LMFDB - Embedded newform 26.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,8,Mod(1,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("26.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$26 = 2 \cdot 13$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	26.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:	$8.12201066259$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	26.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -245.000 q^{5} -216.000 q^{6} -587.000 q^{7} +512.000 q^{8} -1458.00 q^{9} -1960.00 q^{10} -3874.00 q^{11} -1728.00 q^{12} -2197.00 q^{13} -4696.00 q^{14} +6615.00 q^{15} +4096.00 q^{16} +5229.00 q^{17} -11664.0 q^{18} -6522.00 q^{19} -15680.0 q^{20} +15849.0 q^{21} -30992.0 q^{22} -500.000 q^{23} -13824.0 q^{24} -18100.0 q^{25} -17576.0 q^{26} +98415.0 q^{27} -37568.0 q^{28} +226954. q^{29} +52920.0 q^{30} +130156. q^{31} +32768.0 q^{32} +104598. q^{33} +41832.0 q^{34} +143815. q^{35} -93312.0 q^{36} -377769. q^{37} -52176.0 q^{38} +59319.0 q^{39} -125440. q^{40} -539760. q^{41} +126792. q^{42} +13987.0 q^{43} -247936. q^{44} +357210. q^{45} -4000.00 q^{46} -526879. q^{47} -110592. q^{48} -478974. q^{49} -144800. q^{50} -141183. q^{51} -140608. q^{52} -1.64994e6 q^{53} +787320. q^{54} +949130. q^{55} -300544. q^{56} +176094. q^{57} +1.81563e6 q^{58} -81194.0 q^{59} +423360. q^{60} -1.12695e6 q^{61} +1.04125e6 q^{62} +855846. q^{63} +262144. q^{64} +538265. q^{65} +836784. q^{66} +478798. q^{67} +334656. q^{68} +13500.0 q^{69} +1.15052e6 q^{70} +940007. q^{71} -746496. q^{72} +1.67193e6 q^{73} -3.02215e6 q^{74} +488700. q^{75} -417408. q^{76} +2.27404e6 q^{77} +474552. q^{78} -5.80119e6 q^{79} -1.00352e6 q^{80} +531441. q^{81} -4.31808e6 q^{82} +7.39882e6 q^{83} +1.01434e6 q^{84} -1.28110e6 q^{85} +111896. q^{86} -6.12776e6 q^{87} -1.98349e6 q^{88} -953754. q^{89} +2.85768e6 q^{90} +1.28964e6 q^{91} -32000.0 q^{92} -3.51421e6 q^{93} -4.21503e6 q^{94} +1.59789e6 q^{95} -884736. q^{96} -1.03187e7 q^{97} -3.83179e6 q^{98} +5.64829e6 q^{99} +O(q^{100})

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	−27.0000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−27.0000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	64.0000	0.500000
$5$	−245.000	−0.876539	−0.438269	−	0.898844i	$-0.644408\pi$
$5$	−245.000	−0.876539	−0.438269	+	0.898844i	$0.644408\pi$
$6$	−216.000	−0.408248
$7$	−587.000	−0.646837	−0.323419	−	0.946256i	$-0.604832\pi$
$7$	−587.000	−0.646837	−0.323419	+	0.946256i	$0.604832\pi$
$8$	512.000	0.353553
$9$	−1458.00	−0.666667
$10$	−1960.00	−0.619806
$11$	−3874.00	−0.877577	−0.438788	−	0.898590i	$-0.644592\pi$
$11$	−3874.00	−0.877577	−0.438788	+	0.898590i	$0.644592\pi$
$12$	−1728.00	−0.288675
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	−4696.00	−0.457383
$15$	6615.00	0.506070
$16$	4096.00	0.250000
$17$	5229.00	0.258135	0.129068	−	0.991636i	$-0.458802\pi$
$17$	5229.00	0.258135	0.129068	+	0.991636i	$0.458802\pi$
$18$	−11664.0	−0.471405
$19$	−6522.00	−0.218144	−0.109072	−	0.994034i	$-0.534788\pi$
$19$	−6522.00	−0.218144	−0.109072	+	0.994034i	$0.534788\pi$
$20$	−15680.0	−0.438269
$21$	15849.0	0.373452
$22$	−30992.0	−0.620541
$23$	−500.000	−0.00856885	−0.00428443	−	0.999991i	$-0.501364\pi$
$23$	−500.000	−0.00856885	−0.00428443	+	0.999991i	$0.501364\pi$
$24$	−13824.0	−0.204124
$25$	−18100.0	−0.231680
$26$	−17576.0	−0.196116
$27$	98415.0	0.962250
$28$	−37568.0	−0.323419
$29$	226954.	1.72800	0.864002	−	0.503488i	$-0.167950\pi$
$29$	226954.	1.72800	0.864002	+	0.503488i	$0.167950\pi$
$30$	52920.0	0.357845
$31$	130156.	0.784690	0.392345	−	0.919818i	$-0.371664\pi$
$31$	130156.	0.784690	0.392345	+	0.919818i	$0.371664\pi$
$32$	32768.0	0.176777
$33$	104598.	0.506669
$34$	41832.0	0.182529
$35$	143815.	0.566978
$36$	−93312.0	−0.333333
$37$	−377769.	−1.22608	−0.613042	−	0.790050i	$-0.710054\pi$
$37$	−377769.	−1.22608	−0.613042	+	0.790050i	$0.710054\pi$
$38$	−52176.0	−0.154251
$39$	59319.0	0.160128
$40$	−125440.	−0.309903
$41$	−539760.	−1.22309	−0.611543	−	0.791211i	$-0.709451\pi$
$41$	−539760.	−1.22309	−0.611543	+	0.791211i	$0.709451\pi$
$42$	126792.	0.264070
$43$	13987.0	0.0268278	0.0134139	−	0.999910i	$-0.495730\pi$
$43$	13987.0	0.0268278	0.0134139	+	0.999910i	$0.495730\pi$
$44$	−247936.	−0.438788
$45$	357210.	0.584359
$46$	−4000.00	−0.00605909
$47$	−526879.	−0.740233	−0.370116	−	0.928985i	$-0.620682\pi$
$47$	−526879.	−0.740233	−0.370116	+	0.928985i	$0.620682\pi$
$48$	−110592.	−0.144338
$49$	−478974.	−0.581602
$50$	−144800.	−0.163822
$51$	−141183.	−0.149034
$52$	−140608.	−0.138675
$53$	−1.64994e6	−1.52231	−0.761154	−	0.648571i	$-0.775367\pi$
$53$	−1.64994e6	−1.52231	−0.761154	+	0.648571i	$0.775367\pi$
$54$	787320.	0.680414
$55$	949130.	0.769230
$56$	−300544.	−0.228691
$57$	176094.	0.125945
$58$	1.81563e6	1.22188
$59$	−81194.0	−0.0514685	−0.0257343	−	0.999669i	$-0.508192\pi$
$59$	−81194.0	−0.0514685	−0.0257343	+	0.999669i	$0.508192\pi$
$60$	423360.	0.253035
$61$	−1.12695e6	−0.635698	−0.317849	−	0.948141i	$-0.602961\pi$
$61$	−1.12695e6	−0.635698	−0.317849	+	0.948141i	$0.602961\pi$
$62$	1.04125e6	0.554860
$63$	855846.	0.431225
$64$	262144.	0.125000
$65$	538265.	0.243108
$66$	836784.	0.358269
$67$	478798.	0.194487	0.0972435	−	0.995261i	$-0.468997\pi$
$67$	478798.	0.194487	0.0972435	+	0.995261i	$0.468997\pi$
$68$	334656.	0.129068
$69$	13500.0	0.00494723
$70$	1.15052e6	0.400914
$71$	940007.	0.311693	0.155846	−	0.987781i	$-0.450190\pi$
$71$	940007.	0.311693	0.155846	+	0.987781i	$0.450190\pi$
$72$	−746496.	−0.235702
$73$	1.67193e6	0.503022	0.251511	−	0.967854i	$-0.419073\pi$
$73$	1.67193e6	0.503022	0.251511	+	0.967854i	$0.419073\pi$
$74$	−3.02215e6	−0.866972
$75$	488700.	0.133761
$76$	−417408.	−0.109072
$77$	2.27404e6	0.567649
$78$	474552.	0.113228
$79$	−5.80119e6	−1.32380	−0.661900	−	0.749592i	$-0.730249\pi$
$79$	−5.80119e6	−1.32380	−0.661900	+	0.749592i	$0.730249\pi$
$80$	−1.00352e6	−0.219135
$81$	531441.	0.111111
$82$	−4.31808e6	−0.864853
$83$	7.39882e6	1.42033	0.710164	−	0.704036i	$-0.248621\pi$
$83$	7.39882e6	1.42033	0.710164	+	0.704036i	$0.248621\pi$
$84$	1.01434e6	0.186726
$85$	−1.28110e6	−0.226266
$86$	111896.	0.0189701
$87$	−6.12776e6	−0.997664
$88$	−1.98349e6	−0.310270
$89$	−953754.	−0.143407	−0.0717037	−	0.997426i	$-0.522844\pi$
$89$	−953754.	−0.143407	−0.0717037	+	0.997426i	$0.522844\pi$
$90$	2.85768e6	0.413204
$91$	1.28964e6	0.179400
$92$	−32000.0	−0.00428443
$93$	−3.51421e6	−0.453041
$94$	−4.21503e6	−0.523424
$95$	1.59789e6	0.191212
$96$	−884736.	−0.102062
$97$	−1.03187e7	−1.14795	−0.573976	−	0.818872i	$-0.694600\pi$
$97$	−1.03187e7	−1.14795	−0.573976	+	0.818872i	$0.694600\pi$
$98$	−3.83179e6	−0.411254
$99$	5.64829e6	0.585051
$100$	−1.15840e6	−0.115840
$101$	4.73503e6	0.457296	0.228648	−	0.973509i	$-0.426569\pi$
$101$	4.73503e6	0.457296	0.228648	+	0.973509i	$0.426569\pi$
$102$	−1.12946e6	−0.105383
$103$	1.60974e7	1.45153	0.725763	−	0.687945i	$-0.241487\pi$
$103$	1.60974e7	1.45153	0.725763	+	0.687945i	$0.241487\pi$
$104$	−1.12486e6	−0.0980581
$105$	−3.88301e6	−0.327345
$106$	−1.31995e7	−1.07643
$107$	1.77823e7	1.40328	0.701642	−	0.712530i	$-0.252451\pi$
$107$	1.77823e7	1.40328	0.701642	+	0.712530i	$0.252451\pi$
$108$	6.29856e6	0.481125
$109$	−2.20506e7	−1.63090	−0.815450	−	0.578827i	$-0.803511\pi$
$109$	−2.20506e7	−1.63090	−0.815450	+	0.578827i	$0.803511\pi$
$110$	7.59304e6	0.543928
$111$	1.01998e7	0.707880
$112$	−2.40435e6	−0.161709
$113$	−1.80880e7	−1.17928	−0.589639	−	0.807667i	$-0.700730\pi$
$113$	−1.80880e7	−1.17928	−0.589639	+	0.807667i	$0.700730\pi$
$114$	1.40875e6	0.0890569
$115$	122500.	0.00751093
$116$	1.45251e7	0.864002
$117$	3.20323e6	0.184900
$118$	−649552.	−0.0363938
$119$	−3.06942e6	−0.166972
$120$	3.38688e6	0.178923
$121$	−4.47930e6	−0.229859
$122$	−9.01562e6	−0.449507
$123$	1.45735e7	0.706149
$124$	8.32998e6	0.392345
$125$	2.35751e7	1.07962
$126$	6.84677e6	0.304922
$127$	−2.52728e6	−0.109481	−0.0547407	−	0.998501i	$-0.517433\pi$
$127$	−2.52728e6	−0.109481	−0.0547407	+	0.998501i	$0.517433\pi$
$128$	2.09715e6	0.0883883
$129$	−377649.	−0.0154890
$130$	4.30612e6	0.171903
$131$	4.19410e7	1.63001	0.815004	−	0.579456i	$-0.196735\pi$
$131$	4.19410e7	1.63001	0.815004	+	0.579456i	$0.196735\pi$
$132$	6.69427e6	0.253335
$133$	3.82841e6	0.141104
$134$	3.83038e6	0.137523
$135$	−2.41117e7	−0.843450
$136$	2.67725e6	0.0912646
$137$	−2.45947e7	−0.817182	−0.408591	−	0.912718i	$-0.633980\pi$
$137$	−2.45947e7	−0.817182	−0.408591	+	0.912718i	$0.633980\pi$
$138$	108000.	0.00349822
$139$	−5.79349e7	−1.82974	−0.914869	−	0.403752i	$-0.867706\pi$
$139$	−5.79349e7	−1.82974	−0.914869	+	0.403752i	$0.867706\pi$
$140$	9.20416e6	0.283489
$141$	1.42257e7	0.427374
$142$	7.52006e6	0.220400
$143$	8.51118e6	0.243396
$144$	−5.97197e6	−0.166667
$145$	−5.56037e7	−1.51466
$146$	1.33754e7	0.355690
$147$	1.29323e7	0.335788
$148$	−2.41772e7	−0.613042
$149$	−2.90993e7	−0.720660	−0.360330	−	0.932825i	$-0.617336\pi$
$149$	−2.90993e7	−0.720660	−0.360330	+	0.932825i	$0.617336\pi$
$150$	3.90960e6	0.0945830
$151$	−4.13849e7	−0.978187	−0.489094	−	0.872231i	$-0.662672\pi$
$151$	−4.13849e7	−0.978187	−0.489094	+	0.872231i	$0.662672\pi$
$152$	−3.33926e6	−0.0771255
$153$	−7.62388e6	−0.172090
$154$	1.81923e7	0.401389
$155$	−3.18882e7	−0.687811
$156$	3.79642e6	0.0800641
$157$	8.60728e6	0.177508	0.0887538	−	0.996054i	$-0.471712\pi$
$157$	8.60728e6	0.177508	0.0887538	+	0.996054i	$0.471712\pi$
$158$	−4.64095e7	−0.936067
$159$	4.45484e7	0.878905
$160$	−8.02816e6	−0.154952
$161$	293500.	0.00554265
$162$	4.25153e6	0.0785674
$163$	−8.32173e7	−1.50507	−0.752535	−	0.658552i	$-0.771169\pi$
$163$	−8.32173e7	−1.50507	−0.752535	+	0.658552i	$0.771169\pi$
$164$	−3.45446e7	−0.611543
$165$	−2.56265e7	−0.444115
$166$	5.91905e7	1.00432
$167$	1.15768e8	1.92346	0.961729	−	0.274004i	$-0.0883481\pi$
$167$	1.15768e8	1.92346	0.961729	+	0.274004i	$0.0883481\pi$
$168$	8.11469e6	0.132035
$169$	4.82681e6	0.0769231
$170$	−1.02488e7	−0.159994
$171$	9.50908e6	0.145429
$172$	895168.	0.0134139
$173$	9.10083e7	1.33635	0.668174	−	0.744005i	$-0.267076\pi$
$173$	9.10083e7	1.33635	0.668174	+	0.744005i	$0.267076\pi$
$174$	−4.90221e7	−0.705455
$175$	1.06247e7	0.149859
$176$	−1.58679e7	−0.219394
$177$	2.19224e6	0.0297154
$178$	−7.63003e6	−0.101404
$179$	1.38177e8	1.80074	0.900370	−	0.435124i	$-0.143296\pi$
$179$	1.38177e8	1.80074	0.900370	+	0.435124i	$0.143296\pi$
$180$	2.28614e7	0.292180
$181$	1.05517e8	1.32266	0.661332	−	0.750094i	$-0.269992\pi$
$181$	1.05517e8	1.32266	0.661332	+	0.750094i	$0.269992\pi$
$182$	1.03171e7	0.126855
$183$	3.04277e7	0.367021
$184$	−256000.	−0.00302955
$185$	9.25534e7	1.07471
$186$	−2.81137e7	−0.320348
$187$	−2.02571e7	−0.226534
$188$	−3.37203e7	−0.370116
$189$	−5.77696e7	−0.622419
$190$	1.27831e7	0.135207
$191$	−9.36485e7	−0.972487	−0.486244	−	0.873823i	$-0.661633\pi$
$191$	−9.36485e7	−0.972487	−0.486244	+	0.873823i	$0.661633\pi$
$192$	−7.07789e6	−0.0721688
$193$	1.31216e8	1.31382	0.656910	−	0.753969i	$-0.271863\pi$
$193$	1.31216e8	1.31382	0.656910	+	0.753969i	$0.271863\pi$
$194$	−8.25495e7	−0.811724
$195$	−1.45332e7	−0.140359
$196$	−3.06543e7	−0.290801
$197$	−1.35325e8	−1.26109	−0.630546	−	0.776152i	$-0.717169\pi$
$197$	−1.35325e8	−1.26109	−0.630546	+	0.776152i	$0.717169\pi$
$198$	4.51863e7	0.413694
$199$	−1.24505e8	−1.11996	−0.559980	−	0.828506i	$-0.689191\pi$
$199$	−1.24505e8	−1.11996	−0.559980	+	0.828506i	$0.689191\pi$
$200$	−9.26720e6	−0.0819112
$201$	−1.29275e7	−0.112287
$202$	3.78802e7	0.323357
$203$	−1.33222e8	−1.11774
$204$	−9.03571e6	−0.0745172
$205$	1.32241e8	1.07208
$206$	1.28779e8	1.02638
$207$	729000.	0.00571257
$208$	−8.99891e6	−0.0693375
$209$	2.52662e7	0.191438
$210$	−3.10640e7	−0.231468
$211$	8.06435e7	0.590991	0.295496	−	0.955344i	$-0.404515\pi$
$211$	8.06435e7	0.590991	0.295496	+	0.955344i	$0.404515\pi$
$212$	−1.05596e8	−0.761154
$213$	−2.53802e7	−0.179956
$214$	1.42259e8	0.992271
$215$	−3.42682e6	−0.0235156
$216$	5.03885e7	0.340207
$217$	−7.64016e7	−0.507567
$218$	−1.76405e8	−1.15322
$219$	−4.51420e7	−0.290420
$220$	6.07443e7	0.384615
$221$	−1.14881e7	−0.0715939
$222$	8.15981e7	0.500547
$223$	−1.30612e8	−0.788706	−0.394353	−	0.918959i	$-0.629031\pi$
$223$	−1.30612e8	−0.788706	−0.394353	+	0.918959i	$0.629031\pi$
$224$	−1.92348e7	−0.114346
$225$	2.63898e7	0.154453
$226$	−1.44704e8	−0.833875
$227$	−1.44302e8	−0.818809	−0.409405	−	0.912353i	$-0.634264\pi$
$227$	−1.44302e8	−0.818809	−0.409405	+	0.912353i	$0.634264\pi$
$228$	1.12700e7	0.0629727
$229$	−1.03229e8	−0.568039	−0.284019	−	0.958819i	$-0.591668\pi$
$229$	−1.03229e8	−0.568039	−0.284019	+	0.958819i	$0.591668\pi$
$230$	980000.	0.00531103
$231$	−6.13990e7	−0.327733
$232$	1.16200e8	0.610942
$233$	−3.19575e7	−0.165511	−0.0827556	−	0.996570i	$-0.526372\pi$
$233$	−3.19575e7	−0.165511	−0.0827556	+	0.996570i	$0.526372\pi$
$234$	2.56258e7	0.130744
$235$	1.29085e8	0.648843
$236$	−5.19642e6	−0.0257343
$237$	1.56632e8	0.764296
$238$	−2.45554e7	−0.118067
$239$	−1.50183e8	−0.711588	−0.355794	−	0.934564i	$-0.615790\pi$
$239$	−1.50183e8	−0.711588	−0.355794	+	0.934564i	$0.615790\pi$
$240$	2.70950e7	0.126517
$241$	1.36186e8	0.626718	0.313359	−	0.949635i	$-0.398546\pi$
$241$	1.36186e8	0.626718	0.313359	+	0.949635i	$0.398546\pi$
$242$	−3.58344e7	−0.162535
$243$	−2.29583e8	−1.02640
$244$	−7.21249e7	−0.317849
$245$	1.17349e8	0.509796
$246$	1.16588e8	0.499323
$247$	1.43288e7	0.0605022
$248$	6.66399e7	0.277430
$249$	−1.99768e8	−0.820027
$250$	1.88601e8	0.763403
$251$	1.89898e8	0.757989	0.378995	−	0.925399i	$-0.376270\pi$
$251$	1.89898e8	0.757989	0.378995	+	0.925399i	$0.376270\pi$
$252$	5.47741e7	0.215612
$253$	1.93700e6	0.00751983
$254$	−2.02182e7	−0.0774150
$255$	3.45898e7	0.130634
$256$	1.67772e7	0.0625000
$257$	1.72850e8	0.635190	0.317595	−	0.948227i	$-0.397125\pi$
$257$	1.72850e8	0.635190	0.317595	+	0.948227i	$0.397125\pi$
$258$	−3.02119e6	−0.0109524
$259$	2.21750e8	0.793077
$260$	3.44490e7	0.121554
$261$	−3.30899e8	−1.15200
$262$	3.35528e8	1.15259
$263$	−1.54338e8	−0.523151	−0.261576	−	0.965183i	$-0.584242\pi$
$263$	−1.54338e8	−0.523151	−0.261576	+	0.965183i	$0.584242\pi$
$264$	5.35542e7	0.179135
$265$	4.04235e8	1.33436
$266$	3.06273e7	0.0997753
$267$	2.57514e7	0.0827963
$268$	3.06431e7	0.0972435
$269$	−1.34962e7	−0.0422745	−0.0211372	−	0.999777i	$-0.506729\pi$
$269$	−1.34962e7	−0.0422745	−0.0211372	+	0.999777i	$0.506729\pi$
$270$	−1.92893e8	−0.596409
$271$	−6.48939e8	−1.98067	−0.990333	−	0.138707i	$-0.955705\pi$
$271$	−6.48939e8	−1.98067	−0.990333	+	0.138707i	$0.955705\pi$
$272$	2.14180e7	0.0645338
$273$	−3.48203e7	−0.103577
$274$	−1.96757e8	−0.577835
$275$	7.01194e7	0.203317
$276$	864000.	0.00247361
$277$	5.55195e8	1.56952	0.784760	−	0.619800i	$-0.212786\pi$
$277$	5.55195e8	1.56952	0.784760	+	0.619800i	$0.212786\pi$
$278$	−4.63479e8	−1.29382
$279$	−1.89767e8	−0.523127
$280$	7.36333e7	0.200457
$281$	3.34956e8	0.900565	0.450283	−	0.892886i	$-0.351323\pi$
$281$	3.34956e8	0.900565	0.450283	+	0.892886i	$0.351323\pi$
$282$	1.13806e8	0.302199
$283$	5.76535e8	1.51207	0.756037	−	0.654529i	$-0.227133\pi$
$283$	5.76535e8	1.51207	0.756037	+	0.654529i	$0.227133\pi$
$284$	6.01604e7	0.155846
$285$	−4.31430e7	−0.110396
$286$	6.80894e7	0.172107
$287$	3.16839e8	0.791138
$288$	−4.77757e7	−0.117851
$289$	−3.82996e8	−0.933366
$290$	−4.44830e8	−1.07103
$291$	2.78605e8	0.662770
$292$	1.07003e8	0.251511
$293$	−1.77399e8	−0.412016	−0.206008	−	0.978550i	$-0.566047\pi$
$293$	−1.77399e8	−0.412016	−0.206008	+	0.978550i	$0.566047\pi$
$294$	1.03458e8	0.237438
$295$	1.98925e7	0.0451142
$296$	−1.93418e8	−0.433486
$297$	−3.81260e8	−0.844449
$298$	−2.32794e8	−0.509583
$299$	1.09850e6	0.00237657
$300$	3.12768e7	0.0668803
$301$	−8.21037e6	−0.0173532
$302$	−3.31079e8	−0.691683
$303$	−1.27846e8	−0.264020
$304$	−2.67141e7	−0.0545360
$305$	2.76103e8	0.557214
$306$	−6.09911e7	−0.121686
$307$	2.94122e8	0.580155	0.290077	−	0.957003i	$-0.406319\pi$
$307$	2.94122e8	0.580155	0.290077	+	0.957003i	$0.406319\pi$
$308$	1.45538e8	0.283825
$309$	−4.34629e8	−0.838038
$310$	−2.55106e8	−0.486356
$311$	−2.86005e8	−0.539154	−0.269577	−	0.962979i	$-0.586884\pi$
$311$	−2.86005e8	−0.539154	−0.269577	+	0.962979i	$0.586884\pi$
$312$	3.03713e7	0.0566139
$313$	−2.76110e8	−0.508951	−0.254476	−	0.967079i	$-0.581903\pi$
$313$	−2.76110e8	−0.508951	−0.254476	+	0.967079i	$0.581903\pi$
$314$	6.88582e7	0.125517
$315$	−2.09682e8	−0.377985
$316$	−3.71276e8	−0.661900
$317$	5.45989e8	0.962667	0.481334	−	0.876537i	$-0.340153\pi$
$317$	5.45989e8	0.962667	0.481334	+	0.876537i	$0.340153\pi$
$318$	3.56387e8	0.621480
$319$	−8.79220e8	−1.51646
$320$	−6.42253e7	−0.109567
$321$	−4.80123e8	−0.810186
$322$	2.34800e6	0.00391925
$323$	−3.41035e7	−0.0563107
$324$	3.40122e7	0.0555556
$325$	3.97657e7	0.0642565
$326$	−6.65738e8	−1.06425
$327$	5.95366e8	0.941601
$328$	−2.76357e8	−0.432426
$329$	3.09278e8	0.478810
$330$	−2.05012e8	−0.314037
$331$	−3.60921e8	−0.547034	−0.273517	−	0.961867i	$-0.588187\pi$
$331$	−3.60921e8	−0.547034	−0.273517	+	0.961867i	$0.588187\pi$
$332$	4.73524e8	0.710164
$333$	5.50787e8	0.817389
$334$	9.26148e8	1.36009
$335$	−1.17306e8	−0.170475
$336$	6.49175e7	0.0933629
$337$	4.27270e8	0.608132	0.304066	−	0.952651i	$-0.401656\pi$
$337$	4.27270e8	0.608132	0.304066	+	0.952651i	$0.401656\pi$
$338$	3.86145e7	0.0543928
$339$	4.88376e8	0.680856
$340$	−8.19907e7	−0.113133
$341$	−5.04224e8	−0.688626
$342$	7.60726e7	0.102834
$343$	7.64577e8	1.02304
$344$	7.16134e6	0.00948506
$345$	−3.30750e6	−0.00433644
$346$	7.28067e8	0.944941
$347$	1.31637e9	1.69131	0.845656	−	0.533728i	$-0.179209\pi$
$347$	1.31637e9	1.69131	0.845656	+	0.533728i	$0.179209\pi$
$348$	−3.92177e8	−0.498832
$349$	−1.26585e9	−1.59402	−0.797012	−	0.603963i	$-0.793587\pi$
$349$	−1.26585e9	−1.59402	−0.797012	+	0.603963i	$0.793587\pi$
$350$	8.49976e7	0.105966
$351$	−2.16218e8	−0.266880
$352$	−1.26943e8	−0.155135
$353$	1.10618e9	1.33848	0.669241	−	0.743046i	$-0.266619\pi$
$353$	1.10618e9	1.33848	0.669241	+	0.743046i	$0.266619\pi$
$354$	1.75379e7	0.0210119
$355$	−2.30302e8	−0.273211
$356$	−6.10403e7	−0.0717037
$357$	8.28744e7	0.0964010
$358$	1.10542e9	1.27332
$359$	−1.02959e9	−1.17445	−0.587223	−	0.809425i	$-0.699779\pi$
$359$	−1.02959e9	−1.17445	−0.587223	+	0.809425i	$0.699779\pi$
$360$	1.82892e8	0.206602
$361$	−8.51335e8	−0.952413
$362$	8.44140e8	0.935264
$363$	1.20941e8	0.132709
$364$	8.25369e7	0.0897002
$365$	−4.09622e8	−0.440918
$366$	2.43422e8	0.259523
$367$	7.84516e8	0.828458	0.414229	−	0.910173i	$-0.364051\pi$
$367$	7.84516e8	0.828458	0.414229	+	0.910173i	$0.364051\pi$
$368$	−2.04800e6	−0.00214221
$369$	7.86970e8	0.815391
$370$	7.40427e8	0.759935
$371$	9.68515e8	0.984686
$372$	−2.24910e8	−0.226521
$373$	−5.52719e8	−0.551472	−0.275736	−	0.961233i	$-0.588922\pi$
$373$	−5.52719e8	−0.551472	−0.275736	+	0.961233i	$0.588922\pi$
$374$	−1.62057e8	−0.160183
$375$	−6.36528e8	−0.623316
$376$	−2.69762e8	−0.261712
$377$	−4.98618e8	−0.479262
$378$	−4.62157e8	−0.440117
$379$	6.93760e8	0.654594	0.327297	−	0.944921i	$-0.393862\pi$
$379$	6.93760e8	0.654594	0.327297	+	0.944921i	$0.393862\pi$
$380$	1.02265e8	0.0956058
$381$	6.82366e7	0.0632091
$382$	−7.49188e8	−0.687652
$383$	−1.56765e9	−1.42578	−0.712890	−	0.701276i	$-0.752614\pi$
$383$	−1.56765e9	−1.42578	−0.712890	+	0.701276i	$0.752614\pi$
$384$	−5.66231e7	−0.0510310
$385$	−5.57139e8	−0.497567
$386$	1.04973e9	0.929010
$387$	−2.03930e7	−0.0178852
$388$	−6.60396e8	−0.573976
$389$	−2.11817e9	−1.82447	−0.912235	−	0.409668i	$-0.865645\pi$
$389$	−2.11817e9	−1.82447	−0.912235	+	0.409668i	$0.865645\pi$
$390$	−1.16265e8	−0.0992485
$391$	−2.61450e6	−0.00221192
$392$	−2.45235e8	−0.205627
$393$	−1.13241e9	−0.941085
$394$	−1.08260e9	−0.891727
$395$	1.42129e9	1.16036
$396$	3.61491e8	0.292526
$397$	3.65973e8	0.293550	0.146775	−	0.989170i	$-0.453111\pi$
$397$	3.65973e8	0.293550	0.146775	+	0.989170i	$0.453111\pi$
$398$	−9.96043e8	−0.791931
$399$	−1.03367e8	−0.0814662
$400$	−7.41376e7	−0.0579200
$401$	−5.66697e8	−0.438880	−0.219440	−	0.975626i	$-0.570423\pi$
$401$	−5.66697e8	−0.438880	−0.219440	+	0.975626i	$0.570423\pi$
$402$	−1.03420e8	−0.0793990
$403$	−2.85953e8	−0.217634
$404$	3.03042e8	0.228648
$405$	−1.30203e8	−0.0973932
$406$	−1.06578e9	−0.790360
$407$	1.46348e9	1.07598
$408$	−7.22857e7	−0.0526916
$409$	−1.64659e9	−1.19002	−0.595011	−	0.803717i	$-0.702852\pi$
$409$	−1.64659e9	−1.19002	−0.595011	+	0.803717i	$0.702852\pi$
$410$	1.05793e9	0.758077
$411$	6.64056e8	0.471800
$412$	1.03023e9	0.725763
$413$	4.76609e7	0.0332918
$414$	5.83200e6	0.00403939
$415$	−1.81271e9	−1.24497
$416$	−7.19913e7	−0.0490290
$417$	1.56424e9	1.05640
$418$	2.02130e8	0.135367
$419$	−2.07538e8	−0.137831	−0.0689157	−	0.997622i	$-0.521954\pi$
$419$	−2.07538e8	−0.137831	−0.0689157	+	0.997622i	$0.521954\pi$
$420$	−2.48512e8	−0.163672
$421$	1.28086e9	0.836590	0.418295	−	0.908311i	$-0.362628\pi$
$421$	1.28086e9	0.836590	0.418295	+	0.908311i	$0.362628\pi$
$422$	6.45148e8	0.417894
$423$	7.68190e8	0.493489
$424$	−8.44769e8	−0.538217
$425$	−9.46449e7	−0.0598048
$426$	−2.03042e8	−0.127248
$427$	6.61521e8	0.411193
$428$	1.13807e9	0.701642
$429$	−2.29802e8	−0.140525
$430$	−2.74145e7	−0.0166280
$431$	−1.30893e8	−0.0787494	−0.0393747	−	0.999225i	$-0.512537\pi$
$431$	−1.30893e8	−0.0787494	−0.0393747	+	0.999225i	$0.512537\pi$
$432$	4.03108e8	0.240563
$433$	−3.01643e8	−0.178560	−0.0892802	−	0.996007i	$-0.528457\pi$
$433$	−3.01643e8	−0.178560	−0.0892802	+	0.996007i	$0.528457\pi$
$434$	−6.11213e8	−0.358904
$435$	1.50130e9	0.874491
$436$	−1.41124e9	−0.815450
$437$	3.26100e6	0.00186924
$438$	−3.61136e8	−0.205358
$439$	8.03991e8	0.453550	0.226775	−	0.973947i	$-0.427182\pi$
$439$	8.03991e8	0.453550	0.226775	+	0.973947i	$0.427182\pi$
$440$	4.85955e8	0.271964
$441$	6.98344e8	0.387734
$442$	−9.19049e7	−0.0506245
$443$	1.78013e8	0.0972832	0.0486416	−	0.998816i	$-0.484511\pi$
$443$	1.78013e8	0.0972832	0.0486416	+	0.998816i	$0.484511\pi$
$444$	6.52785e8	0.353940
$445$	2.33670e8	0.125702
$446$	−1.04489e9	−0.557699
$447$	7.85680e8	0.416073
$448$	−1.53879e8	−0.0808546
$449$	−1.61622e9	−0.842634	−0.421317	−	0.906914i	$-0.638432\pi$
$449$	−1.61622e9	−0.842634	−0.421317	+	0.906914i	$0.638432\pi$
$450$	2.11118e8	0.109215
$451$	2.09103e9	1.07335
$452$	−1.15763e9	−0.589639
$453$	1.11739e9	0.564757
$454$	−1.15442e9	−0.578986
$455$	−3.15962e8	−0.157251
$456$	9.01601e7	0.0445284
$457$	1.23751e9	0.606517	0.303258	−	0.952908i	$-0.401926\pi$
$457$	1.23751e9	0.606517	0.303258	+	0.952908i	$0.401926\pi$
$458$	−8.25833e8	−0.401664
$459$	5.14612e8	0.248391
$460$	7.84000e6	0.00375546
$461$	1.25678e8	0.0597457	0.0298729	−	0.999554i	$-0.490490\pi$
$461$	1.25678e8	0.0597457	0.0298729	+	0.999554i	$0.490490\pi$
$462$	−4.91192e8	−0.231742
$463$	−1.01176e9	−0.473743	−0.236872	−	0.971541i	$-0.576122\pi$
$463$	−1.01176e9	−0.473743	−0.236872	+	0.971541i	$0.576122\pi$
$464$	9.29604e8	0.432001
$465$	8.60982e8	0.397108
$466$	−2.55660e8	−0.117034
$467$	−1.61515e9	−0.733846	−0.366923	−	0.930251i	$-0.619589\pi$
$467$	−1.61515e9	−0.733846	−0.366923	+	0.930251i	$0.619589\pi$
$468$	2.05006e8	0.0924500
$469$	−2.81054e8	−0.125801
$470$	1.03268e9	0.458801
$471$	−2.32397e8	−0.102484
$472$	−4.15713e7	−0.0181969
$473$	−5.41856e7	−0.0235435
$474$	1.25306e9	0.540439
$475$	1.18048e8	0.0505396
$476$	−1.96443e8	−0.0834858
$477$	2.40561e9	1.01487
$478$	−1.20147e9	−0.503169
$479$	4.98553e8	0.207270	0.103635	−	0.994615i	$-0.466953\pi$
$479$	4.98553e8	0.207270	0.103635	+	0.994615i	$0.466953\pi$
$480$	2.16760e8	0.0894614
$481$	8.29958e8	0.340055
$482$	1.08949e9	0.443157
$483$	−7.92450e6	−0.00320005
$484$	−2.86675e8	−0.114929
$485$	2.52808e9	1.00622
$486$	−1.83666e9	−0.725775
$487$	−4.81147e9	−1.88767	−0.943836	−	0.330413i	$-0.892812\pi$
$487$	−4.81147e9	−1.88767	−0.943836	+	0.330413i	$0.892812\pi$
$488$	−5.76999e8	−0.224753
$489$	2.24687e9	0.868953
$490$	9.38789e8	0.360480
$491$	1.50724e9	0.574643	0.287322	−	0.957834i	$-0.407235\pi$
$491$	1.50724e9	0.574643	0.287322	+	0.957834i	$0.407235\pi$
$492$	9.32705e8	0.353075
$493$	1.18674e9	0.446059
$494$	1.14631e8	0.0427816
$495$	−1.38383e9	−0.512820
$496$	5.33119e8	0.196173
$497$	−5.51784e8	−0.201615
$498$	−1.59814e9	−0.579847
$499$	−3.47961e9	−1.25366	−0.626829	−	0.779157i	$-0.715647\pi$
$499$	−3.47961e9	−1.25366	−0.626829	+	0.779157i	$0.715647\pi$
$500$	1.50881e9	0.539808
$501$	−3.12575e9	−1.11051
$502$	1.51919e9	0.535979
$503$	−1.65804e9	−0.580907	−0.290454	−	0.956889i	$-0.593806\pi$
$503$	−1.65804e9	−0.580907	−0.290454	+	0.956889i	$0.593806\pi$
$504$	4.38193e8	0.152461
$505$	−1.16008e9	−0.400838
$506$	1.54960e7	0.00531732
$507$	−1.30324e8	−0.0444116
$508$	−1.61746e8	−0.0547407
$509$	−8.62799e8	−0.289999	−0.145000	−	0.989432i	$-0.546318\pi$
$509$	−8.62799e8	−0.289999	−0.145000	+	0.989432i	$0.546318\pi$
$510$	2.76719e8	0.0923725
$511$	−9.81421e8	−0.325373
$512$	1.34218e8	0.0441942
$513$	−6.41863e8	−0.209909
$514$	1.38280e9	0.449147
$515$	−3.94386e9	−1.27232
$516$	−2.41695e7	−0.00774452
$517$	2.04113e9	0.649611
$518$	1.77400e9	0.560790
$519$	−2.45723e9	−0.771541
$520$	2.75592e8	0.0859517
$521$	−1.33343e9	−0.413085	−0.206542	−	0.978438i	$-0.566221\pi$
$521$	−1.33343e9	−0.413085	−0.206542	+	0.978438i	$0.566221\pi$
$522$	−2.64719e9	−0.814589
$523$	942900.	0.000288210 0	0.000144105	−	1.00000i	$-0.499954\pi$
$523$	942900.	0.000288210 0	0.000144105		1.00000i	$0.499954\pi$
$524$	2.68423e9	0.815004
$525$	−2.86867e8	−0.0865213
$526$	−1.23470e9	−0.369924
$527$	6.80586e8	0.202556
$528$	4.28433e8	0.126667
$529$	−3.40458e9	−0.999927
$530$	3.23388e9	0.943536
$531$	1.18381e8	0.0343124
$532$	2.45018e8	0.0705518
$533$	1.18585e9	0.339223
$534$	2.06011e8	0.0585458
$535$	−4.35667e9	−1.23003
$536$	2.45145e8	0.0687615
$537$	−3.73079e9	−1.03966
$538$	−1.07969e8	−0.0298926
$539$	1.85555e9	0.510400
$540$	−1.54315e9	−0.421725
$541$	3.42111e9	0.928916	0.464458	−	0.885595i	$-0.346249\pi$
$541$	3.42111e9	0.928916	0.464458	+	0.885595i	$0.346249\pi$
$542$	−5.19151e9	−1.40054
$543$	−2.84897e9	−0.763640
$544$	1.71344e8	0.0456323
$545$	5.40240e9	1.42955
$546$	−2.78562e8	−0.0732399
$547$	4.07726e9	1.06515	0.532577	−	0.846381i	$-0.321224\pi$
$547$	4.07726e9	1.06515	0.532577	+	0.846381i	$0.321224\pi$
$548$	−1.57406e9	−0.408591
$549$	1.64310e9	0.423799
$550$	5.60955e8	0.143767
$551$	−1.48019e9	−0.376954
$552$	6.91200e6	0.00174911
$553$	3.40530e9	0.856283
$554$	4.44156e9	1.10982
$555$	−2.49894e9	−0.620484
$556$	−3.70784e9	−0.914869
$557$	−5.23587e9	−1.28379	−0.641897	−	0.766791i	$-0.721852\pi$
$557$	−5.23587e9	−1.28379	−0.641897	+	0.766791i	$0.721852\pi$
$558$	−1.51814e9	−0.369907
$559$	−3.07294e7	−0.00744069
$560$	5.89066e8	0.141744
$561$	5.46943e8	0.130789
$562$	2.67965e9	0.636796
$563$	7.04236e9	1.66318	0.831589	−	0.555392i	$-0.187432\pi$
$563$	7.04236e9	1.66318	0.831589	+	0.555392i	$0.187432\pi$
$564$	9.10447e8	0.213687
$565$	4.43156e9	1.03368
$566$	4.61228e9	1.06920
$567$	−3.11956e8	−0.0718708
$568$	4.81284e8	0.110200
$569$	8.80025e8	0.200264	0.100132	−	0.994974i	$-0.468074\pi$
$569$	8.80025e8	0.200264	0.100132	+	0.994974i	$0.468074\pi$
$570$	−3.45144e8	−0.0780618
$571$	8.46430e9	1.90267	0.951337	−	0.308151i	$-0.0997102\pi$
$571$	8.46430e9	1.90267	0.951337	+	0.308151i	$0.0997102\pi$
$572$	5.44715e8	0.121698
$573$	2.52851e9	0.561466
$574$	2.53471e9	0.559419
$575$	9.05000e6	0.00198523
$576$	−3.82206e8	−0.0833333
$577$	−2.74152e8	−0.0594122	−0.0297061	−	0.999559i	$-0.509457\pi$
$577$	−2.74152e8	−0.0594122	−0.0297061	+	0.999559i	$0.509457\pi$
$578$	−3.06397e9	−0.659990
$579$	−3.54283e9	−0.758534
$580$	−3.55864e9	−0.757331
$581$	−4.34310e9	−0.918721
$582$	2.22884e9	0.468649
$583$	6.39187e9	1.33594
$584$	8.56026e8	0.177845
$585$	−7.84790e8	−0.162072
$586$	−1.41919e9	−0.291339
$587$	−4.71162e9	−0.961472	−0.480736	−	0.876865i	$-0.659631\pi$
$587$	−4.71162e9	−0.961472	−0.480736	+	0.876865i	$0.659631\pi$
$588$	8.27667e8	0.167894
$589$	−8.48877e8	−0.171175
$590$	1.59140e8	0.0319005
$591$	3.65378e9	0.728092
$592$	−1.54734e9	−0.306521
$593$	−4.59365e9	−0.904620	−0.452310	−	0.891861i	$-0.649400\pi$
$593$	−4.59365e9	−0.904620	−0.452310	+	0.891861i	$0.649400\pi$
$594$	−3.05008e9	−0.597115
$595$	7.52009e8	0.146357
$596$	−1.86235e9	−0.360330
$597$	3.36164e9	0.646609
$598$	8.78800e6	0.00168049
$599$	−2.06624e9	−0.392815	−0.196407	−	0.980522i	$-0.562927\pi$
$599$	−2.06624e9	−0.392815	−0.196407	+	0.980522i	$0.562927\pi$
$600$	2.50214e8	0.0472915
$601$	8.75128e9	1.64441	0.822206	−	0.569190i	$-0.192743\pi$
$601$	8.75128e9	1.64441	0.822206	+	0.569190i	$0.192743\pi$
$602$	−6.56830e7	−0.0122706
$603$	−6.98087e8	−0.129658
$604$	−2.64863e9	−0.489094
$605$	1.09743e9	0.201480
$606$	−1.02277e9	−0.186690
$607$	−5.99457e9	−1.08792	−0.543961	−	0.839111i	$-0.683076\pi$
$607$	−5.99457e9	−1.08792	−0.543961	+	0.839111i	$0.683076\pi$
$608$	−2.13713e8	−0.0385628
$609$	3.59699e9	0.645326
$610$	2.20883e9	0.394010
$611$	1.15755e9	0.205304
$612$	−4.87928e8	−0.0860451
$613$	−3.28777e9	−0.576487	−0.288244	−	0.957557i	$-0.593071\pi$
$613$	−3.28777e9	−0.576487	−0.288244	+	0.957557i	$0.593071\pi$
$614$	2.35298e9	0.410231
$615$	−3.57051e9	−0.618967
$616$	1.16431e9	0.200694
$617$	2.90178e9	0.497355	0.248677	−	0.968586i	$-0.420004\pi$
$617$	2.90178e9	0.497355	0.248677	+	0.968586i	$0.420004\pi$
$618$	−3.47703e9	−0.592583
$619$	6.02255e9	1.02062	0.510309	−	0.859991i	$-0.329531\pi$
$619$	6.02255e9	1.02062	0.510309	+	0.859991i	$0.329531\pi$
$620$	−2.04085e9	−0.343906
$621$	−4.92075e7	−0.00824538
$622$	−2.28804e9	−0.381239
$623$	5.59854e8	0.0927612
$624$	2.42971e8	0.0400320
$625$	−4.36184e9	−0.714644
$626$	−2.20888e9	−0.359883
$627$	−6.82188e8	−0.110527
$628$	5.50866e8	0.0887538
$629$	−1.97535e9	−0.316496
$630$	−1.67746e9	−0.267276
$631$	−2.50727e8	−0.0397282	−0.0198641	−	0.999803i	$-0.506323\pi$
$631$	−2.50727e8	−0.0397282	−0.0198641	+	0.999803i	$0.506323\pi$
$632$	−2.97021e9	−0.468034
$633$	−2.17738e9	−0.341209
$634$	4.36791e9	0.680709
$635$	6.19184e8	0.0959647
$636$	2.85110e9	0.439453
$637$	1.05231e9	0.161307
$638$	−7.03376e9	−1.07230
$639$	−1.37053e9	−0.207795
$640$	−5.13802e8	−0.0774758
$641$	−1.25739e10	−1.88568	−0.942839	−	0.333248i	$-0.891856\pi$
$641$	−1.25739e10	−1.88568	−0.942839	+	0.333248i	$0.891856\pi$
$642$	−3.84098e9	−0.572888
$643$	1.95354e9	0.289790	0.144895	−	0.989447i	$-0.453716\pi$
$643$	1.95354e9	0.289790	0.144895	+	0.989447i	$0.453716\pi$
$644$	1.87840e7	0.00277133
$645$	9.25240e7	0.0135767
$646$	−2.72828e8	−0.0398176
$647$	−7.96393e9	−1.15601	−0.578006	−	0.816032i	$-0.696169\pi$
$647$	−7.96393e9	−1.15601	−0.578006	+	0.816032i	$0.696169\pi$
$648$	2.72098e8	0.0392837
$649$	3.14546e8	0.0451676
$650$	3.18126e8	0.0454362
$651$	2.06284e9	0.293044
$652$	−5.32591e9	−0.752535
$653$	7.42841e8	0.104400	0.0521999	−	0.998637i	$-0.483377\pi$
$653$	7.42841e8	0.104400	0.0521999	+	0.998637i	$0.483377\pi$
$654$	4.76293e9	0.665812
$655$	−1.02756e10	−1.42876
$656$	−2.21086e9	−0.305772
$657$	−2.43767e9	−0.335348
$658$	2.47422e9	0.338570
$659$	−4.38274e9	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$659$	−4.38274e9	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$660$	−1.64010e9	−0.222058
$661$	1.01201e10	1.36296	0.681478	−	0.731839i	$-0.261338\pi$
$661$	1.01201e10	1.36296	0.681478	+	0.731839i	$0.261338\pi$
$662$	−2.88737e9	−0.386812
$663$	3.10179e8	0.0413347
$664$	3.78819e9	0.502162
$665$	−9.37961e8	−0.123683
$666$	4.40630e9	0.577982
$667$	−1.13477e8	−0.0148070
$668$	7.40918e9	0.961729
$669$	3.52651e9	0.455359
$670$	−9.38444e8	−0.120544
$671$	4.36581e9	0.557874
$672$	5.19340e8	0.0660175
$673$	1.41044e10	1.78362	0.891808	−	0.452415i	$-0.149437\pi$
$673$	1.41044e10	1.78362	0.891808	+	0.452415i	$0.149437\pi$
$674$	3.41816e9	0.430014
$675$	−1.78131e9	−0.222934
$676$	3.08916e8	0.0384615
$677$	−1.31550e10	−1.62942	−0.814708	−	0.579872i	$-0.803103\pi$
$677$	−1.31550e10	−1.62942	−0.814708	+	0.579872i	$0.803103\pi$
$678$	3.90701e9	0.481438
$679$	6.05707e9	0.742538
$680$	−6.55926e8	−0.0799970
$681$	3.89616e9	0.472740
$682$	−4.03379e9	−0.486932
$683$	8.18437e9	0.982907	0.491454	−	0.870904i	$-0.336466\pi$
$683$	8.18437e9	0.982907	0.491454	+	0.870904i	$0.336466\pi$
$684$	6.08581e8	0.0727147
$685$	6.02569e9	0.716292
$686$	6.11662e9	0.723398
$687$	2.78718e9	0.327957
$688$	5.72908e7	0.00670695
$689$	3.62492e9	0.422212
$690$	−2.64600e7	−0.00306632
$691$	−6.59230e9	−0.760088	−0.380044	−	0.924968i	$-0.624091\pi$
$691$	−6.59230e9	−0.760088	−0.380044	+	0.924968i	$0.624091\pi$
$692$	5.82453e9	0.668174
$693$	−3.31555e9	−0.378433
$694$	1.05309e10	1.19594
$695$	1.41941e10	1.60384
$696$	−3.13741e9	−0.352727
$697$	−2.82241e9	−0.315722
$698$	−1.01268e10	−1.12715
$699$	8.62853e8	0.0955580
$700$	6.79981e8	0.0749296
$701$	−7.42154e9	−0.813731	−0.406866	−	0.913488i	$-0.633378\pi$
$701$	−7.42154e9	−0.813731	−0.406866	+	0.913488i	$0.633378\pi$
$702$	−1.72974e9	−0.188713
$703$	2.46381e9	0.267463
$704$	−1.01555e9	−0.109697
$705$	−3.48530e9	−0.374610
$706$	8.84940e9	0.946449
$707$	−2.77946e9	−0.295796
$708$	1.40303e8	0.0148577
$709$	−1.28256e10	−1.35150	−0.675748	−	0.737133i	$-0.736179\pi$
$709$	−1.28256e10	−1.35150	−0.675748	+	0.737133i	$0.736179\pi$
$710$	−1.84241e9	−0.193189
$711$	8.45813e9	0.882533
$712$	−4.88322e8	−0.0507021
$713$	−6.50780e7	−0.00672389
$714$	6.62995e8	0.0681658
$715$	−2.08524e9	−0.213346
$716$	8.84335e9	0.900370
$717$	4.05495e9	0.410836
$718$	−8.23671e9	−0.830459
$719$	−1.34066e8	−0.0134514	−0.00672568	−	0.999977i	$-0.502141\pi$
$719$	−1.34066e8	−0.0134514	−0.00672568	+	0.999977i	$0.502141\pi$
$720$	1.46313e9	0.146090
$721$	−9.44916e9	−0.938900
$722$	−6.81068e9	−0.673458
$723$	−3.67702e9	−0.361836
$724$	6.75312e9	0.661332
$725$	−4.10787e9	−0.400344
$726$	9.67528e8	0.0938394
$727$	8.04352e8	0.0776383	0.0388191	−	0.999246i	$-0.487640\pi$
$727$	8.04352e8	0.0776383	0.0388191	+	0.999246i	$0.487640\pi$
$728$	6.60295e8	0.0634276
$729$	5.03647e9	0.481481
$730$	−3.27697e9	−0.311776
$731$	7.31380e7	0.00692520
$732$	1.94737e9	0.183510
$733$	−1.64250e10	−1.54043	−0.770213	−	0.637786i	$-0.779850\pi$
$733$	−1.64250e10	−1.54043	−0.770213	+	0.637786i	$0.779850\pi$
$734$	6.27613e9	0.585808
$735$	−3.16841e9	−0.294331
$736$	−1.63840e7	−0.00151477
$737$	−1.85486e9	−0.170677
$738$	6.29576e9	0.576569
$739$	−3.94761e9	−0.359815	−0.179907	−	0.983684i	$-0.557580\pi$
$739$	−3.94761e9	−0.359815	−0.179907	+	0.983684i	$0.557580\pi$
$740$	5.92342e9	0.537355
$741$	−3.86879e8	−0.0349310
$742$	7.74812e9	0.696278
$743$	1.81094e10	1.61974	0.809868	−	0.586612i	$-0.199539\pi$
$743$	1.81094e10	1.61974	0.809868	+	0.586612i	$0.199539\pi$
$744$	−1.79928e9	−0.160174
$745$	7.12932e9	0.631686
$746$	−4.42175e9	−0.389950
$747$	−1.07875e10	−0.946886
$748$	−1.29646e9	−0.113267
$749$	−1.04382e10	−0.907696
$750$	−5.09223e9	−0.440751
$751$	−7.45449e9	−0.642211	−0.321106	−	0.947043i	$-0.604054\pi$
$751$	−7.45449e9	−0.642211	−0.321106	+	0.947043i	$0.604054\pi$
$752$	−2.15810e9	−0.185058
$753$	−5.12726e9	−0.437625
$754$	−3.98894e9	−0.338890
$755$	1.01393e10	0.857419
$756$	−3.69725e9	−0.311210
$757$	−1.20015e10	−1.00554	−0.502770	−	0.864420i	$-0.667686\pi$
$757$	−1.20015e10	−1.00554	−0.502770	+	0.864420i	$0.667686\pi$
$758$	5.55008e9	0.462868
$759$	−5.22990e7	−0.00434157
$760$	8.18120e8	0.0676035
$761$	1.21973e10	1.00327	0.501635	−	0.865080i	$-0.332732\pi$
$761$	1.21973e10	1.00327	0.501635	+	0.865080i	$0.332732\pi$
$762$	5.45892e8	0.0446956
$763$	1.29437e10	1.05493
$764$	−5.99350e9	−0.486244
$765$	1.86785e9	0.150844
$766$	−1.25412e10	−1.00818
$767$	1.78383e8	0.0142748
$768$	−4.52985e8	−0.0360844
$769$	−7.65065e8	−0.0606675	−0.0303338	−	0.999540i	$-0.509657\pi$
$769$	−7.65065e8	−0.0606675	−0.0303338	+	0.999540i	$0.509657\pi$
$770$	−4.45711e9	−0.351833
$771$	−4.66695e9	−0.366727
$772$	8.39781e9	0.656910
$773$	−5.75818e8	−0.0448391	−0.0224195	−	0.999749i	$-0.507137\pi$
$773$	−5.75818e8	−0.0448391	−0.0224195	+	0.999749i	$0.507137\pi$
$774$	−1.63144e8	−0.0126467
$775$	−2.35582e9	−0.181797
$776$	−5.28317e9	−0.405862
$777$	−5.98726e9	−0.457883
$778$	−1.69453e10	−1.29009
$779$	3.52031e9	0.266809
$780$	−9.30122e8	−0.0701793
$781$	−3.64159e9	−0.273534
$782$	−2.09160e7	−0.00156407
$783$	2.23357e10	1.66277
$784$	−1.96188e9	−0.145400
$785$	−2.10878e9	−0.155592
$786$	−9.05926e9	−0.665448
$787$	5.85027e9	0.427823	0.213912	−	0.976853i	$-0.431380\pi$
$787$	5.85027e9	0.427823	0.213912	+	0.976853i	$0.431380\pi$
$788$	−8.66081e9	−0.630546
$789$	4.16712e9	0.302041
$790$	1.13703e10	0.820499
$791$	1.06177e10	0.762800
$792$	2.89193e9	0.206847
$793$	2.47591e9	0.176311
$794$	2.92779e9	0.207571
$795$	−1.09144e10	−0.770394
$796$	−7.96834e9	−0.559980
$797$	7.90313e9	0.552962	0.276481	−	0.961019i	$-0.410832\pi$
$797$	7.90313e9	0.552962	0.276481	+	0.961019i	$0.410832\pi$
$798$	−8.26937e8	−0.0576053
$799$	−2.75505e9	−0.191080
$800$	−5.93101e8	−0.0409556
$801$	1.39057e9	0.0956049
$802$	−4.53358e9	−0.310335
$803$	−6.47704e9	−0.441441
$804$	−8.27363e8	−0.0561436
$805$	−7.19075e7	−0.00485835
$806$	−2.28762e9	−0.153890
$807$	3.64397e8	0.0244072
$808$	2.42433e9	0.161679
$809$	−1.36831e10	−0.908582	−0.454291	−	0.890853i	$-0.650107\pi$
$809$	−1.36831e10	−0.908582	−0.454291	+	0.890853i	$0.650107\pi$
$810$	−1.04162e9	−0.0688674
$811$	−2.02619e10	−1.33385	−0.666924	−	0.745126i	$-0.732389\pi$
$811$	−2.02619e10	−1.33385	−0.666924	+	0.745126i	$0.732389\pi$
$812$	−8.52621e9	−0.558869
$813$	1.75214e10	1.14354
$814$	1.17078e10	0.760835
$815$	2.03882e10	1.31925
$816$	−5.78286e8	−0.0372586
$817$	−9.12232e7	−0.00585232
$818$	−1.31728e10	−0.841473
$819$	−1.88029e9	−0.119600
$820$	8.46344e9	0.536041
$821$	1.27444e10	0.803742	0.401871	−	0.915696i	$-0.368360\pi$
$821$	1.27444e10	0.803742	0.401871	+	0.915696i	$0.368360\pi$
$822$	5.31245e9	0.333613
$823$	1.30392e9	0.0815366	0.0407683	−	0.999169i	$-0.487019\pi$
$823$	1.30392e9	0.0815366	0.0407683	+	0.999169i	$0.487019\pi$
$824$	8.24185e9	0.513192
$825$	−1.89322e9	−0.117385
$826$	3.81287e8	0.0235408
$827$	6.58079e9	0.404584	0.202292	−	0.979325i	$-0.435161\pi$
$827$	6.58079e9	0.404584	0.202292	+	0.979325i	$0.435161\pi$
$828$	4.66560e7	0.00285628
$829$	−1.84124e10	−1.12246	−0.561229	−	0.827661i	$-0.689671\pi$
$829$	−1.84124e10	−1.12246	−0.561229	+	0.827661i	$0.689671\pi$
$830$	−1.45017e10	−0.880329
$831$	−1.49903e10	−0.906162
$832$	−5.75930e8	−0.0346688
$833$	−2.50456e9	−0.150132
$834$	1.25139e10	0.746987
$835$	−2.83633e10	−1.68598
$836$	1.61704e9	0.0957191
$837$	1.28093e10	0.755069
$838$	−1.66030e9	−0.0974616
$839$	3.25291e10	1.90154	0.950769	−	0.309901i	$-0.100296\pi$
$839$	3.25291e10	1.90154	0.950769	+	0.309901i	$0.100296\pi$
$840$	−1.98810e9	−0.115734
$841$	3.42582e10	1.98600
$842$	1.02468e10	0.591559
$843$	−9.04380e9	−0.519942
$844$	5.16119e9	0.295496
$845$	−1.18257e9	−0.0674260
$846$	6.14552e9	0.348949
$847$	2.62935e9	0.148681
$848$	−6.75815e9	−0.380577
$849$	−1.55664e10	−0.872996
$850$	−7.57159e8	−0.0422884
$851$	1.88884e8	0.0105061
$852$	−1.62433e9	−0.0899780
$853$	2.08665e10	1.15114	0.575569	−	0.817753i	$-0.304780\pi$
$853$	2.08665e10	1.15114	0.575569	+	0.817753i	$0.304780\pi$
$854$	5.29217e9	0.290758
$855$	−2.32972e9	−0.127474
$856$	9.10455e9	0.496136
$857$	−4.27453e9	−0.231983	−0.115991	−	0.993250i	$-0.537004\pi$
$857$	−4.27453e9	−0.231983	−0.115991	+	0.993250i	$0.537004\pi$
$858$	−1.83841e9	−0.0993660
$859$	−3.16915e8	−0.0170595	−0.00852976	−	0.999964i	$-0.502715\pi$
$859$	−3.16915e8	−0.0170595	−0.00852976	+	0.999964i	$0.502715\pi$
$860$	−2.19316e8	−0.0117578
$861$	−8.55466e9	−0.456764
$862$	−1.04715e9	−0.0556842
$863$	2.64396e10	1.40029	0.700144	−	0.714002i	$-0.253119\pi$
$863$	2.64396e10	1.40029	0.700144	+	0.714002i	$0.253119\pi$
$864$	3.22486e9	0.170103
$865$	−2.22970e10	−1.17136
$866$	−2.41314e9	−0.126261
$867$	1.03409e10	0.538879
$868$	−4.88970e9	−0.253783
$869$	2.24738e10	1.16174
$870$	1.20104e10	0.618358
$871$	−1.05192e9	−0.0539410
$872$	−1.12899e10	−0.576610
$873$	1.50447e10	0.765301
$874$	2.60880e7	0.00132175
$875$	−1.38386e10	−0.698335
$876$	−2.88909e9	−0.145210
$877$	−1.09380e9	−0.0547568	−0.0273784	−	0.999625i	$-0.508716\pi$
$877$	−1.09380e9	−0.0547568	−0.0273784	+	0.999625i	$0.508716\pi$
$878$	6.43193e9	0.320708
$879$	4.78977e9	0.237878
$880$	3.88764e9	0.192308
$881$	−5.62114e8	−0.0276955	−0.0138477	−	0.999904i	$-0.504408\pi$
$881$	−5.62114e8	−0.0276955	−0.0138477	+	0.999904i	$0.504408\pi$
$882$	5.58675e9	0.274170
$883$	6.92318e8	0.0338410	0.0169205	−	0.999857i	$-0.494614\pi$
$883$	6.92318e8	0.0338410	0.0169205	+	0.999857i	$0.494614\pi$
$884$	−7.35239e8	−0.0357969
$885$	−5.37098e8	−0.0260467
$886$	1.42410e9	0.0687896
$887$	9.12313e9	0.438946	0.219473	−	0.975619i	$-0.429566\pi$
$887$	9.12313e9	0.438946	0.219473	+	0.975619i	$0.429566\pi$
$888$	5.22228e9	0.250273
$889$	1.48351e9	0.0708166
$890$	1.86936e9	0.0888848
$891$	−2.05880e9	−0.0975086
$892$	−8.35915e9	−0.394353
$893$	3.43630e9	0.161477
$894$	6.28544e9	0.294208
$895$	−3.38534e10	−1.57842
$896$	−1.23103e9	−0.0571729
$897$	−2.96595e7	−0.00137211
$898$	−1.29298e10	−0.595832
$899$	2.95394e10	1.35595
$900$	1.68895e9	0.0772267
$901$	−8.62754e9	−0.392962
$902$	1.67282e10	0.758975
$903$	2.21680e8	0.0100189
$904$	−9.26105e9	−0.416937
$905$	−2.58518e10	−1.15937
$906$	8.93913e9	0.399343
$907$	−3.52627e10	−1.56924	−0.784622	−	0.619975i	$-0.787143\pi$
$907$	−3.52627e10	−1.56924	−0.784622	+	0.619975i	$0.787143\pi$
$908$	−9.23535e9	−0.409405
$909$	−6.90367e9	−0.304864
$910$	−2.52769e9	−0.111193
$911$	−2.42397e10	−1.06222	−0.531108	−	0.847304i	$-0.678224\pi$
$911$	−2.42397e10	−1.06222	−0.531108	+	0.847304i	$0.678224\pi$
$912$	7.21281e8	0.0314864
$913$	−2.86630e10	−1.24645
$914$	9.90009e9	0.428872
$915$	−7.45479e9	−0.321708
$916$	−6.60666e9	−0.284019
$917$	−2.46194e10	−1.05435
$918$	4.11690e9	0.175639
$919$	−2.87487e10	−1.22184	−0.610920	−	0.791692i	$-0.709200\pi$
$919$	−2.87487e10	−1.22184	−0.610920	+	0.791692i	$0.709200\pi$
$920$	6.27200e7	0.00265551
$921$	−7.94131e9	−0.334952
$922$	1.00543e9	0.0422466
$923$	−2.06520e9	−0.0864480
$924$	−3.92954e9	−0.163866
$925$	6.83762e9	0.284059
$926$	−8.09406e9	−0.334987
$927$	−2.34700e10	−0.967683
$928$	7.43683e9	0.305471
$929$	1.67658e10	0.686072	0.343036	−	0.939322i	$-0.388545\pi$
$929$	1.67658e10	0.686072	0.343036	+	0.939322i	$0.388545\pi$
$930$	6.88786e9	0.280798
$931$	3.12387e9	0.126873
$932$	−2.04528e9	−0.0827556
$933$	7.72214e9	0.311281
$934$	−1.29212e10	−0.518907
$935$	4.96300e9	0.198565
$936$	1.64005e9	0.0653720
$937$	4.14519e9	0.164610	0.0823049	−	0.996607i	$-0.473772\pi$
$937$	4.14519e9	0.164610	0.0823049	+	0.996607i	$0.473772\pi$
$938$	−2.24844e9	−0.0889550
$939$	7.45496e9	0.293843
$940$	8.26146e9	0.324421
$941$	2.40106e10	0.939376	0.469688	−	0.882833i	$-0.344366\pi$
$941$	2.40106e10	0.939376	0.469688	+	0.882833i	$0.344366\pi$
$942$	−1.85917e9	−0.0724672
$943$	2.69880e8	0.0104804
$944$	−3.32571e8	−0.0128671
$945$	1.41536e10	0.545575
$946$	−4.33485e8	−0.0166477
$947$	−5.10040e10	−1.95155	−0.975774	−	0.218783i	$-0.929791\pi$
$947$	−5.10040e10	−1.95155	−0.975774	+	0.218783i	$0.929791\pi$
$948$	1.00245e10	0.382148
$949$	−3.67322e9	−0.139513
$950$	9.44386e8	0.0357369
$951$	−1.47417e10	−0.555796
$952$	−1.57154e9	−0.0590333
$953$	1.53610e10	0.574902	0.287451	−	0.957795i	$-0.407192\pi$
$953$	1.53610e10	0.574902	0.287451	+	0.957795i	$0.407192\pi$
$954$	1.92449e10	0.717623
$955$	2.29439e10	0.852423
$956$	−9.61172e9	−0.355794
$957$	2.37389e10	0.875527
$958$	3.98842e9	0.146562
$959$	1.44371e10	0.528584
$960$	1.73408e9	0.0632587
$961$	−1.05720e10	−0.384261
$962$	6.63967e9	0.240455
$963$	−2.59266e10	−0.935522
$964$	8.71589e9	0.313359
$965$	−3.21479e10	−1.15161
$966$	−6.33960e7	−0.00226278
$967$	3.46233e10	1.23133	0.615666	−	0.788007i	$-0.288887\pi$
$967$	3.46233e10	1.23133	0.615666	+	0.788007i	$0.288887\pi$
$968$	−2.29340e9	−0.0812673
$969$	9.20796e8	0.0325110
$970$	2.02246e10	0.711508
$971$	−2.01887e10	−0.707688	−0.353844	−	0.935304i	$-0.615126\pi$
$971$	−2.01887e10	−0.707688	−0.353844	+	0.935304i	$0.615126\pi$
$972$	−1.46933e10	−0.513200
$973$	3.40078e10	1.18354
$974$	−3.84918e10	−1.33479
$975$	−1.07367e9	−0.0370985
$976$	−4.61600e9	−0.158925
$977$	3.10610e10	1.06558	0.532788	−	0.846249i	$-0.321144\pi$
$977$	3.10610e10	1.06558	0.532788	+	0.846249i	$0.321144\pi$
$978$	1.79749e10	0.614442
$979$	3.69484e9	0.125851
$980$	7.51031e9	0.254898
$981$	3.21498e10	1.08727
$982$	1.20580e10	0.406334
$983$	−3.95227e10	−1.32712	−0.663559	−	0.748124i	$-0.730955\pi$
$983$	−3.95227e10	−1.32712	−0.663559	+	0.748124i	$0.730955\pi$
$984$	7.46164e9	0.249662
$985$	3.31547e10	1.10540
$986$	9.49394e9	0.315411
$987$	−8.35051e9	−0.276441
$988$	9.17045e8	0.0302511
$989$	−6.99350e6	−0.000229883 0
$990$	−1.10707e10	−0.362619
$991$	3.60450e10	1.17649	0.588244	−	0.808684i	$-0.299820\pi$
$991$	3.60450e10	1.17649	0.588244	+	0.808684i	$0.299820\pi$
$992$	4.26495e9	0.138715
$993$	9.74488e9	0.315830
$994$	−4.41427e9	−0.142563
$995$	3.05038e10	0.981688
$996$	−1.27852e10	−0.410014
$997$	1.64165e10	0.524623	0.262311	−	0.964983i	$-0.415515\pi$
$997$	1.64165e10	0.524623	0.262311	+	0.964983i	$0.415515\pi$
$998$	−2.78369e10	−0.886470
$999$	−3.71781e10	−1.17980

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	26.8.a.c.1.1	✓	1
3.2	odd	2		234.8.a.b.1.1		1
4.3	odd	2		208.8.a.a.1.1		1
13.5	odd	4		338.8.b.c.337.1		2
13.8	odd	4		338.8.b.c.337.2		2
13.12	even	2		338.8.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.a.c.1.1	✓	1	1.1	even	1	trivial
208.8.a.a.1.1		1	4.3	odd	2
234.8.a.b.1.1		1	3.2	odd	2
338.8.a.b.1.1		1	13.12	even	2
338.8.b.c.337.1		2	13.5	odd	4
338.8.b.c.337.2		2	13.8	odd	4

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	26.8.a.c.1.1

Level	$26$
Weight	$8$
Character	26.1
Self dual	yes
Analytic conductor	$8.122$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$