LMFDB - Embedded newform 784.5.c.a.97.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,5,Mod(97,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("784.97");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	784.c (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:	$81.0420510577$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.1
Root			$0.765367i$ of defining polynomial
Character	$\chi$	$=$	784.97
Dual form			784.5.c.a.97.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.9958i q^{3} -27.8477i q^{5} -174.865 q^{9} +O(q^{10})$	$q-15.9958i q^{3} -27.8477i q^{5} -174.865 q^{9} +139.664 q^{11} -101.931i q^{13} -445.446 q^{15} -542.887i q^{17} -139.953i q^{19} -229.647 q^{23} -150.495 q^{25} +1501.44i q^{27} -383.113 q^{29} +397.492i q^{31} -2234.03i q^{33} -898.912 q^{37} -1630.46 q^{39} -657.862i q^{41} -1155.70 q^{43} +4869.59i q^{45} -1294.51i q^{47} -8683.90 q^{51} -5168.43 q^{53} -3889.32i q^{55} -2238.66 q^{57} +4155.24i q^{59} -1840.09i q^{61} -2838.53 q^{65} +6315.57 q^{67} +3673.38i q^{69} -1263.28 q^{71} +3995.44i q^{73} +2407.28i q^{75} +10614.3 q^{79} +9852.71 q^{81} +5750.31i q^{83} -15118.2 q^{85} +6128.19i q^{87} -4187.05i q^{89} +6358.19 q^{93} -3897.38 q^{95} +3814.28i q^{97} -24422.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 196 q^{9}+O(q^{10}) $ 4 * q - 196 * q^9	$ 4 q - 196 q^{9} - 24 q^{11} - 888 q^{15} - 104 q^{23} + 948 q^{25} - 1408 q^{29} - 3392 q^{37} - 2200 q^{39} + 2024 q^{43} - 18936 q^{51} - 16680 q^{53} - 6064 q^{57} - 6048 q^{65} + 20816 q^{67} + 1984 q^{71} + 29616 q^{79} + 30852 q^{81} - 29688 q^{85} + 18192 q^{93} - 7240 q^{95} - 72160 q^{99}+O(q^{100}) $ 4 * q - 196 * q^9 - 24 * q^11 - 888 * q^15 - 104 * q^23 + 948 * q^25 - 1408 * q^29 - 3392 * q^37 - 2200 * q^39 + 2024 * q^43 - 18936 * q^51 - 16680 * q^53 - 6064 * q^57 - 6048 * q^65 + 20816 * q^67 + 1984 * q^71 + 29616 * q^79 + 30852 * q^81 - 29688 * q^85 + 18192 * q^93 - 7240 * q^95 - 72160 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 15.9958i	− 1.77731i	−0.458577	−	0.888654i	$-0.651641\pi$
$3$	− 15.9958i	− 1.77731i	0.458577	−	0.888654i	$-0.348359\pi$
$4$	0	0
$5$	− 27.8477i	− 1.11391i	−0.830543	−	0.556954i	$-0.811970\pi$
$5$	− 27.8477i	− 1.11391i	0.830543	−	0.556954i	$-0.188030\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−174.865	−2.15883
$10$	0	0
$11$	139.664	1.15425	0.577124	−	0.816657i	$-0.304175\pi$
$11$	139.664	1.15425	0.577124	+	0.816657i	$0.304175\pi$
$12$	0	0
$13$	− 101.931i	− 0.603140i	−0.953444	−	0.301570i	$-0.902489\pi$
$13$	− 101.931i	− 0.603140i	0.953444	−	0.301570i	$-0.0975106\pi$
$14$	0	0
$15$	−445.446	−1.97976
$16$	0	0
$17$	− 542.887i	− 1.87850i	−0.343232	−	0.939251i	$-0.611522\pi$
$17$	− 542.887i	− 1.87850i	0.343232	−	0.939251i	$-0.388478\pi$
$18$	0	0
$19$	− 139.953i	− 0.387682i	−0.981033	−	0.193841i	$-0.937905\pi$
$19$	− 139.953i	− 0.387682i	0.981033	−	0.193841i	$-0.0620947\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−229.647	−0.434115	−0.217057	−	0.976159i	$-0.569646\pi$
$23$	−229.647	−0.434115	−0.217057	+	0.976159i	$0.569646\pi$
$24$	0	0
$25$	−150.495	−0.240791
$26$	0	0
$27$	1501.44i	2.05959i
$28$	0	0
$29$	−383.113	−0.455544	−0.227772	−	0.973714i	$-0.573144\pi$
$29$	−383.113	−0.455544	−0.227772	+	0.973714i	$0.573144\pi$
$30$	0	0
$31$	397.492i	0.413623i	0.978381	+	0.206812i	$0.0663087\pi$
$31$	397.492i	0.413623i	−0.978381	+	0.206812i	$0.933691\pi$
$32$	0	0
$33$	− 2234.03i	− 2.05146i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−898.912	−0.656619	−0.328310	−	0.944570i	$-0.606479\pi$
$37$	−898.912	−0.656619	−0.328310	+	0.944570i	$0.606479\pi$
$38$	0	0
$39$	−1630.46	−1.07197
$40$	0	0
$41$	− 657.862i	− 0.391352i	−0.980669	−	0.195676i	$-0.937310\pi$
$41$	− 657.862i	− 0.391352i	0.980669	−	0.195676i	$-0.0626900\pi$
$42$	0	0
$43$	−1155.70	−0.625041	−0.312521	−	0.949911i	$-0.601173\pi$
$43$	−1155.70	−0.625041	−0.312521	+	0.949911i	$0.601173\pi$
$44$	0	0
$45$	4869.59i	2.40474i
$46$	0	0
$47$	− 1294.51i	− 0.586015i	−0.956110	−	0.293007i	$-0.905344\pi$
$47$	− 1294.51i	− 0.586015i	0.956110	−	0.293007i	$-0.0946561\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−8683.90	−3.33868
$52$	0	0
$53$	−5168.43	−1.83996	−0.919978	−	0.391971i	$-0.871793\pi$
$53$	−5168.43	−1.83996	−0.919978	+	0.391971i	$0.871793\pi$
$54$	0	0
$55$	− 3889.32i	− 1.28573i
$56$	0	0
$57$	−2238.66	−0.689031
$58$	0	0
$59$	4155.24i	1.19369i	0.802356	+	0.596846i	$0.203580\pi$
$59$	4155.24i	1.19369i	−0.802356	+	0.596846i	$0.796420\pi$
$60$	0	0
$61$	− 1840.09i	− 0.494514i	−0.968950	−	0.247257i	$-0.920471\pi$
$61$	− 1840.09i	− 0.494514i	0.968950	−	0.247257i	$-0.0795292\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2838.53	−0.671842
$66$	0	0
$67$	6315.57	1.40690	0.703450	−	0.710745i	$-0.251642\pi$
$67$	6315.57	1.40690	0.703450	+	0.710745i	$0.251642\pi$
$68$	0	0
$69$	3673.38i	0.771556i
$70$	0	0
$71$	−1263.28	−0.250601	−0.125301	−	0.992119i	$-0.539990\pi$
$71$	−1263.28	−0.250601	−0.125301	+	0.992119i	$0.539990\pi$
$72$	0	0
$73$	3995.44i	0.749753i	0.927075	+	0.374877i	$0.122315\pi$
$73$	3995.44i	0.749753i	−0.927075	+	0.374877i	$0.877685\pi$
$74$	0	0
$75$	2407.28i	0.427960i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10614.3	1.70073	0.850366	−	0.526192i	$-0.176381\pi$
$79$	10614.3	1.70073	0.850366	+	0.526192i	$0.176381\pi$
$80$	0	0
$81$	9852.71	1.50171
$82$	0	0
$83$	5750.31i	0.834709i	0.908744	+	0.417355i	$0.137043\pi$
$83$	5750.31i	0.834709i	−0.908744	+	0.417355i	$0.862957\pi$
$84$	0	0
$85$	−15118.2	−2.09248
$86$	0	0
$87$	6128.19i	0.809643i
$88$	0	0
$89$	− 4187.05i	− 0.528601i	−0.964440	−	0.264301i	$-0.914859\pi$
$89$	− 4187.05i	− 0.528601i	0.964440	−	0.264301i	$-0.0851411\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6358.19	0.735136
$94$	0	0
$95$	−3897.38	−0.431843
$96$	0	0
$97$	3814.28i	0.405386i	0.979242	+	0.202693i	$0.0649694\pi$
$97$	3814.28i	0.405386i	−0.979242	+	0.202693i	$0.935031\pi$
$98$	0	0
$99$	−24422.3	−2.49182
$100$	0	0
$101$	− 4925.41i	− 0.482836i	−0.970421	−	0.241418i	$-0.922387\pi$
$101$	− 4925.41i	− 0.482836i	0.970421	−	0.241418i	$-0.0776126\pi$
$102$	0	0
$103$	− 18892.6i	− 1.78081i	−0.455166	−	0.890407i	$-0.650420\pi$
$103$	− 18892.6i	− 1.78081i	0.455166	−	0.890407i	$-0.349580\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8479.39	0.740623	0.370311	−	0.928908i	$-0.379251\pi$
$107$	8479.39	0.740623	0.370311	+	0.928908i	$0.379251\pi$
$108$	0	0
$109$	11678.0	0.982911	0.491455	−	0.870903i	$-0.336465\pi$
$109$	11678.0	0.982911	0.491455	+	0.870903i	$0.336465\pi$
$110$	0	0
$111$	14378.8i	1.16702i
$112$	0	0
$113$	19683.8	1.54153	0.770763	−	0.637121i	$-0.219875\pi$
$113$	19683.8	1.54153	0.770763	+	0.637121i	$0.219875\pi$
$114$	0	0
$115$	6395.13i	0.483564i
$116$	0	0
$117$	17824.1i	1.30207i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	4865.03	0.332288
$122$	0	0
$123$	−10523.0	−0.695553
$124$	0	0
$125$	− 13213.9i	− 0.845689i
$126$	0	0
$127$	27965.5	1.73386	0.866931	−	0.498428i	$-0.166089\pi$
$127$	27965.5	1.73386	0.866931	+	0.498428i	$0.166089\pi$
$128$	0	0
$129$	18486.3i	1.11089i
$130$	0	0
$131$	1169.73i	0.0681622i	0.999419	+	0.0340811i	$0.0108505\pi$
$131$	1169.73i	0.0681622i	−0.999419	+	0.0340811i	$0.989150\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	41811.8	2.29420
$136$	0	0
$137$	−20884.6	−1.11272	−0.556359	−	0.830942i	$-0.687802\pi$
$137$	−20884.6	−1.11272	−0.556359	+	0.830942i	$0.687802\pi$
$138$	0	0
$139$	− 3499.87i	− 0.181143i	−0.995890	−	0.0905716i	$-0.971131\pi$
$139$	− 3499.87i	− 0.181143i	0.995890	−	0.0905716i	$-0.0288694\pi$
$140$	0	0
$141$	−20706.6	−1.04153
$142$	0	0
$143$	− 14236.0i	− 0.696173i
$144$	0	0
$145$	10668.8i	0.507434i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	29533.8	1.33029	0.665146	−	0.746713i	$-0.268369\pi$
$149$	29533.8	1.33029	0.665146	+	0.746713i	$0.268369\pi$
$150$	0	0
$151$	−25780.8	−1.13069	−0.565344	−	0.824856i	$-0.691256\pi$
$151$	−25780.8	−1.13069	−0.565344	+	0.824856i	$0.691256\pi$
$152$	0	0
$153$	94931.9i	4.05536i
$154$	0	0
$155$	11069.2	0.460738
$156$	0	0
$157$	− 25629.4i	− 1.03977i	−0.854235	−	0.519887i	$-0.825974\pi$
$157$	− 25629.4i	− 1.03977i	0.854235	−	0.519887i	$-0.174026\pi$
$158$	0	0
$159$	82673.2i	3.27017i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6182.78	0.232707	0.116353	−	0.993208i	$-0.462880\pi$
$163$	6182.78	0.232707	0.116353	+	0.993208i	$0.462880\pi$
$164$	0	0
$165$	−62212.7	−2.28513
$166$	0	0
$167$	39270.9i	1.40811i	0.710144	+	0.704056i	$0.248630\pi$
$167$	39270.9i	1.40811i	−0.710144	+	0.704056i	$0.751370\pi$
$168$	0	0
$169$	18171.2	0.636223
$170$	0	0
$171$	24472.9i	0.836939i
$172$	0	0
$173$	32905.9i	1.09946i	0.835341	+	0.549732i	$0.185270\pi$
$173$	32905.9i	1.09946i	−0.835341	+	0.549732i	$0.814730\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	66466.3	2.12156
$178$	0	0
$179$	4767.88	0.148806	0.0744029	−	0.997228i	$-0.476295\pi$
$179$	4767.88	0.148806	0.0744029	+	0.997228i	$0.476295\pi$
$180$	0	0
$181$	35195.4i	1.07431i	0.843484	+	0.537154i	$0.180501\pi$
$181$	35195.4i	1.07431i	−0.843484	+	0.537154i	$0.819499\pi$
$182$	0	0
$183$	−29433.6	−0.878904
$184$	0	0
$185$	25032.6i	0.731413i
$186$	0	0
$187$	− 75821.7i	− 2.16826i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−23315.1	−0.639101	−0.319551	−	0.947569i	$-0.603532\pi$
$191$	−23315.1	−0.639101	−0.319551	+	0.947569i	$0.603532\pi$
$192$	0	0
$193$	59183.8	1.58887	0.794434	−	0.607350i	$-0.207768\pi$
$193$	59183.8	1.58887	0.794434	+	0.607350i	$0.207768\pi$
$194$	0	0
$195$	45404.5i	1.19407i
$196$	0	0
$197$	−60902.2	−1.56928	−0.784640	−	0.619952i	$-0.787152\pi$
$197$	−60902.2	−1.56928	−0.784640	+	0.619952i	$0.787152\pi$
$198$	0	0
$199$	− 40116.7i	− 1.01302i	−0.862233	−	0.506512i	$-0.830935\pi$
$199$	− 40116.7i	− 1.01302i	0.862233	−	0.506512i	$-0.169065\pi$
$200$	0	0
$201$	− 101023.i	− 2.50050i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−18319.9	−0.435930
$206$	0	0
$207$	40157.2	0.937179
$208$	0	0
$209$	− 19546.4i	− 0.447482i
$210$	0	0
$211$	−21929.0	−0.492554	−0.246277	−	0.969200i	$-0.579207\pi$
$211$	−21929.0	−0.492554	−0.246277	+	0.969200i	$0.579207\pi$
$212$	0	0
$213$	20207.2i	0.445396i
$214$	0	0
$215$	32183.6i	0.696238i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	63910.1	1.33254
$220$	0	0
$221$	−55336.8	−1.13300
$222$	0	0
$223$	64358.0i	1.29418i	0.762416	+	0.647088i	$0.224013\pi$
$223$	64358.0i	1.29418i	−0.762416	+	0.647088i	$0.775987\pi$
$224$	0	0
$225$	26316.2	0.519827
$226$	0	0
$227$	− 43774.8i	− 0.849517i	−0.905307	−	0.424759i	$-0.860359\pi$
$227$	− 43774.8i	− 0.849517i	0.905307	−	0.424759i	$-0.139641\pi$
$228$	0	0
$229$	28506.4i	0.543590i	0.962355	+	0.271795i	$0.0876173\pi$
$229$	28506.4i	0.543590i	−0.962355	+	0.271795i	$0.912383\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2194.89	−0.0404298	−0.0202149	−	0.999796i	$-0.506435\pi$
$233$	−2194.89	−0.0404298	−0.0202149	+	0.999796i	$0.506435\pi$
$234$	0	0
$235$	−36049.0	−0.652767
$236$	0	0
$237$	− 169783.i	− 3.02273i
$238$	0	0
$239$	−80942.6	−1.41704	−0.708519	−	0.705692i	$-0.750636\pi$
$239$	−80942.6	−1.41704	−0.708519	+	0.705692i	$0.750636\pi$
$240$	0	0
$241$	− 43016.6i	− 0.740631i	−0.928906	−	0.370316i	$-0.879249\pi$
$241$	− 43016.6i	− 0.740631i	0.928906	−	0.370316i	$-0.120751\pi$
$242$	0	0
$243$	− 35984.7i	− 0.609405i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−14265.5	−0.233827
$248$	0	0
$249$	91980.7	1.48354
$250$	0	0
$251$	− 31568.5i	− 0.501080i	−0.968106	−	0.250540i	$-0.919392\pi$
$251$	− 31568.5i	− 0.501080i	0.968106	−	0.250540i	$-0.0806082\pi$
$252$	0	0
$253$	−32073.4	−0.501076
$254$	0	0
$255$	241827.i	3.71898i
$256$	0	0
$257$	− 68105.9i	− 1.03114i	−0.856847	−	0.515571i	$-0.827580\pi$
$257$	− 68105.9i	− 1.03114i	0.856847	−	0.515571i	$-0.172420\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	66993.0	0.983441
$262$	0	0
$263$	8347.56	0.120684	0.0603418	−	0.998178i	$-0.480781\pi$
$263$	8347.56	0.120684	0.0603418	+	0.998178i	$0.480781\pi$
$264$	0	0
$265$	143929.i	2.04954i
$266$	0	0
$267$	−66975.1	−0.939488
$268$	0	0
$269$	91921.0i	1.27031i	0.772384	+	0.635156i	$0.219064\pi$
$269$	91921.0i	1.27031i	−0.772384	+	0.635156i	$0.780936\pi$
$270$	0	0
$271$	5134.36i	0.0699114i	0.999389	+	0.0349557i	$0.0111290\pi$
$271$	5134.36i	0.0699114i	−0.999389	+	0.0349557i	$0.988871\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−21018.7	−0.277933
$276$	0	0
$277$	−66254.2	−0.863483	−0.431741	−	0.901997i	$-0.642101\pi$
$277$	−66254.2	−0.863483	−0.431741	+	0.901997i	$0.642101\pi$
$278$	0	0
$279$	− 69507.4i	− 0.892941i
$280$	0	0
$281$	38156.4	0.483230	0.241615	−	0.970372i	$-0.422323\pi$
$281$	38156.4	0.483230	0.241615	+	0.970372i	$0.422323\pi$
$282$	0	0
$283$	110233.i	1.37638i	0.725528	+	0.688192i	$0.241595\pi$
$283$	110233.i	1.37638i	−0.725528	+	0.688192i	$0.758405\pi$
$284$	0	0
$285$	62341.6i	0.767518i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−211205.	−2.52877
$290$	0	0
$291$	61012.4	0.720497
$292$	0	0
$293$	79011.7i	0.920357i	0.887826	+	0.460179i	$0.152215\pi$
$293$	79011.7i	0.920357i	−0.887826	+	0.460179i	$0.847785\pi$
$294$	0	0
$295$	115714.	1.32966
$296$	0	0
$297$	209698.i	2.37728i
$298$	0	0
$299$	23408.0i	0.261832i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−78785.9	−0.858150
$304$	0	0
$305$	−51242.2	−0.550843
$306$	0	0
$307$	2801.57i	0.0297252i	0.999890	+	0.0148626i	$0.00473109\pi$
$307$	2801.57i	0.0297252i	−0.999890	+	0.0148626i	$0.995269\pi$
$308$	0	0
$309$	−302203.	−3.16506
$310$	0	0
$311$	− 20659.3i	− 0.213597i	−0.994281	−	0.106799i	$-0.965940\pi$
$311$	− 20659.3i	− 0.213597i	0.994281	−	0.106799i	$-0.0340600\pi$
$312$	0	0
$313$	− 37376.1i	− 0.381509i	−0.981638	−	0.190755i	$-0.938907\pi$
$313$	− 37376.1i	− 0.381509i	0.981638	−	0.190755i	$-0.0610935\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−106666.	−1.06147	−0.530735	−	0.847538i	$-0.678084\pi$
$317$	−106666.	−1.06147	−0.530735	+	0.847538i	$0.678084\pi$
$318$	0	0
$319$	−53507.1	−0.525811
$320$	0	0
$321$	− 135634.i	− 1.31632i
$322$	0	0
$323$	−75978.8	−0.728262
$324$	0	0
$325$	15340.0i	0.145231i
$326$	0	0
$327$	− 186798.i	− 1.74694i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−108363.	−0.989063	−0.494532	−	0.869160i	$-0.664660\pi$
$331$	−108363.	−0.989063	−0.494532	+	0.869160i	$0.664660\pi$
$332$	0	0
$333$	157188.	1.41753
$334$	0	0
$335$	− 175874.i	− 1.56716i
$336$	0	0
$337$	186946.	1.64610	0.823050	−	0.567968i	$-0.192270\pi$
$337$	186946.	1.64610	0.823050	+	0.567968i	$0.192270\pi$
$338$	0	0
$339$	− 314857.i	− 2.73977i
$340$	0	0
$341$	55515.3i	0.477424i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	102295.	0.859443
$346$	0	0
$347$	63227.3	0.525105	0.262552	−	0.964918i	$-0.415436\pi$
$347$	63227.3	0.525105	0.262552	+	0.964918i	$0.415436\pi$
$348$	0	0
$349$	93605.0i	0.768508i	0.923227	+	0.384254i	$0.125541\pi$
$349$	93605.0i	0.768508i	−0.923227	+	0.384254i	$0.874459\pi$
$350$	0	0
$351$	153043.	1.24222
$352$	0	0
$353$	− 143251.i	− 1.14960i	−0.818294	−	0.574800i	$-0.805080\pi$
$353$	− 143251.i	− 1.14960i	0.818294	−	0.574800i	$-0.194920\pi$
$354$	0	0
$355$	35179.5i	0.279147i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−102340.	−0.794065	−0.397032	−	0.917805i	$-0.629960\pi$
$359$	−102340.	−0.794065	−0.397032	+	0.917805i	$0.629960\pi$
$360$	0	0
$361$	110734.	0.849702
$362$	0	0
$363$	− 77820.0i	− 0.590579i
$364$	0	0
$365$	111264.	0.835156
$366$	0	0
$367$	15366.4i	0.114088i	0.998372	+	0.0570442i	$0.0181676\pi$
$367$	15366.4i	0.114088i	−0.998372	+	0.0570442i	$0.981832\pi$
$368$	0	0
$369$	115037.i	0.844861i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	97490.8	0.700722	0.350361	−	0.936615i	$-0.386059\pi$
$373$	97490.8	0.700722	0.350361	+	0.936615i	$0.386059\pi$
$374$	0	0
$375$	−211366.	−1.50305
$376$	0	0
$377$	39050.9i	0.274757i
$378$	0	0
$379$	66090.7	0.460110	0.230055	−	0.973178i	$-0.426109\pi$
$379$	66090.7	0.460110	0.230055	+	0.973178i	$0.426109\pi$
$380$	0	0
$381$	− 447329.i	− 3.08161i
$382$	0	0
$383$	− 208901.i	− 1.42411i	−0.702126	−	0.712053i	$-0.747766\pi$
$383$	− 208901.i	− 1.42411i	0.702126	−	0.712053i	$-0.252234\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	202092.	1.34936
$388$	0	0
$389$	−143999.	−0.951614	−0.475807	−	0.879550i	$-0.657844\pi$
$389$	−143999.	−0.951614	−0.475807	+	0.879550i	$0.657844\pi$
$390$	0	0
$391$	124672.i	0.815485i
$392$	0	0
$393$	18710.8	0.121145
$394$	0	0
$395$	− 295583.i	− 1.89446i
$396$	0	0
$397$	− 284577.i	− 1.80559i	−0.430070	−	0.902796i	$-0.641511\pi$
$397$	− 284577.i	− 1.80559i	0.430070	−	0.902796i	$-0.358489\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−107846.	−0.670680	−0.335340	−	0.942097i	$-0.608851\pi$
$401$	−107846.	−0.670680	−0.335340	+	0.942097i	$0.608851\pi$
$402$	0	0
$403$	40516.6	0.249472
$404$	0	0
$405$	− 274375.i	− 1.67276i
$406$	0	0
$407$	−125546.	−0.757901
$408$	0	0
$409$	− 84932.2i	− 0.507722i	−0.967241	−	0.253861i	$-0.918299\pi$
$409$	− 84932.2i	− 0.507722i	0.967241	−	0.253861i	$-0.0817005\pi$
$410$	0	0
$411$	334065.i	1.97764i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	160133.	0.929789
$416$	0	0
$417$	−55983.1	−0.321947
$418$	0	0
$419$	331905.i	1.89054i	0.326292	+	0.945269i	$0.394201\pi$
$419$	331905.i	1.89054i	−0.326292	+	0.945269i	$0.605799\pi$
$420$	0	0
$421$	101490.	0.572609	0.286304	−	0.958139i	$-0.407573\pi$
$421$	101490.	0.572609	0.286304	+	0.958139i	$0.407573\pi$
$422$	0	0
$423$	226364.i	1.26510i
$424$	0	0
$425$	81701.5i	0.452327i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−227716.	−1.23731
$430$	0	0
$431$	−229798.	−1.23706	−0.618530	−	0.785761i	$-0.712272\pi$
$431$	−229798.	−1.23706	−0.618530	+	0.785761i	$0.712272\pi$
$432$	0	0
$433$	− 114264.i	− 0.609442i	−0.952442	−	0.304721i	$-0.901437\pi$
$433$	− 114264.i	− 0.609442i	0.952442	−	0.304721i	$-0.0985632\pi$
$434$	0	0
$435$	170656.	0.901868
$436$	0	0
$437$	32139.8i	0.168299i
$438$	0	0
$439$	352339.i	1.82823i	0.405453	+	0.914116i	$0.367114\pi$
$439$	352339.i	1.82823i	−0.405453	+	0.914116i	$0.632886\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	196316.	1.00034	0.500172	−	0.865926i	$-0.333270\pi$
$443$	196316.	1.00034	0.500172	+	0.865926i	$0.333270\pi$
$444$	0	0
$445$	−116600.	−0.588813
$446$	0	0
$447$	− 472416.i	− 2.36434i
$448$	0	0
$449$	77520.3	0.384523	0.192262	−	0.981344i	$-0.438418\pi$
$449$	77520.3	0.384523	0.192262	+	0.981344i	$0.438418\pi$
$450$	0	0
$451$	− 91879.6i	− 0.451717i
$452$	0	0
$453$	412384.i	2.00958i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	189872.	0.909137	0.454568	−	0.890712i	$-0.349794\pi$
$457$	189872.	0.909137	0.454568	+	0.890712i	$0.349794\pi$
$458$	0	0
$459$	815114.	3.86895
$460$	0	0
$461$	− 52599.9i	− 0.247505i	−0.992313	−	0.123752i	$-0.960507\pi$
$461$	− 52599.9i	− 0.247505i	0.992313	−	0.123752i	$-0.0394928\pi$
$462$	0	0
$463$	244268.	1.13947	0.569736	−	0.821828i	$-0.307045\pi$
$463$	244268.	1.13947	0.569736	+	0.821828i	$0.307045\pi$
$464$	0	0
$465$	− 177061.i	− 0.818874i
$466$	0	0
$467$	− 297615.i	− 1.36465i	−0.731050	−	0.682324i	$-0.760969\pi$
$467$	− 297615.i	− 1.36465i	0.731050	−	0.682324i	$-0.239031\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−409962.	−1.84800
$472$	0	0
$473$	−161410.	−0.721452
$474$	0	0
$475$	21062.2i	0.0933505i
$476$	0	0
$477$	903778.	3.97215
$478$	0	0
$479$	− 46870.8i	− 0.204283i	−0.994770	−	0.102141i	$-0.967431\pi$
$479$	− 46870.8i	− 0.204283i	0.994770	−	0.102141i	$-0.0325694\pi$
$480$	0	0
$481$	91626.6i	0.396033i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	106219.	0.451563
$486$	0	0
$487$	−107184.	−0.451930	−0.225965	−	0.974135i	$-0.572554\pi$
$487$	−107184.	−0.451930	−0.225965	+	0.974135i	$0.572554\pi$
$488$	0	0
$489$	− 98898.4i	− 0.413592i
$490$	0	0
$491$	116367.	0.482687	0.241344	−	0.970440i	$-0.422412\pi$
$491$	116367.	0.482687	0.241344	+	0.970440i	$0.422412\pi$
$492$	0	0
$493$	207987.i	0.855740i
$494$	0	0
$495$	680106.i	2.77566i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−442908.	−1.77874	−0.889369	−	0.457190i	$-0.848856\pi$
$499$	−442908.	−1.77874	−0.889369	+	0.457190i	$0.848856\pi$
$500$	0	0
$501$	628168.	2.50265
$502$	0	0
$503$	68955.1i	0.272540i	0.990672	+	0.136270i	$0.0435115\pi$
$503$	68955.1i	0.272540i	−0.990672	+	0.136270i	$0.956489\pi$
$504$	0	0
$505$	−137161.	−0.537835
$506$	0	0
$507$	− 290662.i	− 1.13076i
$508$	0	0
$509$	− 92773.5i	− 0.358087i	−0.983841	−	0.179044i	$-0.942700\pi$
$509$	− 92773.5i	− 0.358087i	0.983841	−	0.179044i	$-0.0573003\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	210132.	0.798468
$514$	0	0
$515$	−526117.	−1.98366
$516$	0	0
$517$	− 180796.i	− 0.676406i
$518$	0	0
$519$	526355.	1.95409
$520$	0	0
$521$	− 142912.i	− 0.526495i	−0.964728	−	0.263248i	$-0.915206\pi$
$521$	− 142912.i	− 0.526495i	0.964728	−	0.263248i	$-0.0847936\pi$
$522$	0	0
$523$	− 22203.5i	− 0.0811742i	−0.999176	−	0.0405871i	$-0.987077\pi$
$523$	− 22203.5i	− 0.0811742i	0.999176	−	0.0405871i	$-0.0129228\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	215793.	0.776992
$528$	0	0
$529$	−227103.	−0.811544
$530$	0	0
$531$	− 726606.i	− 2.57697i
$532$	0	0
$533$	−67056.3	−0.236040
$534$	0	0
$535$	− 236132.i	− 0.824986i
$536$	0	0
$537$	− 76266.0i	− 0.264474i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	211209.	0.721636	0.360818	−	0.932636i	$-0.382498\pi$
$541$	211209.	0.721636	0.360818	+	0.932636i	$0.382498\pi$
$542$	0	0
$543$	562978.	1.90938
$544$	0	0
$545$	− 325204.i	− 1.09487i
$546$	0	0
$547$	−254227.	−0.849664	−0.424832	−	0.905272i	$-0.639667\pi$
$547$	−254227.	−0.849664	−0.424832	+	0.905272i	$0.639667\pi$
$548$	0	0
$549$	321767.i	1.06757i
$550$	0	0
$551$	53617.9i	0.176606i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	400416.	1.29995
$556$	0	0
$557$	46139.2	0.148717	0.0743584	−	0.997232i	$-0.476309\pi$
$557$	46139.2	0.148717	0.0743584	+	0.997232i	$0.476309\pi$
$558$	0	0
$559$	117801.i	0.376987i
$560$	0	0
$561$	−1.21283e6	−3.85366
$562$	0	0
$563$	− 349629.i	− 1.10304i	−0.834162	−	0.551519i	$-0.814048\pi$
$563$	− 349629.i	− 1.10304i	0.834162	−	0.551519i	$-0.185952\pi$
$564$	0	0
$565$	− 548147.i	− 1.71712i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−176196.	−0.544217	−0.272109	−	0.962267i	$-0.587721\pi$
$569$	−176196.	−0.544217	−0.272109	+	0.962267i	$0.587721\pi$
$570$	0	0
$571$	176527.	0.541426	0.270713	−	0.962660i	$-0.412741\pi$
$571$	176527.	0.541426	0.270713	+	0.962660i	$0.412741\pi$
$572$	0	0
$573$	372943.i	1.13588i
$574$	0	0
$575$	34560.6	0.104531
$576$	0	0
$577$	− 226863.i	− 0.681416i	−0.940169	−	0.340708i	$-0.889333\pi$
$577$	− 226863.i	− 0.681416i	0.940169	−	0.340708i	$-0.110667\pi$
$578$	0	0
$579$	− 946690.i	− 2.82391i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−721844.	−2.12376
$584$	0	0
$585$	496360.	1.45039
$586$	0	0
$587$	− 349659.i	− 1.01477i	−0.861719	−	0.507386i	$-0.830612\pi$
$587$	− 349659.i	− 1.01477i	0.861719	−	0.507386i	$-0.169388\pi$
$588$	0	0
$589$	55630.3	0.160354
$590$	0	0
$591$	974178.i	2.78910i
$592$	0	0
$593$	137755.i	0.391739i	0.980630	+	0.195870i	$0.0627529\pi$
$593$	137755.i	0.391739i	−0.980630	+	0.195870i	$0.937247\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−641699.	−1.80046
$598$	0	0
$599$	−80877.2	−0.225410	−0.112705	−	0.993629i	$-0.535951\pi$
$599$	−80877.2	−0.225410	−0.112705	+	0.993629i	$0.535951\pi$
$600$	0	0
$601$	− 228798.i	− 0.633438i	−0.948519	−	0.316719i	$-0.897419\pi$
$601$	− 228798.i	− 0.633438i	0.948519	−	0.316719i	$-0.102581\pi$
$602$	0	0
$603$	−1.10437e6	−3.03725
$604$	0	0
$605$	− 135480.i	− 0.370139i
$606$	0	0
$607$	− 265951.i	− 0.721812i	−0.932602	−	0.360906i	$-0.882468\pi$
$607$	− 265951.i	− 0.721812i	0.932602	−	0.360906i	$-0.117532\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−131950.	−0.353449
$612$	0	0
$613$	−306522.	−0.815718	−0.407859	−	0.913045i	$-0.633725\pi$
$613$	−306522.	−0.815718	−0.407859	+	0.913045i	$0.633725\pi$
$614$	0	0
$615$	293042.i	0.774782i
$616$	0	0
$617$	491906.	1.29215	0.646074	−	0.763275i	$-0.276410\pi$
$617$	491906.	1.29215	0.646074	+	0.763275i	$0.276410\pi$
$618$	0	0
$619$	115230.i	0.300735i	0.988630	+	0.150368i	$0.0480457\pi$
$619$	115230.i	0.300735i	−0.988630	+	0.150368i	$0.951954\pi$
$620$	0	0
$621$	− 344802.i	− 0.894100i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−462035.	−1.18281
$626$	0	0
$627$	−312661.	−0.795313
$628$	0	0
$629$	488007.i	1.23346i
$630$	0	0
$631$	−72815.9	−0.182880	−0.0914402	−	0.995811i	$-0.529147\pi$
$631$	−72815.9	−0.182880	−0.0914402	+	0.995811i	$0.529147\pi$
$632$	0	0
$633$	350771.i	0.875420i
$634$	0	0
$635$	− 778774.i	− 1.93136i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	220904.	0.541005
$640$	0	0
$641$	−757794.	−1.84431	−0.922157	−	0.386815i	$-0.873575\pi$
$641$	−757794.	−1.84431	−0.922157	+	0.386815i	$0.873575\pi$
$642$	0	0
$643$	353082.i	0.853992i	0.904254	+	0.426996i	$0.140428\pi$
$643$	353082.i	0.853992i	−0.904254	+	0.426996i	$0.859572\pi$
$644$	0	0
$645$	514802.	1.23743
$646$	0	0
$647$	214561.i	0.512557i	0.966603	+	0.256279i	$0.0824965\pi$
$647$	214561.i	0.512557i	−0.966603	+	0.256279i	$0.917504\pi$
$648$	0	0
$649$	580337.i	1.37782i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−497591.	−1.16693	−0.583467	−	0.812137i	$-0.698304\pi$
$653$	−497591.	−1.16693	−0.583467	+	0.812137i	$0.698304\pi$
$654$	0	0
$655$	32574.3	0.0759264
$656$	0	0
$657$	− 698662.i	− 1.61859i
$658$	0	0
$659$	197464.	0.454692	0.227346	−	0.973814i	$-0.426995\pi$
$659$	197464.	0.454692	0.227346	+	0.973814i	$0.426995\pi$
$660$	0	0
$661$	− 668423.i	− 1.52985i	−0.644121	−	0.764924i	$-0.722777\pi$
$661$	− 668423.i	− 1.52985i	0.644121	−	0.764924i	$-0.277223\pi$
$662$	0	0
$663$	885155.i	2.01369i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	87980.6	0.197759
$668$	0	0
$669$	1.02946e6	2.30015
$670$	0	0
$671$	− 256994.i	− 0.570792i
$672$	0	0
$673$	−404534.	−0.893151	−0.446575	−	0.894746i	$-0.647357\pi$
$673$	−404534.	−0.893151	−0.446575	+	0.894746i	$0.647357\pi$
$674$	0	0
$675$	− 225959.i	− 0.495932i
$676$	0	0
$677$	− 211169.i	− 0.460737i	−0.973103	−	0.230369i	$-0.926007\pi$
$677$	− 211169.i	− 0.460737i	0.973103	−	0.230369i	$-0.0739932\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−700212.	−1.50985
$682$	0	0
$683$	170109.	0.364657	0.182329	−	0.983238i	$-0.441637\pi$
$683$	170109.	0.364657	0.182329	+	0.983238i	$0.441637\pi$
$684$	0	0
$685$	581588.i	1.23946i
$686$	0	0
$687$	455983.	0.966128
$688$	0	0
$689$	526822.i	1.10975i
$690$	0	0
$691$	168630.i	0.353167i	0.984286	+	0.176583i	$0.0565045\pi$
$691$	168630.i	0.353167i	−0.984286	+	0.176583i	$0.943496\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−97463.3	−0.201777
$696$	0	0
$697$	−357145.	−0.735154
$698$	0	0
$699$	35109.0i	0.0718562i
$700$	0	0
$701$	−487646.	−0.992359	−0.496179	−	0.868220i	$-0.665264\pi$
$701$	−487646.	−0.992359	−0.496179	+	0.868220i	$0.665264\pi$
$702$	0	0
$703$	125806.i	0.254560i
$704$	0	0
$705$	576633.i	1.16017i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−386929.	−0.769731	−0.384865	−	0.922973i	$-0.625752\pi$
$709$	−386929.	−0.769731	−0.384865	+	0.922973i	$0.625752\pi$
$710$	0	0
$711$	−1.85606e6	−3.67159
$712$	0	0
$713$	− 91282.7i	− 0.179560i
$714$	0	0
$715$	−396441.	−0.775472
$716$	0	0
$717$	1.29474e6i	2.51851i
$718$	0	0
$719$	188374.i	0.364388i	0.983263	+	0.182194i	$0.0583199\pi$
$719$	188374.i	0.364388i	−0.983263	+	0.182194i	$0.941680\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−688084.	−1.31633
$724$	0	0
$725$	57656.4	0.109691
$726$	0	0
$727$	901538.i	1.70575i	0.522115	+	0.852875i	$0.325143\pi$
$727$	901538.i	1.70575i	−0.522115	+	0.852875i	$0.674857\pi$
$728$	0	0
$729$	222465.	0.418607
$730$	0	0
$731$	627415.i	1.17414i
$732$	0	0
$733$	− 402482.i	− 0.749098i	−0.927207	−	0.374549i	$-0.877798\pi$
$733$	− 402482.i	− 0.749098i	0.927207	−	0.374549i	$-0.122202\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	882058.	1.62391
$738$	0	0
$739$	37301.1	0.0683020	0.0341510	−	0.999417i	$-0.489127\pi$
$739$	37301.1	0.0683020	0.0341510	+	0.999417i	$0.489127\pi$
$740$	0	0
$741$	228188.i	0.415582i
$742$	0	0
$743$	831221.	1.50570	0.752851	−	0.658191i	$-0.228678\pi$
$743$	831221.	1.50570	0.752851	+	0.658191i	$0.228678\pi$
$744$	0	0
$745$	− 822449.i	− 1.48182i
$746$	0	0
$747$	− 1.00553e6i	− 1.80199i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	251098.	0.445209	0.222605	−	0.974909i	$-0.428544\pi$
$751$	251098.	0.445209	0.222605	+	0.974909i	$0.428544\pi$
$752$	0	0
$753$	−504964.	−0.890574
$754$	0	0
$755$	717936.i	1.25948i
$756$	0	0
$757$	−422399.	−0.737108	−0.368554	−	0.929606i	$-0.620147\pi$
$757$	−422399.	−0.737108	−0.368554	+	0.929606i	$0.620147\pi$
$758$	0	0
$759$	513039.i	0.890567i
$760$	0	0
$761$	− 247203.i	− 0.426859i	−0.976958	−	0.213430i	$-0.931537\pi$
$761$	− 247203.i	− 0.426859i	0.976958	−	0.213430i	$-0.0684634\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.64364e6	4.51730
$766$	0	0
$767$	423546.	0.719962
$768$	0	0
$769$	903767.i	1.52828i	0.645049	+	0.764141i	$0.276837\pi$
$769$	903767.i	1.52828i	−0.645049	+	0.764141i	$0.723163\pi$
$770$	0	0
$771$	−1.08941e6	−1.83266
$772$	0	0
$773$	244667.i	0.409464i	0.978818	+	0.204732i	$0.0656323\pi$
$773$	244667.i	0.409464i	−0.978818	+	0.204732i	$0.934368\pi$
$774$	0	0
$775$	− 59820.4i	− 0.0995968i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−92070.0	−0.151720
$780$	0	0
$781$	−176435.	−0.289256
$782$	0	0
$783$	− 575222.i	− 0.938236i
$784$	0	0
$785$	−713719.	−1.15821
$786$	0	0
$787$	− 950431.i	− 1.53452i	−0.641339	−	0.767258i	$-0.721621\pi$
$787$	− 950431.i	− 1.53452i	0.641339	−	0.767258i	$-0.278379\pi$
$788$	0	0
$789$	− 133526.i	− 0.214492i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−187561.	−0.298261
$794$	0	0
$795$	2.30226e6	3.64267
$796$	0	0
$797$	560412.i	0.882248i	0.897446	+	0.441124i	$0.145420\pi$
$797$	560412.i	0.882248i	−0.897446	+	0.441124i	$0.854580\pi$
$798$	0	0
$799$	−702771.	−1.10083
$800$	0	0
$801$	732168.i	1.14116i
$802$	0	0
$803$	558019.i	0.865401i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.47035e6	2.25774
$808$	0	0
$809$	−51864.2	−0.0792447	−0.0396224	−	0.999215i	$-0.512615\pi$
$809$	−51864.2	−0.0792447	−0.0396224	+	0.999215i	$0.512615\pi$
$810$	0	0
$811$	− 978879.i	− 1.48829i	−0.668019	−	0.744145i	$-0.732857\pi$
$811$	− 978879.i	− 1.48829i	0.668019	−	0.744145i	$-0.267143\pi$
$812$	0	0
$813$	82128.1	0.124254
$814$	0	0
$815$	− 172176.i	− 0.259214i
$816$	0	0
$817$	161744.i	0.242317i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	148798.	0.220755	0.110377	−	0.993890i	$-0.464794\pi$
$821$	148798.	0.220755	0.110377	+	0.993890i	$0.464794\pi$
$822$	0	0
$823$	−1.33907e6	−1.97698	−0.988490	−	0.151287i	$-0.951658\pi$
$823$	−1.33907e6	−1.97698	−0.988490	+	0.151287i	$0.951658\pi$
$824$	0	0
$825$	336210.i	0.493972i
$826$	0	0
$827$	127284.	0.186108	0.0930538	−	0.995661i	$-0.470337\pi$
$827$	127284.	0.186108	0.0930538	+	0.995661i	$0.470337\pi$
$828$	0	0
$829$	445598.i	0.648386i	0.945991	+	0.324193i	$0.105093\pi$
$829$	445598.i	0.648386i	−0.945991	+	0.324193i	$0.894907\pi$
$830$	0	0
$831$	1.05979e6i	1.53468i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.09360e6	1.56851
$836$	0	0
$837$	−596812.	−0.851896
$838$	0	0
$839$	− 552232.i	− 0.784509i	−0.919857	−	0.392254i	$-0.871695\pi$
$839$	− 552232.i	− 0.784509i	0.919857	−	0.392254i	$-0.128305\pi$
$840$	0	0
$841$	−560506.	−0.792479
$842$	0	0
$843$	− 610341.i	− 0.858850i
$844$	0	0
$845$	− 506025.i	− 0.708694i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.76327e6	2.44626
$850$	0	0
$851$	206432.	0.285048
$852$	0	0
$853$	− 753308.i	− 1.03532i	−0.855586	−	0.517660i	$-0.826803\pi$
$853$	− 753308.i	− 1.03532i	0.855586	−	0.517660i	$-0.173197\pi$
$854$	0	0
$855$	681515.	0.932274
$856$	0	0
$857$	− 1.30117e6i	− 1.77163i	−0.464037	−	0.885816i	$-0.653600\pi$
$857$	− 1.30117e6i	− 1.77163i	0.464037	−	0.885816i	$-0.346400\pi$
$858$	0	0
$859$	982372.i	1.33134i	0.746245	+	0.665671i	$0.231855\pi$
$859$	982372.i	1.33134i	−0.746245	+	0.665671i	$0.768145\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	721435.	0.968670	0.484335	−	0.874883i	$-0.339062\pi$
$863$	721435.	0.968670	0.484335	+	0.874883i	$0.339062\pi$
$864$	0	0
$865$	916353.	1.22470
$866$	0	0
$867$	3.37839e6i	4.49440i
$868$	0	0
$869$	1.48243e6	1.96307
$870$	0	0
$871$	− 643750.i	− 0.848557i
$872$	0	0
$873$	− 666984.i	− 0.875159i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−717008.	−0.932234	−0.466117	−	0.884723i	$-0.654347\pi$
$877$	−717008.	−0.932234	−0.466117	+	0.884723i	$0.654347\pi$
$878$	0	0
$879$	1.26385e6	1.63576
$880$	0	0
$881$	− 79128.9i	− 0.101949i	−0.998700	−	0.0509745i	$-0.983767\pi$
$881$	− 79128.9i	− 0.101949i	0.998700	−	0.0509745i	$-0.0162327\pi$
$882$	0	0
$883$	286591.	0.367571	0.183786	−	0.982966i	$-0.441165\pi$
$883$	286591.	0.367571	0.183786	+	0.982966i	$0.441165\pi$
$884$	0	0
$885$	− 1.85093e6i	− 2.36322i
$886$	0	0
$887$	− 837465.i	− 1.06444i	−0.846607	−	0.532218i	$-0.821359\pi$
$887$	− 837465.i	− 1.06444i	0.846607	−	0.532218i	$-0.178641\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.37607e6	1.73334
$892$	0	0
$893$	−181171.	−0.227188
$894$	0	0
$895$	− 132775.i	− 0.165756i
$896$	0	0
$897$	374430.	0.465356
$898$	0	0
$899$	− 152284.i	− 0.188424i
$900$	0	0
$901$	2.80588e6i	3.45636i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	980112.	1.19668
$906$	0	0
$907$	846569.	1.02908	0.514538	−	0.857467i	$-0.327963\pi$
$907$	846569.	1.02908	0.514538	+	0.857467i	$0.327963\pi$
$908$	0	0
$909$	861283.i	1.04236i
$910$	0	0
$911$	−167017.	−0.201244	−0.100622	−	0.994925i	$-0.532083\pi$
$911$	−167017.	−0.201244	−0.100622	+	0.994925i	$0.532083\pi$
$912$	0	0
$913$	803112.i	0.963461i
$914$	0	0
$915$	819658.i	0.979018i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.12203e6	−1.32854	−0.664268	−	0.747494i	$-0.731257\pi$
$919$	−1.12203e6	−1.32854	−0.664268	+	0.747494i	$0.731257\pi$
$920$	0	0
$921$	44813.3	0.0528309
$922$	0	0
$923$	128767.i	0.151148i
$924$	0	0
$925$	135281.	0.158108
$926$	0	0
$927$	3.30366e6i	3.84447i
$928$	0	0
$929$	− 1.42663e6i	− 1.65302i	−0.562921	−	0.826511i	$-0.690323\pi$
$929$	− 1.42663e6i	− 1.65302i	0.562921	−	0.826511i	$-0.309677\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−330462.	−0.379628
$934$	0	0
$935$	−2.11146e6	−2.41524
$936$	0	0
$937$	265587.i	0.302502i	0.988495	+	0.151251i	$0.0483301\pi$
$937$	265587.i	0.302502i	−0.988495	+	0.151251i	$0.951670\pi$
$938$	0	0
$939$	−597860.	−0.678060
$940$	0	0
$941$	598591.i	0.676007i	0.941145	+	0.338003i	$0.109751\pi$
$941$	598591.i	0.676007i	−0.941145	+	0.338003i	$0.890249\pi$
$942$	0	0
$943$	151076.i	0.169892i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.15085e6	−1.28328	−0.641639	−	0.767007i	$-0.721745\pi$
$947$	−1.15085e6	−1.28328	−0.641639	+	0.767007i	$0.721745\pi$
$948$	0	0
$949$	407257.	0.452206
$950$	0	0
$951$	1.70621e6i	1.88656i
$952$	0	0
$953$	464604.	0.511561	0.255781	−	0.966735i	$-0.417668\pi$
$953$	464604.	0.511561	0.255781	+	0.966735i	$0.417668\pi$
$954$	0	0
$955$	649271.i	0.711900i
$956$	0	0
$957$	855887.i	0.934529i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	765521.	0.828916
$962$	0	0
$963$	−1.48275e6	−1.59888
$964$	0	0
$965$	− 1.64813e6i	− 1.76985i
$966$	0	0
$967$	−101298.	−0.108330	−0.0541648	−	0.998532i	$-0.517250\pi$
$967$	−101298.	−0.108330	−0.0541648	+	0.998532i	$0.517250\pi$
$968$	0	0
$969$	1.21534e6i	1.29435i
$970$	0	0
$971$	326280.i	0.346060i	0.984917	+	0.173030i	$0.0553558\pi$
$971$	326280.i	0.346060i	−0.984917	+	0.173030i	$0.944644\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	245375.	0.258120
$976$	0	0
$977$	741928.	0.777272	0.388636	−	0.921391i	$-0.372946\pi$
$977$	741928.	0.777272	0.388636	+	0.921391i	$0.372946\pi$
$978$	0	0
$979$	− 584780.i	− 0.610137i
$980$	0	0
$981$	−2.04207e6	−2.12193
$982$	0	0
$983$	584497.i	0.604888i	0.953167	+	0.302444i	$0.0978026\pi$
$983$	584497.i	0.604888i	−0.953167	+	0.302444i	$0.902197\pi$
$984$	0	0
$985$	1.69599e6i	1.74803i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	265403.	0.271340
$990$	0	0
$991$	−582343.	−0.592969	−0.296484	−	0.955038i	$-0.595814\pi$
$991$	−582343.	−0.592969	−0.296484	+	0.955038i	$0.595814\pi$
$992$	0	0
$993$	1.73335e6i	1.75787i
$994$	0	0
$995$	−1.11716e6	−1.12842
$996$	0	0
$997$	1.36934e6i	1.37759i	0.724954	+	0.688797i	$0.241861\pi$
$997$	1.36934e6i	1.37759i	−0.724954	+	0.688797i	$0.758139\pi$
$998$	0	0
$999$	− 1.34967e6i	− 1.35237i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.5.c.a.97.1		4
4.3	odd	2		98.5.b.a.97.2	yes	4
7.6	odd	2	inner	784.5.c.a.97.4		4
12.11	even	2		882.5.c.c.685.4		4
28.3	even	6		98.5.d.c.19.3		8
28.11	odd	6		98.5.d.c.19.4		8
28.19	even	6		98.5.d.c.31.4		8
28.23	odd	6		98.5.d.c.31.3		8
28.27	even	2		98.5.b.a.97.1	✓	4
84.83	odd	2		882.5.c.c.685.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.5.b.a.97.1	✓	4	28.27	even	2
98.5.b.a.97.2	yes	4	4.3	odd	2
98.5.d.c.19.3		8	28.3	even	6
98.5.d.c.19.4		8	28.11	odd	6
98.5.d.c.31.3		8	28.23	odd	6
98.5.d.c.31.4		8	28.19	even	6
784.5.c.a.97.1		4	1.1	even	1	trivial
784.5.c.a.97.4		4	7.6	odd	2	inner
882.5.c.c.685.3		4	84.83	odd	2
882.5.c.c.685.4		4	12.11	even	2

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-15.9958i q^{3} -27.8477i q^{5} -174.865 q^{9} +O(q^{10})\)	\(q-15.9958i q^{3} -27.8477i q^{5} -174.865 q^{9} +139.664 q^{11} -101.931i q^{13} -445.446 q^{15} -542.887i q^{17} -139.953i q^{19} -229.647 q^{23} -150.495 q^{25} +1501.44i q^{27} -383.113 q^{29} +397.492i q^{31} -2234.03i q^{33} -898.912 q^{37} -1630.46 q^{39} -657.862i q^{41} -1155.70 q^{43} +4869.59i q^{45} -1294.51i q^{47} -8683.90 q^{51} -5168.43 q^{53} -3889.32i q^{55} -2238.66 q^{57} +4155.24i q^{59} -1840.09i q^{61} -2838.53 q^{65} +6315.57 q^{67} +3673.38i q^{69} -1263.28 q^{71} +3995.44i q^{73} +2407.28i q^{75} +10614.3 q^{79} +9852.71 q^{81} +5750.31i q^{83} -15118.2 q^{85} +6128.19i q^{87} -4187.05i q^{89} +6358.19 q^{93} -3897.38 q^{95} +3814.28i q^{97} -24422.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 196 q^{9}+O(q^{10}) \) 4 * q - 196 * q^9	\( 4 q - 196 q^{9} - 24 q^{11} - 888 q^{15} - 104 q^{23} + 948 q^{25} - 1408 q^{29} - 3392 q^{37} - 2200 q^{39} + 2024 q^{43} - 18936 q^{51} - 16680 q^{53} - 6064 q^{57} - 6048 q^{65} + 20816 q^{67} + 1984 q^{71} + 29616 q^{79} + 30852 q^{81} - 29688 q^{85} + 18192 q^{93} - 7240 q^{95} - 72160 q^{99}+O(q^{100}) \) 4 * q - 196 * q^9 - 24 * q^11 - 888 * q^15 - 104 * q^23 + 948 * q^25 - 1408 * q^29 - 3392 * q^37 - 2200 * q^39 + 2024 * q^43 - 18936 * q^51 - 16680 * q^53 - 6064 * q^57 - 6048 * q^65 + 20816 * q^67 + 1984 * q^71 + 29616 * q^79 + 30852 * q^81 - 29688 * q^85 + 18192 * q^93 - 7240 * q^95 - 72160 * q^99

Introduction

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