LMFDB - Embedded newform 75.14.b.b.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,14,Mod(49,75)]

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, 1]))

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

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	75.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$80.4231967139$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.14.b.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+12.0000i q^{2} -729.000i q^{3} +8048.00 q^{4} +8748.00 q^{6} -235088. i q^{7} +194880. i q^{8} -531441. q^{9} -1.11829e7 q^{11} -5.86699e6i q^{12} +8.04961e6i q^{13} +2.82106e6 q^{14} +6.35907e7 q^{16} +1.17495e8i q^{17} -6.37729e6i q^{18} +2.14061e8 q^{19} -1.71379e8 q^{21} -1.34195e8i q^{22} +8.30556e8i q^{23} +1.42068e8 q^{24} -9.65954e7 q^{26} +3.87420e8i q^{27} -1.89199e9i q^{28} +1.25240e9 q^{29} +6.15935e9 q^{31} +2.35954e9i q^{32} +8.15234e9i q^{33} -1.40994e9 q^{34} -4.27704e9 q^{36} +5.49819e9i q^{37} +2.56874e9i q^{38} +5.86817e9 q^{39} -4.67869e9 q^{41} -2.05655e9i q^{42} +7.11501e9i q^{43} -9.00000e10 q^{44} -9.96667e9 q^{46} +2.95288e10i q^{47} -4.63576e10i q^{48} +4.16226e10 q^{49} +8.56536e10 q^{51} +6.47833e10i q^{52} -2.04125e11i q^{53} -4.64905e9 q^{54} +4.58139e10 q^{56} -1.56051e11i q^{57} +1.50288e10i q^{58} +2.99098e10 q^{59} -1.34392e11 q^{61} +7.39122e10i q^{62} +1.24935e11i q^{63} +4.92620e11 q^{64} -9.78281e10 q^{66} -3.48519e11i q^{67} +9.45597e11i q^{68} +6.05475e11 q^{69} +1.31434e12 q^{71} -1.03567e11i q^{72} -1.17888e12i q^{73} -6.59783e10 q^{74} +1.72277e12 q^{76} +2.62897e12i q^{77} +7.04180e10i q^{78} +1.07242e12 q^{79} +2.82430e11 q^{81} -5.61443e10i q^{82} +1.12403e12i q^{83} -1.37926e12 q^{84} -8.53802e10 q^{86} -9.13000e11i q^{87} -2.17933e12i q^{88} -2.23561e12 q^{89} +1.89237e12 q^{91} +6.68431e12i q^{92} -4.49017e12i q^{93} -3.54345e11 q^{94} +1.72011e12 q^{96} +1.42153e13i q^{97} +4.99472e11i q^{98} +5.94306e12 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16096 q^{4} + 17496 q^{6} - 1062882 q^{9} - 22365816 q^{11} + 5642112 q^{14} + 127181312 q^{16} + 428122760 q^{19} - 342758304 q^{21} + 284135040 q^{24} - 193190736 q^{26} + 2504800500 q^{29} + 12318701104 q^{31}+ \cdots + 11886111620856 q^{99}+O(q^{100}) $ 2 * q + 16096 * q^4 + 17496 * q^6 - 1062882 * q^9 - 22365816 * q^11 + 5642112 * q^14 + 127181312 * q^16 + 428122760 * q^19 - 342758304 * q^21 + 284135040 * q^24 - 193190736 * q^26 + 2504800500 * q^29 + 12318701104 * q^31 - 2819870928 * q^34 - 8554074336 * q^36 + 11736337212 * q^39 - 9357375756 * q^41 - 180000087168 * q^44 - 19933333056 * q^46 + 83245285326 * q^49 + 171307158876 * q^51 - 9298091736 * q^54 + 91627898880 * q^56 + 59819642040 * q^59 - 268784013476 * q^61 + 985240231936 * q^64 - 195656158368 * q^66 + 1210949983152 * q^69 + 2628670818384 * q^71 - 131956593648 * q^74 + 3445531972480 * q^76 + 2144841319280 * q^79 + 564859072962 * q^81 - 2758518830592 * q^84 - 170760330336 * q^86 - 4471221819060 * q^89 + 3784735312064 * q^91 - 708690647808 * q^94 + 3440216365056 * q^96 + 11886111620856 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

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 $

$2$	12.0000i	0.132583i	0.997800	+	0.0662913i	$0.0211166\pi$
$2$	12.0000i	0.132583i	−0.997800	+	0.0662913i	$0.978883\pi$
$3$	− 729.000i	− 0.577350i
$3$	− 729.000i	− 0.577350i
$4$	8048.00	0.982422
$5$	0	0
$5$	0	0
$6$	8748.00	0.0765466
$7$	− 235088.i	− 0.755254i	−0.925958	−	0.377627i	$-0.876740\pi$
$7$	− 235088.i	− 0.755254i	0.925958	−	0.377627i	$-0.123260\pi$
$8$	194880.i	0.262834i
$9$	−531441.	−0.333333
$10$	0	0
$11$	−1.11829e7	−1.90328	−0.951639	−	0.307218i	$-0.900602\pi$
$11$	−1.11829e7	−1.90328	−0.951639	+	0.307218i	$0.900602\pi$
$12$	− 5.86699e6i	− 0.567202i
$13$	8.04961e6i	0.462534i	0.972890	+	0.231267i	$0.0742870\pi$
$13$	8.04961e6i	0.462534i	−0.972890	+	0.231267i	$0.925713\pi$
$14$	2.82106e6	0.100134
$15$	0	0
$16$	6.35907e7	0.947575
$17$	1.17495e8i	1.18059i	0.807187	+	0.590296i	$0.200989\pi$
$17$	1.17495e8i	1.18059i	−0.807187	+	0.590296i	$0.799011\pi$
$18$	− 6.37729e6i	− 0.0441942i
$19$	2.14061e8	1.04385	0.521927	−	0.852990i	$-0.325213\pi$
$19$	2.14061e8	1.04385	0.521927	+	0.852990i	$0.325213\pi$
$20$	0	0
$21$	−1.71379e8	−0.436046
$22$	− 1.34195e8i	− 0.252341i
$23$	8.30556e8i	1.16987i	0.811080	+	0.584935i	$0.198880\pi$
$23$	8.30556e8i	1.16987i	−0.811080	+	0.584935i	$0.801120\pi$
$24$	1.42068e8	0.151748
$25$	0	0
$26$	−9.65954e7	−0.0613239
$27$	3.87420e8i	0.192450i
$28$	− 1.89199e9i	− 0.741978i
$29$	1.25240e9	0.390981	0.195491	−	0.980706i	$-0.437370\pi$
$29$	1.25240e9	0.390981	0.195491	+	0.980706i	$0.437370\pi$
$30$	0	0
$31$	6.15935e9	1.24648	0.623238	−	0.782032i	$-0.285817\pi$
$31$	6.15935e9	1.24648	0.623238	+	0.782032i	$0.285817\pi$
$32$	2.35954e9i	0.388466i
$33$	8.15234e9i	1.09886i
$34$	−1.40994e9	−0.156526
$35$	0	0
$36$	−4.27704e9	−0.327474
$37$	5.49819e9i	0.352297i	0.984364	+	0.176148i	$0.0563639\pi$
$37$	5.49819e9i	0.352297i	−0.984364	+	0.176148i	$0.943636\pi$
$38$	2.56874e9i	0.138397i
$39$	5.86817e9	0.267044
$40$	0	0
$41$	−4.67869e9	−0.153826	−0.0769129	−	0.997038i	$-0.524506\pi$
$41$	−4.67869e9	−0.153826	−0.0769129	+	0.997038i	$0.524506\pi$
$42$	− 2.05655e9i	− 0.0578121i
$43$	7.11501e9i	0.171645i	0.996310	+	0.0858224i	$0.0273518\pi$
$43$	7.11501e9i	0.171645i	−0.996310	+	0.0858224i	$0.972648\pi$
$44$	−9.00000e10	−1.86982
$45$	0	0
$46$	−9.96667e9	−0.155104
$47$	2.95288e10i	0.399585i	0.979838	+	0.199793i	$0.0640268\pi$
$47$	2.95288e10i	0.399585i	−0.979838	+	0.199793i	$0.935973\pi$
$48$	− 4.63576e10i	− 0.547082i
$49$	4.16226e10	0.429591
$50$	0	0
$51$	8.56536e10	0.681615
$52$	6.47833e10i	0.454403i
$53$	− 2.04125e11i	− 1.26504i	−0.774545	−	0.632518i	$-0.782021\pi$
$53$	− 2.04125e11i	− 1.26504i	0.774545	−	0.632518i	$-0.217979\pi$
$54$	−4.64905e9	−0.0255155
$55$	0	0
$56$	4.58139e10	0.198507
$57$	− 1.56051e11i	− 0.602670i
$58$	1.50288e10i	0.0518373i
$59$	2.99098e10	0.0923157	0.0461579	−	0.998934i	$-0.485302\pi$
$59$	2.99098e10	0.0923157	0.0461579	+	0.998934i	$0.485302\pi$
$60$	0	0
$61$	−1.34392e11	−0.333987	−0.166993	−	0.985958i	$-0.553406\pi$
$61$	−1.34392e11	−0.333987	−0.166993	+	0.985958i	$0.553406\pi$
$62$	7.39122e10i	0.165261i
$63$	1.24935e11i	0.251751i
$64$	4.92620e11	0.896071
$65$	0	0
$66$	−9.78281e10	−0.145689
$67$	− 3.48519e11i	− 0.470695i	−0.971911	−	0.235348i	$-0.924377\pi$
$67$	− 3.48519e11i	− 0.470695i	0.971911	−	0.235348i	$-0.0756229\pi$
$68$	9.45597e11i	1.15984i
$69$	6.05475e11	0.675425
$70$	0	0
$71$	1.31434e12	1.21766	0.608831	−	0.793300i	$-0.291639\pi$
$71$	1.31434e12	1.21766	0.608831	+	0.793300i	$0.291639\pi$
$72$	− 1.03567e11i	− 0.0876115i
$73$	− 1.17888e12i	− 0.911737i	−0.890047	−	0.455868i	$-0.849329\pi$
$73$	− 1.17888e12i	− 0.911737i	0.890047	−	0.455868i	$-0.150671\pi$
$74$	−6.59783e10	−0.0467084
$75$	0	0
$76$	1.72277e12	1.02551
$77$	2.62897e12i	1.43746i
$78$	7.04180e10i	0.0354054i
$79$	1.07242e12	0.496351	0.248176	−	0.968715i	$-0.420169\pi$
$79$	1.07242e12	0.496351	0.248176	+	0.968715i	$0.420169\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	− 5.61443e10i	− 0.0203946i
$83$	1.12403e12i	0.377371i	0.982038	+	0.188685i	$0.0604227\pi$
$83$	1.12403e12i	0.377371i	−0.982038	+	0.188685i	$0.939577\pi$
$84$	−1.37926e12	−0.428381
$85$	0	0
$86$	−8.53802e10	−0.0227571
$87$	− 9.13000e11i	− 0.225733i
$88$	− 2.17933e12i	− 0.500247i
$89$	−2.23561e12	−0.476827	−0.238414	−	0.971164i	$-0.576627\pi$
$89$	−2.23561e12	−0.476827	−0.238414	+	0.971164i	$0.576627\pi$
$90$	0	0
$91$	1.89237e12	0.349330
$92$	6.68431e12i	1.14931i
$93$	− 4.49017e12i	− 0.719653i
$94$	−3.54345e11	−0.0529780
$95$	0	0
$96$	1.72011e12	0.224281
$97$	1.42153e13i	1.73276i	0.499385	+	0.866380i	$0.333559\pi$
$97$	1.42153e13i	1.73276i	−0.499385	+	0.866380i	$0.666441\pi$
$98$	4.99472e11i	0.0569563i
$99$	5.94306e12	0.634426
$100$	0	0
$101$	1.70194e13	1.59535	0.797675	−	0.603088i	$-0.206063\pi$
$101$	1.70194e13	1.59535	0.797675	+	0.603088i	$0.206063\pi$
$102$	1.02784e12i	0.0903703i
$103$	1.09904e13i	0.906928i	0.891275	+	0.453464i	$0.149812\pi$
$103$	1.09904e13i	0.906928i	−0.891275	+	0.453464i	$0.850188\pi$
$104$	−1.56871e12	−0.121570
$105$	0	0
$106$	2.44950e12	0.167722
$107$	1.96403e13i	1.26519i	0.774485	+	0.632593i	$0.218009\pi$
$107$	1.96403e13i	1.26519i	−0.774485	+	0.632593i	$0.781991\pi$
$108$	3.11796e12i	0.189067i
$109$	9.82099e12	0.560897	0.280448	−	0.959869i	$-0.409517\pi$
$109$	9.82099e12	0.560897	0.280448	+	0.959869i	$0.409517\pi$
$110$	0	0
$111$	4.00818e12	0.203399
$112$	− 1.49494e13i	− 0.715660i
$113$	− 1.70267e13i	− 0.769344i	−0.923053	−	0.384672i	$-0.874315\pi$
$113$	− 1.70267e13i	− 0.769344i	0.923053	−	0.384672i	$-0.125685\pi$
$114$	1.87261e12	0.0799035
$115$	0	0
$116$	1.00793e13	0.384109
$117$	− 4.27789e12i	− 0.154178i
$118$	3.58918e11i	0.0122395i
$119$	2.76216e13	0.891648
$120$	0	0
$121$	9.05347e13	2.62247
$122$	− 1.61270e12i	− 0.0442808i
$123$	3.41076e12i	0.0888113i
$124$	4.95705e13	1.22457
$125$	0	0
$126$	−1.49922e12	−0.0333778
$127$	4.49347e13i	0.950292i	0.879907	+	0.475146i	$0.157605\pi$
$127$	4.49347e13i	0.950292i	−0.879907	+	0.475146i	$0.842395\pi$
$128$	2.52408e13i	0.507270i
$129$	5.18685e12	0.0990992
$130$	0	0
$131$	−1.20182e12	−0.0207768	−0.0103884	−	0.999946i	$-0.503307\pi$
$131$	−1.20182e12	−0.0207768	−0.0103884	+	0.999946i	$0.503307\pi$
$132$	6.56100e13i	1.07954i
$133$	− 5.03233e13i	− 0.788375i
$134$	4.18223e12	0.0624060
$135$	0	0
$136$	−2.28974e13	−0.310300
$137$	− 1.71562e13i	− 0.221685i	−0.993838	−	0.110842i	$-0.964645\pi$
$137$	− 1.71562e13i	− 0.221685i	0.993838	−	0.110842i	$-0.0353549\pi$
$138$	7.26570e12i	0.0895495i
$139$	−1.05644e14	−1.24236	−0.621182	−	0.783666i	$-0.713347\pi$
$139$	−1.05644e14	−1.24236	−0.621182	+	0.783666i	$0.713347\pi$
$140$	0	0
$141$	2.15265e13	0.230701
$142$	1.57720e13i	0.161441i
$143$	− 9.00181e13i	− 0.880330i
$144$	−3.37947e13	−0.315858
$145$	0	0
$146$	1.41465e13	0.120880
$147$	− 3.03429e13i	− 0.248024i
$148$	4.42494e13i	0.346104i
$149$	8.53533e13	0.639012	0.319506	−	0.947584i	$-0.396483\pi$
$149$	8.53533e13	0.639012	0.319506	+	0.947584i	$0.396483\pi$
$150$	0	0
$151$	−6.16414e13	−0.423177	−0.211589	−	0.977359i	$-0.567864\pi$
$151$	−6.16414e13	−0.423177	−0.211589	+	0.977359i	$0.567864\pi$
$152$	4.17163e13i	0.274361i
$153$	− 6.24415e13i	− 0.393531i
$154$	−3.15476e13	−0.190582
$155$	0	0
$156$	4.72270e13	0.262350
$157$	1.18021e14i	0.628942i	0.949267	+	0.314471i	$0.101827\pi$
$157$	1.18021e14i	0.628942i	−0.949267	+	0.314471i	$0.898173\pi$
$158$	1.28690e13i	0.0658075i
$159$	−1.48807e14	−0.730369
$160$	0	0
$161$	1.95254e14	0.883550
$162$	3.38915e12i	0.0147314i
$163$	1.54710e14i	0.646099i	0.946382	+	0.323050i	$0.104708\pi$
$163$	1.54710e14i	0.646099i	−0.946382	+	0.323050i	$0.895292\pi$
$164$	−3.76541e13	−0.151122
$165$	0	0
$166$	−1.34883e13	−0.0500328
$167$	− 3.76012e14i	− 1.34136i	−0.741748	−	0.670679i	$-0.766003\pi$
$167$	− 3.76012e14i	− 1.34136i	0.741748	−	0.670679i	$-0.233997\pi$
$168$	− 3.33984e13i	− 0.114608i
$169$	2.38079e14	0.786063
$170$	0	0
$171$	−1.13761e14	−0.347951
$172$	5.72616e13i	0.168628i
$173$	3.73562e14i	1.05941i	0.848182	+	0.529704i	$0.177697\pi$
$173$	3.73562e14i	1.05941i	−0.848182	+	0.529704i	$0.822303\pi$
$174$	1.09560e13	0.0299283
$175$	0	0
$176$	−7.11128e14	−1.80350
$177$	− 2.18043e13i	− 0.0532985i
$178$	− 2.68273e13i	− 0.0632190i
$179$	−4.23349e13	−0.0961952	−0.0480976	−	0.998843i	$-0.515316\pi$
$179$	−4.23349e13	−0.0961952	−0.0480976	+	0.998843i	$0.515316\pi$
$180$	0	0
$181$	−3.10447e14	−0.656261	−0.328130	−	0.944632i	$-0.606419\pi$
$181$	−3.10447e14	−0.656261	−0.328130	+	0.944632i	$0.606419\pi$
$182$	2.27084e13i	0.0463151i
$183$	9.79718e13i	0.192827i
$184$	−1.61859e14	−0.307482
$185$	0	0
$186$	5.38820e13	0.0954134
$187$	− 1.31393e15i	− 2.24700i
$188$	2.37648e14i	0.392561i
$189$	9.10779e13	0.145349
$190$	0	0
$191$	−8.62273e14	−1.28507	−0.642537	−	0.766255i	$-0.722118\pi$
$191$	−8.62273e14	−1.28507	−0.642537	+	0.766255i	$0.722118\pi$
$192$	− 3.59120e14i	− 0.517347i
$193$	− 9.37837e14i	− 1.30618i	−0.757278	−	0.653092i	$-0.773471\pi$
$193$	− 9.37837e14i	− 1.30618i	0.757278	−	0.653092i	$-0.226529\pi$
$194$	−1.70583e14	−0.229734
$195$	0	0
$196$	3.34979e14	0.422040
$197$	6.71715e14i	0.818756i	0.912365	+	0.409378i	$0.134254\pi$
$197$	6.71715e14i	0.818756i	−0.912365	+	0.409378i	$0.865746\pi$
$198$	7.13167e13i	0.0841138i
$199$	4.36451e13	0.0498185	0.0249093	−	0.999690i	$-0.492070\pi$
$199$	4.36451e13	0.0498185	0.0249093	+	0.999690i	$0.492070\pi$
$200$	0	0
$201$	−2.54070e14	−0.271756
$202$	2.04233e14i	0.211515i
$203$	− 2.94424e14i	− 0.295290i
$204$	6.89340e14	0.669634
$205$	0	0
$206$	−1.31885e14	−0.120243
$207$	− 4.41391e14i	− 0.389957i
$208$	5.11880e14i	0.438285i
$209$	−2.39383e15	−1.98675
$210$	0	0
$211$	−1.62162e15	−1.26507	−0.632534	−	0.774533i	$-0.717985\pi$
$211$	−1.62162e15	−1.26507	−0.632534	+	0.774533i	$0.717985\pi$
$212$	− 1.64280e15i	− 1.24280i
$213$	− 9.58151e14i	− 0.703018i
$214$	−2.35684e14	−0.167742
$215$	0	0
$216$	−7.55005e13	−0.0505825
$217$	− 1.44799e15i	− 0.941406i
$218$	1.17852e14i	0.0743651i
$219$	−8.59401e14	−0.526392
$220$	0	0
$221$	−9.45786e14	−0.546064
$222$	4.80982e13i	0.0269671i
$223$	1.47333e15i	0.802266i	0.916020	+	0.401133i	$0.131383\pi$
$223$	1.47333e15i	0.802266i	−0.916020	+	0.401133i	$0.868617\pi$
$224$	5.54701e14	0.293391
$225$	0	0
$226$	2.04320e14	0.102002
$227$	3.74889e15i	1.81859i	0.416153	+	0.909294i	$0.363378\pi$
$227$	3.74889e15i	1.81859i	−0.416153	+	0.909294i	$0.636622\pi$
$228$	− 1.25590e15i	− 0.592076i
$229$	1.47993e13	0.00678126	0.00339063	−	0.999994i	$-0.498921\pi$
$229$	1.47993e13	0.00678126	0.00339063	+	0.999994i	$0.498921\pi$
$230$	0	0
$231$	1.91652e15	0.829917
$232$	2.44068e14i	0.102763i
$233$	3.63053e15i	1.48647i	0.669030	+	0.743236i	$0.266710\pi$
$233$	3.63053e15i	1.48647i	−0.669030	+	0.743236i	$0.733290\pi$
$234$	5.13347e13	0.0204413
$235$	0	0
$236$	2.40714e14	0.0906930
$237$	− 7.81795e14i	− 0.286569i
$238$	3.31459e14i	0.118217i
$239$	4.33900e15	1.50592	0.752962	−	0.658063i	$-0.228624\pi$
$239$	4.33900e15	1.50592	0.752962	+	0.658063i	$0.228624\pi$
$240$	0	0
$241$	3.02372e15	0.994103	0.497051	−	0.867721i	$-0.334416\pi$
$241$	3.02372e15	0.994103	0.497051	+	0.867721i	$0.334416\pi$
$242$	1.08642e15i	0.347693i
$243$	− 2.05891e14i	− 0.0641500i
$244$	−1.08159e15	−0.328116
$245$	0	0
$246$	−4.09292e13	−0.0117748
$247$	1.72311e15i	0.482818i
$248$	1.20033e15i	0.327617i
$249$	8.19414e14	0.217875
$250$	0	0
$251$	−1.75146e15	−0.442099	−0.221050	−	0.975263i	$-0.570948\pi$
$251$	−1.75146e15	−0.442099	−0.221050	+	0.975263i	$0.570948\pi$
$252$	1.00548e15i	0.247326i
$253$	− 9.28803e15i	− 2.22659i
$254$	−5.39216e14	−0.125992
$255$	0	0
$256$	3.73265e15	0.828816
$257$	− 4.87604e15i	− 1.05561i	−0.849367	−	0.527803i	$-0.823016\pi$
$257$	− 4.87604e15i	− 1.05561i	0.849367	−	0.527803i	$-0.176984\pi$
$258$	6.22421e13i	0.0131388i
$259$	1.29256e15	0.266074
$260$	0	0
$261$	−6.65577e14	−0.130327
$262$	− 1.44219e13i	− 0.00275463i
$263$	4.67882e15i	0.871815i	0.899992	+	0.435907i	$0.143572\pi$
$263$	4.67882e15i	0.871815i	−0.899992	+	0.435907i	$0.856428\pi$
$264$	−1.58873e15	−0.288818
$265$	0	0
$266$	6.03879e14	0.104525
$267$	1.62976e15i	0.275296i
$268$	− 2.80488e15i	− 0.462422i
$269$	1.80262e15	0.290078	0.145039	−	0.989426i	$-0.453669\pi$
$269$	1.80262e15	0.290078	0.145039	+	0.989426i	$0.453669\pi$
$270$	0	0
$271$	6.10016e15	0.935494	0.467747	−	0.883862i	$-0.345066\pi$
$271$	6.10016e15	0.935494	0.467747	+	0.883862i	$0.345066\pi$
$272$	7.47156e15i	1.11870i
$273$	− 1.37954e15i	− 0.201686i
$274$	2.05874e14	0.0293915
$275$	0	0
$276$	4.87286e15	0.663552
$277$	1.07023e16i	1.42351i	0.702428	+	0.711754i	$0.252099\pi$
$277$	1.07023e16i	1.42351i	−0.702428	+	0.711754i	$0.747901\pi$
$278$	− 1.26773e15i	− 0.164716i
$279$	−3.27333e15	−0.415492
$280$	0	0
$281$	−2.45460e15	−0.297433	−0.148717	−	0.988880i	$-0.547514\pi$
$281$	−2.45460e15	−0.297433	−0.148717	+	0.988880i	$0.547514\pi$
$282$	2.58318e14i	0.0305869i
$283$	4.01155e15i	0.464195i	0.972692	+	0.232098i	$0.0745589\pi$
$283$	4.01155e15i	0.464195i	−0.972692	+	0.232098i	$0.925441\pi$
$284$	1.05778e16	1.19626
$285$	0	0
$286$	1.08022e15	0.116716
$287$	1.09990e15i	0.116178i
$288$	− 1.25396e15i	− 0.129489i
$289$	−3.90041e15	−0.393799
$290$	0	0
$291$	1.03629e16	1.00041
$292$	− 9.48759e15i	− 0.895710i
$293$	2.08187e15i	0.192227i	0.995370	+	0.0961133i	$0.0306411\pi$
$293$	2.08187e15i	0.192227i	−0.995370	+	0.0961133i	$0.969359\pi$
$294$	3.64115e14	0.0328837
$295$	0	0
$296$	−1.07149e15	−0.0925957
$297$	− 4.33249e15i	− 0.366286i
$298$	1.02424e15i	0.0847219i
$299$	−6.68565e15	−0.541104
$300$	0	0
$301$	1.67265e15	0.129636
$302$	− 7.39697e14i	− 0.0561059i
$303$	− 1.24072e16i	− 0.921075i
$304$	1.36123e16	0.989130
$305$	0	0
$306$	7.49298e14	0.0521753
$307$	1.32352e16i	0.902260i	0.892458	+	0.451130i	$0.148979\pi$
$307$	1.32352e16i	0.902260i	−0.892458	+	0.451130i	$0.851021\pi$
$308$	2.11579e16i	1.41219i
$309$	8.01202e15	0.523615
$310$	0	0
$311$	−8.09301e15	−0.507187	−0.253593	−	0.967311i	$-0.581612\pi$
$311$	−8.09301e15	−0.507187	−0.253593	+	0.967311i	$0.581612\pi$
$312$	1.14359e15i	0.0701883i
$313$	− 1.48181e16i	− 0.890748i	−0.895345	−	0.445374i	$-0.853071\pi$
$313$	− 1.48181e16i	− 0.890748i	0.895345	−	0.445374i	$-0.146929\pi$
$314$	−1.41625e15	−0.0833868
$315$	0	0
$316$	8.63084e15	0.487626
$317$	2.43171e16i	1.34594i	0.739670	+	0.672970i	$0.234982\pi$
$317$	2.43171e16i	1.34594i	−0.739670	+	0.672970i	$0.765018\pi$
$318$	− 1.78569e15i	− 0.0968342i
$319$	−1.40055e16	−0.744147
$320$	0	0
$321$	1.43178e16	0.730455
$322$	2.34304e15i	0.117143i
$323$	2.51511e16i	1.23237i
$324$	2.27299e15	0.109158
$325$	0	0
$326$	−1.85652e15	−0.0856615
$327$	− 7.15950e15i	− 0.323834i
$328$	− 9.11783e14i	− 0.0404307i
$329$	6.94186e15	0.301788
$330$	0	0
$331$	1.16232e16	0.485783	0.242892	−	0.970053i	$-0.421904\pi$
$331$	1.16232e16	0.485783	0.242892	+	0.970053i	$0.421904\pi$
$332$	9.04615e15i	0.370737i
$333$	− 2.92196e15i	− 0.117432i
$334$	4.51214e15	0.177841
$335$	0	0
$336$	−1.08981e16	−0.413186
$337$	− 4.62652e16i	− 1.72052i	−0.509853	−	0.860262i	$-0.670300\pi$
$337$	− 4.62652e16i	− 1.72052i	0.509853	−	0.860262i	$-0.329700\pi$
$338$	2.85695e15i	0.104218i
$339$	−1.24125e16	−0.444181
$340$	0	0
$341$	−6.88795e16	−2.37239
$342$	− 1.36513e15i	− 0.0461323i
$343$	− 3.25624e16i	− 1.07970i
$344$	−1.38657e15	−0.0451142
$345$	0	0
$346$	−4.48275e15	−0.140459
$347$	− 4.79404e15i	− 0.147421i	−0.997280	−	0.0737106i	$-0.976516\pi$
$347$	− 4.79404e15i	− 0.147421i	0.997280	−	0.0737106i	$-0.0234841\pi$
$348$	− 7.34782e15i	− 0.221765i
$349$	−3.76900e16	−1.11651	−0.558253	−	0.829671i	$-0.688528\pi$
$349$	−3.76900e16	−1.11651	−0.558253	+	0.829671i	$0.688528\pi$
$350$	0	0
$351$	−3.11859e15	−0.0890146
$352$	− 2.63866e16i	− 0.739360i
$353$	4.80179e16i	1.32089i	0.750873	+	0.660446i	$0.229633\pi$
$353$	4.80179e16i	1.32089i	−0.750873	+	0.660446i	$0.770367\pi$
$354$	2.61651e14	0.00706645
$355$	0	0
$356$	−1.79922e16	−0.468446
$357$	− 2.01361e16i	− 0.514793i
$358$	− 5.08018e14i	− 0.0127538i
$359$	−4.06616e16	−1.00247	−0.501234	−	0.865312i	$-0.667120\pi$
$359$	−4.06616e16	−1.00247	−0.501234	+	0.865312i	$0.667120\pi$
$360$	0	0
$361$	3.76929e15	0.0896320
$362$	− 3.72536e15i	− 0.0870087i
$363$	− 6.59998e16i	− 1.51408i
$364$	1.52298e16	0.343190
$365$	0	0
$366$	−1.17566e15	−0.0255656
$367$	− 2.96733e16i	− 0.633923i	−0.948438	−	0.316961i	$-0.897337\pi$
$367$	− 2.96733e16i	− 0.633923i	0.948438	−	0.316961i	$-0.102663\pi$
$368$	5.28156e16i	1.10854i
$369$	2.48645e15	0.0512752
$370$	0	0
$371$	−4.79873e16	−0.955424
$372$	− 3.61369e16i	− 0.707003i
$373$	− 9.01346e16i	− 1.73294i	−0.499227	−	0.866471i	$-0.666383\pi$
$373$	− 9.01346e16i	− 1.73294i	0.499227	−	0.866471i	$-0.333617\pi$
$374$	1.57672e16	0.297912
$375$	0	0
$376$	−5.75457e15	−0.105025
$377$	1.00813e16i	0.180842i
$378$	1.09293e15i	0.0192707i
$379$	1.54841e16	0.268369	0.134184	−	0.990956i	$-0.457159\pi$
$379$	1.54841e16	0.268369	0.134184	+	0.990956i	$0.457159\pi$
$380$	0	0
$381$	3.27574e16	0.548652
$382$	− 1.03473e16i	− 0.170378i
$383$	9.37088e15i	0.151701i	0.997119	+	0.0758505i	$0.0241672\pi$
$383$	9.37088e15i	0.151701i	−0.997119	+	0.0758505i	$0.975833\pi$
$384$	1.84006e16	0.292872
$385$	0	0
$386$	1.12540e16	0.173177
$387$	− 3.78121e15i	− 0.0572150i
$388$	1.14404e17i	1.70230i
$389$	−2.95806e16	−0.432847	−0.216423	−	0.976300i	$-0.569439\pi$
$389$	−2.95806e16	−0.432847	−0.216423	+	0.976300i	$0.569439\pi$
$390$	0	0
$391$	−9.75858e16	−1.38114
$392$	8.11142e15i	0.112911i
$393$	8.76130e14i	0.0119955i
$394$	−8.06058e15	−0.108553
$395$	0	0
$396$	4.78297e16	0.623274
$397$	− 1.80617e16i	− 0.231538i	−0.993276	−	0.115769i	$-0.963067\pi$
$397$	− 1.80617e16i	− 0.231538i	0.993276	−	0.115769i	$-0.0369332\pi$
$398$	5.23742e14i	0.00660507i
$399$	−3.66857e16	−0.455169
$400$	0	0
$401$	−1.20412e17	−1.44621	−0.723107	−	0.690736i	$-0.757287\pi$
$401$	−1.20412e17	−1.44621	−0.723107	+	0.690736i	$0.757287\pi$
$402$	− 3.04884e15i	− 0.0360301i
$403$	4.95804e16i	0.576537i
$404$	1.36972e17	1.56731
$405$	0	0
$406$	3.53309e15	0.0391503
$407$	− 6.14858e16i	− 0.670519i
$408$	1.66922e16i	0.179152i
$409$	1.77522e16	0.187521	0.0937606	−	0.995595i	$-0.470111\pi$
$409$	1.77522e16	0.187521	0.0937606	+	0.995595i	$0.470111\pi$
$410$	0	0
$411$	−1.25068e16	−0.127990
$412$	8.84510e16i	0.890986i
$413$	− 7.03144e15i	− 0.0697219i
$414$	5.29670e15	0.0517015
$415$	0	0
$416$	−1.89934e16	−0.179679
$417$	7.70145e16i	0.717279i
$418$	− 2.87259e16i	− 0.263408i
$419$	−1.75670e17	−1.58602	−0.793008	−	0.609212i	$-0.791486\pi$
$419$	−1.75670e17	−1.58602	−0.793008	+	0.609212i	$0.791486\pi$
$420$	0	0
$421$	1.84473e17	1.61473	0.807365	−	0.590052i	$-0.200893\pi$
$421$	1.84473e17	1.61473	0.807365	+	0.590052i	$0.200893\pi$
$422$	− 1.94595e16i	− 0.167726i
$423$	− 1.56928e16i	− 0.133195i
$424$	3.97799e16	0.332495
$425$	0	0
$426$	1.14978e16	0.0932079
$427$	3.15939e16i	0.252245i
$428$	1.58065e17i	1.24295i
$429$	−6.56232e16	−0.508259
$430$	0	0
$431$	8.05532e16	0.605314	0.302657	−	0.953100i	$-0.402126\pi$
$431$	8.05532e16	0.605314	0.302657	+	0.953100i	$0.402126\pi$
$432$	2.46363e16i	0.182361i
$433$	− 1.97092e17i	− 1.43714i	−0.695457	−	0.718568i	$-0.744798\pi$
$433$	− 1.97092e17i	− 1.43714i	0.695457	−	0.718568i	$-0.255202\pi$
$434$	1.73759e16	0.124814
$435$	0	0
$436$	7.90393e16	0.551037
$437$	1.77790e17i	1.22117i
$438$	− 1.03128e16i	− 0.0697903i
$439$	−9.89007e16	−0.659447	−0.329724	−	0.944078i	$-0.606956\pi$
$439$	−9.89007e16	−0.659447	−0.329724	+	0.944078i	$0.606956\pi$
$440$	0	0
$441$	−2.21200e16	−0.143197
$442$	− 1.13494e16i	− 0.0723985i
$443$	− 1.25104e17i	− 0.786404i	−0.919452	−	0.393202i	$-0.871367\pi$
$443$	− 1.25104e17i	− 0.786404i	0.919452	−	0.393202i	$-0.128633\pi$
$444$	3.22578e16	0.199823
$445$	0	0
$446$	−1.76800e16	−0.106366
$447$	− 6.22225e16i	− 0.368934i
$448$	− 1.15809e17i	− 0.676761i
$449$	1.80095e17	1.03729	0.518645	−	0.854990i	$-0.326437\pi$
$449$	1.80095e17	1.03729	0.518645	+	0.854990i	$0.326437\pi$
$450$	0	0
$451$	5.23213e16	0.292773
$452$	− 1.37031e17i	− 0.755820i
$453$	4.49366e16i	0.244322i
$454$	−4.49867e16	−0.241113
$455$	0	0
$456$	3.04112e16	0.158402
$457$	9.43597e16i	0.484542i	0.970209	+	0.242271i	$0.0778923\pi$
$457$	9.43597e16i	0.484542i	−0.970209	+	0.242271i	$0.922108\pi$
$458$	1.77592e14i	0 0.000899076i
$459$	−4.55198e16	−0.227205
$460$	0	0
$461$	8.00500e16	0.388423	0.194212	−	0.980960i	$-0.437785\pi$
$461$	8.00500e16	0.388423	0.194212	+	0.980960i	$0.437785\pi$
$462$	2.29982e16i	0.110033i
$463$	2.14174e17i	1.01039i	0.863004	+	0.505196i	$0.168580\pi$
$463$	2.14174e17i	1.01039i	−0.863004	+	0.505196i	$0.831420\pi$
$464$	7.96410e16	0.370484
$465$	0	0
$466$	−4.35663e16	−0.197080
$467$	1.80681e17i	0.806031i	0.915193	+	0.403015i	$0.132038\pi$
$467$	1.80681e17i	0.806031i	−0.915193	+	0.403015i	$0.867962\pi$
$468$	− 3.44285e16i	− 0.151468i
$469$	−8.19326e16	−0.355495
$470$	0	0
$471$	8.60372e16	0.363120
$472$	5.82883e15i	0.0242638i
$473$	− 7.95665e16i	− 0.326688i
$474$	9.38154e15	0.0379940
$475$	0	0
$476$	2.22298e17	0.875974
$477$	1.08480e17i	0.421679i
$478$	5.20680e16i	0.199659i
$479$	2.66712e17	1.00893	0.504466	−	0.863431i	$-0.331689\pi$
$479$	2.66712e17	1.00893	0.504466	+	0.863431i	$0.331689\pi$
$480$	0	0
$481$	−4.42583e16	−0.162949
$482$	3.62847e16i	0.131801i
$483$	− 1.42340e17i	− 0.510118i
$484$	7.28623e17	2.57637
$485$	0	0
$486$	2.47069e15	0.00850517
$487$	2.63552e17i	0.895216i	0.894230	+	0.447608i	$0.147724\pi$
$487$	2.63552e17i	0.895216i	−0.894230	+	0.447608i	$0.852276\pi$
$488$	− 2.61903e16i	− 0.0877833i
$489$	1.12784e17	0.373026
$490$	0	0
$491$	4.11733e17	1.32613	0.663065	−	0.748562i	$-0.269255\pi$
$491$	4.11733e17	1.32613	0.663065	+	0.748562i	$0.269255\pi$
$492$	2.74498e16i	0.0872502i
$493$	1.47150e17i	0.461590i
$494$	−2.06773e16	−0.0640132
$495$	0	0
$496$	3.91677e17	1.18113
$497$	− 3.08984e17i	− 0.919645i
$498$	9.83297e15i	0.0288864i
$499$	−3.99658e17	−1.15887	−0.579435	−	0.815018i	$-0.696727\pi$
$499$	−3.99658e17	−1.15887	−0.579435	+	0.815018i	$0.696727\pi$
$500$	0	0
$501$	−2.74113e17	−0.774433
$502$	− 2.10175e16i	− 0.0586146i
$503$	− 2.83581e17i	− 0.780702i	−0.920666	−	0.390351i	$-0.872354\pi$
$503$	− 2.83581e17i	− 0.780702i	0.920666	−	0.390351i	$-0.127646\pi$
$504$	−2.43474e16	−0.0661690
$505$	0	0
$506$	1.11456e17	0.295207
$507$	− 1.73559e17i	− 0.453834i
$508$	3.61634e17i	0.933588i
$509$	−6.40327e17	−1.63206	−0.816030	−	0.578009i	$-0.803830\pi$
$509$	−6.40327e17	−1.63206	−0.816030	+	0.578009i	$0.803830\pi$
$510$	0	0
$511$	−2.77140e17	−0.688593
$512$	2.51565e17i	0.617156i
$513$	8.29318e16i	0.200890i
$514$	5.85124e16	0.139955
$515$	0	0
$516$	4.17437e16	0.0973572
$517$	− 3.30218e17i	− 0.760522i
$518$	1.55107e16i	0.0352767i
$519$	2.72327e17	0.611650
$520$	0	0
$521$	−4.01348e17	−0.879175	−0.439588	−	0.898200i	$-0.644875\pi$
$521$	−4.01348e17	−0.879175	−0.439588	+	0.898200i	$0.644875\pi$
$522$	− 7.98692e15i	− 0.0172791i
$523$	− 5.05985e17i	− 1.08113i	−0.841303	−	0.540564i	$-0.818211\pi$
$523$	− 5.05985e17i	− 1.08113i	0.841303	−	0.540564i	$-0.181789\pi$
$524$	−9.67228e15	−0.0204115
$525$	0	0
$526$	−5.61459e16	−0.115587
$527$	7.23691e17i	1.47158i
$528$	5.18413e17i	1.04125i
$529$	−1.85786e17	−0.368597
$530$	0	0
$531$	−1.58953e16	−0.0307719
$532$	− 4.05002e17i	− 0.774517i
$533$	− 3.76616e16i	− 0.0711496i
$534$	−1.95571e16	−0.0364995
$535$	0	0
$536$	6.79193e16	0.123715
$537$	3.08621e16i	0.0555383i
$538$	2.16314e16i	0.0384592i
$539$	−4.65462e17	−0.817631
$540$	0	0
$541$	−1.69124e17	−0.290017	−0.145009	−	0.989430i	$-0.546321\pi$
$541$	−1.69124e17	−0.290017	−0.145009	+	0.989430i	$0.546321\pi$
$542$	7.32020e16i	0.124030i
$543$	2.26316e17i	0.378892i
$544$	−2.77234e17	−0.458620
$545$	0	0
$546$	1.65544e16	0.0267400
$547$	4.32104e17i	0.689717i	0.938655	+	0.344858i	$0.112073\pi$
$547$	4.32104e17i	0.689717i	−0.938655	+	0.344858i	$0.887927\pi$
$548$	− 1.38073e17i	− 0.217788i
$549$	7.14214e16	0.111329
$550$	0	0
$551$	2.68091e17	0.408128
$552$	1.17995e17i	0.177525i
$553$	− 2.52113e17i	− 0.374871i
$554$	−1.28428e17	−0.188732
$555$	0	0
$556$	−8.50224e17	−1.22053
$557$	− 1.36804e18i	− 1.94107i	−0.240966	−	0.970534i	$-0.577464\pi$
$557$	− 1.36804e18i	− 1.94107i	0.240966	−	0.970534i	$-0.422536\pi$
$558$	− 3.92800e16i	− 0.0550870i
$559$	−5.72731e16	−0.0793915
$560$	0	0
$561$	−9.57856e17	−1.29730
$562$	− 2.94552e16i	− 0.0394345i
$563$	− 9.52405e17i	− 1.26043i	−0.776422	−	0.630213i	$-0.782968\pi$
$563$	− 9.52405e17i	− 1.26043i	0.776422	−	0.630213i	$-0.217032\pi$
$564$	1.73245e17	0.226645
$565$	0	0
$566$	−4.81386e16	−0.0615442
$567$	− 6.63958e16i	− 0.0839171i
$568$	2.56138e17i	0.320044i
$569$	−1.53632e17	−0.189780	−0.0948902	−	0.995488i	$-0.530250\pi$
$569$	−1.53632e17	−0.189780	−0.0948902	+	0.995488i	$0.530250\pi$
$570$	0	0
$571$	−1.27956e18	−1.54500	−0.772498	−	0.635017i	$-0.780993\pi$
$571$	−1.27956e18	−1.54500	−0.772498	+	0.635017i	$0.780993\pi$
$572$	− 7.24466e17i	− 0.864856i
$573$	6.28597e17i	0.741938i
$574$	−1.31988e16	−0.0154031
$575$	0	0
$576$	−2.61799e17	−0.298690
$577$	− 3.56770e17i	− 0.402481i	−0.979542	−	0.201241i	$-0.935503\pi$
$577$	− 3.56770e17i	− 0.402481i	0.979542	−	0.201241i	$-0.0644973\pi$
$578$	− 4.68049e16i	− 0.0522108i
$579$	−6.83683e17	−0.754126
$580$	0	0
$581$	2.64245e17	0.285011
$582$	1.24355e17i	0.132637i
$583$	2.28271e18i	2.40772i
$584$	2.29739e17	0.239636
$585$	0	0
$586$	−2.49824e16	−0.0254859
$587$	− 1.28968e18i	− 1.30118i	−0.759431	−	0.650588i	$-0.774523\pi$
$587$	− 1.28968e18i	− 1.30118i	0.759431	−	0.650588i	$-0.225477\pi$
$588$	− 2.44200e17i	− 0.243665i
$589$	1.31848e18	1.30114
$590$	0	0
$591$	4.89680e17	0.472709
$592$	3.49634e17i	0.333827i
$593$	− 1.88640e18i	− 1.78147i	−0.454523	−	0.890735i	$-0.650190\pi$
$593$	− 1.88640e18i	− 1.78147i	0.454523	−	0.890735i	$-0.349810\pi$
$594$	5.19899e16	0.0485631
$595$	0	0
$596$	6.86923e17	0.627780
$597$	− 3.18173e16i	− 0.0287627i
$598$	− 8.02278e16i	− 0.0717410i
$599$	1.44668e18	1.27967	0.639834	−	0.768513i	$-0.279003\pi$
$599$	1.44668e18	1.27967	0.639834	+	0.768513i	$0.279003\pi$
$600$	0	0
$601$	−4.44358e16	−0.0384635	−0.0192317	−	0.999815i	$-0.506122\pi$
$601$	−4.44358e16	−0.0384635	−0.0192317	+	0.999815i	$0.506122\pi$
$602$	2.00719e16i	0.0171874i
$603$	1.85217e17i	0.156898i
$604$	−4.96090e17	−0.415739
$605$	0	0
$606$	1.48886e17	0.122118
$607$	− 2.98050e16i	− 0.0241860i	−0.999927	−	0.0120930i	$-0.996151\pi$
$607$	− 2.98050e16i	− 0.0241860i	0.999927	−	0.0120930i	$-0.00384941\pi$
$608$	5.05087e17i	0.405502i
$609$	−2.14635e17	−0.170486
$610$	0	0
$611$	−2.37695e17	−0.184822
$612$	− 5.02529e17i	− 0.386613i
$613$	− 8.84082e17i	− 0.672976i	−0.941688	−	0.336488i	$-0.890761\pi$
$613$	− 8.84082e17i	− 0.672976i	0.941688	−	0.336488i	$-0.109239\pi$
$614$	−1.58823e17	−0.119624
$615$	0	0
$616$	−5.12333e17	−0.377814
$617$	− 1.43684e18i	− 1.04846i	−0.851575	−	0.524232i	$-0.824352\pi$
$617$	− 1.43684e18i	− 1.04846i	0.851575	−	0.524232i	$-0.175648\pi$
$618$	9.61443e16i	0.0694222i
$619$	−1.68862e18	−1.20654	−0.603272	−	0.797535i	$-0.706137\pi$
$619$	−1.68862e18	−1.20654	−0.603272	+	0.797535i	$0.706137\pi$
$620$	0	0
$621$	−3.21774e17	−0.225142
$622$	− 9.71162e16i	− 0.0672441i
$623$	5.25565e17i	0.360126i
$624$	3.73161e17	0.253044
$625$	0	0
$626$	1.77817e17	0.118098
$627$	1.74510e18i	1.14705i
$628$	9.49831e17i	0.617887i
$629$	−6.46008e17	−0.415919
$630$	0	0
$631$	−3.53490e17	−0.222939	−0.111470	−	0.993768i	$-0.535556\pi$
$631$	−3.53490e17	−0.222939	−0.111470	+	0.993768i	$0.535556\pi$
$632$	2.08993e17i	0.130458i
$633$	1.18216e18i	0.730387i
$634$	−2.91805e17	−0.178448
$635$	0	0
$636$	−1.19760e18	−0.717531
$637$	3.35046e17i	0.198700i
$638$	− 1.68066e17i	− 0.0986608i
$639$	−6.98492e17	−0.405888
$640$	0	0
$641$	1.61802e18	0.921313	0.460656	−	0.887579i	$-0.347614\pi$
$641$	1.61802e18	0.921313	0.460656	+	0.887579i	$0.347614\pi$
$642$	1.71814e17i	0.0968456i
$643$	− 1.96065e18i	− 1.09403i	−0.837123	−	0.547015i	$-0.815764\pi$
$643$	− 1.96065e18i	− 1.09403i	0.837123	−	0.547015i	$-0.184236\pi$
$644$	1.57140e18	0.868018
$645$	0	0
$646$	−3.01813e17	−0.163390
$647$	5.96114e17i	0.319486i	0.987159	+	0.159743i	$0.0510665\pi$
$647$	5.96114e17i	0.319486i	−0.987159	+	0.159743i	$0.948933\pi$
$648$	5.50399e16i	0.0292038i
$649$	−3.34479e17	−0.175703
$650$	0	0
$651$	−1.05558e18	−0.543521
$652$	1.24511e18i	0.634742i
$653$	− 2.58318e18i	− 1.30382i	−0.758295	−	0.651912i	$-0.773967\pi$
$653$	− 2.58318e18i	− 1.30382i	0.758295	−	0.651912i	$-0.226033\pi$
$654$	8.59140e16	0.0429347
$655$	0	0
$656$	−2.97521e17	−0.145761
$657$	6.26503e17i	0.303912i
$658$	8.33023e16i	0.0400119i
$659$	−2.64137e18	−1.25624	−0.628121	−	0.778116i	$-0.716176\pi$
$659$	−2.64137e18	−1.25624	−0.628121	+	0.778116i	$0.716176\pi$
$660$	0	0
$661$	4.12451e18	1.92337	0.961685	−	0.274156i	$-0.0883983\pi$
$661$	4.12451e18	1.92337	0.961685	+	0.274156i	$0.0883983\pi$
$662$	1.39478e17i	0.0644064i
$663$	6.89478e17i	0.315270i
$664$	−2.19050e17	−0.0991861
$665$	0	0
$666$	3.50636e16	0.0155695
$667$	1.04019e18i	0.457398i
$668$	− 3.02614e18i	− 1.31778i
$669$	1.07406e18	0.463189
$670$	0	0
$671$	1.50289e18	0.635670
$672$	− 4.04377e17i	− 0.169389i
$673$	2.79726e18i	1.16047i	0.814449	+	0.580236i	$0.197039\pi$
$673$	2.79726e18i	1.16047i	−0.814449	+	0.580236i	$0.802961\pi$
$674$	5.55183e17	0.228111
$675$	0	0
$676$	1.91606e18	0.772245
$677$	4.25553e18i	1.69874i	0.527795	+	0.849372i	$0.323019\pi$
$677$	4.25553e18i	1.69874i	−0.527795	+	0.849372i	$0.676981\pi$
$678$	− 1.48950e17i	− 0.0588906i
$679$	3.34184e18	1.30867
$680$	0	0
$681$	2.73294e18	1.04996
$682$	− 8.26553e17i	− 0.314538i
$683$	1.60893e18i	0.606461i	0.952917	+	0.303230i	$0.0980652\pi$
$683$	1.60893e18i	0.606461i	−0.952917	+	0.303230i	$0.901935\pi$
$684$	−9.15548e17	−0.341835
$685$	0	0
$686$	3.90749e17	0.143150
$687$	− 1.07887e16i	− 0.00391516i
$688$	4.52448e17i	0.162646i
$689$	1.64313e18	0.585122
$690$	0	0
$691$	−3.06331e18	−1.07049	−0.535247	−	0.844696i	$-0.679782\pi$
$691$	−3.06331e18	−1.07049	−0.535247	+	0.844696i	$0.679782\pi$
$692$	3.00643e18i	1.04079i
$693$	− 1.39714e18i	− 0.479153i
$694$	5.75285e16	0.0195455
$695$	0	0
$696$	1.77925e17	0.0593305
$697$	− 5.49721e17i	− 0.181606i
$698$	− 4.52280e17i	− 0.148029i
$699$	2.64666e18	0.858214
$700$	0	0
$701$	2.99144e18	0.952166	0.476083	−	0.879400i	$-0.342056\pi$
$701$	2.99144e18	0.952166	0.476083	+	0.879400i	$0.342056\pi$
$702$	− 3.74230e16i	− 0.0118018i
$703$	1.17695e18i	0.367747i
$704$	−5.50893e18	−1.70547
$705$	0	0
$706$	−5.76215e17	−0.175127
$707$	− 4.00106e18i	− 1.20489i
$708$	− 1.75481e17i	− 0.0523616i
$709$	5.31694e18	1.57203	0.786015	−	0.618207i	$-0.212141\pi$
$709$	5.31694e18	1.57203	0.786015	+	0.618207i	$0.212141\pi$
$710$	0	0
$711$	−5.69928e17	−0.165450
$712$	− 4.35676e17i	− 0.125327i
$713$	5.11568e18i	1.45822i
$714$	2.41634e17	0.0682525
$715$	0	0
$716$	−3.40711e17	−0.0945043
$717$	− 3.16313e18i	− 0.869446i
$718$	− 4.87939e17i	− 0.132910i
$719$	4.03153e18	1.08826	0.544129	−	0.839001i	$-0.316860\pi$
$719$	4.03153e18	1.08826	0.544129	+	0.839001i	$0.316860\pi$
$720$	0	0
$721$	2.58372e18	0.684961
$722$	4.52315e16i	0.0118836i
$723$	− 2.20429e18i	− 0.573945i
$724$	−2.49848e18	−0.644725
$725$	0	0
$726$	7.91998e17	0.200741
$727$	4.77643e18i	1.19986i	0.800054	+	0.599928i	$0.204804\pi$
$727$	4.77643e18i	1.19986i	−0.800054	+	0.599928i	$0.795196\pi$
$728$	3.68785e17i	0.0918161i
$729$	−1.50095e17	−0.0370370
$730$	0	0
$731$	−8.35976e17	−0.202643
$732$	7.88477e17i	0.189438i
$733$	1.71668e18i	0.408803i	0.978887	+	0.204401i	$0.0655248\pi$
$733$	1.71668e18i	0.408803i	−0.978887	+	0.204401i	$0.934475\pi$
$734$	3.56080e17	0.0840471
$735$	0	0
$736$	−1.95973e18	−0.454455
$737$	3.89745e18i	0.895864i
$738$	2.98374e16i	0.00679820i
$739$	8.69723e17	0.196423	0.0982114	−	0.995166i	$-0.468688\pi$
$739$	8.69723e17	0.196423	0.0982114	+	0.995166i	$0.468688\pi$
$740$	0	0
$741$	1.25615e18	0.278755
$742$	− 5.75848e17i	− 0.126673i
$743$	2.40272e18i	0.523933i	0.965077	+	0.261966i	$0.0843710\pi$
$743$	2.40272e18i	0.523933i	−0.965077	+	0.261966i	$0.915629\pi$
$744$	8.75044e17	0.189150
$745$	0	0
$746$	1.08161e18	0.229758
$747$	− 5.97353e17i	− 0.125790i
$748$	− 1.05745e19i	− 2.20750i
$749$	4.61721e18	0.955537
$750$	0	0
$751$	9.37175e18	1.90617	0.953084	−	0.302706i	$-0.0978899\pi$
$751$	9.37175e18	1.90617	0.953084	+	0.302706i	$0.0978899\pi$
$752$	1.87775e18i	0.378637i
$753$	1.27681e18i	0.255246i
$754$	−1.20976e17	−0.0239765
$755$	0	0
$756$	7.32995e17	0.142794
$757$	− 3.09120e18i	− 0.597040i	−0.954403	−	0.298520i	$-0.903507\pi$
$757$	− 3.09120e18i	− 0.597040i	0.954403	−	0.298520i	$-0.0964930\pi$
$758$	1.85810e17i	0.0355810i
$759$	−6.77097e18	−1.28552
$760$	0	0
$761$	−7.97787e18	−1.48897	−0.744486	−	0.667638i	$-0.767306\pi$
$761$	−7.97787e18	−1.48897	−0.744486	+	0.667638i	$0.767306\pi$
$762$	3.93088e17i	0.0727416i
$763$	− 2.30880e18i	− 0.423620i
$764$	−6.93957e18	−1.26248
$765$	0	0
$766$	−1.12451e17	−0.0201129
$767$	2.40763e17i	0.0426991i
$768$	− 2.72110e18i	− 0.478517i
$769$	−7.37344e18	−1.28573	−0.642863	−	0.765981i	$-0.722254\pi$
$769$	−7.37344e18	−1.28573	−0.642863	+	0.765981i	$0.722254\pi$
$770$	0	0
$771$	−3.55463e18	−0.609454
$772$	− 7.54771e18i	− 1.28322i
$773$	− 1.67335e18i	− 0.282111i	−0.990002	−	0.141056i	$-0.954950\pi$
$773$	− 1.67335e18i	− 0.282111i	0.990002	−	0.141056i	$-0.0450497\pi$
$774$	4.53745e16	0.00758570
$775$	0	0
$776$	−2.77027e18	−0.455429
$777$	− 9.42275e17i	− 0.153618i
$778$	− 3.54967e17i	− 0.0573879i
$779$	−1.00153e18	−0.160572
$780$	0	0
$781$	−1.46981e19	−2.31755
$782$	− 1.17103e18i	− 0.183115i
$783$	4.85206e17i	0.0752444i
$784$	2.64681e18	0.407069
$785$	0	0
$786$	−1.05136e16	−0.00159039
$787$	3.75359e18i	0.563133i	0.959542	+	0.281566i	$0.0908540\pi$
$787$	3.75359e18i	0.563133i	−0.959542	+	0.281566i	$0.909146\pi$
$788$	5.40596e18i	0.804363i
$789$	3.41086e18	0.503342
$790$	0	0
$791$	−4.00277e18	−0.581050
$792$	1.15818e18i	0.166749i
$793$	− 1.08180e18i	− 0.154480i
$794$	2.16741e17	0.0306978
$795$	0	0
$796$	3.51256e17	0.0489428
$797$	− 3.38853e18i	− 0.468309i	−0.972199	−	0.234154i	$-0.924768\pi$
$797$	− 3.38853e18i	− 0.468309i	0.972199	−	0.234154i	$-0.0752321\pi$
$798$	− 4.40228e17i	− 0.0603474i
$799$	−3.46947e18	−0.471747
$800$	0	0
$801$	1.18810e18	0.158942
$802$	− 1.44495e18i	− 0.191743i
$803$	1.31833e19i	1.73529i
$804$	−2.04476e18	−0.266979
$805$	0	0
$806$	−5.94965e17	−0.0764387
$807$	− 1.31411e18i	− 0.167476i
$808$	3.31674e18i	0.419313i
$809$	3.13119e18	0.392685	0.196343	−	0.980535i	$-0.437094\pi$
$809$	3.13119e18	0.392685	0.196343	+	0.980535i	$0.437094\pi$
$810$	0	0
$811$	1.04731e19	1.29253	0.646264	−	0.763114i	$-0.276331\pi$
$811$	1.04731e19	1.29253	0.646264	+	0.763114i	$0.276331\pi$
$812$	− 2.36953e18i	− 0.290100i
$813$	− 4.44702e18i	− 0.540108i
$814$	7.37829e17	0.0888991
$815$	0	0
$816$	5.44677e18	0.645881
$817$	1.52305e18i	0.179172i
$818$	2.13026e17i	0.0248620i
$819$	−1.00568e18	−0.116443
$820$	0	0
$821$	−6.85162e18	−0.780841	−0.390421	−	0.920637i	$-0.627670\pi$
$821$	−6.85162e18	−0.780841	−0.390421	+	0.920637i	$0.627670\pi$
$822$	− 1.50082e17i	− 0.0169692i
$823$	3.06934e17i	0.0344308i	0.999852	+	0.0172154i	$0.00548010\pi$
$823$	3.06934e17i	0.0344308i	−0.999852	+	0.0172154i	$0.994520\pi$
$824$	−2.14181e18	−0.238372
$825$	0	0
$826$	8.43773e16	0.00924390
$827$	− 7.75365e18i	− 0.842792i	−0.906877	−	0.421396i	$-0.861540\pi$
$827$	− 7.75365e18i	− 0.842792i	0.906877	−	0.421396i	$-0.138460\pi$
$828$	− 3.55232e18i	− 0.383102i
$829$	−2.34336e18	−0.250747	−0.125373	−	0.992110i	$-0.540013\pi$
$829$	−2.34336e18	−0.250747	−0.125373	+	0.992110i	$0.540013\pi$
$830$	0	0
$831$	7.80200e18	0.821863
$832$	3.96540e18i	0.414463i
$833$	4.89044e18i	0.507172i
$834$	−9.24174e17	−0.0950987
$835$	0	0
$836$	−1.92655e19	−1.95182
$837$	2.38626e18i	0.239884i
$838$	− 2.10804e18i	− 0.210278i
$839$	1.63297e19	1.61632	0.808158	−	0.588965i	$-0.200465\pi$
$839$	1.63297e19	1.61632	0.808158	+	0.588965i	$0.200465\pi$
$840$	0	0
$841$	−8.69212e18	−0.847134
$842$	2.21368e18i	0.214085i
$843$	1.78940e18i	0.171723i
$844$	−1.30508e19	−1.24283
$845$	0	0
$846$	1.88314e17	0.0176593
$847$	− 2.12836e19i	− 1.98063i
$848$	− 1.29804e19i	− 1.19872i
$849$	2.92442e18	0.268003
$850$	0	0
$851$	−4.56655e18	−0.412142
$852$	− 7.71120e18i	− 0.690660i
$853$	1.93794e19i	1.72255i	0.508137	+	0.861276i	$0.330334\pi$
$853$	1.93794e19i	1.72255i	−0.508137	+	0.861276i	$0.669666\pi$
$854$	−3.79127e17	−0.0334433
$855$	0	0
$856$	−3.82751e18	−0.332534
$857$	− 1.20537e19i	− 1.03931i	−0.854376	−	0.519656i	$-0.826060\pi$
$857$	− 1.20537e19i	− 1.03931i	0.854376	−	0.519656i	$-0.173940\pi$
$858$	− 7.87478e17i	− 0.0673862i
$859$	1.00612e19	0.854465	0.427232	−	0.904142i	$-0.359489\pi$
$859$	1.00612e19	0.854465	0.427232	+	0.904142i	$0.359489\pi$
$860$	0	0
$861$	8.01830e17	0.0670751
$862$	9.66639e17i	0.0802541i
$863$	8.01407e18i	0.660363i	0.943917	+	0.330182i	$0.107110\pi$
$863$	8.01407e18i	0.660363i	−0.943917	+	0.330182i	$0.892890\pi$
$864$	−9.14136e17	−0.0747604
$865$	0	0
$866$	2.36510e18	0.190539
$867$	2.84340e18i	0.227360i
$868$	− 1.16534e19i	− 0.924858i
$869$	−1.19928e19	−0.944694
$870$	0	0
$871$	2.80544e18	0.217712
$872$	1.91391e18i	0.147423i
$873$	− 7.55457e18i	− 0.577587i
$874$	−2.13348e18	−0.161906
$875$	0	0
$876$	−6.91646e18	−0.517139
$877$	− 8.87791e17i	− 0.0658891i	−0.999457	−	0.0329445i	$-0.989512\pi$
$877$	− 8.87791e17i	− 0.0658891i	0.999457	−	0.0329445i	$-0.0104885\pi$
$878$	− 1.18681e18i	− 0.0874312i
$879$	1.51768e18	0.110982
$880$	0	0
$881$	3.78774e18	0.272921	0.136460	−	0.990646i	$-0.456427\pi$
$881$	3.78774e18	0.272921	0.136460	+	0.990646i	$0.456427\pi$
$882$	− 2.65440e17i	− 0.0189854i
$883$	− 2.75428e19i	− 1.95553i	−0.209712	−	0.977763i	$-0.567253\pi$
$883$	− 2.75428e19i	− 1.95553i	0.209712	−	0.977763i	$-0.432747\pi$
$884$	−7.61169e18	−0.536465
$885$	0	0
$886$	1.50124e18	0.104263
$887$	2.84165e19i	1.95914i	0.201092	+	0.979572i	$0.435551\pi$
$887$	2.84165e19i	1.95914i	−0.201092	+	0.979572i	$0.564449\pi$
$888$	7.81114e17i	0.0534602i
$889$	1.05636e19	0.717712
$890$	0	0
$891$	−3.15838e18	−0.211475
$892$	1.18574e19i	0.788164i
$893$	6.32097e18i	0.417109i
$894$	7.46670e17	0.0489142
$895$	0	0
$896$	5.93382e18	0.383118
$897$	4.87384e18i	0.312407i
$898$	2.16114e18i	0.137527i
$899$	7.71397e18	0.487349
$900$	0	0
$901$	2.39836e19	1.49349
$902$	6.27856e17i	0.0388166i
$903$	− 1.21937e18i	− 0.0748451i
$904$	3.31816e18	0.202210
$905$	0	0
$906$	−5.39239e17	−0.0323928
$907$	− 7.51023e18i	− 0.447926i	−0.974598	−	0.223963i	$-0.928101\pi$
$907$	− 7.51023e18i	− 0.447926i	0.974598	−	0.223963i	$-0.0718994\pi$
$908$	3.01711e19i	1.78662i
$909$	−9.04482e18	−0.531783
$910$	0	0
$911$	−2.34028e19	−1.35643	−0.678216	−	0.734863i	$-0.737247\pi$
$911$	−2.34028e19	−1.35643	−0.678216	+	0.734863i	$0.737247\pi$
$912$	− 9.92337e18i	− 0.571074i
$913$	− 1.25699e19i	− 0.718242i
$914$	−1.13232e18	−0.0642418
$915$	0	0
$916$	1.19105e17	0.00666206
$917$	2.82535e17i	0.0156917i
$918$	− 5.46238e17i	− 0.0301234i
$919$	−2.20881e19	−1.20950	−0.604752	−	0.796414i	$-0.706728\pi$
$919$	−2.20881e19	−1.20950	−0.604752	+	0.796414i	$0.706728\pi$
$920$	0	0
$921$	9.64848e18	0.520920
$922$	9.60600e17i	0.0514982i
$923$	1.05799e19i	0.563210i
$924$	1.54241e19	0.815329
$925$	0	0
$926$	−2.57009e18	−0.133960
$927$	− 5.84077e18i	− 0.302309i
$928$	2.95509e18i	0.151883i
$929$	1.05632e19	0.539132	0.269566	−	0.962982i	$-0.413120\pi$
$929$	1.05632e19	0.539132	0.269566	+	0.962982i	$0.413120\pi$
$930$	0	0
$931$	8.90980e18	0.448430
$932$	2.92185e19i	1.46034i
$933$	5.89981e18i	0.292824i
$934$	−2.16817e18	−0.106866
$935$	0	0
$936$	8.33676e17	0.0405233
$937$	1.72833e19i	0.834294i	0.908839	+	0.417147i	$0.136970\pi$
$937$	1.72833e19i	0.834294i	−0.908839	+	0.417147i	$0.863030\pi$
$938$	− 9.83191e17i	− 0.0471324i
$939$	−1.08024e19	−0.514273
$940$	0	0
$941$	2.14038e19	1.00498	0.502490	−	0.864583i	$-0.332417\pi$
$941$	2.14038e19	1.00498	0.502490	+	0.864583i	$0.332417\pi$
$942$	1.03245e18i	0.0481434i
$943$	− 3.88591e18i	− 0.179956i
$944$	1.90199e18	0.0874760
$945$	0	0
$946$	9.54799e17	0.0433131
$947$	5.91308e17i	0.0266403i	0.999911	+	0.0133202i	$0.00424006\pi$
$947$	5.91308e17i	0.0266403i	−0.999911	+	0.0133202i	$0.995760\pi$
$948$	− 6.29188e18i	− 0.281531i
$949$	9.48950e18	0.421709
$950$	0	0
$951$	1.77271e19	0.777079
$952$	5.38289e18i	0.234356i
$953$	− 2.39396e19i	− 1.03517i	−0.855631	−	0.517587i	$-0.826830\pi$
$953$	− 2.39396e19i	− 1.03517i	0.855631	−	0.517587i	$-0.173170\pi$
$954$	−1.30177e18	−0.0559073
$955$	0	0
$956$	3.49203e19	1.47945
$957$	1.02100e19i	0.429633i
$958$	3.20055e18i	0.133767i
$959$	−4.03321e18	−0.167429
$960$	0	0
$961$	1.35201e19	0.553702
$962$	− 5.31100e17i	− 0.0216042i
$963$	− 1.04377e19i	− 0.421729i
$964$	2.43349e19	0.976628
$965$	0	0
$966$	1.70808e18	0.0676327
$967$	− 1.99045e18i	− 0.0782853i	−0.999234	−	0.0391427i	$-0.987537\pi$
$967$	− 1.99045e18i	− 0.0782853i	0.999234	−	0.0391427i	$-0.0124627\pi$
$968$	1.76434e19i	0.689275i
$969$	1.83351e19	0.711507
$970$	0	0
$971$	4.12385e19	1.57898	0.789492	−	0.613761i	$-0.210344\pi$
$971$	4.12385e19	1.57898	0.789492	+	0.613761i	$0.210344\pi$
$972$	− 1.65701e18i	− 0.0630224i
$973$	2.48357e19i	0.938301i
$974$	−3.16262e18	−0.118690
$975$	0	0
$976$	−8.54608e18	−0.316478
$977$	1.25374e19i	0.461202i	0.973048	+	0.230601i	$0.0740693\pi$
$977$	1.25374e19i	0.461202i	−0.973048	+	0.230601i	$0.925931\pi$
$978$	1.35340e18i	0.0494567i
$979$	2.50006e19	0.907535
$980$	0	0
$981$	−5.21928e18	−0.186966
$982$	4.94080e18i	0.175822i
$983$	− 1.83966e19i	− 0.650340i	−0.945656	−	0.325170i	$-0.894578\pi$
$983$	− 1.83966e19i	− 0.650340i	0.945656	−	0.325170i	$-0.105422\pi$
$984$	−6.64690e17	−0.0233427
$985$	0	0
$986$	−1.76580e18	−0.0611987
$987$	− 5.06062e18i	− 0.174238i
$988$	1.38676e19i	0.474331i
$989$	−5.90941e18	−0.200802
$990$	0	0
$991$	−3.43064e19	−1.15053	−0.575263	−	0.817969i	$-0.695100\pi$
$991$	−3.43064e19	−1.15053	−0.575263	+	0.817969i	$0.695100\pi$
$992$	1.45333e19i	0.484214i
$993$	− 8.47329e18i	− 0.280467i
$994$	3.70781e18	0.121929
$995$	0	0
$996$	6.59465e18	0.214045
$997$	5.72458e19i	1.84597i	0.384835	+	0.922985i	$0.374258\pi$
$997$	5.72458e19i	1.84597i	−0.384835	+	0.922985i	$0.625742\pi$
$998$	− 4.79589e18i	− 0.153646i
$999$	−2.13011e18	−0.0677995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.14.b.b.49.2		2
5.2	odd	4		3.14.a.a.1.1	✓	1
5.3	odd	4		75.14.a.a.1.1		1
5.4	even	2	inner	75.14.b.b.49.1		2
15.2	even	4		9.14.a.a.1.1		1
20.7	even	4		48.14.a.c.1.1		1
35.27	even	4		147.14.a.a.1.1		1
40.27	even	4		192.14.a.e.1.1		1
40.37	odd	4		192.14.a.j.1.1		1
60.47	odd	4		144.14.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.14.a.a.1.1	✓	1	5.2	odd	4
9.14.a.a.1.1		1	15.2	even	4
48.14.a.c.1.1		1	20.7	even	4
75.14.a.a.1.1		1	5.3	odd	4
75.14.b.b.49.1		2	5.4	even	2	inner
75.14.b.b.49.2		2	1.1	even	1	trivial
144.14.a.k.1.1		1	60.47	odd	4
147.14.a.a.1.1		1	35.27	even	4
192.14.a.e.1.1		1	40.27	even	4
192.14.a.j.1.1		1	40.37	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+12.0000i q^{2} -729.000i q^{3} +8048.00 q^{4} +8748.00 q^{6} -235088. i q^{7} +194880. i q^{8} -531441. q^{9} -1.11829e7 q^{11} -5.86699e6i q^{12} +8.04961e6i q^{13} +2.82106e6 q^{14} +6.35907e7 q^{16} +1.17495e8i q^{17} -6.37729e6i q^{18} +2.14061e8 q^{19} -1.71379e8 q^{21} -1.34195e8i q^{22} +8.30556e8i q^{23} +1.42068e8 q^{24} -9.65954e7 q^{26} +3.87420e8i q^{27} -1.89199e9i q^{28} +1.25240e9 q^{29} +6.15935e9 q^{31} +2.35954e9i q^{32} +8.15234e9i q^{33} -1.40994e9 q^{34} -4.27704e9 q^{36} +5.49819e9i q^{37} +2.56874e9i q^{38} +5.86817e9 q^{39} -4.67869e9 q^{41} -2.05655e9i q^{42} +7.11501e9i q^{43} -9.00000e10 q^{44} -9.96667e9 q^{46} +2.95288e10i q^{47} -4.63576e10i q^{48} +4.16226e10 q^{49} +8.56536e10 q^{51} +6.47833e10i q^{52} -2.04125e11i q^{53} -4.64905e9 q^{54} +4.58139e10 q^{56} -1.56051e11i q^{57} +1.50288e10i q^{58} +2.99098e10 q^{59} -1.34392e11 q^{61} +7.39122e10i q^{62} +1.24935e11i q^{63} +4.92620e11 q^{64} -9.78281e10 q^{66} -3.48519e11i q^{67} +9.45597e11i q^{68} +6.05475e11 q^{69} +1.31434e12 q^{71} -1.03567e11i q^{72} -1.17888e12i q^{73} -6.59783e10 q^{74} +1.72277e12 q^{76} +2.62897e12i q^{77} +7.04180e10i q^{78} +1.07242e12 q^{79} +2.82430e11 q^{81} -5.61443e10i q^{82} +1.12403e12i q^{83} -1.37926e12 q^{84} -8.53802e10 q^{86} -9.13000e11i q^{87} -2.17933e12i q^{88} -2.23561e12 q^{89} +1.89237e12 q^{91} +6.68431e12i q^{92} -4.49017e12i q^{93} -3.54345e11 q^{94} +1.72011e12 q^{96} +1.42153e13i q^{97} +4.99472e11i q^{98} +5.94306e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16096 q^{4} + 17496 q^{6} - 1062882 q^{9} - 22365816 q^{11} + 5642112 q^{14} + 127181312 q^{16} + 428122760 q^{19} - 342758304 q^{21} + 284135040 q^{24} - 193190736 q^{26} + 2504800500 q^{29} + 12318701104 q^{31}+ \cdots + 11886111620856 q^{99}+O(q^{100}) \) 2 * q + 16096 * q^4 + 17496 * q^6 - 1062882 * q^9 - 22365816 * q^11 + 5642112 * q^14 + 127181312 * q^16 + 428122760 * q^19 - 342758304 * q^21 + 284135040 * q^24 - 193190736 * q^26 + 2504800500 * q^29 + 12318701104 * q^31 - 2819870928 * q^34 - 8554074336 * q^36 + 11736337212 * q^39 - 9357375756 * q^41 - 180000087168 * q^44 - 19933333056 * q^46 + 83245285326 * q^49 + 171307158876 * q^51 - 9298091736 * q^54 + 91627898880 * q^56 + 59819642040 * q^59 - 268784013476 * q^61 + 985240231936 * q^64 - 195656158368 * q^66 + 1210949983152 * q^69 + 2628670818384 * q^71 - 131956593648 * q^74 + 3445531972480 * q^76 + 2144841319280 * q^79 + 564859072962 * q^81 - 2758518830592 * q^84 - 170760330336 * q^86 - 4471221819060 * q^89 + 3784735312064 * q^91 - 708690647808 * q^94 + 3440216365056 * q^96 + 11886111620856 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more