LMFDB - Embedded newform 1104.6.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 75x^{2} - 42x + 736 $ x^4 - 75x^2 - 42x + 736
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.04157$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +42.3660 q^{5} -191.647 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +42.3660 q^{5} -191.647 q^{7} +81.0000 q^{9} +655.589 q^{11} -932.384 q^{13} +381.294 q^{15} -344.188 q^{17} +548.438 q^{19} -1724.83 q^{21} +529.000 q^{23} -1330.12 q^{25} +729.000 q^{27} +4399.54 q^{29} +4434.84 q^{31} +5900.30 q^{33} -8119.33 q^{35} -11414.8 q^{37} -8391.46 q^{39} -6661.33 q^{41} +22878.8 q^{43} +3431.65 q^{45} -17936.0 q^{47} +19921.7 q^{49} -3097.69 q^{51} +4562.43 q^{53} +27774.7 q^{55} +4935.95 q^{57} +23244.6 q^{59} +25147.4 q^{61} -15523.4 q^{63} -39501.4 q^{65} +6212.65 q^{67} +4761.00 q^{69} -6800.71 q^{71} -32996.3 q^{73} -11971.1 q^{75} -125642. q^{77} +21087.6 q^{79} +6561.00 q^{81} -28330.8 q^{83} -14581.9 q^{85} +39595.8 q^{87} +137589. q^{89} +178689. q^{91} +39913.5 q^{93} +23235.1 q^{95} -32286.5 q^{97} +53102.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{3} + 22 q^{5} + 62 q^{7} + 324 q^{9}+O(q^{10}) $ 4 * q + 36 * q^3 + 22 * q^5 + 62 * q^7 + 324 * q^9	$ 4 q + 36 q^{3} + 22 q^{5} + 62 q^{7} + 324 q^{9} + 1076 q^{11} - 396 q^{13} + 198 q^{15} + 70 q^{17} + 6366 q^{19} + 558 q^{21} + 2116 q^{23} + 1264 q^{25} + 2916 q^{27} + 3948 q^{29} - 3092 q^{31} + 9684 q^{33} - 1304 q^{35} - 17464 q^{37} - 3564 q^{39} + 18680 q^{41} + 25846 q^{43} + 1782 q^{45} - 18392 q^{47} + 7952 q^{49} + 630 q^{51} - 26518 q^{53} + 40848 q^{55} + 57294 q^{57} + 14520 q^{59} - 13688 q^{61} + 5022 q^{63} + 38324 q^{65} + 11098 q^{67} + 19044 q^{69} + 57496 q^{71} - 112272 q^{73} + 11376 q^{75} - 4792 q^{77} + 240754 q^{79} + 26244 q^{81} + 93268 q^{83} - 323204 q^{85} + 35532 q^{87} - 107582 q^{89} + 301532 q^{91} - 27828 q^{93} + 18640 q^{95} - 53076 q^{97} + 87156 q^{99}+O(q^{100}) $ 4 * q + 36 * q^3 + 22 * q^5 + 62 * q^7 + 324 * q^9 + 1076 * q^11 - 396 * q^13 + 198 * q^15 + 70 * q^17 + 6366 * q^19 + 558 * q^21 + 2116 * q^23 + 1264 * q^25 + 2916 * q^27 + 3948 * q^29 - 3092 * q^31 + 9684 * q^33 - 1304 * q^35 - 17464 * q^37 - 3564 * q^39 + 18680 * q^41 + 25846 * q^43 + 1782 * q^45 - 18392 * q^47 + 7952 * q^49 + 630 * q^51 - 26518 * q^53 + 40848 * q^55 + 57294 * q^57 + 14520 * q^59 - 13688 * q^61 + 5022 * q^63 + 38324 * q^65 + 11098 * q^67 + 19044 * q^69 + 57496 * q^71 - 112272 * q^73 + 11376 * q^75 - 4792 * q^77 + 240754 * q^79 + 26244 * q^81 + 93268 * q^83 - 323204 * q^85 + 35532 * q^87 - 107582 * q^89 + 301532 * q^91 - 27828 * q^93 + 18640 * q^95 - 53076 * q^97 + 87156 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	42.3660	0.757866	0.378933	−	0.925424i	$-0.376291\pi$
$5$	42.3660	0.757866	0.378933	+	0.925424i	$0.376291\pi$
$6$	0	0
$7$	−191.647	−1.47828	−0.739142	−	0.673550i	$-0.764769\pi$
$7$	−191.647	−1.47828	−0.739142	+	0.673550i	$0.764769\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	655.589	1.63362	0.816808	−	0.576909i	$-0.195741\pi$
$11$	655.589	1.63362	0.816808	+	0.576909i	$0.195741\pi$
$12$	0	0
$13$	−932.384	−1.53016	−0.765079	−	0.643936i	$-0.777300\pi$
$13$	−932.384	−1.53016	−0.765079	+	0.643936i	$0.777300\pi$
$14$	0	0
$15$	381.294	0.437554
$16$	0	0
$17$	−344.188	−0.288850	−0.144425	−	0.989516i	$-0.546133\pi$
$17$	−344.188	−0.288850	−0.144425	+	0.989516i	$0.546133\pi$
$18$	0	0
$19$	548.438	0.348533	0.174266	−	0.984699i	$-0.444245\pi$
$19$	548.438	0.348533	0.174266	+	0.984699i	$0.444245\pi$
$20$	0	0
$21$	−1724.83	−0.853487
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−1330.12	−0.425639
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	4399.54	0.971431	0.485715	−	0.874117i	$-0.338559\pi$
$29$	4399.54	0.971431	0.485715	+	0.874117i	$0.338559\pi$
$30$	0	0
$31$	4434.84	0.828845	0.414422	−	0.910085i	$-0.363984\pi$
$31$	4434.84	0.828845	0.414422	+	0.910085i	$0.363984\pi$
$32$	0	0
$33$	5900.30	0.943169
$34$	0	0
$35$	−8119.33	−1.12034
$36$	0	0
$37$	−11414.8	−1.37077	−0.685384	−	0.728182i	$-0.740366\pi$
$37$	−11414.8	−1.37077	−0.685384	+	0.728182i	$0.740366\pi$
$38$	0	0
$39$	−8391.46	−0.883437
$40$	0	0
$41$	−6661.33	−0.618872	−0.309436	−	0.950920i	$-0.600140\pi$
$41$	−6661.33	−0.618872	−0.309436	+	0.950920i	$0.600140\pi$
$42$	0	0
$43$	22878.8	1.88696	0.943478	−	0.331434i	$-0.107533\pi$
$43$	22878.8	1.88696	0.943478	+	0.331434i	$0.107533\pi$
$44$	0	0
$45$	3431.65	0.252622
$46$	0	0
$47$	−17936.0	−1.18435	−0.592176	−	0.805808i	$-0.701731\pi$
$47$	−17936.0	−1.18435	−0.592176	+	0.805808i	$0.701731\pi$
$48$	0	0
$49$	19921.7	1.18532
$50$	0	0
$51$	−3097.69	−0.166768
$52$	0	0
$53$	4562.43	0.223103	0.111552	−	0.993759i	$-0.464418\pi$
$53$	4562.43	0.223103	0.111552	+	0.993759i	$0.464418\pi$
$54$	0	0
$55$	27774.7	1.23806
$56$	0	0
$57$	4935.95	0.201226
$58$	0	0
$59$	23244.6	0.869345	0.434673	−	0.900589i	$-0.356864\pi$
$59$	23244.6	0.869345	0.434673	+	0.900589i	$0.356864\pi$
$60$	0	0
$61$	25147.4	0.865304	0.432652	−	0.901561i	$-0.357578\pi$
$61$	25147.4	0.865304	0.432652	+	0.901561i	$0.357578\pi$
$62$	0	0
$63$	−15523.4	−0.492761
$64$	0	0
$65$	−39501.4	−1.15966
$66$	0	0
$67$	6212.65	0.169079	0.0845396	−	0.996420i	$-0.473058\pi$
$67$	6212.65	0.169079	0.0845396	+	0.996420i	$0.473058\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	−6800.71	−0.160106	−0.0800531	−	0.996791i	$-0.525509\pi$
$71$	−6800.71	−0.160106	−0.0800531	+	0.996791i	$0.525509\pi$
$72$	0	0
$73$	−32996.3	−0.724700	−0.362350	−	0.932042i	$-0.618026\pi$
$73$	−32996.3	−0.724700	−0.362350	+	0.932042i	$0.618026\pi$
$74$	0	0
$75$	−11971.1	−0.245743
$76$	0	0
$77$	−125642.	−2.41495
$78$	0	0
$79$	21087.6	0.380154	0.190077	−	0.981769i	$-0.439126\pi$
$79$	21087.6	0.380154	0.190077	+	0.981769i	$0.439126\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−28330.8	−0.451402	−0.225701	−	0.974197i	$-0.572467\pi$
$83$	−28330.8	−0.451402	−0.225701	+	0.974197i	$0.572467\pi$
$84$	0	0
$85$	−14581.9	−0.218910
$86$	0	0
$87$	39595.8	0.560856
$88$	0	0
$89$	137589.	1.84123	0.920616	−	0.390469i	$-0.127687\pi$
$89$	137589.	1.84123	0.920616	+	0.390469i	$0.127687\pi$
$90$	0	0
$91$	178689.	2.26201
$92$	0	0
$93$	39913.5	0.478534
$94$	0	0
$95$	23235.1	0.264141
$96$	0	0
$97$	−32286.5	−0.348411	−0.174206	−	0.984709i	$-0.555736\pi$
$97$	−32286.5	−0.348411	−0.174206	+	0.984709i	$0.555736\pi$
$98$	0	0
$99$	53102.7	0.544539
$100$	0	0
$101$	−163233.	−1.59222	−0.796112	−	0.605149i	$-0.793113\pi$
$101$	−163233.	−1.59222	−0.796112	+	0.605149i	$0.793113\pi$
$102$	0	0
$103$	68514.0	0.636336	0.318168	−	0.948034i	$-0.396932\pi$
$103$	68514.0	0.636336	0.318168	+	0.948034i	$0.396932\pi$
$104$	0	0
$105$	−73074.0	−0.646829
$106$	0	0
$107$	20369.5	0.171997	0.0859985	−	0.996295i	$-0.472592\pi$
$107$	20369.5	0.171997	0.0859985	+	0.996295i	$0.472592\pi$
$108$	0	0
$109$	245848.	1.98198	0.990992	−	0.133920i	$-0.0427567\pi$
$109$	245848.	1.98198	0.990992	+	0.133920i	$0.0427567\pi$
$110$	0	0
$111$	−102733.	−0.791413
$112$	0	0
$113$	258501.	1.90444	0.952218	−	0.305419i	$-0.0987966\pi$
$113$	258501.	1.90444	0.952218	+	0.305419i	$0.0987966\pi$
$114$	0	0
$115$	22411.6	0.158026
$116$	0	0
$117$	−75523.1	−0.510053
$118$	0	0
$119$	65962.6	0.427003
$120$	0	0
$121$	268746.	1.66870
$122$	0	0
$123$	−59951.9	−0.357306
$124$	0	0
$125$	−188746.	−1.08044
$126$	0	0
$127$	151439.	0.833160	0.416580	−	0.909099i	$-0.363229\pi$
$127$	151439.	0.833160	0.416580	+	0.909099i	$0.363229\pi$
$128$	0	0
$129$	205909.	1.08943
$130$	0	0
$131$	−54236.2	−0.276128	−0.138064	−	0.990423i	$-0.544088\pi$
$131$	−54236.2	−0.276128	−0.138064	+	0.990423i	$0.544088\pi$
$132$	0	0
$133$	−105107.	−0.515230
$134$	0	0
$135$	30884.8	0.145851
$136$	0	0
$137$	210827.	0.959677	0.479839	−	0.877357i	$-0.340695\pi$
$137$	210827.	0.959677	0.479839	+	0.877357i	$0.340695\pi$
$138$	0	0
$139$	319153.	1.40108	0.700539	−	0.713614i	$-0.252943\pi$
$139$	319153.	1.40108	0.700539	+	0.713614i	$0.252943\pi$
$140$	0	0
$141$	−161424.	−0.683786
$142$	0	0
$143$	−611261.	−2.49969
$144$	0	0
$145$	186391.	0.736215
$146$	0	0
$147$	179295.	0.684346
$148$	0	0
$149$	−319057.	−1.17734	−0.588671	−	0.808373i	$-0.700349\pi$
$149$	−319057.	−1.17734	−0.588671	+	0.808373i	$0.700349\pi$
$150$	0	0
$151$	−100255.	−0.357820	−0.178910	−	0.983865i	$-0.557257\pi$
$151$	−100255.	−0.357820	−0.178910	+	0.983865i	$0.557257\pi$
$152$	0	0
$153$	−27879.2	−0.0962835
$154$	0	0
$155$	187886.	0.628154
$156$	0	0
$157$	334886.	1.08430	0.542149	−	0.840283i	$-0.317611\pi$
$157$	334886.	1.08430	0.542149	+	0.840283i	$0.317611\pi$
$158$	0	0
$159$	41061.8	0.128809
$160$	0	0
$161$	−101381.	−0.308243
$162$	0	0
$163$	−268471.	−0.791460	−0.395730	−	0.918367i	$-0.629508\pi$
$163$	−268471.	−0.791460	−0.395730	+	0.918367i	$0.629508\pi$
$164$	0	0
$165$	249972.	0.714796
$166$	0	0
$167$	250738.	0.695711	0.347855	−	0.937548i	$-0.386910\pi$
$167$	250738.	0.695711	0.347855	+	0.937548i	$0.386910\pi$
$168$	0	0
$169$	498047.	1.34138
$170$	0	0
$171$	44423.5	0.116178
$172$	0	0
$173$	478595.	1.21577	0.607887	−	0.794023i	$-0.292017\pi$
$173$	478595.	1.21577	0.607887	+	0.794023i	$0.292017\pi$
$174$	0	0
$175$	254914.	0.629215
$176$	0	0
$177$	209202.	0.501917
$178$	0	0
$179$	416312.	0.971150	0.485575	−	0.874195i	$-0.338610\pi$
$179$	416312.	0.971150	0.485575	+	0.874195i	$0.338610\pi$
$180$	0	0
$181$	−517731.	−1.17465	−0.587324	−	0.809352i	$-0.699818\pi$
$181$	−517731.	−1.17465	−0.587324	+	0.809352i	$0.699818\pi$
$182$	0	0
$183$	226327.	0.499583
$184$	0	0
$185$	−483600.	−1.03886
$186$	0	0
$187$	−225646.	−0.471871
$188$	0	0
$189$	−139711.	−0.284496
$190$	0	0
$191$	162153.	0.321618	0.160809	−	0.986986i	$-0.448590\pi$
$191$	162153.	0.321618	0.160809	+	0.986986i	$0.448590\pi$
$192$	0	0
$193$	−385440.	−0.744840	−0.372420	−	0.928064i	$-0.621472\pi$
$193$	−385440.	−0.744840	−0.372420	+	0.928064i	$0.621472\pi$
$194$	0	0
$195$	−355512.	−0.669527
$196$	0	0
$197$	−788562.	−1.44767	−0.723836	−	0.689972i	$-0.757623\pi$
$197$	−788562.	−1.44767	−0.723836	+	0.689972i	$0.757623\pi$
$198$	0	0
$199$	556981.	0.997028	0.498514	−	0.866882i	$-0.333879\pi$
$199$	556981.	0.997028	0.498514	+	0.866882i	$0.333879\pi$
$200$	0	0
$201$	55913.9	0.0976179
$202$	0	0
$203$	−843159.	−1.43605
$204$	0	0
$205$	−282214.	−0.469022
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	359550.	0.569369
$210$	0	0
$211$	−617009.	−0.954081	−0.477041	−	0.878881i	$-0.658291\pi$
$211$	−617009.	−0.954081	−0.477041	+	0.878881i	$0.658291\pi$
$212$	0	0
$213$	−61206.4	−0.0924374
$214$	0	0
$215$	969283.	1.43006
$216$	0	0
$217$	−849925.	−1.22527
$218$	0	0
$219$	−296967.	−0.418406
$220$	0	0
$221$	320915.	0.441987
$222$	0	0
$223$	−373644.	−0.503148	−0.251574	−	0.967838i	$-0.580948\pi$
$223$	−373644.	−0.503148	−0.251574	+	0.967838i	$0.580948\pi$
$224$	0	0
$225$	−107740.	−0.141880
$226$	0	0
$227$	−550317.	−0.708841	−0.354420	−	0.935086i	$-0.615322\pi$
$227$	−550317.	−0.708841	−0.354420	+	0.935086i	$0.615322\pi$
$228$	0	0
$229$	−830028.	−1.04593	−0.522966	−	0.852353i	$-0.675175\pi$
$229$	−830028.	−1.04593	−0.522966	+	0.852353i	$0.675175\pi$
$230$	0	0
$231$	−1.13078e6	−1.39427
$232$	0	0
$233$	1.54010e6	1.85848	0.929242	−	0.369472i	$-0.120461\pi$
$233$	1.54010e6	1.85848	0.929242	+	0.369472i	$0.120461\pi$
$234$	0	0
$235$	−759877.	−0.897581
$236$	0	0
$237$	189788.	0.219482
$238$	0	0
$239$	−1.35520e6	−1.53465	−0.767323	−	0.641261i	$-0.778412\pi$
$239$	−1.35520e6	−1.53465	−0.767323	+	0.641261i	$0.778412\pi$
$240$	0	0
$241$	1.20119e6	1.33220	0.666099	−	0.745864i	$-0.267963\pi$
$241$	1.20119e6	1.33220	0.666099	+	0.745864i	$0.267963\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	844003.	0.898315
$246$	0	0
$247$	−511355.	−0.533311
$248$	0	0
$249$	−254977.	−0.260617
$250$	0	0
$251$	−523235.	−0.524218	−0.262109	−	0.965038i	$-0.584418\pi$
$251$	−523235.	−0.524218	−0.262109	+	0.965038i	$0.584418\pi$
$252$	0	0
$253$	346807.	0.340633
$254$	0	0
$255$	−131237.	−0.126388
$256$	0	0
$257$	667770.	0.630658	0.315329	−	0.948982i	$-0.397885\pi$
$257$	667770.	0.630658	0.315329	+	0.948982i	$0.397885\pi$
$258$	0	0
$259$	2.18762e6	2.02638
$260$	0	0
$261$	356362.	0.323810
$262$	0	0
$263$	2.15898e6	1.92469	0.962343	−	0.271837i	$-0.0876311\pi$
$263$	2.15898e6	1.92469	0.962343	+	0.271837i	$0.0876311\pi$
$264$	0	0
$265$	193292.	0.169083
$266$	0	0
$267$	1.23830e6	1.06304
$268$	0	0
$269$	−89814.4	−0.0756772	−0.0378386	−	0.999284i	$-0.512047\pi$
$269$	−89814.4	−0.0756772	−0.0378386	+	0.999284i	$0.512047\pi$
$270$	0	0
$271$	1.74478e6	1.44317	0.721583	−	0.692328i	$-0.243415\pi$
$271$	1.74478e6	1.44317	0.721583	+	0.692328i	$0.243415\pi$
$272$	0	0
$273$	1.60820e6	1.30597
$274$	0	0
$275$	−872013.	−0.695331
$276$	0	0
$277$	2.19896e6	1.72194	0.860971	−	0.508654i	$-0.169857\pi$
$277$	2.19896e6	1.72194	0.860971	+	0.508654i	$0.169857\pi$
$278$	0	0
$279$	359222.	0.276282
$280$	0	0
$281$	857174.	0.647595	0.323797	−	0.946126i	$-0.395040\pi$
$281$	857174.	0.647595	0.323797	+	0.946126i	$0.395040\pi$
$282$	0	0
$283$	−500871.	−0.371757	−0.185879	−	0.982573i	$-0.559513\pi$
$283$	−500871.	−0.371757	−0.185879	+	0.982573i	$0.559513\pi$
$284$	0	0
$285$	209116.	0.152502
$286$	0	0
$287$	1.27663e6	0.914869
$288$	0	0
$289$	−1.30139e6	−0.916565
$290$	0	0
$291$	−290579.	−0.201155
$292$	0	0
$293$	543998.	0.370193	0.185096	−	0.982720i	$-0.440740\pi$
$293$	543998.	0.370193	0.185096	+	0.982720i	$0.440740\pi$
$294$	0	0
$295$	984782.	0.658847
$296$	0	0
$297$	477925.	0.314390
$298$	0	0
$299$	−493231.	−0.319060
$300$	0	0
$301$	−4.38466e6	−2.78946
$302$	0	0
$303$	−1.46910e6	−0.919271
$304$	0	0
$305$	1.06540e6	0.655785
$306$	0	0
$307$	2.32077e6	1.40536	0.702678	−	0.711508i	$-0.251987\pi$
$307$	2.32077e6	1.40536	0.702678	+	0.711508i	$0.251987\pi$
$308$	0	0
$309$	616626.	0.367389
$310$	0	0
$311$	1.61095e6	0.944455	0.472227	−	0.881477i	$-0.343450\pi$
$311$	1.61095e6	0.944455	0.472227	+	0.881477i	$0.343450\pi$
$312$	0	0
$313$	458852.	0.264735	0.132368	−	0.991201i	$-0.457742\pi$
$313$	458852.	0.264735	0.132368	+	0.991201i	$0.457742\pi$
$314$	0	0
$315$	−657666.	−0.373447
$316$	0	0
$317$	−1.32737e6	−0.741900	−0.370950	−	0.928653i	$-0.620968\pi$
$317$	−1.32737e6	−0.741900	−0.370950	+	0.928653i	$0.620968\pi$
$318$	0	0
$319$	2.88429e6	1.58695
$320$	0	0
$321$	183325.	0.0993025
$322$	0	0
$323$	−188766.	−0.100674
$324$	0	0
$325$	1.24018e6	0.651295
$326$	0	0
$327$	2.21263e6	1.14430
$328$	0	0
$329$	3.43739e6	1.75081
$330$	0	0
$331$	1.53447e6	0.769819	0.384910	−	0.922954i	$-0.374233\pi$
$331$	1.53447e6	0.769819	0.384910	+	0.922954i	$0.374233\pi$
$332$	0	0
$333$	−924599.	−0.456923
$334$	0	0
$335$	263205.	0.128139
$336$	0	0
$337$	−1.62882e6	−0.781265	−0.390632	−	0.920547i	$-0.627744\pi$
$337$	−1.62882e6	−0.781265	−0.390632	+	0.920547i	$0.627744\pi$
$338$	0	0
$339$	2.32651e6	1.09953
$340$	0	0
$341$	2.90743e6	1.35401
$342$	0	0
$343$	−596925.	−0.273958
$344$	0	0
$345$	201705.	0.0912364
$346$	0	0
$347$	645289.	0.287694	0.143847	−	0.989600i	$-0.454053\pi$
$347$	645289.	0.287694	0.143847	+	0.989600i	$0.454053\pi$
$348$	0	0
$349$	3.55973e6	1.56442	0.782211	−	0.623013i	$-0.214092\pi$
$349$	3.55973e6	1.56442	0.782211	+	0.623013i	$0.214092\pi$
$350$	0	0
$351$	−679708.	−0.294479
$352$	0	0
$353$	−1.45690e6	−0.622292	−0.311146	−	0.950362i	$-0.600713\pi$
$353$	−1.45690e6	−0.622292	−0.311146	+	0.950362i	$0.600713\pi$
$354$	0	0
$355$	−288119.	−0.121339
$356$	0	0
$357$	593664.	0.246530
$358$	0	0
$359$	1.36735e6	0.559942	0.279971	−	0.960008i	$-0.409675\pi$
$359$	1.36735e6	0.559942	0.279971	+	0.960008i	$0.409675\pi$
$360$	0	0
$361$	−2.17531e6	−0.878525
$362$	0	0
$363$	2.41872e6	0.963426
$364$	0	0
$365$	−1.39792e6	−0.549226
$366$	0	0
$367$	469454.	0.181940	0.0909700	−	0.995854i	$-0.471003\pi$
$367$	469454.	0.181940	0.0909700	+	0.995854i	$0.471003\pi$
$368$	0	0
$369$	−539567.	−0.206291
$370$	0	0
$371$	−874377.	−0.329810
$372$	0	0
$373$	3.72580e6	1.38659	0.693295	−	0.720654i	$-0.256159\pi$
$373$	3.72580e6	1.38659	0.693295	+	0.720654i	$0.256159\pi$
$374$	0	0
$375$	−1.69871e6	−0.623794
$376$	0	0
$377$	−4.10206e6	−1.48644
$378$	0	0
$379$	−298202.	−0.106638	−0.0533191	−	0.998578i	$-0.516980\pi$
$379$	−298202.	−0.106638	−0.0533191	+	0.998578i	$0.516980\pi$
$380$	0	0
$381$	1.36295e6	0.481025
$382$	0	0
$383$	666922.	0.232315	0.116158	−	0.993231i	$-0.462942\pi$
$383$	666922.	0.232315	0.116158	+	0.993231i	$0.462942\pi$
$384$	0	0
$385$	−5.32295e6	−1.83021
$386$	0	0
$387$	1.85318e6	0.628986
$388$	0	0
$389$	784906.	0.262993	0.131496	−	0.991317i	$-0.458022\pi$
$389$	784906.	0.262993	0.131496	+	0.991317i	$0.458022\pi$
$390$	0	0
$391$	−182075.	−0.0602295
$392$	0	0
$393$	−488126.	−0.159423
$394$	0	0
$395$	893397.	0.288106
$396$	0	0
$397$	−940631.	−0.299532	−0.149766	−	0.988721i	$-0.547852\pi$
$397$	−940631.	−0.299532	−0.149766	+	0.988721i	$0.547852\pi$
$398$	0	0
$399$	−945961.	−0.297468
$400$	0	0
$401$	−1.47321e6	−0.457514	−0.228757	−	0.973484i	$-0.573466\pi$
$401$	−1.47321e6	−0.457514	−0.228757	+	0.973484i	$0.573466\pi$
$402$	0	0
$403$	−4.13497e6	−1.26826
$404$	0	0
$405$	277963.	0.0842074
$406$	0	0
$407$	−7.48342e6	−2.23931
$408$	0	0
$409$	2.05532e6	0.607533	0.303767	−	0.952746i	$-0.401756\pi$
$409$	2.05532e6	0.607533	0.303767	+	0.952746i	$0.401756\pi$
$410$	0	0
$411$	1.89744e6	0.554070
$412$	0	0
$413$	−4.45477e6	−1.28514
$414$	0	0
$415$	−1.20026e6	−0.342103
$416$	0	0
$417$	2.87238e6	0.808913
$418$	0	0
$419$	2.01044e6	0.559442	0.279721	−	0.960081i	$-0.409758\pi$
$419$	2.01044e6	0.559442	0.279721	+	0.960081i	$0.409758\pi$
$420$	0	0
$421$	1.78234e6	0.490100	0.245050	−	0.969510i	$-0.421196\pi$
$421$	1.78234e6	0.490100	0.245050	+	0.969510i	$0.421196\pi$
$422$	0	0
$423$	−1.45282e6	−0.394784
$424$	0	0
$425$	457811.	0.122946
$426$	0	0
$427$	−4.81943e6	−1.27916
$428$	0	0
$429$	−5.50135e6	−1.44320
$430$	0	0
$431$	5.62870e6	1.45954	0.729768	−	0.683695i	$-0.239628\pi$
$431$	5.62870e6	1.45954	0.729768	+	0.683695i	$0.239628\pi$
$432$	0	0
$433$	−432214.	−0.110785	−0.0553923	−	0.998465i	$-0.517641\pi$
$433$	−432214.	−0.110785	−0.0553923	+	0.998465i	$0.517641\pi$
$434$	0	0
$435$	1.67752e6	0.425054
$436$	0	0
$437$	290124.	0.0726741
$438$	0	0
$439$	2.99306e6	0.741233	0.370616	−	0.928786i	$-0.379146\pi$
$439$	2.99306e6	0.741233	0.370616	+	0.928786i	$0.379146\pi$
$440$	0	0
$441$	1.61366e6	0.395107
$442$	0	0
$443$	1.63968e6	0.396963	0.198481	−	0.980105i	$-0.436399\pi$
$443$	1.63968e6	0.396963	0.198481	+	0.980105i	$0.436399\pi$
$444$	0	0
$445$	5.82909e6	1.39541
$446$	0	0
$447$	−2.87151e6	−0.679739
$448$	0	0
$449$	−3.99192e6	−0.934471	−0.467236	−	0.884133i	$-0.654750\pi$
$449$	−3.99192e6	−0.934471	−0.467236	+	0.884133i	$0.654750\pi$
$450$	0	0
$451$	−4.36709e6	−1.01100
$452$	0	0
$453$	−902297.	−0.206587
$454$	0	0
$455$	7.57034e6	1.71430
$456$	0	0
$457$	−1.36122e6	−0.304887	−0.152444	−	0.988312i	$-0.548714\pi$
$457$	−1.36122e6	−0.304887	−0.152444	+	0.988312i	$0.548714\pi$
$458$	0	0
$459$	−250913.	−0.0555893
$460$	0	0
$461$	5.00721e6	1.09734	0.548672	−	0.836037i	$-0.315133\pi$
$461$	5.00721e6	1.09734	0.548672	+	0.836037i	$0.315133\pi$
$462$	0	0
$463$	−7.87132e6	−1.70646	−0.853228	−	0.521538i	$-0.825359\pi$
$463$	−7.87132e6	−1.70646	−0.853228	+	0.521538i	$0.825359\pi$
$464$	0	0
$465$	1.69098e6	0.362665
$466$	0	0
$467$	1.70051e6	0.360817	0.180409	−	0.983592i	$-0.442258\pi$
$467$	1.70051e6	0.360817	0.180409	+	0.983592i	$0.442258\pi$
$468$	0	0
$469$	−1.19064e6	−0.249947
$470$	0	0
$471$	3.01398e6	0.626019
$472$	0	0
$473$	1.49991e7	3.08256
$474$	0	0
$475$	−729490.	−0.148349
$476$	0	0
$477$	369557.	0.0743678
$478$	0	0
$479$	−4.15702e6	−0.827834	−0.413917	−	0.910315i	$-0.635840\pi$
$479$	−4.15702e6	−0.827834	−0.413917	+	0.910315i	$0.635840\pi$
$480$	0	0
$481$	1.06430e7	2.09749
$482$	0	0
$483$	−912433.	−0.177964
$484$	0	0
$485$	−1.36785e6	−0.264049
$486$	0	0
$487$	2.37037e6	0.452891	0.226445	−	0.974024i	$-0.427290\pi$
$487$	2.37037e6	0.452891	0.226445	+	0.974024i	$0.427290\pi$
$488$	0	0
$489$	−2.41624e6	−0.456950
$490$	0	0
$491$	5.11264e6	0.957064	0.478532	−	0.878070i	$-0.341169\pi$
$491$	5.11264e6	0.957064	0.478532	+	0.878070i	$0.341169\pi$
$492$	0	0
$493$	−1.51427e6	−0.280598
$494$	0	0
$495$	2.24975e6	0.412688
$496$	0	0
$497$	1.30334e6	0.236682
$498$	0	0
$499$	−1.81109e6	−0.325603	−0.162802	−	0.986659i	$-0.552053\pi$
$499$	−1.81109e6	−0.325603	−0.162802	+	0.986659i	$0.552053\pi$
$500$	0	0
$501$	2.25664e6	0.401669
$502$	0	0
$503$	−1.30925e6	−0.230730	−0.115365	−	0.993323i	$-0.536804\pi$
$503$	−1.30925e6	−0.230730	−0.115365	+	0.993323i	$0.536804\pi$
$504$	0	0
$505$	−6.91553e6	−1.20669
$506$	0	0
$507$	4.48242e6	0.774449
$508$	0	0
$509$	−6.70636e6	−1.14734	−0.573671	−	0.819086i	$-0.694481\pi$
$509$	−6.70636e6	−1.14734	−0.573671	+	0.819086i	$0.694481\pi$
$510$	0	0
$511$	6.32366e6	1.07131
$512$	0	0
$513$	399812.	0.0670752
$514$	0	0
$515$	2.90267e6	0.482257
$516$	0	0
$517$	−1.17586e7	−1.93478
$518$	0	0
$519$	4.30736e6	0.701928
$520$	0	0
$521$	6.03711e6	0.974394	0.487197	−	0.873292i	$-0.338019\pi$
$521$	6.03711e6	0.974394	0.487197	+	0.873292i	$0.338019\pi$
$522$	0	0
$523$	200223.	0.0320081	0.0160041	−	0.999872i	$-0.494906\pi$
$523$	200223.	0.0320081	0.0160041	+	0.999872i	$0.494906\pi$
$524$	0	0
$525$	2.29423e6	0.363277
$526$	0	0
$527$	−1.52642e6	−0.239412
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	1.88281e6	0.289782
$532$	0	0
$533$	6.21091e6	0.946973
$534$	0	0
$535$	862974.	0.130351
$536$	0	0
$537$	3.74681e6	0.560693
$538$	0	0
$539$	1.30605e7	1.93636
$540$	0	0
$541$	−7.38505e6	−1.08483	−0.542413	−	0.840112i	$-0.682489\pi$
$541$	−7.38505e6	−1.08483	−0.542413	+	0.840112i	$0.682489\pi$
$542$	0	0
$543$	−4.65958e6	−0.678183
$544$	0	0
$545$	1.04156e7	1.50208
$546$	0	0
$547$	5.16128e6	0.737546	0.368773	−	0.929519i	$-0.379778\pi$
$547$	5.16128e6	0.737546	0.368773	+	0.929519i	$0.379778\pi$
$548$	0	0
$549$	2.03694e6	0.288435
$550$	0	0
$551$	2.41287e6	0.338576
$552$	0	0
$553$	−4.04138e6	−0.561975
$554$	0	0
$555$	−4.35240e6	−0.599786
$556$	0	0
$557$	−4.53224e6	−0.618978	−0.309489	−	0.950903i	$-0.600158\pi$
$557$	−4.53224e6	−0.618978	−0.309489	+	0.950903i	$0.600158\pi$
$558$	0	0
$559$	−2.13318e7	−2.88734
$560$	0	0
$561$	−2.03081e6	−0.272435
$562$	0	0
$563$	−6.52075e6	−0.867014	−0.433507	−	0.901150i	$-0.642724\pi$
$563$	−6.52075e6	−0.867014	−0.433507	+	0.901150i	$0.642724\pi$
$564$	0	0
$565$	1.09517e7	1.44331
$566$	0	0
$567$	−1.25740e6	−0.164254
$568$	0	0
$569$	9.01923e6	1.16785	0.583927	−	0.811806i	$-0.301515\pi$
$569$	9.01923e6	1.16785	0.583927	+	0.811806i	$0.301515\pi$
$570$	0	0
$571$	1.28625e7	1.65095	0.825476	−	0.564437i	$-0.190907\pi$
$571$	1.28625e7	1.65095	0.825476	+	0.564437i	$0.190907\pi$
$572$	0	0
$573$	1.45937e6	0.185686
$574$	0	0
$575$	−703634.	−0.0887518
$576$	0	0
$577$	−5.08026e6	−0.635252	−0.317626	−	0.948216i	$-0.602886\pi$
$577$	−5.08026e6	−0.635252	−0.317626	+	0.948216i	$0.602886\pi$
$578$	0	0
$579$	−3.46896e6	−0.430034
$580$	0	0
$581$	5.42953e6	0.667301
$582$	0	0
$583$	2.99108e6	0.364465
$584$	0	0
$585$	−3.19961e6	−0.386552
$586$	0	0
$587$	−6.83820e6	−0.819118	−0.409559	−	0.912284i	$-0.634317\pi$
$587$	−6.83820e6	−0.819118	−0.409559	+	0.912284i	$0.634317\pi$
$588$	0	0
$589$	2.43223e6	0.288880
$590$	0	0
$591$	−7.09706e6	−0.835814
$592$	0	0
$593$	−1.32165e7	−1.54341	−0.771705	−	0.635981i	$-0.780596\pi$
$593$	−1.32165e7	−1.54341	−0.771705	+	0.635981i	$0.780596\pi$
$594$	0	0
$595$	2.79457e6	0.323611
$596$	0	0
$597$	5.01283e6	0.575635
$598$	0	0
$599$	7.22555e6	0.822818	0.411409	−	0.911451i	$-0.365037\pi$
$599$	7.22555e6	0.822818	0.411409	+	0.911451i	$0.365037\pi$
$600$	0	0
$601$	1.03172e7	1.16514	0.582568	−	0.812782i	$-0.302048\pi$
$601$	1.03172e7	1.16514	0.582568	+	0.812782i	$0.302048\pi$
$602$	0	0
$603$	503225.	0.0563597
$604$	0	0
$605$	1.13857e7	1.26465
$606$	0	0
$607$	−1.41293e7	−1.55650	−0.778248	−	0.627957i	$-0.783891\pi$
$607$	−1.41293e7	−1.55650	−0.778248	+	0.627957i	$0.783891\pi$
$608$	0	0
$609$	−7.58843e6	−0.829104
$610$	0	0
$611$	1.67232e7	1.81225
$612$	0	0
$613$	1.24624e7	1.33952	0.669760	−	0.742578i	$-0.266397\pi$
$613$	1.24624e7	1.33952	0.669760	+	0.742578i	$0.266397\pi$
$614$	0	0
$615$	−2.53992e6	−0.270790
$616$	0	0
$617$	5.63363e6	0.595765	0.297883	−	0.954602i	$-0.403720\pi$
$617$	5.63363e6	0.595765	0.297883	+	0.954602i	$0.403720\pi$
$618$	0	0
$619$	−7.52500e6	−0.789369	−0.394684	−	0.918817i	$-0.629146\pi$
$619$	−7.52500e6	−0.789369	−0.394684	+	0.918817i	$0.629146\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	−2.63686e7	−2.72186
$624$	0	0
$625$	−3.83977e6	−0.393193
$626$	0	0
$627$	3.23595e6	0.328725
$628$	0	0
$629$	3.92883e6	0.395947
$630$	0	0
$631$	8.39408e6	0.839267	0.419633	−	0.907694i	$-0.362159\pi$
$631$	8.39408e6	0.839267	0.419633	+	0.907694i	$0.362159\pi$
$632$	0	0
$633$	−5.55308e6	−0.550839
$634$	0	0
$635$	6.41587e6	0.631424
$636$	0	0
$637$	−1.85747e7	−1.81373
$638$	0	0
$639$	−550857.	−0.0533688
$640$	0	0
$641$	−8.76329e6	−0.842407	−0.421204	−	0.906966i	$-0.638392\pi$
$641$	−8.76329e6	−0.842407	−0.421204	+	0.906966i	$0.638392\pi$
$642$	0	0
$643$	4.80058e6	0.457896	0.228948	−	0.973439i	$-0.426471\pi$
$643$	4.80058e6	0.457896	0.228948	+	0.973439i	$0.426471\pi$
$644$	0	0
$645$	8.72355e6	0.825646
$646$	0	0
$647$	−1.74670e7	−1.64043	−0.820217	−	0.572053i	$-0.806147\pi$
$647$	−1.74670e7	−1.64043	−0.820217	+	0.572053i	$0.806147\pi$
$648$	0	0
$649$	1.52389e7	1.42018
$650$	0	0
$651$	−7.64932e6	−0.707409
$652$	0	0
$653$	−1.60101e7	−1.46930	−0.734651	−	0.678446i	$-0.762654\pi$
$653$	−1.60101e7	−1.46930	−0.734651	+	0.678446i	$0.762654\pi$
$654$	0	0
$655$	−2.29777e6	−0.209268
$656$	0	0
$657$	−2.67270e6	−0.241567
$658$	0	0
$659$	−1.08861e7	−0.976472	−0.488236	−	0.872712i	$-0.662360\pi$
$659$	−1.08861e7	−0.976472	−0.488236	+	0.872712i	$0.662360\pi$
$660$	0	0
$661$	−1.86000e7	−1.65580	−0.827902	−	0.560873i	$-0.810466\pi$
$661$	−1.86000e7	−1.65580	−0.827902	+	0.560873i	$0.810466\pi$
$662$	0	0
$663$	2.88824e6	0.255181
$664$	0	0
$665$	−4.45295e6	−0.390476
$666$	0	0
$667$	2.32735e6	0.202557
$668$	0	0
$669$	−3.36279e6	−0.290493
$670$	0	0
$671$	1.64864e7	1.41357
$672$	0	0
$673$	2.01949e7	1.71872	0.859358	−	0.511374i	$-0.170863\pi$
$673$	2.01949e7	1.71872	0.859358	+	0.511374i	$0.170863\pi$
$674$	0	0
$675$	−969658.	−0.0819142
$676$	0	0
$677$	−1.55789e7	−1.30637	−0.653184	−	0.757200i	$-0.726567\pi$
$677$	−1.55789e7	−1.30637	−0.653184	+	0.757200i	$0.726567\pi$
$678$	0	0
$679$	6.18763e6	0.515050
$680$	0	0
$681$	−4.95286e6	−0.409249
$682$	0	0
$683$	−9.66927e6	−0.793126	−0.396563	−	0.918008i	$-0.629797\pi$
$683$	−9.66927e6	−0.793126	−0.396563	+	0.918008i	$0.629797\pi$
$684$	0	0
$685$	8.93191e6	0.727307
$686$	0	0
$687$	−7.47025e6	−0.603870
$688$	0	0
$689$	−4.25393e6	−0.341384
$690$	0	0
$691$	−6.67472e6	−0.531787	−0.265894	−	0.964002i	$-0.585667\pi$
$691$	−6.67472e6	−0.531787	−0.265894	+	0.964002i	$0.585667\pi$
$692$	0	0
$693$	−1.01770e7	−0.804983
$694$	0	0
$695$	1.35213e7	1.06183
$696$	0	0
$697$	2.29275e6	0.178762
$698$	0	0
$699$	1.38609e7	1.07300
$700$	0	0
$701$	−1.95292e6	−0.150103	−0.0750516	−	0.997180i	$-0.523912\pi$
$701$	−1.95292e6	−0.150103	−0.0750516	+	0.997180i	$0.523912\pi$
$702$	0	0
$703$	−6.26031e6	−0.477758
$704$	0	0
$705$	−6.83889e6	−0.518218
$706$	0	0
$707$	3.12832e7	2.35376
$708$	0	0
$709$	−1.62124e6	−0.121125	−0.0605623	−	0.998164i	$-0.519289\pi$
$709$	−1.62124e6	−0.121125	−0.0605623	+	0.998164i	$0.519289\pi$
$710$	0	0
$711$	1.70809e6	0.126718
$712$	0	0
$713$	2.34603e6	0.172826
$714$	0	0
$715$	−2.58967e7	−1.89443
$716$	0	0
$717$	−1.21968e7	−0.886028
$718$	0	0
$719$	6.94284e6	0.500858	0.250429	−	0.968135i	$-0.419428\pi$
$719$	6.94284e6	0.500858	0.250429	+	0.968135i	$0.419428\pi$
$720$	0	0
$721$	−1.31305e7	−0.940685
$722$	0	0
$723$	1.08107e7	0.769145
$724$	0	0
$725$	−5.85192e6	−0.413479
$726$	0	0
$727$	7.93350e6	0.556710	0.278355	−	0.960478i	$-0.410211\pi$
$727$	7.93350e6	0.556710	0.278355	+	0.960478i	$0.410211\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−7.87460e6	−0.545048
$732$	0	0
$733$	−364053.	−0.0250267	−0.0125134	−	0.999922i	$-0.503983\pi$
$733$	−364053.	−0.0250267	−0.0125134	+	0.999922i	$0.503983\pi$
$734$	0	0
$735$	7.59603e6	0.518643
$736$	0	0
$737$	4.07295e6	0.276211
$738$	0	0
$739$	5.17273e6	0.348424	0.174212	−	0.984708i	$-0.444262\pi$
$739$	5.17273e6	0.348424	0.174212	+	0.984708i	$0.444262\pi$
$740$	0	0
$741$	−4.60220e6	−0.307907
$742$	0	0
$743$	1.68796e7	1.12173	0.560866	−	0.827907i	$-0.310468\pi$
$743$	1.68796e7	1.12173	0.560866	+	0.827907i	$0.310468\pi$
$744$	0	0
$745$	−1.35172e7	−0.892268
$746$	0	0
$747$	−2.29480e6	−0.150467
$748$	0	0
$749$	−3.90376e6	−0.254260
$750$	0	0
$751$	−4.65656e6	−0.301277	−0.150638	−	0.988589i	$-0.548133\pi$
$751$	−4.65656e6	−0.301277	−0.150638	+	0.988589i	$0.548133\pi$
$752$	0	0
$753$	−4.70911e6	−0.302658
$754$	0	0
$755$	−4.24741e6	−0.271180
$756$	0	0
$757$	−1.10890e6	−0.0703322	−0.0351661	−	0.999381i	$-0.511196\pi$
$757$	−1.10890e6	−0.0703322	−0.0351661	+	0.999381i	$0.511196\pi$
$758$	0	0
$759$	3.12126e6	0.196664
$760$	0	0
$761$	−1.61024e7	−1.00793	−0.503964	−	0.863725i	$-0.668126\pi$
$761$	−1.61024e7	−1.00793	−0.503964	+	0.863725i	$0.668126\pi$
$762$	0	0
$763$	−4.71161e7	−2.92993
$764$	0	0
$765$	−1.18113e6	−0.0729700
$766$	0	0
$767$	−2.16729e7	−1.33024
$768$	0	0
$769$	−1.10052e7	−0.671091	−0.335545	−	0.942024i	$-0.608921\pi$
$769$	−1.10052e7	−0.671091	−0.335545	+	0.942024i	$0.608921\pi$
$770$	0	0
$771$	6.00993e6	0.364111
$772$	0	0
$773$	−1.33435e7	−0.803193	−0.401596	−	0.915817i	$-0.631545\pi$
$773$	−1.33435e7	−0.803193	−0.401596	+	0.915817i	$0.631545\pi$
$774$	0	0
$775$	−5.89887e6	−0.352789
$776$	0	0
$777$	1.96885e7	1.16993
$778$	0	0
$779$	−3.65333e6	−0.215697
$780$	0	0
$781$	−4.45847e6	−0.261552
$782$	0	0
$783$	3.20726e6	0.186952
$784$	0	0
$785$	1.41878e7	0.821752
$786$	0	0
$787$	−8.81426e6	−0.507281	−0.253641	−	0.967299i	$-0.581628\pi$
$787$	−8.81426e6	−0.507281	−0.253641	+	0.967299i	$0.581628\pi$
$788$	0	0
$789$	1.94309e7	1.11122
$790$	0	0
$791$	−4.95411e7	−2.81530
$792$	0	0
$793$	−2.34470e7	−1.32405
$794$	0	0
$795$	1.73963e6	0.0976198
$796$	0	0
$797$	3.20515e6	0.178732	0.0893660	−	0.995999i	$-0.471516\pi$
$797$	3.20515e6	0.178732	0.0893660	+	0.995999i	$0.471516\pi$
$798$	0	0
$799$	6.17335e6	0.342101
$800$	0	0
$801$	1.11447e7	0.613744
$802$	0	0
$803$	−2.16320e7	−1.18388
$804$	0	0
$805$	−4.29513e6	−0.233607
$806$	0	0
$807$	−808330.	−0.0436923
$808$	0	0
$809$	−2.41440e7	−1.29700	−0.648498	−	0.761216i	$-0.724603\pi$
$809$	−2.41440e7	−1.29700	−0.648498	+	0.761216i	$0.724603\pi$
$810$	0	0
$811$	1.49460e7	0.797946	0.398973	−	0.916963i	$-0.369367\pi$
$811$	1.49460e7	0.797946	0.398973	+	0.916963i	$0.369367\pi$
$812$	0	0
$813$	1.57030e7	0.833212
$814$	0	0
$815$	−1.13741e7	−0.599821
$816$	0	0
$817$	1.25476e7	0.657666
$818$	0	0
$819$	1.44738e7	0.754003
$820$	0	0
$821$	−3.27314e7	−1.69476	−0.847378	−	0.530991i	$-0.821820\pi$
$821$	−3.27314e7	−1.69476	−0.847378	+	0.530991i	$0.821820\pi$
$822$	0	0
$823$	−4.46307e6	−0.229686	−0.114843	−	0.993384i	$-0.536636\pi$
$823$	−4.46307e6	−0.229686	−0.114843	+	0.993384i	$0.536636\pi$
$824$	0	0
$825$	−7.84812e6	−0.401449
$826$	0	0
$827$	2.80337e7	1.42533	0.712667	−	0.701502i	$-0.247487\pi$
$827$	2.80337e7	1.42533	0.712667	+	0.701502i	$0.247487\pi$
$828$	0	0
$829$	−3.09291e6	−0.156308	−0.0781540	−	0.996941i	$-0.524903\pi$
$829$	−3.09291e6	−0.156308	−0.0781540	+	0.996941i	$0.524903\pi$
$830$	0	0
$831$	1.97907e7	0.994164
$832$	0	0
$833$	−6.85680e6	−0.342381
$834$	0	0
$835$	1.06228e7	0.527256
$836$	0	0
$837$	3.23300e6	0.159511
$838$	0	0
$839$	−1.38702e7	−0.680263	−0.340131	−	0.940378i	$-0.610472\pi$
$839$	−1.38702e7	−0.680263	−0.340131	+	0.940378i	$0.610472\pi$
$840$	0	0
$841$	−1.15523e6	−0.0563222
$842$	0	0
$843$	7.71457e6	0.373889
$844$	0	0
$845$	2.11003e7	1.01659
$846$	0	0
$847$	−5.15045e7	−2.46682
$848$	0	0
$849$	−4.50784e6	−0.214634
$850$	0	0
$851$	−6.03843e6	−0.285825
$852$	0	0
$853$	−4.04588e7	−1.90388	−0.951942	−	0.306278i	$-0.900916\pi$
$853$	−4.04588e7	−1.90388	−0.951942	+	0.306278i	$0.900916\pi$
$854$	0	0
$855$	1.88205e6	0.0880471
$856$	0	0
$857$	1.82715e6	0.0849808	0.0424904	−	0.999097i	$-0.486471\pi$
$857$	1.82715e6	0.0849808	0.0424904	+	0.999097i	$0.486471\pi$
$858$	0	0
$859$	−2.08415e7	−0.963710	−0.481855	−	0.876251i	$-0.660037\pi$
$859$	−2.08415e7	−0.963710	−0.481855	+	0.876251i	$0.660037\pi$
$860$	0	0
$861$	1.14896e7	0.528200
$862$	0	0
$863$	−4.06174e7	−1.85646	−0.928230	−	0.372007i	$-0.878670\pi$
$863$	−4.06174e7	−1.85646	−0.928230	+	0.372007i	$0.878670\pi$
$864$	0	0
$865$	2.02762e7	0.921394
$866$	0	0
$867$	−1.17125e7	−0.529179
$868$	0	0
$869$	1.38248e7	0.621025
$870$	0	0
$871$	−5.79258e6	−0.258718
$872$	0	0
$873$	−2.61521e6	−0.116137
$874$	0	0
$875$	3.61726e7	1.59720
$876$	0	0
$877$	−2.96392e7	−1.30127	−0.650636	−	0.759390i	$-0.725498\pi$
$877$	−2.96392e7	−1.30127	−0.650636	+	0.759390i	$0.725498\pi$
$878$	0	0
$879$	4.89598e6	0.213731
$880$	0	0
$881$	−3.53245e7	−1.53333	−0.766665	−	0.642047i	$-0.778085\pi$
$881$	−3.53245e7	−1.53333	−0.766665	+	0.642047i	$0.778085\pi$
$882$	0	0
$883$	−2.32881e7	−1.00515	−0.502576	−	0.864533i	$-0.667614\pi$
$883$	−2.32881e7	−1.00515	−0.502576	+	0.864533i	$0.667614\pi$
$884$	0	0
$885$	8.86303e6	0.380386
$886$	0	0
$887$	−1.05116e7	−0.448600	−0.224300	−	0.974520i	$-0.572009\pi$
$887$	−1.05116e7	−0.448600	−0.224300	+	0.974520i	$0.572009\pi$
$888$	0	0
$889$	−2.90229e7	−1.23165
$890$	0	0
$891$	4.30132e6	0.181513
$892$	0	0
$893$	−9.83679e6	−0.412786
$894$	0	0
$895$	1.76375e7	0.736001
$896$	0	0
$897$	−4.43908e6	−0.184209
$898$	0	0
$899$	1.95112e7	0.805166
$900$	0	0
$901$	−1.57033e6	−0.0644435
$902$	0	0
$903$	−3.94619e7	−1.61049
$904$	0	0
$905$	−2.19342e7	−0.890225
$906$	0	0
$907$	−3.13327e7	−1.26468	−0.632339	−	0.774692i	$-0.717905\pi$
$907$	−3.13327e7	−1.26468	−0.632339	+	0.774692i	$0.717905\pi$
$908$	0	0
$909$	−1.32219e7	−0.530741
$910$	0	0
$911$	2.74109e7	1.09428	0.547139	−	0.837042i	$-0.315717\pi$
$911$	2.74109e7	1.09428	0.547139	+	0.837042i	$0.315717\pi$
$912$	0	0
$913$	−1.85734e7	−0.737418
$914$	0	0
$915$	9.58856e6	0.378617
$916$	0	0
$917$	1.03942e7	0.408196
$918$	0	0
$919$	6.27955e6	0.245267	0.122634	−	0.992452i	$-0.460866\pi$
$919$	6.27955e6	0.245267	0.122634	+	0.992452i	$0.460866\pi$
$920$	0	0
$921$	2.08869e7	0.811383
$922$	0	0
$923$	6.34087e6	0.244988
$924$	0	0
$925$	1.51831e7	0.583452
$926$	0	0
$927$	5.54964e6	0.212112
$928$	0	0
$929$	6.33332e6	0.240764	0.120382	−	0.992728i	$-0.461588\pi$
$929$	6.33332e6	0.240764	0.120382	+	0.992728i	$0.461588\pi$
$930$	0	0
$931$	1.09258e7	0.413124
$932$	0	0
$933$	1.44986e7	0.545281
$934$	0	0
$935$	−9.55971e6	−0.357615
$936$	0	0
$937$	−2.00147e7	−0.744732	−0.372366	−	0.928086i	$-0.621453\pi$
$937$	−2.00147e7	−0.744732	−0.372366	+	0.928086i	$0.621453\pi$
$938$	0	0
$939$	4.12967e6	0.152845
$940$	0	0
$941$	2.61878e7	0.964106	0.482053	−	0.876142i	$-0.339891\pi$
$941$	2.61878e7	0.964106	0.482053	+	0.876142i	$0.339891\pi$
$942$	0	0
$943$	−3.52384e6	−0.129044
$944$	0	0
$945$	−5.91899e6	−0.215610
$946$	0	0
$947$	2.31722e7	0.839640	0.419820	−	0.907607i	$-0.362093\pi$
$947$	2.31722e7	0.839640	0.419820	+	0.907607i	$0.362093\pi$
$948$	0	0
$949$	3.07652e7	1.10891
$950$	0	0
$951$	−1.19464e7	−0.428336
$952$	0	0
$953$	2.57957e7	0.920057	0.460028	−	0.887904i	$-0.347839\pi$
$953$	2.57957e7	0.920057	0.460028	+	0.887904i	$0.347839\pi$
$954$	0	0
$955$	6.86976e6	0.243743
$956$	0	0
$957$	2.59586e7	0.916223
$958$	0	0
$959$	−4.04045e7	−1.41867
$960$	0	0
$961$	−8.96138e6	−0.313016
$962$	0	0
$963$	1.64993e6	0.0573323
$964$	0	0
$965$	−1.63295e7	−0.564489
$966$	0	0
$967$	1.19166e7	0.409814	0.204907	−	0.978781i	$-0.434311\pi$
$967$	1.19166e7	0.409814	0.204907	+	0.978781i	$0.434311\pi$
$968$	0	0
$969$	−1.69889e6	−0.0581241
$970$	0	0
$971$	2.11925e7	0.721331	0.360666	−	0.932695i	$-0.382550\pi$
$971$	2.11925e7	0.721331	0.360666	+	0.932695i	$0.382550\pi$
$972$	0	0
$973$	−6.11649e7	−2.07119
$974$	0	0
$975$	1.11617e7	0.376025
$976$	0	0
$977$	4.08942e7	1.37064	0.685322	−	0.728240i	$-0.259661\pi$
$977$	4.08942e7	1.37064	0.685322	+	0.728240i	$0.259661\pi$
$978$	0	0
$979$	9.02018e7	3.00787
$980$	0	0
$981$	1.99137e7	0.660661
$982$	0	0
$983$	−3.71260e7	−1.22545	−0.612724	−	0.790297i	$-0.709926\pi$
$983$	−3.71260e7	−1.22545	−0.612724	+	0.790297i	$0.709926\pi$
$984$	0	0
$985$	−3.34082e7	−1.09714
$986$	0	0
$987$	3.09365e7	1.01083
$988$	0	0
$989$	1.21029e7	0.393458
$990$	0	0
$991$	−3.44781e7	−1.11522	−0.557609	−	0.830104i	$-0.688281\pi$
$991$	−3.44781e7	−1.11522	−0.557609	+	0.830104i	$0.688281\pi$
$992$	0	0
$993$	1.38102e7	0.444455
$994$	0	0
$995$	2.35971e7	0.755614
$996$	0	0
$997$	624230.	0.0198887	0.00994435	−	0.999951i	$-0.496835\pi$
$997$	624230.	0.0198887	0.00994435	+	0.999951i	$0.496835\pi$
$998$	0	0
$999$	−8.32139e6	−0.263804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.o.1.3		4
4.3	odd	2		69.6.a.d.1.2	✓	4
12.11	even	2		207.6.a.e.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.d.1.2	✓	4	4.3	odd	2
207.6.a.e.1.3		4	12.11	even	2
1104.6.a.o.1.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +42.3660 q^{5} -191.647 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +42.3660 q^{5} -191.647 q^{7} +81.0000 q^{9} +655.589 q^{11} -932.384 q^{13} +381.294 q^{15} -344.188 q^{17} +548.438 q^{19} -1724.83 q^{21} +529.000 q^{23} -1330.12 q^{25} +729.000 q^{27} +4399.54 q^{29} +4434.84 q^{31} +5900.30 q^{33} -8119.33 q^{35} -11414.8 q^{37} -8391.46 q^{39} -6661.33 q^{41} +22878.8 q^{43} +3431.65 q^{45} -17936.0 q^{47} +19921.7 q^{49} -3097.69 q^{51} +4562.43 q^{53} +27774.7 q^{55} +4935.95 q^{57} +23244.6 q^{59} +25147.4 q^{61} -15523.4 q^{63} -39501.4 q^{65} +6212.65 q^{67} +4761.00 q^{69} -6800.71 q^{71} -32996.3 q^{73} -11971.1 q^{75} -125642. q^{77} +21087.6 q^{79} +6561.00 q^{81} -28330.8 q^{83} -14581.9 q^{85} +39595.8 q^{87} +137589. q^{89} +178689. q^{91} +39913.5 q^{93} +23235.1 q^{95} -32286.5 q^{97} +53102.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{3} + 22 q^{5} + 62 q^{7} + 324 q^{9}+O(q^{10}) \) 4 * q + 36 * q^3 + 22 * q^5 + 62 * q^7 + 324 * q^9	\( 4 q + 36 q^{3} + 22 q^{5} + 62 q^{7} + 324 q^{9} + 1076 q^{11} - 396 q^{13} + 198 q^{15} + 70 q^{17} + 6366 q^{19} + 558 q^{21} + 2116 q^{23} + 1264 q^{25} + 2916 q^{27} + 3948 q^{29} - 3092 q^{31} + 9684 q^{33} - 1304 q^{35} - 17464 q^{37} - 3564 q^{39} + 18680 q^{41} + 25846 q^{43} + 1782 q^{45} - 18392 q^{47} + 7952 q^{49} + 630 q^{51} - 26518 q^{53} + 40848 q^{55} + 57294 q^{57} + 14520 q^{59} - 13688 q^{61} + 5022 q^{63} + 38324 q^{65} + 11098 q^{67} + 19044 q^{69} + 57496 q^{71} - 112272 q^{73} + 11376 q^{75} - 4792 q^{77} + 240754 q^{79} + 26244 q^{81} + 93268 q^{83} - 323204 q^{85} + 35532 q^{87} - 107582 q^{89} + 301532 q^{91} - 27828 q^{93} + 18640 q^{95} - 53076 q^{97} + 87156 q^{99}+O(q^{100}) \) 4 * q + 36 * q^3 + 22 * q^5 + 62 * q^7 + 324 * q^9 + 1076 * q^11 - 396 * q^13 + 198 * q^15 + 70 * q^17 + 6366 * q^19 + 558 * q^21 + 2116 * q^23 + 1264 * q^25 + 2916 * q^27 + 3948 * q^29 - 3092 * q^31 + 9684 * q^33 - 1304 * q^35 - 17464 * q^37 - 3564 * q^39 + 18680 * q^41 + 25846 * q^43 + 1782 * q^45 - 18392 * q^47 + 7952 * q^49 + 630 * q^51 - 26518 * q^53 + 40848 * q^55 + 57294 * q^57 + 14520 * q^59 - 13688 * q^61 + 5022 * q^63 + 38324 * q^65 + 11098 * q^67 + 19044 * q^69 + 57496 * q^71 - 112272 * q^73 + 11376 * q^75 - 4792 * q^77 + 240754 * q^79 + 26244 * q^81 + 93268 * q^83 - 323204 * q^85 + 35532 * q^87 - 107582 * q^89 + 301532 * q^91 - 27828 * q^93 + 18640 * q^95 - 53076 * q^97 + 87156 * q^99

Introduction

L-functions

Modular forms

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Database

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