LMFDB - Embedded newform 68.11.d.b.67.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,11,Mod(67,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("68.67");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 68 = 2^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	68.d (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$43.2042931818$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			67.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	68.67

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+32.0000 q^{2} -451.134 q^{3} +1024.00 q^{4} -14436.3 q^{6} -18340.9 q^{7} +32768.0 q^{8} +144473. q^{9} +O(q^{10})$	\(q+32.0000 q^{2} -451.134 q^{3} +1024.00 q^{4} -14436.3 q^{6} -18340.9 q^{7} +32768.0 q^{8} +144473. q^{9} -73291.6 q^{11} -461961. q^{12} -742568. q^{13} -586910. q^{14} +1.04858e6 q^{16} -1.41986e6 q^{17} +4.62314e6 q^{18} +8.27422e6 q^{21} -2.34533e6 q^{22} +8.72828e6 q^{23} -1.47828e7 q^{24} +9.76562e6 q^{25} -2.37622e7 q^{26} -3.85377e7 q^{27} -1.87811e7 q^{28} +5.15227e7 q^{31} +3.35544e7 q^{32} +3.30644e7 q^{33} -4.54354e7 q^{34} +1.47940e8 q^{36} +3.34998e8 q^{39} +2.64775e8 q^{42} -7.50506e7 q^{44} +2.79305e8 q^{46} -4.73048e8 q^{48} +5.39147e7 q^{49} +3.12500e8 q^{50} +6.40546e8 q^{51} -7.60390e8 q^{52} +8.07749e8 q^{53} -1.23321e9 q^{54} -6.00996e8 q^{56} +1.64873e9 q^{62} -2.64977e9 q^{63} +1.07374e9 q^{64} +1.05806e9 q^{66} -1.45393e9 q^{68} -3.93762e9 q^{69} +3.58911e9 q^{71} +4.73409e9 q^{72} -4.40561e9 q^{75} +1.34424e9 q^{77} +1.07199e10 q^{78} -5.96381e9 q^{79} +8.85468e9 q^{81} +8.47280e9 q^{84} -2.40162e9 q^{88} +8.54293e9 q^{89} +1.36194e10 q^{91} +8.93776e9 q^{92} -2.32436e10 q^{93} -1.51375e10 q^{96} +1.72527e9 q^{98} -1.05887e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 64 q^{2} + 2048 q^{4} + 65536 q^{8} + 288946 q^{9}+O(q^{10}) $ 2 * q + 64 * q^2 + 2048 * q^4 + 65536 * q^8 + 288946 * q^9	$ 2 q + 64 q^{2} + 2048 q^{4} + 65536 q^{8} + 288946 q^{9} - 1485136 q^{13} + 2097152 q^{16} - 2839714 q^{17} + 9246272 q^{18} + 16548444 q^{21} + 19531250 q^{25} - 47524352 q^{26} + 67108864 q^{32} + 66128700 q^{33} - 90870848 q^{34} + 295880704 q^{36} + 529550208 q^{42} + 107829346 q^{49} + 625000000 q^{50} - 1520779264 q^{52} + 1615498636 q^{53} + 2147483648 q^{64} + 2116118400 q^{66} - 2907867136 q^{68} - 7875248700 q^{69} + 9468182528 q^{72} + 2688473700 q^{77} + 17709354302 q^{81} + 16945606656 q^{84} + 17085864704 q^{89} - 46487274108 q^{93} + 3450539072 q^{98}+O(q^{100}) $ 2 * q + 64 * q^2 + 2048 * q^4 + 65536 * q^8 + 288946 * q^9 - 1485136 * q^13 + 2097152 * q^16 - 2839714 * q^17 + 9246272 * q^18 + 16548444 * q^21 + 19531250 * q^25 - 47524352 * q^26 + 67108864 * q^32 + 66128700 * q^33 - 90870848 * q^34 + 295880704 * q^36 + 529550208 * q^42 + 107829346 * q^49 + 625000000 * q^50 - 1520779264 * q^52 + 1615498636 * q^53 + 2147483648 * q^64 + 2116118400 * q^66 - 2907867136 * q^68 - 7875248700 * q^69 + 9468182528 * q^72 + 2688473700 * q^77 + 17709354302 * q^81 + 16945606656 * q^84 + 17085864704 * q^89 - 46487274108 * q^93 + 3450539072 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$-1$	$-1$

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 $

$2$	32.0000	1.00000
$2$	32.0000	1.00000
$3$	−451.134	−1.85652	−0.928260	−	0.371933i	$-0.878695\pi$
$3$	−451.134	−1.85652	−0.928260	+	0.371933i	$0.878695\pi$
$4$	1024.00	1.00000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	−14436.3	−1.85652
$7$	−18340.9	−1.09127	−0.545634	−	0.838024i	$-0.683711\pi$
$7$	−18340.9	−1.09127	−0.545634	+	0.838024i	$0.683711\pi$
$8$	32768.0	1.00000
$9$	144473.	2.44666
$10$	0	0
$11$	−73291.6	−0.455083	−0.227542	−	0.973768i	$-0.573069\pi$
$11$	−73291.6	−0.455083	−0.227542	+	0.973768i	$0.573069\pi$
$12$	−461961.	−1.85652
$13$	−742568.	−1.99995	−0.999976	−	0.00696266i	$-0.997784\pi$
$13$	−742568.	−1.99995	−0.999976	+	0.00696266i	$0.997784\pi$
$14$	−586910.	−1.09127
$15$	0	0
$16$	1.04858e6	1.00000
$17$	−1.41986e6	−1.00000
$17$	−1.41986e6	−1.00000
$18$	4.62314e6	2.44666
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	8.27422e6	2.02596
$22$	−2.34533e6	−0.455083
$23$	8.72828e6	1.35609	0.678046	−	0.735019i	$-0.262827\pi$
$23$	8.72828e6	1.35609	0.678046	+	0.735019i	$0.262827\pi$
$24$	−1.47828e7	−1.85652
$25$	9.76562e6	1.00000
$26$	−2.37622e7	−1.99995
$27$	−3.85377e7	−2.68576
$28$	−1.87811e7	−1.09127
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	5.15227e7	1.79966	0.899829	−	0.436243i	$-0.143691\pi$
$31$	5.15227e7	1.79966	0.899829	+	0.436243i	$0.143691\pi$
$32$	3.35544e7	1.00000
$33$	3.30644e7	0.844871
$34$	−4.54354e7	−1.00000
$35$	0	0
$36$	1.47940e8	2.44666
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	3.34998e8	3.71295
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	2.64775e8	2.02596
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−7.50506e7	−0.455083
$45$	0	0
$46$	2.79305e8	1.35609
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−4.73048e8	−1.85652
$49$	5.39147e7	0.190865
$50$	3.12500e8	1.00000
$51$	6.40546e8	1.85652
$52$	−7.60390e8	−1.99995
$53$	8.07749e8	1.93151	0.965756	−	0.259453i	$-0.0835424\pi$
$53$	8.07749e8	1.93151	0.965756	+	0.259453i	$0.0835424\pi$
$54$	−1.23321e9	−2.68576
$55$	0	0
$56$	−6.00996e8	−1.09127
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	1.64873e9	1.79966
$63$	−2.64977e9	−2.66996
$64$	1.07374e9	1.00000
$65$	0	0
$66$	1.05806e9	0.844871
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.45393e9	−1.00000
$69$	−3.93762e9	−2.51761
$70$	0	0
$71$	3.58911e9	1.98928	0.994638	−	0.103415i	$-0.0329771\pi$
$71$	3.58911e9	1.98928	0.994638	+	0.103415i	$0.0329771\pi$
$72$	4.73409e9	2.44666
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−4.40561e9	−1.85652
$76$	0	0
$77$	1.34424e9	0.496618
$78$	1.07199e10	3.71295
$79$	−5.96381e9	−1.93815	−0.969077	−	0.246758i	$-0.920635\pi$
$79$	−5.96381e9	−1.93815	−0.969077	+	0.246758i	$0.920635\pi$
$80$	0	0
$81$	8.85468e9	2.53950
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	8.47280e9	2.02596
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−2.40162e9	−0.455083
$89$	8.54293e9	1.52988	0.764939	−	0.644102i	$-0.222769\pi$
$89$	8.54293e9	1.52988	0.764939	+	0.644102i	$0.222769\pi$
$90$	0	0
$91$	1.36194e10	2.18248
$92$	8.93776e9	1.35609
$93$	−2.32436e10	−3.34110
$94$	0	0
$95$	0	0
$96$	−1.51375e10	−1.85652
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	1.72527e9	0.190865
$99$	−1.05887e10	−1.11344
$100$	1.00000e10	1.00000
$101$	−8.90425e9	−0.847209	−0.423604	−	0.905847i	$-0.639235\pi$
$101$	−8.90425e9	−0.847209	−0.423604	+	0.905847i	$0.639235\pi$
$102$	2.04975e10	1.85652
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−2.43325e10	−1.99995
$105$	0	0
$106$	2.58480e10	1.93151
$107$	−8.75355e9	−0.624116	−0.312058	−	0.950063i	$-0.601018\pi$
$107$	−8.75355e9	−0.624116	−0.312058	+	0.950063i	$0.601018\pi$
$108$	−3.94626e10	−2.68576
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	−1.92319e10	−1.09127
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.07281e11	−4.89321
$118$	0	0
$119$	2.60415e10	1.09127
$120$	0	0
$121$	−2.05658e10	−0.792899
$122$	0	0
$123$	0	0
$124$	5.27592e10	1.79966
$125$	0	0
$126$	−8.47926e10	−2.66996
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	3.43597e10	1.00000
$129$	0	0
$130$	0	0
$131$	5.47377e10	1.41883	0.709414	−	0.704792i	$-0.248960\pi$
$131$	5.47377e10	1.41883	0.709414	+	0.704792i	$0.248960\pi$
$132$	3.38579e10	0.844871
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−4.65259e10	−1.00000
$137$	−5.33225e10	−1.10486	−0.552431	−	0.833559i	$-0.686300\pi$
$137$	−5.33225e10	−1.10486	−0.552431	+	0.833559i	$0.686300\pi$
$138$	−1.26004e11	−2.51761
$139$	7.75100e10	1.49377	0.746885	−	0.664954i	$-0.231549\pi$
$139$	7.75100e10	1.49377	0.746885	+	0.664954i	$0.231549\pi$
$140$	0	0
$141$	0	0
$142$	1.14852e11	1.98928
$143$	5.44240e10	0.910145
$144$	1.51491e11	2.44666
$145$	0	0
$146$	0	0
$147$	−2.43227e10	−0.354345
$148$	0	0
$149$	6.21358e10	0.846079	0.423039	−	0.906111i	$-0.360963\pi$
$149$	6.21358e10	0.846079	0.423039	+	0.906111i	$0.360963\pi$
$150$	−1.40979e11	−1.85652
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−2.05131e11	−2.44666
$154$	4.30156e10	0.496618
$155$	0	0
$156$	3.43038e11	3.71295
$157$	6.14850e9	0.0644571	0.0322286	−	0.999481i	$-0.489740\pi$
$157$	6.14850e9	0.0644571	0.0322286	+	0.999481i	$0.489740\pi$
$158$	−1.90842e11	−1.93815
$159$	−3.64403e11	−3.58589
$160$	0	0
$161$	−1.60085e11	−1.47986
$162$	2.83350e11	2.53950
$163$	−1.35506e11	−1.17766	−0.588832	−	0.808255i	$-0.700412\pi$
$163$	−1.35506e11	−1.17766	−0.588832	+	0.808255i	$0.700412\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.57385e10	−0.198153	−0.0990766	−	0.995080i	$-0.531589\pi$
$167$	−2.57385e10	−0.198153	−0.0990766	+	0.995080i	$0.531589\pi$
$168$	2.71130e11	2.02596
$169$	4.13549e11	2.99981
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−1.79111e11	−1.09127
$176$	−7.68518e10	−0.455083
$177$	0	0
$178$	2.73374e11	1.52988
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	4.35821e11	2.18248
$183$	0	0
$184$	2.86008e11	1.35609
$185$	0	0
$186$	−7.43796e11	−3.34110
$187$	1.04064e11	0.455083
$188$	0	0
$189$	7.06817e11	2.93088
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−4.84402e11	−1.85652
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	5.52086e10	0.190865
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−3.38837e11	−1.11344
$199$	6.22156e11	1.99358	0.996791	−	0.0800532i	$-0.0255090\pi$
$199$	6.22156e11	1.99358	0.996791	+	0.0800532i	$0.0255090\pi$
$200$	3.20000e11	1.00000
$201$	0	0
$202$	−2.84936e11	−0.847209
$203$	0	0
$204$	6.55919e11	1.85652
$205$	0	0
$206$	0	0
$207$	1.26100e12	3.31790
$208$	−7.78639e11	−1.99995
$209$	0	0
$210$	0	0
$211$	−7.40237e11	−1.76994	−0.884970	−	0.465648i	$-0.845821\pi$
$211$	−7.40237e11	−1.76994	−0.884970	+	0.465648i	$0.845821\pi$
$212$	8.27135e11	1.93151
$213$	−1.61917e12	−3.69313
$214$	−2.80114e11	−0.624116
$215$	0	0
$216$	−1.26280e12	−2.68576
$217$	−9.44974e11	−1.96391
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.05434e12	1.99995
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−6.15420e11	−1.09127
$225$	1.41087e12	2.44666
$226$	0	0
$227$	1.20299e12	1.99588	0.997939	−	0.0641752i	$-0.0204416\pi$
$227$	1.20299e12	1.99588	0.997939	+	0.0641752i	$0.0204416\pi$
$228$	0	0
$229$	−4.53029e11	−0.719363	−0.359682	−	0.933075i	$-0.617115\pi$
$229$	−4.53029e11	−0.719363	−0.359682	+	0.933075i	$0.617115\pi$
$230$	0	0
$231$	−6.06431e11	−0.921980
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	−3.43299e12	−4.89321
$235$	0	0
$236$	0	0
$237$	2.69048e12	3.59822
$238$	8.33328e11	1.09127
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−6.58104e11	−0.792899
$243$	−1.71904e12	−2.02887
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	1.68830e12	1.79966
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−2.71336e12	−2.66996
$253$	−6.39710e11	−0.617135
$254$	0	0
$255$	0	0
$256$	1.09951e12	1.00000
$257$	1.95445e12	1.74325	0.871625	−	0.490174i	$-0.163067\pi$
$257$	1.95445e12	1.74325	0.871625	+	0.490174i	$0.163067\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	1.75161e12	1.41883
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	1.08345e12	0.844871
$265$	0	0
$266$	0	0
$267$	−3.85401e12	−2.84025
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−1.48883e12	−1.00000
$273$	−6.14417e12	−4.05182
$274$	−1.70632e12	−1.10486
$275$	−7.15738e11	−0.455083
$276$	−4.03213e12	−2.51761
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	2.48032e12	1.49377
$279$	7.44364e12	4.40316
$280$	0	0
$281$	7.93872e11	0.453126	0.226563	−	0.973997i	$-0.427251\pi$
$281$	7.93872e11	0.453126	0.226563	+	0.973997i	$0.427251\pi$
$282$	0	0
$283$	−6.76515e11	−0.372688	−0.186344	−	0.982485i	$-0.559664\pi$
$283$	−6.76515e11	−0.372688	−0.186344	+	0.982485i	$0.559664\pi$
$284$	3.67525e12	1.98928
$285$	0	0
$286$	1.74157e12	0.910145
$287$	0	0
$288$	4.84771e12	2.44666
$289$	2.01599e12	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.33437e12	0.617928	0.308964	−	0.951074i	$-0.400018\pi$
$293$	1.33437e12	0.617928	0.308964	+	0.951074i	$0.400018\pi$
$294$	−7.78328e11	−0.354345
$295$	0	0
$296$	0	0
$297$	2.82449e12	1.22224
$298$	1.98835e12	0.846079
$299$	−6.48134e12	−2.71212
$300$	−4.51134e12	−1.85652
$301$	0	0
$302$	0	0
$303$	4.01701e12	1.57286
$304$	0	0
$305$	0	0
$306$	−6.56419e12	−2.44666
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	1.37650e12	0.496618
$309$	0	0
$310$	0	0
$311$	−4.88355e12	−1.67855	−0.839275	−	0.543708i	$-0.817020\pi$
$311$	−4.88355e12	−1.67855	−0.839275	+	0.543708i	$0.817020\pi$
$312$	1.09772e13	3.71295
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	1.96752e11	0.0644571
$315$	0	0
$316$	−6.10694e12	−1.93815
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−1.16609e13	−3.58589
$319$	0	0
$320$	0	0
$321$	3.94903e12	1.15868
$322$	−5.12271e12	−1.47986
$323$	0	0
$324$	9.06719e12	2.53950
$325$	−7.25164e12	−1.99995
$326$	−4.33620e12	−1.17766
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	−8.23632e11	−0.198153
$335$	0	0
$336$	8.67615e12	2.02596
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	1.32336e13	2.99981
$339$	0	0
$340$	0	0
$341$	−3.77618e12	−0.818994
$342$	0	0
$343$	4.19201e12	0.882983
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.21427e11	0.183153	0.0915764	−	0.995798i	$-0.470809\pi$
$347$	9.21427e11	0.183153	0.0915764	+	0.995798i	$0.470809\pi$
$348$	0	0
$349$	5.99660e12	1.15819	0.579093	−	0.815262i	$-0.303407\pi$
$349$	5.99660e12	1.15819	0.579093	+	0.815262i	$0.303407\pi$
$350$	−5.73154e12	−1.09127
$351$	2.86168e13	5.37138
$352$	−2.45926e12	−0.455083
$353$	−3.06120e12	−0.558493	−0.279247	−	0.960219i	$-0.590085\pi$
$353$	−3.06120e12	−0.558493	−0.279247	+	0.960219i	$0.590085\pi$
$354$	0	0
$355$	0	0
$356$	8.74796e12	1.52988
$357$	−1.17482e13	−2.02596
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	6.13107e12	1.00000
$362$	0	0
$363$	9.27792e12	1.47203
$364$	1.39463e13	2.18248
$365$	0	0
$366$	0	0
$367$	−1.33051e13	−1.99842	−0.999211	−	0.0397104i	$-0.987356\pi$
$367$	−1.33051e13	−1.99842	−0.999211	+	0.0397104i	$0.987356\pi$
$368$	9.15226e12	1.35609
$369$	0	0
$370$	0	0
$371$	−1.48149e13	−2.10780
$372$	−2.38015e13	−3.34110
$373$	7.99970e11	0.110797	0.0553987	−	0.998464i	$-0.482357\pi$
$373$	7.99970e11	0.110797	0.0553987	+	0.998464i	$0.482357\pi$
$374$	3.33004e12	0.455083
$375$	0	0
$376$	0	0
$377$	0	0
$378$	2.26181e13	2.93088
$379$	−2.17117e12	−0.277650	−0.138825	−	0.990317i	$-0.544333\pi$
$379$	−2.17117e12	−0.277650	−0.138825	+	0.990317i	$0.544333\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−1.55009e13	−1.85652
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.09405e12	−0.235093	−0.117546	−	0.993067i	$-0.537503\pi$
$389$	−2.09405e12	−0.235093	−0.117546	+	0.993067i	$0.537503\pi$
$390$	0	0
$391$	−1.23929e13	−1.35609
$392$	1.76668e12	0.190865
$393$	−2.46940e13	−2.63408
$394$	0	0
$395$	0	0
$396$	−1.08428e13	−1.11344
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	1.99090e13	1.99358
$399$	0	0
$400$	1.02400e13	1.00000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−3.82591e13	−3.59923
$404$	−9.11795e12	−0.847209
$405$	0	0
$406$	0	0
$407$	0	0
$408$	2.09894e13	1.85652
$409$	2.18817e13	1.91190	0.955948	−	0.293536i	$-0.0948319\pi$
$409$	2.18817e13	1.91190	0.955948	+	0.293536i	$0.0948319\pi$
$410$	0	0
$411$	2.40556e13	2.05120
$412$	0	0
$413$	0	0
$414$	4.03520e13	3.31790
$415$	0	0
$416$	−2.49164e13	−1.99995
$417$	−3.49674e13	−2.77321
$418$	0	0
$419$	−2.51390e13	−1.94660	−0.973302	−	0.229527i	$-0.926282\pi$
$419$	−2.51390e13	−1.94660	−0.973302	+	0.229527i	$0.926282\pi$
$420$	0	0
$421$	−1.34875e13	−1.01982	−0.509908	−	0.860229i	$-0.670320\pi$
$421$	−1.34875e13	−1.01982	−0.509908	+	0.860229i	$0.670320\pi$
$422$	−2.36876e13	−1.76994
$423$	0	0
$424$	2.64683e13	1.93151
$425$	−1.38658e13	−1.00000
$426$	−5.18135e13	−3.69313
$427$	0	0
$428$	−8.96364e12	−0.624116
$429$	−2.45525e13	−1.68970
$430$	0	0
$431$	1.05334e13	0.708244	0.354122	−	0.935199i	$-0.384780\pi$
$431$	1.05334e13	0.708244	0.354122	+	0.935199i	$0.384780\pi$
$432$	−4.04097e13	−2.68576
$433$	2.80833e12	0.184505	0.0922527	−	0.995736i	$-0.470593\pi$
$433$	2.80833e12	0.184505	0.0922527	+	0.995736i	$0.470593\pi$
$434$	−3.02392e13	−1.96391
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	2.67122e13	1.63828	0.819140	−	0.573594i	$-0.194451\pi$
$439$	2.67122e13	1.63828	0.819140	+	0.573594i	$0.194451\pi$
$440$	0	0
$441$	7.78921e12	0.466983
$442$	3.37389e13	1.99995
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.80316e13	−1.57076
$448$	−1.96934e13	−1.09127
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	4.51478e13	2.44666
$451$	0	0
$452$	0	0
$453$	0	0
$454$	3.84958e13	1.99588
$455$	0	0
$456$	0	0
$457$	3.95856e13	1.98589	0.992947	−	0.118556i	$-0.0378264\pi$
$457$	3.95856e13	1.98589	0.992947	+	0.118556i	$0.0378264\pi$
$458$	−1.44969e13	−0.719363
$459$	5.47180e13	2.68576
$460$	0	0
$461$	3.96649e13	1.90503	0.952516	−	0.304490i	$-0.0984859\pi$
$461$	3.96649e13	1.90503	0.952516	+	0.304490i	$0.0984859\pi$
$462$	−1.94058e13	−0.921980
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	−1.09856e14	−4.89321
$469$	0	0
$470$	0	0
$471$	−2.77380e12	−0.119666
$472$	0	0
$473$	0	0
$474$	8.60953e13	3.59822
$475$	0	0
$476$	2.66665e13	1.09127
$477$	1.16698e14	4.72576
$478$	0	0
$479$	4.25231e13	1.68635	0.843175	−	0.537639i	$-0.180684\pi$
$479$	4.25231e13	1.68635	0.843175	+	0.537639i	$0.180684\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	7.22197e13	2.74739
$484$	−2.10593e13	−0.792899
$485$	0	0
$486$	−5.50091e13	−2.02887
$487$	−5.95481e12	−0.217382	−0.108691	−	0.994076i	$-0.534666\pi$
$487$	−5.95481e12	−0.217382	−0.108691	+	0.994076i	$0.534666\pi$
$488$	0	0
$489$	6.11315e13	2.18636
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	5.40254e13	1.79966
$497$	−6.58277e13	−2.17083
$498$	0	0
$499$	−4.46575e12	−0.144342	−0.0721709	−	0.997392i	$-0.522993\pi$
$499$	−4.46575e12	−0.144342	−0.0721709	+	0.997392i	$0.522993\pi$
$500$	0	0
$501$	1.16115e13	0.367875
$502$	0	0
$503$	5.05342e13	1.56944	0.784721	−	0.619849i	$-0.212806\pi$
$503$	5.05342e13	1.56944	0.784721	+	0.619849i	$0.212806\pi$
$504$	−8.68277e13	−2.66996
$505$	0	0
$506$	−2.04707e13	−0.617135
$507$	−1.86566e14	−5.56920
$508$	0	0
$509$	−5.63892e13	−1.65047	−0.825234	−	0.564791i	$-0.808957\pi$
$509$	−5.63892e13	−1.65047	−0.825234	+	0.564791i	$0.808957\pi$
$510$	0	0
$511$	0	0
$512$	3.51844e13	1.00000
$513$	0	0
$514$	6.25425e13	1.74325
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	5.60514e13	1.41883
$525$	8.08029e13	2.02596
$526$	0	0
$527$	−7.31548e13	−1.79966
$528$	3.46705e13	0.844871
$529$	3.47563e13	0.838988
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−1.23328e14	−2.84025
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.95149e12	−0.0868595
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	−4.76425e13	−1.00000
$545$	0	0
$546$	−1.96614e14	−4.05182
$547$	7.67454e13	1.56717	0.783584	−	0.621286i	$-0.213389\pi$
$547$	7.67454e13	1.56717	0.783584	+	0.621286i	$0.213389\pi$
$548$	−5.46022e13	−1.10486
$549$	0	0
$550$	−2.29036e13	−0.455083
$551$	0	0
$552$	−1.29028e14	−2.51761
$553$	1.09382e14	2.11504
$554$	0	0
$555$	0	0
$556$	7.93702e13	1.49377
$557$	−8.50407e13	−1.58617	−0.793087	−	0.609108i	$-0.791528\pi$
$557$	−8.50407e13	−1.58617	−0.793087	+	0.609108i	$0.791528\pi$
$558$	2.38196e14	4.40316
$559$	0	0
$560$	0	0
$561$	−4.69466e13	−0.844871
$562$	2.54039e13	0.453126
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	−2.16485e13	−0.372688
$567$	−1.62403e14	−2.77127
$568$	1.17608e14	1.98928
$569$	−6.32663e13	−1.06075	−0.530373	−	0.847765i	$-0.677948\pi$
$569$	−6.32663e13	−1.06075	−0.530373	+	0.847765i	$0.677948\pi$
$570$	0	0
$571$	1.19464e14	1.96814	0.984068	−	0.177795i	$-0.0568963\pi$
$571$	1.19464e14	1.96814	0.984068	+	0.177795i	$0.0568963\pi$
$572$	5.57302e13	0.910145
$573$	0	0
$574$	0	0
$575$	8.52371e13	1.35609
$576$	1.55127e14	2.44666
$577$	1.24308e14	1.94367	0.971833	−	0.235670i	$-0.0757285\pi$
$577$	1.24308e14	1.94367	0.971833	+	0.235670i	$0.0757285\pi$
$578$	6.45118e13	1.00000
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.92013e13	−0.878999
$584$	0	0
$585$	0	0
$586$	4.26998e13	0.617928
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−2.49065e13	−0.354345
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.90927e13	−0.260371	−0.130186	−	0.991490i	$-0.541557\pi$
$593$	−1.90927e13	−0.260371	−0.130186	+	0.991490i	$0.541557\pi$
$594$	9.03836e13	1.22224
$595$	0	0
$596$	6.36271e13	0.846079
$597$	−2.80676e14	−3.70112
$598$	−2.07403e14	−2.71212
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−1.44363e14	−1.85652
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	1.28544e14	1.57286
$607$	−1.00430e14	−1.21876	−0.609382	−	0.792877i	$-0.708582\pi$
$607$	−1.00430e14	−1.21876	−0.609382	+	0.792877i	$0.708582\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−2.10054e14	−2.44666
$613$	−1.59177e14	−1.83899	−0.919494	−	0.393105i	$-0.871401\pi$
$613$	−1.59177e14	−1.83899	−0.919494	+	0.393105i	$0.871401\pi$
$614$	0	0
$615$	0	0
$616$	4.40480e13	0.496618
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−2.42985e13	−0.267378	−0.133689	−	0.991023i	$-0.542682\pi$
$619$	−2.42985e13	−0.267378	−0.133689	+	0.991023i	$0.542682\pi$
$620$	0	0
$621$	−3.36368e14	−3.64214
$622$	−1.56274e14	−1.67855
$623$	−1.56685e14	−1.66951
$624$	3.51271e14	3.71295
$625$	9.53674e13	1.00000
$626$	0	0
$627$	0	0
$628$	6.29607e12	0.0644571
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	−1.95422e14	−1.93815
$633$	3.33946e14	3.28593
$634$	0	0
$635$	0	0
$636$	−3.73149e14	−3.58589
$637$	−4.00353e13	−0.381721
$638$	0	0
$639$	5.18530e14	4.86709
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	1.26369e14	1.15868
$643$	5.67049e13	0.515900	0.257950	−	0.966158i	$-0.416953\pi$
$643$	5.67049e13	0.515900	0.257950	+	0.966158i	$0.416953\pi$
$644$	−1.63927e14	−1.47986
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	2.90150e14	2.53950
$649$	0	0
$650$	−2.32052e14	−1.99995
$651$	4.26310e14	3.64603
$652$	−1.38759e14	−1.17766
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	2.33982e14	1.85428	0.927138	−	0.374720i	$-0.122261\pi$
$661$	2.33982e14	1.85428	0.927138	+	0.374720i	$0.122261\pi$
$662$	0	0
$663$	−4.75649e14	−3.71295
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−2.63562e13	−0.198153
$669$	0	0
$670$	0	0
$671$	0	0
$672$	2.77637e14	2.02596
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−3.76345e14	−2.68576
$676$	4.23474e14	2.99981
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5.42711e14	−3.70538
$682$	−1.20838e14	−0.818994
$683$	1.89490e14	1.27492	0.637460	−	0.770484i	$-0.279985\pi$
$683$	1.89490e14	1.27492	0.637460	+	0.770484i	$0.279985\pi$
$684$	0	0
$685$	0	0
$686$	1.34144e14	0.882983
$687$	2.04377e14	1.33551
$688$	0	0
$689$	−5.99809e14	−3.86293
$690$	0	0
$691$	2.76722e14	1.75652	0.878262	−	0.478180i	$-0.158703\pi$
$691$	2.76722e14	1.75652	0.878262	+	0.478180i	$0.158703\pi$
$692$	0	0
$693$	1.94206e14	1.21506
$694$	2.94857e13	0.183153
$695$	0	0
$696$	0	0
$697$	0	0
$698$	1.91891e14	1.15819
$699$	0	0
$700$	−1.83409e14	−1.09127
$701$	−7.76568e13	−0.458764	−0.229382	−	0.973336i	$-0.573671\pi$
$701$	−7.76568e13	−0.458764	−0.229382	+	0.973336i	$0.573671\pi$
$702$	9.15739e14	5.37138
$703$	0	0
$704$	−7.86963e13	−0.455083
$705$	0	0
$706$	−9.79584e13	−0.558493
$707$	1.63312e14	0.924532
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−8.61609e14	−4.74201
$712$	2.79935e14	1.52988
$713$	4.49704e14	2.44050
$714$	−3.75943e14	−2.02596
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.32138e13	−0.433063	−0.216531	−	0.976276i	$-0.569474\pi$
$719$	−8.32138e13	−0.433063	−0.216531	+	0.976276i	$0.569474\pi$
$720$	0	0
$721$	0	0
$722$	1.96194e14	1.00000
$723$	0	0
$724$	0	0
$725$	0	0
$726$	2.96893e14	1.47203
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	4.46280e14	2.18248
$729$	2.52656e14	1.22713
$730$	0	0
$731$	0	0
$732$	0	0
$733$	2.13374e14	1.00837	0.504187	−	0.863594i	$-0.331792\pi$
$733$	2.13374e14	1.00837	0.504187	+	0.863594i	$0.331792\pi$
$734$	−4.25763e14	−1.99842
$735$	0	0
$736$	2.92872e14	1.35609
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	−4.74076e14	−2.10780
$743$	−2.31708e12	−0.0102329	−0.00511643	−	0.999987i	$-0.501629\pi$
$743$	−2.31708e12	−0.0102329	−0.00511643	+	0.999987i	$0.501629\pi$
$744$	−7.61647e14	−3.34110
$745$	0	0
$746$	2.55990e13	0.110797
$747$	0	0
$748$	1.06561e14	0.455083
$749$	1.60548e14	0.681078
$750$	0	0
$751$	−4.64814e14	−1.94571	−0.972857	−	0.231406i	$-0.925667\pi$
$751$	−4.64814e14	−1.94571	−0.972857	+	0.231406i	$0.925667\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	7.23781e14	2.93088
$757$	3.80499e14	1.53064	0.765321	−	0.643649i	$-0.222580\pi$
$757$	3.80499e14	1.53064	0.765321	+	0.643649i	$0.222580\pi$
$758$	−6.94775e13	−0.277650
$759$	2.88595e14	1.14572
$760$	0	0
$761$	−5.08622e14	−1.99284	−0.996418	−	0.0845620i	$-0.973051\pi$
$761$	−5.08622e14	−1.99284	−0.996418	+	0.0845620i	$0.973051\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−4.96027e14	−1.85652
$769$	2.66081e14	0.989422	0.494711	−	0.869058i	$-0.335274\pi$
$769$	2.66081e14	0.989422	0.494711	+	0.869058i	$0.335274\pi$
$770$	0	0
$771$	−8.81720e14	−3.23638
$772$	0	0
$773$	5.37689e14	1.94820	0.974100	−	0.226116i	$-0.0726028\pi$
$773$	5.37689e14	1.94820	0.974100	+	0.226116i	$0.0726028\pi$
$774$	0	0
$775$	5.03151e14	1.79966
$776$	0	0
$777$	0	0
$778$	−6.70097e13	−0.235093
$779$	0	0
$780$	0	0
$781$	−2.63052e14	−0.905286
$782$	−3.96573e14	−1.35609
$783$	0	0
$784$	5.65336e13	0.190865
$785$	0	0
$786$	−7.90209e14	−2.63408
$787$	3.94447e14	1.30652	0.653258	−	0.757136i	$-0.273402\pi$
$787$	3.94447e14	1.30652	0.653258	+	0.757136i	$0.273402\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−3.46969e14	−1.11344
$793$	0	0
$794$	0	0
$795$	0	0
$796$	6.37088e14	1.99358
$797$	−6.33068e14	−1.96861	−0.984303	−	0.176489i	$-0.943526\pi$
$797$	−6.33068e14	−1.96861	−0.984303	+	0.176489i	$0.943526\pi$
$798$	0	0
$799$	0	0
$800$	3.27680e14	1.00000
$801$	1.23422e15	3.74310
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−1.22429e15	−3.59923
$807$	0	0
$808$	−2.91774e14	−0.847209
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	−6.42426e14	−1.83113	−0.915564	−	0.402172i	$-0.868255\pi$
$811$	−6.42426e14	−1.83113	−0.915564	+	0.402172i	$0.868255\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	6.71661e14	1.85652
$817$	0	0
$818$	7.00214e14	1.91190
$819$	1.96763e15	5.33980
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	7.69779e14	2.05120
$823$	3.47082e14	0.919248	0.459624	−	0.888114i	$-0.347984\pi$
$823$	3.47082e14	0.919248	0.459624	+	0.888114i	$0.347984\pi$
$824$	0	0
$825$	3.22894e14	0.844871
$826$	0	0
$827$	6.32235e14	1.63437	0.817186	−	0.576374i	$-0.195533\pi$
$827$	6.32235e14	1.63437	0.817186	+	0.576374i	$0.195533\pi$
$828$	1.29126e15	3.31790
$829$	−2.10263e14	−0.537019	−0.268510	−	0.963277i	$-0.586531\pi$
$829$	−2.10263e14	−0.537019	−0.268510	+	0.963277i	$0.586531\pi$
$830$	0	0
$831$	0	0
$832$	−7.97326e14	−1.99995
$833$	−7.65511e13	−0.190865
$834$	−1.11896e15	−2.77321
$835$	0	0
$836$	0	0
$837$	−1.98556e15	−4.83344
$838$	−8.04448e14	−1.94660
$839$	−2.98299e14	−0.717533	−0.358766	−	0.933427i	$-0.616803\pi$
$839$	−2.98299e14	−0.717533	−0.358766	+	0.933427i	$0.616803\pi$
$840$	0	0
$841$	4.20707e14	1.00000
$842$	−4.31600e14	−1.01982
$843$	−3.58143e14	−0.841236
$844$	−7.58003e14	−1.76994
$845$	0	0
$846$	0	0
$847$	3.77195e14	0.865265
$848$	8.46987e14	1.93151
$849$	3.05199e14	0.691903
$850$	−4.43705e14	−1.00000
$851$	0	0
$852$	−1.65803e15	−3.69313
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−2.86836e14	−0.624116
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	−7.85681e14	−1.68970
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	3.37069e14	0.708244
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	−1.29311e15	−2.68576
$865$	0	0
$866$	8.98666e13	0.184505
$867$	−9.09484e14	−1.85652
$868$	−9.67653e14	−1.96391
$869$	4.37097e14	0.882022
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	8.54792e14	1.63828
$879$	−6.01980e14	−1.14720
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	2.49255e14	0.466983
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	1.07964e15	1.99995
$885$	0	0
$886$	0	0
$887$	9.39135e14	1.71045	0.855224	−	0.518258i	$-0.173419\pi$
$887$	9.39135e14	1.71045	0.855224	+	0.518258i	$0.173419\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.48974e14	−1.15568
$892$	0	0
$893$	0	0
$894$	−8.97011e14	−1.57076
$895$	0	0
$896$	−6.30190e14	−1.09127
$897$	2.92395e15	5.03510
$898$	0	0
$899$	0	0
$900$	1.44473e15	2.44666
$901$	−1.14689e15	−1.93151
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.12580e15	−1.83412	−0.917058	−	0.398755i	$-0.869442\pi$
$907$	−1.12580e15	−1.83412	−0.917058	+	0.398755i	$0.869442\pi$
$908$	1.23186e15	1.99588
$909$	−1.28642e15	−2.07283
$910$	0	0
$911$	9.62417e14	1.53381	0.766904	−	0.641761i	$-0.221796\pi$
$911$	9.62417e14	1.53381	0.766904	+	0.641761i	$0.221796\pi$
$912$	0	0
$913$	0	0
$914$	1.26674e15	1.98589
$915$	0	0
$916$	−4.63901e14	−0.719363
$917$	−1.00394e15	−1.54832
$918$	1.75098e15	2.68576
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	1.26928e15	1.90503
$923$	−2.66516e15	−3.97846
$924$	−6.20985e14	−0.921980
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.20314e15	3.11626
$934$	0	0
$935$	0	0
$936$	−3.51538e15	−4.89321
$937$	−7.58207e14	−1.04976	−0.524879	−	0.851177i	$-0.675890\pi$
$937$	−7.58207e14	−1.04976	−0.524879	+	0.851177i	$0.675890\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	−8.87616e13	−0.119666
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.45254e14	0.847190	0.423595	−	0.905852i	$-0.360768\pi$
$947$	6.45254e14	0.847190	0.423595	+	0.905852i	$0.360768\pi$
$948$	2.75505e15	3.59822
$949$	0	0
$950$	0	0
$951$	0	0
$952$	8.53328e14	1.09127
$953$	−1.16672e15	−1.48423	−0.742114	−	0.670273i	$-0.766177\pi$
$953$	−1.16672e15	−1.48423	−0.742114	+	0.670273i	$0.766177\pi$
$954$	3.73433e15	4.72576
$955$	0	0
$956$	0	0
$957$	0	0
$958$	1.36074e15	1.68635
$959$	9.77985e14	1.20570
$960$	0	0
$961$	1.83496e15	2.23877
$962$	0	0
$963$	−1.26465e15	−1.52700
$964$	0	0
$965$	0	0
$966$	2.31103e15	2.74739
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−6.73899e14	−0.792899
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.76029e15	−2.02887
$973$	−1.42161e15	−1.63010
$974$	−1.90554e14	−0.217382
$975$	3.27146e15	3.71295
$976$	0	0
$977$	1.24201e15	1.39525	0.697623	−	0.716465i	$-0.254241\pi$
$977$	1.24201e15	1.39525	0.697623	+	0.716465i	$0.254241\pi$
$978$	1.95621e15	2.18636
$979$	−6.26125e14	−0.696222
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8.96078e14	−0.976289	−0.488144	−	0.872763i	$-0.662326\pi$
$983$	−8.96078e14	−0.976289	−0.488144	+	0.872763i	$0.662326\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.16409e15	−1.21792	−0.608960	−	0.793201i	$-0.708413\pi$
$991$	−1.16409e15	−1.21792	−0.608960	+	0.793201i	$0.708413\pi$
$992$	1.72881e15	1.79966
$993$	0	0
$994$	−2.10648e15	−2.17083
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−1.42904e14	−0.144342
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.11.d.b.67.1	✓	2
4.3	odd	2	inner	68.11.d.b.67.2	yes	2
17.16	even	2	inner	68.11.d.b.67.2	yes	2
68.67	odd	2	CM	68.11.d.b.67.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.11.d.b.67.1	✓	2	1.1	even	1	trivial
68.11.d.b.67.1	✓	2	68.67	odd	2	CM
68.11.d.b.67.2	yes	2	4.3	odd	2	inner
68.11.d.b.67.2	yes	2	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	68.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+32.0000 q^{2} -451.134 q^{3} +1024.00 q^{4} -14436.3 q^{6} -18340.9 q^{7} +32768.0 q^{8} +144473. q^{9} +O(q^{10})\)	\(q+32.0000 q^{2} -451.134 q^{3} +1024.00 q^{4} -14436.3 q^{6} -18340.9 q^{7} +32768.0 q^{8} +144473. q^{9} -73291.6 q^{11} -461961. q^{12} -742568. q^{13} -586910. q^{14} +1.04858e6 q^{16} -1.41986e6 q^{17} +4.62314e6 q^{18} +8.27422e6 q^{21} -2.34533e6 q^{22} +8.72828e6 q^{23} -1.47828e7 q^{24} +9.76562e6 q^{25} -2.37622e7 q^{26} -3.85377e7 q^{27} -1.87811e7 q^{28} +5.15227e7 q^{31} +3.35544e7 q^{32} +3.30644e7 q^{33} -4.54354e7 q^{34} +1.47940e8 q^{36} +3.34998e8 q^{39} +2.64775e8 q^{42} -7.50506e7 q^{44} +2.79305e8 q^{46} -4.73048e8 q^{48} +5.39147e7 q^{49} +3.12500e8 q^{50} +6.40546e8 q^{51} -7.60390e8 q^{52} +8.07749e8 q^{53} -1.23321e9 q^{54} -6.00996e8 q^{56} +1.64873e9 q^{62} -2.64977e9 q^{63} +1.07374e9 q^{64} +1.05806e9 q^{66} -1.45393e9 q^{68} -3.93762e9 q^{69} +3.58911e9 q^{71} +4.73409e9 q^{72} -4.40561e9 q^{75} +1.34424e9 q^{77} +1.07199e10 q^{78} -5.96381e9 q^{79} +8.85468e9 q^{81} +8.47280e9 q^{84} -2.40162e9 q^{88} +8.54293e9 q^{89} +1.36194e10 q^{91} +8.93776e9 q^{92} -2.32436e10 q^{93} -1.51375e10 q^{96} +1.72527e9 q^{98} -1.05887e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 64 q^{2} + 2048 q^{4} + 65536 q^{8} + 288946 q^{9}+O(q^{10}) \) 2 * q + 64 * q^2 + 2048 * q^4 + 65536 * q^8 + 288946 * q^9	\( 2 q + 64 q^{2} + 2048 q^{4} + 65536 q^{8} + 288946 q^{9} - 1485136 q^{13} + 2097152 q^{16} - 2839714 q^{17} + 9246272 q^{18} + 16548444 q^{21} + 19531250 q^{25} - 47524352 q^{26} + 67108864 q^{32} + 66128700 q^{33} - 90870848 q^{34} + 295880704 q^{36} + 529550208 q^{42} + 107829346 q^{49} + 625000000 q^{50} - 1520779264 q^{52} + 1615498636 q^{53} + 2147483648 q^{64} + 2116118400 q^{66} - 2907867136 q^{68} - 7875248700 q^{69} + 9468182528 q^{72} + 2688473700 q^{77} + 17709354302 q^{81} + 16945606656 q^{84} + 17085864704 q^{89} - 46487274108 q^{93} + 3450539072 q^{98}+O(q^{100}) \) 2 * q + 64 * q^2 + 2048 * q^4 + 65536 * q^8 + 288946 * q^9 - 1485136 * q^13 + 2097152 * q^16 - 2839714 * q^17 + 9246272 * q^18 + 16548444 * q^21 + 19531250 * q^25 - 47524352 * q^26 + 67108864 * q^32 + 66128700 * q^33 - 90870848 * q^34 + 295880704 * q^36 + 529550208 * q^42 + 107829346 * q^49 + 625000000 * q^50 - 1520779264 * q^52 + 1615498636 * q^53 + 2147483648 * q^64 + 2116118400 * q^66 - 2907867136 * q^68 - 7875248700 * q^69 + 9468182528 * q^72 + 2688473700 * q^77 + 17709354302 * q^81 + 16945606656 * q^84 + 17085864704 * q^89 - 46487274108 * q^93 + 3450539072 * q^98

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