e11 q^{67} +1.61427e11 q^{69} -1.46786e12 q^{71} -2.18020e12 q^{73} +2.15380e12 q^{77} +1.48519e12 q^{79} +1.99772e12 q^{81} -5.36718e11 q^{83} -1.09712e11 q^{87} -4.39392e12 q^{89} -8.55588e12 q^{91} +2.80434e12 q^{93} -3.00844e12 q^{97} -6.12893e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 936 q^{3} - 112424 q^{7} + 2737236 q^{9}+O(q^{10}) \) 4 * q - 936 * q^3 - 112424 * q^7 + 2737236 * q^9	\( 4 q - 936 q^{3} - 112424 q^{7} + 2737236 q^{9} - 85280 q^{11} - 22276696 q^{13} - 92966696 q^{17} - 52003280 q^{19} - 124736832 q^{21} + 498172904 q^{23} - 1984218768 q^{27} + 5868734200 q^{29} + 1204156144 q^{31} + 2413789344 q^{33} + 1227431400 q^{37} + 47183929776 q^{39} + 17822906952 q^{41} - 92192384488 q^{43} - 187731312904 q^{47} + 218812853892 q^{49} + 436920457680 q^{51} - 617748697656 q^{53} - 962043447456 q^{57} + 155305995248 q^{59} + 918726826936 q^{61} - 453684626952 q^{63} - 1172014952696 q^{67} + 2007735561792 q^{69} + 1499341271376 q^{71} - 3669705031176 q^{73} - 2543134824608 q^{77} + 1474589017632 q^{79} + 1358685231876 q^{81} + 10007128523992 q^{83} + 6056521948368 q^{87} - 2840927507160 q^{89} - 16479691138768 q^{91} + 4603403842560 q^{93} - 7763871939496 q^{97} - 50298890417952 q^{99}+O(q^{100}) \) 4 * q - 936 * q^3 - 112424 * q^7 + 2737236 * q^9 - 85280 * q^11 - 22276696 * q^13 - 92966696 * q^17 - 52003280 * q^19 - 124736832 * q^21 + 498172904 * q^23 - 1984218768 * q^27 + 5868734200 * q^29 + 1204156144 * q^31 + 2413789344 * q^33 + 1227431400 * q^37 + 47183929776 * q^39 + 17822906952 * q^41 - 92192384488 * q^43 - 187731312904 * q^47 + 218812853892 * q^49 + 436920457680 * q^51 - 617748697656 * q^53 - 962043447456 * q^57 + 155305995248 * q^59 + 918726826936 * q^61 - 453684626952 * q^63 - 1172014952696 * q^67 + 2007735561792 * q^69 + 1499341271376 * q^71 - 3669705031176 * q^73 - 2543134824608 * q^77 + 1474589017632 * q^79 + 1358685231876 * q^81 + 10007128523992 * q^83 + 6056521948368 * q^87 - 2840927507160 * q^89 - 16479691138768 * q^91 + 4603403842560 * q^93 - 7763871939496 * q^97 - 50298890417952 * q^99

Introduction

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