LMFDB - Embedded newform 1904.2.c.f.1121.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1904,2,Mod(1121,1904)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1904, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1904.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1904 = 2^{4} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1904.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:	$15.2035165449$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.980441344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 8x^{6} + 18x^{4} + 9x^{2} + 1 $ x^8 + 8x^6 + 18x^4 + 9*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 476)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.4
Root			$-2.06150i$ of defining polynomial
Character	$\chi$	$=$	1904.1121
Dual form			1904.2.c.f.1121.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.811721i q^{3} +1.15845i q^{5} -1.00000i q^{7} +2.34111 q^{9} +O(q^{10})$	$q-0.811721i q^{3} +1.15845i q^{5} -1.00000i q^{7} +2.34111 q^{9} -2.12300i q^{11} -1.15283 q^{13} +0.940336 q^{15} +(-4.12300 - 0.0298321i) q^{17} -0.653275 q^{19} -0.811721 q^{21} -7.65238i q^{23} +3.65800 q^{25} -4.33549i q^{27} +2.49956i q^{29} -0.841553i q^{31} -1.72328 q^{33} +1.15845 q^{35} +2.97017i q^{37} +0.935777i q^{39} -9.81083i q^{41} +7.03456 q^{43} +2.71205i q^{45} -6.55922 q^{47} -1.00000 q^{49} +(-0.0242154 + 3.34673i) q^{51} +6.90489 q^{53} +2.45938 q^{55} +0.530277i q^{57} +0.929992 q^{59} -5.49394i q^{61} -2.34111i q^{63} -1.33549i q^{65} -11.5871 q^{67} -6.21160 q^{69} -5.72328i q^{71} -6.99805i q^{73} -2.96928i q^{75} -2.12300 q^{77} +12.4883i q^{79} +3.50411 q^{81} +11.3988 q^{83} +(0.0345589 - 4.77627i) q^{85} +2.02894 q^{87} +6.75678 q^{89} +1.15283i q^{91} -0.683107 q^{93} -0.756784i q^{95} -13.7810i q^{97} -4.97017i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{9}+O(q^{10}) $ 8 * q - 12 * q^9	$ 8 q - 12 q^{9} + 20 q^{13} - 12 q^{19} - 8 q^{21} - 12 q^{25} - 4 q^{33} + 4 q^{35} + 12 q^{43} - 24 q^{47} - 8 q^{49} + 20 q^{51} + 20 q^{53} + 4 q^{55} + 24 q^{59} + 4 q^{67} + 44 q^{69} + 16 q^{77} + 16 q^{81} - 4 q^{83} - 44 q^{85} - 52 q^{87} - 16 q^{93}+O(q^{100}) $ 8 * q - 12 * q^9 + 20 * q^13 - 12 * q^19 - 8 * q^21 - 12 * q^25 - 4 * q^33 + 4 * q^35 + 12 * q^43 - 24 * q^47 - 8 * q^49 + 20 * q^51 + 20 * q^53 + 4 * q^55 + 24 * q^59 + 4 * q^67 + 44 * q^69 + 16 * q^77 + 16 * q^81 - 4 * q^83 - 44 * q^85 - 52 * q^87 - 16 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1904\mathbb{Z}\right)^\times$.

$n$	$239$	$785$	$1361$	$1429$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		−	0.811721i		−	0.468648i	−0.972159	−	0.234324i	$-0.924712\pi$
$3$		−	0.811721i		−	0.468648i	0.972159	−	0.234324i	$-0.0752876\pi$
$4$		0			0
$5$			1.15845i			0.518073i	0.965868	+	0.259037i	$0.0834050\pi$
$5$			1.15845i			0.518073i	−0.965868	+	0.259037i	$0.916595\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	2.34111			0.780369
$10$		0			0
$11$		−	2.12300i		−	0.640108i	−0.947399	−	0.320054i	$-0.896299\pi$
$11$		−	2.12300i		−	0.640108i	0.947399	−	0.320054i	$-0.103701\pi$
$12$		0			0
$13$	−1.15283			−0.319737			−0.159869	−	0.987138i	$-0.551107\pi$
$13$	−1.15283			−0.319737			−0.159869	+	0.987138i	$0.551107\pi$
$14$		0			0
$15$	0.940336			0.242794
$16$		0			0
$17$	−4.12300	−	0.0298321i	−0.999974	−	0.00723535i
$17$	−4.12300	−	0.0298321i	−0.999974	−	0.00723535i
$18$		0			0
$19$	−0.653275			−0.149872			−0.0749358	−	0.997188i	$-0.523875\pi$
$19$	−0.653275			−0.149872			−0.0749358	+	0.997188i	$0.523875\pi$
$20$		0			0
$21$	−0.811721			−0.177132
$22$		0			0
$23$		−	7.65238i		−	1.59563i	−0.602901	−	0.797816i	$-0.705988\pi$
$23$		−	7.65238i		−	1.59563i	0.602901	−	0.797816i	$-0.294012\pi$
$24$		0			0
$25$	3.65800			0.731600
$26$		0			0
$27$		−	4.33549i		−	0.834366i
$28$		0			0
$29$			2.49956i			0.464156i	0.972697	+	0.232078i	$0.0745524\pi$
$29$			2.49956i			0.464156i	−0.972697	+	0.232078i	$0.925448\pi$
$30$		0			0
$31$		−	0.841553i		−	0.151147i	−0.997140	−	0.0755737i	$-0.975921\pi$
$31$		−	0.841553i		−	0.151147i	0.997140	−	0.0755737i	$-0.0240788\pi$
$32$		0			0
$33$	−1.72328			−0.299985
$34$		0			0
$35$	1.15845			0.195813
$36$		0			0
$37$			2.97017i			0.488293i	0.969738	+	0.244146i	$0.0785077\pi$
$37$			2.97017i			0.488293i	−0.969738	+	0.244146i	$0.921492\pi$
$38$		0			0
$39$			0.935777i			0.149844i
$40$		0			0
$41$		−	9.81083i		−	1.53219i	−0.642725	−	0.766097i	$-0.722196\pi$
$41$		−	9.81083i		−	1.53219i	0.642725	−	0.766097i	$-0.277804\pi$
$42$		0			0
$43$	7.03456			1.07276			0.536380	−	0.843977i	$-0.319791\pi$
$43$	7.03456			1.07276			0.536380	+	0.843977i	$0.319791\pi$
$44$		0			0
$45$			2.71205i			0.404288i
$46$		0			0
$47$	−6.55922			−0.956760			−0.478380	−	0.878153i	$-0.658776\pi$
$47$	−6.55922			−0.956760			−0.478380	+	0.878153i	$0.658776\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−0.0242154	+	3.34673i	−0.00339083	+	0.468635i
$52$		0			0
$53$	6.90489			0.948459			0.474230	−	0.880401i	$-0.342727\pi$
$53$	6.90489			0.948459			0.474230	+	0.880401i	$0.342727\pi$
$54$		0			0
$55$	2.45938			0.331623
$56$		0			0
$57$			0.530277i			0.0702369i
$58$		0			0
$59$	0.929992			0.121075			0.0605373	−	0.998166i	$-0.480719\pi$
$59$	0.929992			0.121075			0.0605373	+	0.998166i	$0.480719\pi$
$60$		0			0
$61$		−	5.49394i		−	0.703427i	−0.936108	−	0.351713i	$-0.885599\pi$
$61$		−	5.49394i		−	0.703427i	0.936108	−	0.351713i	$-0.114401\pi$
$62$		0			0
$63$		−	2.34111i		−	0.294952i
$64$		0			0
$65$		−	1.33549i		−	0.165647i
$66$		0			0
$67$	−11.5871			−1.41559			−0.707795	−	0.706418i	$-0.750310\pi$
$67$	−11.5871			−1.41559			−0.707795	+	0.706418i	$0.750310\pi$
$68$		0			0
$69$	−6.21160			−0.747789
$70$		0			0
$71$		−	5.72328i		−	0.679229i	−0.940565	−	0.339614i	$-0.889703\pi$
$71$		−	5.72328i		−	0.679229i	0.940565	−	0.339614i	$-0.110297\pi$
$72$		0			0
$73$		−	6.99805i		−	0.819060i	−0.912297	−	0.409530i	$-0.865693\pi$
$73$		−	6.99805i		−	0.819060i	0.912297	−	0.409530i	$-0.134307\pi$
$74$		0			0
$75$		−	2.96928i		−	0.342863i
$76$		0			0
$77$	−2.12300			−0.241938
$78$		0			0
$79$			12.4883i			1.40505i	0.711661	+	0.702523i	$0.247943\pi$
$79$			12.4883i			1.40505i	−0.711661	+	0.702523i	$0.752057\pi$
$80$		0			0
$81$	3.50411			0.389346
$82$		0			0
$83$	11.3988			1.25118			0.625592	−	0.780151i	$-0.284858\pi$
$83$	11.3988			1.25118			0.625592	+	0.780151i	$0.284858\pi$
$84$		0			0
$85$	0.0345589	−	4.77627i	0.00374844	−	0.518059i
$86$		0			0
$87$	2.02894			0.217525
$88$		0			0
$89$	6.75678			0.716218			0.358109	−	0.933680i	$-0.383422\pi$
$89$	6.75678			0.716218			0.358109	+	0.933680i	$0.383422\pi$
$90$		0			0
$91$			1.15283i			0.120849i
$92$		0			0
$93$	−0.683107			−0.0708349
$94$		0			0
$95$		−	0.756784i		−	0.0776444i
$96$		0			0
$97$		−	13.7810i		−	1.39925i	−0.714511	−	0.699624i	$-0.753351\pi$
$97$		−	13.7810i		−	1.39925i	0.714511	−	0.699624i	$-0.246649\pi$
$98$		0			0
$99$		−	4.97017i		−	0.499521i
$100$		0			0
$101$	−3.28706			−0.327075			−0.163537	−	0.986537i	$-0.552290\pi$
$101$	−3.28706			−0.327075			−0.163537	+	0.986537i	$0.552290\pi$
$102$		0			0
$103$	−13.2795			−1.30847			−0.654234	−	0.756292i	$-0.727009\pi$
$103$	−13.2795			−1.30847			−0.654234	+	0.756292i	$0.727009\pi$
$104$		0			0
$105$		−	0.940336i		−	0.0917674i
$106$		0			0
$107$		−	7.52939i		−	0.727893i	−0.931420	−	0.363947i	$-0.881429\pi$
$107$		−	7.52939i		−	0.727893i	0.931420	−	0.363947i	$-0.118571\pi$
$108$		0			0
$109$		−	15.9291i		−	1.52573i	−0.646557	−	0.762866i	$-0.723792\pi$
$109$		−	15.9291i		−	1.52573i	0.646557	−	0.762866i	$-0.276208\pi$
$110$		0			0
$111$	2.41095			0.228837
$112$		0			0
$113$		−	17.4390i		−	1.64052i	−0.571989	−	0.820262i	$-0.693828\pi$
$113$		−	17.4390i		−	1.64052i	0.571989	−	0.820262i	$-0.306172\pi$
$114$		0			0
$115$	8.86488			0.826654
$116$		0			0
$117$	−2.69890			−0.249513
$118$		0			0
$119$	−0.0298321	+	4.12300i	−0.00273470	+	0.377955i
$120$		0			0
$121$	6.49288			0.590262
$122$		0			0
$123$	−7.96366			−0.718059
$124$		0			0
$125$			10.0298i			0.897095i
$126$		0			0
$127$	8.51710			0.755770			0.377885	−	0.925853i	$-0.376651\pi$
$127$	8.51710			0.755770			0.377885	+	0.925853i	$0.376651\pi$
$128$		0			0
$129$		−	5.71010i		−	0.502746i
$130$		0			0
$131$			2.69345i			0.235328i	0.993053	+	0.117664i	$0.0375406\pi$
$131$			2.69345i			0.235328i	−0.993053	+	0.117664i	$0.962459\pi$
$132$		0			0
$133$			0.653275i			0.0566461i
$134$		0			0
$135$	5.02244			0.432262
$136$		0			0
$137$	4.89454			0.418169			0.209085	−	0.977898i	$-0.432952\pi$
$137$	4.89454			0.418169			0.209085	+	0.977898i	$0.432952\pi$
$138$		0			0
$139$			1.34478i			0.114063i	0.998372	+	0.0570313i	$0.0181635\pi$
$139$			1.34478i			0.114063i	−0.998372	+	0.0570313i	$0.981837\pi$
$140$		0			0
$141$			5.32426i			0.448383i
$142$		0			0
$143$			2.44746i			0.204666i
$144$		0			0
$145$	−2.89560			−0.240467
$146$		0			0
$147$			0.811721i			0.0669496i
$148$		0			0
$149$	12.8331			1.05133			0.525664	−	0.850692i	$-0.323817\pi$
$149$	12.8331			1.05133			0.525664	+	0.850692i	$0.323817\pi$
$150$		0			0
$151$	−12.1509			−0.988825			−0.494412	−	0.869227i	$-0.664617\pi$
$151$	−12.1509			−0.988825			−0.494412	+	0.869227i	$0.664617\pi$
$152$		0			0
$153$	−9.65238	−	0.0698402i	−0.780349	−	0.00564625i
$154$		0			0
$155$	0.974895			0.0783054
$156$		0			0
$157$	−15.3793			−1.22740			−0.613702	−	0.789537i	$-0.710321\pi$
$157$	−15.3793			−1.22740			−0.613702	+	0.789537i	$0.710321\pi$
$158$		0			0
$159$		−	5.60484i		−	0.444493i
$160$		0			0
$161$	−7.65238			−0.603092
$162$		0			0
$163$			7.41006i			0.580401i	0.956966	+	0.290200i	$0.0937219\pi$
$163$			7.41006i			0.580401i	−0.956966	+	0.290200i	$0.906278\pi$
$164$		0			0
$165$		−	1.99633i		−	0.155414i
$166$		0			0
$167$		−	15.1174i		−	1.16982i	−0.811099	−	0.584909i	$-0.801130\pi$
$167$		−	15.1174i		−	1.16982i	0.811099	−	0.584909i	$-0.198870\pi$
$168$		0			0
$169$	−11.6710			−0.897768
$170$		0			0
$171$	−1.52939			−0.116955
$172$		0			0
$173$			10.0474i			0.763888i	0.924186	+	0.381944i	$0.124745\pi$
$173$			10.0474i			0.763888i	−0.924186	+	0.381944i	$0.875255\pi$
$174$		0			0
$175$		−	3.65800i		−	0.276519i
$176$		0			0
$177$		−	0.754895i		−	0.0567413i
$178$		0			0
$179$	−15.9533			−1.19241			−0.596203	−	0.802833i	$-0.703325\pi$
$179$	−15.9533			−1.19241			−0.596203	+	0.802833i	$0.703325\pi$
$180$		0			0
$181$			5.69256i			0.423125i	0.977365	+	0.211562i	$0.0678551\pi$
$181$			5.69256i			0.423125i	−0.977365	+	0.211562i	$0.932145\pi$
$182$		0			0
$183$	−4.45955			−0.329659
$184$		0			0
$185$	−3.44078			−0.252971
$186$		0			0
$187$	−0.0633335	+	8.75311i	−0.00463140	+	0.640091i
$188$		0			0
$189$	−4.33549			−0.315361
$190$		0			0
$191$	8.15177			0.589842			0.294921	−	0.955522i	$-0.404707\pi$
$191$	8.15177			0.589842			0.294921	+	0.955522i	$0.404707\pi$
$192$		0			0
$193$			2.91807i			0.210047i	0.994470	+	0.105024i	$0.0334918\pi$
$193$			2.91807i			0.210047i	−0.994470	+	0.105024i	$0.966508\pi$
$194$		0			0
$195$	−1.08405			−0.0776302
$196$		0			0
$197$			9.33916i			0.665388i	0.943035	+	0.332694i	$0.107958\pi$
$197$			9.33916i			0.665388i	−0.943035	+	0.332694i	$0.892042\pi$
$198$		0			0
$199$			12.9868i			0.920611i	0.887761	+	0.460306i	$0.152260\pi$
$199$			12.9868i			0.920611i	−0.887761	+	0.460306i	$0.847740\pi$
$200$		0			0
$201$			9.40550i			0.663413i
$202$		0			0
$203$	2.49956			0.175434
$204$		0			0
$205$	11.3653			0.793789
$206$		0			0
$207$		−	17.9151i		−	1.24518i
$208$		0			0
$209$			1.38690i			0.0959339i
$210$		0			0
$211$			8.57412i			0.590267i	0.955456	+	0.295133i	$0.0953641\pi$
$211$			8.57412i			0.590267i	−0.955456	+	0.295133i	$0.904636\pi$
$212$		0			0
$213$	−4.64571			−0.318319
$214$		0			0
$215$			8.14916i			0.555768i
$216$		0			0
$217$	−0.841553			−0.0571284
$218$		0			0
$219$	−5.68047			−0.383850
$220$		0			0
$221$	4.75311	+	0.0343913i	0.319729	+	0.00231341i
$222$		0			0
$223$	0.981401			0.0657195			0.0328597	−	0.999460i	$-0.489539\pi$
$223$	0.981401			0.0657195			0.0328597	+	0.999460i	$0.489539\pi$
$224$		0			0
$225$	8.56378			0.570919
$226$		0			0
$227$		−	27.2626i		−	1.80949i	−0.425958	−	0.904743i	$-0.640063\pi$
$227$		−	27.2626i		−	1.80949i	0.425958	−	0.904743i	$-0.359937\pi$
$228$		0			0
$229$	11.8770			0.784854			0.392427	−	0.919783i	$-0.371635\pi$
$229$	11.8770			0.784854			0.392427	+	0.919783i	$0.371635\pi$
$230$		0			0
$231$			1.72328i			0.113384i
$232$		0			0
$233$		−	6.97017i		−	0.456631i	−0.973587	−	0.228315i	$-0.926678\pi$
$233$		−	6.97017i		−	0.456631i	0.973587	−	0.228315i	$-0.0733217\pi$
$234$		0			0
$235$		−	7.59850i		−	0.495672i
$236$		0			0
$237$	10.1370			0.658471
$238$		0			0
$239$	0.777330			0.0502813			0.0251406	−	0.999684i	$-0.491997\pi$
$239$	0.777330			0.0502813			0.0251406	+	0.999684i	$0.491997\pi$
$240$		0			0
$241$			23.9098i			1.54016i	0.637945	+	0.770082i	$0.279785\pi$
$241$			23.9098i			1.54016i	−0.637945	+	0.770082i	$0.720215\pi$
$242$		0			0
$243$		−	15.8508i		−	1.01683i
$244$		0			0
$245$		−	1.15845i		−	0.0740104i
$246$		0			0
$247$	0.753115			0.0479195
$248$		0			0
$249$		−	9.25267i		−	0.586364i
$250$		0			0
$251$	21.9675			1.38658			0.693288	−	0.720661i	$-0.256161\pi$
$251$	21.9675			1.38658			0.693288	+	0.720661i	$0.256161\pi$
$252$		0			0
$253$	−16.2460			−1.02138
$254$		0			0
$255$	−3.87700	−	0.0280522i	−0.242787	−	0.00175670i
$256$		0			0
$257$	−2.58238			−0.161084			−0.0805421	−	0.996751i	$-0.525665\pi$
$257$	−2.58238			−0.161084			−0.0805421	+	0.996751i	$0.525665\pi$
$258$		0			0
$259$	2.97017			0.184557
$260$		0			0
$261$			5.85173i			0.362213i
$262$		0			0
$263$	2.08213			0.128390			0.0641949	−	0.997937i	$-0.479552\pi$
$263$	2.08213			0.128390			0.0641949	+	0.997937i	$0.479552\pi$
$264$		0			0
$265$			7.99894i			0.491371i
$266$		0			0
$267$		−	5.48463i		−	0.335654i
$268$		0			0
$269$			19.3065i			1.17714i	0.808446	+	0.588571i	$0.200309\pi$
$269$			19.3065i			1.17714i	−0.808446	+	0.588571i	$0.799691\pi$
$270$		0			0
$271$	−9.90961			−0.601966			−0.300983	−	0.953629i	$-0.597315\pi$
$271$	−9.90961			−0.601966			−0.300983	+	0.953629i	$0.597315\pi$
$272$		0			0
$273$	0.935777			0.0566358
$274$		0			0
$275$		−	7.76593i		−	0.468303i
$276$		0			0
$277$			20.7575i			1.24720i	0.781745	+	0.623598i	$0.214330\pi$
$277$			20.7575i			1.24720i	−0.781745	+	0.623598i	$0.785670\pi$
$278$		0			0
$279$		−	1.97017i		−	0.117951i
$280$		0			0
$281$	4.42235			0.263815			0.131908	−	0.991262i	$-0.457890\pi$
$281$	4.42235			0.263815			0.131908	+	0.991262i	$0.457890\pi$
$282$		0			0
$283$			29.1567i			1.73318i	0.499017	+	0.866592i	$0.333694\pi$
$283$			29.1567i			1.73318i	−0.499017	+	0.866592i	$0.666306\pi$
$284$		0			0
$285$	−0.614298			−0.0363879
$286$		0			0
$287$	−9.81083			−0.579115
$288$		0			0
$289$	16.9982	+	0.245995i	0.999895	+	0.0144703i
$290$		0			0
$291$	−11.1863			−0.655754
$292$		0			0
$293$	0.0194886			0.00113853			0.000569267	−	1.00000i	$-0.499819\pi$
$293$	0.0194886			0.00113853			0.000569267		1.00000i	$0.499819\pi$
$294$		0			0
$295$			1.07735i			0.0627255i
$296$		0			0
$297$	−9.20424			−0.534084
$298$		0			0
$299$			8.82190i			0.510183i
$300$		0			0
$301$		−	7.03456i		−	0.405465i
$302$		0			0
$303$			2.66818i			0.153283i
$304$		0			0
$305$	6.36443			0.364426
$306$		0			0
$307$	12.3979			0.707588			0.353794	−	0.935323i	$-0.384891\pi$
$307$	12.3979			0.707588			0.353794	+	0.935323i	$0.384891\pi$
$308$		0			0
$309$			10.7793i			0.613210i
$310$		0			0
$311$			8.02699i			0.455169i	0.973758	+	0.227585i	$0.0730828\pi$
$311$			8.02699i			0.455169i	−0.973758	+	0.227585i	$0.926917\pi$
$312$		0			0
$313$		−	31.7101i		−	1.79236i	−0.443690	−	0.896180i	$-0.646331\pi$
$313$		−	31.7101i		−	1.79236i	0.443690	−	0.896180i	$-0.353669\pi$
$314$		0			0
$315$	2.71205			0.152807
$316$		0			0
$317$			19.7678i			1.11027i	0.831760	+	0.555136i	$0.187334\pi$
$317$			19.7678i			1.11027i	−0.831760	+	0.555136i	$0.812666\pi$
$318$		0			0
$319$	5.30655			0.297110
$320$		0			0
$321$	−6.11176			−0.341125
$322$		0			0
$323$	2.69345	+	0.0194886i	0.149868	+	0.00108437i
$324$		0			0
$325$	−4.21705			−0.233920
$326$		0			0
$327$	−12.9300			−0.715030
$328$		0			0
$329$			6.55922i			0.361621i
$330$		0			0
$331$	27.0224			1.48529			0.742644	−	0.669687i	$-0.233572\pi$
$331$	27.0224			1.48529			0.742644	+	0.669687i	$0.233572\pi$
$332$		0			0
$333$			6.95349i			0.381049i
$334$		0			0
$335$		−	13.4230i		−	0.733379i
$336$		0			0
$337$			6.55343i			0.356988i	0.983941	+	0.178494i	$0.0571226\pi$
$337$			6.55343i			0.356988i	−0.983941	+	0.178494i	$0.942877\pi$
$338$		0			0
$339$	−14.1556			−0.768827
$340$		0			0
$341$	−1.78662			−0.0967507
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	7.19581i		−	0.387409i
$346$		0			0
$347$			18.0463i			0.968777i	0.874853	+	0.484388i	$0.160958\pi$
$347$			18.0463i			0.968777i	−0.874853	+	0.484388i	$0.839042\pi$
$348$		0			0
$349$	12.6439			0.676814			0.338407	−	0.941000i	$-0.390112\pi$
$349$	12.6439			0.676814			0.338407	+	0.941000i	$0.390112\pi$
$350$		0			0
$351$			4.99808i			0.266778i
$352$		0			0
$353$	3.25267			0.173122			0.0865611	−	0.996247i	$-0.472412\pi$
$353$	3.25267			0.173122			0.0865611	+	0.996247i	$0.472412\pi$
$354$		0			0
$355$	6.63012			0.351890
$356$		0			0
$357$	3.34673	+	0.0242154i	0.177127	+	0.00128161i
$358$		0			0
$359$	10.2114			0.538939			0.269470	−	0.963009i	$-0.413152\pi$
$359$	10.2114			0.538939			0.269470	+	0.963009i	$0.413152\pi$
$360$		0			0
$361$	−18.5732			−0.977539
$362$		0			0
$363$		−	5.27041i		−	0.276625i
$364$		0			0
$365$	8.10687			0.424333
$366$		0			0
$367$		−	3.18828i		−	0.166427i	−0.996532	−	0.0832134i	$-0.973482\pi$
$367$		−	3.18828i		−	0.166427i	0.996532	−	0.0832134i	$-0.0265183\pi$
$368$		0			0
$369$		−	22.9682i		−	1.19568i
$370$		0			0
$371$		−	6.90489i		−	0.358484i
$372$		0			0
$373$	2.23301			0.115621			0.0578105	−	0.998328i	$-0.481588\pi$
$373$	2.23301			0.115621			0.0578105	+	0.998328i	$0.481588\pi$
$374$		0			0
$375$	8.14143			0.420422
$376$		0			0
$377$		−	2.88156i		−	0.148408i
$378$		0			0
$379$			25.5657i			1.31322i	0.754230	+	0.656610i	$0.228010\pi$
$379$			25.5657i			1.31322i	−0.754230	+	0.656610i	$0.771990\pi$
$380$		0			0
$381$		−	6.91351i		−	0.354190i
$382$		0			0
$383$	15.9946			0.817283			0.408642	−	0.912695i	$-0.366003\pi$
$383$	15.9946			0.817283			0.408642	+	0.912695i	$0.366003\pi$
$384$		0			0
$385$		−	2.45938i		−	0.125342i
$386$		0			0
$387$	16.4687			0.837149
$388$		0			0
$389$	7.04668			0.357281			0.178640	−	0.983914i	$-0.442830\pi$
$389$	7.04668			0.357281			0.178640	+	0.983914i	$0.442830\pi$
$390$		0			0
$391$	−0.228287	+	31.5508i	−0.0115450	+	1.59559i
$392$		0			0
$393$	2.18633			0.110286
$394$		0			0
$395$	−14.4671			−0.727916
$396$		0			0
$397$			39.0840i			1.96157i	0.195094	+	0.980785i	$0.437499\pi$
$397$			39.0840i			1.96157i	−0.195094	+	0.980785i	$0.562501\pi$
$398$		0			0
$399$	0.530277			0.0265471
$400$		0			0
$401$		−	10.5115i		−	0.524918i	−0.964943	−	0.262459i	$-0.915467\pi$
$401$		−	10.5115i		−	0.524918i	0.964943	−	0.262459i	$-0.0845335\pi$
$402$		0			0
$403$			0.970168i			0.0483275i
$404$		0			0
$405$			4.05933i			0.201710i
$406$		0			0
$407$	6.30566			0.312560
$408$		0			0
$409$	−24.6208			−1.21742			−0.608709	−	0.793393i	$-0.708312\pi$
$409$	−24.6208			−1.21742			−0.608709	+	0.793393i	$0.708312\pi$
$410$		0			0
$411$		−	3.97301i		−	0.195974i
$412$		0			0
$413$		−	0.929992i		−	0.0457619i
$414$		0			0
$415$			13.2049i			0.648205i
$416$		0			0
$417$	1.09158			0.0534551
$418$		0			0
$419$			27.9579i			1.36583i	0.730497	+	0.682916i	$0.239288\pi$
$419$			27.9579i			1.36583i	−0.730497	+	0.682916i	$0.760712\pi$
$420$		0			0
$421$	3.41290			0.166334			0.0831672	−	0.996536i	$-0.473496\pi$
$421$	3.41290			0.166334			0.0831672	+	0.996536i	$0.473496\pi$
$422$		0			0
$423$	−15.3558			−0.746627
$424$		0			0
$425$	−15.0819	−	0.109126i	−0.731581	−	0.00529338i
$426$		0			0
$427$	−5.49394			−0.265870
$428$		0			0
$429$	1.98665			0.0959164
$430$		0			0
$431$		−	7.27216i		−	0.350288i	−0.984543	−	0.175144i	$-0.943961\pi$
$431$		−	7.27216i		−	0.350288i	0.984543	−	0.175144i	$-0.0560390\pi$
$432$		0			0
$433$	29.4853			1.41697			0.708487	−	0.705724i	$-0.249378\pi$
$433$	29.4853			1.41697			0.708487	+	0.705724i	$0.249378\pi$
$434$		0			0
$435$			2.35042i			0.112694i
$436$		0			0
$437$			4.99911i			0.239140i
$438$		0			0
$439$			26.6681i			1.27280i	0.771359	+	0.636401i	$0.219577\pi$
$439$			26.6681i			1.27280i	−0.771359	+	0.636401i	$0.780423\pi$
$440$		0			0
$441$	−2.34111			−0.111481
$442$		0			0
$443$	16.7994			0.798165			0.399083	−	0.916915i	$-0.369329\pi$
$443$	16.7994			0.798165			0.399083	+	0.916915i	$0.369329\pi$
$444$		0			0
$445$			7.82737i			0.371053i
$446$		0			0
$447$		−	10.4169i		−	0.492702i
$448$		0			0
$449$		−	30.8938i		−	1.45797i	−0.684530	−	0.728985i	$-0.739992\pi$
$449$		−	30.8938i		−	1.45797i	0.684530	−	0.728985i	$-0.260008\pi$
$450$		0			0
$451$	−20.8284			−0.980770
$452$		0			0
$453$			9.86313i			0.463410i
$454$		0			0
$455$	−1.33549			−0.0626088
$456$		0			0
$457$	21.3186			0.997244			0.498622	−	0.866820i	$-0.333840\pi$
$457$	21.3186			0.997244			0.498622	+	0.866820i	$0.333840\pi$
$458$		0			0
$459$	−0.129337	+	17.8752i	−0.00603693	+	0.834344i
$460$		0			0
$461$	6.63957			0.309236			0.154618	−	0.987974i	$-0.450585\pi$
$461$	6.63957			0.309236			0.154618	+	0.987974i	$0.450585\pi$
$462$		0			0
$463$	12.0316			0.559154			0.279577	−	0.960123i	$-0.409806\pi$
$463$	12.0316			0.559154			0.279577	+	0.960123i	$0.409806\pi$
$464$		0			0
$465$		−	0.791343i		−	0.0366976i
$466$		0			0
$467$	−20.1230			−0.931181			−0.465591	−	0.885000i	$-0.654158\pi$
$467$	−20.1230			−0.931181			−0.465591	+	0.885000i	$0.654158\pi$
$468$		0			0
$469$			11.5871i			0.535043i
$470$		0			0
$471$			12.4837i			0.575220i
$472$		0			0
$473$		−	14.9344i		−	0.686682i
$474$		0			0
$475$	−2.38968			−0.109646
$476$		0			0
$477$	16.1651			0.740149
$478$		0			0
$479$			7.79014i			0.355941i	0.984036	+	0.177970i	$0.0569531\pi$
$479$			7.79014i			0.355941i	−0.984036	+	0.177970i	$0.943047\pi$
$480$		0			0
$481$		−	3.42410i		−	0.156125i
$482$		0			0
$483$			6.21160i			0.282638i
$484$		0			0
$485$	15.9646			0.724913
$486$		0			0
$487$			19.5388i			0.885389i	0.896672	+	0.442695i	$0.145977\pi$
$487$			19.5388i			0.885389i	−0.896672	+	0.442695i	$0.854023\pi$
$488$		0			0
$489$	6.01490			0.272003
$490$		0			0
$491$	−20.0903			−0.906664			−0.453332	−	0.891342i	$-0.649765\pi$
$491$	−20.0903			−0.906664			−0.453332	+	0.891342i	$0.649765\pi$
$492$		0			0
$493$	0.0745670	−	10.3057i	0.00335833	−	0.464144i
$494$		0			0
$495$	5.75767			0.258788
$496$		0			0
$497$	−5.72328			−0.256724
$498$		0			0
$499$		−	4.29374i		−	0.192214i	−0.995371	−	0.0961070i	$-0.969361\pi$
$499$		−	4.29374i		−	0.192214i	0.995371	−	0.0961070i	$-0.0306391\pi$
$500$		0			0
$501$	−12.2711			−0.548232
$502$		0			0
$503$			30.2863i			1.35040i	0.737634	+	0.675201i	$0.235943\pi$
$503$			30.2863i			1.35040i	−0.737634	+	0.675201i	$0.764057\pi$
$504$		0			0
$505$		−	3.80788i		−	0.169449i
$506$		0			0
$507$			9.47359i			0.420737i
$508$		0			0
$509$	−41.6448			−1.84587			−0.922937	−	0.384951i	$-0.874218\pi$
$509$	−41.6448			−1.84587			−0.922937	+	0.384951i	$0.874218\pi$
$510$		0			0
$511$	−6.99805			−0.309576
$512$		0			0
$513$			2.83227i			0.125048i
$514$		0			0
$515$		−	15.3836i		−	0.677882i
$516$		0			0
$517$			13.9252i			0.612430i
$518$		0			0
$519$	8.15567			0.357994
$520$		0			0
$521$		−	7.74171i		−	0.339171i	−0.985516	−	0.169585i	$-0.945757\pi$
$521$		−	7.74171i		−	0.339171i	0.985516	−	0.169585i	$-0.0542428\pi$
$522$		0			0
$523$	−21.6134			−0.945088			−0.472544	−	0.881307i	$-0.656664\pi$
$523$	−21.6134			−0.945088			−0.472544	+	0.881307i	$0.656664\pi$
$524$		0			0
$525$	−2.96928			−0.129590
$526$		0			0
$527$	−0.0251053	+	3.46972i	−0.00109360	+	0.151144i
$528$		0			0
$529$	−35.5590			−1.54604
$530$		0			0
$531$	2.17721			0.0944830
$532$		0			0
$533$			11.3102i			0.489900i
$534$		0			0
$535$	8.72239			0.377102
$536$		0			0
$537$			12.9496i			0.558819i
$538$		0			0
$539$			2.12300i			0.0914440i
$540$		0			0
$541$			34.9666i			1.50333i	0.659544	+	0.751666i	$0.270749\pi$
$541$			34.9666i			1.50333i	−0.659544	+	0.751666i	$0.729251\pi$
$542$		0			0
$543$	4.62077			0.198296
$544$		0			0
$545$	18.4530			0.790440
$546$		0			0
$547$		−	26.8856i		−	1.14954i	−0.818314	−	0.574772i	$-0.805091\pi$
$547$		−	26.8856i		−	1.14954i	0.818314	−	0.574772i	$-0.194909\pi$
$548$		0			0
$549$		−	12.8619i		−	0.548933i
$550$		0			0
$551$		−	1.63290i		−	0.0695637i
$552$		0			0
$553$	12.4883			0.531057
$554$		0			0
$555$			2.79296i			0.118554i
$556$		0			0
$557$	−39.4297			−1.67069			−0.835344	−	0.549727i	$-0.814732\pi$
$557$	−39.4297			−1.67069			−0.835344	+	0.549727i	$0.814732\pi$
$558$		0			0
$559$	−8.10965			−0.343002
$560$		0			0
$561$	7.10509	+	0.0514091i	0.299977	+	0.00217050i
$562$		0			0
$563$	−13.3197			−0.561357			−0.280679	−	0.959802i	$-0.590560\pi$
$563$	−13.3197			−0.561357			−0.280679	+	0.959802i	$0.590560\pi$
$564$		0			0
$565$	20.2022			0.849911
$566$		0			0
$567$		−	3.50411i		−	0.147159i
$568$		0			0
$569$	1.88420			0.0789898			0.0394949	−	0.999220i	$-0.487425\pi$
$569$	1.88420			0.0789898			0.0394949	+	0.999220i	$0.487425\pi$
$570$		0			0
$571$			32.6866i			1.36789i	0.729533	+	0.683945i	$0.239737\pi$
$571$			32.6866i			1.36789i	−0.729533	+	0.683945i	$0.760263\pi$
$572$		0			0
$573$		−	6.61697i		−	0.276428i
$574$		0			0
$575$		−	27.9924i		−	1.16737i
$576$		0			0
$577$	7.84840			0.326733			0.163366	−	0.986565i	$-0.447765\pi$
$577$	7.84840			0.326733			0.163366	+	0.986565i	$0.447765\pi$
$578$		0			0
$579$	2.36866			0.0984381
$580$		0			0
$581$		−	11.3988i		−	0.472903i
$582$		0			0
$583$		−	14.6591i		−	0.607116i
$584$		0			0
$585$		−	3.12653i		−	0.129266i
$586$		0			0
$587$	1.68500			0.0695472			0.0347736	−	0.999395i	$-0.488929\pi$
$587$	1.68500			0.0695472			0.0347736	+	0.999395i	$0.488929\pi$
$588$		0			0
$589$			0.549766i			0.0226527i
$590$		0			0
$591$	7.58080			0.311832
$592$		0			0
$593$	−40.0065			−1.64287			−0.821435	−	0.570303i	$-0.806826\pi$
$593$	−40.0065			−1.64287			−0.821435	+	0.570303i	$0.806826\pi$
$594$		0			0
$595$	−4.77627	−	0.0345589i	−0.195808	−	0.00141678i
$596$		0			0
$597$	10.5417			0.431442
$598$		0			0
$599$	35.3774			1.44548			0.722741	−	0.691119i	$-0.242882\pi$
$599$	35.3774			1.44548			0.722741	+	0.691119i	$0.242882\pi$
$600$		0			0
$601$			20.8664i			0.851160i	0.904921	+	0.425580i	$0.139930\pi$
$601$			20.8664i			0.851160i	−0.904921	+	0.425580i	$0.860070\pi$
$602$		0			0
$603$	−27.1267			−1.10468
$604$		0			0
$605$			7.52166i			0.305799i
$606$		0			0
$607$		−	13.3223i		−	0.540736i	−0.962757	−	0.270368i	$-0.912855\pi$
$607$		−	13.3223i		−	0.540736i	0.962757	−	0.270368i	$-0.0871454\pi$
$608$		0			0
$609$		−	2.02894i		−	0.0822169i
$610$		0			0
$611$	7.56166			0.305912
$612$		0			0
$613$	−21.3299			−0.861506			−0.430753	−	0.902470i	$-0.641752\pi$
$613$	−21.3299			−0.861506			−0.430753	+	0.902470i	$0.641752\pi$
$614$		0			0
$615$		−	9.22548i		−	0.372007i
$616$		0			0
$617$		−	10.8277i		−	0.435906i	−0.975959	−	0.217953i	$-0.930062\pi$
$617$		−	10.8277i		−	0.435906i	0.975959	−	0.217953i	$-0.0699380\pi$
$618$		0			0
$619$		−	16.5534i		−	0.665339i	−0.943044	−	0.332669i	$-0.892051\pi$
$619$		−	16.5534i		−	0.665339i	0.943044	−	0.332669i	$-0.107949\pi$
$620$		0			0
$621$	−33.1769			−1.33134
$622$		0			0
$623$		−	6.75678i		−	0.270705i
$624$		0			0
$625$	6.67098			0.266839
$626$		0			0
$627$	1.12578			0.0449592
$628$		0			0
$629$	0.0886064	−	12.2460i	0.00353297	−	0.488280i
$630$		0			0
$631$	31.6334			1.25931			0.629653	−	0.776876i	$-0.283197\pi$
$631$	31.6334			1.25931			0.629653	+	0.776876i	$0.283197\pi$
$632$		0			0
$633$	6.95980			0.276627
$634$		0			0
$635$			9.86660i			0.391544i
$636$		0			0
$637$	1.15283			0.0456768
$638$		0			0
$639$		−	13.3988i		−	0.530049i
$640$		0			0
$641$			9.84628i			0.388905i	0.980912	+	0.194452i	$0.0622930\pi$
$641$			9.84628i			0.388905i	−0.980912	+	0.194452i	$0.937707\pi$
$642$		0			0
$643$			2.31795i			0.0914110i	0.998955	+	0.0457055i	$0.0145536\pi$
$643$			2.31795i			0.0914110i	−0.998955	+	0.0457055i	$0.985446\pi$
$644$		0			0
$645$	6.61485			0.260459
$646$		0			0
$647$	−18.7231			−0.736080			−0.368040	−	0.929810i	$-0.619971\pi$
$647$	−18.7231			−0.736080			−0.368040	+	0.929810i	$0.619971\pi$
$648$		0			0
$649$		−	1.97437i		−	0.0775009i
$650$		0			0
$651$			0.683107i			0.0267731i
$652$		0			0
$653$		−	38.9282i		−	1.52338i	−0.647942	−	0.761689i	$-0.724370\pi$
$653$		−	38.9282i		−	1.52338i	0.647942	−	0.761689i	$-0.275630\pi$
$654$		0			0
$655$	−3.12022			−0.121917
$656$		0			0
$657$		−	16.3832i		−	0.639170i
$658$		0			0
$659$	−32.6904			−1.27344			−0.636719	−	0.771096i	$-0.719709\pi$
$659$	−32.6904			−1.27344			−0.636719	+	0.771096i	$0.719709\pi$
$660$		0			0
$661$	17.9507			0.698202			0.349101	−	0.937085i	$-0.386487\pi$
$661$	17.9507			0.698202			0.349101	+	0.937085i	$0.386487\pi$
$662$		0			0
$663$	0.0279162	−	3.85820i	0.00108417	−	0.149840i
$664$		0			0
$665$	−0.756784			−0.0293468
$666$		0			0
$667$	19.1276			0.740622
$668$		0			0
$669$		−	0.796624i		−	0.0307993i
$670$		0			0
$671$	−11.6636			−0.450269
$672$		0			0
$673$			42.8631i			1.65225i	0.563486	+	0.826126i	$0.309460\pi$
$673$			42.8631i			1.65225i	−0.563486	+	0.826126i	$0.690540\pi$
$674$		0			0
$675$		−	15.8592i		−	0.610422i
$676$		0			0
$677$			22.8110i			0.876698i	0.898805	+	0.438349i	$0.144437\pi$
$677$			22.8110i			0.876698i	−0.898805	+	0.438349i	$0.855563\pi$
$678$		0			0
$679$	−13.7810			−0.528866
$680$		0			0
$681$	−22.1297			−0.848011
$682$		0			0
$683$		−	37.2068i		−	1.42368i	−0.702342	−	0.711840i	$-0.747862\pi$
$683$		−	37.2068i		−	1.42368i	0.702342	−	0.711840i	$-0.252138\pi$
$684$		0			0
$685$			5.67007i			0.216642i
$686$		0			0
$687$		−	9.64082i		−	0.367820i
$688$		0			0
$689$	−7.96016			−0.303258
$690$		0			0
$691$		−	15.9263i		−	0.605864i	−0.953012	−	0.302932i	$-0.902035\pi$
$691$		−	15.9263i		−	0.605864i	0.953012	−	0.302932i	$-0.0979655\pi$
$692$		0			0
$693$	−4.97017			−0.188801
$694$		0			0
$695$	−1.55785			−0.0590927
$696$		0			0
$697$	−0.292678	+	40.4500i	−0.0110860	+	1.53215i
$698$		0			0
$699$	−5.65783			−0.213999
$700$		0			0
$701$	36.1249			1.36442			0.682209	−	0.731157i	$-0.261019\pi$
$701$	36.1249			1.36442			0.682209	+	0.731157i	$0.261019\pi$
$702$		0			0
$703$		−	1.94034i		−	0.0731812i
$704$		0			0
$705$	−6.16787			−0.232295
$706$		0			0
$707$			3.28706i			0.123623i
$708$		0			0
$709$		−	24.3906i		−	0.916008i	−0.888950	−	0.458004i	$-0.848565\pi$
$709$		−	24.3906i		−	0.916008i	0.888950	−	0.458004i	$-0.151435\pi$
$710$		0			0
$711$			29.2365i			1.09645i
$712$		0			0
$713$	−6.43989			−0.241176
$714$		0			0
$715$	−2.83525			−0.106032
$716$		0			0
$717$		−	0.630975i		−	0.0235642i
$718$		0			0
$719$		−	26.6103i		−	0.992395i	−0.868210	−	0.496198i	$-0.834729\pi$
$719$		−	26.6103i		−	0.992395i	0.868210	−	0.496198i	$-0.165271\pi$
$720$		0			0
$721$			13.2795i			0.494554i
$722$		0			0
$723$	19.4081			0.721794
$724$		0			0
$725$			9.14338i			0.339576i
$726$		0			0
$727$	−45.7943			−1.69842			−0.849208	−	0.528058i	$-0.822920\pi$
$727$	−45.7943			−1.69842			−0.849208	+	0.528058i	$0.822920\pi$
$728$		0			0
$729$	−2.35412			−0.0871897
$730$		0			0
$731$	−29.0035	−	0.209856i	−1.07273	−	0.00776179i
$732$		0			0
$733$	0.380227			0.0140440			0.00702199	−	0.999975i	$-0.497765\pi$
$733$	0.380227			0.0140440			0.00702199	+	0.999975i	$0.497765\pi$
$734$		0			0
$735$	−0.940336			−0.0346848
$736$		0			0
$737$			24.5994i			0.906130i
$738$		0			0
$739$	29.7613			1.09479			0.547394	−	0.836875i	$-0.315620\pi$
$739$	29.7613			1.09479			0.547394	+	0.836875i	$0.315620\pi$
$740$		0			0
$741$		−	0.611319i		−	0.0224574i
$742$		0			0
$743$		−	8.09336i		−	0.296917i	−0.988919	−	0.148458i	$-0.952569\pi$
$743$		−	8.09336i		−	0.296917i	0.988919	−	0.148458i	$-0.0474311\pi$
$744$		0			0
$745$			14.8665i			0.544665i
$746$		0			0
$747$	26.6859			0.976386
$748$		0			0
$749$	−7.52939			−0.275118
$750$		0			0
$751$		−	32.2728i		−	1.17765i	−0.808260	−	0.588826i	$-0.799590\pi$
$751$		−	32.2728i		−	1.17765i	0.808260	−	0.588826i	$-0.200410\pi$
$752$		0			0
$753$		−	17.8315i		−	0.649816i
$754$		0			0
$755$		−	14.0761i		−	0.512283i
$756$		0			0
$757$	−31.1707			−1.13292			−0.566459	−	0.824090i	$-0.691687\pi$
$757$	−31.1707			−1.13292			−0.566459	+	0.824090i	$0.691687\pi$
$758$		0			0
$759$			13.1872i			0.478666i
$760$		0			0
$761$	30.2779			1.09757			0.548787	−	0.835962i	$-0.315090\pi$
$761$	30.2779			1.09757			0.548787	+	0.835962i	$0.315090\pi$
$762$		0			0
$763$	−15.9291			−0.576672
$764$		0			0
$765$	0.0809061	−	11.1818i	0.00292517	−	0.404278i
$766$		0			0
$767$	−1.07212			−0.0387121
$768$		0			0
$769$	44.7810			1.61484			0.807421	−	0.589975i	$-0.200862\pi$
$769$	44.7810			1.61484			0.807421	+	0.589975i	$0.200862\pi$
$770$		0			0
$771$			2.09617i			0.0754917i
$772$		0			0
$773$	21.2600			0.764669			0.382335	−	0.924024i	$-0.375120\pi$
$773$	21.2600			0.764669			0.382335	+	0.924024i	$0.375120\pi$
$774$		0			0
$775$		−	3.07840i		−	0.110580i
$776$		0			0
$777$		−	2.41095i		−	0.0864923i
$778$		0			0
$779$			6.40917i			0.229632i
$780$		0			0
$781$	−12.1505			−0.434780
$782$		0			0
$783$	10.8368			0.387276
$784$		0			0
$785$		−	17.8161i		−	0.635885i
$786$		0			0
$787$		−	30.1828i		−	1.07590i	−0.842977	−	0.537950i	$-0.819199\pi$
$787$		−	30.1828i		−	1.07590i	0.842977	−	0.537950i	$-0.180801\pi$
$788$		0			0
$789$		−	1.69011i		−	0.0601695i
$790$		0			0
$791$	−17.4390			−0.620059
$792$		0			0
$793$			6.33358i			0.224912i
$794$		0			0
$795$	6.49291			0.230280
$796$		0			0
$797$	12.8320			0.454534			0.227267	−	0.973832i	$-0.427021\pi$
$797$	12.8320			0.454534			0.227267	+	0.973832i	$0.427021\pi$
$798$		0			0
$799$	27.0436	+	0.195675i	0.956735	+	0.00692250i
$800$		0			0
$801$	15.8184			0.558914
$802$		0			0
$803$	−14.8568			−0.524287
$804$		0			0
$805$		−	8.86488i		−	0.312446i
$806$		0			0
$807$	15.6715			0.551664
$808$		0			0
$809$		−	10.4667i		−	0.367990i	−0.982927	−	0.183995i	$-0.941097\pi$
$809$		−	10.4667i		−	0.367990i	0.982927	−	0.183995i	$-0.0589031\pi$
$810$		0			0
$811$			1.42304i			0.0499697i	0.999688	+	0.0249849i	$0.00795375\pi$
$811$			1.42304i			0.0499697i	−0.999688	+	0.0249849i	$0.992046\pi$
$812$		0			0
$813$			8.04384i			0.282110i
$814$		0			0
$815$	−8.58416			−0.300690
$816$		0			0
$817$	−4.59550			−0.160776
$818$		0			0
$819$			2.69890i			0.0943072i
$820$		0			0
$821$			8.39215i			0.292888i	0.989219	+	0.146444i	$0.0467828\pi$
$821$			8.39215i			0.292888i	−0.989219	+	0.146444i	$0.953217\pi$
$822$		0			0
$823$		−	9.64624i		−	0.336247i	−0.985766	−	0.168123i	$-0.946229\pi$
$823$		−	9.64624i		−	0.336247i	0.985766	−	0.168123i	$-0.0537707\pi$
$824$		0			0
$825$	−6.30377			−0.219469
$826$		0			0
$827$			30.0944i			1.04648i	0.852184	+	0.523242i	$0.175278\pi$
$827$			30.0944i			1.04648i	−0.852184	+	0.523242i	$0.824722\pi$
$828$		0			0
$829$	−10.1537			−0.352653			−0.176327	−	0.984332i	$-0.556421\pi$
$829$	−10.1537			−0.352653			−0.176327	+	0.984332i	$0.556421\pi$
$830$		0			0
$831$	16.8493			0.584495
$832$		0			0
$833$	4.12300	+	0.0298321i	0.142853	+	0.00103362i
$834$		0			0
$835$	17.5127			0.606051
$836$		0			0
$837$	−3.64855			−0.126112
$838$		0			0
$839$		−	1.57501i		−	0.0543755i	−0.999630	−	0.0271877i	$-0.991345\pi$
$839$		−	1.57501i		−	0.0543755i	0.999630	−	0.0271877i	$-0.00865519\pi$
$840$		0			0
$841$	22.7522			0.784559
$842$		0			0
$843$		−	3.58972i		−	0.123636i
$844$		0			0
$845$		−	13.5202i		−	0.465109i
$846$		0			0
$847$		−	6.49288i		−	0.223098i
$848$		0			0
$849$	23.6671			0.812252
$850$		0			0
$851$	22.7289			0.779136
$852$		0			0
$853$		−	1.14407i		−	0.0391721i	−0.999808	−	0.0195861i	$-0.993765\pi$
$853$		−	1.14407i		−	0.0391721i	0.999808	−	0.0195861i	$-0.00623483\pi$
$854$		0			0
$855$		−	1.77171i		−	0.0605913i
$856$		0			0
$857$		−	15.0875i		−	0.515381i	−0.966228	−	0.257690i	$-0.917039\pi$
$857$		−	15.0875i		−	0.515381i	0.966228	−	0.257690i	$-0.0829615\pi$
$858$		0			0
$859$	19.5808			0.668088			0.334044	−	0.942557i	$-0.391587\pi$
$859$	19.5808			0.668088			0.334044	+	0.942557i	$0.391587\pi$
$860$		0			0
$861$			7.96366i			0.271401i
$862$		0			0
$863$	54.6083			1.85889			0.929444	−	0.368964i	$-0.120287\pi$
$863$	54.6083			1.85889			0.929444	+	0.368964i	$0.120287\pi$
$864$		0			0
$865$	−11.6393			−0.395750
$866$		0			0
$867$	0.199680	−	13.7978i	0.00678148	−	0.468598i
$868$		0			0
$869$	26.5127			0.899381
$870$		0			0
$871$	13.3580			0.452617
$872$		0			0
$873$		−	32.2628i		−	1.09193i
$874$		0			0
$875$	10.0298			0.339070
$876$		0			0
$877$			0.753805i			0.0254542i	0.999919	+	0.0127271i	$0.00405127\pi$
$877$			0.753805i			0.0254542i	−0.999919	+	0.0127271i	$0.995949\pi$
$878$		0			0
$879$		−	0.0158193i		−	0.000533571i
$880$		0			0
$881$			17.7585i			0.598300i	0.954206	+	0.299150i	$0.0967031\pi$
$881$			17.7585i			0.598300i	−0.954206	+	0.299150i	$0.903297\pi$
$882$		0			0
$883$	6.97578			0.234754			0.117377	−	0.993087i	$-0.462551\pi$
$883$	6.97578			0.234754			0.117377	+	0.993087i	$0.462551\pi$
$884$		0			0
$885$	0.874505			0.0293962
$886$		0			0
$887$			19.6504i			0.659797i	0.944016	+	0.329899i	$0.107015\pi$
$887$			19.6504i			0.659797i	−0.944016	+	0.329899i	$0.892985\pi$
$888$		0			0
$889$		−	8.51710i		−	0.285654i
$890$		0			0
$891$		−	7.43923i		−	0.249223i
$892$		0			0
$893$	4.28497			0.143391
$894$		0			0
$895$		−	18.4811i		−	0.617754i
$896$		0			0
$897$	7.16092			0.239096
$898$		0			0
$899$	2.10351			0.0701560
$900$		0			0
$901$	−28.4688	−	0.205987i	−0.948434	−	0.00686243i
$902$		0			0
$903$	−5.71010			−0.190020
$904$		0			0
$905$	−6.59453			−0.219209
$906$		0			0
$907$			17.8828i			0.593788i	0.954910	+	0.296894i	$0.0959508\pi$
$907$			17.8828i			0.593788i	−0.954910	+	0.296894i	$0.904049\pi$
$908$		0			0
$909$	−7.69537			−0.255239
$910$		0			0
$911$			38.2247i			1.26644i	0.773971	+	0.633221i	$0.218268\pi$
$911$			38.2247i			1.26644i	−0.773971	+	0.633221i	$0.781732\pi$
$912$		0			0
$913$		−	24.1997i		−	0.800893i
$914$		0			0
$915$		−	5.16615i		−	0.170788i
$916$		0			0
$917$	2.69345			0.0889456
$918$		0			0
$919$	−28.6429			−0.944841			−0.472421	−	0.881373i	$-0.656620\pi$
$919$	−28.6429			−0.944841			−0.472421	+	0.881373i	$0.656620\pi$
$920$		0			0
$921$		−	10.0637i		−	0.331609i
$922$		0			0
$923$			6.59797i			0.217175i
$924$		0			0
$925$			10.8649i			0.357235i
$926$		0			0
$927$	−31.0887			−1.02109
$928$		0			0
$929$		−	8.78923i		−	0.288365i	−0.989551	−	0.144183i	$-0.953945\pi$
$929$		−	8.78923i		−	0.288365i	0.989551	−	0.144183i	$-0.0460553\pi$
$930$		0			0
$931$	0.653275			0.0214102
$932$		0			0
$933$	6.51568			0.213314
$934$		0			0
$935$	−10.1400	−	0.0733685i	−0.331614	−	0.00239941i
$936$		0			0
$937$	−18.4664			−0.603270			−0.301635	−	0.953424i	$-0.597532\pi$
$937$	−18.4664			−0.603270			−0.301635	+	0.953424i	$0.597532\pi$
$938$		0			0
$939$	−25.7398			−0.839986
$940$		0			0
$941$		−	45.0296i		−	1.46792i	−0.679192	−	0.733961i	$-0.737669\pi$
$941$		−	45.0296i		−	1.46792i	0.679192	−	0.733961i	$-0.262331\pi$
$942$		0			0
$943$	−75.0763			−2.44482
$944$		0			0
$945$		−	5.02244i		−	0.163380i
$946$		0			0
$947$		−	34.7361i		−	1.12877i	−0.825511	−	0.564386i	$-0.809113\pi$
$947$		−	34.7361i		−	1.12877i	0.825511	−	0.564386i	$-0.190887\pi$
$948$		0			0
$949$			8.06756i			0.261884i
$950$		0			0
$951$	16.0460			0.520326
$952$		0			0
$953$	11.2590			0.364714			0.182357	−	0.983232i	$-0.441627\pi$
$953$	11.2590			0.364714			0.182357	+	0.983232i	$0.441627\pi$
$954$		0			0
$955$			9.44339i			0.305581i
$956$		0			0
$957$		−	4.30744i		−	0.139240i
$958$		0			0
$959$		−	4.89454i		−	0.158053i
$960$		0			0
$961$	30.2918			0.977154
$962$		0			0
$963$		−	17.6271i		−	0.568026i
$964$		0			0
$965$	−3.38043			−0.108820
$966$		0			0
$967$	47.7038			1.53405			0.767025	−	0.641617i	$-0.221736\pi$
$967$	47.7038			1.53405			0.767025	+	0.641617i	$0.221736\pi$
$968$		0			0
$969$	0.0158193	−	2.18633i	0.000508189	−	0.0702351i
$970$		0			0
$971$	18.7589			0.602002			0.301001	−	0.953624i	$-0.402679\pi$
$971$	18.7589			0.602002			0.301001	+	0.953624i	$0.402679\pi$
$972$		0			0
$973$	1.34478			0.0431116
$974$		0			0
$975$			3.42307i			0.109626i
$976$		0			0
$977$	−42.4708			−1.35876			−0.679380	−	0.733786i	$-0.737751\pi$
$977$	−42.4708			−1.35876			−0.679380	+	0.733786i	$0.737751\pi$
$978$		0			0
$979$		−	14.3446i		−	0.458457i
$980$		0			0
$981$		−	37.2918i		−	1.19063i
$982$		0			0
$983$		−	16.4611i		−	0.525028i	−0.964928	−	0.262514i	$-0.915448\pi$
$983$		−	16.4611i		−	0.525028i	0.964928	−	0.262514i	$-0.0845515\pi$
$984$		0			0
$985$	−10.8189			−0.344719
$986$		0			0
$987$	5.32426			0.169473
$988$		0			0
$989$		−	53.8312i		−	1.71173i
$990$		0			0
$991$		−	38.2811i		−	1.21604i	−0.793922	−	0.608019i	$-0.791964\pi$
$991$		−	38.2811i		−	1.21604i	0.793922	−	0.608019i	$-0.208036\pi$
$992$		0			0
$993$		−	21.9347i		−	0.696076i
$994$		0			0
$995$	−15.0445			−0.476944
$996$		0			0
$997$			32.7205i			1.03627i	0.855300	+	0.518134i	$0.173373\pi$
$997$			32.7205i			1.03627i	−0.855300	+	0.518134i	$0.826627\pi$
$998$		0			0
$999$	12.8771			0.407415

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1904.2.c.f.1121.4		8
4.3	odd	2		476.2.b.a.169.5	yes	8
12.11	even	2		4284.2.d.e.3025.4		8
17.16	even	2	inner	1904.2.c.f.1121.5		8
28.27	even	2		3332.2.b.b.2549.4		8
68.47	odd	4		8092.2.a.q.1.3		4
68.55	odd	4		8092.2.a.r.1.2		4
68.67	odd	2		476.2.b.a.169.4	✓	8
204.203	even	2		4284.2.d.e.3025.5		8
476.475	even	2		3332.2.b.b.2549.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.b.a.169.4	✓	8	68.67	odd	2
476.2.b.a.169.5	yes	8	4.3	odd	2
1904.2.c.f.1121.4		8	1.1	even	1	trivial
1904.2.c.f.1121.5		8	17.16	even	2	inner
3332.2.b.b.2549.4		8	28.27	even	2
3332.2.b.b.2549.5		8	476.475	even	2
4284.2.d.e.3025.4		8	12.11	even	2
4284.2.d.e.3025.5		8	204.203	even	2
8092.2.a.q.1.3		4	68.47	odd	4
8092.2.a.r.1.2		4	68.55	odd	4

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:	\( N \)	\(=\)	\( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1904.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.811721i q^{3} +1.15845i q^{5} -1.00000i q^{7} +2.34111 q^{9} +O(q^{10})\)	\(q-0.811721i q^{3} +1.15845i q^{5} -1.00000i q^{7} +2.34111 q^{9} -2.12300i q^{11} -1.15283 q^{13} +0.940336 q^{15} +(-4.12300 - 0.0298321i) q^{17} -0.653275 q^{19} -0.811721 q^{21} -7.65238i q^{23} +3.65800 q^{25} -4.33549i q^{27} +2.49956i q^{29} -0.841553i q^{31} -1.72328 q^{33} +1.15845 q^{35} +2.97017i q^{37} +0.935777i q^{39} -9.81083i q^{41} +7.03456 q^{43} +2.71205i q^{45} -6.55922 q^{47} -1.00000 q^{49} +(-0.0242154 + 3.34673i) q^{51} +6.90489 q^{53} +2.45938 q^{55} +0.530277i q^{57} +0.929992 q^{59} -5.49394i q^{61} -2.34111i q^{63} -1.33549i q^{65} -11.5871 q^{67} -6.21160 q^{69} -5.72328i q^{71} -6.99805i q^{73} -2.96928i q^{75} -2.12300 q^{77} +12.4883i q^{79} +3.50411 q^{81} +11.3988 q^{83} +(0.0345589 - 4.77627i) q^{85} +2.02894 q^{87} +6.75678 q^{89} +1.15283i q^{91} -0.683107 q^{93} -0.756784i q^{95} -13.7810i q^{97} -4.97017i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{9}+O(q^{10}) \) 8 * q - 12 * q^9	\( 8 q - 12 q^{9} + 20 q^{13} - 12 q^{19} - 8 q^{21} - 12 q^{25} - 4 q^{33} + 4 q^{35} + 12 q^{43} - 24 q^{47} - 8 q^{49} + 20 q^{51} + 20 q^{53} + 4 q^{55} + 24 q^{59} + 4 q^{67} + 44 q^{69} + 16 q^{77} + 16 q^{81} - 4 q^{83} - 44 q^{85} - 52 q^{87} - 16 q^{93}+O(q^{100}) \) 8 * q - 12 * q^9 + 20 * q^13 - 12 * q^19 - 8 * q^21 - 12 * q^25 - 4 * q^33 + 4 * q^35 + 12 * q^43 - 24 * q^47 - 8 * q^49 + 20 * q^51 + 20 * q^53 + 4 * q^55 + 24 * q^59 + 4 * q^67 + 44 * q^69 + 16 * q^77 + 16 * q^81 - 4 * q^83 - 44 * q^85 - 52 * q^87 - 16 * q^93

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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