LMFDB - Embedded newform 308.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [308,4,Mod(1,308)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(308, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("308.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 308 = 2^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	308.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:	$18.1725882818$
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} - x^{3} - 43x^{2} - 11x + 222 $ x^4 - x^3 - 43x^2 - 11x + 222
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.25471$ of defining polynomial
Character	$\chi$	$=$	308.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.25471 q^{3} +17.1213 q^{5} -7.00000 q^{7} +12.1213 q^{9} +11.0000 q^{11} +51.7778 q^{13} -107.089 q^{15} -104.255 q^{17} +10.6685 q^{19} +43.7829 q^{21} -9.26667 q^{23} +168.140 q^{25} +93.0616 q^{27} +223.275 q^{29} -237.641 q^{31} -68.8018 q^{33} -119.849 q^{35} +361.973 q^{37} -323.855 q^{39} +273.492 q^{41} +149.666 q^{43} +207.534 q^{45} +531.944 q^{47} +49.0000 q^{49} +652.083 q^{51} +408.170 q^{53} +188.335 q^{55} -66.7280 q^{57} +416.905 q^{59} +393.132 q^{61} -84.8494 q^{63} +886.505 q^{65} -1042.79 q^{67} +57.9603 q^{69} -552.909 q^{71} +724.799 q^{73} -1051.67 q^{75} -77.0000 q^{77} -516.261 q^{79} -909.349 q^{81} +1396.82 q^{83} -1784.98 q^{85} -1396.52 q^{87} -329.372 q^{89} -362.444 q^{91} +1486.37 q^{93} +182.658 q^{95} -10.6082 q^{97} +133.335 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + q^{5} - 28 q^{7} - 19 q^{9} + 44 q^{11} + 98 q^{13} - 33 q^{15} + 64 q^{17} + 114 q^{19} + 21 q^{21} + 231 q^{23} + 417 q^{25} + 63 q^{27} + 268 q^{29} + 33 q^{31} - 33 q^{33} - 7 q^{35}+ \cdots - 209 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + q^5 - 28 * q^7 - 19 * q^9 + 44 * q^11 + 98 * q^13 - 33 * q^15 + 64 * q^17 + 114 * q^19 + 21 * q^21 + 231 * q^23 + 417 * q^25 + 63 * q^27 + 268 * q^29 + 33 * q^31 - 33 * q^33 - 7 * q^35 + 357 * q^37 + 140 * q^39 + 364 * q^41 + 44 * q^43 + 912 * q^45 + 720 * q^47 + 196 * q^49 + 794 * q^51 + 740 * q^53 + 11 * q^55 + 138 * q^57 + 787 * q^59 + 1020 * q^61 + 133 * q^63 + 296 * q^65 - 995 * q^67 + 1059 * q^69 - 1011 * q^71 + 1592 * q^73 - 324 * q^75 - 308 * q^77 - 178 * q^79 - 1396 * q^81 + 324 * q^83 - 94 * q^85 - 1262 * q^87 + 19 * q^89 - 686 * q^91 + 637 * q^93 - 2418 * q^95 - 555 * q^97 - 209 * q^99

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
$3$	−6.25471	−1.20372	−0.601859	−	0.798602i	$-0.705573\pi$
$3$	−6.25471	−1.20372	−0.601859	+	0.798602i	$0.705573\pi$
$4$	0	0
$5$	17.1213	1.53138	0.765690	−	0.643210i	$-0.222398\pi$
$5$	17.1213	1.53138	0.765690	+	0.643210i	$0.222398\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	12.1213	0.448939
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	51.7778	1.10466	0.552330	−	0.833626i	$-0.313739\pi$
$13$	51.7778	1.10466	0.552330	+	0.833626i	$0.313739\pi$
$14$	0	0
$15$	−107.089	−1.84335
$16$	0	0
$17$	−104.255	−1.48738	−0.743691	−	0.668523i	$-0.766927\pi$
$17$	−104.255	−1.48738	−0.743691	+	0.668523i	$0.766927\pi$
$18$	0	0
$19$	10.6685	0.128816	0.0644082	−	0.997924i	$-0.479484\pi$
$19$	10.6685	0.128816	0.0644082	+	0.997924i	$0.479484\pi$
$20$	0	0
$21$	43.7829	0.454963
$22$	0	0
$23$	−9.26667	−0.0840102	−0.0420051	−	0.999117i	$-0.513375\pi$
$23$	−9.26667	−0.0840102	−0.0420051	+	0.999117i	$0.513375\pi$
$24$	0	0
$25$	168.140	1.34512
$26$	0	0
$27$	93.0616	0.663323
$28$	0	0
$29$	223.275	1.42969	0.714847	−	0.699281i	$-0.246496\pi$
$29$	223.275	1.42969	0.714847	+	0.699281i	$0.246496\pi$
$30$	0	0
$31$	−237.641	−1.37682	−0.688412	−	0.725320i	$-0.741692\pi$
$31$	−237.641	−1.37682	−0.688412	+	0.725320i	$0.741692\pi$
$32$	0	0
$33$	−68.8018	−0.362935
$34$	0	0
$35$	−119.849	−0.578807
$36$	0	0
$37$	361.973	1.60832	0.804161	−	0.594411i	$-0.202615\pi$
$37$	361.973	1.60832	0.804161	+	0.594411i	$0.202615\pi$
$38$	0	0
$39$	−323.855	−1.32970
$40$	0	0
$41$	273.492	1.04176	0.520881	−	0.853629i	$-0.325603\pi$
$41$	273.492	1.04176	0.520881	+	0.853629i	$0.325603\pi$
$42$	0	0
$43$	149.666	0.530789	0.265394	−	0.964140i	$-0.414498\pi$
$43$	149.666	0.530789	0.265394	+	0.964140i	$0.414498\pi$
$44$	0	0
$45$	207.534	0.687495
$46$	0	0
$47$	531.944	1.65089	0.825447	−	0.564480i	$-0.190923\pi$
$47$	531.944	1.65089	0.825447	+	0.564480i	$0.190923\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	652.083	1.79039
$52$	0	0
$53$	408.170	1.05786	0.528928	−	0.848667i	$-0.322594\pi$
$53$	408.170	1.05786	0.528928	+	0.848667i	$0.322594\pi$
$54$	0	0
$55$	188.335	0.461728
$56$	0	0
$57$	−66.7280	−0.155059
$58$	0	0
$59$	416.905	0.919938	0.459969	−	0.887935i	$-0.347860\pi$
$59$	416.905	0.919938	0.459969	+	0.887935i	$0.347860\pi$
$60$	0	0
$61$	393.132	0.825170	0.412585	−	0.910919i	$-0.364626\pi$
$61$	393.132	0.825170	0.412585	+	0.910919i	$0.364626\pi$
$62$	0	0
$63$	−84.8494	−0.169683
$64$	0	0
$65$	886.505	1.69165
$66$	0	0
$67$	−1042.79	−1.90145	−0.950723	−	0.310043i	$-0.899657\pi$
$67$	−1042.79	−1.90145	−0.950723	+	0.310043i	$0.899657\pi$
$68$	0	0
$69$	57.9603	0.101125
$70$	0	0
$71$	−552.909	−0.924200	−0.462100	−	0.886828i	$-0.652904\pi$
$71$	−552.909	−0.924200	−0.462100	+	0.886828i	$0.652904\pi$
$72$	0	0
$73$	724.799	1.16207	0.581036	−	0.813878i	$-0.302647\pi$
$73$	724.799	1.16207	0.581036	+	0.813878i	$0.302647\pi$
$74$	0	0
$75$	−1051.67	−1.61915
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−516.261	−0.735239	−0.367620	−	0.929976i	$-0.619827\pi$
$79$	−516.261	−0.735239	−0.367620	+	0.929976i	$0.619827\pi$
$80$	0	0
$81$	−909.349	−1.24739
$82$	0	0
$83$	1396.82	1.84724	0.923621	−	0.383306i	$-0.125215\pi$
$83$	1396.82	1.84724	0.923621	+	0.383306i	$0.125215\pi$
$84$	0	0
$85$	−1784.98	−2.27775
$86$	0	0
$87$	−1396.52	−1.72095
$88$	0	0
$89$	−329.372	−0.392285	−0.196142	−	0.980575i	$-0.562841\pi$
$89$	−329.372	−0.392285	−0.196142	+	0.980575i	$0.562841\pi$
$90$	0	0
$91$	−362.444	−0.417522
$92$	0	0
$93$	1486.37	1.65731
$94$	0	0
$95$	182.658	0.197267
$96$	0	0
$97$	−10.6082	−0.0111042	−0.00555208	−	0.999985i	$-0.501767\pi$
$97$	−10.6082	−0.0111042	−0.00555208	+	0.999985i	$0.501767\pi$
$98$	0	0
$99$	133.335	0.135360
$100$	0	0
$101$	−1303.55	−1.28424	−0.642118	−	0.766606i	$-0.721944\pi$
$101$	−1303.55	−1.28424	−0.642118	+	0.766606i	$0.721944\pi$
$102$	0	0
$103$	27.0841	0.0259095	0.0129547	−	0.999916i	$-0.495876\pi$
$103$	27.0841	0.0259095	0.0129547	+	0.999916i	$0.495876\pi$
$104$	0	0
$105$	749.623	0.696721
$106$	0	0
$107$	−284.914	−0.257417	−0.128709	−	0.991682i	$-0.541083\pi$
$107$	−284.914	−0.257417	−0.128709	+	0.991682i	$0.541083\pi$
$108$	0	0
$109$	−357.655	−0.314286	−0.157143	−	0.987576i	$-0.550228\pi$
$109$	−357.655	−0.314286	−0.157143	+	0.987576i	$0.550228\pi$
$110$	0	0
$111$	−2264.03	−1.93597
$112$	0	0
$113$	−1354.20	−1.12737	−0.563685	−	0.825990i	$-0.690617\pi$
$113$	−1354.20	−1.12737	−0.563685	+	0.825990i	$0.690617\pi$
$114$	0	0
$115$	−158.658	−0.128651
$116$	0	0
$117$	627.616	0.495924
$118$	0	0
$119$	729.783	0.562178
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−1710.61	−1.25399
$124$	0	0
$125$	738.620	0.528513
$126$	0	0
$127$	1253.58	0.875886	0.437943	−	0.899003i	$-0.355707\pi$
$127$	1253.58	0.875886	0.437943	+	0.899003i	$0.355707\pi$
$128$	0	0
$129$	−936.120	−0.638921
$130$	0	0
$131$	1998.97	1.33321	0.666605	−	0.745411i	$-0.267747\pi$
$131$	1998.97	1.33321	0.666605	+	0.745411i	$0.267747\pi$
$132$	0	0
$133$	−74.6792	−0.0486880
$134$	0	0
$135$	1593.34	1.01580
$136$	0	0
$137$	1885.33	1.17573	0.587863	−	0.808960i	$-0.299969\pi$
$137$	1885.33	1.17573	0.587863	+	0.808960i	$0.299969\pi$
$138$	0	0
$139$	2008.67	1.22571	0.612853	−	0.790197i	$-0.290022\pi$
$139$	2008.67	1.22571	0.612853	+	0.790197i	$0.290022\pi$
$140$	0	0
$141$	−3327.15	−1.98721
$142$	0	0
$143$	569.555	0.333067
$144$	0	0
$145$	3822.77	2.18940
$146$	0	0
$147$	−306.481	−0.171960
$148$	0	0
$149$	−2222.24	−1.22183	−0.610916	−	0.791695i	$-0.709199\pi$
$149$	−2222.24	−1.22183	−0.610916	+	0.791695i	$0.709199\pi$
$150$	0	0
$151$	1174.25	0.632840	0.316420	−	0.948619i	$-0.397519\pi$
$151$	1174.25	0.632840	0.316420	+	0.948619i	$0.397519\pi$
$152$	0	0
$153$	−1263.71	−0.667743
$154$	0	0
$155$	−4068.73	−2.10844
$156$	0	0
$157$	2737.25	1.39144	0.695720	−	0.718313i	$-0.255085\pi$
$157$	2737.25	1.39144	0.695720	+	0.718313i	$0.255085\pi$
$158$	0	0
$159$	−2552.98	−1.27336
$160$	0	0
$161$	64.8667	0.0317529
$162$	0	0
$163$	−310.189	−0.149055	−0.0745273	−	0.997219i	$-0.523745\pi$
$163$	−310.189	−0.149055	−0.0745273	+	0.997219i	$0.523745\pi$
$164$	0	0
$165$	−1177.98	−0.555791
$166$	0	0
$167$	−2151.91	−0.997124	−0.498562	−	0.866854i	$-0.666138\pi$
$167$	−2151.91	−0.997124	−0.498562	+	0.866854i	$0.666138\pi$
$168$	0	0
$169$	483.938	0.220272
$170$	0	0
$171$	129.316	0.0578306
$172$	0	0
$173$	−2368.70	−1.04098	−0.520489	−	0.853869i	$-0.674250\pi$
$173$	−2368.70	−1.04098	−0.520489	+	0.853869i	$0.674250\pi$
$174$	0	0
$175$	−1176.98	−0.508408
$176$	0	0
$177$	−2607.62	−1.10735
$178$	0	0
$179$	1269.04	0.529901	0.264950	−	0.964262i	$-0.414644\pi$
$179$	1269.04	0.529901	0.264950	+	0.964262i	$0.414644\pi$
$180$	0	0
$181$	−357.969	−0.147003	−0.0735017	−	0.997295i	$-0.523417\pi$
$181$	−357.969	−0.147003	−0.0735017	+	0.997295i	$0.523417\pi$
$182$	0	0
$183$	−2458.92	−0.993273
$184$	0	0
$185$	6197.46	2.46295
$186$	0	0
$187$	−1146.80	−0.448463
$188$	0	0
$189$	−651.431	−0.250713
$190$	0	0
$191$	−2230.57	−0.845016	−0.422508	−	0.906359i	$-0.638850\pi$
$191$	−2230.57	−0.845016	−0.422508	+	0.906359i	$0.638850\pi$
$192$	0	0
$193$	−4323.07	−1.61234	−0.806169	−	0.591685i	$-0.798463\pi$
$193$	−4323.07	−1.61234	−0.806169	+	0.591685i	$0.798463\pi$
$194$	0	0
$195$	−5544.83	−2.03627
$196$	0	0
$197$	−553.415	−0.200148	−0.100074	−	0.994980i	$-0.531908\pi$
$197$	−553.415	−0.200148	−0.100074	+	0.994980i	$0.531908\pi$
$198$	0	0
$199$	1669.85	0.594836	0.297418	−	0.954747i	$-0.403874\pi$
$199$	1669.85	0.594836	0.297418	+	0.954747i	$0.403874\pi$
$200$	0	0
$201$	6522.33	2.28881
$202$	0	0
$203$	−1562.93	−0.540374
$204$	0	0
$205$	4682.55	1.59533
$206$	0	0
$207$	−112.324	−0.0377154
$208$	0	0
$209$	117.353	0.0388396
$210$	0	0
$211$	3562.75	1.16242	0.581208	−	0.813755i	$-0.302580\pi$
$211$	3562.75	1.16242	0.581208	+	0.813755i	$0.302580\pi$
$212$	0	0
$213$	3458.28	1.11248
$214$	0	0
$215$	2562.49	0.812839
$216$	0	0
$217$	1663.49	0.520390
$218$	0	0
$219$	−4533.40	−1.39881
$220$	0	0
$221$	−5398.08	−1.64305
$222$	0	0
$223$	−1334.49	−0.400735	−0.200368	−	0.979721i	$-0.564214\pi$
$223$	−1334.49	−0.400735	−0.200368	+	0.979721i	$0.564214\pi$
$224$	0	0
$225$	2038.09	0.603877
$226$	0	0
$227$	2334.36	0.682541	0.341271	−	0.939965i	$-0.389143\pi$
$227$	2334.36	0.682541	0.341271	+	0.939965i	$0.389143\pi$
$228$	0	0
$229$	−4512.13	−1.30205	−0.651026	−	0.759055i	$-0.725661\pi$
$229$	−4512.13	−1.30205	−0.651026	+	0.759055i	$0.725661\pi$
$230$	0	0
$231$	481.612	0.137176
$232$	0	0
$233$	−173.678	−0.0488327	−0.0244164	−	0.999702i	$-0.507773\pi$
$233$	−173.678	−0.0488327	−0.0244164	+	0.999702i	$0.507773\pi$
$234$	0	0
$235$	9107.59	2.52814
$236$	0	0
$237$	3229.06	0.885021
$238$	0	0
$239$	−4781.56	−1.29411	−0.647057	−	0.762442i	$-0.724000\pi$
$239$	−4781.56	−1.29411	−0.647057	+	0.762442i	$0.724000\pi$
$240$	0	0
$241$	1574.30	0.420787	0.210394	−	0.977617i	$-0.432525\pi$
$241$	1574.30	0.420787	0.210394	+	0.977617i	$0.432525\pi$
$242$	0	0
$243$	3175.05	0.838187
$244$	0	0
$245$	838.946	0.218768
$246$	0	0
$247$	552.389	0.142298
$248$	0	0
$249$	−8736.71	−2.22356
$250$	0	0
$251$	−3835.27	−0.964462	−0.482231	−	0.876044i	$-0.660173\pi$
$251$	−3835.27	−0.964462	−0.482231	+	0.876044i	$0.660173\pi$
$252$	0	0
$253$	−101.933	−0.0253300
$254$	0	0
$255$	11164.5	2.74177
$256$	0	0
$257$	6643.79	1.61256	0.806281	−	0.591533i	$-0.201477\pi$
$257$	6643.79	1.61256	0.806281	+	0.591533i	$0.201477\pi$
$258$	0	0
$259$	−2533.81	−0.607889
$260$	0	0
$261$	2706.39	0.641845
$262$	0	0
$263$	−4597.37	−1.07789	−0.538947	−	0.842340i	$-0.681178\pi$
$263$	−4597.37	−1.07789	−0.538947	+	0.842340i	$0.681178\pi$
$264$	0	0
$265$	6988.41	1.61998
$266$	0	0
$267$	2060.12	0.472201
$268$	0	0
$269$	−1944.55	−0.440749	−0.220375	−	0.975415i	$-0.570728\pi$
$269$	−1944.55	−0.440749	−0.220375	+	0.975415i	$0.570728\pi$
$270$	0	0
$271$	−1524.87	−0.341806	−0.170903	−	0.985288i	$-0.554668\pi$
$271$	−1524.87	−0.341806	−0.170903	+	0.985288i	$0.554668\pi$
$272$	0	0
$273$	2266.98	0.502579
$274$	0	0
$275$	1849.54	0.405570
$276$	0	0
$277$	2785.41	0.604184	0.302092	−	0.953279i	$-0.402315\pi$
$277$	2785.41	0.604184	0.302092	+	0.953279i	$0.402315\pi$
$278$	0	0
$279$	−2880.52	−0.618109
$280$	0	0
$281$	−1731.31	−0.367549	−0.183775	−	0.982968i	$-0.558832\pi$
$281$	−1731.31	−0.367549	−0.183775	+	0.982968i	$0.558832\pi$
$282$	0	0
$283$	2587.78	0.543561	0.271780	−	0.962359i	$-0.412388\pi$
$283$	2587.78	0.543561	0.271780	+	0.962359i	$0.412388\pi$
$284$	0	0
$285$	−1142.47	−0.237454
$286$	0	0
$287$	−1914.44	−0.393749
$288$	0	0
$289$	5956.06	1.21231
$290$	0	0
$291$	66.3515	0.0133663
$292$	0	0
$293$	−2948.77	−0.587949	−0.293974	−	0.955813i	$-0.594978\pi$
$293$	−2948.77	−0.587949	−0.293974	+	0.955813i	$0.594978\pi$
$294$	0	0
$295$	7137.97	1.40877
$296$	0	0
$297$	1023.68	0.199999
$298$	0	0
$299$	−479.808	−0.0928026
$300$	0	0
$301$	−1047.67	−0.200619
$302$	0	0
$303$	8153.31	1.54586
$304$	0	0
$305$	6730.95	1.26365
$306$	0	0
$307$	674.370	0.125369	0.0626846	−	0.998033i	$-0.480034\pi$
$307$	674.370	0.125369	0.0626846	+	0.998033i	$0.480034\pi$
$308$	0	0
$309$	−169.403	−0.0311877
$310$	0	0
$311$	−6705.01	−1.22253	−0.611264	−	0.791426i	$-0.709339\pi$
$311$	−6705.01	−1.22253	−0.611264	+	0.791426i	$0.709339\pi$
$312$	0	0
$313$	−9621.90	−1.73758	−0.868789	−	0.495182i	$-0.835101\pi$
$313$	−9621.90	−1.73758	−0.868789	+	0.495182i	$0.835101\pi$
$314$	0	0
$315$	−1452.74	−0.259849
$316$	0	0
$317$	4107.58	0.727774	0.363887	−	0.931443i	$-0.381449\pi$
$317$	4107.58	0.727774	0.363887	+	0.931443i	$0.381449\pi$
$318$	0	0
$319$	2456.03	0.431069
$320$	0	0
$321$	1782.05	0.309858
$322$	0	0
$323$	−1112.24	−0.191599
$324$	0	0
$325$	8705.93	1.48590
$326$	0	0
$327$	2237.03	0.378312
$328$	0	0
$329$	−3723.61	−0.623979
$330$	0	0
$331$	−8663.71	−1.43867	−0.719336	−	0.694662i	$-0.755554\pi$
$331$	−8663.71	−1.43867	−0.719336	+	0.694662i	$0.755554\pi$
$332$	0	0
$333$	4387.59	0.722038
$334$	0	0
$335$	−17853.9	−2.91183
$336$	0	0
$337$	1631.87	0.263780	0.131890	−	0.991264i	$-0.457895\pi$
$337$	1631.87	0.263780	0.131890	+	0.991264i	$0.457895\pi$
$338$	0	0
$339$	8470.15	1.35704
$340$	0	0
$341$	−2614.05	−0.415128
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	992.358	0.154860
$346$	0	0
$347$	4472.09	0.691858	0.345929	−	0.938261i	$-0.387564\pi$
$347$	4472.09	0.691858	0.345929	+	0.938261i	$0.387564\pi$
$348$	0	0
$349$	3460.47	0.530758	0.265379	−	0.964144i	$-0.414503\pi$
$349$	3460.47	0.530758	0.265379	+	0.964144i	$0.414503\pi$
$350$	0	0
$351$	4818.52	0.732746
$352$	0	0
$353$	−1926.77	−0.290514	−0.145257	−	0.989394i	$-0.546401\pi$
$353$	−1926.77	−0.290514	−0.145257	+	0.989394i	$0.546401\pi$
$354$	0	0
$355$	−9466.54	−1.41530
$356$	0	0
$357$	−4564.58	−0.676704
$358$	0	0
$359$	1256.66	0.184747	0.0923733	−	0.995724i	$-0.470555\pi$
$359$	1256.66	0.184747	0.0923733	+	0.995724i	$0.470555\pi$
$360$	0	0
$361$	−6745.18	−0.983406
$362$	0	0
$363$	−756.819	−0.109429
$364$	0	0
$365$	12409.5	1.77957
$366$	0	0
$367$	6976.14	0.992238	0.496119	−	0.868254i	$-0.334758\pi$
$367$	6976.14	0.992238	0.496119	+	0.868254i	$0.334758\pi$
$368$	0	0
$369$	3315.09	0.467687
$370$	0	0
$371$	−2857.19	−0.399832
$372$	0	0
$373$	−1727.72	−0.239834	−0.119917	−	0.992784i	$-0.538263\pi$
$373$	−1727.72	−0.239834	−0.119917	+	0.992784i	$0.538263\pi$
$374$	0	0
$375$	−4619.85	−0.636181
$376$	0	0
$377$	11560.7	1.57933
$378$	0	0
$379$	−439.486	−0.0595643	−0.0297822	−	0.999556i	$-0.509481\pi$
$379$	−439.486	−0.0595643	−0.0297822	+	0.999556i	$0.509481\pi$
$380$	0	0
$381$	−7840.79	−1.05432
$382$	0	0
$383$	5054.81	0.674382	0.337191	−	0.941436i	$-0.390523\pi$
$383$	5054.81	0.674382	0.337191	+	0.941436i	$0.390523\pi$
$384$	0	0
$385$	−1318.34	−0.174517
$386$	0	0
$387$	1814.16	0.238292
$388$	0	0
$389$	−8651.92	−1.12769	−0.563843	−	0.825882i	$-0.690678\pi$
$389$	−8651.92	−1.12769	−0.563843	+	0.825882i	$0.690678\pi$
$390$	0	0
$391$	966.095	0.124955
$392$	0	0
$393$	−12502.9	−1.60481
$394$	0	0
$395$	−8839.08	−1.12593
$396$	0	0
$397$	−5525.88	−0.698579	−0.349290	−	0.937015i	$-0.613577\pi$
$397$	−5525.88	−0.698579	−0.349290	+	0.937015i	$0.613577\pi$
$398$	0	0
$399$	467.096	0.0586067
$400$	0	0
$401$	4581.53	0.570551	0.285275	−	0.958446i	$-0.407915\pi$
$401$	4581.53	0.570551	0.285275	+	0.958446i	$0.407915\pi$
$402$	0	0
$403$	−12304.5	−1.52092
$404$	0	0
$405$	−15569.3	−1.91023
$406$	0	0
$407$	3981.70	0.484928
$408$	0	0
$409$	15452.5	1.86816	0.934079	−	0.357067i	$-0.116223\pi$
$409$	15452.5	1.86816	0.934079	+	0.357067i	$0.116223\pi$
$410$	0	0
$411$	−11792.2	−1.41524
$412$	0	0
$413$	−2918.33	−0.347704
$414$	0	0
$415$	23915.5	2.82883
$416$	0	0
$417$	−12563.6	−1.47540
$418$	0	0
$419$	−12626.7	−1.47221	−0.736105	−	0.676867i	$-0.763337\pi$
$419$	−12626.7	−1.47221	−0.736105	+	0.676867i	$0.763337\pi$
$420$	0	0
$421$	−7718.68	−0.893553	−0.446776	−	0.894646i	$-0.647428\pi$
$421$	−7718.68	−0.893553	−0.446776	+	0.894646i	$0.647428\pi$
$422$	0	0
$423$	6447.87	0.741150
$424$	0	0
$425$	−17529.4	−2.00071
$426$	0	0
$427$	−2751.92	−0.311885
$428$	0	0
$429$	−3562.40	−0.400919
$430$	0	0
$431$	7452.18	0.832852	0.416426	−	0.909170i	$-0.363283\pi$
$431$	7452.18	0.832852	0.416426	+	0.909170i	$0.363283\pi$
$432$	0	0
$433$	−6247.78	−0.693416	−0.346708	−	0.937973i	$-0.612700\pi$
$433$	−6247.78	−0.693416	−0.346708	+	0.937973i	$0.612700\pi$
$434$	0	0
$435$	−23910.3	−2.63543
$436$	0	0
$437$	−98.8611	−0.0108219
$438$	0	0
$439$	15177.2	1.65004	0.825020	−	0.565104i	$-0.191164\pi$
$439$	15177.2	1.65004	0.825020	+	0.565104i	$0.191164\pi$
$440$	0	0
$441$	593.946	0.0641341
$442$	0	0
$443$	6242.85	0.669541	0.334770	−	0.942300i	$-0.391341\pi$
$443$	6242.85	0.669541	0.334770	+	0.942300i	$0.391341\pi$
$444$	0	0
$445$	−5639.29	−0.600737
$446$	0	0
$447$	13899.5	1.47074
$448$	0	0
$449$	−15058.3	−1.58273	−0.791366	−	0.611343i	$-0.790630\pi$
$449$	−15058.3	−1.58273	−0.791366	+	0.611343i	$0.790630\pi$
$450$	0	0
$451$	3008.41	0.314103
$452$	0	0
$453$	−7344.57	−0.761762
$454$	0	0
$455$	−6205.53	−0.639384
$456$	0	0
$457$	2560.84	0.262125	0.131063	−	0.991374i	$-0.458161\pi$
$457$	2560.84	0.262125	0.131063	+	0.991374i	$0.458161\pi$
$458$	0	0
$459$	−9702.12	−0.986615
$460$	0	0
$461$	−9820.30	−0.992141	−0.496071	−	0.868282i	$-0.665224\pi$
$461$	−9820.30	−0.992141	−0.496071	+	0.868282i	$0.665224\pi$
$462$	0	0
$463$	14118.1	1.41711	0.708557	−	0.705653i	$-0.249346\pi$
$463$	14118.1	1.41711	0.708557	+	0.705653i	$0.249346\pi$
$464$	0	0
$465$	25448.7	2.53797
$466$	0	0
$467$	2199.62	0.217958	0.108979	−	0.994044i	$-0.465242\pi$
$467$	2199.62	0.217958	0.108979	+	0.994044i	$0.465242\pi$
$468$	0	0
$469$	7299.52	0.718679
$470$	0	0
$471$	−17120.7	−1.67490
$472$	0	0
$473$	1646.33	0.160039
$474$	0	0
$475$	1793.80	0.173274
$476$	0	0
$477$	4947.56	0.474913
$478$	0	0
$479$	−10629.7	−1.01395	−0.506977	−	0.861960i	$-0.669237\pi$
$479$	−10629.7	−1.01395	−0.506977	+	0.861960i	$0.669237\pi$
$480$	0	0
$481$	18742.1	1.77665
$482$	0	0
$483$	−405.722	−0.0382215
$484$	0	0
$485$	−181.627	−0.0170047
$486$	0	0
$487$	−13822.7	−1.28617	−0.643086	−	0.765794i	$-0.722346\pi$
$487$	−13822.7	−1.28617	−0.643086	+	0.765794i	$0.722346\pi$
$488$	0	0
$489$	1940.14	0.179420
$490$	0	0
$491$	−4112.53	−0.377996	−0.188998	−	0.981977i	$-0.560524\pi$
$491$	−4112.53	−0.377996	−0.188998	+	0.981977i	$0.560524\pi$
$492$	0	0
$493$	−23277.5	−2.12650
$494$	0	0
$495$	2282.87	0.207288
$496$	0	0
$497$	3870.36	0.349315
$498$	0	0
$499$	−14392.5	−1.29117	−0.645587	−	0.763687i	$-0.723387\pi$
$499$	−14392.5	−1.29117	−0.645587	+	0.763687i	$0.723387\pi$
$500$	0	0
$501$	13459.6	1.20026
$502$	0	0
$503$	18788.3	1.66546	0.832732	−	0.553676i	$-0.186775\pi$
$503$	18788.3	1.66546	0.832732	+	0.553676i	$0.186775\pi$
$504$	0	0
$505$	−22318.5	−1.96665
$506$	0	0
$507$	−3026.89	−0.265145
$508$	0	0
$509$	15738.9	1.37056	0.685278	−	0.728282i	$-0.259681\pi$
$509$	15738.9	1.37056	0.685278	+	0.728282i	$0.259681\pi$
$510$	0	0
$511$	−5073.59	−0.439222
$512$	0	0
$513$	992.824	0.0854469
$514$	0	0
$515$	463.716	0.0396772
$516$	0	0
$517$	5851.38	0.497763
$518$	0	0
$519$	14815.5	1.25304
$520$	0	0
$521$	−3426.41	−0.288126	−0.144063	−	0.989569i	$-0.546017\pi$
$521$	−3426.41	−0.288126	−0.144063	+	0.989569i	$0.546017\pi$
$522$	0	0
$523$	17110.3	1.43056	0.715279	−	0.698839i	$-0.246300\pi$
$523$	17110.3	1.43056	0.715279	+	0.698839i	$0.246300\pi$
$524$	0	0
$525$	7361.68	0.611981
$526$	0	0
$527$	24775.2	2.04786
$528$	0	0
$529$	−12081.1	−0.992942
$530$	0	0
$531$	5053.44	0.412996
$532$	0	0
$533$	14160.8	1.15079
$534$	0	0
$535$	−4878.10	−0.394203
$536$	0	0
$537$	−7937.45	−0.637851
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	4624.29	0.367493	0.183746	−	0.982974i	$-0.441177\pi$
$541$	4624.29	0.367493	0.183746	+	0.982974i	$0.441177\pi$
$542$	0	0
$543$	2238.99	0.176951
$544$	0	0
$545$	−6123.53	−0.481291
$546$	0	0
$547$	−2710.72	−0.211887	−0.105943	−	0.994372i	$-0.533786\pi$
$547$	−2710.72	−0.211887	−0.105943	+	0.994372i	$0.533786\pi$
$548$	0	0
$549$	4765.29	0.370451
$550$	0	0
$551$	2382.00	0.184168
$552$	0	0
$553$	3613.83	0.277894
$554$	0	0
$555$	−38763.3	−2.96470
$556$	0	0
$557$	11574.4	0.880476	0.440238	−	0.897881i	$-0.354894\pi$
$557$	11574.4	0.880476	0.440238	+	0.897881i	$0.354894\pi$
$558$	0	0
$559$	7749.40	0.586341
$560$	0	0
$561$	7172.91	0.539823
$562$	0	0
$563$	−3321.46	−0.248637	−0.124319	−	0.992242i	$-0.539674\pi$
$563$	−3321.46	−0.248637	−0.124319	+	0.992242i	$0.539674\pi$
$564$	0	0
$565$	−23185.8	−1.72643
$566$	0	0
$567$	6365.45	0.471470
$568$	0	0
$569$	−11684.6	−0.860889	−0.430444	−	0.902617i	$-0.641643\pi$
$569$	−11684.6	−0.860889	−0.430444	+	0.902617i	$0.641643\pi$
$570$	0	0
$571$	16714.8	1.22503	0.612517	−	0.790458i	$-0.290157\pi$
$571$	16714.8	1.22503	0.612517	+	0.790458i	$0.290157\pi$
$572$	0	0
$573$	13951.5	1.01716
$574$	0	0
$575$	−1558.10	−0.113004
$576$	0	0
$577$	26798.6	1.93352	0.966759	−	0.255690i	$-0.0823025\pi$
$577$	26798.6	1.93352	0.966759	+	0.255690i	$0.0823025\pi$
$578$	0	0
$579$	27039.5	1.94080
$580$	0	0
$581$	−9777.75	−0.698192
$582$	0	0
$583$	4489.86	0.318956
$584$	0	0
$585$	10745.6	0.759448
$586$	0	0
$587$	10893.3	0.765955	0.382978	−	0.923758i	$-0.374899\pi$
$587$	10893.3	0.765955	0.382978	+	0.923758i	$0.374899\pi$
$588$	0	0
$589$	−2535.26	−0.177357
$590$	0	0
$591$	3461.45	0.240922
$592$	0	0
$593$	−27442.2	−1.90037	−0.950183	−	0.311693i	$-0.899104\pi$
$593$	−27442.2	−1.90037	−0.950183	+	0.311693i	$0.899104\pi$
$594$	0	0
$595$	12494.9	0.860907
$596$	0	0
$597$	−10444.4	−0.716016
$598$	0	0
$599$	14171.3	0.966650	0.483325	−	0.875441i	$-0.339429\pi$
$599$	14171.3	0.966650	0.483325	+	0.875441i	$0.339429\pi$
$600$	0	0
$601$	−7536.43	−0.511510	−0.255755	−	0.966742i	$-0.582324\pi$
$601$	−7536.43	−0.511510	−0.255755	+	0.966742i	$0.582324\pi$
$602$	0	0
$603$	−12640.0	−0.853632
$604$	0	0
$605$	2071.68	0.139216
$606$	0	0
$607$	18503.3	1.23727	0.618637	−	0.785677i	$-0.287685\pi$
$607$	18503.3	1.23727	0.618637	+	0.785677i	$0.287685\pi$
$608$	0	0
$609$	9775.64	0.650458
$610$	0	0
$611$	27542.9	1.82367
$612$	0	0
$613$	−6228.56	−0.410390	−0.205195	−	0.978721i	$-0.565783\pi$
$613$	−6228.56	−0.410390	−0.205195	+	0.978721i	$0.565783\pi$
$614$	0	0
$615$	−29288.0	−1.92033
$616$	0	0
$617$	−25238.3	−1.64677	−0.823383	−	0.567487i	$-0.807916\pi$
$617$	−25238.3	−1.64677	−0.823383	+	0.567487i	$0.807916\pi$
$618$	0	0
$619$	−12499.1	−0.811601	−0.405801	−	0.913962i	$-0.633007\pi$
$619$	−12499.1	−0.811601	−0.405801	+	0.913962i	$0.633007\pi$
$620$	0	0
$621$	−862.371	−0.0557259
$622$	0	0
$623$	2305.60	0.148270
$624$	0	0
$625$	−8371.38	−0.535768
$626$	0	0
$627$	−734.008	−0.0467520
$628$	0	0
$629$	−37737.4	−2.39219
$630$	0	0
$631$	−16328.7	−1.03017	−0.515083	−	0.857140i	$-0.672239\pi$
$631$	−16328.7	−1.03017	−0.515083	+	0.857140i	$0.672239\pi$
$632$	0	0
$633$	−22283.9	−1.39922
$634$	0	0
$635$	21463.0	1.34131
$636$	0	0
$637$	2537.11	0.157808
$638$	0	0
$639$	−6702.00	−0.414909
$640$	0	0
$641$	−20978.0	−1.29264	−0.646320	−	0.763067i	$-0.723693\pi$
$641$	−20978.0	−1.29264	−0.646320	+	0.763067i	$0.723693\pi$
$642$	0	0
$643$	−15007.0	−0.920401	−0.460200	−	0.887815i	$-0.652222\pi$
$643$	−15007.0	−0.920401	−0.460200	+	0.887815i	$0.652222\pi$
$644$	0	0
$645$	−16027.6	−0.978430
$646$	0	0
$647$	−22590.1	−1.37266	−0.686330	−	0.727291i	$-0.740779\pi$
$647$	−22590.1	−1.37266	−0.686330	+	0.727291i	$0.740779\pi$
$648$	0	0
$649$	4585.95	0.277372
$650$	0	0
$651$	−10404.6	−0.626404
$652$	0	0
$653$	2700.87	0.161858	0.0809289	−	0.996720i	$-0.474211\pi$
$653$	2700.87	0.161858	0.0809289	+	0.996720i	$0.474211\pi$
$654$	0	0
$655$	34225.0	2.04165
$656$	0	0
$657$	8785.53	0.521699
$658$	0	0
$659$	−1350.92	−0.0798548	−0.0399274	−	0.999203i	$-0.512713\pi$
$659$	−1350.92	−0.0798548	−0.0399274	+	0.999203i	$0.512713\pi$
$660$	0	0
$661$	18705.0	1.10066	0.550332	−	0.834946i	$-0.314501\pi$
$661$	18705.0	1.10066	0.550332	+	0.834946i	$0.314501\pi$
$662$	0	0
$663$	33763.4	1.97777
$664$	0	0
$665$	−1278.61	−0.0745598
$666$	0	0
$667$	−2069.02	−0.120109
$668$	0	0
$669$	8346.83	0.482372
$670$	0	0
$671$	4324.45	0.248798
$672$	0	0
$673$	20297.5	1.16257	0.581287	−	0.813699i	$-0.302549\pi$
$673$	20297.5	1.16257	0.581287	+	0.813699i	$0.302549\pi$
$674$	0	0
$675$	15647.4	0.892251
$676$	0	0
$677$	27278.3	1.54858	0.774292	−	0.632828i	$-0.218106\pi$
$677$	27278.3	1.54858	0.774292	+	0.632828i	$0.218106\pi$
$678$	0	0
$679$	74.2577	0.00419698
$680$	0	0
$681$	−14600.7	−0.821587
$682$	0	0
$683$	−24957.7	−1.39821	−0.699107	−	0.715017i	$-0.746419\pi$
$683$	−24957.7	−1.39821	−0.699107	+	0.715017i	$0.746419\pi$
$684$	0	0
$685$	32279.4	1.80048
$686$	0	0
$687$	28222.0	1.56730
$688$	0	0
$689$	21134.1	1.16857
$690$	0	0
$691$	21736.3	1.19665	0.598327	−	0.801252i	$-0.295832\pi$
$691$	21736.3	1.19665	0.598327	+	0.801252i	$0.295832\pi$
$692$	0	0
$693$	−933.343	−0.0511613
$694$	0	0
$695$	34391.1	1.87702
$696$	0	0
$697$	−28512.8	−1.54950
$698$	0	0
$699$	1086.30	0.0587809
$700$	0	0
$701$	−31392.4	−1.69141	−0.845703	−	0.533654i	$-0.820819\pi$
$701$	−31392.4	−1.69141	−0.845703	+	0.533654i	$0.820819\pi$
$702$	0	0
$703$	3861.69	0.207178
$704$	0	0
$705$	−56965.3	−3.04317
$706$	0	0
$707$	9124.84	0.485396
$708$	0	0
$709$	18502.6	0.980087	0.490043	−	0.871698i	$-0.336981\pi$
$709$	18502.6	0.980087	0.490043	+	0.871698i	$0.336981\pi$
$710$	0	0
$711$	−6257.77	−0.330077
$712$	0	0
$713$	2202.14	0.115667
$714$	0	0
$715$	9751.55	0.510052
$716$	0	0
$717$	29907.2	1.55775
$718$	0	0
$719$	−7460.88	−0.386987	−0.193494	−	0.981102i	$-0.561982\pi$
$719$	−7460.88	−0.386987	−0.193494	+	0.981102i	$0.561982\pi$
$720$	0	0
$721$	−189.589	−0.00979286
$722$	0	0
$723$	−9846.79	−0.506509
$724$	0	0
$725$	37541.5	1.92311
$726$	0	0
$727$	29929.1	1.52684	0.763418	−	0.645905i	$-0.223520\pi$
$727$	29929.1	1.52684	0.763418	+	0.645905i	$0.223520\pi$
$728$	0	0
$729$	4693.44	0.238452
$730$	0	0
$731$	−15603.4	−0.789486
$732$	0	0
$733$	14889.8	0.750297	0.375148	−	0.926965i	$-0.377592\pi$
$733$	14889.8	0.750297	0.375148	+	0.926965i	$0.377592\pi$
$734$	0	0
$735$	−5247.36	−0.263336
$736$	0	0
$737$	−11470.7	−0.573307
$738$	0	0
$739$	27119.4	1.34994	0.674969	−	0.737846i	$-0.264157\pi$
$739$	27119.4	1.34994	0.674969	+	0.737846i	$0.264157\pi$
$740$	0	0
$741$	−3455.03	−0.171287
$742$	0	0
$743$	−19717.6	−0.973580	−0.486790	−	0.873519i	$-0.661832\pi$
$743$	−19717.6	−0.973580	−0.486790	+	0.873519i	$0.661832\pi$
$744$	0	0
$745$	−38047.7	−1.87109
$746$	0	0
$747$	16931.4	0.829298
$748$	0	0
$749$	1994.40	0.0972946
$750$	0	0
$751$	986.613	0.0479388	0.0239694	−	0.999713i	$-0.492370\pi$
$751$	986.613	0.0479388	0.0239694	+	0.999713i	$0.492370\pi$
$752$	0	0
$753$	23988.5	1.16094
$754$	0	0
$755$	20104.7	0.969118
$756$	0	0
$757$	34780.1	1.66989	0.834944	−	0.550335i	$-0.185500\pi$
$757$	34780.1	1.66989	0.834944	+	0.550335i	$0.185500\pi$
$758$	0	0
$759$	637.563	0.0304902
$760$	0	0
$761$	−39587.1	−1.88572	−0.942858	−	0.333195i	$-0.891873\pi$
$761$	−39587.1	−1.88572	−0.942858	+	0.333195i	$0.891873\pi$
$762$	0	0
$763$	2503.59	0.118789
$764$	0	0
$765$	−21636.4	−1.02257
$766$	0	0
$767$	21586.4	1.01622
$768$	0	0
$769$	30842.8	1.44632	0.723160	−	0.690681i	$-0.242689\pi$
$769$	30842.8	1.44632	0.723160	+	0.690681i	$0.242689\pi$
$770$	0	0
$771$	−41555.0	−1.94107
$772$	0	0
$773$	−17396.5	−0.809455	−0.404728	−	0.914437i	$-0.632634\pi$
$773$	−17396.5	−0.809455	−0.404728	+	0.914437i	$0.632634\pi$
$774$	0	0
$775$	−39957.0	−1.85200
$776$	0	0
$777$	15848.2	0.731727
$778$	0	0
$779$	2917.74	0.134196
$780$	0	0
$781$	−6082.00	−0.278657
$782$	0	0
$783$	20778.3	0.948349
$784$	0	0
$785$	46865.4	2.13082
$786$	0	0
$787$	−17731.1	−0.803106	−0.401553	−	0.915836i	$-0.631529\pi$
$787$	−17731.1	−0.803106	−0.401553	+	0.915836i	$0.631529\pi$
$788$	0	0
$789$	28755.2	1.29748
$790$	0	0
$791$	9479.43	0.426106
$792$	0	0
$793$	20355.5	0.911532
$794$	0	0
$795$	−43710.4	−1.95000
$796$	0	0
$797$	−17460.0	−0.775992	−0.387996	−	0.921661i	$-0.626833\pi$
$797$	−17460.0	−0.775992	−0.387996	+	0.921661i	$0.626833\pi$
$798$	0	0
$799$	−55457.7	−2.45551
$800$	0	0
$801$	−3992.43	−0.176112
$802$	0	0
$803$	7972.79	0.350378
$804$	0	0
$805$	1110.60	0.0486257
$806$	0	0
$807$	12162.6	0.530538
$808$	0	0
$809$	30027.9	1.30498	0.652488	−	0.757799i	$-0.273725\pi$
$809$	30027.9	1.30498	0.652488	+	0.757799i	$0.273725\pi$
$810$	0	0
$811$	25186.5	1.09053	0.545263	−	0.838265i	$-0.316430\pi$
$811$	25186.5	1.09053	0.545263	+	0.838265i	$0.316430\pi$
$812$	0	0
$813$	9537.62	0.411438
$814$	0	0
$815$	−5310.86	−0.228259
$816$	0	0
$817$	1596.71	0.0683743
$818$	0	0
$819$	−4393.31	−0.187442
$820$	0	0
$821$	614.667	0.0261291	0.0130646	−	0.999915i	$-0.495841\pi$
$821$	614.667	0.0261291	0.0130646	+	0.999915i	$0.495841\pi$
$822$	0	0
$823$	−29936.2	−1.26794	−0.633968	−	0.773359i	$-0.718575\pi$
$823$	−29936.2	−1.26794	−0.633968	+	0.773359i	$0.718575\pi$
$824$	0	0
$825$	−11568.3	−0.488192
$826$	0	0
$827$	25253.2	1.06184	0.530918	−	0.847423i	$-0.321847\pi$
$827$	25253.2	1.06184	0.530918	+	0.847423i	$0.321847\pi$
$828$	0	0
$829$	−19136.9	−0.801751	−0.400876	−	0.916132i	$-0.631294\pi$
$829$	−19136.9	−0.801751	−0.400876	+	0.916132i	$0.631294\pi$
$830$	0	0
$831$	−17421.9	−0.727267
$832$	0	0
$833$	−5108.48	−0.212483
$834$	0	0
$835$	−36843.6	−1.52698
$836$	0	0
$837$	−22115.2	−0.913279
$838$	0	0
$839$	20495.3	0.843359	0.421679	−	0.906745i	$-0.361441\pi$
$839$	20495.3	0.843359	0.421679	+	0.906745i	$0.361441\pi$
$840$	0	0
$841$	25462.8	1.04403
$842$	0	0
$843$	10828.8	0.442426
$844$	0	0
$845$	8285.66	0.337320
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	−16185.8	−0.654294
$850$	0	0
$851$	−3354.28	−0.135115
$852$	0	0
$853$	−18030.4	−0.723739	−0.361870	−	0.932229i	$-0.617861\pi$
$853$	−18030.4	−0.723739	−0.361870	+	0.932229i	$0.617861\pi$
$854$	0	0
$855$	2214.06	0.0885606
$856$	0	0
$857$	−48876.2	−1.94817	−0.974083	−	0.226192i	$-0.927372\pi$
$857$	−48876.2	−1.94817	−0.974083	+	0.226192i	$0.927372\pi$
$858$	0	0
$859$	−31747.6	−1.26102	−0.630508	−	0.776182i	$-0.717154\pi$
$859$	−31747.6	−1.26102	−0.630508	+	0.776182i	$0.717154\pi$
$860$	0	0
$861$	11974.3	0.473963
$862$	0	0
$863$	−36175.9	−1.42693	−0.713466	−	0.700689i	$-0.752876\pi$
$863$	−36175.9	−1.42693	−0.713466	+	0.700689i	$0.752876\pi$
$864$	0	0
$865$	−40555.3	−1.59413
$866$	0	0
$867$	−37253.4	−1.45928
$868$	0	0
$869$	−5678.87	−0.221683
$870$	0	0
$871$	−53993.2	−2.10045
$872$	0	0
$873$	−128.586	−0.00498509
$874$	0	0
$875$	−5170.34	−0.199759
$876$	0	0
$877$	22724.7	0.874980	0.437490	−	0.899223i	$-0.355868\pi$
$877$	22724.7	0.874980	0.437490	+	0.899223i	$0.355868\pi$
$878$	0	0
$879$	18443.7	0.707725
$880$	0	0
$881$	−6148.28	−0.235120	−0.117560	−	0.993066i	$-0.537507\pi$
$881$	−6148.28	−0.235120	−0.117560	+	0.993066i	$0.537507\pi$
$882$	0	0
$883$	34001.7	1.29586	0.647932	−	0.761698i	$-0.275634\pi$
$883$	34001.7	1.29586	0.647932	+	0.761698i	$0.275634\pi$
$884$	0	0
$885$	−44645.9	−1.69577
$886$	0	0
$887$	−38642.9	−1.46280	−0.731398	−	0.681950i	$-0.761132\pi$
$887$	−38642.9	−1.46280	−0.731398	+	0.681950i	$0.761132\pi$
$888$	0	0
$889$	−8775.08	−0.331054
$890$	0	0
$891$	−10002.8	−0.376103
$892$	0	0
$893$	5675.02	0.212662
$894$	0	0
$895$	21727.6	0.811479
$896$	0	0
$897$	3001.05	0.111708
$898$	0	0
$899$	−53059.3	−1.96844
$900$	0	0
$901$	−42553.6	−1.57344
$902$	0	0
$903$	6552.84	0.241489
$904$	0	0
$905$	−6128.91	−0.225118
$906$	0	0
$907$	5093.73	0.186477	0.0932385	−	0.995644i	$-0.470278\pi$
$907$	5093.73	0.186477	0.0932385	+	0.995644i	$0.470278\pi$
$908$	0	0
$909$	−15800.8	−0.576543
$910$	0	0
$911$	19515.5	0.709744	0.354872	−	0.934915i	$-0.384524\pi$
$911$	19515.5	0.709744	0.354872	+	0.934915i	$0.384524\pi$
$912$	0	0
$913$	15365.0	0.556965
$914$	0	0
$915$	−42100.1	−1.52108
$916$	0	0
$917$	−13992.8	−0.503906
$918$	0	0
$919$	3173.14	0.113898	0.0569490	−	0.998377i	$-0.481863\pi$
$919$	3173.14	0.113898	0.0569490	+	0.998377i	$0.481863\pi$
$920$	0	0
$921$	−4217.99	−0.150909
$922$	0	0
$923$	−28628.4	−1.02093
$924$	0	0
$925$	60862.2	2.16339
$926$	0	0
$927$	328.296	0.0116318
$928$	0	0
$929$	−2919.60	−0.103110	−0.0515548	−	0.998670i	$-0.516418\pi$
$929$	−2919.60	−0.103110	−0.0515548	+	0.998670i	$0.516418\pi$
$930$	0	0
$931$	522.754	0.0184023
$932$	0	0
$933$	41937.9	1.47158
$934$	0	0
$935$	−19634.8	−0.686766
$936$	0	0
$937$	36268.3	1.26450	0.632248	−	0.774766i	$-0.282132\pi$
$937$	36268.3	1.26450	0.632248	+	0.774766i	$0.282132\pi$
$938$	0	0
$939$	60182.2	2.09156
$940$	0	0
$941$	−3009.02	−0.104242	−0.0521208	−	0.998641i	$-0.516598\pi$
$941$	−3009.02	−0.104242	−0.0521208	+	0.998641i	$0.516598\pi$
$942$	0	0
$943$	−2534.36	−0.0875187
$944$	0	0
$945$	−11153.4	−0.383936
$946$	0	0
$947$	6520.30	0.223739	0.111870	−	0.993723i	$-0.464316\pi$
$947$	6520.30	0.223739	0.111870	+	0.993723i	$0.464316\pi$
$948$	0	0
$949$	37528.5	1.28369
$950$	0	0
$951$	−25691.7	−0.876036
$952$	0	0
$953$	56034.5	1.90465	0.952326	−	0.305081i	$-0.0986835\pi$
$953$	56034.5	1.90465	0.952326	+	0.305081i	$0.0986835\pi$
$954$	0	0
$955$	−38190.3	−1.29404
$956$	0	0
$957$	−15361.7	−0.518886
$958$	0	0
$959$	−13197.3	−0.444383
$960$	0	0
$961$	26682.1	0.895644
$962$	0	0
$963$	−3453.54	−0.115565
$964$	0	0
$965$	−74016.7	−2.46910
$966$	0	0
$967$	−18669.7	−0.620865	−0.310433	−	0.950595i	$-0.600474\pi$
$967$	−18669.7	−0.620865	−0.310433	+	0.950595i	$0.600474\pi$
$968$	0	0
$969$	6956.72	0.230632
$970$	0	0
$971$	5467.04	0.180686	0.0903428	−	0.995911i	$-0.471204\pi$
$971$	5467.04	0.180686	0.0903428	+	0.995911i	$0.471204\pi$
$972$	0	0
$973$	−14060.7	−0.463273
$974$	0	0
$975$	−54453.0	−1.78861
$976$	0	0
$977$	46112.2	1.50999	0.754996	−	0.655730i	$-0.227639\pi$
$977$	46112.2	1.50999	0.754996	+	0.655730i	$0.227639\pi$
$978$	0	0
$979$	−3623.09	−0.118278
$980$	0	0
$981$	−4335.26	−0.141095
$982$	0	0
$983$	28357.6	0.920109	0.460055	−	0.887891i	$-0.347830\pi$
$983$	28357.6	0.920109	0.460055	+	0.887891i	$0.347830\pi$
$984$	0	0
$985$	−9475.20	−0.306503
$986$	0	0
$987$	23290.1	0.751095
$988$	0	0
$989$	−1386.91	−0.0445917
$990$	0	0
$991$	−34281.1	−1.09886	−0.549432	−	0.835539i	$-0.685156\pi$
$991$	−34281.1	−1.09886	−0.549432	+	0.835539i	$0.685156\pi$
$992$	0	0
$993$	54189.0	1.73176
$994$	0	0
$995$	28590.0	0.910920
$996$	0	0
$997$	20833.1	0.661775	0.330888	−	0.943670i	$-0.392652\pi$
$997$	20833.1	0.661775	0.330888	+	0.943670i	$0.392652\pi$
$998$	0	0
$999$	33685.8	1.06684

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	308.4.a.d.1.1	✓	4
4.3	odd	2		1232.4.a.v.1.4		4
7.6	odd	2		2156.4.a.f.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.4.a.d.1.1	✓	4	1.1	even	1	trivial
1232.4.a.v.1.4		4	4.3	odd	2
2156.4.a.f.1.4		4	7.6	odd	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:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	308.a (trivial)

\(f(q)\)	\(=\)	\(q-6.25471 q^{3} +17.1213 q^{5} -7.00000 q^{7} +12.1213 q^{9} +11.0000 q^{11} +51.7778 q^{13} -107.089 q^{15} -104.255 q^{17} +10.6685 q^{19} +43.7829 q^{21} -9.26667 q^{23} +168.140 q^{25} +93.0616 q^{27} +223.275 q^{29} -237.641 q^{31} -68.8018 q^{33} -119.849 q^{35} +361.973 q^{37} -323.855 q^{39} +273.492 q^{41} +149.666 q^{43} +207.534 q^{45} +531.944 q^{47} +49.0000 q^{49} +652.083 q^{51} +408.170 q^{53} +188.335 q^{55} -66.7280 q^{57} +416.905 q^{59} +393.132 q^{61} -84.8494 q^{63} +886.505 q^{65} -1042.79 q^{67} +57.9603 q^{69} -552.909 q^{71} +724.799 q^{73} -1051.67 q^{75} -77.0000 q^{77} -516.261 q^{79} -909.349 q^{81} +1396.82 q^{83} -1784.98 q^{85} -1396.52 q^{87} -329.372 q^{89} -362.444 q^{91} +1486.37 q^{93} +182.658 q^{95} -10.6082 q^{97} +133.335 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + q^{5} - 28 q^{7} - 19 q^{9} + 44 q^{11} + 98 q^{13} - 33 q^{15} + 64 q^{17} + 114 q^{19} + 21 q^{21} + 231 q^{23} + 417 q^{25} + 63 q^{27} + 268 q^{29} + 33 q^{31} - 33 q^{33} - 7 q^{35}+ \cdots - 209 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + q^5 - 28 * q^7 - 19 * q^9 + 44 * q^11 + 98 * q^13 - 33 * q^15 + 64 * q^17 + 114 * q^19 + 21 * q^21 + 231 * q^23 + 417 * q^25 + 63 * q^27 + 268 * q^29 + 33 * q^31 - 33 * q^33 - 7 * q^35 + 357 * q^37 + 140 * q^39 + 364 * q^41 + 44 * q^43 + 912 * q^45 + 720 * q^47 + 196 * q^49 + 794 * q^51 + 740 * q^53 + 11 * q^55 + 138 * q^57 + 787 * q^59 + 1020 * q^61 + 133 * q^63 + 296 * q^65 - 995 * q^67 + 1059 * q^69 - 1011 * q^71 + 1592 * q^73 - 324 * q^75 - 308 * q^77 - 178 * q^79 - 1396 * q^81 + 324 * q^83 - 94 * q^85 - 1262 * q^87 + 19 * q^89 - 686 * q^91 + 637 * q^93 - 2418 * q^95 - 555 * q^97 - 209 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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