LMFDB - Embedded newform 75.14.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,14,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("75.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	75.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:	$80.4231967139$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+76.0000 q^{2} -729.000 q^{3} -2416.00 q^{4} -55404.0 q^{6} +224160. q^{7} -806208. q^{8} +531441. q^{9} +O(q^{10})$

\(q+76.0000 q^{2} -729.000 q^{3} -2416.00 q^{4} -55404.0 q^{6} +224160. q^{7} -806208. q^{8} +531441. q^{9} +2.31384e6 q^{11} +1.76126e6 q^{12} -1.05373e7 q^{13} +1.70362e7 q^{14} -4.14799e7 q^{16} +1.86661e8 q^{17} +4.03895e7 q^{18} -2.90441e8 q^{19} -1.63413e8 q^{21} +1.75852e8 q^{22} +8.66819e8 q^{23} +5.87726e8 q^{24} -8.00836e8 q^{26} -3.87420e8 q^{27} -5.41571e8 q^{28} -1.56698e9 q^{29} +1.20062e9 q^{31} +3.45198e9 q^{32} -1.68679e9 q^{33} +1.41862e10 q^{34} -1.28396e9 q^{36} +1.21822e10 q^{37} -2.20735e10 q^{38} +7.68170e9 q^{39} -2.91678e10 q^{41} -1.24194e10 q^{42} -4.93618e10 q^{43} -5.59023e9 q^{44} +6.58782e10 q^{46} +1.16715e10 q^{47} +3.02389e10 q^{48} -4.66413e10 q^{49} -1.36076e11 q^{51} +2.54582e10 q^{52} -1.00929e11 q^{53} -2.94440e10 q^{54} -1.80720e11 q^{56} +2.11731e11 q^{57} -1.19091e11 q^{58} -2.65190e11 q^{59} -5.66434e11 q^{61} +9.12474e10 q^{62} +1.19128e11 q^{63} +6.02154e11 q^{64} -1.28196e11 q^{66} -1.44118e12 q^{67} -4.50972e11 q^{68} -6.31911e11 q^{69} -5.02945e11 q^{71} -4.28452e11 q^{72} -1.57491e12 q^{73} +9.25851e11 q^{74} +7.01705e11 q^{76} +5.18669e11 q^{77} +5.83810e11 q^{78} +3.38387e11 q^{79} +2.82430e11 q^{81} -2.21676e12 q^{82} +4.77181e12 q^{83} +3.94805e11 q^{84} -3.75149e12 q^{86} +1.14233e12 q^{87} -1.86543e12 q^{88} +2.74648e12 q^{89} -2.36205e12 q^{91} -2.09423e12 q^{92} -8.75254e11 q^{93} +8.87036e11 q^{94} -2.51649e12 q^{96} -1.97907e12 q^{97} -3.54474e12 q^{98} +1.22967e12 q^{99} +O(q^{100})\)

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 $

$2$	76.0000	0.839689	0.419845	−	0.907596i	$-0.362085\pi$
$2$	76.0000	0.839689	0.419845	+	0.907596i	$0.362085\pi$
$3$	−729.000	−0.577350
$3$	−729.000	−0.577350
$4$	−2416.00	−0.294922
$5$	0	0
$5$	0	0
$6$	−55404.0	−0.484795
$7$	224160.	0.720147	0.360073	−	0.932924i	$-0.382752\pi$
$7$	224160.	0.720147	0.360073	+	0.932924i	$0.382752\pi$
$8$	−806208.	−1.08733
$9$	531441.	0.333333
$10$	0	0
$11$	2.31384e6	0.393804	0.196902	−	0.980423i	$-0.436912\pi$
$11$	2.31384e6	0.393804	0.196902	+	0.980423i	$0.436912\pi$
$12$	1.76126e6	0.170273
$13$	−1.05373e7	−0.605478	−0.302739	−	0.953074i	$-0.597901\pi$
$13$	−1.05373e7	−0.605478	−0.302739	+	0.953074i	$0.597901\pi$
$14$	1.70362e7	0.604699
$15$	0	0
$16$	−4.14799e7	−0.618099
$17$	1.86661e8	1.87558	0.937788	−	0.347208i	$-0.112870\pi$
$17$	1.86661e8	1.87558	0.937788	+	0.347208i	$0.112870\pi$
$18$	4.03895e7	0.279896
$19$	−2.90441e8	−1.41631	−0.708156	−	0.706056i	$-0.750473\pi$
$19$	−2.90441e8	−1.41631	−0.708156	+	0.706056i	$0.750473\pi$
$20$	0	0
$21$	−1.63413e8	−0.415777
$22$	1.75852e8	0.330673
$23$	8.66819e8	1.22095	0.610474	−	0.792036i	$-0.290979\pi$
$23$	8.66819e8	1.22095	0.610474	+	0.792036i	$0.290979\pi$
$24$	5.87726e8	0.627771
$25$	0	0
$26$	−8.00836e8	−0.508413
$27$	−3.87420e8	−0.192450
$28$	−5.41571e8	−0.212387
$29$	−1.56698e9	−0.489189	−0.244595	−	0.969625i	$-0.578655\pi$
$29$	−1.56698e9	−0.489189	−0.244595	+	0.969625i	$0.578655\pi$
$30$	0	0
$31$	1.20062e9	0.242972	0.121486	−	0.992593i	$-0.461234\pi$
$31$	1.20062e9	0.242972	0.121486	+	0.992593i	$0.461234\pi$
$32$	3.45198e9	0.568321
$33$	−1.68679e9	−0.227363
$34$	1.41862e10	1.57490
$35$	0	0
$36$	−1.28396e9	−0.0983073
$37$	1.21822e10	0.780578	0.390289	−	0.920692i	$-0.372375\pi$
$37$	1.21822e10	0.780578	0.390289	+	0.920692i	$0.372375\pi$
$38$	−2.20735e10	−1.18926
$39$	7.68170e9	0.349573
$40$	0	0
$41$	−2.91678e10	−0.958979	−0.479490	−	0.877548i	$-0.659178\pi$
$41$	−2.91678e10	−0.958979	−0.479490	+	0.877548i	$0.659178\pi$
$42$	−1.24194e10	−0.349123
$43$	−4.93618e10	−1.19082	−0.595409	−	0.803422i	$-0.703010\pi$
$43$	−4.93618e10	−1.19082	−0.595409	+	0.803422i	$0.703010\pi$
$44$	−5.59023e9	−0.116141
$45$	0	0
$46$	6.58782e10	1.02522
$47$	1.16715e10	0.157940	0.0789699	−	0.996877i	$-0.474837\pi$
$47$	1.16715e10	0.157940	0.0789699	+	0.996877i	$0.474837\pi$
$48$	3.02389e10	0.356860
$49$	−4.66413e10	−0.481389
$50$	0	0
$51$	−1.36076e11	−1.08286
$52$	2.54582e10	0.178569
$53$	−1.00929e11	−0.625496	−0.312748	−	0.949836i	$-0.601250\pi$
$53$	−1.00929e11	−0.625496	−0.312748	+	0.949836i	$0.601250\pi$
$54$	−2.94440e10	−0.161598
$55$	0	0
$56$	−1.80720e11	−0.783038
$57$	2.11731e11	0.817708
$58$	−1.19091e11	−0.410767
$59$	−2.65190e11	−0.818500	−0.409250	−	0.912422i	$-0.634210\pi$
$59$	−2.65190e11	−0.818500	−0.409250	+	0.912422i	$0.634210\pi$
$60$	0	0
$61$	−5.66434e11	−1.40768	−0.703842	−	0.710357i	$-0.748534\pi$
$61$	−5.66434e11	−1.40768	−0.703842	+	0.710357i	$0.748534\pi$
$62$	9.12474e10	0.204021
$63$	1.19128e11	0.240049
$64$	6.02154e11	1.09531
$65$	0	0
$66$	−1.28196e11	−0.190914
$67$	−1.44118e12	−1.94640	−0.973200	−	0.229959i	$-0.926141\pi$
$67$	−1.44118e12	−1.94640	−0.973200	+	0.229959i	$0.926141\pi$
$68$	−4.50972e11	−0.553148
$69$	−6.31911e11	−0.704915
$70$	0	0
$71$	−5.02945e11	−0.465952	−0.232976	−	0.972483i	$-0.574846\pi$
$71$	−5.02945e11	−0.465952	−0.232976	+	0.972483i	$0.574846\pi$
$72$	−4.28452e11	−0.362444
$73$	−1.57491e12	−1.21803	−0.609014	−	0.793159i	$-0.708435\pi$
$73$	−1.57491e12	−1.21803	−0.609014	+	0.793159i	$0.708435\pi$
$74$	9.25851e11	0.655443
$75$	0	0
$76$	7.01705e11	0.417701
$77$	5.18669e11	0.283597
$78$	5.83810e11	0.293533
$79$	3.38387e11	0.156617	0.0783083	−	0.996929i	$-0.475048\pi$
$79$	3.38387e11	0.156617	0.0783083	+	0.996929i	$0.475048\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	−2.21676e12	−0.805244
$83$	4.77181e12	1.60205	0.801024	−	0.598632i	$-0.204289\pi$
$83$	4.77181e12	1.60205	0.801024	+	0.598632i	$0.204289\pi$
$84$	3.94805e11	0.122622
$85$	0	0
$86$	−3.75149e12	−0.999918
$87$	1.14233e12	0.282434
$88$	−1.86543e12	−0.428196
$89$	2.74648e12	0.585790	0.292895	−	0.956145i	$-0.405381\pi$
$89$	2.74648e12	0.585790	0.292895	+	0.956145i	$0.405381\pi$
$90$	0	0
$91$	−2.36205e12	−0.436033
$92$	−2.09423e12	−0.360084
$93$	−8.75254e11	−0.140280
$94$	8.87036e11	0.132620
$95$	0	0
$96$	−2.51649e12	−0.328120
$97$	−1.97907e12	−0.241238	−0.120619	−	0.992699i	$-0.538488\pi$
$97$	−1.97907e12	−0.241238	−0.120619	+	0.992699i	$0.538488\pi$
$98$	−3.54474e12	−0.404217
$99$	1.22967e12	0.131268
$100$	0	0
$101$	−6.24314e12	−0.585213	−0.292607	−	0.956233i	$-0.594523\pi$
$101$	−6.24314e12	−0.585213	−0.292607	+	0.956233i	$0.594523\pi$
$102$	−1.03417e13	−0.909270
$103$	1.97186e13	1.62718	0.813588	−	0.581442i	$-0.197511\pi$
$103$	1.97186e13	1.62718	0.813588	+	0.581442i	$0.197511\pi$
$104$	8.49527e12	0.658356
$105$	0	0
$106$	−7.67064e12	−0.525222
$107$	−1.92281e13	−1.23863	−0.619316	−	0.785142i	$-0.712590\pi$
$107$	−1.92281e13	−1.23863	−0.619316	+	0.785142i	$0.712590\pi$
$108$	9.36008e11	0.0567577
$109$	−1.36347e13	−0.778707	−0.389353	−	0.921088i	$-0.627301\pi$
$109$	−1.36347e13	−0.778707	−0.389353	+	0.921088i	$0.627301\pi$
$110$	0	0
$111$	−8.88086e12	−0.450667
$112$	−9.29814e12	−0.445122
$113$	−1.55051e13	−0.700589	−0.350295	−	0.936640i	$-0.613919\pi$
$113$	−1.55051e13	−0.700589	−0.350295	+	0.936640i	$0.613919\pi$
$114$	1.60916e13	0.686621
$115$	0	0
$116$	3.78583e12	0.144273
$117$	−5.59996e12	−0.201826
$118$	−2.01544e13	−0.687286
$119$	4.18418e13	1.35069
$120$	0	0
$121$	−2.91689e13	−0.844918
$122$	−4.30490e13	−1.18202
$123$	2.12634e13	0.553667
$124$	−2.90071e12	−0.0716577
$125$	0	0
$126$	9.05371e12	0.201566
$127$	−1.26331e13	−0.267169	−0.133584	−	0.991037i	$-0.542649\pi$
$127$	−1.26331e13	−0.267169	−0.133584	+	0.991037i	$0.542649\pi$
$128$	1.74851e13	0.351401
$129$	3.59847e13	0.687520
$130$	0	0
$131$	−8.86667e13	−1.53284	−0.766420	−	0.642339i	$-0.777964\pi$
$131$	−8.86667e13	−1.53284	−0.766420	+	0.642339i	$0.777964\pi$
$132$	4.07528e12	0.0670543
$133$	−6.51052e13	−1.01995
$134$	−1.09530e14	−1.63437
$135$	0	0
$136$	−1.50487e14	−2.03937
$137$	−1.20273e13	−0.155412	−0.0777058	−	0.996976i	$-0.524759\pi$
$137$	−1.20273e13	−0.155412	−0.0777058	+	0.996976i	$0.524759\pi$
$138$	−4.80252e13	−0.591909
$139$	−6.56716e13	−0.772292	−0.386146	−	0.922438i	$-0.626194\pi$
$139$	−6.56716e13	−0.772292	−0.386146	+	0.922438i	$0.626194\pi$
$140$	0	0
$141$	−8.50854e12	−0.0911866
$142$	−3.82238e13	−0.391255
$143$	−2.43816e13	−0.238440
$144$	−2.20441e13	−0.206033
$145$	0	0
$146$	−1.19693e14	−1.02277
$147$	3.40015e13	0.277930
$148$	−2.94323e13	−0.230210
$149$	1.98217e14	1.48399	0.741993	−	0.670408i	$-0.233881\pi$
$149$	1.98217e14	1.48399	0.741993	+	0.670408i	$0.233881\pi$
$150$	0	0
$151$	2.19190e14	1.50477	0.752384	−	0.658725i	$-0.228904\pi$
$151$	2.19190e14	1.50477	0.752384	+	0.658725i	$0.228904\pi$
$152$	2.34156e14	1.54000
$153$	9.91991e13	0.625192
$154$	3.94189e13	0.238133
$155$	0	0
$156$	−1.85590e13	−0.103097
$157$	−6.82286e13	−0.363596	−0.181798	−	0.983336i	$-0.558192\pi$
$157$	−6.82286e13	−0.363596	−0.181798	+	0.983336i	$0.558192\pi$
$158$	2.57174e13	0.131509
$159$	7.35775e13	0.361130
$160$	0	0
$161$	1.94306e14	0.879262
$162$	2.14646e13	0.0932988
$163$	−4.37509e14	−1.82712	−0.913561	−	0.406701i	$-0.866679\pi$
$163$	−4.37509e14	−1.82712	−0.913561	+	0.406701i	$0.866679\pi$
$164$	7.04695e13	0.282824
$165$	0	0
$166$	3.62658e14	1.34522
$167$	−1.43875e14	−0.513248	−0.256624	−	0.966511i	$-0.582610\pi$
$167$	−1.43875e14	−0.513248	−0.256624	+	0.966511i	$0.582610\pi$
$168$	1.31745e14	0.452087
$169$	−1.91840e14	−0.633397
$170$	0	0
$171$	−1.54352e14	−0.472104
$172$	1.19258e14	0.351199
$173$	4.16197e14	1.18032	0.590160	−	0.807286i	$-0.299065\pi$
$173$	4.16197e14	1.18032	0.590160	+	0.807286i	$0.299065\pi$
$174$	8.68170e13	0.237156
$175$	0	0
$176$	−9.59778e13	−0.243410
$177$	1.93323e14	0.472561
$178$	2.08733e14	0.491882
$179$	−8.72967e13	−0.198359	−0.0991797	−	0.995070i	$-0.531622\pi$
$179$	−8.72967e13	−0.198359	−0.0991797	+	0.995070i	$0.531622\pi$
$180$	0	0
$181$	7.75682e14	1.63973	0.819866	−	0.572555i	$-0.194048\pi$
$181$	7.75682e14	1.63973	0.819866	+	0.572555i	$0.194048\pi$
$182$	−1.79515e14	−0.366132
$183$	4.12930e14	0.812727
$184$	−6.98836e14	−1.32758
$185$	0	0
$186$	−6.65193e13	−0.117791
$187$	4.31902e14	0.738609
$188$	−2.81984e13	−0.0465799
$189$	−8.68442e13	−0.138592
$190$	0	0
$191$	1.99752e14	0.297696	0.148848	−	0.988860i	$-0.452443\pi$
$191$	1.99752e14	0.297696	0.148848	+	0.988860i	$0.452443\pi$
$192$	−4.38970e14	−0.632379
$193$	4.79306e14	0.667560	0.333780	−	0.942651i	$-0.391676\pi$
$193$	4.79306e14	0.667560	0.333780	+	0.942651i	$0.391676\pi$
$194$	−1.50410e14	−0.202565
$195$	0	0
$196$	1.12685e14	0.141972
$197$	1.54863e15	1.88763	0.943815	−	0.330475i	$-0.107209\pi$
$197$	1.54863e15	1.88763	0.943815	+	0.330475i	$0.107209\pi$
$198$	9.34547e13	0.110224
$199$	3.01022e13	0.0343600	0.0171800	−	0.999852i	$-0.494531\pi$
$199$	3.01022e13	0.0343600	0.0171800	+	0.999852i	$0.494531\pi$
$200$	0	0
$201$	1.05062e15	1.12375
$202$	−4.74479e14	−0.491398
$203$	−3.51255e14	−0.352288
$204$	3.28759e14	0.319360
$205$	0	0
$206$	1.49862e15	1.36632
$207$	4.60663e14	0.406983
$208$	4.37087e14	0.374245
$209$	−6.72032e14	−0.557749
$210$	0	0
$211$	6.52015e14	0.508653	0.254327	−	0.967118i	$-0.418146\pi$
$211$	6.52015e14	0.508653	0.254327	+	0.967118i	$0.418146\pi$
$212$	2.43845e14	0.184472
$213$	3.66647e14	0.269017
$214$	−1.46134e15	−1.04007
$215$	0	0
$216$	3.12341e14	0.209257
$217$	2.69132e14	0.174975
$218$	−1.03624e15	−0.653872
$219$	1.14811e15	0.703229
$220$	0	0
$221$	−1.96690e15	−1.13562
$222$	−6.74945e14	−0.378420
$223$	−2.16690e15	−1.17994	−0.589968	−	0.807427i	$-0.700859\pi$
$223$	−2.16690e15	−1.17994	−0.589968	+	0.807427i	$0.700859\pi$
$224$	7.73796e14	0.409274
$225$	0	0
$226$	−1.17838e15	−0.588277
$227$	1.22242e15	0.592995	0.296497	−	0.955034i	$-0.404181\pi$
$227$	1.22242e15	0.592995	0.296497	+	0.955034i	$0.404181\pi$
$228$	−5.11543e14	−0.241160
$229$	2.72237e14	0.124743	0.0623715	−	0.998053i	$-0.480134\pi$
$229$	2.72237e14	0.124743	0.0623715	+	0.998053i	$0.480134\pi$
$230$	0	0
$231$	−3.78110e14	−0.163735
$232$	1.26331e15	0.531911
$233$	−3.40064e15	−1.39234	−0.696172	−	0.717875i	$-0.745115\pi$
$233$	−3.40064e15	−1.39234	−0.696172	+	0.717875i	$0.745115\pi$
$234$	−4.25597e14	−0.169471
$235$	0	0
$236$	6.40698e14	0.241394
$237$	−2.46684e14	−0.0904226
$238$	3.17998e15	1.13416
$239$	2.89833e15	1.00592	0.502958	−	0.864311i	$-0.332245\pi$
$239$	2.89833e15	1.00592	0.502958	+	0.864311i	$0.332245\pi$
$240$	0	0
$241$	−8.76934e14	−0.288308	−0.144154	−	0.989555i	$-0.546046\pi$
$241$	−8.76934e14	−0.288308	−0.144154	+	0.989555i	$0.546046\pi$
$242$	−2.21683e15	−0.709469
$243$	−2.05891e14	−0.0641500
$244$	1.36850e15	0.415157
$245$	0	0
$246$	1.61601e15	0.464908
$247$	3.06047e15	0.857546
$248$	−9.67952e14	−0.264191
$249$	−3.47865e15	−0.924943
$250$	0	0
$251$	−7.75232e15	−1.95683	−0.978414	−	0.206654i	$-0.933743\pi$
$251$	−7.75232e15	−1.95683	−0.978414	+	0.206654i	$0.933743\pi$
$252$	−2.87813e14	−0.0707957
$253$	2.00568e15	0.480814
$254$	−9.60115e14	−0.224339
$255$	0	0
$256$	−3.60398e15	−0.800244
$257$	2.94392e15	0.637325	0.318662	−	0.947868i	$-0.396766\pi$
$257$	2.94392e15	0.637325	0.318662	+	0.947868i	$0.396766\pi$
$258$	2.73484e15	0.577303
$259$	2.73077e15	0.562130
$260$	0	0
$261$	−8.32758e14	−0.163063
$262$	−6.73867e15	−1.28711
$263$	−2.53948e15	−0.473186	−0.236593	−	0.971609i	$-0.576031\pi$
$263$	−2.53948e15	−0.473186	−0.236593	+	0.971609i	$0.576031\pi$
$264$	1.35990e15	0.247219
$265$	0	0
$266$	−4.94799e15	−0.856443
$267$	−2.00219e15	−0.338206
$268$	3.48189e15	0.574036
$269$	−7.89845e15	−1.27102	−0.635509	−	0.772093i	$-0.719210\pi$
$269$	−7.89845e15	−1.27102	−0.635509	+	0.772093i	$0.719210\pi$
$270$	0	0
$271$	1.12886e16	1.73117	0.865586	−	0.500760i	$-0.166946\pi$
$271$	1.12886e16	1.73117	0.865586	+	0.500760i	$0.166946\pi$
$272$	−7.74267e15	−1.15929
$273$	1.72193e15	0.251744
$274$	−9.14072e14	−0.130497
$275$	0	0
$276$	1.52670e15	0.207895
$277$	−1.42530e16	−1.89578	−0.947890	−	0.318598i	$-0.896788\pi$
$277$	−1.42530e16	−1.89578	−0.947890	+	0.318598i	$0.896788\pi$
$278$	−4.99104e15	−0.648485
$279$	6.38061e14	0.0809906
$280$	0	0
$281$	−6.07190e15	−0.735756	−0.367878	−	0.929874i	$-0.619916\pi$
$281$	−6.07190e15	−0.735756	−0.367878	+	0.929874i	$0.619916\pi$
$282$	−6.46649e14	−0.0765684
$283$	−2.92078e15	−0.337977	−0.168988	−	0.985618i	$-0.554050\pi$
$283$	−2.92078e15	−0.337977	−0.168988	+	0.985618i	$0.554050\pi$
$284$	1.21511e15	0.137419
$285$	0	0
$286$	−1.85300e15	−0.200215
$287$	−6.53826e15	−0.690605
$288$	1.83452e15	0.189440
$289$	2.49376e16	2.51779
$290$	0	0
$291$	1.44275e15	0.139279
$292$	3.80498e15	0.359223
$293$	−7.24813e15	−0.669247	−0.334624	−	0.942352i	$-0.608609\pi$
$293$	−7.24813e15	−0.669247	−0.334624	+	0.942352i	$0.608609\pi$
$294$	2.58411e15	0.233375
$295$	0	0
$296$	−9.82143e15	−0.848747
$297$	−8.96427e14	−0.0757876
$298$	1.50645e16	1.24609
$299$	−9.13394e15	−0.739257
$300$	0	0
$301$	−1.10649e16	−0.857564
$302$	1.66584e16	1.26354
$303$	4.55125e15	0.337873
$304$	1.20475e16	0.875422
$305$	0	0
$306$	7.53913e15	0.524967
$307$	−1.83875e16	−1.25349	−0.626747	−	0.779223i	$-0.715614\pi$
$307$	−1.83875e16	−1.25349	−0.626747	+	0.779223i	$0.715614\pi$
$308$	−1.25311e15	−0.0836388
$309$	−1.43749e16	−0.939451
$310$	0	0
$311$	8.18283e15	0.512815	0.256408	−	0.966569i	$-0.417461\pi$
$311$	8.18283e15	0.512815	0.256408	+	0.966569i	$0.417461\pi$
$312$	−6.19305e15	−0.380102
$313$	2.27733e16	1.36895	0.684476	−	0.729036i	$-0.260031\pi$
$313$	2.27733e16	1.36895	0.684476	+	0.729036i	$0.260031\pi$
$314$	−5.18537e15	−0.305307
$315$	0	0
$316$	−8.17543e14	−0.0461896
$317$	1.52985e16	0.846765	0.423383	−	0.905951i	$-0.360843\pi$
$317$	1.52985e16	0.846765	0.423383	+	0.905951i	$0.360843\pi$
$318$	5.59189e15	0.303237
$319$	−3.62574e15	−0.192645
$320$	0	0
$321$	1.40173e16	0.715124
$322$	1.47673e16	0.738307
$323$	−5.42138e16	−2.65640
$324$	−6.82350e14	−0.0327691
$325$	0	0
$326$	−3.32507e16	−1.53421
$327$	9.93971e15	0.449586
$328$	2.35153e16	1.04273
$329$	2.61629e15	0.113740
$330$	0	0
$331$	−1.66583e15	−0.0696224	−0.0348112	−	0.999394i	$-0.511083\pi$
$331$	−1.66583e15	−0.0696224	−0.0348112	+	0.999394i	$0.511083\pi$
$332$	−1.15287e16	−0.472479
$333$	6.47415e15	0.260193
$334$	−1.09345e16	−0.430969
$335$	0	0
$336$	6.77835e15	0.256991
$337$	3.84382e16	1.42945	0.714724	−	0.699407i	$-0.246552\pi$
$337$	3.84382e16	1.42945	0.714724	+	0.699407i	$0.246552\pi$
$338$	−1.45798e16	−0.531856
$339$	1.13032e16	0.404485
$340$	0	0
$341$	2.77805e15	0.0956832
$342$	−1.17308e16	−0.396421
$343$	−3.21738e16	−1.06682
$344$	3.97959e16	1.29482
$345$	0	0
$346$	3.16310e16	0.991102
$347$	1.19339e16	0.366979	0.183490	−	0.983022i	$-0.441261\pi$
$347$	1.19339e16	0.366979	0.183490	+	0.983022i	$0.441261\pi$
$348$	−2.75987e15	−0.0832958
$349$	−5.62271e16	−1.66564	−0.832819	−	0.553545i	$-0.813275\pi$
$349$	−5.62271e16	−1.66564	−0.832819	+	0.553545i	$0.813275\pi$
$350$	0	0
$351$	4.08237e15	0.116524
$352$	7.98732e15	0.223807
$353$	−2.64668e16	−0.728057	−0.364028	−	0.931388i	$-0.618599\pi$
$353$	−2.64668e16	−0.728057	−0.364028	+	0.931388i	$0.618599\pi$
$354$	1.46926e16	0.396805
$355$	0	0
$356$	−6.63551e15	−0.172762
$357$	−3.05027e16	−0.779821
$358$	−6.63455e15	−0.166560
$359$	−4.25772e16	−1.04970	−0.524848	−	0.851196i	$-0.675878\pi$
$359$	−4.25772e16	−1.04970	−0.524848	+	0.851196i	$0.675878\pi$
$360$	0	0
$361$	4.23028e16	1.00594
$362$	5.89518e16	1.37687
$363$	2.12641e16	0.487814
$364$	5.70670e15	0.128596
$365$	0	0
$366$	3.13827e16	0.682438
$367$	−3.00601e16	−0.642187	−0.321094	−	0.947047i	$-0.604050\pi$
$367$	−3.00601e16	−0.642187	−0.321094	+	0.947047i	$0.604050\pi$
$368$	−3.59556e16	−0.754667
$369$	−1.55010e16	−0.319660
$370$	0	0
$371$	−2.26243e16	−0.450449
$372$	2.11461e15	0.0413716
$373$	7.76847e16	1.49358	0.746790	−	0.665060i	$-0.231594\pi$
$373$	7.76847e16	1.49358	0.746790	+	0.665060i	$0.231594\pi$
$374$	3.28246e16	0.620202
$375$	0	0
$376$	−9.40968e15	−0.171733
$377$	1.65118e16	0.296193
$378$	−6.60016e15	−0.116374
$379$	−4.93660e16	−0.855604	−0.427802	−	0.903872i	$-0.640712\pi$
$379$	−4.93660e16	−0.855604	−0.427802	+	0.903872i	$0.640712\pi$
$380$	0	0
$381$	9.20952e15	0.154250
$382$	1.51811e16	0.249972
$383$	3.05868e16	0.495155	0.247578	−	0.968868i	$-0.420365\pi$
$383$	3.05868e16	0.495155	0.247578	+	0.968868i	$0.420365\pi$
$384$	−1.27466e16	−0.202882
$385$	0	0
$386$	3.64272e16	0.560543
$387$	−2.62329e16	−0.396940
$388$	4.78144e15	0.0711464
$389$	−1.21076e17	−1.77168	−0.885838	−	0.463995i	$-0.846416\pi$
$389$	−1.21076e17	−1.77168	−0.885838	+	0.463995i	$0.846416\pi$
$390$	0	0
$391$	1.61801e17	2.28998
$392$	3.76026e16	0.523430
$393$	6.46380e16	0.884986
$394$	1.17696e17	1.58502
$395$	0	0
$396$	−2.97088e15	−0.0387138
$397$	2.01910e16	0.258833	0.129416	−	0.991590i	$-0.458690\pi$
$397$	2.01910e16	0.258833	0.129416	+	0.991590i	$0.458690\pi$
$398$	2.28777e15	0.0288517
$399$	4.74617e16	0.588870
$400$	0	0
$401$	−9.81485e16	−1.17881	−0.589407	−	0.807836i	$-0.700638\pi$
$401$	−9.81485e16	−1.17881	−0.589407	+	0.807836i	$0.700638\pi$
$402$	7.98472e16	0.943605
$403$	−1.26514e16	−0.147114
$404$	1.50834e16	0.172592
$405$	0	0
$406$	−2.66953e16	−0.295812
$407$	2.81877e16	0.307395
$408$	1.09705e17	1.17743
$409$	−1.41320e17	−1.49281	−0.746403	−	0.665494i	$-0.768221\pi$
$409$	−1.41320e17	−1.49281	−0.746403	+	0.665494i	$0.768221\pi$
$410$	0	0
$411$	8.76788e15	0.0897269
$412$	−4.76402e16	−0.479890
$413$	−5.94449e16	−0.589440
$414$	3.50104e16	0.341739
$415$	0	0
$416$	−3.63746e16	−0.344106
$417$	4.78746e16	0.445883
$418$	−5.10744e16	−0.468336
$419$	1.45438e17	1.31307	0.656535	−	0.754295i	$-0.272021\pi$
$419$	1.45438e17	1.31307	0.656535	+	0.754295i	$0.272021\pi$
$420$	0	0
$421$	1.78877e16	0.156575	0.0782873	−	0.996931i	$-0.475055\pi$
$421$	1.78877e16	0.156575	0.0782873	+	0.996931i	$0.475055\pi$
$422$	4.95532e16	0.427111
$423$	6.20273e15	0.0526466
$424$	8.13701e16	0.680122
$425$	0	0
$426$	2.78652e16	0.225891
$427$	−1.26972e17	−1.01374
$428$	4.64551e16	0.365300
$429$	1.77742e16	0.137663
$430$	0	0
$431$	1.70547e17	1.28157	0.640784	−	0.767721i	$-0.278610\pi$
$431$	1.70547e17	1.28157	0.640784	+	0.767721i	$0.278610\pi$
$432$	1.60702e16	0.118953
$433$	−1.66302e17	−1.21262	−0.606310	−	0.795228i	$-0.707351\pi$
$433$	−1.66302e17	−1.21262	−0.606310	+	0.795228i	$0.707351\pi$
$434$	2.04540e16	0.146925
$435$	0	0
$436$	3.29415e16	0.229658
$437$	−2.51759e17	−1.72924
$438$	8.72564e16	0.590494
$439$	−1.99838e16	−0.133248	−0.0666238	−	0.997778i	$-0.521223\pi$
$439$	−1.99838e16	−0.133248	−0.0666238	+	0.997778i	$0.521223\pi$
$440$	0	0
$441$	−2.47871e16	−0.160463
$442$	−1.49485e17	−0.953568
$443$	−4.47997e16	−0.281612	−0.140806	−	0.990037i	$-0.544969\pi$
$443$	−4.47997e16	−0.281612	−0.140806	+	0.990037i	$0.544969\pi$
$444$	2.14562e16	0.132912
$445$	0	0
$446$	−1.64685e17	−0.990779
$447$	−1.44500e17	−0.856780
$448$	1.34979e17	0.788785
$449$	−6.95599e16	−0.400643	−0.200321	−	0.979730i	$-0.564199\pi$
$449$	−6.95599e16	−0.400643	−0.200321	+	0.979730i	$0.564199\pi$
$450$	0	0
$451$	−6.74896e16	−0.377650
$452$	3.74602e16	0.206619
$453$	−1.59789e17	−0.868778
$454$	9.29036e16	0.497931
$455$	0	0
$456$	−1.70699e17	−0.889120
$457$	7.70855e16	0.395838	0.197919	−	0.980218i	$-0.436582\pi$
$457$	7.70855e16	0.395838	0.197919	+	0.980218i	$0.436582\pi$
$458$	2.06900e16	0.104745
$459$	−7.23161e16	−0.360955
$460$	0	0
$461$	−3.52847e17	−1.71211	−0.856053	−	0.516889i	$-0.827090\pi$
$461$	−3.52847e17	−1.71211	−0.856053	+	0.516889i	$0.827090\pi$
$462$	−2.87364e16	−0.137486
$463$	−1.88653e17	−0.889996	−0.444998	−	0.895532i	$-0.646796\pi$
$463$	−1.88653e17	−0.889996	−0.444998	+	0.895532i	$0.646796\pi$
$464$	6.49983e16	0.302367
$465$	0	0
$466$	−2.58448e17	−1.16914
$467$	−3.48352e16	−0.155403	−0.0777013	−	0.996977i	$-0.524758\pi$
$467$	−3.48352e16	−0.155403	−0.0777013	+	0.996977i	$0.524758\pi$
$468$	1.35295e16	0.0595229
$469$	−3.23055e17	−1.40169
$470$	0	0
$471$	4.97386e16	0.209922
$472$	2.13798e17	0.889981
$473$	−1.14215e17	−0.468949
$474$	−1.87480e16	−0.0759269
$475$	0	0
$476$	−1.01090e17	−0.398348
$477$	−5.36380e16	−0.208499
$478$	2.20273e17	0.844657
$479$	−2.64103e16	−0.0999064	−0.0499532	−	0.998752i	$-0.515907\pi$
$479$	−2.64103e16	−0.0999064	−0.0499532	+	0.998752i	$0.515907\pi$
$480$	0	0
$481$	−1.28368e17	−0.472623
$482$	−6.66470e16	−0.242089
$483$	−1.41649e17	−0.507642
$484$	7.04720e16	0.249185
$485$	0	0
$486$	−1.56477e16	−0.0538661
$487$	2.18673e17	0.742775	0.371388	−	0.928478i	$-0.378882\pi$
$487$	2.18673e17	0.742775	0.371388	+	0.928478i	$0.378882\pi$
$488$	4.56663e17	1.53062
$489$	3.18944e17	1.05489
$490$	0	0
$491$	−1.68155e17	−0.541603	−0.270801	−	0.962635i	$-0.587289\pi$
$491$	−1.68155e17	−0.541603	−0.270801	+	0.962635i	$0.587289\pi$
$492$	−5.13723e16	−0.163288
$493$	−2.92494e17	−0.917511
$494$	2.32595e17	0.720072
$495$	0	0
$496$	−4.98018e16	−0.150181
$497$	−1.12740e17	−0.335554
$498$	−2.64377e17	−0.776665
$499$	−1.27608e17	−0.370020	−0.185010	−	0.982737i	$-0.559232\pi$
$499$	−1.27608e17	−0.370020	−0.185010	+	0.982737i	$0.559232\pi$
$500$	0	0
$501$	1.04885e17	0.296324
$502$	−5.89177e17	−1.64313
$503$	−5.31062e17	−1.46202	−0.731009	−	0.682367i	$-0.760950\pi$
$503$	−5.31062e17	−1.46202	−0.731009	+	0.682367i	$0.760950\pi$
$504$	−9.60418e16	−0.261013
$505$	0	0
$506$	1.52431e17	0.403735
$507$	1.39851e17	0.365692
$508$	3.05215e16	0.0787939
$509$	3.24473e16	0.0827013	0.0413507	−	0.999145i	$-0.486834\pi$
$509$	3.24473e16	0.0827013	0.0413507	+	0.999145i	$0.486834\pi$
$510$	0	0
$511$	−3.53032e17	−0.877159
$512$	−4.17140e17	−1.02336
$513$	1.12523e17	0.272569
$514$	2.23738e17	0.535155
$515$	0	0
$516$	−8.69391e16	−0.202765
$517$	2.70060e16	0.0621973
$518$	2.07539e17	0.472015
$519$	−3.03408e17	−0.681458
$520$	0	0
$521$	8.64612e17	1.89398	0.946992	−	0.321257i	$-0.104105\pi$
$521$	8.64612e17	1.89398	0.946992	+	0.321257i	$0.104105\pi$
$522$	−6.32896e16	−0.136922
$523$	7.78772e17	1.66398	0.831992	−	0.554787i	$-0.187200\pi$
$523$	7.78772e17	1.66398	0.831992	+	0.554787i	$0.187200\pi$
$524$	2.14219e17	0.452068
$525$	0	0
$526$	−1.93000e17	−0.397329
$527$	2.24109e17	0.455712
$528$	6.99678e16	0.140533
$529$	2.47338e17	0.490715
$530$	0	0
$531$	−1.40933e17	−0.272833
$532$	1.57294e17	0.300806
$533$	3.07351e17	0.580641
$534$	−1.52166e17	−0.283988
$535$	0	0
$536$	1.16189e18	2.11638
$537$	6.36393e16	0.114523
$538$	−6.00282e17	−1.06726
$539$	−1.07920e17	−0.189573
$540$	0	0
$541$	5.10492e15	0.00875400	0.00437700	−	0.999990i	$-0.498607\pi$
$541$	5.10492e15	0.00875400	0.00437700	+	0.999990i	$0.498607\pi$
$542$	8.57935e17	1.45365
$543$	−5.65472e17	−0.946700
$544$	6.44349e17	1.06593
$545$	0	0
$546$	1.30867e17	0.211386
$547$	1.05077e18	1.67722	0.838610	−	0.544732i	$-0.183369\pi$
$547$	1.05077e18	1.67722	0.838610	+	0.544732i	$0.183369\pi$
$548$	2.90579e16	0.0458343
$549$	−3.01026e17	−0.469228
$550$	0	0
$551$	4.55115e17	0.692845
$552$	5.09451e17	0.766476
$553$	7.58528e16	0.112787
$554$	−1.08323e18	−1.59187
$555$	0	0
$556$	1.58663e17	0.227766
$557$	−7.17372e17	−1.01785	−0.508927	−	0.860810i	$-0.669958\pi$
$557$	−7.17372e17	−1.01785	−0.508927	+	0.860810i	$0.669958\pi$
$558$	4.84926e16	0.0680069
$559$	5.20141e17	0.721015
$560$	0	0
$561$	−3.14857e17	−0.426436
$562$	−4.61465e17	−0.617806
$563$	7.40487e17	0.979971	0.489985	−	0.871731i	$-0.337002\pi$
$563$	7.40487e17	0.979971	0.489985	+	0.871731i	$0.337002\pi$
$564$	2.05566e16	0.0268929
$565$	0	0
$566$	−2.21979e17	−0.283796
$567$	6.33094e16	0.0800163
$568$	4.05478e17	0.506644
$569$	−1.21604e18	−1.50217	−0.751083	−	0.660208i	$-0.770468\pi$
$569$	−1.21604e18	−1.50217	−0.751083	+	0.660208i	$0.770468\pi$
$570$	0	0
$571$	1.34263e18	1.62114	0.810569	−	0.585643i	$-0.199158\pi$
$571$	1.34263e18	1.62114	0.810569	+	0.585643i	$0.199158\pi$
$572$	5.89060e16	0.0703211
$573$	−1.45619e17	−0.171875
$574$	−4.96908e17	−0.579894
$575$	0	0
$576$	3.20009e17	0.365104
$577$	1.71687e17	0.193685	0.0968423	−	0.995300i	$-0.469126\pi$
$577$	1.71687e17	0.193685	0.0968423	+	0.995300i	$0.469126\pi$
$578$	1.89526e18	2.11416
$579$	−3.49414e17	−0.385416
$580$	0	0
$581$	1.06965e18	1.15371
$582$	1.09649e17	0.116951
$583$	−2.33534e17	−0.246323
$584$	1.26971e18	1.32440
$585$	0	0
$586$	−5.50858e17	−0.561960
$587$	6.89908e17	0.696055	0.348027	−	0.937484i	$-0.386852\pi$
$587$	6.89908e17	0.696055	0.348027	+	0.937484i	$0.386852\pi$
$588$	−8.21477e16	−0.0819677
$589$	−3.48710e17	−0.344124
$590$	0	0
$591$	−1.12895e18	−1.08982
$592$	−5.05319e17	−0.482475
$593$	6.35159e17	0.599829	0.299914	−	0.953966i	$-0.403042\pi$
$593$	6.35159e17	0.599829	0.299914	+	0.953966i	$0.403042\pi$
$594$	−6.81285e16	−0.0636380
$595$	0	0
$596$	−4.78892e17	−0.437660
$597$	−2.19445e16	−0.0198378
$598$	−6.94180e17	−0.620746
$599$	5.29451e17	0.468329	0.234165	−	0.972197i	$-0.424765\pi$
$599$	5.29451e17	0.468329	0.234165	+	0.972197i	$0.424765\pi$
$600$	0	0
$601$	1.28661e18	1.11368	0.556842	−	0.830618i	$-0.312013\pi$
$601$	1.28661e18	1.11368	0.556842	+	0.830618i	$0.312013\pi$
$602$	−8.40935e17	−0.720087
$603$	−7.65903e17	−0.648800
$604$	−5.29562e17	−0.443789
$605$	0	0
$606$	3.45895e17	0.283708
$607$	1.15039e18	0.933512	0.466756	−	0.884386i	$-0.345423\pi$
$607$	1.15039e18	0.933512	0.466756	+	0.884386i	$0.345423\pi$
$608$	−1.00260e18	−0.804920
$609$	2.56065e17	0.203394
$610$	0	0
$611$	−1.22987e17	−0.0956291
$612$	−2.39665e17	−0.184383
$613$	−1.27801e18	−0.972837	−0.486419	−	0.873726i	$-0.661697\pi$
$613$	−1.27801e18	−0.972837	−0.486419	+	0.873726i	$0.661697\pi$
$614$	−1.39745e18	−1.05255
$615$	0	0
$616$	−4.18155e17	−0.308364
$617$	−2.98140e17	−0.217554	−0.108777	−	0.994066i	$-0.534693\pi$
$617$	−2.98140e17	−0.217554	−0.108777	+	0.994066i	$0.534693\pi$
$618$	−1.09249e18	−0.788847
$619$	3.57940e17	0.255753	0.127877	−	0.991790i	$-0.459184\pi$
$619$	3.57940e17	0.255753	0.127877	+	0.991790i	$0.459184\pi$
$620$	0	0
$621$	−3.35823e17	−0.234972
$622$	6.21895e17	0.430605
$623$	6.15652e17	0.421855
$624$	−3.18637e17	−0.216071
$625$	0	0
$626$	1.73077e18	1.14949
$627$	4.89911e17	0.322017
$628$	1.64840e17	0.107232
$629$	2.27395e18	1.46403
$630$	0	0
$631$	−7.96707e17	−0.502467	−0.251234	−	0.967926i	$-0.580836\pi$
$631$	−7.96707e17	−0.502467	−0.251234	+	0.967926i	$0.580836\pi$
$632$	−2.72810e17	−0.170294
$633$	−4.75319e17	−0.293671
$634$	1.16268e18	0.711020
$635$	0	0
$636$	−1.77763e17	−0.106505
$637$	4.91474e17	0.291470
$638$	−2.75556e17	−0.161762
$639$	−2.67285e17	−0.155317
$640$	0	0
$641$	1.04983e17	0.0597781	0.0298891	−	0.999553i	$-0.490485\pi$
$641$	1.04983e17	0.0597781	0.0298891	+	0.999553i	$0.490485\pi$
$642$	1.06531e18	0.600482
$643$	−1.43963e18	−0.803301	−0.401650	−	0.915793i	$-0.631563\pi$
$643$	−1.43963e18	−0.803301	−0.401650	+	0.915793i	$0.631563\pi$
$644$	−4.69443e17	−0.259313
$645$	0	0
$646$	−4.12025e18	−2.23055
$647$	−5.11523e17	−0.274149	−0.137075	−	0.990561i	$-0.543770\pi$
$647$	−5.11523e17	−0.274149	−0.137075	+	0.990561i	$0.543770\pi$
$648$	−2.27697e17	−0.120815
$649$	−6.13606e17	−0.322329
$650$	0	0
$651$	−1.96197e17	−0.101022
$652$	1.05702e18	0.538858
$653$	3.66087e16	0.0184777	0.00923887	−	0.999957i	$-0.497059\pi$
$653$	3.66087e16	0.0184777	0.00923887	+	0.999957i	$0.497059\pi$
$654$	7.55418e17	0.377513
$655$	0	0
$656$	1.20988e18	0.592744
$657$	−8.36972e17	−0.406009
$658$	1.98838e17	0.0955061
$659$	−1.45385e18	−0.691455	−0.345727	−	0.938335i	$-0.612368\pi$
$659$	−1.45385e18	−0.691455	−0.345727	+	0.938335i	$0.612368\pi$
$660$	0	0
$661$	−2.41400e18	−1.12571	−0.562856	−	0.826555i	$-0.690298\pi$
$661$	−2.41400e18	−1.12571	−0.562856	+	0.826555i	$0.690298\pi$
$662$	−1.26603e17	−0.0584612
$663$	1.43387e18	0.655650
$664$	−3.84707e18	−1.74196
$665$	0	0
$666$	4.92035e17	0.218481
$667$	−1.35829e18	−0.597275
$668$	3.47601e17	0.151368
$669$	1.57967e18	0.681236
$670$	0	0
$671$	−1.31063e18	−0.554351
$672$	−5.64097e17	−0.236295
$673$	5.97994e17	0.248084	0.124042	−	0.992277i	$-0.460414\pi$
$673$	5.97994e17	0.248084	0.124042	+	0.992277i	$0.460414\pi$
$674$	2.92130e18	1.20029
$675$	0	0
$676$	4.63486e17	0.186802
$677$	−1.33881e18	−0.534433	−0.267217	−	0.963636i	$-0.586104\pi$
$677$	−1.33881e18	−0.534433	−0.267217	+	0.963636i	$0.586104\pi$
$678$	8.59042e17	0.339642
$679$	−4.43629e17	−0.173727
$680$	0	0
$681$	−8.91141e17	−0.342366
$682$	2.11131e17	0.0803442
$683$	3.55434e18	1.33975	0.669877	−	0.742472i	$-0.266347\pi$
$683$	3.55434e18	1.33975	0.669877	+	0.742472i	$0.266347\pi$
$684$	3.72915e17	0.139234
$685$	0	0
$686$	−2.44521e18	−0.895795
$687$	−1.98461e17	−0.0720204
$688$	2.04752e18	0.736044
$689$	1.06353e18	0.378724
$690$	0	0
$691$	2.39415e18	0.836652	0.418326	−	0.908297i	$-0.362617\pi$
$691$	2.39415e18	0.836652	0.418326	+	0.908297i	$0.362617\pi$
$692$	−1.00553e18	−0.348102
$693$	2.75642e17	0.0945322
$694$	9.06978e17	0.308149
$695$	0	0
$696$	−9.20955e17	−0.307099
$697$	−5.44449e18	−1.79864
$698$	−4.27326e18	−1.39862
$699$	2.47906e18	0.803870
$700$	0	0
$701$	5.32639e17	0.169538	0.0847688	−	0.996401i	$-0.472985\pi$
$701$	5.32639e17	0.169538	0.0847688	+	0.996401i	$0.472985\pi$
$702$	3.10260e17	0.0978442
$703$	−3.53822e18	−1.10554
$704$	1.39329e18	0.431338
$705$	0	0
$706$	−2.01147e18	−0.611341
$707$	−1.39946e18	−0.421439
$708$	−4.67069e17	−0.139369
$709$	2.79282e18	0.825739	0.412869	−	0.910790i	$-0.364527\pi$
$709$	2.79282e18	0.825739	0.412869	+	0.910790i	$0.364527\pi$
$710$	0	0
$711$	1.79833e17	0.0522055
$712$	−2.21424e18	−0.636948
$713$	1.04072e18	0.296656
$714$	−2.31821e18	−0.654807
$715$	0	0
$716$	2.10909e17	0.0585006
$717$	−2.11288e18	−0.580766
$718$	−3.23587e18	−0.881418
$719$	1.84208e18	0.497246	0.248623	−	0.968600i	$-0.420022\pi$
$719$	1.84208e18	0.497246	0.248623	+	0.968600i	$0.420022\pi$
$720$	0	0
$721$	4.42013e18	1.17181
$722$	3.21501e18	0.844678
$723$	6.39285e17	0.166454
$724$	−1.87405e18	−0.483593
$725$	0	0
$726$	1.61607e18	0.409612
$727$	1.69067e17	0.0424702	0.0212351	−	0.999775i	$-0.493240\pi$
$727$	1.69067e17	0.0424702	0.0212351	+	0.999775i	$0.493240\pi$
$728$	1.90430e18	0.474112
$729$	1.50095e17	0.0370370
$730$	0	0
$731$	−9.21390e18	−2.23347
$732$	−9.97639e17	−0.239691
$733$	7.04063e18	1.67662	0.838312	−	0.545190i	$-0.183542\pi$
$733$	7.04063e18	1.67662	0.838312	+	0.545190i	$0.183542\pi$
$734$	−2.28457e18	−0.539238
$735$	0	0
$736$	2.99224e18	0.693890
$737$	−3.33466e18	−0.766500
$738$	−1.17807e18	−0.268415
$739$	−1.57737e18	−0.356242	−0.178121	−	0.984009i	$-0.557002\pi$
$739$	−1.57737e18	−0.356242	−0.178121	+	0.984009i	$0.557002\pi$
$740$	0	0
$741$	−2.23108e18	−0.495104
$742$	−1.71945e18	−0.378237
$743$	5.07897e18	1.10751	0.553756	−	0.832679i	$-0.313194\pi$
$743$	5.07897e18	1.10751	0.553756	+	0.832679i	$0.313194\pi$
$744$	7.05637e17	0.152531
$745$	0	0
$746$	5.90404e18	1.25414
$747$	2.53594e18	0.534016
$748$	−1.04348e18	−0.217832
$749$	−4.31017e18	−0.891996
$750$	0	0
$751$	1.17184e18	0.238347	0.119174	−	0.992873i	$-0.461975\pi$
$751$	1.17184e18	0.238347	0.119174	+	0.992873i	$0.461975\pi$
$752$	−4.84134e17	−0.0976225
$753$	5.65144e18	1.12978
$754$	1.25490e18	0.248710
$755$	0	0
$756$	2.09816e17	0.0408739
$757$	3.65200e18	0.705355	0.352677	−	0.935745i	$-0.385271\pi$
$757$	3.65200e18	0.705355	0.352677	+	0.935745i	$0.385271\pi$
$758$	−3.75182e18	−0.718442
$759$	−1.46214e18	−0.277598
$760$	0	0
$761$	−6.37367e18	−1.18957	−0.594784	−	0.803885i	$-0.702763\pi$
$761$	−6.37367e18	−1.18957	−0.594784	+	0.803885i	$0.702763\pi$
$762$	6.99924e17	0.129522
$763$	−3.05636e18	−0.560783
$764$	−4.82600e17	−0.0877971
$765$	0	0
$766$	2.32459e18	0.415777
$767$	2.79439e18	0.495584
$768$	2.62730e18	0.462021
$769$	−5.20322e18	−0.907300	−0.453650	−	0.891180i	$-0.649878\pi$
$769$	−5.20322e18	−0.907300	−0.453650	+	0.891180i	$0.649878\pi$
$770$	0	0
$771$	−2.14612e18	−0.367960
$772$	−1.15800e18	−0.196878
$773$	−9.15563e18	−1.54355	−0.771777	−	0.635894i	$-0.780632\pi$
$773$	−9.15563e18	−1.54355	−0.771777	+	0.635894i	$0.780632\pi$
$774$	−1.99370e18	−0.333306
$775$	0	0
$776$	1.59555e18	0.262306
$777$	−1.99073e18	−0.324546
$778$	−9.20175e18	−1.48766
$779$	8.47153e18	1.35821
$780$	0	0
$781$	−1.16373e18	−0.183494
$782$	1.22969e19	1.92287
$783$	6.07081e17	0.0941445
$784$	1.93468e18	0.297546
$785$	0	0
$786$	4.91249e18	0.743113
$787$	7.56940e18	1.13560	0.567800	−	0.823166i	$-0.307795\pi$
$787$	7.56940e18	1.13560	0.567800	+	0.823166i	$0.307795\pi$
$788$	−3.74149e18	−0.556703
$789$	1.85128e18	0.273194
$790$	0	0
$791$	−3.47561e18	−0.504527
$792$	−9.91368e17	−0.142732
$793$	5.96869e18	0.852321
$794$	1.53451e18	0.217339
$795$	0	0
$796$	−7.27269e16	−0.0101335
$797$	5.93050e18	0.819619	0.409810	−	0.912171i	$-0.365595\pi$
$797$	5.93050e18	0.819619	0.409810	+	0.912171i	$0.365595\pi$
$798$	3.60709e18	0.494468
$799$	2.17861e18	0.296228
$800$	0	0
$801$	1.45959e18	0.195263
$802$	−7.45928e18	−0.989837
$803$	−3.64409e18	−0.479664
$804$	−2.53830e18	−0.331420
$805$	0	0
$806$	−9.61503e17	−0.123530
$807$	5.75797e18	0.733823
$808$	5.03327e18	0.636321
$809$	−7.25995e18	−0.910475	−0.455237	−	0.890370i	$-0.650446\pi$
$809$	−7.25995e18	−0.910475	−0.455237	+	0.890370i	$0.650446\pi$
$810$	0	0
$811$	7.02013e18	0.866382	0.433191	−	0.901302i	$-0.357388\pi$
$811$	7.02013e18	0.866382	0.433191	+	0.901302i	$0.357388\pi$
$812$	8.48631e17	0.103897
$813$	−8.22940e18	−0.999493
$814$	2.14227e18	0.258116
$815$	0	0
$816$	5.64441e18	0.669318
$817$	1.43367e19	1.68657
$818$	−1.07404e19	−1.25349
$819$	−1.25529e18	−0.145344
$820$	0	0
$821$	5.94385e18	0.677388	0.338694	−	0.940897i	$-0.390015\pi$
$821$	5.94385e18	0.677388	0.338694	+	0.940897i	$0.390015\pi$
$822$	6.66359e17	0.0753427
$823$	7.27929e18	0.816563	0.408282	−	0.912856i	$-0.366128\pi$
$823$	7.27929e18	0.816563	0.408282	+	0.912856i	$0.366128\pi$
$824$	−1.58973e19	−1.76928
$825$	0	0
$826$	−4.51782e18	−0.494946
$827$	−6.35419e18	−0.690676	−0.345338	−	0.938478i	$-0.612236\pi$
$827$	−6.35419e18	−0.690676	−0.345338	+	0.938478i	$0.612236\pi$
$828$	−1.11296e18	−0.120028
$829$	1.19116e19	1.27458	0.637291	−	0.770624i	$-0.280055\pi$
$829$	1.19116e19	1.27458	0.637291	+	0.770624i	$0.280055\pi$
$830$	0	0
$831$	1.03904e19	1.09453
$832$	−6.34509e18	−0.663187
$833$	−8.70609e18	−0.902882
$834$	3.63847e18	0.374403
$835$	0	0
$836$	1.62363e18	0.164493
$837$	−4.65146e17	−0.0467599
$838$	1.10533e19	1.10257
$839$	1.86898e18	0.184991	0.0924957	−	0.995713i	$-0.470516\pi$
$839$	1.86898e18	0.184991	0.0924957	+	0.995713i	$0.470516\pi$
$840$	0	0
$841$	−7.80520e18	−0.760694
$842$	1.35947e18	0.131474
$843$	4.42642e18	0.424789
$844$	−1.57527e18	−0.150013
$845$	0	0
$846$	4.71407e17	0.0442068
$847$	−6.53850e18	−0.608465
$848$	4.18655e18	0.386619
$849$	2.12925e18	0.195131
$850$	0	0
$851$	1.05598e19	0.953045
$852$	−8.85818e17	−0.0793391
$853$	1.70066e19	1.51164	0.755822	−	0.654778i	$-0.227238\pi$
$853$	1.70066e19	1.51164	0.755822	+	0.654778i	$0.227238\pi$
$854$	−9.64985e18	−0.851225
$855$	0	0
$856$	1.55019e19	1.34680
$857$	−3.81051e18	−0.328555	−0.164277	−	0.986414i	$-0.552529\pi$
$857$	−3.81051e18	−0.328555	−0.164277	+	0.986414i	$0.552529\pi$
$858$	1.35084e18	0.115594
$859$	3.41465e18	0.289995	0.144998	−	0.989432i	$-0.453683\pi$
$859$	3.41465e18	0.289995	0.144998	+	0.989432i	$0.453683\pi$
$860$	0	0
$861$	4.76639e18	0.398721
$862$	1.29616e19	1.07612
$863$	−6.16691e18	−0.508157	−0.254078	−	0.967184i	$-0.581772\pi$
$863$	−6.16691e18	−0.508157	−0.254078	+	0.967184i	$0.581772\pi$
$864$	−1.33737e18	−0.109373
$865$	0	0
$866$	−1.26389e19	−1.01822
$867$	−1.81795e19	−1.45364
$868$	−6.50222e17	−0.0516040
$869$	7.82972e17	0.0616762
$870$	0	0
$871$	1.51862e19	1.17850
$872$	1.09924e19	0.846713
$873$	−1.05176e18	−0.0804127
$874$	−1.91337e19	−1.45203
$875$	0	0
$876$	−2.77383e18	−0.207398
$877$	8.31159e18	0.616860	0.308430	−	0.951247i	$-0.400196\pi$
$877$	8.31159e18	0.616860	0.308430	+	0.951247i	$0.400196\pi$
$878$	−1.51877e18	−0.111887
$879$	5.28389e18	0.386390
$880$	0	0
$881$	−2.05193e19	−1.47849	−0.739246	−	0.673436i	$-0.764818\pi$
$881$	−2.05193e19	−1.47849	−0.739246	+	0.673436i	$0.764818\pi$
$882$	−1.88382e18	−0.134739
$883$	1.29497e19	0.919423	0.459711	−	0.888068i	$-0.347953\pi$
$883$	1.29497e19	0.919423	0.459711	+	0.888068i	$0.347953\pi$
$884$	4.75204e18	0.334919
$885$	0	0
$886$	−3.40478e18	−0.236466
$887$	3.62066e18	0.249622	0.124811	−	0.992181i	$-0.460167\pi$
$887$	3.62066e18	0.249622	0.124811	+	0.992181i	$0.460167\pi$
$888$	7.15982e18	0.490025
$889$	−2.83183e18	−0.192401
$890$	0	0
$891$	6.53496e17	0.0437560
$892$	5.23524e18	0.347989
$893$	−3.38989e18	−0.223692
$894$	−1.09820e19	−0.719429
$895$	0	0
$896$	3.91946e18	0.253060
$897$	6.65864e18	0.426810
$898$	−5.28655e18	−0.336416
$899$	−1.88135e18	−0.118859
$900$	0	0
$901$	−1.88395e19	−1.17317
$902$	−5.12921e18	−0.317108
$903$	8.06634e18	0.495115
$904$	1.25003e19	0.761773
$905$	0	0
$906$	−1.21440e19	−0.729504
$907$	−2.31466e19	−1.38051	−0.690255	−	0.723566i	$-0.742502\pi$
$907$	−2.31466e19	−1.38051	−0.690255	+	0.723566i	$0.742502\pi$
$908$	−2.95336e18	−0.174887
$909$	−3.31786e18	−0.195071
$910$	0	0
$911$	1.06092e19	0.614915	0.307457	−	0.951562i	$-0.400522\pi$
$911$	1.06092e19	0.614915	0.307457	+	0.951562i	$0.400522\pi$
$912$	−8.78260e18	−0.505425
$913$	1.10412e19	0.630893
$914$	5.85850e18	0.332381
$915$	0	0
$916$	−6.57724e17	−0.0367894
$917$	−1.98755e19	−1.10387
$918$	−5.49603e18	−0.303090
$919$	2.22434e19	1.21801	0.609003	−	0.793168i	$-0.291570\pi$
$919$	2.22434e19	1.21801	0.609003	+	0.793168i	$0.291570\pi$
$920$	0	0
$921$	1.34045e19	0.723705
$922$	−2.68164e19	−1.43764
$923$	5.29969e18	0.282124
$924$	9.13514e17	0.0482889
$925$	0	0
$926$	−1.43377e19	−0.747320
$927$	1.04793e19	0.542392
$928$	−5.40919e18	−0.278016
$929$	1.55649e19	0.794411	0.397205	−	0.917730i	$-0.369980\pi$
$929$	1.55649e19	0.794411	0.397205	+	0.917730i	$0.369980\pi$
$930$	0	0
$931$	1.35465e19	0.681797
$932$	8.21593e18	0.410633
$933$	−5.96528e18	−0.296074
$934$	−2.64747e18	−0.130490
$935$	0	0
$936$	4.51473e18	0.219452
$937$	2.37848e19	1.14813	0.574067	−	0.818808i	$-0.305365\pi$
$937$	2.37848e19	1.14813	0.574067	+	0.818808i	$0.305365\pi$
$938$	−2.45522e19	−1.17699
$939$	−1.66017e19	−0.790364
$940$	0	0
$941$	−2.66324e18	−0.125048	−0.0625242	−	0.998043i	$-0.519915\pi$
$941$	−2.66324e18	−0.125048	−0.0625242	+	0.998043i	$0.519915\pi$
$942$	3.78014e18	0.176269
$943$	−2.52832e19	−1.17086
$944$	1.10001e19	0.505914
$945$	0	0
$946$	−8.68034e18	−0.393772
$947$	−7.85906e18	−0.354075	−0.177038	−	0.984204i	$-0.556651\pi$
$947$	−7.85906e18	−0.354075	−0.177038	+	0.984204i	$0.556651\pi$
$948$	5.95989e17	0.0266676
$949$	1.65953e19	0.737489
$950$	0	0
$951$	−1.11526e19	−0.488880
$952$	−3.37332e19	−1.46865
$953$	3.28659e18	0.142115	0.0710577	−	0.997472i	$-0.477363\pi$
$953$	3.28659e18	0.142115	0.0710577	+	0.997472i	$0.477363\pi$
$954$	−4.07649e18	−0.175074
$955$	0	0
$956$	−7.00237e18	−0.296667
$957$	2.64316e18	0.111223
$958$	−2.00719e18	−0.0838903
$959$	−2.69603e18	−0.111919
$960$	0	0
$961$	−2.29760e19	−0.940965
$962$	−9.75599e18	−0.396856
$963$	−1.02186e19	−0.412877
$964$	2.11867e18	0.0850282
$965$	0	0
$966$	−1.07653e19	−0.426262
$967$	7.84714e18	0.308631	0.154315	−	0.988022i	$-0.450683\pi$
$967$	7.84714e18	0.308631	0.154315	+	0.988022i	$0.450683\pi$
$968$	2.35162e19	0.918707
$969$	3.95219e19	1.53367
$970$	0	0
$971$	−3.28236e19	−1.25679	−0.628394	−	0.777895i	$-0.716287\pi$
$971$	−3.28236e19	−1.25679	−0.628394	+	0.777895i	$0.716287\pi$
$972$	4.97433e17	0.0189192
$973$	−1.47209e19	−0.556163
$974$	1.66192e19	0.623701
$975$	0	0
$976$	2.34956e19	0.870088
$977$	3.29960e19	1.21380	0.606900	−	0.794778i	$-0.292413\pi$
$977$	3.29960e19	1.21380	0.606900	+	0.794778i	$0.292413\pi$
$978$	2.42398e19	0.885779
$979$	6.35491e18	0.230686
$980$	0	0
$981$	−7.24605e18	−0.259569
$982$	−1.27798e19	−0.454778
$983$	8.71759e17	0.0308176	0.0154088	−	0.999881i	$-0.495095\pi$
$983$	8.71759e17	0.0308176	0.0154088	+	0.999881i	$0.495095\pi$
$984$	−1.71427e19	−0.602020
$985$	0	0
$986$	−2.22295e19	−0.770425
$987$	−1.90728e18	−0.0656677
$988$	−7.39409e18	−0.252909
$989$	−4.27877e19	−1.45393
$990$	0	0
$991$	−4.25946e19	−1.42849	−0.714243	−	0.699898i	$-0.753229\pi$
$991$	−4.25946e19	−1.42849	−0.714243	+	0.699898i	$0.753229\pi$
$992$	4.14453e18	0.138086
$993$	1.21439e18	0.0401965
$994$	−8.56825e18	−0.281761
$995$	0	0
$996$	8.40442e18	0.272786
$997$	−1.97490e19	−0.636836	−0.318418	−	0.947950i	$-0.603152\pi$
$997$	−1.97490e19	−0.636836	−0.318418	+	0.947950i	$0.603152\pi$
$998$	−9.69822e18	−0.310701
$999$	−4.71965e18	−0.150222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.14.a.b.1.1		1
5.2	odd	4		75.14.b.a.49.2		2
5.3	odd	4		75.14.b.a.49.1		2
5.4	even	2		15.14.a.a.1.1	✓	1
15.14	odd	2		45.14.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.14.a.a.1.1	✓	1	5.4	even	2
45.14.a.a.1.1		1	15.14	odd	2
75.14.a.b.1.1		1	1.1	even	1	trivial
75.14.b.a.49.1		2	5.3	odd	4
75.14.b.a.49.2		2	5.2	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more